LMFDB - Embedded newform 1764.2.b.g.1567.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1567,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1764.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.b (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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1567.1
Root			$-0.765367i$ of defining polynomial
Character	$\chi$	$=$	1764.1567
Dual form			1764.2.b.g.1567.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -4.46088i q^{5} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -4.46088i q^{5} -2.82843 q^{8} +6.30864i q^{10} +5.99162i q^{13} +4.00000 q^{16} +4.90923i q^{17} -8.92177i q^{20} -14.8995 q^{25} -8.47343i q^{26} -4.24264 q^{29} -5.65685 q^{32} -6.94269i q^{34} -9.89949 q^{37} +12.6173i q^{40} +3.56420i q^{41} +21.0711 q^{50} +11.9832i q^{52} -14.0000 q^{53} +6.00000 q^{58} +7.25972i q^{61} +8.00000 q^{64} +26.7279 q^{65} +9.81845i q^{68} +11.6662i q^{73} +14.0000 q^{74} -17.8435i q^{80} -5.04054i q^{82} +21.8995 q^{85} +3.11586i q^{89} -13.5684i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} - 20 q^{25} + 56 q^{50} - 56 q^{53} + 24 q^{58} + 32 q^{64} + 56 q^{65} + 56 q^{74} + 48 q^{85}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 - 20 * q^25 + 56 * q^50 - 56 * q^53 + 24 * q^58 + 32 * q^64 + 56 * q^65 + 56 * q^74 + 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.41421	−1.00000
$2$	−1.41421	−1.00000
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	− 4.46088i	− 1.99497i	−0.0708890	−	0.997484i	$-0.522584\pi$
$5$	− 4.46088i	− 1.99497i	0.0708890	−	0.997484i	$-0.477416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	0	0
$10$	6.30864i	1.99497i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	5.99162i	1.66178i	0.556440	+	0.830888i	$0.312167\pi$
$13$	5.99162i	1.66178i	−0.556440	+	0.830888i	$0.687833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	4.90923i	1.19066i	0.803480	+	0.595331i	$0.202979\pi$
$17$	4.90923i	1.19066i	−0.803480	+	0.595331i	$0.797021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	− 8.92177i	− 1.99497i
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−14.8995	−2.97990
$26$	− 8.47343i	− 1.66178i
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	− 6.94269i	− 1.19066i
$35$	0	0
$36$	0	0
$37$	−9.89949	−1.62747	−0.813733	−	0.581238i	$-0.802568\pi$
$37$	−9.89949	−1.62747	−0.813733	+	0.581238i	$0.802568\pi$
$38$	0	0
$39$	0	0
$40$	12.6173i	1.99497i
$41$	3.56420i	0.556635i	0.960489	+	0.278317i	$0.0897767\pi$
$41$	3.56420i	0.556635i	−0.960489	+	0.278317i	$0.910223\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	21.0711	2.97990
$51$	0	0
$52$	11.9832i	1.66178i
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.25972i	0.929512i	0.885439	+	0.464756i	$0.153858\pi$
$61$	7.25972i	0.929512i	−0.885439	+	0.464756i	$0.846142\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	26.7279	3.31519
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	9.81845i	1.19066i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	11.6662i	1.36543i	0.730686	+	0.682713i	$0.239200\pi$
$73$	11.6662i	1.36543i	−0.730686	+	0.682713i	$0.760800\pi$
$74$	14.0000	1.62747
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 17.8435i	− 1.99497i
$81$	0	0
$82$	− 5.04054i	− 0.556635i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	21.8995	2.37533
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.11586i	0.330281i	0.986270	+	0.165140i	$0.0528077\pi$
$89$	3.11586i	0.330281i	−0.986270	+	0.165140i	$0.947192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.5684i	− 1.37766i	−0.724924	−	0.688829i	$-0.758125\pi$
$97$	− 13.5684i	− 1.37766i	0.724924	−	0.688829i	$-0.241875\pi$
$98$	0	0
$99$	0	0
$100$	−29.7990	−2.97990
$101$	5.80591i	0.577710i	0.957373	+	0.288855i	$0.0932745\pi$
