LMFDB - Embedded newform 1216.2.t.a.1185.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(353,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1216.353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.t (of order $6$, degree $2$, 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:	$9.70980888579$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			1185.3
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1216.1185
Dual form			1216.2.t.a.353.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.25529 - 0.724745i) q^{3} +(-0.449490 + 0.778539i) q^{9} +O(q^{10})$	$q+(1.25529 - 0.724745i) q^{3} +(-0.449490 + 0.778539i) q^{9} +5.44949i q^{11} +(3.00000 + 5.19615i) q^{17} +(1.25529 - 4.17423i) q^{19} +(-2.50000 + 4.33013i) q^{25} +5.65153i q^{27} +(3.94949 + 6.84072i) q^{33} +(3.39898 + 5.88721i) q^{41} +(8.66025 - 5.00000i) q^{43} -7.00000 q^{49} +(7.53177 + 4.34847i) q^{51} +(-1.44949 - 6.14966i) q^{57} +(13.2047 - 7.62372i) q^{59} +(0.301783 + 0.174235i) q^{67} +(-6.84847 - 11.8619i) q^{73} +7.24745i q^{75} +(2.74745 + 4.75872i) q^{81} -11.4495i q^{83} +(-9.00000 + 15.5885i) q^{89} +(9.84847 + 17.0580i) q^{97} +(-4.24264 - 2.44949i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 24 q^{17} - 20 q^{25} + 12 q^{33} - 12 q^{41} - 56 q^{49} + 8 q^{57} + 4 q^{73} - 76 q^{81} - 72 q^{89} + 20 q^{97}+O(q^{100}) $ 8 * q + 16 * q^9 + 24 * q^17 - 20 * q^25 + 12 * q^33 - 12 * q^41 - 56 * q^49 + 8 * q^57 + 4 * q^73 - 76 * q^81 - 72 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

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
$3$	1.25529	−	0.724745i	0.724745	−	0.418432i	−0.0917517	−	0.995782i	$-0.529247\pi$
$3$	1.25529	−	0.724745i	0.724745	−	0.418432i	0.816497	+	0.577350i	$0.195913\pi$
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−0.449490	+	0.778539i	−0.149830	+	0.259513i
$10$		0			0
$11$			5.44949i			1.64308i	0.570149	+	0.821541i	$0.306886\pi$
$11$			5.44949i			1.64308i	−0.570149	+	0.821541i	$0.693114\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	1.25529	−	4.17423i	0.287984	−	0.957635i
$19$	1.25529	−	4.17423i	0.287984	−	0.957635i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$			5.65153i			1.08764i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	3.94949	+	6.84072i	0.687518	+	1.19082i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.39898	+	5.88721i	0.530831	+	0.919427i	0.999353	+	0.0359748i	$0.0114536\pi$
$41$	3.39898	+	5.88721i	0.530831	+	0.919427i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	8.66025	−	5.00000i	1.32068	−	0.762493i	0.336840	−	0.941562i	$-0.390642\pi$
$43$	8.66025	−	5.00000i	1.32068	−	0.762493i	0.983836	+	0.179069i	$0.0573086\pi$
$44$		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
$49$	−7.00000			−1.00000
$50$		0			0
$51$	7.53177	+	4.34847i	1.05466	+	0.608907i
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.44949	−	6.14966i	−0.191990	−	0.814543i
$58$		0			0
$59$	13.2047	−	7.62372i	1.71910	−	0.992524i	0.798528	−	0.601958i	$-0.205612\pi$
$59$	13.2047	−	7.62372i	1.71910	−	0.992524i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.301783	+	0.174235i	0.0368687	+	0.0212861i	0.518321	−	0.855186i	$-0.326557\pi$
$67$	0.301783	+	0.174235i	0.0368687	+	0.0212861i	−0.481452	+	0.876472i	$0.659891\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	−6.84847	−	11.8619i	−0.801553	−	1.38833i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−6.84847	−	11.8619i	−0.801553	−	1.38833i	0.117041	−	0.993127i	$-0.462659\pi$
$74$		0			0
