Space of modular forms of level 3136, weight 2, and trivial character

• Label: 3136.2.a
• Level: $3136$
• Weight: $2$
• Character orbit: 3136.a
• Rep. character: $\chi_{3136}(1,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $52$
• Sturm bound: $896$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 52 \)
Sturm bound:			\(896\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3136))\).

	Total	New	Old
Modular forms	496	87	409
Cusp forms	401	77	324
Eisenstein series	95	10	85

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(120\)	\(19\)	\(101\)		\(97\)	\(17\)	\(80\)		\(23\)	\(2\)	\(21\)
\(+\)	\(-\)	\(-\)		\(128\)	\(24\)	\(104\)		\(104\)	\(21\)	\(83\)		\(24\)	\(3\)	\(21\)
\(-\)	\(+\)	\(-\)		\(128\)	\(23\)	\(105\)		\(104\)	\(21\)	\(83\)		\(24\)	\(2\)	\(22\)
\(-\)	\(-\)	\(+\)		\(120\)	\(21\)	\(99\)		\(96\)	\(18\)	\(78\)		\(24\)	\(3\)	\(21\)
Plus space		\(+\)		\(240\)	\(40\)	\(200\)		\(193\)	\(35\)	\(158\)		\(47\)	\(5\)	\(42\)
Minus space		\(-\)		\(256\)	\(47\)	\(209\)		\(208\)	\(42\)	\(166\)		\(48\)	\(5\)	\(43\)

Trace form

77 * q - 2 * q^5 + 69 * q^9 + 6 * q^13 + 6 * q^17 + 63 * q^25 + 2 * q^29 + 16 * q^33 - 2 * q^37 + 14 * q^41 - 42 * q^45 - 34 * q^53 + 4 * q^57 + 22 * q^61 + 12 * q^65 - 16 * q^69 - 2 * q^73 + 37 * q^81 + 40 * q^85 - 18 * q^89 + 36 * q^93 - 10 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\) 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
3136.2.a.a	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+6q^{9}-q^{11}+2q^{13}+\cdots\)
3136.2.a.b	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.a	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
3136.2.a.c	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
3136.2.a.d	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				1568.2.a.a	$1$	$2$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3136.2.a.e	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				14.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-4q^{13}-6q^{17}+2q^{19}+\cdots\)
3136.2.a.f	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				224.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}+4q^{11}-4q^{13}+2q^{17}+\cdots\)
3136.2.a.g	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				1568.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
3136.2.a.h	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				28.2.e.a	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
3136.2.a.i	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
3136.2.a.j	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.b	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{9}+3q^{11}+6q^{13}+\cdots\)
3136.2.a.k	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				28.2.e.a	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}-2q^{9}-3q^{11}+2q^{13}+\cdots\)
3136.2.a.l	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1568.2.a.d	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-3q^{9}+4q^{13}+8q^{17}+11q^{25}+\cdots\)
3136.2.a.m	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{5}-3q^{9}+6q^{13}-2q^{17}-q^{25}+\cdots\)
3136.2.a.n	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-4q^{11}+8q^{23}-5q^{25}-2q^{29}+\cdots\)
3136.2.a.o	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+4q^{11}-8q^{23}-5q^{25}-2q^{29}+\cdots\)
3136.2.a.p	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
3136.2.a.q	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
3136.2.a.r	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				1568.2.a.d	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-3q^{9}-4q^{13}-8q^{17}+11q^{25}+\cdots\)
3136.2.a.s	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				28.2.e.a	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-2q^{9}-3q^{11}-2q^{13}+\cdots\)
3136.2.a.t	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{9}+3q^{11}-6q^{13}+\cdots\)
3136.2.a.u	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
3136.2.a.v	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				28.2.e.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
3136.2.a.w	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{5}+q^{9}-8q^{15}+2q^{17}+\cdots\)
3136.2.a.x	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				1568.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
3136.2.a.y	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				224.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-4q^{11}-4q^{13}+2q^{17}+\cdots\)
3136.2.a.z	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				14.2.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-4q^{13}-6q^{17}-2q^{19}+\cdots\)
3136.2.a.ba	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				1568.2.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
3136.2.a.bb	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.a	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
3136.2.a.bc	$3136$	$2$	3136.a	1.a	$1$	$1$	$1$	$25.041$	\(\Q\)	None	✓				56.2.i.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}+6q^{9}-q^{11}-2q^{13}+\cdots\)
3136.2.a.bd	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				224.2.i.a	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots\)
3136.2.a.be	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				224.2.i.a	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots\)
3136.2.a.bf	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{5}) \)	None	✓				224.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1-\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
3136.2.a.bg	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{7}) \)	None	✓				224.2.i.c	$2$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3q^{5}+4q^{9}+\beta q^{11}-4q^{13}+\cdots\)
3136.2.a.bh	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{3}) \)	None	✓				224.2.i.b	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+3\beta q^{11}-\beta q^{15}+5q^{17}+\cdots\)
3136.2.a.bi	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				1568.2.a.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{5}-3q^{9}-\beta q^{13}+5\beta q^{17}-3q^{25}+\cdots\)
3136.2.a.bj	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				1568.2.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3\beta q^{5}-3q^{9}-5\beta q^{13}-3\beta q^{17}+\cdots\)
3136.2.a.bk	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				392.2.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-q^{9}-6q^{11}-4\beta q^{13}+\cdots\)
3136.2.a.bl	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				1568.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{9}-2q^{11}+2\beta q^{13}-3\beta q^{17}+\cdots\)
3136.2.a.bm	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				98.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}-q^{9}-2q^{11}+4q^{15}+\cdots\)
3136.2.a.bn	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				98.2.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-q^{9}+2q^{11}-4q^{15}+\cdots\)
3136.2.a.bo	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				1568.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{9}+2q^{11}-2\beta q^{13}+3\beta q^{17}+\cdots\)
3136.2.a.bp	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				392.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-q^{9}+6q^{11}+4\beta q^{13}+\cdots\)
3136.2.a.bq	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				392.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}+5q^{9}-4q^{11}+\beta q^{13}+\cdots\)
3136.2.a.br	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				196.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-\beta q^{5}+5q^{9}-4q^{11}-3\beta q^{13}+\cdots\)
3136.2.a.bs	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				196.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+\beta q^{5}+5q^{9}+4q^{11}+3\beta q^{13}+\cdots\)
3136.2.a.bt	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				392.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+5q^{9}+4q^{11}-\beta q^{13}+\cdots\)
3136.2.a.bu	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{3}) \)	None	✓				224.2.i.b	$2$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-3\beta q^{11}+\beta q^{15}-5q^{17}+\cdots\)
3136.2.a.bv	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{7}) \)	None	✓				224.2.i.c	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+3q^{5}+4q^{9}-\beta q^{11}+4q^{13}+\cdots\)
3136.2.a.bw	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				224.2.i.a	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots\)
3136.2.a.bx	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{2}) \)	None	✓				224.2.i.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(1+\cdots)q^{11}+\cdots\)
3136.2.a.by	$3136$	$2$	3136.a	1.a	$1$	$2$	$2$	$25.041$	\(\Q(\sqrt{5}) \)	None	✓				224.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
3136.2.a.bz	$3136$	$2$	3136.a	1.a	$1$	$4$	$4$	$25.041$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓		✓		1568.2.a.x	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3136))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3136)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\)\(^{\oplus 2}\)