LMFDB - Embedded newform 8624.2.a.cu.1.2

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:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 539)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.13578$ 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)

$f(q)$	$=$	$q-1.19935 q^{3} +3.07221 q^{5} -1.56155 q^{9} +O(q^{10})$	$q-1.19935 q^{3} +3.07221 q^{5} -1.56155 q^{9} +1.00000 q^{11} +6.67026 q^{13} -3.68466 q^{15} +4.27156 q^{17} +2.39871 q^{19} +1.43845 q^{23} +4.43845 q^{25} +5.47091 q^{27} +2.00000 q^{29} +7.34376 q^{31} -1.19935 q^{33} +8.56155 q^{37} -8.00000 q^{39} +4.27156 q^{41} -6.24621 q^{43} -4.79741 q^{45} -4.27156 q^{47} -5.12311 q^{51} -2.00000 q^{53} +3.07221 q^{55} -2.87689 q^{57} -5.99676 q^{59} -8.01726 q^{61} +20.4924 q^{65} -7.68466 q^{67} -1.72521 q^{69} +16.8078 q^{71} +12.8147 q^{73} -5.32326 q^{75} -13.1231 q^{79} -1.87689 q^{81} +8.54312 q^{83} +13.1231 q^{85} -2.39871 q^{87} -12.6670 q^{89} -8.80776 q^{93} +7.36932 q^{95} +4.12391 q^{97} -1.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} + 4 q^{11} + 10 q^{15} + 14 q^{23} + 26 q^{25} + 8 q^{29} + 26 q^{37} - 32 q^{39} + 8 q^{43} - 4 q^{51} - 8 q^{53} - 28 q^{57} + 16 q^{65} - 6 q^{67} + 26 q^{71} - 36 q^{79} - 24 q^{81} + 36 q^{85} + 6 q^{93} - 20 q^{95} + 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 4 * q^11 + 10 * q^15 + 14 * q^23 + 26 * q^25 + 8 * q^29 + 26 * q^37 - 32 * q^39 + 8 * q^43 - 4 * q^51 - 8 * q^53 - 28 * q^57 + 16 * q^65 - 6 * q^67 + 26 * q^71 - 36 * q^79 - 24 * q^81 + 36 * q^85 + 6 * q^93 - 20 * q^95 + 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
$3$	−1.19935	−0.692447	−0.346223	−	0.938152i	$-0.612536\pi$
$3$	−1.19935	−0.692447	−0.346223	+	0.938152i	$0.612536\pi$
$4$	0	0
$5$	3.07221	1.37393	0.686966	−	0.726690i	$-0.258942\pi$
$5$	3.07221	1.37393	0.686966	+	0.726690i	$0.258942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.56155	−0.520518
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.67026	1.85000	0.924999	−	0.379969i	$-0.124065\pi$
$13$	6.67026	1.85000	0.924999	+	0.379969i	$0.124065\pi$
$14$	0	0
$15$	−3.68466	−0.951375
$16$	0	0
$17$	4.27156	1.03601	0.518003	−	0.855379i	$-0.326676\pi$
$17$	4.27156	1.03601	0.518003	+	0.855379i	$0.326676\pi$
$18$	0	0
$19$	2.39871	0.550301	0.275150	−	0.961401i	$-0.411272\pi$
$19$	2.39871	0.550301	0.275150	+	0.961401i	$0.411272\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	4.43845	0.887689
$26$	0	0
$27$	5.47091	1.05288
$28$	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
$31$	7.34376	1.31898	0.659489	−	0.751714i	$-0.270773\pi$
$31$	7.34376	1.31898	0.659489	+	0.751714i	$0.270773\pi$
$32$	0	0
$33$	−1.19935	−0.208781
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56155	1.40751	0.703755	−	0.710442i	$-0.251505\pi$
$37$	8.56155	1.40751	0.703755	+	0.710442i	$0.251505\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	4.27156	0.667105	0.333553	−	0.942731i	$-0.391752\pi$
$41$	4.27156	0.667105	0.333553	+	0.942731i	$0.391752\pi$
$42$	0	0
$43$	−6.24621	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$44$	0	0
$45$	−4.79741	−0.715156
$46$	0	0
$47$	−4.27156	−0.623071	−0.311535	−	0.950235i	$-0.600843\pi$
$47$	−4.27156	−0.623071	−0.311535	+	0.950235i	$0.600843\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.12311	−0.717378
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	3.07221	0.414256
$56$	0	0
$57$	−2.87689	−0.381054
$58$	0	0
$59$	−5.99676	−0.780712	−0.390356	−	0.920664i	$-0.627648\pi$
$59$	−5.99676	−0.780712	−0.390356	+	0.920664i	$0.627648\pi$
$60$	0	0
$61$	−8.01726	−1.02651	−0.513253	−	0.858238i	$-0.671560\pi$
$61$	−8.01726	−1.02651	−0.513253	+	0.858238i	$0.671560\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.4924	2.54177
$66$	0	0
$67$	−7.68466	−0.938830	−0.469415	−	0.882978i	$-0.655535\pi$
$67$	−7.68466	−0.938830	−0.469415	+	0.882978i	$0.655535\pi$
$68$	0	0
$69$	−1.72521	−0.207690
$70$	0	0
$71$	16.8078	1.99471	0.997357	−	0.0726526i	$-0.0231464\pi$
$71$	16.8078	1.99471	0.997357	+	0.0726526i	$0.0231464\pi$
$72$	0	0
$73$	12.8147	1.49984	0.749922	−	0.661526i	$-0.230091\pi$
$73$	12.8147	1.49984	0.749922	+	0.661526i	$0.230091\pi$
$74$	0	0
$75$	−5.32326	−0.614678
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	0	0
$81$	−1.87689	−0.208544
$82$	0	0
$83$	8.54312	0.937729	0.468864	−	0.883270i	$-0.344663\pi$
$83$	8.54312	0.937729	0.468864	+	0.883270i	$0.344663\pi$
$84$	0	0
$85$	13.1231	1.42340
