LMFDB - Embedded newform 1116.2.g.b.991.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1116,2,Mod(991,1116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("1116.991");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1116 = 2^{2} \cdot 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1116.g (of order $2$, degree $1$, 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:	$8.91130486557$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 58 $ x^4 - 2x^3 - 7x^2 + 8*x + 58
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			991.3
Root			$-1.94949 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	1116.991
Dual form			1116.2.g.b.991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} +2.00000 q^{5} +(2.50000 - 1.32288i) q^{8} +O(q^{10})$	\(q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} +2.00000 q^{5} +(2.50000 - 1.32288i) q^{8} +(-1.00000 + 2.64575i) q^{10} -2.44949 q^{11} +6.48074i q^{13} +(0.500000 + 3.96863i) q^{16} -5.29150i q^{19} +(-3.00000 - 2.64575i) q^{20} +(1.22474 - 3.24037i) q^{22} +4.89898 q^{23} -1.00000 q^{25} +(-8.57321 - 3.24037i) q^{26} +6.48074i q^{29} +(4.89898 + 2.64575i) q^{31} +(-5.50000 - 1.32288i) q^{32} +6.48074i q^{37} +(7.00000 + 2.64575i) q^{38} +(5.00000 - 2.64575i) q^{40} +8.00000 q^{41} +2.44949 q^{43} +(3.67423 + 3.24037i) q^{44} +(-2.44949 + 6.48074i) q^{46} +10.5830i q^{47} +7.00000 q^{49} +(0.500000 - 1.32288i) q^{50} +(8.57321 - 9.72111i) q^{52} +6.48074i q^{53} -4.89898 q^{55} +(-8.57321 - 3.24037i) q^{58} -5.29150i q^{59} -6.48074i q^{61} +(-5.94949 + 5.15787i) q^{62} +(4.50000 - 6.61438i) q^{64} +12.9615i q^{65} -5.29150i q^{67} +5.29150i q^{71} +12.9615i q^{73} +(-8.57321 - 3.24037i) q^{74} +(-7.00000 + 7.93725i) q^{76} -9.79796 q^{79} +(1.00000 + 7.93725i) q^{80} +(-4.00000 + 10.5830i) q^{82} -12.2474 q^{83} +(-1.22474 + 3.24037i) q^{86} +(-6.12372 + 3.24037i) q^{88} -12.9615i q^{89} +(-7.34847 - 6.48074i) q^{92} +(-14.0000 - 5.29150i) q^{94} -10.5830i q^{95} +4.00000 q^{97} +(-3.50000 + 9.26013i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 6 * q^4 + 8 * q^5 + 10 * q^8	$ 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8} - 4 q^{10} + 2 q^{16} - 12 q^{20} - 4 q^{25} - 22 q^{32} + 28 q^{38} + 20 q^{40} + 32 q^{41} + 28 q^{49} + 2 q^{50} - 14 q^{62} + 18 q^{64} - 28 q^{76} + 4 q^{80} - 16 q^{82} - 56 q^{94} + 16 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 6 * q^4 + 8 * q^5 + 10 * q^8 - 4 * q^10 + 2 * q^16 - 12 * q^20 - 4 * q^25 - 22 * q^32 + 28 * q^38 + 20 * q^40 + 32 * q^41 + 28 * q^49 + 2 * q^50 - 14 * q^62 + 18 * q^64 - 28 * q^76 + 4 * q^80 - 16 * q^82 - 56 * q^94 + 16 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times$.

$n$	$497$	$559$	$685$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	+	1.32288i	−0.353553	+	0.935414i
$2$	−0.500000	+	1.32288i	−0.353553	+	0.935414i
$3$		0			0
$3$		0			0
$4$	−1.50000	−	1.32288i	−0.750000	−	0.661438i
$5$	2.00000			0.894427			0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000			0.894427			0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	2.50000	−	1.32288i	0.883883	−	0.467707i
$9$		0			0
$10$	−1.00000	+	2.64575i	−0.316228	+	0.836660i
$11$	−2.44949			−0.738549			−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949			−0.738549			−0.369274	+	0.929320i	$0.620394\pi$
$12$		0			0
$13$			6.48074i			1.79743i	0.438529	+	0.898717i	$0.355500\pi$
$13$			6.48074i			1.79743i	−0.438529	+	0.898717i	$0.644500\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	+	3.96863i	0.125000	+	0.992157i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	5.29150i		−	1.21395i	−0.794719	−	0.606977i	$-0.792382\pi$
$19$		−	5.29150i		−	1.21395i	0.794719	−	0.606977i	$-0.207618\pi$
$20$	−3.00000	−	2.64575i	−0.670820	−	0.591608i
$21$		0			0
$22$	1.22474	−	3.24037i	0.261116	−	0.690849i
$23$	4.89898			1.02151			0.510754	−	0.859727i	$-0.329366\pi$
$23$	4.89898			1.02151			0.510754	+	0.859727i	$0.329366\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−8.57321	−	3.24037i	−1.68135	−	0.635489i
$27$		0			0
$28$		0			0
$29$			6.48074i			1.20344i	0.798706	+	0.601722i	$0.205518\pi$
$29$			6.48074i			1.20344i	−0.798706	+	0.601722i	$0.794482\pi$
$30$		0			0
$31$	4.89898	+	2.64575i	0.879883	+	0.475191i
$31$	4.89898	+	2.64575i	0.879883	+	0.475191i
$32$	−5.50000	−	1.32288i	−0.972272	−	0.233854i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.48074i			1.06543i	0.846296	+	0.532714i	$0.178828\pi$
$37$			6.48074i			1.06543i	−0.846296	+	0.532714i	$0.821172\pi$
$38$	7.00000	+	2.64575i	1.13555	+	0.429198i
$39$		0			0
$40$	5.00000	−	2.64575i	0.790569	−	0.418330i
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$	2.44949			0.373544			0.186772	−	0.982403i	$-0.440197\pi$
$43$	2.44949			0.373544			0.186772	+	0.982403i	$0.440197\pi$
$44$	3.67423	+	3.24037i	0.553912	+	0.488504i
$45$		0			0
$46$	−2.44949	+	6.48074i	−0.361158	+	0.955533i
$47$			10.5830i			1.54369i	0.635811	+	0.771845i	$0.280666\pi$
$47$			10.5830i			1.54369i	−0.635811	+	0.771845i	$0.719334\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	0.500000	−	1.32288i	0.0707107	−	0.187083i
$51$		0			0
$52$	8.57321	−	9.72111i	1.18889	−	1.34808i
$53$			6.48074i			0.890198i	0.895481	+	0.445099i	$0.146832\pi$
$53$			6.48074i			0.890198i	−0.895481	+	0.445099i	$0.853168\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$		0			0
$57$		0			0
$58$	−8.57321	−	3.24037i	−1.12572	−	0.425481i
$59$		−	5.29150i		−	0.688895i	−0.938806	−	0.344447i	$-0.888066\pi$
$59$		−	5.29150i		−	0.688895i	0.938806	−	0.344447i	$-0.111934\pi$
$60$		0			0
$61$		−	6.48074i		−	0.829774i	−0.909873	−	0.414887i	$-0.863821\pi$
$61$		−	6.48074i		−	0.829774i	0.909873	−	0.414887i	$-0.136179\pi$
$62$	−5.94949	+	5.15787i	−0.755586	+	0.655050i
$63$		0			0
$64$	4.50000	−	6.61438i	0.562500	−	0.826797i
$65$			12.9615i			1.60767i
$66$		0			0
$67$		−	5.29150i		−	0.646460i	−0.946320	−	0.323230i	$-0.895231\pi$
