Properties

Label 352.2.a
Level $352$
Weight $2$
Character orbit 352.a
Rep. character $\chi_{352}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $8$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(352))\).

Total New Old
Modular forms 56 10 46
Cusp forms 41 10 31
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.

\(2\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(6\)

Trace form

\( 10q + 4q^{5} + 10q^{9} + O(q^{10}) \) \( 10q + 4q^{5} + 10q^{9} - 12q^{13} - 12q^{17} - 2q^{25} + 20q^{29} + 4q^{37} - 12q^{41} + 12q^{45} - 6q^{49} - 20q^{53} + 16q^{57} + 4q^{61} + 8q^{65} - 8q^{69} - 28q^{73} - 6q^{81} + 24q^{85} - 52q^{89} - 40q^{93} - 20q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(352))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11
352.2.a.a \(1\) \(2.811\) \(\Q\) None \(0\) \(-3\) \(1\) \(0\) \(-\) \(-\) \(q-3q^{3}+q^{5}+6q^{9}+q^{11}-6q^{13}+\cdots\)
352.2.a.b \(1\) \(2.811\) \(\Q\) None \(0\) \(-1\) \(-3\) \(4\) \(+\) \(+\) \(q-q^{3}-3q^{5}+4q^{7}-2q^{9}-q^{11}+\cdots\)
352.2.a.c \(1\) \(2.811\) \(\Q\) None \(0\) \(-1\) \(1\) \(-4\) \(+\) \(+\) \(q-q^{3}+q^{5}-4q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
352.2.a.d \(1\) \(2.811\) \(\Q\) None \(0\) \(1\) \(-3\) \(-4\) \(-\) \(-\) \(q+q^{3}-3q^{5}-4q^{7}-2q^{9}+q^{11}+\cdots\)
352.2.a.e \(1\) \(2.811\) \(\Q\) None \(0\) \(1\) \(1\) \(4\) \(+\) \(-\) \(q+q^{3}+q^{5}+4q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
352.2.a.f \(1\) \(2.811\) \(\Q\) None \(0\) \(3\) \(1\) \(0\) \(-\) \(+\) \(q+3q^{3}+q^{5}+6q^{9}-q^{11}-6q^{13}+\cdots\)
352.2.a.g \(2\) \(2.811\) \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(3\) \(0\) \(-\) \(+\) \(q-\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots\)
352.2.a.h \(2\) \(2.811\) \(\Q(\sqrt{17}) \) None \(0\) \(1\) \(3\) \(0\) \(+\) \(-\) \(q+\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(352))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(352)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 - T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - T + 5 T^{2} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))
$11$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))
$17$ (\( 1 + 4 T + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 - 6 T + 26 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 26 T^{2} - 102 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 3 T + 23 T^{2} \))(\( 1 + 5 T + 23 T^{2} \))(\( 1 + 9 T + 23 T^{2} \))(\( 1 - 5 T + 23 T^{2} \))(\( 1 - 9 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 - 3 T + 10 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 3 T + 10 T^{2} + 69 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 - 4 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 6 T + 50 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 6 T + 50 T^{2} - 174 T^{3} + 841 T^{4} \))
$31$ (\( 1 - 9 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 31 T^{2} \))(\( 1 - 15 T + 114 T^{2} - 465 T^{3} + 961 T^{4} \))(\( 1 + 15 T + 114 T^{2} + 465 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 7 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 9 T + 37 T^{2} \))(\( 1 - 7 T + 37 T^{2} \))(\( 1 - T + 36 T^{2} - 37 T^{3} + 1369 T^{4} \))(\( 1 - T + 36 T^{2} - 37 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 14 T^{2} + 1681 T^{4} \))(\( 1 + 14 T^{2} + 1681 T^{4} \))
$43$ (\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 78 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( ( 1 - 4 T + 47 T^{2} )^{2} \))(\( ( 1 + 4 T + 47 T^{2} )^{2} \))
$53$ (\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 + 3 T + 59 T^{2} \))(\( 1 - 5 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 - 13 T + 122 T^{2} - 767 T^{3} + 3481 T^{4} \))(\( 1 + 13 T + 122 T^{2} + 767 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 6 T - 22 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 6 T - 22 T^{2} + 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 15 T + 67 T^{2} \))(\( 1 + 11 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 + 13 T + 67 T^{2} \))(\( 1 + 15 T + 67 T^{2} \))(\( 1 + 3 T + 98 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 3 T + 98 T^{2} - 201 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 3 T + 71 T^{2} \))(\( 1 - 5 T + 71 T^{2} \))(\( 1 - T + 71 T^{2} \))(\( 1 + 5 T + 71 T^{2} \))(\( 1 + T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 19 T + 194 T^{2} + 1349 T^{3} + 5041 T^{4} \))(\( 1 - 19 T + 194 T^{2} - 1349 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 - 2 T + 79 T^{2} \))(\( 1 - 10 T + 79 T^{2} \))(\( 1 + 2 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 6 T + 79 T^{2} \))(\( 1 - 18 T + 222 T^{2} - 1422 T^{3} + 6241 T^{4} \))(\( 1 + 18 T + 222 T^{2} + 1422 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 14 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 2 T + 14 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 14 T^{2} - 166 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 5 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 13 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 13 T + 89 T^{2} \))(\( 1 + 5 T + 89 T^{2} \))(\( 1 + 3 T + 176 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 3 T + 176 T^{2} + 267 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 3 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 - 13 T + 97 T^{2} \))(\( 1 + 19 T + 97 T^{2} \))(\( 1 + 3 T + 97 T^{2} \))(\( 1 + T + 156 T^{2} + 97 T^{3} + 9409 T^{4} \))(\( 1 + T + 156 T^{2} + 97 T^{3} + 9409 T^{4} \))
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