$86$	0	0
$87$	−2.39871	−0.257168
$88$	0	0
$89$	−12.6670	−1.34270	−0.671351	−	0.741139i	$-0.734286\pi$
$89$	−12.6670	−1.34270	−0.671351	+	0.741139i	$0.734286\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.80776	−0.913323
$94$	0	0
$95$	7.36932	0.756076
$96$	0	0
$97$	4.12391	0.418720	0.209360	−	0.977839i	$-0.432862\pi$
$97$	4.12391	0.418720	0.209360	+	0.977839i	$0.432862\pi$
$98$	0	0
$99$	−1.56155	−0.156942
$100$	0	0
$101$	−15.2134	−1.51379	−0.756894	−	0.653538i	$-0.773284\pi$
$101$	−15.2134	−1.51379	−0.756894	+	0.653538i	$0.773284\pi$
$102$	0	0
$103$	−9.06897	−0.893592	−0.446796	−	0.894636i	$-0.647435\pi$
$103$	−9.06897	−0.893592	−0.446796	+	0.894636i	$0.647435\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.12311	0.108575	0.0542874	−	0.998525i	$-0.482711\pi$
$107$	1.12311	0.108575	0.0542874	+	0.998525i	$0.482711\pi$
$108$	0	0
$109$	−5.36932	−0.514287	−0.257144	−	0.966373i	$-0.582781\pi$
$109$	−5.36932	−0.514287	−0.257144	+	0.966373i	$0.582781\pi$
$110$	0	0
$111$	−10.2683	−0.974626
$112$	0	0
$113$	−14.8078	−1.39300	−0.696499	−	0.717558i	$-0.745260\pi$
$113$	−14.8078	−1.39300	−0.696499	+	0.717558i	$0.745260\pi$
$114$	0	0
$115$	4.41921	0.412093
$116$	0	0
$117$	−10.4160	−0.962957
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.12311	−0.461935
$124$	0	0
$125$	−1.72521	−0.154307
$126$	0	0
$127$	13.1231	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$127$	13.1231	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$128$	0	0
$129$	7.49141	0.659582
$130$	0	0
$131$	−6.14441	−0.536840	−0.268420	−	0.963302i	$-0.586501\pi$
$131$	−6.14441	−0.536840	−0.268420	+	0.963302i	$0.586501\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.8078	1.44658
$136$	0	0
$137$	−8.56155	−0.731463	−0.365731	−	0.930720i	$-0.619181\pi$
$137$	−8.56155	−0.731463	−0.365731	+	0.930720i	$0.619181\pi$
$138$	0	0
$139$	2.39871	0.203456	0.101728	−	0.994812i	$-0.467563\pi$
$139$	2.39871	0.203456	0.101728	+	0.994812i	$0.467563\pi$
$140$	0	0
$141$	5.12311	0.431443
$142$	0	0
$143$	6.67026	0.557796
$144$	0	0
$145$	6.14441	0.510266
$146$	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
$151$	7.36932	0.599707	0.299853	−	0.953985i	$-0.403062\pi$
$151$	7.36932	0.599707	0.299853	+	0.953985i	$0.403062\pi$
$152$	0	0
$153$	−6.67026	−0.539259
$154$	0	0
$155$	22.5616	1.81219
$156$	0	0
$157$	−7.86962	−0.628064	−0.314032	−	0.949412i	$-0.601680\pi$
$157$	−7.86962	−0.628064	−0.314032	+	0.949412i	$0.601680\pi$
$158$	0	0
$159$	2.39871	0.190230
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.4924	1.91839	0.959197	−	0.282738i	$-0.0912426\pi$
$163$	24.4924	1.91839	0.959197	+	0.282738i	$0.0912426\pi$
$164$	0	0
$165$	−3.68466	−0.286850
$166$	0	0
$167$	23.2306	1.79764	0.898821	−	0.438317i	$-0.144425\pi$
$167$	23.2306	1.79764	0.898821	+	0.438317i	$0.144425\pi$
$168$	0	0
$169$	31.4924	2.42249
$170$	0	0
$171$	−3.74571	−0.286441
$172$	0	0
$173$	−2.92456	−0.222350	−0.111175	−	0.993801i	$-0.535461\pi$
$173$	−2.92456	−0.222350	−0.111175	+	0.993801i	$0.535461\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.19224	0.540602
$178$	0	0
$179$	−4.80776	−0.359349	−0.179675	−	0.983726i	$-0.557504\pi$
$179$	−4.80776	−0.359349	−0.179675	+	0.983726i	$0.557504\pi$
$180$	0	0
$181$	−10.2683	−0.763238	−0.381619	−	0.924320i	$-0.624633\pi$
$181$	−10.2683	−0.763238	−0.381619	+	0.924320i	$0.624633\pi$
$182$	0	0
$183$	9.61553	0.710800
$184$	0	0
$185$	26.3029	1.93382
$186$	0	0
$187$	4.27156	0.312367
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.80776	−0.637307	−0.318654	−	0.947871i	$-0.603231\pi$
$191$	−8.80776	−0.637307	−0.318654	+	0.947871i	$0.603231\pi$
$192$	0	0
$193$	6.49242	0.467335	0.233667	−	0.972317i	$-0.424927\pi$
$193$	6.49242	0.467335	0.233667	+	0.972317i	$0.424927\pi$
$194$	0	0
$195$	−24.5776	−1.76004
$196$	0	0
$197$	−4.24621	−0.302530	−0.151265	−	0.988493i	$-0.548335\pi$
$197$	−4.24621	−0.302530	−0.151265	+	0.988493i	$0.548335\pi$
$198$	0	0
$199$	5.32326	0.377356	0.188678	−	0.982039i	$-0.439580\pi$
$199$	5.32326	0.377356	0.188678	+	0.982039i	$0.439580\pi$
$200$	0	0
$201$	9.21662	0.650090
$202$	0	0
$203$	0	0
$204$	0	0
$205$	13.1231	0.916557
$206$	0	0
$207$	−2.24621	−0.156122
$208$	0	0
$209$	2.39871	0.165922
$210$	0	0
$211$	3.36932	0.231953	0.115977	−	0.993252i	$-0.463000\pi$
$211$	3.36932	0.231953	0.115977	+	0.993252i	$0.463000\pi$
$212$	0	0
$213$	−20.1584	−1.38123
$214$	0	0
$215$	−19.1896	−1.30872
