Properties

Label 546.2.s.a
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 0.732051i 0.866025 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i 0.366025 0.633975i
43.2 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
127.1 −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
127.2 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
\(n\): 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.a 4
3.b odd 2 1 1638.2.bj.b 4
13.e even 6 1 inner 546.2.s.a 4
13.f odd 12 1 7098.2.a.bp 2
13.f odd 12 1 7098.2.a.bx 2
39.h odd 6 1 1638.2.bj.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.a 4 1.a even 1 1 trivial
546.2.s.a 4 13.e even 6 1 inner
1638.2.bj.b 4 3.b odd 2 1
1638.2.bj.b 4 39.h odd 6 1
7098.2.a.bp 2 13.f odd 12 1
7098.2.a.bx 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 15 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$31$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484 \) Copy content Toggle raw display
$41$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$47$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$53$ \( (T - 7)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$67$ \( T^{4} - 24 T^{3} + 236 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T^{2} - 18 T + 33)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 11 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$97$ \( T^{4} - 30 T^{3} + 326 T^{2} + \cdots + 676 \) Copy content Toggle raw display
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