LMFDB - Embedded newform 252.12.a.g.1.3

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} - 7374950x + 3293545152 $ x^3 - x^2 - 7374950*x + 3293545152
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2457.02$ 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)

$f(q)$	$=$	$q+11462.1 q^{5} -16807.0 q^{7} +O(q^{10})$	$q+11462.1 q^{5} -16807.0 q^{7} +521496. q^{11} -216325. q^{13} -4.69820e6 q^{17} -1.95420e7 q^{19} +3.37634e7 q^{23} +8.25511e7 q^{25} -3.46253e7 q^{29} +2.80410e8 q^{31} -1.92643e8 q^{35} +3.26690e8 q^{37} -7.09150e8 q^{41} -7.33108e8 q^{43} +7.72527e8 q^{47} +2.82475e8 q^{49} +3.45010e9 q^{53} +5.97743e9 q^{55} +4.54344e9 q^{59} +5.97352e9 q^{61} -2.47953e9 q^{65} +1.20958e10 q^{67} -2.88580e10 q^{71} +1.84700e10 q^{73} -8.76479e9 q^{77} +4.27541e10 q^{79} +3.54084e9 q^{83} -5.38511e10 q^{85} +7.61542e10 q^{89} +3.63577e9 q^{91} -2.23992e11 q^{95} +8.60520e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4906 q^{5} - 50421 q^{7}+O(q^{10}) $ 3 * q + 4906 * q^5 - 50421 * q^7	$ 3 q + 4906 q^{5} - 50421 q^{7} + 224648 q^{11} - 780126 q^{13} - 2637706 q^{17} - 6259572 q^{19} - 30125524 q^{23} + 97536981 q^{25} - 201167450 q^{29} + 138368808 q^{31} - 82455142 q^{35} + 342460074 q^{37} - 1248586074 q^{41} - 208844388 q^{43} - 341757384 q^{47} + 847425747 q^{49} - 398683482 q^{53} + 2599266576 q^{55} + 5742813868 q^{59} + 2163574362 q^{61} - 8719808676 q^{65} + 15756982092 q^{67} - 37445810188 q^{71} + 35819108550 q^{73} - 3775658936 q^{77} + 110751115992 q^{79} - 96804724516 q^{83} + 48367693524 q^{85} - 24437643210 q^{89} + 13111577682 q^{91} - 234084845848 q^{95} + 106628361774 q^{97}+O(q^{100}) $ 3 * q + 4906 * q^5 - 50421 * q^7 + 224648 * q^11 - 780126 * q^13 - 2637706 * q^17 - 6259572 * q^19 - 30125524 * q^23 + 97536981 * q^25 - 201167450 * q^29 + 138368808 * q^31 - 82455142 * q^35 + 342460074 * q^37 - 1248586074 * q^41 - 208844388 * q^43 - 341757384 * q^47 + 847425747 * q^49 - 398683482 * q^53 + 2599266576 * q^55 + 5742813868 * q^59 + 2163574362 * q^61 - 8719808676 * q^65 + 15756982092 * q^67 - 37445810188 * q^71 + 35819108550 * q^73 - 3775658936 * q^77 + 110751115992 * q^79 - 96804724516 * q^83 + 48367693524 * q^85 - 24437643210 * q^89 + 13111577682 * q^91 - 234084845848 * q^95 + 106628361774 * 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
$5$	11462.1	1.64032	0.820159	−	0.572135i	$-0.193885\pi$
$5$	11462.1	1.64032	0.820159	+	0.572135i	$0.193885\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	521496.	0.976318	0.488159	−	0.872755i	$-0.337669\pi$
$11$	521496.	0.976318	0.488159	+	0.872755i	$0.337669\pi$
$12$	0	0
$13$	−216325.	−0.161591	−0.0807957	−	0.996731i	$-0.525746\pi$
$13$	−216325.	−0.161591	−0.0807957	+	0.996731i	$0.525746\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.69820e6	−0.802532	−0.401266	−	0.915962i	$-0.631430\pi$
$17$	−4.69820e6	−0.802532	−0.401266	+	0.915962i	$0.631430\pi$
$18$	0	0
$19$	−1.95420e7	−1.81061	−0.905303	−	0.424767i	$-0.860356\pi$
$19$	−1.95420e7	−1.81061	−0.905303	+	0.424767i	$0.860356\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.37634e7	1.09381	0.546907	−	0.837193i	$-0.315805\pi$
$23$	3.37634e7	1.09381	0.546907	+	0.837193i	$0.315805\pi$
$24$	0	0
$25$	8.25511e7	1.69065
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.46253e7	−0.313477	−0.156738	−	0.987640i	$-0.550098\pi$
$29$	−3.46253e7	−0.313477	−0.156738	+	0.987640i	$0.550098\pi$
$30$	0	0
$31$	2.80410e8	1.75915	0.879577	−	0.475757i	$-0.157826\pi$
$31$	2.80410e8	1.75915	0.879577	+	0.475757i	$0.157826\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.92643e8	−0.619982
$36$	0	0
$37$	3.26690e8	0.774509	0.387254	−	0.921973i	$-0.373424\pi$
$37$	3.26690e8	0.774509	0.387254	+	0.921973i	$0.373424\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.09150e8	−0.955932	−0.477966	−	0.878378i	$-0.658626\pi$
$41$	−7.09150e8	−0.955932	−0.477966	+	0.878378i	$0.658626\pi$
$42$	0	0
$43$	−7.33108e8	−0.760486	−0.380243	−	0.924887i	$-0.624160\pi$
$43$	−7.33108e8	−0.760486	−0.380243	+	0.924887i	$0.624160\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.72527e8	0.491333	0.245666	−	0.969354i	$-0.420993\pi$
$47$	7.72527e8	0.491333	0.245666	+	0.969354i	$0.420993\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.45010e9	1.13322	0.566610	−	0.823986i	$-0.308255\pi$
$53$	3.45010e9	1.13322	0.566610	+	0.823986i	$0.308255\pi$
$54$	0	0
$55$	5.97743e9	1.60147
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.54344e9	0.827368	0.413684	−	0.910421i	$-0.364242\pi$
$59$	4.54344e9	0.827368	0.413684	+	0.910421i	$0.364242\pi$
$60$	0	0
$61$	5.97352e9	0.905558	0.452779	−	0.891623i	$-0.350433\pi$
$61$	5.97352e9	0.905558	0.452779	+	0.891623i	$0.350433\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.47953e9	−0.265061
$66$	0	0
$67$	1.20958e10	1.09452	0.547261	−	0.836962i	$-0.315670\pi$
