LMFDB - Embedded newform 8624.2.a.df.1.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 $ x^10 - 26x^8 + 245x^6 - 1038x^4 + 1884x^2 - 968
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 539)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.903205$ 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)

$f(q)$	$=$	$q-0.903205 q^{3} -0.337987 q^{5} -2.18422 q^{9} +O(q^{10})$	$q-0.903205 q^{3} -0.337987 q^{5} -2.18422 q^{9} -1.00000 q^{11} -6.09040 q^{13} +0.305272 q^{15} -3.93795 q^{17} -8.10936 q^{19} -1.82729 q^{23} -4.88576 q^{25} +4.68241 q^{27} -6.61313 q^{29} -6.18413 q^{31} +0.903205 q^{33} -0.706234 q^{37} +5.50088 q^{39} +1.45556 q^{41} -2.45686 q^{43} +0.738239 q^{45} -4.70607 q^{47} +3.55678 q^{51} +6.15627 q^{53} +0.337987 q^{55} +7.32441 q^{57} -6.77443 q^{59} +4.28399 q^{61} +2.05848 q^{65} +8.68510 q^{67} +1.65041 q^{69} +4.68510 q^{71} -11.0713 q^{73} +4.41285 q^{75} -12.8253 q^{79} +2.32348 q^{81} +12.1808 q^{83} +1.33098 q^{85} +5.97301 q^{87} -0.337987 q^{89} +5.58554 q^{93} +2.74086 q^{95} -6.74935 q^{97} +2.18422 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 22 q^{9}+O(q^{10}) $ 10 * q + 22 * q^9	$ 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93} + 64 q^{95} - 22 q^{99}+O(q^{100}) $ 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * 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
$3$	−0.903205	−0.521466	−0.260733	−	0.965411i	$-0.583964\pi$
$3$	−0.903205	−0.521466	−0.260733	+	0.965411i	$0.583964\pi$
$4$	0	0
$5$	−0.337987	−0.151153	−0.0755763	−	0.997140i	$-0.524080\pi$
$5$	−0.337987	−0.151153	−0.0755763	+	0.997140i	$0.524080\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.18422	−0.728073
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.09040	−1.68917	−0.844587	−	0.535419i	$-0.820154\pi$
$13$	−6.09040	−1.68917	−0.844587	+	0.535419i	$0.820154\pi$
$14$	0	0
$15$	0.305272	0.0788209
$16$	0	0
$17$	−3.93795	−0.955093	−0.477547	−	0.878606i	$-0.658474\pi$
$17$	−3.93795	−0.955093	−0.477547	+	0.878606i	$0.658474\pi$
$18$	0	0
$19$	−8.10936	−1.86041	−0.930207	−	0.367035i	$-0.880373\pi$
$19$	−8.10936	−1.86041	−0.930207	+	0.367035i	$0.880373\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.82729	−0.381015	−0.190508	−	0.981686i	$-0.561013\pi$
$23$	−1.82729	−0.381015	−0.190508	+	0.981686i	$0.561013\pi$
$24$	0	0
$25$	−4.88576	−0.977153
$26$	0	0
$27$	4.68241	0.901131
$28$	0	0
$29$	−6.61313	−1.22803	−0.614014	−	0.789295i	$-0.710446\pi$
$29$	−6.61313	−1.22803	−0.614014	+	0.789295i	$0.710446\pi$
$30$	0	0
$31$	−6.18413	−1.11070	−0.555352	−	0.831616i	$-0.687416\pi$
$31$	−6.18413	−1.11070	−0.555352	+	0.831616i	$0.687416\pi$
$32$	0	0
$33$	0.903205	0.157228
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.706234	−0.116104	−0.0580521	−	0.998314i	$-0.518489\pi$
$37$	−0.706234	−0.116104	−0.0580521	+	0.998314i	$0.518489\pi$
$38$	0	0
$39$	5.50088	0.880846
$40$	0	0
$41$	1.45556	0.227321	0.113660	−	0.993520i	$-0.463742\pi$
$41$	1.45556	0.227321	0.113660	+	0.993520i	$0.463742\pi$
$42$	0	0
$43$	−2.45686	−0.374667	−0.187333	−	0.982296i	$-0.559985\pi$
$43$	−2.45686	−0.374667	−0.187333	+	0.982296i	$0.559985\pi$
$44$	0	0
$45$	0.738239	0.110050
$46$	0	0
$47$	−4.70607	−0.686451	−0.343226	−	0.939253i	$-0.611520\pi$
$47$	−4.70607	−0.686451	−0.343226	+	0.939253i	$0.611520\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.55678	0.498048
$52$	0	0
$53$	6.15627	0.845629	0.422815	−	0.906216i	$-0.361042\pi$
$53$	6.15627	0.845629	0.422815	+	0.906216i	$0.361042\pi$
$54$	0	0
$55$	0.337987	0.0455742
$56$	0	0
$57$	7.32441	0.970142
$58$	0	0
$59$	−6.77443	−0.881956	−0.440978	−	0.897518i	$-0.645368\pi$
$59$	−6.77443	−0.881956	−0.440978	+	0.897518i	$0.645368\pi$
$60$	0	0
$61$	4.28399	0.548509	0.274254	−	0.961657i	$-0.411569\pi$
$61$	4.28399	0.548509	0.274254	+	0.961657i	$0.411569\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.05848	0.255323
$66$	0	0
$67$	8.68510	1.06105	0.530527	−	0.847668i	$-0.321994\pi$
$67$	8.68510	1.06105	0.530527	+	0.847668i	$0.321994\pi$
$68$	0	0
$69$	1.65041	0.198686
$70$	0	0
$71$	4.68510	0.556019	0.278010	−	0.960578i	$-0.410325\pi$
$71$	4.68510	0.556019	0.278010	+	0.960578i	$0.410325\pi$
$72$	0	0
$73$	−11.0713	−1.29580	−0.647898	−	0.761727i	$-0.724352\pi$
$73$	−11.0713	−1.29580	−0.647898	+	0.761727i	$0.724352\pi$
$74$	0	0
$75$	4.41285	0.509552
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.8253	−1.44296	−0.721479	−	0.692436i	$-0.756537\pi$
$79$	−12.8253	−1.44296	−0.721479	+	0.692436i	$0.756537\pi$
$80$	0	0
$81$	2.32348	0.258164
$82$	0	0
$83$	12.1808	1.33702	0.668508	−	0.743705i	$-0.266933\pi$
