Properties

Label 1764.2.a
Level 1764
Weight 2
Character orbit a
Rep. character \(\chi_{1764}(1,\cdot)\)
Character field \(\Q\)
Dimension 17
Newform subspaces 13
Sturm bound 672
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(672\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1764))\).

Total New Old
Modular forms 384 17 367
Cusp forms 289 17 272
Eisenstein series 95 0 95

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

\(2\)\(3\)\(7\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(6\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17q + 4q^{5} + O(q^{10}) \) \( 17q + 4q^{5} - 6q^{11} + 2q^{13} - 4q^{17} + 2q^{23} + 25q^{25} + 4q^{29} - 4q^{31} + 24q^{37} + 12q^{41} + 20q^{43} + 24q^{47} + 6q^{53} - 8q^{55} - 8q^{59} - 10q^{61} + 4q^{65} + 38q^{67} + 20q^{71} + 22q^{73} + 14q^{79} - 16q^{83} + 54q^{85} + 12q^{89} - 22q^{95} - 2q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
1764.2.a.a \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-3\) \(0\) \(-\) \(-\) \(+\) \(q-3q^{5}+3q^{11}+2q^{13}-3q^{17}-q^{19}+\cdots\)
1764.2.a.b \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q-2q^{5}-2q^{11}-4q^{13}-6q^{17}+8q^{19}+\cdots\)
1764.2.a.c \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(-\) \(-\) \(-\) \(q-2q^{5}-2q^{11}+3q^{13}+8q^{17}+q^{19}+\cdots\)
1764.2.a.d \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-5q^{13}+q^{19}-5q^{25}-11q^{31}+\cdots\)
1764.2.a.e \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(-\) \(q-2q^{13}-8q^{19}-5q^{25}+4q^{31}+\cdots\)
1764.2.a.f \(1\) \(14.086\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+5q^{13}-q^{19}-5q^{25}+11q^{31}+\cdots\)
1764.2.a.g \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(-\) \(q+6q^{11}-2q^{13}+4q^{19}+6q^{23}+\cdots\)
1764.2.a.h \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(+\) \(q+2q^{5}-2q^{11}-3q^{13}-8q^{17}-q^{19}+\cdots\)
1764.2.a.i \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}-2q^{11}+4q^{13}+6q^{17}-8q^{19}+\cdots\)
1764.2.a.j \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(3\) \(0\) \(-\) \(-\) \(-\) \(q+3q^{5}+3q^{11}-2q^{13}+3q^{17}+q^{19}+\cdots\)
1764.2.a.k \(1\) \(14.086\) \(\Q\) None \(0\) \(0\) \(4\) \(0\) \(-\) \(-\) \(-\) \(q+4q^{5}-2q^{11}+6q^{13}-4q^{17}+4q^{19}+\cdots\)
1764.2.a.l \(2\) \(14.086\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(+\) \(q+\beta q^{5}-4q^{11}-3\beta q^{13}+\beta q^{17}+\cdots\)
1764.2.a.m \(4\) \(14.086\) \(\Q(\sqrt{2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta _{2}q^{5}-\beta _{3}q^{11}-3\beta _{1}q^{13}+\beta _{2}q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1764))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1764)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(882))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( ( 1 - 4 T^{2} + 25 T^{4} )^{2} \))
$7$ 1
$11$ (\( 1 - 3 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 6 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( 1 + 2 T + 11 T^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 + 5 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 5 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 3 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 8 T^{2} + 169 T^{4} \))(\( ( 1 + 8 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 8 T + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 17 T^{2} \))(\( 1 + 8 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 32 T^{2} + 289 T^{4} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 8 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 30 T^{2} + 361 T^{4} \))(\( ( 1 + 30 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 3 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 + 3 T + 23 T^{2} \))(\( 1 + 2 T + 23 T^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 4 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 30 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 7 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( 1 + 11 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 11 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 3 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 + T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{4} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 - 12 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 32 T^{2} + 1681 T^{4} \))(\( ( 1 + 68 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 + 13 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 13 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - 11 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{4} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 6 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 6 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 62 T^{2} + 2209 T^{4} \))(\( ( 1 + 38 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 3 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 - 12 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( ( 1 + 10 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( 1 + 9 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 9 T + 59 T^{2} \))(\( 1 + 8 T + 59 T^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( ( 1 + 62 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + T + 61 T^{2} \))(\( 1 + 8 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( 1 - T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 + 72 T^{2} + 3721 T^{4} \))(\( ( 1 + 24 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 + 16 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 13 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( ( 1 + 67 T^{2} )^{2} \))(\( ( 1 - 12 T + 67 T^{2} )^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 6 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 - 10 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 14 T + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( 1 + 17 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 - 17 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 11 T + 73 T^{2} \))(\( 1 + 4 T + 73 T^{2} \))(\( 1 - T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 96 T^{2} + 5329 T^{4} \))(\( ( 1 + 144 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + 13 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 - 17 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 - 17 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 + 13 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{4} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 2 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 4 T + 83 T^{2} \))(\( 1 - 34 T^{2} + 6889 T^{4} \))(\( ( 1 - 58 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 - 14 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 14 T + 89 T^{2} \))(\( 1 - 15 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 128 T^{2} + 7921 T^{4} \))(\( ( 1 + 164 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 - 4 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( 1 - 14 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 + 4 T + 97 T^{2} \))(\( 1 - 10 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 192 T^{2} + 9409 T^{4} \))(\( ( 1 + 96 T^{2} + 9409 T^{4} )^{2} \))
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