LMFDB - Embedded newform 252.12.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.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:	$193.622481501$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 522x + 2520 $ x^3 - x^2 - 522*x + 2520
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.02190$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-648.744 q^{5} -16807.0 q^{7} +O(q^{10})$	$q-648.744 q^{5} -16807.0 q^{7} -633226. q^{11} -2.22191e6 q^{13} -7.45019e6 q^{17} -1.88026e7 q^{19} -7.49522e6 q^{23} -4.84073e7 q^{25} +1.89254e8 q^{29} -1.19478e8 q^{31} +1.09034e7 q^{35} -5.47837e8 q^{37} +1.14527e7 q^{41} +1.17708e9 q^{43} +1.21053e9 q^{47} +2.82475e8 q^{49} -4.42637e9 q^{53} +4.10802e8 q^{55} -6.59069e9 q^{59} +1.02763e10 q^{61} +1.44145e9 q^{65} -1.04610e10 q^{67} -2.83389e9 q^{71} -1.42498e9 q^{73} +1.06426e10 q^{77} -2.73845e10 q^{79} +5.48937e10 q^{83} +4.83327e9 q^{85} +9.31872e9 q^{89} +3.73436e10 q^{91} +1.21981e10 q^{95} +6.34089e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) $ 3 * q - 4762 * q^5 - 50421 * q^7	$ 3 q - 4762 q^{5} - 50421 q^{7} - 80468 q^{11} + 996546 q^{13} + 7924090 q^{17} - 15261732 q^{19} + 61428352 q^{23} - 109205331 q^{25} + 160359182 q^{29} - 342148464 q^{31} + 80034934 q^{35} - 722039190 q^{37} - 285288606 q^{41} - 476523612 q^{43} - 1225105680 q^{47} + 847425747 q^{49} - 4030571514 q^{53} + 4822100040 q^{55} - 5222175892 q^{59} + 5684837106 q^{61} - 8153014116 q^{65} + 2837154348 q^{67} + 16928678200 q^{71} - 14560325442 q^{73} + 1352425676 q^{77} - 32832340032 q^{79} + 115451875348 q^{83} - 36782181276 q^{85} + 15661530882 q^{89} - 16748948622 q^{91} + 117828203728 q^{95} + 40693412742 q^{97}+O(q^{100}) $ 3 * q - 4762 * q^5 - 50421 * q^7 - 80468 * q^11 + 996546 * q^13 + 7924090 * q^17 - 15261732 * q^19 + 61428352 * q^23 - 109205331 * q^25 + 160359182 * q^29 - 342148464 * q^31 + 80034934 * q^35 - 722039190 * q^37 - 285288606 * q^41 - 476523612 * q^43 - 1225105680 * q^47 + 847425747 * q^49 - 4030571514 * q^53 + 4822100040 * q^55 - 5222175892 * q^59 + 5684837106 * q^61 - 8153014116 * q^65 + 2837154348 * q^67 + 16928678200 * q^71 - 14560325442 * q^73 + 1352425676 * q^77 - 32832340032 * q^79 + 115451875348 * q^83 - 36782181276 * q^85 + 15661530882 * q^89 - 16748948622 * q^91 + 117828203728 * q^95 + 40693412742 * 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	0
$3$	0	0
$4$	0	0
$5$	−648.744	−0.0928408	−0.0464204	−	0.998922i	$-0.514781\pi$
$5$	−648.744	−0.0928408	−0.0464204	+	0.998922i	$0.514781\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−633226.	−1.18549	−0.592746	−	0.805389i	$-0.701956\pi$
$11$	−633226.	−1.18549	−0.592746	+	0.805389i	$0.701956\pi$
$12$	0	0
$13$	−2.22191e6	−1.65973	−0.829864	−	0.557965i	$-0.811582\pi$
$13$	−2.22191e6	−1.65973	−0.829864	+	0.557965i	$0.811582\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.45019e6	−1.27262	−0.636310	−	0.771434i	$-0.719540\pi$
$17$	−7.45019e6	−1.27262	−0.636310	+	0.771434i	$0.719540\pi$
$18$	0	0
$19$	−1.88026e7	−1.74210	−0.871050	−	0.491195i	$-0.836560\pi$
$19$	−1.88026e7	−1.74210	−0.871050	+	0.491195i	$0.836560\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.49522e6	−0.242818	−0.121409	−	0.992603i	$-0.538741\pi$
$23$	−7.49522e6	−0.242818	−0.121409	+	0.992603i	$0.538741\pi$
$24$	0	0
$25$	−4.84073e7	−0.991381
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.89254e8	1.71339	0.856693	−	0.515827i	$-0.172515\pi$
$29$	1.89254e8	1.71339	0.856693	+	0.515827i	$0.172515\pi$
$30$	0	0
$31$	−1.19478e8	−0.749546	−0.374773	−	0.927117i	$-0.622279\pi$
$31$	−1.19478e8	−0.749546	−0.374773	+	0.927117i	$0.622279\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.09034e7	0.0350905
$36$	0	0
$37$	−5.47837e8	−1.29880	−0.649400	−	0.760447i	$-0.724980\pi$
$37$	−5.47837e8	−1.29880	−0.649400	+	0.760447i	$0.724980\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.14527e7	0.0154382	0.00771909	−	0.999970i	$-0.497543\pi$
$41$	1.14527e7	0.0154382	0.00771909	+	0.999970i	$0.497543\pi$
$42$	0	0
$43$	1.17708e9	1.22104	0.610519	−	0.792002i	$-0.290961\pi$
$43$	1.17708e9	1.22104	0.610519	+	0.792002i	$0.290961\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.21053e9	0.769906	0.384953	−	0.922936i	$-0.374218\pi$
$47$	1.21053e9	0.769906	0.384953	+	0.922936i	$0.374218\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.42637e9	−1.45389	−0.726943	−	0.686698i	$-0.759060\pi$
$53$	−4.42637e9	−1.45389	−0.726943	+	0.686698i	$0.759060\pi$
$54$	0	0
$55$	4.10802e8	0.110062
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.59069e9	−1.20018	−0.600088	−	0.799934i	$-0.704868\pi$
$59$	−6.59069e9	−1.20018	−0.600088	+	0.799934i	$0.704868\pi$
$60$	0	0
$61$	1.02763e10	1.55783	0.778917	−	0.627127i	$-0.215769\pi$
$61$	1.02763e10	1.55783	0.778917	+	0.627127i	$0.215769\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.44145e9	0.154090
$66$	0	0
$67$	−1.04610e10	−0.946593	−0.473297	−	0.880903i	$-0.656936\pi$
