LMFDB - Embedded newform 912.2.bb.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(31,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))

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

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

chi := DirichletCharacter("912.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.bb (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.28235666434$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	912.31
Dual form			912.2.bb.a.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +1.73205i q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{3} +1.73205i q^{7} +(-0.500000 + 0.866025i) q^{9} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(-3.00000 - 5.19615i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(1.50000 - 0.866025i) q^{21} +(-3.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +(-3.00000 - 1.73205i) q^{29} +1.00000 q^{31} +(3.00000 - 1.73205i) q^{33} +8.66025i q^{37} +5.19615i q^{39} +(-6.00000 + 3.46410i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(-9.00000 - 5.19615i) q^{47} +4.00000 q^{49} +(-3.00000 + 5.19615i) q^{51} +(-9.00000 - 5.19615i) q^{53} +(3.50000 + 2.59808i) q^{57} +(6.00000 + 10.3923i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-1.50000 - 0.866025i) q^{63} +(-6.50000 + 11.2583i) q^{67} +3.46410i q^{69} +(2.50000 + 4.33013i) q^{73} -5.00000 q^{75} -6.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -17.3205i q^{83} +3.46410i q^{87} +(4.50000 - 7.79423i) q^{91} +(-0.500000 - 0.866025i) q^{93} +(12.0000 - 6.92820i) q^{97} +(-3.00000 - 1.73205i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{9}+O(q^{10}) $ 2 * q - q^3 - q^9	$ 2 q - q^{3} - q^{9} - 9 q^{13} - 6 q^{17} - 8 q^{19} + 3 q^{21} - 6 q^{23} + 5 q^{25} + 2 q^{27} - 6 q^{29} + 2 q^{31} + 6 q^{33} - 12 q^{41} - 9 q^{43} - 18 q^{47} + 8 q^{49} - 6 q^{51} - 18 q^{53} + 7 q^{57} + 12 q^{59} + 5 q^{61} - 3 q^{63} - 13 q^{67} + 5 q^{73} - 10 q^{75} - 12 q^{77} + q^{79} - q^{81} + 9 q^{91} - q^{93} + 24 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^9 - 9 * q^13 - 6 * q^17 - 8 * q^19 + 3 * q^21 - 6 * q^23 + 5 * q^25 + 2 * q^27 - 6 * q^29 + 2 * q^31 + 6 * q^33 - 12 * q^41 - 9 * q^43 - 18 * q^47 + 8 * q^49 - 6 * q^51 - 18 * q^53 + 7 * q^57 + 12 * q^59 + 5 * q^61 - 3 * q^63 - 13 * q^67 + 5 * q^73 - 10 * q^75 - 12 * q^77 + q^79 - q^81 + 9 * q^91 - q^93 + 24 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$-1$

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
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$			1.73205i			0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$			1.73205i			0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	0.230285	−	0.973123i	$-0.426034\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$	1.50000	−	0.866025i	0.327327	−	0.188982i
$22$		0			0
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	0.153481	−	0.988152i	$-0.450952\pi$
$23$	−3.00000	−	1.73205i	−0.625543	−	0.361158i	−0.779024	+	0.626994i	$0.784285\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−3.00000	−	1.73205i	−0.557086	−	0.321634i	0.194889	−	0.980825i	$-0.437565\pi$
$29$	−3.00000	−	1.73205i	−0.557086	−	0.321634i	−0.751975	+	0.659192i	$0.770899\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$	3.00000	−	1.73205i	0.522233	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.66025i			1.42374i	0.702313	+	0.711868i	$0.252151\pi$
$37$			8.66025i			1.42374i	−0.702313	+	0.711868i	$0.747849\pi$
$38$		0			0
$39$			5.19615i			0.832050i
$40$		0			0
$41$	−6.00000	+	3.46410i	−0.937043	+	0.541002i	−0.889032	−	0.457845i	$-0.848621\pi$
$41$	−6.00000	+	3.46410i	−0.937043	+	0.541002i	−0.0480106	+	0.998847i	$0.515288\pi$
$42$		0			0
$43$	−4.50000	+	2.59808i	−0.686244	+	0.396203i	−0.802203	−	0.597051i	$-0.796339\pi$
$43$	−4.50000	+	2.59808i	−0.686244	+	0.396203i	0.115960	+	0.993254i	$0.463006\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.00000	−	5.19615i	−1.31278	−	0.757937i	−0.330228	−	0.943901i	$-0.607126\pi$
$47$	−9.00000	−	5.19615i	−1.31278	−	0.757937i	−0.982556	+	0.185964i	$0.940459\pi$
$48$		0			0
$49$	4.00000			0.571429
$50$		0			0
$51$	−3.00000	+	5.19615i	−0.420084	+	0.727607i
