Properties

Label 9248.2.a.f
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.8456517893\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} - 3q^{9} + O(q^{10}) \) \( q + 2q^{5} - 3q^{9} + 6q^{13} - q^{25} + 10q^{29} + 2q^{37} - 10q^{41} - 6q^{45} - 7q^{49} + 14q^{53} + 10q^{61} + 12q^{65} + 6q^{73} + 9q^{81} + 10q^{89} - 18q^{97} + O(q^{100}) \)

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
0
0 0 0 2.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9248.2.a.f 1
4.b odd 2 1 CM 9248.2.a.f 1
17.b even 2 1 32.2.a.a 1
51.c odd 2 1 288.2.a.d 1
68.d odd 2 1 32.2.a.a 1
85.c even 2 1 800.2.a.d 1
85.g odd 4 2 800.2.c.e 2
119.d odd 2 1 1568.2.a.e 1
119.h odd 6 2 1568.2.i.f 2
119.j even 6 2 1568.2.i.g 2
136.e odd 2 1 64.2.a.a 1
136.h even 2 1 64.2.a.a 1
153.h even 6 2 2592.2.i.t 2
153.i odd 6 2 2592.2.i.e 2
187.b odd 2 1 3872.2.a.f 1
204.h even 2 1 288.2.a.d 1
221.b even 2 1 5408.2.a.g 1
255.h odd 2 1 7200.2.a.v 1
255.o even 4 2 7200.2.f.m 2
272.k odd 4 2 256.2.b.b 2
272.r even 4 2 256.2.b.b 2
340.d odd 2 1 800.2.a.d 1
340.r even 4 2 800.2.c.e 2
408.b odd 2 1 576.2.a.c 1
408.h even 2 1 576.2.a.c 1
476.e even 2 1 1568.2.a.e 1
476.o odd 6 2 1568.2.i.g 2
476.q even 6 2 1568.2.i.f 2
544.bc even 8 4 1024.2.e.j 4
544.bj odd 8 4 1024.2.e.j 4
612.n even 6 2 2592.2.i.e 2
612.q odd 6 2 2592.2.i.t 2
680.h even 2 1 1600.2.a.n 1
680.k odd 2 1 1600.2.a.n 1
680.u even 4 2 1600.2.c.l 2
680.bi odd 4 2 1600.2.c.l 2
748.f even 2 1 3872.2.a.f 1
816.w even 4 2 2304.2.d.j 2
816.bg odd 4 2 2304.2.d.j 2
884.h odd 2 1 5408.2.a.g 1
952.e odd 2 1 3136.2.a.m 1
952.k even 2 1 3136.2.a.m 1
1020.b even 2 1 7200.2.a.v 1
1020.x odd 4 2 7200.2.f.m 2
1496.g even 2 1 7744.2.a.v 1
1496.p odd 2 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 17.b even 2 1
32.2.a.a 1 68.d odd 2 1
64.2.a.a 1 136.e odd 2 1
64.2.a.a 1 136.h even 2 1
256.2.b.b 2 272.k odd 4 2
256.2.b.b 2 272.r even 4 2
288.2.a.d 1 51.c odd 2 1
288.2.a.d 1 204.h even 2 1
576.2.a.c 1 408.b odd 2 1
576.2.a.c 1 408.h even 2 1
800.2.a.d 1 85.c even 2 1
800.2.a.d 1 340.d odd 2 1
800.2.c.e 2 85.g odd 4 2
800.2.c.e 2 340.r even 4 2
1024.2.e.j 4 544.bc even 8 4
1024.2.e.j 4 544.bj odd 8 4
1568.2.a.e 1 119.d odd 2 1
1568.2.a.e 1 476.e even 2 1
1568.2.i.f 2 119.h odd 6 2
1568.2.i.f 2 476.q even 6 2
1568.2.i.g 2 119.j even 6 2
1568.2.i.g 2 476.o odd 6 2
1600.2.a.n 1 680.h even 2 1
1600.2.a.n 1 680.k odd 2 1
1600.2.c.l 2 680.u even 4 2
1600.2.c.l 2 680.bi odd 4 2
2304.2.d.j 2 816.w even 4 2
2304.2.d.j 2 816.bg odd 4 2
2592.2.i.e 2 153.i odd 6 2
2592.2.i.e 2 612.n even 6 2
2592.2.i.t 2 153.h even 6 2
2592.2.i.t 2 612.q odd 6 2
3136.2.a.m 1 952.e odd 2 1
3136.2.a.m 1 952.k even 2 1
3872.2.a.f 1 187.b odd 2 1
3872.2.a.f 1 748.f even 2 1
5408.2.a.g 1 221.b even 2 1
5408.2.a.g 1 884.h odd 2 1
7200.2.a.v 1 255.h odd 2 1
7200.2.a.v 1 1020.b even 2 1
7200.2.f.m 2 255.o even 4 2
7200.2.f.m 2 1020.x odd 4 2
7744.2.a.v 1 1496.g even 2 1
7744.2.a.v 1 1496.p odd 2 1
9248.2.a.f 1 1.a even 1 1 trivial
9248.2.a.f 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3} \)
\( T_{5} - 2 \)
\( T_{7} \)
\( T_{19} \)
\( T_{43} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( -2 + T \)
$41$ \( 10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -14 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -10 + T \)
$97$ \( 18 + T \)
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