LMFDB - Embedded newform 912.2.cc.a.401.1

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 57)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			401.1
Root			$0.939693 + 0.342020i$ of defining polynomial
Character	$\chi$	$=$	912.401
Dual form			912.2.cc.a.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11334 - 1.32683i) q^{3} +(0.418748 + 0.725293i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})$	$q+(1.11334 - 1.32683i) q^{3} +(0.418748 + 0.725293i) q^{7} +(-0.520945 - 2.95442i) q^{9} +(-4.62449 - 5.51125i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(1.42855 + 0.251892i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(4.97431 - 2.87192i) q^{31} +8.64501i q^{37} -12.4611 q^{39} +(10.9226 - 3.97551i) q^{43} +(3.14930 - 5.45475i) q^{49} +(-6.30200 - 4.15749i) q^{57} +(-10.1356 - 3.68907i) q^{61} +(1.92468 - 1.61500i) q^{63} +(-12.7417 + 2.24670i) q^{67} +(10.8289 + 9.08651i) q^{73} -8.66025i q^{75} +(-10.1814 + 12.1337i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(2.06077 - 5.66193i) q^{91} +(1.72756 - 9.79747i) q^{93} +(5.11721 + 0.902302i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} - 27 q^{27} + 39 q^{43} - 21 q^{49} - 42 q^{61} + 36 q^{63} - 33 q^{67} + 51 q^{73} - 12 q^{79} - 48 q^{91} + 54 q^{93}+O(q^{100}) $ 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 - 27 * q^27 + 39 * q^43 - 21 * q^49 - 42 * q^61 + 36 * q^63 - 33 * q^67 + 51 * 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 $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

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

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
$3$	1.11334	−	1.32683i	0.642788	−	0.766044i
$3$	1.11334	−	1.32683i	0.642788	−	0.766044i
$4$		0			0
$5$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$5$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$6$		0			0
$7$	0.418748	+	0.725293i	0.158272	+	0.274135i	0.934246	−	0.356630i	$-0.116074\pi$
$7$	0.418748	+	0.725293i	0.158272	+	0.274135i	−0.775974	+	0.630765i	$0.782741\pi$
$8$		0			0
$9$	−0.520945	−	2.95442i	−0.173648	−	0.984808i
$10$		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
$13$	−4.62449	−	5.51125i	−1.28260	−	1.52854i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−4.62449	−	5.51125i	−1.28260	−	1.52854i	−0.589226	−	0.807968i	$-0.700567\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$17$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$18$		0			0
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$19$	−0.500000	−	4.33013i	−0.114708	−	0.993399i
$20$		0			0
$21$	1.42855	+	0.251892i	0.311735	+	0.0549673i
$22$		0			0
$23$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$23$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$24$		0			0
$25$	3.83022	−	3.21394i	0.766044	−	0.642788i
$26$		0			0
$27$	−4.50000	−	2.59808i	−0.866025	−	0.500000i
$28$		0			0
$29$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$29$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$30$		0			0
$31$	4.97431	−	2.87192i	0.893412	−	0.515812i	0.0183550	−	0.999832i	$-0.494157\pi$
$31$	4.97431	−	2.87192i	0.893412	−	0.515812i	0.875057	+	0.484020i	$0.160824\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.64501i			1.42123i	0.703581	+	0.710615i	$0.251583\pi$
$37$			8.64501i			1.42123i	−0.703581	+	0.710615i	$0.748417\pi$
$38$		0			0
$39$	−12.4611			−1.99537
$40$		0			0
$41$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$41$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$42$		0			0
$43$	10.9226	−	3.97551i	1.66568	−	0.606259i	0.674443	−	0.738327i	$-0.264384\pi$
$43$	10.9226	−	3.97551i	1.66568	−	0.606259i	0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$47$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$48$		0			0
$49$	3.14930	−	5.45475i	0.449900	−	0.779250i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$53$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.30200	−	4.15749i	−0.834721	−	0.550673i
$58$		0			0
$59$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$59$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$60$		0			0
