LMFDB - Embedded newform 8036.2.a.s.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.476040$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.476040 q^{3} +2.17736 q^{5} -2.77339 q^{9} +O(q^{10})$	$q-0.476040 q^{3} +2.17736 q^{5} -2.77339 q^{9} -0.849919 q^{11} -0.662286 q^{13} -1.03651 q^{15} +5.15506 q^{17} +6.79204 q^{19} -0.144635 q^{23} -0.259093 q^{25} +2.74836 q^{27} -9.46489 q^{29} -8.50663 q^{31} +0.404596 q^{33} -7.75956 q^{37} +0.315275 q^{39} +1.00000 q^{41} -2.77075 q^{43} -6.03867 q^{45} -3.45840 q^{47} -2.45401 q^{51} +7.28924 q^{53} -1.85058 q^{55} -3.23329 q^{57} -9.84084 q^{59} -9.68365 q^{61} -1.44204 q^{65} +2.94318 q^{67} +0.0688519 q^{69} +3.88837 q^{71} +11.8207 q^{73} +0.123339 q^{75} +5.63075 q^{79} +7.01183 q^{81} -8.76939 q^{83} +11.2244 q^{85} +4.50567 q^{87} +5.94828 q^{89} +4.04950 q^{93} +14.7887 q^{95} -2.25908 q^{97} +2.35715 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

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
$3$	−0.476040	−0.274842	−0.137421	−	0.990513i	$-0.543881\pi$
$3$	−0.476040	−0.274842	−0.137421	+	0.990513i	$0.543881\pi$
$4$	0	0
$5$	2.17736	0.973746	0.486873	−	0.873473i	$-0.338137\pi$
$5$	2.17736	0.973746	0.486873	+	0.873473i	$0.338137\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.77339	−0.924462
$10$	0	0
$11$	−0.849919	−0.256260	−0.128130	−	0.991757i	$-0.540897\pi$
$11$	−0.849919	−0.256260	−0.128130	+	0.991757i	$0.540897\pi$
$12$	0	0
$13$	−0.662286	−0.183685	−0.0918426	−	0.995774i	$-0.529276\pi$
$13$	−0.662286	−0.183685	−0.0918426	+	0.995774i	$0.529276\pi$
$14$	0	0
$15$	−1.03651	−0.267626
$16$	0	0
$17$	5.15506	1.25028	0.625142	−	0.780511i	$-0.285041\pi$
$17$	5.15506	1.25028	0.625142	+	0.780511i	$0.285041\pi$
$18$	0	0
$19$	6.79204	1.55820	0.779101	−	0.626899i	$-0.215676\pi$
$19$	6.79204	1.55820	0.779101	+	0.626899i	$0.215676\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.144635	−0.0301584	−0.0150792	−	0.999886i	$-0.504800\pi$
$23$	−0.144635	−0.0301584	−0.0150792	+	0.999886i	$0.504800\pi$
$24$	0	0
$25$	−0.259093	−0.0518187
$26$	0	0
$27$	2.74836	0.528923
$28$	0	0
$29$	−9.46489	−1.75759	−0.878793	−	0.477203i	$-0.841651\pi$
$29$	−9.46489	−1.75759	−0.878793	+	0.477203i	$0.841651\pi$
$30$	0	0
$31$	−8.50663	−1.52784	−0.763918	−	0.645314i	$-0.776727\pi$
$31$	−8.50663	−1.52784	−0.763918	+	0.645314i	$0.776727\pi$
$32$	0	0
$33$	0.404596	0.0704310
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.75956	−1.27566	−0.637832	−	0.770175i	$-0.720169\pi$
$37$	−7.75956	−1.27566	−0.637832	+	0.770175i	$0.720169\pi$
$38$	0	0
$39$	0.315275	0.0504844
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.77075	−0.422536	−0.211268	−	0.977428i	$-0.567759\pi$
$43$	−2.77075	−0.422536	−0.211268	+	0.977428i	$0.567759\pi$
$44$	0	0
$45$	−6.03867	−0.900191
$46$	0	0
$47$	−3.45840	−0.504459	−0.252230	−	0.967667i	$-0.581164\pi$
$47$	−3.45840	−0.504459	−0.252230	+	0.967667i	$0.581164\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.45401	−0.343631
$52$	0	0
$53$	7.28924	1.00125	0.500627	−	0.865663i	$-0.333103\pi$
$53$	7.28924	1.00125	0.500627	+	0.865663i	$0.333103\pi$
$54$	0	0
$55$	−1.85058	−0.249532
$56$	0	0
$57$	−3.23329	−0.428259
$58$	0	0
$59$	−9.84084	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$59$	−9.84084	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$60$	0	0
$61$	−9.68365	−1.23986	−0.619932	−	0.784656i	$-0.712840\pi$
$61$	−9.68365	−1.23986	−0.619932	+	0.784656i	$0.712840\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.44204	−0.178863
$66$	0	0
$67$	2.94318	0.359566	0.179783	−	0.983706i	$-0.442460\pi$
$67$	2.94318	0.359566	0.179783	+	0.983706i	$0.442460\pi$
$68$	0	0
$69$	0.0688519	0.00828879
$70$	0	0
$71$	3.88837	0.461465	0.230732	−	0.973017i	$-0.425888\pi$
$71$	3.88837	0.461465	0.230732	+	0.973017i	$0.425888\pi$
$72$	0	0
$73$	11.8207	1.38350	0.691752	−	0.722135i	$-0.256839\pi$
$73$	11.8207	1.38350	0.691752	+	0.722135i	$0.256839\pi$
$74$	0	0
$75$	0.123339	0.0142419
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.63075	0.633509	0.316754	−	0.948508i	$-0.397407\pi$
$79$	5.63075	0.633509	0.316754	+	0.948508i	$0.397407\pi$
$80$	0	0
$81$	7.01183	0.779092
$82$	0	0
$83$	−8.76939	−0.962566	−0.481283	−	0.876565i	$-0.659829\pi$
$83$	−8.76939	−0.962566	−0.481283	+	0.876565i	$0.659829\pi$
$84$	0	0
$85$	11.2244	1.21746
$86$	0	0
$87$	4.50567	0.483058
$88$	0	0
$89$	5.94828	0.630516	0.315258	−	0.949006i	$-0.397909\pi$
