LMFDB - Embedded newform 672.2.k.b.545.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,2,Mod(545,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [8,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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			545.5
Root			$1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	672.545
Dual form			672.2.k.b.545.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -4.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} -3.58258i q^{11} -7.58258i q^{15} +2.55040 q^{17} -5.29150i q^{19} -4.58258 q^{21} +0.417424i q^{23} +14.1652 q^{25} -5.19615i q^{27} -3.46410i q^{31} +6.20520 q^{33} -11.5826i q^{35} -9.16515 q^{37} -11.3060 q^{41} +13.1334 q^{45} -7.00000 q^{49} +4.41742i q^{51} +15.6838i q^{55} +9.16515 q^{57} -7.93725i q^{63} -0.723000 q^{69} -15.5826i q^{71} +24.5348i q^{75} +9.47860 q^{77} +9.00000 q^{81} -11.1652 q^{85} -14.9608 q^{89} +6.00000 q^{93} +23.1652i q^{95} +10.7477i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	−4.37780	−1.95781	−0.978906	−	0.204310i	$-0.934505\pi$
$5$	−4.37780	−1.95781	−0.978906	+	0.204310i	$0.934505\pi$
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 3.58258i	− 1.08019i	−0.841605	−	0.540094i	$-0.818389\pi$
$11$	− 3.58258i	− 1.08019i	0.841605	−	0.540094i	$-0.181611\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 7.58258i	− 1.95781i
$16$	0	0
$17$	2.55040	0.618563	0.309282	−	0.950971i	$-0.399911\pi$
$17$	2.55040	0.618563	0.309282	+	0.950971i	$0.399911\pi$
$18$	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
$23$	0.417424i	0.0870390i	0.999053	+	0.0435195i	$0.0138571\pi$
$23$	0.417424i	0.0870390i	−0.999053	+	0.0435195i	$0.986143\pi$
$24$	0	0
$25$	14.1652	2.83303
$26$	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	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	6.20520	1.08019
$34$	0	0
$35$	− 11.5826i	− 1.95781i
$36$	0	0
$37$	−9.16515	−1.50674	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	−9.16515	−1.50674	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.3060	−1.76570	−0.882851	−	0.469654i	$-0.844379\pi$
$41$	−11.3060	−1.76570	−0.882851	+	0.469654i	$0.844379\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	13.1334	1.95781
$46$	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
$51$	4.41742i	0.618563i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	15.6838i	2.11480i
$56$	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
$69$	−0.723000	−0.0870390
$70$	0	0
$71$	− 15.5826i	− 1.84931i	−0.380804	−	0.924656i	$-0.624353\pi$
$71$	− 15.5826i	− 1.84931i	0.380804	−	0.924656i	$-0.375647\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	24.5348i	2.83303i
$76$	0	0
$77$	9.47860	1.08019
$78$	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
$85$	−11.1652	−1.21103
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.9608	−1.58584	−0.792921	−	0.609324i	$-0.791441\pi$
$89$	−14.9608	−1.58584	−0.792921	+	0.609324i	$0.791441\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	23.1652i	2.37669i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	10.7477i	1.08019i
$100$	0	0
$101$	−8.03260	−0.799274	−0.399637	−	0.916673i	$-0.630864\pi$
$101$	−8.03260	−0.799274	−0.399637	+	0.916673i	$0.630864\pi$
$102$	0	0
$103$	− 17.3205i	− 1.70664i	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	− 17.3205i	− 1.70664i	0.521387	−	0.853320i	$-0.325415\pi$
$104$	0	0
$105$	20.0616	1.95781
$106$	0	0
$107$	18.7477i	1.81241i	0.422837	+	0.906206i	$0.361034\pi$
$107$	18.7477i	1.81241i	−0.422837	+	0.906206i	$0.638966\pi$
$108$	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
$115$	− 1.82740i	− 0.170406i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.74773i	0.618563i
$120$	0	0
$121$	−1.83485	−0.166804
$122$	0	0
$123$	− 19.5826i	− 1.76570i
$124$	0	0
$125$	−40.1232	−3.58873
$126$	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
$135$	22.7477i	1.95781i
$136$	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	$-0.854719\pi$
$139$	− 10.3923i	− 0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$	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
