LMFDB - Embedded newform 224.3.h.b.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(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
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			$-2.64575$ 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)

$f(q)$	$=$	$q-5.29150 q^{3} -5.29150 q^{5} -7.00000 q^{7} +19.0000 q^{9} +O(q^{10})$	$q-5.29150 q^{3} -5.29150 q^{5} -7.00000 q^{7} +19.0000 q^{9} -5.29150 q^{13} +28.0000 q^{15} +37.0405 q^{19} +37.0405 q^{21} +10.0000 q^{23} +3.00000 q^{25} -52.9150 q^{27} +37.0405 q^{35} +28.0000 q^{39} -100.539 q^{45} +49.0000 q^{49} -196.000 q^{57} -89.9555 q^{59} +121.705 q^{61} -133.000 q^{63} +28.0000 q^{65} -52.9150 q^{69} -110.000 q^{71} -15.8745 q^{75} +130.000 q^{79} +109.000 q^{81} +164.037 q^{83} +37.0405 q^{91} -196.000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7} + 38 q^{9}+O(q^{10}) $ 2 * q - 14 * q^7 + 38 * q^9	$ 2 q - 14 q^{7} + 38 q^{9} + 56 q^{15} + 20 q^{23} + 6 q^{25} + 56 q^{39} + 98 q^{49} - 392 q^{57} - 266 q^{63} + 56 q^{65} - 220 q^{71} + 260 q^{79} + 218 q^{81} - 392 q^{95}+O(q^{100}) $ 2 * q - 14 * q^7 + 38 * q^9 + 56 * q^15 + 20 * q^23 + 6 * q^25 + 56 * q^39 + 98 * q^49 - 392 * q^57 - 266 * q^63 + 56 * q^65 - 220 * q^71 + 260 * q^79 + 218 * q^81 - 392 * 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
$3$	−5.29150	−1.76383	−0.881917	−	0.471405i	$-0.843747\pi$
$3$	−5.29150	−1.76383	−0.881917	+	0.471405i	$0.843747\pi$
$4$	0	0
$5$	−5.29150	−1.05830	−0.529150	−	0.848528i	$-0.677489\pi$
$5$	−5.29150	−1.05830	−0.529150	+	0.848528i	$0.677489\pi$
$6$	0	0
$7$	−7.00000	−1.00000
$7$	−7.00000	−1.00000
$8$	0	0
$9$	19.0000	2.11111
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−5.29150	−0.407039	−0.203519	−	0.979071i	$-0.565238\pi$
$13$	−5.29150	−0.407039	−0.203519	+	0.979071i	$0.565238\pi$
$14$	0	0
$15$	28.0000	1.86667
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	37.0405	1.94950	0.974750	−	0.223297i	$-0.0716819\pi$
$19$	37.0405	1.94950	0.974750	+	0.223297i	$0.0716819\pi$
$20$	0	0
$21$	37.0405	1.76383
$22$	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
$25$	3.00000	0.120000
$26$	0	0
$27$	−52.9150	−1.95982
$28$	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
$35$	37.0405	1.05830
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	28.0000	0.717949
$40$	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
$45$	−100.539	−2.23419
$46$	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
$57$	−196.000	−3.43860
$58$	0	0
$59$	−89.9555	−1.52467	−0.762335	−	0.647182i	$-0.775947\pi$
$59$	−89.9555	−1.52467	−0.762335	+	0.647182i	$0.775947\pi$
$60$	0	0
$61$	121.705	1.99516	0.997578	−	0.0695515i	$-0.0221568\pi$
$61$	121.705	1.99516	0.997578	+	0.0695515i	$0.0221568\pi$
$62$	0	0
$63$	−133.000	−2.11111
$64$	0	0
$65$	28.0000	0.430769
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−52.9150	−0.766884
$70$	0	0
$71$	−110.000	−1.54930	−0.774648	−	0.632393i	$-0.782073\pi$
$71$	−110.000	−1.54930	−0.774648	+	0.632393i	$0.782073\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−15.8745	−0.211660
$76$	0	0
$77$	0	0
$78$	0	0
$79$	130.000	1.64557	0.822785	−	0.568353i	$-0.192419\pi$
$79$	130.000	1.64557	0.822785	+	0.568353i	$0.192419\pi$
$80$	0	0
$81$	109.000	1.34568
$82$	0	0
$83$	164.037	1.97634	0.988172	−	0.153348i	$-0.0490057\pi$
$83$	164.037	1.97634	0.988172	+	0.153348i	$0.0490057\pi$
$84$	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
$91$	37.0405	0.407039
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−196.000	−2.06316
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	121.705	1.20500	0.602498	−	0.798121i	$-0.294172\pi$
$101$	121.705	1.20500	0.602498	+	0.798121i	$0.294172\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−196.000	−1.86667
$106$	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	$-0.463299\pi$
$113$	26.0000	0.230088	0.115044	+	0.993360i	$0.463299\pi$
$114$	0	0
$115$	−52.9150	−0.460131
$116$	0	0
$117$	−100.539	−0.859304
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	116.413	0.931304
