LMFDB - Embedded newform 1764.2.b.e.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			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	1764.1567
Dual form			1764.2.b.e.1567.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -0.317025i q^{5} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -0.317025i q^{5} -2.82843 q^{8} +0.448342i q^{10} -4.01254i q^{13} +4.00000 q^{16} +6.62567i q^{17} -0.634051i q^{20} +4.89949 q^{25} +5.67459i q^{26} -4.24264 q^{29} -5.65685 q^{32} -9.37011i q^{34} +9.89949 q^{37} +0.896683i q^{40} -12.3003i q^{41} -6.92893 q^{50} -8.02509i q^{52} +14.0000 q^{53} +6.00000 q^{58} +13.8310i q^{61} +8.00000 q^{64} -1.27208 q^{65} +13.2513i q^{68} -12.4860i q^{73} -14.0000 q^{74} -1.26810i q^{80} +17.3952i q^{82} +2.10051 q^{85} -18.6089i q^{89} -14.2793i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 0.317025i	− 0.141778i	−0.997484	−	0.0708890i	$-0.977416\pi$
$5$	− 0.317025i	− 0.141778i	0.997484	−	0.0708890i	$-0.0225836\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0.448342i	0.141778i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 4.01254i	− 1.11288i	−0.830888	−	0.556440i	$-0.812167\pi$
$13$	− 4.01254i	− 1.11288i	0.830888	−	0.556440i	$-0.187833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	6.62567i	1.60696i	0.595331	+	0.803480i	$0.297021\pi$
$17$	6.62567i	1.60696i	−0.595331	+	0.803480i	$0.702979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	− 0.634051i	− 0.141778i
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	4.89949	0.979899
$26$	5.67459i	1.11288i
$27$	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
$34$	− 9.37011i	− 1.60696i
$35$	0	0
$36$	0	0
$37$	9.89949	1.62747	0.813733	−	0.581238i	$-0.197432\pi$
$37$	9.89949	1.62747	0.813733	+	0.581238i	$0.197432\pi$
$38$	0	0
$39$	0	0
$40$	0.896683i	0.141778i
$41$	− 12.3003i	− 1.92098i	−0.278317	−	0.960489i	$-0.589777\pi$
$41$	− 12.3003i	− 1.92098i	0.278317	−	0.960489i	$-0.410223\pi$
$42$	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
$50$	−6.92893	−0.979899
$51$	0	0
$52$	− 8.02509i	− 1.11288i
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	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
$61$	13.8310i	1.77088i	0.464756	+	0.885439i	$0.346142\pi$
$61$	13.8310i	1.77088i	−0.464756	+	0.885439i	$0.653858\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.27208	−0.157782
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	13.2513i	1.60696i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 12.4860i	− 1.46137i	−0.682713	−	0.730686i	$-0.739200\pi$
$73$	− 12.4860i	− 1.46137i	0.682713	−	0.730686i	$-0.260800\pi$
$74$	−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			$\pi$
$80$	− 1.26810i	− 0.141778i
$81$	0	0
$82$	17.3952i	1.92098i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	2.10051	0.227832
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 18.6089i	− 1.97254i	−0.165140	−	0.986270i	$-0.552808\pi$
$89$	− 18.6089i	− 1.97254i	0.165140	−	0.986270i	$-0.447192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.2793i	− 1.44985i	−0.688829	−	0.724924i	$-0.741875\pi$
$97$	− 14.2793i	− 1.44985i	0.688829	−	0.724924i	$-0.258125\pi$
$98$	0	0
$99$	0	0
$100$	9.79899	0.979899
$101$	19.2430i	1.91475i	0.288855	+	0.957373i	$0.406726\pi$
$101$	19.2430i	1.91475i	−0.288855	+	0.957373i	$0.593274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	11.3492i	1.11288i
$105$	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	$-0.657226\pi$
$109$	−9.89949	−0.948200	−0.474100	+	0.880471i	$0.657226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	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
$122$	− 19.5600i	− 1.77088i
$123$	0	0
$124$	0	0
$125$	− 3.13839i	− 0.280706i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	1.79899	0.157782
$131$	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
$136$	− 18.7402i	− 1.60696i
$137$	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
$145$	1.34502i	0.111698i
$146$	17.6578i	1.46137i
$147$	0	0
$148$	19.7990	1.62747
