Properties

Label 378.2.e
Level 378
Weight 2
Character orbit e
Rep. character \(\chi_{378}(37,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 16
Newform subspaces 4
Sturm bound 144
Trace bound 2

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(2\)
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 16 152
Cusp forms 120 16 104
Eisenstein series 48 0 48

Trace form

\( 16q + 16q^{4} + 4q^{5} - 2q^{7} + O(q^{10}) \) \( 16q + 16q^{4} + 4q^{5} - 2q^{7} - 4q^{11} + 2q^{13} - 2q^{14} + 16q^{16} + 14q^{17} - 4q^{19} + 4q^{20} + 2q^{23} - 8q^{25} + 16q^{26} - 2q^{28} + 10q^{29} - 4q^{31} + 14q^{35} + 2q^{37} + 12q^{38} + 6q^{41} + 2q^{43} - 4q^{44} - 6q^{46} - 12q^{47} + 4q^{49} + 4q^{50} + 2q^{52} - 24q^{53} - 12q^{55} - 2q^{56} + 6q^{58} - 44q^{59} - 16q^{61} - 44q^{62} + 16q^{64} - 12q^{65} - 28q^{67} + 14q^{68} - 18q^{70} - 52q^{71} - 28q^{73} + 6q^{74} - 4q^{76} - 50q^{77} - 40q^{79} + 4q^{80} - 16q^{83} + 12q^{85} - 12q^{86} + 36q^{89} - 16q^{91} + 2q^{92} - 24q^{94} + 68q^{95} + 2q^{97} + 24q^{98} + O(q^{100}) \)

