LMFDB - Embedded newform 6762.2.a.m.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)

$f(q)$

$=$

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

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

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
$5$	−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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	4.00000	0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−4.00000	−0.685994
$35$	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	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	4.00000	0.640513
$40$	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
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	4.00000	0.554700
$53$	−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
$55$	8.00000	1.07872
$56$	0	0
$57$	2.00000	0.264906
$58$	6.00000	0.787839
$59$	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
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	4.00000	0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	4.00000	0.485071
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	2.00000	0.229416
$77$	0	0
$78$	−4.00000	−0.452911
$79$	−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
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−8.00000	−0.862662
$87$	−6.00000	−0.643268
$88$	4.00000	0.426401
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	10.0000	1.03142
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	−4.00000	−0.396059
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.00000	−0.187317
$115$	−2.00000	−0.186501
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	−12.0000	−1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	10.0000	0.901670
$124$	−6.00000	−0.538816
$125$	12.0000	1.07331
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	8.00000	0.701646
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−2.00000	−0.172133
$136$	−4.00000	−0.342997
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−1.00000	−0.0851257
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	4.00000	0.335673
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−2.00000	−0.162221
$153$	4.00000	0.323381
$154$	0	0
$155$	12.0000	0.963863
$156$	4.00000	0.320256
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	2.00000	0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	10.0000	0.780869
$165$	8.00000	0.622799
$166$	−6.00000	−0.465690
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	8.00000	0.613572
$171$	2.00000	0.152944
$172$	8.00000	0.609994
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	12.0000	0.901975
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−2.00000	−0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	4.00000	0.294086
$186$	6.00000	0.439941
$187$	−16.0000	−1.17004
$188$	−10.0000	−0.729325
$189$	0	0
$190$	4.00000	0.290191
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	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	$0.430717\pi$
$194$	−8.00000	−0.574367
$195$	−8.00000	−0.572892
$196$	0	0
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	4.00000	0.284268
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	16.0000	1.12576
$203$	0	0
$204$	4.00000	0.280056
$205$	−20.0000	−1.39686
$206$	12.0000	0.836080
$207$	1.00000	0.0695048
$208$	4.00000	0.277350
$209$	−8.00000	−0.553372
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−6.00000	−0.412082
$213$	−4.00000	−0.274075
$214$	20.0000	1.36717
$215$	−16.0000	−1.09119
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	8.00000	0.539360
$221$	16.0000	1.07628
$222$	2.00000	0.134231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	2.00000	0.132453
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−4.00000	−0.261488
$235$	20.0000	1.30466
$236$	12.0000	0.781133
$237$	−8.00000	−0.519656
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−2.00000	−0.129099
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	−10.0000	−0.637577
$247$	8.00000	0.509028
$248$	6.00000	0.381000
$249$	6.00000	0.380235
$250$	−12.0000	−0.758947
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−20.0000	−1.25491
$255$	−8.00000	−0.500979
$256$	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	$0.519869\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	−8.00000	−0.496139