$67$	1.20958e10	1.09452	0.547261	+	0.836962i	$0.315670\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.88580e10	−1.89821	−0.949106	−	0.314956i	$-0.898010\pi$
$71$	−2.88580e10	−1.89821	−0.949106	+	0.314956i	$0.898010\pi$
$72$	0	0
$73$	1.84700e10	1.04278	0.521388	−	0.853319i	$-0.325414\pi$
$73$	1.84700e10	1.04278	0.521388	+	0.853319i	$0.325414\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.76479e9	−0.369014
$78$	0	0
$79$	4.27541e10	1.56325	0.781626	−	0.623747i	$-0.214390\pi$
$79$	4.27541e10	1.56325	0.781626	+	0.623747i	$0.214390\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.54084e9	0.0986681	0.0493341	−	0.998782i	$-0.484290\pi$
$83$	3.54084e9	0.0986681	0.0493341	+	0.998782i	$0.484290\pi$
$84$	0	0
$85$	−5.38511e10	−1.31641
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.61542e10	1.44560	0.722801	−	0.691056i	$-0.242854\pi$
$89$	7.61542e10	1.44560	0.722801	+	0.691056i	$0.242854\pi$
$90$	0	0
$91$	3.63577e9	0.0610758
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.23992e11	−2.96997
$96$	0	0
$97$	8.60520e10	1.01746	0.508729	−	0.860927i	$-0.330116\pi$
$97$	8.60520e10	1.01746	0.508729	+	0.860927i	$0.330116\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.08373e11	−1.02602	−0.513008	−	0.858384i	$-0.671469\pi$
$101$	−1.08373e11	−1.02602	−0.513008	+	0.858384i	$0.671469\pi$
$102$	0	0
$103$	−1.89496e11	−1.61063	−0.805313	−	0.592850i	$-0.798003\pi$
$103$	−1.89496e11	−1.61063	−0.805313	+	0.592850i	$0.798003\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.60474e10	−0.593099	−0.296550	−	0.955017i	$-0.595836\pi$
$107$	−8.60474e10	−0.593099	−0.296550	+	0.955017i	$0.595836\pi$
$108$	0	0
$109$	−5.80676e10	−0.361483	−0.180742	−	0.983531i	$-0.557850\pi$
$109$	−5.80676e10	−0.361483	−0.180742	+	0.983531i	$0.557850\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.26354e11	−1.15573	−0.577865	−	0.816132i	$-0.696114\pi$
$113$	−2.26354e11	−1.15573	−0.577865	+	0.816132i	$0.696114\pi$
$114$	0	0
$115$	3.86999e11	1.79420
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.89626e10	0.303328
$120$	0	0
$121$	−1.33533e10	−0.0468025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.86535e11	1.13288
$126$	0	0
$127$	−1.32369e11	−0.355522	−0.177761	−	0.984074i	$-0.556885\pi$
$127$	−1.32369e11	−0.355522	−0.177761	+	0.984074i	$0.556885\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.71951e11	1.52176	0.760879	−	0.648893i	$-0.224768\pi$
$131$	6.71951e11	1.52176	0.760879	+	0.648893i	$0.224768\pi$
$132$	0	0
$133$	3.28442e11	0.684345
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.02604e11	−0.535688	−0.267844	−	0.963462i	$-0.586311\pi$
$137$	−3.02604e11	−0.535688	−0.267844	+	0.963462i	$0.586311\pi$
$138$	0	0
$139$	−1.15510e11	−0.188816	−0.0944079	−	0.995534i	$-0.530096\pi$
$139$	−1.15510e11	−0.188816	−0.0944079	+	0.995534i	$0.530096\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.12813e11	−0.157765
$144$	0	0
$145$	−3.96878e11	−0.514202
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.76820e11	−0.308797	−0.154398	−	0.988009i	$-0.549344\pi$
$149$	−2.76820e11	−0.308797	−0.154398	+	0.988009i	$0.549344\pi$
$150$	0	0
$151$	9.21440e11	0.955199	0.477600	−	0.878578i	$-0.341507\pi$
$151$	9.21440e11	0.955199	0.477600	+	0.878578i	$0.341507\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.21408e12	2.88557
$156$	0	0
$157$	5.32196e11	0.445270	0.222635	−	0.974902i	$-0.428534\pi$
$157$	5.32196e11	0.445270	0.222635	+	0.974902i	$0.428534\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.67462e11	−0.413423
$162$	0	0
$163$	1.97444e12	1.34404	0.672018	−	0.740535i	$-0.265428\pi$
$163$	1.97444e12	1.34404	0.672018	+	0.740535i	$0.265428\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.07988e12	−0.643332	−0.321666	−	0.946853i	$-0.604243\pi$
$167$	−1.07988e12	−0.643332	−0.321666	+	0.946853i	$0.604243\pi$
$168$	0	0
$169$	−1.74536e12	−0.973888
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.61962e12	1.28524	0.642621	−	0.766184i	$-0.277847\pi$
$173$	2.61962e12	1.28524	0.642621	+	0.766184i	$0.277847\pi$
$174$	0	0
$175$	−1.38744e12	−0.639004
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.14941e12	0.874235	0.437118	−	0.899404i	$-0.355999\pi$
$179$	2.14941e12	0.874235	0.437118	+	0.899404i	$0.355999\pi$
$180$	0	0
$181$	−2.45208e12	−0.938217	−0.469108	−	0.883141i	$-0.655425\pi$
$181$	−2.45208e12	−0.938217	−0.469108	+	0.883141i	$0.655425\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.74455e12	1.27044
$186$	0	0
$187$	−2.45009e12	−0.783526
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.28525e12	1.50446	0.752232	−	0.658898i	$-0.228977\pi$
$191$	5.28525e12	1.50446	0.752232	+	0.658898i	$0.228977\pi$
$192$	0	0
$193$	−1.95578e12	−0.525720	−0.262860	−	0.964834i	$-0.584666\pi$
$193$	−1.95578e12	−0.525720	−0.262860	+	0.964834i	$0.584666\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.19029e11	0.0525941	0.0262970	−	0.999654i	$-0.491628\pi$
