LMFDB - Embedded newform 5040.2.t.v.1009.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5040,2,Mod(1009,5040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, 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("5040.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.t (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-2,0,0,0,0,0,12,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.v.1009.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} +2.00000 q^{11} -6.42864i q^{13} +4.42864i q^{17} -2.42864 q^{19} +1.37778i q^{23} +(-4.80642 + 1.37778i) q^{25} +0.755569 q^{29} -5.18421 q^{31} +(-2.21432 + 0.311108i) q^{35} -7.61285i q^{37} +8.23506 q^{41} -10.1017i q^{43} -2.75557i q^{47} -1.00000 q^{49} -9.18421i q^{53} +(-0.622216 - 4.42864i) q^{55} -14.1017 q^{59} +6.85728 q^{61} +(-14.2351 + 2.00000i) q^{65} +2.75557i q^{67} +2.00000 q^{71} -1.57136i q^{73} -2.00000i q^{77} -4.85728 q^{79} +11.6128i q^{83} +(9.80642 - 1.37778i) q^{85} +4.62222 q^{89} -6.42864 q^{91} +(0.755569 + 5.37778i) q^{95} +11.9398i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) $ 6 * q - 2 * q^5 + 12 * q^11 + 12 * q^19 - 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^41 - 6 * q^49 - 4 * q^55 - 32 * q^59 - 12 * q^61 - 32 * q^65 + 12 * q^71 + 24 * q^79 + 32 * q^85 + 28 * q^89 - 12 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	6.42864i		−	1.78298i	−0.453037	−	0.891492i	$-0.649659\pi$
$13$		−	6.42864i		−	1.78298i	0.453037	−	0.891492i	$-0.350341\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.42864i			1.07410i	0.843550	+	0.537051i	$0.180462\pi$
$17$			4.42864i			1.07410i	−0.843550	+	0.537051i	$0.819538\pi$
$18$		0			0
$19$	−2.42864			−0.557168			−0.278584	−	0.960412i	$-0.589865\pi$
$19$	−2.42864			−0.557168			−0.278584	+	0.960412i	$0.589865\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.37778i			0.287288i	0.989629	+	0.143644i	$0.0458820\pi$
$23$			1.37778i			0.287288i	−0.989629	+	0.143644i	$0.954118\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.755569			0.140306			0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569			0.140306			0.0701528	+	0.997536i	$0.477651\pi$
$30$		0			0
$31$	−5.18421			−0.931111			−0.465556	−	0.885019i	$-0.654145\pi$
$31$	−5.18421			−0.931111			−0.465556	+	0.885019i	$0.654145\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.21432	+	0.311108i	−0.374288	+	0.0525868i
$36$		0			0
$37$		−	7.61285i		−	1.25154i	−0.780006	−	0.625772i	$-0.784784\pi$
$37$		−	7.61285i		−	1.25154i	0.780006	−	0.625772i	$-0.215216\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.23506			1.28610			0.643050	−	0.765824i	$-0.277669\pi$
$41$	8.23506			1.28610			0.643050	+	0.765824i	$0.277669\pi$
$42$		0			0
$43$		−	10.1017i		−	1.54050i	−0.637744	−	0.770248i	$-0.720132\pi$
$43$		−	10.1017i		−	1.54050i	0.637744	−	0.770248i	$-0.279868\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.75557i		−	0.401941i	−0.979597	−	0.200971i	$-0.935590\pi$
$47$		−	2.75557i		−	0.401941i	0.979597	−	0.200971i	$-0.0644095\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.18421i		−	1.26155i	−0.775967	−	0.630774i	$-0.782737\pi$
$53$		−	9.18421i		−	1.26155i	0.775967	−	0.630774i	$-0.217263\pi$
$54$		0			0
$55$	−0.622216	−	4.42864i	−0.0838995	−	0.597158i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.1017			−1.83589			−0.917943	−	0.396712i	$-0.870151\pi$
$59$	−14.1017			−1.83589			−0.917943	+	0.396712i	$0.870151\pi$
$60$		0			0
$61$	6.85728			0.877985			0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728			0.877985			0.438992	+	0.898491i	$0.355336\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.2351	+	2.00000i	−1.76564	+	0.248069i
$66$		0			0
$67$			2.75557i			0.336646i	0.985732	+	0.168323i	$0.0538352\pi$
$67$			2.75557i			0.336646i	−0.985732	+	0.168323i	$0.946165\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$		−	1.57136i		−	0.183914i	−0.995763	−	0.0919569i	$-0.970688\pi$
$73$		−	1.57136i		−	0.183914i	0.995763	−	0.0919569i	$-0.0293122\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	−4.85728			−0.546487			−0.273243	−	0.961945i	$-0.588096\pi$
$79$	−4.85728			−0.546487			−0.273243	+	0.961945i	$0.588096\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			11.6128i			1.27468i	0.770585	+	0.637338i	$0.219964\pi$
$83$			11.6128i			1.27468i	−0.770585	+	0.637338i	$0.780036\pi$
$84$		0			0
$85$	9.80642	−	1.37778i	1.06366	−	0.149442i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.62222			0.489954			0.244977	−	0.969529i	$-0.421220\pi$
$89$	4.62222			0.489954			0.244977	+	0.969529i	$0.421220\pi$
$90$		0			0
$91$	−6.42864			−0.673905
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.755569	+	5.37778i	0.0775197	+	0.551749i
$96$		0			0
$97$			11.9398i			1.21230i	0.795350	+	0.606150i	$0.207287\pi$
$97$			11.9398i			1.21230i	−0.795350	+	0.606150i	$0.792713\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.47949			−0.147215			−0.0736076	−	0.997287i	$-0.523451\pi$
