LMFDB - Embedded newform 765.2.b.a.154.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [765,2,Mod(154,765)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("765.154"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			154.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	765.154
Dual form			765.2.b.a.154.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -3.00000i q^{8} +(1.00000 - 2.00000i) q^{10} -6.00000 q^{11} -4.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -1.00000i q^{17} -4.00000 q^{19} +(2.00000 + 1.00000i) q^{20} +6.00000i q^{22} +(3.00000 + 4.00000i) q^{25} -4.00000 q^{26} -4.00000i q^{28} +8.00000 q^{31} -5.00000i q^{32} -1.00000 q^{34} +(4.00000 - 8.00000i) q^{35} +6.00000i q^{37} +4.00000i q^{38} +(3.00000 - 6.00000i) q^{40} +12.0000 q^{41} +6.00000i q^{43} -6.00000 q^{44} -4.00000i q^{47} -9.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -4.00000i q^{52} +6.00000i q^{53} +(-12.0000 - 6.00000i) q^{55} -12.0000 q^{56} +8.00000 q^{59} +10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +(4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -1.00000i q^{68} +(-8.00000 - 4.00000i) q^{70} -2.00000 q^{71} +6.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +24.0000i q^{77} -4.00000 q^{79} +(-2.00000 - 1.00000i) q^{80} -12.0000i q^{82} -8.00000i q^{83} +(1.00000 - 2.00000i) q^{85} +6.00000 q^{86} +18.0000i q^{88} -6.00000 q^{89} -16.0000 q^{91} -4.00000 q^{94} +(-8.00000 - 4.00000i) q^{95} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{5} + 2 q^{10} - 12 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{19} + 4 q^{20} + 6 q^{25} - 8 q^{26} + 16 q^{31} - 2 q^{34} + 8 q^{35} + 6 q^{40} + 24 q^{41} - 12 q^{44} - 18 q^{49} + 8 q^{50}+ \cdots - 16 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^5 + 2 * q^10 - 12 * q^11 - 8 * q^14 - 2 * q^16 - 8 * q^19 + 4 * q^20 + 6 * q^25 - 8 * q^26 + 16 * q^31 - 2 * q^34 + 8 * q^35 + 6 * q^40 + 24 * q^41 - 12 * q^44 - 18 * q^49 + 8 * q^50 - 24 * q^55 - 24 * q^56 + 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^65 - 16 * q^70 - 4 * q^71 + 12 * q^74 - 8 * q^76 - 8 * q^79 - 4 * q^80 + 2 * q^85 + 12 * q^86 - 12 * q^89 - 32 * q^91 - 8 * q^94 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/765\mathbb{Z}\right)^\times$.

$n$	$307$	$496$	$596$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		−	3.00000i		−	1.06066i
$9$		0			0
$10$	1.00000	−	2.00000i	0.316228	−	0.632456i
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−4.00000			−1.06904
$15$		0			0
$16$	−1.00000			−0.250000
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.00000	+	1.00000i	0.447214	+	0.223607i
$21$		0			0
$22$			6.00000i			1.27920i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		−	5.00000i		−	0.883883i
$33$		0			0
$34$	−1.00000			−0.171499
$35$	4.00000	−	8.00000i	0.676123	−	1.35225i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$			4.00000i			0.648886i
$39$		0			0
$40$	3.00000	−	6.00000i	0.474342	−	0.948683i
$41$	12.0000			1.87409			0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000			1.87409			0.937043	+	0.349215i	$0.113552\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$		−	4.00000i		−	0.554700i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	−12.0000	−	6.00000i	−1.61808	−	0.809040i
$56$	−12.0000			−1.60357
$57$		0			0
$58$		0			0
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		−	8.00000i		−	1.01600i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	4.00000	−	8.00000i	0.496139	−	0.992278i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		−	1.00000i		−	0.121268i
$69$		0			0
$70$	−8.00000	−	4.00000i	−0.956183	−	0.478091i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$	−4.00000			−0.458831
$77$			24.0000i			2.73505i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−	1.00000i	−0.223607	−	0.111803i
$81$		0			0
$82$		−	12.0000i		−	1.32518i
$83$		−	8.00000i		−	0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$		−	8.00000i		−	0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$		0			0
$85$	1.00000	−	2.00000i	0.108465	−	0.216930i
$86$	6.00000			0.646997
$87$		0			0
$88$			18.0000i			1.91881i
$89$	−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
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$	−4.00000			−0.412568
$95$	−8.00000	−	4.00000i	−0.820783	−	0.410391i
