Properties

Label 960.2.f
Level $960$
Weight $2$
Character orbit 960.f
Rep. character $\chi_{960}(769,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $11$
Sturm bound $384$
Trace bound $15$

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(384\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(960, [\chi])\).

Total New Old
Modular forms 216 24 192
Cusp forms 168 24 144
Eisenstein series 48 0 48

Trace form

\( 24q - 24q^{9} + O(q^{10}) \) \( 24q - 24q^{9} - 8q^{25} + 16q^{41} - 24q^{49} - 16q^{61} - 16q^{65} + 16q^{69} + 24q^{81} + 48q^{85} + 16q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(960, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
960.2.f.a \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-iq^{3}+(-2+i)q^{5}+2iq^{7}-q^{9}+\cdots\)
960.2.f.b \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-iq^{3}+(-2-i)q^{5}+2iq^{7}-q^{9}+\cdots\)
960.2.f.c \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{3}+(-1-2i)q^{5}+4iq^{7}-q^{9}+\cdots\)
960.2.f.d \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-iq^{3}+(-1-2i)q^{5}+4iq^{7}-q^{9}+\cdots\)
960.2.f.e \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-iq^{3}+(-1+2i)q^{5}+4iq^{7}-q^{9}+\cdots\)
960.2.f.f \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{3}+(-1+2i)q^{5}+4iq^{7}-q^{9}+\cdots\)
960.2.f.g \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q+iq^{3}+(2-i)q^{5}+2iq^{7}-q^{9}-6q^{11}+\cdots\)
960.2.f.h \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q-iq^{3}+(2+i)q^{5}+2iq^{7}-q^{9}-2q^{11}+\cdots\)
960.2.f.i \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q-iq^{3}+(2-i)q^{5}+2iq^{7}-q^{9}+2q^{11}+\cdots\)
960.2.f.j \(2\) \(7.666\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q+iq^{3}+(2+i)q^{5}+2iq^{7}-q^{9}+6q^{11}+\cdots\)
960.2.f.k \(4\) \(7.666\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+2\beta _{1}q^{7}-q^{9}+2\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(960, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(960, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))
$7$ (\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 + 2 T^{2} + 289 T^{4} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 4 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))
$23$ (\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{4} \))
$31$ (\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 18 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 54 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{4} \))
$43$ (\( 1 + 58 T^{2} + 1849 T^{4} \))(\( 1 + 58 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 - 47 T^{2} )^{2} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( ( 1 - 30 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 + 38 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 86 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 62 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{4} \))
$67$ (\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 118 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 62 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 82 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 82 T^{2} + 5329 T^{4} \))(\( 1 - 2 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 2 T^{2} + 5329 T^{4} \))(\( ( 1 - 66 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 150 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))
$97$ (\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 + 126 T^{2} + 9409 T^{4} )^{2} \))
show more
show less