LMFDB - Embedded newform 966.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.a (trivial)

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:	yes
Analytic conductor:	$7.71354883526$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

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

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	1.00000	0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.00000	−0.973329
$39$	−4.00000	−0.640513
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−1.00000	−0.154303
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	1.00000	0.147442
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−6.00000	−0.794719
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	10.0000	1.27000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	−2.00000	−0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	6.00000	0.688247
$77$	0	0
$78$	4.00000	0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−12.0000	−1.29399
$87$	6.00000	0.643268
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	2.00000	0.210819
$91$	−4.00000	−0.419314
$92$	−1.00000	−0.104257
$93$	10.0000	1.03695
$94$	−10.0000	−1.03142
$95$	−12.0000	−1.23117
$96$	1.00000	0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	10.0000	0.971286
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	6.00000	0.561951
$115$	2.00000	0.186501
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	12.0000	1.10469
$119$	0	0
$120$	−2.00000	−0.182574
$121$	−11.0000	−1.00000
$122$	14.0000	1.26750
$123$	2.00000	0.180334
$124$	−10.0000	−0.898027
$125$	12.0000	1.07331
$126$	1.00000	0.0890871
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	8.00000	0.701646
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	12.0000	1.03664
$135$	2.00000	0.172133
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−1.00000	−0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	2.00000	0.169031
$141$	−10.0000	−0.842152
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	−14.0000	−1.15865
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−1.00000	−0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	20.0000	1.60644
$156$	−4.00000	−0.320256
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	2.00000	0.158114
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−2.00000	−0.155230
$167$	14.0000	1.08335	0.541676	−	0.840587i	$-0.317790\pi$
$167$	14.0000	1.08335	0.541676	+	0.840587i	$0.317790\pi$
$168$	−1.00000	−0.0771517
$169$	3.00000	0.230769
$170$	0	0
$171$	6.00000	0.458831
$172$	12.0000	0.914991
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	−6.00000	−0.454859
$175$	1.00000	0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	12.0000	0.899438
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.00000	−0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000	0.296500
$183$	14.0000	1.03491
$184$	1.00000	0.0737210
$185$	12.0000	0.882258
$186$	−10.0000	−0.733236
$187$	0	0
$188$	10.0000	0.729325
$189$	1.00000	0.0727393
$190$	12.0000	0.870572
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−12.0000	−0.861550
$195$	8.00000	0.572892
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	1.00000	0.0707107
$201$	12.0000	0.846415
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	4.00000	0.279372
$206$	4.00000	0.278693
$207$	−1.00000	−0.0695048
$208$	4.00000	0.277350
$209$	0	0
$210$	2.00000	0.138013
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	−10.0000	−0.686803
$213$	12.0000	0.822226
$214$	−8.00000	−0.546869
$215$	−24.0000	−1.63679
$216$	1.00000	0.0680414
$217$	10.0000	0.678844
$218$	10.0000	0.677285
$219$	−14.0000	−0.946032
$220$	0	0
$221$	0	0
$222$	−6.00000	−0.402694
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	14.0000	0.931266
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−6.00000	−0.397360
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	6.00000	0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−4.00000	−0.261488
$235$	−20.0000	−1.30466
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	2.00000	0.129099
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	−2.00000	−0.127775
$246$	−2.00000	−0.127515
$247$	24.0000	1.52708
$248$	10.0000	0.635001
$249$	−2.00000	−0.126745
$250$	−12.0000	−0.758947
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	12.0000	0.747087
$259$	6.00000	0.372822
