LMFDB - Embedded newform 75.3.d.a.74.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	75.d (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:	$2.04360198270$
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			74.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.3.d.a.74.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -4.00000 q^{4} -11.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -4.00000 q^{4} -11.0000i q^{7} -9.00000 q^{9} +12.0000i q^{12} -1.00000i q^{13} +16.0000 q^{16} +37.0000 q^{19} -33.0000 q^{21} +27.0000i q^{27} +44.0000i q^{28} -13.0000 q^{31} +36.0000 q^{36} -26.0000i q^{37} -3.00000 q^{39} -61.0000i q^{43} -48.0000i q^{48} -72.0000 q^{49} +4.00000i q^{52} -111.000i q^{57} +47.0000 q^{61} +99.0000i q^{63} -64.0000 q^{64} +109.000i q^{67} -46.0000i q^{73} -148.000 q^{76} +142.000 q^{79} +81.0000 q^{81} +132.000 q^{84} -11.0000 q^{91} +39.0000i q^{93} +169.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 - 18 * q^9	$ 2 q - 8 q^{4} - 18 q^{9} + 32 q^{16} + 74 q^{19} - 66 q^{21} - 26 q^{31} + 72 q^{36} - 6 q^{39} - 144 q^{49} + 94 q^{61} - 128 q^{64} - 296 q^{76} + 284 q^{79} + 162 q^{81} + 264 q^{84} - 22 q^{91}+O(q^{100}) $ 2 * q - 8 * q^4 - 18 * q^9 + 32 * q^16 + 74 * q^19 - 66 * q^21 - 26 * q^31 + 72 * q^36 - 6 * q^39 - 144 * q^49 + 94 * q^61 - 128 * q^64 - 296 * q^76 + 284 * q^79 + 162 * q^81 + 264 * q^84 - 22 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	− 3.00000i	− 1.00000i
$3$	− 3.00000i	− 1.00000i
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 11.0000i	− 1.57143i	−0.618590	−	0.785714i	$-0.712296\pi$
$7$	− 11.0000i	− 1.57143i	0.618590	−	0.785714i	$-0.287704\pi$
$8$	0	0
$9$	−9.00000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	12.0000i	1.00000i
$13$	− 1.00000i	− 0.0769231i	−0.999260	−	0.0384615i	$-0.987754\pi$
$13$	− 1.00000i	− 0.0769231i	0.999260	−	0.0384615i	$-0.0122457\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	37.0000	1.94737	0.973684	−	0.227901i	$-0.0731864\pi$
$19$	37.0000	1.94737	0.973684	+	0.227901i	$0.0731864\pi$
$20$	0	0
$21$	−33.0000	−1.57143
$22$	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
$27$	27.0000i	1.00000i
$28$	44.0000i	1.57143i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−13.0000	−0.419355	−0.209677	−	0.977771i	$-0.567241\pi$
$31$	−13.0000	−0.419355	−0.209677	+	0.977771i	$0.567241\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	36.0000	1.00000
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	−3.00000	−0.0769231
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 61.0000i	− 1.41860i	−0.704904	−	0.709302i	$-0.749010\pi$
$43$	− 61.0000i	− 1.41860i	0.704904	−	0.709302i	$-0.250990\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 48.0000i	− 1.00000i
$49$	−72.0000	−1.46939
$50$	0	0
$51$	0	0
$52$	4.00000i	0.0769231i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 111.000i	− 1.94737i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	47.0000	0.770492	0.385246	−	0.922814i	$-0.374117\pi$
$61$	47.0000	0.770492	0.385246	+	0.922814i	$0.374117\pi$
$62$	0	0
$63$	99.0000i	1.57143i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	109.000i	1.62687i	0.581659	+	0.813433i	$0.302404\pi$
$67$	109.000i	1.62687i	−0.581659	+	0.813433i	$0.697596\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	− 46.0000i	− 0.630137i	−0.949069	−	0.315068i	$-0.897973\pi$
$73$	− 46.0000i	− 0.630137i	0.949069	−	0.315068i	$-0.102027\pi$
$74$	0	0
$75$	0	0
$76$	−148.000	−1.94737
$77$	0	0
$78$	0	0
$79$	142.000	1.79747	0.898734	−	0.438494i	$-0.144488\pi$
$79$	142.000	1.79747	0.898734	+	0.438494i	$0.144488\pi$
$80$	0	0
$81$	81.0000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	132.000	1.57143
