Properties

Label 546.2.l.d
Level $546$
Weight $2$
Character orbit 546.l
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -2 q^{5} + \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -2 q^{5} + \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + q^{12} + ( -3 - \zeta_{6} ) q^{13} + q^{14} + ( 2 - 2 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -7 \zeta_{6} q^{17} - q^{18} -2 \zeta_{6} q^{19} + 2 \zeta_{6} q^{20} - q^{21} -4 \zeta_{6} q^{22} + ( 1 - \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} - q^{25} + ( -4 + 3 \zeta_{6} ) q^{26} + q^{27} + ( 1 - \zeta_{6} ) q^{28} + ( 2 - 2 \zeta_{6} ) q^{29} -2 \zeta_{6} q^{30} -9 q^{31} + \zeta_{6} q^{32} + 4 \zeta_{6} q^{33} -7 q^{34} -2 \zeta_{6} q^{35} + ( -1 + \zeta_{6} ) q^{36} + ( 2 - 2 \zeta_{6} ) q^{37} -2 q^{38} + ( 4 - 3 \zeta_{6} ) q^{39} + 2 q^{40} + ( -2 + 2 \zeta_{6} ) q^{41} + ( -1 + \zeta_{6} ) q^{42} + 5 \zeta_{6} q^{43} -4 q^{44} + 2 \zeta_{6} q^{45} -\zeta_{6} q^{46} + 6 q^{47} -\zeta_{6} q^{48} + ( -1 + \zeta_{6} ) q^{49} + ( -1 + \zeta_{6} ) q^{50} + 7 q^{51} + ( -1 + 4 \zeta_{6} ) q^{52} + 3 q^{53} + ( 1 - \zeta_{6} ) q^{54} + ( -8 + 8 \zeta_{6} ) q^{55} -\zeta_{6} q^{56} + 2 q^{57} -2 \zeta_{6} q^{58} -15 \zeta_{6} q^{59} -2 q^{60} + 7 \zeta_{6} q^{61} + ( -9 + 9 \zeta_{6} ) q^{62} + ( 1 - \zeta_{6} ) q^{63} + q^{64} + ( 6 + 2 \zeta_{6} ) q^{65} + 4 q^{66} + ( 5 - 5 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{68} + \zeta_{6} q^{69} -2 q^{70} -\zeta_{6} q^{71} + \zeta_{6} q^{72} + 12 q^{73} -2 \zeta_{6} q^{74} + ( 1 - \zeta_{6} ) q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} + 4 q^{77} + ( 1 - 4 \zeta_{6} ) q^{78} -4 q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} - q^{83} + \zeta_{6} q^{84} + 14 \zeta_{6} q^{85} + 5 q^{86} + 2 \zeta_{6} q^{87} + ( -4 + 4 \zeta_{6} ) q^{88} + ( -3 + 3 \zeta_{6} ) q^{89} + 2 q^{90} + ( 1 - 4 \zeta_{6} ) q^{91} - q^{92} + ( 9 - 9 \zeta_{6} ) q^{93} + ( 6 - 6 \zeta_{6} ) q^{94} + 4 \zeta_{6} q^{95} - q^{96} + 16 \zeta_{6} q^{97} + \zeta_{6} q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} - q^{4} - 4q^{5} + q^{6} + q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} - q^{4} - 4q^{5} + q^{6} + q^{7} - 2q^{8} - q^{9} - 2q^{10} + 4q^{11} + 2q^{12} - 7q^{13} + 2q^{14} + 2q^{15} - q^{16} - 7q^{17} - 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} - 4q^{22} + q^{23} + q^{24} - 2q^{25} - 5q^{26} + 2q^{27} + q^{28} + 2q^{29} - 2q^{30} - 18q^{31} + q^{32} + 4q^{33} - 14q^{34} - 2q^{35} - q^{36} + 2q^{37} - 4q^{38} + 5q^{39} + 4q^{40} - 2q^{41} - q^{42} + 5q^{43} - 8q^{44} + 2q^{45} - q^{46} + 12q^{47} - q^{48} - q^{49} - q^{50} + 14q^{51} + 2q^{52} + 6q^{53} + q^{54} - 8q^{55} - q^{56} + 4q^{57} - 2q^{58} - 15q^{59} - 4q^{60} + 7q^{61} - 9q^{62} + q^{63} + 2q^{64} + 14q^{65} + 8q^{66} + 5q^{67} - 7q^{68} + q^{69} - 4q^{70} - q^{71} + q^{72} + 24q^{73} - 2q^{74} + q^{75} - 2q^{76} + 8q^{77} - 2q^{78} - 8q^{79} + 2q^{80} - q^{81} + 2q^{82} - 2q^{83} + q^{84} + 14q^{85} + 10q^{86} + 2q^{87} - 4q^{88} - 3q^{89} + 4q^{90} - 2q^{91} - 2q^{92} + 9q^{93} + 6q^{94} + 4q^{95} - 2q^{96} + 16q^{97} + q^{98} - 8q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 0.866025i −1.00000 1.73205i
295.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −2.00000 0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 0.866025i −1.00000 + 1.73205i
\(n\): 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.d 2
3.b odd 2 1 1638.2.r.k 2
13.c even 3 1 inner 546.2.l.d 2
13.c even 3 1 7098.2.a.h 1
13.e even 6 1 7098.2.a.bf 1
39.i odd 6 1 1638.2.r.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.d 2 1.a even 1 1 trivial
546.2.l.d 2 13.c even 3 1 inner
1638.2.r.k 2 3.b odd 2 1
1638.2.r.k 2 39.i odd 6 1
7098.2.a.h 1 13.c even 3 1
7098.2.a.bf 1 13.e even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( 13 + 7 T + T^{2} \)
$17$ \( 49 + 7 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( 4 - 2 T + T^{2} \)
$31$ \( ( 9 + T )^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( 4 + 2 T + T^{2} \)
$43$ \( 25 - 5 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( ( -3 + T )^{2} \)
$59$ \( 225 + 15 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 25 - 5 T + T^{2} \)
$71$ \( 1 + T + T^{2} \)
$73$ \( ( -12 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( ( 1 + T )^{2} \)
$89$ \( 9 + 3 T + T^{2} \)
$97$ \( 256 - 16 T + T^{2} \)
show more
show less