LMFDB - Embedded newform 1104.2.m.a.689.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,2,Mod(689,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1104.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.m (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:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 69)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			689.4
Root			$1.33454 - 0.467979i$ of defining polynomial
Character	$\chi$	$=$	1104.689
Dual form			1104.2.m.a.689.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.227452 + 1.71705i) q^{3} +(-2.89653 + 0.781094i) q^{9} +O(q^{10})$	$q+(0.227452 + 1.71705i) q^{3} +(-2.89653 + 0.781094i) q^{9} -4.88325 q^{13} +4.79583i q^{23} -5.00000 q^{25} +(-2.00000 - 4.79583i) q^{27} +8.43039i q^{29} -11.1312 q^{31} +(-1.11071 - 8.38480i) q^{39} -12.1742i q^{41} +6.55848i q^{47} +7.00000 q^{49} +9.59166i q^{59} +(-8.23469 + 1.09082i) q^{69} -14.0461i q^{71} -7.61268 q^{73} +(-1.13726 - 8.58526i) q^{75} +(7.77979 - 4.52492i) q^{81} +(-14.4754 + 1.91751i) q^{87} +(-2.53182 - 19.1129i) q^{93} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 30 q^{25} - 12 q^{27} + 24 q^{39} + 42 q^{49} - 48 q^{87} - 6 q^{93}+O(q^{100}) $ 6 * q - 30 * q^25 - 12 * q^27 + 24 * q^39 + 42 * q^49 - 48 * q^87 - 6 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	0.227452	+	1.71705i	0.131319	+	0.991340i
$3$	0.227452	+	1.71705i	0.131319	+	0.991340i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.89653	+	0.781094i	−0.965510	+	0.260365i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−4.88325			−1.35437			−0.677185	−	0.735812i	$-0.736801\pi$
$13$	−4.88325			−1.35437			−0.677185	+	0.735812i	$0.736801\pi$
$14$		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
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.79583i			1.00000i
$23$			4.79583i			1.00000i
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	−2.00000	−	4.79583i	−0.384900	−	0.922958i
$28$		0			0
$29$			8.43039i			1.56548i	0.622346	+	0.782742i	$0.286180\pi$
$29$			8.43039i			1.56548i	−0.622346	+	0.782742i	$0.713820\pi$
$30$		0			0
$31$	−11.1312			−1.99923			−0.999613	−	0.0278144i	$-0.991145\pi$
$31$	−11.1312			−1.99923			−0.999613	+	0.0278144i	$0.991145\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−1.11071	−	8.38480i	−0.177855	−	1.34264i
$40$		0			0
$41$		−	12.1742i		−	1.90129i	−0.310274	−	0.950647i	$-0.600421\pi$
$41$		−	12.1742i		−	1.90129i	0.310274	−	0.950647i	$-0.399579\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.55848i			0.956652i	0.878182	+	0.478326i	$0.158756\pi$
$47$			6.55848i			0.956652i	−0.878182	+	0.478326i	$0.841244\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		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
$57$		0			0
$58$		0			0
$59$			9.59166i			1.24873i	0.781133	+	0.624364i	$0.214642\pi$
$59$			9.59166i			1.24873i	−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−8.23469	+	1.09082i	−0.991340	+	0.131319i
$70$		0			0
$71$		−	14.0461i		−	1.66697i	−0.552542	−	0.833485i	$-0.686342\pi$
$71$		−	14.0461i		−	1.66697i	0.552542	−	0.833485i	$-0.313658\pi$
$72$		0			0
$73$	−7.61268			−0.890997			−0.445498	−	0.895283i	$-0.646973\pi$
$73$	−7.61268			−0.890997			−0.445498	+	0.895283i	$0.646973\pi$
$74$		0			0
$75$	−1.13726	−	8.58526i	−0.131319	−	0.991340i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.77979	−	4.52492i	0.864421	−	0.502769i
$82$		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
$87$	−14.4754	+	1.91751i	−1.55193	+	0.205579i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.53182	−	19.1129i	−0.262537	−	1.98191i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			19.1833i			1.90881i	0.298511	+	0.954406i	$0.403510\pi$
$101$			19.1833i			1.90881i	−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		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
$117$	14.1445	−	3.81428i	1.30766	−	0.352630i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	20.9038	−	2.76905i	1.88483	−	0.249677i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.40180			−0.745539			−0.372769	−	0.927924i	$-0.621592\pi$
$127$	−8.40180			−0.745539			−0.372769	+	0.927924i	$0.621592\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.81465i		−	0.245917i	−0.992412	−	0.122958i	$-0.960762\pi$
$131$		−	2.81465i		−	0.245917i	0.992412	−	0.122958i	$-0.0392382\pi$
$132$		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
$139$	20.8977			1.77252			0.886261	−	0.463186i	$-0.153294\pi$
