Properties

Label 546.2.i.b
Level $546$
Weight $2$
Character orbit 546.i
Analytic conductor $4.360$
Analytic rank $0$
Dimension $2$
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(79,546)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.i (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.

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 2.59808i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
235.1 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −0.500000 2.59808i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.i.b 2
3.b odd 2 1 1638.2.j.h 2
7.c even 3 1 inner 546.2.i.b 2
7.c even 3 1 3822.2.a.be 1
7.d odd 6 1 3822.2.a.v 1
21.h odd 6 1 1638.2.j.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.b 2 1.a even 1 1 trivial
546.2.i.b 2 7.c even 3 1 inner
1638.2.j.h 2 3.b odd 2 1
1638.2.j.h 2 21.h odd 6 1
3822.2.a.v 1 7.d odd 6 1
3822.2.a.be 1 7.c even 3 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 36 \) 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
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$59$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$83$ \( (T + 17)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$97$ \( (T - 13)^{2} \) Copy content Toggle raw display
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