LMFDB - Embedded newform 6762.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -4.00000 q^{20} -5.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} +4.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +5.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} -4.00000 q^{40} +6.00000 q^{41} -6.00000 q^{43} -5.00000 q^{44} -4.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +11.0000 q^{50} +3.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +20.0000 q^{55} +6.00000 q^{57} +1.00000 q^{58} -8.00000 q^{59} +4.00000 q^{60} -2.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +5.00000 q^{66} -2.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} -7.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} -6.00000 q^{74} -11.0000 q^{75} -6.00000 q^{76} +2.00000 q^{78} -1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +16.0000 q^{83} +12.0000 q^{85} -6.00000 q^{86} -1.00000 q^{87} -5.00000 q^{88} +18.0000 q^{89} -4.00000 q^{90} +1.00000 q^{92} +10.0000 q^{93} +9.00000 q^{94} +24.0000 q^{95} -1.00000 q^{96} -4.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−5.00000	−1.06600
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	4.00000	0.730297
$31$	−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
$33$	5.00000	0.870388
$34$	−3.00000	−0.514496
$35$	0	0
$36$	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
$39$	2.00000	0.320256
$40$	−4.00000	−0.632456
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−5.00000	−0.753778
$45$	−4.00000	−0.596285
$46$	1.00000	0.147442
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	11.0000	1.55563
$51$	3.00000	0.420084
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	20.0000	2.69680
$56$	0	0
$57$	6.00000	0.794719
$58$	1.00000	0.131306
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	4.00000	0.516398
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	5.00000	0.615457
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−3.00000	−0.363803
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−6.00000	−0.697486
$75$	−11.0000	−1.27017
$76$	−6.00000	−0.688247
$77$	0	0
$78$	2.00000	0.226455
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	−6.00000	−0.646997
$87$	−1.00000	−0.107211
$88$	−5.00000	−0.533002
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	1.00000	0.104257
$93$	10.0000	1.03695
$94$	9.00000	0.928279
$95$	24.0000	2.46235
$96$	−1.00000	−0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	11.0000	1.10000
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	3.00000	0.297044
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−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
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	20.0000	1.90693
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	6.00000	0.561951
$115$	−4.00000	−0.373002
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	−8.00000	−0.736460
$119$	0	0
$120$	4.00000	0.365148
$121$	14.0000	1.27273
$122$	−2.00000	−0.181071
$123$	−6.00000	−0.541002
$124$	−10.0000	−0.898027
$125$	−24.0000	−2.14663
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	1.00000	0.0883883
$129$	6.00000	0.528271
$130$	8.00000	0.701646
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	5.00000	0.435194
$133$	0	0
$134$	−2.00000	−0.172774
$135$	4.00000	0.344265
$136$	−3.00000	−0.257248
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	−1.00000	−0.0851257
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	−7.00000	−0.587427
$143$	10.0000	0.836242
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−11.0000	−0.910366
$147$	0	0
$148$	−6.00000	−0.493197
$149$	24.0000	1.96616	0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000	1.96616	0.983078	+	0.183186i	$0.0586410\pi$
$150$	−11.0000	−0.898146
$151$	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
$153$	−3.00000	−0.242536
$154$	0	0
$155$	40.0000	3.21288
$156$	2.00000	0.160128
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	−1.00000	−0.0795557
$159$	6.00000	0.475831
$160$	−4.00000	−0.316228
$161$	0	0
$162$	1.00000	0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	6.00000	0.468521
$165$	−20.0000	−1.55700
$166$	16.0000	1.24184
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	12.0000	0.920358
$171$	−6.00000	−0.458831
$172$	−6.00000	−0.457496
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	−5.00000	−0.376889
