LMFDB - Embedded newform 224.3.h.c.209.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(209,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("224.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.h (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:	yes
Analytic conductor:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			209.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	224.209

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} -8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{3} -8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} +25.4558 q^{13} +24.0000 q^{15} +8.48528 q^{19} -19.7990 q^{21} +10.0000 q^{23} +47.0000 q^{25} +28.2843 q^{27} -59.3970 q^{35} -72.0000 q^{39} +8.48528 q^{45} +49.0000 q^{49} -24.0000 q^{57} +76.3675 q^{59} -8.48528 q^{61} -7.00000 q^{63} -216.000 q^{65} -28.2843 q^{69} +110.000 q^{71} -132.936 q^{75} -130.000 q^{79} -71.0000 q^{81} -25.4558 q^{83} +178.191 q^{91} -72.0000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 14 * q^7 - 2 * q^9	$ 2 q + 14 q^{7} - 2 q^{9} + 48 q^{15} + 20 q^{23} + 94 q^{25} - 144 q^{39} + 98 q^{49} - 48 q^{57} - 14 q^{63} - 432 q^{65} + 220 q^{71} - 260 q^{79} - 142 q^{81} - 144 q^{95}+O(q^{100}) $ 2 * q + 14 * q^7 - 2 * q^9 + 48 * q^15 + 20 * q^23 + 94 * q^25 - 144 * q^39 + 98 * q^49 - 48 * q^57 - 14 * q^63 - 432 * q^65 + 220 * q^71 - 260 * q^79 - 142 * q^81 - 144 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.82843	−0.942809	−0.471405	−	0.881917i	$-0.656253\pi$
$3$	−2.82843	−0.942809	−0.471405	+	0.881917i	$0.656253\pi$
$4$	0	0
$5$	−8.48528	−1.69706	−0.848528	−	0.529150i	$-0.822511\pi$
$5$	−8.48528	−1.69706	−0.848528	+	0.529150i	$0.822511\pi$
$6$	0	0
$7$	7.00000	1.00000
$7$	7.00000	1.00000
$8$	0	0
$9$	−1.00000	−0.111111
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	25.4558	1.95814	0.979071	−	0.203519i	$-0.0652380\pi$
$13$	25.4558	1.95814	0.979071	+	0.203519i	$0.0652380\pi$
$14$	0	0
$15$	24.0000	1.60000
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	8.48528	0.446594	0.223297	−	0.974750i	$-0.428318\pi$
$19$	8.48528	0.446594	0.223297	+	0.974750i	$0.428318\pi$
$20$	0	0
$21$	−19.7990	−0.942809
$22$	0	0
$23$	10.0000	0.434783	0.217391	−	0.976085i	$-0.430245\pi$
$23$	10.0000	0.434783	0.217391	+	0.976085i	$0.430245\pi$
$24$	0	0
$25$	47.0000	1.88000
$26$	0	0
$27$	28.2843	1.04757
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−59.3970	−1.69706
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−72.0000	−1.84615
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	8.48528	0.188562
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−24.0000	−0.421053
$58$	0	0
$59$	76.3675	1.29436	0.647182	−	0.762335i	$-0.275947\pi$
$59$	76.3675	1.29436	0.647182	+	0.762335i	$0.275947\pi$
$60$	0	0
$61$	−8.48528	−0.139103	−0.0695515	−	0.997578i	$-0.522157\pi$
$61$	−8.48528	−0.139103	−0.0695515	+	0.997578i	$0.522157\pi$
$62$	0	0
$63$	−7.00000	−0.111111
$64$	0	0
$65$	−216.000	−3.32308
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−28.2843	−0.409917
$70$	0	0
$71$	110.000	1.54930	0.774648	−	0.632393i	$-0.217927\pi$
$71$	110.000	1.54930	0.774648	+	0.632393i	$0.217927\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−132.936	−1.77248
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−130.000	−1.64557	−0.822785	−	0.568353i	$-0.807581\pi$
$79$	−130.000	−1.64557	−0.822785	+	0.568353i	$0.807581\pi$
$80$	0	0
$81$	−71.0000	−0.876543
$82$	0	0
$83$	−25.4558	−0.306697	−0.153348	−	0.988172i	$-0.549006\pi$
