LMFDB - Embedded newform 912.2.cc.b.545.1

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			545.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	912.545
Dual form			912.2.cc.b.497.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.592396 + 1.62760i) q^{3} +(-1.28699 + 2.22913i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})$	$q+(-0.592396 + 1.62760i) q^{3} +(-1.28699 + 2.22913i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(0.262174 + 0.720317i) q^{13} +(-3.50000 + 2.59808i) q^{19} +(-2.86571 - 3.41523i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(4.50000 - 2.59808i) q^{27} +(-9.12108 - 5.26606i) q^{31} +1.12056i q^{37} -1.32770 q^{39} +(-2.22416 - 12.6138i) q^{43} +(0.187319 + 0.324446i) q^{49} +(-2.15523 - 7.23567i) q^{57} +(-1.60694 + 9.11343i) q^{61} +(7.25624 - 2.64106i) q^{63} +(5.60994 - 6.68566i) q^{67} +(-11.1912 - 4.07326i) q^{73} -8.66025i q^{75} +(-6.03462 + 16.5800i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-1.94310 - 0.342620i) q^{91} +(13.9743 - 11.7258i) q^{93} +(-12.2467 - 14.5951i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{5}{18}\right)$	$1$	$-1$	$1$

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
$3$	−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
$7$	−1.28699	+	2.22913i	−0.486436	+	0.842532i	−0.999878	−	0.0155920i	$-0.995037\pi$
$7$	−1.28699	+	2.22913i	−0.486436	+	0.842532i	0.513442	+	0.858124i	$0.328370\pi$
$8$		0			0
$9$	−2.29813	−	1.92836i	−0.766044	−	0.642788i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	0.262174	+	0.720317i	0.0727140	+	0.199780i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	0.262174	+	0.720317i	0.0727140	+	0.199780i	−0.898011	+	0.439972i	$0.854988\pi$
$14$		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
$21$	−2.86571	−	3.41523i	−0.625350	−	0.745263i
$22$		0			0
$23$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$23$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$24$		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		0.642788	−	0.766044i	$-0.277778\pi$
$29$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$30$		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
$37$			1.12056i			0.184220i	0.995749	+	0.0921098i	$0.0293611\pi$
$37$			1.12056i			0.184220i	−0.995749	+	0.0921098i	$0.970639\pi$
$38$		0			0
$39$	−1.32770			−0.212602
$40$		0			0
$41$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$41$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$42$		0			0
$43$	−2.22416	−	12.6138i	−0.339181	−	1.92359i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−2.22416	−	12.6138i	−0.339181	−	1.92359i	0.0420659	−	0.999115i	$-0.486606\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$47$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$48$		0			0
$49$	0.187319	+	0.324446i	0.0267598	+	0.0463494i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$53$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$54$		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	$0.257386\pi$
$61$	−1.60694	+	9.11343i	−0.205748	+	1.16686i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	7.25624	−	2.64106i	0.914201	−	0.332742i
$64$		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	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−11.1912	−	4.07326i	−1.30983	−	0.476739i	−0.900186	+	0.435506i	$0.856569\pi$
$74$		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	$0.572244\pi$
$79$	−6.03462	+	16.5800i	−0.678947	+	1.86539i	−0.453930	+	0.891038i	$0.649978\pi$
$80$		0			0
$81$	1.56283	+	8.86327i	0.173648	+	0.984808i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$89$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$90$		0			0
$91$	−1.94310	−	0.342620i	−0.203692	−	0.0359164i
$92$		0			0
$93$	13.9743	−	11.7258i	1.44907	−	1.21591i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.2467	−	14.5951i	−1.24347	−	1.48191i	−0.816185	−	0.577791i	$-0.803915\pi$
$97$	−12.2467	−	14.5951i	−1.24347	−	1.48191i	−0.427284	−	0.904117i	$-0.640530\pi$
$98$		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	$0.723187\pi$
