Properties

Label 546.2.e.f
Level $546$
Weight $2$
Character orbit 546.e
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(545,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.545");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.e (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{7} q^{3} + q^{4} + (\beta_{7} + \beta_1) q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{2}) q^{7} - q^{8} + (\beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{7} q^{3} + q^{4} + (\beta_{7} + \beta_1) q^{5} + \beta_{7} q^{6} + (\beta_{6} + \beta_{2}) q^{7} - q^{8} + (\beta_{4} - \beta_{2}) q^{9} + ( - \beta_{7} - \beta_1) q^{10} - \beta_{7} q^{12} + (\beta_{3} + 2 \beta_1) q^{13} + ( - \beta_{6} - \beta_{2}) q^{14} + ( - \beta_{4} + \beta_{2} + 3) q^{15} + q^{16} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{4} + \beta_{2}) q^{18} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{19} + (\beta_{7} + \beta_1) q^{20} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{21} + (\beta_{6} + \beta_{5} + 3 \beta_{2}) q^{23} + \beta_{7} q^{24} + (2 \beta_{4} - 1) q^{25} + ( - \beta_{3} - 2 \beta_1) q^{26} + ( - 2 \beta_{3} + \beta_1) q^{27} + (\beta_{6} + \beta_{2}) q^{28} + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{29} + (\beta_{4} - \beta_{2} - 3) q^{30} + (2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - 2 \beta_1) q^{31} - q^{32} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{34} + (\beta_{7} + 2 \beta_{5} - 4 \beta_{2} - \beta_1) q^{35} + (\beta_{4} - \beta_{2}) q^{36} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{37} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{38} + (\beta_{6} + \beta_{5} + 5) q^{39} + ( - \beta_{7} - \beta_1) q^{40} + (4 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{42} + 8 q^{43} + ( - 3 \beta_{7} + 2 \beta_{3} - \beta_1) q^{45} + ( - \beta_{6} - \beta_{5} - 3 \beta_{2}) q^{46} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{47} - \beta_{7} q^{48} + (2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{49} + ( - 2 \beta_{4} + 1) q^{50} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 2) q^{51} + (\beta_{3} + 2 \beta_1) q^{52} + (\beta_{6} + \beta_{5} + 3 \beta_{2}) q^{53} + (2 \beta_{3} - \beta_1) q^{54} + ( - \beta_{6} - \beta_{2}) q^{56} + (\beta_{6} + \beta_{5} + \beta_{4} - 4 \beta_{2} + 5) q^{57} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{58} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{59} + ( - \beta_{4} + \beta_{2} + 3) q^{60} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - 3 \beta_1) q^{61} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{62} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 1) q^{63} + q^{64} + ( - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{2} - 5) q^{65} + (2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2}) q^{67} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1) q^{68} + ( - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} + 6 \beta_1) q^{69} + ( - \beta_{7} - 2 \beta_{5} + 4 \beta_{2} + \beta_1) q^{70} + (4 \beta_{4} + 4) q^{71} + ( - \beta_{4} + \beta_{2}) q^{72} + ( - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{2} + 2 \beta_1) q^{73} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{74} + (\beta_{7} - 2 \beta_{3} + 4 \beta_1) q^{75} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{76} + ( - \beta_{6} - \beta_{5} - 5) q^{78} + ( - 2 \beta_{4} - 6) q^{79} + (\beta_{7} + \beta_1) q^{80} + ( - 2 \beta_{6} - 2 \beta_{5} + 5) q^{81} + ( - 4 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{82} + ( - 3 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - 5 \beta_1) q^{83} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{84} + ( - 4 \beta_{6} - 4 \beta_{5} + 8 \beta_{2}) q^{85} - 8 q^{86} + (2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{87} + ( - 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{89} + (3 \beta_{7} - 2 \beta_{3} + \beta_1) q^{90} + (\beta_{7} + \beta_{6} + \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{91} + (\beta_{6} + \beta_{5} + 3 \beta_{2}) q^{92} + (\beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{2} - 