$52$		0			0
$53$	−9.00000	−	5.19615i	−1.23625	−	0.713746i	−0.267920	−	0.963441i	$-0.586336\pi$
$53$	−9.00000	−	5.19615i	−1.23625	−	0.713746i	−0.968325	+	0.249695i	$0.919670\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.50000	+	2.59808i	0.463586	+	0.344124i
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$	−1.50000	−	0.866025i	−0.188982	−	0.109109i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$		0			0
$69$			3.46410i			0.417029i
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	0.974424	−	0.224716i	$-0.0721453\pi$
$73$	2.50000	+	4.33013i	0.292603	+	0.506803i	−0.681822	+	0.731519i	$0.738812\pi$
$74$		0			0
$75$	−5.00000			−0.577350
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	17.3205i		−	1.90117i	−0.310460	−	0.950586i	$-0.600483\pi$
$83$		−	17.3205i		−	1.90117i	0.310460	−	0.950586i	$-0.399517\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			3.46410i			0.371391i
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	4.50000	−	7.79423i	0.471728	−	0.817057i
$92$		0			0
$93$	−0.500000	−	0.866025i	−0.0518476	−	0.0898027i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000	−	6.92820i	1.21842	−	0.703452i	0.253837	−	0.967247i	$-0.418307\pi$
$97$	12.0000	−	6.92820i	1.21842	−	0.703452i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$	−3.00000	−	1.73205i	−0.301511	−	0.174078i
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−12.0000	+	6.92820i	−1.14939	+	0.663602i	−0.948739	−	0.316061i	$-0.897640\pi$
$109$	−12.0000	+	6.92820i	−1.14939	+	0.663602i	−0.200653	+	0.979662i	$0.564306\pi$
$110$		0			0
$111$	7.50000	−	4.33013i	0.711868	−	0.410997i
$112$		0			0
$113$			6.92820i			0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$			6.92820i			0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.50000	−	2.59808i	0.416025	−	0.240192i
$118$		0			0
$119$	9.00000	−	5.19615i	0.825029	−	0.476331i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	6.00000	+	3.46410i	0.541002	+	0.312348i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	0.255014	+	0.966937i	$0.417920\pi$
$127$	−8.00000	+	13.8564i	−0.709885	+	1.22956i	−0.964899	+	0.262620i	$0.915413\pi$
$128$		0			0
$129$	4.50000	+	2.59808i	0.396203	+	0.228748i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−3.00000	−	6.92820i	−0.260133	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	7.50000	+	4.33013i	0.636142	+	0.367277i	0.783127	−	0.621862i	$-0.213624\pi$
$139$	7.50000	+	4.33013i	0.636142	+	0.367277i	−0.146985	+	0.989139i	$0.546957\pi$
$140$		0			0
$141$			10.3923i			0.875190i
$142$		0			0
$143$	9.00000	−	15.5885i	0.752618	−	1.30357i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.00000	−	3.46410i	−0.164957	−	0.285714i
$148$		0			0
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.50000	+	14.7224i	0.678374	+	1.17498i	0.975470	+	0.220131i	$0.0706483\pi$
$157$	8.50000	+	14.7224i	0.678374	+	1.17498i	−0.297097	+	0.954847i	$0.596018\pi$
$158$		0			0
$159$			10.3923i			0.824163i
$160$		0			0
$161$	3.00000	−	5.19615i	0.236433	−	0.409514i
$162$		0			0
$163$		−	22.5167i		−	1.76364i	−0.471585	−	0.881820i	$-0.656318\pi$
$163$		−	22.5167i		−	1.76364i	0.471585	−	0.881820i	$-0.343682\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	7.00000	+	12.1244i	0.538462	+	0.932643i
$170$		0			0
$171$	0.500000	−	4.33013i	0.0382360	−	0.331133i
$172$		0			0
$173$	−21.0000	+	12.1244i	−1.59660	+	0.921798i	−0.604465	+	0.796632i	$0.706613\pi$
$173$	−21.0000	+	12.1244i	−1.59660	+	0.921798i	−0.992136	+	0.125166i	$0.960054\pi$
$174$		0			0
$175$	7.50000	+	4.33013i	0.566947	+	0.327327i
$176$		0			0
$177$	6.00000	−	10.3923i	0.450988	−	0.781133i
$178$		0			0
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$	−5.00000			−0.369611
$184$		0			0
$185$		0			0
$186$		0			0
$187$	18.0000	−	10.3923i	1.31629	−	0.759961i
$188$		0			0
$189$			1.73205i			0.125988i
$190$		0			0
$191$			6.92820i			0.501307i	0.968077	+	0.250654i	$0.0806455\pi$
$191$			6.92820i			0.501307i	−0.968077	+	0.250654i	$0.919354\pi$
$192$		0			0
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−10.5000	−	6.06218i	−0.744325	−	0.429736i	0.0793146	−	0.996850i	$-0.474727\pi$
