Properties

Label 2816.2.c.h
Level $2816$
Weight $2$
Character orbit 2816.c
Analytic conductor $22.486$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2816,2,Mod(1409,2816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2816, 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("2816.1409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.c (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: \(22.4858732092\)
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: no (minimal twist has level 352)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} - i q^{5} - 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - i q^{5} - 6 q^{9} + i q^{11} - 6 i q^{13} + 3 q^{15} - 4 q^{17} + 6 i q^{19} - 3 q^{23} + 4 q^{25} - 9 i q^{27} - 4 i q^{29} - 9 q^{31} - 3 q^{33} - 7 i q^{37} + 18 q^{39} + 2 q^{41} - 6 i q^{43} + 6 i q^{45} + 12 q^{47} - 7 q^{49} - 12 i q^{51} - 2 i q^{53} + q^{55} - 18 q^{57} - 9 i q^{59} + 8 i q^{61} - 6 q^{65} - 15 i q^{67} - 9 i q^{69} + 3 q^{71} + 6 q^{73} + 12 i q^{75} - 6 q^{79} + 9 q^{81} - 6 i q^{83} + 4 i q^{85} + 12 q^{87} + 5 q^{89} - 27 i q^{93} + 6 q^{95} - 3 q^{97} - 6 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 6 q^{15} - 8 q^{17} - 6 q^{23} + 8 q^{25} - 18 q^{31} - 6 q^{33} + 36 q^{39} + 4 q^{41} + 24 q^{47} - 14 q^{49} + 2 q^{55} - 36 q^{57} - 12 q^{65} + 6 q^{71} + 12 q^{73} - 12 q^{79} + 18 q^{81} + 24 q^{87} + 10 q^{89} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1409.1
1.00000i
1.00000i
0 3.00000i 0 1.00000i 0 0 0 −6.00000 0
1409.2 0 3.00000i 0 1.00000i 0 0 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
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 2816.2.c.h 2
4.b odd 2 1 2816.2.c.g 2
8.b even 2 1 inner 2816.2.c.h 2
8.d odd 2 1 2816.2.c.g 2
16.e even 4 1 352.2.a.f yes 1
16.e even 4 1 704.2.a.a 1
16.f odd 4 1 352.2.a.a 1
16.f odd 4 1 704.2.a.k 1
48.i odd 4 1 3168.2.a.i 1
48.i odd 4 1 6336.2.a.bs 1
48.k even 4 1 3168.2.a.h 1
48.k even 4 1 6336.2.a.bt 1
80.k odd 4 1 8800.2.a.bb 1
80.q even 4 1 8800.2.a.a 1
176.i even 4 1 3872.2.a.a 1
176.i even 4 1 7744.2.a.bj 1
176.l odd 4 1 3872.2.a.m 1
176.l odd 4 1 7744.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.a.a 1 16.f odd 4 1
352.2.a.f yes 1 16.e even 4 1
704.2.a.a 1 16.e even 4 1
704.2.a.k 1 16.f odd 4 1
2816.2.c.g 2 4.b odd 2 1
2816.2.c.g 2 8.d odd 2 1
2816.2.c.h 2 1.a even 1 1 trivial
2816.2.c.h 2 8.b even 2 1 inner
3168.2.a.h 1 48.k even 4 1
3168.2.a.i 1 48.i odd 4 1
3872.2.a.a 1 176.i even 4 1
3872.2.a.m 1 176.l odd 4 1
6336.2.a.bs 1 48.i odd 4 1
6336.2.a.bt 1 48.k even 4 1
7744.2.a.a 1 176.l odd 4 1
7744.2.a.bj 1 176.i even 4 1
8800.2.a.a 1 80.q even 4 1
8800.2.a.bb 1 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2816, [\chi])\):

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{23} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T + 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 81 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 225 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 5)^{2} \) Copy content Toggle raw display
$97$ \( (T + 3)^{2} \) Copy content Toggle raw display
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