Properties

Label 175.2.f
Level 175
Weight 2
Character orbit f
Rep. character \(\chi_{175}(118,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 20
Newform subspaces 4
Sturm bound 40
Trace bound 11

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Defining parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 52 28 24
Cusp forms 28 20 8
Eisenstein series 24 8 16

Trace form

\( 20q + 4q^{2} - 4q^{7} + 8q^{8} + O(q^{10}) \) \( 20q + 4q^{2} - 4q^{7} + 8q^{8} - 16q^{11} - 48q^{16} - 8q^{18} + 8q^{21} - 4q^{22} - 8q^{23} + 104q^{36} + 24q^{37} - 20q^{42} + 12q^{43} - 52q^{46} - 64q^{51} - 4q^{53} + 68q^{56} - 20q^{57} - 12q^{58} + 8q^{63} + 4q^{67} + 24q^{71} + 16q^{72} + 4q^{77} + 20q^{78} - 44q^{81} - 132q^{86} - 8q^{88} + 48q^{91} - 20q^{93} + 12q^{98} + O(q^{100}) \)

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.f.a \(4\) \(1.397\) \(\Q(i, \sqrt{14})\) \(\Q(\sqrt{-35}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-2\beta _{2}q^{4}-\beta _{3}q^{7}+4\beta _{2}q^{9}+\cdots\)
175.2.f.b \(4\) \(1.397\) \(\Q(i, \sqrt{14})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}-\beta _{1}q^{7}+3\beta _{3}q^{8}+\cdots\)
175.2.f.c \(4\) \(1.397\) \(\Q(i, \sqrt{10})\) None \(4\) \(0\) \(0\) \(-4\) \(q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{6}+\cdots\)
175.2.f.d \(8\) \(1.397\) 8.0.\(\cdots\).1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{4}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + T^{4} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))(\( 1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16} \))
$3$ (\( 1 - 17 T^{4} + 81 T^{8} \))(\( ( 1 + 9 T^{4} )^{2} \))(\( 1 - 17 T^{4} + 81 T^{8} \))(\( ( 1 + 9 T^{4} )^{4} \))
$5$ 1
$7$ (\( 1 + 49 T^{4} \))(\( 1 + 49 T^{4} \))(\( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} \))(\( ( 1 + 49 T^{4} )^{2} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{4} \))(\( ( 1 - 4 T + 11 T^{2} )^{4} \))(\( ( 1 + T + 11 T^{2} )^{4} \))(\( ( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{4} \))
$13$ (\( 1 + 23 T^{4} + 28561 T^{8} \))(\( ( 1 + 169 T^{4} )^{2} \))(\( 1 + 103 T^{4} + 28561 T^{8} \))(\( ( 1 + 169 T^{4} )^{4} \))
$17$ (\( 1 + 263 T^{4} + 83521 T^{8} \))(\( ( 1 + 289 T^{4} )^{2} \))(\( 1 + 263 T^{4} + 83521 T^{8} \))(\( ( 1 + 289 T^{4} )^{4} \))
$19$ (\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} )^{8} \))
$23$ (\( ( 1 + 529 T^{4} )^{2} \))(\( 1 - 734 T^{4} + 279841 T^{8} \))(\( ( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( 1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 23 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 54 T^{2} + 2075 T^{4} + 45414 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 52 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{8} \))
$37$ (\( ( 1 + 1369 T^{4} )^{2} \))(\( 1 - 1294 T^{4} + 1874161 T^{8} \))(\( ( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2} \))(\( 1 + 1294 T^{4} - 199725 T^{8} + 2425164334 T^{12} + 3512479453921 T^{16} \))
$41$ (\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 + 8 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{8} \))
$43$ (\( ( 1 + 1849 T^{4} )^{2} \))(\( 1 - 334 T^{4} + 3418801 T^{8} \))(\( ( 1 - 6 T + 18 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \))(\( 1 + 334 T^{4} - 3307245 T^{8} + 1141879534 T^{12} + 11688200277601 T^{16} \))
$47$ (\( 1 - 3457 T^{4} + 4879681 T^{8} \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( 1 - 2017 T^{4} + 4879681 T^{8} \))(\( ( 1 + 2209 T^{4} )^{4} \))
$53$ (\( ( 1 + 2809 T^{4} )^{2} \))(\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( ( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 28 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 59 T^{2} )^{8} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 82 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} )^{8} \))
$67$ (\( ( 1 + 4489 T^{4} )^{2} \))(\( 1 + 4946 T^{4} + 20151121 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 134 T^{3} + 4489 T^{4} )^{2} \))(\( 1 - 4946 T^{4} + 4311795 T^{8} - 99667444466 T^{12} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 12 T + 71 T^{2} )^{4} \))(\( ( 1 + 16 T + 71 T^{2} )^{4} \))(\( ( 1 + 6 T + 71 T^{2} )^{4} \))(\( ( 1 - 16 T + 185 T^{2} - 1136 T^{3} + 5041 T^{4} )^{4} \))
$73$ (\( 1 - 9502 T^{4} + 28398241 T^{8} \))(\( ( 1 + 5329 T^{4} )^{2} \))(\( ( 1 + 5329 T^{4} )^{2} \))(\( ( 1 + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 157 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 11 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 94 T^{2} + 2595 T^{4} + 586654 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 6382 T^{4} + 47458321 T^{8} \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( 1 + 7538 T^{4} + 47458321 T^{8} \))(\( ( 1 + 6889 T^{4} )^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 138 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{8} \))
$97$ (\( 1 + 3383 T^{4} + 88529281 T^{8} \))(\( ( 1 + 9409 T^{4} )^{2} \))(\( 1 + 16903 T^{4} + 88529281 T^{8} \))(\( ( 1 + 9409 T^{4} )^{4} \))
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