LMFDB - Embedded newform 8624.2.a.cu.1.4

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			1.4
Root			$0.662153$ 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)

$f(q)$	$=$	$q+2.35829 q^{3} +3.68260 q^{5} +2.56155 q^{9} +1.00000 q^{11} -3.39228 q^{13} +8.68466 q^{15} +1.32431 q^{17} -4.71659 q^{19} +5.56155 q^{23} +8.56155 q^{25} -1.03399 q^{27} +2.00000 q^{29} +5.00691 q^{31} +2.35829 q^{33} +4.43845 q^{37} -8.00000 q^{39} +1.32431 q^{41} +10.2462 q^{43} +9.43318 q^{45} -1.32431 q^{47} +3.12311 q^{51} -2.00000 q^{53} +3.68260 q^{55} -11.1231 q^{57} +11.7915 q^{59} -13.4061 q^{61} -12.4924 q^{65} +4.68466 q^{67} +13.1158 q^{69} -3.80776 q^{71} +3.97292 q^{73} +20.1907 q^{75} -4.87689 q^{79} -10.1231 q^{81} +2.64861 q^{83} +4.87689 q^{85} +4.71659 q^{87} +15.1838 q^{89} +11.8078 q^{93} -17.3693 q^{95} -17.8324 q^{97} +2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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
$3$	2.35829	1.36156	0.680781	−	0.732487i	$-0.261641\pi$
$3$	2.35829	1.36156	0.680781	+	0.732487i	$0.261641\pi$
$4$	0	0
$5$	3.68260	1.64691	0.823455	−	0.567382i	$-0.192044\pi$
$5$	3.68260	1.64691	0.823455	+	0.567382i	$0.192044\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.56155	0.853851
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.39228	−0.940850	−0.470425	−	0.882440i	$-0.655899\pi$
$13$	−3.39228	−0.940850	−0.470425	+	0.882440i	$0.655899\pi$
$14$	0	0
$15$	8.68466	2.24237
$16$	0	0
$17$	1.32431	0.321192	0.160596	−	0.987020i	$-0.448658\pi$
$17$	1.32431	0.321192	0.160596	+	0.987020i	$0.448658\pi$
$18$	0	0
$19$	−4.71659	−1.08206	−0.541030	−	0.841003i	$-0.681965\pi$
$19$	−4.71659	−1.08206	−0.541030	+	0.841003i	$0.681965\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	8.56155	1.71231
$26$	0	0
$27$	−1.03399	−0.198991
$28$	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
$31$	5.00691	0.899267	0.449634	−	0.893213i	$-0.351555\pi$
$31$	5.00691	0.899267	0.449634	+	0.893213i	$0.351555\pi$
$32$	0	0
$33$	2.35829	0.410526
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.43845	0.729676	0.364838	−	0.931071i	$-0.381124\pi$
$37$	4.43845	0.729676	0.364838	+	0.931071i	$0.381124\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	1.32431	0.206822	0.103411	−	0.994639i	$-0.467024\pi$
$41$	1.32431	0.206822	0.103411	+	0.994639i	$0.467024\pi$
$42$	0	0
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	0	0
$45$	9.43318	1.40622
$46$	0	0
$47$	−1.32431	−0.193170	−0.0965850	−	0.995325i	$-0.530792\pi$
$47$	−1.32431	−0.193170	−0.0965850	+	0.995325i	$0.530792\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.12311	0.437322
$52$	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
$55$	3.68260	0.496562
$56$	0	0
$57$	−11.1231	−1.47329
$58$	0	0
$59$	11.7915	1.53512	0.767559	−	0.640978i	$-0.221471\pi$
$59$	11.7915	1.53512	0.767559	+	0.640978i	$0.221471\pi$
$60$	0	0
$61$	−13.4061	−1.71648	−0.858238	−	0.513253i	$-0.828440\pi$
$61$	−13.4061	−1.71648	−0.858238	+	0.513253i	$0.828440\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.4924	−1.54949
$66$	0	0
$67$	4.68466	0.572322	0.286161	−	0.958182i	$-0.407621\pi$
$67$	4.68466	0.572322	0.286161	+	0.958182i	$0.407621\pi$
$68$	0	0
$69$	13.1158	1.57895
$70$	0	0
$71$	−3.80776	−0.451898	−0.225949	−	0.974139i	$-0.572548\pi$
$71$	−3.80776	−0.451898	−0.225949	+	0.974139i	$0.572548\pi$
$72$	0	0
$73$	3.97292	0.464995	0.232498	−	0.972597i	$-0.425310\pi$
$73$	3.97292	0.464995	0.232498	+	0.972597i	$0.425310\pi$
$74$	0	0
$75$	20.1907	2.33142
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	0	0
$81$	−10.1231	−1.12479
$82$	0	0
$83$	2.64861	0.290723	0.145362	−	0.989379i	$-0.453565\pi$
$83$	2.64861	0.290723	0.145362	+	0.989379i	$0.453565\pi$
$84$	0	0
$85$	4.87689	0.528973
$86$	0	0
$87$	4.71659	0.505671
$88$	0	0
$89$	15.1838	1.60947	0.804737	−	0.593631i	$-0.202306\pi$
$89$	15.1838	1.60947	0.804737	+	0.593631i	$0.202306\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.8078	1.22441
$94$	0	0
$95$	−17.3693	−1.78205
$96$	0	0
$97$	−17.8324	−1.81060	−0.905301	−	0.424770i	$-0.860355\pi$
$97$	−17.8324	−1.81060	−0.905301	+	0.424770i	$0.860355\pi$
$98$	0	0
$99$	2.56155	0.257446
$100$	0	0
$101$	0.743668	0.0739978	0.0369989	−	0.999315i	$-0.488220\pi$
$101$	0.743668	0.0739978	0.0369989	+	0.999315i	$0.488220\pi$
$102$	0	0
