LMFDB - Embedded newform 8032.2.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77983 q^{3} +3.93815 q^{5} -3.71151 q^{7} +0.167788 q^{9} +O(q^{10})$	$q-1.77983 q^{3} +3.93815 q^{5} -3.71151 q^{7} +0.167788 q^{9} +3.12933 q^{11} -5.91725 q^{13} -7.00923 q^{15} -0.159877 q^{17} -2.26671 q^{19} +6.60584 q^{21} -9.26486 q^{23} +10.5090 q^{25} +5.04085 q^{27} +7.78893 q^{29} +5.11771 q^{31} -5.56966 q^{33} -14.6165 q^{35} +11.0153 q^{37} +10.5317 q^{39} -0.215999 q^{41} +1.14556 q^{43} +0.660773 q^{45} +0.412356 q^{47} +6.77528 q^{49} +0.284554 q^{51} -3.01511 q^{53} +12.3238 q^{55} +4.03436 q^{57} +8.18882 q^{59} +12.9495 q^{61} -0.622745 q^{63} -23.3030 q^{65} -4.37117 q^{67} +16.4898 q^{69} -8.95328 q^{71} -7.82405 q^{73} -18.7043 q^{75} -11.6145 q^{77} +5.80781 q^{79} -9.47521 q^{81} -8.05565 q^{83} -0.629620 q^{85} -13.8630 q^{87} +1.45451 q^{89} +21.9619 q^{91} -9.10865 q^{93} -8.92666 q^{95} -12.4420 q^{97} +0.525063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * 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
$3$	−1.77983	−1.02758	−0.513792	−	0.857915i	$-0.671760\pi$
$3$	−1.77983	−1.02758	−0.513792	+	0.857915i	$0.671760\pi$
$4$	0	0
$5$	3.93815	1.76119	0.880597	−	0.473866i	$-0.157142\pi$
$5$	3.93815	1.76119	0.880597	+	0.473866i	$0.157142\pi$
$6$	0	0
$7$	−3.71151	−1.40282	−0.701409	−	0.712759i	$-0.747445\pi$
$7$	−3.71151	−1.40282	−0.701409	+	0.712759i	$0.747445\pi$
$8$	0	0
$9$	0.167788	0.0559293
$10$	0	0
$11$	3.12933	0.943528	0.471764	−	0.881725i	$-0.343617\pi$
$11$	3.12933	0.943528	0.471764	+	0.881725i	$0.343617\pi$
$12$	0	0
$13$	−5.91725	−1.64115	−0.820576	−	0.571538i	$-0.806347\pi$
$13$	−5.91725	−1.64115	−0.820576	+	0.571538i	$0.806347\pi$
$14$	0	0
$15$	−7.00923	−1.80978
$16$	0	0
$17$	−0.159877	−0.0387759	−0.0193880	−	0.999812i	$-0.506172\pi$
$17$	−0.159877	−0.0387759	−0.0193880	+	0.999812i	$0.506172\pi$
$18$	0	0
$19$	−2.26671	−0.520020	−0.260010	−	0.965606i	$-0.583726\pi$
$19$	−2.26671	−0.520020	−0.260010	+	0.965606i	$0.583726\pi$
$20$	0	0
$21$	6.60584	1.44151
$22$	0	0
$23$	−9.26486	−1.93186	−0.965928	−	0.258811i	$-0.916669\pi$
$23$	−9.26486	−1.93186	−0.965928	+	0.258811i	$0.916669\pi$
$24$	0	0
$25$	10.5090	2.10180
$26$	0	0
$27$	5.04085	0.970112
$28$	0	0
$29$	7.78893	1.44637	0.723184	−	0.690655i	$-0.242678\pi$
$29$	7.78893	1.44637	0.723184	+	0.690655i	$0.242678\pi$
$30$	0	0
$31$	5.11771	0.919168	0.459584	−	0.888134i	$-0.347998\pi$
$31$	5.11771	0.919168	0.459584	+	0.888134i	$0.347998\pi$
$32$	0	0
$33$	−5.56966	−0.969554
$34$	0	0
$35$	−14.6165	−2.47063
$36$	0	0
$37$	11.0153	1.81090	0.905451	−	0.424450i	$-0.139533\pi$
$37$	11.0153	1.81090	0.905451	+	0.424450i	$0.139533\pi$
$38$	0	0
$39$	10.5317	1.68642
$40$	0	0
$41$	−0.215999	−0.0337333	−0.0168667	−	0.999858i	$-0.505369\pi$
$41$	−0.215999	−0.0337333	−0.0168667	+	0.999858i	$0.505369\pi$
$42$	0	0
$43$	1.14556	0.174696	0.0873481	−	0.996178i	$-0.472161\pi$
$43$	1.14556	0.174696	0.0873481	+	0.996178i	$0.472161\pi$
$44$	0	0
$45$	0.660773	0.0985023
$46$	0	0
$47$	0.412356	0.0601483	0.0300741	−	0.999548i	$-0.490426\pi$
$47$	0.412356	0.0601483	0.0300741	+	0.999548i	$0.490426\pi$
$48$	0	0
$49$	6.77528	0.967898
$50$	0	0
$51$	0.284554	0.0398455
$52$	0	0
$53$	−3.01511	−0.414157	−0.207079	−	0.978324i	$-0.566396\pi$
$53$	−3.01511	−0.414157	−0.207079	+	0.978324i	$0.566396\pi$
$54$	0	0
$55$	12.3238	1.66174
$56$	0	0
$57$	4.03436	0.534364
$58$	0	0
$59$	8.18882	1.06609	0.533047	−	0.846086i	$-0.321047\pi$
$59$	8.18882	1.06609	0.533047	+	0.846086i	$0.321047\pi$
$60$	0	0
$61$	12.9495	1.65802	0.829009	−	0.559235i	$-0.188905\pi$
$61$	12.9495	1.65802	0.829009	+	0.559235i	$0.188905\pi$
$62$	0	0
$63$	−0.622745	−0.0784586
$64$	0	0
$65$	−23.3030	−2.89039
$66$	0	0
$67$	−4.37117	−0.534023	−0.267011	−	0.963693i	$-0.586036\pi$
$67$	−4.37117	−0.534023	−0.267011	+	0.963693i	$0.586036\pi$
$68$	0	0
$69$	16.4898	1.98514
$70$	0	0
$71$	−8.95328	−1.06256	−0.531279	−	0.847197i	$-0.678288\pi$
$71$	−8.95328	−1.06256	−0.531279	+	0.847197i	$0.678288\pi$
$72$	0	0
$73$	−7.82405	−0.915736	−0.457868	−	0.889020i	$-0.651387\pi$
$73$	−7.82405	−0.915736	−0.457868	+	0.889020i	$0.651387\pi$
$74$	0	0
$75$	−18.7043	−2.15978
$76$	0	0
$77$	−11.6145	−1.32360
$78$	0	0
$79$	5.80781	0.653430	0.326715	−	0.945123i	$-0.394058\pi$
$79$	5.80781	0.653430	0.326715	+	0.945123i	$0.394058\pi$
$80$	0	0
$81$	−9.47521	−1.05280
$82$	0	0
$83$	−8.05565	−0.884222	−0.442111	−	0.896960i	$-0.645770\pi$
