LMFDB - Newform orbit 600.2.bp.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(53,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 10, 10, 7]))

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

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

chi := DirichletCharacter("600.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	600.bp (of order $20$ , degree $8$ , 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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	16.0.6879707136000000000000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561$ x^16 - 9x^12 + 81x^8 - 729*x^4 + 6561
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

$q$ -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 basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{12} + \beta_{8} - \beta_{6} + \cdots + 1) q^{2} + (\beta_{13} - \beta_{9} + \beta_{5} - \beta_1) q^{3} - 2 \beta_{2} q^{4} + (\beta_{12} - \beta_{11} - \beta_{8} + \cdots - 1) q^{5} + (\beta_{13} + \beta_{3}) q^{6}+ \cdots + ( - 3 \beta_{15} + 6 \beta_{14} + \cdots - 6) q^{99}+O(q^{100})$ q + (-b12 + b8 - b6 - b4 + 1) * q^2 + (b13 - b9 + b5 - b1) * q^3 - 2b2 q^4 + (b12 - b11 - b8 - b6 + b4 - 1) * q^5 + (b13 + b3) * q^6 + (b14 - 2b13 + b12 - b10 + b9 + b7 + b6 - b5 - b3 - b2 + b1) q^7 + (2b14 - 2b10 + 2b8 + 2b6 - 2b2) q^8 - 3b14 q^9 + (b13 + 2b12 - b9 - b7 + b5 - b1) q^10 + (2b12 + 2b10 - b7 + b5 - 2b2 - 2) q^11 + (-2b15 + 2b11 - 2b7 + 2b3) * q^12 + (2b15 - 2b13 - 2b11 + b10 + b8 + 2b7 - b6 + b4 - 2b3 + b2) q^14 + (-b13 + 3b8 + b3) q^15 + 4b4 q^16 + (-3b10 - 3) q^18 + (-2b14 + 2b13 + 2b10 + 2b8 - 2b6 + 2b2) * q^20 + (-b15 + 3b14 + 3b10 - b9 - 3b4 + 3) q^21 + (b13 - b11 + 4b8 + 4b6 - b3 + b1) * q^22 + (-2b15 - 2b5) * q^24 + (2b13 - 2b9 + 2b7 + 2b5 - b2 - 2b1) q^25 + 3b11 q^27 + (4b15 - 2b14 - 2b11 - 2b9 + 2b7 + 2b5 - 2b3 + 2) q^28 + (-3b13 + 3b9 + 3b7 - 3b5 - 2b2 + 3b1) * q^29 + (-3b14 - 2b9 + 3b4) q^30 + (-b15 + b11 + 5b10 + 5b6 + b5 - b1) * q^31 + (-4b10 + 4) q^32 + (-2b15 - 2b13 + 2b11 - 4b7 - 2b5 + 3b4 + 2b3 - 3b2 + 2b1) q^33 + (-2b14 + 4b10 + 3b9 - 4b8 - 4b6 - 2b4 - b3 + 4b2) q^35 + (6b12 - 6b8 + 6b4 - 6) q^36 + (-2b15 - 4b14 + 2b11 + 2b9 - 2b7 + 2b3) * q^40 + (b15 - 6b12 - b11 + 6b10 + 6b8 - b5 - 6b4 - b1 + 6) * q^42 + (-4b14 - 4b12 + 2b9 - 2b7 + 4b4 + 4b2) * q^44 + (3b10 - 3b5 - 3) * q^45 - 4b1 q^48 + (2b15 + 5b12 - 7b10 + 2b9 - 5b8 + 2b7 + 10b4 - 2b3 + 2b1 - 5) q^49 + (b14 - b10 + b8 + b6 + 4b3 - b2) q^50 + (5b14 - b13 + 5b12 - 5b10 - 10b8 + 10b6 + 5b4 - 5b2 + b1 - 5) q^53 + (-3b13 + 3b9 + 3b7 - 3b5 + 3b1) q^54 + (-b14 + 3b13 - 7b12 + b11 + b10 + 7b8 - b6 - 7b4 + 3b3 + b2 + b1 + 7) q^55 + (4b15 - 2b12 - 2b10 + 2b8 - 2b6 - 2b4 + 4b1) q^56 + (2b14 - 6b13 - 2b10 + 2b8 + 2b6 - 2b2) * q^58 + (3b15 + 4b14 + 3b13 - 4b8 + 4b6 - 4b2 + 4) * q^59 + (2b15 - 6b10 - 2b5) q^60 + (-10b12 - 2b11 + 5b8 + 2b7 + 5b6 - 5b4 + 5b2 + 5) q^62 + (3b13 + 3b12 - 3b11 - 3b9 - 3b7 + 3b6 + 3b5) q^63 - 8b6 q^64 + (3b14 - 6b10 + 3b8 + 3b6 - 4b5 - 4b3 - 3b2 + 3) q^66 + (-2b15 - b11 + 4b10 + b9 + b7 + 3b5 - 8b4 - b3) * q^70 + (6b12 + 6b2) * q^72 + (7b12 - 7b8 - 7b6 + 7b4 - 4b1 - 7) q^73 + (-b15 - 6b14 + b11 - b7 - 6b4 + b3) * q^75 + (3b15 - 10b14 + 5b13 + b12 - 3b11 + 6b10 - 5b9 - 5b8 + 3b7 - 5b6 + 10b5 - 5b4 + 6b2 - 5b1 - 1) q^77 + (3b13 - 5b12 - 3b11 + 10b8 - 5b4 - 3b3 - 3b1 + 5) q^79 + (-4b15 - 4b10 - 4) * q^80 - 9b8 q^81 + (-5b15 + 2b12 - 5b11 + 2b10 - 2b8 - 2b6 + 2b4) q^83 + (2b13 - 12b12 + 2b11 - 2b9 + 6b8 + 6b6 + 2b5 - 6b4 - 6b2 - 2b1 + 6) * q^84 + (-2b15 + 9b14 + 2b11 - 2b7 - 9b4 + 2b3) * q^87 + (-2b15 + 2b13 - 8b10 - 8b8 + 2b5 - 2b3) * q^88 + (3b11 + 6b6 - 3b1) q^90 + (-3b14 + 3b12 + 3b10 - 3b8 - 5b7 - 3b6 - 5b3) q^93 + (4b13 - 4b9 + 4b7 + 4b5 - 4b1) q^96 + (-b14 - 4b11 - 4b9 + 4b7 - 4b5 - 4b3 - 1) q^97 + (-4b13 + 12b12 - 5b10 + 4b9 - 7b8 - 7b6 + 7b4 + 5b2 + 4b1 - 2) q^98 + (-3b15 + 6b14 + 6b12 + 3b11 - 6b8 - 3b7 + 6b6 + 12b4 + 3b3 - 3b1 - 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 4 q^{2} - 4 q^{5} + 4 q^{7} - 8 q^{8} + 8 q^{10} - 24 q^{11} - 12 q^{15} + 16 q^{16} - 48 q^{18} - 8 q^{20} + 36 q^{21} - 16 q^{22} + 32 q^{28} + 12 q^{30} + 64 q^{32} + 12 q^{33} + 8 q^{35} - 24 q^{36}+ \cdots + 72 q^{98}+O(q^{100})$ 16 * q + 4 * q^2 - 4 * q^5 + 4 * q^7 - 8 * q^8 + 8 * q^10 - 24 * q^11 - 12 * q^15 + 16 * q^16 - 48 * q^18 - 8 * q^20 + 36 * q^21 - 16 * q^22 + 32 * q^28 + 12 * q^30 + 64 * q^32 + 12 * q^33 + 8 * q^35 - 24 * q^36 + 24 * q^42 - 48 * q^45 - 4 * q^50 + 28 * q^55 - 24 * q^56 - 8 * q^58 + 80 * q^59 + 12 * q^63 + 36 * q^66 - 32 * q^70 + 24 * q^72 - 28 * q^73 - 24 * q^75 - 12 * q^77 - 64 * q^80 + 36 * q^81 + 24 * q^83 - 36 * q^87 + 32 * q^88 + 24 * q^93 - 16 * q^97 + 72 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561$ x^16 - 9*x^12 + 81*x^8 - 729*x^4 + 6561 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} ) / 3$ (v^2) / 3
$\beta_{3}$	$=$	$( \nu^{3} ) / 3$ (v^3) / 3
$\beta_{4}$	$=$	$( \nu^{4} ) / 9$ (v^4) / 9
$\beta_{5}$	$=$	$( \nu^{5} ) / 9$ (v^5) / 9
$\beta_{6}$	$=$	$( \nu^{6} ) / 27$ (v^6) / 27
$\beta_{7}$	$=$	$( \nu^{7} ) / 27$ (v^7) / 27
$\beta_{8}$	$=$	$( \nu^{8} ) / 81$ (v^8) / 81
$\beta_{9}$	$=$	$( \nu^{9} ) / 81$ (v^9) / 81
$\beta_{10}$	$=$	$( \nu^{10} ) / 243$ (v^10) / 243
$\beta_{11}$	$=$	$( \nu^{11} ) / 243$ (v^11) / 243
$\beta_{12}$	$=$	$( \nu^{12} ) / 729$ (v^12) / 729
$\beta_{13}$	$=$	$( \nu^{13} ) / 729$ (v^13) / 729
$\beta_{14}$	$=$	$( \nu^{14} ) / 2187$ (v^14) / 2187
$\beta_{15}$	$=$	$( \nu^{15} ) / 2187$ (v^15) / 2187

