LMFDB - Embedded newform 1600.2.n.b.1343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, 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:	no
Analytic conductor:	$12.7760643234$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			1343.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.1343
Dual form			1600.2.n.b.1407.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{3} +(-3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(-1.00000 + 1.00000i) q^{3} +(-3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +6.00000 q^{21} +(1.00000 - 1.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} +6.00000i q^{29} +12.0000 q^{41} +(9.00000 - 9.00000i) q^{43} +(7.00000 + 7.00000i) q^{47} +11.0000i q^{49} +8.00000 q^{61} +(3.00000 - 3.00000i) q^{63} +(3.00000 + 3.00000i) q^{67} +2.00000i q^{69} +5.00000 q^{81} +(-11.0000 + 11.0000i) q^{83} +(-6.00000 - 6.00000i) q^{87} -6.00000i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 6 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 - 6 * q^7	$ 2 q - 2 q^{3} - 6 q^{7} + 12 q^{21} + 2 q^{23} - 8 q^{27} + 24 q^{41} + 18 q^{43} + 14 q^{47} + 16 q^{61} + 6 q^{63} + 6 q^{67} + 10 q^{81} - 22 q^{83} - 12 q^{87}+O(q^{100}) $ 2 * q - 2 * q^3 - 6 * q^7 + 12 * q^21 + 2 * q^23 - 8 * q^27 + 24 * q^41 + 18 * q^43 + 14 * q^47 + 16 * q^61 + 6 * q^63 + 6 * q^67 + 10 * q^81 - 22 * q^83 - 12 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.989524	−	0.144370i	$-0.953885\pi$
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.144370	−	0.989524i	$-0.546115\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$		0			0
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	0.803636	+	0.595121i	$0.202896\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.0000			1.87409			0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000			1.87409			0.937043	+	0.349215i	$0.113552\pi$
$42$		0			0
$43$	9.00000	−	9.00000i	1.37249	−	1.37249i	0.515745	−	0.856742i	$-0.327515\pi$
$43$	9.00000	−	9.00000i	1.37249	−	1.37249i	0.856742	−	0.515745i	$-0.172485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.00000	+	7.00000i	1.02105	+	1.02105i	0.999774	+	0.0212814i	$0.00677460\pi$
$47$	7.00000	+	7.00000i	1.02105	+	1.02105i	0.0212814	+	0.999774i	$0.493225\pi$
$48$		0			0
$49$			11.0000i			1.57143i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		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
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$	3.00000	−	3.00000i	0.377964	−	0.377964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	0.866202	−	0.499694i	$-0.166554\pi$
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	−0.499694	+	0.866202i	$0.666554\pi$
$68$		0			0
$69$			2.00000i			0.240772i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		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
$81$	5.00000			0.555556
$82$		0			0
$83$	−11.0000	+	11.0000i	−1.20741	+	1.20741i	−0.235543	+	0.971864i	$0.575687\pi$
$83$	−11.0000	+	11.0000i	−1.20741	+	1.20741i	−0.971864	+	0.235543i	$0.924313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.00000	−	6.00000i	−0.643268	−	0.643268i
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		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
$103$	−9.00000	+	9.00000i	−0.886796	+	0.886796i	−0.994214	−	0.107418i	$-0.965742\pi$
$103$	−9.00000	+	9.00000i	−0.886796	+	0.886796i	0.107418	+	0.994214i	$0.465742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.0000	+	13.0000i	1.25676	+	1.25676i	0.952632	+	0.304125i	$0.0983642\pi$
$107$	13.0000	+	13.0000i	1.25676	+	1.25676i	0.304125	+	0.952632i	$0.401636\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		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
$123$	−12.0000	+	12.0000i	−1.08200	+	1.08200i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.00000	−	3.00000i	−0.266207	−	0.266207i	0.561363	−	0.827570i	$-0.310277\pi$
$127$	−3.00000	−	3.00000i	−0.266207	−	0.266207i	−0.827570	+	0.561363i	$0.810277\pi$
$128$		0			0
$129$			18.0000i			1.58481i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−14.0000			−1.17901
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.0000	−	11.0000i	−0.907265	−	0.907265i
$148$		0			0
$149$		−	24.0000i		−	1.96616i	−0.183186	−	0.983078i	$-0.558641\pi$
