Newform orbit 1875.2.b.g

• Label: 1875.2.b.g
• Level: $1875$
• Weight: $2$
• Character orbit: 1875.b
• Analytic conductor: $14.972$
• Analytic rank: $0$
• Dimension: $16$
• 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:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{3} - 1) q^{4} + ( - \beta_{13} + \beta_{9} + \beta_{6} + 1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{8} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 25\nu^{13} + 246\nu^{11} + 1220\nu^{9} + 3281\nu^{7} + 4880\nu^{5} + 3936\nu^{3} + 1472\nu ) / 128 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{14} + 25\nu^{12} + 246\nu^{10} + 1220\nu^{8} + 3281\nu^{6} + 4880\nu^{4} + 3872\nu^{2} + 1216 ) / 64 \)
\(\beta_{4}\)	\(=\)	\( ( -3\nu^{15} - 75\nu^{13} - 738\nu^{11} - 3660\nu^{9} - 9843\nu^{7} - 14640\nu^{5} - 11680\nu^{3} - 3904\nu ) / 128 \)
\(\beta_{5}\)	\(=\)	\( ( 4\nu^{15} + 99\nu^{13} + 959\nu^{11} + 4634\nu^{9} + 11904\nu^{7} + 16239\nu^{5} + 10896\nu^{3} + 2752\nu ) / 32 \)
\(\beta_{6}\)	\(=\)	\( ( -13\nu^{14} - 321\nu^{12} - 3098\nu^{10} - 14876\nu^{8} - 37773\nu^{6} - 50380\nu^{4} - 32416\nu^{2} - 7744 ) / 64 \)
\(\beta_{7}\)	\(=\)	\( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18592\nu^{2} + 4032 ) / 32 \)
\(\beta_{8}\)	\(=\)	\( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18624\nu^{2} + 4128 ) / 32 \)
\(\beta_{9}\)	\(=\)	\( ( -5\nu^{14} - 123\nu^{12} - 1180\nu^{10} - 5608\nu^{8} - 13981\nu^{6} - 18078\nu^{4} - 11120\nu^{2} - 2528 ) / 16 \)
\(\beta_{10}\)	\(=\)	\( ( 21\nu^{15} + 519\nu^{13} + 5016\nu^{11} + 24144\nu^{9} + 61581\nu^{7} + 82858\nu^{5} + 54208\nu^{3} + 13248\nu ) / 64 \)
\(\beta_{11}\)	\(=\)	\( ( 79 \nu^{15} + 1935 \nu^{13} + 18450 \nu^{11} + 86940 \nu^{9} + 214335 \nu^{7} + 273672 \nu^{5} + 166320 \nu^{3} + 37440 \nu ) / 128 \)
\(\beta_{12}\)	\(=\)	\( ( 23\nu^{15} + 561\nu^{13} + 5316\nu^{11} + 24816\nu^{9} + 60319\nu^{7} + 75490\nu^{5} + 44712\nu^{3} + 9792\nu ) / 32 \)
\(\beta_{13}\)	\(=\)	\( ( 23\nu^{14} + 561\nu^{12} + 5316\nu^{10} + 24816\nu^{8} + 60319\nu^{6} + 75490\nu^{4} + 44712\nu^{2} + 9824 ) / 32 \)
\(\beta_{14}\)	\(=\)	\( ( 151 \nu^{15} + 3683 \nu^{13} + 34910 \nu^{11} + 163124 \nu^{9} + 397463 \nu^{7} + 500052 \nu^{5} + 298912 \nu^{3} + 66304 \nu ) / 128 \)
\(\beta_{15}\)	\(=\)	\( ( -23\nu^{14} - 560\nu^{12} - 5295\nu^{10} - 24654\nu^{8} - 59763\nu^{6} - 74673\nu^{4} - 44268\nu^{2} - 9744 ) / 16 \)