$67$		−	5.29150i		−	0.646460i	0.946320	−	0.323230i	$-0.104769\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.29150i			0.627986i	0.949425	+	0.313993i	$0.101667\pi$
$71$			5.29150i			0.627986i	−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$			12.9615i			1.51703i	0.651658	+	0.758513i	$0.274074\pi$
$73$			12.9615i			1.51703i	−0.651658	+	0.758513i	$0.725926\pi$
$74$	−8.57321	−	3.24037i	−0.996616	−	0.376685i
$75$		0			0
$76$	−7.00000	+	7.93725i	−0.802955	+	0.910465i
$77$		0			0
$78$		0			0
$79$	−9.79796			−1.10236			−0.551178	−	0.834388i	$-0.685822\pi$
$79$	−9.79796			−1.10236			−0.551178	+	0.834388i	$0.685822\pi$
$80$	1.00000	+	7.93725i	0.111803	+	0.887412i
$81$		0			0
$82$	−4.00000	+	10.5830i	−0.441726	+	1.16870i
$83$	−12.2474			−1.34433			−0.672166	−	0.740400i	$-0.734636\pi$
$83$	−12.2474			−1.34433			−0.672166	+	0.740400i	$0.734636\pi$
$84$		0			0
$85$		0			0
$86$	−1.22474	+	3.24037i	−0.132068	+	0.349418i
$87$		0			0
$88$	−6.12372	+	3.24037i	−0.652791	+	0.345425i
$89$		−	12.9615i		−	1.37391i	−0.726698	−	0.686957i	$-0.758946\pi$
$89$		−	12.9615i		−	1.37391i	0.726698	−	0.686957i	$-0.241054\pi$
$90$		0			0
$91$		0			0
$92$	−7.34847	−	6.48074i	−0.766131	−	0.675664i
$93$		0			0
$94$	−14.0000	−	5.29150i	−1.44399	−	0.545777i
$95$		−	10.5830i		−	1.08579i
$96$		0			0
$97$	4.00000			0.406138			0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000			0.406138			0.203069	+	0.979164i	$0.434908\pi$
$98$	−3.50000	+	9.26013i	−0.353553	+	0.935414i
$99$		0			0
$100$	1.50000	+	1.32288i	0.150000	+	0.132288i
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		0			0
$103$			5.29150i			0.521387i	0.965422	+	0.260694i	$0.0839512\pi$
$103$			5.29150i			0.521387i	−0.965422	+	0.260694i	$0.916049\pi$
$104$	8.57321	+	16.2019i	0.840673	+	1.58872i
$105$		0			0
$106$	−8.57321	−	3.24037i	−0.832704	−	0.314733i
$107$			5.29150i			0.511549i	0.966736	+	0.255774i	$0.0823304\pi$
$107$			5.29150i			0.511549i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	2.44949	−	6.48074i	0.233550	−	0.617914i
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	9.79796			0.913664
$116$	8.57321	−	9.72111i	0.796003	−	0.902583i
$117$		0			0
$118$	7.00000	+	2.64575i	0.644402	+	0.243561i
$119$		0			0
$120$		0			0
$121$	−5.00000			−0.454545
$122$	8.57321	+	3.24037i	0.776182	+	0.293369i
$123$		0			0
$124$	−3.84847	−	10.4494i	−0.345603	−	0.938381i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−9.79796			−0.869428			−0.434714	−	0.900568i	$-0.643151\pi$
$127$	−9.79796			−0.869428			−0.434714	+	0.900568i	$0.643151\pi$
$128$	6.50000	+	9.26013i	0.574524	+	0.818488i
$129$		0			0
$130$	−17.1464	−	6.48074i	−1.50384	−	0.568399i
$131$			5.29150i			0.462321i	0.972916	+	0.231160i	$0.0742522\pi$
$131$			5.29150i			0.462321i	−0.972916	+	0.231160i	$0.925748\pi$
$132$		0			0
$133$		0			0
$134$	7.00000	+	2.64575i	0.604708	+	0.228558i
$135$		0			0
$136$		0			0
$137$		−	12.9615i		−	1.10737i	−0.832725	−	0.553687i	$-0.813220\pi$
$137$		−	12.9615i		−	1.10737i	0.832725	−	0.553687i	$-0.186780\pi$
$138$		0			0
$139$	7.34847			0.623289			0.311645	−	0.950199i	$-0.399120\pi$
$139$	7.34847			0.623289			0.311645	+	0.950199i	$0.399120\pi$
$140$		0			0
$141$		0			0
$142$	−7.00000	−	2.64575i	−0.587427	−	0.222027i
$143$		−	15.8745i		−	1.32749i
$144$		0			0
$145$			12.9615i			1.07639i
$146$	−17.1464	−	6.48074i	−1.41905	−	0.536350i
$147$		0			0
$148$	8.57321	−	9.72111i	0.704714	−	0.799070i
$149$	−2.00000			−0.163846			−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000			−0.163846			−0.0819232	+	0.996639i	$0.526106\pi$
$150$		0			0
$151$	4.89898			0.398673			0.199337	−	0.979931i	$-0.436121\pi$
$151$	4.89898			0.398673			0.199337	+	0.979931i	$0.436121\pi$
$152$	−7.00000	−	13.2288i	−0.567775	−	1.07299i
$153$		0			0
$154$		0			0
$155$	9.79796	+	5.29150i	0.786991	+	0.425024i
$156$		0			0
$157$	6.00000			0.478852			0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000			0.478852			0.239426	+	0.970915i	$0.423041\pi$
$158$	4.89898	−	12.9615i	0.389742	−	1.03116i
$159$		0			0
$160$	−11.0000	−	2.64575i	−0.869626	−	0.209165i
$161$		0			0
$162$		0			0
$163$			15.8745i			1.24339i	0.783260	+	0.621694i	$0.213555\pi$
$163$			15.8745i			1.24339i	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−12.0000	−	10.5830i	−0.937043	−	0.826394i
$165$		0			0
$166$	6.12372	−	16.2019i	0.475293	−	1.25751i
$167$	4.89898			0.379094			0.189547	−	0.981872i	$-0.439298\pi$
$167$	4.89898			0.379094			0.189547	+	0.981872i	$0.439298\pi$
$168$		0			0
$169$	−29.0000			−2.23077
$170$		0			0
$171$		0			0
$172$	−3.67423	−	3.24037i	−0.280158	−	0.247076i
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$		0			0
$176$	−1.22474	−	9.72111i	−0.0923186	−	0.732756i
$177$		0			0
$178$	17.1464	+	6.48074i	1.28518	+	0.485752i
$179$	−12.2474			−0.915417			−0.457709	−	0.889102i	$-0.651330\pi$
$179$	−12.2474			−0.915417			−0.457709	+	0.889102i	$0.651330\pi$
$180$		0			0
$181$		−	6.48074i		−	0.481710i	−0.970561	−	0.240855i	$-0.922572\pi$
$181$		−	6.48074i		−	0.481710i	0.970561	−	0.240855i	$-0.0774278\pi$
$182$		0			0
$183$		0			0
$184$	12.2474	−	6.48074i	0.902894	−	0.477767i
$185$			12.9615i			0.952947i
$186$		0			0
$187$		0			0
$188$	14.0000	−	15.8745i	1.02105	−	1.15777i
$189$		0			0
$190$	14.0000	+	5.29150i	1.01567	+	0.383886i
$191$		−	10.5830i		−	0.765759i	−0.923798	−	0.382880i	$-0.874932\pi$
$191$		−	10.5830i		−	0.765759i	0.923798	−	0.382880i	$-0.125068\pi$
$192$		0			0
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$	−2.00000	+	5.29150i	−0.143592	+	0.379908i
$195$		0			0
$196$	−10.5000	−	9.26013i	−0.750000	−	0.661438i
$197$		−	6.48074i		−	0.461734i	−0.972985	−	0.230867i	$-0.925844\pi$
$197$		−	6.48074i		−	0.461734i	0.972985	−	0.230867i	$-0.0741562\pi$
$198$		0			0
$199$	14.6969			1.04184			0.520919	−	0.853606i	$-0.325589\pi$
$199$	14.6969			1.04184			0.520919	+	0.853606i	$0.325589\pi$
$200$	−2.50000	+	1.32288i	−0.176777	+	0.0935414i
$201$		0			0
