LMFDB - Embedded newform 4080.2.m.k.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)

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);

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")

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,-4,0,0,0,-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 2040)
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.k.2449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +1.00000i q^{17} -4.00000 q^{19} +4.00000 q^{21} -8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -8.00000 q^{29} +2.00000i q^{33} +(4.00000 + 8.00000i) q^{35} -2.00000i q^{37} -4.00000 q^{39} -4.00000 q^{41} +6.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} +12.0000i q^{47} -9.00000 q^{49} +1.00000 q^{51} -14.0000i q^{53} +(-4.00000 + 2.00000i) q^{55} +4.00000i q^{57} +2.00000 q^{61} -4.00000i q^{63} +(-4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -8.00000 q^{69} -14.0000 q^{71} -2.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} -8.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +(1.00000 + 2.00000i) q^{85} +8.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} +(-8.00000 + 4.00000i) q^{95} -14.0000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{19} + 8 q^{21} + 6 q^{25} - 16 q^{29} + 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} - 18 q^{49} + 2 q^{51} - 8 q^{55} + 4 q^{61} - 8 q^{65} - 16 q^{69}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^19 + 8 * q^21 + 6 * q^25 - 16 * q^29 + 8 * q^35 - 8 * q^39 - 8 * q^41 - 4 * q^45 - 18 * q^49 + 2 * q^51 - 8 * q^55 + 4 * q^61 - 8 * q^65 - 16 * q^69 - 28 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 + 2 * q^85 + 12 * q^89 + 32 * q^91 - 16 * q^95 + 4 * 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	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		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	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−8.00000			−1.48556			−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000			−1.48556			−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	4.00000	+	8.00000i	0.676123	+	1.35225i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$	−4.00000			−0.640513
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		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
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	1.00000			0.140028
$52$		0			0
$53$		−	14.0000i		−	1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$		−	14.0000i		−	1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$			4.00000i			0.529813i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		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
$69$	−8.00000			−0.963087
$70$		0			0
$71$	−14.0000			−1.66149			−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000			−1.66149			−0.830747	+	0.556650i	$0.812086\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$		−	8.00000i		−	0.911685i
$78$		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
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	1.00000	+	2.00000i	0.108465	+	0.216930i
$86$		0			0
$87$			8.00000i			0.857690i
$88$		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
$93$		0			0
$94$		0			0
$95$	−8.00000	+	4.00000i	−0.820783	+	0.410391i
$96$		0			0
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	10.0000i		−	0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$		−	10.0000i		−	0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$		0			0
$105$	8.00000	−	4.00000i	0.780720	−	0.390360i
$106$		0			0
$107$		−	20.0000i		−	1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$		−	20.0000i		−	1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$	−2.00000			−0.189832
$112$		0			0
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$	−8.00000	−	16.0000i	−0.746004	−	1.49201i
$116$		0			0
$117$			4.00000i			0.369800i
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		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	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		0			0
$139$	16.0000			1.35710			0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000			1.35710			0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$	−16.0000	+	8.00000i	−1.32873	+	0.664364i
$146$		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
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		−	1.00000i		−	0.0808452i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$	−14.0000			−1.11027
$160$		0			0
