LMFDB - Embedded newform 392.3.j.a.325.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,3,Mod(117,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("392.117");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	392.j (of order $6$, 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:	$10.6812263629$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			325.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	392.325
Dual form			392.3.j.a.117.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(1.41421 - 2.44949i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-4.24264 - 7.34847i) q^{5} -5.65685 q^{6} +8.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(1.41421 - 2.44949i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-4.24264 - 7.34847i) q^{5} -5.65685 q^{6} +8.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +(-8.48528 + 14.6969i) q^{10} +(5.65685 + 9.79796i) q^{12} -25.4558 q^{13} -24.0000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(1.00000 - 1.73205i) q^{18} +(-4.24264 - 7.34847i) q^{19} +33.9411 q^{20} +(5.00000 + 8.66025i) q^{23} +(11.3137 - 19.5959i) q^{24} +(-23.5000 + 40.7032i) q^{25} +(25.4558 + 44.0908i) q^{26} +28.2843 q^{27} +(24.0000 + 41.5692i) q^{30} +(-16.0000 + 27.7128i) q^{32} -4.00000 q^{36} +(-8.48528 + 14.6969i) q^{38} +(-36.0000 + 62.3538i) q^{39} +(-33.9411 - 58.7878i) q^{40} +(4.24264 - 7.34847i) q^{45} +(10.0000 - 17.3205i) q^{46} -45.2548 q^{48} +94.0000 q^{50} +(50.9117 - 88.1816i) q^{52} +(-28.2843 - 48.9898i) q^{54} -24.0000 q^{57} +(-38.1838 + 66.1362i) q^{59} +(48.0000 - 83.1384i) q^{60} +(-4.24264 - 7.34847i) q^{61} +64.0000 q^{64} +(108.000 + 187.061i) q^{65} +28.2843 q^{69} -110.000 q^{71} +(4.00000 + 6.92820i) q^{72} +(66.4680 + 115.126i) q^{75} +33.9411 q^{76} +144.000 q^{78} +(-65.0000 - 112.583i) q^{79} +(-67.8823 + 117.576i) q^{80} +(35.5000 - 61.4878i) q^{81} -25.4558 q^{83} -16.9706 q^{90} -40.0000 q^{92} +(-36.0000 + 62.3538i) q^{95} +(45.2548 + 78.3837i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 + 2 * q^9	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} + 2 q^{9} - 96 q^{15} - 32 q^{16} + 4 q^{18} + 20 q^{23} - 94 q^{25} + 96 q^{30} - 64 q^{32} - 16 q^{36} - 144 q^{39} + 40 q^{46} + 376 q^{50} - 96 q^{57} + 192 q^{60} + 256 q^{64} + 432 q^{65} - 440 q^{71} + 16 q^{72} + 576 q^{78} - 260 q^{79} + 142 q^{81} - 160 q^{92} - 144 q^{95}+O(q^{100}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 + 2 * q^9 - 96 * q^15 - 32 * q^16 + 4 * q^18 + 20 * q^23 - 94 * q^25 + 96 * q^30 - 64 * q^32 - 16 * q^36 - 144 * q^39 + 40 * q^46 + 376 * q^50 - 96 * q^57 + 192 * q^60 + 256 * q^64 + 432 * q^65 - 440 * q^71 + 16 * q^72 + 576 * q^78 - 260 * q^79 + 142 * q^81 - 160 * q^92 - 144 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$3$	1.41421	−	2.44949i	0.471405	−	0.816497i	−0.528060	−	0.849207i	$-0.677081\pi$
$3$	1.41421	−	2.44949i	0.471405	−	0.816497i	0.999465	+	0.0327103i	$0.0104139\pi$
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$	−4.24264	−	7.34847i	−0.848528	−	1.46969i	−0.882522	−	0.470272i	$-0.844156\pi$
$5$	−4.24264	−	7.34847i	−0.848528	−	1.46969i	0.0339935	−	0.999422i	$-0.489177\pi$
$6$	−5.65685			−0.942809
$7$		0			0
$7$		0			0
$8$	8.00000			1.00000
$9$	0.500000	+	0.866025i	0.0555556	+	0.0962250i
$10$	−8.48528	+	14.6969i	−0.848528	+	1.46969i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	5.65685	+	9.79796i	0.471405	+	0.816497i
$13$	−25.4558			−1.95814			−0.979071	−	0.203519i	$-0.934762\pi$
$13$	−25.4558			−1.95814			−0.979071	+	0.203519i	$0.934762\pi$
$14$		0			0
$15$	−24.0000			−1.60000
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	1.00000	−	1.73205i	0.0555556	−	0.0962250i
$19$	−4.24264	−	7.34847i	−0.223297	−	0.386762i	0.732510	−	0.680756i	$-0.238349\pi$
$19$	−4.24264	−	7.34847i	−0.223297	−	0.386762i	−0.955807	+	0.293994i	$0.905015\pi$
$20$	33.9411			1.69706
$21$		0			0
$22$		0			0
$23$	5.00000	+	8.66025i	0.217391	+	0.376533i	0.954010	−	0.299776i	$-0.0969119\pi$
$23$	5.00000	+	8.66025i	0.217391	+	0.376533i	−0.736618	+	0.676309i	$0.763579\pi$
$24$	11.3137	−	19.5959i	0.471405	−	0.816497i
$25$	−23.5000	+	40.7032i	−0.940000	+	1.62813i
$26$	25.4558	+	44.0908i	0.979071	+	1.69580i
$27$	28.2843			1.04757
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	24.0000	+	41.5692i	0.800000	+	1.38564i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−16.0000	+	27.7128i	−0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.00000			−0.111111
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	−8.48528	+	14.6969i	−0.223297	+	0.386762i
$39$	−36.0000	+	62.3538i	−0.923077	+	1.59882i
$40$	−33.9411	−	58.7878i	−0.848528	−	1.46969i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	4.24264	−	7.34847i	0.0942809	−	0.163299i
$46$	10.0000	−	17.3205i	0.217391	−	0.376533i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−45.2548			−0.942809
$49$		0			0
$50$	94.0000			1.88000
$51$		0			0
$52$	50.9117	−	88.1816i	0.979071	−	1.69580i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	−28.2843	−	48.9898i	−0.523783	−	0.907218i
