Properties

Label 1296.4.a
Level $1296$
Weight $4$
Character orbit 1296.a
Rep. character $\chi_{1296}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $30$
Sturm bound $864$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(864\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1296))\).

Total New Old
Modular forms 684 74 610
Cusp forms 612 70 542
Eisenstein series 72 4 68

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(19\)
\(+\)\(-\)\(-\)\(17\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(18\)
Plus space\(+\)\(37\)
Minus space\(-\)\(33\)

Trace form

\( 70 q - 2 q^{7} + O(q^{10}) \) \( 70 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 1552 q^{25} - 182 q^{31} - 400 q^{37} + 250 q^{43} + 2844 q^{49} - 246 q^{55} + 470 q^{61} - 1226 q^{67} + 824 q^{73} - 362 q^{79} + 1584 q^{85} + 1298 q^{91} - 34 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\) 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
1296.4.a.a 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(-21\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q-21q^{5}-8q^{7}+6^{2}q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.b 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(-9\) \(31\) $-$ $+$ $\mathrm{SU}(2)$ \(q-9q^{5}+31q^{7}+15q^{11}-37q^{13}+\cdots\)
1296.4.a.c 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(-5\) \(-36\) $+$ $+$ $\mathrm{SU}(2)$ \(q-5q^{5}-6^{2}q^{7}-2^{6}q^{11}-65q^{13}+\cdots\)
1296.4.a.d 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(-3\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}+4q^{7}-24q^{11}-5^{2}q^{13}+\cdots\)
1296.4.a.e 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(3\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+4q^{7}+24q^{11}-5^{2}q^{13}+\cdots\)
1296.4.a.f 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(5\) \(-36\) $+$ $+$ $\mathrm{SU}(2)$ \(q+5q^{5}-6^{2}q^{7}+2^{6}q^{11}-65q^{13}+\cdots\)
1296.4.a.g 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(9\) \(31\) $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{5}+31q^{7}-15q^{11}-37q^{13}+\cdots\)
1296.4.a.h 1296.a 1.a $1$ $76.466$ \(\Q\) None \(0\) \(0\) \(21\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+21q^{5}-8q^{7}-6^{2}q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.i 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-15\) \(-7\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{5}+(-5+3\beta )q^{7}+(29+\cdots)q^{11}+\cdots\)
1296.4.a.j 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-12\) \(-16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-6+\beta )q^{5}+(-8+2\beta )q^{7}+(18+\cdots)q^{11}+\cdots\)
1296.4.a.k 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(-12\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-6-\beta )q^{5}+(-5-3\beta )q^{7}+(21+\cdots)q^{11}+\cdots\)
1296.4.a.l 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{105}) \) None \(0\) \(0\) \(-9\) \(-19\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-5-\beta )q^{5}+(-9+\beta )q^{7}+(11-2\beta )q^{11}+\cdots\)
1296.4.a.m 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(-8\) \(30\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{5}+(15-\beta )q^{7}+(-23+\cdots)q^{11}+\cdots\)
1296.4.a.n 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(-4\) \(6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{5}+(3+\beta )q^{7}+(5+3\beta )q^{11}+\cdots\)
1296.4.a.o 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+7\beta q^{5}+22q^{7}+34\beta q^{11}-7^{2}q^{13}+\cdots\)
1296.4.a.p 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{129}) \) None \(0\) \(0\) \(4\) \(6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{5}+(3+\beta )q^{7}+(-5-3\beta )q^{11}+\cdots\)
1296.4.a.q 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{201}) \) None \(0\) \(0\) \(8\) \(30\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(4+\beta )q^{5}+(15-\beta )q^{7}+(23+3\beta )q^{11}+\cdots\)
1296.4.a.r 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{105}) \) None \(0\) \(0\) \(9\) \(-19\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(5+\beta )q^{5}+(-9+\beta )q^{7}+(-11+2\beta )q^{11}+\cdots\)
1296.4.a.s 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(12\) \(-16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(6+\beta )q^{5}+(-8-2\beta )q^{7}+(-18+\cdots)q^{11}+\cdots\)
1296.4.a.t 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(12\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(6-\beta )q^{5}+(-5+3\beta )q^{7}+(-21+\cdots)q^{11}+\cdots\)
1296.4.a.u 1296.a 1.a $2$ $76.466$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(15\) \(-7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(8-\beta )q^{5}+(-2-3\beta )q^{7}+(-37+\cdots)q^{11}+\cdots\)
1296.4.a.v 1296.a 1.a $3$ $76.466$ 3.3.1509.1 None \(0\) \(0\) \(-6\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1296.4.a.w 1296.a 1.a $3$ $76.466$ 3.3.1509.1 None \(0\) \(0\) \(6\) \(-6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1296.4.a.x 1296.a 1.a $4$ $76.466$ 4.4.29952.1 None \(0\) \(0\) \(-8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2+\cdots)q^{11}+\cdots\)
1296.4.a.y 1296.a 1.a $4$ $76.466$ 4.4.72153.1 None \(0\) \(0\) \(-5\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots\)
1296.4.a.z 1296.a 1.a $4$ $76.466$ \(\Q(\sqrt{3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{5}+(4-\beta _{3})q^{7}+(\beta _{1}-\beta _{2})q^{11}+\cdots\)
1296.4.a.ba 1296.a 1.a $4$ $76.466$ 4.4.72153.1 None \(0\) \(0\) \(5\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{5}+(1+\beta _{3})q^{7}+(4-\beta _{1}+\cdots)q^{11}+\cdots\)
1296.4.a.bb 1296.a 1.a $4$ $76.466$ 4.4.29952.1 None \(0\) \(0\) \(8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
1296.4.a.bc 1296.a 1.a $5$ $76.466$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-5\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(-5+\cdots)q^{11}+\cdots\)
1296.4.a.bd 1296.a 1.a $5$ $76.466$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(5\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+(5-2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1296))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1296)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 2}\)