$153$	−7.65120	−0.618563
$154$	0	0
$155$	15.1652i	1.21809i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.10440	−0.0870390
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	−27.1652	−2.11480
$166$	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
$173$	16.7882	1.27638	0.638192	−	0.769877i	$-0.279683\pi$
$173$	16.7882	1.27638	0.638192	+	0.769877i	$0.279683\pi$
$174$	0	0
$175$	37.4775i	2.83303i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	26.7477i	1.99922i	0.0279439	+	0.999609i	$0.491104\pi$
$179$	26.7477i	1.99922i	−0.0279439	+	0.999609i	$0.508896\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	40.1232	2.94992
$186$	0	0
$187$	− 9.13701i	− 0.668164i
$188$	0	0
$189$	13.7477	1.00000
$190$	0	0
$191$	− 23.5826i	− 1.70638i	−0.521604	−	0.853188i	$-0.674666\pi$
$191$	− 23.5826i	− 1.70638i	0.521604	−	0.853188i	$-0.325334\pi$
$192$	0	0
$193$	−27.4955	−1.97917	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	−27.4955	−1.97917	−0.989583	+	0.143963i	$0.954015\pi$
$194$	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
$205$	49.4955	3.45691
$206$	0	0
$207$	− 1.25227i	− 0.0870390i
$208$	0	0
$209$	−18.9572	−1.31130
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	26.9898	1.84931
$214$	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
$225$	−42.4955	−2.83303
$226$	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
$231$	16.4174i	1.08019i
$232$	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
$239$	30.7477i	1.98891i	0.105186	+	0.994453i	$0.466456\pi$
$239$	30.7477i	1.98891i	−0.105186	+	0.994453i	$0.533544\pi$
$240$	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
$245$	30.6446	1.95781
$246$	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
$253$	1.49545	0.0940184
$254$	0	0
$255$	− 19.3386i	− 1.21103i
$256$	0	0
$257$	−25.1624	−1.56959	−0.784794	−	0.619757i	$-0.787231\pi$
$257$	−25.1624	−1.56959	−0.784794	+	0.619757i	$0.787231\pi$
$258$	0	0
$259$	− 24.2487i	− 1.50674i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 21.9129i	− 1.35121i	−0.737266	−	0.675603i	$-0.763883\pi$
$263$	− 21.9129i	− 1.35121i	0.737266	−	0.675603i	$-0.236117\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 25.9129i	− 1.58584i
$268$	0	0
$269$	5.82380	0.355083	0.177542	−	0.984113i	$-0.443186\pi$
$269$	5.82380	0.355083	0.177542	+	0.984113i	$0.443186\pi$
$270$	0	0
$271$	− 31.1769i	− 1.89386i	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	− 31.1769i	− 1.89386i	0.321436	−	0.946931i	$-0.395835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 50.7477i	− 3.06020i
$276$	0	0
$277$	−9.16515	−0.550681	−0.275340	−	0.961347i	$-0.588791\pi$
$277$	−9.16515	−0.550681	−0.275340	+	0.961347i	$0.588791\pi$
$278$	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
$285$	−40.1232	−2.37669
$286$	0	0
$287$	− 29.9129i	− 1.76570i
$288$	0	0
$289$	−10.4955	−0.617380
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.93180	0.171278	0.0856388	−	0.996326i	$-0.472707\pi$
$293$	2.93180	0.171278	0.0856388	+	0.996326i	$0.472707\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−18.6156	−1.08019
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 13.9129i	− 0.799274i
$304$	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
$315$	34.7477i	1.95781i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−32.4720	−1.81241
$322$	0	0
$323$	− 13.4955i	− 0.750907i
$324$	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
$341$	−12.4104	−0.672061
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	0	0
$345$	3.16515	0.170406
$346$	0	0
$347$	− 9.91288i	− 0.532151i	−0.963952	−	0.266076i	$-0.914273\pi$
$347$	− 9.91288i	− 0.532151i	0.963952	−	0.266076i	$-0.0857271\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	37.5728	1.99980	0.999900	−	0.0141657i	$-0.00450924\pi$
$353$	37.5728	1.99980	0.999900	+	0.0141657i	$0.00450924\pi$
$354$	0	0
$355$	68.2174i	3.62061i
$356$	0	0
$357$	−11.6874	−0.618563
$358$	0	0
$359$	− 9.25227i	− 0.488316i	−0.969735	−	0.244158i	$-0.921488\pi$