$260$	−8.00000	−0.496139
$261$	−6.00000	−0.371391
$262$	−4.00000	−0.247121
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	6.00000	0.367884
$267$	12.0000	0.734388
$268$	−12.0000	−0.733017
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−2.00000	−0.121716
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	18.0000	1.08742
$275$	0	0
$276$	1.00000	0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	−2.00000	−0.119523
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	10.0000	0.595491
$283$	−18.0000	−1.06999	−0.534994	−	0.844856i	$-0.679686\pi$
$283$	−18.0000	−1.06999	−0.534994	+	0.844856i	$0.679686\pi$
$284$	−12.0000	−0.712069
$285$	12.0000	0.710819
$286$	0	0
$287$	2.00000	0.118056
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−12.0000	−0.704664
$291$	−12.0000	−0.703452
$292$	14.0000	0.819288
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	1.00000	0.0583212
$295$	24.0000	1.39733
$296$	6.00000	0.348743
$297$	0	0
$298$	−18.0000	−1.04271
$299$	−4.00000	−0.231326
$300$	1.00000	0.0577350
$301$	−12.0000	−0.691669
$302$	−12.0000	−0.690522
$303$	0	0
$304$	6.00000	0.344124
$305$	28.0000	1.60328
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	−20.0000	−1.13592
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	4.00000	0.226455
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	2.00000	0.112867
$315$	2.00000	0.112687
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	−10.0000	−0.560772
$319$	0	0
$320$	−2.00000	−0.111803
$321$	−8.00000	−0.446516
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	12.0000	0.664619
$327$	10.0000	0.553001
$328$	2.00000	0.110432
$329$	−10.0000	−0.551318
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	2.00000	0.109764
$333$	−6.00000	−0.328798
$334$	−14.0000	−0.766046
$335$	24.0000	1.31126
$336$	1.00000	0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−3.00000	−0.163178
$339$	14.0000	0.760376
$340$	0	0
$341$	0	0
$342$	−6.00000	−0.324443
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	−2.00000	−0.107676
$346$	24.0000	1.29025
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	6.00000	0.321634
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−1.00000	−0.0534522
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	−12.0000	−0.637793
$355$	24.0000	1.27379
$356$	−12.0000	−0.635999
$357$	0	0
$358$	4.00000	0.211407
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	2.00000	0.105409
$361$	17.0000	0.894737
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	−4.00000	−0.209657
$365$	−28.0000	−1.46559
$366$	−14.0000	−0.731792
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−12.0000	−0.623850
$371$	10.0000	0.519174
$372$	10.0000	0.518476
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	−10.0000	−0.515711
$377$	−24.0000	−1.23606
$378$	−1.00000	−0.0514344
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	−12.0000	−0.615587
$381$	12.0000	0.614779
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	12.0000	0.609994
$388$	12.0000	0.609208
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−4.00000	−0.201773
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	−20.0000	−1.00251
$399$	6.00000	0.300376
$400$	−1.00000	−0.0500000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	−12.0000	−0.598506
$403$	−40.0000	−1.99254
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	−6.00000	−0.297775
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−4.00000	−0.197546
$411$	18.0000	0.887875
$412$	−4.00000	−0.197066
$413$	12.0000	0.590481
$414$	1.00000	0.0491473
$415$	−4.00000	−0.196352
$416$	−4.00000	−0.196116
$417$	0	0
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	−2.00000	−0.0975900
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	28.0000	1.36302
$423$	10.0000	0.486217
$424$	10.0000	0.485643
$425$	0	0
$426$	−12.0000	−0.581402
$427$	14.0000	0.677507
$428$	8.00000	0.386695
$429$	0	0
$430$	24.0000	1.15738
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	−10.0000	−0.480015
$435$	−12.0000	−0.575356
$436$	−10.0000	−0.478913
$437$	−6.00000	−0.287019
$438$	14.0000	0.668946
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	6.00000	0.284747
$445$	24.0000	1.13771
$446$	−18.0000	−0.852325
$447$	−18.0000	−0.851371
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−14.0000	−0.658505
$453$	−12.0000	−0.563809
$454$	−14.0000	−0.657053
