Space of modular forms of level 13520, weight 2, and trivial character

• Label: 13520.2.a
• Level: $13520$
• Weight: $2$
• Character orbit: 13520.a
• Rep. character: $\chi_{13520}(1,\cdot)$
• Character field: $\Q$
• Dimension: $310$
• Newform subspaces: $99$
• Sturm bound: $4368$

Defining parameters

Level:	\( N \)	\(=\)	\( 13520 = 2^{4} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	13520.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 99 \)
Sturm bound:			\(4368\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(13520))\).

	Total	New	Old
Modular forms	2268	310	1958
Cusp forms	2101	310	1791
Eisenstein series	167	0	167

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

\(2\)	\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(273\)	\(35\)	\(238\)		\(253\)	\(35\)	\(218\)		\(20\)	\(0\)	\(20\)
\(+\)	\(+\)	\(-\)	\(-\)		\(293\)	\(42\)	\(251\)		\(272\)	\(42\)	\(230\)		\(21\)	\(0\)	\(21\)
\(+\)	\(-\)	\(+\)	\(-\)		\(287\)	\(42\)	\(245\)		\(266\)	\(42\)	\(224\)		\(21\)	\(0\)	\(21\)
\(+\)	\(-\)	\(-\)	\(+\)		\(279\)	\(36\)	\(243\)		\(258\)	\(36\)	\(222\)		\(21\)	\(0\)	\(21\)
\(-\)	\(+\)	\(+\)	\(-\)		\(294\)	\(42\)	\(252\)		\(273\)	\(42\)	\(231\)		\(21\)	\(0\)	\(21\)
\(-\)	\(+\)	\(-\)	\(+\)		\(274\)	\(36\)	\(238\)		\(253\)	\(36\)	\(217\)		\(21\)	\(0\)	\(21\)
\(-\)	\(-\)	\(+\)	\(+\)		\(280\)	\(35\)	\(245\)		\(259\)	\(35\)	\(224\)		\(21\)	\(0\)	\(21\)
\(-\)	\(-\)	\(-\)	\(-\)		\(288\)	\(42\)	\(246\)		\(267\)	\(42\)	\(225\)		\(21\)	\(0\)	\(21\)
Plus space			\(+\)		\(1106\)	\(142\)	\(964\)		\(1023\)	\(142\)	\(881\)		\(83\)	\(0\)	\(83\)
Minus space			\(-\)		\(1162\)	\(168\)	\(994\)		\(1078\)	\(168\)	\(910\)		\(84\)	\(0\)	\(84\)

