LMFDB - Embedded newform 672.2.k.b.545.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(545,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, 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("672.545");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			545.3
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	672.545
Dual form			672.2.k.b.545.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +0.913701 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +0.913701 q^{5} +2.64575i q^{7} -3.00000 q^{9} -5.58258i q^{11} -1.58258i q^{15} +7.84190 q^{17} -5.29150i q^{19} +4.58258 q^{21} -9.58258i q^{23} -4.16515 q^{25} +5.19615i q^{27} +3.46410i q^{31} -9.66930 q^{33} +2.41742i q^{35} +9.16515 q^{37} -6.01450 q^{41} -2.74110 q^{45} -7.00000 q^{49} -13.5826i q^{51} -5.10080i q^{55} -9.16515 q^{57} -7.93725i q^{63} -16.5975 q^{69} +6.41742i q^{71} +7.21425i q^{75} +14.7701 q^{77} +9.00000 q^{81} +7.16515 q^{85} +11.4967 q^{89} +6.00000 q^{93} -4.83485i q^{95} +16.7477i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81} - 16 q^{85} + 48 q^{93}+O(q^{100}) $ 8 * q - 24 * q^9 + 40 * q^25 - 56 * q^49 + 72 * q^81 - 16 * q^85 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	− 1.73205i	− 1.00000i
$3$	− 1.73205i	− 1.00000i
$4$	0	0
$5$	0.913701	0.408619	0.204310	−	0.978906i	$-0.434505\pi$
$5$	0.913701	0.408619	0.204310	+	0.978906i	$0.434505\pi$
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 5.58258i	− 1.68321i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	− 5.58258i	− 1.68321i	0.540094	−	0.841605i	$-0.318389\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 1.58258i	− 0.408619i
$16$	0	0
$17$	7.84190	1.90194	0.950971	−	0.309282i	$-0.100089\pi$
$17$	7.84190	1.90194	0.950971	+	0.309282i	$0.100089\pi$
$18$	0	0
$19$	− 5.29150i	− 1.21395i	−0.794719	−	0.606977i	$-0.792382\pi$
$19$	− 5.29150i	− 1.21395i	0.794719	−	0.606977i	$-0.207618\pi$
$20$	0	0
$21$	4.58258	1.00000
$22$	0	0
$23$	− 9.58258i	− 1.99811i	−0.0435195	−	0.999053i	$-0.513857\pi$
$23$	− 9.58258i	− 1.99811i	0.0435195	−	0.999053i	$-0.486143\pi$
$24$	0	0
$25$	−4.16515	−0.833030
$26$	0	0
$27$	5.19615i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	3.46410i	0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$	3.46410i	0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	0	0
$33$	−9.66930	−1.68321
$34$	0	0
$35$	2.41742i	0.408619i
$36$	0	0
$37$	9.16515	1.50674	0.753371	−	0.657596i	$-0.228427\pi$
$37$	9.16515	1.50674	0.753371	+	0.657596i	$0.228427\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.01450	−0.939308	−0.469654	−	0.882851i	$-0.655621\pi$
$41$	−6.01450	−0.939308	−0.469654	+	0.882851i	$0.655621\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−2.74110	−0.408619
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 13.5826i	− 1.90194i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 5.10080i	− 0.687792i
$56$	0	0
$57$	−9.16515	−1.21395
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 7.93725i	− 1.00000i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−16.5975	−1.99811
$70$	0	0
$71$	6.41742i	0.761608i	0.924656	+	0.380804i	$0.124353\pi$
$71$	6.41742i	0.761608i	−0.924656	+	0.380804i	$0.875647\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	7.21425i	0.833030i
$76$	0	0
$77$	14.7701	1.68321
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	7.16515	0.777170
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.4967	1.21865	0.609324	−	0.792921i	$-0.291441\pi$
$89$	11.4967	1.21865	0.609324	+	0.792921i	$0.291441\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	− 4.83485i	− 0.496045i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	16.7477i	1.68321i
$100$	0	0
$101$	18.4249	1.83335	0.916673	−	0.399637i	$-0.130864\pi$
$101$	18.4249	1.83335	0.916673	+	0.399637i	$0.130864\pi$
$102$	0	0
$103$	17.3205i	1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$	17.3205i	1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$	0	0
