Properties

Label 7600.2.a.cj
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.6.56310016.1
Defining polynomial: \(x^{6} - 9 x^{4} + 14 x^{2} - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{4} q^{7} + ( 2 - \beta_{1} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{4} q^{7} + ( 2 - \beta_{1} + \beta_{5} ) q^{9} + ( -3 + \beta_{1} ) q^{11} -\beta_{2} q^{13} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{17} - q^{19} -2 \beta_{5} q^{21} + ( \beta_{2} - \beta_{3} ) q^{23} + ( -\beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{29} + ( -2 - 2 \beta_{1} - 2 \beta_{5} ) q^{31} + ( \beta_{2} - 5 \beta_{3} ) q^{33} -\beta_{3} q^{37} + ( 1 - \beta_{1} + \beta_{5} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{41} + ( -2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{43} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( 1 - 2 \beta_{1} - 3 \beta_{5} ) q^{49} + ( -6 + 2 \beta_{1} ) q^{51} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} -\beta_{3} q^{57} + ( -4 + 2 \beta_{5} ) q^{59} + ( 5 - \beta_{1} ) q^{61} + ( -4 \beta_{3} - \beta_{4} ) q^{63} + ( \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{67} + ( -6 + 2 \beta_{1} - 2 \beta_{5} ) q^{69} + ( -8 - 2 \beta_{5} ) q^{71} + ( 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{73} + ( \beta_{2} + \beta_{3} + 5 \beta_{4} ) q^{77} -4 q^{79} + ( 10 - \beta_{1} + 5 \beta_{5} ) q^{81} + ( -\beta_{2} + \beta_{3} ) q^{83} + ( 2 \beta_{2} + 4 \beta_{4} ) q^{87} + ( 2 + 2 \beta_{5} ) q^{89} -4 q^{91} + ( -2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{93} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{97} + ( -17 + 3 \beta_{1} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 10q^{9} + O(q^{10}) \) \( 6q + 10q^{9} - 18q^{11} - 6q^{19} + 4q^{21} - 4q^{29} - 8q^{31} + 4q^{39} + 4q^{41} + 12q^{49} - 36q^{51} - 28q^{59} + 30q^{61} - 32q^{69} - 44q^{71} - 24q^{79} + 50q^{81} + 8q^{89} - 24q^{91} - 90q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 9 x^{4} + 14 x^{2} - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 9 \nu^{3} - 10 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 9 \nu^{3} + 14 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{5} - 8 \nu^{3} + 6 \nu \)
\(\beta_{5}\)\(=\)\( \nu^{4} - 8 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 4 \beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 8 \beta_{1} + 18\)
\(\nu^{5}\)\(=\)\(9 \beta_{4} + 13 \beta_{3} + 29 \beta_{2}\)

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
−0.608430
−1.23277
−2.66648
2.66648
1.23277
0.608430
0 −3.28715 0 0 0 1.93210 0 7.80536 0
1.2 0 −1.62236 0 0 0 −4.74397 0 −0.367938 0
1.3 0 −0.750054 0 0 0 −0.872810 0 −2.43742 0
1.4 0 0.750054 0 0 0 0.872810 0 −2.43742 0
1.5 0 1.62236 0 0 0 4.74397 0 −0.367938 0
1.6 0 3.28715 0 0 0 −1.93210 0 7.80536 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7600.2.a.cj 6
4.b odd 2 1 1900.2.a.k 6
5.b even 2 1 inner 7600.2.a.cj 6
5.c odd 4 2 1520.2.d.i 6
20.d odd 2 1 1900.2.a.k 6
20.e even 4 2 380.2.c.b 6
60.l odd 4 2 3420.2.f.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.b 6 20.e even 4 2
1520.2.d.i 6 5.c odd 4 2
1900.2.a.k 6 4.b odd 2 1
1900.2.a.k 6 20.d odd 2 1
3420.2.f.c 6 60.l odd 4 2
7600.2.a.cj 6 1.a even 1 1 trivial
7600.2.a.cj 6 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{6} - 14 T_{3}^{4} + 36 T_{3}^{2} - 16 \)
\( T_{7}^{6} - 27 T_{7}^{4} + 104 T_{7}^{2} - 64 \)
\( T_{11}^{3} + 9 T_{11}^{2} + 14 T_{11} - 28 \)
\( T_{13}^{6} - 26 T_{13}^{4} + 108 T_{13}^{2} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -16 + 36 T^{2} - 14 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( -64 + 104 T^{2} - 27 T^{4} + T^{6} \)
$11$ \( ( -28 + 14 T + 9 T^{2} + T^{3} )^{2} \)
$13$ \( -64 + 108 T^{2} - 26 T^{4} + T^{6} \)
$17$ \( -3136 + 728 T^{2} - 51 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( -256 + 448 T^{2} - 44 T^{4} + T^{6} \)
$29$ \( ( 88 - 84 T + 2 T^{2} + T^{3} )^{2} \)
$31$ \( ( -256 - 80 T + 4 T^{2} + T^{3} )^{2} \)
$37$ \( -16 + 36 T^{2} - 14 T^{4} + T^{6} \)
$41$ \( ( 488 - 116 T - 2 T^{2} + T^{3} )^{2} \)
$43$ \( -3136 + 12520 T^{2} - 227 T^{4} + T^{6} \)
$47$ \( -118336 + 14024 T^{2} - 243 T^{4} + T^{6} \)
$53$ \( -23104 + 3500 T^{2} - 114 T^{4} + T^{6} \)
$59$ \( ( -128 + 16 T + 14 T^{2} + T^{3} )^{2} \)
$61$ \( ( -44 + 62 T - 15 T^{2} + T^{3} )^{2} \)
$67$ \( -222784 + 35660 T^{2} - 378 T^{4} + T^{6} \)
$71$ \( ( 32 + 112 T + 22 T^{2} + T^{3} )^{2} \)
$73$ \( -118336 + 16344 T^{2} - 251 T^{4} + T^{6} \)
$79$ \( ( 4 + T )^{6} \)
$83$ \( -256 + 448 T^{2} - 44 T^{4} + T^{6} \)
$89$ \( ( 64 - 44 T - 4 T^{2} + T^{3} )^{2} \)
$97$ \( -23104 + 3500 T^{2} - 114 T^{4} + T^{6} \)
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