LMFDB - Embedded newform 4080.2.m.a.2449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4080,2,Mod(2449,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("4080.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-4,0,0,0,-2,0,-12,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$32.5789640247$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4080.2449
Dual form			4080.2.m.a.2449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +4.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -1.00000i q^{17} +4.00000 q^{19} -4.00000 q^{21} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -8.00000 q^{31} +6.00000i q^{33} +(4.00000 + 8.00000i) q^{35} -6.00000i q^{37} +4.00000 q^{39} -12.0000 q^{41} +6.00000i q^{43} +(2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} +6.00000i q^{53} +(12.0000 - 6.00000i) q^{55} -4.00000i q^{57} +8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{63} +(-4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{71} -6.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} +24.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +8.00000i q^{83} +(1.00000 + 2.00000i) q^{85} +6.00000 q^{89} +16.0000 q^{91} +8.00000i q^{93} +(-8.00000 + 4.00000i) q^{95} +6.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 2 q^{9} - 12 q^{11} + 2 q^{15} + 8 q^{19} - 8 q^{21} + 6 q^{25} - 16 q^{31} + 8 q^{35} + 8 q^{39} - 24 q^{41} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 24 q^{55} + 16 q^{59} + 20 q^{61} - 8 q^{65}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 - 2 * q^9 - 12 * q^11 + 2 * q^15 + 8 * q^19 - 8 * q^21 + 6 * q^25 - 16 * q^31 + 8 * q^35 + 8 * q^39 - 24 * q^41 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 24 * q^55 + 16 * q^59 + 20 * q^61 - 8 * q^65 - 4 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 + 2 * q^85 + 12 * q^89 + 32 * q^91 - 16 * q^95 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−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	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−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	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	1.00000	+	2.00000i	0.258199	+	0.516398i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		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	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$			6.00000i			1.04447i
$34$		0			0
$35$	4.00000	+	8.00000i	0.676123	+	1.35225i
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$	2.00000	−	1.00000i	0.298142	−	0.149071i
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−1.00000			−0.140028
$52$		0			0
$53$			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
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		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	$0.278858\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$	−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	$-0.0389851\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−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	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			24.0000i			2.73505i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$	1.00000	+	2.00000i	0.108465	+	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	16.0000			1.67726
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$	−8.00000	+	4.00000i	−0.820783	+	0.410391i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$	8.00000	−	4.00000i	0.780720	−	0.390360i
$106$		0			0
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$		−	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
$117$		−	4.00000i		−	0.369800i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$			12.0000i			1.08200i
$124$		0			0
$125$	−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	$-0.0857705\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$	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
$134$		0			0
$135$	−1.00000	−	2.00000i	−0.0860663	−	0.172133i
$136$		0			0
$137$		−	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	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$		−	24.0000i		−	2.00698i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			1.00000i			0.0808452i
$154$		0			0
$155$	16.0000	−	8.00000i	1.28515	−	0.642575i
$156$		0			0
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$		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	$-0.0500701\pi$
$164$		0			0
$165$	−6.00000	−	12.0000i	−0.467099	−	0.934199i
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			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
$176$		0			0
$177$		−	8.00000i		−	0.601317i
$178$		0			0
$179$	−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	$0.266738\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$	6.00000	+	12.0000i	0.441129	+	0.882258i
$186$		0			0
$187$			6.00000i			0.438763i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$		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	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	−8.00000	+	4.00000i	−0.572892	+	0.286446i
