LMFDB - Embedded newform 6762.2.a.be.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:	yes
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +5.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +3.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -6.00000 q^{38} -5.00000 q^{39} +3.00000 q^{40} -1.00000 q^{41} +1.00000 q^{43} +2.00000 q^{44} +3.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -4.00000 q^{51} +5.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +6.00000 q^{57} +1.00000 q^{58} -8.00000 q^{59} -3.00000 q^{60} +12.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -2.00000 q^{66} -16.0000 q^{67} +4.00000 q^{68} -1.00000 q^{69} +1.00000 q^{72} +10.0000 q^{73} +1.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} -5.00000 q^{78} +6.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -12.0000 q^{83} +12.0000 q^{85} +1.00000 q^{86} -1.00000 q^{87} +2.00000 q^{88} +4.00000 q^{89} +3.00000 q^{90} +1.00000 q^{92} -4.00000 q^{93} +9.00000 q^{94} -18.0000 q^{95} -1.00000 q^{96} +3.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.00000	−0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	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	$0.741620\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	2.00000	0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	−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	$0.470403\pi$
$30$	−3.00000	−0.547723
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−2.00000	−0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−6.00000	−0.973329
$39$	−5.00000	−0.800641
$40$	3.00000	0.474342
$41$	−1.00000	−0.156174	−0.0780869	−	0.996947i	$-0.524881\pi$
$41$	−1.00000	−0.156174	−0.0780869	+	0.996947i	$0.524881\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	2.00000	0.301511
$45$	3.00000	0.447214
$46$	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
$50$	4.00000	0.565685
$51$	−4.00000	−0.560112
$52$	5.00000	0.693375
$53$	−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
$55$	6.00000	0.809040
$56$	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	$0.674350\pi$
$60$	−3.00000	−0.387298
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	15.0000	1.86052
$66$	−2.00000	−0.246183
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	1.00000	0.116248
$75$	−4.00000	−0.461880
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−5.00000	−0.566139
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−1.00000	−0.110432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	1.00000	0.107833
$87$	−1.00000	−0.107211
$88$	2.00000	0.213201
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	1.00000	0.104257
$93$	−4.00000	−0.414781
$94$	9.00000	0.928279
$95$	−18.0000	−1.84676
$96$	−1.00000	−0.102062
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	4.00000	0.400000
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	−4.00000	−0.396059
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	5.00000	0.490290
$105$	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
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	6.00000	0.572078
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	6.00000	0.561951
$115$	3.00000	0.279751
$116$	1.00000	0.0928477
$117$	5.00000	0.462250
$118$	−8.00000	−0.736460
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	12.0000	1.08643
$123$	1.00000	0.0901670
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	15.0000	1.31559
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	−16.0000	−1.38219
$135$	−3.00000	−0.258199
$136$	4.00000	0.342997
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−1.00000	−0.0851257
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	10.0000	0.836242
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	10.0000	0.827606