$359$	− 9.25227i	− 0.488316i	0.969735	−	0.244158i	$-0.0785116\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	− 3.17805i	− 0.166804i
$364$	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
$369$	33.9180	1.76570
$370$	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
$375$	− 69.4955i	− 3.58873i
$376$	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
$385$	−41.4955	−2.11480
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	1.06460i	0.0538391i
$392$	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
$405$	−39.4002	−1.95781
$406$	0	0
$407$	32.8348i	1.62756i
$408$	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	$-0.571696\pi$
$421$	−9.16515	−0.446682	−0.223341	+	0.974740i	$0.571696\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	36.1268	1.75241
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.4174i	1.17615i	0.808808	+	0.588073i	$0.200113\pi$
$431$	24.4174i	1.17615i	−0.808808	+	0.588073i	$0.799887\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.20880	0.105661
$438$	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
$443$	− 41.9129i	− 1.99134i	−0.0929532	−	0.995670i	$-0.529631\pi$
$443$	− 41.9129i	− 1.99134i	0.0929532	−	0.995670i	$-0.470369\pi$
$444$	0	0
$445$	65.4955	3.10478
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	40.5046i	1.90729i
$452$	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
$459$	− 13.2523i	− 0.618563i
$460$	0	0
$461$	20.4430	0.952126	0.476063	−	0.879411i	$-0.342063\pi$
$461$	20.4430	0.952126	0.476063	+	0.879411i	$0.342063\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−26.2668	−1.21809
$466$	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
$475$	− 74.9549i	− 3.43917i
$476$	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
$483$	− 1.91288i	− 0.0870390i
$484$	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
$491$	− 35.5826i	− 1.60582i	−0.596101	−	0.802910i	$-0.703284\pi$
$491$	− 35.5826i	− 1.60582i	0.596101	−	0.802910i	$-0.296716\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 47.0514i	− 2.11480i
$496$	0	0
$497$	41.2276	1.84931
$498$	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
$505$	35.1652	1.56483
$506$	0	0
$507$	22.5167i	1.00000i
$508$	0	0
$509$	44.5010	1.97247	0.986237	−	0.165340i	$-0.0528723\pi$
$509$	44.5010	1.97247	0.986237	+	0.165340i	$0.0528723\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−27.4955	−1.21395
$514$	0	0
$515$	75.8258i	3.34128i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	29.0780i	1.27638i
$520$	0	0
$521$	−42.6736	−1.86957	−0.934783	−	0.355220i	$-0.884406\pi$
$521$	−42.6736	−1.86957	−0.934783	+	0.355220i	$0.884406\pi$
$522$	0	0
$523$	17.3205i	0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$	17.3205i	0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	−64.9129	−2.83303
$526$	0	0
$527$	− 8.83485i	− 0.384852i
$528$	0	0
$529$	22.8258	0.992424
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 82.0738i	− 3.54836i
$536$	0	0
$537$	−46.3284	−1.99922
$538$	0	0
$539$	25.0780i	1.08019i
$540$	0	0
$541$	−45.8258	−1.97020	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	−45.8258	−1.97020	−0.985102	+	0.171973i	$0.944986\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	43.7780	1.87524
$546$	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
$555$	69.4955i	2.94992i
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	15.8258	0.668164
$562$	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
$573$	40.8462	1.70638
$574$	0	0
$575$	5.91288i	0.246584i
$576$	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
$593$	−36.1268	−1.48355	−0.741775	−	0.670648i	$-0.766016\pi$
$593$	−36.1268	−1.48355	−0.741775	+	0.670648i	$0.766016\pi$
$594$	0	0
$595$	− 29.5402i	− 1.21103i
$596$	0	0
$597$	−9.16515	−0.375105
$598$	0	0
$599$	45.0780i	1.84184i	0.389754	+	0.920919i	$0.372560\pi$
$599$	45.0780i	1.84184i	−0.389754	+	0.920919i	$0.627440\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.03260	0.326572
$606$	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
$615$	85.7286i	3.45691i
$616$	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	$0.360145\pi$
