Properties

Label 7600.2.a.cg
Level $7600$
Weight $2$
Character orbit 7600.a
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.253565184.1
Defining polynomial: \(x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 22 x^{2} - 32 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{13} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} - q^{19} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{27} + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -2 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{31} + ( 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{33} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 6 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{39} + ( \beta_{2} + \beta_{4} - \beta_{5} ) q^{41} + ( -1 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( -1 + \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{5} ) q^{49} + ( 4 - \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{51} + ( -2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{53} + \beta_{1} q^{57} + ( -\beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{59} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{61} + ( -5 - 4 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{63} + ( -6 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{69} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{71} + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{73} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( 4 - 2 \beta_{1} ) q^{79} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -4 + 2 \beta_{2} + 2 \beta_{4} ) q^{83} + ( -8 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{87} + ( 8 - 3 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{89} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{91} + ( 2 + 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{93} + ( 8 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{97} + ( 6 - 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{3} - 6q^{7} + 8q^{9} + O(q^{10}) \) \( 6q - 2q^{3} - 6q^{7} + 8q^{9} + 2q^{11} + 14q^{13} + 10q^{17} - 6q^{19} + 18q^{21} - 2q^{23} - 2q^{27} - 2q^{29} - 8q^{31} + 8q^{33} + 4q^{37} + 18q^{39} + 4q^{41} - 4q^{43} - 4q^{47} + 2q^{49} + 34q^{51} - 14q^{53} + 2q^{57} + 2q^{59} + 10q^{61} - 44q^{63} - 42q^{67} + 18q^{69} + 8q^{71} - 2q^{73} + 24q^{77} + 20q^{79} + 6q^{81} - 16q^{83} - 30q^{87} + 32q^{89} - 10q^{91} + 12q^{93} + 40q^{97} + 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 22 x^{2} - 32 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} + 11 \nu^{3} - 16 \nu^{2} - 26 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} + 22 \nu - 28 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 11 \nu^{3} + 9 \nu^{2} + 24 \nu - 10 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 4 \nu^{4} + 37 \nu^{3} - 38 \nu^{2} - 98 \nu + 48 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 9 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(11 \beta_{5} + 15 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 53 \beta_{1}\)

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
2.97875
2.43031
0.664406
0.366738
−1.59277
−2.84742
0 −2.97875 0 0 0 −4.30732 0 5.87292 0
1.2 0 −2.43031 0 0 0 −3.60737 0 2.90640 0
1.3 0 −0.664406 0 0 0 0.345799 0 −2.55856 0
1.4 0 −0.366738 0 0 0 3.08675 0 −2.86550 0
1.5 0 1.59277 0 0 0 −1.66290 0 −0.463073 0
1.6 0 2.84742 0 0 0 0.145034 0 5.10782 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

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.cg 6
4.b odd 2 1 3800.2.a.be 6
5.b even 2 1 7600.2.a.cn 6
5.c odd 4 2 1520.2.d.k 12
20.d odd 2 1 3800.2.a.z 6
20.e even 4 2 760.2.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.e 12 20.e even 4 2
1520.2.d.k 12 5.c odd 4 2
3800.2.a.z 6 20.d odd 2 1
3800.2.a.be 6 4.b odd 2 1
7600.2.a.cg 6 1.a even 1 1 trivial
7600.2.a.cn 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):

\( T_{3}^{6} + 2 T_{3}^{5} - 11 T_{3}^{4} - 20 T_{3}^{3} + 22 T_{3}^{2} + 32 T_{3} + 8 \)
\( T_{7}^{6} + 6 T_{7}^{5} - 4 T_{7}^{4} - 62 T_{7}^{3} - 49 T_{7}^{2} + 36 T_{7} - 4 \)
\( T_{11}^{6} - 2 T_{11}^{5} - 53 T_{11}^{4} + 76 T_{11}^{3} + 762 T_{11}^{2} - 456 T_{11} - 2584 \)
\( T_{13}^{6} - 14 T_{13}^{5} + 27 T_{13}^{4} + 352 T_{13}^{3} - 1270 T_{13}^{2} - 2080 T_{13} + 9376 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 8 + 32 T + 22 T^{2} - 20 T^{3} - 11 T^{4} + 2 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( -4 + 36 T - 49 T^{2} - 62 T^{3} - 4 T^{4} + 6 T^{5} + T^{6} \)
$11$ \( -2584 - 456 T + 762 T^{2} + 76 T^{3} - 53 T^{4} - 2 T^{5} + T^{6} \)
$13$ \( 9376 - 2080 T - 1270 T^{2} + 352 T^{3} + 27 T^{4} - 14 T^{5} + T^{6} \)
$17$ \( 15864 - 6576 T - 715 T^{2} + 574 T^{3} - 34 T^{4} - 10 T^{5} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 272 - 784 T + 608 T^{2} - 44 T^{3} - 61 T^{4} + 2 T^{5} + T^{6} \)
$29$ \( 10064 + 8960 T + 1432 T^{2} - 392 T^{3} - 95 T^{4} + 2 T^{5} + T^{6} \)
$31$ \( -2304 + 3072 T + 464 T^{2} - 320 T^{3} - 38 T^{4} + 8 T^{5} + T^{6} \)
$37$ \( -12352 - 1600 T + 3640 T^{2} + 304 T^{3} - 122 T^{4} - 4 T^{5} + T^{6} \)
$41$ \( -4832 - 960 T + 1072 T^{2} + 144 T^{3} - 58 T^{4} - 4 T^{5} + T^{6} \)
$43$ \( -4632 - 5448 T + 3574 T^{2} - 148 T^{3} - 117 T^{4} + 4 T^{5} + T^{6} \)
$47$ \( -1152 + 960 T + 436 T^{2} - 124 T^{3} - 39 T^{4} + 4 T^{5} + T^{6} \)
$53$ \( 2752 + 25712 T + 666 T^{2} - 1248 T^{3} - 69 T^{4} + 14 T^{5} + T^{6} \)
$59$ \( -374496 - 22752 T + 19924 T^{2} + 476 T^{3} - 265 T^{4} - 2 T^{5} + T^{6} \)
$61$ \( 40256 - 44960 T + 3810 T^{2} + 1340 T^{3} - 133 T^{4} - 10 T^{5} + T^{6} \)
$67$ \( 2752 + 20176 T + 16546 T^{2} + 4900 T^{3} + 665 T^{4} + 42 T^{5} + T^{6} \)
$71$ \( -64 - 128 T + 424 T^{2} + 104 T^{3} - 62 T^{4} - 8 T^{5} + T^{6} \)
$73$ \( 3208 - 5192 T + 1061 T^{2} + 626 T^{3} - 162 T^{4} + 2 T^{5} + T^{6} \)
$79$ \( 1024 + 768 T - 672 T^{2} - 96 T^{3} + 116 T^{4} - 20 T^{5} + T^{6} \)
$83$ \( 6208 + 2560 T - 1488 T^{2} - 576 T^{3} + 12 T^{4} + 16 T^{5} + T^{6} \)
$89$ \( 69504 + 81024 T - 42760 T^{2} + 4424 T^{3} + 122 T^{4} - 32 T^{5} + T^{6} \)
$97$ \( -196992 + 131328 T - 22968 T^{2} - 600 T^{3} + 478 T^{4} - 40 T^{5} + T^{6} \)
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