LMFDB - Embedded newform 8624.2.a.by.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:	$0$
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:	$ 2 $
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)

$f(q)$	$=$	$q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{3} +5.00000 q^{9} +1.00000 q^{11} -2.82843 q^{13} +2.82843 q^{17} -5.65685 q^{19} -8.00000 q^{23} -5.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -8.48528 q^{31} -2.82843 q^{33} +2.00000 q^{37} +8.00000 q^{39} +2.82843 q^{41} +4.00000 q^{43} -2.82843 q^{47} -8.00000 q^{51} +14.0000 q^{53} +16.0000 q^{57} +8.48528 q^{59} -8.48528 q^{61} -4.00000 q^{67} +22.6274 q^{69} -14.1421 q^{73} +14.1421 q^{75} +16.0000 q^{79} +1.00000 q^{81} +16.9706 q^{83} -5.65685 q^{87} -11.3137 q^{89} +24.0000 q^{93} -16.9706 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{9}+O(q^{10}) $ 2 * q + 10 * q^9	$ 2 q + 10 q^{9} + 2 q^{11} - 16 q^{23} - 10 q^{25} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 8 q^{43} - 16 q^{51} + 28 q^{53} + 32 q^{57} - 8 q^{67} + 32 q^{79} + 2 q^{81} + 48 q^{93} + 10 q^{99}+O(q^{100}) $ 2 * q + 10 * q^9 + 2 * q^11 - 16 * q^23 - 10 * q^25 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 8 * q^43 - 16 * q^51 + 28 * q^53 + 32 * q^57 - 8 * q^67 + 32 * q^79 + 2 * q^81 + 48 * q^93 + 10 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	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
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.0000	2.11925
$58$	0	0
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	22.6274	2.72402
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.1421	−1.65521	−0.827606	−	0.561310i	$-0.810298\pi$
$73$	−14.1421	−1.65521	−0.827606	+	0.561310i	$0.810298\pi$
$74$	0	0
$75$	14.1421	1.63299
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$85$	0	0
$86$	0	0
$87$	−5.65685	−0.606478
$88$	0	0
$89$	−11.3137	−1.19925	−0.599625	−	0.800281i	$-0.704684\pi$
$89$	−11.3137	−1.19925	−0.599625	+	0.800281i	$0.704684\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.0000	2.48868
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.9706	−1.72310	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−16.9706	−1.72310	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−2.82843	−0.281439	−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843	−0.281439	−0.140720	+	0.990050i	$0.544942\pi$
$102$	0	0
$103$	−14.1421	−1.39347	−0.696733	−	0.717331i	$-0.745364\pi$
$103$	−14.1421	−1.39347	−0.696733	+	0.717331i	$0.745364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−14.1421	−1.30744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−2.82843	−0.236525
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	14.1421	1.14332
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.3137	−0.902932	−0.451466	−	0.892288i	$-0.649099\pi$
$157$	−11.3137	−0.902932	−0.451466	+	0.892288i	$0.649099\pi$
$158$	0	0
$159$	−39.5980	−3.14032
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−28.2843	−2.16295
$172$	0	0
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.0000	−1.80395
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	5.65685	0.420471	0.210235	−	0.977651i	$-0.432577\pi$
$181$	5.65685	0.420471	0.210235	+	0.977651i	$0.432577\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.82843	0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	19.7990	1.40351	0.701757	−	0.712417i	$-0.252399\pi$
$199$	19.7990	1.40351	0.701757	+	0.712417i	$0.252399\pi$
$200$	0	0
$201$	11.3137	0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−40.0000	−2.78019
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	40.0000	2.70295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	0	0
$225$	−25.0000	−1.66667
$226$	0	0
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	0	0
$229$	−5.65685	−0.373815	−0.186908	−	0.982377i	$-0.559847\pi$
$229$	−5.65685	−0.373815	−0.186908	+	0.982377i	$0.559847\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−45.2548	−2.93962
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	14.1421	0.910975	0.455488	−	0.890242i	$-0.349465\pi$
$241$	14.1421	0.910975	0.455488	+	0.890242i	$0.349465\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	−48.0000	−3.04188
$250$	0	0
$251$	14.1421	0.892644	0.446322	−	0.894873i	$-0.352734\pi$
$251$	14.1421	0.892644	0.446322	+	0.894873i	$0.352734\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.3137	−0.705730	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	−11.3137	−0.705730	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	32.0000	1.95837
$268$	0	0
$269$	−5.65685	−0.344904	−0.172452	−	0.985018i	$-0.555169\pi$
$269$	−5.65685	−0.344904	−0.172452	+	0.985018i	$0.555169\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−42.4264	−2.54000
