LMFDB - Embedded newform 1216.2.t.a.353.1

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			353.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1216.353
Dual form			1216.2.t.a.1185.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.98735 - 1.72474i) q^{3} +(4.44949 + 7.70674i) q^{9} +O(q^{10})$	$q+(-2.98735 - 1.72474i) q^{3} +(4.44949 + 7.70674i) q^{9} -0.550510i q^{11} +(3.00000 - 5.19615i) q^{17} +(-2.98735 - 3.17423i) q^{19} +(-2.50000 - 4.33013i) q^{25} -20.3485i q^{27} +(-0.949490 + 1.64456i) q^{33} +(-6.39898 + 11.0834i) q^{41} +(8.66025 + 5.00000i) q^{43} -7.00000 q^{49} +(-17.9241 + 10.3485i) q^{51} +(3.44949 + 14.6349i) q^{57} +(-8.00853 - 4.62372i) q^{59} +(-12.4261 + 7.17423i) q^{67} +(7.84847 - 13.5939i) q^{73} +17.2474i q^{75} +(-21.7474 + 37.6677i) q^{81} +6.55051i q^{83} +(-9.00000 - 15.5885i) q^{89} +(-4.84847 + 8.39780i) q^{97} +(4.24264 - 2.44949i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$

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
$3$	−2.98735	−	1.72474i	−1.72474	−	0.995782i	−0.908248	−	0.418432i	$-0.862580\pi$
$3$	−2.98735	−	1.72474i	−1.72474	−	0.995782i	−0.816497	−	0.577350i	$-0.804087\pi$
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	4.44949	+	7.70674i	1.48316	+	2.56891i
$10$		0			0
$11$		−	0.550510i		−	0.165985i	−0.996550	−	0.0829925i	$-0.973552\pi$
$11$		−	0.550510i		−	0.165985i	0.996550	−	0.0829925i	$-0.0264478\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−2.98735	−	3.17423i	−0.685344	−	0.728219i
$19$	−2.98735	−	3.17423i	−0.685344	−	0.728219i
$20$		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
$25$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$26$		0			0
$27$		−	20.3485i		−	3.91606i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	−0.949490	+	1.64456i	−0.165285	+	0.286282i
$34$		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
$41$	−6.39898	+	11.0834i	−0.999353	+	1.73093i	−0.468521	+	0.883452i	$0.655213\pi$
$41$	−6.39898	+	11.0834i	−0.999353	+	1.73093i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	8.66025	+	5.00000i	1.32068	+	0.762493i	0.983836	−	0.179069i	$-0.0573086\pi$
$43$	8.66025	+	5.00000i	1.32068	+	0.762493i	0.336840	+	0.941562i	$0.390642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−17.9241	+	10.3485i	−2.50987	+	1.44908i
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.44949	+	14.6349i	0.456896	+	1.93845i
$58$		0			0
$59$	−8.00853	−	4.62372i	−1.04262	−	0.601958i	−0.122047	−	0.992524i	$-0.538946\pi$
$59$	−8.00853	−	4.62372i	−1.04262	−	0.601958i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.4261	+	7.17423i	−1.51809	+	0.876472i	−0.518321	+	0.855186i	$0.673443\pi$
$67$	−12.4261	+	7.17423i	−1.51809	+	0.876472i	−0.999773	+	0.0212861i	$0.993224\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	7.84847	−	13.5939i	0.918594	−	1.59105i	0.117041	−	0.993127i	$-0.462659\pi$
$73$	7.84847	−	13.5939i	0.918594	−	1.59105i	0.801553	−	0.597924i	$-0.204008\pi$
$74$		0			0
$75$			17.2474i			1.99156i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$	−21.7474	+	37.6677i	−2.41638	+	4.18530i
$82$		0			0
$83$			6.55051i			0.719012i	0.933143	+	0.359506i	$0.117055\pi$
$83$			6.55051i			0.719012i	−0.933143	+	0.359506i	$0.882945\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.736644	−	0.676280i	$-0.763591\pi$
$89$	−9.00000	−	15.5885i	−0.953998	−	1.65237i	−0.217354	−	0.976093i	$-0.569742\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.84847	+	8.39780i	−0.492287	+	0.852667i	−0.999961	−	0.00888289i	$-0.997172\pi$
$97$	−4.84847	+	8.39780i	−0.492287	+	0.852667i	0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	4.24264	−	2.44949i	0.426401	−	0.246183i
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		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	$0.0936623\pi$
$107$			6.00000i			0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.7980			−1.76836			−0.884182	−	0.467143i	$-0.845283\pi$
$113$	−18.7980			−1.76836			−0.884182	+	0.467143i	$0.845283\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.6969			0.972449