$101$	−1.47949			−0.147215			−0.0736076	+	0.997287i	$0.523451\pi$
$102$		0			0
$103$			8.85728i			0.872734i	0.899769	+	0.436367i	$0.143735\pi$
$103$			8.85728i			0.872734i	−0.899769	+	0.436367i	$0.856265\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.76494i		−	0.170623i	−0.996354	−	0.0853114i	$-0.972811\pi$
$107$		−	1.76494i		−	0.170623i	0.996354	−	0.0853114i	$-0.0271885\pi$
$108$		0			0
$109$	−5.61285			−0.537613			−0.268807	−	0.963194i	$-0.586629\pi$
$109$	−5.61285			−0.537613			−0.268807	+	0.963194i	$0.586629\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.2859i			1.06169i	0.847469	+	0.530845i	$0.178125\pi$
$113$			11.2859i			1.06169i	−0.847469	+	0.530845i	$0.821875\pi$
$114$		0			0
$115$	3.05086	−	0.428639i	0.284494	−	0.0399708i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.42864			0.405973
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.54617	+	10.2143i	0.406622	+	0.913597i
$126$		0			0
$127$		−	12.8573i		−	1.14090i	−0.821333	−	0.570450i	$-0.806769\pi$
$127$		−	12.8573i		−	1.14090i	0.821333	−	0.570450i	$-0.193231\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.10171			−0.183627			−0.0918136	−	0.995776i	$-0.529266\pi$
$131$	−2.10171			−0.183627			−0.0918136	+	0.995776i	$0.529266\pi$
$132$		0			0
$133$			2.42864i			0.210590i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			15.9398i			1.36183i	0.732364	+	0.680914i	$0.238417\pi$
$137$			15.9398i			1.36183i	−0.732364	+	0.680914i	$0.761583\pi$
$138$		0			0
$139$	11.6731			0.990097			0.495048	−	0.868865i	$-0.335150\pi$
$139$	11.6731			0.990097			0.495048	+	0.868865i	$0.335150\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.8573i		−	1.07518i
$144$		0			0
$145$	−0.235063	−	1.67307i	−0.0195209	−	0.138941i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.2257			−1.73888			−0.869438	−	0.494041i	$-0.835519\pi$
$149$	−21.2257			−1.73888			−0.869438	+	0.494041i	$0.835519\pi$
$150$		0			0
$151$	−16.8573			−1.37183			−0.685913	−	0.727684i	$-0.740597\pi$
$151$	−16.8573			−1.37183			−0.685913	+	0.727684i	$0.740597\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.61285	+	11.4795i	0.129547	+	0.922055i
$156$		0			0
$157$			10.4286i			0.832296i	0.909297	+	0.416148i	$0.136620\pi$
$157$			10.4286i			0.832296i	−0.909297	+	0.416148i	$0.863380\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.37778			0.108585
$162$		0			0
$163$		−	20.8573i		−	1.63367i	−0.576873	−	0.816834i	$-0.695727\pi$
$163$		−	20.8573i		−	1.63367i	0.576873	−	0.816834i	$-0.304273\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.3461i		−	1.18752i	−0.804642	−	0.593760i	$-0.797643\pi$
$167$		−	15.3461i		−	1.18752i	0.804642	−	0.593760i	$-0.202357\pi$
$168$		0			0
$169$	−28.3274			−2.17903
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2.06022i		−	0.156636i	−0.996928	−	0.0783179i	$-0.975045\pi$
$173$		−	2.06022i		−	0.156636i	0.996928	−	0.0783179i	$-0.0249549\pi$
$174$		0			0
$175$	1.37778	+	4.80642i	0.104151	+	0.363331i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	−12.1017			−0.899513			−0.449757	−	0.893151i	$-0.648489\pi$
$181$	−12.1017			−0.899513			−0.449757	+	0.893151i	$0.648489\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.8573	+	2.36842i	−1.23937	+	0.174129i
$186$		0			0
$187$			8.85728i			0.647708i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.488863			0.0353729			0.0176864	−	0.999844i	$-0.494370\pi$
$191$	0.488863			0.0353729			0.0176864	+	0.999844i	$0.494370\pi$
$192$		0			0
$193$		−	22.9590i		−	1.65262i	−0.563212	−	0.826312i	$-0.690435\pi$
$193$		−	22.9590i		−	1.65262i	0.563212	−	0.826312i	$-0.309565\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.18421i		−	0.0843713i	−0.999110	−	0.0421857i	$-0.986568\pi$
$197$		−	1.18421i		−	0.0843713i	0.999110	−	0.0421857i	$-0.0134321\pi$
$198$		0			0
$199$	8.79706			0.623607			0.311803	−	0.950147i	$-0.399067\pi$
$199$	8.79706			0.623607			0.311803	+	0.950147i	$0.399067\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	0.755569i		−	0.0530305i
$204$		0			0
$205$	−2.56199	−	18.2351i	−0.178937	−	1.27359i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.85728			−0.335985
$210$		0			0
$211$	−23.2257			−1.59892			−0.799461	−	0.600717i	$-0.794882\pi$
$211$	−23.2257			−1.59892			−0.799461	+	0.600717i	$0.794882\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−22.3684	+	3.14272i	−1.52551	+	0.214332i
$216$		0			0
$217$			5.18421i			0.351927i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.4701			1.91511
$222$		0			0
$223$			15.2257i			1.01959i	0.860297	+	0.509794i	$0.170278\pi$
$223$			15.2257i			1.01959i	−0.860297	+	0.509794i	$0.829722\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			14.3684i			0.953665i	0.878994	+	0.476833i	$0.158215\pi$
$227$			14.3684i			0.953665i	−0.878994	+	0.476833i	$0.841785\pi$
$228$		0			0
$229$	5.61285			0.370907			0.185454	−	0.982653i	$-0.440625\pi$
$229$	5.61285			0.370907			0.185454	+	0.982653i	$0.440625\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	23.2859i		−	1.52551i	−0.646687	−	0.762756i	$-0.723846\pi$
