LMFDB - Embedded newform 1110.2.h.a.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(961,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1110.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.h (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:	$8.86339462436$
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			961.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1110.961
Dual form			1110.2.h.a.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000i q^{13} -1.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.00000i q^{17} +1.00000i q^{18} -3.00000i q^{19} -1.00000i q^{20} +1.00000 q^{21} -1.00000i q^{22} -3.00000i q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +10.0000i q^{29} +1.00000 q^{30} -2.00000i q^{31} +1.00000i q^{32} +1.00000 q^{33} +5.00000 q^{34} -1.00000i q^{35} -1.00000 q^{36} +(6.00000 + 1.00000i) q^{37} +3.00000 q^{38} +1.00000i q^{39} +1.00000 q^{40} -2.00000 q^{41} +1.00000i q^{42} -12.0000i q^{43} +1.00000 q^{44} +1.00000i q^{45} +3.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -1.00000i q^{50} +5.00000i q^{51} +1.00000i q^{52} +11.0000 q^{53} -1.00000i q^{54} -1.00000i q^{55} +1.00000i q^{56} +3.00000i q^{57} -10.0000 q^{58} -10.0000i q^{59} +1.00000i q^{60} -14.0000i q^{61} +2.00000 q^{62} -1.00000 q^{63} -1.00000 q^{64} +1.00000 q^{65} +1.00000i q^{66} +12.0000 q^{67} +5.00000i q^{68} +3.00000i q^{69} +1.00000 q^{70} +6.00000 q^{71} -1.00000i q^{72} -1.00000 q^{73} +(-1.00000 + 6.00000i) q^{74} +1.00000 q^{75} +3.00000i q^{76} +1.00000 q^{77} -1.00000 q^{78} +10.0000i q^{79} +1.00000i q^{80} +1.00000 q^{81} -2.00000i q^{82} -3.00000 q^{83} -1.00000 q^{84} +5.00000 q^{85} +12.0000 q^{86} -10.0000i q^{87} +1.00000i q^{88} +9.00000i q^{89} -1.00000 q^{90} +1.00000i q^{91} +3.00000i q^{92} +2.00000i q^{93} +12.0000i q^{94} +3.00000 q^{95} -1.00000i q^{96} +12.0000i q^{97} -6.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{21} - 2 q^{25} + 2 q^{26} - 2 q^{27} + 2 q^{28} + 2 q^{30} + 2 q^{33} + 10 q^{34} - 2 q^{36} + 12 q^{37} + 6 q^{38} + 2 q^{40} - 4 q^{41} + 2 q^{44} + 6 q^{46} + 24 q^{47} - 2 q^{48} - 12 q^{49} + 22 q^{53} - 20 q^{58} + 4 q^{62} - 2 q^{63} - 2 q^{64} + 2 q^{65} + 24 q^{67} + 2 q^{70} + 12 q^{71} - 2 q^{73} - 2 q^{74} + 2 q^{75} + 2 q^{77} - 2 q^{78} + 2 q^{81} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 24 q^{86} - 2 q^{90} + 6 q^{95} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^16 + 2 * q^21 - 2 * q^25 + 2 * q^26 - 2 * q^27 + 2 * q^28 + 2 * q^30 + 2 * q^33 + 10 * q^34 - 2 * q^36 + 12 * q^37 + 6 * q^38 + 2 * q^40 - 4 * q^41 + 2 * q^44 + 6 * q^46 + 24 * q^47 - 2 * q^48 - 12 * q^49 + 22 * q^53 - 20 * q^58 + 4 * q^62 - 2 * q^63 - 2 * q^64 + 2 * q^65 + 24 * q^67 + 2 * q^70 + 12 * q^71 - 2 * q^73 - 2 * q^74 + 2 * q^75 + 2 * q^77 - 2 * q^78 + 2 * q^81 - 6 * q^83 - 2 * q^84 + 10 * q^85 + 24 * q^86 - 2 * q^90 + 6 * q^95 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		−	1.00000i		−	0.408248i
$7$	−1.00000			−0.377964			−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000			−0.377964			−0.188982	+	0.981981i	$0.560519\pi$
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	−1.00000			−0.316228
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000			0.288675
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		−	1.00000i		−	0.267261i
$15$		−	1.00000i		−	0.258199i
$16$	1.00000			0.250000
$17$		−	5.00000i		−	1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$		−	5.00000i		−	1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$			1.00000i			0.235702i
$19$		−	3.00000i		−	0.688247i	−0.938924	−	0.344124i	$-0.888176\pi$
$19$		−	3.00000i		−	0.688247i	0.938924	−	0.344124i	$-0.111824\pi$
$20$		−	1.00000i		−	0.223607i
$21$	1.00000			0.218218
$22$		−	1.00000i		−	0.213201i
$23$		−	3.00000i		−	0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$		−	3.00000i		−	0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$			1.00000i			0.204124i
$25$	−1.00000			−0.200000
$26$	1.00000			0.196116
$27$	−1.00000			−0.192450
$28$	1.00000			0.188982
$29$			10.0000i			1.85695i	0.371391	+	0.928477i	$0.378881\pi$
$29$			10.0000i			1.85695i	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000			0.182574
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$			1.00000i			0.176777i
$33$	1.00000			0.174078
$34$	5.00000			0.857493
$35$		−	1.00000i		−	0.169031i
$36$	−1.00000			−0.166667
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$38$	3.00000			0.486664
$39$			1.00000i			0.160128i
$40$	1.00000			0.158114
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$			1.00000i			0.154303i
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	1.00000			0.150756
$45$			1.00000i			0.149071i
$46$	3.00000			0.442326
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000			−0.144338
$49$	−6.00000			−0.857143
