LMFDB - Embedded newform 6762.2.a.o.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} +1.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} +5.00000 q^{50} -3.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{57} -3.00000 q^{58} +2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +3.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} -1.00000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} -2.00000 q^{78} +11.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -2.00000 q^{86} +3.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} -1.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	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
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	5.00000	0.707107
$51$	−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
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	−3.00000	−0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−3.00000	−0.363803
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−1.00000	−0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−2.00000	−0.232495
$75$	−5.00000	−0.577350
$76$	2.00000	0.229416
$77$	0	0
$78$	−2.00000	−0.226455
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	3.00000	0.321634
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	2.00000	0.207390
$94$	3.00000	0.309426
$95$	0	0
$96$	−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
$99$	−3.00000	−0.301511
$100$	−5.00000	−0.500000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	3.00000	0.297044
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	3.00000	0.278543
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	3.00000	0.257248
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	1.00000	0.0851257
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	−3.00000	−0.251754
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	0	0
$146$	−11.0000	−0.910366
$147$	0	0
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	5.00000	0.408248
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−2.00000	−0.162221
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	−11.0000	−0.875113
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	23.0000	1.80150	0.900750	−	0.434339i	$-0.143018\pi$
$163$	23.0000	1.80150	0.900750	+	0.434339i	$0.143018\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	2.00000	0.152499
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	1.00000	0.0737210
$185$	0	0
$186$	−2.00000	−0.146647
$187$	9.00000	0.658145
$188$	−3.00000	−0.218797
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	0	0
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	3.00000	0.213201
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	5.00000	0.353553
$201$	2.00000	0.141069
$202$	3.00000	0.211079
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−5.00000	−0.348367
$207$	−1.00000	−0.0695048
$208$	2.00000	0.138675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	−6.00000	−0.412082
$213$	3.00000	0.205557
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	1.00000	0.0677285
$219$	11.0000	0.743311
$220$	0	0
$221$	−6.00000	−0.403604
$222$	−2.00000	−0.134231
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−18.0000	−1.19734
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	2.00000	0.132453
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	11.0000	0.714527
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	4.00000	0.254514
$248$	−2.00000	−0.127000
$249$	0	0
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	2.00000	0.122169
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	21.0000	1.26866
$275$	15.0000	0.904534
$276$	−1.00000	−0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	13.0000	0.779688
$279$	2.00000	0.119737
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	3.00000	0.178647
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	8.00000	0.468968
$292$	11.0000	0.643726
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−3.00000	−0.174078
$298$	−12.0000	−0.695141
$299$	−2.00000	−0.115663
$300$	−5.00000	−0.288675
$301$	0	0
$302$	4.00000	0.230174
$303$	−3.00000	−0.172345
$304$	2.00000	0.114708
$305$	0	0
$306$	3.00000	0.171499
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	5.00000	0.284440
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	−2.00000	−0.113228
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−23.0000	−1.29797
$315$	0	0
$316$	11.0000	0.618798
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	6.00000	0.336463
$319$	−9.00000	−0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−6.00000	−0.333849
$324$	1.00000	0.0555556
$325$	−10.0000	−0.554700
$326$	−23.0000	−1.27385
$327$	−1.00000	−0.0553001
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	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
$339$	18.0000	0.977626
$340$	0	0
$341$	−6.00000	−0.324918
$342$	−2.00000	−0.108148
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	15.0000	0.806405
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	3.00000	0.160817
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	3.00000	0.159901
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−11.0000	−0.578147
$363$	−2.00000	−0.104973
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−1.00000	−0.0521286
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	3.00000	0.154713
$377$	6.00000	0.309016
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−12.0000	−0.613973
$383$	−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
$386$	−14.0000	−0.712581
$387$	2.00000	0.101666
$388$	8.00000	0.406138
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	0	0
$394$	−15.0000	−0.755689
$395$	0	0
$396$	−3.00000	−0.150756
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	−2.00000	−0.0997509
$403$	4.00000	0.199254
