LMFDB - Newform orbit 820.2.cc.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [820,2,Mod(19,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 20, 9])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.19"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	820.cc (of order $40$, degree $16$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,-4,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$6.54773296574$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{40}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{40}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{2} + (\zeta_{40}^{15} + \zeta_{40}^{13} + \cdots - 1) q^{3} + 2 \zeta_{40}^{6} q^{4} + ( - 2 \zeta_{40}^{13} + \cdots + \zeta_{40}) q^{5} + (\zeta_{40}^{15} - \zeta_{40}^{12} + \cdots + 2) q^{6}+ \cdots + ( - 7 \zeta_{40}^{13} + 10 \zeta_{40}^{12} + \cdots - 12) q^{98}+O(q^{100}) $ q + (-z^14 + z^10 + z^8 - z^6 + z^2) * q^2 + (z^15 + z^13 - z^11 - z^9 - z^8 + z^7 - z^6 + z^2 - z - 1) * q^3 + 2z^6 q^4 + (-2z^13 + 2z^9 + z) * q^5 + (z^15 - z^12 + z^11 - z^9 - z^7 - 2z^4 - z^3 - z^2 + 2) q^6 + (-z^14 + z^13 + z^12 - z^11 + z^10 - 2z^9 - z^7 + 2z^5 - z^4 - z^3 + 2z^2 - z + 1) q^7 + (2z^14 + 2z^4) * q^8 + (-z^8 + 2z^6 + 3z^5 + 2z^4 - z^2) q^9 + (-z^15 + 2z^13 - z^11 - z^9 + z^7 + 2z^5 + z^3) * q^10 + (-2z^14 - 2z^12 - 2z^11 + 2z^9 + 2z^8 - 2z^6 - 2z^5 - 2z^3) * q^12 + (-z^15 - 2z^12 + 3z^10 + 2z^8 - 2z^7 - 3z^5 + 2z^3 + 1) * q^14 + (-2z^15 + z^14 + 2z^12 - z^10 - z^9 + z^7 - 2z^5 - z^4 - z^3 - 3z^2 - z + 1) * q^15 + 4z^12 q^16 + (2z^14 + 3z^13 + z^12 - z^10 + z^8 - z^6 + z^4 + 3z^3 + 2z^2) * q^18 + (4z^11 - 2z^7 + 4z^3) q^20 + (8z^15 - z^14 + z^13 - z^12 - 6z^11 - z^10 + 3z^8 + 4z^7 - 3z^6 + z^4 - 2z^3 + z^2 - z + 1) * q^21 + (2z^15 - 3z^13 - z^11 + z^5 + 3z^3 - 2z) * q^23 + (-2z^15 - 2z^14 - 2z^10 - 4z^9 - 2z^8 + 2z^6 + 2z^5 + 2z^2 - 4z) q^24 + 5z^2 q^25 + (-z^15 - z^14 + z^13 - 3z^12 - 2z^11 - 2z^10 + 3z^8 + 3z^7 - 2z^5 - 2z^4 - 3z^3 + z^2 - z - 1) * q^27 + (-2z^15 + 2z^14 - 4z^13 + 2z^12 + 2z^11 - 4z^10 + 2z^8 + 4z^6 - 2z^5 + 2z^4 - 2z^3 - 2z^2 + 2z) q^28 + (-2z^13 - 4z^12 - 3z^10 - 2z^9 + 4z^8 + 3z^7 + 2z^5 + 2z + 2) * q^29 + (2z^15 - 2z^14 - 5z^13 - 2z^11 + z^10 - 2z^6 - z^3 - 5) q^30 + (4z^10 - 4) q^32 + (-3z^15 - z^14 - z^13 - 3z^12 + 3z^11 + 4z^10 - z^8 - 4z^6 + 3z^5 - 3z^4 + 4z^3 + z^2 - 3z) q^35 + (-2z^14 + 4z^12 + 6z^11 + 4z^10 - 2z^8) q^36 + (2z^15 + 4z^9 + 4z^7 - 2z^5 - 4z^3 + 4z) * q^40 + (6z^13 + 2z^11 - 6z^9 - z^7 + 6z^5 + 2z^3 - 6z) * q^41 + (-z^15 - 5z^14 + 8z^13 + 3z^12 + 4z^11 + 2z^10 - 7z^9 - 3z^8 - 5z^7 + z^6 + 4z^5 - 3z^3 + 4z^2 - 3z - 1) * q^42 + (-5z^15 - 5z^13 + 5z^11 + 10z^9 - 6z^5 + 6z^3 + 4z) q^43 + (2z^15 - 2z^13 + 2z^11 + 6z^10 + 5z^9 - 2z^7 - 3z^6 - 4z^5 + 3z^3 + 6z^2 + 4z) q^45 + (z^15 + 3z^13 - z^11 - 3z^9 + z^7 - 2z^3 + 6z) * q^46 + (5z^15 - 5z^12 - 3z^10 - 3z^9 + 2z^8 + 3z^7 + 5z^6 + 5z^5 - 2z^4 - 3z^3 - 3z^2 - 3z + 5) * q^47 + (4z^15 - 4z^13 - 4z^12 - 4z^11 + 4z^10 - 4z^6 - 4z^5 + 4z^2 + 4z + 4) q^48 + (5z^14 - 7z^9 + 6z^8 + 6z^6 - z^4 - 6z^2 + 6) q^49 + (5z^10 + 5) q^50 + (2z^15 - z^14 - 3z^13 - z^11 - z^10 - 3z^9 - 6z^8 - z^7 + 2z^6 + 3z^5 + 3z^4 - 4z + 1) * q^54 + (-4z^13 + 6z^12 - 6z^11 + 4z^10 + 4z^9 - 6z^8 - 2z^6 + 6z^4 + 4z^2 + 2z - 6) * q^56 + (z^15 - 5z^14 + 4z^12 + z^10 + 4z^9 - 5z^8 - 4z^7 - z^6 + z^5 + 4z^4 + 4z^3 + 5z^2 + 4z) q^58 + (-2z^15 + 4z^14 + 2z^13 - 2z^12 - 4z^11 - 6z^10 - 2z^9 - 4z^8 - 2z^7 + 6z^6 - 2z^4 - 4z^2 + 4z) q^60 + (-6z^13 + 4z^11 - 3z^9 - 4z^7 - 3z^5 + 4z^3 - 6z) q^61 + (-3z^15 - 3z^14 - 2z^13 - z^12 + 5z^11 - 4z^9 - 4z^8 + 6z^6 + 7z^5 + 2z^4 + z^3 - 4z^2 - 4z + 1) q^63 + (8z^14 - 8z^10 + 8z^6 - 8z^2) * q^64 + (-5z^14 - 4z^13 + 5z^12 + 5z^11 + z^10 - z^9 - 5z^8 - 4z^7 - 5z^6 + z^5 + 5z^3 + 4z - 1) q^67 + (-z^15 + 4z^13 - z^12 - 7z^11 - 4z^9 + z^8 + 7z^7 + 8z^6 + 5z^5 - 5z^4 - 7z^3 - 7z^2 - 2z + 3) * q^69 + (6z^15 - 5z^12 - 3z^11 - 6z^10 + 5z^8 + 6z^7 + 3z^6 - 5z^4 - 6z^2 + 5z + 5) * q^70 + (6z^15 + 4z^14 - 4z^12 - 6z^11 + 6z^9 + 6z^8 + 6z^7 + 2z^6 - 2z^4 - 6z^3 - 2z^2 - 2) q^72 + (5z^15 - 5z^11 - 5z^10 - 5z^8 + 5z^5 + 5z^4 - 5z^3 - 5z^2 - 5z) q^75 + (4z^13 + 8z^5 - 8z) q^80 + (-4z^14 - 3z^13 + z^12 + 6z^11 + z^10 + 6z^9 - z^8 - 3z^7 - 4z^6 + 2z^4 - 1) q^81 + (7z^15 + 2z^9 + 2z^7 - 7z^5 - 2z^3 + 2z) * q^82 + (-z^15 + 11z^11 - 7z^9 - 4z^7 - 3z^5 + 4z^3 + 4z) * q^83 + (2z^15 + 4z^14 - 4z^13 - 8z^12 - 2z^11 + 4z^10 + 8z^9 + 4z^8 - 12z^5 - 2z^4 - 2z^3 + 2z^2 - 4z + 4) q^84 + (z^15 - z^13 - z^9 + 11z^7 + 10z^5 - 2z^3 + z) q^86 + (5z^14 - 2z^13 - 5z^12 - z^10 + 2z^9 + z^8 + 6z^6 - 6z^4 - 11z^2 - 12z - 5) * q^87 + (3z^15 + 3z^14 + 8z^9 - 8z^8 - 4z^5 + 4z^4 + 8z - 8) q^89 + (-4z^15 + 3z^14 + 3z^13 + 3z^11 + z^9 + 6z^8 + 3z^7 + 6z^6 + 3z^5 - 3z^4 - 4z^3 - 6z^2 + 6) q^90 + (-6z^15 - 2z^13 + 8z^11 + 8z^9 - 10z^7 - 2z^5 + 6z^3 - 2z) * q^92 + (6z^15 - 3z^14 + 7z^13 - 6z^11 - 6z^6 + 7z^4 - 3z^3 + 6z^2) * q^94 + (-8z^15 - 4z^14 + 4z^11 + 8z^8 - 8z^7 - 4z^4 + 4z + 8) q^96 + (-7z^13 + 10z^12 + 7z^9 - 7z^7 - 7z^5 + 12z^4 + 7z - 12) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} - 12 q^{3} + 20 q^{6} + 16 q^{7} + 8 q^{8} + 12 q^{9} - 16 q^{12} + 20 q^{15} + 16 q^{16} + 4 q^{18} + 4 q^{21} + 8 q^{24} - 48 q^{27} + 8 q^{28} - 80 q^{30} - 64 q^{32} - 20 q^{35} + 24 q^{36}+ \cdots - 104 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 - 12 * q^3 + 20 * q^6 + 16 * q^7 + 8 * q^8 + 12 * q^9 - 16 * q^12 + 20 * q^15 + 16 * q^16 + 4 * q^18 + 4 * q^21 + 8 * q^24 - 48 * q^27 + 8 * q^28 - 80 * q^30 - 64 * q^32 - 20 * q^35 + 24 * q^36 + 8 * q^42 + 44 * q^47 + 48 * q^48 + 68 * q^49 + 80 * q^50 + 52 * q^54 - 24 * q^56 + 52 * q^58 + 36 * q^63 + 24 * q^67 + 20 * q^69 + 20 * q^70 - 80 * q^72 + 40 * q^75 + 8 * q^84 - 128 * q^87 - 80 * q^89 + 60 * q^90 + 28 * q^94 + 80 * q^96 - 104 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$\zeta_{40}^{11}$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

