LMFDB - Embedded newform 1600.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.a (trivial)

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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} +0.763932 q^{7} +7.47214 q^{9} +O(q^{10})$	$q-3.23607 q^{3} +0.763932 q^{7} +7.47214 q^{9} -2.47214 q^{21} -5.70820 q^{23} -14.4721 q^{27} +6.00000 q^{29} -4.47214 q^{41} -11.2361 q^{43} +13.7082 q^{47} -6.41641 q^{49} +13.4164 q^{61} +5.70820 q^{63} +8.18034 q^{67} +18.4721 q^{69} +24.4164 q^{81} +17.7082 q^{83} -19.4164 q^{87} +6.00000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} + 6 q^{7} + 6 q^{9} + 4 q^{21} + 2 q^{23} - 20 q^{27} + 12 q^{29} - 18 q^{43} + 14 q^{47} + 14 q^{49} - 2 q^{63} - 6 q^{67} + 28 q^{69} + 22 q^{81} + 22 q^{83} - 12 q^{87} + 12 q^{89}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^7 + 6 * q^9 + 4 * q^21 + 2 * q^23 - 20 * q^27 + 12 * q^29 - 18 * q^43 + 14 * q^47 + 14 * q^49 - 2 * q^63 - 6 * q^67 + 28 * q^69 + 22 * q^81 + 22 * q^83 - 12 * q^87 + 12 * q^89

Display 100 coefficients 10 coefficients

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
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.763932	0.288739	0.144370	−	0.989524i	$-0.453885\pi$
$7$	0.763932	0.288739	0.144370	+	0.989524i	$0.453885\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.47214	−0.539464
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−11.2361	−1.71348	−0.856742	−	0.515745i	$-0.827515\pi$
$43$	−11.2361	−1.71348	−0.856742	+	0.515745i	$0.827515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.7082	1.99955	0.999774	−	0.0212814i	$-0.00677460\pi$
$47$	13.7082	1.99955	0.999774	+	0.0212814i	$0.00677460\pi$
$48$	0	0
$49$	−6.41641	−0.916630
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	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
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	0	0
$63$	5.70820	0.719166
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.18034	0.999388	0.499694	−	0.866202i	$-0.333446\pi$
$67$	8.18034	0.999388	0.499694	+	0.866202i	$0.333446\pi$
$68$	0	0
$69$	18.4721	2.22378
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	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
$81$	24.4164	2.71293
$82$	0	0
$83$	17.7082	1.94373	0.971864	−	0.235543i	$-0.0756868\pi$
$83$	17.7082	1.94373	0.971864	+	0.235543i	$0.0756868\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19.4164	−2.08166
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	20.1803	1.98843	0.994214	−	0.107418i	$-0.0342582\pi$
$103$	20.1803	1.98843	0.994214	+	0.107418i	$0.0342582\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.29180	0.608251	0.304125	−	0.952632i	$-0.401636\pi$
$107$	6.29180	0.608251	0.304125	+	0.952632i	$0.401636\pi$
$108$	0	0
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	14.4721	1.30491
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.6525	1.65514	0.827570	−	0.561363i	$-0.189723\pi$
$127$	18.6525	1.65514	0.827570	+	0.561363i	$0.189723\pi$
$128$	0	0
$129$	36.3607	3.20138
$130$	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
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−44.3607	−3.73584
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.7639	1.71258
$148$	0	0
$149$	4.47214	0.366372	0.183186	−	0.983078i	$-0.441359\pi$
$149$	4.47214	0.366372	0.183186	+	0.983078i	$0.441359\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.36068	−0.343670
$162$	0	0
$163$	6.65248	0.521062	0.260531	−	0.965465i	$-0.416102\pi$
$163$	6.65248	0.521062	0.260531	+	0.965465i	$0.416102\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.2918	−0.796403	−0.398202	−	0.917298i	$-0.630366\pi$
$167$	−10.2918	−0.796403	−0.398202	+	0.917298i	$0.630366\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−43.4164	−3.20943
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−11.0557	−0.804186
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−26.4721	−1.86720
$202$	0	0
$203$	4.58359	0.321705
