Properties

Label 294.2.a
Level 294
Weight 2
Character orbit a
Rep. character \(\chi_{294}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 7
Sturm bound 112
Trace bound 5

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Defining parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(112\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(294))\).

Total New Old
Modular forms 72 7 65
Cusp forms 41 7 34
Eisenstein series 31 0 31

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(6\)

Trace form

\( 7q - q^{2} + q^{3} + 7q^{4} + 2q^{5} + q^{6} - q^{8} + 7q^{9} + O(q^{10}) \) \( 7q - q^{2} + q^{3} + 7q^{4} + 2q^{5} + q^{6} - q^{8} + 7q^{9} + 2q^{10} + 4q^{11} + q^{12} - 6q^{13} - 2q^{15} + 7q^{16} - 2q^{17} - q^{18} + 4q^{19} + 2q^{20} + 8q^{22} + q^{24} + 21q^{25} - 6q^{26} + q^{27} + 10q^{29} + 2q^{30} - q^{32} - 4q^{33} - 2q^{34} + 7q^{36} - 14q^{37} + 4q^{38} - 22q^{39} + 2q^{40} + 6q^{41} - 12q^{43} + 4q^{44} + 2q^{45} + q^{48} - 39q^{50} + 6q^{51} - 6q^{52} - 38q^{53} + q^{54} - 8q^{55} - 28q^{57} - 34q^{58} - 4q^{59} - 2q^{60} - 6q^{61} + 7q^{64} - 4q^{65} - 4q^{66} - 12q^{67} - 2q^{68} + 8q^{69} + 16q^{71} - q^{72} - 10q^{73} - 22q^{74} - q^{75} + 4q^{76} + 10q^{78} - 12q^{79} + 2q^{80} + 7q^{81} + 6q^{82} + 4q^{83} - 12q^{85} + 12q^{86} - 2q^{87} + 8q^{88} + 6q^{89} + 2q^{90} - 24q^{93} + 32q^{95} + q^{96} + 14q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(294))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
294.2.a.a \(1\) \(2.348\) \(\Q\) None \(-1\) \(-1\) \(-3\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-3q^{5}+q^{6}-q^{8}+\cdots\)
294.2.a.b \(1\) \(2.348\) \(\Q\) None \(-1\) \(-1\) \(4\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}-q^{8}+\cdots\)
294.2.a.c \(1\) \(2.348\) \(\Q\) None \(-1\) \(1\) \(-4\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}-q^{8}+\cdots\)
294.2.a.d \(1\) \(2.348\) \(\Q\) None \(-1\) \(1\) \(3\) \(0\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-q^{8}+\cdots\)
294.2.a.e \(1\) \(2.348\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
294.2.a.f \(1\) \(2.348\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
294.2.a.g \(1\) \(2.348\) \(\Q\) None \(1\) \(1\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(294))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(294)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))
$7$ 1
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 - 5 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))
$29$ (\( 1 - 9 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))
$53$ (\( 1 + 3 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))
$59$ (\( 1 + 3 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 11 T + 59 T^{2} \))(\( 1 - 11 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 4 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))
$67$ (\( 1 + 10 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 10 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 16 T + 73 T^{2} \))(\( 1 - 16 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 - 3 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))
$83$ (\( 1 - 9 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 + 7 T + 83 T^{2} \))(\( 1 - 7 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 8 T + 89 T^{2} \))(\( 1 + 8 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))
$97$ (\( 1 - T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))
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