LMFDB - Embedded newform 975.2.n.i.824.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(749,975)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("975.749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			824.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	975.824
Dual form			975.2.n.i.749.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} +3.00000i q^{6} +(-0.224745 + 0.224745i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} +3.00000i q^{6} +(-0.224745 + 0.224745i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} +(-4.22474 + 4.22474i) q^{11} +(-1.22474 - 1.22474i) q^{12} +(2.00000 - 3.00000i) q^{13} -0.550510i q^{14} +5.00000 q^{16} +3.00000i q^{17} +(3.67423 + 3.67423i) q^{18} +(-4.44949 + 4.44949i) q^{19} +0.550510i q^{21} -10.3485i q^{22} +6.00000i q^{23} -3.00000 q^{24} +(1.22474 + 6.12372i) q^{26} +(-3.67423 - 3.67423i) q^{27} +(0.224745 + 0.224745i) q^{28} +1.89898i q^{29} +(-5.12372 + 5.12372i) q^{31} +(-3.67423 + 3.67423i) q^{32} +10.3485i q^{33} +(-3.67423 - 3.67423i) q^{34} -3.00000 q^{36} +(4.44949 - 4.44949i) q^{37} -10.8990i q^{38} +(-1.22474 - 6.12372i) q^{39} +(5.44949 + 5.44949i) q^{41} +(-0.674235 - 0.674235i) q^{42} +1.10102 q^{43} +(4.22474 + 4.22474i) q^{44} +(-7.34847 - 7.34847i) q^{46} +(1.77526 + 1.77526i) q^{47} +(6.12372 - 6.12372i) q^{48} +6.89898i q^{49} +(3.67423 + 3.67423i) q^{51} +(-3.00000 - 2.00000i) q^{52} -1.89898 q^{53} +9.00000 q^{54} +0.550510 q^{56} +10.8990i q^{57} +(-2.32577 - 2.32577i) q^{58} +(-6.67423 + 6.67423i) q^{59} +12.7980 q^{61} -12.5505i q^{62} +(0.674235 + 0.674235i) q^{63} +1.00000i q^{64} +(-12.6742 - 12.6742i) q^{66} +(-4.67423 - 4.67423i) q^{67} +3.00000 q^{68} +(7.34847 + 7.34847i) q^{69} +(-6.00000 - 6.00000i) q^{71} +(-3.67423 + 3.67423i) q^{72} +(-6.44949 + 6.44949i) q^{73} +10.8990i q^{74} +(4.44949 + 4.44949i) q^{76} -1.89898i q^{77} +(9.00000 + 6.00000i) q^{78} -3.55051 q^{79} -9.00000 q^{81} -13.3485 q^{82} +(1.77526 - 1.77526i) q^{83} +0.550510 q^{84} +(-1.34847 + 1.34847i) q^{86} +(2.32577 + 2.32577i) q^{87} +10.3485 q^{88} +(-0.550510 + 0.550510i) q^{89} +(0.224745 + 1.12372i) q^{91} +6.00000 q^{92} +12.5505i q^{93} -4.34847 q^{94} +9.00000i q^{96} +(6.44949 + 6.44949i) q^{97} +(-8.44949 - 8.44949i) q^{98} +(12.6742 + 12.6742i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 12 q^{11} + 8 q^{13} + 20 q^{16} - 8 q^{19} - 12 q^{24} - 4 q^{28} + 4 q^{31} - 12 q^{36} + 8 q^{37} + 12 q^{41} + 12 q^{42} + 24 q^{43} + 12 q^{44} + 12 q^{47} - 12 q^{52} + 12 q^{53} + 36 q^{54} + 12 q^{56} - 24 q^{58} - 12 q^{59} + 12 q^{61} - 12 q^{63} - 36 q^{66} - 4 q^{67} + 12 q^{68} - 24 q^{71} - 16 q^{73} + 8 q^{76} + 36 q^{78} - 24 q^{79} - 36 q^{81} - 24 q^{82} + 12 q^{83} + 12 q^{84} + 24 q^{86} + 24 q^{87} + 12 q^{88} - 12 q^{89} - 4 q^{91} + 24 q^{92} + 12 q^{94} + 16 q^{97} - 24 q^{98} + 36 q^{99}+O(q^{100}) $ 4 * q + 4 * q^7 - 12 * q^11 + 8 * q^13 + 20 * q^16 - 8 * q^19 - 12 * q^24 - 4 * q^28 + 4 * q^31 - 12 * q^36 + 8 * q^37 + 12 * q^41 + 12 * q^42 + 24 * q^43 + 12 * q^44 + 12 * q^47 - 12 * q^52 + 12 * q^53 + 36 * q^54 + 12 * q^56 - 24 * q^58 - 12 * q^59 + 12 * q^61 - 12 * q^63 - 36 * q^66 - 4 * q^67 + 12 * q^68 - 24 * q^71 - 16 * q^73 + 8 * q^76 + 36 * q^78 - 24 * q^79 - 36 * q^81 - 24 * q^82 + 12 * q^83 + 12 * q^84 + 24 * q^86 + 24 * q^87 + 12 * q^88 - 12 * q^89 - 4 * q^91 + 24 * q^92 + 12 * q^94 + 16 * q^97 - 24 * q^98 + 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	−0.992030	−	0.126004i	$-0.959785\pi$
$2$	−1.22474	+	1.22474i	−0.866025	+	0.866025i	0.126004	+	0.992030i	$0.459785\pi$
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$3$	1.22474	−	1.22474i	0.707107	−	0.707107i
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$			3.00000i			1.22474i
$7$	−0.224745	+	0.224745i	−0.0849456	+	0.0849456i	−0.748303	−	0.663357i	$-0.769131\pi$
$7$	−0.224745	+	0.224745i	−0.0849456	+	0.0849456i	0.663357	+	0.748303i	$0.269131\pi$
$8$	−1.22474	−	1.22474i	−0.433013	−	0.433013i
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−4.22474	+	4.22474i	−1.27381	+	1.27381i	−0.329735	+	0.944074i	$0.606959\pi$
$11$	−4.22474	+	4.22474i	−1.27381	+	1.27381i	−0.944074	+	0.329735i	$0.893041\pi$
$12$	−1.22474	−	1.22474i	−0.353553	−	0.353553i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		−	0.550510i		−	0.147130i
$15$		0			0
$16$	5.00000			1.25000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	3.67423	+	3.67423i	0.866025	+	0.866025i
$19$	−4.44949	+	4.44949i	−1.02078	+	1.02078i	−0.0210036	+	0.999779i	$0.506686\pi$
$19$	−4.44949	+	4.44949i	−1.02078	+	1.02078i	−0.999779	+	0.0210036i	$0.993314\pi$
$20$		0			0
$21$			0.550510i			0.120131i
$22$		−	10.3485i		−	2.20630i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−3.00000			−0.612372
$25$		0			0
$26$	1.22474	+	6.12372i	0.240192	+	1.20096i
$27$	−3.67423	−	3.67423i	−0.707107	−	0.707107i
$28$	0.224745	+	0.224745i	0.0424728	+	0.0424728i
$29$			1.89898i			0.352632i	0.984334	+	0.176316i	$0.0564180\pi$
$29$			1.89898i			0.352632i	−0.984334	+	0.176316i	$0.943582\pi$
$30$		0			0
$31$	−5.12372	+	5.12372i	−0.920248	+	0.920248i	−0.997047	−	0.0767986i	$-0.975530\pi$
$31$	−5.12372	+	5.12372i	−0.920248	+	0.920248i	0.0767986	+	0.997047i	$0.475530\pi$
$32$	−3.67423	+	3.67423i	−0.649519	+	0.649519i
$33$			10.3485i			1.80144i
$34$	−3.67423	−	3.67423i	−0.630126	−	0.630126i
$35$		0			0
$36$	−3.00000			−0.500000
$37$	4.44949	−	4.44949i	0.731492	−	0.731492i	−0.239424	−	0.970915i	$-0.576958\pi$
$37$	4.44949	−	4.44949i	0.731492	−	0.731492i	0.970915	+	0.239424i	$0.0769585\pi$
$38$		−	10.8990i		−	1.76805i
$39$	−1.22474	−	6.12372i	−0.196116	−	0.980581i
$40$		0			0
$41$	5.44949	+	5.44949i	0.851067	+	0.851067i	0.990265	−	0.139197i	$-0.0444522\pi$
$41$	5.44949	+	5.44949i	0.851067	+	0.851067i	−0.139197	+	0.990265i	$0.544452\pi$
$42$	−0.674235	−	0.674235i	−0.104037	−	0.104037i
$43$	1.10102			0.167904			0.0839520	−	0.996470i	$-0.473246\pi$
$43$	1.10102			0.167904			0.0839520	+	0.996470i	$0.473246\pi$
$44$	4.22474	+	4.22474i	0.636904	+	0.636904i
$45$		0			0
$46$	−7.34847	−	7.34847i	−1.08347	−	1.08347i
$47$	1.77526	+	1.77526i	0.258948	+	0.258948i	0.824626	−	0.565678i	$-0.191386\pi$
$47$	1.77526	+	1.77526i	0.258948	+	0.258948i	−0.565678	+	0.824626i	$0.691386\pi$
$48$	6.12372	−	6.12372i	0.883883	−	0.883883i
$49$			6.89898i			0.985568i
$50$		0			0
$51$	3.67423	+	3.67423i	0.514496	+	0.514496i
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$	−1.89898			−0.260845			−0.130422	−	0.991459i	$-0.541633\pi$
$53$	−1.89898			−0.260845			−0.130422	+	0.991459i	$0.541633\pi$
$54$	9.00000			1.22474
$55$		0			0
$56$	0.550510			0.0735650
$57$			10.8990i			1.44361i
$58$	−2.32577	−	2.32577i	−0.305388	−	0.305388i
$59$	−6.67423	+	6.67423i	−0.868911	+	0.868911i	−0.992352	−	0.123441i	$-0.960607\pi$
$59$	−6.67423	+	6.67423i	−0.868911	+	0.868911i	0.123441	+	0.992352i	$0.460607\pi$
$60$		0			0
$61$	12.7980			1.63861			0.819305	−	0.573357i	$-0.194359\pi$
$61$	12.7980			1.63861			0.819305	+	0.573357i	$0.194359\pi$
$62$		−	12.5505i		−	1.59392i
$63$	0.674235	+	0.674235i	0.0849456	+	0.0849456i
$64$			1.00000i			0.125000i
$65$		0			0
$66$	−12.6742	−	12.6742i	−1.56009	−	1.56009i
$67$	−4.67423	−	4.67423i	−0.571049	−	0.571049i	0.361373	−	0.932421i	$-0.382308\pi$
$67$	−4.67423	−	4.67423i	−0.571049	−	0.571049i	−0.932421	+	0.361373i	$0.882308\pi$
$68$	3.00000			0.363803
$69$	7.34847	+	7.34847i	0.884652	+	0.884652i
$70$		0			0
