LMFDB - Embedded newform 912.2.cc.b.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(257,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 9, 17]))

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

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

chi := DirichletCharacter("912.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.cc (of order $18$, degree $6$, 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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			641.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	912.641
Dual form			912.2.cc.b.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.70574 + 0.300767i) q^{3} +(-1.35844 - 2.35289i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})$	$q+(1.70574 + 0.300767i) q^{3} +(-1.35844 - 2.35289i) q^{7} +(2.81908 + 1.02606i) q^{9} +(6.49273 - 1.14484i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(-1.60947 - 4.42198i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.27244 - 4.19875i) q^{31} -9.93058i q^{37} +11.4192 q^{39} +(3.03596 + 2.54747i) q^{43} +(-0.190722 + 0.330341i) q^{49} +(-5.18866 - 5.48432i) q^{57} +(-4.80406 + 4.03109i) q^{61} +(-1.41534 - 8.02682i) q^{63} +(-5.59492 + 15.3719i) q^{67} +(-2.87686 + 16.3155i) q^{73} +8.66025i q^{75} +(-6.84137 - 1.20632i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-11.5137 - 13.7215i) q^{91} +(13.6677 - 4.97464i) q^{93} +(-6.51636 - 17.9035i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} + 27 q^{27} - 15 q^{43} - 21 q^{49} - 42 q^{61} + 36 q^{63} - 15 q^{67} - 21 q^{73} - 12 q^{79} - 48 q^{91} + 54 q^{93}+O(q^{100}) $ 6 * q + 21 * q^13 - 21 * q^19 - 27 * q^21 + 27 * q^27 - 15 * q^43 - 21 * q^49 - 42 * q^61 + 36 * q^63 - 15 * q^67 - 21 * q^73 - 12 * q^79 - 48 * q^91 + 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{7}{18}\right)$	$1$	$-1$	$1$

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
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$3$	1.70574	+	0.300767i	0.984808	+	0.173648i
$4$		0			0
$5$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$5$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$6$		0			0
$7$	−1.35844	−	2.35289i	−0.513442	−	0.889308i	−0.999878	−	0.0155920i	$-0.995037\pi$
$7$	−1.35844	−	2.35289i	−0.513442	−	0.889308i	0.486436	−	0.873716i	$-0.338297\pi$
$8$		0			0
$9$	2.81908	+	1.02606i	0.939693	+	0.342020i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	6.49273	−	1.14484i	1.80076	−	0.317522i	0.830033	−	0.557714i	$-0.188322\pi$
$13$	6.49273	−	1.14484i	1.80076	−	0.317522i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$17$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$18$		0			0
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$20$		0			0
$21$	−1.60947	−	4.42198i	−0.351215	−	0.964956i
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		0			0
$25$	0.868241	+	4.92404i	0.173648	+	0.984808i
$26$		0			0
$27$	4.50000	+	2.59808i	0.866025	+	0.500000i
$28$		0			0
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$	7.27244	−	4.19875i	1.30617	−	0.754117i	0.324714	−	0.945812i	$-0.394732\pi$
$31$	7.27244	−	4.19875i	1.30617	−	0.754117i	0.981455	+	0.191695i	$0.0613985\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.93058i		−	1.63258i	−0.577644	−	0.816289i	$-0.696028\pi$
$37$		−	9.93058i		−	1.63258i	0.577644	−	0.816289i	$-0.303972\pi$
$38$		0			0
$39$	11.4192			1.82854
$40$		0			0
$41$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$41$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$42$		0			0
$43$	3.03596	+	2.54747i	0.462979	+	0.388486i	0.844226	−	0.535988i	$-0.180061\pi$
$43$	3.03596	+	2.54747i	0.462979	+	0.388486i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$47$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$48$		0			0
$49$	−0.190722	+	0.330341i	−0.0272460	+	0.0471915i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$53$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.18866	−	5.48432i	−0.687255	−	0.726416i
$58$		0			0
$59$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$59$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$60$		0			0