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−42.6525	−2.96455
$208$	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
$221$	0	0
$222$	0	0
$223$	23.2361	1.55600	0.778001	−	0.628263i	$-0.216234\pi$
$223$	23.2361	1.55600	0.778001	+	0.628263i	$0.216234\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.1246	0.871111	0.435556	−	0.900162i	$-0.356552\pi$
$227$	13.1246	0.871111	0.435556	+	0.900162i	$0.356552\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	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
$241$	13.4164	0.864227	0.432113	−	0.901819i	$-0.357768\pi$
$241$	13.4164	0.864227	0.432113	+	0.901819i	$0.357768\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−57.3050	−3.63155
$250$	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
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	44.8328	2.77508
$262$	0	0
$263$	9.12461	0.562648	0.281324	−	0.959613i	$-0.409226\pi$
$263$	9.12461	0.562648	0.281324	+	0.959613i	$0.409226\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−19.4164	−1.18826
$268$	0	0
$269$	22.3607	1.36335	0.681677	−	0.731653i	$-0.261251\pi$
$269$	22.3607	1.36335	0.681677	+	0.731653i	$0.261251\pi$
$270$	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
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.3050	1.86750	0.933748	−	0.357930i	$-0.116517\pi$
$281$	31.3050	1.86750	0.933748	+	0.357930i	$0.116517\pi$
$282$	0	0
$283$	−32.1803	−1.91292	−0.956461	−	0.291859i	$-0.905726\pi$
$283$	−32.1803	−1.91292	−0.956461	+	0.291859i	$0.905726\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.41641	−0.201664
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.58359	−0.494750
$302$	0	0
$303$	−58.2492	−3.34633
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−27.5967	−1.57503	−0.787515	−	0.616296i	$-0.788633\pi$
$307$	−27.5967	−1.57503	−0.787515	+	0.616296i	$0.788633\pi$
$308$	0	0
$309$	−65.3050	−3.71507
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−20.3607	−1.13642
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	43.4164	2.40093
$328$	0	0
$329$	10.4721	0.577348
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−10.2492	−0.553406
$344$	0	0
$345$	0	0
$346$	0	0
$347$	37.1246	1.99295	0.996477	−	0.0838690i	$-0.0267277\pi$
$347$	37.1246	1.99295	0.996477	+	0.0838690i	$0.0267277\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$361$	−19.0000	−1.00000
$362$	0	0
$363$	35.5967	1.86834
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.7639	1.29267	0.646333	−	0.763055i	$-0.276302\pi$
$367$	24.7639	1.29267	0.646333	+	0.763055i	$0.276302\pi$
$368$	0	0
$369$	−33.4164	−1.73959
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−60.3607	−3.09237
$382$	0	0
$383$	1.12461	0.0574650	0.0287325	−	0.999587i	$-0.490853\pi$
$383$	1.12461	0.0574650	0.0287325	+	0.999587i	$0.490853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−83.9574	−4.26780
$388$	0	0
$389$	−31.3050	−1.58722	−0.793612	−	0.608424i	$-0.791802\pi$
$389$	−31.3050	−1.58722	−0.793612	+	0.608424i	$0.791802\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	40.2492	1.99020	0.995098	−	0.0988936i	$-0.0315304\pi$
$409$	40.2492	1.99020	0.995098	+	0.0988936i	$0.0315304\pi$
$410$	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
$421$	−40.2492	−1.96163	−0.980814	−	0.194948i	$-0.937546\pi$
$421$	−40.2492	−1.96163	−0.980814	+	0.194948i	$0.937546\pi$
$422$	0	0
$423$	102.430	4.98030
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.2492	0.495995
$428$	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
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	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
$441$	−47.9443	−2.28306
$442$	0	0
$443$	−22.2918	−1.05912	−0.529558	−	0.848274i	$-0.677642\pi$
$443$	−22.2918	−1.05912	−0.529558	+	0.848274i	$0.677642\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.4721	−0.684509
$448$	0	0
$449$	22.3607	1.05527	0.527633	−	0.849473i	$-0.323080\pi$
