LMFDB - Embedded newform 912.2.ch.c.671.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(47,912)]

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([9, 0, 9, 8]))

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

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

chi := DirichletCharacter("912.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.ch (of order $18$, degree $6$, 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			671.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	912.671
Dual form			912.2.ch.c.719.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.592396 - 1.62760i) q^{3} +(-4.00387 - 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})$	$q+(0.592396 - 1.62760i) q^{3} +(-4.00387 - 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(6.73783 - 2.45237i) q^{13} +(-3.50000 + 2.59808i) q^{19} +(-6.13429 + 5.14728i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(-4.50000 + 2.59808i) q^{27} +(-9.12108 - 5.26606i) q^{31} -12.1138 q^{37} -12.4192i q^{39} +(2.77584 - 0.489456i) q^{43} +(7.18732 + 12.4488i) q^{49} +(2.15523 + 7.23567i) q^{57} +(1.60694 - 9.11343i) q^{61} +(4.74376 + 13.0334i) q^{63} +(5.60994 - 6.68566i) q^{67} +(11.1912 + 4.07326i) q^{73} +8.66025i q^{75} +(6.03462 - 16.5800i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-32.6464 - 5.75643i) q^{91} +(-13.9743 + 11.7258i) q^{93} +(-3.83022 + 3.21394i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$1$	$-1$	$-1$

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
$3$	0.592396	−	1.62760i	0.342020	−	0.939693i
$3$	0.592396	−	1.62760i	0.342020	−	0.939693i
$4$		0			0
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$		0			0
$7$	−4.00387	−	2.31164i	−1.51332	−	0.873716i	−0.999878	−	0.0155920i	$-0.995037\pi$
$7$	−4.00387	−	2.31164i	−1.51332	−	0.873716i	−0.513442	−	0.858124i	$-0.671630\pi$
$8$		0			0
$9$	−2.29813	−	1.92836i	−0.766044	−	0.642788i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	6.73783	−	2.45237i	1.86874	−	0.680165i	0.898011	−	0.439972i	$-0.145012\pi$
$13$	6.73783	−	2.45237i	1.86874	−	0.680165i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$17$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$		0			0
$21$	−6.13429	+	5.14728i	−1.33861	+	1.12323i
$22$		0			0
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$		0			0
$25$	−4.69846	+	1.71010i	−0.939693	+	0.342020i
$26$		0			0
$27$	−4.50000	+	2.59808i	−0.866025	+	0.500000i
$28$		0			0
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$	−9.12108	−	5.26606i	−1.63819	−	0.945812i	−0.981455	−	0.191695i	$-0.938602\pi$
$31$	−9.12108	−	5.26606i	−1.63819	−	0.945812i	−0.656740	−	0.754117i	$-0.728065\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−12.1138			−1.99150			−0.995749	−	0.0921098i	$-0.970639\pi$
$37$	−12.1138			−1.99150			−0.995749	+	0.0921098i	$0.970639\pi$
$38$		0			0
$39$		−	12.4192i		−	1.98867i
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	2.77584	−	0.489456i	0.423312	−	0.0746414i	0.0420659	−	0.999115i	$-0.486606\pi$
$43$	2.77584	−	0.489456i	0.423312	−	0.0746414i	0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$47$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$48$		0			0
$49$	7.18732	+	12.4488i	1.02676	+	1.77840i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$53$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.15523	+	7.23567i	0.285467	+	0.958388i
$58$		0			0
$59$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$59$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$60$		0			0
$61$	1.60694	−	9.11343i	0.205748	−	1.16686i	−0.690510	−	0.723323i	$-0.742614\pi$
$61$	1.60694	−	9.11343i	0.205748	−	1.16686i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$	4.74376	+	13.0334i	0.597657	+	1.64205i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.60994	−	6.68566i	0.685363	−	0.816784i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.60994	−	6.68566i	0.685363	−	0.816784i	0.990787	+	0.135433i	$0.0432425\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$71$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$72$		0			0
