Properties

Label 648.4.i
Level $648$
Weight $4$
Character orbit 648.i
Rep. character $\chi_{648}(217,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $72$
Newform subspaces $22$
Sturm bound $432$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 22 \)
Sturm bound: \(432\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(648, [\chi])\).

Total New Old
Modular forms 696 72 624
Cusp forms 600 72 528
Eisenstein series 96 0 96

Trace form

\( 72q + O(q^{10}) \) \( 72q + 180q^{19} - 900q^{25} - 450q^{31} + 972q^{43} - 2160q^{49} - 3060q^{55} + 36q^{61} + 774q^{67} - 2484q^{73} - 630q^{79} + 414q^{85} - 9900q^{91} + 144q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(648, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
648.4.i.a \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-16\) \(12\) \(q-2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.b \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-14\) \(24\) \(q-14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots\)
648.4.i.c \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-5\) \(-36\) \(q-5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.d \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(-3\) \(q-4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots\)
648.4.i.e \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(-24\) \(q-2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(-44+\cdots)q^{11}+\cdots\)
648.4.i.f \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(9\) \(q-\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(17-17\zeta_{6})q^{11}+\cdots\)
648.4.i.g \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(9\) \(q+\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(-17+17\zeta_{6})q^{11}+\cdots\)
648.4.i.h \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(-24\) \(q+2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(44+\cdots)q^{11}+\cdots\)
648.4.i.i \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(-3\) \(q+4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots\)
648.4.i.j \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(5\) \(-36\) \(q+5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.k \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(14\) \(24\) \(q+14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots\)
648.4.i.l \(2\) \(38.233\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(16\) \(12\) \(q+2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(2^{6}-2^{6}\zeta_{6})q^{11}+\cdots\)
648.4.i.m \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(-8\) \(-24\) \(q+(-4-4\beta _{1}-\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.n \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-67})\) None \(0\) \(0\) \(-8\) \(30\) \(q+(-4-4\beta _{1}+\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.o \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(0\) \(-4\) \(6\) \(q+(-2\beta _{1}-\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.p \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-43})\) None \(0\) \(0\) \(-4\) \(6\) \(q+(-2-2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.q \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-43})\) None \(0\) \(0\) \(4\) \(6\) \(q+(2+2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.r \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(0\) \(0\) \(4\) \(6\) \(q+(2\beta _{1}+\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+2\beta _{3})q^{7}+\cdots\)
648.4.i.s \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(8\) \(-24\) \(q+(4+4\beta _{1}+\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
648.4.i.t \(4\) \(38.233\) \(\Q(\sqrt{-3}, \sqrt{-67})\) None \(0\) \(0\) \(8\) \(30\) \(q+(4+4\beta _{1}-\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.u \(8\) \(38.233\) 8.0.897122304.10 None \(0\) \(0\) \(-8\) \(0\) \(q+(2\beta _{1}+\beta _{7})q^{5}+(\beta _{2}+\beta _{3}-\beta _{5}-\beta _{7})q^{7}+\cdots\)
648.4.i.v \(8\) \(38.233\) 8.0.897122304.10 None \(0\) \(0\) \(8\) \(0\) \(q+(2\beta _{1}+\beta _{7})q^{5}+(-\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(648, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)