$147$	0	0
$148$	1.00000	0.0821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−4.00000	−0.326599
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−6.00000	−0.486664
$153$	4.00000	0.323381
$154$	0	0
$155$	12.0000	0.963863
$156$	−5.00000	−0.400320
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	6.00000	0.477334
$159$	6.00000	0.475831
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−1.00000	−0.0780869
$165$	−6.00000	−0.467099
$166$	−12.0000	−0.931381
$167$	−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
$169$	12.0000	0.923077
$170$	12.0000	0.920358
$171$	−6.00000	−0.458831
$172$	1.00000	0.0762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	2.00000	0.150756
$177$	8.00000	0.601317
$178$	4.00000	0.299813
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	3.00000	0.223607
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	1.00000	0.0737210
$185$	3.00000	0.220564
$186$	−4.00000	−0.293294
$187$	8.00000	0.585018
$188$	9.00000	0.656392
$189$	0	0
$190$	−18.0000	−1.30586
$191$	−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
$193$	27.0000	1.94350	0.971751	−	0.236007i	$-0.0758390\pi$
$193$	27.0000	1.94350	0.971751	+	0.236007i	$0.0758390\pi$
$194$	3.00000	0.215387
$195$	−15.0000	−1.07417
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	2.00000	0.142134
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	4.00000	0.282843
$201$	16.0000	1.12855
$202$	−4.00000	−0.281439
$203$	0	0
$204$	−4.00000	−0.280056
$205$	−3.00000	−0.209529
$206$	5.00000	0.348367
$207$	1.00000	0.0695048
$208$	5.00000	0.346688
$209$	−12.0000	−0.830057
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	3.00000	0.204598
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−19.0000	−1.28684
$219$	−10.0000	−0.675737
$220$	6.00000	0.404520
$221$	20.0000	1.34535
$222$	−1.00000	−0.0671156
$223$	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
$225$	4.00000	0.266667
$226$	13.0000	0.864747
$227$	−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
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	3.00000	0.197814
$231$	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	$0.324403\pi$
$234$	5.00000	0.326860
$235$	27.0000	1.76129
$236$	−8.00000	−0.520756
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−3.00000	−0.193649
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	12.0000	0.768221
$245$	0	0
$246$	1.00000	0.0637577
$247$	−30.0000	−1.90885
$248$	4.00000	0.254000
$249$	12.0000	0.760469
$250$	−3.00000	−0.189737
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−11.0000	−0.690201
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−1.00000	−0.0622573
$259$	0	0
$260$	15.0000	0.930261
$261$	1.00000	0.0618984
$262$	12.0000	0.741362
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	−2.00000	−0.123091
$265$	−18.0000	−1.10573
$266$	0	0
$267$	−4.00000	−0.244796
$268$	−16.0000	−0.977356
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−3.00000	−0.182574
$271$	−18.0000	−1.09342	−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000	−1.09342	−0.546711	+	0.837321i	$0.684120\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−15.0000	−0.906183
$275$	8.00000	0.482418
$276$	−1.00000	−0.0601929
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	19.0000	1.13954
$279$	4.00000	0.239474
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	−9.00000	−0.535942
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	18.0000	1.06623
$286$	10.0000	0.591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	3.00000	0.176166
$291$	−3.00000	−0.175863
$292$	10.0000	0.585206
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	1.00000	0.0581238
$297$	−2.00000	−0.116052
$298$	10.0000	0.579284
$299$	5.00000	0.289157