$103$	8.10887	0.798991	0.399495	−	0.916735i	$-0.369185\pi$
$103$	8.10887	0.798991	0.399495	+	0.916735i	$0.369185\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.12311	−0.688617	−0.344308	−	0.938857i	$-0.611887\pi$
$107$	−7.12311	−0.688617	−0.344308	+	0.938857i	$0.611887\pi$
$108$	0	0
$109$	19.3693	1.85524	0.927622	−	0.373520i	$-0.121849\pi$
$109$	19.3693	1.85524	0.927622	+	0.373520i	$0.121849\pi$
$110$	0	0
$111$	10.4672	0.993499
$112$	0	0
$113$	5.80776	0.546348	0.273174	−	0.961965i	$-0.411926\pi$
$113$	5.80776	0.546348	0.273174	+	0.961965i	$0.411926\pi$
$114$	0	0
$115$	20.4810	1.90986
$116$	0	0
$117$	−8.68951	−0.803345
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.12311	0.281601
$124$	0	0
$125$	13.1158	1.17311
$126$	0	0
$127$	4.87689	0.432754	0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689	0.432754	0.216377	+	0.976310i	$0.430576\pi$
$128$	0	0
$129$	24.1636	2.12748
$130$	0	0
$131$	−7.36520	−0.643501	−0.321750	−	0.946825i	$-0.604271\pi$
$131$	−7.36520	−0.643501	−0.321750	+	0.946825i	$0.604271\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.80776	−0.327720
$136$	0	0
$137$	−4.43845	−0.379202	−0.189601	−	0.981861i	$-0.560719\pi$
$137$	−4.43845	−0.379202	−0.189601	+	0.981861i	$0.560719\pi$
$138$	0	0
$139$	−4.71659	−0.400056	−0.200028	−	0.979790i	$-0.564103\pi$
$139$	−4.71659	−0.400056	−0.200028	+	0.979790i	$0.564103\pi$
$140$	0	0
$141$	−3.12311	−0.263013
$142$	0	0
$143$	−3.39228	−0.283677
$144$	0	0
$145$	7.36520	0.611647
$146$	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
$151$	−17.3693	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$151$	−17.3693	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$152$	0	0
$153$	3.39228	0.274250
$154$	0	0
$155$	18.4384	1.48101
$156$	0	0
$157$	5.75058	0.458946	0.229473	−	0.973315i	$-0.426300\pi$
$157$	5.75058	0.458946	0.229473	+	0.973315i	$0.426300\pi$
$158$	0	0
$159$	−4.71659	−0.374050
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.49242	−0.665178	−0.332589	−	0.943072i	$-0.607922\pi$
$163$	−8.49242	−0.665178	−0.332589	+	0.943072i	$0.607922\pi$
$164$	0	0
$165$	8.68466	0.676100
$166$	0	0
$167$	12.6624	0.979848	0.489924	−	0.871765i	$-0.337024\pi$
$167$	12.6624	0.979848	0.489924	+	0.871765i	$0.337024\pi$
$168$	0	0
$169$	−1.49242	−0.114802
$170$	0	0
$171$	−12.0818	−0.923918
$172$	0	0
$173$	15.4741	1.17647	0.588236	−	0.808689i	$-0.299823\pi$
$173$	15.4741	1.17647	0.588236	+	0.808689i	$0.299823\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	27.8078	2.09016
$178$	0	0
$179$	15.8078	1.18153	0.590764	−	0.806844i	$-0.298826\pi$
$179$	15.8078	1.18153	0.590764	+	0.806844i	$0.298826\pi$
$180$	0	0
$181$	10.4672	0.778018	0.389009	−	0.921234i	$-0.372817\pi$
$181$	10.4672	0.778018	0.389009	+	0.921234i	$0.372817\pi$
$182$	0	0
$183$	−31.6155	−2.33709
$184$	0	0
$185$	16.3450	1.20171
$186$	0	0
$187$	1.32431	0.0968429
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.8078	0.854380	0.427190	−	0.904162i	$-0.359504\pi$
$191$	11.8078	0.854380	0.427190	+	0.904162i	$0.359504\pi$
$192$	0	0
$193$	−26.4924	−1.90697	−0.953483	−	0.301446i	$-0.902531\pi$
$193$	−26.4924	−1.90697	−0.953483	+	0.301446i	$0.902531\pi$
$194$	0	0
$195$	−29.4608	−2.10973
$196$	0	0
$197$	12.2462	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$197$	12.2462	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$198$	0	0
$199$	−20.1907	−1.43128	−0.715639	−	0.698470i	$-0.753864\pi$
$199$	−20.1907	−1.43128	−0.715639	+	0.698470i	$0.753864\pi$
$200$	0	0
$201$	11.0478	0.779252
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.87689	0.340617
$206$	0	0
$207$	14.2462	0.990180
$208$	0	0
$209$	−4.71659	−0.326253
$210$	0	0
$211$	−21.3693	−1.47112	−0.735562	−	0.677457i	$-0.763082\pi$
$211$	−21.3693	−1.47112	−0.735562	+	0.677457i	$0.763082\pi$
$212$	0	0
$213$	−8.97983	−0.615288
$214$	0	0
$215$	37.7327	2.57335
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.36932	0.633120
$220$	0	0
$221$	−4.49242	−0.302193
$222$	0	0
$223$	−19.7373	−1.32171	−0.660854	−	0.750514i	$-0.729806\pi$
$223$	−19.7373	−1.32171	−0.660854	+	0.750514i	$0.729806\pi$
$224$	0	0
$225$	21.9309	1.46206
$226$	0	0
$227$	−18.8664	−1.25220	−0.626102	−	0.779741i	$-0.715351\pi$
$227$	−18.8664	−1.25220	−0.626102	+	0.779741i	$0.715351\pi$
$228$	0	0
$229$	8.39919	0.555034	0.277517	−	0.960721i	$-0.410488\pi$