$50$		−	1.00000i		−	0.141421i
$51$			5.00000i			0.700140i
$52$			1.00000i			0.138675i
$53$	11.0000			1.51097			0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000			1.51097			0.755483	+	0.655168i	$0.227402\pi$
$54$		−	1.00000i		−	0.136083i
$55$		−	1.00000i		−	0.134840i
$56$			1.00000i			0.133631i
$57$			3.00000i			0.397360i
$58$	−10.0000			−1.31306
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$			1.00000i			0.129099i
$61$		−	14.0000i		−	1.79252i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$		−	14.0000i		−	1.79252i	0.443533	−	0.896258i	$-0.353725\pi$
$62$	2.00000			0.254000
$63$	−1.00000			−0.125988
$64$	−1.00000			−0.125000
$65$	1.00000			0.124035
$66$			1.00000i			0.123091i
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$			5.00000i			0.606339i
$69$			3.00000i			0.361158i
$70$	1.00000			0.119523
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		−	1.00000i		−	0.117851i
$73$	−1.00000			−0.117041			−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000			−0.117041			−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−1.00000	+	6.00000i	−0.116248	+	0.697486i
$75$	1.00000			0.115470
$76$			3.00000i			0.344124i
$77$	1.00000			0.113961
$78$	−1.00000			−0.113228
$79$			10.0000i			1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$			10.0000i			1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$			1.00000i			0.111803i
$81$	1.00000			0.111111
$82$		−	2.00000i		−	0.220863i
$83$	−3.00000			−0.329293			−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000			−0.329293			−0.164646	+	0.986353i	$0.552648\pi$
$84$	−1.00000			−0.109109
$85$	5.00000			0.542326
$86$	12.0000			1.29399
$87$		−	10.0000i		−	1.07211i
$88$			1.00000i			0.106600i
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	−1.00000			−0.105409
$91$			1.00000i			0.104828i
$92$			3.00000i			0.312772i
$93$			2.00000i			0.207390i
$94$			12.0000i			1.23771i
$95$	3.00000			0.307794
$96$		−	1.00000i		−	0.102062i
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		−	6.00000i		−	0.606092i
$99$	−1.00000			−0.100504
$100$	1.00000			0.100000
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$	−5.00000			−0.495074
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−1.00000			−0.0980581
$105$			1.00000i			0.0975900i
$106$			11.0000i			1.06841i
$107$	−9.00000			−0.870063			−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000			−0.870063			−0.435031	+	0.900415i	$0.643263\pi$
$108$	1.00000			0.0962250
$109$		−	11.0000i		−	1.05361i	−0.849987	−	0.526804i	$-0.823390\pi$
$109$		−	11.0000i		−	1.05361i	0.849987	−	0.526804i	$-0.176610\pi$
$110$	1.00000			0.0953463
$111$	−6.00000	−	1.00000i	−0.569495	−	0.0949158i
$112$	−1.00000			−0.0944911
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−3.00000			−0.280976
$115$	3.00000			0.279751
$116$		−	10.0000i		−	0.928477i
$117$		−	1.00000i		−	0.0924500i
$118$	10.0000			0.920575
$119$			5.00000i			0.458349i
$120$	−1.00000			−0.0912871
$121$	−10.0000			−0.909091
$122$	14.0000			1.26750
$123$	2.00000			0.180334
$124$			2.00000i			0.179605i
$125$		−	1.00000i		−	0.0894427i
$126$		−	1.00000i		−	0.0890871i
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$		−	1.00000i		−	0.0883883i
$129$			12.0000i			1.05654i
$130$			1.00000i			0.0877058i
$131$		−	2.00000i		−	0.174741i	−0.996176	−	0.0873704i	$-0.972154\pi$
$131$		−	2.00000i		−	0.174741i	0.996176	−	0.0873704i	$-0.0278464\pi$
$132$	−1.00000			−0.0870388
$133$			3.00000i			0.260133i
$134$			12.0000i			1.03664i
$135$		−	1.00000i		−	0.0860663i
$136$	−5.00000			−0.428746
$137$	12.0000			1.02523			0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000			1.02523			0.512615	+	0.858619i	$0.328677\pi$
$138$	−3.00000			−0.255377
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$			1.00000i			0.0845154i
$141$	−12.0000			−1.01058
$142$			6.00000i			0.503509i
$143$			1.00000i			0.0836242i
$144$	1.00000			0.0833333
$145$	−10.0000			−0.830455
$146$		−	1.00000i		−	0.0827606i
$147$	6.00000			0.494872
$148$	−6.00000	−	1.00000i	−0.493197	−	0.0821995i
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$			1.00000i			0.0816497i
$151$	−17.0000			−1.38344			−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000			−1.38344			−0.691720	+	0.722166i	$0.743147\pi$
$152$	−3.00000			−0.243332
$153$		−	5.00000i		−	0.404226i
$154$			1.00000i			0.0805823i
$155$	2.00000			0.160644
$156$		−	1.00000i		−	0.0800641i
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$	−10.0000			−0.795557
$159$	−11.0000			−0.872357
$160$	−1.00000			−0.0790569
$161$			3.00000i			0.236433i
$162$			1.00000i			0.0785674i
$163$		−	19.0000i		−	1.48819i	−0.668071	−	0.744097i	$-0.732880\pi$
$163$		−	19.0000i		−	1.48819i	0.668071	−	0.744097i	$-0.267120\pi$
$164$	2.00000			0.156174
$165$			1.00000i			0.0778499i
$166$		−	3.00000i		−	0.232845i