$233$		−	23.2859i		−	1.52551i	0.646687	−	0.762756i	$-0.276154\pi$
$234$		0			0
$235$	−6.10171	+	0.857279i	−0.398032	+	0.0559227i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.48886			0.549099			0.274549	−	0.961573i	$-0.411471\pi$
$239$	8.48886			0.549099			0.274549	+	0.961573i	$0.411471\pi$
$240$		0			0
$241$	−7.24443			−0.466655			−0.233327	−	0.972398i	$-0.574961\pi$
$241$	−7.24443			−0.466655			−0.233327	+	0.972398i	$0.574961\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.311108	+	2.21432i	0.0198759	+	0.141468i
$246$		0			0
$247$			15.6128i			0.993422i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.6128			1.74291			0.871454	−	0.490478i	$-0.163178\pi$
$251$	27.6128			1.74291			0.871454	+	0.490478i	$0.163178\pi$
$252$		0			0
$253$			2.75557i			0.173241i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	0.428639i		−	0.0267378i	−0.999911	−	0.0133689i	$-0.995744\pi$
$257$		−	0.428639i		−	0.0267378i	0.999911	−	0.0133689i	$-0.00425558\pi$
$258$		0			0
$259$	−7.61285			−0.473039
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.37778i		−	0.578259i	−0.957290	−	0.289129i	$-0.906634\pi$
$263$		−	9.37778i		−	0.578259i	0.957290	−	0.289129i	$-0.0933658\pi$
$264$		0			0
$265$	−20.3368	+	2.85728i	−1.24928	+	0.175521i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.74620			0.106468			0.0532339	−	0.998582i	$-0.483047\pi$
$269$	1.74620			0.106468			0.0532339	+	0.998582i	$0.483047\pi$
$270$		0			0
$271$	2.69535			0.163731			0.0818653	−	0.996643i	$-0.473912\pi$
$271$	2.69535			0.163731			0.0818653	+	0.996643i	$0.473912\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.61285	+	2.75557i	−0.579677	+	0.166167i
$276$		0			0
$277$			5.12399i			0.307870i	0.988081	+	0.153935i	$0.0491947\pi$
$277$			5.12399i			0.307870i	−0.988081	+	0.153935i	$0.950805\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−23.9813			−1.43060			−0.715301	−	0.698816i	$-0.753710\pi$
$281$	−23.9813			−1.43060			−0.715301	+	0.698816i	$0.753710\pi$
$282$		0			0
$283$		−	2.36842i		−	0.140788i	−0.997519	−	0.0703939i	$-0.977574\pi$
$283$		−	2.36842i		−	0.140788i	0.997519	−	0.0703939i	$-0.0224256\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.23506i		−	0.486100i
$288$		0			0
$289$	−2.61285			−0.153697
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.42864i		−	0.492406i	−0.969218	−	0.246203i	$-0.920817\pi$
$293$		−	8.42864i		−	0.492406i	0.969218	−	0.246203i	$-0.0791831\pi$
$294$		0			0
$295$	4.38715	+	31.2257i	0.255430	+	1.81803i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.85728			0.512230
$300$		0			0
$301$	−10.1017			−0.582253
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.13335	−	15.1842i	−0.122155	−	0.869445i
$306$		0			0
$307$		−	22.5718i		−	1.28824i	−0.764923	−	0.644121i	$-0.777223\pi$
$307$		−	22.5718i		−	1.28824i	0.764923	−	0.644121i	$-0.222777\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.0830			−1.36562			−0.682810	−	0.730596i	$-0.739242\pi$
$311$	−24.0830			−1.36562			−0.682810	+	0.730596i	$0.739242\pi$
$312$		0			0
$313$			9.65433i			0.545695i	0.962057	+	0.272848i	$0.0879655\pi$
$313$			9.65433i			0.545695i	−0.962057	+	0.272848i	$0.912035\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.04149i		−	0.339324i	−0.985502	−	0.169662i	$-0.945732\pi$
$317$		−	6.04149i		−	0.339324i	0.985502	−	0.169662i	$-0.0542676\pi$
$318$		0			0
$319$	1.51114			0.0846075
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10.7556i		−	0.598456i
$324$		0			0
$325$	8.85728	+	30.8988i	0.491313	+	1.71396i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.75557			−0.151919
$330$		0			0
$331$	−13.5111			−0.742639			−0.371320	−	0.928505i	$-0.621095\pi$
$331$	−13.5111			−0.742639			−0.371320	+	0.928505i	$0.621095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.10171	−	0.857279i	0.333372	−	0.0468382i
$336$		0			0
$337$		−	10.4889i		−	0.571365i	−0.958324	−	0.285682i	$-0.907780\pi$
$337$		−	10.4889i		−	0.571365i	0.958324	−	0.285682i	$-0.0922202\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.3684			−0.561481
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.7239i			0.897787i	0.893585	+	0.448894i	$0.148182\pi$
$347$			16.7239i			0.897787i	−0.893585	+	0.448894i	$0.851818\pi$
$348$		0			0
$349$	16.3684			0.876181			0.438091	−	0.898931i	$-0.355655\pi$
$349$	16.3684			0.876181			0.438091	+	0.898931i	$0.355655\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	0.549086i		−	0.0292249i	−0.999893	−	0.0146124i	$-0.995349\pi$
$353$		−	0.549086i		−	0.0292249i	0.999893	−	0.0146124i	$-0.00465145\pi$
$354$		0			0
$355$	−0.622216	−	4.42864i	−0.0330238	−	0.235048i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.285442			0.0150651			0.00753253	−	0.999972i	$-0.497602\pi$
$359$	0.285442			0.0150651			0.00753253	+	0.999972i	$0.497602\pi$
$360$		0			0
$361$	−13.1017			−0.689564
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.47949	+	0.488863i	−0.182125	+	0.0255882i
$366$		0			0