$83$	−25.4558	−0.306697	−0.153348	+	0.988172i	$0.549006\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	178.191	1.95814
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−72.0000	−0.757895
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	161.220	1.59624	0.798121	−	0.602498i	$-0.205828\pi$
$101$	161.220	1.59624	0.798121	+	0.602498i	$0.205828\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	168.000	1.60000
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−26.0000	−0.230088	−0.115044	−	0.993360i	$-0.536701\pi$
$113$	−26.0000	−0.230088	−0.115044	+	0.993360i	$0.536701\pi$
$114$	0	0
$115$	−84.8528	−0.737851
$116$	0	0
$117$	−25.4558	−0.217571
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−186.676	−1.49341
$126$	0	0
$127$	250.000	1.96850	0.984252	−	0.176771i	$-0.0565653\pi$
$127$	250.000	1.96850	0.984252	+	0.176771i	$0.0565653\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	246.073	1.87842	0.939211	−	0.343342i	$-0.111559\pi$
$131$	246.073	1.87842	0.939211	+	0.343342i	$0.111559\pi$
$132$	0	0
$133$	59.3970	0.446594
$134$	0	0
$135$	−240.000	−1.77778
$136$	0	0
$137$	50.0000	0.364964	0.182482	−	0.983209i	$-0.441587\pi$
$137$	50.0000	0.364964	0.182482	+	0.983209i	$0.441587\pi$
$138$	0	0
$139$	−263.044	−1.89240	−0.946200	−	0.323581i	$-0.895113\pi$
$139$	−263.044	−1.89240	−0.946200	+	0.323581i	$0.895113\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−138.593	−0.942809
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	202.000	1.33775	0.668874	−	0.743376i	$-0.266776\pi$
$151$	202.000	1.33775	0.668874	+	0.743376i	$0.266776\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−313.955	−1.99972	−0.999858	−	0.0168519i	$-0.994636\pi$
$157$	−313.955	−1.99972	−0.999858	+	0.0168519i	$0.994636\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	70.0000	0.434783
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	479.000	2.83432
$170$	0	0
$171$	−8.48528	−0.0496215
$172$	0	0
$173$	229.103	1.32429	0.662146	−	0.749375i	$-0.269646\pi$
$173$	229.103	1.32429	0.662146	+	0.749375i	$0.269646\pi$
$174$	0	0
$175$	329.000	1.88000
$176$	0	0
$177$	−216.000	−1.22034
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	263.044	1.45328	0.726640	−	0.687018i	$-0.241081\pi$
$181$	263.044	1.45328	0.726640	+	0.687018i	$0.241081\pi$
$182$	0	0
$183$	24.0000	0.131148
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	197.990	1.04757
$190$	0	0
$191$	−130.000	−0.680628	−0.340314	−	0.940312i	$-0.610533\pi$
$191$	−130.000	−0.680628	−0.340314	+	0.940312i	$0.610533\pi$
$192$	0	0
$193$	−314.000	−1.62694	−0.813472	−	0.581605i	$-0.802425\pi$
$193$	−314.000	−1.62694	−0.813472	+	0.581605i	$0.802425\pi$
$194$	0	0
$195$	610.940	3.13303
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	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
$207$	−10.0000	−0.0483092
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−311.127	−1.46069
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−47.0000	−0.208889
$226$	0	0
$227$	−398.808	−1.75686	−0.878432	−	0.477867i	$-0.841410\pi$
$227$	−398.808	−1.75686	−0.878432	+	0.477867i	$0.841410\pi$
$228$	0	0
$229$	−246.073	−1.07456	−0.537278	−	0.843405i	$-0.680547\pi$
$229$	−246.073	−1.07456	−0.537278	+	0.843405i	$0.680547\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−430.000	−1.84549	−0.922747	−	0.385407i	$-0.874061\pi$
$233$	−430.000	−1.84549	−0.922747	+	0.385407i	$0.874061\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	367.696	1.55146