19.1

−0.987688	+	0.156434i
0.891007	+	0.453990i
0.453990	+	0.891007i
0.156434	−	0.987688i
0.156434	+	0.987688i
−0.987688	−	0.156434i
−0.891007	+	0.453990i
0.453990	−	0.891007i
−0.453990	+	0.891007i
0.891007	−	0.453990i
0.987688	+	0.156434i
−0.156434	−	0.987688i
−0.156434	+	0.987688i
−0.453990	−	0.891007i
−0.891007	−	0.453990i
0.987688	−	0.156434i

1.26007

−

0.642040i

0.749049

1.80837i

1.17557

−

1.61803i

−2.20854

0.349798i

2.10490

1.79775i

3.90076

−

3.33156i

0.442463

−

2.79360i

−0.587789

0.587789i

−2.55834

1.85874i

99.1

−0.221232

−

1.39680i

1.16396

2.81005i

−1.90211

0.618034i

−1.99235

−

1.01515i

3.66757

−

2.24749i

2.31026

1.41573i

1.28408

2.52015i

−4.42024

4.42024i

−0.977198

3.00750i

179.1

−1.39680

−

0.221232i

−2.89092

1.19746i

1.90211

0.618034i

−1.01515

−

1.99235i

4.30296

−

1.03305i

4.41957

1.06104i

−2.52015

−

1.28408i

4.80220

−

4.80220i

0.977198

3.00750i

199.1

−0.642040

1.26007i

−2.53583

1.05037i

−1.17557

−

1.61803i

0.349798

−

2.20854i

0.304553

−

3.86971i

−0.101838

−

1.29397i

2.79360

−

0.442463i

3.20582

−

3.20582i

2.55834

1.85874i

239.1

−0.642040

−

1.26007i

−2.53583

−

1.05037i

−1.17557

1.61803i

0.349798

2.20854i

0.304553

3.86971i

−0.101838

1.29397i

2.79360

0.442463i

3.20582

3.20582i

2.55834

−

1.85874i

259.1

1.26007

0.642040i

0.749049

−

1.80837i

1.17557

1.61803i

−2.20854

−

0.349798i

2.10490

−

1.79775i

3.90076

3.33156i

0.442463

2.79360i

−0.587789

−

0.587789i

−2.55834

−

1.85874i

299.1

−0.221232

1.39680i

1.53176

0.634475i

−1.90211

−

0.618034i

1.99235

−

1.01515i

−1.22511

1.99920i

2.37484

3.87538i

1.28408

−

2.52015i

−0.177595

−

0.177595i

0.977198

3.00750i

339.1

−1.39680

0.221232i

−2.89092

−

1.19746i

1.90211

−

0.618034i

−1.01515

1.99235i

4.30296

1.03305i

4.41957

−

1.06104i

−2.52015

1.28408i

4.80220

4.80220i

0.977198

−

3.00750i

399.1

−1.39680

0.221232i

−0.568727

1.37303i

1.90211

−

0.618034i

1.01515

−

1.99235i

0.490642

−

2.04367i

−0.632529

−

2.63467i

−2.52015

1.28408i

0.559561

0.559561i

−0.977198

3.00750i

439.1

−0.221232

1.39680i

1.16396

−

2.81005i

−1.90211

−

0.618034i

−1.99235

1.01515i

3.66757

2.24749i

2.31026

−

1.41573i

1.28408

−

2.52015i

−4.42024

−

4.42024i

−0.977198

