Properties

Label 4620.2.f.c
Level $4620$
Weight $2$
Character orbit 4620.f
Analytic conductor $36.891$
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] = [4620,2,Mod(769,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4620.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.f (of order \(2\), degree \(1\), 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: \(36.8908857338\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{2} - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} + (\beta_{2} - 2) q^{7} + q^{9} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{11} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{13} + (\beta_{3} - \beta_{2} + 1) q^{15} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} + (\beta_{3} + \beta_1 + 4) q^{19} + (\beta_{2} - 2) q^{21} + ( - 2 \beta_{3} + 2 \beta_1) q^{23} + (\beta_{2} + 3 \beta_1 + 1) q^{25} + q^{27} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{29} - 4 \beta_{2} q^{31} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{33} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{35} + (3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{37} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{3} - 2 \beta_1 - 4) q^{41} - 4 q^{43} + (\beta_{3} - \beta_{2} + 1) q^{45} + (\beta_{3} + \beta_1 + 8) q^{47} + ( - 4 \beta_{2} + 1) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{51} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{53} + (3 \beta_{3} + 2 \beta_{2} - 2) q^{55} + (\beta_{3} + \beta_1 + 4) q^{57} + (\beta_{3} - 6 \beta_{2} - \beta_1) q^{59} + ( - 2 \beta_{3} - 2 \beta_1 - 8) q^{61} + (\beta_{2} - 2) q^{63} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{65} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{67} + ( - 2 \beta_{3} + 2 \beta_1) q^{69} + (2 \beta_{3} + 2 \beta_1 + 4) q^{71} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{73} + (\beta_{2} + 3 \beta_1 + 1) q^{75} + ( - 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{77} + 6 \beta_{2} q^{79} + q^{81} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{83} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{85} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{87} + (4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{89} + (9 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{91} - 4 \beta_{2} q^{93} + (2 \beta_{3} - \beta_{2} + 3 \beta_1 + 8) q^{95} + (6 \beta_{3} + 6 \beta_1 + 2) q^{97} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 3 q^{5} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 3 q^{5} - 8 q^{7} + 4 q^{9} + 3 q^{15} + 14 q^{19} - 8 q^{21} + q^{25} + 4 q^{27} + 3 q^{35} - 12 q^{41} - 16 q^{43} + 3 q^{45} + 30 q^{47} + 4 q^{49} - 11 q^{55} + 14 q^{57} - 28 q^{61} - 8 q^{63} + 21 q^{65} + 12 q^{71} + q^{75} + 4 q^{81} + 2 q^{85} - 6 q^{91} + 27 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(1541\) \(2311\) \(2521\) \(3697\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
−1.18614 + 1.26217i
−1.18614 1.26217i
1.68614 0.396143i
1.68614 + 0.396143i
0 1.00000 0 −0.686141 2.12819i 0 −2.00000 + 1.73205i 0 1.00000 0
769.2 0 1.00000 0 −0.686141 + 2.12819i 0 −2.00000 1.73205i 0 1.00000 0
769.3 0 1.00000 0 2.18614 0.469882i 0 −2.00000 + 1.73205i 0 1.00000 0
769.4 0 1.00000 0 2.18614 + 0.469882i 0 −2.00000 1.73205i 0 1.00000 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
385.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4620.2.f.c yes 4
5.b even 2 1 4620.2.f.b yes 4
7.b odd 2 1 4620.2.f.a 4
11.b odd 2 1 4620.2.f.d yes 4
35.c odd 2 1 4620.2.f.d yes 4
55.d odd 2 1 4620.2.f.a 4
77.b even 2 1 4620.2.f.b yes 4
385.h even 2 1 inner 4620.2.f.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.f.a 4 7.b odd 2 1
4620.2.f.a 4 55.d odd 2 1
4620.2.f.b yes 4 5.b even 2 1
4620.2.f.b yes 4 77.b even 2 1
4620.2.f.c yes 4 1.a even 1 1 trivial
4620.2.f.c yes 4 385.h even 2 1 inner
4620.2.f.d yes 4 11.b odd 2 1
4620.2.f.d yes 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4620, [\chi])\):

\( T_{13}^{4} + 51T_{13}^{2} + 576 \) Copy content Toggle raw display
\( T_{19}^{2} - 7T_{19} + 4 \) Copy content Toggle raw display
\( T_{43} + 4 \) Copy content Toggle raw display
\( T_{47}^{2} - 15T_{47} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 4 T^{2} - 15 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 24)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 15 T + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( T^{4} + 187T^{2} + 7744 \) Copy content Toggle raw display
$61$ \( (T^{2} + 14 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 51T^{2} + 576 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 171T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 184T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 304T^{2} + 4096 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 296)^{2} \) Copy content Toggle raw display
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