$73$	11.1912	+	4.07326i	1.30983	+	0.476739i	0.900186	−	0.435506i	$-0.143431\pi$
$73$	11.1912	+	4.07326i	1.30983	+	0.476739i	0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.03462	−	16.5800i	0.678947	−	1.86539i	0.225018	−	0.974355i	$-0.427756\pi$
$79$	6.03462	−	16.5800i	0.678947	−	1.86539i	0.453930	−	0.891038i	$-0.350022\pi$
$80$		0			0
$81$	1.56283	+	8.86327i	0.173648	+	0.984808i
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$89$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$	−32.6464	−	5.75643i	−3.42227	−	0.603438i
$92$		0			0
$93$	−13.9743	+	11.7258i	−1.44907	+	1.21591i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.83022	+	3.21394i	−0.388900	+	0.326326i	−0.816185	−	0.577791i	$-0.803915\pi$
$97$	−3.83022	+	3.21394i	−0.388900	+	0.326326i	0.427284	+	0.904117i	$0.359470\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$101$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	16.5364	−	9.54731i	1.62938	−	0.940724i	0.645105	−	0.764094i	$-0.276813\pi$
$103$	16.5364	−	9.54731i	1.62938	−	0.940724i	0.984277	−	0.176631i	$-0.0565198\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	−2.95202	−	16.7417i	−0.282752	−	1.60357i	−0.713206	−	0.700954i	$-0.752758\pi$
$109$	−2.95202	−	16.7417i	−0.282752	−	1.60357i	0.430454	−	0.902613i	$-0.358354\pi$
$110$		0			0
$111$	−7.17617	+	19.7164i	−0.681132	+	1.87140i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−20.2135	−	7.35710i	−1.86874	−	0.680165i
$118$		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
$127$	7.70115	+	21.1587i	0.683367	+	1.87753i	0.383375	+	0.923593i	$0.374762\pi$
$127$	7.70115	+	21.1587i	0.683367	+	1.87753i	0.299991	+	0.953942i	$0.403016\pi$
$128$		0			0
$129$	0.847763	−	4.80790i	0.0746414	−	0.423312i
$130$		0			0
$131$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$131$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$132$		0			0
$133$	20.0194	−	2.31164i	1.73590	−	0.200444i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$137$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$138$		0			0
$139$	4.36050	+	11.9804i	0.369853	+	1.01616i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	4.36050	+	11.9804i	0.369853	+	1.01616i	−0.605564	+	0.795796i	$0.707053\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	24.5194	−	4.32342i	2.02232	−	0.356590i
$148$		0			0
$149$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$149$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$150$		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
$157$	−3.51889	−	19.9566i	−0.280838	−	1.59271i	−0.719785	−	0.694197i	$-0.755760\pi$
$157$	−3.51889	−	19.9566i	−0.280838	−	1.59271i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.4696	+	6.04460i	−0.820039	+	0.473450i	−0.850430	−	0.526088i	$-0.823658\pi$
$163$	−10.4696	+	6.04460i	−0.820039	+	0.473450i	0.0303908	+	0.999538i	$0.490325\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$167$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$	29.4256	−	24.6910i	2.26351	−	1.89931i
$170$		0			0
$171$	13.0535	+	0.778544i	0.998226	+	0.0595368i
$172$		0			0
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$	22.7652	+	4.01411i	1.72088	+	0.303438i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	5.36231	+	4.49951i	0.398577	+	0.334446i	0.819943	−	0.572444i	$-0.194005\pi$
$181$	5.36231	+	4.49951i	0.398577	+	0.334446i	−0.421366	+	0.906891i	$0.638449\pi$
$182$		0			0
$183$	−13.8810	−	8.01422i	−1.02612	−	0.592428i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	24.0232			1.74743
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−18.1789	−	6.61656i	−1.30854	−	0.476271i	−0.408773	−	0.912636i	$-0.634043\pi$
$193$	−18.1789	−	6.61656i	−1.30854	−	0.476271i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−9.86319	+	11.7545i	−0.699183	+	0.833254i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−9.86319	+	11.7545i	−0.699183	+	0.833254i	0.293251	+	0.956036i	$0.405263\pi$