$202$	1.00000	−	2.64575i	0.0703598	−	0.186154i
$203$		0			0
$204$		0			0
$205$	16.0000			1.11749
$206$	−7.00000	−	2.64575i	−0.487713	−	0.184338i
$207$		0			0
$208$	−25.7196	+	3.24037i	−1.78334	+	0.224679i
$209$			12.9615i			0.896564i
$210$		0			0
$211$		−	15.8745i		−	1.09285i	−0.837509	−	0.546423i	$-0.815989\pi$
$211$		−	15.8745i		−	1.09285i	0.837509	−	0.546423i	$-0.184011\pi$
$212$	8.57321	−	9.72111i	0.588811	−	0.667649i
$213$		0			0
$214$	−7.00000	−	2.64575i	−0.478510	−	0.180860i
$215$	4.89898			0.334108
$216$		0			0
$217$		0			0
$218$	−7.00000	+	18.5203i	−0.474100	+	1.25435i
$219$		0			0
$220$	7.34847	+	6.48074i	0.495434	+	0.436931i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	−2.00000	+	5.29150i	−0.133038	+	0.351986i
$227$		−	26.4575i		−	1.75605i	−0.478618	−	0.878023i	$-0.658862\pi$
$227$		−	26.4575i		−	1.75605i	0.478618	−	0.878023i	$-0.341138\pi$
$228$		0			0
$229$			6.48074i			0.428259i	0.976805	+	0.214130i	$0.0686915\pi$
$229$			6.48074i			0.428259i	−0.976805	+	0.214130i	$0.931308\pi$
$230$	−4.89898	+	12.9615i	−0.323029	+	0.854655i
$231$		0			0
$232$	8.57321	+	16.2019i	0.562859	+	1.06370i
$233$	16.0000			1.04819			0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000			1.04819			0.524097	+	0.851658i	$0.324403\pi$
$234$		0			0
$235$			21.1660i			1.38072i
$236$	−7.00000	+	7.93725i	−0.455661	+	0.516671i
$237$		0			0
$238$		0			0
$239$	19.5959			1.26755			0.633777	−	0.773516i	$-0.281504\pi$
$239$	19.5959			1.26755			0.633777	+	0.773516i	$0.281504\pi$
$240$		0			0
$241$		−	25.9230i		−	1.66984i	−0.550368	−	0.834922i	$-0.685512\pi$
$241$		−	25.9230i		−	1.66984i	0.550368	−	0.834922i	$-0.314488\pi$
$242$	2.50000	−	6.61438i	0.160706	−	0.425188i
$243$		0			0
$244$	−8.57321	+	9.72111i	−0.548844	+	0.622330i
$245$	14.0000			0.894427
$246$		0			0
$247$	34.2929			2.18200
$248$	15.7474	+	0.133638i	0.999964	+	0.00848599i
$249$		0			0
$250$	6.00000	−	15.8745i	0.379473	−	1.00399i
$251$	7.34847			0.463831			0.231916	−	0.972736i	$-0.425501\pi$
$251$	7.34847			0.463831			0.231916	+	0.972736i	$0.425501\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	4.89898	−	12.9615i	0.307389	−	0.813276i
$255$		0			0
$256$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$257$	−10.0000			−0.623783			−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000			−0.623783			−0.311891	+	0.950118i	$0.600963\pi$
$258$		0			0
$259$		0			0
$260$	17.1464	−	19.4422i	1.06338	−	1.20576i
$261$		0			0
$262$	−7.00000	−	2.64575i	−0.432461	−	0.163455i
$263$	4.89898			0.302084			0.151042	−	0.988527i	$-0.451737\pi$
$263$	4.89898			0.302084			0.151042	+	0.988527i	$0.451737\pi$
$264$		0			0
$265$			12.9615i			0.796217i
$266$		0			0
$267$		0			0
$268$	−7.00000	+	7.93725i	−0.427593	+	0.484845i
$269$		−	6.48074i		−	0.395138i	−0.980289	−	0.197569i	$-0.936695\pi$
$269$		−	6.48074i		−	0.395138i	0.980289	−	0.197569i	$-0.0633046\pi$
$270$		0			0
$271$	−29.3939			−1.78555			−0.892775	−	0.450502i	$-0.851245\pi$
$271$	−29.3939			−1.78555			−0.892775	+	0.450502i	$0.851245\pi$
$272$		0			0
$273$		0			0
$274$	17.1464	+	6.48074i	1.03585	+	0.391516i
$275$	2.44949			0.147710
$276$		0			0
$277$		−	6.48074i		−	0.389390i	−0.980864	−	0.194695i	$-0.937628\pi$
$277$		−	6.48074i		−	0.389390i	0.980864	−	0.194695i	$-0.0623717\pi$
$278$	−3.67423	+	9.72111i	−0.220366	+	0.583033i
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		−	15.8745i		−	0.943642i	−0.881694	−	0.471821i	$-0.843597\pi$
$283$		−	15.8745i		−	0.943642i	0.881694	−	0.471821i	$-0.156403\pi$
$284$	7.00000	−	7.93725i	0.415374	−	0.470989i
$285$		0			0
$286$	21.0000	+	7.93725i	1.24176	+	0.469340i
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$	−17.1464	−	6.48074i	−1.00687	−	0.380562i
$291$		0			0
$292$	17.1464	−	19.4422i	1.00342	−	1.13777i
$293$	−22.0000			−1.28525			−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000			−1.28525			−0.642627	+	0.766179i	$0.722155\pi$
$294$		0			0
$295$		−	10.5830i		−	0.616166i
$296$	8.57321	+	16.2019i	0.498308	+	0.941714i
$297$		0			0
$298$	1.00000	−	2.64575i	0.0579284	−	0.153264i
$299$			31.7490i			1.83609i
$300$		0			0
$301$		0			0
$302$	−2.44949	+	6.48074i	−0.140952	+	0.372925i
$303$		0			0
$304$	21.0000	−	2.64575i	1.20443	−	0.151744i
$305$		−	12.9615i		−	0.742172i
$306$		0			0
$307$		−	26.4575i		−	1.51001i	−0.655719	−	0.755005i	$-0.727634\pi$
$307$		−	26.4575i		−	1.51001i	0.655719	−	0.755005i	$-0.272366\pi$
$308$		0			0
$309$		0			0
$310$	−11.8990	+	10.3157i	−0.675817	+	0.585894i
$311$		−	5.29150i		−	0.300054i	−0.988682	−	0.150027i	$-0.952064\pi$
$311$		−	5.29150i		−	0.300054i	0.988682	−	0.150027i	$-0.0479360\pi$
$312$		0			0
$313$			12.9615i			0.732626i	0.930492	+	0.366313i	$0.119380\pi$
$313$			12.9615i			0.732626i	−0.930492	+	0.366313i	$0.880620\pi$
$314$	−3.00000	+	7.93725i	−0.169300	+	0.447925i
$315$		0			0
$316$	14.6969	+	12.9615i	0.826767	+	0.729140i
$317$	14.0000			0.786318			0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000			0.786318			0.393159	+	0.919470i	$0.371382\pi$
$318$		0			0
$319$		−	15.8745i		−	0.888802i
$320$	9.00000	−	13.2288i	0.503115	−	0.739510i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	6.48074i		−	0.359487i
$326$	−21.0000	−	7.93725i	−1.16308	−	0.439604i
$327$		0			0
$328$	20.0000	−	10.5830i	1.10432	−	0.584349i
$329$		0			0
$330$		0			0
$331$	−12.2474			−0.673181			−0.336590	−	0.941651i	$-0.609274\pi$
$331$	−12.2474			−0.673181			−0.336590	+	0.941651i	$0.609274\pi$
$332$	18.3712	+	16.2019i	1.00825	+	0.889192i
$333$		0			0
$334$	−2.44949	+	6.48074i	−0.134030	+	0.354610i
$335$		−	10.5830i		−	0.578211i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	14.5000	−	38.3634i	0.788696	−	2.08669i
$339$		0			0
$340$		0			0
$341$	−12.0000	−	6.48074i	−0.649836	−	0.350952i
$342$		0			0
$343$		0			0
$344$	6.12372	−	3.24037i	0.330169	−	0.174709i
$345$		0			0
$346$	1.00000	−	2.64575i	0.0537603	−	0.142236i
$347$	22.0454			1.18346			0.591730	−	0.806136i	$-0.298445\pi$
