LMFDB - Embedded newform 252.12.a.f.1.1

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			1.1
Root			$20.4801$ 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)

$f(q)$	$=$	$q-5824.83 q^{5} -16807.0 q^{7} +O(q^{10})$	$q-5824.83 q^{5} -16807.0 q^{7} -459796. q^{11} +2.00402e6 q^{13} +9.01354e6 q^{17} -1.32118e7 q^{19} +3.15740e7 q^{23} -1.48994e7 q^{25} -9.34680e7 q^{29} -1.75565e8 q^{31} +9.78980e7 q^{35} -1.50741e8 q^{37} -7.88902e8 q^{41} -7.82932e8 q^{43} -3.20342e7 q^{47} +2.82475e8 q^{49} +4.42177e8 q^{53} +2.67823e9 q^{55} +7.07526e9 q^{59} -1.12030e10 q^{61} -1.16731e10 q^{65} -2.23264e8 q^{67} -4.59939e9 q^{71} -2.90802e10 q^{73} +7.72779e9 q^{77} +4.04457e10 q^{79} +4.93939e10 q^{83} -5.25024e10 q^{85} +7.09317e10 q^{89} -3.36815e10 q^{91} +7.69566e10 q^{95} +1.74774e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−5824.83	−0.833582	−0.416791	−	0.909002i	$-0.636845\pi$
$5$	−5824.83	−0.833582	−0.416791	+	0.909002i	$0.636845\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−459796.	−0.860806	−0.430403	−	0.902637i	$-0.641629\pi$
$11$	−459796.	−0.860806	−0.430403	+	0.902637i	$0.641629\pi$
$12$	0	0
$13$	2.00402e6	1.49697	0.748485	−	0.663152i	$-0.230782\pi$
$13$	2.00402e6	1.49697	0.748485	+	0.663152i	$0.230782\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.01354e6	1.53967	0.769833	−	0.638246i	$-0.220340\pi$
$17$	9.01354e6	1.53967	0.769833	+	0.638246i	$0.220340\pi$
$18$	0	0
$19$	−1.32118e7	−1.22410	−0.612051	−	0.790818i	$-0.709655\pi$
$19$	−1.32118e7	−1.22410	−0.612051	+	0.790818i	$0.709655\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.15740e7	1.02288	0.511442	−	0.859318i	$-0.329112\pi$
$23$	3.15740e7	1.02288	0.511442	+	0.859318i	$0.329112\pi$
$24$	0	0
$25$	−1.48994e7	−0.305141
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.34680e7	−0.846201	−0.423101	−	0.906083i	$-0.639058\pi$
$29$	−9.34680e7	−0.846201	−0.423101	+	0.906083i	$0.639058\pi$
$30$	0	0
$31$	−1.75565e8	−1.10141	−0.550705	−	0.834700i	$-0.685641\pi$
$31$	−1.75565e8	−1.10141	−0.550705	+	0.834700i	$0.685641\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.78980e7	0.315064
$36$	0	0
$37$	−1.50741e8	−0.357374	−0.178687	−	0.983906i	$-0.557185\pi$
$37$	−1.50741e8	−0.357374	−0.178687	+	0.983906i	$0.557185\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.88902e8	−1.06344	−0.531718	−	0.846921i	$-0.678454\pi$
$41$	−7.88902e8	−1.06344	−0.531718	+	0.846921i	$0.678454\pi$
$42$	0	0
$43$	−7.82932e8	−0.812171	−0.406086	−	0.913835i	$-0.633107\pi$
$43$	−7.82932e8	−0.812171	−0.406086	+	0.913835i	$0.633107\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.20342e7	−0.0203740	−0.0101870	−	0.999948i	$-0.503243\pi$
$47$	−3.20342e7	−0.0203740	−0.0101870	+	0.999948i	$0.503243\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.42177e8	0.145238	0.0726188	−	0.997360i	$-0.476864\pi$
$53$	4.42177e8	0.145238	0.0726188	+	0.997360i	$0.476864\pi$
$54$	0	0
$55$	2.67823e9	0.717553
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.07526e9	1.28842	0.644208	−	0.764850i	$-0.277187\pi$
$59$	7.07526e9	1.28842	0.644208	+	0.764850i	$0.277187\pi$
$60$	0	0
$61$	−1.12030e10	−1.69832	−0.849162	−	0.528132i	$-0.822893\pi$
$61$	−1.12030e10	−1.69832	−0.849162	+	0.528132i	$0.822893\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.16731e10	−1.24785
$66$	0	0
$67$	−2.23264e8	−0.0202026	−0.0101013	−	0.999949i	$-0.503215\pi$
$67$	−2.23264e8	−0.0202026	−0.0101013	+	0.999949i	$0.503215\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.59939e9	−0.302538	−0.151269	−	0.988493i	$-0.548336\pi$
$71$	−4.59939e9	−0.302538	−0.151269	+	0.988493i	$0.548336\pi$
$72$	0	0
$73$	−2.90802e10	−1.64180	−0.820902	−	0.571069i	$-0.806529\pi$
$73$	−2.90802e10	−1.64180	−0.820902	+	0.571069i	$0.806529\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.72779e9	0.325354
$78$	0	0
$79$	4.04457e10	1.47885	0.739424	−	0.673240i	$-0.235098\pi$
$79$	4.04457e10	1.47885	0.739424	+	0.673240i	$0.235098\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.93939e10	1.37640	0.688199	−	0.725522i	$-0.258402\pi$
$83$	4.93939e10	1.37640	0.688199	+	0.725522i	$0.258402\pi$
$84$	0	0
$85$	−5.25024e10	−1.28344
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.09317e10	1.34647	0.673233	−	0.739430i	$-0.264905\pi$
$89$	7.09317e10	1.34647	0.673233	+	0.739430i	$0.264905\pi$
$90$	0	0
$91$	−3.36815e10	−0.565801
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.69566e10	1.02039
$96$	0	0
$97$	1.74774e10	0.206648	0.103324	−	0.994648i	$-0.467052\pi$