$177$	8.00000	0.601317
$178$	18.0000	1.34916
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−4.00000	−0.298142
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	1.00000	0.0737210
$185$	24.0000	1.76452
$186$	10.0000	0.733236
$187$	15.0000	1.09691
$188$	9.00000	0.656392
$189$	0	0
$190$	24.0000	1.74114
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−1.00000	−0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−4.00000	−0.287183
$195$	−8.00000	−0.572892
$196$	0	0
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	−5.00000	−0.355335
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	11.0000	0.777817
$201$	2.00000	0.141069
$202$	17.0000	1.19612
$203$	0	0
$204$	3.00000	0.210042
$205$	−24.0000	−1.67623
$206$	−9.00000	−0.627060
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	30.0000	2.07514
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	−6.00000	−0.412082
$213$	7.00000	0.479632
$214$	−12.0000	−0.820303
$215$	24.0000	1.63679
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−5.00000	−0.338643
$219$	11.0000	0.743311
$220$	20.0000	1.34840
$221$	6.00000	0.403604
$222$	6.00000	0.402694
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	6.00000	0.399114
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	6.00000	0.397360
$229$	−3.00000	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$229$	−3.00000	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	1.00000	0.0656532
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	−2.00000	−0.130744
$235$	−36.0000	−2.34838
$236$	−8.00000	−0.520756
$237$	1.00000	0.0649570
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	4.00000	0.258199
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	12.0000	0.763542
$248$	−10.0000	−0.635001
$249$	−16.0000	−1.01396
$250$	−24.0000	−1.51789
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	10.0000	0.627456
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	8.00000	0.496139
$261$	1.00000	0.0618984
$262$	−16.0000	−0.988483
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	5.00000	0.307729
$265$	24.0000	1.47431
$266$	0	0
$267$	−18.0000	−1.10158
$268$	−2.00000	−0.122169
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	4.00000	0.243432
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	13.0000	0.785359
$275$	−55.0000	−3.31662
$276$	−1.00000	−0.0601929
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	5.00000	0.299880
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	−9.00000	−0.535942
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−7.00000	−0.415374
$285$	−24.0000	−1.42164
$286$	10.0000	0.591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−4.00000	−0.234888
$291$	4.00000	0.234484
$292$	−11.0000	−0.643726
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	−6.00000	−0.348743
$297$	5.00000	0.290129
$298$	24.0000	1.39028
$299$	−2.00000	−0.115663
$300$	−11.0000	−0.635085
$301$	0	0
$302$	12.0000	0.690522
$303$	−17.0000	−0.976624
$304$	−6.00000	−0.344124
$305$	8.00000	0.458079
$306$	−3.00000	−0.171499
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	0	0
$309$	9.00000	0.511992
$310$	40.0000	2.27185
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	2.00000	0.113228
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	−3.00000	−0.169300
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	6.00000	0.336463
$319$	−5.00000	−0.279946
$320$	−4.00000	−0.223607
$321$	12.0000	0.669775
$322$	0	0
$323$	18.0000	1.00155
$324$	1.00000	0.0555556
$325$	−22.0000	−1.22034
$326$	−1.00000	−0.0553849
$327$	5.00000	0.276501
$328$	6.00000	0.331295
$329$	0	0
$330$	−20.0000	−1.10096
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	16.0000	0.878114
$333$	−6.00000	−0.328798
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	12.0000	0.650791
$341$	50.0000	2.70765
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−6.00000	−0.323498
$345$	4.00000	0.215353
$346$	−3.00000	−0.161281
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−1.00000	−0.0536056
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−5.00000	−0.266501
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	8.00000	0.425195
$355$	28.0000	1.48609
$356$	18.0000	0.953998
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−4.00000	−0.210819
$361$	17.0000	0.894737
$362$	−7.00000	−0.367912
