LMFDB - Embedded newform 585.2.h.e.64.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(64,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("585.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.h (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:	$4.67124851824$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 5x^{2} - 4x + 69 $ x^4 - 2x^3 + 5x^2 - 4*x + 69
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			64.2
Root			$2.30278 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	585.64
Dual form			585.2.h.e.64.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -2.23607i q^{5} +3.60555 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -2.23607i q^{5} +3.60555 q^{7} +2.23607i q^{11} -3.60555 q^{13} +4.00000 q^{16} -8.06226i q^{17} +4.47214i q^{20} -8.06226i q^{23} -5.00000 q^{25} -7.21110 q^{28} -8.06226i q^{35} +3.60555 q^{37} -11.1803i q^{41} -4.47214i q^{44} +6.00000 q^{49} +7.21110 q^{52} -8.06226i q^{53} +5.00000 q^{55} +8.94427i q^{59} -7.00000 q^{61} -8.00000 q^{64} +8.06226i q^{65} +14.4222 q^{67} +16.1245i q^{68} +15.6525i q^{71} -7.21110 q^{73} +8.06226i q^{77} +11.0000 q^{79} -8.94427i q^{80} -18.0278 q^{85} +2.23607i q^{89} -13.0000 q^{91} +16.1245i q^{92} +3.60555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 24 * q^49 + 20 * q^55 - 28 * q^61 - 32 * q^64 + 44 * q^79 - 52 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	3.60555	1.36277	0.681385	−	0.731925i	$-0.261378\pi$
$7$	3.60555	1.36277	0.681385	+	0.731925i	$0.261378\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.23607i	0.674200i	0.941469	+	0.337100i	$0.109446\pi$
$11$	2.23607i	0.674200i	−0.941469	+	0.337100i	$0.890554\pi$
$12$	0	0
$13$	−3.60555	−1.00000
$13$	−3.60555	−1.00000
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 8.06226i	− 1.95538i	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	− 8.06226i	− 1.95538i	0.210042	−	0.977692i	$-0.432640\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	4.47214i	1.00000i
$21$	0	0
$22$	0	0
$23$	− 8.06226i	− 1.68110i	−0.541736	−	0.840548i	$-0.682233\pi$
$23$	− 8.06226i	− 1.68110i	0.541736	−	0.840548i	$-0.317767\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	−7.21110	−1.36277
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 8.06226i	− 1.36277i
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 11.1803i	− 1.74608i	−0.487652	−	0.873038i	$-0.662147\pi$
$41$	− 11.1803i	− 1.74608i	0.487652	−	0.873038i	$-0.337853\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 4.47214i	− 0.674200i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	7.21110	1.00000
$53$	− 8.06226i	− 1.10744i	−0.832704	−	0.553718i	$-0.813209\pi$
$53$	− 8.06226i	− 1.10744i	0.832704	−	0.553718i	$-0.186791\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.94427i	1.16445i	0.813029	+	0.582223i	$0.197817\pi$
$59$	8.94427i	1.16445i	−0.813029	+	0.582223i	$0.802183\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	8.06226i	1.00000i
$66$	0	0
$67$	14.4222	1.76195	0.880976	−	0.473160i	$-0.156887\pi$
$67$	14.4222	1.76195	0.880976	+	0.473160i	$0.156887\pi$
$68$	16.1245i	1.95538i
$69$	0	0
$70$	0	0
$71$	15.6525i	1.85761i	0.370572	+	0.928804i	$0.379162\pi$
$71$	15.6525i	1.85761i	−0.370572	+	0.928804i	$0.620838\pi$
$72$	0	0
$73$	−7.21110	−0.843996	−0.421998	−	0.906597i	$-0.638671\pi$
$73$	−7.21110	−0.843996	−0.421998	+	0.906597i	$0.638671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.06226i	0.918780i
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	− 8.94427i	− 1.00000i
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−18.0278	−1.95538
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.23607i	0.237023i	0.992953	+	0.118511i	$0.0378122\pi$
$89$	2.23607i	0.237023i	−0.992953	+	0.118511i	$0.962188\pi$
