LMFDB - Embedded newform 6036.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.244833 q^{5} +2.69533 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.244833 q^{5} +2.69533 q^{7} +1.00000 q^{9} +4.17525 q^{11} -4.50565 q^{13} +0.244833 q^{15} +1.19780 q^{17} +2.55525 q^{19} -2.69533 q^{21} +4.82577 q^{23} -4.94006 q^{25} -1.00000 q^{27} -2.33354 q^{29} +8.83643 q^{31} -4.17525 q^{33} -0.659904 q^{35} +6.02074 q^{37} +4.50565 q^{39} -9.45996 q^{41} +7.71148 q^{43} -0.244833 q^{45} +2.80134 q^{47} +0.264777 q^{49} -1.19780 q^{51} +3.72490 q^{53} -1.02224 q^{55} -2.55525 q^{57} +6.71732 q^{59} -10.0509 q^{61} +2.69533 q^{63} +1.10313 q^{65} -5.48628 q^{67} -4.82577 q^{69} +2.35862 q^{71} +10.1415 q^{73} +4.94006 q^{75} +11.2536 q^{77} +1.84247 q^{79} +1.00000 q^{81} -6.03637 q^{83} -0.293261 q^{85} +2.33354 q^{87} -0.866842 q^{89} -12.1442 q^{91} -8.83643 q^{93} -0.625608 q^{95} +7.72995 q^{97} +4.17525 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.244833	−0.109493	−0.0547463	−	0.998500i	$-0.517435\pi$
$5$	−0.244833	−0.109493	−0.0547463	+	0.998500i	$0.517435\pi$
$6$	0	0
$7$	2.69533	1.01874	0.509369	−	0.860548i	$-0.329879\pi$
$7$	2.69533	1.01874	0.509369	+	0.860548i	$0.329879\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.17525	1.25888	0.629442	−	0.777047i	$-0.283283\pi$
$11$	4.17525	1.25888	0.629442	+	0.777047i	$0.283283\pi$
$12$	0	0
$13$	−4.50565	−1.24964	−0.624822	−	0.780768i	$-0.714828\pi$
$13$	−4.50565	−1.24964	−0.624822	+	0.780768i	$0.714828\pi$
$14$	0	0
$15$	0.244833	0.0632156
$16$	0	0
$17$	1.19780	0.290509	0.145255	−	0.989394i	$-0.453600\pi$
$17$	1.19780	0.290509	0.145255	+	0.989394i	$0.453600\pi$
$18$	0	0
$19$	2.55525	0.586214	0.293107	−	0.956080i	$-0.405311\pi$
$19$	2.55525	0.586214	0.293107	+	0.956080i	$0.405311\pi$
$20$	0	0
$21$	−2.69533	−0.588168
$22$	0	0
$23$	4.82577	1.00624	0.503122	−	0.864216i	$-0.332185\pi$
$23$	4.82577	1.00624	0.503122	+	0.864216i	$0.332185\pi$
$24$	0	0
$25$	−4.94006	−0.988011
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.33354	−0.433328	−0.216664	−	0.976246i	$-0.569518\pi$
$29$	−2.33354	−0.433328	−0.216664	+	0.976246i	$0.569518\pi$
$30$	0	0
$31$	8.83643	1.58707	0.793535	−	0.608525i	$-0.208238\pi$
$31$	8.83643	1.58707	0.793535	+	0.608525i	$0.208238\pi$
$32$	0	0
$33$	−4.17525	−0.726817
$34$	0	0
$35$	−0.659904	−0.111544
$36$	0	0
$37$	6.02074	0.989803	0.494902	−	0.868949i	$-0.335204\pi$
$37$	6.02074	0.989803	0.494902	+	0.868949i	$0.335204\pi$
$38$	0	0
$39$	4.50565	0.721482
$40$	0	0
$41$	−9.45996	−1.47740	−0.738699	−	0.674036i	$-0.764559\pi$
$41$	−9.45996	−1.47740	−0.738699	+	0.674036i	$0.764559\pi$
$42$	0	0
$43$	7.71148	1.17599	0.587995	−	0.808864i	$-0.299917\pi$
$43$	7.71148	1.17599	0.587995	+	0.808864i	$0.299917\pi$
$44$	0	0
$45$	−0.244833	−0.0364975
$46$	0	0
$47$	2.80134	0.408618	0.204309	−	0.978906i	$-0.434505\pi$
$47$	2.80134	0.408618	0.204309	+	0.978906i	$0.434505\pi$
$48$	0	0
$49$	0.264777	0.0378253
$50$	0	0
$51$	−1.19780	−0.167725
$52$	0	0
$53$	3.72490	0.511655	0.255827	−	0.966722i	$-0.417652\pi$
$53$	3.72490	0.511655	0.255827	+	0.966722i	$0.417652\pi$
$54$	0	0
$55$	−1.02224	−0.137839
$56$	0	0
$57$	−2.55525	−0.338451
$58$	0	0
$59$	6.71732	0.874520	0.437260	−	0.899335i	$-0.355949\pi$
$59$	6.71732	0.874520	0.437260	+	0.899335i	$0.355949\pi$
$60$	0	0
$61$	−10.0509	−1.28689	−0.643443	−	0.765494i	$-0.722495\pi$
$61$	−10.0509	−1.28689	−0.643443	+	0.765494i	$0.722495\pi$
$62$	0	0
$63$	2.69533	0.339579
$64$	0	0
$65$	1.10313	0.136827
$66$	0	0
$67$	−5.48628	−0.670256	−0.335128	−	0.942173i	$-0.608780\pi$
$67$	−5.48628	−0.670256	−0.335128	+	0.942173i	$0.608780\pi$
$68$	0	0
$69$	−4.82577	−0.580955
$70$	0	0
$71$	2.35862	0.279916	0.139958	−	0.990157i	$-0.455303\pi$
$71$	2.35862	0.279916	0.139958	+	0.990157i	$0.455303\pi$
$72$	0	0
$73$	10.1415	1.18697	0.593485	−	0.804845i	$-0.297752\pi$
$73$	10.1415	1.18697	0.593485	+	0.804845i	$0.297752\pi$
$74$	0	0
$75$	4.94006	0.570429
$76$	0	0
$77$	11.2536	1.28247
$78$	0	0
$79$	1.84247	0.207294	0.103647	−	0.994614i	$-0.466949\pi$
$79$	1.84247	0.207294	0.103647	+	0.994614i	$0.466949\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.03637	−0.662578	−0.331289	−	0.943529i	$-0.607483\pi$
$83$	−6.03637	−0.662578	−0.331289	+	0.943529i	$0.607483\pi$
$84$	0	0
$85$	−0.293261	−0.0318086
$86$	0	0
$87$	2.33354	0.250182
$88$	0	0
$89$	−0.866842	−0.0918851	−0.0459425	−	0.998944i	$-0.514629\pi$