$101$	5.80591i	0.577710i	−0.957373	+	0.288855i	$0.906726\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	− 16.9469i	− 1.66178i
$105$	0	0
$106$	19.7990	1.92305
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	9.89949	0.948200	0.474100	−	0.880471i	$-0.342774\pi$
$109$	9.89949	0.948200	0.474100	+	0.880471i	$0.342774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−8.48528	−0.787839
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	− 10.2668i	− 0.929512i
$123$	0	0
$124$	0	0
$125$	44.1605i	3.94984i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	−37.7990	−3.31519
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	− 13.8854i	− 1.19066i
$137$	21.2132	1.81237	0.906183	−	0.422885i	$-0.138983\pi$
$137$	21.2132	1.81237	0.906183	+	0.422885i	$0.138983\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	18.9259i	1.57171i
$146$	− 16.4985i	− 1.36543i
$147$	0	0
$148$	−19.7990	−1.62747
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.9176i	1.98864i	0.106446	+	0.994319i	$0.466053\pi$
$157$	24.9176i	1.98864i	−0.106446	+	0.994319i	$0.533947\pi$
$158$	0	0
$159$	0	0
$160$	25.2346i	1.99497i
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	7.12840i	0.556635i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−22.8995	−1.76150
$170$	−30.9706	−2.37533
$171$	0	0
$172$	0	0
$173$	− 6.25425i	− 0.475502i	−0.971326	−	0.237751i	$-0.923590\pi$
$173$	− 6.25425i	− 0.475502i	0.971326	−	0.237751i	$-0.0764102\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 4.40649i	− 0.330281i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 24.2835i	− 1.80498i	−0.430713	−	0.902489i	$-0.641738\pi$
$181$	− 24.2835i	− 1.80498i	0.430713	−	0.902489i	$-0.358262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	44.1605i	3.24675i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	19.1886i	1.37766i
$195$	0	0
$196$	0	0
$197$	−28.0000	−1.99492	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−28.0000	−1.99492	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	42.1421	2.97990
$201$	0	0
$202$	− 8.21080i	− 0.577710i
$203$	0	0
$204$	0	0
$205$	15.8995	1.11047
$206$	0	0
$207$	0	0
$208$	23.9665i	1.66178i
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−28.0000	−1.92305
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−14.0000	−0.948200
$219$	0	0
$220$	0	0
$221$	−29.4142	−1.97861
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	19.7990	1.31701
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	− 26.1857i	− 1.73040i	−0.501430	−	0.865198i	$-0.667192\pi$
$229$	− 26.1857i	− 1.73040i	0.501430	−	0.865198i	$-0.332808\pi$
$230$	0	0
$231$	0	0
$232$	12.0000	0.787839
$233$	7.07107	0.463241	0.231621	−	0.972806i	$-0.425597\pi$
$233$	7.07107	0.463241	0.231621	+	0.972806i	$0.425597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	− 4.08947i	− 0.263426i	−0.991288	−	0.131713i	$-0.957952\pi$
$241$	− 4.08947i	− 0.263426i	0.991288	−	0.131713i	$-0.0420477\pi$
$242$	−15.5563	−1.00000
$243$	0	0
$244$	14.5194i	0.929512i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	− 62.4524i	− 3.94984i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	30.3295i	1.89190i	0.324308	+	0.945951i	$0.394869\pi$
$257$	30.3295i	1.89190i	−0.324308	+	0.945951i	$0.605131\pi$
$258$	0	0
$259$	0	0
$260$	53.4558	3.31519
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	62.4524i	3.83642i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 31.6745i	− 1.93123i	−0.259975	−	0.965615i	$-0.583714\pi$
$269$	− 31.6745i	− 1.93123i	0.259975	−	0.965615i	$-0.416286\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	19.6369i	1.19066i
$273$	0	0
$274$	−30.0000	−1.81237
$275$	0	0
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.5563	−0.928014	−0.464007	−	0.885832i	$-0.653589\pi$
$281$	−15.5563	−0.928014	−0.464007	+	0.885832i	$0.653589\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.10051	−0.417677
$290$	− 26.7653i	− 1.57171i
$291$	0	0