$75$			7.24745i			0.836863i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	2.74745	+	4.75872i	0.305272	+	0.528747i
$82$		0			0
$83$		−	11.4495i		−	1.25674i	−0.777913	−	0.628372i	$-0.783721\pi$
$83$		−	11.4495i		−	1.25674i	0.777913	−	0.628372i	$-0.216279\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.217354	+	0.976093i	$0.569742\pi$
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.736644	+	0.676280i	$0.763591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.84847	+	17.0580i	0.999961	+	1.73198i	0.507673	+	0.861550i	$0.330506\pi$
$97$	9.84847	+	17.0580i	0.999961	+	1.73198i	0.492287	+	0.870433i	$0.336161\pi$
$98$		0			0
$99$	−4.24264	−	2.44949i	−0.426401	−	0.246183i
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.797959			0.0750657			0.0375328	−	0.999295i	$-0.488050\pi$
$113$	0.797959			0.0750657			0.0375328	+	0.999295i	$0.488050\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−18.6969			−1.69972
$122$		0			0
$123$	8.53344	+	4.92679i	0.769435	+	0.444233i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$		0			0
$129$	7.24745	−	12.5529i	0.638102	−	1.10523i
$130$		0			0
$131$	2.81237	−	1.62372i	0.245718	−	0.141865i	−0.372084	−	0.928199i	$-0.621357\pi$
$131$	2.81237	−	1.62372i	0.245718	−	0.141865i	0.617802	+	0.786334i	$0.288023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.29796	−	14.3725i	0.708942	−	1.22792i	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	8.29796	−	14.3725i	0.708942	−	1.22792i	0.965250	−	0.261329i	$-0.0841608\pi$
$138$		0			0
$139$	−3.16232	−	1.82577i	−0.268224	−	0.154859i	0.359856	−	0.933008i	$-0.382826\pi$
$139$	−3.16232	−	1.82577i	−0.268224	−	0.154859i	−0.628080	+	0.778148i	$0.716159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.78706	+	5.07321i	−0.724745	+	0.418432i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−5.39388			−0.436069
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			21.0454i			1.64840i	0.566296	+	0.824202i	$0.308376\pi$
$163$			21.0454i			1.64840i	−0.566296	+	0.824202i	$0.691624\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$	2.68556	+	2.85357i	0.205370	+	0.218218i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	11.0505	−	19.1400i	0.830607	−	1.43865i
$178$		0			0
$179$		−	26.1464i		−	1.95428i	−0.212607	−	0.977138i	$-0.568195\pi$
$179$		−	26.1464i		−	1.95428i	0.212607	−	0.977138i	$-0.431805\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−28.3164	+	16.3485i	−2.07070	+	1.19552i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	11.0000	+	19.0526i	0.791797	+	1.37143i	0.924853	+	0.380325i	$0.124188\pi$
$193$	11.0000	+	19.0526i	0.791797	+	1.37143i	−0.133056	+	0.991109i	$0.542479\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$	0.505103			0.0356272
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.7474	+	6.84072i	1.57347	+	0.473182i
$210$		0			0
$211$	−12.1244	+	7.00000i	−0.834675	+	0.481900i	−0.855451	−	0.517884i	$-0.826720\pi$
$211$	−12.1244	+	7.00000i	−0.834675	+	0.481900i	0.0207756	+	0.999784i	$0.493386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−17.1937	−	9.92679i	−1.16184	−	0.670790i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$	−2.24745	−	3.89270i	−0.149830	−	0.259513i
$226$		0			0
$227$			12.5505i			0.833007i	0.909134	+	0.416503i	$0.136745\pi$
$227$			12.5505i			0.833007i	−0.909134	+	0.416503i	$0.863255\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.05051	−	8.74774i	−0.330870	−	0.573084i	0.651813	−	0.758380i	$-0.274009\pi$
