LMFDB - Embedded newform 1104.2.m.a.689.6

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			689.6
Root			$-1.07255 + 0.921756i$ of defining polynomial
Character	$\chi$	$=$	1104.689
Dual form			1104.2.m.a.689.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.37328 + 1.05550i) q^{3} +(0.771819 + 2.89902i) q^{9} +O(q^{10})$	$q+(1.37328 + 1.05550i) q^{3} +(0.771819 + 2.89902i) q^{9} +7.03677 q^{13} -4.79583i q^{23} -5.00000 q^{25} +(-2.00000 + 4.79583i) q^{27} +10.0201i q^{29} +5.83384 q^{31} +(9.66349 + 7.42735i) q^{39} -2.64601i q^{41} +13.7071i q^{47} +7.00000 q^{49} -9.59166i q^{59} +(5.06202 - 6.58604i) q^{69} +1.04102i q^{71} -9.44264 q^{73} +(-6.86642 - 5.27752i) q^{75} +(-7.80859 + 4.47503i) q^{81} +(-10.5762 + 13.7604i) q^{87} +(8.01152 + 6.15765i) q^{93} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.37328	+	1.05550i	0.792866	+	0.609396i
$3$	1.37328	+	1.05550i	0.792866	+	0.609396i
$4$		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
$9$	0.771819	+	2.89902i	0.257273	+	0.966339i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	7.03677			1.95165			0.975825	−	0.218554i	$-0.0701339\pi$
$13$	7.03677			1.95165			0.975825	+	0.218554i	$0.0701339\pi$
$14$		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
$29$			10.0201i			1.86068i	0.366702	+	0.930339i	$0.380487\pi$
$29$			10.0201i			1.86068i	−0.366702	+	0.930339i	$0.619513\pi$
$30$		0			0
$31$	5.83384			1.04779			0.523895	−	0.851783i	$-0.324479\pi$
$31$	5.83384			1.04779			0.523895	+	0.851783i	$0.324479\pi$
$32$		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
$39$	9.66349	+	7.42735i	1.54740	+	1.18933i
$40$		0			0
$41$		−	2.64601i		−	0.413237i	−0.978422	−	0.206618i	$-0.933754\pi$
$41$		−	2.64601i		−	0.413237i	0.978422	−	0.206618i	$-0.0662459\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			13.7071i			1.99938i	0.0248485	+	0.999691i	$0.492090\pi$
$47$			13.7071i			1.99938i	−0.0248485	+	0.999691i	$0.507910\pi$
$48$		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	$-0.785358\pi$
$59$		−	9.59166i		−	1.24873i	0.781133	−	0.624364i	$-0.214642\pi$
$60$		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
$69$	5.06202	−	6.58604i	0.609396	−	0.792866i
$70$		0			0
$71$			1.04102i			0.123546i	0.998090	+	0.0617729i	$0.0196755\pi$
$71$			1.04102i			0.123546i	−0.998090	+	0.0617729i	$0.980325\pi$
$72$		0			0
$73$	−9.44264			−1.10518			−0.552588	−	0.833454i	$-0.686360\pi$
$73$	−9.44264			−1.10518			−0.552588	+	0.833454i	$0.686360\pi$
$74$		0			0
$75$	−6.86642	−	5.27752i	−0.792866	−	0.609396i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−7.80859	+	4.47503i	−0.867621	+	0.497225i
$82$		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
$87$	−10.5762	+	13.7604i	−1.13389	+	1.47527i
$88$		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
$93$	8.01152	+	6.15765i	0.830756	+	0.638519i
$94$		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	$-0.596490\pi$
$101$		−	19.1833i		−	1.90881i	0.298511	−	0.954406i	$-0.403510\pi$
$102$		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
$117$	5.43111	+	20.3997i	0.502107	+	1.88595i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	2.79287	−	3.63372i	0.251825	−	0.327641i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	22.3133			1.97998			0.989990	−	0.141134i	$-0.0450749\pi$
$127$	22.3133			1.97998			0.989990	+	0.141134i	$0.0450749\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	21.0811i		−	1.84187i	−0.389721	−	0.920933i	$-0.627428\pi$
$131$		−	21.0811i		−	1.84187i	0.389721	−	0.920933i	$-0.372572\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	−19.9074			−1.68852			−0.844261	−	0.535932i	$-0.819960\pi$
$139$	−19.9074			−1.68852			−0.844261	+	0.535932i	$0.819960\pi$
$140$		0			0
$141$	−14.4679	+	18.8237i	−1.21842	+	1.58524i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	9.61299	+	7.38853i	0.792866	+	0.609396i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−10.6456			−0.866324			−0.433162	−	0.901316i	$-0.642602\pi$