$197$	2.19029e11	0.0525941	0.0262970	+	0.999654i	$0.491628\pi$
$198$	0	0
$199$	4.85157e12	1.10202	0.551011	−	0.834498i	$-0.314242\pi$
$199$	4.85157e12	1.10202	0.551011	+	0.834498i	$0.314242\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.81948e11	0.118483
$204$	0	0
$205$	−8.12834e12	−1.56803
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.01911e13	−1.76773
$210$	0	0
$211$	4.65891e12	0.766886	0.383443	−	0.923564i	$-0.374738\pi$
$211$	4.65891e12	0.766886	0.383443	+	0.923564i	$0.374738\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.40294e12	−1.24744
$216$	0	0
$217$	−4.71285e12	−0.664897
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.01634e12	0.129682
$222$	0	0
$223$	4.98414e12	0.605220	0.302610	−	0.953114i	$-0.402142\pi$
$223$	4.98414e12	0.605220	0.302610	+	0.953114i	$0.402142\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.70925e13	1.88219	0.941093	−	0.338148i	$-0.109800\pi$
$227$	1.70925e13	1.88219	0.941093	+	0.338148i	$0.109800\pi$
$228$	0	0
$229$	1.19187e13	1.25064	0.625322	−	0.780367i	$-0.284968\pi$
$229$	1.19187e13	1.25064	0.625322	+	0.780367i	$0.284968\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.23184e12	0.212915	0.106458	−	0.994317i	$-0.466049\pi$
$233$	2.23184e12	0.212915	0.106458	+	0.994317i	$0.466049\pi$
$234$	0	0
$235$	8.85477e12	0.805942
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.51458e11	−0.0789226	−0.0394613	−	0.999221i	$-0.512564\pi$
$239$	−9.51458e11	−0.0789226	−0.0394613	+	0.999221i	$0.512564\pi$
$240$	0	0
$241$	1.14944e13	0.910734	0.455367	−	0.890304i	$-0.349508\pi$
$241$	1.14944e13	0.910734	0.455367	+	0.890304i	$0.349508\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.23775e12	0.234331
$246$	0	0
$247$	4.22742e12	0.292578
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.87270e13	1.82006	0.910030	−	0.414543i	$-0.136059\pi$
$251$	2.87270e13	1.82006	0.910030	+	0.414543i	$0.136059\pi$
$252$	0	0
$253$	1.76075e13	1.06791
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.59506e12	0.144383	0.0721914	−	0.997391i	$-0.477001\pi$
$257$	2.59506e12	0.144383	0.0721914	+	0.997391i	$0.477001\pi$
$258$	0	0
$259$	−5.49068e12	−0.292737
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.86392e12	0.140347	0.0701737	−	0.997535i	$-0.477645\pi$
$263$	2.86392e12	0.140347	0.0701737	+	0.997535i	$0.477645\pi$
$264$	0	0
$265$	3.95453e13	1.85884
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.46444e13	−1.06679	−0.533397	−	0.845865i	$-0.679085\pi$
$269$	−2.46444e13	−1.06679	−0.533397	+	0.845865i	$0.679085\pi$
$270$	0	0
$271$	2.09412e13	0.870304	0.435152	−	0.900357i	$-0.356695\pi$
$271$	2.09412e13	0.870304	0.435152	+	0.900357i	$0.356695\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.30501e13	1.65061
$276$	0	0
$277$	2.86147e13	1.05427	0.527133	−	0.849783i	$-0.323267\pi$
$277$	2.86147e13	1.05427	0.527133	+	0.849783i	$0.323267\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.66627e13	1.24836	0.624180	−	0.781281i	$-0.285433\pi$
$281$	3.66627e13	1.24836	0.624180	+	0.781281i	$0.285433\pi$
$282$	0	0
$283$	−8.34377e12	−0.273235	−0.136618	−	0.990624i	$-0.543623\pi$
$283$	−8.34377e12	−0.273235	−0.136618	+	0.990624i	$0.543623\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.19187e13	0.361308
$288$	0	0
$289$	−1.21988e13	−0.355943
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.22086e13	−1.14190	−0.570951	−	0.820984i	$-0.693425\pi$
$293$	−4.22086e13	−1.14190	−0.570951	+	0.820984i	$0.693425\pi$
$294$	0	0
$295$	5.20773e13	1.35715
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.30387e12	−0.176751
$300$	0	0
$301$	1.23213e13	0.287437
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.84690e13	1.48540
$306$	0	0
$307$	−7.18404e13	−1.50352	−0.751758	−	0.659440i	$-0.770794\pi$
$307$	−7.18404e13	−1.50352	−0.751758	+	0.659440i	$0.770794\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.66872e13	0.520142	0.260071	−	0.965590i	$-0.416254\pi$
$311$	2.66872e13	0.520142	0.260071	+	0.965590i	$0.416254\pi$
$312$	0	0
$313$	−7.18999e13	−1.35280	−0.676401	−	0.736534i	$-0.736461\pi$
$313$	−7.18999e13	−1.35280	−0.676401	+	0.736534i	$0.736461\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.90302e13	−1.03573	−0.517867	−	0.855461i	$-0.673274\pi$
$317$	−5.90302e13	−1.03573	−0.517867	+	0.855461i	$0.673274\pi$
$318$	0	0
$319$	−1.80570e13	−0.306053
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.18121e13	1.45307
$324$	0	0
$325$	−1.78579e13	−0.273194
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.29839e13	−0.185706
$330$	0	0
$331$	−5.24280e12	−0.0725286	−0.0362643	−	0.999342i	$-0.511546\pi$
$331$	−5.24280e12	−0.0725286	−0.0362643	+	0.999342i	$0.511546\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.38643e14	1.79537
$336$	0	0
$337$	1.22762e14	1.53850	0.769251	−	0.638947i	$-0.220630\pi$