$89$	5.94828	0.630516	0.315258	+	0.949006i	$0.397909\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.04950	0.419913
$94$	0	0
$95$	14.7887	1.51729
$96$	0	0
$97$	−2.25908	−0.229375	−0.114687	−	0.993402i	$-0.536587\pi$
$97$	−2.25908	−0.229375	−0.114687	+	0.993402i	$0.536587\pi$
$98$	0	0
$99$	2.35715	0.236903
$100$	0	0
$101$	−1.07288	−0.106756	−0.0533778	−	0.998574i	$-0.516999\pi$
$101$	−1.07288	−0.106756	−0.0533778	+	0.998574i	$0.516999\pi$
$102$	0	0
$103$	−0.439620	−0.0433171	−0.0216585	−	0.999765i	$-0.506895\pi$
$103$	−0.439620	−0.0433171	−0.0216585	+	0.999765i	$0.506895\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.7443	−1.81208	−0.906038	−	0.423196i	$-0.860908\pi$
$107$	−18.7443	−1.81208	−0.906038	+	0.423196i	$0.860908\pi$
$108$	0	0
$109$	7.68173	0.735777	0.367888	−	0.929870i	$-0.380081\pi$
$109$	7.68173	0.735777	0.367888	+	0.929870i	$0.380081\pi$
$110$	0	0
$111$	3.69387	0.350606
$112$	0	0
$113$	−6.90767	−0.649819	−0.324910	−	0.945745i	$-0.605334\pi$
$113$	−6.90767	−0.649819	−0.324910	+	0.945745i	$0.605334\pi$
$114$	0	0
$115$	−0.314922	−0.0293666
$116$	0	0
$117$	1.83678	0.169810
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.2776	−0.934331
$122$	0	0
$123$	−0.476040	−0.0429231
$124$	0	0
$125$	−11.4510	−1.02420
$126$	0	0
$127$	4.95127	0.439354	0.219677	−	0.975573i	$-0.429500\pi$
$127$	4.95127	0.439354	0.219677	+	0.975573i	$0.429500\pi$
$128$	0	0
$129$	1.31899	0.116131
$130$	0	0
$131$	8.52845	0.745135	0.372567	−	0.928005i	$-0.378478\pi$
$131$	8.52845	0.745135	0.372567	+	0.928005i	$0.378478\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.98418	0.515037
$136$	0	0
$137$	−8.32942	−0.711630	−0.355815	−	0.934556i	$-0.615797\pi$
$137$	−8.32942	−0.711630	−0.355815	+	0.934556i	$0.615797\pi$
$138$	0	0
$139$	−16.0675	−1.36282	−0.681412	−	0.731900i	$-0.738634\pi$
$139$	−16.0675	−1.36282	−0.681412	+	0.731900i	$0.738634\pi$
$140$	0	0
$141$	1.64634	0.138647
$142$	0	0
$143$	0.562890	0.0470712
$144$	0	0
$145$	−20.6085	−1.71144
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.8778	1.13691	0.568456	−	0.822714i	$-0.307541\pi$
$149$	13.8778	1.13691	0.568456	+	0.822714i	$0.307541\pi$
$150$	0	0
$151$	1.96940	0.160267	0.0801337	−	0.996784i	$-0.474465\pi$
$151$	1.96940	0.160267	0.0801337	+	0.996784i	$0.474465\pi$
$152$	0	0
$153$	−14.2970	−1.15584
$154$	0	0
$155$	−18.5220	−1.48772
$156$	0	0
$157$	0.492399	0.0392977	0.0196488	−	0.999807i	$-0.493745\pi$
$157$	0.492399	0.0392977	0.0196488	+	0.999807i	$0.493745\pi$
$158$	0	0
$159$	−3.46997	−0.275187
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3.53172	−0.276626	−0.138313	−	0.990389i	$-0.544168\pi$
$163$	−3.53172	−0.276626	−0.138313	+	0.990389i	$0.544168\pi$
$164$	0	0
$165$	0.880951	0.0685819
$166$	0	0
$167$	10.2126	0.790278	0.395139	−	0.918621i	$-0.370696\pi$
$167$	10.2126	0.790278	0.395139	+	0.918621i	$0.370696\pi$
$168$	0	0
$169$	−12.5614	−0.966260
$170$	0	0
$171$	−18.8370	−1.44050
$172$	0	0
$173$	−5.73071	−0.435698	−0.217849	−	0.975983i	$-0.569904\pi$
$173$	−5.73071	−0.435698	−0.217849	+	0.975983i	$0.569904\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.68464	0.352119
$178$	0	0
$179$	16.5102	1.23403	0.617016	−	0.786951i	$-0.288341\pi$
$179$	16.5102	1.23403	0.617016	+	0.786951i	$0.288341\pi$
$180$	0	0
$181$	12.2094	0.907515	0.453757	−	0.891125i	$-0.350083\pi$
$181$	12.2094	0.907515	0.453757	+	0.891125i	$0.350083\pi$
$182$	0	0
$183$	4.60981	0.340767
$184$	0	0
$185$	−16.8954	−1.24217
$186$	0	0
$187$	−4.38138	−0.320398
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.93337	−0.356966	−0.178483	−	0.983943i	$-0.557119\pi$
$191$	−4.93337	−0.356966	−0.178483	+	0.983943i	$0.557119\pi$
$192$	0	0
$193$	−6.12016	−0.440539	−0.220270	−	0.975439i	$-0.570694\pi$
$193$	−6.12016	−0.440539	−0.220270	+	0.975439i	$0.570694\pi$
$194$	0	0
$195$	0.686468	0.0491590
$196$	0	0
$197$	−26.6268	−1.89708	−0.948539	−	0.316660i	$-0.897439\pi$
$197$	−26.6268	−1.89708	−0.948539	+	0.316660i	$0.897439\pi$
$198$	0	0
$199$	19.5721	1.38743	0.693716	−	0.720249i	$-0.255972\pi$
$199$	19.5721	1.38743	0.693716	+	0.720249i	$0.255972\pi$
$200$	0	0
$201$	−1.40107	−0.0988239
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.17736	0.152074
$206$	0	0
$207$	0.401127	0.0278803
$208$	0	0
$209$	−5.77268	−0.399305
$210$	0	0
$211$	20.2622	1.39490	0.697452	−	0.716631i	$-0.254317\pi$
$211$	20.2622	1.39490	0.697452	+	0.716631i	$0.254317\pi$