$97$	1.74774e10	0.206648	0.103324	+	0.994648i	$0.467052\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.65108e11	−1.56315	−0.781577	−	0.623809i	$-0.785584\pi$
$101$	−1.65108e11	−1.56315	−0.781577	+	0.623809i	$0.785584\pi$
$102$	0	0
$103$	−1.21775e10	−0.103503	−0.0517517	−	0.998660i	$-0.516480\pi$
$103$	−1.21775e10	−0.103503	−0.0517517	+	0.998660i	$0.516480\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.27048e11	1.56497	0.782486	−	0.622669i	$-0.213952\pi$
$107$	2.27048e11	1.56497	0.782486	+	0.622669i	$0.213952\pi$
$108$	0	0
$109$	2.51327e11	1.56457	0.782284	−	0.622922i	$-0.214055\pi$
$109$	2.51327e11	1.56457	0.782284	+	0.622922i	$0.214055\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.02066e11	−0.521132	−0.260566	−	0.965456i	$-0.583909\pi$
$113$	−1.02066e11	−0.521132	−0.260566	+	0.965456i	$0.583909\pi$
$114$	0	0
$115$	−1.83913e11	−0.852657
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.51491e11	−0.581939
$120$	0	0
$121$	−7.38993e10	−0.259013
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.71202e11	1.08794
$126$	0	0
$127$	−3.65160e10	−0.0980759	−0.0490380	−	0.998797i	$-0.515616\pi$
$127$	−3.65160e10	−0.0980759	−0.0490380	+	0.998797i	$0.515616\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.37130e11	0.310556	0.155278	−	0.987871i	$-0.450373\pi$
$131$	1.37130e11	0.310556	0.155278	+	0.987871i	$0.450373\pi$
$132$	0	0
$133$	2.22051e11	0.462667
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.94373e11	−0.521116	−0.260558	−	0.965458i	$-0.583906\pi$
$137$	−2.94373e11	−0.521116	−0.260558	+	0.965458i	$0.583906\pi$
$138$	0	0
$139$	−9.32207e11	−1.52381	−0.761906	−	0.647688i	$-0.775736\pi$
$139$	−9.32207e11	−1.52381	−0.761906	+	0.647688i	$0.775736\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.21439e11	−1.28860
$144$	0	0
$145$	5.44435e11	0.705379
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.25841e12	1.40377	0.701886	−	0.712289i	$-0.252342\pi$
$149$	1.25841e12	1.40377	0.701886	+	0.712289i	$0.252342\pi$
$150$	0	0
$151$	4.77048e11	0.494525	0.247263	−	0.968948i	$-0.420469\pi$
$151$	4.77048e11	0.494525	0.247263	+	0.968948i	$0.420469\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.02264e12	0.918116
$156$	0	0
$157$	−1.45655e12	−1.21864	−0.609321	−	0.792924i	$-0.708558\pi$
$157$	−1.45655e12	−1.21864	−0.609321	+	0.792924i	$0.708558\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.30664e11	−0.386613
$162$	0	0
$163$	2.33506e12	1.58952	0.794762	−	0.606922i	$-0.207596\pi$
$163$	2.33506e12	1.58952	0.794762	+	0.606922i	$0.207596\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.34128e12	0.799057	0.399528	−	0.916721i	$-0.369174\pi$
$167$	1.34128e12	0.799057	0.399528	+	0.916721i	$0.369174\pi$
$168$	0	0
$169$	2.22392e12	1.24092
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.16609e12	−1.55335	−0.776676	−	0.629900i	$-0.783096\pi$
$173$	−3.16609e12	−1.55335	−0.776676	+	0.629900i	$0.783096\pi$
$174$	0	0
$175$	2.50415e11	0.115332
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.69433e11	−0.190934	−0.0954668	−	0.995433i	$-0.530434\pi$
$179$	−4.69433e11	−0.190934	−0.0954668	+	0.995433i	$0.530434\pi$
$180$	0	0
$181$	3.44383e12	1.31768	0.658840	−	0.752283i	$-0.271047\pi$
$181$	3.44383e12	1.31768	0.658840	+	0.752283i	$0.271047\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.78043e11	0.297901
$186$	0	0
$187$	−4.14439e12	−1.32535
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.17817e12	−1.47398	−0.736992	−	0.675901i	$-0.763755\pi$
$191$	−5.17817e12	−1.47398	−0.736992	+	0.675901i	$0.763755\pi$
$192$	0	0
$193$	−3.62581e12	−0.974630	−0.487315	−	0.873226i	$-0.662024\pi$
$193$	−3.62581e12	−0.974630	−0.487315	+	0.873226i	$0.662024\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.54002e12	0.850044	0.425022	−	0.905183i	$-0.360266\pi$
$197$	3.54002e12	0.850044	0.425022	+	0.905183i	$0.360266\pi$
$198$	0	0
$199$	2.18578e12	0.496494	0.248247	−	0.968697i	$-0.420146\pi$
$199$	2.18578e12	0.496494	0.248247	+	0.968697i	$0.420146\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.57092e12	0.319834
$204$	0	0
$205$	4.59522e12	0.886462
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.07474e12	1.05371
$210$	0	0
$211$	9.81277e12	1.61524	0.807622	−	0.589700i	$-0.200754\pi$
$211$	9.81277e12	1.61524	0.807622	+	0.589700i	$0.200754\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.56045e12	0.677011
$216$	0	0
$217$	2.95073e12	0.416294
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.80633e13	2.30483
$222$	0	0
$223$	4.39483e10	0.00533661	0.00266831	−	0.999996i	$-0.499151\pi$
