Properties

Label 128.2.e
Level 128
Weight 2
Character orbit e
Rep. character \(\chi_{128}(33,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 2
Sturm bound 32
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(128, [\chi])\).

Total New Old
Modular forms 48 12 36
Cusp forms 16 4 12
Eisenstein series 32 8 24

Trace form

\( 4q + 4q^{5} + O(q^{10}) \) \( 4q + 4q^{5} + 4q^{13} - 8q^{17} - 8q^{21} - 12q^{29} - 8q^{33} - 12q^{37} - 4q^{45} + 12q^{49} + 20q^{53} + 36q^{61} + 8q^{65} + 24q^{69} - 8q^{77} + 20q^{81} - 8q^{85} - 32q^{93} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(128, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
128.2.e.a \(2\) \(1.022\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(2\) \(0\) \(q+(-1+i)q^{3}+(1+i)q^{5}+2iq^{7}+\cdots\)
128.2.e.b \(2\) \(1.022\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(2\) \(0\) \(q+(1-i)q^{3}+(1+i)q^{5}-2iq^{7}+iq^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(128, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(128, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))
$7$ (\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))
$11$ (\( 1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))
$17$ (\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 10 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))
$31$ (\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 222 T^{3} + 1369 T^{4} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))
$43$ (\( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} \))
$47$ (\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))
$53$ (\( ( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))
$59$ (\( 1 + 6 T + 18 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 18 T + 162 T^{2} - 1098 T^{3} + 3721 T^{4} \))(\( 1 - 18 T + 162 T^{2} - 1098 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 42 T^{2} + 5041 T^{4} \))(\( 1 - 42 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))
$79$ (\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 + 2 T + 2 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 162 T^{2} + 7921 T^{4} \))(\( 1 - 162 T^{2} + 7921 T^{4} \))
$97$ (\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))
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