$55$		0			0
$56$		0			0
$57$	−24.0000			−0.421053
$58$		0			0
$59$	−38.1838	+	66.1362i	−0.647182	+	1.12095i	0.336610	+	0.941644i	$0.390719\pi$
$59$	−38.1838	+	66.1362i	−0.647182	+	1.12095i	−0.983793	+	0.179309i	$0.942614\pi$
$60$	48.0000	−	83.1384i	0.800000	−	1.38564i
$61$	−4.24264	−	7.34847i	−0.0695515	−	0.120467i	0.829152	−	0.559023i	$-0.188823\pi$
$61$	−4.24264	−	7.34847i	−0.0695515	−	0.120467i	−0.898704	+	0.438556i	$0.855490\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$	108.000	+	187.061i	1.66154	+	2.87787i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	28.2843			0.409917
$70$		0			0
$71$	−110.000			−1.54930			−0.774648	−	0.632393i	$-0.782073\pi$
$71$	−110.000			−1.54930			−0.774648	+	0.632393i	$0.782073\pi$
$72$	4.00000	+	6.92820i	0.0555556	+	0.0962250i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	66.4680	+	115.126i	0.886240	+	1.53501i
$76$	33.9411			0.446594
$77$		0			0
$78$	144.000			1.84615
$79$	−65.0000	−	112.583i	−0.822785	−	1.42511i	−0.903601	−	0.428376i	$-0.859086\pi$
$79$	−65.0000	−	112.583i	−0.822785	−	1.42511i	0.0808157	−	0.996729i	$-0.474247\pi$
$80$	−67.8823	+	117.576i	−0.848528	+	1.46969i
$81$	35.5000	−	61.4878i	0.438272	−	0.759109i
$82$		0			0
$83$	−25.4558			−0.306697			−0.153348	−	0.988172i	$-0.549006\pi$
$83$	−25.4558			−0.306697			−0.153348	+	0.988172i	$0.549006\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	−16.9706			−0.188562
$91$		0			0
$92$	−40.0000			−0.434783
$93$		0			0
$94$		0			0
$95$	−36.0000	+	62.3538i	−0.378947	+	0.656356i
$96$	45.2548	+	78.3837i	0.471405	+	0.816497i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−94.0000	−	162.813i	−0.940000	−	1.62813i
$101$	80.6102	−	139.621i	0.798121	−	1.38239i	−0.122718	−	0.992442i	$-0.539161\pi$
$101$	80.6102	−	139.621i	0.798121	−	1.38239i	0.920839	−	0.389944i	$-0.127506\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	−203.647			−1.95814
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−56.5685	+	97.9796i	−0.523783	+	0.907218i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−26.0000			−0.230088			−0.115044	−	0.993360i	$-0.536701\pi$
$113$	−26.0000			−0.230088			−0.115044	+	0.993360i	$0.536701\pi$
$114$	24.0000	+	41.5692i	0.210526	+	0.364642i
$115$	42.4264	−	73.4847i	0.368925	−	0.638997i
$116$		0			0
$117$	−12.7279	−	22.0454i	−0.108786	−	0.188422i
$118$	152.735			1.29436
$119$		0			0
$120$	−192.000			−1.60000
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$	−8.48528	+	14.6969i	−0.0695515	+	0.120467i
$123$		0			0
$124$		0			0
$125$	186.676			1.49341
$126$		0			0
$127$	−250.000			−1.96850			−0.984252	−	0.176771i	$-0.943435\pi$
$127$	−250.000			−1.96850			−0.984252	+	0.176771i	$0.943435\pi$
$128$	−64.0000	−	110.851i	−0.500000	−	0.866025i
$129$		0			0
$130$	216.000	−	374.123i	1.66154	−	2.87787i
$131$	−123.037	−	213.106i	−0.939211	−	1.62676i	−0.766948	−	0.641709i	$-0.778226\pi$
$131$	−123.037	−	213.106i	−0.939211	−	1.62676i	−0.172263	−	0.985051i	$-0.555108\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−120.000	−	207.846i	−0.888889	−	1.53960i
$136$		0			0
$137$	−25.0000	+	43.3013i	−0.182482	+	0.316068i	−0.942725	−	0.333571i	$-0.891746\pi$
$137$	−25.0000	+	43.3013i	−0.182482	+	0.316068i	0.760243	+	0.649638i	$0.225080\pi$
$138$	−28.2843	−	48.9898i	−0.204958	−	0.354999i
$139$	−263.044			−1.89240			−0.946200	−	0.323581i	$-0.895113\pi$
$139$	−263.044			−1.89240			−0.946200	+	0.323581i	$0.895113\pi$
$140$		0			0
$141$		0			0
$142$	110.000	+	190.526i	0.774648	+	1.34173i
$143$		0			0
$144$	8.00000	−	13.8564i	0.0555556	−	0.0962250i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	132.936	−	230.252i	0.886240	−	1.53501i
$151$	101.000	−	174.937i	0.668874	−	1.15852i	−0.309345	−	0.950950i	$-0.600110\pi$
$151$	101.000	−	174.937i	0.668874	−	1.15852i	0.978219	−	0.207574i	$-0.0665568\pi$
$152$	−33.9411	−	58.7878i	−0.223297	−	0.386762i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−144.000	−	249.415i	−0.923077	−	1.59882i
$157$	−156.978	+	271.893i	−0.999858	+	1.73180i	−0.485335	+	0.874328i	$0.661302\pi$
$157$	−156.978	+	271.893i	−0.999858	+	1.73180i	−0.514523	+	0.857476i	$0.672031\pi$
$158$	−130.000	+	225.167i	−0.822785	+	1.42511i
$159$		0			0
$160$	271.529			1.69706
$161$		0			0
$162$	−142.000			−0.876543
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	25.4558	+	44.0908i	0.153348	+	0.265607i
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	479.000			2.83432
$170$		0			0
$171$	4.24264	−	7.34847i	0.0248108	−	0.0429735i
$172$		0			0
$173$	114.551	+	198.409i	0.662146	+	1.14687i	0.980051	+	0.198748i	$0.0636876\pi$
$173$	114.551	+	198.409i	0.662146	+	1.14687i	−0.317904	+	0.948123i	$0.602979\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	108.000	+	187.061i	0.610169	+	1.05684i
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	16.9706	+	29.3939i	0.0942809	+	0.163299i
$181$	−263.044			−1.45328			−0.726640	−	0.687018i	$-0.758919\pi$
$181$	−263.044			−1.45328			−0.726640	+	0.687018i	$0.758919\pi$
$182$		0			0
$183$	−24.0000			−0.131148
