Properties

Label 115.2.a
Level 115
Weight 2
Character orbit a
Rep. character \(\chi_{115}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 3
Sturm bound 24
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(115))\).

Total New Old
Modular forms 14 7 7
Cusp forms 11 7 4
Eisenstein series 3 0 3

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

\(5\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(4\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7q + q^{2} - 4q^{3} + 9q^{4} + q^{5} + 2q^{6} - 4q^{7} + 3q^{8} - q^{9} + O(q^{10}) \) \( 7q + q^{2} - 4q^{3} + 9q^{4} + q^{5} + 2q^{6} - 4q^{7} + 3q^{8} - q^{9} + 3q^{10} + 4q^{11} - 22q^{12} - 10q^{13} - 12q^{14} + 17q^{16} - 2q^{17} + 3q^{18} - 4q^{19} - q^{20} + 12q^{21} - 8q^{22} - 5q^{23} - 24q^{24} + 7q^{25} + 12q^{26} - 4q^{27} - 8q^{28} + 16q^{29} - 4q^{30} - 2q^{31} - 3q^{32} - 10q^{34} - 2q^{35} + 11q^{36} + 2q^{37} + 28q^{38} + 8q^{39} + 15q^{40} + 8q^{41} + 8q^{42} - 12q^{43} - 32q^{44} + 13q^{45} + 3q^{46} + 16q^{47} - 34q^{48} + q^{49} + q^{50} - 4q^{51} - 32q^{52} + 18q^{53} - 22q^{54} + 4q^{55} - 24q^{56} + 40q^{58} + 10q^{59} - 16q^{60} - 6q^{61} - 34q^{62} - 12q^{63} + 23q^{64} + 10q^{65} + 36q^{66} + 8q^{67} + 26q^{68} + 4q^{69} - 12q^{70} - 6q^{71} + 51q^{72} - 26q^{73} + 34q^{74} - 4q^{75} - 28q^{76} + 12q^{77} + 26q^{78} + 12q^{79} - q^{80} - 17q^{81} + 16q^{82} - 16q^{83} + 16q^{84} - 8q^{86} - 8q^{87} + 12q^{88} + 14q^{89} + 3q^{90} - 44q^{91} - 5q^{92} - 12q^{93} + 22q^{94} - 4q^{95} - 46q^{96} - 22q^{97} + 17q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(115))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 23
115.2.a.a \(1\) \(0.918\) \(\Q\) None \(2\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(q+2q^{2}+2q^{4}-q^{5}+q^{7}-3q^{9}-2q^{10}+\cdots\)
115.2.a.b \(2\) \(0.918\) \(\Q(\sqrt{5}) \) None \(-3\) \(-2\) \(-2\) \(-2\) \(+\) \(+\) \(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}-q^{5}+\cdots\)
115.2.a.c \(4\) \(0.918\) 4.4.15317.1 None \(2\) \(-2\) \(4\) \(-3\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(115))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(115)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 2 T^{2} \))(\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 4 T^{2} - 7 T^{3} + 10 T^{4} - 14 T^{5} + 16 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( ( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} )^{2} \))
$5$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{4} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 + 3 T + 14 T^{2} + 11 T^{3} + 66 T^{4} + 77 T^{5} + 686 T^{6} + 1029 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 2 T + 18 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 28 T^{2} - 92 T^{3} + 406 T^{4} - 1012 T^{5} + 3388 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4} \))(\( 1 + 11 T^{2} + 160 T^{4} + 1859 T^{6} + 28561 T^{8} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + T + 50 T^{2} + 27 T^{3} + 1154 T^{4} + 459 T^{5} + 14450 T^{6} + 4913 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 60 T^{2} + 188 T^{3} + 1590 T^{4} + 3572 T^{5} + 21660 T^{6} + 27436 T^{7} + 130321 T^{8} \))
$23$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{4} \))
