Properties

Label 7600.2.a.bx
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + ( 2 + \beta_{1} + \beta_{2} ) q^{11} + ( -3 + \beta_{2} ) q^{13} -2 \beta_{2} q^{17} + q^{19} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{21} + ( -2 + \beta_{1} + \beta_{2} ) q^{23} + ( 2 + 2 \beta_{2} ) q^{27} + ( -4 + 2 \beta_{1} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{31} -4 \beta_{1} q^{33} + ( -7 + \beta_{2} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} ) q^{39} -2 \beta_{2} q^{41} + ( -5 \beta_{1} + \beta_{2} ) q^{43} + ( \beta_{1} - \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} ) q^{49} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{51} + ( -5 - \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{57} + ( 6 + 2 \beta_{1} ) q^{59} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{61} + ( -12 + 5 \beta_{1} - \beta_{2} ) q^{63} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} -4 q^{69} + 4 \beta_{1} q^{71} -2 \beta_{1} q^{73} + ( -4 + 4 \beta_{1} ) q^{77} + ( 2 - 6 \beta_{1} ) q^{79} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{81} + ( -10 - \beta_{1} - \beta_{2} ) q^{83} + ( -10 + 4 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -8 + 4 \beta_{2} ) q^{91} + ( 4 + 4 \beta_{1} ) q^{93} + ( -9 + 6 \beta_{1} + \beta_{2} ) q^{97} + ( 6 - 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{3} + 3q^{9} + O(q^{10}) \) \( 3q + 2q^{3} + 3q^{9} + 8q^{11} - 8q^{13} - 2q^{17} + 3q^{19} - 12q^{21} - 4q^{23} + 8q^{27} - 10q^{29} - 4q^{31} - 4q^{33} - 20q^{37} - 4q^{39} - 2q^{41} - 4q^{43} + 11q^{49} - 4q^{51} - 16q^{53} + 2q^{57} + 20q^{59} - 2q^{61} - 32q^{63} + 2q^{67} - 12q^{69} + 4q^{71} - 2q^{73} - 8q^{77} - q^{81} - 32q^{83} - 28q^{87} + 2q^{89} - 20q^{91} + 16q^{93} - 20q^{97} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
2.17009
−1.48119
0.311108
0 −1.70928 0 0 0 1.07838 0 −0.0783777 0
1.2 0 0.806063 0 0 0 3.35026 0 −2.35026 0
1.3 0 2.90321 0 0 0 −4.42864 0 5.42864 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.bx 3
4.b odd 2 1 475.2.a.f 3
5.b even 2 1 1520.2.a.p 3
12.b even 2 1 4275.2.a.bk 3
20.d odd 2 1 95.2.a.a 3
20.e even 4 2 475.2.b.d 6
40.e odd 2 1 6080.2.a.bo 3
40.f even 2 1 6080.2.a.by 3
60.h even 2 1 855.2.a.i 3
76.d even 2 1 9025.2.a.bb 3
140.c even 2 1 4655.2.a.u 3
380.d even 2 1 1805.2.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 20.d odd 2 1
475.2.a.f 3 4.b odd 2 1
475.2.b.d 6 20.e even 4 2
855.2.a.i 3 60.h even 2 1
1520.2.a.p 3 5.b even 2 1
1805.2.a.f 3 380.d even 2 1
4275.2.a.bk 3 12.b even 2 1
4655.2.a.u 3 140.c even 2 1
6080.2.a.bo 3 40.e odd 2 1
6080.2.a.by 3 40.f even 2 1
7600.2.a.bx 3 1.a even 1 1 trivial
9025.2.a.bb 3 76.d even 2 1

Hecke kernels

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

\( T_{3}^{3} - 2 T_{3}^{2} - 4 T_{3} + 4 \)
\( T_{7}^{3} - 16 T_{7} + 16 \)
\( T_{11}^{3} - 8 T_{11}^{2} + 8 T_{11} + 16 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 12 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 4 - 4 T - 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 16 - 16 T + T^{3} \)
$11$ \( 16 + 8 T - 8 T^{2} + T^{3} \)
$13$ \( 4 + 12 T + 8 T^{2} + T^{3} \)
$17$ \( -104 - 36 T + 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -16 - 8 T + 4 T^{2} + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$37$ \( 244 + 124 T + 20 T^{2} + T^{3} \)
$41$ \( -104 - 36 T + 2 T^{2} + T^{3} \)
$43$ \( -592 - 144 T + 4 T^{2} + T^{3} \)
$47$ \( 16 - 16 T + T^{3} \)
$53$ \( 92 + 76 T + 16 T^{2} + T^{3} \)
$59$ \( -160 + 112 T - 20 T^{2} + T^{3} \)
$61$ \( 232 - 84 T + 2 T^{2} + T^{3} \)
$67$ \( -116 - 76 T - 2 T^{2} + T^{3} \)
$71$ \( 64 - 80 T - 4 T^{2} + T^{3} \)
$73$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$79$ \( 160 - 192 T + T^{3} \)
$83$ \( 1072 + 328 T + 32 T^{2} + T^{3} \)
$89$ \( 680 - 132 T - 2 T^{2} + T^{3} \)
$97$ \( -1748 - 60 T + 20 T^{2} + T^{3} \)
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