LMFDB - Embedded newform 1280.3.h.g.1279.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(1279,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1280.1279");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1279.1
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	1280.1279

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.99070 q^{3} +5.00000 q^{5} -6.65841 q^{7} +26.8885 q^{9} -29.9535 q^{15} +39.8885 q^{21} +40.5995 q^{23} +25.0000 q^{25} -107.165 q^{27} -53.6656 q^{29} -33.2920 q^{35} -53.6656 q^{41} -25.2796 q^{43} +134.443 q^{45} +69.2363 q^{47} -4.66563 q^{49} -58.0000 q^{61} -179.035 q^{63} +34.5902 q^{67} -243.220 q^{69} -149.768 q^{75} +399.997 q^{81} +50.6150 q^{83} +321.495 q^{87} +142.000 q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{5} + 36 q^{9} + 88 q^{21} + 100 q^{25} + 180 q^{45} + 196 q^{49} - 232 q^{61} - 472 q^{69} + 956 q^{81} + 568 q^{89}+O(q^{100}) $ 4 * q + 20 * q^5 + 36 * q^9 + 88 * q^21 + 100 * q^25 + 180 * q^45 + 196 * q^49 - 232 * q^61 - 472 * q^69 + 956 * q^81 + 568 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−5.99070	−1.99690	−0.998451	−	0.0556418i	$-0.982280\pi$
$3$	−5.99070	−1.99690	−0.998451	+	0.0556418i	$0.982280\pi$
$4$	0	0
$5$	5.00000	1.00000
$5$	5.00000	1.00000
$6$	0	0
$7$	−6.65841	−0.951201	−0.475600	−	0.879661i	$-0.657769\pi$
$7$	−6.65841	−0.951201	−0.475600	+	0.879661i	$0.657769\pi$
$8$	0	0
$9$	26.8885	2.98762
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−29.9535	−1.99690
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	39.8885	1.89945
$22$	0	0
$23$	40.5995	1.76520	0.882599	−	0.470128i	$-0.155792\pi$
$23$	40.5995	1.76520	0.882599	+	0.470128i	$0.155792\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	0	0
$27$	−107.165	−3.96907
$28$	0	0
$29$	−53.6656	−1.85054	−0.925270	−	0.379310i	$-0.876161\pi$
$29$	−53.6656	−1.85054	−0.925270	+	0.379310i	$0.876161\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−33.2920	−0.951201
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−53.6656	−1.30892	−0.654459	−	0.756098i	$-0.727104\pi$
$41$	−53.6656	−1.30892	−0.654459	+	0.756098i	$0.727104\pi$
$42$	0	0
$43$	−25.2796	−0.587898	−0.293949	−	0.955821i	$-0.594970\pi$
$43$	−25.2796	−0.587898	−0.293949	+	0.955821i	$0.594970\pi$
$44$	0	0
$45$	134.443	2.98762
$46$	0	0
$47$	69.2363	1.47311	0.736556	−	0.676377i	$-0.236451\pi$
$47$	69.2363	1.47311	0.736556	+	0.676377i	$0.236451\pi$
$48$	0	0
$49$	−4.66563	−0.0952170
$50$	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
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−58.0000	−0.950820	−0.475410	−	0.879764i	$-0.657700\pi$
$61$	−58.0000	−0.950820	−0.475410	+	0.879764i	$0.657700\pi$
$62$	0	0
$63$	−179.035	−2.84182
$64$	0	0
$65$	0	0
$66$	0	0
$67$	34.5902	0.516272	0.258136	−	0.966109i	$-0.416892\pi$
$67$	34.5902	0.516272	0.258136	+	0.966109i	$0.416892\pi$
$68$	0	0
$69$	−243.220	−3.52492
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−149.768	−1.99690
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	399.997	4.93823
$82$	0	0
$83$	50.6150	0.609820	0.304910	−	0.952381i	$-0.401374\pi$
$83$	50.6150	0.609820	0.304910	+	0.952381i	$0.401374\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	321.495	3.69534
$88$	0	0
$89$	142.000	1.59551	0.797753	−	0.602985i	$-0.206022\pi$
$89$	142.000	1.59551	0.797753	+	0.602985i	$0.206022\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	160.997	1.59403	0.797014	−	0.603960i	$-0.206411\pi$
$101$	160.997	1.59403	0.797014	+	0.603960i	$0.206411\pi$
$102$	0	0
$103$	−111.190	−1.07951	−0.539756	−	0.841821i	$-0.681484\pi$
$103$	−111.190	−1.07951	−0.539756	+	0.841821i	$0.681484\pi$
$104$	0	0
$105$	199.443	1.89945
$106$	0	0
$107$	211.010	1.97206	0.986028	−	0.166578i	$-0.0532716\pi$