$199$	−10.5000	−	6.06218i	−0.744325	−	0.429736i	−0.823640	+	0.567113i	$0.808060\pi$
$200$		0			0
$201$	13.0000			0.916949
$202$		0			0
$203$	3.00000	−	5.19615i	0.210559	−	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.00000	−	1.73205i	0.208514	−	0.120386i
$208$		0			0
$209$	−6.00000	−	13.8564i	−0.415029	−	0.958468i
$210$		0			0
$211$	2.50000	+	4.33013i	0.172107	+	0.298098i	0.939156	−	0.343490i	$-0.111609\pi$
$211$	2.50000	+	4.33013i	0.172107	+	0.298098i	−0.767049	+	0.641588i	$0.778276\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			1.73205i			0.117579i
$218$		0			0
$219$	2.50000	−	4.33013i	0.168934	−	0.292603i
$220$		0			0
$221$			31.1769i			2.09719i
$222$		0			0
$223$	−5.50000	−	9.52628i	−0.368307	−	0.637927i	0.620994	−	0.783815i	$-0.286729\pi$
$223$	−5.50000	−	9.52628i	−0.368307	−	0.637927i	−0.989301	+	0.145889i	$0.953396\pi$
$224$		0			0
$225$	2.50000	+	4.33013i	0.166667	+	0.288675i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	23.0000			1.51988			0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000			1.51988			0.759941	+	0.649992i	$0.225228\pi$
$230$		0			0
$231$	3.00000	+	5.19615i	0.197386	+	0.341882i
$232$		0			0
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	0.928312	+	0.371802i	$0.121260\pi$
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	−0.142166	+	0.989843i	$0.545407\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.500000	−	0.866025i	0.0324785	−	0.0562544i
$238$		0			0
$239$		−	6.92820i		−	0.448148i	−0.974572	−	0.224074i	$-0.928064\pi$
$239$		−	6.92820i		−	0.448148i	0.974572	−	0.224074i	$-0.0719358\pi$
$240$		0			0
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	0.721715	−	0.692191i	$-0.243354\pi$
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	−0.238597	+	0.971119i	$0.576688\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	22.5000	+	2.59808i	1.43164	+	0.165312i
$248$		0			0
$249$	−15.0000	+	8.66025i	−0.950586	+	0.548821i
$250$		0			0
$251$	−15.0000	−	8.66025i	−0.946792	−	0.546630i	−0.0547088	−	0.998502i	$-0.517423\pi$
$251$	−15.0000	−	8.66025i	−0.946792	−	0.546630i	−0.892083	+	0.451872i	$0.850756\pi$
$252$		0			0
$253$	6.00000	−	10.3923i	0.377217	−	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000	−	1.73205i	−0.187135	−	0.108042i	0.403506	−	0.914977i	$-0.367792\pi$
$257$	−3.00000	−	1.73205i	−0.187135	−	0.108042i	−0.590641	+	0.806935i	$0.701125\pi$
$258$		0			0
$259$	−15.0000			−0.932055
$260$		0			0
$261$	3.00000	−	1.73205i	0.185695	−	0.107211i
$262$		0			0
$263$	−18.0000	+	10.3923i	−1.10993	+	0.640817i	−0.938811	−	0.344434i	$-0.888071\pi$
$263$	−18.0000	+	10.3923i	−1.10993	+	0.640817i	−0.171117	+	0.985251i	$0.554738\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000	−	3.46410i	0.365826	−	0.211210i	−0.305807	−	0.952093i	$-0.598926\pi$
$269$	6.00000	−	3.46410i	0.365826	−	0.211210i	0.671634	+	0.740883i	$0.265593\pi$
$270$		0			0
$271$	3.00000	−	1.73205i	0.182237	−	0.105215i	−0.406106	−	0.913826i	$-0.633114\pi$
$271$	3.00000	−	1.73205i	0.182237	−	0.105215i	0.588343	+	0.808611i	$0.299780\pi$
$272$		0			0
$273$	−9.00000			−0.544705
$274$		0			0
$275$	15.0000	+	8.66025i	0.904534	+	0.522233i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$	−0.500000	+	0.866025i	−0.0299342	+	0.0518476i
$280$		0			0
$281$	−6.00000	−	3.46410i	−0.357930	−	0.206651i	0.310242	−	0.950657i	$-0.399590\pi$
$281$	−6.00000	−	3.46410i	−0.357930	−	0.206651i	−0.668172	+	0.744007i	$0.732923\pi$
$282$		0			0
$283$	3.00000	−	1.73205i	0.178331	−	0.102960i	−0.408177	−	0.912903i	$-0.633835\pi$
$283$	3.00000	−	1.73205i	0.178331	−	0.102960i	0.586509	+	0.809943i	$0.300502\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	−	10.3923i	−0.354169	−	0.613438i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$	−12.0000	−	6.92820i	−0.703452	−	0.406138i
$292$		0			0
$293$		−	24.2487i		−	1.41662i	−0.705899	−	0.708312i	$-0.749457\pi$
$293$		−	24.2487i		−	1.41662i	0.705899	−	0.708312i	$-0.250543\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			3.46410i			0.201008i
$298$		0			0
$299$	9.00000	+	15.5885i	0.520483	+	0.901504i
$300$		0			0
$301$	−4.50000	−	7.79423i	−0.259376	−	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2.00000	+	3.46410i	0.114146	+	0.197707i	0.917438	−	0.397879i	$-0.130253\pi$