$229$	8.39919	0.555034	0.277517	+	0.960721i	$0.410488\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.2462	0.802276	0.401138	−	0.916018i	$-0.368615\pi$
$233$	12.2462	0.802276	0.401138	+	0.916018i	$0.368615\pi$
$234$	0	0
$235$	−4.87689	−0.318134
$236$	0	0
$237$	−11.5012	−0.747080
$238$	0	0
$239$	6.24621	0.404034	0.202017	−	0.979382i	$-0.435250\pi$
$239$	6.24621	0.404034	0.202017	+	0.979382i	$0.435250\pi$
$240$	0	0
$241$	−11.3381	−0.730353	−0.365176	−	0.930938i	$-0.618991\pi$
$241$	−11.3381	−0.730353	−0.365176	+	0.930938i	$0.618991\pi$
$242$	0	0
$243$	−20.7713	−1.33248
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	6.24621	0.395838
$250$	0	0
$251$	−15.9274	−1.00533	−0.502665	−	0.864481i	$-0.667647\pi$
$251$	−15.9274	−1.00533	−0.502665	+	0.864481i	$0.667647\pi$
$252$	0	0
$253$	5.56155	0.349652
$254$	0	0
$255$	11.5012	0.720230
$256$	0	0
$257$	24.1636	1.50728	0.753641	−	0.657286i	$-0.228296\pi$
$257$	24.1636	1.50728	0.753641	+	0.657286i	$0.228296\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.12311	0.317112
$262$	0	0
$263$	1.36932	0.0844357	0.0422178	−	0.999108i	$-0.486558\pi$
$263$	1.36932	0.0844357	0.0422178	+	0.999108i	$0.486558\pi$
$264$	0	0
$265$	−7.36520	−0.452441
$266$	0	0
$267$	35.8078	2.19140
$268$	0	0
$269$	−26.8122	−1.63477	−0.817384	−	0.576093i	$-0.804577\pi$
$269$	−26.8122	−1.63477	−0.817384	+	0.576093i	$0.804577\pi$
$270$	0	0
$271$	−6.78456	−0.412133	−0.206066	−	0.978538i	$-0.566066\pi$
$271$	−6.78456	−0.412133	−0.206066	+	0.978538i	$0.566066\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.56155	0.516281
$276$	0	0
$277$	−17.6155	−1.05841	−0.529207	−	0.848493i	$-0.677511\pi$
$277$	−17.6155	−1.05841	−0.529207	+	0.848493i	$0.677511\pi$
$278$	0	0
$279$	12.8255	0.767840
$280$	0	0
$281$	29.6155	1.76671	0.883357	−	0.468700i	$-0.155278\pi$
$281$	29.6155	1.76671	0.883357	+	0.468700i	$0.155278\pi$
$282$	0	0
$283$	18.8664	1.12149	0.560744	−	0.827989i	$-0.310515\pi$
$283$	18.8664	1.12149	0.560744	+	0.827989i	$0.310515\pi$
$284$	0	0
$285$	−40.9620	−2.42638
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.2462	−0.896836
$290$	0	0
$291$	−42.0540	−2.46525
$292$	0	0
$293$	−17.5420	−1.02482	−0.512409	−	0.858742i	$-0.671247\pi$
$293$	−17.5420	−1.02482	−0.512409	+	0.858742i	$0.671247\pi$
$294$	0	0
$295$	43.4233	2.52820
$296$	0	0
$297$	−1.03399	−0.0599980
$298$	0	0
$299$	−18.8664	−1.09107
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.75379	0.100753
$304$	0	0
$305$	−49.3693	−2.82688
$306$	0	0
$307$	−9.43318	−0.538380	−0.269190	−	0.963087i	$-0.586756\pi$
$307$	−9.43318	−0.538380	−0.269190	+	0.963087i	$0.586756\pi$
$308$	0	0
$309$	19.1231	1.08788
$310$	0	0
$311$	12.2448	0.694340	0.347170	−	0.937802i	$-0.387143\pi$
$311$	12.2448	0.694340	0.347170	+	0.937802i	$0.387143\pi$
$312$	0	0
$313$	−0.453349	−0.0256248	−0.0128124	−	0.999918i	$-0.504078\pi$
$313$	−0.453349	−0.0256248	−0.0128124	+	0.999918i	$0.504078\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.80776	0.101534	0.0507671	−	0.998711i	$-0.483833\pi$
$317$	1.80776	0.101534	0.0507671	+	0.998711i	$0.483833\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−16.7984	−0.937594
$322$	0	0
$323$	−6.24621	−0.347548
$324$	0	0
$325$	−29.0432	−1.61103
$326$	0	0
$327$	45.6786	2.52603
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.80776	−0.429154	−0.214577	−	0.976707i	$-0.568837\pi$
$331$	−7.80776	−0.429154	−0.214577	+	0.976707i	$0.568837\pi$
$332$	0	0
$333$	11.3693	0.623035
$334$	0	0
$335$	17.2517	0.942563
$336$	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
$339$	13.6964	0.743887
$340$	0	0
$341$	5.00691	0.271139
$342$	0	0
$343$	0	0
$344$	0	0
$345$	48.3002	2.60039
$346$	0	0
$347$	−7.12311	−0.382388	−0.191194	−	0.981552i	$-0.561236\pi$
$347$	−7.12311	−0.382388	−0.191194	+	0.981552i	$0.561236\pi$
$348$	0	0
$349$	14.8934	0.797227	0.398614	−	0.917119i	$-0.369491\pi$
$349$	14.8934	0.797227	0.398614	+	0.917119i	$0.369491\pi$
$350$	0	0
$351$	3.50758	0.187221
$352$	0	0
$353$	−11.0478	−0.588015	−0.294008	−	0.955803i	$-0.594989\pi$
$353$	−11.0478	−0.588015	−0.294008	+	0.955803i	$0.594989\pi$
$354$	0	0
$355$	−14.0225	−0.744236
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.24621	0.329662	0.164831	−	0.986322i	$-0.447292\pi$
$359$	6.24621	0.329662	0.164831	+	0.986322i	$0.447292\pi$
$360$	0	0
