Properties

Label 320.4.a.q
Level $320$
Weight $4$
Character orbit 320.a
Self dual yes
Analytic conductor $18.881$
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] = [320,4,Mod(1,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, 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("320.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.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: \(18.8806112018\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \beta q^{3} + 5 q^{5} - 7 \beta q^{7} - 7 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 5 q^{5} - 7 \beta q^{7} - 7 q^{9} + 2 \beta q^{11} + 62 q^{13} - 5 \beta q^{15} - 46 q^{17} - 24 \beta q^{19} + 140 q^{21} + 43 \beta q^{23} + 25 q^{25} + 34 \beta q^{27} + 90 q^{29} - 34 \beta q^{31} - 40 q^{33} - 35 \beta q^{35} + 214 q^{37} - 62 \beta q^{39} - 10 q^{41} + 15 \beta q^{43} - 35 q^{45} + 89 \beta q^{47} + 637 q^{49} + 46 \beta q^{51} + 678 q^{53} + 10 \beta q^{55} + 480 q^{57} + 92 \beta q^{59} - 250 q^{61} + 49 \beta q^{63} + 310 q^{65} - 11 \beta q^{67} - 860 q^{69} - 82 \beta q^{71} + 522 q^{73} - 25 \beta q^{75} - 280 q^{77} + 196 \beta q^{79} - 491 q^{81} - 85 \beta q^{83} - 230 q^{85} - 90 \beta q^{87} + 970 q^{89} - 434 \beta q^{91} + 680 q^{93} - 120 \beta q^{95} - 934 q^{97} - 14 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 14 q^{9} + 124 q^{13} - 92 q^{17} + 280 q^{21} + 50 q^{25} + 180 q^{29} - 80 q^{33} + 428 q^{37} - 20 q^{41} - 70 q^{45} + 1274 q^{49} + 1356 q^{53} + 960 q^{57} - 500 q^{61} + 620 q^{65} - 1720 q^{69} + 1044 q^{73} - 560 q^{77} - 982 q^{81} - 460 q^{85} + 1940 q^{89} + 1360 q^{93} - 1868 q^{97}+O(q^{100}) \) 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 −4.47214 0 5.00000 0 −31.3050 0 −7.00000 0
1.2 0 4.47214 0 5.00000 0 31.3050 0 −7.00000 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 320.4.a.q 2
4.b odd 2 1 inner 320.4.a.q 2
5.b even 2 1 1600.4.a.cg 2
8.b even 2 1 160.4.a.d 2
8.d odd 2 1 160.4.a.d 2
16.e even 4 2 1280.4.d.v 4
16.f odd 4 2 1280.4.d.v 4
20.d odd 2 1 1600.4.a.cg 2
24.f even 2 1 1440.4.a.bb 2
24.h odd 2 1 1440.4.a.bb 2
40.e odd 2 1 800.4.a.o 2
40.f even 2 1 800.4.a.o 2
40.i odd 4 2 800.4.c.l 4
40.k even 4 2 800.4.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 8.b even 2 1
160.4.a.d 2 8.d odd 2 1
320.4.a.q 2 1.a even 1 1 trivial
320.4.a.q 2 4.b odd 2 1 inner
800.4.a.o 2 40.e odd 2 1
800.4.a.o 2 40.f even 2 1
800.4.c.l 4 40.i odd 4 2
800.4.c.l 4 40.k even 4 2
1280.4.d.v 4 16.e even 4 2
1280.4.d.v 4 16.f odd 4 2
1440.4.a.bb 2 24.f even 2 1
1440.4.a.bb 2 24.h odd 2 1
1600.4.a.cg 2 5.b even 2 1
1600.4.a.cg 2 20.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3}^{2} - 20 \) Copy content Toggle raw display
\( T_{7}^{2} - 980 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 20 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 980 \) Copy content Toggle raw display
$11$ \( T^{2} - 80 \) Copy content Toggle raw display
$13$ \( (T - 62)^{2} \) Copy content Toggle raw display
$17$ \( (T + 46)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 11520 \) Copy content Toggle raw display
$23$ \( T^{2} - 36980 \) Copy content Toggle raw display
$29$ \( (T - 90)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 23120 \) Copy content Toggle raw display
$37$ \( (T - 214)^{2} \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 4500 \) Copy content Toggle raw display
$47$ \( T^{2} - 158420 \) Copy content Toggle raw display
$53$ \( (T - 678)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 169280 \) Copy content Toggle raw display
$61$ \( (T + 250)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 2420 \) Copy content Toggle raw display
$71$ \( T^{2} - 134480 \) Copy content Toggle raw display
$73$ \( (T - 522)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 768320 \) Copy content Toggle raw display
$83$ \( T^{2} - 144500 \) Copy content Toggle raw display
$89$ \( (T - 970)^{2} \) Copy content Toggle raw display
$97$ \( (T + 934)^{2} \) Copy content Toggle raw display
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