Space of modular forms of level 1305, weight 2, and trivial character

• Label: 1305.2.a
• Level: $1305$
• Weight: $2$
• Character orbit: 1305.a
• Rep. character: $\chi_{1305}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $20$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1305))\).

	Total	New	Old
Modular forms	188	48	140
Cusp forms	173	48	125
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.

\(3\)	\(5\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(7\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(18\)
Minus space			\(-\)	\(30\)

Trace form

48 * q + 2 * q^2 + 56 * q^4 + 4 * q^7 + 6 * q^8 + 2 * q^10 - 4 * q^11 + 8 * q^13 + 8 * q^14 + 64 * q^16 + 16 * q^17 - 4 * q^19 - 8 * q^20 - 4 * q^23 + 48 * q^25 + 20 * q^26 + 20 * q^28 - 4 * q^31 + 14 * q^32 - 32 * q^34 - 4 * q^35 + 16 * q^37 + 16 * q^38 - 6 * q^40 + 8 * q^41 - 24 * q^43 - 28 * q^44 - 16 * q^46 + 80 * q^49 + 2 * q^50 + 8 * q^53 + 72 * q^56 - 2 * q^58 - 32 * q^59 + 16 * q^61 - 56 * q^62 + 64 * q^64 - 16 * q^65 - 20 * q^67 + 84 * q^68 - 12 * q^70 - 24 * q^71 - 36 * q^73 + 8 * q^74 - 28 * q^76 + 36 * q^77 - 12 * q^79 - 4 * q^82 + 20 * q^83 + 12 * q^85 + 56 * q^86 + 16 * q^88 - 40 * q^89 + 8 * q^91 + 20 * q^92 + 16 * q^94 + 16 * q^95 - 20 * q^97 - 34 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\) 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	29
1305.2.a.a	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓	✓			1305.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{5}-2q^{7}-2q^{10}+\cdots\)
1305.2.a.b	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓				435.2.a.d	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-q^{5}+4q^{7}+3q^{8}+q^{10}+\cdots\)
1305.2.a.c	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓				435.2.a.b	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-2q^{7}-q^{11}+6q^{13}+\cdots\)
1305.2.a.d	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓				435.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}+2q^{7}-3q^{11}+2q^{13}+\cdots\)
1305.2.a.e	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓				435.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-q^{5}-4q^{7}-3q^{8}-q^{10}+\cdots\)
1305.2.a.f	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓				145.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+q^{5}-2q^{7}-3q^{8}+q^{10}+\cdots\)
1305.2.a.g	$1305$	$2$	1305.a	1.a	$1$	$1$	$1$	$10.420$	\(\Q\)	None	✓	✓			1305.2.a.a	$1$	$1$	\(2\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)
1305.2.a.h	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1305.2.a.h	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+(-1-2\beta )q^{7}+\cdots\)
1305.2.a.i	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{17}) \)	None	✓				435.2.a.h	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}-q^{5}+(2-2\beta )q^{7}+\cdots\)
1305.2.a.j	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{5}) \)	None	✓				435.2.a.g	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}+q^{5}+2q^{7}-\beta q^{8}+\cdots\)
1305.2.a.k	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{5}) \)	None	✓				435.2.a.e	$1$	$1$	\(1\)	\(0\)	\(2\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+q^{5}-3q^{7}+\cdots\)
1305.2.a.l	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1305.2.a.h	$1$	$1$	\(1\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+q^{5}+(-1-2\beta )q^{7}+\cdots\)
1305.2.a.m	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{21}) \)	None	✓				435.2.a.f	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(3+\beta )q^{4}-q^{5}+q^{7}+(5+2\beta )q^{8}+\cdots\)
1305.2.a.n	$1305$	$2$	1305.a	1.a	$1$	$2$	$2$	$10.420$	\(\Q(\sqrt{2}) \)	None	✓				145.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(-2+\cdots)q^{7}+\cdots\)
1305.2.a.o	$1305$	$2$	1305.a	1.a	$1$	$3$	$3$	$10.420$	3.3.148.1	None	✓		✓		145.2.a.d	$1$	$1$	\(-3\)	\(0\)	\(3\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1305.2.a.p	$1305$	$2$	1305.a	1.a	$1$	$3$	$3$	$10.420$	3.3.148.1	None	✓				145.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1305.2.a.q	$1305$	$2$	1305.a	1.a	$1$	$3$	$3$	$10.420$	3.3.469.1	None	✓				435.2.a.i	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
1305.2.a.r	$1305$	$2$	1305.a	1.a	$1$	$4$	$4$	$10.420$	4.4.2225.1	None	✓		✓		435.2.a.j	$1$	$0$	\(3\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
1305.2.a.s	$1305$	$2$	1305.a	1.a	$1$	$7$	$7$	$10.420$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1305.2.a.s	$1$	$0$	\(-1\)	\(0\)	\(-7\)	\(10\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}+(1-\beta _{5}+\cdots)q^{7}+\cdots\)
1305.2.a.t	$1305$	$2$	1305.a	1.a	$1$	$7$	$7$	$10.420$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1305.2.a.s	$1$	$0$	\(1\)	\(0\)	\(7\)	\(10\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}+(1-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1305))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1305)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 2}\)