$149$		−	24.0000i		−	1.96616i	0.183186	−	0.983078i	$-0.441359\pi$
$150$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	−0.260531	−	0.965465i	$-0.583898\pi$
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	0.965465	+	0.260531i	$0.0838976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	17.0000	+	17.0000i	1.31550	+	1.31550i	0.917298	+	0.398202i	$0.130366\pi$
$167$	17.0000	+	17.0000i	1.31550	+	1.31550i	0.398202	+	0.917298i	$0.369634\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		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	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	−8.00000	+	8.00000i	−0.591377	+	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			24.0000i			1.74574i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−6.00000			−0.423207
$202$		0			0
$203$	18.0000	−	18.0000i	1.26335	−	1.26335i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.00000	+	1.00000i	0.0695048	+	0.0695048i
$208$		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
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	21.0000	−	21.0000i	1.40626	−	1.40626i	0.628263	−	0.778001i	$-0.283766\pi$
$223$	21.0000	−	21.0000i	1.40626	−	1.40626i	0.778001	−	0.628263i	$-0.216234\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.00000	−	7.00000i	−0.464606	−	0.464606i	0.435556	−	0.900162i	$-0.356552\pi$
$227$	−7.00000	−	7.00000i	−0.464606	−	0.464606i	−0.900162	+	0.435556i	$0.856552\pi$
$228$		0			0
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		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
$241$	−28.0000			−1.80364			−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000			−1.80364			−0.901819	+	0.432113i	$0.857768\pi$
$242$		0			0
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	22.0000i		−	1.39419i
$250$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	11.0000	−	11.0000i	0.678289	−	0.678289i	−0.281324	−	0.959613i	$-0.590774\pi$
$263$	11.0000	−	11.0000i	0.678289	−	0.678289i	0.959613	+	0.281324i	$0.0907735\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	6.00000	+	6.00000i	0.367194	+	0.367194i
$268$		0			0
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	−21.0000	+	21.0000i	−1.24832	+	1.24832i	−0.291859	+	0.956461i	$0.594274\pi$
$283$	−21.0000	+	21.0000i	−1.24832	+	1.24832i	−0.956461	+	0.291859i	$0.905726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−36.0000	−	36.0000i	−2.12501	−	2.12501i
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−54.0000			−3.11251
$302$		0			0
$303$	−18.0000	+	18.0000i	−1.03407	+	1.03407i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	0.787515	−	0.616296i	$-0.211367\pi$
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$		−	18.0000i		−	1.02398i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−26.0000			−1.45118
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−16.0000	−	16.0000i	−0.884802	−	0.884802i
$328$		0			0
$329$		−	42.0000i		−	2.31553i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.0000	−	17.0000i	−0.912608	−	0.912608i	0.0838690	−	0.996477i	$-0.473272\pi$
$347$	−17.0000	−	17.0000i	−0.912608	−	0.912608i	−0.996477	+	0.0838690i	$0.973272\pi$
$348$		0			0
$349$			26.0000i			1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$			26.0000i			1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		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
$363$	−11.0000	+	11.0000i	−0.577350	+	0.577350i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	27.0000	+	27.0000i	1.40939	+	1.40939i	0.763055	+	0.646333i	$0.223698\pi$
$367$	27.0000	+	27.0000i	1.40939	+	1.40939i	0.646333	+	0.763055i	$0.276302\pi$
$368$		0			0
$369$			12.0000i			0.624695i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		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
$381$	6.00000			0.307389
$382$		0			0
$383$	−19.0000	+	19.0000i	−0.970855	+	0.970855i	−0.999587	−	0.0287325i	$-0.990853\pi$
$383$	−19.0000	+	19.0000i	−0.970855	+	0.970855i	0.0287325	+	0.999587i	$0.490853\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.00000	+	9.00000i	0.457496	+	0.457496i
$388$		0			0
$389$		−	24.0000i		−	1.21685i	−0.793612	−	0.608424i	$-0.791802\pi$
$389$		−	24.0000i		−	1.21685i	0.793612	−	0.608424i	$-0.208198\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		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
