Properties

Label 320.4.a.p
Level $320$
Weight $4$
Character orbit 320.a
Self dual yes
Analytic conductor $18.881$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) 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 \(\beta = 2\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 5 q^{5} - 3 \beta q^{7} + 13 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 5 q^{5} - 3 \beta q^{7} + 13 q^{9} - 2 \beta q^{11} - 38 q^{13} - 5 \beta q^{15} + 34 q^{17} - 16 \beta q^{19} - 120 q^{21} - 13 \beta q^{23} + 25 q^{25} - 14 \beta q^{27} - 270 q^{29} + 54 \beta q^{31} - 80 q^{33} + 15 \beta q^{35} - 206 q^{37} - 38 \beta q^{39} - 270 q^{41} + 85 \beta q^{43} - 65 q^{45} + 21 \beta q^{47} + 17 q^{49} + 34 \beta q^{51} + 258 q^{53} + 10 \beta q^{55} - 640 q^{57} - 12 \beta q^{59} + 250 q^{61} - 39 \beta q^{63} + 190 q^{65} - 129 \beta q^{67} - 520 q^{69} + 102 \beta q^{71} - 1078 q^{73} + 25 \beta q^{75} + 240 q^{77} + 44 \beta q^{79} - 911 q^{81} - 175 \beta q^{83} - 170 q^{85} - 270 \beta q^{87} + 890 q^{89} + 114 \beta q^{91} + 2160 q^{93} + 80 \beta q^{95} - 254 q^{97} - 26 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} + 26 q^{9} - 76 q^{13} + 68 q^{17} - 240 q^{21} + 50 q^{25} - 540 q^{29} - 160 q^{33} - 412 q^{37} - 540 q^{41} - 130 q^{45} + 34 q^{49} + 516 q^{53} - 1280 q^{57} + 500 q^{61} + 380 q^{65} - 1040 q^{69} - 2156 q^{73} + 480 q^{77} - 1822 q^{81} - 340 q^{85} + 1780 q^{89} + 4320 q^{93} - 508 q^{97}+O(q^{100}) \) 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
−3.16228
3.16228
0 −6.32456 0 −5.00000 0 18.9737 0 13.0000 0
1.2 0 6.32456 0 −5.00000 0 −18.9737 0 13.0000 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 320.4.a.p 2
4.b odd 2 1 inner 320.4.a.p 2
5.b even 2 1 1600.4.a.ch 2
8.b even 2 1 160.4.a.f 2
8.d odd 2 1 160.4.a.f 2
16.e even 4 2 1280.4.d.u 4
16.f odd 4 2 1280.4.d.u 4
20.d odd 2 1 1600.4.a.ch 2
24.f even 2 1 1440.4.a.v 2
24.h odd 2 1 1440.4.a.v 2
40.e odd 2 1 800.4.a.p 2
40.f even 2 1 800.4.a.p 2
40.i odd 4 2 800.4.c.j 4
40.k even 4 2 800.4.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.f 2 8.b even 2 1
160.4.a.f 2 8.d odd 2 1
320.4.a.p 2 1.a even 1 1 trivial
320.4.a.p 2 4.b odd 2 1 inner
800.4.a.p 2 40.e odd 2 1
800.4.a.p 2 40.f even 2 1
800.4.c.j 4 40.i odd 4 2
800.4.c.j 4 40.k even 4 2
1280.4.d.u 4 16.e even 4 2
1280.4.d.u 4 16.f odd 4 2
1440.4.a.v 2 24.f even 2 1
1440.4.a.v 2 24.h odd 2 1
1600.4.a.ch 2 5.b even 2 1
1600.4.a.ch 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(320))\):

\( T_{3}^{2} - 40 \) Copy content Toggle raw display
\( T_{7}^{2} - 360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 40 \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 360 \) Copy content Toggle raw display
$11$ \( T^{2} - 160 \) Copy content Toggle raw display
$13$ \( (T + 38)^{2} \) Copy content Toggle raw display
$17$ \( (T - 34)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 10240 \) Copy content Toggle raw display
$23$ \( T^{2} - 6760 \) Copy content Toggle raw display
$29$ \( (T + 270)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 116640 \) Copy content Toggle raw display
$37$ \( (T + 206)^{2} \) Copy content Toggle raw display
$41$ \( (T + 270)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 289000 \) Copy content Toggle raw display
$47$ \( T^{2} - 17640 \) Copy content Toggle raw display
$53$ \( (T - 258)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 5760 \) Copy content Toggle raw display
$61$ \( (T - 250)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 665640 \) Copy content Toggle raw display
$71$ \( T^{2} - 416160 \) Copy content Toggle raw display
$73$ \( (T + 1078)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 77440 \) Copy content Toggle raw display
$83$ \( T^{2} - 1225000 \) Copy content Toggle raw display
$89$ \( (T - 890)^{2} \) Copy content Toggle raw display
$97$ \( (T + 254)^{2} \) Copy content Toggle raw display
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