$361$	3.24621	0.170853
$362$	0	0
$363$	2.35829	0.123778
$364$	0	0
$365$	14.6307	0.765805
$366$	0	0
$367$	−23.8733	−1.24617	−0.623087	−	0.782152i	$-0.714122\pi$
$367$	−23.8733	−1.24617	−0.623087	+	0.782152i	$0.714122\pi$
$368$	0	0
$369$	3.39228	0.176595
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.36932	0.381569	0.190784	−	0.981632i	$-0.438897\pi$
$373$	7.36932	0.381569	0.190784	+	0.981632i	$0.438897\pi$
$374$	0	0
$375$	30.9309	1.59726
$376$	0	0
$377$	−6.78456	−0.349423
$378$	0	0
$379$	−28.6847	−1.47343	−0.736716	−	0.676202i	$-0.763625\pi$
$379$	−28.6847	−1.47343	−0.736716	+	0.676202i	$0.763625\pi$
$380$	0	0
$381$	11.5012	0.589222
$382$	0	0
$383$	10.3041	0.526517	0.263258	−	0.964725i	$-0.415203\pi$
$383$	10.3041	0.526517	0.263258	+	0.964725i	$0.415203\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.2462	1.33417
$388$	0	0
$389$	−32.0540	−1.62520	−0.812601	−	0.582821i	$-0.801949\pi$
$389$	−32.0540	−1.62520	−0.812601	+	0.582821i	$0.801949\pi$
$390$	0	0
$391$	7.36520	0.372474
$392$	0	0
$393$	−17.3693	−0.876166
$394$	0	0
$395$	−17.9597	−0.903648
$396$	0	0
$397$	−20.0276	−1.00516	−0.502579	−	0.864531i	$-0.667615\pi$
$397$	−20.0276	−1.00516	−0.502579	+	0.864531i	$0.667615\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.4924	−0.723717	−0.361859	−	0.932233i	$-0.617858\pi$
$401$	−14.4924	−0.723717	−0.361859	+	0.932233i	$0.617858\pi$
$402$	0	0
$403$	−16.9848	−0.846075
$404$	0	0
$405$	−37.2794	−1.85243
$406$	0	0
$407$	4.43845	0.220006
$408$	0	0
$409$	−25.4879	−1.26030	−0.630148	−	0.776475i	$-0.717006\pi$
$409$	−25.4879	−1.26030	−0.630148	+	0.776475i	$0.717006\pi$
$410$	0	0
$411$	−10.4672	−0.516307
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.75379	0.478795
$416$	0	0
$417$	−11.1231	−0.544701
$418$	0	0
$419$	−10.7575	−0.525538	−0.262769	−	0.964859i	$-0.584636\pi$
$419$	−10.7575	−0.525538	−0.262769	+	0.964859i	$0.584636\pi$
$420$	0	0
$421$	20.2462	0.986740	0.493370	−	0.869820i	$-0.335765\pi$
$421$	20.2462	0.986740	0.493370	+	0.869820i	$0.335765\pi$
$422$	0	0
$423$	−3.39228	−0.164938
$424$	0	0
$425$	11.3381	0.549980
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	20.4924	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$431$	20.4924	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$432$	0	0
$433$	5.16994	0.248451	0.124226	−	0.992254i	$-0.460355\pi$
$433$	5.16994	0.248451	0.124226	+	0.992254i	$0.460355\pi$
$434$	0	0
$435$	17.3693	0.832795
$436$	0	0
$437$	−26.2316	−1.25483
$438$	0	0
$439$	−34.1774	−1.63120	−0.815599	−	0.578617i	$-0.803592\pi$
$439$	−34.1774	−1.63120	−0.815599	+	0.578617i	$0.803592\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.3153	0.537608	0.268804	−	0.963195i	$-0.413372\pi$
$443$	11.3153	0.537608	0.268804	+	0.963195i	$0.413372\pi$
$444$	0	0
$445$	55.9157	2.65066
$446$	0	0
$447$	14.1498	0.669261
$448$	0	0
$449$	−27.1771	−1.28257	−0.641283	−	0.767305i	$-0.721597\pi$
$449$	−27.1771	−1.28257	−0.641283	+	0.767305i	$0.721597\pi$
$450$	0	0
$451$	1.32431	0.0623592
$452$	0	0
$453$	−40.9620	−1.92456
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.8617	−0.929093	−0.464546	−	0.885549i	$-0.653783\pi$
$457$	−19.8617	−0.929093	−0.464546	+	0.885549i	$0.653783\pi$
$458$	0	0
$459$	−1.36932	−0.0639142
$460$	0	0
$461$	6.62153	0.308396	0.154198	−	0.988040i	$-0.450721\pi$
$461$	6.62153	0.308396	0.154198	+	0.988040i	$0.450721\pi$
$462$	0	0
$463$	6.93087	0.322105	0.161052	−	0.986946i	$-0.448511\pi$
$463$	6.93087	0.322105	0.161052	+	0.986946i	$0.448511\pi$
$464$	0	0
$465$	43.4833	2.01649
$466$	0	0
$467$	16.5081	0.763902	0.381951	−	0.924183i	$-0.375252\pi$
$467$	16.5081	0.763902	0.381951	+	0.924183i	$0.375252\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	13.5616	0.624883
$472$	0	0
$473$	10.2462	0.471121
$474$	0	0
$475$	−40.3813	−1.85282
$476$	0	0
$477$	−5.12311	−0.234571
$478$	0	0
$479$	−40.3813	−1.84507	−0.922535	−	0.385914i	$-0.873886\pi$
$479$	−40.3813	−1.84507	−0.922535	+	0.385914i	$0.873886\pi$
$480$	0	0
$481$	−15.0565	−0.686516
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−65.6695	−2.98190
$486$	0	0
$487$	−10.4384	−0.473011	−0.236506	−	0.971630i	$-0.576002\pi$
$487$	−10.4384	−0.473011	−0.236506	+	0.971630i	$0.576002\pi$
$488$	0	0
$489$	−20.0276	−0.905681
$490$	0	0
$491$	−5.36932	−0.242314	−0.121157	−	0.992633i	$-0.538660\pi$
