Properties

Label 9702.2.a.ds
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1386)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} + q^{8} + ( 1 + \beta ) q^{10} + q^{11} + ( -4 - 2 \beta ) q^{13} + q^{16} + ( -5 - 2 \beta ) q^{17} + ( -2 - 2 \beta ) q^{19} + ( 1 + \beta ) q^{20} + q^{22} + ( 1 + 3 \beta ) q^{23} + ( -2 + 2 \beta ) q^{25} + ( -4 - 2 \beta ) q^{26} -2 \beta q^{29} + ( 2 + 6 \beta ) q^{31} + q^{32} + ( -5 - 2 \beta ) q^{34} + ( 4 - 4 \beta ) q^{37} + ( -2 - 2 \beta ) q^{38} + ( 1 + \beta ) q^{40} + ( -1 - 4 \beta ) q^{41} + 2 \beta q^{43} + q^{44} + ( 1 + 3 \beta ) q^{46} + ( -5 + 3 \beta ) q^{47} + ( -2 + 2 \beta ) q^{50} + ( -4 - 2 \beta ) q^{52} + ( -8 - 2 \beta ) q^{53} + ( 1 + \beta ) q^{55} -2 \beta q^{58} + ( 2 + 4 \beta ) q^{59} + ( -3 - \beta ) q^{61} + ( 2 + 6 \beta ) q^{62} + q^{64} + ( -8 - 6 \beta ) q^{65} + ( -5 + 6 \beta ) q^{67} + ( -5 - 2 \beta ) q^{68} + ( 2 - 8 \beta ) q^{71} + ( -2 + 2 \beta ) q^{73} + ( 4 - 4 \beta ) q^{74} + ( -2 - 2 \beta ) q^{76} + ( 9 + 3 \beta ) q^{79} + ( 1 + \beta ) q^{80} + ( -1 - 4 \beta ) q^{82} + ( -7 + 2 \beta ) q^{83} + ( -9 - 7 \beta ) q^{85} + 2 \beta q^{86} + q^{88} + ( 4 - 6 \beta ) q^{89} + ( 1 + 3 \beta ) q^{92} + ( -5 + 3 \beta ) q^{94} + ( -6 - 4 \beta ) q^{95} + ( -13 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + 2q^{10} + 2q^{11} - 8q^{13} + 2q^{16} - 10q^{17} - 4q^{19} + 2q^{20} + 2q^{22} + 2q^{23} - 4q^{25} - 8q^{26} + 4q^{31} + 2q^{32} - 10q^{34} + 8q^{37} - 4q^{38} + 2q^{40} - 2q^{41} + 2q^{44} + 2q^{46} - 10q^{47} - 4q^{50} - 8q^{52} - 16q^{53} + 2q^{55} + 4q^{59} - 6q^{61} + 4q^{62} + 2q^{64} - 16q^{65} - 10q^{67} - 10q^{68} + 4q^{71} - 4q^{73} + 8q^{74} - 4q^{76} + 18q^{79} + 2q^{80} - 2q^{82} - 14q^{83} - 18q^{85} + 2q^{88} + 8q^{89} + 2q^{92} - 10q^{94} - 12q^{95} - 26q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 0 1.00000 −0.414214 0 0 1.00000 0 −0.414214
1.2 1.00000 0 1.00000 2.41421 0 0 1.00000 0 2.41421
\(n\): 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.ds 2
3.b odd 2 1 9702.2.a.cj 2
7.b odd 2 1 9702.2.a.da 2
7.c even 3 2 1386.2.k.q 4
21.c even 2 1 9702.2.a.cv 2
21.h odd 6 2 1386.2.k.u yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.k.q 4 7.c even 3 2
1386.2.k.u yes 4 21.h odd 6 2
9702.2.a.cj 2 3.b odd 2 1
9702.2.a.cv 2 21.c even 2 1
9702.2.a.da 2 7.b odd 2 1
9702.2.a.ds 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} - 2 T_{5} - 1 \)
\( T_{13}^{2} + 8 T_{13} + 8 \)
\( T_{17}^{2} + 10 T_{17} + 17 \)
\( T_{19}^{2} + 4 T_{19} - 4 \)
\( T_{23}^{2} - 2 T_{23} - 17 \)
\( T_{29}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -1 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 8 + 8 T + T^{2} \)
$17$ \( 17 + 10 T + T^{2} \)
$19$ \( -4 + 4 T + T^{2} \)
$23$ \( -17 - 2 T + T^{2} \)
$29$ \( -8 + T^{2} \)
$31$ \( -68 - 4 T + T^{2} \)
$37$ \( -16 - 8 T + T^{2} \)
$41$ \( -31 + 2 T + T^{2} \)
$43$ \( -8 + T^{2} \)
$47$ \( 7 + 10 T + T^{2} \)
$53$ \( 56 + 16 T + T^{2} \)
$59$ \( -28 - 4 T + T^{2} \)
$61$ \( 7 + 6 T + T^{2} \)
$67$ \( -47 + 10 T + T^{2} \)
$71$ \( -124 - 4 T + T^{2} \)
$73$ \( -4 + 4 T + T^{2} \)
$79$ \( 63 - 18 T + T^{2} \)
$83$ \( 41 + 14 T + T^{2} \)
$89$ \( -56 - 8 T + T^{2} \)
$97$ \( 161 + 26 T + T^{2} \)
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