$167$		−	21.0000i		−	1.62503i	−0.582941	−	0.812514i	$-0.698098\pi$
$167$		−	21.0000i		−	1.62503i	0.582941	−	0.812514i	$-0.301902\pi$
$168$		−	1.00000i		−	0.0771517i
$169$	12.0000			0.923077
$170$			5.00000i			0.383482i
$171$		−	3.00000i		−	0.229416i
$172$			12.0000i			0.914991i
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$	10.0000			0.758098
$175$	1.00000			0.0755929
$176$	−1.00000			−0.0753778
$177$			10.0000i			0.751646i
$178$	−9.00000			−0.674579
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$		−	1.00000i		−	0.0745356i
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−1.00000			−0.0741249
$183$			14.0000i			1.03491i
$184$	−3.00000			−0.221163
$185$	−1.00000	+	6.00000i	−0.0735215	+	0.441129i
$186$	−2.00000			−0.146647
$187$			5.00000i			0.365636i
$188$	−12.0000			−0.875190
$189$	1.00000			0.0727393
$190$			3.00000i			0.217643i
$191$		−	3.00000i		−	0.217072i	−0.994092	−	0.108536i	$-0.965384\pi$
$191$		−	3.00000i		−	0.217072i	0.994092	−	0.108536i	$-0.0346163\pi$
$192$	1.00000			0.0721688
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−12.0000			−0.861550
$195$	−1.00000			−0.0716115
$196$	6.00000			0.428571
$197$	−23.0000			−1.63868			−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000			−1.63868			−0.819341	+	0.573306i	$0.805660\pi$
$198$		−	1.00000i		−	0.0710669i
$199$			2.00000i			0.141776i	0.997484	+	0.0708881i	$0.0225833\pi$
$199$			2.00000i			0.141776i	−0.997484	+	0.0708881i	$0.977417\pi$
$200$			1.00000i			0.0707107i
$201$	−12.0000			−0.846415
$202$			10.0000i			0.703598i
$203$		−	10.0000i		−	0.701862i
$204$		−	5.00000i		−	0.350070i
$205$		−	2.00000i		−	0.139686i
$206$	16.0000			1.11477
$207$		−	3.00000i		−	0.208514i
$208$		−	1.00000i		−	0.0693375i
$209$			3.00000i			0.207514i
$210$	−1.00000			−0.0690066
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$	−11.0000			−0.755483
$213$	−6.00000			−0.411113
$214$		−	9.00000i		−	0.615227i
$215$	12.0000			0.818393
$216$			1.00000i			0.0680414i
$217$			2.00000i			0.135769i
$218$	11.0000			0.745014
$219$	1.00000			0.0675737
$220$			1.00000i			0.0674200i
$221$	−5.00000			−0.336336
$222$	1.00000	−	6.00000i	0.0671156	−	0.402694i
$223$	−4.00000			−0.267860			−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000			−0.267860			−0.133930	+	0.990991i	$0.542760\pi$
$224$		−	1.00000i		−	0.0668153i
$225$	−1.00000			−0.0666667
$226$	6.00000			0.399114
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		−	3.00000i		−	0.198680i
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$			3.00000i			0.197814i
$231$	−1.00000			−0.0657952
$232$	10.0000			0.656532
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$	1.00000			0.0653720
$235$			12.0000i			0.782794i
$236$			10.0000i			0.650945i
$237$		−	10.0000i		−	0.649570i
$238$	−5.00000			−0.324102
$239$			8.00000i			0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$			8.00000i			0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$		−	1.00000i		−	0.0645497i
$241$			4.00000i			0.257663i	0.991667	+	0.128831i	$0.0411226\pi$
$241$			4.00000i			0.257663i	−0.991667	+	0.128831i	$0.958877\pi$
$242$		−	10.0000i		−	0.642824i
$243$	−1.00000			−0.0641500
$244$			14.0000i			0.896258i
$245$		−	6.00000i		−	0.383326i
$246$			2.00000i			0.127515i
$247$	−3.00000			−0.190885
$248$	−2.00000			−0.127000
$249$	3.00000			0.190117
$250$	1.00000			0.0632456
$251$		−	8.00000i		−	0.504956i	−0.967603	−	0.252478i	$-0.918755\pi$
$251$		−	8.00000i		−	0.504956i	0.967603	−	0.252478i	$-0.0812455\pi$
$252$	1.00000			0.0629941
$253$			3.00000i			0.188608i
$254$		−	19.0000i		−	1.19217i
$255$	−5.00000			−0.313112
$256$	1.00000			0.0625000
$257$			27.0000i			1.68421i	0.539311	+	0.842107i	$0.318685\pi$
$257$			27.0000i			1.68421i	−0.539311	+	0.842107i	$0.681315\pi$
$258$	−12.0000			−0.747087
$259$	−6.00000	−	1.00000i	−0.372822	−	0.0621370i
$260$	−1.00000			−0.0620174
$261$			10.0000i			0.618984i
$262$	2.00000			0.123560
$263$	4.00000			0.246651			0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000			0.246651			0.123325	+	0.992366i	$0.460644\pi$
$264$		−	1.00000i		−	0.0615457i
$265$			11.0000i			0.675725i
$266$	−3.00000			−0.183942
$267$		−	9.00000i		−	0.550791i
$268$	−12.0000			−0.733017
$269$	−11.0000			−0.670682			−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000			−0.670682			−0.335341	+	0.942097i	$0.608852\pi$
$270$	1.00000			0.0608581
$271$	−4.00000			−0.242983			−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000			−0.242983			−0.121491	+	0.992592i	$0.538768\pi$
$272$		−	5.00000i		−	0.303170i
$273$		−	1.00000i		−	0.0605228i
$274$			12.0000i			0.724947i
$275$	1.00000			0.0603023
$276$		−	3.00000i		−	0.180579i
$277$		−	25.0000i		−	1.50210i	−0.660243	−	0.751052i	$-0.729547\pi$
$277$		−	25.0000i		−	1.50210i	0.660243	−	0.751052i	$-0.270453\pi$
$278$		−	12.0000i		−	0.719712i