$347$	22.0454			1.18346			0.591730	+	0.806136i	$0.298445\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$		0			0
$351$		0			0
$352$	13.4722	+	3.24037i	0.718070	+	0.172712i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$			10.5830i			0.561688i
$356$	−17.1464	+	19.4422i	−0.908759	+	1.03044i
$357$		0			0
$358$	6.12372	−	16.2019i	0.323649	−	0.856294i
$359$			21.1660i			1.11710i	0.829471	+	0.558550i	$0.188642\pi$
$359$			21.1660i			1.11710i	−0.829471	+	0.558550i	$0.811358\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$	8.57321	+	3.24037i	0.450598	+	0.170310i
$363$		0			0
$364$		0			0
$365$			25.9230i			1.35687i
$366$		0			0
$367$	19.5959			1.02290			0.511449	−	0.859313i	$-0.329109\pi$
$367$	19.5959			1.02290			0.511449	+	0.859313i	$0.329109\pi$
$368$	2.44949	+	19.4422i	0.127688	+	1.01350i
$369$		0			0
$370$	−17.1464	−	6.48074i	−0.891400	−	0.336918i
$371$		0			0
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$		0			0
$376$	14.0000	+	26.4575i	0.721995	+	1.36444i
$377$	−42.0000			−2.16311
$378$		0			0
$379$			15.8745i			0.815419i	0.913112	+	0.407709i	$0.133672\pi$
$379$			15.8745i			0.815419i	−0.913112	+	0.407709i	$0.866328\pi$
$380$	−14.0000	+	15.8745i	−0.718185	+	0.814345i
$381$		0			0
$382$	14.0000	+	5.29150i	0.716302	+	0.270737i
$383$	−29.3939			−1.50196			−0.750978	−	0.660327i	$-0.770418\pi$
$383$	−29.3939			−1.50196			−0.750978	+	0.660327i	$0.770418\pi$
$384$		0			0
$385$		0			0
$386$	−3.00000	+	7.93725i	−0.152696	+	0.403996i
$387$		0			0
$388$	−6.00000	−	5.29150i	−0.304604	−	0.268635i
$389$		−	19.4422i		−	0.985760i	−0.870097	−	0.492880i	$-0.835944\pi$
$389$		−	19.4422i		−	0.985760i	0.870097	−	0.492880i	$-0.164056\pi$
$390$		0			0
$391$		0			0
$392$	17.5000	−	9.26013i	0.883883	−	0.467707i
$393$		0			0
$394$	8.57321	+	3.24037i	0.431912	+	0.163247i
$395$	−19.5959			−0.985978
$396$		0			0
$397$	10.0000			0.501886			0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000			0.501886			0.250943	+	0.968002i	$0.419259\pi$
$398$	−7.34847	+	19.4422i	−0.368345	+	0.974551i
$399$		0			0
$400$	−0.500000	−	3.96863i	−0.0250000	−	0.198431i
$401$		−	25.9230i		−	1.29453i	−0.762265	−	0.647265i	$-0.775913\pi$
$401$		−	25.9230i		−	1.29453i	0.762265	−	0.647265i	$-0.224087\pi$
$402$		0			0
$403$	−17.1464	+	31.7490i	−0.854124	+	1.58153i
$404$	3.00000	+	2.64575i	0.149256	+	0.131631i
$405$		0			0
$406$		0			0
$407$		−	15.8745i		−	0.786870i
$408$		0			0
$409$		−	38.8844i		−	1.92271i	−0.275308	−	0.961356i	$-0.588780\pi$
$409$		−	38.8844i		−	1.92271i	0.275308	−	0.961356i	$-0.411220\pi$
$410$	−8.00000	+	21.1660i	−0.395092	+	1.04531i
$411$		0			0
$412$	7.00000	−	7.93725i	0.344865	−	0.391040i
$413$		0			0
$414$		0			0
$415$	−24.4949			−1.20241
$416$	8.57321	−	35.6441i	0.420336	−	1.74759i
$417$		0			0
$418$	−17.1464	−	6.48074i	−0.838659	−	0.316983i
$419$			37.0405i			1.80955i	0.425892	+	0.904774i	$0.359960\pi$
$419$			37.0405i			1.80955i	−0.425892	+	0.904774i	$0.640040\pi$
$420$		0			0
$421$	18.0000			0.877266			0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000			0.877266			0.438633	+	0.898666i	$0.355463\pi$
$422$	21.0000	+	7.93725i	1.02226	+	0.386379i
$423$		0			0
$424$	8.57321	+	16.2019i	0.416352	+	0.786831i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	7.00000	−	7.93725i	0.338358	−	0.383662i
$429$		0			0
$430$	−2.44949	+	6.48074i	−0.118125	+	0.312529i
$431$			10.5830i			0.509765i	0.966972	+	0.254883i	$0.0820369\pi$
$431$			10.5830i			0.509765i	−0.966972	+	0.254883i	$0.917963\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−21.0000	−	18.5203i	−1.00572	−	0.886960i
$437$		−	25.9230i		−	1.24006i
$438$		0			0
$439$		−	21.1660i		−	1.01020i	−0.863061	−	0.505099i	$-0.831456\pi$
$439$		−	21.1660i		−	1.01020i	0.863061	−	0.505099i	$-0.168544\pi$
$440$	−12.2474	+	6.48074i	−0.583874	+	0.308957i
$441$		0			0
$442$		0			0
$443$			5.29150i			0.251407i	0.992068	+	0.125703i	$0.0401188\pi$
$443$			5.29150i			0.251407i	−0.992068	+	0.125703i	$0.959881\pi$
$444$		0			0
$445$		−	25.9230i		−	1.22887i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9230i		−	1.22338i	−0.791097	−	0.611690i	$-0.790490\pi$
$449$		−	25.9230i		−	1.22338i	0.791097	−	0.611690i	$-0.209510\pi$
$450$		0			0
$451$	−19.5959			−0.922736
$452$	−6.00000	−	5.29150i	−0.282216	−	0.248891i
$453$		0			0
$454$	35.0000	+	13.2288i	1.64263	+	0.620856i
$455$		0			0
$456$		0			0
$457$		−	12.9615i		−	0.606313i	−0.952941	−	0.303156i	$-0.901960\pi$
$457$		−	12.9615i		−	0.606313i	0.952941	−	0.303156i	$-0.0980404\pi$
$458$	−8.57321	−	3.24037i	−0.400600	−	0.151413i
$459$		0			0
$460$	−14.6969	−	12.9615i	−0.685248	−	0.604332i
$461$		−	6.48074i		−	0.301838i	−0.988546	−	0.150919i	$-0.951777\pi$
$461$		−	6.48074i		−	0.301838i	0.988546	−	0.150919i	$-0.0482233\pi$
$462$		0			0
$463$	9.79796			0.455350			0.227675	−	0.973737i	$-0.426888\pi$
$463$	9.79796			0.455350			0.227675	+	0.973737i	$0.426888\pi$
$464$	−25.7196	+	3.24037i	−1.19400	+	0.150430i
$465$		0			0
$466$	−8.00000	+	21.1660i	−0.370593	+	0.980497i
$467$		−	5.29150i		−	0.244862i	−0.992477	−	0.122431i	$-0.960931\pi$
$467$		−	5.29150i		−	0.244862i	0.992477	−	0.122431i	$-0.0390689\pi$
$468$		0			0
$469$		0			0
$470$	−28.0000	−	10.5830i	−1.29154	−	0.488158i
$471$		0			0
$472$	−7.00000	−	13.2288i	−0.322201	−	0.608903i
$473$	−6.00000			−0.275880
$474$		0			0
$475$			5.29150i			0.242791i
$476$		0			0
$477$		0			0
$478$	−9.79796	+	25.9230i	−0.448148	+	1.18569i
$479$		−	5.29150i		−	0.241775i	−0.992666	−	0.120887i	$-0.961426\pi$
$479$		−	5.29150i		−	0.241775i	0.992666	−	0.120887i	$-0.0385740\pi$
$480$		0			0
$481$	−42.0000			−1.91504
$482$	34.2929	+	12.9615i	1.56200	+	0.590379i
$483$		0			0
$484$	7.50000	+	6.61438i	0.340909	+	0.300654i
$485$	8.00000			0.363261
$486$		0			0
$487$	4.89898			0.221994			0.110997	−	0.993821i	$-0.464596\pi$
$487$	4.89898			0.221994			0.110997	+	0.993821i	$0.464596\pi$
$488$	−8.57321	−	16.2019i	−0.388091	−	0.733423i
$489$		0			0
