LMFDB - Embedded newform 8624.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1078)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{15} -5.65685 q^{17} -6.00000 q^{23} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +1.41421 q^{31} -1.41421 q^{33} -10.0000 q^{37} -11.3137 q^{41} +8.00000 q^{43} -4.24264 q^{45} -4.24264 q^{47} +8.00000 q^{51} +8.00000 q^{53} +4.24264 q^{55} -1.41421 q^{59} -2.82843 q^{61} -2.00000 q^{67} +8.48528 q^{69} +2.00000 q^{71} +8.48528 q^{73} -18.3848 q^{75} -16.0000 q^{79} -5.00000 q^{81} +16.9706 q^{83} -24.0000 q^{85} -2.82843 q^{87} -7.07107 q^{89} -2.00000 q^{93} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^11 - 12 * q^15 - 12 * q^23 + 26 * q^25 + 4 * q^29 - 20 * q^37 + 16 * q^43 + 16 * q^51 + 16 * q^53 - 4 * q^67 + 4 * q^71 - 32 * q^79 - 10 * q^81 - 48 * q^85 - 4 * q^93 - 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	4.24264	1.89737	0.948683	−	0.316228i	$-0.102416\pi$
$5$	4.24264	1.89737	0.948683	+	0.316228i	$0.102416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−6.00000	−1.54919
$16$	0	0
$17$	−5.65685	−1.37199	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685	−1.37199	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	13.0000	2.60000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.3137	−1.76690	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	−11.3137	−1.76690	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−4.24264	−0.632456
$46$	0	0
$47$	−4.24264	−0.618853	−0.309426	−	0.950923i	$-0.600137\pi$
$47$	−4.24264	−0.618853	−0.309426	+	0.950923i	$0.600137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	−2.82843	−0.362143	−0.181071	−	0.983470i	$-0.557957\pi$
$61$	−2.82843	−0.362143	−0.181071	+	0.983470i	$0.557957\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	8.48528	1.02151
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	−18.3848	−2.12289
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	16.9706	1.86276	0.931381	−	0.364047i	$-0.118605\pi$
$83$	16.9706	1.86276	0.931381	+	0.364047i	$0.118605\pi$
$84$	0	0
$85$	−24.0000	−2.60317
$86$	0	0
$87$	−2.82843	−0.303239
$88$	0	0
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	5.65685	0.562878	0.281439	−	0.959579i	$-0.409188\pi$
$101$	5.65685	0.562878	0.281439	+	0.959579i	$0.409188\pi$
$102$	0	0
$103$	−18.3848	−1.81151	−0.905753	−	0.423806i	$-0.860694\pi$
$103$	−18.3848	−1.81151	−0.905753	+	0.423806i	$0.860694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	14.1421	1.34231
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−25.4558	−2.37377
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.0000	1.44267
$124$	0	0
$125$	33.9411	3.03579
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	19.7990	1.72985	0.864923	−	0.501905i	$-0.167367\pi$
$131$	19.7990	1.72985	0.864923	+	0.501905i	$0.167367\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	24.0000	2.06559
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.48528	0.704664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	4.24264	0.338600	0.169300	−	0.985565i	$-0.445849\pi$
$157$	4.24264	0.338600	0.169300	+	0.985565i	$0.445849\pi$
$158$	0	0
$159$	−11.3137	−0.897235
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−7.07107	−0.525588	−0.262794	−	0.964852i	$-0.584644\pi$
$181$	−7.07107	−0.525588	−0.262794	+	0.964852i	$0.584644\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−42.4264	−3.11925
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	1.41421	0.100251	0.0501255	−	0.998743i	$-0.484038\pi$
$199$	1.41421	0.100251	0.0501255	+	0.998743i	$0.484038\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−48.0000	−3.35247
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	−2.82843	−0.193801
$214$	0	0
$215$	33.9411	2.31477
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.0000	−0.810885
$220$	0	0
$221$	0	0
$222$	0	0
$223$	21.2132	1.42054	0.710271	−	0.703929i	$-0.248573\pi$
$223$	21.2132	1.42054	0.710271	+	0.703929i	$0.248573\pi$
$224$	0	0
$225$	−13.0000	−0.866667
$226$	0	0
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	9.89949	0.654177	0.327089	−	0.944994i	$-0.393932\pi$