$200$		0			0
$201$	−7.55825	−	13.0913i	−0.533118	−	0.923387i
$202$		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
$211$	−5.32934	−	6.35127i	−0.366887	−	0.437239i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	−5.32934	−	6.35127i	−0.366887	−	0.437239i	−0.917630	+	0.397436i	$0.869900\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.3464	+	42.1692i	1.65274	+	2.86263i
$218$		0			0
$219$	13.2592	−	15.8017i	0.895976	−	1.06778i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.186137	−	0.0328209i	0.0124646	−	0.00219785i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	0.186137	−	0.0328209i	0.0124646	−	0.00219785i	0.179877	+	0.983689i	$0.442430\pi$
$224$		0			0
$225$	14.0954	+	5.13030i	0.939693	+	0.342020i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	3.49289			0.230817			0.115408	−	0.993318i	$-0.463182\pi$
$229$	3.49289			0.230817			0.115408	+	0.993318i	$0.463182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$233$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−23.4106	−	19.6438i	−1.52068	−	1.27600i
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−3.45558	+	1.25773i	−0.222594	+	0.0810175i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−3.45558	+	1.25773i	−0.222594	+	0.0810175i	0.228316	+	0.973587i	$0.426678\pi$
$242$		0			0
$243$	15.3516	+	2.70691i	0.984808	+	0.173648i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−17.2110	+	26.0887i	−1.09511	+	1.65998i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$251$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$257$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$	48.5021	+	28.0027i	3.01377	+	1.74000i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$269$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$	15.3516	−	2.70691i	0.932545	−	0.164433i	0.313321	−	0.949647i	$-0.398558\pi$
$271$	15.3516	−	2.70691i	0.932545	−	0.164433i	0.619224	+	0.785214i	$0.287447\pi$
$272$		0			0
$273$	−28.7087	+	49.7250i	−1.73753	+	3.00949i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.5000	−	26.8468i	−0.931305	−	1.61307i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−15.5000	−	26.8468i	−0.931305	−	1.61307i	−0.150210	−	0.988654i	$-0.547995\pi$
$278$		0			0
$279$	10.8066	+	29.6909i	0.646974	+	1.77755i
$280$		0			0
$281$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$281$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	−14.4734	−	17.2488i	−0.860356	−	1.02533i	−0.999386	−	0.0350443i	$-0.988843\pi$
$283$	−14.4734	−	17.2488i	−0.860356	−	1.02533i	0.139030	−	0.990288i	$-0.455602\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.95202	−	16.7417i	0.173648	−	0.984808i
$290$		0			0
$291$	2.96198	+	8.13798i	0.173634	+	0.477057i
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.2456	−	4.45702i	−0.705823	−	0.256898i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.592396	+	1.62760i	−0.0338098	+	0.0928918i	−0.955449	−	0.295156i	$-0.904628\pi$
$307$	−0.592396	+	1.62760i	−0.0338098	+	0.0928918i	0.921639	+	0.388048i	$0.126851\pi$
$308$		0			0
$309$	−5.74304	−	32.5704i	−0.326710	−	1.85287i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	26.8116	+	22.4976i	1.51548	+	1.27164i	0.852134	+	0.523324i	$0.175308\pi$
$313$	26.8116	+	22.4976i	1.51548	+	1.27164i	0.663345	+	0.748314i	$0.269136\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$317$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−27.4636	+	23.0447i	−1.52341	+	1.27829i
$326$		0			0
$327$	−28.9975	−	5.11305i	−1.60357	−	0.282752i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	29.3041	−	16.9187i	1.61070	−	0.929938i	0.621492	−	0.783420i	$-0.286527\pi$
$331$	29.3041	−	16.9187i	1.61070	−	0.929938i	0.989208	−	0.146518i	$-0.0468065\pi$
$332$		0			0
$333$	27.8391	+	23.3598i	1.52558	+	1.28011i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.34436	−	7.62424i	−0.0732319	−	0.415319i	−0.999281	−	0.0379157i	$-0.987928\pi$