$83$	12.1808	1.33702	0.668508	+	0.743705i	$0.266933\pi$
$84$	0	0
$85$	1.33098	0.144365
$86$	0	0
$87$	5.97301	0.640374
$88$	0	0
$89$	−0.337987	−0.0358266	−0.0179133	−	0.999840i	$-0.505702\pi$
$89$	−0.337987	−0.0358266	−0.0179133	+	0.999840i	$0.505702\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.58554	0.579194
$94$	0	0
$95$	2.74086	0.281206
$96$	0	0
$97$	−6.74935	−0.685293	−0.342646	−	0.939464i	$-0.611323\pi$
$97$	−6.74935	−0.685293	−0.342646	+	0.939464i	$0.611323\pi$
$98$	0	0
$99$	2.18422	0.219522
$100$	0	0
$101$	14.1699	1.40996	0.704978	−	0.709230i	$-0.250957\pi$
$101$	14.1699	1.40996	0.704978	+	0.709230i	$0.250957\pi$
$102$	0	0
$103$	−5.35216	−0.527364	−0.263682	−	0.964610i	$-0.584937\pi$
$103$	−5.35216	−0.527364	−0.263682	+	0.964610i	$0.584937\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.76976	0.847805	0.423902	−	0.905708i	$-0.360660\pi$
$107$	8.76976	0.847805	0.423902	+	0.905708i	$0.360660\pi$
$108$	0	0
$109$	10.7993	1.03439	0.517195	−	0.855868i	$-0.326976\pi$
$109$	10.7993	1.03439	0.517195	+	0.855868i	$0.326976\pi$
$110$	0	0
$111$	0.637874	0.0605443
$112$	0	0
$113$	9.26383	0.871468	0.435734	−	0.900076i	$-0.356489\pi$
$113$	9.26383	0.871468	0.435734	+	0.900076i	$0.356489\pi$
$114$	0	0
$115$	0.617599	0.0575914
$116$	0	0
$117$	13.3028	1.22984
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.31467	−0.118540
$124$	0	0
$125$	3.34126	0.298852
$126$	0	0
$127$	0.345429	0.0306518	0.0153259	−	0.999883i	$-0.495121\pi$
$127$	0.345429	0.0306518	0.0153259	+	0.999883i	$0.495121\pi$
$128$	0	0
$129$	2.21904	0.195376
$130$	0	0
$131$	−2.59497	−0.226724	−0.113362	−	0.993554i	$-0.536162\pi$
$131$	−2.59497	−0.226724	−0.113362	+	0.993554i	$0.536162\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.58260	−0.136208
$136$	0	0
$137$	−13.6199	−1.16363	−0.581815	−	0.813321i	$-0.697657\pi$
$137$	−13.6199	−1.16363	−0.581815	+	0.813321i	$0.697657\pi$
$138$	0	0
$139$	8.59787	0.729261	0.364631	−	0.931152i	$-0.381195\pi$
$139$	8.59787	0.729261	0.364631	+	0.931152i	$0.381195\pi$
$140$	0	0
$141$	4.25055	0.357961
$142$	0	0
$143$	6.09040	0.509305
$144$	0	0
$145$	2.23515	0.185619
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.5777	−0.866556	−0.433278	−	0.901260i	$-0.642643\pi$
$149$	−10.5777	−0.866556	−0.433278	+	0.901260i	$0.642643\pi$
$150$	0	0
$151$	3.91158	0.318320	0.159160	−	0.987253i	$-0.449121\pi$
$151$	3.91158	0.318320	0.159160	+	0.987253i	$0.449121\pi$
$152$	0	0
$153$	8.60135	0.695378
$154$	0	0
$155$	2.09016	0.167886
$156$	0	0
$157$	−8.43655	−0.673310	−0.336655	−	0.941628i	$-0.609296\pi$
$157$	−8.43655	−0.673310	−0.336655	+	0.941628i	$0.609296\pi$
$158$	0	0
$159$	−5.56038	−0.440967
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−23.6802	−1.85477	−0.927387	−	0.374103i	$-0.877950\pi$
$163$	−23.6802	−1.85477	−0.927387	+	0.374103i	$0.877950\pi$
$164$	0	0
$165$	−0.305272	−0.0237654
$166$	0	0
$167$	−3.58294	−0.277256	−0.138628	−	0.990345i	$-0.544269\pi$
$167$	−3.58294	−0.277256	−0.138628	+	0.990345i	$0.544269\pi$
$168$	0	0
$169$	24.0930	1.85331
$170$	0	0
$171$	17.7126	1.35452
$172$	0	0
$173$	−24.3066	−1.84800	−0.924000	−	0.382392i	$-0.875100\pi$
$173$	−24.3066	−1.84800	−0.924000	+	0.382392i	$0.875100\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.11870	0.459910
$178$	0	0
$179$	−11.8744	−0.887532	−0.443766	−	0.896143i	$-0.646358\pi$
$179$	−11.8744	−0.887532	−0.443766	+	0.896143i	$0.646358\pi$
$180$	0	0
$181$	7.41869	0.551427	0.275714	−	0.961240i	$-0.411086\pi$
$181$	7.41869	0.551427	0.275714	+	0.961240i	$0.411086\pi$
$182$	0	0
$183$	−3.86932	−0.286029
$184$	0	0
$185$	0.238698	0.0175494
$186$	0	0
$187$	3.93795	0.287971
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.1486	−0.734330	−0.367165	−	0.930156i	$-0.619672\pi$
$191$	−10.1486	−0.734330	−0.367165	+	0.930156i	$0.619672\pi$
$192$	0	0
$193$	24.1822	1.74068	0.870338	−	0.492456i	$-0.163901\pi$
$193$	24.1822	1.74068	0.870338	+	0.492456i	$0.163901\pi$
$194$	0	0
$195$	−1.85923	−0.133142
$196$	0	0
$197$	5.20325	0.370716	0.185358	−	0.982671i	$-0.440656\pi$
$197$	5.20325	0.370716	0.185358	+	0.982671i	$0.440656\pi$
$198$	0	0
$199$	−16.5234	−1.17131	−0.585657	−	0.810559i	$-0.699163\pi$
$199$	−16.5234	−1.17131	−0.585657	+	0.810559i	$0.699163\pi$
$200$	0	0
$201$	−7.84443	−0.553303
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.491962	−0.0343601
$206$	0	0
$207$	3.99119	0.277407
$208$	0	0
$209$	8.10936	0.560936
$210$	0	0
$211$	10.8350	0.745915	0.372957	−	0.927848i	$-0.378344\pi$
$211$	10.8350	0.745915	0.372957	+	0.927848i	$0.378344\pi$
$212$	0	0
$213$	−4.23161	−0.289945
$214$	0	0
$215$	0.830386	0.0566319
