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Properties

Label 1416.2.a
Level $1416$
Weight $2$
Character orbit 1416.a
Rep. character $\chi_{1416}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $8$
Sturm bound $480$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1416 = 2^{3} \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1416.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1416))\).

Total New Old
Modular forms 248 28 220
Cusp forms 233 28 205
Eisenstein series 15 0 15

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

\(2\)\(3\)\(59\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(6\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(18\)

Trace form

\( 28 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 28 q^{9} + O(q^{10}) \) \( 28 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 28 q^{9} + 8 q^{13} + 4 q^{17} + 16 q^{19} + 16 q^{23} + 20 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{31} + 24 q^{35} + 4 q^{39} - 4 q^{41} + 4 q^{43} + 4 q^{45} - 24 q^{47} + 24 q^{49} + 12 q^{51} - 12 q^{53} + 32 q^{55} - 8 q^{57} - 8 q^{61} + 4 q^{63} - 8 q^{65} + 12 q^{67} - 24 q^{71} - 2 q^{75} + 24 q^{77} + 20 q^{79} + 28 q^{81} + 24 q^{83} + 24 q^{85} + 16 q^{87} + 4 q^{89} + 16 q^{91} + 20 q^{93} - 48 q^{95} + 32 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1416))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 59
1416.2.a.a 1416.a 1.a $2$ $11.307$ \(\Q(\sqrt{5}) \) None 1416.2.a.a \(0\) \(2\) \(-2\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}+(-3+\cdots)q^{11}+\cdots\)
1416.2.a.b 1416.a 1.a $2$ $11.307$ \(\Q(\sqrt{5}) \) None 1416.2.a.b \(0\) \(2\) \(-2\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}-q^{11}+\cdots\)
1416.2.a.c 1416.a 1.a $3$ $11.307$ 3.3.229.1 None 1416.2.a.c \(0\) \(-3\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}-\beta _{1}q^{7}+\cdots\)
1416.2.a.d 1416.a 1.a $3$ $11.307$ 3.3.621.1 None 1416.2.a.d \(0\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
1416.2.a.e 1416.a 1.a $3$ $11.307$ 3.3.229.1 None 1416.2.a.e \(0\) \(-3\) \(2\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{1}+2\beta _{2})q^{7}+\cdots\)
1416.2.a.f 1416.a 1.a $4$ $11.307$ 4.4.27004.1 None 1416.2.a.f \(0\) \(-4\) \(4\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1-\beta _{2})q^{5}+\beta _{1}q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1416.2.a.g 1416.a 1.a $5$ $11.307$ 5.5.5755900.1 None 1416.2.a.g \(0\) \(5\) \(0\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta _{1}q^{5}+(1-\beta _{3})q^{7}+q^{9}+(\beta _{3}+\cdots)q^{11}+\cdots\)
1416.2.a.h 1416.a 1.a $6$ $11.307$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 1416.2.a.h \(0\) \(6\) \(6\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(1-\beta _{1})q^{5}+(1-\beta _{3})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1416))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1416)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(236))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(354))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(472))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(708))\)\(^{\oplus 2}\)