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

	−	0.741379i
	−	0.770071i
	−	0.895394i
	−	1.31354i
	−	1.52260i
	−	2.23365i
	−	2.59716i
	−	2.69767i
		2.69767i
		2.59716i
		2.23365i
		1.52260i
		1.31354i
		0.895394i
		0.770071i
		0.741379i

−

2.69767i

−

1.00000i

−5.27745

0

−2.69767

−

3.56649i

8.84149i

−1.00000

0

1249.2

−

2.59716i

1.00000i

−4.74525

0

2.59716

−

3.28414i

7.12986i

−1.00000

0

1249.3

−

2.23365i

1.00000i

−2.98921

0

2.23365

1.03143i

2.20956i

−1.00000

0

1249.4

−

1.52260i

−

1.00000i

−0.318310

0

−1.52260

−

0.990985i

−

2.56054i

−1.00000

0

1249.5

−

1.31354i

1.00000i

0.274605

0

1.31354

−

4.19091i

−

2.98779i

−1.00000

0

1249.6

−

0.895394i

−

1.00000i

1.19827

0

−0.895394

5.08992i

−

2.86371i

−1.00000

0

1249.7

−

0.770071i

−

1.00000i

1.40699

0

−0.770071

3.98808i

−

2.62363i

−1.00000

0

1249.8

−

0.741379i

1.00000i

1.45036

0

0.741379

−

1.03586i

−

2.55802i

−1.00000

0

1249.9

0.741379i

−

1.00000i

1.45036

0

0.741379

1.03586i

2.55802i

−1.00000

0

1249.10

0.770071i

1.00000i

1.40699

0

−0.770071

−

3.98808i

2.62363i

−1.00000

0

1249.11

0.895394i

1.00000i

1.19827

0

−0.895394

−

5.08992i

2.86371i

−1.00000

0

1249.12

1.31354i

−

1.00000i

0.274605

0

1.31354

4.19091i

2.98779i

−1.00000

0

1249.13

1.52260i

1.00000i

−0.318310

0

−1.52260

0.990985i

2.56054i

−1.00000

0

1249.14

2.23365i

−

1.00000i

−2.98921

0

2.23365

−

1.03143i

−

2.20956i

−1.00000

0

1249.15

2.59716i

−

1.00000i

−4.74525

0

2.59716

3.28414i

−

7.12986i

−1.00000

0

1249.16

2.69767i

1.00000i

−5.27745

0

−2.69767

3.56649i

−

8.84149i

−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.g		16
5.b	even	2	1	inner	1875.2.b.g		16
5.c	odd	4	1		1875.2.a.n	✓	8
5.c	odd	4	1		1875.2.a.o	yes	8
15.e	even	4	1		5625.2.a.u		8
15.e	even	4	1		5625.2.a.bc		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1875.2.a.n	✓	8	5.c	odd	4	1
1875.2.a.o	yes	8	5.c	odd	4	1
1875.2.b.g		16	1.a	even	1	1	trivial
1875.2.b.g		16	5.b	even	2	1	inner
5625.2.a.u		8	15.e	even	4	1
5625.2.a.bc		8	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 25T_{2}^{14} + 246T_{2}^{12} + 1220T_{2}^{10} + 3281T_{2}^{8} + 4880T_{2}^{6} + 3936T_{2}^{4} + 1600T_{2}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^16 + 25*T^14 + 246*T^12 + 1220*T^10 + 3281*T^8 + 4880*T^6 + 3936*T^4 + 1600*T^2 + 256
$3$	\( (T^{2} + 1)^{8} \)
$5$	\( T^{16} \)
$7$	T^16 + 86*T^14 + 2941*T^12 + 50956*T^10 + 468186*T^8 + 2154110*T^6 + 4204840*T^4 + 3583900*T^2 + 1113025
$11$	(T^8 - 12*T^7 + 2*T^6 + 455*T^5 - 1749*T^4 + 1160*T^3 + 1728*T^2 + 344*T + 16)^2