$71$	−6.00000	−	6.00000i	−0.712069	−	0.712069i	0.254899	−	0.966968i	$-0.417958\pi$
$71$	−6.00000	−	6.00000i	−0.712069	−	0.712069i	−0.966968	+	0.254899i	$0.917958\pi$
$72$	−3.67423	+	3.67423i	−0.433013	+	0.433013i
$73$	−6.44949	+	6.44949i	−0.754856	+	0.754856i	−0.975381	−	0.220526i	$-0.929223\pi$
$73$	−6.44949	+	6.44949i	−0.754856	+	0.754856i	0.220526	+	0.975381i	$0.429223\pi$
$74$			10.8990i			1.26698i
$75$		0			0
$76$	4.44949	+	4.44949i	0.510391	+	0.510391i
$77$		−	1.89898i		−	0.216409i
$78$	9.00000	+	6.00000i	1.01905	+	0.679366i
$79$	−3.55051			−0.399464			−0.199732	−	0.979851i	$-0.564007\pi$
$79$	−3.55051			−0.399464			−0.199732	+	0.979851i	$0.564007\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	−13.3485			−1.47409
$83$	1.77526	−	1.77526i	0.194860	−	0.194860i	−0.602933	−	0.797792i	$-0.706001\pi$
$83$	1.77526	−	1.77526i	0.194860	−	0.194860i	0.797792	+	0.602933i	$0.206001\pi$
$84$	0.550510			0.0600656
$85$		0			0
$86$	−1.34847	+	1.34847i	−0.145409	+	0.145409i
$87$	2.32577	+	2.32577i	0.249348	+	0.249348i
$88$	10.3485			1.10315
$89$	−0.550510	+	0.550510i	−0.0583540	+	0.0583540i	−0.735682	−	0.677328i	$-0.763138\pi$
$89$	−0.550510	+	0.550510i	−0.0583540	+	0.0583540i	0.677328	+	0.735682i	$0.263138\pi$
$90$		0			0
$91$	0.224745	+	1.12372i	0.0235597	+	0.117798i
$92$	6.00000			0.625543
$93$			12.5505i			1.30143i
$94$	−4.34847			−0.448510
$95$		0			0
$96$			9.00000i			0.918559i
$97$	6.44949	+	6.44949i	0.654846	+	0.654846i	0.954156	−	0.299310i	$-0.0967563\pi$
$97$	6.44949	+	6.44949i	0.654846	+	0.654846i	−0.299310	+	0.954156i	$0.596756\pi$
$98$	−8.44949	−	8.44949i	−0.853527	−	0.853527i
$99$	12.6742	+	12.6742i	1.27381	+	1.27381i
$100$		0			0
$101$	3.00000			0.298511			0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000			0.298511			0.149256	+	0.988799i	$0.452312\pi$
$102$	−9.00000			−0.891133
$103$	−9.55051			−0.941040			−0.470520	−	0.882389i	$-0.655934\pi$
$103$	−9.55051			−0.941040			−0.470520	+	0.882389i	$0.655934\pi$
$104$	−6.12372	+	1.22474i	−0.600481	+	0.120096i
$105$		0			0
$106$	2.32577	−	2.32577i	0.225898	−	0.225898i
$107$	1.34847			0.130361			0.0651807	−	0.997873i	$-0.479238\pi$
$107$	1.34847			0.130361			0.0651807	+	0.997873i	$0.479238\pi$
$108$	−3.67423	+	3.67423i	−0.353553	+	0.353553i
$109$	7.79796	−	7.79796i	0.746909	−	0.746909i	−0.226988	−	0.973897i	$-0.572888\pi$
$109$	7.79796	−	7.79796i	0.746909	−	0.746909i	0.973897	+	0.226988i	$0.0728880\pi$
$110$		0			0
$111$		−	10.8990i		−	1.03449i
$112$	−1.12372	+	1.12372i	−0.106182	+	0.106182i
$113$	−16.8990			−1.58972			−0.794861	−	0.606791i	$-0.792456\pi$
$113$	−16.8990			−1.58972			−0.794861	+	0.606791i	$0.792456\pi$
$114$	−13.3485	−	13.3485i	−1.25020	−	1.25020i
$115$		0			0
$116$	1.89898			0.176316
$117$	−9.00000	−	6.00000i	−0.832050	−	0.554700i
$118$		−	16.3485i		−	1.50500i
$119$	−0.674235	−	0.674235i	−0.0618070	−	0.0618070i
$120$		0			0
$121$		−	24.6969i		−	2.24518i
$122$	−15.6742	+	15.6742i	−1.41908	+	1.41908i
$123$	13.3485			1.20359
$124$	5.12372	+	5.12372i	0.460124	+	0.460124i
$125$		0			0
$126$	−1.65153			−0.147130
$127$	−2.65153			−0.235285			−0.117643	−	0.993056i	$-0.537534\pi$
$127$	−2.65153			−0.235285			−0.117643	+	0.993056i	$0.537534\pi$
$128$	−8.57321	−	8.57321i	−0.757772	−	0.757772i
$129$	1.34847	−	1.34847i	0.118726	−	0.118726i
$130$		0			0
$131$		−	7.10102i		−	0.620419i	−0.950668	−	0.310210i	$-0.899601\pi$
$131$		−	7.10102i		−	0.620419i	0.950668	−	0.310210i	$-0.100399\pi$
$132$	10.3485			0.900719
$133$		−	2.00000i		−	0.173422i
$134$	11.4495			0.989085
$135$		0			0
$136$	3.67423	−	3.67423i	0.315063	−	0.315063i
$137$	4.34847	+	4.34847i	0.371515	+	0.371515i	0.868029	−	0.496514i	$-0.165387\pi$
$137$	4.34847	+	4.34847i	0.371515	+	0.371515i	−0.496514	+	0.868029i	$0.665387\pi$
$138$	−18.0000			−1.53226
$139$	−9.34847			−0.792927			−0.396463	−	0.918051i	$-0.629763\pi$
$139$	−9.34847			−0.792927			−0.396463	+	0.918051i	$0.629763\pi$
$140$		0			0
$141$	4.34847			0.366207
$142$	14.6969			1.23334
$143$	4.22474	+	21.1237i	0.353291	+	1.76645i
$144$		−	15.0000i		−	1.25000i
$145$		0			0
$146$		−	15.7980i		−	1.30745i
$147$	8.44949	+	8.44949i	0.696902	+	0.696902i
$148$	−4.44949	−	4.44949i	−0.365746	−	0.365746i
$149$	−14.4495	−	14.4495i	−1.18375	−	1.18375i	−0.978766	−	0.204983i	$-0.934286\pi$
$149$	−14.4495	−	14.4495i	−1.18375	−	1.18375i	−0.204983	−	0.978766i	$-0.565714\pi$
$150$		0			0
$151$	−1.57321	−	1.57321i	−0.128026	−	0.128026i	0.640190	−	0.768216i	$-0.278856\pi$
$151$	−1.57321	−	1.57321i	−0.128026	−	0.128026i	−0.768216	+	0.640190i	$0.778856\pi$
$152$	10.8990			0.884024
$153$	9.00000			0.727607
$154$	2.32577	+	2.32577i	0.187416	+	0.187416i
$155$		0			0
$156$	−6.12372	+	1.22474i	−0.490290	+	0.0980581i
$157$			7.00000i			0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$			7.00000i			0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	4.34847	−	4.34847i	0.345946	−	0.345946i
$159$	−2.32577	+	2.32577i	−0.184445	+	0.184445i
$160$		0			0
$161$	−1.34847	−	1.34847i	−0.106274	−	0.106274i
$162$	11.0227	−	11.0227i	0.866025	−	0.866025i
$163$	−9.34847	+	9.34847i	−0.732229	+	0.732229i	−0.971061	−	0.238832i	$-0.923235\pi$
$163$	−9.34847	+	9.34847i	−0.732229	+	0.732229i	0.238832	+	0.971061i	$0.423235\pi$
$164$	5.44949	−	5.44949i	0.425534	−	0.425534i
$165$		0			0
$166$			4.34847i			0.337507i
$167$	−12.2474	−	12.2474i	−0.947736	−	0.947736i	0.0509644	−	0.998700i	$-0.483770\pi$
$167$	−12.2474	−	12.2474i	−0.947736	−	0.947736i	−0.998700	+	0.0509644i	$0.983770\pi$
$168$	0.674235	−	0.674235i	0.0520183	−	0.0520183i
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$	13.3485	+	13.3485i	1.02078	+	1.02078i
$172$		−	1.10102i		−	0.0839520i
$173$			6.79796i			0.516839i	0.966033	+	0.258420i	$0.0832017\pi$
$173$			6.79796i			0.516839i	−0.966033	+	0.258420i	$0.916798\pi$
$174$	−5.69694			−0.431884
$175$		0			0
$176$	−21.1237	+	21.1237i	−1.59226	+	1.59226i
$177$			16.3485i			1.22883i
$178$		−	1.34847i		−	0.101072i
$179$	12.2474			0.915417			0.457709	−	0.889102i	$-0.348670\pi$
$179$	12.2474			0.915417			0.457709	+	0.889102i	$0.348670\pi$
$180$		0			0
$181$			23.6969i			1.76138i	0.473693	+	0.880690i	$0.342920\pi$
$181$			23.6969i			1.76138i	−0.473693	+	0.880690i	$0.657080\pi$
$182$	−1.65153	−	1.10102i	−0.122420	−	0.0816131i
$183$	15.6742	−	15.6742i	1.15867	−	1.15867i
$184$	7.34847	−	7.34847i	0.541736	−	0.541736i
$185$		0			0
$186$	−15.3712	−	15.3712i	−1.12707	−	1.12707i
$187$	−12.6742	−	12.6742i	−0.926832	−	0.926832i
$188$	1.77526	−	1.77526i	0.129474	−	0.129474i
$189$	1.65153			0.120131
$190$		0			0
$191$		−	8.69694i		−	0.629288i	−0.949210	−	0.314644i	$-0.898115\pi$
$191$		−	8.69694i		−	0.629288i	0.949210	−	0.314644i	$-0.101885\pi$
$192$	1.22474	+	1.22474i	0.0883883	+	0.0883883i
$193$	1.44949	−	1.44949i	0.104337	−	0.104337i	−0.653011	−	0.757348i	$-0.726495\pi$
$193$	1.44949	−	1.44949i	0.104337	−	0.104337i	0.757348	+	0.653011i	$0.226495\pi$
$194$	−15.7980			−1.13423
$195$		0			0
$196$	6.89898			0.492784
$197$	9.24745	−	9.24745i	0.658853	−	0.658853i	−0.296255	−	0.955109i	$-0.595738\pi$
$197$	9.24745	−	9.24745i	0.658853	−	0.658853i	0.955109	+	0.296255i	$0.0957379\pi$
$198$	−31.0454			−2.20630
$199$			4.69694i			0.332957i	0.986045	+	0.166479i	$0.0532397\pi$
$199$			4.69694i			0.332957i	−0.986045	+	0.166479i	$0.946760\pi$
$200$		0			0
$201$	−11.4495			−0.807585
$202$	−3.67423	+	3.67423i	−0.258518	+	0.258518i
$203$	−0.426786	−	0.426786i	−0.0299545	−	0.0299545i
