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Properties

Label 329.1.f.a
Level $329$
Weight $1$
Character orbit 329.f
Analytic conductor $0.164$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -47
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [329,1,Mod(46,329)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(329, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("329.46");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 329 = 7 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 329.f (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: \(0.164192389156\)
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 3.1.2303.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.5087327.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + 2 \zeta_{6}^{2} q^{3} - 2 q^{6} + \zeta_{6}^{2} q^{7} + q^{8} - 3 \zeta_{6} q^{9} - q^{14} + \zeta_{6} q^{16} - \zeta_{6}^{2} q^{17} - 3 \zeta_{6}^{2} q^{18} - 2 \zeta_{6} q^{21} + \cdots - \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - 4 q^{6} - q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{14} + q^{16} + q^{17} + 3 q^{18} - 2 q^{21} - 2 q^{24} - q^{25} + 8 q^{27} + 2 q^{34} + q^{37} + 2 q^{42} - q^{47} - 4 q^{48} - q^{49}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/329\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(283\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 + 0.866025i −1.00000 + 1.73205i 0 0 −2.00000 −0.500000 + 0.866025i 1.00000 −1.50000 − 2.59808i 0
93.1 0.500000 − 0.866025i −1.00000 − 1.73205i 0 0 −2.00000 −0.500000 − 0.866025i 1.00000 −1.50000 + 2.59808i 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
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
7.c even 3 1 inner
329.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 329.1.f.a ✓ 2
3.b odd 2 1 2961.1.x.a 2
7.b odd 2 1 2303.1.f.a 2
7.c even 3 1 inner 329.1.f.a ✓ 2
7.c even 3 1 2303.1.d.b 1
7.d odd 6 1 2303.1.d.a 1
7.d odd 6 1 2303.1.f.a 2
21.h odd 6 1 2961.1.x.a 2
47.b odd 2 1 CM 329.1.f.a ✓ 2
141.c even 2 1 2961.1.x.a 2
329.c even 2 1 2303.1.f.a 2
329.f odd 6 1 inner 329.1.f.a ✓ 2
329.f odd 6 1 2303.1.d.b 1
329.g even 6 1 2303.1.d.a 1
329.g even 6 1 2303.1.f.a 2
987.m even 6 1 2961.1.x.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
329.1.f.a ✓ 2 1.a even 1 1 trivial
329.1.f.a ✓ 2 7.c even 3 1 inner
329.1.f.a ✓ 2 47.b odd 2 1 CM
329.1.f.a ✓ 2 329.f odd 6 1 inner
2303.1.d.a 1 7.d odd 6 1
2303.1.d.a 1 329.g even 6 1
2303.1.d.b 1 7.c even 3 1
2303.1.d.b 1 329.f odd 6 1
2303.1.f.a 2 7.b odd 2 1
2303.1.f.a 2 7.d odd 6 1
2303.1.f.a 2 329.c even 2 1
2303.1.f.a 2 329.g even 6 1
2961.1.x.a 2 3.b odd 2 1
2961.1.x.a 2 21.h odd 6 1
2961.1.x.a 2 141.c even 2 1
2961.1.x.a 2 987.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(329, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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