$212$	0	0
$213$	−1.85102	−0.126830
$214$	0	0
$215$	−6.03293	−0.411443
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.62711	−0.380245
$220$	0	0
$221$	−3.41412	−0.229659
$222$	0	0
$223$	−25.8696	−1.73236	−0.866180	−	0.499733i	$-0.833432\pi$
$223$	−25.8696	−1.73236	−0.866180	+	0.499733i	$0.833432\pi$
$224$	0	0
$225$	0.718566	0.0479044
$226$	0	0
$227$	0.992803	0.0658947	0.0329473	−	0.999457i	$-0.489511\pi$
$227$	0.992803	0.0658947	0.0329473	+	0.999457i	$0.489511\pi$
$228$	0	0
$229$	−9.34395	−0.617465	−0.308733	−	0.951149i	$-0.599905\pi$
$229$	−9.34395	−0.617465	−0.308733	+	0.951149i	$0.599905\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.5022	−1.67070	−0.835352	−	0.549715i	$-0.814736\pi$
$233$	−25.5022	−1.67070	−0.835352	+	0.549715i	$0.814736\pi$
$234$	0	0
$235$	−7.53019	−0.491215
$236$	0	0
$237$	−2.68046	−0.174115
$238$	0	0
$239$	−0.342630	−0.0221629	−0.0110815	−	0.999939i	$-0.503527\pi$
$239$	−0.342630	−0.0221629	−0.0110815	+	0.999939i	$0.503527\pi$
$240$	0	0
$241$	−24.2427	−1.56161	−0.780805	−	0.624774i	$-0.785191\pi$
$241$	−24.2427	−1.56161	−0.780805	+	0.624774i	$0.785191\pi$
$242$	0	0
$243$	−11.5830	−0.743050
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.49828	−0.286219
$248$	0	0
$249$	4.17458	0.264553
$250$	0	0
$251$	12.3023	0.776517	0.388258	−	0.921551i	$-0.373077\pi$
$251$	12.3023	0.776517	0.388258	+	0.921551i	$0.373077\pi$
$252$	0	0
$253$	0.122928	0.00772839
$254$	0	0
$255$	−5.34328	−0.334609
$256$	0	0
$257$	−13.5864	−0.847496	−0.423748	−	0.905780i	$-0.639286\pi$
$257$	−13.5864	−0.847496	−0.423748	+	0.905780i	$0.639286\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	26.2498	1.62482
$262$	0	0
$263$	30.2101	1.86284	0.931419	−	0.363949i	$-0.118572\pi$
$263$	30.2101	1.86284	0.931419	+	0.363949i	$0.118572\pi$
$264$	0	0
$265$	15.8713	0.974967
$266$	0	0
$267$	−2.83162	−0.173292
$268$	0	0
$269$	−10.7141	−0.653248	−0.326624	−	0.945154i	$-0.605911\pi$
$269$	−10.7141	−0.653248	−0.326624	+	0.945154i	$0.605911\pi$
$270$	0	0
$271$	−8.84303	−0.537176	−0.268588	−	0.963255i	$-0.586557\pi$
$271$	−8.84303	−0.537176	−0.268588	+	0.963255i	$0.586557\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.220208	0.0132791
$276$	0	0
$277$	14.2115	0.853888	0.426944	−	0.904278i	$-0.359590\pi$
$277$	14.2115	0.853888	0.426944	+	0.904278i	$0.359590\pi$
$278$	0	0
$279$	23.5922	1.41243
$280$	0	0
$281$	−30.8179	−1.83844	−0.919220	−	0.393743i	$-0.871180\pi$
$281$	−30.8179	−1.83844	−0.919220	+	0.393743i	$0.871180\pi$
$282$	0	0
$283$	−29.1151	−1.73071	−0.865357	−	0.501156i	$-0.832908\pi$
$283$	−29.1151	−1.73071	−0.865357	+	0.501156i	$0.832908\pi$
$284$	0	0
$285$	−7.04003	−0.417016
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.57460	0.563212
$290$	0	0
$291$	1.07541	0.0630418
$292$	0	0
$293$	−10.9061	−0.637142	−0.318571	−	0.947899i	$-0.603203\pi$
$293$	−10.9061	−0.637142	−0.318571	+	0.947899i	$0.603203\pi$
$294$	0	0
$295$	−21.4271	−1.24753
$296$	0	0
$297$	−2.33589	−0.135542
$298$	0	0
$299$	0.0957895	0.00553965
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.510735	0.0293409
$304$	0	0
$305$	−21.0848	−1.20731
$306$	0	0
$307$	−4.34502	−0.247983	−0.123992	−	0.992283i	$-0.539570\pi$
$307$	−4.34502	−0.247983	−0.123992	+	0.992283i	$0.539570\pi$
$308$	0	0
$309$	0.209277	0.0119054
$310$	0	0
$311$	13.0198	0.738288	0.369144	−	0.929372i	$-0.379651\pi$
$311$	13.0198	0.738288	0.369144	+	0.929372i	$0.379651\pi$
$312$	0	0
$313$	−35.0520	−1.98125	−0.990627	−	0.136595i	$-0.956384\pi$
$313$	−35.0520	−1.98125	−0.990627	+	0.136595i	$0.956384\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.3848	−1.36958	−0.684792	−	0.728738i	$-0.740107\pi$
$317$	−24.3848	−1.36958	−0.684792	+	0.728738i	$0.740107\pi$
$318$	0	0
$319$	8.04439	0.450399
$320$	0	0
$321$	8.92302	0.498035
$322$	0	0
$323$	35.0134	1.94820
$324$	0	0
$325$	0.171594	0.00951832
$326$	0	0
$327$	−3.65681	−0.202222
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.0042	−0.879671	−0.439835	−	0.898078i	$-0.644963\pi$
$331$	−16.0042	−0.879671	−0.439835	+	0.898078i	$0.644963\pi$
$332$	0	0
$333$	21.5203	1.17930
$334$	0	0
$335$	6.40836	0.350126
$336$	0	0
$337$	−2.87190	−0.156443	−0.0782213	−	0.996936i	$-0.524924\pi$
$337$	−2.87190	−0.156443	−0.0782213	+	0.996936i	$0.524924\pi$
$338$	0	0
$339$	3.28833	0.178598
$340$	0	0
$341$	7.22994	0.391523
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.149915	0.00807118