$89$	−0.866842	−0.0918851	−0.0459425	+	0.998944i	$0.514629\pi$
$90$	0	0
$91$	−12.1442	−1.27306
$92$	0	0
$93$	−8.83643	−0.916295
$94$	0	0
$95$	−0.625608	−0.0641861
$96$	0	0
$97$	7.72995	0.784858	0.392429	−	0.919782i	$-0.371635\pi$
$97$	7.72995	0.784858	0.392429	+	0.919782i	$0.371635\pi$
$98$	0	0
$99$	4.17525	0.419628
$100$	0	0
$101$	−0.545900	−0.0543191	−0.0271595	−	0.999631i	$-0.508646\pi$
$101$	−0.545900	−0.0543191	−0.0271595	+	0.999631i	$0.508646\pi$
$102$	0	0
$103$	−3.90518	−0.384789	−0.192394	−	0.981318i	$-0.561625\pi$
$103$	−3.90518	−0.384789	−0.192394	+	0.981318i	$0.561625\pi$
$104$	0	0
$105$	0.659904	0.0644001
$106$	0	0
$107$	9.18330	0.887783	0.443891	−	0.896081i	$-0.353598\pi$
$107$	9.18330	0.887783	0.443891	+	0.896081i	$0.353598\pi$
$108$	0	0
$109$	−9.00728	−0.862741	−0.431371	−	0.902175i	$-0.641970\pi$
$109$	−9.00728	−0.862741	−0.431371	+	0.902175i	$0.641970\pi$
$110$	0	0
$111$	−6.02074	−0.571463
$112$	0	0
$113$	4.71190	0.443258	0.221629	−	0.975131i	$-0.428863\pi$
$113$	4.71190	0.443258	0.221629	+	0.975131i	$0.428863\pi$
$114$	0	0
$115$	−1.18151	−0.110176
$116$	0	0
$117$	−4.50565	−0.416548
$118$	0	0
$119$	3.22846	0.295952
$120$	0	0
$121$	6.43268	0.584790
$122$	0	0
$123$	9.45996	0.852976
$124$	0	0
$125$	2.43365	0.217673
$126$	0	0
$127$	−11.6498	−1.03375	−0.516877	−	0.856060i	$-0.672906\pi$
$127$	−11.6498	−1.03375	−0.516877	+	0.856060i	$0.672906\pi$
$128$	0	0
$129$	−7.71148	−0.678958
$130$	0	0
$131$	−1.83085	−0.159962	−0.0799809	−	0.996796i	$-0.525486\pi$
$131$	−1.83085	−0.159962	−0.0799809	+	0.996796i	$0.525486\pi$
$132$	0	0
$133$	6.88722	0.597198
$134$	0	0
$135$	0.244833	0.0210719
$136$	0	0
$137$	13.4853	1.15213	0.576065	−	0.817404i	$-0.304587\pi$
$137$	13.4853	1.15213	0.576065	+	0.817404i	$0.304587\pi$
$138$	0	0
$139$	11.1740	0.947769	0.473885	−	0.880587i	$-0.342851\pi$
$139$	11.1740	0.947769	0.473885	+	0.880587i	$0.342851\pi$
$140$	0	0
$141$	−2.80134	−0.235916
$142$	0	0
$143$	−18.8122	−1.57316
$144$	0	0
$145$	0.571328	0.0474462
$146$	0	0
$147$	−0.264777	−0.0218385
$148$	0	0
$149$	−23.8442	−1.95339	−0.976696	−	0.214629i	$-0.931146\pi$
$149$	−23.8442	−1.95339	−0.976696	+	0.214629i	$0.931146\pi$
$150$	0	0
$151$	−7.56968	−0.616011	−0.308006	−	0.951385i	$-0.599662\pi$
$151$	−7.56968	−0.616011	−0.308006	+	0.951385i	$0.599662\pi$
$152$	0	0
$153$	1.19780	0.0968363
$154$	0	0
$155$	−2.16345	−0.173772
$156$	0	0
$157$	21.6508	1.72792	0.863960	−	0.503561i	$-0.167977\pi$
$157$	21.6508	1.72792	0.863960	+	0.503561i	$0.167977\pi$
$158$	0	0
$159$	−3.72490	−0.295404
$160$	0	0
$161$	13.0070	1.02510
$162$	0	0
$163$	−16.0454	−1.25677	−0.628386	−	0.777902i	$-0.716284\pi$
$163$	−16.0454	−1.25677	−0.628386	+	0.777902i	$0.716284\pi$
$164$	0	0
$165$	1.02224	0.0795811
$166$	0	0
$167$	9.20314	0.712160	0.356080	−	0.934455i	$-0.384113\pi$
$167$	9.20314	0.712160	0.356080	+	0.934455i	$0.384113\pi$
$168$	0	0
$169$	7.30091	0.561608
$170$	0	0
$171$	2.55525	0.195405
$172$	0	0
$173$	−9.14157	−0.695021	−0.347510	−	0.937676i	$-0.612973\pi$
$173$	−9.14157	−0.695021	−0.347510	+	0.937676i	$0.612973\pi$
$174$	0	0
$175$	−13.3151	−1.00652
$176$	0	0
$177$	−6.71732	−0.504904
$178$	0	0
$179$	−5.68254	−0.424733	−0.212366	−	0.977190i	$-0.568117\pi$
$179$	−5.68254	−0.424733	−0.212366	+	0.977190i	$0.568117\pi$
$180$	0	0
$181$	22.5449	1.67575	0.837874	−	0.545864i	$-0.183798\pi$
$181$	22.5449	1.67575	0.837874	+	0.545864i	$0.183798\pi$
$182$	0	0
$183$	10.0509	0.742984
$184$	0	0
$185$	−1.47407	−0.108376
$186$	0	0
$187$	5.00111	0.365717
$188$	0	0
$189$	−2.69533	−0.196056
$190$	0	0
$191$	−13.8900	−1.00504	−0.502521	−	0.864565i	$-0.667594\pi$
$191$	−13.8900	−1.00504	−0.502521	+	0.864565i	$0.667594\pi$
$192$	0	0
$193$	−11.8571	−0.853491	−0.426745	−	0.904372i	$-0.640340\pi$
$193$	−11.8571	−0.853491	−0.426745	+	0.904372i	$0.640340\pi$
$194$	0	0
$195$	−1.10313	−0.0789969
$196$	0	0
$197$	−7.71344	−0.549560	−0.274780	−	0.961507i	$-0.588605\pi$
$197$	−7.71344	−0.549560	−0.274780	+	0.961507i	$0.588605\pi$
$198$	0	0
$199$	11.8647	0.841068	0.420534	−	0.907277i	$-0.361843\pi$
$199$	11.8647	0.841068	0.420534	+	0.907277i	$0.361843\pi$
$200$	0	0
$201$	5.48628	0.386972
$202$	0	0
$203$	−6.28965	−0.441447
$204$	0	0
$205$	2.31611	0.161764
$206$	0	0
$207$	4.82577	0.335415
$208$	0	0
$209$	10.6688	0.737975
$210$	0	0
$211$	12.8342	0.883545	0.441773	−	0.897127i	$-0.354350\pi$
$211$	12.8342	0.883545	0.441773	+	0.897127i	$0.354350\pi$