$455$	8.00000	0.375046
$456$	6.00000	0.280976
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	2.00000	0.0932505
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−6.00000	−0.278543
$465$	−20.0000	−0.927478
$466$	−22.0000	−1.01913
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	4.00000	0.184900
$469$	12.0000	0.554109
$470$	20.0000	0.922531
$471$	2.00000	0.0921551
$472$	12.0000	0.552345
$473$	0	0
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−10.0000	−0.457869
$478$	−20.0000	−0.914779
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	−2.00000	−0.0912871
$481$	−24.0000	−1.09431
$482$	−24.0000	−1.09317
$483$	−1.00000	−0.0455016
$484$	−11.0000	−0.500000
$485$	−24.0000	−1.08978
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	14.0000	0.633750
$489$	12.0000	0.542659
$490$	2.00000	0.0903508
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	−24.0000	−1.07981
$495$	0	0
$496$	−10.0000	−0.449013
$497$	12.0000	0.538274
$498$	2.00000	0.0896221
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	12.0000	0.536656
$501$	−14.0000	−0.625474
$502$	−18.0000	−0.803379
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−12.0000	−0.532414
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	−12.0000	−0.528271
$517$	0	0
$518$	−6.00000	−0.263625
$519$	24.0000	1.05348
$520$	8.00000	0.350823
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	6.00000	0.262613
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	4.00000	0.174741
$525$	−1.00000	−0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−20.0000	−0.868744
$531$	−12.0000	−0.520756
$532$	−6.00000	−0.260133
$533$	−8.00000	−0.346518
$534$	−12.0000	−0.519291
$535$	−16.0000	−0.691740
$536$	12.0000	0.518321
$537$	4.00000	0.172613
$538$	24.0000	1.03471
$539$	0	0
$540$	2.00000	0.0860663
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	2.00000	0.0859074
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	20.0000	0.856706
$546$	−4.00000	−0.171184
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−18.0000	−0.768922
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−36.0000	−1.53365
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	10.0000	0.424859
$555$	−12.0000	−0.509372
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	10.0000	0.423334
$559$	48.0000	2.03018
$560$	2.00000	0.0845154
$561$	0	0
$562$	2.00000	0.0843649
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	−10.0000	−0.421076
$565$	28.0000	1.17797
$566$	18.0000	0.756596
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	−12.0000	−0.502625
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	17.0000	0.707107
$579$	10.0000	0.415586
$580$	12.0000	0.498273
$581$	−2.00000	−0.0829740
$582$	12.0000	0.497416
$583$	0	0
$584$	−14.0000	−0.579324
$585$	−8.00000	−0.330759
$586$	18.0000	0.743573
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	−1.00000	−0.0412393
$589$	−60.0000	−2.47226
$590$	−24.0000	−0.988064
$591$	−6.00000	−0.246807
$592$	−6.00000	−0.246598
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	−20.0000	−0.818546
$598$	4.00000	0.163572
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−1.00000	−0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	12.0000	0.489083
$603$	−12.0000	−0.488678
$604$	12.0000	0.488273
$605$	22.0000	0.894427
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−6.00000	−0.243332
$609$	−6.00000	−0.243132
$610$	−28.0000	−1.13369
$611$	40.0000	1.61823
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−16.0000	−0.645707
$615$	−4.00000	−0.161296
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	−4.00000	−0.160904
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	20.0000	0.803219
$621$	1.00000	0.0401286
$622$	−10.0000	−0.400963
$623$	12.0000	0.480770
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	8.00000	0.319744
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	−2.00000	−0.0796819
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	28.0000	1.11290
$634$	2.00000	0.0794301
$635$	24.0000	0.952411
$636$	10.0000	0.396526
$637$	4.00000	0.158486
$638$	0	0
$639$	−12.0000	−0.474713
$640$	2.00000	0.0790569
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	8.00000	0.315735
$643$	−46.0000	−1.81406	−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000	−1.81406	−0.907031	+	0.421063i	$0.861657\pi$
