Properties

Label 378.2.k
Level 378
Weight 2
Character orbit k
Rep. character \(\chi_{378}(215,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 20
Newform subspaces 4
Sturm bound 144
Trace bound 10

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.k (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(10\)
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 20 148
Cusp forms 120 20 100
Eisenstein series 48 0 48

Trace form

\( 20q + 10q^{4} + 10q^{7} + O(q^{10}) \) \( 20q + 10q^{4} + 10q^{7} - 6q^{10} - 10q^{16} + 12q^{19} + 12q^{22} - 4q^{25} + 8q^{28} + 24q^{31} + 14q^{37} - 6q^{40} + 44q^{43} - 58q^{49} - 18q^{52} - 24q^{58} - 30q^{61} - 20q^{64} - 26q^{67} - 30q^{70} - 120q^{73} - 2q^{79} - 24q^{82} - 72q^{85} + 6q^{88} + 48q^{91} + 120q^{94} + O(q^{100}) \)

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.k.a \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(2\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-1+3\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
378.2.k.b \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-2\zeta_{12}+4\zeta_{12}^{3})q^{5}+\cdots\)
378.2.k.c \(4\) \(3.018\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(8\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(\zeta_{12}-2\zeta_{12}^{3})q^{5}+\cdots\)
378.2.k.d \(8\) \(3.018\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\zeta_{24}q^{2}+\zeta_{24}^{2}q^{4}+(\zeta_{24}^{5}-\zeta_{24}^{6}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( 1 - T^{2} + T^{4} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))
$3$ 1
$5$ (\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \))(\( ( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 4 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{4} \))
$11$ (\( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 - 34 T^{2} + 555 T^{4} - 5746 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 14 T^{2} - 93 T^{4} + 4046 T^{6} + 83521 T^{8} \))(\( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \))(\( ( 1 - 28 T^{2} + 495 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 + 19 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} - 165 T^{4} + 5054 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \))(\( ( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 23 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 + 23 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} - 894 T^{4} - 6728 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 11 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 6 T + 23 T^{2} + 66 T^{3} - 468 T^{4} + 2046 T^{5} + 22103 T^{6} + 178746 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 1258 T^{5} - 72557 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 76 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 4 T + 43 T^{2} )^{4} \))(\( ( 1 - T + 43 T^{2} )^{4} \))(\( ( 1 + 8 T + 43 T^{2} )^{4} \))(\( ( 1 - 7 T + 43 T^{2} )^{8} \))
$47$ (\( 1 + 14 T^{2} - 2013 T^{4} + 30926 T^{6} + 4879681 T^{8} \))(\( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \))(\( 1 - 82 T^{2} + 4515 T^{4} - 181138 T^{6} + 4879681 T^{8} \))(\( 1 + 40 T^{2} - 626 T^{4} - 87680 T^{6} - 3459005 T^{8} - 193685120 T^{10} - 3054680306 T^{12} + 431168613160 T^{14} + 23811286661761 T^{16} \))
$53$ (\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \))(\( 1 + 97 T^{2} + 6600 T^{4} + 272473 T^{6} + 7890481 T^{8} \))(\( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 116449904 T^{10} + 37574470522 T^{12} - 88657444516 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \))(\( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} \))(\( 1 + 29 T^{2} - 2640 T^{4} + 100949 T^{6} + 12117361 T^{8} \))(\( ( 1 - 112 T^{2} + 9063 T^{4} - 389872 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 6 T + 131 T^{2} - 714 T^{3} + 11172 T^{4} - 43554 T^{5} + 487451 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 10 T + 13 T^{2} - 470 T^{3} - 3620 T^{4} - 31490 T^{5} + 58357 T^{6} + 3007630 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 - 9 T + 100 T^{2} - 657 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 604440 T^{5} + 3527798 T^{6} + 14004612 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 5530 T^{5} - 405665 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 139 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 154 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 68 T^{2} + 4566 T^{4} + 468452 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \))(\( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 5848105984 T^{10} + 517874457214 T^{12} - 15903401310752 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 - 119 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 191 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 334 T^{2} + 46419 T^{4} - 3142606 T^{6} + 88529281 T^{8} )^{2} \))
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