$139$	20.8977			1.77252			0.886261	+	0.463186i	$0.153294\pi$
$140$		0			0
$141$	−11.2612	+	1.49174i	−0.948368	+	0.125627i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.59216	+	12.0194i	0.131319	+	0.991340i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−13.8606			−1.12796			−0.563982	−	0.825787i	$-0.690731\pi$
$151$	−13.8606			−1.12796			−0.563982	+	0.825787i	$0.690731\pi$
$152$		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
$161$		0			0
$162$		0			0
$163$	23.6272			1.85062			0.925311	−	0.379210i	$-0.123804\pi$
$163$	23.6272			1.85062			0.925311	+	0.379210i	$0.123804\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	10.8462			0.834321
$170$		0			0
$171$		0			0
$172$		0			0
$173$			19.1833i			1.45848i	0.684257	+	0.729241i	$0.260127\pi$
$173$			19.1833i			1.45848i	−0.684257	+	0.729241i	$0.739873\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−16.4694	+	2.18164i	−1.23791	+	0.163982i
$178$		0			0
$179$			17.7900i			1.32968i	0.746984	+	0.664842i	$0.231501\pi$
$179$			17.7900i			1.32968i	−0.746984	+	0.664842i	$0.768499\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		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
$193$	27.1457			1.95399			0.976995	−	0.213262i	$-0.0684089\pi$
$193$	27.1457			1.95399			0.976995	+	0.213262i	$0.0684089\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	0.942731i		−	0.0671668i	−0.999436	−	0.0335834i	$-0.989308\pi$
$197$		−	0.942731i		−	0.0671668i	0.999436	−	0.0335834i	$-0.0106919\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.74599	−	13.8913i	−0.260365	−	0.965510i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	24.1179	−	3.19482i	1.65253	−	0.218906i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.73152	−	13.0714i	−0.117005	−	0.883281i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	14.4827	−	3.90547i	0.965510	−	0.260365i
$226$		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
$233$			29.0350i			1.90215i	0.308965	+	0.951073i	$0.400017\pi$
$233$			29.0350i			1.90215i	−0.308965	+	0.951073i	$0.599983\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			27.1631i			1.75703i	0.477711	+	0.878517i	$0.341467\pi$
$239$			27.1631i			1.75703i	−0.477711	+	0.878517i	$0.658533\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	9.53906	+	12.3291i	0.611931	+	0.790911i
$244$		0			0
$245$		0			0
$246$		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
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			19.6619i			1.22647i	0.789899	+	0.613237i	$0.210133\pi$
$257$			19.6619i			1.22647i	−0.789899	+	0.613237i	$0.789867\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.58493	−	24.4189i	−0.407597	−	1.51149i
$262$		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
$269$		−	32.7788i		−	1.99856i	−0.0379247	−	0.999281i	$-0.512075\pi$
$269$		−	32.7788i		−	1.99856i	0.0379247	−	0.999281i	$-0.487925\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	29.8751			1.79502			0.897511	−	0.440992i	$-0.145373\pi$
$277$	29.8751			1.79502			0.897511	+	0.440992i	$0.145373\pi$
$278$		0			0
$279$	32.2419	−	8.69453i	1.93027	−	0.520528i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		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
$299$		−	23.4193i		−	1.35437i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−32.9388	+	4.36329i	−1.89228	+	0.250664i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	34.6508i		−	1.96486i	−0.186621	−	0.982432i	$-0.559754\pi$
$311$		−	34.6508i		−	1.96486i	0.186621	−	0.982432i	$-0.440246\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			19.1833i			1.07744i	0.842484	+	0.538721i	$0.181092\pi$
$317$			19.1833i			1.07744i	−0.842484	+	0.538721i	$0.818908\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	24.4163			1.35437
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.1683			0.998620			0.499310	−	0.866423i	$-0.333587\pi$
$331$	18.1683			0.998620			0.499310	+	0.866423i	$0.333587\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$			9.59166i			0.514907i	0.966291	+	0.257454i	$0.0828835\pi$
$347$			9.59166i			0.514907i	−0.966291	+	0.257454i	$0.917117\pi$
$348$		0			0
$349$	−10.3421			−0.553600			−0.276800	−	0.960928i	$-0.589274\pi$
$349$	−10.3421			−0.553600			−0.276800	+	0.960928i	$0.589274\pi$
$350$		0			0
$351$	9.76651	+	23.4193i	0.521298	+	1.25003i
$352$		0			0
$353$		−	21.5473i		−	1.14685i	−0.819258	−	0.573425i	$-0.805614\pi$
