LMFDB - Embedded newform 756.2.i.a.613.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(37,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

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

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

chi := DirichletCharacter("756.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.i (of order $3$, degree $2$, 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:	$6.03669039281$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			613.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	756.613
Dual form			756.2.i.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(2.00000 + 3.46410i) q^{11} +(-1.50000 - 2.59808i) q^{13} +(3.50000 - 6.06218i) q^{17} +(-2.50000 - 4.33013i) q^{19} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{29} -3.00000 q^{31} +(-4.00000 + 3.46410i) q^{35} +(-5.50000 - 9.52628i) q^{37} +(-4.50000 - 7.79423i) q^{41} +(-2.50000 + 4.33013i) q^{43} -3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(1.50000 - 2.59808i) q^{53} +8.00000 q^{55} +7.00000 q^{59} +3.00000 q^{61} -6.00000 q^{65} +13.0000 q^{67} +8.00000 q^{71} +(-3.50000 + 6.06218i) q^{73} +(-2.00000 - 10.3923i) q^{77} -9.00000 q^{79} +(0.500000 - 0.866025i) q^{83} +(-7.00000 - 12.1244i) q^{85} +(7.50000 + 12.9904i) q^{89} +(1.50000 + 7.79423i) q^{91} -10.0000 q^{95} +(8.50000 - 14.7224i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 5 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 5 * q^7	$ 2 q + 2 q^{5} - 5 q^{7} + 4 q^{11} - 3 q^{13} + 7 q^{17} - 5 q^{19} + 4 q^{23} + q^{25} - q^{29} - 6 q^{31} - 8 q^{35} - 11 q^{37} - 9 q^{41} - 5 q^{43} - 6 q^{47} + 11 q^{49} + 3 q^{53} + 16 q^{55} + 14 q^{59} + 6 q^{61} - 12 q^{65} + 26 q^{67} + 16 q^{71} - 7 q^{73} - 4 q^{77} - 18 q^{79} + q^{83} - 14 q^{85} + 15 q^{89} + 3 q^{91} - 20 q^{95} + 17 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 5 * q^7 + 4 * q^11 - 3 * q^13 + 7 * q^17 - 5 * q^19 + 4 * q^23 + q^25 - q^29 - 6 * q^31 - 8 * q^35 - 11 * q^37 - 9 * q^41 - 5 * q^43 - 6 * q^47 + 11 * q^49 + 3 * q^53 + 16 * q^55 + 14 * q^59 + 6 * q^61 - 12 * q^65 + 26 * q^67 + 16 * q^71 - 7 * q^73 - 4 * q^77 - 18 * q^79 + q^83 - 14 * q^85 + 15 * q^89 + 3 * q^91 - 20 * q^95 + 17 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	−1.50000	−	2.59808i	−0.416025	−	0.720577i	0.579510	−	0.814965i	$-0.303244\pi$
$13$	−1.50000	−	2.59808i	−0.416025	−	0.720577i	−0.995535	+	0.0943882i	$0.969911\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.50000	−	6.06218i	0.848875	−	1.47029i	−0.0333386	−	0.999444i	$-0.510614\pi$
$17$	3.50000	−	6.06218i	0.848875	−	1.47029i	0.882213	−	0.470850i	$-0.156053\pi$
$18$		0			0
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	−0.996199	−	0.0871106i	$-0.972237\pi$
$19$	−2.50000	−	4.33013i	−0.573539	−	0.993399i	0.422659	−	0.906289i	$-0.361097\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	−0.908708	−	0.417432i	$-0.862930\pi$
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	0.815861	+	0.578249i	$0.196264\pi$
$30$		0			0
$31$	−3.00000			−0.538816			−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000			−0.538816			−0.269408	+	0.963026i	$0.586828\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000	+	3.46410i	−0.676123	+	0.585540i
$36$		0			0
$37$	−5.50000	−	9.52628i	−0.904194	−	1.56611i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−5.50000	−	9.52628i	−0.904194	−	1.56611i	−0.0821995	−	0.996616i	$-0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	−0.967486	−	0.252924i	$-0.918608\pi$
$41$	−4.50000	−	7.79423i	−0.702782	−	1.21725i	0.264704	−	0.964330i	$-0.414726\pi$
$42$		0			0
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000			−0.437595			−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000			−0.437595			−0.218797	+	0.975770i	$0.570213\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.50000	−	2.59808i	0.206041	−	0.356873i	−0.744423	−	0.667708i	$-0.767275\pi$
$53$	1.50000	−	2.59808i	0.206041	−	0.356873i	0.950464	+	0.310835i	$0.100609\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.00000			0.911322			0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000			0.911322			0.455661	+	0.890153i	$0.349403\pi$
$60$		0			0
$61$	3.00000			0.384111			0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000			0.384111			0.192055	+	0.981384i	$0.438485\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$	13.0000			1.58820			0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000			1.58820			0.794101	+	0.607785i	$0.207942\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	−3.50000	+	6.06218i	−0.409644	+	0.709524i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−3.50000	+	6.06218i	−0.409644	+	0.709524i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	−	10.3923i	−0.227921	−	1.18431i
$78$		0			0
$79$	−9.00000			−1.01258			−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000			−1.01258			−0.506290	+	0.862364i	$0.668983\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.500000	−	0.866025i	0.0548821	−	0.0950586i	−0.837279	−	0.546776i	$-0.815855\pi$
$83$	0.500000	−	0.866025i	0.0548821	−	0.0950586i	0.892161	+	0.451717i	$0.149188\pi$
