Properties

Label 7600.2.a.bk
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.993.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 + \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 + \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} -2 \beta_{1} q^{11} + ( -2 + \beta_{1} - \beta_{2} ) q^{13} + ( 4 - \beta_{1} + \beta_{2} ) q^{17} - q^{19} + ( 2 + \beta_{1} ) q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( -6 + 3 \beta_{1} - 2 \beta_{2} ) q^{27} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( -1 + 2 \beta_{1} ) q^{37} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{39} + ( 4 + 2 \beta_{1} ) q^{43} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{49} + ( -7 + 7 \beta_{1} - \beta_{2} ) q^{51} + ( -5 + 4 \beta_{1} + \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{57} + ( -1 + \beta_{1} - 4 \beta_{2} ) q^{59} -2 \beta_{2} q^{61} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{63} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{67} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{69} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{73} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -8 + 4 \beta_{1} ) q^{79} + ( 10 - 7 \beta_{1} ) q^{81} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} ) q^{87} + ( 6 - 4 \beta_{1} ) q^{89} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{91} + ( -6 - 2 \beta_{2} ) q^{93} + ( -8 - 6 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 14 - 8 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} - 2q^{7} + 5q^{9} + O(q^{10}) \) \( 3q - 2q^{3} - 2q^{7} + 5q^{9} - 2q^{11} - 6q^{13} + 12q^{17} - 3q^{19} + 7q^{21} + 2q^{23} - 17q^{27} + 6q^{29} - 8q^{31} - 24q^{33} - q^{37} + 11q^{39} + 14q^{43} - 3q^{47} + 9q^{49} - 15q^{51} - 10q^{53} + 2q^{57} - 6q^{59} - 2q^{61} + 14q^{63} - 4q^{67} + 6q^{71} - 12q^{73} - 10q^{77} - 20q^{79} + 23q^{81} + 4q^{83} + 3q^{87} + 14q^{89} - 19q^{91} - 20q^{93} - 28q^{97} + 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
−2.25342
0.480031
2.77339
0 −3.25342 0 0 0 0.0778929 0 7.58473 0
1.2 0 −0.519969 0 0 0 −4.76957 0 −2.72963 0
1.3 0 1.77339 0 0 0 2.69168 0 0.144903 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.bk 3
4.b odd 2 1 950.2.a.l yes 3
5.b even 2 1 7600.2.a.bz 3
12.b even 2 1 8550.2.a.ci 3
20.d odd 2 1 950.2.a.j 3
20.e even 4 2 950.2.b.h 6
60.h even 2 1 8550.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 20.d odd 2 1
950.2.a.l yes 3 4.b odd 2 1
950.2.b.h 6 20.e even 4 2
7600.2.a.bk 3 1.a even 1 1 trivial
7600.2.a.bz 3 5.b even 2 1
8550.2.a.ci 3 12.b even 2 1
8550.2.a.cp 3 60.h even 2 1

Hecke kernels

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

\( T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 3 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 13 T_{7} + 1 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 24 T_{11} - 24 \)
\( T_{13}^{3} + 6 T_{13}^{2} - 3 T_{13} - 35 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -3 - 5 T + 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 1 - 13 T + 2 T^{2} + T^{3} \)
$11$ \( -24 - 24 T + 2 T^{2} + T^{3} \)
$13$ \( -35 - 3 T + 6 T^{2} + T^{3} \)
$17$ \( 9 + 33 T - 12 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -111 - 51 T - 2 T^{2} + T^{3} \)
$29$ \( 9 - 3 T - 6 T^{2} + T^{3} \)
$31$ \( -56 - 4 T + 8 T^{2} + T^{3} \)
$37$ \( -1 - 25 T + T^{2} + T^{3} \)
$41$ \( T^{3} \)
$43$ \( 24 + 40 T - 14 T^{2} + T^{3} \)
$47$ \( 45 - 57 T + 3 T^{2} + T^{3} \)
$53$ \( -867 - 105 T + 10 T^{2} + T^{3} \)
$59$ \( -1431 - 201 T + 6 T^{2} + T^{3} \)
$61$ \( -120 - 56 T + 2 T^{2} + T^{3} \)
$67$ \( -75 - 23 T + 4 T^{2} + T^{3} \)
$71$ \( 1512 - 192 T - 6 T^{2} + T^{3} \)
$73$ \( -317 - 27 T + 12 T^{2} + T^{3} \)
$79$ \( -320 + 32 T + 20 T^{2} + T^{3} \)
$83$ \( -168 - 108 T - 4 T^{2} + T^{3} \)
$89$ \( 312 - 36 T - 14 T^{2} + T^{3} \)
$97$ \( -2440 + 44 T + 28 T^{2} + T^{3} \)
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