LMFDB - Embedded newform 160.4.c.a.129.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [160,4,Mod(129,160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("160.129"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.a.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.0000 + 2.00000i) q^{5} +27.0000 q^{9} -92.0000i q^{13} -104.000i q^{17} +(117.000 - 44.0000i) q^{25} -130.000 q^{29} -396.000i q^{37} +230.000 q^{41} +(-297.000 + 54.0000i) q^{45} +343.000 q^{49} +572.000i q^{53} -830.000 q^{61} +(184.000 + 1012.00i) q^{65} +592.000i q^{73} +729.000 q^{81} +(208.000 + 1144.00i) q^{85} -1670.00 q^{89} -1816.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 22 q^{5} + 54 q^{9} + 234 q^{25} - 260 q^{29} + 460 q^{41} - 594 q^{45} + 686 q^{49} - 1660 q^{61} + 368 q^{65} + 1458 q^{81} + 416 q^{85} - 3340 q^{89}+O(q^{100}) $ 2 * q - 22 * q^5 + 54 * q^9 + 234 * q^25 - 260 * q^29 + 460 * q^41 - 594 * q^45 + 686 * q^49 - 1660 * q^61 + 368 * q^65 + 1458 * q^81 + 416 * q^85 - 3340 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		0			0
$5$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$5$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	27.0000			1.00000
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	92.0000i		−	1.96279i	−0.192012	−	0.981393i	$-0.561501\pi$
$13$		−	92.0000i		−	1.96279i	0.192012	−	0.981393i	$-0.438499\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	104.000i		−	1.48375i	−0.670540	−	0.741874i	$-0.733937\pi$
$17$		−	104.000i		−	1.48375i	0.670540	−	0.741874i	$-0.266063\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	117.000	−	44.0000i	0.936000	−	0.352000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−130.000			−0.832427			−0.416214	−	0.909267i	$-0.636643\pi$
$29$	−130.000			−0.832427			−0.416214	+	0.909267i	$0.636643\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	396.000i		−	1.75951i	−0.475424	−	0.879757i	$-0.657705\pi$
$37$		−	396.000i		−	1.75951i	0.475424	−	0.879757i	$-0.342295\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	230.000			0.876097			0.438048	−	0.898951i	$-0.355670\pi$
$41$	230.000			0.876097			0.438048	+	0.898951i	$0.355670\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−297.000	+	54.0000i	−0.983870	+	0.178885i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	343.000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			572.000i			1.48246i	0.671253	+	0.741229i	$0.265757\pi$
$53$			572.000i			1.48246i	−0.671253	+	0.741229i	$0.734243\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−830.000			−1.74214			−0.871071	−	0.491158i	$-0.836574\pi$
$61$	−830.000			−1.74214			−0.871071	+	0.491158i	$0.836574\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	184.000	+	1012.00i	0.351114	+	1.93113i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			592.000i			0.949156i	0.880214	+	0.474578i	$0.157399\pi$
$73$			592.000i			0.949156i	−0.880214	+	0.474578i	$0.842601\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	729.000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	208.000	+	1144.00i	0.265421	+	1.45981i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1670.00			−1.98898			−0.994492	−	0.104809i	$-0.966577\pi$
$89$	−1670.00			−1.98898			−0.994492	+	0.104809i	$0.966577\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1816.00i		−	1.90090i	−0.310884	−	0.950448i	$-0.600625\pi$
$97$		−	1816.00i		−	1.90090i	0.310884	−	0.950448i	$-0.399375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	598.000			0.589141			0.294570	−	0.955630i	$-0.404823\pi$
$101$	598.000			0.589141			0.294570	+	0.955630i	$0.404823\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	1746.00			1.53428			0.767140	−	0.641480i	$-0.221679\pi$
$109$	1746.00			1.53428			0.767140	+	0.641480i	$0.221679\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1328.00i			1.10556i	0.833329	+	0.552778i	$0.186432\pi$