$204$	3.67423	−	3.67423i	0.257248	−	0.257248i
$205$		0			0
$206$	11.6969	−	11.6969i	0.814964	−	0.814964i
$207$	18.0000			1.25109
$208$	10.0000	−	15.0000i	0.693375	−	1.04006i
$209$		−	37.5959i		−	2.60056i
$210$		0			0
$211$	20.6969			1.42484			0.712418	−	0.701755i	$-0.247600\pi$
$211$	20.6969			1.42484			0.712418	+	0.701755i	$0.247600\pi$
$212$			1.89898i			0.130422i
$213$	−14.6969			−1.00702
$214$	−1.65153	+	1.65153i	−0.112896	+	0.112896i
$215$		0			0
$216$			9.00000i			0.612372i
$217$		−	2.30306i		−	0.156342i
$218$			19.1010i			1.29368i
$219$			15.7980i			1.06753i
$220$		0			0
$221$	9.00000	+	6.00000i	0.605406	+	0.403604i
$222$	13.3485	+	13.3485i	0.895891	+	0.895891i
$223$	7.55051	+	7.55051i	0.505620	+	0.505620i	0.913179	−	0.407559i	$-0.133620\pi$
$223$	7.55051	+	7.55051i	0.505620	+	0.505620i	−0.407559	+	0.913179i	$0.633620\pi$
$224$		−	1.65153i		−	0.110348i
$225$		0			0
$226$	20.6969	−	20.6969i	1.37674	−	1.37674i
$227$	10.2247	−	10.2247i	0.678640	−	0.678640i	−0.281052	−	0.959692i	$-0.590683\pi$
$227$	10.2247	−	10.2247i	0.678640	−	0.678640i	0.959692	+	0.281052i	$0.0906834\pi$
$228$	10.8990			0.721803
$229$	10.2474	+	10.2474i	0.677170	+	0.677170i	0.959359	−	0.282189i	$-0.0910604\pi$
$229$	10.2474	+	10.2474i	0.677170	+	0.677170i	−0.282189	+	0.959359i	$0.591060\pi$
$230$		0			0
$231$	−2.32577	−	2.32577i	−0.153024	−	0.153024i
$232$	2.32577	−	2.32577i	0.152694	−	0.152694i
$233$		−	8.20204i		−	0.537334i	−0.963233	−	0.268667i	$-0.913417\pi$
$233$		−	8.20204i		−	0.537334i	0.963233	−	0.268667i	$-0.0865830\pi$
$234$	18.3712	−	3.67423i	1.20096	−	0.240192i
$235$		0			0
$236$	6.67423	+	6.67423i	0.434456	+	0.434456i
$237$	−4.34847	+	4.34847i	−0.282463	+	0.282463i
$238$	1.65153			0.107053
$239$	10.2247	+	10.2247i	0.661384	+	0.661384i	0.955706	−	0.294322i	$-0.0950940\pi$
$239$	10.2247	+	10.2247i	0.661384	+	0.661384i	−0.294322	+	0.955706i	$0.595094\pi$
$240$		0			0
$241$	20.3485	+	20.3485i	1.31076	+	1.31076i	0.920855	+	0.389905i	$0.127492\pi$
$241$	20.3485	+	20.3485i	1.31076	+	1.31076i	0.389905	+	0.920855i	$0.372508\pi$
$242$	30.2474	+	30.2474i	1.94438	+	1.94438i
$243$	−11.0227	+	11.0227i	−0.707107	+	0.707107i
$244$		−	12.7980i		−	0.819305i
$245$		0			0
$246$	−16.3485	+	16.3485i	−1.04234	+	1.04234i
$247$	4.44949	+	22.2474i	0.283114	+	1.41557i
$248$	12.5505			0.796958
$249$		−	4.34847i		−	0.275573i
$250$		0			0
$251$	14.4495			0.912044			0.456022	−	0.889969i	$-0.349274\pi$
$251$	14.4495			0.912044			0.456022	+	0.889969i	$0.349274\pi$
$252$	0.674235	−	0.674235i	0.0424728	−	0.0424728i
$253$	−25.3485	−	25.3485i	−1.59364	−	1.59364i
$254$	3.24745	−	3.24745i	0.203763	−	0.203763i
$255$		0			0
$256$	19.0000			1.18750
$257$		−	28.5959i		−	1.78376i	−0.452268	−	0.891882i	$-0.649385\pi$
$257$		−	28.5959i		−	1.78376i	0.452268	−	0.891882i	$-0.350615\pi$
$258$			3.30306i			0.205640i
$259$			2.00000i			0.124274i
$260$		0			0
$261$	5.69694			0.352632
$262$	8.69694	+	8.69694i	0.537299	+	0.537299i
$263$	21.7980			1.34412			0.672060	−	0.740497i	$-0.265410\pi$
$263$	21.7980			1.34412			0.672060	+	0.740497i	$0.265410\pi$
$264$	12.6742	−	12.6742i	0.780045	−	0.780045i
$265$		0			0
$266$	2.44949	+	2.44949i	0.150188	+	0.150188i
$267$			1.34847i			0.0825250i
$268$	−4.67423	+	4.67423i	−0.285524	+	0.285524i
$269$		−	6.79796i		−	0.414479i	−0.978290	−	0.207239i	$-0.933552\pi$
$269$		−	6.79796i		−	0.414479i	0.978290	−	0.207239i	$-0.0664479\pi$
$270$		0			0
$271$	7.12372	+	7.12372i	0.432735	+	0.432735i	0.889558	−	0.456822i	$-0.151013\pi$
$271$	7.12372	+	7.12372i	0.432735	+	0.432735i	−0.456822	+	0.889558i	$0.651013\pi$
$272$			15.0000i			0.909509i
$273$	1.65153	+	1.10102i	0.0999552	+	0.0666368i
$274$	−10.6515			−0.643483
$275$		0			0
$276$	7.34847	−	7.34847i	0.442326	−	0.442326i
$277$	−4.89898			−0.294351			−0.147176	−	0.989110i	$-0.547018\pi$
$277$	−4.89898			−0.294351			−0.147176	+	0.989110i	$0.547018\pi$
$278$	11.4495	−	11.4495i	0.686695	−	0.686695i
$279$	15.3712	+	15.3712i	0.920248	+	0.920248i
$280$		0			0
$281$	−9.55051	+	9.55051i	−0.569736	+	0.569736i	−0.932054	−	0.362319i	$-0.881985\pi$
$281$	−9.55051	+	9.55051i	−0.569736	+	0.569736i	0.362319	+	0.932054i	$0.381985\pi$
$282$	−5.32577	+	5.32577i	−0.317145	+	0.317145i
$283$	−5.34847			−0.317933			−0.158967	−	0.987284i	$-0.550816\pi$
$283$	−5.34847			−0.317933			−0.158967	+	0.987284i	$0.550816\pi$
$284$	−6.00000	+	6.00000i	−0.356034	+	0.356034i
$285$		0			0
$286$	−31.0454	−	20.6969i	−1.83575	−	1.22384i
$287$	−2.44949			−0.144589
$288$	11.0227	+	11.0227i	0.649519	+	0.649519i
$289$	8.00000			0.470588
$290$		0			0
$291$	15.7980			0.926093
$292$	6.44949	+	6.44949i	0.377428	+	0.377428i
$293$	18.2474	+	18.2474i	1.06603	+	1.06603i	0.997660	+	0.0683671i	$0.0217789\pi$
$293$	18.2474	+	18.2474i	1.06603	+	1.06603i	0.0683671	+	0.997660i	$0.478221\pi$
$294$	−20.6969			−1.20707
$295$		0			0
$296$	−10.8990			−0.633490
$297$	31.0454			1.80144
$298$	35.3939			2.05031
$299$	18.0000	+	12.0000i	1.04097	+	0.693978i
$300$		0			0
$301$	−0.247449	+	0.247449i	−0.0142627	+	0.0142627i
$302$	3.85357			0.221748
$303$	3.67423	−	3.67423i	0.211079	−	0.211079i
$304$	−22.2474	+	22.2474i	−1.27598	+	1.27598i
$305$		0			0
$306$	−11.0227	+	11.0227i	−0.630126	+	0.630126i
$307$	20.2474	−	20.2474i	1.15558	−	1.15558i	0.170168	−	0.985415i	$-0.445569\pi$
$307$	20.2474	−	20.2474i	1.15558	−	1.15558i	0.985415	−	0.170168i	$-0.0544310\pi$
$308$	−1.89898			−0.108204
$309$	−11.6969	+	11.6969i	−0.665416	+	0.665416i
$310$		0			0
$311$	−15.5505			−0.881789			−0.440894	−	0.897559i	$-0.645339\pi$
$311$	−15.5505			−0.881789			−0.440894	+	0.897559i	$0.645339\pi$
$312$	−6.00000	+	9.00000i	−0.339683	+	0.509525i
$313$		−	21.0000i		−	1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$		−	21.0000i		−	1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$	−8.57321	−	8.57321i	−0.483814	−	0.483814i
$315$		0			0
$316$			3.55051i			0.199732i
$317$	12.7980	−	12.7980i	0.718805	−	0.718805i	−0.249556	−	0.968360i	$-0.580285\pi$
$317$	12.7980	−	12.7980i	0.718805	−	0.718805i	0.968360	+	0.249556i	$0.0802845\pi$
$318$		−	5.69694i		−	0.319468i
$319$	−8.02270	−	8.02270i	−0.449185	−	0.449185i
$320$		0			0
$321$	1.65153	−	1.65153i	0.0921795	−	0.0921795i
$322$	3.30306			0.184072
$323$	−13.3485	−	13.3485i	−0.742729	−	0.742729i
$324$			9.00000i			0.500000i
$325$		0			0
$326$		−	22.8990i		−	1.26826i
$327$		−	19.1010i		−	1.05629i
$328$		−	13.3485i		−	0.737046i
$329$	−0.797959			−0.0439929
$330$		0			0
$331$	−19.7980	+	19.7980i	−1.08819	+	1.08819i	−0.0924797	+	0.995715i	$0.529479\pi$
$331$	−19.7980	+	19.7980i	−1.08819	+	1.08819i	−0.995715	+	0.0924797i	$0.970521\pi$
$332$	−1.77526	−	1.77526i	−0.0974298	−	0.0974298i
$333$	−13.3485	−	13.3485i	−0.731492	−	0.731492i
$334$	30.0000			1.64153
$335$		0			0
$336$			2.75255i			0.150164i
$337$	−17.0000			−0.926049			−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000			−0.926049			−0.463025	+	0.886345i	$0.653236\pi$
$338$	20.8207	+	8.57321i	1.13249	+	0.466321i
$339$	−20.6969	+	20.6969i	−1.12410	+	1.12410i
$340$		0			0
$341$		−	43.2929i		−	2.34444i
$342$	−32.6969			−1.76805
$343$	−3.12372	−	3.12372i	−0.168665	−	0.168665i
$344$	−1.34847	−	1.34847i	−0.0727046	−	0.0727046i
$345$		0			0
$346$	−8.32577	−	8.32577i	−0.447596	−	0.447596i
$347$	−16.0454			−0.861363			−0.430681	−	0.902504i	$-0.641727\pi$
$347$	−16.0454			−0.861363			−0.430681	+	0.902504i	$0.641727\pi$
$348$	2.32577	−	2.32577i	0.124674	−	0.124674i
$349$	−9.89898	−	9.89898i	−0.529880	−	0.529880i	0.390656	−	0.920537i	$-0.372248\pi$
