Properties

Label 546.2.bn.a
Level $546$
Weight $2$
Character orbit 546.bn
Analytic conductor $4.360$
Analytic rank $1$
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(101,546)]
 
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([3, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bn (of order \(6\), 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: \(1\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i −1.50000 0.866025i −0.500000 0.866025i −1.50000 + 0.866025i 1.50000 0.866025i 0.500000 2.59808i 1.00000 1.50000 + 2.59808i 1.73205i
173.1 −0.500000 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i −1.50000 0.866025i 1.50000 + 0.866025i 0.500000 + 2.59808i 1.00000 1.50000 2.59808i 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.y 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.bn.a yes 2
3.b odd 2 1 546.2.bn.d yes 2
7.d odd 6 1 546.2.bi.d yes 2
13.e even 6 1 546.2.bi.b 2
21.g even 6 1 546.2.bi.b 2
39.h odd 6 1 546.2.bi.d yes 2
91.p odd 6 1 546.2.bn.d yes 2
273.y even 6 1 inner 546.2.bn.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bi.b 2 13.e even 6 1
546.2.bi.b 2 21.g even 6 1
546.2.bi.d yes 2 7.d odd 6 1
546.2.bi.d yes 2 39.h odd 6 1
546.2.bn.a yes 2 1.a even 1 1 trivial
546.2.bn.a yes 2 273.y even 6 1 inner
546.2.bn.d yes 2 3.b odd 2 1
546.2.bn.d yes 2 91.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). 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
$5$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$61$ \( T^{2} + 3 \) Copy content Toggle raw display
$67$ \( T^{2} + 3 \) Copy content Toggle raw display
$71$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$73$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
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