$84$		0			0
$85$	−7.00000	−	12.1244i	−0.759257	−	1.31507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	0.922840	+	0.385183i	$0.125862\pi$
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	−0.127842	+	0.991795i	$0.540805\pi$
$90$		0			0
$91$	1.50000	+	7.79423i	0.157243	+	0.817057i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.0000			−1.02598
$96$		0			0
$97$	8.50000	−	14.7224i	0.863044	−	1.49484i	−0.00593185	−	0.999982i	$-0.501888\pi$
$97$	8.50000	−	14.7224i	0.863044	−	1.49484i	0.868976	−	0.494854i	$-0.164778\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	0.784366	−	0.620298i	$-0.212988\pi$
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	−0.929377	+	0.369132i	$0.879655\pi$
$108$		0			0
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	−0.983531	−	0.180741i	$-0.942150\pi$
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	0.648292	+	0.761392i	$0.275484\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.500000	−	0.866025i	−0.0470360	−	0.0814688i	0.841549	−	0.540181i	$-0.181644\pi$
$113$	−0.500000	−	0.866025i	−0.0470360	−	0.0814688i	−0.888585	+	0.458712i	$0.848311\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.0000	+	12.1244i	−1.28338	+	1.11144i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$	2.50000	+	12.9904i	0.216777	+	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	0.993109	+	0.117198i	$0.0373911\pi$
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	−0.395058	+	0.918656i	$0.629276\pi$
$138$		0			0
$139$	2.50000	+	4.33013i	0.212047	+	0.367277i	0.952355	−	0.304991i	$-0.0986536\pi$
$139$	2.50000	+	4.33013i	0.212047	+	0.367277i	−0.740308	+	0.672268i	$0.765320\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	10.3923i	0.501745	−	0.869048i
$144$		0			0
$145$	1.00000	+	1.73205i	0.0830455	+	0.143839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	+	5.19615i	−0.240966	+	0.417365i
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.00000	+	6.92820i	−0.630488	+	0.546019i
$162$		0			0
$163$	9.50000	+	16.4545i	0.744097	+	1.28881i	0.950615	+	0.310372i	$0.100454\pi$
$163$	9.50000	+	16.4545i	0.744097	+	1.28881i	−0.206518	+	0.978443i	$0.566213\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.5000	+	19.9186i	0.889897	+	1.54135i	0.839996	+	0.542592i	$0.182557\pi$
$167$	11.5000	+	19.9186i	0.889897	+	1.54135i	0.0499004	+	0.998754i	$0.484110\pi$
$168$		0			0
$169$	2.00000	−	3.46410i	0.153846	−	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.00000			0.0760286			0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000			0.0760286			0.0380143	+	0.999277i	$0.487897\pi$
$174$		0			0
$175$	−0.500000	−	2.59808i	−0.0377964	−	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5000	+	18.1865i	−0.784807	+	1.35933i	0.144308	+	0.989533i	$0.453905\pi$
$179$	−10.5000	+	18.1865i	−0.784807	+	1.35933i	−0.929114	+	0.369792i	$0.879429\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−22.0000			−1.61747
$186$		0			0
$187$	28.0000			2.04756
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−15.0000			−1.08536			−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000			−1.08536			−0.542681	+	0.839939i	$0.682591\pi$
$192$		0			0
$193$	−1.00000			−0.0719816			−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000			−0.0719816			−0.0359908	+	0.999352i	$0.511459\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	6.50000	−	11.2583i	0.460773	−	0.798082i	−0.538227	−	0.842800i	$-0.680906\pi$
$199$	6.50000	−	11.2583i	0.460773	−	0.798082i	0.999000	+	0.0447181i	$0.0142390\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.00000	−	1.73205i	0.140372	−	0.121566i
$204$		0			0
$205$	−18.0000			−1.25717
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.0000	−	17.3205i	0.691714	−	1.19808i
$210$		0			0
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	0.998221	−	0.0596196i	$-0.0189888\pi$
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.00000	+	8.66025i	0.340997	+	0.590624i
$216$		0			0
$217$	7.50000	+	2.59808i	0.509133	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−21.0000			−1.41261
$222$		0			0
$223$	3.50000	−	6.06218i	0.234377	−	0.405953i	−0.724714	−	0.689050i	$-0.758028\pi$
$223$	3.50000	−	6.06218i	0.234377	−	0.405953i	0.959092	+	0.283096i	$0.0913615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	−0.536515	−	0.843891i	$-0.680260\pi$
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	0.999088	+	0.0426906i	$0.0135930\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.5000	−	25.1147i	−0.949927	−	1.64532i	−0.745573	−	0.666424i	$-0.767824\pi$
$233$	−14.5000	−	25.1147i	−0.949927	−	1.64532i	−0.204354	−	0.978897i	$-0.565509\pi$
$234$		0			0
$235$	−3.00000	+	5.19615i	−0.195698	+	0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5000	−	18.1865i	−0.679189	−	1.17639i	−0.975226	−	0.221213i	$-0.928999\pi$
$239$	−10.5000	−	18.1865i	−0.679189	−	1.17639i	0.296037	−	0.955176i	$-0.404335\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.0000	−	5.19615i	0.830540	−	0.331970i
$246$		0			0
$247$	−7.50000	+	12.9904i	−0.477214	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	−0.435970	−	0.899961i	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		0			0