$349$	−9.89898	−	9.89898i	−0.529880	−	0.529880i	−0.920537	+	0.390656i	$0.872248\pi$
$350$		0			0
$351$	−18.3712	+	3.67423i	−0.980581	+	0.196116i
$352$		−	31.0454i		−	1.65473i
$353$	−7.10102	+	7.10102i	−0.377949	+	0.377949i	−0.870362	−	0.492413i	$-0.836115\pi$
$353$	−7.10102	+	7.10102i	−0.377949	+	0.377949i	0.492413	+	0.870362i	$0.336115\pi$
$354$	−20.0227	−	20.0227i	−1.06419	−	1.06419i
$355$		0			0
$356$	0.550510	+	0.550510i	0.0291770	+	0.0291770i
$357$	−1.65153			−0.0874083
$358$	−15.0000	+	15.0000i	−0.792775	+	0.792775i
$359$	2.02270	−	2.02270i	0.106754	−	0.106754i	−0.651712	−	0.758466i	$-0.725949\pi$
$359$	2.02270	−	2.02270i	0.106754	−	0.106754i	0.758466	+	0.651712i	$0.225949\pi$
$360$		0			0
$361$		−	20.5959i		−	1.08400i
$362$	−29.0227	−	29.0227i	−1.52540	−	1.52540i
$363$	−30.2474	−	30.2474i	−1.58758	−	1.58758i
$364$	1.12372	−	0.224745i	0.0588992	−	0.0117798i
$365$		0			0
$366$			38.3939i			2.00688i
$367$		−	6.04541i		−	0.315568i	−0.987474	−	0.157784i	$-0.949565\pi$
$367$		−	6.04541i		−	0.315568i	0.987474	−	0.157784i	$-0.0504349\pi$
$368$			30.0000i			1.56386i
$369$	16.3485	−	16.3485i	0.851067	−	0.851067i
$370$		0			0
$371$	0.426786	−	0.426786i	0.0221576	−	0.0221576i
$372$	12.5505			0.650714
$373$			31.8990i			1.65167i	0.563914	+	0.825833i	$0.309295\pi$
$373$			31.8990i			1.65167i	−0.563914	+	0.825833i	$0.690705\pi$
$374$	31.0454			1.60532
$375$		0			0
$376$		−	4.34847i		−	0.224255i
$377$	5.69694	+	3.79796i	0.293407	+	0.195605i
$378$	−2.02270	+	2.02270i	−0.104037	+	0.104037i
$379$	26.6742	−	26.6742i	1.37016	−	1.37016i	0.509973	−	0.860190i	$-0.329655\pi$
$379$	26.6742	−	26.6742i	1.37016	−	1.37016i	0.860190	−	0.509973i	$-0.170345\pi$
$380$		0			0
$381$	−3.24745	+	3.24745i	−0.166372	+	0.166372i
$382$	10.6515	+	10.6515i	0.544980	+	0.544980i
$383$	10.6515	−	10.6515i	0.544268	−	0.544268i	−0.380509	−	0.924777i	$-0.624251\pi$
$383$	10.6515	−	10.6515i	0.544268	−	0.544268i	0.924777	+	0.380509i	$0.124251\pi$
$384$	−21.0000			−1.07165
$385$		0			0
$386$			3.55051i			0.180716i
$387$		−	3.30306i		−	0.167904i
$388$	6.44949	−	6.44949i	0.327423	−	0.327423i
$389$	−31.5959			−1.60198			−0.800988	−	0.598680i	$-0.795692\pi$
$389$	−31.5959			−1.60198			−0.800988	+	0.598680i	$0.795692\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$	8.44949	−	8.44949i	0.426764	−	0.426764i
$393$	−8.69694	−	8.69694i	−0.438703	−	0.438703i
$394$			22.6515i			1.14117i
$395$		0			0
$396$	12.6742	−	12.6742i	0.636904	−	0.636904i
$397$	−9.89898	+	9.89898i	−0.496816	+	0.496816i	−0.910445	−	0.413630i	$-0.864261\pi$
$397$	−9.89898	+	9.89898i	−0.496816	+	0.496816i	0.413630	+	0.910445i	$0.364261\pi$
$398$	−5.75255	−	5.75255i	−0.288349	−	0.288349i
$399$	−2.44949	−	2.44949i	−0.122628	−	0.122628i
$400$		0			0
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	−0.445729	−	0.895168i	$-0.647056\pi$
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	0.895168	+	0.445729i	$0.147056\pi$
$402$	14.0227	−	14.0227i	0.699389	−	0.699389i
$403$	5.12372	+	25.6186i	0.255231	+	1.27615i
$404$		−	3.00000i		−	0.149256i
$405$		0			0
$406$	1.04541			0.0518827
$407$			37.5959i			1.86356i
$408$		−	9.00000i		−	0.445566i
$409$	−8.00000	+	8.00000i	−0.395575	+	0.395575i	−0.876669	−	0.481094i	$-0.840239\pi$
$409$	−8.00000	+	8.00000i	−0.395575	+	0.395575i	0.481094	+	0.876669i	$0.340239\pi$
$410$		0			0
$411$	10.6515			0.525401
$412$			9.55051i			0.470520i
$413$		−	3.00000i		−	0.147620i
$414$	−22.0454	+	22.0454i	−1.08347	+	1.08347i
$415$		0			0
$416$	3.67423	+	18.3712i	0.180144	+	0.900721i
$417$	−11.4495	+	11.4495i	−0.560684	+	0.560684i
$418$	46.0454	+	46.0454i	2.25215	+	2.25215i
$419$			34.2929i			1.67532i	0.546195	+	0.837658i	$0.316076\pi$
$419$			34.2929i			1.67532i	−0.546195	+	0.837658i	$0.683924\pi$
$420$		0			0
$421$	−11.1010	+	11.1010i	−0.541031	+	0.541031i	−0.923831	−	0.382800i	$-0.874960\pi$
$421$	−11.1010	+	11.1010i	−0.541031	+	0.541031i	0.382800	+	0.923831i	$0.374960\pi$
$422$	−25.3485	+	25.3485i	−1.23394	+	1.23394i
$423$	5.32577	−	5.32577i	0.258948	−	0.258948i
$424$	2.32577	+	2.32577i	0.112949	+	0.112949i
$425$		0			0
$426$	18.0000	−	18.0000i	0.872103	−	0.872103i
$427$	−2.87628	+	2.87628i	−0.139193	+	0.139193i
$428$		−	1.34847i		−	0.0651807i
$429$	31.0454	+	20.6969i	1.49889	+	0.999258i
$430$		0			0
$431$	22.0454	+	22.0454i	1.06189	+	1.06189i	0.997954	+	0.0639359i	$0.0203653\pi$
$431$	22.0454	+	22.0454i	1.06189	+	1.06189i	0.0639359	+	0.997954i	$0.479635\pi$
$432$	−18.3712	−	18.3712i	−0.883883	−	0.883883i
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$	2.82066	+	2.82066i	0.135396	+	0.135396i
$435$		0			0
$436$	−7.79796	−	7.79796i	−0.373455	−	0.373455i
$437$	−26.6969	−	26.6969i	−1.27709	−	1.27709i
$438$	−19.3485	−	19.3485i	−0.924506	−	0.924506i
$439$		−	20.4495i		−	0.976001i	−0.872843	−	0.488000i	$-0.837726\pi$
$439$		−	20.4495i		−	0.976001i	0.872843	−	0.488000i	$-0.162274\pi$
$440$		0			0
$441$	20.6969			0.985568
$442$	−18.3712	+	3.67423i	−0.873828	+	0.174766i
$443$	−1.10102			−0.0523111			−0.0261555	−	0.999658i	$-0.508327\pi$
$443$	−1.10102			−0.0523111			−0.0261555	+	0.999658i	$0.508327\pi$
$444$	−10.8990			−0.517243
$445$		0			0
$446$	−18.4949			−0.875759
$447$	−35.3939			−1.67407
$448$	−0.224745	−	0.224745i	−0.0106182	−	0.0106182i
$449$	15.2474	−	15.2474i	0.719572	−	0.719572i	−0.248946	−	0.968517i	$-0.580084\pi$
$449$	15.2474	−	15.2474i	0.719572	−	0.719572i	0.968517	+	0.248946i	$0.0800841\pi$
$450$		0			0
$451$	−46.0454			−2.16819
$452$			16.8990i			0.794861i
$453$	−3.85357			−0.181057
$454$			25.0454i			1.17544i
$455$		0			0
$456$	13.3485	−	13.3485i	0.625099	−	0.625099i
$457$	18.6969	+	18.6969i	0.874606	+	0.874606i	0.992970	−	0.118364i	$-0.0377651\pi$
$457$	18.6969	+	18.6969i	0.874606	+	0.874606i	−0.118364	+	0.992970i	$0.537765\pi$
$458$	−25.1010			−1.17289
$459$	11.0227	−	11.0227i	0.514496	−	0.514496i
$460$		0			0
$461$	−9.55051	−	9.55051i	−0.444812	−	0.444812i	0.448814	−	0.893625i	$-0.351847\pi$
$461$	−9.55051	−	9.55051i	−0.444812	−	0.444812i	−0.893625	+	0.448814i	$0.851847\pi$
$462$	5.69694			0.265046
$463$	−2.42679	+	2.42679i	−0.112782	+	0.112782i	−0.761246	−	0.648463i	$-0.775412\pi$
$463$	−2.42679	+	2.42679i	−0.112782	+	0.112782i	0.648463	+	0.761246i	$0.275412\pi$
$464$			9.49490i			0.440790i
$465$		0			0
$466$	10.0454	+	10.0454i	0.465345	+	0.465345i
$467$			0.247449i			0.0114506i	0.999984	+	0.00572528i	$0.00182242\pi$
$467$			0.247449i			0.0114506i	−0.999984	+	0.00572528i	$0.998178\pi$
$468$	−6.00000	+	9.00000i	−0.277350	+	0.416025i
$469$	2.10102			0.0970161
$470$		0			0
$471$	8.57321	+	8.57321i	0.395033	+	0.395033i
$472$	16.3485			0.752499
$473$	−4.65153	+	4.65153i	−0.213878	+	0.213878i
$474$		−	10.6515i		−	0.489241i
$475$		0			0
$476$	−0.674235	+	0.674235i	−0.0309035	+	0.0309035i
$477$			5.69694i			0.260845i
$478$	−25.0454			−1.14555
$479$	9.12372	−	9.12372i	0.416874	−	0.416874i	−0.467251	−	0.884125i	$-0.654756\pi$
$479$	9.12372	−	9.12372i	0.416874	−	0.416874i	0.884125	+	0.467251i	$0.154756\pi$
$480$		0			0
$481$	−4.44949	−	22.2474i	−0.202879	−	1.01440i
$482$	−49.8434			−2.27030
$483$	−3.30306			−0.150295
$484$	−24.6969			−1.12259
$485$		0			0
$486$		−	27.0000i		−	1.22474i
$487$	−23.3712	−	23.3712i	−1.05905	−	1.05905i	−0.998144	−	0.0609054i	$-0.980601\pi$
$487$	−23.3712	−	23.3712i	−1.05905	−	1.05905i	−0.0609054	−	0.998144i	$-0.519399\pi$
$488$	−15.6742	−	15.6742i	−0.709539	−	0.709539i
$489$			22.8990i			1.03553i
