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Properties

Label 1008.2.df.a
Level $1008$
Weight $2$
Character orbit 1008.df
Analytic conductor $8.049$
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] = [1008,2,Mod(689,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5, 1])) 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("1008.689"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.df (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,-6,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 2) q^{3} - 3 q^{5} + ( - \zeta_{6} + 3) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + (2 \zeta_{6} - 1) q^{11} + ( - \zeta_{6} - 1) q^{13} + (3 \zeta_{6} - 6) q^{15} + ( - 3 \zeta_{6} + 3) q^{17}+ \cdots + (3 \zeta_{6} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 6 q^{5} + 5 q^{7} + 3 q^{9} - 3 q^{13} - 9 q^{15} + 3 q^{17} + 9 q^{19} + 6 q^{21} + 8 q^{25} - 9 q^{29} + 6 q^{31} + 3 q^{33} - 15 q^{35} - 7 q^{37} - 6 q^{39} + 3 q^{41} + q^{43} - 9 q^{45}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(\zeta_{6}\) \(1\) \(1 - \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 1.50000 − 0.866025i 0 −3.00000 0 2.50000 − 0.866025i 0 1.50000 − 2.59808i 0
929.1 0 1.50000 + 0.866025i 0 −3.00000 0 2.50000 + 0.866025i 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
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.df.a 2
3.b odd 2 1 3024.2.df.a 2
4.b odd 2 1 63.2.s.a yes 2
7.d odd 6 1 1008.2.ca.a 2
9.c even 3 1 3024.2.ca.a 2
9.d odd 6 1 1008.2.ca.a 2
12.b even 2 1 189.2.s.a 2
21.g even 6 1 3024.2.ca.a 2
28.d even 2 1 441.2.s.a 2
28.f even 6 1 63.2.i.a ✓ 2
28.f even 6 1 441.2.o.a 2
28.g odd 6 1 441.2.i.a 2
28.g odd 6 1 441.2.o.b 2
36.f odd 6 1 189.2.i.a 2
36.f odd 6 1 567.2.p.b 2
36.h even 6 1 63.2.i.a ✓ 2
36.h even 6 1 567.2.p.a 2
63.k odd 6 1 3024.2.df.a 2
63.s even 6 1 inner 1008.2.df.a 2
84.h odd 2 1 1323.2.s.a 2
84.j odd 6 1 189.2.i.a 2
84.j odd 6 1 1323.2.o.b 2
84.n even 6 1 1323.2.i.a 2
84.n even 6 1 1323.2.o.a 2
252.n even 6 1 189.2.s.a 2
252.o even 6 1 441.2.s.a 2
252.r odd 6 1 441.2.o.b 2
252.r odd 6 1 567.2.p.b 2
252.s odd 6 1 441.2.i.a 2
252.u odd 6 1 1323.2.o.b 2
252.bb even 6 1 441.2.o.a 2
252.bi even 6 1 1323.2.i.a 2
252.bj even 6 1 567.2.p.a 2
252.bj even 6 1 1323.2.o.a 2
252.bl odd 6 1 1323.2.s.a 2
252.bn odd 6 1 63.2.s.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.a ✓ 2 28.f even 6 1
63.2.i.a ✓ 2 36.h even 6 1
63.2.s.a yes 2 4.b odd 2 1
63.2.s.a yes 2 252.bn odd 6 1
189.2.i.a 2 36.f odd 6 1
189.2.i.a 2 84.j odd 6 1
189.2.s.a 2 12.b even 2 1
189.2.s.a 2 252.n even 6 1
441.2.i.a 2 28.g odd 6 1
441.2.i.a 2 252.s odd 6 1
441.2.o.a 2 28.f even 6 1
441.2.o.a 2 252.bb even 6 1
441.2.o.b 2 28.g odd 6 1
441.2.o.b 2 252.r odd 6 1
441.2.s.a 2 28.d even 2 1
441.2.s.a 2 252.o even 6 1
567.2.p.a 2 36.h even 6 1
567.2.p.a 2 252.bj even 6 1
567.2.p.b 2 36.f odd 6 1
567.2.p.b 2 252.r odd 6 1
1008.2.ca.a 2 7.d odd 6 1
1008.2.ca.a 2 9.d odd 6 1
1008.2.df.a 2 1.a even 1 1 trivial
1008.2.df.a 2 63.s even 6 1 inner
1323.2.i.a 2 84.n even 6 1
1323.2.i.a 2 252.bi even 6 1
1323.2.o.a 2 84.n even 6 1
1323.2.o.a 2 252.bj even 6 1
1323.2.o.b 2 84.j odd 6 1
1323.2.o.b 2 252.u odd 6 1
1323.2.s.a 2 84.h odd 2 1
1323.2.s.a 2 252.bl odd 6 1
3024.2.ca.a 2 9.c even 3 1
3024.2.ca.a 2 21.g even 6 1
3024.2.df.a 2 3.b odd 2 1
3024.2.df.a 2 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$23$ \( T^{2} + 27 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
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