Trace form

310 * q - 2 * q^3 - 6 * q^7 + 306 * q^9 - 4 * q^11 - 2 * q^15 - 4 * q^17 - 8 * q^19 - 4 * q^21 + 2 * q^23 + 310 * q^25 + 4 * q^27 + 4 * q^29 - 20 * q^31 - 6 * q^35 + 8 * q^37 + 8 * q^41 - 26 * q^43 - 4 * q^45 - 22 * q^47 + 306 * q^49 - 36 * q^51 + 16 * q^53 + 4 * q^55 - 8 * q^57 + 8 * q^59 + 16 * q^61 - 46 * q^63 - 18 * q^67 + 20 * q^69 + 4 * q^71 - 20 * q^73 - 2 * q^75 + 32 * q^77 - 40 * q^79 + 306 * q^81 + 10 * q^83 + 8 * q^85 + 60 * q^87 + 12 * q^89 - 8 * q^93 + 8 * q^95 - 12 * q^97 + 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(13520))\) 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	5	13
13520.2.a.a	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-3q^{7}+q^{9}+5q^{11}+\cdots\)
13520.2.a.b	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}+2q^{15}+6q^{17}+\cdots\)
13520.2.a.c	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}+2q^{11}+2q^{15}+\cdots\)
13520.2.a.d	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
13520.2.a.e	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-2q^{15}+6q^{17}+\cdots\)
13520.2.a.f	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(1\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
13520.2.a.g	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-2\)	\(1\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+3q^{7}+q^{9}-5q^{11}+\cdots\)
13520.2.a.h	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}-2q^{9}+3q^{11}+q^{15}+\cdots\)
13520.2.a.i	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}-2q^{9}-3q^{11}+q^{15}+\cdots\)
13520.2.a.j	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}-2q^{9}+3q^{11}-q^{15}+\cdots\)
13520.2.a.k	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}-q^{15}+\cdots\)
13520.2.a.l	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{7}-3q^{9}+4q^{11}+2q^{17}+\cdots\)
13520.2.a.m	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}-3q^{9}-3q^{11}-4q^{17}+\cdots\)
13520.2.a.n	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}+2q^{17}-8q^{19}+4q^{23}+\cdots\)
13520.2.a.o	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}-4q^{11}-6q^{17}+4q^{19}+\cdots\)
13520.2.a.p	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-3q^{9}+3q^{11}-4q^{17}+\cdots\)
13520.2.a.q	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-1\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-5q^{7}-2q^{9}+5q^{11}+\cdots\)
13520.2.a.r	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}-2q^{9}+5q^{11}+\cdots\)
13520.2.a.s	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}-2q^{9}-q^{11}-q^{15}+\cdots\)
13520.2.a.t	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}-2q^{9}+3q^{11}+\cdots\)
13520.2.a.u	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(1\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}-2q^{9}-3q^{11}+\cdots\)
13520.2.a.v	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(1\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}-2q^{9}+q^{11}+q^{15}+\cdots\)
13520.2.a.w	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(1\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}-2q^{9}-5q^{11}+\cdots\)
13520.2.a.x	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(1\)	\(1\)	\(5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+5q^{7}-2q^{9}-5q^{11}+\cdots\)
13520.2.a.y	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(2\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-4q^{7}+q^{9}-6q^{11}+\cdots\)
13520.2.a.z	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(2\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+q^{7}+q^{9}-3q^{11}-2q^{15}+\cdots\)
13520.2.a.ba	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(2\)	\(1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-4q^{7}+q^{9}+2q^{11}+\cdots\)
13520.2.a.bb	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(2\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-q^{7}+q^{9}+3q^{11}+2q^{15}+\cdots\)
13520.2.a.bc	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(2\)	\(1\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+2q^{7}+q^{9}+2q^{15}+\cdots\)
13520.2.a.bd	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(3\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}-3q^{7}+6q^{9}-5q^{11}+\cdots\)
13520.2.a.be	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(3\)	\(-1\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+3q^{7}+6q^{9}+3q^{11}+\cdots\)
13520.2.a.bf	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(3\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}-3q^{7}+6q^{9}-3q^{11}+\cdots\)
13520.2.a.bg	$13520$	$2$	13520.a	1.a	$1$	$1$	$1$	$107.958$	\(\Q\)	None	✓	?				$-1$		\(0\)	\(3\)	\(1\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+3q^{7}+6q^{9}+5q^{11}+\cdots\)
13520.2.a.bh	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bi	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{13}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bj	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{13}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bk	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bl	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bm	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(-2\)	\(2\)	\(4\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bn	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{10}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bo	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{2}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bp	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bq	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{6}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.br	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{5}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bs	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{10}) \)	None	✓	?				$-1$		\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bt	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{17}) \)	None	✓	?				$-1$		\(0\)	\(1\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.bu	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{17}) \)	None	✓	?				$-1$		\(0\)	\(1\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bv	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(2\)	\(-2\)	\(6\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.bw	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(2\)	\(2\)	\(-6\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.bx	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.by	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{3}) \)	None	✓	?				$-1$		\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.bz	$13520$	$2$	13520.a	1.a	$1$	$2$	$2$	$107.958$	\(\Q(\sqrt{2}) \)	None	✓	?				$-1$		\(0\)	\(4\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.ca	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(-3\)	\(-3\)	\(1\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.cb	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(-3\)	\(3\)	\(-1\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cc	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.564.