$490$		0			0
$491$	−1.34847			−0.0608556			−0.0304278	−	0.999537i	$-0.509687\pi$
$491$	−1.34847			−0.0608556			−0.0304278	+	0.999537i	$0.509687\pi$
$492$		−	13.3485i		−	0.601795i
$493$	−5.69694			−0.256577
$494$	−32.6969	−	21.7980i	−1.47110	−	0.980737i
$495$		0			0
$496$	−25.6186	+	25.6186i	−1.15031	+	1.15031i
$497$	2.69694			0.120974
$498$	5.32577	+	5.32577i	0.238653	+	0.238653i
$499$	−15.5732	+	15.5732i	−0.697153	+	0.697153i	−0.963795	−	0.266643i	$-0.914086\pi$
$499$	−15.5732	+	15.5732i	−0.697153	+	0.697153i	0.266643	+	0.963795i	$0.414086\pi$
$500$		0			0
$501$	−30.0000			−1.34030
$502$	−17.6969	+	17.6969i	−0.789853	+	0.789853i
$503$	2.44949			0.109217			0.0546087	−	0.998508i	$-0.482609\pi$
$503$	2.44949			0.109217			0.0546087	+	0.998508i	$0.482609\pi$
$504$		−	1.65153i		−	0.0735650i
$505$		0			0
$506$	62.0908			2.76027
$507$	−20.8207	−	8.57321i	−0.924678	−	0.380750i
$508$			2.65153i			0.117643i
$509$	23.4495	+	23.4495i	1.03938	+	1.03938i	0.999192	+	0.0401882i	$0.0127958\pi$
$509$	23.4495	+	23.4495i	1.03938	+	1.03938i	0.0401882	+	0.999192i	$0.487204\pi$
$510$		0			0
$511$		−	2.89898i		−	0.128243i
$512$	−6.12372	+	6.12372i	−0.270633	+	0.270633i
$513$	32.6969			1.44361
$514$	35.0227	+	35.0227i	1.54479	+	1.54479i
$515$		0			0
$516$	−1.34847	−	1.34847i	−0.0593630	−	0.0593630i
$517$	−15.0000			−0.659699
$518$	−2.44949	−	2.44949i	−0.107624	−	0.107624i
$519$	8.32577	+	8.32577i	0.365461	+	0.365461i
$520$		0			0
$521$			7.10102i			0.311101i	0.987828	+	0.155551i	$0.0497152\pi$
$521$			7.10102i			0.311101i	−0.987828	+	0.155551i	$0.950285\pi$
$522$	−6.97730	+	6.97730i	−0.305388	+	0.305388i
$523$		−	24.6969i		−	1.07992i	−0.841690	−	0.539961i	$-0.818439\pi$
$523$		−	24.6969i		−	1.07992i	0.841690	−	0.539961i	$-0.181561\pi$
$524$	−7.10102			−0.310210
$525$		0			0
$526$	−26.6969	+	26.6969i	−1.16404	+	1.16404i
$527$	−15.3712	−	15.3712i	−0.669579	−	0.669579i
$528$			51.7423i			2.25180i
$529$	−13.0000			−0.565217
$530$		0			0
$531$	20.0227	+	20.0227i	0.868911	+	0.868911i
$532$	−2.00000			−0.0867110
$533$	27.2474	−	5.44949i	1.18022	−	0.236044i
$534$	−1.65153	−	1.65153i	−0.0714687	−	0.0714687i
$535$		0			0
$536$			11.4495i			0.494543i
$537$	15.0000	−	15.0000i	0.647298	−	0.647298i
$538$	8.32577	+	8.32577i	0.358949	+	0.358949i
$539$	−29.1464	−	29.1464i	−1.25543	−	1.25543i
$540$		0			0
$541$	3.89898	+	3.89898i	0.167630	+	0.167630i	0.785937	−	0.618307i	$-0.212181\pi$
$541$	3.89898	+	3.89898i	0.167630	+	0.167630i	−0.618307	+	0.785937i	$0.712181\pi$
$542$	−17.4495			−0.749520
$543$	29.0227	+	29.0227i	1.24548	+	1.24548i
$544$	−11.0227	−	11.0227i	−0.472595	−	0.472595i
$545$		0			0
$546$	−3.37117	+	0.674235i	−0.144273	+	0.0288546i
$547$			32.0000i			1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$			32.0000i			1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	4.34847	−	4.34847i	0.185757	−	0.185757i
$549$		−	38.3939i		−	1.63861i
$550$		0			0
$551$	−8.44949	−	8.44949i	−0.359960	−	0.359960i
$552$		−	18.0000i		−	0.766131i
$553$	0.797959	−	0.797959i	0.0339327	−	0.0339327i
$554$	6.00000	−	6.00000i	0.254916	−	0.254916i
$555$		0			0
$556$			9.34847i			0.396463i
$557$	18.5505	+	18.5505i	0.786010	+	0.786010i	0.980838	−	0.194827i	$-0.0624146\pi$
$557$	18.5505	+	18.5505i	0.786010	+	0.786010i	−0.194827	+	0.980838i	$0.562415\pi$
$558$	−37.6515			−1.59392
$559$	2.20204	−	3.30306i	0.0931364	−	0.139705i
$560$		0			0
$561$	−31.0454			−1.31074
$562$		−	23.3939i		−	0.986811i
$563$			4.04541i			0.170494i	0.996360	+	0.0852468i	$0.0271679\pi$
$563$			4.04541i			0.170494i	−0.996360	+	0.0852468i	$0.972832\pi$
$564$		−	4.34847i		−	0.183104i
$565$		0			0
$566$	6.55051	−	6.55051i	0.275338	−	0.275338i
$567$	2.02270	−	2.02270i	0.0849456	−	0.0849456i
$568$			14.6969i			0.616670i
$569$	−7.89898			−0.331142			−0.165571	−	0.986198i	$-0.552947\pi$
$569$	−7.89898			−0.331142			−0.165571	+	0.986198i	$0.552947\pi$
$570$		0			0
$571$		−	9.34847i		−	0.391221i	−0.980682	−	0.195611i	$-0.937331\pi$
$571$		−	9.34847i		−	0.391221i	0.980682	−	0.195611i	$-0.0626689\pi$
$572$	21.1237	−	4.22474i	0.883227	−	0.176645i
$573$	−10.6515	−	10.6515i	−0.444974	−	0.444974i
$574$	3.00000	−	3.00000i	0.125218	−	0.125218i
$575$		0			0
$576$	3.00000			0.125000
$577$	−16.4495	−	16.4495i	−0.684801	−	0.684801i	0.276277	−	0.961078i	$-0.410899\pi$
$577$	−16.4495	−	16.4495i	−0.684801	−	0.684801i	−0.961078	+	0.276277i	$0.910899\pi$
$578$	−9.79796	+	9.79796i	−0.407541	+	0.407541i
$579$		−	3.55051i		−	0.147554i
$580$		0			0
$581$			0.797959i			0.0331049i
$582$	−19.3485	+	19.3485i	−0.802020	+	0.802020i
$583$	8.02270	−	8.02270i	0.332266	−	0.332266i
$584$	15.7980			0.653724
$585$		0			0
$586$	−44.6969			−1.84641
$587$	13.7753	−	13.7753i	0.568566	−	0.568566i	−0.363161	−	0.931726i	$-0.618302\pi$
$587$	13.7753	−	13.7753i	0.568566	−	0.568566i	0.931726	+	0.363161i	$0.118302\pi$
$588$	8.44949	−	8.44949i	0.348451	−	0.348451i
$589$		−	45.5959i		−	1.87875i
$590$		0			0
$591$		−	22.6515i		−	0.931759i
$592$	22.2474	−	22.2474i	0.914365	−	0.914365i
$593$	−10.5959	−	10.5959i	−0.435122	−	0.435122i	0.455244	−	0.890367i	$-0.349552\pi$
$593$	−10.5959	−	10.5959i	−0.435122	−	0.435122i	−0.890367	+	0.455244i	$0.849552\pi$
$594$	−38.0227	+	38.0227i	−1.56009	+	1.56009i
$595$		0			0
$596$	−14.4495	+	14.4495i	−0.591874	+	0.591874i
$597$	5.75255	+	5.75255i	0.235436	+	0.235436i
$598$	−36.7423	+	7.34847i	−1.50251	+	0.300501i
$599$			41.3939i			1.69131i	0.533732	+	0.845654i	$0.320789\pi$
$599$			41.3939i			1.69131i	−0.533732	+	0.845654i	$0.679211\pi$
$600$		0			0
$601$	19.8990			0.811696			0.405848	−	0.913941i	$-0.366976\pi$
$601$	19.8990			0.811696			0.405848	+	0.913941i	$0.366976\pi$
$602$		−	0.606123i		−	0.0247037i
$603$	−14.0227	+	14.0227i	−0.571049	+	0.571049i
$604$	−1.57321	+	1.57321i	−0.0640132	+	0.0640132i
$605$		0			0
$606$			9.00000i			0.365600i
$607$		−	36.0454i		−	1.46304i	−0.681821	−	0.731519i	$-0.738812\pi$
$607$		−	36.0454i		−	1.46304i	0.681821	−	0.731519i	$-0.261188\pi$
$608$		−	32.6969i		−	1.32604i
$609$	−1.04541			−0.0423621
$610$		0			0
$611$	8.87628	−	1.77526i	0.359096	−	0.0718191i
$612$		−	9.00000i		−	0.363803i
$613$	19.1464	+	19.1464i	0.773317	+	0.773317i	0.978685	−	0.205368i	$-0.0658391\pi$
$613$	19.1464	+	19.1464i	0.773317	+	0.773317i	−0.205368	+	0.978685i	$0.565839\pi$
$614$			49.5959i			2.00153i
$615$		0			0
$616$	−2.32577	+	2.32577i	−0.0937078	+	0.0937078i
$617$	−11.6969	+	11.6969i	−0.470901	+	0.470901i	−0.902206	−	0.431305i	$-0.858053\pi$
$617$	−11.6969	+	11.6969i	−0.470901	+	0.470901i	0.431305	+	0.902206i	$0.358053\pi$
$618$		−	28.6515i		−	1.15253i
$619$	10.2474	+	10.2474i	0.411880	+	0.411880i	0.882393	−	0.470513i	$-0.155931\pi$
$619$	10.2474	+	10.2474i	0.411880	+	0.411880i	−0.470513	+	0.882393i	$0.655931\pi$
$620$		0			0
$621$	22.0454	−	22.0454i	0.884652	−	0.884652i
$622$	19.0454	−	19.0454i	0.763651	−	0.763651i
$623$		−	0.247449i		−	0.00991382i
$624$	−6.12372	−	30.6186i	−0.245145	−	1.22573i
$625$		0			0
$626$	25.7196	+	25.7196i	1.02796	+	1.02796i
$627$	−46.0454	−	46.0454i	−1.83888	−	1.83888i
$628$	7.00000			0.279330
$629$	13.3485	+	13.3485i	0.532238	+	0.532238i
$630$		0			0
$631$	−12.6969	−	12.6969i	−0.505457	−	0.505457i	0.407672	−	0.913129i	$-0.366341\pi$
$631$	−12.6969	−	12.6969i	−0.505457	−	0.505457i	−0.913129	+	0.407672i	$0.866341\pi$
$632$	4.34847	+	4.34847i	0.172973	+	0.172973i
$633$	25.3485	−	25.3485i	1.00751	−	1.00751i
$634$			31.3485i			1.24501i
$635$		0			0
$636$	2.32577	+	2.32577i	0.0922226	+	0.0922226i