$161$	32.0000			2.52195
$162$		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
$165$	2.00000	+	4.00000i	0.155700	+	0.311400i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		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	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	16.0000	+	12.0000i	1.20949	+	0.907115i
$176$		0			0
$177$		0			0
$178$		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
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$		−	2.00000i		−	0.146254i
$188$		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
$193$			10.0000i			0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$			10.0000i			0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$		0			0
$195$	−8.00000	+	4.00000i	−0.572892	+	0.286446i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$	−2.00000			−0.141069
$202$		0			0
$203$		−	32.0000i		−	2.24596i
$204$		0			0
$205$	−8.00000	+	4.00000i	−0.558744	+	0.279372i
$206$		0			0
$207$			8.00000i			0.556038i
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$		0			0
$213$			14.0000i			0.959264i
$214$		0			0
$215$	6.00000	+	12.0000i	0.409197	+	0.818393i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.00000			−0.135147
$220$		0			0
$221$	4.00000			0.269069
$222$		0			0
$223$		−	18.0000i		−	1.20537i	−0.797980	−	0.602685i	$-0.794098\pi$
$223$		−	18.0000i		−	1.20537i	0.797980	−	0.602685i	$-0.205902\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	−8.00000			−0.526361
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	12.0000	+	24.0000i	0.782794	+	1.56559i
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	−28.0000			−1.81117			−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000			−1.81117			−0.905585	+	0.424165i	$0.860568\pi$
$240$		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
$249$	−16.0000			−1.01396
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$	2.00000	−	1.00000i	0.125245	−	0.0626224i
$256$		0			0
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	8.00000			0.495188
$262$		0			0
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$		0			0
$265$	−14.0000	−	28.0000i	−0.860013	−	1.72003i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		−	16.0000i		−	0.968364i
$274$		0			0
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		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
$287$		−	16.0000i		−	0.944450i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−14.0000			−0.820695
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$	−32.0000			−1.85061
$300$		0			0
$301$	−24.0000			−1.38334
$302$		0			0
$303$			6.00000i			0.344691i
$304$		0			0
$305$	4.00000	−	2.00000i	0.229039	−	0.114520i
$306$		0			0
$307$			2.00000i			0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		0			0
$309$	−10.0000			−0.568880
$310$		0			0
$311$	30.0000			1.70114			0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000			1.70114			0.850572	+	0.525859i	$0.176256\pi$
$312$		0			0
$313$			34.0000i			1.92179i	0.276907	+	0.960897i	$0.410691\pi$
$313$			34.0000i			1.92179i	−0.276907	+	0.960897i	$0.589309\pi$
$314$		0			0
$315$	−4.00000	−	8.00000i	−0.225374	−	0.450749i
$316$		0			0
$317$			30.0000i			1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$			30.0000i			1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$		0			0
$319$	16.0000			0.895828
$320$		0			0
$321$	−20.0000			−1.11629
$322$		0			0
$323$		−	4.00000i		−	0.222566i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$		0			0
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	−48.0000			−2.64633
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$			2.00000i			0.109599i
$334$		0			0
$335$	−2.00000	−	4.00000i	−0.109272	−	0.218543i
$336$		0			0
$337$		−	22.0000i		−	1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$		−	22.0000i		−	1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$		0			0
$339$	2.00000			0.108625
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$	−16.0000	+	8.00000i	−0.861411	+	0.430706i
$346$		0			0
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		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	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$		0			0
$355$	−28.0000	+	14.0000i	−1.48609	+	0.743043i
$356$		0			0
$357$			4.00000i			0.211702i
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−2.00000	−	4.00000i	−0.104685	−	0.209370i
$366$		0			0
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$		0			0
$369$	4.00000			0.208232
$370$		0			0
$371$	56.0000			2.90738