$29$ (\( 1 - 7 T + 29 T^{2} \))(\( 1 + 10 T + 63 T^{2} + 290 T^{3} + 841 T^{4} \))(\( 1 - 19 T + 233 T^{2} - 1922 T^{3} + 12034 T^{4} - 55738 T^{5} + 195953 T^{6} - 463391 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 5 T + 31 T^{2} \))(\( 1 - 4 T + 61 T^{2} - 124 T^{3} + 961 T^{4} \))(\( 1 + T + 23 T^{2} + 104 T^{3} + 1648 T^{4} + 3224 T^{5} + 22103 T^{6} + 29791 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 11 T + 37 T^{2} \))(\( 1 + 6 T + 38 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 3 T + 32 T^{2} + 349 T^{3} + 1638 T^{4} + 12913 T^{5} + 43808 T^{6} + 151959 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - T + 41 T^{2} \))(\( 1 + 6 T + 71 T^{2} + 246 T^{3} + 1681 T^{4} \))(\( 1 - 13 T + 209 T^{2} - 1602 T^{3} + 13682 T^{4} - 65682 T^{5} + 351329 T^{6} - 895973 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + 43 T^{2} \))(\( 1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 + 6 T + 136 T^{2} + 758 T^{3} + 8126 T^{4} + 32594 T^{5} + 251464 T^{6} + 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 10 T + 99 T^{2} - 470 T^{3} + 2209 T^{4} \))(\( 1 - 6 T + 105 T^{2} - 298 T^{3} + 5324 T^{4} - 14006 T^{5} + 231945 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 11 T + 53 T^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( 1 - 19 T + 178 T^{2} - 929 T^{3} + 4474 T^{4} - 49237 T^{5} + 500002 T^{6} - 2828663 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 13 T + 59 T^{2} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 - 23 T + 336 T^{2} - 3511 T^{3} + 29550 T^{4} - 207149 T^{5} + 1169616 T^{6} - 4723717 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 2 T - 2 T^{2} - 122 T^{3} + 3721 T^{4} \))(\( 1 + 188 T^{2} + 136 T^{3} + 15462 T^{4} + 8296 T^{5} + 699548 T^{6} + 13845841 T^{8} \))
$67$ (\( 1 - 5 T + 67 T^{2} \))(\( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 3 T + 170 T^{2} + 391 T^{3} + 15834 T^{4} + 26197 T^{5} + 763130 T^{6} + 902289 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 - 5 T + 71 T^{2} \))(\( 1 + 8 T + 153 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 + 3 T + 135 T^{2} + 104 T^{3} + 9080 T^{4} + 7384 T^{5} + 680535 T^{6} + 1073733 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 - 6 T + 73 T^{2} \))(\( 1 + 101 T^{2} + 5329 T^{4} \))(\( 1 + 32 T + 635 T^{2} + 8400 T^{3} + 83736 T^{4} + 613200 T^{5} + 3383915 T^{6} + 12448544 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 12 T + 79 T^{2} \))(\( 1 - 22 T + 274 T^{2} - 1738 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 176 T^{2} - 826 T^{3} + 15838 T^{4} - 65254 T^{5} + 1098416 T^{6} - 986078 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 9 T + 83 T^{2} \))(\( 1 + 4 T + 150 T^{2} + 332 T^{3} + 6889 T^{4} \))(\( 1 + 21 T + 428 T^{2} + 5005 T^{3} + 56054 T^{4} + 415415 T^{5} + 2948492 T^{6} + 12007527 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 - 4 T + 89 T^{2} \))(\( 1 - 10 T + 198 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 + 140 T^{2} - 1496 T^{3} + 6326 T^{4} - 133144 T^{5} + 1108940 T^{6} + 62742241 T^{8} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 94 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 + 18 T + 460 T^{2} + 5038 T^{3} + 69350 T^{4} + 488686 T^{5} + 4328140 T^{6} + 16428114 T^{7} + 88529281 T^{8} \))
show more
show less