Properties

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

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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: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{3} - 2 \beta_{3} q^{5} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{3} - 2 \beta_{3} q^{5} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9} + (\beta_{2} - 3 \beta_1 + 1) q^{11} + (4 \beta_{2} + 1) q^{13} + (4 \beta_{2} + 2 \beta_1 + 4) q^{15} + ( - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{17} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{19} + (4 \beta_{3} - 5) q^{21} + (3 \beta_{2} - 3 \beta_1 + 3) q^{23} + 3 q^{25} + ( - \beta_{3} - 1) q^{27} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{29} - 4 \beta_{3} q^{31} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{33} + (6 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{35} + ( - \beta_{2} + 6 \beta_1 - 1) q^{37} + (4 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{39} + (3 \beta_{2} + 2 \beta_1 + 3) q^{41} + (5 \beta_{3} - 3 \beta_{2} + 5 \beta_1) q^{43} + 8 \beta_{2} q^{45} + 6 q^{47} + ( - 4 \beta_{2} - 6 \beta_1 - 4) q^{49} + ( - 5 \beta_{3} + 7) q^{51} + 2 \beta_{3} q^{53} + ( - 12 \beta_{2} + 2 \beta_1 - 12) q^{55} + 3 q^{57} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{59} + 7 \beta_{2} q^{61} + ( - 4 \beta_{2} - 6 \beta_1 - 4) q^{63} + (6 \beta_{3} + 8 \beta_1) q^{65} + ( - 9 \beta_{2} - 3 \beta_1 - 9) q^{67} - 3 \beta_{2} q^{69} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{71} + ( - 6 \beta_{3} + 4) q^{73} + (3 \beta_{2} + 3 \beta_1 + 3) q^{75} + ( - 8 \beta_{3} + 3) q^{77} - 6 q^{79} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + 4 q^{83} + ( - 6 \beta_{3} - 8 \beta_{2} - 6 \beta_1) q^{85} + ( - 7 \beta_{3} - 11 \beta_{2} - 7 \beta_1) q^{87} + (9 \beta_{2} - 4 \beta_1 + 9) q^{89} + (\beta_{3} - 9 \beta_{2} - 3 \beta_1 - 12) q^{91} + (8 \beta_{2} + 4 \beta_1 + 8) q^{93} + (6 \beta_{3} - 12 \beta_{2} + 6 \beta_1) q^{95} - 9 \beta_{2} q^{97} + (2 \beta_{3} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 6 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 6 q^{29} + 10 q^{33} - 8 q^{35} - 2 q^{37} - 14 q^{39} + 6 q^{41} + 6 q^{43} - 16 q^{45} + 24 q^{47} - 8 q^{49} + 28 q^{51} - 24 q^{55} + 12 q^{57} - 6 q^{59} - 14 q^{61} - 8 q^{63} - 18 q^{67} + 6 q^{69} + 6 q^{71} + 16 q^{73} + 6 q^{75} + 12 q^{77} - 24 q^{79} + 2 q^{81} + 16 q^{83} + 16 q^{85} + 22 q^{87} + 18 q^{89} - 30 q^{91} + 16 q^{93} + 24 q^{95} + 18 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) 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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.207107 0.358719i 0 −2.82843 0 −0.792893 + 1.37333i 0 1.41421 2.44949i 0
289.2 0 1.20711 + 2.09077i 0 2.82843 0 −2.20711 + 3.82282i 0 −1.41421 + 2.44949i 0
321.1 0 −0.207107 + 0.358719i 0 −2.82843 0 −0.792893 1.37333i 0 1.41421 + 2.44949i 0
321.2 0 1.20711 2.09077i 0 2.82843 0 −2.20711 3.82282i 0 −1.41421 2.44949i 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.f yes 4
4.b odd 2 1 416.2.i.c 4
8.b even 2 1 832.2.i.k 4
8.d odd 2 1 832.2.i.p 4
13.c even 3 1 inner 416.2.i.f yes 4
13.c even 3 1 5408.2.a.o 2
13.e even 6 1 5408.2.a.n 2
52.i odd 6 1 5408.2.a.bf 2
52.j odd 6 1 416.2.i.c 4
52.j odd 6 1 5408.2.a.be 2
104.n odd 6 1 832.2.i.p 4
104.r even 6 1 832.2.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.i.c 4 4.b odd 2 1
416.2.i.c 4 52.j odd 6 1
416.2.i.f yes 4 1.a even 1 1 trivial
416.2.i.f yes 4 13.c even 3 1 inner
832.2.i.k 4 8.b even 2 1
832.2.i.k 4 104.r even 6 1
832.2.i.p 4 8.d odd 2 1
832.2.i.p 4 104.n odd 6 1
5408.2.a.n 2 13.e even 6 1
5408.2.a.o 2 13.c even 3 1
5408.2.a.be 2 52.j odd 6 1
5408.2.a.bf 2 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1 \) 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
$3$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( (T - 6)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 18 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( (T - 4)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$97$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
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