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.3693	−1.03856
$220$	0	0
$221$	28.4924	1.91661
$222$	0	0
$223$	−19.6326	−1.31470	−0.657348	−	0.753587i	$-0.728322\pi$
$223$	−19.6326	−1.31470	−0.657348	+	0.753587i	$0.728322\pi$
$224$	0	0
$225$	−6.93087	−0.462058
$226$	0	0
$227$	9.59482	0.636831	0.318415	−	0.947951i	$-0.396849\pi$
$227$	9.59482	0.636831	0.318415	+	0.947951i	$0.396849\pi$
$228$	0	0
$229$	0.673500	0.0445061	0.0222531	−	0.999752i	$-0.492916\pi$
$229$	0.673500	0.0445061	0.0222531	+	0.999752i	$0.492916\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.24621	−0.278179	−0.139089	−	0.990280i	$-0.544417\pi$
$233$	−4.24621	−0.278179	−0.139089	+	0.990280i	$0.544417\pi$
$234$	0	0
$235$	−13.1231	−0.856057
$236$	0	0
$237$	15.7392	1.02237
$238$	0	0
$239$	−10.2462	−0.662772	−0.331386	−	0.943495i	$-0.607516\pi$
$239$	−10.2462	−0.662772	−0.331386	+	0.943495i	$0.607516\pi$
$240$	0	0
$241$	−18.9591	−1.22126	−0.610631	−	0.791915i	$-0.709084\pi$
$241$	−18.9591	−1.22126	−0.610631	+	0.791915i	$0.709084\pi$
$242$	0	0
$243$	−14.1617	−0.908472
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	27.8804	1.75980	0.879898	−	0.475163i	$-0.157611\pi$
$251$	27.8804	1.75980	0.879898	+	0.475163i	$0.157611\pi$
$252$	0	0
$253$	1.43845	0.0904344
$254$	0	0
$255$	−15.7392	−0.985629
$256$	0	0
$257$	7.49141	0.467301	0.233651	−	0.972321i	$-0.424933\pi$
$257$	7.49141	0.467301	0.233651	+	0.972321i	$0.424933\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.12311	−0.193315
$262$	0	0
$263$	−23.3693	−1.44101	−0.720507	−	0.693448i	$-0.756091\pi$
$263$	−23.3693	−1.44101	−0.720507	+	0.693448i	$0.756091\pi$
$264$	0	0
$265$	−6.14441	−0.377448
$266$	0	0
$267$	15.1922	0.929750
$268$	0	0
$269$	−16.0345	−0.977642	−0.488821	−	0.872384i	$-0.662573\pi$
$269$	−16.0345	−0.977642	−0.488821	+	0.872384i	$0.662573\pi$
$270$	0	0
$271$	13.3405	0.810379	0.405190	−	0.914233i	$-0.367205\pi$
$271$	13.3405	0.810379	0.405190	+	0.914233i	$0.367205\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.43845	0.267648
$276$	0	0
$277$	23.6155	1.41892	0.709460	−	0.704746i	$-0.248939\pi$
$277$	23.6155	1.41892	0.709460	+	0.704746i	$0.248939\pi$
$278$	0	0
$279$	−11.4677	−0.686552
$280$	0	0
$281$	−11.6155	−0.692924	−0.346462	−	0.938064i	$-0.612617\pi$
$281$	−11.6155	−0.692924	−0.346462	+	0.938064i	$0.612617\pi$
$282$	0	0
$283$	−9.59482	−0.570353	−0.285176	−	0.958475i	$-0.592052\pi$
$283$	−9.59482	−0.570353	−0.285176	+	0.958475i	$0.592052\pi$
$284$	0	0
$285$	−8.83841	−0.523542
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.24621	0.0733065
$290$	0	0
$291$	−4.94602	−0.289941
$292$	0	0
$293$	13.8664	0.810083	0.405041	−	0.914298i	$-0.367257\pi$
$293$	13.8664	0.810083	0.405041	+	0.914298i	$0.367257\pi$
$294$	0	0
$295$	−18.4233	−1.07265
$296$	0	0
$297$	5.47091	0.317454
$298$	0	0
$299$	9.59482	0.554883
$300$	0	0
$301$	0	0
$302$	0	0
$303$	18.2462	1.04822
$304$	0	0
$305$	−24.6307	−1.41035
$306$	0	0
$307$	4.79741	0.273803	0.136901	−	0.990585i	$-0.456286\pi$
$307$	4.79741	0.273803	0.136901	+	0.990585i	$0.456286\pi$
$308$	0	0
$309$	10.8769	0.618765
$310$	0	0
$311$	−30.9526	−1.75516	−0.877581	−	0.479429i	$-0.840844\pi$
$311$	−30.9526	−1.75516	−0.877581	+	0.479429i	$0.840844\pi$
$312$	0	0
$313$	24.9559	1.41059	0.705294	−	0.708915i	$-0.250815\pi$
$313$	24.9559	1.41059	0.705294	+	0.708915i	$0.250815\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.8078	−1.05635	−0.528175	−	0.849136i	$-0.677123\pi$
$317$	−18.8078	−1.05635	−0.528175	+	0.849136i	$0.677123\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−1.34700	−0.0751822
$322$	0	0
$323$	10.2462	0.570114
$324$	0	0
$325$	29.6056	1.64222
$326$	0	0
$327$	6.43971	0.356117
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.8078	0.703978	0.351989	−	0.936004i	$-0.385505\pi$
$331$	12.8078	0.703978	0.351989	+	0.936004i	$0.385505\pi$
$332$	0	0
$333$	−13.3693	−0.732634
$334$	0	0
$335$	−23.6089	−1.28989
$336$	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
$339$	17.7597	0.964576
$340$	0	0
$341$	7.34376	0.397687
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.30019	−0.285352
$346$	0	0
$347$	1.12311	0.0602915	0.0301457	−	0.999546i	$-0.490403\pi$
$347$	1.12311	0.0602915	0.0301457	+	0.999546i	$0.490403\pi$
$348$	0	0
$349$	−22.4095	−1.19955	−0.599776	−	0.800168i	$-0.704744\pi$
$349$	−22.4095	−1.19955	−0.599776	+	0.800168i	$0.704744\pi$
$350$	0	0
$351$	36.4924	1.94782
$352$	0	0
$353$	−9.21662	−0.490551	−0.245276	−	0.969453i	$-0.578878\pi$
