Properties

Label 5312.2.a
Level $5312$
Weight $2$
Character orbit 5312.a
Rep. character $\chi_{5312}(1,\cdot)$
Character field $\Q$
Dimension $164$
Newform subspaces $48$
Sturm bound $1344$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 5312 = 2^{6} \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5312.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 48 \)
Sturm bound: \(1344\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5312))\).

Total New Old
Modular forms 684 164 520
Cusp forms 661 164 497
Eisenstein series 23 0 23

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

\(2\)\(83\)FrickeDim.
\(+\)\(+\)\(+\)\(37\)
\(+\)\(-\)\(-\)\(46\)
\(-\)\(+\)\(-\)\(45\)
\(-\)\(-\)\(+\)\(36\)
Plus space\(+\)\(73\)
Minus space\(-\)\(91\)

Trace form

\( 164 q + 164 q^{9} + O(q^{10}) \) \( 164 q + 164 q^{9} + 16 q^{13} - 8 q^{17} + 16 q^{21} + 156 q^{25} - 16 q^{29} - 8 q^{41} + 48 q^{45} + 164 q^{49} + 32 q^{53} + 16 q^{61} + 48 q^{69} + 8 q^{73} - 16 q^{77} + 164 q^{81} + 8 q^{89} + 16 q^{93} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\) 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 83
5312.2.a.a 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-3\) \(2\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{5}+3q^{7}+6q^{9}-3q^{11}+\cdots\)
5312.2.a.b 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-3\) \(4\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{5}+5q^{7}+6q^{9}-3q^{11}+\cdots\)
5312.2.a.c 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(-2\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}-3q^{7}-2q^{9}-q^{11}+\cdots\)
5312.2.a.d 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}-2q^{9}-q^{11}-6q^{13}+\cdots\)
5312.2.a.e 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}-2q^{9}-q^{11}+4q^{13}+\cdots\)
5312.2.a.f 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}-2q^{9}-5q^{11}+\cdots\)
5312.2.a.g 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}-2q^{9}+3q^{11}+\cdots\)
5312.2.a.h 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(-1\) \(2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+3q^{7}-2q^{9}+3q^{11}+\cdots\)
5312.2.a.i 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(-2\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+3q^{7}-2q^{9}+q^{11}+\cdots\)
5312.2.a.j 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}+q^{11}+4q^{13}+\cdots\)
5312.2.a.k 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(0\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{7}-2q^{9}+q^{11}-6q^{13}+\cdots\)
5312.2.a.l 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(2\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-3q^{7}-2q^{9}-3q^{11}+\cdots\)
5312.2.a.m 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(2\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}-2q^{9}-3q^{11}+\cdots\)
5312.2.a.n 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(1\) \(2\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}-2q^{9}+5q^{11}+\cdots\)
5312.2.a.o 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(3\) \(2\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}-3q^{7}+6q^{9}+3q^{11}+\cdots\)
5312.2.a.p 5312.a 1.a $1$ $42.417$ \(\Q\) None \(0\) \(3\) \(4\) \(-5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{5}-5q^{7}+6q^{9}+3q^{11}+\cdots\)
5312.2.a.q 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-3\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{7}+\cdots\)
5312.2.a.r 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-1\) \(1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2\beta q^{3}-\beta q^{5}+(1-\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
5312.2.a.s 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+\beta q^{5}+(3+\beta )q^{7}-2\beta q^{9}+\cdots\)
5312.2.a.t 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1+\beta )q^{5}+\beta q^{7}+4q^{9}+(-2+\cdots)q^{11}+\cdots\)
5312.2.a.u 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1-\beta )q^{5}+\beta q^{7}+4q^{9}+(2+\cdots)q^{11}+\cdots\)
5312.2.a.v 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-3\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+(-2+\beta )q^{5}+(-1-\beta )q^{7}+\cdots\)
5312.2.a.w 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-1\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}-\beta q^{5}+(-1+\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
5312.2.a.x 5312.a 1.a $2$ $42.417$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-\beta q^{5}+(-3+\beta )q^{7}+2\beta q^{9}+\cdots\)
