Properties

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

Related objects

Downloads

Learn more

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 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 3.00000 0 −2.00000 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.h 1
4.b odd 2 1 950.2.a.b 1
5.b even 2 1 304.2.a.d 1
12.b even 2 1 8550.2.a.u 1
15.d odd 2 1 2736.2.a.w 1
20.d odd 2 1 38.2.a.b 1
20.e even 4 2 950.2.b.c 2
40.e odd 2 1 1216.2.a.n 1
40.f even 2 1 1216.2.a.g 1
60.h even 2 1 342.2.a.d 1
95.d odd 2 1 5776.2.a.d 1
140.c even 2 1 1862.2.a.f 1
220.g even 2 1 4598.2.a.a 1
260.g odd 2 1 6422.2.a.b 1
380.d even 2 1 722.2.a.b 1
380.p odd 6 2 722.2.c.d 2
380.s even 6 2 722.2.c.f 2
380.ba odd 18 6 722.2.e.c 6
380.bb even 18 6 722.2.e.d 6
1140.p odd 2 1 6498.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 20.d odd 2 1
304.2.a.d 1 5.b even 2 1
342.2.a.d 1 60.h even 2 1
722.2.a.b 1 380.d even 2 1
722.2.c.d 2 380.p odd 6 2
722.2.c.f 2 380.s even 6 2
722.2.e.c 6 380.ba odd 18 6
722.2.e.d 6 380.bb even 18 6
950.2.a.b 1 4.b odd 2 1
950.2.b.c 2 20.e even 4 2
1216.2.a.g 1 40.f even 2 1
1216.2.a.n 1 40.e odd 2 1
1862.2.a.f 1 140.c even 2 1
2736.2.a.w 1 15.d odd 2 1
4598.2.a.a 1 220.g even 2 1
5776.2.a.d 1 95.d odd 2 1
6422.2.a.b 1 260.g odd 2 1
6498.2.a.y 1 1140.p odd 2 1
7600.2.a.h 1 1.a even 1 1 trivial
8550.2.a.u 1 12.b even 2 1

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{7} - 3 \)
\( T_{11} + 2 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( -3 + T \)
$11$ \( 2 + T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( 1 + T \)
$29$ \( 5 + T \)
$31$ \( -8 + T \)
$37$ \( -2 + T \)
$41$ \( 8 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( -1 + T \)
$59$ \( 15 + T \)
$61$ \( -2 + T \)
$67$ \( -3 + T \)
$71$ \( 2 + T \)
$73$ \( 9 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( T \)
$97$ \( -2 + T \)
show more
show less