LMFDB - Embedded newform 252.12.a.i.1.6

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 5681627x^{4} + 7706355585775x^{2} - 2456393975347843125 $ x^6 - 5681627x^4 + 7706355585775x^2 - 2456393975347843125
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{10}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1960.20$ 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)

$f(q)$	$=$	$q+11761.2 q^{5} +16807.0 q^{7} +O(q^{10})$	$q+11761.2 q^{5} +16807.0 q^{7} +559834. q^{11} +479009. q^{13} +1.05643e7 q^{17} -1.66358e7 q^{19} +1.69805e7 q^{23} +8.94979e7 q^{25} +1.71183e7 q^{29} +1.46333e8 q^{31} +1.97671e8 q^{35} -7.01691e8 q^{37} +6.75780e8 q^{41} +4.83617e8 q^{43} +1.93367e9 q^{47} +2.82475e8 q^{49} +1.39043e9 q^{53} +6.58433e9 q^{55} +6.21591e9 q^{59} -1.08515e10 q^{61} +5.63372e9 q^{65} +6.71819e9 q^{67} +1.98389e10 q^{71} -1.83844e10 q^{73} +9.40914e9 q^{77} -4.21665e10 q^{79} +1.13043e10 q^{83} +1.24249e11 q^{85} -7.17093e10 q^{89} +8.05070e9 q^{91} -1.95657e11 q^{95} -1.35800e11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 100842 q^{7}+O(q^{10}) $ 6 * q + 100842 * q^7	$ 6 q + 100842 q^{7} + 1227468 q^{13} - 6532728 q^{19} + 116108394 q^{25} + 293608488 q^{31} - 74878260 q^{37} + 1432813080 q^{43} + 1694851494 q^{49} + 6150010104 q^{55} - 11105239740 q^{61} - 3754055112 q^{67} - 31466324076 q^{73} - 58630542864 q^{79} + 94916392200 q^{85} + 20630054676 q^{91} - 6046820556 q^{97}+O(q^{100}) $ 6 * q + 100842 * q^7 + 1227468 * q^13 - 6532728 * q^19 + 116108394 * q^25 + 293608488 * q^31 - 74878260 * q^37 + 1432813080 * q^43 + 1694851494 * q^49 + 6150010104 * q^55 - 11105239740 * q^61 - 3754055112 * q^67 - 31466324076 * q^73 - 58630542864 * q^79 + 94916392200 * q^85 + 20630054676 * q^91 - 6046820556 * 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
$5$	11761.2	1.68313	0.841564	−	0.540158i	$-0.181636\pi$
$5$	11761.2	1.68313	0.841564	+	0.540158i	$0.181636\pi$
$6$	0	0
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	559834.	1.04809	0.524047	−	0.851690i	$-0.324422\pi$
$11$	559834.	1.04809	0.524047	+	0.851690i	$0.324422\pi$
$12$	0	0
$13$	479009.	0.357812	0.178906	−	0.983866i	$-0.442744\pi$
$13$	479009.	0.357812	0.178906	+	0.983866i	$0.442744\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.05643e7	1.80456	0.902280	−	0.431150i	$-0.141892\pi$
$17$	1.05643e7	1.80456	0.902280	+	0.431150i	$0.141892\pi$
$18$	0	0
$19$	−1.66358e7	−1.54134	−0.770670	−	0.637234i	$-0.780078\pi$
$19$	−1.66358e7	−1.54134	−0.770670	+	0.637234i	$0.780078\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.69805e7	0.550107	0.275053	−	0.961429i	$-0.411304\pi$
$23$	1.69805e7	0.550107	0.275053	+	0.961429i	$0.411304\pi$
$24$	0	0
$25$	8.94979e7	1.83292
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.71183e7	0.154979	0.0774893	−	0.996993i	$-0.475310\pi$
$29$	1.71183e7	0.154979	0.0774893	+	0.996993i	$0.475310\pi$
$30$	0	0
$31$	1.46333e8	0.918023	0.459011	−	0.888430i	$-0.348204\pi$
$31$	1.46333e8	0.918023	0.459011	+	0.888430i	$0.348204\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.97671e8	0.636162
$36$	0	0
$37$	−7.01691e8	−1.66355	−0.831776	−	0.555112i	$-0.812676\pi$
$37$	−7.01691e8	−1.66355	−0.831776	+	0.555112i	$0.812676\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.75780e8	0.910950	0.455475	−	0.890249i	$-0.349470\pi$
$41$	6.75780e8	0.910950	0.455475	+	0.890249i	$0.349470\pi$
$42$	0	0
$43$	4.83617e8	0.501678	0.250839	−	0.968029i	$-0.419294\pi$
$43$	4.83617e8	0.501678	0.250839	+	0.968029i	$0.419294\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.93367e9	1.22983	0.614914	−	0.788594i	$-0.289191\pi$
$47$	1.93367e9	1.22983	0.614914	+	0.788594i	$0.289191\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.39043e9	0.456699	0.228350	−	0.973579i	$-0.426667\pi$
$53$	1.39043e9	0.456699	0.228350	+	0.973579i	$0.426667\pi$
$54$	0	0
$55$	6.58433e9	1.76407
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.21591e9	1.13193	0.565964	−	0.824430i	$-0.308504\pi$
$59$	6.21591e9	1.13193	0.565964	+	0.824430i	$0.308504\pi$
$60$	0	0
$61$	−1.08515e10	−1.64504	−0.822520	−	0.568737i	$-0.807432\pi$
$61$	−1.08515e10	−1.64504	−0.822520	+	0.568737i	$0.807432\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.63372e9	0.602244
$66$	0	0
$67$	6.71819e9	0.607913	0.303956	−	0.952686i	$-0.401692\pi$
$67$	6.71819e9	0.607913	0.303956	+	0.952686i	$0.401692\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.98389e10	1.30496	0.652481	−	0.757805i	$-0.273728\pi$
$71$	1.98389e10	1.30496	0.652481	+	0.757805i	$0.273728\pi$
$72$	0	0
$73$	−1.83844e10	−1.03794	−0.518971	−	0.854792i	$-0.673685\pi$
$73$	−1.83844e10	−1.03794	−0.518971	+	0.854792i	$0.673685\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.40914e9	0.396142