$196$		0			0
$197$			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	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$	−2.00000			−0.141069
$202$		0			0
$203$		0			0
$204$		0			0
$205$	24.0000	−	12.0000i	1.67623	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$			2.00000i			0.137038i
$214$		0			0
$215$	−6.00000	−	12.0000i	−0.409197	−	0.818393i
$216$		0			0
$217$			32.0000i			2.17230i
$218$		0			0
$219$	−6.00000			−0.405442
$220$		0			0
$221$	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	$0.736423\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		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
$231$	24.0000			1.57908
$232$		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
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	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	$0.750660\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$			16.0000i			1.01806i
$248$		0			0
$249$	8.00000			0.506979
$250$		0			0
$251$	−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
$254$		0			0
$255$	2.00000	−	1.00000i	0.125245	−	0.0626224i
$256$		0			0
$257$		−	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
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$		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	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		−	16.0000i		−	0.968364i
$274$		0			0
$275$	−18.0000	+	24.0000i	−1.08544	+	1.44725i
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$		0			0
$285$	4.00000	+	8.00000i	0.236940	+	0.473879i
$286$		0			0
$287$			48.0000i			2.83335i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	6.00000			0.351726
$292$		0			0
$293$			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
$296$		0			0
$297$		−	6.00000i		−	0.348155i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	24.0000			1.38334
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	−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	$-0.327116\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	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	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$	−4.00000	−	8.00000i	−0.225374	−	0.450749i
$316$		0			0
$317$			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
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		−	4.00000i		−	0.222566i
$324$		0			0
$325$	16.0000	+	12.0000i	0.887520	+	0.665640i
$326$		0			0
$327$		−	14.0000i		−	0.774202i
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$	2.00000	+	4.00000i	0.109272	+	0.218543i
$336$		0			0
$337$			14.0000i			0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$			14.0000i			0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$		0			0
$339$	−2.00000			−0.108625
$340$		0			0
$341$	48.0000			2.59935
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−4.00000			−0.213504
$352$		0			0
$353$		−	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
$356$		0			0
$357$			4.00000i			0.211702i
$358$		0			0
$359$	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
$362$		0			0
$363$		−	25.0000i		−	1.31216i
$364$		0			0
$365$	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
$369$	12.0000			0.624695
$370$		0			0
$371$	24.0000			1.24602
$372$		0			0
$373$		−	16.0000i		−	0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$		−	16.0000i		−	0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$		0			0
$375$	11.0000	+	2.00000i	0.568038	+	0.103280i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−6.00000			−0.307389
$382$		0			0
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$	−24.0000	−	48.0000i	−1.22315	−	2.44631i
$386$		0			0
$387$		−	6.00000i		−	0.304997i
$388$		0			0
$389$	34.0000			1.72387			0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000			1.72387			0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	10.0000i		−	0.504433i
$394$		0			0
$395$	−8.00000	+	4.00000i	−0.402524	+	0.201262i
$396$		0			0
$397$			26.0000i			1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$			26.0000i			1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$	−4.00000			−0.199750			−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000			−0.199750			−0.0998752	+	0.995000i	$0.531844\pi$
$402$		0			0
$403$		−	32.0000i		−	1.59403i
$404$		0			0
$405$	−2.00000	+	1.00000i	−0.0993808	+	0.0496904i
$406$		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	$0.452607\pi$
$410$		0			0
$411$	−14.0000			−0.690569
$412$		0			0
$413$		−	32.0000i		−	1.57462i
$414$		0			0
$415$	−8.00000	−	16.0000i	−0.392705	−	0.785409i
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−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	$0.760972\pi$
$422$		0			0
$423$		−	4.00000i		−	0.194487i
$424$		0			0
$425$	−4.00000	−	3.00000i	−0.194029	−	0.145521i
$426$		0			0
$427$		−	40.0000i		−	1.93574i
$428$		0			0
$429$	−24.0000			−1.15873
$430$		0			0
