Space of modular forms of level 308, weight 4, and trivial character

• Label: 308.4.a
• Level: $308$
• Weight: $4$
• Character orbit: 308.a
• Rep. character: $\chi_{308}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(308))\).

	Total	New	Old
Modular forms	150	14	136
Cusp forms	138	14	124
Eisenstein series	12	0	12

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

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

Trace form

14 * q + 4 * q^3 + 12 * q^5 + 58 * q^9 - 22 * q^11 + 76 * q^13 - 152 * q^15 + 132 * q^17 + 40 * q^19 - 8 * q^23 + 870 * q^25 + 256 * q^27 - 52 * q^29 + 56 * q^31 - 176 * q^33 - 56 * q^35 - 512 * q^37 + 1096 * q^39 + 820 * q^41 - 328 * q^43 + 1472 * q^45 + 1152 * q^47 + 686 * q^49 + 832 * q^51 - 308 * q^53 - 396 * q^55 - 992 * q^57 - 1100 * q^59 + 1556 * q^61 + 1176 * q^63 - 128 * q^65 - 336 * q^67 + 332 * q^69 + 40 * q^71 + 3252 * q^73 + 932 * q^75 - 308 * q^77 - 1040 * q^79 - 1522 * q^81 + 1808 * q^83 + 848 * q^85 - 4736 * q^87 + 2464 * q^89 + 476 * q^91 - 812 * q^93 - 2296 * q^95 - 3008 * q^97 - 814 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\) 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	7	11
308.4.a.a	$308$	$4$	308.a	1.a	$1$	$1$	$1$	$18.173$	\(\Q\)	None	✓	✓			308.4.a.a	$1$	$1$	\(0\)	\(-7\)	\(-1\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-q^{5}+7q^{7}+22q^{9}+11q^{11}+\cdots\)
308.4.a.b	$308$	$4$	308.a	1.a	$1$	$1$	$1$	$18.173$	\(\Q\)	None	✓	✓			308.4.a.b	$1$	$1$	\(0\)	\(4\)	\(-12\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-12q^{5}+7q^{7}-11q^{9}+11q^{11}+\cdots\)
308.4.a.c	$308$	$4$	308.a	1.a	$1$	$3$	$3$	$18.173$	3.3.5925.1	None	✓	✓	✓	✓	308.4.a.c	$1$	$1$	\(0\)	\(5\)	\(9\)	\(-21\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(2+3\beta _{1})q^{5}-7q^{7}+\cdots\)
308.4.a.d	$308$	$4$	308.a	1.a	$1$	$4$	$4$	$18.173$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	308.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(1\)	\(-28\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{2}q^{5}-7q^{7}+(-5+\cdots)q^{9}+\cdots\)
308.4.a.e	$308$	$4$	308.a	1.a	$1$	$5$	$5$	$18.173$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	308.4.a.e	$1$	$0$	\(0\)	\(5\)	\(15\)	\(35\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(3+\beta _{2})q^{5}+7q^{7}+(20+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(308))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(308)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)