Properties

Label 3856.2.a
Level $3856$
Weight $2$
Character orbit 3856.a
Rep. character $\chi_{3856}(1,\cdot)$
Character field $\Q$
Dimension $120$
Newform subspaces $17$
Sturm bound $968$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(968\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3856))\).

Total New Old
Modular forms 490 120 370
Cusp forms 479 120 359
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.

\(2\)\(241\)FrickeDim
\(+\)\(+\)$+$\(27\)
\(+\)\(-\)$-$\(33\)
\(-\)\(+\)$-$\(33\)
\(-\)\(-\)$+$\(27\)
Plus space\(+\)\(54\)
Minus space\(-\)\(66\)

Trace form

\( 120 q - 2 q^{7} + 120 q^{9} + O(q^{10}) \) \( 120 q - 2 q^{7} + 120 q^{9} + 2 q^{11} + 12 q^{15} + 6 q^{19} + 2 q^{23} + 120 q^{25} + 8 q^{29} - 6 q^{31} + 12 q^{35} + 8 q^{37} + 4 q^{39} - 14 q^{43} + 12 q^{47} + 120 q^{49} + 36 q^{51} + 12 q^{55} - 16 q^{57} - 16 q^{59} - 8 q^{61} - 10 q^{63} - 16 q^{65} - 8 q^{67} - 16 q^{69} - 10 q^{71} - 32 q^{75} - 16 q^{77} - 8 q^{79} + 104 q^{81} + 12 q^{83} - 16 q^{85} - 16 q^{87} - 8 q^{89} - 16 q^{91} - 8 q^{93} - 28 q^{95} - 8 q^{97} - 10 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3856))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 241
3856.2.a.a 3856.a 1.a $1$ $30.790$ \(\Q\) None \(0\) \(-2\) \(3\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+3q^{5}-3q^{7}+q^{9}-4q^{11}+\cdots\)
3856.2.a.b 3856.a 1.a $1$ $30.790$ \(\Q\) None \(0\) \(0\) \(2\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-2q^{7}-3q^{9}+2q^{11}-2q^{13}+\cdots\)
3856.2.a.c 3856.a 1.a $1$ $30.790$ \(\Q\) None \(0\) \(2\) \(-1\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
3856.2.a.d 3856.a 1.a $1$ $30.790$ \(\Q\) None \(0\) \(2\) \(1\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
3856.2.a.e 3856.a 1.a $2$ $30.790$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-3\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{5}+(3-2\beta )q^{7}-2q^{9}+\cdots\)
3856.2.a.f 3856.a 1.a $2$ $30.790$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(1\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(2-3\beta )q^{5}-3q^{7}-2q^{9}+(-1+\cdots)q^{11}+\cdots\)
3856.2.a.g 3856.a 1.a $3$ $30.790$ 3.3.229.1 None \(0\) \(2\) \(2\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1}+\beta _{2})q^{3}+(1+\beta _{2})q^{5}+3q^{7}+\cdots\)
3856.2.a.h 3856.a 1.a $6$ $30.790$ 6.6.131357120.1 None \(0\) \(-2\) \(-5\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1+\beta _{5})q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
3856.2.a.i 3856.a 1.a $7$ $30.790$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(1\) \(-8\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1-\beta _{2})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3856.2.a.j 3856.a 1.a $7$ $30.790$ 7.7.31056073.1 None \(0\) \(3\) \(-8\) \(7\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{6}q^{3}+(-1-\beta _{1}-\beta _{2}-\beta _{6})q^{5}+\cdots\)
3856.2.a.k 3856.a 1.a $9$ $30.790$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-4\) \(5\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{4})q^{5}+(-1-\beta _{3})q^{7}+\cdots\)
3856.2.a.l 3856.a 1.a $11$ $30.790$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(1\) \(-7\) \(9\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1-\beta _{10})q^{5}+(1-\beta _{8}+\cdots)q^{7}+\cdots\)
3856.2.a.m 3856.a 1.a $12$ $30.790$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-5\) \(11\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{7})q^{5}+\beta _{3}q^{7}+(3+\beta _{2}+\cdots)q^{9}+\cdots\)
3856.2.a.n 3856.a 1.a $12$ $30.790$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-1\) \(6\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{8}q^{3}+(1-\beta _{9})q^{5}+(-\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)
3856.2.a.o 3856.a 1.a $13$ $30.790$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(-7\) \(-7\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{10})q^{5}+\beta _{6}q^{7}+\cdots\)
3856.2.a.p 3856.a 1.a $13$ $30.790$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(7\) \(1\) \(11\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}-\beta _{4}q^{5}+(1-\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
3856.2.a.q 3856.a 1.a $19$ $30.790$ \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None \(0\) \(-1\) \(7\) \(-9\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{6}q^{5}+\beta _{12}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3856))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3856)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(482))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(964))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1928))\)\(^{\oplus 2}\)