Properties

Label 378.2.d
Level 378
Weight 2
Character orbit d
Rep. character \(\chi_{378}(377,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 3
Sturm bound 144
Trace bound 22

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(144\)
Trace bound: \(22\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(378, [\chi])\).

Total New Old
Modular forms 84 12 72
Cusp forms 60 12 48
Eisenstein series 24 0 24

Trace form

\( 12q - 12q^{4} + 6q^{7} + O(q^{10}) \) \( 12q - 12q^{4} + 6q^{7} + 12q^{16} + 12q^{25} - 6q^{28} + 12q^{37} - 60q^{43} + 30q^{49} - 36q^{58} - 12q^{64} - 12q^{67} - 18q^{70} + 24q^{79} + 72q^{85} - 36q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
378.2.d.a \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) \(q-\zeta_{12}^{3}q^{2}-q^{4}+(-2\zeta_{12}+\zeta_{12}^{3})q^{5}+\cdots\)
378.2.d.b \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) \(q-\zeta_{12}q^{2}-q^{4}-\zeta_{12}^{3}q^{5}+(2-\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
378.2.d.c \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{12}q^{2}-q^{4}-2\zeta_{12}^{3}q^{5}+(2-\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$3$ 1
$5$ (\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))
$11$ (\( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))
$17$ (\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 31 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 35 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{4} \))(\( ( 1 - 7 T + 37 T^{2} )^{4} \))(\( ( 1 + 2 T + 37 T^{2} )^{4} \))
$41$ (\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 65 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 2 T + 43 T^{2} )^{4} \))(\( ( 1 + 11 T + 43 T^{2} )^{4} \))
$47$ (\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 46 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 97 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 38 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 - 2 T + 67 T^{2} )^{4} \))(\( ( 1 + 7 T + 67 T^{2} )^{4} \))
$71$ (\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 133 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 133 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 98 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 10 T + 79 T^{2} )^{4} \))(\( ( 1 - 8 T + 79 T^{2} )^{4} \))
$83$ (\( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 154 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 151 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 47 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 146 T^{2} + 9409 T^{4} )^{2} \))
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