Newform orbit 966.2.i.m

• Label: 966.2.i.m
• Level: $966$
• Weight: $2$
• Character orbit: 966.i
• Analytic conductor: $7.714$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.71354883526\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 2x^{7} + 13x^{6} + 10x^{5} + 47x^{4} + 180x^{3} + 220x^{2} + 768x + 1164 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} + 10x^{5} + 47x^{4} + 180x^{3} + 220x^{2} + 768x + 1164 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} - 19\nu^{5} + 8\nu^{4} - 145\nu^{3} - 142\nu^{2} - 254\nu - 1064 ) / 784 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} + 6\nu^{6} - 37\nu^{5} + 106\nu^{4} - 183\nu^{3} - 108\nu^{2} - 742\nu - 2228 ) / 784 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{7} + 33\nu^{6} - 11\nu^{5} + 275\nu^{4} + 215\nu^{3} + 2227\nu^{2} + 4228\nu + 6758 ) / 2744 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 3\nu^{6} - 9\nu^{5} + 49\nu^{4} - 19\nu^{3} + 17\nu^{2} + 148\nu - 582 ) / 392 \)
\(\beta_{5}\)	\(=\)	\( ( -13\nu^{7} + 41\nu^{6} - 107\nu^{5} + 99\nu^{4} - 337\nu^{3} + 235\nu^{2} + 476\nu - 3138 ) / 2744 \)
\(\beta_{6}\)	\(=\)	\( ( 2\nu^{7} - 9\nu^{6} + 24\nu^{5} - 40\nu^{4} - 100\nu^{3} + 120\nu^{2} - 840\nu - 529 ) / 343 \)
\(\beta_{7}\)	\(=\)	\( ( -39\nu^{7} + 32\nu^{6} - 517\nu^{5} - 648\nu^{4} - 3223\nu^{3} - 3474\nu^{2} - 16786\nu - 19592 ) / 5488 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(829\)	\(925\)
\(\chi(n)\)	\(1\)	\(-1 - \beta_{1}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

277.1

1.76655	+	2.53227i
1.26426	−	2.63252i
−0.415888	+	2.23501i
−1.61492	−	0.402708i
1.76655	−	2.53227i
1.26426	+	2.63252i
−0.415888	−	2.23501i
−1.61492	+	0.402708i

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

−1.76655

−

3.05976i

−1.00000

1.80974

+

1.92999i

1.00000

−0.500000

−

0.866025i

−1.76655

+

3.05976i

277.2

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

−1.26426

−

2.18976i

−1.00000

−2.41196

−

1.08740i

1.00000

−0.500000

−

0.866025i

−1.26426

+

2.18976i

277.3

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

0.415888

+

0.720339i

−1.00000

2.64352

−

0.108691i

1.00000

−0.500000

−

0.866025i

0.415888

−

0.720339i

277.4

−0.500000

−

0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

1.61492

+

2.79713i

−1.00000

0.958707

−

2.46594i

1.00000

−0.500000

−

0.866025i

1.61492

−

2.79713i

415.1

−0.500000

+

0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

−1.76655

+

3.05976i

−1.00000

1.80974

−

1.92999i

1.00000

−0.500000

+

0.866025i

−1.76655

−

3.05976i

415.2

−0.500000

+

0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

−1.26426

+

2.18976i

−1.00000

−2.41196

+

1.08740i

1.00000

−0.500000

+

0.866025i

−1.26426

−

2.18976i

415.3

−0.500000

+

0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

0.415888

−

0.720339i

−1.00000

2.64352

+

0.108691i

1.00000

−0.500000

+

0.866025i

0.415888

+

0.720339i

415.4

−0.500000

+

0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

1.61492

−

2.79713i

−1.00000

0.958707

+

2.46594i

1.00000

−0.500000

+

0.866025i

1.61492

+

2.79713i

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	966.2.i.m	✓	8
7.c	even	3	1	inner	966.2.i.m	✓	8
7.c	even	3	1		6762.2.a.cl		4
7.d	odd	6	1		6762.2.a.cr		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.i.m	✓	8	1.a	even	1	1	trivial
966.2.i.m	✓	8	7.c	even	3	1	inner
6762.2.a.cl		4	7.c	even	3	1
6762.2.a.cr		4	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 17T_{5}^{6} + 14T_{5}^{5} + 185T_{5}^{4} + 164T_{5}^{3} + 712T_{5}^{2} - 480T_{5} + 576 \)

\( T_{11}^{8} + 31T_{11}^{6} + 24T_{11}^{5} + 781T_{11}^{4} + 372T_{11}^{3} + 5724T_{11}^{2} - 2160T_{11} + 32400 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{4} \)
$3$	\( (T^{2} - T + 1)^{4} \)
$5$	T^8 + 2*T^7 + 17*T^6 + 14*T^5 + 185*T^4 + 164*T^3 + 712*T^2 - 480*T + 576
$7$	T^8 - 6*T^7 + 12*T^6 + 12*T^5 - 107*T^4 + 84*T^3 + 588*T^2 - 2058*T + 2401
$11$	T^8 + 31*T^6 + 24*T^5 + 781*T^4 + 372*T^3 + 5724*T^2 - 2160*T + 32400