LMFDB - Embedded newform 1134.2.f.f.379.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(379,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1134.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.f (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:	$9.05503558921$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			379.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1134.379
Dual form			1134.2.f.f.757.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(2.00000 + 3.46410i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -6.00000 q^{17} +2.00000 q^{19} +(2.50000 - 4.33013i) q^{25} -4.00000 q^{26} +1.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(3.00000 - 5.19615i) q^{34} +2.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(3.00000 + 5.19615i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(-6.00000 + 10.3923i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(2.50000 + 4.33013i) q^{50} +(2.00000 - 3.46410i) q^{52} -6.00000 q^{53} +(-0.500000 + 0.866025i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(-3.00000 - 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(2.00000 + 3.46410i) q^{67} +(3.00000 + 5.19615i) q^{68} +2.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(-4.00000 + 6.92820i) q^{79} -6.00000 q^{82} +(-3.00000 + 5.19615i) q^{83} +(-4.00000 - 6.92820i) q^{86} +6.00000 q^{89} -4.00000 q^{91} +(-6.00000 - 10.3923i) q^{94} +(5.00000 - 8.66025i) q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} + 4 q^{13} - q^{14} - q^{16} - 12 q^{17} + 4 q^{19} + 5 q^{25} - 8 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 6 q^{41} - 8 q^{43} - 12 q^{47} - q^{49} + 5 q^{50} + 4 q^{52} - 12 q^{53} - q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{67} + 6 q^{68} + 4 q^{73} - 2 q^{74} - 2 q^{76} - 8 q^{79} - 12 q^{82} - 6 q^{83} - 8 q^{86} + 12 q^{89} - 8 q^{91} - 12 q^{94} + 10 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - q^7 + 2 * q^8 + 4 * q^13 - q^14 - q^16 - 12 * q^17 + 4 * q^19 + 5 * q^25 - 8 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 6 * q^41 - 8 * q^43 - 12 * q^47 - q^49 + 5 * q^50 + 4 * q^52 - 12 * q^53 - q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^67 + 6 * q^68 + 4 * q^73 - 2 * q^74 - 2 * q^76 - 8 * q^79 - 12 * q^82 - 6 * q^83 - 8 * q^86 + 12 * q^89 - 8 * q^91 - 12 * q^94 + 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	−4.00000			−0.784465
$27$		0			0
$28$	1.00000			0.188982
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	3.00000	−	5.19615i	0.514496	−	0.891133i
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.00000	+	1.73205i	−0.162221	+	0.280976i
$39$		0			0
$40$		0			0
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	0.381246	+	0.924473i	$0.375495\pi$
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.856560	+	0.516047i	$0.827403\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$	2.50000	+	4.33013i	0.353553	+	0.612372i
$51$		0			0
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	−3.00000	−	5.19615i	−0.393919	−	0.682288i
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	$-0.294388\pi$
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	−0.992524	+	0.122047i	$0.961054\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	−4.00000			−0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$	3.00000	+	5.19615i	0.363803	+	0.630126i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$		0			0
$82$	−6.00000			−0.662589
