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 $1$

Related objects

Downloads

Learn more

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: \( x^{2} - 7 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} - q^{11} + 4 q^{13} + q^{16} - 3 q^{17} - 2 \beta q^{19} + \beta q^{20} - q^{22} + \beta q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} + (2 \beta - 4) q^{37} - 2 \beta q^{38} + \beta q^{40} + 9 q^{41} + (2 \beta + 4) q^{43} - q^{44} + \beta q^{46} + ( - 3 \beta + 4) q^{47} + 2 q^{50} + 4 q^{52} - 4 q^{53} - \beta q^{55} - 2 q^{58} + (4 \beta - 4) q^{59} + ( - 3 \beta - 4) q^{61} + 4 q^{62} + q^{64} + 4 \beta q^{65} + (4 \beta - 3) q^{67} - 3 q^{68} + ( - 2 \beta + 8) q^{71} + (2 \beta + 10) q^{73} + (2 \beta - 4) q^{74} - 2 \beta q^{76} + (\beta + 8) q^{79} + \beta q^{80} + 9 q^{82} + (4 \beta + 5) q^{83} - 3 \beta q^{85} + (2 \beta + 4) q^{86} - q^{88} + ( - 2 \beta + 8) q^{89} + \beta q^{92} + ( - 3 \beta + 4) q^{94} - 14 q^{95} + (4 \beta + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{22} + 4 q^{25} + 8 q^{26} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 6 q^{34} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 2 q^{44} + 8 q^{47} + 4 q^{50} + 8 q^{52} - 8 q^{53} - 4 q^{58} - 8 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 6 q^{67} - 6 q^{68} + 16 q^{71} + 20 q^{73} - 8 q^{74} + 16 q^{79} + 18 q^{82} + 10 q^{83} + 8 q^{86} - 2 q^{88} + 16 q^{89} + 8 q^{94} - 28 q^{95} + 2 q^{97}+O(q^{100}) \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

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