Properties

Label 546.2.s.b
Level $546$
Weight $2$
Character orbit 546.s
Analytic conductor $4.360$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(43,546)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.s (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: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.73205i 0.866025 0.500000i −0.866025 + 0.500000i 1.00000i −0.500000 0.866025i −1.36603 + 2.36603i
43.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.732051i −0.866025 + 0.500000i 0.866025 0.500000i 1.00000i −0.500000 0.866025i 0.366025 0.633975i
127.1 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 2.73205i 0.866025 + 0.500000i −0.866025 0.500000i 1.00000i −0.500000 + 0.866025i −1.36603 2.36603i
127.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.732051i −0.866025 0.500000i 0.866025 + 0.500000i 1.00000i −0.500000 + 0.866025i 0.366025 + 0.633975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.s.b 4
3.b odd 2 1 1638.2.bj.a 4
13.e even 6 1 inner 546.2.s.b 4
13.f odd 12 1 7098.2.a.bo 2
13.f odd 12 1 7098.2.a.by 2
39.h odd 6 1 1638.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.b 4 1.a even 1 1 trivial
546.2.s.b 4 13.e even 6 1 inner
1638.2.bj.a 4 3.b odd 2 1
1638.2.bj.a 4 39.h odd 6 1
7098.2.a.bo 2 13.f odd 12 1
7098.2.a.by 2 13.f odd 12 1

Hecke kernels

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

\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 9T_{11} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 51 T^{2} + 104 T + 169 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + 60 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} - 14 T^{3} + 159 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$31$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 10 T^{2} + 132 T + 484 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 35 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 150 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$47$ \( T^{4} + 114T^{2} + 1521 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 24 T^{3} + 236 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} - 66 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$73$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T - 9)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 248 T^{2} + 13924 \) Copy content Toggle raw display
$89$ \( T^{4} + 48 T^{3} + 951 T^{2} + \cdots + 33489 \) Copy content Toggle raw display
$97$ \( T^{4} + 18 T^{3} + 54 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
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