LMFDB - Embedded newform 63.6.c.a.62.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,6,Mod(62,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.62");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.1041806482$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			62.2
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	63.62
Dual form			63.6.c.a.62.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.90735i q^{2} -2.89674 q^{4} +129.642 q^{7} -171.923i q^{8} +O(q^{10})$	$q-5.90735i q^{2} -2.89674 q^{4} +129.642 q^{7} -171.923i q^{8} -511.250i q^{11} -765.839i q^{14} -1108.30 q^{16} -3020.13 q^{22} -2719.59i q^{23} -3125.00 q^{25} -375.538 q^{28} +1340.14i q^{29} +1045.60i q^{32} +14086.0 q^{37} +21213.6 q^{43} +1480.96i q^{44} -16065.5 q^{46} +16807.0 q^{49} +18460.5i q^{50} +40490.4i q^{53} -22288.4i q^{56} +7916.70 q^{58} -29289.0 q^{64} -69364.0 q^{67} +61622.5i q^{71} -83210.8i q^{74} -66279.4i q^{77} +80168.0 q^{79} -125316. i q^{86} -87895.7 q^{88} +7877.92i q^{92} -99284.8i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 128 q^{4}+O(q^{10}) $ 4 * q - 128 * q^4	$ 4 q - 128 q^{4} - 708 q^{16} + 5900 q^{22} - 12500 q^{25} + 15092 q^{28} - 114796 q^{46} + 67228 q^{49} + 188708 q^{58} + 22656 q^{64} - 277456 q^{67} + 320672 q^{79} - 523292 q^{88}+O(q^{100}) $ 4 * q - 128 * q^4 - 708 * q^16 + 5900 * q^22 - 12500 * q^25 + 15092 * q^28 - 114796 * q^46 + 67228 * q^49 + 188708 * q^58 + 22656 * q^64 - 277456 * q^67 + 320672 * q^79 - 523292 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 5.90735i	− 1.04428i	−0.852859	−	0.522141i	$-0.825134\pi$
$2$	− 5.90735i	− 1.04428i	0.852859	−	0.522141i	$-0.174866\pi$
$3$	0	0
$3$	0	0
$4$	−2.89674	−0.0905230
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	129.642	1.00000
$7$	129.642	1.00000
$8$	− 171.923i	− 0.949750i
$9$	0	0
$10$	0	0
$11$	− 511.250i	− 1.27395i	−0.770885	−	0.636974i	$-0.780186\pi$
$11$	− 511.250i	− 1.27395i	0.770885	−	0.636974i	$-0.219814\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	− 765.839i	− 1.04428i
$15$	0	0
$16$	−1108.30	−1.08233
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	−3020.13	−1.33036
$23$	− 2719.59i	− 1.07197i	−0.844227	−	0.535986i	$-0.819940\pi$
$23$	− 2719.59i	− 1.07197i	0.844227	−	0.535986i	$-0.180060\pi$
$24$	0	0
$25$	−3125.00	−1.00000
$26$	0	0
$27$	0	0
$28$	−375.538	−0.0905230
$29$	1340.14i	0.295908i	0.988994	+	0.147954i	$0.0472687\pi$
$29$	1340.14i	0.295908i	−0.988994	+	0.147954i	$0.952731\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1045.60i	0.180506i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14086.0	1.69154	0.845771	−	0.533546i	$-0.179141\pi$
$37$	14086.0	1.69154	0.845771	+	0.533546i	$0.179141\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	21213.6	1.74962	0.874810	−	0.484465i	$-0.160986\pi$
$43$	21213.6	1.74962	0.874810	+	0.484465i	$0.160986\pi$
$44$	1480.96i	0.115322i
$45$	0	0
$46$	−16065.5	−1.11944
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	16807.0	1.00000
$50$	18460.5i	1.04428i
$51$	0	0
$52$	0	0
$53$	40490.4i	1.97999i	0.141113	+	0.989994i	$0.454932\pi$
$53$	40490.4i	1.97999i	−0.141113	+	0.989994i	$0.545068\pi$
$54$	0	0
$55$	0	0
$56$	− 22288.4i	− 0.949750i
$57$	0	0
