Properties

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

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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.

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

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.f 2
3.b odd 2 1 1638.2.r.j 2
13.c even 3 1 inner 546.2.l.f 2
13.c even 3 1 7098.2.a.c 1
13.e even 6 1 7098.2.a.u 1
39.i odd 6 1 1638.2.r.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.f 2 1.a even 1 1 trivial
546.2.l.f 2 13.c even 3 1 inner
1638.2.r.j 2 3.b odd 2 1
1638.2.r.j 2 39.i odd 6 1
7098.2.a.c 1 13.c even 3 1
7098.2.a.u 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5} + 2 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 5 T + T^{2} \)
$17$ \( 1 + T + T^{2} \)
$19$ \( 36 + 6 T + T^{2} \)
$23$ \( 9 + 3 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( 100 - 10 T + T^{2} \)
$43$ \( 9 + 3 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( ( -7 + T )^{2} \)
$59$ \( 49 - 7 T + T^{2} \)
$61$ \( 121 + 11 T + T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( 9 - 3 T + T^{2} \)
$73$ \( ( -12 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( ( 15 + T )^{2} \)
$89$ \( 121 - 11 T + T^{2} \)
$97$ \( 144 + 12 T + T^{2} \)
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