$337$	1.22762e14	1.53850	0.769251	+	0.638947i	$0.220630\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.46233e14	1.71749
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.53108e13	−0.910315	−0.455158	−	0.890411i	$-0.650417\pi$
$347$	−8.53108e13	−0.910315	−0.455158	+	0.890411i	$0.650417\pi$
$348$	0	0
$349$	−1.68939e14	−1.74658	−0.873292	−	0.487197i	$-0.838019\pi$
$349$	−1.68939e14	−1.74658	−0.873292	+	0.487197i	$0.838019\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.60735e13	0.835812	0.417906	−	0.908490i	$-0.362764\pi$
$353$	8.60735e13	0.835812	0.417906	+	0.908490i	$0.362764\pi$
$354$	0	0
$355$	−3.30772e14	−3.11367
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.85522e13	−0.252709	−0.126355	−	0.991985i	$-0.540328\pi$
$359$	−2.85522e13	−0.252709	−0.126355	+	0.991985i	$0.540328\pi$
$360$	0	0
$361$	2.65399e14	2.27829
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.11705e14	1.71049
$366$	0	0
$367$	1.59707e14	1.25216	0.626082	−	0.779758i	$-0.284658\pi$
$367$	1.59707e14	1.25216	0.626082	+	0.779758i	$0.284658\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.79858e13	−0.428317
$372$	0	0
$373$	−4.08909e13	−0.293244	−0.146622	−	0.989193i	$-0.546840\pi$
$373$	−4.08909e13	−0.293244	−0.146622	+	0.989193i	$0.546840\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.49032e12	0.0506551
$378$	0	0
$379$	1.26417e14	0.830403	0.415202	−	0.909729i	$-0.363711\pi$
$379$	1.26417e14	0.830403	0.415202	+	0.909729i	$0.363711\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.31828e14	−1.43738	−0.718691	−	0.695329i	$-0.755259\pi$
$383$	−2.31828e14	−1.43738	−0.718691	+	0.695329i	$0.755259\pi$
$384$	0	0
$385$	−1.00463e14	−0.605300
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.62173e14	−0.923115	−0.461557	−	0.887110i	$-0.652709\pi$
$389$	−1.62173e14	−0.923115	−0.461557	+	0.887110i	$0.652709\pi$
$390$	0	0
$391$	−1.58627e14	−0.877821
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.90051e14	2.56423
$396$	0	0
$397$	2.20030e14	1.11978	0.559892	−	0.828566i	$-0.310843\pi$
$397$	2.20030e14	1.11978	0.559892	+	0.828566i	$0.310843\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.18617e14	−1.05291	−0.526454	−	0.850204i	$-0.676479\pi$
$401$	−2.18617e14	−1.05291	−0.526454	+	0.850204i	$0.676479\pi$
$402$	0	0
$403$	−6.06596e13	−0.284264
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.70368e14	0.756167
$408$	0	0
$409$	1.50462e14	0.650055	0.325027	−	0.945705i	$-0.394626\pi$
$409$	1.50462e14	0.650055	0.325027	+	0.945705i	$0.394626\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.63616e13	−0.312716
$414$	0	0
$415$	4.05854e13	0.161847
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.64327e11	0.00326965	0.00163482	−	0.999999i	$-0.499480\pi$
$419$	8.64327e11	0.00326965	0.00163482	+	0.999999i	$0.499480\pi$
$420$	0	0
$421$	4.40714e14	1.62407	0.812037	−	0.583606i	$-0.198359\pi$
$421$	4.40714e14	1.62407	0.812037	+	0.583606i	$0.198359\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.87841e14	−1.35680
$426$	0	0
$427$	−1.00397e14	−0.342269
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.68307e14	1.19285	0.596424	−	0.802669i	$-0.296588\pi$
$431$	3.68307e14	1.19285	0.596424	+	0.802669i	$0.296588\pi$
$432$	0	0
$433$	−2.00831e13	−0.0634084	−0.0317042	−	0.999497i	$-0.510093\pi$
$433$	−2.00831e13	−0.0634084	−0.0317042	+	0.999497i	$0.510093\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.59805e14	−1.98047
$438$	0	0
$439$	1.67733e14	0.490980	0.245490	−	0.969399i	$-0.421051\pi$
$439$	1.67733e14	0.490980	0.245490	+	0.969399i	$0.421051\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.89295e14	−1.36254	−0.681272	−	0.732031i	$-0.738573\pi$
$443$	−4.89295e14	−1.36254	−0.681272	+	0.732031i	$0.738573\pi$
$444$	0	0
$445$	8.72885e14	2.37125
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.45098e14	−0.633849	−0.316924	−	0.948451i	$-0.602650\pi$
$449$	−2.45098e14	−0.633849	−0.316924	+	0.948451i	$0.602650\pi$
$450$	0	0
$451$	−3.69819e14	−0.933294
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.16735e13	0.100184
$456$	0	0
$457$	−1.03215e14	−0.242216	−0.121108	−	0.992639i	$-0.538645\pi$
$457$	−1.03215e14	−0.242216	−0.121108	+	0.992639i	$0.538645\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.39700e14	−1.43094	−0.715470	−	0.698644i	$-0.753787\pi$
$461$	−6.39700e14	−1.43094	−0.715470	+	0.698644i	$0.753787\pi$
$462$	0	0
$463$	2.17103e14	0.474210	0.237105	−	0.971484i	$-0.423801\pi$
$463$	2.17103e14	0.474210	0.237105	+	0.971484i	$0.423801\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.76625e14	−0.992964	−0.496482	−	0.868047i	$-0.665375\pi$
$467$	−4.76625e14	−0.992964	−0.496482	+	0.868047i	$0.665375\pi$
$468$	0	0
$469$	−2.03295e14	−0.413691
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.82313e14	−0.742477
$474$	0	0
$475$	−1.61321e15	−3.06109