$644$	1.00000	0.0394055
$645$	24.0000	0.944999
$646$	0	0
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	4.00000	0.156893
$651$	−10.0000	−0.391931
$652$	−12.0000	−0.469956
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	−10.0000	−0.391031
$655$	−8.00000	−0.312586
$656$	−2.00000	−0.0780869
$657$	14.0000	0.546192
$658$	10.0000	0.389841
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	12.0000	0.465340
$666$	6.00000	0.232495
$667$	6.00000	0.232321
$668$	14.0000	0.541676
$669$	−18.0000	−0.695920
$670$	−24.0000	−0.927201
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−14.0000	−0.539260
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−14.0000	−0.537667
$679$	−12.0000	−0.460518
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	6.00000	0.229416
$685$	36.0000	1.37549
$686$	1.00000	0.0381802
$687$	6.00000	0.228914
$688$	12.0000	0.457496
$689$	−40.0000	−1.52388
$690$	2.00000	0.0761387
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	16.0000	0.605609
$699$	−22.0000	−0.832116
$700$	1.00000	0.0377964
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	4.00000	0.150970
$703$	−36.0000	−1.35777
$704$	0	0
$705$	20.0000	0.753244
$706$	26.0000	0.978523
$707$	0	0
$708$	12.0000	0.450988
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	12.0000	0.449719
$713$	10.0000	0.374503
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−20.0000	−0.746914
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	−2.00000	−0.0745356
$721$	4.00000	0.148968
$722$	−17.0000	−0.632674
$723$	−24.0000	−0.892570
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	−11.0000	−0.408248
$727$	36.0000	1.33517	0.667583	−	0.744535i	$-0.267329\pi$
$727$	36.0000	1.33517	0.667583	+	0.744535i	$0.267329\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	0	0
$732$	14.0000	0.517455
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	20.0000	0.738213
$735$	2.00000	0.0737711
$736$	1.00000	0.0368605
$737$	0	0
$738$	2.00000	0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	12.0000	0.441129
$741$	−24.0000	−0.881662
$742$	−10.0000	−0.367112
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−10.0000	−0.366618
$745$	−36.0000	−1.31894
$746$	14.0000	0.512576
$747$	2.00000	0.0731762
$748$	0	0
$749$	−8.00000	−0.292314
$750$	12.0000	0.438178
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	10.0000	0.364662
$753$	−18.0000	−0.655956
$754$	24.0000	0.874028
$755$	−24.0000	−0.873449
$756$	1.00000	0.0363696
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	12.0000	0.435286
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−12.0000	−0.434714
$763$	10.0000	0.362024
$764$	0	0
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−48.0000	−1.73318
$768$	−1.00000	−0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−10.0000	−0.359908
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−12.0000	−0.431331
$775$	10.0000	0.359211
$776$	−12.0000	−0.430775
$777$	−6.00000	−0.215249
$778$	18.0000	0.645331
$779$	−12.0000	−0.429945
$780$	8.00000	0.286446
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	1.00000	0.0357143
$785$	4.00000	0.142766
$786$	4.00000	0.142675
$787$	2.00000	0.0712923	0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000	0.0712923	0.0356462	+	0.999364i	$0.488651\pi$
$788$	6.00000	0.213741
$789$	−16.0000	−0.569615
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	−56.0000	−1.98862
$794$	12.0000	0.425864
$795$	−20.0000	−0.709327
$796$	20.0000	0.708881
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−6.00000	−0.212398
$799$	0	0
$800$	1.00000	0.0353553
$801$	−12.0000	−0.423999
$802$	−22.0000	−0.776847
$803$	0	0
$804$	12.0000	0.423207
$805$	−2.00000	−0.0704907
$806$	40.0000	1.40894
$807$	24.0000	0.844840
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	2.00000	0.0702728
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	6.00000	0.210559
$813$	2.00000	0.0701431
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	72.0000	2.51896
$818$	22.0000	0.769212
$819$	−4.00000	−0.139771
$820$	4.00000	0.139686
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−18.0000	−0.627822
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	−1.00000	−0.0347524
$829$	−12.0000	−0.416777	−0.208389	−	0.978046i	$-0.566822\pi$
$829$	−12.0000	−0.416777	−0.208389	+	0.978046i	$0.566822\pi$
$830$	4.00000	0.138842
$831$	10.0000	0.346896