$212$	0	0
$213$	−2.35862	−0.161610
$214$	0	0
$215$	−1.88803	−0.128762
$216$	0	0
$217$	23.8171	1.61681
$218$	0	0
$219$	−10.1415	−0.685298
$220$	0	0
$221$	−5.39687	−0.363033
$222$	0	0
$223$	20.5846	1.37844	0.689222	−	0.724550i	$-0.257953\pi$
$223$	20.5846	1.37844	0.689222	+	0.724550i	$0.257953\pi$
$224$	0	0
$225$	−4.94006	−0.329337
$226$	0	0
$227$	−25.9870	−1.72482	−0.862410	−	0.506211i	$-0.831046\pi$
$227$	−25.9870	−1.72482	−0.862410	+	0.506211i	$0.831046\pi$
$228$	0	0
$229$	16.2462	1.07358	0.536791	−	0.843715i	$-0.319636\pi$
$229$	16.2462	1.07358	0.536791	+	0.843715i	$0.319636\pi$
$230$	0	0
$231$	−11.2536	−0.740436
$232$	0	0
$233$	8.61228	0.564209	0.282104	−	0.959384i	$-0.408968\pi$
$233$	8.61228	0.564209	0.282104	+	0.959384i	$0.408968\pi$
$234$	0	0
$235$	−0.685861	−0.0447406
$236$	0	0
$237$	−1.84247	−0.119681
$238$	0	0
$239$	27.0577	1.75022	0.875110	−	0.483924i	$-0.160789\pi$
$239$	27.0577	1.75022	0.875110	+	0.483924i	$0.160789\pi$
$240$	0	0
$241$	4.99854	0.321985	0.160992	−	0.986956i	$-0.448531\pi$
$241$	4.99854	0.321985	0.160992	+	0.986956i	$0.448531\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.0648262	−0.00414160
$246$	0	0
$247$	−11.5131	−0.732558
$248$	0	0
$249$	6.03637	0.382539
$250$	0	0
$251$	8.14507	0.514113	0.257056	−	0.966396i	$-0.417247\pi$
$251$	8.14507	0.514113	0.257056	+	0.966396i	$0.417247\pi$
$252$	0	0
$253$	20.1488	1.26674
$254$	0	0
$255$	0.293261	0.0183647
$256$	0	0
$257$	8.67847	0.541348	0.270674	−	0.962671i	$-0.412753\pi$
$257$	8.67847	0.541348	0.270674	+	0.962671i	$0.412753\pi$
$258$	0	0
$259$	16.2278	1.00835
$260$	0	0
$261$	−2.33354	−0.144443
$262$	0	0
$263$	−15.7470	−0.971003	−0.485501	−	0.874236i	$-0.661363\pi$
$263$	−15.7470	−0.971003	−0.485501	+	0.874236i	$0.661363\pi$
$264$	0	0
$265$	−0.911979	−0.0560224
$266$	0	0
$267$	0.866842	0.0530499
$268$	0	0
$269$	13.5710	0.827440	0.413720	−	0.910404i	$-0.364229\pi$
$269$	13.5710	0.827440	0.413720	+	0.910404i	$0.364229\pi$
$270$	0	0
$271$	25.8147	1.56813	0.784064	−	0.620679i	$-0.213143\pi$
$271$	25.8147	1.56813	0.784064	+	0.620679i	$0.213143\pi$
$272$	0	0
$273$	12.1442	0.735000
$274$	0	0
$275$	−20.6260	−1.24379
$276$	0	0
$277$	−15.9744	−0.959809	−0.479905	−	0.877321i	$-0.659329\pi$
$277$	−15.9744	−0.959809	−0.479905	+	0.877321i	$0.659329\pi$
$278$	0	0
$279$	8.83643	0.529023
$280$	0	0
$281$	−3.06969	−0.183122	−0.0915611	−	0.995799i	$-0.529186\pi$
$281$	−3.06969	−0.183122	−0.0915611	+	0.995799i	$0.529186\pi$
$282$	0	0
$283$	31.9714	1.90050	0.950250	−	0.311487i	$-0.100827\pi$
$283$	31.9714	1.90050	0.950250	+	0.311487i	$0.100827\pi$
$284$	0	0
$285$	0.625608	0.0370578
$286$	0	0
$287$	−25.4977	−1.50508
$288$	0	0
$289$	−15.5653	−0.915605
$290$	0	0
$291$	−7.72995	−0.453138
$292$	0	0
$293$	−19.2704	−1.12579	−0.562893	−	0.826530i	$-0.690312\pi$
$293$	−19.2704	−1.12579	−0.562893	+	0.826530i	$0.690312\pi$
$294$	0	0
$295$	−1.64462	−0.0957535
$296$	0	0
$297$	−4.17525	−0.242272
$298$	0	0
$299$	−21.7433	−1.25745
$300$	0	0
$301$	20.7850	1.19803
$302$	0	0
$303$	0.545900	0.0313611
$304$	0	0
$305$	2.46079	0.140905
$306$	0	0
$307$	6.36581	0.363316	0.181658	−	0.983362i	$-0.441854\pi$
$307$	6.36581	0.363316	0.181658	+	0.983362i	$0.441854\pi$
$308$	0	0
$309$	3.90518	0.222158
$310$	0	0
$311$	29.4669	1.67092	0.835458	−	0.549554i	$-0.185202\pi$
$311$	29.4669	1.67092	0.835458	+	0.549554i	$0.185202\pi$
$312$	0	0
$313$	22.9018	1.29448	0.647242	−	0.762285i	$-0.275922\pi$
$313$	22.9018	1.29448	0.647242	+	0.762285i	$0.275922\pi$
$314$	0	0
$315$	−0.659904	−0.0371814
$316$	0	0
$317$	−29.7613	−1.67156	−0.835780	−	0.549065i	$-0.814984\pi$
$317$	−29.7613	−1.67156	−0.835780	+	0.549065i	$0.814984\pi$
$318$	0	0
$319$	−9.74311	−0.545510
$320$	0	0
$321$	−9.18330	−0.512562
$322$	0	0
$323$	3.06067	0.170300
$324$	0	0
$325$	22.2582	1.23466
$326$	0	0
$327$	9.00728	0.498104
$328$	0	0
$329$	7.55053	0.416274
$330$	0	0
$331$	−6.56891	−0.361060	−0.180530	−	0.983569i	$-0.557781\pi$
$331$	−6.56891	−0.361060	−0.180530	+	0.983569i	$0.557781\pi$
$332$	0	0
$333$	6.02074	0.329934
$334$	0	0
$335$	1.34322	0.0733881
$336$	0	0
$337$	7.65801	0.417159	0.208579	−	0.978005i	$-0.433116\pi$
$337$	7.65801	0.417159	0.208579	+	0.978005i	$0.433116\pi$
$338$	0	0
$339$	−4.71190	−0.255915
$340$	0	0
$341$	36.8943	1.99794
$342$	0	0
$343$	−18.1536	−0.980203
$344$	0	0
$345$	1.18151	0.0636103
$346$	0	0
$347$	−13.1422	−0.705512	−0.352756	−	0.935715i	$-0.614755\pi$