$122$		0			0
$123$	38.2319	−	22.0732i	3.44726	−	1.99027i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$		0			0
$129$	−17.2474	−	29.8735i	−1.51855	−	2.63021i
$130$		0			0
$131$	−18.4008	−	10.6237i	−1.60769	−	0.928199i	−0.989886	−	0.141865i	$-0.954690\pi$
$131$	−18.4008	−	10.6237i	−1.60769	−	0.928199i	−0.617802	−	0.786334i	$-0.711977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.2980	−	19.5686i	−0.965250	−	1.67186i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	−11.2980	−	19.5686i	−0.965250	−	1.67186i	−0.256307	−	0.966595i	$-0.582506\pi$
$138$		0			0
$139$	−15.8902	+	9.17423i	−1.34779	+	0.778148i	−0.987937	−	0.154859i	$-0.950508\pi$
$139$	−15.8902	+	9.17423i	−1.34779	+	0.778148i	−0.359856	+	0.933008i	$0.617174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.9114	+	12.0732i	1.72474	+	0.995782i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	53.3939			4.31664
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			23.0454i			1.80506i	0.430632	+	0.902528i	$0.358291\pi$
$163$			23.0454i			1.80506i	−0.430632	+	0.902528i	$0.641709\pi$
$164$		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
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$	11.1708	−	37.1464i	0.854256	−	2.84066i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	15.9495	+	27.6253i	1.19884	+	2.07645i
$178$		0			0
$179$		−	8.14643i		−	0.608893i	−0.952529	−	0.304446i	$-0.901529\pi$
$179$		−	8.14643i		−	0.608893i	0.952529	−	0.304446i	$-0.0984714\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.86054	−	1.65153i	−0.209183	−	0.120772i
$188$		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	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	0.924853	−	0.380325i	$-0.124188\pi$
$194$		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	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$	49.4949			3.49110
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.74745	+	1.64456i	−0.120874	+	0.113757i
$210$		0			0
$211$	−12.1244	−	7.00000i	−0.834675	−	0.481900i	0.0207756	−	0.999784i	$-0.493386\pi$
$211$	−12.1244	−	7.00000i	−0.834675	−	0.481900i	−0.855451	+	0.517884i	$0.826720\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−46.8922	+	27.0732i	−3.16868	+	1.82944i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	22.2474	−	38.5337i	1.48316	−	2.56891i
$226$		0			0
$227$		−	17.4495i		−	1.15816i	−0.815270	−	0.579082i	$-0.803411\pi$
$227$		−	17.4495i		−	1.15816i	0.815270	−	0.579082i	$-0.196589\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.94949	+	17.2330i	−0.651813	+	1.12897i	0.330870	+	0.943676i	$0.392658\pi$
$233$	−9.94949	+	17.2330i	−0.651813	+	1.12897i	−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
$241$	−0.848469	−	1.46959i	−0.0546547	−	0.0946647i	0.837404	−	0.546585i	$-0.184072\pi$
$241$	−0.848469	−	1.46959i	−0.0546547	−	0.0946647i	−0.892058	+	0.451920i	$0.850739\pi$
$242$		0			0
$243$	77.0674	−	44.4949i	4.94388	−	2.85435i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	11.2980	−	19.5686i	0.715979	−	1.24011i
$250$		0			0
$251$	20.7364	−	11.9722i	1.30887	−	0.755678i	0.326965	−	0.945036i	$-0.393974\pi$
$251$	20.7364	−	11.9722i	1.30887	−	0.755678i	0.981908	+	0.189358i	$0.0606408\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.60102	−	4.50510i	−0.162247	−	0.281020i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	−2.60102	−	4.50510i	−0.162247	−	0.281020i	−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	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			62.0908i			3.79990i
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.38378	+	1.37628i	−0.143747	+	0.0829925i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.74745	−	13.4190i	−0.462174	−	0.800509i	0.536895	−	0.843649i	$-0.319597\pi$
$281$	−7.74745	−	13.4190i	−0.462174	−	0.800509i	−0.999069	+	0.0431402i	$0.986264\pi$
$282$		0			0
$283$	28.6182	+	16.5227i	1.70117	+	0.982173i	0.944577	+	0.328291i	$0.106473\pi$
$283$	28.6182	+	16.5227i	1.70117	+	0.982173i	0.756596	+	0.653882i	$0.226861\pi$
$284$		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