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.42792e14	0.258736	0.129368	−	0.991597i	$-0.458705\pi$
$479$	1.42792e14	0.258736	0.129368	+	0.991597i	$0.458705\pi$
$480$	0	0
$481$	−7.06712e13	−0.125154
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.86334e14	1.66895
$486$	0	0
$487$	−1.01487e15	−1.67880	−0.839401	−	0.543512i	$-0.817094\pi$
$487$	−1.01487e15	−1.67880	−0.839401	+	0.543512i	$0.817094\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.71010e14	−1.21930	−0.609652	−	0.792669i	$-0.708691\pi$
$491$	−7.71010e14	−1.21930	−0.609652	+	0.792669i	$0.708691\pi$
$492$	0	0
$493$	1.62677e14	0.251575
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.85016e14	0.717457
$498$	0	0
$499$	−1.71366e14	−0.247954	−0.123977	−	0.992285i	$-0.539565\pi$
$499$	−1.71366e14	−0.247954	−0.123977	+	0.992285i	$0.539565\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.57307e14	1.32565	0.662823	−	0.748776i	$-0.269358\pi$
$503$	9.57307e14	1.32565	0.662823	+	0.748776i	$0.269358\pi$
$504$	0	0
$505$	−1.24218e15	−1.68299
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.18387e15	−1.53588	−0.767938	−	0.640525i	$-0.778717\pi$
$509$	−1.18387e15	−1.53588	−0.767938	+	0.640525i	$0.778717\pi$
$510$	0	0
$511$	−3.10425e14	−0.394133
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.17201e15	−2.64194
$516$	0	0
$517$	4.02870e14	0.479697
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.02501e15	1.16982	0.584911	−	0.811098i	$-0.301129\pi$
$521$	1.02501e15	1.16982	0.584911	+	0.811098i	$0.301129\pi$
$522$	0	0
$523$	1.25136e15	1.39837	0.699186	−	0.714940i	$-0.253546\pi$
$523$	1.25136e15	1.39837	0.699186	+	0.714940i	$0.253546\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.31742e15	−1.41178
$528$	0	0
$529$	1.87160e14	0.196430
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.53407e14	0.154470
$534$	0	0
$535$	−9.86282e14	−0.972872
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.47310e14	0.139474
$540$	0	0
$541$	−8.05229e14	−0.747024	−0.373512	−	0.927625i	$-0.621846\pi$
$541$	−8.05229e14	−0.747024	−0.373512	+	0.927625i	$0.621846\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.65575e14	−0.592948
$546$	0	0
$547$	2.25608e15	1.96981	0.984904	−	0.173103i	$-0.0553793\pi$
$547$	2.25608e15	1.96981	0.984904	+	0.173103i	$0.0553793\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.76648e14	0.567583
$552$	0	0
$553$	−7.18569e14	−0.590854
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.03811e15	−0.820430	−0.410215	−	0.911989i	$-0.634546\pi$
$557$	−1.03811e15	−0.820430	−0.410215	+	0.911989i	$0.634546\pi$
$558$	0	0
$559$	1.58589e14	0.122888
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.19994e15	0.894050	0.447025	−	0.894521i	$-0.352483\pi$
$563$	1.19994e15	0.894050	0.447025	+	0.894521i	$0.352483\pi$
$564$	0	0
$565$	−2.59448e15	−1.89577
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.64560e13	0.0677972	0.0338986	−	0.999425i	$-0.489208\pi$
$569$	9.64560e13	0.0677972	0.0338986	+	0.999425i	$0.489208\pi$
$570$	0	0
$571$	1.76977e15	1.22016	0.610082	−	0.792338i	$-0.291136\pi$
$571$	1.76977e15	1.22016	0.610082	+	0.792338i	$0.291136\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.78721e15	1.84925
$576$	0	0
$577$	9.30564e14	0.605731	0.302865	−	0.953033i	$-0.402057\pi$
$577$	9.30564e14	0.605731	0.302865	+	0.953033i	$0.402057\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.95109e13	−0.0372930
$582$	0	0
$583$	1.79921e15	1.10638
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.05599e15	1.21762	0.608808	−	0.793317i	$-0.291648\pi$
$587$	2.05599e15	1.21762	0.608808	+	0.793317i	$0.291648\pi$
$588$	0	0
$589$	−5.47976e15	−3.18513
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.01289e14	0.0567234	0.0283617	−	0.999598i	$-0.490971\pi$
$593$	1.01289e14	0.0567234	0.0283617	+	0.999598i	$0.490971\pi$
$594$	0	0
$595$	9.05075e14	0.497555
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.37396e15	−1.25784	−0.628920	−	0.777470i	$-0.716503\pi$
$599$	−2.37396e15	−1.25784	−0.628920	+	0.777470i	$0.716503\pi$
$600$	0	0
$601$	−2.22499e15	−1.15749	−0.578746	−	0.815508i	$-0.696458\pi$
$601$	−2.22499e15	−1.15749	−0.578746	+	0.815508i	$0.696458\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.53056e14	−0.0767710
$606$	0	0
$607$	5.79566e14	0.285473	0.142736	−	0.989761i	$-0.454410\pi$
$607$	5.79566e14	0.285473	0.142736	+	0.989761i	$0.454410\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.67117e14	−0.0793951
$612$	0	0
$613$	3.13049e15	1.46076	0.730381	−	0.683040i	$-0.239343\pi$
$613$	3.13049e15	1.46076	0.730381	+	0.683040i	$0.239343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.73178e15	0.779694	0.389847	−	0.920880i	$-0.372528\pi$
$617$	1.73178e15	0.779694	0.389847	+	0.920880i	$0.372528\pi$
$618$	0	0
$619$	−2.84777e15	−1.25953	−0.629763	−	0.776787i	$-0.716848\pi$
$619$	−2.84777e15	−1.25953	−0.629763	+	0.776787i	$0.716848\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.27992e15	−0.546386
