Properties

Label 546.2.q.f
Level $546$
Weight $2$
Character orbit 546.q
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.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.500000 + 0.866025i −1.18614 + 1.26217i −0.500000 0.866025i 0.792287i −0.500000 1.65831i 2.50000 + 0.866025i 1.00000 −0.186141 2.99422i −0.686141 0.396143i
251.2 −0.500000 + 0.866025i 1.68614 0.396143i −0.500000 0.866025i 2.52434i −0.500000 + 1.65831i 2.50000 + 0.866025i 1.00000 2.68614 1.33591i 2.18614 + 1.26217i
335.1 −0.500000 0.866025i −1.18614 1.26217i −0.500000 + 0.866025i 0.792287i −0.500000 + 1.65831i 2.50000 0.866025i 1.00000 −0.186141 + 2.99422i −0.686141 + 0.396143i
335.2 −0.500000 0.866025i 1.68614 + 0.396143i −0.500000 + 0.866025i 2.52434i −0.500000 1.65831i 2.50000 0.866025i 1.00000 2.68614 + 1.33591i 2.18614 1.26217i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
273.u 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.q.f yes 4
3.b odd 2 1 546.2.q.h yes 4
7.b odd 2 1 546.2.q.e 4
13.e even 6 1 546.2.q.g yes 4
21.c even 2 1 546.2.q.g yes 4
39.h odd 6 1 546.2.q.e 4
91.t odd 6 1 546.2.q.h yes 4
273.u even 6 1 inner 546.2.q.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.e 4 7.b odd 2 1
546.2.q.e 4 39.h odd 6 1
546.2.q.f yes 4 1.a even 1 1 trivial
546.2.q.f yes 4 273.u even 6 1 inner
546.2.q.g yes 4 13.e even 6 1
546.2.q.g yes 4 21.c even 2 1
546.2.q.h yes 4 3.b odd 2 1
546.2.q.h yes 4 91.t odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 7 T_{5}^{2} + 4 \)
\( T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36 \)
\( T_{17}^{4} - 3 T_{17}^{3} + 15 T_{17}^{2} + 18 T_{17} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( 9 - 3 T - 2 T^{2} - T^{3} + T^{4} \)
$5$ \( 4 + 7 T^{2} + T^{4} \)
$7$ \( ( 7 - 5 T + T^{2} )^{2} \)
$11$ \( 36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4} \)
$13$ \( ( 13 + 7 T + T^{2} )^{2} \)
$17$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$19$ \( 64 - 8 T + 9 T^{2} + T^{3} + T^{4} \)
$23$ \( 16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$29$ \( 4 - 6 T + T^{2} + 3 T^{3} + T^{4} \)
$31$ \( ( -32 + 2 T + T^{2} )^{2} \)
$37$ \( 9216 - 576 T - 84 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( 3364 + 1566 T + 301 T^{2} + 27 T^{3} + T^{4} \)
$43$ \( ( 16 - 4 T + T^{2} )^{2} \)
$47$ \( 16 + 19 T^{2} + T^{4} \)
$53$ \( 1024 + 211 T^{2} + T^{4} \)
$59$ \( 3364 + 1566 T + 301 T^{2} + 27 T^{3} + T^{4} \)
$61$ \( ( 27 + 9 T + T^{2} )^{2} \)
$67$ \( 9216 + 576 T - 84 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( 324 + 270 T + 243 T^{2} - 15 T^{3} + T^{4} \)
$73$ \( ( -8 - 10 T + T^{2} )^{2} \)
$79$ \( ( 148 - 25 T + T^{2} )^{2} \)
$83$ \( 64 + 28 T^{2} + T^{4} \)
$89$ \( 17956 + 402 T - 131 T^{2} - 3 T^{3} + T^{4} \)
$97$ \( 64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4} \)
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