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	−7.00000	+	7.00000i	−0.340352	+	0.340352i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.0000	−	24.0000i	−1.16144	−	1.16144i
$428$		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		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		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
$441$	−11.0000			−0.523810
$442$		0			0
$443$	29.0000	−	29.0000i	1.37783	−	1.37783i	0.529558	−	0.848274i	$-0.322358\pi$
$443$	29.0000	−	29.0000i	1.37783	−	1.37783i	0.848274	−	0.529558i	$-0.177642\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	24.0000	+	24.0000i	1.13516	+	1.13516i
$448$		0			0
$449$		−	36.0000i		−	1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$		−	36.0000i		−	1.69895i	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		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−42.0000			−1.95614			−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000			−1.95614			−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$	−9.00000	+	9.00000i	−0.418265	+	0.418265i	−0.884606	−	0.466340i	$-0.845572\pi$
$463$	−9.00000	+	9.00000i	−0.418265	+	0.418265i	0.466340	+	0.884606i	$0.345572\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.0000	+	23.0000i	1.06431	+	1.06431i	0.997785	+	0.0665285i	$0.0211923\pi$
$467$	23.0000	+	23.0000i	1.06431	+	1.06431i	0.0665285	+	0.997785i	$0.478808\pi$
$468$		0			0
$469$		−	18.0000i		−	0.831163i
$470$		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
$483$	6.00000	−	6.00000i	0.273009	−	0.273009i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	27.0000	+	27.0000i	1.22349	+	1.22349i	0.966384	+	0.257103i	$0.0827679\pi$
$487$	27.0000	+	27.0000i	1.22349	+	1.22349i	0.257103	+	0.966384i	$0.417232\pi$
$488$		0			0
$489$			18.0000i			0.813988i
$490$		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			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−34.0000			−1.51901
$502$		0			0
$503$	31.0000	−	31.0000i	1.38222	−	1.38222i	0.541559	−	0.840663i	$-0.317834\pi$
$503$	31.0000	−	31.0000i	1.38222	−	1.38222i	0.840663	−	0.541559i	$-0.182166\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	13.0000	+	13.0000i	0.577350	+	0.577350i
$508$		0			0
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		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
$523$	−21.0000	+	21.0000i	−0.918266	+	0.918266i	−0.996903	−	0.0786374i	$-0.974943\pi$
$523$	−21.0000	+	21.0000i	−0.918266	+	0.918266i	0.0786374	+	0.996903i	$0.474943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			21.0000i			0.913043i
$530$		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	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	2.00000	−	2.00000i	0.0858282	−	0.0858282i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	33.0000	+	33.0000i	1.41098	+	1.41098i	0.753293	+	0.657685i	$0.228464\pi$
$547$	33.0000	+	33.0000i	1.41098	+	1.41098i	0.657685	+	0.753293i	$0.271536\pi$
$548$		0			0
$549$			8.00000i			0.341432i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.00000	+	1.00000i	−0.0421450	+	0.0421450i	−0.727865	−	0.685720i	$-0.759487\pi$
$563$	−1.00000	+	1.00000i	−0.0421450	+	0.0421450i	0.685720	+	0.727865i	$0.259487\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−15.0000	−	15.0000i	−0.629941	−	0.629941i
$568$		0			0
$569$		−	36.0000i		−	1.50920i	−0.656186	−	0.754599i	$-0.727831\pi$
$569$		−	36.0000i		−	1.50920i	0.656186	−	0.754599i	$-0.272169\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	66.0000			2.73814
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.00000	−	7.00000i	−0.288921	−	0.288921i	0.547733	−	0.836653i	$-0.315491\pi$
$587$	−7.00000	−	7.00000i	−0.288921	−	0.288921i	−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		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		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
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$		0			0
$603$	−3.00000	+	3.00000i	−0.122169	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−33.0000	−	33.0000i	−1.33943	−	1.33943i	−0.896612	−	0.442816i	$-0.853979\pi$
$607$	−33.0000	−	33.0000i	−1.33943	−	1.33943i	−0.442816	−	0.896612i	$-0.646021\pi$
$608$		0			0