$67$	−1.04610e10	−0.946593	−0.473297	+	0.880903i	$0.656936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.83389e9	−0.186407	−0.0932036	−	0.995647i	$-0.529711\pi$
$71$	−2.83389e9	−0.186407	−0.0932036	+	0.995647i	$0.529711\pi$
$72$	0	0
$73$	−1.42498e9	−0.0804510	−0.0402255	−	0.999191i	$-0.512808\pi$
$73$	−1.42498e9	−0.0804510	−0.0402255	+	0.999191i	$0.512808\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.06426e10	0.448074
$78$	0	0
$79$	−2.73845e10	−1.00128	−0.500641	−	0.865655i	$-0.666902\pi$
$79$	−2.73845e10	−1.00128	−0.500641	+	0.865655i	$0.666902\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.48937e10	1.52965	0.764827	−	0.644236i	$-0.222825\pi$
$83$	5.48937e10	1.52965	0.764827	+	0.644236i	$0.222825\pi$
$84$	0	0
$85$	4.83327e9	0.118151
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.31872e9	0.176893	0.0884466	−	0.996081i	$-0.471810\pi$
$89$	9.31872e9	0.176893	0.0884466	+	0.996081i	$0.471810\pi$
$90$	0	0
$91$	3.73436e10	0.627318
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.21981e10	0.161738
$96$	0	0
$97$	6.34089e10	0.749731	0.374866	−	0.927079i	$-0.377689\pi$
$97$	6.34089e10	0.749731	0.374866	+	0.927079i	$0.377689\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.34493e10	−0.411354	−0.205677	−	0.978620i	$-0.565940\pi$
$101$	−4.34493e10	−0.411354	−0.205677	+	0.978620i	$0.565940\pi$
$102$	0	0
$103$	−9.18931e8	−0.00781049	−0.00390524	−	0.999992i	$-0.501243\pi$
$103$	−9.18931e8	−0.00781049	−0.00390524	+	0.999992i	$0.501243\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.79888e11	−1.23991	−0.619956	−	0.784636i	$-0.712850\pi$
$107$	−1.79888e11	−1.23991	−0.619956	+	0.784636i	$0.712850\pi$
$108$	0	0
$109$	−4.60258e10	−0.286520	−0.143260	−	0.989685i	$-0.545759\pi$
$109$	−4.60258e10	−0.286520	−0.143260	+	0.989685i	$0.545759\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.34722e11	−1.70904	−0.854521	−	0.519418i	$-0.826149\pi$
$113$	−3.34722e11	−1.70904	−0.854521	+	0.519418i	$0.826149\pi$
$114$	0	0
$115$	4.86249e9	0.0225434
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.25215e11	0.481005
$120$	0	0
$121$	1.15663e11	0.405392
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.30809e10	0.184881
$126$	0	0
$127$	9.84461e10	0.264410	0.132205	−	0.991222i	$-0.457794\pi$
$127$	9.84461e10	0.264410	0.132205	+	0.991222i	$0.457794\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.74026e11	1.07352	0.536760	−	0.843735i	$-0.319648\pi$
$131$	4.74026e11	1.07352	0.536760	+	0.843735i	$0.319648\pi$
$132$	0	0
$133$	3.16015e11	0.658452
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.30338e11	0.761810	0.380905	−	0.924614i	$-0.375612\pi$
$137$	4.30338e11	0.761810	0.380905	+	0.924614i	$0.375612\pi$
$138$	0	0
$139$	−3.82543e11	−0.625315	−0.312658	−	0.949866i	$-0.601219\pi$
$139$	−3.82543e11	−0.625315	−0.312658	+	0.949866i	$0.601219\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.40697e12	1.96760
$144$	0	0
$145$	−1.22777e11	−0.159072
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.48678e12	−1.65853	−0.829263	−	0.558859i	$-0.811239\pi$
$149$	−1.48678e12	−1.65853	−0.829263	+	0.558859i	$0.811239\pi$
$150$	0	0
$151$	−4.19057e11	−0.434410	−0.217205	−	0.976126i	$-0.569694\pi$
$151$	−4.19057e11	−0.434410	−0.217205	+	0.976126i	$0.569694\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.75106e10	0.0695884
$156$	0	0
$157$	−6.42896e11	−0.537889	−0.268945	−	0.963156i	$-0.586675\pi$
$157$	−6.42896e11	−0.537889	−0.268945	+	0.963156i	$0.586675\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.25972e11	0.0917767
$162$	0	0
$163$	−8.36446e11	−0.569385	−0.284693	−	0.958619i	$-0.591892\pi$
$163$	−8.36446e11	−0.569385	−0.284693	+	0.958619i	$0.591892\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.84011e12	1.09623	0.548117	−	0.836402i	$-0.315345\pi$
$167$	1.84011e12	1.09623	0.548117	+	0.836402i	$0.315345\pi$
$168$	0	0
$169$	3.14470e12	1.75470
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.21348e12	−0.595361	−0.297681	−	0.954666i	$-0.596213\pi$
$173$	−1.21348e12	−0.595361	−0.297681	+	0.954666i	$0.596213\pi$
$174$	0	0
$175$	8.13581e11	0.374707
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.87530e12	1.57621	0.788104	−	0.615542i	$-0.211063\pi$
$179$	3.87530e12	1.57621	0.788104	+	0.615542i	$0.211063\pi$
$180$	0	0
$181$	−3.47120e12	−1.32815	−0.664076	−	0.747665i	$-0.731175\pi$
$181$	−3.47120e12	−1.32815	−0.664076	+	0.747665i	$0.731175\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.55406e11	0.120582
$186$	0	0
$187$	4.71765e12	1.50868
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.55611e12	−1.86622	−0.933109	−	0.359593i	$-0.882915\pi$
$191$	−6.55611e12	−1.86622	−0.933109	+	0.359593i	$0.882915\pi$
$192$	0	0
$193$	3.76100e12	1.01097	0.505485	−	0.862835i	$-0.331313\pi$
$193$	3.76100e12	1.01097	0.505485	+	0.862835i	$0.331313\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.74135e12	0.658264	0.329132	−	0.944284i	$-0.393244\pi$
