Properties

Label 546.8.a
Level $546$
Weight $8$
Character orbit 546.a
Rep. character $\chi_{546}(1,\cdot)$
Character field $\Q$
Dimension $84$
Newform subspaces $19$
Sturm bound $896$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(896\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(546))\).

Total New Old
Modular forms 792 84 708
Cusp forms 776 84 692
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.
\(+\)\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(48\)
Minus space\(-\)\(36\)

Trace form

\( 84 q + 5376 q^{4} + 61236 q^{9} + O(q^{10}) \) \( 84 q + 5376 q^{4} + 61236 q^{9} + 344064 q^{16} + 52208 q^{17} - 90704 q^{19} + 37044 q^{21} + 5824 q^{22} - 102072 q^{23} + 1110900 q^{25} - 91992 q^{29} - 157248 q^{30} + 386864 q^{31} - 163728 q^{33} - 117992 q^{35} + 3919104 q^{36} + 777264 q^{37} + 371072 q^{38} + 239488 q^{41} - 626440 q^{43} + 846528 q^{46} + 2910272 q^{47} + 9882516 q^{49} - 1763072 q^{50} + 162864 q^{51} - 2494600 q^{53} - 921024 q^{55} + 4923504 q^{57} + 3464128 q^{58} + 6413152 q^{59} - 4173936 q^{61} - 7740928 q^{62} + 22020096 q^{64} + 3567928 q^{65} - 4358016 q^{66} - 8343144 q^{67} + 3341312 q^{68} + 1578096 q^{69} + 6717312 q^{70} + 1505008 q^{71} + 21211344 q^{73} - 7584512 q^{74} - 7525440 q^{75} - 5805056 q^{76} + 1898208 q^{78} - 13005104 q^{79} + 44641044 q^{81} + 2168768 q^{82} + 18519072 q^{83} + 2370816 q^{84} + 5307248 q^{85} + 18457088 q^{86} + 2312280 q^{87} + 372736 q^{88} - 8864928 q^{89} + 3014284 q^{91} - 6532608 q^{92} - 25970112 q^{93} - 14377536 q^{94} + 26140928 q^{95} + 48598944 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\) 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
546.8.a.a 546.a 1.a $1$ $170.562$ \(\Q\) None \(8\) \(-27\) \(-135\) \(343\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-135q^{5}+\cdots\)
546.8.a.b 546.a 1.a $1$ $170.562$ \(\Q\) None \(8\) \(27\) \(-390\) \(-343\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-390q^{5}+\cdots\)
546.8.a.c 546.a 1.a $1$ $170.562$ \(\Q\) None \(8\) \(27\) \(-10\) \(-343\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-10q^{5}+6^{3}q^{6}+\cdots\)
546.8.a.d 546.a 1.a $3$ $170.562$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(24\) \(-81\) \(351\) \(1029\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(117+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.e 546.a 1.a $3$ $170.562$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(24\) \(81\) \(-378\) \(-1029\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-126+\beta _{2})q^{5}+\cdots\)
546.8.a.f 546.a 1.a $4$ $170.562$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(-108\) \(-134\) \(1372\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-34-2\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.g 546.a 1.a $4$ $170.562$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(108\) \(-328\) \(1372\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-82+\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.h 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-40\) \(-135\) \(168\) \(-1715\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(34-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.i 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-40\) \(135\) \(-340\) \(1715\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-68-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.j 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-40\) \(135\) \(56\) \(-1715\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(11-\beta _{2}+\cdots)q^{5}+\cdots\)
546.8.a.k 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-40\) \(135\) \(509\) \(-1715\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(102-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.l 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(40\) \(-135\) \(-250\) \(-1715\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-50+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.m 546.a 1.a $5$ $170.562$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(40\) \(135\) \(299\) \(-1715\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(60-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.n 546.a 1.a $6$ $170.562$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-48\) \(-162\) \(-181\) \(2058\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-30-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.o 546.a 1.a $6$ $170.562$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-48\) \(-162\) \(-35\) \(-2058\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-6+\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.p 546.a 1.a $6$ $170.562$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-48\) \(162\) \(-43\) \(2058\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-7-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.q 546.a 1.a $6$ $170.562$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(48\) \(-162\) \(13\) \(2058\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(2+\beta _{1})q^{5}+\cdots\)
546.8.a.r 546.a 1.a $6$ $170.562$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(48\) \(-162\) \(203\) \(-2058\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(34-\beta _{1}+\cdots)q^{5}+\cdots\)
546.8.a.s 546.a 1.a $7$ $170.562$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(56\) \(189\) \(625\) \(2401\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(89+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(546))\) into lower level spaces

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