LMFDB - Space of Modular Forms of Level 448, Weight 2, and Character 448.f

Defining parameters

Level:	$ N $	=	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	448.f (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 28 $
Character field:			$\Q$
Newform subspaces:			$ 4 $
Sturm bound:			$128$
Trace bound:			$3$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(448, [\chi])$.

	Total	New	Old
Modular forms	76	18	58
Cusp forms	52	14	38
Eisenstein series	24	4	20

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(448, [\chi])$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				$q$-expansion
Label	Dim.	$A$	Field	CM	$a_2$	$a_3$	$a_5$	$a_7$	$q$-expansion
448.2.f.a	$2$	$3.577$	$\Q(\sqrt{-3}) $	None	$0$	$-4$	$0$	$-4$	$q-2q^{3}-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+q^{9}+\cdots$
448.2.f.b	$2$	$3.577$	$\Q(\sqrt{-7}) $	$\Q(\sqrt{-7}) $	$0$	$0$	$0$	$0$	$q-\beta q^{7}-3q^{9}-2\beta q^{11}-2\beta q^{23}+\cdots$
448.2.f.c	$2$	$3.577$	$\Q(\sqrt{-3}) $	None	$0$	$4$	$0$	$4$	$q+2q^{3}+2\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}+q^{9}+\cdots$
448.2.f.d	$8$	$3.577$	$\Q(\zeta_{16})$	None	$0$	$0$	$0$	$0$	$q-\zeta_{16}^{2}q^{3}-\zeta_{16}^{5}q^{5}-\zeta_{16}^{4}q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(448, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(448, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(28, [\chi])$$^{\oplus 5}$$\oplus$$S_{2}^{\mathrm{new}}(112, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(224, [\chi])$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	()()()()
$3$	($ ( 1 + 2 T + 3 T^{2} )^{2} $)($ ( 1 + 3 T^{2} )^{2} $)($ ( 1 - 2 T + 3 T^{2} )^{2} $)($ ( 1 + 4 T^{2} + 14 T^{4} + 36 T^{6} + 81 T^{8} )^{2} $)
$5$	($ 1 + 2 T^{2} + 25 T^{4} $)($ ( 1 - 5 T^{2} )^{2} $)($ 1 + 2 T^{2} + 25 T^{4} $)($ ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} $)
$7$	($ 1 + 4 T + 7 T^{2} $)($ 1 + 7 T^{2} $)($ 1 - 4 T + 7 T^{2} $)($ 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} $)
$11$	($ 1 - 10 T^{2} + 121 T^{4} $)($ ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) $)($ 1 - 10 T^{2} + 121 T^{4} $)($ ( 1 - 18 T^{2} + 121 T^{4} )^{4} $)
$13$	($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 13 T^{2} )^{2} $)($ 1 - 14 T^{2} + 169 T^{4} $)($ ( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2} $)
$17$	($ ( 1 - 17 T^{2} )^{2} $)($ ( 1 - 17 T^{2} )^{2} $)($ ( 1 - 17 T^{2} )^{2} $)($ ( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2} $)
$19$	($ ( 1 + 2 T + 19 T^{2} )^{2} $)($ ( 1 + 19 T^{2} )^{2} $)($ ( 1 - 2 T + 19 T^{2} )^{2} $)($ ( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2} $)
$23$	($ 1 - 34 T^{2} + 529 T^{4} $)($ ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) $)($ 1 - 34 T^{2} + 529 T^{4} $)($ ( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} $)
$29$	($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 - 2 T + 29 T^{2} )^{2} $)($ ( 1 + 6 T + 29 T^{2} )^{2} $)($ ( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} )^{4} $)
$31$	($ ( 1 + 8 T + 31 T^{2} )^{2} $)($ ( 1 + 31 T^{2} )^{2} $)($ ( 1 - 8 T + 31 T^{2} )^{2} $)($ ( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2} $)
$37$	($ ( 1 - 2 T + 37 T^{2} )^{2} $)($ ( 1 + 6 T + 37 T^{2} )^{2} $)($ ( 1 - 2 T + 37 T^{2} )^{2} $)($ ( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} )^{4} $)
$41$	($ 1 - 34 T^{2} + 1681 T^{4} $)($ ( 1 - 41 T^{2} )^{2} $)($ 1 - 34 T^{2} + 1681 T^{4} $)($ ( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2} $)
$43$	($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) $)($ ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) $)($ ( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2} $)
$47$	($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 47 T^{2} )^{2} $)($ ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} $)
$53$	($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 - 10 T + 53 T^{2} )^{2} $)($ ( 1 + 6 T + 53 T^{2} )^{2} $)($ ( 1 - 2 T + 53 T^{2} )^{8} $)
$59$	($ ( 1 + 6 T + 59 T^{2} )^{2} $)($ ( 1 + 59 T^{2} )^{2} $)($ ( 1 - 6 T + 59 T^{2} )^{2} $)($ ( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} $)
$61$	($ 1 - 110 T^{2} + 3721 T^{4} $)($ ( 1 - 61 T^{2} )^{2} $)($ 1 - 110 T^{2} + 3721 T^{4} $)($ ( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2} $)
$67$	($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) $)($ ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) $)($ ( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2} $)
$71$	($ 1 - 130 T^{2} + 5041 T^{4} $)($ ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) $)($ 1 - 130 T^{2} + 5041 T^{4} $)($ ( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2} $)
$73$	($ 1 - 98 T^{2} + 5329 T^{4} $)($ ( 1 - 73 T^{2} )^{2} $)($ 1 - 98 T^{2} + 5329 T^{4} $)($ ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} $)
$79$	($ 1 - 146 T^{2} + 6241 T^{4} $)($ ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) $)($ 1 - 146 T^{2} + 6241 T^{4} $)($ ( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2} $)
$83$	($ ( 1 - 6 T + 83 T^{2} )^{2} $)($ ( 1 + 83 T^{2} )^{2} $)($ ( 1 + 6 T + 83 T^{2} )^{2} $)($ ( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2} $)
$89$	($ 1 - 130 T^{2} + 7921 T^{4} $)($ ( 1 - 89 T^{2} )^{2} $)($ 1 - 130 T^{2} + 7921 T^{4} $)($ ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} $)
$97$	($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 97 T^{2} )^{2} $)($ ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) $)($ ( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	448.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(128\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
448.2.f.a	\(2\)	\(3.577\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-4\)	\(0\)	\(-4\)	\(q-2q^{3}-2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+q^{9}+\cdots\)
448.2.f.b	\(2\)	\(3.577\)	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\beta q^{7}-3q^{9}-2\beta q^{11}-2\beta q^{23}+\cdots\)
448.2.f.c	\(2\)	\(3.577\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(4\)	\(0\)	\(4\)	\(q+2q^{3}+2\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}+q^{9}+\cdots\)
448.2.f.d	\(8\)	\(3.577\)	\(\Q(\zeta_{16})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\zeta_{16}^{2}q^{3}-\zeta_{16}^{5}q^{5}-\zeta_{16}^{4}q^{7}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(448, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

