Properties

Label 416.2.b.b
Level $416$
Weight $2$
Character orbit 416.b
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(209,416)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.b (of order \(2\), degree \(1\), not 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{3} - 2 \beta_{2} q^{5} + ( - \beta_{3} + 3) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{3} - 2 \beta_{2} q^{5} + ( - \beta_{3} + 3) q^{7} - q^{9} + ( - \beta_{2} - 3 \beta_1) q^{11} - \beta_1 q^{13} + 4 \beta_{3} q^{15} + (2 \beta_{3} + 2) q^{17} + (\beta_{2} - \beta_1) q^{19} + ( - 2 \beta_{2} + 6 \beta_1) q^{21} - 4 q^{23} - 7 q^{25} + 4 \beta_1 q^{27} - 2 \beta_1 q^{29} + (\beta_{3} + 5) q^{31} + (2 \beta_{3} + 6) q^{33} + ( - 6 \beta_{2} + 6 \beta_1) q^{35} + (4 \beta_{2} + 2 \beta_1) q^{37} + 2 q^{39} + (4 \beta_{3} + 2) q^{41} + (2 \beta_{2} - 4 \beta_1) q^{43} + 2 \beta_{2} q^{45} + ( - \beta_{3} - 5) q^{47} + ( - 6 \beta_{3} + 5) q^{49} + (4 \beta_{2} + 4 \beta_1) q^{51} + (4 \beta_{2} - 4 \beta_1) q^{53} + ( - 6 \beta_{3} - 6) q^{55} + ( - 2 \beta_{3} + 2) q^{57} + ( - 3 \beta_{2} - 5 \beta_1) q^{59} + ( - 4 \beta_{2} + 4 \beta_1) q^{61} + (\beta_{3} - 3) q^{63} - 2 \beta_{3} q^{65} + (\beta_{2} - \beta_1) q^{67} - 8 \beta_1 q^{69} + (3 \beta_{3} + 3) q^{71} + ( - 2 \beta_{3} - 4) q^{73} - 14 \beta_1 q^{75} - 6 \beta_1 q^{77} + ( - 2 \beta_{3} - 2) q^{79} - 11 q^{81} + ( - \beta_{2} + 5 \beta_1) q^{83} + ( - 4 \beta_{2} - 12 \beta_1) q^{85} + 4 q^{87} - 10 \beta_{3} q^{89} + (\beta_{2} - 3 \beta_1) q^{91} + (2 \beta_{2} + 10 \beta_1) q^{93} + ( - 2 \beta_{3} + 6) q^{95} + (6 \beta_{3} - 4) q^{97} + (\beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 2.00000i 0 3.46410i 0 4.73205 0 −1.00000 0
209.2 0 2.00000i 0 3.46410i 0 1.26795 0 −1.00000 0
209.3 0 2.00000i 0 3.46410i 0 1.26795 0 −1.00000 0
209.4 0 2.00000i 0 3.46410i 0 4.73205 0 −1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.b.b 4
3.b odd 2 1 3744.2.g.b 4
4.b odd 2 1 104.2.b.b 4
8.b even 2 1 inner 416.2.b.b 4
8.d odd 2 1 104.2.b.b 4
12.b even 2 1 936.2.g.b 4
16.e even 4 1 3328.2.a.m 2
16.e even 4 1 3328.2.a.bc 2
16.f odd 4 1 3328.2.a.n 2
16.f odd 4 1 3328.2.a.bd 2
24.f even 2 1 936.2.g.b 4
24.h odd 2 1 3744.2.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.b.b 4 4.b odd 2 1
104.2.b.b 4 8.d odd 2 1
416.2.b.b 4 1.a even 1 1 trivial
416.2.b.b 4 8.b even 2 1 inner
936.2.g.b 4 12.b even 2 1
936.2.g.b 4 24.f even 2 1
3328.2.a.m 2 16.e even 4 1
3328.2.a.n 2 16.f odd 4 1
3328.2.a.bc 2 16.e even 4 1
3328.2.a.bd 2 16.f odd 4 1
3744.2.g.b 4 3.b odd 2 1
3744.2.g.b 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) 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
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 22)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 22)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$89$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 92)^{2} \) Copy content Toggle raw display
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