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Properties

Label 8048.2.a
Level $8048$
Weight $2$
Character orbit 8048.a
Rep. character $\chi_{8048}(1,\cdot)$
Character field $\Q$
Dimension $251$
Newform subspaces $25$
Sturm bound $2016$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(2016\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8048))\).

Total New Old
Modular forms 1014 251 763
Cusp forms 1003 251 752
Eisenstein series 11 0 11

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

\(2\)\(503\)FrickeDim
\(+\)\(+\)$+$\(63\)
\(+\)\(-\)$-$\(63\)
\(-\)\(+\)$-$\(73\)
\(-\)\(-\)$+$\(52\)
Plus space\(+\)\(115\)
Minus space\(-\)\(136\)

Trace form

\( 251 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 247 q^{9} + O(q^{10}) \) \( 251 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 247 q^{9} + 2 q^{11} - 2 q^{13} + 12 q^{15} - 2 q^{17} + 4 q^{19} - 10 q^{23} + 245 q^{25} + 8 q^{27} + 6 q^{29} + 16 q^{31} - 8 q^{33} - 2 q^{37} - 16 q^{39} + 6 q^{41} + 2 q^{43} - 10 q^{45} + 14 q^{47} + 251 q^{49} - 20 q^{51} - 2 q^{53} - 16 q^{55} - 8 q^{57} + 18 q^{59} - 18 q^{61} + 22 q^{63} - 4 q^{65} - 10 q^{67} + 8 q^{71} - 2 q^{73} + 46 q^{75} + 10 q^{79} + 227 q^{81} - 10 q^{83} - 28 q^{85} + 24 q^{87} - 2 q^{89} + 56 q^{91} + 8 q^{93} + 24 q^{95} - 2 q^{97} + 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 503
8048.2.a.a 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(-3\) \(-2\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{5}-3q^{7}+6q^{9}-3q^{11}+\cdots\)
8048.2.a.b 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(-1\) \(-4\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-q^{7}-2q^{9}+3q^{11}+\cdots\)
8048.2.a.c 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(-1\) \(-4\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}+3q^{7}-2q^{9}-5q^{11}+\cdots\)
8048.2.a.d 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(-1\) \(-2\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}-q^{7}-2q^{9}-q^{11}-3q^{13}+\cdots\)
8048.2.a.e 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(-1\) \(-2\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+3q^{7}-2q^{9}-q^{11}+\cdots\)
8048.2.a.f 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{9}-4q^{11}+2q^{13}-2q^{17}+2q^{19}+\cdots\)
8048.2.a.g 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+5q^{11}-5q^{13}+\cdots\)
8048.2.a.h 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}-2q^{9}+5q^{11}+3q^{13}+\cdots\)
8048.2.a.i 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(1\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+5q^{7}-2q^{9}+5q^{11}+q^{13}+\cdots\)
8048.2.a.j 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(1\) \(2\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-q^{7}-2q^{9}-3q^{11}+\cdots\)
8048.2.a.k 8048.a 1.a $1$ $64.264$ \(\Q\) None \(0\) \(3\) \(0\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+3q^{7}+6q^{9}-q^{11}+5q^{13}+\cdots\)
8048.2.a.l 8048.a 1.a $2$ $64.264$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-1+\beta )q^{5}+(2+\beta )q^{7}+2q^{9}+\cdots\)
8048.2.a.m 8048.a 1.a $3$ $64.264$ 3.3.257.1 None \(0\) \(-1\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{7}+\beta _{2}q^{9}+\cdots\)
8048.2.a.n 8048.a 1.a $5$ $64.264$ 5.5.36497.1 None \(0\) \(0\) \(-1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{3}-\beta _{4})q^{3}+(\beta _{2}+\beta _{3})q^{5}+\cdots\)
8048.2.a.o 8048.a 1.a $5$ $64.264$ 5.5.205225.1 None \(0\) \(4\) \(-3\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{3}-\beta _{4})q^{5}+\cdots\)
8048.2.a.p 8048.a 1.a $10$ $64.264$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(8\) \(-1\) \(5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}-\beta _{2}q^{5}+(\beta _{1}-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
8048.2.a.q 8048.a 1.a $12$ $64.264$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-5\) \(5\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{9}q^{5}+(-1-\beta _{10})q^{7}+\cdots\)
8048.2.a.r 8048.a 1.a $12$ $64.264$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(3\) \(7\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{3}+(\beta _{7}-\beta _{10})q^{5}+(-\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)
8048.2.a.s 8048.a 1.a $21$ $64.264$ None \(0\) \(-10\) \(3\) \(-13\) $-$ $-$ $\mathrm{SU}(2)$
8048.2.a.t 8048.a 1.a $21$ $64.264$ None \(0\) \(10\) \(-3\) \(15\) $-$ $+$ $\mathrm{SU}(2)$
8048.2.a.u 8048.a 1.a $26$ $64.264$ None \(0\) \(-4\) \(9\) \(-11\) $-$ $+$ $\mathrm{SU}(2)$
8048.2.a.v 8048.a 1.a $28$ $64.264$ None \(0\) \(-2\) \(-12\) \(0\) $+$ $+$ $\mathrm{SU}(2)$
8048.2.a.w 8048.a 1.a $29$ $64.264$ None \(0\) \(7\) \(-4\) \(13\) $+$ $-$ $\mathrm{SU}(2)$
8048.2.a.x 8048.a 1.a $33$ $64.264$ None \(0\) \(-10\) \(0\) \(-12\) $+$ $+$ $\mathrm{SU}(2)$
8048.2.a.y 8048.a 1.a $33$ $64.264$ None \(0\) \(2\) \(12\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8048))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8048)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2012))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\)\(^{\oplus 2}\)