$83$	−3.00000	+	5.19615i	−0.329293	+	0.570352i	−0.982372	−	0.186938i	$-0.940144\pi$
$83$	−3.00000	+	5.19615i	−0.329293	+	0.570352i	0.653079	+	0.757290i	$0.273477\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$	−6.00000	−	10.3923i	−0.618853	−	1.07188i
$95$		0			0
$96$		0			0
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	−0.492287	−	0.870433i	$-0.663839\pi$
$97$	5.00000	−	8.66025i	0.507673	−	0.879316i	0.999961	−	0.00888289i	$-0.00282755\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$	−5.00000			−0.500000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$	2.00000	+	3.46410i	0.196116	+	0.339683i
$105$		0			0
$106$	3.00000	−	5.19615i	0.291386	−	0.504695i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	−	0.866025i	−0.0472456	−	0.0818317i
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$	6.00000			0.552345
$119$	3.00000	−	5.19615i	0.275010	−	0.476331i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$	−4.00000	−	6.92820i	−0.362143	−	0.627250i
$123$		0			0
$124$	2.00000	−	3.46410i	0.179605	−	0.311086i
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	9.00000	+	15.5885i	0.786334	+	1.36197i	0.928199	+	0.372084i	$0.121357\pi$
$131$	9.00000	+	15.5885i	0.786334	+	1.36197i	−0.141865	+	0.989886i	$0.545310\pi$
$132$		0			0
$133$	−1.00000	+	1.73205i	−0.0867110	+	0.150188i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−6.00000			−0.514496
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	−0.993724	−	0.111856i	$-0.964321\pi$
$139$	−7.00000	−	12.1244i	−0.593732	−	1.02837i	0.399992	−	0.916519i	$-0.369013\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−1.00000	+	1.73205i	−0.0827606	+	0.143346i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$	2.00000			0.162221
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	−0.775113	+	0.631822i	$0.782307\pi$
$158$	−4.00000	−	6.92820i	−0.318223	−	0.551178i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	3.00000	−	5.19615i	0.234261	−	0.405751i
$165$		0			0
$166$	−3.00000	−	5.19615i	−0.232845	−	0.403300i
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	−0.998755	−	0.0498898i	$-0.984113\pi$
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	0.542583	+	0.840002i	$0.317446\pi$
$174$		0			0
$175$	2.50000	+	4.33013i	0.188982	+	0.327327i
$176$		0			0
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$	2.00000	−	3.46410i	0.148250	−	0.256776i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	12.0000			0.875190
$189$		0			0
$190$		0			0
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.00454614	−	0.999990i	$-0.498553\pi$
$191$	12.0000	−	20.7846i	0.868290	−	1.50392i	0.863743	−	0.503932i	$-0.168114\pi$
$192$		0			0
$193$	−7.00000	−	12.1244i	−0.503871	−	0.872730i	−0.999990	−	0.00447566i	$-0.998575\pi$
$193$	−7.00000	−	12.1244i	−0.503871	−	0.872730i	0.496119	−	0.868255i	$-0.334758\pi$
$194$	5.00000	+	8.66025i	0.358979	+	0.621770i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.0357143	+	0.0618590i
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$	2.50000	−	4.33013i	0.176777	−	0.306186i
$201$		0			0
$202$		0			0
$203$	−3.00000	−	5.19615i	−0.210559	−	0.364698i
$204$		0			0
$205$		0			0
$206$	−4.00000			−0.278693
$207$		0			0
$208$	−4.00000			−0.277350
$209$		0			0
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$		0			0
$216$		0			0
$217$	−4.00000			−0.271538
$218$	−1.00000	+	1.73205i	−0.0677285	+	0.117309i
$219$		0			0
$220$		0			0
$221$	−12.0000	−	20.7846i	−0.807207	−	1.39812i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−6.00000			−0.399114
