Properties

Label 1575.2.a
Level $1575$
Weight $2$
Character orbit 1575.a
Rep. character $\chi_{1575}(1,\cdot)$
Character field $\Q$
Dimension $47$
Newform subspaces $26$
Sturm bound $480$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(480\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1575))\).

Total New Old
Modular forms 264 47 217
Cusp forms 217 47 170
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.

\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(16\)
Minus space\(-\)\(31\)

Trace form

\( 47 q - q^{2} + 51 q^{4} + q^{7} - 9 q^{8} + O(q^{10}) \) \( 47 q - q^{2} + 51 q^{4} + q^{7} - 9 q^{8} + 8 q^{11} + 2 q^{13} + 3 q^{14} + 67 q^{16} - 10 q^{17} + 20 q^{19} + 12 q^{22} + 8 q^{23} + 18 q^{26} + 3 q^{28} + 18 q^{29} + 12 q^{31} - 9 q^{32} + 34 q^{34} - 2 q^{37} + 28 q^{38} + 10 q^{41} + 24 q^{43} + 50 q^{44} + 2 q^{46} + 20 q^{47} + 47 q^{49} + 34 q^{52} + 10 q^{53} + 21 q^{56} - 6 q^{58} - 24 q^{59} + 22 q^{61} + 24 q^{62} + 133 q^{64} + 24 q^{67} - 18 q^{68} - 16 q^{71} - 6 q^{73} + 60 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} - 34 q^{82} - 12 q^{83} - 66 q^{86} + 8 q^{88} - 62 q^{89} - 14 q^{91} + 40 q^{92} - 92 q^{94} - 2 q^{97} - q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7
1575.2.a.a 1575.a 1.a $1$ $12.576$ \(\Q\) None 35.2.b.a \(-2\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{7}+3q^{11}+q^{13}+\cdots\)
1575.2.a.b 1575.a 1.a $1$ $12.576$ \(\Q\) None 315.2.d.b \(-1\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{7}+3q^{8}+4q^{13}+q^{14}+\cdots\)
1575.2.a.c 1575.a 1.a $1$ $12.576$ \(\Q\) None 21.2.a.a \(-1\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{11}+2q^{13}+\cdots\)
1575.2.a.d 1575.a 1.a $1$ $12.576$ \(\Q\) None 315.2.d.b \(-1\) \(0\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{7}+3q^{8}-4q^{13}-q^{14}+\cdots\)
1575.2.a.e 1575.a 1.a $1$ $12.576$ \(\Q\) None 105.2.d.a \(-1\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{7}+3q^{8}+6q^{11}+2q^{13}+\cdots\)
1575.2.a.f 1575.a 1.a $1$ $12.576$ \(\Q\) None 35.2.a.a \(0\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{7}+3q^{11}-5q^{13}+4q^{16}+\cdots\)
1575.2.a.g 1575.a 1.a $1$ $12.576$ \(\Q\) None 315.2.d.b \(1\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{7}-3q^{8}+4q^{13}-q^{14}+\cdots\)
1575.2.a.h 1575.a 1.a $1$ $12.576$ \(\Q\) None 105.2.a.a \(1\) \(0\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{13}-q^{14}+\cdots\)
1575.2.a.i 1575.a 1.a $1$ $12.576$ \(\Q\) None 105.2.d.a \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{7}-3q^{8}+6q^{11}-2q^{13}+\cdots\)
1575.2.a.j 1575.a 1.a $1$ $12.576$ \(\Q\) None 315.2.d.b \(1\) \(0\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{7}-3q^{8}-4q^{13}+q^{14}+\cdots\)
1575.2.a.k 1575.a 1.a $1$ $12.576$ \(\Q\) None 35.2.b.a \(2\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}+q^{7}+3q^{11}-q^{13}+\cdots\)
1575.2.a.l 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{5}) \) None 525.2.a.e \(-3\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{7}+(-1+\cdots)q^{8}+\cdots\)
1575.2.a.m 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{2}) \) None 315.2.a.c \(-2\) \(0\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{7}+(-3+\cdots)q^{8}+\cdots\)
1575.2.a.n 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{5}) \) None 175.2.a.d \(-1\) \(0\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+q^{7}+(-1+2\beta )q^{8}+\cdots\)
1575.2.a.o 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{13}) \) None 525.2.a.f \(-1\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{4}+q^{7}-3q^{8}+3q^{11}+\cdots\)
1575.2.a.p 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{17}) \) None 35.2.a.b \(-1\) \(0\) \(0\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}+q^{7}+(-4-\beta )q^{8}+\cdots\)
1575.2.a.q 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{3}) \) None 63.2.a.b \(0\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
1575.2.a.r 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{5}) \) None 105.2.a.b \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}-q^{7}-\beta q^{8}+(-2-2\beta )q^{11}+\cdots\)
1575.2.a.s 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{5}) \) None 175.2.a.d \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}-q^{7}+(1-2\beta )q^{8}+\cdots\)
1575.2.a.t 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{13}) \) None 525.2.a.f \(1\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(1+\beta )q^{4}-q^{7}+3q^{8}+3q^{11}+\cdots\)
1575.2.a.u 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{2}) \) None 315.2.a.c \(2\) \(0\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{7}+(3+\beta )q^{8}+\cdots\)
1575.2.a.v 1575.a 1.a $2$ $12.576$ \(\Q(\sqrt{5}) \) None 525.2.a.e \(3\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3\beta q^{4}+q^{7}+(1+4\beta )q^{8}+\cdots\)
1575.2.a.w 1575.a 1.a $3$ $12.576$ 3.3.148.1 None 105.2.d.b \(-1\) \(0\) \(0\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.2.a.x 1575.a 1.a $3$ $12.576$ 3.3.148.1 None 105.2.d.b \(1\) \(0\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+q^{7}+(2+\cdots)q^{8}+\cdots\)
1575.2.a.y 1575.a 1.a $4$ $12.576$ 4.4.174928.1 None 1575.2.a.y \(0\) \(0\) \(0\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}-q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1575.2.a.z 1575.a 1.a $4$ $12.576$ 4.4.174928.1 None 1575.2.a.y \(0\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{7}+(2\beta _{1}+\beta _{3})q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1575)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)