$347$	−13.1422	−0.705512	−0.352756	+	0.935715i	$0.614755\pi$
$348$	0	0
$349$	5.85668	0.313501	0.156750	−	0.987638i	$-0.449898\pi$
$349$	5.85668	0.313501	0.156750	+	0.987638i	$0.449898\pi$
$350$	0	0
$351$	4.50565	0.240494
$352$	0	0
$353$	−9.81238	−0.522260	−0.261130	−	0.965304i	$-0.584095\pi$
$353$	−9.81238	−0.522260	−0.261130	+	0.965304i	$0.584095\pi$
$354$	0	0
$355$	−0.577467	−0.0306488
$356$	0	0
$357$	−3.22846	−0.170868
$358$	0	0
$359$	23.9807	1.26565	0.632825	−	0.774295i	$-0.281895\pi$
$359$	23.9807	1.26565	0.632825	+	0.774295i	$0.281895\pi$
$360$	0	0
$361$	−12.4707	−0.656354
$362$	0	0
$363$	−6.43268	−0.337628
$364$	0	0
$365$	−2.48297	−0.129964
$366$	0	0
$367$	27.4318	1.43193	0.715963	−	0.698138i	$-0.245988\pi$
$367$	27.4318	1.43193	0.715963	+	0.698138i	$0.245988\pi$
$368$	0	0
$369$	−9.45996	−0.492466
$370$	0	0
$371$	10.0398	0.521242
$372$	0	0
$373$	16.9189	0.876028	0.438014	−	0.898968i	$-0.355682\pi$
$373$	16.9189	0.876028	0.438014	+	0.898968i	$0.355682\pi$
$374$	0	0
$375$	−2.43365	−0.125673
$376$	0	0
$377$	10.5141	0.541505
$378$	0	0
$379$	36.6865	1.88446	0.942229	−	0.334968i	$-0.108726\pi$
$379$	36.6865	1.88446	0.942229	+	0.334968i	$0.108726\pi$
$380$	0	0
$381$	11.6498	0.596838
$382$	0	0
$383$	−1.57033	−0.0802403	−0.0401201	−	0.999195i	$-0.512774\pi$
$383$	−1.57033	−0.0802403	−0.0401201	+	0.999195i	$0.512774\pi$
$384$	0	0
$385$	−2.75526	−0.140421
$386$	0	0
$387$	7.71148	0.391997
$388$	0	0
$389$	9.54858	0.484132	0.242066	−	0.970260i	$-0.422175\pi$
$389$	9.54858	0.484132	0.242066	+	0.970260i	$0.422175\pi$
$390$	0	0
$391$	5.78031	0.292323
$392$	0	0
$393$	1.83085	0.0923540
$394$	0	0
$395$	−0.451096	−0.0226971
$396$	0	0
$397$	5.85597	0.293903	0.146951	−	0.989144i	$-0.453054\pi$
$397$	5.85597	0.293903	0.146951	+	0.989144i	$0.453054\pi$
$398$	0	0
$399$	−6.88722	−0.344792
$400$	0	0
$401$	7.88455	0.393735	0.196868	−	0.980430i	$-0.436923\pi$
$401$	7.88455	0.393735	0.196868	+	0.980430i	$0.436923\pi$
$402$	0	0
$403$	−39.8139	−1.98327
$404$	0	0
$405$	−0.244833	−0.0121658
$406$	0	0
$407$	25.1381	1.24605
$408$	0	0
$409$	30.8972	1.52777	0.763884	−	0.645353i	$-0.223290\pi$
$409$	30.8972	1.52777	0.763884	+	0.645353i	$0.223290\pi$
$410$	0	0
$411$	−13.4853	−0.665183
$412$	0	0
$413$	18.1054	0.890906
$414$	0	0
$415$	1.47790	0.0725474
$416$	0	0
$417$	−11.1740	−0.547195
$418$	0	0
$419$	34.8669	1.70336	0.851679	−	0.524063i	$-0.175585\pi$
$419$	34.8669	1.70336	0.851679	+	0.524063i	$0.175585\pi$
$420$	0	0
$421$	24.2800	1.18334	0.591668	−	0.806182i	$-0.298470\pi$
$421$	24.2800	1.18334	0.591668	+	0.806182i	$0.298470\pi$
$422$	0	0
$423$	2.80134	0.136206
$424$	0	0
$425$	−5.91720	−0.287026
$426$	0	0
$427$	−27.0905	−1.31100
$428$	0	0
$429$	18.8122	0.908262
$430$	0	0
$431$	−31.0586	−1.49604	−0.748020	−	0.663676i	$-0.768995\pi$
$431$	−31.0586	−1.49604	−0.748020	+	0.663676i	$0.768995\pi$
$432$	0	0
$433$	2.40142	0.115405	0.0577025	−	0.998334i	$-0.481623\pi$
$433$	2.40142	0.115405	0.0577025	+	0.998334i	$0.481623\pi$
$434$	0	0
$435$	−0.571328	−0.0273931
$436$	0	0
$437$	12.3310	0.589874
$438$	0	0
$439$	15.0664	0.719081	0.359540	−	0.933129i	$-0.382933\pi$
$439$	15.0664	0.719081	0.359540	+	0.933129i	$0.382933\pi$
$440$	0	0
$441$	0.264777	0.0126084
$442$	0	0
$443$	−3.31651	−0.157572	−0.0787861	−	0.996892i	$-0.525104\pi$
$443$	−3.31651	−0.157572	−0.0787861	+	0.996892i	$0.525104\pi$
$444$	0	0
$445$	0.212231	0.0100607
$446$	0	0
$447$	23.8442	1.12779
$448$	0	0
$449$	20.9427	0.988347	0.494174	−	0.869363i	$-0.335471\pi$
$449$	20.9427	0.988347	0.494174	+	0.869363i	$0.335471\pi$
$450$	0	0
$451$	−39.4977	−1.85987
$452$	0	0
$453$	7.56968	0.355654
$454$	0	0
$455$	2.97330	0.139390
$456$	0	0
$457$	10.5175	0.491988	0.245994	−	0.969271i	$-0.420886\pi$
$457$	10.5175	0.491988	0.245994	+	0.969271i	$0.420886\pi$
$458$	0	0
$459$	−1.19780	−0.0559085
$460$	0	0
$461$	−17.6364	−0.821408	−0.410704	−	0.911769i	$-0.634717\pi$
$461$	−17.6364	−0.821408	−0.410704	+	0.911769i	$0.634717\pi$
$462$	0	0
$463$	3.71983	0.172875	0.0864375	−	0.996257i	$-0.472452\pi$
$463$	3.71983	0.172875	0.0864375	+	0.996257i	$0.472452\pi$
$464$	0	0
$465$	2.16345	0.100328
$466$	0	0
$467$	−29.4817	−1.36425	−0.682125	−	0.731236i	$-0.738944\pi$
$467$	−29.4817	−1.36425	−0.682125	+	0.731236i	$0.738944\pi$
$468$	0	0
$469$	−14.7873	−0.682815
$470$	0	0
$471$	−21.6508	−0.997615
$472$	0	0
$473$	32.1974	1.48044
$474$	0	0
$475$	−12.6231	−0.579186
