Properties

Label 7600.2.a.bt
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: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{2} q^{7} + ( \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{2} q^{7} + ( \beta_{1} - \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{11} -\beta_{1} q^{13} + ( 2 \beta_{1} - \beta_{2} ) q^{17} + q^{19} + ( -3 - \beta_{1} + \beta_{2} ) q^{21} + ( -1 + \beta_{1} - \beta_{2} ) q^{23} + ( 3 + \beta_{2} ) q^{27} + ( 1 - 4 \beta_{1} - \beta_{2} ) q^{29} + 3 \beta_{1} q^{31} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{2} ) q^{39} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{41} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{47} + ( -4 + \beta_{1} - \beta_{2} ) q^{49} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{51} + ( 3 - 2 \beta_{2} ) q^{53} -\beta_{2} q^{57} + ( 4 - 2 \beta_{1} ) q^{59} + ( -8 + 3 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -3 + 2 \beta_{2} ) q^{63} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{67} + ( 3 - \beta_{2} ) q^{69} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + 5 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -3 - \beta_{1} + 2 \beta_{2} ) q^{77} + ( 8 - 4 \beta_{1} - \beta_{2} ) q^{79} + ( -3 - 4 \beta_{1} + \beta_{2} ) q^{81} + ( -2 + 5 \beta_{1} + \beta_{2} ) q^{83} + ( 3 + 5 \beta_{1} + 2 \beta_{2} ) q^{87} + ( 1 + \beta_{1} - 5 \beta_{2} ) q^{89} + ( -\beta_{1} - \beta_{2} ) q^{91} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 4 + 3 \beta_{1} ) q^{97} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{9} + O(q^{10}) \) \( 3q + q^{9} + 3q^{11} - q^{13} + 2q^{17} + 3q^{19} - 10q^{21} - 2q^{23} + 9q^{27} - q^{29} + 3q^{31} + 10q^{33} + q^{37} + q^{39} - 9q^{41} + q^{43} + 13q^{47} - 11q^{49} + 8q^{51} + 9q^{53} + 10q^{59} - 21q^{61} - 9q^{63} + 13q^{67} + 9q^{69} - q^{71} + 17q^{73} - 10q^{77} + 20q^{79} - 13q^{81} - q^{83} + 14q^{87} + 4q^{89} - q^{91} - 3q^{93} + 15q^{97} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
2.19869
−1.91223
0.713538
0 −1.83424 0 0 0 1.83424 0 0.364448 0
1.2 0 −0.656620 0 0 0 0.656620 0 −2.56885 0
1.3 0 2.49086 0 0 0 −2.49086 0 3.20440 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.bt 3
4.b odd 2 1 3800.2.a.u 3
5.b even 2 1 7600.2.a.bu 3
20.d odd 2 1 3800.2.a.v yes 3
20.e even 4 2 3800.2.d.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.u 3 4.b odd 2 1
3800.2.a.v yes 3 20.d odd 2 1
3800.2.d.m 6 20.e even 4 2
7600.2.a.bt 3 1.a even 1 1 trivial
7600.2.a.bu 3 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{3} - 5 T_{3} - 3 \)
\( T_{7}^{3} - 5 T_{7} + 3 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 2 T_{11} + 1 \)
\( T_{13}^{3} + T_{13}^{2} - 4 T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -3 - 5 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 3 - 5 T + T^{3} \)
$11$ \( 1 - 2 T - 3 T^{2} + T^{3} \)
$13$ \( -3 - 4 T + T^{2} + T^{3} \)
$17$ \( 45 - 19 T - 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -5 - 7 T + 2 T^{2} + T^{3} \)
$29$ \( 49 - 78 T + T^{2} + T^{3} \)
$31$ \( 81 - 36 T - 3 T^{2} + T^{3} \)
$37$ \( 45 - 24 T - T^{2} + T^{3} \)
$41$ \( -175 - 20 T + 9 T^{2} + T^{3} \)
$43$ \( 113 - 41 T - T^{2} + T^{3} \)
$47$ \( 9 + 30 T - 13 T^{2} + T^{3} \)
$53$ \( 9 + 7 T - 9 T^{2} + T^{3} \)
$59$ \( 8 + 16 T - 10 T^{2} + T^{3} \)
$61$ \( -305 + 82 T + 21 T^{2} + T^{3} \)
$67$ \( -25 + 36 T - 13 T^{2} + T^{3} \)
$71$ \( -281 - 81 T + T^{2} + T^{3} \)
$73$ \( 1075 - 42 T - 17 T^{2} + T^{3} \)
$79$ \( 301 + 55 T - 20 T^{2} + T^{3} \)
$83$ \( -109 - 118 T + T^{2} + T^{3} \)
$89$ \( -355 - 119 T - 4 T^{2} + T^{3} \)
$97$ \( 113 + 36 T - 15 T^{2} + T^{3} \)
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