Properties

Label 546.2.bn.d
Level $546$
Weight $2$
Character orbit 546.bn
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.bn (of order \(6\), 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 1.50000 0.866025i −0.500000 0.866025i 1.50000 0.866025i 1.73205i 0.500000 2.59808i −1.00000 1.50000 2.59808i 1.73205i
173.1 0.500000 + 0.866025i 1.50000 + 0.866025i −0.500000 + 0.866025i 1.50000 + 0.866025i 1.73205i 0.500000 + 2.59808i −1.00000 1.50000 + 2.59808i 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
273.y even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bn.d yes 2
3.b odd 2 1 546.2.bn.a yes 2
7.d odd 6 1 546.2.bi.b 2
13.e even 6 1 546.2.bi.d yes 2
21.g even 6 1 546.2.bi.d yes 2
39.h odd 6 1 546.2.bi.b 2
91.p odd 6 1 546.2.bn.a yes 2
273.y even 6 1 inner 546.2.bn.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bi.b 2 7.d odd 6 1
546.2.bi.b 2 39.h odd 6 1
546.2.bi.d yes 2 13.e even 6 1
546.2.bi.d yes 2 21.g even 6 1
546.2.bn.a yes 2 3.b odd 2 1
546.2.bn.a yes 2 91.p odd 6 1
546.2.bn.d yes 2 1.a even 1 1 trivial
546.2.bn.d yes 2 273.y even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( 3 - 3 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( 12 - 6 T + T^{2} \)
$29$ \( 75 - 15 T + T^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( T^{2} \)
$41$ \( 27 - 9 T + T^{2} \)
$43$ \( 1 + T + T^{2} \)
$47$ \( 3 + 3 T + T^{2} \)
$53$ \( 3 + 3 T + T^{2} \)
$59$ \( 108 - 18 T + T^{2} \)
$61$ \( 3 + T^{2} \)
$67$ \( 3 + T^{2} \)
$71$ \( 81 + 9 T + T^{2} \)
$73$ \( 169 + 13 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 48 - 12 T + T^{2} \)
$97$ \( 361 - 19 T + T^{2} \)
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