$300$	−4.00000	−0.230940
$301$	0	0
$302$	5.00000	0.287718
$303$	4.00000	0.229794
$304$	−6.00000	−0.344124
$305$	36.0000	2.06135
$306$	4.00000	0.228665
$307$	1.00000	0.0570730	0.0285365	−	0.999593i	$-0.490915\pi$
$307$	1.00000	0.0570730	0.0285365	+	0.999593i	$0.490915\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	12.0000	0.681554
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−5.00000	−0.283069
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	6.00000	0.337526
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	6.00000	0.336463
$319$	2.00000	0.111979
$320$	3.00000	0.167705
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	−8.00000	−0.443079
$327$	19.0000	1.05070
$328$	−1.00000	−0.0552158
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−12.0000	−0.658586
$333$	1.00000	0.0547997
$334$	−8.00000	−0.437741
$335$	−48.0000	−2.62252
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	12.0000	0.652714
$339$	−13.0000	−0.706063
$340$	12.0000	0.650791
$341$	8.00000	0.433224
$342$	−6.00000	−0.324443
$343$	0	0
$344$	1.00000	0.0539164
$345$	−3.00000	−0.161515
$346$	18.0000	0.967686
$347$	−13.0000	−0.697877	−0.348938	−	0.937146i	$-0.613458\pi$
$347$	−13.0000	−0.697877	−0.348938	+	0.937146i	$0.613458\pi$
$348$	−1.00000	−0.0536056
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	2.00000	0.106600
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	4.00000	0.212000
$357$	0	0
$358$	−9.00000	−0.475665
$359$	17.0000	0.897226	0.448613	−	0.893726i	$-0.351918\pi$
$359$	17.0000	0.897226	0.448613	+	0.893726i	$0.351918\pi$
$360$	3.00000	0.158114
$361$	17.0000	0.894737
$362$	−14.0000	−0.735824
$363$	7.00000	0.367405
$364$	0	0
$365$	30.0000	1.57027
$366$	−12.0000	−0.627250
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	1.00000	0.0521286
$369$	−1.00000	−0.0520579
$370$	3.00000	0.155963
$371$	0	0
$372$	−4.00000	−0.207390
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	8.00000	0.413670
$375$	3.00000	0.154919
$376$	9.00000	0.464140
$377$	5.00000	0.257513
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	−18.0000	−0.923381
$381$	11.0000	0.563547
$382$	−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
$386$	27.0000	1.37426
$387$	1.00000	0.0508329
$388$	3.00000	0.152302
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−15.0000	−0.759555
$391$	4.00000	0.202289
$392$	0	0
$393$	−12.0000	−0.605320
$394$	−21.0000	−1.05796
$395$	18.0000	0.905678
$396$	2.00000	0.100504
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	4.00000	0.200000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	16.0000	0.798007
$403$	20.0000	0.996271
$404$	−4.00000	−0.199007
$405$	3.00000	0.149071
$406$	0	0
$407$	2.00000	0.0991363
$408$	−4.00000	−0.198030
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−3.00000	−0.148159
$411$	15.0000	0.739895
$412$	5.00000	0.246332
$413$	0	0
$414$	1.00000	0.0491473
$415$	−36.0000	−1.76717
$416$	5.00000	0.245145
$417$	−19.0000	−0.930434
$418$	−12.0000	−0.586939
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	21.0000	1.02348	0.511739	−	0.859141i	$-0.329002\pi$
$421$	21.0000	1.02348	0.511739	+	0.859141i	$0.329002\pi$
$422$	10.0000	0.486792
$423$	9.00000	0.437595
$424$	−6.00000	−0.291386
$425$	16.0000	0.776114
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−10.0000	−0.482805
$430$	3.00000	0.144673
$431$	−37.0000	−1.78223	−0.891114	−	0.453780i	$-0.850075\pi$
$431$	−37.0000	−1.78223	−0.891114	+	0.453780i	$0.850075\pi$
$432$	−1.00000	−0.0481125
$433$	−15.0000	−0.720854	−0.360427	−	0.932787i	$-0.617369\pi$
$433$	−15.0000	−0.720854	−0.360427	+	0.932787i	$0.617369\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	−19.0000	−0.909935
$437$	−6.00000	−0.287019
$438$	−10.0000	−0.477818