$229$	9.89949	0.654177	0.327089	+	0.944994i	$0.393932\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	0	0
$237$	22.6274	1.46981
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−24.0000	−1.52094
$250$	0	0
$251$	−18.3848	−1.16044	−0.580218	−	0.814461i	$-0.697033\pi$
$251$	−18.3848	−1.16044	−0.580218	+	0.814461i	$0.697033\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	33.9411	2.12548
$256$	0	0
$257$	1.41421	0.0882162	0.0441081	−	0.999027i	$-0.485955\pi$
$257$	1.41421	0.0882162	0.0441081	+	0.999027i	$0.485955\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	33.9411	2.08499
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	18.3848	1.12094	0.560470	−	0.828175i	$-0.310621\pi$
$269$	18.3848	1.12094	0.560470	+	0.828175i	$0.310621\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.0000	0.783929
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	19.7990	1.17693	0.588464	−	0.808523i	$-0.299733\pi$
$283$	19.7990	1.17693	0.588464	+	0.808523i	$0.299733\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	25.4558	1.45284	0.726421	−	0.687250i	$-0.241182\pi$
$307$	25.4558	1.45284	0.726421	+	0.687250i	$0.241182\pi$
$308$	0	0
$309$	26.0000	1.47909
$310$	0	0
$311$	−18.3848	−1.04251	−0.521253	−	0.853402i	$-0.674535\pi$
$311$	−18.3848	−1.04251	−0.521253	+	0.853402i	$0.674535\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	22.6274	1.26294
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.82843	0.156412
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	−8.48528	−0.463600
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	2.82843	0.153619
$340$	0	0
$341$	1.41421	0.0765840
$342$	0	0
$343$	0	0
$344$	0	0
$345$	36.0000	1.93817
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	14.1421	0.757011	0.378506	−	0.925599i	$-0.376438\pi$
$349$	14.1421	0.757011	0.378506	+	0.925599i	$0.376438\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	0	0
$355$	8.48528	0.450352
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.41421	−0.0742270
$364$	0	0
$365$	36.0000	1.88433
$366$	0	0
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	0	0
$369$	11.3137	0.588968
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	−48.0000	−2.47871
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	0	0
$381$	22.6274	1.15924
$382$	0	0
$383$	−15.5563	−0.794892	−0.397446	−	0.917625i	$-0.630103\pi$
$383$	−15.5563	−0.794892	−0.397446	+	0.917625i	$0.630103\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	33.9411	1.71648
$392$	0	0
$393$	−28.0000	−1.41241
$394$	0	0
$395$	−67.8823	−3.41553
$396$	0	0
$397$	12.7279	0.638796	0.319398	−	0.947621i	$-0.396519\pi$
$397$	12.7279	0.638796	0.319398	+	0.947621i	$0.396519\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−21.2132	−1.05409
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	−2.82843	−0.139857	−0.0699284	−	0.997552i	$-0.522277\pi$
$409$	−2.82843	−0.139857	−0.0699284	+	0.997552i	$0.522277\pi$
$410$	0	0
$411$	25.4558	1.25564
$412$	0	0
$413$	0	0
$414$	0	0
$415$	72.0000	3.53434
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	4.24264	0.206284
$424$	0	0
$425$	−73.5391	−3.56717
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−12.7279	−0.611665	−0.305832	−	0.952085i	$-0.598935\pi$
$433$	−12.7279	−0.611665	−0.305832	+	0.952085i	$0.598935\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	0	0
$437$	0	0
$438$	0	0
$439$	25.4558	1.21494	0.607471	−	0.794342i	$-0.292184\pi$
$439$	25.4558	1.21494	0.607471	+	0.794342i	$0.292184\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	−30.0000	−1.42214
$446$	0	0
$447$	14.1421	0.668900
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	0	0
$453$	5.65685	0.265782
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	−32.0000	−1.49363
$460$	0	0
$461$	2.82843	0.131733	0.0658665	−	0.997828i	$-0.479019\pi$
$461$	2.82843	0.131733	0.0658665	+	0.997828i	$0.479019\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	0	0
$465$	−8.48528	−0.393496
$466$	0	0
$467$	4.24264	0.196326	0.0981630	−	0.995170i	$-0.468703\pi$
$467$	4.24264	0.196326	0.0981630	+	0.995170i	$0.468703\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.00000	−0.366295