$78$	0	0
$79$	−4.21665e10	−1.54176	−0.770882	−	0.636978i	$-0.780184\pi$
$79$	−4.21665e10	−1.54176	−0.770882	+	0.636978i	$0.780184\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.13043e10	0.315004	0.157502	−	0.987519i	$-0.449656\pi$
$83$	1.13043e10	0.315004	0.157502	+	0.987519i	$0.449656\pi$
$84$	0	0
$85$	1.24249e11	3.03730
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.17093e10	−1.36123	−0.680614	−	0.732643i	$-0.738287\pi$
$89$	−7.17093e10	−1.36123	−0.680614	+	0.732643i	$0.738287\pi$
$90$	0	0
$91$	8.05070e9	0.135240
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.95657e11	−2.59427
$96$	0	0
$97$	−1.35800e11	−1.60567	−0.802835	−	0.596202i	$-0.796676\pi$
$97$	−1.35800e11	−1.60567	−0.802835	+	0.596202i	$0.796676\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.42721e10	−0.419143	−0.209572	−	0.977793i	$-0.567207\pi$
$101$	−4.42721e10	−0.419143	−0.209572	+	0.977793i	$0.567207\pi$
$102$	0	0
$103$	−4.24972e10	−0.361207	−0.180603	−	0.983556i	$-0.557805\pi$
$103$	−4.24972e10	−0.361207	−0.180603	+	0.983556i	$0.557805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.63217e11	−1.12501	−0.562504	−	0.826795i	$-0.690162\pi$
$107$	−1.63217e11	−1.12501	−0.562504	+	0.826795i	$0.690162\pi$
$108$	0	0
$109$	1.75638e11	1.09339	0.546693	−	0.837333i	$-0.315887\pi$
$109$	1.75638e11	1.09339	0.546693	+	0.837333i	$0.315887\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.28405e10	0.371913	0.185956	−	0.982558i	$-0.440462\pi$
$113$	7.28405e10	0.371913	0.185956	+	0.982558i	$0.440462\pi$
$114$	0	0
$115$	1.99711e11	0.925900
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.77554e11	0.682060
$120$	0	0
$121$	2.81029e10	0.0984990
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.78326e11	1.40190
$126$	0	0
$127$	−3.03680e10	−0.0815635	−0.0407818	−	0.999168i	$-0.512985\pi$
$127$	−3.03680e10	−0.0815635	−0.0407818	+	0.999168i	$0.512985\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.16737e11	−0.490840	−0.245420	−	0.969417i	$-0.578926\pi$
$131$	−2.16737e11	−0.490840	−0.245420	+	0.969417i	$0.578926\pi$
$132$	0	0
$133$	−2.79598e11	−0.582572
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.56530e11	−0.277098	−0.138549	−	0.990356i	$-0.544244\pi$
$137$	−1.56530e11	−0.277098	−0.138549	+	0.990356i	$0.544244\pi$
$138$	0	0
$139$	3.51961e10	0.0575325	0.0287662	−	0.999586i	$-0.490842\pi$
$139$	3.51961e10	0.0575325	0.0287662	+	0.999586i	$0.490842\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.68166e11	0.375021
$144$	0	0
$145$	2.01332e11	0.260849
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.69512e11	−0.523748	−0.261874	−	0.965102i	$-0.584340\pi$
$149$	−4.69512e11	−0.523748	−0.261874	+	0.965102i	$0.584340\pi$
$150$	0	0
$151$	−3.82637e11	−0.396656	−0.198328	−	0.980136i	$-0.563551\pi$
$151$	−3.82637e11	−0.396656	−0.198328	+	0.980136i	$0.563551\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.72106e12	1.54515
$156$	0	0
$157$	2.29966e12	1.92405	0.962023	−	0.272969i	$-0.0880057\pi$
$157$	2.29966e12	1.92405	0.962023	+	0.272969i	$0.0880057\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.85391e11	0.207921
$162$	0	0
$163$	−2.58960e11	−0.176279	−0.0881397	−	0.996108i	$-0.528092\pi$
$163$	−2.58960e11	−0.176279	−0.0881397	+	0.996108i	$0.528092\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.29855e12	1.36934	0.684672	−	0.728851i	$-0.259945\pi$
$167$	2.29855e12	1.36934	0.684672	+	0.728851i	$0.259945\pi$
$168$	0	0
$169$	−1.56271e12	−0.871970
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.31468e11	−0.0645008	−0.0322504	−	0.999480i	$-0.510267\pi$
$173$	−1.31468e11	−0.0645008	−0.0322504	+	0.999480i	$0.510267\pi$
$174$	0	0
$175$	1.50419e12	0.692777
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.04632e12	1.64577	0.822884	−	0.568209i	$-0.192364\pi$
$179$	4.04632e12	1.64577	0.822884	+	0.568209i	$0.192364\pi$
$180$	0	0
$181$	2.53787e12	0.971038	0.485519	−	0.874226i	$-0.338631\pi$
$181$	2.53787e12	0.971038	0.485519	+	0.874226i	$0.338631\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.25273e12	−2.79997
$186$	0	0
$187$	5.91426e12	1.89135
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.06158e11	0.144080	0.0720398	−	0.997402i	$-0.477049\pi$
$191$	5.06158e11	0.144080	0.0720398	+	0.997402i	$0.477049\pi$
$192$	0	0
$193$	9.62802e11	0.258804	0.129402	−	0.991592i	$-0.458694\pi$
$193$	9.62802e11	0.258804	0.129402	+	0.991592i	$0.458694\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.58528e12	−1.82141	−0.910704	−	0.413060i	$-0.864460\pi$
$197$	−7.58528e12	−1.82141	−0.910704	+	0.413060i	$0.864460\pi$
$198$	0	0
$199$	−8.12267e12	−1.84504	−0.922522	−	0.385945i	$-0.873875\pi$