$624$	0	0
$625$	3.99680e14	0.167638
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.53485e15	−0.621568
$630$	0	0
$631$	−2.72340e15	−1.08380	−0.541900	−	0.840443i	$-0.682295\pi$
$631$	−2.72340e15	−1.08380	−0.541900	+	0.840443i	$0.682295\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.51723e15	−0.583170
$636$	0	0
$637$	−6.11064e13	−0.0230845
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.89436e15	−1.05641	−0.528206	−	0.849116i	$-0.677135\pi$
$641$	−2.89436e15	−1.05641	−0.528206	+	0.849116i	$0.677135\pi$
$642$	0	0
$643$	4.08007e15	1.46389	0.731943	−	0.681366i	$-0.238614\pi$
$643$	4.08007e15	1.46389	0.731943	+	0.681366i	$0.238614\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.08478e15	−1.06967	−0.534836	−	0.844956i	$-0.679626\pi$
$647$	−3.08478e15	−1.06967	−0.534836	+	0.844956i	$0.679626\pi$
$648$	0	0
$649$	2.36939e15	0.807775
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.61118e15	−1.19022	−0.595110	−	0.803644i	$-0.702892\pi$
$653$	−3.61118e15	−1.19022	−0.595110	+	0.803644i	$0.702892\pi$
$654$	0	0
$655$	7.70196e15	2.49617
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.26708e14	0.0710553	0.0355276	−	0.999369i	$-0.488689\pi$
$659$	2.26708e14	0.0710553	0.0355276	+	0.999369i	$0.488689\pi$
$660$	0	0
$661$	−4.22422e15	−1.30208	−0.651042	−	0.759042i	$-0.725668\pi$
$661$	−4.22422e15	−1.30208	−0.651042	+	0.759042i	$0.725668\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.76463e15	1.12254
$666$	0	0
$667$	−1.16907e15	−0.342885
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.11517e15	0.884113
$672$	0	0
$673$	−3.74585e14	−0.104585	−0.0522923	−	0.998632i	$-0.516653\pi$
$673$	−3.74585e14	−0.104585	−0.0522923	+	0.998632i	$0.516653\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.88728e15	0.780281	0.390140	−	0.920755i	$-0.372426\pi$
$677$	2.88728e15	0.780281	0.390140	+	0.920755i	$0.372426\pi$
$678$	0	0
$679$	−1.44628e15	−0.384563
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.82863e15	−0.470775	−0.235387	−	0.971902i	$-0.575636\pi$
$683$	−1.82863e15	−0.470775	−0.235387	+	0.971902i	$0.575636\pi$
$684$	0	0
$685$	−3.46847e15	−0.878699
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.46342e14	−0.183118
$690$	0	0
$691$	7.54776e15	1.82259	0.911294	−	0.411755i	$-0.135084\pi$
$691$	7.54776e15	1.82259	0.911294	+	0.411755i	$0.135084\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.32398e15	−0.309718
$696$	0	0
$697$	3.33173e15	0.767166
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.81626e15	−0.851509	−0.425755	−	0.904839i	$-0.639991\pi$
$701$	−3.81626e15	−0.851509	−0.425755	+	0.904839i	$0.639991\pi$
$702$	0	0
$703$	−6.38417e15	−1.40233
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.82143e15	0.387798
$708$	0	0
$709$	6.43362e15	1.34866	0.674328	−	0.738432i	$-0.264433\pi$
$709$	6.43362e15	1.34866	0.674328	+	0.738432i	$0.264433\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.46760e15	1.92419
$714$	0	0
$715$	−1.29307e15	−0.258784
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.57652e14	0.127640	0.0638200	−	0.997961i	$-0.479672\pi$
$719$	6.57652e14	0.127640	0.0638200	+	0.997961i	$0.479672\pi$
$720$	0	0
$721$	3.18485e15	0.608759
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.85836e15	−0.529978
$726$	0	0
$727$	−1.36387e15	−0.249077	−0.124539	−	0.992215i	$-0.539745\pi$
$727$	−1.36387e15	−0.249077	−0.124539	+	0.992215i	$0.539745\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.44429e15	0.610314
$732$	0	0
$733$	3.20327e15	0.559141	0.279571	−	0.960125i	$-0.409808\pi$
$733$	3.20327e15	0.559141	0.279571	+	0.960125i	$0.409808\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.30794e15	1.06860
$738$	0	0
$739$	3.33543e14	0.0556682	0.0278341	−	0.999613i	$-0.491139\pi$
$739$	3.33543e14	0.0556682	0.0278341	+	0.999613i	$0.491139\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.87467e15	−0.627764	−0.313882	−	0.949462i	$-0.601630\pi$
$743$	−3.87467e15	−0.627764	−0.313882	+	0.949462i	$0.601630\pi$
$744$	0	0
$745$	−3.17293e15	−0.506525
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.44620e15	0.224170
$750$	0	0
$751$	1.08434e16	1.65633	0.828165	−	0.560485i	$-0.189385\pi$
$751$	1.08434e16	1.65633	0.828165	+	0.560485i	$0.189385\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.05616e16	1.56683
$756$	0	0
$757$	−9.21826e15	−1.34779	−0.673894	−	0.738828i	$-0.735380\pi$
$757$	−9.21826e15	−1.34779	−0.673894	+	0.738828i	$0.735380\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.31856e15	−0.471340	−0.235670	−	0.971833i	$-0.575729\pi$
$761$	−3.31856e15	−0.471340	−0.235670	+	0.971833i	$0.575729\pi$
$762$	0	0
$763$	9.75942e14	0.136628
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.82859e14	−0.133695
$768$	0	0
$769$	1.20004e16	1.60916	0.804580	−	0.593844i	$-0.202390\pi$