$279$		−	2.00000i		−	0.119737i
$280$	−1.00000			−0.0597614
$281$		−	1.00000i		−	0.0596550i	−0.999555	−	0.0298275i	$-0.990504\pi$
$281$		−	1.00000i		−	0.0596550i	0.999555	−	0.0298275i	$-0.00949580\pi$
$282$		−	12.0000i		−	0.714590i
$283$		−	7.00000i		−	0.416107i	−0.978117	−	0.208053i	$-0.933287\pi$
$283$		−	7.00000i		−	0.416107i	0.978117	−	0.208053i	$-0.0667128\pi$
$284$	−6.00000			−0.356034
$285$	−3.00000			−0.177705
$286$	−1.00000			−0.0591312
$287$	2.00000			0.118056
$288$			1.00000i			0.0589256i
$289$	−8.00000			−0.470588
$290$		−	10.0000i		−	0.587220i
$291$		−	12.0000i		−	0.703452i
$292$	1.00000			0.0585206
$293$	15.0000			0.876309			0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000			0.876309			0.438155	+	0.898900i	$0.355632\pi$
$294$			6.00000i			0.349927i
$295$	10.0000			0.582223
$296$	1.00000	−	6.00000i	0.0581238	−	0.348743i
$297$	1.00000			0.0580259
$298$			18.0000i			1.04271i
$299$	−3.00000			−0.173494
$300$	−1.00000			−0.0577350
$301$			12.0000i			0.691669i
$302$		−	17.0000i		−	0.978240i
$303$	−10.0000			−0.574485
$304$		−	3.00000i		−	0.172062i
$305$	14.0000			0.801638
$306$	5.00000			0.285831
$307$	8.00000			0.456584			0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000			0.456584			0.228292	+	0.973593i	$0.426686\pi$
$308$	−1.00000			−0.0569803
$309$			16.0000i			0.910208i
$310$			2.00000i			0.113592i
$311$		−	20.0000i		−	1.13410i	−0.823685	−	0.567048i	$-0.808085\pi$
$311$		−	20.0000i		−	1.13410i	0.823685	−	0.567048i	$-0.191915\pi$
$312$	1.00000			0.0566139
$313$		−	26.0000i		−	1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$		−	26.0000i		−	1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$			12.0000i			0.677199i
$315$		−	1.00000i		−	0.0563436i
$316$		−	10.0000i		−	0.562544i
$317$	−30.0000			−1.68497			−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000			−1.68497			−0.842484	+	0.538721i	$0.818908\pi$
$318$		−	11.0000i		−	0.616849i
$319$		−	10.0000i		−	0.559893i
$320$		−	1.00000i		−	0.0559017i
$321$	9.00000			0.502331
$322$	−3.00000			−0.167183
$323$	−15.0000			−0.834622
$324$	−1.00000			−0.0555556
$325$			1.00000i			0.0554700i
$326$	19.0000			1.05231
$327$			11.0000i			0.608301i
$328$			2.00000i			0.110432i
$329$	−12.0000			−0.661581
$330$	−1.00000			−0.0550482
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	3.00000			0.164646
$333$	6.00000	+	1.00000i	0.328798	+	0.0547997i
$334$	21.0000			1.14907
$335$			12.0000i			0.655630i
$336$	1.00000			0.0545545
$337$	−29.0000			−1.57973			−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000			−1.57973			−0.789865	+	0.613280i	$0.789850\pi$
$338$			12.0000i			0.652714i
$339$			6.00000i			0.325875i
$340$	−5.00000			−0.271163
$341$			2.00000i			0.108306i
$342$	3.00000			0.162221
$343$	13.0000			0.701934
$344$	−12.0000			−0.646997
$345$	−3.00000			−0.161515
$346$		−	9.00000i		−	0.483843i
$347$			2.00000i			0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$			2.00000i			0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$			10.0000i			0.536056i
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$			1.00000i			0.0534522i
$351$			1.00000i			0.0533761i
$352$		−	1.00000i		−	0.0533002i
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−10.0000			−0.531494
$355$			6.00000i			0.318447i
$356$		−	9.00000i		−	0.476999i
$357$		−	5.00000i		−	0.264628i
$358$	−12.0000			−0.634220
$359$	34.0000			1.79445			0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000			1.79445			0.897226	+	0.441572i	$0.145579\pi$
$360$	1.00000			0.0527046
$361$	10.0000			0.526316
$362$		−	10.0000i		−	0.525588i
$363$	10.0000			0.524864
$364$		−	1.00000i		−	0.0524142i
$365$		−	1.00000i		−	0.0523424i
$366$	−14.0000			−0.731792
$367$	27.0000			1.40939			0.704694	−	0.709511i	$-0.251084\pi$
$367$	27.0000			1.40939			0.704694	+	0.709511i	$0.251084\pi$
$368$		−	3.00000i		−	0.156386i
$369$	−2.00000			−0.104116
$370$	−6.00000	−	1.00000i	−0.311925	−	0.0519875i
$371$	−11.0000			−0.571092
$372$		−	2.00000i		−	0.103695i
$373$	−36.0000			−1.86401			−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000			−1.86401			−0.932005	+	0.362446i	$0.881942\pi$
$374$	−5.00000			−0.258544
$375$			1.00000i			0.0516398i
$376$		−	12.0000i		−	0.618853i
$377$	10.0000			0.515026
$378$			1.00000i			0.0514344i
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$	−3.00000			−0.153897
$381$	19.0000			0.973399
$382$	3.00000			0.153493
$383$		−	25.0000i		−	1.27744i	−0.769439	−	0.638720i	$-0.779464\pi$
$383$		−	25.0000i		−	1.27744i	0.769439	−	0.638720i	$-0.220536\pi$
$384$			1.00000i			0.0510310i
$385$			1.00000i			0.0509647i
$386$	4.00000			0.203595
$387$		−	12.0000i		−	0.609994i
$388$		−	12.0000i		−	0.609208i
$389$			20.0000i			1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$389$			20.0000i			1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$390$		−	1.00000i		−	0.0506370i
