LMFDB - Embedded newform 930.2.d.b.559.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(559,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("930.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.d (of order $2$, degree $1$, 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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	930.559
Dual form			930.2.d.b.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 - 1.00000i) q^{10} -5.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} -1.00000 q^{14} +(-2.00000 + 1.00000i) q^{15} +1.00000 q^{16} -1.00000i q^{18} +5.00000 q^{19} +(1.00000 + 2.00000i) q^{20} +1.00000 q^{21} -5.00000i q^{22} +9.00000i q^{23} -1.00000 q^{24} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} +(-1.00000 - 2.00000i) q^{30} +1.00000 q^{31} +1.00000i q^{32} +5.00000i q^{33} +(2.00000 - 1.00000i) q^{35} +1.00000 q^{36} +8.00000i q^{37} +5.00000i q^{38} +4.00000 q^{39} +(-2.00000 + 1.00000i) q^{40} +6.00000 q^{41} +1.00000i q^{42} -1.00000i q^{43} +5.00000 q^{44} +(1.00000 + 2.00000i) q^{45} -9.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -4.00000i q^{52} +13.0000i q^{53} -1.00000 q^{54} +(5.00000 + 10.0000i) q^{55} +1.00000 q^{56} -5.00000i q^{57} +2.00000i q^{58} -10.0000 q^{59} +(2.00000 - 1.00000i) q^{60} -14.0000 q^{61} +1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -5.00000 q^{66} +14.0000i q^{67} +9.00000 q^{69} +(1.00000 + 2.00000i) q^{70} -9.00000 q^{71} +1.00000i q^{72} +9.00000i q^{73} -8.00000 q^{74} +(4.00000 + 3.00000i) q^{75} -5.00000 q^{76} -5.00000i q^{77} +4.00000i q^{78} -5.00000 q^{79} +(-1.00000 - 2.00000i) q^{80} +1.00000 q^{81} +6.00000i q^{82} -6.00000i q^{83} -1.00000 q^{84} +1.00000 q^{86} -2.00000i q^{87} +5.00000i q^{88} -3.00000 q^{89} +(-2.00000 + 1.00000i) q^{90} -4.00000 q^{91} -9.00000i q^{92} -1.00000i q^{93} +12.0000 q^{94} +(-5.00000 - 10.0000i) q^{95} +1.00000 q^{96} -18.0000i q^{97} +6.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} - 10 q^{11} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{21} - 2 q^{24} - 6 q^{25} - 8 q^{26} + 4 q^{29} - 2 q^{30} + 2 q^{31} + 4 q^{35} + 2 q^{36} + 8 q^{39} - 4 q^{40} + 12 q^{41} + 10 q^{44} + 2 q^{45} - 18 q^{46} + 12 q^{49} - 8 q^{50} - 2 q^{54} + 10 q^{55} + 2 q^{56} - 20 q^{59} + 4 q^{60} - 28 q^{61} - 2 q^{64} + 16 q^{65} - 10 q^{66} + 18 q^{69} + 2 q^{70} - 18 q^{71} - 16 q^{74} + 8 q^{75} - 10 q^{76} - 10 q^{79} - 2 q^{80} + 2 q^{81} - 2 q^{84} + 2 q^{86} - 6 q^{89} - 4 q^{90} - 8 q^{91} + 24 q^{94} - 10 q^{95} + 2 q^{96} + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 - 10 * q^11 - 2 * q^14 - 4 * q^15 + 2 * q^16 + 10 * q^19 + 2 * q^20 + 2 * q^21 - 2 * q^24 - 6 * q^25 - 8 * q^26 + 4 * q^29 - 2 * q^30 + 2 * q^31 + 4 * q^35 + 2 * q^36 + 8 * q^39 - 4 * q^40 + 12 * q^41 + 10 * q^44 + 2 * q^45 - 18 * q^46 + 12 * q^49 - 8 * q^50 - 2 * q^54 + 10 * q^55 + 2 * q^56 - 20 * q^59 + 4 * q^60 - 28 * q^61 - 2 * q^64 + 16 * q^65 - 10 * q^66 + 18 * q^69 + 2 * q^70 - 18 * q^71 - 16 * q^74 + 8 * q^75 - 10 * q^76 - 10 * q^79 - 2 * q^80 + 2 * q^81 - 2 * q^84 + 2 * q^86 - 6 * q^89 - 4 * q^90 - 8 * q^91 + 24 * q^94 - 10 * q^95 + 2 * q^96 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$	1.00000			0.408248
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$	−5.00000			−1.50756			−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000			−1.50756			−0.753778	+	0.657129i	$0.771771\pi$
$12$			1.00000i			0.288675i
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000			−0.267261
$15$	−2.00000	+	1.00000i	−0.516398	+	0.258199i
$16$	1.00000			0.250000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		−	1.00000i		−	0.235702i
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$	1.00000	+	2.00000i	0.223607	+	0.447214i
$21$	1.00000			0.218218
$22$		−	5.00000i		−	1.06600i
$23$			9.00000i			1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$			9.00000i			1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	−1.00000			−0.204124
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−4.00000			−0.784465
$27$			1.00000i			0.192450i
$28$		−	1.00000i		−	0.188982i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−	2.00000i	−0.182574	−	0.365148i
$31$	1.00000			0.179605
$31$	1.00000			0.179605
$32$			1.00000i			0.176777i
$33$			5.00000i			0.870388i
$34$		0			0
$35$	2.00000	−	1.00000i	0.338062	−	0.169031i
$36$	1.00000			0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$			5.00000i			0.811107i
$39$	4.00000			0.640513
$40$	−2.00000	+	1.00000i	−0.316228	+	0.158114i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$			1.00000i			0.154303i
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	5.00000			0.753778
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$	−9.00000			−1.32698
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		−	1.00000i		−	0.144338i
$49$	6.00000			0.857143
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$		−	4.00000i		−	0.554700i
