Properties

Label 98.2.a
Level 98
Weight 2
Character orbit a
Rep. character \(\chi_{98}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 28
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 98.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(98))\).

Total New Old
Modular forms 22 3 19
Cusp forms 7 3 4
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.

\(2\)\(7\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{6} + q^{8} - q^{9} + O(q^{10}) \) \( 3q + q^{2} + 2q^{3} + 3q^{4} - 2q^{6} + q^{8} - q^{9} - 4q^{11} + 2q^{12} + 4q^{13} - 8q^{15} + 3q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 4q^{22} - 8q^{23} - 2q^{24} + q^{25} - 4q^{26} - 4q^{27} - 2q^{29} - 8q^{30} + 4q^{31} + q^{32} + 6q^{34} - q^{36} + 22q^{37} + 2q^{38} + 8q^{39} - 6q^{41} + 12q^{43} - 4q^{44} - 8q^{46} + 12q^{47} + 2q^{48} + 11q^{50} - 8q^{51} + 4q^{52} + 2q^{53} + 4q^{54} + 16q^{57} + 10q^{58} + 6q^{59} - 8q^{60} - 8q^{61} - 4q^{62} + 3q^{64} + 20q^{67} - 6q^{68} - 24q^{71} - 3q^{72} - 2q^{73} + 18q^{74} - 10q^{75} - 2q^{76} - 8q^{78} - 21q^{81} + 6q^{82} + 6q^{83} - 8q^{85} - 4q^{86} - 12q^{87} - 4q^{88} + 6q^{89} - 8q^{92} - 16q^{93} - 12q^{94} - 40q^{95} - 2q^{96} + 10q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(98))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
98.2.a.a \(1\) \(0.783\) \(\Q\) None \(-1\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{8}+q^{9}+\cdots\)
98.2.a.b \(2\) \(0.783\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-2\beta q^{5}+\beta q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(98))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(98)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 - 2 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))
$19$ (\( 1 + 2 T + 19 T^{2} \))(\( 1 - 12 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 10 T^{2} + 961 T^{4} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 10 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 16 T^{2} + 1681 T^{4} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + 2 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 116 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 + 114 T^{2} + 3721 T^{4} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 144 T^{2} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 68 T^{2} + 6889 T^{4} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 96 T^{2} + 9409 T^{4} \))
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