$367$			1.71456i			0.0894992i	0.998998	+	0.0447496i	$0.0142490\pi$
$367$			1.71456i			0.0894992i	−0.998998	+	0.0447496i	$0.985751\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.18421			−0.476820
$372$		0			0
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	4.85728i		−	0.250163i
$378$		0			0
$379$	−4.85728			−0.249502			−0.124751	−	0.992188i	$-0.539813\pi$
$379$	−4.85728			−0.249502			−0.124751	+	0.992188i	$0.539813\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	8.38715i		−	0.428563i	−0.976772	−	0.214282i	$-0.931259\pi$
$383$		−	8.38715i		−	0.428563i	0.976772	−	0.214282i	$-0.0687411\pi$
$384$		0			0
$385$	−4.42864	+	0.622216i	−0.225704	+	0.0317110i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.95899			0.454239			0.227119	−	0.973867i	$-0.427069\pi$
$389$	8.95899			0.454239			0.227119	+	0.973867i	$0.427069\pi$
$390$		0			0
$391$	−6.10171			−0.308577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.51114	+	10.7556i	0.0760336	+	0.541171i
$396$		0			0
$397$			2.54909i			0.127935i	0.997952	+	0.0639675i	$0.0203754\pi$
$397$			2.54909i			0.127935i	−0.997952	+	0.0639675i	$0.979625\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.958989			−0.0478896			−0.0239448	−	0.999713i	$-0.507623\pi$
$401$	−0.958989			−0.0478896			−0.0239448	+	0.999713i	$0.507623\pi$
$402$		0			0
$403$			33.3274i			1.66016i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	15.2257i		−	0.754710i
$408$		0			0
$409$	31.9813			1.58137			0.790686	−	0.612222i	$-0.209724\pi$
$409$	31.9813			1.58137			0.790686	+	0.612222i	$0.209724\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			14.1017i			0.693900i
$414$		0			0
$415$	25.7146	−	3.61285i	1.26228	−	0.177348i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.470127			0.0229672			0.0114836	−	0.999934i	$-0.496345\pi$
$419$	0.470127			0.0229672			0.0114836	+	0.999934i	$0.496345\pi$
$420$		0			0
$421$	−33.6128			−1.63819			−0.819095	−	0.573658i	$-0.805524\pi$
$421$	−33.6128			−1.63819			−0.819095	+	0.573658i	$0.805524\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.10171	−	21.2859i	−0.295976	−	1.03252i
$426$		0			0
$427$		−	6.85728i		−	0.331847i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.7146			0.564270			0.282135	−	0.959375i	$-0.408957\pi$
$431$	11.7146			0.564270			0.282135	+	0.959375i	$0.408957\pi$
$432$		0			0
$433$		−	0.0602231i		−	0.00289414i	−0.999999	−	0.00144707i	$-0.999539\pi$
$433$		−	0.0602231i		−	0.00289414i	0.999999	−	0.00144707i	$-0.000460616\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.34614i		−	0.160068i
$438$		0			0
$439$	−22.4286			−1.07046			−0.535230	−	0.844706i	$-0.679775\pi$
$439$	−22.4286			−1.07046			−0.535230	+	0.844706i	$0.679775\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	23.9496i		−	1.13788i	−0.822379	−	0.568940i	$-0.807353\pi$
$443$		−	23.9496i		−	1.13788i	0.822379	−	0.568940i	$-0.192647\pi$
$444$		0			0
$445$	−1.43801	−	10.2351i	−0.0681681	−	0.485189i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	29.4291			1.38885			0.694423	−	0.719567i	$-0.255660\pi$
$449$	29.4291			1.38885			0.694423	+	0.719567i	$0.255660\pi$
$450$		0			0
$451$	16.4701			0.775548
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	+	14.2351i	0.0937614	+	0.667350i
$456$		0			0
$457$			3.14272i			0.147010i	0.997295	+	0.0735051i	$0.0234185\pi$
$457$			3.14272i			0.147010i	−0.997295	+	0.0735051i	$0.976581\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.37778			0.157319			0.0786596	−	0.996902i	$-0.474936\pi$
$461$	3.37778			0.157319			0.0786596	+	0.996902i	$0.474936\pi$
$462$		0			0
$463$		−	20.8573i		−	0.969320i	−0.874703	−	0.484660i	$-0.838943\pi$
$463$		−	20.8573i		−	0.969320i	0.874703	−	0.484660i	$-0.161057\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14.3684i			0.664891i	0.943122	+	0.332446i	$0.107874\pi$
$467$			14.3684i			0.664891i	−0.943122	+	0.332446i	$0.892126\pi$
$468$		0			0
$469$	2.75557			0.127240
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.2034i		−	0.928954i
$474$		0			0
$475$	11.6731	−	3.34614i	0.535597	−	0.153532i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.36842			−0.290980			−0.145490	−	0.989360i	$-0.546476\pi$
$479$	−6.36842			−0.290980			−0.145490	+	0.989360i	$0.546476\pi$
$480$		0			0
$481$	−48.9403			−2.23148
$482$		0			0
$483$		0			0
$484$		0			0
$485$	26.4385	−	3.71456i	1.20051	−	0.168669i
$486$		0			0
$487$			17.3274i			0.785180i	0.919714	+	0.392590i	$0.128421\pi$
$487$			17.3274i			0.785180i	−0.919714	+	0.392590i	$0.871579\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$			3.34614i			0.150703i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.00000i		−	0.0897123i
$498$		0			0
$499$	23.3461			1.04512			0.522558	−	0.852603i	$-0.324978\pi$
$499$	23.3461			1.04512			0.522558	+	0.852603i	$0.324978\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			0.387152i			0.0172623i	0.999963	+	0.00863113i	$0.00274741\pi$
$503$			0.387152i			0.0172623i	−0.999963	+	0.00863113i	$0.997253\pi$
$504$		0			0
