LMFDB - Embedded newform 6664.2.a.y.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 8x^{5} + 21x^{4} + 25x^{3} - 41x^{2} - 28x + 19 $ x^7 - 3x^6 - 8x^5 + 21x^4 + 25x^3 - 41x^2 - 28x + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.01208$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.01208 q^{3} -0.157432 q^{5} +6.07263 q^{9} -0.866090 q^{11} -6.43337 q^{13} -0.474197 q^{15} -1.00000 q^{17} -4.73982 q^{19} -0.446225 q^{23} -4.97522 q^{25} +9.25501 q^{27} -9.62945 q^{29} +9.90871 q^{31} -2.60873 q^{33} -6.33360 q^{37} -19.3778 q^{39} -7.13219 q^{41} -8.63691 q^{43} -0.956024 q^{45} +5.49339 q^{47} -3.01208 q^{51} +13.5800 q^{53} +0.136350 q^{55} -14.2767 q^{57} -7.24800 q^{59} -3.14997 q^{61} +1.01282 q^{65} -7.23290 q^{67} -1.34406 q^{69} +5.46149 q^{71} -9.86082 q^{73} -14.9857 q^{75} +10.1993 q^{79} +9.65894 q^{81} +3.01109 q^{83} +0.157432 q^{85} -29.0047 q^{87} +0.419919 q^{89} +29.8458 q^{93} +0.746198 q^{95} +8.30943 q^{97} -5.25945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 12 * q^11 - 2 * q^13 - 6 * q^15 - 7 * q^17 + 3 * q^19 - 18 * q^23 + 15 * q^25 + 18 * q^27 + 5 * q^29 + 10 * q^31 - 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 - 13 * q^45 - 15 * q^47 - 3 * q^51 + 8 * q^53 + 9 * q^55 - 3 * q^57 - 9 * q^59 + 4 * q^61 - 38 * q^65 - 30 * q^67 + 28 * q^69 - 27 * q^71 + 5 * q^73 + 2 * q^75 + 3 * q^79 - 17 * q^81 + 12 * q^83 + 2 * q^85 - 19 * q^87 - 32 * q^89 - 9 * q^93 - 42 * q^95 + 39 * q^97 - 20 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	3.01208	1.73903	0.869513	−	0.493910i	$-0.164433\pi$
$3$	3.01208	1.73903	0.869513	+	0.493910i	$0.164433\pi$
$4$	0	0
$5$	−0.157432	−0.0704056	−0.0352028	−	0.999380i	$-0.511208\pi$
$5$	−0.157432	−0.0704056	−0.0352028	+	0.999380i	$0.511208\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.07263	2.02421
$10$	0	0
$11$	−0.866090	−0.261136	−0.130568	−	0.991439i	$-0.541680\pi$
$11$	−0.866090	−0.261136	−0.130568	+	0.991439i	$0.541680\pi$
$12$	0	0
$13$	−6.43337	−1.78430	−0.892148	−	0.451743i	$-0.850802\pi$
$13$	−6.43337	−1.78430	−0.892148	+	0.451743i	$0.850802\pi$
$14$	0	0
$15$	−0.474197	−0.122437
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.73982	−1.08739	−0.543695	−	0.839283i	$-0.682975\pi$
$19$	−4.73982	−1.08739	−0.543695	+	0.839283i	$0.682975\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.446225	−0.0930443	−0.0465221	−	0.998917i	$-0.514814\pi$
$23$	−0.446225	−0.0930443	−0.0465221	+	0.998917i	$0.514814\pi$
$24$	0	0
$25$	−4.97522	−0.995043
$26$	0	0
$27$	9.25501	1.78113
$28$	0	0
$29$	−9.62945	−1.78814	−0.894072	−	0.447924i	$-0.852164\pi$
$29$	−9.62945	−1.78814	−0.894072	+	0.447924i	$0.852164\pi$
$30$	0	0
$31$	9.90871	1.77966	0.889828	−	0.456296i	$-0.150824\pi$
$31$	9.90871	1.77966	0.889828	+	0.456296i	$0.150824\pi$
$32$	0	0
$33$	−2.60873	−0.454122
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.33360	−1.04124	−0.520618	−	0.853790i	$-0.674299\pi$
$37$	−6.33360	−1.04124	−0.520618	+	0.853790i	$0.674299\pi$
$38$	0	0
$39$	−19.3778	−3.10294
$40$	0	0
$41$	−7.13219	−1.11386	−0.556930	−	0.830559i	$-0.688021\pi$
$41$	−7.13219	−1.11386	−0.556930	+	0.830559i	$0.688021\pi$
$42$	0	0
$43$	−8.63691	−1.31712	−0.658558	−	0.752530i	$-0.728833\pi$
$43$	−8.63691	−1.31712	−0.658558	+	0.752530i	$0.728833\pi$
$44$	0	0
$45$	−0.956024	−0.142516
$46$	0	0
$47$	5.49339	0.801293	0.400647	−	0.916233i	$-0.368786\pi$
$47$	5.49339	0.801293	0.400647	+	0.916233i	$0.368786\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.01208	−0.421776
$52$	0	0
$53$	13.5800	1.86536	0.932681	−	0.360702i	$-0.117463\pi$
$53$	13.5800	1.86536	0.932681	+	0.360702i	$0.117463\pi$
$54$	0	0
$55$	0.136350	0.0183854
$56$	0	0
$57$	−14.2767	−1.89100
$58$	0	0
$59$	−7.24800	−0.943610	−0.471805	−	0.881703i	$-0.656397\pi$
$59$	−7.24800	−0.943610	−0.471805	+	0.881703i	$0.656397\pi$
$60$	0	0
$61$	−3.14997	−0.403312	−0.201656	−	0.979456i	$-0.564632\pi$
$61$	−3.14997	−0.403312	−0.201656	+	0.979456i	$0.564632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.01282	0.125624
$66$	0	0
$67$	−7.23290	−0.883639	−0.441820	−	0.897104i	$-0.645667\pi$
$67$	−7.23290	−0.883639	−0.441820	+	0.897104i	$0.645667\pi$
$68$	0	0
$69$	−1.34406	−0.161806
$70$	0	0
$71$	5.46149	0.648160	0.324080	−	0.946030i	$-0.394945\pi$
$71$	5.46149	0.648160	0.324080	+	0.946030i	$0.394945\pi$
$72$	0	0
$73$	−9.86082	−1.15412	−0.577061	−	0.816701i	$-0.695801\pi$
$73$	−9.86082	−1.15412	−0.577061	+	0.816701i	$0.695801\pi$
$74$	0	0
$75$	−14.9857	−1.73041
