Space of modular forms of level 1305, weight 2, and Character 1305.f

• Label: 1305.2.f
• Level: $1305$
• Weight: $2$
• Character orbit: 1305.f
• Rep. character: $\chi_{1305}(289,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $13$
• Sturm bound: $360$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 145 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(360\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1305, [\chi])\).

	Total	New	Old
Modular forms	188	76	112
Cusp forms	172	72	100
Eisenstein series	16	4	12

Trace form

\( 72 q + 64 q^{4} + 6 q^{5} + 48 q^{16} + 38 q^{20} + 10 q^{25} - 32 q^{34} - 40 q^{49} + 16 q^{64} - 26 q^{65} - 56 q^{71} + 56 q^{74} + 74 q^{80} - 20 q^{86} + 8 q^{91} - 28 q^{94}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1305, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1305.2.f.a	$1305$	$2$	1305.f	145.d	$2$	$2$	$2$	$10.420$	\(\Q(\sqrt{-1}) \)	None					435.2.f.a	$2$	$1$	\(-4\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 q^{2}+2 q^{4}+(-i-2)q^{5}+4 i q^{7}+\cdots\)
1305.2.f.b	$1305$	$2$	1305.f	145.d	$2$	$2$	$2$	$10.420$	\(\Q(\sqrt{-1}) \)	None					435.2.f.b	$2$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-q^{4}+(-\beta+1)q^{5}+\beta q^{7}+\cdots\)
1305.2.f.c	$1305$	$2$	1305.f	145.d	$2$	$2$	$2$	$10.420$	\(\Q(\sqrt{-1}) \)	None					435.2.f.b	$2$	$0$	\(2\)	\(0\)	\(2\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-q^{4}+(-\beta+1)q^{5}+\beta q^{7}+\cdots\)
1305.2.f.d	$1305$	$2$	1305.f	145.d	$2$	$2$	$2$	$10.420$	\(\Q(\sqrt{-1}) \)	None					435.2.f.a	$2$	$0$	\(4\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 q^{2}+2 q^{4}+(-i-2)q^{5}+4 i q^{7}+\cdots\)
1305.2.f.e	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					145.2.d.a	$2$	$0$	\(-4\)	\(0\)	\(-3\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-q^{4}+(-1+\beta _{1})q^{5}-\beta _{3}q^{7}+\cdots\)
1305.2.f.f	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(\sqrt{5}, \sqrt{-6})\)	None		✓			1305.2.f.f	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}-q^{4}+\beta _{2}q^{5}+\beta _{1}q^{7}+3q^{8}+\cdots\)
1305.2.f.g	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(\sqrt{-5}, \sqrt{-29})\)	\(\Q(\sqrt{-435}) \)		✓			1305.2.f.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{4}-\beta _{1}q^{5}-\beta _{2}q^{11}+4q^{16}+\cdots\)
1305.2.f.h	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(i, \sqrt{5})\)	None					145.2.d.b	$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+3q^{4}+(1+\beta _{1})q^{5}+\beta _{1}q^{7}+\cdots\)
1305.2.f.i	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None					145.2.d.a	$2$	$0$	\(4\)	\(0\)	\(-3\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-q^{4}+(-1+\beta _{1})q^{5}-\beta _{3}q^{7}+\cdots\)
1305.2.f.j	$1305$	$2$	1305.f	145.d	$2$	$4$	$4$	$10.420$	\(\Q(\sqrt{5}, \sqrt{-6})\)	None		✓			1305.2.f.f	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{2}-q^{4}-\beta _{2}q^{5}-\beta _{1}q^{7}-3q^{8}+\cdots\)
1305.2.f.k	$1305$	$2$	1305.f	145.d	$2$	$12$	$12$	$10.420$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					435.2.f.e	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+(1-\beta _{3})q^{4}+(1-\beta _{1})q^{5}+\cdots\)
1305.2.f.l	$1305$	$2$	1305.f	145.d	$2$	$12$	$12$	$10.420$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					435.2.f.e	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(1-\beta _{3})q^{4}+(1-\beta _{1})q^{5}+\cdots\)
1305.2.f.m	$1305$	$2$	1305.f	145.d	$2$	$16$	$16$	$10.420$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		1305.2.f.m	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(2+\beta _{3})q^{4}-\beta _{14}q^{5}+\beta _{5}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1305, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1305, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(145, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(435, [\chi])\)\(^{\oplus 2}\)