Properties

Label 448.2.a
Level 448
Weight 2
Character orbit a
Rep. character \(\chi_{448}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 10
Sturm bound 128
Trace bound 7

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Defining parameters

Level: \( N \) = \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 448.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(128\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(448))\).

Total New Old
Modular forms 76 12 64
Cusp forms 53 12 41
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.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(5\)
Minus space\(-\)\(7\)

Trace form

\( 12q + 12q^{9} + O(q^{10}) \) \( 12q + 12q^{9} - 8q^{17} + 4q^{25} - 8q^{29} - 16q^{33} - 8q^{37} - 24q^{41} + 48q^{45} + 12q^{49} + 40q^{53} - 16q^{57} - 32q^{61} + 16q^{69} + 8q^{73} - 8q^{77} - 4q^{81} - 32q^{85} + 8q^{89} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(448))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
448.2.a.a \(1\) \(3.577\) \(\Q\) None \(0\) \(-2\) \(0\) \(-1\) \(-\) \(+\) \(q-2q^{3}-q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
448.2.a.b \(1\) \(3.577\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(-\) \(q-2q^{3}+q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
448.2.a.c \(1\) \(3.577\) \(\Q\) None \(0\) \(-2\) \(4\) \(1\) \(+\) \(-\) \(q-2q^{3}+4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)
448.2.a.d \(1\) \(3.577\) \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) \(+\) \(+\) \(q-2q^{5}-q^{7}-3q^{9}+4q^{11}-2q^{13}+\cdots\)
448.2.a.e \(1\) \(3.577\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(-\) \(-\) \(q-2q^{5}+q^{7}-3q^{9}-4q^{11}-2q^{13}+\cdots\)
448.2.a.f \(1\) \(3.577\) \(\Q\) None \(0\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(q+2q^{3}-q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
448.2.a.g \(1\) \(3.577\) \(\Q\) None \(0\) \(2\) \(0\) \(1\) \(+\) \(-\) \(q+2q^{3}+q^{7}+q^{9}+4q^{13}+6q^{17}+\cdots\)
448.2.a.h \(1\) \(3.577\) \(\Q\) None \(0\) \(2\) \(4\) \(-1\) \(-\) \(+\) \(q+2q^{3}+4q^{5}-q^{7}+q^{9}+8q^{15}+\cdots\)
448.2.a.i \(2\) \(3.577\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(-2\) \(+\) \(+\) \(q+(-1-\beta )q^{3}+(-1+\beta )q^{5}-q^{7}+\cdots\)
448.2.a.j \(2\) \(3.577\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(2\) \(+\) \(-\) \(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(448))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(448)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 + 14 T^{2} + 289 T^{4} \))(\( 1 + 14 T^{2} + 289 T^{4} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - 6 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 + 2 T + 19 T^{2} \))(\( 1 - 2 T + 34 T^{2} - 38 T^{3} + 361 T^{4} \))(\( 1 + 2 T + 34 T^{2} + 38 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 38 T^{2} + 841 T^{4} \))(\( 1 + 38 T^{2} + 841 T^{4} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 - 4 T + 46 T^{2} - 124 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 10 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 8 T + 78 T^{2} + 328 T^{3} + 1681 T^{4} \))(\( 1 + 8 T + 78 T^{2} + 328 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 4 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 12 T + 110 T^{2} + 564 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 110 T^{2} - 564 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 6 T + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))(\( 1 + 14 T + 162 T^{2} + 826 T^{3} + 3481 T^{4} \))(\( 1 - 14 T + 162 T^{2} - 826 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 4 T + 61 T^{2} \))(\( 1 + 18 T + 198 T^{2} + 1098 T^{3} + 3721 T^{4} \))(\( 1 + 18 T + 198 T^{2} + 1098 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 12 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4} \))(\( 1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 - 12 T + 102 T^{2} - 876 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 102 T^{2} - 876 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 16 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 8 T + 83 T^{2} \))(\( 1 - 8 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 + 14 T + 210 T^{2} + 1162 T^{3} + 6889 T^{4} \))(\( 1 - 14 T + 210 T^{2} - 1162 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))(\( 1 - 16 T + 238 T^{2} - 1552 T^{3} + 9409 T^{4} \))
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