LMFDB - Embedded newform 1296.2.s.k.431.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(431,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1296.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.s (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			431.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1296.431
Dual form			1296.2.s.k.863.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.89778 + 1.67303i) q^{5} +O(q^{10})$	$q+(-2.89778 + 1.67303i) q^{5} +(3.59808 + 6.23205i) q^{13} -4.00240i q^{17} +(3.09808 - 5.36603i) q^{25} +(-9.26174 - 5.34727i) q^{29} -11.3923 q^{37} +(-11.0227 + 6.36396i) q^{41} +(-3.50000 - 6.06218i) q^{49} +12.7279i q^{53} +(2.69615 - 4.66987i) q^{61} +(-20.8528 - 12.0394i) q^{65} -13.1962 q^{73} +(6.69615 + 11.5981i) q^{85} -18.0430i q^{89} +(4.00000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} + 4 q^{25} - 8 q^{37} - 28 q^{49} - 20 q^{61} - 64 q^{73} + 12 q^{85} + 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 + 4 * q^25 - 8 * q^37 - 28 * q^49 - 20 * q^61 - 64 * q^73 + 12 * q^85 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$	−2.89778	+	1.67303i	−1.29593	+	0.748203i	−0.979698	−	0.200480i	$-0.935750\pi$
$5$	−2.89778	+	1.67303i	−1.29593	+	0.748203i	−0.316228	+	0.948683i	$0.602416\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	3.59808	+	6.23205i	0.997927	+	1.72846i	0.554700	+	0.832050i	$0.312833\pi$
$13$	3.59808	+	6.23205i	0.997927	+	1.72846i	0.443227	+	0.896410i	$0.353834\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.00240i		−	0.970726i	−0.874313	−	0.485363i	$-0.838688\pi$
$17$		−	4.00240i		−	0.970726i	0.874313	−	0.485363i	$-0.161312\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	3.09808	−	5.36603i	0.619615	−	1.07321i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.26174	−	5.34727i	−1.71986	−	0.992963i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	−9.26174	−	5.34727i	−1.71986	−	0.992963i	−0.800717	−	0.599043i	$-0.795548\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−11.3923			−1.87288			−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−11.3923			−1.87288			−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.0227	+	6.36396i	−1.72146	+	0.993884i	−0.805513	+	0.592578i	$0.798110\pi$
$41$	−11.0227	+	6.36396i	−1.72146	+	0.993884i	−0.915944	+	0.401305i	$0.868557\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.7279i			1.74831i	0.485643	+	0.874157i	$0.338586\pi$
$53$			12.7279i			1.74831i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	2.69615	−	4.66987i	0.345207	−	0.597916i	0.985391	+	0.170305i	$0.0544754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−20.8528	−	12.0394i	−2.58648	−	1.49330i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−13.1962			−1.54449			−0.772246	−	0.635323i	$-0.780867\pi$
$73$	−13.1962			−1.54449			−0.772246	+	0.635323i	$0.780867\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$	6.69615	+	11.5981i	0.726300	+	1.25799i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.0430i		−	1.91255i	−0.292462	−	0.956277i	$-0.594474\pi$
$89$		−	18.0430i		−	1.91255i	0.292462	−	0.956277i	$-0.405526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	−0.588315	−	0.808632i	$-0.700208\pi$
$97$	4.00000	−	6.92820i	0.406138	−	0.703452i	0.994453	+	0.105180i	$0.0335417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.0227	+	6.36396i	1.09680	+	0.633238i	0.935379	−	0.353648i	$-0.115059\pi$
$101$	11.0227	+	6.36396i	1.09680	+	0.633238i	0.161421	+	0.986886i	$0.448392\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	4.80385			0.460125			0.230063	−	0.973176i	$-0.426107\pi$
$109$	4.80385			0.460125			0.230063	+	0.973176i	$0.426107\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.12493	+	4.69093i	−0.764329	+	0.441285i	−0.830848	−	0.556500i	$-0.812144\pi$
$113$	−8.12493	+	4.69093i	−0.764329	+	0.441285i	0.0665190	+	0.997785i	$0.478811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			4.00240i			0.357986i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.76097	−	1.01669i	−0.150449	−	0.0868620i	0.422885	−	0.906183i	$-0.361017\pi$
$137$	−1.76097	−	1.01669i	−0.150449	−	0.0868620i	−0.573335	+	0.819321i	$0.694351\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	35.7846			2.97175