$391$	−15.0000			−0.758583
$392$			6.00000i			0.303046i
$393$			2.00000i			0.100887i
$394$		−	23.0000i		−	1.15872i
$395$	−10.0000			−0.503155
$396$	1.00000			0.0502519
$397$	4.00000			0.200754			0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000			0.200754			0.100377	+	0.994949i	$0.467995\pi$
$398$	−2.00000			−0.100251
$399$		−	3.00000i		−	0.150188i
$400$	−1.00000			−0.0500000
$401$			9.00000i			0.449439i	0.974424	+	0.224719i	$0.0721465\pi$
$401$			9.00000i			0.449439i	−0.974424	+	0.224719i	$0.927853\pi$
$402$		−	12.0000i		−	0.598506i
$403$	−2.00000			−0.0996271
$404$	−10.0000			−0.497519
$405$			1.00000i			0.0496904i
$406$	10.0000			0.496292
$407$	−6.00000	−	1.00000i	−0.297409	−	0.0495682i
$408$	5.00000			0.247537
$409$			16.0000i			0.791149i	0.918434	+	0.395575i	$0.129455\pi$
$409$			16.0000i			0.791149i	−0.918434	+	0.395575i	$0.870545\pi$
$410$	2.00000			0.0987730
$411$	−12.0000			−0.591916
$412$			16.0000i			0.788263i
$413$			10.0000i			0.492068i
$414$	3.00000			0.147442
$415$		−	3.00000i		−	0.147264i
$416$	1.00000			0.0490290
$417$	12.0000			0.587643
$418$	−3.00000			−0.146735
$419$	7.00000			0.341972			0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000			0.341972			0.170986	+	0.985273i	$0.445305\pi$
$420$		−	1.00000i		−	0.0487950i
$421$			6.00000i			0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$			6.00000i			0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$		−	10.0000i		−	0.486792i
$423$	12.0000			0.583460
$424$		−	11.0000i		−	0.534207i
$425$			5.00000i			0.242536i
$426$		−	6.00000i		−	0.290701i
$427$			14.0000i			0.677507i
$428$	9.00000			0.435031
$429$		−	1.00000i		−	0.0482805i
$430$			12.0000i			0.578691i
$431$		−	3.00000i		−	0.144505i	−0.997386	−	0.0722525i	$-0.976981\pi$
$431$		−	3.00000i		−	0.144505i	0.997386	−	0.0722525i	$-0.0230187\pi$
$432$	−1.00000			−0.0481125
$433$	39.0000			1.87422			0.937110	−	0.349034i	$-0.113490\pi$
$433$	39.0000			1.87422			0.937110	+	0.349034i	$0.113490\pi$
$434$	−2.00000			−0.0960031
$435$	10.0000			0.479463
$436$			11.0000i			0.526804i
$437$	−9.00000			−0.430528
$438$			1.00000i			0.0477818i
$439$			10.0000i			0.477274i	0.971109	+	0.238637i	$0.0767006\pi$
$439$			10.0000i			0.477274i	−0.971109	+	0.238637i	$0.923299\pi$
$440$	−1.00000			−0.0476731
$441$	−6.00000			−0.285714
$442$		−	5.00000i		−	0.237826i
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$	6.00000	+	1.00000i	0.284747	+	0.0474579i
$445$	−9.00000			−0.426641
$446$		−	4.00000i		−	0.189405i
$447$	−18.0000			−0.851371
$448$	1.00000			0.0472456
$449$			10.0000i			0.471929i	0.971762	+	0.235965i	$0.0758249\pi$
$449$			10.0000i			0.471929i	−0.971762	+	0.235965i	$0.924175\pi$
$450$		−	1.00000i		−	0.0471405i
$451$	2.00000			0.0941763
$452$			6.00000i			0.282216i
$453$	17.0000			0.798730
$454$	4.00000			0.187729
$455$	−1.00000			−0.0468807
$456$	3.00000			0.140488
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		−	20.0000i		−	0.934539i
$459$			5.00000i			0.233380i
$460$	−3.00000			−0.139876
$461$			16.0000i			0.745194i	0.927993	+	0.372597i	$0.121533\pi$
$461$			16.0000i			0.745194i	−0.927993	+	0.372597i	$0.878467\pi$
$462$		−	1.00000i		−	0.0465242i
$463$			14.0000i			0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$			14.0000i			0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$			10.0000i			0.464238i
$465$	−2.00000			−0.0927478
$466$			12.0000i			0.555889i
$467$		−	40.0000i		−	1.85098i	−0.378773	−	0.925490i	$-0.623654\pi$
$467$		−	40.0000i		−	1.85098i	0.378773	−	0.925490i	$-0.376346\pi$
$468$			1.00000i			0.0462250i
$469$	−12.0000			−0.554109
$470$	−12.0000			−0.553519
$471$	−12.0000			−0.552931
$472$	−10.0000			−0.460287
$473$			12.0000i			0.551761i
$474$	10.0000			0.459315
$475$			3.00000i			0.137649i
$476$		−	5.00000i		−	0.229175i
$477$	11.0000			0.503655
$478$	−8.00000			−0.365911
$479$			15.0000i			0.685367i	0.939451	+	0.342684i	$0.111336\pi$
$479$			15.0000i			0.685367i	−0.939451	+	0.342684i	$0.888664\pi$
$480$	1.00000			0.0456435
$481$	1.00000	−	6.00000i	0.0455961	−	0.273576i
$482$	−4.00000			−0.182195
$483$		−	3.00000i		−	0.136505i
$484$	10.0000			0.454545
$485$	−12.0000			−0.544892
$486$		−	1.00000i		−	0.0453609i
$487$			28.0000i			1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$			28.0000i			1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	−14.0000			−0.633750
$489$			19.0000i			0.859210i
$490$	6.00000			0.271052
$491$	−7.00000			−0.315906			−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000			−0.315906			−0.157953	+	0.987447i	$0.550489\pi$
$492$	−2.00000			−0.0901670
$493$	50.0000			2.25189
$494$		−	3.00000i		−	0.134976i
$495$		−	1.00000i		−	0.0449467i
$496$		−	2.00000i		−	0.0898027i
$497$	−6.00000			−0.269137
$498$			3.00000i			0.134433i
$499$		−	19.0000i		−	0.850557i	−0.905063	−	0.425278i	$-0.860176\pi$