$58$	7916.70	0.309011
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−29289.0	−0.893830
$65$	0	0
$66$	0	0
$67$	−69364.0	−1.88776	−0.943881	−	0.330286i	$-0.892855\pi$
$67$	−69364.0	−1.88776	−0.943881	+	0.330286i	$0.892855\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	61622.5i	1.45075i	0.688353	+	0.725376i	$0.258334\pi$
$71$	61622.5i	1.45075i	−0.688353	+	0.725376i	$0.741666\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	− 83210.8i	− 1.76645i
$75$	0	0
$76$	0	0
$77$	− 66279.4i	− 1.27395i
$78$	0	0
$79$	80168.0	1.44522	0.722609	−	0.691257i	$-0.242943\pi$
$79$	80168.0	1.44522	0.722609	+	0.691257i	$0.242943\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	− 125316.i	− 1.82710i
$87$	0	0
$88$	−87895.7	−1.20993
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	7877.92i	0.0970381i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 99284.8i	− 1.04428i
$99$	0	0
$100$	9052.30	0.0905230
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	239191.	2.06766
$107$	206966.i	1.74759i	0.486296	+	0.873794i	$0.338348\pi$
$107$	206966.i	1.74759i	−0.486296	+	0.873794i	$0.661652\pi$
$108$	0	0
$109$	−115450.	−0.930739	−0.465369	−	0.885117i	$-0.654078\pi$
$109$	−115450.	−0.930739	−0.465369	+	0.885117i	$0.654078\pi$
$110$	0	0
$111$	0	0
$112$	−143683.	−1.08233
$113$	258171.i	1.90200i	0.309190	+	0.951000i	$0.399942\pi$
$113$	258171.i	1.90200i	−0.309190	+	0.951000i	$0.600058\pi$
$114$	0	0
$115$	0	0
$116$	− 3882.04i	− 0.0267865i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−100326.	−0.622944
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	262064.	1.44178	0.720888	−	0.693051i	$-0.243734\pi$
$127$	262064.	1.44178	0.720888	+	0.693051i	$0.243734\pi$
$128$	206480.i	1.11392i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	409757.i	1.97135i
$135$	0	0
$136$	0	0
$137$	− 434480.i	− 1.97774i	−0.148789	−	0.988869i	$-0.547537\pi$
$137$	− 434480.i	− 1.97774i	0.148789	−	0.988869i	$-0.452463\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	364025.	1.51499
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−40803.4	−0.153123
$149$	− 61840.6i	− 0.228196i	−0.993470	−	0.114098i	$-0.963602\pi$
$149$	− 61840.6i	− 0.228196i	0.993470	−	0.114098i	$-0.0363978\pi$
$150$	0	0
$151$	495544.	1.76864	0.884321	−	0.466880i	$-0.154622\pi$
$151$	495544.	1.76864	0.884321	+	0.466880i	$0.154622\pi$
$152$	0	0
$153$	0	0
$154$	−391535.	−1.33036
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 473580.i	− 1.50921i
$159$	0	0
$160$	0	0
$161$	− 352572.i	− 1.07197i
$162$	0	0
$163$	−663100.	−1.95483	−0.977417	−	0.211318i	$-0.932224\pi$
$163$	−663100.	−1.95483	−0.977417	+	0.211318i	$0.932224\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	371293.	1.00000
$170$	0	0
$171$	0	0
$172$	−61450.3	−0.158381
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−405131.	−1.00000
$176$	566621.i	1.37883i
$177$	0	0
$178$	0	0
$179$	− 856829.i	− 1.99876i	−0.0351437	−	0.999382i	$-0.511189\pi$
$179$	− 856829.i	− 1.99876i	0.0351437	−	0.999382i	$-0.488811\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−467560.	−1.01810
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	636206.i	1.26187i	0.775836	+	0.630935i	$0.217328\pi$
$191$	636206.i	1.26187i	−0.775836	+	0.630935i	$0.782672\pi$