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.1993	1.14751	0.573756	−	0.819026i	$-0.305486\pi$
$79$	10.1993	1.14751	0.573756	+	0.819026i	$0.305486\pi$
$80$	0	0
$81$	9.65894	1.07322
$82$	0	0
$83$	3.01109	0.330510	0.165255	−	0.986251i	$-0.447155\pi$
$83$	3.01109	0.330510	0.165255	+	0.986251i	$0.447155\pi$
$84$	0	0
$85$	0.157432	0.0170759
$86$	0	0
$87$	−29.0047	−3.10963
$88$	0	0
$89$	0.419919	0.0445114	0.0222557	−	0.999752i	$-0.492915\pi$
$89$	0.419919	0.0445114	0.0222557	+	0.999752i	$0.492915\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	29.8458	3.09487
$94$	0	0
$95$	0.746198	0.0765583
$96$	0	0
$97$	8.30943	0.843695	0.421848	−	0.906667i	$-0.361382\pi$
$97$	8.30943	0.843695	0.421848	+	0.906667i	$0.361382\pi$
$98$	0	0
$99$	−5.25945	−0.528594
$100$	0	0
$101$	−1.64162	−0.163347	−0.0816736	−	0.996659i	$-0.526027\pi$
$101$	−1.64162	−0.163347	−0.0816736	+	0.996659i	$0.526027\pi$
$102$	0	0
$103$	1.04940	0.103401	0.0517005	−	0.998663i	$-0.483536\pi$
$103$	1.04940	0.103401	0.0517005	+	0.998663i	$0.483536\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4034	−1.58577	−0.792886	−	0.609370i	$-0.791423\pi$
$107$	−16.4034	−1.58577	−0.792886	+	0.609370i	$0.791423\pi$
$108$	0	0
$109$	−6.03398	−0.577951	−0.288975	−	0.957337i	$-0.593315\pi$
$109$	−6.03398	−0.577951	−0.288975	+	0.957337i	$0.593315\pi$
$110$	0	0
$111$	−19.0773	−1.81074
$112$	0	0
$113$	7.18018	0.675454	0.337727	−	0.941244i	$-0.390342\pi$
$113$	7.18018	0.675454	0.337727	+	0.941244i	$0.390342\pi$
$114$	0	0
$115$	0.0702499	0.00655083
$116$	0	0
$117$	−39.0675	−3.61179
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.2499	−0.931808
$122$	0	0
$123$	−21.4827	−1.93703
$124$	0	0
$125$	1.57041	0.140462
$126$	0	0
$127$	18.7810	1.66654	0.833272	−	0.552864i	$-0.186465\pi$
$127$	18.7810	1.66654	0.833272	+	0.552864i	$0.186465\pi$
$128$	0	0
$129$	−26.0151	−2.29050
$130$	0	0
$131$	8.85946	0.774054	0.387027	−	0.922068i	$-0.373502\pi$
$131$	8.85946	0.774054	0.387027	+	0.922068i	$0.373502\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.45703	−0.125401
$136$	0	0
$137$	−3.66704	−0.313296	−0.156648	−	0.987654i	$-0.550069\pi$
$137$	−3.66704	−0.313296	−0.156648	+	0.987654i	$0.550069\pi$
$138$	0	0
$139$	4.09899	0.347672	0.173836	−	0.984775i	$-0.444384\pi$
$139$	4.09899	0.347672	0.173836	+	0.984775i	$0.444384\pi$
$140$	0	0
$141$	16.5465	1.39347
$142$	0	0
$143$	5.57188	0.465944
$144$	0	0
$145$	1.51598	0.125895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.57821	−0.456985	−0.228492	−	0.973546i	$-0.573380\pi$
$149$	−5.57821	−0.456985	−0.228492	+	0.973546i	$0.573380\pi$
$150$	0	0
$151$	−12.4699	−1.01479	−0.507395	−	0.861714i	$-0.669391\pi$
$151$	−12.4699	−1.01479	−0.507395	+	0.861714i	$0.669391\pi$
$152$	0	0
$153$	−6.07263	−0.490943
$154$	0	0
$155$	−1.55994	−0.125298
$156$	0	0
$157$	−17.2914	−1.38001	−0.690004	−	0.723806i	$-0.742391\pi$
$157$	−17.2914	−1.38001	−0.690004	+	0.723806i	$0.742391\pi$
$158$	0	0
$159$	40.9042	3.24391
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.17251	0.0918382	0.0459191	−	0.998945i	$-0.485378\pi$
$163$	1.17251	0.0918382	0.0459191	+	0.998945i	$0.485378\pi$
$164$	0	0
$165$	0.410697	0.0319727
$166$	0	0
$167$	9.68349	0.749331	0.374666	−	0.927160i	$-0.377758\pi$
$167$	9.68349	0.749331	0.374666	+	0.927160i	$0.377758\pi$
$168$	0	0
$169$	28.3883	2.18371
$170$	0	0
$171$	−28.7832	−2.20111
$172$	0	0
$173$	14.5075	1.10299	0.551493	−	0.834179i	$-0.314058\pi$
$173$	14.5075	1.10299	0.551493	+	0.834179i	$0.314058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.8316	−1.64096
$178$	0	0
$179$	−16.3656	−1.22322	−0.611610	−	0.791160i	$-0.709478\pi$
$179$	−16.3656	−1.22322	−0.611610	+	0.791160i	$0.709478\pi$
$180$	0	0
$181$	1.21667	0.0904343	0.0452171	−	0.998977i	$-0.485602\pi$
$181$	1.21667	0.0904343	0.0452171	+	0.998977i	$0.485602\pi$
$182$	0	0
$183$	−9.48796	−0.701370
$184$	0	0
$185$	0.997108	0.0733089
$186$	0	0
$187$	0.866090	0.0633348
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0207	1.30393	0.651967	−	0.758248i	$-0.273944\pi$
$191$	18.0207	1.30393	0.651967	+	0.758248i	$0.273944\pi$
$192$	0	0
$193$	18.1289	1.30495	0.652475	−	0.757810i	$-0.273731\pi$
$193$	18.1289	1.30495	0.652475	+	0.757810i	$0.273731\pi$
$194$	0	0
$195$	3.05068	0.218464
$196$	0	0
$197$	−21.6162	−1.54009	−0.770044	−	0.637990i	$-0.779766\pi$
$197$	−21.6162	−1.54009	−0.770044	+	0.637990i	$0.779766\pi$
$198$	0	0
$199$	15.9115	1.12794	0.563968	−	0.825797i	$-0.309274\pi$
$199$	15.9115	1.12794	0.563968	+	0.825797i	$0.309274\pi$
$200$	0	0
$201$	−21.7861	−1.53667
