Properties

Label 7600.2.a.cd
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: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} + ( -3 - \beta_{2} ) q^{13} + ( 1 + \beta_{2} ) q^{17} - q^{19} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{21} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{27} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + ( -4 - 2 \beta_{1} ) q^{37} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 3 \beta_{1} + \beta_{2} ) q^{41} + ( 6 + \beta_{1} + \beta_{2} ) q^{43} + ( -4 + 2 \beta_{2} ) q^{47} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{49} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{51} + ( -5 + \beta_{2} ) q^{53} + ( -1 + \beta_{1} ) q^{57} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + 2 \beta_{1} q^{63} + ( 1 - 3 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -9 - 5 \beta_{2} ) q^{69} + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{71} + ( -5 + 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{77} + ( 2 - 6 \beta_{1} ) q^{79} + ( -5 - 2 \beta_{1} ) q^{81} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -1 + 5 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -4 - \beta_{1} + \beta_{2} ) q^{89} + ( -1 - 3 \beta_{1} ) q^{91} + ( -4 + 3 \beta_{1} - 5 \beta_{2} ) q^{93} + ( -2 + 4 \beta_{2} ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{3} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 3q + 2q^{3} + 4q^{7} + 5q^{9} - 8q^{13} + 2q^{17} - 3q^{19} - 10q^{21} + 2q^{27} - 8q^{29} - 4q^{31} - 8q^{33} - 14q^{37} - 10q^{39} + 2q^{41} + 18q^{43} - 14q^{47} - 3q^{49} + 6q^{51} - 16q^{53} - 2q^{57} - 2q^{59} - 30q^{61} + 2q^{63} + 2q^{67} - 22q^{69} + 8q^{71} - 10q^{73} + 8q^{77} - 17q^{81} - 6q^{83} + 6q^{87} - 14q^{89} - 6q^{91} - 4q^{93} - 10q^{97} + 12q^{99} + O(q^{100}) \)

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

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

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
3.12489
−0.363328
−1.76156
0 −2.12489 0 0 0 4.12489 0 1.51514 0
1.2 0 1.36333 0 0 0 0.636672 0 −1.14134 0
1.3 0 2.76156 0 0 0 −0.761557 0 4.62620 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.cd 3
4.b odd 2 1 950.2.a.i 3
5.b even 2 1 7600.2.a.bi 3
5.c odd 4 2 1520.2.d.j 6
12.b even 2 1 8550.2.a.cl 3
20.d odd 2 1 950.2.a.n 3
20.e even 4 2 190.2.b.b 6
60.h even 2 1 8550.2.a.ck 3
60.l odd 4 2 1710.2.d.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 20.e even 4 2
950.2.a.i 3 4.b odd 2 1
950.2.a.n 3 20.d odd 2 1
1520.2.d.j 6 5.c odd 4 2
1710.2.d.d 6 60.l odd 4 2
7600.2.a.bi 3 5.b even 2 1
7600.2.a.cd 3 1.a even 1 1 trivial
8550.2.a.ck 3 60.h even 2 1
8550.2.a.cl 3 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 8 - 5 T - 2 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 2 - T - 4 T^{2} + T^{3} \)
$11$ \( 8 - 10 T + T^{3} \)
$13$ \( -2 + 13 T + 8 T^{2} + T^{3} \)
$17$ \( 4 - 7 T - 2 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 122 - 49 T + T^{3} \)
$29$ \( -410 - 51 T + 8 T^{2} + T^{3} \)
$31$ \( -232 - 62 T + 4 T^{2} + T^{3} \)
$37$ \( 16 + 40 T + 14 T^{2} + T^{3} \)
$41$ \( -100 - 50 T - 2 T^{2} + T^{3} \)
$43$ \( -148 + 98 T - 18 T^{2} + T^{3} \)
$47$ \( -64 + 32 T + 14 T^{2} + T^{3} \)
$53$ \( 106 + 77 T + 16 T^{2} + T^{3} \)
$59$ \( -80 - 29 T + 2 T^{2} + T^{3} \)
$61$ \( 892 + 290 T + 30 T^{2} + T^{3} \)
$67$ \( 64 - 61 T - 2 T^{2} + T^{3} \)
$71$ \( 1016 - 122 T - 8 T^{2} + T^{3} \)
$73$ \( 164 - 95 T + 10 T^{2} + T^{3} \)
$79$ \( 880 - 228 T + T^{3} \)
$83$ \( -8 - 28 T + 6 T^{2} + T^{3} \)
$89$ \( -20 + 46 T + 14 T^{2} + T^{3} \)
$97$ \( -488 - 100 T + 10 T^{2} + T^{3} \)
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