Properties

Label 7600.2.a.bv
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
Defining polynomial: \(x^{3} - x^{2} - 10 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} + ( 2 + \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} + q^{19} + ( -4 + 3 \beta_{1} ) q^{21} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{23} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + 2 q^{37} + ( 8 - 3 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} ) q^{41} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{49} + ( -4 + \beta_{1} ) q^{51} + ( -2 + \beta_{1} - 4 \beta_{2} ) q^{53} + \beta_{1} q^{57} + ( 8 - \beta_{1} ) q^{59} + ( 2 \beta_{1} - \beta_{2} ) q^{61} + ( 18 - 4 \beta_{1} + 3 \beta_{2} ) q^{63} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{67} + ( -16 + 3 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -3 \beta_{1} + \beta_{2} ) q^{73} + ( 6 + \beta_{2} ) q^{77} + ( -8 - 2 \beta_{1} ) q^{79} + ( 17 + 4 \beta_{2} ) q^{81} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -\beta_{1} + 2 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -8 + 5 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 2 - 2 \beta_{2} ) q^{97} + ( 18 + 2 \beta_{1} + 5 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 3q + q^{3} + 4q^{7} + 12q^{9} + 5q^{11} - 5q^{13} - 2q^{17} + 3q^{19} - 9q^{21} - 5q^{23} + q^{27} - 9q^{29} + 12q^{33} + 6q^{37} + 19q^{39} + 8q^{41} + 17q^{43} - q^{47} + 5q^{49} - 11q^{51} - q^{53} + q^{57} + 23q^{59} + 3q^{61} + 47q^{63} + 15q^{67} - 43q^{69} + 12q^{71} - 4q^{73} + 17q^{77} - 26q^{79} + 47q^{81} - 6q^{83} - 3q^{87} + 18q^{89} - 17q^{91} + 8q^{97} + 51q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 10 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} - \beta_{1} + 8\)

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
−3.08387
0.786802
3.29707
0 −3.08387 0 0 0 4.29707 0 6.51027 0
1.2 0 0.786802 0 0 0 −2.08387 0 −2.38094 0
1.3 0 3.29707 0 0 0 1.78680 0 7.87067 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.bv 3
4.b odd 2 1 3800.2.a.r 3
5.b even 2 1 304.2.a.g 3
15.d odd 2 1 2736.2.a.bd 3
20.d odd 2 1 152.2.a.c 3
20.e even 4 2 3800.2.d.j 6
40.e odd 2 1 1216.2.a.u 3
40.f even 2 1 1216.2.a.v 3
60.h even 2 1 1368.2.a.n 3
95.d odd 2 1 5776.2.a.bp 3
140.c even 2 1 7448.2.a.bf 3
380.d even 2 1 2888.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 20.d odd 2 1
304.2.a.g 3 5.b even 2 1
1216.2.a.u 3 40.e odd 2 1
1216.2.a.v 3 40.f even 2 1
1368.2.a.n 3 60.h even 2 1
2736.2.a.bd 3 15.d odd 2 1
2888.2.a.o 3 380.d even 2 1
3800.2.a.r 3 4.b odd 2 1
3800.2.d.j 6 20.e even 4 2
5776.2.a.bp 3 95.d odd 2 1
7448.2.a.bf 3 140.c even 2 1
7600.2.a.bv 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{3} - T_{3}^{2} - 10 T_{3} + 8 \)
\( T_{7}^{3} - 4 T_{7}^{2} - 5 T_{7} + 16 \)
\( T_{11}^{3} - 5 T_{11}^{2} - 2 T_{11} + 8 \)
\( T_{13}^{3} + 5 T_{13}^{2} - 2 T_{13} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 8 - 10 T - T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 16 - 5 T - 4 T^{2} + T^{3} \)
$11$ \( 8 - 2 T - 5 T^{2} + T^{3} \)
$13$ \( -8 - 2 T + 5 T^{2} + T^{3} \)
$17$ \( -2 - 9 T + 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -256 - 64 T + 5 T^{2} + T^{3} \)
$29$ \( -4 - 4 T + 9 T^{2} + T^{3} \)
$31$ \( T^{3} \)
$37$ \( ( -2 + T )^{3} \)
$41$ \( 128 - 20 T - 8 T^{2} + T^{3} \)
$43$ \( 368 + 24 T - 17 T^{2} + T^{3} \)
$47$ \( -256 - 72 T + T^{2} + T^{3} \)
$53$ \( -256 - 134 T + T^{2} + T^{3} \)
$59$ \( -376 + 166 T - 23 T^{2} + T^{3} \)
$61$ \( 92 - 28 T - 3 T^{2} + T^{3} \)
$67$ \( -32 + 44 T - 15 T^{2} + T^{3} \)
$71$ \( 928 - 76 T - 12 T^{2} + T^{3} \)
$73$ \( -326 - 67 T + 4 T^{2} + T^{3} \)
$79$ \( 256 + 184 T + 26 T^{2} + T^{3} \)
$83$ \( -736 - 112 T + 6 T^{2} + T^{3} \)
$89$ \( 1024 - 16 T - 18 T^{2} + T^{3} \)
$97$ \( 128 - 20 T - 8 T^{2} + T^{3} \)
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