Properties

Label 4032.2.b.h
Level $4032$
Weight $2$
Character orbit 4032.b
Analytic conductor $32.196$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 112)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{5} + ( - \beta + 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta q^{5} + ( - \beta + 2) q^{7} - 2 \beta q^{11} - 2 \beta q^{13} + 2 q^{19} - 2 \beta q^{23} - 7 q^{25} + 6 q^{29} + 8 q^{31} + ( - 4 \beta - 6) q^{35} + 2 q^{37} + 4 \beta q^{41} - 6 \beta q^{43} + ( - 4 \beta + 1) q^{49} + 6 q^{53} - 12 q^{55} - 6 q^{59} - 2 \beta q^{61} - 12 q^{65} + 2 \beta q^{67} + 2 \beta q^{71} + 4 \beta q^{73} + ( - 4 \beta - 6) q^{77} - 2 \beta q^{79} + 6 q^{83} + 4 \beta q^{89} + ( - 4 \beta - 6) q^{91} - 4 \beta q^{95} + 8 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} + 4 q^{19} - 14 q^{25} + 12 q^{29} + 16 q^{31} - 12 q^{35} + 4 q^{37} + 2 q^{49} + 12 q^{53} - 24 q^{55} - 12 q^{59} - 24 q^{65} - 12 q^{77} + 12 q^{83} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.46410i 0 2.00000 1.73205i 0 0 0
3583.2 0 0 0 3.46410i 0 2.00000 + 1.73205i 0 0 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
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.b.h 2
3.b odd 2 1 448.2.f.c 2
4.b odd 2 1 4032.2.b.b 2
7.b odd 2 1 4032.2.b.b 2
8.b even 2 1 1008.2.b.g 2
8.d odd 2 1 1008.2.b.b 2
12.b even 2 1 448.2.f.a 2
21.c even 2 1 448.2.f.a 2
24.f even 2 1 112.2.f.b yes 2
24.h odd 2 1 112.2.f.a 2
28.d even 2 1 inner 4032.2.b.h 2
48.i odd 4 2 1792.2.e.a 4
48.k even 4 2 1792.2.e.c 4
56.e even 2 1 1008.2.b.g 2
56.h odd 2 1 1008.2.b.b 2
84.h odd 2 1 448.2.f.c 2
120.i odd 2 1 2800.2.k.e 2
120.m even 2 1 2800.2.k.b 2
120.q odd 4 2 2800.2.e.b 4
120.w even 4 2 2800.2.e.c 4
168.e odd 2 1 112.2.f.a 2
168.i even 2 1 112.2.f.b yes 2
168.s odd 6 1 784.2.p.e 2
168.s odd 6 1 784.2.p.f 2
168.v even 6 1 784.2.p.a 2
168.v even 6 1 784.2.p.b 2
168.ba even 6 1 784.2.p.a 2
168.ba even 6 1 784.2.p.b 2
168.be odd 6 1 784.2.p.e 2
168.be odd 6 1 784.2.p.f 2
336.v odd 4 2 1792.2.e.a 4
336.y even 4 2 1792.2.e.c 4
840.b odd 2 1 2800.2.k.e 2
840.u even 2 1 2800.2.k.b 2
840.bm even 4 2 2800.2.e.c 4
840.bp odd 4 2 2800.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 24.h odd 2 1
112.2.f.a 2 168.e odd 2 1
112.2.f.b yes 2 24.f even 2 1
112.2.f.b yes 2 168.i even 2 1
448.2.f.a 2 12.b even 2 1
448.2.f.a 2 21.c even 2 1
448.2.f.c 2 3.b odd 2 1
448.2.f.c 2 84.h odd 2 1
784.2.p.a 2 168.v even 6 1
784.2.p.a 2 168.ba even 6 1
784.2.p.b 2 168.v even 6 1
784.2.p.b 2 168.ba even 6 1
784.2.p.e 2 168.s odd 6 1
784.2.p.e 2 168.be odd 6 1
784.2.p.f 2 168.s odd 6 1
784.2.p.f 2 168.be odd 6 1
1008.2.b.b 2 8.d odd 2 1
1008.2.b.b 2 56.h odd 2 1
1008.2.b.g 2 8.b even 2 1
1008.2.b.g 2 56.e even 2 1
1792.2.e.a 4 48.i odd 4 2
1792.2.e.a 4 336.v odd 4 2
1792.2.e.c 4 48.k even 4 2
1792.2.e.c 4 336.y even 4 2
2800.2.e.b 4 120.q odd 4 2
2800.2.e.b 4 840.bp odd 4 2
2800.2.e.c 4 120.w even 4 2
2800.2.e.c 4 840.bm even 4 2
2800.2.k.b 2 120.m even 2 1
2800.2.k.b 2 840.u even 2 1
2800.2.k.e 2 120.i odd 2 1
2800.2.k.e 2 840.b odd 2 1
4032.2.b.b 2 4.b odd 2 1
4032.2.b.b 2 7.b odd 2 1
4032.2.b.h 2 1.a even 1 1 trivial
4032.2.b.h 2 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 12 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 12 \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( T^{2} + 12 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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