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Properties

Label 1040.2.q.o
Level $1040$
Weight $2$
Character orbit 1040.q
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(81,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.q (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - 2 \beta_1 + 2) q^{9} + ( - \beta_{2} - 2 \beta_1) q^{11} - \beta_{3} q^{13} - \beta_1 q^{15} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 - 4) q^{17} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{19} - q^{21} - 3 \beta_1 q^{23} + q^{25} + 5 q^{27} + ( - 2 \beta_{2} - \beta_1) q^{29} + 4 q^{31} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{33} + ( - \beta_1 + 1) q^{35} + \beta_{2} q^{37} + \beta_{2} q^{39} - 3 \beta_1 q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{43}+ \cdots + (2 \beta_{3} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{17} - 4 q^{19} - 4 q^{21} - 6 q^{23} + 4 q^{25} + 20 q^{27} - 2 q^{29} + 16 q^{31} + 4 q^{33} + 2 q^{35} - 6 q^{41} - 6 q^{43} - 4 q^{45} - 8 q^{47} + 12 q^{49} - 16 q^{51} + 16 q^{53} + 4 q^{55} - 8 q^{57} + 2 q^{61} + 4 q^{63} - 14 q^{67} + 6 q^{69} - 12 q^{71} - 32 q^{73} + 2 q^{75} + 8 q^{77} + 8 q^{79} - 2 q^{81} + 8 q^{83} + 8 q^{85} + 2 q^{87} - 2 q^{89} + 8 q^{93} + 4 q^{95} + 24 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
1.15139 + 1.99426i
−0.651388 − 1.12824i
1.15139 − 1.99426i
−0.651388 + 1.12824i
0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0 1.00000 − 1.73205i 0
81.2 0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0 1.00000 − 1.73205i 0
321.1 0 0.500000 − 0.866025i 0 −1.00000 0 −0.500000 − 0.866025i 0 1.00000 + 1.73205i 0
321.2 0 0.500000 − 0.866025i 0 −1.00000 0 −0.500000 − 0.866025i 0 1.00000 + 1.73205i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.q.o 4
4.b odd 2 1 65.2.e.b ✓ 4
12.b even 2 1 585.2.j.d 4
13.c even 3 1 inner 1040.2.q.o 4
20.d odd 2 1 325.2.e.a 4
20.e even 4 2 325.2.o.b 8
52.b odd 2 1 845.2.e.d 4
52.f even 4 2 845.2.m.d 8
52.i odd 6 1 845.2.a.f 2
52.i odd 6 1 845.2.e.d 4
52.j odd 6 1 65.2.e.b ✓ 4
52.j odd 6 1 845.2.a.c 2
52.l even 12 2 845.2.c.d 4
52.l even 12 2 845.2.m.d 8
156.p even 6 1 585.2.j.d 4
156.p even 6 1 7605.2.a.bg 2
156.r even 6 1 7605.2.a.bb 2
260.v odd 6 1 325.2.e.a 4
260.v odd 6 1 4225.2.a.x 2
260.w odd 6 1 4225.2.a.t 2
260.bj even 12 2 325.2.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b ✓ 4 4.b odd 2 1
65.2.e.b ✓ 4 52.j odd 6 1
325.2.e.a 4 20.d odd 2 1
325.2.e.a 4 260.v odd 6 1
325.2.o.b 8 20.e even 4 2
325.2.o.b 8 260.bj even 12 2
585.2.j.d 4 12.b even 2 1
585.2.j.d 4 156.p even 6 1
845.2.a.c 2 52.j odd 6 1
845.2.a.f 2 52.i odd 6 1
845.2.c.d 4 52.l even 12 2
845.2.e.d 4 52.b odd 2 1
845.2.e.d 4 52.i odd 6 1
845.2.m.d 8 52.f even 4 2
845.2.m.d 8 52.l even 12 2
1040.2.q.o 4 1.a even 1 1 trivial
1040.2.q.o 4 13.c even 3 1 inner
4225.2.a.t 2 260.w odd 6 1
4225.2.a.x 2 260.v odd 6 1
7605.2.a.bb 2 156.r even 6 1
7605.2.a.bg 2 156.p even 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 25T_{11}^{2} - 36T_{11} + 81 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} + 25T_{19}^{2} - 36T_{19} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 117 T^{2} + 13689 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T + 12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$97$ \( T^{4} - 24 T^{3} + \cdots + 17161 \) Copy content Toggle raw display
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