Space of modular forms of level 1395, weight 2, and trivial character

• Label: 1395.2.a
• Level: $1395$
• Weight: $2$
• Character orbit: 1395.a
• Rep. character: $\chi_{1395}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $17$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1395.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1395))\).

	Total	New	Old
Modular forms	200	50	150
Cusp forms	185	50	135
Eisenstein series	15	0	15

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

\(3\)	\(5\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(10\)
Plus space			\(+\)	\(21\)
Minus space			\(-\)	\(29\)

Trace form

50 * q - 2 * q^2 + 46 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8 - 2 * q^10 - 4 * q^11 + 16 * q^14 + 38 * q^16 + 8 * q^17 - 20 * q^19 + 6 * q^20 + 4 * q^22 - 12 * q^23 + 50 * q^25 - 8 * q^26 - 8 * q^28 + 8 * q^29 - 14 * q^32 + 28 * q^34 + 4 * q^35 - 4 * q^37 + 36 * q^38 - 6 * q^40 + 20 * q^41 - 24 * q^43 + 8 * q^44 - 12 * q^46 + 40 * q^47 + 42 * q^49 - 2 * q^50 + 56 * q^52 - 28 * q^53 + 8 * q^55 + 48 * q^56 + 40 * q^58 + 8 * q^59 - 12 * q^61 + 8 * q^62 + 70 * q^64 - 12 * q^65 - 28 * q^67 + 12 * q^68 + 8 * q^70 - 20 * q^71 - 44 * q^73 + 44 * q^74 - 24 * q^76 - 28 * q^77 - 12 * q^79 + 14 * q^80 - 8 * q^82 - 4 * q^83 - 12 * q^85 - 32 * q^86 - 8 * q^88 - 8 * q^89 + 28 * q^92 + 40 * q^94 + 4 * q^95 - 28 * q^97 - 50 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1395))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	31
1395.2.a.a	$1395$	$2$	1395.a	1.a	$1$	$1$	$1$	$11.139$	\(\Q\)	None	✓				465.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}-2q^{7}+3q^{8}+q^{10}+\cdots\)
1395.2.a.b	$1395$	$2$	1395.a	1.a	$1$	$1$	$1$	$11.139$	\(\Q\)	None	✓				155.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+4q^{11}-6q^{13}+4q^{16}+\cdots\)
1395.2.a.c	$1395$	$2$	1395.a	1.a	$1$	$1$	$1$	$11.139$	\(\Q\)	None	✓				465.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-4q^{7}-3q^{8}-q^{10}+\cdots\)
1395.2.a.d	$1395$	$2$	1395.a	1.a	$1$	$1$	$1$	$11.139$	\(\Q\)	None	✓				155.2.a.b	$1$	$0$	\(1\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}+4q^{7}-3q^{8}+q^{10}+\cdots\)
1395.2.a.e	$1395$	$2$	1395.a	1.a	$1$	$1$	$1$	$11.139$	\(\Q\)	None	✓				155.2.a.a	$1$	$1$	\(2\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)
1395.2.a.f	$1395$	$2$	1395.a	1.a	$1$	$2$	$2$	$11.139$	\(\Q(\sqrt{3}) \)	None	✓				465.2.a.d	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+q^{5}+(-3+\beta )q^{7}-\beta q^{8}+\cdots\)
1395.2.a.g	$1395$	$2$	1395.a	1.a	$1$	$2$	$2$	$11.139$	\(\Q(\sqrt{2}) \)	None	✓				465.2.a.c	$1$	$0$	\(2\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}+(-2+\cdots)q^{7}+\cdots\)
1395.2.a.h	$1395$	$2$	1395.a	1.a	$1$	$3$	$3$	$11.139$	3.3.148.1	None	✓		✓		465.2.a.g	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1395.2.a.i	$1395$	$2$	1395.a	1.a	$1$	$3$	$3$	$11.139$	3.3.148.1	None	✓		✓		465.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
1395.2.a.j	$1395$	$2$	1395.a	1.a	$1$	$3$	$3$	$11.139$	3.3.564.1	None	✓				465.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
1395.2.a.k	$1395$	$2$	1395.a	1.a	$1$	$4$	$4$	$11.139$	4.4.8468.1	None	✓				465.2.a.h	$1$	$0$	\(-2\)	\(0\)	\(-4\)	\(4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(2-\beta _{1})q^{4}-q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
1395.2.a.l	$1395$	$2$	1395.a	1.a	$1$	$4$	$4$	$11.139$	4.4.8468.1	None	✓				155.2.a.e	$1$	$0$	\(-1\)	\(0\)	\(-4\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{4}-q^{5}+\cdots\)
1395.2.a.m	$1395$	$2$	1395.a	1.a	$1$	$4$	$4$	$11.139$	4.4.20308.1	None	✓		✓		155.2.a.d	$1$	$0$	\(1\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+q^{5}+\beta _{2}q^{7}+\cdots\)
1395.2.a.n	$1395$	$2$	1395.a	1.a	$1$	$5$	$5$	$11.139$	5.5.223824.1	None	✓	✓	✓	✓	1395.2.a.n	$1$	$1$	\(-3\)	\(0\)	\(5\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1395.2.a.o	$1395$	$2$	1395.a	1.a	$1$	$5$	$5$	$11.139$	5.5.582992.1	None	✓	✓	✓	✓	1395.2.a.o	$1$	$1$	\(-1\)	\(0\)	\(-5\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots\)
1395.2.a.p	$1395$	$2$	1395.a	1.a	$1$	$5$	$5$	$11.139$	5.5.582992.1	None	✓	✓	✓	✓	1395.2.a.o	$1$	$0$	\(1\)	\(0\)	\(5\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots\)
1395.2.a.q	$1395$	$2$	1395.a	1.a	$1$	$5$	$5$	$11.139$	5.5.223824.1	None	✓	✓	✓	✓	1395.2.a.n	$1$	$0$	\(3\)	\(0\)	\(-5\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1395))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1395)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(279))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 2}\)