$83$	−8.05565	−0.884222	−0.442111	+	0.896960i	$0.645770\pi$
$84$	0	0
$85$	−0.629620	−0.0682919
$86$	0	0
$87$	−13.8630	−1.48627
$88$	0	0
$89$	1.45451	0.154178	0.0770888	−	0.997024i	$-0.475438\pi$
$89$	1.45451	0.154178	0.0770888	+	0.997024i	$0.475438\pi$
$90$	0	0
$91$	21.9619	2.30224
$92$	0	0
$93$	−9.10865	−0.944523
$94$	0	0
$95$	−8.92666	−0.915856
$96$	0	0
$97$	−12.4420	−1.26329	−0.631644	−	0.775258i	$-0.717620\pi$
$97$	−12.4420	−1.26329	−0.631644	+	0.775258i	$0.717620\pi$
$98$	0	0
$99$	0.525063	0.0527708
$100$	0	0
$101$	9.25684	0.921090	0.460545	−	0.887636i	$-0.347654\pi$
$101$	9.25684	0.921090	0.460545	+	0.887636i	$0.347654\pi$
$102$	0	0
$103$	−14.2058	−1.39974	−0.699869	−	0.714271i	$-0.746758\pi$
$103$	−14.2058	−1.39974	−0.699869	+	0.714271i	$0.746758\pi$
$104$	0	0
$105$	26.0148	2.53878
$106$	0	0
$107$	−1.86091	−0.179901	−0.0899506	−	0.995946i	$-0.528671\pi$
$107$	−1.86091	−0.179901	−0.0899506	+	0.995946i	$0.528671\pi$
$108$	0	0
$109$	−1.25909	−0.120599	−0.0602993	−	0.998180i	$-0.519206\pi$
$109$	−1.25909	−0.120599	−0.0602993	+	0.998180i	$0.519206\pi$
$110$	0	0
$111$	−19.6053	−1.86085
$112$	0	0
$113$	−5.10279	−0.480030	−0.240015	−	0.970769i	$-0.577152\pi$
$113$	−5.10279	−0.480030	−0.240015	+	0.970769i	$0.577152\pi$
$114$	0	0
$115$	−36.4864	−3.40237
$116$	0	0
$117$	−0.992843	−0.0917884
$118$	0	0
$119$	0.593385	0.0543955
$120$	0	0
$121$	−1.20731	−0.109755
$122$	0	0
$123$	0.384441	0.0346638
$124$	0	0
$125$	21.6954	1.94049
$126$	0	0
$127$	−8.35149	−0.741075	−0.370538	−	0.928817i	$-0.620827\pi$
$127$	−8.35149	−0.741075	−0.370538	+	0.928817i	$0.620827\pi$
$128$	0	0
$129$	−2.03890	−0.179515
$130$	0	0
$131$	13.2432	1.15706	0.578532	−	0.815659i	$-0.303626\pi$
$131$	13.2432	1.15706	0.578532	+	0.815659i	$0.303626\pi$
$132$	0	0
$133$	8.41292	0.729493
$134$	0	0
$135$	19.8516	1.70856
$136$	0	0
$137$	−21.2456	−1.81514	−0.907568	−	0.419905i	$-0.862063\pi$
$137$	−21.2456	−1.81514	−0.907568	+	0.419905i	$0.862063\pi$
$138$	0	0
$139$	−21.2335	−1.80100	−0.900500	−	0.434855i	$-0.856800\pi$
$139$	−21.2335	−1.80100	−0.900500	+	0.434855i	$0.856800\pi$
$140$	0	0
$141$	−0.733923	−0.0618074
$142$	0	0
$143$	−18.5170	−1.54847
$144$	0	0
$145$	30.6740	2.54734
$146$	0	0
$147$	−12.0588	−0.994596
$148$	0	0
$149$	−16.1322	−1.32160	−0.660802	−	0.750560i	$-0.729784\pi$
$149$	−16.1322	−1.32160	−0.660802	+	0.750560i	$0.729784\pi$
$150$	0	0
$151$	16.0957	1.30985	0.654926	−	0.755693i	$-0.272700\pi$
$151$	16.0957	1.30985	0.654926	+	0.755693i	$0.272700\pi$
$152$	0	0
$153$	−0.0268254	−0.00216871
$154$	0	0
$155$	20.1543	1.61883
$156$	0	0
$157$	2.87675	0.229590	0.114795	−	0.993389i	$-0.463379\pi$
$157$	2.87675	0.229590	0.114795	+	0.993389i	$0.463379\pi$
$158$	0	0
$159$	5.36638	0.425582
$160$	0	0
$161$	34.3866	2.71004
$162$	0	0
$163$	−14.5648	−1.14081	−0.570403	−	0.821365i	$-0.693213\pi$
$163$	−14.5648	−1.14081	−0.570403	+	0.821365i	$0.693213\pi$
$164$	0	0
$165$	−21.9342	−1.70757
$166$	0	0
$167$	−3.58057	−0.277073	−0.138536	−	0.990357i	$-0.544240\pi$
$167$	−3.58057	−0.277073	−0.138536	+	0.990357i	$0.544240\pi$
$168$	0	0
$169$	22.0139	1.69338
$170$	0	0
$171$	−0.380327	−0.0290843
$172$	0	0
$173$	−18.4950	−1.40615	−0.703073	−	0.711117i	$-0.748190\pi$
$173$	−18.4950	−1.40615	−0.703073	+	0.711117i	$0.748190\pi$
$174$	0	0
$175$	−39.0043	−2.94845
$176$	0	0
$177$	−14.5747	−1.09550
$178$	0	0
$179$	−13.8023	−1.03163	−0.515817	−	0.856699i	$-0.672512\pi$
$179$	−13.8023	−1.03163	−0.515817	+	0.856699i	$0.672512\pi$
$180$	0	0
$181$	−18.6071	−1.38305	−0.691527	−	0.722351i	$-0.743061\pi$
$181$	−18.6071	−1.38305	−0.691527	+	0.722351i	$0.743061\pi$
$182$	0	0
$183$	−23.0480	−1.70375
$184$	0	0
$185$	43.3799	3.18935
$186$	0	0
$187$	−0.500308	−0.0365862
$188$	0	0
$189$	−18.7092	−1.36089
$190$	0	0
$191$	11.0564	0.800012	0.400006	−	0.916512i	$-0.369008\pi$
$191$	11.0564	0.800012	0.400006	+	0.916512i	$0.369008\pi$
$192$	0	0
$193$	−0.733864	−0.0528247	−0.0264123	−	0.999651i	$-0.508408\pi$
$193$	−0.733864	−0.0528247	−0.0264123	+	0.999651i	$0.508408\pi$
$194$	0	0
$195$	41.4754	2.97011
$196$	0	0
$197$	23.0815	1.64449	0.822246	−	0.569132i	$-0.192721\pi$
$197$	23.0815	1.64449	0.822246	+	0.569132i	$0.192721\pi$
$198$	0	0
$199$	−26.5412	−1.88146	−0.940728	−	0.339163i	$-0.889856\pi$
$199$	−26.5412	−1.88146	−0.940728	+	0.339163i	$0.889856\pi$
$200$	0	0
$201$	7.77992	0.548754
$202$	0	0
$203$	−28.9087	−2.02899
$204$	0	0
$205$	−0.850635	−0.0594109
$206$	0	0
$207$	−1.55453	−0.108047
$208$	0	0