$491$	−5.36932	−0.242314	−0.121157	+	0.992633i	$0.538660\pi$
$492$	0	0
$493$	2.64861	0.119288
$494$	0	0
$495$	9.43318	0.423990
$496$	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
$501$	29.8617	1.33412
$502$	0	0
$503$	26.8122	1.19550	0.597748	−	0.801684i	$-0.296062\pi$
$503$	26.8122	1.19550	0.597748	+	0.801684i	$0.296062\pi$
$504$	0	0
$505$	2.73863	0.121868
$506$	0	0
$507$	−3.51957	−0.156310
$508$	0	0
$509$	6.33122	0.280626	0.140313	−	0.990107i	$-0.455189\pi$
$509$	6.33122	0.280626	0.140313	+	0.990107i	$0.455189\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.87689	0.215320
$514$	0	0
$515$	29.8617	1.31587
$516$	0	0
$517$	−1.32431	−0.0582430
$518$	0	0
$519$	36.4924	1.60184
$520$	0	0
$521$	−1.03399	−0.0452998	−0.0226499	−	0.999743i	$-0.507210\pi$
$521$	−1.03399	−0.0452998	−0.0226499	+	0.999743i	$0.507210\pi$
$522$	0	0
$523$	−14.7304	−0.644116	−0.322058	−	0.946720i	$-0.604375\pi$
$523$	−14.7304	−0.644116	−0.322058	+	0.946720i	$0.604375\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.63068	0.288837
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	30.2045	1.31076
$532$	0	0
$533$	−4.49242	−0.194588
$534$	0	0
$535$	−26.2316	−1.13409
$536$	0	0
$537$	37.2794	1.60872
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.1231	1.76802	0.884010	−	0.467468i	$-0.154834\pi$
$541$	41.1231	1.76802	0.884010	+	0.467468i	$0.154834\pi$
$542$	0	0
$543$	24.6847	1.05932
$544$	0	0
$545$	71.3295	3.05542
$546$	0	0
$547$	−18.2462	−0.780152	−0.390076	−	0.920783i	$-0.627551\pi$
$547$	−18.2462	−0.780152	−0.390076	+	0.920783i	$0.627551\pi$
$548$	0	0
$549$	−34.3404	−1.46561
$550$	0	0
$551$	−9.43318	−0.401867
$552$	0	0
$553$	0	0
$554$	0	0
$555$	38.5464	1.63620
$556$	0	0
$557$	−8.24621	−0.349403	−0.174702	−	0.984621i	$-0.555896\pi$
$557$	−8.24621	−0.349403	−0.174702	+	0.984621i	$0.555896\pi$
$558$	0	0
$559$	−34.7580	−1.47011
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	−16.2177	−0.683496	−0.341748	−	0.939792i	$-0.611019\pi$
$563$	−16.2177	−0.683496	−0.341748	+	0.939792i	$0.611019\pi$
$564$	0	0
$565$	21.3877	0.899786
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.7538	0.660433	0.330217	−	0.943905i	$-0.392878\pi$
$569$	15.7538	0.660433	0.330217	+	0.943905i	$0.392878\pi$
$570$	0	0
$571$	−26.2462	−1.09837	−0.549185	−	0.835701i	$-0.685062\pi$
$571$	−26.2462	−1.09837	−0.549185	+	0.835701i	$0.685062\pi$
$572$	0	0
$573$	27.8462	1.16329
$574$	0	0
$575$	47.6155	1.98570
$576$	0	0
$577$	21.0616	0.876807	0.438403	−	0.898778i	$-0.355544\pi$
$577$	21.0616	0.876807	0.438403	+	0.898778i	$0.355544\pi$
$578$	0	0
$579$	−62.4769	−2.59645
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−32.0000	−1.32304
$586$	0	0
$587$	39.0570	1.61206	0.806028	−	0.591878i	$-0.201613\pi$
$587$	39.0570	1.61206	0.806028	+	0.591878i	$0.201613\pi$
$588$	0	0
$589$	−23.6155	−0.973061
$590$	0	0
$591$	28.8802	1.18797
$592$	0	0
$593$	−34.9211	−1.43404	−0.717018	−	0.697054i	$-0.754494\pi$
$593$	−34.9211	−1.43404	−0.717018	+	0.697054i	$0.754494\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−47.6155	−1.94877
$598$	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
$601$	−24.9073	−1.01599	−0.507994	−	0.861361i	$-0.669613\pi$
$601$	−24.9073	−1.01599	−0.507994	+	0.861361i	$0.669613\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	3.68260	0.149719
$606$	0	0
$607$	−8.52648	−0.346079	−0.173040	−	0.984915i	$-0.555359\pi$
$607$	−8.52648	−0.346079	−0.173040	+	0.984915i	$0.555359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.49242	0.181744
$612$	0	0
$613$	8.63068	0.348590	0.174295	−	0.984693i	$-0.444235\pi$
$613$	8.63068	0.348590	0.174295	+	0.984693i	$0.444235\pi$
$614$	0	0
$615$	11.5012	0.463771
$616$	0	0
$617$	−36.7386	−1.47904	−0.739521	−	0.673134i	$-0.764948\pi$
$617$	−36.7386	−1.47904	−0.739521	+	0.673134i	$0.764948\pi$
$618$	0	0
$619$	−40.0910	−1.61139	−0.805697	−	0.592328i	$-0.798209\pi$
$619$	−40.0910	−1.61139	−0.805697	+	0.592328i	$0.798209\pi$
$620$	0	0
$621$	−5.75058	−0.230763
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.49242	0.219697
$626$	0	0
$627$	−11.1231	−0.444214
$628$	0	0
$629$	5.87787	0.234366
$630$	0	0
$631$	−19.8078	−0.788535	−0.394267	−	0.918996i	$-0.629002\pi$
$631$	−19.8078	−0.788535	−0.394267	+	0.918996i	$0.629002\pi$
$632$	0	0
$633$	−50.3951	−2.00303
$634$	0	0
$635$	17.9597	0.712707
