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Properties

Label 355.4.a
Level $355$
Weight $4$
Character orbit 355.a
Rep. character $\chi_{355}(1,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $4$
Sturm bound $144$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 355 = 5 \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 355.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(355))\).

Total New Old
Modular forms 110 70 40
Cusp forms 106 70 36
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.

\(5\)\(71\)FrickeDim
\(+\)\(+\)\(+\)\(19\)
\(+\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(16\)
\(-\)\(-\)\(+\)\(20\)
Plus space\(+\)\(39\)
Minus space\(-\)\(31\)

Trace form

\( 70 q + 4 q^{2} - 4 q^{3} + 292 q^{4} + 10 q^{5} - 44 q^{6} - 40 q^{7} + 48 q^{8} + 662 q^{9} + O(q^{10}) \) \( 70 q + 4 q^{2} - 4 q^{3} + 292 q^{4} + 10 q^{5} - 44 q^{6} - 40 q^{7} + 48 q^{8} + 662 q^{9} - 40 q^{10} - 32 q^{11} - 96 q^{12} + 24 q^{13} - 96 q^{14} + 60 q^{15} + 1084 q^{16} - 260 q^{17} - 76 q^{18} - 240 q^{19} + 120 q^{20} + 8 q^{21} + 576 q^{22} + 64 q^{23} + 20 q^{24} + 1750 q^{25} + 620 q^{26} - 64 q^{27} + 100 q^{28} + 156 q^{29} - 80 q^{30} - 72 q^{31} + 588 q^{32} - 88 q^{33} - 396 q^{34} - 260 q^{35} + 4368 q^{36} - 804 q^{37} + 1244 q^{38} + 408 q^{39} - 428 q^{41} + 2172 q^{42} - 788 q^{43} + 724 q^{44} - 390 q^{45} - 1192 q^{46} + 720 q^{47} - 52 q^{48} + 4886 q^{49} + 100 q^{50} - 1976 q^{51} - 2136 q^{52} - 328 q^{53} - 1320 q^{54} - 360 q^{55} - 596 q^{56} - 3016 q^{57} + 2468 q^{58} - 16 q^{59} + 800 q^{60} + 692 q^{61} + 264 q^{62} - 3816 q^{63} + 760 q^{64} - 420 q^{65} - 888 q^{66} - 2300 q^{67} - 1060 q^{68} - 2464 q^{69} - 360 q^{70} + 3352 q^{72} + 2364 q^{73} + 2236 q^{74} - 100 q^{75} - 2520 q^{76} + 80 q^{77} + 2436 q^{78} - 4776 q^{79} - 1200 q^{80} + 7550 q^{81} + 1668 q^{82} - 1636 q^{83} + 1680 q^{84} - 240 q^{85} - 1184 q^{86} + 2928 q^{87} + 9480 q^{88} - 2508 q^{89} - 1260 q^{90} + 1776 q^{91} - 404 q^{92} - 5096 q^{93} - 1516 q^{94} + 1800 q^{95} - 5836 q^{96} - 1444 q^{97} - 7036 q^{98} + 7016 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(355))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 71
355.4.a.a 355.a 1.a $15$ $20.946$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 355.4.a.a \(-3\) \(-4\) \(-75\) \(24\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{8}q^{3}+(3+\beta _{2})q^{4}-5q^{5}+\cdots\)
355.4.a.b 355.a 1.a $16$ $20.946$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 355.4.a.b \(-11\) \(-10\) \(80\) \(-86\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1+\beta _{6})q^{3}+(4+\cdots)q^{4}+\cdots\)
355.4.a.c 355.a 1.a $19$ $20.946$ \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None 355.4.a.c \(9\) \(-4\) \(-95\) \(-18\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(5+\beta _{2})q^{4}-5q^{5}+\cdots\)
355.4.a.d 355.a 1.a $20$ $20.946$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 355.4.a.d \(9\) \(14\) \(100\) \(40\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(5+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(355))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(355)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(71))\)\(^{\oplus 2}\)