Properties

Label 800.4.a.r
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $1$
Dimension $2$
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(1,800)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

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: yes
Analytic conductor: \(47.2015280046\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{3} - 4 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{3} - 4 q^{7} - 11 q^{9} - 2 \beta q^{11} + \beta q^{13} + 2 \beta q^{17} + 6 \beta q^{19} - 16 q^{21} + 52 q^{23} - 152 q^{27} - 158 q^{29} - 8 \beta q^{31} - 8 \beta q^{33} - 13 \beta q^{37} + 4 \beta q^{39} - 170 q^{41} - 316 q^{43} - 244 q^{47} - 327 q^{49} + 8 \beta q^{51} + 23 \beta q^{53} + 24 \beta q^{57} - 30 \beta q^{59} + 82 q^{61} + 44 q^{63} - 692 q^{67} + 208 q^{69} + 44 \beta q^{71} - 20 \beta q^{73} + 8 \beta q^{77} - 16 \beta q^{79} - 311 q^{81} - 940 q^{83} - 632 q^{87} + 6 q^{89} - 4 \beta q^{91} - 32 \beta q^{93} - 50 \beta q^{97} + 22 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} - 8 q^{7} - 22 q^{9} - 32 q^{21} + 104 q^{23} - 304 q^{27} - 316 q^{29} - 340 q^{41} - 632 q^{43} - 488 q^{47} - 654 q^{49} + 164 q^{61} + 88 q^{63} - 1384 q^{67} + 416 q^{69} - 622 q^{81} - 1880 q^{83} - 1264 q^{87} + 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.19258
−2.19258
0 4.00000 0 0 0 −4.00000 0 −11.0000 0
1.2 0 4.00000 0 0 0 −4.00000 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a.r 2
4.b odd 2 1 800.4.a.n 2
5.b even 2 1 800.4.a.n 2
5.c odd 4 2 160.4.c.b 4
8.b even 2 1 1600.4.a.cc 2
8.d odd 2 1 1600.4.a.co 2
15.e even 4 2 1440.4.f.h 4
20.d odd 2 1 inner 800.4.a.r 2
20.e even 4 2 160.4.c.b 4
40.e odd 2 1 1600.4.a.cc 2
40.f even 2 1 1600.4.a.co 2
40.i odd 4 2 320.4.c.i 4
40.k even 4 2 320.4.c.i 4
60.l odd 4 2 1440.4.f.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.b 4 5.c odd 4 2
160.4.c.b 4 20.e even 4 2
320.4.c.i 4 40.i odd 4 2
320.4.c.i 4 40.k even 4 2
800.4.a.n 2 4.b odd 2 1
800.4.a.n 2 5.b even 2 1
800.4.a.r 2 1.a even 1 1 trivial
800.4.a.r 2 20.d odd 2 1 inner
1440.4.f.h 4 15.e even 4 2
1440.4.f.h 4 60.l odd 4 2
1600.4.a.cc 2 8.b even 2 1
1600.4.a.cc 2 40.e odd 2 1
1600.4.a.co 2 8.d odd 2 1
1600.4.a.co 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 1856 \) Copy content Toggle raw display
\( T_{13}^{2} - 464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 1856 \) Copy content Toggle raw display
$13$ \( T^{2} - 464 \) Copy content Toggle raw display
$17$ \( T^{2} - 1856 \) Copy content Toggle raw display
$19$ \( T^{2} - 16704 \) Copy content Toggle raw display
$23$ \( (T - 52)^{2} \) Copy content Toggle raw display
$29$ \( (T + 158)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 29696 \) Copy content Toggle raw display
$37$ \( T^{2} - 78416 \) Copy content Toggle raw display
$41$ \( (T + 170)^{2} \) Copy content Toggle raw display
$43$ \( (T + 316)^{2} \) Copy content Toggle raw display
$47$ \( (T + 244)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 245456 \) Copy content Toggle raw display
$59$ \( T^{2} - 417600 \) Copy content Toggle raw display
$61$ \( (T - 82)^{2} \) Copy content Toggle raw display
$67$ \( (T + 692)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 898304 \) Copy content Toggle raw display
$73$ \( T^{2} - 185600 \) Copy content Toggle raw display
$79$ \( T^{2} - 118784 \) Copy content Toggle raw display
$83$ \( (T + 940)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1160000 \) Copy content Toggle raw display
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