$151$	−10.6456			−0.866324			−0.433162	+	0.901316i	$0.642602\pi$
$152$		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
$163$	−3.42798			−0.268500			−0.134250	−	0.990947i	$-0.542863\pi$
$163$	−3.42798			−0.268500			−0.134250	+	0.990947i	$0.542863\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.59166i		−	0.742225i	−0.928588	−	0.371113i	$-0.878976\pi$
$167$		−	9.59166i		−	0.742225i	0.928588	−	0.371113i	$-0.121024\pi$
$168$		0			0
$169$	36.5162			2.80894
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	19.1833i		−	1.45848i	−0.684257	−	0.729241i	$-0.739873\pi$
$173$		−	19.1833i		−	1.45848i	0.684257	−	0.729241i	$-0.260127\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	10.1240	−	13.1721i	0.760970	−	0.990074i
$178$		0			0
$179$		−	8.41506i		−	0.628971i	−0.949262	−	0.314486i	$-0.898168\pi$
$179$		−	8.41506i		−	0.628971i	0.949262	−	0.314486i	$-0.101832\pi$
$180$		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
$193$	−18.7045			−1.34638			−0.673188	−	0.739471i	$-0.735076\pi$
$193$	−18.7045			−1.34638			−0.673188	+	0.739471i	$0.735076\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	24.7681i		−	1.76466i	−0.470634	−	0.882329i	$-0.655975\pi$
$197$		−	24.7681i		−	1.76466i	0.470634	−	0.882329i	$-0.344025\pi$
$198$		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
$207$	13.9032	−	3.70151i	0.966339	−	0.257273i
$208$		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
$213$	−1.09880	+	1.42961i	−0.0752884	+	0.0979553i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−12.9674	−	9.96675i	−0.876257	−	0.673490i
$220$		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
$225$	−3.85909	−	14.4951i	−0.257273	−	0.966339i
$226$		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
$233$			22.6861i			1.48622i	0.669171	+	0.743108i	$0.266649\pi$
$233$			22.6861i			1.48622i	−0.669171	+	0.743108i	$0.733351\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.3731i			1.70594i	0.521963	+	0.852968i	$0.325200\pi$
$239$			26.3731i			1.70594i	−0.521963	+	0.852968i	$0.674800\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−15.4468	−	2.09652i	−0.990915	−	0.134492i
$244$		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
$257$		−	12.1021i		−	0.754907i	−0.926028	−	0.377454i	$-0.876800\pi$
$257$		−	12.1021i		−	0.754907i	0.926028	−	0.377454i	$-0.123200\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−29.0483	+	7.73366i	−1.79804	+	0.478702i
$262$		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
$269$		−	15.3121i		−	0.933593i	−0.884365	−	0.466797i	$-0.845408\pi$
$269$		−	15.3121i		−	0.933593i	0.884365	−	0.466797i	$-0.154592\pi$
$270$		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
$277$	−2.22505			−0.133690			−0.0668451	−	0.997763i	$-0.521293\pi$
$277$	−2.22505			−0.133690			−0.0668451	+	0.997763i	$0.521293\pi$
$278$		0			0
$279$	4.50267	+	16.9124i	0.269568	+	1.01252i
$280$		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
$299$		−	33.7472i		−	1.95165i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	20.2481	−	26.3442i	1.16322	−	1.51343i
$304$		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
$311$		−	11.6250i		−	0.659196i	−0.944121	−	0.329598i	$-0.893087\pi$
$311$		−	11.6250i		−	0.659196i	0.944121	−	0.329598i	$-0.106913\pi$
$312$		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	$-0.818908\pi$
$317$		−	19.1833i		−	1.07744i	0.842484	−	0.538721i	$-0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−35.1839			−1.95165
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−36.3868			−2.00000			−1.00000		0.000796384i	$-0.999747\pi$
$331$	−36.3868			−2.00000			−1.00000		0.000796384i	$0.999747\pi$
$332$		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	$-0.917117\pi$
$347$		−	9.59166i		−	0.514907i	0.966291	−	0.257454i	$-0.0828835\pi$
$348$		0			0
$349$	−25.9220			−1.38758			−0.693788	−	0.720180i	$-0.744059\pi$
$349$	−25.9220			−1.38758			−0.693788	+	0.720180i	$0.744059\pi$