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$			9.00000i			0.909137i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	−12.0000			−1.17670
$105$		0			0
$106$	6.00000			0.582772
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	−6.00000	+	12.0000i	−0.572078	+	1.14416i
$111$		0			0
$112$			4.00000i			0.377964i
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	8.00000i		−	0.736460i
$119$	−4.00000			−0.366679
$120$		0			0
$121$	25.0000			2.27273
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$	8.00000			0.718421
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	6.00000i		−	0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$		−	6.00000i		−	0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$		−	3.00000i		−	0.265165i
$129$		0			0
$130$	−8.00000	−	4.00000i	−0.701646	−	0.350823i
$131$	10.0000			0.873704			0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000			0.873704			0.436852	+	0.899533i	$0.356093\pi$
$132$		0			0
$133$			16.0000i			1.38738i
$134$	−2.00000			−0.172774
$135$		0			0
$136$	−3.00000			−0.257248
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$	4.00000	−	8.00000i	0.338062	−	0.676123i
$141$		0			0
$142$			2.00000i			0.167836i
$143$			24.0000i			2.00698i
$144$		0			0
$145$		0			0
$146$	6.00000			0.496564
$147$		0			0
$148$			6.00000i			0.493197i
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$			12.0000i			0.973329i
$153$		0			0
$154$	24.0000			1.93398
$155$	16.0000	+	8.00000i	1.28515	+	0.642575i
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$			4.00000i			0.318223i
$159$		0			0
$160$	5.00000	−	10.0000i	0.395285	−	0.790569i
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	12.0000			0.937043
$165$		0			0
$166$	−8.00000			−0.620920
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	−2.00000	−	1.00000i	−0.153393	−	0.0766965i
$171$		0			0
$172$			6.00000i			0.457496i
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$	6.00000			0.452267
$177$		0			0
$178$			6.00000i			0.449719i
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$			16.0000i			1.18600i
$183$		0			0
$184$		0			0
$185$	−6.00000	+	12.0000i	−0.441129	+	0.882258i
$186$		0			0
$187$			6.00000i			0.438763i
$188$		−	4.00000i		−	0.291730i
$189$		0			0
$190$	−4.00000	+	8.00000i	−0.290191	+	0.580381i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−6.00000			−0.430775
$195$		0			0
$196$	−9.00000			−0.642857
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$	12.0000	−	9.00000i	0.848528	−	0.636396i
$201$		0			0
$202$			10.0000i			0.703598i
$203$		0			0
$204$		0			0
$205$	24.0000	+	12.0000i	1.67623	+	0.838116i
$206$	14.0000			0.975426
$207$		0			0
$208$			4.00000i			0.277350i
$209$	24.0000			1.66011
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	4.00000			0.273434
$215$	−6.00000	+	12.0000i	−0.409197	+	0.818393i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		−	14.0000i		−	0.948200i
$219$		0			0
$220$	−12.0000	−	6.00000i	−0.809040	−	0.404520i
$221$	−4.00000			−0.269069
$222$		0			0
$223$			22.0000i			1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$			22.0000i			1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	−20.0000			−1.33631
$225$		0			0
$226$	−2.00000			−0.133038
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.0000i			1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$			22.0000i			1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$		0			0
$235$	4.00000	−	8.00000i	0.260931	−	0.521862i
$236$	8.00000			0.520756
$237$		0			0
$238$			4.00000i			0.259281i
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		−	25.0000i		−	1.60706i
$243$		0			0
$244$	10.0000			0.640184
$245$	−18.0000	−	9.00000i	−1.14998	−	0.574989i
$246$		0			0
$247$			16.0000i			1.01806i
$248$		−	24.0000i		−	1.52400i
$249$		0			0
$250$	11.0000	−	2.00000i	0.695701	−	0.126491i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		0			0
$254$	−6.00000			−0.376473
$255$		0			0
$256$	−17.0000			−1.06250
$257$		−	14.0000i		−	0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$		−	14.0000i		−	0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$	4.00000	−	8.00000i	0.248069	−	0.496139i
$261$		0			0
$262$		−	10.0000i		−	0.617802i
$263$		−	4.00000i		−	0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$		−	4.00000i		−	0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$		0			0
$265$	−6.00000	+	12.0000i	−0.368577	+	0.737154i
