Properties

Label 416.2.a.f
Level $416$
Weight $2$
Character orbit 416.a
Self dual yes
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(1,416)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.a (trivial)

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: yes
Analytic conductor: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{3} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 3) q^{9} - \beta_1 q^{11} + q^{13} + (3 \beta_{2} - 2 \beta_1) q^{15} + (\beta_{3} + 4) q^{17} + \beta_1 q^{19} + ( - 3 \beta_{3} + 2) q^{21} + ( - \beta_{3} + 5) q^{25} + ( - 3 \beta_{2} + 2 \beta_1) q^{27} - 2 q^{29} + (2 \beta_{2} + \beta_1) q^{31} + (2 \beta_{3} + 4) q^{33} + 5 \beta_{2} q^{35} + \beta_{3} q^{37} - \beta_{2} q^{39} + (2 \beta_{3} - 2) q^{41} + (\beta_{2} - 2 \beta_1) q^{43} + (4 \beta_{3} - 10) q^{45} + ( - \beta_{2} + \beta_1) q^{47} + ( - \beta_{3} + 7) q^{49} + ( - \beta_{2} - 2 \beta_1) q^{51} + ( - 2 \beta_{3} - 6) q^{53} + ( - 2 \beta_{2} - 2 \beta_1) q^{55} + ( - 2 \beta_{3} - 4) q^{57} + (4 \beta_{2} - \beta_1) q^{59} + ( - 2 \beta_{3} + 2) q^{61} + ( - 8 \beta_{2} + 3 \beta_1) q^{63} + \beta_{3} q^{65} + (4 \beta_{2} + \beta_1) q^{67} + ( - 3 \beta_{2} - \beta_1) q^{71} - 6 q^{73} + ( - 8 \beta_{2} + 2 \beta_1) q^{75} + ( - 2 \beta_{3} - 12) q^{77} + (4 \beta_{2} - 2 \beta_1) q^{79} + ( - 4 \beta_{3} + 1) q^{81} + ( - 2 \beta_{2} + \beta_1) q^{83} + (3 \beta_{3} + 10) q^{85} + 2 \beta_{2} q^{87} - 6 q^{89} + ( - \beta_{2} + \beta_1) q^{91} - 16 q^{93} + (2 \beta_{2} + 2 \beta_1) q^{95} + ( - 4 \beta_{3} - 6) q^{97} + (2 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 14 q^{9} + 4 q^{13} + 14 q^{17} + 14 q^{21} + 22 q^{25} - 8 q^{29} + 12 q^{33} - 2 q^{37} - 12 q^{41} - 48 q^{45} + 30 q^{49} - 20 q^{53} - 12 q^{57} + 12 q^{61} - 2 q^{65} - 24 q^{73} - 44 q^{77} + 12 q^{81} + 34 q^{85} - 24 q^{89} - 64 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.546295
2.58874
−2.58874
0.546295
0 −3.11473 0 −3.70156 0 −4.20732 0 6.70156 0
1.2 0 −1.81616 0 2.70156 0 3.36131 0 0.298438 0
1.3 0 1.81616 0 2.70156 0 −3.36131 0 0.298438 0
1.4 0 3.11473 0 −3.70156 0 4.20732 0 6.70156 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.a.f 4
3.b odd 2 1 3744.2.a.be 4
4.b odd 2 1 inner 416.2.a.f 4
8.b even 2 1 832.2.a.p 4
8.d odd 2 1 832.2.a.p 4
12.b even 2 1 3744.2.a.be 4
13.b even 2 1 5408.2.a.bj 4
16.e even 4 2 3328.2.b.bb 8
16.f odd 4 2 3328.2.b.bb 8
24.f even 2 1 7488.2.a.da 4
24.h odd 2 1 7488.2.a.da 4
52.b odd 2 1 5408.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.f 4 1.a even 1 1 trivial
416.2.a.f 4 4.b odd 2 1 inner
832.2.a.p 4 8.b even 2 1
832.2.a.p 4 8.d odd 2 1
3328.2.b.bb 8 16.e even 4 2
3328.2.b.bb 8 16.f odd 4 2
3744.2.a.be 4 3.b odd 2 1
3744.2.a.be 4 12.b even 2 1
5408.2.a.bj 4 13.b even 2 1
5408.2.a.bj 4 52.b odd 2 1
7488.2.a.da 4 24.f even 2 1
7488.2.a.da 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 13T_{3}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(416))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 13T^{2} + 32 \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 10)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 29T^{2} + 200 \) Copy content Toggle raw display
$11$ \( T^{4} - 28T^{2} + 32 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 7 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 28T^{2} + 32 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 104T^{2} + 2048 \) Copy content Toggle raw display
$37$ \( (T^{2} + T - 10)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 101T^{2} + 2048 \) Copy content Toggle raw display
$47$ \( T^{4} - 29T^{2} + 200 \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 188T^{2} + 800 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 284 T^{2} + 20000 \) Copy content Toggle raw display
$71$ \( T^{4} - 181T^{2} + 7688 \) Copy content Toggle raw display
$73$ \( (T + 6)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 224T^{2} + 2048 \) Copy content Toggle raw display
$83$ \( T^{4} - 56T^{2} + 128 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 148)^{2} \) Copy content Toggle raw display
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