$107$	211.010	1.97206	0.986028	+	0.166578i	$0.0532716\pi$
$108$	0	0
$109$	−38.0000	−0.348624	−0.174312	−	0.984690i	$-0.555770\pi$
$109$	−38.0000	−0.348624	−0.174312	+	0.984690i	$0.555770\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	202.998	1.76520
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	321.495	2.61378
$124$	0	0
$125$	125.000	1.00000
$126$	0	0
$127$	100.469	0.791097	0.395549	−	0.918445i	$-0.370554\pi$
$127$	100.469	0.791097	0.395549	+	0.918445i	$0.370554\pi$
$128$	0	0
$129$	151.443	1.17397
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−535.825	−3.96907
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−414.774	−2.94166
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−268.328	−1.85054
$146$	0	0
$147$	27.9504	0.190139
$148$	0	0
$149$	278.000	1.86577	0.932886	−	0.360172i	$-0.117282\pi$
$149$	278.000	1.86577	0.932886	+	0.360172i	$0.117282\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−270.328	−1.67906
$162$	0	0
$163$	83.2580	0.510785	0.255393	−	0.966837i	$-0.417795\pi$
$163$	83.2580	0.510785	0.255393	+	0.966837i	$0.417795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.2579	−0.0674125	−0.0337063	−	0.999432i	$-0.510731\pi$
$167$	−11.2579	−0.0674125	−0.0337063	+	0.999432i	$0.510731\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−166.460	−0.951201
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−53.6656	−0.296495	−0.148248	−	0.988950i	$-0.547363\pi$
$181$	−53.6656	−0.296495	−0.148248	+	0.988950i	$0.547363\pi$
$182$	0	0
$183$	347.461	1.89869
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	713.548	3.77539
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−207.220	−1.03094
$202$	0	0
$203$	357.328	1.76023
$204$	0	0
$205$	−268.328	−1.30892
$206$	0	0
$207$	1091.66	5.27373
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−126.398	−0.587898
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−374.706	−1.68030	−0.840149	−	0.542356i	$-0.817532\pi$
$223$	−374.706	−1.68030	−0.840149	+	0.542356i	$0.817532\pi$
$224$	0	0
$225$	672.214	2.98762
$226$	0	0
$227$	−52.5065	−0.231306	−0.115653	−	0.993290i	$-0.536896\pi$
$227$	−52.5065	−0.231306	−0.115653	+	0.993290i	$0.536896\pi$
$228$	0	0
$229$	375.659	1.64043	0.820217	−	0.572052i	$-0.193853\pi$
$229$	375.659	1.64043	0.820217	+	0.572052i	$0.193853\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	346.181	1.47311
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	375.659	1.55875	0.779376	−	0.626556i	$-0.215536\pi$
$241$	375.659	1.55875	0.779376	+	0.626556i	$0.215536\pi$
$242$	0	0
$243$	−1431.78	−5.89209
$244$	0	0
$245$	−23.3282	−0.0952170
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−303.220	−1.21775
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	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
$261$	−1442.99	−5.52870
$262$	0	0
$263$	−513.884	−1.95393	−0.976965	−	0.213398i	$-0.931547\pi$
$263$	−513.884	−1.95393	−0.976965	+	0.213398i	$0.931547\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−850.680	−3.18607
$268$	0	0
$269$	38.0000	0.141264	0.0706320	−	0.997502i	$-0.477498\pi$
$269$	38.0000	0.141264	0.0706320	+	0.997502i	$0.477498\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	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
$281$	375.659	1.33687	0.668433	−	0.743772i	$-0.266965\pi$
$281$	375.659	1.33687	0.668433	+	0.743772i	$0.266965\pi$
$282$	0	0
$283$	−108.593	−0.383722	−0.191861	−	0.981422i	$-0.561452\pi$
$283$	−108.593	−0.383722	−0.191861	+	0.981422i	$0.561452\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	357.328	1.24504
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	168.322	0.559209
$302$	0	0
$303$	−964.485	−3.18312
$304$	0	0
$305$	−290.000	−0.950820