$233$	−5.05051	−	8.74774i	−0.330870	−	0.573084i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	13.8485	−	23.9863i	0.892058	−	1.54509i	0.0546547	−	0.998505i	$-0.482594\pi$
$241$	13.8485	−	23.9863i	0.892058	−	1.54509i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$	−7.78539	−	4.49490i	−0.499433	−	0.288348i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−8.29796	−	14.3725i	−0.525862	−	0.910819i
$250$		0			0
$251$	−25.9326	−	14.9722i	−1.63685	−	0.945036i	−0.981908	−	0.189358i	$-0.939359\pi$
$251$	−25.9326	−	14.9722i	−1.63685	−	0.945036i	−0.654943	−	0.755678i	$-0.727307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.3990	+	21.4757i	−0.773427	+	1.33962i	0.162247	+	0.986750i	$0.448126\pi$
$257$	−12.3990	+	21.4757i	−0.773427	+	1.33962i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			26.0908i			1.59673i
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.5970	−	13.6237i	−1.42295	−	0.821541i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.7474	−	29.0074i	0.999069	−	1.73044i	0.462174	−	0.886789i	$-0.347070\pi$
$281$	16.7474	−	29.0074i	0.999069	−	1.73044i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$	−9.56560	+	5.52270i	−0.568616	+	0.328291i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	−9.56560	+	5.52270i	−0.568616	+	0.328291i	0.187980	+	0.982173i	$0.439806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$	24.7255	+	14.2753i	1.44943	+	0.836830i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−30.7980			−1.78708
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−21.0864	+	12.1742i	−1.20346	+	0.694820i	−0.961324	−	0.275421i	$-0.911183\pi$
$307$	−21.0864	+	12.1742i	−1.20346	+	0.694820i	−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	17.1969	−	29.7860i	0.972028	−	1.68360i	0.282617	−	0.959233i	$-0.408798\pi$
$313$	17.1969	−	29.7860i	0.972028	−	1.68360i	0.689412	−	0.724370i	$-0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.34847	−	7.53177i	−0.242708	−	0.420382i
$322$		0			0
$323$	25.4558	−	6.00000i	1.41640	−	0.333849i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	35.0454i		−	1.92627i	−0.269019	−	0.963135i	$-0.586699\pi$
$331$		−	35.0454i		−	1.92627i	0.269019	−	0.963135i	$-0.413301\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.609936	−	0.792451i	$-0.708805\pi$
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.381314	−	0.924445i	$-0.624528\pi$
$338$		0			0
$339$	1.00167	−	0.578317i	0.0544035	−	0.0314099i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	30.1752	−	17.4217i	1.61989	−	0.935245i	0.632945	−	0.774197i	$-0.281846\pi$
$347$	30.1752	−	17.4217i	1.61989	−	0.935245i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	34.5959			1.84135			0.920677	−	0.390324i	$-0.127637\pi$
$353$	34.5959			1.84135			0.920677	+	0.390324i	$0.127637\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−15.8485	−	10.4798i	−0.834130	−	0.551568i
$362$		0			0
$363$	−23.4702	+	13.5505i	−1.23186	+	0.711217i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	−6.11123			−0.318138
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	38.0000i		−	1.95193i	−0.217930	−	0.975964i	$-0.569930\pi$
$379$		−	38.0000i		−	1.95193i	0.217930	−	0.975964i	$-0.430070\pi$
$380$		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
$387$			8.98979i			0.456977i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	2.35357	−	4.07651i	0.118722	−	0.205633i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.6464	+	27.1004i	0.781345	+	1.35333i	0.931158	+	0.364615i	$0.118800\pi$