$103$	−16.5364	+	9.54731i	−1.62938	+	0.940724i	−0.984277	+	0.176631i	$0.943480\pi$
$104$		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
$109$	11.9402	−	2.10537i	1.14366	−	0.201658i	0.430454	−	0.902613i	$-0.358354\pi$
$109$	11.9402	−	2.10537i	1.14366	−	0.201658i	0.713206	+	0.700954i	$0.247242\pi$
$110$		0			0
$111$	−1.82383	−	0.663818i	−0.173110	−	0.0630068i
$112$		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
$117$	0.786522	−	2.16095i	0.0727140	−	0.199780i
$118$		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
$129$	21.8478	+	3.85235i	1.92359	+	0.339181i
$130$		0			0
$131$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$131$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$132$		0			0
$133$	−1.28699	−	11.1457i	−0.111596	−	0.966451i
$134$		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
$139$	18.6395	−	6.78422i	1.58098	−	0.575430i	0.605564	−	0.795796i	$-0.292947\pi$
$139$	18.6395	−	6.78422i	1.58098	−	0.575430i	0.975417	+	0.220366i	$0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.639033	+	0.112679i	−0.0527066	+	0.00929359i
$148$		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	$0.114628\pi$
$151$			8.66025i			0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		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	$0.244240\pi$
$157$	3.51889	+	19.9566i	0.280838	+	1.59271i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.2456	+	19.4779i	0.880821	+	1.52563i	0.850430	+	0.526088i	$0.176342\pi$
$163$	11.2456	+	19.4779i	0.880821	+	1.52563i	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
$169$	9.50846	−	7.97854i	0.731420	−	0.613734i
$170$		0			0
$171$	13.0535	+	0.778544i	0.998226	+	0.0595368i
$172$		0			0
$173$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$173$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$174$		0			0
$175$	2.23483	−	12.6744i	0.168937	−	0.958092i
$176$		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
$181$	−16.7001	+	19.9024i	−1.24131	+	1.47934i	−0.421366	+	0.906891i	$0.638449\pi$
$181$	−16.7001	+	19.9024i	−1.24131	+	1.47934i	−0.819943	+	0.572444i	$0.805995\pi$
$182$		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
$189$			13.3748i			0.972872i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−6.82114	+	18.7409i	−0.490996	+	1.34900i	0.408773	+	0.912636i	$0.365957\pi$
$193$	−6.82114	+	18.7409i	−0.490996	+	1.34900i	−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
$199$	18.1368	+	15.2186i	1.28568	+	1.07882i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	18.1368	+	15.2186i	1.28568	+	1.07882i	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
$217$	23.4775	−	13.5547i	1.59375	−	0.920154i
$218$		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	−0.179877	−	0.983689i	$-0.557570\pi$
$223$	−0.186137	+	0.0328209i	−0.0124646	+	0.00219785i	0.167412	+	0.985887i	$0.446459\pi$
$224$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	10.5444	+	28.9705i	0.679225	+	1.86616i	0.450910	+	0.892570i	$0.351100\pi$
$241$	10.5444	+	28.9705i	0.679225	+	1.86616i	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
$247$	−2.78905	−	1.83996i	−0.177463	−	0.117074i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$251$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$257$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$258$		0			0
$259$	−2.49788	−	1.44215i	−0.155211	−	0.0896111i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$263$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$269$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$270$		0			0
$271$	−5.03580	−	28.5594i	−0.305903	−	1.73486i	−0.619224	−	0.785214i	$-0.712553\pi$
$271$	−5.03580	−	28.5594i	−0.305903	−	1.73486i	0.313321	−	0.949647i	$-0.398558\pi$
$272$		0			0
$273$	1.70873	−	2.95961i	0.103417	−	0.179124i
$274$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$281$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$282$		0			0
$283$	19.1511	−	16.0697i	1.13842	−	0.955244i	0.139030	−	0.990288i	$-0.455602\pi$
$283$	19.1511	−	16.0697i	1.13842	−	0.955244i	0.999386	+	0.0350443i	$0.0111572\pi$
$284$		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
