Properties

Label 175.2.a
Level 175
Weight 2
Character orbit a
Rep. character \(\chi_{175}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 6
Sturm bound 40
Trace bound 3

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Defining parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(40\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(175))\).

Total New Old
Modular forms 26 9 17
Cusp forms 15 9 6
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(7\)

Trace form

\( 9q + q^{2} + 5q^{4} + 4q^{6} + q^{7} + 9q^{8} + 9q^{9} + O(q^{10}) \) \( 9q + q^{2} + 5q^{4} + 4q^{6} + q^{7} + 9q^{8} + 9q^{9} - 4q^{12} - 10q^{13} - q^{14} - 7q^{16} + 2q^{17} - 7q^{18} - 4q^{19} - 4q^{21} - 8q^{22} + 8q^{23} - 24q^{24} - 2q^{26} + 12q^{27} + 7q^{28} + 14q^{29} - 12q^{31} + 9q^{32} + 12q^{33} - 2q^{34} - 31q^{36} - 14q^{37} - 20q^{38} - 20q^{39} + 22q^{41} - 8q^{42} - 6q^{44} + 6q^{46} - 4q^{47} - 28q^{48} + 9q^{49} - 4q^{51} + 6q^{52} - 10q^{53} - 8q^{54} + 9q^{56} + 12q^{57} + 26q^{58} + 32q^{59} - 14q^{61} + 5q^{63} - 9q^{64} + 10q^{68} - 12q^{69} + 8q^{71} + 5q^{72} + 6q^{73} + 8q^{74} - 16q^{76} + 4q^{77} + 20q^{78} - 20q^{79} + 33q^{81} + 18q^{82} - 20q^{83} + 14q^{86} - 28q^{87} - 4q^{88} + 54q^{89} - 6q^{91} - 24q^{92} + 4q^{93} + 12q^{94} + 8q^{96} + 10q^{97} + q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(175))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
175.2.a.a \(1\) \(1.397\) \(\Q\) None \(-2\) \(-1\) \(0\) \(1\) \(-\) \(-\) \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{7}-2q^{9}+\cdots\)
175.2.a.b \(1\) \(1.397\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(q-q^{3}-2q^{4}-q^{7}-2q^{9}-3q^{11}+\cdots\)
175.2.a.c \(1\) \(1.397\) \(\Q\) None \(2\) \(1\) \(0\) \(-1\) \(-\) \(+\) \(q+2q^{2}+q^{3}+2q^{4}+2q^{6}-q^{7}-2q^{9}+\cdots\)
175.2.a.d \(2\) \(1.397\) \(\Q(\sqrt{5}) \) None \(-1\) \(2\) \(0\) \(-2\) \(-\) \(+\) \(q-\beta q^{2}+(2-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
175.2.a.e \(2\) \(1.397\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(0\) \(2\) \(+\) \(-\) \(q+\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
175.2.a.f \(2\) \(1.397\) \(\Q(\sqrt{17}) \) None \(1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(q+\beta q^{2}+(1-\beta )q^{3}+(2+\beta )q^{4}-4q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(175)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T - 2 T^{3} + 4 T^{4} \))
$3$ (\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4} \))(\( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ 1
$7$ (\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 + 3 T + 11 T^{2} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 5 T + 28 T^{2} + 65 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 7 T + 17 T^{2} \))(\( 1 + 3 T + 17 T^{2} \))(\( 1 - 7 T + 17 T^{2} \))(\( 1 - 4 T + 18 T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 + 18 T^{2} + 361 T^{4} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \))(\( ( 1 + 31 T^{2} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 3 T + 37 T^{2} )^{2} \))(\( ( 1 - 3 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 + 12 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4} \))(\( 1 - 14 T + 126 T^{2} - 574 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 10 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 + 10 T + 94 T^{2} + 430 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 3 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( ( 1 - 2 T + 47 T^{2} )^{2} \))(\( ( 1 + 2 T + 47 T^{2} )^{2} \))(\( 1 - 5 T + 62 T^{2} - 235 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 - 2 T + 90 T^{2} - 106 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 10 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 10 T + 59 T^{2} \))(\( 1 - 10 T + 98 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( 1 - 10 T + 98 T^{2} - 590 T^{3} + 3481 T^{4} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 + 6 T + 86 T^{2} + 366 T^{3} + 3721 T^{4} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 - 4 T + 133 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( 1 - 22 T + 262 T^{2} - 1606 T^{3} + 5329 T^{4} \))(\( 1 + 22 T + 262 T^{2} + 1606 T^{3} + 5329 T^{4} \))(\( 1 - 8 T + 94 T^{2} - 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 5 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 + 5 T + 79 T^{2} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( 1 + 33 T^{2} + 6241 T^{4} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 2 T + 122 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 + 2 T + 122 T^{2} + 166 T^{3} + 6889 T^{4} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 + 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 30 T + 398 T^{2} - 2670 T^{3} + 7921 T^{4} \))(\( 1 - 30 T + 398 T^{2} - 2670 T^{3} + 7921 T^{4} \))(\( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 7 T + 97 T^{2} \))(\( 1 - T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 198 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( 1 - 9 T + 108 T^{2} - 873 T^{3} + 9409 T^{4} \))
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