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Properties

Label 546.4.a
Level $546$
Weight $4$
Character orbit 546.a
Rep. character $\chi_{546}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $17$
Sturm bound $448$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(546))\).

Total New Old
Modular forms 344 36 308
Cusp forms 328 36 292
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
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(+\)\(-\)\(-\)$+$\(3\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(2\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(1\)
\(-\)\(-\)\(-\)\(+\)$-$\(1\)
\(-\)\(-\)\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(24\)
Minus space\(-\)\(12\)

Trace form

\( 36 q + 144 q^{4} + 324 q^{9} + O(q^{10}) \) \( 36 q + 144 q^{4} + 324 q^{9} + 576 q^{16} + 256 q^{17} + 160 q^{19} + 84 q^{21} + 144 q^{22} + 144 q^{23} + 1100 q^{25} - 288 q^{29} - 48 q^{30} + 480 q^{31} + 384 q^{33} - 112 q^{35} + 1296 q^{36} - 432 q^{37} + 544 q^{38} + 1040 q^{41} + 144 q^{43} + 464 q^{46} + 544 q^{47} + 1764 q^{49} + 1088 q^{50} - 480 q^{51} - 656 q^{53} + 2336 q^{55} + 1296 q^{57} + 1328 q^{58} - 880 q^{59} + 592 q^{61} + 64 q^{62} + 2304 q^{64} - 832 q^{65} + 480 q^{66} + 2664 q^{67} + 1024 q^{68} + 960 q^{69} - 448 q^{70} + 224 q^{71} - 1888 q^{73} + 1088 q^{74} + 1920 q^{75} + 640 q^{76} + 312 q^{78} + 2328 q^{79} + 2916 q^{81} - 1744 q^{82} + 2592 q^{83} + 336 q^{84} + 4288 q^{85} + 1216 q^{86} - 1704 q^{87} + 576 q^{88} + 2208 q^{89} + 364 q^{91} + 576 q^{92} + 768 q^{93} + 2736 q^{94} + 2488 q^{95} + 784 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\) 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
546.4.a.a 546.a 1.a $1$ $32.215$ \(\Q\) None \(-2\) \(-3\) \(12\) \(7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+12q^{5}+6q^{6}+\cdots\)
546.4.a.b 546.a 1.a $1$ $32.215$ \(\Q\) None \(2\) \(-3\) \(-9\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-9q^{5}-6q^{6}+\cdots\)
546.4.a.c 546.a 1.a $1$ $32.215$ \(\Q\) None \(2\) \(3\) \(-14\) \(-7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-14q^{5}+6q^{6}+\cdots\)
546.4.a.d 546.a 1.a $1$ $32.215$ \(\Q\) None \(2\) \(3\) \(-12\) \(7\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-12q^{5}+6q^{6}+\cdots\)
546.4.a.e 546.a 1.a $1$ $32.215$ \(\Q\) None \(2\) \(3\) \(9\) \(-7\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+9q^{5}+6q^{6}+\cdots\)
546.4.a.f 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{21}) \) None \(-4\) \(-6\) \(-8\) \(-14\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-4-\beta )q^{5}+\cdots\)
546.4.a.g 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{65}) \) None \(-4\) \(6\) \(-15\) \(14\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-8-\beta )q^{5}+\cdots\)
546.4.a.h 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{129}) \) None \(-4\) \(6\) \(1\) \(-14\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+\beta q^{5}-6q^{6}+\cdots\)
546.4.a.i 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{673}) \) None \(-4\) \(6\) \(3\) \(-14\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(1+\beta )q^{5}-6q^{6}+\cdots\)
546.4.a.j 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{105}) \) None \(4\) \(-6\) \(5\) \(-14\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(3-\beta )q^{5}-6q^{6}+\cdots\)
546.4.a.k 546.a 1.a $2$ $32.215$ \(\Q(\sqrt{1401}) \) None \(4\) \(6\) \(5\) \(-14\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(3-\beta )q^{5}+6q^{6}+\cdots\)
546.4.a.l 546.a 1.a $3$ $32.215$ 3.3.7441.1 None \(-6\) \(-9\) \(-6\) \(21\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-2+\beta _{1})q^{5}+\cdots\)
546.4.a.m 546.a 1.a $3$ $32.215$ 3.3.1600113.1 None \(-6\) \(-9\) \(0\) \(-21\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+\beta _{1}q^{5}+6q^{6}+\cdots\)
546.4.a.n 546.a 1.a $3$ $32.215$ 3.3.360321.1 None \(-6\) \(9\) \(13\) \(21\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(4-\beta _{2})q^{5}+\cdots\)
546.4.a.o 546.a 1.a $3$ $32.215$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(6\) \(-9\) \(-1\) \(21\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-\beta _{1}q^{5}-6q^{6}+\cdots\)
546.4.a.p 546.a 1.a $3$ $32.215$ 3.3.118088.1 None \(6\) \(-9\) \(7\) \(-21\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
546.4.a.q 546.a 1.a $4$ $32.215$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(12\) \(10\) \(28\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(3+\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(546))\) into lower level spaces

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