$637$	20.6969	+	13.7980i	0.820043	+	0.546695i
$638$	19.6515			0.778012
$639$	−18.0000	+	18.0000i	−0.712069	+	0.712069i
$640$		0			0
$641$	−35.6969			−1.40994			−0.704972	−	0.709235i	$-0.749041\pi$
$641$	−35.6969			−1.40994			−0.704972	+	0.709235i	$0.749041\pi$
$642$			4.04541i			0.159660i
$643$	12.6969	+	12.6969i	0.500718	+	0.500718i	0.911661	−	0.410943i	$-0.134800\pi$
$643$	12.6969	+	12.6969i	0.500718	+	0.500718i	−0.410943	+	0.911661i	$0.634800\pi$
$644$	−1.34847	+	1.34847i	−0.0531371	+	0.0531371i
$645$		0			0
$646$	32.6969			1.28644
$647$			21.5505i			0.847238i	0.905840	+	0.423619i	$0.139240\pi$
$647$			21.5505i			0.847238i	−0.905840	+	0.423619i	$0.860760\pi$
$648$	11.0227	+	11.0227i	0.433013	+	0.433013i
$649$		−	56.3939i		−	2.21365i
$650$		0			0
$651$	−2.82066	−	2.82066i	−0.110550	−	0.110550i
$652$	9.34847	+	9.34847i	0.366114	+	0.366114i
$653$	5.20204			0.203572			0.101786	−	0.994806i	$-0.467544\pi$
$653$	5.20204			0.203572			0.101786	+	0.994806i	$0.467544\pi$
$654$	23.3939	+	23.3939i	0.914773	+	0.914773i
$655$		0			0
$656$	27.2474	+	27.2474i	1.06383	+	1.06383i
$657$	19.3485	+	19.3485i	0.754856	+	0.754856i
$658$	0.977296	−	0.977296i	0.0380990	−	0.0380990i
$659$			6.24745i			0.243366i	0.992569	+	0.121683i	$0.0388291\pi$
$659$			6.24745i			0.243366i	−0.992569	+	0.121683i	$0.961171\pi$
$660$		0			0
$661$	8.89898	+	8.89898i	0.346130	+	0.346130i	0.858666	−	0.512536i	$-0.171294\pi$
$661$	8.89898	+	8.89898i	0.346130	+	0.346130i	−0.512536	+	0.858666i	$0.671294\pi$
$662$		−	48.4949i		−	1.88481i
$663$	18.3712	−	3.67423i	0.713477	−	0.142695i
$664$	−4.34847			−0.168753
$665$		0			0
$666$	32.6969			1.26698
$667$	−11.3939			−0.441173
$668$	−12.2474	+	12.2474i	−0.473868	+	0.473868i
$669$	18.4949			0.715054
$670$		0			0
$671$	−54.0681	+	54.0681i	−2.08728	+	2.08728i
$672$	−2.02270	−	2.02270i	−0.0780275	−	0.0780275i
$673$	10.5959			0.408443			0.204221	−	0.978925i	$-0.434534\pi$
$673$	10.5959			0.408443			0.204221	+	0.978925i	$0.434534\pi$
$674$	20.8207	−	20.8207i	0.801982	−	0.801982i
$675$		0			0
$676$	−12.0000	+	5.00000i	−0.461538	+	0.192308i
$677$	36.4949			1.40261			0.701306	−	0.712860i	$-0.252600\pi$
$677$	36.4949			1.40261			0.701306	+	0.712860i	$0.252600\pi$
$678$		−	50.6969i		−	1.94700i
$679$	−2.89898			−0.111253
$680$		0			0
$681$		−	25.0454i		−	0.959742i
$682$	53.0227	+	53.0227i	2.03034	+	2.03034i
$683$	−0.921683	−	0.921683i	−0.0352672	−	0.0352672i	0.689253	−	0.724520i	$-0.257939\pi$
$683$	−0.921683	−	0.921683i	−0.0352672	−	0.0352672i	−0.724520	+	0.689253i	$0.757939\pi$
$684$	13.3485	−	13.3485i	0.510391	−	0.510391i
$685$		0			0
$686$	7.65153			0.292137
$687$	25.1010			0.957664
$688$	5.50510			0.209880
$689$	−3.79796	+	5.69694i	−0.144691	+	0.217036i
$690$		0			0
$691$	−8.67423	+	8.67423i	−0.329983	+	0.329983i	−0.852580	−	0.522597i	$-0.824963\pi$
$691$	−8.67423	+	8.67423i	−0.329983	+	0.329983i	0.522597	+	0.852580i	$0.324963\pi$
$692$	6.79796			0.258420
$693$	−5.69694			−0.216409
$694$	19.6515	−	19.6515i	0.745962	−	0.745962i
$695$		0			0
$696$		−	5.69694i		−	0.215942i
$697$	−16.3485	+	16.3485i	−0.619242	+	0.619242i
$698$	24.2474			0.917779
$699$	−10.0454	−	10.0454i	−0.379952	−	0.379952i
$700$		0			0
$701$	−11.2020			−0.423095			−0.211548	−	0.977368i	$-0.567850\pi$
$701$	−11.2020			−0.423095			−0.211548	+	0.977368i	$0.567850\pi$
$702$	18.0000	−	27.0000i	0.679366	−	1.01905i
$703$			39.5959i			1.49339i
$704$	−4.22474	−	4.22474i	−0.159226	−	0.159226i
$705$		0			0
$706$		−	17.3939i		−	0.654627i
$707$	−0.674235	+	0.674235i	−0.0253572	+	0.0253572i
$708$	16.3485			0.614413
$709$	5.24745	+	5.24745i	0.197072	+	0.197072i	0.798744	−	0.601672i	$-0.205498\pi$
$709$	5.24745	+	5.24745i	0.197072	+	0.197072i	−0.601672	+	0.798744i	$0.705498\pi$
$710$		0			0
$711$			10.6515i			0.399464i
$712$	1.34847			0.0505360
$713$	−30.7423	−	30.7423i	−1.15131	−	1.15131i
$714$	2.02270	−	2.02270i	0.0756978	−	0.0756978i
$715$		0			0
$716$		−	12.2474i		−	0.457709i
$717$	25.0454			0.935338
$718$			4.95459i			0.184904i
$719$	−17.3939			−0.648682			−0.324341	−	0.945940i	$-0.605143\pi$
$719$	−17.3939			−0.648682			−0.324341	+	0.945940i	$0.605143\pi$
$720$		0			0
$721$	2.14643	−	2.14643i	0.0799372	−	0.0799372i
$722$	25.2247	+	25.2247i	0.938768	+	0.938768i
$723$	49.8434			1.85369
$724$	23.6969			0.880690
$725$		0			0
$726$	74.0908			2.74977
$727$	−30.0454			−1.11432			−0.557161	−	0.830404i	$-0.688109\pi$
$727$	−30.0454			−1.11432			−0.557161	+	0.830404i	$0.688109\pi$
$728$	1.10102	−	1.65153i	0.0408065	−	0.0612098i
$729$			27.0000i			1.00000i
$730$		0			0
$731$			3.30306i			0.122168i
$732$	−15.6742	−	15.6742i	−0.579336	−	0.579336i
$733$	−20.3485	−	20.3485i	−0.751588	−	0.751588i	0.223188	−	0.974776i	$-0.428354\pi$
$733$	−20.3485	−	20.3485i	−0.751588	−	0.751588i	−0.974776	+	0.223188i	$0.928354\pi$
$734$	7.40408	+	7.40408i	0.273290	+	0.273290i
$735$		0			0
$736$	−22.0454	−	22.0454i	−0.812605	−	0.812605i
$737$	39.4949			1.45481
$738$			40.0454i			1.47409i
$739$	−16.0227	−	16.0227i	−0.589405	−	0.589405i	0.348066	−	0.937470i	$-0.386839\pi$
$739$	−16.0227	−	16.0227i	−0.589405	−	0.589405i	−0.937470	+	0.348066i	$0.886839\pi$
$740$		0			0
$741$	32.6969	+	21.7980i	1.20115	+	0.800768i
$742$			1.04541i			0.0383781i
$743$	6.92168	−	6.92168i	0.253932	−	0.253932i	−0.568649	−	0.822581i	$-0.692534\pi$
$743$	6.92168	−	6.92168i	0.253932	−	0.253932i	0.822581	+	0.568649i	$0.192534\pi$
$744$	15.3712	−	15.3712i	0.563535	−	0.563535i
$745$		0			0
$746$	−39.0681	−	39.0681i	−1.43039	−	1.43039i
$747$	−5.32577	−	5.32577i	−0.194860	−	0.194860i
$748$	−12.6742	+	12.6742i	−0.463416	+	0.463416i
$749$	−0.303062	+	0.303062i	−0.0110736	+	0.0110736i
$750$		0			0
$751$			54.4949i			1.98855i	0.106865	+	0.994274i	$0.465919\pi$
$751$			54.4949i			1.98855i	−0.106865	+	0.994274i	$0.534081\pi$
$752$	8.87628	+	8.87628i	0.323684	+	0.323684i
$753$	17.6969	−	17.6969i	0.644912	−	0.644912i
$754$	−11.6288	+	2.32577i	−0.423497	+	0.0846994i
$755$		0			0
$756$		−	1.65153i		−	0.0600656i
$757$			21.4949i			0.781245i	0.920551	+	0.390623i	$0.127740\pi$
$757$			21.4949i			0.781245i	−0.920551	+	0.390623i	$0.872260\pi$
$758$			65.3383i			2.37319i
$759$	−62.0908			−2.25375
$760$		0			0
$761$	−15.4949	+	15.4949i	−0.561690	+	0.561690i	−0.929787	−	0.368098i	$-0.880009\pi$
$761$	−15.4949	+	15.4949i	−0.561690	+	0.561690i	0.368098	+	0.929787i	$0.380009\pi$
$762$		−	7.95459i		−	0.288164i
$763$			3.50510i			0.126893i
$764$	−8.69694			−0.314644
$765$		0			0
$766$			26.0908i			0.942699i
$767$	6.67423	+	33.3712i	0.240993	+	1.20496i
$768$	23.2702	−	23.2702i	0.839689	−	0.839689i
$769$	10.6969	−	10.6969i	0.385741	−	0.385741i	−0.487424	−	0.873165i	$-0.662063\pi$
$769$	10.6969	−	10.6969i	0.385741	−	0.385741i	0.873165	+	0.487424i	$0.162063\pi$
$770$		0			0
$771$	−35.0227	−	35.0227i	−1.26131	−	1.26131i
$772$	−1.44949	−	1.44949i	−0.0521683	−	0.0521683i
$773$	−35.1464	+	35.1464i	−1.26413	+	1.26413i	−0.315056	+	0.949073i	$0.602023\pi$
$773$	−35.1464	+	35.1464i	−1.26413	+	1.26413i	−0.949073	+	0.315056i	$0.897977\pi$
$774$	4.04541	+	4.04541i	0.145409	+	0.145409i
$775$		0			0
$776$		−	15.7980i		−	0.567114i
$777$	2.44949	+	2.44949i	0.0878750	+	0.0878750i
$778$	38.6969	−	38.6969i	1.38735	−	1.38735i
$779$	−48.4949			−1.73751
$780$		0			0
$781$	50.6969			1.81408
$782$	22.0454	−	22.0454i	0.788342	−	0.788342i
$783$	6.97730	−	6.97730i	0.249348	−	0.249348i
$784$			34.4949i			1.23196i
$785$		0			0
$786$	21.3031			0.759855