$292$	23.3324i	1.36543i
$293$	29.8812i	1.74568i	0.488010	+	0.872838i	$0.337723\pi$
$293$	29.8812i	1.74568i	−0.488010	+	0.872838i	$0.662277\pi$
$294$	0	0
$295$	0	0
$296$	28.0000	1.62747
$297$	0	0
$298$	19.7990	1.14692
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	32.3848	1.85435
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 14.8365i	− 0.838607i	−0.907846	−	0.419303i	$-0.862274\pi$
$313$	− 14.8365i	− 0.838607i	0.907846	−	0.419303i	$-0.137726\pi$
$314$	− 35.2387i	− 1.98864i
$315$	0	0
$316$	0	0
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	0	0
$319$	0	0
$320$	− 35.6871i	− 1.99497i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 89.2721i	− 4.95192i
$326$	0	0
$327$	0	0
$328$	− 10.0811i	− 0.556635i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.89949	−0.539260	−0.269630	−	0.962964i	$-0.586901\pi$
$337$	−9.89949	−0.539260	−0.269630	+	0.962964i	$0.586901\pi$
$338$	32.3848	1.76150
$339$	0	0
$340$	43.7990	2.37533
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	8.84485i	0.475502i
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	− 23.0154i	− 1.23199i	−0.787752	−	0.615993i	$-0.788755\pi$
$349$	− 23.0154i	− 1.23199i	0.787752	−	0.615993i	$-0.211245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.77084i	− 0.0942521i	−0.998889	−	0.0471260i	$-0.984994\pi$
$353$	− 1.77084i	− 0.0942521i	0.998889	−	0.0471260i	$-0.0150062\pi$
$354$	0	0
$355$	0	0
$356$	6.23172i	0.330281i
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	34.3421i	1.80498i
$363$	0	0
$364$	0	0
$365$	52.0416	2.72398
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	− 62.4524i	− 3.24675i
$371$	0	0
$372$	0	0
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 25.4203i	− 1.30921i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−33.9411	−1.72756
$387$	0	0
$388$	− 27.1367i	− 1.37766i
$389$	−38.1838	−1.93599	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	−38.1838	−1.93599	−0.967997	+	0.250962i	$0.919253\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	39.5980	1.99492
$395$	0	0
$396$	0	0
$397$	− 3.45542i	− 0.173422i	−0.996233	−	0.0867112i	$-0.972364\pi$
$397$	− 3.45542i	− 0.173422i	0.996233	−	0.0867112i	$-0.0276357\pi$
$398$	0	0
$399$	0	0
$400$	−59.5980	−2.97990
$401$	26.8701	1.34183	0.670913	−	0.741536i	$-0.265902\pi$
$401$	26.8701	1.34183	0.670913	+	0.741536i	$0.265902\pi$
$402$	0	0
$403$	0	0
$404$	11.6118i	0.577710i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 9.76406i	− 0.482802i	−0.970425	−	0.241401i	$-0.922393\pi$
$409$	− 9.76406i	− 0.482802i	0.970425	−	0.241401i	$-0.0776069\pi$
$410$	−22.4853	−1.11047
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	− 33.8937i	− 1.66178i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	0	0
$423$	0	0
$424$	39.5980	1.92305
$425$	− 73.1450i	− 3.54805i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	9.16187i	0.440291i	0.975467	+	0.220146i	$0.0706533\pi$
$433$	9.16187i	0.440291i	−0.975467	+	0.220146i	$0.929347\pi$
$434$	0	0
$435$	0	0
$436$	19.7990	0.948200
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	41.5980	1.97861
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	13.8995	0.658899
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	0	0
$452$	−28.0000	−1.31701
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	37.0321i	1.73040i
$459$	0	0
$460$	0	0
$461$	− 33.0196i	− 1.53787i	−0.639324	−	0.768937i	$-0.720786\pi$
$461$	− 33.0196i	− 1.53787i	0.639324	−	0.768937i	$-0.279214\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	−16.9706	−0.787839
$465$	0	0
$466$	−10.0000	−0.463241
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	− 59.3140i	− 2.70449i
$482$	5.78338i	0.263426i
$483$	0	0
$484$	22.0000	1.00000
$485$	−60.5269	−2.74839
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 20.5336i	− 0.929512i
