Properties

Label 177.2.a
Level 177
Weight 2
Character orbit a
Rep. character \(\chi_{177}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 4
Sturm bound 40
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(177))\).

Total New Old
Modular forms 22 9 13
Cusp forms 19 9 10
Eisenstein series 3 0 3

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

\(3\)\(59\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(5\)

Trace form

\( 9q - 3q^{2} - q^{3} + 3q^{4} - 6q^{5} - q^{6} - 4q^{7} - 3q^{8} + 9q^{9} + O(q^{10}) \) \( 9q - 3q^{2} - q^{3} + 3q^{4} - 6q^{5} - q^{6} - 4q^{7} - 3q^{8} + 9q^{9} + 10q^{10} + q^{12} - 6q^{13} + 20q^{14} - 2q^{15} + 3q^{16} - 2q^{17} - 3q^{18} - 8q^{19} - 14q^{20} - 8q^{21} - 16q^{22} - 8q^{23} - 9q^{24} - q^{25} + 2q^{26} - q^{27} + 4q^{28} - 2q^{29} + 10q^{30} + 4q^{31} - 19q^{32} + 4q^{33} + 2q^{34} + 16q^{35} + 3q^{36} - 22q^{37} + 32q^{38} + 2q^{39} + 26q^{40} + 2q^{41} - 8q^{42} + 12q^{43} - 6q^{45} - 4q^{46} + 4q^{47} + 17q^{48} + 29q^{49} - 37q^{50} - 2q^{51} - 46q^{52} + 10q^{53} - q^{54} + 4q^{55} + 20q^{56} - 12q^{57} + 18q^{58} + q^{59} - 6q^{60} - 2q^{61} - 24q^{62} - 4q^{63} - 9q^{64} + 8q^{65} - 4q^{66} + 20q^{67} + 2q^{68} + 4q^{69} - 4q^{70} + 36q^{71} - 3q^{72} + 10q^{73} - 26q^{74} + q^{75} - 36q^{76} + 12q^{77} + 26q^{78} - 34q^{80} + 9q^{81} + 54q^{82} - 16q^{84} - 40q^{85} + 4q^{86} - 10q^{87} - 36q^{88} - 30q^{89} + 10q^{90} + 8q^{91} - 4q^{92} - 20q^{93} + 24q^{94} - 12q^{95} - 29q^{96} + 6q^{97} + 17q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(177))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 59
177.2.a.a \(2\) \(1.413\) \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(-6\) \(-7\) \(-\) \(-\) \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}-3q^{5}+\cdots\)
177.2.a.b \(2\) \(1.413\) \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-7\) \(+\) \(+\) \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
177.2.a.c \(2\) \(1.413\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(2\) \(1\) \(-\) \(+\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
177.2.a.d \(3\) \(1.413\) 3.3.229.1 None \(0\) \(-3\) \(-2\) \(9\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(177))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(177)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} - T^{3} + 4 T^{4} + 8 T^{6} \))
$3$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 25 T^{4} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( 1 + 2 T + 10 T^{2} + 18 T^{3} + 50 T^{4} + 50 T^{5} + 125 T^{6} \))
$7$ (\( 1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4} \))(\( 1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4} \))(\( 1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4} \))(\( 1 - 9 T + 44 T^{2} - 142 T^{3} + 308 T^{4} - 441 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 + 17 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 48 T^{3} + 242 T^{4} + 242 T^{5} + 1331 T^{6} \))
$13$ (\( 1 - 19 T^{2} + 169 T^{4} \))(\( 1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( 1 - 4 T + 32 T^{2} - 78 T^{3} + 416 T^{4} - 676 T^{5} + 2197 T^{6} \))
$17$ (\( 1 + 3 T + 35 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T + 25 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - T + 23 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 - 3 T + 8 T^{2} - 4 T^{3} + 136 T^{4} - 867 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 13 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 7 T + 68 T^{2} - 270 T^{3} + 1292 T^{4} - 2527 T^{5} + 6859 T^{6} \))