$238$	0	0
$239$	−422.000	−1.76569	−0.882845	−	0.469664i	$-0.844375\pi$
$239$	−422.000	−1.76569	−0.882845	+	0.469664i	$0.844375\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	−53.7401	−0.221153
$244$	0	0
$245$	−415.779	−1.69706
$246$	0	0
$247$	216.000	0.874494
$248$	0	0
$249$	72.0000	0.289157
$250$	0	0
$251$	−500.632	−1.99455	−0.997274	−	0.0737859i	$-0.976492\pi$
$251$	−500.632	−1.99455	−0.997274	+	0.0737859i	$0.976492\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−274.000	−1.04183	−0.520913	−	0.853610i	$-0.674408\pi$
$263$	−274.000	−1.04183	−0.520913	+	0.853610i	$0.674408\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	161.220	0.599332	0.299666	−	0.954044i	$-0.403125\pi$
$269$	161.220	0.599332	0.299666	+	0.954044i	$0.403125\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	−504.000	−1.84615
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	338.000	1.20285	0.601423	−	0.798930i	$-0.294600\pi$
$281$	338.000	1.20285	0.601423	+	0.798930i	$0.294600\pi$
$282$	0	0
$283$	313.955	1.10938	0.554692	−	0.832056i	$-0.312836\pi$
$283$	313.955	1.10938	0.554692	+	0.832056i	$0.312836\pi$
$284$	0	0
$285$	203.647	0.714550
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−449.720	−1.53488	−0.767440	−	0.641121i	$-0.778470\pi$
$293$	−449.720	−1.53488	−0.767440	+	0.641121i	$0.778470\pi$
$294$	0	0
$295$	−648.000	−2.19661
$296$	0	0
$297$	0	0
$298$	0	0
$299$	254.558	0.851366
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−456.000	−1.50495
$304$	0	0
$305$	72.0000	0.236066
$306$	0	0
$307$	449.720	1.46489	0.732443	−	0.680829i	$-0.238380\pi$
$307$	449.720	1.46489	0.732443	+	0.680829i	$0.238380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	59.3970	0.188562
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1196.42	3.68131
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.0000	−0.0771513	−0.0385757	−	0.999256i	$-0.512282\pi$
$337$	−26.0000	−0.0771513	−0.0385757	+	0.999256i	$0.512282\pi$
$338$	0	0
$339$	73.5391	0.216930
$340$	0	0
$341$	0	0
$342$	0	0
$343$	343.000	1.00000
$344$	0	0
$345$	240.000	0.695652
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−687.308	−1.96936	−0.984682	−	0.174362i	$-0.944214\pi$
$349$	−687.308	−1.96936	−0.984682	+	0.174362i	$0.944214\pi$
$350$	0	0
$351$	720.000	2.05128
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−933.381	−2.62924
$356$	0	0
$357$	0	0
$358$	0	0
$359$	682.000	1.89972	0.949861	−	0.312673i	$-0.101225\pi$
$359$	682.000	1.89972	0.949861	+	0.312673i	$0.101225\pi$
$360$	0	0
$361$	−289.000	−0.800554
$362$	0	0
$363$	−342.240	−0.942809
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	528.000	1.40800
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−707.107	−1.85592
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−696.000	−1.77099
$394$	0	0
$395$	1103.09	2.79262
$396$	0	0
$397$	−483.661	−1.21829	−0.609145	−	0.793059i	$-0.708487\pi$
$397$	−483.661	−1.21829	−0.609145	+	0.793059i	$0.708487\pi$
$398$	0	0
$399$	−168.000	−0.421053
$400$	0	0
$401$	550.000	1.37157	0.685786	−	0.727804i	$-0.259459\pi$
$401$	550.000	1.37157	0.685786	+	0.727804i	$0.259459\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	602.455	1.48754
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−141.421	−0.344091
$412$	0	0
$413$	534.573	1.29436
$414$	0	0
$415$	216.000	0.520482
$416$	0	0
$417$	744.000	1.78417
$418$	0	0
$419$	−500.632	−1.19482	−0.597412	−	0.801934i	$-0.703804\pi$