$787$	−9.57321	+	9.57321i	−0.341248	+	0.341248i	−0.856836	−	0.515588i	$-0.827573\pi$
$787$	−9.57321	+	9.57321i	−0.341248	+	0.341248i	0.515588	+	0.856836i	$0.327573\pi$
$788$	−9.24745	−	9.24745i	−0.329427	−	0.329427i
$789$	26.6969	−	26.6969i	0.950436	−	0.950436i
$790$		0			0
$791$	3.79796	−	3.79796i	0.135040	−	0.135040i
$792$		−	31.0454i		−	1.10315i
$793$	25.5959	−	38.3939i	0.908938	−	1.36341i
$794$		−	24.2474i		−	0.860510i
$795$		0			0
$796$	4.69694			0.166479
$797$			18.7980i			0.665858i	0.942952	+	0.332929i	$0.108037\pi$
$797$			18.7980i			0.665858i	−0.942952	+	0.332929i	$0.891963\pi$
$798$	6.00000			0.212398
$799$	−5.32577	+	5.32577i	−0.188412	+	0.188412i
$800$		0			0
$801$	1.65153	+	1.65153i	0.0583540	+	0.0583540i
$802$			22.0454i			0.778450i
$803$		−	54.4949i		−	1.92308i
$804$			11.4495i			0.403792i
$805$		0			0
$806$	−37.6515	−	25.1010i	−1.32622	−	0.884146i
$807$	−8.32577	−	8.32577i	−0.293081	−	0.293081i
$808$	−3.67423	−	3.67423i	−0.129259	−	0.129259i
$809$			34.2929i			1.20567i	0.797865	+	0.602836i	$0.205963\pi$
$809$			34.2929i			1.20567i	−0.797865	+	0.602836i	$0.794037\pi$
$810$		0			0
$811$	−5.77526	+	5.77526i	−0.202797	+	0.202797i	−0.801197	−	0.598400i	$-0.795803\pi$
$811$	−5.77526	+	5.77526i	−0.202797	+	0.202797i	0.598400	+	0.801197i	$0.295803\pi$
$812$	−0.426786	+	0.426786i	−0.0149772	+	0.0149772i
$813$	17.4495			0.611980
$814$	−46.0454	−	46.0454i	−1.61389	−	1.61389i
$815$		0			0
$816$	18.3712	+	18.3712i	0.643120	+	0.643120i
$817$	−4.89898	+	4.89898i	−0.171394	+	0.171394i
$818$		−	19.5959i		−	0.685155i
$819$	3.37117	−	0.674235i	0.117798	−	0.0235597i
$820$		0			0
$821$	0.303062	+	0.303062i	0.0105769	+	0.0105769i	0.712375	−	0.701799i	$-0.247619\pi$
$821$	0.303062	+	0.303062i	0.0105769	+	0.0105769i	−0.701799	+	0.712375i	$0.747619\pi$
$822$	−13.0454	+	13.0454i	−0.455011	+	0.455011i
$823$	−14.6969			−0.512303			−0.256152	−	0.966637i	$-0.582455\pi$
$823$	−14.6969			−0.512303			−0.256152	+	0.966637i	$0.582455\pi$
$824$	11.6969	+	11.6969i	0.407482	+	0.407482i
$825$		0			0
$826$	3.67423	+	3.67423i	0.127843	+	0.127843i
$827$	−15.9773	−	15.9773i	−0.555585	−	0.555585i	0.372462	−	0.928047i	$-0.378514\pi$
$827$	−15.9773	−	15.9773i	−0.555585	−	0.555585i	−0.928047	+	0.372462i	$0.878514\pi$
$828$		−	18.0000i		−	0.625543i
$829$			39.6969i			1.37873i	0.724413	+	0.689366i	$0.242111\pi$
$829$			39.6969i			1.37873i	−0.724413	+	0.689366i	$0.757889\pi$
$830$		0			0
$831$	−6.00000	+	6.00000i	−0.208138	+	0.208138i
$832$	3.00000	+	2.00000i	0.104006	+	0.0693375i
$833$	−20.6969			−0.717106
$834$		−	28.0454i		−	0.971133i
$835$		0			0
$836$	−37.5959			−1.30028
$837$	37.6515			1.30143
$838$	−42.0000	−	42.0000i	−1.45087	−	1.45087i
$839$	37.3485	−	37.3485i	1.28941	−	1.28941i	0.354269	−	0.935144i	$-0.384730\pi$
$839$	37.3485	−	37.3485i	1.28941	−	1.28941i	0.935144	−	0.354269i	$-0.115270\pi$
$840$		0			0
$841$	25.3939			0.875651
$842$		−	27.1918i		−	0.937093i
$843$			23.3939i			0.805728i
$844$		−	20.6969i		−	0.712418i
$845$		0			0
$846$			13.0454i			0.448510i
$847$	5.55051	+	5.55051i	0.190718	+	0.190718i
$848$	−9.49490			−0.326056
$849$	−6.55051	+	6.55051i	−0.224813	+	0.224813i
$850$		0			0
$851$	26.6969	+	26.6969i	0.915159	+	0.915159i
$852$			14.6969i			0.503509i
$853$	5.65153	−	5.65153i	0.193505	−	0.193505i	−0.603704	−	0.797209i	$-0.706309\pi$
$853$	5.65153	−	5.65153i	0.193505	−	0.193505i	0.797209	+	0.603704i	$0.206309\pi$
$854$		−	7.04541i		−	0.241089i
$855$		0			0
$856$	−1.65153	−	1.65153i	−0.0564482	−	0.0564482i
$857$			49.5959i			1.69416i	0.531462	+	0.847082i	$0.321643\pi$
$857$			49.5959i			1.69416i	−0.531462	+	0.847082i	$0.678357\pi$
$858$	−63.3712	+	12.6742i	−2.16346	+	0.432691i
$859$	−1.59592			−0.0544520			−0.0272260	−	0.999629i	$-0.508667\pi$
$859$	−1.59592			−0.0544520			−0.0272260	+	0.999629i	$0.508667\pi$
$860$		0			0
$861$	−3.00000	+	3.00000i	−0.102240	+	0.102240i
$862$	−54.0000			−1.83925
$863$	−8.87628	+	8.87628i	−0.302152	+	0.302152i	−0.841855	−	0.539703i	$-0.818536\pi$
$863$	−8.87628	+	8.87628i	−0.302152	+	0.302152i	0.539703	+	0.841855i	$0.318536\pi$
$864$	27.0000			0.918559
$865$		0			0
$866$	19.5959	−	19.5959i	0.665896	−	0.665896i
$867$	9.79796	−	9.79796i	0.332756	−	0.332756i
$868$	−2.30306			−0.0781710
$869$	15.0000	−	15.0000i	0.508840	−	0.508840i
$870$		0			0
$871$	−23.3712	+	4.67423i	−0.791902	+	0.158380i
$872$	−19.1010			−0.646842
$873$	19.3485	−	19.3485i	0.654846	−	0.654846i
$874$	65.3939			2.21198
$875$		0			0
$876$	15.7980			0.533764
$877$	15.9444	+	15.9444i	0.538404	+	0.538404i	0.923060	−	0.384656i	$-0.125680\pi$
$877$	15.9444	+	15.9444i	0.538404	+	0.538404i	−0.384656	+	0.923060i	$0.625680\pi$
$878$	25.0454	+	25.0454i	0.845242	+	0.845242i
$879$	44.6969			1.50759
$880$		0			0
$881$	−4.10102			−0.138167			−0.0690834	−	0.997611i	$-0.522007\pi$
$881$	−4.10102			−0.138167			−0.0690834	+	0.997611i	$0.522007\pi$
$882$	−25.3485	+	25.3485i	−0.853527	+	0.853527i
$883$	42.6969			1.43687			0.718433	−	0.695596i	$-0.244860\pi$
$883$	42.6969			1.43687			0.718433	+	0.695596i	$0.244860\pi$
$884$	6.00000	−	9.00000i	0.201802	−	0.302703i
$885$		0			0
$886$	1.34847	−	1.34847i	0.0453027	−	0.0453027i
$887$	1.34847			0.0452772			0.0226386	−	0.999744i	$-0.492793\pi$
$887$	1.34847			0.0452772			0.0226386	+	0.999744i	$0.492793\pi$
$888$	−13.3485	+	13.3485i	−0.447945	+	0.447945i
$889$	0.595918	−	0.595918i	0.0199864	−	0.0199864i
$890$		0			0
$891$	38.0227	−	38.0227i	1.27381	−	1.27381i
$892$	7.55051	−	7.55051i	0.252810	−	0.252810i
$893$	−15.7980			−0.528659
$894$	43.3485	−	43.3485i	1.44979	−	1.44979i
$895$		0			0
$896$	3.85357			0.128739
$897$	36.7423	−	7.34847i	1.22679	−	0.245358i
$898$			37.3485i			1.24633i
$899$	−9.72985	−	9.72985i	−0.324509	−	0.324509i
$900$		0			0
$901$		−	5.69694i		−	0.189793i
$902$	56.3939	−	56.3939i	1.87771	−	1.87771i
$903$			0.606123i			0.0201705i
$904$	20.6969	+	20.6969i	0.688370	+	0.688370i
$905$		0			0
$906$	4.71964	−	4.71964i	0.156800	−	0.156800i
$907$	46.6969			1.55055			0.775273	−	0.631626i	$-0.217612\pi$
$907$	46.6969			1.55055			0.775273	+	0.631626i	$0.217612\pi$
$908$	−10.2247	−	10.2247i	−0.339320	−	0.339320i
$909$		−	9.00000i		−	0.298511i
$910$		0			0
$911$			35.1464i			1.16445i	0.813027	+	0.582227i	$0.197818\pi$
$911$			35.1464i			1.16445i	−0.813027	+	0.582227i	$0.802182\pi$
$912$			54.4949i			1.80451i
$913$			15.0000i			0.496428i
$914$	−45.7980			−1.51486
$915$		0			0
$916$	10.2474	−	10.2474i	0.338585	−	0.338585i
$917$	1.59592	+	1.59592i	0.0527019	+	0.0527019i
$918$			27.0000i			0.891133i
$919$	15.7980			0.521127			0.260563	−	0.965457i	$-0.416092\pi$
$919$	15.7980			0.521127			0.260563	+	0.965457i	$0.416092\pi$
$920$		0			0
$921$		−	49.5959i		−	1.63424i
$922$	23.3939			0.770436
$923$	−30.0000	+	6.00000i	−0.987462	+	0.197492i
$924$	−2.32577	+	2.32577i	−0.0765121	+	0.0765121i
$925$		0			0
$926$		−	5.94439i		−	0.195345i
$927$			28.6515i			0.941040i
$928$	−6.97730	−	6.97730i	−0.229041	−	0.229041i
$929$	−20.3939	−	20.3939i	−0.669101	−	0.669101i	0.288407	−	0.957508i	$-0.406875\pi$
$929$	−20.3939	−	20.3939i	−0.669101	−	0.669101i	−0.957508	+	0.288407i	$0.906875\pi$
$930$		0			0
$931$	−30.6969	−	30.6969i	−1.00605	−	1.00605i
$932$	−8.20204			−0.268667
$933$	−19.0454	+	19.0454i	−0.623519	+	0.623519i
$934$	−0.303062	−	0.303062i	−0.00991648	−	0.00991648i
$935$		0			0
$936$	3.67423	+	18.3712i	0.120096	+	0.600481i
$937$		−	1.40408i		−	0.0458694i	−0.999737	−	0.0229347i	$-0.992699\pi$