$197$	2.74135e12	0.658264	0.329132	+	0.944284i	$0.393244\pi$
$198$	0	0
$199$	−4.97228e12	−1.12944	−0.564720	−	0.825283i	$-0.691016\pi$
$199$	−4.97228e12	−1.12944	−0.564720	+	0.825283i	$0.691016\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.18079e12	−0.647599
$204$	0	0
$205$	−7.42987e9	−0.00143329
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.19063e13	2.06525
$210$	0	0
$211$	8.39964e12	1.38263	0.691317	−	0.722551i	$-0.257031\pi$
$211$	8.39964e12	1.38263	0.691317	+	0.722551i	$0.257031\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.63624e11	−0.113362
$216$	0	0
$217$	2.00807e12	0.283302
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.65536e13	2.11220
$222$	0	0
$223$	−1.45288e13	−1.76423	−0.882113	−	0.471039i	$-0.843879\pi$
$223$	−1.45288e13	−1.76423	−0.882113	+	0.471039i	$0.843879\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.62449e12	0.289003	0.144502	−	0.989505i	$-0.453842\pi$
$227$	2.62449e12	0.289003	0.144502	+	0.989505i	$0.453842\pi$
$228$	0	0
$229$	3.04342e12	0.319349	0.159675	−	0.987170i	$-0.448955\pi$
$229$	3.04342e12	0.319349	0.159675	+	0.987170i	$0.448955\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.67763e12	0.732436	0.366218	−	0.930529i	$-0.380652\pi$
$233$	7.67763e12	0.732436	0.366218	+	0.930529i	$0.380652\pi$
$234$	0	0
$235$	−7.85325e11	−0.0714787
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.45586e11	−0.0535507	−0.0267754	−	0.999641i	$-0.508524\pi$
$239$	−6.45586e11	−0.0535507	−0.0267754	+	0.999641i	$0.508524\pi$
$240$	0	0
$241$	−8.23641e12	−0.652596	−0.326298	−	0.945267i	$-0.605801\pi$
$241$	−8.23641e12	−0.652596	−0.326298	+	0.945267i	$0.605801\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.83254e11	−0.0132630
$246$	0	0
$247$	4.17776e13	2.89141
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.04687e13	1.29684	0.648419	−	0.761284i	$-0.275431\pi$
$251$	2.04687e13	1.29684	0.648419	+	0.761284i	$0.275431\pi$
$252$	0	0
$253$	4.74617e12	0.287859
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.02452e11	−0.0446464	−0.0223232	−	0.999751i	$-0.507106\pi$
$257$	−8.02452e11	−0.0446464	−0.0223232	+	0.999751i	$0.507106\pi$
$258$	0	0
$259$	9.20750e12	0.490900
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.29048e12	−0.308267	−0.154134	−	0.988050i	$-0.549259\pi$
$263$	−6.29048e12	−0.308267	−0.154134	+	0.988050i	$0.549259\pi$
$264$	0	0
$265$	2.87158e12	0.134980
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.15063e12	0.136383	0.0681914	−	0.997672i	$-0.478277\pi$
$269$	3.15063e12	0.136383	0.0681914	+	0.997672i	$0.478277\pi$
$270$	0	0
$271$	2.15924e12	0.0897366	0.0448683	−	0.998993i	$-0.485713\pi$
$271$	2.15924e12	0.0897366	0.0448683	+	0.998993i	$0.485713\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.06527e13	1.17527
$276$	0	0
$277$	−8.87758e12	−0.327081	−0.163541	−	0.986537i	$-0.552291\pi$
$277$	−8.87758e12	−0.327081	−0.163541	+	0.986537i	$0.552291\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.92587e13	−1.67725	−0.838627	−	0.544707i	$-0.816641\pi$
$281$	−4.92587e13	−1.67725	−0.838627	+	0.544707i	$0.816641\pi$
$282$	0	0
$283$	1.33777e13	0.438084	0.219042	−	0.975715i	$-0.429707\pi$
$283$	1.33777e13	0.438084	0.219042	+	0.975715i	$0.429707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.92485e11	−0.00583509
$288$	0	0
$289$	2.12335e13	0.619560
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.32771e13	1.98242	0.991212	−	0.132286i	$-0.0422316\pi$
$293$	7.32771e13	1.98242	0.991212	+	0.132286i	$0.0422316\pi$
$294$	0	0
$295$	4.27567e12	0.111425
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.66537e13	0.403013
$300$	0	0
$301$	−1.97832e13	−0.461509
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.66667e12	−0.144630
$306$	0	0
$307$	3.63396e13	0.760535	0.380267	−	0.924877i	$-0.375832\pi$
$307$	3.63396e13	0.760535	0.380267	+	0.924877i	$0.375832\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.82944e13	1.13617	0.568086	−	0.822969i	$-0.307684\pi$
$311$	5.82944e13	1.13617	0.568086	+	0.822969i	$0.307684\pi$
$312$	0	0
$313$	3.65978e13	0.688590	0.344295	−	0.938861i	$-0.388118\pi$
$313$	3.65978e13	0.688590	0.344295	+	0.938861i	$0.388118\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.42092e13	0.951146	0.475573	−	0.879676i	$-0.342241\pi$
$317$	5.42092e13	0.951146	0.475573	+	0.879676i	$0.342241\pi$
$318$	0	0
$319$	−1.19840e14	−2.03121
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.40083e14	2.21703
$324$	0	0
$325$	1.07556e14	1.64542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.03454e13	−0.290997
$330$	0	0
$331$	−4.49124e13	−0.621316	−0.310658	−	0.950522i	$-0.600549\pi$
$331$	−4.49124e13	−0.621316	−0.310658	+	0.950522i	$0.600549\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.78654e12	0.0878824
$336$	0	0
$337$	−4.36251e13	−0.546729	−0.273364	−	0.961911i	$-0.588136\pi$
