Properties

Label 476.2.i.c
Level $476$
Weight $2$
Character orbit 476.i
Analytic conductor $3.801$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [476,2,Mod(137,476)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(476, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.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: \(3.80087913621\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} - 4 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - 6 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{3} - 4 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - 6 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{11} + 3 q^{13} - 12 q^{15} + ( - \zeta_{6} + 1) q^{17} + 2 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 9) q^{21} - 4 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - 9 q^{27} + (8 \zeta_{6} - 8) q^{31} + 3 \zeta_{6} q^{33} + ( - 12 \zeta_{6} + 8) q^{35} - 8 \zeta_{6} q^{37} + ( - 9 \zeta_{6} + 9) q^{39} + 10 q^{43} + (24 \zeta_{6} - 24) q^{45} + 10 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - 3 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + 4 q^{55} + 6 q^{57} + ( - 14 \zeta_{6} + 14) q^{59} + 8 \zeta_{6} q^{61} + ( - 18 \zeta_{6} + 12) q^{63} - 12 \zeta_{6} q^{65} + ( - 10 \zeta_{6} + 10) q^{67} - 12 q^{69} - 5 q^{71} + ( - 16 \zeta_{6} + 16) q^{73} + 33 \zeta_{6} q^{75} + (\zeta_{6} - 3) q^{77} - 11 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + 12 q^{83} - 4 q^{85} - 9 \zeta_{6} q^{89} + (6 \zeta_{6} + 3) q^{91} + 24 \zeta_{6} q^{93} + ( - 8 \zeta_{6} + 8) q^{95} + 10 q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 4 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 4 q^{5} + 4 q^{7} - 6 q^{9} - q^{11} + 6 q^{13} - 24 q^{15} + q^{17} + 2 q^{19} + 15 q^{21} - 4 q^{23} - 11 q^{25} - 18 q^{27} - 8 q^{31} + 3 q^{33} + 4 q^{35} - 8 q^{37} + 9 q^{39} + 20 q^{43} - 24 q^{45} + 10 q^{47} + 2 q^{49} - 3 q^{51} - 3 q^{53} + 8 q^{55} + 12 q^{57} + 14 q^{59} + 8 q^{61} + 6 q^{63} - 12 q^{65} + 10 q^{67} - 24 q^{69} - 10 q^{71} + 16 q^{73} + 33 q^{75} - 5 q^{77} - 11 q^{79} - 9 q^{81} + 24 q^{83} - 8 q^{85} - 9 q^{89} + 12 q^{91} + 24 q^{93} + 8 q^{95} + 20 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/476\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(309\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 2.59808i 0 −2.00000 3.46410i 0 2.00000 + 1.73205i 0 −3.00000 5.19615i 0
205.1 0 1.50000 + 2.59808i 0 −2.00000 + 3.46410i 0 2.00000 1.73205i 0 −3.00000 + 5.19615i 0
\(n\): e.g. 2-40 or 990-1000
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 476.2.i.c 2
7.c even 3 1 inner 476.2.i.c 2
7.c even 3 1 3332.2.a.a 1
7.d odd 6 1 3332.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.i.c 2 1.a even 1 1 trivial
476.2.i.c 2 7.c even 3 1 inner
3332.2.a.a 1 7.c even 3 1
3332.2.a.f 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( (T + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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