$476$	0	0
$477$	3.72490	0.170552
$478$	0	0
$479$	35.4820	1.62122	0.810608	−	0.585590i	$-0.199137\pi$
$479$	35.4820	1.62122	0.810608	+	0.585590i	$0.199137\pi$
$480$	0	0
$481$	−27.1274	−1.23690
$482$	0	0
$483$	−13.0070	−0.591840
$484$	0	0
$485$	−1.89255	−0.0859361
$486$	0	0
$487$	2.19988	0.0996862	0.0498431	−	0.998757i	$-0.484128\pi$
$487$	2.19988	0.0996862	0.0498431	+	0.998757i	$0.484128\pi$
$488$	0	0
$489$	16.0454	0.725598
$490$	0	0
$491$	2.81164	0.126888	0.0634438	−	0.997985i	$-0.479792\pi$
$491$	2.81164	0.126888	0.0634438	+	0.997985i	$0.479792\pi$
$492$	0	0
$493$	−2.79511	−0.125886
$494$	0	0
$495$	−1.02224	−0.0459462
$496$	0	0
$497$	6.35724	0.285161
$498$	0	0
$499$	25.4483	1.13922	0.569612	−	0.821914i	$-0.307094\pi$
$499$	25.4483	1.13922	0.569612	+	0.821914i	$0.307094\pi$
$500$	0	0
$501$	−9.20314	−0.411166
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0.133654	0.00594754
$506$	0	0
$507$	−7.30091	−0.324245
$508$	0	0
$509$	3.89029	0.172434	0.0862170	−	0.996276i	$-0.472522\pi$
$509$	3.89029	0.172434	0.0862170	+	0.996276i	$0.472522\pi$
$510$	0	0
$511$	27.3346	1.20921
$512$	0	0
$513$	−2.55525	−0.112817
$514$	0	0
$515$	0.956116	0.0421315
$516$	0	0
$517$	11.6963	0.514402
$518$	0	0
$519$	9.14157	0.401270
$520$	0	0
$521$	−2.72429	−0.119353	−0.0596767	−	0.998218i	$-0.519007\pi$
$521$	−2.72429	−0.119353	−0.0596767	+	0.998218i	$0.519007\pi$
$522$	0	0
$523$	−27.6255	−1.20798	−0.603990	−	0.796992i	$-0.706423\pi$
$523$	−27.6255	−1.20798	−0.603990	+	0.796992i	$0.706423\pi$
$524$	0	0
$525$	13.3151	0.581117
$526$	0	0
$527$	10.5843	0.461058
$528$	0	0
$529$	0.288101	0.0125261
$530$	0	0
$531$	6.71732	0.291507
$532$	0	0
$533$	42.6233	1.84622
$534$	0	0
$535$	−2.24837	−0.0972057
$536$	0	0
$537$	5.68254	0.245220
$538$	0	0
$539$	1.10551	0.0476177
$540$	0	0
$541$	−21.0276	−0.904048	−0.452024	−	0.892006i	$-0.649298\pi$
$541$	−21.0276	−0.904048	−0.452024	+	0.892006i	$0.649298\pi$
$542$	0	0
$543$	−22.5449	−0.967493
$544$	0	0
$545$	2.20528	0.0944638
$546$	0	0
$547$	−16.1019	−0.688470	−0.344235	−	0.938884i	$-0.611862\pi$
$547$	−16.1019	−0.688470	−0.344235	+	0.938884i	$0.611862\pi$
$548$	0	0
$549$	−10.0509	−0.428962
$550$	0	0
$551$	−5.96277	−0.254023
$552$	0	0
$553$	4.96605	0.211178
$554$	0	0
$555$	1.47407	0.0625710
$556$	0	0
$557$	24.8272	1.05196	0.525981	−	0.850496i	$-0.323698\pi$
$557$	24.8272	1.05196	0.525981	+	0.850496i	$0.323698\pi$
$558$	0	0
$559$	−34.7453	−1.46957
$560$	0	0
$561$	−5.00111	−0.211147
$562$	0	0
$563$	−18.5378	−0.781276	−0.390638	−	0.920544i	$-0.627746\pi$
$563$	−18.5378	−0.781276	−0.390638	+	0.920544i	$0.627746\pi$
$564$	0	0
$565$	−1.15363	−0.0485335
$566$	0	0
$567$	2.69533	0.113193
$568$	0	0
$569$	1.86399	0.0781424	0.0390712	−	0.999236i	$-0.487560\pi$
$569$	1.86399	0.0781424	0.0390712	+	0.999236i	$0.487560\pi$
$570$	0	0
$571$	4.01062	0.167839	0.0839197	−	0.996473i	$-0.473256\pi$
$571$	4.01062	0.167839	0.0839197	+	0.996473i	$0.473256\pi$
$572$	0	0
$573$	13.8900	0.580262
$574$	0	0
$575$	−23.8396	−0.994180
$576$	0	0
$577$	1.58184	0.0658527	0.0329264	−	0.999458i	$-0.489517\pi$
$577$	1.58184	0.0658527	0.0329264	+	0.999458i	$0.489517\pi$
$578$	0	0
$579$	11.8571	0.492763
$580$	0	0
$581$	−16.2700	−0.674993
$582$	0	0
$583$	15.5524	0.644114
$584$	0	0
$585$	1.10313	0.0456089
$586$	0	0
$587$	−29.5921	−1.22140	−0.610698	−	0.791863i	$-0.709111\pi$
$587$	−29.5921	−1.22140	−0.610698	+	0.791863i	$0.709111\pi$
$588$	0	0
$589$	22.5793	0.930362
$590$	0	0
$591$	7.71344	0.317289
$592$	0	0
$593$	−25.9349	−1.06502	−0.532510	−	0.846424i	$-0.678751\pi$
$593$	−25.9349	−1.06502	−0.532510	+	0.846424i	$0.678751\pi$
$594$	0	0
$595$	−0.790433	−0.0324046
$596$	0	0
$597$	−11.8647	−0.485591
$598$	0	0
$599$	35.9091	1.46721	0.733604	−	0.679578i	$-0.237837\pi$
$599$	35.9091	1.46721	0.733604	+	0.679578i	$0.237837\pi$
$600$	0	0
$601$	9.35909	0.381765	0.190883	−	0.981613i	$-0.438865\pi$
$601$	9.35909	0.381765	0.190883	+	0.981613i	$0.438865\pi$
$602$	0	0
$603$	−5.48628	−0.223419
$604$	0	0
$605$	−1.57493	−0.0640301
$606$	0	0
$607$	24.5857	0.997902	0.498951	−	0.866630i	$-0.333719\pi$
$607$	24.5857	0.997902	0.498951	+	0.866630i	$0.333719\pi$
$608$	0	0
$609$	6.28965	0.254870
$610$	0	0
$611$	−12.6219	−0.510626
$612$	0	0
$613$	−26.5654	−1.07297	−0.536483	−	0.843911i	$-0.680248\pi$
$613$	−26.5654	−1.07297	−0.536483	+	0.843911i	$0.680248\pi$
$614$	0	0
$615$	−2.31611	−0.0933945
$616$	0	0