$419$	−500.632	−1.19482	−0.597412	+	0.801934i	$0.703804\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−59.3970	−0.139103
$428$	0	0
$429$	0	0
$430$	0	0
$431$	538.000	1.24826	0.624130	−	0.781321i	$-0.285454\pi$
$431$	538.000	1.24826	0.624130	+	0.781321i	$0.285454\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	84.8528	0.194171
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−49.0000	−0.111111
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.00000	0.00445434	0.00222717	−	0.999998i	$-0.499291\pi$
$449$	2.00000	0.00445434	0.00222717	+	0.999998i	$0.499291\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−571.342	−1.26124
$454$	0	0
$455$	−1512.00	−3.32308
$456$	0	0
$457$	886.000	1.93873	0.969365	−	0.245623i	$-0.0789925\pi$
$457$	886.000	1.93873	0.969365	+	0.245623i	$0.0789925\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	263.044	0.570594	0.285297	−	0.958439i	$-0.407908\pi$
$461$	263.044	0.570594	0.285297	+	0.958439i	$0.407908\pi$
$462$	0	0
$463$	−226.000	−0.488121	−0.244060	−	0.969760i	$-0.578480\pi$
$463$	−226.000	−0.488121	−0.244060	+	0.969760i	$0.578480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	483.661	1.03568	0.517838	−	0.855478i	$-0.326737\pi$
$467$	483.661	1.03568	0.517838	+	0.855478i	$0.326737\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	888.000	1.88535
$472$	0	0
$473$	0	0
$474$	0	0
$475$	398.808	0.839596
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−197.990	−0.409917
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−470.000	−0.965092	−0.482546	−	0.875871i	$-0.660288\pi$
$487$	−470.000	−0.965092	−0.482546	+	0.875871i	$0.660288\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	770.000	1.54930
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1368.00	−2.70891
$506$	0	0
$507$	−1354.82	−2.67222
$508$	0	0
$509$	941.866	1.85042	0.925212	−	0.379450i	$-0.123887\pi$
$509$	941.866	1.85042	0.925212	+	0.379450i	$0.123887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	240.000	0.467836
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−648.000	−1.24855
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	958.837	1.83334	0.916670	−	0.399645i	$-0.130867\pi$
$523$	958.837	1.83334	0.916670	+	0.399645i	$0.130867\pi$
$524$	0	0
$525$	−930.553	−1.77248
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−429.000	−0.810964
$530$	0	0
$531$	−76.3675	−0.143818
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−744.000	−1.37017
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	8.48528	0.0154559
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−910.000	−1.64557
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−398.808	−0.708363	−0.354181	−	0.935177i	$-0.615240\pi$
$563$	−398.808	−0.708363	−0.354181	+	0.935177i	$0.615240\pi$
$564$	0	0
$565$	220.617	0.390473
$566$	0	0
$567$	−497.000	−0.876543
$568$	0	0
$569$	−1130.00	−1.98594	−0.992970	−	0.118365i	$-0.962235\pi$
$569$	−1130.00	−1.98594	−0.992970	+	0.118365i	$0.962235\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	367.696	0.641702
$574$	0	0
$575$	470.000	0.817391
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	888.126	1.53390
$580$	0	0
$581$	−178.191	−0.306697
$582$	0	0
$583$	0	0
$584$	0	0
$585$	216.000	0.369231
$586$	0	0
$587$	1162.48	1.98038	0.990190	−	0.139725i	$-0.0446217\pi$
$587$	1162.48	1.98038	0.990190	+	0.139725i	$0.0446217\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1070.00	1.78631	0.893155	−	0.449748i	$-0.148486\pi$
$599$	1070.00	1.78631	0.893155	+	0.449748i	$0.148486\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1026.72	−1.69706
