LMFDB - Embedded newform 300.5.b.a.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,5,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not 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:	$31.0109889252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			149.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.5.b.a.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} -94.0000i q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} -94.0000i q^{7} -81.0000 q^{9} -146.000i q^{13} +46.0000 q^{19} -846.000 q^{21} +729.000i q^{27} +194.000 q^{31} -2062.00i q^{37} -1314.00 q^{39} +3214.00i q^{43} -6435.00 q^{49} -414.000i q^{57} -1966.00 q^{61} +7614.00i q^{63} +5906.00i q^{67} +8542.00i q^{73} -7682.00 q^{79} +6561.00 q^{81} -13724.0 q^{91} -1746.00i q^{93} -18814.0i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{9}+O(q^{10}) $ 2 * q - 162 * q^9	$ 2 q - 162 q^{9} + 92 q^{19} - 1692 q^{21} + 388 q^{31} - 2628 q^{39} - 12870 q^{49} - 3932 q^{61} - 15364 q^{79} + 13122 q^{81} - 27448 q^{91}+O(q^{100}) $ 2 * q - 162 * q^9 + 92 * q^19 - 1692 * q^21 + 388 * q^31 - 2628 * q^39 - 12870 * q^49 - 3932 * q^61 - 15364 * q^79 + 13122 * q^81 - 27448 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$-1$

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
$2$	0	0
$3$	− 9.00000i	− 1.00000i
$3$	− 9.00000i	− 1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 94.0000i	− 1.91837i	−0.282784	−	0.959184i	$-0.591258\pi$
$7$	− 94.0000i	− 1.91837i	0.282784	−	0.959184i	$-0.408742\pi$
$8$	0	0
$9$	−81.0000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 146.000i	− 0.863905i	−0.901896	−	0.431953i	$-0.857825\pi$
$13$	− 146.000i	− 0.863905i	0.901896	−	0.431953i	$-0.142175\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	46.0000	0.127424	0.0637119	−	0.997968i	$-0.479706\pi$
$19$	46.0000	0.127424	0.0637119	+	0.997968i	$0.479706\pi$
$20$	0	0
$21$	−846.000	−1.91837
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	729.000i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	194.000	0.201873	0.100937	−	0.994893i	$-0.467816\pi$
$31$	194.000	0.201873	0.100937	+	0.994893i	$0.467816\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2062.00i	− 1.50621i	−0.657901	−	0.753104i	$-0.728555\pi$
$37$	− 2062.00i	− 1.50621i	0.657901	−	0.753104i	$-0.271445\pi$
$38$	0	0
$39$	−1314.00	−0.863905
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	3214.00i	1.73824i	0.494604	+	0.869118i	$0.335313\pi$
$43$	3214.00i	1.73824i	−0.494604	+	0.869118i	$0.664687\pi$
$44$	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
$49$	−6435.00	−2.68013
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 414.000i	− 0.127424i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−1966.00	−0.528353	−0.264176	−	0.964474i	$-0.585100\pi$
$61$	−1966.00	−0.528353	−0.264176	+	0.964474i	$0.585100\pi$
$62$	0	0
$63$	7614.00i	1.91837i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5906.00i	1.31566i	0.753166	+	0.657830i	$0.228526\pi$
$67$	5906.00i	1.31566i	−0.753166	+	0.657830i	$0.771474\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	8542.00i	1.60293i	0.598043	+	0.801464i	$0.295945\pi$
$73$	8542.00i	1.60293i	−0.598043	+	0.801464i	$0.704055\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7682.00	−1.23089	−0.615446	−	0.788179i	$-0.711024\pi$
$79$	−7682.00	−1.23089	−0.615446	+	0.788179i	$0.711024\pi$
$80$	0	0
$81$	6561.00	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−13724.0	−1.65729
$92$	0	0
$93$	− 1746.00i	− 0.201873i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 18814.0i	− 1.99957i	−0.0206175	−	0.999787i	$-0.506563\pi$
$97$	− 18814.0i	− 1.99957i	0.0206175	−	0.999787i	$-0.493437\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 16418.0i	− 1.54755i	−0.633458	−	0.773777i	$-0.718365\pi$