$53$			13.0000i			1.78569i	0.450367	+	0.892844i	$0.351293\pi$
$53$			13.0000i			1.78569i	−0.450367	+	0.892844i	$0.648707\pi$
$54$	−1.00000			−0.136083
$55$	5.00000	+	10.0000i	0.674200	+	1.34840i
$56$	1.00000			0.133631
$57$		−	5.00000i		−	0.662266i
$58$			2.00000i			0.262613i
$59$	−10.0000			−1.30189			−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$	2.00000	−	1.00000i	0.258199	−	0.129099i
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$			1.00000i			0.127000i
$63$		−	1.00000i		−	0.125988i
$64$	−1.00000			−0.125000
$65$	8.00000	−	4.00000i	0.992278	−	0.496139i
$66$	−5.00000			−0.615457
$67$			14.0000i			1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$			14.0000i			1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$		0			0
$69$	9.00000			1.08347
$70$	1.00000	+	2.00000i	0.119523	+	0.239046i
$71$	−9.00000			−1.06810			−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000			−1.06810			−0.534052	+	0.845452i	$0.679331\pi$
$72$			1.00000i			0.117851i
$73$			9.00000i			1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$			9.00000i			1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	−8.00000			−0.929981
$75$	4.00000	+	3.00000i	0.461880	+	0.346410i
$76$	−5.00000			−0.573539
$77$		−	5.00000i		−	0.569803i
$78$			4.00000i			0.452911i
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$	−1.00000	−	2.00000i	−0.111803	−	0.223607i
$81$	1.00000			0.111111
$82$			6.00000i			0.662589i
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	−1.00000			−0.109109
$85$		0			0
$86$	1.00000			0.107833
$87$		−	2.00000i		−	0.214423i
$88$			5.00000i			0.533002i
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$	−2.00000	+	1.00000i	−0.210819	+	0.105409i
$91$	−4.00000			−0.419314
$92$		−	9.00000i		−	0.938315i
$93$		−	1.00000i		−	0.103695i
$94$	12.0000			1.23771
$95$	−5.00000	−	10.0000i	−0.512989	−	1.02598i
$96$	1.00000			0.102062
$97$		−	18.0000i		−	1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$		−	18.0000i		−	1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$			6.00000i			0.606092i
$99$	5.00000			0.502519
$100$	3.00000	−	4.00000i	0.300000	−	0.400000i
$101$	−7.00000			−0.696526			−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000			−0.696526			−0.348263	+	0.937397i	$0.613228\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	4.00000			0.392232
$105$	−1.00000	−	2.00000i	−0.0975900	−	0.195180i
$106$	−13.0000			−1.26267
$107$			11.0000i			1.06341i	0.846930	+	0.531705i	$0.178449\pi$
$107$			11.0000i			1.06341i	−0.846930	+	0.531705i	$0.821551\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	−10.0000	+	5.00000i	−0.953463	+	0.476731i
$111$	8.00000			0.759326
$112$			1.00000i			0.0944911i
$113$		−	13.0000i		−	1.22294i	−0.791269	−	0.611469i	$-0.790579\pi$
$113$		−	13.0000i		−	1.22294i	0.791269	−	0.611469i	$-0.209421\pi$
$114$	5.00000			0.468293
$115$	18.0000	−	9.00000i	1.67851	−	0.839254i
$116$	−2.00000			−0.185695
$117$		−	4.00000i		−	0.369800i
$118$		−	10.0000i		−	0.920575i
$119$		0			0
$120$	1.00000	+	2.00000i	0.0912871	+	0.182574i
$121$	14.0000			1.27273
$122$		−	14.0000i		−	1.26750i
$123$		−	6.00000i		−	0.541002i
$124$	−1.00000			−0.0898027
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$	1.00000			0.0890871
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−1.00000			−0.0880451
$130$	4.00000	+	8.00000i	0.350823	+	0.701646i
$131$	−6.00000			−0.524222			−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000			−0.524222			−0.262111	+	0.965038i	$0.584419\pi$
$132$		−	5.00000i		−	0.435194i
$133$			5.00000i			0.433555i
$134$	−14.0000			−1.20942
$135$	2.00000	−	1.00000i	0.172133	−	0.0860663i
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$			9.00000i			0.766131i
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	−2.00000	+	1.00000i	−0.169031	+	0.0845154i
$141$	−12.0000			−1.01058
$142$		−	9.00000i		−	0.755263i
$143$		−	20.0000i		−	1.67248i
$144$	−1.00000			−0.0833333
$145$	−2.00000	−	4.00000i	−0.166091	−	0.332182i
$146$	−9.00000			−0.744845
$147$		−	6.00000i		−	0.494872i
$148$		−	8.00000i		−	0.657596i
$149$	−7.00000			−0.573462			−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000			−0.573462			−0.286731	+	0.958011i	$0.592569\pi$
$150$	−3.00000	+	4.00000i	−0.244949	+	0.326599i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	5.00000i		−	0.405554i
$153$		0			0
$154$	5.00000			0.402911
$155$	−1.00000	−	2.00000i	−0.0803219	−	0.160644i
$156$	−4.00000			−0.320256
$157$			13.0000i			1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$			13.0000i			1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$		−	5.00000i		−	0.397779i
$159$	13.0000			1.03097
$160$	2.00000	−	1.00000i	0.158114	−	0.0790569i
$161$	−9.00000			−0.709299
$162$			1.00000i			0.0785674i
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−6.00000			−0.468521
$165$	10.0000	−	5.00000i	0.778499	−	0.389249i
$166$	6.00000			0.465690