$346$	0	0
$347$	16.6309	0.892795	0.446397	−	0.894835i	$-0.352707\pi$
$347$	16.6309	0.892795	0.446397	+	0.894835i	$0.352707\pi$
$348$	0	0
$349$	13.7125	0.734014	0.367007	−	0.930218i	$-0.380382\pi$
$349$	13.7125	0.734014	0.367007	+	0.930218i	$0.380382\pi$
$350$	0	0
$351$	−1.82020	−0.0971553
$352$	0	0
$353$	23.3413	1.24233	0.621166	−	0.783679i	$-0.286659\pi$
$353$	23.3413	1.24233	0.621166	+	0.783679i	$0.286659\pi$
$354$	0	0
$355$	8.46639	0.449349
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.9339	−0.893737	−0.446868	−	0.894600i	$-0.647461\pi$
$359$	−16.9339	−0.893737	−0.446868	+	0.894600i	$0.647461\pi$
$360$	0	0
$361$	27.1318	1.42799
$362$	0	0
$363$	4.89257	0.256793
$364$	0	0
$365$	25.7379	1.34718
$366$	0	0
$367$	29.4979	1.53978	0.769890	−	0.638177i	$-0.220311\pi$
$367$	29.4979	1.53978	0.769890	+	0.638177i	$0.220311\pi$
$368$	0	0
$369$	−2.77339	−0.144377
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.5701	1.53108	0.765540	−	0.643389i	$-0.222472\pi$
$373$	29.5701	1.53108	0.765540	+	0.643389i	$0.222472\pi$
$374$	0	0
$375$	5.45111	0.281494
$376$	0	0
$377$	6.26847	0.322843
$378$	0	0
$379$	35.0216	1.79894	0.899470	−	0.436983i	$-0.143953\pi$
$379$	35.0216	1.79894	0.899470	+	0.436983i	$0.143953\pi$
$380$	0	0
$381$	−2.35700	−0.120753
$382$	0	0
$383$	−12.4678	−0.637077	−0.318538	−	0.947910i	$-0.603192\pi$
$383$	−12.4678	−0.637077	−0.318538	+	0.947910i	$0.603192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.68437	0.390618
$388$	0	0
$389$	−21.9851	−1.11469	−0.557345	−	0.830281i	$-0.688180\pi$
$389$	−21.9851	−1.11469	−0.557345	+	0.830281i	$0.688180\pi$
$390$	0	0
$391$	−0.745599	−0.0377066
$392$	0	0
$393$	−4.05989	−0.204794
$394$	0	0
$395$	12.2602	0.616877
$396$	0	0
$397$	−20.1115	−1.00937	−0.504683	−	0.863304i	$-0.668391\pi$
$397$	−20.1115	−1.00937	−0.504683	+	0.863304i	$0.668391\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.2650	0.662425	0.331212	−	0.943556i	$-0.392542\pi$
$401$	13.2650	0.662425	0.331212	+	0.943556i	$0.392542\pi$
$402$	0	0
$403$	5.63382	0.280641
$404$	0	0
$405$	15.2673	0.758637
$406$	0	0
$407$	6.59500	0.326902
$408$	0	0
$409$	−25.6563	−1.26862	−0.634312	−	0.773077i	$-0.718717\pi$
$409$	−25.6563	−1.26862	−0.634312	+	0.773077i	$0.718717\pi$
$410$	0	0
$411$	3.96514	0.195586
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−19.0941	−0.937295
$416$	0	0
$417$	7.64876	0.374561
$418$	0	0
$419$	−36.6937	−1.79260	−0.896301	−	0.443445i	$-0.853756\pi$
$419$	−36.6937	−1.79260	−0.896301	+	0.443445i	$0.853756\pi$
$420$	0	0
$421$	4.48285	0.218481	0.109240	−	0.994015i	$-0.465158\pi$
$421$	4.48285	0.218481	0.109240	+	0.994015i	$0.465158\pi$
$422$	0	0
$423$	9.59147	0.466353
$424$	0	0
$425$	−1.33564	−0.0647881
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.267958	−0.0129371
$430$	0	0
$431$	18.4278	0.887635	0.443818	−	0.896117i	$-0.353624\pi$
$431$	18.4278	0.887635	0.443818	+	0.896117i	$0.353624\pi$
$432$	0	0
$433$	−19.7652	−0.949857	−0.474928	−	0.880024i	$-0.657526\pi$
$433$	−19.7652	−0.949857	−0.474928	+	0.880024i	$0.657526\pi$
$434$	0	0
$435$	9.81047	0.470376
$436$	0	0
$437$	−0.982364	−0.0469928
$438$	0	0
$439$	−14.9960	−0.715719	−0.357859	−	0.933775i	$-0.616493\pi$
$439$	−14.9960	−0.715719	−0.357859	+	0.933775i	$0.616493\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.5822	1.69056	0.845280	−	0.534323i	$-0.179433\pi$
$443$	35.5822	1.69056	0.845280	+	0.534323i	$0.179433\pi$
$444$	0	0
$445$	12.9516	0.613963
$446$	0	0
$447$	−6.60638	−0.312471
$448$	0	0
$449$	9.40746	0.443966	0.221983	−	0.975051i	$-0.428747\pi$
$449$	9.40746	0.443966	0.221983	+	0.975051i	$0.428747\pi$
$450$	0	0
$451$	−0.849919	−0.0400211
$452$	0	0
$453$	−0.937513	−0.0440482
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.0885	−1.40748	−0.703739	−	0.710458i	$-0.748488\pi$
$457$	−30.0885	−1.40748	−0.703739	+	0.710458i	$0.748488\pi$
$458$	0	0
$459$	14.1680	0.661304
$460$	0	0
$461$	27.7681	1.29329	0.646645	−	0.762791i	$-0.276172\pi$
$461$	27.7681	1.29329	0.646645	+	0.762791i	$0.276172\pi$
$462$	0	0
$463$	−5.94632	−0.276349	−0.138175	−	0.990408i	$-0.544123\pi$
$463$	−5.94632	−0.276349	−0.138175	+	0.990408i	$0.544123\pi$
$464$	0	0
$465$	8.81722	0.408889
$466$	0	0
$467$	−21.5875	−0.998952	−0.499476	−	0.866328i	$-0.666474\pi$
$467$	−21.5875	−0.998952	−0.499476	+	0.866328i	$0.666474\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.234402	−0.0108007
$472$	0	0
