Properties

Label 9702.2.a.ea
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{3} + 1) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{3} + 1) q^{5} - q^{8} + ( - \beta_{3} - 1) q^{10} - q^{11} + (\beta_{3} - \beta_{2} - 2) q^{13} + q^{16} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{17} + ( - \beta_{2} - 3) q^{19} + (\beta_{3} + 1) q^{20} + q^{22} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{3} + \beta_{2} + 2) q^{26} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + (3 \beta_{3} + \beta_1 - 1) q^{31} - q^{32} + (\beta_{3} + 2 \beta_{2} - 1) q^{34} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{37} + (\beta_{2} + 3) q^{38} + ( - \beta_{3} - 1) q^{40} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 + 3) q^{41} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{43} - q^{44} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{46} + ( - \beta_{2} - 2 \beta_1 + 1) q^{47} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{50} + (\beta_{3} - \beta_{2} - 2) q^{52} + (3 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{3} - 1) q^{55} + (2 \beta_{3} - 2 \beta_1 - 2) q^{58} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{59} + ( - \beta_{3} - 3 \beta_{2} - 6) q^{61} + ( - 3 \beta_{3} - \beta_1 + 1) q^{62} + q^{64} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{65} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{67} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{68} + (4 \beta_{3} - 2 \beta_{2} - 2) q^{71} + ( - \beta_{3} - 4 \beta_1 - 1) q^{73} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{74} + ( - \beta_{2} - 3) q^{76} + (\beta_{3} + 3 \beta_{2} - 3 \beta_1 - 2) q^{79} + (\beta_{3} + 1) q^{80} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 3) q^{82} + (\beta_{3} + 5 \beta_1 + 1) q^{83} + ( - 2 \beta_{2} - 8 \beta_1 - 2) q^{85} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{86} + q^{88} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 6) q^{89} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{92} + (\beta_{2} + 2 \beta_1 - 1) q^{94} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{95} + (4 \beta_{3} + 5 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} + 4 q^{16} + 4 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} + 8 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 4 q^{50} - 8 q^{52} + 8 q^{53} - 4 q^{55} - 8 q^{58} - 24 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} + 4 q^{68} - 8 q^{71} - 4 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \) Copy content Toggle raw display

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
−1.27133
0.334904
−1.74912
2.68554
−1.00000 0 1.00000 −1.79793 0 0 −1.00000 0 1.79793
1.2 −1.00000 0 1.00000 −0.473626 0 0 −1.00000 0 0.473626
1.3 −1.00000 0 1.00000 2.47363 0 0 −1.00000 0 −2.47363
1.4 −1.00000 0 1.00000 3.79793 0 0 −1.00000 0 −3.79793
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 9702.2.a.ea 4
3.b odd 2 1 3234.2.a.bl 4
7.b odd 2 1 9702.2.a.dz 4
21.c even 2 1 3234.2.a.bm yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bl 4 3.b odd 2 1
3234.2.a.bm yes 4 21.c even 2 1
9702.2.a.dz 4 7.b odd 2 1
9702.2.a.ea 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{4} - 4T_{5}^{3} - 4T_{5}^{2} + 16T_{5} + 8 \) Copy content Toggle raw display
\( T_{13}^{4} + 8T_{13}^{3} - 4T_{13}^{2} - 80T_{13} - 28 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 52T_{17}^{2} + 112T_{17} + 776 \) Copy content Toggle raw display
\( T_{19}^{4} + 12T_{19}^{3} + 40T_{19}^{2} + 24T_{19} - 28 \) Copy content Toggle raw display
\( T_{23}^{4} - 8T_{23}^{3} + 32T_{23} + 16 \) Copy content Toggle raw display
\( T_{29}^{4} - 8T_{29}^{3} - 32T_{29}^{2} + 64T_{29} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} - 4 T^{2} + 16 T + 8 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} - 4 T^{2} - 80 T - 28 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 52 T^{2} + 112 T + 776 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 32 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} - 32 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 88 T^{2} - 328 T + 964 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} - 32 T^{2} + 96 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} - 84 T^{2} + \cdots - 4984 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} - 32 T^{2} - 96 T + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} - 24 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$53$ \( T^{4} - 8 T^{3} - 160 T^{2} + \cdots + 3856 \) Copy content Toggle raw display
$59$ \( T^{4} - 144 T^{2} + 512 T - 448 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + 92 T^{2} + \cdots + 164 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} - 40 T^{2} - 32 T + 32 \) Copy content Toggle raw display
$71$ \( T^{4} + 8 T^{3} - 224 T^{2} + \cdots + 12352 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} - 68 T^{2} - 80 T + 712 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 136 T^{2} - 992 T + 32 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} - 104 T^{2} + \cdots + 1988 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + 124 T^{2} + \cdots - 284 \) Copy content Toggle raw display
$97$ \( T^{4} - 260 T^{2} - 1280 T - 1148 \) Copy content Toggle raw display
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