Properties

Label 800.4.c.g
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.449");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 q^{9} - 9 \beta q^{13} + 47 \beta q^{17} + 130 q^{29} - 107 \beta q^{37} - 230 q^{41} + 343 q^{49} + 259 \beta q^{53} + 830 q^{61} + 549 \beta q^{73} + 729 q^{81} + 1670 q^{89} - 297 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 54 q^{9} + 260 q^{29} - 460 q^{41} + 686 q^{49} + 1660 q^{61} + 1458 q^{81} + 3340 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 0 0 0 0 0 0 27.0000 0
449.2 0 0 0 0 0 0 0 27.0000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.c.g 2
4.b odd 2 1 CM 800.4.c.g 2
5.b even 2 1 inner 800.4.c.g 2
5.c odd 4 1 32.4.a.b 1
5.c odd 4 1 800.4.a.f 1
15.e even 4 1 288.4.a.a 1
20.d odd 2 1 inner 800.4.c.g 2
20.e even 4 1 32.4.a.b 1
20.e even 4 1 800.4.a.f 1
35.f even 4 1 1568.4.a.g 1
40.i odd 4 1 64.4.a.c 1
40.i odd 4 1 1600.4.a.ba 1
40.k even 4 1 64.4.a.c 1
40.k even 4 1 1600.4.a.ba 1
60.l odd 4 1 288.4.a.a 1
80.i odd 4 1 256.4.b.d 2
80.j even 4 1 256.4.b.d 2
80.s even 4 1 256.4.b.d 2
80.t odd 4 1 256.4.b.d 2
120.q odd 4 1 576.4.a.y 1
120.w even 4 1 576.4.a.y 1
140.j odd 4 1 1568.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.b 1 5.c odd 4 1
32.4.a.b 1 20.e even 4 1
64.4.a.c 1 40.i odd 4 1
64.4.a.c 1 40.k even 4 1
256.4.b.d 2 80.i odd 4 1
256.4.b.d 2 80.j even 4 1
256.4.b.d 2 80.s even 4 1
256.4.b.d 2 80.t odd 4 1
288.4.a.a 1 15.e even 4 1
288.4.a.a 1 60.l odd 4 1
576.4.a.y 1 120.q odd 4 1
576.4.a.y 1 120.w even 4 1
800.4.a.f 1 5.c odd 4 1
800.4.a.f 1 20.e even 4 1
800.4.c.g 2 1.a even 1 1 trivial
800.4.c.g 2 4.b odd 2 1 CM
800.4.c.g 2 5.b even 2 1 inner
800.4.c.g 2 20.d odd 2 1 inner
1568.4.a.g 1 35.f even 4 1
1568.4.a.g 1 140.j odd 4 1
1600.4.a.ba 1 40.i odd 4 1
1600.4.a.ba 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 324 \) Copy content Toggle raw display
$17$ \( T^{2} + 8836 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 130)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 45796 \) Copy content Toggle raw display
$41$ \( (T + 230)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 268324 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 830)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1205604 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 1670)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 352836 \) Copy content Toggle raw display
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