LMFDB - Embedded newform 63.2.c.a.62.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,2,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, 2, names="a")

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

chi := DirichletCharacter("63.62");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
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:	$0.503057532734$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			62.2
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	63.62
Dual form			63.2.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)

$f(q)$	$=$	$q-1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} -3.07892i q^{8} +O(q^{10})$	$q-1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} -3.07892i q^{8} +6.57008i q^{11} +3.07892i q^{14} -2.29150 q^{16} +7.64575 q^{22} -1.91520i q^{23} -5.00000 q^{25} -1.70850 q^{28} -8.89753i q^{29} -3.49117i q^{32} +10.5830 q^{37} -5.29150 q^{43} +4.24264i q^{44} -2.22876 q^{46} +7.00000 q^{49} +5.81861i q^{50} -0.412247i q^{53} +8.14605i q^{56} -10.3542 q^{58} -8.64575 q^{64} -4.00000 q^{67} +15.0554i q^{71} -12.3157i q^{74} -17.3828i q^{77} +8.00000 q^{79} +6.15784i q^{86} +20.2288 q^{88} -1.23674i q^{92} -8.14605i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^16 + 20 * q^22 - 20 * q^25 - 28 * q^28 + 44 * q^46 + 28 * q^49 - 52 * q^58 - 24 * q^64 - 16 * q^67 + 32 * q^79 + 28 * 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 $

$2$	− 1.16372i	− 0.822876i	−0.911438	−	0.411438i	$-0.865027\pi$
$2$	− 1.16372i	− 0.822876i	0.911438	−	0.411438i	$-0.134973\pi$
$3$	0	0
$3$	0	0
$4$	0.645751	0.322876
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−2.64575	−1.00000
$7$	−2.64575	−1.00000
$8$	− 3.07892i	− 1.08856i
$9$	0	0
$10$	0	0
$11$	6.57008i	1.98096i	0.137675	+	0.990478i	$0.456037\pi$
$11$	6.57008i	1.98096i	−0.137675	+	0.990478i	$0.543963\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	3.07892i	0.822876i
$15$	0	0
$16$	−2.29150	−0.572876
$17$	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
$22$	7.64575	1.63008
$23$	− 1.91520i	− 0.399346i	−0.979863	−	0.199673i	$-0.936012\pi$
$23$	− 1.91520i	− 0.399346i	0.979863	−	0.199673i	$-0.0639880\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−1.70850	−0.322876
$29$	− 8.89753i	− 1.65223i	−0.563502	−	0.826115i	$-0.690546\pi$
$29$	− 8.89753i	− 1.65223i	0.563502	−	0.826115i	$-0.309454\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 3.49117i	− 0.617157i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.5830	1.73984	0.869918	−	0.493197i	$-0.164172\pi$
$37$	10.5830	1.73984	0.869918	+	0.493197i	$0.164172\pi$
$38$	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
$43$	−5.29150	−0.806947	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150	−0.806947	−0.403473	+	0.914991i	$0.632197\pi$
$44$	4.24264i	0.639602i
$45$	0	0
$46$	−2.22876	−0.328612
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	5.81861i	0.822876i
$51$	0	0
$52$	0	0
$53$	− 0.412247i	− 0.0566265i	−0.999599	−	0.0283132i	$-0.990986\pi$
$53$	− 0.412247i	− 0.0566265i	0.999599	−	0.0283132i	$-0.00901359\pi$
$54$	0	0
$55$	0	0
$56$	8.14605i	1.08856i
$57$	0	0
$58$	−10.3542	−1.35958
$59$	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
$64$	−8.64575	−1.08072
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.0554i	1.78674i	0.449319	+	0.893372i	$0.351667\pi$
$71$	15.0554i	1.78674i	−0.449319	+	0.893372i	$0.648333\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	− 12.3157i	− 1.43167i
$75$	0	0
$76$	0	0
$77$	− 17.3828i	− 1.98096i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	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
$86$	6.15784i	0.664017i
$87$	0	0
$88$	20.2288	2.15639
