Properties

Label 175.2.o
Level 175
Weight 2
Character orbit o
Rep. character \(\chi_{175}(68,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 40
Newform subspaces 4
Sturm bound 40
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.o (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(175, [\chi])\).

Total New Old
Modular forms 104 56 48
Cusp forms 56 40 16
Eisenstein series 48 16 32

Trace form

\( 40q + 2q^{2} + 6q^{3} + 10q^{7} + 4q^{8} + O(q^{10}) \) \( 40q + 2q^{2} + 6q^{3} + 10q^{7} + 4q^{8} - 20q^{11} - 6q^{12} - 12q^{16} - 12q^{17} - 4q^{18} - 44q^{21} - 8q^{22} - 10q^{23} + 12q^{26} - 18q^{28} + 12q^{31} + 18q^{32} - 32q^{36} + 12q^{38} + 26q^{42} + 12q^{43} - 8q^{46} - 24q^{47} + 4q^{51} - 24q^{52} - 20q^{53} - 104q^{56} - 16q^{57} + 6q^{58} + 72q^{61} + 4q^{63} + 204q^{66} + 14q^{67} + 12q^{68} - 96q^{71} + 8q^{72} + 24q^{73} + 20q^{77} - 32q^{78} - 64q^{81} + 6q^{82} - 24q^{86} - 18q^{87} + 8q^{88} + 36q^{91} + 36q^{92} - 4q^{93} + 60q^{96} - 18q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(175, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
175.2.o.a \(4\) \(1.397\) \(\Q(\zeta_{12})\) None \(-2\) \(4\) \(0\) \(10\) \(q+(-1+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(1+\zeta_{12}+\cdots)q^{3}+\cdots\)
175.2.o.b \(4\) \(1.397\) \(\Q(\zeta_{12})\) None \(4\) \(2\) \(0\) \(0\) \(q+(1-\zeta_{12})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots\)
175.2.o.c \(8\) \(1.397\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{2}+(-2\zeta_{24}+\zeta_{24}^{5})q^{3}+\cdots\)
175.2.o.d \(24\) \(1.397\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(175, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8} \))(\( 1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16} \))
$3$ (\( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 12 T^{5} + 45 T^{6} - 108 T^{7} + 81 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 30 T^{5} + 45 T^{6} - 54 T^{7} + 81 T^{8} \))(\( ( 1 + 9 T^{4} )^{2}( 1 - 9 T^{4} + 81 T^{8} ) \))
$5$ 1
$7$ (\( ( 1 - 5 T + 7 T^{2} )^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 23 T^{4} + 2401 T^{8} \))
$11$ (\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( ( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 142 T^{4} + 28561 T^{8} )^{2} \))
$17$ (\( 1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 884 T^{5} + 5780 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 1700 T^{5} + 5780 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 94 T^{4} - 74685 T^{8} + 7850974 T^{12} + 6975757441 T^{16} \))
$19$ (\( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 11 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 5198 T^{5} + 28037 T^{6} + 170338 T^{7} + 279841 T^{8} \))(\( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 5612 T^{5} + 28037 T^{6} - 48668 T^{7} + 279841 T^{8} \))(\( 1 + 1049 T^{4} + 820560 T^{8} + 293553209 T^{12} + 78310985281 T^{16} \))
$29$ (\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 57 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( ( 1 - 11 T + 31 T^{2} )^{4}( 1 - 7 T + 31 T^{2} )^{4} \))
$37$ (\( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 626 T^{4} - 1482285 T^{8} - 1173224786 T^{12} + 3512479453921 T^{16} \))
$41$ (\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))(\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))(\( ( 1 - 7 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 2580 T^{5} + 33282 T^{6} - 477042 T^{7} + 3418801 T^{8} \))(\( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 2580 T^{5} + 33282 T^{6} - 477042 T^{7} + 3418801 T^{8} \))(\( ( 1 + 23 T^{4} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 31584 T^{5} + 198810 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 24816 T^{5} + 198810 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 2306 T^{4} + 437955 T^{8} - 11252544386 T^{12} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))(\( 1 - 4786 T^{4} + 15015315 T^{8} - 37763842066 T^{12} + 62259690411361 T^{16} \))
$59$ (\( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( ( 1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( ( 1 - 3 T + 64 T^{2} - 183 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( 1 + 8 T + 137 T^{2} + 1224 T^{3} + 11492 T^{4} + 82008 T^{5} + 614993 T^{6} + 2406104 T^{7} + 20151121 T^{8} \))(\( 1 - 22 T + 137 T^{2} + 834 T^{3} - 16648 T^{4} + 55878 T^{5} + 614993 T^{6} - 6616786 T^{7} + 20151121 T^{8} \))(\( 1 + 6169 T^{4} + 17905440 T^{8} + 124312265449 T^{12} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 2 T + 71 T^{2} )^{8} \))
$73$ (\( 1 + 144 T^{2} - 600 T^{3} + 10991 T^{4} - 43800 T^{5} + 767376 T^{6} + 28398241 T^{8} \))(\( 1 - 24 T + 144 T^{2} + 1752 T^{3} - 31057 T^{4} + 127896 T^{5} + 767376 T^{6} - 9336408 T^{7} + 28398241 T^{8} \))(\( 1 + 9214 T^{4} + 56499555 T^{8} + 261661392574 T^{12} + 806460091894081 T^{16} \))
$79$ (\( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 64464 T^{5} + 923668 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( ( 1 + 154 T^{2} + 17475 T^{4} + 961114 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))(\( ( 1 + 5543 T^{4} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} ) \))(\( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} ) \))(\( ( 1 - 31 T^{2} - 6960 T^{4} - 245551 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))(\( ( 1 + 2498 T^{4} + 88529281 T^{8} )^{2} \))
show more
show less