Properties

Label 1350.4.a
Level $1350$
Weight $4$
Character orbit 1350.a
Rep. character $\chi_{1350}(1,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $48$
Sturm bound $1080$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 48 \)
Sturm bound: \(1080\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1350))\).

Total New Old
Modular forms 846 76 770
Cusp forms 774 76 698
Eisenstein series 72 0 72

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.
\(+\)\(+\)\(+\)\(+\)\(9\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(41\)
Minus space\(-\)\(35\)

Trace form

\( 76 q + 304 q^{4} + 56 q^{7} + O(q^{10}) \) \( 76 q + 304 q^{4} + 56 q^{7} + 26 q^{13} + 1216 q^{16} - 310 q^{19} - 12 q^{22} + 224 q^{28} + 146 q^{31} + 480 q^{34} - 1078 q^{37} - 1900 q^{43} - 408 q^{46} + 3252 q^{49} + 104 q^{52} + 408 q^{58} + 2102 q^{61} + 4864 q^{64} + 1130 q^{67} + 4232 q^{73} - 1240 q^{76} - 1702 q^{79} - 216 q^{82} - 48 q^{88} - 2426 q^{91} - 3672 q^{94} + 1436 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\) 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
1350.4.a.a 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-29\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-29q^{7}-8q^{8}-57q^{11}+\cdots\)
1350.4.a.b 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-23\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-23q^{7}-8q^{8}-30q^{11}+\cdots\)
1350.4.a.c 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-19\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-19q^{7}-8q^{8}-12q^{11}+\cdots\)
1350.4.a.d 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-22q^{11}+\cdots\)
1350.4.a.e 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-3q^{11}+\cdots\)
1350.4.a.f 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(-8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}-8q^{7}-8q^{8}+18q^{11}+\cdots\)
1350.4.a.g 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+4q^{7}-8q^{8}+42q^{11}+\cdots\)
1350.4.a.h 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+7q^{7}-8q^{8}+60q^{11}+\cdots\)
1350.4.a.i 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(13\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+13q^{7}-8q^{8}-30q^{11}+\cdots\)
1350.4.a.j 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+14q^{7}-8q^{8}+22q^{11}+\cdots\)
1350.4.a.k 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(19\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+19q^{7}-8q^{8}+12q^{11}+\cdots\)
1350.4.a.l 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(22\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+22q^{7}-8q^{8}-12q^{11}+\cdots\)
1350.4.a.m 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(23\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+23q^{7}-8q^{8}+30q^{11}+\cdots\)
1350.4.a.n 1350.a 1.a $1$ $79.653$ \(\Q\) None \(-2\) \(0\) \(0\) \(34\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+34q^{7}-8q^{8}-48q^{11}+\cdots\)
1350.4.a.o 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-29\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-29q^{7}+8q^{8}+57q^{11}+\cdots\)
1350.4.a.p 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-23\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-23q^{7}+8q^{8}+30q^{11}+\cdots\)
1350.4.a.q 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-19\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-19q^{7}+8q^{8}+12q^{11}+\cdots\)
1350.4.a.r 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+3q^{11}+\cdots\)
1350.4.a.s 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-14q^{7}+8q^{8}+22q^{11}+\cdots\)
1350.4.a.t 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(-8\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-8q^{7}+8q^{8}-18q^{11}+\cdots\)
1350.4.a.u 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+4q^{7}+8q^{8}-42q^{11}+\cdots\)
1350.4.a.v 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+7q^{7}+8q^{8}-60q^{11}+\cdots\)
1350.4.a.w 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+13q^{7}+8q^{8}+30q^{11}+\cdots\)
1350.4.a.x 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+14q^{7}+8q^{8}-22q^{11}+\cdots\)
1350.4.a.y 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(19\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+19q^{7}+8q^{8}-12q^{11}+\cdots\)
1350.4.a.z 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+22q^{7}+8q^{8}+12q^{11}+\cdots\)
1350.4.a.ba 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(23\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+23q^{7}+8q^{8}-30q^{11}+\cdots\)
1350.4.a.bb 1350.a 1.a $1$ $79.653$ \(\Q\) None \(2\) \(0\) \(0\) \(34\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+34q^{7}+8q^{8}+48q^{11}+\cdots\)
1350.4.a.bc 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{6}) \) None \(-4\) \(0\) \(0\) \(-20\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-10+\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bd 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{209}) \) None \(-4\) \(0\) \(0\) \(-13\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-6-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.be 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{21}) \) None \(-4\) \(0\) \(0\) \(-10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-5-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bf 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{401}) \) None \(-4\) \(0\) \(0\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-1-\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bg 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{21}) \) None \(-4\) \(0\) \(0\) \(10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(5+\beta )q^{7}-8q^{8}+(-15+\cdots)q^{11}+\cdots\)
1350.4.a.bh 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{209}) \) None \(-4\) \(0\) \(0\) \(13\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(7-\beta )q^{7}-8q^{8}+(-10+\cdots)q^{11}+\cdots\)
1350.4.a.bi 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{6}) \) None \(-4\) \(0\) \(0\) \(20\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(10+\beta )q^{7}-8q^{8}+\cdots\)
1350.4.a.bj 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{6}) \) None \(4\) \(0\) \(0\) \(-20\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-10+\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bk 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{209}) \) None \(4\) \(0\) \(0\) \(-13\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-6-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bl 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{21}) \) None \(4\) \(0\) \(0\) \(-10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-5-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bm 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{401}) \) None \(4\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-1-\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bn 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{21}) \) None \(4\) \(0\) \(0\) \(10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(5+\beta )q^{7}+8q^{8}+(15+\cdots)q^{11}+\cdots\)
1350.4.a.bo 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{209}) \) None \(4\) \(0\) \(0\) \(13\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(7-\beta )q^{7}+8q^{8}+(10+\cdots)q^{11}+\cdots\)
1350.4.a.bp 1350.a 1.a $2$ $79.653$ \(\Q(\sqrt{6}) \) None \(4\) \(0\) \(0\) \(20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(10+\beta )q^{7}+8q^{8}+\cdots\)
1350.4.a.bq 1350.a 1.a $3$ $79.653$ 3.3.46616.1 None \(-6\) \(0\) \(0\) \(-22\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-7+\beta _{2})q^{7}-8q^{8}+\cdots\)
1350.4.a.br 1350.a 1.a $3$ $79.653$ 3.3.46616.1 None \(-6\) \(0\) \(0\) \(22\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(7-\beta _{2})q^{7}-8q^{8}+\cdots\)
1350.4.a.bs 1350.a 1.a $3$ $79.653$ 3.3.46616.1 None \(6\) \(0\) \(0\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-7+\beta _{2})q^{7}+8q^{8}+\cdots\)
1350.4.a.bt 1350.a 1.a $3$ $79.653$ 3.3.46616.1 None \(6\) \(0\) \(0\) \(22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(7-\beta _{2})q^{7}+8q^{8}+\cdots\)
1350.4.a.bu 1350.a 1.a $4$ $79.653$ 4.4.29021904.1 None \(-8\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+\beta _{1}q^{7}-8q^{8}-\beta _{1}q^{11}+\cdots\)
1350.4.a.bv 1350.a 1.a $4$ $79.653$ 4.4.29021904.1 None \(8\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+\beta _{1}q^{7}+8q^{8}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1350))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1350)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 2}\)