$259$	5.50000	+	28.5788i	0.341753	+	1.77580i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−3.00000	−	5.19615i	−0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.500000	+	0.866025i	−0.0304855	+	0.0528025i	−0.880866	−	0.473366i	$-0.843039\pi$
$269$	−0.500000	+	0.866025i	−0.0304855	+	0.0528025i	0.850380	+	0.526169i	$0.176372\pi$
$270$		0			0
$271$	1.50000	+	2.59808i	0.0911185	+	0.157822i	0.907982	−	0.419009i	$-0.137622\pi$
$271$	1.50000	+	2.59808i	0.0911185	+	0.157822i	−0.816864	+	0.576831i	$0.804289\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.00000	+	3.46410i	−0.120605	+	0.208893i
$276$		0			0
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	0.834419	−	0.551131i	$-0.185804\pi$
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	−0.894503	+	0.447062i	$0.852470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.50000	+	14.7224i	−0.507067	+	0.878267i	0.492899	+	0.870087i	$0.335937\pi$
$281$	−8.50000	+	14.7224i	−0.507067	+	0.878267i	−0.999967	+	0.00818015i	$0.997396\pi$
$282$		0			0
$283$	1.00000			0.0594438			0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000			0.0594438			0.0297219	+	0.999558i	$0.490538\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.50000	+	23.3827i	0.265627	+	1.38024i
$288$		0			0
$289$	−16.0000	−	27.7128i	−0.941176	−	1.63017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.5000	+	23.3827i	0.788678	+	1.36603i	0.926777	+	0.375613i	$0.122568\pi$
$293$	13.5000	+	23.3827i	0.788678	+	1.36603i	−0.138098	+	0.990419i	$0.544099\pi$
$294$		0			0
$295$	7.00000	−	12.1244i	0.407556	−	0.705907i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000			−0.693978
$300$		0			0
$301$	10.0000	−	8.66025i	0.576390	−	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000	−	5.19615i	0.171780	−	0.297531i
$306$		0			0
$307$	−8.00000			−0.456584			−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000			−0.456584			−0.228292	+	0.973593i	$0.573314\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	31.0000			1.75785			0.878924	−	0.476961i	$-0.158262\pi$
$311$	31.0000			1.75785			0.878924	+	0.476961i	$0.158262\pi$
$312$		0			0
$313$	−17.0000			−0.960897			−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000			−0.960897			−0.480448	+	0.877023i	$0.659526\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000			0.505490			0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000			0.505490			0.252745	+	0.967533i	$0.418667\pi$
$318$		0			0
$319$	−4.00000			−0.223957
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−35.0000			−1.94745
$324$		0			0
$325$	1.50000	−	2.59808i	0.0832050	−	0.144115i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.50000	+	2.59808i	0.413488	+	0.143237i
$330$		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
$333$		0			0
$334$		0			0
$335$	13.0000	−	22.5167i	0.710266	−	1.23022i
$336$		0			0
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	0.822274	−	0.569091i	$-0.192705\pi$
$337$	−1.50000	−	2.59808i	−0.0817102	−	0.141526i	−0.903985	+	0.427565i	$0.859372\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000	−	10.3923i	−0.324918	−	0.562775i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0000			0.805242			0.402621	−	0.915367i	$-0.368099\pi$
$347$	15.0000			0.805242			0.402621	+	0.915367i	$0.368099\pi$
$348$		0			0
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	−0.791325	−	0.611396i	$-0.790608\pi$
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	0.925147	+	0.379610i	$0.123942\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.00000	+	5.19615i	0.159674	+	0.276563i	0.934751	−	0.355303i	$-0.115622\pi$
$353$	3.00000	+	5.19615i	0.159674	+	0.276563i	−0.775077	+	0.631867i	$0.782289\pi$
$354$		0			0
$355$	8.00000	−	13.8564i	0.424596	−	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.5000	−	26.8468i	−0.818059	−	1.41692i	−0.907111	−	0.420892i	$-0.861717\pi$
$359$	−15.5000	−	26.8468i	−0.818059	−	1.41692i	0.0890519	−	0.996027i	$-0.471616\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.00000	+	12.1244i	0.366397	+	0.634618i
$366$		0			0
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	−0.893873	−	0.448320i	$-0.852022\pi$
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	0.0586798	−	0.998277i	$-0.481311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000	+	5.19615i	−0.311504	+	0.269771i
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.00000			0.154508
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	−0.585973	−	0.810331i	$-0.699287\pi$
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	0.994753	+	0.102302i	$0.0326207\pi$
$384$		0			0
$385$	−20.0000	−	6.92820i	−1.01929	−	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	0.980842	+	0.194804i	$0.0624070\pi$
$389$	13.0000	+	22.5167i	0.659126	+	1.14164i	−0.321716	+	0.946836i	$0.604260\pi$
$390$		0			0
$391$	−14.0000	−	24.2487i	−0.708010	−	1.22631i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.00000	+	15.5885i	−0.452839	+	0.784340i
$396$		0			0
$397$	10.5000	+	18.1865i	0.526980	+	0.912756i	0.999506	+	0.0314391i	$0.0100090\pi$
