Space of modular forms of level 5408, weight 2, and trivial character

• Label: 5408.2.a
• Level: $5408$
• Weight: $2$
• Character orbit: 5408.a
• Rep. character: $\chi_{5408}(1,\cdot)$
• Character field: $\Q$
• Dimension: $155$
• Newform subspaces: $47$
• Sturm bound: $1456$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 47 \)
Sturm bound:			\(1456\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5408))\).

	Total	New	Old
Modular forms	784	155	629
Cusp forms	673	155	518
Eisenstein series	111	0	111

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

\(2\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(35\)
\(+\)	\(-\)	\(-\)	\(42\)
\(-\)	\(+\)	\(-\)	\(42\)
\(-\)	\(-\)	\(+\)	\(36\)
Plus space		\(+\)	\(71\)
Minus space		\(-\)	\(84\)

Trace form

155 * q + 2 * q^5 + 151 * q^9 - 10 * q^17 - 16 * q^21 + 141 * q^25 - 6 * q^29 + 2 * q^37 + 14 * q^41 + 26 * q^45 + 147 * q^49 + 34 * q^53 + 16 * q^57 + 10 * q^61 + 32 * q^69 - 34 * q^73 + 48 * q^77 + 195 * q^81 + 20 * q^85 + 30 * q^89 + 16 * q^93 + 6 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\) 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	13
5408.2.a.a	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.f.a	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
5408.2.a.b	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.f.a	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
5408.2.a.c	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.i.a	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}+4q^{11}+2q^{15}+\cdots\)
5408.2.a.d	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.i.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
5408.2.a.e	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+3q^{7}-2q^{9}+2q^{11}+\cdots\)
5408.2.a.f	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				416.2.f.b	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-3q^{9}+2q^{17}+11q^{25}+10q^{29}+\cdots\)
5408.2.a.g	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{5}-3q^{9}+2q^{17}-q^{25}-10q^{29}+\cdots\)
5408.2.a.h	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				416.2.f.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-3q^{9}+2q^{17}+11q^{25}+10q^{29}+\cdots\)
5408.2.a.i	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
5408.2.a.j	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.i.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+q^{9}-4q^{11}-2q^{15}+\cdots\)
5408.2.a.k	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.i.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+4q^{11}+2q^{15}+\cdots\)
5408.2.a.l	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.f.a	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}+q^{7}+6q^{9}-4q^{11}+\cdots\)
5408.2.a.m	$5408$	$2$	5408.a	1.a	$1$	$1$	$1$	$43.183$	\(\Q\)	None	✓				416.2.f.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
5408.2.a.n	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{2}) \)	None	✓				416.2.i.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+2\beta q^{5}+(-3+\beta )q^{7}+\cdots\)
5408.2.a.o	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{2}) \)	None	✓				416.2.i.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{5}+(3-\beta )q^{7}+\cdots\)
5408.2.a.p	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{17}) \)	None	✓				416.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(2-\beta )q^{5}+(2-\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
5408.2.a.q	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{5}) \)	None	✓				416.2.a.d	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-3q^{5}+\beta q^{7}+2q^{9}-2\beta q^{11}+\cdots\)
5408.2.a.r	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓				416.2.w.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-2-\beta )q^{5}-3q^{9}+(-1-4\beta )q^{17}+\cdots\)
5408.2.a.s	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓				416.2.i.e	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{5}-3q^{9}+(-1+2\beta )q^{17}+\cdots\)
5408.2.a.t	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{5}) \)	None	✓				416.2.f.e	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{5}-\beta q^{7}+2q^{9}+\beta q^{15}+\cdots\)
5408.2.a.u	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	None	✓				416.2.w.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-3q^{7}-5q^{11}+6q^{15}+\cdots\)
5408.2.a.v	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{13}) \)	\(\Q(\sqrt{-13}) \)	✓				416.2.f.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{7}-3q^{9}+(3-\beta )q^{11}+\cdots\)
5408.2.a.w	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{11}) \)	None	✓				416.2.i.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+8q^{9}-\beta q^{11}-3q^{17}+\cdots\)