$167$		−	9.00000i		−	0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$		−	9.00000i		−	0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$		−	1.00000i		−	0.0771517i
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−5.00000			−0.382360
$172$			1.00000i			0.0762493i
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	2.00000			0.151620
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$	−5.00000			−0.376889
$177$			10.0000i			0.751646i
$178$		−	3.00000i		−	0.224860i
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$	−1.00000	−	2.00000i	−0.0745356	−	0.149071i
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$		−	4.00000i		−	0.296500i
$183$			14.0000i			1.03491i
$184$	9.00000			0.663489
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$	1.00000			0.0733236
$187$		0			0
$188$			12.0000i			0.875190i
$189$	−1.00000			−0.0727393
$190$	10.0000	−	5.00000i	0.725476	−	0.362738i
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$			1.00000i			0.0721688i
$193$			22.0000i			1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$			22.0000i			1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	18.0000			1.29232
$195$	−4.00000	−	8.00000i	−0.286446	−	0.572892i
$196$	−6.00000			−0.428571
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$			5.00000i			0.355335i
$199$	11.0000			0.779769			0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000			0.779769			0.389885	+	0.920864i	$0.372515\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$	14.0000			0.987484
$202$		−	7.00000i		−	0.492518i
$203$			2.00000i			0.140372i
$204$		0			0
$205$	−6.00000	−	12.0000i	−0.419058	−	0.838116i
$206$	8.00000			0.557386
$207$		−	9.00000i		−	0.625543i
$208$			4.00000i			0.277350i
$209$	−25.0000			−1.72929
$210$	2.00000	−	1.00000i	0.138013	−	0.0690066i
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$		−	13.0000i		−	0.892844i
$213$			9.00000i			0.616670i
$214$	−11.0000			−0.751945
$215$	−2.00000	+	1.00000i	−0.136399	+	0.0681994i
$216$	1.00000			0.0680414
$217$			1.00000i			0.0678844i
$218$			2.00000i			0.135457i
$219$	9.00000			0.608164
$220$	−5.00000	−	10.0000i	−0.337100	−	0.674200i
$221$		0			0
$222$			8.00000i			0.536925i
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−1.00000			−0.0668153
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$	13.0000			0.864747
$227$			15.0000i			0.995585i	0.867296	+	0.497792i	$0.165856\pi$
$227$			15.0000i			0.995585i	−0.867296	+	0.497792i	$0.834144\pi$
$228$			5.00000i			0.331133i
$229$	19.0000			1.25556			0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000			1.25556			0.627778	+	0.778393i	$0.283965\pi$
$230$	9.00000	+	18.0000i	0.593442	+	1.18688i
$231$	−5.00000			−0.328976
$232$		−	2.00000i		−	0.131306i
$233$			1.00000i			0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$			1.00000i			0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$	4.00000			0.261488
$235$	−24.0000	+	12.0000i	−1.56559	+	0.782794i
$236$	10.0000			0.650945
$237$			5.00000i			0.324785i
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$	−2.00000	+	1.00000i	−0.129099	+	0.0645497i
$241$	24.0000			1.54598			0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000			1.54598			0.772988	+	0.634421i	$0.218761\pi$
$242$			14.0000i			0.899954i
$243$		−	1.00000i		−	0.0641500i
$244$	14.0000			0.896258
$245$	−6.00000	−	12.0000i	−0.383326	−	0.766652i
$246$	6.00000			0.382546
$247$			20.0000i			1.27257i
$248$		−	1.00000i		−	0.0635001i
$249$	−6.00000			−0.380235
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$			1.00000i			0.0629941i
$253$		−	45.0000i		−	2.82913i
$254$	−20.0000			−1.25491
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	7.00000i		−	0.436648i	−0.975876	−	0.218324i	$-0.929941\pi$
$257$		−	7.00000i		−	0.436648i	0.975876	−	0.218324i	$-0.0700590\pi$
$258$		−	1.00000i		−	0.0622573i
$259$	−8.00000			−0.497096
$260$	−8.00000	+	4.00000i	−0.496139	+	0.248069i
$261$	−2.00000			−0.123797
$262$		−	6.00000i		−	0.370681i
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	5.00000			0.307729
$265$	26.0000	−	13.0000i	1.59717	−	0.798584i
$266$	−5.00000			−0.306570
$267$			3.00000i			0.183597i
$268$		−	14.0000i		−	0.855186i
$269$	22.0000			1.34136			0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000			1.34136			0.670682	+	0.741745i	$0.266002\pi$
$270$	1.00000	+	2.00000i	0.0608581	+	0.121716i
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$		0			0
$273$			4.00000i			0.242091i
$274$	−2.00000			−0.120824
$275$	15.0000	−	20.0000i	0.904534	−	1.20605i
$276$	−9.00000			−0.541736
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		−	16.0000i		−	0.959616i
$279$	−1.00000			−0.0598684
$280$	−1.00000	−	2.00000i	−0.0597614	−	0.119523i
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		−	12.0000i		−	0.714590i
$283$		−	26.0000i		−	1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$		−	26.0000i		−	1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	9.00000			0.534052
