Properties

Label 170.2.c
Level $170$
Weight $2$
Character orbit 170.c
Rep. character $\chi_{170}(69,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $54$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(54\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 32 8 24
Cusp forms 24 8 16
Eisenstein series 8 0 8

Trace form

\( 8q - 8q^{4} + 4q^{5} - 12q^{9} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{5} - 12q^{9} - 4q^{10} + 8q^{15} + 8q^{16} + 12q^{19} - 4q^{20} - 24q^{21} + 4q^{25} - 4q^{26} - 24q^{29} + 20q^{30} + 16q^{31} - 4q^{34} + 20q^{35} + 12q^{36} - 40q^{39} + 4q^{40} + 32q^{41} + 12q^{45} - 8q^{46} + 16q^{49} - 12q^{50} + 4q^{51} - 24q^{54} - 8q^{55} - 28q^{59} - 8q^{60} - 24q^{61} - 8q^{64} + 16q^{66} + 48q^{69} + 4q^{70} - 32q^{71} + 8q^{75} - 12q^{76} - 24q^{79} + 4q^{80} + 40q^{81} + 24q^{84} + 4q^{85} + 16q^{86} + 20q^{89} + 20q^{90} - 24q^{91} + 36q^{94} - 56q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
170.2.c.a \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}+iq^{3}-q^{4}+(1+2i)q^{5}-q^{6}+\cdots\)
170.2.c.b \(6\) \(1.357\) 6.0.5161984.1 None \(0\) \(0\) \(2\) \(0\) \(q+\beta _{4}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{3} \))
$3$ (\( 1 - 5 T^{2} + 9 T^{4} \))(\( 1 - T^{2} + 3 T^{4} - 38 T^{6} + 27 T^{8} - 81 T^{10} + 729 T^{12} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T - 3 T^{2} + 24 T^{3} - 15 T^{4} - 50 T^{5} + 125 T^{6} \))
$7$ (\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 275 T^{4} - 2316 T^{6} + 13475 T^{8} - 52822 T^{10} + 117649 T^{12} \))
$11$ (\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{6} \))
$13$ (\( 1 - 17 T^{2} + 169 T^{4} \))(\( 1 - 61 T^{2} + 1723 T^{4} - 28558 T^{6} + 291187 T^{8} - 1742221 T^{10} + 4826809 T^{12} \))
$17$ (\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{3} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )^{2} \))(\( ( 1 + T + 33 T^{2} + 58 T^{3} + 627 T^{4} + 361 T^{5} + 6859 T^{6} )^{2} \))
$23$ (\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 70 T^{2} + 2355 T^{4} - 57580 T^{6} + 1245795 T^{8} - 19588870 T^{10} + 148035889 T^{12} \))
$29$ (\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 + 17 T + 153 T^{2} + 936 T^{3} + 4437 T^{4} + 14297 T^{5} + 24389 T^{6} )^{2} \))
$31$ (\( ( 1 - 5 T + 31 T^{2} )^{2} \))(\( ( 1 - 3 T + 47 T^{2} - 260 T^{3} + 1457 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 10 T + 93 T^{2} - 472 T^{3} + 3441 T^{4} - 13690 T^{5} + 50653 T^{6} )( 1 + 10 T + 93 T^{2} + 472 T^{3} + 3441 T^{4} + 13690 T^{5} + 50653 T^{6} ) \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 16 T + 183 T^{2} - 1296 T^{3} + 7503 T^{4} - 26896 T^{5} + 68921 T^{6} )^{2} \))
$43$ (\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 186 T^{2} + 16391 T^{4} - 879788 T^{6} + 30306959 T^{8} - 635896986 T^{10} + 6321363049 T^{12} \))
$47$ (\( 1 - 85 T^{2} + 2209 T^{4} \))(\( 1 - 25 T^{2} + 4995 T^{4} - 125830 T^{6} + 11033955 T^{8} - 121992025 T^{10} + 10779215329 T^{12} \))
$53$ (\( 1 - 25 T^{2} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 7995 T^{4} - 263774 T^{6} + 22457955 T^{8} - 418195493 T^{10} + 22164361129 T^{12} \))
$59$ (\( ( 1 + 5 T + 59 T^{2} )^{2} \))(\( ( 1 + 9 T + 161 T^{2} + 902 T^{3} + 9499 T^{4} + 31329 T^{5} + 205379 T^{6} )^{2} \))
$61$ (\( ( 1 + 3 T + 61 T^{2} )^{2} \))(\( ( 1 + 9 T + 153 T^{2} + 1064 T^{3} + 9333 T^{4} + 33489 T^{5} + 226981 T^{6} )^{2} \))
$67$ (\( 1 - 130 T^{2} + 4489 T^{4} \))(\( 1 - 270 T^{2} + 34775 T^{4} - 2833220 T^{6} + 156104975 T^{8} - 5440802670 T^{10} + 90458382169 T^{12} \))
$71$ (\( ( 1 + 15 T + 71 T^{2} )^{2} \))(\( ( 1 + T + 95 T^{2} + 476 T^{3} + 6745 T^{4} + 5041 T^{5} + 357911 T^{6} )^{2} \))
$73$ (\( 1 - 25 T^{2} + 5329 T^{4} \))(\( 1 - 421 T^{2} + 75043 T^{4} - 7247278 T^{6} + 399904147 T^{8} - 11955659461 T^{10} + 151334226289 T^{12} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 163 T^{2} + 472 T^{3} + 12877 T^{4} + 24964 T^{5} + 493039 T^{6} )^{2} \))
$83$ (\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 170 T^{2} + 855 T^{4} + 1073140 T^{6} + 5890095 T^{8} - 8067914570 T^{10} + 326940373369 T^{12} \))
$89$ (\( ( 1 - T + 89 T^{2} )^{2} \))(\( ( 1 - 9 T + 251 T^{2} - 1442 T^{3} + 22339 T^{4} - 71289 T^{5} + 704969 T^{6} )^{2} \))
$97$ (\( 1 - 113 T^{2} + 9409 T^{4} \))(\( 1 - 493 T^{2} + 108803 T^{4} - 13645854 T^{6} + 1023727427 T^{8} - 43644935533 T^{10} + 832972004929 T^{12} \))
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