$223$	4.39483e10	0.00533661	0.00266831	+	0.999996i	$0.499151\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.18709e13	1.30720	0.653598	−	0.756842i	$-0.273259\pi$
$227$	1.18709e13	1.30720	0.653598	+	0.756842i	$0.273259\pi$
$228$	0	0
$229$	1.14287e13	1.19922	0.599612	−	0.800291i	$-0.295322\pi$
$229$	1.14287e13	1.19922	0.599612	+	0.800291i	$0.295322\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.85518e12	0.844773	0.422386	−	0.906416i	$-0.361193\pi$
$233$	8.85518e12	0.844773	0.422386	+	0.906416i	$0.361193\pi$
$234$	0	0
$235$	1.86594e11	0.0169834
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.50794e13	−1.25082	−0.625411	−	0.780295i	$-0.715069\pi$
$239$	−1.50794e13	−1.25082	−0.625411	+	0.780295i	$0.715069\pi$
$240$	0	0
$241$	−1.33435e13	−1.05725	−0.528624	−	0.848856i	$-0.677292\pi$
$241$	−1.33435e13	−1.05725	−0.528624	+	0.848856i	$0.677292\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.64537e12	−0.119083
$246$	0	0
$247$	−2.64767e13	−1.83244
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.52359e13	1.59887	0.799434	−	0.600754i	$-0.205133\pi$
$251$	2.52359e13	1.59887	0.799434	+	0.600754i	$0.205133\pi$
$252$	0	0
$253$	−1.45176e13	−0.880504
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.83412e12	0.491509	0.245754	−	0.969332i	$-0.420964\pi$
$257$	8.83412e12	0.491509	0.245754	+	0.969332i	$0.420964\pi$
$258$	0	0
$259$	2.53351e12	0.135075
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.79742e12	−0.333110	−0.166555	−	0.986032i	$-0.553264\pi$
$263$	−6.79742e12	−0.333110	−0.166555	+	0.986032i	$0.553264\pi$
$264$	0	0
$265$	−2.57561e12	−0.121067
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.64129e12	0.287485	0.143742	−	0.989615i	$-0.454086\pi$
$269$	6.64129e12	0.287485	0.143742	+	0.989615i	$0.454086\pi$
$270$	0	0
$271$	3.76214e13	1.56352	0.781761	−	0.623578i	$-0.214322\pi$
$271$	3.76214e13	1.56352	0.781761	+	0.623578i	$0.214322\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.85071e12	0.262667
$276$	0	0
$277$	−1.38040e13	−0.508586	−0.254293	−	0.967127i	$-0.581843\pi$
$277$	−1.38040e13	−0.508586	−0.254293	+	0.967127i	$0.581843\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.04373e13	1.03639	0.518193	−	0.855264i	$-0.326605\pi$
$281$	3.04373e13	1.03639	0.518193	+	0.855264i	$0.326605\pi$
$282$	0	0
$283$	4.35779e13	1.42706	0.713528	−	0.700627i	$-0.247096\pi$
$283$	4.35779e13	1.42706	0.713528	+	0.700627i	$0.247096\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.32591e13	0.401941
$288$	0	0
$289$	4.69720e13	1.37057
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.37023e12	−0.172339	−0.0861694	−	0.996280i	$-0.527463\pi$
$293$	−6.37023e12	−0.172339	−0.0861694	+	0.996280i	$0.527463\pi$
$294$	0	0
$295$	−4.12122e13	−1.07400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.32747e13	1.53122
$300$	0	0
$301$	1.31587e13	0.306972
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.52557e13	1.41569
$306$	0	0
$307$	−2.35323e13	−0.492497	−0.246248	−	0.969207i	$-0.579198\pi$
$307$	−2.35323e13	−0.492497	−0.246248	+	0.969207i	$0.579198\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.06147e13	0.596689	0.298344	−	0.954458i	$-0.403566\pi$
$311$	3.06147e13	0.596689	0.298344	+	0.954458i	$0.403566\pi$
$312$	0	0
$313$	2.38039e13	0.447873	0.223936	−	0.974604i	$-0.428109\pi$
$313$	2.38039e13	0.447873	0.223936	+	0.974604i	$0.428109\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.24265e13	0.218033	0.109017	−	0.994040i	$-0.465230\pi$
$317$	1.24265e13	0.218033	0.109017	+	0.994040i	$0.465230\pi$
$318$	0	0
$319$	4.29762e13	0.728415
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.19085e14	−1.88471
$324$	0	0
$325$	−2.98587e13	−0.456786
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.38399e11	0.00770064
$330$	0	0
$331$	1.12449e14	1.55561	0.777805	−	0.628505i	$-0.216333\pi$
$331$	1.12449e14	1.55561	0.777805	+	0.628505i	$0.216333\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.30048e12	0.0168406
$336$	0	0
$337$	−1.64768e13	−0.206494	−0.103247	−	0.994656i	$-0.532923\pi$
$337$	−1.64768e13	−0.206494	−0.103247	+	0.994656i	$0.532923\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.07242e13	0.948101
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.43388e12	0.0153004	0.00765019	−	0.999971i	$-0.497565\pi$
$347$	1.43388e12	0.0153004	0.00765019	+	0.999971i	$0.497565\pi$
$348$	0	0
$349$	1.15151e14	1.19049	0.595247	−	0.803543i	$-0.297054\pi$
$349$	1.15151e14	1.19049	0.595247	+	0.803543i	$0.297054\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.42785e13	0.235755	0.117877	−	0.993028i	$-0.462391\pi$
$353$	2.42785e13	0.235755	0.117877	+	0.993028i	$0.462391\pi$