$85$	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
$91$	−11.0000	−0.120879
$92$	0	0
$93$	39.0000i	0.419355i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	169.000i	1.74227i	0.491045	+	0.871134i	$0.336615\pi$
$97$	169.000i	1.74227i	−0.491045	+	0.871134i	$0.663385\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	194.000i	1.88350i	0.336321	+	0.941748i	$0.390817\pi$
$103$	194.000i	1.88350i	−0.336321	+	0.941748i	$0.609183\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 108.000i	− 1.00000i
$109$	−143.000	−1.31193	−0.655963	−	0.754793i	$-0.727737\pi$
$109$	−143.000	−1.31193	−0.655963	+	0.754793i	$0.727737\pi$
$110$	0	0
$111$	−78.0000	−0.702703
$112$	− 176.000i	− 1.57143i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.00000i	0.0769231i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	52.0000	0.419355
$125$	0	0
$126$	0	0
$127$	− 146.000i	− 1.14961i	−0.818292	−	0.574803i	$-0.805079\pi$
$127$	− 146.000i	− 1.14961i	0.818292	−	0.574803i	$-0.194921\pi$
$128$	0	0
$129$	−183.000	−1.41860
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 407.000i	− 3.06015i
$134$	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
$139$	22.0000	0.158273	0.0791367	−	0.996864i	$-0.474784\pi$
$139$	22.0000	0.158273	0.0791367	+	0.996864i	$0.474784\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−144.000	−1.00000
$145$	0	0
$146$	0	0
$147$	216.000i	1.46939i
$148$	104.000i	0.702703i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	227.000	1.50331	0.751656	−	0.659556i	$-0.229256\pi$
$151$	227.000	1.50331	0.751656	+	0.659556i	$0.229256\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	12.0000	0.0769231
$157$	− 311.000i	− 1.98089i	−0.137902	−	0.990446i	$-0.544036\pi$
$157$	− 311.000i	− 1.98089i	0.137902	−	0.990446i	$-0.455964\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	299.000i	1.83436i	0.398478	+	0.917178i	$0.369539\pi$
$163$	299.000i	1.83436i	−0.398478	+	0.917178i	$0.630461\pi$
$164$	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
$169$	168.000	0.994083
$170$	0	0
$171$	−333.000	−1.94737
$172$	244.000i	1.41860i
$173$	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
$181$	−313.000	−1.72928	−0.864641	−	0.502390i	$-0.832454\pi$
$181$	−313.000	−1.72928	−0.864641	+	0.502390i	$0.832454\pi$
$182$	0	0
$183$	− 141.000i	− 0.770492i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	297.000	1.57143
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	192.000i	1.00000i
$193$	239.000i	1.23834i	0.785256	+	0.619171i	$0.212531\pi$
$193$	239.000i	1.23834i	−0.785256	+	0.619171i	$0.787469\pi$
$194$	0	0
$195$	0	0
$196$	288.000	1.46939
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	277.000	1.39196	0.695980	−	0.718061i	$-0.254970\pi$
$199$	277.000	1.39196	0.695980	+	0.718061i	$0.254970\pi$
$200$	0	0
$201$	327.000	1.62687
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 16.0000i	− 0.0769231i
$209$	0	0
$210$	0	0
$211$	−253.000	−1.19905	−0.599526	−	0.800355i	$-0.704644\pi$
$211$	−253.000	−1.19905	−0.599526	+	0.800355i	$0.704644\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	143.000i	0.658986i
$218$	0	0
$219$	−138.000	−0.630137
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 421.000i	− 1.88789i	−0.330099	−	0.943946i	$-0.607082\pi$
$223$	− 421.000i	− 1.88789i	0.330099	−	0.943946i	$-0.392918\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	444.000i	1.94737i
$229$	−383.000	−1.67249	−0.836245	−	0.548357i	$-0.815254\pi$
$229$	−383.000	−1.67249	−0.836245	+	0.548357i	$0.815254\pi$
$230$	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