$353$	−9.21662	−0.490551	−0.245276	+	0.969453i	$0.578878\pi$
$354$	0	0
$355$	51.6369	2.74060
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.2462	−0.540774	−0.270387	−	0.962752i	$-0.587152\pi$
$359$	−10.2462	−0.540774	−0.270387	+	0.962752i	$0.587152\pi$
$360$	0	0
$361$	−13.2462	−0.697169
$362$	0	0
$363$	−1.19935	−0.0629497
$364$	0	0
$365$	39.3693	2.06068
$366$	0	0
$367$	2.25106	0.117504	0.0587522	−	0.998273i	$-0.481288\pi$
$367$	2.25106	0.117504	0.0587522	+	0.998273i	$0.481288\pi$
$368$	0	0
$369$	−6.67026	−0.347240
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.3693	−0.899349	−0.449675	−	0.893192i	$-0.648460\pi$
$373$	−17.3693	−0.899349	−0.449675	+	0.893192i	$0.648460\pi$
$374$	0	0
$375$	2.06913	0.106849
$376$	0	0
$377$	13.3405	0.687072
$378$	0	0
$379$	−16.3153	−0.838063	−0.419031	−	0.907972i	$-0.637630\pi$
$379$	−16.3153	−0.838063	−0.419031	+	0.907972i	$0.637630\pi$
$380$	0	0
$381$	−15.7392	−0.806345
$382$	0	0
$383$	24.4300	1.24831	0.624157	−	0.781299i	$-0.285442\pi$
$383$	24.4300	1.24831	0.624157	+	0.781299i	$0.285442\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.75379	0.495813
$388$	0	0
$389$	5.05398	0.256247	0.128123	−	0.991758i	$-0.459105\pi$
$389$	5.05398	0.256247	0.128123	+	0.991758i	$0.459105\pi$
$390$	0	0
$391$	6.14441	0.310736
$392$	0	0
$393$	7.36932	0.371733
$394$	0	0
$395$	−40.3169	−2.02856
$396$	0	0
$397$	−29.3751	−1.47429	−0.737146	−	0.675734i	$-0.763827\pi$
$397$	−29.3751	−1.47429	−0.737146	+	0.675734i	$0.763827\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.4924	0.923468	0.461734	−	0.887019i	$-0.347228\pi$
$401$	18.4924	0.923468	0.461734	+	0.887019i	$0.347228\pi$
$402$	0	0
$403$	48.9848	2.44011
$404$	0	0
$405$	−5.76621	−0.286525
$406$	0	0
$407$	8.56155	0.424380
$408$	0	0
$409$	−11.7630	−0.581641	−0.290821	−	0.956778i	$-0.593928\pi$
$409$	−11.7630	−0.581641	−0.290821	+	0.956778i	$0.593928\pi$
$410$	0	0
$411$	10.2683	0.506499
$412$	0	0
$413$	0	0
$414$	0	0
$415$	26.2462	1.28838
$416$	0	0
$417$	−2.87689	−0.140882
$418$	0	0
$419$	0.525853	0.0256896	0.0128448	−	0.999918i	$-0.495911\pi$
$419$	0.525853	0.0256896	0.0128448	+	0.999918i	$0.495911\pi$
$420$	0	0
$421$	3.75379	0.182948	0.0914742	−	0.995807i	$-0.470842\pi$
$421$	3.75379	0.182948	0.0914742	+	0.995807i	$0.470842\pi$
$422$	0	0
$423$	6.67026	0.324319
$424$	0	0
$425$	18.9591	0.919651
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−12.4924	−0.601739	−0.300869	−	0.953665i	$-0.597277\pi$
$431$	−12.4924	−0.601739	−0.300869	+	0.953665i	$0.597277\pi$
$432$	0	0
$433$	−27.3546	−1.31458	−0.657288	−	0.753639i	$-0.728296\pi$
$433$	−27.3546	−1.31458	−0.657288	+	0.753639i	$0.728296\pi$
$434$	0	0
$435$	−7.36932	−0.353332
$436$	0	0
$437$	3.45041	0.165056
$438$	0	0
$439$	−22.1789	−1.05854	−0.529272	−	0.848453i	$-0.677535\pi$
$439$	−22.1789	−1.05854	−0.529272	+	0.848453i	$0.677535\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.6847	1.12529	0.562646	−	0.826698i	$-0.309783\pi$
$443$	23.6847	1.12529	0.562646	+	0.826698i	$0.309783\pi$
$444$	0	0
$445$	−38.9157	−1.84478
$446$	0	0
$447$	−7.19612	−0.340365
$448$	0	0
$449$	18.1771	0.857829	0.428915	−	0.903345i	$-0.358896\pi$
$449$	18.1771	0.857829	0.428915	+	0.903345i	$0.358896\pi$
$450$	0	0
$451$	4.27156	0.201140
$452$	0	0
$453$	−8.83841	−0.415265
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.8617	1.77110	0.885549	−	0.464546i	$-0.153783\pi$
$457$	37.8617	1.77110	0.885549	+	0.464546i	$0.153783\pi$
$458$	0	0
$459$	23.3693	1.09079
$460$	0	0
$461$	21.3578	0.994732	0.497366	−	0.867541i	$-0.334301\pi$
$461$	21.3578	0.994732	0.497366	+	0.867541i	$0.334301\pi$
$462$	0	0
$463$	−21.9309	−1.01921	−0.509607	−	0.860407i	$-0.670209\pi$
$463$	−21.9309	−1.01921	−0.509607	+	0.860407i	$0.670209\pi$
$464$	0	0
$465$	−27.0593	−1.25484
$466$	0	0
$467$	−8.39547	−0.388496	−0.194248	−	0.980952i	$-0.562227\pi$
$467$	−8.39547	−0.388496	−0.194248	+	0.980952i	$0.562227\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.43845	0.434901
$472$	0	0
$473$	−6.24621	−0.287201
$474$	0	0
$475$	10.6465	0.488496
$476$	0	0
$477$	3.12311	0.142997
$478$	0	0
$479$	10.6465	0.486452	0.243226	−	0.969970i	$-0.421794\pi$
$479$	10.6465	0.486452	0.243226	+	0.969970i	$0.421794\pi$
$480$	0	0
$481$	57.1078	2.60389
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.6695	0.575293
$486$	0	0
$487$	−14.5616	−0.659847	−0.329923	−	0.944008i	$-0.607023\pi$