$307$	2.00000	+	3.46410i	0.114146	+	0.197707i	−0.803292	+	0.595585i	$0.796920\pi$
$308$		0			0
$309$	0.500000	+	0.866025i	0.0284440	+	0.0492665i
$310$		0			0
$311$		−	20.7846i		−	1.17859i	−0.807919	−	0.589294i	$-0.799406\pi$
$311$		−	20.7846i		−	1.17859i	0.807919	−	0.589294i	$-0.200594\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	24.0000	+	13.8564i	1.34797	+	0.778253i	0.987962	−	0.154694i	$-0.0494393\pi$
$317$	24.0000	+	13.8564i	1.34797	+	0.778253i	0.360012	+	0.932948i	$0.382773\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	21.0000	+	15.5885i	1.16847	+	0.867365i
$324$		0			0
$325$	−22.5000	+	12.9904i	−1.24808	+	0.720577i
$326$		0			0
$327$	12.0000	+	6.92820i	0.663602	+	0.383131i
$328$		0			0
$329$	9.00000	−	15.5885i	0.496186	−	0.859419i
$330$		0			0
$331$	−25.0000			−1.37412			−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000			−1.37412			−0.687062	+	0.726599i	$0.741100\pi$
$332$		0			0
$333$	−7.50000	−	4.33013i	−0.410997	−	0.237289i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.50000	+	4.33013i	−0.408551	+	0.235877i	−0.690167	−	0.723650i	$-0.742463\pi$
$337$	−7.50000	+	4.33013i	−0.408551	+	0.235877i	0.281616	+	0.959527i	$0.409130\pi$
$338$		0			0
$339$	6.00000	−	3.46410i	0.325875	−	0.188144i
$340$		0			0
$341$			3.46410i			0.187592i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	+	6.92820i	−0.644194	+	0.371925i	−0.786228	−	0.617936i	$-0.787969\pi$
$347$	−12.0000	+	6.92820i	−0.644194	+	0.371925i	0.142034	+	0.989862i	$0.454636\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$	−4.50000	−	2.59808i	−0.240192	−	0.138675i
$352$		0			0
$353$	24.0000			1.27739			0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000			1.27739			0.638696	+	0.769460i	$0.279474\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−9.00000	−	5.19615i	−0.476331	−	0.275010i
$358$		0			0
$359$	−18.0000	+	10.3923i	−0.950004	+	0.548485i	−0.893082	−	0.449894i	$-0.851462\pi$
$359$	−18.0000	+	10.3923i	−0.950004	+	0.548485i	−0.0569216	+	0.998379i	$0.518129\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	0.500000	+	0.866025i	0.0262432	+	0.0454545i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.50000	+	0.866025i	0.0782994	+	0.0452062i	0.538639	−	0.842537i	$-0.318939\pi$
$367$	1.50000	+	0.866025i	0.0782994	+	0.0452062i	−0.460339	+	0.887743i	$0.652272\pi$
$368$		0			0
$369$		−	6.92820i		−	0.360668i
$370$		0			0
$371$	9.00000	−	15.5885i	0.467257	−	0.809312i
$372$		0			0
$373$		−	13.8564i		−	0.717458i	−0.933442	−	0.358729i	$-0.883210\pi$
$373$		−	13.8564i		−	0.717458i	0.933442	−	0.358729i	$-0.116790\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.00000	+	15.5885i	0.463524	+	0.802846i
$378$		0			0
$379$	−25.0000			−1.28416			−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000			−1.28416			−0.642082	+	0.766636i	$0.721929\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	5.19615i		−	0.264135i
$388$		0			0
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$		0			0
$391$			20.7846i			1.05112i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	0.940389	−	0.340099i	$-0.110461\pi$
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	−0.764730	+	0.644351i	$0.777127\pi$
$398$		0			0
$399$	−4.50000	+	6.06218i	−0.225282	+	0.303488i
$400$		0			0
$401$	24.0000	−	13.8564i	1.19850	−	0.691956i	0.238282	−	0.971196i	$-0.423416\pi$
$401$	24.0000	−	13.8564i	1.19850	−	0.691956i	0.960221	+	0.279240i	$0.0900826\pi$
$402$		0			0
$403$	−4.50000	−	2.59808i	−0.224161	−	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−30.0000			−1.48704
$408$		0			0
$409$	−18.0000	−	10.3923i	−0.890043	−	0.513866i	−0.0160862	−	0.999871i	$-0.505121\pi$
$409$	−18.0000	−	10.3923i	−0.890043	−	0.513866i	−0.873956	+	0.486004i	$0.838454\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$	−18.0000	+	10.3923i	−0.885722	+	0.511372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	8.66025i		−	0.424094i
$418$		0			0
$419$			31.1769i			1.52309i	0.648111	+	0.761546i	$0.275559\pi$
$419$			31.1769i			1.52309i	−0.648111	+	0.761546i	$0.724441\pi$
$420$		0			0
$421$	18.0000	−	10.3923i	0.877266	−	0.506490i	0.00751023	−	0.999972i	$-0.497609\pi$