$404$	−3.00000	−0.149256
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	3.00000	0.148522
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	0	0
$411$	−21.0000	−1.03585
$412$	5.00000	0.246332
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−13.0000	−0.636613
$418$	6.00000	0.293470
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	19.0000	0.924906
$423$	−3.00000	−0.145865
$424$	6.00000	0.291386
$425$	15.0000	0.727607
$426$	−3.00000	−0.145350
$427$	0	0
$428$	12.0000	0.580042
$429$	−6.00000	−0.289683
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	−2.00000	−0.0956730
$438$	−11.0000	−0.525600
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	6.00000	0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	10.0000	0.473514
$447$	12.0000	0.567581
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	5.00000	0.235702
$451$	−18.0000	−0.847587
$452$	18.0000	0.846649
$453$	−4.00000	−0.187936
$454$	3.00000	0.140797
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	1.00000	0.0467269
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	0	0
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	23.0000	1.05978
$472$	0	0
$473$	−6.00000	−0.275880
$474$	−11.0000	−0.505247
$475$	−10.0000	−0.458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−15.0000	−0.686084
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−2.00000	−0.0905357
$489$	23.0000	1.04010
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000	0.270501
$493$	−9.00000	−0.405340
$494$	−4.00000	−0.179969
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	35.0000	1.56682	0.783408	−	0.621508i	$-0.213480\pi$
$499$	35.0000	1.56682	0.783408	+	0.621508i	$0.213480\pi$
$500$	0	0
$501$	0	0
$502$	−3.00000	−0.133897
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	0	0
$506$	−3.00000	−0.133366
$507$	−9.00000	−0.399704
$508$	2.00000	0.0887357
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	−6.00000	−0.264649
$515$	0	0
$516$	2.00000	0.0880451
$517$	9.00000	0.395820
$518$	0	0
$519$	−15.0000	−0.658427
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−3.00000	−0.131306
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−6.00000	−0.261364
$528$	−3.00000	−0.130558
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000	0.519778
$534$	6.00000	0.259645
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	6.00000	0.258919
$538$	−15.0000	−0.646696
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	22.0000	0.944981
$543$	11.0000	0.472055
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	−21.0000	−0.897076
$549$	2.00000	0.0853579
$550$	−15.0000	−0.639602
$551$	6.00000	0.255609
$552$	1.00000	0.0425628
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−13.0000	−0.551323
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−2.00000	−0.0846668
$559$	4.00000	0.169182
$560$	0	0
$561$	9.00000	0.379980
$562$	3.00000	0.126547
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−8.00000	−0.336265
$567$	0	0
$568$	−3.00000	−0.125877
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−6.00000	−0.250873
$573$	12.0000	0.501307
$574$	0	0
$575$	5.00000	0.208514
$576$	1.00000	0.0416667
$577$	−31.0000	−1.29055	−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000	−1.29055	−0.645273	+	0.763952i	$0.723257\pi$
$578$	8.00000	0.332756
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	−8.00000	−0.331611
$583$	18.0000	0.745484
$584$	−11.0000	−0.455183
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	15.0000	0.617018
$592$	2.00000	0.0821995
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	12.0000	0.491539
$597$	17.0000	0.695764
$598$	2.00000	0.0817861
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	5.00000	0.204124
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	−4.00000	−0.162758
$605$	0	0
$606$	3.00000	0.121867
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	−3.00000	−0.121268
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	−11.0000	−0.443924
$615$	0	0
$616$	0	0
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	−5.00000	−0.201129
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	3.00000	0.120289
$623$	0	0
$624$	2.00000	0.0800641
$625$	25.0000	1.00000
$626$	−8.00000	−0.319744
$627$	−6.00000	−0.239617
$628$	23.0000	0.917800
$629$	−6.00000	−0.239236
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	−11.0000	−0.437557
$633$	−19.0000	−0.755182
$634$	18.0000	0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	9.00000	0.356313
$639$	3.00000	0.118678
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	−12.0000	−0.473602
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	10.0000	0.392232
$651$	0	0
$652$	23.0000	0.900750
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	1.00000	0.0391031
$655$	0	0
$656$	6.00000	0.234261
$657$	11.0000	0.429151
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	−8.00000	−0.310929
$663$	−6.00000	−0.233021
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−3.00000	−0.116160
$668$	0	0
$669$	−10.0000	−0.386622
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	−14.0000	−0.539260
$675$	−5.00000	−0.192450
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−18.0000	−0.691286
$679$	0	0
$680$	0	0
$681$	−3.00000	−0.114960
$682$	6.00000	0.229752
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	0	0
$687$	−1.00000	−0.0381524
$688$	2.00000	0.0762493
$689$	−12.0000	−0.457164
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−3.00000	−0.113715
$697$	−18.0000	−0.681799
$698$	22.0000	0.832712
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	4.00000	0.150863
$704$	−3.00000	−0.113067
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	6.00000	0.224860
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	15.0000	0.560185
$718$	12.0000	0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	0	0