$105$	4.18710	0.408619
$106$	0	0
$107$	8.74773i	0.845675i	0.906206	+	0.422837i	$0.138966\pi$
$107$	8.74773i	0.845675i	−0.906206	+	0.422837i	$0.861034\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	− 15.8745i	− 1.50674i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	− 8.75560i	− 0.816464i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.7477i	1.90194i
$120$	0	0
$121$	−20.1652	−1.83320
$122$	0	0
$123$	10.4174i	0.939308i
$124$	0	0
$125$	−8.37420	−0.749012
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	14.0000	1.21395
$134$	0	0
$135$	4.74773i	0.408619i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	10.3923i	0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$	10.3923i	0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.1244i	1.00000i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−23.5257	−1.90194
$154$	0	0
$155$	3.16515i	0.254231i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	25.3531	1.99811
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	−8.83485	−0.687792
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	15.8745i	1.21395i
$172$	0	0
$173$	−20.2523	−1.53975	−0.769877	−	0.638192i	$-0.779683\pi$
$173$	−20.2523	−1.53975	−0.769877	+	0.638192i	$0.779683\pi$
$174$	0	0
$175$	− 11.0200i	− 0.833030i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.747727i	0.0558877i	0.999609	+	0.0279439i	$0.00889597\pi$
$179$	0.747727i	0.0558877i	−0.999609	+	0.0279439i	$0.991104\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.37420	0.615684
$186$	0	0
$187$	− 43.7780i	− 3.20137i
$188$	0	0
$189$	−13.7477	−1.00000
$190$	0	0
$191$	14.4174i	1.04321i	0.853188	+	0.521604i	$0.174666\pi$
$191$	14.4174i	1.04321i	−0.853188	+	0.521604i	$0.825334\pi$
$192$	0	0
$193$	27.4955	1.97917	0.989583	−	0.143963i	$-0.0459847\pi$
$193$	27.4955	1.97917	0.989583	+	0.143963i	$0.0459847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	5.29150i	0.375105i	0.982255	+	0.187552i	$0.0600554\pi$
$199$	5.29150i	0.375105i	−0.982255	+	0.187552i	$0.939945\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5.49545	−0.383819
$206$	0	0
$207$	28.7477i	1.99811i
$208$	0	0
$209$	−29.5402	−2.04334
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	11.1153	0.761608
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.16515	−0.622171
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.4575i	1.77173i	0.463947	+	0.885863i	$0.346433\pi$
$223$	26.4575i	1.77173i	−0.463947	+	0.885863i	$0.653567\pi$
$224$	0	0
$225$	12.4955	0.833030
$226$	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
$231$	− 25.5826i	− 1.68321i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 3.25227i	− 0.210372i	−0.994453	−	0.105186i	$-0.966456\pi$
$239$	− 3.25227i	− 0.210372i	0.994453	−	0.105186i	$-0.0335438\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 15.5885i	− 1.00000i
$244$	0	0
$245$	−6.39590	−0.408619
$246$	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
$253$	−53.4955	−3.36323
$254$	0	0
$255$	− 12.4104i	− 0.777170i
$256$	0	0
$257$	−19.8709	−1.23951	−0.619757	−	0.784794i	$-0.712769\pi$
$257$	−19.8709	−1.23951	−0.619757	+	0.784794i	$0.712769\pi$
$258$	0	0
$259$	24.2487i	1.50674i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 23.9129i	− 1.47453i	−0.675603	−	0.737266i	$-0.736117\pi$
$263$	− 23.9129i	− 1.47453i	0.675603	−	0.737266i	$-0.263883\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 19.9129i	− 1.21865i
$268$	0	0
$269$	32.2813	1.96823	0.984113	−	0.177542i	$-0.0568145\pi$
$269$	32.2813	1.96823	0.984113	+	0.177542i	$0.0568145\pi$
$270$	0	0
$271$	31.1769i	1.89386i	0.321436	+	0.946931i	$0.395835\pi$
$271$	31.1769i	1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.2523i	1.40216i
$276$	0	0
$277$	9.16515	0.550681	0.275340	−	0.961347i	$-0.411209\pi$
$277$	9.16515	0.550681	0.275340	+	0.961347i	$0.411209\pi$
$278$	0	0
