Properties

Label 75.2.e
Level 75
Weight 2
Character orbit e
Rep. character \(\chi_{75}(32,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 2
Sturm bound 20
Trace bound 6

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 75.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(75, [\chi])\).

Total New Old
Modular forms 32 16 16
Cusp forms 8 8 0
Eisenstein series 24 8 16

Trace form

\( 8q - 12q^{6} + O(q^{10}) \) \( 8q - 12q^{6} + 4q^{16} - 12q^{21} - 4q^{31} + 36q^{36} - 24q^{46} + 48q^{51} - 44q^{61} - 8q^{76} - 72q^{81} + 36q^{91} - 36q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.2.e.a \(4\) \(0.599\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}-\beta _{3}q^{8}+\cdots\)
75.2.e.b \(4\) \(0.599\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-2\beta _{2}q^{4}+\beta _{3}q^{7}+3\beta _{2}q^{9}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 7 T^{4} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2} \))
$3$ (\( 1 + 9 T^{4} \))(\( 1 + 9 T^{4} \))
$5$ 1
$7$ (\( ( 1 + 49 T^{4} )^{2} \))(\( 1 + 23 T^{4} + 2401 T^{8} \))
$11$ (\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} )^{4} \))
$13$ (\( ( 1 + 169 T^{4} )^{2} \))(\( 1 - 337 T^{4} + 28561 T^{8} \))
$17$ (\( 1 - 382 T^{4} + 83521 T^{8} \))(\( ( 1 + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 98 T^{4} + 279841 T^{8} \))(\( ( 1 + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 8 T + 31 T^{2} )^{4} \))(\( ( 1 - 7 T + 31 T^{2} )^{4} \))
$37$ (\( ( 1 + 1369 T^{4} )^{2} \))(\( 1 - 2062 T^{4} + 1874161 T^{8} \))
$41$ (\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 41 T^{2} )^{4} \))
$43$ (\( ( 1 + 1849 T^{4} )^{2} \))(\( 1 + 23 T^{4} + 3418801 T^{8} \))
$47$ (\( 1 - 4222 T^{4} + 4879681 T^{8} \))(\( ( 1 + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 1778 T^{4} + 7890481 T^{8} \))(\( ( 1 + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))
$61$ (\( ( 1 - 2 T + 61 T^{2} )^{4} \))(\( ( 1 + 13 T + 61 T^{2} )^{4} \))
$67$ (\( ( 1 + 4489 T^{4} )^{2} \))(\( 1 + 2903 T^{4} + 20151121 T^{8} \))
$71$ (\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 5329 T^{4} )^{2} \))(\( 1 - 8542 T^{4} + 28398241 T^{8} \))
$79$ (\( ( 1 + 98 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 9938 T^{4} + 47458321 T^{8} \))(\( ( 1 + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))
$97$ (\( ( 1 + 9409 T^{4} )^{2} \))(\( 1 + 9743 T^{4} + 88529281 T^{8} \))
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