Properties

Label 980.2.e.d
Level $980$
Weight $2$
Character orbit 980.e
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{-21})\)
Defining polynomial: \(x^{4} + 13 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( -4 + \beta_{3} ) q^{9} + ( 1 + \beta_{3} ) q^{11} + ( \beta_{1} + 2 \beta_{2} ) q^{13} + ( 3 - \beta_{3} ) q^{15} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{17} -5 q^{25} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{27} + ( -5 + \beta_{3} ) q^{29} + ( -\beta_{1} + 4 \beta_{2} ) q^{33} + ( -1 - \beta_{3} ) q^{39} + ( 5 \beta_{1} - \beta_{2} ) q^{45} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 15 - \beta_{3} ) q^{51} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{55} + ( -7 - \beta_{3} ) q^{65} -12 q^{71} -6 \beta_{2} q^{73} -5 \beta_{1} q^{75} + ( 1 - 3 \beta_{3} ) q^{79} + ( 21 - 4 \beta_{3} ) q^{81} + 4 \beta_{2} q^{83} + ( 1 + 3 \beta_{3} ) q^{85} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -7 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 22 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{9} + O(q^{10}) \) \( 4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 13 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 7\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 9 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
3.40932i
1.17325i
1.17325i
3.40932i
0 3.40932i 0 2.23607i 0 0 0 −8.62348 0
589.2 0 1.17325i 0 2.23607i 0 0 0 1.62348 0
589.3 0 1.17325i 0 2.23607i 0 0 0 1.62348 0
589.4 0 3.40932i 0 2.23607i 0 0 0 −8.62348 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.e.d 4
5.b even 2 1 inner 980.2.e.d 4
5.c odd 4 2 4900.2.a.bj 4
7.b odd 2 1 inner 980.2.e.d 4
7.c even 3 2 980.2.q.i 8
7.d odd 6 2 980.2.q.i 8
35.c odd 2 1 CM 980.2.e.d 4
35.f even 4 2 4900.2.a.bj 4
35.i odd 6 2 980.2.q.i 8
35.j even 6 2 980.2.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.d 4 1.a even 1 1 trivial
980.2.e.d 4 5.b even 2 1 inner
980.2.e.d 4 7.b odd 2 1 inner
980.2.e.d 4 35.c odd 2 1 CM
980.2.q.i 8 7.c even 3 2
980.2.q.i 8 7.d odd 6 2
980.2.q.i 8 35.i odd 6 2
980.2.q.i 8 35.j even 6 2
4900.2.a.bj 4 5.c odd 4 2
4900.2.a.bj 4 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{4} + 13 T_{3}^{2} + 16 \)
\( T_{11}^{2} - 3 T_{11} - 24 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 13 T^{2} + T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -24 - 3 T + T^{2} )^{2} \)
$13$ \( 36 + 33 T^{2} + T^{4} \)
$17$ \( 2116 + 97 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -6 + 9 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( 256 + 157 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( ( 180 + T^{2} )^{2} \)
$79$ \( ( -236 + T + T^{2} )^{2} \)
$83$ \( ( 80 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 60516 + 537 T^{2} + T^{4} \)
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