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.99964	0.675713
$220$	0	0
$221$	23.9837	1.61332
$222$	0	0
$223$	−6.91273	−0.462911	−0.231455	−	0.972846i	$-0.574349\pi$
$223$	−6.91273	−0.462911	−0.231455	+	0.972846i	$0.574349\pi$
$224$	0	0
$225$	10.6716	0.711439
$226$	0	0
$227$	9.85600	0.654166	0.327083	−	0.944996i	$-0.393934\pi$
$227$	9.85600	0.654166	0.327083	+	0.944996i	$0.393934\pi$
$228$	0	0
$229$	−18.6793	−1.23436	−0.617180	−	0.786822i	$-0.711725\pi$
$229$	−18.6793	−1.23436	−0.617180	+	0.786822i	$0.711725\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.83222	−0.185545	−0.0927725	−	0.995687i	$-0.529573\pi$
$233$	−2.83222	−0.185545	−0.0927725	+	0.995687i	$0.529573\pi$
$234$	0	0
$235$	1.59059	0.103759
$236$	0	0
$237$	11.5839	0.752454
$238$	0	0
$239$	−1.10930	−0.0717547	−0.0358773	−	0.999356i	$-0.511423\pi$
$239$	−1.10930	−0.0717547	−0.0358773	+	0.999356i	$0.511423\pi$
$240$	0	0
$241$	−19.1346	−1.23257	−0.616285	−	0.787523i	$-0.711363\pi$
$241$	−19.1346	−1.23257	−0.616285	+	0.787523i	$0.711363\pi$
$242$	0	0
$243$	−16.1458	−1.03576
$244$	0	0
$245$	0	0
$246$	0	0
$247$	49.3892	3.14256
$248$	0	0
$249$	−11.0018	−0.697209
$250$	0	0
$251$	−5.24482	−0.331050	−0.165525	−	0.986206i	$-0.552932\pi$
$251$	−5.24482	−0.331050	−0.165525	+	0.986206i	$0.552932\pi$
$252$	0	0
$253$	1.82729	0.114880
$254$	0	0
$255$	−1.20215	−0.0752813
$256$	0	0
$257$	−13.2315	−0.825361	−0.412681	−	0.910876i	$-0.635407\pi$
$257$	−13.2315	−0.825361	−0.412681	+	0.910876i	$0.635407\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	14.4445	0.894094
$262$	0	0
$263$	1.14218	0.0704300	0.0352150	−	0.999380i	$-0.488788\pi$
$263$	1.14218	0.0704300	0.0352150	+	0.999380i	$0.488788\pi$
$264$	0	0
$265$	−2.08074	−0.127819
$266$	0	0
$267$	0.305272	0.0186823
$268$	0	0
$269$	−24.1569	−1.47287	−0.736435	−	0.676508i	$-0.763492\pi$
$269$	−24.1569	−1.47287	−0.736435	+	0.676508i	$0.763492\pi$
$270$	0	0
$271$	−3.26678	−0.198443	−0.0992213	−	0.995065i	$-0.531635\pi$
$271$	−3.26678	−0.198443	−0.0992213	+	0.995065i	$0.531635\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.88576	0.294623
$276$	0	0
$277$	13.8079	0.829636	0.414818	−	0.909904i	$-0.363845\pi$
$277$	13.8079	0.829636	0.414818	+	0.909904i	$0.363845\pi$
$278$	0	0
$279$	13.5075	0.808674
$280$	0	0
$281$	−14.9163	−0.889832	−0.444916	−	0.895572i	$-0.646766\pi$
$281$	−14.9163	−0.889832	−0.444916	+	0.895572i	$0.646766\pi$
$282$	0	0
$283$	−12.4483	−0.739975	−0.369988	−	0.929037i	$-0.620638\pi$
$283$	−12.4483	−0.739975	−0.369988	+	0.929037i	$0.620638\pi$
$284$	0	0
$285$	−2.47556	−0.146639
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.49255	−0.0877972
$290$	0	0
$291$	6.09605	0.357357
$292$	0	0
$293$	28.4204	1.66034	0.830170	−	0.557511i	$-0.188244\pi$
$293$	28.4204	1.66034	0.830170	+	0.557511i	$0.188244\pi$
$294$	0	0
$295$	2.28967	0.133310
$296$	0	0
$297$	−4.68241	−0.271701
$298$	0	0
$299$	11.1289	0.643601
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.7983	−0.735243
$304$	0	0
$305$	−1.44793	−0.0829085
$306$	0	0
$307$	−11.0964	−0.633303	−0.316651	−	0.948542i	$-0.602559\pi$
$307$	−11.0964	−0.633303	−0.316651	+	0.948542i	$0.602559\pi$
$308$	0	0
$309$	4.83410	0.275002
$310$	0	0
$311$	−21.5709	−1.22317	−0.611586	−	0.791178i	$-0.709468\pi$
$311$	−21.5709	−1.22317	−0.611586	+	0.791178i	$0.709468\pi$
$312$	0	0
$313$	11.8986	0.672549	0.336275	−	0.941764i	$-0.390833\pi$
$313$	11.8986	0.672549	0.336275	+	0.941764i	$0.390833\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.96066	0.503281	0.251640	−	0.967821i	$-0.419030\pi$
$317$	8.96066	0.503281	0.251640	+	0.967821i	$0.419030\pi$
$318$	0	0
$319$	6.61313	0.370264
$320$	0	0
$321$	−7.92090	−0.442101
$322$	0	0
$323$	31.9342	1.77687
$324$	0	0
$325$	29.7563	1.65058
$326$	0	0
$327$	−9.75402	−0.539399
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−30.3961	−1.67072	−0.835362	−	0.549701i	$-0.814742\pi$
$331$	−30.3961	−1.67072	−0.835362	+	0.549701i	$0.814742\pi$
$332$	0	0
$333$	1.54257	0.0845323
$334$	0	0
$335$	−2.93546	−0.160381
$336$	0	0
$337$	−23.6574	−1.28870	−0.644350	−	0.764731i	$-0.722872\pi$
$337$	−23.6574	−1.28870	−0.644350	+	0.764731i	$0.722872\pi$
$338$	0	0
$339$	−8.36714	−0.454441
$340$	0	0
$341$	6.18413	0.334890
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.557819	−0.0300320
$346$	0	0
$347$	19.7160	1.05841	0.529205	−	0.848494i	$-0.322490\pi$
$347$	19.7160	1.05841	0.529205	+	0.848494i	$0.322490\pi$
$348$	0	0
$349$	−12.4053	−0.664040	−0.332020	−	0.943272i	$-0.607730\pi$
$349$	−12.4053	−0.664040	−0.332020	+	0.943272i	$0.607730\pi$
$350$	0	0
$351$	−28.5178	−1.52217
$352$	0	0
