Properties

Label 7448.2.a.g
Level $7448$
Weight $2$
Character orbit 7448.a
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{9} + 2q^{11} - q^{13} + 5q^{17} - q^{19} - q^{23} - 5q^{25} + 5q^{27} - 3q^{29} - 4q^{31} - 2q^{33} + 2q^{37} + q^{39} + 8q^{41} - 8q^{43} + 8q^{47} - 5q^{51} + 9q^{53} + q^{57} - q^{59} - 14q^{61} + 13q^{67} + q^{69} + 10q^{71} - 9q^{73} + 5q^{75} - 10q^{79} + q^{81} - 10q^{83} + 3q^{87} + 12q^{89} + 4q^{93} - 14q^{97} - 4q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 7448.2.a.g 1
7.b odd 2 1 152.2.a.b 1
21.c even 2 1 1368.2.a.g 1
28.d even 2 1 304.2.a.b 1
35.c odd 2 1 3800.2.a.d 1
35.f even 4 2 3800.2.d.f 2
56.e even 2 1 1216.2.a.l 1
56.h odd 2 1 1216.2.a.f 1
84.h odd 2 1 2736.2.a.k 1
133.c even 2 1 2888.2.a.b 1
140.c even 2 1 7600.2.a.o 1
532.b odd 2 1 5776.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 7.b odd 2 1
304.2.a.b 1 28.d even 2 1
1216.2.a.f 1 56.h odd 2 1
1216.2.a.l 1 56.e even 2 1
1368.2.a.g 1 21.c even 2 1
2736.2.a.k 1 84.h odd 2 1
2888.2.a.b 1 133.c even 2 1
3800.2.a.d 1 35.c odd 2 1
3800.2.d.f 2 35.f even 4 2
5776.2.a.l 1 532.b odd 2 1
7448.2.a.g 1 1.a even 1 1 trivial
7600.2.a.o 1 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):

\( T_{3} + 1 \)
\( T_{5} \)
\( T_{11} - 2 \)
\( T_{13} + 1 \)
\( T_{17} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( 1 + T \)
$17$ \( -5 + T \)
$19$ \( 1 + T \)
$23$ \( 1 + T \)
$29$ \( 3 + T \)
$31$ \( 4 + T \)
$37$ \( -2 + T \)
$41$ \( -8 + T \)
$43$ \( 8 + T \)
$47$ \( -8 + T \)
$53$ \( -9 + T \)
$59$ \( 1 + T \)
$61$ \( 14 + T \)
$67$ \( -13 + T \)
$71$ \( -10 + T \)
$73$ \( 9 + T \)
$79$ \( 10 + T \)
$83$ \( 10 + T \)
$89$ \( -12 + T \)
$97$ \( 14 + T \)
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