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	− 1.23674i	− 0.128939i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 8.14605i	− 0.822876i
$99$	0	0
$100$	−3.22876	−0.322876
$101$	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
$106$	−0.479741	−0.0465965
$107$	− 10.4005i	− 1.00545i	−0.864446	−	0.502726i	$-0.832330\pi$
$107$	− 10.4005i	− 1.00545i	0.864446	−	0.502726i	$-0.167670\pi$
$108$	0	0
$109$	10.5830	1.01367	0.506834	−	0.862044i	$-0.330816\pi$
$109$	10.5830	1.01367	0.506834	+	0.862044i	$0.330816\pi$
$110$	0	0
$111$	0	0
$112$	6.06275	0.572876
$113$	13.5524i	1.27490i	0.770490	+	0.637452i	$0.220012\pi$
$113$	13.5524i	1.27490i	−0.770490	+	0.637452i	$0.779988\pi$
$114$	0	0
$115$	0	0
$116$	− 5.74559i	− 0.533465i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−32.1660	−2.92418
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	3.07892i	0.272141i
$129$	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
$134$	4.65489i	0.402121i
$135$	0	0
$136$	0	0
$137$	22.0377i	1.88281i	0.337282	+	0.941404i	$0.390493\pi$
$137$	22.0377i	1.88281i	−0.337282	+	0.941404i	$0.609507\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	17.5203	1.47027
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	6.83399	0.561750
$149$	8.07303i	0.661369i	0.943741	+	0.330684i	$0.107280\pi$
$149$	8.07303i	0.661369i	−0.943741	+	0.330684i	$0.892720\pi$
$150$	0	0
$151$	−5.29150	−0.430616	−0.215308	−	0.976546i	$-0.569076\pi$
$151$	−5.29150	−0.430616	−0.215308	+	0.976546i	$0.569076\pi$
$152$	0	0
$153$	0	0
$154$	−20.2288	−1.63008
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	− 9.30978i	− 0.740646i
$159$	0	0
$160$	0	0
$161$	5.06713i	0.399346i
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	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
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	−3.41699	−0.260543
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	13.2288	1.00000
$176$	− 15.0554i	− 1.13484i
$177$	0	0
$178$	0	0
$179$	− 15.8799i	− 1.18692i	−0.804865	−	0.593458i	$-0.797762\pi$
$179$	− 15.8799i	− 1.18692i	0.804865	−	0.593458i	$-0.202238\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−5.89674	−0.434713
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 24.3651i	− 1.76300i	−0.472184	−	0.881500i	$-0.656534\pi$
$191$	− 24.3651i	− 1.76300i	0.472184	−	0.881500i	$-0.343466\pi$
$192$	0	0
$193$	−21.1660	−1.52356	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	−21.1660	−1.52356	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	4.52026	0.322876
$197$	− 25.8681i	− 1.84303i	−0.388348	−	0.921513i	$-0.626954\pi$
$197$	− 25.8681i	− 1.84303i	0.388348	−	0.921513i	$-0.373046\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	15.3946i	1.08856i
$201$	0	0
$202$	0	0
$203$	23.5406i	1.65223i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	26.4575	1.82141	0.910705	−	0.413057i	$-0.135539\pi$
$211$	26.4575	1.82141	0.910705	+	0.413057i	$0.135539\pi$
$212$	− 0.266209i	− 0.0182833i
$213$	0	0
$214$	−12.1033	−0.827362
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 12.3157i	− 0.834123i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	9.23676i	0.617157i
$225$	0	0
$226$	15.7712	1.04909
$227$	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
$232$	−27.3948	−1.79855
$233$	30.5230i	1.99963i	0.0193169	+	0.999813i	$0.493851\pi$
$233$	30.5230i	1.99963i	−0.0193169	+	0.999813i	$0.506149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 7.39458i	− 0.478316i	−0.970981	−	0.239158i	$-0.923129\pi$
$239$	− 7.39458i	− 0.478316i	0.970981	−	0.239158i	$-0.0768713\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	37.4323i	2.40624i
$243$	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
$253$	12.5830	0.791087
$254$	18.6196i	1.16829i
$255$	0	0