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.1941	+	9.34967i	−1.32667	+	0.765955i	−0.984784	−	0.173785i	$-0.944400\pi$
$149$	−16.1941	+	9.34967i	−1.32667	+	0.765955i	−0.341889	+	0.939740i	$0.611067\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.303848	−	0.526279i	−0.0242497	−	0.0420017i	0.853646	−	0.520854i	$-0.174386\pi$
$157$	−0.303848	−	0.526279i	−0.0242497	−	0.0420017i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−19.3923	+	33.5885i	−1.49172	+	2.58373i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	20.8528	+	12.0394i	1.58541	+	0.915338i	0.994049	+	0.108933i	$0.0347435\pi$
$173$	20.8528	+	12.0394i	1.58541	+	0.915338i	0.591364	+	0.806405i	$0.298590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	33.0124	−	19.0597i	2.42712	−	1.40130i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−6.89230	−	11.9378i	−0.496119	−	0.859303i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	−6.89230	−	11.9378i	−0.496119	−	0.859303i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.31508i			0.378684i	0.981911	+	0.189342i	$0.0606355\pi$
$197$			5.31508i			0.378684i	−0.981911	+	0.189342i	$0.939365\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	21.2942	−	36.8827i	1.48725	−	2.57600i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	24.9432	−	14.4010i	1.67786	−	0.968713i
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−11.9904	−	20.7679i	−0.792347	−	1.37238i	−0.924510	−	0.381157i	$-0.875526\pi$
$229$	−11.9904	−	20.7679i	−0.792347	−	1.37238i	0.132164	−	0.991228i	$-0.457808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.72552i		−	0.571628i	−0.958285	−	0.285814i	$-0.907736\pi$
$233$		−	8.72552i		−	0.571628i	0.958285	−	0.285814i	$-0.0922639\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−14.9904	+	25.9641i	−0.965615	+	1.67249i	−0.257663	+	0.966235i	$0.582952\pi$
$241$	−14.9904	+	25.9641i	−0.965615	+	1.67249i	−0.707953	+	0.706260i	$0.750381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	20.2844	+	11.7112i	1.29593	+	0.748203i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	27.2168	−	15.7136i	1.69774	−	0.980189i	0.749838	−	0.661622i	$-0.230131\pi$
$257$	27.2168	−	15.7136i	1.69774	−	0.980189i	0.947900	−	0.318568i	$-0.103202\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−21.2942	−	36.8827i	−1.30809	−	2.26569i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			26.0478i			1.58816i	0.607811	+	0.794082i	$0.292048\pi$
$269$			26.0478i			1.58816i	−0.607811	+	0.794082i	$0.707952\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	0.0477206	+	0.998861i	$0.484804\pi$
$277$	−14.0000	+	24.2487i	−0.841178	+	1.45696i	−0.888899	+	0.458103i	$0.848529\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.19256	−	0.688524i	−0.0711421	−	0.0410739i	0.464007	−	0.885832i	$-0.346411\pi$
$281$	−1.19256	−	0.688524i	−0.0711421	−	0.0410739i	−0.535149	+	0.844758i	$0.679745\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.980762			0.0576919
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.9670	+	6.33178i	−0.640696	+	0.369906i	−0.784883	−	0.619644i	$-0.787277\pi$
$293$	−10.9670	+	6.33178i	−0.640696	+	0.369906i	0.144186	+	0.989551i	$0.453944\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			18.0430i			1.03314i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−3.89230	+	6.74167i	−0.220006	+	0.381062i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−3.89230	+	6.74167i	−0.220006	+	0.381062i	0.734803	+	0.678280i	$0.237274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	28.3536	+	16.3700i	1.59250	+	0.919429i	0.992877	+	0.119145i	$0.0380154\pi$
$317$	28.3536	+	16.3700i	1.59250	+	0.919429i	0.599621	+	0.800284i	$0.295318\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	44.5885			2.47332
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	16.0000	+	27.7128i	0.871576	+	1.50961i	0.860366	+	0.509676i	$0.170235\pi$
$337$	16.0000	+	27.7128i	0.871576	+	1.50961i	0.0112091	+	0.999937i	$0.496432\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	0.968253	+	0.249973i	$0.0804216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.0227	+	6.36396i	0.586679	+	0.338719i	0.763783	−	0.645473i	$-0.223340\pi$
$353$	11.0227	+	6.36396i	0.586679	+	0.338719i	−0.177104	+	0.984192i	$0.556673\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	38.2395	−	22.0776i	2.00155	−	1.15559i