$401$	15.6464	+	27.1004i	0.781345	+	1.35333i	−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−20.1969	+	34.9821i	−0.998674	+	1.72975i	−0.454759	+	0.890614i	$0.650275\pi$
$409$	−20.1969	+	34.9821i	−0.998674	+	1.72975i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$		−	24.0556i		−	1.18658i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.29286			−0.259192
$418$		0			0
$419$			18.0000i			0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$419$			18.0000i			0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−30.0000			−1.45521
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	19.0000	−	32.9090i	0.913082	−	1.58150i	0.103396	−	0.994640i	$-0.467029\pi$
$433$	19.0000	−	32.9090i	0.913082	−	1.58150i	0.809686	−	0.586864i	$-0.199638\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	3.14643	−	5.44977i	0.149830	−	0.259513i
$442$		0			0
$443$	20.3079	+	11.7247i	0.964855	+	0.557059i	0.897664	−	0.440681i	$-0.145263\pi$
$443$	20.3079	+	11.7247i	0.964855	+	0.557059i	0.0671913	+	0.997740i	$0.478596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.1010			−0.759854			−0.379927	−	0.925016i	$-0.624051\pi$
$449$	−16.1010			−0.759854			−0.379927	+	0.925016i	$0.624051\pi$
$450$		0			0
$451$	−32.0823	+	18.5227i	−1.51069	+	0.872200i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−42.3939			−1.98310			−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−42.3939			−1.98310			−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	−29.3662	+	16.9546i	−1.37070	+	0.791373i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	11.9444i		−	0.552720i	−0.961054	−	0.276360i	$-0.910872\pi$
$467$		−	11.9444i		−	0.552720i	0.961054	−	0.276360i	$-0.0891283\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	27.2474	+	47.1940i	1.25284	+	2.16998i
$474$		0			0
$475$	14.9367	+	15.8712i	0.685344	+	0.728219i
$476$		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
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	15.2526	+	26.4182i	0.689744	+	1.19467i
$490$		0			0
$491$	36.3731	−	21.0000i	1.64149	−	0.947717i	0.661190	−	0.750218i	$-0.270052\pi$
$491$	36.3731	−	21.0000i	1.64149	−	0.947717i	0.980303	−	0.197499i	$-0.0632818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	37.8820	−	21.8712i	1.69583	−	0.979088i	0.746197	−	0.665725i	$-0.231878\pi$
$499$	37.8820	−	21.8712i	1.69583	−	0.979088i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−16.3188	−	9.42168i	−0.724745	−	0.418432i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	23.5908	+	7.09434i	1.04156	+	0.313223i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.1918			1.84846			0.924229	−	0.381839i	$-0.124709\pi$
$521$	42.1918			1.84846			0.924229	+	0.381839i	$0.124709\pi$
$522$		0			0
$523$	32.9090	+	19.0000i	1.43901	+	0.830812i	0.997781	−	0.0665832i	$-0.0212098\pi$
$523$	32.9090	+	19.0000i	1.43901	+	0.830812i	0.441228	+	0.897395i	$0.354543\pi$
$524$		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
$531$			13.7071i			0.594839i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−18.9495	−	32.8215i	−0.817731	−	1.41635i
$538$		0			0
$539$		−	38.1464i		−	1.64308i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	39.8372	+	23.0000i	1.70331	+	0.983409i	0.942358	+	0.334606i	$0.108603\pi$
$547$	39.8372	+	23.0000i	1.70331	+	0.983409i	0.760956	+	0.648803i	$0.224730\pi$
$548$		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
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−23.6969	+	41.0443i	−1.00049	+	1.73289i
$562$		0			0
$563$		−	16.8434i		−	0.709863i	−0.934892	−	0.354932i	$-0.884504\pi$