$279$	− 10.3923i	− 0.622171i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 26.4575i	− 1.57274i	−0.617758	−	0.786368i	$-0.711959\pi$
$283$	− 26.4575i	− 1.57274i	0.617758	−	0.786368i	$-0.288041\pi$
$284$	0	0
$285$	−8.37420	−0.496045
$286$	0	0
$287$	− 15.9129i	− 0.939308i
$288$	0	0
$289$	44.4955	2.61738
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−34.1087	−1.99265	−0.996326	−	0.0856388i	$-0.972707\pi$
$293$	−34.1087	−1.99265	−0.996326	+	0.0856388i	$0.972707\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	29.0079	1.68321
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 31.9129i	− 1.83335i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.29150i	− 0.302002i	−0.988534	−	0.151001i	$-0.951750\pi$
$307$	− 5.29150i	− 0.302002i	0.988534	−	0.151001i	$-0.0482497\pi$
$308$	0	0
$309$	30.0000	1.70664
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	− 7.25227i	− 0.408619i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	15.1515	0.845675
$322$	0	0
$323$	− 41.4955i	− 2.30887i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.3205i	0.957826i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−27.4955	−1.50674
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.3386	1.04724
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	0	0
$345$	−15.1652	−0.816464
$346$	0	0
$347$	− 35.9129i	− 1.92790i	−0.266076	−	0.963952i	$-0.585727\pi$
$347$	− 35.9129i	− 1.92790i	0.266076	−	0.963952i	$-0.414273\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.532300	0.0283315	0.0141657	−	0.999900i	$-0.495491\pi$
$353$	0.532300	0.0283315	0.0141657	+	0.999900i	$0.495491\pi$
$354$	0	0
$355$	5.86360i	0.311208i
$356$	0	0
$357$	35.9361	1.90194
$358$	0	0
$359$	36.7477i	1.93947i	0.244158	+	0.969735i	$0.421488\pi$
$359$	36.7477i	1.93947i	−0.244158	+	0.969735i	$0.578512\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	34.9271i	1.83320i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.4575i	1.38107i	0.723299	+	0.690535i	$0.242625\pi$
$367$	26.4575i	1.38107i	−0.723299	+	0.690535i	$0.757375\pi$
$368$	0	0
$369$	18.0435	0.939308
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	14.5045i	0.749012i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	13.4955	0.687792
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	− 75.1456i	− 3.80028i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	− 24.2487i	− 1.21395i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.22330	0.408619
$406$	0	0
$407$	− 51.1652i	− 2.53616i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.0000	0.881464
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	9.16515	0.446682	0.223341	−	0.974740i	$-0.428304\pi$
$421$	9.16515	0.446682	0.223341	+	0.974740i	$0.428304\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−32.6627	−1.58437
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 33.5826i	− 1.61762i	−0.588073	−	0.808808i	$-0.700113\pi$
$431$	− 33.5826i	− 1.61762i	0.588073	−	0.808808i	$-0.299887\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−50.7062	−2.42561
$438$	0	0
$439$	5.29150i	0.252550i	0.991995	+	0.126275i	$0.0403021\pi$
$439$	5.29150i	0.252550i	−0.991995	+	0.126275i	$0.959698\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	− 3.91288i	− 0.185906i	−0.995670	−	0.0929532i	$-0.970369\pi$
$443$	− 3.91288i	− 0.185906i	0.995670	−	0.0929532i	$-0.0296307\pi$
$444$	0	0
$445$	10.5045	0.497963
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	33.5764i	1.58105i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	40.7477i	1.90194i
$460$	0	0
$461$	−37.7635	−1.75882	−0.879411	−	0.476063i	$-0.842063\pi$
$461$	−37.7635	−1.75882	−0.879411	+	0.476063i	$0.842063\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	5.48220	0.254231
$466$	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
$475$	22.0399i	1.01126i
$476$	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
$483$	− 43.9129i	− 1.99811i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.4174i	1.19220i	0.802910	+	0.596101i	$0.203284\pi$