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	20.0000	0.951303
$443$	−19.0000	−0.902717	−0.451359	−	0.892343i	$-0.649060\pi$
$443$	−19.0000	−0.902717	−0.451359	+	0.892343i	$0.649060\pi$
$444$	−1.00000	−0.0474579
$445$	12.0000	0.568855
$446$	22.0000	1.04173
$447$	−10.0000	−0.472984
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	4.00000	0.188562
$451$	−2.00000	−0.0941763
$452$	13.0000	0.611469
$453$	−5.00000	−0.234920
$454$	−3.00000	−0.140797
$455$	0	0
$456$	6.00000	0.280976
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	4.00000	0.186908
$459$	−4.00000	−0.186704
$460$	3.00000	0.139876
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	1.00000	0.0464238
$465$	−12.0000	−0.556487
$466$	16.0000	0.741186
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	5.00000	0.231125
$469$	0	0
$470$	27.0000	1.24542
$471$	10.0000	0.460776
$472$	−8.00000	−0.368230
$473$	2.00000	0.0919601
$474$	−6.00000	−0.275589
$475$	−24.0000	−1.10120
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−16.0000	−0.731823
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	−3.00000	−0.136931
$481$	5.00000	0.227980
$482$	−11.0000	−0.501036
$483$	0	0
$484$	−7.00000	−0.318182
$485$	9.00000	0.408669
$486$	−1.00000	−0.0453609
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	12.0000	0.543214
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	1.00000	0.0450835
$493$	4.00000	0.180151
$494$	−30.0000	−1.34976
$495$	6.00000	0.269680
$496$	4.00000	0.179605
$497$	0	0
$498$	12.0000	0.537733
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	−3.00000	−0.134164
$501$	8.00000	0.357414
$502$	13.0000	0.580218
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	2.00000	0.0889108
$507$	−12.0000	−0.532939
$508$	−11.0000	−0.488046
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	−2.00000	−0.0882162
$515$	15.0000	0.660979
$516$	−1.00000	−0.0440225
$517$	18.0000	0.791639
$518$	0	0
$519$	−18.0000	−0.790112
$520$	15.0000	0.657794
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	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	$0.400980\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	11.0000	0.479623
$527$	16.0000	0.696971
$528$	−2.00000	−0.0870388
$529$	1.00000	0.0434783
$530$	−18.0000	−0.781870
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−5.00000	−0.216574
$534$	−4.00000	−0.173097
$535$	−36.0000	−1.55642
$536$	−16.0000	−0.691095
$537$	9.00000	0.388379
$538$	0	0
$539$	0	0
$540$	−3.00000	−0.129099
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−18.0000	−0.773166
$543$	14.0000	0.600798
$544$	4.00000	0.171499
$545$	−57.0000	−2.44161
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	−15.0000	−0.640768
$549$	12.0000	0.512148
$550$	8.00000	0.341121
$551$	−6.00000	−0.255609
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−18.0000	−0.764747
$555$	−3.00000	−0.127343
$556$	19.0000	0.805779
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	4.00000	0.169334
$559$	5.00000	0.211477
$560$	0	0
$561$	−8.00000	−0.337760
$562$	15.0000	0.632737
$563$	33.0000	1.39078	0.695392	−	0.718631i	$-0.255231\pi$
$563$	33.0000	1.39078	0.695392	+	0.718631i	$0.255231\pi$
$564$	−9.00000	−0.378968
$565$	39.0000	1.64074
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	18.0000	0.753937
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	10.0000	0.418121
$573$	4.00000	0.167102
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−1.00000	−0.0415945
$579$	−27.0000	−1.12208
$580$	3.00000	0.124568
$581$	0	0
$582$	−3.00000	−0.124354
$583$	−12.0000	−0.496989
$584$	10.0000	0.413803
$585$	15.0000	0.620174
$586$	−10.0000	−0.413096
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	−24.0000	−0.988064
$591$	21.0000	0.863825
$592$	1.00000	0.0410997