$563$		−	16.8434i		−	0.709863i	0.934892	−	0.354932i	$-0.115496\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$			47.7423i			1.99796i	0.0452101	+	0.998978i	$0.485604\pi$
$571$			47.7423i			1.99796i	−0.0452101	+	0.998978i	$0.514396\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−46.3939			−1.93140			−0.965701	−	0.259656i	$-0.916391\pi$
$577$	−46.3939			−1.93140			−0.965701	+	0.259656i	$0.916391\pi$
$578$		0			0
$579$	27.6165	+	15.9444i	1.14770	+	0.662626i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.19615	+	3.00000i	−0.214468	+	0.123823i	−0.603386	−	0.797449i	$-0.706182\pi$
$587$	−5.19615	+	3.00000i	−0.214468	+	0.123823i	0.388918	+	0.921272i	$0.372849\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.0959	+	26.1469i	−0.619915	+	1.07372i	0.369586	+	0.929197i	$0.379500\pi$
$593$	−15.0959	+	26.1469i	−0.619915	+	1.07372i	−0.989501	+	0.144528i	$0.953834\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	8.30306			0.338689			0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306			0.338689			0.169344	+	0.985557i	$0.445835\pi$
$602$		0			0
$603$	−0.271297	+	0.156633i	−0.0110481	+	0.00637860i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.64643	−	16.7081i	0.388351	−	0.672643i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	9.64643	−	16.7081i	0.388351	−	0.672643i	0.992228	+	0.124434i	$0.0397116\pi$
$618$		0			0
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	33.5125	−	7.89898i	1.33836	−	0.315455i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$631$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$632$		0			0
$633$	−10.1464	+	17.5741i	−0.403284	+	0.698509i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.7474	+	39.3997i	0.898470	+	1.55620i	0.829450	+	0.558581i	$0.188654\pi$
$641$	22.7474	+	39.3997i	0.898470	+	1.55620i	0.0690201	+	0.997615i	$0.478013\pi$
$642$		0			0
$643$	−28.0146	+	16.1742i	−1.10479	+	0.637850i	−0.937474	−	0.348054i	$-0.886843\pi$
$643$	−28.0146	+	16.1742i	−1.10479	+	0.637850i	−0.167313	+	0.985904i	$0.553509\pi$
$644$		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
$649$	41.5454	+	71.9588i	1.63080	+	2.82463i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	12.3133			0.480386
$658$		0			0
$659$	−15.5885	−	9.00000i	−0.607240	−	0.350590i	0.164644	−	0.986353i	$-0.447352\pi$
$659$	−15.5885	−	9.00000i	−0.607240	−	0.350590i	−0.771885	+	0.635763i	$0.780686\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		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
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$	−24.4718	−	14.1288i	−0.941922	−	0.543819i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	9.09592	+	15.7546i	0.348556	+	0.603717i
$682$		0			0
$683$			42.0000i			1.60709i	0.595247	+	0.803543i	$0.297054\pi$
$683$			42.0000i			1.60709i	−0.595247	+	0.803543i	$0.702946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	46.0000i		−	1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$		−	46.0000i		−	1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−20.3939	+	35.3232i	−0.772473	+	1.33796i
$698$		0			0
$699$	−12.6798	−	7.32066i	−0.479593	−	0.276893i
$700$		0			0
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		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		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	40.1464i		−	1.49306i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$	−29.5153			−1.09316
$730$		0			0
$731$	51.9615	+	30.0000i	1.92187	+	1.10959i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.949490	+	1.64456i	−0.0349749	+	0.0605783i