$421$	18.0000	−	10.3923i	0.877266	−	0.506490i	0.869756	+	0.493482i	$0.164276\pi$
$422$		0			0
$423$	9.00000	−	5.19615i	0.437595	−	0.252646i
$424$		0			0
$425$	−30.0000			−1.45521
$426$		0			0
$427$	7.50000	+	4.33013i	0.362950	+	0.209550i
$428$		0			0
$429$	−18.0000			−0.869048
$430$		0			0
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	0.237460	+	0.971397i	$0.423685\pi$
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	−0.959985	+	0.280052i	$0.909648\pi$
$432$		0			0
$433$	−22.5000	−	12.9904i	−1.08128	−	0.624278i	−0.150039	−	0.988680i	$-0.547940\pi$
$433$	−22.5000	−	12.9904i	−1.08128	−	0.624278i	−0.931242	+	0.364402i	$0.881273\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.0000	+	1.73205i	0.717547	+	0.0828552i
$438$		0			0
$439$	9.50000	+	16.4545i	0.453410	+	0.785330i	0.998595	−	0.0529862i	$-0.0168739\pi$
$439$	9.50000	+	16.4545i	0.453410	+	0.785330i	−0.545185	+	0.838316i	$0.683541\pi$
$440$		0			0
$441$	−2.00000	+	3.46410i	−0.0952381	+	0.164957i
$442$		0			0
$443$	21.0000	+	12.1244i	0.997740	+	0.576046i	0.907579	−	0.419882i	$-0.137928\pi$
$443$	21.0000	+	12.1244i	0.997740	+	0.576046i	0.0901612	+	0.995927i	$0.471262\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.00000	+	15.5885i	−0.425685	+	0.737309i
$448$		0			0
$449$			10.3923i			0.490443i	0.969467	+	0.245222i	$0.0788607\pi$
$449$			10.3923i			0.490443i	−0.969467	+	0.245222i	$0.921139\pi$
$450$		0			0
$451$	−12.0000	−	20.7846i	−0.565058	−	0.978709i
$452$		0			0
$453$	−4.00000	−	6.92820i	−0.187936	−	0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.00000			−0.233890			−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000			−0.233890			−0.116945	+	0.993138i	$0.537310\pi$
$458$		0			0
$459$	−3.00000	−	5.19615i	−0.140028	−	0.242536i
$460$		0			0
$461$	−6.00000	−	10.3923i	−0.279448	−	0.484018i	0.691800	−	0.722089i	$-0.256818\pi$
$461$	−6.00000	−	10.3923i	−0.279448	−	0.484018i	−0.971248	+	0.238071i	$0.923485\pi$
$462$		0			0
$463$			5.19615i			0.241486i	0.992684	+	0.120743i	$0.0385276\pi$
$463$			5.19615i			0.241486i	−0.992684	+	0.120743i	$0.961472\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−19.5000	−	11.2583i	−0.900426	−	0.519861i
$470$		0			0
$471$	8.50000	−	14.7224i	0.391659	−	0.678374i
$472$		0			0
$473$	−9.00000	−	15.5885i	−0.413820	−	0.716758i
$474$		0			0
$475$	−2.50000	+	21.6506i	−0.114708	+	0.993399i
$476$		0			0
$477$	9.00000	−	5.19615i	0.412082	−	0.237915i
$478$		0			0
$479$	12.0000	+	6.92820i	0.548294	+	0.316558i	0.748434	−	0.663210i	$-0.230806\pi$
$479$	12.0000	+	6.92820i	0.548294	+	0.316558i	−0.200140	+	0.979767i	$0.564140\pi$
$480$		0			0
$481$	22.5000	−	38.9711i	1.02591	−	1.77693i
$482$		0			0
$483$	−6.00000			−0.273009
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	−19.5000	+	11.2583i	−0.881820	+	0.509119i
$490$		0			0
$491$	−12.0000	+	6.92820i	−0.541552	+	0.312665i	−0.745708	−	0.666273i	$-0.767889\pi$
$491$	−12.0000	+	6.92820i	−0.541552	+	0.312665i	0.204155	+	0.978938i	$0.434555\pi$
$492$		0			0
$493$			20.7846i			0.936092i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7.50000	+	4.33013i	−0.335746	+	0.193843i	−0.658389	−	0.752678i	$-0.728762\pi$
$499$	−7.50000	+	4.33013i	−0.335746	+	0.193843i	0.322643	+	0.946521i	$0.395429\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$		0			0
$503$	−15.0000	−	8.66025i	−0.668817	−	0.386142i	0.126811	−	0.991927i	$-0.459526\pi$
$503$	−15.0000	−	8.66025i	−0.668817	−	0.386142i	−0.795628	+	0.605785i	$0.792859\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	7.00000	−	12.1244i	0.310881	−	0.538462i
$508$		0			0
$509$	−21.0000	−	12.1244i	−0.930809	−	0.537403i	−0.0437414	−	0.999043i	$-0.513928\pi$
$509$	−21.0000	−	12.1244i	−0.930809	−	0.537403i	−0.887067	+	0.461640i	$0.847261\pi$
$510$		0			0
$511$	−7.50000	+	4.33013i	−0.331780	+	0.191554i
$512$		0			0
$513$	−4.00000	+	1.73205i	−0.176604	+	0.0764719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	18.0000	−	31.1769i	0.791639	−	1.37116i
$518$		0			0
$519$	21.0000	+	12.1244i	0.921798	+	0.532200i
$520$		0			0
$521$			3.46410i			0.151765i	0.997117	+	0.0758825i	$0.0241774\pi$
$521$			3.46410i			0.151765i	−0.997117	+	0.0758825i	$0.975823\pi$