$103$	− 16418.0i	− 1.54755i	0.633458	−	0.773777i	$-0.281635\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−22034.0	−1.85456	−0.927279	−	0.374371i	$-0.877859\pi$
$109$	−22034.0	−1.85456	−0.927279	+	0.374371i	$0.877859\pi$
$110$	0	0
$111$	−18558.0	−1.50621
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11826.0i	0.863905i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14641.0	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10942.0i	− 0.678405i	−0.940713	−	0.339203i	$-0.889843\pi$
$127$	− 10942.0i	− 0.678405i	0.940713	−	0.339203i	$-0.110157\pi$
$128$	0	0
$129$	28926.0	1.73824
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 4324.00i	− 0.244446i
$134$	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
$139$	38158.0	1.97495	0.987475	−	0.157777i	$-0.0504327\pi$
$139$	38158.0	1.97495	0.987475	+	0.157777i	$0.0504327\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	57915.0i	2.68013i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	36194.0	1.58739	0.793693	−	0.608318i	$-0.208156\pi$
$151$	36194.0	1.58739	0.793693	+	0.608318i	$0.208156\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 35374.0i	− 1.43511i	−0.696502	−	0.717554i	$-0.745261\pi$
$157$	− 35374.0i	− 1.43511i	0.696502	−	0.717554i	$-0.254739\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 15506.0i	− 0.583612i	−0.956477	−	0.291806i	$-0.905744\pi$
$163$	− 15506.0i	− 0.583612i	0.956477	−	0.291806i	$-0.0942562\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	7245.00	0.253668
$170$	0	0
$171$	−3726.00	−0.127424
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	33074.0	1.00955	0.504777	−	0.863250i	$-0.331575\pi$
$181$	33074.0	1.00955	0.504777	+	0.863250i	$0.331575\pi$
$182$	0	0
$183$	17694.0i	0.528353i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	68526.0	1.91837
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	− 71426.0i	− 1.91753i	−0.284203	−	0.958764i	$-0.591729\pi$
$193$	− 71426.0i	− 1.91753i	0.284203	−	0.958764i	$-0.408271\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−69794.0	−1.76243	−0.881215	−	0.472715i	$-0.843274\pi$
$199$	−69794.0	−1.76243	−0.881215	+	0.472715i	$0.843274\pi$
$200$	0	0
$201$	53154.0	1.31566
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−61486.0	−1.38106	−0.690528	−	0.723306i	$-0.742622\pi$
$211$	−61486.0	−1.38106	−0.690528	+	0.723306i	$0.742622\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 18236.0i	− 0.387267i
$218$	0	0
$219$	76878.0	1.60293
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 14786.0i	− 0.297332i	−0.988888	−	0.148666i	$-0.952502\pi$
$223$	− 14786.0i	− 0.297332i	0.988888	−	0.148666i	$-0.0474979\pi$
$224$	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
$229$	104206.	1.98711	0.993555	−	0.113354i	$-0.0361594\pi$
$229$	104206.	1.98711	0.993555	+	0.113354i	$0.0361594\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	69138.0i	1.23089i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−34366.0	−0.591691	−0.295845	−	0.955236i	$-0.595601\pi$
$241$	−34366.0	−0.591691	−0.295845	+	0.955236i	$0.595601\pi$
$242$	0	0
$243$	− 59049.0i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6716.00i	− 0.110082i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−193828.	−2.88946
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−88318.0	−1.20257	−0.601285	−	0.799034i	$-0.705345\pi$
$271$	−88318.0	−1.20257	−0.601285	+	0.799034i	$0.705345\pi$
$272$	0	0
$273$	123516.i	1.65729i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 138574.i	− 1.80602i	−0.429621	−	0.903009i	$-0.641353\pi$
$277$	− 138574.i	− 1.80602i	0.429621	−	0.903009i	$-0.358647\pi$
$278$	0	0
$279$	−15714.0	−0.201873
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 49586.0i	− 0.619136i	−0.950877	−	0.309568i	$-0.899816\pi$
$283$	− 49586.0i	− 0.619136i	0.950877	−	0.309568i	$-0.100184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−83521.0	−1.00000
$290$	0	0
$291$	−169326.	−1.99957