$61$	−4.80406	+	4.03109i	−0.615097	+	0.516128i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−4.80406	+	4.03109i	−0.615097	+	0.516128i	0.281161	+	0.959661i	$0.409281\pi$
$62$		0			0
$63$	−1.41534	−	8.02682i	−0.178317	−	1.01128i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.59492	+	15.3719i	−0.683529	+	1.87798i	−0.305424	+	0.952217i	$0.598798\pi$
$67$	−5.59492	+	15.3719i	−0.683529	+	1.87798i	−0.378105	+	0.925763i	$0.623424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$71$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$72$		0			0
$73$	−2.87686	+	16.3155i	−0.336711	+	1.90958i	0.0729331	+	0.997337i	$0.476764\pi$
$73$	−2.87686	+	16.3155i	−0.336711	+	1.90958i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.84137	−	1.20632i	−0.769714	−	0.135721i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−6.84137	−	1.20632i	−0.769714	−	0.135721i	−0.544696	+	0.838633i	$0.683355\pi$
$80$		0			0
$81$	6.89440	+	5.78509i	0.766044	+	0.642788i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$89$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$90$		0			0
$91$	−11.5137	−	13.7215i	−1.20696	−	1.43840i
$92$		0			0
$93$	13.6677	−	4.97464i	1.41728	−	0.515846i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.51636	−	17.9035i	−0.661636	−	1.81783i	−0.569346	−	0.822098i	$-0.692804\pi$
$97$	−6.51636	−	17.9035i	−0.661636	−	1.81783i	−0.0922897	−	0.995732i	$-0.529419\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$101$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$102$		0			0
$103$	13.4315	+	7.75470i	1.32345	+	0.764094i	0.984277	−	0.176631i	$-0.0565198\pi$
$103$	13.4315	+	7.75470i	1.32345	+	0.764094i	0.339172	+	0.940724i	$0.389853\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	−7.79339	+	9.28780i	−0.746471	+	0.889609i	−0.996912	−	0.0785223i	$-0.974980\pi$
$109$	−7.79339	+	9.28780i	−0.746471	+	0.889609i	0.250441	+	0.968132i	$0.419424\pi$
$110$		0			0
$111$	2.98680	−	16.9390i	0.283494	−	1.60778i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	19.4782	+	3.43453i	1.80076	+	0.317522i
$118$		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
$125$		0			0
$126$		0			0
$127$	−22.1746	+	3.90998i	−1.96768	+	0.346954i	−0.976134	+	0.217171i	$0.930317\pi$
$127$	−22.1746	+	3.90998i	−1.96768	+	0.346954i	−0.991542	+	0.129783i	$0.958572\pi$
$128$		0			0
$129$	4.41235	+	5.25844i	0.388486	+	0.462979i
$130$		0			0
$131$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$131$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$132$		0			0
$133$	−1.35844	+	11.7644i	−0.117792	+	1.02011i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$137$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$138$		0			0
$139$	−0.195060	−	1.10624i	−0.0165447	−	0.0938298i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	−0.195060	−	1.10624i	−0.0165447	−	0.0938298i	−0.991962	+	0.126536i	$0.959614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.424678	+	0.506111i	−0.0350268	+	0.0417434i
$148$		0			0
$149$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$149$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$150$		0			0
$151$		−	8.66025i		−	0.704761i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$		−	8.66025i		−	0.704761i	0.935857	−	0.352381i	$-0.114628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−17.5424	−	14.7198i	−1.40003	−	1.17477i	−0.961085	−	0.276254i	$-0.910907\pi$
$157$	−17.5424	−	14.7198i	−1.40003	−	1.17477i	−0.438948	−	0.898513i	$-0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.8576	+	18.8059i	−0.850430	+	1.47299i	0.0303908	+	0.999538i	$0.490325\pi$
$163$	−10.8576	+	18.8059i	−0.850430	+	1.47299i	−0.880821	+	0.473450i	$0.843009\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	28.6288	−	10.4200i	2.20222	−	0.801541i
$170$		0			0
$171$	−7.20099	−	10.9154i	−0.550673	−	0.834721i
$172$		0			0
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$		0			0
$175$	10.4063	−	8.73189i	0.786639	−	0.660069i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−8.88594	+	24.4139i	−0.660487	+	1.81467i	−0.0857797	+	0.996314i	$0.527338\pi$