$291$	31.0099	−	11.2867i	1.81783	−	0.661636i
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	30.9803	+	11.2759i	1.78567	+	0.649932i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.592396	−	1.62760i	0.0338098	−	0.0928918i	−0.921639	−	0.388048i	$-0.873149\pi$
$307$	0.592396	−	1.62760i	0.0338098	−	0.0928918i	0.955449	+	0.295156i	$0.0953717\pi$
$308$		0			0
$309$	−5.74304	−	32.5704i	−0.326710	−	1.85287i
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−26.8116	−	22.4976i	−1.51548	−	1.27164i	−0.852134	−	0.523324i	$-0.824692\pi$
$313$	−26.8116	−	22.4976i	−1.51548	−	1.27164i	−0.663345	−	0.748314i	$-0.730864\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$317$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.46363	−	2.93604i	−0.136658	−	0.162862i
$326$		0			0
$327$	−3.64661	+	20.6810i	−0.201658	+	1.14366i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−29.3041	+	16.9187i	−1.61070	+	0.929938i	−0.621492	+	0.783420i	$0.713473\pi$
$331$	−29.3041	+	16.9187i	−1.61070	+	0.929938i	−0.989208	+	0.146518i	$0.953193\pi$
$332$		0			0
$333$	2.16085	−	2.57521i	0.118414	−	0.141120i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	35.3444	−	6.23216i	1.92533	−	0.339488i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	35.3444	−	6.23216i	1.92533	−	0.339488i	0.999281	+	0.0379157i	$0.0120718\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.9822			−1.02494
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$347$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$348$		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
$351$	3.05122	+	2.56028i	0.162862	+	0.136658i
$352$		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		0.766044	−	0.642788i	$-0.222222\pi$
$359$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$360$		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
$367$	−25.8957	+	9.42528i	−1.35175	+	0.491996i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−25.8957	+	9.42528i	−1.35175	+	0.491996i	−0.438254	+	0.898851i	$0.644403\pi$
$368$		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	$-0.275929\pi$
$373$	6.00000	+	3.46410i	0.310668	+	0.179364i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			14.4953i			0.744576i	0.928117	+	0.372288i	$0.121427\pi$
$379$			14.4953i			0.744576i	−0.928117	+	0.372288i	$0.878573\pi$
$380$		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
$387$	−19.2126	+	33.2772i	−0.976631	+	1.69158i
$388$		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
$399$	18.9030	+	4.50795i	0.946335	+	0.225680i
$400$		0			0
$401$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$401$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$402$		0			0
$403$	1.40192	−	7.95070i	0.0698347	−	0.396052i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.90673	+	10.6146i	−0.440409	+	0.524859i	−0.939895	−	0.341463i	$-0.889078\pi$
$409$	−8.90673	+	10.6146i	−0.440409	+	0.524859i	0.499486	+	0.866322i	$0.333522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			34.3565i			1.68245i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−0.592396	+	1.62760i	−0.0288716	+	0.0793241i	−0.953291	−	0.302053i	$-0.902328\pi$
$421$	−0.592396	+	1.62760i	−0.0288716	+	0.0793241i	0.924419	+	0.381377i	$0.124550\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−18.2469	−	15.3110i	−0.883030	−	0.740950i
$428$		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
$433$	−39.1425	−	6.90188i	−1.88107	−	0.331683i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−39.1425	−	6.90188i	−1.88107	−	0.331683i	−0.992015	+	0.126121i	$0.959747\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−14.2447	−	16.9762i	−0.679862	−	0.810228i	0.310228	−	0.950662i	$-0.399595\pi$
$439$	−14.2447	−	16.9762i	−0.679862	−	0.810228i	−0.990090	+	0.140434i	$0.955150\pi$
$440$		0			0
$441$	0.195165	−	1.10684i	0.00929359	−	0.0527066i
$442$		0			0
$443$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$443$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$449$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$450$		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	$-0.535966\pi$
