Properties

Label 546.2.k.a
Level $546$
Weight $2$
Character orbit 546.k
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.k (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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −2.50000 0.866025i 1.00000 1.00000 0
445.1 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 0 0.500000 0.866025i −2.50000 + 0.866025i 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.k.a yes 2
3.b odd 2 1 1638.2.p.d 2
7.c even 3 1 546.2.j.a 2
13.c even 3 1 546.2.j.a 2
21.h odd 6 1 1638.2.m.a 2
39.i odd 6 1 1638.2.m.a 2
91.g even 3 1 inner 546.2.k.a yes 2
273.bm odd 6 1 1638.2.p.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.a 2 7.c even 3 1
546.2.j.a 2 13.c even 3 1
546.2.k.a yes 2 1.a even 1 1 trivial
546.2.k.a yes 2 91.g even 3 1 inner
1638.2.m.a 2 21.h odd 6 1
1638.2.m.a 2 39.i odd 6 1
1638.2.p.d 2 3.b odd 2 1
1638.2.p.d 2 273.bm odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 13 - 7 T + T^{2} \)
$17$ \( 16 + 4 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 121 - 11 T + T^{2} \)
$41$ \( 4 - 2 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( 16 + 4 T + T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( ( -12 + T )^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( 49 - 7 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( ( 14 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 81 + 9 T + T^{2} \)
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