Properties

Label 270.4.a
Level $270$
Weight $4$
Character orbit 270.a
Rep. character $\chi_{270}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $14$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(270))\).

Total New Old
Modular forms 174 16 158
Cusp forms 150 16 134
Eisenstein series 24 0 24

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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(10\)
Minus space\(-\)\(6\)

Trace form

\( 16 q + 64 q^{4} - 100 q^{7} + O(q^{10}) \) \( 16 q + 64 q^{4} - 100 q^{7} + 92 q^{13} + 256 q^{16} + 284 q^{19} + 400 q^{25} - 400 q^{28} + 44 q^{31} + 312 q^{34} + 1412 q^{37} + 488 q^{43} + 168 q^{46} + 2292 q^{49} + 368 q^{52} + 60 q^{55} + 48 q^{58} - 988 q^{61} + 1024 q^{64} - 940 q^{67} - 892 q^{73} + 1136 q^{76} + 1520 q^{79} - 528 q^{82} - 1196 q^{91} - 2184 q^{94} - 3028 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(270))\) 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 5
270.4.a.a 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(-5\) \(-34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}-34q^{7}-8q^{8}+\cdots\)
270.4.a.b 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(-5\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+8q^{7}-8q^{8}+\cdots\)
270.4.a.c 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(5\) \(-22\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-22q^{7}-8q^{8}+\cdots\)
270.4.a.d 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(5\) \(-13\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-13q^{7}-8q^{8}+\cdots\)
270.4.a.e 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(5\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}-4q^{7}-8q^{8}+\cdots\)
270.4.a.f 270.a 1.a $1$ $15.931$ \(\Q\) None \(-2\) \(0\) \(5\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+5q^{5}+14q^{7}-8q^{8}+\cdots\)
270.4.a.g 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(-5\) \(-22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-22q^{7}+8q^{8}+\cdots\)
270.4.a.h 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(-5\) \(-13\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-13q^{7}+8q^{8}+\cdots\)
270.4.a.i 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(-5\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}-4q^{7}+8q^{8}+\cdots\)
270.4.a.j 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(-5\) \(14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-5q^{5}+14q^{7}+8q^{8}+\cdots\)
270.4.a.k 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(5\) \(-34\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}-34q^{7}+8q^{8}+\cdots\)
270.4.a.l 270.a 1.a $1$ $15.931$ \(\Q\) None \(2\) \(0\) \(5\) \(8\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+8q^{7}+8q^{8}+\cdots\)
270.4.a.m 270.a 1.a $2$ $15.931$ \(\Q(\sqrt{401}) \) None \(-4\) \(0\) \(-10\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-5q^{5}+(1+\beta )q^{7}-8q^{8}+\cdots\)
270.4.a.n 270.a 1.a $2$ $15.931$ \(\Q(\sqrt{401}) \) None \(4\) \(0\) \(10\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+5q^{5}+(1+\beta )q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(270))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(270)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)