$291$	28.9681	−	16.7247i	1.69814	−	0.980422i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−11.2020			−0.650008
$298$		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
$307$	−8.35847	−	4.82577i	−0.477043	−	0.275421i	0.242140	−	0.970241i	$-0.422151\pi$
$307$	−8.35847	−	4.82577i	−0.477043	−	0.275421i	−0.719183	+	0.694820i	$0.755484\pi$
$308$		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
$313$	−12.1969	−	21.1257i	−0.689412	−	1.19410i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−12.1969	−	21.1257i	−0.689412	−	1.19410i	0.282617	−	0.959233i	$-0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	10.3485	−	17.9241i	0.577595	−	1.00042i
$322$		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
$331$		−	9.04541i		−	0.497181i	−0.968609	−	0.248590i	$-0.920033\pi$
$331$		−	9.04541i		−	0.497181i	0.968609	−	0.248590i	$-0.0799673\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.1969	−	19.3937i	0.609936	−	1.05644i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	11.1969	−	19.3937i	0.609936	−	1.05644i	0.991250	−	0.131995i	$-0.0421382\pi$
$338$		0			0
$339$	56.1560	+	32.4217i	3.04998	+	1.76090i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−24.9791	−	14.4217i	−1.34095	−	0.774197i	−0.354001	−	0.935245i	$-0.615179\pi$
$347$	−24.9791	−	14.4217i	−1.34095	−	0.774197i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.59592			−0.244616			−0.122308	−	0.992492i	$-0.539030\pi$
$353$	−4.59592			−0.244616			−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−1.15153	+	18.9651i	−0.0606069	+	0.998162i
$362$		0			0
$363$	−31.9555	−	18.4495i	−1.67723	−	0.968347i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$	−113.889			−5.92881
$370$		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	$0.430070\pi$
$379$			38.0000i			1.95193i	−0.217930	+	0.975964i	$0.569930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			88.9898i			4.52361i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	36.6464	+	63.4735i	1.84857	+	3.20181i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.6464	+	32.2966i	−0.931158	+	1.61281i	−0.149813	+	0.988714i	$0.547867\pi$
$401$	−18.6464	+	32.2966i	−0.931158	+	1.61281i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$			77.9444i			3.84471i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	63.2929			3.09946
$418$		0			0
$419$		−	18.0000i		−	0.879358i	−0.898155	−	0.439679i	$-0.855092\pi$
$419$		−	18.0000i		−	0.879358i	0.898155	−	0.439679i	$-0.144908\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		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	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	19.0000	+	32.9090i	0.913082	+	1.58150i	0.809686	+	0.586864i	$0.199638\pi$
$433$	19.0000	+	32.9090i	0.913082	+	1.58150i	0.103396	+	0.994640i	$0.467029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	−31.1464	−	53.9472i	−1.48316	−	2.56891i
$442$		0			0
$443$	16.0652	−	9.27526i	0.763281	−	0.440681i	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	16.0652	−	9.27526i	0.763281	−	0.440681i	0.830473	+	0.557059i	$0.188070\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−25.8990			−1.22225			−0.611124	−	0.791535i	$-0.709282\pi$
$449$	−25.8990			−1.22225			−0.611124	+	0.791535i	$0.709282\pi$
$450$		0			0
$451$	6.10150	+	3.52270i	0.287309	+	0.165878i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.3939			0.766873			0.383437	−	0.923567i	$-0.374740\pi$
$457$	16.3939			0.766873			0.383437	+	0.923567i	$0.374740\pi$
$458$		0			0
$459$	−105.734	−	61.0454i	−4.93523	−	2.84936i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		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
$467$		−	41.9444i		−	1.94095i	−0.241192	−	0.970477i	$-0.577538\pi$
$467$		−	41.9444i		−	1.94095i	0.241192	−	0.970477i	$-0.422462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.75255	−	4.76756i	0.126562	−	0.219213i
$474$		0			0
$475$	−6.27647	+	20.8712i	−0.287984	+	0.957635i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		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
$489$	39.7474	−	68.8446i	1.79744	−	3.11326i
