Properties

Label 416.1.bb.a
Level $416$
Weight $1$
Character orbit 416.bb
Analytic conductor $0.208$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,1,Mod(159,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.159");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 416.bb (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: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.10816.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12}^{5} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} - \zeta_{12}^{5} q^{7} - \zeta_{12} q^{11} + q^{13} - \zeta_{12}^{2} q^{17} - \zeta_{12}^{5} q^{19} - q^{21} + \zeta_{12} q^{23} - q^{25} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{29} + \zeta_{12}^{2} q^{33} + \zeta_{12}^{4} q^{37} - \zeta_{12} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} + 2 \zeta_{12}^{3} q^{47} + \zeta_{12}^{3} q^{51} - q^{57} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{2} q^{61} + \zeta_{12} q^{67} - \zeta_{12}^{2} q^{69} + \zeta_{12}^{5} q^{71} + \zeta_{12} q^{75} - q^{77} - 2 \zeta_{12}^{3} q^{79} - \zeta_{12}^{4} q^{81} + \zeta_{12}^{5} q^{87} - \zeta_{12}^{4} q^{89} - \zeta_{12}^{5} q^{91} + \zeta_{12}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{13} - 2 q^{17} - 4 q^{21} - 4 q^{25} + 2 q^{29} + 2 q^{33} - 2 q^{37} - 2 q^{41} - 4 q^{57} + 2 q^{61} - 2 q^{69} - 4 q^{77} + 2 q^{81} + 2 q^{89} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 0.500000i 0 0 0 0.866025 + 0.500000i 0 0 0
159.2 0 0.866025 0.500000i 0 0 0 −0.866025 0.500000i 0 0 0
191.1 0 −0.866025 0.500000i 0 0 0 0.866025 0.500000i 0 0 0
191.2 0 0.866025 + 0.500000i 0 0 0 −0.866025 + 0.500000i 0 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
4.b odd 2 1 inner
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.bb.a 4
3.b odd 2 1 3744.1.dy.a 4
4.b odd 2 1 inner 416.1.bb.a 4
8.b even 2 1 832.1.bb.b 4
8.d odd 2 1 832.1.bb.b 4
12.b even 2 1 3744.1.dy.a 4
13.c even 3 1 inner 416.1.bb.a 4
16.e even 4 1 3328.1.v.a 4
16.e even 4 1 3328.1.v.c 4
16.f odd 4 1 3328.1.v.a 4
16.f odd 4 1 3328.1.v.c 4
39.i odd 6 1 3744.1.dy.a 4
52.j odd 6 1 inner 416.1.bb.a 4
104.n odd 6 1 832.1.bb.b 4
104.r even 6 1 832.1.bb.b 4
156.p even 6 1 3744.1.dy.a 4
208.bg odd 12 1 3328.1.v.a 4
208.bg odd 12 1 3328.1.v.c 4
208.bj even 12 1 3328.1.v.a 4
208.bj even 12 1 3328.1.v.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bb.a 4 1.a even 1 1 trivial
416.1.bb.a 4 4.b odd 2 1 inner
416.1.bb.a 4 13.c even 3 1 inner
416.1.bb.a 4 52.j odd 6 1 inner
832.1.bb.b 4 8.b even 2 1
832.1.bb.b 4 8.d odd 2 1
832.1.bb.b 4 104.n odd 6 1
832.1.bb.b 4 104.r even 6 1
3328.1.v.a 4 16.e even 4 1
3328.1.v.a 4 16.f odd 4 1
3328.1.v.a 4 208.bg odd 12 1
3328.1.v.a 4 208.bj even 12 1
3328.1.v.c 4 16.e even 4 1
3328.1.v.c 4 16.f odd 4 1
3328.1.v.c 4 208.bg odd 12 1
3328.1.v.c 4 208.bj even 12 1
3744.1.dy.a 4 3.b odd 2 1
3744.1.dy.a 4 12.b even 2 1
3744.1.dy.a 4 39.i odd 6 1
3744.1.dy.a 4 156.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(416, [\chi])\).

Hecke characteristic polynomials

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