Properties

Label 1232.4.a.w
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.509800.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 5x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 77)
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 3) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{5} + 7 q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 3) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 - 5) q^{5} + 7 q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} + 15) q^{9} - 11 q^{11} + (5 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 31) q^{13} + ( - 11 \beta_{3} - \beta_{2} + \cdots + 10) q^{15}+ \cdots + (33 \beta_{3} + 55 \beta_{2} - 165) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} - 18 q^{5} + 28 q^{7} + 66 q^{9} - 44 q^{11} - 134 q^{13} + 62 q^{15} - 74 q^{17} + 164 q^{19} + 84 q^{21} - 194 q^{23} + 38 q^{25} + 510 q^{27} - 108 q^{29} + 412 q^{31} - 132 q^{33} - 126 q^{35} + 286 q^{37} + 256 q^{39} - 18 q^{41} + 496 q^{43} + 580 q^{45} - 62 q^{47} + 196 q^{49} + 508 q^{51} - 828 q^{53} + 198 q^{55} + 700 q^{57} + 1224 q^{59} - 350 q^{61} + 462 q^{63} - 396 q^{65} + 1498 q^{67} - 386 q^{69} - 2326 q^{71} - 1630 q^{73} + 1362 q^{75} - 308 q^{77} + 1020 q^{79} + 1128 q^{81} + 1920 q^{83} + 2008 q^{85} - 1640 q^{87} + 1550 q^{89} - 938 q^{91} + 6046 q^{93} - 2332 q^{95} - 2202 q^{97} - 726 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 10x^{2} + 5x + 15 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 7\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 4\beta_{2} + 7\beta _1 + 17 ) / 4 \) 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
−1.11082
3.18303
−2.66444
1.59222
0 −5.17115 0 −10.0822 0 7.00000 0 −0.259212 0
1.2 0 0.163384 0 −5.36789 0 7.00000 0 −26.9733 0
1.3 0 7.36360 0 −15.4926 0 7.00000 0 27.2227 0
1.4 0 9.64416 0 12.9427 0 7.00000 0 66.0098 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.4.a.w 4
4.b odd 2 1 77.4.a.c 4
12.b even 2 1 693.4.a.m 4
20.d odd 2 1 1925.4.a.q 4
28.d even 2 1 539.4.a.f 4
44.c even 2 1 847.4.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.c 4 4.b odd 2 1
539.4.a.f 4 28.d even 2 1
693.4.a.m 4 12.b even 2 1
847.4.a.e 4 44.c even 2 1
1232.4.a.w 4 1.a even 1 1 trivial
1925.4.a.q 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{4} - 12T_{3}^{3} - 15T_{3}^{2} + 370T_{3} - 60 \) Copy content Toggle raw display
\( T_{5}^{4} + 18T_{5}^{3} - 107T_{5}^{2} - 2960T_{5} - 10852 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 12 T^{3} + \cdots - 60 \) Copy content Toggle raw display
$5$ \( T^{4} + 18 T^{3} + \cdots - 10852 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 134 T^{3} + \cdots - 9904192 \) Copy content Toggle raw display
$17$ \( T^{4} + 74 T^{3} + \cdots - 4708304 \) Copy content Toggle raw display
$19$ \( T^{4} - 164 T^{3} + \cdots - 85552320 \) Copy content Toggle raw display
$23$ \( T^{4} + 194 T^{3} + \cdots - 39720496 \) Copy content Toggle raw display
$29$ \( T^{4} + 108 T^{3} + \cdots - 365881040 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 5207968724 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1094639996 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + \cdots - 410971280 \) Copy content Toggle raw display
$43$ \( T^{4} - 496 T^{3} + \cdots - 998066176 \) Copy content Toggle raw display
$47$ \( T^{4} + 62 T^{3} + \cdots + 463480064 \) Copy content Toggle raw display
$53$ \( T^{4} + 828 T^{3} + \cdots - 394495824 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1674727140 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 3730099088 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 57482107536 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 109860635344 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 34532794928 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 48577598400 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 42421669632 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 926653158300 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 78194289572 \) Copy content Toggle raw display
show more
show less