$431$	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	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$	−12.0000	+	6.00000i	−0.568855	+	0.284427i
$446$		0			0
$447$			6.00000i			0.283790i
$448$		0			0
$449$	−36.0000			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$450$		0			0
$451$	72.0000			3.39035
$452$		0			0
$453$		−	8.00000i		−	0.375873i
$454$		0			0
$455$	−32.0000	+	16.0000i	−1.50018	+	0.750092i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	1.00000			0.0466760
$460$		0			0
$461$	26.0000			1.21094			0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000			1.21094			0.605470	+	0.795868i	$0.292985\pi$
$462$		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
$465$	−8.00000	−	16.0000i	−0.370991	−	0.741982i
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	4.00000			0.184310
$472$		0			0
$473$		−	36.0000i		−	1.65528i
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	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
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−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	$0.816996\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−12.0000	+	6.00000i	−0.539360	+	0.269680i
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		0			0
$503$			40.0000i			1.78351i	0.452517	+	0.891756i	$0.350526\pi$
$503$			40.0000i			1.78351i	−0.452517	+	0.891756i	$0.649474\pi$
$504$		0			0
$505$	−20.0000	+	10.0000i	−0.889988	+	0.444994i
$506$		0			0
$507$			3.00000i			0.133235i
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$	−24.0000			−1.06170
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	−14.0000	−	28.0000i	−0.616914	−	1.23383i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$			30.0000i			1.31181i	0.754844	+	0.655904i	$0.227712\pi$
$523$			30.0000i			1.31181i	−0.754844	+	0.655904i	$0.772288\pi$
$524$		0			0
$525$	−12.0000	+	16.0000i	−0.523723	+	0.698297i
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−8.00000			−0.347170
$532$		0			0
$533$		−	48.0000i		−	2.07911i
$534$		0			0
$535$	4.00000	+	8.00000i	0.172935	+	0.345870i
$536$		0			0
$537$			4.00000i			0.172613i
$538$		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	$0.486311\pi$
$542$		0			0
$543$		−	18.0000i		−	0.772454i
$544$		0			0
$545$	−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	$-0.140630\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$	12.0000	−	6.00000i	0.509372	−	0.254686i
$556$		0			0
$557$			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
$560$		0			0
$561$	6.00000			0.253320
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	2.00000	+	4.00000i	0.0841406	+	0.168281i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	20.0000i		−	0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$		−	20.0000i		−	0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$		0			0
$579$	14.0000			0.581820
$580$		0			0
$581$	32.0000			1.32758
$582$		0			0
$583$		−	36.0000i		−	1.49097i
$584$		0			0
$585$	4.00000	+	8.00000i	0.165380	+	0.330759i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$	6.00000			0.246807
$592$		0			0
$593$			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
$596$		0			0
$597$		−	20.0000i		−	0.818546i
$598$		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	$0.619655\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$	−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	$-0.224988\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$			28.0000i			1.13091i	0.824779	+	0.565455i	$0.191299\pi$
$613$			28.0000i			1.13091i	−0.824779	+	0.565455i	$0.808701\pi$
$614$		0			0
$615$	−12.0000	−	24.0000i	−0.483887	−	0.967773i
$616$		0			0
$617$		−	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	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	24.0000i		−	0.961540i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			24.0000i			0.958468i
$628$		0			0
$629$	−6.00000			−0.239236
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		−	20.0000i		−	0.794929i
$634$		0			0
$635$	6.00000	+	12.0000i	0.238103	+	0.476205i
$636$		0			0
$637$		−	36.0000i		−	1.42637i
$638$		0			0
$639$	2.00000			0.0791188
$640$		0			0
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		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
$645$	−12.0000	+	6.00000i	−0.472500	+	0.236250i
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$	32.0000			1.25418
$652$		0			0
$653$		−	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
$656$		0			0
$657$			6.00000i			0.234082i
$658$		0			0
$659$	−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	$0.562301\pi$
$662$		0			0
$663$		−	4.00000i		−	0.155347i
$664$		0			0
$665$	16.0000	+	32.0000i	0.620453	+	1.24091i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	22.0000			0.850569
$670$		0			0
$671$	−60.0000			−2.31627
$672$		0			0
$673$			22.0000i			0.848038i	0.905653	+	0.424019i	$0.139381\pi$
$673$			22.0000i			0.848038i	−0.905653	+	0.424019i	$0.860619\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$			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
