Properties

Label 378.2.f
Level 378
Weight 2
Character orbit f
Rep. character \(\chi_{378}(127,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 12
Newform subspaces 4
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.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 168 12 156
Cusp forms 120 12 108
Eisenstein series 48 0 48

Trace form

\( 12q - 2q^{2} - 6q^{4} + 4q^{5} + 4q^{8} + O(q^{10}) \) \( 12q - 2q^{2} - 6q^{4} + 4q^{5} + 4q^{8} + 2q^{11} - 4q^{14} - 6q^{16} - 4q^{17} + 12q^{19} + 4q^{20} - 6q^{22} - 4q^{23} - 18q^{25} - 8q^{29} - 12q^{31} - 2q^{32} - 6q^{34} + 8q^{35} + 24q^{37} + 14q^{38} + 18q^{41} - 6q^{43} - 4q^{44} - 6q^{49} - 10q^{50} + 24q^{55} - 4q^{56} - 14q^{59} - 24q^{62} + 12q^{64} - 36q^{65} - 18q^{67} + 2q^{68} - 16q^{71} + 12q^{73} - 4q^{74} - 6q^{76} - 8q^{77} - 12q^{79} - 8q^{80} - 36q^{82} + 44q^{83} + 12q^{85} + 2q^{86} - 6q^{88} + 24q^{89} + 24q^{91} - 4q^{92} + 44q^{95} + 6q^{97} + 4q^{98} + O(q^{100}) \)

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.f.a \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-1\) \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-3\zeta_{6}q^{5}+\cdots\)
378.2.f.b \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+2\zeta_{6}q^{5}+\cdots\)
378.2.f.c \(4\) \(3.018\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(0\) \(3\) \(-2\) \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
378.2.f.d \(4\) \(3.018\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(0\) \(2\) \(2\) \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( ( 1 - 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \))(\( 1 - 2 T - T^{2} + 10 T^{3} - 20 T^{4} + 50 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))
$7$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$11$ (\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 198 T^{5} - 847 T^{6} - 3993 T^{7} + 14641 T^{8} \))(\( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 5 T + 17 T^{2} )^{2} \))(\( ( 1 - 3 T + 28 T^{2} - 51 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{4} \))
$19$ (\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{4} \))(\( ( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( ( 1 + 9 T + 23 T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} ) \))(\( ( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( 1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 4176 T^{5} + 1682 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 2320 T^{5} - 18502 T^{6} + 97556 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} \))(\( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{4} \))(\( ( 1 - 4 T - 18 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 - 41 T^{2} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 29520 T^{5} + 159695 T^{6} - 1033815 T^{7} + 2825761 T^{8} \))(\( 1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 3182 T^{5} - 20339 T^{6} + 79507 T^{7} + 3418801 T^{8} \))(\( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 2205 T^{4} + 4418 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 + 12 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 82 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 12 T + 118 T^{2} - 636 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 12744 T^{5} - 128797 T^{6} + 616137 T^{7} + 12117361 T^{8} \))(\( ( 1 + 2 T - 55 T^{2} + 118 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 29524 T^{5} + 160003 T^{6} - 2496791 T^{7} + 13845841 T^{8} \))(\( 1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 82350 T^{5} + 472567 T^{6} + 4085658 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( ( 1 + 13 T + 67 T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} ) \))(\( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 42880 T^{5} + 368098 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 3 T + 70 T^{2} + 213 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 10 T + 143 T^{2} + 710 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 - T + 73 T^{2} )^{2} \))(\( ( 1 - 7 T + 84 T^{2} - 511 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \))(\( 1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4} \))(\( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 34286 T^{5} - 293327 T^{6} + 3451273 T^{7} + 38950081 T^{8} \))(\( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 7110 T^{5} - 667787 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 16 T + 173 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 95616 T^{5} + 509786 T^{6} - 6861444 T^{7} + 47458321 T^{8} \))(\( ( 1 - 2 T - 79 T^{2} - 166 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 24 T + 298 T^{2} - 2136 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( 1 + T - 119 T^{2} - 74 T^{3} + 4894 T^{4} - 7178 T^{5} - 1119671 T^{6} + 912673 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))
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