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	302116.	3.33458
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 60334.0i	− 0.640155i	−0.947391	−	0.320078i	$-0.896291\pi$
$307$	− 60334.0i	− 0.640155i	0.947391	−	0.320078i	$-0.103709\pi$
$308$	0	0
$309$	−147762.	−1.54755
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	175774.i	1.79418i	0.441848	+	0.897090i	$0.354323\pi$
$313$	175774.i	1.79418i	−0.441848	+	0.897090i	$0.645677\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	198306.i	1.85456i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−88078.0	−0.803917	−0.401959	−	0.915658i	$-0.631670\pi$
$331$	−88078.0	−0.803917	−0.401959	+	0.915658i	$0.631670\pi$
$332$	0	0
$333$	167022.i	1.50621i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5186.00i	0.0456639i	0.999739	+	0.0228319i	$0.00726826\pi$
$337$	5186.00i	0.0456639i	−0.999739	+	0.0228319i	$0.992732\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	379196.i	3.22311i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−8402.00	−0.0689814	−0.0344907	−	0.999405i	$-0.510981\pi$
$349$	−8402.00	−0.0689814	−0.0344907	+	0.999405i	$0.510981\pi$
$350$	0	0
$351$	106434.	0.863905
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−128205.	−0.983763
$362$	0	0
$363$	− 131769.i	− 1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	246146.i	1.82751i	0.406261	+	0.913757i	$0.366832\pi$
$367$	246146.i	1.82751i	−0.406261	+	0.913757i	$0.633168\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 208946.i	− 1.50181i	−0.660407	−	0.750907i	$-0.729616\pi$
$373$	− 208946.i	− 1.50181i	0.660407	−	0.750907i	$-0.270384\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−194354.	−1.35305	−0.676527	−	0.736418i	$-0.736516\pi$
$379$	−194354.	−1.35305	−0.676527	+	0.736418i	$0.736516\pi$
$380$	0	0
$381$	−98478.0	−0.678405
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 260334.i	− 1.73824i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 184174.i	− 1.16855i	−0.811556	−	0.584275i	$-0.801379\pi$
$397$	− 184174.i	− 1.16855i	0.811556	−	0.584275i	$-0.198621\pi$
$398$	0	0
$399$	−38916.0	−0.244446
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 28324.0i	− 0.174399i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−57314.0	−0.342621	−0.171311	−	0.985217i	$-0.554800\pi$
$409$	−57314.0	−0.342621	−0.171311	+	0.985217i	$0.554800\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 343422.i	− 1.97495i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−226318.	−1.27689	−0.638447	−	0.769666i	$-0.720423\pi$
$421$	−226318.	−1.27689	−0.638447	+	0.769666i	$0.720423\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	184804.i	1.01357i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	− 368066.i	− 1.96313i	−0.191119	−	0.981567i	$-0.561212\pi$
$433$	− 368066.i	− 1.96313i	0.191119	−	0.981567i	$-0.438788\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	376606.	1.95415	0.977076	−	0.212892i	$-0.0682884\pi$
$439$	376606.	1.95415	0.977076	+	0.212892i	$0.0682884\pi$
$440$	0	0
$441$	521235.	2.68013
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 325746.i	− 1.58739i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	244898.i	1.17261i	0.810091	+	0.586304i	$0.199418\pi$
$457$	244898.i	1.17261i	−0.810091	+	0.586304i	$0.800582\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	152062.i	0.709347i	0.934990	+	0.354673i	$0.115408\pi$
$463$	152062.i	0.709347i	−0.934990	+	0.354673i	$0.884592\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	555164.	2.52392
$470$	0	0
$471$	−318366.	−1.43511
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−301052.	−1.30122
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	451106.i	1.90204i	0.309122	+	0.951022i	$0.399965\pi$
$487$	451106.i	1.90204i	−0.309122	+	0.951022i	$0.600035\pi$
$488$	0	0
$489$	−139554.	−0.583612
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	497326.	1.99729	0.998643	−	0.0520865i	$-0.0165872\pi$