$285$	−10.0000	+	5.00000i	−0.592349	+	0.296174i
$286$	20.0000			1.18262
$287$			6.00000i			0.354169i
$288$		−	1.00000i		−	0.0589256i
$289$	17.0000			1.00000
$290$	4.00000	−	2.00000i	0.234888	−	0.117444i
$291$	−18.0000			−1.05518
$292$		−	9.00000i		−	0.526685i
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	6.00000			0.349927
$295$	10.0000	+	20.0000i	0.582223	+	1.16445i
$296$	8.00000			0.464991
$297$		−	5.00000i		−	0.290129i
$298$		−	7.00000i		−	0.405499i
$299$	−36.0000			−2.08193
$300$	−4.00000	−	3.00000i	−0.230940	−	0.173205i
$301$	1.00000			0.0576390
$302$		0			0
$303$			7.00000i			0.402139i
$304$	5.00000			0.286770
$305$	14.0000	+	28.0000i	0.801638	+	1.60328i
$306$		0			0
$307$		−	8.00000i		−	0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$		−	8.00000i		−	0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$			5.00000i			0.284901i
$309$	−8.00000			−0.455104
$310$	2.00000	−	1.00000i	0.113592	−	0.0567962i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		−	4.00000i		−	0.226455i
$313$		−	6.00000i		−	0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$		−	6.00000i		−	0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−13.0000			−0.733632
$315$	−2.00000	+	1.00000i	−0.112687	+	0.0563436i
$316$	5.00000			0.281272
$317$			34.0000i			1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$			34.0000i			1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$			13.0000i			0.729004i
$319$	−10.0000			−0.559893
$320$	1.00000	+	2.00000i	0.0559017	+	0.111803i
$321$	11.0000			0.613960
$322$		−	9.00000i		−	0.501550i
$323$		0			0
$324$	−1.00000			−0.0555556
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	4.00000			0.221540
$327$		−	2.00000i		−	0.110600i
$328$		−	6.00000i		−	0.331295i
$329$	12.0000			0.661581
$330$	5.00000	+	10.0000i	0.275241	+	0.550482i
$331$	−24.0000			−1.31916			−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000			−1.31916			−0.659580	+	0.751635i	$0.729266\pi$
$332$			6.00000i			0.329293i
$333$		−	8.00000i		−	0.438397i
$334$	9.00000			0.492458
$335$	28.0000	−	14.0000i	1.52980	−	0.764902i
$336$	1.00000			0.0545545
$337$			2.00000i			0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$			2.00000i			0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$		−	3.00000i		−	0.163178i
$339$	−13.0000			−0.706063
$340$		0			0
$341$	−5.00000			−0.270765
$342$		−	5.00000i		−	0.270369i
$343$			13.0000i			0.701934i
$344$	−1.00000			−0.0539164
$345$	−9.00000	−	18.0000i	−0.484544	−	0.969087i
$346$	−6.00000			−0.322562
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$			2.00000i			0.107211i
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$	3.00000	−	4.00000i	0.160357	−	0.213809i
$351$	−4.00000			−0.213504
$352$		−	5.00000i		−	0.266501i
$353$			4.00000i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$			4.00000i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$	−10.0000			−0.531494
$355$	9.00000	+	18.0000i	0.477670	+	0.955341i
$356$	3.00000			0.159000
$357$		0			0
$358$			16.0000i			0.845626i
$359$	9.00000			0.475002			0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000			0.475002			0.237501	+	0.971387i	$0.423672\pi$
$360$	2.00000	−	1.00000i	0.105409	−	0.0527046i
$361$	6.00000			0.315789
$362$			17.0000i			0.893500i
$363$		−	14.0000i		−	0.734809i
$364$	4.00000			0.209657
$365$	18.0000	−	9.00000i	0.942163	−	0.471082i
$366$	−14.0000			−0.731792
$367$			18.0000i			0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$			18.0000i			0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$			9.00000i			0.469157i
$369$	−6.00000			−0.312348
$370$	8.00000	+	16.0000i	0.415900	+	0.831800i
$371$	−13.0000			−0.674926
$372$			1.00000i			0.0518476i
$373$		−	1.00000i		−	0.0517780i	−0.999665	−	0.0258890i	$-0.991758\pi$
$373$		−	1.00000i		−	0.0517780i	0.999665	−	0.0258890i	$-0.00824165\pi$
$374$		0			0
$375$	2.00000	−	11.0000i	0.103280	−	0.568038i
$376$	−12.0000			−0.618853
$377$			8.00000i			0.412021i
$378$		−	1.00000i		−	0.0514344i
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$	5.00000	+	10.0000i	0.256495	+	0.512989i
$381$	20.0000			1.02463
$382$			8.00000i			0.409316i
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	−1.00000			−0.0510310
$385$	−10.0000	+	5.00000i	−0.509647	+	0.254824i
$386$	−22.0000			−1.11977
$387$			1.00000i			0.0508329i
$388$			18.0000i			0.913812i
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$	8.00000	−	4.00000i	0.405096	−	0.202548i
$391$		0			0
$392$		−	6.00000i		−	0.303046i
$393$			6.00000i			0.302660i
$394$	10.0000			0.503793
$395$	5.00000	+	10.0000i	0.251577	+	0.503155i
$396$	−5.00000			−0.251259
$397$		−	3.00000i		−	0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$		−	3.00000i		−	0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$			11.0000i			0.551380i
$399$	5.00000			0.250313
$400$	−3.00000	+	4.00000i	−0.150000	+	0.200000i
