Properties

Label 546.2.s.c
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{2} q^{4} + ( - 4 \zeta_{12}^{2} + 2) q^{5} + (\zeta_{12}^{3} - \zeta_{12}) q^{6} + (\zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + (\zeta_{12}^{2} - 1) q^{3} + \zeta_{12}^{2} q^{4} + ( - 4 \zeta_{12}^{2} + 2) 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} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{10} + 4 \zeta_{12} q^{11} - q^{12} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}) q^{13} - q^{14} + (2 \zeta_{12}^{2} + 2) q^{15} + (\zeta_{12}^{2} - 1) q^{16} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{17} - \zeta_{12}^{3} q^{18} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{19} + ( - 2 \zeta_{12}^{2} + 4) q^{20} - \zeta_{12}^{3} q^{21} + 4 \zeta_{12}^{2} q^{22} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + \zeta_{12} + 4) q^{23} - \zeta_{12} q^{24} - 7 q^{25} + ( - \zeta_{12}^{2} + 4) q^{26} + q^{27} - \zeta_{12} q^{28} + (8 \zeta_{12}^{2} - 8) q^{29} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{30} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{33} + (3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{34} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + ( - \zeta_{12}^{2} + 1) q^{36} + ( - 4 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{37} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} + 4) q^{38} + (3 \zeta_{12}^{3} + \zeta_{12}) q^{39} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{40} + ( - 4 \zeta_{12}^{2} - 4) q^{41} + ( - \zeta_{12}^{2} + 1) q^{42} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12}) q^{43} + 4 \zeta_{12}^{3} q^{44} + (2 \zeta_{12}^{2} - 4) q^{45} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{46} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{47} - \zeta_{12}^{2} q^{48} + ( - \zeta_{12}^{2} + 1) q^{49} - 7 \zeta_{12} q^{50} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 3) q^{51} + ( - \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 10) q^{53} + \zeta_{12} q^{54} + ( - 16 \zeta_{12}^{3} + 8 \zeta_{12}) q^{55} - \zeta_{12}^{2} q^{56} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{57} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{58} + (8 \zeta_{12}^{3} - \zeta_{12}^{2} - 8 \zeta_{12} + 2) q^{59} + (4 \zeta_{12}^{2} - 2) q^{60} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{61} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{62} + \zeta_{12} q^{63} - q^{64} + ( - 4 \zeta_{12}^{3} - 10 \zeta_{12}) q^{65} - 4 q^{66} + (3 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{67} + (4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{68} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12}) q^{69} + (4 \zeta_{12}^{2} - 2) q^{70} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{71} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{72} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{73} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12}) q^{74} + ( - 7 \zeta_{12}^{2} + 7) q^{75} + (2 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{76} - 4 q^{77} + (4 \zeta_{12}^{2} - 3) q^{78} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 6) q^{79} + (2 \zeta_{12}^{2} + 2) q^{80} + (\zeta_{12}^{2} - 1) q^{81} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{82} + (2 \zeta_{12}^{2} - 1) q^{83} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{84} + ( - 12 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{85} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{86} - 8 \zeta_{12}^{2} q^{87} + (4 \zeta_{12}^{2} - 4) q^{88} + ( - \zeta_{12}^{2} - 10 \zeta_{12} - 1) q^{89} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{90} + (4 \zeta_{12}^{2} - 3) q^{91} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 4) q^{92} + (2 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{93} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{94} + ( - 8 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 8 \zeta_{12}) q^{95} - \zeta_{12}^{3} q^{96} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 12) q^{97} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{98} - 4 \zeta_{12}^{3} 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 \( 4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} - 4 q^{12} - 4 q^{14} + 12 q^{15} - 2 q^{16} + 6 q^{17} + 12 q^{19} + 12 q^{20} + 8 q^{22} + 8 q^{23} - 28 q^{25} + 14 q^{26} + 4 q^{27} - 16 q^{29} + 2 q^{36} - 24 q^{37} + 16 q^{38} - 24 q^{41} + 2 q^{42} + 2 q^{43} - 12 q^{45} + 6 q^{46} - 2 q^{48} + 2 q^{49} - 12 q^{51} - 40 q^{53} - 2 q^{56} + 6 q^{59} + 6 q^{62} - 4 q^{64} - 16 q^{66} + 18 q^{67} - 6 q^{68} + 8 q^{69} + 24 q^{71} + 4 q^{74} + 14 q^{75} + 12 q^{76} - 16 q^{77} - 4 q^{78} - 24 q^{79} + 12 q^{80} - 2 q^{81} + 36 q^{85} - 16 q^{87} - 8 q^{88} - 6 q^{89} - 4 q^{91} + 16 q^{92} + 12 q^{93} - 8 q^{94} - 24 q^{95} + 36 q^{97}+O(q^{100}) \) 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 3.46410i 0.866025 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i −1.73205 + 3.00000i
43.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 3.46410i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i 1.73205 3.00000i
127.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.46410i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i −1.73205 3.00000i
127.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 3.46410i −0.866025 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i 1.73205 + 3.00000i
\(n\): 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.c 4
3.b odd 2 1 1638.2.bj.e 4
13.e even 6 1 inner 546.2.s.c 4
13.f odd 12 1 7098.2.a.bn 2
13.f odd 12 1 7098.2.a.bz 2
39.h odd 6 1 1638.2.bj.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 1.a even 1 1 trivial
546.2.s.c 4 13.e even 6 1 inner
1638.2.bj.e 4 3.b odd 2 1
1638.2.bj.e 4 39.h odd 6 1
7098.2.a.bn 2 13.f odd 12 1
7098.2.a.bz 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{4} - 16T_{11}^{2} + 256 \) 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
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 236 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 15 T^{2} + 22 T + 121 \) Copy content Toggle raw display
$47$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 20 T + 97)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 49 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$61$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 131 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + 215 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$73$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} - 85 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
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