$372$		0			0
$373$			32.0000i			1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$			32.0000i			1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$		0			0
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$		0			0
$377$			32.0000i			1.64808i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	18.0000			0.922168
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$	−8.00000	−	16.0000i	−0.407718	−	0.815436i
$386$		0			0
$387$		−	6.00000i		−	0.304997i
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$			18.0000i			0.907980i
$394$		0			0
$395$	8.00000	−	4.00000i	0.402524	−	0.201262i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$	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
$403$		0			0
$404$		0			0
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$		0			0
$407$			4.00000i			0.198273i
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	14.0000			0.690569
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−16.0000	−	32.0000i	−0.785409	−	1.57082i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$		−	12.0000i		−	0.583460i
$424$		0			0
$425$	4.00000	+	3.00000i	0.194029	+	0.145521i
$426$		0			0
$427$			8.00000i			0.387147i
$428$		0			0
$429$	8.00000			0.386244
$430$		0			0
$431$	2.00000			0.0963366			0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000			0.0963366			0.0481683	+	0.998839i	$0.484662\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	8.00000	+	16.0000i	0.383571	+	0.767141i
$436$		0			0
$437$			32.0000i			1.53077i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$		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	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$	32.0000	−	16.0000i	1.50018	−	0.750092i
$456$		0			0
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$		0			0
$459$	−1.00000			−0.0466760
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$		−	14.0000i		−	0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$		−	14.0000i		−	0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	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
$470$		0			0
$471$	−4.00000			−0.184310
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$	−12.0000	+	16.0000i	−0.550598	+	0.734130i
$476$		0			0
$477$			14.0000i			0.641016i
$478$		0			0
$479$	14.0000			0.639676			0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000			0.639676			0.319838	+	0.947472i	$0.396371\pi$
$480$		0			0
$481$	−8.00000			−0.364769
$482$		0			0
$483$		−	32.0000i		−	1.45605i
$484$		0			0
$485$	−14.0000	−	28.0000i	−0.635707	−	1.27141i
$486$		0			0
$487$			16.0000i			0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$			16.0000i			0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$	4.00000	−	2.00000i	0.179787	−	0.0898933i
$496$		0			0
$497$		−	56.0000i		−	2.51194i
$498$		0			0
$499$	−40.0000			−1.79065			−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000			−1.79065			−0.895323	+	0.445418i	$0.853055\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	16.0000i		−	0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$		−	16.0000i		−	0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$		0			0
$505$	−12.0000	+	6.00000i	−0.533993	+	0.266996i
$506$		0			0
$507$			3.00000i			0.133235i
$508$		0			0
$509$	38.0000			1.68432			0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000			1.68432			0.842160	+	0.539227i	$0.181284\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		−	4.00000i		−	0.176604i
$514$		0			0
$515$	−10.0000	−	20.0000i	−0.440653	−	0.881305i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	28.0000			1.22670			0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000			1.22670			0.613351	+	0.789810i	$0.289821\pi$
$522$		0			0
$523$		−	2.00000i		−	0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$		−	2.00000i		−	0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$		0			0
$525$	12.0000	−	16.0000i	0.523723	−	0.698297i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$			16.0000i			0.693037i
$534$		0			0
$535$	−20.0000	−	40.0000i	−0.864675	−	1.72935i
$536$		0			0
$537$			4.00000i			0.172613i
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$		−	10.0000i		−	0.429141i
$544$		0			0
$545$	12.0000	−	6.00000i	0.514024	−	0.257012i
$546$		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
$549$	−2.00000			−0.0853579
$550$		0			0
$551$	32.0000			1.36325
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	−4.00000	+	2.00000i	−0.169791	+	0.0848953i
$556$		0			0
