Properties

Label 9800.2.a.cl
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( -2 + \beta_{1} - 2 \beta_{3} ) q^{17} + ( -\beta_{2} - \beta_{3} ) q^{19} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{27} + ( -\beta_{1} - 3 \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{31} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -2 \beta_{2} - \beta_{3} ) q^{37} + ( 2 + 3 \beta_{1} - 3 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + ( \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{53} + ( 2 + 4 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 2 + 5 \beta_{2} ) q^{59} + ( 6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 2 + 6 \beta_{2} - \beta_{3} ) q^{67} + ( 6 - 2 \beta_{2} + \beta_{3} ) q^{69} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{71} + ( -2 - 4 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{73} + ( 6 - \beta_{1} + 5 \beta_{2} ) q^{79} + ( -5 + 2 \beta_{3} ) q^{81} + ( -8 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{87} + ( 10 - 3 \beta_{2} + \beta_{3} ) q^{89} + ( 6 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{93} + ( -6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 2 + 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 6q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 6q^{9} + 2q^{11} - 10q^{13} - 6q^{17} + 4q^{23} - 14q^{27} - 2q^{29} - 12q^{31} - 18q^{33} + 14q^{39} + 12q^{41} + 8q^{43} + 2q^{47} + 2q^{51} + 4q^{53} + 8q^{57} + 8q^{59} + 20q^{61} + 8q^{67} + 24q^{69} + 4q^{71} - 16q^{73} + 22q^{79} - 20q^{81} - 36q^{83} + 18q^{87} + 40q^{89} + 32q^{93} - 26q^{97} + 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.87996
2.18398
−1.18398
−1.87996
0 −2.87996 0 0 0 0 0 5.29417 0
1.2 0 −2.18398 0 0 0 0 0 1.76977 0
1.3 0 1.18398 0 0 0 0 0 −1.59819 0
1.4 0 1.87996 0 0 0 0 0 0.534253 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9800.2.a.cl 4
5.b even 2 1 1960.2.a.y yes 4
7.b odd 2 1 9800.2.a.cs 4
20.d odd 2 1 3920.2.a.cd 4
35.c odd 2 1 1960.2.a.x 4
35.i odd 6 2 1960.2.q.y 8
35.j even 6 2 1960.2.q.x 8
140.c even 2 1 3920.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x 4 35.c odd 2 1
1960.2.a.y yes 4 5.b even 2 1
1960.2.q.x 8 35.j even 6 2
1960.2.q.y 8 35.i odd 6 2
3920.2.a.cd 4 20.d odd 2 1
3920.2.a.ce 4 140.c even 2 1
9800.2.a.cl 4 1.a even 1 1 trivial
9800.2.a.cs 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{4} + 2 T_{3}^{3} - 7 T_{3}^{2} - 8 T_{3} + 14 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 11 T_{11}^{2} + 20 T_{11} - 4 \)
\( T_{13}^{4} + 10 T_{13}^{3} + 19 T_{13}^{2} - 52 T_{13} - 124 \)
\( T_{19}^{4} - 26 T_{19}^{2} + 24 T_{19} + 8 \)
\( T_{23}^{4} - 4 T_{23}^{3} - 38 T_{23}^{2} - 48 T_{23} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 14 - 8 T - 7 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( -4 + 20 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( -124 - 52 T + 19 T^{2} + 10 T^{3} + T^{4} \)
$17$ \( 434 - 244 T - 51 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 8 + 24 T - 26 T^{2} + T^{4} \)
$23$ \( 16 - 48 T - 38 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 188 - 20 T - 43 T^{2} + 2 T^{3} + T^{4} \)
$31$ \( -784 - 288 T + 10 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 8 + 40 T - 42 T^{2} + T^{4} \)
$41$ \( -4228 + 1528 T - 96 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 752 + 192 T - 48 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( -56 - 96 T - 41 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( -568 + 424 T - 70 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( ( -46 - 4 T + T^{2} )^{2} \)
$61$ \( -5344 + 1088 T + 44 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( 2336 + 432 T - 114 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( 10976 - 252 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( -14192 - 3744 T - 170 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( -56 + 488 T + 73 T^{2} - 22 T^{3} + T^{4} \)
$83$ \( 256 + 1600 T + 418 T^{2} + 36 T^{3} + T^{4} \)
$89$ \( 5408 - 3120 T + 558 T^{2} - 40 T^{3} + T^{4} \)
$97$ \( -1022 + 36 T + 157 T^{2} + 26 T^{3} + T^{4} \)
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