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Properties

Label 342.6.a
Level $342$
Weight $6$
Character orbit 342.a
Rep. character $\chi_{342}(1,\cdot)$
Character field $\Q$
Dimension $37$
Newform subspaces $17$
Sturm bound $360$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(360\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(342))\).

Total New Old
Modular forms 308 37 271
Cusp forms 292 37 255
Eisenstein series 16 0 16

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

\(2\)\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(17\)
Minus space\(-\)\(20\)

Trace form

\( 37 q + 4 q^{2} + 592 q^{4} - 110 q^{5} + 42 q^{7} + 64 q^{8} + O(q^{10}) \) \( 37 q + 4 q^{2} + 592 q^{4} - 110 q^{5} + 42 q^{7} + 64 q^{8} - 808 q^{10} - 716 q^{11} + 542 q^{13} - 80 q^{14} + 9472 q^{16} + 996 q^{17} - 361 q^{19} - 1760 q^{20} + 1424 q^{22} + 8314 q^{23} + 15263 q^{25} - 7408 q^{26} + 672 q^{28} - 7846 q^{29} + 25708 q^{31} + 1024 q^{32} + 3384 q^{34} - 13764 q^{35} + 1306 q^{37} + 4332 q^{38} - 12928 q^{40} + 22058 q^{41} - 15992 q^{43} - 11456 q^{44} - 38608 q^{46} + 220 q^{47} + 118043 q^{49} - 16260 q^{50} + 8672 q^{52} - 89558 q^{53} + 60208 q^{55} - 1280 q^{56} - 25328 q^{58} + 83220 q^{59} - 58522 q^{61} - 37584 q^{62} + 151552 q^{64} + 209552 q^{65} + 67764 q^{67} + 15936 q^{68} + 220560 q^{70} + 13180 q^{71} - 17508 q^{73} + 60552 q^{74} - 5776 q^{76} + 179712 q^{77} - 152476 q^{79} - 28160 q^{80} - 237320 q^{82} - 169476 q^{83} + 78612 q^{85} + 127200 q^{86} + 22784 q^{88} + 76590 q^{89} - 113400 q^{91} + 133024 q^{92} + 219776 q^{94} - 72922 q^{95} - 8078 q^{97} + 168388 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(342))\) 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 19
342.6.a.a 342.a 1.a $1$ $54.851$ \(\Q\) None \(-4\) \(0\) \(-21\) \(-143\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-21q^{5}-143q^{7}+\cdots\)
342.6.a.b 342.a 1.a $1$ $54.851$ \(\Q\) None \(-4\) \(0\) \(45\) \(-121\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+45q^{5}-11^{2}q^{7}+\cdots\)
342.6.a.c 342.a 1.a $1$ $54.851$ \(\Q\) None \(-4\) \(0\) \(91\) \(-33\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+91q^{5}-33q^{7}-2^{6}q^{8}+\cdots\)
342.6.a.d 342.a 1.a $1$ $54.851$ \(\Q\) None \(4\) \(0\) \(-81\) \(-247\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-3^{4}q^{5}-247q^{7}+\cdots\)
342.6.a.e 342.a 1.a $1$ $54.851$ \(\Q\) None \(4\) \(0\) \(-31\) \(-27\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}-31q^{5}-3^{3}q^{7}+2^{6}q^{8}+\cdots\)
342.6.a.f 342.a 1.a $1$ $54.851$ \(\Q\) None \(4\) \(0\) \(54\) \(104\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+54q^{5}+104q^{7}+\cdots\)
342.6.a.g 342.a 1.a $2$ $54.851$ \(\Q(\sqrt{2441}) \) None \(-8\) \(0\) \(5\) \(-105\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(3+\beta )q^{5}+(-53+\cdots)q^{7}+\cdots\)
342.6.a.h 342.a 1.a $2$ $54.851$ \(\Q(\sqrt{201}) \) None \(8\) \(0\) \(13\) \(-33\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(4+5\beta )q^{5}+(-22+\cdots)q^{7}+\cdots\)
342.6.a.i 342.a 1.a $2$ $54.851$ \(\Q(\sqrt{1441}) \) None \(8\) \(0\) \(45\) \(114\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(24-3\beta )q^{5}+(59+\cdots)q^{7}+\cdots\)
342.6.a.j 342.a 1.a $2$ $54.851$ \(\Q(\sqrt{4089}) \) None \(8\) \(0\) \(49\) \(-105\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(5^{2}-\beta )q^{5}+(-55+\cdots)q^{7}+\cdots\)
342.6.a.k 342.a 1.a $3$ $54.851$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(0\) \(-135\) \(125\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(-45+\beta _{1})q^{5}+(42+\cdots)q^{7}+\cdots\)
342.6.a.l 342.a 1.a $3$ $54.851$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(0\) \(-81\) \(228\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(-3^{3}-\beta _{1})q^{5}+(76+\cdots)q^{7}+\cdots\)
342.6.a.m 342.a 1.a $3$ $54.851$ 3.3.364092.1 None \(-12\) \(0\) \(96\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(33-3\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.6.a.n 342.a 1.a $3$ $54.851$ 3.3.364092.1 None \(12\) \(0\) \(-96\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-33+3\beta _{1}-\beta _{2})q^{5}+\cdots\)
342.6.a.o 342.a 1.a $3$ $54.851$ 3.3.2922585.1 None \(12\) \(0\) \(-63\) \(125\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-21-\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.6.a.p 342.a 1.a $4$ $54.851$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(0\) \(46\) \(78\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(11+\beta _{2})q^{5}+(20+\cdots)q^{7}+\cdots\)
342.6.a.q 342.a 1.a $4$ $54.851$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(0\) \(-46\) \(78\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-11-\beta _{2})q^{5}+(20+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(342)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)