$490$	−7.00000	+	18.5203i	−0.316228	+	0.836660i
$491$	−7.34847			−0.331632			−0.165816	−	0.986157i	$-0.553026\pi$
$491$	−7.34847			−0.331632			−0.165816	+	0.986157i	$0.553026\pi$
$492$		0			0
$493$		0			0
$494$	−17.1464	+	45.3652i	−0.771454	+	2.04108i
$495$		0			0
$496$	−8.05051	+	20.7651i	−0.361479	+	0.932380i
$497$		0			0
$498$		0			0
$499$	−22.0454			−0.986888			−0.493444	−	0.869778i	$-0.664262\pi$
$499$	−22.0454			−0.986888			−0.493444	+	0.869778i	$0.664262\pi$
$500$	18.0000	+	15.8745i	0.804984	+	0.709930i
$501$		0			0
$502$	−3.67423	+	9.72111i	−0.163989	+	0.433874i
$503$			5.29150i			0.235936i	0.993017	+	0.117968i	$0.0376381\pi$
$503$			5.29150i			0.235936i	−0.993017	+	0.117968i	$0.962362\pi$
$504$		0			0
$505$	−4.00000			−0.177998
$506$	6.00000	−	15.8745i	0.266733	−	0.705708i
$507$		0			0
$508$	14.6969	+	12.9615i	0.652071	+	0.575073i
$509$			19.4422i			0.861761i	0.902409	+	0.430881i	$0.141797\pi$
$509$			19.4422i			0.861761i	−0.902409	+	0.430881i	$0.858203\pi$
$510$		0			0
$511$		0			0
$512$	2.50000	−	22.4889i	0.110485	−	0.993878i
$513$		0			0
$514$	5.00000	−	13.2288i	0.220541	−	0.583495i
$515$			10.5830i			0.466343i
$516$		0			0
$517$		−	25.9230i		−	1.14009i
$518$		0			0
$519$		0			0
$520$	17.1464	+	32.4037i	0.751921	+	1.42100i
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$	2.44949			0.107109			0.0535544	−	0.998565i	$-0.482945\pi$
$523$	2.44949			0.107109			0.0535544	+	0.998565i	$0.482945\pi$
$524$	7.00000	−	7.93725i	0.305796	−	0.346741i
$525$		0			0
$526$	−2.44949	+	6.48074i	−0.106803	+	0.282574i
$527$		0			0
$528$		0			0
$529$	1.00000			0.0434783
$530$	−17.1464	−	6.48074i	−0.744793	−	0.281505i
$531$		0			0
$532$		0			0
$533$			51.8459i			2.24570i
$534$		0			0
$535$			10.5830i			0.457543i
$536$	−7.00000	−	13.2288i	−0.302354	−	0.571395i
$537$		0			0
$538$	8.57321	+	3.24037i	0.369618	+	0.139702i
$539$	−17.1464			−0.738549
$540$		0			0
$541$	−6.00000			−0.257960			−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000			−0.257960			−0.128980	+	0.991647i	$0.541170\pi$
$542$	14.6969	−	38.8844i	0.631288	−	1.67023i
$543$		0			0
$544$		0			0
$545$	28.0000			1.19939
$546$		0			0
$547$		−	5.29150i		−	0.226248i	−0.993581	−	0.113124i	$-0.963914\pi$
$547$		−	5.29150i		−	0.226248i	0.993581	−	0.113124i	$-0.0360858\pi$
$548$	−17.1464	+	19.4422i	−0.732459	+	0.830531i
$549$		0			0
$550$	−1.22474	+	3.24037i	−0.0522233	+	0.138170i
$551$	34.2929			1.46092
$552$		0			0
$553$		0			0
$554$	8.57321	+	3.24037i	0.364241	+	0.137670i
$555$		0			0
$556$	−11.0227	−	9.72111i	−0.467467	−	0.412267i
$557$		−	19.4422i		−	0.823793i	−0.911231	−	0.411897i	$-0.864866\pi$
$557$		−	19.4422i		−	0.823793i	0.911231	−	0.411897i	$-0.135134\pi$
$558$		0			0
$559$			15.8745i			0.671420i
$560$		0			0
$561$		0			0
$562$	−5.00000	+	13.2288i	−0.210912	+	0.558021i
$563$		−	26.4575i		−	1.11505i	−0.830160	−	0.557526i	$-0.811751\pi$
$563$		−	26.4575i		−	1.11505i	0.830160	−	0.557526i	$-0.188249\pi$
$564$		0			0
$565$	8.00000			0.336563
$566$	21.0000	+	7.93725i	0.882696	+	0.333628i
$567$		0			0
$568$	7.00000	+	13.2288i	0.293713	+	0.555066i
$569$			12.9615i			0.543374i	0.962386	+	0.271687i	$0.0875815\pi$
$569$			12.9615i			0.543374i	−0.962386	+	0.271687i	$0.912419\pi$
$570$		0			0
$571$	−46.5403			−1.94765			−0.973826	−	0.227297i	$-0.927011\pi$
$571$	−46.5403			−1.94765			−0.973826	+	0.227297i	$0.927011\pi$
$572$	−21.0000	+	23.8118i	−0.878054	+	0.995620i
$573$		0			0
$574$		0			0
$575$	−4.89898			−0.204302
$576$		0			0
$577$	−12.0000			−0.499567			−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000			−0.499567			−0.249783	+	0.968302i	$0.580359\pi$
$578$	−8.50000	+	22.4889i	−0.353553	+	0.935414i
$579$		0			0
$580$	17.1464	−	19.4422i	0.711967	−	0.807294i
$581$		0			0
$582$		0			0
$583$		−	15.8745i		−	0.657455i
$584$	17.1464	+	32.4037i	0.709524	+	1.34087i
$585$		0			0
$586$	11.0000	−	29.1033i	0.454406	−	1.20224i
$587$	−41.6413			−1.71872			−0.859361	−	0.511370i	$-0.829138\pi$
$587$	−41.6413			−1.71872			−0.859361	+	0.511370i	$0.829138\pi$
$588$		0			0
$589$	14.0000	−	25.9230i	0.576860	−	1.06814i
$590$	14.0000	+	5.29150i	0.576371	+	0.217848i
$591$		0			0
$592$	−25.7196	+	3.24037i	−1.05707	+	0.133178i
$593$	4.00000			0.164260			0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000			0.164260			0.0821302	+	0.996622i	$0.473828\pi$
$594$		0			0
$595$		0			0
$596$	3.00000	+	2.64575i	0.122885	+	0.108374i
$597$		0			0
$598$	−42.0000	−	15.8745i	−1.71751	−	0.649157i
$599$		−	21.1660i		−	0.864820i	−0.901677	−	0.432410i	$-0.857663\pi$
$599$		−	21.1660i		−	0.864820i	0.901677	−	0.432410i	$-0.142337\pi$
$600$		0			0
$601$			12.9615i			0.528710i	0.964425	+	0.264355i	$0.0851590\pi$
$601$			12.9615i			0.528710i	−0.964425	+	0.264355i	$0.914841\pi$
$602$		0			0
$603$		0			0
$604$	−7.34847	−	6.48074i	−0.299005	−	0.263698i
$605$	−10.0000			−0.406558
$606$		0			0
$607$			37.0405i			1.50343i	0.659489	+	0.751714i	$0.270773\pi$
$607$			37.0405i			1.50343i	−0.659489	+	0.751714i	$0.729227\pi$
$608$	−7.00000	+	29.1033i	−0.283887	+	1.18029i
$609$		0			0
$610$	17.1464	+	6.48074i	0.694239	+	0.262398i
$611$	−68.5857			−2.77468
$612$		0			0
$613$			6.48074i			0.261755i	0.991399	+	0.130877i	$0.0417794\pi$
$613$			6.48074i			0.261755i	−0.991399	+	0.130877i	$0.958221\pi$
$614$	35.0000	+	13.2288i	1.41249	+	0.533869i
$615$		0			0
$616$		0			0
$617$	10.0000			0.402585			0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000			0.402585			0.201292	+	0.979531i	$0.435486\pi$
$618$		0			0
$619$	−17.1464			−0.689173			−0.344587	−	0.938755i	$-0.611981\pi$
$619$	−17.1464			−0.689173			−0.344587	+	0.938755i	$0.611981\pi$
$620$	−7.69694	−	20.8987i	−0.309116	−	0.839313i
$621$		0			0
$622$	7.00000	+	2.64575i	0.280674	+	0.106085i
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−17.1464	−	6.48074i	−0.685309	−	0.259022i
$627$		0			0
$628$	−9.00000	−	7.93725i	−0.359139	−	0.316731i
$629$		0			0
$630$		0			0
$631$	14.6969			0.585076			0.292538	−	0.956254i	$-0.405500\pi$
