Properties

Label 1584.4.a.bc
Level $1584$
Weight $4$
Character orbit 1584.a
Self dual yes
Analytic conductor $93.459$
Analytic rank $0$
Dimension $2$
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] = [1584,4,Mod(1,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, 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("1584.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.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: \(93.4590254491\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
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 \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta - 1) q^{5} + ( - \beta - 10) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta - 1) q^{5} + ( - \beta - 10) q^{7} - 11 q^{11} + (5 \beta + 40) q^{13} + (3 \beta + 62) q^{17} + (15 \beta - 36) q^{19} + (9 \beta - 49) q^{23} + ( - 4 \beta + 68) q^{25} + ( - 14 \beta - 72) q^{29} + (7 \beta + 17) q^{31} + ( - 19 \beta - 86) q^{35} + (2 \beta + 27) q^{37} + ( - \beta - 268) q^{41} + ( - 4 \beta + 30) q^{43} + (30 \beta - 136) q^{47} + (20 \beta - 195) q^{49} + ( - 14 \beta + 246) q^{53} + ( - 22 \beta + 11) q^{55} + (33 \beta + 317) q^{59} + ( - 46 \beta + 420) q^{61} + (75 \beta + 440) q^{65} + ( - 5 \beta - 377) q^{67} + ( - 19 \beta - 339) q^{71} + (117 \beta - 200) q^{73} + (11 \beta + 110) q^{77} + (164 \beta - 158) q^{79} + ( - 30 \beta + 234) q^{83} + (121 \beta + 226) q^{85} + ( - 82 \beta + 921) q^{89} + ( - 90 \beta - 640) q^{91} + ( - 87 \beta + 1476) q^{95} + ( - 36 \beta + 1097) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 20 q^{7} - 22 q^{11} + 80 q^{13} + 124 q^{17} - 72 q^{19} - 98 q^{23} + 136 q^{25} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} - 536 q^{41} + 60 q^{43} - 272 q^{47} - 390 q^{49} + 492 q^{53} + 22 q^{55} + 634 q^{59} + 840 q^{61} + 880 q^{65} - 754 q^{67} - 678 q^{71} - 400 q^{73} + 220 q^{77} - 316 q^{79} + 468 q^{83} + 452 q^{85} + 1842 q^{89} - 1280 q^{91} + 2952 q^{95} + 2194 q^{97}+O(q^{100}) \) 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 0 0 −14.8564 0 −3.07180 0 0 0
1.2 0 0 0 12.8564 0 −16.9282 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)

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 1584.4.a.bc 2
3.b odd 2 1 176.4.a.i 2
4.b odd 2 1 99.4.a.c 2
12.b even 2 1 11.4.a.a 2
20.d odd 2 1 2475.4.a.q 2
24.f even 2 1 704.4.a.p 2
24.h odd 2 1 704.4.a.n 2
33.d even 2 1 1936.4.a.w 2
44.c even 2 1 1089.4.a.v 2
60.h even 2 1 275.4.a.b 2
60.l odd 4 2 275.4.b.c 4
84.h odd 2 1 539.4.a.e 2
132.d odd 2 1 121.4.a.c 2
132.n odd 10 4 121.4.c.f 8
132.o even 10 4 121.4.c.c 8
156.h even 2 1 1859.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 12.b even 2 1
99.4.a.c 2 4.b odd 2 1
121.4.a.c 2 132.d odd 2 1
121.4.c.c 8 132.o even 10 4
121.4.c.f 8 132.n odd 10 4
176.4.a.i 2 3.b odd 2 1
275.4.a.b 2 60.h even 2 1
275.4.b.c 4 60.l odd 4 2
539.4.a.e 2 84.h odd 2 1
704.4.a.n 2 24.h odd 2 1
704.4.a.p 2 24.f even 2 1
1089.4.a.v 2 44.c even 2 1
1584.4.a.bc 2 1.a even 1 1 trivial
1859.4.a.a 2 156.h even 2 1
1936.4.a.w 2 33.d even 2 1
2475.4.a.q 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\):

\( T_{5}^{2} + 2T_{5} - 191 \) Copy content Toggle raw display
\( T_{7}^{2} + 20T_{7} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 52 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 80T + 400 \) Copy content Toggle raw display
$17$ \( T^{2} - 124T + 3412 \) Copy content Toggle raw display
$19$ \( T^{2} + 72T - 9504 \) Copy content Toggle raw display
$23$ \( T^{2} + 98T - 1487 \) Copy content Toggle raw display
$29$ \( T^{2} + 144T - 4224 \) Copy content Toggle raw display
$31$ \( T^{2} - 34T - 2063 \) Copy content Toggle raw display
$37$ \( T^{2} - 54T + 537 \) Copy content Toggle raw display
$41$ \( T^{2} + 536T + 71776 \) Copy content Toggle raw display
$43$ \( T^{2} - 60T + 132 \) Copy content Toggle raw display
$47$ \( T^{2} + 272T - 24704 \) Copy content Toggle raw display
$53$ \( T^{2} - 492T + 51108 \) Copy content Toggle raw display
$59$ \( T^{2} - 634T + 48217 \) Copy content Toggle raw display
$61$ \( T^{2} - 840T + 74832 \) Copy content Toggle raw display
$67$ \( T^{2} + 754T + 140929 \) Copy content Toggle raw display
$71$ \( T^{2} + 678T + 97593 \) Copy content Toggle raw display
$73$ \( T^{2} + 400T - 617072 \) Copy content Toggle raw display
$79$ \( T^{2} + 316 T - 1266044 \) Copy content Toggle raw display
$83$ \( T^{2} - 468T + 11556 \) Copy content Toggle raw display
$89$ \( T^{2} - 1842 T + 525489 \) Copy content Toggle raw display
$97$ \( T^{2} - 2194 T + 1141201 \) Copy content Toggle raw display
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