$491$	26.4174i	1.19220i	−0.802910	+	0.596101i	$0.796716\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	15.3024i	0.687792i
$496$	0	0
$497$	−16.9789	−0.761608
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$505$	16.8348	0.749141
$506$	0	0
$507$	− 22.5167i	− 1.00000i
$508$	0	0
$509$	7.46050	0.330681	0.165340	−	0.986237i	$-0.447128\pi$
$509$	7.46050	0.330681	0.165340	+	0.986237i	$0.447128\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	27.4955	1.21395
$514$	0	0
$515$	15.8258i	0.697366i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	35.0780i	1.53975i
$520$	0	0
$521$	−16.2161	−0.710441	−0.355220	−	0.934783i	$-0.615594\pi$
$521$	−16.2161	−0.710441	−0.355220	+	0.934783i	$0.615594\pi$
$522$	0	0
$523$	− 17.3205i	− 0.757373i	−0.925525	−	0.378686i	$-0.876376\pi$
$523$	− 17.3205i	− 0.757373i	0.925525	−	0.378686i	$-0.123624\pi$
$524$	0	0
$525$	−19.0871	−0.833030
$526$	0	0
$527$	27.1652i	1.18333i
$528$	0	0
$529$	−68.8258	−2.99242
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	7.99280i	0.345559i
$536$	0	0
$537$	1.29510	0.0558877
$538$	0	0
$539$	39.0780i	1.68321i
$540$	0	0
$541$	45.8258	1.97020	0.985102	−	0.171973i	$-0.0550143\pi$
$541$	45.8258	1.97020	0.985102	+	0.171973i	$0.0550143\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.13701	−0.391386
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 14.5045i	− 0.615684i
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−75.8258	−3.20137
$562$	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
$567$	23.8118i	1.00000i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	24.9717	1.04321
$574$	0	0
$575$	39.9129i	1.66448i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	− 47.6235i	− 1.97917i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	18.3303	0.755287
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.6627	1.34130	0.670648	−	0.741775i	$-0.266016\pi$
$593$	32.6627	1.34130	0.670648	+	0.741775i	$0.266016\pi$
$594$	0	0
$595$	18.9572i	0.777170i
$596$	0	0
$597$	9.16515	0.375105
$598$	0	0
$599$	19.0780i	0.779507i	0.920919	+	0.389754i	$0.127440\pi$
$599$	19.0780i	0.779507i	−0.920919	+	0.389754i	$0.872560\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18.4249	−0.749079
$606$	0	0
$607$	26.4575i	1.07388i	0.843621	+	0.536939i	$0.180419\pi$
$607$	26.4575i	1.07388i	−0.843621	+	0.536939i	$0.819581\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	0	0
$615$	9.51841i	0.383819i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 45.0333i	− 1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$	− 45.0333i	− 1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$	0	0
$621$	49.7925	1.99811
$622$	0	0
$623$	30.4174i	1.21865i
$624$	0	0
$625$	13.1742	0.526970
$626$	0	0
$627$	51.1652i	2.04334i
$628$	0	0
$629$	71.8722	2.86573
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 19.2523i	− 0.761608i
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	37.0405i	1.46074i	0.683054	+	0.730368i	$0.260651\pi$
$643$	37.0405i	1.46074i	−0.683054	+	0.730368i	$0.739349\pi$
$644$	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
$651$	15.8745i	0.622171i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0780i	0.587357i	0.955904	+	0.293678i	$0.0948794\pi$
$659$	15.0780i	0.587357i	−0.955904	+	0.293678i	$0.905121\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.7918	0.496045
$666$	0	0
$667$	0	0
$668$	0	0
$669$	45.8258	1.77173
$670$	0	0
$671$	0	0
$672$	0	0
$673$	27.4955	1.05987	0.529936	−	0.848038i	$-0.322216\pi$
$673$	27.4955	1.05987	0.529936	+	0.848038i	$0.322216\pi$
$674$	0	0
$675$	− 21.6428i	− 0.833030i
$676$	0	0
$677$	−51.6199	−1.98391	−0.991957	−	0.126572i	$-0.959603\pi$
$677$	−51.6199	−1.98391	−0.991957	+	0.126572i	$0.959603\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 51.9129i	− 1.98639i	−0.116459	−	0.993196i	$-0.537154\pi$
