Properties

Label 32.6.a.d
Level $32$
Weight $6$
Character orbit 32.a
Self dual yes
Analytic conductor $5.132$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,6,Mod(1,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)

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: yes
Analytic conductor: \(5.13228223402\)
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: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 16\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 46 q^{5} - 6 \beta q^{7} + 525 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 46 q^{5} - 6 \beta q^{7} + 525 q^{9} + 3 \beta q^{11} - 42 q^{13} + 46 \beta q^{15} + 962 q^{17} - 75 \beta q^{19} - 4608 q^{21} - 114 \beta q^{23} - 1009 q^{25} + 282 \beta q^{27} - 2554 q^{29} + 72 \beta q^{31} + 2304 q^{33} - 276 \beta q^{35} + 11950 q^{37} - 42 \beta q^{39} - 5078 q^{41} - 453 \beta q^{43} + 24150 q^{45} + 444 \beta q^{47} + 10841 q^{49} + 962 \beta q^{51} - 19714 q^{53} + 138 \beta q^{55} - 57600 q^{57} - 321 \beta q^{59} + 29318 q^{61} - 3150 \beta q^{63} - 1932 q^{65} + 609 \beta q^{67} - 87552 q^{69} + 2922 \beta q^{71} + 37914 q^{73} - 1009 \beta q^{75} - 13824 q^{77} + 3204 \beta q^{79} + 89001 q^{81} - 1419 \beta q^{83} + 44252 q^{85} - 2554 \beta q^{87} + 13930 q^{89} + 252 \beta q^{91} + 55296 q^{93} - 3450 \beta q^{95} + 163602 q^{97} + 1575 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 92 q^{5} + 1050 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 92 q^{5} + 1050 q^{9} - 84 q^{13} + 1924 q^{17} - 9216 q^{21} - 2018 q^{25} - 5108 q^{29} + 4608 q^{33} + 23900 q^{37} - 10156 q^{41} + 48300 q^{45} + 21682 q^{49} - 39428 q^{53} - 115200 q^{57} + 58636 q^{61} - 3864 q^{65} - 175104 q^{69} + 75828 q^{73} - 27648 q^{77} + 178002 q^{81} + 88504 q^{85} + 27860 q^{89} + 110592 q^{93} + 327204 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −27.7128 0 46.0000 0 166.277 0 525.000 0
1.2 0 27.7128 0 46.0000 0 −166.277 0 525.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.6.a.d 2
3.b odd 2 1 288.6.a.l 2
4.b odd 2 1 inner 32.6.a.d 2
5.b even 2 1 800.6.a.k 2
5.c odd 4 2 800.6.c.d 4
8.b even 2 1 64.6.a.h 2
8.d odd 2 1 64.6.a.h 2
12.b even 2 1 288.6.a.l 2
16.e even 4 2 256.6.b.l 4
16.f odd 4 2 256.6.b.l 4
20.d odd 2 1 800.6.a.k 2
20.e even 4 2 800.6.c.d 4
24.f even 2 1 576.6.a.bp 2
24.h odd 2 1 576.6.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.d 2 1.a even 1 1 trivial
32.6.a.d 2 4.b odd 2 1 inner
64.6.a.h 2 8.b even 2 1
64.6.a.h 2 8.d odd 2 1
256.6.b.l 4 16.e even 4 2
256.6.b.l 4 16.f odd 4 2
288.6.a.l 2 3.b odd 2 1
288.6.a.l 2 12.b even 2 1
576.6.a.bp 2 24.f even 2 1
576.6.a.bp 2 24.h odd 2 1
800.6.a.k 2 5.b even 2 1
800.6.a.k 2 20.d odd 2 1
800.6.c.d 4 5.c odd 4 2
800.6.c.d 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 768 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(32))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 768 \) Copy content Toggle raw display
$5$ \( (T - 46)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 27648 \) Copy content Toggle raw display
$11$ \( T^{2} - 6912 \) Copy content Toggle raw display
$13$ \( (T + 42)^{2} \) Copy content Toggle raw display
$17$ \( (T - 962)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4320000 \) Copy content Toggle raw display
$23$ \( T^{2} - 9980928 \) Copy content Toggle raw display
$29$ \( (T + 2554)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3981312 \) Copy content Toggle raw display
$37$ \( (T - 11950)^{2} \) Copy content Toggle raw display
$41$ \( (T + 5078)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 157600512 \) Copy content Toggle raw display
$47$ \( T^{2} - 151400448 \) Copy content Toggle raw display
$53$ \( (T + 19714)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 79135488 \) Copy content Toggle raw display
$61$ \( (T - 29318)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 284836608 \) Copy content Toggle raw display
$71$ \( T^{2} - 6557248512 \) Copy content Toggle raw display
$73$ \( (T - 37914)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 7883993088 \) Copy content Toggle raw display
$83$ \( T^{2} - 1546414848 \) Copy content Toggle raw display
$89$ \( (T - 13930)^{2} \) Copy content Toggle raw display
$97$ \( (T - 163602)^{2} \) Copy content Toggle raw display
show more
show less