LMFDB - Space of Modular Forms of Level 225, Weight 2, and Character 225.f

Defining parameters

Level:	$ N $	=	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	225.f (of order $4$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	=	$ 15 $
Character field:			$\Q(i)$
Newform subspaces:			$ 2 $
Sturm bound:			$60$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(225, [\chi])$.

	Total	New	Old
Modular forms	84	12	72
Cusp forms	36	12	24
Eisenstein series	48	0	48

Trace form

Decomposition of $S_{2}^{\mathrm{new}}(225, [\chi])$ 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
225.2.f.a	$4$	$1.797$	$\Q(\zeta_{8})$	None	$0$	$0$	$0$	$8$	$q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(2-2\zeta_{8}^{2})q^{7}+\cdots$
225.2.f.b	$8$	$1.797$	$\Q(\zeta_{24})$	None	$0$	$0$	$0$	$0$	$q-\zeta_{24}^{6}q^{2}-4\zeta_{24}^{2}q^{4}+\zeta_{24}q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(225, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(225, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(45, [\chi])$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(75, [\chi])$$^{\oplus 2}$

Hecke Characteristic Polynomials

$p$	$F_p(T)$
$2$	($ 1 + T^{4} + 16 T^{8} $)($ ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} $)
$3$	1
$5$	1
$7$	($ ( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} )^{2} $)($ ( 1 + 23 T^{4} + 2401 T^{8} )^{2} $)
$11$	($ ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} $)($ ( 1 - 4 T^{2} + 121 T^{4} )^{4} $)
$13$	($ ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} $)($ ( 1 - 337 T^{4} + 28561 T^{8} )^{2} $)
$17$	($ 1 - 254 T^{4} + 83521 T^{8} $)($ ( 1 + 206 T^{4} + 83521 T^{8} )^{2} $)
$19$	($ ( 1 - 19 T^{2} )^{4} $)($ ( 1 - 13 T^{2} + 361 T^{4} )^{4} $)
$23$	($ 1 - 158 T^{4} + 279841 T^{8} $)($ ( 1 + 542 T^{4} + 279841 T^{8} )^{2} $)
$29$	($ ( 1 + 40 T^{2} + 841 T^{4} )^{2} $)($ ( 1 + 40 T^{2} + 841 T^{4} )^{4} $)
$31$	($ ( 1 + 4 T + 31 T^{2} )^{4} $)($ ( 1 - T + 31 T^{2} )^{8} $)
$37$	($ ( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} $)($ ( 1 + 1106 T^{4} + 1874161 T^{8} )^{2} $)
$41$	($ ( 1 - 80 T^{2} + 1681 T^{4} )^{2} $)($ ( 1 - 10 T^{2} + 1681 T^{4} )^{4} $)
$43$	($ ( 1 - 16 T + 128 T^{2} - 688 T^{3} + 1849 T^{4} )^{2} $)($ ( 1 + 3191 T^{4} + 3418801 T^{8} )^{2} $)
$47$	($ 1 - 3518 T^{4} + 4879681 T^{8} $)($ ( 1 - 2818 T^{4} + 4879681 T^{8} )^{2} $)
$53$	($ ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) $)($ ( 1 - 5518 T^{4} + 7890481 T^{8} )^{2} $)
$59$	($ ( 1 + 46 T^{2} + 3481 T^{4} )^{2} $)($ ( 1 - 44 T^{2} + 3481 T^{4} )^{4} $)
$61$	($ ( 1 - 8 T + 61 T^{2} )^{4} $)($ ( 1 + 7 T + 61 T^{2} )^{8} $)
$67$	($ ( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} $)($ ( 1 + 2471 T^{4} + 20151121 T^{8} )^{2} $)
$71$	($ ( 1 - 110 T^{2} + 5041 T^{4} )^{2} $)($ ( 1 - 70 T^{2} + 5041 T^{4} )^{4} $)
$73$	($ ( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} $)($ ( 1 + 7298 T^{4} + 28398241 T^{8} )^{2} $)
$79$	($ ( 1 - 14 T^{2} + 6241 T^{4} )^{2} $)($ ( 1 - 154 T^{2} + 6241 T^{4} )^{4} $)
$83$	($ 1 + 8722 T^{4} + 47458321 T^{8} $)($ ( 1 + 11822 T^{4} + 47458321 T^{8} )^{2} $)
$89$	($ ( 1 + 16 T^{2} + 7921 T^{4} )^{2} $)($ ( 1 + 106 T^{2} + 7921 T^{4} )^{4} $)
$97$	($ ( 1 - 22 T + 242 T^{2} - 2134 T^{3} + 9409 T^{4} )^{2} $)($ ( 1 - 16609 T^{4} + 88529281 T^{8} )^{2} $)
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Introduction	Features
Universe	News