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.12283	0.0784220
$206$	0	0
$207$	−2.70976	−0.188341
$208$	0	0
$209$	4.10511	0.283957
$210$	0	0
$211$	−7.88674	−0.542946	−0.271473	−	0.962446i	$-0.587511\pi$
$211$	−7.88674	−0.542946	−0.271473	+	0.962446i	$0.587511\pi$
$212$	0	0
$213$	16.4505	1.12717
$214$	0	0
$215$	1.35972	0.0927323
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−29.7016	−2.00705
$220$	0	0
$221$	6.43337	0.432755
$222$	0	0
$223$	−1.32251	−0.0885615	−0.0442808	−	0.999019i	$-0.514100\pi$
$223$	−1.32251	−0.0885615	−0.0442808	+	0.999019i	$0.514100\pi$
$224$	0	0
$225$	−30.2126	−2.01418
$226$	0	0
$227$	7.58076	0.503152	0.251576	−	0.967837i	$-0.419051\pi$
$227$	7.58076	0.503152	0.251576	+	0.967837i	$0.419051\pi$
$228$	0	0
$229$	9.64847	0.637589	0.318794	−	0.947824i	$-0.396722\pi$
$229$	9.64847	0.637589	0.318794	+	0.947824i	$0.396722\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.35399	−0.416263	−0.208132	−	0.978101i	$-0.566738\pi$
$233$	−6.35399	−0.416263	−0.208132	+	0.978101i	$0.566738\pi$
$234$	0	0
$235$	−0.864833	−0.0564155
$236$	0	0
$237$	30.7211	1.99555
$238$	0	0
$239$	7.67304	0.496328	0.248164	−	0.968718i	$-0.420173\pi$
$239$	7.67304	0.496328	0.248164	+	0.968718i	$0.420173\pi$
$240$	0	0
$241$	−19.3831	−1.24858	−0.624288	−	0.781195i	$-0.714611\pi$
$241$	−19.3831	−1.24858	−0.624288	+	0.781195i	$0.714611\pi$
$242$	0	0
$243$	1.32848	0.0852222
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.4930	1.94023
$248$	0	0
$249$	9.06965	0.574766
$250$	0	0
$251$	17.1926	1.08519	0.542594	−	0.839995i	$-0.317442\pi$
$251$	17.1926	1.08519	0.542594	+	0.839995i	$0.317442\pi$
$252$	0	0
$253$	0.386471	0.0242972
$254$	0	0
$255$	0.474197	0.0296954
$256$	0	0
$257$	9.53142	0.594554	0.297277	−	0.954791i	$-0.403922\pi$
$257$	9.53142	0.594554	0.297277	+	0.954791i	$0.403922\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−58.4761	−3.61958
$262$	0	0
$263$	10.4063	0.641682	0.320841	−	0.947133i	$-0.396034\pi$
$263$	10.4063	0.641682	0.320841	+	0.947133i	$0.396034\pi$
$264$	0	0
$265$	−2.13793	−0.131332
$266$	0	0
$267$	1.26483	0.0774064
$268$	0	0
$269$	17.9246	1.09288	0.546441	−	0.837498i	$-0.315982\pi$
$269$	17.9246	1.09288	0.546441	+	0.837498i	$0.315982\pi$
$270$	0	0
$271$	19.8979	1.20871	0.604355	−	0.796715i	$-0.293431\pi$
$271$	19.8979	1.20871	0.604355	+	0.796715i	$0.293431\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.30899	0.259842
$276$	0	0
$277$	16.6356	0.999539	0.499770	−	0.866158i	$-0.333418\pi$
$277$	16.6356	0.999539	0.499770	+	0.866158i	$0.333418\pi$
$278$	0	0
$279$	60.1719	3.60240
$280$	0	0
$281$	−10.8530	−0.647437	−0.323719	−	0.946153i	$-0.604933\pi$
$281$	−10.8530	−0.647437	−0.323719	+	0.946153i	$0.604933\pi$
$282$	0	0
$283$	−21.0669	−1.25230	−0.626149	−	0.779703i	$-0.715370\pi$
$283$	−21.0669	−1.25230	−0.626149	+	0.779703i	$0.715370\pi$
$284$	0	0
$285$	2.24761	0.133137
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	25.0287	1.46721
$292$	0	0
$293$	1.62455	0.0949070	0.0474535	−	0.998873i	$-0.484889\pi$
$293$	1.62455	0.0949070	0.0474535	+	0.998873i	$0.484889\pi$
$294$	0	0
$295$	1.14107	0.0664354
$296$	0	0
$297$	−8.01567	−0.465116
$298$	0	0
$299$	2.87073	0.166019
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.94469	−0.284065
$304$	0	0
$305$	0.495905	0.0283954
$306$	0	0
$307$	−9.76306	−0.557207	−0.278604	−	0.960406i	$-0.589872\pi$
$307$	−9.76306	−0.557207	−0.278604	+	0.960406i	$0.589872\pi$
$308$	0	0
$309$	3.16089	0.179817
$310$	0	0
$311$	−25.1977	−1.42883	−0.714415	−	0.699722i	$-0.753307\pi$
$311$	−25.1977	−1.42883	−0.714415	+	0.699722i	$0.753307\pi$
$312$	0	0
$313$	−4.20842	−0.237874	−0.118937	−	0.992902i	$-0.537949\pi$
$313$	−4.20842	−0.237874	−0.118937	+	0.992902i	$0.537949\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.1203	−1.46706	−0.733532	−	0.679655i	$-0.762129\pi$
$317$	−26.1203	−1.46706	−0.733532	+	0.679655i	$0.762129\pi$
$318$	0	0
$319$	8.33997	0.466949
$320$	0	0
$321$	−49.4082	−2.75770
$322$	0	0
$323$	4.73982	0.263731
$324$	0	0
$325$	32.0074	1.77545
$326$	0	0
$327$	−18.1748	−1.00507
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.2337	1.05718	0.528590	−	0.848877i	$-0.322721\pi$
$331$	19.2337	1.05718	0.528590	+	0.848877i	$0.322721\pi$
$332$	0	0
$333$	−38.4616	−2.10768
$334$	0	0
$335$	1.13869	0.0622131
$336$	0	0
$337$	−2.47114	−0.134612	−0.0673058	−	0.997732i	$-0.521440\pi$
$337$	−2.47114	−0.134612	−0.0673058	+	0.997732i	$0.521440\pi$
$338$	0	0
$339$	21.6273	1.17463
$340$	0	0
$341$	−8.58184	−0.464732