$266$	16.0000			0.981023
$267$		0			0
$268$		−	2.00000i		−	0.122169i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$			1.00000i			0.0606339i
$273$		0			0
$274$	−14.0000			−0.845771
$275$	−18.0000	−	24.0000i	−1.08544	−	1.44725i
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		−	8.00000i		−	0.479808i
$279$		0			0
$280$	−24.0000	−	12.0000i	−1.43427	−	0.717137i
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	−2.00000			−0.118678
$285$		0			0
$286$	24.0000			1.41915
$287$		−	48.0000i		−	2.83335i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$			6.00000i			0.351123i
$293$			26.0000i			1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	16.0000	+	8.00000i	0.931556	+	0.465778i
$296$	18.0000			1.04623
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$		0			0
$300$		0			0
$301$	24.0000			1.38334
$302$			8.00000i			0.460348i
$303$		0			0
$304$	4.00000			0.229416
$305$	20.0000	+	10.0000i	1.14520	+	0.572598i
$306$		0			0
$307$		−	30.0000i		−	1.71219i	−0.516818	−	0.856095i	$-0.672884\pi$
$307$		−	30.0000i		−	1.71219i	0.516818	−	0.856095i	$-0.327116\pi$
$308$			24.0000i			1.36753i
$309$		0			0
$310$	8.00000	−	16.0000i	0.454369	−	0.908739i
$311$	2.00000			0.113410			0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000			0.113410			0.0567048	+	0.998391i	$0.481941\pi$
$312$		0			0
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	−4.00000			−0.225018
$317$			34.0000i			1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$			34.0000i			1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$		0			0
$319$		0			0
$320$	−14.0000	−	7.00000i	−0.782624	−	0.391312i
$321$		0			0
$322$		0			0
$323$			4.00000i			0.222566i
$324$		0			0
$325$	16.0000	−	12.0000i	0.887520	−	0.665640i
$326$	−4.00000			−0.221540
$327$		0			0
$328$		−	36.0000i		−	1.98777i
$329$	−16.0000			−0.882109
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		−	8.00000i		−	0.439057i
$333$		0			0
$334$	−8.00000			−0.437741
$335$	2.00000	−	4.00000i	0.109272	−	0.218543i
$336$		0			0
$337$		−	14.0000i		−	0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$		−	14.0000i		−	0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$			3.00000i			0.163178i
$339$		0			0
$340$	1.00000	−	2.00000i	0.0542326	−	0.108465i
$341$	−48.0000			−2.59935
$342$		0			0
$343$			8.00000i			0.431959i
$344$	18.0000			0.970495
$345$		0			0
$346$	6.00000			0.322562
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$	−12.0000	−	16.0000i	−0.641427	−	0.855236i
$351$		0			0
$352$			30.0000i			1.59901i
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$	−4.00000	−	2.00000i	−0.212298	−	0.106149i
$356$	−6.00000			−0.317999
$357$		0			0
$358$			4.00000i			0.211407i
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		−	18.0000i		−	0.946059i
$363$		0			0
$364$	−16.0000			−0.838628
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$		0			0
$370$	12.0000	+	6.00000i	0.623850	+	0.311925i
$371$	24.0000			1.24602
$372$		0			0
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	6.00000			0.310253
$375$		0			0
$376$	−12.0000			−0.618853
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	−8.00000	−	4.00000i	−0.410391	−	0.205196i
$381$		0			0
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$	−24.0000	+	48.0000i	−1.22315	+	2.44631i
$386$	−14.0000			−0.712581
$387$		0			0
$388$		−	6.00000i		−	0.304604i
$389$	−34.0000			−1.72387			−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000			−1.72387			−0.861934	+	0.507020i	$0.830747\pi$
$390$		0			0
$391$		0			0
$392$			27.0000i			1.36371i
$393$		0			0
$394$	6.00000			0.302276
$395$	−8.00000	−	4.00000i	−0.402524	−	0.201262i
$396$		0			0
$397$		−	26.0000i		−	1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$		−	26.0000i		−	1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$			20.0000i			1.00251i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	4.00000			0.199750			0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000			0.199750			0.0998752	+	0.995000i	$0.468156\pi$
$402$		0			0
$403$		−	32.0000i		−	1.59403i
$404$	−10.0000			−0.497519
$405$		0			0
$406$		0			0
$407$		−	36.0000i		−	1.78445i
$408$		0			0
$409$	6.00000			0.296681			0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000			0.296681			0.148340	+	0.988936i	$0.452607\pi$
$410$	12.0000	−	24.0000i	0.592638	−	1.18528i
$411$		0			0
$412$			14.0000i			0.689730i