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	10.0000	0.409616
$597$	−7.00000	−0.286491
$598$	5.00000	0.204465
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	−4.00000	−0.163299
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	−16.0000	−0.651570
$604$	5.00000	0.203447
$605$	−21.0000	−0.853771
$606$	4.00000	0.162489
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	36.0000	1.45760
$611$	45.0000	1.82051
$612$	4.00000	0.161690
$613$	−21.0000	−0.848182	−0.424091	−	0.905620i	$-0.639406\pi$
$613$	−21.0000	−0.848182	−0.424091	+	0.905620i	$0.639406\pi$
$614$	1.00000	0.0403567
$615$	3.00000	0.120972
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−5.00000	−0.201129
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	12.0000	0.481932
$621$	−1.00000	−0.0401286
$622$	12.0000	0.481156
$623$	0	0
$624$	−5.00000	−0.200160
$625$	−29.0000	−1.16000
$626$	22.0000	0.879297
$627$	12.0000	0.479234
$628$	−10.0000	−0.399043
$629$	4.00000	0.159490
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	6.00000	0.238667
$633$	−10.0000	−0.397464
$634$	23.0000	0.913447
$635$	−33.0000	−1.30957
$636$	6.00000	0.237915
$637$	0	0
$638$	2.00000	0.0791808
$639$	0	0
$640$	3.00000	0.118585
$641$	−25.0000	−0.987441	−0.493720	−	0.869621i	$-0.664363\pi$
$641$	−25.0000	−0.987441	−0.493720	+	0.869621i	$0.664363\pi$
$642$	12.0000	0.473602
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	−24.0000	−0.944267
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	20.0000	0.784465
$651$	0	0
$652$	−8.00000	−0.313304
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	19.0000	0.742959
$655$	36.0000	1.40664
$656$	−1.00000	−0.0390434
$657$	10.0000	0.390137
$658$	0	0
$659$	50.0000	1.94772	0.973862	−	0.227142i	$-0.0729380\pi$
$659$	50.0000	1.94772	0.973862	+	0.227142i	$0.0729380\pi$
$660$	−6.00000	−0.233550
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	−8.00000	−0.310929
$663$	−20.0000	−0.776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	1.00000	0.0387492
$667$	1.00000	0.0387202
$668$	−8.00000	−0.309529
$669$	−22.0000	−0.850569
$670$	−48.0000	−1.85440
$671$	24.0000	0.926510
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	14.0000	0.539260
$675$	−4.00000	−0.153960
$676$	12.0000	0.461538
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	−13.0000	−0.499262
$679$	0	0
$680$	12.0000	0.460179
$681$	3.00000	0.114960
$682$	8.00000	0.306336
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	−6.00000	−0.229416
$685$	−45.0000	−1.71936
$686$	0	0
$687$	−4.00000	−0.152610
$688$	1.00000	0.0381246
$689$	−30.0000	−1.14291
$690$	−3.00000	−0.114208
$691$	21.0000	0.798878	0.399439	−	0.916760i	$-0.369205\pi$
$691$	21.0000	0.798878	0.399439	+	0.916760i	$0.369205\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−13.0000	−0.493473
$695$	57.0000	2.16213
$696$	−1.00000	−0.0379049
$697$	−4.00000	−0.151511
$698$	−10.0000	−0.378506
$699$	−16.0000	−0.605176
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	−5.00000	−0.188713
$703$	−6.00000	−0.226294
$704$	2.00000	0.0753778
$705$	−27.0000	−1.01688
$706$	−3.00000	−0.112906
$707$	0	0
$708$	8.00000	0.300658
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	4.00000	0.149906
$713$	4.00000	0.149801
$714$	0	0
$715$	30.0000	1.12194
$716$	−9.00000	−0.336346
$717$	16.0000	0.597531
$718$	17.0000	0.634434
$719$	−19.0000	−0.708580	−0.354290	−	0.935136i	$-0.615277\pi$
$719$	−19.0000	−0.708580	−0.354290	+	0.935136i	$0.615277\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	17.0000	0.632674
$723$	11.0000	0.409094
$724$	−14.0000	−0.520306
$725$	4.00000	0.148556
$726$	7.00000	0.259794
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	30.0000	1.11035
$731$	4.00000	0.147945
$732$	−12.0000	−0.443533
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	1.00000	0.0368605