$192$	0	0
$193$	−960323.	−1.85577	−0.927885	−	0.372867i	$-0.878375\pi$
$193$	−960323.	−1.85577	−0.927885	+	0.372867i	$0.878375\pi$
$194$	0	0
$195$	0	0
$196$	−48685.4	−0.0905230
$197$	− 993223.i	− 1.82340i	−0.410860	−	0.911698i	$-0.634772\pi$
$197$	− 993223.i	− 1.82340i	0.410860	−	0.911698i	$-0.365228\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	537260.i	0.949750i
$201$	0	0
$202$	0	0
$203$	173739.i	0.295908i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	685117.	1.05940	0.529699	−	0.848186i	$-0.322305\pi$
$211$	685117.	1.05940	0.529699	+	0.848186i	$0.322305\pi$
$212$	− 117290.i	− 0.179234i
$213$	0	0
$214$	1.22262e6	1.82497
$215$	0	0
$216$	0	0
$217$	0	0
$218$	682003.i	0.971953i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	135554.i	0.180506i
$225$	0	0
$226$	1.52510e6	1.98622
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	230402.	0.281038
$233$	159838.i	0.192881i	0.995339	+	0.0964404i	$0.0307457\pi$
$233$	159838.i	0.192881i	−0.995339	+	0.0964404i	$0.969254\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	627624.i	0.710730i	0.934728	+	0.355365i	$0.115643\pi$
$239$	627624.i	0.710730i	−0.934728	+	0.355365i	$0.884357\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	592659.i	0.650528i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−1.39039e6	−1.36564
$254$	− 1.54810e6i	− 1.50562i
$255$	0	0
$256$	282498.	0.269411
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	1.82613e6	1.69154
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.24217e6i	1.99885i	0.0339779	+	0.999423i	$0.489182\pi$
$263$	2.24217e6i	1.99885i	−0.0339779	+	0.999423i	$0.510818\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	200929.	0.170886
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−2.56663e6	−2.06531
$275$	1.59766e6i	1.27395i
$276$	0	0
$277$	−2.55145e6	−1.99796	−0.998982	−	0.0451116i	$-0.985636\pi$
$277$	−2.55145e6	−1.99796	−0.998982	+	0.0451116i	$0.985636\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.30952e6i	− 1.74484i	−0.488754	−	0.872422i	$-0.662548\pi$
$281$	− 2.30952e6i	− 1.74484i	0.488754	−	0.872422i	$-0.337452\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	− 178504.i	− 0.131326i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41986e6	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	− 2.42170e6i	− 1.60654i
$297$	0	0
$298$	−365314.	−0.238301
$299$	0	0
$300$	0	0
$301$	2.75017e6	1.74962
$302$	− 2.92735e6i	− 1.84696i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	191994.i	0.115322i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	−232225.	−0.130825
$317$	− 2.68216e6i	− 1.49912i	−0.661935	−	0.749561i	$-0.730265\pi$
$317$	− 2.68216e6i	− 1.49912i	0.661935	−	0.749561i	$-0.269735\pi$
$318$	0	0
$319$	685149.	0.376971
$320$	0	0
$321$	0	0
$322$	−2.08277e6	−1.11944
$323$	0	0
$324$	0	0
$325$	0	0
$326$	3.91716e6i	2.04140i
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.65765e6	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$331$	2.65765e6	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	289318.	0.138772	0.0693859	−	0.997590i	$-0.477896\pi$
$337$	289318.	0.138772	0.0693859	+	0.997590i	$0.477896\pi$
$338$	− 2.19336e6i	− 1.04428i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.17889e6	1.00000
