Properties

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 $31$

Related objects

Downloads

Learn more

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.

\(2\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(34\)
\(+\)\(-\)\(-\)\(40\)
\(-\)\(+\)\(-\)\(36\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(70\)
Minus space\(-\)\(76\)

Trace form

\( 146 q + 130 q^{9} + O(q^{10}) \) \( 146 q + 130 q^{9} + 4 q^{17} - 8 q^{33} + 12 q^{41} + 82 q^{49} - 24 q^{57} - 52 q^{73} + 74 q^{81} - 20 q^{89} - 28 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\) into newform subspaces

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

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

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