5312.2.a.y 5312.a 1.a $3$ $42.417$ \(\Q(\zeta_{14})^+\) None \(0\) \(-4\) \(-2\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+2\beta _{2}q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.z 5312.a 1.a $3$ $42.417$ 3.3.229.1 None \(0\) \(-1\) \(1\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}+\beta _{2})q^{3}-\beta _{2}q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.ba 5312.a 1.a $3$ $42.417$ 3.3.148.1 None \(0\) \(-1\) \(4\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1+\beta _{1}+2\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
5312.2.a.bb 5312.a 1.a $3$ $42.417$ 3.3.229.1 None \(0\) \(1\) \(1\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}-\beta _{2})q^{3}-\beta _{2}q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bc 5312.a 1.a $3$ $42.417$ 3.3.148.1 None \(0\) \(1\) \(4\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{1}+2\beta _{2})q^{5}+(2-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bd 5312.a 1.a $3$ $42.417$ \(\Q(\zeta_{14})^+\) None \(0\) \(4\) \(-2\) \(-8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+2\beta _{2}q^{5}+(-2-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.be 5312.a 1.a $4$ $42.417$ 4.4.15529.1 None \(0\) \(-3\) \(0\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
5312.2.a.bf 5312.a 1.a $4$ $42.417$ 4.4.2225.1 None \(0\) \(-2\) \(-1\) \(5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}-\beta _{1}q^{5}+(1+\beta _{1})q^{7}+\cdots\)
5312.2.a.bg 5312.a 1.a $4$ $42.417$ 4.4.2225.1 None \(0\) \(2\) \(-1\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}-\beta _{1}q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
5312.2.a.bh 5312.a 1.a $4$ $42.417$ 4.4.15529.1 None \(0\) \(3\) \(0\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3})q^{7}+(1+\cdots)q^{9}+\cdots\)
5312.2.a.bi 5312.a 1.a $5$ $42.417$ 5.5.265504.1 None \(0\) \(-3\) \(-2\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{3}-\beta _{3}q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bj 5312.a 1.a $5$ $42.417$ 5.5.356173.1 None \(0\) \(-2\) \(1\) \(7\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{2}+\beta _{4})q^{5}+(1-\beta _{4})q^{7}+\cdots\)
5312.2.a.bk 5312.a 1.a $5$ $42.417$ 5.5.356173.1 None \(0\) \(2\) \(1\) \(-7\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{2}+\beta _{4})q^{5}+(-1+\beta _{4})q^{7}+\cdots\)
5312.2.a.bl 5312.a 1.a $5$ $42.417$ 5.5.265504.1 None \(0\) \(3\) \(-2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{3})q^{3}-\beta _{3}q^{5}+(1+\beta _{1})q^{7}+\cdots\)
5312.2.a.bm 5312.a 1.a $6$ $42.417$ 6.6.905177.1 None \(0\) \(-3\) \(4\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{3}+(1-\beta _{2}-\beta _{3})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bn 5312.a 1.a $6$ $42.417$ 6.6.9059636.1 None \(0\) \(-1\) \(-2\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{3}+(-1-\beta _{2}-\beta _{3}-\beta _{5})q^{5}+\cdots\)
5312.2.a.bo 5312.a 1.a $6$ $42.417$ 6.6.9059636.1 None \(0\) \(1\) \(-2\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{3}+(-1-\beta _{2}-\beta _{3}-\beta _{5})q^{5}+\cdots\)
5312.2.a.bp 5312.a 1.a $6$ $42.417$ 6.6.905177.1 None \(0\) \(3\) \(4\) \(-5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{3}+(1-\beta _{2}-\beta _{3})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5312.2.a.bq 5312.a 1.a $8$ $42.417$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-3\) \(1\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
5312.2.a.br 5312.a 1.a $8$ $42.417$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-2\) \(-7\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-1-\beta _{7})q^{5}-\beta _{2}q^{7}+(2+\cdots)q^{9}+\cdots\)
5312.2.a.bs 5312.a 1.a $8$ $42.417$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(2\) \(-7\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1-\beta _{7})q^{5}+\beta _{2}q^{7}+(2+\cdots)q^{9}+\cdots\)
5312.2.a.bt 5312.a 1.a $8$ $42.417$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(3\) \(1\) \(10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(1+\beta _{1}-\beta _{3}+\beta _{6}+\cdots)q^{7}+\cdots\)
5312.2.a.bu 5312.a 1.a $11$ $42.417$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(-1\) \(-5\) \(-10\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-1-\beta _{10})q^{7}+\cdots\)
5312.2.a.bv 5312.a 1.a $11$ $42.417$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(1\) \(-5\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(1+\beta _{10})q^{7}+(2+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5312)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(83))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(166))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(332))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(664))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1328))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2656))\)\(^{\oplus 2}\)