$126$	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
$131$	−89.9555	−0.686684	−0.343342	−	0.939211i	$-0.611559\pi$
$131$	−89.9555	−0.686684	−0.343342	+	0.939211i	$0.611559\pi$
$132$	0	0
$133$	−259.284	−1.94950
$134$	0	0
$135$	280.000	2.07407
$136$	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
$139$	−89.9555	−0.647162	−0.323581	−	0.946200i	$-0.604887\pi$
$139$	−89.9555	−0.647162	−0.323581	+	0.946200i	$0.604887\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−259.284	−1.76383
$148$	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
$157$	−5.29150	−0.0337038	−0.0168519	−	0.999858i	$-0.505364\pi$
$157$	−5.29150	−0.0337038	−0.0168519	+	0.999858i	$0.505364\pi$
$158$	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
$169$	−141.000	−0.834320
$170$	0	0
$171$	703.770	4.11561
$172$	0	0
$173$	−259.284	−1.49875	−0.749375	−	0.662146i	$-0.769646\pi$
$173$	−259.284	−1.49875	−0.749375	+	0.662146i	$0.769646\pi$
$174$	0	0
$175$	−21.0000	−0.120000
$176$	0	0
$177$	476.000	2.68927
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	248.701	1.37404	0.687018	−	0.726640i	$-0.258919\pi$
$181$	248.701	1.37404	0.687018	+	0.726640i	$0.258919\pi$
$182$	0	0
$183$	−644.000	−3.51913
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	370.405	1.95982
$190$	0	0
$191$	130.000	0.680628	0.340314	−	0.940312i	$-0.389467\pi$
$191$	130.000	0.680628	0.340314	+	0.940312i	$0.389467\pi$
$192$	0	0
$193$	314.000	1.62694	0.813472	−	0.581605i	$-0.197575\pi$
$193$	314.000	1.62694	0.813472	+	0.581605i	$0.197575\pi$
$194$	0	0
$195$	−148.162	−0.759806
$196$	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
$207$	190.000	0.917874
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	582.065	2.73270
$214$	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
$225$	57.0000	0.253333
$226$	0	0
$227$	−216.952	−0.955734	−0.477867	−	0.878432i	$-0.658590\pi$
$227$	−216.952	−0.955734	−0.477867	+	0.878432i	$0.658590\pi$
$228$	0	0
$229$	−386.280	−1.68681	−0.843405	−	0.537278i	$-0.819453\pi$
$229$	−386.280	−1.68681	−0.843405	+	0.537278i	$0.819453\pi$
$230$	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
$237$	−687.895	−2.90251
$238$	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
$243$	−100.539	−0.413739
$244$	0	0
$245$	−259.284	−1.05830
$246$	0	0
$247$	−196.000	−0.793522
$248$	0	0
$249$	−868.000	−3.48594
$250$	0	0
$251$	37.0405	0.147572	0.0737859	−	0.997274i	$-0.476492\pi$
$251$	37.0405	0.147572	0.0737859	+	0.997274i	$0.476492\pi$
$252$	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	$-0.325592\pi$
$263$	274.000	1.04183	0.520913	+	0.853610i	$0.325592\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−513.276	−1.90809	−0.954044	−	0.299666i	$-0.903125\pi$
$269$	−513.276	−1.90809	−0.954044	+	0.299666i	$0.903125\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	−196.000	−0.717949
$274$	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
$283$	−470.944	−1.66411	−0.832056	−	0.554692i	$-0.812836\pi$
$283$	−470.944	−1.66411	−0.832056	+	0.554692i	$0.812836\pi$
$284$	0	0
$285$	1037.13	3.63907
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	375.697	1.28224	0.641121	−	0.767440i	$-0.278470\pi$
$293$	375.697	1.28224	0.641121	+	0.767440i	$0.278470\pi$
$294$	0	0
$295$	476.000	1.61356
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−52.9150	−0.176973
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−644.000	−2.12541
$304$	0	0
$305$	−644.000	−2.11148
$306$	0	0
$307$	418.029	1.36166	0.680829	−	0.732443i	$-0.261620\pi$
$307$	418.029	1.36166	0.680829	+	0.732443i	$0.261620\pi$
$308$	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
$315$	703.770	2.23419
$316$	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
$325$	−15.8745	−0.0488446
$326$	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	$-0.487718\pi$
$337$	26.0000	0.0771513	0.0385757	+	0.999256i	$0.487718\pi$
$338$	0	0
$339$	−137.579	−0.405838
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−343.000	−1.00000
$344$	0	0
$345$	280.000	0.811594