$449$	22.3607	1.05527	0.527633	+	0.849473i	$0.323080\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	38.0689	1.76921	0.884606	−	0.466340i	$-0.154428\pi$
$463$	38.0689	1.76921	0.884606	+	0.466340i	$0.154428\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.87539	−0.133057	−0.0665285	−	0.997785i	$-0.521192\pi$
$467$	−2.87539	−0.133057	−0.0665285	+	0.997785i	$0.521192\pi$
$468$	0	0
$469$	6.24922	0.288562
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.1115	0.642093
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.6525	1.93277	0.966384	−	0.257103i	$-0.0827679\pi$
$487$	42.6525	1.93277	0.966384	+	0.257103i	$0.0827679\pi$
$488$	0	0
$489$	−21.5279	−0.973524
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	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
$501$	33.3050	1.48796
$502$	0	0
$503$	−37.7082	−1.68133	−0.840663	−	0.541559i	$-0.817834\pi$
$503$	−37.7082	−1.68133	−0.840663	+	0.541559i	$0.817834\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	42.0689	1.86834
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	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
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	3.59675	0.157275	0.0786374	−	0.996903i	$-0.474943\pi$
$523$	3.59675	0.157275	0.0786374	+	0.996903i	$0.474943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	0	0
$532$	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
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	0	0
$543$	−6.47214	−0.277746
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.2361	1.50659	0.753293	−	0.657685i	$-0.228464\pi$
$547$	35.2361	1.50659	0.753293	+	0.657685i	$0.228464\pi$
$548$	0	0
$549$	100.249	4.27853
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.5410	1.37144	0.685720	−	0.727865i	$-0.259487\pi$
$563$	32.5410	1.37144	0.685720	+	0.727865i	$0.259487\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.6525	0.783330
$568$	0	0
$569$	−31.3050	−1.31237	−0.656186	−	0.754599i	$-0.727831\pi$
$569$	−31.3050	−1.31237	−0.656186	+	0.754599i	$0.727831\pi$
$570$	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
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.5279	0.561230
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.5410	−1.67331	−0.836653	−	0.547733i	$-0.815491\pi$
$587$	−40.5410	−1.67331	−0.836653	+	0.547733i	$0.815491\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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
$601$	−40.2492	−1.64180	−0.820900	−	0.571072i	$-0.806528\pi$
$601$	−40.2492	−1.64180	−0.820900	+	0.571072i	$0.806528\pi$
$602$	0	0
$603$	61.1246	2.48919
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−44.1803	−1.79322	−0.896612	−	0.442816i	$-0.853979\pi$
$607$	−44.1803	−1.79322	−0.896612	+	0.442816i	$0.853979\pi$
$608$	0	0
$609$	−14.8328	−0.601056
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	82.6099	3.31502
$622$	0	0
$623$	4.58359	0.183638
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	49.1935	1.94303	0.971513	−	0.236986i	$-0.0761595\pi$
$641$	49.1935	1.94303	0.971513	+	0.236986i	$0.0761595\pi$
$642$	0	0
$643$	−50.0689	−1.97452	−0.987262	−	0.159103i	$-0.949140\pi$
$643$	−50.0689	−1.97452	−0.987262	+	0.159103i	$0.949140\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.5410	0.807551	0.403775	−	0.914858i	$-0.367698\pi$
$647$	20.5410	0.807551	0.403775	+	0.914858i	$0.367698\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−40.2492	−1.56551	−0.782757	−	0.622328i	$-0.786187\pi$
$661$	−40.2492	−1.56551	−0.782757	+	0.622328i	$0.786187\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.2492	−1.32614
$668$	0	0
$669$	−75.1935	−2.90715
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−42.4721	−1.62754
$682$	0	0
$683$	10.8754	0.416135	0.208068	−	0.978114i	$-0.433283\pi$
$683$	10.8754	0.416135	0.208068	+	0.978114i	$0.433283\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−45.3050	−1.72849
$688$	0	0
$689$	0	0
$690$	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