$181$	−8.88594	+	24.4139i	−0.660487	+	1.81467i	−0.574707	+	0.818359i	$0.694884\pi$
$182$		0			0
$183$	−9.40689	+	5.43107i	−0.695377	+	0.401476i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	14.1173i		−	1.02688i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−26.3195	−	4.64085i	−1.89452	−	0.334055i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−26.3195	−	4.64085i	−1.89452	−	0.334055i	−0.994753	+	0.102310i	$0.967377\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	23.6113	+	8.59380i	1.67376	+	0.609199i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	23.6113	+	8.59380i	1.67376	+	0.609199i	0.681326	+	0.731980i	$0.261404\pi$
$200$		0			0
$201$	−14.1668	+	24.5377i	−0.999252	+	1.73076i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	9.66503	+	26.5545i	0.665368	+	1.82808i	0.550743	+	0.834675i	$0.314345\pi$
$211$	9.66503	+	26.5545i	0.665368	+	1.82808i	0.114625	+	0.993409i	$0.463433\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−19.7584	−	11.4075i	−1.34128	−	0.774391i
$218$		0			0
$219$	−9.81433	+	26.9647i	−0.663191	+	1.82210i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.5646	−	19.7410i	1.10925	−	1.32195i	0.167412	−	0.985887i	$-0.446459\pi$
$223$	16.5646	−	19.7410i	1.10925	−	1.32195i	0.941838	−	0.336066i	$-0.109097\pi$
$224$		0			0
$225$	−2.60472	+	14.7721i	−0.173648	+	0.984808i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−27.7820			−1.83589			−0.917943	−	0.396713i	$-0.870151\pi$
$229$	−27.7820			−1.83589			−0.917943	+	0.396713i	$0.870151\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$233$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−11.3068	−	4.11532i	−0.734452	−	0.267319i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	18.3170	−	3.22978i	1.17990	−	0.208049i	0.450910	−	0.892570i	$-0.351100\pi$
$241$	18.3170	−	3.22978i	1.17990	−	0.208049i	0.728993	+	0.684521i	$0.239989\pi$
$242$		0			0
$243$	10.0201	+	11.9415i	0.642788	+	0.766044i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−25.6989	−	12.8616i	−1.63518	−	0.818367i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$251$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$257$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$258$		0			0
$259$	−23.3656	+	13.4901i	−1.45186	+	0.838235i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$269$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$270$		0			0
$271$	−22.2153	−	18.6408i	−1.34948	−	1.13235i	−0.979079	−	0.203479i	$-0.934775\pi$
$271$	−22.2153	−	18.6408i	−1.34948	−	1.13235i	−0.370403	−	0.928871i	$-0.620781\pi$
$272$		0			0
$273$	−15.5123	−	26.8681i	−0.938849	−	1.62613i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.150210	+	0.988654i	$0.547995\pi$
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	24.8097	−	4.37463i	1.48532	−	0.261902i
$280$		0			0
$281$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$281$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$282$		0			0
$283$	−23.4923	+	8.55050i	−1.39647	+	0.508275i	−0.927130	−	0.374741i	$-0.877732\pi$
$283$	−23.4923	+	8.55050i	−1.39647	+	0.508275i	−0.469344	+	0.883016i	$0.655509\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0228	−	10.9274i	0.766044	−	0.642788i
$290$		0			0
$291$	−5.73039	−	32.4987i	−0.335921	−	1.90510i
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.86975	−	10.6039i	0.107770	−	0.611197i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.70574	−	0.300767i	−0.0973516	−	0.0171657i	0.124760	−	0.992187i	$-0.460184\pi$
$307$	−1.70574	−	0.300767i	−0.0973516	−	0.0171657i	−0.222112	+	0.975021i	$0.571295\pi$
$308$		0			0
$309$	20.5783	+	17.2673i	1.17066	+	0.982300i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	32.8892	+	11.9707i	1.85901	+	0.676624i	0.979731	+	0.200316i	$0.0641970\pi$
$313$	32.8892	+	11.9707i	1.85901	+	0.676624i	0.879279	+	0.476308i	$0.158025\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$317$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	11.2745	+	30.9764i	0.625397	+	1.71826i