$306$	0	0
$307$	525.791	1.71267	0.856337	−	0.516418i	$-0.172735\pi$
$307$	525.791	1.71267	0.856337	+	0.516418i	$0.172735\pi$
$308$	0	0
$309$	666.105	2.15568
$310$	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
$315$	−895.174	−2.84182
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1264.10	−3.93800
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	227.647	0.696168
$328$	0	0
$329$	−461.003	−1.40123
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	172.951	0.516272
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	357.328	1.04177
$344$	0	0
$345$	−1216.10	−3.52492
$346$	0	0
$347$	565.853	1.63070	0.815350	−	0.578968i	$-0.196545\pi$
$347$	565.853	1.63070	0.815350	+	0.578968i	$0.196545\pi$
$348$	0	0
$349$	−697.653	−1.99901	−0.999503	−	0.0315186i	$-0.989966\pi$
$349$	−697.653	−1.99901	−0.999503	+	0.0315186i	$0.989966\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	−724.875	−1.99690
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−426.564	−1.16230	−0.581150	−	0.813797i	$-0.697397\pi$
$367$	−426.564	−1.16230	−0.581150	+	0.813797i	$0.697397\pi$
$368$	0	0
$369$	−1442.99	−3.91054
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−748.838	−1.99690
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−601.882	−1.57974
$382$	0	0
$383$	571.862	1.49311	0.746556	−	0.665322i	$-0.231706\pi$
$383$	571.862	1.49311	0.746556	+	0.665322i	$0.231706\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−679.732	−1.75641
$388$	0	0
$389$	−202.000	−0.519280	−0.259640	−	0.965705i	$-0.583604\pi$
$389$	−202.000	−0.519280	−0.259640	+	0.965705i	$0.583604\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−478.000	−1.19202	−0.596010	−	0.802977i	$-0.703248\pi$
$401$	−478.000	−1.19202	−0.596010	+	0.802977i	$0.703248\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1999.98	4.93823
$406$	0	0
$407$	0	0
$408$	0	0
$409$	160.997	0.393635	0.196818	−	0.980440i	$-0.436939\pi$
$409$	160.997	0.393635	0.196818	+	0.980440i	$0.436939\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	253.075	0.609820
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−778.000	−1.84798	−0.923990	−	0.382415i	$-0.875092\pi$
$421$	−778.000	−1.84798	−0.923990	+	0.382415i	$0.875092\pi$
$422$	0	0
$423$	1861.66	4.40109
$424$	0	0
$425$	0	0
$426$	0	0
$427$	386.188	0.904420
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	1607.47	3.69534
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−125.452	−0.284472
$442$	0	0
$443$	837.975	1.89159	0.945796	−	0.324762i	$-0.105284\pi$
$443$	837.975	1.89159	0.945796	+	0.324762i	$0.105284\pi$
$444$	0	0
$445$	710.000	1.59551
$446$	0	0
$447$	−1665.42	−3.72576
$448$	0	0
$449$	804.984	1.79284	0.896419	−	0.443207i	$-0.146159\pi$
$449$	804.984	1.79284	0.896419	+	0.443207i	$0.146159\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	375.659	0.814879	0.407440	−	0.913232i	$-0.366422\pi$
$461$	375.659	0.814879	0.407440	+	0.913232i	$0.366422\pi$
$462$	0	0
$463$	−170.243	−0.367696	−0.183848	−	0.982955i	$-0.558855\pi$
$463$	−170.243	−0.367696	−0.183848	+	0.982955i	$0.558855\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	742.273	1.58945	0.794725	−	0.606970i	$-0.207615\pi$
$467$	742.273	1.58945	0.794725	+	0.606970i	$0.207615\pi$
$468$	0	0
$469$	−230.316	−0.491078
$470$	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	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1619.46	3.35291
$484$	0	0
$485$	0	0
$486$	0	0
$487$	939.910	1.93000	0.965000	−	0.262249i	$-0.0844643\pi$
$487$	939.910	1.93000	0.965000	+	0.262249i	$0.0844643\pi$
$488$	0	0
$489$	−498.774	−1.01999
$490$	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
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	67.4427	0.134616
$502$	0	0
