Properties

Label 441.4.e
Level $441$
Weight $4$
Character orbit 441.e
Rep. character $\chi_{441}(226,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $26$
Sturm bound $224$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 26 \)
Sturm bound: \(224\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(441, [\chi])\).

Total New Old
Modular forms 368 104 264
Cusp forms 304 96 208
Eisenstein series 64 8 56

Trace form

\( 96q + q^{2} - 193q^{4} - 15q^{5} + 102q^{8} + O(q^{10}) \) \( 96q + q^{2} - 193q^{4} - 15q^{5} + 102q^{8} - 10q^{10} - 29q^{11} - 172q^{13} - 741q^{16} - 111q^{17} + 53q^{19} + 816q^{20} + 372q^{22} + 41q^{23} - 825q^{25} - 438q^{26} - 784q^{29} + 19q^{31} - 1327q^{32} - 1404q^{34} + 407q^{37} + 252q^{38} + 612q^{40} + 1908q^{41} + 1504q^{43} + 548q^{44} + 922q^{46} - 285q^{47} - 2234q^{50} + 276q^{52} - 2299q^{53} - 2978q^{55} - 3314q^{58} - 1023q^{59} + 731q^{61} + 4224q^{62} + 6578q^{64} + 378q^{65} + 1017q^{67} - 2112q^{68} - 2312q^{71} + 1987q^{73} - 3138q^{74} + 192q^{76} + 351q^{79} - 276q^{80} + 2224q^{82} + 4176q^{83} - 3118q^{85} + 546q^{86} - 2760q^{88} - 2343q^{89} + 4768q^{92} - 3570q^{94} - 57q^{95} - 5300q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(441, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
441.4.e.a \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(-5\) \(0\) \(0\) \(0\) \(q-5\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}+45q^{8}+\cdots\)
441.4.e.b \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(-18\) \(0\) \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots\)
441.4.e.c \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(3\) \(0\) \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
441.4.e.d \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(18\) \(0\) \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots\)
441.4.e.e \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-16\) \(0\) \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-2^{4}\zeta_{6}q^{5}+\cdots\)
441.4.e.f \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-12\) \(0\) \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-12\zeta_{6}q^{5}+\cdots\)
441.4.e.g \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(12\) \(0\) \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+12\zeta_{6}q^{5}+\cdots\)
441.4.e.h \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(16\) \(0\) \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+2^{4}\zeta_{6}q^{5}+\cdots\)
441.4.e.i \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(8-8\zeta_{6})q^{4}-70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots\)
441.4.e.j \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+(8-8\zeta_{6})q^{4}+70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots\)
441.4.e.k \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-7\) \(0\) \(q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{4}-7\zeta_{6}q^{5}+\cdots\)
441.4.e.l \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(-18\) \(0\) \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots\)
441.4.e.m \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(-4\) \(0\) \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots\)
441.4.e.n \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(4\) \(0\) \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+4\zeta_{6}q^{5}+\cdots\)
441.4.e.o \(2\) \(26.020\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(18\) \(0\) \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots\)
441.4.e.p \(4\) \(26.020\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(-3\) \(0\) \(-6\) \(0\) \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
441.4.e.q \(4\) \(26.020\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(-3\) \(0\) \(6\) \(0\) \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
441.4.e.r \(4\) \(26.020\) \(\Q(\sqrt{-3}, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}+2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots\)
441.4.e.s \(4\) \(26.020\) \(\Q(\sqrt{-3}, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}-2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots\)
441.4.e.t \(4\) \(26.020\) \(\Q(\sqrt{-3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{2}q^{4}-9\beta _{3}q^{8}+(-10\beta _{1}+\cdots)q^{11}+\cdots\)
441.4.e.u \(4\) \(26.020\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(-20\) \(0\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots\)
441.4.e.v \(4\) \(26.020\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(0\) \(20\) \(0\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots\)
441.4.e.w \(6\) \(26.020\) 6.0.9924270768.1 None \(1\) \(0\) \(-11\) \(0\) \(q+\beta _{1}q^{2}+(-8+\beta _{1}+\beta _{2}+8\beta _{4}+\beta _{5})q^{4}+\cdots\)
441.4.e.x \(8\) \(26.020\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{3})q^{2}+(-2-2\beta _{2}-\beta _{6})q^{4}+\cdots\)
441.4.e.y \(8\) \(26.020\) 8.0.\(\cdots\).19 None \(2\) \(0\) \(0\) \(0\) \(q+(1-\beta _{2}+\beta _{6})q^{2}+(-8-8\beta _{1}+\beta _{6}+\cdots)q^{4}+\cdots\)
441.4.e.z \(16\) \(26.020\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{2}+(-9+9\beta _{2}+\beta _{7})q^{4}-\beta _{6}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)