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	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
$157$	− 2.66752i	− 0.212891i	−0.994319	−	0.106446i	$-0.966053\pi$
$157$	− 2.66752i	− 0.212891i	0.994319	−	0.106446i	$-0.0339470\pi$
$158$	0	0
$159$	0	0
$160$	1.79337i	0.141778i
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 24.6005i	− 1.92098i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−3.10051	−0.238500
$170$	−2.97056	−0.227832
$171$	0	0
$172$	0	0
$173$	− 25.5516i	− 1.94265i	−0.237751	−	0.971326i	$-0.576410\pi$
$173$	− 25.5516i	− 1.94265i	0.237751	−	0.971326i	$-0.423590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	26.3170i	1.97254i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	11.5893i	0.861425i	0.902489	+	0.430713i	$0.141738\pi$
$181$	11.5893i	0.861425i	−0.902489	+	0.430713i	$0.858262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.13839i	− 0.230739i
$186$	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	$0.168091\pi$
$194$	20.1940i	1.44985i
$195$	0	0
$196$	0	0
$197$	28.0000	1.99492	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	28.0000	1.99492	0.997459	+	0.0712470i	$0.0226979\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−13.8579	−0.979899
$201$	0	0
$202$	− 27.2137i	− 1.91475i
$203$	0	0
$204$	0	0
$205$	−3.89949	−0.272353
$206$	0	0
$207$	0	0
$208$	− 16.0502i	− 1.11288i
$209$	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
$221$	26.5858	1.78835
$222$	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
$229$	− 15.1760i	− 1.00286i	−0.865198	−	0.501430i	$-0.832808\pi$
$229$	− 15.1760i	− 1.00286i	0.865198	−	0.501430i	$-0.167192\pi$
$230$	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
$241$	30.7779i	1.98258i	0.131713	+	0.991288i	$0.457952\pi$
$241$	30.7779i	1.98258i	−0.131713	+	0.991288i	$0.542048\pi$
$242$	−15.5563	−1.00000
$243$	0	0
$244$	27.6620i	1.77088i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	4.43835i	0.280706i
$251$	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
$257$	− 10.3981i	− 0.648616i	−0.945951	−	0.324308i	$-0.894869\pi$
$257$	− 10.3981i	− 0.648616i	0.945951	−	0.324308i	$-0.105131\pi$
$258$	0	0
$259$	0	0
$260$	−2.54416	−0.157782
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	− 4.43835i	− 0.272646i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.52782i	− 0.519950i	−0.965615	−	0.259975i	$-0.916286\pi$
$269$	− 8.52782i	− 0.519950i	0.965615	−	0.259975i	$-0.0837144\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	26.5027i	1.60696i
$273$	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
$289$	−26.8995	−1.58232
$290$	− 1.90215i	− 0.111698i
$291$	0	0
$292$	− 24.9719i	− 1.46137i
$293$	− 16.7068i	− 0.976019i	−0.872838	−	0.488010i	$-0.837723\pi$
$293$	− 16.7068i	− 0.976019i	0.872838	−	0.488010i	$-0.162277\pi$
$294$	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
$305$	4.38478	0.251072
$306$	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
$313$	− 32.1229i	− 1.81569i	−0.419303	−	0.907846i	$-0.637726\pi$
$313$	− 32.1229i	− 1.81569i	0.419303	−	0.907846i	$-0.362274\pi$
$314$	3.77244i	0.212891i
$315$	0	0
$316$	0	0
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	0	0
$320$	− 2.53620i	− 0.141778i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 19.6594i	− 1.09051i
$326$	0	0
$327$	0	0
$328$	34.7904i	1.92098i
$329$	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	$-0.413099\pi$
$337$	9.89949	0.539260	0.269630	+	0.962964i	$0.413099\pi$
$338$	4.38478	0.238500
$339$	0	0
$340$	4.20101	0.227832
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	36.1354i	1.94265i
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	29.4328i	1.57550i	0.615993	+	0.787752i	$0.288755\pi$
$349$	29.4328i	1.57550i	−0.615993	+	0.787752i	$0.711245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	37.5348i	1.99778i	0.0471260	+	0.998889i	$0.484994\pi$
$353$	37.5348i	1.99778i	−0.0471260	+	0.998889i	$0.515006\pi$
$354$	0	0
$355$	0	0
$356$	− 37.2178i	− 1.97254i
$357$	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
