Properties

Label 392.2.a
Level 392
Weight 2
Character orbit a
Rep. character \(\chi_{392}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 8
Sturm bound 112
Trace bound 9

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(112\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(392))\).

Total New Old
Modular forms 72 10 62
Cusp forms 41 10 31
Eisenstein series 31 0 31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(7\)

Trace form

\( 10q - 2q^{3} + 2q^{5} + 14q^{9} + O(q^{10}) \) \( 10q - 2q^{3} + 2q^{5} + 14q^{9} + 4q^{11} - 2q^{13} + 12q^{15} + 8q^{17} - 6q^{19} - 4q^{23} + 6q^{25} + 4q^{27} - 8q^{29} - 12q^{31} + 12q^{37} - 8q^{39} + 10q^{45} + 12q^{47} - 4q^{51} - 16q^{53} + 8q^{55} - 4q^{57} - 6q^{59} + 2q^{61} - 36q^{65} + 20q^{67} - 16q^{69} + 4q^{73} - 22q^{75} - 4q^{79} + 10q^{81} - 14q^{83} - 24q^{85} - 4q^{87} - 4q^{89} - 12q^{93} - 28q^{95} + 8q^{97} - 64q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
392.2.a.a \(1\) \(3.130\) \(\Q\) None \(0\) \(-3\) \(1\) \(0\) \(-\) \(-\) \(q-3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots\)
392.2.a.b \(1\) \(3.130\) \(\Q\) None \(0\) \(-2\) \(4\) \(0\) \(+\) \(-\) \(q-2q^{3}+4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots\)
392.2.a.c \(1\) \(3.130\) \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(q-q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots\)
392.2.a.d \(1\) \(3.130\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(q-2q^{5}-3q^{9}-4q^{11}-2q^{13}+6q^{17}+\cdots\)
392.2.a.e \(1\) \(3.130\) \(\Q\) None \(0\) \(1\) \(1\) \(0\) \(+\) \(-\) \(q+q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots\)
392.2.a.f \(1\) \(3.130\) \(\Q\) None \(0\) \(3\) \(-1\) \(0\) \(-\) \(+\) \(q+3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots\)
392.2.a.g \(2\) \(3.130\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta q^{3}+2\beta q^{5}-q^{9}+6q^{11}-4\beta q^{13}+\cdots\)
392.2.a.h \(2\) \(3.130\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(392)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 9 T^{4} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 + T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T^{2} + 169 T^{4} \))(\( 1 + 18 T^{2} + 169 T^{4} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 + 20 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 3 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 54 T^{2} + 961 T^{4} \))(\( 1 + 30 T^{2} + 961 T^{4} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 5 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 + 50 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 - 5 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 5 T + 47 T^{2} \))(\( 1 - T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 9 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 15 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))(\( 1 + 110 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 3 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 5 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 5 T + 61 T^{2} \))(\( 1 - 3 T + 61 T^{2} \))(\( 1 + 50 T^{2} + 3721 T^{4} \))(\( 1 - 78 T^{2} + 3721 T^{4} \))
$67$ (\( 1 - 11 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 9 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 9 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 16 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 16 T + 71 T^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 + 48 T^{2} + 5329 T^{4} \))(\( ( 1 + 73 T^{2} )^{2} \))
$79$ (\( 1 + 11 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 - T + 79 T^{2} \))(\( 1 + 11 T + 79 T^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 164 T^{2} + 6889 T^{4} \))(\( 1 - 34 T^{2} + 6889 T^{4} \))
$89$ (\( 1 - 9 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 - 7 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 7 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))
$97$ (\( 1 + 6 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 6 T + 97 T^{2} \))(\( 1 + 32 T^{2} + 9409 T^{4} \))(\( 1 + 162 T^{2} + 9409 T^{4} \))
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