$337$	−1.34436	−	7.62424i	−0.0732319	−	0.415319i	0.926049	−	0.377403i	$-0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	34.0949i		−	1.84095i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$347$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$348$		0			0
$349$	−16.4957	+	28.5714i	−0.882996	+	1.52939i	−0.0350017	+	0.999387i	$0.511144\pi$
$349$	−16.4957	+	28.5714i	−0.882996	+	1.52939i	−0.847994	+	0.530006i	$0.822190\pi$
$350$		0			0
$351$	−23.9488	+	28.5410i	−1.27829	+	1.52341i
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$359$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		0			0
$363$	−12.2467	−	14.5951i	−0.642788	−	0.766044i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.10426	−	25.0137i	−0.475238	−	1.30571i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−9.10426	−	25.0137i	−0.475238	−	1.30571i	0.438254	−	0.898851i	$-0.355597\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	14.4953i		−	0.744576i	−0.928117	−	0.372288i	$-0.878573\pi$
$379$		−	14.4953i		−	0.744576i	0.928117	−	0.372288i	$-0.121427\pi$
$380$		0			0
$381$	39.0000			1.99803
$382$		0			0
$383$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$383$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.32311	−	4.22800i	−0.372255	−	0.214921i
$388$		0			0
$389$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$389$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.1573	−	9.36208i	0.559968	−	0.469869i	−0.318332	−	0.947979i	$-0.603123\pi$
$397$	11.1573	−	9.36208i	0.559968	−	0.469869i	0.878300	+	0.478110i	$0.158678\pi$
$398$		0			0
$399$	8.09698	−	33.9528i	0.405356	−	1.69977i
$400$		0			0
$401$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$401$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$402$		0			0
$403$	−74.3706	−	13.1135i	−3.70466	−	0.653232i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−29.1097	−	24.4259i	−1.43938	−	1.20778i	−0.939895	−	0.341463i	$-0.889078\pi$
$409$	−29.1097	−	24.4259i	−1.43938	−	1.20778i	−0.499486	−	0.866322i	$-0.666478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	22.0823			1.08138
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	38.5274	+	14.0228i	1.87771	+	0.683431i	0.953291	+	0.302053i	$0.0976721\pi$
$421$	38.5274	+	14.0228i	1.87771	+	0.683431i	0.924419	+	0.381377i	$0.124550\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−27.5009	+	32.7743i	−1.33086	+	1.58606i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$431$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$432$		0			0
$433$	2.14249	−	12.1507i	0.102962	−	0.583924i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	2.14249	−	12.1507i	0.102962	−	0.583924i	0.992015	−	0.126121i	$-0.0402527\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	14.2447	+	16.9762i	0.679862	+	0.810228i	0.990090	−	0.140434i	$-0.0448499\pi$
$439$	14.2447	+	16.9762i	0.679862	+	0.810228i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	7.48839	−	42.4688i	0.356590	−	2.02232i
$442$		0			0
$443$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$443$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−14.0954	−	5.13030i	−0.662259	−	0.241043i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.82058			0.225497			0.112749	−	0.993624i	$-0.464034\pi$
$457$	4.82058			0.225497			0.112749	+	0.993624i	$0.464034\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$461$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$462$		0			0
$463$	11.3270	+	6.53964i	0.526409	+	0.303923i	0.739553	−	0.673098i	$-0.235037\pi$
$463$	11.3270	+	6.53964i	0.526409	+	0.303923i	−0.213144	+	0.977021i	$0.568370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	−37.9163	+	13.8004i	−1.75081	+	0.637243i
$470$		0			0
$471$	−34.5658	−	6.09489i	−1.59271	−	0.280838i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.0016	−	18.1923i	0.550673	−	0.834721i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$479$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$480$		0			0