$61$	−10.1356	−	3.68907i	−1.29773	−	0.472337i	−0.401476	−	0.915869i	$-0.631503\pi$
$61$	−10.1356	−	3.68907i	−1.29773	−	0.472337i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	1.92468	−	1.61500i	0.242487	−	0.203470i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.7417	+	2.24670i	−1.55665	+	0.274479i	−0.884714	−	0.466134i	$-0.845646\pi$
$67$	−12.7417	+	2.24670i	−1.55665	+	0.274479i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$71$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$72$		0			0
$73$	10.8289	+	9.08651i	1.26742	+	1.06350i	0.994850	+	0.101361i	$0.0323196\pi$
$73$	10.8289	+	9.08651i	1.26742	+	1.06350i	0.272575	+	0.962135i	$0.412125\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.1814	+	12.1337i	−1.14550	+	1.36515i	−0.225018	+	0.974355i	$0.572244\pi$
$79$	−10.1814	+	12.1337i	−1.14550	+	1.36515i	−0.920478	+	0.390794i	$0.872200\pi$
$80$		0			0
$81$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$82$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$89$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$90$		0			0
$91$	2.06077	−	5.66193i	0.216028	−	0.593532i
$92$		0			0
$93$	1.72756	−	9.79747i	0.179140	−	1.01595i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.11721	+	0.902302i	0.519574	+	0.0916149i	0.427284	−	0.904117i	$-0.359470\pi$
$97$	5.11721	+	0.902302i	0.519574	+	0.0916149i	0.0922897	+	0.995732i	$0.470581\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$	8.74304	+	5.04780i	0.861477	+	0.497374i	0.864507	−	0.502621i	$-0.167631\pi$
$103$	8.74304	+	5.04780i	0.861477	+	0.497374i	−0.00302937	+	0.999995i	$0.500964\pi$
$104$		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
$109$	2.96198	+	8.13798i	0.283706	+	0.779477i	0.996912	+	0.0785223i	$0.0250202\pi$
$109$	2.96198	+	8.13798i	0.283706	+	0.779477i	−0.713206	+	0.700954i	$0.752758\pi$
$110$		0			0
$111$	11.4704	+	9.62484i	1.08873	+	0.913549i
$112$		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
$117$	−13.8735	+	16.5337i	−1.28260	+	1.52854i
$118$		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
$127$	7.79339	+	9.28780i	0.691551	+	0.824159i	0.991542	−	0.129783i	$-0.0414282\pi$
$127$	7.79339	+	9.28780i	0.691551	+	0.824159i	−0.299991	+	0.953942i	$0.596984\pi$
$128$		0			0
$129$	6.88578	−	18.9185i	0.606259	−	1.66568i
$130$		0			0
$131$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$131$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$132$		0			0
$133$	2.93124	−	2.17588i	0.254170	−	0.188673i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$137$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$138$		0			0
$139$	6.97952	−	5.85651i	0.591995	−	0.496743i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	6.97952	−	5.85651i	0.591995	−	0.496743i	0.888861	+	0.458176i	$0.151497\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.73127	−	10.2516i	−0.307750	−	0.845535i
$148$		0			0
$149$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$149$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$150$		0			0
$151$			15.5885i			1.26857i	0.773099	+	0.634285i	$0.218706\pi$
$151$			15.5885i			1.26857i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	22.6322	−	8.23746i	1.80625	−	0.657421i	0.808640	−	0.588304i	$-0.200204\pi$
$157$	22.6322	−	8.23746i	1.80625	−	0.657421i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.90554	−	13.6928i	0.619210	−	1.07250i	−0.370420	−	0.928864i	$-0.620786\pi$
$163$	7.90554	−	13.6928i	0.619210	−	1.07250i	0.989630	−	0.143639i	$-0.0458804\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$167$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$168$		0			0
$169$	−6.73055	+	38.1709i	−0.517735	+	2.93622i
$170$		0			0
$171$	−12.5326	+	3.73297i	−0.958388	+	0.285467i
$172$		0			0
$173$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$173$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$174$		0			0
$175$	3.93494	+	1.43220i	0.297454	+	0.108264i
$176$		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
$181$	18.7631	−	3.30844i	1.39465	−	0.245915i	0.574707	−	0.818359i	$-0.305116\pi$