−

3.00750i

479.1

1.26007

0.642040i

−2.64054

−

1.09375i

1.17557

1.61803i

2.20854

0.349798i

−2.62505

−

3.07353i

0.842968

−

0.986988i

0.442463

2.79360i

3.65485

3.65485i

2.55834

1.85874i

499.1

−0.642040

−

1.26007i

−0.808747

1.95249i

−1.17557

1.61803i

−0.349798

−

2.20854i

2.97953

−

0.234494i

−5.11403

−

0.402483i

2.79360

0.442463i

−1.03682

−

1.03682i

−2.55834

1.85874i

539.1

−0.642040

1.26007i

−0.808747

−

1.95249i

−1.17557

−

1.61803i

−0.349798

2.20854i

2.97953

0.234494i

−5.11403

0.402483i

2.79360

−

0.442463i

−1.03682

1.03682i

−2.55834

−

1.85874i

559.1

−1.39680

−

0.221232i

−0.568727

−

1.37303i

1.90211

0.618034i

1.01515

1.99235i

0.490642

2.04367i

−0.632529

2.63467i

−2.52015

−

1.28408i

0.559561

−

0.559561i

−0.977198

−

3.00750i

639.1

−0.221232

−

1.39680i

1.53176

−

0.634475i

−1.90211

0.618034i

1.99235

1.01515i

−1.22511

−

1.99920i

2.37484

−

3.87538i

1.28408

2.52015i

−0.177595

0.177595i

0.977198

−

3.00750i

719.1

1.26007

−

0.642040i

−2.64054

1.09375i

1.17557

−

1.61803i

2.20854

−

0.349798i

−2.62505

3.07353i

0.842968

0.986988i

0.442463

−

2.79360i

3.65485

−

3.65485i

2.55834

−

1.85874i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
41.h	odd	40	1	inner
820.cc	even	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	820.2.cc.a	✓	16
4.b	odd	2	1		820.2.cc.f	yes	16
5.b	even	2	1		820.2.cc.f	yes	16
20.d	odd	2	1	CM	820.2.cc.a	✓	16
41.h	odd	40	1	inner	820.2.cc.a	✓	16
164.o	even	40	1		820.2.cc.f	yes	16
205.ba	odd	40	1		820.2.cc.f	yes	16
820.cc	even	40	1	inner	820.2.cc.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
820.2.cc.a	✓	16	1.a	even	1	1	trivial
820.2.cc.a	✓	16	20.d	odd	2	1	CM
820.2.cc.a	✓	16	41.h	odd	40	1	inner
820.2.cc.a	✓	16	820.cc	even	40	1	inner
820.2.cc.f	yes	16	4.b	odd	2	1
820.2.cc.f	yes	16	5.b	even	2	1
820.2.cc.f	yes	16	164.o	even	40	1
820.2.cc.f	yes	16	205.ba	odd	40	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(820, [\chi])$:

$ T_{3}^{16} + 12 T_{3}^{15} + 66 T_{3}^{14} + 248 T_{3}^{13} + 834 T_{3}^{12} + 2488 T_{3}^{11} + \cdots + 579121 $

T3^16 + 12*T3^15 + 66*T3^14 + 248*T3^13 + 834*T3^12 + 2488*T3^11 + 5652*T3^10 + 9264*T3^9 + 11251*T3^8 + 9840*T3^7 + 6672*T3^6 + 1344*T3^5 + 9634*T3^4 + 142180*T3^3 + 440290*T3^2 + 596624*T3 + 579121