$344$	− 3.64711e6i	− 1.66170i
$345$	0	0
$346$	0	0
$347$	1.09911e6i	0.490026i	0.969520	+	0.245013i	$0.0787923\pi$
$347$	1.09911e6i	0.490026i	−0.969520	+	0.245013i	$0.921208\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.39325e6i	1.04428i
$351$	0	0
$352$	534563.	0.229955
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−5.06159e6	−2.08727
$359$	4.69616e6i	1.92312i	0.274590	+	0.961561i	$0.411458\pi$
$359$	4.69616e6i	1.92312i	−0.274590	+	0.961561i	$0.588542\pi$
$360$	0	0
$361$	2.47610e6	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	3.01413e6i	1.16023i
$369$	0	0
$370$	0	0
$371$	5.24925e6i	1.97999i
$372$	0	0
$373$	−599302.	−0.223035	−0.111518	−	0.993762i	$-0.535571\pi$
$373$	−599302.	−0.223035	−0.111518	+	0.993762i	$0.535571\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24039.3	0.00859654	0.00429827	−	0.999991i	$-0.498632\pi$
$379$	24039.3	0.00859654	0.00429827	+	0.999991i	$0.498632\pi$
$380$	0	0
$381$	0	0
$382$	3.75829e6	1.31775
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	5.67296e6i	1.93795i
$387$	0	0
$388$	0	0
$389$	− 3.23458e6i	− 1.08379i	−0.840447	−	0.541894i	$-0.817708\pi$
$389$	− 3.23458e6i	− 1.08379i	0.840447	−	0.541894i	$-0.182292\pi$
$390$	0	0
$391$	0	0
$392$	− 2.88951e6i	− 0.949750i
$393$	0	0
$394$	−5.86731e6	−1.90414
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	3.46345e6	1.08233
$401$	5.85940e6i	1.81967i	0.414970	+	0.909835i	$0.363792\pi$
$401$	5.85940e6i	1.81967i	−0.414970	+	0.909835i	$0.636208\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	1.02633e6	0.309011
$407$	− 7.20146e6i	− 2.15494i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−2.07477e6	−0.570513	−0.285257	−	0.958451i	$-0.592079\pi$
$421$	−2.07477e6	−0.570513	−0.285257	+	0.958451i	$0.592079\pi$
$422$	− 4.04722e6i	− 1.10631i
$423$	0	0
$424$	6.96123e6	1.88049
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 599525.i	− 0.158197i
$429$	0	0
$430$	0	0
$431$	− 3.51131e6i	− 0.910492i	−0.890366	−	0.455246i	$-0.849551\pi$
$431$	− 3.51131e6i	− 0.910492i	0.890366	−	0.455246i	$-0.150449\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	334428.	0.0842532
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 983014.i	− 0.237985i	−0.992895	−	0.118993i	$-0.962033\pi$
$443$	− 983014.i	− 0.237985i	0.992895	−	0.118993i	$-0.0379665\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−3.79708e6	−0.893830
$449$	− 4.46092e6i	− 1.04426i	−0.852866	−	0.522129i	$-0.825138\pi$
$449$	− 4.46092e6i	− 1.04426i	0.852866	−	0.522129i	$-0.174862\pi$
$450$	0	0
$451$	0	0
$452$	− 747852.i	− 0.172175i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.80652e6	−1.52453	−0.762263	−	0.647267i	$-0.775912\pi$
$457$	−6.80652e6	−1.52453	−0.762263	+	0.647267i	$0.775912\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	2.88620e6	0.625711	0.312856	−	0.949801i	$-0.398714\pi$
$463$	2.88620e6	0.625711	0.312856	+	0.949801i	$0.398714\pi$
$464$	− 1.48529e6i	− 0.320270i
$465$	0	0
$466$	944216.	0.201422
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−8.99247e6	−1.88776
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.08455e7i	− 2.22893i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	3.70759e6	0.742202
