Space of modular forms of level 306, weight 4, and trivial character

• Label: 306.4.a
• Level: $306$
• Weight: $4$
• Character orbit: 306.a
• Rep. character: $\chi_{306}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $13$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	306.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(216\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(306))\).

	Total	New	Old
Modular forms	170	20	150
Cusp forms	154	20	134
Eisenstein series	16	0	16

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
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(13\)
Minus space			\(-\)	\(7\)

Trace form

20 * q + 80 * q^4 + 6 * q^5 + 40 * q^7 + 44 * q^10 + 6 * q^11 - 44 * q^13 + 24 * q^14 + 320 * q^16 - 34 * q^17 - 232 * q^19 + 24 * q^20 + 212 * q^22 + 588 * q^23 + 928 * q^25 - 72 * q^26 + 160 * q^28 - 354 * q^29 + 656 * q^31 + 68 * q^34 - 360 * q^35 - 210 * q^37 + 96 * q^38 + 176 * q^40 + 684 * q^41 + 116 * q^43 + 24 * q^44 + 336 * q^46 + 1176 * q^47 + 2084 * q^49 + 744 * q^50 - 176 * q^52 + 1020 * q^53 + 872 * q^55 + 96 * q^56 - 324 * q^58 + 84 * q^59 + 2374 * q^61 + 216 * q^62 + 1280 * q^64 - 348 * q^65 - 324 * q^67 - 136 * q^68 + 544 * q^70 - 1716 * q^71 + 1392 * q^73 - 1476 * q^74 - 928 * q^76 - 1200 * q^77 - 2500 * q^79 + 96 * q^80 - 184 * q^82 - 108 * q^83 + 986 * q^85 + 552 * q^86 + 848 * q^88 + 576 * q^89 - 1752 * q^91 + 2352 * q^92 - 1120 * q^94 - 2856 * q^95 - 3692 * q^97 + 2016 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(306))\) 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
306.4.a.a	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				102.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+12q^{7}-8q^{8}+\cdots\)
306.4.a.b	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓	✓	✓	✓	306.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(9\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+9q^{5}-10q^{7}-8q^{8}+\cdots\)
306.4.a.c	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				102.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(12\)	\(-22\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+12q^{5}-22q^{7}-8q^{8}+\cdots\)
306.4.a.d	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				34.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-16\)	\(24\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-2^{4}q^{5}+24q^{7}+8q^{8}+\cdots\)
306.4.a.e	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓	✓	✓	✓	306.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-9\)	\(-10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-9q^{5}-10q^{7}+8q^{8}+\cdots\)
306.4.a.f	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				102.4.a.a	$1$	$0$	\(2\)	\(0\)	\(3\)	\(20\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+3q^{5}+20q^{7}+8q^{8}+\cdots\)
306.4.a.g	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				102.4.a.b	$1$	$1$	\(2\)	\(0\)	\(5\)	\(-32\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-2^{5}q^{7}+8q^{8}+\cdots\)
306.4.a.h	$306$	$4$	306.a	1.a	$1$	$1$	$1$	$18.055$	\(\Q\)	None	✓				34.4.a.a	$1$	$0$	\(2\)	\(0\)	\(18\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+18q^{5}-10q^{7}+8q^{8}+\cdots\)
306.4.a.i	$306$	$4$	306.a	1.a	$1$	$2$	$2$	$18.055$	\(\Q(\sqrt{393}) \)	None	✓		✓		102.4.a.f	$1$	$0$	\(-4\)	\(0\)	\(-3\)	\(22\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-1-\beta )q^{5}+(12-2\beta )q^{7}+\cdots\)
306.4.a.j	$306$	$4$	306.a	1.a	$1$	$2$	$2$	$18.055$	\(\Q(\sqrt{13}) \)	None	✓		✓		34.4.a.c	$1$	$1$	\(-4\)	\(0\)	\(4\)	\(-6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2+4\beta )q^{5}+(-3-\beta )q^{7}+\cdots\)
306.4.a.k	$306$	$4$	306.a	1.a	$1$	$2$	$2$	$18.055$	\(\Q(\sqrt{15}) \)	None	✓		✓		102.4.a.e	$1$	$0$	\(4\)	\(0\)	\(-12\)	\(16\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-6+\beta )q^{5}+(8+\beta )q^{7}+\cdots\)
306.4.a.l	$306$	$4$	306.a	1.a	$1$	$3$	$3$	$18.055$	3.3.2747992.1	None	✓	✓	✓	✓	306.4.a.l	$1$	$0$	\(-6\)	\(0\)	\(-25\)	\(18\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-8-\beta _{1})q^{5}+(6+\beta _{2})q^{7}+\cdots\)
306.4.a.m	$306$	$4$	306.a	1.a	$1$	$3$	$3$	$18.055$	3.3.2747992.1	None	✓	✓	✓	✓	306.4.a.l	$1$	$0$	\(6\)	\(0\)	\(25\)	\(18\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(8+\beta _{1})q^{5}+(6+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(306))\) into lower level spaces

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