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.211598	0.0113921
$346$	0	0
$347$	−13.8854	−0.745407	−0.372703	−	0.927951i	$-0.621569\pi$
$347$	−13.8854	−0.745407	−0.372703	+	0.927951i	$0.621569\pi$
$348$	0	0
$349$	−19.9724	−1.06910	−0.534549	−	0.845137i	$-0.679519\pi$
$349$	−19.9724	−1.06910	−0.534549	+	0.845137i	$0.679519\pi$
$350$	0	0
$351$	−59.5409	−3.17806
$352$	0	0
$353$	−13.7945	−0.734207	−0.367104	−	0.930180i	$-0.619651\pi$
$353$	−13.7945	−0.734207	−0.367104	+	0.930180i	$0.619651\pi$
$354$	0	0
$355$	−0.859812	−0.0456341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.9015	−0.839251	−0.419625	−	0.907697i	$-0.637839\pi$
$359$	−15.9015	−0.839251	−0.419625	+	0.907697i	$0.637839\pi$
$360$	0	0
$361$	3.46592	0.182417
$362$	0	0
$363$	−30.8735	−1.62044
$364$	0	0
$365$	1.55241	0.0812566
$366$	0	0
$367$	−13.9926	−0.730410	−0.365205	−	0.930927i	$-0.619001\pi$
$367$	−13.9926	−0.730410	−0.365205	+	0.930927i	$0.619001\pi$
$368$	0	0
$369$	−43.3111	−2.25469
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.3235	−1.77720	−0.888602	−	0.458680i	$-0.848322\pi$
$373$	−34.3235	−1.77720	−0.888602	+	0.458680i	$0.848322\pi$
$374$	0	0
$375$	4.73021	0.244267
$376$	0	0
$377$	61.9498	3.19058
$378$	0	0
$379$	−12.9188	−0.663593	−0.331797	−	0.943351i	$-0.607655\pi$
$379$	−12.9188	−0.663593	−0.331797	+	0.943351i	$0.607655\pi$
$380$	0	0
$381$	56.5699	2.89816
$382$	0	0
$383$	−6.97490	−0.356401	−0.178200	−	0.983994i	$-0.557028\pi$
$383$	−6.97490	−0.356401	−0.178200	+	0.983994i	$0.557028\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−52.4488	−2.66612
$388$	0	0
$389$	−11.5090	−0.583527	−0.291764	−	0.956490i	$-0.594242\pi$
$389$	−11.5090	−0.583527	−0.291764	+	0.956490i	$0.594242\pi$
$390$	0	0
$391$	0.446225	0.0225665
$392$	0	0
$393$	26.6854	1.34610
$394$	0	0
$395$	−1.60569	−0.0807912
$396$	0	0
$397$	−13.7726	−0.691229	−0.345615	−	0.938377i	$-0.612330\pi$
$397$	−13.7726	−0.691229	−0.345615	+	0.938377i	$0.612330\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.866796	0.0432857	0.0216429	−	0.999766i	$-0.493110\pi$
$401$	0.866796	0.0432857	0.0216429	+	0.999766i	$0.493110\pi$
$402$	0	0
$403$	−63.7464	−3.17543
$404$	0	0
$405$	−1.52062	−0.0755603
$406$	0	0
$407$	5.48547	0.271904
$408$	0	0
$409$	22.5603	1.11554	0.557768	−	0.829997i	$-0.311658\pi$
$409$	22.5603	1.11554	0.557768	+	0.829997i	$0.311658\pi$
$410$	0	0
$411$	−11.0454	−0.544830
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.474041	−0.0232698
$416$	0	0
$417$	12.3465	0.604610
$418$	0	0
$419$	25.4532	1.24347	0.621736	−	0.783227i	$-0.286428\pi$
$419$	25.4532	1.24347	0.621736	+	0.783227i	$0.286428\pi$
$420$	0	0
$421$	15.4477	0.752874	0.376437	−	0.926442i	$-0.377149\pi$
$421$	15.4477	0.752874	0.376437	+	0.926442i	$0.377149\pi$
$422$	0	0
$423$	33.3593	1.62199
$424$	0	0
$425$	4.97522	0.241333
$426$	0	0
$427$	0	0
$428$	0	0
$429$	16.7830	0.810289
$430$	0	0
$431$	3.01545	0.145249	0.0726246	−	0.997359i	$-0.476862\pi$
$431$	3.01545	0.145249	0.0726246	+	0.997359i	$0.476862\pi$
$432$	0	0
$433$	10.6679	0.512666	0.256333	−	0.966589i	$-0.417486\pi$
$433$	10.6679	0.512666	0.256333	+	0.966589i	$0.417486\pi$
$434$	0	0
$435$	4.56625	0.218935
$436$	0	0
$437$	2.11503	0.101175
$438$	0	0
$439$	−9.55550	−0.456059	−0.228030	−	0.973654i	$-0.573228\pi$
$439$	−9.55550	−0.456059	−0.228030	+	0.973654i	$0.573228\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.57659	0.122418	0.0612088	−	0.998125i	$-0.480504\pi$
$443$	2.57659	0.122418	0.0612088	+	0.998125i	$0.480504\pi$
$444$	0	0
$445$	−0.0661086	−0.00313385
$446$	0	0
$447$	−16.8020	−0.794708
$448$	0	0
$449$	−26.9006	−1.26952	−0.634759	−	0.772710i	$-0.718901\pi$
$449$	−26.9006	−1.26952	−0.634759	+	0.772710i	$0.718901\pi$
$450$	0	0
$451$	6.17712	0.290869
$452$	0	0
$453$	−37.5605	−1.76475
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.9665	0.887216	0.443608	−	0.896221i	$-0.353698\pi$
$457$	18.9665	0.887216	0.443608	+	0.896221i	$0.353698\pi$
$458$	0	0
$459$	−9.25501	−0.431987
$460$	0	0
$461$	35.8551	1.66994	0.834969	−	0.550297i	$-0.185486\pi$
$461$	35.8551	1.66994	0.834969	+	0.550297i	$0.185486\pi$
$462$	0	0
$463$	9.04951	0.420566	0.210283	−	0.977641i	$-0.432561\pi$
$463$	9.04951	0.420566	0.210283	+	0.977641i	$0.432561\pi$
$464$	0	0
$465$	−4.69868	−0.217896
$466$	0	0
$467$	−3.55696	−0.164597	−0.0822983	−	0.996608i	$-0.526226\pi$
$467$	−3.55696	−0.164597	−0.0822983	+	0.996608i	$0.526226\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−52.0832	−2.39987
$472$	0	0
$473$	7.48034	0.343947
$474$	0	0
$475$	23.5816	1.08200