$181$	18.7631	−	3.30844i	1.39465	−	0.245915i	0.819943	+	0.572444i	$0.194005\pi$
$182$		0			0
$183$	−16.1792	+	9.34105i	−1.19600	+	0.690510i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	4.35176i		−	0.316544i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	1.82723	−	2.17761i	0.131527	−	0.156748i	−0.696261	−	0.717788i	$-0.745155\pi$
$193$	1.82723	−	2.17761i	0.131527	−	0.156748i	0.827788	+	0.561041i	$0.189599\pi$
$194$		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
$199$	1.43670	+	8.14793i	0.101845	+	0.577591i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	1.43670	+	8.14793i	0.101845	+	0.577591i	−0.890589	+	0.454809i	$0.849707\pi$
$200$		0			0
$201$	−11.2049	+	19.4074i	−0.790330	+	1.36889i
$202$		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
$211$	11.3708	+	2.00497i	0.782796	+	0.138028i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	11.3708	+	2.00497i	0.782796	+	0.138028i	0.232053	+	0.972703i	$0.425456\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.16596	+	2.40522i	0.282804	+	0.163277i
$218$		0			0
$219$	24.1125	−	4.25168i	1.62937	−	0.287302i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.43289	−	9.43178i	−0.229883	−	0.631598i	0.770097	−	0.637927i	$-0.220208\pi$
$223$	−3.43289	−	9.43178i	−0.229883	−	0.631598i	−0.999980	+	0.00632846i	$0.997986\pi$
$224$		0			0
$225$	−11.4907	−	9.64181i	−0.766044	−	0.642788i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−16.6486			−1.10017			−0.550085	−	0.835109i	$-0.685405\pi$
$229$	−16.6486			−1.10017			−0.550085	+	0.835109i	$0.685405\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$233$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.76399	+	27.0179i	0.309454	+	1.75500i
$238$		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
$241$	19.4304	+	23.1562i	1.25162	+	1.49162i	0.800710	+	0.599052i	$0.204456\pi$
$241$	19.4304	+	23.1562i	1.25162	+	1.49162i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−5.33157	+	14.6484i	−0.342020	+	0.939693i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−21.5522	+	22.7802i	−1.37133	+	1.44947i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$251$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$257$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$258$		0			0
$259$	−6.27016	+	3.62008i	−0.389609	+	0.224941i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$269$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$270$		0			0
$271$	−0.939693	+	0.342020i	−0.0570823	+	0.0207762i	−0.370403	−	0.928871i	$-0.620781\pi$
$271$	−0.939693	+	0.342020i	−0.0570823	+	0.0207762i	0.313321	+	0.949647i	$0.398558\pi$
$272$		0			0
$273$	−5.21806	−	9.03795i	−0.315812	−	0.547002i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	0.931305	+	0.364241i	$0.118672\pi$
$278$		0			0
$279$	−11.0762	−	13.2001i	−0.663115	−	0.790269i
$280$		0			0
$281$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$281$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$282$		0			0
$283$	1.21554	−	6.89365i	0.0722562	−	0.409785i	−0.927130	−	0.374741i	$-0.877732\pi$
$283$	1.21554	−	6.89365i	0.0722562	−	0.409785i	0.999386	−	0.0350443i	$-0.0111572\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.9748	−	5.81434i	−0.939693	−	0.342020i
$290$		0			0
$291$	6.89440	−	5.78509i	0.404157	−	0.339128i
$292$		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
$301$	7.45723	+	6.25736i	0.429828	+	0.360668i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−18.9268	+	22.5561i	−1.08021	+	1.28734i	−0.124760	+	0.992187i	$0.539816\pi$
$307$	−18.9268	+	22.5561i	−1.08021	+	1.28734i	−0.955449	+	0.295156i	$0.904628\pi$
$308$		0			0
$309$	16.4315	−	5.98059i	0.934758	−	0.340224i
$310$		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
$313$	2.25743	+	12.8025i	0.127597	+	0.723640i	0.979731	+	0.200316i	$0.0641970\pi$
$313$	2.25743	+	12.8025i	0.127597	+	0.723640i	−0.852134	+	0.523324i	$0.824692\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−35.4256	−	6.24649i	−1.96506	−	0.346493i
$326$		0			0
