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Properties

Label 1584.3.j.f
Level $1584$
Weight $3$
Character orbit 1584.j
Analytic conductor $43.161$
Analytic rank $0$
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] = [1584,3,Mod(1297,1584)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1584.1297");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1584.j (of order \(2\), degree \(1\), not 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: \(43.1608738747\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.39744.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
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 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.

\(f(q)\) \(=\) \( q + ( - 3 \beta_1 - 1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{11} + (6 \beta_{3} - \beta_{2}) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{17} + (2 \beta_{3} + 4 \beta_{2}) q^{19} + (5 \beta_1 - 23) q^{23} + (6 \beta_1 + 3) q^{25} + (8 \beta_{3} - 4 \beta_{2}) q^{29} + (18 \beta_1 - 20) q^{31} + ( - 6 \beta_{3} - 2 \beta_{2}) q^{35} + (2 \beta_1 - 16) q^{37} + ( - 4 \beta_{3} - 6 \beta_{2}) q^{41} + (2 \beta_{3} + 12 \beta_{2}) q^{43} + ( - 3 \beta_1 + 25) q^{47} + (10 \beta_1 + 25) q^{49} + (11 \beta_1 - 7) q^{53} + (16 \beta_{3} + \beta_{2} - 16 \beta_1 - 14) q^{55} + ( - 42 \beta_1 + 10) q^{59} + (2 \beta_{3} + 11 \beta_{2}) q^{61} + ( - 30 \beta_{3} + 16 \beta_{2}) q^{65} + 34 q^{67} + (\beta_1 - 71) q^{71} + ( - 4 \beta_{3} + 10 \beta_{2}) q^{73} + (2 \beta_{3} - 4 \beta_{2} - 13 \beta_1 + 45) q^{77} + ( - 24 \beta_{3} + 5 \beta_{2}) q^{79} + ( - 16 \beta_{3} + 6 \beta_{2}) q^{83} + (20 \beta_{3} - 2 \beta_{2}) q^{85} + ( - 18 \beta_1 - 76) q^{89} + ( - 32 \beta_1 - 6) q^{91} + (16 \beta_{3} + 14 \beta_{2}) q^{95} + ( - 42 \beta_1 - 94) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 20 q^{11} - 92 q^{23} + 12 q^{25} - 80 q^{31} - 64 q^{37} + 100 q^{47} + 100 q^{49} - 28 q^{53} - 56 q^{55} + 40 q^{59} + 136 q^{67} - 284 q^{71} + 180 q^{77} - 304 q^{89} - 24 q^{91} - 376 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} + 2\nu + 23 ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 28\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 10\nu^{2} - 32\nu + 9 ) / 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} - 2\beta_{2} + 12\beta _1 - 19 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1
−0.366025 + 1.29224i
−0.366025 − 1.29224i
1.36603 + 3.21405i
1.36603 − 3.21405i
0 0 0 −6.19615 0 − 2.58447i 0 0 0
1297.2 0 0 0 −6.19615 0 2.58447i 0 0 0
1297.3 0 0 0 4.19615 0 − 6.42810i 0 0 0
1297.4 0 0 0 4.19615 0 6.42810i 0 0 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
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.3.j.f 4
3.b odd 2 1 528.3.j.c 4
4.b odd 2 1 99.3.c.b 4
11.b odd 2 1 inner 1584.3.j.f 4
12.b even 2 1 33.3.c.a ✓ 4
24.f even 2 1 2112.3.j.a 4
24.h odd 2 1 2112.3.j.d 4
33.d even 2 1 528.3.j.c 4
44.c even 2 1 99.3.c.b 4
60.h even 2 1 825.3.b.a 4
60.l odd 4 2 825.3.h.a 8
132.d odd 2 1 33.3.c.a ✓ 4
132.n odd 10 4 363.3.g.e 16
132.o even 10 4 363.3.g.e 16
264.m even 2 1 2112.3.j.d 4
264.p odd 2 1 2112.3.j.a 4
660.g odd 2 1 825.3.b.a 4
660.q even 4 2 825.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a ✓ 4 12.b even 2 1
33.3.c.a ✓ 4 132.d odd 2 1
99.3.c.b 4 4.b odd 2 1
99.3.c.b 4 44.c even 2 1
363.3.g.e 16 132.n odd 10 4
363.3.g.e 16 132.o even 10 4
528.3.j.c 4 3.b odd 2 1
528.3.j.c 4 33.d even 2 1
825.3.b.a 4 60.h even 2 1
825.3.b.a 4 660.g odd 2 1
825.3.h.a 8 60.l odd 4 2
825.3.h.a 8 660.q even 4 2
1584.3.j.f 4 1.a even 1 1 trivial
1584.3.j.f 4 11.b odd 2 1 inner
2112.3.j.a 4 24.f even 2 1
2112.3.j.a 4 264.p odd 2 1
2112.3.j.d 4 24.h odd 2 1
2112.3.j.d 4 264.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1584, [\chi])\):

\( T_{5}^{2} + 2T_{5} - 26 \) Copy content Toggle raw display
\( T_{7}^{4} + 48T_{7}^{2} + 276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 48T^{2} + 276 \) Copy content Toggle raw display
$11$ \( T^{4} - 20 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 624 T^{2} + 33396 \) Copy content Toggle raw display
$17$ \( T^{4} + 216T^{2} + 9936 \) Copy content Toggle raw display
$19$ \( T^{4} + 936T^{2} + 9936 \) Copy content Toggle raw display
$23$ \( (T^{2} + 46 T + 454)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 1536 T^{2} + 70656 \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T - 572)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 32 T + 244)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2304 T^{2} + 4416 \) Copy content Toggle raw display
$43$ \( T^{4} + 7272 T^{2} + 3843024 \) Copy content Toggle raw display
$47$ \( (T^{2} - 50 T + 598)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T - 314)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T - 5192)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 6144 T^{2} + 2596884 \) Copy content Toggle raw display
$67$ \( (T - 34)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 142 T + 5038)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4608 T^{2} + 4809024 \) Copy content Toggle raw display
$79$ \( T^{4} + 10128 T^{2} + 5643924 \) Copy content Toggle raw display
$83$ \( T^{4} + 5184 T^{2} + 4416 \) Copy content Toggle raw display
$89$ \( (T^{2} + 152 T + 4804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 188 T + 3544)^{2} \) Copy content Toggle raw display
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