Properties

Label 1584.4.a
Level $1584$
Weight $4$
Character orbit 1584.a
Rep. character $\chi_{1584}(1,\cdot)$
Character field $\Q$
Dimension $75$
Newform subspaces $44$
Sturm bound $1152$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 44 \)
Sturm bound: \(1152\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1584))\).

Total New Old
Modular forms 888 75 813
Cusp forms 840 75 765
Eisenstein series 48 0 48

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
\(+\)\(+\)\(+\)\(+\)\(7\)
\(+\)\(+\)\(-\)\(-\)\(7\)
\(+\)\(-\)\(+\)\(-\)\(10\)
\(+\)\(-\)\(-\)\(+\)\(13\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(8\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(11\)
Plus space\(+\)\(39\)
Minus space\(-\)\(36\)

Trace form

\( 75 q + 2 q^{5} - 36 q^{7} + O(q^{10}) \) \( 75 q + 2 q^{5} - 36 q^{7} + 33 q^{11} - 46 q^{13} - 26 q^{17} - 204 q^{19} - 138 q^{23} + 1897 q^{25} - 86 q^{29} + 330 q^{31} + 228 q^{35} + 66 q^{37} + 118 q^{41} + 168 q^{43} - 1256 q^{47} + 4035 q^{49} - 286 q^{53} + 220 q^{55} + 2426 q^{59} + 834 q^{61} - 812 q^{65} - 2082 q^{67} - 1598 q^{71} - 586 q^{73} + 3180 q^{79} + 992 q^{83} - 940 q^{85} + 874 q^{89} + 560 q^{91} - 5592 q^{95} - 94 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 11
1584.4.a.a 1584.a 1.a $1$ $93.459$ \(\Q\) None 132.4.a.c \(0\) \(0\) \(-22\) \(20\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-22q^{5}+20q^{7}+11q^{11}+22q^{13}+\cdots\)
1584.4.a.b 1584.a 1.a $1$ $93.459$ \(\Q\) None 22.4.a.b \(0\) \(0\) \(-14\) \(8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-14q^{5}+8q^{7}-11q^{11}-50q^{13}+\cdots\)
1584.4.a.c 1584.a 1.a $1$ $93.459$ \(\Q\) None 396.4.a.b \(0\) \(0\) \(-12\) \(-26\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-12q^{5}-26q^{7}+11q^{11}-34q^{13}+\cdots\)
1584.4.a.d 1584.a 1.a $1$ $93.459$ \(\Q\) None 264.4.a.b \(0\) \(0\) \(-12\) \(-22\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-12q^{5}-22q^{7}+11q^{11}-48q^{13}+\cdots\)
1584.4.a.e 1584.a 1.a $1$ $93.459$ \(\Q\) None 66.4.a.b \(0\) \(0\) \(-10\) \(-16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-10q^{5}-2^{4}q^{7}+11q^{11}+10q^{13}+\cdots\)
1584.4.a.f 1584.a 1.a $1$ $93.459$ \(\Q\) None 132.4.a.d \(0\) \(0\) \(-10\) \(-8\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-10q^{5}-8q^{7}-11q^{11}+18q^{13}+\cdots\)
1584.4.a.g 1584.a 1.a $1$ $93.459$ \(\Q\) None 88.4.a.b \(0\) \(0\) \(-9\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-9q^{5}-2q^{7}-11q^{11}+38q^{17}+\cdots\)
1584.4.a.h 1584.a 1.a $1$ $93.459$ \(\Q\) None 198.4.a.c \(0\) \(0\) \(-8\) \(22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{5}+22q^{7}+11q^{11}-54q^{13}+\cdots\)
1584.4.a.i 1584.a 1.a $1$ $93.459$ \(\Q\) None 66.4.a.a \(0\) \(0\) \(0\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{7}+11q^{11}+80q^{13}-30q^{17}+\cdots\)
1584.4.a.j 1584.a 1.a $1$ $93.459$ \(\Q\) None 132.4.a.b \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}-11q^{11}-88q^{13}+66q^{17}+\cdots\)
1584.4.a.k 1584.a 1.a $1$ $93.459$ \(\Q\) None 22.4.a.c \(0\) \(0\) \(3\) \(10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}+10q^{7}+11q^{11}-2^{4}q^{13}+\cdots\)
1584.4.a.l 1584.a 1.a $1$ $93.459$ \(\Q\) None 33.4.a.b \(0\) \(0\) \(4\) \(26\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{5}+26q^{7}+11q^{11}-2^{5}q^{13}+\cdots\)
1584.4.a.m 1584.a 1.a $1$ $93.459$ \(\Q\) None 264.4.a.d \(0\) \(0\) \(6\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+6q^{5}+8q^{7}-11q^{11}-30q^{13}+\cdots\)
1584.4.a.n 1584.a 1.a $1$ $93.459$ \(\Q\) None 264.4.a.c \(0\) \(0\) \(6\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+6q^{5}+14q^{7}+11q^{11}+6q^{13}+\cdots\)
1584.4.a.o 1584.a 1.a $1$ $93.459$ \(\Q\) None 88.4.a.a \(0\) \(0\) \(7\) \(6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{5}+6q^{7}-11q^{11}-40q^{13}+\cdots\)
1584.4.a.p 1584.a 1.a $1$ $93.459$ \(\Q\) None 44.4.a.a \(0\) \(0\) \(7\) \(26\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{5}+26q^{7}-11q^{11}+52q^{13}+\cdots\)
1584.4.a.q 1584.a 1.a $1$ $93.459$ \(\Q\) None 198.4.a.c \(0\) \(0\) \(8\) \(22\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{5}+22q^{7}-11q^{11}-54q^{13}+\cdots\)
1584.4.a.r 1584.a 1.a $1$ $93.459$ \(\Q\) None 396.4.a.b \(0\) \(0\) \(12\) \(-26\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+12q^{5}-26q^{7}-11q^{11}-34q^{13}+\cdots\)
1584.4.a.s 1584.a 1.a $1$ $93.459$ \(\Q\) None 132.4.a.a \(0\) \(0\) \(12\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+12q^{5}-14q^{7}+11q^{11}+56q^{13}+\cdots\)
1584.4.a.t 1584.a 1.a $1$ $93.459$ \(\Q\) None 33.4.a.a \(0\) \(0\) \(14\) \(32\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+14q^{5}+2^{5}q^{7}-11q^{11}-38q^{13}+\cdots\)
1584.4.a.u 1584.a 1.a $1$ $93.459$ \(\Q\) None 264.4.a.a \(0\) \(0\) \(18\) \(28\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+18q^{5}+28q^{7}+11q^{11}-18q^{13}+\cdots\)
