Properties

Label 592.2.w.e
Level $592$
Weight $2$
Character orbit 592.w
Analytic conductor $4.727$
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] = [592,2,Mod(529,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.w (of order \(6\), degree \(2\), 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_1 + 2) q^{5} - 2 \beta_1 q^{7} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_1 + 2) q^{5} - 2 \beta_1 q^{7} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{3} + 3) q^{11} + (2 \beta_1 - 4) q^{13} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{21} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{23} + (2 \beta_1 - 2) q^{25} - 4 q^{27} + ( - 10 \beta_1 + 5) q^{29} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{31} + ( - 4 \beta_{2} + 6 \beta_1) q^{33} + ( - 2 \beta_1 - 2) q^{35} + (4 \beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{39} + (4 \beta_{2} - 3 \beta_1) q^{41} + (2 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{43} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{45} + (\beta_{3} - 3) q^{47} + ( - 3 \beta_1 + 3) q^{49} + (3 \beta_{3} - 6 \beta_{2} + 14 \beta_1 - 7) q^{51} + ( - 2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{53} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{55} + ( - 2 \beta_1 - 2) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - \beta_1 + 2) q^{61} + ( - 4 \beta_{3} + 2) q^{63} + (6 \beta_1 - 6) q^{65} + (3 \beta_{2} - 5 \beta_1) q^{67} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 8) q^{69} - 2 \beta_{2} q^{71} + 4 q^{73} + (2 \beta_{3} - 2) q^{75} + (2 \beta_{2} - 6 \beta_1) q^{77} + ( - 10 \beta_{3} + 5 \beta_{2} + \beta_1 - 2) q^{79} + ( - 2 \beta_{2} - \beta_1) q^{81} + (5 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{83} + ( - 6 \beta_{3} + 3) q^{85} + ( - 10 \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 10) q^{87} + (4 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 3) q^{89} + (4 \beta_1 + 4) q^{91} + ( - 2 \beta_1 + 4) q^{93} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{95} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{97} + (7 \beta_{3} - 7 \beta_{2} + 9 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} - 2 q^{9} + 12 q^{11} - 12 q^{13} + 6 q^{15} + 6 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{25} - 16 q^{27} + 12 q^{33} - 12 q^{35} - 2 q^{37} - 12 q^{39} - 6 q^{41} - 12 q^{47} + 6 q^{49} - 12 q^{53} + 18 q^{55} - 12 q^{57} + 12 q^{59} + 6 q^{61} + 8 q^{63} - 12 q^{65} - 10 q^{67} - 24 q^{69} + 16 q^{73} - 8 q^{75} - 12 q^{77} - 6 q^{79} - 2 q^{81} + 6 q^{83} + 12 q^{85} + 30 q^{87} + 18 q^{89} + 24 q^{91} + 12 q^{93} + 6 q^{95} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.366025 + 0.633975i 0 1.50000 + 0.866025i 0 −1.00000 + 1.73205i 0 1.23205 + 2.13397i 0
529.2 0 1.36603 2.36603i 0 1.50000 + 0.866025i 0 −1.00000 + 1.73205i 0 −2.23205 3.86603i 0
545.1 0 −0.366025 0.633975i 0 1.50000 0.866025i 0 −1.00000 1.73205i 0 1.23205 2.13397i 0
545.2 0 1.36603 + 2.36603i 0 1.50000 0.866025i 0 −1.00000 1.73205i 0 −2.23205 + 3.86603i 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
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.w.e 4
4.b odd 2 1 74.2.e.b 4
12.b even 2 1 666.2.s.a 4
37.e even 6 1 inner 592.2.w.e 4
148.j odd 6 1 74.2.e.b 4
148.l even 12 1 2738.2.a.e 2
148.l even 12 1 2738.2.a.i 2
444.p even 6 1 666.2.s.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.b 4 4.b odd 2 1
74.2.e.b 4 148.j odd 6 1
592.2.w.e 4 1.a even 1 1 trivial
592.2.w.e 4 37.e even 6 1 inner
666.2.s.a 4 12.b even 2 1
666.2.s.a 4 444.p even 6 1
2738.2.a.e 2 148.l even 12 1
2738.2.a.i 2 148.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\):

\( T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 33 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 75 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$43$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4 \) Copy content Toggle raw display
$71$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} - 210 T^{2} + \cdots + 49284 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} - 9 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
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