Properties

Label 354.6.a
Level 354
Weight 6
Character orbit a
Rep. character \(\chi_{354}(1,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 9
Sturm bound 360
Trace bound 5

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Defining parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 354.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(360\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(354))\).

Total New Old
Modular forms 304 48 256
Cusp forms 296 48 248
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.
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(20\)
Minus space\(-\)\(28\)

Trace form

\( 48q - 8q^{2} + 18q^{3} + 768q^{4} + 132q^{5} - 276q^{7} - 128q^{8} + 3888q^{9} + O(q^{10}) \) \( 48q - 8q^{2} + 18q^{3} + 768q^{4} + 132q^{5} - 276q^{7} - 128q^{8} + 3888q^{9} + 624q^{10} - 632q^{11} + 288q^{12} + 1448q^{13} - 792q^{15} + 12288q^{16} + 3972q^{17} - 648q^{18} + 1816q^{19} + 2112q^{20} + 3168q^{21} + 3824q^{22} - 400q^{23} + 26928q^{25} + 7024q^{26} + 1458q^{27} - 4416q^{28} + 1332q^{29} - 8352q^{30} - 7492q^{31} - 2048q^{32} - 720q^{33} - 10288q^{34} + 41000q^{35} + 62208q^{36} + 21136q^{37} - 14048q^{38} - 684q^{39} + 9984q^{40} - 28172q^{41} + 11788q^{43} - 10112q^{44} + 10692q^{45} + 3456q^{46} + 6272q^{47} + 4608q^{48} + 113700q^{49} + 38216q^{50} - 15444q^{51} + 23168q^{52} + 8852q^{53} - 103128q^{55} - 11448q^{57} + 304q^{58} - 12672q^{60} + 44800q^{61} - 63424q^{62} - 22356q^{63} + 196608q^{64} - 132912q^{65} + 23040q^{66} + 1412q^{67} + 63552q^{68} + 70488q^{69} + 45984q^{70} - 135200q^{71} - 10368q^{72} - 65528q^{73} - 117232q^{74} + 59886q^{75} + 29056q^{76} + 249296q^{77} + 24336q^{78} + 147604q^{79} + 33792q^{80} + 314928q^{81} - 169360q^{82} + 206720q^{83} + 50688q^{84} + 259968q^{85} + 157152q^{86} - 82728q^{87} + 61184q^{88} + 374524q^{89} + 50544q^{90} + 241072q^{91} - 6400q^{92} + 18972q^{93} + 100128q^{94} + 639464q^{95} + 52528q^{97} - 7880q^{98} - 51192q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 59
354.6.a.a \(1\) \(56.776\) \(\Q\) None \(4\) \(-9\) \(10\) \(144\) \(-\) \(+\) \(+\) \(q+4q^{2}-9q^{3}+2^{4}q^{4}+10q^{5}-6^{2}q^{6}+\cdots\)
354.6.a.b \(4\) \(56.776\) 4.4.32832.1 None \(16\) \(36\) \(-104\) \(-162\) \(-\) \(-\) \(+\) \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-5^{2}+2\beta _{1}+\cdots)q^{5}+\cdots\)
354.6.a.c \(5\) \(56.776\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-20\) \(45\) \(-10\) \(-162\) \(+\) \(-\) \(-\) \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
354.6.a.d \(5\) \(56.776\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(20\) \(-45\) \(-24\) \(-103\) \(-\) \(+\) \(-\) \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-5+\beta _{3}+\cdots)q^{5}+\cdots\)
354.6.a.e \(5\) \(56.776\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(20\) \(-45\) \(166\) \(-198\) \(-\) \(+\) \(+\) \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(33-\beta _{2})q^{5}+\cdots\)
354.6.a.f \(6\) \(56.776\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-54\) \(-46\) \(-103\) \(+\) \(+\) \(+\) \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-8+\beta _{5})q^{5}+\cdots\)
354.6.a.g \(6\) \(56.776\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-54\) \(4\) \(-54\) \(+\) \(+\) \(-\) \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(1-\beta _{5})q^{5}+\cdots\)
354.6.a.h \(8\) \(56.776\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-32\) \(72\) \(40\) \(181\) \(+\) \(-\) \(+\) \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(5+\beta _{1})q^{5}+\cdots\)
354.6.a.i \(8\) \(56.776\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(32\) \(72\) \(96\) \(181\) \(-\) \(-\) \(-\) \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(12-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(354))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(354)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)