$363$	−14.0000	−0.734809
$364$	0	0
$365$	44.0000	2.30307
$366$	2.00000	0.104542
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	1.00000	0.0521286
$369$	6.00000	0.312348
$370$	24.0000	1.24770
$371$	0	0
$372$	10.0000	0.518476
$373$	−17.0000	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$374$	15.0000	0.775632
$375$	24.0000	1.23935
$376$	9.00000	0.464140
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	24.0000	1.23117
$381$	−10.0000	−0.512316
$382$	−4.00000	−0.204658
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	−6.00000	−0.304997
$388$	−4.00000	−0.203069
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	−8.00000	−0.405096
$391$	−3.00000	−0.151717
$392$	0	0
$393$	16.0000	0.807093
$394$	21.0000	1.05796
$395$	4.00000	0.201262
$396$	−5.00000	−0.251259
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−21.0000	−1.05263
$399$	0	0
$400$	11.0000	0.550000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	2.00000	0.0997509
$403$	20.0000	0.996271
$404$	17.0000	0.845782
$405$	−4.00000	−0.198762
$406$	0	0
$407$	30.0000	1.48704
$408$	3.00000	0.148522
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	−24.0000	−1.18528
$411$	−13.0000	−0.641243
$412$	−9.00000	−0.443398
$413$	0	0
$414$	1.00000	0.0491473
$415$	−64.0000	−3.14164
$416$	−2.00000	−0.0980581
$417$	−5.00000	−0.244851
$418$	30.0000	1.46735
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	−11.0000	−0.535472
$423$	9.00000	0.437595
$424$	−6.00000	−0.291386
$425$	−33.0000	−1.60074
$426$	7.00000	0.339151
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−10.0000	−0.482805
$430$	24.0000	1.15738
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	−1.00000	−0.0481125
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	−5.00000	−0.239457
$437$	−6.00000	−0.287019
$438$	11.0000	0.525600
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	20.0000	0.953463
$441$	0	0
$442$	6.00000	0.285391
$443$	−40.0000	−1.90046	−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000	−1.90046	−0.950229	+	0.311553i	$0.899151\pi$
$444$	6.00000	0.284747
$445$	−72.0000	−3.41313
$446$	22.0000	1.04173
$447$	−24.0000	−1.13516
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	11.0000	0.518545
$451$	−30.0000	−1.41264
$452$	6.00000	0.282216
$453$	−12.0000	−0.563809
$454$	−3.00000	−0.140797
$455$	0	0
$456$	6.00000	0.280976
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	−3.00000	−0.140181
$459$	3.00000	0.140028
$460$	−4.00000	−0.186501
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	1.00000	0.0464238
$465$	−40.0000	−1.85496
$466$	16.0000	0.741186
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−36.0000	−1.66056
$471$	3.00000	0.138233
$472$	−8.00000	−0.368230
$473$	30.0000	1.37940
$474$	1.00000	0.0459315
$475$	−66.0000	−3.02829
$476$	0	0
$477$	−6.00000	−0.274721
$478$	5.00000	0.228695
$479$	26.0000	1.18797	0.593985	−	0.804476i	$-0.297554\pi$
$479$	26.0000	1.18797	0.593985	+	0.804476i	$0.297554\pi$
$480$	4.00000	0.182574
$481$	12.0000	0.547153
$482$	10.0000	0.455488
$483$	0	0
$484$	14.0000	0.636364
$485$	16.0000	0.726523
$486$	−1.00000	−0.0453609
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	−2.00000	−0.0905357
$489$	1.00000	0.0452216
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	−3.00000	−0.135113
$494$	12.0000	0.539906
$495$	20.0000	0.898933
$496$	−10.0000	−0.449013
$497$	0	0
$498$	−16.0000	−0.716977
$499$	27.0000	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$499$	27.0000	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$500$	−24.0000	−1.07331
$501$	8.00000	0.357414
$502$	27.0000	1.20507
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	−68.0000	−3.02596
$506$	−5.00000	−0.222277
$507$	9.00000	0.399704
$508$	10.0000	0.443678
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	−30.0000	−1.32324
$515$	36.0000	1.58635
$516$	6.00000	0.264135
$517$	−45.0000	−1.97910
$518$	0	0
$519$	3.00000	0.131685
$520$	8.00000	0.350823
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	1.00000	0.0437688
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	−24.0000	−1.04645
$527$	30.0000	1.30682
$528$	5.00000	0.217597
$529$	1.00000	0.0434783
$530$	24.0000	1.04249
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−18.0000	−0.778936
$535$	48.0000	2.07522
$536$	−2.00000	−0.0863868
$537$	2.00000	0.0863064
$538$	−21.0000	−0.905374
$539$	0	0