$199$	−8.12267e12	−1.84504	−0.922522	+	0.385945i	$0.873875\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.87707e11	0.0585764
$204$	0	0
$205$	7.94799e12	1.53324
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.31329e12	−1.61547
$210$	0	0
$211$	6.90148e12	1.13603	0.568014	−	0.823019i	$-0.307712\pi$
$211$	6.90148e12	1.13603	0.568014	+	0.823019i	$0.307712\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.68791e12	0.844387
$216$	0	0
$217$	2.45942e12	0.346980
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.06039e12	0.645694
$222$	0	0
$223$	7.09212e12	0.861191	0.430595	−	0.902545i	$-0.358304\pi$
$223$	7.09212e12	0.861191	0.430595	+	0.902545i	$0.358304\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.14480e13	−1.26063	−0.630317	−	0.776338i	$-0.717075\pi$
$227$	−1.14480e13	−1.26063	−0.630317	+	0.776338i	$0.717075\pi$
$228$	0	0
$229$	−4.48122e12	−0.470221	−0.235110	−	0.971969i	$-0.575545\pi$
$229$	−4.48122e12	−0.470221	−0.235110	+	0.971969i	$0.575545\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.26218e12	−0.215809	−0.107904	−	0.994161i	$-0.534414\pi$
$233$	−2.26218e12	−0.215809	−0.107904	+	0.994161i	$0.534414\pi$
$234$	0	0
$235$	2.27423e13	2.06996
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.59425e12	0.132242	0.0661208	−	0.997812i	$-0.478938\pi$
$239$	1.59425e12	0.132242	0.0661208	+	0.997812i	$0.478938\pi$
$240$	0	0
$241$	1.68606e13	1.33591	0.667957	−	0.744199i	$-0.267169\pi$
$241$	1.68606e13	1.33591	0.667957	+	0.744199i	$0.267169\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.32225e12	0.240447
$246$	0	0
$247$	−7.96869e12	−0.551510
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.94078e13	1.22962	0.614810	−	0.788675i	$-0.289233\pi$
$251$	1.94078e13	1.22962	0.614810	+	0.788675i	$0.289233\pi$
$252$	0	0
$253$	9.50626e12	0.576563
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.21039e13	1.22981	0.614903	−	0.788603i	$-0.289195\pi$
$257$	2.21039e13	1.22981	0.614903	+	0.788603i	$0.289195\pi$
$258$	0	0
$259$	−1.17933e13	−0.628763
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.78357e11	0.0185415	0.00927076	−	0.999957i	$-0.497049\pi$
$263$	3.78357e11	0.0185415	0.00927076	+	0.999957i	$0.497049\pi$
$264$	0	0
$265$	1.63531e13	0.768683
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.57287e13	0.680856	0.340428	−	0.940271i	$-0.389428\pi$
$269$	1.57287e13	0.680856	0.340428	+	0.940271i	$0.389428\pi$
$270$	0	0
$271$	3.36536e13	1.39862	0.699311	−	0.714817i	$-0.253490\pi$
$271$	3.36536e13	1.39862	0.699311	+	0.714817i	$0.253490\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.01040e13	1.92107
$276$	0	0
$277$	−1.01293e13	−0.373200	−0.186600	−	0.982436i	$-0.559747\pi$
$277$	−1.01293e13	−0.373200	−0.186600	+	0.982436i	$0.559747\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.24670e13	−1.78649	−0.893246	−	0.449568i	$-0.851578\pi$
$281$	−5.24670e13	−1.78649	−0.893246	+	0.449568i	$0.851578\pi$
$282$	0	0
$283$	5.89400e13	1.93012	0.965062	−	0.262023i	$-0.0843897\pi$
$283$	5.89400e13	1.93012	0.965062	+	0.262023i	$0.0843897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.13578e13	0.344307
$288$	0	0
$289$	7.73324e13	2.25644
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.34640e13	1.44640	0.723202	−	0.690636i	$-0.242669\pi$
$293$	5.34640e13	1.44640	0.723202	+	0.690636i	$0.242669\pi$
$294$	0	0
$295$	7.31066e13	1.90518
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.13381e12	0.196835
$300$	0	0
$301$	8.12814e12	0.189616
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.27627e14	−2.76881
$306$	0	0
$307$	−1.63226e13	−0.341609	−0.170805	−	0.985305i	$-0.554637\pi$
$307$	−1.63226e13	−0.341609	−0.170805	+	0.985305i	$0.554637\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.51950e12	−0.107576	−0.0537882	−	0.998552i	$-0.517130\pi$
$311$	−5.51950e12	−0.107576	−0.0537882	+	0.998552i	$0.517130\pi$
$312$	0	0
$313$	−5.18455e13	−0.975478	−0.487739	−	0.872990i	$-0.662178\pi$
$313$	−5.18455e13	−0.975478	−0.487739	+	0.872990i	$0.662178\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.65937e13	−0.642066	−0.321033	−	0.947068i	$-0.604030\pi$
$317$	−3.65937e13	−0.642066	−0.321033	+	0.947068i	$0.604030\pi$
$318$	0	0
$319$	9.58342e12	0.162432
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.75745e14	−2.78144
$324$	0	0
$325$	4.28703e13	0.655840
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.24992e13	0.464831
$330$	0	0
$331$	2.09702e13	0.290101	0.145050	−	0.989424i	$-0.453666\pi$
$331$	2.09702e13	0.290101	0.145050	+	0.989424i	$0.453666\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.90141e13	1.02319
$336$	0	0
$337$	1.56098e13	0.195629	0.0978147	−	0.995205i	$-0.468815\pi$
$337$	1.56098e13	0.195629	0.0978147	+	0.995205i	$0.468815\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.19224e13	0.962173