$499$		−	19.0000i		−	0.850557i	0.905063	−	0.425278i	$-0.139824\pi$
$500$			1.00000i			0.0447214i
$501$			21.0000i			0.938211i
$502$	8.00000			0.357057
$503$			32.0000i			1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$			32.0000i			1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$			1.00000i			0.0445435i
$505$			10.0000i			0.444994i
$506$	−3.00000			−0.133366
$507$	−12.0000			−0.532939
$508$	19.0000			0.842989
$509$	29.0000			1.28540			0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000			1.28540			0.642701	+	0.766117i	$0.277814\pi$
$510$		−	5.00000i		−	0.221404i
$511$	1.00000			0.0442374
$512$			1.00000i			0.0441942i
$513$			3.00000i			0.132453i
$514$	−27.0000			−1.19092
$515$	16.0000			0.705044
$516$		−	12.0000i		−	0.528271i
$517$	−12.0000			−0.527759
$518$	1.00000	−	6.00000i	0.0439375	−	0.263625i
$519$	9.00000			0.395056
$520$		−	1.00000i		−	0.0438529i
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$	−10.0000			−0.437688
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$			2.00000i			0.0873704i
$525$	−1.00000			−0.0436436
$526$			4.00000i			0.174408i
$527$	−10.0000			−0.435607
$528$	1.00000			0.0435194
$529$	14.0000			0.608696
$530$	−11.0000			−0.477809
$531$		−	10.0000i		−	0.433963i
$532$		−	3.00000i		−	0.130066i
$533$			2.00000i			0.0866296i
$534$	9.00000			0.389468
$535$		−	9.00000i		−	0.389104i
$536$		−	12.0000i		−	0.518321i
$537$		−	12.0000i		−	0.517838i
$538$		−	11.0000i		−	0.474244i
$539$	6.00000			0.258438
$540$			1.00000i			0.0430331i
$541$		−	29.0000i		−	1.24681i	−0.781900	−	0.623404i	$-0.785749\pi$
$541$		−	29.0000i		−	1.24681i	0.781900	−	0.623404i	$-0.214251\pi$
$542$		−	4.00000i		−	0.171815i
$543$	10.0000			0.429141
$544$	5.00000			0.214373
$545$	11.0000			0.471188
$546$	1.00000			0.0427960
$547$			13.0000i			0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$			13.0000i			0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$	−12.0000			−0.512615
$549$		−	14.0000i		−	0.597505i
$550$			1.00000i			0.0426401i
$551$	30.0000			1.27804
$552$	3.00000			0.127688
$553$		−	10.0000i		−	0.425243i
$554$	25.0000			1.06215
$555$	1.00000	−	6.00000i	0.0424476	−	0.254686i
$556$	12.0000			0.508913
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	2.00000			0.0846668
$559$	−12.0000			−0.507546
$560$		−	1.00000i		−	0.0422577i
$561$		−	5.00000i		−	0.211100i
$562$	1.00000			0.0421825
$563$			30.0000i			1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$			30.0000i			1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	12.0000			0.505291
$565$	6.00000			0.252422
$566$	7.00000			0.294232
$567$	−1.00000			−0.0419961
$568$		−	6.00000i		−	0.251754i
$569$			27.0000i			1.13190i	0.824440	+	0.565949i	$0.191490\pi$
$569$			27.0000i			1.13190i	−0.824440	+	0.565949i	$0.808510\pi$
$570$		−	3.00000i		−	0.125656i
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		−	1.00000i		−	0.0418121i
$573$			3.00000i			0.125327i
$574$			2.00000i			0.0834784i
$575$			3.00000i			0.125109i
$576$	−1.00000			−0.0416667
$577$			20.0000i			0.832611i	0.909225	+	0.416305i	$0.136675\pi$
$577$			20.0000i			0.832611i	−0.909225	+	0.416305i	$0.863325\pi$
$578$		−	8.00000i		−	0.332756i
$579$			4.00000i			0.166234i
$580$	10.0000			0.415227
$581$	3.00000			0.124461
$582$	12.0000			0.497416
$583$	−11.0000			−0.455573
$584$			1.00000i			0.0413803i
$585$	1.00000			0.0413449
$586$			15.0000i			0.619644i
$587$			6.00000i			0.247647i	0.992304	+	0.123823i	$0.0395156\pi$
$587$			6.00000i			0.247647i	−0.992304	+	0.123823i	$0.960484\pi$
$588$	−6.00000			−0.247436
$589$	−6.00000			−0.247226
$590$			10.0000i			0.411693i
$591$	23.0000			0.946094
$592$	6.00000	+	1.00000i	0.246598	+	0.0410997i
$593$	−14.0000			−0.574911			−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$			1.00000i			0.0410305i
$595$	−5.00000			−0.204980
$596$	−18.0000			−0.737309
$597$		−	2.00000i		−	0.0818546i
$598$		−	3.00000i		−	0.122679i
$599$	−38.0000			−1.55264			−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000			−1.55264			−0.776319	+	0.630340i	$0.782915\pi$
$600$		−	1.00000i		−	0.0408248i
$601$	15.0000			0.611863			0.305931	−	0.952054i	$-0.401032\pi$
$601$	15.0000			0.611863			0.305931	+	0.952054i	$0.401032\pi$
$602$	−12.0000			−0.489083
$603$	12.0000			0.488678
$604$	17.0000			0.691720
$605$		−	10.0000i		−	0.406558i
$606$		−	10.0000i		−	0.406222i
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	3.00000			0.121666
$609$			10.0000i			0.405220i
$610$			14.0000i			0.566843i
$611$		−	12.0000i		−	0.485468i
$612$			5.00000i			0.202113i
$613$	32.0000			1.29247			0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000			1.29247			0.646234	+	0.763139i	$0.276343\pi$
$614$			8.00000i			0.322854i
$615$			2.00000i			0.0806478i
$616$		−	1.00000i		−	0.0402911i