$476$	0	0
$477$	82.4666	3.77588
$478$	0	0
$479$	1.08454	0.0495538	0.0247769	−	0.999693i	$-0.492112\pi$
$479$	1.08454	0.0495538	0.0247769	+	0.999693i	$0.492112\pi$
$480$	0	0
$481$	40.7464	1.85787
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.30817	−0.0594008
$486$	0	0
$487$	−21.3850	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$487$	−21.3850	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$488$	0	0
$489$	3.53170	0.159709
$490$	0	0
$491$	−35.0702	−1.58270	−0.791348	−	0.611366i	$-0.790620\pi$
$491$	−35.0702	−1.58270	−0.791348	+	0.611366i	$0.790620\pi$
$492$	0	0
$493$	9.62945	0.433688
$494$	0	0
$495$	0.828003	0.0372160
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.82153	0.260607	0.130304	−	0.991474i	$-0.458405\pi$
$499$	5.82153	0.260607	0.130304	+	0.991474i	$0.458405\pi$
$500$	0	0
$501$	29.1675	1.30311
$502$	0	0
$503$	−2.77794	−0.123862	−0.0619312	−	0.998080i	$-0.519726\pi$
$503$	−2.77794	−0.123862	−0.0619312	+	0.998080i	$0.519726\pi$
$504$	0	0
$505$	0.258443	0.0115006
$506$	0	0
$507$	85.5077	3.79753
$508$	0	0
$509$	−18.0637	−0.800658	−0.400329	−	0.916371i	$-0.631104\pi$
$509$	−18.0637	−0.800658	−0.400329	+	0.916371i	$0.631104\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−43.8671	−1.93678
$514$	0	0
$515$	−0.165210	−0.00728000
$516$	0	0
$517$	−4.75777	−0.209247
$518$	0	0
$519$	43.6978	1.91812
$520$	0	0
$521$	12.5772	0.551016	0.275508	−	0.961299i	$-0.411154\pi$
$521$	12.5772	0.551016	0.275508	+	0.961299i	$0.411154\pi$
$522$	0	0
$523$	−21.8069	−0.953551	−0.476775	−	0.879025i	$-0.658194\pi$
$523$	−21.8069	−0.953551	−0.476775	+	0.879025i	$0.658194\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.90871	−0.431630
$528$	0	0
$529$	−22.8009	−0.991343
$530$	0	0
$531$	−44.0144	−1.91006
$532$	0	0
$533$	45.8840	1.98746
$534$	0	0
$535$	2.58241	0.111647
$536$	0	0
$537$	−49.2944	−2.12721
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.8211	−1.32510	−0.662550	−	0.749017i	$-0.730526\pi$
$541$	−30.8211	−1.32510	−0.662550	+	0.749017i	$0.730526\pi$
$542$	0	0
$543$	3.66470	0.157267
$544$	0	0
$545$	0.949940	0.0406910
$546$	0	0
$547$	27.1369	1.16029	0.580146	−	0.814513i	$-0.302996\pi$
$547$	27.1369	1.16029	0.580146	+	0.814513i	$0.302996\pi$
$548$	0	0
$549$	−19.1286	−0.816388
$550$	0	0
$551$	45.6419	1.94441
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.00337	0.127486
$556$	0	0
$557$	−38.7938	−1.64375	−0.821874	−	0.569669i	$-0.807071\pi$
$557$	−38.7938	−1.64375	−0.821874	+	0.569669i	$0.807071\pi$
$558$	0	0
$559$	55.5644	2.35013
$560$	0	0
$561$	2.60873	0.110141
$562$	0	0
$563$	33.1559	1.39736	0.698678	−	0.715436i	$-0.253772\pi$
$563$	33.1559	1.39736	0.698678	+	0.715436i	$0.253772\pi$
$564$	0	0
$565$	−1.13039	−0.0475557
$566$	0	0
$567$	0	0
$568$	0	0
$569$	43.7216	1.83290	0.916452	−	0.400144i	$-0.131040\pi$
$569$	43.7216	1.83290	0.916452	+	0.400144i	$0.131040\pi$
$570$	0	0
$571$	20.6019	0.862163	0.431082	−	0.902313i	$-0.358132\pi$
$571$	20.6019	0.862163	0.431082	+	0.902313i	$0.358132\pi$
$572$	0	0
$573$	54.2799	2.26757
$574$	0	0
$575$	2.22006	0.0925831
$576$	0	0
$577$	20.8344	0.867349	0.433674	−	0.901070i	$-0.357217\pi$
$577$	20.8344	0.867349	0.433674	+	0.901070i	$0.357217\pi$
$578$	0	0
$579$	54.6058	2.26934
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.7615	−0.487113
$584$	0	0
$585$	6.15046	0.254290
$586$	0	0
$587$	−16.7766	−0.692444	−0.346222	−	0.938153i	$-0.612536\pi$
$587$	−16.7766	−0.692444	−0.346222	+	0.938153i	$0.612536\pi$
$588$	0	0
$589$	−46.9655	−1.93518
$590$	0	0
$591$	−65.1097	−2.67825
$592$	0	0
$593$	35.6166	1.46260	0.731299	−	0.682057i	$-0.238914\pi$
$593$	35.6166	1.46260	0.731299	+	0.682057i	$0.238914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	47.9267	1.96151
$598$	0	0
$599$	−41.6702	−1.70260	−0.851299	−	0.524681i	$-0.824185\pi$
$599$	−41.6702	−1.70260	−0.851299	+	0.524681i	$0.824185\pi$
$600$	0	0
$601$	−4.64938	−0.189652	−0.0948261	−	0.995494i	$-0.530230\pi$
$601$	−4.64938	−0.189652	−0.0948261	+	0.995494i	$0.530230\pi$
$602$	0	0
$603$	−43.9227	−1.78867
$604$	0	0
$605$	1.61366	0.0656045
$606$	0	0
$607$	7.50788	0.304735	0.152368	−	0.988324i	$-0.451310\pi$
$607$	7.50788	0.304735	0.152368	+	0.988324i	$0.451310\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.3410	−1.42974
$612$	0	0
$613$	2.34377	0.0946640	0.0473320	−	0.998879i	$-0.484928\pi$
$613$	2.34377	0.0946640	0.0473320	+	0.998879i	$0.484928\pi$
$614$	0	0
$615$	3.38206	0.136378
$616$	0	0
$617$	1.91036	0.0769083	0.0384542	−	0.999260i	$-0.487757\pi$
