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Properties

Label 320.2.n.a
Level $320$
Weight $2$
Character orbit 320.n
Analytic conductor $2.555$
Analytic rank $0$
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] = [320,2,Mod(63,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.n (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 i - 2) q^{3} + ( - i + 2) q^{5} + ( - 2 i + 2) q^{7} + 5 i q^{9} + ( - i + 1) q^{13} + ( - 2 i - 6) q^{15} + ( - 5 i - 5) q^{17} - 4 q^{19} - 8 q^{21} + (2 i + 2) q^{23} + ( - 4 i + 3) q^{25}+ \cdots + ( - 3 i - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{13} - 12 q^{15} - 10 q^{17} - 8 q^{19} - 16 q^{21} + 4 q^{23} + 6 q^{25} + 8 q^{27} + 4 q^{35} - 2 q^{37} - 8 q^{39} + 12 q^{43} + 10 q^{45} - 4 q^{47} + 14 q^{53}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
− 1.00000i
1.00000i
0 −2.00000 + 2.00000i 0 2.00000 + 1.00000i 0 2.00000 + 2.00000i 0 − 5.00000i 0
127.1 0 −2.00000 − 2.00000i 0 2.00000 − 1.00000i 0 2.00000 − 2.00000i 0 5.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.n.a 2
4.b odd 2 1 320.2.n.h 2
5.b even 2 1 1600.2.n.n 2
5.c odd 4 1 320.2.n.h 2
5.c odd 4 1 1600.2.n.a 2
8.b even 2 1 160.2.n.f yes 2
8.d odd 2 1 160.2.n.a ✓ 2
16.e even 4 1 1280.2.o.b 2
16.e even 4 1 1280.2.o.o 2
16.f odd 4 1 1280.2.o.a 2
16.f odd 4 1 1280.2.o.p 2
20.d odd 2 1 1600.2.n.a 2
20.e even 4 1 inner 320.2.n.a 2
20.e even 4 1 1600.2.n.n 2
24.f even 2 1 1440.2.x.i 2
24.h odd 2 1 1440.2.x.j 2
40.e odd 2 1 800.2.n.j 2
40.f even 2 1 800.2.n.a 2
40.i odd 4 1 160.2.n.a ✓ 2
40.i odd 4 1 800.2.n.j 2
40.k even 4 1 160.2.n.f yes 2
40.k even 4 1 800.2.n.a 2
80.i odd 4 1 1280.2.o.p 2
80.j even 4 1 1280.2.o.o 2
80.s even 4 1 1280.2.o.b 2
80.t odd 4 1 1280.2.o.a 2
120.q odd 4 1 1440.2.x.j 2
120.w even 4 1 1440.2.x.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.a ✓ 2 8.d odd 2 1
160.2.n.a ✓ 2 40.i odd 4 1
160.2.n.f yes 2 8.b even 2 1
160.2.n.f yes 2 40.k even 4 1
320.2.n.a 2 1.a even 1 1 trivial
320.2.n.a 2 20.e even 4 1 inner
320.2.n.h 2 4.b odd 2 1
320.2.n.h 2 5.c odd 4 1
800.2.n.a 2 40.f even 2 1
800.2.n.a 2 40.k even 4 1
800.2.n.j 2 40.e odd 2 1
800.2.n.j 2 40.i odd 4 1
1280.2.o.a 2 16.f odd 4 1
1280.2.o.a 2 80.t odd 4 1
1280.2.o.b 2 16.e even 4 1
1280.2.o.b 2 80.s even 4 1
1280.2.o.o 2 16.e even 4 1
1280.2.o.o 2 80.j even 4 1
1280.2.o.p 2 16.f odd 4 1
1280.2.o.p 2 80.i odd 4 1
1440.2.x.i 2 24.f even 2 1
1440.2.x.i 2 120.w even 4 1
1440.2.x.j 2 24.h odd 2 1
1440.2.x.j 2 120.q odd 4 1
1600.2.n.a 2 5.c odd 4 1
1600.2.n.a 2 20.d odd 2 1
1600.2.n.n 2 5.b even 2 1
1600.2.n.n 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 200 \) Copy content Toggle raw display
$71$ \( T^{2} + 144 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
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