Properties

Label 546.2.bq.b
Level $546$
Weight $2$
Character orbit 546.bq
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bq (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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −0.866025 1.50000i 0.500000 0.866025i −3.46410 1.50000 + 0.866025i −2.50000 + 0.866025i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
419.2 0.866025 0.500000i 0.866025 + 1.50000i 0.500000 0.866025i 3.46410 1.50000 + 0.866025i −2.50000 + 0.866025i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
503.1 −0.866025 0.500000i −0.866025 + 1.50000i 0.500000 + 0.866025i −3.46410 1.50000 0.866025i −2.50000 0.866025i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
503.2 0.866025 + 0.500000i 0.866025 1.50000i 0.500000 + 0.866025i 3.46410 1.50000 0.866025i −2.50000 0.866025i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.n odd 6 1 inner
273.bn 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.bq.b yes 4
3.b odd 2 1 inner 546.2.bq.b yes 4
7.b odd 2 1 546.2.bq.a 4
13.c even 3 1 546.2.bq.a 4
21.c even 2 1 546.2.bq.a 4
39.i odd 6 1 546.2.bq.a 4
91.n odd 6 1 inner 546.2.bq.b yes 4
273.bn even 6 1 inner 546.2.bq.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bq.a 4 7.b odd 2 1
546.2.bq.a 4 13.c even 3 1
546.2.bq.a 4 21.c even 2 1
546.2.bq.a 4 39.i odd 6 1
546.2.bq.b yes 4 1.a even 1 1 trivial
546.2.bq.b yes 4 3.b odd 2 1 inner
546.2.bq.b yes 4 91.n odd 6 1 inner
546.2.bq.b yes 4 273.bn even 6 1 inner

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{61}^{2} - 21 T_{61} + 147 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( ( -12 + T^{2} )^{2} \)
$7$ \( ( 7 + 5 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 13 - 7 T + T^{2} )^{2} \)
$17$ \( 9 + 3 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( 1296 - 36 T^{2} + T^{4} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{2} \)
$41$ \( 2304 + 48 T^{2} + T^{4} \)
$43$ \( ( 1 + T + T^{2} )^{2} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( 81 + T^{2} )^{2} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( ( 147 - 21 T + T^{2} )^{2} \)
$67$ \( ( 49 - 7 T + T^{2} )^{2} \)
$71$ \( 50625 - 225 T^{2} + T^{4} \)
$73$ \( ( 192 + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( ( -147 + T^{2} )^{2} \)
$89$ \( 59049 + 243 T^{2} + T^{4} \)
$97$ \( ( 300 + 30 T + T^{2} )^{2} \)
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