Properties

Label 980.2.e.e
Level $980$
Weight $2$
Character orbit 980.e
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} - q^{11} -\beta_{2} q^{13} + ( 3 + \beta_{3} ) q^{15} -3 \beta_{2} q^{17} + 2 \beta_{1} q^{19} -\beta_{3} q^{23} + ( -1 - 2 \beta_{3} ) q^{25} -3 \beta_{2} q^{27} + 7 q^{29} + 5 \beta_{1} q^{31} + \beta_{2} q^{33} -3 \beta_{3} q^{37} -3 q^{39} + 5 \beta_{1} q^{41} + 4 \beta_{3} q^{43} -7 \beta_{2} q^{47} -9 q^{51} -5 \beta_{3} q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + 2 \beta_{3} q^{57} + 5 \beta_{1} q^{59} -10 \beta_{1} q^{61} + ( 3 + \beta_{3} ) q^{65} + 5 \beta_{3} q^{67} + 3 \beta_{1} q^{69} -10 q^{71} + ( 6 \beta_{1} + \beta_{2} ) q^{75} -3 q^{79} -9 q^{81} + ( 9 + 3 \beta_{3} ) q^{85} -7 \beta_{2} q^{87} + 5 \beta_{3} q^{93} + ( 4 - 2 \beta_{3} ) q^{95} + 3 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{11} + 12q^{15} - 4q^{25} + 28q^{29} - 12q^{39} - 36q^{51} + 12q^{65} - 40q^{71} - 12q^{79} - 36q^{81} + 36q^{85} + 16q^{95} + O(q^{100}) \)

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

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

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
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 1.73205i 0 −1.41421 + 1.73205i 0 0 0 0 0
589.2 0 1.73205i 0 1.41421 + 1.73205i 0 0 0 0 0
589.3 0 1.73205i 0 −1.41421 1.73205i 0 0 0 0 0
589.4 0 1.73205i 0 1.41421 1.73205i 0 0 0 0 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
5.b even 2 1 inner
7.b odd 2 1 inner
35.c 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.e 4
5.b even 2 1 inner 980.2.e.e 4
5.c odd 4 2 4900.2.a.bh 4
7.b odd 2 1 inner 980.2.e.e 4
7.c even 3 1 980.2.q.a 4
7.c even 3 1 980.2.q.h 4
7.d odd 6 1 980.2.q.a 4
7.d odd 6 1 980.2.q.h 4
35.c odd 2 1 inner 980.2.e.e 4
35.f even 4 2 4900.2.a.bh 4
35.i odd 6 1 980.2.q.a 4
35.i odd 6 1 980.2.q.h 4
35.j even 6 1 980.2.q.a 4
35.j even 6 1 980.2.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.e.e 4 1.a even 1 1 trivial
980.2.e.e 4 5.b even 2 1 inner
980.2.e.e 4 7.b odd 2 1 inner
980.2.e.e 4 35.c odd 2 1 inner
980.2.q.a 4 7.c even 3 1
980.2.q.a 4 7.d odd 6 1
980.2.q.a 4 35.i odd 6 1
980.2.q.a 4 35.j even 6 1
980.2.q.h 4 7.c even 3 1
980.2.q.h 4 7.d odd 6 1
980.2.q.h 4 35.i odd 6 1
980.2.q.h 4 35.j even 6 1
4900.2.a.bh 4 5.c odd 4 2
4900.2.a.bh 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 \(S_{2}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} + 3 \)
\( T_{11} + 1 \)
\( T_{19}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 25 + 2 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( ( 27 + T^{2} )^{2} \)
$19$ \( ( -8 + T^{2} )^{2} \)
$23$ \( ( 6 + T^{2} )^{2} \)
$29$ \( ( -7 + T )^{4} \)
$31$ \( ( -50 + T^{2} )^{2} \)
$37$ \( ( 54 + T^{2} )^{2} \)
$41$ \( ( -50 + T^{2} )^{2} \)
$43$ \( ( 96 + T^{2} )^{2} \)
$47$ \( ( 147 + T^{2} )^{2} \)
$53$ \( ( 150 + T^{2} )^{2} \)
$59$ \( ( -50 + T^{2} )^{2} \)
$61$ \( ( -200 + T^{2} )^{2} \)
$67$ \( ( 150 + T^{2} )^{2} \)
$71$ \( ( 10 + T )^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 3 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 27 + T^{2} )^{2} \)
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