Hecke Characteristic Polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	448.2.f

Level	448
Weight	2
Character orbit	f
Rep. character	\(\chi_{448}(447,\cdot)\)
Character field	\(\Q\)
Dimension	14
Newform subspaces	4
Sturm bound	128
Trace bound	3

$p$	$F_p(T)$
$2$	(\( \))(\( \))(\( \))(\( \))
$3$	(\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))(\( ( 1 + 4 T^{2} + 14 T^{4} + 36 T^{6} + 81 T^{8} )^{2} \))
$5$	(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))(\( ( 1 - 12 T^{2} + 78 T^{4} - 300 T^{6} + 625 T^{8} )^{2} \))
$7$	(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))
$11$	(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} ) \))(\( 1 - 10 T^{2} + 121 T^{4} \))(\( ( 1 - 18 T^{2} + 121 T^{4} )^{4} \))
$13$	(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 12 T^{2} + 366 T^{4} - 2028 T^{6} + 28561 T^{8} )^{2} \))
$17$	(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4} )^{2} \))
$19$	(\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 36 T^{2} + 1038 T^{4} + 12996 T^{6} + 130321 T^{8} )^{2} \))
$23$	(\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} ) \))(\( 1 - 34 T^{2} + 529 T^{4} \))(\( ( 1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \))
$29$	(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} )^{4} \))
$31$	(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 60 T^{2} + 2310 T^{4} + 57660 T^{6} + 923521 T^{8} )^{2} \))
$37$	(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 2 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 46 T^{2} - 148 T^{3} + 1369 T^{4} )^{4} \))
$41$	(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))(\( ( 1 - 4 T^{2} + 3238 T^{4} - 6724 T^{6} + 2825761 T^{8} )^{2} \))
$43$	(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 100 T^{2} + 5686 T^{4} - 184900 T^{6} + 3418801 T^{8} )^{2} \))
$47$	(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 124 T^{2} + 7750 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{2} \))
$53$	(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 2 T + 53 T^{2} )^{8} \))
$59$	(\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))(\( ( 1 + 132 T^{2} + 10926 T^{4} + 459492 T^{6} + 12117361 T^{8} )^{2} \))
$61$	(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))(\( ( 1 - 236 T^{2} + 21358 T^{4} - 878156 T^{6} + 13845841 T^{8} )^{2} \))
$67$	(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 4 T + 67 T^{2} )( 1 + 4 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 4 T^{2} - 3818 T^{4} - 17956 T^{6} + 20151121 T^{8} )^{2} \))
$71$	(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} ) \))(\( 1 - 130 T^{2} + 5041 T^{4} \))(\( ( 1 - 196 T^{2} + 18534 T^{4} - 988036 T^{6} + 25411681 T^{8} )^{2} \))
$73$	(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 73 T^{2} )^{2} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))(\( ( 1 - 132 T^{2} + 8742 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2} \))
$79$	(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( 1 - 146 T^{2} + 6241 T^{4} \))(\( ( 1 - 100 T^{2} + 11782 T^{4} - 624100 T^{6} + 38950081 T^{8} )^{2} \))
$83$	(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 196 T^{2} + 19150 T^{4} + 1350244 T^{6} + 47458321 T^{8} )^{2} \))
$89$	(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))(\( ( 1 - 324 T^{2} + 41958 T^{4} - 2566404 T^{6} + 62742241 T^{8} )^{2} \))
$97$	(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 68 T^{2} + 19462 T^{4} - 639812 T^{6} + 88529281 T^{8} )^{2} \))
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