LMFDB - Embedded newform 1134.2.f.l.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.l.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	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		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	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		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	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		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	$-0.494070\pi$
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.856560	−	0.516047i	$-0.172597\pi$
$48$		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	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		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	0.992524	−	0.122047i	$-0.0389457\pi$
$59$	3.00000	+	5.19615i	0.390567	+	0.676481i	−0.601958	+	0.798528i	$0.705612\pi$
$60$		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	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		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	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		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	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		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	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		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	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		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	$0.445873\pi$
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	−0.938148	+	0.346235i	$0.887460\pi$
$138$		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	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		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	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		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	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		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	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		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	$0.501447\pi$
$191$	−12.0000	+	20.7846i	−0.868290	+	1.50392i	−0.863743	+	0.503932i	$0.831886\pi$
$192$		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	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		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	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$		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	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		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	$-0.883804\pi$
$239$	−12.0000	−	20.7846i	−0.776215	−	1.34444i	0.157893	−	0.987456i	$-0.449530\pi$
$240$		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	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		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	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		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	$-0.619214\pi$
$269$	−12.0000			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$270$		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	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		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	$-0.919383\pi$
$293$	−12.0000	−	20.7846i	−0.701047	−	1.21425i	0.267052	−	0.963682i	$-0.413951\pi$
$294$		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	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		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	−0.937892	−	0.346929i	$-0.887225\pi$
$317$	−3.00000	+	5.19615i	−0.168497	+	0.291845i	0.769395	+	0.638774i	$0.220558\pi$
$318$		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	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		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	−0.999711	−	0.0240566i	$-0.992342\pi$
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	0.520689	+	0.853746i	$0.325675\pi$
$354$		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	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		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	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		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	−0.998763	−	0.0497253i	$-0.984165\pi$
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	0.542445	+	0.840091i	$0.317499\pi$
$390$		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	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		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	0.783394	−	0.621525i	$-0.213487\pi$
$419$	−3.00000	−	5.19615i	−0.146560	−	0.253849i	−0.929954	+	0.367677i	$0.880153\pi$
$420$		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	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		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	−0.687557	−	0.726130i	$-0.741317\pi$
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	0.972626	+	0.232377i	$0.0746503\pi$
$444$		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	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		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	−0.971248	−	0.238071i	$-0.923485\pi$
$461$	−6.00000	+	10.3923i	−0.279448	+	0.484018i	0.691800	+	0.722089i	$0.256818\pi$
$462$		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	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		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	$-0.525945\pi$
$479$	18.0000	−	31.1769i	0.822441	−	1.42451i	0.903859	−	0.427830i	$-0.140722\pi$
$480$		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	0.969061	−	0.246822i	$-0.0793863\pi$
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	−0.698285	+	0.715820i	$0.746053\pi$
$492$		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	$-0.872645\pi$
$509$	−18.0000	−	31.1769i	−0.797836	−	1.38189i	0.123187	−	0.992384i	$-0.460689\pi$
$510$		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	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		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	$-0.459429\pi$
$557$	6.00000			0.254228			0.127114	+	0.991888i	$0.459429\pi$
$558$		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	$-0.948829\pi$
$563$	−15.0000	−	25.9808i	−0.632175	−	1.09496i	0.354932	−	0.934892i	$-0.384504\pi$
$564$		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	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		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	$-0.499530\pi$
$587$	21.0000	−	36.3731i	0.866763	−	1.50128i	0.865286	−	0.501278i	$-0.167137\pi$
$588$		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	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		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	0.999938	−	0.0111569i	$-0.00355143\pi$
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	−0.509631	+	0.860393i	$0.670218\pi$
$600$		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	0.799298	−	0.600935i	$-0.205205\pi$
$617$	−3.00000	−	5.19615i	−0.120775	−	0.209189i	−0.920074	+	0.391745i	$0.871871\pi$
$618$		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	−0.631721	−	0.775196i	$-0.717651\pi$
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	0.987200	+	0.159489i	$0.0509845\pi$
$642$		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	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		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	0.634437	−	0.772975i	$-0.281232\pi$
$653$	−9.00000	−	15.5885i	−0.352197	−	0.610023i	−0.986634	+	0.162951i	$0.947899\pi$
$654$		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	−0.531855	−	0.846836i	$-0.678505\pi$
$659$	12.0000	−	20.7846i	0.467454	−	0.809653i	0.999309	+	0.0371821i	$0.0118382\pi$
$660$		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	−0.727386	−	0.686229i	$-0.759265\pi$
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	0.957984	+	0.286820i	$0.0925982\pi$
$678$		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	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		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	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		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	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		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	0.557470	−	0.830197i	$-0.311772\pi$
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	−0.997707	+	0.0676840i	$0.978439\pi$
$744$		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	0.981764	−	0.190101i	$-0.0608816\pi$
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	−0.655515	+	0.755182i	$0.727548\pi$
$762$		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	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		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			−1.13635

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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