$401$	−25.0000			−1.24844			−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000			−1.24844			−0.624220	+	0.781248i	$0.714583\pi$
$402$			14.0000i			0.698257i
$403$			4.00000i			0.199254i
$404$	7.00000			0.348263
$405$	−1.00000	−	2.00000i	−0.0496904	−	0.0993808i
$406$	−2.00000			−0.0992583
$407$		−	40.0000i		−	1.98273i
$408$		0			0
$409$	28.0000			1.38451			0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000			1.38451			0.692255	+	0.721653i	$0.256617\pi$
$410$	12.0000	−	6.00000i	0.592638	−	0.296319i
$411$	2.00000			0.0986527
$412$			8.00000i			0.394132i
$413$		−	10.0000i		−	0.492068i
$414$	9.00000			0.442326
$415$	−12.0000	+	6.00000i	−0.589057	+	0.294528i
$416$	−4.00000			−0.196116
$417$			16.0000i			0.783523i
$418$		−	25.0000i		−	1.22279i
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$	1.00000	+	2.00000i	0.0487950	+	0.0975900i
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		−	13.0000i		−	0.632830i
$423$			12.0000i			0.583460i
$424$	13.0000			0.631336
$425$		0			0
$426$	−9.00000			−0.436051
$427$		−	14.0000i		−	0.677507i
$428$		−	11.0000i		−	0.531705i
$429$	−20.0000			−0.965609
$430$	−1.00000	−	2.00000i	−0.0482243	−	0.0964486i
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$			1.00000i			0.0481125i
$433$			11.0000i			0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$			11.0000i			0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	−1.00000			−0.0480015
$435$	−4.00000	+	2.00000i	−0.191785	+	0.0958927i
$436$	−2.00000			−0.0957826
$437$			45.0000i			2.15264i
$438$			9.00000i			0.430037i
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$	10.0000	−	5.00000i	0.476731	−	0.238366i
$441$	−6.00000			−0.285714
$442$		0			0
$443$		−	13.0000i		−	0.617649i	−0.951119	−	0.308824i	$-0.900064\pi$
$443$		−	13.0000i		−	0.617649i	0.951119	−	0.308824i	$-0.0999355\pi$
$444$	−8.00000			−0.379663
$445$	3.00000	+	6.00000i	0.142214	+	0.284427i
$446$	16.0000			0.757622
$447$			7.00000i			0.331089i
$448$		−	1.00000i		−	0.0472456i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$	4.00000	+	3.00000i	0.188562	+	0.141421i
$451$	−30.0000			−1.41264
$452$			13.0000i			0.611469i
$453$		0			0
$454$	−15.0000			−0.703985
$455$	4.00000	+	8.00000i	0.187523	+	0.375046i
$456$	−5.00000			−0.234146
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$			19.0000i			0.887812i
$459$		0			0
$460$	−18.0000	+	9.00000i	−0.839254	+	0.419627i
$461$	36.0000			1.67669			0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000			1.67669			0.838344	+	0.545142i	$0.183524\pi$
$462$		−	5.00000i		−	0.232621i
$463$		−	30.0000i		−	1.39422i	−0.716965	−	0.697109i	$-0.754469\pi$
$463$		−	30.0000i		−	1.39422i	0.716965	−	0.697109i	$-0.245531\pi$
$464$	2.00000			0.0928477
$465$	−2.00000	+	1.00000i	−0.0927478	+	0.0463739i
$466$	−1.00000			−0.0463241
$467$			20.0000i			0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$			20.0000i			0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$			4.00000i			0.184900i
$469$	−14.0000			−0.646460
$470$	−12.0000	−	24.0000i	−0.553519	−	1.10704i
$471$	13.0000			0.599008
$472$			10.0000i			0.460287i
$473$			5.00000i			0.229900i
$474$	−5.00000			−0.229658
$475$	−15.0000	+	20.0000i	−0.688247	+	0.917663i
$476$		0			0
$477$		−	13.0000i		−	0.595229i
$478$			6.00000i			0.274434i
$479$	21.0000			0.959514			0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000			0.959514			0.479757	+	0.877401i	$0.340725\pi$
$480$	−1.00000	−	2.00000i	−0.0456435	−	0.0912871i
$481$	−32.0000			−1.45907
$482$			24.0000i			1.09317i
$483$			9.00000i			0.409514i
$484$	−14.0000			−0.636364
$485$	−36.0000	+	18.0000i	−1.63468	+	0.817338i
$486$	1.00000			0.0453609
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$			14.0000i			0.633750i
$489$	−4.00000			−0.180886
$490$	12.0000	−	6.00000i	0.542105	−	0.271052i
$491$	15.0000			0.676941			0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000			0.676941			0.338470	+	0.940977i	$0.390091\pi$
$492$			6.00000i			0.270501i
$493$		0			0
$494$	−20.0000			−0.899843
$495$	−5.00000	−	10.0000i	−0.224733	−	0.449467i
$496$	1.00000			0.0449013
$497$		−	9.00000i		−	0.403705i
$498$		−	6.00000i		−	0.268866i
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$	−11.0000	−	2.00000i	−0.491935	−	0.0894427i
$501$	−9.00000			−0.402090
$502$		−	28.0000i		−	1.24970i
$503$		−	22.0000i		−	0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$		−	22.0000i		−	0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$	−1.00000			−0.0445435
$505$	7.00000	+	14.0000i	0.311496	+	0.622992i
$506$	45.0000			2.00049
$507$			3.00000i			0.133235i
$508$		−	20.0000i		−	0.887357i
$509$	20.0000			0.886484			0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000			0.886484			0.443242	+	0.896402i	$0.353828\pi$
$510$		0			0
$511$	−9.00000			−0.398137
$512$			1.00000i			0.0441942i
$513$			5.00000i			0.220755i
$514$	7.00000			0.308757
$515$	−16.0000	+	8.00000i	−0.705044	+	0.352522i
$516$	1.00000			0.0440225
