Space of modular forms of level 845, weight 2, and trivial character

• Label: 845.2.a
• Level: $845$
• Weight: $2$
• Character orbit: 845.a
• Rep. character: $\chi_{845}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $15$
• Sturm bound: $182$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(182\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(845))\).

	Total	New	Old
Modular forms	104	51	53
Cusp forms	77	51	26
Eisenstein series	27	0	27

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

\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(9\)
\(+\)	\(-\)	\(-\)	\(16\)
\(-\)	\(+\)	\(-\)	\(16\)
\(-\)	\(-\)	\(+\)	\(10\)
Plus space		\(+\)	\(19\)
Minus space		\(-\)	\(32\)

Trace form

51 * q + 3 * q^2 + 53 * q^4 + q^5 - 4 * q^7 + 3 * q^8 + 55 * q^9 + q^10 + 4 * q^12 + 4 * q^14 + 61 * q^16 - 2 * q^17 + 11 * q^18 + 4 * q^19 - q^20 - 4 * q^21 - 4 * q^24 + 51 * q^25 - 12 * q^27 - 28 * q^28 + 6 * q^29 - 8 * q^30 - 12 * q^31 - q^32 + 20 * q^33 - 6 * q^34 - 4 * q^35 + 37 * q^36 + 10 * q^37 - 48 * q^38 + 9 * q^40 + 18 * q^41 - 16 * q^42 - 12 * q^43 - 4 * q^44 + 5 * q^45 - 8 * q^46 - 12 * q^47 - 12 * q^48 + 35 * q^49 + 3 * q^50 + 20 * q^51 - 2 * q^53 + 20 * q^54 - 12 * q^55 + 4 * q^57 - 2 * q^58 - 12 * q^59 + 12 * q^60 - 6 * q^61 - 8 * q^62 + 4 * q^63 + 49 * q^64 - 68 * q^66 + 16 * q^67 - 34 * q^68 - 20 * q^69 + 16 * q^70 - 16 * q^71 - 21 * q^72 + 14 * q^73 - 46 * q^74 - 12 * q^77 - 4 * q^79 - 17 * q^80 + 67 * q^81 - 18 * q^82 + 40 * q^83 + 28 * q^84 + 6 * q^85 - 36 * q^86 - 20 * q^87 + 4 * q^88 - 2 * q^89 - 23 * q^90 - 52 * q^92 - 60 * q^93 - 20 * q^94 - 12 * q^95 - 32 * q^96 + 2 * q^97 + 51 * q^98 + 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(845))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	13
845.2.a.a	$845$	$2$	845.a	1.a	$1$	$1$	$1$	$6.747$	\(\Q\)	None	✓				65.2.a.a	$1$	$0$	\(1\)	\(-2\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}+q^{5}-2q^{6}+4q^{7}+\cdots\)
845.2.a.b	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{5}) \)	None	✓				65.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
845.2.a.c	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{13}) \)	None	✓				65.2.e.b	$1$	$1$	\(-1\)	\(2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
845.2.a.d	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{3}) \)	None	✓				65.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(3+\beta )q^{6}+\cdots\)
845.2.a.e	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{5}) \)	None	✓				65.2.e.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
845.2.a.f	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{13}) \)	None	✓				65.2.e.b	$1$	$0$	\(1\)	\(2\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(1+\beta )q^{4}+q^{5}+\beta q^{6}+\cdots\)
845.2.a.g	$845$	$2$	845.a	1.a	$1$	$2$	$2$	$6.747$	\(\Q(\sqrt{2}) \)	None	✓				65.2.a.b	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)
845.2.a.h	$845$	$2$	845.a	1.a	$1$	$3$	$3$	$6.747$	\(\Q(\zeta_{14})^+\)	None	✓	✓			845.2.a.h	$1$	$1$	\(-1\)	\(-5\)	\(3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
845.2.a.i	$845$	$2$	845.a	1.a	$1$	$3$	$3$	$6.747$	3.3.564.1	None	✓				65.2.c.a	$1$	$1$	\(-1\)	\(-2\)	\(3\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
845.2.a.j	$845$	$2$	845.a	1.a	$1$	$3$	$3$	$6.747$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		845.2.a.h	$1$	$1$	\(1\)	\(-5\)	\(-3\)	\(5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
845.2.a.k	$845$	$2$	845.a	1.a	$1$	$3$	$3$	$6.747$	3.3.564.1	None	✓				65.2.c.a	$1$	$0$	\(1\)	\(-2\)	\(-3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
845.2.a.l	$845$	$2$	845.a	1.a	$1$	$4$	$4$	$6.747$	4.4.4752.1	None	✓		✓		65.2.m.a	$1$	$1$	\(-2\)	\(-2\)	\(4\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
845.2.a.m	$845$	$2$	845.a	1.a	$1$	$4$	$4$	$6.747$	4.4.4752.1	None	✓				65.2.m.a	$1$	$0$	\(2\)	\(-2\)	\(-4\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
845.2.a.n	$845$	$2$	845.a	1.a	$1$	$9$	$9$	$6.747$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		845.2.a.n	$1$	$0$	\(-3\)	\(7\)	\(-9\)	\(-7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{6})q^{2}+(1-\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)
845.2.a.o	$845$	$2$	845.a	1.a	$1$	$9$	$9$	$6.747$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		845.2.a.n	$1$	$0$	\(3\)	\(7\)	\(9\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{6})q^{2}+(1-\beta _{4})q^{3}+(2-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(845))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(845)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)