$209$	−7.09329	−0.490653
$210$	0	0
$211$	−4.59805	−0.316543	−0.158271	−	0.987396i	$-0.550592\pi$
$211$	−4.59805	−0.316543	−0.158271	+	0.987396i	$0.550592\pi$
$212$	0	0
$213$	15.9353	1.09187
$214$	0	0
$215$	4.51138	0.307674
$216$	0	0
$217$	−18.9944	−1.28943
$218$	0	0
$219$	13.9255	0.940996
$220$	0	0
$221$	0.946034	0.0636371
$222$	0	0
$223$	−0.401142	−0.0268625	−0.0134312	−	0.999910i	$-0.504275\pi$
$223$	−0.401142	−0.0268625	−0.0134312	+	0.999910i	$0.504275\pi$
$224$	0	0
$225$	1.76329	0.117552
$226$	0	0
$227$	1.16875	0.0775728	0.0387864	−	0.999248i	$-0.487651\pi$
$227$	1.16875	0.0775728	0.0387864	+	0.999248i	$0.487651\pi$
$228$	0	0
$229$	−21.5518	−1.42418	−0.712092	−	0.702086i	$-0.752252\pi$
$229$	−21.5518	−1.42418	−0.712092	+	0.702086i	$0.752252\pi$
$230$	0	0
$231$	20.6718	1.36011
$232$	0	0
$233$	16.3056	1.06822	0.534108	−	0.845416i	$-0.320648\pi$
$233$	16.3056	1.06822	0.534108	+	0.845416i	$0.320648\pi$
$234$	0	0
$235$	1.62392	0.105933
$236$	0	0
$237$	−10.3369	−0.671454
$238$	0	0
$239$	−19.5505	−1.26462	−0.632309	−	0.774717i	$-0.717893\pi$
$239$	−19.5505	−1.26462	−0.632309	+	0.774717i	$0.717893\pi$
$240$	0	0
$241$	14.8577	0.957072	0.478536	−	0.878068i	$-0.341168\pi$
$241$	14.8577	0.957072	0.478536	+	0.878068i	$0.341168\pi$
$242$	0	0
$243$	1.74169	0.111730
$244$	0	0
$245$	26.6821	1.70466
$246$	0	0
$247$	13.4127	0.853431
$248$	0	0
$249$	14.3377	0.908613
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−28.9928	−1.82276
$254$	0	0
$255$	1.12062	0.0701757
$256$	0	0
$257$	−2.77263	−0.172952	−0.0864758	−	0.996254i	$-0.527561\pi$
$257$	−2.77263	−0.172952	−0.0864758	+	0.996254i	$0.527561\pi$
$258$	0	0
$259$	−40.8833	−2.54037
$260$	0	0
$261$	1.30689	0.0808943
$262$	0	0
$263$	22.3138	1.37593	0.687964	−	0.725745i	$-0.258505\pi$
$263$	22.3138	1.37593	0.687964	+	0.725745i	$0.258505\pi$
$264$	0	0
$265$	−11.8740	−0.729412
$266$	0	0
$267$	−2.58878	−0.158430
$268$	0	0
$269$	−15.0819	−0.919559	−0.459779	−	0.888033i	$-0.652072\pi$
$269$	−15.0819	−0.919559	−0.459779	+	0.888033i	$0.652072\pi$
$270$	0	0
$271$	12.5716	0.763668	0.381834	−	0.924231i	$-0.375293\pi$
$271$	12.5716	0.763668	0.381834	+	0.924231i	$0.375293\pi$
$272$	0	0
$273$	−39.0885	−2.36574
$274$	0	0
$275$	32.8862	1.98311
$276$	0	0
$277$	−11.7186	−0.704103	−0.352051	−	0.935981i	$-0.614516\pi$
$277$	−11.7186	−0.704103	−0.352051	+	0.935981i	$0.614516\pi$
$278$	0	0
$279$	0.858689	0.0514084
$280$	0	0
$281$	−17.7910	−1.06132	−0.530660	−	0.847585i	$-0.678056\pi$
$281$	−17.7910	−1.06132	−0.530660	+	0.847585i	$0.678056\pi$
$282$	0	0
$283$	−9.79620	−0.582324	−0.291162	−	0.956674i	$-0.594042\pi$
$283$	−9.79620	−0.582324	−0.291162	+	0.956674i	$0.594042\pi$
$284$	0	0
$285$	15.8879	0.941119
$286$	0	0
$287$	0.801681	0.0473217
$288$	0	0
$289$	−16.9744	−0.998496
$290$	0	0
$291$	22.1445	1.29814
$292$	0	0
$293$	−24.5040	−1.43154	−0.715769	−	0.698337i	$-0.753924\pi$
$293$	−24.5040	−1.43154	−0.715769	+	0.698337i	$0.753924\pi$
$294$	0	0
$295$	32.2488	1.87760
$296$	0	0
$297$	15.7745	0.915328
$298$	0	0
$299$	54.8225	3.17047
$300$	0	0
$301$	−4.25175	−0.245067
$302$	0	0
$303$	−16.4756	−0.946498
$304$	0	0
$305$	50.9972	2.92009
$306$	0	0
$307$	7.47772	0.426776	0.213388	−	0.976968i	$-0.431550\pi$
$307$	7.47772	0.426776	0.213388	+	0.976968i	$0.431550\pi$
$308$	0	0
$309$	25.2839	1.43835
$310$	0	0
$311$	14.4991	0.822166	0.411083	−	0.911598i	$-0.365151\pi$
$311$	14.4991	0.822166	0.411083	+	0.911598i	$0.365151\pi$
$312$	0	0
$313$	21.6860	1.22577	0.612883	−	0.790174i	$-0.290010\pi$
$313$	21.6860	1.22577	0.612883	+	0.790174i	$0.290010\pi$
$314$	0	0
$315$	−2.45246	−0.138181
$316$	0	0
$317$	24.0549	1.35106	0.675529	−	0.737333i	$-0.263915\pi$
$317$	24.0549	1.35106	0.675529	+	0.737333i	$0.263915\pi$
$318$	0	0
$319$	24.3741	1.36469
$320$	0	0
$321$	3.31210	0.184864
$322$	0	0
$323$	0.362396	0.0201642
$324$	0	0
$325$	−62.1846	−3.44938
$326$	0	0
$327$	2.24096	0.123925
$328$	0	0
$329$	−1.53046	−0.0843771
$330$	0	0
$331$	−17.5953	−0.967125	−0.483563	−	0.875310i	$-0.660657\pi$
$331$	−17.5953	−0.967125	−0.483563	+	0.875310i	$0.660657\pi$
$332$	0	0
$333$	1.84823	0.101282
$334$	0	0
$335$	−17.2143	−0.940518
$336$	0	0
$337$	−15.5479	−0.846948	−0.423474	−	0.905908i	$-0.639189\pi$
$337$	−15.5479	−0.846948	−0.423474	+	0.905908i	$0.639189\pi$
$338$	0	0
$339$	9.08209	0.493271
$340$	0	0
$341$	16.0150	0.867261
$342$	0	0
$343$	0.834035	0.0450336
$344$	0	0
$345$	64.9395	3.49622
$346$	0	0