$503$	−941.802	−1.87237	−0.936185	−	0.351509i	$-0.885669\pi$
$503$	−941.802	−1.87237	−0.936185	+	0.351509i	$0.885669\pi$
$504$	0	0
$505$	804.984	1.59403
$506$	0	0
$507$	−1012.43	−1.99690
$508$	0	0
$509$	−268.328	−0.527167	−0.263584	−	0.964637i	$-0.584905\pi$
$509$	−268.328	−0.527167	−0.263584	+	0.964637i	$0.584905\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−555.949	−1.07951
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−722.000	−1.38580	−0.692898	−	0.721035i	$-0.743667\pi$
$521$	−722.000	−1.38580	−0.692898	+	0.721035i	$0.743667\pi$
$522$	0	0
$523$	614.521	1.17499	0.587496	−	0.809227i	$-0.300114\pi$
$523$	614.521	1.17499	0.587496	+	0.809227i	$0.300114\pi$
$524$	0	0
$525$	997.214	1.89945
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1119.32	2.11592
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1055.05	1.97206
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1019.65	1.88474	0.942372	−	0.334566i	$-0.108590\pi$
$541$	1019.65	1.88474	0.942372	+	0.334566i	$0.108590\pi$
$542$	0	0
$543$	321.495	0.592072
$544$	0	0
$545$	−190.000	−0.348624
$546$	0	0
$547$	662.149	1.21051	0.605255	−	0.796032i	$-0.293071\pi$
$547$	662.149	1.21051	0.605255	+	0.796032i	$0.293071\pi$
$548$	0	0
$549$	−1559.54	−2.84068
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−747.354	−1.32745	−0.663725	−	0.747977i	$-0.731025\pi$
$563$	−747.354	−1.32745	−0.663725	+	0.747977i	$0.731025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2663.34	−4.69725
$568$	0	0
$569$	−1126.98	−1.98063	−0.990315	−	0.138840i	$-0.955663\pi$
$569$	−1126.98	−1.98063	−0.990315	+	0.138840i	$0.955663\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1014.99	1.76520
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−337.016	−0.580061
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1092.89	1.86182	0.930908	−	0.365254i	$-0.119018\pi$
$587$	1092.89	1.86182	0.930908	+	0.365254i	$0.119018\pi$
$588$	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
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1126.98	−1.87517	−0.937586	−	0.347754i	$-0.886945\pi$
$601$	−1126.98	−1.87517	−0.937586	+	0.347754i	$0.886945\pi$
$602$	0	0
$603$	930.081	1.54242
$604$	0	0
$605$	605.000	1.00000
$606$	0	0
$607$	1203.43	1.98258	0.991291	−	0.131693i	$-0.0420413\pi$
$607$	1203.43	1.98258	0.991291	+	0.131693i	$0.0420413\pi$
$608$	0	0
$609$	−2140.64	−3.51501
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	1607.47	2.61378
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−4350.85	−7.00620
$622$	0	0
$623$	−945.494	−1.51765
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	502.347	0.791097
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	590.322	0.920939	0.460470	−	0.887676i	$-0.347681\pi$
$641$	590.322	0.920939	0.460470	+	0.887676i	$0.347681\pi$
$642$	0	0
$643$	1148.97	1.78689	0.893447	−	0.449168i	$-0.148280\pi$
$643$	1148.97	1.78689	0.893447	+	0.449168i	$0.148280\pi$
$644$	0	0
$645$	757.214	1.17397
$646$	0	0
$647$	−1292.64	−1.99790	−0.998948	−	0.0458655i	$-0.985395\pi$
$647$	−1292.64	−1.99790	−0.998948	+	0.0458655i	$0.985395\pi$
$648$	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
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	298.000	0.450832	0.225416	−	0.974263i	$-0.427626\pi$
$661$	298.000	0.450832	0.225416	+	0.974263i	$0.427626\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2178.80	−3.26657
$668$	0	0
$669$	2244.76	3.35539
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−2679.12	−3.96907
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	314.551	0.461896
$682$	0	0
$683$	−1275.43	−1.86739	−0.933695	−	0.358070i	$-0.883435\pi$
$683$	−1275.43	−1.86739	−0.933695	+	0.358070i	$0.883435\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2250.46	−3.27579
