Properties

Label 2450.4.a
Level $2450$
Weight $4$
Character orbit 2450.a
Rep. character $\chi_{2450}(1,\cdot)$
Character field $\Q$
Dimension $195$
Newform subspaces $81$
Sturm bound $1680$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 81 \)
Sturm bound: \(1680\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2450))\).

Total New Old
Modular forms 1308 195 1113
Cusp forms 1212 195 1017
Eisenstein series 96 0 96

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

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(25\)
\(+\)\(+\)\(-\)\(-\)\(22\)
\(+\)\(-\)\(+\)\(-\)\(24\)
\(+\)\(-\)\(-\)\(+\)\(27\)
\(-\)\(+\)\(+\)\(-\)\(19\)
\(-\)\(+\)\(-\)\(+\)\(26\)
\(-\)\(-\)\(+\)\(+\)\(28\)
\(-\)\(-\)\(-\)\(-\)\(24\)
Plus space\(+\)\(106\)
Minus space\(-\)\(89\)

Trace form

\( 195 q - 2 q^{2} + 2 q^{3} + 780 q^{4} - 8 q^{8} + 1709 q^{9} + O(q^{10}) \) \( 195 q - 2 q^{2} + 2 q^{3} + 780 q^{4} - 8 q^{8} + 1709 q^{9} + 22 q^{11} + 8 q^{12} - 16 q^{13} + 3120 q^{16} + 38 q^{17} + 22 q^{18} - 28 q^{19} - 56 q^{22} - 64 q^{23} - 264 q^{26} - 124 q^{27} - 102 q^{29} + 8 q^{31} - 32 q^{32} - 840 q^{33} + 64 q^{34} + 6836 q^{36} - 1122 q^{37} + 4 q^{38} - 208 q^{39} + 88 q^{41} - 928 q^{43} + 88 q^{44} + 280 q^{46} + 1216 q^{47} + 32 q^{48} + 554 q^{51} - 64 q^{52} + 1722 q^{53} - 900 q^{54} + 2812 q^{57} - 612 q^{58} + 798 q^{59} + 732 q^{61} - 56 q^{62} + 12480 q^{64} + 116 q^{66} + 196 q^{67} + 152 q^{68} - 2396 q^{69} - 56 q^{71} + 88 q^{72} + 1330 q^{73} + 1132 q^{74} - 112 q^{76} + 704 q^{78} + 2152 q^{79} + 11939 q^{81} + 372 q^{82} + 3702 q^{83} + 3224 q^{86} - 1780 q^{87} - 224 q^{88} + 1800 q^{89} - 256 q^{92} - 2796 q^{93} - 1752 q^{94} - 3050 q^{97} - 1664 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\) 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 7
2450.4.a.a 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-10\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-10q^{3}+4q^{4}+20q^{6}-8q^{8}+\cdots\)
2450.4.a.b 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-8\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-8q^{3}+4q^{4}+2^{4}q^{6}-8q^{8}+\cdots\)
2450.4.a.c 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-7\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}-8q^{8}+\cdots\)
2450.4.a.d 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-5\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-5q^{3}+4q^{4}+10q^{6}-8q^{8}+\cdots\)
2450.4.a.e 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-4\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{3}+4q^{4}+8q^{6}-8q^{8}+\cdots\)
2450.4.a.f 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-4\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{3}+4q^{4}+8q^{6}-8q^{8}+\cdots\)
2450.4.a.g 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-8q^{8}+\cdots\)
2450.4.a.h 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-8q^{8}+\cdots\)
2450.4.a.i 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-8q^{8}+\cdots\)
2450.4.a.j 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+4q^{4}+2q^{6}-8q^{8}+\cdots\)
2450.4.a.k 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+4q^{4}+2q^{6}-8q^{8}+\cdots\)
2450.4.a.l 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots\)
2450.4.a.m 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots\)
2450.4.a.n 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}-2q^{6}-8q^{8}+\cdots\)
2450.4.a.o 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}-4q^{6}-8q^{8}+\cdots\)
2450.4.a.p 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(4\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{3}+4q^{4}-8q^{6}-8q^{8}+\cdots\)
2450.4.a.q 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(5\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+5q^{3}+4q^{4}-10q^{6}-8q^{8}+\cdots\)
2450.4.a.r 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(5\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+5q^{3}+4q^{4}-10q^{6}-8q^{8}+\cdots\)
2450.4.a.s 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(7\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-8q^{8}+\cdots\)
2450.4.a.t 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(7\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+7q^{3}+4q^{4}-14q^{6}-8q^{8}+\cdots\)
2450.4.a.u 2450.a 1.a $1$ $144.555$ \(\Q\) None \(-2\) \(8\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+8q^{3}+4q^{4}-2^{4}q^{6}-8q^{8}+\cdots\)
2450.4.a.v 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-10\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-10q^{3}+4q^{4}-20q^{6}+8q^{8}+\cdots\)