$90$	0	0
$91$	−13.0000	−1.36277
$92$	16.1245i	1.68110i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.60555	0.366088	0.183044	−	0.983105i	$-0.441405\pi$
$97$	3.60555	0.366088	0.183044	+	0.983105i	$0.441405\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	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	0
$107$	− 8.06226i	− 0.779408i	−0.920940	−	0.389704i	$-0.872577\pi$
$107$	− 8.06226i	− 0.779408i	0.920940	−	0.389704i	$-0.127423\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	14.4222	1.36277
$113$	16.1245i	1.51687i	0.651751	+	0.758433i	$0.274035\pi$
$113$	16.1245i	1.51687i	−0.651751	+	0.758433i	$0.725965\pi$
$114$	0	0
$115$	−18.0278	−1.68110
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 29.0689i	− 2.66474i
$120$	0	0
$121$	6.00000	0.545455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	16.1245i	1.36277i
$141$	0	0
$142$	0	0
$143$	− 8.06226i	− 0.674200i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−7.21110	−0.592749
$149$	15.6525i	1.28230i	0.767415	+	0.641150i	$0.221543\pi$
$149$	15.6525i	1.28230i	−0.767415	+	0.641150i	$0.778457\pi$
$150$	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
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 29.0689i	− 2.29095i
$162$	0	0
$163$	25.2389	1.97686	0.988430	−	0.151678i	$-0.0484676\pi$
$163$	25.2389	1.97686	0.988430	+	0.151678i	$0.0484676\pi$
$164$	22.3607i	1.74608i
$165$	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
$172$	0	0
$173$	16.1245i	1.22592i	0.790112	+	0.612962i	$0.210022\pi$
$173$	16.1245i	1.22592i	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	−18.0278	−1.36277
$176$	8.94427i	0.674200i
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 8.06226i	− 0.592749i
$186$	0	0
$187$	18.0278	1.31832
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	25.2389	1.81673	0.908366	−	0.418175i	$-0.137330\pi$
$193$	25.2389	1.81673	0.908366	+	0.418175i	$0.137330\pi$
$194$	0	0
$195$	0	0
$196$	−12.0000	−0.857143
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−25.0000	−1.74608
$206$	0	0
$207$	0	0
$208$	−14.4222	−1.00000
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	16.1245i	1.10744i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−10.0000	−0.674200
$221$	29.0689i	1.95538i
$222$	0	0
$223$	−28.8444	−1.93156	−0.965782	−	0.259354i	$-0.916490\pi$
$223$	−28.8444	−1.93156	−0.965782	+	0.259354i	$0.916490\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	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
$232$	0	0
$233$	− 8.06226i	− 0.528176i	−0.964499	−	0.264088i	$-0.914929\pi$
$233$	− 8.06226i	− 0.528176i	0.964499	−	0.264088i	$-0.0850709\pi$
$234$	0	0
$235$	0	0
$236$	− 17.8885i	− 1.16445i
$237$	0	0
$238$	0	0
$239$	− 24.5967i	− 1.59103i	−0.605933	−	0.795516i	$-0.707200\pi$
$239$	− 24.5967i	− 1.59103i	0.605933	−	0.795516i	$-0.292800\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	14.0000	0.896258
$245$	− 13.4164i	− 0.857143i
$246$	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
$253$	18.0278	1.13340
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	16.1245i	1.00582i	0.864339	+	0.502910i	$0.167737\pi$
$257$	16.1245i	1.00582i	−0.864339	+	0.502910i	$0.832263\pi$
$258$	0	0
$259$	13.0000	0.807781
$260$	− 16.1245i	− 1.00000i
$261$	0	0
$262$	0	0
$263$	− 32.2490i	− 1.98856i	−0.106803	−	0.994280i	$-0.534061\pi$
$263$	− 32.2490i	− 1.98856i	0.106803	−	0.994280i	$-0.465939\pi$
$264$	0	0
$265$	−18.0278	−1.10744
$266$	0	0
$267$	0	0
$268$	−28.8444	−1.76195
$269$	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			$\pi$
$272$	− 32.2490i	− 1.95538i
$273$	0	0
$274$	0	0
$275$	− 11.1803i	− 0.674200i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.3607i	1.33393i	0.745091	+	0.666963i	$0.232406\pi$