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	4.00000	0.246183
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	2.00000	0.121716
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−16.0000	−0.959616
$279$	−6.00000	−0.359211
$280$	0	0
$281$	14.0000	0.835170	0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000	0.835170	0.417585	+	0.908638i	$0.362877\pi$
$282$	10.0000	0.595491
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	−4.00000	−0.237356
$285$	−4.00000	−0.236940
$286$	16.0000	0.946100
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−12.0000	−0.704664
$291$	8.00000	0.468968
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	2.00000	0.116248
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	4.00000	0.231326
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−16.0000	−0.919176
$304$	2.00000	0.114708
$305$	−20.0000	−1.14520
$306$	−4.00000	−0.228665
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	−12.0000	−0.681554
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	−4.00000	−0.226455
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	6.00000	0.336463
$319$	24.0000	1.34374
$320$	−2.00000	−0.111803
$321$	−20.0000	−1.11629
$322$	0	0
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−12.0000	−0.664619
$327$	2.00000	0.110600
$328$	−10.0000	−0.552158
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	6.00000	0.329293
$333$	−2.00000	−0.109599
$334$	−18.0000	−0.984916
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	−8.00000	−0.433861
$341$	24.0000	1.29967
$342$	−2.00000	−0.108148
$343$	0	0
$344$	−8.00000	−0.431331
$345$	−2.00000	−0.107676
$346$	16.0000	0.860165
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−6.00000	−0.321634
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	4.00000	0.213201
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	−12.0000	−0.637793
$355$	8.00000	0.424596
$356$	0	0
$357$	0	0
$358$	−4.00000	−0.211407
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	−2.00000	−0.105118
$363$	5.00000	0.262432
$364$	0	0
$365$	−4.00000	−0.209370
$366$	−10.0000	−0.522708
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	−4.00000	−0.207950
$371$	0	0
$372$	−6.00000	−0.311086
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	16.0000	0.827340
$375$	12.0000	0.619677
$376$	10.0000	0.515711
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	−4.00000	−0.205196
$381$	20.0000	1.02463
$382$	−16.0000	−0.818631
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	8.00000	0.406663
$388$	8.00000	0.406138
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	8.00000	0.405096
$391$	4.00000	0.202289
$392$	0	0
$393$	12.0000	0.605320
$394$	26.0000	1.30986
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	−8.00000	−0.399004
$403$	−24.0000	−1.19553
$404$	−16.0000	−0.796030
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	8.00000	0.396545
$408$	−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	$0.234019\pi$
$410$	20.0000	0.987730
$411$	6.00000	0.295958
$412$	−12.0000	−0.591198
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−12.0000	−0.589057
$416$	−4.00000	−0.196116
$417$	16.0000	0.783523
$418$	8.00000	0.391293
$419$	22.0000	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−4.00000	−0.194717
$423$	−10.0000	−0.486217
$424$	6.00000	0.291386
$425$	−4.00000	−0.194029
$426$	4.00000	0.193801
$427$	0	0
$428$	−20.0000	−0.966736
$429$	−16.0000	−0.772487
$430$	16.0000	0.771589
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	2.00000	0.0957826
$437$	2.00000	0.0956730
$438$	−2.00000	−0.0955637
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	−16.0000	−0.761042
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−14.0000	−0.662919
$447$	6.00000	0.283790
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	−40.0000	−1.88353
$452$	−6.00000	−0.282216
$453$	4.00000	0.187936
$454$	−10.0000	−0.469323
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	6.00000	0.280362
$459$	4.00000	0.186704
$460$	−2.00000	−0.0932505
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−6.00000	−0.278543
$465$	12.0000	0.556487
$466$	−6.00000	−0.277945
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−20.0000	−0.922531
$471$	−10.0000	−0.460776