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.75379	−0.385854
$640$	0	0
$641$	20.9309	0.826720	0.413360	−	0.910568i	$-0.364355\pi$
$641$	20.9309	0.826720	0.413360	+	0.910568i	$0.364355\pi$
$642$	0	0
$643$	−31.8191	−1.25482	−0.627412	−	0.778688i	$-0.715886\pi$
$643$	−31.8191	−1.25482	−0.627412	+	0.778688i	$0.715886\pi$
$644$	0	0
$645$	88.9848	3.50377
$646$	0	0
$647$	15.6014	0.613353	0.306676	−	0.951814i	$-0.400783\pi$
$647$	15.6014	0.613353	0.306676	+	0.951814i	$0.400783\pi$
$648$	0	0
$649$	11.7915	0.462856
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.6847	−0.887719	−0.443860	−	0.896096i	$-0.646391\pi$
$653$	−22.6847	−0.887719	−0.443860	+	0.896096i	$0.646391\pi$
$654$	0	0
$655$	−27.1231	−1.05979
$656$	0	0
$657$	10.1768	0.397037
$658$	0	0
$659$	−5.36932	−0.209159	−0.104579	−	0.994517i	$-0.533350\pi$
$659$	−5.36932	−0.209159	−0.104579	+	0.994517i	$0.533350\pi$
$660$	0	0
$661$	−22.5490	−0.877053	−0.438526	−	0.898718i	$-0.644499\pi$
$661$	−22.5490	−0.877053	−0.438526	+	0.898718i	$0.644499\pi$
$662$	0	0
$663$	−10.5945	−0.411455
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.1231	0.430688
$668$	0	0
$669$	−46.5464	−1.79959
$670$	0	0
$671$	−13.4061	−0.517537
$672$	0	0
$673$	−25.6155	−0.987406	−0.493703	−	0.869631i	$-0.664357\pi$
$673$	−25.6155	−0.987406	−0.493703	+	0.869631i	$0.664357\pi$
$674$	0	0
$675$	−8.85254	−0.340734
$676$	0	0
$677$	−6.62153	−0.254486	−0.127243	−	0.991872i	$-0.540613\pi$
$677$	−6.62153	−0.254486	−0.127243	+	0.991872i	$0.540613\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−44.4924	−1.70495
$682$	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
$685$	−16.3450	−0.624512
$686$	0	0
$687$	19.8078	0.755713
$688$	0	0
$689$	6.78456	0.258471
$690$	0	0
$691$	−11.2108	−0.426480	−0.213240	−	0.977000i	$-0.568402\pi$
$691$	−11.2108	−0.426480	−0.213240	+	0.977000i	$0.568402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.3693	−0.658856
$696$	0	0
$697$	1.75379	0.0664295
$698$	0	0
$699$	28.8802	1.09235
$700$	0	0
$701$	35.3693	1.33588	0.667940	−	0.744215i	$-0.267176\pi$
$701$	35.3693	1.33588	0.667940	+	0.744215i	$0.267176\pi$
$702$	0	0
$703$	−20.9343	−0.789553
$704$	0	0
$705$	−11.5012	−0.433158
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29.8078	1.11945	0.559727	−	0.828677i	$-0.310906\pi$
$709$	29.8078	1.11945	0.559727	+	0.828677i	$0.310906\pi$
$710$	0	0
$711$	−12.4924	−0.468502
$712$	0	0
$713$	27.8462	1.04285
$714$	0	0
$715$	−12.4924	−0.467190
$716$	0	0
$717$	14.7304	0.550117
$718$	0	0
$719$	14.4401	0.538524	0.269262	−	0.963067i	$-0.413220\pi$
$719$	14.4401	0.538524	0.269262	+	0.963067i	$0.413220\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−26.7386	−0.994420
$724$	0	0
$725$	17.1231	0.635936
$726$	0	0
$727$	43.9009	1.62819	0.814097	−	0.580729i	$-0.197232\pi$
$727$	43.9009	1.62819	0.814097	+	0.580729i	$0.197232\pi$
$728$	0	0
$729$	−18.6155	−0.689464
$730$	0	0
$731$	13.5691	0.501872
$732$	0	0
$733$	3.39228	0.125297	0.0626484	−	0.998036i	$-0.480045\pi$
$733$	3.39228	0.125297	0.0626484	+	0.998036i	$0.480045\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.68466	0.172562
$738$	0	0
$739$	5.36932	0.197514	0.0987568	−	0.995112i	$-0.468513\pi$
$739$	5.36932	0.197514	0.0987568	+	0.995112i	$0.468513\pi$
$740$	0	0
$741$	37.7327	1.38615
$742$	0	0
$743$	−46.2462	−1.69661	−0.848304	−	0.529509i	$-0.822376\pi$
$743$	−46.2462	−1.69661	−0.848304	+	0.529509i	$0.822376\pi$
$744$	0	0
$745$	22.0956	0.809520
$746$	0	0
$747$	6.78456	0.248234
$748$	0	0
$749$	0	0
$750$	0	0
$751$	21.1771	0.772763	0.386381	−	0.922339i	$-0.373725\pi$
$751$	21.1771	0.772763	0.386381	+	0.922339i	$0.373725\pi$
$752$	0	0
$753$	−37.5616	−1.36882
$754$	0	0
$755$	−63.9643	−2.32790
$756$	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
$759$	13.1158	0.476073
$760$	0	0
$761$	−51.7194	−1.87483	−0.937414	−	0.348216i	$-0.886788\pi$
$761$	−51.7194	−1.87483	−0.937414	+	0.348216i	$0.886788\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	12.4924	0.451664
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	−12.8255	−0.462498	−0.231249	−	0.972895i	$-0.574281\pi$
$769$	−12.8255	−0.462498	−0.231249	+	0.972895i	$0.574281\pi$
$770$	0	0
$771$	56.9848	2.05226
$772$	0	0
$773$	−44.5173	−1.60118	−0.800588	−	0.599216i	$-0.795479\pi$
$773$	−44.5173	−1.60118	−0.800588	+	0.599216i	$0.795479\pi$
