Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + 3 \beta q^{5} - q^{8} -3 \beta q^{10} - q^{11} -4 \beta q^{13} + q^{16} -5 \beta q^{17} -\beta q^{19} + 3 \beta q^{20} + q^{22} + 8 q^{23} + 13 q^{25} + 4 \beta q^{26} + 8 q^{29} -3 \beta q^{31} - q^{32} + 5 \beta q^{34} + 2 q^{37} + \beta q^{38} -3 \beta q^{40} + \beta q^{41} + 8 q^{43} - q^{44} -8 q^{46} + 7 \beta q^{47} -13 q^{50} -4 \beta q^{52} + 2 q^{53} -3 \beta q^{55} -8 q^{58} -6 \beta q^{59} + 3 \beta q^{62} + q^{64} -24 q^{65} -2 q^{67} -5 \beta q^{68} -12 q^{71} -5 \beta q^{73} -2 q^{74} -\beta q^{76} + 14 q^{79} + 3 \beta q^{80} -\beta q^{82} + 9 \beta q^{83} -30 q^{85} -8 q^{86} + q^{88} + 2 \beta q^{89} + 8 q^{92} -7 \beta q^{94} -6 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{11} + 2 q^{16} + 2 q^{22} + 16 q^{23} + 26 q^{25} + 16 q^{29} - 2 q^{32} + 4 q^{37} + 16 q^{43} - 2 q^{44} - 16 q^{46} - 26 q^{50} + 4 q^{53} - 16 q^{58} + 2 q^{64} - 48 q^{65} - 4 q^{67} - 24 q^{71} - 4 q^{74} + 28 q^{79} - 60 q^{85} - 16 q^{86} + 2 q^{88} + 16 q^{92} - 12 q^{95} + O(q^{100}) \)

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 −4.24264 0 0 −1.00000 0 4.24264
1.2 −1.00000 0 1.00000 4.24264 0 0 −1.00000 0 −4.24264
\(n\): 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

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 9702.2.a.cm 2
3.b odd 2 1 9702.2.a.do yes 2
7.b odd 2 1 inner 9702.2.a.cm 2
21.c even 2 1 9702.2.a.do yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9702.2.a.cm 2 1.a even 1 1 trivial
9702.2.a.cm 2 7.b odd 2 1 inner
9702.2.a.do yes 2 3.b odd 2 1
9702.2.a.do yes 2 21.c even 2 1

Hecke kernels

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

\( T_{5}^{2} - 18 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 50 \)
\( T_{19}^{2} - 2 \)
\( T_{23} - 8 \)
\( T_{29} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -18 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -50 + T^{2} \)
$19$ \( -2 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( -18 + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -2 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( -98 + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -72 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( -50 + T^{2} \)
$79$ \( ( -14 + T )^{2} \)
$83$ \( -162 + T^{2} \)
$89$ \( -8 + T^{2} \)
$97$ \( T^{2} \)
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