Properties

Label 169.2.a
Level 169
Weight 2
Character orbit a
Rep. character \(\chi_{169}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 3
Sturm bound 30
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(30\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(169))\).

Total New Old
Modular forms 21 19 2
Cusp forms 8 8 0
Eisenstein series 13 11 2

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

\(13\)Dim.
\(+\)\(3\)
\(-\)\(5\)

Trace form

\( 8q + 2q^{4} - 4q^{9} + O(q^{10}) \) \( 8q + 2q^{4} - 4q^{9} + 4q^{10} + 4q^{12} - 10q^{14} - 6q^{16} + 2q^{17} + 6q^{22} + 2q^{23} - 14q^{25} - 6q^{27} + 4q^{29} - 14q^{30} + 8q^{35} - 12q^{36} + 12q^{38} + 16q^{42} + 10q^{43} - 18q^{48} - 22q^{49} + 10q^{51} - 4q^{53} + 12q^{55} - 8q^{56} + 10q^{61} + 14q^{62} - 20q^{64} + 10q^{66} - 36q^{68} + 12q^{69} + 14q^{74} + 22q^{75} + 16q^{77} - 2q^{79} - 24q^{81} - 10q^{82} - 24q^{87} + 30q^{88} - 30q^{90} + 12q^{92} + 22q^{94} + 18q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 13
169.2.a.a \(2\) \(1.349\) \(\Q(\sqrt{3}) \) None \(0\) \(4\) \(0\) \(0\) \(-\) \(q+\beta q^{2}+2q^{3}+q^{4}-\beta q^{5}+2\beta q^{6}+\cdots\)
169.2.a.b \(3\) \(1.349\) \(\Q(\zeta_{14})^+\) None \(-2\) \(-2\) \(-4\) \(-3\) \(+\) \(q+(-1-\beta _{2})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
169.2.a.c \(3\) \(1.349\) \(\Q(\zeta_{14})^+\) None \(2\) \(-2\) \(4\) \(3\) \(-\) \(q+(1-\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} + 4 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 - 2 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 8 T^{5} + 8 T^{6} \))
$3$ (\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( 1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6} \))
$5$ (\( 1 + 7 T^{2} + 25 T^{4} \))(\( 1 + 4 T + 18 T^{2} + 39 T^{3} + 90 T^{4} + 100 T^{5} + 125 T^{6} \))(\( 1 - 4 T + 18 T^{2} - 39 T^{3} + 90 T^{4} - 100 T^{5} + 125 T^{6} \))
$7$ (\( ( 1 + 7 T^{2} )^{2} \))(\( 1 + 3 T + 17 T^{2} + 29 T^{3} + 119 T^{4} + 147 T^{5} + 343 T^{6} \))(\( 1 - 3 T + 17 T^{2} - 29 T^{3} + 119 T^{4} - 147 T^{5} + 343 T^{6} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( 1 + 8 T + 52 T^{2} + 189 T^{3} + 572 T^{4} + 968 T^{5} + 1331 T^{6} \))(\( 1 - 8 T + 52 T^{2} - 189 T^{3} + 572 T^{4} - 968 T^{5} + 1331 T^{6} \))
$13$ 1
$17$ (\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( 1 + 2 T + 36 T^{2} + 81 T^{3} + 612 T^{4} + 578 T^{5} + 4913 T^{6} \))(\( 1 + 2 T + 36 T^{2} + 81 T^{3} + 612 T^{4} + 578 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 26 T^{2} + 361 T^{4} \))(\( 1 + 4 T + 46 T^{2} + 151 T^{3} + 874 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 - 4 T + 46 T^{2} - 151 T^{3} + 874 T^{4} - 1444 T^{5} + 6859 T^{6} \))
$23$ (\( ( 1 - 6 T + 23 T^{2} )^{2} \))(\( 1 + 5 T + 68 T^{2} + 217 T^{3} + 1564 T^{4} + 2645 T^{5} + 12167 T^{6} \))(\( 1 + 5 T + 68 T^{2} + 217 T^{3} + 1564 T^{4} + 2645 T^{5} + 12167 T^{6} \))
$29$ (\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( 1 + T + 43 T^{2} + 141 T^{3} + 1247 T^{4} + 841 T^{5} + 24389 T^{6} \))(\( 1 + T + 43 T^{2} + 141 T^{3} + 1247 T^{4} + 841 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 50 T^{2} + 961 T^{4} \))(\( 1 + 5 T + 57 T^{2} + 143 T^{3} + 1767 T^{4} + 4805 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 57 T^{2} - 143 T^{3} + 1767 T^{4} - 4805 T^{5} + 29791 T^{6} \))