$617$	−46.0000			−1.85189			−0.925945	−	0.377658i	$-0.876729\pi$
$617$	−46.0000			−1.85189			−0.925945	+	0.377658i	$0.876729\pi$
$618$	−16.0000			−0.643614
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$	−2.00000			−0.0803219
$621$			3.00000i			0.120386i
$622$	20.0000			0.801927
$623$		−	9.00000i		−	0.360577i
$624$			1.00000i			0.0400320i
$625$	1.00000			0.0400000
$626$	26.0000			1.03917
$627$		−	3.00000i		−	0.119808i
$628$	−12.0000			−0.478852
$629$	5.00000	−	30.0000i	0.199363	−	1.19618i
$630$	1.00000			0.0398410
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	10.0000			0.397779
$633$	10.0000			0.397464
$634$		−	30.0000i		−	1.19145i
$635$		−	19.0000i		−	0.753992i
$636$	11.0000			0.436178
$637$			6.00000i			0.237729i
$638$	10.0000			0.395904
$639$	6.00000			0.237356
$640$	1.00000			0.0395285
$641$	20.0000			0.789953			0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000			0.789953			0.394976	+	0.918691i	$0.370753\pi$
$642$			9.00000i			0.355202i
$643$			31.0000i			1.22252i	0.791430	+	0.611260i	$0.209337\pi$
$643$			31.0000i			1.22252i	−0.791430	+	0.611260i	$0.790663\pi$
$644$		−	3.00000i		−	0.118217i
$645$	−12.0000			−0.472500
$646$		−	15.0000i		−	0.590167i
$647$		−	9.00000i		−	0.353827i	−0.984226	−	0.176913i	$-0.943389\pi$
$647$		−	9.00000i		−	0.353827i	0.984226	−	0.176913i	$-0.0566112\pi$
$648$		−	1.00000i		−	0.0392837i
$649$			10.0000i			0.392534i
$650$	−1.00000			−0.0392232
$651$		−	2.00000i		−	0.0783862i
$652$			19.0000i			0.744097i
$653$		−	38.0000i		−	1.48705i	−0.668705	−	0.743527i	$-0.733151\pi$
$653$		−	38.0000i		−	1.48705i	0.668705	−	0.743527i	$-0.266849\pi$
$654$	−11.0000			−0.430134
$655$	2.00000			0.0781465
$656$	−2.00000			−0.0780869
$657$	−1.00000			−0.0390137
$658$		−	12.0000i		−	0.467809i
$659$	24.0000			0.934907			0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000			0.934907			0.467454	+	0.884018i	$0.345171\pi$
$660$		−	1.00000i		−	0.0389249i
$661$		−	7.00000i		−	0.272268i	−0.990690	−	0.136134i	$-0.956532\pi$
$661$		−	7.00000i		−	0.272268i	0.990690	−	0.136134i	$-0.0434678\pi$
$662$	−4.00000			−0.155464
$663$	5.00000			0.194184
$664$			3.00000i			0.116423i
$665$	−3.00000			−0.116335
$666$	−1.00000	+	6.00000i	−0.0387492	+	0.232495i
$667$	30.0000			1.16160
$668$			21.0000i			0.812514i
$669$	4.00000			0.154649
$670$	−12.0000			−0.463600
$671$			14.0000i			0.540464i
$672$			1.00000i			0.0385758i
$673$	37.0000			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$674$		−	29.0000i		−	1.11704i
$675$	1.00000			0.0384900
$676$	−12.0000			−0.461538
$677$	31.0000			1.19143			0.595713	−	0.803197i	$-0.296869\pi$
$677$	31.0000			1.19143			0.595713	+	0.803197i	$0.296869\pi$
$678$	−6.00000			−0.230429
$679$		−	12.0000i		−	0.460518i
$680$		−	5.00000i		−	0.191741i
$681$			4.00000i			0.153280i
$682$	−2.00000			−0.0765840
$683$		−	16.0000i		−	0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$		−	16.0000i		−	0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$			3.00000i			0.114708i
$685$			12.0000i			0.458496i
$686$			13.0000i			0.496342i
$687$	20.0000			0.763048
$688$		−	12.0000i		−	0.457496i
$689$		−	11.0000i		−	0.419067i
$690$		−	3.00000i		−	0.114208i
$691$	14.0000			0.532585			0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000			0.532585			0.266293	+	0.963892i	$0.414201\pi$
$692$	9.00000			0.342129
$693$	1.00000			0.0379869
$694$	−2.00000			−0.0759190
$695$		−	12.0000i		−	0.455186i
$696$	−10.0000			−0.379049
$697$			10.0000i			0.378777i
$698$			30.0000i			1.13552i
$699$	−12.0000			−0.453882
$700$	−1.00000			−0.0377964
$701$			24.0000i			0.906467i	0.891392	+	0.453234i	$0.149730\pi$
$701$			24.0000i			0.906467i	−0.891392	+	0.453234i	$0.850270\pi$
$702$	−1.00000			−0.0377426
$703$	3.00000	−	18.0000i	0.113147	−	0.678883i
$704$	1.00000			0.0376889
$705$		−	12.0000i		−	0.451946i
$706$	−14.0000			−0.526897
$707$	−10.0000			−0.376089
$708$		−	10.0000i		−	0.375823i
$709$		−	33.0000i		−	1.23934i	−0.784862	−	0.619671i	$-0.787266\pi$
$709$		−	33.0000i		−	1.23934i	0.784862	−	0.619671i	$-0.212734\pi$
$710$	−6.00000			−0.225176
$711$			10.0000i			0.375029i
$712$	9.00000			0.337289
$713$	−6.00000			−0.224702
$714$	5.00000			0.187120
$715$	−1.00000			−0.0373979
$716$		−	12.0000i		−	0.448461i
$717$		−	8.00000i		−	0.298765i
$718$			34.0000i			1.26887i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$			1.00000i			0.0372678i
$721$			16.0000i			0.595871i
$722$			10.0000i			0.372161i
$723$		−	4.00000i		−	0.148762i
$724$	10.0000			0.371647
$725$		−	10.0000i		−	0.371391i
$726$			10.0000i			0.371135i
$727$		−	38.0000i		−	1.40934i	−0.709534	−	0.704671i	$-0.751095\pi$
$727$		−	38.0000i		−	1.40934i	0.709534	−	0.704671i	$-0.248905\pi$
$728$	1.00000			0.0370625
$729$	1.00000			0.0370370
$730$	1.00000			0.0370117
$731$	−60.0000			−2.21918
$732$		−	14.0000i		−	0.517455i