$347$	35.9032	1.92739	0.963693	−	0.267011i	$-0.0860360\pi$
$347$	35.9032	1.92739	0.963693	+	0.267011i	$0.0860360\pi$
$348$	0	0
$349$	−11.0105	−0.589377	−0.294688	−	0.955593i	$-0.595216\pi$
$349$	−11.0105	−0.589377	−0.294688	+	0.955593i	$0.595216\pi$
$350$	0	0
$351$	−29.8280	−1.59210
$352$	0	0
$353$	17.8792	0.951614	0.475807	−	0.879550i	$-0.342156\pi$
$353$	17.8792	0.951614	0.475807	+	0.879550i	$0.342156\pi$
$354$	0	0
$355$	−35.2593	−1.87137
$356$	0	0
$357$	−1.05612	−0.0558960
$358$	0	0
$359$	−2.80919	−0.148264	−0.0741318	−	0.997248i	$-0.523619\pi$
$359$	−2.80919	−0.148264	−0.0741318	+	0.997248i	$0.523619\pi$
$360$	0	0
$361$	−13.8620	−0.729580
$362$	0	0
$363$	2.14880	0.112783
$364$	0	0
$365$	−30.8123	−1.61279
$366$	0	0
$367$	26.2600	1.37076	0.685381	−	0.728185i	$-0.259636\pi$
$367$	26.2600	1.37076	0.685381	+	0.728185i	$0.259636\pi$
$368$	0	0
$369$	−0.0362419	−0.00188668
$370$	0	0
$371$	11.1906	0.580987
$372$	0	0
$373$	23.7358	1.22899	0.614495	−	0.788920i	$-0.289360\pi$
$373$	23.7358	1.22899	0.614495	+	0.788920i	$0.289360\pi$
$374$	0	0
$375$	−38.6140	−1.99402
$376$	0	0
$377$	−46.0891	−2.37371
$378$	0	0
$379$	25.8622	1.32845	0.664225	−	0.747533i	$-0.268762\pi$
$379$	25.8622	1.32845	0.664225	+	0.747533i	$0.268762\pi$
$380$	0	0
$381$	14.8642	0.761517
$382$	0	0
$383$	12.6910	0.648479	0.324240	−	0.945975i	$-0.394892\pi$
$383$	12.6910	0.648479	0.324240	+	0.945975i	$0.394892\pi$
$384$	0	0
$385$	−45.7397	−2.33111
$386$	0	0
$387$	0.192211	0.00977063
$388$	0	0
$389$	−24.6459	−1.24960	−0.624799	−	0.780785i	$-0.714819\pi$
$389$	−24.6459	−1.24960	−0.624799	+	0.780785i	$0.714819\pi$
$390$	0	0
$391$	1.48124	0.0749095
$392$	0	0
$393$	−23.5706	−1.18898
$394$	0	0
$395$	22.8720	1.15082
$396$	0	0
$397$	0.400579	0.0201045	0.0100523	−	0.999949i	$-0.496800\pi$
$397$	0.400579	0.0201045	0.0100523	+	0.999949i	$0.496800\pi$
$398$	0	0
$399$	−14.9736	−0.749615
$400$	0	0
$401$	−21.3934	−1.06834	−0.534168	−	0.845378i	$-0.679375\pi$
$401$	−21.3934	−1.06834	−0.534168	+	0.845378i	$0.679375\pi$
$402$	0	0
$403$	−30.2828	−1.50849
$404$	0	0
$405$	−37.3148	−1.85419
$406$	0	0
$407$	34.4704	1.70864
$408$	0	0
$409$	−20.5775	−1.01749	−0.508747	−	0.860916i	$-0.669891\pi$
$409$	−20.5775	−1.01749	−0.508747	+	0.860916i	$0.669891\pi$
$410$	0	0
$411$	37.8136	1.86521
$412$	0	0
$413$	−30.3929	−1.49554
$414$	0	0
$415$	−31.7243	−1.55729
$416$	0	0
$417$	37.7920	1.85068
$418$	0	0
$419$	−25.8610	−1.26339	−0.631695	−	0.775217i	$-0.717641\pi$
$419$	−25.8610	−1.26339	−0.631695	+	0.775217i	$0.717641\pi$
$420$	0	0
$421$	−4.73774	−0.230903	−0.115452	−	0.993313i	$-0.536832\pi$
$421$	−4.73774	−0.230903	−0.115452	+	0.993313i	$0.536832\pi$
$422$	0	0
$423$	0.0691883	0.00336405
$424$	0	0
$425$	−1.68015	−0.0814994
$426$	0	0
$427$	−48.0623	−2.32590
$428$	0	0
$429$	32.9571	1.59118
$430$	0	0
$431$	−23.5255	−1.13318	−0.566591	−	0.823999i	$-0.691738\pi$
$431$	−23.5255	−1.13318	−0.566591	+	0.823999i	$0.691738\pi$
$432$	0	0
$433$	−32.5922	−1.56628	−0.783141	−	0.621844i	$-0.786384\pi$
$433$	−32.5922	−1.56628	−0.783141	+	0.621844i	$0.786384\pi$
$434$	0	0
$435$	−54.5944	−2.61760
$436$	0	0
$437$	21.0008	1.00460
$438$	0	0
$439$	16.7110	0.797574	0.398787	−	0.917044i	$-0.369431\pi$
$439$	16.7110	0.797574	0.398787	+	0.917044i	$0.369431\pi$
$440$	0	0
$441$	1.13681	0.0541338
$442$	0	0
$443$	−32.2988	−1.53456	−0.767280	−	0.641312i	$-0.778390\pi$
$443$	−32.2988	−1.53456	−0.767280	+	0.641312i	$0.778390\pi$
$444$	0	0
$445$	5.72807	0.271537
$446$	0	0
$447$	28.7126	1.35806
$448$	0	0
$449$	−6.33653	−0.299039	−0.149520	−	0.988759i	$-0.547773\pi$
$449$	−6.33653	−0.299039	−0.149520	+	0.988759i	$0.547773\pi$
$450$	0	0
$451$	−0.675931	−0.0318283
$452$	0	0
$453$	−28.6476	−1.34598
$454$	0	0
$455$	86.4894	4.05468
$456$	0	0
$457$	−33.8982	−1.58569	−0.792845	−	0.609423i	$-0.791401\pi$
$457$	−33.8982	−1.58569	−0.792845	+	0.609423i	$0.791401\pi$
$458$	0	0
$459$	−0.805917	−0.0376170
$460$	0	0
$461$	11.6701	0.543530	0.271765	−	0.962364i	$-0.412393\pi$
$461$	11.6701	0.543530	0.271765	+	0.962364i	$0.412393\pi$
$462$	0	0
$463$	21.5873	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$463$	21.5873	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$464$	0	0
$465$	−35.8712	−1.66349
$466$	0	0
$467$	18.3591	0.849559	0.424780	−	0.905297i	$-0.360352\pi$
$467$	18.3591	0.849559	0.424780	+	0.905297i	$0.360352\pi$
$468$	0	0
$469$	16.2236	0.749137
$470$	0	0
$471$	−5.12013	−0.235923
$472$	0	0
$473$	3.58483	0.164831
$474$	0	0