$683$	− 51.9129i	− 1.98639i	0.116459	−	0.993196i	$-0.462846\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 31.1769i	− 1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	− 31.1769i	− 1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$	0	0
$693$	−44.3103	−1.68321
$694$	0	0
$695$	9.49545i	0.360183i
$696$	0	0
$697$	−47.1652	−1.78651
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 48.4974i	− 1.82911i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.7477i	1.83335i
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	33.1950	1.24316
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.63310	−0.210372
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−45.8258	−1.70664
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 10.3923i	− 0.385429i	−0.981255	−	0.192715i	$-0.938271\pi$
$727$	− 10.3923i	− 0.385429i	0.981255	−	0.192715i	$-0.0617292\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	11.0780i	0.408619i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.08712i	0.296688i	0.988936	+	0.148344i	$0.0473942\pi$
$743$	8.08712i	0.296688i	−0.988936	+	0.148344i	$0.952606\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−23.1443	−0.845675
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	92.6568i	3.36323i
$760$	0	0
$761$	−3.12250	−0.113191	−0.0565953	−	0.998397i	$-0.518024\pi$
$761$	−3.12250	−0.113191	−0.0565953	+	0.998397i	$0.518024\pi$
$762$	0	0
$763$	− 26.4575i	− 0.957826i
$764$	0	0
$765$	−21.4955	−0.777170
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	34.4174i	1.23951i
$772$	0	0
$773$	53.4473	1.92237	0.961183	−	0.275910i	$-0.0889793\pi$
$773$	53.4473	1.92237	0.961183	+	0.275910i	$0.0889793\pi$
$774$	0	0
$775$	− 14.4285i	− 0.518287i
$776$	0	0
$777$	42.0000	1.50674
$778$	0	0
$779$	31.8258i	1.14028i
$780$	0	0
$781$	35.8258	1.28195
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	51.9615i	1.85223i	0.377243	+	0.926114i	$0.376872\pi$
$787$	51.9615i	1.85223i	−0.377243	+	0.926114i	$0.623128\pi$
$788$	0	0
$789$	−41.4183	−1.47453
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.4829	1.50482	0.752411	−	0.658694i	$-0.228891\pi$
$797$	42.4829	1.50482	0.752411	+	0.658694i	$0.228891\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−34.4901	−1.21865
$802$	0	0
$803$	0	0
$804$	0	0
$805$	23.1652	0.816464
$806$	0	0
$807$	− 55.9129i	− 1.96823i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	38.1051i	1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$	38.1051i	1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	0	0
$813$	54.0000	1.89386
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	40.2741	1.40216
$826$	0	0
$827$	7.07803i	0.246127i	0.992399	+	0.123064i	$0.0392719\pi$
$827$	7.07803i	0.246127i	−0.992399	+	0.123064i	$0.960728\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	− 15.8745i	− 0.550681i
$832$	0	0
$833$	−54.8933	−1.90194
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−18.0000	−0.622171
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.8781	0.408619
$846$	0	0
$847$	− 53.3520i	− 1.83320i
$848$	0	0
$849$	−45.8258	−1.57274
$850$	0	0
$851$	− 87.8258i	− 3.01063i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	14.5045i	0.496045i
$856$	0	0
$857$	−58.5481	−1.99997	−0.999983	−	0.00583441i	$-0.998143\pi$
$857$	−58.5481	−1.99997	−0.999983	+	0.00583441i	$0.998143\pi$
$858$	0	0
$859$	58.2065i	1.98598i	0.118194	+	0.992991i	$0.462290\pi$
$859$	58.2065i	1.98598i	−0.118194	+	0.992991i	$0.537710\pi$
$860$	0	0
$861$	−27.5619	−0.939308
$862$	0	0
$863$	− 46.2432i	− 1.57414i	−0.616866	−	0.787068i	$-0.711598\pi$
$863$	− 46.2432i	− 1.57414i	0.616866	−	0.787068i	$-0.288402\pi$
$864$	0	0
$865$	−18.5045	−0.629173
$866$	0	0
$867$	− 77.0684i	− 2.61738i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 22.1561i	− 0.749012i
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	59.0780i	1.99265i
$880$	0	0
$881$	53.0659	1.78784	0.893918	−	0.448230i	$-0.147945\pi$
$881$	53.0659	1.78784	0.893918	+	0.448230i	$0.147945\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$889$	0	0