$113$			1328.00i			1.10556i	−0.833329	+	0.552778i	$0.813568\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	2484.00i		−	1.96279i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1199.00	+	718.000i	−0.857935	+	0.513759i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		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
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2776.00i		−	1.73117i	−0.500766	−	0.865583i	$-0.666948\pi$
$137$		−	2776.00i		−	1.73117i	0.500766	−	0.865583i	$-0.333052\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1430.00	−	260.000i	0.819000	−	0.148909i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3514.00			1.93207			0.966034	−	0.258415i	$-0.0832003\pi$
$149$	3514.00			1.93207			0.966034	+	0.258415i	$0.0832003\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	2808.00i		−	1.48375i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	3924.00i		−	1.99471i	−0.0726920	−	0.997354i	$-0.523159\pi$
$157$		−	3924.00i		−	1.99471i	0.0726920	−	0.997354i	$-0.476841\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		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
$169$	−6267.00			−2.85253
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2012.00i			0.884217i	0.896962	+	0.442108i	$0.145769\pi$
$173$			2012.00i			0.884217i	−0.896962	+	0.442108i	$0.854231\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	3942.00			1.61882			0.809410	−	0.587243i	$-0.199787\pi$
$181$	3942.00			1.61882			0.809410	+	0.587243i	$0.199787\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	792.000	+	4356.00i	0.314751	+	1.73113i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	72.0000i		−	0.0268532i	−0.999910	−	0.0134266i	$-0.995726\pi$
$193$		−	72.0000i		−	0.0268532i	0.999910	−	0.0134266i	$-0.00427395\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5404.00i			1.95441i	0.212295	+	0.977206i	$0.431906\pi$
$197$			5404.00i			1.95441i	−0.212295	+	0.977206i	$0.568094\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2530.00	+	460.000i	−0.861965	+	0.156721i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9568.00			−2.91228
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	3159.00	−	1188.00i	0.936000	−	0.352000i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	6390.00			1.84394			0.921972	−	0.387257i	$-0.126577\pi$
$229$	6390.00			1.84394			0.921972	+	0.387257i	$0.126577\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7088.00i			1.99292i	0.0840693	+	0.996460i	$0.473208\pi$
$233$			7088.00i			1.99292i	−0.0840693	+	0.996460i	$0.526792\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	5310.00			1.41928			0.709641	−	0.704563i	$-0.248857\pi$
$241$	5310.00			1.41928			0.709641	+	0.704563i	$0.248857\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3773.00	+	686.000i	−0.983870	+	0.178885i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			8096.00i			1.96504i	0.186164	+	0.982519i	$0.440394\pi$
$257$			8096.00i			1.96504i	−0.186164	+	0.982519i	$0.559606\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3510.00			−0.832427
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−1144.00	−	6292.00i	−0.265190	−	1.45855i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3406.00			−0.771998			−0.385999	−	0.922499i	$-0.626143\pi$
$269$	−3406.00			−0.771998			−0.385999	+	0.922499i	$0.626143\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			1316.00i			0.285454i	0.989762	+	0.142727i	$0.0455871\pi$
$277$			1316.00i			0.285454i	−0.989762	+	0.142727i	$0.954413\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7430.00			1.57735			0.788677	−	0.614807i	$-0.210766\pi$
$281$	7430.00			1.57735			0.788677	+	0.614807i	$0.210766\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5903.00			−1.20151
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	3452.00i		−	0.688287i	−0.938917	−	0.344143i	$-0.888169\pi$
