Properties

Label 231.2.a
Level 231
Weight 2
Character orbit a
Rep. character \(\chi_{231}(1,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 5
Sturm bound 64
Trace bound 2

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Defining parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(231))\).

Total New Old
Modular forms 36 11 25
Cusp forms 29 11 18
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(11\)

Trace form

\( 11q + q^{2} - q^{3} + 17q^{4} + 10q^{5} + 5q^{6} - q^{7} - 3q^{8} + 11q^{9} + O(q^{10}) \) \( 11q + q^{2} - q^{3} + 17q^{4} + 10q^{5} + 5q^{6} - q^{7} - 3q^{8} + 11q^{9} - 2q^{10} - q^{11} - 7q^{12} + 2q^{13} - 3q^{14} + 2q^{15} + 17q^{16} + 22q^{17} + q^{18} + 4q^{19} - 2q^{20} - q^{21} + q^{22} + 9q^{24} + 29q^{25} - 10q^{26} - q^{27} - 7q^{28} + 18q^{29} - 18q^{30} - 8q^{31} - 35q^{32} - q^{33} - 14q^{34} + 2q^{35} + 17q^{36} - 6q^{37} - 36q^{38} - 14q^{39} - 42q^{40} + 30q^{41} + q^{42} - 20q^{43} - 7q^{44} + 10q^{45} - 32q^{46} - 24q^{47} - 31q^{48} + 11q^{49} - 65q^{50} + 6q^{51} - 58q^{52} - 6q^{53} + 5q^{54} - 6q^{55} - 15q^{56} - 20q^{57} - 46q^{58} - 20q^{59} - 6q^{60} + 26q^{61} - 16q^{62} - q^{63} + 29q^{64} + 36q^{65} - 3q^{66} - 20q^{67} + 26q^{68} + 16q^{69} + 2q^{70} + 8q^{71} - 3q^{72} + 38q^{73} + 54q^{74} - 15q^{75} + 4q^{76} - q^{77} - 14q^{78} + 16q^{79} + 6q^{80} + 11q^{81} - 14q^{82} - 4q^{83} - 7q^{84} - 4q^{85} + 12q^{86} - 6q^{87} + 9q^{88} + 62q^{89} - 2q^{90} + 10q^{91} + 8q^{92} - 8q^{93} + 24q^{94} - 16q^{95} + 33q^{96} + 14q^{97} + q^{98} - q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 11
231.2.a.a \(1\) \(1.845\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+q^{7}+\cdots\)
231.2.a.b \(2\) \(1.845\) \(\Q(\sqrt{21}) \) None \(-1\) \(-2\) \(6\) \(2\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+(3+\beta )q^{4}+3q^{5}+\beta q^{6}+\cdots\)
231.2.a.c \(2\) \(1.845\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(2\) \(2\) \(-\) \(-\) \(-\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
231.2.a.d \(3\) \(1.845\) 3.3.837.1 None \(0\) \(-3\) \(0\) \(-3\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
231.2.a.e \(3\) \(1.845\) 3.3.229.1 None \(2\) \(3\) \(4\) \(-3\) \(-\) \(+\) \(+\) \(q+(1+\beta _{2})q^{2}+q^{3}+(2+\beta _{1})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(231))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(231)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 - T^{3} + 8 T^{6} \))(\( 1 - 2 T + 2 T^{2} - T^{3} + 4 T^{4} - 8 T^{5} + 8 T^{6} \))
$3$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( 1 + 2 T^{3} + 125 T^{6} \))(\( 1 - 4 T + 8 T^{2} - 14 T^{3} + 40 T^{4} - 100 T^{5} + 125 T^{6} \))
$7$ (\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{3} \))
$11$ (\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{3} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( ( 1 - T + 13 T^{2} )^{2} \))(\( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 24 T^{2} + 2 T^{3} + 312 T^{4} + 2197 T^{6} \))(\( 1 + 4 T + 12 T^{2} + 10 T^{3} + 156 T^{4} + 676 T^{5} + 2197 T^{6} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 - 6 T + 22 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 27 T^{2} + 8 T^{3} + 459 T^{4} + 4913 T^{6} \))(\( 1 - 8 T + 11 T^{2} + 56 T^{3} + 187 T^{4} - 2312 T^{5} + 4913 T^{6} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 21 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 7 T^{2} + 361 T^{4} \))(\( 1 - 12 T + 84 T^{2} - 420 T^{3} + 1596 T^{4} - 4332 T^{5} + 6859 T^{6} \))(\( 1 + 8 T + 72 T^{2} + 308 T^{3} + 1368 T^{4} + 2888 T^{5} + 6859 T^{6} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 2 T + 26 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 57 T^{2} + 244 T^{3} + 1311 T^{4} + 3174 T^{5} + 12167 T^{6} \))(\( 1 - 10 T + 81 T^{2} - 396 T^{3} + 1863 T^{4} - 5290 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 2 T - 25 T^{2} - 58 T^{3} + 841 T^{4} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( 1 - 12 T + 120 T^{2} - 702 T^{3} + 3480 T^{4} - 10092 T^{5} + 24389 T^{6} \))(\( 1 + 4 T + 60 T^{2} + 138 T^{3} + 1740 T^{4} + 3364 T^{5} + 24389 T^{6} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( 1 + 2 T + 42 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 66 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 6 T + 57 T^{2} + 404 T^{3} + 1767 T^{4} + 5766 T^{5} + 29791 T^{6} \))(\( 1 + 2 T + 17 T^{2} - 132 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6} \))