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	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
$701$	−902.000	−1.28673	−0.643367	−	0.765558i	$-0.722463\pi$
$701$	−902.000	−1.28673	−0.643367	+	0.765558i	$0.722463\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−2073.87	−2.94166
$706$	0	0
$707$	−1071.98	−1.51624
$708$	0	0
$709$	1234.31	1.74092	0.870458	−	0.492243i	$-0.163823\pi$
$709$	1234.31	1.74092	0.870458	+	0.492243i	$0.163823\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	740.347	1.02683
$722$	0	0
$723$	−2250.46	−3.11268
$724$	0	0
$725$	−1341.64	−1.85054
$726$	0	0
$727$	−1176.68	−1.61854	−0.809272	−	0.587434i	$-0.800138\pi$
$727$	−1176.68	−1.61854	−0.809272	+	0.587434i	$0.800138\pi$
$728$	0	0
$729$	4977.39	6.82770
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	139.752	0.190139
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	361.020	0.485894	0.242947	−	0.970040i	$-0.421886\pi$
$743$	361.020	0.485894	0.242947	+	0.970040i	$0.421886\pi$
$744$	0	0
$745$	1390.00	1.86577
$746$	0	0
$747$	1360.96	1.82191
$748$	0	0
$749$	−1404.99	−1.87582
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	242.000	0.318003	0.159001	−	0.987278i	$-0.449173\pi$
$761$	242.000	0.318003	0.159001	+	0.987278i	$0.449173\pi$
$762$	0	0
$763$	253.019	0.331611
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1342.00	1.74512	0.872562	−	0.488504i	$-0.162457\pi$
$769$	1342.00	1.74512	0.872562	+	0.488504i	$0.162457\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	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
$783$	5751.08	7.34493
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1027.94	−1.30614	−0.653072	−	0.757296i	$-0.726520\pi$
$787$	−1027.94	−1.30614	−0.653072	+	0.757296i	$0.726520\pi$
$788$	0	0
$789$	3078.53	3.90181
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	3818.17	4.76676
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1351.64	−1.67906
$806$	0	0
$807$	−227.647	−0.282090
$808$	0	0
$809$	−1298.00	−1.60445	−0.802225	−	0.597022i	$-0.796351\pi$
$809$	−1298.00	−1.60445	−0.802225	+	0.597022i	$0.796351\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	416.290	0.510785
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−662.000	−0.806334	−0.403167	−	0.915126i	$-0.632091\pi$
$821$	−662.000	−0.806334	−0.403167	+	0.915126i	$0.632091\pi$
$822$	0	0
$823$	−370.477	−0.450154	−0.225077	−	0.974341i	$-0.572263\pi$
$823$	−370.477	−0.450154	−0.225077	+	0.974341i	$0.572263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1512.42	1.82880	0.914402	−	0.404807i	$-0.132661\pi$
$827$	1512.42	1.82880	0.914402	+	0.404807i	$0.132661\pi$
$828$	0	0
$829$	1478.00	1.78287	0.891435	−	0.453148i	$-0.149699\pi$
$829$	1478.00	1.78287	0.891435	+	0.453148i	$0.149699\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−56.2895	−0.0674125
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2039.00	2.42449
$842$	0	0
$843$	−2250.46	−2.66959
$844$	0	0
$845$	845.000	1.00000
$846$	0	0
$847$	−805.667	−0.951201
$848$	0	0
$849$	650.551	0.766256
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	−2140.64	−2.48623
$862$	0	0
$863$	767.867	0.889764	0.444882	−	0.895589i	$-0.353246\pi$
$863$	767.867	0.889764	0.444882	+	0.895589i	$0.353246\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1731.31	−1.99690
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−832.301	−0.951201
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−697.653	−0.791888	−0.395944	−	0.918275i	$-0.629583\pi$
$881$	−697.653	−0.791888	−0.395944	+	0.918275i	$0.629583\pi$
$882$	0	0
$883$	38.9665	0.0441296	0.0220648	−	0.999757i	$-0.492976\pi$
$883$	38.9665	0.0441296	0.0220648	+	0.999757i	$0.492976\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1734.69	1.95568	0.977841	−	0.209350i	$-0.0671349\pi$
