Properties

Label 546.2.l.l
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: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
1.28078 2.21837i
−0.780776 + 1.35234i
1.28078 + 2.21837i
−0.780776 1.35234i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −2.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.28078 + 2.21837i
211.2 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.56155 0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.780776 1.35234i
295.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i −2.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 1.28078 2.21837i
295.2 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.56155 0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −0.780776 + 1.35234i
\(n\): 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.l 4
3.b odd 2 1 1638.2.r.y 4
13.c even 3 1 inner 546.2.l.l 4
13.c even 3 1 7098.2.a.bt 2
13.e even 6 1 7098.2.a.bi 2
39.i odd 6 1 1638.2.r.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.l 4 1.a even 1 1 trivial
546.2.l.l 4 13.c even 3 1 inner
1638.2.r.y 4 3.b odd 2 1
1638.2.r.y 4 39.i odd 6 1
7098.2.a.bi 2 13.e even 6 1
7098.2.a.bt 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} + T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} + 5T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + 5 T^{2} + 4 T + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 65 T^{2} + 8 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + 5 T^{2} + 4 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 29 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 95 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 13 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T - 7)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 2 T^{3} + 20 T^{2} - 32 T + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 95 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + 92 T^{2} + 80 T + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + 156 T^{2} + \cdots + 23104 \) Copy content Toggle raw display
$73$ \( (T^{2} + 9 T + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$97$ \( T^{4} - 10 T^{3} + 92 T^{2} - 80 T + 64 \) Copy content Toggle raw display
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