$362$	− 16.3897i	− 0.861425i
$363$	0	0
$364$	0	0
$365$	−3.95837	−0.207191
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	4.43835i	0.230739i
$371$	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
$377$	17.0238i	0.876769i
$378$	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
$388$	− 28.5587i	− 1.44985i
$389$	−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
$397$	39.6996i	1.99247i	0.0867112	+	0.996233i	$0.472364\pi$
$397$	39.6996i	1.99247i	−0.0867112	+	0.996233i	$0.527636\pi$
$398$	0	0
$399$	0	0
$400$	19.5980	0.979899
$401$	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
$404$	38.4859i	1.91475i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	39.2513i	1.94085i	0.241401	+	0.970425i	$0.422393\pi$
$409$	39.2513i	1.94085i	−0.241401	+	0.970425i	$0.577607\pi$
$410$	5.51472	0.272353
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	22.6984i	1.11288i
$417$	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
$425$	32.4624i	1.57466i
$426$	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
$433$	40.5963i	1.95093i	0.220146	+	0.975467i	$0.429347\pi$
$433$	40.5963i	1.95093i	−0.220146	+	0.975467i	$0.570653\pi$
$434$	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
$442$	−37.5980	−1.78835
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−5.89949	−0.279663
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	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	$0.440087\pi$
$458$	21.4621i	1.00286i
$459$	0	0
$460$	0	0
$461$	− 27.4538i	− 1.27865i	−0.768937	−	0.639324i	$-0.779214\pi$
$461$	− 27.4538i	− 1.27865i	0.768937	−	0.639324i	$-0.220786\pi$
$462$	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
$481$	− 39.7222i	− 1.81117i
$482$	− 43.5265i	− 1.98258i
$483$	0	0
$484$	22.0000	1.00000
$485$	−4.52691	−0.205556
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 39.1200i	− 1.77088i
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 28.1103i	− 1.26603i
$494$	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			$\pi$
$500$	− 6.27678i	− 0.280706i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	6.10051	0.271469
$506$	0	0
$507$	0	0
$508$	0	0
$509$	44.4775i	1.97143i	0.168416	+	0.985716i	$0.446135\pi$
$509$	44.4775i	1.97143i	−0.168416	+	0.985716i	$0.553865\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−1.00000
$513$	0	0
$514$	14.7051i	0.648616i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	3.59798	0.157782
$521$	− 35.6327i	− 1.56110i	−0.625096	−	0.780548i	$-0.714940\pi$
$521$	− 35.6327i	− 1.56110i	0.625096	−	0.780548i	$-0.285060\pi$
$522$	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
$530$	6.27678i	0.272646i
$531$	0	0
$532$	0	0
$533$	−49.3553	−2.13782
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	12.0602i	0.519950i
$539$	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
$544$	− 37.4804i	− 1.60696i
$545$	3.13839i	0.134434i
$546$	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	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	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
$565$	− 4.43835i	− 0.186723i
$566$	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
$577$	16.5210i	0.687780i	0.939010	+	0.343890i	$0.111745\pi$
$577$	16.5210i	0.687780i	−0.939010	+	0.343890i	$0.888255\pi$
$578$	38.0416	1.58232
$579$	0	0
$580$	2.69005i	0.111698i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	35.3157i	1.46137i
$585$	0	0
$586$	23.6269i	0.976019i
$587$	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
$593$	− 2.82137i	− 0.115860i	−0.998321	−	0.0579298i	$-0.981550\pi$
$593$	− 2.82137i	− 0.115860i	0.998321	−	0.0579298i	$-0.0184500\pi$
$594$	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
$601$	− 48.1731i	− 1.96502i	−0.186210	−	0.982510i	$-0.559621\pi$
$601$	− 48.1731i	− 1.96502i	0.186210	−	0.982510i	$-0.440379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.48728i	− 0.141778i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	−6.20101	−0.251072
$611$	0	0
$612$	0	0