$237$	− 426.000i	− 1.79747i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−193.000	−0.800830	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	−193.000	−0.800830	−0.400415	+	0.916334i	$0.631134\pi$
$242$	0	0
$243$	− 243.000i	− 1.00000i
$244$	−188.000	−0.770492
$245$	0	0
$246$	0	0
$247$	− 37.0000i	− 0.149798i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 396.000i	− 1.57143i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−286.000	−1.10425
$260$	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
$268$	− 436.000i	− 1.62687i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	242.000	0.892989	0.446494	−	0.894786i	$-0.352672\pi$
$271$	242.000	0.892989	0.446494	+	0.894786i	$0.352672\pi$
$272$	0	0
$273$	33.0000i	0.120879i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	529.000i	1.90975i	0.297012	+	0.954874i	$0.404010\pi$
$277$	529.000i	1.90975i	−0.297012	+	0.954874i	$0.595990\pi$
$278$	0	0
$279$	117.000	0.419355
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	59.0000i	0.208481i	0.994552	+	0.104240i	$0.0332411\pi$
$283$	59.0000i	0.208481i	−0.994552	+	0.104240i	$0.966759\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−289.000	−1.00000
$290$	0	0
$291$	507.000	1.74227
$292$	184.000i	0.630137i
$293$	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
$301$	−671.000	−2.22924
$302$	0	0
$303$	0	0
$304$	592.000	1.94737
$305$	0	0
$306$	0	0
$307$	− 611.000i	− 1.99023i	−0.0987325	−	0.995114i	$-0.531479\pi$
$307$	− 611.000i	− 1.99023i	0.0987325	−	0.995114i	$-0.468521\pi$
$308$	0	0
$309$	582.000	1.88350
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	599.000i	1.91374i	0.290520	+	0.956869i	$0.406172\pi$
$313$	599.000i	1.91374i	−0.290520	+	0.956869i	$0.593828\pi$
$314$	0	0
$315$	0	0
$316$	−568.000	−1.79747
$317$	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
$324$	−324.000	−1.00000
$325$	0	0
$326$	0	0
$327$	429.000i	1.31193i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	362.000	1.09366	0.546828	−	0.837245i	$-0.315835\pi$
$331$	362.000	1.09366	0.546828	+	0.837245i	$0.315835\pi$
$332$	0	0
$333$	234.000i	0.702703i
$334$	0	0
$335$	0	0
$336$	−528.000	−1.57143
$337$	649.000i	1.92582i	0.269830	+	0.962908i	$0.413033\pi$
$337$	649.000i	1.92582i	−0.269830	+	0.962908i	$0.586967\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	253.000i	0.737609i
$344$	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
$349$	502.000	1.43840	0.719198	−	0.694805i	$-0.244510\pi$
$349$	502.000	1.43840	0.719198	+	0.694805i	$0.244510\pi$
$350$	0	0
$351$	27.0000	0.0769231
$352$	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
$361$	1008.00	2.79224
$362$	0	0
$363$	− 363.000i	− 1.00000i
$364$	44.0000	0.120879
$365$	0	0
$366$	0	0
$367$	− 491.000i	− 1.33787i	−0.743319	−	0.668937i	$-0.766749\pi$
$367$	− 491.000i	− 1.33787i	0.743319	−	0.668937i	$-0.233251\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	− 156.000i	− 0.419355i
$373$	− 121.000i	− 0.324397i	−0.986758	−	0.162198i	$-0.948142\pi$
$373$	− 121.000i	− 0.324397i	0.986758	−	0.162198i	$-0.0518585\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−83.0000	−0.218997	−0.109499	−	0.993987i	$-0.534925\pi$
$379$	−83.0000	−0.218997	−0.109499	+	0.993987i	$0.534925\pi$
$380$	0	0
$381$	−438.000	−1.14961
$382$	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
$387$	549.000i	1.41860i
$388$	− 676.000i	− 1.74227i
$389$	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
$397$	− 431.000i	− 1.08564i	−0.839848	−	0.542821i	$-0.817356\pi$
$397$	− 431.000i	− 1.08564i	0.839848	−	0.542821i	$-0.182644\pi$
$398$	0	0
$399$	−1221.00	−3.06015
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	13.0000i	0.0322581i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−143.000	−0.349633	−0.174817	−	0.984601i	$-0.555933\pi$