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$3\beta_{2}$ 3*b2
$\nu^{3}$	$=$	$3\beta_{3}$ 3*b3
$\nu^{4}$	$=$	$9\beta_{4}$ 9*b4
$\nu^{5}$	$=$	$9\beta_{5}$ 9*b5
$\nu^{6}$	$=$	$27\beta_{6}$ 27*b6
$\nu^{7}$	$=$	$27\beta_{7}$ 27*b7
$\nu^{8}$	$=$	$81\beta_{8}$ 81*b8
$\nu^{9}$	$=$	$81\beta_{9}$ 81*b9
$\nu^{10}$	$=$	$243\beta_{10}$ 243*b10
$\nu^{11}$	$=$	$243\beta_{11}$ 243*b11
$\nu^{12}$	$=$	$729\beta_{12}$ 729*b12
$\nu^{13}$	$=$	$729\beta_{13}$ 729*b13
$\nu^{14}$	$=$	$2187\beta_{14}$ 2187*b14
$\nu^{15}$	$=$	$2187\beta_{15}$ 2187*b15

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$ .

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-\beta_{2}$

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}

53.1

1.54327	−	0.786335i
−1.54327	+	0.786335i
−0.270952	+	1.71073i
0.270952	−	1.71073i
1.71073	+	0.270952i
−1.71073	−	0.270952i
−0.786335	−	1.54327i
0.786335	+	1.54327i
1.54327	+	0.786335i
−1.54327	−	0.786335i
−0.270952	−	1.71073i
0.270952	+	1.71073i
1.71073	−	0.270952i
−1.71073	+	0.270952i
−0.786335	+	1.54327i
0.786335	−	1.54327i