$774$	0	0
$775$	42.8669	1.53982
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.24621	−0.223794
$780$	0	0
$781$	−3.80776	−0.136253
$782$	0	0
$783$	−2.06798	−0.0739034
$784$	0	0
$785$	21.1771	0.755842
$786$	0	0
$787$	24.7442	0.882036	0.441018	−	0.897498i	$-0.354617\pi$
$787$	24.7442	0.882036	0.441018	+	0.897498i	$0.354617\pi$
$788$	0	0
$789$	3.22925	0.114964
$790$	0	0
$791$	0	0
$792$	0	0
$793$	45.4773	1.61495
$794$	0	0
$795$	−17.3693	−0.616026
$796$	0	0
$797$	17.2517	0.611088	0.305544	−	0.952178i	$-0.401162\pi$
$797$	17.2517	0.611088	0.305544	+	0.952178i	$0.401162\pi$
$798$	0	0
$799$	−1.75379	−0.0620446
$800$	0	0
$801$	38.8940	1.37425
$802$	0	0
$803$	3.97292	0.140201
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−63.2311	−2.22584
$808$	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
$811$	2.06798	0.0726164	0.0363082	−	0.999341i	$-0.488440\pi$
$811$	2.06798	0.0726164	0.0363082	+	0.999341i	$0.488440\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	−31.2742	−1.09549
$816$	0	0
$817$	−48.3272	−1.69075
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.7386	−1.00299	−0.501493	−	0.865162i	$-0.667216\pi$
$821$	−28.7386	−1.00299	−0.501493	+	0.865162i	$0.667216\pi$
$822$	0	0
$823$	−13.5616	−0.472726	−0.236363	−	0.971665i	$-0.575955\pi$
$823$	−13.5616	−0.472726	−0.236363	+	0.971665i	$0.575955\pi$
$824$	0	0
$825$	20.1907	0.702949
$826$	0	0
$827$	43.2311	1.50329	0.751646	−	0.659567i	$-0.229260\pi$
$827$	43.2311	1.50329	0.751646	+	0.659567i	$0.229260\pi$
$828$	0	0
$829$	23.7102	0.823490	0.411745	−	0.911299i	$-0.364919\pi$
$829$	23.7102	0.823490	0.411745	+	0.911299i	$0.364919\pi$
$830$	0	0
$831$	−41.5426	−1.44110
$832$	0	0
$833$	0	0
$834$	0	0
$835$	46.6307	1.61372
$836$	0	0
$837$	−5.17708	−0.178946
$838$	0	0
$839$	33.3064	1.14987	0.574933	−	0.818200i	$-0.305028\pi$
$839$	33.3064	1.14987	0.574933	+	0.818200i	$0.305028\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	69.8421	2.40549
$844$	0	0
$845$	−5.49600	−0.189068
$846$	0	0
$847$	0	0
$848$	0	0
$849$	44.4924	1.52698
$850$	0	0
$851$	24.6847	0.846179
$852$	0	0
$853$	42.8669	1.46774	0.733868	−	0.679293i	$-0.237713\pi$
$853$	42.8669	1.46774	0.733868	+	0.679293i	$0.237713\pi$
$854$	0	0
$855$	−44.4924	−1.52161
$856$	0	0
$857$	−4.55356	−0.155547	−0.0777733	−	0.996971i	$-0.524781\pi$
$857$	−4.55356	−0.155547	−0.0777733	+	0.996971i	$0.524781\pi$
$858$	0	0
$859$	20.6440	0.704365	0.352182	−	0.935931i	$-0.385440\pi$
$859$	20.6440	0.704365	0.352182	+	0.935931i	$0.385440\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.7386	0.365547	0.182774	−	0.983155i	$-0.441492\pi$
$863$	10.7386	0.365547	0.182774	+	0.983155i	$0.441492\pi$
$864$	0	0
$865$	56.9848	1.93754
$866$	0	0
$867$	−35.9551	−1.22110
$868$	0	0
$869$	−4.87689	−0.165437
$870$	0	0
$871$	−15.8917	−0.538469
$872$	0	0
$873$	−45.6786	−1.54598
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.4924	1.02966	0.514828	−	0.857294i	$-0.327856\pi$
$877$	30.4924	1.02966	0.514828	+	0.857294i	$0.327856\pi$
$878$	0	0
$879$	−41.3693	−1.39535
$880$	0	0
$881$	32.5628	1.09707	0.548534	−	0.836128i	$-0.315186\pi$
$881$	32.5628	1.09707	0.548534	+	0.836128i	$0.315186\pi$
$882$	0	0
$883$	15.5076	0.521872	0.260936	−	0.965356i	$-0.415969\pi$
$883$	15.5076	0.521872	0.260936	+	0.965356i	$0.415969\pi$
$884$	0	0
$885$	102.405	3.44230
$886$	0	0
$887$	30.0414	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$887$	30.0414	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10.1231	−0.339137
$892$	0	0
$893$	6.24621	0.209021
$894$	0	0
$895$	58.2137	1.94587
$896$	0	0
$897$	−44.4924	−1.48556
$898$	0	0
$899$	10.0138	0.333979
$900$	0	0
$901$	−2.64861	−0.0882381
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38.5464	1.28133
$906$	0	0
$907$	−5.75379	−0.191051	−0.0955257	−	0.995427i	$-0.530453\pi$
$907$	−5.75379	−0.191051	−0.0955257	+	0.995427i	$0.530453\pi$
$908$	0	0
$909$	1.90495	0.0631831
$910$	0	0
$911$	3.50758	0.116211	0.0581056	−	0.998310i	$-0.481494\pi$
$911$	3.50758	0.116211	0.0581056	+	0.998310i	$0.481494\pi$
$912$	0	0
$913$	2.64861	0.0876563
$914$	0	0
$915$	−116.427	−3.84897
$916$	0	0
$917$	0	0
$918$	0	0
$919$	54.2462	1.78942	0.894709	−	0.446650i	$-0.147383\pi$
$919$	54.2462	1.78942	0.894709	+	0.446650i	$0.147383\pi$
$920$	0	0
$921$	−22.2462	−0.733038
$922$	0	0
$923$	12.9170	0.425169
$924$	0	0