$457$	−4.82058			−0.225497			−0.112749	+	0.993624i	$0.535966\pi$
$458$		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
$463$	20.4996	−	35.5063i	0.952697	−	1.65012i	0.213144	−	0.977021i	$-0.431630\pi$
$463$	20.4996	−	35.5063i	0.952697	−	1.65012i	0.739553	−	0.673098i	$-0.235037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	7.68329	+	21.1097i	0.354781	+	0.974753i
$470$		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		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$	−0.807162	+	0.293783i	−0.0368034	+	0.0133953i
$482$		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
$489$	−38.3640	+	6.76460i	−1.73488	+	0.305906i
$490$		0			0
$491$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$491$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.36665	−	24.7645i	−0.195478	−	1.10861i	−0.911736	−	0.410776i	$-0.865258\pi$
$499$	−4.36665	−	24.7645i	−0.195478	−	1.10861i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$503$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.35306	+	20.2024i	0.326561	+	0.897219i
$508$		0			0
$509$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$509$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$510$		0			0
$511$	23.4828	−	19.7044i	1.03882	−	0.871670i
$512$		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	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		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
$525$	19.3048	+	11.1457i	0.842532	+	0.486436i
$526$		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	$0.0635271\pi$
$541$	31.2977	+	26.2619i	1.34559	+	1.12909i	0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−22.5000	−	38.9711i	−0.965567	−	1.67241i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.96720	−	1.58116i	−0.383410	−	0.0676055i	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−8.96720	−	1.58116i	−0.383410	−	0.0676055i	−0.362031	+	0.932166i	$0.617917\pi$
$548$		0			0
$549$	21.2670	−	17.8451i	0.907653	−	0.761611i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−29.1924	−	34.7902i	−1.24139	−	1.47943i
$554$		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
$559$	8.50283	−	4.90911i	0.359631	−	0.207633i
$560$		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
$567$	−21.7687	−	7.92317i	−0.914201	−	0.332742i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−16.6901			−0.698461			−0.349231	−	0.937037i	$-0.613557\pi$
$571$	−16.6901			−0.698461			−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	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	5.50000	−	9.52628i	0.228968	−	0.396584i	0.957503	+	0.288425i	$0.0931316\pi$
$578$		0			0
$579$	−26.4618	−	22.2041i	−1.09972	−	0.922771i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$587$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$588$		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
$597$	−35.5139	+	20.5040i	−1.45349	+	0.839171i
$598$		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
$601$	−28.4259	−	16.4117i	−1.15952	−	0.669448i	−0.208329	−	0.978059i	$-0.566802\pi$
$601$	−28.4259	−	16.4117i	−1.15952	−	0.669448i	−0.951188	+	0.308611i	$0.900136\pi$
$602$		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	$-0.657495\pi$
$613$	−8.16146	−	46.2860i	−0.329638	−	1.86947i	0.145204	−	0.989402i	$-0.453616\pi$
$614$		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
$619$	21.5514	+	37.3282i	0.866226	+	1.50035i	0.865825	+	0.500347i	$0.166794\pi$
$619$	21.5514	+	37.3282i	0.866226	+	1.50035i	0.000400419		1.00000i	$0.499873\pi$
$620$		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
$631$	8.55959	−	48.5439i	0.340752	−	1.93250i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	8.55959	−	48.5439i	0.340752	−	1.93250i	0.360657	−	0.932699i	$-0.382553\pi$
$632$		0			0
$633$	13.4944	−	4.91155i	0.536353	−	0.195217i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.184594	+	0.219990i	−0.00731387	+	0.00871633i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$641$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$642$		0			0
$643$	45.3410	+	16.5028i	1.78807	+	0.650805i	0.999350	+	0.0360565i	$0.0114796\pi$