4) q^{93} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{94} + ( - 2 \beta_{6} - 2 \beta_{5} + 8 \beta_{2}) q^{95} + \beta_{7} q^{96} + (4 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} - 4 \beta_1) q^{97} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 24 q^{15} + 8 q^{16} - 8 q^{21} - 8 q^{25} - 24 q^{30} - 8 q^{32} + 40 q^{39} + 8 q^{42} + 64 q^{43} - 8 q^{49} + 8 q^{50} - 16 q^{51} + 40 q^{57} + 24 q^{60} + 8 q^{63} + 8 q^{64} - 40 q^{65} + 32 q^{71} - 40 q^{78} - 48 q^{79} + 40 q^{81} - 8 q^{84} - 64 q^{86} - 32 q^{91} - 32 q^{93} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 7\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 19\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 3\nu^{5} + 9\nu^{4} + \nu^{3} - \nu^{2} - 3\nu - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 3\nu^{5} + 9\nu^{4} - \nu^{3} + \nu^{2} + 3\nu - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 10\nu^{3} ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{4} + 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{7} + 20\beta_{6} - 20\beta_{5} + 20\beta_{3} + 20\beta_{2} + 20\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
−1.68014 + 0.420861i
−1.68014 0.420861i
−0.420861 + 1.68014i
−0.420861 1.68014i
0.420861 + 1.68014i
0.420861 1.68014i
1.68014 + 0.420861i
1.68014 0.420861i
−1.00000 −1.68014 0.420861i 1.00000 0.841723i 1.68014 + 0.420861i 0.595188 2.57794i −1.00000 2.64575 + 1.41421i 0.841723i
545.2 −1.00000 −1.68014 + 0.420861i 1.00000 0.841723i 1.68014 0.420861i 0.595188 + 2.57794i −1.00000 2.64575 1.41421i 0.841723i
545.3 −1.00000 −0.420861 1.68014i 1.00000 3.36028i 0.420861 + 1.68014i 2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 3.36028i
545.4 −1.00000 −0.420861 + 1.68014i 1.00000 3.36028i 0.420861 1.68014i 2.37608 1.16372i −1.00000 −2.64575 1.41421i 3.36028i
545.5 −1.00000 0.420861 1.68014i 1.00000 3.36028i −0.420861 + 1.68014i −2.37608 1.16372i −1.00000 −2.64575 1.41421i 3.36028i
545.6 −1.00000 0.420861 + 1.68014i 1.00000 3.36028i −0.420861 1.68014i −2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 3.36028i
545.7 −1.00000 1.68014 0.420861i 1.00000 0.841723i −1.68014 + 0.420861i −0.595188 + 2.57794i −1.00000 2.64575 1.41421i 0.841723i
545.8 −1.00000 1.68014 + 0.420861i 1.00000 0.841723i −1.68014 0.420861i −0.595188 2.57794i −1.00000 2.64575 + 1.41421i 0.841723i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
39.d odd 2 1 inner
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.e.f 8
3.b odd 2 1 546.2.e.h yes 8
7.b odd 2 1 inner 546.2.e.f 8
13.b even 2 1 546.2.e.h yes 8
21.c even 2 1 546.2.e.h yes 8
39.d odd 2 1 inner 546.2.e.f 8
91.b odd 2 1 546.2.e.h yes 8
273.g even 2 1 inner 546.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.e.f 8 1.a even 1 1 trivial
546.2.e.f 8 7.b odd 2 1 inner
546.2.e.f 8 39.d odd 2 1 inner
546.2.e.f 8 273.g even 2 1 inner
546.2.e.h yes 8 3.b odd 2 1
546.2.e.h yes 8 13.b even 2 1
546.2.e.h yes 8 21.c even 2 1
546.2.e.h yes 8 91.b odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 12T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{17}^{4} - 80T_{17}^{2} + 1152 \) Copy content Toggle raw display
\( T_{19}^{4} - 52T_{19}^{2} + 648 \) Copy content Toggle raw display
\( T_{71}^{2} - 8T_{71} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 10T^{4} + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{6} - 10 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 310 T^{4} + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} - 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 64 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 216 T^{2} + 11552)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 64 T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 176 T^{2} + 576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 96)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 168 T^{2} + 1568)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 180 T^{2} + 6728)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 168 T^{2} + 1568)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 216 T^{2} + 11552)^{2} \) Copy content Toggle raw display
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