$472$	−12.0000	−0.552345
$473$	−32.0000	−1.47136
$474$	8.00000	0.367452
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−20.0000	−0.914779
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	2.00000	0.0912871
$481$	−8.00000	−0.364769
$482$	12.0000	0.546585
$483$	0	0
$484$	5.00000	0.227273
$485$	−16.0000	−0.726523
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−10.0000	−0.452679
$489$	12.0000	0.542659
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	10.0000	0.450835
$493$	−24.0000	−1.08091
$494$	−8.00000	−0.359937
$495$	8.00000	0.359573
$496$	−6.00000	−0.269408
$497$	0	0
$498$	−6.00000	−0.268866
$499$	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
$501$	18.0000	0.804181
$502$	26.0000	1.16044
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	32.0000	1.42398
$506$	4.00000	0.177822
$507$	3.00000	0.133235
$508$	20.0000	0.887357
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	8.00000	0.354246
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	2.00000	0.0882162
$515$	24.0000	1.05757
$516$	8.00000	0.352180
$517$	40.0000	1.75920
$518$	0	0
$519$	−16.0000	−0.702322
$520$	8.00000	0.350823
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	6.00000	0.262613
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−24.0000	−1.04546
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	12.0000	0.520756
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	40.0000	1.72935
$536$	−8.00000	−0.345547
$537$	4.00000	0.172613
$538$	0	0
$539$	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	$0.276899\pi$
$542$	−26.0000	−1.11680
$543$	2.00000	0.0858282
$544$	−4.00000	−0.171499
$545$	−4.00000	−0.171341
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	6.00000	0.256307
$549$	10.0000	0.426790
$550$	−4.00000	−0.170561
$551$	−12.0000	−0.511217
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−14.0000	−0.594803
$555$	4.00000	0.169791
$556$	16.0000	0.678551
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	6.00000	0.254000
$559$	32.0000	1.35346
$560$	0	0
$561$	−16.0000	−0.675521
$562$	−14.0000	−0.590554
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	−10.0000	−0.421076
$565$	12.0000	0.504844
$566$	−18.0000	−0.756596
$567$	0	0
$568$	4.00000	0.167836
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	4.00000	0.167542
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	−16.0000	−0.668994
$573$	16.0000	0.668410
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	1.00000	0.0415945
$579$	6.00000	0.249351
$580$	12.0000	0.498273
$581$	0	0
$582$	−8.00000	−0.331611
$583$	24.0000	0.993978
$584$	−2.00000	−0.0827606
$585$	−8.00000	−0.330759
$586$	−6.00000	−0.247858
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	24.0000	0.988064
$591$	−26.0000	−1.06950
$592$	−2.00000	−0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	12.0000	0.491127
$598$	−4.00000	−0.163572
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	1.00000	0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	4.00000	0.162758
$605$	−10.0000	−0.406558
$606$	16.0000	0.649956
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	20.0000	0.809776
$611$	−40.0000	−1.61823
$612$	4.00000	0.161690
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	−20.0000	−0.806478
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	12.0000	0.482711
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	12.0000	0.481932
$621$	1.00000	0.0401286
$622$	−30.0000	−1.20289
$623$	0	0
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	−20.0000	−0.799361
$627$	−8.00000	−0.319489
$628$	−10.0000	−0.399043
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	8.00000	0.318223
$633$	4.00000	0.158986
$634$	−30.0000	−1.19145
$635$	−40.0000	−1.58735
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−24.0000	−0.950169
$639$	−4.00000	−0.158238
$640$	2.00000	0.0790569
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	20.0000	0.789337
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	−8.00000	−0.314756
$647$	−14.0000	−0.550397	−0.275198	−	0.961387i	$-0.588744\pi$
$647$	−14.0000	−0.550397	−0.275198	+	0.961387i	$0.588744\pi$
$648$	−1.00000	−0.0392837
$649$	−48.0000	−1.88416
$650$	4.00000	0.156893
$651$	0	0
$652$	12.0000	0.469956
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	−2.00000	−0.0782062
