Properties

Label 1350.2.a
Level 1350
Weight 2
Character orbit a
Rep. character \(\chi_{1350}(1,\cdot)\)
Character field \(\Q\)
Dimension 26
Newforms 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: \(\Q\)
Newforms: \( 24 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\) into irreducible Hecke orbits

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

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