5408.2.a.x	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{11}) \)	None	✓				416.2.i.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+8q^{9}+\beta q^{11}-3q^{17}+\cdots\)
5408.2.a.y	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{13}) \)	\(\Q(\sqrt{-13}) \)	✓				416.2.f.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{7}-3q^{9}+(-3+\beta )q^{11}+\cdots\)
5408.2.a.z	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	None	✓				416.2.w.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}+3q^{7}+5q^{11}-6q^{15}+\cdots\)
5408.2.a.ba	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓				416.2.i.e	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(1-\beta )q^{5}-3q^{9}+(-1-2\beta )q^{17}+\cdots\)
5408.2.a.bb	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{5}) \)	None	✓				416.2.f.e	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+q^{5}+\beta q^{7}+2q^{9}-\beta q^{15}+\cdots\)
5408.2.a.bc	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓				416.2.w.b	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(2-\beta )q^{5}-3q^{9}+(-1+4\beta )q^{17}+\cdots\)
5408.2.a.bd	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{17}) \)	None	✓				416.2.a.c	$1$	$1$	\(0\)	\(1\)	\(3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(-2+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
5408.2.a.be	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{2}) \)	None	✓				416.2.i.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+2\beta q^{5}+(-3-\beta )q^{7}+\cdots\)
5408.2.a.bf	$5408$	$2$	5408.a	1.a	$1$	$2$	$2$	$43.183$	\(\Q(\sqrt{2}) \)	None	✓				416.2.i.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2\beta q^{5}+(3+\beta )q^{7}+2\beta q^{9}+\cdots\)
5408.2.a.bg	$5408$	$2$	5408.a	1.a	$1$	$4$	$4$	$43.183$	4.4.7488.1	None	✓				416.2.w.c	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-2q^{5}+(\beta _{1}+\beta _{3})q^{7}+(2-2\beta _{2}+\cdots)q^{9}+\cdots\)
5408.2.a.bh	$5408$	$2$	5408.a	1.a	$1$	$4$	$4$	$43.183$	4.4.9248.1	None	✓				416.2.i.g	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5408.2.a.bi	$5408$	$2$	5408.a	1.a	$1$	$4$	$4$	$43.183$	4.4.9248.1	None	✓				416.2.i.g	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(\beta _{1}-\beta _{2})q^{7}-\beta _{3}q^{9}+\cdots\)
5408.2.a.bj	$5408$	$2$	5408.a	1.a	$1$	$4$	$4$	$43.183$	4.4.13448.1	None	✓				416.2.a.f	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
5408.2.a.bk	$5408$	$2$	5408.a	1.a	$1$	$4$	$4$	$43.183$	4.4.7488.1	None	✓				416.2.w.c	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+2q^{5}+(-\beta _{1}-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
5408.2.a.bl	$5408$	$2$	5408.a	1.a	$1$	$6$	$6$	$43.183$	6.6.1997632.1	None	✓	✓			5408.2.a.bl	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
5408.2.a.bm	$5408$	$2$	5408.a	1.a	$1$	$6$	$6$	$43.183$	6.6.134509248.1	None	✓				416.2.w.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-6\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
5408.2.a.bn	$5408$	$2$	5408.a	1.a	$1$	$6$	$6$	$43.183$	6.6.134509248.1	None	✓				416.2.w.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{5}+(1+\beta _{2})q^{7}+\cdots\)
5408.2.a.bo	$5408$	$2$	5408.a	1.a	$1$	$6$	$6$	$43.183$	6.6.1997632.1	None	✓	✓			5408.2.a.bl	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{4})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
5408.2.a.bp	$5408$	$2$	5408.a	1.a	$1$	$9$	$9$	$43.183$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		5408.2.a.bp	$1$	$1$	\(0\)	\(-7\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(-\beta _{4}+\cdots)q^{7}+\cdots\)
5408.2.a.bq	$5408$	$2$	5408.a	1.a	$1$	$9$	$9$	$43.183$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		5408.2.a.bp	$1$	$1$	\(0\)	\(-7\)	\(6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.br	$5408$	$2$	5408.a	1.a	$1$	$9$	$9$	$43.183$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			5408.2.a.bp	$1$	$0$	\(0\)	\(7\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.bs	$5408$	$2$	5408.a	1.a	$1$	$9$	$9$	$43.183$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			5408.2.a.bp	$1$	$0$	\(0\)	\(7\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{3})q^{5}+(-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
5408.2.a.bt	$5408$	$2$	5408.a	1.a	$1$	$12$	$12$	$43.183$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		5408.2.a.bt	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4}+\beta _{5}+\beta _{7})q^{5}+\cdots\)
5408.2.a.bu	$5408$	$2$	5408.a	1.a	$1$	$12$	$12$	$43.183$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		5408.2.a.bt	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{4}-\beta _{5}-\beta _{7})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5408))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5408)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\)\(^{\oplus 2}\)