$353$	−15.4831	−0.824080	−0.412040	−	0.911166i	$-0.635184\pi$
$353$	−15.4831	−0.824080	−0.412040	+	0.911166i	$0.635184\pi$
$354$	0	0
$355$	−1.58351	−0.0840438
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.3206	1.07248	0.536241	−	0.844065i	$-0.319844\pi$
$359$	20.3206	1.07248	0.536241	+	0.844065i	$0.319844\pi$
$360$	0	0
$361$	46.7617	2.46114
$362$	0	0
$363$	−0.903205	−0.0474060
$364$	0	0
$365$	3.74195	0.195863
$366$	0	0
$367$	−18.9552	−0.989455	−0.494728	−	0.869048i	$-0.664732\pi$
$367$	−18.9552	−0.989455	−0.494728	+	0.869048i	$0.664732\pi$
$368$	0	0
$369$	−3.17927	−0.165506
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.634964	0.0328772	0.0164386	−	0.999865i	$-0.494767\pi$
$373$	0.634964	0.0328772	0.0164386	+	0.999865i	$0.494767\pi$
$374$	0	0
$375$	−3.01785	−0.155841
$376$	0	0
$377$	40.2766	2.07435
$378$	0	0
$379$	3.77139	0.193723	0.0968617	−	0.995298i	$-0.469120\pi$
$379$	3.77139	0.193723	0.0968617	+	0.995298i	$0.469120\pi$
$380$	0	0
$381$	−0.311993	−0.0159839
$382$	0	0
$383$	14.4360	0.737643	0.368822	−	0.929500i	$-0.379761\pi$
$383$	14.4360	0.737643	0.368822	+	0.929500i	$0.379761\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.36631	0.272785
$388$	0	0
$389$	−6.02971	−0.305719	−0.152859	−	0.988248i	$-0.548848\pi$
$389$	−6.02971	−0.305719	−0.152859	+	0.988248i	$0.548848\pi$
$390$	0	0
$391$	7.19576	0.363905
$392$	0	0
$393$	2.34379	0.118229
$394$	0	0
$395$	4.33479	0.218107
$396$	0	0
$397$	−7.19976	−0.361346	−0.180673	−	0.983543i	$-0.557828\pi$
$397$	−7.19976	−0.361346	−0.180673	+	0.983543i	$0.557828\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.0018	−0.749152	−0.374576	−	0.927196i	$-0.622212\pi$
$401$	−15.0018	−0.749152	−0.374576	+	0.927196i	$0.622212\pi$
$402$	0	0
$403$	37.6639	1.87617
$404$	0	0
$405$	−0.785307	−0.0390222
$406$	0	0
$407$	0.706234	0.0350067
$408$	0	0
$409$	−14.4208	−0.713061	−0.356530	−	0.934284i	$-0.616040\pi$
$409$	−14.4208	−0.713061	−0.356530	+	0.934284i	$0.616040\pi$
$410$	0	0
$411$	12.3016	0.606793
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.11696	−0.202094
$416$	0	0
$417$	−7.76564	−0.380285
$418$	0	0
$419$	−0.872247	−0.0426120	−0.0213060	−	0.999773i	$-0.506782\pi$
$419$	−0.872247	−0.0426120	−0.0213060	+	0.999773i	$0.506782\pi$
$420$	0	0
$421$	5.41939	0.264125	0.132063	−	0.991241i	$-0.457840\pi$
$421$	5.41939	0.264125	0.132063	+	0.991241i	$0.457840\pi$
$422$	0	0
$423$	10.2791	0.499787
$424$	0	0
$425$	19.2399	0.933272
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.50088	−0.265585
$430$	0	0
$431$	25.4358	1.22520	0.612601	−	0.790393i	$-0.290123\pi$
$431$	25.4358	1.22520	0.612601	+	0.790393i	$0.290123\pi$
$432$	0	0
$433$	−8.34258	−0.400919	−0.200459	−	0.979702i	$-0.564243\pi$
$433$	−8.34258	−0.400919	−0.200459	+	0.979702i	$0.564243\pi$
$434$	0	0
$435$	−2.01880	−0.0967942
$436$	0	0
$437$	14.8181	0.708846
$438$	0	0
$439$	−0.0528184	−0.00252089	−0.00126044	−	0.999999i	$-0.500401\pi$
$439$	−0.0528184	−0.00252089	−0.00126044	+	0.999999i	$0.500401\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.24258	0.391617	0.195808	−	0.980642i	$-0.437267\pi$
$443$	8.24258	0.391617	0.195808	+	0.980642i	$0.437267\pi$
$444$	0	0
$445$	0.114235	0.00541528
$446$	0	0
$447$	9.55380	0.451879
$448$	0	0
$449$	−30.4894	−1.43888	−0.719441	−	0.694553i	$-0.755602\pi$
$449$	−30.4894	−1.43888	−0.719441	+	0.694553i	$0.755602\pi$
$450$	0	0
$451$	−1.45556	−0.0685399
$452$	0	0
$453$	−3.53296	−0.165993
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.503010	0.0235298	0.0117649	−	0.999931i	$-0.496255\pi$
$457$	0.503010	0.0235298	0.0117649	+	0.999931i	$0.496255\pi$
$458$	0	0
$459$	−18.4391	−0.860664
$460$	0	0
$461$	2.29972	0.107109	0.0535544	−	0.998565i	$-0.482945\pi$
$461$	2.29972	0.107109	0.0535544	+	0.998565i	$0.482945\pi$
$462$	0	0
$463$	0.601027	0.0279321	0.0139661	−	0.999902i	$-0.495554\pi$
$463$	0.601027	0.0279321	0.0139661	+	0.999902i	$0.495554\pi$
$464$	0	0
$465$	−1.88784	−0.0875466
$466$	0	0
$467$	−12.9090	−0.597357	−0.298679	−	0.954354i	$-0.596546\pi$
$467$	−12.9090	−0.597357	−0.298679	+	0.954354i	$0.596546\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.61993	0.351108
$472$	0	0
$473$	2.45686	0.112966
$474$	0	0
$475$	39.6204	1.81791
$476$	0	0
$477$	−13.4467	−0.615680
$478$	0	0
$479$	12.3982	0.566486	0.283243	−	0.959048i	$-0.408590\pi$
$479$	12.3982	0.566486	0.283243	+	0.959048i	$0.408590\pi$
$480$	0	0
$481$	4.30125	0.196120
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.28120	0.103584
$486$	0	0
$487$	−25.2264	−1.14312	−0.571559	−	0.820561i	$-0.693661\pi$
$487$	−25.2264	−1.14312	−0.571559	+	0.820561i	$0.693661\pi$
$488$	0	0