$737$	−32.0000	−1.17874
$738$	−1.00000	−0.0368105
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	3.00000	0.110282
$741$	30.0000	1.10208
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	−4.00000	−0.146647
$745$	30.0000	1.09911
$746$	−10.0000	−0.366126
$747$	−12.0000	−0.439057
$748$	8.00000	0.292509
$749$	0	0
$750$	3.00000	0.109545
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	9.00000	0.328196
$753$	−13.0000	−0.473746
$754$	5.00000	0.182089
$755$	15.0000	0.545906
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	19.0000	0.690111
$759$	−2.00000	−0.0725954
$760$	−18.0000	−0.652929
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	11.0000	0.398488
$763$	0	0
$764$	−4.00000	−0.144715
$765$	12.0000	0.433861
$766$	−6.00000	−0.216789
$767$	−40.0000	−1.44432
$768$	−1.00000	−0.0360844
$769$	7.00000	0.252426	0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000	0.252426	0.126213	+	0.992003i	$0.459718\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	27.0000	0.971751
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	1.00000	0.0359443
$775$	16.0000	0.574737
$776$	3.00000	0.107694
$777$	0	0
$778$	−6.00000	−0.215110
$779$	6.00000	0.214972
$780$	−15.0000	−0.537086
$781$	0	0
$782$	4.00000	0.143040
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−30.0000	−1.07075
$786$	−12.0000	−0.428026
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	−21.0000	−0.748094
$789$	−11.0000	−0.391610
$790$	18.0000	0.640411
$791$	0	0
$792$	2.00000	0.0710669
$793$	60.0000	2.13066
$794$	26.0000	0.922705
$795$	18.0000	0.638394
$796$	7.00000	0.248108
$797$	−37.0000	−1.31061	−0.655304	−	0.755366i	$-0.727459\pi$
$797$	−37.0000	−1.31061	−0.655304	+	0.755366i	$0.727459\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	4.00000	0.141421
$801$	4.00000	0.141333
$802$	22.0000	0.776847
$803$	20.0000	0.705785
$804$	16.0000	0.564276
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	−4.00000	−0.140720
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	3.00000	0.105409
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	18.0000	0.631288
$814$	2.00000	0.0701000
$815$	−24.0000	−0.840683
$816$	−4.00000	−0.140028
$817$	−6.00000	−0.209913
$818$	30.0000	1.04893
$819$	0	0
$820$	−3.00000	−0.104765
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	15.0000	0.523185
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	5.00000	0.174183
$825$	−8.00000	−0.278524
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	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	$0.466774\pi$
$830$	−36.0000	−1.24958
$831$	18.0000	0.624413
$832$	5.00000	0.173344
$833$	0	0
$834$	−19.0000	−0.657916
$835$	−24.0000	−0.830554
$836$	−12.0000	−0.415029
$837$	−4.00000	−0.138260
$838$	−28.0000	−0.967244
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	21.0000	0.723708
$843$	−15.0000	−0.516627
$844$	10.0000	0.344214
$845$	36.0000	1.23844
$846$	9.00000	0.309426
$847$	0	0
$848$	−6.00000	−0.206041
$849$	16.0000	0.549119
$850$	16.0000	0.548795
$851$	1.00000	0.0342796
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	0	0
$855$	−18.0000	−0.615587
$856$	−12.0000	−0.410152
$857$	47.0000	1.60549	0.802745	−	0.596323i	$-0.203372\pi$
$857$	47.0000	1.60549	0.802745	+	0.596323i	$0.203372\pi$
$858$	−10.0000	−0.341394
$859$	−9.00000	−0.307076	−0.153538	−	0.988143i	$-0.549067\pi$
$859$	−9.00000	−0.307076	−0.153538	+	0.988143i	$0.549067\pi$
$860$	3.00000	0.102299
$861$	0	0
$862$	−37.0000	−1.26023
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	−1.00000	−0.0340207
$865$	54.0000	1.83606
$866$	−15.0000	−0.509721
$867$	1.00000	0.0339618
$868$	0	0
$869$	12.0000	0.407072
$870$	−3.00000	−0.101710
$871$	−80.0000	−2.71070
$872$	−19.0000	−0.643421