$481$	−81.6207	+	29.7075i	−3.72158	+	1.35455i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	31.5000	+	18.1865i	1.42740	+	0.824110i	0.996915	−	0.0784867i	$-0.0250088\pi$
$487$	31.5000	+	18.1865i	1.42740	+	0.824110i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$	3.63604	+	20.6210i	0.164427	+	0.932514i
$490$		0			0
$491$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$491$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−36.3666	+	6.41242i	−1.62799	+	0.287059i	−0.911736	−	0.410776i	$-0.865258\pi$
$499$	−36.3666	+	6.41242i	−1.62799	+	0.287059i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$503$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−22.7554	−	62.5199i	−1.01060	−	2.77660i
$508$		0			0
$509$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$509$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$	−35.3922	−	42.1788i	−1.56566	−	1.86588i
$512$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−18.8692	+	22.4874i	−0.825091	+	0.983306i	−0.999999	−	0.00127919i	$-0.999593\pi$
$523$	−18.8692	+	22.4874i	−0.825091	+	0.983306i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$	20.0194	−	34.6745i	0.873716	−	1.51332i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.6129	+	7.86646i	0.939693	+	0.342020i
$530$		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
$541$	−31.2977	−	26.2619i	−1.34559	−	1.12909i	−0.980151	−	0.198254i	$-0.936473\pi$
$541$	−31.2977	−	26.2619i	−1.34559	−	1.12909i	−0.365444	−	0.930834i	$-0.619083\pi$
$542$		0			0
$543$	10.5000	−	6.06218i	0.450598	−	0.260153i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.96720	+	1.58116i	0.383410	+	0.0676055i	0.362031	−	0.932166i	$-0.382083\pi$
$547$	8.96720	+	1.58116i	0.383410	+	0.0676055i	0.0213785	+	0.999771i	$0.493195\pi$
$548$		0			0
$549$	−21.2670	+	17.8451i	−0.907653	+	0.761611i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−62.4887	+	52.4342i	−2.65729	+	2.22973i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$557$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$558$		0			0
$559$	17.5028	−	10.1053i	0.740291	−	0.427407i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	14.2313	−	39.1001i	0.597657	−	1.64205i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$			44.7821i			1.87407i	0.349231	+	0.937037i	$0.386443\pi$
$571$			44.7821i			1.87407i	−0.349231	+	0.937037i	$0.613557\pi$
$572$		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	$-0.906868\pi$
$577$	−5.50000	+	9.52628i	−0.228968	+	0.396584i	0.728535	+	0.685009i	$0.240202\pi$
$578$		0			0
$579$	−21.5382	+	25.6682i	−0.895096	+	1.06673i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$587$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$588$		0			0
$589$	45.6054	−	5.26606i	1.87914	−	0.216984i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$593$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	13.2886	+	23.0166i	0.543868	+	0.942007i
$598$		0			0
$599$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$599$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$600$		0			0
$601$	18.2114	−	31.5431i	0.742859	−	1.28667i	−0.208329	−	0.978059i	$-0.566802\pi$
$601$	18.2114	−	31.5431i	0.742859	−	1.28667i	0.951188	−	0.308611i	$-0.0998642\pi$
$602$		0			0
$603$	−25.7848	+	4.54655i	−1.05004	+	0.185150i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			35.4771i			1.43997i	0.693990	+	0.719985i	$0.255851\pi$
$607$			35.4771i			1.43997i	−0.693990	+	0.719985i	$0.744149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	8.16146	+	46.2860i	0.329638	+	1.86947i	0.474843	+	0.880071i	$0.342505\pi$
$613$	8.16146	+	46.2860i	0.329638	+	1.86947i	−0.145204	+	0.989402i	$0.546384\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$617$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$618$		0			0
$619$	21.5315	−	12.4312i	0.865425	−	0.499653i	−0.000400419	−	1.00000i	$-0.500127\pi$
