Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(129,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.f (of order \(2\), degree \(1\), 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: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} - 3 q^{9} + (\beta - 3) q^{13} - 2 q^{17} - 11 q^{25} + 10 q^{29} + 6 \beta q^{37} + 4 \beta q^{41} - 6 \beta q^{45} + 7 q^{49} + 14 q^{53} - 10 q^{61} + ( - 6 \beta - 8) q^{65} - 8 \beta q^{73} + 9 q^{81} - 4 \beta q^{85} - 8 \beta q^{89} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{9} - 6 q^{13} - 4 q^{17} - 22 q^{25} + 20 q^{29} + 14 q^{49} + 28 q^{53} - 20 q^{61} - 16 q^{65} + 18 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 0 0 4.00000i 0 0 0 −3.00000 0
129.2 0 0 0 4.00000i 0 0 0 −3.00000 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 CM by \(\Q(\sqrt{-1}) \)
13.b even 2 1 inner
52.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.f.b 2
3.b odd 2 1 3744.2.c.a 2
4.b odd 2 1 CM 416.2.f.b 2
8.b even 2 1 832.2.f.c 2
8.d odd 2 1 832.2.f.c 2
12.b even 2 1 3744.2.c.a 2
13.b even 2 1 inner 416.2.f.b 2
13.d odd 4 1 5408.2.a.f 1
13.d odd 4 1 5408.2.a.h 1
39.d odd 2 1 3744.2.c.a 2
52.b odd 2 1 inner 416.2.f.b 2
52.f even 4 1 5408.2.a.f 1
52.f even 4 1 5408.2.a.h 1
104.e even 2 1 832.2.f.c 2
104.h odd 2 1 832.2.f.c 2
156.h even 2 1 3744.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.f.b 2 1.a even 1 1 trivial
416.2.f.b 2 4.b odd 2 1 CM
416.2.f.b 2 13.b even 2 1 inner
416.2.f.b 2 52.b odd 2 1 inner
832.2.f.c 2 8.b even 2 1
832.2.f.c 2 8.d odd 2 1
832.2.f.c 2 104.e even 2 1
832.2.f.c 2 104.h odd 2 1
3744.2.c.a 2 3.b odd 2 1
3744.2.c.a 2 12.b even 2 1
3744.2.c.a 2 39.d odd 2 1
3744.2.c.a 2 156.h even 2 1
5408.2.a.f 1 13.d odd 4 1
5408.2.a.f 1 52.f even 4 1
5408.2.a.h 1 13.d odd 4 1
5408.2.a.h 1 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 144 \) Copy content Toggle raw display
$41$ \( T^{2} + 64 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 14)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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