$489$	21.3881	0.967201
$490$	0	0
$491$	3.77515	0.170370	0.0851850	−	0.996365i	$-0.472852\pi$
$491$	3.77515	0.170370	0.0851850	+	0.996365i	$0.472852\pi$
$492$	0	0
$493$	26.0422	1.17288
$494$	0	0
$495$	−0.738239	−0.0331814
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.5058	−0.649370	−0.324685	−	0.945822i	$-0.605258\pi$
$499$	−14.5058	−0.649370	−0.324685	+	0.945822i	$0.605258\pi$
$500$	0	0
$501$	3.23613	0.144580
$502$	0	0
$503$	39.9709	1.78222	0.891108	−	0.453791i	$-0.149929\pi$
$503$	39.9709	1.78222	0.891108	+	0.453791i	$0.149929\pi$
$504$	0	0
$505$	−4.78924	−0.213118
$506$	0	0
$507$	−21.7609	−0.966436
$508$	0	0
$509$	−34.7930	−1.54217	−0.771087	−	0.636730i	$-0.780287\pi$
$509$	−34.7930	−1.54217	−0.771087	+	0.636730i	$0.780287\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−37.9714	−1.67648
$514$	0	0
$515$	1.80896	0.0797125
$516$	0	0
$517$	4.70607	0.206973
$518$	0	0
$519$	21.9539	0.963669
$520$	0	0
$521$	25.9277	1.13591	0.567956	−	0.823059i	$-0.307735\pi$
$521$	25.9277	1.13591	0.567956	+	0.823059i	$0.307735\pi$
$522$	0	0
$523$	6.44125	0.281656	0.140828	−	0.990034i	$-0.455024\pi$
$523$	6.44125	0.281656	0.140828	+	0.990034i	$0.455024\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.3528	1.06083
$528$	0	0
$529$	−19.6610	−0.854827
$530$	0	0
$531$	14.7969	0.642129
$532$	0	0
$533$	−8.86497	−0.383985
$534$	0	0
$535$	−2.96407	−0.128148
$536$	0	0
$537$	10.7250	0.462818
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.2550	1.30076	0.650382	−	0.759607i	$-0.274609\pi$
$541$	30.2550	1.30076	0.650382	+	0.759607i	$0.274609\pi$
$542$	0	0
$543$	−6.70060	−0.287550
$544$	0	0
$545$	−3.65004	−0.156351
$546$	0	0
$547$	23.0018	0.983485	0.491742	−	0.870741i	$-0.336360\pi$
$547$	23.0018	0.983485	0.491742	+	0.870741i	$0.336360\pi$
$548$	0	0
$549$	−9.35718	−0.399355
$550$	0	0
$551$	53.6282	2.28464
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.215593	−0.00915143
$556$	0	0
$557$	−33.5794	−1.42281	−0.711403	−	0.702784i	$-0.751940\pi$
$557$	−33.5794	−1.42281	−0.711403	+	0.702784i	$0.751940\pi$
$558$	0	0
$559$	14.9632	0.632878
$560$	0	0
$561$	−3.55678	−0.150167
$562$	0	0
$563$	−42.4359	−1.78846	−0.894230	−	0.447608i	$-0.852276\pi$
$563$	−42.4359	−1.78846	−0.894230	+	0.447608i	$0.852276\pi$
$564$	0	0
$565$	−3.13106	−0.131725
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.1943	−1.72696	−0.863478	−	0.504387i	$-0.831719\pi$
$569$	−41.1943	−1.72696	−0.863478	+	0.504387i	$0.831719\pi$
$570$	0	0
$571$	14.9842	0.627067	0.313534	−	0.949577i	$-0.398487\pi$
$571$	14.9842	0.627067	0.313534	+	0.949577i	$0.398487\pi$
$572$	0	0
$573$	9.16630	0.382928
$574$	0	0
$575$	8.92769	0.372310
$576$	0	0
$577$	−8.33425	−0.346959	−0.173480	−	0.984837i	$-0.555501\pi$
$577$	−8.33425	−0.346959	−0.173480	+	0.984837i	$0.555501\pi$
$578$	0	0
$579$	−21.8415	−0.907703
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.15627	−0.254967
$584$	0	0
$585$	−4.49617	−0.185894
$586$	0	0
$587$	21.5300	0.888638	0.444319	−	0.895869i	$-0.353446\pi$
$587$	21.5300	0.888638	0.444319	+	0.895869i	$0.353446\pi$
$588$	0	0
$589$	50.1493	2.06637
$590$	0	0
$591$	−4.69960	−0.193316
$592$	0	0
$593$	−1.16929	−0.0480169	−0.0240085	−	0.999712i	$-0.507643\pi$
$593$	−1.16929	−0.0480169	−0.0240085	+	0.999712i	$0.507643\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.9240	0.610800
$598$	0	0
$599$	30.6169	1.25097	0.625486	−	0.780235i	$-0.284901\pi$
$599$	30.6169	1.25097	0.625486	+	0.780235i	$0.284901\pi$
$600$	0	0
$601$	0.608559	0.0248237	0.0124118	−	0.999923i	$-0.496049\pi$
$601$	0.608559	0.0248237	0.0124118	+	0.999923i	$0.496049\pi$
$602$	0	0
$603$	−18.9702	−0.772525
$604$	0	0
$605$	−0.337987	−0.0137411
$606$	0	0
$607$	−7.67100	−0.311356	−0.155678	−	0.987808i	$-0.549756\pi$
$607$	−7.67100	−0.311356	−0.155678	+	0.987808i	$0.549756\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.6619	1.15954
$612$	0	0
$613$	−23.4712	−0.947992	−0.473996	−	0.880527i	$-0.657189\pi$
$613$	−23.4712	−0.947992	−0.473996	+	0.880527i	$0.657189\pi$
$614$	0	0
$615$	0.444343	0.0179176
$616$	0	0
$617$	−5.37736	−0.216484	−0.108242	−	0.994125i	$-0.534522\pi$
$617$	−5.37736	−0.216484	−0.108242	+	0.994125i	$0.534522\pi$
$618$	0	0
$619$	41.7944	1.67986	0.839929	−	0.542697i	$-0.182597\pi$
$619$	41.7944	1.67986	0.839929	+	0.542697i	$0.182597\pi$
$620$	0	0
$621$	−8.55611	−0.343345
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.2995	0.931981
$626$	0	0
$627$	−7.32441	−0.292509
$628$	0	0
$629$	2.78111	0.110890
$630$	0	0
$631$	−0.0411038	−0.00163632	−0.000818159	−	1.00000i	$-0.500260\pi$
$631$	−0.0411038	−0.00163632	−0.000818159		1.00000i	$0.500260\pi$
