Properties

Label 315.2.cg
Level 315
Weight 2
Character orbit cg
Rep. character \(\chi_{315}(157,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 176
Newform subspaces 5
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.cg (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 315 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 208 208 0
Cusp forms 176 176 0
Eisenstein series 32 32 0

Trace form

\( 176q + 2q^{2} - 6q^{3} - 2q^{7} - 8q^{8} + O(q^{10}) \) \( 176q + 2q^{2} - 6q^{3} - 2q^{7} - 8q^{8} - 12q^{10} - 6q^{12} - 2q^{15} + 68q^{16} - 30q^{17} - 26q^{18} - 48q^{20} - 36q^{21} - 12q^{22} + 12q^{23} - 4q^{25} - 24q^{26} + 36q^{27} + 4q^{28} - 44q^{30} - 12q^{31} + 30q^{32} + 24q^{33} - 44q^{35} + 40q^{36} - 4q^{37} - 12q^{41} - 10q^{42} - 4q^{43} + 12q^{45} - 20q^{46} - 6q^{47} - 18q^{48} - 28q^{50} - 28q^{51} + 16q^{53} - 88q^{56} + 8q^{57} - 44q^{58} - 10q^{60} - 48q^{61} + 54q^{63} + 26q^{65} - 84q^{66} + 2q^{67} - 42q^{70} - 64q^{71} + 10q^{72} - 12q^{73} - 6q^{75} - 48q^{76} - 106q^{77} - 62q^{78} + 12q^{80} + 20q^{81} - 24q^{82} + 84q^{83} - 4q^{85} + 80q^{86} - 24q^{87} - 84q^{88} - 36q^{90} - 16q^{91} - 8q^{92} + 28q^{93} - 50q^{95} + 72q^{96} + 120q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.cg.a \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-8\) \(0\) \(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
315.2.cg.b \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(4\) \(0\) \(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
315.2.cg.c \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(4\) \(0\) \(8\) \(-10\) \(q+(1+\zeta_{12}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
315.2.cg.d \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(4\) \(6\) \(-4\) \(2\) \(q+(1+\zeta_{12}-\zeta_{12}^{3})q^{2}+(1+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
315.2.cg.e \(160\) \(2.515\) None \(-2\) \(-12\) \(0\) \(6\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8} \))
$3$ (\( ( 1 + 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )^{2} \))
$5$ (\( ( 1 + 4 T + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )^{2} \))(\( ( 1 + 2 T + 5 T^{2} )^{2} \))
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 - 13 T^{2} + 49 T^{4} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))
$11$ (\( ( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{4} \))(\( ( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{4} \))
$13$ (\( 1 + 4 T + 20 T^{2} + 84 T^{3} + 263 T^{4} + 1092 T^{5} + 3380 T^{6} + 8788 T^{7} + 28561 T^{8} \))(\( 1 - 2 T + 26 T^{2} - 84 T^{3} + 407 T^{4} - 1092 T^{5} + 4394 T^{6} - 4394 T^{7} + 28561 T^{8} \))(\( 1 + 8 T + 20 T^{2} - 36 T^{3} - 361 T^{4} - 468 T^{5} + 3380 T^{6} + 17576 T^{7} + 28561 T^{8} \))(\( 1 - 10 T + 26 T^{2} + 108 T^{3} - 841 T^{4} + 1404 T^{5} + 4394 T^{6} - 21970 T^{7} + 28561 T^{8} \))
$17$ (\( 1 + 4 T + 8 T^{2} - 104 T^{3} - 497 T^{4} - 1768 T^{5} + 2312 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 16 T + 80 T^{2} - 8 T^{3} - 1121 T^{4} - 136 T^{5} + 23120 T^{6} + 78608 T^{7} + 83521 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 104 T^{3} - 497 T^{4} + 1768 T^{5} + 2312 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 + 8 T + 80 T^{2} + 392 T^{3} + 2143 T^{4} + 6664 T^{5} + 23120 T^{6} + 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 10 T + 40 T^{2} + 220 T^{3} + 