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	290617.	0.0563907
$485$	0	0
$486$	0	0
$487$	−1.00969e7	−1.92915	−0.964575	−	0.263807i	$-0.915022\pi$
$487$	−1.00969e7	−1.92915	−0.964575	+	0.263807i	$0.915022\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.04978e7i	1.96515i	0.185878	+	0.982573i	$0.440487\pi$
$491$	1.04978e7i	1.96515i	−0.185878	+	0.982573i	$0.559513\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.98885e6i	1.45075i
$498$	0	0
$499$	304129.	0.0546772	0.0273386	−	0.999626i	$-0.491297\pi$
$499$	304129.	0.0546772	0.0273386	+	0.999626i	$0.491297\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	8.21351e6i	1.42611i
$507$	0	0
$508$	−759130.	−0.130514
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	4.93854e6i	0.832575i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 1.07876e7i	− 1.76645i
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	1.32453e7	2.08736
$527$	0	0
$528$	0	0
$529$	−959807.	−0.149123
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	1.19253e7i	1.79290i
$537$	0	0
$538$	0	0
$539$	− 8.59258e6i	− 1.27395i
$540$	0	0
$541$	1.11261e7	1.63437	0.817186	−	0.576374i	$-0.195533\pi$
$541$	1.11261e7	1.63437	0.817186	+	0.576374i	$0.195533\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.23604e6	−0.319529	−0.159765	−	0.987155i	$-0.551074\pi$
$547$	−2.23604e6	−0.319529	−0.159765	+	0.987155i	$0.551074\pi$
$548$	1.25857e6i	0.179031i
$549$	0	0
$550$	9.43791e6	1.33036
$551$	0	0
$552$	0	0
$553$	1.03931e7	1.44522
$554$	1.50723e7i	2.08644i
$555$	0	0
$556$	0	0
$557$	− 1.37801e7i	− 1.88198i	−0.338436	−	0.940990i	$-0.609898\pi$
$557$	− 1.37801e7i	− 1.88198i	0.338436	−	0.940990i	$-0.390102\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−1.36431e7	−1.82211
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1.05943e7	1.37785
$569$	621150.i	0.0804296i	0.999191	+	0.0402148i	$0.0128042\pi$
$569$	621150.i	0.0804296i	−0.999191	+	0.0402148i	$0.987196\pi$
$570$	0	0
$571$	−6.33912e6	−0.813653	−0.406826	−	0.913506i	$-0.633365\pi$
$571$	−6.33912e6	−0.813653	−0.406826	+	0.913506i	$0.633365\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.49871e6i	1.07197i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	8.38759e6i	1.04428i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.07007e7	2.52240
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.56116e7	−1.83080
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	179136.i	0.0206570i
$597$	0	0
$598$	0	0
$599$	− 1.64097e7i	− 1.86868i	−0.356383	−	0.934340i	$-0.615990\pi$
$599$	− 1.64097e7i	− 1.86868i	0.356383	−	0.934340i	$-0.384010\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	− 1.62462e7i	− 1.82710i
$603$	0	0
$604$	−1.43546e6	−0.160103
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.75514e7	−1.88652	−0.943258	−	0.332060i	$-0.892256\pi$
$613$	−1.75514e7	−1.88652	−0.943258	+	0.332060i	$0.892256\pi$
$614$	0	0
$615$	0	0
$616$	−1.13950e7	−1.20993
$617$	1.74358e7i	1.84387i	0.387349	+	0.921933i	$0.373391\pi$
$617$	1.74358e7i	1.84387i	−0.387349	+	0.921933i	$0.626609\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9.76562e6	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.99786e7	−1.99752	−0.998760	−	0.0497844i	$-0.984147\pi$
