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Properties

Label 2816.2.c.p
Level $2816$
Weight $2$
Character orbit 2816.c
Analytic conductor $22.486$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 2) q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 2 \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 2) q^{7} + (\beta_{3} - 2) q^{9} - \beta_{2} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{3} + 3) q^{15} + 2 q^{17} + 4 \beta_{2} q^{19} + (8 \beta_{2} - 2 \beta_1) q^{21} + ( - \beta_{3} + 5) q^{23} - 3 \beta_{3} q^{25} + (4 \beta_{2} + \beta_1) q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_{3} + 3) q^{31} + (\beta_{3} - 1) q^{33} + ( - 8 \beta_{2} - 2 \beta_1) q^{35} + (6 \beta_{2} + \beta_1) q^{37} + 8 q^{39} + (2 \beta_{3} - 4) q^{41} + ( - 4 \beta_{2} - 2 \beta_1) q^{43} - 2 \beta_{2} q^{45} - 8 q^{47} + ( - 4 \beta_{3} + 13) q^{49} + 2 \beta_1 q^{51} + ( - 6 \beta_{2} - 4 \beta_1) q^{53} + ( - \beta_{3} - 1) q^{55} + ( - 4 \beta_{3} + 4) q^{57} + 5 \beta_1 q^{59} + ( - 2 \beta_{2} + 2 \beta_1) q^{61} + ( - 4 \beta_{3} + 12) q^{63} + ( - 4 \beta_{3} - 8) q^{65} + ( - 8 \beta_{2} - \beta_1) q^{67} + ( - 4 \beta_{2} + 5 \beta_1) q^{69} + ( - 3 \beta_{3} - 1) q^{71} - 2 \beta_{3} q^{73} - 12 \beta_{2} q^{75} - 2 \beta_1 q^{77} + (2 \beta_{3} + 6) q^{79} - 7 q^{81} + ( - 4 \beta_{2} + 2 \beta_1) q^{83} + ( - 4 \beta_{2} - 2 \beta_1) q^{85} + 8 q^{87} + ( - 3 \beta_{3} + 5) q^{89} - 16 \beta_{2} q^{91} + (4 \beta_{2} + 3 \beta_1) q^{93} + (4 \beta_{3} + 4) q^{95} + (\beta_{3} + 13) q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 6 q^{9} + 14 q^{15} + 8 q^{17} + 18 q^{23} - 6 q^{25} + 14 q^{31} - 2 q^{33} + 32 q^{39} - 12 q^{41} - 32 q^{47} + 44 q^{49} - 6 q^{55} + 8 q^{57} + 40 q^{63} - 40 q^{65} - 10 q^{71} - 4 q^{73} + 28 q^{79} - 28 q^{81} + 32 q^{87} + 14 q^{89} + 24 q^{95} + 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 2.56155i
− 1.56155i
1.56155i
2.56155i
0 − 2.56155i 0 0.561553i 0 −5.12311 0 −3.56155 0
1409.2 0 − 1.56155i 0 3.56155i 0 3.12311 0 0.561553 0
1409.3 0 1.56155i 0 − 3.56155i 0 3.12311 0 0.561553 0
1409.4 0 2.56155i 0 − 0.561553i 0 −5.12311 0 −3.56155 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.p 4
4.b odd 2 1 2816.2.c.w 4
8.b even 2 1 inner 2816.2.c.p 4
8.d odd 2 1 2816.2.c.w 4
16.e even 4 1 176.2.a.d 2
16.e even 4 1 704.2.a.p 2
16.f odd 4 1 88.2.a.b ✓ 2
16.f odd 4 1 704.2.a.m 2
48.i odd 4 1 1584.2.a.t 2
48.i odd 4 1 6336.2.a.cx 2
48.k even 4 1 792.2.a.h 2
48.k even 4 1 6336.2.a.cu 2
80.i odd 4 1 4400.2.b.v 4
80.j even 4 1 2200.2.b.g 4
80.k odd 4 1 2200.2.a.o 2
80.q even 4 1 4400.2.a.bp 2
80.s even 4 1 2200.2.b.g 4
80.t odd 4 1 4400.2.b.v 4
112.j even 4 1 4312.2.a.n 2
112.l odd 4 1 8624.2.a.cb 2
176.i even 4 1 968.2.a.j 2
176.i even 4 1 7744.2.a.by 2
176.l odd 4 1 1936.2.a.r 2
176.l odd 4 1 7744.2.a.cl 2
176.v odd 20 4 968.2.i.r 8
176.x even 20 4 968.2.i.q 8
528.s odd 4 1 8712.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b ✓ 2 16.f odd 4 1
176.2.a.d 2 16.e even 4 1
704.2.a.m 2 16.f odd 4 1
704.2.a.p 2 16.e even 4 1
792.2.a.h 2 48.k even 4 1
968.2.a.j 2 176.i even 4 1
968.2.i.q 8 176.x even 20 4
968.2.i.r 8 176.v odd 20 4
1584.2.a.t 2 48.i odd 4 1
1936.2.a.r 2 176.l odd 4 1
2200.2.a.o 2 80.k odd 4 1
2200.2.b.g 4 80.j even 4 1
2200.2.b.g 4 80.s even 4 1
2816.2.c.p 4 1.a even 1 1 trivial
2816.2.c.p 4 8.b even 2 1 inner
2816.2.c.w 4 4.b odd 2 1
2816.2.c.w 4 8.d odd 2 1
4312.2.a.n 2 112.j even 4 1
4400.2.a.bp 2 80.q even 4 1
4400.2.b.v 4 80.i odd 4 1
4400.2.b.v 4 80.t odd 4 1
6336.2.a.cu 2 48.k even 4 1
6336.2.a.cx 2 48.i odd 4 1
7744.2.a.by 2 176.i even 4 1
7744.2.a.cl 2 176.l odd 4 1
8624.2.a.cb 2 112.l odd 4 1
8712.2.a.bb 2 528.s odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 9T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 16 \) Copy content Toggle raw display
\( T_{23}^{2} - 9T_{23} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$31$ \( (T^{2} - 7 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 69T^{2} + 676 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T + 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$59$ \( T^{4} + 225 T^{2} + 10000 \) Copy content Toggle raw display
$61$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$71$ \( (T^{2} + 5 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} - 7 T - 26)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 27 T + 178)^{2} \) Copy content Toggle raw display
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