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Properties

Label 546.6.a
Level $546$
Weight $6$
Character orbit 546.a
Rep. character $\chi_{546}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $24$
Sturm bound $672$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(672\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(546))\).

Total New Old
Modular forms 568 60 508
Cusp forms 552 60 492
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.

\(2\)\(3\)\(7\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(+\)\(-\)\(-\)$+$\(3\)
\(+\)\(-\)\(+\)\(+\)$-$\(4\)
\(+\)\(-\)\(+\)\(-\)$+$\(4\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(4\)
\(-\)\(+\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(5\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(5\)
\(-\)\(-\)\(-\)\(+\)$-$\(5\)
\(-\)\(-\)\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(24\)
Minus space\(-\)\(36\)

Trace form

\( 60 q + 960 q^{4} + 4860 q^{9} + O(q^{10}) \) \( 60 q + 960 q^{4} + 4860 q^{9} + 15360 q^{16} - 3824 q^{17} + 1136 q^{19} - 1764 q^{21} - 1888 q^{22} + 5448 q^{23} + 39804 q^{25} + 22008 q^{29} + 8352 q^{30} - 13904 q^{31} - 4464 q^{33} + 12152 q^{35} + 77760 q^{36} + 31632 q^{37} - 16064 q^{38} - 24448 q^{41} + 37144 q^{43} + 4512 q^{46} - 38720 q^{47} + 144060 q^{49} - 11392 q^{50} + 51696 q^{51} + 107944 q^{53} - 39744 q^{55} - 78192 q^{57} - 69856 q^{58} - 78688 q^{59} + 123696 q^{61} + 184960 q^{62} + 245760 q^{64} + 1352 q^{65} + 23616 q^{66} + 13368 q^{67} - 61184 q^{68} - 47664 q^{69} - 56448 q^{70} + 200240 q^{71} - 67152 q^{73} + 39296 q^{74} + 75456 q^{75} + 18176 q^{76} - 24336 q^{78} - 37456 q^{79} + 393660 q^{81} + 356512 q^{82} - 5280 q^{83} - 28224 q^{84} - 86288 q^{85} - 47744 q^{86} + 124920 q^{87} - 30208 q^{88} - 443232 q^{89} - 33124 q^{91} + 87168 q^{92} + 340704 q^{93} + 94560 q^{94} + 4096 q^{95} + 214368 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\) 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 7 13
546.6.a.a 546.a 1.a $1$ $87.570$ \(\Q\) None \(-4\) \(9\) \(81\) \(-49\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+3^{4}q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.b 546.a 1.a $1$ $87.570$ \(\Q\) None \(-4\) \(9\) \(99\) \(49\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+99q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.c 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(-9\) \(-61\) \(-49\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-61q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.d 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(-9\) \(33\) \(49\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+33q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.e 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(-9\) \(93\) \(-49\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+93q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.f 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(9\) \(-54\) \(49\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-54q^{5}+6^{2}q^{6}+\cdots\)
546.6.a.g 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(9\) \(-21\) \(49\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-21q^{5}+6^{2}q^{6}+\cdots\)
546.6.a.h 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(9\) \(-16\) \(-49\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-2^{4}q^{5}+6^{2}q^{6}+\cdots\)
546.6.a.i 546.a 1.a $1$ $87.570$ \(\Q\) None \(4\) \(9\) \(24\) \(49\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}+6^{2}q^{6}+\cdots\)
546.6.a.j 546.a 1.a $2$ $87.570$ \(\Q(\sqrt{105}) \) None \(8\) \(-18\) \(-38\) \(98\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-19-5\beta )q^{5}+\cdots\)
546.6.a.k 546.a 1.a $2$ $87.570$ \(\Q(\sqrt{69001}) \) None \(8\) \(-18\) \(-32\) \(-98\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-2^{4}q^{5}-6^{2}q^{6}+\cdots\)
546.6.a.l 546.a 1.a $2$ $87.570$ \(\Q(\sqrt{1401}) \) None \(8\) \(18\) \(-5\) \(-98\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-3-\beta )q^{5}+\cdots\)
546.6.a.m 546.a 1.a $3$ $87.570$ 3.3.1875993.1 None \(-12\) \(-27\) \(-7\) \(-147\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-4+5\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.n 546.a 1.a $3$ $87.570$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(-27\) \(61\) \(147\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(20-\beta _{1}-\beta _{2})q^{5}+\cdots\)
546.6.a.o 546.a 1.a $3$ $87.570$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(27\) \(-117\) \(-147\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-39-\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.p 546.a 1.a $3$ $87.570$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(27\) \(-19\) \(147\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-7+\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.q 546.a 1.a $3$ $87.570$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(27\) \(-17\) \(147\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-6+\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.r 546.a 1.a $3$ $87.570$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(-27\) \(-49\) \(-147\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-2^{4}-\beta _{2})q^{5}+\cdots\)
546.6.a.s 546.a 1.a $4$ $87.570$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(-36\) \(-6\) \(-196\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
546.6.a.t 546.a 1.a $4$ $87.570$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(36\) \(-85\) \(-196\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-21-\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.u 546.a 1.a $4$ $87.570$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(36\) \(50\) \(196\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(13-\beta _{1})q^{5}+\cdots\)
546.6.a.v 546.a 1.a $5$ $87.570$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-20\) \(-45\) \(10\) \(245\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(2-\beta _{1})q^{5}+\cdots\)
546.6.a.w 546.a 1.a $5$ $87.570$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(20\) \(-45\) \(-4\) \(245\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
546.6.a.x 546.a 1.a $5$ $87.570$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(20\) \(45\) \(80\) \(-245\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(2^{4}-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(546))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(546)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 2}\)