Newform orbit 399.2.j.d

• Label: 399.2.j.d
• Level: $399$
• Weight: $2$
• Character orbit: 399.j
• Analytic conductor: $3.186$
• Analytic rank: $0$
• Dimension: $8$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18603104065\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.310217769.2
Defining polynomial:	\( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b1 * q^2 + (-b5 + 1) * q^3 + (b7 + b6 + b4 - b2 - b1) * q^4 + (-b5 - b4 + b2 + b1) * q^5 + (b3 - b1) * q^6 + (-b6 - b4 + b3 + b2 - b1) * q^7 + (b7 + b3 - b1 - 1) * q^8 - b5 * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 4 q^{5} + 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -49\nu^{7} + 64\nu^{6} - 256\nu^{5} + 338\nu^{4} - 1088\nu^{3} + 1216\nu^{2} - 1385\nu + 256 ) / 289 \)
\(\beta_{3}\)	\(=\)	\( ( 60\nu^{7} + 16\nu^{6} + 225\nu^{5} - 60\nu^{4} + 884\nu^{3} + 15\nu^{2} + 304\nu + 64 ) / 289 \)
\(\beta_{4}\)	\(=\)	\( ( 63\nu^{7} - 41\nu^{6} + 164\nu^{5} - 352\nu^{4} + 697\nu^{3} - 779\nu^{2} - 490\nu - 164 ) / 289 \)
\(\beta_{5}\)	\(=\)	\( ( -64\nu^{7} + 60\nu^{6} - 240\nu^{5} + 353\nu^{4} - 1020\nu^{3} + 1140\nu^{2} - 305\nu + 240 ) / 289 \)
\(\beta_{6}\)	\(=\)	\( ( -164\nu^{7} - 63\nu^{6} - 615\nu^{5} + 164\nu^{4} - 2108\nu^{3} - 41\nu^{2} - 41\nu + 37 ) / 289 \)
\(\beta_{7}\)	\(=\)	\( ( 180\nu^{7} + 48\nu^{6} + 675\nu^{5} - 180\nu^{4} + 2363\nu^{3} + 45\nu^{2} + 45\nu + 481 ) / 289 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\)	\(115\)	\(134\)	\(211\)
\(\chi(n)\)	\(-\beta_{5}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

58.1

0.882007	−	1.52768i
0.346911	−	0.600868i
−0.198169	+	0.343239i
−1.03075	+	1.78531i
0.882007	+	1.52768i
0.346911	+	0.600868i
−0.198169	−	0.343239i
−1.03075	−	1.78531i

−0.882007

+

1.52768i

0.500000

+

0.866025i

−0.555874

−

0.962803i

−1.05587

+

1.82883i

−1.76401

−0.893022

+

2.49048i

−1.56689

−0.500000

+

0.866025i

−1.86258

−

3.22608i

58.2

−0.346911

+

0.600868i

0.500000

+

0.866025i

0.759305

+

1.31516i

0.259305

−

0.449130i

−0.693822

−2.54751

−

0.714287i

−2.44129

−0.500000

+

0.866025i

0.179912

+

0.311616i

58.3

0.198169

−

0.343239i

0.500000

+

0.866025i

0.921458

+

1.59601i

0.421458

−

0.729986i

0.396339

1.79981

−

1.93925i

1.52310

−0.500000

+

0.866025i

−0.167040

−

0.289322i

58.4

1.03075

−

1.78531i

0.500000

+

0.866025i

−1.12489

−

1.94836i

−1.62489

+

2.81439i

2.06150

2.64072

+

0.163054i

−0.514916

−0.500000

+

0.866025i

3.34971

+

5.80186i

172.1

−0.882007

−

1.52768i

0.500000

−

0.866025i

−0.555874

+

0.962803i

−1.05587

−

1.82883i

−1.76401

−0.893022

−

2.49048i

−1.56689

−0.500000

−

0.866025i

−1.86258

+

3.22608i

172.2

−0.346911

−

0.600868i

0.500000

−

0.866025i

0.759305

−

1.31516i

0.259305

+

0.449130i

−0.693822

−2.54751

+

0.714287i

−2.44129

−0.500000

−

0.866025i

0.179912

−

0.311616i

172.3

0.198169

+

0.343239i

0.500000

−

0.866025i

0.921458

−

1.59601i

0.421458

+

0.729986i

0.396339

1.79981

+

1.93925i

1.52310

−0.500000

−

0.866025i

−0.167040

+

0.289322i

172.4

1.03075

+

1.78531i

0.500000

−

0.866025i

−1.12489

+

1.94836i

−1.62489

−

2.81439i

2.06150

2.64072

−

0.163054i

−0.514916

−0.500000

−

0.866025i

3.34971

−

5.80186i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.2.j.d	✓	8
3.b	odd	2	1		1197.2.j.k		8
7.c	even	3	1	inner	399.2.j.d	✓	8
7.c	even	3	1		2793.2.a.bc		4
7.d	odd	6	1		2793.2.a.bd		4
21.g	even	6	1		8379.2.a.bt		4
21.h	odd	6	1		1197.2.j.k		8
21.h	odd	6	1		8379.2.a.br		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.j.d	✓	8	1.a	even	1	1	trivial
399.2.j.d	✓	8	7.c	even	3	1	inner
1197.2.j.k		8	3.b	odd	2	1
1197.2.j.k		8	21.h	odd	6	1
2793.2.a.bc		4	7.c	even	3	1
2793.2.a.bd		4	7.d	odd	6	1
8379.2.a.br		4	21.h	odd	6	1
8379.2.a.bt		4	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{6} + 2T_{2}^{5} + 15T_{2}^{4} + 4T_{2}^{3} + 5T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 4*T^6 + 2*T^5 + 15*T^4 + 4*T^3 + 5*T^2 - T + 1
$3$	\( (T^{2} - T + 1)^{4} \)
$5$	T^8 + 4*T^7 + 16*T^6 + 14*T^5 + 25*T^4 - 24*T^3 + 49*T^2 - 21*T + 9
$7$	T^8 - 2*T^7 - 5*T^6 + 8*T^5 + 5*T^4 + 56*T^3 - 245*T^2 - 686*T + 2401
$11$	T^8 - 2*T^7 + 26*T^6 + 98*T^5 + 429*T^4 + 598*T^3 + 751*T^2 - 27*T + 1