$327$	14.0954	+	5.13030i	0.779477	+	0.283706i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	29.8965	+	17.2608i	1.64326	+	0.948737i	0.979663	+	0.200648i	$0.0643046\pi$
$331$	29.8965	+	17.2608i	1.64326	+	0.948737i	0.663598	+	0.748090i	$0.269029\pi$
$332$		0			0
$333$	25.5410	−	4.50357i	1.39964	−	0.246794i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.476119	−	1.30813i	−0.0259358	−	0.0712581i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	−0.476119	−	1.30813i	−0.0259358	−	0.0712581i	−0.951985	+	0.306145i	$0.900961\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	11.1375			0.601370
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$347$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$	−18.2729	−	31.6496i	−0.978126	−	1.69416i	−0.669207	−	0.743076i	$-0.733366\pi$
$349$	−18.2729	−	31.6496i	−0.978126	−	1.69416i	−0.308920	−	0.951088i	$-0.599967\pi$
$350$		0			0
$351$	6.49154	+	36.8154i	0.346493	+	1.96506i
$352$		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		0.173648	−	0.984808i	$-0.444444\pi$
$359$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$360$		0			0
$361$	−18.5000	+	4.33013i	−0.973684	+	0.227901i
$362$		0			0
$363$	−18.7631	−	3.30844i	−0.984808	−	0.173648i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	29.2788	−	24.5679i	1.52834	−	1.28243i	0.719249	−	0.694752i	$-0.244486\pi$
$367$	29.2788	−	24.5679i	1.52834	−	1.28243i	0.809093	−	0.587680i	$-0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	33.0709i		−	1.69874i	−0.527798	−	0.849370i	$-0.676982\pi$
$379$		−	33.0709i		−	1.69874i	0.527798	−	0.849370i	$-0.323018\pi$
$380$		0			0
$381$	21.0000			1.07586
$382$		0			0
$383$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$383$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−17.4354	−	30.1990i	−0.886292	−	1.53510i
$388$		0			0
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.78240	−	38.4649i	0.340399	−	1.93050i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	6.78240	−	38.4649i	0.340399	−	1.93050i	0.365493	−	0.930814i	$-0.380900\pi$
$398$		0			0
$399$	0.376449	−	6.31174i	0.0188460	−	0.315982i
$400$		0			0
$401$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$401$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$402$		0			0
$403$	−38.8315	−	14.1335i	−1.93433	−	0.704040i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.6459	+	2.40614i	−0.674746	+	0.118976i	−0.500514	−	0.865729i	$-0.666856\pi$
$409$	−13.6459	+	2.40614i	−0.674746	+	0.118976i	−0.174232	+	0.984705i	$0.555744\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	15.7809i		−	0.772795i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−23.3802	+	27.8634i	−1.13948	+	1.35798i	−0.215060	+	0.976601i	$0.568995\pi$
$421$	−23.3802	+	27.8634i	−1.13948	+	1.35798i	−0.924419	+	0.381377i	$0.875450\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.56862	−	8.89609i	−0.0759109	−	0.430512i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$431$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$432$		0			0
$433$	11.3309	−	31.1315i	0.544530	−	1.49608i	−0.296466	−	0.955043i	$-0.595808\pi$
$433$	11.3309	−	31.1315i	0.544530	−	1.49608i	0.840996	−	0.541041i	$-0.181970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−38.2827	−	6.75027i	−1.82713	−	0.322173i	−0.848722	−	0.528839i	$-0.822628\pi$
$439$	−38.2827	−	6.75027i	−1.82713	−	0.322173i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$	−17.7562	−	6.46274i	−0.845535	−	0.307750i
$442$		0			0
$443$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$443$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$444$		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
$453$	20.6832	+	17.3553i	0.971782	+	0.815422i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.0597			0.891577			0.445788	−	0.895138i	$-0.352923\pi$
$457$	19.0597			0.891577			0.445788	+	0.895138i	$0.352923\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$	−2.88057	−	4.98929i	−0.133871	−	0.231872i	0.791294	−	0.611435i	$-0.209408\pi$
$463$	−2.88057	−	4.98929i	−0.133871	−	0.231872i	−0.925166	+	0.379563i	$0.876074\pi$