$925$	38.0000	1.24943
$926$	0	0
$927$	20.7713	0.682219
$928$	0	0
$929$	5.62329	0.184494	0.0922470	−	0.995736i	$-0.470595\pi$
$929$	5.62329	0.184494	0.0922470	+	0.995736i	$0.470595\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	28.8769	0.945387
$934$	0	0
$935$	4.87689	0.159492
$936$	0	0
$937$	−31.6918	−1.03533	−0.517663	−	0.855584i	$-0.673198\pi$
$937$	−31.6918	−1.03533	−0.517663	+	0.855584i	$0.673198\pi$
$938$	0	0
$939$	−1.06913	−0.0348897
$940$	0	0
$941$	−14.5674	−0.474883	−0.237441	−	0.971402i	$-0.576309\pi$
$941$	−14.5674	−0.474883	−0.237441	+	0.971402i	$0.576309\pi$
$942$	0	0
$943$	7.36520	0.239844
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.94602	−0.0632373	−0.0316187	−	0.999500i	$-0.510066\pi$
$947$	−1.94602	−0.0632373	−0.0316187	+	0.999500i	$0.510066\pi$
$948$	0	0
$949$	−13.4773	−0.437491
$950$	0	0
$951$	4.26324	0.138245
$952$	0	0
$953$	27.3693	0.886579	0.443290	−	0.896378i	$-0.353811\pi$
$953$	27.3693	0.886579	0.443290	+	0.896378i	$0.353811\pi$
$954$	0	0
$955$	43.4833	1.40709
$956$	0	0
$957$	4.71659	0.152466
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.93087	−0.191318
$962$	0	0
$963$	−18.2462	−0.587976
$964$	0	0
$965$	−97.5610	−3.14060
$966$	0	0
$967$	4.87689	0.156830	0.0784152	−	0.996921i	$-0.475014\pi$
$967$	4.87689	0.156830	0.0784152	+	0.996921i	$0.475014\pi$
$968$	0	0
$969$	−14.7304	−0.473209
$970$	0	0
$971$	36.5357	1.17249	0.586243	−	0.810135i	$-0.300606\pi$
$971$	36.5357	1.17249	0.586243	+	0.810135i	$0.300606\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−68.4924	−2.19351
$976$	0	0
$977$	−46.7926	−1.49703	−0.748514	−	0.663119i	$-0.769232\pi$
$977$	−46.7926	−1.49703	−0.748514	+	0.663119i	$0.769232\pi$
$978$	0	0
$979$	15.1838	0.485275
$980$	0	0
$981$	49.6155	1.58410
$982$	0	0
$983$	−39.7649	−1.26830	−0.634152	−	0.773208i	$-0.718651\pi$
$983$	−39.7649	−1.26830	−0.634152	+	0.773208i	$0.718651\pi$
$984$	0	0
$985$	45.0979	1.43694
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.9848	1.81201
$990$	0	0
$991$	−4.49242	−0.142707	−0.0713533	−	0.997451i	$-0.522732\pi$
$991$	−4.49242	−0.142707	−0.0713533	+	0.997451i	$0.522732\pi$
$992$	0	0
$993$	−18.4130	−0.584319
$994$	0	0
$995$	−74.3542	−2.35719
$996$	0	0
$997$	−22.2586	−0.704938	−0.352469	−	0.935823i	$-0.614658\pi$
$997$	−22.2586	−0.704938	−0.352469	+	0.935823i	$0.614658\pi$
$998$	0	0
$999$	−4.58930	−0.145199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cu.1.4		4
4.3	odd	2		539.2.a.k.1.3	✓	4
7.6	odd	2	inner	8624.2.a.cu.1.1		4
12.11	even	2		4851.2.a.bv.1.1		4
28.3	even	6		539.2.e.n.177.1		8
28.11	odd	6		539.2.e.n.177.2		8
28.19	even	6		539.2.e.n.67.1		8
28.23	odd	6		539.2.e.n.67.2		8
28.27	even	2		539.2.a.k.1.4	yes	4
44.43	even	2		5929.2.a.ba.1.1		4
84.83	odd	2		4851.2.a.bv.1.2		4
308.307	odd	2		5929.2.a.ba.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.k.1.3	✓	4	4.3	odd	2
539.2.a.k.1.4	yes	4	28.27	even	2
539.2.e.n.67.1		8	28.19	even	6
539.2.e.n.67.2		8	28.23	odd	6
539.2.e.n.177.1		8	28.3	even	6
539.2.e.n.177.2		8	28.11	odd	6
4851.2.a.bv.1.1		4	12.11	even	2
4851.2.a.bv.1.2		4	84.83	odd	2
5929.2.a.ba.1.1		4	44.43	even	2
5929.2.a.ba.1.2		4	308.307	odd	2
8624.2.a.cu.1.1		4	7.6	odd	2	inner
8624.2.a.cu.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.35829 q^{3} +3.68260 q^{5} +2.56155 q^{9} +1.00000 q^{11} -3.39228 q^{13} +8.68466 q^{15} +1.32431 q^{17} -4.71659 q^{19} +5.56155 q^{23} +8.56155 q^{25} -1.03399 q^{27} +2.00000 q^{29} +5.00691 q^{31} +2.35829 q^{33} +4.43845 q^{37} -8.00000 q^{39} +1.32431 q^{41} +10.2462 q^{43} +9.43318 q^{45} -1.32431 q^{47} +3.12311 q^{51} -2.00000 q^{53} +3.68260 q^{55} -11.1231 q^{57} +11.7915 q^{59} -13.4061 q^{61} -12.4924 q^{65} +4.68466 q^{67} +13.1158 q^{69} -3.80776 q^{71} +3.97292 q^{73} +20.1907 q^{75} -4.87689 q^{79} -10.1231 q^{81} +2.64861 q^{83} +4.87689 q^{85} +4.71659 q^{87} +15.1838 q^{89} +11.8078 q^{93} -17.3693 q^{95} -17.8324 q^{97} +2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9} + 4 q^{11} + 10 q^{15} + 14 q^{23} + 26 q^{25} + 8 q^{29} + 26 q^{37} - 32 q^{39} + 8 q^{43} - 4 q^{51} - 8 q^{53} - 28 q^{57} + 16 q^{65} - 6 q^{67} + 26 q^{71} - 36 q^{79} - 24 q^{81} + 36 q^{85}+ \cdots + 2 q^{99}+O(q^{100}) \) 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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