Properties

Label 175.2.b
Level 175
Weight 2
Character orbit b
Rep. character \(\chi_{175}(99,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 3
Sturm bound 40
Trace bound 4

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Defining parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(40\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(175, [\chi])\).

Total New Old
Modular forms 26 10 16
Cusp forms 14 10 4
Eisenstein series 12 0 12

Trace form

\( 10q - 4q^{4} - 8q^{6} - 14q^{9} + O(q^{10}) \) \( 10q - 4q^{4} - 8q^{6} - 14q^{9} + 4q^{11} - 4q^{14} + 8q^{16} + 8q^{19} + 28q^{24} - 28q^{29} - 20q^{31} + 36q^{34} - 20q^{36} + 28q^{39} + 8q^{41} - 6q^{44} - 34q^{46} - 10q^{49} - 4q^{51} - 24q^{54} + 18q^{56} - 4q^{59} + 16q^{61} - 6q^{64} + 8q^{66} + 12q^{69} + 40q^{71} + 6q^{74} - 52q^{76} + 20q^{79} + 18q^{81} + 4q^{84} + 42q^{86} - 48q^{89} - 4q^{91} - 52q^{94} + 20q^{96} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(175, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
175.2.b.a \(2\) \(1.397\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{3}+2q^{4}-iq^{7}+2q^{9}-3q^{11}+\cdots\)
175.2.b.b \(4\) \(1.397\) \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
175.2.b.c \(4\) \(1.397\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2\beta _{1}-2\beta _{3})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(175, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} )^{2} \))(\( 1 + T^{2} + 4 T^{4} + 4 T^{6} + 16 T^{8} \))(\( 1 - 5 T^{2} + 13 T^{4} - 20 T^{6} + 16 T^{8} \))
$3$ (\( 1 - 5 T^{2} + 9 T^{4} \))(\( 1 - 3 T^{2} + 16 T^{4} - 27 T^{6} + 81 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} )( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} ) \))
$5$ 1
$7$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( 1 - T^{2} + 169 T^{4} \))(\( 1 - 31 T^{2} + 472 T^{4} - 5239 T^{6} + 28561 T^{8} \))(\( 1 - 40 T^{2} + 718 T^{4} - 6760 T^{6} + 28561 T^{8} \))
$17$ (\( 1 - 25 T^{2} + 289 T^{4} \))(\( 1 - 47 T^{2} + 1024 T^{4} - 13583 T^{6} + 83521 T^{8} \))(\( 1 - 20 T^{2} + 358 T^{4} - 5780 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 18 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 - 10 T^{2} + 529 T^{4} \))(\( 1 - 56 T^{2} + 1774 T^{4} - 29624 T^{6} + 279841 T^{8} \))(\( 1 - 50 T^{2} + 1363 T^{4} - 26450 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{4} \))(\( ( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 65 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 - 88 T^{2} + 3934 T^{4} - 162712 T^{6} + 3418801 T^{8} \))(\( 1 - 130 T^{2} + 7603 T^{4} - 240370 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 - 13 T^{2} + 2209 T^{4} \))(\( 1 - 99 T^{2} + 5912 T^{4} - 218691 T^{6} + 4879681 T^{8} \))(\( ( 1 - 90 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 176 T^{2} + 13294 T^{4} - 494384 T^{6} + 7890481 T^{8} \))(\( 1 - 140 T^{2} + 9238 T^{4} - 393260 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{4} \))(\( ( 1 + 10 T + 98 T^{2} + 590 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 - 124 T^{2} + 11734 T^{4} - 556636 T^{6} + 20151121 T^{8} \))(\( 1 - 250 T^{2} + 24523 T^{4} - 1122250 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{4} \))(\( ( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 124 T^{2} + 10150 T^{4} - 660796 T^{6} + 28398241 T^{8} \))(\( 1 - 40 T^{2} + 8638 T^{4} - 213160 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 - T + 79 T^{2} )^{2} \))(\( ( 1 - 9 T + 174 T^{2} - 711 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 33 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 150 T^{2} + 6889 T^{4} )^{2} \))(\( 1 - 240 T^{2} + 27998 T^{4} - 1653360 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 - 12 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 193 T^{2} + 9409 T^{4} \))(\( 1 - 135 T^{2} + 14768 T^{4} - 1270215 T^{6} + 88529281 T^{8} \))(\( 1 - 360 T^{2} + 51038 T^{4} - 3387240 T^{6} + 88529281 T^{8} \))
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