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	−28.0000	−0.968980
$836$	0	0
$837$	10.0000	0.345651
$838$	14.0000	0.483622
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	2.00000	0.0688837
$844$	−28.0000	−0.963800
$845$	−6.00000	−0.206406
$846$	−10.0000	−0.343807
$847$	11.0000	0.377964
$848$	−10.0000	−0.343401
$849$	18.0000	0.617758
$850$	0	0
$851$	6.00000	0.205677
$852$	12.0000	0.411113
$853$	−12.0000	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$853$	−12.0000	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$854$	−14.0000	−0.479070
$855$	−12.0000	−0.410391
$856$	−8.00000	−0.273434
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	56.0000	1.91070	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	56.0000	1.91070	0.955348	+	0.295484i	$0.0954809\pi$
$860$	−24.0000	−0.818393
$861$	−2.00000	−0.0681598
$862$	24.0000	0.817443
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	1.00000	0.0340207
$865$	48.0000	1.63205
$866$	−4.00000	−0.135926
$867$	17.0000	0.577350
$868$	10.0000	0.339422
$869$	0	0
$870$	12.0000	0.406838
$871$	−48.0000	−1.62642
$872$	10.0000	0.338643
$873$	12.0000	0.406138
$874$	6.00000	0.202953
$875$	−12.0000	−0.405674
$876$	−14.0000	−0.473016
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	22.0000	0.742464
$879$	18.0000	0.607125
$880$	0	0
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	−1.00000	−0.0336718
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	−36.0000	−1.20944
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	−6.00000	−0.201347
$889$	12.0000	0.402467
$890$	−24.0000	−0.804482
$891$	0	0
$892$	18.0000	0.602685
$893$	60.0000	2.00782
$894$	18.0000	0.602010
$895$	8.00000	0.267411
$896$	1.00000	0.0334077
$897$	4.00000	0.133556
$898$	−14.0000	−0.467186
$899$	60.0000	2.00111
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	0	0
$903$	12.0000	0.399335
$904$	14.0000	0.465633
$905$	−4.00000	−0.132964
$906$	12.0000	0.398673
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	−8.00000	−0.265197
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	−6.00000	−0.198680
$913$	0	0
$914$	−26.0000	−0.860004
$915$	−28.0000	−0.925651
$916$	−6.00000	−0.198246
$917$	−4.00000	−0.132092
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−2.00000	−0.0659380
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	6.00000	0.197279
$926$	16.0000	0.525793
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	20.0000	0.655826
$931$	6.00000	0.196642
$932$	22.0000	0.720634
$933$	−10.0000	−0.327385
$934$	−6.00000	−0.196326
$935$	0	0
$936$	−4.00000	−0.130744
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	−12.0000	−0.391814
$939$	8.00000	0.261070
$940$	−20.0000	−0.652328
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−2.00000	−0.0651635
$943$	2.00000	0.0651290
$944$	−12.0000	−0.390567
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	56.0000	1.81784
$950$	6.00000	0.194666
$951$	2.00000	0.0648544
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	−36.0000	−1.16311
$959$	18.0000	0.581250
$960$	2.00000	0.0645497
$961$	69.0000	2.22581
$962$	24.0000	0.773791
$963$	8.00000	0.257796
$964$	24.0000	0.772988
$965$	20.0000	0.643823
$966$	1.00000	0.0321745
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	24.0000	0.770594
$971$	−22.0000	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	16.0000	0.512673
$975$	4.00000	0.128103
$976$	−14.0000	−0.448129
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	−10.0000	−0.319275
$982$	20.0000	0.638226
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−2.00000	−0.0637577
$985$	−12.0000	−0.382352
$986$	0	0
$987$	10.0000	0.318304
$988$	24.0000	0.763542
$989$	−12.0000	−0.381578
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	10.0000	0.317500
$993$	20.0000	0.634681
$994$	−12.0000	−0.380617
$995$	−40.0000	−1.26809
$996$	−2.00000	−0.0633724
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	44.0000	1.39280
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.a.b.1.1	✓	1
3.2	odd	2		2898.2.a.r.1.1		1
4.3	odd	2		7728.2.a.p.1.1		1
7.6	odd	2		6762.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.b.1.1	✓	1	1.1	even	1	trivial
2898.2.a.r.1.1		1	3.2	odd	2
6762.2.a.r.1.1		1	7.6	odd	2
7728.2.a.p.1.1		1	4.3	odd	2

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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