$256$	−13.7085	−0.856781
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−28.0000	−1.73984
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 18.8858i	− 1.16455i	−0.812993	−	0.582273i	$-0.802164\pi$
$263$	− 18.8858i	− 1.16455i	0.812993	−	0.582273i	$-0.197836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−2.58301	−0.157782
$269$	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
$274$	25.6458	1.54932
$275$	− 32.8504i	− 1.98096i
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.41815i	− 0.203910i	−0.994789	−	0.101955i	$-0.967490\pi$
$281$	− 3.41815i	− 0.203910i	0.994789	−	0.101955i	$-0.0325097\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	9.72202i	0.576896i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−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
$296$	− 32.5842i	− 1.89392i
$297$	0	0
$298$	9.39477	0.544224
$299$	0	0
$300$	0	0
$301$	14.0000	0.806947
$302$	6.15784i	0.354344i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 11.2250i	− 0.639602i
$309$	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
$316$	5.16601	0.290611
$317$	16.5583i	0.930008i	0.885309	+	0.465004i	$0.153947\pi$
$317$	16.5583i	0.930008i	−0.885309	+	0.465004i	$0.846053\pi$
$318$	0	0
$319$	58.4575	3.27299
$320$	0	0
$321$	0	0
$322$	5.89674	0.328612
$323$	0	0
$324$	0	0
$325$	0	0
$326$	− 23.2744i	− 1.28905i
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5.29150	−0.290847	−0.145424	−	0.989369i	$-0.546455\pi$
$331$	−5.29150	−0.290847	−0.145424	+	0.989369i	$0.546455\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.1660	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$338$	− 15.1284i	− 0.822876i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.5203	−1.00000
$344$	16.2921i	0.878412i
$345$	0	0
$346$	0	0
$347$	29.0200i	1.55788i	0.627100	+	0.778938i	$0.284242\pi$
$347$	29.0200i	1.55788i	−0.627100	+	0.778938i	$0.715758\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	− 15.3946i	− 0.822876i
$351$	0	0
$352$	22.9373	1.22256
$353$	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
$358$	−18.4797	−0.976685
$359$	20.5347i	1.08378i	0.840449	+	0.541891i	$0.182292\pi$
$359$	20.5347i	1.08378i	−0.840449	+	0.541891i	$0.817708\pi$
$360$	0	0
$361$	19.0000	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			$\pi$
$368$	4.38868i	0.228776i
$369$	0	0
$370$	0	0
$371$	1.09070i	0.0566265i
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−37.0405	−1.90264	−0.951322	−	0.308199i	$-0.900274\pi$
$379$	−37.0405	−1.90264	−0.951322	+	0.308199i	$0.900274\pi$
$380$	0	0
$381$	0	0
$382$	−28.3542	−1.45073
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	24.6314i	1.25370i
$387$	0	0
$388$	0	0
$389$	− 34.3534i	− 1.74179i	−0.491473	−	0.870893i	$-0.663542\pi$
$389$	− 34.3534i	− 1.74179i	0.491473	−	0.870893i	$-0.336458\pi$
$390$	0	0
$391$	0	0
$392$	− 21.5524i	− 1.08856i
$393$	0	0
$394$	−30.1033	−1.51658
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	11.4575	0.572876
$401$	39.0083i	1.94798i	0.226592	+	0.973990i	$0.427242\pi$
$401$	39.0083i	1.94798i	−0.226592	+	0.973990i	$0.572758\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	27.3948	1.35958
$407$	69.5312i	3.44654i
$408$	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
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	− 30.7892i	− 1.49879i
$423$	0	0
$424$	−1.26927	−0.0616414
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 6.71612i	− 0.324636i
$429$	0	0
$430$	0	0
$431$	− 41.3357i	− 1.99107i	−0.0943889	−	0.995535i	$-0.530090\pi$
$431$	− 41.3357i	− 1.99107i	0.0943889	−	0.995535i	$-0.469910\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	6.83399	0.327289
$437$	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