$413$		−	32.0000i		−	1.57462i
$414$		0			0
$415$	8.00000	−	16.0000i	0.392705	−	0.785409i
$416$	−20.0000			−0.980581
$417$		0			0
$418$		−	24.0000i		−	1.17388i
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$			20.0000i			0.973585i
$423$		0			0
$424$	18.0000			0.874157
$425$	4.00000	−	3.00000i	0.194029	−	0.145521i
$426$		0			0
$427$		−	40.0000i		−	1.93574i
$428$			4.00000i			0.193347i
$429$		0			0
$430$	12.0000	+	6.00000i	0.578691	+	0.289346i
$431$	14.0000			0.674356			0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000			0.674356			0.337178	+	0.941441i	$0.390528\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−32.0000			−1.53605
$435$		0			0
$436$	14.0000			0.670478
$437$		0			0
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$	−18.0000	+	36.0000i	−0.858116	+	1.71623i
$441$		0			0
$442$			4.00000i			0.190261i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$	−12.0000	−	6.00000i	−0.568855	−	0.284427i
$446$	22.0000			1.04173
$447$		0			0
$448$			28.0000i			1.32288i
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	−72.0000			−3.39035
$452$		−	2.00000i		−	0.0940721i
$453$		0			0
$454$	20.0000			0.938647
$455$	−32.0000	−	16.0000i	−1.50018	−	0.750092i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$			26.0000i			1.21490i
$459$		0			0
$460$		0			0
$461$	−26.0000			−1.21094			−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000			−1.21094			−0.605470	+	0.795868i	$0.707015\pi$
$462$		0			0
$463$		−	22.0000i		−	1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$		−	22.0000i		−	1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$		0			0
$465$		0			0
$466$	22.0000			1.01913
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$	−8.00000	−	4.00000i	−0.369012	−	0.184506i
$471$		0			0
$472$		−	24.0000i		−	1.10469i
$473$		−	36.0000i		−	1.65528i
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$	−4.00000			−0.183340
$477$		0			0
$478$		−	12.0000i		−	0.548867i
$479$	18.0000			0.822441			0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000			0.822441			0.411220	+	0.911536i	$0.365103\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$			22.0000i			1.00207i
$483$		0			0
$484$	25.0000			1.13636
$485$	6.00000	−	12.0000i	0.272446	−	0.544892i
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		−	30.0000i		−	1.35804i
$489$		0			0
$490$	−9.00000	+	18.0000i	−0.406579	+	0.813157i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$	16.0000			0.719874
$495$		0			0
$496$	−8.00000			−0.359211
$497$			8.00000i			0.358849i
$498$		0			0
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$	2.00000	+	11.0000i	0.0894427	+	0.491935i
$501$		0			0
$502$			12.0000i			0.535586i
$503$		−	40.0000i		−	1.78351i	−0.452517	−	0.891756i	$-0.649474\pi$
$503$		−	40.0000i		−	1.78351i	0.452517	−	0.891756i	$-0.350526\pi$
$504$		0			0
$505$	−20.0000	−	10.0000i	−0.889988	−	0.444994i
$506$		0			0
$507$		0			0
$508$		−	6.00000i		−	0.266207i
$509$	10.0000			0.443242			0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000			0.443242			0.221621	+	0.975133i	$0.428865\pi$
$510$		0			0
$511$	24.0000			1.06170
$512$			11.0000i			0.486136i
$513$		0			0
$514$	−14.0000			−0.617514
$515$	−14.0000	+	28.0000i	−0.616914	+	1.23383i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		−	24.0000i		−	1.05450i
$519$		0			0
$520$	−24.0000	−	12.0000i	−1.05247	−	0.526235i
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$			30.0000i			1.31181i	0.754844	+	0.655904i	$0.227712\pi$
$523$			30.0000i			1.31181i	−0.754844	+	0.655904i	$0.772288\pi$
$524$	10.0000			0.436852
$525$		0			0
$526$	−4.00000			−0.174408
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	23.0000			1.00000
$530$	12.0000	+	6.00000i	0.521247	+	0.260623i
$531$		0			0
$532$			16.0000i			0.693688i
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	−4.00000	+	8.00000i	−0.172935	+	0.345870i
$536$	−6.00000			−0.259161
$537$		0			0
$538$		0			0
$539$	54.0000			2.32594
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$			8.00000i			0.343629i
$543$		0			0
$544$	−5.00000			−0.214373
$545$	28.0000	+	14.0000i	1.19939	+	0.599694i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		−	14.0000i		−	0.598050i
$549$		0			0
$550$	−24.0000	+	18.0000i	−1.02336	+	0.767523i
$551$		0			0
$552$		0			0
$553$			16.0000i			0.680389i
$554$	10.0000			0.424859
$555$		0			0
$556$	8.00000			0.339276
