LMFDB - Space of modular forms of level 225, weight 6, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 22 $
Sturm bound:			$180$
Trace bound:			$7$
Distinguishing $T_p$:			$2$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(\Gamma_0(225))$.

	Total	New	Old
Modular forms	162	41	121
Cusp forms	138	38	100
Eisenstein series	24	3	21

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

$3$	$5$	Fricke	Dim.
$+$	$+$	$+$	$7$
$+$	$-$	$-$	$9$
$-$	$+$	$-$	$11$
$-$	$-$	$+$	$11$
Plus space		$+$	$18$
Minus space		$-$	$20$

Trace form

$ 38 q + 578 q^{4} - 80 q^{7} - 180 q^{8} + O(q^{10}) $

$ 38 q + 578 q^{4} - 80 q^{7} - 180 q^{8} + 324 q^{11} + 100 q^{13} - 1896 q^{14} + 8330 q^{16} + 1860 q^{17} + 4036 q^{19} + 2640 q^{22} + 2400 q^{23} - 13524 q^{26} - 9920 q^{28} + 8916 q^{29} - 164 q^{31} - 20460 q^{32} + 10650 q^{34} + 12340 q^{37} + 61560 q^{38} - 22428 q^{41} - 11240 q^{43} + 14862 q^{44} + 39948 q^{46} - 31920 q^{47} + 48330 q^{49} + 4840 q^{52} + 77340 q^{53} - 118260 q^{56} - 114000 q^{58} - 19788 q^{59} + 25264 q^{61} - 51000 q^{62} + 149786 q^{64} - 18200 q^{67} + 291240 q^{68} - 109584 q^{71} - 177380 q^{73} + 179124 q^{74} + 265450 q^{76} + 24000 q^{77} + 79336 q^{79} + 210000 q^{82} + 120120 q^{83} - 297756 q^{86} + 87360 q^{88} - 210492 q^{89} + 216188 q^{91} - 256320 q^{92} + 343512 q^{94} - 35540 q^{97} + 367440 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{6}^{\mathrm{new}}(\Gamma_0(225))$ into newform subspaces

Label	Dim.	$A$	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	$A$	Field	CM	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	3	5	$q$-expansion
225.6.a.a	$1$	$36.086$	$\Q$	None	$-6$	$0$	$0$	$40$	$-$	$+$	$q-6q^{2}+4q^{4}+40q^{7}+168q^{8}+564q^{11}+\cdots$
225.6.a.b	$1$	$36.086$	$\Q$	None	$-4$	$0$	$0$	$-225$	$-$	$-$	$q-4q^{2}-2^{4}q^{4}-15^{2}q^{7}+192q^{8}+\cdots$
225.6.a.c	$1$	$36.086$	$\Q$	None	$-2$	$0$	$0$	$132$	$-$	$+$	$q-2q^{2}-28q^{4}+132q^{7}+120q^{8}+\cdots$
225.6.a.d	$1$	$36.086$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$-25$	$+$	$-$	$q-2^{5}q^{4}-5^{2}q^{7}-775q^{13}+2^{10}q^{16}+\cdots$
225.6.a.e	$1$	$36.086$	$\Q$	$\Q(\sqrt{-3}) $	$0$	$0$	$0$	$25$	$+$	$+$	$q-2^{5}q^{4}+5^{2}q^{7}+775q^{13}+2^{10}q^{16}+\cdots$
225.6.a.f	$1$	$36.086$	$\Q$	None	$2$	$0$	$0$	$-192$	$-$	$+$	$q+2q^{2}-28q^{4}-192q^{7}-120q^{8}+\cdots$
