Properties

Label 9702.2.a.ci
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} -2 q^{5} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} -2 q^{5} - q^{8} + 2 q^{10} + q^{11} + ( 4 + \beta ) q^{13} + q^{16} -2 q^{17} + ( 2 - 3 \beta ) q^{19} -2 q^{20} - q^{22} + ( 2 + 2 \beta ) q^{23} - q^{25} + ( -4 - \beta ) q^{26} + ( -4 - 4 \beta ) q^{29} + ( 2 + 3 \beta ) q^{31} - q^{32} + 2 q^{34} + ( -2 + 2 \beta ) q^{37} + ( -2 + 3 \beta ) q^{38} + 2 q^{40} + ( -6 - 4 \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} + q^{44} + ( -2 - 2 \beta ) q^{46} + ( -2 + 7 \beta ) q^{47} + q^{50} + ( 4 + \beta ) q^{52} + ( 2 + 6 \beta ) q^{53} -2 q^{55} + ( 4 + 4 \beta ) q^{58} + ( -4 - 2 \beta ) q^{59} + ( 4 - \beta ) q^{61} + ( -2 - 3 \beta ) q^{62} + q^{64} + ( -8 - 2 \beta ) q^{65} + ( -4 + 2 \beta ) q^{67} -2 q^{68} -4 \beta q^{71} + ( 6 + 4 \beta ) q^{73} + ( 2 - 2 \beta ) q^{74} + ( 2 - 3 \beta ) q^{76} + ( -8 - 6 \beta ) q^{79} -2 q^{80} + ( 6 + 4 \beta ) q^{82} + ( 2 - 5 \beta ) q^{83} + 4 q^{85} + ( 2 + 2 \beta ) q^{86} - q^{88} + ( -8 + 3 \beta ) q^{89} + ( 2 + 2 \beta ) q^{92} + ( 2 - 7 \beta ) q^{94} + ( -4 + 6 \beta ) q^{95} -7 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 8 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} - 8 q^{26} - 8 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} - 4 q^{47} + 2 q^{50} + 8 q^{52} + 4 q^{53} - 4 q^{55} + 8 q^{58} - 8 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} - 4 q^{68} + 12 q^{73} + 4 q^{74} + 4 q^{76} - 16 q^{79} - 4 q^{80} + 12 q^{82} + 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} - 16 q^{89} + 4 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 0 1.00000 −2.00000 0 0 −1.00000 0 2.00000
1.2 −1.00000 0 1.00000 −2.00000 0 0 −1.00000 0 2.00000
\(n\): 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.ci 2
3.b odd 2 1 3234.2.a.bc 2
7.b odd 2 1 9702.2.a.cy 2
21.c even 2 1 3234.2.a.bd yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 3.b odd 2 1
3234.2.a.bd yes 2 21.c even 2 1
9702.2.a.ci 2 1.a even 1 1 trivial
9702.2.a.cy 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5} + 2 \)
\( T_{13}^{2} - 8 T_{13} + 14 \)
\( T_{17} + 2 \)
\( T_{19}^{2} - 4 T_{19} - 14 \)
\( T_{23}^{2} - 4 T_{23} - 4 \)
\( T_{29}^{2} + 8 T_{29} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 14 - 8 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( -14 - 4 T + T^{2} \)
$23$ \( -4 - 4 T + T^{2} \)
$29$ \( -16 + 8 T + T^{2} \)
$31$ \( -14 - 4 T + T^{2} \)
$37$ \( -4 + 4 T + T^{2} \)
$41$ \( 4 + 12 T + T^{2} \)
$43$ \( -4 + 4 T + T^{2} \)
$47$ \( -94 + 4 T + T^{2} \)
$53$ \( -68 - 4 T + T^{2} \)
$59$ \( 8 + 8 T + T^{2} \)
$61$ \( 14 - 8 T + T^{2} \)
$67$ \( 8 + 8 T + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( 4 - 12 T + T^{2} \)
$79$ \( -8 + 16 T + T^{2} \)
$83$ \( -46 - 4 T + T^{2} \)
$89$ \( 46 + 16 T + T^{2} \)
$97$ \( -98 + T^{2} \)
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