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.e.a \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-3\) \(-4\) \(q-q^{2}+q^{4}+(-3+3\zeta_{6})q^{5}+(-3+\cdots)q^{7}+\cdots\)
378.2.e.b \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(3\) \(-4\) \(q+q^{2}+q^{4}+(3-3\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
378.2.e.c \(6\) \(3.018\) 6.0.309123.1 None \(-6\) \(0\) \(5\) \(4\) \(q-q^{2}+q^{4}+(2-2\beta _{4}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots\)
378.2.e.d \(6\) \(3.018\) 6.0.309123.1 None \(6\) \(0\) \(-1\) \(2\) \(q+q^{2}+q^{4}-\beta _{2}q^{5}+(\beta _{3}+\beta _{4}-\beta _{5})q^{7}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))
$3$ 1
$5$ (\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 5 T + 6 T^{2} - T^{3} + 31 T^{4} - 68 T^{5} + 29 T^{6} - 340 T^{7} + 775 T^{8} - 125 T^{9} + 3750 T^{10} - 15625 T^{11} + 15625 T^{12} \))(\( 1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 260 T^{7} + 575 T^{8} - 2125 T^{9} - 5000 T^{10} + 3125 T^{11} + 15625 T^{12} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 4 T + 14 T^{2} - 55 T^{3} + 98 T^{4} - 196 T^{5} + 343 T^{6} \))(\( 1 - 2 T - 4 T^{2} + 31 T^{3} - 28 T^{4} - 98 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - T - 6 T^{2} + 103 T^{3} - 83 T^{4} - 352 T^{5} + 5027 T^{6} - 3872 T^{7} - 10043 T^{8} + 137093 T^{9} - 87846 T^{10} - 161051 T^{11} + 1771561 T^{12} \))(\( 1 - T - 26 T^{2} + 23 T^{3} + 407 T^{4} - 202 T^{5} - 4853 T^{6} - 2222 T^{7} + 49247 T^{8} + 30613 T^{9} - 380666 T^{10} - 161051 T^{11} + 1771561 T^{12} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 9 T^{2} - 92 T^{3} + 58 T^{4} + 20 T^{5} + 5393 T^{6} + 340 T^{7} + 16762 T^{8} - 451996 T^{9} + 751689 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))(\( 1 - 4 T - 23 T^{2} + 68 T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 3740 T^{7} + 118490 T^{8} + 334084 T^{9} - 1920983 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 3 T - 42 T^{2} - 61 T^{3} + 1311 T^{4} + 726 T^{5} - 27501 T^{6} + 13794 T^{7} + 473271 T^{8} - 418399 T^{9} - 5473482 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 1026 T^{7} - 55233 T^{8} - 459553 T^{9} - 1563852 T^{10} + 7428297 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} \))(\( 1 - 7 T - 24 T^{2} + 127 T^{3} + 1417 T^{4} - 3484 T^{5} - 22393 T^{6} - 80132 T^{7} + 749593 T^{8} + 1545209 T^{9} - 6716184 T^{10} - 45054401 T^{11} + 148035889 T^{12} \))(\( 1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 90758 T^{7} + 1281767 T^{8} + 1009861 T^{9} - 8954912 T^{10} - 45054401 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 174580 T^{7} - 155585 T^{8} + 9048319 T^{9} - 21218430 T^{10} - 102555745 T^{11} + 594823321 T^{12} \))(\( 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 99122 T^{7} + 165677 T^{8} - 6121639 T^{9} + 2829124 T^{10} - 102555745 T^{11} + 594823321 T^{12} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 14 T + 138 T^{2} - 841 T^{3} + 4278 T^{4} - 13454 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 6634 T^{4} + 19220 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \))
$41$ (\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 12 T - 18 T^{2} + 78 T^{3} + 7470 T^{4} - 24546 T^{5} - 158105 T^{6} - 1006386 T^{7} + 12557070 T^{8} + 5375838 T^{9} - 50863698 T^{10} - 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \))
$43$ (\( 1 + 11 T + 78 T^{2} + 473 T^{3} + 1849 T^{4} \))(\( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \))(\( 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 1440156 T^{7} + 13290612 T^{8} - 54223774 T^{9} + 389743314 T^{10} - 2646151974 T^{11} + 6321363049 T^{12} \))(\( ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 258 T^{4} - 22188 T^{5} + 79507 T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 8514 T^{4} + 33282 T^{5} + 79507 T^{6} ) \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 3 T + 117 T^{2} - 309 T^{3} + 5499 T^{4} - 6627 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 9 T + 87 T^{2} + 657 T^{3} + 4089 T^{4} + 19881 T^{5} + 103823 T^{6} )^{2} \))
$53$ (\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 + 9 T - 36 T^{2} - 873 T^{3} - 1179 T^{4} + 26334 T^{5} + 272077 T^{6} + 1395702 T^{7} - 3311811 T^{8} - 129969621 T^{9} - 284057316 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 3199080 T^{7} + 38300715 T^{8} + 4912941 T^{9} + 6272932395 T^{11} + 22164361129 T^{12} \))
$59$ (\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 76 T^{2} - 649 T^{3} + 4484 T^{4} - 13924 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 12744 T^{4} + 48734 T^{5} + 205379 T^{6} )^{2} \))
$61$ (\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 2928 T^{4} + 14884 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2} \))
$67$ (\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 9581 T^{4} + 22445 T^{5} + 300763 T^{6} )^{2} \))(\( ( 1 + T + 89 T^{2} - 77 T^{3} + 5963 T^{4} + 4489 T^{5} + 300763 T^{6} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 11573 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \))(\( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 7347 T^{4} + 43687 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 5 T + 163 T^{2} + 469 T^{3} + 12877 T^{4} + 31205 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 2690860 T^{7} + 202564156 T^{8} - 296185666 T^{9} - 8542497780 T^{10} + 31512325144 T^{11} + 326940373369 T^{12} \))(\( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 1125812 T^{7} + 129430532 T^{8} + 1143574 T^{9} - 8637414422 T^{10} + 7878081286 T^{11} + 326940373369 T^{12} \))
$89$ (\( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 - 9 T - 180 T^{2} + 729 T^{3} + 31041 T^{4} - 54846 T^{5} - 2925911 T^{6} - 4881294 T^{7} + 245875761 T^{8} + 513922401 T^{9} - 11293603380 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 6524946 T^{7} + 127536021 T^{8} + 843847893 T^{9} - 9034882704 T^{10} - 50256535041 T^{11} + 496981290961 T^{12} \))
$97$ (\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \))
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