$619$	45.0333i	1.81004i	−0.425367	+	0.905021i	$0.639855\pi$
$620$	0	0
$621$	2.16900	0.0870390
$622$	0	0
$623$	− 39.5826i	− 1.58584i
$624$	0	0
$625$	104.826	4.19303
$626$	0	0
$627$	− 32.8348i	− 1.31130i
$628$	0	0
$629$	−23.3748	−0.932015
$630$	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
$639$	46.7477i	1.84931i
$640$	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
$659$	49.0780i	1.91181i	0.293678	+	0.955904i	$0.405121\pi$
$659$	49.0780i	1.91181i	−0.293678	+	0.955904i	$0.594879\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−61.2892	−2.37669
$666$	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	$-0.677784\pi$
$673$	−27.4955	−1.05987	−0.529936	+	0.848038i	$0.677784\pi$
$674$	0	0
$675$	− 73.6043i	− 2.83303i
$676$	0	0
$677$	6.58660	0.253144	0.126572	−	0.991957i	$-0.459603\pi$
$677$	6.58660	0.253144	0.126572	+	0.991957i	$0.459603\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.08712i	0.232917i	0.993196	+	0.116459i	$0.0371542\pi$
$683$	6.08712i	0.232917i	−0.993196	+	0.116459i	$0.962846\pi$
$684$	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	$0.202062\pi$
$691$	31.1769i	1.18603i	−0.805193	+	0.593013i	$0.797938\pi$
$692$	0	0
$693$	−28.4358	−1.08019
$694$	0	0
$695$	45.4955i	1.72574i
$696$	0	0
$697$	−28.8348	−1.09220
$698$	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
$707$	− 21.2523i	− 0.799274i
$708$	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
$713$	1.44600	0.0541531
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−53.2566	−1.98891
$718$	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	$0.0617292\pi$
$727$	10.3923i	0.385429i	−0.981255	+	0.192715i	$0.938271\pi$
$728$	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
$735$	53.0780i	1.95781i
$736$	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
$743$	− 53.9129i	− 1.97787i	−0.148344	−	0.988936i	$-0.547394\pi$
$743$	− 53.9129i	− 1.97787i	0.148344	−	0.988936i	$-0.452606\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−49.6018	−1.81241
$750$	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
$759$	2.59020i	0.0940184i
$760$	0	0
$761$	55.0840	1.99679	0.998397	−	0.0565953i	$-0.0180245\pi$
$761$	55.0840	1.99679	0.998397	+	0.0565953i	$0.0180245\pi$
$762$	0	0
$763$	− 26.4575i	− 0.957826i
$764$	0	0
$765$	33.4955	1.21103
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	− 43.5826i	− 1.56959i
$772$	0	0
$773$	−15.3422	−0.551821	−0.275910	−	0.961183i	$-0.588979\pi$
$773$	−15.3422	−0.551821	−0.275910	+	0.961183i	$0.588979\pi$
$774$	0	0
$775$	− 49.0695i	− 1.76263i
$776$	0	0
$777$	42.0000	1.50674
$778$	0	0
$779$	59.8258i	2.14348i
$780$	0	0
$781$	−55.8258	−1.99760
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 51.9615i	− 1.85223i	−0.377243	−	0.926114i	$-0.623128\pi$
$787$	− 51.9615i	− 1.85223i	0.377243	−	0.926114i	$-0.376872\pi$
$788$	0	0
$789$	37.9542	1.35121
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.1914	1.31739	0.658694	−	0.752411i	$-0.271109\pi$
$797$	37.1914	1.31739	0.658694	+	0.752411i	$0.271109\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	44.8824	1.58584
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.83485	0.170406
$806$	0	0
$807$	10.0871i	0.355083i
$808$	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	$-0.766712\pi$
$811$	− 38.1051i	− 1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	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
$825$	87.8976	3.06020
$826$	0	0
$827$	57.0780i	1.98480i	0.123064	+	0.992399i	$0.460728\pi$
$827$	57.0780i	1.98480i	−0.123064	+	0.992399i	$0.539272\pi$
$828$	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
$833$	−17.8528	−0.618563
$834$	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
$845$	−56.9114	−1.95781
$846$	0	0
$847$	− 4.85455i	− 0.166804i
$848$	0	0
$849$	45.8258	1.57274
$850$	0	0
$851$	− 3.82576i	− 0.131145i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	− 69.4955i	− 2.37669i
$856$	0	0
$857$	−0.341599	−0.0116688	−0.00583441	−	0.999983i	$-0.501857\pi$
$857$	−0.341599	−0.0116688	−0.00583441	+	0.999983i	$0.501857\pi$
