Properties

Label 546.2.i.f
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 \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + \zeta_{6} q^{12} - q^{13} + ( - 2 \zeta_{6} + 3) q^{14} - q^{15} - \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + ( - \zeta_{6} + 1) q^{18} + q^{20} + ( - \zeta_{6} - 2) q^{21} + 3 q^{22} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + ( - 4 \zeta_{6} + 4) q^{25} - \zeta_{6} q^{26} - q^{27} + (\zeta_{6} + 2) q^{28} + 9 q^{29} - \zeta_{6} q^{30} + (5 \zeta_{6} - 5) q^{31} + ( - \zeta_{6} + 1) q^{32} - 3 \zeta_{6} q^{33} + 2 q^{34} + (2 \zeta_{6} - 3) q^{35} + q^{36} + 8 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{39} + \zeta_{6} q^{40} - 4 q^{41} + ( - 3 \zeta_{6} + 1) q^{42} + 3 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + ( - 6 \zeta_{6} + 6) q^{46} - q^{48} + (3 \zeta_{6} - 8) q^{49} + 4 q^{50} - 2 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + (\zeta_{6} - 1) q^{53} - \zeta_{6} q^{54} - 3 q^{55} + (3 \zeta_{6} - 1) q^{56} + 9 \zeta_{6} q^{58} + ( - 7 \zeta_{6} + 7) q^{59} + ( - \zeta_{6} + 1) q^{60} + 4 \zeta_{6} q^{61} - 5 q^{62} + (2 \zeta_{6} - 3) q^{63} + q^{64} + \zeta_{6} q^{65} + ( - 3 \zeta_{6} + 3) q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + 2 \zeta_{6} q^{68} - 6 q^{69} + ( - \zeta_{6} - 2) q^{70} + 6 q^{71} + \zeta_{6} q^{72} + (6 \zeta_{6} - 6) q^{73} + (8 \zeta_{6} - 8) q^{74} - 4 \zeta_{6} q^{75} + ( - 3 \zeta_{6} - 6) q^{77} - q^{78} + 13 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (\zeta_{6} - 1) q^{81} - 4 \zeta_{6} q^{82} - 3 q^{83} + ( - 2 \zeta_{6} + 3) q^{84} - 2 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} + (3 \zeta_{6} - 3) q^{88} + 8 \zeta_{6} q^{89} - q^{90} + (3 \zeta_{6} - 1) q^{91} + 6 q^{92} + 5 \zeta_{6} q^{93} - \zeta_{6} q^{96} - 15 q^{97} + ( - 5 \zeta_{6} - 3) q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} - q^{4} - q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} - q^{4} - q^{5} + 2 q^{6} - q^{7} - 2 q^{8} - q^{9} + q^{10} + 3 q^{11} + q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{15} - q^{16} + 2 q^{17} + q^{18} + 2 q^{20} - 5 q^{21} + 6 q^{22} - 6 q^{23} - q^{24} + 4 q^{25} - q^{26} - 2 q^{27} + 5 q^{28} + 18 q^{29} - q^{30} - 5 q^{31} + q^{32} - 3 q^{33} + 4 q^{34} - 4 q^{35} + 2 q^{36} + 8 q^{37} - q^{39} + q^{40} - 8 q^{41} - q^{42} + 3 q^{44} - q^{45} + 6 q^{46} - 2 q^{48} - 13 q^{49} + 8 q^{50} - 2 q^{51} + q^{52} - q^{53} - q^{54} - 6 q^{55} + q^{56} + 9 q^{58} + 7 q^{59} + q^{60} + 4 q^{61} - 10 q^{62} - 4 q^{63} + 2 q^{64} + q^{65} + 3 q^{66} + 4 q^{67} + 2 q^{68} - 12 q^{69} - 5 q^{70} + 12 q^{71} + q^{72} - 6 q^{73} - 8 q^{74} - 4 q^{75} - 15 q^{77} - 2 q^{78} + 13 q^{79} - q^{80} - q^{81} - 4 q^{82} - 6 q^{83} + 4 q^{84} - 4 q^{85} + 9 q^{87} - 3 q^{88} + 8 q^{89} - 2 q^{90} + q^{91} + 12 q^{92} + 5 q^{93} - q^{96} - 30 q^{97} - 11 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 4 \) 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
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$97$ \( (T + 15)^{2} \) Copy content Toggle raw display
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