$350$		0			0
$351$	−14.0735	+	33.7472i	−0.751190	+	1.80129i
$352$		0			0
$353$		−	37.4342i		−	1.99242i	−0.0869714	−	0.996211i	$-0.527719\pi$
$353$		−	37.4342i		−	1.99242i	0.0869714	−	0.996211i	$-0.472281\pi$
$354$		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
$363$	−15.1061	−	11.6106i	−0.792866	−	0.609396i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	7.67082	−	2.04224i	0.399327	−	0.106315i
$370$		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
$377$			70.5088i			3.63139i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	30.6424	+	23.5517i	1.56986	+	1.20659i
$382$		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
$393$	22.2512	−	28.9504i	1.12243	−	1.46035i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.2986			0.818003			0.409002	−	0.912534i	$-0.365877\pi$
$397$	16.2986			0.818003			0.409002	+	0.912534i	$0.365877\pi$
$398$		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
$403$	41.0514			2.04492
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	32.7780			1.62077			0.810384	−	0.585899i	$-0.199258\pi$
$409$	32.7780			1.62077			0.810384	+	0.585899i	$0.199258\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−27.3385	−	21.0123i	−1.33877	−	1.02898i
$418$		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
$423$	−39.7370	+	10.5794i	−1.93208	+	0.514387i
$424$		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
$439$	38.7927			1.85147			0.925736	−	0.378170i	$-0.123446\pi$
$439$	38.7927			1.85147			0.925736	+	0.378170i	$0.123446\pi$
$440$		0			0
$441$	5.40273	+	20.2931i	0.257273	+	0.966339i
$442$		0			0
$443$			4.25100i			0.201971i	0.994888	+	0.100985i	$0.0321996\pi$
$443$			4.25100i			0.201971i	−0.994888	+	0.100985i	$0.967800\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			38.3667i			1.81063i	0.424736	+	0.905317i	$0.360367\pi$
$449$			38.3667i			1.81063i	−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−14.6194	−	11.2364i	−0.686879	−	0.527934i
$454$		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
$461$			19.4761i			0.907094i	0.891232	+	0.453547i	$0.149842\pi$
$461$			19.4761i			0.907094i	−0.891232	+	0.453547i	$0.850158\pi$
$462$		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
$487$	−27.1250			−1.22915			−0.614575	−	0.788858i	$-0.710672\pi$
$487$	−27.1250			−1.22915			−0.614575	+	0.788858i	$0.710672\pi$
$488$		0			0
$489$	−4.70759	−	3.61825i	−0.212885	−	0.163623i
$490$		0			0
$491$			35.8292i			1.61695i	0.588531	+	0.808475i	$0.299707\pi$
$491$			35.8292i			1.61695i	−0.588531	+	0.808475i	$0.700293\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	31.5751			1.41349			0.706747	−	0.707466i	$-0.250162\pi$
$499$	31.5751			1.41349			0.706747	+	0.707466i	$0.250162\pi$
$500$		0			0
$501$	10.1240	−	13.1721i	0.452309	−	0.588485i
$502$		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
$507$	50.1471	+	38.5430i	2.22711	+	1.71176i
$508$		0			0
$509$			44.8082i			1.98609i	0.117732	+	0.993045i	$0.462438\pi$
$509$			44.8082i			1.98609i	−0.117732	+	0.993045i	$0.537562\pi$
$510$		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
$519$	20.2481	−	26.3442i	0.888793	−	1.15638i
$520$		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
$531$	27.8064	−	7.40302i	1.20669	−	0.321264i
$532$		0			0
$533$		−	18.6194i		−	0.806494i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	8.88214	−	11.5563i	0.383293	−	0.498690i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	39.9956			1.71954			0.859772	−	0.510677i	$-0.170605\pi$
$541$	39.9956			1.71954			0.859772	+	0.510677i	$0.170605\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.0957			0.645444			0.322722	−	0.946494i	$-0.395402\pi$
$547$	15.0957			0.645444			0.322722	+	0.946494i	$0.395402\pi$
$548$		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
$577$	14.2544			0.593417			0.296708	−	0.954968i	$-0.404111\pi$
$577$	14.2544			0.593417			0.296708	+	0.954968i	$0.404111\pi$
$578$		0			0