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 20.8281i	− 0.938050i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	88.3210i	3.94984i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	25.8995	1.15251
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.59928i	0.336832i	0.985716	+	0.168416i	$0.0538652\pi$
$509$	7.59928i	0.336832i	−0.985716	+	0.168416i	$0.946135\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	0	0
$514$	− 42.8924i	− 1.89190i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−75.5980	−3.31519
$521$	28.5361i	1.25019i	0.780548	+	0.625096i	$0.214940\pi$
$521$	28.5361i	1.25019i	−0.780548	+	0.625096i	$0.785060\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	− 88.3210i	− 3.83642i
$531$	0	0
$532$	0	0
$533$	−21.3553	−0.925002
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	44.7946i	1.93123i
$539$	0	0
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	0	0
$544$	− 27.7708i	− 1.19066i
$545$	− 44.1605i	− 1.89163i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	42.4264	1.81237
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	25.4558	1.08152
$555$	0	0
$556$	0	0
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	22.0000	0.928014
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	62.4524i	2.62739i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.6690	−1.95647	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	−46.6690	−1.95647	−0.978234	+	0.207504i	$0.933466\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.1116i	1.87802i	0.343890	+	0.939010i	$0.388255\pi$
$577$	45.1116i	1.87802i	−0.343890	+	0.939010i	$0.611745\pi$
$578$	10.0416	0.417677
$579$	0	0
$580$	37.8519i	1.57171i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	− 32.9970i	− 1.36543i
$585$	0	0
$586$	− 42.2584i	− 1.74568i
$587$	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
$592$	−39.5980	−1.62747
$593$	48.6214i	1.99664i	0.0579298	+	0.998321i	$0.481550\pi$
$593$	48.6214i	1.99664i	−0.0579298	+	0.998321i	$0.518450\pi$
$594$	0	0
$595$	0	0
$596$	−28.0000	−1.14692
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	9.13001i	0.372421i	0.982510	+	0.186210i	$0.0596206\pi$
$601$	9.13001i	0.372421i	−0.982510	+	0.186210i	$0.940379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 49.0697i	− 1.99497i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	−45.7990	−1.85435
$611$	0	0
$612$	0	0
$613$	49.4975	1.99918	0.999592	−	0.0285598i	$-0.00909209\pi$
$613$	49.4975	1.99918	0.999592	+	0.0285598i	$0.00909209\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.24264	−0.170802	−0.0854011	−	0.996347i	$-0.527217\pi$
$617$	−4.24264	−0.170802	−0.0854011	+	0.996347i	$0.527217\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	122.497	4.89990
$626$	20.9819i	0.838607i
$627$	0	0
$628$	49.8351i	1.98864i
$629$	− 48.5989i	− 1.93776i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−39.5980	−1.57264
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	50.4692i	1.99497i
$641$	−41.0122	−1.61988	−0.809942	−	0.586510i	$-0.800502\pi$
$641$	−41.0122	−1.61988	−0.809942	+	0.586510i	$0.800502\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	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
$650$	126.250i	4.95192i
$651$	0	0
$652$	0	0
$653$	12.7279	0.498082	0.249041	−	0.968493i	$-0.419885\pi$
$653$	12.7279	0.498082	0.249041	+	0.968493i	$0.419885\pi$
$654$	0	0
$655$	0	0
$656$	14.2568i	0.556635i
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	50.7862i	1.97535i	0.156508	+	0.987677i	$0.449976\pi$
$661$	50.7862i	1.97535i	−0.156508	+	0.987677i	$0.550024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−49.4975	−1.90799	−0.953994	−	0.299827i	$-0.903071\pi$
$673$	−49.4975	−1.90799	−0.953994	+	0.299827i	$0.903071\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−45.7990	−1.76150
$677$	− 47.2764i	− 1.81698i	−0.417908	−	0.908489i	$-0.637237\pi$
$677$	− 47.2764i	− 1.81698i	0.417908	−	0.908489i	$-0.362763\pi$