$873$	3.00000	0.101535
$874$	−6.00000	−0.202953
$875$	0	0
$876$	−10.0000	−0.337869
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	40.0000	1.34993
$879$	10.0000	0.337292
$880$	6.00000	0.202260
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	20.0000	0.672673
$885$	24.0000	0.806751
$886$	−19.0000	−0.638317
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	−1.00000	−0.0335578
$889$	0	0
$890$	12.0000	0.402241
$891$	2.00000	0.0670025
$892$	22.0000	0.736614
$893$	−54.0000	−1.80704
$894$	−10.0000	−0.334450
$895$	−27.0000	−0.902510
$896$	0	0
$897$	−5.00000	−0.166945
$898$	26.0000	0.867631
$899$	4.00000	0.133407
$900$	4.00000	0.133333
$901$	−24.0000	−0.799556
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	13.0000	0.432374
$905$	−42.0000	−1.39613
$906$	−5.00000	−0.166114
$907$	35.0000	1.16216	0.581078	−	0.813848i	$-0.302631\pi$
$907$	35.0000	1.16216	0.581078	+	0.813848i	$0.302631\pi$
$908$	−3.00000	−0.0995585
$909$	−4.00000	−0.132672
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	6.00000	0.198680
$913$	−24.0000	−0.794284
$914$	−16.0000	−0.529233
$915$	−36.0000	−1.19012
$916$	4.00000	0.132164
$917$	0	0
$918$	−4.00000	−0.132020
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	3.00000	0.0989071
$921$	−1.00000	−0.0329511
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.00000	0.131519
$926$	−41.0000	−1.34734
$927$	5.00000	0.164222
$928$	1.00000	0.0328266
$929$	3.00000	0.0984268	0.0492134	−	0.998788i	$-0.484329\pi$
$929$	3.00000	0.0984268	0.0492134	+	0.998788i	$0.484329\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	16.0000	0.524097
$933$	−12.0000	−0.392862
$934$	−33.0000	−1.07979
$935$	24.0000	0.784884
$936$	5.00000	0.163430
$937$	−5.00000	−0.163343	−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000	−0.163343	−0.0816714	+	0.996659i	$0.526026\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	27.0000	0.880643
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	10.0000	0.325818
$943$	−1.00000	−0.0325645
$944$	−8.00000	−0.260378
$945$	0	0
$946$	2.00000	0.0650256
$947$	−51.0000	−1.65728	−0.828639	−	0.559784i	$-0.810884\pi$
$947$	−51.0000	−1.65728	−0.828639	+	0.559784i	$0.810884\pi$
$948$	−6.00000	−0.194871
$949$	50.0000	1.62307
$950$	−24.0000	−0.778663
$951$	−23.0000	−0.745826
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−6.00000	−0.194257
$955$	−12.0000	−0.388311
$956$	−16.0000	−0.517477
$957$	−2.00000	−0.0646508
$958$	−2.00000	−0.0646171
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	−15.0000	−0.483871
$962$	5.00000	0.161206
$963$	−12.0000	−0.386695
$964$	−11.0000	−0.354286
$965$	81.0000	2.60748
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−7.00000	−0.224989
$969$	24.0000	0.770991
$970$	9.00000	0.288973
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	7.00000	0.224294
$975$	−20.0000	−0.640513
$976$	12.0000	0.384111
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	8.00000	0.255812
$979$	8.00000	0.255681
$980$	0	0
$981$	−19.0000	−0.606623
$982$	−12.0000	−0.382935
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	1.00000	0.0318788
$985$	−63.0000	−2.00735
$986$	4.00000	0.127386
$987$	0	0
$988$	−30.0000	−0.954427
$989$	1.00000	0.0317982
$990$	6.00000	0.190693
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000	0.127000
$993$	8.00000	0.253872
$994$	0	0
$995$	21.0000	0.665745
$996$	12.0000	0.380235
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	34.0000	1.07625
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.be.1.1	✓	1
7.6	odd	2		6762.2.a.bf.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6762.2.a.be.1.1	✓	1	1.1	even	1	trivial
6762.2.a.bf.1.1	yes	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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