Properties

Label 416.2.k.b
Level $416$
Weight $2$
Character orbit 416.k
Analytic conductor $3.322$
Analytic rank $0$
Dimension $2$
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(31,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.k (of order \(4\), 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: \(2\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + ( - i - 1) q^{5} + (i + 1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} + ( - i - 1) q^{5} + (i + 1) q^{7} - q^{9} + (3 i + 3) q^{11} + (2 i - 3) q^{13} + ( - 2 i + 2) q^{15} + 4 i q^{17} + (3 i - 3) q^{19} + (2 i - 2) q^{21} - 3 i q^{25} + 4 i q^{27} - 6 q^{29} + ( - 3 i + 3) q^{31} + (6 i - 6) q^{33} - 2 i q^{35} + ( - 3 i + 3) q^{37} + ( - 6 i - 4) q^{39} + (i + 1) q^{41} - 4 q^{43} + (i + 1) q^{45} + (5 i + 5) q^{47} - 5 i q^{49} - 8 q^{51} + 6 q^{53} - 6 i q^{55} + ( - 6 i - 6) q^{57} + (7 i + 7) q^{59} + 14 q^{61} + ( - i - 1) q^{63} + (i + 5) q^{65} + ( - 5 i + 5) q^{67} + (5 i - 5) q^{71} + ( - 9 i + 9) q^{73} + 6 q^{75} + 6 i q^{77} - 6 i q^{79} - 11 q^{81} + (7 i - 7) q^{83} + ( - 4 i + 4) q^{85} - 12 i q^{87} + ( - 5 i + 5) q^{89} + ( - i - 5) q^{91} + (6 i + 6) q^{93} + 6 q^{95} + (13 i + 13) q^{97} + ( - 3 i - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{9} + 6 q^{11} - 6 q^{13} + 4 q^{15} - 6 q^{19} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 12 q^{33} + 6 q^{37} - 8 q^{39} + 2 q^{41} - 8 q^{43} + 2 q^{45} + 10 q^{47} - 16 q^{51} + 12 q^{53} - 12 q^{57} + 14 q^{59} + 28 q^{61} - 2 q^{63} + 10 q^{65} + 10 q^{67} - 10 q^{71} + 18 q^{73} + 12 q^{75} - 22 q^{81} - 14 q^{83} + 8 q^{85} + 10 q^{89} - 10 q^{91} + 12 q^{93} + 12 q^{95} + 26 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.00000i
1.00000i
0 2.00000i 0 −1.00000 + 1.00000i 0 1.00000 1.00000i 0 −1.00000 0
255.1 0 2.00000i 0 −1.00000 1.00000i 0 1.00000 + 1.00000i 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
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.k.b yes 2
4.b odd 2 1 416.2.k.a 2
8.b even 2 1 832.2.k.c 2
8.d odd 2 1 832.2.k.b 2
13.d odd 4 1 416.2.k.a 2
52.f even 4 1 inner 416.2.k.b yes 2
104.j odd 4 1 832.2.k.b 2
104.m even 4 1 832.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.k.a 2 4.b odd 2 1
416.2.k.a 2 13.d odd 4 1
416.2.k.b yes 2 1.a even 1 1 trivial
416.2.k.b yes 2 52.f even 4 1 inner
832.2.k.b 2 8.d odd 2 1
832.2.k.b 2 104.j odd 4 1
832.2.k.c 2 8.b even 2 1
832.2.k.c 2 104.m even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$79$ \( T^{2} + 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 26T + 338 \) Copy content Toggle raw display
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