Properties

Label 1344.4.a
Level $1344$
Weight $4$
Character orbit 1344.a
Rep. character $\chi_{1344}(1,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $48$
Sturm bound $1024$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 48 \)
Sturm bound: \(1024\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1344))\).

Total New Old
Modular forms 792 72 720
Cusp forms 744 72 672
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.

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(10\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(11\)
\(-\)\(+\)\(+\)\(-\)\(8\)
\(-\)\(+\)\(-\)\(+\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(40\)
Minus space\(-\)\(32\)

Trace form

\( 72 q + 648 q^{9} + O(q^{10}) \) \( 72 q + 648 q^{9} - 208 q^{17} + 1976 q^{25} - 400 q^{29} - 16 q^{37} + 944 q^{41} + 3528 q^{49} + 752 q^{53} - 1824 q^{61} + 3072 q^{65} - 1056 q^{69} + 592 q^{73} + 1904 q^{77} + 5832 q^{81} + 480 q^{85} - 176 q^{89} + 3632 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\) 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 7
1344.4.a.a 1344.a 1.a $1$ $79.299$ \(\Q\) None 42.4.a.a \(0\) \(-3\) \(-18\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-18q^{5}-7q^{7}+9q^{9}-72q^{11}+\cdots\)
1344.4.a.b 1344.a 1.a $1$ $79.299$ \(\Q\) None 84.4.a.b \(0\) \(-3\) \(-14\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-14q^{5}-7q^{7}+9q^{9}-4q^{11}+\cdots\)
1344.4.a.c 1344.a 1.a $1$ $79.299$ \(\Q\) None 672.4.a.b \(0\) \(-3\) \(-6\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-6q^{5}-7q^{7}+9q^{9}-4q^{11}+\cdots\)
1344.4.a.d 1344.a 1.a $1$ $79.299$ \(\Q\) None 84.4.a.a \(0\) \(-3\) \(-6\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-6q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1344.4.a.e 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.c \(0\) \(-3\) \(-4\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-4q^{5}+7q^{7}+9q^{9}-26q^{11}+\cdots\)
1344.4.a.f 1344.a 1.a $1$ $79.299$ \(\Q\) None 42.4.a.b \(0\) \(-3\) \(-2\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{5}-7q^{7}+9q^{9}+8q^{11}+\cdots\)
1344.4.a.g 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.f \(0\) \(-3\) \(2\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{5}-7q^{7}+9q^{9}-52q^{11}+\cdots\)
1344.4.a.h 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.b \(0\) \(-3\) \(2\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{5}-7q^{7}+9q^{9}+12q^{11}+\cdots\)
1344.4.a.i 1344.a 1.a $1$ $79.299$ \(\Q\) None 21.4.a.b \(0\) \(-3\) \(4\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{5}+7q^{7}+9q^{9}+62q^{11}+\cdots\)
1344.4.a.j 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.a \(0\) \(-3\) \(10\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+10q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
1344.4.a.k 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.e \(0\) \(-3\) \(10\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+10q^{5}-7q^{7}+9q^{9}+52q^{11}+\cdots\)
1344.4.a.l 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.d \(0\) \(-3\) \(16\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+2^{4}q^{5}+7q^{7}+9q^{9}+18q^{11}+\cdots\)
1344.4.a.m 1344.a 1.a $1$ $79.299$ \(\Q\) None 672.4.a.a \(0\) \(-3\) \(18\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+18q^{5}-7q^{7}+9q^{9}-44q^{11}+\cdots\)
1344.4.a.n 1344.a 1.a $1$ $79.299$ \(\Q\) None 21.4.a.a \(0\) \(-3\) \(18\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+18q^{5}-7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
1344.4.a.o 1344.a 1.a $1$ $79.299$ \(\Q\) None 42.4.a.a \(0\) \(3\) \(-18\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-18q^{5}+7q^{7}+9q^{9}+72q^{11}+\cdots\)
1344.4.a.p 1344.a 1.a $1$ $79.299$ \(\Q\) None 84.4.a.b \(0\) \(3\) \(-14\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-14q^{5}+7q^{7}+9q^{9}+4q^{11}+\cdots\)
1344.4.a.q 1344.a 1.a $1$ $79.299$ \(\Q\) None 84.4.a.a \(0\) \(3\) \(-6\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-6q^{5}+7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
1344.4.a.r 1344.a 1.a $1$ $79.299$ \(\Q\) None 672.4.a.b \(0\) \(3\) \(-6\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-6q^{5}+7q^{7}+9q^{9}+4q^{11}+\cdots\)
1344.4.a.s 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.c \(0\) \(3\) \(-4\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{5}-7q^{7}+9q^{9}+26q^{11}+\cdots\)
1344.4.a.t 1344.a 1.a $1$ $79.299$ \(\Q\) None 42.4.a.b \(0\) \(3\) \(-2\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{5}+7q^{7}+9q^{9}-8q^{11}+\cdots\)
1344.4.a.u 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.b \(0\) \(3\) \(2\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}+7q^{7}+9q^{9}-12q^{11}+\cdots\)
1344.4.a.v 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.f \(0\) \(3\) \(2\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}+7q^{7}+9q^{9}+52q^{11}+\cdots\)
1344.4.a.w 1344.a 1.a $1$ $79.299$ \(\Q\) None 21.4.a.b \(0\) \(3\) \(4\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{5}-7q^{7}+9q^{9}-62q^{11}+\cdots\)
1344.4.a.x 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.e \(0\) \(3\) \(10\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+10q^{5}+7q^{7}+9q^{9}-52q^{11}+\cdots\)