$409$	−143.000	−0.349633	−0.174817	+	0.984601i	$0.555933\pi$
$410$	0	0
$411$	0	0
$412$	− 776.000i	− 1.88350i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 66.0000i	− 0.158273i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−358.000	−0.850356	−0.425178	−	0.905110i	$-0.639789\pi$
$421$	−358.000	−0.850356	−0.425178	+	0.905110i	$0.639789\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 517.000i	− 1.21077i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	432.000i	1.00000i
$433$	359.000i	0.829099i	0.910027	+	0.414550i	$0.136061\pi$
$433$	359.000i	0.829099i	−0.910027	+	0.414550i	$0.863939\pi$
$434$	0	0
$435$	0	0
$436$	572.000	1.31193
$437$	0	0
$438$	0	0
$439$	−803.000	−1.82916	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−803.000	−1.82916	−0.914579	+	0.404408i	$0.867478\pi$
$440$	0	0
$441$	648.000	1.46939
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	312.000	0.702703
$445$	0	0
$446$	0	0
$447$	0	0
$448$	704.000i	1.57143i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 681.000i	− 1.50331i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	814.000i	1.78118i	0.454805	+	0.890591i	$0.349709\pi$
$457$	814.000i	1.78118i	−0.454805	+	0.890591i	$0.650291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 526.000i	− 1.13607i	−0.823005	−	0.568035i	$-0.807704\pi$
$463$	− 526.000i	− 1.13607i	0.823005	−	0.568035i	$-0.192296\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 36.0000i	− 0.0769231i
$469$	1199.00	2.55650
$470$	0	0
$471$	−933.000	−1.98089
$472$	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
$481$	−26.0000	−0.0540541
$482$	0	0
$483$	0	0
$484$	−484.000	−1.00000
$485$	0	0
$486$	0	0
$487$	349.000i	0.716632i	0.933600	+	0.358316i	$0.116649\pi$
$487$	349.000i	0.716632i	−0.933600	+	0.358316i	$0.883351\pi$
$488$	0	0
$489$	897.000	1.83436
$490$	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
$496$	−208.000	−0.419355
$497$	0	0
$498$	0	0
$499$	877.000	1.75752	0.878758	−	0.477269i	$-0.158373\pi$
$499$	877.000	1.75752	0.878758	+	0.477269i	$0.158373\pi$
$500$	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
$507$	− 504.000i	− 0.994083i
$508$	584.000i	1.14961i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−506.000	−0.990215
$512$	0	0
$513$	999.000i	1.94737i
$514$	0	0
$515$	0	0
$516$	732.000	1.41860
$517$	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
$523$	179.000i	0.342256i	0.985249	+	0.171128i	$0.0547412\pi$
$523$	179.000i	0.342256i	−0.985249	+	0.171128i	$0.945259\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	1628.00i	3.06015i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−793.000	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−793.000	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$542$	0	0
$543$	939.000i	1.72928i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 506.000i	− 0.925046i	−0.886607	−	0.462523i	$-0.846944\pi$
$547$	− 506.000i	− 0.925046i	0.886607	−	0.462523i	$-0.153056\pi$
$548$	0	0
$549$	−423.000	−0.770492
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 1562.00i	− 2.82459i
$554$	0	0
$555$	0	0
$556$	−88.0000	−0.158273
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−61.0000	−0.109123
$560$	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
$567$	− 891.000i	− 1.57143i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1067.00	1.86865	0.934326	−	0.356420i	$-0.116003\pi$
$571$	1067.00	1.86865	0.934326	+	0.356420i	$0.116003\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	576.000	1.00000
$577$	− 71.0000i	− 0.123050i	−0.998106	−	0.0615251i	$-0.980404\pi$
$577$	− 71.0000i	− 0.123050i	0.998106	−	0.0615251i	$-0.0195964\pi$
$578$	0	0
$579$	717.000	1.23834
