Properties

Label 800.6.a.i
Level $800$
Weight $6$
Character orbit 800.a
Self dual yes
Analytic conductor $128.307$
Analytic rank $1$
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] = [800,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(128.307055850\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta q^{3} - 31 \beta q^{7} - 63 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{3} - 31 \beta q^{7} - 63 q^{9} + 58 \beta q^{11} - 154 q^{13} - 178 q^{17} + 216 \beta q^{19} + 1860 q^{21} - 589 \beta q^{23} + 918 \beta q^{27} + 4110 q^{29} + 706 \beta q^{31} - 3480 q^{33} - 7442 q^{37} + 462 \beta q^{39} + 7270 q^{41} + 4005 \beta q^{43} + 1657 \beta q^{47} + 2413 q^{49} + 534 \beta q^{51} - 32226 q^{53} - 12960 q^{57} + 7612 \beta q^{59} + 26770 q^{61} + 1953 \beta q^{63} - 11137 \beta q^{67} + 35340 q^{69} + 12098 \beta q^{71} + 18534 q^{73} - 35960 q^{77} - 19396 \beta q^{79} - 39771 q^{81} + 17585 \beta q^{83} - 12330 \beta q^{87} - 107590 q^{89} + 4774 \beta q^{91} - 42360 q^{93} + 108838 q^{97} - 3654 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 126 q^{9} - 308 q^{13} - 356 q^{17} + 3720 q^{21} + 8220 q^{29} - 6960 q^{33} - 14884 q^{37} + 14540 q^{41} + 4826 q^{49} - 64452 q^{53} - 25920 q^{57} + 53540 q^{61} + 70680 q^{69} + 37068 q^{73} - 71920 q^{77} - 79542 q^{81} - 215180 q^{89} - 84720 q^{93} + 217676 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 −13.4164 0 0 0 −138.636 0 −63.0000 0
1.2 0 13.4164 0 0 0 138.636 0 −63.0000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.a.i 2
4.b odd 2 1 inner 800.6.a.i 2
5.b even 2 1 160.6.a.b 2
5.c odd 4 2 800.6.c.h 4
20.d odd 2 1 160.6.a.b 2
20.e even 4 2 800.6.c.h 4
40.e odd 2 1 320.6.a.t 2
40.f even 2 1 320.6.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.b 2 5.b even 2 1
160.6.a.b 2 20.d odd 2 1
320.6.a.t 2 40.e odd 2 1
320.6.a.t 2 40.f even 2 1
800.6.a.i 2 1.a even 1 1 trivial
800.6.a.i 2 4.b odd 2 1 inner
800.6.c.h 4 5.c odd 4 2
800.6.c.h 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{2} - 180 \) Copy content Toggle raw display
\( T_{11}^{2} - 67280 \) Copy content Toggle raw display
\( T_{13} + 154 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 180 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 19220 \) Copy content Toggle raw display
$11$ \( T^{2} - 67280 \) Copy content Toggle raw display
$13$ \( (T + 154)^{2} \) Copy content Toggle raw display
$17$ \( (T + 178)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 933120 \) Copy content Toggle raw display
$23$ \( T^{2} - 6938420 \) Copy content Toggle raw display
$29$ \( (T - 4110)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 9968720 \) Copy content Toggle raw display
$37$ \( (T + 7442)^{2} \) Copy content Toggle raw display
$41$ \( (T - 7270)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 320800500 \) Copy content Toggle raw display
$47$ \( T^{2} - 54912980 \) Copy content Toggle raw display
$53$ \( (T + 32226)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 1158850880 \) Copy content Toggle raw display
$61$ \( (T - 26770)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 2480655380 \) Copy content Toggle raw display
$71$ \( T^{2} - 2927232080 \) Copy content Toggle raw display
$73$ \( (T - 18534)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 7524096320 \) Copy content Toggle raw display
$83$ \( T^{2} - 6184644500 \) Copy content Toggle raw display
$89$ \( (T + 107590)^{2} \) Copy content Toggle raw display
$97$ \( (T - 108838)^{2} \) Copy content Toggle raw display
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