1344.4.a.y 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.a \(0\) \(3\) \(10\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+10q^{5}+7q^{7}+9q^{9}+12q^{11}+\cdots\)
1344.4.a.z 1344.a 1.a $1$ $79.299$ \(\Q\) None 168.4.a.d \(0\) \(3\) \(16\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2^{4}q^{5}-7q^{7}+9q^{9}-18q^{11}+\cdots\)
1344.4.a.ba 1344.a 1.a $1$ $79.299$ \(\Q\) None 21.4.a.a \(0\) \(3\) \(18\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+18q^{5}+7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1344.4.a.bb 1344.a 1.a $1$ $79.299$ \(\Q\) None 672.4.a.a \(0\) \(3\) \(18\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+18q^{5}+7q^{7}+9q^{9}+44q^{11}+\cdots\)
1344.4.a.bc 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{43}) \) None 672.4.a.i \(0\) \(-6\) \(-16\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-8+\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bd 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{177}) \) None 168.4.a.h \(0\) \(-6\) \(-14\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-7-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.be 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{137}) \) None 672.4.a.h \(0\) \(-6\) \(-10\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-5-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bf 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{11}) \) None 672.4.a.g \(0\) \(-6\) \(-8\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-4+\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bg 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{57}) \) None 21.4.a.c \(0\) \(-6\) \(-6\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-3-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bh 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{37}) \) None 672.4.a.f \(0\) \(-6\) \(4\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(2+\beta )q^{5}+7q^{7}+9q^{9}+(24+\cdots)q^{11}+\cdots\)
1344.4.a.bi 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{337}) \) None 168.4.a.g \(0\) \(-6\) \(6\) \(14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(3+\beta )q^{5}+7q^{7}+9q^{9}+(13+\cdots)q^{11}+\cdots\)
1344.4.a.bj 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{17}) \) None 672.4.a.e \(0\) \(-6\) \(10\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(5-\beta )q^{5}+7q^{7}+9q^{9}+(11+\cdots)q^{11}+\cdots\)
1344.4.a.bk 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{43}) \) None 672.4.a.i \(0\) \(6\) \(-16\) \(14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-8+\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bl 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{177}) \) None 168.4.a.h \(0\) \(6\) \(-14\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-7-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bm 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{137}) \) None 672.4.a.h \(0\) \(6\) \(-10\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-5-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bn 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{11}) \) None 672.4.a.g \(0\) \(6\) \(-8\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-4+\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bo 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{57}) \) None 21.4.a.c \(0\) \(6\) \(-6\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-3-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bp 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{37}) \) None 672.4.a.f \(0\) \(6\) \(4\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(2+\beta )q^{5}-7q^{7}+9q^{9}+(-24+\cdots)q^{11}+\cdots\)
1344.4.a.bq 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{337}) \) None 168.4.a.g \(0\) \(6\) \(6\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(3+\beta )q^{5}-7q^{7}+9q^{9}+(-13+\cdots)q^{11}+\cdots\)
1344.4.a.br 1344.a 1.a $2$ $79.299$ \(\Q(\sqrt{17}) \) None 672.4.a.e \(0\) \(6\) \(10\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(5-\beta )q^{5}-7q^{7}+9q^{9}+(-11+\cdots)q^{11}+\cdots\)
1344.4.a.bs 1344.a 1.a $3$ $79.299$ 3.3.37341.1 None 672.4.a.p \(0\) \(-9\) \(-6\) \(21\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2+\beta _{1})q^{5}+7q^{7}+9q^{9}+\cdots\)
1344.4.a.bt 1344.a 1.a $3$ $79.299$ 3.3.22700.1 None 672.4.a.o \(0\) \(-9\) \(10\) \(-21\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(3+\beta _{2})q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bu 1344.a 1.a $3$ $79.299$ 3.3.37341.1 None 672.4.a.p \(0\) \(9\) \(-6\) \(-21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-2+\beta _{1})q^{5}-7q^{7}+9q^{9}+\cdots\)
1344.4.a.bv 1344.a 1.a $3$ $79.299$ 3.3.22700.1 None 672.4.a.o \(0\) \(9\) \(10\) \(21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(3+\beta _{2})q^{5}+7q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1344))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1344)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(448))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(672))\)\(^{\oplus 2}\)