$738$		0			0
$739$	17.0974	−	9.87117i	0.628937	−	0.363117i	−0.151403	−	0.988472i	$-0.548379\pi$
$739$	17.0974	−	9.87117i	0.628937	−	0.363117i	0.780340	+	0.625355i	$0.215046\pi$
$740$		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
$745$		0			0
$746$		0			0
$747$	8.91388	+	5.14643i	0.326142	+	0.188298i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$	−43.4041			−1.58173
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	36.7980			1.33392			0.666962	−	0.745091i	$-0.267594\pi$
$761$	36.7980			1.33392			0.666962	+	0.745091i	$0.267594\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	−0.596643	−	0.802507i	$-0.703499\pi$
$769$	11.0000	−	19.0526i	0.396670	−	0.687053i	0.993313	+	0.115454i	$0.0368323\pi$
$770$		0			0
$771$			35.9444i			1.29451i
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.8413	−	6.79796i	1.03335	−	0.243562i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	2.95459i		−	0.105320i	−0.998613	−	0.0526599i	$-0.983230\pi$
$787$		−	2.95459i		−	0.105320i	0.998613	−	0.0526599i	$-0.0167699\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−8.09082	−	14.0137i	−0.285875	−	0.495150i
$802$		0			0
$803$	64.6413	−	37.3207i	2.28114	−	1.31702i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.9898			1.82786			0.913932	−	0.405868i	$-0.133031\pi$
$809$	51.9898			1.82786			0.913932	+	0.405868i	$0.133031\pi$
$810$		0			0
$811$	32.9090	+	19.0000i	1.15559	+	0.667180i	0.950243	−	0.311509i	$-0.100834\pi$
$811$	32.9090	+	19.0000i	1.15559	+	0.667180i	0.205347	+	0.978689i	$0.434168\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.0000	−	42.4264i	−0.349856	−	1.48431i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$	−39.4949			−1.37504
$826$		0			0
$827$	−38.2319	−	22.0732i	−1.32945	−	0.767561i	−0.344239	−	0.938882i	$-0.611863\pi$
$827$	−38.2319	−	22.0732i	−1.32945	−	0.767561i	−0.985215	+	0.171321i	$0.945196\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.0000	−	36.3731i	−0.727607	−	1.26025i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		−	48.5505i		−	1.67217i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−8.00510	+	13.8652i	−0.274734	+	0.475854i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.70204	−	6.41212i	−0.126459	−	0.219034i	0.795843	−	0.605503i	$-0.207028\pi$
$857$	−3.70204	−	6.41212i	−0.126459	−	0.219034i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	18.7508	+	10.8258i	0.639768	+	0.369370i	0.784525	−	0.620097i	$-0.212907\pi$
$859$	18.7508	+	10.8258i	0.639768	+	0.369370i	−0.144757	+	0.989467i	$0.546240\pi$
$860$		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
$865$		0			0
$866$		0			0
$867$			27.5403i			0.935318i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−17.7071			−0.599296
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−57.9898			−1.95373			−0.976863	−	0.213866i	$-0.931394\pi$
$881$	−57.9898			−1.95373			−0.976863	+	0.213866i	$0.931394\pi$
$882$		0			0
$883$	−43.6817	−	25.2196i	−1.47001	−	0.848709i	−0.470573	−	0.882361i	$-0.655953\pi$
$883$	−43.6817	−	25.2196i	−1.47001	−	0.848709i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−25.9326	+	14.9722i	−0.868775	+	0.501587i
$892$		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
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−48.8779	+	28.2196i	−1.62296	+	0.937018i	−0.636841	+	0.770996i	$0.719759\pi$
$907$	−48.8779	+	28.2196i	−1.62296	+	0.937018i	−0.986122	+	0.166022i	$0.946908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	62.3939			2.06494