$499$	497326.	1.99729	0.998643	+	0.0520865i	$0.0165872\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 65205.0i	− 0.253668i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	802948.	3.07500
$512$	0	0
$513$	33534.0i	0.127424i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 417266.i	− 1.52549i	−0.646699	−	0.762745i	$-0.723851\pi$
$523$	− 417266.i	− 1.52549i	0.646699	−	0.762745i	$-0.276149\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−279841.	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	483794.	1.65297	0.826487	−	0.562956i	$-0.190336\pi$
$541$	483794.	1.65297	0.826487	+	0.562956i	$0.190336\pi$
$542$	0	0
$543$	− 297666.i	− 1.00955i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 342382.i	− 1.14429i	−0.820152	−	0.572145i	$-0.806111\pi$
$547$	− 342382.i	− 1.14429i	0.820152	−	0.572145i	$-0.193889\pi$
$548$	0	0
$549$	159246.	0.528353
$550$	0	0
$551$	0	0
$552$	0	0
$553$	722108.i	2.36130i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	469244.	1.50167
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 616734.i	− 1.91837i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	132914.	0.407660	0.203830	−	0.979006i	$-0.434661\pi$
$571$	132914.	0.407660	0.203830	+	0.979006i	$0.434661\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	259586.i	0.779704i	0.920878	+	0.389852i	$0.127474\pi$
$577$	259586.i	0.779704i	−0.920878	+	0.389852i	$0.872526\pi$
$578$	0	0
$579$	−642834.	−1.91753
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	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
$589$	8924.00	0.0257234
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	628146.i	1.76243i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−445726.	−1.23401	−0.617005	−	0.786959i	$-0.711654\pi$
$601$	−445726.	−1.23401	−0.617005	+	0.786959i	$0.711654\pi$
$602$	0	0
$603$	− 478386.i	− 1.31566i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 74302.0i	− 0.201662i	−0.994904	−	0.100831i	$-0.967850\pi$
$607$	− 74302.0i	− 0.201662i	0.994904	−	0.100831i	$-0.0321501\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 516338.i	− 1.37408i	−0.726618	−	0.687042i	$-0.758909\pi$
$613$	− 516338.i	− 1.37408i	0.726618	−	0.687042i	$-0.241091\pi$
$614$	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
$619$	720526.	1.88048	0.940239	−	0.340515i	$-0.110601\pi$
$619$	720526.	1.88048	0.940239	+	0.340515i	$0.110601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−342046.	−0.859065	−0.429532	−	0.903052i	$-0.641322\pi$
$631$	−342046.	−0.859065	−0.429532	+	0.903052i	$0.641322\pi$
$632$	0	0
$633$	553374.i	1.38106i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	939510.i	2.31538i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	728302.i	1.76153i	0.473555	+	0.880764i	$0.342970\pi$
$643$	728302.i	1.76153i	−0.473555	+	0.880764i	$0.657030\pi$
$644$	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
$650$	0	0
$651$	−164124.	−0.387267
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 691902.i	− 1.60293i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−858958.	−1.96593	−0.982967	−	0.183781i	$-0.941166\pi$
$661$	−858958.	−1.96593	−0.982967	+	0.183781i	$0.941166\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−133074.	−0.297332
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 425858.i	− 0.940231i	−0.882605	−	0.470116i	$-0.844212\pi$
$673$	− 425858.i	− 0.940231i	0.882605	−	0.470116i	$-0.155788\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−1.76852e6	−3.83592
$680$	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
$687$	− 937854.i	− 1.98711i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	782162.	1.63810	0.819050	−	0.573722i	$-0.194501\pi$
$691$	782162.	1.63810	0.819050	+	0.573722i	$0.194501\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 94852.0i	− 0.191927i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	133006.	0.264593	0.132297	−	0.991210i	$-0.457765\pi$