$517$			60.0000i			2.63880i
$518$		−	8.00000i		−	0.351500i
$519$	6.00000			0.263371
$520$	−4.00000	−	8.00000i	−0.175412	−	0.350823i
$521$	−36.0000			−1.57719			−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000			−1.57719			−0.788594	+	0.614914i	$0.789191\pi$
$522$		−	2.00000i		−	0.0875376i
$523$		−	17.0000i		−	0.743358i	−0.928361	−	0.371679i	$-0.878782\pi$
$523$		−	17.0000i		−	0.743358i	0.928361	−	0.371679i	$-0.121218\pi$
$524$	6.00000			0.262111
$525$	−3.00000	+	4.00000i	−0.130931	+	0.174574i
$526$	−8.00000			−0.348817
$527$		0			0
$528$			5.00000i			0.217597i
$529$	−58.0000			−2.52174
$530$	13.0000	+	26.0000i	0.564684	+	1.12937i
$531$	10.0000			0.433963
$532$		−	5.00000i		−	0.216777i
$533$			24.0000i			1.03956i
$534$	−3.00000			−0.129823
$535$	22.0000	−	11.0000i	0.951143	−	0.475571i
$536$	14.0000			0.604708
$537$		−	16.0000i		−	0.690451i
$538$			22.0000i			0.948487i
$539$	−30.0000			−1.29219
$540$	−2.00000	+	1.00000i	−0.0860663	+	0.0430331i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		−	25.0000i		−	1.07384i
$543$		−	17.0000i		−	0.729540i
$544$		0			0
$545$	−2.00000	−	4.00000i	−0.0856706	−	0.171341i
$546$	−4.00000			−0.171184
$547$			16.0000i			0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$			16.0000i			0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$		−	2.00000i		−	0.0854358i
$549$	14.0000			0.597505
$550$	20.0000	+	15.0000i	0.852803	+	0.639602i
$551$	10.0000			0.426014
$552$		−	9.00000i		−	0.383065i
$553$		−	5.00000i		−	0.212622i
$554$	2.00000			0.0849719
$555$	−8.00000	−	16.0000i	−0.339581	−	0.679162i
$556$	16.0000			0.678551
$557$			7.00000i			0.296600i	0.988942	+	0.148300i	$0.0473800\pi$
$557$			7.00000i			0.296600i	−0.988942	+	0.148300i	$0.952620\pi$
$558$		−	1.00000i		−	0.0423334i
$559$	4.00000			0.169182
$560$	2.00000	−	1.00000i	0.0845154	−	0.0422577i
$561$		0			0
$562$			12.0000i			0.506189i
$563$			44.0000i			1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$			44.0000i			1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	12.0000			0.505291
$565$	−26.0000	+	13.0000i	−1.09383	+	0.546914i
$566$	26.0000			1.09286
$567$			1.00000i			0.0419961i
$568$			9.00000i			0.377632i
$569$	21.0000			0.880366			0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000			0.880366			0.440183	+	0.897908i	$0.354914\pi$
$570$	−5.00000	−	10.0000i	−0.209427	−	0.418854i
$571$	−10.0000			−0.418487			−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000			−0.418487			−0.209243	+	0.977864i	$0.567100\pi$
$572$			20.0000i			0.836242i
$573$		−	8.00000i		−	0.334205i
$574$	−6.00000			−0.250435
$575$	−36.0000	−	27.0000i	−1.50130	−	1.12598i
$576$	1.00000			0.0416667
$577$		−	2.00000i		−	0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$		−	2.00000i		−	0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$			17.0000i			0.707107i
$579$	22.0000			0.914289
$580$	2.00000	+	4.00000i	0.0830455	+	0.166091i
$581$	6.00000			0.248922
$582$		−	18.0000i		−	0.746124i
$583$		−	65.0000i		−	2.69202i
$584$	9.00000			0.372423
$585$	−8.00000	+	4.00000i	−0.330759	+	0.165380i
$586$	−14.0000			−0.578335
$587$		−	6.00000i		−	0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$		−	6.00000i		−	0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$			6.00000i			0.247436i
$589$	5.00000			0.206021
$590$	−20.0000	+	10.0000i	−0.823387	+	0.411693i
$591$	−10.0000			−0.411345
$592$			8.00000i			0.328798i
$593$			10.0000i			0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$			10.0000i			0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$	5.00000			0.205152
$595$		0			0
$596$	7.00000			0.286731
$597$		−	11.0000i		−	0.450200i
$598$		−	36.0000i		−	1.47215i
$599$	39.0000			1.59350			0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000			1.59350			0.796748	+	0.604311i	$0.206552\pi$
$600$	3.00000	−	4.00000i	0.122474	−	0.163299i
$601$	−12.0000			−0.489490			−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000			−0.489490			−0.244745	+	0.969587i	$0.578704\pi$
$602$			1.00000i			0.0407570i
$603$		−	14.0000i		−	0.570124i
$604$		0			0
$605$	−14.0000	−	28.0000i	−0.569181	−	1.13836i
$606$	−7.00000			−0.284356
$607$			13.0000i			0.527654i	0.964570	+	0.263827i	$0.0849848\pi$
$607$			13.0000i			0.527654i	−0.964570	+	0.263827i	$0.915015\pi$
$608$			5.00000i			0.202777i
$609$	2.00000			0.0810441
$610$	−28.0000	+	14.0000i	−1.13369	+	0.566843i
$611$	48.0000			1.94187
$612$		0			0
$613$		−	4.00000i		−	0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$		−	4.00000i		−	0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	8.00000			0.322854
$615$	−12.0000	+	6.00000i	−0.483887	+	0.241943i
$616$	−5.00000			−0.201456
$617$			9.00000i			0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$			9.00000i			0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$		−	8.00000i		−	0.321807i
$619$	−16.0000			−0.643094			−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000			−0.643094			−0.321547	+	0.946894i	$0.604203\pi$