$631$	−1.99786e7	−1.99752	−0.998760	+	0.0497844i	$0.984147\pi$
$632$	− 1.37827e7i	− 1.37260i
$633$	0	0
$634$	−1.58445e7	−1.56550
$635$	0	0
$636$	0	0
$637$	0	0
$638$	− 4.04741e6i	− 0.393664i
$639$	0	0
$640$	0	0
$641$	2.03908e7i	1.96015i	0.198632	+	0.980074i	$0.436350\pi$
$641$	2.03908e7i	1.96015i	−0.198632	+	0.980074i	$0.563650\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	1.02131e6i	0.0970381i
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1.92083e6	0.176957
$653$	− 5.55005e6i	− 0.509347i	−0.967027	−	0.254674i	$-0.918032\pi$
$653$	− 5.55005e6i	− 0.509347i	0.967027	−	0.254674i	$-0.0819680\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.09397e7i	− 1.87826i	−0.343561	−	0.939130i	$-0.611633\pi$
$659$	− 2.09397e7i	− 1.87826i	0.343561	−	0.939130i	$-0.388367\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 1.56997e7i	− 1.39234i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.64464e6	0.317205
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.34592e7	1.99653	0.998263	−	0.0589085i	$-0.0187620\pi$
$673$	2.34592e7	1.99653	0.998263	+	0.0589085i	$0.0187620\pi$
$674$	− 1.70910e6i	− 0.144917i
$675$	0	0
$676$	−1.07554e6	−0.0905230
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.69951e6i	− 0.549530i	−0.961511	−	0.274765i	$-0.911400\pi$
$683$	− 6.69951e6i	− 0.549530i	0.961511	−	0.274765i	$-0.0886000\pi$
$684$	0	0
$685$	0	0
$686$	− 1.28715e7i	− 1.04428i
$687$	0	0
$688$	−2.35112e7	−1.89366
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	6.49285e6	0.511725
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1.17356e6	0.0905230
$701$	2.15273e7i	1.65461i	0.561755	+	0.827304i	$0.310126\pi$
$701$	2.15273e7i	1.65461i	−0.561755	+	0.827304i	$0.689874\pi$
$702$	0	0
$703$	0	0
$704$	1.49740e7i	1.13869i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.26161e7	1.68967	0.844837	−	0.535023i	$-0.179697\pi$
$709$	2.26161e7	1.68967	0.844837	+	0.535023i	$0.179697\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.48201e6i	0.180934i
$717$	0	0
$718$	2.77419e7	2.00828
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 1.46272e7i	− 1.04428i
$723$	0	0
$724$	0	0
$725$	− 4.18795e6i	− 0.295908i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	2.84360e6	0.193497
$737$	3.54624e7i	2.40491i
$738$	0	0
$739$	−2.64893e6	−0.178427	−0.0892133	−	0.996013i	$-0.528435\pi$
$739$	−2.64893e6	−0.178427	−0.0892133	+	0.996013i	$0.528435\pi$
$740$	0	0
$741$	0	0
$742$	3.10091e7	2.06766
$743$	5.74228e6i	0.381603i	0.981629	+	0.190802i	$0.0611087\pi$
$743$	5.74228e6i	0.381603i	−0.981629	+	0.190802i	$0.938891\pi$
$744$	0	0
$745$	0	0
$746$	3.54028e6i	0.232912i
$747$	0	0
$748$	0	0
$749$	2.68314e7i	1.74759i
$750$	0	0
$751$	−1.80309e7	−1.16659	−0.583295	−	0.812260i	$-0.698237\pi$
$751$	−1.80309e7	−1.16659	−0.583295	+	0.812260i	$0.698237\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.59693e7	−1.64710	−0.823551	−	0.567242i	$-0.808011\pi$
$757$	−2.59693e7	−1.64710	−0.823551	+	0.567242i	$0.808011\pi$
$758$	− 142008.i	− 0.00897720i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−1.49671e7	−0.930739
$764$	− 1.84292e6i	− 0.114228i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	2.78180e6	0.167990
