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Properties

Label 1040.2.q.h
Level $1040$
Weight $2$
Character orbit 1040.q
Analytic conductor $8.304$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,2,Mod(81,1040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-3,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - 3 \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + (\zeta_{6} - 4) q^{13} + 4 \zeta_{6} q^{17} + 7 \zeta_{6} q^{19} + (4 \zeta_{6} - 4) q^{23} + q^{25} + ( - 8 \zeta_{6} + 8) q^{29} + \cdots - 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 7 q^{13} + 4 q^{17} + 7 q^{19} - 4 q^{23} + 2 q^{25} + 8 q^{29} + 20 q^{31} - 3 q^{35} - 3 q^{37} + 2 q^{41} + 6 q^{43} + 3 q^{45} + 2 q^{47} - 2 q^{49} - 18 q^{53}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 1.00000 0 −1.50000 + 2.59808i 0 1.50000 − 2.59808i 0
321.1 0 0 0 1.00000 0 −1.50000 − 2.59808i 0 1.50000 + 2.59808i 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.h 2
4.b odd 2 1 130.2.e.a ✓ 2
12.b even 2 1 1170.2.i.i 2
13.c even 3 1 inner 1040.2.q.h 2
20.d odd 2 1 650.2.e.b 2
20.e even 4 2 650.2.o.d 4
52.b odd 2 1 1690.2.e.h 2
52.f even 4 2 1690.2.l.e 4
52.i odd 6 1 1690.2.a.c 1
52.i odd 6 1 1690.2.e.h 2
52.j odd 6 1 130.2.e.a ✓ 2
52.j odd 6 1 1690.2.a.h 1
52.l even 12 2 1690.2.d.c 2
52.l even 12 2 1690.2.l.e 4
156.p even 6 1 1170.2.i.i 2
260.v odd 6 1 650.2.e.b 2
260.v odd 6 1 8450.2.a.g 1
260.w odd 6 1 8450.2.a.q 1
260.bj even 12 2 650.2.o.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.a ✓ 2 4.b odd 2 1
130.2.e.a ✓ 2 52.j odd 6 1
650.2.e.b 2 20.d odd 2 1
650.2.e.b 2 260.v odd 6 1
650.2.o.d 4 20.e even 4 2
650.2.o.d 4 260.bj even 12 2
1040.2.q.h 2 1.a even 1 1 trivial
1040.2.q.h 2 13.c even 3 1 inner
1170.2.i.i 2 12.b even 2 1
1170.2.i.i 2 156.p even 6 1
1690.2.a.c 1 52.i odd 6 1
1690.2.a.h 1 52.j odd 6 1
1690.2.d.c 2 52.l even 12 2
1690.2.e.h 2 52.b odd 2 1
1690.2.e.h 2 52.i odd 6 1
1690.2.l.e 4 52.f even 4 2
1690.2.l.e 4 52.l even 12 2
8450.2.a.g 1 260.v odd 6 1
8450.2.a.q 1 260.w odd 6 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19}^{2} - 7T_{19} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$47$ \( (T - 1)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
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