$722$	15.0000	0.558242
$723$	2.00000	0.0743808
$724$	11.0000	0.408812
$725$	−15.0000	−0.557086
$726$	2.00000	0.0742270
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.00000	−0.221918
$732$	2.00000	0.0739221
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	1.00000	0.0368605
$737$	−6.00000	−0.221013
$738$	−6.00000	−0.220863
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	13.0000	0.475964
$747$	0	0
$748$	9.00000	0.329073
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−3.00000	−0.109399
$753$	3.00000	0.109326
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−20.0000	−0.726433
$759$	3.00000	0.108893
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	1.00000	0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	14.0000	0.503871
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−2.00000	−0.0718885
$775$	−10.0000	−0.359211
$776$	−8.00000	−0.287183
$777$	0	0
$778$	−24.0000	−0.860442
$779$	12.0000	0.429945
$780$	0	0
$781$	−9.00000	−0.322045
$782$	−3.00000	−0.107280
$783$	3.00000	0.107211
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	15.0000	0.534353
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	3.00000	0.106600
$793$	4.00000	0.142044
$794$	34.0000	1.20661
$795$	0	0
$796$	17.0000	0.602549
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	15.0000	0.529668
$803$	−33.0000	−1.16454
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	15.0000	0.528025
$808$	3.00000	0.105540
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	−22.0000	−0.771574
$814$	6.00000	0.210300
$815$	0	0
$816$	−3.00000	−0.105021
$817$	4.00000	0.139942
$818$	−29.0000	−1.01396
$819$	0	0
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	21.0000	0.732459
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	−5.00000	−0.174183
$825$	15.0000	0.522233
$826$	0	0
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	−1.00000	−0.0347524
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	2.00000	0.0693375
$833$	0	0
$834$	13.0000	0.450153
$835$	0	0
$836$	−6.00000	−0.207514
$837$	2.00000	0.0691301
$838$	21.0000	0.725433
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−17.0000	−0.585859
$843$	−3.00000	−0.103325
$844$	−19.0000	−0.654007
$845$	0	0
$846$	3.00000	0.103142
$847$	0	0
$848$	−6.00000	−0.206041
$849$	8.00000	0.274559
$850$	−15.0000	−0.514496
$851$	−2.00000	−0.0685591
$852$	3.00000	0.102778
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	6.00000	0.204837
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	0	0
$862$	−30.0000	−1.02180
$863$	3.00000	0.102121	0.0510606	−	0.998696i	$-0.483740\pi$
$863$	3.00000	0.102121	0.0510606	+	0.998696i	$0.483740\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−33.0000	−1.11945
$870$	0	0
$871$	4.00000	0.135535
$872$	1.00000	0.0338643
$873$	8.00000	0.270759
$874$	2.00000	0.0676510
$875$	0	0
$876$	11.0000	0.371656
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	24.0000	0.806296
$887$	27.0000	0.906571	0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000	0.906571	0.453286	+	0.891365i	$0.350252\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−10.0000	−0.334825
$893$	−6.00000	−0.200782
$894$	−12.0000	−0.401340
$895$	0	0
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	−12.0000	−0.400445
$899$	6.00000	0.200111
$900$	−5.00000	−0.166667
$901$	18.0000	0.599667
$902$	18.0000	0.599334
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	4.00000	0.132891
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−3.00000	−0.0995585
$909$	−3.00000	−0.0995037
$910$	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
$913$	0	0
$914$	−20.0000	−0.661541
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	0	0
$918$	3.00000	0.0990148
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	11.0000	0.362462
$922$	30.0000	0.987997
$923$	6.00000	0.197492
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−32.0000	−1.05159
$927$	5.00000	0.164222
$928$	−3.00000	−0.0984798
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.00000	−0.0982156
$934$	21.0000	0.687141
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	−23.0000	−0.749380
$943$	−6.00000	−0.195387
$944$	0	0
$945$	0	0
$946$	6.00000	0.195077
$947$	−6.00000	−0.194974	−0.0974869	−	0.995237i	$-0.531080\pi$
$947$	−6.00000	−0.194974	−0.0974869	+	0.995237i	$0.531080\pi$
$948$	11.0000	0.357263
$949$	22.0000	0.714150
$950$	10.0000	0.324443
$951$	−18.0000	−0.583690
$952$	0	0
$953$	33.0000	1.06897	0.534487	−	0.845176i	$-0.320505\pi$
$953$	33.0000	1.06897	0.534487	+	0.845176i	$0.320505\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	15.0000	0.485135
$957$	−9.00000	−0.290929
$958$	−30.0000	−0.969256
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	12.0000	0.386695
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	2.00000	0.0642824
$969$	−6.00000	−0.192748
$970$	0	0
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	−10.0000	−0.320256
$976$	2.00000	0.0640184
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−23.0000	−0.735459
$979$	18.0000	0.575282
$980$	0	0
$981$	−1.00000	−0.0319275
$982$	0	0
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	9.00000	0.286618
$987$	0	0
$988$	4.00000	0.127257
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−2.00000	−0.0635001
$993$	8.00000	0.253872
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	−35.0000	−1.10791
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.o.1.1		1
7.2	even	3		966.2.i.f.277.1	✓	2
7.4	even	3		966.2.i.f.415.1	yes	2
7.6	odd	2		6762.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.i.f.277.1	✓	2	7.2	even	3
966.2.i.f.415.1	yes	2	7.4	even	3
6762.2.a.e.1.1		1	7.6	odd	2
6762.2.a.o.1.1		1	1.1	even	1	trivial

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

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