$ T_{13} $

T13

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} $ (T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16)^2
$3$	$ T^{16} + 12 T^{15} + \cdots + 579121 $ T^16 + 12T^15 + 66T^14 + 248T^13 + 834T^12 + 2488T^11 + 5652T^10 + 9264T^9 + 11251T^8 + 9840T^7 + 6672T^6 + 1344T^5 + 9634T^4 + 142180T^3 + 440290T^2 + 596624*T + 579121
$5$	$ T^{16} - 25 T^{12} + \cdots + 390625 $ T^16 - 25T^12 + 625T^8 - 15625*T^4 + 390625
$7$	$ T^{16} - 16 T^{15} + \cdots + 45212176 $ T^16 - 16T^15 + 94T^14 + 4T^13 - 3326T^12 + 20776T^11 - 51832T^10 - 52608T^9 + 944796T^8 - 4493680T^7 + 15422448T^6 - 39842288T^5 + 74721224T^4 - 101778400T^3 + 112290800T^2 - 86013408*T + 45212176
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ T^{16} $ T^16
$19$	$ T^{16} $ T^16
$23$	$ T^{16} + \cdots + 24343800625 $ T^16 + 40T^14 + 6705T^12 - 11600T^10 + 6993150T^8 - 284837000T^6 + 4930344500T^4 - 17271967500*T^2 + 24343800625
$29$	$ T^{16} + \cdots + 14170283521 $ T^16 - 208T^14 - 348T^13 + 24627T^12 + 84448T^11 - 1576046T^10 - 4559844T^9 + 4574440T^8 + 31689344T^7 + 3711977712T^6 + 12243410356T^5 + 65287726402T^4 + 112212286568T^3 + 122234524852T^2 + 49627835256T + 14170283521
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + \cdots + 7984925229121 $ T^16 + 120T^14 + 6959T^12 + 345600T^10 + 15961441T^8 + 580953600T^6 + 19664470799T^4 + 570012508920*T^2 + 7984925229121
$43$	$ T^{16} + \cdots + 29\!\cdots\!41 $ T^16 - 16756T^12 + 4689580T^10 + 369861686T^8 - 78578602480T^6 + 1656187563069T^4 + 69526908482060T^2 + 2900998810968241
$47$	$ T^{16} + \cdots + 47284340602321 $ T^16 - 44T^15 + 774T^14 - 6104T^13 + 17679T^12 - 342376T^11 + 13354758T^10 - 260294952T^9 + 3330646856T^8 - 28981168420T^7 + 177573180298T^6 - 818007861172T^5 + 2572814455774T^4 - 6641717442920T^3 + 33562321454420T^2 - 50929987790052*T + 47284340602321
$53$	$ T^{16} $ T^16
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + \cdots + 102389324175121 $ T^16 + 240T^14 + 14509T^12 + 9474240T^10 + 3316553206T^8 + 378515570160T^6 + 24150960417964T^4 - 95867570216640*T^2 + 102389324175121
$67$	$ T^{16} + \cdots + 25\!\cdots\!41 $ T^16 - 24T^15 + 172T^14 + 2768T^13 - 56573T^12 - 167544T^11 + 13181982T^10 - 198342036T^9 + 402702080T^8 + 20893227196T^7 - 268865434176T^6 + 1547944342168T^5 + 12121169627562T^4 - 243154196196828T^3 + 2258437390273792T^2 - 10307500058611900*T + 25221162726981841
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ T^{16} $ T^16
$83$	$ (T^{8} - 740 T^{6} + \cdots + 26988025)^{2} $ (T^8 - 740T^6 + 174515T^4 - 12883000*T^2 + 26988025)^2
$89$	$ T^{16} + \cdots + 31\!\cdots\!01 $ T^16 + 80T^15 + 3342T^14 + 94212T^13 + 1983447T^12 + 32615088T^11 + 427166054T^10 + 4535548776T^9 + 40582671760T^8 + 338485465784T^7 + 3097707252582T^6 + 34300810817036T^5 + 426002606027582T^4 + 4549454229857888T^3 + 33488679108061732T^2 + 149871036025959436*T + 319201497562598401