$353$		−	21.5473i		−	1.14685i	0.819258	−	0.573425i	$-0.194386\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	−2.50197	−	18.8876i	−0.131319	−	0.991340i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	9.50921	+	35.2630i	0.495030	+	1.83572i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	41.1678i		−	2.12025i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.91101	−	14.4263i	−0.0979038	−	0.739083i
$382$		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
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	4.83289	−	0.640197i	0.243787	−	0.0322937i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−39.6416			−1.98956			−0.994778	−	0.102061i	$-0.967456\pi$
$397$	−39.6416			−1.98956			−0.994778	+	0.102061i	$0.967456\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	54.3566			2.70769
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−36.9122			−1.82519			−0.912595	−	0.408864i	$-0.865925\pi$
$409$	−36.9122			−1.82519			−0.912595	+	0.408864i	$0.865925\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.75323	+	35.8825i	0.232767	+	1.75717i
$418$		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
$423$	−5.12279	−	18.9968i	−0.249078	−	0.923658i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		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
$439$	−5.67237			−0.270728			−0.135364	−	0.990796i	$-0.543220\pi$
$439$	−5.67237			−0.270728			−0.135364	+	0.990796i	$0.543220\pi$
$440$		0			0
$441$	−20.2757	+	5.46766i	−0.965510	+	0.260365i
$442$		0			0
$443$			38.3946i			1.82418i	0.409988	+	0.912091i	$0.365533\pi$
$443$			38.3946i			1.82418i	−0.409988	+	0.912091i	$0.634467\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	38.3667i		−	1.81063i	−0.424736	−	0.905317i	$-0.639633\pi$
$449$		−	38.3667i		−	1.81063i	0.424736	−	0.905317i	$-0.360367\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−3.15263	−	23.7994i	−0.148124	−	1.11820i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	23.4057i		−	1.09011i	−0.838399	−	0.545056i	$-0.816508\pi$
$461$		−	23.4057i		−	1.09011i	0.838399	−	0.545056i	$-0.183492\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$		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
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−16.5901			−0.751768			−0.375884	−	0.926667i	$-0.622661\pi$
$487$	−16.5901			−0.751768			−0.375884	+	0.926667i	$0.622661\pi$
$488$		0			0
$489$	5.37404	+	40.5690i	0.243023	+	1.83460i
$490$		0			0
$491$		−	4.67301i		−	0.210890i	−0.994425	−	0.105445i	$-0.966373\pi$
$491$		−	4.67301i		−	0.210890i	0.994425	−	0.105445i	$-0.0336267\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−43.1602			−1.93211			−0.966057	−	0.258328i	$-0.916828\pi$
$499$	−43.1602			−1.93211			−0.966057	+	0.258328i	$0.916828\pi$
$500$		0			0
$501$	−16.4694	+	2.18164i	−0.735798	+	0.0974686i
$502$		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
$507$	2.46698	+	18.6234i	0.109563	+	0.827096i
$508$		0			0
$509$			17.8035i			0.789127i	0.918869	+	0.394564i	$0.129104\pi$
$509$			17.8035i			0.789127i	−0.918869	+	0.394564i	$0.870896\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−32.9388	+	4.36329i	−1.44585	+	0.191527i
$520$		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
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−7.49199	−	27.7826i	−0.325125	−	1.20566i
$532$		0			0
$533$			59.4498i			2.57506i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−30.5463	+	4.04637i	−1.31817	+	0.174614i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.575595			0.0247468			0.0123734	−	0.999923i	$-0.496061\pi$
$541$	0.575595			0.0247468			0.0123734	+	0.999923i	$0.496061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−45.8896			−1.96210			−0.981049	−	0.193761i	$-0.937931\pi$
$547$	−45.8896			−1.96210			−0.981049	+	0.193761i	$0.937931\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		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
$559$		0			0
$560$		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
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	23.9792i		−	1.00000i
$576$		0			0
$577$	32.6045			1.35734			0.678672	−	0.734441i	$-0.262556\pi$
$577$	32.6045			1.35734			0.678672	+	0.734441i	$0.262556\pi$
$578$		0			0
$579$	6.17434	+	46.6106i	0.256597	+	1.93707i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	25.2776i		−	1.04332i	−0.853154	−	0.521660i	$-0.825313\pi$
$587$		−	25.2776i		−	1.04332i	0.853154	−	0.521660i	$-0.174687\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	1.61872	−	0.214426i	0.0665852	−	0.00882032i