$337$	−4.36251e13	−0.546729	−0.273364	+	0.961911i	$0.588136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.56565e13	0.888581
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.37183e13	−0.786617	−0.393308	−	0.919407i	$-0.628670\pi$
$347$	−7.37183e13	−0.786617	−0.393308	+	0.919407i	$0.628670\pi$
$348$	0	0
$349$	6.50944e13	0.672983	0.336491	−	0.941687i	$-0.390760\pi$
$349$	6.50944e13	0.672983	0.336491	+	0.941687i	$0.390760\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.57756e12	−0.0444501	−0.0222251	−	0.999753i	$-0.507075\pi$
$353$	−4.57756e12	−0.0444501	−0.0222251	+	0.999753i	$0.507075\pi$
$354$	0	0
$355$	1.83847e12	0.0173062
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.93563e13	0.259826	0.129913	−	0.991525i	$-0.458530\pi$
$359$	2.93563e13	0.259826	0.129913	+	0.991525i	$0.458530\pi$
$360$	0	0
$361$	2.37047e14	2.03491
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.24445e11	0.00746913
$366$	0	0
$367$	7.92308e12	0.0621199	0.0310599	−	0.999518i	$-0.490112\pi$
$367$	7.92308e12	0.0621199	0.0310599	+	0.999518i	$0.490112\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.43940e13	0.549517
$372$	0	0
$373$	−1.00653e14	−0.721818	−0.360909	−	0.932601i	$-0.617534\pi$
$373$	−1.00653e14	−0.721818	−0.360909	+	0.932601i	$0.617534\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.20504e14	−2.84376
$378$	0	0
$379$	−5.71613e13	−0.375479	−0.187740	−	0.982219i	$-0.560116\pi$
$379$	−5.71613e13	−0.375479	−0.187740	+	0.982219i	$0.560116\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.16877e13	−0.444479	−0.222240	−	0.974992i	$-0.571337\pi$
$383$	−7.16877e13	−0.444479	−0.222240	+	0.974992i	$0.571337\pi$
$384$	0	0
$385$	−6.90434e12	−0.0415995
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.23853e14	1.27421	0.637103	−	0.770779i	$-0.280133\pi$
$389$	2.23853e14	1.27421	0.637103	+	0.770779i	$0.280133\pi$
$390$	0	0
$391$	5.58409e13	0.309015
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.77656e13	0.0929597
$396$	0	0
$397$	−1.16498e14	−0.592888	−0.296444	−	0.955050i	$-0.595801\pi$
$397$	−1.16498e14	−0.592888	−0.296444	+	0.955050i	$0.595801\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.51118e12	0.00727818	0.00363909	−	0.999993i	$-0.498842\pi$
$401$	1.51118e12	0.00727818	0.00363909	+	0.999993i	$0.498842\pi$
$402$	0	0
$403$	2.65469e14	1.24404
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.46905e14	1.53972
$408$	0	0
$409$	1.71162e13	0.0739483	0.0369741	−	0.999316i	$-0.488228\pi$
$409$	1.71162e13	0.0739483	0.0369741	+	0.999316i	$0.488228\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.10770e14	0.453624
$414$	0	0
$415$	−3.56120e13	−0.142014
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.88901e14	−1.09288	−0.546439	−	0.837499i	$-0.684017\pi$
$419$	−2.88901e14	−1.09288	−0.546439	+	0.837499i	$0.684017\pi$
$420$	0	0
$421$	−3.06255e14	−1.12858	−0.564289	−	0.825577i	$-0.690850\pi$
$421$	−3.06255e14	−1.12858	−0.564289	+	0.825577i	$0.690850\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.60643e14	1.26165
$426$	0	0
$427$	−1.72713e14	−0.588806
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.06437e14	−0.668593	−0.334297	−	0.942468i	$-0.608499\pi$
$431$	−2.06437e14	−0.668593	−0.334297	+	0.942468i	$0.608499\pi$
$432$	0	0
$433$	−5.11116e13	−0.161375	−0.0806875	−	0.996739i	$-0.525712\pi$
$433$	−5.11116e13	−0.161375	−0.0806875	+	0.996739i	$0.525712\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.40930e14	0.423014
$438$	0	0
$439$	−3.07547e14	−0.900235	−0.450118	−	0.892969i	$-0.648618\pi$
$439$	−3.07547e14	−0.900235	−0.450118	+	0.892969i	$0.648618\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.15446e14	−1.15689	−0.578447	−	0.815720i	$-0.696341\pi$
$443$	−4.15446e14	−1.15689	−0.578447	+	0.815720i	$0.696341\pi$
$444$	0	0
$445$	−6.04547e12	−0.0164229
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.19339e14	−1.60167	−0.800835	−	0.598885i	$-0.795611\pi$
$449$	−6.19339e14	−1.60167	−0.800835	+	0.598885i	$0.795611\pi$
$450$	0	0
$451$	−7.25214e12	−0.0183018
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.42264e13	−0.0582407
$456$	0	0
$457$	−5.13217e14	−1.20438	−0.602188	−	0.798354i	$-0.705704\pi$
$457$	−5.13217e14	−1.20438	−0.602188	+	0.798354i	$0.705704\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.46977e14	−0.328773	−0.164386	−	0.986396i	$-0.552564\pi$
$461$	−1.46977e14	−0.328773	−0.164386	+	0.986396i	$0.552564\pi$
$462$	0	0
$463$	6.63853e13	0.145003	0.0725014	−	0.997368i	$-0.476902\pi$
$463$	6.63853e13	0.145003	0.0725014	+	0.997368i	$0.476902\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.19055e14	−0.456364	−0.228182	−	0.973619i	$-0.573278\pi$
$467$	−2.19055e14	−0.456364	−0.228182	+	0.973619i	$0.573278\pi$
$468$	0	0
$469$	1.75819e14	0.357779
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.45357e14	−1.44753
$474$	0	0
$475$	9.10182e14	1.72708
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.99705e14	−1.26786	−0.633928	−	0.773392i	$-0.718558\pi$
