Properties

Label 5175.2.a
Level $5175$
Weight $2$
Character orbit 5175.a
Rep. character $\chi_{5175}(1,\cdot)$
Character field $\Q$
Dimension $173$
Newform subspaces $62$
Sturm bound $1440$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 62 \)
Sturm bound: \(1440\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5175))\).

Total New Old
Modular forms 744 173 571
Cusp forms 697 173 524
Eisenstein series 47 0 47

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

\(3\)\(5\)\(23\)FrickeDim
\(+\)\(+\)\(+\)$+$\(17\)
\(+\)\(+\)\(-\)$-$\(17\)
\(+\)\(-\)\(+\)$-$\(17\)
\(+\)\(-\)\(-\)$+$\(17\)
\(-\)\(+\)\(+\)$-$\(29\)
\(-\)\(+\)\(-\)$+$\(20\)
\(-\)\(-\)\(+\)$+$\(25\)
\(-\)\(-\)\(-\)$-$\(31\)
Plus space\(+\)\(79\)
Minus space\(-\)\(94\)

Trace form

\( 173 q - 2 q^{2} + 170 q^{4} + 2 q^{7} - 15 q^{8} + O(q^{10}) \) \( 173 q - 2 q^{2} + 170 q^{4} + 2 q^{7} - 15 q^{8} + 2 q^{11} - 8 q^{13} - 20 q^{14} + 164 q^{16} - 4 q^{17} - 4 q^{19} + 10 q^{22} - 3 q^{23} + 25 q^{26} - 10 q^{28} + 24 q^{29} - 12 q^{31} - 12 q^{32} + 10 q^{34} - 24 q^{37} + 18 q^{38} - 4 q^{41} + 20 q^{43} - 8 q^{44} - 2 q^{46} - 16 q^{47} + 161 q^{49} - q^{52} + 10 q^{53} - 2 q^{56} + 17 q^{58} - 36 q^{59} + 22 q^{61} + 37 q^{62} + 191 q^{64} + 2 q^{67} - 28 q^{68} - 16 q^{71} + 8 q^{73} + 40 q^{74} - 6 q^{76} + 36 q^{77} - 68 q^{79} - 17 q^{82} - 22 q^{83} + 12 q^{86} + 10 q^{88} - 2 q^{89} - 90 q^{91} - 6 q^{92} - 25 q^{94} + 12 q^{97} - 46 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 23
5175.2.a.a 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-2\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+q^{7}+2q^{13}-2q^{14}+\cdots\)
5175.2.a.b 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-2\) \(0\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+5q^{7}+2q^{11}+6q^{13}+\cdots\)
5175.2.a.c 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-4q^{7}+3q^{8}+4q^{11}+\cdots\)
5175.2.a.d 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-3q^{7}+3q^{8}-5q^{11}+\cdots\)
5175.2.a.e 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{7}+3q^{8}+q^{11}-q^{13}+\cdots\)
5175.2.a.f 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{8}+4q^{13}-q^{16}+\cdots\)
5175.2.a.g 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{7}+3q^{8}+4q^{13}+\cdots\)
5175.2.a.h 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+3q^{7}+3q^{8}+q^{11}-q^{13}+\cdots\)
5175.2.a.i 5175.a 1.a $1$ $41.323$ \(\Q\) None \(-1\) \(0\) \(0\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+5q^{7}+3q^{8}-5q^{11}+\cdots\)
5175.2.a.j 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-3q^{7}-2q^{11}+3q^{13}+4q^{16}+\cdots\)
5175.2.a.k 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{7}-4q^{11}+4q^{16}+5q^{17}+\cdots\)
5175.2.a.l 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{7}+6q^{11}+5q^{13}+4q^{16}+\cdots\)
5175.2.a.m 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{7}+6q^{11}-5q^{13}+4q^{16}+\cdots\)
5175.2.a.n 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}+3q^{7}-2q^{11}-3q^{13}+4q^{16}+\cdots\)
5175.2.a.o 5175.a 1.a $1$ $41.323$ \(\Q\) None \(0\) \(0\) \(0\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+3q^{7}+4q^{11}+4q^{16}-3q^{17}+\cdots\)
5175.2.a.p 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-5q^{7}-3q^{8}-5q^{11}+\cdots\)
5175.2.a.q 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-4q^{7}-3q^{8}-4q^{11}+\cdots\)
5175.2.a.r 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{7}-3q^{8}+q^{11}+q^{13}+\cdots\)
5175.2.a.s 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-2q^{7}-3q^{8}-4q^{13}+\cdots\)
5175.2.a.t 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{8}-4q^{13}-q^{16}+\cdots\)
5175.2.a.u 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{7}-3q^{8}+q^{11}+q^{13}+\cdots\)
5175.2.a.v 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{7}-3q^{8}-4q^{11}+\cdots\)
5175.2.a.w 5175.a 1.a $1$ $41.323$ \(\Q\) None \(1\) \(0\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+3q^{7}-3q^{8}-5q^{11}+\cdots\)
5175.2.a.x 5175.a 1.a $1$ $41.323$ \(\Q\) None \(2\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-3q^{7}-2q^{11}+2q^{13}+\cdots\)
5175.2.a.y 5175.a 1.a $1$ $41.323$ \(\Q\) None \(2\) \(0\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{7}-2q^{11}+2q^{13}+\cdots\)
5175.2.a.z 5175.a 1.a $1$ $41.323$ \(\Q\) None \(2\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{7}-2q^{13}-2q^{14}+\cdots\)
5175.2.a.ba 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}+\beta q^{7}+(-1+\cdots)q^{8}+\cdots\)
5175.2.a.bb 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}+(2-2\beta )q^{7}+\cdots\)
5175.2.a.bc 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2-\beta )q^{7}+\cdots\)
5175.2.a.bd 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+3q^{7}+\cdots\)
5175.2.a.be 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{5}) \) None \(-1\) \(0\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(-2+2\beta )q^{7}+\cdots\)
