Properties

Label 9702.2.a.cy
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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 2 q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + 2 q^{5} - q^{8} - 2 q^{10} + q^{11} + (\beta - 4) q^{13} + q^{16} + 2 q^{17} + ( - 3 \beta - 2) q^{19} + 2 q^{20} - q^{22} + ( - 2 \beta + 2) q^{23} - q^{25} + ( - \beta + 4) q^{26} + (4 \beta - 4) q^{29} + (3 \beta - 2) q^{31} - q^{32} - 2 q^{34} + ( - 2 \beta - 2) q^{37} + (3 \beta + 2) q^{38} - 2 q^{40} + ( - 4 \beta + 6) q^{41} + (2 \beta - 2) q^{43} + q^{44} + (2 \beta - 2) q^{46} + (7 \beta + 2) q^{47} + q^{50} + (\beta - 4) q^{52} + ( - 6 \beta + 2) q^{53} + 2 q^{55} + ( - 4 \beta + 4) q^{58} + ( - 2 \beta + 4) q^{59} + ( - \beta - 4) q^{61} + ( - 3 \beta + 2) q^{62} + q^{64} + (2 \beta - 8) q^{65} + ( - 2 \beta - 4) q^{67} + 2 q^{68} + 4 \beta q^{71} + (4 \beta - 6) q^{73} + (2 \beta + 2) q^{74} + ( - 3 \beta - 2) q^{76} + (6 \beta - 8) q^{79} + 2 q^{80} + (4 \beta - 6) q^{82} + ( - 5 \beta - 2) q^{83} + 4 q^{85} + ( - 2 \beta + 2) q^{86} - q^{88} + (3 \beta + 8) q^{89} + ( - 2 \beta + 2) q^{92} + ( - 7 \beta - 2) q^{94} + ( - 6 \beta - 4) q^{95} - 7 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} - 8 q^{29} - 4 q^{31} - 2 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} + 4 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} + 4 q^{55} + 8 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} + 4 q^{68} - 12 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{79} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} + 16 q^{89} + 4 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) \) 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
−1.41421
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.cy 2
3.b odd 2 1 3234.2.a.bd yes 2
7.b odd 2 1 9702.2.a.ci 2
21.c even 2 1 3234.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 21.c even 2 1
3234.2.a.bd yes 2 3.b odd 2 1
9702.2.a.ci 2 7.b odd 2 1
9702.2.a.cy 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

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

Hecke characteristic polynomials

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