$619$	21.5315	−	12.4312i	0.865425	−	0.499653i	0.865825	+	0.500347i	$0.166794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.1511	−	16.0697i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−9.55959	−	1.68561i	−0.380561	−	0.0671032i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−9.55959	−	1.68561i	−0.380561	−	0.0671032i	−0.360657	+	0.932699i	$0.617447\pi$
$632$		0			0
$633$	−13.4944	+	4.91155i	−0.536353	+	0.195217i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	78.9559	+	66.2519i	3.12835	+	2.62500i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$641$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$642$		0			0
$643$	−5.34096	+	14.6742i	−0.210627	+	0.578692i	−0.999350	−	0.0360565i	$-0.988520\pi$
$643$	−5.34096	+	14.6742i	−0.210627	+	0.578692i	0.788723	+	0.614749i	$0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	83.0572	−	14.6452i	3.25527	−	0.573991i
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−17.8641	−	30.9416i	−0.696946	−	1.20715i
$658$		0			0
$659$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$659$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$660$		0			0
$661$	6.59863	−	37.4227i	0.256657	−	1.45557i	−0.535126	−	0.844772i	$-0.679736\pi$
$661$	6.59863	−	37.4227i	0.256657	−	1.45557i	0.791783	−	0.610802i	$-0.209153\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.0568475	−	0.322398i	0.00219785	−	0.0124646i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.8619	+	44.7941i	0.996903	+	1.72669i	0.566557	+	0.824023i	$0.308275\pi$
$673$	25.8619	+	44.7941i	0.996903	+	1.72669i	0.430346	+	0.902664i	$0.358391\pi$
$674$		0			0
$675$	16.7001	−	19.9024i	0.642788	−	0.766044i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	22.7652	−	4.01411i	0.873647	−	0.154048i
$680$		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
$687$	2.06917	−	5.68501i	0.0789439	−	0.216897i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−16.5000	−	9.52628i	−0.627690	−	0.362397i	0.152167	−	0.988355i	$-0.451375\pi$
$691$	−16.5000	−	9.52628i	−0.627690	−	0.362397i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		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		0.766044	−	0.642788i	$-0.222222\pi$
$701$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$702$		0			0
$703$	42.3983	−	31.4726i	1.59908	−	1.18701i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−48.4702	+	17.6417i	−1.82034	+	0.662548i	−0.825108	+	0.564975i	$0.808886\pi$
$709$	−48.4702	+	17.6417i	−1.82034	+	0.662548i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	−45.8405	+	26.4661i	−1.71915	+	0.992554i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$719$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$720$		0			0
$721$	−88.2796			−3.28770
$722$		0			0
$723$			6.36937i			0.236879i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−37.3417	+	6.58434i	−1.38493	+	0.244200i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−37.3417	+	6.58434i	−1.38493	+	0.244200i	−0.568991	+	0.822344i	$0.692666\pi$
$728$		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	$-0.207932\pi$
$733$	−3.50000	−	6.06218i	−0.129275	−	0.223912i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	27.8684	+	33.2123i	1.02516	+	1.22173i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	27.8684	+	33.2123i	1.02516	+	1.22173i	0.0503375	+	0.998732i	$0.483970\pi$
$740$		0			0
$741$	32.2661	+	43.4673i	1.18532	+	1.59681i
$742$		0			0
$743$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$743$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	34.8505	−	41.5332i	1.27171	−	1.51557i	0.523655	−	0.851930i	$-0.324568\pi$
$751$	34.8505	−	41.5332i	1.27171	−	1.51557i	0.748056	−	0.663636i	$-0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.5775	−	3.84991i	−0.384447	−	0.139927i	0.142564	−	0.989786i	$-0.454465\pi$