$490$		0			0
$491$	36.3731	+	21.0000i	1.64149	+	0.947717i	0.980303	+	0.197499i	$0.0632818\pi$
$491$	36.3731	+	21.0000i	1.64149	+	0.947717i	0.661190	+	0.750218i	$0.270052\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−25.7576	−	14.8712i	−1.15307	−	0.665725i	−0.203436	−	0.979088i	$-0.565211\pi$
$499$	−25.7576	−	14.8712i	−1.15307	−	0.665725i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	38.8355	−	22.4217i	1.72474	−	0.995782i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−64.5908	+	60.7879i	−2.85175	+	2.68385i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−36.1918			−1.58559			−0.792797	−	0.609486i	$-0.791376\pi$
$521$	−36.1918			−1.58559			−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	32.9090	−	19.0000i	1.43901	−	0.830812i	0.441228	−	0.897395i	$-0.354543\pi$
$523$	32.9090	−	19.0000i	1.43901	−	0.830812i	0.997781	+	0.0665832i	$0.0212098\pi$
$524$		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
$531$		−	82.2929i		−	3.57121i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−14.0505	+	24.3362i	−0.606324	+	1.05018i
$538$		0			0
$539$			3.85357i			0.165985i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	39.8372	−	23.0000i	1.70331	−	0.983409i	0.760956	−	0.648803i	$-0.224730\pi$
$547$	39.8372	−	23.0000i	1.70331	−	0.983409i	0.942358	−	0.334606i	$-0.108603\pi$
$548$		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	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.69694	+	9.86739i	0.240525	+	0.416601i
$562$		0			0
$563$		−	46.8434i		−	1.97421i	−0.160066	−	0.987106i	$-0.551171\pi$
$563$		−	46.8434i		−	1.97421i	0.160066	−	0.987106i	$-0.448829\pi$
$564$		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
$571$			25.7423i			1.07728i	0.842535	+	0.538642i	$0.181062\pi$
$571$			25.7423i			1.07728i	−0.842535	+	0.538642i	$0.818938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	12.3939			0.515964			0.257982	−	0.966150i	$-0.416942\pi$
$577$	12.3939			0.515964			0.257982	+	0.966150i	$0.416942\pi$
$578$		0			0
$579$	−65.7216	+	37.9444i	−2.73130	+	1.57691i
$580$		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	0.388918	−	0.921272i	$-0.372849\pi$
$587$	−5.19615	−	3.00000i	−0.214468	−	0.123823i	−0.603386	+	0.797449i	$0.706182\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.0959	+	41.7354i	0.989501	+	1.71387i	0.619915	+	0.784669i	$0.287167\pi$
$593$	24.0959	+	41.7354i	0.989501	+	1.71387i	0.369586	+	0.929197i	$0.379500\pi$
$594$		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
$601$	37.6969			1.53769			0.768845	−	0.639435i	$-0.220832\pi$
$601$	37.6969			1.53769			0.768845	+	0.639435i	$0.220832\pi$
$602$		0			0
$603$	−110.580	−	63.8434i	−4.50316	−	2.59990i
$604$		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	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.6464	−	42.6889i	−0.992228	−	1.71859i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−24.6464	−	42.6889i	−0.992228	−	1.71859i	−0.388351	−	0.921512i	$-0.626955\pi$
$618$		0			0
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$		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
$627$	8.05669	−	1.89898i	0.321753	−	0.0758379i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$	24.1464	+	41.8228i	0.959734	+	1.66231i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.74745	+	3.02667i	−0.0690201	+	0.119546i	−0.898470	−	0.439034i	$-0.855321\pi$
$641$	−1.74745	+	3.02667i	−0.0690201	+	0.119546i	0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$	−15.2867	−	8.82577i	−0.602848	−	0.348054i	0.167313	−	0.985904i	$-0.446491\pi$
$643$	−15.2867	−	8.82577i	−0.602848	−	0.348054i	−0.770161	+	0.637850i	$0.779824\pi$
$644$		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
$649$	−2.54541	+	4.40878i	−0.0999160	+	0.173060i
$650$		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
$657$	139.687			5.44970
$658$		0			0
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	−0.771885	−	0.635763i	$-0.780686\pi$
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	0.164644	+	0.986353i	$0.447352\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		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