2450.4.a.w 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-8\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+8q^{8}+\cdots\)
2450.4.a.x 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-8\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-8q^{3}+4q^{4}-2^{4}q^{6}+8q^{8}+\cdots\)
2450.4.a.y 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-7\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}-14q^{6}+8q^{8}+\cdots\)
2450.4.a.z 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-4\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-4q^{3}+4q^{4}-8q^{6}+8q^{8}+\cdots\)
2450.4.a.ba 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+8q^{8}+\cdots\)
2450.4.a.bb 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-2q^{3}+4q^{4}-4q^{6}+8q^{8}+\cdots\)
2450.4.a.bc 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots\)
2450.4.a.bd 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots\)
2450.4.a.be 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots\)
2450.4.a.bf 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}-2q^{6}+8q^{8}+\cdots\)
2450.4.a.bg 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}+2q^{6}+8q^{8}+\cdots\)
2450.4.a.bh 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(1\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}+2q^{6}+8q^{8}+\cdots\)
2450.4.a.bi 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(2\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+8q^{8}+\cdots\)
2450.4.a.bj 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+8q^{8}+\cdots\)
2450.4.a.bk 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(4\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots\)
2450.4.a.bl 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(4\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots\)
2450.4.a.bm 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(4\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{3}+4q^{4}+8q^{6}+8q^{8}+\cdots\)
2450.4.a.bn 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(7\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+7q^{3}+4q^{4}+14q^{6}+8q^{8}+\cdots\)
2450.4.a.bo 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(8\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+8q^{3}+4q^{4}+2^{4}q^{6}+8q^{8}+\cdots\)
2450.4.a.bp 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(10\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+10q^{3}+4q^{4}+20q^{6}+8q^{8}+\cdots\)
2450.4.a.bq 2450.a 1.a $1$ $144.555$ \(\Q\) None \(2\) \(10\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+10q^{3}+4q^{4}+20q^{6}+8q^{8}+\cdots\)
2450.4.a.br 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{7}) \) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-8q^{8}+\cdots\)
2450.4.a.bs 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{22}) \) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-8q^{8}+\cdots\)
2450.4.a.bt 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{177}) \) None \(4\) \(-5\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta )q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots\)
2450.4.a.bu 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{2}) \) None \(4\) \(-2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+3\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
2450.4.a.bv 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{46}) \) None \(4\) \(-2\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
2450.4.a.bw 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{7}) \) None \(4\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta q^{3}+4q^{4}+2\beta q^{6}+8q^{8}+\cdots\)
2450.4.a.bx 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{2}) \) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+5\beta q^{3}+4q^{4}+10\beta q^{6}+8q^{8}+\cdots\)
2450.4.a.by 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{2}) \) None \(4\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+3\beta )q^{3}+4q^{4}+(2+6\beta )q^{6}+\cdots\)
2450.4.a.bz 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{46}) \) None \(4\) \(2\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta )q^{3}+4q^{4}+(2+2\beta )q^{6}+\cdots\)
2450.4.a.ca 2450.a 1.a $2$ $144.555$ \(\Q(\sqrt{177}) \) None \(4\) \(5\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(3-\beta )q^{3}+4q^{4}+(6-2\beta )q^{6}+\cdots\)