$592$		0			0
$593$		−	38.3667i		−	1.57553i	−0.615976	−	0.787765i	$-0.711238\pi$
$593$		−	38.3667i		−	1.57553i	0.615976	−	0.787765i	$-0.288762\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	−42.3711			−1.72835			−0.864176	−	0.503190i	$-0.832159\pi$
$601$	−42.3711			−1.72835			−0.864176	+	0.503190i	$0.832159\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	40.0000			1.62355			0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000			1.62355			0.811775	+	0.583970i	$0.198502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	32.0267i		−	1.29566i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		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
$621$	23.0000	−	9.59166i	0.922958	−	0.384900i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	0.909808	+	6.86821i	0.0361616	+	0.272987i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−34.1828			−1.35437
$638$		0			0
$639$	10.9714	+	40.6851i	0.434020	+	1.60948i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.7677i			1.87794i	0.343996	+	0.938971i	$0.388219\pi$
$647$			47.7677i			1.87794i	−0.343996	+	0.938971i	$0.611781\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			49.6396i			1.94255i	0.237962	+	0.971274i	$0.423520\pi$
$653$			49.6396i			1.94255i	−0.237962	+	0.971274i	$0.576480\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	22.0504	−	5.94622i	0.860267	−	0.231984i
$658$		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
$667$	−40.4307			−1.56548
$668$		0			0
$669$	−1.81962	−	13.7364i	−0.0703504	−	0.531080i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	21.6868			0.835966			0.417983	−	0.908455i	$-0.362737\pi$
$673$	21.6868			0.835966			0.417983	+	0.908455i	$0.362737\pi$
$674$		0			0
$675$	10.0000	+	23.9792i	0.384900	+	0.922958i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.0239i		−	1.68453i	−0.539066	−	0.842263i	$-0.681223\pi$
$683$		−	44.0239i		−	1.68453i	0.539066	−	0.842263i	$-0.318777\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−44.0000			−1.67384			−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000			−1.67384			−0.836919	+	0.547326i	$0.815646\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−49.8546	+	6.60407i	−1.88567	+	0.249789i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		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
$713$		−	53.3835i		−	1.99923i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−46.6404	+	6.17830i	−1.74182	+	0.230733i
$718$		0			0
$719$		−	47.9583i		−	1.78854i	−0.447524	−	0.894272i	$-0.647694\pi$
$719$		−	47.9583i		−	1.78854i	0.447524	−	0.894272i	$-0.352306\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	42.1520i		−	1.56548i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−19.0000	+	19.1833i	−0.703704	+	0.710494i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	15.4389			0.567928			0.283964	−	0.958835i	$-0.408350\pi$
$739$	15.4389			0.567928			0.283964	+	0.958835i	$0.408350\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		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
$761$		−	44.0103i		−	1.59537i	−0.603072	−	0.797687i	$-0.706057\pi$
$761$		−	44.0103i		−	1.59537i	0.603072	−	0.797687i	$-0.293943\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	46.8385i		−	1.69124i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−33.7605	+	4.47214i	−1.21585	+	0.161060i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	55.6561			1.99923
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	40.4307	−	16.8608i	1.44488	−	0.602555i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		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
$807$	56.2830	−	7.45561i	1.98125	−	0.262450i
$808$		0			0
$809$		−	38.3667i		−	1.34890i	−0.738321	−	0.674450i	$-0.764381\pi$
$809$		−	38.3667i		−	1.34890i	0.738321	−	0.674450i	$-0.235619\pi$
$810$		0			0
$811$	−48.6190			−1.70724			−0.853622	−	0.520892i	$-0.825599\pi$
$811$	−48.6190			−1.70724			−0.853622	+	0.520892i	$0.825599\pi$
$812$		0			0
$813$	3.63923	+	27.4728i	0.127633	+	0.963514i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19.1833i			0.669503i	0.942306	+	0.334751i	$0.108652\pi$
$821$			19.1833i			0.669503i	−0.942306	+	0.334751i	$0.891348\pi$
$822$		0			0
$823$	52.9267			1.84491			0.922454	−	0.386107i	$-0.126180\pi$
$823$	52.9267			1.84491			0.922454	+	0.386107i	$0.126180\pi$
$824$		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