$479$	−6.99705e14	−1.26786	−0.633928	+	0.773392i	$0.718558\pi$
$480$	0	0
$481$	1.21724e15	2.15565
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.11362e13	−0.0696056
$486$	0	0
$487$	1.76436e14	0.291861	0.145931	−	0.989295i	$-0.453382\pi$
$487$	1.76436e14	0.291861	0.145931	+	0.989295i	$0.453382\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.06991e14	−0.485487	−0.242744	−	0.970090i	$-0.578047\pi$
$491$	−3.06991e14	−0.485487	−0.242744	+	0.970090i	$0.578047\pi$
$492$	0	0
$493$	−1.40998e15	−2.18049
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.76293e13	0.0704553
$498$	0	0
$499$	3.09634e14	0.448018	0.224009	−	0.974587i	$-0.428086\pi$
$499$	3.09634e14	0.448018	0.224009	+	0.974587i	$0.428086\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.14385e15	1.58396	0.791981	−	0.610545i	$-0.209050\pi$
$503$	1.14385e15	1.58396	0.791981	+	0.610545i	$0.209050\pi$
$504$	0	0
$505$	2.81875e13	0.0381904
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.44622e14	0.317357	0.158678	−	0.987330i	$-0.449277\pi$
$509$	2.44622e14	0.317357	0.158678	+	0.987330i	$0.449277\pi$
$510$	0	0
$511$	2.39496e13	0.0304076
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.96151e11	0.000725132 0
$516$	0	0
$517$	−7.66539e14	−0.912718
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.34294e15	−1.53267	−0.766335	−	0.642441i	$-0.777922\pi$
$521$	−1.34294e15	−1.53267	−0.766335	+	0.642441i	$0.777922\pi$
$522$	0	0
$523$	−1.54133e15	−1.72241	−0.861205	−	0.508258i	$-0.830290\pi$
$523$	−1.54133e15	−1.72241	−0.861205	+	0.508258i	$0.830290\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.90134e14	0.953887
$528$	0	0
$529$	−8.96631e14	−0.941039
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.54468e13	−0.0256232
$534$	0	0
$535$	1.16701e14	0.115114
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.78871e14	−0.169356
$540$	0	0
$541$	−5.10304e14	−0.473417	−0.236709	−	0.971581i	$-0.576069\pi$
$541$	−5.10304e14	−0.473417	−0.236709	+	0.971581i	$0.576069\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.98590e13	0.0266008
$546$	0	0
$547$	1.07989e14	0.0942867	0.0471434	−	0.998888i	$-0.484988\pi$
$547$	1.07989e14	0.0942867	0.0471434	+	0.998888i	$0.484988\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.55846e15	−2.98489
$552$	0	0
$553$	4.60252e14	0.378449
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.62684e14	−0.365663	−0.182831	−	0.983144i	$-0.558526\pi$
$557$	−4.62684e14	−0.365663	−0.182831	+	0.983144i	$0.558526\pi$
$558$	0	0
$559$	−2.61536e15	−2.02659
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.22525e15	0.912909	0.456455	−	0.889747i	$-0.349119\pi$
$563$	1.22525e15	0.912909	0.456455	+	0.889747i	$0.349119\pi$
$564$	0	0
$565$	2.17149e14	0.158669
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.21282e15	−0.852473	−0.426236	−	0.904612i	$-0.640161\pi$
$569$	−1.21282e15	−0.852473	−0.426236	+	0.904612i	$0.640161\pi$
$570$	0	0
$571$	−1.44825e15	−0.998496	−0.499248	−	0.866459i	$-0.666390\pi$
$571$	−1.44825e15	−0.998496	−0.499248	+	0.866459i	$0.666390\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.62823e14	0.240725
$576$	0	0
$577$	−4.22166e14	−0.274800	−0.137400	−	0.990516i	$-0.543875\pi$
$577$	−4.22166e14	−0.274800	−0.137400	+	0.990516i	$0.543875\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.22599e14	−0.578155
$582$	0	0
$583$	2.80289e15	1.72357
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.46912e15	0.870054	0.435027	−	0.900417i	$-0.356739\pi$
$587$	1.46912e15	0.870054	0.435027	+	0.900417i	$0.356739\pi$
$588$	0	0
$589$	2.24650e15	1.30578
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.64017e13	−0.0371859	−0.0185930	−	0.999827i	$-0.505919\pi$
$593$	−6.64017e13	−0.0371859	−0.0185930	+	0.999827i	$0.505919\pi$
$594$	0	0
$595$	−8.12328e13	−0.0446569
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.80522e14	0.519529	0.259765	−	0.965672i	$-0.416355\pi$
$599$	9.80522e14	0.519529	0.259765	+	0.965672i	$0.416355\pi$
$600$	0	0
$601$	6.23880e14	0.324558	0.162279	−	0.986745i	$-0.448116\pi$
$601$	6.23880e14	0.324558	0.162279	+	0.986745i	$0.448116\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.50357e13	−0.0376369
$606$	0	0
$607$	−1.16794e15	−0.575282	−0.287641	−	0.957738i	$-0.592871\pi$
$607$	−1.16794e15	−0.575282	−0.287641	+	0.957738i	$0.592871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.68969e15	−1.27783
$612$	0	0
$613$	7.66521e14	0.357677	0.178839	−	0.983878i	$-0.442766\pi$
$613$	7.66521e14	0.357677	0.178839	+	0.983878i	$0.442766\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.00148e15	−0.901119	−0.450560	−	0.892746i	$-0.648776\pi$
$617$	−2.00148e15	−0.901119	−0.450560	+	0.892746i	$0.648776\pi$
$618$	0	0
$619$	2.56891e15	1.13619	0.568093	−	0.822964i	$-0.307681\pi$
$619$	2.56891e15	1.13619	0.568093	+	0.822964i	$0.307681\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.56620e14	−0.0668594
$624$	0	0
$625$	2.32271e15	0.974216