$293$		−	3452.00i		−	0.688287i	0.938917	−	0.344143i	$-0.111831\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9130.00	−	1660.00i	1.71404	−	0.311644i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	8712.00i		−	1.57326i	−0.617423	−	0.786632i	$-0.711823\pi$
$313$		−	8712.00i		−	1.57326i	0.617423	−	0.786632i	$-0.288177\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4676.00i			0.828487i	0.910166	+	0.414243i	$0.135954\pi$
$317$			4676.00i			0.828487i	−0.910166	+	0.414243i	$0.864046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4048.00	−	10764.0i	−0.690900	−	1.83717i
$326$		0			0
$327$		0			0
$328$		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
$333$		−	10692.0i		−	1.75951i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	416.000i		−	0.0672432i	−0.999435	−	0.0336216i	$-0.989296\pi$
$337$		−	416.000i		−	0.0672432i	0.999435	−	0.0336216i	$-0.0107041\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	9470.00			1.45249			0.726243	−	0.687438i	$-0.241265\pi$
$349$	9470.00			1.45249			0.726243	+	0.687438i	$0.241265\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	12848.0i		−	1.93720i	−0.248633	−	0.968598i	$-0.579981\pi$
$353$		−	12848.0i		−	1.93720i	0.248633	−	0.968598i	$-0.420019\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−6859.00			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1184.00	−	6512.00i	−0.169790	−	0.933846i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	6210.00			0.876097
$370$		0			0
$371$		0			0
$372$		0			0
$373$			6372.00i			0.884530i	0.896884	+	0.442265i	$0.145825\pi$
$373$			6372.00i			0.884530i	−0.896884	+	0.442265i	$0.854175\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			11960.0i			1.63388i
$378$		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		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−374.000			−0.0487469			−0.0243735	−	0.999703i	$-0.507759\pi$
$389$	−374.000			−0.0487469			−0.0243735	+	0.999703i	$0.507759\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			12564.0i			1.58834i	0.607699	+	0.794168i	$0.292093\pi$
$397$			12564.0i			1.58834i	−0.607699	+	0.794168i	$0.707907\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2398.00			−0.298629			−0.149315	−	0.988790i	$-0.547707\pi$
$401$	−2398.00			−0.298629			−0.149315	+	0.988790i	$0.547707\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−8019.00	+	1458.00i	−0.983870	+	0.178885i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7146.00			−0.863929			−0.431964	−	0.901891i	$-0.642179\pi$
$409$	−7146.00			−0.863929			−0.431964	+	0.901891i	$0.642179\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	10890.0			1.26068			0.630340	−	0.776319i	$-0.282916\pi$
$421$	10890.0			1.26068			0.630340	+	0.776319i	$0.282916\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4576.00	−	12168.0i	−0.522279	−	1.38879i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			17352.0i			1.92583i	0.269807	+	0.962914i	$0.413040\pi$
$433$			17352.0i			1.92583i	−0.269807	+	0.962914i	$0.586960\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	9261.00			1.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	18370.0	−	3340.00i	1.95690	−	0.355800i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16114.0			−1.69369			−0.846845	−	0.531840i	$-0.821501\pi$
$449$	−16114.0			−1.69369			−0.846845	+	0.531840i	$0.821501\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			10456.0i			1.07026i	0.844768	+	0.535132i	$0.179738\pi$
$457$			10456.0i			1.07026i	−0.844768	+	0.535132i	$0.820262\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2318.00			0.234187			0.117093	−	0.993121i	$-0.462642\pi$
$461$	2318.00			0.234187			0.117093	+	0.993121i	$0.462642\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			15444.0i			1.48246i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−36432.0			−3.45355