$342$	0	0
$343$	4.74756e12	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.31391e14	−1.40202	−0.701011	−	0.713151i	$-0.747267\pi$
$347$	−1.31391e14	−1.40202	−0.701011	+	0.713151i	$0.747267\pi$
$348$	0	0
$349$	1.05452e14	1.09023	0.545113	−	0.838363i	$-0.316487\pi$
$349$	1.05452e14	1.09023	0.545113	+	0.838363i	$0.316487\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.47706e13	0.823160	0.411580	−	0.911374i	$-0.364977\pi$
$353$	8.47706e13	0.823160	0.411580	+	0.911374i	$0.364977\pi$
$354$	0	0
$355$	2.33330e14	2.19642
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.26032e14	1.11548	0.557740	−	0.830016i	$-0.311669\pi$
$359$	1.26032e14	1.11548	0.557740	+	0.830016i	$0.311669\pi$
$360$	0	0
$361$	1.60259e14	1.37573
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.16222e14	−1.74699
$366$	0	0
$367$	−2.20545e14	−1.72915	−0.864577	−	0.502500i	$-0.832414\pi$
$367$	−2.20545e14	−1.72915	−0.864577	+	0.502500i	$0.832414\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.33689e13	0.172616
$372$	0	0
$373$	−1.19511e14	−0.857056	−0.428528	−	0.903528i	$-0.640968\pi$
$373$	−1.19511e14	−0.857056	−0.428528	+	0.903528i	$0.640968\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.19982e12	0.0554533
$378$	0	0
$379$	−2.18665e14	−1.43636	−0.718179	−	0.695858i	$-0.755024\pi$
$379$	−2.18665e14	−1.43636	−0.718179	+	0.695858i	$0.755024\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.13562e14	1.32413	0.662064	−	0.749447i	$-0.269681\pi$
$383$	2.13562e14	1.32413	0.662064	+	0.749447i	$0.269681\pi$
$384$	0	0
$385$	1.10663e14	0.666757
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.45220e14	1.39583	0.697916	−	0.716180i	$-0.254111\pi$
$389$	2.45220e14	1.39583	0.697916	+	0.716180i	$0.254111\pi$
$390$	0	0
$391$	1.79387e14	0.992701
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.95928e14	−2.59499
$396$	0	0
$397$	−1.14761e14	−0.584044	−0.292022	−	0.956412i	$-0.594328\pi$
$397$	−1.14761e14	−0.584044	−0.292022	+	0.956412i	$0.594328\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.31715e14	1.59761	0.798806	−	0.601589i	$-0.205466\pi$
$401$	3.31715e14	1.59761	0.798806	+	0.601589i	$0.205466\pi$
$402$	0	0
$403$	7.00949e13	0.328480
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.92831e14	−1.74356
$408$	0	0
$409$	2.18896e13	0.0945716	0.0472858	−	0.998881i	$-0.484943\pi$
$409$	2.18896e13	0.0945716	0.0472858	+	0.998881i	$0.484943\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.04471e14	0.427828
$414$	0	0
$415$	1.32953e14	0.530191
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.43108e14	1.29794	0.648968	−	0.760816i	$-0.275201\pi$
$419$	3.43108e14	1.29794	0.648968	+	0.760816i	$0.275201\pi$
$420$	0	0
$421$	1.16935e14	0.430918	0.215459	−	0.976513i	$-0.430875\pi$
$421$	1.16935e14	0.430918	0.215459	+	0.976513i	$0.430875\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.45482e14	3.30761
$426$	0	0
$427$	−1.82381e14	−0.621766
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.22483e13	−0.104444	−0.0522218	−	0.998636i	$-0.516630\pi$
$431$	−3.22483e13	−0.104444	−0.0522218	+	0.998636i	$0.516630\pi$
$432$	0	0
$433$	−2.54817e14	−0.804535	−0.402268	−	0.915522i	$-0.631778\pi$
$433$	−2.54817e14	−0.804535	−0.402268	+	0.915522i	$0.631778\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.82484e14	−0.847902
$438$	0	0
$439$	−8.78779e13	−0.257232	−0.128616	−	0.991694i	$-0.541053\pi$
$439$	−8.78779e13	−0.257232	−0.128616	+	0.991694i	$0.541053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.15257e14	1.15637	0.578184	−	0.815906i	$-0.303761\pi$
$443$	4.15257e14	1.15637	0.578184	+	0.815906i	$0.303761\pi$
$444$	0	0
$445$	−8.43388e14	−2.29112
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.83625e14	−1.76792	−0.883961	−	0.467561i	$-0.845133\pi$
$449$	−6.83625e14	−1.76792	−0.883961	+	0.467561i	$0.845133\pi$
$450$	0	0
$451$	3.78325e14	0.954760
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.46860e13	0.227627
$456$	0	0
$457$	−3.26050e14	−0.765147	−0.382573	−	0.923925i	$-0.624962\pi$
$457$	−3.26050e14	−0.765147	−0.382573	+	0.923925i	$0.624962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.21591e13	0.139043	0.0695216	−	0.997580i	$-0.477853\pi$
$461$	6.21591e13	0.139043	0.0695216	+	0.997580i	$0.477853\pi$
$462$	0	0
$463$	−1.11413e14	−0.243354	−0.121677	−	0.992570i	$-0.538827\pi$
$463$	−1.11413e14	−0.243354	−0.121677	+	0.992570i	$0.538827\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.12622e14	−1.90129	−0.950644	−	0.310284i	$-0.899576\pi$
$467$	−9.12622e14	−1.90129	−0.950644	+	0.310284i	$0.899576\pi$
$468$	0	0
$469$	1.12913e14	0.229769
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.70745e14	0.525805
$474$	0	0
