Properties

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

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Defining parameters

Level: \( N \) = \( 35 = 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 35.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(35))\).

Total New Old
Modular forms 6 3 3
Cusp forms 3 3 0
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.

\(5\)\(7\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q - q^{2} + 3q^{4} + q^{5} - 8q^{6} - q^{7} - 9q^{8} + q^{9} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} + q^{5} - 8q^{6} - q^{7} - 9q^{8} + q^{9} - q^{10} - 2q^{11} + 4q^{12} + 10q^{13} + q^{14} - 2q^{15} + 7q^{16} - 2q^{17} + 7q^{18} - 4q^{19} + 7q^{20} + 2q^{21} + 8q^{22} - 8q^{23} - 4q^{24} + 3q^{25} + 6q^{26} - 12q^{27} - 7q^{28} + 4q^{29} - 8q^{30} - 4q^{31} - 9q^{32} - 12q^{33} - 6q^{34} - 3q^{35} + 3q^{36} + 14q^{37} + 20q^{38} - 6q^{39} - 9q^{40} - 10q^{41} + 8q^{42} + 5q^{45} - 16q^{46} + 4q^{47} + 28q^{48} + 3q^{49} - q^{50} + 14q^{51} - 6q^{52} + 10q^{53} + 12q^{54} + 4q^{55} + 9q^{56} - 12q^{57} - 26q^{58} - 8q^{59} + 8q^{60} + 14q^{61} - 5q^{63} - q^{64} - 4q^{66} - 10q^{68} + 12q^{69} + q^{70} + 16q^{71} - 5q^{72} - 6q^{73} - 6q^{74} - 36q^{76} - 4q^{77} - 20q^{78} - 10q^{79} - q^{80} - 13q^{81} - 18q^{82} + 20q^{83} - 8q^{84} - 8q^{85} + 12q^{86} + 28q^{87} + 4q^{88} - 6q^{89} + 7q^{90} + 24q^{92} - 4q^{93} + 28q^{94} - 8q^{95} - 4q^{96} - 10q^{97} - q^{98} + 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7
35.2.a.a \(1\) \(0.279\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(+\) \(-\) \(q+q^{3}-2q^{4}-q^{5}+q^{7}-2q^{9}-3q^{11}+\cdots\)
35.2.a.b \(2\) \(0.279\) \(\Q(\sqrt{17}) \) None \(-1\) \(-1\) \(2\) \(-2\) \(-\) \(+\) \(q-\beta q^{2}+(-1+\beta )q^{3}+(2+\beta )q^{4}+q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$7$ (\( 1 - T \))(\( ( 1 + T )^{2} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 2 T + 19 T^{2} \))(\( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 3 T + 29 T^{2} \))(\( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 12 T + 41 T^{2} \))(\( 1 - 2 T + 66 T^{2} - 82 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 - 10 T + 94 T^{2} - 430 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 5 T + 62 T^{2} + 235 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 12 T + 53 T^{2} \))(\( 1 + 2 T + 90 T^{2} + 106 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 59 T^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + T + 97 T^{2} \))(\( 1 + 9 T + 108 T^{2} + 873 T^{3} + 9409 T^{4} \))
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