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Properties

Label 354.8.a
Level $354$
Weight $8$
Character orbit 354.a
Rep. character $\chi_{354}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $9$
Sturm bound $480$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 354.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(480\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(354))\).

Total New Old
Modular forms 424 68 356
Cusp forms 416 68 348
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
\(+\)\(+\)\(+\)$+$\(9\)
\(+\)\(+\)\(-\)$-$\(9\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(9\)
\(-\)\(-\)\(+\)$+$\(10\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(38\)
Minus space\(-\)\(30\)

Trace form

\( 68 q - 16 q^{2} - 54 q^{3} + 4352 q^{4} + 228 q^{5} + 1804 q^{7} - 1024 q^{8} + 49572 q^{9} + O(q^{10}) \) \( 68 q - 16 q^{2} - 54 q^{3} + 4352 q^{4} + 228 q^{5} + 1804 q^{7} - 1024 q^{8} + 49572 q^{9} + 6240 q^{10} - 15448 q^{11} - 3456 q^{12} - 2752 q^{13} + 27432 q^{15} + 278528 q^{16} - 9516 q^{17} - 11664 q^{18} - 52296 q^{19} + 14592 q^{20} + 85104 q^{21} - 48416 q^{22} + 96208 q^{23} + 1094588 q^{25} + 287456 q^{26} - 39366 q^{27} + 115456 q^{28} + 293652 q^{29} + 157248 q^{30} + 241244 q^{31} - 65536 q^{32} - 380592 q^{33} + 275872 q^{34} - 626936 q^{35} + 3172608 q^{36} + 516792 q^{37} - 131776 q^{38} - 299916 q^{39} + 399360 q^{40} + 995012 q^{41} + 493692 q^{43} - 988672 q^{44} + 166212 q^{45} - 1800576 q^{46} + 3256288 q^{47} - 221184 q^{48} + 7176672 q^{49} - 128240 q^{50} - 612036 q^{51} - 176128 q^{52} + 3301204 q^{53} + 2318184 q^{55} - 828792 q^{57} + 461408 q^{58} + 1755648 q^{60} - 1515768 q^{61} + 3682432 q^{62} + 1315116 q^{63} + 17825792 q^{64} + 2865888 q^{65} - 120960 q^{66} - 174028 q^{67} - 609024 q^{68} + 190728 q^{69} - 3302976 q^{70} - 3428800 q^{71} - 746496 q^{72} + 4073376 q^{73} - 3251552 q^{74} - 245754 q^{75} - 3346944 q^{76} - 12953072 q^{77} - 1898208 q^{78} - 11020428 q^{79} + 933888 q^{80} + 36137988 q^{81} + 14894944 q^{82} - 15696128 q^{83} + 5446656 q^{84} - 16026768 q^{85} - 12878400 q^{86} - 6122952 q^{87} - 3098624 q^{88} - 12700132 q^{89} + 4548960 q^{90} - 30843344 q^{91} + 6157312 q^{92} - 12646908 q^{93} + 2154816 q^{94} + 685672 q^{95} + 16619736 q^{97} - 21439120 q^{98} - 11261592 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(354))\) 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
354.8.a.a 354.a 1.a $1$ $110.584$ \(\Q\) None \(8\) \(27\) \(-320\) \(-505\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-320q^{5}+\cdots\)
354.8.a.b 354.a 1.a $5$ $110.584$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(40\) \(135\) \(164\) \(-76\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(33+\beta _{1}+\cdots)q^{5}+\cdots\)
354.8.a.c 354.a 1.a $7$ $110.584$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-56\) \(189\) \(-158\) \(-581\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-23-\beta _{2}+\cdots)q^{5}+\cdots\)
354.8.a.d 354.a 1.a $8$ $110.584$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(64\) \(-216\) \(-592\) \(-340\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-74+\beta _{3}+\cdots)q^{5}+\cdots\)
354.8.a.e 354.a 1.a $9$ $110.584$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-72\) \(-243\) \(-230\) \(-340\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-26+\beta _{3}+\cdots)q^{5}+\cdots\)
354.8.a.f 354.a 1.a $9$ $110.584$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-72\) \(-243\) \(20\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(2+\beta _{1})q^{5}+\cdots\)
354.8.a.g 354.a 1.a $9$ $110.584$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(72\) \(-243\) \(408\) \(3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(45+\beta _{1}+\cdots)q^{5}+\cdots\)
354.8.a.h 354.a 1.a $10$ $110.584$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-80\) \(270\) \(92\) \(1820\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(9+\beta _{1})q^{5}+\cdots\)
354.8.a.i 354.a 1.a $10$ $110.584$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(80\) \(270\) \(844\) \(1820\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(84+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(354))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(354)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(177))\)\(^{\oplus 2}\)