$97$	$ T^{16} $ T^16
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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:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	820.cc (of order \(40\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{2} + (\zeta_{40}^{15} + \zeta_{40}^{13} + \cdots - 1) q^{3} + 2 \zeta_{40}^{6} q^{4} + ( - 2 \zeta_{40}^{13} + \cdots + \zeta_{40}) q^{5} + (\zeta_{40}^{15} - \zeta_{40}^{12} + \cdots + 2) q^{6}+ \cdots + ( - 7 \zeta_{40}^{13} + 10 \zeta_{40}^{12} + \cdots - 12) q^{98}+O(q^{100}) \) q + (-z^14 + z^10 + z^8 - z^6 + z^2) * q^2 + (z^15 + z^13 - z^11 - z^9 - z^8 + z^7 - z^6 + z^2 - z - 1) * q^3 + 2z^6 q^4 + (-2z^13 + 2z^9 + z) * q^5 + (z^15 - z^12 + z^11 - z^9 - z^7 - 2z^4 - z^3 - z^2 + 2) q^6 + (-z^14 + z^13 + z^12 - z^11 + z^10 - 2z^9 - z^7 + 2z^5 - z^4 - z^3 + 2z^2 - z + 1) q^7 + (2z^14 + 2z^4) * q^8 + (-z^8 + 2z^6 + 3z^5 + 2z^4 - z^2) q^9 + (-z^15 + 2z^13 - z^11 - z^9 + z^7 + 2z^5 + z^3) * q^10 + (-2z^14 - 2z^12 - 2z^11 + 2z^9 + 2z^8 - 2z^6 - 2z^5 - 2z^3) * q^12 + (-z^15 - 2z^12 + 3z^10 + 2z^8 - 2z^7 - 3z^5 + 2z^3 + 1) * q^14 + (-2z^15 + z^14 + 2z^12 - z^10 - z^9 + z^7 - 2z^5 - z^4 - z^3 - 3z^2 - z + 1) * q^15 + 4z^12 q^16 + (2z^14 + 3z^13 + z^12 - z^10 + z^8 - z^6 + z^4 + 3z^3 + 2z^2) * q^18 + (4z^11 - 2z^7 + 4z^3) q^20 + (8z^15 - z^14 + z^13 - z^12 - 6z^11 - z^10 + 3z^8 + 4z^7 - 3z^6 + z^4 - 2z^3 + z^2 - z + 1) * q^21 + (2z^15 - 3z^13 - z^11 + z^5 + 3z^3 - 2z) * q^23 + (-2z^15 - 2z^14 - 2z^10 - 4z^9 - 2z^8 + 2z^6 + 2z^5 + 2z^2 - 4z) q^24 + 5z^2 q^25 + (-z^15 - z^14 + z^13 - 3z^12 - 2z^11 - 2z^10 + 3z^8 + 3z^7 - 2z^5 - 2z^4 - 3z^3 + z^2 - z - 1) * q^27 + (-2z^15 + 2z^14 - 4z^13 + 2z^12 + 2z^11 - 4z^10 + 2z^8 + 4z^6 - 2z^5 + 2z^4 - 2z^3 - 2z^2 + 2z) q^28 + (-2z^13 - 4z^12 - 3z^10 - 2z^9 + 4z^8 + 3z^7 + 2z^5 + 2z + 2) * q^29 + (2z^15 - 2z^14 - 5z^13 - 2z^11 + z^10 - 2z^6 - z^3 - 5) q^30 + (4z^10 - 4) q^32 + (-3z^15 - z^14 - z^13 - 3z^12 + 3z^11 + 4z^10 - z^8 - 4z^6 + 3z^5 - 3z^4 + 4z^3 + z^2 - 3z) q^35 + (-2z^14 + 4z^12 + 6z^11 + 4z^10 - 2z^8) q^36 + (2z^15 + 4z^9 + 4z^7 - 2z^5 - 4z^3 + 4z) * q^40 + (6z^13 + 2z^11 - 6z^9 - z^7 + 6z^5 + 2z^3 - 6z) * q^41 + (-z^15 - 5z^14 + 8z^13 + 3z^12 + 4z^11 + 2z^10 - 7z^9 - 3z^8 - 5z^7 + z^6 + 4z^5 - 3z^3 + 4z^2 - 3z - 1) * q^42 + (-5z^15 - 5z^13 + 5z^11 + 10z^9 - 6z^5 + 6z^3 + 4z) q^43 + (2z^15 - 2z^13 + 2z^11 + 6z^10 + 5z^9 - 2z^7 - 3z^6 - 4z^5 + 3z^3 + 6z^2 + 4z) q^45 + (z^15 + 3z^13 - z^11 - 3z^9 + z^7 - 2z^3 + 6z) * q^46 + (5z^15 - 5z^12 - 3z^10 - 3z^9 + 2z^8 + 3z^7 + 5z^6 + 5z^5 - 2z^4 - 3z^3 - 3z^2 - 3z + 5) * q^47 + (4z^15 - 4z^13 - 4z^12 - 4z^11 + 4z^10 - 4z^6 - 4z^5 + 4z^2 + 4z + 4) q^48 + (5z^14 - 7z^9 + 6z^8 + 6z^6 - z^4 - 6z^2 + 6) q^49 + (5z^10 + 5) q^50 + (2z^15 - z^14 - 3z^13 - z^11 - z^10 - 3z^9 - 6z^8 - z^7 + 2z^6 + 3z^5 + 3z^4 - 4z + 1) * q^54 + (-4z^13 + 6z^12 - 6z^11 + 4z^10 + 4z^9 - 6z^8 - 2z^6 + 6z^4 + 4z^2 + 2z - 6) * q^56 + (z^15 - 5z^14 + 4z^12 + z^10 + 4z^9 - 5z^8 - 4z^7 - z^6 + z^5 + 4z^4 + 4z^3 + 5z^2 + 4z) q^58 + (-2z^15 + 4z^14 + 2z^13 - 2z^12 - 4z^11 - 6z^10 - 2z^9 - 4z^8 - 2z^7 + 6z^6 - 2z^4 - 4z^2 + 4z) q^60 + (-6z^13 + 4z^11 - 3z^9 - 4z^7 - 3z^5 + 4z^3 - 6z) q^61 + (-3z^15 - 3z^14 - 2z^13 - z^12 + 5z^11 - 4z^9 - 4z^8 + 6z^6 + 7z^5 + 2z^4 + z^3 - 4z^2 - 4z + 1) q^63 + (8z^14 - 8z^10 + 8z^6 - 8z^2) * q^64 + (-5z^14 - 4z^13 + 5z^12 + 5z^11 + z^10 - z^9 - 5z^8 - 4z^7 - 5z^6 + z^5 + 5z^3 + 4z - 1) q^67 + (-z^15 + 4z^13 - z^12 - 7z^11 - 4z^9 + z^8 + 7z^7 + 8z^6 + 5z^5 - 5z^4 - 7z^3 - 7z^2 - 2z + 3) * q^69 + (6z^15 - 5z^12 - 3z^11 - 6z^10 + 5z^8 + 6z^7 + 3z^6 - 5z^4 - 6z^2 + 5z + 5) * q^70 + (6z^15 + 4z^14 - 4z^12 - 6z^11 + 6z^9 + 6z^8 + 6z^7 + 2z^6 - 2z^4 - 6z^3 - 2z^2 - 2) q^72 + (5z^15 - 5z^11 - 5z^10 - 5z^8 + 5z^5 + 5z^4 - 5z^3 - 5z^2 - 5z) q^75 + (4z^13 + 8z^5 - 8z) q^80 + (-4z^14 - 3z^13 + z^12 + 6z^11 + z^10 + 6z^9 - z^8 - 3z^7 - 4z^6 + 2z^4 - 1) q^81 + (7z^15 + 2z^9 + 2z^7 - 7z^5 - 2z^3 + 2z) * q^82 + (-z^15 + 11z^11 - 7z^9 - 4z^7 - 3z^5 + 4z^3 + 4z) * q^83 + (2z^15 + 4z^14 - 4z^13 - 8z^12 - 2z^11 + 4z^10 + 8z^9 + 4z^8 - 12z^5 - 2z^4 - 2z^3 + 2z^2 - 4z + 4) q^84 + (z^15 - z^13 - z^9 + 11z^7 + 10z^5 - 2z^3 + z) q^86 + (5z^14 - 2z^13 - 5z^12 - z^10 + 2z^9 + z^8 + 6z^6 - 6z^4 - 11z^2 - 12z - 5) * q^87 + (3z^15 + 3z^14 + 8z^9 - 8z^8 - 4z^5 + 4z^4 + 8z - 8) q^89 + (-4z^15 + 3z^14 + 3z^13 + 3z^11 + z^9 + 6z^8 + 3z^7 + 6z^6 + 3z^5 - 3z^4 - 4z^3 - 6z^2 + 6) q^90 + (-6z^15 - 2z^13 + 8z^11 + 8z^9 - 10z^7 - 2z^5 + 6z^3 - 2z) * q^92 + (6z^15 - 3z^14 + 7z^13 - 6z^11 - 6z^6 + 7z^4 - 3z^3 + 6z^2) * q^94 + (-8z^15 - 4z^14 + 4z^11 + 8z^8 - 8z^7 - 4z^4 + 4z + 8) q^96 + (-7z^13 + 10z^12 + 7z^9 - 7z^7 - 7z^5 + 12z^4 + 7z - 12) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 12 q^{3} + 20 q^{6} + 16 q^{7} + 8 q^{8} + 12 q^{9} - 16 q^{12} + 20 q^{15} + 16 q^{16} + 4 q^{18} + 4 q^{21} + 8 q^{24} - 48 q^{27} + 8 q^{28} - 80 q^{30} - 64 q^{32} - 20 q^{35} + 24 q^{36}+ \cdots - 104 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 - 12 * q^3 + 20 * q^6 + 16 * q^7 + 8 * q^8 + 12 * q^9 - 16 * q^12 + 20 * q^15 + 16 * q^16 + 4 * q^18 + 4 * q^21 + 8 * q^24 - 48 * q^27 + 8 * q^28 - 80 * q^30 - 64 * q^32 - 20 * q^35 + 24 * q^36 + 8 * q^42 + 44 * q^47 + 48 * q^48 + 68 * q^49 + 80 * q^50 + 52 * q^54 - 24 * q^56 + 52 * q^58 + 36 * q^63 + 24 * q^67 + 20 * q^69 + 20 * q^70 - 80 * q^72 + 40 * q^75 + 8 * q^84 - 128 * q^87 - 80 * q^89 + 60 * q^90 + 28 * q^94 + 80 * q^96 - 104 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	820.2.cc.a