$675$	−88.1115	+	50.8712i	−3.39141	+	1.95803i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−30.0959	+	52.1277i	−1.15328	+	1.99754i
$682$		0			0
$683$		−	42.0000i		−	1.60709i	−0.595247	−	0.803543i	$-0.702946\pi$
$683$		−	42.0000i		−	1.60709i	0.595247	−	0.803543i	$-0.297054\pi$
$684$		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	$0.339113\pi$
$691$			46.0000i			1.74992i	−0.484193	+	0.874961i	$0.660887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	38.3939	+	66.5001i	1.45427	+	2.51887i
$698$		0			0
$699$	59.4451	−	34.3207i	2.24842	−	1.29813i
$700$		0			0
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		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	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		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	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			5.85357i			0.217697i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	−176.485			−6.53647
$730$		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
$737$	3.94949	+	6.84072i	0.145481	+	0.251981i
$738$		0			0
$739$	−46.5422	−	26.8712i	−1.71208	−	0.988472i	−0.931744	−	0.363117i	$-0.881713\pi$
$739$	−46.5422	−	26.8712i	−1.71208	−	0.988472i	−0.780340	−	0.625355i	$-0.784954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−50.4831	+	29.1464i	−1.84708	+	1.06641i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	−82.5959			−3.00996
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.2020			0.623574			0.311787	−	0.950152i	$-0.399073\pi$
$761$	17.2020			0.623574			0.311787	+	0.950152i	$0.399073\pi$
$762$		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	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$			17.9444i			0.646251i
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	54.2971	−	12.7980i	1.94540	−	0.458534i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			47.0454i			1.67699i	0.544911	+	0.838494i	$0.316563\pi$
$787$			47.0454i			1.67699i	−0.544911	+	0.838494i	$0.683437\pi$
$788$		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
$801$	80.0908	−	138.721i	2.82987	−	4.90148i
$802$		0			0
$803$	−7.48361	−	4.32066i	−0.264091	−	0.152473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−45.9898			−1.61692			−0.808458	−	0.588555i	$-0.799697\pi$
$809$	−45.9898			−1.61692			−0.808458	+	0.588555i	$0.799697\pi$
$810$		0			0
$811$	32.9090	−	19.0000i	1.15559	−	0.667180i	0.205347	−	0.978689i	$-0.434168\pi$
$811$	32.9090	−	19.0000i	1.15559	−	0.667180i	0.950243	+	0.311509i	$0.100834\pi$
$812$		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	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$	9.49490			0.330570
$826$		0			0
$827$	−8.53344	+	4.92679i	−0.296737	+	0.171321i	−0.640976	−	0.767561i	$-0.721470\pi$
$827$	−8.53344	+	4.92679i	−0.296737	+	0.171321i	0.344239	+	0.938882i	$0.388137\pi$
$828$		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	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$			53.4495i			1.84090i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−56.9949	−	98.7181i	−1.95606	−	3.38799i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.2980	+	40.3532i	−0.795843	+	1.37844i	0.126459	+	0.991972i	$0.459639\pi$
$857$	−23.2980	+	40.3532i	−0.795843	+	1.37844i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	31.4787	−	18.1742i	1.07404	−	0.620097i	0.144757	−	0.989467i	$-0.453760\pi$
$859$	31.4787	−	18.1742i	1.07404	−	0.620097i	0.929282	+	0.369370i	$0.120427\pi$
$860$		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
$867$			65.5403i			2.22587i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−86.2929			−2.92057
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	39.9898			1.34729			0.673645	−	0.739055i	$-0.264728\pi$
$881$	39.9898			1.34729			0.673645	+	0.739055i	$0.264728\pi$
$882$		0			0
$883$	45.4138	−	26.2196i	1.52829	−	0.882361i	0.528861	−	0.848709i	$-0.322619\pi$
$883$	45.4138	−	26.2196i	1.52829	−	0.882361i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$		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