$655$	−24.0000	−0.937758
$656$	10.0000	0.390434
$657$	2.00000	0.0780274
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	8.00000	0.311400
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	4.00000	0.155464
$663$	16.0000	0.621389
$664$	−6.00000	−0.232845
$665$	0	0
$666$	2.00000	0.0774984
$667$	−6.00000	−0.232321
$668$	18.0000	0.696441
$669$	14.0000	0.541271
$670$	16.0000	0.618134
$671$	−40.0000	−1.54418
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	10.0000	0.385186
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	8.00000	0.306786
$681$	10.0000	0.383201
$682$	−24.0000	−0.919007
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	2.00000	0.0764719
$685$	−12.0000	−0.458496
$686$	0	0
$687$	−6.00000	−0.228914
$688$	8.00000	0.304997
$689$	−24.0000	−0.914327
$690$	2.00000	0.0761387
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−16.0000	−0.608229
$693$	0	0
$694$	−28.0000	−1.06287
$695$	−32.0000	−1.21383
$696$	6.00000	0.227429
$697$	40.0000	1.51511
$698$	−16.0000	−0.605609
$699$	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	$0.308281\pi$
$702$	−4.00000	−0.150970
$703$	−4.00000	−0.150863
$704$	−4.00000	−0.150756
$705$	20.0000	0.753244
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	12.0000	0.450988
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	−8.00000	−0.300235
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	32.0000	1.19673
$716$	4.00000	0.149487
$717$	20.0000	0.746914
$718$	−16.0000	−0.597115
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	15.0000	0.558242
$723$	−12.0000	−0.446285
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	32.0000	1.18356
$732$	10.0000	0.369611
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−32.0000	−1.17874
$738$	−10.0000	−0.368105
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	4.00000	0.147043
$741$	8.00000	0.293887
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	6.00000	0.219971
$745$	−12.0000	−0.439646
$746$	−6.00000	−0.219676
$747$	6.00000	0.219529
$748$	−16.0000	−0.585018
$749$	0	0
$750$	−12.0000	−0.438178
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−10.0000	−0.364662
$753$	−26.0000	−0.947493
$754$	24.0000	0.874028
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	28.0000	1.01701
$759$	−4.00000	−0.145191
$760$	4.00000	0.145095
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	−20.0000	−0.724524
$763$	0	0
$764$	16.0000	0.578860
$765$	−8.00000	−0.289241
$766$	32.0000	1.15621
$767$	48.0000	1.73318
$768$	1.00000	0.0360844
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	6.00000	0.215945
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−8.00000	−0.287554
$775$	6.00000	0.215526
$776$	−8.00000	−0.287183
$777$	0	0
$778$	6.00000	0.215110
$779$	20.0000	0.716574
$780$	−8.00000	−0.286446
$781$	16.0000	0.572525
$782$	−4.00000	−0.143040
$783$	−6.00000	−0.214423
$784$	0	0
$785$	20.0000	0.713831
$786$	−12.0000	−0.428026
$787$	6.00000	0.213877	0.106938	−	0.994266i	$-0.465895\pi$
$787$	6.00000	0.213877	0.106938	+	0.994266i	$0.465895\pi$
$788$	−26.0000	−0.926212
$789$	16.0000	0.569615
$790$	−16.0000	−0.569254
$791$	0	0
$792$	4.00000	0.142134
$793$	40.0000	1.42044
$794$	28.0000	0.993683
$795$	12.0000	0.425596
$796$	12.0000	0.425329
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	1.00000	0.0353553
$801$	0	0
$802$	2.00000	0.0706225
$803$	−8.00000	−0.282314
$804$	8.00000	0.282138
$805$	0	0
$806$	24.0000	0.845364
$807$	0	0
$808$	16.0000	0.562878
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	2.00000	0.0702728
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	26.0000	0.911860
$814$	−8.00000	−0.280400
$815$	−24.0000	−0.840683
$816$	4.00000	0.140028
$817$	16.0000	0.559769
$818$	−30.0000	−1.04893
$819$	0	0
$820$	−20.0000	−0.698430
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−6.00000	−0.209274
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	12.0000	0.418040
$825$	4.00000	0.139262
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	1.00000	0.0347524
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	12.0000	0.416526
$831$	14.0000	0.485655
$832$	4.00000	0.138675
$833$	0	0
$834$	−16.0000	−0.554035
$835$	−36.0000	−1.24583
$836$	−8.00000	−0.276686
$837$	−6.00000	−0.207390
$838$	−22.0000	−0.759977
