Properties

Label 7600.2.a.br
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: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{11} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{13} + ( 2 - \beta_{1} ) q^{17} - q^{19} + ( -2 \beta_{1} + \beta_{2} ) q^{21} + ( 2 + 4 \beta_{1} - \beta_{2} ) q^{23} + ( -1 + 3 \beta_{1} ) q^{27} + ( -1 - \beta_{1} + 4 \beta_{2} ) q^{29} + ( 5 + \beta_{1} + \beta_{2} ) q^{31} + ( -3 \beta_{1} + \beta_{2} ) q^{33} + ( 3 - 5 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{39} + ( -3 - 4 \beta_{1} + \beta_{2} ) q^{41} + ( 1 + 2 \beta_{2} ) q^{43} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{47} + ( -1 - 3 \beta_{2} ) q^{49} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{51} + ( -1 + 4 \beta_{1} ) q^{53} + \beta_{1} q^{57} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -3 + \beta_{1} + 5 \beta_{2} ) q^{61} + ( 3 + 2 \beta_{1} - 4 \beta_{2} ) q^{63} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{67} + ( -7 - \beta_{1} - 4 \beta_{2} ) q^{69} + ( 3 + 4 \beta_{2} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{73} + ( 6 - \beta_{1} - \beta_{2} ) q^{77} + ( 4 - 3 \beta_{1} ) q^{79} + ( -3 + \beta_{1} - 6 \beta_{2} ) q^{81} + ( 3 - 6 \beta_{1} + 7 \beta_{2} ) q^{83} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 9 + 2 \beta_{1} - 7 \beta_{2} ) q^{91} + ( -3 - 6 \beta_{1} - \beta_{2} ) q^{93} + ( 7 - 5 \beta_{1} + 3 \beta_{2} ) q^{97} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{9} + O(q^{10}) \) \( 3q - 3q^{9} + 3q^{11} - 3q^{13} + 6q^{17} - 3q^{19} + 6q^{23} - 3q^{27} - 3q^{29} + 15q^{31} + 9q^{37} - 3q^{39} - 9q^{41} + 3q^{43} + 3q^{47} - 3q^{49} + 6q^{51} - 3q^{53} - 6q^{59} - 9q^{61} + 9q^{63} - 3q^{67} - 21q^{69} + 9q^{71} - 9q^{73} + 18q^{77} + 12q^{79} - 9q^{81} + 9q^{83} - 6q^{87} + 6q^{89} + 27q^{91} - 9q^{93} + 21q^{97} + 6q^{99} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
0 −1.87939 0 0 0 1.18479 0 0.532089 0
1.2 0 0.347296 0 0 0 −3.41147 0 −2.87939 0
1.3 0 1.53209 0 0 0 2.22668 0 −0.652704 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.br 3
4.b odd 2 1 3800.2.a.s 3
5.b even 2 1 7600.2.a.bs 3
20.d odd 2 1 3800.2.a.t yes 3
20.e even 4 2 3800.2.d.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.s 3 4.b odd 2 1
3800.2.a.t yes 3 20.d odd 2 1
3800.2.d.o 6 20.e even 4 2
7600.2.a.br 3 1.a even 1 1 trivial
7600.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{3} - 3 T_{3} + 1 \)
\( T_{7}^{3} - 9 T_{7} + 9 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 6 T_{11} + 17 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 18 T_{13} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 1 - 3 T + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 9 - 9 T + T^{3} \)
$11$ \( 17 - 6 T - 3 T^{2} + T^{3} \)
$13$ \( 17 - 18 T + 3 T^{2} + T^{3} \)
$17$ \( -1 + 9 T - 6 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 89 - 27 T - 6 T^{2} + T^{3} \)
$29$ \( 51 - 36 T + 3 T^{2} + T^{3} \)
$31$ \( -89 + 66 T - 15 T^{2} + T^{3} \)
$37$ \( 37 - 30 T - 9 T^{2} + T^{3} \)
$41$ \( -109 - 12 T + 9 T^{2} + T^{3} \)
$43$ \( 19 - 9 T - 3 T^{2} + T^{3} \)
$47$ \( -17 - 18 T - 3 T^{2} + T^{3} \)
$53$ \( -111 - 45 T + 3 T^{2} + T^{3} \)
$59$ \( -296 - 72 T + 6 T^{2} + T^{3} \)
$61$ \( -233 - 66 T + 9 T^{2} + T^{3} \)
$67$ \( -17 - 6 T + 3 T^{2} + T^{3} \)
$71$ \( 181 - 21 T - 9 T^{2} + T^{3} \)
$73$ \( 9 + 18 T + 9 T^{2} + T^{3} \)
$79$ \( 71 + 21 T - 12 T^{2} + T^{3} \)
$83$ \( 289 - 102 T - 9 T^{2} + T^{3} \)
$89$ \( 51 - 9 T - 6 T^{2} + T^{3} \)
$97$ \( -107 + 90 T - 21 T^{2} + T^{3} \)
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