$522$		0			0
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614728\pi$
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	0.986718	−	0.162446i	$-0.0519382\pi$
$524$		0			0
$525$		−	8.66025i		−	0.377964i
$526$		0			0
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−5.50000	−	9.52628i	−0.239130	−	0.414186i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	36.0000			1.55933
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−3.00000	−	5.19615i	−0.129460	−	0.224231i
$538$		0			0
$539$			13.8564i			0.596838i
$540$		0			0
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	0.461625	+	0.887075i	$0.347267\pi$
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	−0.999042	+	0.0437584i	$0.986067\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.50000	−	11.2583i	0.277920	−	0.481371i	−0.692948	−	0.720988i	$-0.743688\pi$
$547$	6.50000	−	11.2583i	0.277920	−	0.481371i	0.970868	+	0.239616i	$0.0770217\pi$
$548$		0			0
$549$	2.50000	+	4.33013i	0.106697	+	0.184805i
$550$		0			0
$551$	15.0000	+	1.73205i	0.639021	+	0.0737878i
$552$		0			0
$553$	−1.50000	+	0.866025i	−0.0637865	+	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.0000	−	20.7846i	0.508456	−	0.880672i	−0.491496	−	0.870880i	$-0.663550\pi$
$557$	12.0000	−	20.7846i	0.508456	−	0.880672i	0.999952	−	0.00979220i	$-0.00311700\pi$
$558$		0			0
$559$	27.0000			1.14198
$560$		0			0
$561$	−18.0000	−	10.3923i	−0.759961	−	0.438763i
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.50000	−	0.866025i	0.0629941	−	0.0363696i
$568$		0			0
$569$			34.6410i			1.45223i	0.687575	+	0.726113i	$0.258675\pi$
$569$			34.6410i			1.45223i	−0.687575	+	0.726113i	$0.741325\pi$
$570$		0			0
$571$		−	32.9090i		−	1.37720i	−0.725143	−	0.688599i	$-0.758226\pi$
$571$		−	32.9090i		−	1.37720i	0.725143	−	0.688599i	$-0.241774\pi$
$572$		0			0
$573$	6.00000	−	3.46410i	0.250654	−	0.144715i
$574$		0			0
$575$	−15.0000	+	8.66025i	−0.625543	+	0.361158i
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$	−13.5000	−	7.79423i	−0.561041	−	0.323917i
$580$		0			0
$581$	30.0000			1.24461
$582$		0			0
$583$	18.0000	−	31.1769i	0.745484	−	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−36.0000	+	20.7846i	−1.48588	+	0.857873i	−0.999871	−	0.0160815i	$-0.994881\pi$
$587$	−36.0000	+	20.7846i	−1.48588	+	0.857873i	−0.486008	+	0.873954i	$0.661548\pi$
$588$		0			0
$589$	−4.00000	+	1.73205i	−0.164817	+	0.0713679i
$590$		0			0
$591$	6.00000	+	10.3923i	0.246807	+	0.427482i
$592$		0			0
$593$	12.0000	−	20.7846i	0.492781	−	0.853522i	−0.507184	−	0.861838i	$-0.669314\pi$
$593$	12.0000	−	20.7846i	0.492781	−	0.853522i	0.999965	+	0.00831589i	$0.00264706\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			12.1244i			0.496217i
$598$		0			0
$599$	−21.0000	+	36.3731i	−0.858037	+	1.48616i	0.0157622	+	0.999876i	$0.494983\pi$
$599$	−21.0000	+	36.3731i	−0.858037	+	1.48616i	−0.873799	+	0.486287i	$0.838351\pi$
$600$		0			0
$601$			46.7654i			1.90760i	0.300443	+	0.953800i	$0.402865\pi$
$601$			46.7654i			1.90760i	−0.300443	+	0.953800i	$0.597135\pi$
$602$		0			0
$603$	−6.50000	−	11.2583i	−0.264700	−	0.458475i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−47.0000			−1.90767			−0.953836	−	0.300329i	$-0.902903\pi$
$607$	−47.0000			−1.90767			−0.953836	+	0.300329i	$0.902903\pi$
$608$		0			0
$609$	−6.00000			−0.243132
$610$		0			0
$611$	27.0000	+	46.7654i	1.09230	+	1.89192i
$612$		0			0
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	0.938967	+	0.344008i	$0.111785\pi$
$613$	19.0000	+	32.9090i	0.767403	+	1.32918i	−0.171564	+	0.985173i	$0.554882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	0.0398207	+	0.999207i	$0.487321\pi$
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	−0.885249	+	0.465118i	$0.846012\pi$
$618$		0			0
$619$			39.8372i			1.60119i	0.599205	+	0.800595i	$0.295483\pi$
$619$			39.8372i			1.60119i	−0.599205	+	0.800595i	$0.704517\pi$
$620$		0			0
$621$	−3.00000	−	1.73205i	−0.120386	−	0.0695048i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	−9.00000	+	12.1244i	−0.359425	+	0.484200i
$628$		0			0
$629$	45.0000	−	25.9808i	1.79427	−	1.03592i
$630$		0			0
$631$	−19.5000	−	11.2583i	−0.776283	−	0.448187i	0.0588285	−	0.998268i	$-0.481263\pi$