$478$	0	0
$479$	−2.82843	−0.129234	−0.0646171	−	0.997910i	$-0.520583\pi$
$479$	−2.82843	−0.129234	−0.0646171	+	0.997910i	$0.520583\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−42.0000	−1.90712
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−14.1421	−0.639529
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−11.3137	−0.509544
$494$	0	0
$495$	−4.24264	−0.190693
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	−31.1127	−1.38725	−0.693623	−	0.720338i	$-0.743987\pi$
$503$	−31.1127	−1.38725	−0.693623	+	0.720338i	$0.743987\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	18.3848	0.816497
$508$	0	0
$509$	12.7279	0.564155	0.282078	−	0.959392i	$-0.408976\pi$
$509$	12.7279	0.564155	0.282078	+	0.959392i	$0.408976\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−78.0000	−3.43709
$516$	0	0
$517$	−4.24264	−0.186591
$518$	0	0
$519$	16.0000	0.702322
$520$	0	0
$521$	1.41421	0.0619578	0.0309789	−	0.999520i	$-0.490138\pi$
$521$	1.41421	0.0619578	0.0309789	+	0.999520i	$0.490138\pi$
$522$	0	0
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−67.8823	−2.93481
$536$	0	0
$537$	16.9706	0.732334
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−8.48528	−0.363470
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	0	0
$549$	2.82843	0.120714
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	60.0000	2.54686
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−22.6274	−0.953632	−0.476816	−	0.879003i	$-0.658209\pi$
$563$	−22.6274	−0.953632	−0.476816	+	0.879003i	$0.658209\pi$
$564$	0	0
$565$	−8.48528	−0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	22.6274	0.945274
$574$	0	0
$575$	−78.0000	−3.25282
$576$	0	0
$577$	−7.07107	−0.294372	−0.147186	−	0.989109i	$-0.547022\pi$
$577$	−7.07107	−0.294372	−0.147186	+	0.989109i	$0.547022\pi$
$578$	0	0
$579$	8.48528	0.352636
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.8701	1.10905	0.554523	−	0.832168i	$-0.312901\pi$
$587$	26.8701	1.10905	0.554523	+	0.832168i	$0.312901\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−31.1127	−1.27981
$592$	0	0
$593$	42.4264	1.74224	0.871122	−	0.491067i	$-0.163393\pi$
$593$	42.4264	1.74224	0.871122	+	0.491067i	$0.163393\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	5.65685	0.230748	0.115374	−	0.993322i	$-0.463193\pi$
$601$	5.65685	0.230748	0.115374	+	0.993322i	$0.463193\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	4.24264	0.172488
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	67.8823	2.73728
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	0	0
$619$	−9.89949	−0.397894	−0.198947	−	0.980010i	$-0.563752\pi$
$619$	−9.89949	−0.397894	−0.198947	+	0.980010i	$0.563752\pi$
$620$	0	0
$621$	−33.9411	−1.36201
$622$	0	0
$623$	0	0
$624$	0	0
$625$	79.0000	3.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	56.5685	2.25554
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	0	0
$633$	11.3137	0.449680
$634$	0	0
$635$	−67.8823	−2.69382
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	38.1838	1.50582	0.752910	−	0.658123i	$-0.228649\pi$
$643$	38.1838	1.50582	0.752910	+	0.658123i	$0.228649\pi$
$644$	0	0
$645$	−48.0000	−1.89000
$646$	0	0
$647$	15.5563	0.611583	0.305792	−	0.952098i	$-0.401079\pi$
$647$	15.5563	0.611583	0.305792	+	0.952098i	$0.401079\pi$
$648$	0	0
$649$	−1.41421	−0.0555127
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	0	0
$655$	84.0000	3.28215
$656$	0	0
$657$	−8.48528	−0.331042
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	12.7279	0.495059	0.247529	−	0.968880i	$-0.420381\pi$
$661$	12.7279	0.495059	0.247529	+	0.968880i	$0.420381\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−30.0000	−1.15987
$670$	0	0
$671$	−2.82843	−0.109190
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	73.5391	2.83052
$676$	0	0
$677$	−39.5980	−1.52187	−0.760937	−	0.648826i	$-0.775260\pi$
$677$	−39.5980	−1.52187	−0.760937	+	0.648826i	$0.775260\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	−76.3675	−2.91785
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−15.5563	−0.591791	−0.295896	−	0.955220i	$-0.595618\pi$