$184$	40.0000	+	69.2820i	0.217391	+	0.376533i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	144.000			0.757895
$191$	−65.0000	−	112.583i	−0.340314	−	0.589441i	0.644177	−	0.764877i	$-0.277200\pi$
$191$	−65.0000	−	112.583i	−0.340314	−	0.589441i	−0.984491	+	0.175435i	$0.943867\pi$
$192$	90.5097	−	156.767i	0.471405	−	0.816497i
$193$	157.000	−	271.932i	0.813472	−	1.40897i	−0.0969488	−	0.995289i	$-0.530908\pi$
$193$	157.000	−	271.932i	0.813472	−	1.40897i	0.910420	−	0.413685i	$-0.135758\pi$
$194$		0			0
$195$	610.940			3.13303
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−188.000	+	325.626i	−0.940000	+	1.62813i
$201$		0			0
$202$	−322.441			−1.59624
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.00000	+	8.66025i	−0.0241546	+	0.0418370i
$208$	203.647	+	352.727i	0.979071	+	1.69580i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−155.563	+	269.444i	−0.730345	+	1.26499i
$214$		0			0
$215$		0			0
$216$	226.274			1.04757
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−47.0000			−0.208889
$226$	26.0000	+	45.0333i	0.115044	+	0.199262i
$227$	199.404	−	345.378i	0.878432	−	1.52149i	0.0253712	−	0.999678i	$-0.491923\pi$
$227$	199.404	−	345.378i	0.878432	−	1.52149i	0.853061	−	0.521811i	$-0.174743\pi$
$228$	48.0000	−	83.1384i	0.210526	−	0.364642i
$229$	−123.037	−	213.106i	−0.537278	−	0.930592i	−0.999049	−	0.0435934i	$-0.986119\pi$
$229$	−123.037	−	213.106i	−0.537278	−	0.930592i	0.461772	−	0.886999i	$-0.347214\pi$
$230$	−169.706			−0.737851
$231$		0			0
$232$		0			0
$233$	215.000	+	372.391i	0.922747	+	1.59824i	0.795145	+	0.606419i	$0.207394\pi$
$233$	215.000	+	372.391i	0.922747	+	1.59824i	0.127601	+	0.991826i	$0.459272\pi$
$234$	−25.4558	+	44.0908i	−0.108786	+	0.188422i
$235$		0			0
$236$	−152.735	−	264.545i	−0.647182	−	1.12095i
$237$	−367.696			−1.55146
$238$		0			0
$239$	422.000			1.76569			0.882845	−	0.469664i	$-0.155625\pi$
$239$	422.000			1.76569			0.882845	+	0.469664i	$0.155625\pi$
$240$	192.000	+	332.554i	0.800000	+	1.38564i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−121.000	+	209.578i	−0.500000	+	0.866025i
$243$	26.8701	+	46.5403i	0.110576	+	0.191524i
$244$	33.9411			0.139103
$245$		0			0
$246$		0			0
$247$	108.000	+	187.061i	0.437247	+	0.757334i
$248$		0			0
$249$	−36.0000	+	62.3538i	−0.144578	+	0.250417i
$250$	−186.676	−	323.333i	−0.746705	−	1.29333i
$251$	−500.632			−1.99455			−0.997274	−	0.0737859i	$-0.976492\pi$
$251$	−500.632			−1.99455			−0.997274	+	0.0737859i	$0.976492\pi$
$252$		0			0
$253$		0			0
$254$	250.000	+	433.013i	0.984252	+	1.70477i
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$	−864.000			−3.32308
$261$		0			0
$262$	−246.073	+	426.211i	−0.939211	+	1.62676i
$263$	−137.000	+	237.291i	−0.520913	+	0.902247i	0.478792	+	0.877929i	$0.341075\pi$
$263$	−137.000	+	237.291i	−0.520913	+	0.902247i	−0.999704	+	0.0243185i	$0.992258\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	80.6102	−	139.621i	0.299666	−	0.519037i	−0.676393	−	0.736541i	$-0.736458\pi$
$269$	80.6102	−	139.621i	0.299666	−	0.519037i	0.976060	+	0.217504i	$0.0697914\pi$
$270$	−240.000	+	415.692i	−0.888889	+	1.53960i
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$	100.000			0.364964
$275$		0			0
$276$	−56.5685	+	97.9796i	−0.204958	+	0.354999i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	263.044	+	455.605i	0.946200	+	1.63887i
$279$		0			0
$280$		0			0
$281$	338.000			1.20285			0.601423	−	0.798930i	$-0.294600\pi$
$281$	338.000			1.20285			0.601423	+	0.798930i	$0.294600\pi$
$282$		0			0
$283$	−156.978	+	271.893i	−0.554692	+	0.960754i	0.443236	+	0.896405i	$0.353830\pi$
$283$	−156.978	+	271.893i	−0.554692	+	0.960754i	−0.997927	+	0.0643489i	$0.979503\pi$
$284$	220.000	−	381.051i	0.774648	−	1.34173i
$285$	101.823	+	176.363i	0.357275	+	0.618818i
$286$		0			0
$287$		0			0
$288$	−32.0000			−0.111111
$289$	−144.500	−	250.281i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	449.720			1.53488			0.767440	−	0.641121i	$-0.221530\pi$
$293$	449.720			1.53488			0.767440	+	0.641121i	$0.221530\pi$
$294$		0			0
$295$	648.000			2.19661
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−127.279	−	220.454i	−0.425683	−	0.737305i
$300$	−531.744			−1.77248
$301$		0			0
$302$	−404.000			−1.33775
$303$	−228.000	−	394.908i	−0.752475	−	1.30333i
$304$	−67.8823	+	117.576i	−0.223297	+	0.386762i
$305$	−36.0000	+	62.3538i	−0.118033	+	0.204439i
$306$		0			0
$307$	449.720			1.46489			0.732443	−	0.680829i	$-0.238380\pi$
$307$	449.720			1.46489			0.732443	+	0.680829i	$0.238380\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$	−288.000	+	498.831i	−0.923077	+	1.59882i
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	627.911			1.99972
$315$		0			0
$316$	520.000			1.64557
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$	−271.529	−	470.302i	−0.848528	−	1.46969i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	142.000	+	245.951i	0.438272	+	0.759109i
$325$	598.212	−	1036.13i	1.84065	−	3.18811i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	50.9117	−	88.1816i	0.153348	−	0.265607i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−26.0000			−0.0771513			−0.0385757	−	0.999256i	$-0.512282\pi$