$464$		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
$469$	−6.96508	−	8.30066i	−0.321618	−	0.383289i
$470$		0			0
$471$	14.2677	−	39.2002i	0.657421	−	1.80625i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−15.8319	−	14.9784i	−0.726416	−	0.687255i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$	47.6448	−	39.9787i	2.17241	−	1.82287i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	34.5000	−	19.9186i	1.56334	−	0.902597i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	34.5000	−	19.9186i	1.56334	−	0.902597i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$	−9.36643	−	25.7341i	−0.423565	−	1.16373i
$490$		0			0
$491$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$491$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	38.2768	−	13.9316i	1.71350	−	0.623664i	0.716258	−	0.697835i	$-0.245853\pi$
$499$	38.2768	−	13.9316i	1.71350	−	0.623664i	0.997246	+	0.0741708i	$0.0236310\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$503$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	43.1528	+	51.4275i	1.91648	+	2.28397i
$508$		0			0
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$	−2.05581	+	11.6591i	−0.0909436	+	0.515767i
$512$		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
$523$	0.0576190	−	0.0101598i	0.00251950	−	0.000444257i	−0.172388	−	0.985029i	$-0.555148\pi$
$523$	0.0576190	−	0.0101598i	0.00251950	−	0.000444257i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$	6.28122	−	3.62646i	0.274135	−	0.158272i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	17.6190	+	14.7841i	0.766044	+	0.642788i
$530$		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
$541$	−5.34570	−	30.3170i	−0.229830	−	1.30343i	−0.853233	−	0.521529i	$-0.825362\pi$
$541$	−5.34570	−	30.3170i	−0.229830	−	1.30343i	0.623404	−	0.781900i	$-0.285749\pi$
$542$		0			0
$543$	16.5000	−	28.5788i	0.708083	−	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.44063	+	6.70557i	−0.104354	+	0.286709i	−0.980870	−	0.194662i	$-0.937639\pi$
$547$	−2.44063	+	6.70557i	−0.104354	+	0.286709i	0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	−5.61897	+	31.8667i	−0.239812	+	1.36004i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−13.0639	−	2.30352i	−0.555535	−	0.0979558i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$557$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$558$		0			0
$559$	−72.4215	−	41.8126i	−3.06310	−	1.76848i
$560$		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
$567$	−5.77403	−	4.84499i	−0.242487	−	0.203470i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−41.5708			−1.73968			−0.869842	−	0.493331i	$-0.835779\pi$
$571$	−41.5708			−1.73968			−0.869842	+	0.493331i	$0.835779\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	−0.854981	−	4.84884i	−0.0355318	−	0.201511i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$587$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$588$		0			0
$589$	−14.9229	−	20.1034i	−0.614888	−	0.828347i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$593$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12.4104	+	7.16517i	0.507925	+	0.293251i
$598$		0			0
$599$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$599$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$600$		0			0
$601$	−26.1278	+	15.0849i	−1.06577	+	0.615325i	−0.927024	−	0.375002i	$-0.877642\pi$
$601$	−26.1278	+	15.0849i	−1.06577	+	0.615325i	−0.138751	+	0.990327i	$0.544309\pi$
$602$		0			0
$603$	13.2754	+	36.4740i	0.540617	+	1.48533i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	21.6418i		−	0.878412i	−0.898386	−	0.439206i	$-0.855260\pi$
$607$		−	21.6418i		−	0.878412i	0.898386	−	0.439206i	$-0.144740\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.7686	+	12.6547i	−1.40429	+	0.511120i	−0.929449	−	0.368950i	$-0.879717\pi$
$613$	−34.7686	+	12.6547i	−1.40429	+	0.511120i	−0.474843	+	0.880071i	$0.657495\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$617$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$	−24.5035	+	42.4412i	−0.984877	+	1.70586i	−0.342396	+	0.939556i	$0.611239\pi$