$691$	−15.5563	−0.591791	−0.295896	+	0.955220i	$0.595618\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−48.0000	−1.82074
$696$	0	0
$697$	64.0000	2.42417
$698$	0	0
$699$	−19.7990	−0.748867
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	25.4558	0.958723
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−8.48528	−0.317776
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.9706	−0.633777
$718$	0	0
$719$	−18.3848	−0.685636	−0.342818	−	0.939402i	$-0.611381\pi$
$719$	−18.3848	−0.685636	−0.342818	+	0.939402i	$0.611381\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26.0000	0.965616
$726$	0	0
$727$	−12.7279	−0.472052	−0.236026	−	0.971747i	$-0.575845\pi$
$727$	−12.7279	−0.472052	−0.236026	+	0.971747i	$0.575845\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−45.2548	−1.67381
$732$	0	0
$733$	14.1421	0.522352	0.261176	−	0.965291i	$-0.415890\pi$
$733$	14.1421	0.522352	0.261176	+	0.965291i	$0.415890\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	−42.4264	−1.55438
$746$	0	0
$747$	−16.9706	−0.620920
$748$	0	0
$749$	0	0
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	0	0
$755$	−16.9706	−0.617622
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	0	0
$759$	8.48528	0.307996
$760$	0	0
$761$	−42.4264	−1.53796	−0.768978	−	0.639275i	$-0.779234\pi$
$761$	−42.4264	−1.53796	−0.768978	+	0.639275i	$0.779234\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	0	0
$768$	0	0
$769$	19.7990	0.713970	0.356985	−	0.934110i	$-0.383805\pi$
$769$	19.7990	0.713970	0.356985	+	0.934110i	$0.383805\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	1.41421	0.0508657	0.0254329	−	0.999677i	$-0.491904\pi$
$773$	1.41421	0.0508657	0.0254329	+	0.999677i	$0.491904\pi$
$774$	0	0
$775$	18.3848	0.660401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	11.3137	0.404319
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	5.65685	0.201645	0.100823	−	0.994904i	$-0.467853\pi$
$787$	5.65685	0.201645	0.100823	+	0.994904i	$0.467853\pi$
$788$	0	0
$789$	−11.3137	−0.402779
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−48.0000	−1.70238
$796$	0	0
$797$	7.07107	0.250470	0.125235	−	0.992127i	$-0.460032\pi$
$797$	7.07107	0.250470	0.125235	+	0.992127i	$0.460032\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	7.07107	0.249844
$802$	0	0
$803$	8.48528	0.299439
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.0000	−0.915243
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	−36.7696	−1.29115	−0.645577	−	0.763695i	$-0.723383\pi$
$811$	−36.7696	−1.29115	−0.645577	+	0.763695i	$0.723383\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	0	0
$815$	42.4264	1.48613
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	−18.3848	−0.640076
$826$	0	0
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	0	0
$829$	−4.24264	−0.147353	−0.0736765	−	0.997282i	$-0.523473\pi$
$829$	−4.24264	−0.147353	−0.0736765	+	0.997282i	$0.523473\pi$
$830$	0	0
$831$	2.82843	0.0981170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	−1.41421	−0.0488241	−0.0244120	−	0.999702i	$-0.507771\pi$
$839$	−1.41421	−0.0488241	−0.0244120	+	0.999702i	$0.507771\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	19.7990	0.681913
$844$	0	0
$845$	−55.1543	−1.89737
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	28.2843	0.968435	0.484218	−	0.874948i	$-0.339104\pi$
$853$	28.2843	0.968435	0.484218	+	0.874948i	$0.339104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.4558	−0.869555	−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558	−0.869555	−0.434778	+	0.900538i	$0.643173\pi$
$858$	0	0
$859$	−12.7279	−0.434271	−0.217136	−	0.976141i	$-0.569671\pi$
$859$	−12.7279	−0.434271	−0.217136	+	0.976141i	$0.569671\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	−21.2132	−0.720438
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	0	0
$872$	0	0
$873$	9.89949	0.335047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	−29.6985	−1.00057	−0.500284	−	0.865862i	$-0.666771\pi$
$881$	−29.6985	−1.00057	−0.500284	+	0.865862i	$0.666771\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	8.48528	0.285230
$886$	0	0