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	14.0000	0.482186
$844$	4.00000	0.137686
$845$	−6.00000	−0.206406
$846$	10.0000	0.343807
$847$	0	0
$848$	−6.00000	−0.206041
$849$	18.0000	0.617758
$850$	4.00000	0.137199
$851$	−2.00000	−0.0685591
$852$	−4.00000	−0.137038
$853$	−12.0000	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$853$	−12.0000	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	20.0000	0.683586
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	16.0000	0.546231
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	−32.0000	−1.08992
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	−1.00000	−0.0340207
$865$	32.0000	1.08803
$866$	−16.0000	−0.543702
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	32.0000	1.08553
$870$	−12.0000	−0.406838
$871$	32.0000	1.08428
$872$	−2.00000	−0.0677285
$873$	8.00000	0.270759
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	2.00000	0.0675737
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−14.0000	−0.472477
$879$	6.00000	0.202375
$880$	8.00000	0.269680
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	16.0000	0.538138
$885$	−24.0000	−0.806751
$886$	4.00000	0.134383
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	14.0000	0.468755
$893$	−20.0000	−0.669274
$894$	−6.00000	−0.200670
$895$	−8.00000	−0.267411
$896$	0	0
$897$	4.00000	0.133556
$898$	−14.0000	−0.467186
$899$	36.0000	1.20067
$900$	−1.00000	−0.0333333
$901$	−24.0000	−0.799556
$902$	40.0000	1.33185
$903$	0	0
$904$	6.00000	0.199557
$905$	−4.00000	−0.132964
$906$	−4.00000	−0.132891
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	10.0000	0.331862
$909$	−16.0000	−0.530687
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	2.00000	0.0662266
$913$	−24.0000	−0.794284
$914$	−18.0000	−0.595387
$915$	−20.0000	−0.661180
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−4.00000	−0.132020
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	2.00000	0.0659380
$921$	0	0
$922$	−24.0000	−0.790398
$923$	−16.0000	−0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	32.0000	1.05159
$927$	−12.0000	−0.394132
$928$	6.00000	0.196960
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	6.00000	0.196537
$933$	30.0000	0.982156
$934$	22.0000	0.719862
$935$	32.0000	1.04651
$936$	−4.00000	−0.130744
$937$	−4.00000	−0.130674	−0.0653372	−	0.997863i	$-0.520812\pi$
$937$	−4.00000	−0.130674	−0.0653372	+	0.997863i	$0.520812\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	20.0000	0.652328
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	10.0000	0.325818
$943$	10.0000	0.325645
$944$	12.0000	0.390567
$945$	0	0
$946$	32.0000	1.04041
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−8.00000	−0.259828
$949$	8.00000	0.259691
$950$	2.00000	0.0648886
$951$	30.0000	0.972817
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	6.00000	0.194257
$955$	−32.0000	−1.03550
$956$	20.0000	0.646846
$957$	24.0000	0.775810
$958$	4.00000	0.129234
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	5.00000	0.161290
$962$	8.00000	0.257930
$963$	−20.0000	−0.644491
$964$	−12.0000	−0.386494
$965$	−12.0000	−0.386294
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−5.00000	−0.160706
$969$	8.00000	0.256997
$970$	16.0000	0.513729
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−4.00000	−0.128103
$976$	10.0000	0.320092
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	28.0000	0.893516
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−10.0000	−0.318788
$985$	52.0000	1.65686
$986$	24.0000	0.764316
$987$	0	0
$988$	8.00000	0.254514
$989$	8.00000	0.254385
$990$	−8.00000	−0.254257
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	6.00000	0.190500
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−24.0000	−0.760851
$996$	6.00000	0.190117
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	−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	6762.2.a.m.1.1		1
7.6	odd	2		966.2.a.c.1.1	✓	1
21.20	even	2		2898.2.a.l.1.1		1
28.27	even	2		7728.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.c.1.1	✓	1	7.6	odd	2
2898.2.a.l.1.1		1	21.20	even	2
6762.2.a.m.1.1		1	1.1	even	1	trivial
7728.2.a.t.1.1		1	28.27	even	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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