$617$	1.91036	0.0769083	0.0384542	+	0.999260i	$0.487757\pi$
$618$	0	0
$619$	−21.5676	−0.866873	−0.433437	−	0.901184i	$-0.642699\pi$
$619$	−21.5676	−0.866873	−0.433437	+	0.901184i	$0.642699\pi$
$620$	0	0
$621$	−4.12981	−0.165724
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.6288	0.985154
$626$	0	0
$627$	12.3649	0.493808
$628$	0	0
$629$	6.33360	0.252537
$630$	0	0
$631$	−22.9980	−0.915535	−0.457767	−	0.889072i	$-0.651351\pi$
$631$	−22.9980	−0.915535	−0.457767	+	0.889072i	$0.651351\pi$
$632$	0	0
$633$	−23.7555	−0.944196
$634$	0	0
$635$	−2.95672	−0.117334
$636$	0	0
$637$	0	0
$638$	0	0
$639$	33.1656	1.31201
$640$	0	0
$641$	−12.5585	−0.496030	−0.248015	−	0.968756i	$-0.579778\pi$
$641$	−12.5585	−0.496030	−0.248015	+	0.968756i	$0.579778\pi$
$642$	0	0
$643$	45.8777	1.80924	0.904620	−	0.426218i	$-0.140154\pi$
$643$	45.8777	1.80924	0.904620	+	0.426218i	$0.140154\pi$
$644$	0	0
$645$	4.09559	0.161264
$646$	0	0
$647$	−25.9405	−1.01983	−0.509913	−	0.860226i	$-0.670322\pi$
$647$	−25.9405	−1.01983	−0.509913	+	0.860226i	$0.670322\pi$
$648$	0	0
$649$	6.27743	0.246411
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.2707	1.14545	0.572725	−	0.819748i	$-0.305886\pi$
$653$	29.2707	1.14545	0.572725	+	0.819748i	$0.305886\pi$
$654$	0	0
$655$	−1.39476	−0.0544977
$656$	0	0
$657$	−59.8811	−2.33619
$658$	0	0
$659$	14.8439	0.578237	0.289118	−	0.957293i	$-0.406638\pi$
$659$	14.8439	0.578237	0.289118	+	0.957293i	$0.406638\pi$
$660$	0	0
$661$	17.7975	0.692243	0.346121	−	0.938190i	$-0.387499\pi$
$661$	17.7975	0.692243	0.346121	+	0.938190i	$0.387499\pi$
$662$	0	0
$663$	19.3778	0.752573
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.29690	0.166376
$668$	0	0
$669$	−3.98349	−0.154011
$670$	0	0
$671$	2.72816	0.105319
$672$	0	0
$673$	1.10707	0.0426743	0.0213371	−	0.999772i	$-0.493208\pi$
$673$	1.10707	0.0426743	0.0213371	+	0.999772i	$0.493208\pi$
$674$	0	0
$675$	−46.0457	−1.77230
$676$	0	0
$677$	7.45141	0.286381	0.143190	−	0.989695i	$-0.454264\pi$
$677$	7.45141	0.286381	0.143190	+	0.989695i	$0.454264\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	22.8339	0.874995
$682$	0	0
$683$	−11.0579	−0.423118	−0.211559	−	0.977365i	$-0.567854\pi$
$683$	−11.0579	−0.423118	−0.211559	+	0.977365i	$0.567854\pi$
$684$	0	0
$685$	0.577308	0.0220578
$686$	0	0
$687$	29.0620	1.10878
$688$	0	0
$689$	−87.3654	−3.32836
$690$	0	0
$691$	−15.1680	−0.577017	−0.288508	−	0.957477i	$-0.593159\pi$
$691$	−15.1680	−0.577017	−0.288508	+	0.957477i	$0.593159\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.645311	−0.0244780
$696$	0	0
$697$	7.13219	0.270151
$698$	0	0
$699$	−19.1387	−0.723893
$700$	0	0
$701$	−43.0469	−1.62586	−0.812930	−	0.582361i	$-0.802129\pi$
$701$	−43.0469	−1.62586	−0.812930	+	0.582361i	$0.802129\pi$
$702$	0	0
$703$	30.0201	1.13223
$704$	0	0
$705$	−2.60495	−0.0981080
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−13.3758	−0.502338	−0.251169	−	0.967943i	$-0.580815\pi$
$709$	−13.3758	−0.502338	−0.251169	+	0.967943i	$0.580815\pi$
$710$	0	0
$711$	61.9366	2.32280
$712$	0	0
$713$	−4.42151	−0.165587
$714$	0	0
$715$	−0.877190	−0.0328051
$716$	0	0
$717$	23.1118	0.863127
$718$	0	0
$719$	−38.0724	−1.41986	−0.709930	−	0.704272i	$-0.751274\pi$
$719$	−38.0724	−1.41986	−0.709930	+	0.704272i	$0.751274\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−58.3835	−2.17130
$724$	0	0
$725$	47.9086	1.77928
$726$	0	0
$727$	−12.8956	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$727$	−12.8956	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$728$	0	0
$729$	−24.9753	−0.925012
$730$	0	0
$731$	8.63691	0.319448
$732$	0	0
$733$	9.83645	0.363318	0.181659	−	0.983362i	$-0.441853\pi$
$733$	9.83645	0.363318	0.181659	+	0.983362i	$0.441853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.26434	0.230750
$738$	0	0
$739$	5.97756	0.219888	0.109944	−	0.993938i	$-0.464933\pi$
$739$	5.97756	0.219888	0.109944	+	0.993938i	$0.464933\pi$
$740$	0	0
$741$	91.8475	3.37410
$742$	0	0
$743$	−0.0446927	−0.00163961	−0.000819807	−	1.00000i	$-0.500261\pi$
$743$	−0.0446927	−0.00163961	−0.000819807		1.00000i	$0.500261\pi$
$744$	0	0
$745$	0.878187	0.0321743
$746$	0	0
$747$	18.2852	0.669022
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.3114	−1.47099	−0.735493	−	0.677533i	$-0.763049\pi$
$751$	−40.3114	−1.47099	−0.735493	+	0.677533i	$0.763049\pi$
$752$	0	0
$753$	51.7855	1.88717
$754$	0	0
$755$	1.96316	0.0714469
$756$	0	0
$757$	13.9993	0.508814	0.254407	−	0.967097i	$-0.418120\pi$
$757$	13.9993	0.508814	0.254407	+	0.967097i	$0.418120\pi$
$758$	0	0