$680$		0			0
$681$	−20.0000			−0.766402
$682$		0			0
$683$		−	44.0000i		−	1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$		−	44.0000i		−	1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$		0			0
$685$	14.0000	+	28.0000i	0.534913	+	1.06983i
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		−	24.0000i		−	0.911685i
$694$		0			0
$695$	16.0000	−	8.00000i	0.606915	−	0.303457i
$696$		0			0
$697$			12.0000i			0.454532i
$698$		0			0
$699$	22.0000			0.832116
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$	−8.00000	+	4.00000i	−0.301297	+	0.150649i
$706$		0			0
$707$		−	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	$0.337644\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000	+	48.0000i	0.897549	+	1.79510i
$716$		0			0
$717$		−	12.0000i		−	0.448148i
$718$		0			0
$719$	−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
$722$		0			0
$723$			22.0000i			0.818189i
$724$		0			0
$725$		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	$0.751095\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	6.00000			0.221918
$732$		0			0
$733$			20.0000i			0.738717i	0.929287	+	0.369358i	$0.120423\pi$
$733$			20.0000i			0.738717i	−0.929287	+	0.369358i	$0.879577\pi$
$734$		0			0
$735$	−9.00000	−	18.0000i	−0.331970	−	0.663940i
$736$		0			0
$737$			12.0000i			0.442026i
$738$		0			0
$739$	28.0000			1.03000			0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000			1.03000			0.514998	+	0.857191i	$0.327793\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	12.0000	−	6.00000i	0.439646	−	0.219823i
$746$		0			0
$747$		−	8.00000i		−	0.292705i
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$		0			0
$755$	−16.0000	+	8.00000i	−0.582300	+	0.291150i
$756$		0			0
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		−	56.0000i		−	2.02734i
$764$		0			0
$765$	−1.00000	−	2.00000i	−0.0361551	−	0.0723102i
$766$		0			0
$767$			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	$0.488519\pi$
$770$		0			0
$771$	−14.0000			−0.504198
$772$		0			0
$773$			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
$776$		0			0
$777$			24.0000i			0.860995i
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−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	$0.883983\pi$
$788$		0			0
$789$	4.00000			0.142404
$790$		0			0
$791$	−8.00000			−0.284447
$792$		0			0
$793$			40.0000i			1.42044i
$794$		0			0
$795$	−12.0000	+	6.00000i	−0.425596	+	0.212798i
$796$		0			0
$797$			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
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$			36.0000i			1.27041i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$	−32.0000			−1.12367			−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000			−1.12367			−0.561836	+	0.827249i	$0.689905\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	4.00000	+	8.00000i	0.140114	+	0.280228i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$	−16.0000			−0.559085
$820$		0			0
$821$	−20.0000			−0.698005			−0.349002	−	0.937122i	$-0.613479\pi$
$821$	−20.0000			−0.698005			−0.349002	+	0.937122i	$0.613479\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		0			0
$825$	24.0000	+	18.0000i	0.835573	+	0.626680i
$826$		0			0
$827$			44.0000i			1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$			44.0000i			1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$	−10.0000			−0.346896
$832$		0			0
$833$			9.00000i			0.311832i
$834$		0			0
$835$	−8.00000	−	16.0000i	−0.276851	−	0.553703i
$836$		0			0
$837$		−	8.00000i		−	0.276520i
$838$		0			0
$839$	−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
$842$		0			0
$843$			6.00000i			0.206651i
$844$		0			0
$845$	6.00000	−	3.00000i	0.206406	−	0.103203i
$846$		0			0
$847$		−	100.000i		−	3.43604i
$848$		0			0
$849$	24.0000			0.823678
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$		0			0
$855$	8.00000	−	4.00000i	0.273594	−	0.136797i
$856$		0			0
$857$		−	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	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$		0			0
$861$	48.0000			1.63584
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	8.00000			0.271070
$872$		0			0
$873$		−	6.00000i		−	0.203069i
$874$		0			0
$875$	44.0000	+	8.00000i	1.48747	+	0.270449i
$876$		0			0
$877$			34.0000i			1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$			34.0000i			1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$		0			0
$879$	26.0000			0.876958
$880$		0			0
$881$	−20.0000			−0.673817			−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000			−0.673817			−0.336909	+	0.941537i	$0.609381\pi$
$882$		0			0
$883$			22.0000i			0.740359i	0.928960	+	0.370179i	$0.120704\pi$
$883$			22.0000i			0.740359i	−0.928960	+	0.370179i	$0.879296\pi$
$884$		0			0
$885$	8.00000	+	16.0000i	0.268917	+	0.537834i
$886$		0			0
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$		0			0
$889$	−24.0000			−0.804934
$890$		0			0
$891$	−6.00000			−0.201008
$892$		0			0
