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Properties

Label 1040.2.bg.k
Level $1040$
Weight $2$
Character orbit 1040.bg
Analytic conductor $8.304$
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] = [1040,2,Mod(577,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bg (of order \(4\), 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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 - 1) q^{3} + ( - \beta_{3} + 1) q^{5} - 3 \beta_{2} q^{7} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 - 1) q^{3} + ( - \beta_{3} + 1) q^{5} - 3 \beta_{2} q^{7} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{11} + (3 \beta_{2} - 2) q^{13} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{15} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} + 2) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{21} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{23} + ( - 3 \beta_{3} - 2) q^{25} + ( - \beta_{3} + 6 \beta_{2} - \beta_1 + 5) q^{27} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{29} + (\beta_{2} - 1) q^{31} + ( - \beta_{2} + 2 \beta_1 - 11) q^{33} + ( - 3 \beta_{2} - 3 \beta_1) q^{35} + 3 \beta_{2} q^{37} + (5 \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{39} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{41} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 1) q^{43} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 7) q^{45} + (2 \beta_{3} + 6 \beta_{2} + 1) q^{47} - 2 q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} - 1) q^{51} + ( - 4 \beta_{3} - 5 \beta_{2} + \cdots + 1) q^{53}+ \cdots + (6 \beta_{3} - 8 \beta_{2} - 6 \beta_1 + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 6 q^{5} - 8 q^{13} - 14 q^{15} + 6 q^{17} + 8 q^{19} + 6 q^{21} - 2 q^{25} + 22 q^{27} - 4 q^{31} - 44 q^{33} - 2 q^{39} - 12 q^{41} - 10 q^{43} + 22 q^{45} - 8 q^{49} + 12 q^{53} - 22 q^{55} - 8 q^{57} - 12 q^{59} + 8 q^{61} - 36 q^{63} - 12 q^{65} - 20 q^{67} - 44 q^{69} + 18 q^{71} - 16 q^{73} - 32 q^{75} - 8 q^{81} - 2 q^{85} - 16 q^{87} + 12 q^{89} + 36 q^{91} + 12 q^{95} - 4 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.65831 + 0.500000i
1.65831 + 0.500000i
−1.65831 − 0.500000i
1.65831 − 0.500000i
0 −2.15831 + 2.15831i 0 1.50000 + 1.65831i 0 − 3.00000i 0 − 6.31662i 0
577.2 0 1.15831 − 1.15831i 0 1.50000 − 1.65831i 0 − 3.00000i 0 0.316625i 0
593.1 0 −2.15831 − 2.15831i 0 1.50000 − 1.65831i 0 3.00000i 0 6.31662i 0
593.2 0 1.15831 + 1.15831i 0 1.50000 + 1.65831i 0 3.00000i 0 − 0.316625i 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
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.bg.k 4
4.b odd 2 1 130.2.g.d ✓ 4
5.c odd 4 1 1040.2.cd.i 4
12.b even 2 1 1170.2.m.e 4
13.d odd 4 1 1040.2.cd.i 4
20.d odd 2 1 650.2.g.g 4
20.e even 4 1 130.2.j.d yes 4
20.e even 4 1 650.2.j.f 4
52.f even 4 1 130.2.j.d yes 4
60.l odd 4 1 1170.2.w.e 4
65.k even 4 1 inner 1040.2.bg.k 4
156.l odd 4 1 1170.2.w.e 4
260.l odd 4 1 650.2.g.g 4
260.s odd 4 1 130.2.g.d ✓ 4
260.u even 4 1 650.2.j.f 4
780.bn even 4 1 1170.2.m.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.d ✓ 4 4.b odd 2 1
130.2.g.d ✓ 4 260.s odd 4 1
130.2.j.d yes 4 20.e even 4 1
130.2.j.d yes 4 52.f even 4 1
650.2.g.g 4 20.d odd 2 1
650.2.g.g 4 260.l odd 4 1
650.2.j.f 4 20.e even 4 1
650.2.j.f 4 260.u even 4 1
1040.2.bg.k 4 1.a even 1 1 trivial
1040.2.bg.k 4 65.k even 4 1 inner
1040.2.cd.i 4 5.c odd 4 1
1040.2.cd.i 4 13.d odd 4 1
1170.2.m.e 4 12.b even 2 1
1170.2.m.e 4 780.bn even 4 1
1170.2.w.e 4 60.l odd 4 1
1170.2.w.e 4 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 10T_{3} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} + 484 \) Copy content Toggle raw display
\( T_{19}^{2} - 4T_{19} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 484 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 484 \) Copy content Toggle raw display
$29$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{4} + 94T^{2} + 625 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 4900 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T - 74)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + \cdots + 1225 \) Copy content Toggle raw display
$73$ \( (T + 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 216T^{2} + 8100 \) Copy content Toggle raw display
$83$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 98)^{2} \) Copy content Toggle raw display
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