$473$	2.35491	0.108279
$474$	0	0
$475$	−1.75977	−0.0807439
$476$	0	0
$477$	−20.2159	−0.925621
$478$	0	0
$479$	15.0166	0.686127	0.343063	−	0.939312i	$-0.388535\pi$
$479$	15.0166	0.686127	0.343063	+	0.939312i	$0.388535\pi$
$480$	0	0
$481$	5.13905	0.234321
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.91884	−0.223353
$486$	0	0
$487$	17.6788	0.801104	0.400552	−	0.916274i	$-0.368818\pi$
$487$	17.6788	0.801104	0.400552	+	0.916274i	$0.368818\pi$
$488$	0	0
$489$	1.68124	0.0760283
$490$	0	0
$491$	−10.4434	−0.471304	−0.235652	−	0.971837i	$-0.575723\pi$
$491$	−10.4434	−0.471304	−0.235652	+	0.971837i	$0.575723\pi$
$492$	0	0
$493$	−48.7920	−2.19748
$494$	0	0
$495$	5.13237	0.230683
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.1815	−0.948213	−0.474106	−	0.880468i	$-0.657229\pi$
$499$	−21.1815	−0.948213	−0.474106	+	0.880468i	$0.657229\pi$
$500$	0	0
$501$	−4.86163	−0.217202
$502$	0	0
$503$	−21.2565	−0.947782	−0.473891	−	0.880584i	$-0.657151\pi$
$503$	−21.2565	−0.947782	−0.473891	+	0.880584i	$0.657151\pi$
$504$	0	0
$505$	−2.33605	−0.103953
$506$	0	0
$507$	5.97972	0.265569
$508$	0	0
$509$	−10.2697	−0.455195	−0.227598	−	0.973755i	$-0.573087\pi$
$509$	−10.2697	−0.455195	−0.227598	+	0.973755i	$0.573087\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	18.6670	0.824168
$514$	0	0
$515$	−0.957213	−0.0421798
$516$	0	0
$517$	2.93936	0.129273
$518$	0	0
$519$	2.72805	0.119748
$520$	0	0
$521$	8.72319	0.382170	0.191085	−	0.981574i	$-0.438799\pi$
$521$	8.72319	0.382170	0.191085	+	0.981574i	$0.438799\pi$
$522$	0	0
$523$	43.6976	1.91076	0.955382	−	0.295374i	$-0.0954444\pi$
$523$	43.6976	1.91076	0.955382	+	0.295374i	$0.0954444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.8521	−1.91023
$528$	0	0
$529$	−22.9791	−0.999090
$530$	0	0
$531$	27.2924	1.18439
$532$	0	0
$533$	−0.662286	−0.0286868
$534$	0	0
$535$	−40.8130	−1.76450
$536$	0	0
$537$	−7.85953	−0.339164
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.2105	−0.438984	−0.219492	−	0.975614i	$-0.570440\pi$
$541$	−10.2105	−0.438984	−0.219492	+	0.975614i	$0.570440\pi$
$542$	0	0
$543$	−5.81215	−0.249423
$544$	0	0
$545$	16.7259	0.716460
$546$	0	0
$547$	−24.7641	−1.05884	−0.529419	−	0.848361i	$-0.677590\pi$
$547$	−24.7641	−1.05884	−0.529419	+	0.848361i	$0.677590\pi$
$548$	0	0
$549$	26.8565	1.14621
$550$	0	0
$551$	−64.2859	−2.73867
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.04288	0.341401
$556$	0	0
$557$	−20.1450	−0.853572	−0.426786	−	0.904353i	$-0.640354\pi$
$557$	−20.1450	−0.853572	−0.426786	+	0.904353i	$0.640354\pi$
$558$	0	0
$559$	1.83503	0.0776136
$560$	0	0
$561$	2.08571	0.0880588
$562$	0	0
$563$	−16.5815	−0.698828	−0.349414	−	0.936968i	$-0.613619\pi$
$563$	−16.5815	−0.698828	−0.349414	+	0.936968i	$0.613619\pi$
$564$	0	0
$565$	−15.0405	−0.632759
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.4448	−0.647481	−0.323741	−	0.946146i	$-0.604941\pi$
$569$	−15.4448	−0.647481	−0.323741	+	0.946146i	$0.604941\pi$
$570$	0	0
$571$	28.4319	1.18984	0.594918	−	0.803786i	$-0.297184\pi$
$571$	28.4319	1.18984	0.594918	+	0.803786i	$0.297184\pi$
$572$	0	0
$573$	2.34848	0.0981093
$574$	0	0
$575$	0.0374738	0.00156277
$576$	0	0
$577$	5.32206	0.221560	0.110780	−	0.993845i	$-0.464665\pi$
$577$	5.32206	0.221560	0.110780	+	0.993845i	$0.464665\pi$
$578$	0	0
$579$	2.91344	0.121079
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.19526	−0.256581
$584$	0	0
$585$	3.99933	0.165352
$586$	0	0
$587$	32.1095	1.32530	0.662650	−	0.748929i	$-0.269432\pi$
$587$	32.1095	1.32530	0.662650	+	0.748929i	$0.269432\pi$
$588$	0	0
$589$	−57.7774	−2.38068
$590$	0	0
$591$	12.6754	0.521397
$592$	0	0
$593$	−2.90368	−0.119240	−0.0596200	−	0.998221i	$-0.518989\pi$
$593$	−2.90368	−0.119240	−0.0596200	+	0.998221i	$0.518989\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.31712	−0.381324
$598$	0	0
$599$	19.0544	0.778541	0.389270	−	0.921124i	$-0.372727\pi$
$599$	19.0544	0.778541	0.389270	+	0.921124i	$0.372727\pi$
$600$	0	0
$601$	−13.6204	−0.555588	−0.277794	−	0.960641i	$-0.589603\pi$
$601$	−13.6204	−0.555588	−0.277794	+	0.960641i	$0.589603\pi$
$602$	0	0
$603$	−8.16257	−0.332405
$604$	0	0
$605$	−22.3781	−0.909801
$606$	0	0
$607$	34.6925	1.40813	0.704063	−	0.710138i	$-0.251367\pi$
$607$	34.6925	1.40813	0.704063	+	0.710138i	$0.251367\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.29045	0.0926617
$612$	0	0
$613$	−3.09004	−0.124805	−0.0624027	−	0.998051i	$-0.519876\pi$