$475$	−1.48887e15	−2.82515
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.05108e15	−1.90454	−0.952271	−	0.305253i	$-0.901259\pi$
$479$	−1.05108e15	−1.90454	−0.952271	+	0.305253i	$0.901259\pi$
$480$	0	0
$481$	−3.36116e14	−0.595239
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.59718e15	−2.70255
$486$	0	0
$487$	2.83229e14	0.468520	0.234260	−	0.972174i	$-0.424733\pi$
$487$	2.83229e14	0.468520	0.234260	+	0.972174i	$0.424733\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.45198e14	−0.387765	−0.193882	−	0.981025i	$-0.562108\pi$
$491$	−2.45198e14	−0.387765	−0.193882	+	0.981025i	$0.562108\pi$
$492$	0	0
$493$	1.80843e14	0.279668
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.33433e14	0.493229
$498$	0	0
$499$	−2.83665e14	−0.410444	−0.205222	−	0.978715i	$-0.565792\pi$
$499$	−2.83665e14	−0.410444	−0.205222	+	0.978715i	$0.565792\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.99052e14	0.552593	0.276297	−	0.961072i	$-0.410893\pi$
$503$	3.99052e14	0.552593	0.276297	+	0.961072i	$0.410893\pi$
$504$	0	0
$505$	−5.20694e14	−0.705472
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.52729e14	0.846808	0.423404	−	0.905941i	$-0.360835\pi$
$509$	6.52729e14	0.846808	0.423404	+	0.905941i	$0.360835\pi$
$510$	0	0
$511$	−3.08986e14	−0.392305
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.99818e14	−0.607957
$516$	0	0
$517$	1.08253e15	1.28897
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.81894e13	0.0207592	0.0103796	−	0.999946i	$-0.496696\pi$
$521$	1.81894e13	0.0207592	0.0103796	+	0.999946i	$0.496696\pi$
$522$	0	0
$523$	−1.62485e15	−1.81574	−0.907868	−	0.419255i	$-0.862291\pi$
$523$	−1.62485e15	−1.81574	−0.907868	+	0.419255i	$0.862291\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.54591e15	1.65663
$528$	0	0
$529$	−6.64473e14	−0.697382
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.23705e14	0.325949
$534$	0	0
$535$	−1.91963e15	−1.89353
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.58139e14	0.149728
$540$	0	0
$541$	−1.15343e15	−1.07006	−0.535028	−	0.844834i	$-0.679699\pi$
$541$	−1.15343e15	−1.07006	−0.535028	+	0.844834i	$0.679699\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.06572e15	1.84031
$546$	0	0
$547$	9.88658e14	0.863209	0.431604	−	0.902063i	$-0.357948\pi$
$547$	9.88658e14	0.863209	0.431604	+	0.902063i	$0.357948\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.84777e14	−0.238875
$552$	0	0
$553$	−7.08692e14	−0.582732
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.94353e15	−1.53599	−0.767995	−	0.640456i	$-0.778745\pi$
$557$	−1.94353e15	−1.53599	−0.767995	+	0.640456i	$0.778745\pi$
$558$	0	0
$559$	2.31657e14	0.179506
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.14115e15	1.59533	0.797666	−	0.603099i	$-0.206068\pi$
$563$	2.14115e15	1.59533	0.797666	+	0.603099i	$0.206068\pi$
$564$	0	0
$565$	8.56692e14	0.625977
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.49802e14	−0.456734	−0.228367	−	0.973575i	$-0.573339\pi$
$569$	−6.49802e14	−0.456734	−0.228367	+	0.973575i	$0.573339\pi$
$570$	0	0
$571$	1.59530e15	1.09987	0.549937	−	0.835206i	$-0.314652\pi$
$571$	1.59530e15	1.09987	0.549937	+	0.835206i	$0.314652\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.51972e15	1.00830
$576$	0	0
$577$	−2.25470e15	−1.46765	−0.733825	−	0.679338i	$-0.762267\pi$
$577$	−2.25470e15	−1.46765	−0.733825	+	0.679338i	$0.762267\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.89992e14	0.119060
$582$	0	0
$583$	7.78408e14	0.478663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.06087e13	0.0299720	0.0149860	−	0.999888i	$-0.495230\pi$
$587$	5.06087e13	0.0299720	0.0149860	+	0.999888i	$0.495230\pi$
$588$	0	0
$589$	−2.43437e15	−1.41499
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.09776e15	−0.614759	−0.307379	−	0.951587i	$-0.599452\pi$
$593$	−1.09776e15	−0.614759	−0.307379	+	0.951587i	$0.599452\pi$
$594$	0	0
$595$	2.08825e15	1.14799
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.60967e15	0.852884	0.426442	−	0.904515i	$-0.359767\pi$
$599$	1.60967e15	0.852884	0.426442	+	0.904515i	$0.359767\pi$
$600$	0	0
$601$	−2.29400e15	−1.19339	−0.596697	−	0.802466i	$-0.703521\pi$
$601$	−2.29400e15	−1.19339	−0.596697	+	0.802466i	$0.703521\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.30524e14	0.165786
$606$	0	0
$607$	−9.23068e14	−0.454669	−0.227335	−	0.973817i	$-0.573001\pi$
$607$	−9.23068e14	−0.454669	−0.227335	+	0.973817i	$0.573001\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.26245e14	0.440047
$612$	0	0
$613$	−2.89423e15	−1.35052	−0.675260	−	0.737580i	$-0.735969\pi$
$613$	−2.89423e15	−1.35052	−0.675260	+	0.737580i	$0.735969\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.35959e15	0.612122	0.306061	−	0.952012i	$-0.400989\pi$
