Properties

Label 684.2.i
Level 684684
Weight 22
Character orbit 684.i
Rep. character χ684(229,)\chi_{684}(229,\cdot)
Character field Q(ζ3)\Q(\zeta_{3})
Dimension 3636
Newform subspaces 44
Sturm bound 240240
Trace bound 55

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Defining parameters

Level: N N == 684=223219 684 = 2^{2} \cdot 3^{2} \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 684.i (of order 33 and degree 22)
Character conductor: cond(χ)\operatorname{cond}(\chi) == 9 9
Character field: Q(ζ3)\Q(\zeta_{3})
Newform subspaces: 4 4
Sturm bound: 240240
Trace bound: 55
Distinguishing TpT_p: 55

Dimensions

The following table gives the dimensions of various subspaces of M2(684,[χ])M_{2}(684, [\chi]).

Total New Old
Modular forms 252 36 216
Cusp forms 228 36 192
Eisenstein series 24 0 24

Trace form

36q2q3+6q5+10q9+8q11+4q1516q1718q21+2q2312q252q2712q29+6q31+14q3312q3512q37+6q3916q4116q45+68q99+O(q100) 36 q - 2 q^{3} + 6 q^{5} + 10 q^{9} + 8 q^{11} + 4 q^{15} - 16 q^{17} - 18 q^{21} + 2 q^{23} - 12 q^{25} - 2 q^{27} - 12 q^{29} + 6 q^{31} + 14 q^{33} - 12 q^{35} - 12 q^{37} + 6 q^{39} - 16 q^{41} - 16 q^{45}+ \cdots - 68 q^{99}+O(q^{100}) Copy content Toggle raw display

Decomposition of S2new(684,[χ])S_{2}^{\mathrm{new}}(684, [\chi]) into newform subspaces

Label Char Prim Dim AA Field CM Minimal twist Traces Sato-Tate qq-expansion
a2a_{2} a3a_{3} a5a_{5} a7a_{7}
684.2.i.a 684.i 9.c 22 5.4625.462 Q(3)\Q(\sqrt{-3}) None 684.2.i.a 00 00 3-3 55 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(12ζ6)q33ζ6q5+(55ζ6)q7+q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(5-5\zeta_{6})q^{7}+\cdots
684.2.i.b 684.i 9.c 22 5.4625.462 Q(3)\Q(\sqrt{-3}) None 684.2.i.b 00 00 11 3-3 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(12ζ6)q3+ζ6q5+(3+3ζ6)q7+q+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots
684.2.i.c 684.i 9.c 1616 5.4625.462 Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots) None 684.2.i.c 00 1-1 22 33 SU(2)[C3]\mathrm{SU}(2)[C_{3}] qβ1q3β13q5+(β3+β8+β11+)q7+q-\beta _{1}q^{3}-\beta _{13}q^{5}+(-\beta _{3}+\beta _{8}+\beta _{11}+\cdots)q^{7}+\cdots
684.2.i.d 684.i 9.c 1616 5.4625.462 Q[x]/(x16+)\mathbb{Q}[x]/(x^{16} + \cdots) None 684.2.i.d 00 1-1 66 5-5 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(β2+β13)q3+(β7+β8)q5+(1+)q7+q+(\beta _{2}+\beta _{13})q^{3}+(\beta _{7}+\beta _{8})q^{5}+(-1+\cdots)q^{7}+\cdots

Decomposition of S2old(684,[χ])S_{2}^{\mathrm{old}}(684, [\chi]) into lower level spaces

S2old(684,[χ]) S_{2}^{\mathrm{old}}(684, [\chi]) \simeq S2new(18,[χ])S_{2}^{\mathrm{new}}(18, [\chi])4^{\oplus 4}\oplusS2new(36,[χ])S_{2}^{\mathrm{new}}(36, [\chi])2^{\oplus 2}\oplusS2new(171,[χ])S_{2}^{\mathrm{new}}(171, [\chi])3^{\oplus 3}\oplusS2new(342,[χ])S_{2}^{\mathrm{new}}(342, [\chi])2^{\oplus 2}