$769$	1.20004e16	1.60916	0.804580	+	0.593844i	$0.202390\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.32556e16	1.72747	0.863735	−	0.503946i	$-0.168119\pi$
$773$	1.32556e16	1.72747	0.863735	+	0.503946i	$0.168119\pi$
$774$	0	0
$775$	2.31481e16	2.97411
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.38582e16	1.73082
$780$	0	0
$781$	−1.50493e16	−1.85326
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.10007e15	0.730385
$786$	0	0
$787$	−1.84000e15	−0.217249	−0.108624	−	0.994083i	$-0.534645\pi$
$787$	−1.84000e15	−0.217249	−0.108624	+	0.994083i	$0.534645\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.80433e15	0.436825
$792$	0	0
$793$	−1.29222e15	−0.146330
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.05915e16	1.16664	0.583319	−	0.812243i	$-0.301754\pi$
$797$	1.05915e16	1.16664	0.583319	+	0.812243i	$0.301754\pi$
$798$	0	0
$799$	−3.62949e15	−0.394310
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.63204e15	1.01808
$804$	0	0
$805$	−6.50429e15	−0.678145
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.12229e15	−0.519693	−0.259847	−	0.965650i	$-0.583672\pi$
$809$	−5.12229e15	−0.519693	−0.259847	+	0.965650i	$0.583672\pi$
$810$	0	0
$811$	−1.11072e16	−1.11170	−0.555852	−	0.831281i	$-0.687608\pi$
$811$	−1.11072e16	−1.11170	−0.555852	+	0.831281i	$0.687608\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.26311e16	2.20465
$816$	0	0
$817$	1.43264e16	1.37694
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.71152e15	−0.347267	−0.173634	−	0.984810i	$-0.555551\pi$
$821$	−3.71152e15	−0.347267	−0.173634	+	0.984810i	$0.555551\pi$
$822$	0	0
$823$	−5.82939e15	−0.538176	−0.269088	−	0.963116i	$-0.586722\pi$
$823$	−5.82939e15	−0.538176	−0.269088	+	0.963116i	$0.586722\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.56636e16	−1.40803	−0.704016	−	0.710184i	$-0.748612\pi$
$827$	−1.56636e16	−1.40803	−0.704016	+	0.710184i	$0.748612\pi$
$828$	0	0
$829$	2.24325e15	0.198988	0.0994942	−	0.995038i	$-0.468278\pi$
$829$	2.24325e15	0.198988	0.0994942	+	0.995038i	$0.468278\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.32712e15	−0.114647
$834$	0	0
$835$	−1.23777e16	−1.05527
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.91290e16	−1.58855	−0.794276	−	0.607557i	$-0.792150\pi$
$839$	−1.91290e16	−1.58855	−0.794276	+	0.607557i	$0.792150\pi$
$840$	0	0
$841$	−1.10016e16	−0.901732
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.00055e16	−1.59749
$846$	0	0
$847$	2.24429e14	0.0176897
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.10302e16	0.847169
$852$	0	0
$853$	−2.12478e16	−1.61099	−0.805497	−	0.592600i	$-0.798101\pi$
$853$	−2.12478e16	−1.61099	−0.805497	+	0.592600i	$0.798101\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.54208e16	−1.13949	−0.569747	−	0.821820i	$-0.692959\pi$
$857$	−1.54208e16	−1.13949	−0.569747	+	0.821820i	$0.692959\pi$
$858$	0	0
$859$	−2.29554e16	−1.67464	−0.837321	−	0.546711i	$-0.815880\pi$
$859$	−2.29554e16	−1.67464	−0.837321	+	0.546711i	$0.815880\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.35310e15	−0.594003	−0.297001	−	0.954877i	$-0.595987\pi$
$863$	−8.35310e15	−0.594003	−0.297001	+	0.954877i	$0.595987\pi$
$864$	0	0
$865$	3.00263e16	2.10821
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.22961e16	1.52623
$870$	0	0
$871$	−2.61663e15	−0.176865
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.49650e15	−0.428188
$876$	0	0
$877$	1.52930e16	0.995394	0.497697	−	0.867351i	$-0.334179\pi$
$877$	1.52930e16	0.995394	0.497697	+	0.867351i	$0.334179\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.02865e16	1.28778	0.643888	−	0.765120i	$-0.277320\pi$
$881$	2.02865e16	1.28778	0.643888	+	0.765120i	$0.277320\pi$
$882$	0	0
$883$	−1.89963e15	−0.119093	−0.0595463	−	0.998226i	$-0.518965\pi$
$883$	−1.89963e15	−0.119093	−0.0595463	+	0.998226i	$0.518965\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.46372e16	0.895114	0.447557	−	0.894255i	$-0.352294\pi$
$887$	1.46372e16	0.895114	0.447557	+	0.894255i	$0.352294\pi$
$888$	0	0
$889$	2.22473e15	0.134375
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.50967e16	−0.889610
$894$	0	0
$895$	2.46367e16	1.43402
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.70928e15	−0.551453
$900$	0	0
$901$	−1.62092e16	−0.909444
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.81060e16	−1.53897
$906$	0	0
$907$	−1.86584e15	−0.100933	−0.0504665	−	0.998726i	$-0.516071\pi$
$907$	−1.86584e15	−0.100933	−0.0504665	+	0.998726i	$0.516071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.00213e16	1.58518	0.792589	−	0.609757i	$-0.208733\pi$
$911$	3.00213e16	1.58518	0.792589	+	0.609757i	$0.208733\pi$
$912$	0	0
$913$	1.84654e15	0.0963315
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.12935e16	−0.575171
$918$	0	0
$919$	−2.27756e16	−1.14613	−0.573066	−	0.819509i	$-0.694246\pi$
$919$	−2.27756e16	−1.14613	−0.573066	+	0.819509i	$0.694246\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.24270e15	0.306735