$354$	0	0
$355$	2.67907e13	0.252190
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.92949e12	−0.0259282	−0.0129641	−	0.999916i	$-0.504127\pi$
$359$	−2.92949e12	−0.0259282	−0.0129641	+	0.999916i	$0.504127\pi$
$360$	0	0
$361$	5.80618e13	0.498426
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.69387e14	1.36858
$366$	0	0
$367$	1.24306e14	0.974604	0.487302	−	0.873233i	$-0.337981\pi$
$367$	1.24306e14	0.974604	0.487302	+	0.873233i	$0.337981\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.43167e12	−0.0548946
$372$	0	0
$373$	1.00386e13	0.0719905	0.0359953	−	0.999352i	$-0.488540\pi$
$373$	1.00386e13	0.0719905	0.0359953	+	0.999352i	$0.488540\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.87311e14	−1.26674
$378$	0	0
$379$	−3.15901e13	−0.207508	−0.103754	−	0.994603i	$-0.533086\pi$
$379$	−3.15901e13	−0.207508	−0.103754	+	0.994603i	$0.533086\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.49442e13	0.0926572	0.0463286	−	0.998926i	$-0.485248\pi$
$383$	1.49442e13	0.0926572	0.0463286	+	0.998926i	$0.485248\pi$
$384$	0	0
$385$	−4.50131e13	−0.271209
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.26360e14	−0.719261	−0.359630	−	0.933095i	$-0.617097\pi$
$389$	−1.26360e14	−0.719261	−0.359630	+	0.933095i	$0.617097\pi$
$390$	0	0
$391$	2.84593e14	1.57490
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.35590e14	−1.23274
$396$	0	0
$397$	−2.56150e14	−1.30360	−0.651802	−	0.758389i	$-0.725987\pi$
$397$	−2.56150e14	−1.30360	−0.651802	+	0.758389i	$0.725987\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.25795e14	1.56910	0.784548	−	0.620068i	$-0.212895\pi$
$401$	3.25795e14	1.56910	0.784548	+	0.620068i	$0.212895\pi$
$402$	0	0
$403$	−3.51836e14	−1.64878
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.93103e13	0.307630
$408$	0	0
$409$	−2.55484e14	−1.10379	−0.551894	−	0.833914i	$-0.686095\pi$
$409$	−2.55484e14	−1.10379	−0.551894	+	0.833914i	$0.686095\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.18914e14	−0.486976
$414$	0	0
$415$	−2.87711e14	−1.14734
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.30810e14	1.25141	0.625707	−	0.780058i	$-0.284811\pi$
$419$	3.30810e14	1.25141	0.625707	+	0.780058i	$0.284811\pi$
$420$	0	0
$421$	−4.00227e14	−1.47487	−0.737436	−	0.675417i	$-0.763964\pi$
$421$	−4.00227e14	−1.47487	−0.737436	+	0.675417i	$0.763964\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.34297e14	−0.469814
$426$	0	0
$427$	1.88289e14	0.641906
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.17730e14	−0.381296	−0.190648	−	0.981658i	$-0.561059\pi$
$431$	−1.17730e14	−0.381296	−0.190648	+	0.981658i	$0.561059\pi$
$432$	0	0
$433$	2.90178e14	0.916180	0.458090	−	0.888906i	$-0.348534\pi$
$433$	2.90178e14	0.916180	0.458090	+	0.888906i	$0.348534\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.17149e14	−1.25211
$438$	0	0
$439$	2.67036e14	0.781655	0.390827	−	0.920464i	$-0.372189\pi$
$439$	2.67036e14	0.781655	0.390827	+	0.920464i	$0.372189\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.00010e14	0.835440	0.417720	−	0.908576i	$-0.362829\pi$
$443$	3.00010e14	0.835440	0.417720	+	0.908576i	$0.362829\pi$
$444$	0	0
$445$	−4.13165e14	−1.12239
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.05247e14	−0.530790	−0.265395	−	0.964140i	$-0.585502\pi$
$449$	−2.05247e14	−0.530790	−0.265395	+	0.964140i	$0.585502\pi$
$450$	0	0
$451$	3.62734e14	0.915413
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.96189e14	0.471642
$456$	0	0
$457$	−3.45801e14	−0.811496	−0.405748	−	0.913985i	$-0.632989\pi$
$457$	−3.45801e14	−0.811496	−0.405748	+	0.913985i	$0.632989\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.13235e13	0.0700673	0.0350337	−	0.999386i	$-0.488846\pi$
$461$	3.13235e13	0.0700673	0.0350337	+	0.999386i	$0.488846\pi$
$462$	0	0
$463$	−1.26760e14	−0.276876	−0.138438	−	0.990371i	$-0.544208\pi$
$463$	−1.26760e14	−0.276876	−0.138438	+	0.990371i	$0.544208\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.96544e14	−0.826129	−0.413065	−	0.910702i	$-0.635542\pi$
$467$	−3.96544e14	−0.826129	−0.413065	+	0.910702i	$0.635542\pi$
$468$	0	0
$469$	3.75241e12	0.00763588
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.59989e14	0.699122
$474$	0	0
$475$	1.96849e14	0.373523
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.68400e13	0.139233	0.0696165	−	0.997574i	$-0.477822\pi$
$479$	7.68400e13	0.139233	0.0696165	+	0.997574i	$0.477822\pi$
$480$	0	0
$481$	−3.02088e14	−0.534978
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.01803e14	−0.172258
$486$	0	0