1584.4.a.v 1584.a 1.a $1$ $93.459$ \(\Q\) None 22.4.a.a \(0\) \(0\) \(19\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+19q^{5}-14q^{7}+11q^{11}-72q^{13}+\cdots\)
1584.4.a.w 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{3}) \) None 99.4.a.d \(0\) \(0\) \(-20\) \(16\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-10+\beta )q^{5}+(8-5\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.x 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{33}) \) None 33.4.a.d \(0\) \(0\) \(-16\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-8-2\beta )q^{5}+(-1-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.y 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{31}) \) None 396.4.a.h \(0\) \(0\) \(-12\) \(24\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-6+\beta )q^{5}+(12-\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.z 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{97}) \) None 44.4.a.b \(0\) \(0\) \(-11\) \(-10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-5-\beta )q^{5}+(-2-6\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.ba 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{97}) \) None 66.4.a.c \(0\) \(0\) \(-10\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta )q^{5}+(1+3\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bb 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{70}) \) None 198.4.a.i \(0\) \(0\) \(-8\) \(-36\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-4+\beta )q^{5}+(-18+\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bc 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{3}) \) None 11.4.a.a \(0\) \(0\) \(-2\) \(-20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+2\beta )q^{5}+(-10-\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bd 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{185}) \) None 264.4.a.g \(0\) \(0\) \(6\) \(-22\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{5}+(-11+\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.be 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{137}) \) None 264.4.a.f \(0\) \(0\) \(6\) \(-16\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{5}+(-8-2\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bf 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{17}) \) None 264.4.a.e \(0\) \(0\) \(6\) \(-10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{5}+(-5-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bg 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{5}) \) None 88.4.a.c \(0\) \(0\) \(6\) \(56\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+4\beta )q^{5}+(28-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bh 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{70}) \) None 198.4.a.i \(0\) \(0\) \(8\) \(-36\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{5}+(-18-\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bi 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{31}) \) None 396.4.a.h \(0\) \(0\) \(12\) \(24\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{5}+(12+\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bj 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{97}) \) None 33.4.a.c \(0\) \(0\) \(14\) \(-24\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{5}+(-12-2\beta )q^{7}-11q^{11}+\cdots\)
1584.4.a.bk 1584.a 1.a $2$ $93.459$ \(\Q(\sqrt{3}) \) None 99.4.a.d \(0\) \(0\) \(20\) \(16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(10+\beta )q^{5}+(8+5\beta )q^{7}+11q^{11}+\cdots\)
1584.4.a.bl 1584.a 1.a $3$ $93.459$ 3.3.11109.1 None 88.4.a.d \(0\) \(0\) \(-8\) \(-24\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{2})q^{5}+(-8+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1584.4.a.bm 1584.a 1.a $3$ $93.459$ 3.3.4364.1 None 792.4.a.k \(0\) \(0\) \(-8\) \(-10\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{5}+(-3+\beta _{2})q^{7}-11q^{11}+\cdots\)
1584.4.a.bn 1584.a 1.a $3$ $93.459$ 3.3.123209.1 None 264.4.a.i \(0\) \(0\) \(-4\) \(-28\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(-10-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1584.4.a.bo 1584.a 1.a $3$ $93.459$ 3.3.142161.1 None 264.4.a.h \(0\) \(0\) \(-4\) \(12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(4-\beta _{2})q^{7}-11q^{11}+\cdots\)
1584.4.a.bp 1584.a 1.a $3$ $93.459$ 3.3.4364.1 None 792.4.a.k \(0\) \(0\) \(8\) \(-10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{5}+(-3+\beta _{2})q^{7}+11q^{11}+\cdots\)
1584.4.a.bq 1584.a 1.a $4$ $93.459$ 4.4.8611212.1 None 792.4.a.p \(0\) \(0\) \(-8\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{5}+(-2-\beta _{2})q^{7}+11q^{11}+\cdots\)
1584.4.a.br 1584.a 1.a $4$ $93.459$ 4.4.8611212.1 None 792.4.a.p \(0\) \(0\) \(8\) \(-8\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{5}+(-2-\beta _{2})q^{7}-11q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1584))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1584)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 2}\)