Properties

Label 1040.2.bg.f
Level $1040$
Weight $2$
Character orbit 1040.bg
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (i + 1) q^{3} + ( - 2 i + 1) q^{5} - 2 i q^{7} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{3} + ( - 2 i + 1) q^{5} - 2 i q^{7} - i q^{9} + (i - 1) q^{11} + (2 i - 3) q^{13} + ( - i + 3) q^{15} + (5 i + 5) q^{17} + ( - 3 i + 3) q^{19} + ( - 2 i + 2) q^{21} + ( - 5 i + 5) q^{23} + ( - 4 i - 3) q^{25} + ( - 4 i + 4) q^{27} - 4 i q^{29} + ( - i - 1) q^{31} - 2 q^{33} + ( - 2 i - 4) q^{35} - 8 i q^{37} + ( - i - 5) q^{39} + (i + 1) q^{41} + (5 i - 5) q^{43} + ( - i - 2) q^{45} - 2 i q^{47} + 3 q^{49} + 10 i q^{51} + ( - i - 1) q^{53} + (3 i + 1) q^{55} + 6 q^{57} + ( - 3 i - 3) q^{59} + 2 q^{61} - 2 q^{63} + (8 i + 1) q^{65} + 12 q^{67} + 10 q^{69} + ( - i - 1) q^{71} - 6 q^{73} + ( - 7 i + 1) q^{75} + (2 i + 2) q^{77} + 14 i q^{79} + 5 q^{81} + 6 i q^{83} + ( - 5 i + 15) q^{85} + ( - 4 i + 4) q^{87} + ( - 7 i - 7) q^{89} + (6 i + 4) q^{91} - 2 i q^{93} + ( - 9 i - 3) q^{95} - 2 q^{97} + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 6 q^{13} + 6 q^{15} + 10 q^{17} + 6 q^{19} + 4 q^{21} + 10 q^{23} - 6 q^{25} + 8 q^{27} - 2 q^{31} - 4 q^{33} - 8 q^{35} - 10 q^{39} + 2 q^{41} - 10 q^{43} - 4 q^{45} + 6 q^{49} - 2 q^{53} + 2 q^{55} + 12 q^{57} - 6 q^{59} + 4 q^{61} - 4 q^{63} + 2 q^{65} + 24 q^{67} + 20 q^{69} - 2 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 10 q^{81} + 30 q^{85} + 8 q^{87} - 14 q^{89} + 8 q^{91} - 6 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(-i\) \(i\) \(1\)

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
1.00000i
1.00000i
0 1.00000 1.00000i 0 1.00000 + 2.00000i 0 2.00000i 0 1.00000i 0
593.1 0 1.00000 + 1.00000i 0 1.00000 2.00000i 0 2.00000i 0 1.00000i 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.f 2
4.b odd 2 1 130.2.g.c 2
5.c odd 4 1 1040.2.cd.e 2
12.b even 2 1 1170.2.m.a 2
13.d odd 4 1 1040.2.cd.e 2
20.d odd 2 1 650.2.g.b 2
20.e even 4 1 130.2.j.b yes 2
20.e even 4 1 650.2.j.d 2
52.f even 4 1 130.2.j.b yes 2
60.l odd 4 1 1170.2.w.c 2
65.k even 4 1 inner 1040.2.bg.f 2
156.l odd 4 1 1170.2.w.c 2
260.l odd 4 1 650.2.g.b 2
260.s odd 4 1 130.2.g.c 2
260.u even 4 1 650.2.j.d 2
780.bn even 4 1 1170.2.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.c 2 4.b odd 2 1
130.2.g.c 2 260.s odd 4 1
130.2.j.b yes 2 20.e even 4 1
130.2.j.b yes 2 52.f even 4 1
650.2.g.b 2 20.d odd 2 1
650.2.g.b 2 260.l odd 4 1
650.2.j.d 2 20.e even 4 1
650.2.j.d 2 260.u even 4 1
1040.2.bg.f 2 1.a even 1 1 trivial
1040.2.bg.f 2 65.k even 4 1 inner
1040.2.cd.e 2 5.c odd 4 1
1040.2.cd.e 2 13.d odd 4 1
1170.2.m.a 2 12.b even 2 1
1170.2.m.a 2 780.bn even 4 1
1170.2.w.c 2 60.l odd 4 1
1170.2.w.c 2 156.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 \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 196 \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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