$487$	−1.53948e14	−0.254663	−0.127332	−	0.991860i	$-0.540641\pi$
$487$	−1.53948e14	−0.254663	−0.127332	+	0.991860i	$0.540641\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.85292e14	−1.40003	−0.700016	−	0.714127i	$-0.746824\pi$
$491$	−8.85292e14	−1.40003	−0.700016	+	0.714127i	$0.746824\pi$
$492$	0	0
$493$	−8.42477e14	−1.30287
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.73019e13	0.114348
$498$	0	0
$499$	6.33667e14	0.916870	0.458435	−	0.888728i	$-0.348410\pi$
$499$	6.33667e14	0.916870	0.458435	+	0.888728i	$0.348410\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.26921e15	−1.75756	−0.878782	−	0.477224i	$-0.841643\pi$
$503$	−1.26921e15	−1.75756	−0.878782	+	0.477224i	$0.841643\pi$
$504$	0	0
$505$	9.61729e14	1.30302
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.97092e14	1.29356	0.646781	−	0.762676i	$-0.276115\pi$
$509$	9.97092e14	1.29356	0.646781	+	0.762676i	$0.276115\pi$
$510$	0	0
$511$	4.88751e14	0.620544
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.09320e13	0.0862785
$516$	0	0
$517$	1.47292e13	0.0175380
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.14464e14	−0.587148	−0.293574	−	0.955936i	$-0.594845\pi$
$521$	−5.14464e14	−0.587148	−0.293574	+	0.955936i	$0.594845\pi$
$522$	0	0
$523$	5.55783e13	0.0621078	0.0310539	−	0.999518i	$-0.490114\pi$
$523$	5.55783e13	0.0621078	0.0310539	+	0.999518i	$0.490114\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.58246e15	−1.69580
$528$	0	0
$529$	4.41053e13	0.0462897
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.58097e15	−1.59193
$534$	0	0
$535$	−1.32251e15	−1.30453
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.29881e14	−0.122972
$540$	0	0
$541$	−3.62253e14	−0.336068	−0.168034	−	0.985781i	$-0.553742\pi$
$541$	−3.62253e14	−0.336068	−0.168034	+	0.985781i	$0.553742\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.46394e15	−1.30420
$546$	0	0
$547$	−1.39107e15	−1.21456	−0.607280	−	0.794488i	$-0.707740\pi$
$547$	−1.39107e15	−1.21456	−0.607280	+	0.794488i	$0.707740\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.23488e15	1.03584
$552$	0	0
$553$	−6.79771e14	−0.558952
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.46032e15	1.15410	0.577049	−	0.816709i	$-0.304204\pi$
$557$	1.46032e15	1.15410	0.577049	+	0.816709i	$0.304204\pi$
$558$	0	0
$559$	−1.56901e15	−1.21580
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.61701e15	−1.20481	−0.602404	−	0.798192i	$-0.705790\pi$
$563$	−1.61701e15	−1.20481	−0.602404	+	0.798192i	$0.705790\pi$
$564$	0	0
$565$	5.94515e14	0.434406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.86975e15	−1.31422	−0.657108	−	0.753796i	$-0.728221\pi$
$569$	−1.86975e15	−1.31422	−0.657108	+	0.753796i	$0.728221\pi$
$570$	0	0
$571$	−8.90087e14	−0.613668	−0.306834	−	0.951763i	$-0.599270\pi$
$571$	−8.90087e14	−0.613668	−0.306834	+	0.951763i	$0.599270\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.70435e14	−0.312123
$576$	0	0
$577$	2.37370e15	1.54511	0.772555	−	0.634948i	$-0.218979\pi$
$577$	2.37370e15	1.54511	0.772555	+	0.634948i	$0.218979\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.30163e14	−0.520229
$582$	0	0
$583$	−2.03311e14	−0.125021
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.78053e14	0.520010	0.260005	−	0.965607i	$-0.416276\pi$
$587$	8.78053e14	0.520010	0.260005	+	0.965607i	$0.416276\pi$
$588$	0	0
$589$	2.31954e15	1.34824
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.94595e15	−1.64978	−0.824888	−	0.565297i	$-0.808762\pi$
$593$	−2.94595e15	−1.64978	−0.824888	+	0.565297i	$0.808762\pi$
$594$	0	0
$595$	8.82407e14	0.485094
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.13040e15	1.65864	0.829320	−	0.558774i	$-0.188728\pi$
$599$	3.13040e15	1.65864	0.829320	+	0.558774i	$0.188728\pi$
$600$	0	0
$601$	7.88766e14	0.410335	0.205168	−	0.978727i	$-0.434226\pi$
$601$	7.88766e14	0.410335	0.205168	+	0.978727i	$0.434226\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.30451e14	0.215908
$606$	0	0
$607$	−9.20107e14	−0.453211	−0.226605	−	0.973987i	$-0.572763\pi$
$607$	−9.20107e14	−0.453211	−0.226605	+	0.973987i	$0.572763\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.41971e13	−0.0304992
$612$	0	0
$613$	−2.04580e15	−0.954619	−0.477309	−	0.878735i	$-0.658388\pi$
$613$	−2.04580e15	−0.954619	−0.477309	+	0.878735i	$0.658388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.63431e15	1.18604	0.593018	−	0.805189i	$-0.297936\pi$
$617$	2.63431e15	1.18604	0.593018	+	0.805189i	$0.297936\pi$
$618$	0	0
$619$	2.93881e15	1.29979	0.649895	−	0.760024i	$-0.274813\pi$