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.08150e15	1.65288
$630$	0	0
$631$	−5.10135e14	−0.203013	−0.101506	−	0.994835i	$-0.532366\pi$
$631$	−5.10135e14	−0.203013	−0.101506	+	0.994835i	$0.532366\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.38664e13	−0.0245480
$636$	0	0
$637$	−6.27633e14	−0.237104
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.87191e14	−0.360315	−0.180157	−	0.983638i	$-0.557661\pi$
$641$	−9.87191e14	−0.360315	−0.180157	+	0.983638i	$0.557661\pi$
$642$	0	0
$643$	−4.71517e15	−1.69175	−0.845877	−	0.533378i	$-0.820922\pi$
$643$	−4.71517e15	−1.69175	−0.845877	+	0.533378i	$0.820922\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.11889e15	0.734741	0.367371	−	0.930075i	$-0.380258\pi$
$647$	2.11889e15	0.734741	0.367371	+	0.930075i	$0.380258\pi$
$648$	0	0
$649$	4.17339e15	1.42280
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.25712e14	−0.107352	−0.0536761	−	0.998558i	$-0.517094\pi$
$653$	−3.25712e14	−0.107352	−0.0536761	+	0.998558i	$0.517094\pi$
$654$	0	0
$655$	−3.07522e14	−0.0996664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.48112e15	1.09106	0.545531	−	0.838090i	$-0.316328\pi$
$659$	3.48112e15	1.09106	0.545531	+	0.838090i	$0.316328\pi$
$660$	0	0
$661$	−5.39055e15	−1.66159	−0.830796	−	0.556576i	$-0.812115\pi$
$661$	−5.39055e15	−1.66159	−0.830796	+	0.556576i	$0.812115\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.05013e14	−0.0611312
$666$	0	0
$667$	−1.41850e15	−0.416042
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.50719e15	−1.84680
$672$	0	0
$673$	−2.04603e15	−0.571254	−0.285627	−	0.958341i	$-0.592202\pi$
$673$	−2.04603e15	−0.571254	−0.285627	+	0.958341i	$0.592202\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.76612e15	0.747537	0.373769	−	0.927522i	$-0.378065\pi$
$677$	2.76612e15	0.747537	0.373769	+	0.927522i	$0.378065\pi$
$678$	0	0
$679$	−1.06571e15	−0.283372
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.85070e15	0.991347	0.495674	−	0.868509i	$-0.334921\pi$
$683$	3.85070e15	0.991347	0.495674	+	0.868509i	$0.334921\pi$
$684$	0	0
$685$	−2.79180e14	−0.0707270
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.83497e15	2.41306
$690$	0	0
$691$	5.28063e15	1.27514	0.637568	−	0.770394i	$-0.279940\pi$
$691$	5.28063e15	1.27514	0.637568	+	0.770394i	$0.279940\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.48173e14	0.0580547
$696$	0	0
$697$	−8.53248e13	−0.0196469
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.85690e15	−1.52995	−0.764977	−	0.644057i	$-0.777250\pi$
$701$	−6.85690e15	−1.52995	−0.764977	+	0.644057i	$0.777250\pi$
$702$	0	0
$703$	1.03008e16	2.26264
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.30253e14	0.155477
$708$	0	0
$709$	−4.79517e15	−1.00519	−0.502597	−	0.864521i	$-0.667622\pi$
$709$	−4.79517e15	−1.00519	−0.502597	+	0.864521i	$0.667622\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.95514e14	0.182004
$714$	0	0
$715$	−9.12762e14	−0.182673
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.52615e15	1.26662	0.633312	−	0.773896i	$-0.281695\pi$
$719$	6.52615e15	1.26662	0.633312	+	0.773896i	$0.281695\pi$
$720$	0	0
$721$	1.54445e13	0.00295209
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.16125e15	−1.69862
$726$	0	0
$727$	1.39058e15	0.253956	0.126978	−	0.991906i	$-0.459472\pi$
$727$	1.39058e15	0.253956	0.126978	+	0.991906i	$0.459472\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.76947e15	−1.55392
$732$	0	0
$733$	−2.71766e15	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$733$	−2.71766e15	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.62420e15	1.12218
$738$	0	0
$739$	1.14128e16	1.90480	0.952399	−	0.304855i	$-0.0986079\pi$
$739$	1.14128e16	1.90480	0.952399	+	0.304855i	$0.0986079\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.85629e15	1.43487	0.717436	−	0.696625i	$-0.245316\pi$
$743$	8.85629e15	1.43487	0.717436	+	0.696625i	$0.245316\pi$
$744$	0	0
$745$	9.64540e14	0.153979
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.02338e15	0.468643
$750$	0	0
$751$	9.04535e15	1.38168	0.690838	−	0.723010i	$-0.257242\pi$
$751$	9.04535e15	1.38168	0.690838	+	0.723010i	$0.257242\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.71861e14	0.0403309
$756$	0	0
$757$	−2.86043e15	−0.418219	−0.209110	−	0.977892i	$-0.567057\pi$
$757$	−2.86043e15	−0.418219	−0.209110	+	0.977892i	$0.567057\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.00044e15	0.994283	0.497141	−	0.867670i	$-0.334383\pi$
$761$	7.00044e15	0.994283	0.497141	+	0.867670i	$0.334383\pi$
$762$	0	0
$763$	7.73555e14	0.108294
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.46439e16	1.99196
$768$	0	0
$769$	1.42620e16	1.91243	0.956214	−	0.292668i	$-0.0945432\pi$
$769$	1.42620e16	1.91243	0.956214	+	0.292668i	$0.0945432\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.33560e15	−0.434697	−0.217349	−	0.976094i	$-0.569741\pi$
$773$	−3.33560e15	−0.434697	−0.217349	+	0.976094i	$0.569741\pi$