$487$	−14.5616	−0.659847	−0.329923	+	0.944008i	$0.607023\pi$
$488$	0	0
$489$	−29.3751	−1.32839
$490$	0	0
$491$	19.3693	0.874125	0.437063	−	0.899431i	$-0.356019\pi$
$491$	19.3693	0.874125	0.437063	+	0.899431i	$0.356019\pi$
$492$	0	0
$493$	8.54312	0.384763
$494$	0	0
$495$	−4.79741	−0.215628
$496$	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
$501$	−27.8617	−1.24477
$502$	0	0
$503$	16.0345	0.714944	0.357472	−	0.933924i	$-0.383639\pi$
$503$	16.0345	0.714944	0.357472	+	0.933924i	$0.383639\pi$
$504$	0	0
$505$	−46.7386	−2.07984
$506$	0	0
$507$	−37.7705	−1.67745
$508$	0	0
$509$	11.6153	0.514840	0.257420	−	0.966300i	$-0.417128\pi$
$509$	11.6153	0.514840	0.257420	+	0.966300i	$0.417128\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	13.1231	0.579399
$514$	0	0
$515$	−27.8617	−1.22773
$516$	0	0
$517$	−4.27156	−0.187863
$518$	0	0
$519$	3.50758	0.153966
$520$	0	0
$521$	5.47091	0.239685	0.119842	−	0.992793i	$-0.461761\pi$
$521$	5.47091	0.239685	0.119842	+	0.992793i	$0.461761\pi$
$522$	0	0
$523$	−12.2888	−0.537353	−0.268676	−	0.963231i	$-0.586586\pi$
$523$	−12.2888	−0.537353	−0.268676	+	0.963231i	$0.586586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.3693	1.36647
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	9.36426	0.406374
$532$	0	0
$533$	28.4924	1.23414
$534$	0	0
$535$	3.45041	0.149174
$536$	0	0
$537$	5.76621	0.248830
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.8769	1.41349	0.706744	−	0.707469i	$-0.250163\pi$
$541$	32.8769	1.41349	0.706744	+	0.707469i	$0.250163\pi$
$542$	0	0
$543$	12.3153	0.528502
$544$	0	0
$545$	−16.4956	−0.706596
$546$	0	0
$547$	−1.75379	−0.0749866	−0.0374933	−	0.999297i	$-0.511937\pi$
$547$	−1.75379	−0.0749866	−0.0374933	+	0.999297i	$0.511937\pi$
$548$	0	0
$549$	12.5194	0.534314
$550$	0	0
$551$	4.79741	0.204377
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−31.5464	−1.33907
$556$	0	0
$557$	8.24621	0.349403	0.174702	−	0.984621i	$-0.444104\pi$
$557$	8.24621	0.349403	0.174702	+	0.984621i	$0.444104\pi$
$558$	0	0
$559$	−41.6639	−1.76219
$560$	0	0
$561$	−5.12311	−0.216298
$562$	0	0
$563$	18.1379	0.764423	0.382212	−	0.924075i	$-0.375163\pi$
$563$	18.1379	0.764423	0.382212	+	0.924075i	$0.375163\pi$
$564$	0	0
$565$	−45.4925	−1.91388
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.2462	1.35183	0.675916	−	0.736979i	$-0.263748\pi$
$569$	32.2462	1.35183	0.675916	+	0.736979i	$0.263748\pi$
$570$	0	0
$571$	−9.75379	−0.408183	−0.204092	−	0.978952i	$-0.565424\pi$
$571$	−9.75379	−0.408183	−0.204092	+	0.978952i	$0.565424\pi$
$572$	0	0
$573$	10.5636	0.441301
$574$	0	0
$575$	6.38447	0.266251
$576$	0	0
$577$	23.9041	0.995143	0.497571	−	0.867423i	$-0.334225\pi$
$577$	23.9041	0.995143	0.497571	+	0.867423i	$0.334225\pi$
$578$	0	0
$579$	−7.78671	−0.323604
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−32.0000	−1.32304
$586$	0	0
$587$	−14.9181	−0.615735	−0.307868	−	0.951429i	$-0.599615\pi$
$587$	−14.9181	−0.615735	−0.307868	+	0.951429i	$0.599615\pi$
$588$	0	0
$589$	17.6155	0.725835
$590$	0	0
$591$	5.09271	0.209486
$592$	0	0
$593$	−6.96556	−0.286041	−0.143021	−	0.989720i	$-0.545682\pi$
$593$	−6.96556	−0.286041	−0.143021	+	0.989720i	$0.545682\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.38447	−0.261299
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	7.72197	0.314986	0.157493	−	0.987520i	$-0.449659\pi$
$601$	7.72197	0.314986	0.157493	+	0.987520i	$0.449659\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	3.07221	0.124903
$606$	0	0
$607$	−45.1143	−1.83113	−0.915566	−	0.402167i	$-0.868257\pi$
$607$	−45.1143	−1.83113	−0.915566	+	0.402167i	$0.868257\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.4924	−1.15268
$612$	0	0
$613$	33.3693	1.34777	0.673887	−	0.738834i	$-0.264623\pi$
$613$	33.3693	1.34777	0.673887	+	0.738834i	$0.264623\pi$
$614$	0	0
$615$	−15.7392	−0.634667
$616$	0	0
$617$	12.7386	0.512838	0.256419	−	0.966566i	$-0.417457\pi$
$617$	12.7386	0.512838	0.256419	+	0.966566i	$0.417457\pi$
$618$	0	0
$619$	20.3890	0.819503	0.409752	−	0.912197i	$-0.365615\pi$
$619$	20.3890	0.819503	0.409752	+	0.912197i	$0.365615\pi$
$620$	0	0
$621$	7.86962	0.315797
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.4924	−1.09970
$626$	0	0
$627$	−2.87689	−0.114892
$628$	0	0
$629$	36.5712	1.45819
$630$	0	0
$631$	0.807764	0.0321566	0.0160783	−	0.999871i	$-0.494882\pi$
$631$	0.807764	0.0321566	0.0160783	+	0.999871i	$0.494882\pi$
$632$	0	0
$633$	−4.04100	−0.160615
$634$	0	0
