Properties

Label 416.2.ba.a
Level $416$
Weight $2$
Character orbit 416.ba
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(17,416)]
 
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([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.ba (of order \(6\), degree \(2\), not 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: \(3.32177672409\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{3} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{2} + 4) q^{7} + ( - 2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + \zeta_{12} - 1) q^{3} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 2) q^{5} + ( - 2 \zeta_{12}^{2} + 4) q^{7} + ( - 2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{9}+ \cdots + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9} - 4 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{21} + 6 q^{23} + 8 q^{25} + 12 q^{33} + 24 q^{35} + 12 q^{37} - 14 q^{39} - 18 q^{41} + 12 q^{43} - 8 q^{45} + 10 q^{49} - 8 q^{55} - 8 q^{59} + 12 q^{63} - 18 q^{65} - 18 q^{67} + 6 q^{71} + 12 q^{75} - 2 q^{81} + 8 q^{83} + 6 q^{87} + 12 q^{89} - 6 q^{95} - 8 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −2.36603 1.36603i 0 0.267949 0 3.00000 1.73205i 0 2.23205 + 3.86603i 0
17.2 0 −0.633975 0.366025i 0 3.73205 0 3.00000 1.73205i 0 −1.23205 2.13397i 0
49.1 0 −2.36603 + 1.36603i 0 0.267949 0 3.00000 + 1.73205i 0 2.23205 3.86603i 0
49.2 0 −0.633975 + 0.366025i 0 3.73205 0 3.00000 + 1.73205i 0 −1.23205 + 2.13397i 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
104.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.ba.a 4
4.b odd 2 1 104.2.s.b yes 4
8.b even 2 1 416.2.ba.b 4
8.d odd 2 1 104.2.s.a 4
12.b even 2 1 936.2.dg.a 4
13.e even 6 1 416.2.ba.b 4
24.f even 2 1 936.2.dg.b 4
52.i odd 6 1 104.2.s.a 4
104.p odd 6 1 104.2.s.b yes 4
104.s even 6 1 inner 416.2.ba.a 4
156.r even 6 1 936.2.dg.b 4
312.ba even 6 1 936.2.dg.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.s.a 4 8.d odd 2 1
104.2.s.a 4 52.i odd 6 1
104.2.s.b yes 4 4.b odd 2 1
104.2.s.b yes 4 104.p odd 6 1
416.2.ba.a 4 1.a even 1 1 trivial
416.2.ba.a 4 104.s even 6 1 inner
416.2.ba.b 4 8.b even 2 1
416.2.ba.b 4 13.e even 6 1
936.2.dg.a 4 12.b even 2 1
936.2.dg.a 4 312.ba even 6 1
936.2.dg.b 4 24.f even 2 1
936.2.dg.b 4 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6T_{3}^{3} + 14T_{3}^{2} + 12T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$29$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$47$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$53$ \( T^{4} + 114T^{2} + 1521 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$67$ \( T^{4} + 18 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 13924 \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
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