$620$	1.00000	+	2.00000i	0.0401610	+	0.0803219i
$621$	−9.00000			−0.361158
$622$		−	8.00000i		−	0.320771i
$623$		−	3.00000i		−	0.120192i
$624$	4.00000			0.160128
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	6.00000			0.239808
$627$			25.0000i			0.998404i
$628$		−	13.0000i		−	0.518756i
$629$		0			0
$630$	−1.00000	−	2.00000i	−0.0398410	−	0.0796819i
$631$	31.0000			1.23409			0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000			1.23409			0.617045	+	0.786928i	$0.288330\pi$
$632$			5.00000i			0.198889i
$633$			13.0000i			0.516704i
$634$	−34.0000			−1.35031
$635$	40.0000	−	20.0000i	1.58735	−	0.793676i
$636$	−13.0000			−0.515484
$637$			24.0000i			0.950915i
$638$		−	10.0000i		−	0.395904i
$639$	9.00000			0.356034
$640$	−2.00000	+	1.00000i	−0.0790569	+	0.0395285i
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$			11.0000i			0.434135i
$643$			5.00000i			0.197181i	0.995128	+	0.0985904i	$0.0314334\pi$
$643$			5.00000i			0.197181i	−0.995128	+	0.0985904i	$0.968567\pi$
$644$	9.00000			0.354650
$645$	1.00000	+	2.00000i	0.0393750	+	0.0787499i
$646$		0			0
$647$			5.00000i			0.196570i	0.995158	+	0.0982851i	$0.0313357\pi$
$647$			5.00000i			0.196570i	−0.995158	+	0.0982851i	$0.968664\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	50.0000			1.96267
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$	1.00000			0.0391931
$652$			4.00000i			0.156652i
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	2.00000			0.0782062
$655$	6.00000	+	12.0000i	0.234439	+	0.468879i
$656$	6.00000			0.234261
$657$		−	9.00000i		−	0.351123i
$658$			12.0000i			0.467809i
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$	−10.0000	+	5.00000i	−0.389249	+	0.194625i
$661$	28.0000			1.08907			0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000			1.08907			0.544537	+	0.838737i	$0.316705\pi$
$662$		−	24.0000i		−	0.932786i
$663$		0			0
$664$	−6.00000			−0.232845
$665$	10.0000	−	5.00000i	0.387783	−	0.193892i
$666$	8.00000			0.309994
$667$			18.0000i			0.696963i
$668$			9.00000i			0.348220i
$669$	−16.0000			−0.618596
$670$	14.0000	+	28.0000i	0.540867	+	1.08173i
$671$	70.0000			2.70232
$672$			1.00000i			0.0385758i
$673$		−	38.0000i		−	1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$		−	38.0000i		−	1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	−2.00000			−0.0770371
$675$	−4.00000	−	3.00000i	−0.153960	−	0.115470i
$676$	3.00000			0.115385
$677$			51.0000i			1.96009i	0.198778	+	0.980045i	$0.436303\pi$
$677$			51.0000i			1.96009i	−0.198778	+	0.980045i	$0.563697\pi$
$678$		−	13.0000i		−	0.499262i
$679$	18.0000			0.690777
$680$		0			0
$681$	15.0000			0.574801
$682$		−	5.00000i		−	0.191460i
$683$		−	3.00000i		−	0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$683$		−	3.00000i		−	0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$684$	5.00000			0.191180
$685$	4.00000	−	2.00000i	0.152832	−	0.0764161i
$686$	−13.0000			−0.496342
$687$		−	19.0000i		−	0.724895i
$688$		−	1.00000i		−	0.0381246i
$689$	−52.0000			−1.98104
$690$	18.0000	−	9.00000i	0.685248	−	0.342624i
$691$	27.0000			1.02713			0.513564	−	0.858051i	$-0.328325\pi$
$691$	27.0000			1.02713			0.513564	+	0.858051i	$0.328325\pi$
$692$		−	6.00000i		−	0.228086i
$693$			5.00000i			0.189934i
$694$	18.0000			0.683271
$695$	16.0000	+	32.0000i	0.606915	+	1.21383i
$696$	−2.00000			−0.0758098
$697$		0			0
$698$		−	4.00000i		−	0.151402i
$699$	1.00000			0.0378235
$700$	4.00000	+	3.00000i	0.151186	+	0.113389i
$701$	29.0000			1.09531			0.547657	−	0.836703i	$-0.315520\pi$
$701$	29.0000			1.09531			0.547657	+	0.836703i	$0.315520\pi$
$702$		−	4.00000i		−	0.150970i
$703$			40.0000i			1.50863i
$704$	5.00000			0.188445
$705$	12.0000	+	24.0000i	0.451946	+	0.903892i
$706$	−4.00000			−0.150542
$707$		−	7.00000i		−	0.263262i
$708$		−	10.0000i		−	0.375823i
$709$	17.0000			0.638448			0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000			0.638448			0.319224	+	0.947679i	$0.396578\pi$
$710$	−18.0000	+	9.00000i	−0.675528	+	0.337764i
$711$	5.00000			0.187515
$712$			3.00000i			0.112430i
$713$			9.00000i			0.337053i
$714$		0			0
$715$	−40.0000	+	20.0000i	−1.49592	+	0.747958i
$716$	−16.0000			−0.597948
$717$		−	6.00000i		−	0.224074i
$718$			9.00000i			0.335877i
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$	1.00000	+	2.00000i	0.0372678	+	0.0745356i
$721$	8.00000			0.297936
$722$			6.00000i			0.223297i
$723$		−	24.0000i		−	0.892570i
$724$	−17.0000			−0.631800
$725$	−6.00000	+	8.00000i	−0.222834	+	0.297113i
$726$	14.0000			0.519589
$727$			1.00000i			0.0370879i	0.999828	+	0.0185440i	$0.00590307\pi$
$727$			1.00000i			0.0370879i	−0.999828	+	0.0185440i	$0.994097\pi$
$728$			4.00000i			0.148250i
$729$	−1.00000			−0.0370370
$730$	9.00000	+	18.0000i	0.333105	+	0.666210i
$731$		0			0
$732$		−	14.0000i		−	0.517455i
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−18.0000			−0.664392
$735$	−12.0000	+	6.00000i	−0.442627	+	0.221313i
