Properties

Label 980.2.c.b
Level $980$
Weight $2$
Character orbit 980.c
Analytic conductor $7.825$
Analytic rank $0$
Dimension $8$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
Defining polynomial: \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2 q^{4} + \beta_{6} q^{5} + ( -\beta_{5} + \beta_{6} ) q^{6} -2 \beta_{1} q^{8} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{3} q^{3} + 2 q^{4} + \beta_{6} q^{5} + ( -\beta_{5} + \beta_{6} ) q^{6} -2 \beta_{1} q^{8} + ( -1 + \beta_{2} ) q^{9} + ( \beta_{3} + \beta_{7} ) q^{10} + 2 \beta_{3} q^{12} + ( 3 \beta_{1} - \beta_{4} ) q^{15} + 4 q^{16} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{18} + 2 \beta_{6} q^{20} + ( -2 \beta_{1} + 3 \beta_{4} ) q^{23} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{24} -5 q^{25} + ( -2 \beta_{3} + \beta_{7} ) q^{27} + ( -3 + 2 \beta_{2} ) q^{29} + ( -5 + \beta_{2} ) q^{30} -4 \beta_{1} q^{32} + ( -2 + 2 \beta_{2} ) q^{36} + ( 2 \beta_{3} + 2 \beta_{7} ) q^{40} + ( 6 \beta_{5} + \beta_{6} ) q^{41} + ( -4 \beta_{1} - \beta_{4} ) q^{43} + ( 5 \beta_{5} - \beta_{6} ) q^{45} + ( 1 - 3 \beta_{2} ) q^{46} + ( -3 \beta_{3} - 3 \beta_{7} ) q^{47} + 4 \beta_{3} q^{48} + 5 \beta_{1} q^{50} + ( 3 \beta_{5} - \beta_{6} ) q^{54} + ( 5 \beta_{1} - 4 \beta_{4} ) q^{58} + ( 6 \beta_{1} - 2 \beta_{4} ) q^{60} + ( -4 \beta_{5} - 3 \beta_{6} ) q^{61} + 8 q^{64} + ( -\beta_{1} + 5 \beta_{4} ) q^{67} + ( -8 \beta_{5} + 5 \beta_{6} ) q^{69} + ( 4 \beta_{1} - 4 \beta_{4} ) q^{72} -5 \beta_{3} q^{75} + 4 \beta_{6} q^{80} + ( 4 + \beta_{2} ) q^{81} + ( -5 \beta_{3} + 7 \beta_{7} ) q^{82} + ( 4 \beta_{3} - 7 \beta_{7} ) q^{83} + ( 9 + \beta_{2} ) q^{86} + ( -11 \beta_{3} + 2 \beta_{7} ) q^{87} + ( 3 \beta_{5} + 4 \beta_{6} ) q^{89} + ( -6 \beta_{3} + 4 \beta_{7} ) q^{90} + ( -4 \beta_{1} + 6 \beta_{4} ) q^{92} -6 \beta_{6} q^{94} + ( -4 \beta_{5} + 4 \beta_{6} ) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{4} - 8q^{9} + O(q^{10}) \) \( 8q + 16q^{4} - 8q^{9} + 32q^{16} - 40q^{25} - 24q^{29} - 40q^{30} - 16q^{36} + 8q^{46} + 64q^{64} + 32q^{81} + 72q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu \)\()/189\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} + 7 \nu^{3} - 9 \nu \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 7 \nu^{3} + 63 \nu \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225 \)\()/63\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 22 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - \nu^{5} + 4 \nu^{3} - 24 \nu \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 2 \beta_{5} + \beta_{2} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + \beta_{3} + 7 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{6} + \beta_{5} + 4 \beta_{2} + 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{7} + 5 \beta_{4} - 12 \beta_{3} + 7 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-7 \beta_{6} + 22\)
\(\nu^{7}\)\(=\)\((\)\(21 \beta_{7} + 29 \beta_{4} + 8 \beta_{3} - 21 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
979.1
−1.01575 1.40294i
1.72286 0.178197i
1.72286 + 0.178197i
−1.01575 + 1.40294i
1.01575 1.40294i
−1.72286 0.178197i
