Properties

Label 1350.2.a
Level $1350$
Weight $2$
Character orbit 1350.a
Rep. character $\chi_{1350}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $24$
Sturm bound $540$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(540\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1350))\).

Total New Old
Modular forms 306 26 280
Cusp forms 235 26 209
Eisenstein series 71 0 71

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

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(16\)

Trace form

\( 26 q + 26 q^{4} - 6 q^{7} + O(q^{10}) \) \( 26 q + 26 q^{4} - 6 q^{7} + 26 q^{16} - 6 q^{22} - 6 q^{28} + 2 q^{31} + 8 q^{34} + 12 q^{37} + 36 q^{43} - 4 q^{46} + 60 q^{49} + 36 q^{58} + 4 q^{61} + 26 q^{64} - 12 q^{67} + 30 q^{73} + 12 q^{79} + 12 q^{82} - 6 q^{88} + 24 q^{91} + 28 q^{94} - 30 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
1350.2.a.a $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ \(q-q^{2}+q^{4}-4q^{7}-q^{8}+3q^{11}-q^{13}+\cdots\)
1350.2.a.b $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $+$ $+$ $-$ \(q-q^{2}+q^{4}-4q^{7}-q^{8}+5q^{11}+3q^{13}+\cdots\)
1350.2.a.c $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ \(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.d $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) $+$ $-$ $+$ \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.e $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-2\) $+$ $+$ $+$ \(q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-5q^{13}+\cdots\)
1350.2.a.f $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) $+$ $-$ $-$ \(q-q^{2}+q^{4}-q^{7}-q^{8}+2q^{13}+q^{14}+\cdots\)
1350.2.a.g $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(1\) $+$ $+$ $+$ \(q-q^{2}+q^{4}+q^{7}-q^{8}-2q^{13}-q^{14}+\cdots\)
1350.2.a.h $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(1\) $+$ $-$ $+$ \(q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{11}+4q^{13}+\cdots\)
1350.2.a.i $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(2\) $+$ $+$ $-$ \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{11}+5q^{13}+\cdots\)
1350.2.a.j $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(4\) $+$ $-$ $-$ \(q-q^{2}+q^{4}+4q^{7}-q^{8}-5q^{11}-3q^{13}+\cdots\)
1350.2.a.k $1$ $10.780$ \(\Q\) None \(-1\) \(0\) \(0\) \(4\) $+$ $-$ $+$ \(q-q^{2}+q^{4}+4q^{7}-q^{8}-3q^{11}+q^{13}+\cdots\)
1350.2.a.l $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-4\) $-$ $+$ $-$ \(q+q^{2}+q^{4}-4q^{7}+q^{8}-5q^{11}+3q^{13}+\cdots\)
1350.2.a.m $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-4\) $-$ $+$ $-$ \(q+q^{2}+q^{4}-4q^{7}+q^{8}-3q^{11}-q^{13}+\cdots\)
1350.2.a.n $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.o $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{11}-5q^{13}+\cdots\)
1350.2.a.p $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.q $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ \(q+q^{2}+q^{4}-q^{7}+q^{8}+2q^{13}-q^{14}+\cdots\)
1350.2.a.r $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $+$ $+$ \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{11}+4q^{13}+\cdots\)
1350.2.a.s $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $+$ $+$ \(q+q^{2}+q^{4}+q^{7}+q^{8}-2q^{13}+q^{14}+\cdots\)
1350.2.a.t $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $-$ $-$ \(q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+5q^{13}+\cdots\)
1350.2.a.u $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(4\) $-$ $+$ $+$ \(q+q^{2}+q^{4}+4q^{7}+q^{8}+3q^{11}+q^{13}+\cdots\)
1350.2.a.v $1$ $10.780$ \(\Q\) None \(1\) \(0\) \(0\) \(4\) $-$ $-$ $-$ \(q+q^{2}+q^{4}+4q^{7}+q^{8}+5q^{11}-3q^{13}+\cdots\)
1350.2.a.w $2$ $10.780$ \(\Q(\sqrt{19}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $-$ \(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+\beta q^{11}-\beta q^{14}+\cdots\)
1350.2.a.x $2$ $10.780$ \(\Q(\sqrt{19}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $-$ \(q+q^{2}+q^{4}+\beta q^{7}+q^{8}-\beta q^{11}+\beta q^{14}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)