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Properties

Label 476.1.o.a
Level $476$
Weight $1$
Character orbit 476.o
Analytic conductor $0.238$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -68
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [476,1,Mod(67,476)]
 
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([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("476.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 476.o (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: \(0.237554946013\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.3332.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} - q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + q^{6} - q^{7} + q^{8} - \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} - q^{13} - \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} - \zeta_{6} q^{17} + \zeta_{6} q^{21} + q^{22} - \zeta_{6}^{2} q^{23} - \zeta_{6} q^{24} - \zeta_{6} q^{25} - \zeta_{6}^{2} q^{26} - q^{27} + \zeta_{6} q^{28} + \zeta_{6} q^{31} - \zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} + q^{34} + \zeta_{6} q^{39} - q^{42} + \zeta_{6}^{2} q^{44} + 2 \zeta_{6} q^{46} + q^{48} + q^{49} + q^{50} + \zeta_{6}^{2} q^{51} + \zeta_{6} q^{52} + \zeta_{6} q^{53} - \zeta_{6}^{2} q^{54} - q^{56} - 2 q^{62} + q^{64} - \zeta_{6} q^{66} + \zeta_{6}^{2} q^{68} - 2 q^{69} + q^{71} + \zeta_{6}^{2} q^{75} + \zeta_{6} q^{77} - q^{78} + \zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} - \zeta_{6}^{2} q^{84} - \zeta_{6} q^{88} - \zeta_{6}^{2} q^{89} + q^{91} - 2 q^{92} - 2 \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} + \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} - q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{16} - q^{17} + q^{21} + 2 q^{22} + 2 q^{23} - q^{24} - q^{25} + q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - q^{32} - q^{33} + 2 q^{34} + q^{39} - 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - q^{51} + q^{52} + q^{53} + q^{54} - 2 q^{56} - 4 q^{62} + 2 q^{64} - q^{66} - q^{68} - 4 q^{69} + 2 q^{71} - q^{75} + q^{77} - 2 q^{78} - q^{79} + q^{81} + q^{84} - q^{88} + q^{89} + 2 q^{91} - 4 q^{92} + 2 q^{93} - q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(239\) \(309\) \(409\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 −1.00000 1.00000 0 0
135.1 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 0 1.00000 −1.00000 1.00000 0 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
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
7.c even 3 1 inner
476.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.1.o.a ✓ 2
4.b odd 2 1 476.1.o.b yes 2
7.b odd 2 1 3332.1.o.b 2
7.c even 3 1 inner 476.1.o.a ✓ 2
7.c even 3 1 3332.1.g.e 1
7.d odd 6 1 3332.1.g.c 1
7.d odd 6 1 3332.1.o.b 2
17.b even 2 1 476.1.o.b yes 2
28.d even 2 1 3332.1.o.a 2
28.f even 6 1 3332.1.g.d 1
28.f even 6 1 3332.1.o.a 2
28.g odd 6 1 476.1.o.b yes 2
28.g odd 6 1 3332.1.g.b 1
68.d odd 2 1 CM 476.1.o.a ✓ 2
119.d odd 2 1 3332.1.o.a 2
119.h odd 6 1 3332.1.g.d 1
119.h odd 6 1 3332.1.o.a 2
119.j even 6 1 476.1.o.b yes 2
119.j even 6 1 3332.1.g.b 1
476.e even 2 1 3332.1.o.b 2
476.o odd 6 1 inner 476.1.o.a ✓ 2
476.o odd 6 1 3332.1.g.e 1
476.q even 6 1 3332.1.g.c 1
476.q even 6 1 3332.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.a ✓ 2 1.a even 1 1 trivial
476.1.o.a ✓ 2 7.c even 3 1 inner
476.1.o.a ✓ 2 68.d odd 2 1 CM
476.1.o.a ✓ 2 476.o odd 6 1 inner
476.1.o.b yes 2 4.b odd 2 1
476.1.o.b yes 2 17.b even 2 1
476.1.o.b yes 2 28.g odd 6 1
476.1.o.b yes 2 119.j even 6 1
3332.1.g.b 1 28.g odd 6 1
3332.1.g.b 1 119.j even 6 1
3332.1.g.c 1 7.d odd 6 1
3332.1.g.c 1 476.q even 6 1
3332.1.g.d 1 28.f even 6 1
3332.1.g.d 1 119.h odd 6 1
3332.1.g.e 1 7.c even 3 1
3332.1.g.e 1 476.o odd 6 1
3332.1.o.a 2 28.d even 2 1
3332.1.o.a 2 28.f even 6 1
3332.1.o.a 2 119.d odd 2 1
3332.1.o.a 2 119.h odd 6 1
3332.1.o.b 2 7.b odd 2 1
3332.1.o.b 2 7.d odd 6 1
3332.1.o.b 2 476.e even 2 1
3332.1.o.b 2 476.q even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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