$443$	12.0495i	0.572487i	0.958157	+	0.286244i	$0.0924067\pi$
$443$	12.0495i	0.572487i	−0.958157	+	0.286244i	$0.907593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	22.8745	1.08072
$449$	− 31.3475i	− 1.47938i	−0.672948	−	0.739689i	$-0.734972\pi$
$449$	− 31.3475i	− 1.47938i	0.672948	−	0.739689i	$-0.265028\pi$
$450$	0	0
$451$	0	0
$452$	8.75149i	0.411635i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.3320	1.98021	0.990104	−	0.140334i	$-0.0448177\pi$
$457$	42.3320	1.98021	0.990104	+	0.140334i	$0.0448177\pi$
$458$	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
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	20.3887i	0.946522i
$465$	0	0
$466$	35.5203	1.64544
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	10.5830	0.488678
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 34.7656i	− 1.59852i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−8.60523	−0.393594
$479$	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
$484$	−20.7712	−0.944147
$485$	0	0
$486$	0	0
$487$	−37.0405	−1.67847	−0.839233	−	0.543772i	$-0.816996\pi$
$487$	−37.0405	−1.67847	−0.839233	+	0.543772i	$0.816996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 27.3710i	− 1.23524i	−0.786478	−	0.617619i	$-0.788097\pi$
$491$	− 27.3710i	− 1.23524i	0.786478	−	0.617619i	$-0.211903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 39.8328i	− 1.78674i
$498$	0	0
$499$	26.4575	1.18440	0.592200	−	0.805791i	$-0.298259\pi$
$499$	26.4575	1.18440	0.592200	+	0.805791i	$0.298259\pi$
$500$	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
$506$	− 14.6431i	− 0.650966i
$507$	0	0
$508$	−10.3320	−0.458409
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.1107i	0.977165i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	32.5842i	1.43167i
$519$	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
$526$	−21.9778	−0.958276
$527$	0	0
$528$	0	0
$529$	19.3320	0.840523
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	12.3157i	0.531956i
$537$	0	0
$538$	0	0
$539$	45.9906i	1.98096i
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	14.2309i	0.607913i
$549$	0	0
$550$	−38.2288	−1.63008
$551$	0	0
$552$	0	0
$553$	−21.1660	−0.900070
$554$	11.6372i	0.494418i
$555$	0	0
$556$	0	0
$557$	25.0436i	1.06113i	0.847644	+	0.530566i	$0.178020\pi$
$557$	25.0436i	1.06113i	−0.847644	+	0.530566i	$0.821980\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−3.97777	−0.167792
$563$	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
$568$	46.3542	1.94498
$569$	− 14.3769i	− 0.602711i	−0.953512	−	0.301356i	$-0.902561\pi$
$569$	− 14.3769i	− 0.602711i	0.953512	−	0.301356i	$-0.0974392\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.57598i	0.399346i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	19.7833i	0.822876i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.70850	0.112174
$584$	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
$592$	−24.2510	−0.996709
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	5.21317i	0.213540i
$597$	0	0
$598$	0	0
$599$	3.56418i	0.145629i	0.997346	+	0.0728143i	$0.0231980\pi$
$599$	3.56418i	0.145629i	−0.997346	+	0.0728143i	$0.976802\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	− 16.2921i	− 0.664017i
$603$	0	0
$604$	−3.41699	−0.139036
$605$	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
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	0	0
$616$	−53.5203	−2.15639
$617$	− 48.3180i	− 1.94521i	−0.232462	−	0.972605i	$-0.574678\pi$
$617$	− 48.3180i	− 1.94521i	0.232462	−	0.972605i	$-0.425322\pi$
$618$	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
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 24.6314i	− 0.979783i
$633$	0	0
$634$	19.2693	0.765281
$635$	0	0
$636$	0	0
$637$	0	0
$638$	− 68.0283i	− 2.69327i