225.6.a.g	$1$	$36.086$	$\Q$	None	$4$	$0$	$0$	$225$	$-$	$+$	$q+4q^{2}-2^{4}q^{4}+15^{2}q^{7}-192q^{8}+\cdots$
225.6.a.h	$1$	$36.086$	$\Q$	None	$7$	$0$	$0$	$-12$	$-$	$+$	$q+7q^{2}+17q^{4}-12q^{7}-105q^{8}+\cdots$
225.6.a.i	$2$	$36.086$	$\Q(\sqrt{89}) $	None	$-9$	$0$	$0$	$108$	$-$	$-$	$q+(-4-\beta )q^{2}+(6+9\beta )q^{4}+(42+24\beta )q^{7}+\cdots$
225.6.a.j	$2$	$36.086$	$\Q(\sqrt{31}) $	None	$-6$	$0$	$0$	$102$	$-$	$+$	$q+(-3+\beta )q^{2}+(8-6\beta )q^{4}+(51-24\beta )q^{7}+\cdots$
225.6.a.k	$2$	$36.086$	$\Q(\sqrt{145}) $	None	$-5$	$0$	$0$	$-80$	$+$	$+$	$q+(-2-\beta )q^{2}+(8+5\beta )q^{4}+(-30+\cdots)q^{7}+\cdots$
225.6.a.l	$2$	$36.086$	$\Q(\sqrt{241}) $	None	$-5$	$0$	$0$	$200$	$-$	$-$	$q+(-2-\beta )q^{2}+(2^{5}+5\beta )q^{4}+(102+\cdots)q^{7}+\cdots$
225.6.a.m	$2$	$36.086$	$\Q(\sqrt{409}) $	None	$-1$	$0$	$0$	$112$	$-$	$+$	$q-\beta q^{2}+(70+\beta )q^{4}+(2^{6}-2^{4}\beta )q^{7}+\cdots$
225.6.a.n	$2$	$36.086$	$\Q(\sqrt{11}) $	None	$0$	$0$	$0$	$0$	$-$	$-$	$q+\beta q^{2}+12q^{4}+9\beta q^{7}-20\beta q^{8}+\cdots$
225.6.a.o	$2$	$36.086$	$\Q(\sqrt{70}) $	None	$0$	$0$	$0$	$-90$	$+$	$+$	$q+\beta q^{2}+38q^{4}-45q^{7}+6\beta q^{8}-40\beta q^{11}+\cdots$
225.6.a.p	$2$	$36.086$	$\Q(\sqrt{70}) $	None	$0$	$0$	$0$	$90$	$+$	$-$	$q+\beta q^{2}+38q^{4}+45q^{7}+6\beta q^{8}+40\beta q^{11}+\cdots$
225.6.a.q	$2$	$36.086$	$\Q(\sqrt{5}) $	$\Q(\sqrt{-15}) $	$0$	$0$	$0$	$0$	$+$	$-$	$q-\beta q^{2}+93q^{4}-61\beta q^{8}+4649q^{16}+\cdots$
225.6.a.r	$2$	$36.086$	$\Q(\sqrt{145}) $	None	$5$	$0$	$0$	$-80$	$+$	$+$	$q+(3-\beta )q^{2}+(13-5\beta )q^{4}+(-50+20\beta )q^{7}+\cdots$
225.6.a.s	$2$	$36.086$	$\Q(\sqrt{241}) $	None	$5$	$0$	$0$	$-200$	$-$	$+$	$q+(3-\beta )q^{2}+(37-5\beta )q^{4}+(-98-4\beta )q^{7}+\cdots$
225.6.a.t	$2$	$36.086$	$\Q(\sqrt{31}) $	None	$6$	$0$	$0$	$-102$	$-$	$-$	$q+(3+\beta )q^{2}+(8+6\beta )q^{4}+(-51-24\beta )q^{7}+\cdots$
225.6.a.u	$2$	$36.086$	$\Q(\sqrt{89}) $	None	$9$	$0$	$0$	$-108$	$-$	$-$	$q+(5-\beta )q^{2}+(15-9\beta )q^{4}+(-66+24\beta )q^{7}+\cdots$
225.6.a.v	$4$	$36.086$	$\Q(\sqrt{5}, \sqrt{14})$	None	$0$	$0$	$0$	$0$	$+$	$-$	$q-\beta _{1}q^{2}-12q^{4}-\beta _{2}q^{7}+44\beta _{1}q^{8}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(\Gamma_0(225))$ into lower level spaces

