Properties

Label 416.2.b.a
Level $416$
Weight $2$
Character orbit 416.b
Analytic conductor $3.322$
Analytic rank $0$
Dimension $2$
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: \(2\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + 3 i q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + 3 i q^{5} - 3 q^{7} + 2 q^{9} + i q^{13} - 3 q^{15} - 7 q^{17} + 4 i q^{19} - 3 i q^{21} - 4 q^{23} - 4 q^{25} + 5 i q^{27} - 4 i q^{29} + 8 q^{31} - 9 i q^{35} + 7 i q^{37} - q^{39} + 2 q^{41} + i q^{43} + 6 i q^{45} + 7 q^{47} + 2 q^{49} - 7 i q^{51} + 4 i q^{53} - 4 q^{57} + 14 i q^{59} - 10 i q^{61} - 6 q^{63} - 3 q^{65} - 2 i q^{67} - 4 i q^{69} + 3 q^{71} + 14 q^{73} - 4 i q^{75} + 10 q^{79} + q^{81} - 14 i q^{83} - 21 i q^{85} + 4 q^{87} - 3 i q^{91} + 8 i q^{93} - 12 q^{95} + 8 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{7} + 4 q^{9} - 6 q^{15} - 14 q^{17} - 8 q^{23} - 8 q^{25} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 14 q^{47} + 4 q^{49} - 8 q^{57} - 12 q^{63} - 6 q^{65} + 6 q^{71} + 28 q^{73} + 20 q^{79} + 2 q^{81} + 8 q^{87} - 24 q^{95} + 16 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.00000i
1.00000i
0 1.00000i 0 3.00000i 0 −3.00000 0 2.00000 0
209.2 0 1.00000i 0 3.00000i 0 −3.00000 0 2.00000 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.a 2
3.b odd 2 1 3744.2.g.a 2
4.b odd 2 1 104.2.b.a 2
8.b even 2 1 inner 416.2.b.a 2
8.d odd 2 1 104.2.b.a 2
12.b even 2 1 936.2.g.a 2
16.e even 4 1 3328.2.a.f 1
16.e even 4 1 3328.2.a.g 1
16.f odd 4 1 3328.2.a.c 1
16.f odd 4 1 3328.2.a.j 1
24.f even 2 1 936.2.g.a 2
24.h odd 2 1 3744.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.b.a 2 4.b odd 2 1
104.2.b.a 2 8.d odd 2 1
416.2.b.a 2 1.a even 1 1 trivial
416.2.b.a 2 8.b even 2 1 inner
936.2.g.a 2 12.b even 2 1
936.2.g.a 2 24.f even 2 1
3328.2.a.c 1 16.f odd 4 1
3328.2.a.f 1 16.e even 4 1
3328.2.a.g 1 16.e even 4 1
3328.2.a.j 1 16.f odd 4 1
3744.2.g.a 2 3.b odd 2 1
3744.2.g.a 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T - 7)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 196 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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