$613$	−49.4975	−1.99918	−0.999592	−	0.0285598i	$-0.990908\pi$
$613$	−49.4975	−1.99918	−0.999592	+	0.0285598i	$0.990908\pi$
$614$	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
$625$	23.5025	0.940101
$626$	45.4286i	1.81569i
$627$	0	0
$628$	− 5.33504i	− 0.212891i
$629$	65.5908i	2.61528i
$630$	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
$640$	3.58673i	0.141778i
$641$	−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
$650$	27.8026i	1.09051i
$651$	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
$656$	− 49.2011i	− 1.92098i
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	8.04762i	0.313016i	0.987677	+	0.156508i	$0.0500237\pi$
$661$	8.04762i	0.313016i	−0.987677	+	0.156508i	$0.949976\pi$
$662$	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	$-0.0969288\pi$
$673$	49.4975	1.90799	0.953994	+	0.299827i	$0.0969288\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−6.20101	−0.238500
$677$	21.7473i	0.835817i	0.908489	+	0.417908i	$0.137237\pi$
$677$	21.7473i	0.835817i	−0.908489	+	0.417908i	$0.862763\pi$
$678$	0	0
$679$	0	0
$680$	−5.94113	−0.227832
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	− 6.72512i	− 0.256954i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 56.1756i	− 2.14012i
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	− 51.1032i	− 1.94265i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	81.4975	3.08694
$698$	− 41.6243i	− 1.57550i
$699$	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
$706$	− 53.0823i	− 1.99778i
$707$	0	0
$708$	0	0
$709$	9.89949	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$709$	9.89949	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$710$	0	0
$711$	0	0
$712$	52.6339i	1.97254i
$713$	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
$724$	23.1786i	0.861425i
$725$	−20.7868	−0.772002
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	5.59798	0.207191
$731$	0	0
$732$	0	0
$733$	16.9694i	0.626779i	0.949625	+	0.313389i	$0.101464\pi$
$733$	16.9694i	0.626779i	−0.949625	+	0.313389i	$0.898536\pi$
$734$	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			$\pi$
$740$	− 6.27678i	− 0.230739i
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	− 4.43835i	− 0.162609i
$746$	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
$754$	− 24.0753i	− 0.876769i
$755$	0	0
$756$	0	0
$757$	−49.4975	−1.79902	−0.899508	−	0.436904i	$-0.856075\pi$
$757$	−49.4975	−1.79902	−0.899508	+	0.436904i	$0.856075\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.4132i	0.812478i	0.913767	+	0.406239i	$0.133160\pi$
$761$	22.4132i	0.812478i	−0.913767	+	0.406239i	$0.866840\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 37.0096i	− 1.33460i	−0.744789	−	0.667300i	$-0.767450\pi$
$769$	− 37.0096i	− 1.33460i	0.744789	−	0.667300i	$-0.232550\pi$
$770$	0	0
$771$	0	0
$772$	48.0000	1.72756
$773$	− 48.2500i	− 1.73543i	−0.497061	−	0.867715i	$-0.665588\pi$
$773$	− 48.2500i	− 1.73543i	0.497061	−	0.867715i	$-0.334412\pi$
$774$	0	0
$775$	0	0
$776$	40.3881i	1.44985i
$777$	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
$785$	−0.845671	−0.0301833
$786$	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
$793$	55.4975	1.97077
$794$	− 56.1437i	− 1.99247i
$795$	0	0
$796$	0	0
$797$	56.4608i	1.99994i	0.00754150	+	0.999972i	$0.497599\pi$
$797$	56.4608i	1.99994i	−0.00754150	+	0.999972i	$0.502401\pi$
$798$	0	0
$799$	0	0
$800$	−27.7157	−0.979899
$801$	0	0
$802$	−38.0000	−1.34183
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 54.4273i	− 1.91475i
$809$	−56.0000	−1.96886	−0.984428	−	0.175791i	$-0.943752\pi$
$809$	−56.0000	−1.96886	−0.984428	+	0.175791i	$0.943752\pi$
$810$	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
$818$	− 55.5097i	− 1.94085i
$819$	0	0
$820$	−7.79899	−0.272353
$821$	28.0000	0.977207	0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000	0.977207	0.488603	+	0.872506i	$0.337507\pi$
$822$	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
$829$	− 57.5432i	− 1.99856i	−0.0379843	−	0.999278i	$-0.512094\pi$
$829$	− 57.5432i	− 1.99856i	0.0379843	−	0.999278i	$-0.487906\pi$