$ S_{6}^{\mathrm{old}}(\Gamma_0(225)) \cong $ $S_{6}^{\mathrm{new}}(\Gamma_0(3))$$^{\oplus 6}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$^{\oplus 6}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(25))$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 2}$$\oplus$$S_{6}^{\mathrm{new}}(\Gamma_0(75))$$^{\oplus 2}$

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Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(180\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

\(3\)	\(5\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(+\)	\(11\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(20\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\) into newform subspaces

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(225))\) into lower level spaces

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	225.6.a

Level	$225$
Weight	$6$
Character orbit	225.a
Rep. character	$\chi_{225}(1,\cdot)$
Character field	$\Q$
Dimension	$38$
Newform subspaces	$22$
Sturm bound	$180$
Trace bound	$7$

Label	Dim.	\(A\)	Field	CM	Traces				A-L signs		$q$-expansion
Label	Dim.	\(A\)	Field	CM	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	3	5	$q$-expansion
225.6.a.a	$1$	$36.086$	\(\Q\)	None	\(-6\)	\(0\)	\(0\)	\(40\)	$-$	$+$	\(q-6q^{2}+4q^{4}+40q^{7}+168q^{8}+564q^{11}+\cdots\)
225.6.a.b	$1$	$36.086$	\(\Q\)	None	\(-4\)	\(0\)	\(0\)	\(-225\)	$-$	$-$	\(q-4q^{2}-2^{4}q^{4}-15^{2}q^{7}+192q^{8}+\cdots\)
225.6.a.c	$1$	$36.086$	\(\Q\)	None	\(-2\)	\(0\)	\(0\)	\(132\)	$-$	$+$	\(q-2q^{2}-28q^{4}+132q^{7}+120q^{8}+\cdots\)
225.6.a.d	$1$	$36.086$	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(-25\)	$+$	$-$	\(q-2^{5}q^{4}-5^{2}q^{7}-775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.e	$1$	$36.086$	\(\Q\)	\(\Q(\sqrt{-3}) \)	\(0\)	\(0\)	\(0\)	\(25\)	$+$	$+$	\(q-2^{5}q^{4}+5^{2}q^{7}+775q^{13}+2^{10}q^{16}+\cdots\)
225.6.a.f	$1$	$36.086$	\(\Q\)	None	\(2\)	\(0\)	\(0\)	\(-192\)	$-$	$+$	\(q+2q^{2}-28q^{4}-192q^{7}-120q^{8}+\cdots\)
225.6.a.g	$1$	$36.086$	\(\Q\)	None	\(4\)	\(0\)	\(0\)	\(225\)	$-$	$+$	\(q+4q^{2}-2^{4}q^{4}+15^{2}q^{7}-192q^{8}+\cdots\)
225.6.a.h	$1$	$36.086$	\(\Q\)	None	\(7\)	\(0\)	\(0\)	\(-12\)	$-$	$+$	\(q+7q^{2}+17q^{4}-12q^{7}-105q^{8}+\cdots\)
225.6.a.i	$2$	$36.086$	\(\Q(\sqrt{89}) \)	None	\(-9\)	\(0\)	\(0\)	\(108\)	$-$	$-$	\(q+(-4-\beta )q^{2}+(6+9\beta )q^{4}+(42+24\beta )q^{7}+\cdots\)
225.6.a.j	$2$	$36.086$	\(\Q(\sqrt{31}) \)	None	\(-6\)	\(0\)	\(0\)	\(102\)	$-$	$+$	\(q+(-3+\beta )q^{2}+(8-6\beta )q^{4}+(51-24\beta )q^{7}+\cdots\)
225.6.a.k	$2$	$36.086$	\(\Q(\sqrt{145}) \)	None	\(-5\)	\(0\)	\(0\)	\(-80\)	$+$	$+$	\(q+(-2-\beta )q^{2}+(8+5\beta )q^{4}+(-30+\cdots)q^{7}+\cdots\)
225.6.a.l	$2$	$36.086$	\(\Q(\sqrt{241}) \)	None	\(-5\)	\(0\)	\(0\)	\(200\)	$-$	$-$	\(q+(-2-\beta )q^{2}+(2^{5}+5\beta )q^{4}+(102+\cdots)q^{7}+\cdots\)
225.6.a.m	$2$	$36.086$	\(\Q(\sqrt{409}) \)	None	\(-1\)	\(0\)	\(0\)	\(112\)	$-$	$+$	\(q-\beta q^{2}+(70+\beta )q^{4}+(2^{6}-2^{4}\beta )q^{7}+\cdots\)
225.6.a.n	$2$	$36.086$	\(\Q(\sqrt{11}) \)	None	\(0\)	\(0\)	\(0\)	\(0\)	$-$	$-$	\(q+\beta q^{2}+12q^{4}+9\beta q^{7}-20\beta q^{8}+\cdots\)
225.6.a.o	$2$	$36.086$	\(\Q(\sqrt{70}) \)	None	\(0\)	\(0\)	\(0\)	\(-90\)	$+$	$+$	\(q+\beta q^{2}+38q^{4}-45q^{7}+6\beta q^{8}-40\beta q^{11}+\cdots\)
225.6.a.p	$2$	$36.086$	\(\Q(\sqrt{70}) \)	None	\(0\)	\(0\)	\(0\)	\(90\)	$+$	$-$	\(q+\beta q^{2}+38q^{4}+45q^{7}+6\beta q^{8}+40\beta q^{11}+\cdots\)
225.6.a.q	$2$	$36.086$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-15}) \)	\(0\)	\(0\)	\(0\)	\(0\)	$+$	$-$	\(q-\beta q^{2}+93q^{4}-61\beta q^{8}+4649q^{16}+\cdots\)
225.6.a.r	$2$	$36.086$	\(\Q(\sqrt{145}) \)	None	\(5\)	\(0\)	\(0\)	\(-80\)	$+$	$+$	\(q+(3-\beta )q^{2}+(13-5\beta )q^{4}+(-50+20\beta )q^{7}+\cdots\)
225.6.a.s	$2$	$36.086$	\(\Q(\sqrt{241}) \)	None	\(5\)	\(0\)	\(0\)	\(-200\)	$-$	$+$	\(q+(3-\beta )q^{2}+(37-5\beta )q^{4}+(-98-4\beta )q^{7}+\cdots\)
225.6.a.t	$2$	$36.086$	\(\Q(\sqrt{31}) \)	None	\(6\)	\(0\)	\(0\)	\(-102\)	$-$	$-$	\(q+(3+\beta )q^{2}+(8+6\beta )q^{4}+(-51-24\beta )q^{7}+\cdots\)
225.6.a.u	$2$	$36.086$	\(\Q(\sqrt{89}) \)	None	\(9\)	\(0\)	\(0\)	\(-108\)	$-$	$-$	\(q+(5-\beta )q^{2}+(15-9\beta )q^{4}+(-66+24\beta )q^{7}+\cdots\)
225.6.a.v	$4$	$36.086$	\(\Q(\sqrt{5}, \sqrt{14})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	$+$	$-$	\(q-\beta _{1}q^{2}-12q^{4}-\beta _{2}q^{7}+44\beta _{1}q^{8}+\cdots\)