Properties

Label 196.2.a
Level 196
Weight 2
Character orbit a
Rep. character \(\chi_{196}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 3
Sturm bound 56
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(196))\).

Total New Old
Modular forms 40 4 36
Cusp forms 17 4 13
Eisenstein series 23 0 23

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.
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4q + 6q^{9} + O(q^{10}) \) \( 4q + 6q^{9} + 2q^{11} - 2q^{15} - 2q^{23} + 2q^{25} + 4q^{29} - 18q^{37} - 20q^{39} - 16q^{43} - 2q^{51} + 26q^{53} - 18q^{57} + 24q^{65} - 14q^{67} - 10q^{79} + 4q^{81} + 22q^{85} - 14q^{93} + 2q^{95} + 52q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(196))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
196.2.a.a \(1\) \(1.565\) \(\Q\) None \(0\) \(-1\) \(-3\) \(0\) \(-\) \(-\) \(q-q^{3}-3q^{5}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
196.2.a.b \(1\) \(1.565\) \(\Q\) None \(0\) \(1\) \(3\) \(0\) \(-\) \(+\) \(q+q^{3}+3q^{5}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
196.2.a.c \(2\) \(1.565\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+2\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}-3\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(196))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(196)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))
$19$ (\( 1 - T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - 7 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 32 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))
$53$ (\( 1 - 3 T + 53 T^{2} \))(\( 1 - 3 T + 53 T^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))
$61$ (\( 1 - T + 61 T^{2} \))(\( 1 + T + 61 T^{2} \))(\( 1 + 72 T^{2} + 3721 T^{4} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 - T + 73 T^{2} \))(\( 1 + T + 73 T^{2} \))(\( 1 + 96 T^{2} + 5329 T^{4} \))
$79$ (\( 1 + 13 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 34 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 192 T^{2} + 9409 T^{4} \))
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