$893$			16.0000i			0.535420i
$894$		0			0
$895$	8.00000	−	4.00000i	0.267411	−	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	6.00000			0.199889
$902$		0			0
$903$		−	24.0000i		−	0.798670i
$904$		0			0
$905$	−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	$0.869547\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	−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
$914$		0			0
$915$	10.0000	+	20.0000i	0.330590	+	0.661180i
$916$		0			0
$917$		−	40.0000i		−	1.32092i
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$	−30.0000			−0.988534
$922$		0			0
$923$		−	8.00000i		−	0.263323i
$924$		0			0
$925$	−24.0000	−	18.0000i	−0.789115	−	0.591836i
$926$		0			0
$927$		−	14.0000i		−	0.459820i
$928$		0			0
$929$	56.0000			1.83730			0.918650	−	0.395072i	$-0.129280\pi$
$929$	56.0000			1.83730			0.918650	+	0.395072i	$0.129280\pi$
$930$		0			0
$931$	−36.0000			−1.17985
$932$		0			0
$933$		−	2.00000i		−	0.0654771i
$934$		0			0
$935$	−6.00000	−	12.0000i	−0.196221	−	0.392442i
$936$		0			0
$937$		−	60.0000i		−	1.96011i	−0.198715	−	0.980057i	$-0.563677\pi$
$937$		−	60.0000i		−	1.96011i	0.198715	−	0.980057i	$-0.436323\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		0			0
$941$	−16.0000			−0.521585			−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000			−0.521585			−0.260793	+	0.965395i	$0.583984\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−8.00000	+	4.00000i	−0.260240	+	0.130120i
$946$		0			0
$947$		−	44.0000i		−	1.42981i	−0.699223	−	0.714904i	$-0.746470\pi$
$947$		−	44.0000i		−	1.42981i	0.699223	−	0.714904i	$-0.253530\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$	34.0000			1.10253
$952$		0			0
$953$		−	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
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−56.0000			−1.80833
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			4.00000i			0.128898i
$964$		0			0
$965$	−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	$0.969244\pi$
$968$		0			0
$969$	−4.00000			−0.128499
$970$		0			0
$971$	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
$974$		0			0
$975$	12.0000	−	16.0000i	0.384308	−	0.512410i
$976$		0			0
$977$		−	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
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$		0			0
$985$	−6.00000	−	12.0000i	−0.191176	−	0.382352i
$986$		0			0
$987$		−	16.0000i		−	0.509286i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$		−	4.00000i		−	0.126936i
$994$		0			0
$995$	−40.0000	+	20.0000i	−1.26809	+	0.634043i
$996$		0			0
$997$			46.0000i			1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$			46.0000i			1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$		0			0
$999$	6.00000			0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.a.2449.1		2
4.3	odd	2		255.2.b.a.154.1	✓	2
5.4	even	2	inner	4080.2.m.a.2449.2		2
12.11	even	2		765.2.b.a.154.2		2
20.3	even	4		1275.2.a.b.1.1		1
20.7	even	4		1275.2.a.f.1.1		1
20.19	odd	2		255.2.b.a.154.2	yes	2
60.23	odd	4		3825.2.a.m.1.1		1
60.47	odd	4		3825.2.a.c.1.1		1
60.59	even	2		765.2.b.a.154.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.b.a.154.1	✓	2	4.3	odd	2
255.2.b.a.154.2	yes	2	20.19	odd	2
765.2.b.a.154.1		2	60.59	even	2
765.2.b.a.154.2		2	12.11	even	2
1275.2.a.b.1.1		1	20.3	even	4
1275.2.a.f.1.1		1	20.7	even	4
3825.2.a.c.1.1		1	60.47	odd	4
3825.2.a.m.1.1		1	60.23	odd	4
4080.2.m.a.2449.1		2	1.1	even	1	trivial
4080.2.m.a.2449.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +4.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -1.00000i q^{17} +4.00000 q^{19} -4.00000 q^{21} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -8.00000 q^{31} +6.00000i q^{33} +(4.00000 + 8.00000i) q^{35} -6.00000i q^{37} +4.00000 q^{39} -12.0000 q^{41} +6.00000i q^{43} +(2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} -1.00000 q^{51} +6.00000i q^{53} +(12.0000 - 6.00000i) q^{55} -4.00000i q^{57} +8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{63} +(-4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{71} -6.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} +24.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +8.00000i q^{83} +(1.00000 + 2.00000i) q^{85} +6.00000 q^{89} +16.0000 q^{91} +8.00000i q^{93} +(-8.00000 + 4.00000i) q^{95} +6.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 2 q^{9} - 12 q^{11} + 2 q^{15} + 8 q^{19} - 8 q^{21} + 6 q^{25} - 16 q^{31} + 8 q^{35} + 8 q^{39} - 24 q^{41} + 4 q^{45} - 18 q^{49} - 2 q^{51} + 24 q^{55} + 16 q^{59} + 20 q^{61} - 8 q^{65}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 - 2 * q^9 - 12 * q^11 + 2 * q^15 + 8 * q^19 - 8 * q^21 + 6 * q^25 - 16 * q^31 + 8 * q^35 + 8 * q^39 - 24 * q^41 + 4 * q^45 - 18 * q^49 - 2 * q^51 + 24 * q^55 + 16 * q^59 + 20 * q^61 - 8 * q^65 - 4 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 + 2 * q^85 + 12 * q^89 + 32 * q^91 - 16 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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