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$			27.0000i			0.996588i
$735$			6.00000i			0.221313i
$736$	3.00000			0.110581
$737$	−12.0000			−0.442026
$738$		−	2.00000i		−	0.0736210i
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$	1.00000	−	6.00000i	0.0367607	−	0.220564i
$741$	3.00000			0.110208
$742$		−	11.0000i		−	0.403823i
$743$	−34.0000			−1.24734			−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000			−1.24734			−0.623670	+	0.781688i	$0.714359\pi$
$744$	2.00000			0.0733236
$745$			18.0000i			0.659469i
$746$		−	36.0000i		−	1.31805i
$747$	−3.00000			−0.109764
$748$		−	5.00000i		−	0.182818i
$749$	9.00000			0.328853
$750$	−1.00000			−0.0365148
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	12.0000			0.437595
$753$			8.00000i			0.291536i
$754$			10.0000i			0.364179i
$755$		−	17.0000i		−	0.618693i
$756$	−1.00000			−0.0363696
$757$		−	7.00000i		−	0.254419i	−0.991876	−	0.127210i	$-0.959398\pi$
$757$		−	7.00000i		−	0.254419i	0.991876	−	0.127210i	$-0.0406021\pi$
$758$			10.0000i			0.363216i
$759$		−	3.00000i		−	0.108893i
$760$		−	3.00000i		−	0.108821i
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$			19.0000i			0.688297i
$763$			11.0000i			0.398227i
$764$			3.00000i			0.108536i
$765$	5.00000			0.180775
$766$	25.0000			0.903287
$767$	−10.0000			−0.361079
$768$	−1.00000			−0.0360844
$769$		−	38.0000i		−	1.37032i	−0.728395	−	0.685158i	$-0.759733\pi$
$769$		−	38.0000i		−	1.37032i	0.728395	−	0.685158i	$-0.240267\pi$
$770$	−1.00000			−0.0360375
$771$		−	27.0000i		−	0.972381i
$772$			4.00000i			0.143963i
$773$	−7.00000			−0.251773			−0.125886	−	0.992045i	$-0.540177\pi$
$773$	−7.00000			−0.251773			−0.125886	+	0.992045i	$0.540177\pi$
$774$	12.0000			0.431331
$775$			2.00000i			0.0718421i
$776$	12.0000			0.430775
$777$	6.00000	+	1.00000i	0.215249	+	0.0358748i
$778$	−20.0000			−0.717035
$779$			6.00000i			0.214972i
$780$	1.00000			0.0358057
$781$	−6.00000			−0.214697
$782$		−	15.0000i		−	0.536399i
$783$		−	10.0000i		−	0.357371i
$784$	−6.00000			−0.214286
$785$			12.0000i			0.428298i
$786$ </

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000 q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000i q^{13} -1.00000i q^{14} -1.00000i q^{15} +1.00000 q^{16} -5.00000i q^{17} +1.00000i q^{18} -3.00000i q^{19} -1.00000i q^{20} +1.00000 q^{21} -1.00000i q^{22} -3.00000i q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +10.0000i q^{29} +1.00000 q^{30} -2.00000i q^{31} +1.00000i q^{32} +1.00000 q^{33} +5.00000 q^{34} -1.00000i q^{35} -1.00000 q^{36} +(6.00000 + 1.00000i) q^{37} +3.00000 q^{38} +1.00000i q^{39} +1.00000 q^{40} -2.00000 q^{41} +1.00000i q^{42} -12.0000i q^{43} +1.00000 q^{44} +1.00000i q^{45} +3.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -1.00000i q^{50} +5.00000i q^{51} +1.00000i q^{52} +11.0000 q^{53} -1.00000i q^{54} -1.00000i q^{55} +1.00000i q^{56} +3.00000i q^{57} -10.0000 q^{58} -10.0000i q^{59} +1.00000i q^{60} -14.0000i q^{61} +2.00000 q^{62} -1.00000 q^{63} -1.00000 q^{64} +1.00000 q^{65} +1.00000i q^{66} +12.0000 q^{67} +5.00000i q^{68} +3.00000i q^{69} +1.00000 q^{70} +6.00000 q^{71} -1.00000i q^{72} -1.00000 q^{73} +(-1.00000 + 6.00000i) q^{74} +1.00000 q^{75} +3.00000i q^{76} +1.00000 q^{77} -1.00000 q^{78} +10.0000i q^{79} +1.00000i q^{80} +1.00000 q^{81} -2.00000i q^{82} -3.00000 q^{83} -1.00000 q^{84} +5.00000 q^{85} +12.0000 q^{86} -10.0000i q^{87} +1.00000i q^{88} +9.00000i q^{89} -1.00000 q^{90} +1.00000i q^{91} +3.00000i q^{92} +2.00000i q^{93} +12.0000i q^{94} +3.00000 q^{95} -1.00000i q^{96} +12.0000i q^{97} -6.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} - 2 q^{7} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{16} + 2 q^{21} - 2 q^{25} + 2 q^{26} - 2 q^{27} + 2 q^{28} + 2 q^{30} + 2 q^{33} + 10 q^{34} - 2 q^{36} + 12 q^{37} + 6 q^{38} + 2 q^{40} - 4 q^{41} + 2 q^{44} + 6 q^{46} + 24 q^{47} - 2 q^{48} - 12 q^{49} + 22 q^{53} - 20 q^{58} + 4 q^{62} - 2 q^{63} - 2 q^{64} + 2 q^{65} + 24 q^{67} + 2 q^{70} + 12 q^{71} - 2 q^{73} - 2 q^{74} + 2 q^{75} + 2 q^{77} - 2 q^{78} + 2 q^{81} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 24 q^{86} - 2 q^{90} + 6 q^{95} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 - 2 * q^7 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 2 * q^16 + 2 * q^21 - 2 * q^25 + 2 * q^26 - 2 * q^27 + 2 * q^28 + 2 * q^30 + 2 * q^33 + 10 * q^34 - 2 * q^36 + 12 * q^37 + 6 * q^38 + 2 * q^40 - 4 * q^41 + 2 * q^44 + 6 * q^46 + 24 * q^47 - 2 * q^48 - 12 * q^49 + 22 * q^53 - 20 * q^58 + 4 * q^62 - 2 * q^63 - 2 * q^64 + 2 * q^65 + 24 * q^67 + 2 * q^70 + 12 * q^71 - 2 * q^73 - 2 * q^74 + 2 * q^75 + 2 * q^77 - 2 * q^78 + 2 * q^81 - 6 * q^83 - 2 * q^84 + 10 * q^85 + 24 * q^86 - 2 * q^90 + 6 * q^95 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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