Properties

Label 75.2.b
Level 75
Weight 2
Character orbit b
Rep. character \(\chi_{75}(49,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 20
Trace bound 4

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 16 4 12
Cusp forms 4 4 0
Eisenstein series 12 0 12

Trace form

\( 4q - 2q^{4} - 2q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{4} - 2q^{6} - 4q^{9} - 4q^{11} + 12q^{14} - 10q^{16} + 2q^{19} + 6q^{21} + 6q^{24} + 8q^{26} - 16q^{29} - 6q^{31} - 4q^{34} + 2q^{36} - 2q^{39} + 4q^{41} - 16q^{44} + 24q^{46} + 10q^{49} - 8q^{51} + 2q^{54} + 28q^{59} + 10q^{61} + 2q^{64} - 16q^{66} + 12q^{69} - 32q^{71} - 28q^{74} - 28q^{76} + 4q^{81} - 12q^{84} - 4q^{86} + 12q^{89} - 6q^{91} + 8q^{94} + 26q^{96} + 4q^{99} + O(q^{100}) \)

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.b.a \(2\) \(0.599\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}+iq^{3}-2q^{4}-2q^{6}-3iq^{7}+\cdots\)
75.2.b.b \(2\) \(0.599\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}-iq^{3}+q^{4}+q^{6}+3iq^{8}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \))(\( 1 - 3 T^{2} + 4 T^{4} \))
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))
$5$ 1
$7$ (\( 1 - 5 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))
$19$ (\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 10 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))
$29$ (\( ( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))
$41$ (\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 85 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 90 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))
$53$ (\( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 - 6 T^{2} + 2809 T^{4} \))
$59$ (\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))
$61$ (\( ( 1 - 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 125 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))
$71$ (\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))
$73$ (\( 1 + 50 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))
$79$ (\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))
$97$ (\( 1 + 95 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))
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