Properties

Label 320.2.o.a
Level $320$
Weight $2$
Character orbit 320.o
Analytic conductor $2.555$
Analytic rank $1$
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(223,320)]
 
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, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.o (of order \(4\), degree \(2\), 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: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (i - 1) q^{3} + ( - 2 i - 1) q^{5} + (i - 1) q^{7} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{3} + ( - 2 i - 1) q^{5} + (i - 1) q^{7} + i q^{9} - 4 q^{11} + ( - 3 i - 3) q^{13} + (i + 3) q^{15} + ( - 3 i - 3) q^{17} + 6 i q^{19} - 2 i q^{21} + ( - 3 i - 3) q^{23} + (4 i - 3) q^{25} + ( - 4 i - 4) q^{27} + 2 q^{29} - 6 i q^{31} + ( - 4 i + 4) q^{33} + (i + 3) q^{35} + (3 i - 3) q^{37} + 6 q^{39} + 6 q^{41} + ( - 3 i + 3) q^{43} + ( - i + 2) q^{45} + (9 i - 9) q^{47} + 5 i q^{49} + 6 q^{51} + (5 i + 5) q^{53} + (8 i + 4) q^{55} + ( - 6 i - 6) q^{57} - 10 i q^{59} + 12 i q^{61} + ( - i - 1) q^{63} + (9 i - 3) q^{65} + (9 i + 9) q^{67} + 6 q^{69} - 6 i q^{71} + ( - 5 i + 5) q^{73} + ( - 7 i - 1) q^{75} + ( - 4 i + 4) q^{77} + 5 q^{81} + ( - 3 i + 3) q^{83} + (9 i - 3) q^{85} + (2 i - 2) q^{87} + 6 q^{91} + (6 i + 6) q^{93} + ( - 6 i + 12) q^{95} + ( - 7 i - 7) q^{97} - 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} - 8 q^{11} - 6 q^{13} + 6 q^{15} - 6 q^{17} - 6 q^{23} - 6 q^{25} - 8 q^{27} + 4 q^{29} + 8 q^{33} + 6 q^{35} - 6 q^{37} + 12 q^{39} + 12 q^{41} + 6 q^{43} + 4 q^{45} - 18 q^{47} + 12 q^{51} + 10 q^{53} + 8 q^{55} - 12 q^{57} - 2 q^{63} - 6 q^{65} + 18 q^{67} + 12 q^{69} + 10 q^{73} - 2 q^{75} + 8 q^{77} + 10 q^{81} + 6 q^{83} - 6 q^{85} - 4 q^{87} + 12 q^{91} + 12 q^{93} + 24 q^{95} - 14 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
1.00000i
1.00000i
0 −1.00000 1.00000i 0 −1.00000 + 2.00000i 0 −1.00000 1.00000i 0 1.00000i 0
287.1 0 −1.00000 + 1.00000i 0 −1.00000 2.00000i 0 −1.00000 + 1.00000i 0 1.00000i 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
40.k 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.o.a 2
4.b odd 2 1 320.2.o.c yes 2
5.b even 2 1 1600.2.o.d 2
5.c odd 4 1 320.2.o.b yes 2
5.c odd 4 1 1600.2.o.c 2
8.b even 2 1 320.2.o.d yes 2
8.d odd 2 1 320.2.o.b yes 2
16.e even 4 1 1280.2.n.c 2
16.e even 4 1 1280.2.n.j 2
16.f odd 4 1 1280.2.n.d 2
16.f odd 4 1 1280.2.n.i 2
20.d odd 2 1 1600.2.o.a 2
20.e even 4 1 320.2.o.d yes 2
20.e even 4 1 1600.2.o.b 2
40.e odd 2 1 1600.2.o.c 2
40.f even 2 1 1600.2.o.b 2
40.i odd 4 1 320.2.o.c yes 2
40.i odd 4 1 1600.2.o.a 2
40.k even 4 1 inner 320.2.o.a 2
40.k even 4 1 1600.2.o.d 2
80.i odd 4 1 1280.2.n.i 2
80.j even 4 1 1280.2.n.j 2
80.s even 4 1 1280.2.n.c 2
80.t odd 4 1 1280.2.n.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.o.a 2 1.a even 1 1 trivial
320.2.o.a 2 40.k even 4 1 inner
320.2.o.b yes 2 5.c odd 4 1
320.2.o.b yes 2 8.d odd 2 1
320.2.o.c yes 2 4.b odd 2 1
320.2.o.c yes 2 40.i odd 4 1
320.2.o.d yes 2 8.b even 2 1
320.2.o.d yes 2 20.e even 4 1
1280.2.n.c 2 16.e even 4 1
1280.2.n.c 2 80.s even 4 1
1280.2.n.d 2 16.f odd 4 1
1280.2.n.d 2 80.t odd 4 1
1280.2.n.i 2 16.f odd 4 1
1280.2.n.i 2 80.i odd 4 1
1280.2.n.j 2 16.e even 4 1
1280.2.n.j 2 80.j even 4 1
1600.2.o.a 2 20.d odd 2 1
1600.2.o.a 2 40.i odd 4 1
1600.2.o.b 2 20.e even 4 1
1600.2.o.b 2 40.f even 2 1
1600.2.o.c 2 5.c odd 4 1
1600.2.o.c 2 40.e odd 2 1
1600.2.o.d 2 5.b even 2 1
1600.2.o.d 2 40.k even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 144 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
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