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Properties

Label 2178.2.a
Level $2178$
Weight $2$
Character orbit 2178.a
Rep. character $\chi_{2178}(1,\cdot)$
Character field $\Q$
Dimension $45$
Newform subspaces $29$
Sturm bound $792$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(792\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2178))\).

Total New Old
Modular forms 444 45 399
Cusp forms 349 45 304
Eisenstein series 95 0 95

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

\(2\)\(3\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(20\)
Minus space\(-\)\(25\)

Trace form

\( 45 q + q^{2} + 45 q^{4} + q^{8} + O(q^{10}) \) \( 45 q + q^{2} + 45 q^{4} + q^{8} + 2 q^{10} + 2 q^{13} - 8 q^{14} + 45 q^{16} - 6 q^{17} - 4 q^{19} + 4 q^{23} + 49 q^{25} + 22 q^{29} + q^{32} + 10 q^{34} + 6 q^{38} + 2 q^{40} + 2 q^{41} + 4 q^{43} + 8 q^{46} - 24 q^{47} + 53 q^{49} + 15 q^{50} + 2 q^{52} + 4 q^{53} - 8 q^{56} + 4 q^{58} + 10 q^{59} + 34 q^{61} - 16 q^{62} + 45 q^{64} - 28 q^{65} - 6 q^{67} - 6 q^{68} + 8 q^{70} + 4 q^{71} - 18 q^{73} + 14 q^{74} - 4 q^{76} - 24 q^{79} + 10 q^{82} - 4 q^{83} - 12 q^{85} + 6 q^{86} + 18 q^{89} - 80 q^{91} + 4 q^{92} - 16 q^{94} + 8 q^{95} - 78 q^{97} + 9 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2178))\) 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 3 11
2178.2.a.a 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{13}+2q^{14}+\cdots\)
2178.2.a.b 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{7}-q^{8}+4q^{13}+2q^{14}+\cdots\)
2178.2.a.c 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{8}+6q^{13}+q^{16}+6q^{17}+\cdots\)
2178.2.a.d 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
2178.2.a.e 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(1\) \(4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
2178.2.a.f 2178.a 1.a $1$ $17.391$ \(\Q\) None \(-1\) \(0\) \(3\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+3q^{5}-2q^{7}-q^{8}-3q^{10}+\cdots\)
2178.2.a.g 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(-2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}+4q^{7}+q^{8}-2q^{10}+\cdots\)
2178.2.a.h 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}-2q^{13}-2q^{14}+\cdots\)
2178.2.a.i 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{8}-6q^{13}+q^{16}-6q^{17}+\cdots\)
2178.2.a.j 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)
2178.2.a.k 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
2178.2.a.l 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(3\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+3q^{5}+2q^{7}+q^{8}+3q^{10}+\cdots\)
2178.2.a.m 2178.a 1.a $1$ $17.391$ \(\Q\) None \(1\) \(0\) \(4\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{5}+2q^{7}+q^{8}+4q^{10}+\cdots\)
2178.2.a.n 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-5\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2-\beta )q^{5}+(1-3\beta )q^{7}+\cdots\)
2178.2.a.o 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-5\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2-\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
2178.2.a.p 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(-2\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2\beta q^{5}-2q^{7}-q^{8}+2\beta q^{10}+\cdots\)
2178.2.a.q 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}-\beta q^{7}-q^{8}-\beta q^{10}+\cdots\)
2178.2.a.r 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}-q^{8}-\beta q^{10}+\beta q^{13}+\cdots\)
2178.2.a.s 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(0\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}+(3+\beta )q^{7}-q^{8}+\cdots\)
2178.2.a.t 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(1\) \(7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1+3\beta )q^{5}+(4-\beta )q^{7}+\cdots\)
2178.2.a.u 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(5\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(3-\beta )q^{5}+(2-3\beta )q^{7}+\cdots\)
2178.2.a.v 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-5\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2-\beta )q^{5}+(-1+\beta )q^{7}+\cdots\)
2178.2.a.w 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-5\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2-\beta )q^{5}+(-1+3\beta )q^{7}+\cdots\)
2178.2.a.x 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(-2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2\beta q^{5}+2q^{7}+q^{8}-2\beta q^{10}+\cdots\)
2178.2.a.y 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+(-3-\beta )q^{7}+q^{8}+\cdots\)
2178.2.a.z 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+q^{8}+\beta q^{10}-\beta q^{13}+\cdots\)
2178.2.a.ba 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+\beta q^{7}+q^{8}+\beta q^{10}+\cdots\)
2178.2.a.bb 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(1\) \(-7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-1+3\beta )q^{5}+(-4+\beta )q^{7}+\cdots\)
2178.2.a.bc 2178.a 1.a $2$ $17.391$ \(\Q(\sqrt{5}) \) None \(2\) \(0\) \(5\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(3-\beta )q^{5}+(-2+3\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2178))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2178)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(726))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\)\(^{\oplus 2}\)