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

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

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
441.4.e.a 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) 49.4.a.a \(-5\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q-5\zeta_{6}q^{2}+(-17+17\zeta_{6})q^{4}+45q^{8}+\cdots\)
441.4.e.b 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 21.4.a.a \(-3\) \(0\) \(-18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots\)
441.4.e.c 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 21.4.e.a \(-3\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
441.4.e.d 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 21.4.a.a \(-3\) \(0\) \(18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots\)
441.4.e.e 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 7.4.a.a \(-1\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-2^{4}\zeta_{6}q^{5}+\cdots\)
441.4.e.f 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 147.4.a.d \(-1\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}-12\zeta_{6}q^{5}+\cdots\)
441.4.e.g 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 147.4.a.d \(-1\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+12\zeta_{6}q^{5}+\cdots\)
441.4.e.h 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 7.4.a.a \(-1\) \(0\) \(16\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(7-7\zeta_{6})q^{4}+2^{4}\zeta_{6}q^{5}+\cdots\)
441.4.e.i 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 9.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+(8-8\zeta_{6})q^{4}-70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots\)
441.4.e.j 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 9.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+(8-8\zeta_{6})q^{4}+70q^{13}-2^{6}\zeta_{6}q^{16}+\cdots\)
441.4.e.k 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 7.4.c.a \(2\) \(0\) \(-7\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{4}-7\zeta_{6}q^{5}+\cdots\)
441.4.e.l 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 147.4.a.f \(4\) \(0\) \(-18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-18\zeta_{6}q^{5}+\cdots\)
441.4.e.m 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 21.4.a.b \(4\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots\)
441.4.e.n 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 21.4.a.b \(4\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+4\zeta_{6}q^{5}+\cdots\)
441.4.e.o 441.e 7.c $2$ $26.020$ \(\Q(\sqrt{-3}) \) None 147.4.a.f \(4\) \(0\) \(18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{4}+18\zeta_{6}q^{5}+\cdots\)
441.4.e.p 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None 21.4.a.c \(-3\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
441.4.e.q 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None 21.4.a.c \(-3\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
441.4.e.r 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{-3}, \sqrt{19})\) None 63.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}+2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots\)
441.4.e.s 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{-3}, \sqrt{19})\) None 63.4.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+11\beta _{2}q^{4}-2\beta _{1}q^{5}+3\beta _{3}q^{8}+\cdots\)
441.4.e.t 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{-3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) 441.4.a.p \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{4}-9\beta _{3}q^{8}+(-10\beta _{1}+\cdots)q^{11}+\cdots\)
441.4.e.u 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 147.4.a.j \(2\) \(0\) \(-20\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots\)
441.4.e.v 441.e 7.c $4$ $26.020$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 147.4.a.j \(2\) \(0\) \(20\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}-5\beta _{2}+2\beta _{3})q^{4}+\cdots\)
441.4.e.w 441.e 7.c $6$ $26.020$ 6.0.9924270768.1 None 21.4.e.b \(1\) \(0\) \(-11\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-8+\beta _{1}+\beta _{2}+8\beta _{4}+\beta _{5})q^{4}+\cdots\)
441.4.e.x 441.e 7.c $8$ $26.020$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 63.4.e.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(-2-2\beta _{2}-\beta _{6})q^{4}+\cdots\)
441.4.e.y 441.e 7.c $8$ $26.020$ 8.0.\(\cdots\).19 None 49.4.a.e \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2}+\beta _{6})q^{2}+(-8-8\beta _{1}+\beta _{6}+\cdots)q^{4}+\cdots\)
441.4.e.z 441.e 7.c $16$ $26.020$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 441.4.a.x \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{2}+(-9+9\beta _{2}+\beta _{7})q^{4}-\beta _{6}q^{5}+\cdots\)

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

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