$397$	10.5000	+	18.1865i	0.526980	+	0.912756i	−0.472526	+	0.881317i	$0.656658\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	−0.199207	−	0.979957i	$-0.563837\pi$
$401$	15.0000	−	25.9808i	0.749064	−	1.29742i	0.948272	−	0.317460i	$-0.102830\pi$
$402$		0			0
$403$	4.50000	+	7.79423i	0.224161	+	0.388258i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	22.0000	−	38.1051i	1.09050	−	1.88880i
$408$		0			0
$409$	23.0000			1.13728			0.568638	−	0.822588i	$-0.307470\pi$
$409$	23.0000			1.13728			0.568638	+	0.822588i	$0.307470\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−17.5000	−	6.06218i	−0.861119	−	0.298300i
$414$		0			0
$415$	−1.00000	−	1.73205i	−0.0490881	−	0.0850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.50000	+	12.9904i	0.366399	+	0.634622i	0.989000	−	0.147918i	$-0.0472572\pi$
$419$	7.50000	+	12.9904i	0.366399	+	0.634622i	−0.622601	+	0.782540i	$0.713924\pi$
$420$		0			0
$421$	4.50000	−	7.79423i	0.219317	−	0.379867i	−0.735283	−	0.677761i	$-0.762951\pi$
$421$	4.50000	−	7.79423i	0.219317	−	0.379867i	0.954599	+	0.297893i	$0.0962839\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.00000			0.339550
$426$		0			0
$427$	−7.50000	−	2.59808i	−0.362950	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.5000	−	33.7750i	0.939282	−	1.62688i	0.172468	−	0.985015i	$-0.444826\pi$
$431$	19.5000	−	33.7750i	0.939282	−	1.62688i	0.766814	−	0.641869i	$-0.221841\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.0000			−0.956730
$438$		0			0
$439$	−21.0000			−1.00228			−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000			−1.00228			−0.501138	+	0.865368i	$0.667085\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.0000			1.37783			0.688916	−	0.724841i	$-0.258087\pi$
$443$	29.0000			1.37783			0.688916	+	0.724841i	$0.258087\pi$
$444$		0			0
$445$	30.0000			1.42214
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	18.0000	−	31.1769i	0.847587	−	1.46806i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15.0000	+	5.19615i	0.703211	+	0.243599i
$456$		0			0
$457$	31.0000			1.45012			0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000			1.45012			0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.5000	−	19.9186i	0.535608	−	0.927701i	−0.463525	−	0.886084i	$-0.653416\pi$
$461$	11.5000	−	19.9186i	0.535608	−	0.927701i	0.999134	−	0.0416172i	$-0.0132510\pi$
$462$		0			0
$463$	2.50000	+	4.33013i	0.116185	+	0.201238i	0.918253	−	0.395995i	$-0.129600\pi$
$463$	2.50000	+	4.33013i	0.116185	+	0.201238i	−0.802068	+	0.597233i	$0.796267\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.50000	+	14.7224i	0.393333	+	0.681273i	0.992887	−	0.119062i	$-0.0379886\pi$
$467$	8.50000	+	14.7224i	0.393333	+	0.681273i	−0.599554	+	0.800334i	$0.704655\pi$
$468$		0			0
$469$	−32.5000	−	11.2583i	−1.50071	−	0.519861i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−20.0000			−0.919601
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.00000	−	6.92820i	−0.182765	−	0.316558i	0.760056	−	0.649857i	$-0.225171\pi$
$479$	−4.00000	−	6.92820i	−0.182765	−	0.316558i	−0.942821	+	0.333300i	$0.891838\pi$
$480$		0			0
$481$	−16.5000	+	28.5788i	−0.752335	+	1.30308i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.0000	−	29.4449i	−0.771930	−	1.33702i
$486$		0			0
$487$	0.500000	−	0.866025i	0.0226572	−	0.0392434i	−0.854475	−	0.519493i	$-0.826121\pi$
$487$	0.500000	−	0.866025i	0.0226572	−	0.0392434i	0.877132	+	0.480250i	$0.159454\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.50000	−	7.79423i	−0.203082	−	0.351749i	0.746438	−	0.665455i	$-0.231763\pi$
$491$	−4.50000	−	7.79423i	−0.203082	−	0.351749i	−0.949520	+	0.313707i	$0.898429\pi$
$492$		0			0
$493$	3.50000	+	6.06218i	0.157632	+	0.273027i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−20.0000	−	6.92820i	−0.897123	−	0.310772i
$498$		0			0
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	−0.907314	−	0.420455i	$-0.861871\pi$
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	0.817781	+	0.575529i	$0.195204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	4.00000			0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	14.0000	−	12.1244i	0.619324	−	0.536350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	13.8564i	0.352522	+	0.610586i
$516$		0			0
$517$	−6.00000	−	10.3923i	−0.263880	−	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	−0.831295	−	0.555831i	$-0.812400\pi$
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	0.897011	+	0.442007i	$0.145733\pi$
$522$		0			0
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	−0.986718	−	0.162446i	$-0.948062\pi$
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	0.352677	−	0.935745i	$-0.385272\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.5000	+	18.1865i	−0.457387	+	0.792218i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.5000	+	23.3827i	−0.584750	+	1.01282i
$534$		0			0
$535$	−6.00000			−0.259403
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.00000	+	27.7128i	−0.172292	+	1.19368i
$540$		0			0
$541$	16.5000	+	28.5788i	0.709390	+	1.22870i	0.965084	+	0.261942i	$0.0843630\pi$
$541$	16.5000	+	28.5788i	0.709390	+	1.22870i	−0.255693	+	0.966758i	$0.582304\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.00000	+	12.1244i	0.299847	+	0.519350i