$579$	−25.6865	−	19.7426i	−1.06750	−	0.820476i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			23.1632i			0.956046i	0.878347	+	0.478023i	$0.158646\pi$
$587$			23.1632i			0.956046i	−0.878347	+	0.478023i	$0.841354\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	26.1429	−	34.0137i	1.07538	−	1.39914i
$592$		0			0
$593$			38.3667i			1.57553i	0.615976	+	0.787765i	$0.288762\pi$
$593$			38.3667i			1.57553i	−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	9.59166i		−	0.391905i	−0.980613	−	0.195952i	$-0.937220\pi$
$599$		−	9.59166i		−	0.391905i	0.980613	−	0.195952i	$-0.0627798\pi$
$600$		0			0
$601$	−0.180813			−0.00737552			−0.00368776	−	0.999993i	$-0.501174\pi$
$601$	−0.180813			−0.00737552			−0.00368776	+	0.999993i	$0.501174\pi$
$602$		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
$611$			96.4536i			3.90209i
$612$		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
$633$	5.49314	+	4.22202i	0.218333	+	0.167810i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	49.2574			1.95165
$638$		0			0
$639$	−3.01792	+	0.803476i	−0.119387	+	0.0317850i
$640$		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
$647$			39.0392i			1.53479i	0.641175	+	0.767395i	$0.278447\pi$
$647$			39.0392i			1.53479i	−0.641175	+	0.767395i	$0.721553\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			35.3522i			1.38344i	0.722167	+	0.691719i	$0.243146\pi$
$653$			35.3522i			1.38344i	−0.722167	+	0.691719i	$0.756854\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.28800	−	27.3744i	−0.284332	−	1.06798i
$658$		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
$667$	48.0545			1.86068
$668$		0			0
$669$	−10.9863	−	8.44404i	−0.424754	−	0.326465i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−51.6633			−1.99147			−0.995737	−	0.0922433i	$-0.970596\pi$
$673$	−51.6633			−1.99147			−0.995737	+	0.0922433i	$0.970596\pi$
$674$		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
$683$		−	46.4132i		−	1.77595i	−0.459889	−	0.887977i	$-0.652111\pi$
$683$		−	46.4132i		−	1.77595i	0.459889	−	0.887977i	$-0.347889\pi$
$684$		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
$699$	−23.9453	+	31.1545i	−0.905695	+	1.17837i
$700$		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
$713$		−	27.9781i		−	1.04779i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−27.8370	+	36.2178i	−1.03959	+	1.35258i
$718$		0			0
$719$			47.9583i			1.78854i	0.447524	+	0.894272i	$0.352306\pi$
$719$			47.9583i			1.78854i	−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	50.1003i		−	1.86068i
$726$		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
$739$	−52.8662			−1.94471			−0.972357	−	0.233497i	$-0.924983\pi$
$739$	−52.8662			−1.94471			−0.972357	+	0.233497i	$0.924983\pi$
$740$		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
$761$			6.81007i			0.246865i	0.992353	+	0.123432i	$0.0393902\pi$
$761$			6.81007i			0.246865i	−0.992353	+	0.123432i	$0.960610\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	67.4944i		−	2.43708i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	12.7738	−	16.6196i	0.460038	−	0.598540i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−29.1692			−1.04779
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−48.0545	−	20.0401i	−1.71733	−	0.716175i
$784$		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
$807$	16.1620	−	21.0278i	0.568928	−	0.740214i
$808$		0			0
$809$			38.3667i			1.34890i	0.738321	+	0.674450i	$0.235619\pi$
$809$			38.3667i			1.34890i	−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	−1.38374			−0.0485898			−0.0242949	−	0.999705i	$-0.507734\pi$
$811$	−1.38374			−0.0485898			−0.0242949	+	0.999705i	$0.507734\pi$
$812$		0			0
$813$	21.9725	+	16.8881i	0.770611	+	0.592291i
$814$		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	$-0.891348\pi$
$821$		−	19.1833i		−	0.669503i	0.942306	−	0.334751i	$-0.108652\pi$
$822$		0			0
$823$	−45.6486			−1.59121			−0.795605	−	0.605815i	$-0.792847\pi$
$823$	−45.6486			−1.59121			−0.795605	+	0.605815i	$0.792847\pi$