$281$	22.3607i	1.33393i	−0.745091	+	0.666963i	$0.767594\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	− 31.3050i	− 1.85761i
$285$	0	0
$286$	0	0
$287$	− 40.3113i	− 2.37950i
$288$	0	0
$289$	−48.0000	−2.82353
$290$	0	0
$291$	0	0
$292$	14.4222	0.843996
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	20.0000	1.16445
$296$	0	0
$297$	0	0
$298$	0	0
$299$	29.0689i	1.68110i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.6525i	0.896258i
$306$	0	0
$307$	3.60555	0.205780	0.102890	−	0.994693i	$-0.467191\pi$
$307$	3.60555	0.205780	0.102890	+	0.994693i	$0.467191\pi$
$308$	− 16.1245i	− 0.918780i
$309$	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
$316$	−22.0000	−1.23760
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	17.8885i	1.00000i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	18.0278	1.00000
$326$	0	0
$327$	0	0
$328$	0	0
$329$	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
$335$	− 32.2490i	− 1.76195i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	36.0555	1.95538
$341$	0	0
$342$	0	0
$343$	−3.60555	−0.194681
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.06226i	− 0.432805i	−0.976304	−	0.216402i	$-0.930568\pi$
$347$	− 8.06226i	− 0.432805i	0.976304	−	0.216402i	$-0.0694323\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	35.0000	1.85761
$356$	− 4.47214i	− 0.237023i
$357$	0	0
$358$	0	0
$359$	35.7771i	1.88824i	0.329598	+	0.944121i	$0.393087\pi$
$359$	35.7771i	1.88824i	−0.329598	+	0.944121i	$0.606913\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	26.0000	1.36277
$365$	16.1245i	0.843996i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	− 32.2490i	− 1.68110i
$369$	0	0
$370$	0	0
$371$	− 29.0689i	− 1.50918i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	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
$385$	18.0278	0.918780
$386$	0	0
$387$	0	0
$388$	−7.21110	−0.366088
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−65.0000	−3.28719
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 24.5967i	− 1.23760i
$396$	0	0
$397$	−39.6611	−1.99053	−0.995266	−	0.0971897i	$-0.969015\pi$
$397$	−39.6611	−1.99053	−0.995266	+	0.0971897i	$0.969015\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	− 31.3050i	− 1.56329i	−0.623721	−	0.781647i	$-0.714380\pi$
$401$	− 31.3050i	− 1.56329i	0.623721	−	0.781647i	$-0.285620\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.06226i	0.399631i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	32.2490i	1.58687i
$414$	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
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	40.3113i	1.95538i
$426$	0	0
$427$	−25.2389	−1.22139
$428$	16.1245i	0.779408i
$429$	0	0
$430$	0	0
$431$	− 17.8885i	− 0.861661i	−0.902433	−	0.430830i	$-0.858221\pi$
$431$	− 17.8885i	− 0.861661i	0.902433	−	0.430830i	$-0.141779\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.3113i	1.91525i	0.288023	+	0.957624i	$0.407002\pi$
$443$	40.3113i	1.91525i	−0.288023	+	0.957624i	$0.592998\pi$
$444$	0	0
$445$	5.00000	0.237023
$446$	0	0
$447$	0	0
$448$	−28.8444	−1.36277
$449$	− 38.0132i	− 1.79395i	−0.442080	−	0.896976i	$-0.645759\pi$
$449$	− 38.0132i	− 1.79395i	0.442080	−	0.896976i	$-0.354241\pi$
$450$	0	0
$451$	25.0000	1.17720
$452$	− 32.2490i	− 1.51687i
$453$	0	0
$454$	0	0
$455$	29.0689i	1.36277i
$456$	0	0
$457$	3.60555	0.168661	0.0843303	−	0.996438i	$-0.473125\pi$
$457$	3.60555	0.168661	0.0843303	+	0.996438i	$0.473125\pi$
$458$	0	0
$459$	0	0
$460$	36.0555	1.68110
$461$	42.4853i	1.97874i	0.145429	+	0.989369i	$0.453544\pi$
$461$	42.4853i	1.97874i	−0.145429	+	0.989369i	$0.546456\pi$
$462$	0	0
$463$	25.2389	1.17295	0.586475	−	0.809968i	$-0.300515\pi$
$463$	25.2389	1.17295	0.586475	+	0.809968i	$0.300515\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.3113i	1.86538i	0.360674	+	0.932692i	$0.382547\pi$
