Properties

Label 48.6.c.a
Level $48$
Weight $6$
Character orbit 48.c
Analytic conductor $7.698$
Analytic rank $1$
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] = [48,6,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 48.c (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: \(7.69842335102\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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 \(\beta = 3\sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 12) q^{3} - 8 \beta q^{5} + 18 \beta q^{7} + (24 \beta + 45) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 12) q^{3} - 8 \beta q^{5} + 18 \beta q^{7} + (24 \beta + 45) q^{9} - 648 q^{11} - 242 q^{13} + (96 \beta - 792) q^{15} + 32 \beta q^{17} + 126 \beta q^{19} + ( - 216 \beta + 1782) q^{21} - 1296 q^{23} - 3211 q^{25} + ( - 333 \beta + 1836) q^{27} + 200 \beta q^{29} - 342 \beta q^{31} + (648 \beta + 7776) q^{33} + 14256 q^{35} - 12058 q^{37} + (242 \beta + 2904) q^{39} - 1488 \beta q^{41} - 1818 \beta q^{43} + ( - 360 \beta + 19008) q^{45} - 12960 q^{47} - 15269 q^{49} + ( - 384 \beta + 3168) q^{51} + 2712 \beta q^{53} + 5184 \beta q^{55} + ( - 1512 \beta + 12474) q^{57} - 8424 q^{59} - 25762 q^{61} + (810 \beta - 42768) q^{63} + 1936 \beta q^{65} - 1026 \beta q^{67} + (1296 \beta + 15552) q^{69} - 55728 q^{71} + 26026 q^{73} + (3211 \beta + 38532) q^{75} - 11664 \beta q^{77} - 1926 \beta q^{79} + (2160 \beta - 54999) q^{81} + 78408 q^{83} + 25344 q^{85} + ( - 2400 \beta + 19800) q^{87} + 8464 \beta q^{89} - 4356 \beta q^{91} + (4104 \beta - 33858) q^{93} + 99792 q^{95} + 103090 q^{97} + ( - 15552 \beta - 29160) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{3} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 24 q^{3} + 90 q^{9} - 1296 q^{11} - 484 q^{13} - 1584 q^{15} + 3564 q^{21} - 2592 q^{23} - 6422 q^{25} + 3672 q^{27} + 15552 q^{33} + 28512 q^{35} - 24116 q^{37} + 5808 q^{39} + 38016 q^{45} - 25920 q^{47} - 30538 q^{49} + 6336 q^{51} + 24948 q^{57} - 16848 q^{59} - 51524 q^{61} - 85536 q^{63} + 31104 q^{69} - 111456 q^{71} + 52052 q^{73} + 77064 q^{75} - 109998 q^{81} + 156816 q^{83} + 50688 q^{85} + 39600 q^{87} - 67716 q^{93} + 199584 q^{95} + 206180 q^{97} - 58320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 1.65831i
0.500000 1.65831i
0 −12.0000 9.94987i 0 79.5990i 0 179.098i 0 45.0000 + 238.797i 0
47.2 0 −12.0000 + 9.94987i 0 79.5990i 0 179.098i 0 45.0000 238.797i 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.6.c.a 2
3.b odd 2 1 48.6.c.c yes 2
4.b odd 2 1 48.6.c.c yes 2
8.b even 2 1 192.6.c.c 2
8.d odd 2 1 192.6.c.a 2
12.b even 2 1 inner 48.6.c.a 2
24.f even 2 1 192.6.c.c 2
24.h odd 2 1 192.6.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.6.c.a 2 1.a even 1 1 trivial
48.6.c.a 2 12.b even 2 1 inner
48.6.c.c yes 2 3.b odd 2 1
48.6.c.c yes 2 4.b odd 2 1
192.6.c.a 2 8.d odd 2 1
192.6.c.a 2 24.h odd 2 1
192.6.c.c 2 8.b even 2 1
192.6.c.c 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(48, [\chi])\):

\( T_{5}^{2} + 6336 \) Copy content Toggle raw display
\( T_{11} + 648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 24T + 243 \) Copy content Toggle raw display
$5$ \( T^{2} + 6336 \) Copy content Toggle raw display
$7$ \( T^{2} + 32076 \) Copy content Toggle raw display
$11$ \( (T + 648)^{2} \) Copy content Toggle raw display
$13$ \( (T + 242)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 101376 \) Copy content Toggle raw display
$19$ \( T^{2} + 1571724 \) Copy content Toggle raw display
$23$ \( (T + 1296)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3960000 \) Copy content Toggle raw display
$31$ \( T^{2} + 11579436 \) Copy content Toggle raw display
$37$ \( (T + 12058)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 219200256 \) Copy content Toggle raw display
$43$ \( T^{2} + 327207276 \) Copy content Toggle raw display
$47$ \( (T + 12960)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 728139456 \) Copy content Toggle raw display
$59$ \( (T + 8424)^{2} \) Copy content Toggle raw display
$61$ \( (T + 25762)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 104214924 \) Copy content Toggle raw display
$71$ \( (T + 55728)^{2} \) Copy content Toggle raw display
$73$ \( (T - 26026)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 367238124 \) Copy content Toggle raw display
$83$ \( (T - 78408)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 7092290304 \) Copy content Toggle raw display
$97$ \( (T - 103090)^{2} \) Copy content Toggle raw display
show more
show less