Properties

Label 546.2.q.a
Level $546$
Weight $2$
Character orbit 546.q
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
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.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) 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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i 3.46410i 1.50000 0.866025i 0.500000 2.59808i 1.00000 1.50000 + 2.59808i 3.00000 + 1.73205i
335.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i 3.46410i 1.50000 + 0.866025i 0.500000 + 2.59808i 1.00000 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
273.u 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.q.a 2
3.b odd 2 1 546.2.q.c yes 2
7.b odd 2 1 546.2.q.b yes 2
13.e even 6 1 546.2.q.d yes 2
21.c even 2 1 546.2.q.d yes 2
39.h odd 6 1 546.2.q.b yes 2
91.t odd 6 1 546.2.q.c yes 2
273.u even 6 1 inner 546.2.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.q.a 2 1.a even 1 1 trivial
546.2.q.a 2 273.u even 6 1 inner
546.2.q.b yes 2 7.b odd 2 1
546.2.q.b yes 2 39.h odd 6 1
546.2.q.c yes 2 3.b odd 2 1
546.2.q.c yes 2 91.t odd 6 1
546.2.q.d yes 2 13.e even 6 1
546.2.q.d yes 2 21.c even 2 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
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 75 \) Copy content Toggle raw display
$53$ \( T^{2} + 75 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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