1339 T^{4} + 4180 T^{5} + 14440 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 10 T + 40 T^{2} - 220 T^{3} + 1339 T^{4} - 4180 T^{5} + 14440 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 - 10 T + 50 T^{2} - 340 T^{3} + 2191 T^{4} - 7820 T^{5} + 26450 T^{6} - 121670 T^{7} + 279841 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 482 T^{4} - 92 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} \))(\( 1 - 10 T + 50 T^{2} - 340 T^{3} + 2191 T^{4} - 7820 T^{5} + 26450 T^{6} - 121670 T^{7} + 279841 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 4 T^{3} - 482 T^{4} - 92 T^{5} + 4232 T^{6} - 48668 T^{7} + 279841 T^{8} \))
$29$ (\( ( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4} )^{2} \))(\( 1 - 24 T + 289 T^{2} - 2328 T^{3} + 14136 T^{4} - 67512 T^{5} + 243049 T^{6} - 585336 T^{7} + 707281 T^{8} \))(\( ( 1 - 12 T + 77 T^{2} - 348 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 24 T + 289 T^{2} + 2328 T^{3} + 14136 T^{4} + 67512 T^{5} + 243049 T^{6} + 585336 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 35712 T^{5} + 165292 T^{6} + 536238 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 118 T^{2} + 840 T^{3} + 5427 T^{4} + 26040 T^{5} + 113398 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 35712 T^{5} + 165292 T^{6} + 536238 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 118 T^{2} + 840 T^{3} + 5427 T^{4} + 26040 T^{5} + 113398 T^{6} + 357492 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T + 90 T^{2} + 552 T^{3} + 4439 T^{4} + 20424 T^{5} + 123210 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 18 T + 90 T^{2} + 408 T^{3} - 6217 T^{4} + 15096 T^{5} + 123210 T^{6} - 911754 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 28536 T^{5} + 178186 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 28536 T^{5} + 178186 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 12 T + 45 T^{2} - 204 T^{3} - 3316 T^{4} - 8772 T^{5} + 83205 T^{6} + 954084 T^{7} + 3418801 T^{8} \))(\( 1 + 9 T^{2} + 240 T^{3} - 1324 T^{4} + 10320 T^{5} + 16641 T^{6} + 3418801 T^{8} \))(\( 1 - 6 T + 45 T^{2} - 366 T^{3} + 1328 T^{4} - 15738 T^{5} + 83205 T^{6} - 477042 T^{7} + 3418801 T^{8} \))(\( 1 - 6 T + 9 T^{2} + 258 T^{3} - 2872 T^{4} + 11094 T^{5} + 16641 T^{6} - 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 30 T + 369 T^{2} - 2526 T^{3} + 14864 T^{4} - 118722 T^{5} + 815121 T^{6} - 3114690 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 45 T^{2} - 612 T^{3} - 6892 T^{4} - 28764 T^{5} + 99405 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))(\( 1 - 24 T + 369 T^{2} - 3792 T^{3} + 30092 T^{4} - 178224 T^{5} + 815121 T^{6} - 2491752 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T + 45 T^{2} + 378 T^{3} - 1816 T^{4} + 17766 T^{5} + 99405 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 14 T + 74 T^{2} - 8 T^{3} - 2537 T^{4} - 424 T^{5} + 207866 T^{6} - 2084278 T^{7} + 7890481 T^{8} \))(\( 1 - 8 T + 20 T^{2} + 196 T^{3} - 3641 T^{4} + 10388 T^{5} + 56180 T^{6} - 1191016 T^{7} + 7890481 T^{8} \))(\( 1 + 10 T + 74 T^{2} + 520 T^{3} + 2551 T^{4} + 27560 T^{5} + 207866 T^{6} + 1488770 T^{7} + 7890481 T^{8} \))(\( 1 + 4 T + 20 T^{2} + 244 T^{3} - 1097 T^{4} + 12932 T^{5} + 56180 T^{6} + 595508 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8} \))(\( 1 + 6 T + 56 T^{2} - 828 T^{3} - 5205 T^{4} - 48852 T^{5} + 194936 