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	−0.395860	−	0.918311i	$-0.629553\pi$
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	0.993210	−	0.116331i	$-0.0371134\pi$
$228$		0			0
$229$	2.00000	+	3.46410i	0.132164	+	0.228914i	0.924510	−	0.381157i	$-0.124474\pi$
$229$	2.00000	+	3.46410i	0.132164	+	0.228914i	−0.792347	+	0.610071i	$0.791141\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000	+	5.19615i	−0.196960	+	0.341144i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$	−3.00000	+	5.19615i	−0.195283	+	0.338241i
$237$		0			0
$238$	3.00000	+	5.19615i	0.194461	+	0.336817i
$239$	12.0000	+	20.7846i	0.776215	+	1.34444i	0.934109	+	0.356988i	$0.116196\pi$
$239$	12.0000	+	20.7846i	0.776215	+	1.34444i	−0.157893	+	0.987456i	$0.550470\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$	−11.0000			−0.707107
$243$		0			0
$244$	8.00000			0.512148
$245$		0			0
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$	2.00000	+	3.46410i	0.127000	+	0.219971i
$249$		0			0
$250$		0			0
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$		0			0
$254$	8.00000	−	13.8564i	0.501965	−	0.869428i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	−1.00000	+	1.73205i	−0.0621370	+	0.107624i
$260$		0			0
$261$		0			0
$262$	−18.0000			−1.11204
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−1.00000	−	1.73205i	−0.0613139	−	0.106199i
$267$		0			0
$268$	2.00000	−	3.46410i	0.122169	−	0.211604i
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000	−	5.19615i	0.181902	−	0.315063i
$273$		0			0
$274$	9.00000	+	15.5885i	0.543710	+	0.941733i
$275$		0			0
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$	14.0000			0.839664
$279$		0			0
$280$		0			0
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	−0.941526	−	0.336939i	$-0.890608\pi$
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	0.762561	+	0.646916i	$0.223942\pi$
$282$		0			0
$283$	11.0000	+	19.0526i	0.653882	+	1.13256i	0.982173	+	0.187980i	$0.0601941\pi$
$283$	11.0000	+	19.0526i	0.653882	+	1.13256i	−0.328291	+	0.944577i	$0.606473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000			−0.354169
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$	−1.00000	−	1.73205i	−0.0585206	−	0.101361i
$293$	12.0000	+	20.7846i	0.701047	+	1.21425i	0.968099	+	0.250568i	$0.0806172\pi$
$293$	12.0000	+	20.7846i	0.701047	+	1.21425i	−0.267052	+	0.963682i	$0.586049\pi$
$294$		0			0
$295$		0			0
$296$	2.00000			0.116248
$297$		0			0
$298$	18.0000			1.04271
$299$		0			0
$300$		0			0
$301$	−4.00000	−	6.92820i	−0.230556	−	0.399335i
$302$	−4.00000	−	6.92820i	−0.230174	−	0.398673i
$303$		0			0
$304$	−1.00000	+	1.73205i	−0.0573539	+	0.0993399i
$305$		0			0
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	−0.974841	−	0.222900i	$-0.928448\pi$
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	0.294384	−	0.955687i	$-0.404886\pi$
$312$		0			0
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	0.972028	+	0.234863i	$0.0754642\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	8.00000			0.450035
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	−0.769395	−	0.638774i	$-0.779442\pi$
$317$	3.00000	−	5.19615i	0.168497	−	0.291845i	0.937892	+	0.346929i	$0.112775\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	20.0000			1.10940
$326$	8.00000	−	13.8564i	0.443079	−	0.767435i
$327$		0			0
$328$	3.00000	+	5.19615i	0.165647	+	0.286910i
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$	6.00000			0.329293
