Properties

Label 471.4.a
Level $471$
Weight $4$
Character orbit 471.a
Rep. character $\chi_{471}(1,\cdot)$
Character field $\Q$
Dimension $78$
Newform subspaces $4$
Sturm bound $210$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 471.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(210\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(471))\).

Total New Old
Modular forms 160 78 82
Cusp forms 156 78 78
Eisenstein series 4 0 4

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

\(3\)\(157\)FrickeDim
\(+\)\(+\)$+$\(22\)
\(+\)\(-\)$-$\(17\)
\(-\)\(+\)$-$\(14\)
\(-\)\(-\)$+$\(25\)
Plus space\(+\)\(47\)
Minus space\(-\)\(31\)

Trace form

\( 78 q + 4 q^{2} + 304 q^{4} + 40 q^{7} - 36 q^{8} + 702 q^{9} + O(q^{10}) \) \( 78 q + 4 q^{2} + 304 q^{4} + 40 q^{7} - 36 q^{8} + 702 q^{9} - 84 q^{10} + 16 q^{13} + 68 q^{14} - 24 q^{15} + 1224 q^{16} - 32 q^{17} + 36 q^{18} + 24 q^{19} + 300 q^{20} + 84 q^{21} + 328 q^{22} - 104 q^{23} + 2402 q^{25} + 232 q^{26} + 700 q^{28} + 432 q^{29} - 48 q^{30} - 660 q^{31} - 844 q^{32} + 60 q^{33} - 876 q^{34} + 792 q^{35} + 2736 q^{36} + 184 q^{37} + 960 q^{38} + 120 q^{40} + 32 q^{41} - 216 q^{42} - 544 q^{43} + 324 q^{44} + 240 q^{46} + 136 q^{47} + 624 q^{48} + 3594 q^{49} - 916 q^{50} - 528 q^{51} - 80 q^{52} - 1120 q^{53} + 1432 q^{55} - 828 q^{56} - 312 q^{57} + 608 q^{58} + 1512 q^{59} - 384 q^{60} + 2124 q^{61} + 288 q^{62} + 360 q^{63} + 5284 q^{64} + 1208 q^{65} + 1656 q^{66} + 1856 q^{67} - 536 q^{68} + 984 q^{69} + 1152 q^{70} - 472 q^{71} - 324 q^{72} + 116 q^{73} + 2028 q^{74} + 1104 q^{75} - 4224 q^{76} - 360 q^{77} - 852 q^{78} + 200 q^{79} + 4432 q^{80} + 6318 q^{81} - 128 q^{82} - 56 q^{83} + 1080 q^{84} + 1416 q^{85} - 1424 q^{86} - 48 q^{87} + 2824 q^{88} + 5240 q^{89} - 756 q^{90} + 496 q^{91} + 2476 q^{92} + 1464 q^{93} + 144 q^{94} + 5992 q^{95} - 300 q^{96} + 4260 q^{97} + 4376 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(471))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 157
471.4.a.a 471.a 1.a $14$ $27.790$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-2\) \(42\) \(-32\) \(-60\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
471.4.a.b 471.a 1.a $17$ $27.790$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(-2\) \(-51\) \(-28\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
471.4.a.c 471.a 1.a $22$ $27.790$ None \(4\) \(-66\) \(32\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$
471.4.a.d 471.a 1.a $25$ $27.790$ None \(4\) \(75\) \(28\) \(94\) $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(471))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(471)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(157))\)\(^{\oplus 2}\)