5175.2.a.bf 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{13}) \) None \(-1\) \(0\) \(0\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+(-1+\beta )q^{7}-3q^{8}+\cdots\)
5175.2.a.bg 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{21}) \) None \(-1\) \(0\) \(0\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(3+\beta )q^{4}+(1-\beta )q^{7}+(-5+\cdots)q^{8}+\cdots\)
5175.2.a.bh 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1-\beta )q^{7}-2\beta q^{8}+(-2+\beta )q^{11}+\cdots\)
5175.2.a.bi 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{7}-2\beta q^{8}+(2+\beta )q^{11}+\cdots\)
5175.2.a.bj 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+2\beta )q^{7}-2\beta q^{8}+(4+\beta )q^{11}+\cdots\)
5175.2.a.bk 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}+(-1-\beta )q^{7}-\beta q^{8}+\cdots\)
5175.2.a.bl 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(0\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+4q^{4}+q^{7}+2\beta q^{8}+\beta q^{11}+\cdots\)
5175.2.a.bm 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{13}) \) None \(1\) \(0\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}+(1-\beta )q^{7}+3q^{8}+\cdots\)
5175.2.a.bn 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{21}) \) None \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(3+\beta )q^{4}+(-1+\beta )q^{7}+(5+\cdots)q^{8}+\cdots\)
5175.2.a.bo 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2+\beta )q^{7}+\cdots\)
5175.2.a.bp 5175.a 1.a $2$ $41.323$ \(\Q(\sqrt{5}) \) None \(3\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3\beta q^{4}-\beta q^{7}+(1+4\beta )q^{8}+\cdots\)
5175.2.a.bq 5175.a 1.a $3$ $41.323$ 3.3.229.1 None \(-2\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+\beta _{1}q^{7}+\cdots\)
5175.2.a.br 5175.a 1.a $3$ $41.323$ 3.3.316.1 None \(-1\) \(0\) \(0\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-2-\beta _{2})q^{7}+\cdots\)
5175.2.a.bs 5175.a 1.a $3$ $41.323$ 3.3.148.1 None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
5175.2.a.bt 5175.a 1.a $3$ $41.323$ 3.3.148.1 None \(0\) \(0\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5175.2.a.bu 5175.a 1.a $3$ $41.323$ 3.3.229.1 None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+(2+\beta _{1})q^{4}-\beta _{1}q^{7}+\cdots\)
5175.2.a.bv 5175.a 1.a $4$ $41.323$ 4.4.5744.1 None \(0\) \(0\) \(0\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{7}+\cdots\)
5175.2.a.bw 5175.a 1.a $4$ $41.323$ 4.4.5744.1 None \(0\) \(0\) \(0\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
5175.2.a.bx 5175.a 1.a $4$ $41.323$ 4.4.15317.1 None \(2\) \(0\) \(0\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
5175.2.a.by 5175.a 1.a $6$ $41.323$ 6.6.98838128.1 None \(0\) \(0\) \(0\) \(-6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(2+\beta _{4})q^{4}+(-1-\beta _{3})q^{7}+\cdots\)
5175.2.a.bz 5175.a 1.a $6$ $41.323$ 6.6.98838128.1 None \(0\) \(0\) \(0\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(2+\beta _{4})q^{4}+(-1-\beta _{3})q^{7}+\cdots\)
5175.2.a.ca 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-2\) \(0\) \(0\) \(-5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(2-\beta _{3})q^{4}+(-1-\beta _{5})q^{7}+\cdots\)
5175.2.a.cb 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(0\) \(0\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\beta _{6})q^{7}+\cdots\)
5175.2.a.cc 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(0\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots\)
5175.2.a.cd 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots\)
5175.2.a.ce 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(0\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots\)
5175.2.a.cf 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots\)
5175.2.a.cg 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(0\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
5175.2.a.ch 5175.a 1.a $7$ $41.323$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(0\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+(2-\beta _{3})q^{4}+(1+\beta _{5})q^{7}+\cdots\)
5175.2.a.ci 5175.a 1.a $10$ $41.323$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-4\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{6}q^{2}+(-\beta _{6}-\beta _{7})q^{4}-\beta _{2}q^{7}+\cdots\)
5175.2.a.cj 5175.a 1.a $10$ $41.323$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(4\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{6}q^{2}+(-\beta _{6}-\beta _{7})q^{4}+\beta _{2}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5175))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5175)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(575))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1035))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1725))\)\(^{\oplus 2}\)