$540$	4.00000	0.172133
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	10.0000	0.429537
$543$	7.00000	0.300399
$544$	−3.00000	−0.128624
$545$	20.0000	0.856706
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	13.0000	0.555332
$549$	−2.00000	−0.0853579
$550$	−55.0000	−2.34521
$551$	−6.00000	−0.255609
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−4.00000	−0.169944
$555$	−24.0000	−1.01874
$556$	5.00000	0.212047
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−10.0000	−0.423334
$559$	12.0000	0.507546
$560$	0	0
$561$	−15.0000	−0.633300
$562$	−13.0000	−0.548372
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	−9.00000	−0.378968
$565$	−24.0000	−1.00969
$566$	12.0000	0.504398
$567$	0	0
$568$	−7.00000	−0.293713
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−24.0000	−1.00525
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	10.0000	0.418121
$573$	4.00000	0.167102
$574$	0	0
$575$	11.0000	0.458732
$576$	1.00000	0.0416667
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	−8.00000	−0.332756
$579$	−6.00000	−0.249351
$580$	−4.00000	−0.166091
$581$	0	0
$582$	4.00000	0.165805
$583$	30.0000	1.24247
$584$	−11.0000	−0.455183
$585$	8.00000	0.330759
$586$	4.00000	0.165238
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437113\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437113\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	32.0000	1.31742
$591$	−21.0000	−0.863825
$592$	−6.00000	−0.246598
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	5.00000	0.205152
$595$	0	0
$596$	24.0000	0.983078
$597$	21.0000	0.859473
$598$	−2.00000	−0.0817861
$599$	−35.0000	−1.43006	−0.715031	−	0.699093i	$-0.753587\pi$
$599$	−35.0000	−1.43006	−0.715031	+	0.699093i	$0.753587\pi$
$600$	−11.0000	−0.449073
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	12.0000	0.488273
$605$	−56.0000	−2.27672
$606$	−17.0000	−0.690578
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	8.00000	0.323911
$611$	−18.0000	−0.728202
$612$	−3.00000	−0.121268
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	29.0000	1.17034
$615$	24.0000	0.967773
$616$	0	0
$617$	−9.00000	−0.362326	−0.181163	−	0.983453i	$-0.557986\pi$
$617$	−9.00000	−0.362326	−0.181163	+	0.983453i	$0.557986\pi$
$618$	9.00000	0.362033
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	40.0000	1.60644
$621$	−1.00000	−0.0401286
$622$	33.0000	1.32318
$623$	0	0
$624$	2.00000	0.0800641
$625$	41.0000	1.64000
$626$	−20.0000	−0.799361
$627$	−30.0000	−1.19808
$628$	−3.00000	−0.119713
$629$	18.0000	0.717707
$630$	0	0
$631$	−11.0000	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$632$	−1.00000	−0.0397779
$633$	11.0000	0.437211
$634$	2.00000	0.0794301
$635$	−40.0000	−1.58735
$636$	6.00000	0.237915
$637$	0	0
$638$	−5.00000	−0.197952
$639$	−7.00000	−0.276916
$640$	−4.00000	−0.158114
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	12.0000	0.473602
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	18.0000	0.708201
$647$	−27.0000	−1.06148	−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000	−1.06148	−0.530740	+	0.847535i	$0.678086\pi$
$648$	1.00000	0.0392837
$649$	40.0000	1.57014
$650$	−22.0000	−0.862911
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−7.00000	−0.273931	−0.136966	−	0.990576i	$-0.543735\pi$
$653$	−7.00000	−0.273931	−0.136966	+	0.990576i	$0.543735\pi$
$654$	5.00000	0.195515
$655$	64.0000	2.50069
$656$	6.00000	0.234261
$657$	−11.0000	−0.429151
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	−20.0000	−0.778499
$661$	15.0000	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$661$	15.0000	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$662$	−8.00000	−0.310929
$663$	−6.00000	−0.233021
$664$	16.0000	0.620920
$665$	0	0
$666$	−6.00000	−0.232495
$667$	1.00000	0.0387202
$668$	−8.00000	−0.309529
$669$	−22.0000	−0.850569
$670$	8.00000	0.309067
$671$	10.0000	0.386046
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	14.0000	0.539260
$675$	−11.0000	−0.423390
$676$	−9.00000	−0.346154
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	12.0000	0.460179
$681$	3.00000	0.114960
$682$	50.0000	1.91460
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−6.00000	−0.229416
$685$	−52.0000	−1.98682
$686$	0	0
$687$	3.00000	0.114457
$688$	−6.00000	−0.228748
$689$	12.0000	0.457164
$690$	4.00000	0.152277