$505$	0.460282	+	3.27607i	0.0204823	+	0.145783i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.9496			−1.32749			−0.663747	−	0.747957i	$-0.731035\pi$
$509$	−29.9496			−1.32749			−0.663747	+	0.747957i	$0.731035\pi$
$510$		0			0
$511$	−1.57136			−0.0695129
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.6128	−	2.75557i	0.864245	−	0.121425i
$516$		0			0
$517$		−	5.51114i		−	0.242380i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.5205			0.811398			0.405699	−	0.914007i	$-0.367028\pi$
$521$	18.5205			0.811398			0.405699	+	0.914007i	$0.367028\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	22.9590i		−	1.00011i
$528$		0			0
$529$	21.1017			0.917466
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	52.9403i		−	2.29310i
$534$		0			0
$535$	−3.90813	+	0.549086i	−0.168963	+	0.0237390i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	14.5906			0.627298			0.313649	−	0.949539i	$-0.398449\pi$
$541$	14.5906			0.627298			0.313649	+	0.949539i	$0.398449\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.74620	+	12.4286i	0.0747990	+	0.532384i
$546$		0			0
$547$		−	18.7556i		−	0.801930i	−0.916093	−	0.400965i	$-0.868675\pi$
$547$		−	18.7556i		−	0.801930i	0.916093	−	0.400965i	$-0.131325\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.83500			−0.0781738
$552$		0			0
$553$			4.85728i			0.206553i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			31.8765i			1.35065i	0.737520	+	0.675325i	$0.235997\pi$
$557$			31.8765i			1.35065i	−0.737520	+	0.675325i	$0.764003\pi$
$558$		0			0
$559$	−64.9403			−2.74668
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.01874i		−	0.0850796i	−0.999095	−	0.0425398i	$-0.986455\pi$
$563$		−	2.01874i		−	0.0850796i	0.999095	−	0.0425398i	$-0.0135449\pi$
$564$		0			0
$565$	24.9906	−	3.51114i	1.05136	−	0.147715i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.9590			−1.21402			−0.607012	−	0.794693i	$-0.707632\pi$
$569$	−28.9590			−1.21402			−0.607012	+	0.794693i	$0.707632\pi$
$570$		0			0
$571$	−8.97773			−0.375706			−0.187853	−	0.982197i	$-0.560153\pi$
$571$	−8.97773			−0.375706			−0.187853	+	0.982197i	$0.560153\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.89829	−	6.62222i	−0.0791642	−	0.276165i
$576$		0			0
$577$		−	28.6766i		−	1.19382i	−0.802307	−	0.596911i	$-0.796394\pi$
$577$		−	28.6766i		−	1.19382i	0.802307	−	0.596911i	$-0.203606\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11.6128			0.481782
$582$		0			0
$583$		−	18.3684i		−	0.760742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	45.2070i		−	1.86589i	−0.360018	−	0.932945i	$-0.617229\pi$
$587$		−	45.2070i		−	1.86589i	0.360018	−	0.932945i	$-0.382771\pi$
$588$		0			0
$589$	12.5906			0.518786
$590$		0			0
$591$		0			0
$592$		0			0
$593$			18.2636i			0.749998i	0.927025	+	0.374999i	$0.122357\pi$
$593$			18.2636i			0.749998i	−0.927025	+	0.374999i	$0.877643\pi$
$594$		0			0
$595$	−1.37778	−	9.80642i	−0.0564837	−	0.402024i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.7368			0.929002			0.464501	−	0.885573i	$-0.346234\pi$
$599$	22.7368			0.929002			0.464501	+	0.885573i	$0.346234\pi$
$600$		0			0
$601$	0.488863			0.0199411			0.00997056	−	0.999950i	$-0.496826\pi$
$601$	0.488863			0.0199411			0.00997056	+	0.999950i	$0.496826\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.17775	+	15.5002i	0.0885383	+	0.630174i
$606$		0			0
$607$			20.2034i			0.820032i	0.912078	+	0.410016i	$0.134477\pi$
$607$			20.2034i			0.820032i	−0.912078	+	0.410016i	$0.865523\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17.7146			−0.716654
$612$		0			0
$613$		−	10.3684i		−	0.418776i	−0.977833	−	0.209388i	$-0.932853\pi$
$613$		−	10.3684i		−	0.418776i	0.977833	−	0.209388i	$-0.0671472\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			39.2859i			1.58159i	0.612080	+	0.790796i	$0.290333\pi$
$617$			39.2859i			1.58159i	−0.612080	+	0.790796i	$0.709667\pi$
$618$		0			0
$619$	42.8988			1.72425			0.862123	−	0.506698i	$-0.169134\pi$
$619$	42.8988			1.72425			0.862123	+	0.506698i	$0.169134\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	4.62222i		−	0.185185i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	33.7146			1.34429
$630$		0			0
$631$	−15.3461			−0.610920			−0.305460	−	0.952205i	$-0.598810\pi$
$631$	−15.3461			−0.610920			−0.305460	+	0.952205i	$0.598810\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−28.4701	+	4.00000i	−1.12980	+	0.158735i
$636$		0			0
$637$			6.42864i			0.254712i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.6735			−1.21153			−0.605766	−	0.795643i	$-0.707133\pi$
$641$	−30.6735			−1.21153			−0.605766	+	0.795643i	$0.707133\pi$
$642$		0			0
$643$		−	49.0607i		−	1.93477i	−0.253320	−	0.967383i	$-0.581523\pi$
$643$		−	49.0607i		−	1.93477i	0.253320	−	0.967383i	$-0.418477\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	15.3461i		−	0.603319i	−0.953416	−	0.301660i	$-0.902459\pi$
$647$		−	15.3461i		−	0.603319i	0.953416	−	0.301660i	$-0.0975406\pi$
$648$		0			0
$649$	−28.2034			−1.10708
