Newform orbit 1875.2.b.e

• Label: 1875.2.b.e
• Level: $1875$
• Weight: $2$
• Character orbit: 1875.b
• Analytic conductor: $14.972$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9719503790\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{11} q^{6} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{7} + (\beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_1) q^{8} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 4 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} + 20\nu^{9} + 139\nu^{7} + 377\nu^{5} + 268\nu^{3} - 93\nu ) / 72 \)
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{10} + 37\nu^{8} + 227\nu^{6} + 490\nu^{4} + 161\nu^{2} - 9 ) / 36 \)
\(\beta_{5}\)	\(=\)	\( ( 2\nu^{11} + 37\nu^{9} + 227\nu^{7} + 490\nu^{5} + 161\nu^{3} - 45\nu ) / 36 \)
\(\beta_{6}\)	\(=\)	\( ( 5\nu^{11} + 97\nu^{9} + 662\nu^{7} + 1873\nu^{5} + 1937\nu^{3} + 522\nu ) / 108 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{10} + 23\nu^{8} + 190\nu^{6} + 677\nu^{4} + 967\nu^{2} + 342 ) / 36 \)
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} - 16\nu^{9} - 71\nu^{7} - \nu^{5} + 472\nu^{3} + 409\nu ) / 24 \)
\(\beta_{9}\)	\(=\)	\( ( -2\nu^{11} - 55\nu^{9} - 551\nu^{7} - 2434\nu^{5} - 4355\nu^{3} - 1683\nu ) / 108 \)
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} + 18\nu^{8} + 109\nu^{6} + 249\nu^{4} + 154\nu^{2} + 21 ) / 8 \)
\(\beta_{11}\)	\(=\)	\( ( 2\nu^{10} + 37\nu^{8} + 233\nu^{6} + 562\nu^{4} + 377\nu^{2} + 45 ) / 12 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\)	\(626\)	\(1252\)
\(\chi(n)\)	\(1\)	\(-1\)

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.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1249.1

	−	2.78712i
	−	2.38719i
	−	2.13324i
	−	2.02791i
	−	0.858825i
	−	0.364088i
		0.364088i
		0.858825i
		2.02791i
		2.13324i
		2.38719i
		2.78712i

−

2.78712i

−

1.00000i

−5.76803

0

−2.78712

3.15000i

10.5020i

−1.00000

0

1249.2

−

2.38719i

1.00000i

−3.69868

0

2.38719

−

3.31671i

4.05506i

−1.00000

0

1249.3

−

2.13324i

−

1.00000i

−2.55073

0

−2.13324

2.16876i

1.17484i

−1.00000

0

1249.4

−

2.02791i

1.00000i

−2.11242

0

2.02791

0.505614i

0.227977i

−1.00000

0

1249.5

−

0.858825i

−

1.00000i

1.26242

0

−0.858825

−

3.88045i

−

2.80185i

−1.00000

0

1249.6

−

0.364088i

1.00000i

1.86744

0

0.364088

2.24941i

−

1.40809i

−1.00000

0

1249.7

0.364088i

−

1.00000i

1.86744

0

0.364088

−

2.24941i

1.40809i

−1.00000

0

1249.8

0.858825i

1.00000i

1.26242

0

−0.858825

3.88045i

2.80185i

−1.00000

0

1249.9

2.02791i

−

1.00000i

−2.11242

0

2.02791

−

0.505614i

−

0.227977i

−1.00000

0

1249.10

2.13324i

1.00000i

−2.55073

0

−2.13324

−

2.16876i

−

1.17484i

−1.00000

0

1249.11

2.38719i

−

1.00000i

−3.69868

0

2.38719

3.31671i

−

4.05506i

−1.00000

0

1249.12

2.78712i

1.00000i

−5.76803

0

−2.78712

−

3.15000i

−

10.5020i

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1875.2.b.e		12
5.b	even	2	1	inner	1875.2.b.e		12
5.c	odd	4	1		1875.2.a.i	✓	6
5.c	odd	4	1		1875.2.a.l	yes	6
15.e	even	4	1		5625.2.a.o		6
15.e	even	4	1		5625.2.a.r		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1875.2.a.i	✓	6	5.c	odd	4	1
1875.2.a.l	yes	6	5.c	odd	4	1
1875.2.b.e		12	1.a	even	1	1	trivial
1875.2.b.e		12	5.b	even	2	1	inner
5625.2.a.o		6	15.e	even	4	1
5625.2.a.r		6	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 23T_{2}^{10} + 199T_{2}^{8} + 794T_{2}^{6} + 1399T_{2}^{4} + 783T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^12 + 23*T^10 + 199*T^8 + 794*T^6 + 1399*T^4 + 783*T^2 + 81
$3$	\( (T^{2} + 1)^{6} \)
$5$	\( T^{12} \)
$7$	T^12 + 46*T^10 + 811*T^8 + 6846*T^6 + 27841*T^4 + 45800*T^2 + 10000
$11$	(T^6 - 29*T^4 - 8*T^3 + 184*T^2 + 96*T - 144)^2