Properties

Label 405.4.e
Level $405$
Weight $4$
Character orbit 405.e
Rep. character $\chi_{405}(136,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $24$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 24 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(405, [\chi])\).

Total New Old
Modular forms 348 96 252
Cusp forms 300 96 204
Eisenstein series 48 0 48

Trace form

\( 96 q - 192 q^{4} - 60 q^{7} + O(q^{10}) \) \( 96 q - 192 q^{4} - 60 q^{7} + 120 q^{13} - 768 q^{16} - 960 q^{19} + 72 q^{22} - 1200 q^{25} + 1920 q^{28} - 600 q^{31} - 1350 q^{34} - 1680 q^{37} - 450 q^{40} + 156 q^{43} - 1188 q^{46} - 756 q^{49} + 3756 q^{52} + 1188 q^{58} + 84 q^{61} + 1644 q^{64} - 1464 q^{67} - 360 q^{70} - 5064 q^{73} + 9750 q^{76} + 3036 q^{79} + 6984 q^{82} + 720 q^{85} + 864 q^{88} - 14088 q^{91} - 3924 q^{94} + 2532 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(405, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
405.4.e.a $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-5\) \(0\) \(5\) \(-9\) \(q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.b $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-5\) \(0\) \(5\) \(30\) \(q+(-5+5\zeta_{6})q^{2}-17\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.c $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(-5\) \(-6\) \(q+(-4+4\zeta_{6})q^{2}-8\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.d $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(5\) \(-20\) \(q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.e $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(5\) \(0\) \(q+(-2+2\zeta_{6})q^{2}+4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.f $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-5\) \(6\) \(q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.g $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-5\) \(24\) \(q+(-1+\zeta_{6})q^{2}+7\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.h $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(5\) \(6\) \(q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(6+\cdots)q^{7}+\cdots\)
405.4.e.i $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(5\) \(24\) \(q+(1-\zeta_{6})q^{2}+7\zeta_{6}q^{4}+5\zeta_{6}q^{5}+(24+\cdots)q^{7}+\cdots\)
405.4.e.j $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(0\) \(q+(2-2\zeta_{6})q^{2}+4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.k $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(-5\) \(-20\) \(q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}-5\zeta_{6}q^{5}+(-20+\cdots)q^{7}+\cdots\)
405.4.e.l $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(5\) \(-6\) \(q+(4-4\zeta_{6})q^{2}-8\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
405.4.e.m $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(5\) \(0\) \(-5\) \(-9\) \(q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.n $2$ $23.896$ \(\Q(\sqrt{-3}) \) None \(5\) \(0\) \(-5\) \(30\) \(q+(5-5\zeta_{6})q^{2}-17\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.4.e.o $4$ $23.896$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-10\) \(0\) \(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(4-4\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.4.e.p $4$ $23.896$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(10\) \(0\) \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(4-4\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.4.e.q $6$ $23.896$ 6.0.84779568.3 None \(-5\) \(0\) \(-15\) \(4\) \(q+(-\beta _{1}-2\beta _{3}+\beta _{5})q^{2}+(-7+7\beta _{3}+\cdots)q^{4}+\cdots\)
405.4.e.r $6$ $23.896$ 6.0.95327307.1 None \(-1\) \(0\) \(15\) \(-44\) \(q+\beta _{4}q^{2}+(-\beta _{1}-7\beta _{3}-\beta _{4}+\beta _{5})q^{4}+\cdots\)
405.4.e.s $6$ $23.896$ 6.0.148347072.2 None \(-1\) \(0\) \(15\) \(25\) \(q+(-\beta _{1}+\beta _{3})q^{2}+(-2-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
405.4.e.t $6$ $23.896$ 6.0.95327307.1 None \(1\) \(0\) \(-15\) \(-44\) \(q+(\beta _{1}+\beta _{4})q^{2}+(-7-\beta _{2}+7\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
405.4.e.u $6$ $23.896$ 6.0.148347072.2 None \(1\) \(0\) \(-15\) \(25\) \(q-\beta _{1}q^{2}+(\beta _{1}+2\beta _{2}-\beta _{3}+\beta _{4})q^{4}+\cdots\)
405.4.e.v $6$ $23.896$ 6.0.84779568.3 None \(5\) \(0\) \(15\) \(4\) \(q+(2-2\beta _{3}+\beta _{5})q^{2}+(-3\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
405.4.e.w $12$ $23.896$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(0\) \(-30\) \(-40\) \(q+(-\beta _{2}-\beta _{3})q^{2}+(-6+6\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
405.4.e.x $12$ $23.896$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(0\) \(30\) \(-40\) \(q+(1-\beta _{2}-\beta _{9})q^{2}+(-6\beta _{2}-\beta _{3}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(405, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)