−1.72286 + 0.178197i
1.01575 + 1.40294i
−1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 −2.82843 −4.87298 3.16228i
979.2 −1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 −2.82843 2.87298 3.16228i
979.3 −1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 −2.82843 2.87298 3.16228i
979.4 −1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 −2.82843 −4.87298 3.16228i
979.5 1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 2.82843 −4.87298 3.16228i
979.6 1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 2.82843 2.87298 3.16228i
979.7 1.41421 0.356394i 2.00000 2.23607i 0.504017i 0 2.82843 2.87298 3.16228i
979.8 1.41421 2.80588i 2.00000 2.23607i 3.96812i 0 2.82843 −4.87298 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 979.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.c.b 8
4.b odd 2 1 inner 980.2.c.b 8
5.b even 2 1 inner 980.2.c.b 8
7.b odd 2 1 inner 980.2.c.b 8
7.c even 3 1 140.2.s.a 8
7.c even 3 1 980.2.s.a 8
7.d odd 6 1 140.2.s.a 8
7.d odd 6 1 980.2.s.a 8
20.d odd 2 1 CM 980.2.c.b 8
28.d even 2 1 inner 980.2.c.b 8
28.f even 6 1 140.2.s.a 8
28.f even 6 1 980.2.s.a 8
28.g odd 6 1 140.2.s.a 8
28.g odd 6 1 980.2.s.a 8
35.c odd 2 1 inner 980.2.c.b 8
35.i odd 6 1 140.2.s.a 8
35.i odd 6 1 980.2.s.a 8
35.j even 6 1 140.2.s.a 8
35.j even 6 1 980.2.s.a 8
35.k even 12 2 700.2.p.b 8
35.l odd 12 2 700.2.p.b 8
140.c even 2 1 inner 980.2.c.b 8
140.p odd 6 1 140.2.s.a 8
140.p odd 6 1 980.2.s.a 8
140.s even 6 1 140.2.s.a 8
140.s even 6 1 980.2.s.a 8
140.w even 12 2 700.2.p.b 8
140.x odd 12 2 700.2.p.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.s.a 8 7.c even 3 1
140.2.s.a 8 7.d odd 6 1
140.2.s.a 8 28.f even 6 1
140.2.s.a 8 28.g odd 6 1
140.2.s.a 8 35.i odd 6 1
140.2.s.a 8 35.j even 6 1
140.2.s.a 8 140.p odd 6 1
140.2.s.a 8 140.s even 6 1
700.2.p.b 8 35.k even 12 2
700.2.p.b 8 35.l odd 12 2
700.2.p.b 8 140.w even 12 2
700.2.p.b 8 140.x odd 12 2
980.2.c.b 8 1.a even 1 1 trivial
980.2.c.b 8 4.b odd 2 1 inner
980.2.c.b 8 5.b even 2 1 inner
980.2.c.b 8 7.b odd 2 1 inner
980.2.c.b 8 20.d odd 2 1 CM
980.2.c.b 8 28.d even 2 1 inner
980.2.c.b 8 35.c odd 2 1 inner
980.2.c.b 8 140.c even 2 1 inner
980.2.s.a 8 7.c even 3 1
980.2.s.a 8 7.d odd 6 1
980.2.s.a 8 28.f even 6 1
980.2.s.a 8 28.g odd 6 1
980.2.s.a 8 35.i odd 6 1
980.2.s.a 8 35.j even 6 1
980.2.s.a 8 140.p odd 6 1
980.2.s.a 8 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 8 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{4} \)
$3$ \( ( 1 + 8 T^{2} + T^{4} )^{2} \)
$5$ \( ( 5 + T^{2} )^{4} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 4489 - 136 T^{2} + T^{4} )^{2} \)
$29$ \( ( -51 + 6 T + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 10609 + 226 T^{2} + T^{4} )^{2} \)
$43$ \( ( 1089 - 96 T^{2} + T^{4} )^{2} \)
$47$ \( ( 90 + T^{2} )^{4} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 9 + 186 T^{2} + T^{4} )^{2} \)
$67$ \( ( 33489 - 384 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 25281 + 408 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2809 + 214 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
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