$858$	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
$861$	51.8106	1.76570
$862$	0	0
$863$	− 36.2432i	− 1.23373i	−0.787068	−	0.616866i	$-0.788402\pi$
$863$	− 36.2432i	− 1.23373i	0.787068	−	0.616866i	$-0.211598\pi$
$864$	0	0
$865$	−73.4955	−2.49892
$866$	0	0
$867$	− 18.1787i	− 0.617380i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 106.156i	− 3.58873i
$876$	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
$879$	5.07803i	0.171278i
$880$	0	0
$881$	26.6084	0.896460	0.448230	−	0.893918i	$-0.352055\pi$
$881$	26.6084	0.896460	0.448230	+	0.893918i	$0.352055\pi$
$882$	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
$891$	− 32.2432i	− 1.08019i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 117.096i	− 3.91410i
$896$	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
$909$	24.0978	0.799274
$910$	0	0
$911$	− 52.2432i	− 1.73089i	−0.501001	−	0.865447i	$-0.667035\pi$
$911$	− 52.2432i	− 1.73089i	0.501001	−	0.865447i	$-0.332965\pi$
$912$	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
$925$	−129.826	−4.26864
$926$	0	0
$927$	51.9615i	1.70664i
$928$	0	0
$929$	−8.41400	−0.276055	−0.138027	−	0.990428i	$-0.544076\pi$
$929$	−8.41400	−0.276055	−0.138027	+	0.990428i	$0.544076\pi$
$930$	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
$941$	−45.9470	−1.49783	−0.748915	−	0.662666i	$-0.769425\pi$
$941$	−45.9470	−1.49783	−0.748915	+	0.662666i	$0.769425\pi$
$942$	0	0
$943$	− 4.71940i	− 0.153685i
$944$	0	0
$945$	−60.1848	−1.95781
$946$	0	0
$947$	14.0871i	0.457770i	0.973453	+	0.228885i	$0.0735080\pi$
$947$	14.0871i	0.457770i	−0.973453	+	0.228885i	$0.926492\pi$
$948$	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
$955$	103.240i	3.34076i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	− 56.2432i	− 1.81241i
$964$	0	0
$965$	120.370	3.87484
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	23.3748	0.750907
$970$	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
$979$	53.5982i	1.71301i
$980$	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
$995$	− 23.1652i	− 0.734385i
$996$	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.545.5	yes	8
3.2	odd	2	inner	672.2.k.b.545.8	yes	8
4.3	odd	2	inner	672.2.k.b.545.1	✓	8
7.6	odd	2	inner	672.2.k.b.545.4	yes	8
8.3	odd	2		1344.2.k.g.1217.8		8
8.5	even	2		1344.2.k.g.1217.4		8
12.11	even	2	inner	672.2.k.b.545.4	yes	8
21.20	even	2	inner	672.2.k.b.545.1	✓	8
24.5	odd	2		1344.2.k.g.1217.1		8
24.11	even	2		1344.2.k.g.1217.5		8
28.27	even	2	inner	672.2.k.b.545.8	yes	8
56.13	odd	2		1344.2.k.g.1217.5		8
56.27	even	2		1344.2.k.g.1217.1		8
84.83	odd	2	CM	672.2.k.b.545.5	yes	8
168.83	odd	2		1344.2.k.g.1217.4		8
168.125	even	2		1344.2.k.g.1217.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.k.b.545.1	✓	8	4.3	odd	2	inner
672.2.k.b.545.1	✓	8	21.20	even	2	inner
672.2.k.b.545.4	yes	8	7.6	odd	2	inner
672.2.k.b.545.4	yes	8	12.11	even	2	inner
672.2.k.b.545.5	yes	8	1.1	even	1	trivial
672.2.k.b.545.5	yes	8	84.83	odd	2	CM
672.2.k.b.545.8	yes	8	3.2	odd	2	inner
672.2.k.b.545.8	yes	8	28.27	even	2	inner
1344.2.k.g.1217.1		8	24.5	odd	2
1344.2.k.g.1217.1		8	56.27	even	2
1344.2.k.g.1217.4		8	8.5	even	2
1344.2.k.g.1217.4		8	168.83	odd	2
1344.2.k.g.1217.5		8	24.11	even	2
1344.2.k.g.1217.5		8	56.13	odd	2
1344.2.k.g.1217.8		8	8.3	odd	2
1344.2.k.g.1217.8		8	168.125	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -4.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} -3.58258i q^{11} -7.58258i q^{15} +2.55040 q^{17} -5.29150i q^{19} -4.58258 q^{21} +0.417424i q^{23} +14.1652 q^{25} -5.19615i q^{27} -3.46410i q^{31} +6.20520 q^{33} -11.5826i q^{35} -9.16515 q^{37} -11.3060 q^{41} +13.1334 q^{45} -7.00000 q^{49} +4.41742i q^{51} +15.6838i q^{55} +9.16515 q^{57} -7.93725i q^{63} -0.723000 q^{69} -15.5826i q^{71} +24.5348i q^{75} +9.47860 q^{77} +9.00000 q^{81} -11.1652 q^{85} -14.9608 q^{89} +6.00000 q^{93} +23.1652i q^{95} +10.7477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81} - 16 q^{85} + 48 q^{93}+O(q^{100}) \) 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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