Properties

Label 546.2.l.l
Level $546$
Weight $2$
Character orbit 546.l
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} + ( 1 - \beta_{2} ) q^{3} -\beta_{2} q^{4} + \beta_{3} q^{5} + \beta_{2} q^{6} -\beta_{2} q^{7} + q^{8} -\beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} + ( 1 - \beta_{2} ) q^{3} -\beta_{2} q^{4} + \beta_{3} q^{5} + \beta_{2} q^{6} -\beta_{2} q^{7} + q^{8} -\beta_{2} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{10} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} - q^{12} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + q^{14} + ( \beta_{1} + \beta_{3} ) q^{15} + ( -1 + \beta_{2} ) q^{16} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{17} + q^{18} + ( -\beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{20} - q^{21} + ( -\beta_{1} + \beta_{2} ) q^{22} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( 1 - \beta_{2} ) q^{24} + ( -1 - \beta_{3} ) q^{25} + ( 1 - \beta_{1} + \beta_{3} ) q^{26} - q^{27} + ( -1 + \beta_{2} ) q^{28} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{29} -\beta_{1} q^{30} -\beta_{2} q^{32} + ( \beta_{1} - \beta_{2} ) q^{33} + ( -3 + 2 \beta_{3} ) q^{34} + \beta_{1} q^{35} + ( -1 + \beta_{2} ) q^{36} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{37} + ( -1 - \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{3} ) q^{39} + \beta_{3} q^{40} + ( 5 + 4 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 1 - \beta_{2} ) q^{42} + 8 \beta_{2} q^{43} + ( -1 - \beta_{3} ) q^{44} + \beta_{1} q^{45} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( -5 + 3 \beta_{3} ) q^{47} + \beta_{2} q^{48} + ( -1 + \beta_{2} ) q^{49} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( 3 - 2 \beta_{3} ) q^{51} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{52} + 7 q^{53} + ( 1 - \beta_{2} ) q^{54} + ( 4 - 4 \beta_{2} ) q^{55} -\beta_{2} q^{56} + ( 1 + \beta_{3} ) q^{57} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} -\beta_{3} q^{60} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{61} + ( -1 + \beta_{2} ) q^{63} + q^{64} + ( 4 - \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{65} + ( 1 + \beta_{3} ) q^{66} + ( -6 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{67} + ( 3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{68} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{69} + \beta_{3} q^{70} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{71} -\beta_{2} q^{72} + ( -4 + \beta_{3} ) q^{73} + ( \beta_{1} + 4 \beta_{2} ) q^{74} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -1 - \beta_{3} ) q^{77} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{78} + ( -5 - \beta_{3} ) q^{79} + ( -\beta_{1} - \beta_{3} ) q^{80} + ( -1 + \beta_{2} ) q^{81} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{82} + ( -2 - 2 \beta_{3} ) q^{83} + \beta_{2} q^{84} + ( -5 \beta_{1} - 8 \beta_{2} ) q^{85} -8 q^{86} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + ( -5 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{89} + \beta_{3} q^{90} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{91} + ( 2 - 2 \beta_{3} ) q^{92} + ( 5 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} ) q^{94} + 4 \beta_{2} q^{95} - q^{96} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{97} -\beta_{2} q^{98} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 4q^{8} - 2q^{9} + q^{10} + q^{11} - 4q^{12} - 2q^{13} + 4q^{14} - q^{15} - 2q^{16} + 8q^{17} + 4q^{18} + q^{19} + q^{20} - 4q^{21} + q^{22} - 6q^{23} + 2q^{24} - 2q^{25} + q^{26} - 4q^{27} - 2q^{28} + 4q^{29} - q^{30} - 2q^{32} - q^{33} - 16q^{34} + q^{35} - 2q^{36} + 9q^{37} - 2q^{38} - q^{39} - 2q^{40} + 6q^{41} + 2q^{42} + 16q^{43} - 2q^{44} + q^{45} - 6q^{46} - 26q^{47} + 2q^{48} - 2q^{49} + q^{50} + 16q^{51} + q^{52} + 28q^{53} + 2q^{54} + 8q^{55} - 2q^{56} + 2q^{57} + 4q^{58} - 2q^{59} + 2q^{60} + 6q^{61} - 2q^{63} + 4q^{64} + q^{65} + 2q^{66} - 10q^{67} + 8q^{68} + 6q^{69} - 2q^{70} - 2q^{71} - 2q^{72} - 18q^{73} + 9q^{74} - q^{75} + q^{76} - 2q^{77} - q^{78} - 18q^{79} + q^{80} - 2q^{81} + 6q^{82} - 4q^{83} + 2q^{84} - 21q^{85} - 32q^{86} - 4q^{87} + q^{88} - 11q^{89} - 2q^{90} + q^{91} + 12q^{92} + 13q^{94} + 8q^{95} - 4q^{96} + 10q^{97} - 2q^{98} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
1.28078 2.21837i
−0.780776 + 1.35234i
1.28078 + 2.21837i
−0.780776 1.35234i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.28078 + 2.21837i
211.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.780776 1.35234i
295.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 1.28078 2.21837i
295.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.780776 + 1.35234i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.l.l 4
3.b odd 2 1 1638.2.r.y 4
13.c even 3 1 inner 546.2.l.l 4
13.c even 3 1 7098.2.a.bt 2
13.e even 6 1 7098.2.a.bi 2
39.i odd 6 1 1638.2.r.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.l 4 1.a even 1 1 trivial
546.2.l.l 4 13.c even 3 1 inner
1638.2.r.y 4 3.b odd 2 1
1638.2.r.y 4 39.i odd 6 1
7098.2.a.bi 2 13.e even 6 1
7098.2.a.bt 2 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} + T_{5} - 4 \)
\( T_{11}^{4} - T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( -4 + T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$13$ \( ( 13 + T + T^{2} )^{2} \)
$17$ \( 1 + 8 T + 65 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$23$ \( 64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 169 + 52 T + 29 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( 256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4} \)
$41$ \( 3481 + 354 T + 95 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( ( 64 - 8 T + T^{2} )^{2} \)
$47$ \( ( 4 + 13 T + T^{2} )^{2} \)
$53$ \( ( -7 + T )^{4} \)
$59$ \( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} \)
$61$ \( 3481 + 354 T + 95 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4} \)
$71$ \( 23104 - 304 T + 156 T^{2} + 2 T^{3} + T^{4} \)
$73$ \( ( 16 + 9 T + T^{2} )^{2} \)
$79$ \( ( 16 + 9 T + T^{2} )^{2} \)
$83$ \( ( -16 + 2 T + T^{2} )^{2} \)
$89$ \( 676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4} \)
$97$ \( 64 - 80 T + 92 T^{2} - 10 T^{3} + T^{4} \)
show more
show less