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$	−4.00000	+	8.00000i	−0.169031	+	0.338062i
$561$		0			0
$562$		−	6.00000i		−	0.253095i
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	2.00000	−	4.00000i	0.0841406	−	0.168281i
$566$	24.0000			1.00880
$567$		0			0
$568$			6.00000i			0.251754i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$			24.0000i			1.00349i
$573$		0			0
$574$	−48.0000			−2.00348
$575$		0			0
$576$		0			0
$577$			20.0000i			0.832611i	0.909225	+	0.416305i	$0.136675\pi$
$577$			20.0000i			0.832611i	−0.909225	+	0.416305i	$0.863325\pi$
$578$			1.00000i			0.0415945i
$579$		0			0
$580$		0			0
$581$	−32.0000			−1.32758
$582$		0			0
$583$		−	36.0000i		−	1.49097i
$584$	18.0000			0.744845
$585$		0			0
$586$	26.0000			1.07405
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$	8.00000	−	16.0000i	0.329355	−	0.658710i
$591$		0			0
$592$		−	6.00000i		−	0.246598i
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	−8.00000	−	4.00000i	−0.327968	−	0.163984i
$596$	6.00000			0.245770
$597$		0			0
$598$		0			0
$599$	−4.00000			−0.163436			−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000			−0.163436			−0.0817178	+	0.996656i	$0.526041\pi$
$600$		0			0
$601$	−18.0000			−0.734235			−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000			−0.734235			−0.367118	+	0.930175i	$0.619655\pi$
$602$		−	24.0000i		−	0.978167i
$603$		0			0
$604$	−8.00000			−0.325515
$605$	50.0000	+	25.0000i	2.03279	+	1.01639i
$606$		0			0
$607$		−	32.0000i		−	1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$		−	32.0000i		−	1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$			20.0000i			0.811107i
$609$		0			0
$610$	10.0000	−	20.0000i	0.404888	−	0.809776i
$611$	−16.0000			−0.647291
$612$		0			0
$613$		−	28.0000i		−	1.13091i	−0.824779	−	0.565455i	$-0.808701\pi$
$613$		−	28.0000i		−	1.13091i	0.824779	−	0.565455i	$-0.191299\pi$
$614$	−30.0000			−1.21070
$615$		0			0
$616$	72.0000			2.90096
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$	36.0000			1.44696			0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000			1.44696			0.723481	+	0.690344i	$0.242541\pi$
$620$	16.0000	+	8.00000i	0.642575	+	0.321288i
$621$		0			0
$622$		−	2.00000i		−	0.0801927i
$623$			24.0000i			0.961540i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	10.0000			0.399680
$627$		0			0
$628$		−	4.00000i		−	0.159617i
$629$	6.00000			0.239236
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			12.0000i			0.477334i
$633$		0			0
$634$	34.0000			1.35031
$635$	6.00000	−	12.0000i	0.238103	−	0.476205i
$636$		0			0
$637$			36.0000i			1.42637i
$638$		0			0
$639$		0			0
$640$	3.00000	−	6.00000i	0.118585	−	0.237171i
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$			44.0000i			1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$			44.0000i			1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$		0			0
$645$		0			0
$646$	4.00000			0.157378
$647$		−	24.0000i		−	0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$		−	24.0000i		−	0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$	−12.0000	−	16.0000i	−0.470679	−	0.627572i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$		−	6.00000i		−	0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$		−	6.00000i		−	0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$		0			0
$655$	20.0000	+	10.0000i	0.781465	+	0.390732i
$656$	−12.0000			−0.468521
$657$		0			0
$658$			16.0000i			0.623745i
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$			4.00000i			0.155464i
$663$		0			0
$664$	−24.0000			−0.931381
$665$	−16.0000	+	32.0000i	−0.620453	+	1.24091i
$666$		0			0
$667$		0			0
$668$		−	8.00000i		−	0.309529i
$669$		0			0
$670$	−4.00000	−	2.00000i	−0.154533	−	0.0772667i
$671$	−60.0000			−2.31627
$672$		0			0
$673$		−	22.0000i		−	0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$		−	22.0000i		−	0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	−14.0000			−0.539260
$675$		0			0
$676$	−3.00000			−0.115385
$677$			26.0000i			0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$			26.0000i			0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$	−6.00000	−	3.00000i	−0.230089	−	0.115045i
$681$		0			0
$682$			48.0000i			1.83801i
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$	14.0000	−	28.0000i	0.534913	−	1.06983i
$686$	8.00000			0.305441
$687$		0			0
$688$		−	6.00000i		−	0.228748i
