Properties

Label 546.2.j.a
Level $546$
Weight $2$
Character orbit 546.j
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.j (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: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 0.500000 0.866025i 1.00000 0 0.500000 0.866025i 0.500000 + 2.59808i 1.00000 −0.500000 0.866025i 0
529.1 1.00000 0.500000 + 0.866025i 1.00000 0 0.500000 + 0.866025i 0.500000 2.59808i 1.00000 −0.500000 + 0.866025i 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.h 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.j.a 2
3.b odd 2 1 1638.2.m.a 2
7.c even 3 1 546.2.k.a yes 2
13.c even 3 1 546.2.k.a yes 2
21.h odd 6 1 1638.2.p.d 2
39.i odd 6 1 1638.2.p.d 2
91.h even 3 1 inner 546.2.j.a 2
273.s odd 6 1 1638.2.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.a 2 1.a even 1 1 trivial
546.2.j.a 2 91.h even 3 1 inner
546.2.k.a yes 2 7.c even 3 1
546.2.k.a yes 2 13.c even 3 1
1638.2.m.a 2 3.b odd 2 1
1638.2.m.a 2 273.s odd 6 1
1638.2.p.d 2 21.h odd 6 1
1638.2.p.d 2 39.i 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

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( 13 - 7 T + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( 1 + T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 11 + T )^{2} \)
$41$ \( 4 - 2 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 144 + 12 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( 49 - 7 T + T^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( ( 14 + T )^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 81 + 9 T + T^{2} \)
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