$333$		0			0
$334$	12.0000			0.656611
$335$		0			0
$336$		0			0
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	0.609936	−	0.792451i	$-0.291195\pi$
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	−0.991250	+	0.131995i	$0.957862\pi$
$338$	−1.50000	−	2.59808i	−0.0815892	−	0.141317i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.00000			0.0539949
$344$	−4.00000	+	6.92820i	−0.215666	+	0.373544i
$345$		0			0
$346$	−6.00000	−	10.3923i	−0.322562	−	0.558694i
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	−0.984487	−	0.175457i	$-0.943860\pi$
$347$	−12.0000	−	20.7846i	−0.644194	−	1.11578i	0.340293	−	0.940319i	$-0.389474\pi$
$348$		0			0
$349$	14.0000	−	24.2487i	0.749403	−	1.29800i	−0.198706	−	0.980059i	$-0.563674\pi$
$349$	14.0000	−	24.2487i	0.749403	−	1.29800i	0.948109	−	0.317945i	$-0.102993\pi$
$350$	−5.00000			−0.267261
$351$		0			0
$352$		0			0
$353$	9.00000	−	15.5885i	0.479022	−	0.829690i	−0.520689	−	0.853746i	$-0.674325\pi$
$353$	9.00000	−	15.5885i	0.479022	−	0.829690i	0.999711	+	0.0240566i	$0.00765819\pi$
$354$		0			0
$355$		0			0
$356$	−3.00000	−	5.19615i	−0.159000	−	0.275396i
$357$		0			0
$358$	−6.00000	+	10.3923i	−0.317110	+	0.549250i
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	−10.0000	+	17.3205i	−0.525588	+	0.910346i
$363$		0			0
$364$	2.00000	+	3.46410i	0.104828	+	0.181568i
$365$		0			0
$366$		0			0
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	−0.951336	−	0.308155i	$-0.900289\pi$
$367$	−4.00000	+	6.92820i	−0.208798	+	0.361649i	0.742538	+	0.669804i	$0.233622\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	−	5.19615i	0.155752	−	0.269771i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$	−6.00000	+	10.3923i	−0.309426	+	0.535942i
$377$	−24.0000			−1.23606
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$	12.0000	+	20.7846i	0.613973	+	1.06343i
$383$	18.0000	+	31.1769i	0.919757	+	1.59307i	0.799783	+	0.600289i	$0.204948\pi$
$383$	18.0000	+	31.1769i	0.919757	+	1.59307i	0.119974	+	0.992777i	$0.461719\pi$
$384$		0			0
$385$		0			0
$386$	14.0000			0.712581
$387$		0			0
$388$	−10.0000			−0.507673
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	−0.542445	−	0.840091i	$-0.682501\pi$
$389$	9.00000	−	15.5885i	0.456318	−	0.790366i	0.998763	+	0.0497253i	$0.0158346\pi$
$390$		0			0
$391$		0			0
$392$	−0.500000	−	0.866025i	−0.0252538	−	0.0437409i
$393$		0			0
$394$	−9.00000	+	15.5885i	−0.453413	+	0.785335i
$395$		0			0
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$	−10.0000	+	17.3205i	−0.501255	+	0.868199i
$399$		0			0
$400$	2.50000	+	4.33013i	0.125000	+	0.216506i
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−8.00000	+	13.8564i	−0.398508	+	0.690237i
$404$		0			0
$405$		0			0
$406$	6.00000			0.297775
$407$		0			0
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$		0			0
$411$		0			0
$412$	2.00000	−	3.46410i	0.0985329	−	0.170664i
$413$	6.00000			0.295241
$414$		0			0
$415$		0			0
$416$	2.00000	−	3.46410i	0.0980581	−	0.169842i
$417$		0			0
$418$		0			0
$419$	3.00000	+	5.19615i	0.146560	+	0.253849i	0.929954	−	0.367677i	$-0.119847\pi$
$419$	3.00000	+	5.19615i	0.146560	+	0.253849i	−0.783394	+	0.621525i	$0.786513\pi$
$420$		0			0
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	−0.718076	−	0.695965i	$-0.754977\pi$
$421$	5.00000	−	8.66025i	0.243685	−	0.422075i	0.961761	+	0.273890i	$0.0883103\pi$
$422$	−4.00000			−0.194717
$423$		0			0
$424$	−6.00000			−0.291386
$425$	−15.0000	+	25.9808i	−0.727607	+	1.26025i
$426$		0			0