$689$	24.0000			0.914327
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$			6.00000i			0.228086i
$693$		0			0
$694$	−12.0000			−0.455514
$695$	16.0000	+	8.00000i	0.606915	+	0.303457i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$			14.0000i			0.529908i
$699$		0			0
$700$	16.0000	−	12.0000i	0.604743	−	0.453557i
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$	42.0000			1.58293
$705$		0			0
$706$	−6.00000			−0.225813
$707$			40.0000i			1.50435i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$	−2.00000	+	4.00000i	−0.0750587	+	0.150117i
$711$		0			0
$712$			18.0000i			0.674579i
$713$		0			0
$714$		0			0
$715$	−24.0000	+	48.0000i	−0.897549	+	1.79510i
$716$	−4.00000			−0.149487
$717$		0			0
$718$		−	20.0000i		−	0.746393i
$719$	−18.0000			−0.671287			−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000			−0.671287			−0.335643	+	0.941989i	$0.608954\pi$
$720$		0			0
$721$	56.0000			2.08555
$722$			3.00000i			0.111648i
$723$		0			0
$724$	18.0000			0.668965
$725$		0			0
$726$		0			0
$727$			38.0000i			1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$			38.0000i			1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$			48.0000i			1.77900i
$729$		0			0
$730$	12.0000	+	6.00000i	0.444140	+	0.222070i
$731$	6.00000			0.221918
$732$		0			0
$733$		−	20.0000i		−	0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$		−	20.0000i		−	0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$	8.00000			0.295285
$735$		0			0
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$	−28.0000			−1.03000			−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000			−1.03000			−0.514998	+	0.857191i	$0.672207\pi$
$740$	−6.00000	+	12.0000i	−0.220564	+	0.441129i
$741$		0			0
$742$		−	24.0000i		−	0.881068i
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$	12.0000	+	6.00000i	0.439646	+	0.219823i
$746$	16.0000			0.585802
$747$		0			0
$748$			6.00000i			0.219382i
$749$	16.0000			0.584627
$750$		0			0
$751$	−12.0000			−0.437886			−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000			−0.437886			−0.218943	+	0.975738i	$0.570261\pi$
$752$			4.00000i			0.145865i
$753$		0			0
$754$		0			0
$755$	−16.0000	−	8.00000i	−0.582300	−	0.291150i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$		0			0
$760$	−12.0000	+	24.0000i	−0.435286	+	0.870572i
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$		−	56.0000i		−	2.02734i
$764$		0			0
$765$		0			0
$766$	16.0000			0.578103
$767$		−	32.0000i		−	1.15545i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$	48.0000	+	24.0000i	1.72980	+	0.864900i
$771$		0			0
$772$		−	14.0000i		−	0.503871i
$773$			30.0000i			1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$			30.0000i			1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$	−18.0000			−0.646162
$777$		0			0
$778$			34.0000i			1.21896i
$779$	−48.0000			−1.71978
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		0			0
$784$	9.00000			0.321429
$785$	4.00000	−	8.00000i	0.142766	−	0.285532i
$786$		0			0
$787$			20.0000i			0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$			20.0000i			0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$			6.00000i			0.213741i
$789$		0			0
$790$	−4.00000	+	8.00000i	−0.142314	+	0.284627i
$791$	−8.00000			−0.284447
$792$		0			0
$793$		−	40.0000i		−	1.42044i
$794$	−26.0000			−0.922705
$795$		0			0
$796$	−20.0000			−0.708881
$797$			22.0000i			0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$			22.0000i			0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$	−4.00000			−0.141510
$800$	20.0000	−	15.0000i	0.707107	−	0.530330i
$801$		0			0
$802$		−	4.00000i		−	0.141245i
$803$		−	36.0000i		−	1.27041i
$804$		0			0
$805$		0			0
$806$	−32.0000			−1.12715
$807$		0			0
$808$			30.0000i			1.05540i
$809$	−12.0000			−0.421898			−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000			−0.421898			−0.210949	+	0.977497i	$0.567655\pi$
$810$		0			0
$811$	32.0000			1.12367			0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000			1.12367			0.561836	+	0.827249i	$0.310095\pi$
$812$		0			0
$813$		0			0
$814$	−36.0000			−1.26180
$815$	4.00000	−	8.00000i	0.140114	−	0.280228i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$		−	6.00000i		−	0.209785i
$819$		0			0
$820$	24.0000	+	12.0000i	0.838116	+	0.419058i