$635$	40.3169	1.59993
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−26.2462	−1.03828
$640$	0	0
$641$	−7.93087	−0.313251	−0.156625	−	0.987658i	$-0.550061\pi$
$641$	−7.93087	−0.313251	−0.156625	+	0.987658i	$0.550061\pi$
$642$	0	0
$643$	−23.3783	−0.921950	−0.460975	−	0.887413i	$-0.652500\pi$
$643$	−23.3783	−0.921950	−0.460975	+	0.887413i	$0.652500\pi$
$644$	0	0
$645$	23.0152	0.906221
$646$	0	0
$647$	41.5162	1.63217	0.816086	−	0.577931i	$-0.196140\pi$
$647$	41.5162	1.63217	0.816086	+	0.577931i	$0.196140\pi$
$648$	0	0
$649$	−5.99676	−0.235394
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.3153	−0.403671	−0.201835	−	0.979419i	$-0.564691\pi$
$653$	−10.3153	−0.403671	−0.201835	+	0.979419i	$0.564691\pi$
$654$	0	0
$655$	−18.8769	−0.737581
$656$	0	0
$657$	−20.0108	−0.780695
$658$	0	0
$659$	19.3693	0.754521	0.377261	−	0.926107i	$-0.376866\pi$
$659$	19.3693	0.754521	0.377261	+	0.926107i	$0.376866\pi$
$660$	0	0
$661$	6.52262	0.253700	0.126850	−	0.991922i	$-0.459513\pi$
$661$	6.52262	0.253700	0.126850	+	0.991922i	$0.459513\pi$
$662$	0	0
$663$	−34.1725	−1.32715
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.87689	0.111394
$668$	0	0
$669$	23.5464	0.910356
$670$	0	0
$671$	−8.01726	−0.309503
$672$	0	0
$673$	15.6155	0.601934	0.300967	−	0.953634i	$-0.402691\pi$
$673$	15.6155	0.601934	0.300967	+	0.953634i	$0.402691\pi$
$674$	0	0
$675$	24.2824	0.934628
$676$	0	0
$677$	−21.3578	−0.820847	−0.410423	−	0.911895i	$-0.634619\pi$
$677$	−21.3578	−0.820847	−0.410423	+	0.911895i	$0.634619\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.5076	−0.440971
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	−26.3029	−1.00498
$686$	0	0
$687$	−0.807764	−0.0308181
$688$	0	0
$689$	−13.3405	−0.508234
$690$	0	0
$691$	25.4817	0.969370	0.484685	−	0.874689i	$-0.338934\pi$
$691$	25.4817	0.969370	0.484685	+	0.874689i	$0.338934\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.36932	0.279534
$696$	0	0
$697$	18.2462	0.691125
$698$	0	0
$699$	5.09271	0.192624
$700$	0	0
$701$	10.6307	0.401515	0.200758	−	0.979641i	$-0.435660\pi$
$701$	10.6307	0.401515	0.200758	+	0.979641i	$0.435660\pi$
$702$	0	0
$703$	20.5366	0.774554
$704$	0	0
$705$	15.7392	0.592774
$706$	0	0
$707$	0	0
$708$	0	0
$709$	9.19224	0.345222	0.172611	−	0.984990i	$-0.444780\pi$
$709$	9.19224	0.345222	0.172611	+	0.984990i	$0.444780\pi$
$710$	0	0
$711$	20.4924	0.768526
$712$	0	0
$713$	10.5636	0.395611
$714$	0	0
$715$	20.4924	0.766373
$716$	0	0
$717$	12.2888	0.458934
$718$	0	0
$719$	2.54635	0.0949629	0.0474815	−	0.998872i	$-0.484880\pi$
$719$	2.54635	0.0949629	0.0474815	+	0.998872i	$0.484880\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.7386	0.845659
$724$	0	0
$725$	8.87689	0.329680
$726$	0	0
$727$	27.1240	1.00597	0.502987	−	0.864294i	$-0.332234\pi$
$727$	27.1240	1.00597	0.502987	+	0.864294i	$0.332234\pi$
$728$	0	0
$729$	22.6155	0.837612
$730$	0	0
$731$	−26.6811	−0.986835
$732$	0	0
$733$	−6.67026	−0.246372	−0.123186	−	0.992384i	$-0.539311\pi$
$733$	−6.67026	−0.246372	−0.123186	+	0.992384i	$0.539311\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.68466	−0.283068
$738$	0	0
$739$	−19.3693	−0.712512	−0.356256	−	0.934388i	$-0.615947\pi$
$739$	−19.3693	−0.712512	−0.356256	+	0.934388i	$0.615947\pi$
$740$	0	0
$741$	−19.1896	−0.704949
$742$	0	0
$743$	−29.7538	−1.09156	−0.545780	−	0.837928i	$-0.683767\pi$
$743$	−29.7538	−1.09156	−0.545780	+	0.837928i	$0.683767\pi$
$744$	0	0
$745$	18.4332	0.675341
$746$	0	0
$747$	−13.3405	−0.488104
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.1771	−0.882234	−0.441117	−	0.897450i	$-0.645418\pi$
$751$	−24.1771	−0.882234	−0.441117	+	0.897450i	$0.645418\pi$
$752$	0	0
$753$	−33.4384	−1.21856
$754$	0	0
$755$	22.6401	0.823956
$756$	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
$759$	−1.72521	−0.0626210
$760$	0	0
$761$	−8.31256	−0.301330	−0.150665	−	0.988585i	$-0.548141\pi$
$761$	−8.31256	−0.301330	−0.150665	+	0.988585i	$0.548141\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−20.4924	−0.740905
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	11.4677	0.413535	0.206767	−	0.978390i	$-0.433706\pi$
$769$	11.4677	0.413535	0.206767	+	0.978390i	$0.433706\pi$
$770$	0	0
$771$	−8.98485	−0.323581
$772$	0	0
$773$	32.5302	1.17003	0.585015	−	0.811023i	$-0.301089\pi$
$773$	32.5302	1.17003	0.585015	+	0.811023i	$0.301089\pi$
$774$	0	0
$775$	32.5949	1.17084
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.2462	0.367109