$427$	−4.00000	−	6.92820i	−0.193574	−	0.335279i
$428$	6.00000	+	10.3923i	0.290021	+	0.502331i
$429$		0			0
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$	2.00000	−	3.46410i	0.0960031	−	0.166282i
$435$		0			0
$436$	−1.00000	−	1.73205i	−0.0478913	−	0.0829502i
$437$		0			0
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$	24.0000			1.14156
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$		0			0
$446$	−4.00000	−	6.92820i	−0.189405	−	0.328060i
$447$		0			0
$448$	−0.500000	+	0.866025i	−0.0236228	+	0.0409159i
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$	3.00000	−	5.19615i	0.141108	−	0.244406i
$453$		0			0
$454$	9.00000	+	15.5885i	0.422391	+	0.731603i
$455$		0			0
$456$		0			0
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	0.958950	+	0.283577i	$0.0915211\pi$
$458$	−4.00000			−0.186908
$459$		0			0
$460$		0			0
$461$	6.00000	−	10.3923i	0.279448	−	0.484018i	−0.691800	−	0.722089i	$-0.743182\pi$
$461$	6.00000	−	10.3923i	0.279448	−	0.484018i	0.971248	+	0.238071i	$0.0765153\pi$
$462$		0			0
$463$	−16.0000	−	27.7128i	−0.743583	−	1.28792i	−0.950854	−	0.309640i	$-0.899791\pi$
$463$	−16.0000	−	27.7128i	−0.743583	−	1.28792i	0.207271	−	0.978284i	$-0.433542\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	−3.00000	+	5.19615i	−0.138972	+	0.240707i
$467$	6.00000			0.277647			0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000			0.277647			0.138823	+	0.990317i	$0.455668\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$		0			0
$472$	−3.00000	−	5.19615i	−0.138086	−	0.239172i
$473$		0			0
$474$		0			0
$475$	5.00000	−	8.66025i	0.229416	−	0.397360i
$476$	−6.00000			−0.275010
$477$		0			0
$478$	−24.0000			−1.09773
$479$	−18.0000	+	31.1769i	−0.822441	+	1.42451i	0.0814184	+	0.996680i	$0.474055\pi$
$479$	−18.0000	+	31.1769i	−0.822441	+	1.42451i	−0.903859	+	0.427830i	$0.859278\pi$
$480$		0			0
$481$	4.00000	+	6.92820i	0.182384	+	0.315899i
$482$	5.00000	+	8.66025i	0.227744	+	0.394464i
$483$		0			0
$484$	5.50000	−	9.52628i	0.250000	−	0.433013i
$485$		0			0
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−4.00000	+	6.92820i	−0.181071	+	0.313625i
$489$		0			0
$490$		0			0
$491$	−6.00000	−	10.3923i	−0.270776	−	0.468998i	0.698285	−	0.715820i	$-0.253947\pi$
$491$	−6.00000	−	10.3923i	−0.270776	−	0.468998i	−0.969061	+	0.246822i	$0.920614\pi$
$492$		0			0
$493$	18.0000	−	31.1769i	0.810679	−	1.40414i
$494$	−8.00000			−0.359937
$495$		0			0
$496$	−4.00000			−0.179605
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$		0			0
$502$	−9.00000	+	15.5885i	−0.401690	+	0.695747i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	8.00000	+	13.8564i	0.354943	+	0.614779i
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	0.921023	+	0.389509i	$0.127355\pi$
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	−0.123187	+	0.992384i	$0.539311\pi$
$510$		0			0
$511$	−1.00000	+	1.73205i	−0.0442374	+	0.0766214i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−18.0000			−0.793946
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.00000	−	1.73205i	−0.0439375	−	0.0761019i
$519$		0			0
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	2.00000			0.0874539			0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000			0.0874539			0.0437269	+	0.999044i	$0.486077\pi$
$524$	9.00000	−	15.5885i	0.393167	−	0.680985i
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.7846i	−0.522728	−	0.905392i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	2.00000			0.0867110