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(-3\)	\(-2\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.cd	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.1016.1	None	✓	?				$-1$		\(0\)	\(-1\)	\(-3\)	\(-9\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.ce	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(-3\)	\(-5\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.cf	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(3\)	\(5\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cg	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.1016.1	None	✓	?				$-1$		\(0\)	\(-1\)	\(3\)	\(9\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.ch	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.756.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.ci	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.148.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-3\)	\(2\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cj	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.148.1	None	✓	?				$-1$		\(0\)	\(0\)	\(3\)	\(-2\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.ck	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.756.1	None	✓	?				$-1$		\(0\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cl	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cm	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(1\)	\(-3\)	\(1\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cn	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(1\)	\(3\)	\(-1\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.co	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(1\)	\(3\)	\(2\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.cp	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.564.1	None	✓	?				$-1$		\(0\)	\(2\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cq	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	3.3.564.1	None	✓	?				$-1$		\(0\)	\(2\)	\(3\)	\(2\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cr	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(5\)	\(-3\)	\(-5\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.cs	$13520$	$2$	13520.a	1.a	$1$	$3$	$3$	$107.958$	\(\Q(\zeta_{14})^+\)	None	✓	?				$-1$		\(0\)	\(5\)	\(3\)	\(5\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.ct	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cu	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(-4\)	\(6\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cv	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.34868.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(-4\)	\(6\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cw	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(4\)	\(-6\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cx	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.34868.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(4\)	\(-6\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.cy	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.cz	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.25488.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-4\)	\(-4\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.da	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.25488.1	None	✓	?				$-1$		\(0\)	\(0\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.db	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(2\)	\(-4\)	\(-10\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.dc	$13520$	$2$	13520.a	1.a	$1$	$4$	$4$	$107.958$	4.4.4752.1	None	✓	?				$-1$		\(0\)	\(2\)	\(4\)	\(10\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.dd	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.2249737.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-6\)	\(-2\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.de	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.7674048.1	None	✓	?				$-1$		\(0\)	\(0\)	\(-6\)	\(4\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.df	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.7674048.1	None	✓	?				$-1$		\(0\)	\(0\)	\(6\)	\(-4\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.dg	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.2249737.1	None	✓	?				$-1$		\(0\)	\(0\)	\(6\)	\(2\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.dh	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.406193977.1	None	✓	?				$-1$		\(0\)	\(2\)	\(-6\)	\(1\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.di	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.20439713.1	None	✓	?				$-1$		\(0\)	\(2\)	\(-6\)	\(3\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.dj	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.20439713.1	None	✓	?				$-1$		\(0\)	\(2\)	\(6\)	\(-3\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.dk	$13520$	$2$	13520.a	1.a	$1$	$6$	$6$	$107.958$	6.6.406193977.1	None	✓	?				$-1$		\(0\)	\(2\)	\(6\)	\(-1\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.dl	$13520$	$2$	13520.a	1.a	$1$	$8$	$8$	$107.958$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	?				$-1$		\(0\)	\(-4\)	\(-8\)	\(6\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.dm	$13520$	$2$	13520.a	1.a	$1$	$8$	$8$	$107.958$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(-4\)	\(8\)	\(-6\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.dn	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(-7\)	\(-9\)	\(7\)		$-$	$+$	$-$	$+$		$\mathrm{SU}(2)$
13520.2.a.do	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(-7\)	\(9\)	\(-7\)		$-$	$-$	$+$	$+$		$\mathrm{SU}(2)$
13520.2.a.dp	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(-9\)	\(-7\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.dq	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?				$-1$		\(0\)	\(-1\)	\(9\)	\(7\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.dr	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(1\)	\(-9\)	\(1\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
13520.2.a.ds	$13520$	$2$	13520.a	1.a	$1$	$9$	$9$	$107.958$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(1\)	\(9\)	\(-1\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.dt	$13520$	$2$	13520.a	1.a	$1$	$12$	$12$	$107.958$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(0\)	\(-12\)	\(7\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$
13520.2.a.du	$13520$	$2$	13520.a	1.a	$1$	$12$	$12$	$107.958$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	?	✓			$-1$		\(0\)	\(0\)	\(12\)	\(-7\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(13520))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(13520)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1040))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1352))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\)\(^{\oplus 2}\)