$337$	−26.0000			−0.0771513			−0.0385757	+	0.999256i	$0.512282\pi$
$338$	−479.000	−	829.652i	−1.41716	−	2.45459i
$339$	−36.7696	+	63.6867i	−0.108465	+	0.187866i
$340$		0			0
$341$		0			0
$342$	−16.9706			−0.0496215
$343$		0			0
$344$		0			0
$345$	−120.000	−	207.846i	−0.347826	−	0.602452i
$346$	229.103	−	396.817i	0.662146	−	1.14687i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	687.308			1.96936			0.984682	−	0.174362i	$-0.0557862\pi$
$349$	687.308			1.96936			0.984682	+	0.174362i	$0.0557862\pi$
$350$		0			0
$351$	−720.000			−2.05128
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$	216.000	−	374.123i	0.610169	−	1.05684i
$355$	466.690	+	808.332i	1.31462	+	2.27699i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	341.000	+	590.629i	0.949861	+	1.64521i	0.745713	+	0.666267i	$0.232109\pi$
$359$	341.000	+	590.629i	0.949861	+	1.64521i	0.204147	+	0.978940i	$0.434558\pi$
$360$	33.9411	−	58.7878i	0.0942809	−	0.163299i
$361$	144.500	−	250.281i	0.400277	−	0.693300i
$362$	263.044	+	455.605i	0.726640	+	1.25858i
$363$	−342.240			−0.942809
$364$		0			0
$365$		0			0
$366$	24.0000	+	41.5692i	0.0655738	+	0.113577i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	80.0000	−	138.564i	0.217391	−	0.376533i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	264.000	−	457.261i	0.704000	−	1.21936i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−144.000	−	249.415i	−0.378947	−	0.656356i
$381$	−353.553	+	612.372i	−0.927962	+	1.60728i
$382$	−130.000	+	225.167i	−0.340314	+	0.589441i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	−362.039			−0.942809
$385$		0			0
$386$	−628.000			−1.62694
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$	−610.940	−	1058.18i	−1.56651	−	2.71328i
$391$		0			0
$392$		0			0
$393$	−696.000			−1.77099
$394$		0			0
$395$	−551.543	+	955.301i	−1.39631	+	2.41848i
$396$		0			0
$397$	−241.831	−	418.863i	−0.609145	−	1.05507i	−0.991382	−	0.131005i	$-0.958179\pi$
$397$	−241.831	−	418.863i	−0.609145	−	1.05507i	0.382237	−	0.924064i	$-0.375154\pi$
$398$		0			0
$399$		0			0
$400$	752.000			1.88000
$401$	−275.000	−	476.314i	−0.685786	−	1.18782i	−0.973189	−	0.230006i	$-0.926125\pi$
$401$	−275.000	−	476.314i	−0.685786	−	1.18782i	0.287404	−	0.957810i	$-0.407208\pi$
$402$		0			0
$403$		0			0
$404$	322.441	+	558.484i	0.798121	+	1.38239i
$405$	−602.455			−1.48754
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	70.7107	+	122.474i	0.172045	+	0.297991i
$412$		0			0
$413$		0			0
$414$	20.0000			0.0483092
$415$	108.000	+	187.061i	0.260241	+	0.450751i
$416$	407.294	−	705.453i	0.979071	−	1.69580i
$417$	−372.000	+	644.323i	−0.892086	+	1.54514i
$418$		0			0
$419$	−500.632			−1.19482			−0.597412	−	0.801934i	$-0.703804\pi$
$419$	−500.632			−1.19482			−0.597412	+	0.801934i	$0.703804\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	622.254			1.46069
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	269.000	−	465.922i	0.624130	−	1.08102i	−0.364578	−	0.931173i	$-0.618787\pi$
$431$	269.000	−	465.922i	0.624130	−	1.08102i	0.988708	−	0.149852i	$-0.0478798\pi$
$432$	−226.274	−	391.918i	−0.523783	−	0.907218i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	42.4264	−	73.4847i	0.0970856	−	0.168157i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.00000			0.00445434			0.00222717	−	0.999998i	$-0.499291\pi$
$449$	2.00000			0.00445434			0.00222717	+	0.999998i	$0.499291\pi$
$450$	47.0000	+	81.4064i	0.104444	+	0.180903i
$451$		0			0
$452$	52.0000	−	90.0666i	0.115044	−	0.199262i
$453$	−285.671	−	494.797i	−0.630621	−	1.09227i
$454$	−797.616			−1.75686
$455$		0			0
$456$	−192.000			−0.421053
$457$	−443.000	−	767.299i	−0.969365	−	1.67899i	−0.697398	−	0.716684i	$-0.745659\pi$
$457$	−443.000	−	767.299i	−0.969365	−	1.67899i	−0.271967	−	0.962307i	$-0.587674\pi$
$458$	−246.073	+	426.211i	−0.537278	+	0.930592i
$459$		0			0
$460$	169.706	+	293.939i	0.368925	+	0.638997i
$461$	−263.044			−0.570594			−0.285297	−	0.958439i	$-0.592092\pi$
$461$	−263.044			−0.570594			−0.285297	+	0.958439i	$0.592092\pi$
$462$		0			0
$463$	226.000			0.488121			0.244060	−	0.969760i	$-0.421520\pi$
$463$	226.000			0.488121			0.244060	+	0.969760i	$0.421520\pi$
$464$		0			0
$465$		0			0
$466$	430.000	−	744.782i	0.922747	−	1.59824i
$467$	−241.831	−	418.863i	−0.517838	−	0.896922i	−0.999785	−	0.0207219i	$-0.993404\pi$
$467$	−241.831	−	418.863i	−0.517838	−	0.896922i	0.481947	−	0.876200i	$-0.339930\pi$
$468$	101.823			0.217571
$469$		0			0
$470$		0			0
$471$	444.000	+	769.031i	0.942675	+	1.63276i
$472$	−305.470	+	529.090i	−0.647182	+	1.12095i
$473$		0			0
$474$	367.696	+	636.867i	0.775729	+	1.34360i
$475$	398.808			0.839596
$476$		0			0
$477$		0			0
$478$	−422.000	−	730.925i	−0.882845	−	1.52913i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$	384.000	−	665.108i	0.800000	−	1.38564i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	484.000			1.00000
$485$		0			0
$486$	53.7401	−	93.0806i	0.110576	−	0.191524i
