Properties

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

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 121 \) 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{7} + 8 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1) q^{7} + 8 \beta_{2} q^{9} + \beta_1 q^{11} + (4 \beta_{2} + 1) q^{13} - 3 \beta_{2} q^{17} + ( - \beta_{3} - \beta_1) q^{19} + 11 q^{21} - \beta_1 q^{23} - 5 q^{25} + 5 \beta_{3} q^{27} + (5 \beta_{2} + 5) q^{29} + 11 \beta_{2} q^{33} + ( - 9 \beta_{2} - 9) q^{37} + (4 \beta_{3} + \beta_1) q^{39} + (3 \beta_{2} + 3) q^{41} + ( - 3 \beta_{3} - 3 \beta_1) q^{43} - 2 \beta_{3} q^{47} + ( - 4 \beta_{2} - 4) q^{49} - 3 \beta_{3} q^{51} - 8 q^{53} + 11 q^{57} + ( - \beta_{3} - \beta_1) q^{59} - 9 \beta_{2} q^{61} + 8 \beta_1 q^{63} + 3 \beta_1 q^{67} - 11 \beta_{2} q^{69} + ( - 3 \beta_{3} - 3 \beta_1) q^{71} - 4 q^{73} - 5 \beta_1 q^{75} + 11 q^{77} + 2 \beta_{3} q^{79} + ( - 31 \beta_{2} - 31) q^{81} - 4 \beta_{3} q^{83} + (5 \beta_{3} + 5 \beta_1) q^{87} + (\beta_{2} + 1) q^{89} + ( - \beta_{3} + 3 \beta_1) q^{91} + 7 \beta_{2} q^{97} + 8 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{9} - 4 q^{13} + 6 q^{17} + 44 q^{21} - 20 q^{25} + 10 q^{29} - 22 q^{33} - 18 q^{37} + 6 q^{41} - 8 q^{49} - 32 q^{53} + 44 q^{57} + 18 q^{61} + 22 q^{69} - 16 q^{73} + 44 q^{77} - 62 q^{81} + 2 q^{89} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.65831 2.87228i
1.65831 + 2.87228i
−1.65831 + 2.87228i
1.65831 2.87228i
0 −1.65831 2.87228i 0 0 0 −1.65831 + 2.87228i 0 −4.00000 + 6.92820i 0
289.2 0 1.65831 + 2.87228i 0 0 0 1.65831 2.87228i 0 −4.00000 + 6.92820i 0
321.1 0 −1.65831 + 2.87228i 0 0 0 −1.65831 2.87228i 0 −4.00000 6.92820i 0
321.2 0 1.65831 2.87228i 0 0 0 1.65831 + 2.87228i 0 −4.00000 6.92820i 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 416.2.i.d 4
4.b odd 2 1 inner 416.2.i.d 4
8.b even 2 1 832.2.i.n 4
8.d odd 2 1 832.2.i.n 4
13.c even 3 1 inner 416.2.i.d 4
13.c even 3 1 5408.2.a.x 2
13.e even 6 1 5408.2.a.w 2
52.i odd 6 1 5408.2.a.w 2
52.j odd 6 1 inner 416.2.i.d 4
52.j odd 6 1 5408.2.a.x 2
104.n odd 6 1 832.2.i.n 4
104.r even 6 1 832.2.i.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.d 4 1.a even 1 1 trivial
416.2.i.d 4 4.b odd 2 1 inner
416.2.i.d 4 13.c even 3 1 inner
416.2.i.d 4 52.j odd 6 1 inner
832.2.i.n 4 8.b even 2 1
832.2.i.n 4 8.d odd 2 1
832.2.i.n 4 104.n odd 6 1
832.2.i.n 4 104.r even 6 1
5408.2.a.w 2 13.e even 6 1
5408.2.a.w 2 52.i odd 6 1
5408.2.a.x 2 13.c even 3 1
5408.2.a.x 2 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 11T_{3}^{2} + 121 \) 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
$3$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$11$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$23$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 99T^{2} + 9801 \) Copy content Toggle raw display
$47$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$53$ \( (T + 8)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 11T^{2} + 121 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 99T^{2} + 9801 \) Copy content Toggle raw display
$71$ \( T^{4} + 99T^{2} + 9801 \) Copy content Toggle raw display
$73$ \( (T + 4)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 176)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
show more
show less