$546$		0			0
$547$	−16.5000	+	28.5788i	−0.705489	+	1.22194i	0.261026	+	0.965332i	$0.415939\pi$
$547$	−16.5000	+	28.5788i	−0.705489	+	1.22194i	−0.966515	+	0.256611i	$0.917394\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.00000			0.213007
$552$		0			0
$553$	22.5000	+	7.79423i	0.956797	+	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.50000	−	9.52628i	0.233042	−	0.403641i	−0.725660	−	0.688054i	$-0.758465\pi$
$557$	5.50000	−	9.52628i	0.233042	−	0.403641i	0.958702	+	0.284413i	$0.0917985\pi$
$558$		0			0
$559$	15.0000			0.634432
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.0000			−0.547885			−0.273942	−	0.961746i	$-0.588328\pi$
$563$	−13.0000			−0.547885			−0.273942	+	0.961746i	$0.588328\pi$
$564$		0			0
$565$	−2.00000			−0.0841406
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.0000			−1.13190			−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000			−1.13190			−0.565949	+	0.824440i	$0.691490\pi$
$570$		0			0
$571$	−3.00000			−0.125546			−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000			−0.125546			−0.0627730	+	0.998028i	$0.519994\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−3.50000	+	6.06218i	−0.145707	+	0.252372i	−0.929636	−	0.368478i	$-0.879879\pi$
$577$	−3.50000	+	6.06218i	−0.145707	+	0.252372i	0.783930	+	0.620850i	$0.213212\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.00000	+	1.73205i	−0.0829740	+	0.0718576i
$582$		0			0
$583$	12.0000			0.496989
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.5000	−	32.0429i	0.763577	−	1.32255i	−0.177419	−	0.984135i	$-0.556775\pi$
$587$	18.5000	−	32.0429i	0.763577	−	1.32255i	0.940996	−	0.338418i	$-0.109892\pi$
$588$		0			0
$589$	7.50000	+	12.9904i	0.309032	+	0.535259i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.5000	+	23.3827i	0.554379	+	0.960212i	0.997952	+	0.0639736i	$0.0203773\pi$
$593$	13.5000	+	23.3827i	0.554379	+	0.960212i	−0.443573	+	0.896238i	$0.646289\pi$
$594$		0			0
$595$	7.00000	+	36.3731i	0.286972	+	1.49115i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.00000			0.122577			0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000			0.122577			0.0612883	+	0.998120i	$0.480479\pi$
$600$		0			0
$601$	−17.5000	+	30.3109i	−0.713840	+	1.23641i	0.249565	+	0.968358i	$0.419712\pi$
$601$	−17.5000	+	30.3109i	−0.713840	+	1.23641i	−0.963405	+	0.268049i	$0.913621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.00000	+	8.66025i	0.203279	+	0.352089i
$606$		0			0
$607$	24.0000	−	41.5692i	0.974130	−	1.68724i	0.291353	−	0.956616i	$-0.405895\pi$
$607$	24.0000	−	41.5692i	0.974130	−	1.68724i	0.682777	−	0.730627i	$-0.260772\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.50000	+	7.79423i	0.182051	+	0.315321i
$612$		0			0
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	−0.760724	−	0.649075i	$-0.775156\pi$
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	0.942478	+	0.334269i	$0.108489\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.5000	+	37.2391i	0.865557	+	1.49919i	0.866493	+	0.499190i	$0.166369\pi$
$617$	21.5000	+	37.2391i	0.865557	+	1.49919i	−0.000935233		1.00000i	$0.500298\pi$
$618$		0			0
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	0.993959	−	0.109749i	$-0.0350048\pi$
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	−0.592025	+	0.805919i	$0.701671\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.50000	−	38.9711i	−0.300481	−	1.56135i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−77.0000			−3.07019
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.00000	−	20.7846i	0.118864	−	0.823516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.00000	−	1.73205i	−0.0394976	−	0.0684119i	0.845601	−	0.533816i	$-0.179242\pi$
$641$	−1.00000	−	1.73205i	−0.0394976	−	0.0684119i	−0.885098	+	0.465404i	$0.845909\pi$
$642$		0			0
$643$	6.50000	+	11.2583i	0.256335	+	0.443985i	0.965257	−	0.261301i	$-0.0841516\pi$
$643$	6.50000	+	11.2583i	0.256335	+	0.443985i	−0.708922	+	0.705287i	$0.750818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	−0.835033	−	0.550200i	$-0.814551\pi$
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	0.894004	+	0.448059i	$0.147885\pi$
$648$		0			0
$649$	14.0000	+	24.2487i	0.549548	+	0.951845i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	−0.986634	−	0.162951i	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$	4.00000	+	6.92820i	0.156293	+	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.5000	−	21.6506i	0.486931	−	0.843389i	−0.512956	−	0.858415i	$-0.671450\pi$
$659$	12.5000	−	21.6506i	0.486931	−	0.843389i	0.999887	+	0.0150258i	$0.00478303\pi$
$660$		0			0
$661$	−17.0000			−0.661223			−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000			−0.661223			−0.330612	+	0.943767i	$0.607255\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25.0000	+	8.66025i	0.969458	+	0.335830i
$666$		0			0
$667$	2.00000	+	3.46410i	0.0774403	+	0.134131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.00000	+	10.3923i	0.231627	+	0.401190i
$672$		0			0
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	−0.517943	−	0.855415i	$-0.673302\pi$
