Properties

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

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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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Defining polynomial: \(x^{4} - 7 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{2} ) q^{9} + ( 2 + \beta_{2} ) q^{11} -\beta_{1} q^{13} -\beta_{3} q^{17} + ( -3 \beta_{1} + \beta_{3} ) q^{19} + ( -2 - 2 \beta_{2} ) q^{23} + \beta_{3} q^{27} + ( -4 - \beta_{2} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 4 \beta_{1} + \beta_{3} ) q^{33} + ( -2 - 2 \beta_{2} ) q^{37} + ( -4 - \beta_{2} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{41} -4 q^{43} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{47} + ( -2 - \beta_{2} ) q^{51} + ( -6 - 2 \beta_{2} ) q^{53} + ( -10 - 2 \beta_{2} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{59} + ( \beta_{1} - 3 \beta_{3} ) q^{61} -8 q^{67} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{2} ) q^{71} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{73} -3 \beta_{2} q^{79} + ( -1 - 2 \beta_{2} ) q^{81} + ( \beta_{1} - 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} - \beta_{3} ) q^{87} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{89} + 4 q^{93} + ( 4 \beta_{1} - \beta_{3} ) q^{97} + ( 12 + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} + 6q^{11} - 4q^{23} - 14q^{29} - 4q^{37} - 14q^{39} - 16q^{43} - 6q^{51} - 20q^{53} - 36q^{57} - 32q^{67} + 4q^{71} + 6q^{79} + 16q^{93} + 44q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.58874
−0.546295
0.546295
2.58874
0 −2.58874 0 0 0 0 0 3.70156 0
1.2 0 −0.546295 0 0 0 0 0 −2.70156 0
1.3 0 0.546295 0 0 0 0 0 −2.70156 0
1.4 0 2.58874 0 0 0 0 0 3.70156 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 9800.2.a.co 4
5.b even 2 1 9800.2.a.cp 4
5.c odd 4 2 1960.2.g.d 8
7.b odd 2 1 inner 9800.2.a.co 4
35.c odd 2 1 9800.2.a.cp 4
35.f even 4 2 1960.2.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.g.d 8 5.c odd 4 2
1960.2.g.d 8 35.f even 4 2
9800.2.a.co 4 1.a even 1 1 trivial
9800.2.a.co 4 7.b odd 2 1 inner
9800.2.a.cp 4 5.b even 2 1
9800.2.a.cp 4 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{4} - 7 T_{3}^{2} + 2 \)
\( T_{11}^{2} - 3 T_{11} - 8 \)
\( T_{13}^{4} - 7 T_{13}^{2} + 2 \)
\( T_{19}^{4} - 58 T_{19}^{2} + 800 \)
\( T_{23}^{2} + 2 T_{23} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 - 7 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -8 - 3 T + T^{2} )^{2} \)
$13$ \( 2 - 7 T^{2} + T^{4} \)
$17$ \( 32 - 13 T^{2} + T^{4} \)
$19$ \( 800 - 58 T^{2} + T^{4} \)
$23$ \( ( -40 + 2 T + T^{2} )^{2} \)
$29$ \( ( 2 + 7 T + T^{2} )^{2} \)
$31$ \( 128 - 56 T^{2} + T^{4} \)
$37$ \( ( -40 + 2 T + T^{2} )^{2} \)
$41$ \( 128 - 56 T^{2} + T^{4} \)
$43$ \( ( 4 + T )^{4} \)
$47$ \( 7688 - 181 T^{2} + T^{4} \)
$53$ \( ( -16 + 10 T + T^{2} )^{2} \)
$59$ \( 10368 - 234 T^{2} + T^{4} \)
$61$ \( 800 - 106 T^{2} + T^{4} \)
$67$ \( ( 8 + T )^{4} \)
$71$ \( ( -40 - 2 T + T^{2} )^{2} \)
$73$ \( 20000 - 284 T^{2} + T^{4} \)
$79$ \( ( -90 - 3 T + T^{2} )^{2} \)
$83$ \( 800 - 106 T^{2} + T^{4} \)
$89$ \( 51200 - 464 T^{2} + T^{4} \)
$97$ \( 2048 - 101 T^{2} + T^{4} \)
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