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.7508	0.517151
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	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
$721$	15.4164	0.574137
$722$	0	0
$723$	−43.4164	−1.61467
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.0132	−1.29857	−0.649283	−	0.760547i	$-0.724931\pi$
$727$	−35.0132	−1.29857	−0.649283	+	0.760547i	$0.724931\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	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
$743$	−52.5410	−1.92754	−0.963772	−	0.266729i	$-0.914057\pi$
$743$	−52.5410	−1.92754	−0.963772	+	0.266729i	$0.914057\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	132.318	4.84127
$748$	0	0
$749$	4.80650	0.175626
$750$	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
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−10.2492	−0.371047
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	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
$783$	−86.8328	−3.10315
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.06888	0.0737477	0.0368739	−	0.999320i	$-0.488260\pi$
$787$	2.06888	0.0737477	0.0368739	+	0.999320i	$0.488260\pi$
$788$	0	0
$789$	−29.5279	−1.05122
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	44.8328	1.58409
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−72.3607	−2.54722
$808$	0	0
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	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
$821$	31.3050	1.09255	0.546275	−	0.837606i	$-0.316045\pi$
$821$	31.3050	1.09255	0.546275	+	0.837606i	$0.316045\pi$
$822$	0	0
$823$	−27.8197	−0.969732	−0.484866	−	0.874588i	$-0.661132\pi$
$823$	−27.8197	−0.969732	−0.484866	+	0.874588i	$0.661132\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−56.5410	−1.96612	−0.983062	−	0.183274i	$-0.941331\pi$
$827$	−56.5410	−1.96612	−0.983062	+	0.183274i	$0.941331\pi$
$828$	0	0
$829$	−13.4164	−0.465971	−0.232986	−	0.972480i	$-0.574849\pi$
$829$	−13.4164	−0.465971	−0.232986	+	0.972480i	$0.574849\pi$
$830$	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		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−101.305	−3.48913
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.40325	−0.288739
$848$	0	0
$849$	104.138	3.57400
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	11.0557	0.376778
$862$	0	0
$863$	34.2918	1.16731	0.583653	−	0.812003i	$-0.301623\pi$
$863$	34.2918	1.16731	0.583653	+	0.812003i	$0.301623\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	55.0132	1.86834
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−58.1378	−1.95871	−0.979356	−	0.202145i	$-0.935209\pi$
$881$	−58.1378	−1.95871	−0.979356	+	0.202145i	$0.935209\pi$
$882$	0	0
$883$	54.6525	1.83920	0.919601	−	0.392853i	$-0.128512\pi$
$883$	54.6525	1.83920	0.919601	+	0.392853i	$0.128512\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−57.1246	−1.91806	−0.959028	−	0.283310i	$-0.908567\pi$
$887$	−57.1246	−1.91806	−0.959028	+	0.283310i	$0.908567\pi$
$888$	0	0
$889$	14.2492	0.477904
$890$	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
$901$	0	0
$902$	0	0
$903$	27.7771	0.924364
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−45.4853	−1.51031	−0.755157	−	0.655544i	$-0.772439\pi$
$907$	−45.4853	−1.51031	−0.755157	+	0.655544i	$0.772439\pi$
$908$	0	0
$909$	134.498	4.46103
$910$	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
$921$	89.3050	2.94270
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	150.790	4.95260
$928$	0	0
$929$	−49.1935	−1.61399	−0.806993	−	0.590561i	$-0.798907\pi$
$929$	−49.1935	−1.61399	−0.806993	+	0.590561i	$0.798907\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	25.5279	0.831302
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.7082	−1.61530	−0.807650	−	0.589662i	$-0.799261\pi$
$947$	−49.7082	−1.61530	−0.807650	+	0.589662i	$0.799261\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	47.0132	1.51498