$891$	20.7364	+	11.9722i	0.694697	+	0.401084i
$892$		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
$907$	40.2176	+	23.2196i	1.33540	+	0.770996i	0.986122	−	0.166022i	$-0.0530924\pi$
$907$	40.2176	+	23.2196i	1.33540	+	0.770996i	0.349281	+	0.937018i	$0.386426\pi$
$908$		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
$913$	3.60612			0.119345
$914$		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
$921$	16.6464	+	28.8325i	0.548518	+	0.950062i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.7474	+	44.5959i	−0.844746	+	1.46314i	0.0410949	+	0.999155i	$0.486915\pi$
$929$	−25.7474	+	44.5959i	−0.844746	+	1.46314i	−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$	20.9114	+	22.2196i	0.685344	+	0.728219i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−13.5454	−	23.4613i	−0.442509	−	0.766448i	0.555366	−	0.831606i	$-0.312578\pi$
$937$	−13.5454	−	23.4613i	−0.442509	−	0.766448i	−0.997875	+	0.0651578i	$0.979245\pi$
$938$		0			0
$939$			84.1464i			2.74601i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.9808	−	15.0000i	−0.844261	−	0.487435i	0.0144491	−	0.999896i	$-0.495401\pi$
$947$	−25.9808	−	15.0000i	−0.844261	−	0.487435i	−0.858710	+	0.512461i	$0.828734\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.09592	+	15.7546i	−0.294646	+	0.510341i	−0.974902	−	0.222633i	$-0.928535\pi$
$953$	−9.09592	+	15.7546i	−0.294646	+	0.510341i	0.680257	+	0.732974i	$0.261868\pi$
$954$		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
$963$	−46.2405	+	26.6969i	−1.49008	+	0.860297i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$967$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$968$		0			0
$969$	86.3939	+	25.9808i	2.77537	+	0.834622i
$970$		0			0
$971$	0.0481621	+	0.0278064i	0.00154560	+	0.000892350i	0.500773	−	0.865579i	$-0.333049\pi$
$971$	0.0481621	+	0.0278064i	0.00154560	+	0.000892350i	−0.499227	+	0.866471i	$0.666383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	56.8888			1.82003			0.910017	−	0.414572i	$-0.136069\pi$
$977$	56.8888			1.82003			0.910017	+	0.414572i	$0.136069\pi$
$978$		0			0
$979$	−8.58161	+	4.95459i	−0.274269	+	0.158349i
$980$		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
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$991$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$992$		0			0
$993$	−15.6010	+	27.0218i	−0.495083	+	0.857510i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		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.353.1	✓	8
4.3	odd	2	inner	1216.2.t.a.353.4	yes	8
8.3	odd	2	CM	1216.2.t.a.353.1	✓	8
8.5	even	2	inner	1216.2.t.a.353.4	yes	8
19.7	even	3	inner	1216.2.t.a.1185.4	yes	8
76.7	odd	6	inner	1216.2.t.a.1185.1	yes	8
152.45	even	6	inner	1216.2.t.a.1185.1	yes	8
152.83	odd	6	inner	1216.2.t.a.1185.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.t.a.353.1	✓	8	1.1	even	1	trivial
1216.2.t.a.353.1	✓	8	8.3	odd	2	CM
1216.2.t.a.353.4	yes	8	4.3	odd	2	inner
1216.2.t.a.353.4	yes	8	8.5	even	2	inner
1216.2.t.a.1185.1	yes	8	76.7	odd	6	inner
1216.2.t.a.1185.1	yes	8	152.45	even	6	inner
1216.2.t.a.1185.4	yes	8	19.7	even	3	inner
1216.2.t.a.1185.4	yes	8	152.83	odd	6	inner

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)

\(f(q)\)	\(=\)	\(q+(-2.98735 - 1.72474i) q^{3} +(4.44949 + 7.70674i) q^{9} +O(q^{10})\)	\(q+(-2.98735 - 1.72474i) q^{3} +(4.44949 + 7.70674i) q^{9} -0.550510i q^{11} +(3.00000 - 5.19615i) q^{17} +(-2.98735 - 3.17423i) q^{19} +(-2.50000 - 4.33013i) q^{25} -20.3485i q^{27} +(-0.949490 + 1.64456i) q^{33} +(-6.39898 + 11.0834i) q^{41} +(8.66025 + 5.00000i) q^{43} -7.00000 q^{49} +(-17.9241 + 10.3485i) q^{51} +(3.44949 + 14.6349i) q^{57} +(-8.00853 - 4.62372i) q^{59} +(-12.4261 + 7.17423i) q^{67} +(7.84847 - 13.5939i) q^{73} +17.2474i q^{75} +(-21.7474 + 37.6677i) q^{81} +6.55051i q^{83} +(-9.00000 - 15.5885i) q^{89} +(-4.84847 + 8.39780i) q^{97} +(4.24264 - 2.44949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 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

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