$632$	0	0
$633$	−9.78626	−0.388969
$634$	0	0
$635$	−0.116751	−0.00463310
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.2333	−0.404823
$640$	0	0
$641$	−3.02724	−0.119569	−0.0597845	−	0.998211i	$-0.519041\pi$
$641$	−3.02724	−0.119569	−0.0597845	+	0.998211i	$0.519041\pi$
$642$	0	0
$643$	−20.8037	−0.820417	−0.410208	−	0.911992i	$-0.634544\pi$
$643$	−20.8037	−0.820417	−0.410208	+	0.911992i	$0.634544\pi$
$644$	0	0
$645$	−0.750009	−0.0295316
$646$	0	0
$647$	−21.1241	−0.830475	−0.415237	−	0.909713i	$-0.636302\pi$
$647$	−21.1241	−0.830475	−0.415237	+	0.909713i	$0.636302\pi$
$648$	0	0
$649$	6.77443	0.265920
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.6457	1.43406	0.717028	−	0.697045i	$-0.245502\pi$
$653$	36.6457	1.43406	0.717028	+	0.697045i	$0.245502\pi$
$654$	0	0
$655$	0.877067	0.0342698
$656$	0	0
$657$	24.1821	0.943434
$658$	0	0
$659$	47.5488	1.85224	0.926120	−	0.377230i	$-0.123123\pi$
$659$	47.5488	1.85224	0.926120	+	0.377230i	$0.123123\pi$
$660$	0	0
$661$	28.8388	1.12170	0.560849	−	0.827918i	$-0.310475\pi$
$661$	28.8388	1.12170	0.560849	+	0.827918i	$0.310475\pi$
$662$	0	0
$663$	−21.6622	−0.841290
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0841	0.467897
$668$	0	0
$669$	6.24362	0.241392
$670$	0	0
$671$	−4.28399	−0.165382
$672$	0	0
$673$	−0.747654	−0.0288199	−0.0144100	−	0.999896i	$-0.504587\pi$
$673$	−0.747654	−0.0288199	−0.0144100	+	0.999896i	$0.504587\pi$
$674$	0	0
$675$	−22.8772	−0.880543
$676$	0	0
$677$	−10.0797	−0.387393	−0.193696	−	0.981062i	$-0.562048\pi$
$677$	−10.0797	−0.387393	−0.193696	+	0.981062i	$0.562048\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.90199	−0.341125
$682$	0	0
$683$	−37.9385	−1.45168	−0.725838	−	0.687866i	$-0.758548\pi$
$683$	−37.9385	−1.45168	−0.725838	+	0.687866i	$0.758548\pi$
$684$	0	0
$685$	4.60337	0.175886
$686$	0	0
$687$	16.8712	0.643677
$688$	0	0
$689$	−37.4942	−1.42841
$690$	0	0
$691$	32.8262	1.24877	0.624384	−	0.781118i	$-0.285350\pi$
$691$	32.8262	1.24877	0.624384	+	0.781118i	$0.285350\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.90597	−0.110230
$696$	0	0
$697$	−5.73194	−0.217113
$698$	0	0
$699$	2.55808	0.0967553
$700$	0	0
$701$	−23.9841	−0.905865	−0.452933	−	0.891545i	$-0.649622\pi$
$701$	−23.9841	−0.905865	−0.452933	+	0.891545i	$0.649622\pi$
$702$	0	0
$703$	5.72710	0.216002
$704$	0	0
$705$	−1.43663	−0.0541067
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.5116	0.995663	0.497832	−	0.867274i	$-0.334130\pi$
$709$	26.5116	0.995663	0.497832	+	0.867274i	$0.334130\pi$
$710$	0	0
$711$	28.0133	1.05058
$712$	0	0
$713$	11.3002	0.423195
$714$	0	0
$715$	−2.05848	−0.0769827
$716$	0	0
$717$	1.00193	0.0374176
$718$	0	0
$719$	−38.5807	−1.43882	−0.719409	−	0.694587i	$-0.755587\pi$
$719$	−38.5807	−1.43882	−0.719409	+	0.694587i	$0.755587\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	17.2825	0.642744
$724$	0	0
$725$	32.3102	1.19997
$726$	0	0
$727$	26.7675	0.992752	0.496376	−	0.868108i	$-0.334664\pi$
$727$	26.7675	0.992752	0.496376	+	0.868108i	$0.334664\pi$
$728$	0	0
$729$	7.61256	0.281946
$730$	0	0
$731$	9.67497	0.357842
$732$	0	0
$733$	17.4234	0.643548	0.321774	−	0.946817i	$-0.395721\pi$
$733$	17.4234	0.643548	0.321774	+	0.946817i	$0.395721\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.68510	−0.319920
$738$	0	0
$739$	−10.5453	−0.387914	−0.193957	−	0.981010i	$-0.562132\pi$
$739$	−10.5453	−0.387914	−0.193957	+	0.981010i	$0.562132\pi$
$740$	0	0
$741$	−44.6086	−1.63874
$742$	0	0
$743$	8.55301	0.313780	0.156890	−	0.987616i	$-0.449853\pi$
$743$	8.55301	0.313780	0.156890	+	0.987616i	$0.449853\pi$
$744$	0	0
$745$	3.57512	0.130982
$746$	0	0
$747$	−26.6056	−0.973447
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−30.8901	−1.12720	−0.563598	−	0.826049i	$-0.690583\pi$
$751$	−30.8901	−1.12720	−0.563598	+	0.826049i	$0.690583\pi$
$752$	0	0
$753$	4.73715	0.172631
$754$	0	0
$755$	−1.32207	−0.0481149
$756$	0	0
$757$	45.3951	1.64991	0.824957	−	0.565195i	$-0.191199\pi$
$757$	45.3951	1.64991	0.824957	+	0.565195i	$0.191199\pi$
$758$	0	0
$759$	−1.65041	−0.0599062
$760$	0	0
$761$	−7.93220	−0.287542	−0.143771	−	0.989611i	$-0.545923\pi$
$761$	−7.93220	−0.287542	−0.143771	+	0.989611i	$0.545923\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.90715	−0.105108
$766$	0	0
$767$	41.2590	1.48978
$768$	0	0
$769$	22.8803	0.825086	0.412543	−	0.910938i	$-0.364641\pi$
$769$	22.8803	0.825086	0.412543	+	0.910938i	$0.364641\pi$
$770$	0	0
$771$	11.9508	0.430398
$772$	0	0
$773$	−47.5131	−1.70893	−0.854463	−	0.519511i	$-0.826114\pi$
$773$	−47.5131	−1.70893	−0.854463	+	0.519511i	$0.826114\pi$
$774$	0	0
$775$	30.2142	1.08533