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8} \))(\( 1 - 6 T + 56 T^{2} + 828 T^{3} - 5205 T^{4} + 48852 T^{5} + 194936 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 36 T + 661 T^{2} + 8244 T^{3} + 75072 T^{4} + 502884 T^{5} + 2459581 T^{6} + 8171316 T^{7} + 13845841 T^{8} \))(\( 1 - 24 T + 346 T^{2} - 3696 T^{3} + 31707 T^{4} - 225456 T^{5} + 1287466 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))(\( 1 + 36 T + 661 T^{2} + 8244 T^{3} + 75072 T^{4} + 502884 T^{5} + 2459581 T^{6} + 8171316 T^{7} + 13845841 T^{8} \))(\( 1 - 24 T + 346 T^{2} - 3696 T^{3} + 31707 T^{4} - 225456 T^{5} + 1287466 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 2 T + 65 T^{2} - 762 T^{3} + 2600 T^{4} - 51054 T^{5} + 291785 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 14 T + 74 T^{2} + 48 T^{3} - 4273 T^{4} + 3216 T^{5} + 332186 T^{6} - 4210682 T^{7} + 20151121 T^{8} \))(\( 1 + 16 T + 65 T^{2} - 960 T^{3} - 14284 T^{4} - 64320 T^{5} + 291785 T^{6} + 4812208 T^{7} + 20151121 T^{8} \))(\( 1 + 10 T + 74 T^{2} + 576 T^{3} + 2159 T^{4} + 38592 T^{5} + 332186 T^{6} + 3007630 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 6 T + 90 T^{2} + 984 T^{3} + 6095 T^{4} + 71832 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 + 24 T + 144 T^{2} - 1752 T^{3} - 31057 T^{4} - 127896 T^{5} + 767376 T^{6} + 9336408 T^{7} + 28398241 T^{8} \))(\( 1 + 18 T + 90 T^{2} - 840 T^{3} - 14929 T^{4} - 61320 T^{5} + 479610 T^{6} + 7002306 T^{7} + 28398241 T^{8} \))(\( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 - 12 T + 154 T^{2} - 1272 T^{3} + 8787 T^{4} - 100488 T^{5} + 961114 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 - 12 T + 154 T^{2} - 1272 T^{3} + 8787 T^{4} - 100488 T^{5} + 961114 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 8787 T^{4} + 100488 T^{5} + 961114 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T + 154 T^{2} + 1272 T^{3} + 8787 T^{4} + 100488 T^{5} + 961114 T^{6} + 5916468 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 27224 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 28 T + 365 T^{2} + 2980 T^{3} + 23356 T^{4} + 247340 T^{5} + 2514485 T^{6} + 16010036 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 26 T + 365 T^{2} + 3554 T^{3} + 32320 T^{4} + 294982 T^{5} + 2514485 T^{6} + 14866462 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 8 T - 103 T^{2} - 88 T^{3} + 14272 T^{4} - 7832 T^{5} - 815863 T^{6} + 5639752 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 62 T^{2} - 256 T^{3} + 6931 T^{4} - 22784 T^{5} + 491102 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 8 T - 103 T^{2} + 88 T^{3} + 14272 T^{4} + 7832 T^{5} - 815863 T^{6} - 5639752 T^{7} + 62742241 T^{8} \))(\( 1 + 16 T + 62 T^{2} + 256 T^{3} + 6931 T^{4} + 22784 T^{5} + 491102 T^{6} + 11279504 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 - 2 T + 50 T^{2} + 1188 T^{3} - 4465 T^{4} + 115236 T^{5} + 470450 T^{6} - 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - 26 T + 458 T^{2} - 5604 T^{3} + 61583 T^{4} - 543588 T^{5} + 4309322 T^{6} - 23729498 T^{7} + 88529281 T^{8} \))(\( 1 + 14 T + 50 T^{2} - 1500 T^{3} - 23089 T^{4} - 145500 T^{5} + 470450 T^{6} + 12777422 T^{7} + 88529281 T^{8} \))(\( 1 - 34 T + 458 T^{2} - 2916 T^{3} + 15023 T^{4} - 282852 T^{5} + 4309322 T^{6} - 31030882 T^{7} + 88529281 T^{8} \))
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