$617$	−27.2929	−1.09877	−0.549385	−	0.835570i	$-0.685138\pi$
$617$	−27.2929	−1.09877	−0.549385	+	0.835570i	$0.685138\pi$
$618$	0	0
$619$	17.6654	0.710032	0.355016	−	0.934860i	$-0.384475\pi$
$619$	17.6654	0.710032	0.355016	+	0.934860i	$0.384475\pi$
$620$	0	0
$621$	−4.82577	−0.193652
$622$	0	0
$623$	−2.33642	−0.0936067
$624$	0	0
$625$	24.1044	0.964178
$626$	0	0
$627$	−10.6688	−0.426070
$628$	0	0
$629$	7.21164	0.287547
$630$	0	0
$631$	43.3134	1.72428	0.862139	−	0.506672i	$-0.169124\pi$
$631$	43.3134	1.72428	0.862139	+	0.506672i	$0.169124\pi$
$632$	0	0
$633$	−12.8342	−0.510115
$634$	0	0
$635$	2.85226	0.113188
$636$	0	0
$637$	−1.19300	−0.0472682
$638$	0	0
$639$	2.35862	0.0933054
$640$	0	0
$641$	11.0104	0.434884	0.217442	−	0.976073i	$-0.430229\pi$
$641$	11.0104	0.434884	0.217442	+	0.976073i	$0.430229\pi$
$642$	0	0
$643$	22.4543	0.885513	0.442756	−	0.896642i	$-0.354001\pi$
$643$	22.4543	0.885513	0.442756	+	0.896642i	$0.354001\pi$
$644$	0	0
$645$	1.88803	0.0743409
$646$	0	0
$647$	−4.60853	−0.181180	−0.0905900	−	0.995888i	$-0.528875\pi$
$647$	−4.60853	−0.181180	−0.0905900	+	0.995888i	$0.528875\pi$
$648$	0	0
$649$	28.0465	1.10092
$650$	0	0
$651$	−23.8171	−0.933464
$652$	0	0
$653$	6.19941	0.242602	0.121301	−	0.992616i	$-0.461293\pi$
$653$	6.19941	0.242602	0.121301	+	0.992616i	$0.461293\pi$
$654$	0	0
$655$	0.448251	0.0175146
$656$	0	0
$657$	10.1415	0.395657
$658$	0	0
$659$	−45.6675	−1.77895	−0.889477	−	0.456980i	$-0.848931\pi$
$659$	−45.6675	−1.77895	−0.889477	+	0.456980i	$0.848931\pi$
$660$	0	0
$661$	−40.1949	−1.56340	−0.781701	−	0.623654i	$-0.785648\pi$
$661$	−40.1949	−1.56340	−0.781701	+	0.623654i	$0.785648\pi$
$662$	0	0
$663$	5.39687	0.209597
$664$	0	0
$665$	−1.68622	−0.0653887
$666$	0	0
$667$	−11.2611	−0.436033
$668$	0	0
$669$	−20.5846	−0.795845
$670$	0	0
$671$	−41.9650	−1.62004
$672$	0	0
$673$	−13.7682	−0.530727	−0.265363	−	0.964148i	$-0.585492\pi$
$673$	−13.7682	−0.530727	−0.265363	+	0.964148i	$0.585492\pi$
$674$	0	0
$675$	4.94006	0.190143
$676$	0	0
$677$	−48.3502	−1.85825	−0.929124	−	0.369768i	$-0.879437\pi$
$677$	−48.3502	−1.85825	−0.929124	+	0.369768i	$0.879437\pi$
$678$	0	0
$679$	20.8347	0.799564
$680$	0	0
$681$	25.9870	0.995825
$682$	0	0
$683$	−39.3798	−1.50683	−0.753414	−	0.657547i	$-0.771594\pi$
$683$	−39.3798	−1.50683	−0.753414	+	0.657547i	$0.771594\pi$
$684$	0	0
$685$	−3.30166	−0.126150
$686$	0	0
$687$	−16.2462	−0.619833
$688$	0	0
$689$	−16.7831	−0.639386
$690$	0	0
$691$	−39.0610	−1.48595	−0.742975	−	0.669319i	$-0.766586\pi$
$691$	−39.0610	−1.48595	−0.742975	+	0.669319i	$0.766586\pi$
$692$	0	0
$693$	11.2536	0.427491
$694$	0	0
$695$	−2.73577	−0.103774
$696$	0	0
$697$	−11.3311	−0.429197
$698$	0	0
$699$	−8.61228	−0.325746
$700$	0	0
$701$	−47.4853	−1.79349	−0.896747	−	0.442543i	$-0.854076\pi$
$701$	−47.4853	−1.79349	−0.896747	+	0.442543i	$0.854076\pi$
$702$	0	0
$703$	15.3845	0.580236
$704$	0	0
$705$	0.685861	0.0258310
$706$	0	0
$707$	−1.47138	−0.0553369
$708$	0	0
$709$	11.0321	0.414319	0.207160	−	0.978307i	$-0.433578\pi$
$709$	11.0321	0.414319	0.207160	+	0.978307i	$0.433578\pi$
$710$	0	0
$711$	1.84247	0.0690979
$712$	0	0
$713$	42.6426	1.59698
$714$	0	0
$715$	4.60585	0.172249
$716$	0	0
$717$	−27.0577	−1.01049
$718$	0	0
$719$	15.9644	0.595371	0.297686	−	0.954664i	$-0.403785\pi$
$719$	15.9644	0.595371	0.297686	+	0.954664i	$0.403785\pi$
$720$	0	0
$721$	−10.5257	−0.391998
$722$	0	0
$723$	−4.99854	−0.185898
$724$	0	0
$725$	11.5278	0.428133
$726$	0	0
$727$	−44.5597	−1.65263	−0.826315	−	0.563209i	$-0.809567\pi$
$727$	−44.5597	−1.65263	−0.826315	+	0.563209i	$0.809567\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.23681	0.341636
$732$	0	0
$733$	−45.4408	−1.67839	−0.839196	−	0.543828i	$-0.816974\pi$
$733$	−45.4408	−1.67839	−0.839196	+	0.543828i	$0.816974\pi$
$734$	0	0
$735$	0.0648262	0.00239115
$736$	0	0
$737$	−22.9066	−0.843775
$738$	0	0
$739$	7.70406	0.283398	0.141699	−	0.989910i	$-0.454743\pi$
$739$	7.70406	0.283398	0.141699	+	0.989910i	$0.454743\pi$
$740$	0	0
$741$	11.5131	0.422943
$742$	0	0
$743$	−11.0135	−0.404047	−0.202024	−	0.979381i	$-0.564752\pi$
$743$	−11.0135	−0.404047	−0.202024	+	0.979381i	$0.564752\pi$
$744$	0	0
$745$	5.83784	0.213882
$746$	0	0
$747$	−6.03637	−0.220859
$748$	0	0
$749$	24.7520	0.904417
$750$	0	0
$751$	−22.3053	−0.813933	−0.406967	−	0.913443i	$-0.633413\pi$
$751$	−22.3053	−0.813933	−0.406967	+	0.913443i	$0.633413\pi$
$752$	0	0