$613$	−3.09004	−0.124805	−0.0624027	+	0.998051i	$0.519876\pi$
$614$	0	0
$615$	−1.03651	−0.0417962
$616$	0	0
$617$	32.0608	1.29072	0.645360	−	0.763879i	$-0.276707\pi$
$617$	32.0608	1.29072	0.645360	+	0.763879i	$0.276707\pi$
$618$	0	0
$619$	−45.4105	−1.82520	−0.912601	−	0.408852i	$-0.865929\pi$
$619$	−45.4105	−1.82520	−0.912601	+	0.408852i	$0.865929\pi$
$620$	0	0
$621$	−0.397508	−0.0159515
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.6374	−0.945496
$626$	0	0
$627$	2.74803	0.109746
$628$	0	0
$629$	−40.0010	−1.59494
$630$	0	0
$631$	−21.8131	−0.868366	−0.434183	−	0.900825i	$-0.642963\pi$
$631$	−21.8131	−0.868366	−0.434183	+	0.900825i	$0.642963\pi$
$632$	0	0
$633$	−9.64560	−0.383378
$634$	0	0
$635$	10.7807	0.427820
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.7839	−0.426606
$640$	0	0
$641$	−5.25809	−0.207682	−0.103841	−	0.994594i	$-0.533113\pi$
$641$	−5.25809	−0.207682	−0.103841	+	0.994594i	$0.533113\pi$
$642$	0	0
$643$	22.5524	0.889381	0.444690	−	0.895684i	$-0.353314\pi$
$643$	22.5524	0.889381	0.444690	+	0.895684i	$0.353314\pi$
$644$	0	0
$645$	2.87192	0.113082
$646$	0	0
$647$	8.47016	0.332996	0.166498	−	0.986042i	$-0.446754\pi$
$647$	8.47016	0.332996	0.166498	+	0.986042i	$0.446754\pi$
$648$	0	0
$649$	8.36391	0.328312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.8553	−0.424799	−0.212400	−	0.977183i	$-0.568128\pi$
$653$	−10.8553	−0.424799	−0.212400	+	0.977183i	$0.568128\pi$
$654$	0	0
$655$	18.5695	0.725572
$656$	0	0
$657$	−32.7833	−1.27900
$658$	0	0
$659$	−21.6543	−0.843531	−0.421766	−	0.906705i	$-0.638589\pi$
$659$	−21.6543	−0.843531	−0.421766	+	0.906705i	$0.638589\pi$
$660$	0	0
$661$	−2.39901	−0.0933109	−0.0466554	−	0.998911i	$-0.514856\pi$
$661$	−2.39901	−0.0933109	−0.0466554	+	0.998911i	$0.514856\pi$
$662$	0	0
$663$	1.62526	0.0631199
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.36895	0.0530060
$668$	0	0
$669$	12.3150	0.476125
$670$	0	0
$671$	8.23031	0.317728
$672$	0	0
$673$	−1.83891	−0.0708846	−0.0354423	−	0.999372i	$-0.511284\pi$
$673$	−1.83891	−0.0708846	−0.0354423	+	0.999372i	$0.511284\pi$
$674$	0	0
$675$	−0.712083	−0.0274081
$676$	0	0
$677$	−22.5383	−0.866219	−0.433109	−	0.901341i	$-0.642584\pi$
$677$	−22.5383	−0.866219	−0.433109	+	0.901341i	$0.642584\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.472614	−0.0181106
$682$	0	0
$683$	−29.5536	−1.13084	−0.565419	−	0.824803i	$-0.691286\pi$
$683$	−29.5536	−1.13084	−0.565419	+	0.824803i	$0.691286\pi$
$684$	0	0
$685$	−18.1362	−0.692947
$686$	0	0
$687$	4.44809	0.169705
$688$	0	0
$689$	−4.82756	−0.183916
$690$	0	0
$691$	14.0818	0.535697	0.267848	−	0.963461i	$-0.413687\pi$
$691$	14.0818	0.535697	0.267848	+	0.963461i	$0.413687\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−34.9847	−1.32704
$696$	0	0
$697$	5.15506	0.195262
$698$	0	0
$699$	12.1401	0.459180
$700$	0	0
$701$	−38.3917	−1.45003	−0.725017	−	0.688731i	$-0.758168\pi$
$701$	−38.3917	−1.45003	−0.725017	+	0.688731i	$0.758168\pi$
$702$	0	0
$703$	−52.7033	−1.98774
$704$	0	0
$705$	3.58467	0.135007
$706$	0	0
$707$	0	0
$708$	0	0
$709$	38.3374	1.43979	0.719896	−	0.694081i	$-0.244189\pi$
$709$	38.3374	1.43979	0.719896	+	0.694081i	$0.244189\pi$
$710$	0	0
$711$	−15.6162	−0.585655
$712$	0	0
$713$	1.23035	0.0460771
$714$	0	0
$715$	1.22561	0.0458354
$716$	0	0
$717$	0.163106	0.00609130
$718$	0	0
$719$	−19.2051	−0.716230	−0.358115	−	0.933678i	$-0.616580\pi$
$719$	−19.2051	−0.716230	−0.358115	+	0.933678i	$0.616580\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.5405	0.429196
$724$	0	0
$725$	2.45229	0.0910758
$726$	0	0
$727$	6.45574	0.239430	0.119715	−	0.992808i	$-0.461802\pi$
$727$	6.45574	0.239430	0.119715	+	0.992808i	$0.461802\pi$
$728$	0	0
$729$	−15.5215	−0.574870
$730$	0	0
$731$	−14.2834	−0.528290
$732$	0	0
$733$	44.1794	1.63181	0.815903	−	0.578189i	$-0.196241\pi$
$733$	44.1794	1.63181	0.815903	+	0.578189i	$0.196241\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.50146	−0.0921425
$738$	0	0
$739$	−48.3821	−1.77976	−0.889881	−	0.456192i	$-0.849213\pi$
$739$	−48.3821	−1.77976	−0.889881	+	0.456192i	$0.849213\pi$
$740$	0	0
$741$	2.14136	0.0786649
$742$	0	0
$743$	14.3359	0.525932	0.262966	−	0.964805i	$-0.415299\pi$
$743$	14.3359	0.525932	0.262966	+	0.964805i	$0.415299\pi$
$744$	0	0
$745$	30.2169	1.10706
$746$	0	0
$747$	24.3209	0.889855
$748$	0	0
$749$	0	0
$750$	0	0
$751$	43.2473	1.57812	0.789059	−	0.614317i	$-0.210569\pi$