$631$	−19.5000	−	11.2583i	−0.776283	−	0.448187i	−0.835111	+	0.550081i	$0.814597\pi$
$632$		0			0
$633$	2.50000	−	4.33013i	0.0993661	−	0.172107i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−18.0000	−	10.3923i	−0.713186	−	0.411758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.00000	−	1.73205i	0.118493	−	0.0684119i	−0.439582	−	0.898202i	$-0.644873\pi$
$641$	3.00000	−	1.73205i	0.118493	−	0.0684119i	0.558075	+	0.829790i	$0.311540\pi$
$642$		0			0
$643$	−10.5000	+	6.06218i	−0.414080	+	0.239069i	−0.692541	−	0.721378i	$-0.743509\pi$
$643$	−10.5000	+	6.06218i	−0.414080	+	0.239069i	0.278462	+	0.960447i	$0.410176\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	24.2487i		−	0.953315i	−0.879089	−	0.476658i	$-0.841848\pi$
$647$		−	24.2487i		−	0.953315i	0.879089	−	0.476658i	$-0.158152\pi$
$648$		0			0
$649$	−36.0000	+	20.7846i	−1.41312	+	0.815867i
$650$		0			0
$651$	1.50000	−	0.866025i	0.0587896	−	0.0339422i
$652$		0			0
$653$	48.0000			1.87839			0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000			1.87839			0.939193	+	0.343391i	$0.111576\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.00000			−0.195069
$658$		0			0
$659$	−9.00000	+	15.5885i	−0.350590	+	0.607240i	−0.986353	−	0.164644i	$-0.947352\pi$
$659$	−9.00000	+	15.5885i	−0.350590	+	0.607240i	0.635763	+	0.771885i	$0.280686\pi$
$660$		0			0
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	0.378754	−	0.925497i	$-0.376353\pi$
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	−0.612127	+	0.790759i	$0.709686\pi$
$662$		0			0
$663$	27.0000	−	15.5885i	1.04859	−	0.605406i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.00000	+	10.3923i	0.232321	+	0.402392i
$668$		0			0
$669$	−5.50000	+	9.52628i	−0.212642	+	0.368307i
$670$		0			0
$671$	15.0000	+	8.66025i	0.579069	+	0.334325i
$672$		0			0
$673$		−	8.66025i		−	0.333828i	−0.985971	−	0.166914i	$-0.946620\pi$
$673$		−	8.66025i		−	0.333828i	0.985971	−	0.166914i	$-0.0533803\pi$
$674$		0			0
$675$	2.50000	−	4.33013i	0.0962250	−	0.166667i
$676$		0			0
$677$			17.3205i			0.665681i	0.942983	+	0.332841i	$0.108007\pi$
$677$			17.3205i			0.665681i	−0.942983	+	0.332841i	$0.891993\pi$
$678$		0			0
$679$	12.0000	+	20.7846i	0.460518	+	0.797640i
$680$		0			0
$681$	−6.00000	−	10.3923i	−0.229920	−	0.398234i
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−11.5000	−	19.9186i	−0.438752	−	0.759941i
$688$		0			0
$689$	27.0000	+	46.7654i	1.02862	+	1.78162i
$690$		0			0
$691$		−	3.46410i		−	0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$		−	3.46410i		−	0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$		0			0
$693$	3.00000	−	5.19615i	0.113961	−	0.197386i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	36.0000	+	20.7846i	1.36360	+	0.787273i
$698$		0			0
$699$	12.0000	−	20.7846i	0.453882	−	0.786146i
$700$		0			0
$701$	12.0000	+	20.7846i	0.453234	+	0.785024i	0.998585	−	0.0531839i	$-0.0169370\pi$
$701$	12.0000	+	20.7846i	0.453234	+	0.785024i	−0.545351	+	0.838208i	$0.683604\pi$
$702$		0			0
$703$	−15.0000	−	34.6410i	−0.565736	−	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	−0.324104	−	0.946021i	$-0.605063\pi$
$709$	17.5000	−	30.3109i	0.657226	−	1.13835i	0.981331	−	0.192328i	$-0.0616038\pi$
$710$		0			0
$711$	−1.00000			−0.0375029
$712$		0			0
$713$	−3.00000	−	1.73205i	−0.112351	−	0.0648658i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−6.00000	+	3.46410i	−0.224074	+	0.129369i
$718$		0			0
$719$	39.0000	−	22.5167i	1.45445	−	0.839730i	0.455725	−	0.890121i	$-0.349380\pi$
$719$	39.0000	−	22.5167i	1.45445	−	0.839730i	0.998730	+	0.0503909i	$0.0160467\pi$
$720$		0			0
$721$		−	1.73205i		−	0.0645049i
$722$		0			0
$723$		−	8.66025i		−	0.322078i
$724$		0			0
$725$	−15.0000	+	8.66025i	−0.557086	+	0.321634i
$726$		0			0
$727$	16.5000	−	9.52628i	0.611951	−	0.353310i	−0.161778	−	0.986827i	$-0.551723\pi$
$727$	16.5000	−	9.52628i	0.611951	−	0.353310i	0.773729	+	0.633517i	$0.218389\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	27.0000	+	15.5885i	0.998631	+	0.576560i
$732$		0			0
$733$	22.0000			0.812589			0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000			0.812589			0.406294	+	0.913742i	$0.366821\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.0000	−	22.5167i	−1.43658	−	0.829412i