$924$	0	0
$925$	2.69686e16	1.30942
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.57900e15	0.311942	0.155971	−	0.987762i	$-0.450149\pi$
$929$	6.57900e15	0.311942	0.155971	+	0.987762i	$0.450149\pi$
$930$	0	0
$931$	−5.52013e15	−0.258658
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.80831e16	−1.28523
$936$	0	0
$937$	−2.89389e16	−1.30893	−0.654463	−	0.756094i	$-0.727105\pi$
$937$	−2.89389e16	−1.30893	−0.654463	+	0.756094i	$0.727105\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.04455e16	−1.34518	−0.672589	−	0.740016i	$-0.734818\pi$
$941$	−3.04455e16	−1.34518	−0.672589	+	0.740016i	$0.734818\pi$
$942$	0	0
$943$	−2.39434e16	−1.04561
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.59423e14	0.0324010	0.0162005	−	0.999869i	$-0.494843\pi$
$947$	7.59423e14	0.0324010	0.0162005	+	0.999869i	$0.494843\pi$
$948$	0	0
$949$	−3.99552e15	−0.168504
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.22300e16	−1.74024	−0.870121	−	0.492838i	$-0.835960\pi$
$953$	−4.22300e16	−1.74024	−0.870121	+	0.492838i	$0.835960\pi$
$954$	0	0
$955$	6.05799e16	2.46780
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.08587e15	0.202471
$960$	0	0
$961$	5.32211e16	2.09462
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.24173e16	−0.862349
$966$	0	0
$967$	4.33744e16	1.64964	0.824818	−	0.565398i	$-0.191278\pi$
$967$	4.33744e16	1.64964	0.824818	+	0.565398i	$0.191278\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.60053e15	0.208220	0.104110	−	0.994566i	$-0.466801\pi$
$971$	5.60053e15	0.208220	0.104110	+	0.994566i	$0.466801\pi$
$972$	0	0
$973$	1.94138e15	0.0713657
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.17497e16	0.422286	0.211143	−	0.977455i	$-0.432282\pi$
$977$	1.17497e16	0.422286	0.211143	+	0.977455i	$0.432282\pi$
$978$	0	0
$979$	3.97141e16	1.41137
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.52828e16	0.531080	0.265540	−	0.964100i	$-0.414450\pi$
$983$	1.52828e16	0.531080	0.265540	+	0.964100i	$0.414450\pi$
$984$	0	0
$985$	2.51053e15	0.0862711
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.47522e16	−0.831831
$990$	0	0
$991$	2.70143e16	0.897818	0.448909	−	0.893578i	$-0.351813\pi$
$991$	2.70143e16	0.897818	0.448909	+	0.893578i	$0.351813\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.56090e16	1.80767
$996$	0	0
$997$	−3.58961e16	−1.15405	−0.577024	−	0.816727i	$-0.695786\pi$
$997$	−3.58961e16	−1.15405	−0.577024	+	0.816727i	$0.695786\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.g.1.3		3
3.2	odd	2		84.12.a.e.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.a.e.1.1	✓	3	3.2	odd	2
252.12.a.g.1.3		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)

\(f(q)\)	\(=\)	\(q+11462.1 q^{5} -16807.0 q^{7} +O(q^{10})\)	\(q+11462.1 q^{5} -16807.0 q^{7} +521496. q^{11} -216325. q^{13} -4.69820e6 q^{17} -1.95420e7 q^{19} +3.37634e7 q^{23} +8.25511e7 q^{25} -3.46253e7 q^{29} +2.80410e8 q^{31} -1.92643e8 q^{35} +3.26690e8 q^{37} -7.09150e8 q^{41} -7.33108e8 q^{43} +7.72527e8 q^{47} +2.82475e8 q^{49} +3.45010e9 q^{53} +5.97743e9 q^{55} +4.54344e9 q^{59} +5.97352e9 q^{61} -2.47953e9 q^{65} +1.20958e10 q^{67} -2.88580e10 q^{71} +1.84700e10 q^{73} -8.76479e9 q^{77} +4.27541e10 q^{79} +3.54084e9 q^{83} -5.38511e10 q^{85} +7.61542e10 q^{89} +3.63577e9 q^{91} -2.23992e11 q^{95} +8.60520e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4906 q^{5} - 50421 q^{7}+O(q^{10}) \) 3 * q + 4906 * q^5 - 50421 * q^7	\( 3 q + 4906 q^{5} - 50421 q^{7} + 224648 q^{11} - 780126 q^{13} - 2637706 q^{17} - 6259572 q^{19} - 30125524 q^{23} + 97536981 q^{25} - 201167450 q^{29} + 138368808 q^{31} - 82455142 q^{35} + 342460074 q^{37} - 1248586074 q^{41} - 208844388 q^{43} - 341757384 q^{47} + 847425747 q^{49} - 398683482 q^{53} + 2599266576 q^{55} + 5742813868 q^{59} + 2163574362 q^{61} - 8719808676 q^{65} + 15756982092 q^{67} - 37445810188 q^{71} + 35819108550 q^{73} - 3775658936 q^{77} + 110751115992 q^{79} - 96804724516 q^{83} + 48367693524 q^{85} - 24437643210 q^{89} + 13111577682 q^{91} - 234084845848 q^{95} + 106628361774 q^{97}+O(q^{100}) \) 3 * q + 4906 * q^5 - 50421 * q^7 + 224648 * q^11 - 780126 * q^13 - 2637706 * q^17 - 6259572 * q^19 - 30125524 * q^23 + 97536981 * q^25 - 201167450 * q^29 + 138368808 * q^31 - 82455142 * q^35 + 342460074 * q^37 - 1248586074 * q^41 - 208844388 * q^43 - 341757384 * q^47 + 847425747 * q^49 - 398683482 * q^53 + 2599266576 * q^55 + 5742813868 * q^59 + 2163574362 * q^61 - 8719808676 * q^65 + 15756982092 * q^67 - 37445810188 * q^71 + 35819108550 * q^73 - 3775658936 * q^77 + 110751115992 * q^79 - 96804724516 * q^83 + 48367693524 * q^85 - 24437643210 * q^89 + 13111577682 * q^91 - 234084845848 * q^95 + 106628361774 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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