$639$	0	0
$640$	0	0
$641$	47.4935i	1.87588i	0.346795	+	0.937941i	$0.387270\pi$
$641$	47.4935i	1.87588i	−0.346795	+	0.937941i	$0.612730\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	3.27211i	0.128939i
$645$	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
$652$	12.9150	0.505791
$653$	− 42.8387i	− 1.67641i	−0.545358	−	0.838203i	$-0.683606\pi$
$653$	− 42.8387i	− 1.67641i	0.545358	−	0.838203i	$-0.316394\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 49.8210i	− 1.94075i	−0.241604	−	0.970375i	$-0.577673\pi$
$659$	− 49.8210i	− 1.94075i	0.241604	−	0.970375i	$-0.422327\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	6.15784i	0.239331i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.0405	−0.659812
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	42.3320	1.63178	0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320	1.63178	0.815890	+	0.578208i	$0.196248\pi$
$674$	24.6314i	0.948764i
$675$	0	0
$676$	8.39477	0.322876
$677$	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
$683$	40.5112i	1.55012i	0.631889	+	0.775059i	$0.282280\pi$
$683$	40.5112i	1.55012i	−0.631889	+	0.775059i	$0.717720\pi$
$684$	0	0
$685$	0	0
$686$	21.5524i	0.822876i
$687$	0	0
$688$	12.1255	0.462280
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	33.7712	1.28194
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	8.54249	0.322876
$701$	36.0024i	1.35979i	0.733309	+	0.679895i	$0.237975\pi$
$701$	36.0024i	1.35979i	−0.733309	+	0.679895i	$0.762025\pi$
$702$	0	0
$703$	0	0
$704$	− 56.8033i	− 2.14086i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−52.9150	−1.98727	−0.993633	−	0.112667i	$-0.964061\pi$
$709$	−52.9150	−1.98727	−0.993633	+	0.112667i	$0.964061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	− 10.2544i	− 0.383226i
$717$	0	0
$718$	23.8967	0.891818
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 22.1107i	− 0.822876i
$723$	0	0
$724$	0	0
$725$	44.4876i	1.65223i
$726$	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
$736$	−6.68627	−0.246459
$737$	− 26.2803i	− 0.968049i
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	0	0
$742$	1.26927	0.0465965
$743$	54.4759i	1.99853i	0.0383863	+	0.999263i	$0.487778\pi$
$743$	54.4759i	1.99853i	−0.0383863	+	0.999263i	$0.512222\pi$
$744$	0	0
$745$	0	0
$746$	25.6019i	0.937352i
$747$	0	0
$748$	0	0
$749$	27.5171i	1.00545i
$750$	0	0
$751$	26.4575	0.965448	0.482724	−	0.875772i	$-0.339647\pi$
$751$	26.4575	0.965448	0.482724	+	0.875772i	$0.339647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.5830	0.384646	0.192323	−	0.981332i	$-0.438398\pi$
$757$	10.5830	0.384646	0.192323	+	0.981332i	$0.438398\pi$
$758$	43.1049i	1.56564i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−28.0000	−1.01367
$764$	− 15.7338i	− 0.569230i
$765$	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
$772$	−13.6680	−0.491921
$773$	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
$778$	−39.9778	−1.43327
$779$	0	0
$780$	0	0
$781$	−98.9150	−3.53946
$782$	0	0
$783$	0	0
$784$	−16.0405	−0.572876
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 16.7044i	− 0.595068i
$789$	0	0
$790$	0	0
$791$	− 35.8563i	− 1.27490i
$792$	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
$800$	17.4558i	0.617157i
$801$	0	0
$802$	45.3948	1.60294
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 56.8033i	− 1.99710i	−0.0538482	−	0.998549i	$-0.517149\pi$
$809$	− 56.8033i	− 1.99710i	0.0538482	−	0.998549i	$-0.482851\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	15.2014i	0.533465i
$813$	0	0
$814$	80.9150	2.83607
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.9729i	1.84877i	0.381464	+	0.924384i	$0.375420\pi$