$673$	12.5000	−	21.6506i	0.481840	−	0.834571i	0.999783	+	0.0208444i	$0.00663546\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.0000			−1.03769			−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000			−1.03769			−0.518847	+	0.854867i	$0.673639\pi$
$678$		0			0
$679$	−34.0000	+	29.4449i	−1.30480	+	1.12999i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.5000	+	18.1865i	−0.401771	+	0.695888i	−0.993940	−	0.109926i	$-0.964939\pi$
$683$	−10.5000	+	18.1865i	−0.401771	+	0.695888i	0.592168	+	0.805814i	$0.298272\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.00000			−0.342873
$690$		0			0
$691$	11.0000			0.418460			0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000			0.418460			0.209230	+	0.977866i	$0.432904\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.0000			0.379322
$696$		0			0
$697$	−63.0000			−2.38630
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$	−27.5000	+	47.6314i	−1.03718	+	1.79645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.00000	−	5.19615i	−0.0376089	−	0.195421i
$708$		0			0
$709$	−25.0000			−0.938895			−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000			−0.938895			−0.469447	+	0.882960i	$0.655547\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.00000	+	10.3923i	−0.224702	+	0.389195i
$714$		0			0
$715$	−12.0000	−	20.7846i	−0.448775	−	0.777300i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.50000	−	12.9904i	−0.279703	−	0.484459i	0.691608	−	0.722273i	$-0.256903\pi$
$719$	−7.50000	−	12.9904i	−0.279703	−	0.484459i	−0.971311	+	0.237814i	$0.923569\pi$
$720$		0			0
$721$	16.0000	−	13.8564i	0.595871	−	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.00000			−0.0371391
$726$		0			0
$727$	23.5000	−	40.7032i	0.871567	−	1.50960i	0.0111912	−	0.999937i	$-0.496438\pi$
$727$	23.5000	−	40.7032i	0.871567	−	1.50960i	0.860376	−	0.509661i	$-0.170229\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.5000	+	30.3109i	0.647261	+	1.12109i
$732$		0			0
$733$	−15.0000	+	25.9808i	−0.554038	+	0.959621i	0.443940	+	0.896056i	$0.353580\pi$
$733$	−15.0000	+	25.9808i	−0.554038	+	0.959621i	−0.997978	+	0.0635649i	$0.979753\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	26.0000	+	45.0333i	0.957722	+	1.65882i
$738$		0			0
$739$	10.5000	−	18.1865i	0.386249	−	0.669002i	−0.605693	−	0.795699i	$-0.707104\pi$
$739$	10.5000	−	18.1865i	0.386249	−	0.669002i	0.991942	+	0.126696i	$0.0404373\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	0.771598	−	0.636111i	$-0.219458\pi$
$743$	−4.50000	−	7.79423i	−0.165089	−	0.285943i	−0.936687	+	0.350168i	$0.886124\pi$
$744$		0			0
$745$	6.00000	+	10.3923i	0.219823	+	0.380745i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.50000	+	7.79423i	0.0548088	+	0.284795i
$750$		0			0
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	−0.559640	−	0.828736i	$-0.689061\pi$
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	0.997526	+	0.0702946i	$0.0223939\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.00000	+	1.73205i	−0.0362500	+	0.0627868i	−0.883581	−	0.468278i	$-0.844875\pi$
$761$	−1.00000	+	1.73205i	−0.0362500	+	0.0627868i	0.847331	+	0.531065i	$0.178208\pi$
$762$		0			0
$763$	14.0000	−	12.1244i	0.506834	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.5000	−	18.1865i	−0.379133	−	0.656678i
$768$		0			0
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	0.907575	−	0.419890i	$-0.137931\pi$
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	−0.817423	+	0.576038i	$0.804598\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.5000	−	19.9186i	0.413626	−	0.716422i	−0.581657	−	0.813434i	$-0.697595\pi$
$773$	11.5000	−	19.9186i	0.413626	−	0.716422i	0.995283	+	0.0970125i	$0.0309287\pi$
$774$		0			0
$775$	−1.50000	−	2.59808i	−0.0538816	−	0.0933257i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.5000	+	38.9711i	−0.806146	+	1.39629i
$780$		0			0
$781$	16.0000	+	27.7128i	0.572525	+	0.991642i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.0000	+	22.5167i	−0.463990	+	0.803654i
$786$		0			0
$787$	21.0000			0.748569			0.374285	−	0.927314i	$-0.377888\pi$
$787$	21.0000			0.748569			0.374285	+	0.927314i	$0.377888\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.500000	+	2.59808i	0.0177780	+	0.0923770i
$792$		0			0
$793$	−4.50000	−	7.79423i	−0.159800	−	0.276781i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.50000	+	12.9904i	0.265664	+	0.460143i	0.967737	−	0.251961i	$-0.0810756\pi$
$797$	7.50000	+	12.9904i	0.265664	+	0.460143i	−0.702074	+	0.712104i	$0.747742\pi$
$798$		0			0
$799$	−10.5000	+	18.1865i	−0.371463	+	0.643393i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−28.0000			−0.988099
$804$		0			0
$805$	4.00000	+	20.7846i	0.140981	+	0.732561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.500000	+	0.866025i	−0.0175791	+	0.0304478i	−0.874681	−	0.484699i	$-0.838929\pi$
$809$	−0.500000	+	0.866025i	−0.0175791	+	0.0304478i	0.857102	+	0.515147i	$0.172263\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	38.0000			1.33108
$816$		0			0
$817$	25.0000			0.874639
$818$		0			0
$819$		0			0
$820$		0			0
$821$	53.0000			1.84971			0.924856	−	0.380317i	$-0.124185\pi$
$821$	53.0000			1.84971			0.924856	+	0.380317i	$0.124185\pi$
$822$		0			0