$757$	−10.5775	−	3.84991i	−0.384447	−	0.139927i	−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−26.8813	+	73.8557i	−0.973168	+	2.67376i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−41.5474	−	34.8624i	−1.49824	−	1.25717i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−41.5474	−	34.8624i	−1.49824	−	1.25717i	−0.614745	−	0.788726i	$-0.710741\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$773$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$774$		0			0
$775$	51.8606	+	9.14442i	1.86289	+	0.328477i
$776$		0			0
$777$	74.3096	−	62.3531i	2.66584	−	2.23691i
$778$		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
$787$	48.2816	−	27.8754i	1.72105	−	0.993650i	0.804269	−	0.594265i	$-0.202557\pi$
$787$	48.2816	−	27.8754i	1.72105	−	0.993650i	0.916783	−	0.399385i	$-0.130776\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−11.5222	−	65.3455i	−0.409165	−	2.32049i
$794$		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
$801$		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	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	14.8099	+	40.6899i	0.520046	+	1.42882i	0.870469	+	0.492223i	$0.163816\pi$
$811$	14.8099	+	40.6899i	0.520046	+	1.42882i	−0.350423	+	0.936592i	$0.613962\pi$
$812$		0			0
$813$	4.68850	−	26.5898i	0.164433	−	0.932545i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.44381	+	8.92495i	−0.295412	+	0.312245i
$818$		0			0
$819$	63.9252	+	76.1831i	2.23373	+	2.66205i
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$		0			0
$823$	19.5491	+	53.7106i	0.681438	+	1.87224i	0.422608	+	0.906313i	$0.361115\pi$
$823$	19.5491	+	53.7106i	0.681438	+	1.87224i	0.258830	+	0.965923i	$0.416663\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$827$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$828$		0			0
$829$	15.6890	−	27.1741i	0.544900	−	0.943795i	−0.453713	−	0.891148i	$-0.649901\pi$
$829$	15.6890	−	27.1741i	0.544900	−	0.943795i	0.998613	−	0.0526472i	$-0.0167659\pi$
$830$		0			0
$831$	−52.8778	+	9.32379i	−1.83431	+	0.323439i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	54.7265			1.89162
$838$		0			0
$839$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$839$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$840$		0			0
$841$	5.03580	+	28.5594i	0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−44.0426	+	25.4280i	−1.51332	+	0.873716i
$848$		0			0
$849$	−36.6480	+	13.3388i	−1.25776	+	0.457786i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−30.6243	+	25.6969i	−1.04856	+	0.879844i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−30.6243	+	25.6969i	−1.04856	+	0.879844i	−0.0556158	+	0.998452i	$0.517712\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$857$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$858$		0			0
$859$	2.83346	+	0.499616i	0.0966765	+	0.0170467i	0.221777	−	0.975097i	$-0.428814\pi$
$859$	2.83346	+	0.499616i	0.0966765	+	0.0170467i	−0.125101	+	0.992144i	$0.539925\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		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
$871$	21.4031	−	58.8045i	0.725216	−	1.99251i
$872$		0			0
$873$	15.0000			0.507673
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.4360	−	5.25427i	−0.487469	−	0.177424i	0.0865807	−	0.996245i	$-0.472406\pi$
$877$	−14.4360	−	5.25427i	−0.487469	−	0.177424i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	33.8120	−	40.2955i	1.13786	−	1.35605i	0.212415	−	0.977180i	$-0.431867\pi$
$883$	33.8120	−	40.2955i	1.13786	−	1.35605i	0.925449	−	0.378873i	$-0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$887$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$888$		0			0
$889$	18.0769	−	102.519i	0.606279	−	3.43838i
$890$		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
$903$	−14.5085	+	17.2905i	−0.482811	+	0.575392i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.9402	+	2.10537i	−0.396466	+	0.0699077i	−0.368327	−	0.929696i	$-0.620069\pi$
$907$	−11.9402	+	2.10537i	−0.396466	+	0.0699077i	−0.0281394	+	0.999604i	$0.508958\pi$
$908$		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