$821$	52.9729i	1.84877i	−0.381464	+	0.924384i	$0.624580\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.92110i	− 0.171123i	−0.996333	−	0.0855616i	$-0.972732\pi$
$827$	− 4.92110i	− 0.171123i	0.996333	−	0.0855616i	$-0.0272685\pi$
$828$	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
$841$	−50.1660	−1.72986
$842$	− 30.2568i	− 1.04272i
$843$	0	0
$844$	17.0850	0.588089
$845$	0	0
$846$	0	0
$847$	85.1033	2.92418
$848$	0.944665i	0.0324399i
$849$	0	0
$850$	0	0
$851$	− 20.2685i	− 0.694797i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−32.0222	−1.09450
$857$	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
$862$	−48.1033	−1.63840
$863$	− 46.8151i	− 1.59360i	−0.604240	−	0.796802i	$-0.706523\pi$
$863$	− 46.8151i	− 1.59360i	0.604240	−	0.796802i	$-0.293477\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	52.5607i	1.78300i
$870$	0	0
$871$	0	0
$872$	− 32.5842i	− 1.10344i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	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
$883$	58.2065	1.95881	0.979403	−	0.201916i	$-0.0647168\pi$
$883$	58.2065	1.95881	0.979403	+	0.201916i	$0.0647168\pi$
$884$	0	0
$885$	0	0
$886$	14.0222	0.471086
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	42.3320	1.41977
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	− 8.14605i	− 0.272141i
$897$	0	0
$898$	−36.4797	−1.21734
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	41.7268	1.38781
$905$	0	0
$906$	0	0
$907$	−5.29150	−0.175701	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	−5.29150	−0.175701	−0.0878507	+	0.996134i	$0.528000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 29.8445i	− 0.988793i	−0.869236	−	0.494397i	$-0.835389\pi$
$911$	− 29.8445i	− 0.988793i	0.869236	−	0.494397i	$-0.164611\pi$
$912$	0	0
$913$	0	0
$914$	− 49.2627i	− 1.62947i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37.0405	−1.22185	−0.610927	−	0.791687i	$-0.709203\pi$
$919$	−37.0405	−1.22185	−0.610927	+	0.791687i	$0.709203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−52.9150	−1.73984
$926$	46.5489i	1.52969i
$927$	0	0
$928$	−31.0627	−1.01968
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	19.7103i	0.645631i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	− 12.3157i	− 0.402121i
$939$	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
$946$	−40.4575	−1.31539
$947$	− 55.3004i	− 1.79702i	−0.438953	−	0.898510i	$-0.644650\pi$
$947$	− 55.3004i	− 1.79702i	0.438953	−	0.898510i	$-0.355350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	55.9788i	1.81333i	0.421849	+	0.906666i	$0.361381\pi$
$953$	55.9788i	1.81333i	−0.421849	+	0.906666i	$0.638619\pi$
$954$	0	0
$955$	0	0
$956$	− 4.77506i	− 0.154436i
$957$	0	0
$958$	0	0
$959$	− 58.3063i	− 1.88281i
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	99.0365i	3.18315i
$969$	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
$974$	43.1049i	1.38117i
$975$	0	0
$976$	0	0
$977$	2.59365i	0.0829783i	0.999139	+	0.0414892i	$0.0132102\pi$
$977$	2.59365i	0.0829783i	−0.999139	+	0.0414892i	$0.986790\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−31.8523	−1.01645
$983$	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
$989$	10.1343i	0.322251i
$990$	0	0
$991$	58.2065	1.84899	0.924496	−	0.381193i	$-0.124487\pi$
$991$	58.2065	1.84899	0.924496	+	0.381193i	$0.124487\pi$
$992$	0	0
$993$	0	0
$994$	−46.3542	−1.47027
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 30.7892i	− 0.974615i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.2.c.a.62.2	✓	4
3.2	odd	2	inner	63.2.c.a.62.3	yes	4
4.3	odd	2		1008.2.k.a.881.3		4
5.2	odd	4		1575.2.g.d.1574.5		8
5.3	odd	4		1575.2.g.d.1574.4		8
5.4	even	2		1575.2.b.a.251.3		4
7.2	even	3		441.2.p.b.80.2		8
7.3	odd	6		441.2.p.b.215.3		8
7.4	even	3		441.2.p.b.215.3		8
7.5	odd	6		441.2.p.b.80.2		8