$475$	−23.8209	−1.09298
$476$	0	0
$477$	−0.505899	−0.0231635
$478$	0	0
$479$	−8.59873	−0.392886	−0.196443	−	0.980515i	$-0.562939\pi$
$479$	−8.59873	−0.392886	−0.196443	+	0.980515i	$0.562939\pi$
$480$	0	0
$481$	−65.1803	−2.97196
$482$	0	0
$483$	−61.2022	−2.78480
$484$	0	0
$485$	−48.9983	−2.22490
$486$	0	0
$487$	−18.3993	−0.833754	−0.416877	−	0.908963i	$-0.636875\pi$
$487$	−18.3993	−0.833754	−0.416877	+	0.908963i	$0.636875\pi$
$488$	0	0
$489$	25.9229	1.17227
$490$	0	0
$491$	−12.4076	−0.559946	−0.279973	−	0.960008i	$-0.590326\pi$
$491$	−12.4076	−0.559946	−0.279973	+	0.960008i	$0.590326\pi$
$492$	0	0
$493$	−1.24527	−0.0560843
$494$	0	0
$495$	2.06778	0.0929396
$496$	0	0
$497$	33.2301	1.49058
$498$	0	0
$499$	−23.4440	−1.04950	−0.524750	−	0.851257i	$-0.675841\pi$
$499$	−23.4440	−1.04950	−0.524750	+	0.851257i	$0.675841\pi$
$500$	0	0
$501$	6.37279	0.284715
$502$	0	0
$503$	22.2058	0.990109	0.495055	−	0.868862i	$-0.335148\pi$
$503$	22.2058	0.990109	0.495055	+	0.868862i	$0.335148\pi$
$504$	0	0
$505$	36.4548	1.62222
$506$	0	0
$507$	−39.1810	−1.74009
$508$	0	0
$509$	16.7218	0.741183	0.370591	−	0.928796i	$-0.379155\pi$
$509$	16.7218	0.741183	0.370591	+	0.928796i	$0.379155\pi$
$510$	0	0
$511$	29.0390	1.28461
$512$	0	0
$513$	−11.4262	−0.504477
$514$	0	0
$515$	−55.9445	−2.46521
$516$	0	0
$517$	1.29040	0.0567516
$518$	0	0
$519$	32.9179	1.44493
$520$	0	0
$521$	22.0308	0.965186	0.482593	−	0.875845i	$-0.339695\pi$
$521$	22.0308	0.965186	0.482593	+	0.875845i	$0.339695\pi$
$522$	0	0
$523$	−32.5700	−1.42419	−0.712093	−	0.702085i	$-0.752253\pi$
$523$	−32.5700	−1.42419	−0.712093	+	0.702085i	$0.752253\pi$
$524$	0	0
$525$	69.4210	3.02978
$526$	0	0
$527$	−0.818205	−0.0356416
$528$	0	0
$529$	62.8375	2.73207
$530$	0	0
$531$	1.37398	0.0596258
$532$	0	0
$533$	1.27812	0.0553615
$534$	0	0
$535$	−7.32855	−0.316841
$536$	0	0
$537$	24.5658	1.06009
$538$	0	0
$539$	21.2021	0.913238
$540$	0	0
$541$	−32.5894	−1.40113	−0.700565	−	0.713589i	$-0.747068\pi$
$541$	−32.5894	−1.40113	−0.700565	+	0.713589i	$0.747068\pi$
$542$	0	0
$543$	33.1174	1.42120
$544$	0	0
$545$	−4.95847	−0.212397
$546$	0	0
$547$	25.3368	1.08332	0.541662	−	0.840596i	$-0.317795\pi$
$547$	25.3368	1.08332	0.541662	+	0.840596i	$0.317795\pi$
$548$	0	0
$549$	2.17277	0.0927318
$550$	0	0
$551$	−17.6553	−0.752140
$552$	0	0
$553$	−21.5557	−0.916643
$554$	0	0
$555$	−77.2087	−3.27733
$556$	0	0
$557$	12.2075	0.517247	0.258623	−	0.965978i	$-0.416731\pi$
$557$	12.2075	0.517247	0.258623	+	0.965978i	$0.416731\pi$
$558$	0	0
$559$	−6.77857	−0.286703
$560$	0	0
$561$	0.890462	0.0375954
$562$	0	0
$563$	26.5841	1.12039	0.560193	−	0.828362i	$-0.310727\pi$
$563$	26.5841	1.12039	0.560193	+	0.828362i	$0.310727\pi$
$564$	0	0
$565$	−20.0955	−0.845426
$566$	0	0
$567$	35.1673	1.47689
$568$	0	0
$569$	−30.1745	−1.26498	−0.632490	−	0.774568i	$-0.717967\pi$
$569$	−30.1745	−1.26498	−0.632490	+	0.774568i	$0.717967\pi$
$570$	0	0
$571$	−18.3268	−0.766954	−0.383477	−	0.923550i	$-0.625273\pi$
$571$	−18.3268	−0.766954	−0.383477	+	0.923550i	$0.625273\pi$
$572$	0	0
$573$	−19.6785	−0.822080
$574$	0	0
$575$	−97.3646	−4.06038
$576$	0	0
$577$	22.7503	0.947106	0.473553	−	0.880765i	$-0.342971\pi$
$577$	22.7503	0.947106	0.473553	+	0.880765i	$0.342971\pi$
$578$	0	0
$579$	1.30615	0.0542818
$580$	0	0
$581$	29.8986	1.24040
$582$	0	0
$583$	−9.43527	−0.390769
$584$	0	0
$585$	−3.90996	−0.161657
$586$	0	0
$587$	−10.1011	−0.416918	−0.208459	−	0.978031i	$-0.566845\pi$
$587$	−10.1011	−0.416918	−0.208459	+	0.978031i	$0.566845\pi$
$588$	0	0
$589$	−11.6004	−0.477985
$590$	0	0
$591$	−41.0812	−1.68985
$592$	0	0
$593$	9.52360	0.391087	0.195544	−	0.980695i	$-0.437353\pi$
$593$	9.52360	0.391087	0.195544	+	0.980695i	$0.437353\pi$
$594$	0	0
$595$	2.33684	0.0958011
$596$	0	0
$597$	47.2388	1.93335
$598$	0	0
$599$	−39.0068	−1.59378	−0.796888	−	0.604128i	$-0.793522\pi$
$599$	−39.0068	−1.59378	−0.796888	+	0.604128i	$0.793522\pi$
$600$	0	0
$601$	6.89579	0.281285	0.140643	−	0.990060i	$-0.455083\pi$
$601$	6.89579	0.281285	0.140643	+	0.990060i	$0.455083\pi$
$602$	0	0
$603$	−0.733428	−0.0298675
$604$	0	0
$605$	−4.75457	−0.193301
$606$	0	0
$607$	−3.80787	−0.154556	−0.0772782	−	0.997010i	$-0.524623\pi$
$607$	−3.80787	−0.154556	−0.0772782	+	0.997010i	$0.524623\pi$
$608$	0	0
$609$	51.4525	2.08496
$610$	0	0
$611$	−2.44002	−0.0987125
$612$	0	0
$613$	−3.87986	−0.156706	−0.0783531	−	0.996926i	$-0.524966\pi$
$613$	−3.87986	−0.156706	−0.0783531	+	0.996926i	$0.524966\pi$
$614$	0	0
