Properties

Label 315.2.a
Level 315
Weight 2
Character orbit a
Rep. character \(\chi_{315}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 6
Sturm bound 96
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(315))\).

Total New Old
Modular forms 56 10 46
Cusp forms 41 10 31
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(7\)

Trace form

\( 10q + 12q^{4} - 2q^{7} + 12q^{8} + O(q^{10}) \) \( 10q + 12q^{4} - 2q^{7} + 12q^{8} + 4q^{10} - 2q^{11} - 4q^{13} - 2q^{14} + 16q^{16} + 4q^{17} - 8q^{19} - 20q^{22} - 8q^{23} + 10q^{25} + 20q^{26} - 6q^{28} + 2q^{29} + 12q^{31} + 4q^{32} - 28q^{34} + 4q^{35} - 8q^{37} - 32q^{38} + 20q^{41} - 12q^{43} - 12q^{44} + 32q^{46} - 20q^{47} + 10q^{49} - 40q^{52} - 4q^{53} + 8q^{55} - 6q^{56} + 4q^{58} + 4q^{59} + 32q^{61} - 24q^{62} - 48q^{64} + 6q^{65} - 20q^{67} + 24q^{68} - 2q^{70} - 24q^{71} - 16q^{73} - 32q^{74} + 16q^{76} + 6q^{79} + 6q^{85} + 24q^{86} - 8q^{88} + 16q^{89} + 2q^{91} - 40q^{92} - 20q^{94} + 20q^{95} - 44q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
315.2.a.a \(1\) \(2.515\) \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(q-q^{2}-q^{4}-q^{5}+q^{7}+3q^{8}+q^{10}+\cdots\)
315.2.a.b \(1\) \(2.515\) \(\Q\) None \(0\) \(0\) \(1\) \(1\) \(-\) \(-\) \(-\) \(q-2q^{4}+q^{5}+q^{7}+3q^{11}+5q^{13}+\cdots\)
315.2.a.c \(2\) \(2.515\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(+\) \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-q^{7}+\cdots\)
315.2.a.d \(2\) \(2.515\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q-\beta q^{2}+3q^{4}+q^{5}+q^{7}-\beta q^{8}-\beta q^{10}+\cdots\)
315.2.a.e \(2\) \(2.515\) \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(q+\beta q^{2}+(2+\beta )q^{4}-q^{5}-q^{7}+(4+\beta )q^{8}+\cdots\)
315.2.a.f \(2\) \(2.515\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(315)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - T - 2 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$7$ (\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \))(\( 1 + 6 T^{2} + 169 T^{4} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( 1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))(\( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( 1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 - 4 T + 54 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \))(\( 1 + 4 T + 54 T^{2} + 164 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 + 6 T^{2} + 1849 T^{4} \))(\( 1 - 10 T + 94 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 5 T + 62 T^{2} - 235 T^{3} + 2209 T^{4} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 10 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 16 T + 162 T^{2} + 848 T^{3} + 2809 T^{4} \))(\( 1 - 16 T + 150 T^{2} - 848 T^{3} + 2809 T^{4} \))(\( 1 - 2 T + 90 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 - 16 T + 162 T^{2} - 848 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 22 T^{2} + 536 T^{3} + 4489 T^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 8 T + 22 T^{2} + 536 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( 1 - 4 T + 74 T^{2} - 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 4 T - 50 T^{2} - 292 T^{3} + 5329 T^{4} \))(\( 1 + 16 T + 190 T^{2} + 1168 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))(\( 1 - 4 T - 50 T^{2} - 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 126 T^{2} + 6241 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 + 126 T^{2} + 6241 T^{4} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 - 8 T + 83 T^{2} )^{2} \))(\( 1 - 16 T + 150 T^{2} - 1328 T^{3} + 6889 T^{4} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 8 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 12 T + 86 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))(\( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 12 T + 222 T^{2} + 1164 T^{3} + 9409 T^{4} \))(\( 1 - 8 T + 190 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 9 T + 108 T^{2} + 873 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 222 T^{2} + 1164 T^{3} + 9409 T^{4} \))
show more
show less