$609$			36.0000i			1.45879i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$	−18.0000	+	18.0000i	−0.721155	+	0.721155i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		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
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		0			0
$643$	−21.0000	+	21.0000i	−0.828159	+	0.828159i	−0.987262	−	0.159103i	$-0.949140\pi$
$643$	−21.0000	+	21.0000i	−0.828159	+	0.828159i	0.159103	+	0.987262i	$0.449140\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.0000	−	13.0000i	−0.511083	−	0.511083i	0.403775	−	0.914858i	$-0.367698\pi$
$647$	−13.0000	−	13.0000i	−0.511083	−	0.511083i	−0.914858	+	0.403775i	$0.867698\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		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
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.00000	+	6.00000i	0.232321	+	0.232321i
$668$		0			0
$669$			42.0000i			1.62381i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	14.0000			0.536481
$682$		0			0
$683$	−31.0000	+	31.0000i	−1.18618	+	1.18618i	−0.208068	+	0.978114i	$0.566717\pi$
$683$	−31.0000	+	31.0000i	−1.18618	+	1.18618i	−0.978114	+	0.208068i	$0.933283\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	14.0000	+	14.0000i	0.534133	+	0.534133i
$688$		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
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−54.0000	−	54.0000i	−2.03088	−	2.03088i
$708$		0			0
$709$			46.0000i			1.72757i	0.503864	+	0.863783i	$0.331911\pi$
$709$			46.0000i			1.72757i	−0.503864	+	0.863783i	$0.668089\pi$
$710$		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
$721$	54.0000			2.01107
$722$		0			0
$723$	28.0000	−	28.0000i	1.04133	−	1.04133i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−3.00000	−	3.00000i	−0.111264	−	0.111264i	0.649283	−	0.760547i	$-0.275069\pi$
$727$	−3.00000	−	3.00000i	−0.111264	−	0.111264i	−0.760547	+	0.649283i	$0.775069\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		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
$743$	−19.0000	+	19.0000i	−0.697042	+	0.697042i	−0.963772	−	0.266729i	$-0.914057\pi$
$743$	−19.0000	+	19.0000i	−0.697042	+	0.697042i	0.266729	+	0.963772i	$0.414057\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−11.0000	−	11.0000i	−0.402469	−	0.402469i
$748$		0			0
$749$		−	78.0000i		−	2.85006i
$750$		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		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		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
$763$	48.0000	−	48.0000i	1.73772	−	1.73772i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			14.0000i			0.504853i	0.967616	+	0.252426i	$0.0812286\pi$
$769$			14.0000i			0.504853i	−0.967616	+	0.252426i	$0.918771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		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
$783$	24.0000	−	24.0000i	0.857690	−	0.857690i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	0.0368739	−	0.999320i	$-0.488260\pi$
$787$	−27.0000	−	27.0000i	−0.962446	−	0.962446i	−0.999320	+	0.0368739i	$0.988260\pi$
$788$		0			0
$789$			22.0000i			0.783221i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		0			0
$804$		0

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.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(-3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{3} +(-3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +6.00000 q^{21} +(1.00000 - 1.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} +6.00000i q^{29} +12.0000 q^{41} +(9.00000 - 9.00000i) q^{43} +(7.00000 + 7.00000i) q^{47} +11.0000i q^{49} +8.00000 q^{61} +(3.00000 - 3.00000i) q^{63} +(3.00000 + 3.00000i) q^{67} +2.00000i q^{69} +5.00000 q^{81} +(-11.0000 + 11.0000i) q^{83} +(-6.00000 - 6.00000i) q^{87} -6.00000i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 6 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 - 6 * q^7	\( 2 q - 2 q^{3} - 6 q^{7} + 12 q^{21} + 2 q^{23} - 8 q^{27} + 24 q^{41} + 18 q^{43} + 14 q^{47} + 16 q^{61} + 6 q^{63} + 6 q^{67} + 10 q^{81} - 22 q^{83} - 12 q^{87}+O(q^{100}) \) 2 * q - 2 * q^3 - 6 * q^7 + 12 * q^21 + 2 * q^23 - 8 * q^27 + 24 * q^41 + 18 * q^43 + 14 * q^47 + 16 * q^61 + 6 * q^63 + 6 * q^67 + 10 * q^81 - 22 * q^83 - 12 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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