$533$	−12.0000	+	20.7846i	−0.519778	+	0.900281i
$534$		0			0
$535$		0			0
$536$	2.00000	+	3.46410i	0.0863868	+	0.149626i
$537$		0			0
$538$	−6.00000	+	10.3923i	−0.258678	+	0.448044i
$539$		0			0
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$	8.00000	−	13.8564i	0.343629	−	0.595184i
$543$		0			0
$544$	3.00000	+	5.19615i	0.128624	+	0.222783i
$545$		0			0
$546$		0			0
$547$	−4.00000	+	6.92820i	−0.171028	+	0.296229i	−0.938779	−	0.344519i	$-0.888042\pi$
$547$	−4.00000	+	6.92820i	−0.171028	+	0.296229i	0.767752	+	0.640747i	$0.221375\pi$
$548$	−18.0000			−0.768922
$549$		0			0
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$	−4.00000	−	6.92820i	−0.170097	−	0.294617i
$554$	5.00000	+	8.66025i	0.212430	+	0.367939i
$555$		0			0
$556$	−7.00000	+	12.1244i	−0.296866	+	0.514187i
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$		0			0
$561$		0			0
$562$	−3.00000	−	5.19615i	−0.126547	−	0.219186i
$563$	15.0000	+	25.9808i	0.632175	+	1.09496i	0.987106	+	0.160066i	$0.0511708\pi$
$563$	15.0000	+	25.9808i	0.632175	+	1.09496i	−0.354932	+	0.934892i	$0.615496\pi$
$564$		0			0
$565$		0			0
$566$	−22.0000			−0.924729
$567$		0			0
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$		0			0
$573$		0			0
$574$	3.00000	−	5.19615i	0.125218	−	0.216883i
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−9.50000	+	16.4545i	−0.395148	+	0.684416i
$579$		0			0
$580$		0			0
$581$	−3.00000	−	5.19615i	−0.124461	−	0.215573i
$582$		0			0
$583$		0			0
$584$	2.00000			0.0827606
$585$		0			0
$586$	−24.0000			−0.991431
$587$	−21.0000	+	36.3731i	−0.866763	+	1.50128i	−0.00147660	+	0.999999i	$0.500470\pi$
$587$	−21.0000	+	36.3731i	−0.866763	+	1.50128i	−0.865286	+	0.501278i	$0.832863\pi$
$588$		0			0
$589$	4.00000	+	6.92820i	0.164817	+	0.285472i
$590$		0			0
$591$		0			0
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$		0			0
$596$	−9.00000	+	15.5885i	−0.368654	+	0.638528i
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	0.469095	+	0.883148i	$0.344580\pi$
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$	8.00000			0.326056
$603$		0			0
$604$	8.00000			0.325515
$605$		0			0
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$	−1.00000	−	1.73205i	−0.0405554	−	0.0702439i
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$	−1.00000	+	1.73205i	−0.0403567	+	0.0698999i
$615$		0			0
$616$		0			0
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$		0			0
$619$	−13.0000	+	22.5167i	−0.522514	+	0.905021i	0.477143	+	0.878826i	$0.341672\pi$
$619$	−13.0000	+	22.5167i	−0.522514	+	0.905021i	−0.999657	+	0.0261952i	$0.991661\pi$
$620$		0			0
$621$		0			0
$622$	24.0000			0.962312
$623$	−3.00000	+	5.19615i	−0.120192	+	0.208179i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	5.00000	+	8.66025i	0.199840	+	0.346133i
$627$		0			0
$628$	2.00000	−	3.46410i	0.0798087	−	0.138233i
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−4.00000	+	6.92820i	−0.159111	+	0.275589i
$633$		0			0
$634$	3.00000	+	5.19615i	0.119145	+	0.206366i
$635$		0			0
$636$		0			0
$637$	2.00000	−	3.46410i	0.0792429	−	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$		0			0
$643$	−7.00000	−	12.1244i	−0.276053	−	0.478138i	0.694347	−	0.719640i	$-0.255693\pi$
$643$	−7.00000	−	12.1244i	−0.276053	−	0.478138i	−0.970400	+	0.241502i	$0.922360\pi$
$644$		0			0
$645$		0			0
$646$	6.00000	−	10.3923i	0.236067	−	0.408880i
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$		0			0
$650$	−10.0000	+	17.3205i	−0.392232	+	0.679366i
$651$		0			0