$751$	43.2473	1.57812	0.789059	+	0.614317i	$0.210569\pi$
$752$	0	0
$753$	−5.85641	−0.213419
$754$	0	0
$755$	4.28809	0.156060
$756$	0	0
$757$	29.2404	1.06276	0.531380	−	0.847133i	$-0.321673\pi$
$757$	29.2404	1.06276	0.531380	+	0.847133i	$0.321673\pi$
$758$	0	0
$759$	−0.0585185	−0.00212409
$760$	0	0
$761$	19.8163	0.718341	0.359170	−	0.933272i	$-0.383060\pi$
$761$	19.8163	0.718341	0.359170	+	0.933272i	$0.383060\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−31.1297	−1.12550
$766$	0	0
$767$	6.51745	0.235332
$768$	0	0
$769$	11.3692	0.409984	0.204992	−	0.978764i	$-0.434283\pi$
$769$	11.3692	0.409984	0.204992	+	0.978764i	$0.434283\pi$
$770$	0	0
$771$	6.46767	0.232927
$772$	0	0
$773$	28.8925	1.03919	0.519596	−	0.854412i	$-0.326083\pi$
$773$	28.8925	1.03919	0.519596	+	0.854412i	$0.326083\pi$
$774$	0	0
$775$	2.20401	0.0791704
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.79204	0.243350
$780$	0	0
$781$	−3.30480	−0.118255
$782$	0	0
$783$	−26.0130	−0.929627
$784$	0	0
$785$	1.07213	0.0382660
$786$	0	0
$787$	6.69690	0.238719	0.119359	−	0.992851i	$-0.461916\pi$
$787$	6.69690	0.238719	0.119359	+	0.992851i	$0.461916\pi$
$788$	0	0
$789$	−14.3812	−0.511986
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.41335	0.227745
$794$	0	0
$795$	−7.55538	−0.267962
$796$	0	0
$797$	−52.9987	−1.87731	−0.938654	−	0.344859i	$-0.887927\pi$
$797$	−52.9987	−1.87731	−0.938654	+	0.344859i	$0.887927\pi$
$798$	0	0
$799$	−17.8282	−0.630718
$800$	0	0
$801$	−16.4969	−0.582888
$802$	0	0
$803$	−10.0466	−0.354537
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.10033	0.179540
$808$	0	0
$809$	−3.14889	−0.110709	−0.0553545	−	0.998467i	$-0.517629\pi$
$809$	−3.14889	−0.110709	−0.0553545	+	0.998467i	$0.517629\pi$
$810$	0	0
$811$	5.67346	0.199222	0.0996110	−	0.995026i	$-0.468240\pi$
$811$	5.67346	0.199222	0.0996110	+	0.995026i	$0.468240\pi$
$812$	0	0
$813$	4.20964	0.147638
$814$	0	0
$815$	−7.68983	−0.269363
$816$	0	0
$817$	−18.8191	−0.658396
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.0738	−0.526079	−0.263039	−	0.964785i	$-0.584725\pi$
$821$	−15.0738	−0.526079	−0.263039	+	0.964785i	$0.584725\pi$
$822$	0	0
$823$	0.723286	0.0252122	0.0126061	−	0.999921i	$-0.495987\pi$
$823$	0.723286	0.0252122	0.0126061	+	0.999921i	$0.495987\pi$
$824$	0	0
$825$	−0.104828	−0.00364964
$826$	0	0
$827$	−52.4583	−1.82415	−0.912077	−	0.410019i	$-0.865522\pi$
$827$	−52.4583	−1.82415	−0.912077	+	0.410019i	$0.865522\pi$
$828$	0	0
$829$	−11.7102	−0.406713	−0.203356	−	0.979105i	$-0.565185\pi$
$829$	−11.7102	−0.406713	−0.203356	+	0.979105i	$0.565185\pi$
$830$	0	0
$831$	−6.76526	−0.234684
$832$	0	0
$833$	0	0
$834$	0	0
$835$	22.2366	0.769530
$836$	0	0
$837$	−23.3793	−0.808107
$838$	0	0
$839$	32.7489	1.13062	0.565308	−	0.824880i	$-0.308757\pi$
$839$	32.7489	1.13062	0.565308	+	0.824880i	$0.308757\pi$
$840$	0	0
$841$	60.5841	2.08911
$842$	0	0
$843$	14.6706	0.505281
$844$	0	0
$845$	−27.3507	−0.940892
$846$	0	0
$847$	0	0
$848$	0	0
$849$	13.8600	0.475673
$850$	0	0
$851$	1.12230	0.0384720
$852$	0	0
$853$	−50.8909	−1.74247	−0.871236	−	0.490864i	$-0.836681\pi$
$853$	−50.8909	−1.74247	−0.871236	+	0.490864i	$0.836681\pi$
$854$	0	0
$855$	−41.0149	−1.40268
$856$	0	0
$857$	−23.6916	−0.809290	−0.404645	−	0.914474i	$-0.632605\pi$
$857$	−23.6916	−0.809290	−0.404645	+	0.914474i	$0.632605\pi$
$858$	0	0
$859$	6.16836	0.210462	0.105231	−	0.994448i	$-0.466442\pi$
$859$	6.16836	0.210462	0.105231	+	0.994448i	$0.466442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.06291	−0.274465	−0.137232	−	0.990539i	$-0.543821\pi$
$863$	−8.06291	−0.274465	−0.137232	+	0.990539i	$0.543821\pi$
$864$	0	0
$865$	−12.4778	−0.424259
$866$	0	0
$867$	−4.55789	−0.154794
$868$	0	0
$869$	−4.78568	−0.162343
$870$	0	0
$871$	−1.94923	−0.0660470
$872$	0	0
$873$	6.26530	0.212048
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.4028	1.53314	0.766572	−	0.642158i	$-0.221961\pi$
$877$	45.4028	1.53314	0.766572	+	0.642158i	$0.221961\pi$
$878$	0	0
$879$	5.19175	0.175113
$880$	0	0
$881$	−11.4992	−0.387418	−0.193709	−	0.981059i	$-0.562052\pi$
$881$	−11.4992	−0.387418	−0.193709	+	0.981059i	$0.562052\pi$
$882$	0	0
$883$	7.43213	0.250111	0.125056	−	0.992150i	$-0.460089\pi$
$883$	7.43213	0.250111	0.125056	+	0.992150i	$0.460089\pi$
$884$	0	0
$885$	10.2001	0.342874
$886$	0	0
$887$	35.8611	1.20410	0.602050	−	0.798459i	$-0.294351\pi$
