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Properties

Label 1176.2.a
Level $1176$
Weight $2$
Character orbit 1176.a
Rep. character $\chi_{1176}(1,\cdot)$
Character field $\Q$
Dimension $21$
Newform subspaces $15$
Sturm bound $448$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(448\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1176))\).

Total New Old
Modular forms 256 21 235
Cusp forms 193 21 172
Eisenstein series 63 0 63

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

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(8\)
Minus space\(-\)\(13\)

Trace form

\( 21 q + q^{3} - 2 q^{5} + 21 q^{9} + O(q^{10}) \) \( 21 q + q^{3} - 2 q^{5} + 21 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{17} + 4 q^{19} + 16 q^{23} + 23 q^{25} + q^{27} - 2 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{37} + 6 q^{39} + 18 q^{41} - 16 q^{43} - 2 q^{45} - 10 q^{51} + 46 q^{53} + 8 q^{55} - 20 q^{59} + 14 q^{61} + 28 q^{65} - 24 q^{67} - 8 q^{69} - 8 q^{71} + 2 q^{73} - q^{75} + 21 q^{81} - 12 q^{83} + 44 q^{85} - 10 q^{87} + 2 q^{89} + 20 q^{93} + 8 q^{95} - 22 q^{97} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\) 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 7
1176.2.a.a 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.a.b \(0\) \(-1\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+2q^{13}+2q^{15}+\cdots\)
1176.2.a.b 1176.a 1.a $1$ $9.390$ \(\Q\) None 1176.2.a.b \(0\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{9}-4q^{13}-4q^{17}+4q^{19}+\cdots\)
1176.2.a.c 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.q.a \(0\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}+3q^{11}-4q^{13}+\cdots\)
1176.2.a.d 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.q.b \(0\) \(-1\) \(2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{9}-6q^{11}-3q^{13}+\cdots\)
1176.2.a.e 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.q.b \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-6q^{11}+3q^{13}+\cdots\)
1176.2.a.f 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.a.a \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-6q^{13}-2q^{15}+\cdots\)
1176.2.a.g 1176.a 1.a $1$ $9.390$ \(\Q\) None 168.2.q.a \(0\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{9}+3q^{11}+4q^{13}+\cdots\)
1176.2.a.h 1176.a 1.a $1$ $9.390$ \(\Q\) None 1176.2.a.b \(0\) \(1\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+4q^{13}+4q^{17}-4q^{19}+\cdots\)
1176.2.a.i 1176.a 1.a $1$ $9.390$ \(\Q\) None 24.2.a.a \(0\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1176.2.a.j 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{2}) \) None 1176.2.a.j \(0\) \(-2\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-2+\beta )q^{5}+q^{9}+(2-2\beta )q^{11}+\cdots\)
1176.2.a.k 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{57}) \) None 168.2.q.c \(0\) \(-2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}+q^{9}-\beta q^{11}+(3-\beta )q^{13}+\cdots\)
1176.2.a.l 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{2}) \) None 1176.2.a.l \(0\) \(-2\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(2+\beta )q^{5}+q^{9}+(-2+2\beta )q^{11}+\cdots\)
1176.2.a.m 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{2}) \) None 1176.2.a.l \(0\) \(2\) \(-4\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-2+\beta )q^{5}+q^{9}+(-2-2\beta )q^{11}+\cdots\)
1176.2.a.n 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{57}) \) None 168.2.q.c \(0\) \(2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+q^{9}-\beta q^{11}+(-3+\beta )q^{13}+\cdots\)
1176.2.a.o 1176.a 1.a $2$ $9.390$ \(\Q(\sqrt{2}) \) None 1176.2.a.j \(0\) \(2\) \(4\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(2+\beta )q^{5}+q^{9}+(2+2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1176))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1176)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(588))\)\(^{\oplus 2}\)