Properties

Label 336.2.a
Level 336
Weight 2
Character orbit a
Rep. character \(\chi_{336}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 6
Sturm bound 128
Trace bound 5

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Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(336))\).

Total New Old
Modular forms 76 6 70
Cusp forms 53 6 47
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.

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(5\)

Trace form

\( 6q + 4q^{5} + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 4q^{5} + 2q^{7} + 6q^{9} + 4q^{11} + 4q^{13} + 4q^{15} - 4q^{17} + 8q^{19} + 4q^{23} + 2q^{25} - 4q^{29} + 8q^{31} - 4q^{37} + 8q^{39} - 4q^{41} + 8q^{43} + 4q^{45} - 24q^{47} + 6q^{49} - 4q^{51} - 4q^{53} - 8q^{55} - 24q^{59} - 12q^{61} + 2q^{63} - 24q^{65} - 32q^{67} - 16q^{69} - 20q^{71} - 20q^{73} + 16q^{75} - 8q^{77} + 8q^{79} + 6q^{81} + 16q^{83} - 24q^{87} - 4q^{89} - 12q^{91} + 8q^{93} + 16q^{95} + 12q^{97} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(336))\) 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
336.2.a.a \(1\) \(2.683\) \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) \(-\) \(+\) \(-\) \(q-q^{3}-2q^{5}+q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
336.2.a.b \(1\) \(2.683\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q-q^{3}-q^{7}+q^{9}+6q^{11}+2q^{13}+\cdots\)
336.2.a.c \(1\) \(2.683\) \(\Q\) None \(0\) \(-1\) \(2\) \(1\) \(+\) \(+\) \(-\) \(q-q^{3}+2q^{5}+q^{7}+q^{9}-2q^{13}-2q^{15}+\cdots\)
336.2.a.d \(1\) \(2.683\) \(\Q\) None \(0\) \(1\) \(-2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}-2q^{5}+q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
336.2.a.e \(1\) \(2.683\) \(\Q\) None \(0\) \(1\) \(2\) \(-1\) \(+\) \(-\) \(+\) \(q+q^{3}+2q^{5}-q^{7}+q^{9}+6q^{13}+2q^{15}+\cdots\)
336.2.a.f \(1\) \(2.683\) \(\Q\) None \(0\) \(1\) \(4\) \(1\) \(-\) \(-\) \(-\) \(q+q^{3}+4q^{5}+q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(336))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(336)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 10 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 8 T + 59 T^{2} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 4 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 12 T + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 12 T + 79 T^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 83 T^{2} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))
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