$615$	1.51398	0.0610497
$616$	0	0
$617$	−26.4513	−1.06489	−0.532444	−	0.846465i	$-0.678726\pi$
$617$	−26.4513	−1.06489	−0.532444	+	0.846465i	$0.678726\pi$
$618$	0	0
$619$	10.2223	0.410870	0.205435	−	0.978671i	$-0.434139\pi$
$619$	10.2223	0.410870	0.205435	+	0.978671i	$0.434139\pi$
$620$	0	0
$621$	−46.7028	−1.87412
$622$	0	0
$623$	−5.39842	−0.216283
$624$	0	0
$625$	32.8944	1.31578
$626$	0	0
$627$	12.6248	0.504187
$628$	0	0
$629$	−1.76109	−0.0702194
$630$	0	0
$631$	−6.34946	−0.252768	−0.126384	−	0.991981i	$-0.540337\pi$
$631$	−6.34946	−0.252768	−0.126384	+	0.991981i	$0.540337\pi$
$632$	0	0
$633$	8.18374	0.325274
$634$	0	0
$635$	−32.8894	−1.30518
$636$	0	0
$637$	−40.0911	−1.58847
$638$	0	0
$639$	−1.50225	−0.0594281
$640$	0	0
$641$	−10.6226	−0.419566	−0.209783	−	0.977748i	$-0.567276\pi$
$641$	−10.6226	−0.419566	−0.209783	+	0.977748i	$0.567276\pi$
$642$	0	0
$643$	29.8241	1.17615	0.588073	−	0.808808i	$-0.299887\pi$
$643$	29.8241	1.17615	0.588073	+	0.808808i	$0.299887\pi$
$644$	0	0
$645$	−8.02949	−0.316161
$646$	0	0
$647$	21.2500	0.835424	0.417712	−	0.908579i	$-0.362832\pi$
$647$	21.2500	0.835424	0.417712	+	0.908579i	$0.362832\pi$
$648$	0	0
$649$	25.6255	1.00589
$650$	0	0
$651$	33.8068	1.32499
$652$	0	0
$653$	31.9534	1.25043	0.625216	−	0.780451i	$-0.285011\pi$
$653$	31.9534	1.25043	0.625216	+	0.780451i	$0.285011\pi$
$654$	0	0
$655$	52.1538	2.03782
$656$	0	0
$657$	−1.31278	−0.0512165
$658$	0	0
$659$	23.5711	0.918200	0.459100	−	0.888385i	$-0.348172\pi$
$659$	23.5711	0.918200	0.459100	+	0.888385i	$0.348172\pi$
$660$	0	0
$661$	0.891079	0.0346589	0.0173295	−	0.999850i	$-0.494484\pi$
$661$	0.891079	0.0346589	0.0173295	+	0.999850i	$0.494484\pi$
$662$	0	0
$663$	−1.68378	−0.0653925
$664$	0	0
$665$	33.1313	1.28478
$666$	0	0
$667$	−72.1633	−2.79418
$668$	0	0
$669$	0.713964	0.0276034
$670$	0	0
$671$	40.5234	1.56439
$672$	0	0
$673$	−18.6918	−0.720516	−0.360258	−	0.932853i	$-0.617311\pi$
$673$	−18.6918	−0.720516	−0.360258	+	0.932853i	$0.617311\pi$
$674$	0	0
$675$	52.9744	2.03899
$676$	0	0
$677$	−50.6168	−1.94536	−0.972680	−	0.232149i	$-0.925424\pi$
$677$	−50.6168	−1.94536	−0.972680	+	0.232149i	$0.925424\pi$
$678$	0	0
$679$	46.1784	1.77216
$680$	0	0
$681$	−2.08018	−0.0797126
$682$	0	0
$683$	−25.2304	−0.965414	−0.482707	−	0.875782i	$-0.660346\pi$
$683$	−25.2304	−0.965414	−0.482707	+	0.875782i	$0.660346\pi$
$684$	0	0
$685$	−83.6684	−3.19681
$686$	0	0
$687$	38.3585	1.46347
$688$	0	0
$689$	17.8412	0.679695
$690$	0	0
$691$	−27.3991	−1.04231	−0.521156	−	0.853461i	$-0.674499\pi$
$691$	−27.3991	−1.04231	−0.521156	+	0.853461i	$0.674499\pi$
$692$	0	0
$693$	−1.94877	−0.0740278
$694$	0	0
$695$	−83.6206	−3.17191
$696$	0	0
$697$	0.0345333	0.00130804
$698$	0	0
$699$	−29.0212	−1.09768
$700$	0	0
$701$	12.4577	0.470521	0.235260	−	0.971932i	$-0.424406\pi$
$701$	12.4577	0.470521	0.235260	+	0.971932i	$0.424406\pi$
$702$	0	0
$703$	−24.9685	−0.941705
$704$	0	0
$705$	−2.89030	−0.108855
$706$	0	0
$707$	−34.3568	−1.29212
$708$	0	0
$709$	23.3285	0.876121	0.438061	−	0.898945i	$-0.355666\pi$
$709$	23.3285	0.876121	0.438061	+	0.898945i	$0.355666\pi$
$710$	0	0
$711$	0.974479	0.0365458
$712$	0	0
$713$	−47.4149	−1.77570
$714$	0	0
$715$	−72.9228	−2.72716
$716$	0	0
$717$	34.7965	1.29950
$718$	0	0
$719$	22.6939	0.846340	0.423170	−	0.906050i	$-0.360917\pi$
$719$	22.6939	0.846340	0.423170	+	0.906050i	$0.360917\pi$
$720$	0	0
$721$	52.7249	1.96358
$722$	0	0
$723$	−26.4442	−0.983472
$724$	0	0
$725$	81.8541	3.03998
$726$	0	0
$727$	48.4830	1.79813	0.899067	−	0.437811i	$-0.144246\pi$
$727$	48.4830	1.79813	0.899067	+	0.437811i	$0.144246\pi$
$728$	0	0
$729$	25.3257	0.937990
$730$	0	0
$731$	−0.183149	−0.00677400
$732$	0	0
$733$	−1.89258	−0.0699040	−0.0349520	−	0.999389i	$-0.511128\pi$
$733$	−1.89258	−0.0699040	−0.0349520	+	0.999389i	$0.511128\pi$
$734$	0	0
$735$	−47.4895	−1.75168
$736$	0	0
$737$	−13.6788	−0.503865
$738$	0	0
$739$	28.1230	1.03452	0.517260	−	0.855828i	$-0.326952\pi$
$739$	28.1230	1.03452	0.517260	+	0.855828i	$0.326952\pi$
$740$	0	0
$741$	−23.8723	−0.876972
$742$	0	0
$743$	−37.3863	−1.37157	−0.685785	−	0.727804i	$-0.740541\pi$
$743$	−37.3863	−1.37157	−0.685785	+	0.727804i	$0.740541\pi$
$744$	0	0
$745$	−63.5311	−2.32760
$746$	0	0
$747$	−1.35164	−0.0494539
$748$	0	0
$749$	6.90679	0.252368
$750$	0	0
$751$	−36.5397	−1.33335	−0.666677	−	0.745347i	$-0.732284\pi$
$751$	−36.5397	−1.33335	−0.666677	+	0.745347i	$0.732284\pi$
$752$	0	0
$753$	−1.77983	−0.0648605
