Properties

Label 9702.2.a.cz
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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: \(1\)
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 154)
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.

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

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 −3.64575 0 0 1.00000 0 −3.64575
1.2 1.00000 0 1.00000 1.64575 0 0 1.00000 0 1.64575
\(n\): 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.cz 2
3.b odd 2 1 1078.2.a.s 2
7.b odd 2 1 9702.2.a.dr 2
7.c even 3 2 1386.2.k.s 4
12.b even 2 1 8624.2.a.ca 2
21.c even 2 1 1078.2.a.n 2
21.g even 6 2 1078.2.e.v 4
21.h odd 6 2 154.2.e.f 4
84.h odd 2 1 8624.2.a.bk 2
84.n even 6 2 1232.2.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 21.h odd 6 2
1078.2.a.n 2 21.c even 2 1
1078.2.a.s 2 3.b odd 2 1
1078.2.e.v 4 21.g even 6 2
1232.2.q.g 4 84.n even 6 2
1386.2.k.s 4 7.c even 3 2
8624.2.a.bk 2 84.h odd 2 1
8624.2.a.ca 2 12.b even 2 1
9702.2.a.cz 2 1.a even 1 1 trivial
9702.2.a.dr 2 7.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} - 6 \)
\( T_{13} - 5 \)
\( T_{17} + 6 \)
\( T_{19}^{2} + 6 T_{19} + 2 \)
\( T_{23}^{2} - 2 T_{23} - 6 \)
\( T_{29}^{2} + 2 T_{29} - 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 2 + 6 T + T^{2} \)
$23$ \( -6 - 2 T + T^{2} \)
$29$ \( -27 + 2 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( -6 - 2 T + T^{2} \)
$41$ \( -54 + 6 T + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 36 + 16 T + T^{2} \)
$53$ \( -6 - 2 T + T^{2} \)
$59$ \( -3 + 4 T + T^{2} \)
$61$ \( 53 - 18 T + T^{2} \)
$67$ \( -47 + 8 T + T^{2} \)
$71$ \( 42 + 14 T + T^{2} \)
$73$ \( 2 - 6 T + T^{2} \)
$79$ \( -7 + T^{2} \)
$83$ \( 36 + 16 T + T^{2} \)
$89$ \( -96 - 8 T + T^{2} \)
$97$ \( 93 + 22 T + T^{2} \)
show more
show less