$326$		0			0
$327$	−16.0869	+	13.4985i	−0.889609	+	0.746471i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.61705	+	2.66565i	0.253776	+	0.146518i	0.621492	−	0.783420i	$-0.286527\pi$
$331$	4.61705	+	2.66565i	0.253776	+	0.146518i	−0.367716	+	0.929938i	$0.619860\pi$
$332$		0			0
$333$	10.1894	−	27.9951i	0.558374	−	1.53412i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7.22503	−	8.61046i	0.393573	−	0.469042i	−0.532476	−	0.846445i	$-0.678738\pi$
$337$	7.22503	−	8.61046i	0.393573	−	0.469042i	0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.9818			−0.970928
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$347$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$348$		0			0
$349$	0.653886	+	1.13256i	0.0350017	+	0.0606247i	0.882996	−	0.469381i	$-0.155523\pi$
$349$	0.653886	+	1.13256i	0.0350017	+	0.0606247i	−0.847994	+	0.530006i	$0.822190\pi$
$350$		0			0
$351$	32.1917	+	11.7168i	1.71826	+	0.625397i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$359$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$	−6.51636	−	17.9035i	−0.342020	−	0.939693i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.61040	+	9.13306i	0.0840624	+	0.476742i	0.997555	+	0.0698862i	$0.0222636\pi$
$367$	1.61040	+	9.13306i	0.0840624	+	0.476742i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.00000	−	3.46410i	0.310668	−	0.179364i	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	6.00000	−	3.46410i	0.310668	−	0.179364i	0.647225	+	0.762299i	$0.275929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	24.0479i		−	1.23526i	−0.786469	−	0.617629i	$-0.788093\pi$
$379$		−	24.0479i		−	1.23526i	0.786469	−	0.617629i	$-0.211907\pi$
$380$		0			0
$381$	−39.0000			−1.99803
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.94475	+	10.2966i	0.302188	+	0.523406i
$388$		0			0
$389$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$389$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	37.0292	−	13.4775i	1.85844	−	0.676417i	0.878300	−	0.478110i	$-0.158678\pi$
$397$	37.0292	−	13.4775i	1.85844	−	0.676417i	0.980140	−	0.198307i	$-0.0635442\pi$
$398$		0			0
$399$	−5.85550	+	19.6585i	−0.293142	+	0.984154i
$400$		0			0
$401$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$401$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$402$		0			0
$403$	42.4111	−	35.5871i	2.11265	−	1.77272i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.73917	+	13.0208i	−0.234337	+	0.643835i	0.765663	+	0.643242i	$0.222411\pi$
$409$	−4.73917	+	13.0208i	−0.234337	+	0.643835i	−1.00000		0.000593299i	$0.999811\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	1.94562i		−	0.0952773i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.70574	+	0.300767i	0.0831325	+	0.0146585i	0.215060	−	0.976601i	$-0.431005\pi$
$421$	1.70574	+	0.300767i	0.0831325	+	0.0146585i	−0.131927	+	0.991259i	$0.542117\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	16.0107	+	5.82743i	0.774814	+	0.282009i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$431$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$432$		0			0
$433$	−5.90595	−	7.03844i	−0.283822	−	0.338246i	0.605231	−	0.796050i	$-0.293081\pi$
$433$	−5.90595	−	7.03844i	−0.283822	−	0.338246i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	14.3241	+	39.3552i	0.683653	+	1.87832i	0.373425	+	0.927660i	$0.378183\pi$
$439$	14.3241	+	39.3552i	0.683653	+	1.87832i	0.310228	+	0.950662i	$0.399595\pi$
$440$		0			0
$441$	−0.876611	+	0.735564i	−0.0417434	+	0.0350268i
$442$		0			0
$443$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$443$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.60472	−	14.7721i	0.122381	−	0.694055i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	39.2012			1.83376			0.916878	−	0.399169i	$-0.130701\pi$
$457$	39.2012			1.83376			0.916878	+	0.399169i	$0.130701\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$461$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$462$		0			0
$463$	−4.58630	−	7.94371i	−0.213144	−	0.369176i	0.739553	−	0.673098i	$-0.235037\pi$