$919$	49.6301	+	28.6539i	1.63714	+	0.945206i	0.981810	+	0.189867i	$0.0608058\pi$
$919$	49.6301	+	28.6539i	1.63714	+	0.945206i	0.655335	+	0.755338i	$0.272528\pi$
$920$		0			0
$921$	2.29813	+	1.92836i	0.0757261	+	0.0635417i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	56.9163	−	20.7158i	1.87140	−	0.681132i
$926$		0			0
$927$	−56.4136	−	9.94724i	−1.85287	−	0.326710i
$928$		0			0
$929$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$929$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$930$		0			0
$931$	−57.4986	−	24.8976i	−1.88444	−	0.815986i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.14512	−	1.87267i	0.168084	−	0.0611775i	−0.256608	−	0.966516i	$-0.582605\pi$
$937$	5.14512	−	1.87267i	0.168084	−	0.0611775i	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		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$947$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$948$		0			0
$949$	85.3934			2.77199
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$953$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	39.9628	+	69.2175i	1.28912	+	2.23282i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−20.7340	−	24.7098i	−0.666761	−	0.794615i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	−20.7340	−	24.7098i	−0.666761	−	0.794615i	−0.988339	+	0.152268i	$0.951342\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$971$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$972$		0			0
$973$	10.2354	−	58.0478i	0.328131	−	1.86093i
$974$		0			0
$975$	21.2381	+	58.3513i	0.680165	+	1.86874i
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−25.5000	+	44.1673i	−0.814152	+	1.41015i
$982$		0			0
$983$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$983$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−17.4948	+	48.0667i	−0.555742	+	1.52689i	0.270011	+	0.962857i	$0.412973\pi$
$991$	−17.4948	+	48.0667i	−0.555742	+	1.52689i	−0.825753	+	0.564031i	$0.809250\pi$
$992$		0			0
$993$	−10.1772	−	57.7178i	−0.322964	−	1.83162i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−36.5715	−	30.6871i	−1.15823	−	0.971871i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−36.5715	−	30.6871i	−1.15823	−	0.971871i	−0.999880	+	0.0155113i	$0.995062\pi$
$998$		0			0
$999$	54.5121	−	31.4726i	1.72469	−	0.995749i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.ch.c.671.1	✓	6
3.2	odd	2	CM	912.2.ch.c.671.1	✓	6
4.3	odd	2		912.2.ch.d.671.1	yes	6
12.11	even	2		912.2.ch.d.671.1	yes	6
19.16	even	9		912.2.ch.d.719.1	yes	6
57.35	odd	18		912.2.ch.d.719.1	yes	6
76.35	odd	18	inner	912.2.ch.c.719.1	yes	6
228.35	even	18	inner	912.2.ch.c.719.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.ch.c.671.1	✓	6	1.1	even	1	trivial
912.2.ch.c.671.1	✓	6	3.2	odd	2	CM
912.2.ch.c.719.1	yes	6	76.35	odd	18	inner
912.2.ch.c.719.1	yes	6	228.35	even	18	inner
912.2.ch.d.671.1	yes	6	4.3	odd	2
912.2.ch.d.671.1	yes	6	12.11	even	2
912.2.ch.d.719.1	yes	6	19.16	even	9
912.2.ch.d.719.1	yes	6	57.35	odd	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.ch (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.592396 - 1.62760i) q^{3} +(-4.00387 - 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)	\(q+(0.592396 - 1.62760i) q^{3} +(-4.00387 - 2.31164i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(6.73783 - 2.45237i) q^{13} +(-3.50000 + 2.59808i) q^{19} +(-6.13429 + 5.14728i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(-4.50000 + 2.59808i) q^{27} +(-9.12108 - 5.26606i) q^{31} -12.1138 q^{37} -12.4192i q^{39} +(2.77584 - 0.489456i) q^{43} +(7.18732 + 12.4488i) q^{49} +(2.15523 + 7.23567i) q^{57} +(1.60694 - 9.11343i) q^{61} +(4.74376 + 13.0334i) q^{63} +(5.60994 - 6.68566i) q^{67} +(11.1912 + 4.07326i) q^{73} +8.66025i q^{75} +(6.03462 - 16.5800i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-32.6464 - 5.75643i) q^{91} +(-13.9743 + 11.7258i) q^{93} +(-3.83022 + 3.21394i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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