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	76.9595i		−	3.96362i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	33.0681	+	19.0919i	1.67662	+	0.967997i	0.963791	+	0.266659i	$0.0859197\pi$
$389$	33.0681	+	19.0919i	1.67662	+	0.967997i	0.712829	+	0.701338i	$0.247414\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−29.3923			−1.47516			−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−29.3923			−1.47516			−0.737579	+	0.675261i	$0.764031\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−33.0124	+	19.0597i	−1.64856	+	0.951796i	−0.670913	+	0.741536i	$0.734098\pi$
$401$	−33.0124	+	19.0597i	−1.64856	+	0.951796i	−0.977645	+	0.210260i	$0.932569\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	12.5981	+	21.8205i	0.622935	+	1.07895i	0.988936	+	0.148340i	$0.0473931\pi$
$409$	12.5981	+	21.8205i	0.622935	+	1.07895i	−0.366002	+	0.930614i	$0.619274\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−5.99038	+	10.3756i	−0.291953	+	0.505678i	−0.974272	−	0.225377i	$-0.927639\pi$
$421$	−5.99038	+	10.3756i	−0.291953	+	0.505678i	0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−21.4770	−	12.3998i	−1.04179	−	0.601476i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	37.7846			1.81581			0.907906	−	0.419173i	$-0.137680\pi$
$433$	37.7846			1.81581			0.907906	+	0.419173i	$0.137680\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	30.1865	+	52.2846i	1.43098	+	2.47853i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			38.1838i			1.80200i	0.433816	+	0.901002i	$0.357167\pi$
$449$			38.1838i			1.80200i	−0.433816	+	0.901002i	$0.642833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.1865	−	28.0359i	0.757174	−	1.31146i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	16.1865	−	28.0359i	0.757174	−	1.31146i	0.944286	−	0.329125i	$-0.106754\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.0227	−	6.36396i	−0.513378	−	0.296399i	0.220843	−	0.975309i	$-0.429119\pi$
$461$	−11.0227	−	6.36396i	−0.513378	−	0.296399i	−0.734221	+	0.678910i	$0.762453\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−40.9904	−	70.9974i	−1.86900	−	3.23720i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			26.7685i			1.21550i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	−21.4019	+	37.0692i	−0.963894	+	1.66951i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−42.5885			−1.89516
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−33.0681	+	19.0919i	−1.46572	+	0.846233i	−0.999266	−	0.0383134i	$-0.987801\pi$
$509$	−33.0681	+	19.0919i	−1.46572	+	0.846233i	−0.466453	+	0.884546i	$0.654468\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			12.7279i			0.557620i	0.960346	+	0.278810i	$0.0899400\pi$
$521$			12.7279i			0.557620i	−0.960346	+	0.278810i	$0.910060\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−79.3211	−	45.7960i	−3.43578	−	1.98365i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−26.3731			−1.13387			−0.566933	−	0.823764i	$-0.691870\pi$
$541$	−26.3731			−1.13387			−0.566933	+	0.823764i	$0.691870\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.9205	+	8.03699i	−0.596288	+	0.344267i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.4582i		−	1.24819i	−0.781350	−	0.624093i	$-0.785469\pi$
$557$		−	29.4582i		−	1.24819i	0.781350	−	0.624093i	$-0.214531\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	15.6962	−	27.1865i	0.660342	−	1.14375i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7837	+	7.38065i	0.535919	+	0.309413i	0.743423	−	0.668821i	$-0.233201\pi$
$569$	12.7837	+	7.38065i	0.535919	+	0.309413i	−0.207504	+	0.978234i	$0.566534\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−42.5692			−1.77218			−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−42.5692			−1.77218			−0.886090	+	0.463513i	$0.846589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	27.3605i		−	1.12356i	−0.827286	−	0.561780i	$-0.810117\pi$
$593$		−	27.3605i		−	1.12356i	0.827286	−	0.561780i	$-0.189883\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	18.2846	−	31.6699i	0.745845	−	1.29184i	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	18.2846	−	31.6699i	0.745845	−	1.29184i	0.949799	−	0.312861i	$-0.101287\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.8756	−	18.4034i	−1.29593	−	0.748203i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.2860	−	20.3724i	1.42056	−	0.820161i	0.424214	−	0.905562i	$-0.360551\pi$