$678$	0	0
$679$	0	0
$680$	−61.9411	−2.37533
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	− 94.6297i	− 3.61561i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 83.8827i	− 3.19567i
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	− 12.5085i	− 0.475502i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−17.4975	−0.662764
$698$	32.5487i	1.23199i
$699$	0	0
$700$	0	0
$701$	43.8406	1.65584	0.827919	−	0.560848i	$-0.189525\pi$
$701$	43.8406	1.65584	0.827919	+	0.560848i	$0.189525\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	2.50434i	0.0942521i
$707$	0	0
$708$	0	0
$709$	−9.89949	−0.371783	−0.185892	−	0.982570i	$-0.559517\pi$
$709$	−9.89949	−0.371783	−0.185892	+	0.982570i	$0.559517\pi$
$710$	0	0
$711$	0	0
$712$	− 8.81298i	− 0.330281i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	26.8701	1.00000
$723$	0	0
$724$	− 48.5670i	− 1.80498i
$725$	63.2132	2.34768
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	−73.5980	−2.72398
$731$	0	0
$732$	0	0
$733$	51.4202i	1.89925i	0.313389	+	0.949625i	$0.398536\pi$
$733$	51.4202i	1.89925i	−0.313389	+	0.949625i	$0.601464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	88.3210i	3.24675i
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	62.4524i	2.28808i
$746$	50.9117	1.86401
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	35.9497i	1.30921i
$755$	0	0
$756$	0	0
$757$	49.4975	1.79902	0.899508	−	0.436904i	$-0.143925\pi$
$757$	49.4975	1.79902	0.899508	+	0.436904i	$0.143925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.4148i	1.82753i	0.406239	+	0.913767i	$0.366840\pi$
$761$	50.4148i	1.82753i	−0.406239	+	0.913767i	$0.633160\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	41.3073i	1.48958i	0.667300	+	0.744789i	$0.267450\pi$
$769$	41.3073i	1.48958i	−0.667300	+	0.744789i	$0.732550\pi$
$770$	0	0
$771$	0	0
$772$	48.0000	1.72756
$773$	27.6395i	0.994122i	0.867715	+	0.497061i	$0.165588\pi$
$773$	27.6395i	0.994122i	−0.867715	+	0.497061i	$0.834412\pi$
$774$	0	0
$775$	0	0
$776$	38.3771i	1.37766i
$777$	0	0
$778$	54.0000	1.93599
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	111.154	3.96727
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	−56.0000	−1.99492
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−43.4975	−1.54464
$794$	4.88670i	0.173422i
$795$	0	0
$796$	0	0
$797$	− 0.425811i	− 0.0150830i	−0.999972	−	0.00754150i	$-0.997599\pi$
$797$	− 0.425811i	− 0.0150830i	0.999972	−	0.00754150i	$-0.00240056\pi$
$798$	0	0
$799$	0	0
$800$	84.2843	2.97990
$801$	0	0
$802$	−38.0000	−1.34183
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 16.4216i	− 0.577710i
$809$	56.0000	1.96886	0.984428	−	0.175791i	$-0.0562482\pi$
$809$	56.0000	1.96886	0.984428	+	0.175791i	$0.0562482\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	13.8085i	0.482802i
$819$	0	0
$820$	31.7990	1.11047
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	2.18731i	0.0759686i	0.999278	+	0.0379843i	$0.0120937\pi$
$829$	2.18731i	0.0759686i	−0.999278	+	0.0379843i	$0.987906\pi$
$830$	0	0
$831$	0	0
$832$	47.9329i	1.66178i
$833$	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
$841$	−11.0000	−0.379310
$842$	42.4264	1.46211
$843$	0	0
$844$	0	0
$845$	102.152i	3.51414i
$846$	0	0
$847$	0	0
$848$	−56.0000	−1.92305
$849$	0	0
$850$	103.443i	3.54805i
$851$	0	0
$852$	0	0
$853$	28.7219i	0.983418i	0.870760	+	0.491709i	$0.163628\pi$
$853$	28.7219i	0.983418i	−0.870760	+	0.491709i	$0.836372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.6465i	− 1.93501i	−0.252858	−	0.967503i	$-0.581370\pi$
$857$	− 56.6465i	− 1.93501i	0.252858	−	0.967503i	$-0.418630\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−27.8995	−0.948611
$866$	− 12.9568i	− 0.440291i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−28.0000	−0.948200
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.4975	−1.67141	−0.835705	−	0.549178i	$-0.814941\pi$