$643$	45.3410	+	16.5028i	1.78807	+	0.650805i	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
$651$	8.15364	+	46.2416i	0.319566	+	1.81235i
$652$		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
$661$	−34.1147	−	6.01535i	−1.32691	−	0.233970i	−0.535126	−	0.844772i	$-0.679736\pi$
$661$	−34.1147	−	6.01535i	−1.32691	−	0.233970i	−0.791783	+	0.610802i	$0.790847\pi$
$662$		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
$673$	−3.53360	+	2.04012i	−0.136210	+	0.0786409i	−0.566557	−	0.824023i	$-0.691725\pi$
$673$	−3.53360	+	2.04012i	−0.136210	+	0.0786409i	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		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	48.2958	−	8.51586i	1.85342	−	0.326809i
$680$		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
$691$	−24.5000	+	42.4352i	−0.932024	+	1.61431i	−0.152167	+	0.988355i	$0.548625\pi$
$691$	−24.5000	+	42.4352i	−0.932024	+	1.61431i	−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
$703$	−2.91131	−	3.92198i	−0.109802	−	0.147920i
$704$		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	$-0.191114\pi$
$709$	48.4702	−	17.6417i	1.82034	−	0.662548i	0.995228	−	0.0975728i	$-0.0311079\pi$
$710$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$719$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$720$		0			0
$721$		−	49.1491i		−	1.83041i
$722$		0			0
$723$	−53.3988			−1.98592
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.65833	−	37.7613i	−0.246944	−	1.40049i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−6.65833	−	37.7613i	−0.246944	−	1.40049i	0.568991	−	0.822344i	$-0.307334\pi$
$728$		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	0.923396	−	0.383849i	$-0.125402\pi$
$733$	3.50000	+	6.06218i	0.129275	+	0.223912i	−0.794121	+	0.607760i	$0.792068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	25.1316	−	21.0879i	0.924481	−	0.775731i	−0.0503375	−	0.998732i	$-0.516030\pi$
$739$	25.1316	−	21.0879i	0.924481	−	0.775731i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$	4.64694	−	3.44946i	0.170709	−	0.126719i
$742$		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	$0.675432\pi$
$751$	−34.8505	+	41.5332i	−1.27171	+	1.51557i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		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
$763$	−10.6737	+	29.3258i	−0.386414	+	1.06166i
$764$		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	$0.155186\pi$
$769$	41.5474	+	34.8624i	1.49824	+	1.25717i	0.614745	+	0.788726i	$0.289259\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$773$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$774$		0			0
$775$	51.8606	+	9.14442i	1.86289	+	0.328477i
$776$		0			0
$777$	3.82698	−	3.21122i	0.137292	−	0.115202i
$778$		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
$793$	−6.98586	+	1.23180i	−0.248075	+	0.0437424i
$794$		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	$-0.666667\pi$
$809$		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	$-0.836184\pi$
$811$	−14.8099	−	40.6899i	−0.520046	−	1.42882i	0.350423	−	0.936592i	$-0.386038\pi$
$812$		0			0
$813$	49.4664	+	8.72226i	1.73486	+	0.305903i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	40.5562	+	38.3698i	1.41888	+	1.34239i
$818$		0			0
$819$	3.80480	+	4.53438i	0.132950	+	0.158444i
$820$		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
$823$	−4.69846	+	1.71010i	−0.163778	+	0.0596104i	−0.422608	−	0.906313i	$-0.638885\pi$
$823$	−4.69846	+	1.71010i	−0.163778	+	0.0596104i	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
$829$	41.8159	+	24.1424i	1.45233	+	0.838501i	0.998613	−	0.0526472i	$-0.0167659\pi$
$829$	41.8159	+	24.1424i	1.45233	+	0.838501i	0.453713	+	0.891148i	$0.350099\pi$
$830$		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
$847$	−14.1569	−	24.5204i	−0.486436	−	0.842532i
$848$		0			0
$849$	14.8099	+	40.6899i	0.508275	+	1.39647i
$850$		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		−0.642788	−	0.766044i	$-0.722222\pi$
$857$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$858$		0			0
$859$	−10.1665	+	57.6573i	−0.346878	+	1.96724i	−0.125101	+	0.992144i	$0.539925\pi$