7.6	odd	2	CM	63.2.c.a.62.2	✓	4
8.3	odd	2		4032.2.k.b.3905.4		4
8.5	even	2		4032.2.k.c.3905.1		4
9.2	odd	6		567.2.o.f.377.3		8
9.4	even	3		567.2.o.f.188.3		8
9.5	odd	6		567.2.o.f.188.2		8
9.7	even	3		567.2.o.f.377.2		8
12.11	even	2		1008.2.k.a.881.4		4
15.2	even	4		1575.2.g.d.1574.3		8
15.8	even	4		1575.2.g.d.1574.6		8
15.14	odd	2		1575.2.b.a.251.2		4
21.2	odd	6		441.2.p.b.80.3		8
21.5	even	6		441.2.p.b.80.3		8
21.11	odd	6		441.2.p.b.215.2		8
21.17	even	6		441.2.p.b.215.2		8
21.20	even	2	inner	63.2.c.a.62.3	yes	4
24.5	odd	2		4032.2.k.c.3905.2		4
24.11	even	2		4032.2.k.b.3905.3		4
28.27	even	2		1008.2.k.a.881.3		4
35.13	even	4		1575.2.g.d.1574.4		8
35.27	even	4		1575.2.g.d.1574.5		8
35.34	odd	2		1575.2.b.a.251.3		4
56.13	odd	2		4032.2.k.c.3905.1		4
56.27	even	2		4032.2.k.b.3905.4		4
63.13	odd	6		567.2.o.f.188.3		8
63.20	even	6		567.2.o.f.377.3		8
63.34	odd	6		567.2.o.f.377.2		8
63.41	even	6		567.2.o.f.188.2		8
84.83	odd	2		1008.2.k.a.881.4		4
105.62	odd	4		1575.2.g.d.1574.3		8
105.83	odd	4		1575.2.g.d.1574.6		8
105.104	even	2		1575.2.b.a.251.2		4
168.83	odd	2		4032.2.k.b.3905.3		4
168.125	even	2		4032.2.k.c.3905.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.c.a.62.2	✓	4	1.1	even	1	trivial
63.2.c.a.62.2	✓	4	7.6	odd	2	CM
63.2.c.a.62.3	yes	4	3.2	odd	2	inner
63.2.c.a.62.3	yes	4	21.20	even	2	inner
441.2.p.b.80.2		8	7.2	even	3
441.2.p.b.80.2		8	7.5	odd	6
441.2.p.b.80.3		8	21.2	odd	6
441.2.p.b.80.3		8	21.5	even	6
441.2.p.b.215.2		8	21.11	odd	6
441.2.p.b.215.2		8	21.17	even	6
441.2.p.b.215.3		8	7.3	odd	6
441.2.p.b.215.3		8	7.4	even	3
567.2.o.f.188.2		8	9.5	odd	6
567.2.o.f.188.2		8	63.41	even	6
567.2.o.f.188.3		8	9.4	even	3
567.2.o.f.188.3		8	63.13	odd	6
567.2.o.f.377.2		8	9.7	even	3
567.2.o.f.377.2		8	63.34	odd	6
567.2.o.f.377.3		8	9.2	odd	6
567.2.o.f.377.3		8	63.20	even	6
1008.2.k.a.881.3		4	4.3	odd	2
1008.2.k.a.881.3		4	28.27	even	2
1008.2.k.a.881.4		4	12.11	even	2
1008.2.k.a.881.4		4	84.83	odd	2
1575.2.b.a.251.2		4	15.14	odd	2
1575.2.b.a.251.2		4	105.104	even	2
1575.2.b.a.251.3		4	5.4	even	2
1575.2.b.a.251.3		4	35.34	odd	2
1575.2.g.d.1574.3		8	15.2	even	4
1575.2.g.d.1574.3		8	105.62	odd	4
1575.2.g.d.1574.4		8	5.3	odd	4
1575.2.g.d.1574.4		8	35.13	even	4
1575.2.g.d.1574.5		8	5.2	odd	4
1575.2.g.d.1574.5		8	35.27	even	4
1575.2.g.d.1574.6		8	15.8	even	4
1575.2.g.d.1574.6		8	105.83	odd	4
4032.2.k.b.3905.3		4	24.11	even	2
4032.2.k.b.3905.3		4	168.83	odd	2
4032.2.k.b.3905.4		4	8.3	odd	2
4032.2.k.b.3905.4		4	56.27	even	2
4032.2.k.c.3905.1		4	8.5	even	2
4032.2.k.c.3905.1		4	56.13	odd	2
4032.2.k.c.3905.2		4	24.5	odd	2
4032.2.k.c.3905.2		4	168.125	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	63.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} -3.07892i q^{8} +O(q^{10})\)	\(q-1.16372i q^{2} +0.645751 q^{4} -2.64575 q^{7} -3.07892i q^{8} +6.57008i q^{11} +3.07892i q^{14} -2.29150 q^{16} +7.64575 q^{22} -1.91520i q^{23} -5.00000 q^{25} -1.70850 q^{28} -8.89753i q^{29} -3.49117i q^{32} +10.5830 q^{37} -5.29150 q^{43} +4.24264i q^{44} -2.22876 q^{46} +7.00000 q^{49} +5.81861i q^{50} -0.412247i q^{53} +8.14605i q^{56} -10.3542 q^{58} -8.64575 q^{64} -4.00000 q^{67} +15.0554i q^{71} -12.3157i q^{74} -17.3828i q^{77} +8.00000 q^{79} +6.15784i q^{86} +20.2288 q^{88} -1.23674i q^{92} -8.14605i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 12 q^{16} + 20 q^{22} - 20 q^{25} - 28 q^{28} + 44 q^{46} + 28 q^{49} - 52 q^{58} - 24 q^{64} - 16 q^{67} + 32 q^{79} + 28 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^16 + 20 * q^22 - 20 * q^25 - 28 * q^28 + 44 * q^46 + 28 * q^49 - 52 * q^58 - 24 * q^64 - 16 * q^67 + 32 * q^79 + 28 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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