Properties

Label 35.2.k
Level 35
Weight 2
Character orbit k
Rep. character \(\chi_{35}(3,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 8
Newform subspaces 2
Sturm bound 8
Trace bound 2

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Defining parameters

Level: \( N \) = \( 35 = 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 35.k (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(35, [\chi])\).

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

Trace form

\( 8q - 2q^{2} - 6q^{3} - 10q^{7} - 4q^{8} + O(q^{10}) \) \( 8q - 2q^{2} - 6q^{3} - 10q^{7} - 4q^{8} + 6q^{10} + 4q^{11} + 6q^{12} - 12q^{15} - 4q^{16} + 12q^{17} + 4q^{18} + 20q^{21} + 8q^{22} + 10q^{23} - 12q^{25} - 24q^{26} + 18q^{28} + 8q^{30} - 24q^{31} - 18q^{32} - 8q^{35} - 24q^{36} - 12q^{38} + 18q^{40} - 26q^{42} - 12q^{43} + 24q^{45} + 28q^{46} + 24q^{47} + 28q^{50} - 8q^{51} + 24q^{52} + 20q^{53} + 16q^{56} + 16q^{57} - 6q^{58} - 6q^{60} - 24q^{61} - 4q^{63} - 24q^{65} - 12q^{66} - 14q^{67} - 12q^{68} - 40q^{70} + 24q^{71} - 8q^{72} - 24q^{73} - 6q^{75} - 20q^{77} + 32q^{78} - 24q^{80} - 8q^{81} - 6q^{82} + 8q^{85} + 36q^{86} + 18q^{87} - 8q^{88} + 40q^{91} - 36q^{92} + 4q^{93} + 8q^{95} + 60q^{96} + 18q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(35, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
35.2.k.a \(4\) \(0.279\) \(\Q(\zeta_{12})\) None \(-4\) \(-2\) \(-4\) \(0\) \(q+(-1+\zeta_{12})q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
35.2.k.b \(4\) \(0.279\) \(\Q(\zeta_{12})\) None \(2\) \(-4\) \(4\) \(-10\) \(q+(1-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1-\zeta_{12}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T + 5 T^{2} - 2 T^{3} - 11 T^{4} - 4 T^{5} + 20 T^{6} + 32 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 8 T^{3} + 13 T^{4} - 16 T^{5} + 20 T^{6} - 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 30 T^{5} + 45 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 12 T^{5} + 45 T^{6} + 108 T^{7} + 81 T^{8} \))
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))(\( ( 1 + 5 T + 7 T^{2} )^{2} \))
$11$ (\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 44 T^{5} - 1936 T^{6} - 2662 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 4 T + 20 T^{2} - 100 T^{3} + 271 T^{4} - 1700 T^{5} + 5780 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 8 T + 20 T^{2} + 52 T^{3} - 545 T^{4} + 884 T^{5} + 5780 T^{6} - 39304 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 4 T + 53 T^{2} + 244 T^{3} + 1588 T^{4} + 5612 T^{5} + 28037 T^{6} + 48668 T^{7} + 279841 T^{8} \))(\( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 5198 T^{5} + 28037 T^{6} - 170338 T^{7} + 279841 T^{8} \))
$29$ (\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))
$37$ (\( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))(\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 18 T + 90 T^{2} + 528 T^{3} - 8377 T^{4} + 24816 T^{5} + 198810 T^{6} - 1868814 T^{7} + 4879681 T^{8} \))(\( 1 - 6 T + 90 T^{2} - 672 T^{3} + 5159 T^{4} - 31584 T^{5} + 198810 T^{6} - 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} )( 1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4} ) \))
$59$ (\( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))(\( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 6 T + 148 T^{2} - 426 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 24 T + 144 T^{2} - 1752 T^{3} - 31057 T^{4} - 127896 T^{5} + 767376 T^{6} + 9336408 T^{7} + 28398241 T^{8} \))(\( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 64464 T^{5} + 923668 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))(\( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} ) \))(\( ( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} ) \))
$97$ (\( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))
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