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	121.705	0.348724	0.174362	−	0.984682i	$-0.444214\pi$
$349$	121.705	0.348724	0.174362	+	0.984682i	$0.444214\pi$
$350$	0	0
$351$	280.000	0.797721
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	582.065	1.63962
$356$	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
$361$	1011.00	2.80055
$362$	0	0
$363$	−640.272	−1.76383
$364$	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
$375$	−616.000	−1.64267
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1322.88	−3.47211
$382$	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
$393$	476.000	1.21120
$394$	0	0
$395$	−687.895	−1.74151
$396$	0	0
$397$	629.689	1.58612	0.793059	−	0.609145i	$-0.208487\pi$
$397$	629.689	1.58612	0.793059	+	0.609145i	$0.208487\pi$
$398$	0	0
$399$	1372.00	3.43860
$400$	0	0
$401$	−550.000	−1.37157	−0.685786	−	0.727804i	$-0.740541\pi$
$401$	−550.000	−1.37157	−0.685786	+	0.727804i	$0.740541\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−576.774	−1.42413
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−264.575	−0.643735
$412$	0	0
$413$	629.689	1.52467
$414$	0	0
$415$	−868.000	−2.09157
$416$	0	0
$417$	476.000	1.14149
$418$	0	0
$419$	672.021	1.60387	0.801934	−	0.597412i	$-0.203804\pi$
$419$	672.021	1.60387	0.801934	+	0.597412i	$0.203804\pi$
$420$	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
$427$	−851.932	−1.99516
$428$	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
$437$	370.405	0.847609
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	931.000	2.11111
$442$	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
$453$	−1068.88	−2.35957
$454$	0	0
$455$	−196.000	−0.430769
$456$	0	0
$457$	−886.000	−1.93873	−0.969365	−	0.245623i	$-0.921007\pi$
$457$	−886.000	−1.93873	−0.969365	+	0.245623i	$0.921007\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	883.681	1.91688	0.958439	−	0.285297i	$-0.0920922\pi$
$461$	883.681	1.91688	0.958439	+	0.285297i	$0.0920922\pi$
$462$	0	0
$463$	226.000	0.488121	0.244060	−	0.969760i	$-0.421520\pi$
$463$	226.000	0.488121	0.244060	+	0.969760i	$0.421520\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	799.017	1.71096	0.855478	−	0.517838i	$-0.173263\pi$
$467$	799.017	1.71096	0.855478	+	0.517838i	$0.173263\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.0000	0.0594480
$472$	0	0
$473$	0	0
$474$	0	0
$475$	111.122	0.233940
$476$	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
$483$	370.405	0.766884
$484$	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
$505$	−644.000	−1.27525
$506$	0	0
$507$	746.102	1.47160
$508$	0	0
$509$	−386.280	−0.758899	−0.379450	−	0.925212i	$-0.623887\pi$
$509$	−386.280	−0.758899	−0.379450	+	0.925212i	$0.623887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1960.00	−3.82066
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1372.00	2.64355
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	418.029	0.799290	0.399645	−	0.916670i	$-0.369133\pi$
$523$	418.029	0.799290	0.399645	+	0.916670i	$0.369133\pi$
$524$	0	0
$525$	111.122	0.211660
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−429.000	−0.810964
$530$	0	0
$531$	−1709.16	−3.21875
$532$	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
$543$	−1316.00	−2.42357
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	2312.39	4.21200
$550$	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
$563$	1053.01	1.87035	0.935177	−	0.354181i	$-0.115240\pi$
$563$	1053.01	1.87035	0.935177	+	0.354181i	$0.115240\pi$
$564$	0	0
$565$	−137.579	−0.243503
$566$	0	0
$567$	−763.000	−1.34568
$568$	0	0
$569$	1130.00	1.98594	0.992970	−	0.118365i	$-0.0377654\pi$
$569$	1130.00	1.98594	0.992970	+	0.118365i	$0.0377654\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	−687.895	−1.20052
$574$	0	0
$575$	30.0000	0.0521739
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	−1661.53	−2.86966
$580$	0	0
$581$	−1148.26	−1.97634
$582$	0	0
$583$	0	0
$584$	0	0
$585$	532.000	0.909402
$586$	0	0
$587$	164.037	0.279449	0.139725	−	0.990190i	$-0.455378\pi$
$587$	164.037	0.279449	0.139725	+	0.990190i	$0.455378\pi$
