Properties

Label 441.4.a
Level $441$
Weight $4$
Character orbit 441.a
Rep. character $\chi_{441}(1,\cdot)$
Character field $\Q$
Dimension $49$
Newform subspaces $24$
Sturm bound $224$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(224\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(441))\).

Total New Old
Modular forms 184 54 130
Cusp forms 152 49 103
Eisenstein series 32 5 27

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(12\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(13\)
\(-\)\(-\)\(+\)\(15\)
Plus space\(+\)\(27\)
Minus space\(-\)\(22\)

Trace form

\( 49 q - 4 q^{2} + 188 q^{4} - 36 q^{8} + O(q^{10}) \) \( 49 q - 4 q^{2} + 188 q^{4} - 36 q^{8} - 32 q^{10} + 26 q^{11} - 42 q^{13} + 748 q^{16} + 174 q^{17} + 54 q^{19} + 60 q^{20} + 420 q^{22} - 2 q^{23} + 1249 q^{25} + 144 q^{26} - 476 q^{29} - 464 q^{31} - 212 q^{32} + 462 q^{34} - 756 q^{37} + 1470 q^{38} + 456 q^{40} - 834 q^{41} + 64 q^{43} + 1672 q^{44} + 68 q^{46} + 876 q^{47} - 1264 q^{50} - 748 q^{52} - 266 q^{53} - 64 q^{55} - 208 q^{58} + 1602 q^{59} - 1866 q^{61} - 1836 q^{62} + 4148 q^{64} + 672 q^{65} - 130 q^{67} + 114 q^{68} - 1912 q^{71} - 668 q^{73} + 216 q^{74} + 950 q^{76} + 1838 q^{79} + 4140 q^{80} + 134 q^{82} + 906 q^{83} + 454 q^{85} + 4740 q^{86} + 7536 q^{88} + 1650 q^{89} - 1024 q^{92} - 2436 q^{94} - 702 q^{95} + 184 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
441.4.a.a 441.a 1.a $1$ $26.020$ \(\Q\) None \(-4\) \(0\) \(-18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}-18q^{5}+72q^{10}+50q^{11}+\cdots\)
441.4.a.b 441.a 1.a $1$ $26.020$ \(\Q\) None \(-4\) \(0\) \(-4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}-4q^{5}+2^{4}q^{10}-62q^{11}+\cdots\)
441.4.a.c 441.a 1.a $1$ $26.020$ \(\Q\) None \(-4\) \(0\) \(18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+8q^{4}+18q^{5}-72q^{10}+50q^{11}+\cdots\)
441.4.a.d 441.a 1.a $1$ $26.020$ \(\Q\) None \(-2\) \(0\) \(-7\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}-7q^{5}+24q^{8}+14q^{10}+\cdots\)
441.4.a.e 441.a 1.a $1$ $26.020$ \(\Q\) None \(-2\) \(0\) \(7\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{4}+7q^{5}+24q^{8}-14q^{10}+\cdots\)
441.4.a.f 441.a 1.a $1$ $26.020$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-8q^{4}+70q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
441.4.a.g 441.a 1.a $1$ $26.020$ \(\Q\) None \(1\) \(0\) \(-12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}-12q^{5}-15q^{8}-12q^{10}+\cdots\)
441.4.a.h 441.a 1.a $1$ $26.020$ \(\Q\) None \(1\) \(0\) \(12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+12q^{5}-15q^{8}+12q^{10}+\cdots\)
441.4.a.i 441.a 1.a $1$ $26.020$ \(\Q\) None \(1\) \(0\) \(16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-7q^{4}+2^{4}q^{5}-15q^{8}+2^{4}q^{10}+\cdots\)
441.4.a.j 441.a 1.a $1$ $26.020$ \(\Q\) None \(3\) \(0\) \(-18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-18q^{5}-21q^{8}-54q^{10}+\cdots\)
441.4.a.k 441.a 1.a $1$ $26.020$ \(\Q\) None \(3\) \(0\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}-3q^{5}-21q^{8}-9q^{10}+\cdots\)
441.4.a.l 441.a 1.a $1$ $26.020$ \(\Q\) None \(3\) \(0\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{2}+q^{4}+3q^{5}-21q^{8}+9q^{10}+\cdots\)
441.4.a.m 441.a 1.a $1$ $26.020$ \(\Q\) \(\Q(\sqrt{-7}) \) \(5\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+5q^{2}+17q^{4}+45q^{8}+68q^{11}+\cdots\)
441.4.a.n 441.a 1.a $2$ $26.020$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-20\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(-5-2\beta )q^{4}+(-10+\cdots)q^{5}+\cdots\)
441.4.a.o 441.a 1.a $2$ $26.020$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(20\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(-5-2\beta )q^{4}+(10+\cdots)q^{5}+\cdots\)
441.4.a.p 441.a 1.a $2$ $26.020$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{2}-q^{4}-9\beta q^{8}+10\beta q^{11}-55q^{16}+\cdots\)
441.4.a.q 441.a 1.a $2$ $26.020$ \(\Q(\sqrt{19}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+11q^{4}-2\beta q^{5}+3\beta q^{8}-38q^{10}+\cdots\)
441.4.a.r 441.a 1.a $2$ $26.020$ \(\Q(\sqrt{57}) \) None \(3\) \(0\) \(6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(2+2\beta )q^{5}+\cdots\)
441.4.a.s 441.a 1.a $3$ $26.020$ 3.3.57516.1 None \(-1\) \(0\) \(-11\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(8+\beta _{1}+\beta _{2})q^{4}+(-4+\beta _{1}+\cdots)q^{5}+\cdots\)
441.4.a.t 441.a 1.a $3$ $26.020$ 3.3.57516.1 None \(-1\) \(0\) \(11\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(8+\beta _{1}+\beta _{2})q^{4}+(4-\beta _{1}+\cdots)q^{5}+\cdots\)
441.4.a.u 441.a 1.a $4$ $26.020$ \(\Q(\sqrt{2}, \sqrt{65})\) None \(-2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(9+\beta _{1})q^{4}+\beta _{3}q^{5}+\cdots\)
441.4.a.v 441.a 1.a $4$ $26.020$ 4.4.6257832.1 None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
441.4.a.w 441.a 1.a $4$ $26.020$ 4.4.6257832.1 None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
441.4.a.x 441.a 1.a $8$ $26.020$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(8-\beta _{3})q^{4}+\beta _{1}q^{5}+(-11\beta _{2}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(441)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)