$754$	0	0
$755$	63.3873	2.30690
$756$	0	0
$757$	29.9289	1.08779	0.543893	−	0.839155i	$-0.316950\pi$
$757$	29.9289	1.08779	0.543893	+	0.839155i	$0.316950\pi$
$758$	0	0
$759$	51.6021	1.87304
$760$	0	0
$761$	36.8821	1.33697	0.668487	−	0.743724i	$-0.266942\pi$
$761$	36.8821	1.33697	0.668487	+	0.743724i	$0.266942\pi$
$762$	0	0
$763$	4.67311	0.169178
$764$	0	0
$765$	−0.105643	−0.00381952
$766$	0	0
$767$	−48.4554	−1.74962
$768$	0	0
$769$	−40.5378	−1.46183	−0.730914	−	0.682469i	$-0.760906\pi$
$769$	−40.5378	−1.46183	−0.730914	+	0.682469i	$0.760906\pi$
$770$	0	0
$771$	4.93480	0.177722
$772$	0	0
$773$	43.4841	1.56401	0.782007	−	0.623269i	$-0.214196\pi$
$773$	43.4841	1.56401	0.782007	+	0.623269i	$0.214196\pi$
$774$	0	0
$775$	53.7821	1.93191
$776$	0	0
$777$	72.7653	2.61044
$778$	0	0
$779$	0.489607	0.0175420
$780$	0	0
$781$	−28.0177	−1.00255
$782$	0	0
$783$	39.2628	1.40314
$784$	0	0
$785$	11.3291	0.404352
$786$	0	0
$787$	10.9029	0.388646	0.194323	−	0.980938i	$-0.437749\pi$
$787$	10.9029	0.388646	0.194323	+	0.980938i	$0.437749\pi$
$788$	0	0
$789$	−39.7147	−1.41388
$790$	0	0
$791$	18.9390	0.673395
$792$	0	0
$793$	−76.6257	−2.72106
$794$	0	0
$795$	21.1336	0.749532
$796$	0	0
$797$	−35.7203	−1.26528	−0.632639	−	0.774446i	$-0.718028\pi$
$797$	−35.7203	−1.26528	−0.632639	+	0.774446i	$0.718028\pi$
$798$	0	0
$799$	−0.0659263	−0.00233231
$800$	0	0
$801$	0.244049	0.00862304
$802$	0	0
$803$	−24.4840	−0.864023
$804$	0	0
$805$	135.419	4.77291
$806$	0	0
$807$	26.8432	0.944924
$808$	0	0
$809$	14.2361	0.500513	0.250257	−	0.968180i	$-0.419485\pi$
$809$	14.2361	0.500513	0.250257	+	0.968180i	$0.419485\pi$
$810$	0	0
$811$	19.9049	0.698956	0.349478	−	0.936945i	$-0.386359\pi$
$811$	19.9049	0.698956	0.349478	+	0.936945i	$0.386359\pi$
$812$	0	0
$813$	−22.3752	−0.784733
$814$	0	0
$815$	−57.3585	−2.00918
$816$	0	0
$817$	−2.59665	−0.0908454
$818$	0	0
$819$	3.68494	0.128762
$820$	0	0
$821$	−7.04356	−0.245822	−0.122911	−	0.992418i	$-0.539223\pi$
$821$	−7.04356	−0.245822	−0.122911	+	0.992418i	$0.539223\pi$
$822$	0	0
$823$	27.8288	0.970051	0.485025	−	0.874500i	$-0.338810\pi$
$823$	27.8288	0.970051	0.485025	+	0.874500i	$0.338810\pi$
$824$	0	0
$825$	−58.5317	−2.03781
$826$	0	0
$827$	14.3326	0.498393	0.249196	−	0.968453i	$-0.419834\pi$
$827$	14.3326	0.498393	0.249196	+	0.968453i	$0.419834\pi$
$828$	0	0
$829$	−18.8154	−0.653485	−0.326743	−	0.945113i	$-0.605951\pi$
$829$	−18.8154	−0.653485	−0.326743	+	0.945113i	$0.605951\pi$
$830$	0	0
$831$	20.8571	0.723525
$832$	0	0
$833$	−1.08321	−0.0375311
$834$	0	0
$835$	−14.1008	−0.487978
$836$	0	0
$837$	25.7976	0.891696
$838$	0	0
$839$	−8.68577	−0.299866	−0.149933	−	0.988696i	$-0.547906\pi$
$839$	−8.68577	−0.299866	−0.149933	+	0.988696i	$0.547906\pi$
$840$	0	0
$841$	31.6675	1.09198
$842$	0	0
$843$	31.6649	1.09060
$844$	0	0
$845$	86.6940	2.98237
$846$	0	0
$847$	4.48094	0.153967
$848$	0	0
$849$	17.4356	0.598387
$850$	0	0
$851$	−102.055	−3.49840
$852$	0	0
$853$	−22.7384	−0.778549	−0.389275	−	0.921122i	$-0.627274\pi$
$853$	−22.7384	−0.778549	−0.389275	+	0.921122i	$0.627274\pi$
$854$	0	0
$855$	−1.49778	−0.0512231
$856$	0	0
$857$	28.7505	0.982099	0.491049	−	0.871132i	$-0.336613\pi$
$857$	28.7505	0.982099	0.491049	+	0.871132i	$0.336613\pi$
$858$	0	0
$859$	−53.1507	−1.81348	−0.906740	−	0.421690i	$-0.861437\pi$
$859$	−53.1507	−1.81348	−0.906740	+	0.421690i	$0.861437\pi$
$860$	0	0
$861$	−1.42685	−0.0486270
$862$	0	0
$863$	23.2600	0.791778	0.395889	−	0.918298i	$-0.370436\pi$
$863$	23.2600	0.791778	0.395889	+	0.918298i	$0.370436\pi$
$864$	0	0
$865$	−72.8360	−2.47650
$866$	0	0
$867$	30.2116	1.02604
$868$	0	0
$869$	18.1745	0.616529
$870$	0	0
$871$	25.8653	0.876412
$872$	0	0
$873$	−2.08761	−0.0706548
$874$	0	0
$875$	−80.5225	−2.72216
$876$	0	0
$877$	−23.2847	−0.786269	−0.393135	−	0.919481i	$-0.628609\pi$
$877$	−23.2847	−0.786269	−0.393135	+	0.919481i	$0.628609\pi$
$878$	0	0
$879$	43.6129	1.47103
$880$	0	0
$881$	−22.2017	−0.747995	−0.373997	−	0.927430i	$-0.622013\pi$
$881$	−22.2017	−0.747995	−0.373997	+	0.927430i	$0.622013\pi$
$882$	0	0
$883$	−13.1385	−0.442147	−0.221074	−	0.975257i	$-0.570956\pi$
$883$	−13.1385	−0.442147	−0.221074	+	0.975257i	$0.570956\pi$
$884$	0	0
$885$	−57.3973	−1.92939
$886$	0	0
$887$	−11.2484	−0.377685	−0.188842	−	0.982007i	$-0.560474\pi$
$887$	−11.2484	−0.377685	−0.188842	+	0.982007i	$0.560474\pi$
$888$	0	0
$889$	30.9966	1.03959
$890$	0	0
$891$	−29.6510	−0.993347
$892$	0	0
$893$	−0.934693	−0.0312783