$821$	20.0000			0.698005			0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000			0.698005			0.349002	+	0.937122i	$0.386521\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$	42.0000			1.46314
$825$		0			0
$826$	−32.0000			−1.11342
$827$		−	44.0000i		−	1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$		−	44.0000i		−	1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$	−16.0000	−	8.00000i	−0.555368	−	0.277684i
$831$		0			0
$832$			28.0000i			0.970725i
$833$			9.00000i			0.311832i
$834$		0			0
$835$	8.00000	−	16.0000i	0.276851	−	0.553703i
$836$	24.0000			0.830057
$837$		0			0
$838$			18.0000i			0.621800i
$839$	−22.0000			−0.759524			−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000			−0.759524			−0.379762	+	0.925084i	$0.623994\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$			30.0000i			1.03387i
$843$		0			0
$844$	−20.0000			−0.688428
$845$	−6.00000	−	3.00000i	−0.206406	−	0.103203i
$846$		0			0
$847$		−	100.000i		−	3.43604i
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$	−3.00000	−	4.00000i	−0.102899	−	0.137199i
$851$		0			0
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	−40.0000			−1.36877
$855$		0			0
$856$	12.0000			0.410152
$857$		−	30.0000i		−	1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$857$		−	30.0000i		−	1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$858$		0			0
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$	−6.00000	+	12.0000i	−0.204598	+	0.409197i
$861$		0			0
$862$		−	14.0000i		−	0.476842i
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−6.00000	+	12.0000i	−0.204006	+	0.408012i
$866$	−16.0000			−0.543702
$867$		0			0
$868$		−	32.0000i		−	1.08615i
$869$	24.0000			0.814144
$870$		0			0
$871$	−8.00000			−0.271070
$872$		−	42.0000i		−	1.42230i
$873$		0			0
$874$		0			0
$875$	44.0000	−	8.00000i	1.48747	−	0.270449i
$876$		0			0
$877$		−	34.0000i		−	1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$		−	34.0000i		−	1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$		−	8.00000i		−	0.269987i
$879$		0			0
$880$	12.0000	+	6.00000i	0.404520	+	0.202260i
$881$	20.0000			0.673817			0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000			0.673817			0.336909	+	0.941537i	$0.390619\pi$
$882$		0			0
$883$			22.0000i			0.740359i	0.928960	+	0.370179i	$0.120704\pi$
$883$			22.0000i			0.740359i	−0.928960	+	0.370179i	$0.879296\pi$
$884$	−4.00000			−0.134535
$885$		0			0
$886$	−4.00000			−0.134383
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$	−24.0000			−0.804934
$890$	−6.00000	+	12.0000i	−0.201120	+	0.402241i
$891$		0			0
$892$			22.0000i			0.736614i
$893$			16.0000i			0.535420i
$894$		0			0
$895$	−8.00000	−	4.00000i	−0.267411	−	0.133705i
$896$	−12.0000			−0.400892
$897$		0			0
$898$		−	36.0000i		−	1.20134i
$899$		0			0
$900$		0			0
$901$	6.00000			0.199889
$902$			72.0000i			2.39734i
$903$		0			0
$904$	−6.00000			−0.199557
$905$	36.0000	+	18.0000i	1.19668	+	0.598340i
$906$		0			0
$907$			24.0000i			0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$			24.0000i			0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$			20.0000i			0.663723i
$909$		0			0
$910$	−16.0000	+	32.0000i	−0.530395	+	1.06079i
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$			48.0000i			1.58857i
$914$	28.0000			0.926158
$915$		0			0
$916$	−26.0000			−0.859064
$917$		−	40.0000i		−	1.32092i
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		0			0
$922$			26.0000i			0.856264i
$923$			8.00000i			0.263323i
$924$		0			0
$925$	−24.0000	+	18.0000i	−0.789115	+	0.591836i
$926$	−22.0000			−0.722965
$927$		0			0
$928$		0			0
$929$	−56.0000			−1.83730			−0.918650	−	0.395072i	$-0.870720\pi$
$929$	−56.0000			−1.83730			−0.918650	+	0.395072i	$0.870720\pi$
$930$		0			0
$931$	36.0000			1.17985
$932$			22.0000i			0.720634i
$933$		0			0
$934$	−12.0000			−0.392652
$935$	−6.00000	+	12.0000i	−0.196221	+	0.392442i
$936$		0			0
$937$			60.0000i			1.96011i	0.198715	+	0.980057i	$0.436323\pi$
$937$			60.0000i			1.96011i	−0.198715	+	0.980057i	$0.563677\pi$
$938$			8.00000i			0.261209i
$939$		0			0
$940$	4.00000	−	8.00000i	0.130466	−	0.260931i
$941$	16.0000			0.521585			0.260793	−	0.965395i	$-0.416016\pi$
$941$	16.0000			0.521585			0.260793	+	0.965395i	$0.416016\pi$
$942$		0			0
$943$		0			0
$944$	−8.00000			−0.260378
$945$		0			0
$946$	−36.0000			−1.17046
$947$			44.0000i			1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$			44.0000i			1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$	−16.0000	+	12.0000i	−0.519109	+	0.389331i
