Properties

Label 464.1.v.a
Level $464$
Weight $1$
Character orbit 464.v
Analytic conductor $0.232$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [464,1,Mod(63,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.63");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 464.v (of order \(14\), degree \(6\), 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: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{14}^{3} - \zeta_{14}) q^{5} - \zeta_{14}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{14}^{3} - \zeta_{14}) q^{5} - \zeta_{14}^{6} q^{9} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{13} + ( - \zeta_{14}^{5} - \zeta_{14}^{2}) q^{17} + (\zeta_{14}^{6} + \zeta_{14}^{4} + \zeta_{14}^{2}) q^{25} + \zeta_{14} q^{29} + (\zeta_{14}^{3} + \zeta_{14}^{2}) q^{37} + (\zeta_{14}^{4} + \zeta_{14}^{3}) q^{41} + ( - \zeta_{14}^{2} - 1) q^{45} + \zeta_{14}^{6} q^{49} + ( - \zeta_{14}^{4} - 1) q^{53} + ( - \zeta_{14}^{5} + \zeta_{14}) q^{61} + ( - \zeta_{14}^{6} + \zeta_{14}^{5} + \zeta_{14} - 1) q^{65} + ( - \zeta_{14}^{3} - 1) q^{73} - \zeta_{14}^{5} q^{81} + (\zeta_{14}^{6} + \zeta_{14}^{5} + \zeta_{14}^{3} - \zeta_{14}) q^{85} + (\zeta_{14}^{3} - \zeta_{14}) q^{89} + (\zeta_{14} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + q^{9} + 2 q^{13} - 3 q^{25} + q^{29} - 5 q^{45} - q^{49} - 5 q^{53} - 3 q^{65} - 7 q^{73} - q^{81} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{14}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
0.900969 + 0.433884i
−0.623490 0.781831i
0.222521 0.974928i
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0 0 0 −1.12349 1.40881i 0 0 0 0.900969 0.433884i 0
207.1 0 0 0 −0.277479 + 1.21572i 0 0 0 −0.623490 + 0.781831i 0
303.1 0 0 0 0.400969 + 0.193096i 0 0 0 0.222521 + 0.974928i 0
383.1 0 0 0 −1.12349 + 1.40881i 0 0 0 0.900969 + 0.433884i 0
399.1 0 0 0 −0.277479 1.21572i 0 0 0 −0.623490 0.781831i 0
415.1 0 0 0 0.400969 0.193096i 0 0 0 0.222521 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
29.e even 14 1 inner
116.h odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.1.v.a 6
4.b odd 2 1 CM 464.1.v.a 6
8.b even 2 1 1856.1.bd.a 6
8.d odd 2 1 1856.1.bd.a 6
29.e even 14 1 inner 464.1.v.a 6
116.h odd 14 1 inner 464.1.v.a 6
232.o even 14 1 1856.1.bd.a 6
232.t odd 14 1 1856.1.bd.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.1.v.a 6 1.a even 1 1 trivial
464.1.v.a 6 4.b odd 2 1 CM
464.1.v.a 6 29.e even 14 1 inner
464.1.v.a 6 116.h odd 14 1 inner
1856.1.bd.a 6 8.b even 2 1
1856.1.bd.a 6 8.d odd 2 1
1856.1.bd.a 6 232.o even 14 1
1856.1.bd.a 6 232.t odd 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(464, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 4 T^{4} + T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 7 T^{4} + 14 T^{2} + 7 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{3} - 7 T + 7 \) Copy content Toggle raw display
$41$ \( T^{6} + 7 T^{4} + 14 T^{2} + 7 \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 7 T^{3} + 7 T + 7 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} + 21 T^{4} + 35 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{3} + 7 T + 7 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} + 21 T^{4} - 35 T^{3} + \cdots + 7 \) Copy content Toggle raw display
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