$823$	−23.0000			−0.801730			−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000			−0.801730			−0.400865	+	0.916137i	$0.631290\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	−9.50000	+	16.4545i	−0.329949	+	0.571488i	−0.982501	−	0.186256i	$-0.940365\pi$
$829$	−9.50000	+	16.4545i	−0.329949	+	0.571488i	0.652553	+	0.757743i	$0.273698\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	45.5000	−	18.1865i	1.57648	−	0.630126i
$834$		0			0
$835$	46.0000			1.59190
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.5000	+	19.9186i	−0.397024	+	0.687666i	−0.993357	−	0.115071i	$-0.963290\pi$
$839$	−11.5000	+	19.9186i	−0.397024	+	0.687666i	0.596333	+	0.802737i	$0.296624\pi$
$840$		0			0
$841$	14.0000	+	24.2487i	0.482759	+	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.00000	−	6.92820i	−0.137604	−	0.238337i
$846$		0			0
$847$	10.0000	−	8.66025i	0.343604	−	0.297570i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−44.0000			−1.50830
$852$		0			0
$853$	−17.5000	+	30.3109i	−0.599189	+	1.03783i	0.393753	+	0.919216i	$0.371177\pi$
$853$	−17.5000	+	30.3109i	−0.599189	+	1.03783i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	0.670366	−	0.742030i	$-0.266137\pi$
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	−0.977800	+	0.209539i	$0.932804\pi$
$858$		0			0
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	0.196887	+	0.980426i	$0.436917\pi$
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	−0.947518	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.5000	+	35.5070i	0.697828	+	1.20867i	0.969218	+	0.246204i	$0.0791834\pi$
$863$	20.5000	+	35.5070i	0.697828	+	1.20867i	−0.271390	+	0.962470i	$0.587483\pi$
$864$		0			0
$865$	1.00000	−	1.73205i	0.0340010	−	0.0588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.0000	−	31.1769i	−0.610608	−	1.05760i
$870$		0			0
$871$	−19.5000	−	33.7750i	−0.660732	−	1.14442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−30.0000	−	10.3923i	−1.01419	−	0.351324i
$876$		0			0
$877$	19.0000	−	32.9090i	0.641584	−	1.11126i	−0.343495	−	0.939155i	$-0.611611\pi$
$877$	19.0000	−	32.9090i	0.641584	−	1.11126i	0.985079	−	0.172102i	$-0.0550559\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000	−	10.3923i	0.201460	−	0.348939i	−0.747539	−	0.664218i	$-0.768765\pi$
$887$	6.00000	−	10.3923i	0.201460	−	0.348939i	0.948999	+	0.315279i	$0.102098\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.50000	+	12.9904i	0.250978	+	0.434707i
$894$		0			0
$895$	21.0000	+	36.3731i	0.701953	+	1.21582i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.50000	−	2.59808i	0.0500278	−	0.0866507i
$900$		0			0
$901$	−10.5000	−	18.1865i	−0.349806	−	0.605881i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.00000	+	10.3923i	−0.199447	+	0.345452i
$906$		0			0
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	−0.993163	−	0.116739i	$-0.962756\pi$
$907$	−18.0000	−	31.1769i	−0.597680	−	1.03521i	0.395482	−	0.918474i	$-0.370577\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	0.337991	+	0.941149i	$0.390253\pi$
$911$	−19.5000	+	33.7750i	−0.646064	+	1.11902i	−0.984055	+	0.177866i	$0.943081\pi$
$912$		0			0
$913$	4.00000			0.132381
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.00000	−	6.92820i	0.264183	−	0.228789i
$918$		0			0
$919$	17.5000	+	30.3109i	0.577272	+	0.999864i	0.995791	+	0.0916559i	$0.0292160\pi$
$919$	17.5000	+	30.3109i	0.577272	+	0.999864i	−0.418519	+	0.908208i	$0.637451\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.0000	−	20.7846i	−0.394985	−	0.684134i
$924$		0			0
$925$	5.50000	−	9.52628i	0.180839	−	0.313222i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000			−0.0984268			−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000			−0.0984268			−0.0492134	+	0.998788i	$0.515671\pi$
$930$		0			0
$931$	5.00000	−	34.6410i	0.163868	−	1.13531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28.0000	−	48.4974i	0.915698	−	1.58604i
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−59.0000			−1.92335			−0.961673	−	0.274201i	$-0.911587\pi$
$941$	−59.0000			−1.92335			−0.961673	+	0.274201i	$0.911587\pi$
$942$		0			0
$943$	−36.0000			−1.17232
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.0000			0.812391			0.406195	−	0.913786i	$-0.366855\pi$
$947$	25.0000			0.812391			0.406195	+	0.913786i	$0.366855\pi$
$948$		0			0
$949$	21.0000			0.681689
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	−15.0000	+	25.9808i	−0.485389	+	0.840718i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.00000	−	36.3731i	−0.226042	−	1.17455i
$960$		0			0
$961$	−22.0000			−0.709677
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.00000	+	1.73205i	−0.0321911	+	0.0557567i
$966$		0			0
$967$	−23.5000	−	40.7032i	−0.755709	−	1.30893i	−0.945021	−	0.327009i	$-0.893959\pi$
$967$	−23.5000	−	40.7032i	−0.755709	−	1.30893i	0.189312	−	0.981917i	$-0.439374\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	0.720224	−	0.693742i	$-0.244039\pi$
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	−0.960910	+	0.276861i	$0.910706\pi$
$972$		0			0
$973$	−2.50000	−	12.9904i	−0.0801463	−	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.00000			−0.0959785			−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000			−0.0959785			−0.0479893	+	0.998848i	$0.515281\pi$
