Properties

Label 35.2.e
Level 35
Weight 2
Character orbit e
Rep. character \(\chi_{35}(11,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 4
Newform subspaces 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 35 = 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 35.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 12 4 8
Cusp forms 4 4 0
Eisenstein series 8 0 8

Trace form

\( 4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{5} - 4q^{6} + 2q^{7} + 12q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{5} - 4q^{6} + 2q^{7} + 12q^{8} - 2q^{10} - 4q^{11} + 6q^{12} - 8q^{13} - 4q^{14} + 4q^{15} - 6q^{16} - 4q^{17} + 8q^{18} + 4q^{20} + 14q^{21} - 8q^{22} + 2q^{23} - 2q^{24} - 2q^{25} + 12q^{26} + 4q^{27} - 22q^{28} - 4q^{29} + 2q^{30} + 12q^{31} + 6q^{32} - 12q^{33} + 24q^{34} - 4q^{35} - 32q^{36} - 8q^{38} - 4q^{39} - 6q^{40} - 20q^{41} - 2q^{42} + 20q^{43} + 12q^{44} - 2q^{46} - 4q^{47} + 12q^{48} + 10q^{49} + 4q^{50} + 4q^{51} + 20q^{52} + 8q^{53} - 6q^{54} + 8q^{55} + 18q^{56} - 16q^{57} + 2q^{58} + 8q^{59} + 6q^{60} + 6q^{61} - 24q^{62} - 28q^{64} + 4q^{65} + 4q^{66} - 22q^{67} - 20q^{68} - 12q^{69} - 10q^{70} - 16q^{71} + 8q^{72} - 4q^{73} - 2q^{75} + 32q^{76} + 28q^{77} + 8q^{78} - 24q^{79} - 6q^{80} + 2q^{81} + 18q^{82} + 4q^{83} + 12q^{84} + 8q^{85} - 6q^{86} + 2q^{87} - 4q^{88} + 6q^{89} - 16q^{90} - 28q^{91} + 12q^{92} + 12q^{93} - 4q^{94} + 10q^{96} + 24q^{97} - 14q^{98} + 32q^{99} + O(q^{100}) \)

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.e.a \(4\) \(0.279\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(-2\) \(-2\) \(2\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + T^{2} - 2 T^{3} - 3 T^{4} - 4 T^{5} + 4 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 6 T^{5} - 9 T^{6} + 54 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 176 T^{5} - 242 T^{6} + 5324 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 + 4 T - 14 T^{2} - 16 T^{3} + 339 T^{4} - 272 T^{5} - 4046 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} \)
$23$ \( 1 - 2 T - 41 T^{2} + 2 T^{3} + 1404 T^{4} + 46 T^{5} - 21689 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 37 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 10 T + 99 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 10 T + 109 T^{2} - 430 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 8 T - 50 T^{2} - 64 T^{3} + 6795 T^{4} - 3392 T^{5} - 140450 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 8 T + 2 T^{2} + 448 T^{3} - 3413 T^{4} + 26432 T^{5} + 6962 T^{6} - 1643032 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 23058 T^{5} - 85583 T^{6} - 1361886 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 22 T + 231 T^{2} + 2618 T^{3} + 27092 T^{4} + 175406 T^{5} + 1036959 T^{6} + 6616786 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 + 4 T - 126 T^{2} - 16 T^{3} + 13667 T^{4} - 1168 T^{5} - 671454 T^{6} + 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 257856 T^{5} + 1759962 T^{6} + 11832936 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 2 T + 5 T^{2} - 166 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 6 T - 119 T^{2} + 138 T^{3} + 12900 T^{4} + 12282 T^{5} - 942599 T^{6} - 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4} )^{2} \)
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