Properties

Label 546.2.q.g
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: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9} + (\beta_{3} + \beta_{2} - 1) q^{10} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{11} + \beta_1 q^{12} + (\beta_{2} + 3) q^{13} + (2 \beta_{2} - 3) q^{14} + (\beta_1 - 3) q^{15} - \beta_{2} q^{16} + (2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{17} + ( - \beta_{3} - 3 \beta_{2}) q^{18} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{19} + (\beta_1 - 1) q^{20} + (2 \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{22} + (\beta_{3} - \beta_{2} + 3) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{25} + (4 \beta_{2} - 1) q^{26} + ( - 2 \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - \beta_{2} - 2) q^{28} + (\beta_{3} + 1) q^{29} + ( - \beta_{3} - 3 \beta_{2} + \beta_1) q^{30} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} + 1) q^{32} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{33} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{34} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{35} + ( - 3 \beta_{2} - \beta_1 + 3) q^{36} + ( - 6 \beta_{3} - 2 \beta_{2} - 2) q^{37} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{38} + (4 \beta_{3} - 3 \beta_1) q^{39} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{40} + ( - \beta_{3} + 4 \beta_{2} - 9) q^{41} + ( - \beta_{3} + 3 \beta_1) q^{42} + ( - 4 \beta_{2} + 4) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{44} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{45} + (2 \beta_{2} + \beta_1 + 1) q^{46} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{47} - \beta_{3} q^{48} + (3 \beta_{2} - 8) q^{49} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{50} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 3) q^{51} + (3 \beta_{2} - 4) q^{52} + (5 \beta_{3} + 7 \beta_{2} - 5 \beta_1 - 1) q^{53} + ( - 2 \beta_{3} + 3 \beta_{2}) q^{54} + ( - 2 \beta_{2} + 4) q^{55} + ( - 3 \beta_{2} + 1) q^{56} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{57} + (\beta_{2} + \beta_1) q^{58} + ( - 4 \beta_{2} + \beta_1 - 5) q^{59} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{60} + (3 \beta_{2} + 3) q^{61} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{62} + ( - 2 \beta_{3} - 9 \beta_{2} - \beta_1 + 3) q^{63} + q^{64} + (4 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 1) q^{65} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{66} + (6 \beta_{3} + 2 \beta_{2} + 2) q^{67} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{68} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{69} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{70} + ( - 6 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 9) q^{71} + (\beta_{3} - \beta_1 + 3) q^{72} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{73} + ( - 4 \beta_{2} - 6 \beta_1 + 2) q^{74} + (\beta_{3} - 6 \beta_{2} - 2 \beta_1 + 3) q^{75} + (\beta_{3} - 2 \beta_1 + 1) q^{76} + ( - 5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{77} + (3 \beta_{3} + \beta_1) q^{78} + ( - \beta_{3} - \beta_{2} - \beta_1 + 13) q^{79} + ( - \beta_{3} - \beta_{2} + 1) q^{80} + (5 \beta_{3} - 5 \beta_1 + 6) q^{81} + ( - 5 \beta_{2} - \beta_1 - 4) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} + 2 \beta_1) q^{84} + ( - 2 \beta_{2} - 2) q^{85} + 4 q^{86} + ( - 3 \beta_{2} - \beta_1) q^{87} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{88} + ( - 7 \beta_{3} - 4 \beta_{2} + 1) q^{89} + ( - 2 \beta_{3} - \beta_1 + 3) q^{90} + (11 \beta_{2} - 6) q^{91} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{92} + ( - 12 \beta_{2} - 2 \beta_1 + 6) q^{93} + (\beta_{3} + 2 \beta_{2} - 3) q^{94} + (2 \beta_{2} - 2 \beta_1 + 4) q^{95} - \beta_1 q^{96} + ( - 4 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 4) q^{97} + ( - 5 \beta_{2} - 3) q^{98} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - q^{6} + 2 q^{7} - 4 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - q^{6} + 2 q^{7} - 4 q^{8} - 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} + 14 q^{13} - 8 q^{14} - 11 q^{15} - 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} - 3 q^{20} - q^{21} - 3 q^{22} + 9 q^{23} + 2 q^{24} + 6 q^{25} + 4 q^{26} + 16 q^{27} - 10 q^{28} + 3 q^{29} - 4 q^{30} + 4 q^{31} + 2 q^{32} + 15 q^{33} + 6 q^{34} - 9 q^{35} + 5 q^{36} - 6 q^{37} + 2 q^{38} - 7 q^{39} - 27 q^{41} + 4 q^{42} + 8 q^{43} - 6 q^{44} + 11 q^{45} + 9 q^{46} + q^{48} - 26 q^{49} + 3 q^{50} - 18 q^{51} - 10 q^{52} + 8 q^{54} + 12 q^{55} - 2 q^{56} + 16 q^{57} + 3 q^{58} - 27 q^{59} + 7 q^{60} + 18 q^{61} + 2 q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 18 q^{66} + 6 q^{67} + 3 q^{68} - 10 q^{69} - 6 q^{70} - 15 q^{71} + 10 q^{72} - 20 q^{73} - 6 q^{74} - 3 q^{75} + q^{76} - 12 q^{77} - 2 q^{78} + 50 q^{79} + 3 q^{80} + 14 q^{81} - 27 q^{82} + 5 q^{84} - 12 q^{85} + 16 q^{86} - 7 q^{87} - 3 q^{88} + 3 q^{89} + 13 q^{90} - 2 q^{91} - 2 q^{93} - 9 q^{94} + 18 q^{95} - q^{96} + 10 q^{97} - 22 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

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

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

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
1.68614 + 0.396143i
−1.18614 1.26217i
0.500000 0.866025i −0.500000 1.65831i −0.500000 0.866025i 2.52434i −1.68614 0.396143i 0.500000 2.59808i −1.00000 −2.50000 + 1.65831i −2.18614 1.26217i
251.2 0.500000 0.866025i −0.500000 + 1.65831i −0.500000 0.866025i 0.792287i 1.18614 + 1.26217i 0.500000 2.59808i −1.00000 −2.50000 1.65831i 0.686141 + 0.396143i
335.1 0.500000 + 0.866025i −0.500000 1.65831i −0.500000 + 0.866025i 0.792287i 1.18614 1.26217i 0.500000 + 2.59808i −1.00000 −2.50000 + 1.65831i 0.686141 0.396143i
335.2 0.500000 + 0.866025i −0.500000 + 1.65831i −0.500000 + 0.866025i 2.52434i −1.68614 + 0.396143i 0.500000 + 2.59808i −1.00000 −2.50000 1.65831i −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.g yes 4
3.b odd 2 1 546.2.q.e 4
7.b odd 2 1 546.2.q.h yes 4
13.e even 6 1 546.2.q.f yes 4
21.c even 2 1 546.2.q.f yes 4
39.h odd 6 1 546.2.q.h yes 4
91.t odd 6 1 546.2.q.e 4
273.u even 6 1 inner 546.2.q.g yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.e 4 3.b odd 2 1
546.2.q.e 4 91.t odd 6 1
546.2.q.f yes 4 13.e even 6 1
546.2.q.f yes 4 21.c even 2 1
546.2.q.g yes 4 1.a even 1 1 trivial
546.2.q.g yes 4 273.u even 6 1 inner
546.2.q.h yes 4 7.b odd 2 1
546.2.q.h yes 4 39.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 7T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 3T_{11}^{3} + 15T_{11}^{2} + 18T_{11} + 36 \) Copy content Toggle raw display
\( T_{17}^{4} - 3T_{17}^{3} + 15T_{17}^{2} + 18T_{17} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

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