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−1.91078e7	−1.13178
$779$	0	0
$780$	0	0
$781$	3.15045e7	1.84818
$782$	0	0
$783$	0	0
$784$	−1.86273e7	−1.08233
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	2.87710e6i	0.165059i
$789$	0	0
$790$	0	0
$791$	3.34697e7i	1.90200i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	− 3.26750e6i	− 0.180506i
$801$	0	0
$802$	3.46135e7	1.90025
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.90804e6i	0.532251i	0.963938	+	0.266126i	$0.0857436\pi$
$809$	9.90804e6i	0.532251i	−0.963938	+	0.266126i	$0.914256\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 503275.i	− 0.0267865i
$813$	0	0
$814$	−4.25415e7	−2.25036
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.57832e7i	− 1.85277i	−0.376577	−	0.926385i	$-0.622899\pi$
$821$	− 3.57832e7i	− 1.85277i	0.376577	−	0.926385i	$-0.377101\pi$
$822$	0	0
$823$	7.08675e6	0.364710	0.182355	−	0.983233i	$-0.441628\pi$
$823$	7.08675e6	0.364710	0.182355	+	0.983233i	$0.441628\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.57827e7i	− 1.81932i	−0.415354	−	0.909660i	$-0.636342\pi$
$827$	− 3.57827e7i	− 1.81932i	0.415354	−	0.909660i	$-0.363658\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	1.87152e7	0.912439
$842$	1.22564e7i	0.595776i
$843$	0	0
$844$	−1.98460e6	−0.0958998
$845$	0	0
$846$	0	0
$847$	−1.30064e7	−0.622944
$848$	− 4.48757e7i	− 2.14300i
$849$	0	0
$850$	0	0
$851$	− 3.83080e7i	− 1.81328i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	3.55822e7	1.65977
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−2.07425e7	−0.950809
$863$	− 4.47598e6i	− 0.204579i	−0.994755	−	0.102290i	$-0.967383\pi$
$863$	− 4.47598e6i	− 0.204579i	0.994755	−	0.102290i	$-0.0326168\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.09859e7i	− 1.84113i
$870$	0	0
$871$	0	0
$872$	1.98485e7i	0.883969i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.33428e7	−1.90291	−0.951453	−	0.307793i	$-0.900410\pi$
$877$	−4.33428e7	−1.90291	−0.951453	+	0.307793i	$0.900410\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−2.43867e7	−1.05257	−0.526286	−	0.850308i	$-0.676416\pi$
$883$	−2.43867e7	−1.05257	−0.526286	+	0.850308i	$0.676416\pi$
$884$	0	0
$885$	0	0
$886$	−5.80700e6	−0.248524
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	3.39745e7	1.44178
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	2.67684e7i	1.11392i
$897$	0	0
$898$	−2.63522e7	−1.09050
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	4.43855e7	1.80642
$905$	0	0
$906$	0	0
$907$	2.10975e7	0.851554	0.425777	−	0.904828i	$-0.360001\pi$
$907$	2.10975e7	0.851554	0.425777	+	0.904828i	$0.360001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 4.25553e7i	− 1.69886i	−0.527700	−	0.849431i	$-0.676945\pi$
$911$	− 4.25553e7i	− 1.69886i	0.527700	−	0.849431i	$-0.323055\pi$
$912$	0	0
$913$	0	0
$914$	4.02085e7i	1.59203i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−7.37666e6	−0.288118	−0.144059	−	0.989569i	$-0.546016\pi$
$919$	−7.37666e6	−0.288118	−0.144059	+	0.989569i	$0.546016\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.40187e7	−1.69154
$926$	− 1.70498e7i	− 0.653418i
$927$	0	0
$928$	−1.40126e6	−0.0534130
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	− 463007.i	− 0.0174602i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	5.31217e7i	1.97135i
