Properties

Label 1536.4.a.m
Level $1536$
Weight $4$
Character orbit 1536.a
Self dual yes
Analytic conductor $90.627$
Analytic rank $1$
Dimension $4$
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] = [1536,4,Mod(1,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1536.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1536.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: \(90.6269337688\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 12x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 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.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{3} + \beta_1) q^{7} + 9 q^{9} + ( - \beta_{2} - 2) q^{11} - 9 \beta_1 q^{13} + ( - 3 \beta_{3} + 3 \beta_1) q^{15} - 86 q^{17} + (2 \beta_{2} + 4) q^{19} + (3 \beta_{3} + 3 \beta_1) q^{21} + (10 \beta_{3} - 2 \beta_1) q^{23} + (5 \beta_{2} + 101) q^{25} + 27 q^{27} + ( - 5 \beta_{3} + 14 \beta_1) q^{29} + (15 \beta_{3} + 10 \beta_1) q^{31} + ( - 3 \beta_{2} - 6) q^{33} + ( - \beta_{2} - 146) q^{35} + ( - 6 \beta_{3} - 39 \beta_1) q^{37} - 27 \beta_1 q^{39} + (2 \beta_{2} - 226) q^{41} + (4 \beta_{2} - 208) q^{43} + ( - 9 \beta_{3} + 9 \beta_1) q^{45} + (18 \beta_{3} - 44 \beta_1) q^{47} + ( - 3 \beta_{2} - 149) q^{49} - 258 q^{51} + ( - 37 \beta_{3} + 10 \beta_1) q^{53} + (22 \beta_{3} - 85 \beta_1) q^{55} + (6 \beta_{2} + 12) q^{57} + ( - 18 \beta_{2} - 108) q^{59} + (2 \beta_{3} + 7 \beta_1) q^{61} + (9 \beta_{3} + 9 \beta_1) q^{63} + ( - 18 \beta_{2} - 360) q^{65} + ( - 2 \beta_{2} + 104) q^{67} + (30 \beta_{3} - 6 \beta_1) q^{69} + ( - 18 \beta_{3} - 44 \beta_1) q^{71} + ( - 25 \beta_{2} - 264) q^{73} + (15 \beta_{2} + 303) q^{75} + (10 \beta_{3} + 89 \beta_1) q^{77} + (13 \beta_{3} + 75 \beta_1) q^{79} + 81 q^{81} + (9 \beta_{2} + 378) q^{83} + (86 \beta_{3} - 86 \beta_1) q^{85} + ( - 15 \beta_{3} + 42 \beta_1) q^{87} + (24 \beta_{2} + 178) q^{89} + (18 \beta_{2} - 216) q^{91} + (45 \beta_{3} + 30 \beta_1) q^{93} + ( - 44 \beta_{3} + 170 \beta_1) q^{95} + (16 \beta_{2} - 670) q^{97} + ( - 9 \beta_{2} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{3} + 36 q^{9} - 8 q^{11} - 344 q^{17} + 16 q^{19} + 404 q^{25} + 108 q^{27} - 24 q^{33} - 584 q^{35} - 904 q^{41} - 832 q^{43} - 596 q^{49} - 1032 q^{51} + 48 q^{57} - 432 q^{59} - 1440 q^{65} + 416 q^{67} - 1056 q^{73} + 1212 q^{75} + 324 q^{81} + 1512 q^{83} + 712 q^{89} - 864 q^{91} - 2680 q^{97} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 12x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{3} + 28\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 136\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 20\nu^{2} - 7\nu - 120 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + \beta _1 + 96 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{2} - 34\beta_1 ) / 16 \) 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
3.05231
−3.05231
−1.63810
1.63810
0 3.00000 0 −20.3376 0 9.02386 0 9.00000 0
1.2 0 3.00000 0 −6.19543 0 17.5091 0 9.00000 0
1.3 0 3.00000 0 6.19543 0 −17.5091 0 9.00000 0
1.4 0 3.00000 0 20.3376 0 −9.02386 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.4.a.m yes 4
4.b odd 2 1 1536.4.a.h 4
8.b even 2 1 1536.4.a.h 4
8.d odd 2 1 inner 1536.4.a.m yes 4
16.e even 4 2 1536.4.d.e 8
16.f odd 4 2 1536.4.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.4.a.h 4 4.b odd 2 1
1536.4.a.h 4 8.b even 2 1
1536.4.a.m yes 4 1.a even 1 1 trivial
1536.4.a.m yes 4 8.d odd 2 1 inner
1536.4.d.e 8 16.e even 4 2
1536.4.d.e 8 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1536))\):

\( T_{5}^{4} - 452T_{5}^{2} + 15876 \) Copy content Toggle raw display
\( T_{7}^{4} - 388T_{7}^{2} + 24964 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 1404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 452 T^{2} + 15876 \) Copy content Toggle raw display
$7$ \( T^{4} - 388 T^{2} + 24964 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 1404)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2592)^{2} \) Copy content Toggle raw display
$17$ \( (T + 86)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T - 5616)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36496 T^{2} + \cdots + 287370304 \) Copy content Toggle raw display
$29$ \( T^{4} - 23684 T^{2} + \cdots + 9253764 \) Copy content Toggle raw display
$31$ \( T^{4} - 81700 T^{2} + \cdots + 1470722500 \) Copy content Toggle raw display
$37$ \( T^{4} - 102672 T^{2} + \cdots + 1494904896 \) Copy content Toggle raw display
$41$ \( (T^{2} + 452 T + 45444)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 416 T + 20736)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 264592 T^{2} + \cdots + 332989504 \) Copy content Toggle raw display
$53$ \( T^{4} - 505604 T^{2} + \cdots + 52480395396 \) Copy content Toggle raw display
$59$ \( (T^{2} + 216 T - 444528)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 4112 T^{2} + 419904 \) Copy content Toggle raw display
$67$ \( (T^{2} - 208 T + 5184)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 213904 T^{2} + \cdots + 50353216 \) Copy content Toggle raw display
$73$ \( (T^{2} + 528 T - 810304)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 388964 T^{2} + \cdots + 18223380036 \) Copy content Toggle raw display
$83$ \( (T^{2} - 756 T + 28836)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 356 T - 779324)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1340 T + 88452)^{2} \) Copy content Toggle raw display
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