$631$	14.6969			0.585076			0.292538	+	0.956254i	$0.405500\pi$
$632$	−24.4949	+	12.9615i	−0.974355	+	0.515580i
$633$		0			0
$634$	−7.00000	+	18.5203i	−0.278006	+	0.735533i
$635$	−19.5959			−0.777640
$636$		0			0
$637$			45.3652i			1.79743i
$638$	21.0000	+	7.93725i	0.831398	+	0.314239i
$639$		0			0
$640$	13.0000	+	18.5203i	0.513870	+	0.732078i
$641$			25.9230i			1.02390i	0.859017	+	0.511948i	$0.171076\pi$
$641$			25.9230i			1.02390i	−0.859017	+	0.511948i	$0.828924\pi$
$642$		0			0
$643$	22.0454			0.869386			0.434693	−	0.900579i	$-0.356857\pi$
$643$	22.0454			0.869386			0.434693	+	0.900579i	$0.356857\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	34.2929			1.34819			0.674096	−	0.738644i	$-0.264534\pi$
$647$	34.2929			1.34819			0.674096	+	0.738644i	$0.264534\pi$
$648$		0			0
$649$			12.9615i			0.508783i
$650$	8.57321	+	3.24037i	0.336269	+	0.127098i
$651$		0			0
$652$	21.0000	−	23.8118i	0.822423	−	0.932541i
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$		0			0
$655$			10.5830i			0.413512i
$656$	4.00000	+	31.7490i	0.156174	+	1.23959i
$657$		0			0
$658$		0			0
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$	6.12372	−	16.2019i	0.238005	−	0.629703i
$663$		0			0
$664$	−30.6186	+	16.2019i	−1.18823	+	0.628754i
$665$		0			0
$666$		0			0
$667$			31.7490i			1.22933i
$668$	−7.34847	−	6.48074i	−0.284321	−	0.250747i
$669$		0			0
$670$	14.0000	+	5.29150i	0.540867	+	0.204429i
$671$			15.8745i			0.612829i
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	43.5000	+	38.3634i	1.67308	+	1.47552i
$677$			32.4037i			1.24538i	0.782470	+	0.622688i	$0.213959\pi$
$677$			32.4037i			1.24538i	−0.782470	+	0.622688i	$0.786041\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	14.5732	−	12.6341i	0.558037	−	0.483786i
$683$		−	5.29150i		−	0.202474i	−0.994862	−	0.101237i	$-0.967720\pi$
$683$		−	5.29150i		−	0.202474i	0.994862	−	0.101237i	$-0.0322800\pi$
$684$		0			0
$685$		−	25.9230i		−	0.990465i
$686$		0			0
$687$		0			0
$688$	1.22474	+	9.72111i	0.0466930	+	0.370614i
$689$	−42.0000			−1.60007
$690$		0			0
$691$			47.6235i			1.81168i	0.423615	+	0.905842i	$0.360761\pi$
$691$			47.6235i			1.81168i	−0.423615	+	0.905842i	$0.639239\pi$
$692$	3.00000	+	2.64575i	0.114043	+	0.100576i
$693$		0			0
$694$	−11.0227	+	29.1633i	−0.418416	+	1.10702i
$695$	14.6969			0.557487
$696$		0			0
$697$		0			0
$698$	9.00000	−	23.8118i	0.340655	−	0.901288i
$699$		0			0
$700$		0			0
$701$	−38.0000			−1.43524			−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000			−1.43524			−0.717620	+	0.696435i	$0.754769\pi$
$702$		0			0
$703$	34.2929			1.29338
$704$	−11.0227	+	16.2019i	−0.415434	+	0.610630i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			19.4422i			0.730168i	0.930975	+	0.365084i	$0.118960\pi$
$709$			19.4422i			0.730168i	−0.930975	+	0.365084i	$0.881040\pi$
$710$	−14.0000	−	5.29150i	−0.525411	−	0.198587i
$711$		0			0
$712$	−17.1464	−	32.4037i	−0.642590	−	1.21438i
$713$	24.0000	+	12.9615i	0.898807	+	0.485411i
$714$		0			0
$715$		−	31.7490i		−	1.18735i
$716$	18.3712	+	16.2019i	0.686563	+	0.605492i
$717$		0			0
$718$	−28.0000	−	10.5830i	−1.04495	−	0.394954i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	4.50000	−	11.9059i	0.167473	−	0.443091i
$723$		0			0
$724$	−8.57321	+	9.72111i	−0.318621	+	0.361282i
$725$		−	6.48074i		−	0.240689i
$726$		0			0
$727$		−	47.6235i		−	1.76626i	−0.469130	−	0.883129i	$-0.655432\pi$
$727$		−	47.6235i		−	1.76626i	0.469130	−	0.883129i	$-0.344568\pi$
$728$		0			0
$729$		0			0
$730$	−34.2929	−	12.9615i	−1.26924	−	0.479726i
$731$		0			0
$732$		0			0
$733$	30.0000			1.10808			0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000			1.10808			0.554038	+	0.832492i	$0.313086\pi$
$734$	−9.79796	+	25.9230i	−0.361649	+	0.956834i
$735$		0			0
$736$	−26.9444	−	6.48074i	−0.993183	−	0.238883i
$737$			12.9615i			0.477442i
$738$		0			0
$739$	−36.7423			−1.35159			−0.675795	−	0.737090i	$-0.736199\pi$
$739$	−36.7423			−1.35159			−0.675795	+	0.737090i	$0.736199\pi$
$740$	17.1464	−	19.4422i	0.630315	−	0.714710i
$741$		0			0
$742$		0			0
$743$	−14.6969			−0.539178			−0.269589	−	0.962975i	$-0.586888\pi$
$743$	−14.6969			−0.539178			−0.269589	+	0.962975i	$0.586888\pi$
$744$		0			0
$745$	−4.00000			−0.146549
$746$	13.0000	−	34.3948i	0.475964	−	1.25928i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	10.5830i		−	0.386179i	−0.981181	−	0.193090i	$-0.938149\pi$
$751$		−	10.5830i		−	0.386179i	0.981181	−	0.193090i	$-0.0618508\pi$
$752$	−42.0000	+	5.29150i	−1.53158	+	0.192961i
$753$		0			0
$754$	21.0000	−	55.5608i	0.764775	−	2.02340i
$755$	9.79796			0.356584
$756$		0			0
$757$			32.4037i			1.17773i	0.808230	+	0.588866i	$0.200426\pi$
$757$			32.4037i			1.17773i	−0.808230	+	0.588866i	$0.799574\pi$
$758$	−21.0000	−	7.93725i	−0.762754	−	0.288294i
$759$		0			0
$760$	−14.0000	−	26.4575i	−0.507833	−	0.959715i
$761$		−	12.9615i		−	0.469853i	−0.972013	−	0.234927i	$-0.924515\pi$
$761$		−	12.9615i		−	0.469853i	0.972013	−	0.234927i	$-0.0754850\pi$
$762$		0			0
$763$		0			0
$764$	−14.0000	+	15.8745i	−0.506502	+	0.574320i
$765$		0			0
$766$	14.6969	−	38.8844i	0.531022	−	1.40495i
$767$	34.2929			1.23824
$768$		0			0
$769$	36.0000			1.29819			0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000			1.29819			0.649097	+	0.760706i	$0.275147\pi$
$770$		0			0
$771$		0			0
$772$	−9.00000	−	7.93725i	−0.323917	−	0.285668i
$773$		−	19.4422i		−	0.699288i	−0.936883	−	0.349644i	$-0.886303\pi$
$773$		−	19.4422i		−	0.699288i	0.936883	−	0.349644i	$-0.113697\pi$
$774$		0			0
$775$	−4.89898	−	2.64575i	−0.175977	−	0.0950382i
$776$	10.0000	−	5.29150i	0.358979	−	0.189954i
$777$		0			0
$778$	25.7196	+	9.72111i	0.922094	+	0.348519i
$779$		−	42.3320i		−	1.51670i
$780$		0			0
$781$		−	12.9615i		−	0.463798i
$782$		0			0
$783$		0			0
$784$	3.50000	+	27.7804i	0.125000	+	0.992157i
$785$	12.0000			0.428298
$786$		0			0
$787$	31.8434			1.13509			0.567547	−	0.823341i	$-0.307893\pi$