$738$		0			0
$739$	28.5000	−	16.4545i	1.04839	−	0.605288i	0.126191	−	0.992006i	$-0.459725\pi$
$739$	28.5000	−	16.4545i	1.04839	−	0.605288i	0.922198	+	0.386718i	$0.126391\pi$
$740$		0			0
$741$	−9.00000	−	20.7846i	−0.330623	−	0.763542i
$742$		0			0
$743$	−21.0000	−	36.3731i	−0.770415	−	1.33440i	−0.937336	−	0.348428i	$-0.886716\pi$
$743$	−21.0000	−	36.3731i	−0.770415	−	1.33440i	0.166920	−	0.985970i	$-0.446618\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	15.0000	+	8.66025i	0.548821	+	0.316862i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$		0			0
$753$			17.3205i			0.631194i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17.5000	−	30.3109i	−0.636048	−	1.10167i	−0.986292	−	0.165009i	$-0.947235\pi$
$757$	−17.5000	−	30.3109i	−0.636048	−	1.10167i	0.350244	−	0.936659i	$-0.386099\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$	−12.0000	−	20.7846i	−0.434429	−	0.752453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	62.3538i		−	2.25147i
$768$		0			0
$769$	−0.500000	+	0.866025i	−0.0180305	+	0.0312297i	−0.874900	−	0.484304i	$-0.839073\pi$
$769$	−0.500000	+	0.866025i	−0.0180305	+	0.0312297i	0.856869	+	0.515534i	$0.172406\pi$
$770$		0			0
$771$			3.46410i			0.124757i
$772$		0			0
$773$	−3.00000	−	1.73205i	−0.107903	−	0.0622975i	0.445078	−	0.895492i	$-0.353176\pi$
$773$	−3.00000	−	1.73205i	−0.107903	−	0.0622975i	−0.552980	+	0.833194i	$0.686509\pi$
$774$		0			0
$775$	2.50000	−	4.33013i	0.0898027	−	0.155543i
$776$		0			0
$777$	7.50000	+	12.9904i	0.269061	+	0.466027i
$778$		0			0
$779$	18.0000	−	24.2487i	0.644917	−	0.868800i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−3.00000	−	1.73205i	−0.107211	−	0.0618984i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	31.0000			1.10503			0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000			1.10503			0.552515	+	0.833503i	$0.313668\pi$
$788$		0			0
$789$	18.0000	+	10.3923i	0.640817	+	0.369976i
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	−22.5000	+	12.9904i	−0.798998	+	0.461302i

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.bb (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +1.73205i q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +1.73205i q^{7} +(-0.500000 + 0.866025i) q^{9} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(-3.00000 - 5.19615i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(1.50000 - 0.866025i) q^{21} +(-3.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +(-3.00000 - 1.73205i) q^{29} +1.00000 q^{31} +(3.00000 - 1.73205i) q^{33} +8.66025i q^{37} +5.19615i q^{39} +(-6.00000 + 3.46410i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(-9.00000 - 5.19615i) q^{47} +4.00000 q^{49} +(-3.00000 + 5.19615i) q^{51} +(-9.00000 - 5.19615i) q^{53} +(3.50000 + 2.59808i) q^{57} +(6.00000 + 10.3923i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-1.50000 - 0.866025i) q^{63} +(-6.50000 + 11.2583i) q^{67} +3.46410i q^{69} +(2.50000 + 4.33013i) q^{73} -5.00000 q^{75} -6.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -17.3205i q^{83} +3.46410i q^{87} +(4.50000 - 7.79423i) q^{91} +(-0.500000 - 0.866025i) q^{93} +(12.0000 - 6.92820i) q^{97} +(-3.00000 - 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{9}+O(q^{10}) \) 2 * q - q^3 - q^9	\( 2 q - q^{3} - q^{9} - 9 q^{13} - 6 q^{17} - 8 q^{19} + 3 q^{21} - 6 q^{23} + 5 q^{25} + 2 q^{27} - 6 q^{29} + 2 q^{31} + 6 q^{33} - 12 q^{41} - 9 q^{43} - 18 q^{47} + 8 q^{49} - 6 q^{51} - 18 q^{53} + 7 q^{57} + 12 q^{59} + 5 q^{61} - 3 q^{63} - 13 q^{67} + 5 q^{73} - 10 q^{75} - 12 q^{77} + q^{79} - q^{81} + 9 q^{91} - q^{93} + 24 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^9 - 9 * q^13 - 6 * q^17 - 8 * q^19 + 3 * q^21 - 6 * q^23 + 5 * q^25 + 2 * q^27 - 6 * q^29 + 2 * q^31 + 6 * q^33 - 12 * q^41 - 9 * q^43 - 18 * q^47 + 8 * q^49 - 6 * q^51 - 18 * q^53 + 7 * q^57 + 12 * q^59 + 5 * q^61 - 3 * q^63 - 13 * q^67 + 5 * q^73 - 10 * q^75 - 12 * q^77 + q^79 - q^81 + 9 * q^91 - q^93 + 24 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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