Properties

Label 416.2.f.a
Level $416$
Weight $2$
Character orbit 416.f
Analytic conductor $3.322$
Analytic rank $1$
Dimension $2$
CM no
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(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: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 3 q^{3} + 3 i q^{5} - i q^{7} + 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 3 i q^{5} - i q^{7} + 6 q^{9} + 4 i q^{11} + ( - 3 i - 2) q^{13} - 9 i q^{15} - 5 q^{17} - 6 i q^{19} + 3 i q^{21} - 6 q^{23} - 4 q^{25} - 9 q^{27} - 4 q^{29} - 12 i q^{33} + 3 q^{35} - 3 i q^{37} + (9 i + 6) q^{39} - 12 i q^{41} - 3 q^{43} + 18 i q^{45} + 7 i q^{47} + 6 q^{49} + 15 q^{51} - 2 q^{53} - 12 q^{55} + 18 i q^{57} - 2 i q^{59} - 12 q^{61} - 6 i q^{63} + ( - 6 i + 9) q^{65} - 4 i q^{67} + 18 q^{69} + 11 i q^{71} + 6 i q^{73} + 12 q^{75} + 4 q^{77} + 6 q^{79} + 9 q^{81} - 10 i q^{83} - 15 i q^{85} + 12 q^{87} + 6 i q^{89} + (2 i - 3) q^{91} + 18 q^{95} + 24 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 12 q^{9} - 4 q^{13} - 10 q^{17} - 12 q^{23} - 8 q^{25} - 18 q^{27} - 8 q^{29} + 6 q^{35} + 12 q^{39} - 6 q^{43} + 12 q^{49} + 30 q^{51} - 4 q^{53} - 24 q^{55} - 24 q^{61} + 18 q^{65} + 36 q^{69} + 24 q^{75} + 8 q^{77} + 12 q^{79} + 18 q^{81} + 24 q^{87} - 6 q^{91} + 36 q^{95}+O(q^{100}) \) 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 −3.00000 0 3.00000i 0 1.00000i 0 6.00000 0
129.2 0 −3.00000 0 3.00000i 0 1.00000i 0 6.00000 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
13.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.f.a 2
3.b odd 2 1 3744.2.c.c 2
4.b odd 2 1 416.2.f.c yes 2
8.b even 2 1 832.2.f.e 2
8.d odd 2 1 832.2.f.a 2
12.b even 2 1 3744.2.c.b 2
13.b even 2 1 inner 416.2.f.a 2
13.d odd 4 1 5408.2.a.a 1
13.d odd 4 1 5408.2.a.b 1
39.d odd 2 1 3744.2.c.c 2
52.b odd 2 1 416.2.f.c yes 2
52.f even 4 1 5408.2.a.l 1
52.f even 4 1 5408.2.a.m 1
104.e even 2 1 832.2.f.e 2
104.h odd 2 1 832.2.f.a 2
156.h even 2 1 3744.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.f.a 2 1.a even 1 1 trivial
416.2.f.a 2 13.b even 2 1 inner
416.2.f.c yes 2 4.b odd 2 1
416.2.f.c yes 2 52.b odd 2 1
832.2.f.a 2 8.d odd 2 1
832.2.f.a 2 104.h odd 2 1
832.2.f.e 2 8.b even 2 1
832.2.f.e 2 104.e even 2 1
3744.2.c.b 2 12.b even 2 1
3744.2.c.b 2 156.h even 2 1
3744.2.c.c 2 3.b odd 2 1
3744.2.c.c 2 39.d odd 2 1
5408.2.a.a 1 13.d odd 4 1
5408.2.a.b 1 13.d odd 4 1
5408.2.a.l 1 52.f even 4 1
5408.2.a.m 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 \(S_{2}^{\mathrm{new}}(416, [\chi])\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{5}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 13 \) Copy content Toggle raw display
$17$ \( (T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T + 3)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 49 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4 \) Copy content Toggle raw display
$61$ \( (T + 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 121 \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 100 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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