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Properties

Label 363.2.e
Level $363$
Weight $2$
Character orbit 363.e
Rep. character $\chi_{363}(124,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $72$
Newform subspaces $14$
Sturm bound $88$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 14 \)
Sturm bound: \(88\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(363, [\chi])\).

Total New Old
Modular forms 224 72 152
Cusp forms 128 72 56
Eisenstein series 96 0 96

Trace form

\( 72 q + 4 q^{2} - 14 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} - 8 q^{8} - 18 q^{9} + O(q^{10}) \) \( 72 q + 4 q^{2} - 14 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} - 8 q^{8} - 18 q^{9} - 4 q^{10} + 8 q^{12} + 2 q^{13} + 6 q^{14} - 2 q^{15} - 22 q^{16} - 14 q^{17} - 6 q^{18} + 20 q^{19} + 2 q^{20} - 16 q^{21} - 4 q^{23} - 12 q^{24} - 20 q^{25} + 2 q^{26} + 12 q^{28} + 4 q^{29} + 8 q^{31} + 48 q^{32} - 56 q^{34} - 14 q^{36} - 2 q^{37} + 14 q^{38} + 16 q^{39} - 4 q^{40} - 20 q^{41} + 14 q^{42} - 16 q^{43} + 4 q^{45} - 10 q^{46} - 6 q^{47} - 8 q^{48} - 16 q^{49} - 18 q^{50} - 10 q^{52} - 6 q^{53} - 8 q^{54} - 108 q^{56} + 44 q^{58} + 46 q^{59} + 12 q^{60} + 18 q^{61} + 28 q^{62} + 2 q^{63} + 2 q^{64} + 28 q^{65} - 44 q^{67} + 28 q^{68} + 10 q^{69} + 34 q^{70} + 32 q^{71} + 2 q^{72} - 20 q^{73} + 30 q^{74} - 40 q^{76} - 64 q^{78} + 6 q^{79} + 14 q^{80} - 18 q^{81} + 46 q^{82} - 34 q^{83} + 2 q^{85} + 24 q^{86} - 24 q^{87} - 60 q^{89} - 4 q^{90} - 18 q^{91} + 22 q^{92} - 56 q^{93} - 18 q^{94} - 30 q^{95} + 8 q^{96} + 10 q^{97} - 40 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(363, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
363.2.e.a 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(-5\) \(-1\) \(-2\) \(-10\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.b 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(-4\) \(1\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.c 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(4\) \(3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
363.2.e.d 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(-2\) \(1\) \(-4\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+2\zeta_{10}-2\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.e 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.f 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(1\) \(1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.g 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(1\) \(1\) \(2\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.h 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(2\) \(-1\) \(4\) \(-3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
363.2.e.i 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(2\) \(1\) \(-4\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.j 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(3\) \(-1\) \(-1\) \(3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots\)
363.2.e.k 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(4\) \(1\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.l 363.e 11.c $4$ $2.899$ \(\Q(\zeta_{10})\) None \(5\) \(-1\) \(-2\) \(10\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.m 363.e 11.c $8$ $2.899$ 8.0.324000000.3 None \(0\) \(2\) \(6\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{7}q^{2}-\beta _{6}q^{3}+\beta _{4}q^{4}+(3+3\beta _{2}+\cdots)q^{5}+\cdots\)
363.2.e.n 363.e 11.c $16$ $2.899$ 16.0.\(\cdots\).3 None \(0\) \(-4\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{12}q^{2}+\beta _{4}q^{3}+(\beta _{3}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(363, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(363, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(121, [\chi])\)\(^{\oplus 2}\)