$487$	−235.000	+	407.032i	−0.482546	+	0.835795i	−0.999799	−	0.0200380i	$-0.993621\pi$
$487$	−235.000	+	407.032i	−0.482546	+	0.835795i	0.517253	+	0.855833i	$0.326955\pi$
$488$	−33.9411	−	58.7878i	−0.0695515	−	0.120467i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	216.000	−	374.123i	0.437247	−	0.757334i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	144.000			0.289157
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−373.352	+	646.665i	−0.746705	+	1.29333i
$501$		0			0
$502$	500.632	+	867.119i	0.997274	+	1.72733i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−1368.00			−2.70891
$506$		0			0
$507$	677.408	−	1173.31i	1.33611	−	2.31421i
$508$	500.000	−	866.025i	0.984252	−	1.70477i
$509$	470.933	+	815.680i	0.925212	+	1.60251i	0.791219	+	0.611533i	$0.209447\pi$
$509$	470.933	+	815.680i	0.925212	+	1.60251i	0.133993	+	0.990982i	$0.457220\pi$
$510$		0			0
$511$		0			0
$512$	512.000			1.00000
$513$	−120.000	−	207.846i	−0.233918	−	0.405158i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	648.000			1.24855
$520$	864.000	+	1496.49i	1.66154	+	2.87787i
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−479.418	−	830.377i	−0.916670	−	1.58772i	−0.804438	−	0.594037i	$-0.797533\pi$
$523$	−479.418	−	830.377i	−0.916670	−	1.58772i	−0.112232	−	0.993682i	$-0.535800\pi$
$524$	984.293			1.87842
$525$		0			0
$526$	548.000			1.04183
$527$		0			0
$528$		0			0
$529$	214.500	−	371.525i	0.405482	−	0.702315i
$530$		0			0
$531$	−76.3675			−0.143818
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−322.441			−0.599332
$539$		0			0
$540$	960.000			1.77778
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$	−372.000	+	644.323i	−0.685083	+	1.18660i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−100.000	−	173.205i	−0.182482	−	0.316068i
$549$	4.24264	−	7.34847i	0.00772794	−	0.0133852i
$550$		0			0
$551$		0			0
$552$	226.274			0.409917
$553$		0			0
$554$		0			0
$555$		0			0
$556$	526.087	−	911.210i	0.946200	−	1.63887i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−338.000	−	585.433i	−0.601423	−	1.04170i
$563$	199.404	−	345.378i	0.354181	−	0.613460i	−0.632796	−	0.774318i	$-0.718093\pi$
$563$	199.404	−	345.378i	0.354181	−	0.613460i	0.986978	+	0.160858i	$0.0514262\pi$
$564$		0			0
$565$	110.309	+	191.060i	0.195237	+	0.338160i
$566$	627.911			1.10938
$567$		0			0
$568$	−880.000			−1.54930
$569$	565.000	+	978.609i	0.992970	+	1.71987i	0.598992	+	0.800755i	$0.295568\pi$
$569$	565.000	+	978.609i	0.992970	+	1.71987i	0.393978	+	0.919120i	$0.371099\pi$
$570$	203.647	−	352.727i	0.357275	−	0.618818i
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−367.696			−0.641702
$574$		0			0
$575$	−470.000			−0.817391
$576$	32.0000	+	55.4256i	0.0555556	+	0.0962250i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	−289.000	+	500.563i	−0.500000	+	0.866025i
$579$	−444.063	−	769.140i	−0.766948	−	1.32839i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−108.000	+	187.061i	−0.184615	+	0.319763i
$586$	−449.720	−	778.938i	−0.767440	−	1.32925i
$587$	1162.48			1.98038			0.990190	−	0.139725i	$-0.0446217\pi$
$587$	1162.48			1.98038			0.990190	+	0.139725i	$0.0446217\pi$
$588$		0			0
$589$		0			0
$590$	−648.000	−	1122.37i	−1.09831	−	1.90232i
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−254.558	+	440.908i	−0.425683	+	0.737305i
$599$	535.000	−	926.647i	0.893155	−	1.54699i	0.0570840	−	0.998369i	$-0.481820\pi$
$599$	535.000	−	926.647i	0.893155	−	1.54699i	0.836071	−	0.548621i	$-0.184847\pi$
$600$	531.744	+	921.008i	0.886240	+	1.53501i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	404.000	+	699.749i	0.668874	+	1.15852i
$605$	−513.360	+	889.165i	−0.848528	+	1.46969i
$606$	−456.000	+	789.815i	−0.752475	+	1.30333i
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	271.529			0.446594
$609$		0			0
$610$	144.000			0.236066
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−449.720	−	778.938i	−0.732443	−	1.26863i
$615$		0			0
$616$		0			0
$617$	−1034.00			−1.67585			−0.837925	−	0.545785i	$-0.816232\pi$
$617$	−1034.00			−1.67585			−0.837925	+	0.545785i	$0.816232\pi$
$618$		0			0
$619$	555.786	−	962.649i	0.897877	−	1.55517i	0.0676742	−	0.997707i	$-0.478442\pi$
$619$	555.786	−	962.649i	0.897877	−	1.55517i	0.830203	−	0.557461i	$-0.188225\pi$
$620$		0			0
$621$	141.421	+	244.949i	0.227732	+	0.394443i
$622$		0			0
$623$		0			0
$624$	1152.00			1.84615
$625$	−204.500	−	354.204i	−0.327200	−	0.566727i
$626$		0			0
$627$		0			0
$628$	−627.911	−	1087.57i	−0.999858	−	1.73180i
$629$		0			0
$630$		0			0
$631$	−110.000			−0.174326			−0.0871632	−	0.996194i	$-0.527780\pi$
$631$	−110.000			−0.174326			−0.0871632	+	0.996194i	$0.527780\pi$
$632$	−520.000	−	900.666i	−0.822785	−	1.42511i
$633$		0			0
$634$		0			0
$635$	1060.66	+	1837.12i	1.67033	+	2.89310i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−55.0000	−	95.2628i	−0.0860720	−	0.149081i
$640$	−543.058	+	940.604i	−0.848528	+	1.46969i
$641$	−515.000	+	892.006i	−0.803432	+	1.39159i	0.113912	+	0.993491i	$0.463662\pi$
$641$	−515.000	+	892.006i	−0.803432	+	1.39159i	−0.917344	+	0.398094i	$0.869672\pi$
