Space of modular forms of level 612, weight 4, and trivial character

• Label: 612.4.a
• Level: $612$
• Weight: $4$
• Character orbit: 612.a
• Rep. character: $\chi_{612}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $10$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	612.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(612))\).

	Total	New	Old
Modular forms	336	20	316
Cusp forms	312	20	292
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(17\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(9\)

Trace form

20 * q - 18 * q^5 - 20 * q^7 + 6 * q^11 + 4 * q^13 - 34 * q^17 + 104 * q^19 - 24 * q^23 + 160 * q^25 - 42 * q^29 + 116 * q^31 - 24 * q^35 + 342 * q^37 + 156 * q^41 - 100 * q^43 + 192 * q^47 + 1316 * q^49 - 132 * q^53 + 1016 * q^55 - 780 * q^59 - 962 * q^61 + 1140 * q^65 + 708 * q^67 - 1080 * q^71 + 2928 * q^77 - 784 * q^79 - 1188 * q^83 - 238 * q^85 + 168 * q^91 + 3912 * q^95 + 1588 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(612))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	17
612.4.a.a	$612$	$4$	612.a	1.a	$1$	$1$	$1$	$36.109$	\(\Q\)	None	✓	✓			612.4.a.a	$1$	$1$	\(0\)	\(0\)	\(-17\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-17q^{5}+6q^{7}+17q^{11}+43q^{13}+\cdots\)
612.4.a.b	$612$	$4$	612.a	1.a	$1$	$1$	$1$	$36.109$	\(\Q\)	None	✓				204.4.a.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-2^{4}q^{7}+57q^{11}-5^{2}q^{13}+\cdots\)
612.4.a.c	$612$	$4$	612.a	1.a	$1$	$1$	$1$	$36.109$	\(\Q\)	None	✓				68.4.a.a	$1$	$1$	\(0\)	\(0\)	\(8\)	\(-12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{5}-12q^{7}+10q^{11}-38q^{13}+\cdots\)
612.4.a.d	$612$	$4$	612.a	1.a	$1$	$1$	$1$	$36.109$	\(\Q\)	None	✓	✓			612.4.a.a	$1$	$0$	\(0\)	\(0\)	\(17\)	\(6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+17q^{5}+6q^{7}-17q^{11}+43q^{13}+\cdots\)
612.4.a.e	$612$	$4$	612.a	1.a	$1$	$2$	$2$	$36.109$	\(\Q(\sqrt{217}) \)	None	✓				204.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-19\)	\(18\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-9-\beta )q^{5}+(8+2\beta )q^{7}+(-9+3\beta )q^{11}+\cdots\)
612.4.a.f	$612$	$4$	612.a	1.a	$1$	$2$	$2$	$36.109$	\(\Q(\sqrt{201}) \)	None	✓				204.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}-2\beta q^{7}+(5+5\beta )q^{11}+\cdots\)
612.4.a.g	$612$	$4$	612.a	1.a	$1$	$3$	$3$	$36.109$	3.3.1524.1	None	✓				68.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-26\)	\(8\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-9+\beta _{1})q^{5}+(2+3\beta _{1}-\beta _{2})q^{7}+\cdots\)
612.4.a.h	$612$	$4$	612.a	1.a	$1$	$3$	$3$	$36.109$	3.3.104664.1	None	✓	✓	✓		612.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-19\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-6-\beta _{1})q^{5}+(-2-4\beta _{1}+\beta _{2})q^{7}+\cdots\)
612.4.a.i	$612$	$4$	612.a	1.a	$1$	$3$	$3$	$36.109$	3.3.104664.1	None	✓	✓	✓		612.4.a.h	$1$	$1$	\(0\)	\(0\)	\(19\)	\(-10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(6+\beta _{1})q^{5}+(-2-4\beta _{1}+\beta _{2})q^{7}+\cdots\)
612.4.a.j	$612$	$4$	612.a	1.a	$1$	$3$	$3$	$36.109$	3.3.21324.1	None	✓				204.4.a.d	$1$	$0$	\(0\)	\(0\)	\(19\)	\(-8\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(7+\beta _{1}-\beta _{2})q^{5}+(-3+\beta _{1}+2\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(612))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(612)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 2}\)