$759$	1.16408	0.0422535
$760$	0	0
$761$	−3.02492	−0.109653	−0.0548267	−	0.998496i	$-0.517461\pi$
$761$	−3.02492	−0.109653	−0.0548267	+	0.998496i	$0.517461\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.956024	0.0345651
$766$	0	0
$767$	46.6291	1.68368
$768$	0	0
$769$	−35.2845	−1.27239	−0.636196	−	0.771528i	$-0.719493\pi$
$769$	−35.2845	−1.27239	−0.636196	+	0.771528i	$0.719493\pi$
$770$	0	0
$771$	28.7094	1.03394
$772$	0	0
$773$	46.7814	1.68261	0.841305	−	0.540560i	$-0.181788\pi$
$773$	46.7814	1.68261	0.841305	+	0.540560i	$0.181788\pi$
$774$	0	0
$775$	−49.2980	−1.77083
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.8053	1.21120
$780$	0	0
$781$	−4.73015	−0.169258
$782$	0	0
$783$	−89.1206	−3.18491
$784$	0	0
$785$	2.72222	0.0971602
$786$	0	0
$787$	−1.24588	−0.0444109	−0.0222054	−	0.999753i	$-0.507069\pi$
$787$	−1.24588	−0.0444109	−0.0222054	+	0.999753i	$0.507069\pi$
$788$	0	0
$789$	31.3447	1.11590
$790$	0	0
$791$	0	0
$792$	0	0
$793$	20.2649	0.719628
$794$	0	0
$795$	−6.43961	−0.228389
$796$	0	0
$797$	−1.58479	−0.0561361	−0.0280681	−	0.999606i	$-0.508936\pi$
$797$	−1.58479	−0.0561361	−0.0280681	+	0.999606i	$0.508936\pi$
$798$	0	0
$799$	−5.49339	−0.194342
$800$	0	0
$801$	2.55001	0.0901003
$802$	0	0
$803$	8.54037	0.301383
$804$	0	0
$805$	0	0
$806$	0	0
$807$	53.9903	1.90055
$808$	0	0
$809$	−25.4483	−0.894713	−0.447357	−	0.894356i	$-0.647634\pi$
$809$	−25.4483	−0.894713	−0.447357	+	0.894356i	$0.647634\pi$
$810$	0	0
$811$	49.7826	1.74810	0.874052	−	0.485832i	$-0.161483\pi$
$811$	49.7826	1.74810	0.874052	+	0.485832i	$0.161483\pi$
$812$	0	0
$813$	59.9340	2.10198
$814$	0	0
$815$	−0.184590	−0.00646592
$816$	0	0
$817$	40.9374	1.43222
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.3653	1.37386	0.686929	−	0.726725i	$-0.258958\pi$
$821$	39.3653	1.37386	0.686929	+	0.726725i	$0.258958\pi$
$822$	0	0
$823$	8.64074	0.301197	0.150599	−	0.988595i	$-0.451880\pi$
$823$	8.64074	0.301197	0.150599	+	0.988595i	$0.451880\pi$
$824$	0	0
$825$	12.9790	0.451871
$826$	0	0
$827$	−14.6703	−0.510136	−0.255068	−	0.966923i	$-0.582098\pi$
$827$	−14.6703	−0.510136	−0.255068	+	0.966923i	$0.582098\pi$
$828$	0	0
$829$	9.87051	0.342817	0.171409	−	0.985200i	$-0.445168\pi$
$829$	9.87051	0.342817	0.171409	+	0.985200i	$0.445168\pi$
$830$	0	0
$831$	50.1079	1.73822
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.52449	−0.0527571
$836$	0	0
$837$	91.7052	3.16979
$838$	0	0
$839$	−3.34017	−0.115316	−0.0576578	−	0.998336i	$-0.518363\pi$
$839$	−3.34017	−0.115316	−0.0576578	+	0.998336i	$0.518363\pi$
$840$	0	0
$841$	63.7262	2.19746
$842$	0	0
$843$	−32.6902	−1.12591
$844$	0	0
$845$	−4.46921	−0.153745
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−63.4553	−2.17778
$850$	0	0
$851$	2.82621	0.0968811
$852$	0	0
$853$	15.8707	0.543403	0.271702	−	0.962382i	$-0.412414\pi$
$853$	15.8707	0.543403	0.271702	+	0.962382i	$0.412414\pi$
$854$	0	0
$855$	4.53138	0.154970
$856$	0	0
$857$	−14.1592	−0.483668	−0.241834	−	0.970318i	$-0.577749\pi$
$857$	−14.1592	−0.483668	−0.241834	+	0.970318i	$0.577749\pi$
$858$	0	0
$859$	28.9475	0.987675	0.493838	−	0.869554i	$-0.335594\pi$
$859$	28.9475	0.987675	0.493838	+	0.869554i	$0.335594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.4456	−0.730017	−0.365008	−	0.931004i	$-0.618934\pi$
$863$	−21.4456	−0.730017	−0.365008	+	0.931004i	$0.618934\pi$
$864$	0	0
$865$	−2.28394	−0.0776564
$866$	0	0
$867$	3.01208	0.102296
$868$	0	0
$869$	−8.83352	−0.299657
$870$	0	0
$871$	46.5319	1.57667
$872$	0	0
$873$	50.4601	1.70782
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.4319	1.02761	0.513807	−	0.857906i	$-0.328235\pi$
$877$	30.4319	1.02761	0.513807	+	0.857906i	$0.328235\pi$
$878$	0	0
$879$	4.89326	0.165046
$880$	0	0
$881$	−52.2916	−1.76175	−0.880874	−	0.473351i	$-0.843044\pi$
$881$	−52.2916	−1.76175	−0.880874	+	0.473351i	$0.843044\pi$
$882$	0	0
$883$	−25.8601	−0.870262	−0.435131	−	0.900367i	$-0.643298\pi$
$883$	−25.8601	−0.870262	−0.435131	+	0.900367i	$0.643298\pi$
$884$	0	0
$885$	3.43698	0.115533
$886$	0	0
$887$	−37.3535	−1.25421	−0.627104	−	0.778935i	$-0.715760\pi$
$887$	−37.3535	−1.25421	−0.627104	+	0.778935i	$0.715760\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−8.36551	−0.280255
$892$	0	0
$893$	−26.0377	−0.871318
$894$	0	0
$895$	2.57646	0.0861215
$896$	0	0
$897$	8.64687	0.288710
$898$	0	0
$899$	−95.4154	−3.18228
$900$	0	0
$901$	−13.5800	−0.452417
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.191542	−0.00636708
$906$	0	0
$907$	−36.9490	−1.22687	−0.613436	−	0.789744i	$-0.710213\pi$