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3632.00	+	19976.0i	0.340043	+	1.87023i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			13520.0i			1.23511i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		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		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−6578.00	+	1196.00i	−0.579638	+	0.105389i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14270.0			1.24265			0.621323	−	0.783555i	$-0.286596\pi$
$509$	14270.0			1.24265			0.621323	+	0.783555i	$0.286596\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	23738.0			1.99612			0.998062	−	0.0622265i	$-0.0198201\pi$
$521$	23738.0			1.99612			0.998062	+	0.0622265i	$0.0198201\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12167.0			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	21160.0i		−	1.71959i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5922.00			−0.470622			−0.235311	−	0.971920i	$-0.575611\pi$
$541$	−5922.00			−0.470622			−0.235311	+	0.971920i	$0.575611\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19206.0	+	3492.00i	−1.50953	+	0.274460i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	−22410.0			−1.74214
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24836.0i		−	1.88929i	−0.328093	−	0.944646i	$-0.606406\pi$
$557$		−	24836.0i		−	1.88929i	0.328093	−	0.944646i	$-0.393594\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	−2656.00	−	14608.0i	−0.197768	−	1.08772i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26806.0			1.97498			0.987492	−	0.157669i	$-0.0503978\pi$
$569$	26806.0			1.97498			0.987492	+	0.157669i	$0.0503978\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	27504.0i		−	1.98441i	−0.124603	−	0.992207i	$-0.539766\pi$
$577$		−	27504.0i		−	1.98441i	0.124603	−	0.992207i	$-0.460234\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	4968.00	+	27324.0i	0.351114	+	1.93113i
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24368.0i		−	1.68748i	−0.536755	−	0.843738i	$-0.680350\pi$
$593$		−	24368.0i		−	1.68748i	0.536755	−	0.843738i	$-0.319650\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	17030.0			1.15585			0.577927	−	0.816089i	$-0.303862\pi$
$601$	17030.0			1.15585			0.577927	+	0.816089i	$0.303862\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14641.0	−	2662.00i	0.983870	−	0.178885i
$606$		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
$613$			19548.0i			1.28799i	0.765031	+	0.643994i	$0.222724\pi$
$613$			19548.0i			1.28799i	−0.765031	+	0.643994i	$0.777276\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26464.0i		−	1.72674i	−0.504569	−	0.863372i	$-0.668348\pi$
$617$		−	26464.0i		−	1.72674i	0.504569	−	0.863372i	$-0.331652\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11753.0	−	10296.0i	0.752192	−	0.658944i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41184.0			−2.61067
$630$		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
$637$		−	31556.0i		−	1.96279i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−28850.0			−1.77770			−0.888851	−	0.458197i	$-0.848495\pi$
$641$	−28850.0			−1.77770			−0.888851	+	0.458197i	$0.848495\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		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
$653$			1012.00i			0.0606472i	0.999540	+	0.0303236i	$0.00965378\pi$
$653$			1012.00i			0.0606472i	−0.999540	+	0.0303236i	$0.990346\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			15984.0i			0.949156i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	25850.0			1.52110			0.760551	−	0.649278i	$-0.224929\pi$
$661$	25850.0			1.52110			0.760551	+	0.649278i	$0.224929\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			34632.0i			1.98360i	0.127784	+	0.991802i	$0.459214\pi$
$673$			34632.0i			1.98360i	−0.127784	+	0.991802i	$0.540786\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	34996.0i		−	1.98671i	−0.115072	−	0.993357i	$-0.536710\pi$
$677$		−	34996.0i		−	1.98671i	0.115072	−	0.993357i	$-0.463290\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	5552.00	+	30536.0i	0.309680	+	1.70324i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	52624.0			2.90975
