Space of modular forms of level 4304, weight 2, and trivial character

• Label: 4304.2.a
• Level: $4304$
• Weight: $2$
• Character orbit: 4304.a
• Rep. character: $\chi_{4304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $134$
• Newform subspaces: $15$
• Sturm bound: $1080$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(1080\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4304))\).

	Total	New	Old
Modular forms	546	134	412
Cusp forms	535	134	401
Eisenstein series	11	0	11

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

\(2\)	\(269\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(28\)
\(+\)	\(-\)	\(-\)	\(39\)
\(-\)	\(+\)	\(-\)	\(39\)
\(-\)	\(-\)	\(+\)	\(28\)
Plus space		\(+\)	\(56\)
Minus space		\(-\)	\(78\)

Trace form

134 * q + 2 * q^3 + 2 * q^7 + 130 * q^9 - 4 * q^11 - 4 * q^15 - 4 * q^17 + 2 * q^19 + 8 * q^23 + 134 * q^25 + 8 * q^27 + 8 * q^29 + 2 * q^31 + 12 * q^35 + 8 * q^37 + 20 * q^39 - 4 * q^41 - 4 * q^43 - 12 * q^47 + 134 * q^49 + 16 * q^51 + 8 * q^53 - 4 * q^55 + 8 * q^57 + 18 * q^59 + 16 * q^61 + 42 * q^63 - 8 * q^65 - 28 * q^67 + 6 * q^71 - 4 * q^73 + 6 * q^75 - 8 * q^77 + 126 * q^81 + 30 * q^83 + 16 * q^85 + 8 * q^87 - 20 * q^89 + 56 * q^91 - 16 * q^93 + 32 * q^95 - 12 * q^97 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4304))\) 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	269
4304.2.a.a	$4304$	$2$	4304.a	1.a	$1$	$1$	$1$	$34.368$	\(\Q\)	None	✓				269.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{7}-3q^{9}+3q^{11}+2q^{13}+\cdots\)
4304.2.a.b	$4304$	$2$	4304.a	1.a	$1$	$1$	$1$	$34.368$	\(\Q\)	None	✓				2152.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-3q^{5}+2q^{7}+q^{9}+q^{11}+\cdots\)
4304.2.a.c	$4304$	$2$	4304.a	1.a	$1$	$2$	$2$	$34.368$	\(\Q(\sqrt{13}) \)	None	✓				538.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{5}+q^{7}+\beta q^{9}-3q^{11}+\cdots\)
4304.2.a.d	$4304$	$2$	4304.a	1.a	$1$	$2$	$2$	$34.368$	\(\Q(\sqrt{5}) \)	None	✓				538.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-5\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-3+\beta )q^{5}+(3-2\beta )q^{7}+\cdots\)
4304.2.a.e	$4304$	$2$	4304.a	1.a	$1$	$4$	$4$	$34.368$	4.4.4913.1	None	✓				538.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+(1-\beta _{3})q^{5}+\beta _{1}q^{7}+\cdots\)
4304.2.a.f	$4304$	$2$	4304.a	1.a	$1$	$5$	$5$	$34.368$	5.5.65657.1	None	✓				269.2.a.b	$1$	$0$	\(0\)	\(5\)	\(-4\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{3})q^{5}+\cdots\)
4304.2.a.g	$4304$	$2$	4304.a	1.a	$1$	$6$	$6$	$34.368$	6.6.4125937.1	None	✓				1076.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{5})q^{5}+(1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
4304.2.a.h	$4304$	$2$	4304.a	1.a	$1$	$7$	$7$	$34.368$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				538.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(7\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(1+\beta _{4})q^{5}+(-1+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
4304.2.a.i	$4304$	$2$	4304.a	1.a	$1$	$7$	$7$	$34.368$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				538.2.a.d	$1$	$0$	\(0\)	\(4\)	\(-6\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{6})q^{3}+(\beta _{3}-\beta _{4})q^{5}+(1+\cdots)q^{7}+\cdots\)
4304.2.a.j	$4304$	$2$	4304.a	1.a	$1$	$13$	$13$	$34.368$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓				2152.2.a.b	$1$	$1$	\(0\)	\(3\)	\(-9\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{5}-\beta _{9}q^{7}+\beta _{2}q^{9}+\cdots\)
4304.2.a.k	$4304$	$2$	4304.a	1.a	$1$	$15$	$15$	$34.368$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓		✓		2152.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-13\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{8}q^{5}+(-1-\beta _{6})q^{7}+(1+\cdots)q^{9}+\cdots\)
4304.2.a.l	$4304$	$2$	4304.a	1.a	$1$	$16$	$16$	$34.368$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		269.2.a.c	$1$	$1$	\(0\)	\(-5\)	\(-1\)	\(-11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{7}q^{3}+\beta _{12}q^{5}+(-1-\beta _{15})q^{7}+\cdots\)
4304.2.a.m	$4304$	$2$	4304.a	1.a	$1$	$17$	$17$	$34.368$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		1076.2.a.b	$1$	$0$	\(0\)	\(1\)	\(5\)	\(-9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{10}q^{5}+(-1-\beta _{7})q^{7}+\cdots\)
4304.2.a.n	$4304$	$2$	4304.a	1.a	$1$	$18$	$18$	$34.368$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓				2152.2.a.d	$1$	$0$	\(0\)	\(4\)	\(-6\)	\(17\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{6}q^{5}+(1+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
4304.2.a.o	$4304$	$2$	4304.a	1.a	$1$	$20$	$20$	$34.368$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		2152.2.a.e	$1$	$0$	\(0\)	\(-6\)	\(13\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{10})q^{5}-\beta _{9}q^{7}+(1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4304))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4304)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(269))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(538))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1076))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2152))\)\(^{\oplus 2}\)