$877$	−49.4975	−1.67141	−0.835705	+	0.549178i	$0.814941\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	58.4398i	1.96889i	0.175697	+	0.984444i	$0.443782\pi$
$881$	58.4398i	1.96889i	−0.175697	+	0.984444i	$0.556218\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−58.8284	−1.97861
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	−19.6569	−0.658899
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−19.7990	−0.660701
$899$	0	0
$900$	0	0
$901$	− 68.7292i	− 2.28970i
$902$	0	0
$903$	0	0
$904$	39.5980	1.31701
$905$	−108.326	−3.60087
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	−11.3137	−0.374224
$915$	0	0
$916$	− 52.3713i	− 1.73040i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	46.6967i	1.53787i
$923$	0	0
$924$	0	0
$925$	147.497	4.84969
$926$	0	0
$927$	0	0
$928$	24.0000	0.787839
$929$	− 27.1911i	− 0.892112i	−0.895005	−	0.446056i	$-0.852828\pi$
$929$	− 27.1911i	− 0.892112i	0.895005	−	0.446056i	$-0.147172\pi$
$930$	0	0
$931$	0	0
$932$	14.1421	0.463241
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 16.7386i	− 0.546827i	−0.961897	−	0.273413i	$-0.911847\pi$
$937$	− 16.7386i	− 0.546827i	0.961897	−	0.273413i	$-0.0881527\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	45.9313i	1.49732i	0.662955	+	0.748660i	$0.269302\pi$
$941$	45.9313i	1.49732i	−0.662955	+	0.748660i	$0.730698\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−69.8995	−2.26903
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	83.8827i	2.70449i
$963$	0	0
$964$	− 8.17893i	− 0.263426i
$965$	− 107.061i	− 3.44642i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−31.1127	−1.00000
$969$	0	0
$970$	85.5980	2.74839
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	29.0389i	0.929512i
$977$	−38.1838	−1.22161	−0.610803	−	0.791782i	$-0.709153\pi$
$977$	−38.1838	−1.22161	−0.610803	+	0.791782i	$0.709153\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	124.905i	3.97980i
$986$	29.4554i	0.938050i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 52.6883i	− 1.66866i	−0.551268	−	0.834328i	$-0.685856\pi$
$997$	− 52.6883i	− 1.66866i	0.551268	−	0.834328i	$-0.314144\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.b.g.1567.1	yes	4
3.2	odd	2		1764.2.b.e.1567.4	yes	4
4.3	odd	2	CM	1764.2.b.g.1567.1	yes	4
7.6	odd	2	inner	1764.2.b.g.1567.2	yes	4
12.11	even	2		1764.2.b.e.1567.4	yes	4
21.20	even	2		1764.2.b.e.1567.3	✓	4
28.27	even	2	inner	1764.2.b.g.1567.2	yes	4
84.83	odd	2		1764.2.b.e.1567.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.b.e.1567.3	✓	4	21.20	even	2
1764.2.b.e.1567.3	✓	4	84.83	odd	2
1764.2.b.e.1567.4	yes	4	3.2	odd	2
1764.2.b.e.1567.4	yes	4	12.11	even	2
1764.2.b.g.1567.1	yes	4	1.1	even	1	trivial
1764.2.b.g.1567.1	yes	4	4.3	odd	2	CM
1764.2.b.g.1567.2	yes	4	7.6	odd	2	inner
1764.2.b.g.1567.2	yes	4	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.46088i q^{5} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.46088i q^{5} -2.82843 q^{8} +6.30864i q^{10} +5.99162i q^{13} +4.00000 q^{16} +4.90923i q^{17} -8.92177i q^{20} -14.8995 q^{25} -8.47343i q^{26} -4.24264 q^{29} -5.65685 q^{32} -6.94269i q^{34} -9.89949 q^{37} +12.6173i q^{40} +3.56420i q^{41} +21.0711 q^{50} +11.9832i q^{52} -14.0000 q^{53} +6.00000 q^{58} +7.25972i q^{61} +8.00000 q^{64} +26.7279 q^{65} +9.81845i q^{68} +11.6662i q^{73} +14.0000 q^{74} -17.8435i q^{80} -5.04054i q^{82} +21.8995 q^{85} +3.11586i q^{89} -13.5684i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} - 20 q^{25} + 56 q^{50} - 56 q^{53} + 24 q^{58} + 32 q^{64} + 56 q^{65} + 56 q^{74} + 48 q^{85}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 - 20 * q^25 + 56 * q^50 - 56 * q^53 + 24 * q^58 + 32 * q^64 + 56 * q^65 + 56 * q^74 + 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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