$642$		0			0
$643$	−738.219			−1.14809			−0.574043	−	0.818825i	$-0.694626\pi$
$643$	−738.219			−1.14809			−0.574043	+	0.818825i	$0.694626\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	284.000	−	491.902i	0.438272	−	0.759109i
$649$		0			0
$650$	−2392.85			−3.68131
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	−1044.00	+	1808.26i	−1.59389	+	2.76070i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−632.153	+	1094.92i	−0.956359	+	1.65646i	−0.225133	+	0.974328i	$0.572282\pi$
$661$	−632.153	+	1094.92i	−0.956359	+	1.65646i	−0.731227	+	0.682135i	$0.761052\pi$
$662$		0			0
$663$		0			0
$664$	−203.647			−0.306697
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−670.000			−0.995542			−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000			−0.995542			−0.497771	+	0.867308i	$0.665848\pi$
$674$	26.0000	+	45.0333i	0.0385757	+	0.0668150i
$675$	−664.680	+	1151.26i	−0.984712	+	1.70557i
$676$	−958.000	+	1659.30i	−1.41716	+	2.45459i
$677$	−241.831	−	418.863i	−0.357209	−	0.618704i	0.630284	−	0.776364i	$-0.282938\pi$
$677$	−241.831	−	418.863i	−0.357209	−	0.618704i	−0.987493	+	0.157660i	$0.949605\pi$
$678$	147.078			0.216930
$679$		0			0
$680$		0			0
$681$	−564.000	−	976.877i	−0.828194	−	1.43447i
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	16.9706	+	29.3939i	0.0248108	+	0.0429735i
$685$	424.264			0.619364
$686$		0			0
$687$	−696.000			−1.01310
$688$		0			0
$689$		0			0
$690$	−240.000	+	415.692i	−0.347826	+	0.602452i
$691$	−123.037	−	213.106i	−0.178056	−	0.308402i	0.763159	−	0.646211i	$-0.223647\pi$
$691$	−123.037	−	213.106i	−0.178056	−	0.308402i	−0.941215	+	0.337809i	$0.890314\pi$
$692$	−916.410			−1.32429
$693$		0			0
$694$		0			0
$695$	1116.00	+	1932.97i	1.60576	+	2.78125i
$696$		0			0
$697$		0			0
$698$	−687.308	−	1190.45i	−0.984682	−	1.70552i
$699$	1216.22			1.73995
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	720.000	+	1247.08i	1.02564	+	1.77646i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−864.000			−1.22034
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	933.381	−	1616.66i	1.31462	−	2.27699i
$711$	65.0000	−	112.583i	0.0914205	−	0.158345i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	596.798	−	1033.68i	0.832354	−	1.44168i
$718$	682.000	−	1181.26i	0.949861	−	1.64521i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−135.765			−0.188562
$721$		0			0
$722$	−578.000			−0.800554
$723$		0			0
$724$	526.087	−	911.210i	0.726640	−	1.25858i
$725$		0			0
$726$	342.240	+	592.777i	0.471405	+	0.816497i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	791.000			1.08505
$730$		0			0
$731$		0			0
$732$	48.0000	−	83.1384i	0.0655738	−	0.113577i
$733$	708.521	+	1227.19i	0.966604	+	1.67421i	0.705241	+	0.708967i	$0.250839\pi$
$733$	708.521	+	1227.19i	0.966604	+	1.67421i	0.261363	+	0.965241i	$0.415828\pi$
$734$		0			0
$735$		0			0
$736$	−320.000			−0.434783
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$	610.940			0.824481
$742$		0			0
$743$	1430.00			1.92463			0.962315	−	0.271937i	$-0.0876644\pi$
$743$	1430.00			1.92463			0.962315	+	0.271937i	$0.0876644\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−12.7279	−	22.0454i	−0.0170387	−	0.0295119i
$748$		0			0
$749$		0			0
$750$	−1056.00			−1.40800
$751$	−499.000	−	864.293i	−0.664447	−	1.15086i	−0.979435	−	0.201761i	$-0.935334\pi$
$751$	−499.000	−	864.293i	−0.664447	−	1.15086i	0.314987	−	0.949096i	$-0.398000\pi$
$752$		0			0
$753$	−708.000	+	1226.29i	−0.940239	+	1.62854i
$754$		0			0
$755$	−1714.03			−2.27023
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−288.000	+	498.831i	−0.378947	+	0.656356i
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$	1414.21			1.85592
$763$		0			0
$764$	520.000			0.680628
$765$		0			0
$766$		0			0
$767$	972.000	−	1683.55i	1.26728	−	2.19498i
$768$	362.039	+	627.069i	0.471405	+	0.816497i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	628.000	+	1087.73i	0.813472	+	1.40897i
$773$	436.992	−	756.892i	0.565320	−	0.979162i	−0.431700	−	0.902017i	$-0.642086\pi$
$773$	436.992	−	756.892i	0.565320	−	0.979162i	0.997020	−	0.0771451i	$-0.0245805\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−1221.88	+	2116.36i	−1.56651	+	2.71328i
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2664.00			3.39363
$786$	696.000	+	1205.51i	0.885496	+	1.53372i
$787$	−750.947	+	1300.68i	−0.954190	+	1.65271i	−0.217979	+	0.975954i	$0.569946\pi$
$787$	−750.947	+	1300.68i	−0.954190	+	1.65271i	−0.736211	+	0.676752i	$0.763387\pi$
$788$		0			0
$789$	387.495	+	671.160i	0.491121	+	0.850647i
$790$	2206.17			2.79262
$791$		0			0
$792$		0			0
$793$	108.000	+	187.061i	0.136192	+	0.235891i
$794$	−483.661	+	837.725i	−0.609145	+	1.05507i
$795$		0			0
$796$		0			0
$797$	449.720			0.564266			0.282133	−	0.959375i	$-0.408958\pi$
$797$	449.720			0.564266			0.282133	+	0.959375i	$0.408958\pi$
$798$		0			0
$799$		0			0
$800$	−752.000	−	1302.50i	−0.940000	−	1.62813i
$801$		0			0
$802$	−550.000	+	952.628i	−0.685786	+	1.18782i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−228.000	−	394.908i	−0.282528	−	0.489353i