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−62.0689	−1.99600	−0.998000	−	0.0632081i	$-0.979867\pi$
$967$	−62.0689	−1.99600	−0.998000	+	0.0632081i	$0.979867\pi$
$968$	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
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−100.249	−3.20071
$982$	0	0
$983$	−4.54102	−0.144836	−0.0724180	−	0.997374i	$-0.523072\pi$
$983$	−4.54102	−0.144836	−0.0724180	+	0.997374i	$0.523072\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−33.8885	−1.07868
$988$	0	0
$989$	64.1378	2.03946
$990$	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
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.a.z.1.1		2
4.3	odd	2		1600.2.a.bd.1.2		2
5.2	odd	4		320.2.c.d.129.4		4
5.3	odd	4		320.2.c.d.129.1		4
5.4	even	2		1600.2.a.bd.1.2		2
8.3	odd	2		800.2.a.j.1.1		2
8.5	even	2		800.2.a.n.1.2		2
15.2	even	4		2880.2.f.w.1729.4		4
15.8	even	4		2880.2.f.w.1729.3		4
20.3	even	4		320.2.c.d.129.4		4
20.7	even	4		320.2.c.d.129.1		4
20.19	odd	2	CM	1600.2.a.z.1.1		2
24.5	odd	2		7200.2.a.cr.1.1		2
24.11	even	2		7200.2.a.cb.1.2		2
40.3	even	4		160.2.c.b.129.1	✓	4
40.13	odd	4		160.2.c.b.129.4	yes	4
40.19	odd	2		800.2.a.n.1.2		2
40.27	even	4		160.2.c.b.129.4	yes	4
40.29	even	2		800.2.a.j.1.1		2
40.37	odd	4		160.2.c.b.129.1	✓	4
60.23	odd	4		2880.2.f.w.1729.4		4
60.47	odd	4		2880.2.f.w.1729.3		4
80.3	even	4		1280.2.f.g.129.2		4
80.13	odd	4		1280.2.f.h.129.4		4
80.27	even	4		1280.2.f.g.129.1		4
80.37	odd	4		1280.2.f.h.129.3		4
80.43	even	4		1280.2.f.h.129.3		4
80.53	odd	4		1280.2.f.g.129.1		4
80.67	even	4		1280.2.f.h.129.4		4
80.77	odd	4		1280.2.f.g.129.2		4
120.29	odd	2		7200.2.a.cb.1.2		2
120.53	even	4		1440.2.f.i.289.1		4
120.59	even	2		7200.2.a.cr.1.1		2
120.77	even	4		1440.2.f.i.289.2		4
120.83	odd	4		1440.2.f.i.289.2		4
120.107	odd	4		1440.2.f.i.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.c.b.129.1	✓	4	40.3	even	4
160.2.c.b.129.1	✓	4	40.37	odd	4
160.2.c.b.129.4	yes	4	40.13	odd	4
160.2.c.b.129.4	yes	4	40.27	even	4
320.2.c.d.129.1		4	5.3	odd	4
320.2.c.d.129.1		4	20.7	even	4
320.2.c.d.129.4		4	5.2	odd	4
320.2.c.d.129.4		4	20.3	even	4
800.2.a.j.1.1		2	8.3	odd	2
800.2.a.j.1.1		2	40.29	even	2
800.2.a.n.1.2		2	8.5	even	2
800.2.a.n.1.2		2	40.19	odd	2
1280.2.f.g.129.1		4	80.27	even	4
1280.2.f.g.129.1		4	80.53	odd	4
1280.2.f.g.129.2		4	80.3	even	4
1280.2.f.g.129.2		4	80.77	odd	4
1280.2.f.h.129.3		4	80.37	odd	4
1280.2.f.h.129.3		4	80.43	even	4
1280.2.f.h.129.4		4	80.13	odd	4
1280.2.f.h.129.4		4	80.67	even	4
1440.2.f.i.289.1		4	120.53	even	4
1440.2.f.i.289.1		4	120.107	odd	4
1440.2.f.i.289.2		4	120.77	even	4
1440.2.f.i.289.2		4	120.83	odd	4
1600.2.a.z.1.1		2	1.1	even	1	trivial
1600.2.a.z.1.1		2	20.19	odd	2	CM
1600.2.a.bd.1.2		2	4.3	odd	2
1600.2.a.bd.1.2		2	5.4	even	2
2880.2.f.w.1729.3		4	15.8	even	4
2880.2.f.w.1729.3		4	60.47	odd	4
2880.2.f.w.1729.4		4	15.2	even	4
2880.2.f.w.1729.4		4	60.23	odd	4
7200.2.a.cb.1.2		2	24.11	even	2
7200.2.a.cb.1.2		2	120.29	odd	2
7200.2.a.cr.1.1		2	24.5	odd	2
7200.2.a.cr.1.1		2	120.59	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} +0.763932 q^{7} +7.47214 q^{9} +O(q^{10})\)	\(q-3.23607 q^{3} +0.763932 q^{7} +7.47214 q^{9} -2.47214 q^{21} -5.70820 q^{23} -14.4721 q^{27} +6.00000 q^{29} -4.47214 q^{41} -11.2361 q^{43} +13.7082 q^{47} -6.41641 q^{49} +13.4164 q^{61} +5.70820 q^{63} +8.18034 q^{67} +18.4721 q^{69} +24.4164 q^{81} +17.7082 q^{83} -19.4164 q^{87} +6.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} + 6 q^{7} + 6 q^{9} + 4 q^{21} + 2 q^{23} - 20 q^{27} + 12 q^{29} - 18 q^{43} + 14 q^{47} + 14 q^{49} - 2 q^{63} - 6 q^{67} + 28 q^{69} + 22 q^{81} + 22 q^{83} - 12 q^{87} + 12 q^{89}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^7 + 6 * q^9 + 4 * q^21 + 2 * q^23 - 20 * q^27 + 12 * q^29 - 18 * q^43 + 14 * q^47 + 14 * q^49 - 2 * q^63 - 6 * q^67 + 28 * q^69 + 22 * q^81 + 22 * q^83 - 12 * q^87 + 12 * q^89

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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