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1034.00	−1.67585	−0.837925	−	0.545785i	$-0.816232\pi$
$617$	−1034.00	−1.67585	−0.837925	+	0.545785i	$0.816232\pi$
$618$	0	0
$619$	−1111.57	−1.79575	−0.897877	−	0.440246i	$-0.854891\pi$
$619$	−1111.57	−1.79575	−0.897877	+	0.440246i	$0.854891\pi$
$620$	0	0
$621$	282.843	0.455463
$622$	0	0
$623$	0	0
$624$	0	0
$625$	409.000	0.654400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	110.000	0.174326	0.0871632	−	0.996194i	$-0.472220\pi$
$631$	110.000	0.174326	0.0871632	+	0.996194i	$0.472220\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2121.32	−3.34066
$636$	0	0
$637$	1247.34	1.95814
$638$	0	0
$639$	−110.000	−0.172144
$640$	0	0
$641$	1030.00	1.60686	0.803432	−	0.595396i	$-0.203005\pi$
$641$	1030.00	1.60686	0.803432	+	0.595396i	$0.203005\pi$
$642$	0	0
$643$	−738.219	−1.14809	−0.574043	−	0.818825i	$-0.694626\pi$
$643$	−738.219	−1.14809	−0.574043	+	0.818825i	$0.694626\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−2088.00	−3.18779
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1264.31	−1.91272	−0.956359	−	0.292193i	$-0.905615\pi$
$661$	−1264.31	−1.91272	−0.956359	+	0.292193i	$0.905615\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−504.000	−0.757895
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−670.000	−0.995542	−0.497771	−	0.867308i	$-0.665848\pi$
$673$	−670.000	−0.995542	−0.497771	+	0.867308i	$0.665848\pi$
$674$	0	0
$675$	1329.36	1.96942
$676$	0	0
$677$	−483.661	−0.714418	−0.357209	−	0.934024i	$-0.616272\pi$
$677$	−483.661	−0.714418	−0.357209	+	0.934024i	$0.616272\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1128.00	1.65639
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−424.264	−0.619364
$686$	0	0
$687$	696.000	1.01310
$688$	0	0
$689$	0	0
$690$	0	0
$691$	246.073	0.356112	0.178056	−	0.984020i	$-0.443019\pi$
$691$	246.073	0.356112	0.178056	+	0.984020i	$0.443019\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2232.00	3.21151
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1216.22	1.73995
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1128.54	1.59624
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	130.000	0.182841
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1193.60	1.66471
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	791.000	1.08505
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1417.04	1.93321	0.966604	−	0.256273i	$-0.0824947\pi$
$733$	1417.04	1.93321	0.966604	+	0.256273i	$0.0824947\pi$
$734$	0	0
$735$	1176.00	1.60000
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	−610.940	−0.824481
$742$	0	0
$743$	−1430.00	−1.92463	−0.962315	−	0.271937i	$-0.912336\pi$
$743$	−1430.00	−1.92463	−0.962315	+	0.271937i	$0.912336\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	25.4558	0.0340774
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−998.000	−1.32889	−0.664447	−	0.747335i	$-0.731333\pi$
$751$	−998.000	−1.32889	−0.664447	+	0.747335i	$0.731333\pi$
$752$	0	0
$753$	1416.00	1.88048
$754$	0	0
$755$	−1714.03	−2.27023
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1944.00	2.53455
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	873.984	1.13064	0.565320	−	0.824872i	$-0.308753\pi$
$773$	873.984	1.13064	0.565320	+	0.824872i	$0.308753\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2664.00	3.39363
$786$	0	0
$787$	1501.89	1.90838	0.954190	−	0.299202i	$-0.0967204\pi$
$787$	1501.89	1.90838	0.954190	+	0.299202i	$0.0967204\pi$