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	23920.0i		−	1.29991i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20030.0			−1.07920			−0.539602	−	0.841920i	$-0.681425\pi$
$701$	−20030.0			−1.07920			−0.539602	+	0.841920i	$0.681425\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−36810.0			−1.94983			−0.974914	−	0.222580i	$-0.928552\pi$
$709$	−36810.0			−1.94983			−0.974914	+	0.222580i	$0.928552\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15210.0	+	5720.00i	−0.779152	+	0.293014i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	19683.0			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$			8732.00i			0.440005i	0.975499	+	0.220003i	$0.0706066\pi$
$733$			8732.00i			0.440005i	−0.975499	+	0.220003i	$0.929393\pi$
$734$		0			0
$735$		0			0
$736$		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
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−38654.0	+	7028.00i	−1.90090	+	0.345619i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		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
$757$			22516.0i			1.08105i	0.841327	+	0.540527i	$0.181775\pi$
$757$			22516.0i			1.08105i	−0.841327	+	0.540527i	$0.818225\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31882.0			1.51869			0.759344	−	0.650689i	$-0.225520\pi$
$761$	31882.0			1.51869			0.759344	+	0.650689i	$0.225520\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	5616.00	+	30888.0i	0.265421	+	1.45981i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	9650.00			0.452520			0.226260	−	0.974067i	$-0.427350\pi$
$769$	9650.00			0.452520			0.226260	+	0.974067i	$0.427350\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	16852.0i		−	0.784119i	−0.919940	−	0.392060i	$-0.871763\pi$
$773$		−	16852.0i		−	0.784119i	0.919940	−	0.392060i	$-0.128237\pi$
$774$		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
$785$	7848.00	+	43164.0i	0.356824	+	1.96253i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			76360.0i			3.41945i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			16276.0i			0.723370i	0.932300	+	0.361685i	$0.117798\pi$
$797$			16276.0i			0.723370i	−0.932300	+	0.361685i	$0.882202\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−45090.0			−1.98898
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23270.0			−1.01129			−0.505643	−	0.862743i	$-0.668745\pi$
$809$	−23270.0			−1.01129			−0.505643	+	0.862743i	$0.668745\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1850.00			0.0786424			0.0393212	−	0.999227i	$-0.487480\pi$
$821$	1850.00			0.0786424			0.0393212	+	0.999227i	$0.487480\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−23166.0			−0.970553			−0.485276	−	0.874361i	$-0.661281\pi$
$829$	−23166.0			−0.970553			−0.485276	+	0.874361i	$0.661281\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	35672.0i		−	1.48375i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−7489.00			−0.307065
$842$		0			0
$843$		0			0
$844$		0			0
$845$	68937.0	−	12534.0i	2.80651	−	0.510275i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			45468.0i			1.82508i	0.408986	+	0.912541i	$0.365883\pi$
$853$			45468.0i			1.82508i	−0.408986	+	0.912541i	$0.634117\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20056.0i		−	0.799416i	−0.916642	−	0.399708i	$-0.869111\pi$
$857$		−	20056.0i		−	0.799416i	0.916642	−	0.399708i	$-0.130889\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−4024.00	−	22132.0i	−0.158174	−	0.869954i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	49032.0i		−	1.90090i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	29844.0i		−	1.14910i	−0.818470	−	0.574550i	$-0.805177\pi$
$877$		−	29844.0i		−	1.14910i	0.818470	−	0.574550i	$-0.194823\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7150.00			0.273427			0.136714	−	0.990611i	$-0.456346\pi$
$881$	7150.00			0.273427			0.136714	+	0.990611i	$0.456346\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	59488.0			2.19959