$808$	644.881	−	1116.97i	0.798121	−	1.38239i
$809$	325.000	−	562.917i	0.401731	−	0.695818i	−0.592204	−	0.805788i	$-0.701742\pi$
$809$	325.000	−	562.917i	0.401731	−	0.695818i	0.993935	+	0.109970i	$0.0350755\pi$
$810$	602.455	+	1043.48i	0.743772	+	1.28825i
$811$	−1450.98			−1.78913			−0.894564	−	0.446939i	$-0.852514\pi$
$811$	−1450.98			−1.78913			−0.894564	+	0.446939i	$0.852514\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	141.421	−	244.949i	0.172045	−	0.297991i
$823$	−473.000	+	819.260i	−0.574727	+	0.995456i	0.421345	+	0.906901i	$0.361558\pi$
$823$	−473.000	+	819.260i	−0.574727	+	0.995456i	−0.996071	+	0.0885551i	$0.971775\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−20.0000	−	34.6410i	−0.0241546	−	0.0418370i
$829$	−38.1838	+	66.1362i	−0.0460600	+	0.0797783i	−0.888136	−	0.459580i	$-0.848000\pi$
$829$	−38.1838	+	66.1362i	−0.0460600	+	0.0797783i	0.842076	+	0.539358i	$0.181333\pi$
$830$	216.000	−	374.123i	0.260241	−	0.450751i
$831$		0			0
$832$	−1629.17			−1.95814
$833$		0			0
$834$	1488.00			1.78417
$835$		0			0
$836$		0			0
$837$		0			0
$838$	500.632	+	867.119i	0.597412	+	1.03475i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	478.004	−	827.928i	0.567028	−	0.982120i
$844$		0			0
$845$	−2032.22	−	3519.92i	−2.40500	−	4.16558i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	444.000	+	769.031i	0.522968	+	0.905807i
$850$		0			0
$851$		0			0
$852$	−622.254	−	1077.78i	−0.730345	−	1.26499i
$853$	449.720			0.527221			0.263611	−	0.964629i	$-0.415087\pi$
$853$	449.720			0.527221			0.263611	+	0.964629i	$0.415087\pi$
$854$		0			0
$855$	−72.0000			−0.0842105
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−598.212	−	1036.13i	−0.696406	−	1.20621i	−0.969705	−	0.244281i	$-0.921448\pi$
$859$	−598.212	−	1036.13i	−0.696406	−	1.20621i	0.273299	−	0.961929i	$-0.411885\pi$
$860$		0			0
$861$		0			0
$862$	−1076.00			−1.24826
$863$	−737.000	−	1276.52i	−0.853998	−	1.47917i	−0.877572	−	0.479445i	$-0.840838\pi$
$863$	−737.000	−	1276.52i	−0.853998	−	1.47917i	0.0235741	−	0.999722i	$-0.492495\pi$
$864$	−452.548	+	783.837i	−0.523783	+	0.907218i
$865$	972.000	−	1683.55i	1.12370	−	1.94630i
$866$		0			0
$867$	−817.415			−0.942809
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−169.706			−0.194171
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	636.000	−	1101.58i	0.723549	−	1.25322i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	916.410	−	1587.27i	1.03549	−	1.79352i
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−720.000			−0.802676
$898$	−2.00000	−	3.46410i	−0.00222717	−	0.00385757i
$899$		0			0
$900$	94.0000	−	162.813i	0.104444	−	0.180903i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−208.000			−0.230088
$905$	1116.00	+	1932.97i	1.23315	+	2.13588i
$906$	−571.342	+	989.594i	−0.630621	+	1.09227i
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	797.616	+	1381.51i	0.878432	+	1.52149i
$909$	161.220			0.177360
$910$		0			0
$911$	−922.000			−1.01207			−0.506037	−	0.862512i	$-0.668890\pi$
$911$	−922.000			−1.01207			−0.506037	+	0.862512i	$0.668890\pi$
$912$	192.000	+	332.554i	0.210526	+	0.364642i
$913$		0			0
$914$	−886.000	+	1534.60i	−0.969365	+	1.67899i
$915$	101.823	+	176.363i	0.111282	+	0.192747i
$916$	984.293			1.07456
$917$		0			0
$918$		0			0
$919$	775.000	+	1342.34i	0.843308	+	1.46065i	0.887083	+	0.461611i	$0.152728\pi$
$919$	775.000	+	1342.34i	0.843308	+	1.46065i	−0.0437746	+	0.999041i	$0.513938\pi$
$920$	339.411	−	587.878i	0.368925	−	0.638997i
$921$	636.000	−	1101.58i	0.690554	−	1.19607i
$922$	263.044	+	455.605i	0.285297	+	0.494149i
$923$	2800.14			3.03374
$924$		0			0
$925$		0			0
$926$	−226.000	−	391.443i	−0.244060	−	0.422725i
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	−1720.00			−1.84549
$933$		0			0
$934$	−483.661	+	837.725i	−0.517838	+	0.896922i
$935$		0			0
$936$	−101.823	−	176.363i	−0.108786	−	0.188422i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	555.786	−	962.649i	0.590633	−	1.02301i	−0.403514	−	0.914973i	$-0.632211\pi$
$941$	555.786	−	962.649i	0.590633	−	1.02301i	0.994147	−	0.108033i	$-0.0344553\pi$
$942$	888.000	−	1538.06i	0.942675	−	1.63276i
$943$		0			0
$944$	1221.88			1.29436
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	735.391	−	1273.73i	0.775729	−	1.34360i
$949$		0			0
$950$	−398.808	−	690.756i	−0.419798	−	0.727112i
$951$		0			0
$952$		0			0
$953$	1010.00			1.05981			0.529906	−	0.848057i	$-0.322227\pi$
$953$	1010.00			1.05981			0.529906	+	0.848057i	$0.322227\pi$
$954$		0			0
$955$	−551.543	+	955.301i	−0.577532	+	1.00032i
$956$	−844.000	+	1461.85i	−0.882845	+	1.52913i
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1536.00			−1.60000
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2664.38			−2.76101
$966$		0			0
$967$	1430.00			1.47880			0.739400	−	0.673266i	$-0.235109\pi$
$967$	1430.00			1.47880			0.739400	+	0.673266i	$0.235109\pi$
$968$	−484.000	−	838.313i	−0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$	−4.24264	−	7.34847i	−0.00436935	−	0.00756794i	0.863832	−	0.503779i	$-0.168057\pi$