0.642040

1.26007i

−0.270952

−

1.71073i

−1.17557

1.61803i

0.473739

−

2.18531i

1.98168

−

1.43977i

−3.20022

3.20022i

−2.79360

−

0.442463i

−2.85317

0.927051i

3.05781

−

0.806108i

53.2

0.642040

1.26007i

0.270952

1.71073i

−1.17557

1.61803i

2.04641

0.901229i

−1.98168

1.43977i

3.64268

−

3.64268i

−2.79360

−

0.442463i

−2.85317

0.927051i

0.178260

3.15725i

77.1

1.39680

0.221232i

−0.786335

−

1.54327i

1.90211

0.618034i

−1.93196

1.12585i

−0.756934

−

2.32960i

3.72858

3.72858i

2.52015

1.28408i

−1.76336

2.42705i

−2.94764

1.14518i

77.2

1.39680

0.221232i

0.786335

1.54327i

1.90211

0.618034i

1.48949

1.66775i

0.756934

2.32960i

−2.44450

−

2.44450i

2.52015

1.28408i

−1.76336

2.42705i

1.71157

2.65905i

173.1

0.221232

−

1.39680i

−1.54327

0.786335i

−1.90211

−

0.618034i

−1.12585

−

1.93196i

0.756934

2.32960i

0.312596

−

0.312596i

−1.28408

2.52015i

1.76336

−

2.42705i

−2.94764

1.14518i

173.2

0.221232

−

1.39680i

1.54327

−

0.786335i

−1.90211

−

0.618034i

−1.66775

1.48949i

−0.756934

−

2.32960i

−2.83274

2.83274i

−1.28408

2.52015i

1.76336

−

2.42705i

1.71157

2.65905i

197.1

−1.26007

0.642040i

−1.71073

0.270952i

1.17557

−

1.61803i

0.901229

−

2.04641i

1.98168

−

1.43977i

0.854897

0.854897i

−0.442463

2.79360i

2.85317

−

0.927051i

0.178260

3.15725i

197.2

−1.26007

0.642040i

1.71073

−

0.270952i

1.17557

−

1.61803i

−2.18531

−

0.473739i

−1.98168

1.43977i

1.93871

1.93871i

−0.442463

2.79360i

2.85317

−

0.927051i

3.05781

−

0.806108i

317.1

0.642040

−

1.26007i

−0.270952

1.71073i

−1.17557

−

1.61803i

0.473739

2.18531i

1.98168

1.43977i

−3.20022

−

3.20022i

−2.79360

0.442463i

−2.85317

−

0.927051i

3.05781

0.806108i

317.2

0.642040

−

1.26007i

0.270952

−

1.71073i

−1.17557

−

1.61803i

2.04641

−

0.901229i

−1.98168

−

1.43977i

3.64268

3.64268i

−2.79360

0.442463i

−2.85317

−

0.927051i

0.178260

−

3.15725i

413.1

1.39680

−

0.221232i

−0.786335

1.54327i

1.90211

−

0.618034i

−1.93196

−

1.12585i

−0.756934

2.32960i

3.72858

−

3.72858i

2.52015

−

1.28408i

−1.76336

−

2.42705i

−2.94764

−

1.14518i

413.2

1.39680

−

0.221232i

0.786335

−

1.54327i

1.90211

−

0.618034i

1.48949

−

1.66775i

0.756934

−

2.32960i

−2.44450

2.44450i

2.52015

−

1.28408i

−1.76336

−

2.42705i

1.71157

−

2.65905i

437.1

0.221232

1.39680i

−1.54327

−

0.786335i

−1.90211

0.618034i

−1.12585

1.93196i

0.756934

−

2.32960i

0.312596

0.312596i

−1.28408

−

2.52015i

1.76336

2.42705i

−2.94764

−

1.14518i

437.2

0.221232

1.39680i

1.54327

0.786335i

−1.90211

0.618034i

−1.66775

−

1.48949i

−0.756934

2.32960i

−2.83274

−

2.83274i

−1.28408

−

2.52015i

1.76336

2.42705i

1.71157

−

2.65905i

533.1

−1.26007

−

0.642040i

−1.71073

−

0.270952i

1.17557

1.61803i

0.901229

2.04641i

1.98168

1.43977i

0.854897

−

0.854897i

−0.442463

−

2.79360i

2.85317

0.927051i

0.178260

−

3.15725i

533.2

−1.26007

−

0.642040i

1.71073

0.270952i

1.17557

1.61803i

−2.18531

0.473739i

−1.98168

−

1.43977i

1.93871

−

1.93871i

−0.442463

−

2.79360i

2.85317

0.927051i

3.05781

0.806108i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
25.f	odd	20	1	inner
600.bp	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.2.bp.b	yes	16
3.b	odd	2	1		600.2.bp.a	✓	16
8.b	even	2	1		600.2.bp.a	✓	16
24.h	odd	2	1	CM	600.2.bp.b	yes	16
25.f	odd	20	1	inner	600.2.bp.b	yes	16
75.l	even	20	1		600.2.bp.a	✓	16
200.x	odd	20	1		600.2.bp.a	✓	16
600.bp	even	20	1	inner	600.2.bp.b	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.2.bp.a	✓	16	3.b	odd	2	1
600.2.bp.a	✓	16	8.b	even	2	1
600.2.bp.a	✓	16	75.l	even	20	1
600.2.bp.a	✓	16	200.x	odd	20	1
600.2.bp.b	yes	16	1.a	even	1	1	trivial
600.2.bp.b	yes	16	24.h	odd	2	1	CM
600.2.bp.b	yes	16	25.f	odd	20	1	inner
600.2.bp.b	yes	16	600.bp	even	20	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(600, [\chi])$ :

T_{7}^{16} - 4 T_{7}^{15} + 8 T_{7}^{14} + 40 T_{7}^{13} + 986 T_{7}^{12} - 3060 T_{7}^{11} + \cdots + 6225025

T_{11}^{16} + 24 T_{11}^{15} + 332 T_{11}^{14} + 3192 T_{11}^{13} + 23945 T_{11}^{12} + 145128 T_{11}^{11} + \cdots + 515517025