$619$	−24.5035	+	42.4412i	−0.984877	+	1.70586i	−0.642481	+	0.766302i	$0.722095\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29.6199	−	10.7808i	−1.17915	−	0.429175i	−0.323248	−	0.946314i	$-0.604775\pi$
$631$	−29.6199	−	10.7808i	−1.17915	−	0.429175i	−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	15.3198	−	12.8548i	0.608907	−	0.510934i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−44.6264	+	7.86883i	−1.76816	+	0.311774i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$641$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$642$		0			0
$643$	38.8246	+	32.5777i	1.53109	+	1.28474i	0.788723	+	0.614749i	$0.210743\pi$
$643$	38.8246	+	32.5777i	1.53109	+	1.28474i	0.742370	+	0.669991i	$0.233702\pi$
$644$		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
$651$	7.82945	−	2.84969i	0.306860	−	0.111688i
$652$		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
$657$	21.2041	−	36.7267i	0.827252	−	1.43284i
$658$		0			0
$659$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$659$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$660$		0			0
$661$	−11.8479	+	32.5519i	−0.460831	+	1.26612i	0.464031	+	0.885819i	$0.346403\pi$
$661$	−11.8479	+	32.5519i	−0.460831	+	1.26612i	−0.924862	+	0.380303i	$0.875820\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−16.3363	−	5.94593i	−0.631598	−	0.229883i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−18.6410	−	10.7624i	−0.718557	−	0.414859i	0.0956642	−	0.995414i	$-0.469503\pi$
$673$	−18.6410	−	10.7624i	−0.718557	−	0.414859i	−0.814221	+	0.580554i	$0.802836\pi$
$674$		0			0
$675$	−25.5861	+	4.51151i	−0.984808	+	0.173648i
$676$		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
$679$	1.48839	+	4.08931i	0.0571191	+	0.156933i
$680$		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
$687$	−18.5355	+	22.0898i	−0.707175	+	0.842779i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	0.932024	+	0.362397i	$0.118041\pi$
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	−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		0.173648	−	0.984808i	$-0.444444\pi$
$701$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$	37.4340	−	4.32250i	1.41185	−	0.163026i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.13017	−	2.62652i	0.117556	−	0.0986411i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	3.13017	−	2.62652i	0.117556	−	0.0986411i	0.699671	+	0.714466i	$0.253330\pi$
$710$		0			0
$711$	41.1520	+	23.7591i	1.54332	+	0.891038i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$719$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$720$		0			0
$721$			8.45502i			0.314881i
$722$		0			0
$723$	52.3569			1.94717
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−28.8329	+	10.4943i	−1.06935	+	0.389213i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−28.8329	+	10.4943i	−1.06935	+	0.389213i	−0.253419	+	0.967357i	$0.581555\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.92824	+	44.9633i	−0.291645	+	1.65400i	0.388888	+	0.921285i	$0.372859\pi$
$739$	−7.92824	+	44.9633i	−0.291645	+	1.65400i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$	6.23055	+	53.9582i	0.228885	+	1.98220i
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.48617	+	0.262051i	−0.0542310	+	0.00956239i	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−1.48617	+	0.262051i	−0.0542310	+	0.00956239i	0.146467	+	0.989216i	$0.453210\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.1282	+	27.7979i	1.20406	+	1.01033i	0.999505	+	0.0314762i	$0.0100208\pi$
$757$	33.1282	+	27.7979i	1.20406	+	1.01033i	0.204560	+	0.978854i	$0.434424\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−4.66209	+	5.55607i	−0.168779	+	0.201143i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	3.61814	+	20.5195i	0.130474	+	0.739953i	0.977905	+	0.209048i	$0.0670366\pi$
$769$	3.61814	+	20.5195i	0.130474	+	0.739953i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$773$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$774$		0			0
$775$	9.82254	−	26.9872i	0.352836	−	0.969409i
$776$		0			0
$777$	−2.17760	+	12.3498i	−0.0781211	+	0.443047i
