Properties

Label 441.2.a
Level 441
Weight 2
Character orbit a
Rep. character \(\chi_{441}(1,\cdot)\)
Character field \(\Q\)
Dimension 14
Newform subspaces 10
Sturm bound 112
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(112\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(441))\).

Total New Old
Modular forms 72 19 53
Cusp forms 41 14 27
Eisenstein series 31 5 26

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

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(10\)

Trace form

\( 14q + 14q^{4} - 2q^{5} + 12q^{8} + O(q^{10}) \) \( 14q + 14q^{4} - 2q^{5} + 12q^{8} + 10q^{10} + 4q^{11} - 2q^{13} + 22q^{16} - 6q^{17} + 4q^{19} + 2q^{20} - 16q^{22} - 10q^{25} + 2q^{26} + 8q^{29} + 8q^{31} + 4q^{32} - 18q^{34} + 4q^{37} - 4q^{38} - 6q^{40} + 2q^{41} + 20q^{43} + 24q^{44} - 20q^{50} - 6q^{52} - 12q^{53} + 32q^{55} - 36q^{58} + 12q^{59} + 22q^{61} - 18q^{64} - 24q^{65} + 20q^{67} + 6q^{68} + 4q^{71} - 22q^{73} - 16q^{74} + 12q^{76} + 28q^{79} + 2q^{80} - 34q^{82} - 12q^{83} - 20q^{85} - 44q^{86} - 72q^{88} - 14q^{89} - 48q^{92} - 24q^{94} + 28q^{95} - 46q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
441.2.a.a \(1\) \(3.521\) \(\Q\) None \(-2\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(q-2q^{2}+2q^{4}-2q^{5}+4q^{10}+2q^{11}+\cdots\)
441.2.a.b \(1\) \(3.521\) \(\Q\) None \(-2\) \(0\) \(2\) \(0\) \(-\) \(+\) \(q-2q^{2}+2q^{4}+2q^{5}-4q^{10}+2q^{11}+\cdots\)
441.2.a.c \(1\) \(3.521\) \(\Q\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q-q^{2}-q^{4}+3q^{8}-4q^{11}-q^{16}+\cdots\)
441.2.a.d \(1\) \(3.521\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-2q^{4}-7q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
441.2.a.e \(1\) \(3.521\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q-2q^{4}+7q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
441.2.a.f \(1\) \(3.521\) \(\Q\) None \(1\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(q+q^{2}-q^{4}-2q^{5}-3q^{8}-2q^{10}+\cdots\)
441.2.a.g \(2\) \(3.521\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+q^{4}+2\beta q^{5}-\beta q^{8}+6q^{10}+\cdots\)
441.2.a.h \(2\) \(3.521\) \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}+5q^{4}+3\beta q^{8}-2\beta q^{11}+11q^{16}+\cdots\)
441.2.a.i \(2\) \(3.521\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-4\) \(0\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
441.2.a.j \(2\) \(3.521\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(4\) \(0\) \(-\) \(+\) \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2-\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(441)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + T^{2} + 4 T^{4} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ 1
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T^{2} + 25 T^{4} \))(\( ( 1 + 5 T^{2} )^{2} \))(\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ 1
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 + 10 T^{2} + 121 T^{4} \))(\( 1 - 6 T^{2} + 121 T^{4} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + 13 T^{2} \))(\( 1 + 7 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))(\( 1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 22 T^{2} + 289 T^{4} \))(\( ( 1 + 17 T^{2} )^{2} \))(\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T + 20 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 7 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( 1 + 30 T^{2} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 34 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 4 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( 1 - 54 T^{2} + 841 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} \))(\( 1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 9 T + 31 T^{2} \))(\( 1 - 9 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))(\( ( 1 + 4 T + 37 T^{2} )^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 26 T^{2} + 1681 T^{4} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( 1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4} \))(\( 1 - 4 T + 68 T^{2} - 164 T^{3} + 1681 T^{4} \))
$43$ (\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 - 5 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{2} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))(\( 1 + 54 T^{2} + 1849 T^{4} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 46 T^{2} + 2209 T^{4} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 12 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 58 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{2} \))
$59$ (\( 1 + 12 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 70 T^{2} + 3481 T^{4} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( 1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 61 T^{2} )^{2} \))(\( 1 - 16 T + 168 T^{2} - 976 T^{3} + 3721 T^{4} \))(\( 1 + 16 T + 168 T^{2} + 976 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 5 T + 67 T^{2} \))(\( 1 + 5 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 - 11 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))(\( 1 + 102 T^{2} + 4489 T^{4} \))
$71$ (\( 1 - 6 T + 71 T^{2} \))(\( 1 - 6 T + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 34 T^{2} + 5041 T^{4} \))(\( 1 + 114 T^{2} + 5041 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} \))(\( 1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 3 T + 73 T^{2} \))(\( 1 + 3 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 + 7 T + 73 T^{2} \))(\( 1 - 7 T + 73 T^{2} \))(\( 1 - 6 T + 73 T^{2} \))(\( ( 1 + 14 T + 73 T^{2} )^{2} \))(\( ( 1 + 73 T^{2} )^{2} \))(\( 1 - 8 T + 64 T^{2} - 584 T^{3} + 5329 T^{4} \))(\( 1 + 8 T + 64 T^{2} + 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + T + 79 T^{2} \))(\( 1 + T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 + 16 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( 1 - 8 T + 54 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 16 T + 89 T^{2} \))(\( 1 + 16 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 + 166 T^{2} + 7921 T^{4} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \))(\( 1 + 20 T + 260 T^{2} + 1780 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 + 6 T + 97 T^{2} \))(\( 1 + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 18 T + 97 T^{2} \))(\( ( 1 + 14 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( 1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4} \))(\( 1 + 8 T + 208 T^{2} + 776 T^{3} + 9409 T^{4} \))
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