$859$	−10.1665	+	57.6573i	−0.346878	+	1.96724i	−0.221777	+	0.975097i	$0.571186\pi$
$860$		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
$871$	6.28658	+	2.28813i	0.213013	+	0.0775302i
$872$		0			0
$873$			57.1577i			1.93449i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−19.5640	+	53.7517i	−0.660630	+	1.81507i	−0.0865807	+	0.996245i	$0.527594\pi$
$877$	−19.5640	+	53.7517i	−0.660630	+	1.81507i	−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
$883$	−21.1880	−	17.7789i	−0.713034	−	0.598307i	0.212415	−	0.977180i	$-0.431867\pi$
$883$	−21.1880	−	17.7789i	−0.713034	−	0.598307i	−0.925449	+	0.378873i	$0.876312\pi$
$884$		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
$889$	−57.0769	−	10.0642i	−1.91430	−	0.337542i
$890$		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
$903$	−36.7052	+	43.7436i	−1.22147	+	1.45570i
$904$		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
$919$	9.89709	−	17.1423i	0.326475	−	0.565471i	−0.655335	−	0.755338i	$-0.727472\pi$
$919$	9.89709	−	17.1423i	0.326475	−	0.565471i	0.981810	+	0.189867i	$0.0608058\pi$
$920$		0			0
$921$	2.29813	+	1.92836i	0.0757261	+	0.0635417i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.91628	−	5.26493i	−0.0630068	−	0.173110i
$926$		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
$931$	−1.49855	−	0.648891i	−0.0491130	−	0.0212665i
$932$		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		0.642788	−	0.766044i	$-0.277778\pi$
$941$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$947$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$948$		0			0
$949$		−	9.12911i		−	0.296343i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$953$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$954$		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
$967$	40.7340	−	34.1799i	1.30992	−	1.09915i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	40.7340	−	34.1799i	1.30992	−	1.09915i	0.988339	−	0.152268i	$-0.0486578\pi$
$968$		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
$973$	−8.86591	+	50.2811i	−0.284228	+	1.61194i
$974$		0			0
$975$	6.23813	−	2.27049i	0.199780	−	0.0727140i
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−31.5000	−	18.1865i	−1.00572	−	0.580651i
$982$		0			0
$983$		0			0		0.984808	−	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	0.999880	−	0.0155113i	$-0.00493761\pi$
$997$	36.5715	+	30.6871i	1.15823	+	0.971871i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	2.91131	+	5.04254i	0.0921098	+	0.159539i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.cc.b.545.1		6
3.2	odd	2	CM	912.2.cc.b.545.1		6
4.3	odd	2		228.2.t.a.89.1	yes	6
12.11	even	2		228.2.t.a.89.1	yes	6
19.3	odd	18	inner	912.2.cc.b.497.1		6
57.41	even	18	inner	912.2.cc.b.497.1		6
76.3	even	18		228.2.t.a.41.1	✓	6
228.155	odd	18		228.2.t.a.41.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.t.a.41.1	✓	6	76.3	even	18
228.2.t.a.41.1	✓	6	228.155	odd	18
228.2.t.a.89.1	yes	6	4.3	odd	2
228.2.t.a.89.1	yes	6	12.11	even	2
912.2.cc.b.497.1		6	19.3	odd	18	inner
912.2.cc.b.497.1		6	57.41	even	18	inner
912.2.cc.b.545.1		6	1.1	even	1	trivial
912.2.cc.b.545.1		6	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.cc (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.592396 + 1.62760i) q^{3} +(-1.28699 + 2.22913i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)	\(q+(-0.592396 + 1.62760i) q^{3} +(-1.28699 + 2.22913i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(0.262174 + 0.720317i) q^{13} +(-3.50000 + 2.59808i) q^{19} +(-2.86571 - 3.41523i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(4.50000 - 2.59808i) q^{27} +(-9.12108 - 5.26606i) q^{31} +1.12056i q^{37} -1.32770 q^{39} +(-2.22416 - 12.6138i) q^{43} +(0.187319 + 0.324446i) q^{49} +(-2.15523 - 7.23567i) q^{57} +(-1.60694 + 9.11343i) q^{61} +(7.25624 - 2.64106i) q^{63} +(5.60994 - 6.68566i) q^{67} +(-11.1912 - 4.07326i) q^{73} -8.66025i q^{75} +(-6.03462 + 16.5800i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-1.94310 - 0.342620i) q^{91} +(13.9743 - 11.7258i) q^{93} +(-12.2467 - 14.5951i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more