$463$	−4.58630	−	7.94371i	−0.213144	−	0.369176i	−0.952697	+	0.303923i	$0.901704\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	43.7688	−	7.71762i	2.02105	−	0.356367i
$470$		0			0
$471$	−25.4954	−	30.3843i	−1.17477	−	1.40003i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	9.75418	−	19.4899i	0.447553	−	0.894258i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$	−11.3690	−	64.4766i	−0.518380	−	2.93988i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.430486	−	0.902597i	$-0.358342\pi$
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$	−24.1763	+	28.8122i	−1.09329	+	1.30293i
$490$		0			0
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	34.1300	+	28.6385i	1.52787	+	1.28203i	0.811610	+	0.584199i	$0.198591\pi$
$499$	34.1300	+	28.6385i	1.52787	+	1.28203i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$503$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	51.9673	−	9.16323i	2.30795	−	0.406953i
$508$		0			0
$509$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$509$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$510$		0			0
$511$	42.2965	−	15.3947i	1.87109	−	0.681021i
$512$		0			0
$513$	−9.00000	−	20.7846i	−0.397360	−	0.917663i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	15.4093	−	42.3366i	0.673800	−	1.85125i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	15.4093	−	42.3366i	0.673800	−	1.85125i	0.498892	−	0.866664i	$-0.333741\pi$
$524$		0			0
$525$	20.3766	−	11.7644i	0.889308	−	0.513442i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.99391	−	22.6506i	0.173648	−	0.984808i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.09462	+	0.398408i	0.0470613	+	0.0171289i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	1.09462	+	0.398408i	0.0470613	+	0.0171289i	−0.318382	+	0.947962i	$0.603140\pi$
$542$		0			0
$543$	−22.5000	+	38.9711i	−0.965567	+	1.67241i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	22.6143	+	26.9506i	0.966917	+	1.15233i	0.988295	+	0.152555i	$0.0487501\pi$
$547$	22.6143	+	26.9506i	0.966917	+	1.15233i	−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$	−17.6792	+	6.43469i	−0.754528	+	0.274626i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	6.45526	+	17.7357i	0.274506	+	0.754198i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$	22.6281	+	13.0643i	0.957067	+	0.552563i
$560$		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
$565$		0			0
$566$		0			0
$567$	4.24603	−	24.0805i	0.178317	−	1.01128i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−30.4374			−1.27376			−0.636882	−	0.770961i	$-0.719776\pi$
$571$	−30.4374			−1.27376			−0.636882	+	0.770961i	$0.719776\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	5.50000	+	9.52628i	0.228968	+	0.396584i	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	5.50000	+	9.52628i	0.228968	+	0.396584i	−0.728535	+	0.685009i	$0.759798\pi$
$578$		0			0
$579$	−43.4984	−	15.8321i	−1.80773	−	0.657961i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$587$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$588$		0			0
$589$	−36.3622	−	4.19875i	−1.49828	−	0.173006i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$593$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	37.6899	+	21.7603i	1.54254	+	0.890589i
$598$		0			0
$599$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$599$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$600$		0			0
$601$	−13.1042	+	7.56570i	−0.534530	+	0.308611i	−0.742859	−	0.669448i	$-0.766531\pi$
$601$	−13.1042	+	7.56570i	−0.534530	+	0.308611i	0.208329	+	0.978059i	$0.433198\pi$
$602$		0			0
$603$	−31.5450	+	37.5939i	−1.28461	+	1.53094i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	11.8762i		−	0.482040i	−0.970520	−	0.241020i	$-0.922518\pi$
$607$		−	11.8762i		−	0.482040i	0.970520	−	0.241020i	$-0.0774820\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−36.0041	−	30.2110i	−1.45419	−	1.22021i	−0.929449	−	0.368950i	$-0.879717\pi$
$613$	−36.0041	−	30.2110i	−1.45419	−	1.22021i	−0.524742	−	0.851261i	$-0.675838\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$617$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$618$		0			0
