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Properties

Label 1008.2.r.k
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2) q^{5} + \beta_{4} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{5} - 8 \beta_{4} + 2 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 5 q^{5} - 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + 5 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 3 q^{13} - 11 q^{15} - 24 q^{17} + 6 q^{19} - 2 q^{21} - 6 q^{25} + 7 q^{27} - q^{29} - 3 q^{31} + 8 q^{33} - 10 q^{35} - 6 q^{37} - 2 q^{39} + 22 q^{41} - 3 q^{43} + 5 q^{45} - 9 q^{47} - 3 q^{49} - 9 q^{51} - 36 q^{53} + 12 q^{55} + 11 q^{57} - 9 q^{59} + 6 q^{61} + 8 q^{63} + 5 q^{65} - 39 q^{69} - 18 q^{71} + 6 q^{73} - 31 q^{75} - 2 q^{77} + 15 q^{79} + 32 q^{81} - 12 q^{83} - 9 q^{85} + q^{87} - 4 q^{89} + 6 q^{91} + 33 q^{93} + 16 q^{95} - 3 q^{97} + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 + 0.224437i
0.500000 − 1.41036i
0.500000 + 2.05195i
0.500000 − 0.224437i
0.500000 + 1.41036i
0.500000 − 2.05195i
0 −0.349814 − 1.69636i 0 1.79418 − 3.10761i 0 −0.500000 − 0.866025i 0 −2.75526 + 1.18682i 0
337.2 0 0.619562 + 1.61745i 0 −0.590972 + 1.02359i 0 −0.500000 − 0.866025i 0 −2.23229 + 2.00422i 0
337.3 0 1.73025 + 0.0789082i 0 1.29679 − 2.24611i 0 −0.500000 − 0.866025i 0 2.98755 + 0.273062i 0
673.1 0 −0.349814 + 1.69636i 0 1.79418 + 3.10761i 0 −0.500000 + 0.866025i 0 −2.75526 − 1.18682i 0
673.2 0 0.619562 − 1.61745i 0 −0.590972 − 1.02359i 0 −0.500000 + 0.866025i 0 −2.23229 − 2.00422i 0
673.3 0 1.73025 − 0.0789082i 0 1.29679 + 2.24611i 0 −0.500000 + 0.866025i 0 2.98755 − 0.273062i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.k 6
3.b odd 2 1 3024.2.r.g 6
4.b odd 2 1 63.2.f.b ✓ 6
9.c even 3 1 inner 1008.2.r.k 6
9.c even 3 1 9072.2.a.bq 3
9.d odd 6 1 3024.2.r.g 6
9.d odd 6 1 9072.2.a.cd 3
12.b even 2 1 189.2.f.a 6
28.d even 2 1 441.2.f.d 6
28.f even 6 1 441.2.g.d 6
28.f even 6 1 441.2.h.b 6
28.g odd 6 1 441.2.g.e 6
28.g odd 6 1 441.2.h.c 6
36.f odd 6 1 63.2.f.b ✓ 6
36.f odd 6 1 567.2.a.d 3
36.h even 6 1 189.2.f.a 6
36.h even 6 1 567.2.a.g 3
84.h odd 2 1 1323.2.f.c 6
84.j odd 6 1 1323.2.g.b 6
84.j odd 6 1 1323.2.h.e 6
84.n even 6 1 1323.2.g.c 6
84.n even 6 1 1323.2.h.d 6
252.n even 6 1 441.2.h.b 6
252.o even 6 1 1323.2.h.d 6
252.r odd 6 1 1323.2.g.b 6
252.s odd 6 1 1323.2.f.c 6
252.s odd 6 1 3969.2.a.p 3
252.u odd 6 1 441.2.g.e 6
252.bb even 6 1 1323.2.g.c 6
252.bi even 6 1 441.2.f.d 6
252.bi even 6 1 3969.2.a.m 3
252.bj even 6 1 441.2.g.d 6
252.bl odd 6 1 441.2.h.c 6
252.bn odd 6 1 1323.2.h.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b ✓ 6 4.b odd 2 1
63.2.f.b ✓ 6 36.f odd 6 1
189.2.f.a 6 12.b even 2 1
189.2.f.a 6 36.h even 6 1
441.2.f.d 6 28.d even 2 1
441.2.f.d 6 252.bi even 6 1
441.2.g.d 6 28.f even 6 1
441.2.g.d 6 252.bj even 6 1
441.2.g.e 6 28.g odd 6 1
441.2.g.e 6 252.u odd 6 1
441.2.h.b 6 28.f even 6 1
441.2.h.b 6 252.n even 6 1
441.2.h.c 6 28.g odd 6 1
441.2.h.c 6 252.bl odd 6 1
567.2.a.d 3 36.f odd 6 1
567.2.a.g 3 36.h even 6 1
1008.2.r.k 6 1.a even 1 1 trivial
1008.2.r.k 6 9.c even 3 1 inner
1323.2.f.c 6 84.h odd 2 1
1323.2.f.c 6 252.s odd 6 1
1323.2.g.b 6 84.j odd 6 1
1323.2.g.b 6 252.r odd 6 1
1323.2.g.c 6 84.n even 6 1
1323.2.g.c 6 252.bb even 6 1
1323.2.h.d 6 84.n even 6 1
1323.2.h.d 6 252.o even 6 1
1323.2.h.e 6 84.j odd 6 1
1323.2.h.e 6 252.bn odd 6 1
3024.2.r.g 6 3.b odd 2 1
3024.2.r.g 6 9.d odd 6 1
3969.2.a.m 3 252.bi even 6 1
3969.2.a.p 3 252.s odd 6 1
9072.2.a.bq 3 9.c even 3 1
9072.2.a.cd 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{6} - 5T_{5}^{5} + 23T_{5}^{4} - 32T_{5}^{3} + 59T_{5}^{2} + 22T_{5} + 121 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} + 23T_{11}^{4} + 56T_{11}^{3} + 455T_{11}^{2} + 893T_{11} + 2209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$17$ \( (T^{3} + 12 T^{2} + \cdots + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T^{2} - 6 T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 54 T + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 22 T^{5} + \cdots + 124609 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( (T^{3} + 18 T^{2} + 75 T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 4489 \) Copy content Toggle raw display
$67$ \( T^{6} + 207 T^{4} + \cdots + 466489 \) Copy content Toggle raw display
$71$ \( (T^{3} + 9 T^{2} - 6 T - 81)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 3 T^{2} + \cdots - 243)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 15 T^{5} + \cdots + 591361 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} + \cdots + 379)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + \cdots + 363609 \) Copy content Toggle raw display
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