Properties

Label 9702.2.a.dw
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.2700.1
Defining polynomial: \(x^{3} - 15 x - 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} - q^{8} -\beta_{1} q^{10} + q^{11} + q^{16} + ( 1 + 2 \beta_{2} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + \beta_{1} q^{20} - q^{22} + ( -1 - \beta_{1} + \beta_{2} ) q^{23} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{29} + ( 2 + 2 \beta_{2} ) q^{31} - q^{32} + ( -1 - 2 \beta_{2} ) q^{34} + ( 7 + \beta_{2} ) q^{37} + ( -3 - \beta_{2} ) q^{38} -\beta_{1} q^{40} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{41} + ( 1 + 3 \beta_{2} ) q^{43} + q^{44} + ( 1 + \beta_{1} - \beta_{2} ) q^{46} + ( 1 - \beta_{1} - \beta_{2} ) q^{47} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{50} -8 q^{53} + \beta_{1} q^{55} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{59} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 - 2 \beta_{2} ) q^{62} + q^{64} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( 1 + 2 \beta_{2} ) q^{68} + ( -3 + 3 \beta_{2} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -7 - \beta_{2} ) q^{74} + ( 3 + \beta_{2} ) q^{76} + ( 4 - \beta_{1} ) q^{79} + \beta_{1} q^{80} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{82} + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{85} + ( -1 - 3 \beta_{2} ) q^{86} - q^{88} + ( -4 - 2 \beta_{1} ) q^{89} + ( -1 - \beta_{1} + \beta_{2} ) q^{92} + ( -1 + \beta_{1} + \beta_{2} ) q^{94} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{95} -7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{8} + 3q^{11} + 3q^{16} + 3q^{17} + 9q^{19} - 3q^{22} - 3q^{23} + 15q^{25} - 9q^{29} + 6q^{31} - 3q^{32} - 3q^{34} + 21q^{37} - 9q^{38} + 12q^{41} + 3q^{43} + 3q^{44} + 3q^{46} + 3q^{47} - 15q^{50} - 24q^{53} + 9q^{58} - 9q^{59} + 6q^{61} - 6q^{62} + 3q^{64} + 6q^{67} + 3q^{68} - 9q^{71} - 21q^{74} + 9q^{76} + 12q^{79} - 12q^{82} + 18q^{83} - 3q^{86} - 3q^{88} - 12q^{89} - 3q^{92} - 3q^{94} - 21q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 15 x - 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 10\)

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
−2.80560
−1.61323
4.41883
−1.00000 0 1.00000 −2.80560 0 0 −1.00000 0 2.80560
1.2 −1.00000 0 1.00000 −1.61323 0 0 −1.00000 0 1.61323
1.3 −1.00000 0 1.00000 4.41883 0 0 −1.00000 0 −4.41883
\(n\): 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.dw 3
3.b odd 2 1 3234.2.a.bh 3
7.b odd 2 1 9702.2.a.dv 3
7.d odd 6 2 1386.2.k.v 6
21.c even 2 1 3234.2.a.bf 3
21.g even 6 2 462.2.i.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 21.g even 6 2
1386.2.k.v 6 7.d odd 6 2
3234.2.a.bf 3 21.c even 2 1
3234.2.a.bh 3 3.b odd 2 1
9702.2.a.dv 3 7.b odd 2 1
9702.2.a.dw 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{3} - 15 T_{5} - 20 \)
\( T_{13} \)
\( T_{17}^{3} - 3 T_{17}^{2} - 57 T_{17} + 139 \)
\( T_{19}^{3} - 9 T_{19}^{2} + 12 T_{19} + 28 \)
\( T_{23}^{3} + 3 T_{23}^{2} - 27 T_{23} - 89 \)
\( T_{29}^{3} + 9 T_{29}^{2} - 48 T_{29} - 348 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( -20 - 15 T + T^{3} \)
$7$ \( T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( T^{3} \)
$17$ \( 139 - 57 T - 3 T^{2} + T^{3} \)
$19$ \( 28 + 12 T - 9 T^{2} + T^{3} \)
$23$ \( -89 - 27 T + 3 T^{2} + T^{3} \)
$29$ \( -348 - 48 T + 9 T^{2} + T^{3} \)
$31$ \( 192 - 48 T - 6 T^{2} + T^{3} \)
$37$ \( -228 + 132 T - 21 T^{2} + T^{3} \)
$41$ \( 306 - 27 T - 12 T^{2} + T^{3} \)
$43$ \( 404 - 132 T - 3 T^{2} + T^{3} \)
$47$ \( 9 - 27 T - 3 T^{2} + T^{3} \)
$53$ \( ( 8 + T )^{3} \)
$59$ \( -48 - 48 T + 9 T^{2} + T^{3} \)
$61$ \( -98 - 63 T - 6 T^{2} + T^{3} \)
$67$ \( 212 - 63 T - 6 T^{2} + T^{3} \)
$71$ \( -108 - 108 T + 9 T^{2} + T^{3} \)
$73$ \( 480 - 120 T + T^{3} \)
$79$ \( 16 + 33 T - 12 T^{2} + T^{3} \)
$83$ \( 164 + 33 T - 18 T^{2} + T^{3} \)
$89$ \( -16 - 12 T + 12 T^{2} + T^{3} \)
$97$ \( ( 7 + T )^{3} \)
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