Properties

Label 3200.2.d.k
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3200,2,Mod(1601,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3200.1601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), 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: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{3} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{3} q^{7} - 3 q^{9} - 2 \beta_{2} q^{11} - 4 q^{17} - 2 \beta_{2} q^{19} - 3 \beta_1 q^{21} - \beta_{3} q^{23} + 4 \beta_1 q^{29} + 4 \beta_{3} q^{31} + 12 q^{33} + 2 \beta_1 q^{37} + 8 q^{41} - 3 \beta_{2} q^{43} + 5 \beta_{3} q^{47} - q^{49} - 4 \beta_{2} q^{51} + 4 \beta_1 q^{53} + 12 q^{57} - 2 \beta_{2} q^{59} + 3 \beta_1 q^{61} + 3 \beta_{3} q^{63} + \beta_{2} q^{67} - 3 \beta_1 q^{69} + 4 \beta_{3} q^{71} + 4 q^{73} + 6 \beta_1 q^{77} + 4 \beta_{3} q^{79} - 9 q^{81} - \beta_{2} q^{83} - 8 \beta_{3} q^{87} - 2 q^{89} + 12 \beta_1 q^{93} + 4 q^{97} + 6 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} - 16 q^{17} + 48 q^{33} + 32 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{73} - 36 q^{81} - 8 q^{89} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
0 2.44949i 0 0 0 −2.44949 0 −3.00000 0
1601.2 0 2.44949i 0 0 0 2.44949 0 −3.00000 0
1601.3 0 2.44949i 0 0 0 −2.44949 0 −3.00000 0
1601.4 0 2.44949i 0 0 0 2.44949 0 −3.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.d.k 4
4.b odd 2 1 inner 3200.2.d.k 4
5.b even 2 1 3200.2.d.l 4
5.c odd 4 1 640.2.f.c 4
5.c odd 4 1 640.2.f.g yes 4
8.b even 2 1 inner 3200.2.d.k 4
8.d odd 2 1 inner 3200.2.d.k 4
16.e even 4 1 6400.2.a.bu 2
16.e even 4 1 6400.2.a.bv 2
16.f odd 4 1 6400.2.a.bu 2
16.f odd 4 1 6400.2.a.bv 2
20.d odd 2 1 3200.2.d.l 4
20.e even 4 1 640.2.f.c 4
20.e even 4 1 640.2.f.g yes 4
40.e odd 2 1 3200.2.d.l 4
40.f even 2 1 3200.2.d.l 4
40.i odd 4 1 640.2.f.c 4
40.i odd 4 1 640.2.f.g yes 4
40.k even 4 1 640.2.f.c 4
40.k even 4 1 640.2.f.g yes 4
80.i odd 4 1 1280.2.c.e 4
80.i odd 4 1 1280.2.c.m 4
80.j even 4 1 1280.2.c.e 4
80.j even 4 1 1280.2.c.m 4
80.k odd 4 1 6400.2.a.bw 2
80.k odd 4 1 6400.2.a.bx 2
80.q even 4 1 6400.2.a.bw 2
80.q even 4 1 6400.2.a.bx 2
80.s even 4 1 1280.2.c.e 4
80.s even 4 1 1280.2.c.m 4
80.t odd 4 1 1280.2.c.e 4
80.t odd 4 1 1280.2.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.c 4 5.c odd 4 1
640.2.f.c 4 20.e even 4 1
640.2.f.c 4 40.i odd 4 1
640.2.f.c 4 40.k even 4 1
640.2.f.g yes 4 5.c odd 4 1
640.2.f.g yes 4 20.e even 4 1
640.2.f.g yes 4 40.i odd 4 1
640.2.f.g yes 4 40.k even 4 1
1280.2.c.e 4 80.i odd 4 1
1280.2.c.e 4 80.j even 4 1
1280.2.c.e 4 80.s even 4 1
1280.2.c.e 4 80.t odd 4 1
1280.2.c.m 4 80.i odd 4 1
1280.2.c.m 4 80.j even 4 1
1280.2.c.m 4 80.s even 4 1
1280.2.c.m 4 80.t odd 4 1
3200.2.d.k 4 1.a even 1 1 trivial
3200.2.d.k 4 4.b odd 2 1 inner
3200.2.d.k 4 8.b even 2 1 inner
3200.2.d.k 4 8.d odd 2 1 inner
3200.2.d.l 4 5.b even 2 1
3200.2.d.l 4 20.d odd 2 1
3200.2.d.l 4 40.e odd 2 1
3200.2.d.l 4 40.f even 2 1
6400.2.a.bu 2 16.e even 4 1
6400.2.a.bu 2 16.f odd 4 1
6400.2.a.bv 2 16.e even 4 1
6400.2.a.bv 2 16.f odd 4 1
6400.2.a.bw 2 80.k odd 4 1
6400.2.a.bw 2 80.q even 4 1
6400.2.a.bx 2 80.k odd 4 1
6400.2.a.bx 2 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 24 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T - 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( (T - 4)^{4} \) Copy content Toggle raw display
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