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Properties

Label 312.4.a
Level $312$
Weight $4$
Character orbit 312.a
Rep. character $\chi_{312}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $8$
Sturm bound $224$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(224\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(312))\).

Total New Old
Modular forms 176 18 158
Cusp forms 160 18 142
Eisenstein series 16 0 16

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

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(10\)
Minus space\(-\)\(8\)

Trace form

\( 18 q + 12 q^{7} + 162 q^{9} + O(q^{10}) \) \( 18 q + 12 q^{7} + 162 q^{9} + 26 q^{13} - 84 q^{15} + 4 q^{17} - 228 q^{19} + 84 q^{21} - 168 q^{23} + 270 q^{25} - 140 q^{29} - 276 q^{31} + 168 q^{33} + 1192 q^{35} + 100 q^{37} + 200 q^{41} + 112 q^{43} + 688 q^{47} + 994 q^{49} - 676 q^{53} + 2080 q^{55} - 468 q^{57} - 216 q^{59} + 60 q^{61} + 108 q^{63} - 556 q^{67} - 600 q^{69} - 712 q^{71} + 1564 q^{73} - 1216 q^{77} + 904 q^{79} + 1458 q^{81} + 1752 q^{83} - 3808 q^{85} - 744 q^{87} + 1568 q^{89} - 1092 q^{91} - 1404 q^{93} - 3592 q^{95} - 964 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\) 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 13
312.4.a.a 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{3}) \) None 312.4.a.a \(0\) \(-6\) \(-4\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2+\beta )q^{5}+(2-3\beta )q^{7}+\cdots\)
312.4.a.b 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{55}) \) None 312.4.a.b \(0\) \(-6\) \(-4\) \(20\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2+\beta )q^{5}+(10+\beta )q^{7}+\cdots\)
312.4.a.c 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{113}) \) None 312.4.a.c \(0\) \(-6\) \(6\) \(-10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(3+\beta )q^{5}+(-5-\beta )q^{7}+9q^{9}+\cdots\)
312.4.a.d 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{17}) \) None 312.4.a.d \(0\) \(6\) \(-18\) \(-10\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-9-\beta )q^{5}+(-5+5\beta )q^{7}+\cdots\)
312.4.a.e 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{7}) \) None 312.4.a.e \(0\) \(6\) \(-4\) \(-20\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-2+\beta )q^{5}+(-10-3\beta )q^{7}+\cdots\)
312.4.a.f 312.a 1.a $2$ $18.409$ \(\Q(\sqrt{43}) \) None 312.4.a.f \(0\) \(6\) \(12\) \(44\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(6+\beta )q^{5}+(22+\beta )q^{7}+9q^{9}+\cdots\)
312.4.a.g 312.a 1.a $3$ $18.409$ 3.3.36248.1 None 312.4.a.g \(0\) \(-9\) \(16\) \(-22\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(5+\beta _{2})q^{5}+(-8+\beta _{1}+\beta _{2})q^{7}+\cdots\)
312.4.a.h 312.a 1.a $3$ $18.409$ 3.3.13916.1 None 312.4.a.h \(0\) \(9\) \(-4\) \(6\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-1+\beta _{2})q^{5}+(2+\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(312))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(312)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)