$588$	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	$-0.851514\pi$
$599$	−1070.00	−1.78631	−0.893155	+	0.449748i	$0.851514\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−640.272	−1.05830
$606$	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	$-0.183768\pi$
$617$	1034.00	1.67585	0.837925	+	0.545785i	$0.183768\pi$
$618$	0	0
$619$	545.025	0.880492	0.440246	−	0.897877i	$-0.354891\pi$
$619$	545.025	0.880492	0.440246	+	0.897877i	$0.354891\pi$
$620$	0	0
$621$	−529.150	−0.852094
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−691.000	−1.10560
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−110.000	−0.174326	−0.0871632	−	0.996194i	$-0.527780\pi$
$631$	−110.000	−0.174326	−0.0871632	+	0.996194i	$0.527780\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1322.88	−2.08327
$636$	0	0
$637$	−259.284	−0.407039
$638$	0	0
$639$	−2090.00	−3.27074
$640$	0	0
$641$	−1030.00	−1.60686	−0.803432	−	0.595396i	$-0.796995\pi$
$641$	−1030.00	−1.60686	−0.803432	+	0.595396i	$0.796995\pi$
$642$	0	0
$643$	1053.01	1.63765	0.818825	−	0.574043i	$-0.194626\pi$
$643$	1053.01	1.63765	0.818825	+	0.574043i	$0.194626\pi$
$644$	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
$655$	476.000	0.726718
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−386.280	−0.584387	−0.292193	−	0.956359i	$-0.594385\pi$
$661$	−386.280	−0.584387	−0.292193	+	0.956359i	$0.594385\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1372.00	2.06316
$666$	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
$675$	−158.745	−0.235178
$676$	0	0
$677$	1264.67	1.86805	0.934024	−	0.357209i	$-0.116272\pi$
$677$	1264.67	1.86805	0.934024	+	0.357209i	$0.116272\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1148.00	1.68576
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−264.575	−0.386241
$686$	0	0
$687$	2044.00	2.97525
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1359.92	−1.96804	−0.984020	−	0.178056i	$-0.943019\pi$
$691$	−1359.92	−1.96804	−0.984020	+	0.178056i	$0.943019\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	476.000	0.684892
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2275.35	3.25514
$700$	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
$707$	−851.932	−1.20500
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	2470.00	3.47398
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2233.01	3.11439
$718$	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
$729$	−449.000	−0.615912
$730$	0	0
$731$	0	0
$732$	0	0
$733$	375.697	0.512547	0.256273	−	0.966604i	$-0.417505\pi$
$733$	375.697	0.512547	0.256273	+	0.966604i	$0.417505\pi$
$734$	0	0
$735$	1372.00	1.86667
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	1037.13	1.39964
$742$	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
$747$	3116.70	4.17228
$748$	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
$753$	−196.000	−0.260292
$754$	0	0
$755$	−1068.88	−1.41574
$756$	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
$767$	476.000	0.620600
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1275.25	−1.64974	−0.824872	−	0.565320i	$-0.808753\pi$
$773$	−1275.25	−1.64974	−0.824872	+	0.565320i	$0.808753\pi$
$774$	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
$785$	28.0000	0.0356688
$786$	0	0
$787$	−470.944	−0.598404	−0.299202	−	0.954190i	$-0.596720\pi$
$787$	−470.944	−0.598404	−0.299202	+	0.954190i	$0.596720\pi$
$788$	0	0
$789$	−1449.87	−1.83761
$790$	0	0
$791$	−182.000	−0.230088
$792$	0	0
$793$	−644.000	−0.812106
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1529.24	−1.91875	−0.959375	−	0.282133i	$-0.908958\pi$
$797$	−1529.24	−1.91875	−0.959375	+	0.282133i	$0.908958\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	370.405	0.460131
$806$	0	0
$807$	2716.00	3.36555
$808$	0	0
$809$	650.000	0.803461	0.401731	−	0.915758i	$-0.368409\pi$
$809$	650.000	0.803461	0.401731	+	0.915758i	$0.368409\pi$
$810$	0	0
$811$	−724.936	−0.893879	−0.446939	−	0.894564i	$-0.647486\pi$
$811$	−724.936	−0.893879	−0.446939	+	0.894564i	$0.647486\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	703.770	0.859304
