Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(211,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.211");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −2.00000 0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 + 0.866025i −1.00000 1.73205i
295.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −2.00000 0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 0.866025i −1.00000 + 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.l.c 2
3.b odd 2 1 1638.2.r.l 2
13.c even 3 1 inner 546.2.l.c 2
13.c even 3 1 7098.2.a.i 1
13.e even 6 1 7098.2.a.bd 1
39.i odd 6 1 1638.2.r.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.c 2 1.a even 1 1 trivial
546.2.l.c 2 13.c even 3 1 inner
1638.2.r.l 2 3.b odd 2 1
1638.2.r.l 2 39.i odd 6 1
7098.2.a.i 1 13.c even 3 1
7098.2.a.bd 1 13.e even 6 1

Hecke kernels

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

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) 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
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$47$ \( (T + 1)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$73$ \( (T - 12)^{2} \) Copy content Toggle raw display
$79$ \( (T - 3)^{2} \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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