Properties

Label 9702.2.a.dm
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} - q^{11} + 4 q^{13} + q^{16} -3 q^{17} -2 \beta q^{19} + \beta q^{20} - q^{22} + \beta q^{23} + 2 q^{25} + 4 q^{26} -2 q^{29} + 4 q^{31} + q^{32} -3 q^{34} + ( -4 + 2 \beta ) q^{37} -2 \beta q^{38} + \beta q^{40} + 9 q^{41} + ( 4 + 2 \beta ) q^{43} - q^{44} + \beta q^{46} + ( 4 - 3 \beta ) q^{47} + 2 q^{50} + 4 q^{52} -4 q^{53} -\beta q^{55} -2 q^{58} + ( -4 + 4 \beta ) q^{59} + ( -4 - 3 \beta ) q^{61} + 4 q^{62} + q^{64} + 4 \beta q^{65} + ( -3 + 4 \beta ) q^{67} -3 q^{68} + ( 8 - 2 \beta ) q^{71} + ( 10 + 2 \beta ) q^{73} + ( -4 + 2 \beta ) q^{74} -2 \beta q^{76} + ( 8 + \beta ) q^{79} + \beta q^{80} + 9 q^{82} + ( 5 + 4 \beta ) q^{83} -3 \beta q^{85} + ( 4 + 2 \beta ) q^{86} - q^{88} + ( 8 - 2 \beta ) q^{89} + \beta q^{92} + ( 4 - 3 \beta ) q^{94} -14 q^{95} + ( 1 + 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
1.00000 0 1.00000 −2.64575 0 0 1.00000 0 −2.64575
1.2 1.00000 0 1.00000 2.64575 0 0 1.00000 0 2.64575
\(n\): 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.dm 2
3.b odd 2 1 3234.2.a.ba 2
7.b odd 2 1 9702.2.a.db 2
7.d odd 6 2 1386.2.k.r 4
21.c even 2 1 3234.2.a.w 2
21.g even 6 2 462.2.i.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 21.g even 6 2
1386.2.k.r 4 7.d odd 6 2
3234.2.a.w 2 21.c even 2 1
3234.2.a.ba 2 3.b odd 2 1
9702.2.a.db 2 7.b odd 2 1
9702.2.a.dm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} - 7 \)
\( T_{13} - 4 \)
\( T_{17} + 3 \)
\( T_{19}^{2} - 28 \)
\( T_{23}^{2} - 7 \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -7 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( -28 + T^{2} \)
$23$ \( -7 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( -12 + 8 T + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( -12 - 8 T + T^{2} \)
$47$ \( -47 - 8 T + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( -96 + 8 T + T^{2} \)
$61$ \( -47 + 8 T + T^{2} \)
$67$ \( -103 + 6 T + T^{2} \)
$71$ \( 36 - 16 T + T^{2} \)
$73$ \( 72 - 20 T + T^{2} \)
$79$ \( 57 - 16 T + T^{2} \)
$83$ \( -87 - 10 T + T^{2} \)
$89$ \( 36 - 16 T + T^{2} \)
$97$ \( -111 - 2 T + T^{2} \)
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