Properties

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

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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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.522072.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 4) q^{3} + (2 \beta_{2} - \beta_1 + 4) q^{5} + 7 q^{7} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 18) q^{9} + 11 q^{11} + ( - 3 \beta_{2} - 10 \beta_1 + 18) q^{13} + ( - 6 \beta_{3} - 2 \beta_{2} + \cdots - 70) q^{15}+ \cdots + (33 \beta_{3} + 44 \beta_{2} + \cdots + 198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 12\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 2\nu^{2} + 24\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 6\beta _1 + 3 \) 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
3.79597
−3.20317
−0.148103
0.555307
0 −10.1459 0 8.69995 0 7.00000 0 75.9402 0
1.2 0 −6.57251 0 15.5514 0 7.00000 0 16.1978 0
1.3 0 −2.77399 0 1.84418 0 7.00000 0 −19.3050 0
1.4 0 5.49244 0 −16.0955 0 7.00000 0 3.16692 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.s 4
4.b odd 2 1 77.4.a.d 4
12.b even 2 1 693.4.a.l 4
20.d odd 2 1 1925.4.a.p 4
28.d even 2 1 539.4.a.g 4
44.c even 2 1 847.4.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.d 4 4.b odd 2 1
539.4.a.g 4 28.d even 2 1
693.4.a.l 4 12.b even 2 1
847.4.a.d 4 44.c even 2 1
1232.4.a.s 4 1.a even 1 1 trivial
1925.4.a.p 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016 \) Copy content Toggle raw display
\( T_{5}^{4} - 10T_{5}^{3} - 240T_{5}^{2} + 2648T_{5} - 4016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} + \cdots - 1016 \) Copy content Toggle raw display
$5$ \( T^{4} - 10 T^{3} + \cdots - 4016 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 58 T^{3} + \cdots - 4947656 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots - 2705024 \) Copy content Toggle raw display
$19$ \( T^{4} + 258 T^{3} + \cdots - 14423904 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots - 17449856 \) Copy content Toggle raw display
$29$ \( T^{4} + 396 T^{3} + \cdots + 22336464 \) Copy content Toggle raw display
$31$ \( T^{4} - 56 T^{3} + \cdots - 11250248 \) Copy content Toggle raw display
$37$ \( T^{4} - 84 T^{3} + \cdots + 11157312 \) Copy content Toggle raw display
$41$ \( T^{4} - 52 T^{3} + \cdots - 659233664 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1210397376 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots - 318931592 \) Copy content Toggle raw display
$53$ \( T^{4} - 624 T^{3} + \cdots + 403923072 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 17599820728 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 6668930664 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 140865466496 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 72982082688 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 322052228384 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 59537293568 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 21951092064 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 109303561968 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 868634650768 \) Copy content Toggle raw display
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