Properties

Label 170.2.a
Level $170$
Weight $2$
Character orbit 170.a
Rep. character $\chi_{170}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $6$
Sturm bound $54$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(54\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(170))\).

Total New Old
Modular forms 30 7 23
Cusp forms 23 7 16
Eisenstein series 7 0 7

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

\(2\)\(5\)\(17\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(6\)

Trace form

\( 7q - q^{2} + 7q^{4} + q^{5} + 8q^{7} - q^{8} + 7q^{9} + O(q^{10}) \) \( 7q - q^{2} + 7q^{4} + q^{5} + 8q^{7} - q^{8} + 7q^{9} + q^{10} - 8q^{11} + 2q^{13} - 4q^{15} + 7q^{16} + 3q^{17} - 5q^{18} + q^{20} - 8q^{21} - 8q^{22} - 16q^{23} + 7q^{25} + 6q^{26} + 8q^{28} - 2q^{29} - 8q^{31} - q^{32} - 16q^{33} - q^{34} - 4q^{35} + 7q^{36} - 2q^{37} - 4q^{38} - 8q^{39} + q^{40} - 18q^{41} - 24q^{42} - 4q^{43} - 8q^{44} - 3q^{45} + 7q^{49} - q^{50} - 4q^{51} + 2q^{52} - 22q^{53} - 24q^{54} - 12q^{55} + 16q^{57} - 2q^{58} + 8q^{59} - 4q^{60} + 22q^{61} + 24q^{63} + 7q^{64} + 6q^{65} + 24q^{66} + 20q^{67} + 3q^{68} + 16q^{69} + 4q^{70} + 8q^{71} - 5q^{72} + 38q^{73} + 6q^{74} - 16q^{78} - 8q^{79} + q^{80} - 25q^{81} + 14q^{82} + 12q^{83} - 8q^{84} + q^{85} - 4q^{86} + 40q^{87} - 8q^{88} + 2q^{89} + 13q^{90} + 40q^{91} - 16q^{92} + 8q^{93} + 12q^{94} - 20q^{95} - 10q^{97} + 31q^{98} - 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 17
170.2.a.a \(1\) \(1.357\) \(\Q\) None \(-1\) \(-2\) \(-1\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+2q^{7}+\cdots\)
170.2.a.b \(1\) \(1.357\) \(\Q\) None \(-1\) \(-2\) \(1\) \(-2\) \(+\) \(-\) \(-\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots\)
170.2.a.c \(1\) \(1.357\) \(\Q\) None \(-1\) \(1\) \(1\) \(2\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
170.2.a.d \(1\) \(1.357\) \(\Q\) None \(-1\) \(3\) \(-1\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+2q^{7}+\cdots\)
170.2.a.e \(1\) \(1.357\) \(\Q\) None \(1\) \(1\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
170.2.a.f \(2\) \(1.357\) \(\Q(\sqrt{17}) \) None \(2\) \(-1\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}-\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(170))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(170)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \))
$11$ (\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 6 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))
$17$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))
$19$ (\( 1 - 8 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 3 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + T + 34 T^{2} + 19 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 9 T + 29 T^{2} \))(\( 1 + 3 T + 29 T^{2} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 9 T + 78 T^{2} + 279 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 8 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 2 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 10 T + 43 T^{2} \))(\( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 12 T + 47 T^{2} \))(\( 1 - 4 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 + 13 T + 47 T^{2} \))(\( 1 + 3 T + 47 T^{2} \))(\( 1 - 9 T + 110 T^{2} - 423 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 3 T + 104 T^{2} - 159 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))(\( 1 - 3 T + 59 T^{2} \))(\( 1 + 13 T + 122 T^{2} + 767 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 7 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( 1 - 11 T + 61 T^{2} \))(\( 1 - 19 T + 208 T^{2} - 1159 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 2 T + 67 T^{2} \))(\( 1 - 2 T + 67 T^{2} \))(\( 1 + 10 T + 142 T^{2} + 670 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 21 T + 214 T^{2} + 1491 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 - 7 T + 52 T^{2} - 511 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 + 14 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 + 9 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))(\( 1 - 9 T + 176 T^{2} - 873 T^{3} + 9409 T^{4} \))
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