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−11.3137	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$283$	−11.3137	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	48.0000	2.81381
$292$	0	0
$293$	14.1421	0.826192	0.413096	−	0.910687i	$-0.364447\pi$
$293$	14.1421	0.826192	0.413096	+	0.910687i	$0.364447\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	22.6274	1.30858
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.6274	1.29141	0.645707	−	0.763585i	$-0.276563\pi$
$307$	22.6274	1.29141	0.645707	+	0.763585i	$0.276563\pi$
$308$	0	0
$309$	40.0000	2.27552
$310$	0	0
$311$	31.1127	1.76424	0.882120	−	0.471025i	$-0.156116\pi$
$311$	31.1127	1.76424	0.882120	+	0.471025i	$0.156116\pi$
$312$	0	0
$313$	−11.3137	−0.639489	−0.319744	−	0.947504i	$-0.603597\pi$
$313$	−11.3137	−0.639489	−0.319744	+	0.947504i	$0.603597\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	33.9411	1.89441
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	14.1421	0.784465
$326$	0	0
$327$	−5.65685	−0.312825
$328$	0	0
$329$	0	0
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	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	$0.375472\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	25.4558	1.36262	0.681310	−	0.731995i	$-0.261411\pi$
$349$	25.4558	1.36262	0.681310	+	0.731995i	$0.261411\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	−11.3137	−0.602168	−0.301084	−	0.953598i	$-0.597348\pi$
$353$	−11.3137	−0.602168	−0.301084	+	0.953598i	$0.597348\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−2.82843	−0.148454
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.48528	0.442928	0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528	0.442928	0.221464	+	0.975169i	$0.428916\pi$
$368$	0	0
$369$	14.1421	0.736210
$370$	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
$375$	0	0
$376$	0	0
$377$	−5.65685	−0.291343
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.82843	0.144526	0.0722629	−	0.997386i	$-0.476978\pi$
$383$	2.82843	0.144526	0.0722629	+	0.997386i	$0.476978\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000	1.01666
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	32.0000	1.61419
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.6274	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$397$	−22.6274	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	8.48528	0.419570	0.209785	−	0.977748i	$-0.432724\pi$
$409$	8.48528	0.419570	0.209785	+	0.977748i	$0.432724\pi$
$410$	0	0
$411$	5.65685	0.279032
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−48.0000	−2.35057
$418$	0	0
$419$	−25.4558	−1.24360	−0.621800	−	0.783176i	$-0.713598\pi$
$419$	−25.4558	−1.24360	−0.621800	+	0.783176i	$0.713598\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	−14.1421	−0.687614
$424$	0	0
$425$	−14.1421	−0.685994
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−16.9706	−0.815553	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	−16.9706	−0.815553	−0.407777	+	0.913082i	$0.633696\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.2548	2.16483
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.9706	−0.802680
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	2.82843	0.133185
$452$	0	0
$453$	45.2548	2.12626
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	−16.0000	−0.746816
$460$	0	0
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.4558	1.17796	0.588978	−	0.808149i	$-0.299530\pi$
$467$	25.4558	1.17796	0.588978	+	0.808149i	$0.299530\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	32.0000	1.47448
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	28.2843	1.29777
$476$	0	0
$477$	70.0000	3.20508
$478$	0	0
$479$	16.9706	0.775405	0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706	0.775405	0.387702	+	0.921785i	$0.373269\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−11.3137	−0.511624
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−32.0000	−1.42965
$502$	0	0
$503$	−5.65685	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.1421	0.628074
$508$	0	0
$509$	−16.9706	−0.752207	−0.376103	−	0.926578i	$-0.622736\pi$
$509$	−16.9706	−0.752207	−0.376103	+	0.926578i	$0.622736\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	32.0000	1.41283
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.82843	−0.124394
$518$	0	0
$519$	−56.0000	−2.45813
$520$	0	0
$521$	39.5980	1.73482	0.867409	−	0.497595i	$-0.165783\pi$
$521$	39.5980	1.73482	0.867409	+	0.497595i	$0.165783\pi$
$522$	0	0
$523$	−33.9411	−1.48414	−0.742071	−	0.670321i	$-0.766156\pi$
$523$	−33.9411	−1.48414	−0.742071	+	0.670321i	$0.766156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	42.4264	1.84115
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	0	0