$971$	−4.24264	−	7.34847i	−0.00436935	−	0.00756794i	−0.868202	+	0.496211i	$0.834724\pi$
$972$	−214.960			−0.221153
$973$		0			0
$974$	940.000			0.965092
$975$	−1692.00	−	2930.63i	−1.73538	−	3.00577i
$976$	−67.8823	+	117.576i	−0.0695515	+	0.120467i
$977$	815.000	−	1411.62i	0.834186	−	1.44485i	−0.0605049	−	0.998168i	$-0.519271\pi$
$977$	815.000	−	1411.62i	0.834186	−	1.44485i	0.894691	−	0.446685i	$-0.147396\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−864.000			−0.874494
$989$		0			0
$990$		0			0
$991$	−305.000	+	528.275i	−0.307770	+	0.533073i	−0.977874	−	0.209194i	$-0.932916\pi$
$991$	−305.000	+	528.275i	−0.307770	+	0.533073i	0.670104	+	0.742267i	$0.266249\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−144.000	−	249.415i	−0.144578	−	0.250417i
$997$	−988.535	+	1712.19i	−0.991510	+	1.71735i	−0.383144	+	0.923689i	$0.625159\pi$
$997$	−988.535	+	1712.19i	−0.991510	+	1.71735i	−0.608366	+	0.793657i	$0.708175\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.3.j.a.325.2		4
7.2	even	3	inner	392.3.j.a.117.2		4
7.3	odd	6		56.3.h.c.13.2	yes	2
7.4	even	3		56.3.h.c.13.1	✓	2
7.5	odd	6	inner	392.3.j.a.117.1		4
7.6	odd	2	inner	392.3.j.a.325.1		4
8.5	even	2	inner	392.3.j.a.325.1		4
21.11	odd	6		504.3.l.a.181.1		2
21.17	even	6		504.3.l.a.181.2		2
28.3	even	6		224.3.h.c.209.1		2
28.11	odd	6		224.3.h.c.209.2		2
56.3	even	6		224.3.h.c.209.2		2
56.5	odd	6	inner	392.3.j.a.117.2		4
56.11	odd	6		224.3.h.c.209.1		2
56.13	odd	2	CM	392.3.j.a.325.2		4
56.37	even	6	inner	392.3.j.a.117.1		4
56.45	odd	6		56.3.h.c.13.1	✓	2
56.53	even	6		56.3.h.c.13.2	yes	2
84.11	even	6		2016.3.l.c.433.1		2
84.59	odd	6		2016.3.l.c.433.2		2
168.11	even	6		2016.3.l.c.433.2		2
168.53	odd	6		504.3.l.a.181.2		2
168.59	odd	6		2016.3.l.c.433.1		2
168.101	even	6		504.3.l.a.181.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.c.13.1	✓	2	7.4	even	3
56.3.h.c.13.1	✓	2	56.45	odd	6
56.3.h.c.13.2	yes	2	7.3	odd	6
56.3.h.c.13.2	yes	2	56.53	even	6
224.3.h.c.209.1		2	28.3	even	6
224.3.h.c.209.1		2	56.11	odd	6
224.3.h.c.209.2		2	28.11	odd	6
224.3.h.c.209.2		2	56.3	even	6
392.3.j.a.117.1		4	7.5	odd	6	inner
392.3.j.a.117.1		4	56.37	even	6	inner
392.3.j.a.117.2		4	7.2	even	3	inner
392.3.j.a.117.2		4	56.5	odd	6	inner
392.3.j.a.325.1		4	7.6	odd	2	inner
392.3.j.a.325.1		4	8.5	even	2	inner
392.3.j.a.325.2		4	1.1	even	1	trivial
392.3.j.a.325.2		4	56.13	odd	2	CM
504.3.l.a.181.1		2	21.11	odd	6
504.3.l.a.181.1		2	168.101	even	6
504.3.l.a.181.2		2	21.17	even	6
504.3.l.a.181.2		2	168.53	odd	6
2016.3.l.c.433.1		2	84.11	even	6
2016.3.l.c.433.1		2	168.59	odd	6
2016.3.l.c.433.2		2	84.59	odd	6
2016.3.l.c.433.2		2	168.11	even	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	392.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(1.41421 - 2.44949i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-4.24264 - 7.34847i) q^{5} -5.65685 q^{6} +8.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(1.41421 - 2.44949i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-4.24264 - 7.34847i) q^{5} -5.65685 q^{6} +8.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +(-8.48528 + 14.6969i) q^{10} +(5.65685 + 9.79796i) q^{12} -25.4558 q^{13} -24.0000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(1.00000 - 1.73205i) q^{18} +(-4.24264 - 7.34847i) q^{19} +33.9411 q^{20} +(5.00000 + 8.66025i) q^{23} +(11.3137 - 19.5959i) q^{24} +(-23.5000 + 40.7032i) q^{25} +(25.4558 + 44.0908i) q^{26} +28.2843 q^{27} +(24.0000 + 41.5692i) q^{30} +(-16.0000 + 27.7128i) q^{32} -4.00000 q^{36} +(-8.48528 + 14.6969i) q^{38} +(-36.0000 + 62.3538i) q^{39} +(-33.9411 - 58.7878i) q^{40} +(4.24264 - 7.34847i) q^{45} +(10.0000 - 17.3205i) q^{46} -45.2548 q^{48} +94.0000 q^{50} +(50.9117 - 88.1816i) q^{52} +(-28.2843 - 48.9898i) q^{54} -24.0000 q^{57} +(-38.1838 + 66.1362i) q^{59} +(48.0000 - 83.1384i) q^{60} +(-4.24264 - 7.34847i) q^{61} +64.0000 q^{64} +(108.000 + 187.061i) q^{65} +28.2843 q^{69} -110.000 q^{71} +(4.00000 + 6.92820i) q^{72} +(66.4680 + 115.126i) q^{75} +33.9411 q^{76} +144.000 q^{78} +(-65.0000 - 112.583i) q^{79} +(-67.8823 + 117.576i) q^{80} +(35.5000 - 61.4878i) q^{81} -25.4558 q^{83} -16.9706 q^{90} -40.0000 q^{92} +(-36.0000 + 62.3538i) q^{95} +(45.2548 + 78.3837i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 + 2 * q^9	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} + 2 q^{9} - 96 q^{15} - 32 q^{16} + 4 q^{18} + 20 q^{23} - 94 q^{25} + 96 q^{30} - 64 q^{32} - 16 q^{36} - 144 q^{39} + 40 q^{46} + 376 q^{50} - 96 q^{57} + 192 q^{60} + 256 q^{64} + 432 q^{65} - 440 q^{71} + 16 q^{72} + 576 q^{78} - 260 q^{79} + 142 q^{81} - 160 q^{92} - 144 q^{95}+O(q^{100}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 + 2 * q^9 - 96 * q^15 - 32 * q^16 + 4 * q^18 + 20 * q^23 - 94 * q^25 + 96 * q^30 - 64 * q^32 - 16 * q^36 - 144 * q^39 + 40 * q^46 + 376 * q^50 - 96 * q^57 + 192 * q^60 + 256 * q^64 + 432 * q^65 - 440 * q^71 + 16 * q^72 + 576 * q^78 - 260 * q^79 + 142 * q^81 - 160 * q^92 - 144 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more