$617$	1.35959e15	0.612122	0.306061	+	0.952012i	$0.400989\pi$
$618$	0	0
$619$	1.33242e15	0.589310	0.294655	−	0.955604i	$-0.404795\pi$
$619$	1.33242e15	0.589310	0.294655	+	0.955604i	$0.404795\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.20522e15	−0.514495
$624$	0	0
$625$	1.25567e15	0.526667
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.41287e15	−3.00198
$630$	0	0
$631$	−2.13346e15	−0.849030	−0.424515	−	0.905421i	$-0.639555\pi$
$631$	−2.13346e15	−0.849030	−0.424515	+	0.905421i	$0.639555\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.57165e14	−0.137282
$636$	0	0
$637$	1.35308e14	0.0511160
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.80999e14	−0.321556	−0.160778	−	0.986991i	$-0.551400\pi$
$641$	−8.80999e14	−0.321556	−0.160778	+	0.986991i	$0.551400\pi$
$642$	0	0
$643$	4.08569e15	1.46590	0.732952	−	0.680281i	$-0.238142\pi$
$643$	4.08569e15	1.46590	0.732952	+	0.680281i	$0.238142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.01301e15	1.04478	0.522392	−	0.852706i	$-0.325040\pi$
$647$	3.01301e15	1.04478	0.522392	+	0.852706i	$0.325040\pi$
$648$	0	0
$649$	3.47988e15	1.18637
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.56321e15	−1.17441	−0.587205	−	0.809439i	$-0.699772\pi$
$653$	−3.56321e15	−1.17441	−0.587205	+	0.809439i	$0.699772\pi$
$654$	0	0
$655$	−2.54908e15	−0.826147
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.50464e14	−0.0785011	−0.0392505	−	0.999229i	$-0.512497\pi$
$659$	−2.50464e14	−0.0785011	−0.0392505	+	0.999229i	$0.512497\pi$
$660$	0	0
$661$	1.87600e15	0.578263	0.289131	−	0.957289i	$-0.406634\pi$
$661$	1.87600e15	0.578263	0.289131	+	0.957289i	$0.406634\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.28841e15	−0.980542
$666$	0	0
$667$	2.90677e14	0.0852548
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.07505e15	−1.72415
$672$	0	0
$673$	2.96424e15	0.827618	0.413809	−	0.910364i	$-0.364198\pi$
$673$	2.96424e15	0.827618	0.413809	+	0.910364i	$0.364198\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.12731e15	−0.304654	−0.152327	−	0.988330i	$-0.548677\pi$
$677$	−1.12731e15	−0.304654	−0.152327	+	0.988330i	$0.548677\pi$
$678$	0	0
$679$	−2.28240e15	−0.606886
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.69609e15	1.72388	0.861941	−	0.507009i	$-0.169249\pi$
$683$	6.69609e15	1.72388	0.861941	+	0.507009i	$0.169249\pi$
$684$	0	0
$685$	−1.84098e15	−0.466391
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.66026e14	0.163413
$690$	0	0
$691$	8.47911e14	0.204749	0.102374	−	0.994746i	$-0.467356\pi$
$691$	8.47911e14	0.204749	0.102374	+	0.994746i	$0.467356\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.13949e14	0.0968345
$696$	0	0
$697$	7.13914e15	1.64386
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.56382e15	1.68769	0.843844	−	0.536589i	$-0.180287\pi$
$701$	7.56382e15	1.68769	0.843844	+	0.536589i	$0.180287\pi$
$702$	0	0
$703$	1.16732e16	2.56410
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.44081e14	−0.158421
$708$	0	0
$709$	4.10709e15	0.860954	0.430477	−	0.902602i	$-0.358345\pi$
$709$	4.10709e15	0.860954	0.430477	+	0.902602i	$0.358345\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.48481e15	0.505011
$714$	0	0
$715$	3.15395e15	0.631207
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.55176e15	−1.46568	−0.732840	−	0.680401i	$-0.761806\pi$
$719$	−7.55176e15	−1.46568	−0.732840	+	0.680401i	$0.761806\pi$
$720$	0	0
$721$	−7.14251e14	−0.136523
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.53205e15	0.284063
$726$	0	0
$727$	−3.83291e14	−0.0699986	−0.0349993	−	0.999387i	$-0.511143\pi$
$727$	−3.83291e14	−0.0699986	−0.0349993	+	0.999387i	$0.511143\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.10907e15	0.905307
$732$	0	0
$733$	−6.37162e15	−1.11219	−0.556094	−	0.831120i	$-0.687700\pi$
$733$	−6.37162e15	−1.11219	−0.556094	+	0.831120i	$0.687700\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.76108e15	0.637149
$738$	0	0
$739$	6.20162e15	1.03505	0.517524	−	0.855669i	$-0.326854\pi$
$739$	6.20162e15	1.03505	0.517524	+	0.855669i	$0.326854\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.37355e13	0.0103263	0.00516313	−	0.999987i	$-0.498357\pi$
$743$	6.37355e13	0.0103263	0.00516313	+	0.999987i	$0.498357\pi$
$744$	0	0
$745$	−5.52203e15	−0.881534
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.74319e15	−0.425213
$750$	0	0
$751$	−4.40400e15	−0.672710	−0.336355	−	0.941735i	$-0.609194\pi$
$751$	−4.40400e15	−0.672710	−0.336355	+	0.941735i	$0.609194\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.50028e15	−0.667623
$756$	0	0
$757$	2.53533e15	0.370686	0.185343	−	0.982674i	$-0.440660\pi$