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.8037	−0.422911
$780$	0	0
$781$	−4.68510	−0.167646
$782$	0	0
$783$	−30.9654	−1.10661
$784$	0	0
$785$	2.85145	0.101772
$786$	0	0
$787$	24.8735	0.886645	0.443323	−	0.896362i	$-0.353800\pi$
$787$	24.8735	0.886645	0.443323	+	0.896362i	$0.353800\pi$
$788$	0	0
$789$	−1.03163	−0.0367268
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.0912	−0.926527
$794$	0	0
$795$	1.87934	0.0666532
$796$	0	0
$797$	−51.0891	−1.80967	−0.904835	−	0.425763i	$-0.860006\pi$
$797$	−51.0891	−1.80967	−0.904835	+	0.425763i	$0.860006\pi$
$798$	0	0
$799$	18.5323	0.655625
$800$	0	0
$801$	0.738239	0.0260844
$802$	0	0
$803$	11.0713	0.390697
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.8186	0.768052
$808$	0	0
$809$	−7.27441	−0.255755	−0.127877	−	0.991790i	$-0.540816\pi$
$809$	−7.27441	−0.255755	−0.127877	+	0.991790i	$0.540816\pi$
$810$	0	0
$811$	16.0694	0.564274	0.282137	−	0.959374i	$-0.408957\pi$
$811$	16.0694	0.564274	0.282137	+	0.959374i	$0.408957\pi$
$812$	0	0
$813$	2.95057	0.103481
$814$	0	0
$815$	8.00360	0.280354
$816$	0	0
$817$	19.9235	0.697036
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.6506	−0.825411	−0.412706	−	0.910864i	$-0.635416\pi$
$821$	−23.6506	−0.825411	−0.412706	+	0.910864i	$0.635416\pi$
$822$	0	0
$823$	9.58412	0.334081	0.167041	−	0.985950i	$-0.446579\pi$
$823$	9.58412	0.334081	0.167041	+	0.985950i	$0.446579\pi$
$824$	0	0
$825$	−4.41285	−0.153636
$826$	0	0
$827$	−55.7714	−1.93936	−0.969681	−	0.244375i	$-0.921417\pi$
$827$	−55.7714	−1.93936	−0.969681	+	0.244375i	$0.921417\pi$
$828$	0	0
$829$	43.1792	1.49967	0.749837	−	0.661622i	$-0.230132\pi$
$829$	43.1792	1.49967	0.749837	+	0.661622i	$0.230132\pi$
$830$	0	0
$831$	−12.4714	−0.432627
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.21099	0.0419080
$836$	0	0
$837$	−28.9567	−1.00089
$838$	0	0
$839$	−47.6252	−1.64421	−0.822103	−	0.569339i	$-0.807199\pi$
$839$	−47.6252	−1.64421	−0.822103	+	0.569339i	$0.807199\pi$
$840$	0	0
$841$	14.7335	0.508051
$842$	0	0
$843$	13.4725	0.464017
$844$	0	0
$845$	−8.14313	−0.280132
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.2434	0.385872
$850$	0	0
$851$	1.29049	0.0442375
$852$	0	0
$853$	14.5131	0.496918	0.248459	−	0.968642i	$-0.420076\pi$
$853$	14.5131	0.496918	0.248459	+	0.968642i	$0.420076\pi$
$854$	0	0
$855$	−5.98664	−0.204739
$856$	0	0
$857$	−7.82656	−0.267350	−0.133675	−	0.991025i	$-0.542678\pi$
$857$	−7.82656	−0.267350	−0.133675	+	0.991025i	$0.542678\pi$
$858$	0	0
$859$	−20.7714	−0.708713	−0.354356	−	0.935110i	$-0.615300\pi$
$859$	−20.7714	−0.708713	−0.354356	+	0.935110i	$0.615300\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.9401	0.406447	0.203223	−	0.979132i	$-0.434858\pi$
$863$	11.9401	0.406447	0.203223	+	0.979132i	$0.434858\pi$
$864$	0	0
$865$	8.21534	0.279330
$866$	0	0
$867$	1.34808	0.0457833
$868$	0	0
$869$	12.8253	0.435068
$870$	0	0
$871$	−52.8958	−1.79230
$872$	0	0
$873$	14.7421	0.498944
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.3713	1.83599	0.917993	−	0.396596i	$-0.129808\pi$
$877$	54.3713	1.83599	0.917993	+	0.396596i	$0.129808\pi$
$878$	0	0
$879$	−25.6695	−0.865810
$880$	0	0
$881$	47.0818	1.58622	0.793112	−	0.609075i	$-0.208459\pi$
$881$	47.0818	1.58622	0.793112	+	0.609075i	$0.208459\pi$
$882$	0	0
$883$	−40.3391	−1.35752	−0.678759	−	0.734361i	$-0.737482\pi$
$883$	−40.3391	−1.35752	−0.678759	+	0.734361i	$0.737482\pi$
$884$	0	0
$885$	−2.06804	−0.0695165
$886$	0	0
$887$	−31.6268	−1.06192	−0.530962	−	0.847396i	$-0.678169\pi$
$887$	−31.6268	−1.06192	−0.530962	+	0.847396i	$0.678169\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.32348	−0.0778395
$892$	0	0
$893$	38.1632	1.27708
$894$	0	0
$895$	4.01339	0.134153
$896$	0	0
$897$	−10.0517	−0.335616
$898$	0	0
$899$	40.8965	1.36397
$900$	0	0
$901$	−24.2431	−0.807655
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.50743	−0.0833496
$906$	0	0
$907$	−16.6244	−0.552005	−0.276002	−	0.961157i	$-0.589010\pi$
$907$	−16.6244	−0.552005	−0.276002	+	0.961157i	$0.589010\pi$
$908$	0	0
$909$	−30.9501	−1.02655
$910$	0	0
$911$	−35.8816	−1.18881	−0.594406	−	0.804165i	$-0.702613\pi$
$911$	−35.8816	−1.18881	−0.594406	+	0.804165i	$0.702613\pi$
$912$	0	0
$913$	−12.1808	−0.403126
$914$	0	0
$915$	1.30778	0.0432340
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.75613	0.0909164	0.0454582	−	0.998966i	$-0.485525\pi$
$919$	2.75613	0.0909164	0.0454582	+	0.998966i	$0.485525\pi$
$920$	0	0
$921$	10.0223	0.330246
$922$	0	0
$923$	−28.5342	−0.939213
$924$	0	0
$925$	3.45049	0.113452
$926$	0	0
$927$	11.6903	0.383960
$928$	0	0
$929$	2.97025	0.0974506	0.0487253	−	0.998812i	$-0.484484\pi$