$619$	2.93881e15	1.29979	0.649895	+	0.760024i	$0.274813\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.19215e15	−0.508916
$624$	0	0
$625$	−1.43468e15	−0.601748
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.35871e15	−0.550236
$630$	0	0
$631$	−6.59294e14	−0.262372	−0.131186	−	0.991358i	$-0.541879\pi$
$631$	−6.59294e14	−0.262372	−0.131186	+	0.991358i	$0.541879\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.12699e14	0.0817543
$636$	0	0
$637$	5.66085e14	0.213853
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.71339e15	−1.35535	−0.677676	−	0.735361i	$-0.737013\pi$
$641$	−3.71339e15	−1.35535	−0.677676	+	0.735361i	$0.737013\pi$
$642$	0	0
$643$	−7.25342e14	−0.260245	−0.130122	−	0.991498i	$-0.541537\pi$
$643$	−7.25342e14	−0.260245	−0.130122	+	0.991498i	$0.541537\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.49303e15	0.864477	0.432238	−	0.901759i	$-0.357724\pi$
$647$	2.49303e15	0.864477	0.432238	+	0.901759i	$0.357724\pi$
$648$	0	0
$649$	−3.25318e15	−1.10908
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.56228e15	1.17410	0.587051	−	0.809550i	$-0.300289\pi$
$653$	3.56228e15	1.17410	0.587051	+	0.809550i	$0.300289\pi$
$654$	0	0
$655$	−7.98759e14	−0.258874
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.91608e14	−0.0913965	−0.0456982	−	0.998955i	$-0.514551\pi$
$659$	−2.91608e14	−0.0913965	−0.0456982	+	0.998955i	$0.514551\pi$
$660$	0	0
$661$	4.29519e15	1.32396	0.661980	−	0.749522i	$-0.269717\pi$
$661$	4.29519e15	1.32396	0.661980	+	0.749522i	$0.269717\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.29341e15	−0.385671
$666$	0	0
$667$	−2.95115e15	−0.865565
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.15110e15	1.46193
$672$	0	0
$673$	3.36827e15	0.940425	0.470213	−	0.882553i	$-0.344177\pi$
$673$	3.36827e15	0.940425	0.470213	+	0.882553i	$0.344177\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.43874e15	1.46981	0.734904	−	0.678171i	$-0.237227\pi$
$677$	5.43874e15	1.46981	0.734904	+	0.678171i	$0.237227\pi$
$678$	0	0
$679$	−2.93743e14	−0.0781058
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.22850e15	1.08861	0.544305	−	0.838887i	$-0.316793\pi$
$683$	4.22850e15	1.08861	0.544305	+	0.838887i	$0.316793\pi$
$684$	0	0
$685$	1.71467e15	0.434393
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.86130e14	0.217416
$690$	0	0
$691$	−4.29490e15	−1.03711	−0.518553	−	0.855045i	$-0.673529\pi$
$691$	−4.29490e15	−1.03711	−0.518553	+	0.855045i	$0.673529\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.42995e15	1.27022
$696$	0	0
$697$	−7.11080e15	−1.63734
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.78478e15	0.621357	0.310678	−	0.950515i	$-0.399444\pi$
$701$	2.78478e15	0.621357	0.310678	+	0.950515i	$0.399444\pi$
$702$	0	0
$703$	1.99157e15	0.437462
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.77498e15	0.590817
$708$	0	0
$709$	5.04946e15	1.05850	0.529250	−	0.848466i	$-0.322473\pi$
$709$	5.04946e15	1.05850	0.529250	+	0.848466i	$0.322473\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.54329e15	−1.12661
$714$	0	0
$715$	5.36723e15	1.07415
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.50621e15	0.292333	0.146166	−	0.989260i	$-0.453307\pi$
$719$	1.50621e15	0.292333	0.146166	+	0.989260i	$0.453307\pi$
$720$	0	0
$721$	2.04668e14	0.0391206
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.39262e15	0.258210
$726$	0	0
$727$	4.77997e15	0.872943	0.436472	−	0.899718i	$-0.356228\pi$
$727$	4.77997e15	0.872943	0.436472	+	0.899718i	$0.356228\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.05699e15	−1.25047
$732$	0	0
$733$	−1.98875e15	−0.347143	−0.173572	−	0.984821i	$-0.555531\pi$
$733$	−1.98875e15	−0.347143	−0.173572	+	0.984821i	$0.555531\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.02656e14	0.0173906
$738$	0	0
$739$	−4.76486e15	−0.795254	−0.397627	−	0.917547i	$-0.630166\pi$
$739$	−4.76486e15	−0.795254	−0.397627	+	0.917547i	$0.630166\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.92703e15	−0.960281	−0.480140	−	0.877192i	$-0.659414\pi$
$743$	−5.92703e15	−0.960281	−0.480140	+	0.877192i	$0.659414\pi$
$744$	0	0
$745$	−7.33001e15	−1.17016
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.81599e15	−0.591503
$750$	0	0
$751$	1.22509e16	1.87132	0.935661	−	0.352900i	$-0.114805\pi$
$751$	1.22509e16	1.87132	0.935661	+	0.352900i	$0.114805\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.77872e15	−0.412228
$756$	0	0
$757$	7.22296e15	1.05606	0.528029	−	0.849226i	$-0.322931\pi$
$757$	7.22296e15	1.05606	0.528029	+	0.849226i	$0.322931\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.06648e15	−0.293505	−0.146752	−	0.989173i	$-0.546882\pi$
