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Properties

Label 640.2.q.a
Level $640$
Weight $2$
Character orbit 640.q
Analytic conductor $5.110$
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] = [640,2,Mod(289,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.q (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: \(5.11042572936\)
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: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{3} + ( - i - 2) q^{5} - i q^{9} + ( - 3 i - 3) q^{11} + (3 i + 3) q^{13} + (3 i + 1) q^{15} + 4 i q^{17} + (i - 1) q^{19} - 8 q^{23} + (4 i + 3) q^{25} + (4 i - 4) q^{27} + (3 i - 3) q^{29} + \cdots + (3 i - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} - 6 q^{11} + 6 q^{13} + 2 q^{15} - 2 q^{19} - 16 q^{23} + 6 q^{25} - 8 q^{27} - 6 q^{29} + 6 q^{37} - 6 q^{43} - 2 q^{45} - 14 q^{49} + 8 q^{51} - 18 q^{53} + 6 q^{55} + 4 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
− 1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 −2.00000 + 1.00000i 0 0 0 1.00000i 0
609.1 0 −1.00000 − 1.00000i 0 −2.00000 − 1.00000i 0 0 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
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.q.a 2
4.b odd 2 1 640.2.q.c 2
5.b even 2 1 640.2.q.d 2
8.b even 2 1 320.2.q.b 2
8.d odd 2 1 80.2.q.b yes 2
16.e even 4 1 320.2.q.a 2
16.e even 4 1 640.2.q.d 2
16.f odd 4 1 80.2.q.a ✓ 2
16.f odd 4 1 640.2.q.b 2
20.d odd 2 1 640.2.q.b 2
24.f even 2 1 720.2.bm.a 2
40.e odd 2 1 80.2.q.a ✓ 2
40.f even 2 1 320.2.q.a 2
40.i odd 4 1 1600.2.l.b 2
40.i odd 4 1 1600.2.l.c 2
40.k even 4 1 400.2.l.a 2
40.k even 4 1 400.2.l.b 2
48.k even 4 1 720.2.bm.b 2
80.i odd 4 1 1600.2.l.c 2
80.j even 4 1 400.2.l.b 2
80.k odd 4 1 80.2.q.b yes 2
80.k odd 4 1 640.2.q.c 2
80.q even 4 1 320.2.q.b 2
80.q even 4 1 inner 640.2.q.a 2
80.s even 4 1 400.2.l.a 2
80.t odd 4 1 1600.2.l.b 2
120.m even 2 1 720.2.bm.b 2
240.t even 4 1 720.2.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.a ✓ 2 16.f odd 4 1
80.2.q.a ✓ 2 40.e odd 2 1
80.2.q.b yes 2 8.d odd 2 1
80.2.q.b yes 2 80.k odd 4 1
320.2.q.a 2 16.e even 4 1
320.2.q.a 2 40.f even 2 1
320.2.q.b 2 8.b even 2 1
320.2.q.b 2 80.q even 4 1
400.2.l.a 2 40.k even 4 1
400.2.l.a 2 80.s even 4 1
400.2.l.b 2 40.k even 4 1
400.2.l.b 2 80.j even 4 1
640.2.q.a 2 1.a even 1 1 trivial
640.2.q.a 2 80.q even 4 1 inner
640.2.q.b 2 16.f odd 4 1
640.2.q.b 2 20.d odd 2 1
640.2.q.c 2 4.b odd 2 1
640.2.q.c 2 80.k odd 4 1
640.2.q.d 2 5.b even 2 1
640.2.q.d 2 16.e even 4 1
720.2.bm.a 2 24.f even 2 1
720.2.bm.a 2 240.t even 4 1
720.2.bm.b 2 48.k even 4 1
720.2.bm.b 2 120.m even 2 1
1600.2.l.b 2 40.i odd 4 1
1600.2.l.b 2 80.t odd 4 1
1600.2.l.c 2 40.i odd 4 1
1600.2.l.c 2 80.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(640, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 18 \) 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
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$89$ \( T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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