$619$	−21.5415	+	37.3109i	−0.865825	+	1.49965i	0.000400419		1.00000i	$0.499873\pi$
$619$	−21.5415	+	37.3109i	−0.865825	+	1.49965i	−0.866226	+	0.499653i	$0.833461\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−25.3200	+	21.2460i	−1.00797	+	0.845790i	−0.988069	−	0.154011i	$-0.950781\pi$
$631$	−25.3200	+	21.2460i	−1.00797	+	0.845790i	−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$	8.49928	+	48.2018i	0.337816	+	1.91585i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.860120	+	2.36316i	−0.0340792	+	0.0936318i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$641$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$642$		0			0
$643$	6.53771	−	37.0772i	0.257822	−	1.46218i	−0.530901	−	0.847434i	$-0.678146\pi$
$643$	6.53771	−	37.0772i	0.257822	−	1.46218i	0.788723	−	0.614749i	$-0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−30.2716	−	25.4009i	−1.18644	−	0.995538i
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−24.8508	+	43.0428i	−0.969520	+	1.67926i
$658$		0			0
$659$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$659$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$660$		0			0
$661$	22.2668	+	26.5366i	0.866079	+	1.03215i	0.999157	+	0.0410470i	$0.0130693\pi$
$661$	22.2668	+	26.5366i	0.866079	+	1.03215i	−0.133078	+	0.991106i	$0.542486\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	34.1924	−	28.6908i	1.32195	−	1.10925i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	40.5596	+	23.4171i	1.56346	+	0.902664i	0.996903	+	0.0786409i	$0.0250581\pi$
$673$	40.5596	+	23.4171i	1.56346	+	0.902664i	0.566557	+	0.824023i	$0.308275\pi$
$674$		0			0
$675$	−8.88594	+	24.4139i	−0.342020	+	0.939693i
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−33.2730	+	39.6532i	−1.27690	+	1.52175i
$680$		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
$685$		0			0
$686$		0			0
$687$	−47.3888	−	8.35592i	−1.80799	−	0.318798i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−24.5000	−	42.4352i	−0.932024	−	1.61431i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−24.5000	−	42.4352i	−0.932024	−	1.61431i	−0.152167	−	0.988355i	$-0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$701$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$702$		0			0
$703$	−25.8004	+	34.7570i	−0.973081	+	1.31089i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.48674	+	14.1030i	0.0933915	+	0.529649i	0.995228	+	0.0975728i	$0.0311079\pi$
$709$	2.48674	+	14.1030i	0.0933915	+	0.529649i	−0.901837	+	0.432077i	$0.857781\pi$
$710$		0			0
$711$	−18.0486	−	10.4204i	−0.676875	−	0.390794i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		0			0
$721$		−	42.1372i		−	1.56927i
$722$		0			0
$723$	32.2154			1.19810
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−10.4686	−	8.78422i	−0.388260	−	0.325789i	0.427675	−	0.903933i	$-0.359333\pi$
$727$	−10.4686	−	8.78422i	−0.388260	−	0.325789i	−0.815935	+	0.578144i	$0.803777\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	3.50000	−	6.06218i	0.129275	−	0.223912i	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	3.50000	−	6.06218i	0.129275	−	0.223912i	0.923396	+	0.383849i	$0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	50.6969	−	18.4522i	1.86491	−	0.678773i	0.890096	−	0.455773i	$-0.150637\pi$
$739$	50.6969	−	18.4522i	1.86491	−	0.678773i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$	−39.9673	−	29.6680i	−1.46823	−	1.08988i
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.89399	−	18.9411i	0.251565	−	0.691170i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	6.89399	−	18.9411i	0.251565	−	0.691170i	0.999621	−	0.0275338i	$-0.00876539\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	7.12289	−	40.3959i	0.258886	−	1.46822i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	7.12289	−	40.3959i	0.258886	−	1.46822i	0.785897	−	0.618357i	$-0.212201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	32.4400	+	5.72005i	1.17441	+	0.207080i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	34.9180	+	12.7091i	1.25918	+	0.458303i	0.883493	−	0.468445i	$-0.155186\pi$
