Newform orbit 980.2.s.c

• Label: 980.2.s.c
• Level: $980$
• Weight: $2$
• Character orbit: 980.s
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $16$
• CM discriminant: -4
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{48})\)
Defining polynomial:	\( x^{16} - x^{8} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 7^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_{5} q^{2} + ( - 2 \beta_{2} + 2) q^{4} - \beta_{12} q^{5} + ( - 2 \beta_{7} + 2 \beta_{5}) q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( 7\zeta_{48}^{4} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{48}^{8} \)
\(\beta_{3}\)	\(=\)	\( 7\zeta_{48}^{12} \)
\(\beta_{4}\)	\(=\)	\( \zeta_{48}^{13} + 2\zeta_{48} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{48}^{14} + \zeta_{48}^{2} \)
\(\beta_{6}\)	\(=\)	\( -\zeta_{48}^{15} + 3\zeta_{48}^{3} \)
\(\beta_{7}\)	\(=\)	\( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \)
\(\beta_{8}\)	\(=\)	\( -2\zeta_{48}^{15} - \zeta_{48}^{3} \)
\(\beta_{9}\)	\(=\)	\( -3\zeta_{48}^{13} + \zeta_{48} \)
\(\beta_{10}\)	\(=\)	\( -3\zeta_{48}^{13} + \zeta_{48}^{9} + 3\zeta_{48}^{5} \)
\(\beta_{11}\)	\(=\)	\( -\zeta_{48}^{13} - 2\zeta_{48}^{9} + \zeta_{48}^{5} \)
\(\beta_{12}\)	\(=\)	\( 2\zeta_{48}^{15} + \zeta_{48}^{11} - 2\zeta_{48}^{7} \)
\(\beta_{13}\)	\(=\)	\( 4\zeta_{48}^{14} + 3\zeta_{48}^{2} \)
\(\beta_{14}\)	\(=\)	\( 3\zeta_{48}^{11} + \zeta_{48}^{7} - 3\zeta_{48}^{3} \)
\(\beta_{15}\)	\(=\)	\( -4\zeta_{48}^{10} + 3\zeta_{48}^{6} + 4\zeta_{48}^{2} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(\beta_{2}\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

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Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
5.b	even	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner
140.c	even	2	1	inner
140.p	odd	6	1	inner
140.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.s.c		16
4.b	odd	2	1	CM	980.2.s.c		16
5.b	even	2	1	inner	980.2.s.c		16
7.b	odd	2	1	inner	980.2.s.c		16
7.c	even	3	1		980.2.c.a	✓	8
7.c	even	3	1	inner	980.2.s.c		16
7.d	odd	6	1		980.2.c.a	✓	8
7.d	odd	6	1	inner	980.2.s.c		16
20.d	odd	2	1	inner	980.2.s.c		16
28.d	even	2	1	inner	980.2.s.c		16
28.f	even	6	1		980.2.c.a	✓	8
28.f	even	6	1	inner	980.2.s.c		16
28.g	odd	6	1		980.2.c.a	✓	8
28.g	odd	6	1	inner	980.2.s.c		16
35.c	odd	2	1	inner	980.2.s.c		16
35.i	odd	6	1		980.2.c.a	✓	8
35.i	odd	6	1	inner	980.2.s.c		16
35.j	even	6	1		980.2.c.a	✓	8
35.j	even	6	1	inner	980.2.s.c		16
140.c	even	2	1	inner	980.2.s.c		16
140.p	odd	6	1		980.2.c.a	✓	8
140.p	odd	6	1	inner	980.2.s.c		16
140.s	even	6	1		980.2.c.a	✓	8
140.s	even	6	1	inner	980.2.s.c		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.2.c.a	✓	8	7.c	even	3	1
980.2.c.a	✓	8	7.d	odd	6	1
980.2.c.a	✓	8	28.f	even	6	1
980.2.c.a	✓	8	28.g	odd	6	1
980.2.c.a	✓	8	35.i	odd	6	1
980.2.c.a	✓	8	35.j	even	6	1
980.2.c.a	✓	8	140.p	odd	6	1
980.2.c.a	✓	8	140.s	even	6	1
980.2.s.c		16	1.a	even	1	1	trivial
980.2.s.c		16	4.b	odd	2	1	CM
980.2.s.c		16	5.b	even	2	1	inner
980.2.s.c		16	7.b	odd	2	1	inner
980.2.s.c		16	7.c	even	3	1	inner
980.2.s.c		16	7.d	odd	6	1	inner
980.2.s.c		16	20.d	odd	2	1	inner
980.2.s.c		16	28.d	even	2	1	inner
980.2.s.c		16	28.f	even	6	1	inner
980.2.s.c		16	28.g	odd	6	1	inner
980.2.s.c		16	35.c	odd	2	1	inner
980.2.s.c		16	35.i	odd	6	1	inner
980.2.s.c		16	35.j	even	6	1	inner
980.2.s.c		16	140.c	even	2	1	inner
980.2.s.c		16	140.p	odd	6	1	inner
980.2.s.c		16	140.s	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} - 2 T^{2} + 4)^{4} \)
$3$	\( T^{16} \)
$5$	T^16 + 48*T^12 + 1679*T^8 + 30000*T^4 + 390625
$7$	\( T^{16} \)
$11$	\( T^{16} \)