$824$		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
$831$	−3.05562	−	2.34855i	−0.105998	−	0.0814703i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−11.6677	+	27.9781i	−0.403294	+	0.967066i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−71.4015			−2.46212
$842$		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
$857$			57.4743i			1.96328i	0.190730	+	0.981642i	$0.438914\pi$
$857$			57.4743i			1.96328i	−0.190730	+	0.981642i	$0.561086\pi$
$858$		0			0
$859$	29.5308			1.00758			0.503790	−	0.863826i	$-0.331939\pi$
$859$	29.5308			1.00758			0.503790	+	0.863826i	$0.331939\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	55.8693i		−	1.90181i	−0.309477	−	0.950907i	$-0.600154\pi$
$863$		−	55.8693i		−	1.90181i	0.309477	−	0.950907i	$-0.399846\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−23.3458	−	17.9436i	−0.792866	−	0.609396i
$868$		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
$887$		−	30.5372i		−	1.02534i	−0.858586	−	0.512669i	$-0.828657\pi$
$887$		−	30.5372i		−	1.02534i	0.858586	−	0.512669i	$-0.171343\pi$
$888$		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
$897$	35.6203	−	46.3445i	1.18933	−	1.54740i
$898$		0			0
$899$			58.4554i			1.94960i
$900$		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
$909$	55.6128	−	14.8060i	1.84456	−	0.491086i
$910$		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
$921$	−27.4657	−	21.1101i	−0.905025	−	0.695601i
$922$		0			0
$923$			7.32539i			0.241118i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	46.8903i		−	1.53842i	−0.638996	−	0.769210i	$-0.720650\pi$
$929$		−	46.8903i		−	1.53842i	0.638996	−	0.769210i	$-0.279350\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	12.2703	−	15.9645i	0.401711	−	0.522654i
$934$		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
$943$	−12.6898			−0.413237
$944$		0			0
$945$		0			0
$946$		0			0
$947$			61.1613i			1.98748i	0.111734	+	0.993738i	$0.464359\pi$
$947$			61.1613i			1.98748i	−0.111734	+	0.993738i	$0.535641\pi$
$948$		0			0
$949$	−66.4457			−2.15692
$950$		0			0
$951$	20.2481	−	26.3442i	0.656589	−	0.854268i
$952$		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
$961$	3.03372			0.0978619
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	55.2721			1.77743			0.888715	−	0.458460i	$-0.151599\pi$
$967$	55.2721			1.77743			0.888715	+	0.458460i	$0.151599\pi$
$968$		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
$975$	−48.3174	−	37.1367i	−1.54740	−	1.18933i
$976$		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
$993$	−49.9694	−	38.4064i	−1.58573	−	1.21879i
$994$		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.689.6		6
3.2	odd	2	inner	1104.2.m.a.689.5		6
4.3	odd	2		69.2.c.a.68.2	✓	6
12.11	even	2		69.2.c.a.68.5	yes	6
23.22	odd	2	CM	1104.2.m.a.689.6		6
69.68	even	2	inner	1104.2.m.a.689.5		6
92.91	even	2		69.2.c.a.68.2	✓	6
276.275	odd	2		69.2.c.a.68.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.c.a.68.2	✓	6	4.3	odd	2
69.2.c.a.68.2	✓	6	92.91	even	2
69.2.c.a.68.5	yes	6	12.11	even	2
69.2.c.a.68.5	yes	6	276.275	odd	2
1104.2.m.a.689.5		6	3.2	odd	2	inner
1104.2.m.a.689.5		6	69.68	even	2	inner
1104.2.m.a.689.6		6	1.1	even	1	trivial
1104.2.m.a.689.6		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)

\(f(q)\)	\(=\)	\(q+(1.37328 + 1.05550i) q^{3} +(0.771819 + 2.89902i) q^{9} +O(q^{10})\)	\(q+(1.37328 + 1.05550i) q^{3} +(0.771819 + 2.89902i) q^{9} +7.03677 q^{13} -4.79583i q^{23} -5.00000 q^{25} +(-2.00000 + 4.79583i) q^{27} +10.0201i q^{29} +5.83384 q^{31} +(9.66349 + 7.42735i) q^{39} -2.64601i q^{41} +13.7071i q^{47} +7.00000 q^{49} -9.59166i q^{59} +(5.06202 - 6.58604i) q^{69} +1.04102i q^{71} -9.44264 q^{73} +(-6.86642 - 5.27752i) q^{75} +(-7.80859 + 4.47503i) q^{81} +(-10.5762 + 13.7604i) q^{87} +(8.01152 + 6.15765i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 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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