Properties

Label 546.2.s.b
Level $546$
Weight $2$
Character orbit 546.s
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.s (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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.73205i 0.866025 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i −1.36603 + 2.36603i
43.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.732051i −0.866025 + 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i 0.366025 0.633975i
127.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 2.73205i 0.866025 + 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i −1.36603 2.36603i
127.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.732051i −0.866025 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i 0.366025 + 0.633975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.s.b 4
3.b odd 2 1 1638.2.bj.a 4
13.e even 6 1 inner 546.2.s.b 4
13.f odd 12 1 7098.2.a.bo 2
13.f odd 12 1 7098.2.a.by 2
39.h odd 6 1 1638.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.b 4 1.a even 1 1 trivial
546.2.s.b 4 13.e even 6 1 inner
1638.2.bj.a 4 3.b odd 2 1
1638.2.bj.a 4 39.h odd 6 1
7098.2.a.bo 2 13.f odd 12 1
7098.2.a.by 2 13.f odd 12 1

Hecke kernels

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

\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{11}^{2} - 9 T_{11} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( 4 + 8 T^{2} + T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( ( 27 - 9 T + T^{2} )^{2} \)
$13$ \( 169 + 23 T^{2} + T^{4} \)
$17$ \( 169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 1369 - 518 T + 159 T^{2} - 14 T^{3} + T^{4} \)
$31$ \( 484 + 56 T^{2} + T^{4} \)
$37$ \( 484 + 132 T - 10 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( 169 - 156 T + 35 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 2116 + 644 T + 150 T^{2} + 14 T^{3} + T^{4} \)
$47$ \( 1521 + 114 T^{2} + T^{4} \)
$53$ \( ( -39 + 6 T + T^{2} )^{2} \)
$59$ \( 1936 - 1056 T + 236 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( 1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 10000 - 100 T^{2} + T^{4} \)
$71$ \( 6084 - 468 T - 66 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( 64 + 32 T^{2} + T^{4} \)
$79$ \( ( -9 + T )^{4} \)
$83$ \( 13924 + 248 T^{2} + T^{4} \)
$89$ \( 33489 + 8784 T + 951 T^{2} + 48 T^{3} + T^{4} \)
$97$ \( 2916 - 972 T + 54 T^{2} + 18 T^{3} + T^{4} \)
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