$37$ (\( 1 - T^{2} + 1369 T^{4} \))(\( 1 - 12 T + 152 T^{2} - 917 T^{3} + 5624 T^{4} - 16428 T^{5} + 50653 T^{6} \))(\( 1 + 12 T + 152 T^{2} + 917 T^{3} + 5624 T^{4} + 16428 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + 55 T^{2} + 1681 T^{4} \))(\( 1 + 7 T + 74 T^{2} + 623 T^{3} + 3034 T^{4} + 11767 T^{5} + 68921 T^{6} \))(\( 1 - 7 T + 74 T^{2} - 623 T^{3} + 3034 T^{4} - 11767 T^{5} + 68921 T^{6} \))
$43$ (\( ( 1 + 8 T + 43 T^{2} )^{2} \))(\( 1 - 13 T + 169 T^{2} - 1105 T^{3} + 7267 T^{4} - 24037 T^{5} + 79507 T^{6} \))(\( 1 - 13 T + 169 T^{2} - 1105 T^{3} + 7267 T^{4} - 24037 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 82 T^{2} + 2209 T^{4} \))(\( 1 + 18 T + 242 T^{2} + 1859 T^{3} + 11374 T^{4} + 39762 T^{5} + 103823 T^{6} \))(\( 1 - 18 T + 242 T^{2} - 1859 T^{3} + 11374 T^{4} - 39762 T^{5} + 103823 T^{6} \))
$53$ (\( ( 1 + 3 T + 53 T^{2} )^{2} \))(\( 1 - T + 73 T^{2} + 231 T^{3} + 3869 T^{4} - 2809 T^{5} + 148877 T^{6} \))(\( 1 - T + 73 T^{2} + 231 T^{3} + 3869 T^{4} - 2809 T^{5} + 148877 T^{6} \))
$59$ (\( 1 + 70 T^{2} + 3481 T^{4} \))(\( 1 + 19 T + 260 T^{2} + 2243 T^{3} + 15340 T^{4} + 66139 T^{5} + 205379 T^{6} \))(\( 1 - 19 T + 260 T^{2} - 2243 T^{3} + 15340 T^{4} - 66139 T^{5} + 205379 T^{6} \))
$61$ (\( ( 1 - T + 61 T^{2} )^{2} \))(\( 1 - 4 T + 116 T^{2} - 249 T^{3} + 7076 T^{4} - 14884 T^{5} + 226981 T^{6} \))(\( 1 - 4 T + 116 T^{2} - 249 T^{3} + 7076 T^{4} - 14884 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 122 T^{2} + 4489 T^{4} \))(\( 1 + T + 129 T^{2} + 175 T^{3} + 8643 T^{4} + 4489 T^{5} + 300763 T^{6} \))(\( 1 - T + 129 T^{2} - 175 T^{3} + 8643 T^{4} - 4489 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 130 T^{2} + 5041 T^{4} \))(\( 1 + 27 T + 435 T^{2} + 4381 T^{3} + 30885 T^{4} + 136107 T^{5} + 357911 T^{6} \))(\( 1 - 27 T + 435 T^{2} - 4381 T^{3} + 30885 T^{4} - 136107 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 143 T^{2} + 5329 T^{4} \))(\( 1 - 9 T + 99 T^{2} - 403 T^{3} + 7227 T^{4} - 47961 T^{5} + 389017 T^{6} \))(\( 1 + 9 T + 99 T^{2} + 403 T^{3} + 7227 T^{4} + 47961 T^{5} + 389017 T^{6} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )^{2} \))(\( 1 + 5 T + 75 T^{2} + 917 T^{3} + 5925 T^{4} + 31205 T^{5} + 493039 T^{6} \))(\( 1 + 5 T + 75 T^{2} + 917 T^{3} + 5925 T^{4} + 31205 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 26 T^{2} + 6889 T^{4} \))(\( 1 + 7 T + 109 T^{2} + 1365 T^{3} + 9047 T^{4} + 48223 T^{5} + 571787 T^{6} \))(\( 1 - 7 T + 109 T^{2} - 1365 T^{3} + 9047 T^{4} - 48223 T^{5} + 571787 T^{6} \))
$89$ (\( 1 + 130 T^{2} + 7921 T^{4} \))(\( 1 + 11 T + 193 T^{2} + 1677 T^{3} + 17177 T^{4} + 87131 T^{5} + 704969 T^{6} \))(\( 1 - 11 T + 193 T^{2} - 1677 T^{3} + 17177 T^{4} - 87131 T^{5} + 704969 T^{6} \))
$97$ (\( 1 + 146 T^{2} + 9409 T^{4} \))(\( 1 - 7 T + 207 T^{2} - 1057 T^{3} + 20079 T^{4} - 65863 T^{5} + 912673 T^{6} \))(\( 1 + 7 T + 207 T^{2} + 1057 T^{3} + 20079 T^{4} + 65863 T^{5} + 912673 T^{6} \))
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