Properties

Label 800.4.a.s
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{3} + ( - 5 \beta + 4) q^{7} + (8 \beta + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{3} + ( - 5 \beta + 4) q^{7} + (8 \beta + 13) q^{9} + ( - 6 \beta - 32) q^{11} + ( - 8 \beta - 6) q^{13} + (24 \beta - 2) q^{17} + (8 \beta - 104) q^{19} + ( - 16 \beta - 104) q^{21} + ( - 11 \beta + 60) q^{23} + (18 \beta + 136) q^{27} + ( - 16 \beta - 146) q^{29} + (6 \beta - 88) q^{31} + ( - 56 \beta - 272) q^{33} + ( - 16 \beta + 178) q^{37} + ( - 38 \beta - 216) q^{39} + ( - 40 \beta + 50) q^{41} + (37 \beta - 188) q^{43} + ( - 13 \beta - 140) q^{47} + ( - 40 \beta + 273) q^{49} + (94 \beta + 568) q^{51} + ( - 24 \beta - 158) q^{53} + ( - 72 \beta - 224) q^{57} + (20 \beta - 360) q^{59} + ( - 32 \beta - 634) q^{61} + ( - 33 \beta - 908) q^{63} + (47 \beta - 372) q^{67} + (16 \beta - 24) q^{69} + ( - 218 \beta + 24) q^{71} + (120 \beta + 470) q^{73} + (136 \beta + 592) q^{77} + (172 \beta - 16) q^{79} + ( - 8 \beta + 625) q^{81} + ( - 47 \beta - 796) q^{83} + ( - 210 \beta - 968) q^{87} + (48 \beta - 390) q^{89} + ( - 2 \beta + 936) q^{91} + ( - 64 \beta - 208) q^{93} + (40 \beta - 610) q^{97} + ( - 334 \beta - 1568) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{3} + 8 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{3} + 8 q^{7} + 26 q^{9} - 64 q^{11} - 12 q^{13} - 4 q^{17} - 208 q^{19} - 208 q^{21} + 120 q^{23} + 272 q^{27} - 292 q^{29} - 176 q^{31} - 544 q^{33} + 356 q^{37} - 432 q^{39} + 100 q^{41} - 376 q^{43} - 280 q^{47} + 546 q^{49} + 1136 q^{51} - 316 q^{53} - 448 q^{57} - 720 q^{59} - 1268 q^{61} - 1816 q^{63} - 744 q^{67} - 48 q^{69} + 48 q^{71} + 940 q^{73} + 1184 q^{77} - 32 q^{79} + 1250 q^{81} - 1592 q^{83} - 1936 q^{87} - 780 q^{89} + 1872 q^{91} - 416 q^{93} - 1220 q^{97} - 3136 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
0 −0.898979 0 0 0 28.4949 0 −26.1918 0
1.2 0 8.89898 0 0 0 −20.4949 0 52.1918 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.a.s 2
4.b odd 2 1 800.4.a.m 2
5.b even 2 1 160.4.a.c 2
5.c odd 4 2 800.4.c.i 4
8.b even 2 1 1600.4.a.cd 2
8.d odd 2 1 1600.4.a.cn 2
15.d odd 2 1 1440.4.a.t 2
20.d odd 2 1 160.4.a.g yes 2
20.e even 4 2 800.4.c.k 4
40.e odd 2 1 320.4.a.o 2
40.f even 2 1 320.4.a.s 2
60.h even 2 1 1440.4.a.x 2
80.k odd 4 2 1280.4.d.q 4
80.q even 4 2 1280.4.d.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 5.b even 2 1
160.4.a.g yes 2 20.d odd 2 1
320.4.a.o 2 40.e odd 2 1
320.4.a.s 2 40.f even 2 1
800.4.a.m 2 4.b odd 2 1
800.4.a.s 2 1.a even 1 1 trivial
800.4.c.i 4 5.c odd 4 2
800.4.c.k 4 20.e even 4 2
1280.4.d.q 4 80.k odd 4 2
1280.4.d.x 4 80.q even 4 2
1440.4.a.t 2 15.d odd 2 1
1440.4.a.x 2 60.h even 2 1
1600.4.a.cd 2 8.b even 2 1
1600.4.a.cn 2 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 8T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 64T_{11} + 160 \) Copy content Toggle raw display
\( T_{13}^{2} + 12T_{13} - 1500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8T - 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8T - 584 \) Copy content Toggle raw display
$11$ \( T^{2} + 64T + 160 \) Copy content Toggle raw display
$13$ \( T^{2} + 12T - 1500 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 13820 \) Copy content Toggle raw display
$19$ \( T^{2} + 208T + 9280 \) Copy content Toggle raw display
$23$ \( T^{2} - 120T + 696 \) Copy content Toggle raw display
$29$ \( T^{2} + 292T + 15172 \) Copy content Toggle raw display
$31$ \( T^{2} + 176T + 6880 \) Copy content Toggle raw display
$37$ \( T^{2} - 356T + 25540 \) Copy content Toggle raw display
$41$ \( T^{2} - 100T - 35900 \) Copy content Toggle raw display
$43$ \( T^{2} + 376T + 2488 \) Copy content Toggle raw display
$47$ \( T^{2} + 280T + 15544 \) Copy content Toggle raw display
$53$ \( T^{2} + 316T + 11140 \) Copy content Toggle raw display
$59$ \( T^{2} + 720T + 120000 \) Copy content Toggle raw display
$61$ \( T^{2} + 1268 T + 377380 \) Copy content Toggle raw display
$67$ \( T^{2} + 744T + 85368 \) Copy content Toggle raw display
$71$ \( T^{2} - 48T - 1140000 \) Copy content Toggle raw display
$73$ \( T^{2} - 940T - 124700 \) Copy content Toggle raw display
$79$ \( T^{2} + 32T - 709760 \) Copy content Toggle raw display
$83$ \( T^{2} + 1592 T + 580600 \) Copy content Toggle raw display
$89$ \( T^{2} + 780T + 96804 \) Copy content Toggle raw display
$97$ \( T^{2} + 1220 T + 333700 \) Copy content Toggle raw display
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