$761$	−2.06648e15	−0.293505	−0.146752	+	0.989173i	$0.546882\pi$
$762$	0	0
$763$	−4.22406e15	−0.591351
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.41789e16	1.92872
$768$	0	0
$769$	3.97859e15	0.533500	0.266750	−	0.963766i	$-0.414050\pi$
$769$	3.97859e15	0.533500	0.266750	+	0.963766i	$0.414050\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.20926e15	0.548554	0.274277	−	0.961651i	$-0.411562\pi$
$773$	4.20926e15	0.548554	0.274277	+	0.961651i	$0.411562\pi$
$774$	0	0
$775$	2.61583e15	0.336085
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.04228e16	1.30176
$780$	0	0
$781$	2.11478e15	0.260426
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.48414e15	1.01584
$786$	0	0
$787$	7.90410e15	0.933236	0.466618	−	0.884459i	$-0.345472\pi$
$787$	7.90410e15	0.933236	0.466618	+	0.884459i	$0.345472\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.71542e15	0.196969
$792$	0	0
$793$	−2.24510e16	−2.54234
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.97540e15	−0.548033	−0.274017	−	0.961725i	$-0.588352\pi$
$797$	−4.97540e15	−0.548033	−0.274017	+	0.961725i	$0.588352\pi$
$798$	0	0
$799$	−2.88742e14	−0.0313691
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.33710e16	1.41328
$804$	0	0
$805$	3.09103e15	0.322274
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.11832e14	−0.0113462	−0.00567309	−	0.999984i	$-0.501806\pi$
$809$	−1.11832e14	−0.0113462	−0.00567309	+	0.999984i	$0.501806\pi$
$810$	0	0
$811$	5.74284e15	0.574794	0.287397	−	0.957812i	$-0.407210\pi$
$811$	5.74284e15	0.574794	0.287397	+	0.957812i	$0.407210\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.36014e16	−1.32500
$816$	0	0
$817$	1.03440e16	0.994181
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.00968e16	−0.944706	−0.472353	−	0.881409i	$-0.656595\pi$
$821$	−1.00968e16	−0.944706	−0.472353	+	0.881409i	$0.656595\pi$
$822$	0	0
$823$	1.41355e16	1.30501	0.652503	−	0.757786i	$-0.273719\pi$
$823$	1.41355e16	1.30501	0.652503	+	0.757786i	$0.273719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.65399e16	1.48680	0.743402	−	0.668845i	$-0.233211\pi$
$827$	1.65399e16	1.48680	0.743402	+	0.668845i	$0.233211\pi$
$828$	0	0
$829$	7.29180e15	0.646822	0.323411	−	0.946259i	$-0.395170\pi$
$829$	7.29180e15	0.646822	0.323411	+	0.946259i	$0.395170\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.54610e15	0.219952
$834$	0	0
$835$	−7.81271e15	−0.666079
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.26764e15	0.105270	0.0526352	−	0.998614i	$-0.483238\pi$
$839$	1.26764e15	0.105270	0.0526352	+	0.998614i	$0.483238\pi$
$840$	0	0
$841$	−3.46425e15	−0.283943
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.29540e16	−1.03441
$846$	0	0
$847$	1.24203e15	0.0978976
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.75950e15	−0.365552
$852$	0	0
$853$	2.28320e15	0.173111	0.0865556	−	0.996247i	$-0.472414\pi$
$853$	2.28320e15	0.173111	0.0865556	+	0.996247i	$0.472414\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.14991e15	0.602225	0.301112	−	0.953589i	$-0.402642\pi$
$857$	8.14991e15	0.602225	0.301112	+	0.953589i	$0.402642\pi$
$858$	0	0
$859$	−7.53543e15	−0.549725	−0.274862	−	0.961484i	$-0.588632\pi$
$859$	−7.53543e15	−0.549725	−0.274862	+	0.961484i	$0.588632\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.31933e15	0.520489	0.260245	−	0.965543i	$-0.416197\pi$
$863$	7.31933e15	0.520489	0.260245	+	0.965543i	$0.416197\pi$
$864$	0	0
$865$	1.84420e16	1.29485
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.85968e16	−1.27300
$870$	0	0
$871$	−4.47426e14	−0.0302427
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.23880e15	−0.411203
$876$	0	0
$877$	−1.43587e16	−0.934581	−0.467291	−	0.884104i	$-0.654770\pi$
$877$	−1.43587e16	−0.934581	−0.467291	+	0.884104i	$0.654770\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.50615e15	−0.349527	−0.174764	−	0.984610i	$-0.555916\pi$
$881$	−5.50615e15	−0.349527	−0.174764	+	0.984610i	$0.555916\pi$
$882$	0	0
$883$	7.99249e15	0.501070	0.250535	−	0.968108i	$-0.419394\pi$
$883$	7.99249e15	0.501070	0.250535	+	0.968108i	$0.419394\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.70403e16	1.04207	0.521036	−	0.853535i	$-0.325546\pi$
$887$	1.70403e16	1.04207	0.521036	+	0.853535i	$0.325546\pi$
$888$	0	0
$889$	6.13724e14	0.0370692
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.23230e14	0.0249398
$894$	0	0
$895$	2.73437e15	0.159159
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.64097e16	0.932015
$900$	0	0
$901$	3.98558e15	0.223617