$937$		−	1.40408i		−	0.0458694i	0.999737	−	0.0229347i	$-0.00730098\pi$
$938$	−2.57321	+	2.57321i	−0.0840184	+	0.0840184i
$939$	−25.7196	−	25.7196i	−0.839329	−	0.839329i
$940$		0			0
$941$	6.24745	+	6.24745i	0.203661	+	0.203661i	0.801567	−	0.597906i	$-0.204000\pi$
$941$	6.24745	+	6.24745i	0.203661	+	0.203661i	−0.597906	+	0.801567i	$0.704000\pi$
$942$	−21.0000			−0.684217
$943$	−32.6969	+	32.6969i	−1.06476	+	1.06476i
$944$	−33.3712	+	33.3712i	−1.08614	+	1.08614i
$945$		0			0
$946$		−	11.3939i		−	0.370447i
$947$	−33.3712	−	33.3712i	−1.08442	−	1.08442i	−0.996092	−	0.0883258i	$-0.971848\pi$
$947$	−33.3712	−	33.3712i	−1.08442	−	1.08442i	−0.0883258	−	0.996092i	$-0.528152\pi$
$948$	4.34847	+	4.34847i	0.141232	+	0.141232i
$949$	6.44949	+	32.2474i	0.209359	+	1.04680i
$950$		0			0
$951$		−	31.3485i		−	1.01654i
$952$			1.65153i			0.0535264i
$953$		−	47.6969i		−	1.54506i	−0.634981	−	0.772528i	$-0.718992\pi$
$953$		−	47.6969i		−	1.54506i	0.634981	−	0.772528i	$-0.281008\pi$
$954$	−6.97730	−	6.97730i	−0.225898	−	0.225898i
$955$		0			0
$956$	10.2247	−	10.2247i	0.330692	−	0.330692i
$957$	−19.6515			−0.635244
$958$			22.3485i			0.722046i
$959$	−1.95459			−0.0631171
$960$		0			0
$961$		−	21.5051i		−	0.693713i
$962$	32.6969	+	21.7980i	1.05419	+	0.702794i
$963$		−	4.04541i		−	0.130361i
$964$	20.3485	−	20.3485i	0.655380	−	0.655380i
$965$		0			0
$966$	4.04541	−	4.04541i	0.130159	−	0.130159i
$967$	26.9217	+	26.9217i	0.865743	+	0.865743i	0.991998	−	0.126255i	$-0.0402957\pi$
$967$	26.9217	+	26.9217i	0.865743	+	0.865743i	−0.126255	+	0.991998i	$0.540296\pi$
$968$	−30.2474	+	30.2474i	−0.972190	+	0.972190i
$969$	−32.6969			−1.05038
$970$		0			0
$971$		−	42.2474i		−	1.35579i	−0.735161	−	0.677893i	$-0.762893\pi$
$971$		−	42.2474i		−	1.35579i	0.735161	−	0.677893i	$-0.237107\pi$
$972$	11.0227	+	11.0227i	0.353553	+	0.353553i
$973$	2.10102	−	2.10102i	0.0673556	−	0.0673556i
$974$	57.2474			1.83433
$975$		0			0
$976$	63.9898			2.04826
$977$	−13.6515	+	13.6515i	−0.436751	+	0.436751i	−0.890917	−	0.454166i	$-0.849937\pi$
$977$	−13.6515	+	13.6515i	−0.436751	+	0.436751i	0.454166	+	0.890917i	$0.349937\pi$
$978$	−28.0454	−	28.0454i	−0.896793	−	0.896793i
$979$		−	4.65153i		−	0.148664i
$980$		0			0
$981$	−23.3939	−	23.3939i	−0.746909	−	0.746909i
$982$	1.65153	−	1.65153i	0.0527025	−	0.0527025i
$983$	−0.921683	−	0.921683i	−0.0293971	−	0.0293971i	0.692255	−	0.721653i	$-0.256617\pi$
$983$	−0.921683	−	0.921683i	−0.0293971	−	0.0293971i	−0.721653	+	0.692255i	$0.756617\pi$
$984$	−16.3485	−	16.3485i	−0.521170	−	0.521170i
$985$		0			0
$986$	6.97730	−	6.97730i	0.222202	−	0.222202i
$987$	−0.977296	+	0.977296i	−0.0311077	+	0.0311077i
$988$	22.2474	−	4.44949i	0.707786	−	0.141557i
$989$			6.60612i			0.210062i
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$		−	37.6515i		−	1.19544i
$993$			48.4949i			1.53894i
$994$	−3.30306	+	3.30306i	−0.104767	+	0.104767i
$995$		0			0
$996$	−4.34847			−0.137787
$997$		−	49.0908i		−	1.55472i	−0.629055	−	0.777361i	$-0.716558\pi$
$997$		−	49.0908i		−	1.55472i	0.629055	−	0.777361i	$-0.283442\pi$
$998$		−	38.1464i		−	1.20750i
$999$	−32.6969			−1.03449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.n.i.824.1		4
3.2	odd	2		975.2.n.b.824.2		4
5.2	odd	4		975.2.o.g.551.1	yes	4
5.3	odd	4		975.2.o.f.551.2	yes	4
5.4	even	2		975.2.n.f.824.2		4
13.8	odd	4		975.2.n.a.749.1		4
15.2	even	4		975.2.o.d.551.2	yes	4
15.8	even	4		975.2.o.c.551.1	yes	4
15.14	odd	2		975.2.n.a.824.1		4
39.8	even	4		975.2.n.f.749.2		4
65.8	even	4		975.2.o.c.476.1	✓	4
65.34	odd	4		975.2.n.b.749.2		4
65.47	even	4		975.2.o.d.476.2	yes	4
195.8	odd	4		975.2.o.f.476.2	yes	4
195.47	odd	4		975.2.o.g.476.1	yes	4
195.164	even	4	inner	975.2.n.i.749.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.n.a.749.1		4	13.8	odd	4
975.2.n.a.824.1		4	15.14	odd	2
975.2.n.b.749.2		4	65.34	odd	4
975.2.n.b.824.2		4	3.2	odd	2
975.2.n.f.749.2		4	39.8	even	4
975.2.n.f.824.2		4	5.4	even	2
975.2.n.i.749.1		4	195.164	even	4	inner
975.2.n.i.824.1		4	1.1	even	1	trivial
975.2.o.c.476.1	✓	4	65.8	even	4
975.2.o.c.551.1	yes	4	15.8	even	4
975.2.o.d.476.2	yes	4	65.47	even	4
975.2.o.d.551.2	yes	4	15.2	even	4
975.2.o.f.476.2	yes	4	195.8	odd	4
975.2.o.f.551.2	yes	4	5.3	odd	4
975.2.o.g.476.1	yes	4	195.47	odd	4
975.2.o.g.551.1	yes	4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} +3.00000i q^{6} +(-0.224745 + 0.224745i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{2} +(1.22474 - 1.22474i) q^{3} -1.00000i q^{4} +3.00000i q^{6} +(-0.224745 + 0.224745i) q^{7} +(-1.22474 - 1.22474i) q^{8} -3.00000i q^{9} +(-4.22474 + 4.22474i) q^{11} +(-1.22474 - 1.22474i) q^{12} +(2.00000 - 3.00000i) q^{13} -0.550510i q^{14} +5.00000 q^{16} +3.00000i q^{17} +(3.67423 + 3.67423i) q^{18} +(-4.44949 + 4.44949i) q^{19} +0.550510i q^{21} -10.3485i q^{22} +6.00000i q^{23} -3.00000 q^{24} +(1.22474 + 6.12372i) q^{26} +(-3.67423 - 3.67423i) q^{27} +(0.224745 + 0.224745i) q^{28} +1.89898i q^{29} +(-5.12372 + 5.12372i) q^{31} +(-3.67423 + 3.67423i) q^{32} +10.3485i q^{33} +(-3.67423 - 3.67423i) q^{34} -3.00000 q^{36} +(4.44949 - 4.44949i) q^{37} -10.8990i q^{38} +(-1.22474 - 6.12372i) q^{39} +(5.44949 + 5.44949i) q^{41} +(-0.674235 - 0.674235i) q^{42} +1.10102 q^{43} +(4.22474 + 4.22474i) q^{44} +(-7.34847 - 7.34847i) q^{46} +(1.77526 + 1.77526i) q^{47} +(6.12372 - 6.12372i) q^{48} +6.89898i q^{49} +(3.67423 + 3.67423i) q^{51} +(-3.00000 - 2.00000i) q^{52} -1.89898 q^{53} +9.00000 q^{54} +0.550510 q^{56} +10.8990i q^{57} +(-2.32577 - 2.32577i) q^{58} +(-6.67423 + 6.67423i) q^{59} +12.7980 q^{61} -12.5505i q^{62} +(0.674235 + 0.674235i) q^{63} +1.00000i q^{64} +(-12.6742 - 12.6742i) q^{66} +(-4.67423 - 4.67423i) q^{67} +3.00000 q^{68} +(7.34847 + 7.34847i) q^{69} +(-6.00000 - 6.00000i) q^{71} +(-3.67423 + 3.67423i) q^{72} +(-6.44949 + 6.44949i) q^{73} +10.8990i q^{74} +(4.44949 + 4.44949i) q^{76} -1.89898i q^{77} +(9.00000 + 6.00000i) q^{78} -3.55051 q^{79} -9.00000 q^{81} -13.3485 q^{82} +(1.77526 - 1.77526i) q^{83} +0.550510 q^{84} +(-1.34847 + 1.34847i) q^{86} +(2.32577 + 2.32577i) q^{87} +10.3485 q^{88} +(-0.550510 + 0.550510i) q^{89} +(0.224745 + 1.12372i) q^{91} +6.00000 q^{92} +12.5505i q^{93} -4.34847 q^{94} +9.00000i q^{96} +(6.44949 + 6.44949i) q^{97} +(-8.44949 - 8.44949i) q^{98} +(12.6742 + 12.6742i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 12 q^{11} + 8 q^{13} + 20 q^{16} - 8 q^{19} - 12 q^{24} - 4 q^{28} + 4 q^{31} - 12 q^{36} + 8 q^{37} + 12 q^{41} + 12 q^{42} + 24 q^{43} + 12 q^{44} + 12 q^{47} - 12 q^{52} + 12 q^{53} + 36 q^{54} + 12 q^{56} - 24 q^{58} - 12 q^{59} + 12 q^{61} - 12 q^{63} - 36 q^{66} - 4 q^{67} + 12 q^{68} - 24 q^{71} - 16 q^{73} + 8 q^{76} + 36 q^{78} - 24 q^{79} - 36 q^{81} - 24 q^{82} + 12 q^{83} + 12 q^{84} + 24 q^{86} + 24 q^{87} + 12 q^{88} - 12 q^{89} - 4 q^{91} + 24 q^{92} + 12 q^{94} + 16 q^{97} - 24 q^{98} + 36 q^{99}+O(q^{100}) \) 4 * q + 4 * q^7 - 12 * q^11 + 8 * q^13 + 20 * q^16 - 8 * q^19 - 12 * q^24 - 4 * q^28 + 4 * q^31 - 12 * q^36 + 8 * q^37 + 12 * q^41 + 12 * q^42 + 24 * q^43 + 12 * q^44 + 12 * q^47 - 12 * q^52 + 12 * q^53 + 36 * q^54 + 12 * q^56 - 24 * q^58 - 12 * q^59 + 12 * q^61 - 12 * q^63 - 36 * q^66 - 4 * q^67 + 12 * q^68 - 24 * q^71 - 16 * q^73 + 8 * q^76 + 36 * q^78 - 24 * q^79 - 36 * q^81 - 24 * q^82 + 12 * q^83 + 12 * q^84 + 24 * q^86 + 24 * q^87 + 12 * q^88 - 12 * q^89 - 4 * q^91 + 24 * q^92 + 12 * q^94 + 16 * q^97 - 24 * q^98 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more