Space of modular forms of level 6400, weight 2, and trivial character

• Label: 6400.2.a
• Level: $6400$
• Weight: $2$
• Character orbit: 6400.a
• Rep. character: $\chi_{6400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $146$
• Newform subspaces: $74$
• Sturm bound: $1920$
• Trace bound: $31$

Defining parameters

Level:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 74 \)
Sturm bound:			\(1920\)
Trace bound:			\(31\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\), \(17\), \(29\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6400))\).

	Total	New	Old
Modular forms	1032	158	874
Cusp forms	889	146	743
Eisenstein series	143	12	131

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(34\)
\(+\)	\(-\)	\(-\)	\(40\)
\(-\)	\(+\)	\(-\)	\(36\)
\(-\)	\(-\)	\(+\)	\(36\)
Plus space		\(+\)	\(70\)
Minus space		\(-\)	\(76\)

Trace form

\( 146 q + 130 q^{9} + 4 q^{17} - 8 q^{33} + 12 q^{41} + 82 q^{49} - 24 q^{57} - 52 q^{73} + 74 q^{81} - 20 q^{89} - 28 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\) 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
6400.2.a.a	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{3}+q^{9}+6q^{11}+6q^{17}+2q^{19}+\cdots\)
6400.2.a.b	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}-3q^{11}-q^{17}+\cdots\)
6400.2.a.c	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}-2q^{9}+3q^{11}+q^{17}+\cdots\)
6400.2.a.d	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$2$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}-5q^{11}-6q^{13}+\cdots\)
6400.2.a.e	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}+5q^{11}+6q^{13}+\cdots\)
6400.2.a.f	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-5q^{11}+6q^{13}+\cdots\)
6400.2.a.g	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}+5q^{11}-6q^{13}+\cdots\)
6400.2.a.h	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}-2q^{9}-3q^{11}-q^{17}+\cdots\)
6400.2.a.i	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{7}-2q^{9}+3q^{11}+q^{17}+\cdots\)
6400.2.a.j	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				640.2.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-6q^{13}-8q^{17}-4q^{29}-2q^{37}+\cdots\)
6400.2.a.k	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				640.2.f.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-6q^{13}+8q^{17}+4q^{29}-2q^{37}+\cdots\)
6400.2.a.l	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-4q^{13}+2q^{17}+4q^{29}+12q^{37}+\cdots\)
6400.2.a.m	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+4q^{13}+2q^{17}-4q^{29}-12q^{37}+\cdots\)
6400.2.a.n	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				640.2.f.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+6q^{13}-8q^{17}+4q^{29}+2q^{37}+\cdots\)
6400.2.a.o	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				640.2.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+6q^{13}+8q^{17}-4q^{29}+2q^{37}+\cdots\)
6400.2.a.p	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}-2q^{9}-3q^{11}+q^{17}+\cdots\)
6400.2.a.q	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}-2q^{9}+3q^{11}-q^{17}+\cdots\)
6400.2.a.r	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-5q^{11}-6q^{13}+\cdots\)
6400.2.a.s	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}+5q^{11}+6q^{13}+\cdots\)
6400.2.a.t	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}-5q^{11}+6q^{13}+\cdots\)
6400.2.a.u	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				200.2.d.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}+5q^{11}-6q^{13}+\cdots\)
6400.2.a.v	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}-3q^{11}+q^{17}+\cdots\)
6400.2.a.w	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	None	✓				3200.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}-2q^{9}+3q^{11}-q^{17}+\cdots\)
6400.2.a.x	$6400$	$2$	6400.a	1.a	$1$	$1$	$1$	$51.104$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{3}+q^{9}-6q^{11}+6q^{17}-2q^{19}+\cdots\)
6400.2.a.y	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-3+\beta )q^{7}+(1-2\beta )q^{9}+\cdots\)
6400.2.a.z	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(-1-\beta )q^{7}+(1-2\beta )q^{9}+\cdots\)
6400.2.a.ba	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				1600.2.d.e	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{7}-2q^{9}-3q^{11}+\beta q^{13}+\cdots\)
6400.2.a.bb	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				1600.2.d.e	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{7}-2q^{9}+3q^{11}-\beta q^{13}+\cdots\)
6400.2.a.bc	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓				1600.2.d.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+\beta )q^{3}+(4-2\beta )q^{9}+(-3+\beta )q^{11}+\cdots\)
6400.2.a.bd	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓				1600.2.d.c	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+(-1+\beta )q^{3}+(4-2\beta )q^{9}+(3-\beta )q^{11}+\cdots\)
6400.2.a.be	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1+\beta )q^{7}+(1-2\beta )q^{9}+\cdots\)
6400.2.a.bf	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(3-\beta )q^{7}+(1-2\beta )q^{9}+\cdots\)
6400.2.a.bg	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{7}) \)	None	✓				200.2.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-4q^{7}+4q^{9}+\beta q^{11}-3q^{17}+\cdots\)
6400.2.a.bh	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{7}) \)	None	✓				200.2.d.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-4q^{7}+4q^{9}-\beta q^{11}+3q^{17}+\cdots\)
6400.2.a.bi	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-10}) \)	✓				320.2.f.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}-3q^{9}-2q^{11}-\beta q^{13}+6q^{19}+\cdots\)
6400.2.a.bj	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-10}) \)	✓				320.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$N(\mathrm{U}(1))$	\(q+\beta q^{7}-3q^{9}+2q^{11}+\beta q^{13}-6q^{19}+\cdots\)
6400.2.a.bk	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+3\beta q^{7}-q^{9}-2\beta q^{11}-6q^{13}+\cdots\)
6400.2.a.bl	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3\beta q^{7}-q^{9}-4\beta q^{11}-2q^{13}+\cdots\)
6400.2.a.bm	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}-q^{9}+2\beta q^{11}-2q^{13}+\cdots\)
6400.2.a.bn	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+3\beta q^{7}-q^{9}-4\beta q^{11}+2q^{13}+\cdots\)