$709$	133006.	0.264593	0.132297	+	0.991210i	$0.457765\pi$
$710$	0	0
$711$	622242.	1.23089
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−1.54329e6	−2.96878
$722$	0	0
$723$	309294.i	0.591691i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 824734.i	− 1.56043i	−0.625510	−	0.780216i	$-0.715109\pi$
$727$	− 824734.i	− 1.56043i	0.625510	−	0.780216i	$-0.284891\pi$
$728$	0	0
$729$	−531441.	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5422.00i	0.0100914i	0.999987	+	0.00504570i	$0.00160610\pi$
$733$	5422.00i	0.0100914i	−0.999987	+	0.00504570i	$0.998394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−401042.	−0.734346	−0.367173	−	0.930153i	$-0.619674\pi$
$739$	−401042.	−0.734346	−0.367173	+	0.930153i	$0.619674\pi$
$740$	0	0
$741$	−60444.0	−0.110082
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	316802.	0.561705	0.280852	−	0.959751i	$-0.409383\pi$
$751$	316802.	0.561705	0.280852	+	0.959751i	$0.409383\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 443854.i	− 0.774548i	−0.921965	−	0.387274i	$-0.873417\pi$
$757$	− 443854.i	− 0.774548i	0.921965	−	0.387274i	$-0.126583\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	2.07120e6i	3.55772i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.17043e6	−1.97922	−0.989610	−	0.143775i	$-0.954076\pi$
$769$	−1.17043e6	−1.97922	−0.989610	+	0.143775i	$0.954076\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.74445e6i	2.88946i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.20111e6i	1.93924i	0.244613	+	0.969621i	$0.421339\pi$
$787$	1.20111e6i	1.93924i	−0.244613	+	0.969621i	$0.578661\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	287036.i	0.456447i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	976754.	1.48506	0.742529	−	0.669814i	$-0.233626\pi$
$811$	976754.	1.48506	0.742529	+	0.669814i	$0.233626\pi$
$812$	0	0
$813$	794862.i	1.20257i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	147844.i	0.221493i
$818$	0	0
$819$	1.11164e6	1.65729
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	235294.i	0.347385i	0.984800	+	0.173693i	$0.0555699\pi$
$823$	235294.i	0.347385i	−0.984800	+	0.173693i	$0.944430\pi$
$824$	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
$829$	1.16472e6	1.69477	0.847387	−	0.530976i	$-0.178175\pi$
$829$	1.16472e6	1.69477	0.847387	+	0.530976i	$0.178175\pi$
$830$	0	0
$831$	−1.24717e6	−1.80602
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	141426.i	0.201873i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	707281.	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.37625e6i	− 1.91837i
$848$	0	0
$849$	−446274.	−0.619136
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 1.29375e6i	− 1.77808i	−0.457831	−	0.889039i	$-0.651373\pi$
$853$	− 1.29375e6i	− 1.77808i	0.457831	−	0.889039i	$-0.348627\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−534962.	−0.724998	−0.362499	−	0.931984i	$-0.618076\pi$
$859$	−534962.	−0.724998	−0.362499	+	0.931984i	$0.618076\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	751689.i	1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	862276.	1.13661
$872$	0	0
$873$	1.52393e6i	1.99957i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.18065e6i	− 1.53505i	−0.641017	−	0.767527i	$-0.721487\pi$
$877$	− 1.18065e6i	− 1.53505i	0.641017	−	0.767527i	$-0.278513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 1.33743e6i	− 1.71533i	−0.514207	−	0.857666i	$-0.671914\pi$
$883$	− 1.33743e6i	− 1.71533i	0.514207	−	0.857666i	$-0.328086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−1.02855e6	−1.30143
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 2.71904e6i	− 3.33458i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.59950e6i	− 1.94433i	−0.234295	−	0.972166i	$-0.575278\pi$
$907$	− 1.59950e6i	− 1.94433i	0.234295	−	0.972166i	$-0.424722\pi$