$774$	0	0
$775$	5.78360e15	0.743085
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.15340e14	−0.0268949
$780$	0	0
$781$	1.79449e15	0.220984
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.17075e14	0.0499380
$786$	0	0
$787$	−6.05403e15	−0.714798	−0.357399	−	0.933952i	$-0.616336\pi$
$787$	−6.05403e15	−0.714798	−0.357399	+	0.933952i	$0.616336\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.62567e15	0.645957
$792$	0	0
$793$	−2.28329e16	−2.58558
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.60408e14	−0.0947728	−0.0473864	−	0.998877i	$-0.515089\pi$
$797$	−8.60408e14	−0.0947728	−0.0473864	+	0.998877i	$0.515089\pi$
$798$	0	0
$799$	−9.01869e15	−0.979797
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.02331e14	0.0953741
$804$	0	0
$805$	−8.17238e13	−0.00852062
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.50228e16	−1.52417	−0.762085	−	0.647478i	$-0.775824\pi$
$809$	−1.50228e16	−1.52417	−0.762085	+	0.647478i	$0.775824\pi$
$810$	0	0
$811$	3.30697e15	0.330990	0.165495	−	0.986211i	$-0.447078\pi$
$811$	3.30697e15	0.330990	0.165495	+	0.986211i	$0.447078\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.42640e14	0.0528621
$816$	0	0
$817$	−2.21321e16	−2.12717
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.05488e15	0.566524	0.283262	−	0.959043i	$-0.408583\pi$
$821$	6.05488e15	0.566524	0.283262	+	0.959043i	$0.408583\pi$
$822$	0	0
$823$	2.21260e15	0.204270	0.102135	−	0.994771i	$-0.467433\pi$
$823$	2.21260e15	0.204270	0.102135	+	0.994771i	$0.467433\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.78449e16	1.60411	0.802053	−	0.597253i	$-0.203741\pi$
$827$	1.78449e16	1.60411	0.802053	+	0.597253i	$0.203741\pi$
$828$	0	0
$829$	−1.02005e16	−0.904837	−0.452418	−	0.891806i	$-0.649439\pi$
$829$	−1.02005e16	−0.904837	−0.452418	+	0.891806i	$0.649439\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.10450e15	−0.181803
$834$	0	0
$835$	−1.19376e15	−0.101775
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.48904e15	0.538877	0.269439	−	0.963018i	$-0.413162\pi$
$839$	6.48904e15	0.538877	0.269439	+	0.963018i	$0.413162\pi$
$840$	0	0
$841$	2.36164e16	1.93569
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.04011e15	−0.162908
$846$	0	0
$847$	−1.94395e15	−0.153224
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.10616e15	0.315372
$852$	0	0
$853$	2.48935e16	1.88741	0.943703	−	0.330793i	$-0.107316\pi$
$853$	2.48935e16	1.88741	0.943703	+	0.330793i	$0.107316\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.46863e16	1.08522	0.542611	−	0.839984i	$-0.317436\pi$
$857$	1.46863e16	1.08522	0.542611	+	0.839984i	$0.317436\pi$
$858$	0	0
$859$	−1.40018e16	−1.02146	−0.510729	−	0.859742i	$-0.670624\pi$
$859$	−1.40018e16	−1.02146	−0.510729	+	0.859742i	$0.670624\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.22495e15	0.300443	0.150222	−	0.988652i	$-0.452001\pi$
$863$	4.22495e15	0.300443	0.150222	+	0.988652i	$0.452001\pi$
$864$	0	0
$865$	7.87241e14	0.0552738
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.73406e16	1.18701
$870$	0	0
$871$	2.32434e16	1.57109
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.06020e15	−0.0698786
$876$	0	0
$877$	2.56310e16	1.66827	0.834136	−	0.551558i	$-0.185966\pi$
$877$	2.56310e16	1.66827	0.834136	+	0.551558i	$0.185966\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.10805e16	−1.33818	−0.669088	−	0.743183i	$-0.733315\pi$
$881$	−2.10805e16	−1.33818	−0.669088	+	0.743183i	$0.733315\pi$
$882$	0	0
$883$	−4.44887e15	−0.278911	−0.139456	−	0.990228i	$-0.544535\pi$
$883$	−4.44887e15	−0.278911	−0.139456	+	0.990228i	$0.544535\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.63981e16	−1.00280	−0.501399	−	0.865216i	$-0.667181\pi$
$887$	−1.63981e16	−1.00280	−0.501399	+	0.865216i	$0.667181\pi$
$888$	0	0
$889$	−1.65458e15	−0.0999376
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.27611e16	−1.34125
$894$	0	0
$895$	−2.51408e15	−0.146336
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.26116e16	−1.28426
$900$	0	0
$901$	3.29773e16	1.85024
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.25192e15	0.123307
$906$	0	0
$907$	−1.97109e16	−1.06627	−0.533133	−	0.846031i	$-0.678986\pi$
$907$	−1.97109e16	−1.06627	−0.533133	+	0.846031i	$0.678986\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.78243e15	0.252521	0.126261	−	0.991997i	$-0.459702\pi$
$911$	4.78243e15	0.252521	0.126261	+	0.991997i	$0.459702\pi$
$912$	0	0
$913$	−3.47601e16	−1.81339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.96696e15	−0.405753
$918$	0	0
$919$	−1.88714e16	−0.949662	−0.474831	−	0.880077i	$-0.657491\pi$
$919$	−1.88714e16	−0.949662	−0.474831	+	0.880077i	$0.657491\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.29665e15	0.309385
$924$	0	0
$925$	2.65193e16	1.28760
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.58600e15	−0.359688	−0.179844	−	0.983695i	$-0.557559\pi$
$929$	−7.58600e15	−0.359688	−0.179844	+	0.983695i	$0.557559\pi$