$787$	31.8434			1.13509			0.567547	+	0.823341i	$0.307893\pi$
$788$	−8.57321	+	9.72111i	−0.305408	+	0.346300i
$789$		0			0
$790$	9.79796	−	25.9230i	0.348596	−	0.922298i
$791$		0			0
$792$		0			0
$793$	42.0000			1.49146
$794$	−5.00000	+	13.2288i	−0.177443	+	0.469471i
$795$		0			0
$796$	−22.0454	−	19.4422i	−0.781379	−	0.689111i
$797$		−	45.3652i		−	1.60692i	−0.595361	−	0.803459i	$-0.702991\pi$
$797$		−	45.3652i		−	1.60692i	0.595361	−	0.803459i	$-0.297009\pi$
$798$		0			0
$799$		0			0
$800$	5.50000	+	1.32288i	0.194454	+	0.0467707i
$801$		0			0
$802$	34.2929	+	12.9615i	1.21092	+	0.457686i
$803$		−	31.7490i		−	1.12040i
$804$		0			0
$805$		0			0
$806$	−33.4268	−	38.5571i	−1.17741	−	1.35812i
$807$		0			0
$808$	−5.00000	+	2.64575i	−0.175899	+	0.0930772i
$809$		−	12.9615i		−	0.455701i	−0.973696	−	0.227851i	$-0.926830\pi$
$809$		−	12.9615i		−	0.455701i	0.973696	−	0.227851i	$-0.0731698\pi$
$810$		0			0
$811$			5.29150i			0.185810i	0.995675	+	0.0929049i	$0.0296153\pi$
$811$			5.29150i			0.185810i	−0.995675	+	0.0929049i	$0.970385\pi$
$812$		0			0
$813$		0			0
$814$	21.0000	+	7.93725i	0.736050	+	0.278201i
$815$			31.7490i			1.11212i
$816$		0			0
$817$		−	12.9615i		−	0.453465i
$818$	51.4393	+	19.4422i	1.79853	+	0.679781i
$819$		0			0
$820$	−24.0000	−	21.1660i	−0.838116	−	0.739149i
$821$			19.4422i			0.678538i	0.940689	+	0.339269i	$0.110180\pi$
$821$			19.4422i			0.678538i	−0.940689	+	0.339269i	$0.889820\pi$
$822$		0			0
$823$	53.8888			1.87844			0.939222	−	0.343310i	$-0.111548\pi$
$823$	53.8888			1.87844			0.939222	+	0.343310i	$0.111548\pi$
$824$	7.00000	+	13.2288i	0.243857	+	0.460846i
$825$		0			0
$826$		0			0
$827$	26.9444			0.936948			0.468474	−	0.883477i	$-0.344804\pi$
$827$	26.9444			0.936948			0.468474	+	0.883477i	$0.344804\pi$
$828$		0			0
$829$		−	19.4422i		−	0.675256i	−0.941280	−	0.337628i	$-0.890375\pi$
$829$		−	19.4422i		−	0.675256i	0.941280	−	0.337628i	$-0.109625\pi$
$830$	12.2474	−	32.4037i	0.425115	−	1.12475i
$831$		0			0
$832$	42.8661	+	29.1633i	1.48611	+	1.01106i
$833$		0			0
$834$		0			0
$835$	9.79796			0.339072
$836$	17.1464	−	19.4422i	0.593022	−	0.672423i
$837$		0			0
$838$	−49.0000	−	18.5203i	−1.69268	−	0.639772i
$839$		−	42.3320i		−	1.46146i	−0.682665	−	0.730732i	$-0.739179\pi$
$839$		−	42.3320i		−	1.46146i	0.682665	−	0.730732i	$-0.260821\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$	−9.00000	+	23.8118i	−0.310160	+	0.820608i
$843$		0			0
$844$	−21.0000	+	23.8118i	−0.722850	+	0.819635i
$845$	−58.0000			−1.99526
$846$		0			0
$847$		0			0
$848$	−25.7196	+	3.24037i	−0.883216	+	0.111275i
$849$		0			0
$850$		0			0
$851$			31.7490i			1.08834i
$852$		0			0
$853$	18.0000			0.616308			0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000			0.616308			0.308154	+	0.951336i	$0.400289\pi$
$854$		0			0
$855$		0			0
$856$	7.00000	+	13.2288i	0.239255	+	0.452150i
$857$	−22.0000			−0.751506			−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000			−0.751506			−0.375753	+	0.926720i	$0.622616\pi$
$858$		0			0
$859$	12.2474			0.417878			0.208939	−	0.977929i	$-0.432999\pi$
$859$	12.2474			0.417878			0.208939	+	0.977929i	$0.432999\pi$
$860$	−7.34847	−	6.48074i	−0.250581	−	0.220991i
$861$		0			0
$862$	−14.0000	−	5.29150i	−0.476842	−	0.180229i
$863$	−9.79796			−0.333526			−0.166763	−	0.985997i	$-0.553332\pi$
$863$	−9.79796			−0.333526			−0.166763	+	0.985997i	$0.553332\pi$
$864$		0			0
$865$	−4.00000			−0.136004
$866$		0			0
$867$		0			0
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	34.2929			1.16197
$872$	35.0000	−	18.5203i	1.18525	−	0.627175i
$873$		0			0
$874$	34.2929	+	12.9615i	1.15997	+	0.438429i
$875$		0			0
$876$		0			0
$877$	−30.0000			−1.01303			−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000			−1.01303			−0.506514	+	0.862232i	$0.669066\pi$
$878$	28.0000	+	10.5830i	0.944954	+	0.357159i
$879$		0			0
$880$	−2.44949	−	19.4422i	−0.0825723	−	0.655397i
$881$		−	25.9230i		−	0.873367i	−0.899615	−	0.436683i	$-0.856153\pi$
$881$		−	25.9230i		−	0.873367i	0.899615	−	0.436683i	$-0.143847\pi$
$882$		0			0
$883$	−26.9444			−0.906751			−0.453375	−	0.891320i	$-0.649780\pi$
$883$	−26.9444			−0.906751			−0.453375	+	0.891320i	$0.649780\pi$
$884$		0			0
$885$		0			0
$886$	−7.00000	−	2.64575i	−0.235170	−	0.0888858i
$887$		−	21.1660i		−	0.710685i	−0.934736	−	0.355343i	$-0.884364\pi$
$887$		−	21.1660i		−	0.710685i	0.934736	−	0.355343i	$-0.115636\pi$
$888$		0			0
$889$		0			0
$890$	34.2929	+	12.9615i	1.14950	+	0.434470i
$891$		0			0
$892$		0			0
$893$	56.0000			1.87397
$894$		0			0
$895$	−24.4949			−0.818774
$896$		0			0
$897$		0			0
$898$	34.2929	+	12.9615i	1.14437	+	0.432530i
$899$	−17.1464	+	31.7490i	−0.571865	+	1.05889i
$900$		0			0
$901$		0			0
$902$	9.79796	−	25.9230i	0.326236	−	0.863140i
$903$		0			0
$904$	10.0000	−	5.29150i	0.332595	−	0.175993i
$905$		−	12.9615i		−	0.430854i
$906$		0			0
$907$		−	37.0405i		−	1.22991i	−0.788562	−	0.614955i	$-0.789174\pi$
$907$		−	37.0405i		−	1.22991i	0.788562	−	0.614955i	$-0.210826\pi$
$908$	−35.0000	+	39.6863i	−1.16152	+	1.31704i
$909$		0			0
$910$		0			0
$911$	48.9898			1.62310			0.811552	−	0.584280i	$-0.198623\pi$
$911$	48.9898			1.62310			0.811552	+	0.584280i	$0.198623\pi$
$912$		0			0
$913$	30.0000			0.992855
$914$	17.1464	+	6.48074i	0.567153	+	0.214364i
$915$		0			0
$916$	8.57321	−	9.72111i	0.283267	−	0.321195i
$917$		0			0
$918$		0			0
$919$		−	5.29150i		−	0.174551i	−0.996184	−	0.0872753i	$-0.972184\pi$
$919$		−	5.29150i		−	0.174551i	0.996184	−	0.0872753i	$-0.0278160\pi$
$920$	24.4949	−	12.9615i	0.807573	−	0.427327i
$921$		0			0
$922$	8.57321	+	3.24037i	0.282344	+	0.106716i
$923$	−34.2929			−1.12876
$924$		0			0
$925$		−	6.48074i		−	0.213085i
$926$	−4.89898	+	12.9615i	−0.160990	+	0.425941i
$927$		0			0
$928$	8.57321	−	35.6441i	0.281430	−	1.17007i
$929$			51.8459i			1.70101i	0.525967	+	0.850505i	$0.323703\pi$
$929$			51.8459i			1.70101i	−0.525967	+	0.850505i	$0.676297\pi$
$930$		0			0