$652$	8.00000	+	13.8564i	0.313304	+	0.542659i
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$		0			0
$656$	−6.00000			−0.234261
$657$		0			0
$658$	12.0000			0.467809
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	−0.999309	−	0.0371821i	$-0.988162\pi$
$659$	−12.0000	+	20.7846i	−0.467454	+	0.809653i	0.531855	+	0.846836i	$0.321495\pi$
$660$		0			0
$661$	20.0000	+	34.6410i	0.777910	+	1.34738i	0.933144	+	0.359502i	$0.117053\pi$
$661$	20.0000	+	34.6410i	0.777910	+	1.34738i	−0.155235	+	0.987878i	$0.549613\pi$
$662$	−4.00000	−	6.92820i	−0.155464	−	0.269272i
$663$		0			0
$664$	−3.00000	+	5.19615i	−0.116423	+	0.201650i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−6.00000	+	10.3923i	−0.232147	+	0.402090i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	0.498886	+	0.866668i	$0.333743\pi$
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	−0.999999	+	0.00128586i	$0.999591\pi$
$674$	14.0000			0.539260
$675$		0			0
$676$	3.00000			0.115385
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	−0.957984	−	0.286820i	$-0.907402\pi$
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	0.727386	+	0.686229i	$0.240735\pi$
$678$		0			0
$679$	5.00000	+	8.66025i	0.191882	+	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	+	0.866025i	−0.0190901	+	0.0330650i
$687$		0			0
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$	−12.0000	−	20.7846i	−0.457164	−	0.791831i
$690$		0			0
$691$	23.0000	−	39.8372i	0.874961	−	1.51548i	0.0181572	−	0.999835i	$-0.494220\pi$
$691$	23.0000	−	39.8372i	0.874961	−	1.51548i	0.856804	−	0.515642i	$-0.172447\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$	24.0000			0.911028
$695$		0			0
$696$		0			0
$697$	−18.0000	−	31.1769i	−0.681799	−	1.18091i
$698$	14.0000	+	24.2487i	0.529908	+	0.917827i
$699$		0			0
$700$	2.50000	−	4.33013i	0.0944911	−	0.163663i
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$	9.00000	+	15.5885i	0.338719	+	0.586679i
$707$		0			0
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−6.00000	−	10.3923i	−0.224231	−	0.388379i
$717$		0			0
$718$	−12.0000	+	20.7846i	−0.447836	+	0.775675i
$719$	−12.0000			−0.447524			−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000			−0.447524			−0.223762	+	0.974644i	$0.571834\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$	7.50000	−	12.9904i	0.279121	−	0.483452i
$723$		0			0
$724$	−10.0000	−	17.3205i	−0.371647	−	0.643712i
$725$	15.0000	+	25.9808i	0.557086	+	0.964901i
$726$		0			0
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	−0.908655	+	0.417548i	$0.862889\pi$
$728$	−4.00000			−0.148250
$729$		0			0
$730$		0			0
$731$	24.0000	−	41.5692i	0.887672	−	1.53749i
$732$		0			0
$733$	20.0000	+	34.6410i	0.738717	+	1.27950i	0.953073	+	0.302740i	$0.0979013\pi$
$733$	20.0000	+	34.6410i	0.738717	+	1.27950i	−0.214356	+	0.976756i	$0.568765\pi$
$734$	−4.00000	−	6.92820i	−0.147643	−	0.255725i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$	3.00000	+	5.19615i	0.110133	+	0.190757i
$743$	12.0000	+	20.7846i	0.440237	+	0.762513i	0.997707	−	0.0676840i	$-0.0215610\pi$
$743$	12.0000	+	20.7846i	0.440237	+	0.762513i	−0.557470	+	0.830197i	$0.688228\pi$
$744$		0			0
$745$		0			0
$746$	14.0000			0.512576
$747$		0			0
$748$		0			0
$749$	6.00000	−	10.3923i	0.219235	−	0.379727i
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$	−6.00000	−	10.3923i	−0.218797	−	0.378968i
$753$		0			0
$754$	12.0000	−	20.7846i	0.437014	−	0.756931i
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	8.00000	−	13.8564i	0.290573	−	0.503287i