$23$ (\( 1 + 5 T + 51 T^{2} + 115 T^{3} + 529 T^{4} \))(\( 1 + 7 T + 57 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 3 T + 37 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - T + 42 T^{2} + 18 T^{3} + 966 T^{4} - 529 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 11 T + 87 T^{2} + 319 T^{3} + 841 T^{4} \))(\( 1 - 15 T + 113 T^{2} - 435 T^{3} + 841 T^{4} \))(\( 1 - 5 T + 63 T^{2} - 145 T^{3} + 841 T^{4} \))(\( 1 + 11 T + 96 T^{2} + 564 T^{3} + 2784 T^{4} + 9251 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 7 T + 73 T^{2} + 217 T^{3} + 961 T^{4} \))(\( 1 + T - 39 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + T + 51 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 - 13 T + 130 T^{2} - 778 T^{3} + 4030 T^{4} - 12493 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + T + 43 T^{2} + 37 T^{3} + 1369 T^{4} \))(\( 1 + 7 T + 75 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 + 9 T + 93 T^{2} + 333 T^{3} + 1369 T^{4} \))(\( 1 + 5 T + 92 T^{2} + 384 T^{3} + 3404 T^{4} + 6845 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 - 5 T + 57 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4} \))(\( 1 + T + 84 T^{2} + 156 T^{3} + 3444 T^{4} + 1681 T^{5} + 68921 T^{6} \))
$43$ (\( 1 - 12 T + 77 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 + 4 T + 45 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 2 T + 7 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 38 T^{2} + 76 T^{3} + 1634 T^{4} - 11094 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 3 T + 85 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 + 15 T + 139 T^{2} + 705 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 104 T^{2} - 538 T^{3} + 4888 T^{4} - 24299 T^{5} + 103823 T^{6} \))
$53$ (\( 1 - 2 T + 87 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 117 T^{2} - 424 T^{3} + 2809 T^{4} \))(\( 1 + 2 T + 27 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 - 2 T + 70 T^{2} - 270 T^{3} + 3710 T^{4} - 5618 T^{5} + 148877 T^{6} \))
$59$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$61$ (\( 1 - 13 T + 133 T^{2} - 793 T^{3} + 3721 T^{4} \))(\( 1 + 13 T + 153 T^{2} + 793 T^{3} + 3721 T^{4} \))(\( 1 + T + 111 T^{2} + 61 T^{3} + 3721 T^{4} \))(\( 1 + T + 82 T^{2} + 220 T^{3} + 5002 T^{4} + 3721 T^{5} + 226981 T^{6} \))
$67$ (\( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 8 T + 105 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 - 6 T + 63 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 82 T^{2} - 556 T^{3} + 5494 T^{4} - 44890 T^{5} + 300763 T^{6} \))
$71$ (\( 1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 2 T + 63 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 26 T + 406 T^{2} - 4116 T^{3} + 28826 T^{4} - 131066 T^{5} + 357911 T^{6} \))
$73$ (\( 1 - 5 T + 121 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( 1 + 5 T + 141 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 147 T^{2} - 219 T^{3} + 5329 T^{4} \))(\( 1 - 7 T + 78 T^{2} - 304 T^{3} + 5694 T^{4} - 37303 T^{5} + 389017 T^{6} \))
$79$ (\( ( 1 + 3 T + 79 T^{2} )^{2} \))(\( ( 1 + 3 T + 79 T^{2} )^{2} \))(\( 1 - 10 T + 103 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 2 T + 206 T^{2} - 348 T^{3} + 16274 T^{4} - 12482 T^{5} + 493039 T^{6} \))
$83$ (\( 1 + 9 T + 175 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + T + 165 T^{2} + 83 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 197 T^{2} - 1079 T^{3} + 6889 T^{4} \))(\( 1 + 3 T + 50 T^{2} + 350 T^{3} + 4150 T^{4} + 20667 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - T + 147 T^{2} - 89 T^{3} + 7921 T^{4} \))(\( 1 + 3 T + 29 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 5 T + 153 T^{2} + 445 T^{3} + 7921 T^{4} \))(\( 1 + 23 T + 358 T^{2} + 3816 T^{3} + 31862 T^{4} + 182183 T^{5} + 704969 T^{6} \))
$97$ (\( 1 + 10 T + 199 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( ( 1 - 3 T + 97 T^{2} )^{2} \))(\( 1 + 4 T + 153 T^{2} + 388 T^{3} + 9409 T^{4} \))(\( 1 - 14 T + 266 T^{2} - 2514 T^{3} + 25802 T^{4} - 131726 T^{5} + 912673 T^{6} \))
show more
show less