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Properties

Label 320.4.n
Level $320$
Weight $4$
Character orbit 320.n
Rep. character $\chi_{320}(63,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $68$
Newform subspaces $11$
Sturm bound $192$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.n (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 11 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(320, [\chi])\).

Total New Old
Modular forms 312 76 236
Cusp forms 264 68 196
Eisenstein series 48 8 40

Trace form

\( 68 q + 4 q^{5} + O(q^{10}) \) \( 68 q + 4 q^{5} + 4 q^{13} + 100 q^{17} + 8 q^{21} - 92 q^{25} + 104 q^{33} + 4 q^{37} - 8 q^{41} - 496 q^{45} - 812 q^{53} - 112 q^{57} + 1832 q^{61} + 1044 q^{65} + 292 q^{73} + 2136 q^{77} - 5388 q^{81} + 2356 q^{85} - 5048 q^{93} + 4788 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(320, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
320.4.n.a 320.n 20.e $2$ $18.881$ \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(-20\) \(36\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2i)q^{3}+(-10-5i)q^{5}+(18+\cdots)q^{7}+\cdots\)
320.4.n.b 320.n 20.e $2$ $18.881$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2+11i)q^{5}-3^{3}iq^{9}+(-55+\cdots)q^{13}+\cdots\)
320.4.n.c 320.n 20.e $2$ $18.881$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(2+11i)q^{5}-3^{3}iq^{9}+(37-37i)q^{13}+\cdots\)
320.4.n.d 320.n 20.e $2$ $18.881$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(-20\) \(-36\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+(-10-5i)q^{5}+(-18+\cdots)q^{7}+\cdots\)
320.4.n.e 320.n 20.e $4$ $18.881$ \(\Q(i, \sqrt{35})\) None \(0\) \(0\) \(20\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{3}+(5-10\beta _{1})q^{5}-\beta _{3}q^{7}+43\beta _{1}q^{9}+\cdots\)
320.4.n.f 320.n 20.e $8$ $18.881$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(-6\) \(6\) \(70\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1}+\beta _{4})q^{3}+(1+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
320.4.n.g 320.n 20.e $8$ $18.881$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-2\) \(14\) \(10\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}+(2+\beta _{6})q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
320.4.n.h 320.n 20.e $8$ $18.881$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(2\) \(14\) \(-10\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{3}+(2-\beta _{4})q^{5}+(-1-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
320.4.n.i 320.n 20.e $8$ $18.881$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(6\) \(6\) \(-70\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{1}+\beta _{5})q^{3}+(1-\beta _{1}-\beta _{2}-\beta _{5}+\cdots)q^{5}+\cdots\)
320.4.n.j 320.n 20.e $12$ $18.881$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+(-1-\beta _{1}-\beta _{4})q^{5}+(2\beta _{2}+\cdots)q^{7}+\cdots\)
320.4.n.k 320.n 20.e $12$ $18.881$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{7}q^{3}+(-1+2\beta _{1}+2\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(320, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)