$769$	34.9180	+	12.7091i	1.25918	+	0.458303i	0.375684	+	0.926748i	$0.377408\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$773$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$774$		0			0
$775$	26.9890	+	32.1643i	0.969474	+	1.15537i
$776$		0			0
$777$	−43.9129	+	15.9830i	−1.57537	+	0.573386i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−28.8754	−	16.6712i	−1.02930	−	0.594265i	−0.112514	−	0.993650i	$-0.535890\pi$
$787$	−28.8754	−	16.6712i	−1.02930	−	0.594265i	−0.916783	+	0.399385i	$0.869224\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−26.5765	+	31.6726i	−0.943759	+	1.12473i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		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
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	42.6434	−	7.51919i	1.49741	−	0.264034i	0.635901	−	0.771771i	$-0.280629\pi$
$811$	42.6434	−	7.51919i	1.49741	−	0.264034i	0.861512	+	0.507736i	$0.169518\pi$
$812$		0			0
$813$	−32.2869	−	38.4780i	−1.13235	−	1.34948i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.00733	−	16.8038i	−0.140199	−	0.587891i
$818$		0			0
$819$	−18.3789	−	50.4956i	−0.642210	−	1.76446i
$820$		0			0
$821$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$821$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$	0.868241	+	4.92404i	0.0302650	+	0.171641i	0.996194	−	0.0871670i	$-0.0277814\pi$
$823$	0.868241	+	4.92404i	0.0302650	+	0.171641i	−0.965929	+	0.258808i	$0.916670\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$827$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$828$		0			0
$829$	−44.4414	+	25.6583i	−1.54351	+	0.891148i	−0.544900	+	0.838501i	$0.683433\pi$
$829$	−44.4414	+	25.6583i	−1.54351	+	0.891148i	−0.998613	+	0.0526472i	$0.983234\pi$
$830$		0			0
$831$	−34.5136	+	41.1317i	−1.19726	+	1.42684i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	43.6346			1.50823
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−14.9428	+	25.8818i	−0.513442	+	0.889308i
$848$		0			0
$849$	−42.6434	+	7.51919i	−1.46352	+	0.258058i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−53.4420	+	19.4513i	−1.82982	+	0.665999i	−0.836877	+	0.547391i	$0.815621\pi$
$853$	−53.4420	+	19.4513i	−1.82982	+	0.665999i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$857$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$858$		0			0
$859$	20.5159	−	17.2149i	0.699995	−	0.587366i	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	20.5159	−	17.2149i	0.699995	−	0.587366i	0.921772	+	0.387732i	$0.126741\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.5000	−	14.7224i	0.866025	−	0.500000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−18.7279	+	106.211i	−0.634569	+	3.59882i
$872$		0			0
$873$		−	57.1577i		−	1.93449i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−41.2683	−	7.27672i	−1.39353	−	0.245717i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−41.2683	−	7.27672i	−1.39353	−	0.245717i	−0.819483	+	0.573103i	$0.805739\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	−55.8029	−	20.3106i	−1.87792	−	0.683506i	−0.952470	−	0.304633i	$-0.901466\pi$
$883$	−55.8029	−	20.3106i	−1.87792	−	0.683506i	−0.925449	−	0.378873i	$-0.876312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$887$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$888$		0			0
$889$	39.3226	+	46.8628i	1.31884	+	1.57173i
$890$		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
$901$		0			0
$902$		0			0
$903$	6.37859	−	17.5250i	0.212266	−	0.583197i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7.79339	−	9.28780i	0.258775	−	0.308396i	−0.620977	−	0.783829i	$-0.713264\pi$
$907$	7.79339	−	9.28780i	0.258775	−	0.308396i	0.879752	+	0.475433i	$0.157708\pi$
$908$		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
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.7636	−	51.5520i	−0.981810	−	1.70054i	−0.655335	−	0.755338i	$-0.727472\pi$
$919$	−29.7636	−	51.5520i	−0.981810	−	1.70054i	−0.326475	−	0.945206i	$-0.605861\pi$
$920$		0			0