$37$ (\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 7 T + 37 T^{2} )^{2} \))(\( 1 + 36 T^{2} - 246 T^{3} + 1332 T^{4} + 50653 T^{6} \))(\( 1 + 68 T^{2} + 106 T^{3} + 2516 T^{4} + 50653 T^{6} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 4 T + 2 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4} \))(\( 1 - 6 T + 51 T^{2} - 460 T^{3} + 2091 T^{4} - 10086 T^{5} + 68921 T^{6} \))(\( 1 - 14 T + 163 T^{2} - 1116 T^{3} + 6683 T^{4} - 23534 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 + 6 T + 74 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} \))(\( 1 - 6 T + 117 T^{2} - 468 T^{3} + 5031 T^{4} - 11094 T^{5} + 79507 T^{6} \))(\( 1 + 14 T + 85 T^{2} + 356 T^{3} + 3655 T^{4} + 25886 T^{5} + 79507 T^{6} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 - 12 T + 109 T^{2} - 564 T^{3} + 2209 T^{4} \))(\( 1 + 4 T + 93 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 + 24 T + 312 T^{2} + 2584 T^{3} + 14664 T^{4} + 53016 T^{5} + 103823 T^{6} \))(\( 1 + 80 T^{2} + 32 T^{3} + 3760 T^{4} + 103823 T^{6} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 110 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( 1 + 2 T - 18 T^{2} + 106 T^{3} + 2809 T^{4} \))(\( 1 + 111 T^{2} - 120 T^{3} + 5883 T^{4} + 148877 T^{6} \))(\( 1 + 143 T^{2} + 8 T^{3} + 7579 T^{4} + 148877 T^{6} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 97 T^{2} + 3481 T^{4} \))(\( 1 - 7 T^{2} + 3481 T^{4} \))(\( 1 + 24 T + 264 T^{2} + 2116 T^{3} + 15576 T^{4} + 83544 T^{5} + 205379 T^{6} \))(\( 1 + 120 T^{2} - 52 T^{3} + 7080 T^{4} + 205379 T^{6} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{3} \))(\( ( 1 + 2 T + 61 T^{2} )^{3} \))
$67$ (\( 1 + 12 T + 67 T^{2} \))(\( 1 - 8 T + 129 T^{2} - 536 T^{3} + 4489 T^{4} \))(\( 1 + 24 T + 273 T^{2} + 1608 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 228 T^{2} - 1604 T^{3} + 15276 T^{4} - 53868 T^{5} + 300763 T^{6} \))(\( 1 + 4 T + 116 T^{2} + 300 T^{3} + 7772 T^{4} + 17956 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 4 T + 62 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 12 T + 165 T^{2} - 1320 T^{3} + 11715 T^{4} - 60492 T^{5} + 357911 T^{6} \))(\( 1 + 12 T + 197 T^{2} + 1448 T^{3} + 13987 T^{4} + 60492 T^{5} + 357911 T^{6} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( ( 1 - 7 T + 73 T^{2} )^{2} \))(\( 1 - 18 T + 207 T^{2} - 1314 T^{3} + 5329 T^{4} \))(\( 1 - 24 T + 396 T^{2} - 3898 T^{3} + 28908 T^{4} - 127896 T^{5} + 389017 T^{6} \))(\( 1 + 20 T + 320 T^{2} + 3054 T^{3} + 23360 T^{4} + 106580 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 16 T + 79 T^{2} \))(\( 1 + 4 T + 78 T^{2} + 316 T^{3} + 6241 T^{4} \))(\( 1 + 20 T + 238 T^{2} + 1580 T^{3} + 6241 T^{4} \))(\( 1 - 12 T + 189 T^{2} - 1640 T^{3} + 14931 T^{4} - 74892 T^{5} + 493039 T^{6} \))(\( 1 - 12 T + 221 T^{2} - 1640 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 14 T + 194 T^{2} + 1162 T^{3} + 6889 T^{4} \))(\( 1 - 18 T + 242 T^{2} - 1494 T^{3} + 6889 T^{4} \))(\( 1 + 18 T + 309 T^{2} + 3036 T^{3} + 25647 T^{4} + 124002 T^{5} + 571787 T^{6} \))(\( 1 - 6 T + 117 T^{2} - 500 T^{3} + 9711 T^{4} - 41334 T^{5} + 571787 T^{6} \))
$89$ (\( 1 - 18 T + 89 T^{2} \))(\( 1 + 94 T^{2} + 7921 T^{4} \))(\( 1 + 158 T^{2} + 7921 T^{4} \))(\( 1 - 18 T + 183 T^{2} - 1308 T^{3} + 16287 T^{4} - 142578 T^{5} + 704969 T^{6} \))(\( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 36223 T^{4} - 205946 T^{5} + 704969 T^{6} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 14 T + 222 T^{2} + 1358 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4} \))(\( 1 - 24 T + 435 T^{2} - 4664 T^{3} + 42195 T^{4} - 225816 T^{5} + 912673 T^{6} \))(\( 1 + 4 T + 171 T^{2} + 544 T^{3} + 16587 T^{4} + 37636 T^{5} + 912673 T^{6} \))
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