$780$	0	0
$781$	16.8078	0.601429
$782$	0	0
$783$	10.9418	0.391029
$784$	0	0
$785$	−24.1771	−0.862917
$786$	0	0
$787$	26.9764	0.961603	0.480802	−	0.876829i	$-0.340346\pi$
$787$	26.9764	0.961603	0.480802	+	0.876829i	$0.340346\pi$
$788$	0	0
$789$	28.0281	0.997825
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−53.4773	−1.89903
$794$	0	0
$795$	7.36932	0.261363
$796$	0	0
$797$	−23.6089	−0.836268	−0.418134	−	0.908385i	$-0.637316\pi$
$797$	−23.6089	−0.836268	−0.418134	+	0.908385i	$0.637316\pi$
$798$	0	0
$799$	−18.2462	−0.645505
$800$	0	0
$801$	19.7802	0.698900
$802$	0	0
$803$	12.8147	0.452220
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.2311	0.676965
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−10.9418	−0.384219	−0.192110	−	0.981373i	$-0.561533\pi$
$811$	−10.9418	−0.384219	−0.192110	+	0.981373i	$0.561533\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	75.2458	2.63574
$816$	0	0
$817$	−14.9828	−0.524183
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.7386	0.723783	0.361892	−	0.932220i	$-0.382131\pi$
$821$	20.7386	0.723783	0.361892	+	0.932220i	$0.382131\pi$
$822$	0	0
$823$	−9.43845	−0.329004	−0.164502	−	0.986377i	$-0.552602\pi$
$823$	−9.43845	−0.329004	−0.164502	+	0.986377i	$0.552602\pi$
$824$	0	0
$825$	−5.32326	−0.185332
$826$	0	0
$827$	−39.2311	−1.36420	−0.682099	−	0.731260i	$-0.738933\pi$
$827$	−39.2311	−1.36420	−0.682099	+	0.731260i	$0.738933\pi$
$828$	0	0
$829$	32.4473	1.12694	0.563470	−	0.826137i	$-0.309466\pi$
$829$	32.4473	1.12694	0.563470	+	0.826137i	$0.309466\pi$
$830$	0	0
$831$	−28.3234	−0.982526
$832$	0	0
$833$	0	0
$834$	0	0
$835$	71.3693	2.46984
$836$	0	0
$837$	40.1771	1.38872
$838$	0	0
$839$	−7.04847	−0.243340	−0.121670	−	0.992571i	$-0.538825\pi$
$839$	−7.04847	−0.243340	−0.121670	+	0.992571i	$0.538825\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	13.9311	0.479813
$844$	0	0
$845$	96.7512	3.32834
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.5076	0.394939
$850$	0	0
$851$	12.3153	0.422164
$852$	0	0
$853$	32.5949	1.11603	0.558014	−	0.829831i	$-0.311563\pi$
$853$	32.5949	1.11603	0.558014	+	0.829831i	$0.311563\pi$
$854$	0	0
$855$	−11.5076	−0.393551
$856$	0	0
$857$	−32.2996	−1.10333	−0.551667	−	0.834065i	$-0.686008\pi$
$857$	−32.2996	−1.10333	−0.551667	+	0.834065i	$0.686008\pi$
$858$	0	0
$859$	−30.2791	−1.03311	−0.516555	−	0.856254i	$-0.672786\pi$
$859$	−30.2791	−1.03311	−0.516555	+	0.856254i	$0.672786\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.7386	−1.31868	−0.659339	−	0.751846i	$-0.729164\pi$
$863$	−38.7386	−1.31868	−0.659339	+	0.751846i	$0.729164\pi$
$864$	0	0
$865$	−8.98485	−0.305494
$866$	0	0
$867$	−1.49465	−0.0507609
$868$	0	0
$869$	−13.1231	−0.445171
$870$	0	0
$871$	−51.2587	−1.73683
$872$	0	0
$873$	−6.43971	−0.217951
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.49242	−0.0841631	−0.0420816	−	0.999114i	$-0.513399\pi$
$877$	−2.49242	−0.0841631	−0.0420816	+	0.999114i	$0.513399\pi$
$878$	0	0
$879$	−16.6307	−0.560939
$880$	0	0
$881$	8.16491	0.275083	0.137541	−	0.990496i	$-0.456080\pi$
$881$	8.16491	0.275083	0.137541	+	0.990496i	$0.456080\pi$
$882$	0	0
$883$	48.4924	1.63190	0.815950	−	0.578123i	$-0.196214\pi$
$883$	48.4924	1.63190	0.815950	+	0.578123i	$0.196214\pi$
$884$	0	0
$885$	22.0960	0.742750
$886$	0	0
$887$	44.0626	1.47948	0.739738	−	0.672895i	$-0.234949\pi$
$887$	44.0626	1.47948	0.739738	+	0.672895i	$0.234949\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.87689	−0.0628783
$892$	0	0
$893$	−10.2462	−0.342876
$894$	0	0
$895$	−14.7704	−0.493721
$896$	0	0
$897$	−11.5076	−0.384227
$898$	0	0
$899$	14.6875	0.489856
$900$	0	0
$901$	−8.54312	−0.284612
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.5464	−1.04864
$906$	0	0
$907$	−22.2462	−0.738673	−0.369337	−	0.929296i	$-0.620415\pi$
$907$	−22.2462	−0.738673	−0.369337	+	0.929296i	$0.620415\pi$
$908$	0	0
$909$	23.7565	0.787953
$910$	0	0
$911$	36.4924	1.20905	0.604524	−	0.796587i	$-0.293363\pi$
$911$	36.4924	1.20905	0.604524	+	0.796587i	$0.293363\pi$
$912$	0	0
$913$	8.54312	0.282736
$914$	0	0
$915$	29.5409	0.976591
$916$	0	0
$917$	0	0
$918$	0	0
$919$	37.7538	1.24538	0.622691	−	0.782468i	$-0.286039\pi$
$919$	37.7538	1.24538	0.622691	+	0.782468i	$0.286039\pi$
$920$	0	0
$921$	−5.75379	−0.189594
$922$	0	0
$923$	112.112	3.69022
$924$	0	0
$925$	38.0000	1.24943
$926$	0	0
$927$	14.1617	0.465130
$928$	0	0
$929$	−52.3104	−1.71625	−0.858124	−	0.513442i	$-0.828370\pi$
