Properties

Label 570.4.i
Level $570$
Weight $4$
Character orbit 570.i
Rep. character $\chi_{570}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $15$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 570.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(570, [\chi])\).

Total New Old
Modular forms 736 80 656
Cusp forms 704 80 624
Eisenstein series 32 0 32

Trace form

\( 80q - 12q^{3} - 160q^{4} - 56q^{7} - 360q^{9} + O(q^{10}) \) \( 80q - 12q^{3} - 160q^{4} - 56q^{7} - 360q^{9} + 40q^{10} + 56q^{11} + 96q^{12} + 156q^{13} + 168q^{14} - 640q^{16} - 136q^{17} + 164q^{19} + 84q^{21} + 288q^{22} + 320q^{23} - 1000q^{25} + 768q^{26} + 216q^{27} + 112q^{28} - 40q^{29} - 88q^{31} + 264q^{33} - 240q^{34} + 260q^{35} - 1440q^{36} - 1688q^{37} + 192q^{38} + 168q^{39} + 160q^{40} - 348q^{41} - 48q^{42} + 28q^{43} - 112q^{44} - 624q^{46} + 1016q^{47} - 192q^{48} + 6432q^{49} + 624q^{52} - 568q^{53} - 1344q^{56} - 420q^{57} - 2912q^{58} - 216q^{59} + 404q^{61} + 464q^{62} + 252q^{63} + 5120q^{64} + 560q^{65} + 240q^{66} + 1236q^{67} + 1088q^{68} - 1200q^{69} + 560q^{70} + 1632q^{71} - 1500q^{73} + 264q^{74} + 600q^{75} + 560q^{76} - 10528q^{77} + 624q^{78} + 3452q^{79} - 3240q^{81} - 2496q^{82} + 2176q^{83} - 672q^{84} + 2064q^{86} + 816q^{87} - 2304q^{88} + 3204q^{89} + 360q^{90} - 2532q^{91} + 1280q^{92} - 324q^{93} - 4448q^{94} + 1080q^{95} + 3480q^{97} - 1504q^{98} - 252q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(570, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
570.4.i.a \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(-2\) \(-3\) \(-5\) \(26\) \(q+(-2+2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots\)
570.4.i.b \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(-2\) \(3\) \(-5\) \(-68\) \(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.c \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(-2\) \(3\) \(-5\) \(-28\) \(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.d \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(-2\) \(3\) \(-5\) \(2\) \(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.e \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(-2\) \(3\) \(-5\) \(58\) \(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.f \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(2\) \(3\) \(5\) \(-38\) \(q+(2-2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.g \(2\) \(33.631\) \(\Q(\sqrt{-3}) \) None \(2\) \(3\) \(5\) \(40\) \(q+(2-2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
570.4.i.h \(4\) \(33.631\) \(\Q(\sqrt{-3}, \sqrt{385})\) None \(-4\) \(6\) \(10\) \(36\) \(q+(-2+2\beta _{2})q^{2}+(3-3\beta _{2})q^{3}-4\beta _{2}q^{4}+\cdots\)
570.4.i.i \(4\) \(33.631\) \(\Q(\sqrt{-3}, \sqrt{481})\) None \(4\) \(6\) \(10\) \(6\) \(q+2\beta _{2}q^{2}+3\beta _{2}q^{3}+(-4+4\beta _{2})q^{4}+\cdots\)
570.4.i.j \(6\) \(33.631\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-6\) \(9\) \(15\) \(-46\) \(q+(-2+2\beta _{4})q^{2}+(3-3\beta _{4})q^{3}-4\beta _{4}q^{4}+\cdots\)
570.4.i.k \(8\) \(33.631\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(-12\) \(-20\) \(-100\) \(q+(-2-2\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+\cdots\)
570.4.i.l \(10\) \(33.631\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(-15\) \(25\) \(18\) \(q+2\beta _{2}q^{2}-3\beta _{2}q^{3}+(-4+4\beta _{2})q^{4}+\cdots\)
570.4.i.m \(10\) \(33.631\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(15\) \(-25\) \(38\) \(q+(2-2\beta _{3})q^{2}+(3-3\beta _{3})q^{3}-4\beta _{3}q^{4}+\cdots\)
570.4.i.n \(12\) \(33.631\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(-18\) \(30\) \(8\) \(q-2\beta _{3}q^{2}-3\beta _{3}q^{3}+(-4+4\beta _{3})q^{4}+\cdots\)
570.4.i.o \(12\) \(33.631\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(-18\) \(-30\) \(-8\) \(q-2\beta _{4}q^{2}+3\beta _{4}q^{3}+(-4-4\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(570, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(570, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(95, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(190, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(285, [\chi])\)\(^{\oplus 2}\)