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	−2.00000			−0.0844401
$562$		0			0
$563$		−	12.0000i		−	0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$		−	12.0000i		−	0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$		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
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−32.0000	−	24.0000i	−1.33449	−	1.00087i
$576$		0			0
$577$			20.0000i			0.832611i	0.909225	+	0.416305i	$0.136675\pi$
$577$			20.0000i			0.832611i	−0.909225	+	0.416305i	$0.863325\pi$
$578$		0			0
$579$	10.0000			0.415586
$580$		0			0
$581$	64.0000			2.65517
$582$		0			0
$583$			28.0000i			1.15964i
$584$		0			0
$585$	4.00000	+	8.00000i	0.165380	+	0.330759i
$586$		0			0
$587$			40.0000i			1.65098i	0.564419	+	0.825488i	$0.309100\pi$
$587$			40.0000i			1.65098i	−0.564419	+	0.825488i	$0.690900\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$	−8.00000	+	4.00000i	−0.327968	+	0.163984i
$596$		0			0
$597$		−	4.00000i		−	0.163709i
$598$		0			0
$599$	4.00000			0.163436			0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000			0.163436			0.0817178	+	0.996656i	$0.473959\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$	−14.0000	+	7.00000i	−0.569181	+	0.284590i
$606$		0			0
$607$			8.00000i			0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$			8.00000i			0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$		0			0
$609$	−32.0000			−1.29671
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$			20.0000i			0.807792i	0.914805	+	0.403896i	$0.132344\pi$
$613$			20.0000i			0.807792i	−0.914805	+	0.403896i	$0.867656\pi$
$614$		0			0
$615$	4.00000	+	8.00000i	0.161296	+	0.322591i
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$			24.0000i			0.961540i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		−	8.00000i		−	0.319489i
$628$		0			0
$629$	2.00000			0.0797452
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$			28.0000i			1.11290i
$634$		0			0
$635$	18.0000	+	36.0000i	0.714308	+	1.42862i
$636$		0			0
$637$			36.0000i			1.42637i
$638$		0			0
$639$	14.0000			0.553831
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		0			0
$645$	12.0000	−	6.00000i	0.472500	−	0.236250i
$646$		0			0
$647$			16.0000i			0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$			16.0000i			0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	−36.0000	+	18.0000i	−1.40664	+	0.703318i
$656$		0			0
$657$			2.00000i			0.0780274i
$658$		0			0
$659$	24.0000			0.934907			0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000			0.934907			0.467454	+	0.884018i	$0.345171\pi$
$660$		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
$667$			64.0000i			2.47809i
$668$		0			0
$669$	−18.0000			−0.695920
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$			34.0000i			1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$			34.0000i			1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$	56.0000			2.14908
$680$		0			0
$681$	−4.00000			−0.153280
$682$		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
$687$		−	6.00000i		−	0.228914i
$688$		0			0
$689$	−56.0000			−2.13343
$690$		0			0
$691$	−24.0000			−0.913003			−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000			−0.913003			−0.456502	+	0.889723i	$0.650898\pi$
$692$		0			0
$693$			8.00000i			0.303895i
$694$		0			0
$695$	32.0000	−	16.0000i	1.21383	−	0.606915i
$696$		0			0
$697$		−	4.00000i		−	0.151511i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$			8.00000i			0.301726i
$704$		0			0
$705$	24.0000	−	12.0000i	0.903892	−	0.451946i
$706$		0			0
$707$		−	24.0000i		−	0.902613i
$708$		0			0
$709$	50.0000			1.87779			0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000			1.87779			0.938895	+	0.344204i	$0.111851\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$		0			0
$714$		0			0
$715$	8.00000	+	16.0000i	0.299183	+	0.598366i
$716$		0			0
$717$			28.0000i			1.04568i
$718$		0			0
$719$	−14.0000			−0.522112			−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000			−0.522112			−0.261056	+	0.965324i	$0.584071\pi$
$720$		0			0
$721$	40.0000			1.48968
$722$		0			0
$723$			22.0000i			0.818189i
$724$		0			0
$725$	−24.0000	+	32.0000i	−0.891338	+	1.18845i
$726$		0			0
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−6.00000			−0.221918
$732$		0			0
$733$			12.0000i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$			12.0000i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$		0			0
$735$	9.00000	+	18.0000i	0.331970	+	0.663940i
$736$		0			0
$737$			4.00000i			0.147342i
