↑

Properties

Label 1040.2.k.d
Level $1040$
Weight $2$
Character orbit 1040.k
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(961,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (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: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + ( - \beta_{3} + 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + ( - \beta_{3} + 2 \beta_1 + 2) q^{9} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{11} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + (7 \beta_{5} - 8 \beta_{4} - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 6 q^{9} - 8 q^{13} + 8 q^{17} - 16 q^{23} - 6 q^{25} + 28 q^{27} - 4 q^{29} + 4 q^{35} + 16 q^{39} - 24 q^{43} + 2 q^{49} - 8 q^{51} + 12 q^{53} + 12 q^{55} - 12 q^{61} + 2 q^{65} - 32 q^{69} - 4 q^{75} + 36 q^{77} - 24 q^{79} + 30 q^{81} + 4 q^{87} - 32 q^{91} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 6\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.66044 − 1.66044i
1.66044 + 1.66044i
−1.33641 + 1.33641i
−1.33641 − 1.33641i
0.675970 − 0.675970i
0.675970 + 0.675970i
0 −1.51414 0 − 1.00000i 0 3.32088i 0 −0.707389 0
961.2 0 −1.51414 0 1.00000i 0 − 3.32088i 0 −0.707389 0
961.3 0 0.428007 0 − 1.00000i 0 − 2.67282i 0 −2.81681 0
961.4 0 0.428007 0 1.00000i 0 2.67282i 0 −2.81681 0
961.5 0 3.08613 0 − 1.00000i 0 1.35194i 0 6.52420 0
961.6 0 3.08613 0 1.00000i 0 − 1.35194i 0 6.52420 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.k.d 6
4.b odd 2 1 65.2.c.a ✓ 6
12.b even 2 1 585.2.b.g 6
13.b even 2 1 inner 1040.2.k.d 6
20.d odd 2 1 325.2.c.g 6
20.e even 4 1 325.2.d.e 6
20.e even 4 1 325.2.d.f 6
52.b odd 2 1 65.2.c.a ✓ 6
52.f even 4 1 845.2.a.i 3
52.f even 4 1 845.2.a.k 3
52.i odd 6 2 845.2.m.h 12
52.j odd 6 2 845.2.m.h 12
52.l even 12 2 845.2.e.i 6
52.l even 12 2 845.2.e.k 6
156.h even 2 1 585.2.b.g 6
156.l odd 4 1 7605.2.a.bs 3
156.l odd 4 1 7605.2.a.cc 3
260.g odd 2 1 325.2.c.g 6
260.p even 4 1 325.2.d.e 6
260.p even 4 1 325.2.d.f 6
260.u even 4 1 4225.2.a.bc 3
260.u even 4 1 4225.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a ✓ 6 4.b odd 2 1
65.2.c.a ✓ 6 52.b odd 2 1
325.2.c.g 6 20.d odd 2 1
325.2.c.g 6 260.g odd 2 1
325.2.d.e 6 20.e even 4 1
325.2.d.e 6 260.p even 4 1
325.2.d.f 6 20.e even 4 1
325.2.d.f 6 260.p even 4 1
585.2.b.g 6 12.b even 2 1
585.2.b.g 6 156.h even 2 1
845.2.a.i 3 52.f even 4 1
845.2.a.k 3 52.f even 4 1
845.2.e.i 6 52.l even 12 2
845.2.e.k 6 52.l even 12 2
845.2.m.h 12 52.i odd 6 2
845.2.m.h 12 52.j odd 6 2
1040.2.k.d 6 1.a even 1 1 trivial
1040.2.k.d 6 13.b even 2 1 inner
4225.2.a.bc 3 260.u even 4 1
4225.2.a.be 3 260.u even 4 1
7605.2.a.bs 3 156.l odd 4 1
7605.2.a.cc 3 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} + 48 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 4 T^{2} - 32 T + 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 44 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( (T^{3} + 8 T^{2} + 16 T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 8 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 56 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$37$ \( T^{6} + 152 T^{4} + \cdots + 51984 \) Copy content Toggle raw display
$41$ \( T^{6} + 92 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( (T^{3} + 12 T^{2} + 12 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 20 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 60 T - 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 84 T^{4} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 12 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 156 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$71$ \( T^{6} + 44 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$73$ \( T^{6} + 296 T^{4} + \cdots + 266256 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 156 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( T^{6} + 192 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$97$ \( T^{6} + 140 T^{4} + \cdots + 576 \) Copy content Toggle raw display
show more
show less