$978$		0			0
$979$	−30.0000	+	51.9615i	−0.958804	+	1.66070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	−0.996138	−	0.0878058i	$-0.972015\pi$
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	0.422027	−	0.906583i	$-0.361319\pi$
$984$		0			0
$985$	26.0000	−	45.0333i	0.828429	−	1.43488i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.0000	+	17.3205i	0.317982	+	0.550760i
$990$		0			0
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	−0.406865	−	0.913488i	$-0.633378\pi$
$991$	18.5000	−	32.0429i	0.587672	−	1.01788i	0.994537	−	0.104389i	$-0.0332887\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.0000	−	22.5167i	−0.412128	−	0.713826i
$996$		0			0
$997$	11.0000	+	19.0526i	0.348373	+	0.603401i	0.985961	−	0.166978i	$-0.0534008\pi$
$997$	11.0000	+	19.0526i	0.348373	+	0.603401i	−0.637587	+	0.770378i	$0.720067\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.i.a.613.1		2
3.2	odd	2		252.2.i.a.25.1	✓	2
4.3	odd	2		3024.2.q.e.2881.1		2
7.2	even	3		756.2.l.a.289.1		2
7.3	odd	6		5292.2.j.b.3529.1		2
7.4	even	3		5292.2.j.c.3529.1		2
7.5	odd	6		5292.2.l.b.3313.1		2
7.6	odd	2		5292.2.i.b.2125.1		2
9.2	odd	6		2268.2.k.a.1621.1		2
9.4	even	3		756.2.l.a.361.1		2
9.5	odd	6		252.2.l.a.193.1	yes	2
9.7	even	3		2268.2.k.b.1621.1		2
12.11	even	2		1008.2.q.f.529.1		2
21.2	odd	6		252.2.l.a.205.1	yes	2
21.5	even	6		1764.2.l.b.961.1		2
21.11	odd	6		1764.2.j.c.1177.1		2
21.17	even	6		1764.2.j.a.1177.1		2
21.20	even	2		1764.2.i.b.1537.1		2
28.23	odd	6		3024.2.t.b.289.1		2
36.23	even	6		1008.2.t.b.193.1		2
36.31	odd	6		3024.2.t.b.1873.1		2
63.2	odd	6		2268.2.k.a.1297.1		2
63.4	even	3		5292.2.j.c.1765.1		2
63.5	even	6		1764.2.i.b.373.1		2
63.13	odd	6		5292.2.l.b.361.1		2
63.16	even	3		2268.2.k.b.1297.1		2
63.23	odd	6		252.2.i.a.121.1	yes	2
63.31	odd	6		5292.2.j.b.1765.1		2
63.32	odd	6		1764.2.j.c.589.1		2
63.40	odd	6		5292.2.i.b.1549.1		2
63.41	even	6		1764.2.l.b.949.1		2
63.58	even	3	inner	756.2.i.a.37.1		2
63.59	even	6		1764.2.j.a.589.1		2
84.23	even	6		1008.2.t.b.961.1		2
252.23	even	6		1008.2.q.f.625.1		2
252.247	odd	6		3024.2.q.e.2305.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.i.a.25.1	✓	2	3.2	odd	2
252.2.i.a.121.1	yes	2	63.23	odd	6
252.2.l.a.193.1	yes	2	9.5	odd	6
252.2.l.a.205.1	yes	2	21.2	odd	6
756.2.i.a.37.1		2	63.58	even	3	inner
756.2.i.a.613.1		2	1.1	even	1	trivial
756.2.l.a.289.1		2	7.2	even	3
756.2.l.a.361.1		2	9.4	even	3
1008.2.q.f.529.1		2	12.11	even	2
1008.2.q.f.625.1		2	252.23	even	6
1008.2.t.b.193.1		2	36.23	even	6
1008.2.t.b.961.1		2	84.23	even	6
1764.2.i.b.373.1		2	63.5	even	6
1764.2.i.b.1537.1		2	21.20	even	2
1764.2.j.a.589.1		2	63.59	even	6
1764.2.j.a.1177.1		2	21.17	even	6
1764.2.j.c.589.1		2	63.32	odd	6
1764.2.j.c.1177.1		2	21.11	odd	6
1764.2.l.b.949.1		2	63.41	even	6
1764.2.l.b.961.1		2	21.5	even	6
2268.2.k.a.1297.1		2	63.2	odd	6
2268.2.k.a.1621.1		2	9.2	odd	6
2268.2.k.b.1297.1		2	63.16	even	3
2268.2.k.b.1621.1		2	9.7	even	3
3024.2.q.e.2305.1		2	252.247	odd	6
3024.2.q.e.2881.1		2	4.3	odd	2
3024.2.t.b.289.1		2	28.23	odd	6
3024.2.t.b.1873.1		2	36.31	odd	6
5292.2.i.b.1549.1		2	63.40	odd	6
5292.2.i.b.2125.1		2	7.6	odd	2
5292.2.j.b.1765.1		2	63.31	odd	6
5292.2.j.b.3529.1		2	7.3	odd	6
5292.2.j.c.1765.1		2	63.4	even	3
5292.2.j.c.3529.1		2	7.4	even	3
5292.2.l.b.361.1		2	63.13	odd	6
5292.2.l.b.3313.1		2	7.5	odd	6

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(2.00000 + 3.46410i) q^{11} +(-1.50000 - 2.59808i) q^{13} +(3.50000 - 6.06218i) q^{17} +(-2.50000 - 4.33013i) q^{19} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{29} -3.00000 q^{31} +(-4.00000 + 3.46410i) q^{35} +(-5.50000 - 9.52628i) q^{37} +(-4.50000 - 7.79423i) q^{41} +(-2.50000 + 4.33013i) q^{43} -3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(1.50000 - 2.59808i) q^{53} +8.00000 q^{55} +7.00000 q^{59} +3.00000 q^{61} -6.00000 q^{65} +13.0000 q^{67} +8.00000 q^{71} +(-3.50000 + 6.06218i) q^{73} +(-2.00000 - 10.3923i) q^{77} -9.00000 q^{79} +(0.500000 - 0.866025i) q^{83} +(-7.00000 - 12.1244i) q^{85} +(7.50000 + 12.9904i) q^{89} +(1.50000 + 7.79423i) q^{91} -10.0000 q^{95} +(8.50000 - 14.7224i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 5 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 5 * q^7	\( 2 q + 2 q^{5} - 5 q^{7} + 4 q^{11} - 3 q^{13} + 7 q^{17} - 5 q^{19} + 4 q^{23} + q^{25} - q^{29} - 6 q^{31} - 8 q^{35} - 11 q^{37} - 9 q^{41} - 5 q^{43} - 6 q^{47} + 11 q^{49} + 3 q^{53} + 16 q^{55} + 14 q^{59} + 6 q^{61} - 12 q^{65} + 26 q^{67} + 16 q^{71} - 7 q^{73} - 4 q^{77} - 18 q^{79} + q^{83} - 14 q^{85} + 15 q^{89} + 3 q^{91} - 20 q^{95} + 17 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 5 * q^7 + 4 * q^11 - 3 * q^13 + 7 * q^17 - 5 * q^19 + 4 * q^23 + q^25 - q^29 - 6 * q^31 - 8 * q^35 - 11 * q^37 - 9 * q^41 - 5 * q^43 - 6 * q^47 + 11 * q^49 + 3 * q^53 + 16 * q^55 + 14 * q^59 + 6 * q^61 - 12 * q^65 + 26 * q^67 + 16 * q^71 - 7 * q^73 - 4 * q^77 - 18 * q^79 + q^83 - 14 * q^85 + 15 * q^89 + 3 * q^91 - 20 * q^95 + 17 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more