$691$	−49.0000	−1.86405	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−49.0000	−1.86405	−0.932024	+	0.362397i	$0.881959\pi$
$692$	−3.00000	−0.114043
$693$	0	0
$694$	−6.00000	−0.227757
$695$	−20.0000	−0.758643
$696$	−1.00000	−0.0379049
$697$	−18.0000	−0.681799
$698$	18.0000	0.681310
$699$	−16.0000	−0.605176
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	2.00000	0.0754851
$703$	36.0000	1.35777
$704$	−5.00000	−0.188445
$705$	36.0000	1.35584
$706$	−10.0000	−0.376355
$707$	0	0
$708$	8.00000	0.300658
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	28.0000	1.05082
$711$	−1.00000	−0.0375029
$712$	18.0000	0.674579
$713$	−10.0000	−0.374503
$714$	0	0
$715$	−40.0000	−1.49592
$716$	−2.00000	−0.0747435
$717$	−5.00000	−0.186728
$718$	−4.00000	−0.149279
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	17.0000	0.632674
$723$	−10.0000	−0.371904
$724$	−7.00000	−0.260153
$725$	11.0000	0.408530
$726$	−14.0000	−0.519589
$727$	−43.0000	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$727$	−43.0000	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	44.0000	1.62851
$731$	18.0000	0.665754
$732$	2.00000	0.0739221
$733$	43.0000	1.58824	0.794121	−	0.607760i	$-0.207932\pi$
$733$	43.0000	1.58824	0.794121	+	0.607760i	$0.207932\pi$
$734$	0	0
$735$	0	0
$736$	1.00000	0.0368605
$737$	10.0000	0.368355
$738$	6.00000	0.220863
$739$	47.0000	1.72892	0.864461	−	0.502699i	$-0.167660\pi$
$739$	47.0000	1.72892	0.864461	+	0.502699i	$0.167660\pi$
$740$	24.0000	0.882258
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	10.0000	0.366618
$745$	−96.0000	−3.51717
$746$	−17.0000	−0.622414
$747$	16.0000	0.585409
$748$	15.0000	0.548454
$749$	0	0
$750$	24.0000	0.876356
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	9.00000	0.328196
$753$	−27.0000	−0.983935
$754$	−2.00000	−0.0728357
$755$	−48.0000	−1.74690
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−16.0000	−0.581146
$759$	5.00000	0.181489
$760$	24.0000	0.870572
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	−10.0000	−0.362262
$763$	0	0
$764$	−4.00000	−0.144715
$765$	12.0000	0.433861
$766$	−6.00000	−0.216789
$767$	16.0000	0.577727
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	6.00000	0.215945
$773$	32.0000	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$774$	−6.00000	−0.215666
$775$	−110.000	−3.95132
$776$	−4.00000	−0.143592
$777$	0	0
$778$	8.00000	0.286814
$779$	−36.0000	−1.28983
$780$	−8.00000	−0.286446
$781$	35.0000	1.25240
$782$	−3.00000	−0.107280
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	12.0000	0.428298
$786$	16.0000	0.570701
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	21.0000	0.748094
$789$	24.0000	0.854423
$790$	4.00000	0.142314
$791$	0	0
$792$	−5.00000	−0.177667
$793$	4.00000	0.142044
$794$	26.0000	0.922705
$795$	−24.0000	−0.851192
$796$	−21.0000	−0.744325
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	11.0000	0.388909
$801$	18.0000	0.635999
$802$	15.0000	0.529668
$803$	55.0000	1.94091
$804$	2.00000	0.0705346
$805$	0	0
$806$	20.0000	0.704470
$807$	21.0000	0.739235
$808$	17.0000	0.598058
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	−4.00000	−0.140546
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	30.0000	1.05150
$815$	4.00000	0.140114
$816$	3.00000	0.105021
$817$	36.0000	1.25948
$818$	−5.00000	−0.174821
$819$	0	0
$820$	−24.0000	−0.838116
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	−13.0000	−0.453427
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	−9.00000	−0.313530
$825$	55.0000	1.91485
$826$	0	0
$827$	−43.0000	−1.49526	−0.747628	−	0.664117i	$-0.768807\pi$
$827$	−43.0000	−1.49526	−0.747628	+	0.664117i	$0.768807\pi$
$828$	1.00000	0.0347524
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	−64.0000	−2.22147
$831$	4.00000	0.138758
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	−5.00000	−0.173136
$835$	32.0000	1.10741
$836$	30.0000	1.03757
$837$	10.0000	0.345651
$838$	35.0000	1.20905
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−35.0000	−1.20618
$843$	13.0000	0.447744
$844$	−11.0000	−0.378636
$845$	36.0000	1.23844
$846$	9.00000	0.309426
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−12.0000	−0.411839
$850$	−33.0000	−1.13189
$851$	−6.00000	−0.205677
$852$	7.00000	0.239816
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	−12.0000	−0.410152
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	−10.0000	−0.341394