$738$		0			0
$739$	36.0000			1.32428			0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000			1.32428			0.662141	+	0.749380i	$0.269648\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−12.0000	+	6.00000i	−0.439646	+	0.219823i
$746$		0			0
$747$			16.0000i			0.585409i
$748$		0			0
$749$	80.0000			2.92314
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$		−	20.0000i		−	0.728841i
$754$		0			0
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$			40.0000i			1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$			40.0000i			1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	−26.0000			−0.942499			−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000			−0.942499			−0.471250	+	0.882000i	$0.656197\pi$
$762$		0			0
$763$			24.0000i			0.868858i
$764$		0			0
$765$	−1.00000	−	2.00000i	−0.0361551	−	0.0723102i
$766$		0			0
$767$		0			0
$768$		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
$771$	−18.0000			−0.648254
$772$		0			0
$773$			26.0000i			0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$			26.0000i			0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	8.00000i		−	0.286998i
$778$		0			0
$779$	16.0000			0.573259
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$		−	8.00000i		−	0.285897i
$784$		0			0
$785$	−4.00000	−	8.00000i	−0.142766	−	0.285532i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$		0			0
$789$	12.0000			0.427211
$790$		0			0
$791$	−8.00000			−0.284447
$792$		0			0
$793$		−	8.00000i		−	0.284088i
$794$		0			0
$795$	−28.0000	+	14.0000i	−0.993058	+	0.496529i
$796$		0			0
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$			4.00000i			0.141157i
$804$		0			0
$805$	64.0000	−	32.0000i	2.25570	−	1.12785i
$806$		0			0
$807$		−	24.0000i		−	0.844840i
$808$		0			0
$809$	−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
$811$	−24.0000			−0.842754			−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000			−0.842754			−0.421377	+	0.906886i	$0.638453\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		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
$821$	−12.0000			−0.418803			−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000			−0.418803			−0.209401	+	0.977830i	$0.567152\pi$
$822$		0			0
$823$			4.00000i			0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$			4.00000i			0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$		0			0
$825$	8.00000	+	6.00000i	0.278524	+	0.208893i
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$	18.0000			0.624413
$832$		0			0
$833$		−	9.00000i		−	0.311832i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−42.0000			−1.45000			−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000			−1.45000			−0.725001	+	0.688748i	$0.758161\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$			6.00000i			0.206651i
$844$		0			0
$845$	−6.00000	+	3.00000i	−0.206406	+	0.103203i
$846$		0			0
$847$		−	28.0000i		−	0.962091i
$848$		0			0
$849$	24.0000			0.823678
$850$		0			0
$851$	−16.0000			−0.548473
$852$		0			0
$853$			38.0000i			1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$			38.0000i			1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$		0			0
$855$	8.00000	−	4.00000i	0.273594	−	0.136797i
$856$		0			0
$857$		−	50.0000i		−	1.70797i	−0.520300	−	0.853984i	$-0.674180\pi$
$857$		−	50.0000i		−	1.70797i	0.520300	−	0.853984i	$-0.325820\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$	−16.0000			−0.545279
$862$		0			0
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$		0			0
$867$			1.00000i			0.0339618i
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$	−8.00000			−0.271070
$872$		0			0
$873$			14.0000i			0.473828i
$874$		0			0
$875$	44.0000	+	8.00000i	1.48747	+	0.270449i
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$		0			0
$879$	14.0000			0.472208
$880$		0			0
$881$	4.00000			0.134763			0.0673817	−	0.997727i	$-0.478535\pi$
$881$	4.00000			0.134763			0.0673817	+	0.997727i	$0.478535\pi$
$882$		0			0
$883$		−	10.0000i		−	0.336527i	−0.985742	−	0.168263i	$-0.946184\pi$
$883$		−	10.0000i		−	0.336527i	0.985742	−	0.168263i	$-0.0538159\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			24.0000i			0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$			24.0000i			0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$		0			0
$889$	−72.0000			−2.41480
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$	−8.00000	+	4.00000i	−0.267411	+	0.133705i
$896$		0			0
$897$			32.0000i			1.06845i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	14.0000			0.466408
