Properties

Label 354.4.a
Level $354$
Weight $4$
Character orbit 354.a
Rep. character $\chi_{354}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $9$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 354.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(240\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(354))\).

Total New Old
Modular forms 184 28 156
Cusp forms 176 28 148
Eisenstein series 8 0 8

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

\(2\)\(3\)\(59\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(4\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(18\)
Minus space\(-\)\(10\)

Trace form

\( 28 q + 4 q^{2} + 6 q^{3} + 112 q^{4} - 12 q^{5} + 44 q^{7} + 16 q^{8} + 252 q^{9} + O(q^{10}) \) \( 28 q + 4 q^{2} + 6 q^{3} + 112 q^{4} - 12 q^{5} + 44 q^{7} + 16 q^{8} + 252 q^{9} + 40 q^{10} + 112 q^{11} + 24 q^{12} + 8 q^{13} - 48 q^{15} + 448 q^{16} + 324 q^{17} + 36 q^{18} - 96 q^{19} - 48 q^{20} - 96 q^{21} - 232 q^{22} - 208 q^{23} + 1268 q^{25} - 152 q^{26} + 54 q^{27} + 176 q^{28} + 12 q^{29} + 48 q^{30} + 68 q^{31} + 64 q^{32} + 192 q^{33} + 152 q^{34} + 104 q^{35} + 1008 q^{36} - 544 q^{37} + 208 q^{38} + 252 q^{39} + 160 q^{40} - 356 q^{41} + 92 q^{43} + 448 q^{44} - 108 q^{45} + 64 q^{46} + 584 q^{47} + 96 q^{48} + 1560 q^{49} - 580 q^{50} + 36 q^{51} + 32 q^{52} - 412 q^{53} + 1584 q^{55} + 888 q^{57} + 136 q^{58} - 192 q^{60} + 3608 q^{61} + 32 q^{62} + 396 q^{63} + 1792 q^{64} + 1848 q^{65} - 768 q^{66} - 332 q^{67} + 1296 q^{68} + 1248 q^{69} + 144 q^{70} + 1864 q^{71} + 144 q^{72} + 1664 q^{73} + 1688 q^{74} + 426 q^{75} - 384 q^{76} + 1496 q^{77} - 312 q^{78} - 1748 q^{79} - 192 q^{80} + 2268 q^{81} + 648 q^{82} - 1288 q^{83} - 384 q^{84} + 1752 q^{85} + 48 q^{86} + 1152 q^{87} - 928 q^{88} - 5468 q^{89} + 360 q^{90} + 1392 q^{91} - 832 q^{92} + 36 q^{93} - 2672 q^{94} - 4768 q^{95} - 2336 q^{97} - 860 q^{98} + 1008 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(354))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 59
354.4.a.a 354.a 1.a $2$ $20.887$ \(\Q(\sqrt{21}) \) None \(-4\) \(6\) \(-12\) \(-23\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-5-2\beta )q^{5}+\cdots\)
354.4.a.b 354.a 1.a $2$ $20.887$ \(\Q(\sqrt{51}) \) None \(-4\) \(6\) \(26\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(13+\beta )q^{5}+\cdots\)
354.4.a.c 354.a 1.a $2$ $20.887$ \(\Q(\sqrt{13}) \) None \(4\) \(6\) \(-20\) \(-23\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-9-2\beta )q^{5}+\cdots\)
354.4.a.d 354.a 1.a $3$ $20.887$ 3.3.30645.1 None \(-6\) \(-9\) \(-6\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1}-2\beta _{2})q^{5}+\cdots\)
354.4.a.e 354.a 1.a $3$ $20.887$ 3.3.45581.1 None \(-6\) \(-9\) \(4\) \(13\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
354.4.a.f 354.a 1.a $3$ $20.887$ 3.3.18989.1 None \(-6\) \(9\) \(-28\) \(26\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-9+\beta _{1}+\beta _{2})q^{5}+\cdots\)
354.4.a.g 354.a 1.a $3$ $20.887$ 3.3.47277.1 None \(6\) \(-9\) \(-18\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-6+\beta _{1}+\beta _{2})q^{5}+\cdots\)
354.4.a.h 354.a 1.a $4$ $20.887$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-12\) \(22\) \(13\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(5+\beta _{1}-\beta _{3})q^{5}+\cdots\)
354.4.a.i 354.a 1.a $6$ $20.887$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(18\) \(20\) \(26\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(3+\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(354))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(354)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)