Properties

Label 91.2.a
Level 91
Weight 2
Character orbit a
Rep. character \(\chi_{91}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 4
Sturm bound 18
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(18\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(91))\).

Total New Old
Modular forms 10 7 3
Cusp forms 7 7 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.

\(7\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7q - q^{2} - 4q^{3} + 3q^{4} + 2q^{5} - q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 7q - q^{2} - 4q^{3} + 3q^{4} + 2q^{5} - q^{7} + 3q^{8} + 3q^{9} + 2q^{10} - 4q^{11} - 8q^{12} + q^{13} + q^{14} - 4q^{15} - 9q^{16} + 2q^{17} - 9q^{18} - 12q^{19} - 2q^{20} - 4q^{22} + 10q^{23} - 4q^{24} + 15q^{25} + 3q^{26} - 4q^{27} - 7q^{28} + 16q^{29} + 8q^{30} - 4q^{31} + 15q^{32} - 4q^{33} + 2q^{34} + 4q^{35} - 9q^{36} - 6q^{37} - 8q^{38} - 4q^{39} - 10q^{40} + 2q^{41} - 8q^{42} - 2q^{43} + 22q^{45} - 16q^{46} + 8q^{47} + 12q^{48} + 7q^{49} + q^{50} + 8q^{51} - q^{52} - 16q^{53} + 48q^{54} + 12q^{55} - 3q^{56} + 30q^{58} + 16q^{59} - 20q^{60} - 14q^{61} - 8q^{62} - 5q^{63} - 17q^{64} - 4q^{65} + 12q^{66} - 16q^{67} + 42q^{68} - 20q^{69} + 6q^{70} - 32q^{71} - q^{72} - 22q^{73} + 26q^{74} - 48q^{75} + 16q^{76} + 4q^{77} + 8q^{78} + 2q^{79} - 38q^{80} - 9q^{81} + 14q^{82} + 24q^{83} + 16q^{84} - 8q^{85} - 24q^{86} - 16q^{87} + 36q^{88} + 26q^{89} - 50q^{90} - 3q^{91} - 12q^{92} - 20q^{93} - 20q^{94} + 6q^{95} - 4q^{96} - 6q^{97} - q^{98} + 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(91))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7 13
91.2.a.a \(1\) \(0.727\) \(\Q\) None \(-2\) \(0\) \(-3\) \(-1\) \(+\) \(+\) \(q-2q^{2}+2q^{4}-3q^{5}-q^{7}-3q^{9}+\cdots\)
91.2.a.b \(1\) \(0.727\) \(\Q\) None \(0\) \(-2\) \(-3\) \(1\) \(-\) \(-\) \(q-2q^{3}-2q^{4}-3q^{5}+q^{7}+q^{9}+4q^{12}+\cdots\)
91.2.a.c \(2\) \(0.727\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(6\) \(2\) \(-\) \(+\) \(q+\beta q^{2}-\beta q^{3}+(3+\beta )q^{5}-2q^{6}+q^{7}+\cdots\)
91.2.a.d \(3\) \(0.727\) 3.3.316.1 None \(1\) \(-2\) \(2\) \(-3\) \(+\) \(-\) \(q+\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - T + 2 T^{2} - 2 T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 2 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))(\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 9 T^{4} + 18 T^{5} + 27 T^{6} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 3 T + 5 T^{2} \))(\( 1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 12 T^{2} - 18 T^{3} + 60 T^{4} - 50 T^{5} + 125 T^{6} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T^{2} + 121 T^{4} \))(\( 1 - 2 T + 27 T^{2} - 36 T^{3} + 297 T^{4} - 242 T^{5} + 1331 T^{6} \))
$13$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{3} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( 1 - 4 T + 41 T^{2} - 140 T^{3} + 697 T^{4} - 1156 T^{5} + 4913 T^{6} \))
$19$ (\( 1 - 5 T + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 + 6 T + 29 T^{2} + 114 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 58 T^{2} + 148 T^{3} + 1102 T^{4} + 1444 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 6 T + 47 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 10 T + 70 T^{2} - 324 T^{3} + 1610 T^{4} - 5290 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 5 T + 29 T^{2} \))(\( 1 + 9 T + 29 T^{2} \))(\( 1 - 6 T + 59 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 7888 T^{4} - 20184 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 3 T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 + 2 T + 45 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 74 T^{2} + 264 T^{3} + 2294 T^{4} + 3844 T^{5} + 29791 T^{6} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 53 T^{2} - 124 T^{3} + 1961 T^{4} + 50653 T^{6} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 12 T + 110 T^{2} - 492 T^{3} + 1681 T^{4} \))(\( 1 - 2 T + 95 T^{2} - 172 T^{3} + 3895 T^{4} - 3362 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + T + 43 T^{2} \))(\( 1 + T + 43 T^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( 1 - 10 T + 58 T^{2} - 232 T^{3} + 2494 T^{4} - 18490 T^{5} + 79507 T^{6} \))
$47$ (\( 1 - 7 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 - 6 T + 101 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 62 T^{2} + 208 T^{3} + 2914 T^{4} + 17672 T^{5} + 103823 T^{6} \))
$53$ (\( 1 + 9 T + 53 T^{2} \))(\( 1 + 9 T + 53 T^{2} \))(\( 1 + 6 T + 107 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 8 T + 124 T^{2} - 870 T^{3} + 6572 T^{4} - 22472 T^{5} + 148877 T^{6} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 122 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 4 T + 21 T^{2} - 216 T^{3} + 1239 T^{4} + 13924 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{3} \))
$67$ (\( 1 + 6 T + 67 T^{2} \))(\( 1 - 14 T + 67 T^{2} \))(\( 1 + 12 T + 98 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 + 12 T + 77 T^{2} + 632 T^{3} + 5159 T^{4} + 53868 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 + 12 T + 128 T^{2} + 852 T^{3} + 5041 T^{4} \))(\( 1 + 6 T + 191 T^{2} + 868 T^{3} + 13561 T^{4} + 30246 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 13 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4} \))(\( 1 + 10 T + 120 T^{2} + 1186 T^{3} + 8760 T^{4} + 53290 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 3 T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 14 T + 135 T^{2} - 1106 T^{3} + 6241 T^{4} \))(\( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 19118 T^{4} + 87374 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 15 T + 83 T^{2} \))(\( 1 - 3 T + 83 T^{2} \))(\( 1 - 18 T + 229 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 + 12 T - 22 T^{2} - 1276 T^{3} - 1826 T^{4} + 82668 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 3 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 - 6 T + 185 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 2 T + 172 T^{2} + 66 T^{3} + 15308 T^{4} - 15842 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 7 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 2 T + 33 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 10 T + 320 T^{2} + 1962 T^{3} + 31040 T^{4} + 94090 T^{5} + 912673 T^{6} \))
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