$929$	2.97025	0.0974506	0.0487253	+	0.998812i	$0.484484\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19.4829	0.637842
$934$	0	0
$935$	−1.33098	−0.0435276
$936$	0	0
$937$	−32.2479	−1.05349	−0.526747	−	0.850022i	$-0.676589\pi$
$937$	−32.2479	−1.05349	−0.526747	+	0.850022i	$0.676589\pi$
$938$	0	0
$939$	−10.7469	−0.350711
$940$	0	0
$941$	48.7786	1.59014	0.795069	−	0.606519i	$-0.207435\pi$
$941$	48.7786	1.59014	0.795069	+	0.606519i	$0.207435\pi$
$942$	0	0
$943$	−2.65973	−0.0866128
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.5517	−0.765328	−0.382664	−	0.923888i	$-0.624993\pi$
$947$	−23.5517	−0.765328	−0.382664	+	0.923888i	$0.624993\pi$
$948$	0	0
$949$	67.4286	2.18882
$950$	0	0
$951$	−8.09332	−0.262444
$952$	0	0
$953$	−25.5414	−0.827366	−0.413683	−	0.910421i	$-0.635758\pi$
$953$	−25.5414	−0.827366	−0.413683	+	0.910421i	$0.635758\pi$
$954$	0	0
$955$	3.43011	0.110996
$956$	0	0
$957$	−5.97301	−0.193080
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.24352	0.233662
$962$	0	0
$963$	−19.1551	−0.617264
$964$	0	0
$965$	−8.17329	−0.263107
$966$	0	0
$967$	−13.4875	−0.433728	−0.216864	−	0.976202i	$-0.569583\pi$
$967$	−13.4875	−0.433728	−0.216864	+	0.976202i	$0.569583\pi$
$968$	0	0
$969$	−28.8432	−0.926576
$970$	0	0
$971$	35.2268	1.13048	0.565240	−	0.824926i	$-0.308783\pi$
$971$	35.2268	1.13048	0.565240	+	0.824926i	$0.308783\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−26.8760	−0.860721
$976$	0	0
$977$	−16.6842	−0.533773	−0.266887	−	0.963728i	$-0.585995\pi$
$977$	−16.6842	−0.533773	−0.266887	+	0.963728i	$0.585995\pi$
$978$	0	0
$979$	0.337987	0.0108021
$980$	0	0
$981$	−23.5881	−0.753111
$982$	0	0
$983$	30.9021	0.985623	0.492812	−	0.870136i	$-0.335969\pi$
$983$	30.9021	0.985623	0.492812	+	0.870136i	$0.335969\pi$
$984$	0	0
$985$	−1.75863	−0.0560347
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.48938	0.142754
$990$	0	0
$991$	−42.7014	−1.35646	−0.678228	−	0.734852i	$-0.737252\pi$
$991$	−42.7014	−1.35646	−0.678228	+	0.734852i	$0.737252\pi$
$992$	0	0
$993$	27.4540	0.871225
$994$	0	0
$995$	5.58470	0.177047
$996$	0	0
$997$	−30.1349	−0.954380	−0.477190	−	0.878800i	$-0.658345\pi$
$997$	−30.1349	−0.954380	−0.477190	+	0.878800i	$0.658345\pi$
$998$	0	0
$999$	−3.30688	−0.104625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.df.1.5		10
4.3	odd	2		539.2.a.l.1.10	yes	10
7.6	odd	2	inner	8624.2.a.df.1.6		10
12.11	even	2		4851.2.a.cg.1.2		10
28.3	even	6		539.2.e.o.177.2		20
28.11	odd	6		539.2.e.o.177.1		20
28.19	even	6		539.2.e.o.67.2		20
28.23	odd	6		539.2.e.o.67.1		20
28.27	even	2		539.2.a.l.1.9	✓	10
44.43	even	2		5929.2.a.bv.1.2		10
84.83	odd	2		4851.2.a.cg.1.1		10
308.307	odd	2		5929.2.a.bv.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.l.1.9	✓	10	28.27	even	2
539.2.a.l.1.10	yes	10	4.3	odd	2
539.2.e.o.67.1		20	28.23	odd	6
539.2.e.o.67.2		20	28.19	even	6
539.2.e.o.177.1		20	28.11	odd	6
539.2.e.o.177.2		20	28.3	even	6
4851.2.a.cg.1.1		10	84.83	odd	2
4851.2.a.cg.1.2		10	12.11	even	2
5929.2.a.bv.1.1		10	308.307	odd	2
5929.2.a.bv.1.2		10	44.43	even	2
8624.2.a.df.1.5		10	1.1	even	1	trivial
8624.2.a.df.1.6		10	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)

\(f(q)\)	\(=\)	\(q-0.903205 q^{3} -0.337987 q^{5} -2.18422 q^{9} +O(q^{10})\)	\(q-0.903205 q^{3} -0.337987 q^{5} -2.18422 q^{9} -1.00000 q^{11} -6.09040 q^{13} +0.305272 q^{15} -3.93795 q^{17} -8.10936 q^{19} -1.82729 q^{23} -4.88576 q^{25} +4.68241 q^{27} -6.61313 q^{29} -6.18413 q^{31} +0.903205 q^{33} -0.706234 q^{37} +5.50088 q^{39} +1.45556 q^{41} -2.45686 q^{43} +0.738239 q^{45} -4.70607 q^{47} +3.55678 q^{51} +6.15627 q^{53} +0.337987 q^{55} +7.32441 q^{57} -6.77443 q^{59} +4.28399 q^{61} +2.05848 q^{65} +8.68510 q^{67} +1.65041 q^{69} +4.68510 q^{71} -11.0713 q^{73} +4.41285 q^{75} -12.8253 q^{79} +2.32348 q^{81} +12.1808 q^{83} +1.33098 q^{85} +5.97301 q^{87} -0.337987 q^{89} +5.58554 q^{93} +2.74086 q^{95} -6.74935 q^{97} +2.18422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 22 q^{9}+O(q^{10}) \) 10 * q + 22 * q^9	\( 10 q + 22 q^{9} - 10 q^{11} - 8 q^{15} - 4 q^{23} + 18 q^{25} + 12 q^{29} + 40 q^{37} + 16 q^{39} + 8 q^{43} + 16 q^{53} - 8 q^{57} - 32 q^{65} + 4 q^{67} - 36 q^{71} - 8 q^{79} - 6 q^{81} + 88 q^{85} + 44 q^{93} + 64 q^{95} - 22 q^{99}+O(q^{100}) \) 10 * q + 22 * q^9 - 10 * q^11 - 8 * q^15 - 4 * q^23 + 18 * q^25 + 12 * q^29 + 40 * q^37 + 16 * q^39 + 8 * q^43 + 16 * q^53 - 8 * q^57 - 32 * q^65 + 4 * q^67 - 36 * q^71 - 8 * q^79 - 6 * q^81 + 88 * q^85 + 44 * q^93 + 64 * q^95 - 22 * q^99

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