Properties

Label 546.2.i.k
Level $546$
Weight $2$
Character orbit 546.i
Analytic conductor $4.360$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} + q^{6} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + (\beta_{3} - \beta_{2} - \beta_1) q^{5} + q^{6} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{7} - q^{8} + ( - \beta_{2} - 1) q^{9} + (\beta_{4} - \beta_{2} - \beta_1) q^{10} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{2} + 1) q^{12} + q^{13} + (\beta_{3} + \beta_{2} + 1) q^{14} + ( - \beta_{4} + \beta_{3}) q^{15} + ( - \beta_{2} - 1) q^{16} + (2 \beta_{5} - \beta_{4} - 3 \beta_1) q^{17} - \beta_{2} q^{18} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 2) q^{19} + (\beta_{4} - \beta_{3}) q^{20} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{21} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{22}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} + 3 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 6 q^{13} - 6 q^{15} - 3 q^{16} - 6 q^{17} + 3 q^{18} - 6 q^{19} + 6 q^{20} + 3 q^{21} + 6 q^{22} - 3 q^{24} - 18 q^{25} + 3 q^{26} - 6 q^{27} - 3 q^{28} - 18 q^{29} - 3 q^{30} + 15 q^{31} + 3 q^{32} - 3 q^{33} - 12 q^{34} + 30 q^{35} + 6 q^{36} - 6 q^{37} + 6 q^{38} + 3 q^{39} + 3 q^{40} + 36 q^{41} + 3 q^{42} - 24 q^{43} + 3 q^{44} - 3 q^{45} + 6 q^{47} - 6 q^{48} + 21 q^{49} - 36 q^{50} + 6 q^{51} - 3 q^{52} - 3 q^{53} - 3 q^{54} + 54 q^{55} - 3 q^{56} - 12 q^{57} - 9 q^{58} + 3 q^{59} + 3 q^{60} + 30 q^{62} + 6 q^{64} - 3 q^{65} + 3 q^{66} - 6 q^{67} - 6 q^{68} + 3 q^{70} - 72 q^{71} + 3 q^{72} - 24 q^{73} + 6 q^{74} + 18 q^{75} + 12 q^{76} + 33 q^{77} + 6 q^{78} - 15 q^{79} - 3 q^{80} - 3 q^{81} + 18 q^{82} + 18 q^{83} - 48 q^{85} - 12 q^{86} - 9 q^{87} - 3 q^{88} - 30 q^{89} - 6 q^{90} + 3 q^{91} - 15 q^{93} - 6 q^{94} - 6 q^{95} - 3 q^{96} + 6 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139 \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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 + 0.679547i
0.500000 3.05087i
0.500000 + 3.23735i
0.500000 0.679547i
0.500000 + 3.05087i
0.500000 3.23735i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.90280 + 3.29575i 1.00000 −2.64411 0.0932392i −1.00000 −0.500000 + 0.866025i 1.90280 + 3.29575i
79.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.30661 + 2.26312i 1.00000 1.77890 1.95845i −1.00000 −0.500000 + 0.866025i 1.30661 + 2.26312i
79.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.70942 2.96080i 1.00000 2.36521 + 1.18566i −1.00000 −0.500000 + 0.866025i −1.70942 2.96080i
235.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.90280 3.29575i 1.00000 −2.64411 + 0.0932392i −1.00000 −0.500000 0.866025i 1.90280 3.29575i
235.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.30661 2.26312i 1.00000 1.77890 + 1.95845i −1.00000 −0.500000 0.866025i 1.30661 2.26312i
235.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.70942 + 2.96080i 1.00000 2.36521 1.18566i −1.00000 −0.500000 0.866025i −1.70942 + 2.96080i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
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.k 6
3.b odd 2 1 1638.2.j.q 6
7.c even 3 1 inner 546.2.i.k 6
7.c even 3 1 3822.2.a.bv 3
7.d odd 6 1 3822.2.a.bw 3
21.h odd 6 1 1638.2.j.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.k 6 1.a even 1 1 trivial
546.2.i.k 6 7.c even 3 1 inner
1638.2.j.q 6 3.b odd 2 1
1638.2.j.q 6 21.h odd 6 1
3822.2.a.bv 3 7.c even 3 1
3822.2.a.bw 3 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 \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{6} + 3T_{5}^{5} + 21T_{5}^{4} + 32T_{5}^{3} + 246T_{5}^{2} + 408T_{5} + 1156 \) Copy content Toggle raw display
\( T_{17}^{6} + 6T_{17}^{5} + 99T_{17}^{4} + 386T_{17}^{3} + 6261T_{17}^{2} + 24066T_{17} + 145924 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 145924 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{6} + 30 T^{4} + \cdots + 3600 \) Copy content Toggle raw display
$29$ \( (T + 3)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 15 T^{5} + \cdots + 1600 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} - 18 T^{2} + \cdots - 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 12 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$53$ \( T^{6} + 3 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{6} + 75 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{3} + 36 T^{2} + \cdots + 1538)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 24 T^{5} + \cdots + 44944 \) Copy content Toggle raw display
$79$ \( T^{6} + 15 T^{5} + \cdots + 10000 \) Copy content Toggle raw display
$83$ \( (T^{3} - 9 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 30 T^{5} + \cdots + 462400 \) Copy content Toggle raw display
$97$ \( (T^{3} - 3 T^{2} + \cdots - 166)^{2} \) Copy content Toggle raw display
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