$907$	−36.9490	−1.22687	−0.613436	+	0.789744i	$0.710213\pi$
$908$	0	0
$909$	−9.96895	−0.330649
$910$	0	0
$911$	−55.6472	−1.84367	−0.921837	−	0.387577i	$-0.873312\pi$
$911$	−55.6472	−1.84367	−0.921837	+	0.387577i	$0.873312\pi$
$912$	0	0
$913$	−2.60788	−0.0863081
$914$	0	0
$915$	1.49370	0.0493804
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.95039	−0.196285	−0.0981427	−	0.995172i	$-0.531290\pi$
$919$	−5.95039	−0.196285	−0.0981427	+	0.995172i	$0.531290\pi$
$920$	0	0
$921$	−29.4071	−0.968997
$922$	0	0
$923$	−35.1358	−1.15651
$924$	0	0
$925$	31.5110	1.03608
$926$	0	0
$927$	6.37265	0.209305
$928$	0	0
$929$	−44.7738	−1.46898	−0.734490	−	0.678619i	$-0.762579\pi$
$929$	−44.7738	−1.46898	−0.734490	+	0.678619i	$0.762579\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−75.8975	−2.48477
$934$	0	0
$935$	−0.136350	−0.00445912
$936$	0	0
$937$	48.3356	1.57906	0.789528	−	0.613714i	$-0.210325\pi$
$937$	48.3356	1.57906	0.789528	+	0.613714i	$0.210325\pi$
$938$	0	0
$939$	−12.6761	−0.413669
$940$	0	0
$941$	6.66785	0.217366	0.108683	−	0.994076i	$-0.465337\pi$
$941$	6.66785	0.217366	0.108683	+	0.994076i	$0.465337\pi$
$942$	0	0
$943$	3.18256	0.103638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.1077	−0.360951	−0.180476	−	0.983579i	$-0.557764\pi$
$947$	−11.1077	−0.360951	−0.180476	+	0.983579i	$0.557764\pi$
$948$	0	0
$949$	63.4383	2.05930
$950$	0	0
$951$	−78.6765	−2.55126
$952$	0	0
$953$	−44.7160	−1.44849	−0.724247	−	0.689541i	$-0.757812\pi$
$953$	−44.7160	−1.44849	−0.724247	+	0.689541i	$0.757812\pi$
$954$	0	0
$955$	−2.83703	−0.0918042
$956$	0	0
$957$	25.1207	0.812036
$958$	0	0
$959$	0	0
$960$	0	0
$961$	67.1825	2.16718
$962$	0	0
$963$	−99.6115	−3.20994
$964$	0	0
$965$	−2.85407	−0.0918757
$966$	0	0
$967$	15.1807	0.488179	0.244090	−	0.969753i	$-0.421511\pi$
$967$	15.1807	0.488179	0.244090	+	0.969753i	$0.421511\pi$
$968$	0	0
$969$	14.2767	0.458635
$970$	0	0
$971$	30.7239	0.985976	0.492988	−	0.870036i	$-0.335905\pi$
$971$	30.7239	0.985976	0.492988	+	0.870036i	$0.335905\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	96.4089	3.08756
$976$	0	0
$977$	−10.6385	−0.340356	−0.170178	−	0.985413i	$-0.554434\pi$
$977$	−10.6385	−0.340356	−0.170178	+	0.985413i	$0.554434\pi$
$978$	0	0
$979$	−0.363688	−0.0116235
$980$	0	0
$981$	−36.6422	−1.16989
$982$	0	0
$983$	−34.1564	−1.08942	−0.544710	−	0.838624i	$-0.683360\pi$
$983$	−34.1564	−1.08942	−0.544710	+	0.838624i	$0.683360\pi$
$984$	0	0
$985$	3.40307	0.108431
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.85400	0.122550
$990$	0	0
$991$	−7.51472	−0.238713	−0.119356	−	0.992851i	$-0.538083\pi$
$991$	−7.51472	−0.238713	−0.119356	+	0.992851i	$0.538083\pi$
$992$	0	0
$993$	57.9335	1.83846
$994$	0	0
$995$	−2.50497	−0.0794129
$996$	0	0
$997$	10.6128	0.336110	0.168055	−	0.985778i	$-0.446251\pi$
$997$	10.6128	0.336110	0.168055	+	0.985778i	$0.446251\pi$
$998$	0	0
$999$	−58.6175	−1.85457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.y.1.7		7
7.3	odd	6		952.2.q.e.681.7	yes	14
7.5	odd	6		952.2.q.e.137.7	✓	14
7.6	odd	2		6664.2.a.v.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.e.137.7	✓	14	7.5	odd	6
952.2.q.e.681.7	yes	14	7.3	odd	6
6664.2.a.v.1.1		7	7.6	odd	2
6664.2.a.y.1.7		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+3.01208 q^{3} -0.157432 q^{5} +6.07263 q^{9} -0.866090 q^{11} -6.43337 q^{13} -0.474197 q^{15} -1.00000 q^{17} -4.73982 q^{19} -0.446225 q^{23} -4.97522 q^{25} +9.25501 q^{27} -9.62945 q^{29} +9.90871 q^{31} -2.60873 q^{33} -6.33360 q^{37} -19.3778 q^{39} -7.13219 q^{41} -8.63691 q^{43} -0.956024 q^{45} +5.49339 q^{47} -3.01208 q^{51} +13.5800 q^{53} +0.136350 q^{55} -14.2767 q^{57} -7.24800 q^{59} -3.14997 q^{61} +1.01282 q^{65} -7.23290 q^{67} -1.34406 q^{69} +5.46149 q^{71} -9.86082 q^{73} -14.9857 q^{75} +10.1993 q^{79} +9.65894 q^{81} +3.01109 q^{83} +0.157432 q^{85} -29.0047 q^{87} +0.419919 q^{89} +29.8458 q^{93} +0.746198 q^{95} +8.30943 q^{97} -5.25945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 12 * q^11 - 2 * q^13 - 6 * q^15 - 7 * q^17 + 3 * q^19 - 18 * q^23 + 15 * q^25 + 18 * q^27 + 5 * q^29 + 10 * q^31 - 21 * q^33 + 11 * q^37 - 12 * q^39 - 15 * q^43 - 13 * q^45 - 15 * q^47 - 3 * q^51 + 8 * q^53 + 9 * q^55 - 3 * q^57 - 9 * q^59 + 4 * q^61 - 38 * q^65 - 30 * q^67 + 28 * q^69 - 27 * q^71 + 5 * q^73 + 2 * q^75 + 3 * q^79 - 17 * q^81 + 12 * q^83 + 2 * q^85 - 19 * q^87 - 32 * q^89 - 9 * q^93 - 42 * q^95 + 39 * q^97 - 20 * q^99

Introduction

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