Properties

Label 318.2.a
Level 318
Weight 2
Character orbit a
Rep. character \(\chi_{318}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 7
Sturm bound 108
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 318 = 2 \cdot 3 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 318.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(318))\).

Total New Old
Modular forms 58 9 49
Cusp forms 51 9 42
Eisenstein series 7 0 7

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

\(2\)\(3\)\(53\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(7\)

Trace form

\( 9q - q^{2} + q^{3} + 9q^{4} + 2q^{5} - q^{6} + 4q^{7} - q^{8} + 9q^{9} + O(q^{10}) \) \( 9q - q^{2} + q^{3} + 9q^{4} + 2q^{5} - q^{6} + 4q^{7} - q^{8} + 9q^{9} - 6q^{10} - 4q^{11} + q^{12} - 2q^{13} - 8q^{14} + 2q^{15} + 9q^{16} - 2q^{17} - q^{18} + 12q^{19} + 2q^{20} + 8q^{21} - q^{24} + 11q^{25} - 6q^{26} + q^{27} + 4q^{28} - 22q^{29} + 6q^{30} - 4q^{31} - q^{32} + 6q^{34} + 8q^{35} + 9q^{36} + 22q^{37} + 4q^{38} + 14q^{39} - 6q^{40} - 18q^{41} - 4q^{44} + 2q^{45} - 8q^{46} + q^{48} - 11q^{49} - 15q^{50} - 14q^{51} - 2q^{52} - q^{53} - q^{54} - 8q^{55} - 8q^{56} - 10q^{58} - 12q^{59} + 2q^{60} - 14q^{61} + 4q^{63} + 9q^{64} - 20q^{65} - 4q^{66} - 28q^{67} - 2q^{68} - 16q^{71} - q^{72} + 2q^{73} - 14q^{74} - q^{75} + 12q^{76} + 8q^{77} - 14q^{78} - 12q^{79} + 2q^{80} + 9q^{81} - 18q^{82} - 20q^{83} + 8q^{84} - 20q^{85} + 4q^{86} + 10q^{87} - 26q^{89} - 6q^{90} + 20q^{93} - 56q^{95} - q^{96} + 18q^{97} + 7q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(318))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 53
318.2.a.a \(1\) \(2.539\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
318.2.a.b \(1\) \(2.539\) \(\Q\) None \(-1\) \(-1\) \(4\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}+q^{7}+\cdots\)
318.2.a.c \(1\) \(2.539\) \(\Q\) None \(-1\) \(1\) \(0\) \(5\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+5q^{7}-q^{8}+\cdots\)
318.2.a.d \(1\) \(2.539\) \(\Q\) None \(1\) \(-1\) \(-3\) \(-4\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-4q^{7}+\cdots\)
318.2.a.e \(1\) \(2.539\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
318.2.a.f \(2\) \(2.539\) \(\Q(\sqrt{41}) \) None \(-2\) \(2\) \(1\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}-q^{8}+\cdots\)
318.2.a.g \(2\) \(2.539\) \(\Q(\sqrt{17}) \) None \(2\) \(2\) \(1\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+(1-\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(318))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(318)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(159))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - T - 5 T^{3} + 25 T^{4} \))(\( 1 - T + 6 T^{2} - 5 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 - T + 10 T^{2} - 7 T^{3} + 49 T^{4} \))
$11$ (\( 1 + T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \))(\( ( 1 + T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( 1 + 2 T + 10 T^{2} + 26 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 5 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 9 T + 44 T^{2} + 153 T^{3} + 289 T^{4} \))(\( 1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 5 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 2 T - 2 T^{2} + 38 T^{3} + 361 T^{4} \))(\( 1 - 3 T + 36 T^{2} - 57 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 5 T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 7 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 3 T + 38 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 29 T^{2} + 529 T^{4} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 8 T + 29 T^{2} \))(\( 1 + T + 29 T^{2} \))(\( 1 + 2 T + 18 T^{2} + 58 T^{3} + 841 T^{4} \))(\( 1 + 7 T + 66 T^{2} + 203 T^{3} + 841 T^{4} \))
$31$ (\( 1 + T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - T + 58 T^{2} - 31 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 12 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( 1 - 2 T + 58 T^{2} - 74 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 4 T + 41 T^{2} \))(\( 1 - 5 T + 41 T^{2} \))(\( 1 + 3 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 2 T + 42 T^{2} - 82 T^{3} + 1681 T^{4} \))(\( 1 + 13 T + 86 T^{2} + 533 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 - 11 T + 106 T^{2} - 473 T^{3} + 1849 T^{4} \))(\( 1 - 3 T + 50 T^{2} - 129 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 10 T + 78 T^{2} + 470 T^{3} + 2209 T^{4} \))(\( 1 + 26 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 3 T + 110 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 3 T + 82 T^{2} + 177 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( 1 - 5 T + 124 T^{2} - 305 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( 1 - T + 67 T^{2} \))(\( 1 + 14 T + 142 T^{2} + 938 T^{3} + 4489 T^{4} \))(\( 1 - 9 T + 48 T^{2} - 603 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 7 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 + 5 T + 110 T^{2} + 355 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( 1 + 78 T^{2} + 5329 T^{4} \))
$79$ (\( 1 - T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 15 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 15 T + 122 T^{2} + 1185 T^{3} + 6241 T^{4} \))(\( 1 - 15 T + 210 T^{2} - 1185 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 10 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 - 2 T + 126 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 + 18 T + 230 T^{2} + 1494 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + T + 89 T^{2} \))(\( 1 + 4 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 3 T + 88 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + T - 30 T^{2} + 89 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 13 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 - 5 T + 97 T^{2} \))(\( 1 - 19 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 + 3 T + 104 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 + 41 T^{2} + 9409 T^{4} \))
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