$929$	−52.3104	−1.71625	−0.858124	+	0.513442i	$0.828370\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	37.1231	1.21536
$934$	0	0
$935$	13.1231	0.429171
$936$	0	0
$937$	21.0625	0.688082	0.344041	−	0.938955i	$-0.388204\pi$
$937$	21.0625	0.688082	0.344041	+	0.938955i	$0.388204\pi$
$938$	0	0
$939$	−29.9309	−0.976757
$940$	0	0
$941$	−46.9871	−1.53174	−0.765869	−	0.642997i	$-0.777691\pi$
$941$	−46.9871	−1.53174	−0.765869	+	0.642997i	$0.777691\pi$
$942$	0	0
$943$	6.14441	0.200090
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.0540	−1.26908	−0.634542	−	0.772889i	$-0.718811\pi$
$947$	−39.0540	−1.26908	−0.634542	+	0.772889i	$0.718811\pi$
$948$	0	0
$949$	85.4773	2.77471
$950$	0	0
$951$	22.5571	0.731466
$952$	0	0
$953$	2.63068	0.0852162	0.0426081	−	0.999092i	$-0.486433\pi$
$953$	2.63068	0.0852162	0.0426081	+	0.999092i	$0.486433\pi$
$954$	0	0
$955$	−27.0593	−0.875617
$956$	0	0
$957$	−2.39871	−0.0775391
$958$	0	0
$959$	0	0
$960$	0	0
$961$	22.9309	0.739705
$962$	0	0
$963$	−1.75379	−0.0565151
$964$	0	0
$965$	19.9461	0.642086
$966$	0	0
$967$	13.1231	0.422011	0.211005	−	0.977485i	$-0.432326\pi$
$967$	13.1231	0.422011	0.211005	+	0.977485i	$0.432326\pi$
$968$	0	0
$969$	−12.2888	−0.394774
$970$	0	0
$971$	20.9796	0.673267	0.336633	−	0.941636i	$-0.390712\pi$
$971$	20.9796	0.673267	0.336633	+	0.941636i	$0.390712\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−35.5076	−1.13715
$976$	0	0
$977$	39.7926	1.27308	0.636539	−	0.771244i	$-0.280365\pi$
$977$	39.7926	1.27308	0.636539	+	0.771244i	$0.280365\pi$
$978$	0	0
$979$	−12.6670	−0.404840
$980$	0	0
$981$	8.38447	0.267696
$982$	0	0
$983$	−49.0076	−1.56310	−0.781551	−	0.623842i	$-0.785571\pi$
$983$	−49.0076	−1.56310	−0.781551	+	0.623842i	$0.785571\pi$
$984$	0	0
$985$	−13.0452	−0.415656
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.98485	−0.285701
$990$	0	0
$991$	28.4924	0.905092	0.452546	−	0.891741i	$-0.350516\pi$
$991$	28.4924	0.905092	0.452546	+	0.891741i	$0.350516\pi$
$992$	0	0
$993$	−15.3610	−0.487467
$994$	0	0
$995$	16.3542	0.518462
$996$	0	0
$997$	16.2651	0.515120	0.257560	−	0.966262i	$-0.417081\pi$
$997$	16.2651	0.515120	0.257560	+	0.966262i	$0.417081\pi$
$998$	0	0
$999$	46.8395	1.48194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cu.1.2		4
4.3	odd	2		539.2.a.k.1.2	yes	4
7.6	odd	2	inner	8624.2.a.cu.1.3		4
12.11	even	2		4851.2.a.bv.1.3		4
28.3	even	6		539.2.e.n.177.4		8
28.11	odd	6		539.2.e.n.177.3		8
28.19	even	6		539.2.e.n.67.4		8
28.23	odd	6		539.2.e.n.67.3		8
28.27	even	2		539.2.a.k.1.1	✓	4
44.43	even	2		5929.2.a.ba.1.4		4
84.83	odd	2		4851.2.a.bv.1.4		4
308.307	odd	2		5929.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.k.1.1	✓	4	28.27	even	2
539.2.a.k.1.2	yes	4	4.3	odd	2
539.2.e.n.67.3		8	28.23	odd	6
539.2.e.n.67.4		8	28.19	even	6
539.2.e.n.177.3		8	28.11	odd	6
539.2.e.n.177.4		8	28.3	even	6
4851.2.a.bv.1.3		4	12.11	even	2
4851.2.a.bv.1.4		4	84.83	odd	2
5929.2.a.ba.1.3		4	308.307	odd	2
5929.2.a.ba.1.4		4	44.43	even	2
8624.2.a.cu.1.2		4	1.1	even	1	trivial
8624.2.a.cu.1.3		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.19935 q^{3} +3.07221 q^{5} -1.56155 q^{9} +O(q^{10})\)	\(q-1.19935 q^{3} +3.07221 q^{5} -1.56155 q^{9} +1.00000 q^{11} +6.67026 q^{13} -3.68466 q^{15} +4.27156 q^{17} +2.39871 q^{19} +1.43845 q^{23} +4.43845 q^{25} +5.47091 q^{27} +2.00000 q^{29} +7.34376 q^{31} -1.19935 q^{33} +8.56155 q^{37} -8.00000 q^{39} +4.27156 q^{41} -6.24621 q^{43} -4.79741 q^{45} -4.27156 q^{47} -5.12311 q^{51} -2.00000 q^{53} +3.07221 q^{55} -2.87689 q^{57} -5.99676 q^{59} -8.01726 q^{61} +20.4924 q^{65} -7.68466 q^{67} -1.72521 q^{69} +16.8078 q^{71} +12.8147 q^{73} -5.32326 q^{75} -13.1231 q^{79} -1.87689 q^{81} +8.54312 q^{83} +13.1231 q^{85} -2.39871 q^{87} -12.6670 q^{89} -8.80776 q^{93} +7.36932 q^{95} +4.12391 q^{97} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} + 4 q^{11} + 10 q^{15} + 14 q^{23} + 26 q^{25} + 8 q^{29} + 26 q^{37} - 32 q^{39} + 8 q^{43} - 4 q^{51} - 8 q^{53} - 28 q^{57} + 16 q^{65} - 6 q^{67} + 26 q^{71} - 36 q^{79} - 24 q^{81} + 36 q^{85} + 6 q^{93} - 20 q^{95} + 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 4 * q^11 + 10 * q^15 + 14 * q^23 + 26 * q^25 + 8 * q^29 + 26 * q^37 - 32 * q^39 + 8 * q^43 - 4 * q^51 - 8 * q^53 - 28 * q^57 + 16 * q^65 - 6 * q^67 + 26 * q^71 - 36 * q^79 - 24 * q^81 + 36 * q^85 + 6 * q^93 - 20 * q^95 + 2 * q^99

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