$736$	−9.00000			−0.331744
$737$		−	70.0000i		−	2.57848i
$738$		−	6.00000i		−	0.220863i
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$	−16.0000	+	8.00000i	−0.588172	+	0.294086i
$741$	20.0000			0.734718
$742$		−	13.0000i		−	0.477245i
$743$			43.0000i			1.57752i	0.614703	+	0.788759i	$0.289276\pi$
$743$			43.0000i			1.57752i	−0.614703	+	0.788759i	$0.710724\pi$
$744$	−1.00000			−0.0366618
$745$	7.00000	+	14.0000i	0.256460	+	0.512920i
$746$	1.00000			0.0366126
$747$			6.00000i			0.219529i
$748$		0			0
$749$	−11.0000			−0.401931
$750$	11.0000	+	2.00000i	0.401663	+	0.0730297i
$751$	18.0000			0.656829			0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000			0.656829			0.328415	+	0.944534i	$0.393486\pi$
$752$		−	12.0000i		−	0.437595i
$753$			28.0000i			1.02038i
$754$	−8.00000			−0.291343
$755$		0			0
$756$	1.00000			0.0363696
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$			11.0000i			0.399538i
$759$	−45.0000			−1.63340
$760$	−10.0000	+	5.00000i	−0.362738	+	0.181369i
$761$	15.0000			0.543750			0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000			0.543750			0.271875	+	0.962333i	$0.412356\pi$
$762$			20.0000i			0.724524i
$763$			2.00000i			0.0724049i
$764$	−8.00000			−0.289430
$765$		0			0
$766$	12.0000			0.433578
$767$		−	40.0000i		−	1.44432i
$768$		−	1.00000i		−	0.0360844i
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$	−5.00000	−	10.0000i	−0.180187	−	0.360375i
$771$	−7.00000			−0.252099
$772$		−	22.0000i		−	0.791797i
$773$			27.0000i			0.971123i	0.874203	+	0.485561i	$0.161385\pi$
$773$			27.0000i			0.971123i	−0.874203	+	0.485561i	$0.838615\pi$
$774$	−1.00000			−0.0359443
$775$	−3.00000	+	4.00000i	−0.107763	+	0.143684i
$776$	−18.0000			−0.646162
$777$			8.00000i			0.286998i
$778$			16.0000i			0.573628i
$779$	30.0000			1.07486
$780$	4.00000	+	8.00000i	0.143223	+	0.286446i
$781$	45.0000			1.61023
$782$		0			0
$783$			2.00000i			0.0714742i
$784$	6.00000

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 - 1.00000i) q^{10} -5.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} -1.00000 q^{14} +(-2.00000 + 1.00000i) q^{15} +1.00000 q^{16} -1.00000i q^{18} +5.00000 q^{19} +(1.00000 + 2.00000i) q^{20} +1.00000 q^{21} -5.00000i q^{22} +9.00000i q^{23} -1.00000 q^{24} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +2.00000 q^{29} +(-1.00000 - 2.00000i) q^{30} +1.00000 q^{31} +1.00000i q^{32} +5.00000i q^{33} +(2.00000 - 1.00000i) q^{35} +1.00000 q^{36} +8.00000i q^{37} +5.00000i q^{38} +4.00000 q^{39} +(-2.00000 + 1.00000i) q^{40} +6.00000 q^{41} +1.00000i q^{42} -1.00000i q^{43} +5.00000 q^{44} +(1.00000 + 2.00000i) q^{45} -9.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -4.00000i q^{52} +13.0000i q^{53} -1.00000 q^{54} +(5.00000 + 10.0000i) q^{55} +1.00000 q^{56} -5.00000i q^{57} +2.00000i q^{58} -10.0000 q^{59} +(2.00000 - 1.00000i) q^{60} -14.0000 q^{61} +1.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +(8.00000 - 4.00000i) q^{65} -5.00000 q^{66} +14.0000i q^{67} +9.00000 q^{69} +(1.00000 + 2.00000i) q^{70} -9.00000 q^{71} +1.00000i q^{72} +9.00000i q^{73} -8.00000 q^{74} +(4.00000 + 3.00000i) q^{75} -5.00000 q^{76} -5.00000i q^{77} +4.00000i q^{78} -5.00000 q^{79} +(-1.00000 - 2.00000i) q^{80} +1.00000 q^{81} +6.00000i q^{82} -6.00000i q^{83} -1.00000 q^{84} +1.00000 q^{86} -2.00000i q^{87} +5.00000i q^{88} -3.00000 q^{89} +(-2.00000 + 1.00000i) q^{90} -4.00000 q^{91} -9.00000i q^{92} -1.00000i q^{93} +12.0000 q^{94} +(-5.00000 - 10.0000i) q^{95} +1.00000 q^{96} -18.0000i q^{97} +6.00000i q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} - 10 q^{11} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 10 q^{19} + 2 q^{20} + 2 q^{21} - 2 q^{24} - 6 q^{25} - 8 q^{26} + 4 q^{29} - 2 q^{30} + 2 q^{31} + 4 q^{35} + 2 q^{36} + 8 q^{39} - 4 q^{40} + 12 q^{41} + 10 q^{44} + 2 q^{45} - 18 q^{46} + 12 q^{49} - 8 q^{50} - 2 q^{54} + 10 q^{55} + 2 q^{56} - 20 q^{59} + 4 q^{60} - 28 q^{61} - 2 q^{64} + 16 q^{65} - 10 q^{66} + 18 q^{69} + 2 q^{70} - 18 q^{71} - 16 q^{74} + 8 q^{75} - 10 q^{76} - 10 q^{79} - 2 q^{80} + 2 q^{81} - 2 q^{84} + 2 q^{86} - 6 q^{89} - 4 q^{90} - 8 q^{91} + 24 q^{94} - 10 q^{95} + 2 q^{96} + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^9 + 4 * q^10 - 10 * q^11 - 2 * q^14 - 4 * q^15 + 2 * q^16 + 10 * q^19 + 2 * q^20 + 2 * q^21 - 2 * q^24 - 6 * q^25 - 8 * q^26 + 4 * q^29 - 2 * q^30 + 2 * q^31 + 4 * q^35 + 2 * q^36 + 8 * q^39 - 4 * q^40 + 12 * q^41 + 10 * q^44 + 2 * q^45 - 18 * q^46 + 12 * q^49 - 8 * q^50 - 2 * q^54 + 10 * q^55 + 2 * q^56 - 20 * q^59 + 4 * q^60 - 28 * q^61 - 2 * q^64 + 16 * q^65 - 10 * q^66 + 18 * q^69 + 2 * q^70 - 18 * q^71 - 16 * q^74 + 8 * q^75 - 10 * q^76 - 10 * q^79 - 2 * q^80 + 2 * q^81 - 2 * q^84 + 2 * q^86 - 6 * q^89 - 4 * q^90 - 8 * q^91 + 24 * q^94 - 10 * q^95 + 2 * q^96 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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