$788$	0	0
$789$	774.989	0.982242
$790$	0	0
$791$	−182.000	−0.230088
$792$	0	0
$793$	−216.000	−0.272383
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−449.720	−0.564266	−0.282133	−	0.959375i	$-0.591042\pi$
$797$	−449.720	−0.564266	−0.282133	+	0.959375i	$0.591042\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−593.970	−0.737851
$806$	0	0
$807$	−456.000	−0.565056
$808$	0	0
$809$	−650.000	−0.803461	−0.401731	−	0.915758i	$-0.631591\pi$
$809$	−650.000	−0.803461	−0.401731	+	0.915758i	$0.631591\pi$
$810$	0	0
$811$	−1450.98	−1.78913	−0.894564	−	0.446939i	$-0.852514\pi$
$811$	−1450.98	−1.78913	−0.894564	+	0.446939i	$0.852514\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−178.191	−0.217571
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−946.000	−1.14945	−0.574727	−	0.818345i	$-0.694892\pi$
$823$	−946.000	−1.14945	−0.574727	+	0.818345i	$0.694892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−76.3675	−0.0921201	−0.0460600	−	0.998939i	$-0.514667\pi$
$829$	−76.3675	−0.0921201	−0.0460600	+	0.998939i	$0.514667\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	−956.008	−1.13406
$844$	0	0
$845$	−4064.45	−4.81000
$846$	0	0
$847$	847.000	1.00000
$848$	0	0
$849$	−888.000	−1.04594
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−449.720	−0.527221	−0.263611	−	0.964629i	$-0.584913\pi$
$853$	−449.720	−0.527221	−0.263611	+	0.964629i	$0.584913\pi$
$854$	0	0
$855$	72.0000	0.0842105
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1196.42	1.39281	0.696406	−	0.717648i	$-0.254782\pi$
$859$	1196.42	1.39281	0.696406	+	0.717648i	$0.254782\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1474.00	−1.70800	−0.853998	−	0.520277i	$-0.825829\pi$
$863$	−1474.00	−1.70800	−0.853998	+	0.520277i	$0.825829\pi$
$864$	0	0
$865$	−1944.00	−2.24740
$866$	0	0
$867$	−817.415	−0.942809
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1306.73	−1.49341
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	1272.00	1.44710
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	1832.82	2.07098
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	1750.00	1.96850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−720.000	−0.802676
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2232.00	−2.46630
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−161.220	−0.177360
$910$	0	0
$911$	922.000	1.01207	0.506037	−	0.862512i	$-0.331110\pi$
$911$	922.000	1.01207	0.506037	+	0.862512i	$0.331110\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−203.647	−0.222565
$916$	0	0
$917$	1722.51	1.87842
$918$	0	0
$919$	1550.00	1.68662	0.843308	−	0.537431i	$-0.180605\pi$
$919$	1550.00	1.68662	0.843308	+	0.537431i	$0.180605\pi$
$920$	0	0
$921$	−1272.00	−1.38111
$922$	0	0
$923$	2800.14	3.03374
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	415.779	0.446594
$932$	0	0
$933$	0	0
$934$	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
$941$	1111.57	1.18127	0.590633	−	0.806940i	$-0.298878\pi$
$941$	1111.57	1.18127	0.590633	+	0.806940i	$0.298878\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1680.00	−1.77778
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1010.00	1.05981	0.529906	−	0.848057i	$-0.322227\pi$
$953$	1010.00	1.05981	0.529906	+	0.848057i	$0.322227\pi$
$954$	0	0
$955$	1103.09	1.15506
$956$	0	0
$957$	0	0
$958$	0	0
$959$	350.000	0.364964
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2664.38	2.76101
$966$	0	0
$967$	−1430.00	−1.47880	−0.739400	−	0.673266i	$-0.764891\pi$