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−17.6464	+	30.5645i	−0.581470	+	1.00713i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.25255	−	2.16948i	−0.0410949	−	0.0711784i	0.844746	−	0.535167i	$-0.179751\pi$
$929$	−1.25255	−	2.16948i	−0.0410949	−	0.0711784i	−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$	−8.78706	+	29.2196i	−0.287984	+	0.957635i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.5454	−	52.9062i	0.997875	−	1.72837i	0.442509	−	0.896764i	$-0.354088\pi$
$937$	30.5454	−	52.9062i	0.997875	−	1.72837i	0.555366	−	0.831606i	$-0.312578\pi$
$938$		0			0
$939$		−	49.8536i		−	1.62691i
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.9808	+	15.0000i	−0.844261	+	0.487435i	−0.858710	−	0.512461i	$-0.828734\pi$
$947$	−25.9808	+	15.0000i	−0.844261	+	0.487435i	0.0144491	+	0.999896i	$0.495401\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0959	+	52.1277i	0.974902	+	1.68858i	0.680257	+	0.732974i	$0.261868\pi$
$953$	30.0959	+	52.1277i	0.974902	+	1.68858i	0.294646	+	0.955607i	$0.404798\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	4.67123	+	2.69694i	0.150528	+	0.0869076i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$		0			0
$969$	27.6061	−	25.9808i	0.886836	−	0.834622i
$970$		0			0
$971$	46.7172	−	26.9722i	1.49923	−	0.865579i	0.499227	−	0.866471i	$-0.333617\pi$
$971$	46.7172	−	26.9722i	1.49923	−	0.865579i	1.00000		0.000892350i	$0.000284044\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−50.8888			−1.62808			−0.814038	−	0.580812i	$-0.802735\pi$
$977$	−50.8888			−1.62808			−0.814038	+	0.580812i	$0.802735\pi$
$978$		0			0
$979$	−84.9491	−	49.0454i	−2.71499	−	1.56750i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$		0			0
$993$	−25.3990	−	43.9923i	−0.806012	−	1.39605i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.t.a.1185.3	yes	8
4.3	odd	2	inner	1216.2.t.a.1185.2	yes	8
8.3	odd	2	CM	1216.2.t.a.1185.3	yes	8
8.5	even	2	inner	1216.2.t.a.1185.2	yes	8
19.11	even	3	inner	1216.2.t.a.353.2	✓	8
76.11	odd	6	inner	1216.2.t.a.353.3	yes	8
152.11	odd	6	inner	1216.2.t.a.353.2	✓	8
152.125	even	6	inner	1216.2.t.a.353.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.t.a.353.2	✓	8	19.11	even	3	inner
1216.2.t.a.353.2	✓	8	152.11	odd	6	inner
1216.2.t.a.353.3	yes	8	76.11	odd	6	inner
1216.2.t.a.353.3	yes	8	152.125	even	6	inner
1216.2.t.a.1185.2	yes	8	4.3	odd	2	inner
1216.2.t.a.1185.2	yes	8	8.5	even	2	inner
1216.2.t.a.1185.3	yes	8	1.1	even	1	trivial
1216.2.t.a.1185.3	yes	8	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.25529 - 0.724745i) q^{3} +(-0.449490 + 0.778539i) q^{9} +O(q^{10})\)	\(q+(1.25529 - 0.724745i) q^{3} +(-0.449490 + 0.778539i) q^{9} +5.44949i q^{11} +(3.00000 + 5.19615i) q^{17} +(1.25529 - 4.17423i) q^{19} +(-2.50000 + 4.33013i) q^{25} +5.65153i q^{27} +(3.94949 + 6.84072i) q^{33} +(3.39898 + 5.88721i) q^{41} +(8.66025 - 5.00000i) q^{43} -7.00000 q^{49} +(7.53177 + 4.34847i) q^{51} +(-1.44949 - 6.14966i) q^{57} +(13.2047 - 7.62372i) q^{59} +(0.301783 + 0.174235i) q^{67} +(-6.84847 - 11.8619i) q^{73} +7.24745i q^{75} +(2.74745 + 4.75872i) q^{81} -11.4495i q^{83} +(-9.00000 + 15.5885i) q^{89} +(9.84847 + 17.0580i) q^{97} +(-4.24264 - 2.44949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 24 q^{17} - 20 q^{25} + 12 q^{33} - 12 q^{41} - 56 q^{49} + 8 q^{57} + 4 q^{73} - 76 q^{81} - 72 q^{89} + 20 q^{97}+O(q^{100}) \) 8 * q + 16 * q^9 + 24 * q^17 - 20 * q^25 + 12 * q^33 - 12 * q^41 - 56 * q^49 + 8 * q^57 + 4 * q^73 - 76 * q^81 - 72 * q^89 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more