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Properties

Label 1040.2.q.q
Level $1040$
Weight $2$
Character orbit 1040.q
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.27870912.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 520)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{3} - q^{5} + 3 \beta_{4} q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{3} - q^{5} + 3 \beta_{4} q^{7} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{9}+ \cdots + (2 \beta_{3} + \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} - 6 q^{5} + 9 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} - 6 q^{5} + 9 q^{7} - 4 q^{9} + 3 q^{11} + 4 q^{13} - q^{15} - 7 q^{17} + q^{19} + 6 q^{21} + q^{23} + 6 q^{25} - 14 q^{27} - 3 q^{29} + 8 q^{31} - 5 q^{33} - 9 q^{35} - 5 q^{37} - 15 q^{39} - 3 q^{41} + 3 q^{43} + 4 q^{45} - 12 q^{47} - 6 q^{49} - 14 q^{51} + 32 q^{53} - 3 q^{55} - 58 q^{57} + 31 q^{59} - 7 q^{61} + 12 q^{63} - 4 q^{65} - 7 q^{67} - 9 q^{69} - 9 q^{71} + 12 q^{73} + q^{75} + 18 q^{77} - 32 q^{79} + q^{81} + 16 q^{83} + 7 q^{85} - 33 q^{87} + 11 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 3 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 7\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\) \(1\)

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
−1.08870 + 1.88569i
0.160819 − 0.278546i
1.42789 − 2.47317i
−1.08870 − 1.88569i
0.160819 + 0.278546i
1.42789 + 2.47317i
0 −1.08870 − 1.88569i 0 −1.00000 0 1.50000 − 2.59808i 0 −0.870556 + 1.50785i 0
81.2 0 0.160819 + 0.278546i 0 −1.00000 0 1.50000 − 2.59808i 0 1.44827 − 2.50849i 0
81.3 0 1.42789 + 2.47317i 0 −1.00000 0 1.50000 − 2.59808i 0 −2.57772 + 4.46474i 0
321.1 0 −1.08870 + 1.88569i 0 −1.00000 0 1.50000 + 2.59808i 0 −0.870556 − 1.50785i 0
321.2 0 0.160819 − 0.278546i 0 −1.00000 0 1.50000 + 2.59808i 0 1.44827 + 2.50849i 0
321.3 0 1.42789 − 2.47317i 0 −1.00000 0 1.50000 + 2.59808i 0 −2.57772 − 4.46474i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.3
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.q 6
4.b odd 2 1 520.2.q.g ✓ 6
13.c even 3 1 inner 1040.2.q.q 6
52.i odd 6 1 6760.2.a.x 3
52.j odd 6 1 520.2.q.g ✓ 6
52.j odd 6 1 6760.2.a.w 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.q.g ✓ 6 4.b odd 2 1
520.2.q.g ✓ 6 52.j odd 6 1
1040.2.q.q 6 1.a even 1 1 trivial
1040.2.q.q 6 13.c even 3 1 inner
6760.2.a.w 3 52.j odd 6 1
6760.2.a.x 3 52.i odd 6 1

Hecke kernels

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

\( T_{3}^{6} - T_{3}^{5} + 7T_{3}^{4} + 2T_{3}^{3} + 38T_{3}^{2} - 12T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 9 \) Copy content Toggle raw display
\( T_{11}^{6} - 3T_{11}^{5} + 20T_{11}^{4} + 39T_{11}^{3} + 112T_{11}^{2} + 33T_{11} + 9 \) Copy content Toggle raw display
\( T_{19}^{6} - T_{19}^{5} + 48T_{19}^{4} + 265T_{19}^{3} + 2100T_{19}^{2} + 5123T_{19} + 11881 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + 7 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 4 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 7 T^{5} + \cdots + 66564 \) Copy content Toggle raw display
$19$ \( T^{6} - T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$23$ \( T^{6} - T^{5} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 53824 \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 26 T + 96)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} + \cdots + 39601 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 53824 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 126736 \) Copy content Toggle raw display
$47$ \( (T^{3} + 6 T^{2} + \cdots + 116)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 16 T^{2} + \cdots + 218)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 31 T^{5} + \cdots + 678976 \) Copy content Toggle raw display
$61$ \( T^{6} + 7 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$67$ \( T^{6} + 7 T^{5} + \cdots + 602176 \) Copy content Toggle raw display
$71$ \( T^{6} + 9 T^{5} + \cdots + 104976 \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} + \cdots + 1284)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 8 T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 11 T^{5} + \cdots + 744769 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 3364 \) Copy content Toggle raw display
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