$908$	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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	939166.	1.11202	0.556008	−	0.831177i	$-0.312332\pi$
$919$	939166.	1.11202	0.556008	+	0.831177i	$0.312332\pi$
$920$	0	0
$921$	−543006.	−0.640155
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.32986e6i	1.54755i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−296010.	−0.341513
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 320734.i	− 0.365314i	−0.983177	−	0.182657i	$-0.941530\pi$
$937$	− 320734.i	− 0.365314i	0.983177	−	0.182657i	$-0.0584697\pi$
$938$	0	0
$939$	1.58197e6	1.79418
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	1.24713e6	1.38478
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−885885.	−0.959247
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	482978.i	0.516505i	0.966077	+	0.258252i	$0.0831466\pi$
$967$	482978.i	0.516505i	−0.966077	+	0.258252i	$0.916853\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	− 3.58685e6i	− 3.78868i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.78475e6	1.85456
$982$	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
$991$	−1.96205e6	−1.99785	−0.998923	−	0.0464053i	$-0.985223\pi$
$991$	−1.96205e6	−1.99785	−0.998923	+	0.0464053i	$0.985223\pi$
$992$	0	0
$993$	792702.i	0.803917i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.59922e6i	1.60886i	0.594050	+	0.804428i	$0.297528\pi$
$997$	1.59922e6i	1.60886i	−0.594050	+	0.804428i	$0.702472\pi$
$998$	0	0
$999$	1.50320e6	1.50621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.5.b.a.149.1		2
3.2	odd	2	CM	300.5.b.a.149.1		2
5.2	odd	4		300.5.g.b.101.1		1
5.3	odd	4		12.5.c.a.5.1	✓	1
5.4	even	2	inner	300.5.b.a.149.2		2
15.2	even	4		300.5.g.b.101.1		1
15.8	even	4		12.5.c.a.5.1	✓	1
15.14	odd	2	inner	300.5.b.a.149.2		2
20.3	even	4		48.5.e.a.17.1		1
35.13	even	4		588.5.c.a.197.1		1
40.3	even	4		192.5.e.b.65.1		1
40.13	odd	4		192.5.e.a.65.1		1
45.13	odd	12		324.5.g.b.269.1		2
45.23	even	12		324.5.g.b.269.1		2
45.38	even	12		324.5.g.b.53.1		2
45.43	odd	12		324.5.g.b.53.1		2
60.23	odd	4		48.5.e.a.17.1		1
105.83	odd	4		588.5.c.a.197.1		1
120.53	even	4		192.5.e.a.65.1		1
120.83	odd	4		192.5.e.b.65.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.5.c.a.5.1	✓	1	5.3	odd	4
12.5.c.a.5.1	✓	1	15.8	even	4
48.5.e.a.17.1		1	20.3	even	4
48.5.e.a.17.1		1	60.23	odd	4
192.5.e.a.65.1		1	40.13	odd	4
192.5.e.a.65.1		1	120.53	even	4
192.5.e.b.65.1		1	40.3	even	4
192.5.e.b.65.1		1	120.83	odd	4
300.5.b.a.149.1		2	1.1	even	1	trivial
300.5.b.a.149.1		2	3.2	odd	2	CM
300.5.b.a.149.2		2	5.4	even	2	inner
300.5.b.a.149.2		2	15.14	odd	2	inner
300.5.g.b.101.1		1	5.2	odd	4
300.5.g.b.101.1		1	15.2	even	4
324.5.g.b.53.1		2	45.38	even	12
324.5.g.b.53.1		2	45.43	odd	12
324.5.g.b.269.1		2	45.13	odd	12
324.5.g.b.269.1		2	45.23	even	12
588.5.c.a.197.1		1	35.13	even	4
588.5.c.a.197.1		1	105.83	odd	4

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} -94.0000i q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} -94.0000i q^{7} -81.0000 q^{9} -146.000i q^{13} +46.0000 q^{19} -846.000 q^{21} +729.000i q^{27} +194.000 q^{31} -2062.00i q^{37} -1314.00 q^{39} +3214.00i q^{43} -6435.00 q^{49} -414.000i q^{57} -1966.00 q^{61} +7614.00i q^{63} +5906.00i q^{67} +8542.00i q^{73} -7682.00 q^{79} +6561.00 q^{81} -13724.0 q^{91} -1746.00i q^{93} -18814.0i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{9}+O(q^{10}) \) 2 * q - 162 * q^9	\( 2 q - 162 q^{9} + 92 q^{19} - 1692 q^{21} + 388 q^{31} - 2628 q^{39} - 12870 q^{49} - 3932 q^{61} - 15364 q^{79} + 13122 q^{81} - 27448 q^{91}+O(q^{100}) \) 2 * q - 162 * q^9 + 92 * q^19 - 1692 * q^21 + 388 * q^31 - 2628 * q^39 - 12870 * q^49 - 3932 * q^61 - 15364 * q^79 + 13122 * q^81 - 27448 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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