Properties

Label 416.2.i.a
Level $416$
Weight $2$
Character orbit 416.i
Analytic conductor $3.322$
Analytic rank $0$
Dimension $2$
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(289,416)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.i (of order \(3\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} + q^{5} - \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{11} + (3 \zeta_{6} + 1) q^{13} + (2 \zeta_{6} - 2) q^{15} - 3 \zeta_{6} q^{17} + 2 \zeta_{6} q^{19} + (2 \zeta_{6} - 2) q^{23} - 4 q^{25} - 4 q^{27} + (5 \zeta_{6} - 5) q^{29} + 2 q^{31} - 8 \zeta_{6} q^{33} + (5 \zeta_{6} - 5) q^{37} + (2 \zeta_{6} - 8) q^{39} + (3 \zeta_{6} - 3) q^{41} + 4 \zeta_{6} q^{43} - \zeta_{6} q^{45} + 6 q^{47} + ( - 7 \zeta_{6} + 7) q^{49} + 6 q^{51} + 13 q^{53} + (4 \zeta_{6} - 4) q^{55} - 4 q^{57} - 12 \zeta_{6} q^{59} + 7 \zeta_{6} q^{61} + (3 \zeta_{6} + 1) q^{65} + ( - 14 \zeta_{6} + 14) q^{67} - 4 \zeta_{6} q^{69} - 6 \zeta_{6} q^{71} + 7 q^{73} + ( - 8 \zeta_{6} + 8) q^{75} + 8 q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 4 q^{83} - 3 \zeta_{6} q^{85} - 10 \zeta_{6} q^{87} + (14 \zeta_{6} - 14) q^{89} + (4 \zeta_{6} - 4) q^{93} + 2 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - q^{9} - 4 q^{11} + 5 q^{13} - 2 q^{15} - 3 q^{17} + 2 q^{19} - 2 q^{23} - 8 q^{25} - 8 q^{27} - 5 q^{29} + 4 q^{31} - 8 q^{33} - 5 q^{37} - 14 q^{39} - 3 q^{41} + 4 q^{43} - q^{45} + 12 q^{47} + 7 q^{49} + 12 q^{51} + 26 q^{53} - 4 q^{55} - 8 q^{57} - 12 q^{59} + 7 q^{61} + 5 q^{65} + 14 q^{67} - 4 q^{69} - 6 q^{71} + 14 q^{73} + 8 q^{75} + 16 q^{79} + 11 q^{81} + 8 q^{83} - 3 q^{85} - 10 q^{87} - 14 q^{89} - 4 q^{93} + 2 q^{95} + 2 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 0 1.00000 0 0 0 −0.500000 + 0.866025i 0
321.1 0 −1.00000 + 1.73205i 0 1.00000 0 0 0 −0.500000 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.i.a 2
4.b odd 2 1 416.2.i.b yes 2
8.b even 2 1 832.2.i.h 2
8.d odd 2 1 832.2.i.b 2
13.c even 3 1 inner 416.2.i.a 2
13.c even 3 1 5408.2.a.k 1
13.e even 6 1 5408.2.a.j 1
52.i odd 6 1 5408.2.a.c 1
52.j odd 6 1 416.2.i.b yes 2
52.j odd 6 1 5408.2.a.d 1
104.n odd 6 1 832.2.i.b 2
104.r even 6 1 832.2.i.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.a 2 1.a even 1 1 trivial
416.2.i.a 2 13.c even 3 1 inner
416.2.i.b yes 2 4.b odd 2 1
416.2.i.b yes 2 52.j odd 6 1
832.2.i.b 2 8.d odd 2 1
832.2.i.b 2 104.n odd 6 1
832.2.i.h 2 8.b even 2 1
832.2.i.h 2 104.r even 6 1
5408.2.a.c 1 52.i odd 6 1
5408.2.a.d 1 52.j odd 6 1
5408.2.a.j 1 13.e even 6 1
5408.2.a.k 1 13.c even 3 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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