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Properties

Label 546.2.o.c
Level $546$
Weight $2$
Character orbit 546.o
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
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(265,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.265");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.o (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{7} q^{3} - \beta_{7} q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_{2} q^{6} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{7} q^{3} - \beta_{7} q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_{2} q^{6} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \cdots - 1) q^{7}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} - 8 q^{7} - 8 q^{9} - 4 q^{10} + 8 q^{12} - 16 q^{13} - 4 q^{14} - 4 q^{15} - 8 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{20} - 12 q^{22} + 16 q^{26} + 12 q^{29} + 8 q^{31} + 8 q^{34} - 24 q^{35} - 4 q^{37} + 4 q^{38} - 4 q^{39} - 12 q^{41} - 4 q^{42} - 4 q^{45} + 24 q^{46} - 32 q^{49} - 8 q^{50} + 4 q^{52} + 40 q^{53} + 4 q^{56} - 8 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{60} - 8 q^{62} + 8 q^{63} - 12 q^{65} + 32 q^{67} + 28 q^{69} - 12 q^{71} - 20 q^{73} + 20 q^{74} + 12 q^{75} + 8 q^{76} - 8 q^{77} - 4 q^{78} + 24 q^{79} - 4 q^{80} + 8 q^{81} - 40 q^{82} - 44 q^{83} - 8 q^{84} + 20 q^{85} - 20 q^{86} - 16 q^{89} + 4 q^{90} - 12 q^{91} - 28 q^{92} - 8 q^{93} + 8 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 11\nu^{3} + 7\nu^{2} + 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 11\nu^{3} + 7\nu^{2} - 26\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + \nu^{5} + 11\nu^{4} + 7\nu^{3} + 26\nu^{2} + 6\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - \nu^{5} + 11\nu^{4} - 7\nu^{3} + 26\nu^{2} - 6\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 30\nu^{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{6} + 7\beta_{5} + 7\beta_{4} + 4\beta_{3} + 4\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} + 11\beta_{4} + 7\beta_{3} - 7\beta_{2} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51\beta_{6} - 47\beta_{5} - 47\beta_{4} - 44\beta_{3} - 44\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 91\beta_{5} - 91\beta_{4} - 43\beta_{3} + 43\beta_{2} - 181\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1
− 1.65222i
2.06644i
0.222191i
− 2.63640i
1.65222i
− 2.06644i
− 0.222191i
2.63640i
−0.707107 + 0.707107i 1.00000i − 1.00000i −0.461191 − 0.461191i −0.707107 − 0.707107i −0.292893 − 2.62949i 0.707107 + 0.707107i −1.00000 0.652223
265.2 −0.707107 + 0.707107i 1.00000i − 1.00000i 2.16830 + 2.16830i −0.707107 − 0.707107i −0.292893 + 2.62949i 0.707107 + 0.707107i −1.00000 −3.06644
265.3 0.707107 − 0.707107i 1.00000i − 1.00000i −0.864220 − 0.864220i 0.707107 + 0.707107i −1.70711 − 2.02133i −0.707107 − 0.707107i −1.00000 −1.22219
265.4 0.707107 − 0.707107i 1.00000i − 1.00000i 1.15711 + 1.15711i 0.707107 + 0.707107i −1.70711 + 2.02133i −0.707107 − 0.707107i −1.00000 1.63640
307.1 −0.707107 − 0.707107i − 1.00000i 1.00000i −0.461191 + 0.461191i −0.707107 + 0.707107i −0.292893 + 2.62949i 0.707107 − 0.707107i −1.00000 0.652223
307.2 −0.707107 − 0.707107i − 1.00000i 1.00000i 2.16830 − 2.16830i −0.707107 + 0.707107i −0.292893 − 2.62949i 0.707107 − 0.707107i −1.00000 −3.06644
307.3 0.707107 + 0.707107i − 1.00000i 1.00000i −0.864220 + 0.864220i 0.707107 − 0.707107i −1.70711 + 2.02133i −0.707107 + 0.707107i −1.00000 −1.22219
307.4 0.707107 + 0.707107i − 1.00000i 1.00000i 1.15711 − 1.15711i 0.707107 − 0.707107i −1.70711 − 2.02133i −0.707107 + 0.707107i −1.00000 1.63640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.o.c yes 8
3.b odd 2 1 1638.2.x.a 8
7.b odd 2 1 546.2.o.b ✓ 8
13.d odd 4 1 546.2.o.b ✓ 8
21.c even 2 1 1638.2.x.c 8
39.f even 4 1 1638.2.x.c 8
91.i even 4 1 inner 546.2.o.c yes 8
273.o odd 4 1 1638.2.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.o.b ✓ 8 7.b odd 2 1
546.2.o.b ✓ 8 13.d odd 4 1
546.2.o.c yes 8 1.a even 1 1 trivial
546.2.o.c yes 8 91.i even 4 1 inner
1638.2.x.a 8 3.b odd 2 1
1638.2.x.a 8 273.o odd 4 1
1638.2.x.c 8 21.c even 2 1
1638.2.x.c 8 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 4T_{5}^{5} + T_{5}^{4} - 8T_{5}^{3} + 32T_{5}^{2} + 32T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + 16 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 48 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 16 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} - 23 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{8} + 146 T^{6} + \cdots + 454276 \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - 45 T^{2} + \cdots + 98)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{8} + 86 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + \cdots + 1993744 \) Copy content Toggle raw display
$61$ \( T^{8} + 502 T^{6} + \cdots + 134606404 \) Copy content Toggle raw display
$67$ \( T^{8} - 32 T^{7} + \cdots + 10837264 \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} + \cdots + 817216 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + \cdots + 23078416 \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + \cdots - 1264)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 44 T^{7} + \cdots + 1459264 \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} + \cdots + 295936 \) Copy content Toggle raw display
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