$536$	0	0
$537$	56.5685	2.44111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−16.0000	−0.686626
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−42.4264	−1.81071
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−22.6274	−0.945274
$574$	0	0
$575$	40.0000	1.66812
$576$	0	0
$577$	39.5980	1.64849	0.824243	−	0.566237i	$-0.191601\pi$
$577$	39.5980	1.64849	0.824243	+	0.566237i	$0.191601\pi$
$578$	0	0
$579$	39.5980	1.64564
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.0000	0.579821
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1421	0.583708	0.291854	−	0.956463i	$-0.405728\pi$
$587$	14.1421	0.583708	0.291854	+	0.956463i	$0.405728\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	28.2843	1.16346
$592$	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
$597$	−56.0000	−2.29193
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	25.4558	1.03837	0.519183	−	0.854663i	$-0.326236\pi$
$601$	25.4558	1.03837	0.519183	+	0.854663i	$0.326236\pi$
$602$	0	0
$603$	−20.0000	−0.814463
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	2.82843	0.113684	0.0568420	−	0.998383i	$-0.481897\pi$
$619$	2.82843	0.113684	0.0568420	+	0.998383i	$0.481897\pi$
$620$	0	0
$621$	45.2548	1.81601
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	5.65685	0.225554
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	11.3137	0.449680
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	25.4558	1.00388	0.501940	−	0.864902i	$-0.332620\pi$
$643$	25.4558	1.00388	0.501940	+	0.864902i	$0.332620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	0	0
$649$	8.48528	0.333076
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−70.7107	−2.75869
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	22.6274	0.878776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	40.0000	1.54649
$670$	0	0
$671$	−8.48528	−0.327571
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	28.2843	1.08866
$676$	0	0
$677$	−2.82843	−0.108705	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843	−0.108705	−0.0543526	+	0.998522i	$0.517310\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	32.0000	1.22624
$682$	0	0
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.0000	0.610438
$688$	0	0
$689$	−39.5980	−1.50856
$690$	0	0
$691$	8.48528	0.322795	0.161398	−	0.986889i	$-0.448400\pi$
$691$	8.48528	0.322795	0.161398	+	0.986889i	$0.448400\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	−62.2254	−2.35358
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	80.0000	3.00023
$712$	0	0
$713$	67.8823	2.54221
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−45.2548	−1.69007
$718$	0	0
$719$	−19.7990	−0.738378	−0.369189	−	0.929354i	$-0.620364\pi$
$719$	−19.7990	−0.738378	−0.369189	+	0.929354i	$0.620364\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−40.0000	−1.48762
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−19.7990	−0.734304	−0.367152	−	0.930161i	$-0.619667\pi$
$727$	−19.7990	−0.734304	−0.367152	+	0.930161i	$0.619667\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	11.3137	0.418453
$732$	0	0
$733$	48.0833	1.77600	0.887998	−	0.459848i	$-0.152096\pi$
$733$	48.0833	1.77600	0.887998	+	0.459848i	$0.152096\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−45.2548	−1.66248
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	84.8528	3.10460
$748$	0	0
$749$	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	$0.301546\pi$
$752$	0	0
$753$	−40.0000	−1.45768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	22.6274	0.821323
$760$	0	0
$761$	8.48528	0.307591	0.153796	−	0.988103i	$-0.450850\pi$
$761$	8.48528	0.307591	0.153796	+	0.988103i	$0.450850\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−2.82843	−0.101996	−0.0509978	−	0.998699i	$-0.516240\pi$
$769$	−2.82843	−0.101996	−0.0509978	+	0.998699i	$0.516240\pi$
$770$	0	0
$771$	32.0000	1.15245
$772$	0	0
$773$	28.2843	1.01731	0.508657	−	0.860969i	$-0.330142\pi$
$773$	28.2843	1.01731	0.508657	+	0.860969i	$0.330142\pi$
$774$	0	0
$775$	42.4264	1.52400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.5980	1.41152	0.705758	−	0.708453i	$-0.250607\pi$
$787$	39.5980	1.41152	0.705758	+	0.708453i	$0.250607\pi$
$788$	0	0
$789$	−45.2548	−1.61111
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.3137	−0.400752	−0.200376	−	0.979719i	$-0.564216\pi$
$797$	−11.3137	−0.400752	−0.200376	+	0.979719i	$0.564216\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−56.5685	−1.99875
$802$	0	0
$803$	−14.1421	−0.499065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000	0.563227
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	22.6274	0.794556	0.397278	−	0.917698i	$-0.369955\pi$
$811$	22.6274	0.794556	0.397278	+	0.917698i	$0.369955\pi$
$812$	0	0