$580$	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	$0.5\pi$
$588$	− 864.000i	− 1.46939i
$589$	−481.000	−0.816638
$590$	0	0
$591$	0	0
$592$	− 416.000i	− 0.702703i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 831.000i	− 1.39196i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−673.000	−1.11980	−0.559900	−	0.828560i	$-0.689161\pi$
$601$	−673.000	−1.11980	−0.559900	+	0.828560i	$0.689161\pi$
$602$	0	0
$603$	− 981.000i	− 1.62687i
$604$	−908.000	−1.50331
$605$	0	0
$606$	0	0
$607$	814.000i	1.34102i	0.741900	+	0.670511i	$0.233925\pi$
$607$	814.000i	1.34102i	−0.741900	+	0.670511i	$0.766075\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 1126.00i	− 1.83687i	−0.395574	−	0.918434i	$-0.629454\pi$
$613$	− 1126.00i	− 1.83687i	0.395574	−	0.918434i	$-0.370546\pi$
$614$	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
$619$	−1163.00	−1.87884	−0.939418	−	0.342773i	$-0.888634\pi$
$619$	−1163.00	−1.87884	−0.939418	+	0.342773i	$0.888634\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−48.0000	−0.0769231
$625$	0	0
$626$	0	0
$627$	0	0
$628$	1244.00i	1.98089i
$629$	0	0
$630$	0	0
$631$	587.000	0.930269	0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000	0.930269	0.465135	+	0.885240i	$0.346006\pi$
$632$	0	0
$633$	759.000i	1.19905i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	72.0000i	0.113030i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	314.000i	0.488336i	0.969733	+	0.244168i	$0.0785148\pi$
$643$	314.000i	0.488336i	−0.969733	+	0.244168i	$0.921485\pi$
$644$	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
$651$	429.000	0.658986
$652$	− 1196.00i	− 1.83436i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	414.000i	0.630137i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	122.000	0.184569	0.0922844	−	0.995733i	$-0.470583\pi$
$661$	122.000	0.184569	0.0922844	+	0.995733i	$0.470583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−1263.00	−1.88789
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1154.00i	1.71471i	0.514725	+	0.857355i	$0.327894\pi$
$673$	1154.00i	1.71471i	−0.514725	+	0.857355i	$0.672106\pi$
$674$	0	0
$675$	0	0
$676$	−672.000	−0.994083
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	1859.00	2.73785
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	1332.00	1.94737
$685$	0	0
$686$	0	0
$687$	1149.00i	1.67249i
$688$	− 976.000i	− 1.41860i
$689$	0	0
$690$	0	0
$691$	−1318.00	−1.90738	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−1318.00	−1.90738	−0.953690	+	0.300790i	$0.902750\pi$
$692$	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
$703$	− 962.000i	− 1.36842i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	457.000	0.644570	0.322285	−	0.946643i	$-0.395549\pi$
$709$	457.000	0.644570	0.322285	+	0.946643i	$0.395549\pi$
$710$	0	0
$711$	−1278.00	−1.79747
$712$	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
$721$	2134.00	2.95978
$722$	0	0
$723$	579.000i	0.800830i
$724$	1252.00	1.72928
$725$	0	0
$726$	0	0
$727$	1429.00i	1.96561i	0.184641	+	0.982806i	$0.440888\pi$
$727$	1429.00i	1.96561i	−0.184641	+	0.982806i	$0.559112\pi$
$728$	0	0
$729$	−729.000	−1.00000
$730$	0	0
$731$	0	0
$732$	564.000i	0.770492i
$733$	1034.00i	1.41064i	0.708888	+	0.705321i	$0.249197\pi$
$733$	1034.00i	1.41064i	−0.708888	+	0.705321i	$0.750803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1222.00	1.65359	0.826793	−	0.562506i	$-0.190163\pi$
$739$	1222.00	1.65359	0.826793	+	0.562506i	$0.190163\pi$
$740$	0	0
$741$	−111.000	−0.149798
$742$	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
$751$	1202.00	1.60053	0.800266	−	0.599645i	$-0.204691\pi$
$751$	1202.00	1.60053	0.800266	+	0.599645i	$0.204691\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−1188.00	−1.57143