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
$3$	$T^{16} - 9 T^{12} + \cdots + 6561$ T^16 - 9T^12 + 81T^8 - 729*T^4 + 6561
$5$	$T^{16} + 4 T^{15} + \cdots + 390625$ T^16 + 4T^15 + 8T^14 + 20T^13 + 71T^12 + 140T^11 + 192T^10 + 568T^9 + 1761T^8 + 2840T^7 + 4800T^6 + 17500T^5 + 44375T^4 + 62500T^3 + 125000T^2 + 312500*T + 390625
$7$	$T^{16} - 4 T^{15} + \cdots + 6225025$ T^16 - 4T^15 + 8T^14 + 40T^13 + 986T^12 - 3060T^11 + 5152T^10 + 22372T^9 + 308571T^8 - 664020T^7 + 684320T^6 + 229620T^5 + 20932810T^4 - 49897600T^3 + 58969800T^2 - 27095700*T + 6225025
$11$	$T^{16} + \cdots + 515517025$ T^16 + 24T^15 + 332T^14 + 3192T^13 + 23945T^12 + 145128T^11 + 738072T^10 + 3051736T^9 + 9764486T^8 + 21796120T^7 + 35405100T^6 + 66101360T^5 + 299609660T^4 + 1032649000T^3 + 2290509300T^2 + 864606400*T + 515517025
$13$	$T^{16}$ T^16
$17$	$T^{16}$ T^16
$19$	$T^{16}$ T^16
$23$	$T^{16}$ T^16
$29$	$T^{16} + \cdots + 39062500000000$ T^16 - 116T^14 + 10956T^12 - 980896T^10 + 86393936T^8 - 2452240000T^6 + 68475000000T^4 - 1812500000000*T^2 + 39062500000000
$31$	$T^{16} + \cdots + 7611645084241$ T^16 + 238T^14 + 30093T^12 + 2734636T^10 + 282618630T^8 + 17647280446T^6 + 529085728868T^4 - 1038104722512*T^2 + 7611645084241
$37$	$T^{16}$ T^16
$41$	$T^{16}$ T^16
$43$	$T^{16}$ T^16
$47$	$T^{16}$ T^16
$53$	$T^{16} + \cdots + 14\!\cdots\!61$ T^16 - 5300T^13 - 49964T^12 + 14045000T^10 + 190370700T^9 + 952209726T^8 - 19026030100T^7 - 307220330000T^6 - 2321786322300T^5 + 5632142867981T^4 + 207331841964500T^3 + 1702227047645000T^2 + 7071629270763700T + 14688974838082561
$59$	$T^{16} + \cdots + 34\!\cdots\!25$ T^16 - 80T^15 + 2964T^14 - 67760T^13 + 1094441T^12 - 14116240T^11 + 167012904T^10 - 1919282960T^9 + 19345804166T^8 - 144899065040T^7 + 637984151220T^6 - 132476715840T^5 - 16221300793540T^4 + 77526355141200T^3 + 42638901005900T^2 - 1419275064283200*T + 3464064638253025
$61$	$T^{16}$ T^16
$67$	$T^{16}$ T^16
$71$	$T^{16}$ T^16
$73$	$T^{16} + \cdots + 39062500000000$ T^16 + 28T^15 + 392T^14 + 1400T^13 - 77764T^12 - 338800T^11 + 21977088T^10 + 635798464T^9 + 8907428496T^8 + 31789923200T^7 + 54942720000T^6 - 42350000000T^5 - 486025000000T^4 + 437500000000T^3 + 6125000000000T^2 + 21875000000000*T + 39062500000000
$79$	$(T^{8} - 216 T^{6} + \cdots + 55636681)^{2}$ (T^8 - 216T^6 - 3950T^5 + 23746T^4 + 853200T^3 + 7196839T^2 + 29463050T + 55636681)^2
$83$	$T^{16} + \cdots + 24\!\cdots\!41$ T^16 - 24T^15 + 288T^14 - 5368T^13 + 96781T^12 - 823144T^11 + 6290240T^10 + 82719896T^9 + 1487957174T^8 - 96105131608T^7 + 1163958395680T^6 - 11765945706352T^5 - 43530324976404T^4 + 1155378323596024T^3 + 8488238201570432T^2 + 64150380604443568*T + 242410217171661841
$89$	$T^{16}$ T^16
$97$	$T^{16} + \cdots + 22\!\cdots\!25$ T^16 + 16T^15 + 1078T^14 + 14740T^13 + 474921T^12 + 5658460T^11 + 104764472T^10 + 1091402872T^9 + 10222386926T^8 + 80252639600T^7 - 22780683610T^6 - 9014464505020T^5 + 19925238953460T^4 + 562482729010400T^3 + 664571901579700T^2 - 3411016888305400*T + 2259281992929025
show more
show less

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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.2
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	600.2.bp.b

Level	$600$
Weight	$2$
Character orbit	600.bp
Analytic conductor	$4.791$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-24
Inner twists	$4$