$467$	40.3113i	1.86538i	−0.360674	+	0.932692i	$0.617453\pi$
$468$	0	0
$469$	52.0000	2.40114
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	58.1378i	2.66474i
$477$	0	0
$478$	0	0
$479$	2.23607i	0.102169i	0.998694	+	0.0510843i	$0.0162677\pi$
$479$	2.23607i	0.102169i	−0.998694	+	0.0510843i	$0.983732\pi$
$480$	0	0
$481$	−13.0000	−0.592749
$482$	0	0
$483$	0	0
$484$	−12.0000	−0.545455
$485$	− 8.06226i	− 0.366088i
$486$	0	0
$487$	−39.6611	−1.79721	−0.898607	−	0.438754i	$-0.855420\pi$
$487$	−39.6611	−1.79721	−0.898607	+	0.438754i	$0.855420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	56.4358i	2.53149i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 22.3607i	− 1.00000i
$501$	0	0
$502$	0	0
$503$	− 32.2490i	− 1.43791i	−0.695055	−	0.718957i	$-0.744620\pi$
$503$	− 32.2490i	− 1.43791i	0.695055	−	0.718957i	$-0.255380\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 11.1803i	− 0.495560i	−0.968816	−	0.247780i	$-0.920299\pi$
$509$	− 11.1803i	− 0.495560i	0.968816	−	0.247780i	$-0.0797010\pi$
$510$	0	0
$511$	−26.0000	−1.15017
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	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
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−42.0000	−1.82609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	40.3113i	1.74608i
$534$	0	0
$535$	−18.0278	−0.779408
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.4164i	0.577886i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	39.6611	1.68656
$554$	0	0
$555$	0	0
$556$	38.0000	1.61156
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	− 32.2490i	− 1.36277i
$561$	0	0
$562$	0	0
$563$	− 8.06226i	− 0.339784i	−0.985463	−	0.169892i	$-0.945658\pi$
$563$	− 8.06226i	− 0.339784i	0.985463	−	0.169892i	$-0.0543418\pi$
$564$	0	0
$565$	36.0555	1.51687
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	16.1245i	0.674200i
$573$	0	0
$574$	0	0
$575$	40.3113i	1.68110i
$576$	0	0
$577$	−39.6611	−1.65111	−0.825556	−	0.564320i	$-0.809138\pi$
$577$	−39.6611	−1.65111	−0.825556	+	0.564320i	$0.809138\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.0278	0.746633
$584$	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
$592$	14.4222	0.592749
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	−65.0000	−2.66474
$596$	− 31.3050i	− 1.28230i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	47.0000	1.91717	0.958585	−	0.284807i	$-0.0919294\pi$
$601$	47.0000	1.91717	0.958585	+	0.284807i	$0.0919294\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 13.4164i	− 0.545455i
$606$	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
$613$	25.2389	1.01939	0.509694	−	0.860356i	$-0.329759\pi$
$613$	25.2389	1.01939	0.509694	+	0.860356i	$0.329759\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.06226i	0.323008i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 29.0689i	− 1.15905i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−21.6333	−0.857143
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	46.8722	1.84846	0.924229	−	0.381839i	$-0.124709\pi$
$643$	46.8722	1.84846	0.924229	+	0.381839i	$0.124709\pi$
$644$	58.1378i	2.29095i
$645$	0	0
$646$	0	0
$647$	− 8.06226i	− 0.316960i	−0.987362	−	0.158480i	$-0.949341\pi$
$647$	− 8.06226i	− 0.316960i	0.987362	−	0.158480i	$-0.0506593\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	−50.4777	−1.97686
$653$	16.1245i	0.631001i	0.948925	+	0.315501i	$0.102172\pi$
$653$	16.1245i	0.631001i	−0.948925	+	0.315501i	$0.897828\pi$
$654$	0	0
$655$	0	0
$656$	− 44.7214i	− 1.74608i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	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
$671$	− 15.6525i	− 0.604257i
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−26.0000	−1.00000