$778$		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
$787$	−13.7680	−	7.94897i	−0.490777	−	0.283350i	0.234120	−	0.972208i	$-0.424779\pi$
$787$	−13.7680	−	7.94897i	−0.490777	−	0.283350i	−0.724897	+	0.688858i	$0.758113\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	26.5407	+	72.9200i	0.942488	+	2.58946i
$794$		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
$811$	−34.5136	−	41.1317i	−1.21194	−	1.44433i	−0.861512	−	0.507736i	$-0.830482\pi$
$811$	−34.5136	−	41.1317i	−1.21194	−	1.44433i	−0.350423	−	0.936592i	$-0.613962\pi$
$812$		0			0
$813$	−0.592396	+	1.62760i	−0.0207762	+	0.0570823i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−22.6758	−	45.3086i	−0.793325	−	1.58515i
$818$		0			0
$819$	−17.8013	−	3.13885i	−0.622027	−	0.109680i
$820$		0			0
$821$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$821$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$822$		0			0
$823$	36.0041	−	30.2110i	1.25502	−	1.05309i	0.258830	−	0.965923i	$-0.416663\pi$
$823$	36.0041	−	30.2110i	1.25502	−	1.05309i	0.996194	−	0.0871670i	$-0.0277814\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$827$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$828$		0			0
$829$	48.5882	−	28.0524i	1.68754	−	0.974300i	0.731141	−	0.682226i	$-0.238988\pi$
$829$	48.5882	−	28.0524i	1.68754	−	0.974300i	0.956396	−	0.292074i	$-0.0943452\pi$
$830$		0			0
$831$	−2.96198	−	8.13798i	−0.102750	−	0.282303i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−29.8458			−1.03162
$838$		0			0
$839$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$839$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$840$		0			0
$841$	27.2511	−	9.91858i	0.939693	−	0.342020i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.60623	−	7.97822i	0.158272	−	0.274135i
$848$		0			0
$849$	−7.79339	−	9.28780i	−0.267468	−	0.318756i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.564121	+	3.19929i	−0.0193152	+	0.109542i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−0.564121	+	3.19929i	−0.0193152	+	0.109542i	0.973626	+	0.228150i	$0.0732677\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$857$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$858$		0			0
$859$	−50.7734	−	18.4800i	−1.73237	−	0.630529i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−50.7734	−	18.4800i	−1.73237	−	0.630529i	−0.998795	+	0.0490840i	$0.984370\pi$
$860$		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
$871$	71.3059	+	59.8328i	2.41611	+	2.02736i
$872$		0			0
$873$		−	15.5885i		−	0.527589i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−37.9283	+	45.2012i	−1.28075	+	1.52634i	−0.574049	+	0.818821i	$0.694628\pi$
$877$	−37.9283	+	45.2012i	−1.28075	+	1.52634i	−0.706698	+	0.707515i	$0.749816\pi$
$878$		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
$883$	7.63738	+	43.3138i	0.257018	+	1.45762i	0.790838	+	0.612026i	$0.209645\pi$
$883$	7.63738	+	43.3138i	0.257018	+	1.45762i	−0.533820	+	0.845598i	$0.679244\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$887$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$888$		0			0
$889$	−3.47291	+	9.54173i	−0.116478	+	0.320019i
$890$		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
$903$	16.6049	−	2.92789i	0.552576	−	0.0974341i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	19.5491	+	53.7106i	0.649116	+	1.78343i	0.620977	+	0.783829i	$0.286736\pi$
$907$	19.5491	+	53.7106i	0.649116	+	1.78343i	0.0281394	+	0.999604i	$0.491042\pi$
$908$		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
$919$	−10.8368	−	18.7699i	−0.357472	−	0.619160i	0.630065	−	0.776542i	$-0.283028\pi$
$919$	−10.8368	−	18.7699i	−0.357472	−	0.619160i	−0.987538	+	0.157382i	$0.949695\pi$
$920$		0			0
$921$	8.85606	+	50.2252i	0.291817	+	1.65498i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	27.7845	+	33.1123i	0.913549	+	1.08873i
$926$		0			0
$927$	10.3587	−	28.4603i	0.340224	−	0.934758i
$928$		0			0