Level:	\( N \)	=	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	225.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	=	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(60\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
Label	Dim.	\(A\)	Field	CM	\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)	$q$-expansion
225.2.f.a	\(4\)	\(1.797\)	\(\Q(\zeta_{8})\)	None	\(0\)	\(0\)	\(0\)	\(8\)	\(q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(2-2\zeta_{8}^{2})q^{7}+\cdots\)
225.2.f.b	\(8\)	\(1.797\)	\(\Q(\zeta_{24})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q-\zeta_{24}^{6}q^{2}-4\zeta_{24}^{2}q^{4}+\zeta_{24}q^{7}+\cdots\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(225, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(225, [\chi])\) 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	225.2.f

Level	225
Weight	2
Character orbit	f
Rep. character	\(\chi_{225}(107,\cdot)\)
Character field	\(\Q(\zeta_{4})\)
Dimension	12
Newform subspaces	2
Sturm bound	60
Trace bound	1

$p$	$F_p(T)$
$2$	(\( 1 + T^{4} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))
$3$	1
$5$	1
$7$	(\( ( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 + 23 T^{4} + 2401 T^{8} )^{2} \))
$11$	(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T^{2} + 121 T^{4} )^{4} \))
$13$	(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 337 T^{4} + 28561 T^{8} )^{2} \))
$17$	(\( 1 - 254 T^{4} + 83521 T^{8} \))(\( ( 1 + 206 T^{4} + 83521 T^{8} )^{2} \))
$19$	(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 13 T^{2} + 361 T^{4} )^{4} \))
$23$	(\( 1 - 158 T^{4} + 279841 T^{8} \))(\( ( 1 + 542 T^{4} + 279841 T^{8} )^{2} \))
$29$	(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{4} \))
$31$	(\( ( 1 + 4 T + 31 T^{2} )^{4} \))(\( ( 1 - T + 31 T^{2} )^{8} \))
$37$	(\( ( 1 + 2 T + 2 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 1106 T^{4} + 1874161 T^{8} )^{2} \))
$41$	(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 1681 T^{4} )^{4} \))
$43$	(\( ( 1 - 16 T + 128 T^{2} - 688 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 3191 T^{4} + 3418801 T^{8} )^{2} \))
$47$	(\( 1 - 3518 T^{4} + 4879681 T^{8} \))(\( ( 1 - 2818 T^{4} + 4879681 T^{8} )^{2} \))
$53$	(\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))(\( ( 1 - 5518 T^{4} + 7890481 T^{8} )^{2} \))
$59$	(\( ( 1 + 46 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 3481 T^{4} )^{4} \))
$61$	(\( ( 1 - 8 T + 61 T^{2} )^{4} \))(\( ( 1 + 7 T + 61 T^{2} )^{8} \))
$67$	(\( ( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 2471 T^{4} + 20151121 T^{8} )^{2} \))
$71$	(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5041 T^{4} )^{4} \))
$73$	(\( ( 1 + 2 T + 2 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 7298 T^{4} + 28398241 T^{8} )^{2} \))
$79$	(\( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{4} \))
$83$	(\( 1 + 8722 T^{4} + 47458321 T^{8} \))(\( ( 1 + 11822 T^{4} + 47458321 T^{8} )^{2} \))
$89$	(\( ( 1 + 16 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 7921 T^{4} )^{4} \))
$97$	(\( ( 1 - 22 T + 242 T^{2} - 2134 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16609 T^{4} + 88529281 T^{8} )^{2} \))
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