$677$	40.3113i	1.54929i	0.632397	+	0.774644i	$0.282071\pi$
$677$	40.3113i	1.54929i	−0.632397	+	0.774644i	$0.717929\pi$
$678$	0	0
$679$	13.0000	0.498894
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	29.0689i	1.10744i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 32.2490i	− 1.22592i
$693$	0	0
$694$	0	0
$695$	42.4853i	1.61156i
$696$	0	0
$697$	−90.1388	−3.41425
$698$	0	0
$699$	0	0
$700$	36.0555	1.36277
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	− 17.8885i	− 0.674200i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−18.0278	−0.674200
$716$	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
$722$	0	0
$723$	0	0
$724$	−46.0000	−1.70958
$725$	0	0
$726$	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
$733$	46.8722	1.73126	0.865631	−	0.500682i	$-0.166917\pi$
$733$	46.8722	1.73126	0.865631	+	0.500682i	$0.166917\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.2490i	1.18791i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	16.1245i	0.592749i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	35.0000	1.28230
$746$	0	0
$747$	0	0
$748$	−36.0555	−1.31832
$749$	− 29.0689i	− 1.06215i
$750$	0	0
$751$	53.0000	1.93400	0.966999	−	0.254781i	$-0.0820034\pi$
$751$	53.0000	1.93400	0.966999	+	0.254781i	$0.0820034\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.1935i	1.78326i	0.452762	+	0.891631i	$0.350439\pi$
$761$	49.1935i	1.78326i	−0.452762	+	0.891631i	$0.649561\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 32.2490i	− 1.16445i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−50.4777	−1.81673
$773$	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
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−35.0000	−1.25240
$782$	0	0
$783$	0	0
$784$	24.0000	0.857143
$785$	0	0
$786$	0	0
$787$	14.4222	0.514096	0.257048	−	0.966399i	$-0.417250\pi$
$787$	14.4222	0.514096	0.257048	+	0.966399i	$0.417250\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	58.1378i	2.06714i
$792$	0	0
$793$	25.2389	0.896258
$794$	0	0
$795$	0	0
$796$	8.00000	0.283552
$797$	− 56.4358i	− 1.99906i	−0.0306762	−	0.999529i	$-0.509766\pi$
$797$	− 56.4358i	− 1.99906i	0.0306762	−	0.999529i	$-0.490234\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 16.1245i	− 0.569022i
$804$	0	0
$805$	−65.0000	−2.29095
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 56.4358i	− 1.97686i
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	50.0000	1.74608
$821$	− 11.1803i	− 0.390197i	−0.980784	−	0.195098i	$-0.937497\pi$
$821$	− 11.1803i	− 0.390197i	0.980784	−	0.195098i	$-0.0625026\pi$
$822$	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		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	28.8444	1.00000
$833$	− 48.3735i	− 1.67604i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 38.0132i	− 1.31236i	−0.754604	−	0.656180i	$-0.772171\pi$
$839$	− 38.0132i	− 1.31236i	0.754604	−	0.656180i	$-0.227829\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	−16.0000	−0.550743
$845$	− 29.0689i	− 1.00000i
$846$	0	0
$847$	21.6333	0.743329
$848$	− 32.2490i	− 1.10744i
$849$	0	0
$850$	0	0
$851$	− 29.0689i	− 0.996468i
$852$	0	0
$853$	46.8722	1.60487	0.802436	−	0.596738i	$-0.203537\pi$
$853$	46.8722	1.60487	0.802436	+	0.596738i	$0.203537\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 56.4358i	− 1.92781i	−0.266246	−	0.963905i	$-0.585783\pi$
$857$	− 56.4358i	− 1.92781i	0.266246	−	0.963905i	$-0.414217\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	36.0555	1.22592
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.5967i	0.834388i
$870$	0	0
$871$	−52.0000	−1.76195
$872$	0	0
$873$	0	0
$874$	0	0
$875$	40.3113i	1.36277i
$876$	0	0
$877$	−50.4777	−1.70451	−0.852256	−	0.523125i	$-0.824766\pi$
$877$	−50.4777	−1.70451	−0.852256	+	0.523125i	$0.824766\pi$