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	19.4697i		−	0.761906i	−0.924594	−	0.380953i	$-0.875596\pi$
$653$		−	19.4697i		−	0.761906i	0.924594	−	0.380953i	$-0.124404\pi$
$654$		0			0
$655$	0.653858	+	4.65386i	0.0255484	+	0.181841i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.9403			−1.20526			−0.602631	−	0.798020i	$-0.705881\pi$
$659$	−30.9403			−1.20526			−0.602631	+	0.798020i	$0.705881\pi$
$660$		0			0
$661$	47.7975			1.85911			0.929554	−	0.368685i	$-0.120192\pi$
$661$	47.7975			1.85911			0.929554	+	0.368685i	$0.120192\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.37778	−	0.755569i	0.208542	−	0.0292997i
$666$		0			0
$667$			1.04101i			0.0403081i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.7146			0.529445
$672$		0			0
$673$		−	27.8163i		−	1.07224i	−0.844142	−	0.536119i	$-0.819890\pi$
$673$		−	27.8163i		−	1.07224i	0.844142	−	0.536119i	$-0.180110\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			19.0005i			0.730248i	0.930959	+	0.365124i	$0.118973\pi$
$677$			19.0005i			0.730248i	−0.930959	+	0.365124i	$0.881027\pi$
$678$		0			0
$679$	11.9398			0.458207
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	4.52051i		−	0.172972i	−0.996253	−	0.0864862i	$-0.972436\pi$
$683$		−	4.52051i		−	0.172972i	0.996253	−	0.0864862i	$-0.0275638\pi$
$684$		0			0
$685$	35.2958	−	4.95899i	1.34858	−	0.189473i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−59.0420			−2.24932
$690$		0			0
$691$	1.18421			0.0450494			0.0225247	−	0.999746i	$-0.492830\pi$
$691$	1.18421			0.0450494			0.0225247	+	0.999746i	$0.492830\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.63158	−	25.8479i	−0.137754	−	0.980467i
$696$		0			0
$697$			36.4701i			1.38140i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.6735			1.00745			0.503723	−	0.863865i	$-0.331963\pi$
$701$	26.6735			1.00745			0.503723	+	0.863865i	$0.331963\pi$
$702$		0			0
$703$			18.4889i			0.697321i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.47949i			0.0556421i
$708$		0			0
$709$	18.2034			0.683644			0.341822	−	0.939765i	$-0.388956\pi$
$709$	18.2034			0.683644			0.341822	+	0.939765i	$0.388956\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.14272i		−	0.267497i
$714$		0			0
$715$	−28.4701	+	4.00000i	−1.06472	+	0.149592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.85728			−0.181146			−0.0905730	−	0.995890i	$-0.528870\pi$
$719$	−4.85728			−0.181146			−0.0905730	+	0.995890i	$0.528870\pi$
$720$		0			0
$721$	8.85728			0.329862
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.63158	+	1.04101i	−0.134874	+	0.0386622i
$726$		0			0
$727$		−	21.0607i		−	0.781098i	−0.920582	−	0.390549i	$-0.872285\pi$
$727$		−	21.0607i		−	0.781098i	0.920582	−	0.390549i	$-0.127715\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	44.7368			1.65465
$732$		0			0
$733$		−	9.45091i		−	0.349077i	−0.984650	−	0.174539i	$-0.944157\pi$
$733$		−	9.45091i		−	0.349077i	0.984650	−	0.174539i	$-0.0558434\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.51114i			0.203005i
$738$		0			0
$739$	−8.20342			−0.301768			−0.150884	−	0.988551i	$-0.548212\pi$
$739$	−8.20342			−0.301768			−0.150884	+	0.988551i	$0.548212\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	8.33677i		−	0.305847i	−0.988238	−	0.152923i	$-0.951131\pi$
$743$		−	8.33677i		−	0.305847i	0.988238	−	0.152923i	$-0.0488687\pi$
$744$		0			0
$745$	6.60348	+	47.0005i	0.241933	+	1.72196i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.76494			−0.0644894
$750$		0			0
$751$	25.9180			0.945760			0.472880	−	0.881127i	$-0.343214\pi$
$751$	25.9180			0.945760			0.472880	+	0.881127i	$0.343214\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.24443	+	37.3274i	0.190864	+	1.35848i
$756$		0			0
$757$			8.94025i			0.324939i	0.986714	+	0.162470i	$0.0519459\pi$
$757$			8.94025i			0.324939i	−0.986714	+	0.162470i	$0.948054\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.825636			0.0299293			0.0149646	−	0.999888i	$-0.495236\pi$
$761$	0.825636			0.0299293			0.0149646	+	0.999888i	$0.495236\pi$
$762$		0			0
$763$			5.61285i			0.203199i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			90.6548i			3.27336i
$768$		0			0
$769$	−21.2257			−0.765418			−0.382709	−	0.923869i	$-0.625009\pi$
$769$	−21.2257			−0.765418			−0.382709	+	0.923869i	$0.625009\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			29.4893i			1.06066i	0.847792	+	0.530329i	$0.177932\pi$
$773$			29.4893i			1.06066i	−0.847792	+	0.530329i	$0.822068\pi$
$774$		0			0
$775$	24.9175	−	7.14272i	0.895063	−	0.256574i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.0000			−0.716574
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.0923	−	3.24443i	0.824201	−	0.115799i
$786$		0			0
$787$			34.4514i			1.22806i	0.789283	+	0.614030i	$0.210453\pi$
$787$			34.4514i			1.22806i	−0.789283	+	0.614030i	$0.789547\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.2859			0.401281
$792$		0			0
$793$		−	44.0830i		−	1.56543i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	18.9175i		−	0.670092i	−0.942202	−	0.335046i	$-0.891248\pi$
