Properties

Label 378.2.a
Level 378
Weight 2
Character orbit a
Rep. character \(\chi_{378}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 8
Sturm bound 144
Trace bound 5

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(378))\).

Total New Old
Modular forms 84 8 76
Cusp forms 61 8 53
Eisenstein series 23 0 23

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(6\)

Trace form

\( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} + 16q^{10} + 8q^{13} + 8q^{16} - 8q^{19} - 4q^{22} + 12q^{25} - 16q^{31} + 8q^{34} + 4q^{37} + 16q^{40} - 20q^{43} - 16q^{46} + 8q^{49} + 8q^{52} - 40q^{55} + 4q^{58} - 16q^{61} + 8q^{64} - 36q^{67} - 4q^{70} - 24q^{73} - 8q^{76} - 40q^{79} - 16q^{85} - 4q^{88} - 4q^{91} - 24q^{94} + 48q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(378))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
378.2.a.a \(1\) \(3.018\) \(\Q\) None \(-1\) \(0\) \(-4\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}-4q^{5}-q^{7}-q^{8}+4q^{10}+\cdots\)
378.2.a.b \(1\) \(3.018\) \(\Q\) None \(-1\) \(0\) \(-3\) \(1\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{4}-3q^{5}+q^{7}-q^{8}+3q^{10}+\cdots\)
378.2.a.c \(1\) \(3.018\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
378.2.a.d \(1\) \(3.018\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{7}-q^{8}+5q^{13}-q^{14}+\cdots\)
378.2.a.e \(1\) \(3.018\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{13}+q^{14}+\cdots\)
378.2.a.f \(1\) \(3.018\) \(\Q\) None \(1\) \(0\) \(1\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
378.2.a.g \(1\) \(3.018\) \(\Q\) None \(1\) \(0\) \(3\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+3q^{5}+q^{7}+q^{8}+3q^{10}+\cdots\)
378.2.a.h \(1\) \(3.018\) \(\Q\) None \(1\) \(0\) \(4\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}+4q^{5}-q^{7}+q^{8}+4q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(378))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(378)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(189))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$3$ 1
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))
$13$ (\( 1 - 3 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))
$17$ (\( 1 - 7 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 7 T + 17 T^{2} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))
$23$ (\( 1 - T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))
$29$ (\( 1 + T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - T + 29 T^{2} \))
$31$ (\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 + 7 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))
$43$ (\( 1 - 11 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))
$53$ (\( 1 - 9 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))
$59$ (\( 1 - 5 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 14 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))
$67$ (\( 1 - 7 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 7 T + 67 T^{2} \))
$71$ (\( 1 - 7 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 13 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 13 T + 71 T^{2} \))(\( 1 + 9 T + 71 T^{2} \))(\( 1 + 7 T + 71 T^{2} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))
$79$ (\( 1 + 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 9 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))
$97$ (\( 1 + 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 16 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))
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