$878$	0	0
$879$	0	0
$880$	20.0000	0.674200
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	− 58.1378i	− 1.95538i
$885$	0	0
$886$	0	0
$887$	− 56.4358i	− 1.89493i	−0.319861	−	0.947464i	$-0.603636\pi$
$887$	− 56.4358i	− 1.89493i	0.319861	−	0.947464i	$-0.396364\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	57.6888	1.93156
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−65.0000	−2.16546
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 51.4296i	− 1.70958i
$906$	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		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	59.0000	1.94623	0.973115	−	0.230319i	$-0.0739769\pi$
$919$	59.0000	1.94623	0.973115	+	0.230319i	$0.0739769\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 56.4358i	− 1.85761i
$924$	0	0
$925$	−18.0278	−0.592749
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.4853i	1.39390i	0.717121	+	0.696949i	$0.245459\pi$
$929$	42.4853i	1.39390i	−0.717121	+	0.696949i	$0.754541\pi$
$930$	0	0
$931$	0	0
$932$	16.1245i	0.528176i
$933$	0	0
$934$	0	0
$935$	− 40.3113i	− 1.31832i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24.5967i	− 0.801831i	−0.916115	−	0.400916i	$-0.868692\pi$
$941$	− 24.5967i	− 0.801831i	0.916115	−	0.400916i	$-0.131308\pi$
$942$	0	0
$943$	−90.1388	−2.93532
$944$	35.7771i	1.16445i
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	26.0000	0.843996
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.3113i	1.30581i	0.757439	+	0.652905i	$0.226450\pi$
$953$	40.3113i	1.30581i	−0.757439	+	0.652905i	$0.773550\pi$
$954$	0	0
$955$	0	0
$956$	49.1935i	1.59103i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 56.4358i	− 1.81673i
$966$	0	0
$967$	57.6888	1.85515	0.927574	−	0.373640i	$-0.121891\pi$
$967$	57.6888	1.85515	0.927574	+	0.373640i	$0.121891\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−68.5055	−2.19618
$974$	0	0
$975$	0	0
$976$	−28.0000	−0.896258
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−5.00000	−0.159801
$980$	26.8328i	0.857143i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.94427i	0.283552i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.h.e.64.2	yes	4
3.2	odd	2	inner	585.2.h.e.64.4	yes	4
5.4	even	2	inner	585.2.h.e.64.1	✓	4
13.12	even	2	inner	585.2.h.e.64.3	yes	4
15.14	odd	2	inner	585.2.h.e.64.3	yes	4
39.38	odd	2	inner	585.2.h.e.64.1	✓	4
65.64	even	2	inner	585.2.h.e.64.4	yes	4
195.194	odd	2	CM	585.2.h.e.64.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.h.e.64.1	✓	4	5.4	even	2	inner
585.2.h.e.64.1	✓	4	39.38	odd	2	inner
585.2.h.e.64.2	yes	4	1.1	even	1	trivial
585.2.h.e.64.2	yes	4	195.194	odd	2	CM
585.2.h.e.64.3	yes	4	13.12	even	2	inner
585.2.h.e.64.3	yes	4	15.14	odd	2	inner
585.2.h.e.64.4	yes	4	3.2	odd	2	inner
585.2.h.e.64.4	yes	4	65.64	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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -2.23607i q^{5} +3.60555 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -2.23607i q^{5} +3.60555 q^{7} +2.23607i q^{11} -3.60555 q^{13} +4.00000 q^{16} -8.06226i q^{17} +4.47214i q^{20} -8.06226i q^{23} -5.00000 q^{25} -7.21110 q^{28} -8.06226i q^{35} +3.60555 q^{37} -11.1803i q^{41} -4.47214i q^{44} +6.00000 q^{49} +7.21110 q^{52} -8.06226i q^{53} +5.00000 q^{55} +8.94427i q^{59} -7.00000 q^{61} -8.00000 q^{64} +8.06226i q^{65} +14.4222 q^{67} +16.1245i q^{68} +15.6525i q^{71} -7.21110 q^{73} +8.06226i q^{77} +11.0000 q^{79} -8.94427i q^{80} -18.0278 q^{85} +2.23607i q^{89} -13.0000 q^{91} +16.1245i q^{92} +3.60555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 24 * q^49 + 20 * q^55 - 28 * q^61 - 32 * q^64 + 44 * q^79 - 52 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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