$797$		−	18.9175i		−	0.670092i	0.942202	−	0.335046i	$-0.108752\pi$
$798$		0			0
$799$	12.2034			0.431726
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	3.14272i		−	0.110904i
$804$		0			0
$805$	−0.428639	−	3.05086i	−0.0151076	−	0.107529i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21.2257			0.746256			0.373128	−	0.927780i	$-0.378285\pi$
$809$	21.2257			0.746256			0.373128	+	0.927780i	$0.378285\pi$
$810$		0			0
$811$	21.5081			0.755251			0.377625	−	0.925958i	$-0.376741\pi$
$811$	21.5081			0.755251			0.377625	+	0.925958i	$0.376741\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−46.1847	+	6.48886i	−1.61778	+	0.227295i
$816$		0			0
$817$			24.5334i			0.858315i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	46.2034			1.61251			0.806255	−	0.591568i	$-0.201491\pi$
$821$	46.2034			1.61251			0.806255	+	0.591568i	$0.201491\pi$
$822$		0			0
$823$		−	17.8350i		−	0.621689i	−0.950461	−	0.310845i	$-0.899388\pi$
$823$		−	17.8350i		−	0.621689i	0.950461	−	0.310845i	$-0.100612\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			35.2128i			1.22447i	0.790676	+	0.612234i	$0.209729\pi$
$827$			35.2128i			1.22447i	−0.790676	+	0.612234i	$0.790271\pi$
$828$		0			0
$829$	−14.3872			−0.499686			−0.249843	−	0.968286i	$-0.580379\pi$
$829$	−14.3872			−0.499686			−0.249843	+	0.968286i	$0.580379\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	4.42864i		−	0.153443i
$834$		0			0
$835$	−33.9813	+	4.77430i	−1.17597	+	0.165222i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.51114			−0.0521703			−0.0260851	−	0.999660i	$-0.508304\pi$
$839$	−1.51114			−0.0521703			−0.0260851	+	0.999660i	$0.508304\pi$
$840$		0			0
$841$	−28.4291			−0.980314
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.81288	+	62.7259i	0.303172	+	2.15784i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.4889			0.359554
$852$		0			0
$853$			15.4064i			0.527504i	0.964591	+	0.263752i	$0.0849600\pi$
$853$			15.4064i			0.527504i	−0.964591	+	0.263752i	$0.915040\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.8578i		−	0.678328i	−0.940727	−	0.339164i	$-0.889856\pi$
$857$		−	19.8578i		−	0.678328i	0.940727	−	0.339164i	$-0.110144\pi$
$858$		0			0
$859$	2.42864			0.0828641			0.0414321	−	0.999141i	$-0.486808\pi$
$859$	2.42864			0.0828641			0.0414321	+	0.999141i	$0.486808\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			39.2958i			1.33764i	0.743423	+	0.668822i	$0.233201\pi$
$863$			39.2958i			1.33764i	−0.743423	+	0.668822i	$0.766799\pi$
$864$		0			0
$865$	−4.56199	+	0.640951i	−0.155112	+	0.0217930i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.71456			−0.329544
$870$		0			0
$871$	17.7146			0.600235
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.2143	−	4.54617i	0.345307	−	0.153689i
$876$		0			0
$877$		−	56.2864i		−	1.90066i	−0.311249	−	0.950328i	$-0.600747\pi$
$877$		−	56.2864i		−	1.90066i	0.311249	−	0.950328i	$-0.399253\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.33677			0.0787279			0.0393640	−	0.999225i	$-0.487467\pi$
$881$	2.33677			0.0787279			0.0393640	+	0.999225i	$0.487467\pi$
$882$		0			0
$883$			33.7146i			1.13459i	0.823516	+	0.567293i	$0.192009\pi$
$883$			33.7146i			1.13459i	−0.823516	+	0.567293i	$0.807991\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			47.8992i			1.60830i	0.594427	+	0.804150i	$0.297379\pi$
$887$			47.8992i			1.60830i	−0.594427	+	0.804150i	$0.702621\pi$
$888$		0			0
$889$	−12.8573			−0.431219
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6.69228i			0.223949i
$894$		0			0
$895$	3.11108	+	22.1432i	0.103992	+	0.740165i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.91703			−0.130640
$900$		0			0
$901$	40.6735			1.35503
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.76494	+	26.7971i	0.125151	+	0.890764i
$906$		0			0
$907$		−	23.7591i		−	0.788908i	−0.918916	−	0.394454i	$-0.870934\pi$
$907$		−	23.7591i		−	0.788908i	0.918916	−	0.394454i	$-0.129066\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.9403			0.760045			0.380022	−	0.924977i	$-0.375916\pi$
$911$	22.9403			0.760045			0.380022	+	0.924977i	$0.375916\pi$
$912$		0			0
$913$			23.2257i			0.768658i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.10171i			0.0694046i
$918$		0			0
$919$	16.9777			0.560043			0.280022	−	0.959994i	$-0.409658\pi$
$919$	16.9777			0.560043			0.280022	+	0.959994i	$0.409658\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	12.8573i		−	0.423202i
$924$		0			0
$925$	10.4889	+	36.5906i	0.344872	+	1.20309i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	39.3403			1.29071			0.645357	−	0.763881i	$-0.276709\pi$
$929$	39.3403			1.29071			0.645357	+	0.763881i	$0.276709\pi$
$930$		0			0
$931$	2.42864			0.0795954
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.6128	−	2.75557i	0.641409	−	0.0901167i
$936$		0			0
$937$		−	17.7748i		−	0.580677i	−0.956924	−	0.290338i	$-0.906232\pi$
$937$		−	17.7748i		−	0.580677i	0.956924	−	0.290338i	$-0.0937679\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−35.5812			−1.15991			−0.579957	−	0.814647i	$-0.696931\pi$