$921$	−2.81908	−	1.02606i	−0.0928918	−	0.0338098i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	48.8986	−	8.62214i	1.60778	−	0.283494i
$926$		0			0
$927$	29.9078	+	35.6427i	0.982300	+	1.17066i
$928$		0			0
$929$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$929$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$930$		0			0
$931$	1.52578	−	0.660681i	0.0500054	−	0.0216530i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−8.69434	−	49.3081i	−0.284032	−	1.61082i	−0.708723	−	0.705487i	$-0.750729\pi$
$937$	−8.69434	−	49.3081i	−0.284032	−	1.61082i	0.424691	−	0.905338i	$-0.360383\pi$
$938$		0			0
$939$	52.5000	+	30.3109i	1.71327	+	0.989158i
$940$		0			0
$941$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$947$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$948$		0			0
$949$			109.225i			3.54561i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$953$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.7589	−	34.2235i	0.637385	−	1.10398i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.26635	+	0.460913i	−0.0407230	+	0.0148220i	−0.362301	−	0.932061i	$-0.618009\pi$
$967$	−1.26635	+	0.460913i	−0.0407230	+	0.0148220i	0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$971$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$972$		0			0
$973$	−2.33788	+	1.96171i	−0.0749489	+	0.0628896i
$974$		0			0
$975$	9.91463	+	56.2287i	0.317522	+	1.80076i
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−31.5000	+	18.1865i	−1.00572	+	0.580651i
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	6.12045	+	1.07920i	0.194423	+	0.0342819i	0.270011	−	0.962857i	$-0.412973\pi$
$991$	6.12045	+	1.07920i	0.194423	+	0.0342819i	−0.0755888	+	0.997139i	$0.524084\pi$
$992$		0			0
$993$	7.07373	+	5.93556i	0.224478	+	0.188359i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.2099	−	4.08007i	−0.355021	−	0.129217i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	−11.2099	−	4.08007i	−0.355021	−	0.129217i	−0.513373	+	0.858166i	$0.671604\pi$
$998$		0			0
$999$	25.8004	−	44.6876i	0.816289	−	1.41385i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cc.b.641.1		6
3.2	odd	2	CM	912.2.cc.b.641.1		6
4.3	odd	2		228.2.t.a.185.1	yes	6
12.11	even	2		228.2.t.a.185.1	yes	6
19.15	odd	18	inner	912.2.cc.b.737.1		6
57.53	even	18	inner	912.2.cc.b.737.1		6
76.15	even	18		228.2.t.a.53.1	✓	6
228.167	odd	18		228.2.t.a.53.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.t.a.53.1	✓	6	76.15	even	18
228.2.t.a.53.1	✓	6	228.167	odd	18
228.2.t.a.185.1	yes	6	4.3	odd	2
228.2.t.a.185.1	yes	6	12.11	even	2
912.2.cc.b.641.1		6	1.1	even	1	trivial
912.2.cc.b.641.1		6	3.2	odd	2	CM
912.2.cc.b.737.1		6	19.15	odd	18	inner
912.2.cc.b.737.1		6	57.53	even	18	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.cc (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.70574 + 0.300767i) q^{3} +(-1.35844 - 2.35289i) q^{7} +(2.81908 + 1.02606i) q^{9} +O(q^{10})\)	\(q+(1.70574 + 0.300767i) q^{3} +(-1.35844 - 2.35289i) q^{7} +(2.81908 + 1.02606i) q^{9} +(6.49273 - 1.14484i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(-1.60947 - 4.42198i) q^{21} +(0.868241 + 4.92404i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.27244 - 4.19875i) q^{31} -9.93058i q^{37} +11.4192 q^{39} +(3.03596 + 2.54747i) q^{43} +(-0.190722 + 0.330341i) q^{49} +(-5.18866 - 5.48432i) q^{57} +(-4.80406 + 4.03109i) q^{61} +(-1.41534 - 8.02682i) q^{63} +(-5.59492 + 15.3719i) q^{67} +(-2.87686 + 16.3155i) q^{73} +8.66025i q^{75} +(-6.84137 - 1.20632i) q^{79} +(6.89440 + 5.78509i) q^{81} +(-11.5137 - 13.7215i) q^{91} +(13.6677 - 4.97464i) q^{93} +(-6.51636 - 17.9035i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} + 27 q^{27} - 15 q^{43} - 21 q^{49} - 42 q^{61} + 36 q^{63} - 15 q^{67} - 21 q^{73} - 12 q^{79} - 48 q^{91} + 54 q^{93}+O(q^{100}) \) 6 * q + 21 * q^13 - 21 * q^19 - 27 * q^21 + 27 * q^27 - 15 * q^43 - 21 * q^49 - 42 * q^61 + 36 * q^63 - 15 * q^67 - 21 * q^73 - 12 * q^79 - 48 * q^91 + 54 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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