$617$	35.2860	−	20.3724i	1.42056	−	0.820161i	0.996347	+	0.0854011i	$0.0272172\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	8.79423	+	15.2321i	0.351769	+	0.609282i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			45.5966i			1.81806i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	25.1865	−	43.6244i	0.997927	−	1.72846i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.8993	−	10.3342i	−0.706981	−	0.408176i	0.102961	−	0.994685i	$-0.467168\pi$
$641$	−17.8993	−	10.3342i	−0.706981	−	0.408176i	−0.809942	+	0.586510i	$0.800502\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.0227	+	6.36396i	−0.431352	+	0.249041i	−0.699922	−	0.714219i	$-0.746782\pi$
$653$	−11.0227	+	6.36396i	−0.431352	+	0.249041i	0.268571	+	0.963260i	$0.413449\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	17.6962	+	30.6506i	0.688301	+	1.19217i	0.972387	+	0.233373i	$0.0749763\pi$
$661$	17.6962	+	30.6506i	0.688301	+	1.19217i	−0.284087	+	0.958799i	$0.591690\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	0.0426985	+	0.999088i	$0.486405\pi$
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−33.0681	−	19.0919i	−1.27091	−	0.733761i	−0.295751	−	0.955265i	$-0.595570\pi$
$677$	−33.0681	−	19.0919i	−1.27091	−	0.733761i	−0.975159	+	0.221504i	$0.928903\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	6.80385			0.259962
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−79.3211	+	45.7960i	−3.02189	+	1.74469i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	25.4711	+	44.1173i	0.964788	+	1.67106i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	52.8163i		−	1.99485i	−0.0717494	−	0.997423i	$-0.522858\pi$
$701$		−	52.8163i		−	1.99485i	0.0717494	−	0.997423i	$-0.477142\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.9904	+	41.5526i	−0.900978	+	1.56054i	−0.0747503	+	0.997202i	$0.523816\pi$
$709$	−23.9904	+	41.5526i	−0.900978	+	1.56054i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−57.3871	+	33.1325i	−2.13131	+	1.23051i
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	0.826723	−	0.562609i	$-0.190202\pi$
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	−0.900595	+	0.434659i	$0.856869\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$	31.2846	−	54.1865i	1.14618	−	1.98524i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	52.0000			1.88997			0.944986	−	0.327111i	$-0.106075\pi$
$757$	52.0000			1.88997			0.944986	+	0.327111i	$0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−24.9432	+	14.4010i	−0.904190	+	0.522034i	−0.878557	−	0.477637i	$-0.841493\pi$
$761$	−24.9432	+	14.4010i	−0.904190	+	0.522034i	−0.0256326	+	0.999671i	$0.508160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.10770	+	3.65064i	0.0760054	+	0.131645i	0.901523	−	0.432731i	$-0.142450\pi$
$769$	2.10770	+	3.65064i	0.0760054	+	0.131645i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			21.4534i			0.771627i	0.922577	+	0.385813i	$0.126079\pi$
$773$			21.4534i			0.771627i	−0.922577	+	0.385813i	$0.873921\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.76097	+	1.01669i	0.0628515	+	0.0362874i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	38.8038			1.37797
$794$		0			0
$795$		0			0
$796$		0			0
$797$	30.0588	−	17.3545i	1.06474	−	0.614727i	0.137999	−	0.990432i	$-0.455933\pi$
$797$	30.0588	−	17.3545i	1.06474	−	0.614727i	0.926739	+	0.375705i	$0.122599\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			51.5037i			1.81077i	0.424589	+	0.905386i	$0.360418\pi$
$809$			51.5037i			1.81077i	−0.424589	+	0.905386i	$0.639582\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.2153	+	7.05249i	0.426316	+	0.246133i	0.697776	−	0.716316i	$-0.254173\pi$
$821$	12.2153	+	7.05249i	0.426316	+	0.246133i	−0.271460	+	0.962450i	$0.587507\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.2633	+	14.0084i	−0.840673	+	0.485363i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	42.6865	+	73.9352i	1.47195	+	2.54949i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	129.776i		−	4.46442i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	23.0000	−	39.8372i	0.787505	−	1.36400i	−0.139986	−	0.990153i	$-0.544706\pi$
$853$	23.0000	−	39.8372i	0.787505	−	1.36400i	0.927491	−	0.373845i	$-0.121961\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.30821	+	3.64205i	0.215484	+	0.124410i	0.603858	−	0.797092i	$-0.293630\pi$
