# 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

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