$813$	−48.0000	−1.68343
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−22.6274	−0.791633
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	14.1421	0.492366
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	11.3137	0.392941	0.196471	−	0.980510i	$-0.437052\pi$
$829$	11.3137	0.392941	0.196471	+	0.980510i	$0.437052\pi$
$830$	0	0
$831$	−62.2254	−2.15858
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	48.0000	1.65912
$838$	0	0
$839$	2.82843	0.0976481	0.0488241	−	0.998807i	$-0.484453\pi$
$839$	2.82843	0.0976481	0.0488241	+	0.998807i	$0.484453\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	28.2843	0.974162
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	32.0000	1.09824
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	8.48528	0.290531	0.145265	−	0.989393i	$-0.453596\pi$
$853$	8.48528	0.290531	0.145265	+	0.989393i	$0.453596\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.4264	1.44926	0.724629	−	0.689139i	$-0.242011\pi$
$857$	42.4264	1.44926	0.724629	+	0.689139i	$0.242011\pi$
$858$	0	0
$859$	25.4558	0.868542	0.434271	−	0.900782i	$-0.357006\pi$
$859$	25.4558	0.868542	0.434271	+	0.900782i	$0.357006\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0000	1.36162	0.680808	−	0.732462i	$-0.261629\pi$
$863$	40.0000	1.36162	0.680808	+	0.732462i	$0.261629\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	25.4558	0.864526
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	11.3137	0.383350
$872$	0	0
$873$	−84.8528	−2.87183
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	−40.0000	−1.34917
$880$	0	0
$881$	−5.65685	−0.190584	−0.0952921	−	0.995449i	$-0.530379\pi$
$881$	−5.65685	−0.190584	−0.0952921	+	0.995449i	$0.530379\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−64.0000	−2.13690
$898$	0	0
$899$	−16.9706	−0.566000
$900$	0	0
$901$	39.5980	1.31920
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−14.1421	−0.469065
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	16.9706	0.561644
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−64.0000	−2.10887
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	−70.7107	−2.32244
$928$	0	0
$929$	−28.2843	−0.927977	−0.463988	−	0.885841i	$-0.653582\pi$
$929$	−28.2843	−0.927977	−0.463988	+	0.885841i	$0.653582\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−88.0000	−2.88099
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.4264	1.38601	0.693005	−	0.720933i	$-0.256286\pi$
$937$	42.4264	1.38601	0.693005	+	0.720933i	$0.256286\pi$
$938$	0	0
$939$	32.0000	1.04428
$940$	0	0
$941$	2.82843	0.0922041	0.0461020	−	0.998937i	$-0.485320\pi$
$941$	2.82843	0.0922041	0.0461020	+	0.998937i	$0.485320\pi$
$942$	0	0
$943$	−22.6274	−0.736850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	0	0
$951$	5.65685	0.183436
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.65685	−0.182860
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−60.0000	−1.93347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	45.2548	1.45379
$970$	0	0
$971$	36.7696	1.17999	0.589996	−	0.807406i	$-0.299129\pi$
$971$	36.7696	1.17999	0.589996	+	0.807406i	$0.299129\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−40.0000	−1.28103
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−2.82843	−0.0902128	−0.0451064	−	0.998982i	$-0.514363\pi$
$983$	−2.82843	−0.0902128	−0.0451064	+	0.998982i	$0.514363\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	−79.1960	−2.51321
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.4264	1.34366	0.671829	−	0.740706i	$-0.265509\pi$
$997$	42.4264	1.34366	0.671829	+	0.740706i	$0.265509\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{3} +5.00000 q^{9} +1.00000 q^{11} -2.82843 q^{13} +2.82843 q^{17} -5.65685 q^{19} -8.00000 q^{23} -5.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -8.48528 q^{31} -2.82843 q^{33} +2.00000 q^{37} +8.00000 q^{39} +2.82843 q^{41} +4.00000 q^{43} -2.82843 q^{47} -8.00000 q^{51} +14.0000 q^{53} +16.0000 q^{57} +8.48528 q^{59} -8.48528 q^{61} -4.00000 q^{67} +22.6274 q^{69} -14.1421 q^{73} +14.1421 q^{75} +16.0000 q^{79} +1.00000 q^{81} +16.9706 q^{83} -5.65685 q^{87} -11.3137 q^{89} +24.0000 q^{93} -16.9706 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{9}+O(q^{10}) \) 2 * q + 10 * q^9	\( 2 q + 10 q^{9} + 2 q^{11} - 16 q^{23} - 10 q^{25} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 8 q^{43} - 16 q^{51} + 28 q^{53} + 32 q^{57} - 8 q^{67} + 32 q^{79} + 2 q^{81} + 48 q^{93} + 10 q^{99}+O(q^{100}) \) 2 * q + 10 * q^9 + 2 * q^11 - 16 * q^23 - 10 * q^25 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 8 * q^43 - 16 * q^51 + 28 * q^53 + 32 * q^57 - 8 * q^67 + 32 * q^79 + 2 * q^81 + 48 * q^93 + 10 * q^99

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