$890$	0	0
$891$	− 50.2432i	− 1.68321i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0.683199i	0.0228368i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−55.2747	−1.83335
$910$	0	0
$911$	− 30.2432i	− 1.00200i	−0.865447	−	0.501001i	$-0.832965\pi$
$911$	− 30.2432i	− 1.00200i	0.865447	−	0.501001i	$-0.167035\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−9.16515	−0.302002
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−38.1742	−1.25516
$926$	0	0
$927$	− 51.9615i	− 1.70664i
$928$	0	0
$929$	60.3755	1.98086	0.990428	−	0.138027i	$-0.0440762\pi$
$929$	60.3755	1.98086	0.990428	+	0.138027i	$0.0440762\pi$
$930$	0	0
$931$	37.0405i	1.21395i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 40.0000i	− 1.30814i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40.6555	−1.32533	−0.662666	−	0.748915i	$-0.730575\pi$
$941$	−40.6555	−1.32533	−0.662666	+	0.748915i	$0.730575\pi$
$942$	0	0
$943$	57.6344i	1.87684i
$944$	0	0
$945$	−12.5613	−0.408619
$946$	0	0
$947$	− 59.9129i	− 1.94691i	−0.228885	−	0.973453i	$-0.573508\pi$
$947$	− 59.9129i	− 1.94691i	0.228885	−	0.973453i	$-0.426492\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	13.1732i	0.426275i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	− 26.2432i	− 0.845675i
$964$	0	0
$965$	25.1226	0.808725
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−71.8722	−2.30887
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−27.4955	−0.881464
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	− 64.1812i	− 2.05124i
$980$	0	0
$981$	30.0000	0.957826
$982$	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
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.83485i	0.153275i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	47.6235i	1.50674i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.k.b.545.3	yes	8
3.2	odd	2	inner	672.2.k.b.545.2	✓	8
4.3	odd	2	inner	672.2.k.b.545.7	yes	8
7.6	odd	2	inner	672.2.k.b.545.6	yes	8
8.3	odd	2		1344.2.k.g.1217.2		8
8.5	even	2		1344.2.k.g.1217.6		8
12.11	even	2	inner	672.2.k.b.545.6	yes	8
21.20	even	2	inner	672.2.k.b.545.7	yes	8
24.5	odd	2		1344.2.k.g.1217.7		8
24.11	even	2		1344.2.k.g.1217.3		8
28.27	even	2	inner	672.2.k.b.545.2	✓	8
56.13	odd	2		1344.2.k.g.1217.3		8
56.27	even	2		1344.2.k.g.1217.7		8
84.83	odd	2	CM	672.2.k.b.545.3	yes	8
168.83	odd	2		1344.2.k.g.1217.6		8
168.125	even	2		1344.2.k.g.1217.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.k.b.545.2	✓	8	3.2	odd	2	inner
672.2.k.b.545.2	✓	8	28.27	even	2	inner
672.2.k.b.545.3	yes	8	1.1	even	1	trivial
672.2.k.b.545.3	yes	8	84.83	odd	2	CM
672.2.k.b.545.6	yes	8	7.6	odd	2	inner
672.2.k.b.545.6	yes	8	12.11	even	2	inner
672.2.k.b.545.7	yes	8	4.3	odd	2	inner
672.2.k.b.545.7	yes	8	21.20	even	2	inner
1344.2.k.g.1217.2		8	8.3	odd	2
1344.2.k.g.1217.2		8	168.125	even	2
1344.2.k.g.1217.3		8	24.11	even	2
1344.2.k.g.1217.3		8	56.13	odd	2
1344.2.k.g.1217.6		8	8.5	even	2
1344.2.k.g.1217.6		8	168.83	odd	2
1344.2.k.g.1217.7		8	24.5	odd	2
1344.2.k.g.1217.7		8	56.27	even	2

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +0.913701 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +0.913701 q^{5} +2.64575i q^{7} -3.00000 q^{9} -5.58258i q^{11} -1.58258i q^{15} +7.84190 q^{17} -5.29150i q^{19} +4.58258 q^{21} -9.58258i q^{23} -4.16515 q^{25} +5.19615i q^{27} +3.46410i q^{31} -9.66930 q^{33} +2.41742i q^{35} +9.16515 q^{37} -6.01450 q^{41} -2.74110 q^{45} -7.00000 q^{49} -13.5826i q^{51} -5.10080i q^{55} -9.16515 q^{57} -7.93725i q^{63} -16.5975 q^{69} +6.41742i q^{71} +7.21425i q^{75} +14.7701 q^{77} +9.00000 q^{81} +7.16515 q^{85} +11.4967 q^{89} +6.00000 q^{93} -4.83485i q^{95} +16.7477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81} - 16 q^{85} + 48 q^{93}+O(q^{100}) \) 8 * q - 24 * q^9 + 40 * q^25 - 56 * q^49 + 72 * q^81 - 16 * q^85 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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