Properties

Label 1600.3.b.t
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
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] = [1600,3,Mod(1151,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (of order \(2\), degree \(1\), 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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + 3 \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + 3 \beta_1 q^{7} + q^{9} - \beta_{3} q^{11} + \beta_{2} q^{13} - 2 \beta_{2} q^{17} - \beta_{3} q^{19} - 24 q^{21} + 9 \beta_1 q^{23} + 10 \beta_1 q^{27} - 22 q^{29} + 4 \beta_{3} q^{31} + 4 \beta_{2} q^{33} - 5 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + 22 q^{41} + 21 \beta_1 q^{43} + 3 \beta_1 q^{47} - 23 q^{49} - 4 \beta_{3} q^{51} + 3 \beta_{2} q^{53} + 4 \beta_{2} q^{57} + \beta_{3} q^{59} - 46 q^{61} + 3 \beta_1 q^{63} - 21 \beta_1 q^{67} - 72 q^{69} - 2 \beta_{3} q^{71} - 8 \beta_{2} q^{73} + 12 \beta_{2} q^{77} - 71 q^{81} - 27 \beta_1 q^{83} - 22 \beta_1 q^{87} - 146 q^{89} + 6 \beta_{3} q^{91} - 16 \beta_{2} q^{93} - 6 \beta_{2} q^{97} - \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 96 q^{21} - 88 q^{29} + 88 q^{41} - 92 q^{49} - 184 q^{61} - 288 q^{69} - 284 q^{81} - 584 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−1.22474 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
0 2.82843i 0 0 0 8.48528i 0 1.00000 0
1151.2 0 2.82843i 0 0 0 8.48528i 0 1.00000 0
1151.3 0 2.82843i 0 0 0 8.48528i 0 1.00000 0
1151.4 0 2.82843i 0 0 0 8.48528i 0 1.00000 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
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.b.t 4
4.b odd 2 1 inner 1600.3.b.t 4
5.b even 2 1 inner 1600.3.b.t 4
5.c odd 4 2 320.3.h.e 4
8.b even 2 1 400.3.b.h 4
8.d odd 2 1 400.3.b.h 4
20.d odd 2 1 inner 1600.3.b.t 4
20.e even 4 2 320.3.h.e 4
24.f even 2 1 3600.3.e.bd 4
24.h odd 2 1 3600.3.e.bd 4
40.e odd 2 1 400.3.b.h 4
40.f even 2 1 400.3.b.h 4
40.i odd 4 2 80.3.h.b 4
40.k even 4 2 80.3.h.b 4
80.i odd 4 2 1280.3.e.j 8
80.j even 4 2 1280.3.e.j 8
80.s even 4 2 1280.3.e.j 8
80.t odd 4 2 1280.3.e.j 8
120.i odd 2 1 3600.3.e.bd 4
120.m even 2 1 3600.3.e.bd 4
120.q odd 4 2 720.3.j.e 4
120.w even 4 2 720.3.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.h.b 4 40.i odd 4 2
80.3.h.b 4 40.k even 4 2
320.3.h.e 4 5.c odd 4 2
320.3.h.e 4 20.e even 4 2
400.3.b.h 4 8.b even 2 1
400.3.b.h 4 8.d odd 2 1
400.3.b.h 4 40.e odd 2 1
400.3.b.h 4 40.f even 2 1
720.3.j.e 4 120.q odd 4 2
720.3.j.e 4 120.w even 4 2
1280.3.e.j 8 80.i odd 4 2
1280.3.e.j 8 80.j even 4 2
1280.3.e.j 8 80.s even 4 2
1280.3.e.j 8 80.t odd 4 2
1600.3.b.t 4 1.a even 1 1 trivial
1600.3.b.t 4 4.b odd 2 1 inner
1600.3.b.t 4 5.b even 2 1 inner
1600.3.b.t 4 20.d odd 2 1 inner
3600.3.e.bd 4 24.f even 2 1
3600.3.e.bd 4 24.h odd 2 1
3600.3.e.bd 4 120.i odd 2 1
3600.3.e.bd 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 72 \) Copy content Toggle raw display
\( T_{13}^{2} - 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 384)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 648)^{2} \) Copy content Toggle raw display
$29$ \( (T + 22)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3072)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2400)^{2} \) Copy content Toggle raw display
$41$ \( (T - 22)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3528)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 864)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$61$ \( (T + 46)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3528)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6144)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 5832)^{2} \) Copy content Toggle raw display
$89$ \( (T + 146)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3456)^{2} \) Copy content Toggle raw display
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