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.00000	−0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−50.9117	−1.70179
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.82843	0.0943333
$900$	0	0
$901$	−45.2548	−1.50766
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.0000	−0.997234
$906$	0	0
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	0	0
$909$	−5.65685	−0.187626
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	16.9706	0.561644
$914$	0	0
$915$	16.9706	0.561029
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−36.0000	−1.18624
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−130.000	−4.27437
$926$	0	0
$927$	18.3848	0.603835
$928$	0	0
$929$	−32.5269	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$929$	−32.5269	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	26.0000	0.851202
$934$	0	0
$935$	−24.0000	−0.784884
$936$	0	0
$937$	25.4558	0.831606	0.415803	−	0.909455i	$-0.363501\pi$
$937$	25.4558	0.831606	0.415803	+	0.909455i	$0.363501\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	31.1127	1.01424	0.507122	−	0.861874i	$-0.330709\pi$
$941$	31.1127	1.01424	0.507122	+	0.861874i	$0.330709\pi$
$942$	0	0
$943$	67.8823	2.21055
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−42.4264	−1.37577
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	−67.8823	−2.19662
$956$	0	0
$957$	−2.82843	−0.0914301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	−25.4558	−0.819453
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52.3259	−1.67922	−0.839609	−	0.543191i	$-0.817216\pi$
$971$	−52.3259	−1.67922	−0.839609	+	0.543191i	$0.817216\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	0	0
$979$	−7.07107	−0.225992
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−1.41421	−0.0451064	−0.0225532	−	0.999746i	$-0.507180\pi$
$983$	−1.41421	−0.0451064	−0.0225532	+	0.999746i	$0.507180\pi$
$984$	0	0
$985$	93.3381	2.97400
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	46.0000	1.46124	0.730619	−	0.682785i	$-0.239232\pi$
$991$	46.0000	1.46124	0.730619	+	0.682785i	$0.239232\pi$
$992$	0	0
$993$	28.2843	0.897574
$994$	0	0
$995$	6.00000	0.190213
$996$	0	0
$997$	16.9706	0.537463	0.268732	−	0.963215i	$-0.413396\pi$
$997$	16.9706	0.537463	0.268732	+	0.963215i	$0.413396\pi$
$998$	0	0
$999$	−56.5685	−1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bs.1.1		2
4.3	odd	2		1078.2.a.u.1.2	yes	2
7.6	odd	2	inner	8624.2.a.bs.1.2		2
12.11	even	2		9702.2.a.cp.1.1		2
28.3	even	6		1078.2.e.p.177.2		4
28.11	odd	6		1078.2.e.p.177.1		4
28.19	even	6		1078.2.e.p.67.2		4
28.23	odd	6		1078.2.e.p.67.1		4
28.27	even	2		1078.2.a.u.1.1	✓	2
84.83	odd	2		9702.2.a.cp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.a.u.1.1	✓	2	28.27	even	2
1078.2.a.u.1.2	yes	2	4.3	odd	2
1078.2.e.p.67.1		4	28.23	odd	6
1078.2.e.p.67.2		4	28.19	even	6
1078.2.e.p.177.1		4	28.11	odd	6
1078.2.e.p.177.2		4	28.3	even	6
8624.2.a.bs.1.1		2	1.1	even	1	trivial
8624.2.a.bs.1.2		2	7.6	odd	2	inner
9702.2.a.cp.1.1		2	12.11	even	2
9702.2.a.cp.1.2		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +4.24264 q^{5} -1.00000 q^{9} +1.00000 q^{11} -6.00000 q^{15} -5.65685 q^{17} -6.00000 q^{23} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +1.41421 q^{31} -1.41421 q^{33} -10.0000 q^{37} -11.3137 q^{41} +8.00000 q^{43} -4.24264 q^{45} -4.24264 q^{47} +8.00000 q^{51} +8.00000 q^{53} +4.24264 q^{55} -1.41421 q^{59} -2.82843 q^{61} -2.00000 q^{67} +8.48528 q^{69} +2.00000 q^{71} +8.48528 q^{73} -18.3848 q^{75} -16.0000 q^{79} -5.00000 q^{81} +16.9706 q^{83} -24.0000 q^{85} -2.82843 q^{87} -7.07107 q^{89} -2.00000 q^{93} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 2 q^{11} - 12 q^{15} - 12 q^{23} + 26 q^{25} + 4 q^{29} - 20 q^{37} + 16 q^{43} + 16 q^{51} + 16 q^{53} - 4 q^{67} + 4 q^{71} - 32 q^{79} - 10 q^{81} - 48 q^{85} - 4 q^{93} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^11 - 12 * q^15 - 12 * q^23 + 26 * q^25 + 4 * q^29 - 20 * q^37 + 16 * q^43 + 16 * q^51 + 16 * q^53 - 4 * q^67 + 4 * q^71 - 32 * q^79 - 10 * q^81 - 48 * q^85 - 4 * q^93 - 2 * q^99

Introduction

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