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	−6.40680e7	−2.32763
$947$	4.21673e7i	1.52792i	0.645262	+	0.763961i	$0.276748\pi$
$947$	4.21673e7i	1.52792i	−0.645262	+	0.763961i	$0.723252\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.60692e7i	1.64316i	0.570097	+	0.821578i	$0.306906\pi$
$953$	4.60692e7i	1.64316i	−0.570097	+	0.821578i	$0.693094\pi$
$954$	0	0
$955$	0	0
$956$	− 1.81806e6i	− 0.0643374i
$957$	0	0
$958$	0	0
$959$	− 5.63268e7i	− 1.97774i
$960$	0	0
$961$	2.86292e7	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.00222e7	0.688566	0.344283	−	0.938866i	$-0.388122\pi$
$967$	2.00222e7	0.688566	0.344283	+	0.938866i	$0.388122\pi$
$968$	1.72483e7i	0.591641i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	5.96460e7i	2.01458i
$975$	0	0
$976$	0	0
$977$	5.83913e7i	1.95710i	0.206019	+	0.978548i	$0.433949\pi$
$977$	5.83913e7i	1.95710i	−0.206019	+	0.978548i	$0.566051\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	6.20142e7	2.05216
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 5.76923e7i	− 1.87554i
$990$	0	0
$991$	2.32006e7	0.750438	0.375219	−	0.926936i	$-0.377567\pi$
$991$	2.32006e7	0.750438	0.375219	+	0.926936i	$0.377567\pi$
$992$	0	0
$993$	0	0
$994$	4.71929e7	1.51499
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 1.79660e6i	− 0.0570984i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.6.c.a.62.2	✓	4
3.2	odd	2	inner	63.6.c.a.62.3	yes	4
4.3	odd	2		1008.6.k.a.881.2		4
7.6	odd	2	CM	63.6.c.a.62.2	✓	4
12.11	even	2		1008.6.k.a.881.1		4
21.20	even	2	inner	63.6.c.a.62.3	yes	4
28.27	even	2		1008.6.k.a.881.2		4
84.83	odd	2		1008.6.k.a.881.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.c.a.62.2	✓	4	1.1	even	1	trivial
63.6.c.a.62.2	✓	4	7.6	odd	2	CM
63.6.c.a.62.3	yes	4	3.2	odd	2	inner
63.6.c.a.62.3	yes	4	21.20	even	2	inner
1008.6.k.a.881.1		4	12.11	even	2
1008.6.k.a.881.1		4	84.83	odd	2
1008.6.k.a.881.2		4	4.3	odd	2
1008.6.k.a.881.2		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.90735i q^{2} -2.89674 q^{4} +129.642 q^{7} -171.923i q^{8} +O(q^{10})\)	\(q-5.90735i q^{2} -2.89674 q^{4} +129.642 q^{7} -171.923i q^{8} -511.250i q^{11} -765.839i q^{14} -1108.30 q^{16} -3020.13 q^{22} -2719.59i q^{23} -3125.00 q^{25} -375.538 q^{28} +1340.14i q^{29} +1045.60i q^{32} +14086.0 q^{37} +21213.6 q^{43} +1480.96i q^{44} -16065.5 q^{46} +16807.0 q^{49} +18460.5i q^{50} +40490.4i q^{53} -22288.4i q^{56} +7916.70 q^{58} -29289.0 q^{64} -69364.0 q^{67} +61622.5i q^{71} -83210.8i q^{74} -66279.4i q^{77} +80168.0 q^{79} -125316. i q^{86} -87895.7 q^{88} +7877.92i q^{92} -99284.8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 128 q^{4}+O(q^{10}) \) 4 * q - 128 * q^4	\( 4 q - 128 q^{4} - 708 q^{16} + 5900 q^{22} - 12500 q^{25} + 15092 q^{28} - 114796 q^{46} + 67228 q^{49} + 188708 q^{58} + 22656 q^{64} - 277456 q^{67} + 320672 q^{79} - 523292 q^{88}+O(q^{100}) \) 4 * q - 128 * q^4 - 708 * q^16 + 5900 * q^22 - 12500 * q^25 + 15092 * q^28 - 114796 * q^46 + 67228 * q^49 + 188708 * q^58 + 22656 * q^64 - 277456 * q^67 + 320672 * q^79 - 523292 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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