LMFDB - Embedded newform 3234.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} -4.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{93} -4.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	2.00000	0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	−2.00000	−0.316228
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	2.00000	0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	2.00000	0.214423
$88$	1.00000	0.106600
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	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	$0.334245\pi$
$102$	−2.00000	−0.198030
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−2.00000	−0.190693
$111$	2.00000	0.189832
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	−10.0000	−0.901670
$124$	−4.00000	−0.359211
$125$	12.0000	1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	12.0000	1.03664
$135$	2.00000	0.172133
$136$	2.00000	0.171499
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	8.00000	0.671345
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−6.00000	−0.496564
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−8.00000	−0.636446
$159$	2.00000	0.158610
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	10.0000	0.780869
$165$	2.00000	0.155700
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	−12.0000	−0.901975
$178$	14.0000	1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−2.00000	−0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	4.00000	0.294086
$186$	4.00000	0.293294
$187$	2.00000	0.146254
$188$	−4.00000	−0.291730
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	14.0000	1.00514
$195$	4.00000	0.286446
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	1.00000	0.0710669
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−1.00000	−0.0707107
$201$	−12.0000	−0.846415
$202$	10.0000	0.703598
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−20.0000	−1.39686
$206$	−4.00000	−0.278693
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−2.00000	−0.137361
$213$	−8.00000	−0.548151
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	−2.00000	−0.134840
$221$	4.00000	0.269069
$222$	2.00000	0.134231
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	8.00000	0.521862
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	2.00000	0.129099
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−4.00000	−0.254000
$249$	−8.00000	−0.506979
$250$	12.0000	0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−4.00000	−0.248069
$261$	−2.00000	−0.123797
$262$	0	0
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	−1.00000	−0.0615457
$265$	4.00000	0.245718
$266$	0	0
$267$	−14.0000	−0.856786
$268$	12.0000	0.733017
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	2.00000	0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	10.0000	0.604122
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−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
$279$	−4.00000	−0.239474
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	4.00000	0.238197
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	−14.0000	−0.820695
$292$	−6.00000	−0.351123
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−2.00000	−0.116248
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	8.00000	0.460348
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−4.00000	−0.229039
$306$	2.00000	0.114332
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	8.00000	0.454369
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−2.00000	−0.113228
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	2.00000	0.112154
$319$	−2.00000	−0.111979
$320$	−2.00000	−0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	−14.0000	−0.774202
$328$	10.0000	0.552158
$329$	0	0
$330$	2.00000	0.110096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	8.00000	0.439057
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	−18.0000	−0.977626
$340$	−4.00000	−0.216930
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	1.00000	0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−12.0000	−0.637793
$355$	−16.0000	−0.849192
$356$	14.0000	0.741999
$357$	0	0
$358$	12.0000	0.634220
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	−2.00000	−0.105409
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	12.0000	0.628109
$366$	−2.00000	−0.104542
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	4.00000	0.207950
$371$	0	0
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	2.00000	0.103418
$375$	−12.0000	−0.619677
$376$	−4.00000	−0.206284
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	−8.00000	−0.409316
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	4.00000	0.203331
$388$	14.0000	0.710742
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	0	0
$393$	0	0
$394$	6.00000	0.302276
$395$	16.0000	0.805047
$396$	1.00000	0.0502519
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−12.0000	−0.598506
$403$	−8.00000	−0.398508
$404$	10.0000	0.497519
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	−2.00000	−0.0990148
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	−20.0000	−0.987730
$411$	−10.0000	−0.493264
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	−16.0000	−0.785409
$416$	2.00000	0.0980581
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	12.0000	0.584151
$423$	−4.00000	−0.194487
$424$	−2.00000	−0.0971286
$425$	−2.00000	−0.0970143
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−2.00000	−0.0965609
$430$	−8.00000	−0.385794
