Properties

Label 800.4.c.a
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
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] = [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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta q^{3} - 8 \beta q^{7} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{3} - 8 \beta q^{7} - 37 q^{9} - 40 q^{11} - 25 \beta q^{13} + 15 \beta q^{17} - 40 q^{19} + 128 q^{21} + 24 \beta q^{23} - 40 \beta q^{27} + 34 q^{29} + 320 q^{31} - 160 \beta q^{33} - 155 \beta q^{37} + 400 q^{39} + 410 q^{41} + 76 \beta q^{43} + 208 \beta q^{47} + 87 q^{49} - 240 q^{51} - 205 \beta q^{53} - 160 \beta q^{57} + 200 q^{59} + 30 q^{61} + 296 \beta q^{63} - 388 \beta q^{67} - 384 q^{69} + 400 q^{71} - 315 \beta q^{73} + 320 \beta q^{77} + 1120 q^{79} - 359 q^{81} + 276 \beta q^{83} + 136 \beta q^{87} + 326 q^{89} - 800 q^{91} + 1280 \beta q^{93} + 55 \beta q^{97} + 1480 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 74 q^{9} - 80 q^{11} - 80 q^{19} + 256 q^{21} + 68 q^{29} + 640 q^{31} + 800 q^{39} + 820 q^{41} + 174 q^{49} - 480 q^{51} + 400 q^{59} + 60 q^{61} - 768 q^{69} + 800 q^{71} + 2240 q^{79} - 718 q^{81} + 652 q^{89} - 1600 q^{91} + 2960 q^{99}+O(q^{100}) \) 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 8.00000i 0 0 0 16.0000i 0 −37.0000 0
449.2 0 8.00000i 0 0 0 16.0000i 0 −37.0000 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
5.b even 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.a 2
4.b odd 2 1 800.4.c.b 2
5.b even 2 1 inner 800.4.c.a 2
5.c odd 4 1 32.4.a.c yes 1
5.c odd 4 1 800.4.a.a 1
15.e even 4 1 288.4.a.i 1
20.d odd 2 1 800.4.c.b 2
20.e even 4 1 32.4.a.a 1
20.e even 4 1 800.4.a.k 1
35.f even 4 1 1568.4.a.c 1
40.i odd 4 1 64.4.a.a 1
40.i odd 4 1 1600.4.a.bw 1
40.k even 4 1 64.4.a.e 1
40.k even 4 1 1600.4.a.e 1
60.l odd 4 1 288.4.a.h 1
80.i odd 4 1 256.4.b.c 2
80.j even 4 1 256.4.b.e 2
80.s even 4 1 256.4.b.e 2
80.t odd 4 1 256.4.b.c 2
120.q odd 4 1 576.4.a.g 1
120.w even 4 1 576.4.a.h 1
140.j odd 4 1 1568.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 20.e even 4 1
32.4.a.c yes 1 5.c odd 4 1
64.4.a.a 1 40.i odd 4 1
64.4.a.e 1 40.k even 4 1
256.4.b.c 2 80.i odd 4 1
256.4.b.c 2 80.t odd 4 1
256.4.b.e 2 80.j even 4 1
256.4.b.e 2 80.s even 4 1
288.4.a.h 1 60.l odd 4 1
288.4.a.i 1 15.e even 4 1
576.4.a.g 1 120.q odd 4 1
576.4.a.h 1 120.w even 4 1
800.4.a.a 1 5.c odd 4 1
800.4.a.k 1 20.e even 4 1
800.4.c.a 2 1.a even 1 1 trivial
800.4.c.a 2 5.b even 2 1 inner
800.4.c.b 2 4.b odd 2 1
800.4.c.b 2 20.d odd 2 1
1568.4.a.c 1 35.f even 4 1
1568.4.a.o 1 140.j odd 4 1
1600.4.a.e 1 40.k even 4 1
1600.4.a.bw 1 40.i odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{11} + 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T + 40)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2500 \) Copy content Toggle raw display
$17$ \( T^{2} + 900 \) Copy content Toggle raw display
$19$ \( (T + 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2304 \) Copy content Toggle raw display
$29$ \( (T - 34)^{2} \) Copy content Toggle raw display
$31$ \( (T - 320)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 96100 \) Copy content Toggle raw display
$41$ \( (T - 410)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 23104 \) Copy content Toggle raw display
$47$ \( T^{2} + 173056 \) Copy content Toggle raw display
$53$ \( T^{2} + 168100 \) Copy content Toggle raw display
$59$ \( (T - 200)^{2} \) Copy content Toggle raw display
$61$ \( (T - 30)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 602176 \) Copy content Toggle raw display
$71$ \( (T - 400)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 396900 \) Copy content Toggle raw display
$79$ \( (T - 1120)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 304704 \) Copy content Toggle raw display
$89$ \( (T - 326)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12100 \) Copy content Toggle raw display
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