$951$		0			0
$952$			12.0000i			0.388922i
$953$		−	26.0000i		−	0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$		−	26.0000i		−	0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$		0			0
$955$		0			0
$956$	12.0000			0.388108
$957$		0			0
$958$		−	18.0000i		−	0.581554i
$959$	−56.0000			−1.80833
$960$		0			0
$961$	33.0000			1.06452
$962$		−	24.0000i		−	0.773791i
$963$		0			0
$964$	−22.0000			−0.708572
$965$	14.0000	−	28.0000i	0.450676	−	0.901352i
$966$		0			0
$967$			6.00000i			0.192947i	0.995336	+	0.0964735i	$0.0307563\pi$
$967$			6.00000i			0.192947i	−0.995336	+	0.0964735i	$0.969244\pi$
$968$		−	75.0000i		−	2.41059i
$969$		0			0
$970$	−12.0000	−	6.00000i	−0.385297	−	0.192648i
$971$	32.0000			1.02693			0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000			1.02693			0.513464	+	0.858111i	$0.328362\pi$
$972$		0			0
$973$		−	32.0000i		−	1.02587i
$974$	24.0000			0.769010
$975$		0			0
$976$	−10.0000			−0.320092
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	36.0000			1.15056
$980$	−18.0000	−	9.00000i	−0.574989	−	0.287494i
$981$		0			0
$982$		0			0
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$		0			0
$985$	−6.00000	+	12.0000i	−0.191176	+	0.382352i
$986$		0			0
$987$		0			0
$988$			16.0000i			0.509028i
$989$		0			0
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$		−	40.0000i		−	1.27000i
$993$		0			0
$994$	8.00000			0.253745
$995$	−40.0000	−	20.0000i	−1.26809	−	0.634043i
$996$		0			0
$997$		−	46.0000i		−	1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$		−	46.0000i		−	1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$		−	32.0000i		−	1.01294i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.b.a.154.1		2
3.2	odd	2		255.2.b.a.154.2	yes	2
5.2	odd	4		3825.2.a.m.1.1		1
5.3	odd	4		3825.2.a.c.1.1		1
5.4	even	2	inner	765.2.b.a.154.2		2
12.11	even	2		4080.2.m.a.2449.2		2
15.2	even	4		1275.2.a.b.1.1		1
15.8	even	4		1275.2.a.f.1.1		1
15.14	odd	2		255.2.b.a.154.1	✓	2
60.59	even	2		4080.2.m.a.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.b.a.154.1	✓	2	15.14	odd	2
255.2.b.a.154.2	yes	2	3.2	odd	2
765.2.b.a.154.1		2	1.1	even	1	trivial
765.2.b.a.154.2		2	5.4	even	2	inner
1275.2.a.b.1.1		1	15.2	even	4
1275.2.a.f.1.1		1	15.8	even	4
3825.2.a.c.1.1		1	5.3	odd	4
3825.2.a.m.1.1		1	5.2	odd	4
4080.2.m.a.2449.1		2	60.59	even	2
4080.2.m.a.2449.2		2	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -3.00000i q^{8} +(1.00000 - 2.00000i) q^{10} -6.00000 q^{11} -4.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -1.00000i q^{17} -4.00000 q^{19} +(2.00000 + 1.00000i) q^{20} +6.00000i q^{22} +(3.00000 + 4.00000i) q^{25} -4.00000 q^{26} -4.00000i q^{28} +8.00000 q^{31} -5.00000i q^{32} -1.00000 q^{34} +(4.00000 - 8.00000i) q^{35} +6.00000i q^{37} +4.00000i q^{38} +(3.00000 - 6.00000i) q^{40} +12.0000 q^{41} +6.00000i q^{43} -6.00000 q^{44} -4.00000i q^{47} -9.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -4.00000i q^{52} +6.00000i q^{53} +(-12.0000 - 6.00000i) q^{55} -12.0000 q^{56} +8.00000 q^{59} +10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +(4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -1.00000i q^{68} +(-8.00000 - 4.00000i) q^{70} -2.00000 q^{71} +6.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +24.0000i q^{77} -4.00000 q^{79} +(-2.00000 - 1.00000i) q^{80} -12.0000i q^{82} -8.00000i q^{83} +(1.00000 - 2.00000i) q^{85} +6.00000 q^{86} +18.0000i q^{88} -6.00000 q^{89} -16.0000 q^{91} -4.00000 q^{94} +(-8.00000 - 4.00000i) q^{95} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{5} + 2 q^{10} - 12 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{19} + 4 q^{20} + 6 q^{25} - 8 q^{26} + 16 q^{31} - 2 q^{34} + 8 q^{35} + 6 q^{40} + 24 q^{41} - 12 q^{44} - 18 q^{49} + 8 q^{50}+ \cdots - 16 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^5 + 2 * q^10 - 12 * q^11 - 8 * q^14 - 2 * q^16 - 8 * q^19 + 4 * q^20 + 6 * q^25 - 8 * q^26 + 16 * q^31 - 2 * q^34 + 8 * q^35 + 6 * q^40 + 24 * q^41 - 12 * q^44 - 18 * q^49 + 8 * q^50 - 24 * q^55 - 24 * q^56 + 16 * q^59 + 20 * q^61 - 14 * q^64 + 8 * q^65 - 16 * q^70 - 4 * q^71 + 12 * q^74 - 8 * q^76 - 8 * q^79 - 4 * q^80 + 2 * q^85 + 12 * q^86 - 12 * q^89 - 32 * q^91 - 8 * q^94 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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