Properties

Label 7600.2.a.p
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 3q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - 3q^{7} + q^{9} - 5q^{11} + 4q^{13} + 3q^{17} + q^{19} - 6q^{21} + 8q^{23} - 4q^{27} - 2q^{29} - 4q^{31} - 10q^{33} - 10q^{37} + 8q^{39} + 10q^{41} + q^{43} - q^{47} + 2q^{49} + 6q^{51} + 4q^{53} + 2q^{57} - 6q^{59} - 13q^{61} - 3q^{63} - 12q^{67} + 16q^{69} - 2q^{71} - 9q^{73} + 15q^{77} - 8q^{79} - 11q^{81} - 12q^{83} - 4q^{87} + 12q^{89} - 12q^{91} - 8q^{93} + 8q^{97} - 5q^{99} + O(q^{100}) \)

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 2.00000 0 0 0 −3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 7600.2.a.p 1
4.b odd 2 1 1900.2.a.b 1
5.b even 2 1 304.2.a.a 1
15.d odd 2 1 2736.2.a.q 1
20.d odd 2 1 76.2.a.a 1
20.e even 4 2 1900.2.c.b 2
40.e odd 2 1 1216.2.a.c 1
40.f even 2 1 1216.2.a.q 1
60.h even 2 1 684.2.a.b 1
95.d odd 2 1 5776.2.a.p 1
140.c even 2 1 3724.2.a.a 1
220.g even 2 1 9196.2.a.f 1
380.d even 2 1 1444.2.a.a 1
380.p odd 6 2 1444.2.e.a 2
380.s even 6 2 1444.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 20.d odd 2 1
304.2.a.a 1 5.b even 2 1
684.2.a.b 1 60.h even 2 1
1216.2.a.c 1 40.e odd 2 1
1216.2.a.q 1 40.f even 2 1
1444.2.a.a 1 380.d even 2 1
1444.2.e.a 2 380.p odd 6 2
1444.2.e.c 2 380.s even 6 2
1900.2.a.b 1 4.b odd 2 1
1900.2.c.b 2 20.e even 4 2
2736.2.a.q 1 15.d odd 2 1
3724.2.a.a 1 140.c even 2 1
5776.2.a.p 1 95.d odd 2 1
7600.2.a.p 1 1.a even 1 1 trivial
9196.2.a.f 1 220.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3} - 2 \)
\( T_{7} + 3 \)
\( T_{11} + 5 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( 3 + T \)
$11$ \( 5 + T \)
$13$ \( -4 + T \)
$17$ \( -3 + T \)
$19$ \( -1 + T \)
$23$ \( -8 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( -10 + T \)
$43$ \( -1 + T \)
$47$ \( 1 + T \)
$53$ \( -4 + T \)
$59$ \( 6 + T \)
$61$ \( 13 + T \)
$67$ \( 12 + T \)
$71$ \( 2 + T \)
$73$ \( 9 + T \)
$79$ \( 8 + T \)
$83$ \( 12 + T \)
$89$ \( -12 + T \)
$97$ \( -8 + T \)
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