Level	$820$
Weight	$2$
Character orbit	820.cc
Analytic conductor	$6.548$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-20
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.54773296574\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{40}]$

\(n\)	\(411\)	\(621\)	\(657\)
\(\chi(n)\)	\(-1\)	\(\zeta_{40}^{11}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} \) (T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16)^2
$3$	\( T^{16} + 12 T^{15} + \cdots + 579121 \) T^16 + 12T^15 + 66T^14 + 248T^13 + 834T^12 + 2488T^11 + 5652T^10 + 9264T^9 + 11251T^8 + 9840T^7 + 6672T^6 + 1344T^5 + 9634T^4 + 142180T^3 + 440290T^2 + 596624*T + 579121
$5$	\( T^{16} - 25 T^{12} + \cdots + 390625 \) T^16 - 25T^12 + 625T^8 - 15625*T^4 + 390625
$7$	\( T^{16} - 16 T^{15} + \cdots + 45212176 \) T^16 - 16T^15 + 94T^14 + 4T^13 - 3326T^12 + 20776T^11 - 51832T^10 - 52608T^9 + 944796T^8 - 4493680T^7 + 15422448T^6 - 39842288T^5 + 74721224T^4 - 101778400T^3 + 112290800T^2 - 86013408*T + 45212176
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( T^{16} \) T^16
$19$	\( T^{16} \) T^16
$23$	\( T^{16} + \cdots + 24343800625 \) T^16 + 40T^14 + 6705T^12 - 11600T^10 + 6993150T^8 - 284837000T^6 + 4930344500T^4 - 17271967500*T^2 + 24343800625
$29$	\( T^{16} + \cdots + 14170283521 \) T^16 - 208T^14 - 348T^13 + 24627T^12 + 84448T^11 - 1576046T^10 - 4559844T^9 + 4574440T^8 + 31689344T^7 + 3711977712T^6 + 12243410356T^5 + 65287726402T^4 + 112212286568T^3 + 122234524852T^2 + 49627835256T + 14170283521
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + \cdots + 7984925229121 \) T^16 + 120T^14 + 6959T^12 + 345600T^10 + 15961441T^8 + 580953600T^6 + 19664470799T^4 + 570012508920*T^2 + 7984925229121
$43$	\( T^{16} + \cdots + 29\!\cdots\!41 \) T^16 - 16756T^12 + 4689580T^10 + 369861686T^8 - 78578602480T^6 + 1656187563069T^4 + 69526908482060T^2 + 2900998810968241
$47$	\( T^{16} + \cdots + 47284340602321 \) T^16 - 44T^15 + 774T^14 - 6104T^13 + 17679T^12 - 342376T^11 + 13354758T^10 - 260294952T^9 + 3330646856T^8 - 28981168420T^7 + 177573180298T^6 - 818007861172T^5 + 2572814455774T^4 - 6641717442920T^3 + 33562321454420T^2 - 50929987790052*T + 47284340602321
$53$	\( T^{16} \) T^16
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + \cdots + 102389324175121 \) T^16 + 240T^14 + 14509T^12 + 9474240T^10 + 3316553206T^8 + 378515570160T^6 + 24150960417964T^4 - 95867570216640*T^2 + 102389324175121
$67$	\( T^{16} + \cdots + 25\!\cdots\!41 \) T^16 - 24T^15 + 172T^14 + 2768T^13 - 56573T^12 - 167544T^11 + 13181982T^10 - 198342036T^9 + 402702080T^8 + 20893227196T^7 - 268865434176T^6 + 1547944342168T^5 + 12121169627562T^4 - 243154196196828T^3 + 2258437390273792T^2 - 10307500058611900*T + 25221162726981841
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( T^{16} \) T^16
$83$	\( (T^{8} - 740 T^{6} + \cdots + 26988025)^{2} \) (T^8 - 740T^6 + 174515T^4 - 12883000*T^2 + 26988025)^2
$89$	\( T^{16} + \cdots + 31\!\cdots\!01 \) T^16 + 80T^15 + 3342T^14 + 94212T^13 + 1983447T^12 + 32615088T^11 + 427166054T^10 + 4535548776T^9 + 40582671760T^8 + 338485465784T^7 + 3097707252582T^6 + 34300810817036T^5 + 426002606027582T^4 + 4549454229857888T^3 + 33488679108061732T^2 + 149871036025959436*T + 319201497562598401
$97$	\( T^{16} \) T^16
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