$857$	6.30821	+	3.64205i	0.215484	+	0.124410i	−0.388373	+	0.921502i	$0.626963\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−80.5692			−2.73944
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.30385	−	16.1147i	−0.314169	−	0.544156i	0.665092	−	0.746762i	$-0.268392\pi$
$877$	−9.30385	−	16.1147i	−0.314169	−	0.544156i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.7279i			0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$			12.7279i			0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	50.9423			1.69713
$902$		0			0
$903$		0			0
$904$		0			0
$905$	57.9555	−	33.4607i	1.92651	−	1.11227i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−35.2942	+	61.1314i	−1.16047	+	2.00999i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−47.4455	−	27.3927i	−1.55664	−	0.898725i	−0.997575	−	0.0695983i	$-0.977828\pi$
$929$	−47.4455	−	27.3927i	−1.55664	−	0.898725i	−0.559061	−	0.829126i	$-0.688838\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−60.5692			−1.97871			−0.989355	−	0.145522i	$-0.953514\pi$
$937$	−60.5692			−1.97871			−0.989355	+	0.145522i	$0.953514\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	44.0351	−	25.4237i	1.43550	−	0.828788i	0.437969	−	0.898990i	$-0.355698\pi$
$941$	44.0351	−	25.4237i	1.43550	−	0.828788i	0.997533	+	0.0702023i	$0.0223645\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	−47.4808	−	82.2391i	−1.54129	−	2.66959i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			60.8211i			1.97019i	0.172011	+	0.985095i	$0.444974\pi$
$953$			60.8211i			1.97019i	−0.172011	+	0.985095i	$0.555026\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	+	26.8468i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	39.9447	+	23.0621i	1.28587	+	0.742395i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	33.0681	−	19.0919i	1.05794	−	0.610803i	0.133080	−	0.991105i	$-0.457513\pi$
$977$	33.0681	−	19.0919i	1.05794	−	0.610803i	0.924862	+	0.380302i	$0.124180\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−8.89230	−	15.4019i	−0.283332	−	0.490746i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20.6962	−	35.8468i	0.655454	−	1.13528i	−0.326326	−	0.945257i	$-0.605811\pi$
$997$	20.6962	−	35.8468i	0.655454	−	1.13528i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.s.k.431.1		8
3.2	odd	2	inner	1296.2.s.k.431.4		8
4.3	odd	2	CM	1296.2.s.k.431.1		8
9.2	odd	6		1296.2.c.e.1295.4	yes	4
9.4	even	3	inner	1296.2.s.k.863.4		8
9.5	odd	6	inner	1296.2.s.k.863.1		8
9.7	even	3		1296.2.c.e.1295.1	✓	4
12.11	even	2	inner	1296.2.s.k.431.4		8
36.7	odd	6		1296.2.c.e.1295.1	✓	4
36.11	even	6		1296.2.c.e.1295.4	yes	4
36.23	even	6	inner	1296.2.s.k.863.1		8
36.31	odd	6	inner	1296.2.s.k.863.4		8
72.11	even	6		5184.2.c.g.5183.1		4
72.29	odd	6		5184.2.c.g.5183.1		4
72.43	odd	6		5184.2.c.g.5183.4		4
72.61	even	6		5184.2.c.g.5183.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.2.c.e.1295.1	✓	4	9.7	even	3
1296.2.c.e.1295.1	✓	4	36.7	odd	6
1296.2.c.e.1295.4	yes	4	9.2	odd	6
1296.2.c.e.1295.4	yes	4	36.11	even	6
1296.2.s.k.431.1		8	1.1	even	1	trivial
1296.2.s.k.431.1		8	4.3	odd	2	CM
1296.2.s.k.431.4		8	3.2	odd	2	inner
1296.2.s.k.431.4		8	12.11	even	2	inner
1296.2.s.k.863.1		8	9.5	odd	6	inner
1296.2.s.k.863.1		8	36.23	even	6	inner
1296.2.s.k.863.4		8	9.4	even	3	inner
1296.2.s.k.863.4		8	36.31	odd	6	inner
5184.2.c.g.5183.1		4	72.11	even	6
5184.2.c.g.5183.1		4	72.29	odd	6
5184.2.c.g.5183.4		4	72.43	odd	6
5184.2.c.g.5183.4		4	72.61	even	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.89778 + 1.67303i) q^{5} +O(q^{10})\)	\(q+(-2.89778 + 1.67303i) q^{5} +(3.59808 + 6.23205i) q^{13} -4.00240i q^{17} +(3.09808 - 5.36603i) q^{25} +(-9.26174 - 5.34727i) q^{29} -11.3923 q^{37} +(-11.0227 + 6.36396i) q^{41} +(-3.50000 - 6.06218i) q^{49} +12.7279i q^{53} +(2.69615 - 4.66987i) q^{61} +(-20.8528 - 12.0394i) q^{65} -13.1962 q^{73} +(6.69615 + 11.5981i) q^{85} -18.0430i q^{89} +(4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} + 4 q^{25} - 8 q^{37} - 28 q^{49} - 20 q^{61} - 64 q^{73} + 12 q^{85} + 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 + 4 * q^25 - 8 * q^37 - 28 * q^49 - 20 * q^61 - 64 * q^73 + 12 * q^85 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more