$930$	0	0
$931$	−5.31127e15	−0.248871
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.06055e15	−0.140067
$936$	0	0
$937$	−4.15527e16	−1.87945	−0.939727	−	0.341927i	$-0.888921\pi$
$937$	−4.15527e16	−1.87945	−0.939727	+	0.341927i	$0.888921\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.72517e16	−0.762233	−0.381117	−	0.924527i	$-0.624460\pi$
$941$	−1.72517e16	−0.762233	−0.381117	+	0.924527i	$0.624460\pi$
$942$	0	0
$943$	−8.58405e13	−0.00374868
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.73554e15	−0.244709	−0.122354	−	0.992486i	$-0.539044\pi$
$947$	−5.73554e15	−0.244709	−0.122354	+	0.992486i	$0.539044\pi$
$948$	0	0
$949$	3.16616e15	0.133527
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.84589e16	−1.58484	−0.792422	−	0.609973i	$-0.791180\pi$
$953$	−3.84589e16	−1.58484	−0.792422	+	0.609973i	$0.791180\pi$
$954$	0	0
$955$	4.25324e15	0.173261
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.23270e15	−0.287937
$960$	0	0
$961$	−1.11335e16	−0.438181
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.43993e15	−0.0938593
$966$	0	0
$967$	−2.89168e16	−1.09978	−0.549889	−	0.835238i	$-0.685330\pi$
$967$	−2.89168e16	−1.09978	−0.549889	+	0.835238i	$0.685330\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.12803e16	1.53475	0.767373	−	0.641201i	$-0.221563\pi$
$971$	4.12803e16	1.53475	0.767373	+	0.641201i	$0.221563\pi$
$972$	0	0
$973$	6.42940e15	0.236347
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.41113e16	0.866563	0.433281	−	0.901259i	$-0.357356\pi$
$977$	2.41113e16	0.866563	0.433281	+	0.901259i	$0.357356\pi$
$978$	0	0
$979$	−5.90085e15	−0.209706
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.77350e15	0.200630	0.100315	−	0.994956i	$-0.468015\pi$
$983$	5.77350e15	0.200630	0.100315	+	0.994956i	$0.468015\pi$
$984$	0	0
$985$	−1.77844e15	−0.0611137
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.82247e15	−0.296490
$990$	0	0
$991$	1.79872e16	0.597805	0.298902	−	0.954284i	$-0.403380\pi$
$991$	1.79872e16	0.597805	0.298902	+	0.954284i	$0.403380\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.22574e15	0.104858
$996$	0	0
$997$	−2.98101e16	−0.958387	−0.479193	−	0.877709i	$-0.659071\pi$
$997$	−2.98101e16	−0.958387	−0.479193	+	0.877709i	$0.659071\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.f.1.2		3
3.2	odd	2		28.12.a.a.1.2	✓	3
12.11	even	2		112.12.a.g.1.2		3
21.2	odd	6		196.12.e.e.165.2		6
21.5	even	6		196.12.e.d.165.2		6
21.11	odd	6		196.12.e.e.177.2		6
21.17	even	6		196.12.e.d.177.2		6
21.20	even	2		196.12.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.12.a.a.1.2	✓	3	3.2	odd	2
112.12.a.g.1.2		3	12.11	even	2
196.12.a.c.1.2		3	21.20	even	2
196.12.e.d.165.2		6	21.5	even	6
196.12.e.d.177.2		6	21.17	even	6
196.12.e.e.165.2		6	21.2	odd	6
196.12.e.e.177.2		6	21.11	odd	6
252.12.a.f.1.2		3	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q-648.744 q^{5} -16807.0 q^{7} +O(q^{10})\)	\(q-648.744 q^{5} -16807.0 q^{7} -633226. q^{11} -2.22191e6 q^{13} -7.45019e6 q^{17} -1.88026e7 q^{19} -7.49522e6 q^{23} -4.84073e7 q^{25} +1.89254e8 q^{29} -1.19478e8 q^{31} +1.09034e7 q^{35} -5.47837e8 q^{37} +1.14527e7 q^{41} +1.17708e9 q^{43} +1.21053e9 q^{47} +2.82475e8 q^{49} -4.42637e9 q^{53} +4.10802e8 q^{55} -6.59069e9 q^{59} +1.02763e10 q^{61} +1.44145e9 q^{65} -1.04610e10 q^{67} -2.83389e9 q^{71} -1.42498e9 q^{73} +1.06426e10 q^{77} -2.73845e10 q^{79} +5.48937e10 q^{83} +4.83327e9 q^{85} +9.31872e9 q^{89} +3.73436e10 q^{91} +1.21981e10 q^{95} +6.34089e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) \) 3 * q - 4762 * q^5 - 50421 * q^7	\( 3 q - 4762 q^{5} - 50421 q^{7} - 80468 q^{11} + 996546 q^{13} + 7924090 q^{17} - 15261732 q^{19} + 61428352 q^{23} - 109205331 q^{25} + 160359182 q^{29} - 342148464 q^{31} + 80034934 q^{35} - 722039190 q^{37} - 285288606 q^{41} - 476523612 q^{43} - 1225105680 q^{47} + 847425747 q^{49} - 4030571514 q^{53} + 4822100040 q^{55} - 5222175892 q^{59} + 5684837106 q^{61} - 8153014116 q^{65} + 2837154348 q^{67} + 16928678200 q^{71} - 14560325442 q^{73} + 1352425676 q^{77} - 32832340032 q^{79} + 115451875348 q^{83} - 36782181276 q^{85} + 15661530882 q^{89} - 16748948622 q^{91} + 117828203728 q^{95} + 40693412742 q^{97}+O(q^{100}) \) 3 * q - 4762 * q^5 - 50421 * q^7 - 80468 * q^11 + 996546 * q^13 + 7924090 * q^17 - 15261732 * q^19 + 61428352 * q^23 - 109205331 * q^25 + 160359182 * q^29 - 342148464 * q^31 + 80034934 * q^35 - 722039190 * q^37 - 285288606 * q^41 - 476523612 * q^43 - 1225105680 * q^47 + 847425747 * q^49 - 4030571514 * q^53 + 4822100040 * q^55 - 5222175892 * q^59 + 5684837106 * q^61 - 8153014116 * q^65 + 2837154348 * q^67 + 16928678200 * q^71 - 14560325442 * q^73 + 1352425676 * q^77 - 32832340032 * q^79 + 115451875348 * q^83 - 36782181276 * q^85 + 15661530882 * q^89 - 16748948622 * q^91 + 117828203728 * q^95 + 40693412742 * q^97

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