$967$	−1430.00	−1.47880	−0.739400	+	0.673266i	$0.764891\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.48528	0.00873870	0.00436935	−	0.999990i	$-0.498609\pi$
$971$	8.48528	0.00873870	0.00436935	+	0.999990i	$0.498609\pi$
$972$	0	0
$973$	−1841.31	−1.89240
$974$	0	0
$975$	−3384.00	−3.47077
$976$	0	0
$977$	−1630.00	−1.66837	−0.834186	−	0.551483i	$-0.814062\pi$
$977$	−1630.00	−1.66837	−0.834186	+	0.551483i	$0.814062\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−610.000	−0.615540	−0.307770	−	0.951461i	$-0.599583\pi$
$991$	−610.000	−0.615540	−0.307770	+	0.951461i	$0.599583\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1977.07	−1.98302	−0.991510	−	0.130032i	$-0.958492\pi$
$997$	−1977.07	−1.98302	−0.991510	+	0.130032i	$0.958492\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.h.c.209.1		2
3.2	odd	2		2016.3.l.c.433.2		2
4.3	odd	2		56.3.h.c.13.2	yes	2
7.6	odd	2	inner	224.3.h.c.209.2		2
8.3	odd	2		56.3.h.c.13.1	✓	2
8.5	even	2	inner	224.3.h.c.209.2		2
12.11	even	2		504.3.l.a.181.2		2
21.20	even	2		2016.3.l.c.433.1		2
24.5	odd	2		2016.3.l.c.433.1		2
24.11	even	2		504.3.l.a.181.1		2
28.3	even	6		392.3.j.a.117.2		4
28.11	odd	6		392.3.j.a.117.1		4
28.19	even	6		392.3.j.a.325.2		4
28.23	odd	6		392.3.j.a.325.1		4
28.27	even	2		56.3.h.c.13.1	✓	2
56.3	even	6		392.3.j.a.117.1		4
56.11	odd	6		392.3.j.a.117.2		4
56.13	odd	2	CM	224.3.h.c.209.1		2
56.19	even	6		392.3.j.a.325.1		4
56.27	even	2		56.3.h.c.13.2	yes	2
56.51	odd	6		392.3.j.a.325.2		4
84.83	odd	2		504.3.l.a.181.1		2
168.83	odd	2		504.3.l.a.181.2		2
168.125	even	2		2016.3.l.c.433.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.c.13.1	✓	2	8.3	odd	2
56.3.h.c.13.1	✓	2	28.27	even	2
56.3.h.c.13.2	yes	2	4.3	odd	2
56.3.h.c.13.2	yes	2	56.27	even	2
224.3.h.c.209.1		2	1.1	even	1	trivial
224.3.h.c.209.1		2	56.13	odd	2	CM
224.3.h.c.209.2		2	7.6	odd	2	inner
224.3.h.c.209.2		2	8.5	even	2	inner
392.3.j.a.117.1		4	28.11	odd	6
392.3.j.a.117.1		4	56.3	even	6
392.3.j.a.117.2		4	28.3	even	6
392.3.j.a.117.2		4	56.11	odd	6
392.3.j.a.325.1		4	28.23	odd	6
392.3.j.a.325.1		4	56.19	even	6
392.3.j.a.325.2		4	28.19	even	6
392.3.j.a.325.2		4	56.51	odd	6
504.3.l.a.181.1		2	24.11	even	2
504.3.l.a.181.1		2	84.83	odd	2
504.3.l.a.181.2		2	12.11	even	2
504.3.l.a.181.2		2	168.83	odd	2
2016.3.l.c.433.1		2	21.20	even	2
2016.3.l.c.433.1		2	24.5	odd	2
2016.3.l.c.433.2		2	3.2	odd	2
2016.3.l.c.433.2		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} -8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{3} -8.48528 q^{5} +7.00000 q^{7} -1.00000 q^{9} +25.4558 q^{13} +24.0000 q^{15} +8.48528 q^{19} -19.7990 q^{21} +10.0000 q^{23} +47.0000 q^{25} +28.2843 q^{27} -59.3970 q^{35} -72.0000 q^{39} +8.48528 q^{45} +49.0000 q^{49} -24.0000 q^{57} +76.3675 q^{59} -8.48528 q^{61} -7.00000 q^{63} -216.000 q^{65} -28.2843 q^{69} +110.000 q^{71} -132.936 q^{75} -130.000 q^{79} -71.0000 q^{81} -25.4558 q^{83} +178.191 q^{91} -72.0000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 14 * q^7 - 2 * q^9	\( 2 q + 14 q^{7} - 2 q^{9} + 48 q^{15} + 20 q^{23} + 94 q^{25} - 144 q^{39} + 98 q^{49} - 48 q^{57} - 14 q^{63} - 432 q^{65} + 220 q^{71} - 260 q^{79} - 142 q^{81} - 144 q^{95}+O(q^{100}) \) 2 * q + 14 * q^7 - 2 * q^9 + 48 * q^15 + 20 * q^23 + 94 * q^25 - 144 * q^39 + 98 * q^49 - 48 * q^57 - 14 * q^63 - 432 * q^65 + 220 * q^71 - 260 * q^79 - 142 * q^81 - 144 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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