$931$		−	37.0405i		−	1.21395i
$932$	−24.0000	−	21.1660i	−0.786146	−	0.693316i
$933$		0			0
$934$	7.00000	+	2.64575i	0.229047	+	0.0865716i
$935$		0			0
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$		0			0
$940$	28.0000	−	31.7490i	0.913259	−	1.03554i
$941$		−	6.48074i		−	0.211266i	−0.994405	−	0.105633i	$-0.966313\pi$
$941$		−	6.48074i		−	0.211266i	0.994405	−	0.105633i	$-0.0336869\pi$
$942$		0			0
$943$	39.1918			1.27626
$944$	21.0000	−	2.64575i	0.683492	−	0.0861119i
$945$		0			0
$946$	3.00000	−	7.93725i	0.0975384	−	0.258062i
$947$	26.9444			0.875575			0.437787	−	0.899078i	$-0.355762\pi$
$947$	26.9444			0.875575			0.437787	+	0.899078i	$0.355762\pi$
$948$		0			0
$949$	−84.0000			−2.72676
$950$	−7.00000	−	2.64575i	−0.227110	−	0.0858395i
$951$		0			0
$952$		0			0
$953$			12.9615i			0.419864i	0.977716	+	0.209932i	$0.0673242\pi$
$953$			12.9615i			0.419864i	−0.977716	+	0.209932i	$0.932676\pi$
$954$		0			0
$955$		−	21.1660i		−	0.684916i
$956$	−29.3939	−	25.9230i	−0.950666	−	0.838409i
$957$		0			0
$958$	7.00000	+	2.64575i	0.226160	+	0.0854803i
$959$		0			0
$960$		0			0
$961$	17.0000	+	25.9230i	0.548387	+	0.836225i
$962$	21.0000	−	55.5608i	0.677067	−	1.79135i
$963$		0			0
$964$	−34.2929	+	38.8844i	−1.10450	+	1.25238i
$965$	12.0000			0.386294
$966$		0			0
$967$	−14.6969			−0.472622			−0.236311	−	0.971678i	$-0.575938\pi$
$967$	−14.6969			−0.472622			−0.236311	+	0.971678i	$0.575938\pi$
$968$	−12.5000	+	6.61438i	−0.401765	+	0.212594i
$969$		0			0
$970$	−4.00000	+	10.5830i	−0.128432	+	0.339800i
$971$			26.4575i			0.849062i	0.905413	+	0.424531i	$0.139561\pi$
$971$			26.4575i			0.849062i	−0.905413	+	0.424531i	$0.860439\pi$
$972$		0			0
$973$		0			0
$974$	−2.44949	+	6.48074i	−0.0784867	+	0.207656i
$975$		0			0
$976$	25.7196	−	3.24037i	0.823266	−	0.103722i
$977$	−10.0000			−0.319928			−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000			−0.319928			−0.159964	+	0.987123i	$0.551138\pi$
$978$		0			0
$979$			31.7490i			1.01470i
$980$	−21.0000	−	18.5203i	−0.670820	−	0.591608i
$981$		0			0
$982$	3.67423	−	9.72111i	0.117250	−	0.310213i
$983$	53.8888			1.71878			0.859392	−	0.511316i	$-0.170842\pi$
$983$	53.8888			1.71878			0.859392	+	0.511316i	$0.170842\pi$
$984$		0			0
$985$		−	12.9615i		−	0.412987i
$986$		0			0
$987$		0			0
$988$	−51.4393	−	45.3652i	−1.63650	−	1.44326i
$989$	12.0000			0.381578
$990$		0			0
$991$	−29.3939			−0.933727			−0.466864	−	0.884329i	$-0.654616\pi$
$991$	−29.3939			−0.933727			−0.466864	+	0.884329i	$0.654616\pi$
$992$	−23.4444	−	21.0324i	−0.744360	−	0.667779i
$993$		0			0
$994$		0			0
$995$	29.3939			0.931849
$996$		0			0
$997$	−58.0000			−1.83688			−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000			−1.83688			−0.918439	+	0.395562i	$0.870550\pi$
$998$	11.0227	−	29.1633i	0.348918	−	0.923149i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1116.2.g.b.991.3		4
3.2	odd	2		124.2.d.b.123.2	yes	4
4.3	odd	2	inner	1116.2.g.b.991.2		4
12.11	even	2		124.2.d.b.123.3	yes	4
24.5	odd	2		1984.2.h.e.1983.1		4
24.11	even	2		1984.2.h.e.1983.3		4
31.30	odd	2	inner	1116.2.g.b.991.4		4
93.92	even	2		124.2.d.b.123.1	✓	4
124.123	even	2	inner	1116.2.g.b.991.1		4
372.371	odd	2		124.2.d.b.123.4	yes	4
744.371	odd	2		1984.2.h.e.1983.2		4
744.557	even	2		1984.2.h.e.1983.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.2.d.b.123.1	✓	4	93.92	even	2
124.2.d.b.123.2	yes	4	3.2	odd	2
124.2.d.b.123.3	yes	4	12.11	even	2
124.2.d.b.123.4	yes	4	372.371	odd	2
1116.2.g.b.991.1		4	124.123	even	2	inner
1116.2.g.b.991.2		4	4.3	odd	2	inner
1116.2.g.b.991.3		4	1.1	even	1	trivial
1116.2.g.b.991.4		4	31.30	odd	2	inner
1984.2.h.e.1983.1		4	24.5	odd	2
1984.2.h.e.1983.2		4	744.371	odd	2
1984.2.h.e.1983.3		4	24.11	even	2
1984.2.h.e.1983.4		4	744.557	even	2

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:	\( N \)	\(=\)	\( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1116.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} +2.00000 q^{5} +(2.50000 - 1.32288i) q^{8} +O(q^{10})\)	\(q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} +2.00000 q^{5} +(2.50000 - 1.32288i) q^{8} +(-1.00000 + 2.64575i) q^{10} -2.44949 q^{11} +6.48074i q^{13} +(0.500000 + 3.96863i) q^{16} -5.29150i q^{19} +(-3.00000 - 2.64575i) q^{20} +(1.22474 - 3.24037i) q^{22} +4.89898 q^{23} -1.00000 q^{25} +(-8.57321 - 3.24037i) q^{26} +6.48074i q^{29} +(4.89898 + 2.64575i) q^{31} +(-5.50000 - 1.32288i) q^{32} +6.48074i q^{37} +(7.00000 + 2.64575i) q^{38} +(5.00000 - 2.64575i) q^{40} +8.00000 q^{41} +2.44949 q^{43} +(3.67423 + 3.24037i) q^{44} +(-2.44949 + 6.48074i) q^{46} +10.5830i q^{47} +7.00000 q^{49} +(0.500000 - 1.32288i) q^{50} +(8.57321 - 9.72111i) q^{52} +6.48074i q^{53} -4.89898 q^{55} +(-8.57321 - 3.24037i) q^{58} -5.29150i q^{59} -6.48074i q^{61} +(-5.94949 + 5.15787i) q^{62} +(4.50000 - 6.61438i) q^{64} +12.9615i q^{65} -5.29150i q^{67} +5.29150i q^{71} +12.9615i q^{73} +(-8.57321 - 3.24037i) q^{74} +(-7.00000 + 7.93725i) q^{76} -9.79796 q^{79} +(1.00000 + 7.93725i) q^{80} +(-4.00000 + 10.5830i) q^{82} -12.2474 q^{83} +(-1.22474 + 3.24037i) q^{86} +(-6.12372 + 3.24037i) q^{88} -12.9615i q^{89} +(-7.34847 - 6.48074i) q^{92} +(-14.0000 - 5.29150i) q^{94} -10.5830i q^{95} +4.00000 q^{97} +(-3.50000 + 9.26013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 6 * q^4 + 8 * q^5 + 10 * q^8	\( 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8} - 4 q^{10} + 2 q^{16} - 12 q^{20} - 4 q^{25} - 22 q^{32} + 28 q^{38} + 20 q^{40} + 32 q^{41} + 28 q^{49} + 2 q^{50} - 14 q^{62} + 18 q^{64} - 28 q^{76} + 4 q^{80} - 16 q^{82} - 56 q^{94} + 16 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 6 * q^4 + 8 * q^5 + 10 * q^8 - 4 * q^10 + 2 * q^16 - 12 * q^20 - 4 * q^25 - 22 * q^32 + 28 * q^38 + 20 * q^40 + 32 * q^41 + 28 * q^49 + 2 * q^50 - 14 * q^62 + 18 * q^64 - 28 * q^76 + 4 * q^80 - 16 * q^82 - 56 * q^94 + 16 * q^97 - 14 * q^98

Introduction

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