$887$	1734.69	1.95568	0.977841	+	0.209350i	$0.0671349\pi$
$888$	0	0
$889$	−668.966	−0.752493
$890$	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
$903$	−1008.37	−1.11669
$904$	0	0
$905$	−268.328	−0.296495
$906$	0	0
$907$	−1450.21	−1.59891	−0.799456	−	0.600724i	$-0.794879\pi$
$907$	−1450.21	−1.59891	−0.799456	+	0.600724i	$0.794879\pi$
$908$	0	0
$909$	4328.97	4.76235
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1737.30	1.89869
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−3149.86	−3.42004
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2989.73	−3.22517
$928$	0	0
$929$	−1770.97	−1.90631	−0.953157	−	0.302476i	$-0.902187\pi$
$929$	−1770.97	−1.90631	−0.953157	+	0.302476i	$0.902187\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1878.30	1.99606	0.998032	−	0.0626993i	$-0.0199709\pi$
$941$	1878.30	1.99606	0.998032	+	0.0626993i	$0.0199709\pi$
$942$	0	0
$943$	−2178.80	−2.31050
$944$	0	0
$945$	3567.74	3.77539
$946$	0	0
$947$	−867.687	−0.916248	−0.458124	−	0.888888i	$-0.651478\pi$
$947$	−867.687	−0.916248	−0.458124	+	0.888888i	$0.651478\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	5673.75	5.89175
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1529.15	−1.58134	−0.790668	−	0.612246i	$-0.790266\pi$
$967$	−1529.15	−1.58134	−0.790668	+	0.612246i	$0.790266\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1021.76	−1.04155
$982$	0	0
$983$	−1576.41	−1.60367	−0.801836	−	0.597544i	$-0.796143\pi$
$983$	−1576.41	−1.60367	−0.801836	+	0.597544i	$0.796143\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2761.73	2.79811
$988$	0	0
$989$	−1026.34	−1.03776
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.h.g.1279.1		4
4.3	odd	2	inner	1280.3.h.g.1279.4		4
5.4	even	2	inner	1280.3.h.g.1279.4		4
8.3	odd	2		1280.3.h.e.1279.1		4
8.5	even	2		1280.3.h.e.1279.4		4
16.3	odd	4		640.3.e.h.319.7	yes	8
16.5	even	4		640.3.e.h.319.8	yes	8
16.11	odd	4		640.3.e.h.319.2	yes	8
16.13	even	4		640.3.e.h.319.1	✓	8
20.19	odd	2	CM	1280.3.h.g.1279.1		4
40.19	odd	2		1280.3.h.e.1279.4		4
40.29	even	2		1280.3.h.e.1279.1		4
80.19	odd	4		640.3.e.h.319.1	✓	8
80.29	even	4		640.3.e.h.319.7	yes	8
80.59	odd	4		640.3.e.h.319.8	yes	8
80.69	even	4		640.3.e.h.319.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.e.h.319.1	✓	8	16.13	even	4
640.3.e.h.319.1	✓	8	80.19	odd	4
640.3.e.h.319.2	yes	8	16.11	odd	4
640.3.e.h.319.2	yes	8	80.69	even	4
640.3.e.h.319.7	yes	8	16.3	odd	4
640.3.e.h.319.7	yes	8	80.29	even	4
640.3.e.h.319.8	yes	8	16.5	even	4
640.3.e.h.319.8	yes	8	80.59	odd	4
1280.3.h.e.1279.1		4	8.3	odd	2
1280.3.h.e.1279.1		4	40.29	even	2
1280.3.h.e.1279.4		4	8.5	even	2
1280.3.h.e.1279.4		4	40.19	odd	2
1280.3.h.g.1279.1		4	1.1	even	1	trivial
1280.3.h.g.1279.1		4	20.19	odd	2	CM
1280.3.h.g.1279.4		4	4.3	odd	2	inner
1280.3.h.g.1279.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.99070 q^{3} +5.00000 q^{5} -6.65841 q^{7} +26.8885 q^{9} -29.9535 q^{15} +39.8885 q^{21} +40.5995 q^{23} +25.0000 q^{25} -107.165 q^{27} -53.6656 q^{29} -33.2920 q^{35} -53.6656 q^{41} -25.2796 q^{43} +134.443 q^{45} +69.2363 q^{47} -4.66563 q^{49} -58.0000 q^{61} -179.035 q^{63} +34.5902 q^{67} -243.220 q^{69} -149.768 q^{75} +399.997 q^{81} +50.6150 q^{83} +321.495 q^{87} +142.000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{5} + 36 q^{9} + 88 q^{21} + 100 q^{25} + 180 q^{45} + 196 q^{49} - 232 q^{61} - 472 q^{69} + 956 q^{81} + 568 q^{89}+O(q^{100}) \) 4 * q + 20 * q^5 + 36 * q^9 + 88 * q^21 + 100 * q^25 + 180 * q^45 + 196 * q^49 - 232 * q^61 - 472 * q^69 + 956 * q^81 + 568 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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