$830$	0	0
$831$	0	0
$832$	− 32.1003i	− 1.11288i
$833$	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
$845$	0.982939i	0.0338141i
$846$	0	0
$847$	0	0
$848$	56.0000	1.92305
$849$	0	0
$850$	− 45.9088i	− 1.57466i
$851$	0	0
$852$	0	0
$853$	50.8631i	1.74152i	0.491709	+	0.870760i	$0.336372\pi$
$853$	50.8631i	1.74152i	−0.491709	+	0.870760i	$0.663628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.8046i	0.505716i	0.967503	+	0.252858i	$0.0813705\pi$
$857$	14.8046i	0.505716i	−0.967503	+	0.252858i	$0.918630\pi$
$858$	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
$865$	−8.10051	−0.275425
$866$	− 57.4118i	− 1.95093i
$867$	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	$-0.185059\pi$
$877$	49.4975	1.67141	0.835705	+	0.549178i	$0.185059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.4300i	0.351395i	0.984444	+	0.175697i	$0.0562180\pi$
$881$	10.4300i	0.351395i	−0.984444	+	0.175697i	$0.943782\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	53.1716	1.78835
$885$	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
$890$	8.34315	0.279663
$891$	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
$901$	92.7594i	3.09026i
$902$	0	0
$903$	0	0
$904$	−39.5980	−1.31701
$905$	3.67410	0.122131
$906$	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
$916$	− 30.3520i	− 1.00286i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	38.8255i	1.27865i
$923$	0	0
$924$	0	0
$925$	48.5025	1.59475
$926$	0	0
$927$	0	0
$928$	24.0000	0.787839
$929$	54.5586i	1.79001i	0.446056	+	0.895005i	$0.352828\pi$
$929$	54.5586i	1.79001i	−0.446056	+	0.895005i	$0.647172\pi$
$930$	0	0
$931$	0	0
$932$	14.1421	0.463241
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 58.8882i	− 1.92379i	−0.273413	−	0.961897i	$-0.588153\pi$
$937$	− 58.8882i	− 1.92379i	0.273413	−	0.961897i	$-0.411847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 40.6732i	− 1.32591i	−0.748660	−	0.662955i	$-0.769302\pi$
$941$	− 40.6732i	− 1.32591i	0.748660	−	0.662955i	$-0.230698\pi$
$942$	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
$949$	−50.1005	−1.62633
$950$	0	0
$951$	0	0
$952$	0	0
$953$	56.0000	1.81402	0.907009	−	0.421111i	$-0.138360\pi$
$953$	56.0000	1.81402	0.907009	+	0.421111i	$0.138360\pi$
$954$	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
$962$	56.1756i	1.81117i
$963$	0	0
$964$	61.5557i	1.98258i
$965$	− 7.60861i	− 0.244930i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−31.1127	−1.00000
$969$	0	0
$970$	6.40202	0.205556
$971$	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
$976$	55.3240i	1.77088i
$977$	−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
$985$	− 8.87671i	− 0.282835i
$986$	39.7540i	1.26603i
$987$	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
$997$	− 34.8129i	− 1.10254i	−0.834328	−	0.551268i	$-0.814144\pi$
$997$	− 34.8129i	− 1.10254i	0.834328	−	0.551268i	$-0.185856\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.b.e.1567.1	✓	4	1.1	even	1	trivial
1764.2.b.e.1567.1	✓	4	4.3	odd	2	CM
1764.2.b.e.1567.2	yes	4	7.6	odd	2	inner
1764.2.b.e.1567.2	yes	4	28.27	even	2	inner
1764.2.b.g.1567.3	yes	4	21.20	even	2
1764.2.b.g.1567.3	yes	4	84.83	odd	2
1764.2.b.g.1567.4	yes	4	3.2	odd	2
1764.2.b.g.1567.4	yes	4	12.11	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -0.317025i q^{5} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -0.317025i q^{5} -2.82843 q^{8} +0.448342i q^{10} -4.01254i q^{13} +4.00000 q^{16} +6.62567i q^{17} -0.634051i q^{20} +4.89949 q^{25} +5.67459i q^{26} -4.24264 q^{29} -5.65685 q^{32} -9.37011i q^{34} +9.89949 q^{37} +0.896683i q^{40} -12.3003i q^{41} -6.92893 q^{50} -8.02509i q^{52} +14.0000 q^{53} +6.00000 q^{58} +13.8310i q^{61} +8.00000 q^{64} -1.27208 q^{65} +13.2513i q^{68} -12.4860i q^{73} -14.0000 q^{74} -1.26810i q^{80} +17.3952i q^{82} +2.10051 q^{85} -18.6089i q^{89} -14.2793i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 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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