$929$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$929$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$	−25.1944	−	10.9095i	−0.825713	−	0.357544i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−12.0344	+	10.0980i	−0.393146	+	0.329888i	−0.817837	−	0.575450i	$-0.804827\pi$
$937$	−12.0344	+	10.0980i	−0.393146	+	0.329888i	0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$	19.5000	+	11.2583i	0.636358	+	0.367402i
$940$		0			0
$941$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$941$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$947$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$948$		0			0
$949$		−	101.701i		−	3.30136i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$953$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.995825	−	1.72482i	0.0321234	−	0.0556393i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.6738	−	60.5342i	0.343247	−	1.94665i	0.0216683	−	0.999765i	$-0.493102\pi$
$967$	10.6738	−	60.5342i	0.343247	−	1.94665i	0.321578	−	0.946883i	$-0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$971$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$972$		0			0
$973$	7.17035	+	2.60979i	0.229871	+	0.0836662i
$974$		0			0
$975$	−47.7288	+	40.0492i	−1.52854	+	1.28260i
$976$		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
$981$	22.5000	−	12.9904i	0.718370	−	0.414751i
$982$		0			0
$983$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$983$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−17.5279	+	20.8890i	−0.556793	+	0.663559i	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−17.5279	+	20.8890i	−0.556793	+	0.663559i	0.412072	+	0.911151i	$0.364805\pi$
$992$		0			0
$993$	56.1871	−	20.4504i	1.78304	−	0.648974i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.9647	+	62.1837i	0.347255	+	1.96938i	0.188903	+	0.981996i	$0.439507\pi$
$997$	10.9647	+	62.1837i	0.347255	+	1.96938i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	22.4604	−	38.9025i	0.710615	−	1.23082i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cc.a.401.1		6
3.2	odd	2	CM	912.2.cc.a.401.1		6
4.3	odd	2		57.2.j.a.2.1	✓	6
12.11	even	2		57.2.j.a.2.1	✓	6
19.10	odd	18	inner	912.2.cc.a.257.1		6
57.29	even	18	inner	912.2.cc.a.257.1		6
76.3	even	18		1083.2.d.a.1082.2		6
76.35	odd	18		1083.2.d.a.1082.5		6
76.67	even	18		57.2.j.a.29.1	yes	6
228.35	even	18		1083.2.d.a.1082.5		6
228.143	odd	18		57.2.j.a.29.1	yes	6
228.155	odd	18		1083.2.d.a.1082.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.j.a.2.1	✓	6	4.3	odd	2
57.2.j.a.2.1	✓	6	12.11	even	2
57.2.j.a.29.1	yes	6	76.67	even	18
57.2.j.a.29.1	yes	6	228.143	odd	18
912.2.cc.a.257.1		6	19.10	odd	18	inner
912.2.cc.a.257.1		6	57.29	even	18	inner
912.2.cc.a.401.1		6	1.1	even	1	trivial
912.2.cc.a.401.1		6	3.2	odd	2	CM
1083.2.d.a.1082.2		6	76.3	even	18
1083.2.d.a.1082.2		6	228.155	odd	18
1083.2.d.a.1082.5		6	76.35	odd	18
1083.2.d.a.1082.5		6	228.35	even	18

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)

\(f(q)\)	\(=\)	\(q+(1.11334 - 1.32683i) q^{3} +(0.418748 + 0.725293i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})\)	\(q+(1.11334 - 1.32683i) q^{3} +(0.418748 + 0.725293i) q^{7} +(-0.520945 - 2.95442i) q^{9} +(-4.62449 - 5.51125i) q^{13} +(-0.500000 - 4.33013i) q^{19} +(1.42855 + 0.251892i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(4.97431 - 2.87192i) q^{31} +8.64501i q^{37} -12.4611 q^{39} +(10.9226 - 3.97551i) q^{43} +(3.14930 - 5.45475i) q^{49} +(-6.30200 - 4.15749i) q^{57} +(-10.1356 - 3.68907i) q^{61} +(1.92468 - 1.61500i) q^{63} +(-12.7417 + 2.24670i) q^{67} +(10.8289 + 9.08651i) q^{73} -8.66025i q^{75} +(-10.1814 + 12.1337i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(2.06077 - 5.66193i) q^{91} +(1.72756 - 9.79747i) q^{93} +(5.11721 + 0.902302i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} - 27 q^{27} + 39 q^{43} - 21 q^{49} - 42 q^{61} + 36 q^{63} - 33 q^{67} + 51 q^{73} - 12 q^{79} - 48 q^{91} + 54 q^{93}+O(q^{100}) \) 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 - 27 * q^27 + 39 * q^43 - 21 * q^49 - 42 * q^61 + 36 * q^63 - 33 * q^67 + 51 * 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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