$753$	−8.14507	−0.296823
$754$	0	0
$755$	1.85331	0.0674487
$756$	0	0
$757$	6.29421	0.228767	0.114383	−	0.993437i	$-0.463511\pi$
$757$	6.29421	0.228767	0.114383	+	0.993437i	$0.463511\pi$
$758$	0	0
$759$	−20.1488	−0.731355
$760$	0	0
$761$	14.7260	0.533817	0.266909	−	0.963722i	$-0.413998\pi$
$761$	14.7260	0.533817	0.266909	+	0.963722i	$0.413998\pi$
$762$	0	0
$763$	−24.2776	−0.878906
$764$	0	0
$765$	−0.293261	−0.0106029
$766$	0	0
$767$	−30.2659	−1.09284
$768$	0	0
$769$	31.6883	1.14271	0.571354	−	0.820704i	$-0.306418\pi$
$769$	31.6883	1.14271	0.571354	+	0.820704i	$0.306418\pi$
$770$	0	0
$771$	−8.67847	−0.312547
$772$	0	0
$773$	33.7530	1.21401	0.607005	−	0.794698i	$-0.292371\pi$
$773$	33.7530	1.21401	0.607005	+	0.794698i	$0.292371\pi$
$774$	0	0
$775$	−43.6525	−1.56804
$776$	0	0
$777$	−16.2278	−0.582171
$778$	0	0
$779$	−24.1725	−0.866071
$780$	0	0
$781$	9.84781	0.352382
$782$	0	0
$783$	2.33354	0.0833940
$784$	0	0
$785$	−5.30082	−0.189194
$786$	0	0
$787$	9.27299	0.330546	0.165273	−	0.986248i	$-0.447149\pi$
$787$	9.27299	0.330546	0.165273	+	0.986248i	$0.447149\pi$
$788$	0	0
$789$	15.7470	0.560609
$790$	0	0
$791$	12.7001	0.451564
$792$	0	0
$793$	45.2859	1.60815
$794$	0	0
$795$	0.911979	0.0323446
$796$	0	0
$797$	41.7808	1.47995	0.739976	−	0.672633i	$-0.234837\pi$
$797$	41.7808	1.47995	0.739976	+	0.672633i	$0.234837\pi$
$798$	0	0
$799$	3.35545	0.118707
$800$	0	0
$801$	−0.866842	−0.0306284
$802$	0	0
$803$	42.3432	1.49426
$804$	0	0
$805$	−3.18455	−0.112241
$806$	0	0
$807$	−13.5710	−0.477723
$808$	0	0
$809$	5.07013	0.178256	0.0891282	−	0.996020i	$-0.471592\pi$
$809$	5.07013	0.178256	0.0891282	+	0.996020i	$0.471592\pi$
$810$	0	0
$811$	−18.4918	−0.649334	−0.324667	−	0.945828i	$-0.605252\pi$
$811$	−18.4918	−0.649334	−0.324667	+	0.945828i	$0.605252\pi$
$812$	0	0
$813$	−25.8147	−0.905360
$814$	0	0
$815$	3.92844	0.137607
$816$	0	0
$817$	19.7047	0.689382
$818$	0	0
$819$	−12.1442	−0.424353
$820$	0	0
$821$	8.55963	0.298733	0.149367	−	0.988782i	$-0.452277\pi$
$821$	8.55963	0.298733	0.149367	+	0.988782i	$0.452277\pi$
$822$	0	0
$823$	0.229162	0.00798809	0.00399405	−	0.999992i	$-0.498729\pi$
$823$	0.229162	0.00798809	0.00399405	+	0.999992i	$0.498729\pi$
$824$	0	0
$825$	20.6260	0.718104
$826$	0	0
$827$	−10.2949	−0.357987	−0.178994	−	0.983850i	$-0.557284\pi$
$827$	−10.2949	−0.357987	−0.178994	+	0.983850i	$0.557284\pi$
$828$	0	0
$829$	4.42281	0.153611	0.0768053	−	0.997046i	$-0.475528\pi$
$829$	4.42281	0.153611	0.0768053	+	0.997046i	$0.475528\pi$
$830$	0	0
$831$	15.9744	0.554146
$832$	0	0
$833$	0.317150	0.0109886
$834$	0	0
$835$	−2.25323	−0.0779763
$836$	0	0
$837$	−8.83643	−0.305432
$838$	0	0
$839$	−30.2460	−1.04421	−0.522103	−	0.852882i	$-0.674852\pi$
$839$	−30.2460	−1.04421	−0.522103	+	0.852882i	$0.674852\pi$
$840$	0	0
$841$	−23.5546	−0.812227
$842$	0	0
$843$	3.06969	0.105726
$844$	0	0
$845$	−1.78750	−0.0614920
$846$	0	0
$847$	17.3382	0.595747
$848$	0	0
$849$	−31.9714	−1.09725
$850$	0	0
$851$	29.0547	0.995983
$852$	0	0
$853$	23.7512	0.813227	0.406613	−	0.913600i	$-0.366710\pi$
$853$	23.7512	0.813227	0.406613	+	0.913600i	$0.366710\pi$
$854$	0	0
$855$	−0.625608	−0.0213954
$856$	0	0
$857$	−26.2245	−0.895813	−0.447906	−	0.894080i	$-0.647830\pi$
$857$	−26.2245	−0.895813	−0.447906	+	0.894080i	$0.647830\pi$
$858$	0	0
$859$	35.5067	1.21147	0.605736	−	0.795665i	$-0.292879\pi$
$859$	35.5067	1.21147	0.605736	+	0.795665i	$0.292879\pi$
$860$	0	0
$861$	25.4977	0.868958
$862$	0	0
$863$	−24.2621	−0.825891	−0.412945	−	0.910756i	$-0.635500\pi$
$863$	−24.2621	−0.825891	−0.412945	+	0.910756i	$0.635500\pi$
$864$	0	0
$865$	2.23816	0.0760996
$866$	0	0
$867$	15.5653	0.528625
$868$	0	0
$869$	7.69275	0.260959
$870$	0	0
$871$	24.7193	0.837581
$872$	0	0
$873$	7.72995	0.261619
$874$	0	0
$875$	6.55949	0.221751
$876$	0	0
$877$	−9.06096	−0.305967	−0.152983	−	0.988229i	$-0.548888\pi$
$877$	−9.06096	−0.305967	−0.152983	+	0.988229i	$0.548888\pi$
$878$	0	0
$879$	19.2704	0.649973
$880$	0	0
$881$	−45.6812	−1.53904	−0.769519	−	0.638624i	$-0.779504\pi$
$881$	−45.6812	−1.53904	−0.769519	+	0.638624i	$0.779504\pi$
$882$	0	0
$883$	−41.1800	−1.38582	−0.692909	−	0.721025i	$-0.743671\pi$
$883$	−41.1800	−1.38582	−0.692909	+	0.721025i	$0.743671\pi$
$884$	0	0
$885$	1.64462	0.0552833
$886$	0	0
$887$	21.3278	0.716119	0.358059	−	0.933699i	$-0.383438\pi$
$887$	21.3278	0.716119	0.358059	+	0.933699i	$0.383438\pi$
$888$	0	0
$889$	−31.4000	−1.05312
$890$	0	0