2450.4.a.cb 2450.a 1.a $3$ $144.555$ 3.3.115880.1 None \(-6\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cc 2450.a 1.a $3$ $144.555$ 3.3.238585.1 None \(-6\) \(-3\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(2-2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cd 2450.a 1.a $3$ $144.555$ 3.3.238585.1 None \(-6\) \(3\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(-2+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.ce 2450.a 1.a $3$ $144.555$ 3.3.115880.1 None \(-6\) \(4\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2-2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cf 2450.a 1.a $3$ $144.555$ 3.3.51960.1 None \(-6\) \(7\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(-4-2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cg 2450.a 1.a $3$ $144.555$ 3.3.51960.1 None \(6\) \(-7\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta _{1})q^{3}+4q^{4}+(-4+\cdots)q^{6}+\cdots\)
2450.4.a.ch 2450.a 1.a $3$ $144.555$ 3.3.238585.1 None \(6\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
2450.4.a.ci 2450.a 1.a $3$ $144.555$ 3.3.238585.1 None \(6\) \(3\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(2-2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cj 2450.a 1.a $4$ $144.555$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(-8\) \(-10\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-3-\beta _{1})q^{3}+4q^{4}+(6+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.ck 2450.a 1.a $4$ $144.555$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-\beta _{2}q^{3}+4q^{4}+2\beta _{2}q^{6}-8q^{8}+\cdots\)
2450.4.a.cl 2450.a 1.a $4$ $144.555$ \(\Q(\sqrt{2}, \sqrt{29})\) None \(-8\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2\beta _{1}+\beta _{2})q^{3}+4q^{4}+(-4\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cm 2450.a 1.a $4$ $144.555$ 4.4.10197128.1 None \(-8\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots\)
2450.4.a.cn 2450.a 1.a $4$ $144.555$ 4.4.1555279308.1 None \(-8\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots\)
2450.4.a.co 2450.a 1.a $4$ $144.555$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-8\) \(1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{2}q^{3}+4q^{4}-2\beta _{2}q^{6}-8q^{8}+\cdots\)
2450.4.a.cp 2450.a 1.a $4$ $144.555$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(-8\) \(10\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(-4+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cq 2450.a 1.a $4$ $144.555$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-1\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-\beta _{2}q^{3}+4q^{4}-2\beta _{2}q^{6}+8q^{8}+\cdots\)
2450.4.a.cr 2450.a 1.a $4$ $144.555$ \(\Q(\sqrt{2}, \sqrt{29})\) None \(8\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2\beta _{1}+\beta _{2})q^{3}+4q^{4}+(4\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cs 2450.a 1.a $4$ $144.555$ 4.4.10197128.1 None \(8\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots\)
2450.4.a.ct 2450.a 1.a $4$ $144.555$ 4.4.1555279308.1 None \(8\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots\)
2450.4.a.cu 2450.a 1.a $4$ $144.555$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{2}q^{3}+4q^{4}+2\beta _{2}q^{6}+8q^{8}+\cdots\)
2450.4.a.cv 2450.a 1.a $6$ $144.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-7\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cw 2450.a 1.a $6$ $144.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(7\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2-2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cx 2450.a 1.a $6$ $144.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(-7\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
2450.4.a.cy 2450.a 1.a $6$ $144.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(7\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(2+2\beta _{1}+\cdots)q^{6}+\cdots\)
2450.4.a.cz 2450.a 1.a $8$ $144.555$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-\beta _{3}q^{3}+4q^{4}+2\beta _{3}q^{6}-8q^{8}+\cdots\)
2450.4.a.da 2450.a 1.a $8$ $144.555$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-\beta _{3}q^{3}+4q^{4}-2\beta _{3}q^{6}+8q^{8}+\cdots\)
2450.4.a.db 2450.a 1.a $10$ $144.555$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-20\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{1}q^{3}+4q^{4}-2\beta _{1}q^{6}-8q^{8}+\cdots\)
2450.4.a.dc 2450.a 1.a $10$ $144.555$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(20\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+2\beta _{1}q^{6}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2450))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2450)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 2}\)