$902$		0			0
$903$			24.0000i			0.798670i
$904$		0			0
$905$	20.0000	−	10.0000i	0.664822	−	0.332411i
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$			32.0000i			1.05905i
$914$		0			0
$915$	−2.00000	−	4.00000i	−0.0661180	−	0.132236i
$916$		0			0
$917$		−	72.0000i		−	2.37765i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	2.00000			0.0659022
$922$		0			0
$923$			56.0000i			1.84326i
$924$		0			0
$925$	−8.00000	−	6.00000i	−0.263038	−	0.197279i
$926$		0			0
$927$			10.0000i			0.328443i
$928$		0			0
$929$	−32.0000			−1.04989			−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000			−1.04989			−0.524943	+	0.851137i	$0.675913\pi$
$930$		0			0
$931$	36.0000			1.17985
$932$		0			0
$933$		−	30.0000i		−	0.982156i
$934$		0			0
$935$	−2.00000	−	4.00000i	−0.0654070	−	0.130814i
$936$		0			0
$937$		−	4.00000i		−	0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$		−	4.00000i		−	0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$		0			0
$939$	34.0000			1.10955
$940$		0			0
$941$	−40.0000			−1.30396			−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000			−1.30396			−0.651981	+	0.758235i	$0.726062\pi$
$942$		0			0
$943$			32.0000i			1.04206i
$944$		0			0
$945$	−8.00000	+	4.00000i	−0.260240	+	0.130120i
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−8.00000			−0.259691
$950$		0			0
$951$	30.0000			0.972817
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	16.0000i		−	0.517207i
$958$		0			0
$959$	−56.0000			−1.80833
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$			20.0000i			0.644491i
$964$		0			0
$965$	10.0000	+	20.0000i	0.321911	+	0.643823i
$966$		0			0
$967$			46.0000i			1.47926i	0.673014	+	0.739630i	$0.265000\pi$
$967$			46.0000i			1.47926i	−0.673014	+	0.739630i	$0.735000\pi$
$968$		0			0
$969$	−4.00000			−0.128499
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$			64.0000i			2.05175i
$974$		0			0
$975$	−12.0000	+	16.0000i	−0.384308	+	0.512410i
$976$		0			0
$977$			2.00000i			0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$			2.00000i			0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$		−	32.0000i		−	1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$		−	32.0000i		−	1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$		0			0
$985$	−6.00000	−	12.0000i	−0.191176	−	0.382352i
$986$		0			0
$987$			48.0000i			1.52786i
$988$		0			0
$989$	48.0000			1.52631
$990$		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
$995$	8.00000	−	4.00000i	0.253617	−	0.126809i
$996$		0			0
$997$		−	6.00000i		−	0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$		−	6.00000i		−	0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$		0			0
$999$	2.00000			0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.k.2449.1		2
4.3	odd	2		2040.2.m.g.409.2	yes	2
5.4	even	2	inner	4080.2.m.k.2449.2		2
20.19	odd	2		2040.2.m.g.409.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.m.g.409.1	✓	2	20.19	odd	2
2040.2.m.g.409.2	yes	2	4.3	odd	2
4080.2.m.k.2449.1		2	1.1	even	1	trivial
4080.2.m.k.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +1.00000i q^{17} -4.00000 q^{19} +4.00000 q^{21} -8.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -8.00000 q^{29} +2.00000i q^{33} +(4.00000 + 8.00000i) q^{35} -2.00000i q^{37} -4.00000 q^{39} -4.00000 q^{41} +6.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} +12.0000i q^{47} -9.00000 q^{49} +1.00000 q^{51} -14.0000i q^{53} +(-4.00000 + 2.00000i) q^{55} +4.00000i q^{57} +2.00000 q^{61} -4.00000i q^{63} +(-4.00000 - 8.00000i) q^{65} -2.00000i q^{67} -8.00000 q^{69} -14.0000 q^{71} -2.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} -8.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -16.0000i q^{83} +(1.00000 + 2.00000i) q^{85} +8.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} +(-8.00000 + 4.00000i) q^{95} -14.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{19} + 8 q^{21} + 6 q^{25} - 16 q^{29} + 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} - 18 q^{49} + 2 q^{51} - 8 q^{55} + 4 q^{61} - 8 q^{65} - 16 q^{69}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^19 + 8 * q^21 + 6 * q^25 - 16 * q^29 + 8 * q^35 - 8 * q^39 - 8 * q^41 - 4 * q^45 - 18 * q^49 + 2 * q^51 - 8 * q^55 + 4 * q^61 - 8 * q^65 - 16 * q^69 - 28 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 + 2 * q^85 + 12 * q^89 + 32 * q^91 - 16 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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