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	−1.00000	−0.0481125
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	0	0
$438$	6.00000	0.286691
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	4.00000	0.190261
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	2.00000	0.0949158
$445$	−28.0000	−1.32733
$446$	−20.0000	−0.947027
$447$	10.0000	0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.00000	−0.0471405
$451$	10.0000	0.470882
$452$	18.0000	0.846649
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	14.0000	0.654177
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	−8.00000	−0.370991
$466$	−6.00000	−0.277945
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	8.00000	0.369012
$471$	18.0000	0.829396
$472$	12.0000	0.552345
$473$	4.00000	0.183920
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	16.0000	0.731823
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	2.00000	0.0912871
$481$	−4.00000	−0.182384
$482$	−30.0000	−1.36646
$483$	0	0
$484$	1.00000	0.0454545
$485$	−28.0000	−1.27141
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	2.00000	0.0905357
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−10.0000	−0.450835
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−8.00000	−0.358489
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	8.00000	0.352522
$516$	−4.00000	−0.176090
$517$	−4.00000	−0.175920
$518$	0	0
$519$	6.00000	0.263371
$520$	−4.00000	−0.175412
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−2.00000	−0.0875376
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	32.0000	1.39527
$527$	−8.00000	−0.348485
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	4.00000	0.173749
$531$	12.0000	0.520756
$532$	0	0
$533$	20.0000	0.866296
$534$	−14.0000	−0.605839
$535$	24.0000	1.03761
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	−10.0000	−0.431131
$539$	0	0
$540$	2.00000	0.0860663
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	8.00000	0.343629
$543$	2.00000	0.0858282
$544$	2.00000	0.0857493
$545$	−28.0000	−1.19939
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	10.0000	0.427179
$549$	2.00000	0.0853579
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	−4.00000	−0.169791
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	−4.00000	−0.169334
$559$	8.00000	0.338364
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	10.0000	0.421825
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	4.00000	0.168430
$565$	−36.0000	−1.51453
$566$	−16.0000	−0.672530
$567$	0	0
$568$	8.00000	0.335673
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	2.00000	0.0836242
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	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	$0.566745\pi$
$578$	−13.0000	−0.540729
$579$	−2.00000	−0.0831172
$580$	4.00000	0.166091
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−2.00000	−0.0828315
$584$	−6.00000	−0.248282
$585$	−4.00000	−0.165380
$586$	−14.0000	−0.578335
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	0	0
$590$	−24.0000	−0.988064
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	20.0000	0.818546
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	8.00000	0.325515
$605$	−2.00000	−0.0813116
$606$	−10.0000	−0.406222
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	−4.00000	−0.161955
$611$	−8.00000	−0.323645
$612$	2.00000	0.0808452
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−8.00000	−0.322854
$615$	20.0000	0.806478
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	4.00000	0.160904
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−20.0000	−0.801927
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	30.0000	1.19904
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−4.00000	−0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−8.00000	−0.318223
$633$	−12.0000	−0.476957
$634$	22.0000	0.873732
$635$	−16.0000	−0.634941
$636$	2.00000	0.0793052
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	8.00000	0.316475
$640$	−2.00000	−0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	1.00000	0.0392837
$649$	12.0000	0.471041
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	10.0000	0.390434
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	2.00000	0.0778499
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	20.0000	0.777322
$663$	−4.00000	−0.155347
$664$	8.00000	0.310460
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	20.0000	0.773245
$670$	−24.0000	−0.927201
$671$	2.00000	0.0772091
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−22.0000	−0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	−18.0000	−0.691286
$679$	0	0
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	−20.0000	−0.764161
$686$	0	0
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	2.00000	0.0758098
$697$	20.0000	0.757554
$698$	26.0000	0.984115
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	1.00000	0.0376889
$705$	−8.00000	−0.301297
$706$	6.00000	0.225813
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	−16.0000	−0.600469
$711$	−8.00000	−0.300023
$712$	14.0000	0.524672
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	12.0000	0.448461
$717$	−16.0000	−0.597531
$718$	−32.0000	−1.19423
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−19.0000	−0.707107
$723$	30.0000	1.11571
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	−1.00000	−0.0371135
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	8.00000	0.295891
$732$	−2.00000	−0.0739221