$894$	0	0
$895$	−54.3556	−1.81691
$896$	0	0
$897$	−97.5746	−3.25792
$898$	0	0
$899$	39.8615	1.32946
$900$	0	0
$901$	0.482048	0.0160593
$902$	0	0
$903$	7.56739	0.251827
$904$	0	0
$905$	−73.2774	−2.43582
$906$	0	0
$907$	2.03218	0.0674774	0.0337387	−	0.999431i	$-0.489259\pi$
$907$	2.03218	0.0674774	0.0337387	+	0.999431i	$0.489259\pi$
$908$	0	0
$909$	1.55318	0.0515159
$910$	0	0
$911$	32.9856	1.09286	0.546431	−	0.837504i	$-0.315986\pi$
$911$	32.9856	1.09286	0.546431	+	0.837504i	$0.315986\pi$
$912$	0	0
$913$	−25.2088	−0.834288
$914$	0	0
$915$	−90.7663	−3.00064
$916$	0	0
$917$	−49.1523	−1.62315
$918$	0	0
$919$	26.8429	0.885464	0.442732	−	0.896654i	$-0.354009\pi$
$919$	26.8429	0.885464	0.442732	+	0.896654i	$0.354009\pi$
$920$	0	0
$921$	−13.3091	−0.438548
$922$	0	0
$923$	52.9788	1.74382
$924$	0	0
$925$	115.760	3.80616
$926$	0	0
$927$	−2.38356	−0.0782863
$928$	0	0
$929$	−27.1560	−0.890960	−0.445480	−	0.895292i	$-0.646967\pi$
$929$	−27.1560	−0.890960	−0.445480	+	0.895292i	$0.646967\pi$
$930$	0	0
$931$	−15.3576	−0.503326
$932$	0	0
$933$	−25.8058	−0.844845
$934$	0	0
$935$	−1.97029	−0.0644353
$936$	0	0
$937$	−16.1798	−0.528571	−0.264286	−	0.964444i	$-0.585136\pi$
$937$	−16.1798	−0.528571	−0.264286	+	0.964444i	$0.585136\pi$
$938$	0	0
$939$	−38.5974	−1.25958
$940$	0	0
$941$	−14.0269	−0.457265	−0.228633	−	0.973513i	$-0.573425\pi$
$941$	−14.0269	−0.457265	−0.228633	+	0.973513i	$0.573425\pi$
$942$	0	0
$943$	2.00120	0.0651679
$944$	0	0
$945$	−73.6794	−2.39679
$946$	0	0
$947$	−31.1556	−1.01242	−0.506211	−	0.862410i	$-0.668954\pi$
$947$	−31.1556	−1.01242	−0.506211	+	0.862410i	$0.668954\pi$
$948$	0	0
$949$	46.2969	1.50286
$950$	0	0
$951$	−42.8136	−1.38833
$952$	0	0
$953$	−17.6403	−0.571425	−0.285713	−	0.958315i	$-0.592230\pi$
$953$	−17.6403	−0.571425	−0.285713	+	0.958315i	$0.592230\pi$
$954$	0	0
$955$	43.5417	1.40898
$956$	0	0
$957$	−43.3817	−1.40233
$958$	0	0
$959$	78.8533	2.54631
$960$	0	0
$961$	−4.80903	−0.155130
$962$	0	0
$963$	−0.312238	−0.0100617
$964$	0	0
$965$	−2.89006	−0.0930345
$966$	0	0
$967$	−46.3186	−1.48951	−0.744753	−	0.667340i	$-0.767433\pi$
$967$	−46.3186	−1.48951	−0.744753	+	0.667340i	$0.767433\pi$
$968$	0	0
$969$	−0.645002	−0.0207205
$970$	0	0
$971$	32.8704	1.05486	0.527430	−	0.849598i	$-0.323156\pi$
$971$	32.8704	1.05486	0.527430	+	0.849598i	$0.323156\pi$
$972$	0	0
$973$	78.8082	2.52648
$974$	0	0
$975$	110.678	3.54453
$976$	0	0
$977$	18.7137	0.598704	0.299352	−	0.954143i	$-0.403230\pi$
$977$	18.7137	0.598704	0.299352	+	0.954143i	$0.403230\pi$
$978$	0	0
$979$	4.55163	0.145471
$980$	0	0
$981$	−0.211259	−0.00674499
$982$	0	0
$983$	9.52208	0.303707	0.151854	−	0.988403i	$-0.451476\pi$
$983$	9.52208	0.303707	0.151854	+	0.988403i	$0.451476\pi$
$984$	0	0
$985$	90.8986	2.89627
$986$	0	0
$987$	2.72396	0.0867046
$988$	0	0
$989$	−10.6134	−0.337488
$990$	0	0
$991$	38.4936	1.22279	0.611395	−	0.791326i	$-0.290609\pi$
$991$	38.4936	1.22279	0.611395	+	0.791326i	$0.290609\pi$
$992$	0	0
$993$	31.3166	0.993803
$994$	0	0
$995$	−104.523	−3.31361
$996$	0	0
$997$	−21.2332	−0.672463	−0.336232	−	0.941779i	$-0.609152\pi$
$997$	−21.2332	−0.672463	−0.336232	+	0.941779i	$0.609152\pi$
$998$	0	0
$999$	55.5264	1.75678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.g.1.10	✓	30
4.3	odd	2		8032.2.a.j.1.21	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.10	✓	30	1.1	even	1	trivial
8032.2.a.j.1.21	yes	30	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77983 q^{3} +3.93815 q^{5} -3.71151 q^{7} +0.167788 q^{9} +O(q^{10})\)	\(q-1.77983 q^{3} +3.93815 q^{5} -3.71151 q^{7} +0.167788 q^{9} +3.12933 q^{11} -5.91725 q^{13} -7.00923 q^{15} -0.159877 q^{17} -2.26671 q^{19} +6.60584 q^{21} -9.26486 q^{23} +10.5090 q^{25} +5.04085 q^{27} +7.78893 q^{29} +5.11771 q^{31} -5.56966 q^{33} -14.6165 q^{35} +11.0153 q^{37} +10.5317 q^{39} -0.215999 q^{41} +1.14556 q^{43} +0.660773 q^{45} +0.412356 q^{47} +6.77528 q^{49} +0.284554 q^{51} -3.01511 q^{53} +12.3238 q^{55} +4.03436 q^{57} +8.18882 q^{59} +12.9495 q^{61} -0.622745 q^{63} -23.3030 q^{65} -4.37117 q^{67} +16.4898 q^{69} -8.95328 q^{71} -7.82405 q^{73} -18.7043 q^{75} -11.6145 q^{77} +5.80781 q^{79} -9.47521 q^{81} -8.05565 q^{83} -0.629620 q^{85} -13.8630 q^{87} +1.45451 q^{89} +21.9619 q^{91} -9.10865 q^{93} -8.92666 q^{95} -12.4420 q^{97} +0.525063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * q^99

Introduction

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