$757$	− 1511.00i	− 1.99604i	−0.0629213	−	0.998018i	$-0.520042\pi$
$757$	− 1511.00i	− 1.99604i	0.0629213	−	0.998018i	$-0.479958\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1573.00i	2.06160i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 768.000i	− 1.00000i
$769$	−863.000	−1.12224	−0.561118	−	0.827736i	$-0.689629\pi$
$769$	−863.000	−1.12224	−0.561118	+	0.827736i	$0.689629\pi$
$770$	0	0
$771$	0	0
$772$	− 956.000i	− 1.23834i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	858.000i	1.10425i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1152.00	−1.46939
$785$	0	0
$786$	0	0
$787$	949.000i	1.20584i	0.797800	+	0.602922i	$0.205997\pi$
$787$	949.000i	1.20584i	−0.797800	+	0.602922i	$0.794003\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 47.0000i	− 0.0592686i
$794$	0	0
$795$	0	0
$796$	−1108.00	−1.39196
$797$	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
$804$	−1308.00	−1.62687
$805$	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
$811$	−253.000	−0.311961	−0.155980	−	0.987760i	$-0.549854\pi$
$811$	−253.000	−0.311961	−0.155980	+	0.987760i	$0.549854\pi$
$812$	0	0
$813$	− 726.000i	− 0.892989i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2257.00i	− 2.76255i
$818$	0	0
$819$	99.0000	0.120879
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 1621.00i	− 1.96962i	−0.173626	−	0.984812i	$-0.555548\pi$
$823$	− 1621.00i	− 1.96962i	0.173626	−	0.984812i	$-0.444452\pi$
$824$	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
$829$	−458.000	−0.552473	−0.276236	−	0.961090i	$-0.589087\pi$
$829$	−458.000	−0.552473	−0.276236	+	0.961090i	$0.589087\pi$
$830$	0	0
$831$	1587.00	1.90975
$832$	64.0000i	0.0769231i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 351.000i	− 0.419355i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	1012.00	1.19905
$845$	0	0
$846$	0	0
$847$	− 1331.00i	− 1.57143i
$848$	0	0
$849$	177.000	0.208481
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 481.000i	− 0.563892i	−0.959430	−	0.281946i	$-0.909020\pi$
$853$	− 481.000i	− 0.563892i	0.959430	−	0.281946i	$-0.0909799\pi$
$854$	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
$859$	−1418.00	−1.65076	−0.825378	−	0.564580i	$-0.809038\pi$
$859$	−1418.00	−1.65076	−0.825378	+	0.564580i	$0.809038\pi$
$860$	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
$867$	867.000i	1.00000i
$868$	− 572.000i	− 0.658986i
$869$	0	0
$870$	0	0
$871$	109.000	0.125144
$872$	0	0
$873$	− 1521.00i	− 1.74227i
$874$	0	0
$875$	0	0
$876$	552.000	0.630137
$877$	1129.00i	1.28734i	0.765302	+	0.643672i	$0.222590\pi$
$877$	1129.00i	1.28734i	−0.765302	+	0.643672i	$0.777410\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	1259.00i	1.42582i	0.701255	+	0.712911i	$0.252623\pi$
$883$	1259.00i	1.42582i	−0.701255	+	0.712911i	$0.747377\pi$
$884$	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
$889$	−1606.00	−1.80652
$890$	0	0
$891$	0	0
$892$	1684.00i	1.88789i
$893$	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
$903$	2013.00i	2.22924i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	214.000i	0.235943i	0.993017	+	0.117971i	$0.0376391\pi$
$907$	214.000i	0.235943i	−0.993017	+	0.117971i	$0.962361\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 1776.00i	− 1.94737i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1532.00	1.67249
$917$	0	0
$918$	0	0
$919$	1837.00	1.99891	0.999456	−	0.0329825i	$-0.0105006\pi$
$919$	1837.00	1.99891	0.999456	+	0.0329825i	$0.0105006\pi$
$920$	0	0
$921$	−1833.00	−1.99023
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1746.00i	− 1.88350i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−2664.00	−2.86144