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	0	0
$737$	12.0000	0.442026
$738$	10.0000	0.368105
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	4.00000	0.146647
$745$	20.0000	0.732743
$746$	22.0000	0.805477
$747$	8.00000	0.292705
$748$	2.00000	0.0731272
$749$	0	0
$750$	−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	$0.546628\pi$
$752$	−4.00000	−0.145865
$753$	4.00000	0.145768
$754$	−4.00000	−0.145671
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−4.00000	−0.144620
$766$	−12.0000	−0.433578
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	2.00000	0.0719816
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	4.00000	0.143777
$775$	4.00000	0.143684
$776$	14.0000	0.502571
$777$	0	0
$778$	22.0000	0.788738
$779$	0	0
$780$	4.00000	0.143223
$781$	8.00000	0.286263
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	36.0000	1.28490
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	6.00000	0.213741
$789$	−32.0000	−1.13923
$790$	16.0000	0.569254
$791$	0	0
$792$	1.00000	0.0355335
$793$	4.00000	0.142044
$794$	30.0000	1.06466
$795$	−4.00000	−0.141865
$796$	−20.0000	−0.708881
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−1.00000	−0.0353553
$801$	14.0000	0.494666
$802$	18.0000	0.635602
$803$	−6.00000	−0.211735
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−8.00000	−0.281788
$807$	10.0000	0.352017
$808$	10.0000	0.351799
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	−2.00000	−0.0702728
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−2.00000	−0.0701000
$815$	−8.00000	−0.280228
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−14.0000	−0.489499
$819$	0	0
$820$	−20.0000	−0.698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−10.0000	−0.348790
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	−4.00000	−0.139347
$825$	1.00000	0.0348155
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	−16.0000	−0.555368
$831$	10.0000	0.346896
$832$	2.00000	0.0693375
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	12.0000	0.414533
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	−10.0000	−0.344418
$844$	12.0000	0.413057
$845$	18.0000	0.619219
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	16.0000	0.549119
$850$	−2.00000	−0.0685994
$851$	0	0
$852$	−8.00000	−0.274075
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	−2.00000	−0.0682789
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−40.0000	−1.36241
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−1.00000	−0.0340207
$865$	12.0000	0.408012
$866$	−18.0000	−0.611665
$867$	13.0000	0.441503
$868$	0	0
$869$	−8.00000	−0.271381
$870$	−4.00000	−0.135613
$871$	24.0000	0.813209
$872$	14.0000	0.474100
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	6.00000	0.202721
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−16.0000	−0.539974
$879$	14.0000	0.472208
$880$	−2.00000	−0.0674200
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	4.00000	0.134535
$885$	24.0000	0.806751
$886$	−12.0000	−0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−28.0000	−0.938562
$891$	1.00000	0.0335013
$892$	−20.0000	−0.669650
$893$	0	0
$894$	10.0000	0.334450
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	10.0000	0.332964
$903$	0	0
$904$	18.0000	0.598671
$905$	4.00000	0.132964
$906$	−8.00000	−0.265782
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	−38.0000	−1.25693
$915$	4.00000	0.132236
$916$	14.0000	0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	18.0000	0.592798
$923$	16.0000	0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	16.0000	0.525793
$927$	−4.00000	−0.131377
$928$	−2.00000	−0.0656532
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	−6.00000	−0.196537
$933$	20.0000	0.654771
$934$	−4.00000	−0.130884
$935$	−4.00000	−0.130814
$936$	2.00000	0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−30.0000	−0.979013
$940$	8.00000	0.260931
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	8.00000	0.259828
$949$	−12.0000	−0.389536
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−2.00000	−0.0647524
$955$	16.0000	0.517748
$956$	16.0000	0.517477
$957$	2.00000	0.0646508
$958$	32.0000	1.03387
$959$	0	0
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	−12.0000	−0.386695
$964$	−30.0000	−0.966235
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−28.0000	−0.899026
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	2.00000	0.0640513
$976$	2.00000	0.0640184
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−4.00000	−0.127906
$979$	14.0000	0.447442
$980$	0	0
$981$	14.0000	0.446986
$982$	−36.0000	−1.14881
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	−10.0000	−0.318788
$985$	−12.0000	−0.382352
$986$	−4.00000	−0.127386
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−2.00000	−0.0635642
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−4.00000	−0.127000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	40.0000	1.26809
$996$	−8.00000	−0.253490
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	28.0000	0.886325
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.p.1.1		1
3.2	odd	2		9702.2.a.r.1.1		1
7.6	odd	2		462.2.a.g.1.1	✓	1
21.20	even	2		1386.2.a.a.1.1		1
28.27	even	2		3696.2.a.m.1.1		1
77.76	even	2		5082.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.g.1.1	✓	1	7.6	odd	2
1386.2.a.a.1.1		1	21.20	even	2
3234.2.a.p.1.1		1	1.1	even	1	trivial
3696.2.a.m.1.1		1	28.27	even	2
5082.2.a.n.1.1		1	77.76	even	2
9702.2.a.r.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more