$887$	35.8611	1.20410	0.602050	+	0.798459i	$0.294351\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.95948	−0.199650
$892$	0	0
$893$	−23.4896	−0.786049
$894$	0	0
$895$	35.9487	1.20163
$896$	0	0
$897$	−0.0455997	−0.00152253
$898$	0	0
$899$	80.5143	2.68530
$900$	0	0
$901$	37.5764	1.25185
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.5842	0.883689
$906$	0	0
$907$	−22.5584	−0.749040	−0.374520	−	0.927219i	$-0.622192\pi$
$907$	−22.5584	−0.749040	−0.374520	+	0.927219i	$0.622192\pi$
$908$	0	0
$909$	2.97551	0.0986916
$910$	0	0
$911$	−16.4853	−0.546184	−0.273092	−	0.961988i	$-0.588046\pi$
$911$	−16.4853	−0.546184	−0.273092	+	0.961988i	$0.588046\pi$
$912$	0	0
$913$	7.45327	0.246667
$914$	0	0
$915$	10.0372	0.331820
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−41.4145	−1.36614	−0.683070	−	0.730353i	$-0.739356\pi$
$919$	−41.4145	−1.36614	−0.683070	+	0.730353i	$0.739356\pi$
$920$	0	0
$921$	2.06840	0.0681563
$922$	0	0
$923$	−2.57521	−0.0847642
$924$	0	0
$925$	2.01045	0.0661032
$926$	0	0
$927$	1.21924	0.0400450
$928$	0	0
$929$	−29.2462	−0.959536	−0.479768	−	0.877395i	$-0.659279\pi$
$929$	−29.2462	−0.959536	−0.479768	+	0.877395i	$0.659279\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.19797	−0.202912
$934$	0	0
$935$	−9.53985	−0.311986
$936$	0	0
$937$	−32.6481	−1.06657	−0.533283	−	0.845937i	$-0.679042\pi$
$937$	−32.6481	−1.06657	−0.533283	+	0.845937i	$0.679042\pi$
$938$	0	0
$939$	16.6861	0.544532
$940$	0	0
$941$	8.83688	0.288074	0.144037	−	0.989572i	$-0.453992\pi$
$941$	8.83688	0.288074	0.144037	+	0.989572i	$0.453992\pi$
$942$	0	0
$943$	−0.144635	−0.00470995
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.5368	1.54474	0.772369	−	0.635174i	$-0.219072\pi$
$947$	47.5368	1.54474	0.772369	+	0.635174i	$0.219072\pi$
$948$	0	0
$949$	−7.82867	−0.254129
$950$	0	0
$951$	11.6081	0.376419
$952$	0	0
$953$	17.7811	0.575985	0.287992	−	0.957633i	$-0.407012\pi$
$953$	17.7811	0.575985	0.287992	+	0.957633i	$0.407012\pi$
$954$	0	0
$955$	−10.7417	−0.347594
$956$	0	0
$957$	−3.82945	−0.123789
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.3627	1.33428
$962$	0	0
$963$	51.9851	1.67520
$964$	0	0
$965$	−13.3258	−0.428973
$966$	0	0
$967$	28.2771	0.909329	0.454665	−	0.890663i	$-0.349759\pi$
$967$	28.2771	0.909329	0.454665	+	0.890663i	$0.349759\pi$
$968$	0	0
$969$	−16.6678	−0.535446
$970$	0	0
$971$	22.4810	0.721451	0.360725	−	0.932672i	$-0.382529\pi$
$971$	22.4810	0.721451	0.360725	+	0.932672i	$0.382529\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.0816856	−0.00261603
$976$	0	0
$977$	29.1987	0.934150	0.467075	−	0.884218i	$-0.345308\pi$
$977$	29.1987	0.934150	0.467075	+	0.884218i	$0.345308\pi$
$978$	0	0
$979$	−5.05555	−0.161576
$980$	0	0
$981$	−21.3044	−0.680198
$982$	0	0
$983$	21.7693	0.694333	0.347167	−	0.937803i	$-0.387144\pi$
$983$	21.7693	0.694333	0.347167	+	0.937803i	$0.387144\pi$
$984$	0	0
$985$	−57.9761	−1.84727
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.400746	0.0127430
$990$	0	0
$991$	35.6120	1.13125	0.565626	−	0.824662i	$-0.308635\pi$
$991$	35.6120	1.13125	0.565626	+	0.824662i	$0.308635\pi$
$992$	0	0
$993$	7.61865	0.241771
$994$	0	0
$995$	42.6156	1.35101
$996$	0	0
$997$	−16.6525	−0.527389	−0.263695	−	0.964606i	$-0.584941\pi$
$997$	−16.6525	−0.527389	−0.263695	+	0.964606i	$0.584941\pi$
$998$	0	0
$999$	−21.3261	−0.674728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.10	✓	20
7.6	odd	2		8036.2.a.t.1.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.10	✓	20	1.1	even	1	trivial
8036.2.a.t.1.11	yes	20	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-0.476040 q^{3} +2.17736 q^{5} -2.77339 q^{9} +O(q^{10})\)	\(q-0.476040 q^{3} +2.17736 q^{5} -2.77339 q^{9} -0.849919 q^{11} -0.662286 q^{13} -1.03651 q^{15} +5.15506 q^{17} +6.79204 q^{19} -0.144635 q^{23} -0.259093 q^{25} +2.74836 q^{27} -9.46489 q^{29} -8.50663 q^{31} +0.404596 q^{33} -7.75956 q^{37} +0.315275 q^{39} +1.00000 q^{41} -2.77075 q^{43} -6.03867 q^{45} -3.45840 q^{47} -2.45401 q^{51} +7.28924 q^{53} -1.85058 q^{55} -3.23329 q^{57} -9.84084 q^{59} -9.68365 q^{61} -1.44204 q^{65} +2.94318 q^{67} +0.0688519 q^{69} +3.88837 q^{71} +11.8207 q^{73} +0.123339 q^{75} +5.63075 q^{79} +7.01183 q^{81} -8.76939 q^{83} +11.2244 q^{85} +4.50567 q^{87} +5.94828 q^{89} +4.04950 q^{93} +14.7887 q^{95} -2.25908 q^{97} +2.35715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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