Properties

Label 546.2.l.j
Level $546$
Weight $2$
Character orbit 546.l
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 2 \nu^{2} + 8 \nu - 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 13\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 19\)\()/3\)

Character values

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

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

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
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
−0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −3.27492 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.63746 + 2.83616i
211.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 4.27492 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −2.13746 3.70219i
295.1 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −3.27492 −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 1.63746 2.83616i
295.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 4.27492 −0.500000 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −2.13746 + 3.70219i
\(n\): 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.j 4
3.b odd 2 1 1638.2.r.x 4
13.c even 3 1 inner 546.2.l.j 4
13.c even 3 1 7098.2.a.ca 2
13.e even 6 1 7098.2.a.bm 2
39.i odd 6 1 1638.2.r.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.j 4 1.a even 1 1 trivial
546.2.l.j 4 13.c even 3 1 inner
1638.2.r.x 4 3.b odd 2 1
1638.2.r.x 4 39.i odd 6 1
7098.2.a.bm 2 13.e even 6 1
7098.2.a.ca 2 13.c even 3 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} - 14 \)
\( T_{11}^{2} - 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( -14 - T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( ( 16 - 4 T + T^{2} )^{2} \)
$13$ \( ( 13 - 7 T + T^{2} )^{2} \)
$17$ \( ( 9 + 3 T + T^{2} )^{2} \)
$19$ \( 3136 + 112 T + 60 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4} \)
$29$ \( 36 + 54 T + 75 T^{2} + 9 T^{3} + T^{4} \)
$31$ \( ( -8 - 5 T + T^{2} )^{2} \)
$37$ \( 4 + 14 T + 51 T^{2} - 7 T^{3} + T^{4} \)
$41$ \( 36 + 54 T + 75 T^{2} + 9 T^{3} + T^{4} \)
$43$ \( 16384 + 128 T + 129 T^{2} - T^{3} + T^{4} \)
$47$ \( ( -56 - 2 T + T^{2} )^{2} \)
$53$ \( ( -41 - 8 T + T^{2} )^{2} \)
$59$ \( 16384 + 128 T + 129 T^{2} - T^{3} + T^{4} \)
$61$ \( ( 81 - 9 T + T^{2} )^{2} \)
$67$ \( 784 - 364 T + 141 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( 144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4} \)
$73$ \( ( 76 - 19 T + T^{2} )^{2} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( ( -128 + T + T^{2} )^{2} \)
$89$ \( 36 + 54 T + 75 T^{2} + 9 T^{3} + T^{4} \)
$97$ \( 64 - 112 T + 204 T^{2} + 14 T^{3} + T^{4} \)
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