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−43362.0	+	7884.00i	−1.59271	+	0.289583i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	16146.0			0.589141
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−17424.0	−	46332.0i	−0.619349	−	1.64690i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30866.0			−1.09008			−0.545038	−	0.838411i	$-0.683485\pi$
$929$	−30866.0			−1.09008			−0.545038	+	0.838411i	$0.683485\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			24336.0i			0.848476i	0.905551	+	0.424238i	$0.139458\pi$
$937$			24336.0i			0.848476i	−0.905551	+	0.424238i	$0.860542\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31378.0			−1.08703			−0.543514	−	0.839400i	$-0.682907\pi$
$941$	−31378.0			−1.08703			−0.543514	+	0.839400i	$0.682907\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	54464.0			1.86299
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	15512.0i		−	0.527264i	−0.964623	−	0.263632i	$-0.915079\pi$
$953$		−	15512.0i		−	0.527264i	0.964623	−	0.263632i	$-0.0849205\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29791.0			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	144.000	+	792.000i	0.00480365	+	0.0264201i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	22936.0i		−	0.751062i	−0.926810	−	0.375531i	$-0.877460\pi$
$977$		−	22936.0i		−	0.751062i	0.926810	−	0.375531i	$-0.122540\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	47142.0			1.53428
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	−10808.0	−	59444.0i	−0.349616	−	1.92289i
$986$		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
$995$		0			0
$996$		0			0
$997$		−	34164.0i		−	1.08524i	−0.839978	−	0.542620i	$-0.817432\pi$
$997$		−	34164.0i		−	1.08524i	0.839978	−	0.542620i	$-0.182568\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.c.a.129.2	yes	2
3.2	odd	2		1440.4.f.e.289.1		2
4.3	odd	2	CM	160.4.c.a.129.2	yes	2
5.2	odd	4		800.4.a.e.1.1		1
5.3	odd	4		800.4.a.g.1.1		1
5.4	even	2	inner	160.4.c.a.129.1	✓	2
8.3	odd	2		320.4.c.e.129.1		2
8.5	even	2		320.4.c.e.129.1		2
12.11	even	2		1440.4.f.e.289.1		2
15.14	odd	2		1440.4.f.e.289.2		2
20.3	even	4		800.4.a.g.1.1		1
20.7	even	4		800.4.a.e.1.1		1
20.19	odd	2	inner	160.4.c.a.129.1	✓	2
40.3	even	4		1600.4.a.z.1.1		1
40.13	odd	4		1600.4.a.z.1.1		1
40.19	odd	2		320.4.c.e.129.2		2
40.27	even	4		1600.4.a.bb.1.1		1
40.29	even	2		320.4.c.e.129.2		2
40.37	odd	4		1600.4.a.bb.1.1		1
60.59	even	2		1440.4.f.e.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.a.129.1	✓	2	5.4	even	2	inner
160.4.c.a.129.1	✓	2	20.19	odd	2	inner
160.4.c.a.129.2	yes	2	1.1	even	1	trivial
160.4.c.a.129.2	yes	2	4.3	odd	2	CM
320.4.c.e.129.1		2	8.3	odd	2
320.4.c.e.129.1		2	8.5	even	2
320.4.c.e.129.2		2	40.19	odd	2
320.4.c.e.129.2		2	40.29	even	2
800.4.a.e.1.1		1	5.2	odd	4
800.4.a.e.1.1		1	20.7	even	4
800.4.a.g.1.1		1	5.3	odd	4
800.4.a.g.1.1		1	20.3	even	4
1440.4.f.e.289.1		2	3.2	odd	2
1440.4.f.e.289.1		2	12.11	even	2
1440.4.f.e.289.2		2	15.14	odd	2
1440.4.f.e.289.2		2	60.59	even	2
1600.4.a.z.1.1		1	40.3	even	4
1600.4.a.z.1.1		1	40.13	odd	4
1600.4.a.bb.1.1		1	40.27	even	4
1600.4.a.bb.1.1		1	40.37	odd	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-11.0000 + 2.00000i) q^{5} +27.0000 q^{9} -92.0000i q^{13} -104.000i q^{17} +(117.000 - 44.0000i) q^{25} -130.000 q^{29} -396.000i q^{37} +230.000 q^{41} +(-297.000 + 54.0000i) q^{45} +343.000 q^{49} +572.000i q^{53} -830.000 q^{61} +(184.000 + 1012.00i) q^{65} +592.000i q^{73} +729.000 q^{81} +(208.000 + 1144.00i) q^{85} -1670.00 q^{89} -1816.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{5} + 54 q^{9} + 234 q^{25} - 260 q^{29} + 460 q^{41} - 594 q^{45} + 686 q^{49} - 1660 q^{61} + 368 q^{65} + 1458 q^{81} + 416 q^{85} - 3340 q^{89}+O(q^{100}) \) 2 * q - 22 * q^5 + 54 * q^9 + 234 * q^25 - 260 * q^29 + 460 * q^41 - 594 * q^45 + 686 * q^49 - 1660 * q^61 + 368 * q^65 + 1458 * q^81 + 416 * q^85 - 3340 * q^89

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