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	6.79516	+	51.2971i	0.235721	+	1.77948i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.2624	+	53.3835i	0.769503	+	1.84520i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−42.0715			−1.45074
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			38.4081i			1.31200i	0.754762	+	0.655998i	$0.227752\pi$
$857$			38.4081i			1.31200i	−0.754762	+	0.655998i	$0.772248\pi$
$858$		0			0
$859$	29.0860			0.992402			0.496201	−	0.868208i	$-0.334728\pi$
$859$	29.0860			0.992402			0.496201	+	0.868208i	$0.334728\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1878i		−	0.414877i	−0.978248	−	0.207438i	$-0.933487\pi$
$863$		−	12.1878i		−	0.414877i	0.978248	−	0.207438i	$-0.0665126\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.86668	−	29.1899i	−0.131319	−	0.991340i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	52.0000			1.74994			0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000			1.74994			0.874970	+	0.484178i	$0.160881\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			29.0215i			0.974445i	0.873278	+	0.487223i	$0.161990\pi$
$887$			29.0215i			0.974445i	−0.873278	+	0.487223i	$0.838010\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	40.2121	−	5.32676i	1.34264	−	0.177855i
$898$		0			0
$899$		−	93.8406i		−	3.12976i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−14.9840	−	55.5651i	−0.496987	−	1.84298i
$910$		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
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−4.54904	−	34.3410i	−0.149896	−	1.13158i
$922$		0			0
$923$			68.5909i			2.25770i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			10.2888i			0.337563i	0.985653	+	0.168782i	$0.0539833\pi$
$929$			10.2888i			0.337563i	−0.985653	+	0.168782i	$0.946017\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	59.4971	−	7.88139i	1.94785	−	0.258025i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	58.3855			1.90129
$944$		0			0
$945$		0			0
$946$		0			0
$947$			36.5362i			1.18727i	0.804735	+	0.593634i	$0.202307\pi$
$947$			36.5362i			1.18727i	−0.804735	+	0.593634i	$0.797693\pi$
$948$		0			0
$949$	37.1746			1.20674
$950$		0			0
$951$	−32.9388	+	4.36329i	−1.06811	+	0.141489i
$952$		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
$961$	92.9041			2.99691
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−2.94295			−0.0946388			−0.0473194	−	0.998880i	$-0.515068\pi$
$967$	−2.94295			−0.0946388			−0.0473194	+	0.998880i	$0.515068\pi$
$968$		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
$973$		0			0
$974$		0			0
$975$	5.55353	+	41.9240i	0.177855	+	1.34264i
$976$		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
$981$		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
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$		0			0
$993$	4.13242	+	31.1959i	0.131138	+	0.989972i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	26.0000			0.823428			0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000			0.823428			0.411714	+	0.911313i	$0.364930\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.m.a.689.4		6
3.2	odd	2	inner	1104.2.m.a.689.3		6
4.3	odd	2		69.2.c.a.68.4	yes	6
12.11	even	2		69.2.c.a.68.3	✓	6
23.22	odd	2	CM	1104.2.m.a.689.4		6
69.68	even	2	inner	1104.2.m.a.689.3		6
92.91	even	2		69.2.c.a.68.4	yes	6
276.275	odd	2		69.2.c.a.68.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.c.a.68.3	✓	6	12.11	even	2
69.2.c.a.68.3	✓	6	276.275	odd	2
69.2.c.a.68.4	yes	6	4.3	odd	2
69.2.c.a.68.4	yes	6	92.91	even	2
1104.2.m.a.689.3		6	3.2	odd	2	inner
1104.2.m.a.689.3		6	69.68	even	2	inner
1104.2.m.a.689.4		6	1.1	even	1	trivial
1104.2.m.a.689.4		6	23.22	odd	2	CM

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.227452 + 1.71705i) q^{3} +(-2.89653 + 0.781094i) q^{9} +O(q^{10})\)	\(q+(0.227452 + 1.71705i) q^{3} +(-2.89653 + 0.781094i) q^{9} -4.88325 q^{13} +4.79583i q^{23} -5.00000 q^{25} +(-2.00000 - 4.79583i) q^{27} +8.43039i q^{29} -11.1312 q^{31} +(-1.11071 - 8.38480i) q^{39} -12.1742i q^{41} +6.55848i q^{47} +7.00000 q^{49} +9.59166i q^{59} +(-8.23469 + 1.09082i) q^{69} -14.0461i q^{71} -7.61268 q^{73} +(-1.13726 - 8.58526i) q^{75} +(7.77979 - 4.52492i) q^{81} +(-14.4754 + 1.91751i) q^{87} +(-2.53182 - 19.1129i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 30 q^{25} - 12 q^{27} + 24 q^{39} + 42 q^{49} - 48 q^{87} - 6 q^{93}+O(q^{100}) \) 6 * q - 30 * q^25 - 12 * q^27 + 24 * q^39 + 42 * q^49 - 48 * q^87 - 6 * q^93

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