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	649.000i	0.692636i	0.938117	+	0.346318i	$0.112568\pi$
$937$	649.000i	0.692636i	−0.938117	+	0.346318i	$0.887432\pi$
$938$	0	0
$939$	1797.00	1.91374
$940$	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	$0.5\pi$
$948$	1704.00i	1.79747i
$949$	−46.0000	−0.0484721
$950$	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
$961$	−792.000	−0.824142
$962$	0	0
$963$	0	0
$964$	772.000	0.800830
$965$	0	0
$966$	0	0
$967$	1534.00i	1.58635i	0.608994	+	0.793175i	$0.291573\pi$
$967$	1534.00i	1.58635i	−0.608994	+	0.793175i	$0.708427\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	972.000i	1.00000i
$973$	− 242.000i	− 0.248715i
$974$	0	0
$975$	0	0
$976$	752.000	0.770492
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1287.00	1.31193
$982$	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
$988$	148.000i	0.149798i
$989$	0	0
$990$	0	0
$991$	−1693.00	−1.70838	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	−1693.00	−1.70838	−0.854188	+	0.519965i	$0.825945\pi$
$992$	0	0
$993$	− 1086.00i	− 1.09366i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1894.00i	1.89970i	0.312707	+	0.949850i	$0.398764\pi$
$997$	1894.00i	1.89970i	−0.312707	+	0.949850i	$0.601236\pi$
$998$	0	0
$999$	702.000	0.702703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.3.d.a.74.1		2
3.2	odd	2	CM	75.3.d.a.74.1		2
4.3	odd	2		1200.3.c.a.449.2		2
5.2	odd	4		75.3.c.a.26.1	✓	1
5.3	odd	4		75.3.c.b.26.1	yes	1
5.4	even	2	inner	75.3.d.a.74.2		2
12.11	even	2		1200.3.c.a.449.2		2
15.2	even	4		75.3.c.a.26.1	✓	1
15.8	even	4		75.3.c.b.26.1	yes	1
15.14	odd	2	inner	75.3.d.a.74.2		2
20.3	even	4		1200.3.l.c.401.1		1
20.7	even	4		1200.3.l.d.401.1		1
20.19	odd	2		1200.3.c.a.449.1		2
60.23	odd	4		1200.3.l.c.401.1		1
60.47	odd	4		1200.3.l.d.401.1		1
60.59	even	2		1200.3.c.a.449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.c.a.26.1	✓	1	5.2	odd	4
75.3.c.a.26.1	✓	1	15.2	even	4
75.3.c.b.26.1	yes	1	5.3	odd	4
75.3.c.b.26.1	yes	1	15.8	even	4
75.3.d.a.74.1		2	1.1	even	1	trivial
75.3.d.a.74.1		2	3.2	odd	2	CM
75.3.d.a.74.2		2	5.4	even	2	inner
75.3.d.a.74.2		2	15.14	odd	2	inner
1200.3.c.a.449.1		2	20.19	odd	2
1200.3.c.a.449.1		2	60.59	even	2
1200.3.c.a.449.2		2	4.3	odd	2
1200.3.c.a.449.2		2	12.11	even	2
1200.3.l.c.401.1		1	20.3	even	4
1200.3.l.c.401.1		1	60.23	odd	4
1200.3.l.d.401.1		1	20.7	even	4
1200.3.l.d.401.1		1	60.47	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -4.00000 q^{4} -11.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -4.00000 q^{4} -11.0000i q^{7} -9.00000 q^{9} +12.0000i q^{12} -1.00000i q^{13} +16.0000 q^{16} +37.0000 q^{19} -33.0000 q^{21} +27.0000i q^{27} +44.0000i q^{28} -13.0000 q^{31} +36.0000 q^{36} -26.0000i q^{37} -3.00000 q^{39} -61.0000i q^{43} -48.0000i q^{48} -72.0000 q^{49} +4.00000i q^{52} -111.000i q^{57} +47.0000 q^{61} +99.0000i q^{63} -64.0000 q^{64} +109.000i q^{67} -46.0000i q^{73} -148.000 q^{76} +142.000 q^{79} +81.0000 q^{81} +132.000 q^{84} -11.0000 q^{91} +39.0000i q^{93} +169.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 18 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 - 18 * q^9	\( 2 q - 8 q^{4} - 18 q^{9} + 32 q^{16} + 74 q^{19} - 66 q^{21} - 26 q^{31} + 72 q^{36} - 6 q^{39} - 144 q^{49} + 94 q^{61} - 128 q^{64} - 296 q^{76} + 284 q^{79} + 162 q^{81} + 264 q^{84} - 22 q^{91}+O(q^{100}) \) 2 * q - 8 * q^4 - 18 * q^9 + 32 * q^16 + 74 * q^19 - 66 * q^21 - 26 * q^31 + 72 * q^36 - 6 * q^39 - 144 * q^49 + 94 * q^61 - 128 * q^64 - 296 * q^76 + 284 * q^79 + 162 * q^81 + 264 * q^84 - 22 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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