6400.2.a.bo	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}-q^{9}+2\beta q^{11}+2q^{13}+\cdots\)
6400.2.a.bp	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	None	✓				640.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3\beta q^{7}-q^{9}-2\beta q^{11}+6q^{13}+\cdots\)
6400.2.a.bq	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	None	✓				3200.2.d.o	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{9}-\beta q^{11}-4q^{13}-3q^{17}+\cdots\)
6400.2.a.br	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	None	✓				3200.2.d.o	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{9}+\beta q^{11}-4q^{13}+3q^{17}+\cdots\)
6400.2.a.bs	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	None	✓				3200.2.d.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{9}-\beta q^{11}+4q^{13}-3q^{17}+\cdots\)
6400.2.a.bt	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{5}) \)	None	✓				3200.2.d.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{9}+\beta q^{11}+4q^{13}+3q^{17}+\cdots\)
6400.2.a.bu	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	None	✓				640.2.f.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+3q^{9}+2\beta q^{11}-4q^{17}+\cdots\)
6400.2.a.bv	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	None	✓				640.2.f.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+3q^{9}+2\beta q^{11}-4q^{17}+\cdots\)
6400.2.a.bw	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	None	✓				640.2.f.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+3q^{9}-2\beta q^{11}+4q^{17}+\cdots\)
6400.2.a.bx	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	None	✓				640.2.f.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+3q^{9}-2\beta q^{11}+4q^{17}+\cdots\)
6400.2.a.by	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				128.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{3}+5q^{9}+\beta q^{11}-6q^{17}+3\beta q^{19}+\cdots\)
6400.2.a.bz	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{10}) \)	None	✓				640.2.d.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+7q^{9}-6q^{13}+2q^{17}+\cdots\)
6400.2.a.ca	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{10}) \)	None	✓				640.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+7q^{9}+6q^{13}+2q^{17}+\cdots\)
6400.2.a.cb	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{7}) \)	None	✓				200.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+4q^{7}+4q^{9}+\beta q^{11}-3q^{17}+\cdots\)
6400.2.a.cc	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{7}) \)	None	✓				200.2.d.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+4q^{7}+4q^{9}-\beta q^{11}+3q^{17}+\cdots\)
6400.2.a.cd	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-3-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6400.2.a.ce	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6400.2.a.cf	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				1600.2.d.e	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{7}-2q^{9}-3q^{11}+\beta q^{13}+\cdots\)
6400.2.a.cg	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				1600.2.d.e	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{7}-2q^{9}+3q^{11}-\beta q^{13}+\cdots\)
6400.2.a.ch	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓				1600.2.d.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{3}+(4+2\beta )q^{9}+(-3-\beta )q^{11}+\cdots\)
6400.2.a.ci	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓				1600.2.d.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{3}+(4+2\beta )q^{9}+(3+\beta )q^{11}+\cdots\)
6400.2.a.cj	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				40.2.d.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6400.2.a.ck	$6400$	$2$	6400.a	1.a	$1$	$2$	$2$	$51.104$	\(\Q(\sqrt{3}) \)	None	✓				320.2.d.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(3+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
6400.2.a.cl	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	\(\Q(\sqrt{-10}) \)	✓				640.2.f.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{5}$	$N(\mathrm{U}(1))$	\(q-\beta _{1}q^{7}-3q^{9}+\beta _{2}q^{11}+\beta _{3}q^{13}+\cdots\)
6400.2.a.cm	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	None	✓				40.2.f.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}-q^{9}-\beta _{3}q^{11}-2\beta _{2}q^{17}+\cdots\)
6400.2.a.cn	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)	✓				640.2.f.e	$8$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$N(\mathrm{U}(1))$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}-q^{9}+\beta _{3}q^{21}-3\beta _{2}q^{23}+\cdots\)
6400.2.a.co	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	None	✓				40.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}-q^{9}+\beta _{3}q^{11}+2\beta _{2}q^{17}+\cdots\)
6400.2.a.cp	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓		✓		3200.2.d.m	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+\beta _{1}q^{7}+2q^{9}-\beta _{3}q^{11}+\cdots\)
6400.2.a.cq	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-2}) \)	✓				3200.2.d.n	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$N(\mathrm{U}(1))$	\(q+\beta _{1}q^{3}+(2+\beta _{3})q^{9}+(-2\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
6400.2.a.cr	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	\(\Q(\sqrt{-2}) \)	✓		✓		3200.2.d.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$N(\mathrm{U}(1))$	\(q+\beta _{1}q^{3}+(2+\beta _{3})q^{9}+(2\beta _{1}+\beta _{2})q^{11}+\cdots\)
6400.2.a.cs	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				3200.2.d.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+\beta _{1}q^{7}+2q^{9}+\beta _{3}q^{11}+\cdots\)
6400.2.a.ct	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	None	✓				320.2.f.b	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{1}q^{7}+3q^{9}-2q^{11}-4\beta _{1}q^{13}+\cdots\)
6400.2.a.cu	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\zeta_{24})^+\)	None	✓				320.2.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{1}q^{7}+3q^{9}+2q^{11}-4\beta _{1}q^{13}+\cdots\)
6400.2.a.cv	$6400$	$2$	6400.a	1.a	$1$	$4$	$4$	$51.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)	✓				640.2.f.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$N(\mathrm{U}(1))$	\(q+\beta _{2}q^{3}-3\beta _{1}q^{7}+7q^{9}-3\beta _{3}q^{21}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(640))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\)\(^{\oplus 2}\)