$757$	2.53533e15	0.370686	0.185343	+	0.982674i	$0.440660\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.38430e15	0.338646	0.169323	−	0.985561i	$-0.445842\pi$
$761$	2.38430e15	0.338646	0.169323	+	0.985561i	$0.445842\pi$
$762$	0	0
$763$	2.95195e15	0.413261
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.97748e15	0.405017
$768$	0	0
$769$	6.46650e15	0.867110	0.433555	−	0.901127i	$-0.357259\pi$
$769$	6.46650e15	0.867110	0.433555	+	0.901127i	$0.357259\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.49994e15	0.847075	0.423537	−	0.905879i	$-0.360788\pi$
$773$	6.49994e15	0.847075	0.423537	+	0.905879i	$0.360788\pi$
$774$	0	0
$775$	1.30965e16	1.68266
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.12421e16	−1.40408
$780$	0	0
$781$	1.11065e16	1.36772
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.70468e16	3.23841
$786$	0	0
$787$	1.71364e15	0.202329	0.101165	−	0.994870i	$-0.467743\pi$
$787$	1.71364e15	0.202329	0.101165	+	0.994870i	$0.467743\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.22423e15	0.140570
$792$	0	0
$793$	−5.19797e15	−0.588615
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.99335e15	0.880457	0.440228	−	0.897886i	$-0.354897\pi$
$797$	7.99335e15	0.880457	0.440228	+	0.897886i	$0.354897\pi$
$798$	0	0
$799$	2.04279e16	2.21930
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.02922e16	−1.08786
$804$	0	0
$805$	3.35654e15	0.349957
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.25550e14	−0.0736123	−0.0368062	−	0.999322i	$-0.511718\pi$
$809$	−7.25550e14	−0.0736123	−0.0368062	+	0.999322i	$0.511718\pi$
$810$	0	0
$811$	−2.53094e15	−0.253319	−0.126659	−	0.991946i	$-0.540425\pi$
$811$	−2.53094e15	−0.253319	−0.126659	+	0.991946i	$0.540425\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.04569e15	−0.296700
$816$	0	0
$817$	−8.04534e15	−0.773256
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.57805e15	−0.802604	−0.401302	−	0.915946i	$-0.631442\pi$
$821$	−8.57805e15	−0.802604	−0.401302	+	0.915946i	$0.631442\pi$
$822$	0	0
$823$	−7.99435e15	−0.738047	−0.369023	−	0.929420i	$-0.620308\pi$
$823$	−7.99435e15	−0.738047	−0.369023	+	0.929420i	$0.620308\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.63911e16	1.47342	0.736711	−	0.676207i	$-0.236378\pi$
$827$	1.63911e16	1.47342	0.736711	+	0.676207i	$0.236378\pi$
$828$	0	0
$829$	−3.87286e15	−0.343544	−0.171772	−	0.985137i	$-0.554949\pi$
$829$	−3.87286e15	−0.343544	−0.171772	+	0.985137i	$0.554949\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.98415e15	0.257794
$834$	0	0
$835$	2.70337e16	2.30478
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.43100e16	−1.18836	−0.594182	−	0.804331i	$-0.702524\pi$
$839$	−1.43100e16	−1.18836	−0.594182	+	0.804331i	$0.702524\pi$
$840$	0	0
$841$	−1.19075e16	−0.975982
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.83794e16	−1.46764
$846$	0	0
$847$	4.72325e14	0.0372291
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.19151e16	−0.915131
$852$	0	0
$853$	−5.49586e15	−0.416693	−0.208346	−	0.978055i	$-0.566808\pi$
$853$	−5.49586e15	−0.416693	−0.208346	+	0.978055i	$0.566808\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.00345e16	−0.741480	−0.370740	−	0.928737i	$-0.620896\pi$
$857$	−1.00345e16	−0.741480	−0.370740	+	0.928737i	$0.620896\pi$
$858$	0	0
$859$	−8.18473e15	−0.597093	−0.298547	−	0.954395i	$-0.596502\pi$
$859$	−8.18473e15	−0.597093	−0.298547	+	0.954395i	$0.596502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.91848e16	−1.36427	−0.682133	−	0.731228i	$-0.738947\pi$
$863$	−1.91848e16	−1.36427	−0.682133	+	0.731228i	$0.738947\pi$
$864$	0	0
$865$	−1.54622e15	−0.108563
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.36062e16	−1.61591
$870$	0	0
$871$	3.21808e15	0.217519
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.03922e15	0.529870
$876$	0	0
$877$	1.17181e16	0.762710	0.381355	−	0.924429i	$-0.375457\pi$
$877$	1.17181e16	0.762710	0.381355	+	0.924429i	$0.375457\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.44379e16	0.916510	0.458255	−	0.888821i	$-0.348475\pi$
$881$	1.44379e16	0.916510	0.458255	+	0.888821i	$0.348475\pi$
$882$	0	0
$883$	−6.98181e15	−0.437708	−0.218854	−	0.975758i	$-0.570232\pi$
$883$	−6.98181e15	−0.437708	−0.218854	+	0.975758i	$0.570232\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.60566e15	−0.0981913	−0.0490956	−	0.998794i	$-0.515634\pi$
$887$	−1.60566e15	−0.0981913	−0.0490956	+	0.998794i	$0.515634\pi$
$888$	0	0
$889$	−5.10395e14	−0.0308281
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.21681e16	−1.89558
$894$	0	0
$895$	4.75896e16	2.77004
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.50498e15	0.142274
$900$	0	0
$901$	1.46889e16	0.824142
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.98484e16	1.63438
$906$	0	0
$907$	−1.18483e16	−0.640939	−0.320470	−	0.947259i	$-0.603841\pi$