$941$	−35.5812			−1.15991			−0.579957	+	0.814647i	$0.696931\pi$
$942$		0			0
$943$			11.3461i			0.369481i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			30.5018i			0.991174i	0.868558	+	0.495587i	$0.165047\pi$
$947$			30.5018i			0.991174i	−0.868558	+	0.495587i	$0.834953\pi$
$948$		0			0
$949$	−10.1017			−0.327915
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	51.1655i		−	1.65741i	−0.559684	−	0.828706i	$-0.689077\pi$
$953$		−	51.1655i		−	1.65741i	0.559684	−	0.828706i	$-0.310923\pi$
$954$		0			0
$955$	−0.152089	−	1.08250i	−0.00492148	−	0.0350288i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.9398			0.514722
$960$		0			0
$961$	−4.12399			−0.133032
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−50.8385	+	7.14272i	−1.63655	+	0.229932i
$966$		0			0
$967$		−	47.8992i		−	1.54034i	−0.637841	−	0.770168i	$-0.720172\pi$
$967$		−	47.8992i		−	1.54034i	0.637841	−	0.770168i	$-0.279828\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	40.6735			1.30528			0.652638	−	0.757670i	$-0.273662\pi$
$971$	40.6735			1.30528			0.652638	+	0.757670i	$0.273662\pi$
$972$		0			0
$973$		−	11.6731i		−	0.374221i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			27.4893i			0.879462i	0.898130	+	0.439731i	$0.144926\pi$
$977$			27.4893i			0.879462i	−0.898130	+	0.439731i	$0.855074\pi$
$978$		0			0
$979$	9.24443			0.295453
$980$		0			0
$981$		0			0
$982$		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
$985$	−2.62222	+	0.368416i	−0.0835507	+	0.0117387i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.9180			0.442566
$990$		0			0
$991$	34.6923			1.10204			0.551018	−	0.834493i	$-0.314239\pi$
$991$	34.6923			1.10204			0.551018	+	0.834493i	$0.314239\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.73683	−	19.4795i	−0.0867634	−	0.617541i
$996$		0			0
$997$		−	28.6766i		−	0.908197i	−0.890951	−	0.454099i	$-0.849961\pi$
$997$		−	28.6766i		−	0.908197i	0.890951	−	0.454099i	$-0.150039\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.v.1009.3		6
3.2	odd	2		1680.2.t.k.1009.5		6
4.3	odd	2		315.2.d.e.64.2		6
5.4	even	2	inner	5040.2.t.v.1009.4		6
12.11	even	2		105.2.d.b.64.5	yes	6
15.2	even	4		8400.2.a.dj.1.1		3
15.8	even	4		8400.2.a.dg.1.3		3
15.14	odd	2		1680.2.t.k.1009.2		6
20.3	even	4		1575.2.a.x.1.1		3
20.7	even	4		1575.2.a.w.1.3		3
20.19	odd	2		315.2.d.e.64.5		6
28.27	even	2		2205.2.d.l.1324.2		6
60.23	odd	4		525.2.a.j.1.3		3
60.47	odd	4		525.2.a.k.1.1		3
60.59	even	2		105.2.d.b.64.2	✓	6
84.11	even	6		735.2.q.e.79.2		12
84.23	even	6		735.2.q.e.214.5		12
84.47	odd	6		735.2.q.f.214.5		12
84.59	odd	6		735.2.q.f.79.2		12
84.83	odd	2		735.2.d.b.589.5		6
140.139	even	2		2205.2.d.l.1324.5		6
420.59	odd	6		735.2.q.f.79.5		12
420.83	even	4		3675.2.a.bi.1.3		3
420.167	even	4		3675.2.a.bj.1.1		3
420.179	even	6		735.2.q.e.79.5		12
420.299	odd	6		735.2.q.f.214.2		12
420.359	even	6		735.2.q.e.214.2		12
420.419	odd	2		735.2.d.b.589.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.2	✓	6	60.59	even	2
105.2.d.b.64.5	yes	6	12.11	even	2
315.2.d.e.64.2		6	4.3	odd	2
315.2.d.e.64.5		6	20.19	odd	2
525.2.a.j.1.3		3	60.23	odd	4
525.2.a.k.1.1		3	60.47	odd	4
735.2.d.b.589.2		6	420.419	odd	2
735.2.d.b.589.5		6	84.83	odd	2
735.2.q.e.79.2		12	84.11	even	6
735.2.q.e.79.5		12	420.179	even	6
735.2.q.e.214.2		12	420.359	even	6
735.2.q.e.214.5		12	84.23	even	6
735.2.q.f.79.2		12	84.59	odd	6
735.2.q.f.79.5		12	420.59	odd	6
735.2.q.f.214.2		12	420.299	odd	6
735.2.q.f.214.5		12	84.47	odd	6
1575.2.a.w.1.3		3	20.7	even	4
1575.2.a.x.1.1		3	20.3	even	4
1680.2.t.k.1009.2		6	15.14	odd	2
1680.2.t.k.1009.5		6	3.2	odd	2
2205.2.d.l.1324.2		6	28.27	even	2
2205.2.d.l.1324.5		6	140.139	even	2
3675.2.a.bi.1.3		3	420.83	even	4
3675.2.a.bj.1.1		3	420.167	even	4
5040.2.t.v.1009.3		6	1.1	even	1	trivial
5040.2.t.v.1009.4		6	5.4	even	2	inner
8400.2.a.dg.1.3		3	15.8	even	4
8400.2.a.dj.1.1		3	15.2	even	4

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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} +2.00000 q^{11} -6.42864i q^{13} +4.42864i q^{17} -2.42864 q^{19} +1.37778i q^{23} +(-4.80642 + 1.37778i) q^{25} +0.755569 q^{29} -5.18421 q^{31} +(-2.21432 + 0.311108i) q^{35} -7.61285i q^{37} +8.23506 q^{41} -10.1017i q^{43} -2.75557i q^{47} -1.00000 q^{49} -9.18421i q^{53} +(-0.622216 - 4.42864i) q^{55} -14.1017 q^{59} +6.85728 q^{61} +(-14.2351 + 2.00000i) q^{65} +2.75557i q^{67} +2.00000 q^{71} -1.57136i q^{73} -2.00000i q^{77} -4.85728 q^{79} +11.6128i q^{83} +(9.80642 - 1.37778i) q^{85} +4.62222 q^{89} -6.42864 q^{91} +(0.755569 + 5.37778i) q^{95} +11.9398i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) \) 6 * q - 2 * q^5 + 12 * q^11 + 12 * q^19 - 2 * q^25 + 4 * q^29 - 4 * q^31 - 4 * q^41 - 6 * q^49 - 4 * q^55 - 32 * q^59 - 12 * q^61 - 32 * q^65 + 12 * q^71 + 24 * q^79 + 32 * q^85 + 28 * q^89 - 12 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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