$759$		0			0
$760$		0			0
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	0.655515	−	0.755182i	$-0.272452\pi$
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	−0.981764	+	0.190101i	$0.939118\pi$
$762$		0			0
$763$	−1.00000	+	1.73205i	−0.0362024	+	0.0627044i
$764$	−24.0000			−0.868290
$765$		0			0
$766$	−36.0000			−1.30073
$767$	12.0000	−	20.7846i	0.433295	−	0.750489i
$768$		0			0
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	0.711767	−	0.702416i	$-0.247895\pi$
$769$	−7.00000	−	12.1244i	−0.252426	−	0.437215i	−0.964193	+	0.265200i	$0.914562\pi$
$770$		0			0
$771$		0			0
$772$	−7.00000	+	12.1244i	−0.251936	+	0.436365i
$773$	−24.0000			−0.863220			−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000			−0.863220			−0.431610	+	0.902060i	$0.642054\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$	5.00000	−	8.66025i	0.179490	−	0.310885i
$777$		0			0
$778$	9.00000	+	15.5885i	0.322666	+	0.558873i
$779$	6.00000	+	10.3923i	0.214972	+	0.372343i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			0.0357143
$785$		0			0
$786$		0			0
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	0.992727	−	0.120384i	$-0.0384127\pi$
$787$	11.0000	+	19.0526i	0.392108	+	0.679150i	−0.600620	+	0.799535i	$0.705079\pi$
$788$	−9.00000	−	15.5885i	−0.320612	−	0.555316i
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$	−32.0000

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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(2.00000 + 3.46410i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -6.00000 q^{17} +2.00000 q^{19} +(2.50000 - 4.33013i) q^{25} -4.00000 q^{26} +1.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(3.00000 - 5.19615i) q^{34} +2.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(3.00000 + 5.19615i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(-6.00000 + 10.3923i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(2.50000 + 4.33013i) q^{50} +(2.00000 - 3.46410i) q^{52} -6.00000 q^{53} +(-0.500000 + 0.866025i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(-3.00000 - 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} -4.00000 q^{62} +1.00000 q^{64} +(2.00000 + 3.46410i) q^{67} +(3.00000 + 5.19615i) q^{68} +2.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(-4.00000 + 6.92820i) q^{79} -6.00000 q^{82} +(-3.00000 + 5.19615i) q^{83} +(-4.00000 - 6.92820i) q^{86} +6.00000 q^{89} -4.00000 q^{91} +(-6.00000 - 10.3923i) q^{94} +(5.00000 - 8.66025i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} + 4 q^{13} - q^{14} - q^{16} - 12 q^{17} + 4 q^{19} + 5 q^{25} - 8 q^{26} + 2 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 6 q^{41} - 8 q^{43} - 12 q^{47} - q^{49} + 5 q^{50} + 4 q^{52} - 12 q^{53} - q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{67} + 6 q^{68} + 4 q^{73} - 2 q^{74} - 2 q^{76} - 8 q^{79} - 12 q^{82} - 6 q^{83} - 8 q^{86} + 12 q^{89} - 8 q^{91} - 12 q^{94} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - q^7 + 2 * q^8 + 4 * q^13 - q^14 - q^16 - 12 * q^17 + 4 * q^19 + 5 * q^25 - 8 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 6 * q^41 - 8 * q^43 - 12 * q^47 - q^49 + 5 * q^50 + 4 * q^52 - 12 * q^53 - q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^67 + 6 * q^68 + 4 * q^73 - 2 * q^74 - 2 * q^76 - 8 * q^79 - 12 * q^82 - 6 * q^83 - 8 * q^86 + 12 * q^89 - 8 * q^91 - 12 * q^94 + 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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