$891$	4.17525	0.139876
$892$	0	0
$893$	7.15812	0.239537
$894$	0	0
$895$	1.39127	0.0465051
$896$	0	0
$897$	21.7433	0.725987
$898$	0	0
$899$	−20.6202	−0.687721
$900$	0	0
$901$	4.46169	0.148640
$902$	0	0
$903$	−20.7850	−0.691680
$904$	0	0
$905$	−5.51973	−0.183482
$906$	0	0
$907$	52.9012	1.75656	0.878278	−	0.478151i	$-0.158693\pi$
$907$	52.9012	1.75656	0.878278	+	0.478151i	$0.158693\pi$
$908$	0	0
$909$	−0.545900	−0.0181064
$910$	0	0
$911$	−16.5751	−0.549159	−0.274579	−	0.961564i	$-0.588539\pi$
$911$	−16.5751	−0.549159	−0.274579	+	0.961564i	$0.588539\pi$
$912$	0	0
$913$	−25.2033	−0.834109
$914$	0	0
$915$	−2.46079	−0.0813513
$916$	0	0
$917$	−4.93473	−0.162959
$918$	0	0
$919$	−34.8122	−1.14835	−0.574174	−	0.818733i	$-0.694677\pi$
$919$	−34.8122	−1.14835	−0.574174	+	0.818733i	$0.694677\pi$
$920$	0	0
$921$	−6.36581	−0.209761
$922$	0	0
$923$	−10.6271	−0.349796
$924$	0	0
$925$	−29.7428	−0.977937
$926$	0	0
$927$	−3.90518	−0.128263
$928$	0	0
$929$	50.6853	1.66293	0.831466	−	0.555576i	$-0.187502\pi$
$929$	50.6853	1.66293	0.831466	+	0.555576i	$0.187502\pi$
$930$	0	0
$931$	0.676571	0.0221737
$932$	0	0
$933$	−29.4669	−0.964704
$934$	0	0
$935$	−1.22444	−0.0400433
$936$	0	0
$937$	11.2435	0.367310	0.183655	−	0.982991i	$-0.441207\pi$
$937$	11.2435	0.367310	0.183655	+	0.982991i	$0.441207\pi$
$938$	0	0
$939$	−22.9018	−0.747371
$940$	0	0
$941$	−5.98689	−0.195167	−0.0975836	−	0.995227i	$-0.531111\pi$
$941$	−5.98689	−0.195167	−0.0975836	+	0.995227i	$0.531111\pi$
$942$	0	0
$943$	−45.6516	−1.48662
$944$	0	0
$945$	0.659904	0.0214667
$946$	0	0
$947$	5.43436	0.176593	0.0882965	−	0.996094i	$-0.471858\pi$
$947$	5.43436	0.176593	0.0882965	+	0.996094i	$0.471858\pi$
$948$	0	0
$949$	−45.6940	−1.48329
$950$	0	0
$951$	29.7613	0.965075
$952$	0	0
$953$	−6.33785	−0.205303	−0.102652	−	0.994717i	$-0.532733\pi$
$953$	−6.33785	−0.205303	−0.102652	+	0.994717i	$0.532733\pi$
$954$	0	0
$955$	3.40072	0.110045
$956$	0	0
$957$	9.74311	0.314950
$958$	0	0
$959$	36.3474	1.17372
$960$	0	0
$961$	47.0825	1.51879
$962$	0	0
$963$	9.18330	0.295928
$964$	0	0
$965$	2.90300	0.0934509
$966$	0	0
$967$	12.8182	0.412206	0.206103	−	0.978530i	$-0.433922\pi$
$967$	12.8182	0.412206	0.206103	+	0.978530i	$0.433922\pi$
$968$	0	0
$969$	−3.06067	−0.0983230
$970$	0	0
$971$	−44.2204	−1.41910	−0.709549	−	0.704656i	$-0.751101\pi$
$971$	−44.2204	−1.41910	−0.709549	+	0.704656i	$0.751101\pi$
$972$	0	0
$973$	30.1177	0.965528
$974$	0	0
$975$	−22.2582	−0.712832
$976$	0	0
$977$	−11.9512	−0.382354	−0.191177	−	0.981556i	$-0.561230\pi$
$977$	−11.9512	−0.382354	−0.191177	+	0.981556i	$0.561230\pi$
$978$	0	0
$979$	−3.61928	−0.115673
$980$	0	0
$981$	−9.00728	−0.287580
$982$	0	0
$983$	−15.5494	−0.495950	−0.247975	−	0.968766i	$-0.579765\pi$
$983$	−15.5494	−0.495950	−0.247975	+	0.968766i	$0.579765\pi$
$984$	0	0
$985$	1.88850	0.0601728
$986$	0	0
$987$	−7.55053	−0.240336
$988$	0	0
$989$	37.2139	1.18333
$990$	0	0
$991$	14.6255	0.464594	0.232297	−	0.972645i	$-0.425376\pi$
$991$	14.6255	0.464594	0.232297	+	0.972645i	$0.425376\pi$
$992$	0	0
$993$	6.56891	0.208458
$994$	0	0
$995$	−2.90488	−0.0920908
$996$	0	0
$997$	−20.0004	−0.633420	−0.316710	−	0.948522i	$-0.602578\pi$
$997$	−20.0004	−0.633420	−0.316710	+	0.948522i	$0.602578\pi$
$998$	0	0
$999$	−6.02074	−0.190488

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.12	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.12	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.244833 q^{5} +2.69533 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.244833 q^{5} +2.69533 q^{7} +1.00000 q^{9} +4.17525 q^{11} -4.50565 q^{13} +0.244833 q^{15} +1.19780 q^{17} +2.55525 q^{19} -2.69533 q^{21} +4.82577 q^{23} -4.94006 q^{25} -1.00000 q^{27} -2.33354 q^{29} +8.83643 q^{31} -4.17525 q^{33} -0.659904 q^{35} +6.02074 q^{37} +4.50565 q^{39} -9.45996 q^{41} +7.71148 q^{43} -0.244833 q^{45} +2.80134 q^{47} +0.264777 q^{49} -1.19780 q^{51} +3.72490 q^{53} -1.02224 q^{55} -2.55525 q^{57} +6.71732 q^{59} -10.0509 q^{61} +2.69533 q^{63} +1.10313 q^{65} -5.48628 q^{67} -4.82577 q^{69} +2.35862 q^{71} +10.1415 q^{73} +4.94006 q^{75} +11.2536 q^{77} +1.84247 q^{79} +1.00000 q^{81} -6.03637 q^{83} -0.293261 q^{85} +2.33354 q^{87} -0.866842 q^{89} -12.1442 q^{91} -8.83643 q^{93} -0.625608 q^{95} +7.72995 q^{97} +4.17525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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