Properties

Label 350.2.a
Level 350
Weight 2
Character orbit a
Rep. character \(\chi_{350}(1,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 8
Sturm bound 120
Trace bound 3

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(350))\).

Total New Old
Modular forms 72 10 62
Cusp forms 49 10 39
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.

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(8\)

Trace form

\( 10q + 2q^{3} + 10q^{4} - 2q^{6} + 18q^{9} + O(q^{10}) \) \( 10q + 2q^{3} + 10q^{4} - 2q^{6} + 18q^{9} + 2q^{12} + 10q^{13} + 2q^{14} + 10q^{16} - 8q^{17} + 4q^{18} - 2q^{19} + 6q^{21} - 4q^{22} - 2q^{24} - 18q^{26} - 4q^{27} - 16q^{29} + 4q^{31} - 8q^{34} + 18q^{36} + 8q^{37} + 2q^{38} + 24q^{39} - 12q^{41} - 2q^{42} - 12q^{43} + 8q^{46} + 4q^{47} + 2q^{48} + 10q^{49} + 10q^{52} - 4q^{53} - 32q^{54} + 2q^{56} + 4q^{57} - 12q^{58} + 2q^{59} - 6q^{61} - 12q^{62} - 4q^{63} + 10q^{64} - 12q^{66} + 16q^{67} - 8q^{68} - 48q^{69} - 48q^{71} + 4q^{72} - 4q^{73} + 12q^{74} - 2q^{76} + 4q^{77} + 8q^{78} + 32q^{79} - 18q^{81} + 4q^{82} - 2q^{83} + 6q^{84} - 20q^{86} - 12q^{87} - 4q^{88} - 88q^{89} + 10q^{91} - 8q^{93} - 4q^{94} - 2q^{96} + 8q^{97} - 84q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7
350.2.a.a \(1\) \(2.795\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.b \(1\) \(2.795\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
350.2.a.c \(1\) \(2.795\) \(\Q\) None \(-1\) \(3\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots\)
350.2.a.d \(1\) \(2.795\) \(\Q\) None \(1\) \(-3\) \(0\) \(-1\) \(-\) \(-\) \(+\) \(q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.e \(1\) \(2.795\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
350.2.a.f \(1\) \(2.795\) \(\Q\) None \(1\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.g \(2\) \(2.795\) \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.h \(2\) \(2.795\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + 9 T^{4} \))(\( 1 + 9 T^{4} \))
$5$ 1
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 2 T^{2} + 121 T^{4} \))(\( 1 - 2 T^{2} + 121 T^{4} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 24 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - T + 17 T^{2} \))(\( 1 + T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 7 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 + 3 T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 8 T + 48 T^{2} - 152 T^{3} + 361 T^{4} \))(\( 1 - 8 T + 48 T^{2} - 152 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 4 T + 26 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 26 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} \))(\( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 8 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 9 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 11 T + 41 T^{2} \))(\( 1 - 11 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 78 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 78 T^{2} + 344 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 6 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 2 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 8 T + 86 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 86 T^{2} + 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 - 4 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 118 T^{2} - 636 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 7 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 + 9 T + 67 T^{2} \))(\( 1 - 9 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 8 T + 67 T^{2} )^{2} \))(\( ( 1 - 8 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 + 10 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 5 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 5 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 + 4 T + 126 T^{2} + 292 T^{3} + 5329 T^{4} \))(\( 1 - 4 T + 126 T^{2} - 292 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 14 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 - 14 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 4 T + 138 T^{2} - 316 T^{3} + 6241 T^{4} \))(\( 1 - 4 T + 138 T^{2} - 316 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 9 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 + 11 T + 83 T^{2} \))(\( 1 - 11 T + 83 T^{2} \))(\( 1 + 9 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 160 T^{2} + 6889 T^{4} \))(\( 1 + 160 T^{2} + 6889 T^{4} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 11 T + 89 T^{2} \))(\( 1 + 11 T + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 12 T + 134 T^{2} - 1164 T^{3} + 9409 T^{4} \))(\( 1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4} \))
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