$820$	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	$-0.305108\pi$
$823$	946.000	1.14945	0.574727	+	0.818345i	$0.305108\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1656.24	−1.99788	−0.998939	−	0.0460600i	$-0.985333\pi$
$829$	−1656.24	−1.99788	−0.998939	+	0.0460600i	$0.985333\pi$
$830$	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
$843$	−1788.53	−2.12162
$844$	0	0
$845$	746.102	0.882961
$846$	0	0
$847$	−847.000	−1.00000
$848$	0	0
$849$	2492.00	2.93522
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1645.66	1.92926	0.964629	−	0.263611i	$-0.0849134\pi$
$853$	1645.66	1.92926	0.964629	+	0.263611i	$0.0849134\pi$
$854$	0	0
$855$	−3724.00	−4.35556
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1232.92	−1.43530	−0.717648	−	0.696406i	$-0.754782\pi$
$859$	−1232.92	−1.43530	−0.717648	+	0.696406i	$0.754782\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1474.00	1.70800	0.853998	−	0.520277i	$-0.174171\pi$
$863$	1474.00	1.70800	0.853998	+	0.520277i	$0.174171\pi$
$864$	0	0
$865$	1372.00	1.58613
$866$	0	0
$867$	−1529.24	−1.76383
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−814.891	−0.931304
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1988.00	−2.26166
$880$	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
$885$	−2518.76	−2.84605
$886$	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
$897$	280.000	0.312152
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1316.00	−1.45414
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	2312.39	2.54388
$910$	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
$915$	3407.73	3.72429
$916$	0	0
$917$	629.689	0.686684
$918$	0	0
$919$	−1550.00	−1.68662	−0.843308	−	0.537431i	$-0.819395\pi$
$919$	−1550.00	−1.68662	−0.843308	+	0.537431i	$0.819395\pi$
$920$	0	0
$921$	−2212.00	−2.40174
$922$	0	0
$923$	582.065	0.630623
$924$	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
$931$	1814.99	1.94950
$932$	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
$941$	1518.66	1.61388	0.806940	−	0.590633i	$-0.201122\pi$
$941$	1518.66	1.61388	0.806940	+	0.590633i	$0.201122\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1960.00	−2.07407
$946$	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
$955$	−687.895	−0.720309
$956$	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
$965$	−1661.53	−1.72179
$966$	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
$971$	1941.98	1.99998	0.999990	−	0.00436935i	$-0.00139081\pi$
$971$	1941.98	1.99998	0.999990	+	0.00436935i	$0.00139081\pi$
$972$	0	0
$973$	629.689	0.647162
$974$	0	0
$975$	84.0000	0.0861538
$976$	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	$-0.400417\pi$
$991$	610.000	0.615540	0.307770	+	0.951461i	$0.400417\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−259.284	−0.260064	−0.130032	−	0.991510i	$-0.541508\pi$
$997$	−259.284	−0.260064	−0.130032	+	0.991510i	$0.541508\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-5.29150 q^{3} -5.29150 q^{5} -7.00000 q^{7} +19.0000 q^{9} +O(q^{10})\)	\(q-5.29150 q^{3} -5.29150 q^{5} -7.00000 q^{7} +19.0000 q^{9} -5.29150 q^{13} +28.0000 q^{15} +37.0405 q^{19} +37.0405 q^{21} +10.0000 q^{23} +3.00000 q^{25} -52.9150 q^{27} +37.0405 q^{35} +28.0000 q^{39} -100.539 q^{45} +49.0000 q^{49} -196.000 q^{57} -89.9555 q^{59} +121.705 q^{61} -133.000 q^{63} +28.0000 q^{65} -52.9150 q^{69} -110.000 q^{71} -15.8745 q^{75} +130.000 q^{79} +109.000 q^{81} +164.037 q^{83} +37.0405 q^{91} -196.000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7} + 38 q^{9}+O(q^{10}) \) 2 * q - 14 * q^7 + 38 * q^9	\( 2 q - 14 q^{7} + 38 q^{9} + 56 q^{15} + 20 q^{23} + 6 q^{25} + 56 q^{39} + 98 q^{49} - 392 q^{57} - 266 q^{63} + 56 q^{65} - 220 q^{71} + 260 q^{79} + 218 q^{81} - 392 q^{95}+O(q^{100}) \) 2 * q - 14 * q^7 + 38 * q^9 + 56 * q^15 + 20 * q^23 + 6 * q^25 + 56 * q^39 + 98 * q^49 - 392 * q^57 - 266 * q^63 + 56 * q^65 - 220 * q^71 + 260 * q^79 + 218 * q^81 - 392 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more