$907$	−1.18483e16	−0.640939	−0.320470	+	0.947259i	$0.603841\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.50417e16	−0.794228	−0.397114	−	0.917769i	$-0.629988\pi$
$911$	−1.50417e16	−0.794228	−0.397114	+	0.917769i	$0.629988\pi$
$912$	0	0
$913$	6.32856e15	0.330153
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.64269e15	−0.185520
$918$	0	0
$919$	−2.94891e16	−1.48398	−0.741988	−	0.670413i	$-0.766117\pi$
$919$	−2.94891e16	−1.48398	−0.741988	+	0.670413i	$0.766117\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.50303e15	0.466931
$924$	0	0
$925$	−6.27998e16	−3.04915
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.35386e16	1.59022	0.795111	−	0.606463i	$-0.207412\pi$
$929$	3.35386e16	1.59022	0.795111	+	0.606463i	$0.207412\pi$
$930$	0	0
$931$	−4.69920e15	−0.220191
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.95588e16	3.18338
$936$	0	0
$937$	−3.74928e16	−1.69582	−0.847910	−	0.530140i	$-0.822139\pi$
$937$	−3.74928e16	−1.69582	−0.847910	+	0.530140i	$0.822139\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.19965e16	−1.41371	−0.706853	−	0.707360i	$-0.749886\pi$
$941$	−3.19965e16	−1.41371	−0.706853	+	0.707360i	$0.749886\pi$
$942$	0	0
$943$	1.14751e16	0.501120
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.01047e16	0.857773	0.428886	−	0.903358i	$-0.358906\pi$
$947$	2.01047e16	0.857773	0.428886	+	0.903358i	$0.358906\pi$
$948$	0	0
$949$	−8.80628e15	−0.371388
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.92253e16	1.20434	0.602168	−	0.798370i	$-0.294304\pi$
$953$	2.92253e16	1.20434	0.602168	+	0.798370i	$0.294304\pi$
$954$	0	0
$955$	5.95303e15	0.242504
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.63080e15	−0.104733
$960$	0	0
$961$	−3.99507e15	−0.157234
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.13237e16	0.435601
$966$	0	0
$967$	−4.12005e15	−0.156695	−0.0783477	−	0.996926i	$-0.524964\pi$
$967$	−4.12005e15	−0.156695	−0.0783477	+	0.996926i	$0.524964\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.16254e16	1.54758	0.773789	−	0.633444i	$-0.218359\pi$
$971$	4.16254e16	1.54758	0.773789	+	0.633444i	$0.218359\pi$
$972$	0	0
$973$	5.91541e14	0.0217452
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.43878e15	−0.231411	−0.115705	−	0.993284i	$-0.536913\pi$
$977$	−6.43878e15	−0.231411	−0.115705	+	0.993284i	$0.536913\pi$
$978$	0	0
$979$	−4.01453e16	−1.42669
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.02109e16	−0.354829	−0.177414	−	0.984136i	$-0.556773\pi$
$983$	−1.02109e16	−0.354829	−0.177414	+	0.984136i	$0.556773\pi$
$984$	0	0
$985$	−8.92120e16	−3.06566
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.21205e15	0.275976
$990$	0	0
$991$	4.36684e16	1.45132	0.725659	−	0.688054i	$-0.241535\pi$
$991$	4.36684e16	1.45132	0.725659	+	0.688054i	$0.241535\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.55324e16	−3.10544
$996$	0	0
$997$	−1.86792e16	−0.600529	−0.300265	−	0.953856i	$-0.597075\pi$
$997$	−1.86792e16	−0.600529	−0.300265	+	0.953856i	$0.597075\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.i.1.6	yes	6
3.2	odd	2	inner	252.12.a.i.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.a.i.1.1	✓	6	3.2	odd	2	inner
252.12.a.i.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+11761.2 q^{5} +16807.0 q^{7} +O(q^{10})\)	\(q+11761.2 q^{5} +16807.0 q^{7} +559834. q^{11} +479009. q^{13} +1.05643e7 q^{17} -1.66358e7 q^{19} +1.69805e7 q^{23} +8.94979e7 q^{25} +1.71183e7 q^{29} +1.46333e8 q^{31} +1.97671e8 q^{35} -7.01691e8 q^{37} +6.75780e8 q^{41} +4.83617e8 q^{43} +1.93367e9 q^{47} +2.82475e8 q^{49} +1.39043e9 q^{53} +6.58433e9 q^{55} +6.21591e9 q^{59} -1.08515e10 q^{61} +5.63372e9 q^{65} +6.71819e9 q^{67} +1.98389e10 q^{71} -1.83844e10 q^{73} +9.40914e9 q^{77} -4.21665e10 q^{79} +1.13043e10 q^{83} +1.24249e11 q^{85} -7.17093e10 q^{89} +8.05070e9 q^{91} -1.95657e11 q^{95} -1.35800e11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 100842 q^{7}+O(q^{10}) \) 6 * q + 100842 * q^7	\( 6 q + 100842 q^{7} + 1227468 q^{13} - 6532728 q^{19} + 116108394 q^{25} + 293608488 q^{31} - 74878260 q^{37} + 1432813080 q^{43} + 1694851494 q^{49} + 6150010104 q^{55} - 11105239740 q^{61} - 3754055112 q^{67} - 31466324076 q^{73} - 58630542864 q^{79} + 94916392200 q^{85} + 20630054676 q^{91} - 6046820556 q^{97}+O(q^{100}) \) 6 * q + 100842 * q^7 + 1227468 * q^13 - 6532728 * q^19 + 116108394 * q^25 + 293608488 * q^31 - 74878260 * q^37 + 1432813080 * q^43 + 1694851494 * q^49 + 6150010104 * q^55 - 11105239740 * q^61 - 3754055112 * q^67 - 31466324076 * q^73 - 58630542864 * q^79 + 94916392200 * q^85 + 20630054676 * q^91 - 6046820556 * q^97

Introduction

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