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.00598e16	−1.09839
$906$	0	0
$907$	5.92409e15	0.320465	0.160233	−	0.987079i	$-0.448776\pi$
$907$	5.92409e15	0.320465	0.160233	+	0.987079i	$0.448776\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.91137e15	0.523338	0.261669	−	0.965158i	$-0.415727\pi$
$911$	9.91137e15	0.523338	0.261669	+	0.965158i	$0.415727\pi$
$912$	0	0
$913$	−2.27111e16	−1.18481
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.30474e15	−0.117379
$918$	0	0
$919$	8.85446e15	0.445581	0.222791	−	0.974866i	$-0.428483\pi$
$919$	8.85446e15	0.445581	0.222791	+	0.974866i	$0.428483\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.21725e15	−0.452889
$924$	0	0
$925$	2.24596e15	0.109049
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.45731e16	0.690980	0.345490	−	0.938422i	$-0.387713\pi$
$929$	1.45731e16	0.690980	0.345490	+	0.938422i	$0.387713\pi$
$930$	0	0
$931$	−3.73201e15	−0.174872
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.41404e16	1.10479
$936$	0	0
$937$	−1.76320e16	−0.797506	−0.398753	−	0.917058i	$-0.630557\pi$
$937$	−1.76320e16	−0.797506	−0.398753	+	0.917058i	$0.630557\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.31253e16	−0.579920	−0.289960	−	0.957039i	$-0.593642\pi$
$941$	−1.31253e16	−0.579920	−0.289960	+	0.957039i	$0.593642\pi$
$942$	0	0
$943$	−2.49088e16	−1.08777
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.13420e16	−0.910562	−0.455281	−	0.890348i	$-0.650461\pi$
$947$	−2.13420e16	−0.910562	−0.455281	+	0.890348i	$0.650461\pi$
$948$	0	0
$949$	−5.82772e16	−2.45773
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.42837e15	0.182487	0.0912436	−	0.995829i	$-0.470916\pi$
$953$	4.42837e15	0.182487	0.0912436	+	0.995829i	$0.470916\pi$
$954$	0	0
$955$	3.01620e16	1.22869
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.94752e15	0.196963
$960$	0	0
$961$	5.41468e15	0.213105
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.11197e16	0.812434
$966$	0	0
$967$	−2.03109e16	−0.772472	−0.386236	−	0.922400i	$-0.626225\pi$
$967$	−2.03109e16	−0.772472	−0.386236	+	0.922400i	$0.626225\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.00151e16	−0.744135	−0.372067	−	0.928206i	$-0.621351\pi$
$971$	−2.00151e16	−0.744135	−0.372067	+	0.928206i	$0.621351\pi$
$972$	0	0
$973$	1.56676e16	0.575946
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.64523e16	−1.31010	−0.655050	−	0.755585i	$-0.727353\pi$
$977$	−3.64523e16	−1.31010	−0.655050	+	0.755585i	$0.727353\pi$
$978$	0	0
$979$	−3.26141e16	−1.15905
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.62356e16	−0.564187	−0.282093	−	0.959387i	$-0.591029\pi$
$983$	−1.62356e16	−0.564187	−0.282093	+	0.959387i	$0.591029\pi$
$984$	0	0
$985$	−2.06200e16	−0.708582
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.47203e16	−0.830756
$990$	0	0
$991$	−4.71455e16	−1.56688	−0.783439	−	0.621469i	$-0.786536\pi$
$991$	−4.71455e16	−1.56688	−0.783439	+	0.621469i	$0.786536\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.27318e16	−0.413869
$996$	0	0
$997$	−3.48808e16	−1.12141	−0.560703	−	0.828017i	$-0.689469\pi$
$997$	−3.48808e16	−1.12141	−0.560703	+	0.828017i	$0.689469\pi$
$998$	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.1.1		3
3.2	odd	2		28.12.a.a.1.1	✓	3
12.11	even	2		112.12.a.g.1.3		3
21.2	odd	6		196.12.e.e.165.3		6
21.5	even	6		196.12.e.d.165.1		6
21.11	odd	6		196.12.e.e.177.3		6
21.17	even	6		196.12.e.d.177.1		6
21.20	even	2		196.12.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.12.a.a.1.1	✓	3	3.2	odd	2
112.12.a.g.1.3		3	12.11	even	2
196.12.a.c.1.3		3	21.20	even	2
196.12.e.d.165.1		6	21.5	even	6
196.12.e.d.177.1		6	21.17	even	6
196.12.e.e.165.3		6	21.2	odd	6
196.12.e.e.177.3		6	21.11	odd	6
252.12.a.f.1.1		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)

\(f(q)\)	\(=\)	\(q-5824.83 q^{5} -16807.0 q^{7} +O(q^{10})\)	\(q-5824.83 q^{5} -16807.0 q^{7} -459796. q^{11} +2.00402e6 q^{13} +9.01354e6 q^{17} -1.32118e7 q^{19} +3.15740e7 q^{23} -1.48994e7 q^{25} -9.34680e7 q^{29} -1.75565e8 q^{31} +9.78980e7 q^{35} -1.50741e8 q^{37} -7.88902e8 q^{41} -7.82932e8 q^{43} -3.20342e7 q^{47} +2.82475e8 q^{49} +4.42177e8 q^{53} +2.67823e9 q^{55} +7.07526e9 q^{59} -1.12030e10 q^{61} -1.16731e10 q^{65} -2.23264e8 q^{67} -4.59939e9 q^{71} -2.90802e10 q^{73} +7.72779e9 q^{77} +4.04457e10 q^{79} +4.93939e10 q^{83} -5.25024e10 q^{85} +7.09317e10 q^{89} -3.36815e10 q^{91} +7.69566e10 q^{95} +1.74774e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) \) 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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