$859$	−51.0000	−1.74010	−0.870049	−	0.492966i	$-0.835913\pi$
$859$	−51.0000	−1.74010	−0.870049	+	0.492966i	$0.835913\pi$
$860$	24.0000	0.818393
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	1.00000	0.0340404	0.0170202	−	0.999855i	$-0.494582\pi$
$863$	1.00000	0.0340404	0.0170202	+	0.999855i	$0.494582\pi$
$864$	−1.00000	−0.0340207
$865$	12.0000	0.408012
$866$	−22.0000	−0.747590
$867$	8.00000	0.271694
$868$	0	0
$869$	5.00000	0.169613
$870$	4.00000	0.135613
$871$	4.00000	0.135535
$872$	−5.00000	−0.169321
$873$	−4.00000	−0.135379
$874$	−6.00000	−0.202953
$875$	0	0
$876$	11.0000	0.371656
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	40.0000	1.34993
$879$	−4.00000	−0.134917
$880$	20.0000	0.674200
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	6.00000	0.201802
$885$	−32.0000	−1.07567
$886$	−40.0000	−1.34383
$887$	−49.0000	−1.64526	−0.822629	−	0.568578i	$-0.807494\pi$
$887$	−49.0000	−1.64526	−0.822629	+	0.568578i	$0.807494\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	−72.0000	−2.41345
$891$	−5.00000	−0.167506
$892$	22.0000	0.736614
$893$	−54.0000	−1.80704
$894$	−24.0000	−0.802680
$895$	8.00000	0.267411
$896$	0	0
$897$	2.00000	0.0667781
$898$	12.0000	0.400445
$899$	−10.0000	−0.333519
$900$	11.0000	0.366667
$901$	18.0000	0.599667
$902$	−30.0000	−0.998891
$903$	0	0
$904$	6.00000	0.199557
$905$	28.0000	0.930751
$906$	−12.0000	−0.398673
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−3.00000	−0.0995585
$909$	17.0000	0.563854
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	6.00000	0.198680
$913$	−80.0000	−2.64761
$914$	12.0000	0.396925
$915$	−8.00000	−0.264472
$916$	−3.00000	−0.0991228
$917$	0	0
$918$	3.00000	0.0990148
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	−4.00000	−0.131876
$921$	−29.0000	−0.955582
$922$	−14.0000	−0.461065
$923$	14.0000	0.460816
$924$	0	0
$925$	−66.0000	−2.17007
$926$	−20.0000	−0.657241
$927$	−9.00000	−0.295599
$928$	1.00000	0.0328266
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	−40.0000	−1.31165
$931$	0	0
$932$	16.0000	0.524097
$933$	−33.0000	−1.08037
$934$	−5.00000	−0.163605
$935$	−60.0000	−1.96221
$936$	−2.00000	−0.0653720
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	−36.0000	−1.17419
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	3.00000	0.0977453
$943$	6.00000	0.195387
$944$	−8.00000	−0.260378
$945$	0	0
$946$	30.0000	0.975384
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	1.00000	0.0324785
$949$	22.0000	0.714150
$950$	−66.0000	−2.14132
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	7.00000	0.226752	0.113376	−	0.993552i	$-0.463833\pi$
$953$	7.00000	0.226752	0.113376	+	0.993552i	$0.463833\pi$
$954$	−6.00000	−0.194257
$955$	16.0000	0.517748
$956$	5.00000	0.161712
$957$	5.00000	0.161627
$958$	26.0000	0.840022
$959$	0	0
$960$	4.00000	0.129099
$961$	69.0000	2.22581
$962$	12.0000	0.386896
$963$	−12.0000	−0.386695
$964$	10.0000	0.322078
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	14.0000	0.449977
$969$	−18.0000	−0.578243
$970$	16.0000	0.513729
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	14.0000	0.448589
$975$	22.0000	0.704564
$976$	−2.00000	−0.0640184
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	1.00000	0.0319765
$979$	−90.0000	−2.87641
$980$	0	0
$981$	−5.00000	−0.159638
$982$	−12.0000	−0.382935
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	−6.00000	−0.191273
$985$	−84.0000	−2.67646
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	12.0000	0.381771
$989$	−6.00000	−0.190789
$990$	20.0000	0.635642
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−10.0000	−0.317500
$993$	8.00000	0.253872
$994$	0	0
$995$	84.0000	2.66298
$996$	−16.0000	−0.506979
$997$	48.0000	1.52018	0.760088	−	0.649821i	$-0.225156\pi$
$997$	48.0000	1.52018	0.760088	+	0.649821i	$0.225156\pi$
$998$	27.0000	0.854670
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.w.1.1		1
7.3	odd	6		966.2.i.a.415.1	yes	2
7.5	odd	6		966.2.i.a.277.1	✓	2
7.6	odd	2		6762.2.a.bn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.i.a.277.1	✓	2	7.5	odd	6
966.2.i.a.415.1	yes	2	7.3	odd	6
6762.2.a.w.1.1		1	1.1	even	1	trivial
6762.2.a.bn.1.1		1	7.6	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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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