Properties

Label 1008.3.cd.m
Level $1008$
Weight $3$
Character orbit 1008.cd
Analytic conductor $27.466$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(415,1008)]
 
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([3, 0, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cd (of order \(6\), degree \(2\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.2682209403.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 56x^{4} + 784x^{2} + 2883 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{7} + (\beta_{5} - \beta_{2} - \beta_1 + 2) q^{11} + ( - \beta_{5} + \beta_{4} + 2) q^{13} + ( - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{17} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + 6 \beta_1 + 6) q^{19} + ( - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1 + 4) q^{23} + (3 \beta_{5} + 6 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 18 \beta_1) q^{25} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} + 5) q^{29} + (3 \beta_{5} - \beta_{3} - 5 \beta_{2} - \beta_1 + 2) q^{31} + (\beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 16 \beta_1 + 7) q^{35} + ( - 2 \beta_{5} - \beta_{4} - 6 \beta_{3} - 5 \beta_{2} - 10 \beta_1 + 10) q^{37} + (2 \beta_{5} - 2 \beta_{4} - 32) q^{41} + (\beta_{5} + \beta_{4} + 8 \beta_{3} + 4 \beta_{2} - 24 \beta_1 + 12) q^{43} + ( - 6 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} - 14 \beta_1 - 14) q^{47} + (3 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 25 \beta_1 - 3) q^{49} + (2 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 19 \beta_1) q^{53} + ( - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} + 34 \beta_1 - 17) q^{55} + (11 \beta_{5} + 8 \beta_{3} + 5 \beta_{2} + 11 \beta_1 - 22) q^{59} + ( - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 18 \beta_1 - 18) q^{61} + ( - 8 \beta_{5} - 4 \beta_{4} - 10 \beta_{3} - 6 \beta_{2} + 28 \beta_1 - 28) q^{65} + (3 \beta_{5} + 6 \beta_{3} + 9 \beta_{2} + 40 \beta_1 - 80) q^{67} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 76 \beta_1 - 38) q^{71} + ( - \beta_{5} - 2 \beta_{4} + 12 \beta_{3} - \beta_{2} - 16 \beta_1) q^{73} + (2 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 21 \beta_1 + 43) q^{77} + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 11 \beta_1 + 11) q^{79} + ( - 5 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 1) q^{83} + (2 \beta_{5} - 2 \beta_{4} + 22 \beta_{2} - 16) q^{85} + (2 \beta_{3} + 2 \beta_{2} - 86 \beta_1 + 86) q^{89} + ( - 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 54 \beta_1 - 18) q^{91} + (16 \beta_{5} + 14 \beta_{3} + 12 \beta_{2} - 72 \beta_1 + 144) q^{95} + (7 \beta_{5} - 7 \beta_{4} - 10 \beta_{2} + 53) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{5} + 11 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{5} + 11 q^{7} + 12 q^{11} + 10 q^{13} + 28 q^{17} + 51 q^{19} + 36 q^{23} - 59 q^{25} + 32 q^{29} + 21 q^{31} + 90 q^{35} + 33 q^{37} - 188 q^{41} - 102 q^{47} + 61 q^{49} + 56 q^{53} - 90 q^{59} - 62 q^{61} - 86 q^{65} - 369 q^{67} - 33 q^{73} + 196 q^{77} + 105 q^{79} - 136 q^{85} + 256 q^{89} - 253 q^{91} + 654 q^{95} + 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 56x^{4} + 784x^{2} + 2883 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 28\nu + 31 ) / 62 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} - 59\nu^{2} - 589 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 47\nu^{3} + 59\nu^{2} + 439\nu + 589 ) / 62 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 3\nu^{4} - 47\nu^{3} - 115\nu^{2} - 377\nu - 589 ) / 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 47\nu^{3} + 115\nu^{2} - 377\nu + 589 ) / 62 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} - 3\beta_{2} - 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -14\beta_{5} - 14\beta_{4} - 28\beta_{3} - 14\beta_{2} + 62\beta _1 - 31 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 59\beta_{5} - 59\beta_{4} + 115\beta_{2} + 1064 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 877\beta_{5} + 877\beta_{4} + 1878\beta_{3} + 939\beta_{2} - 5828\beta _1 + 2914 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
2.43024i
3.63982i
6.07006i
2.43024i
3.63982i
6.07006i
0 0 0 −3.94231 + 6.82829i 0 6.17532 3.29627i 0 0 0
415.2 0 0 0 0.776320 1.34463i 0 −5.35699 4.50585i 0 0 0
415.3 0 0 0 4.16599 7.21571i 0 4.68167 + 5.20404i 0 0 0
991.1 0 0 0 −3.94231 6.82829i 0 6.17532 + 3.29627i 0 0 0
991.2 0 0 0 0.776320 + 1.34463i 0 −5.35699 + 4.50585i 0 0 0
991.3 0 0 0 4.16599 + 7.21571i 0 4.68167 5.20404i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cd.m 6
3.b odd 2 1 336.3.be.c 6
4.b odd 2 1 1008.3.cd.l 6
7.c even 3 1 1008.3.cd.l 6
12.b even 2 1 336.3.be.e yes 6
21.g even 6 1 2352.3.m.m 6
21.h odd 6 1 336.3.be.e yes 6
21.h odd 6 1 2352.3.m.p 6
28.g odd 6 1 inner 1008.3.cd.m 6
84.j odd 6 1 2352.3.m.m 6
84.n even 6 1 336.3.be.c 6
84.n even 6 1 2352.3.m.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.c 6 3.b odd 2 1
336.3.be.c 6 84.n even 6 1
336.3.be.e yes 6 12.b even 2 1
336.3.be.e yes 6 21.h odd 6 1
1008.3.cd.l 6 4.b odd 2 1
1008.3.cd.l 6 7.c even 3 1
1008.3.cd.m 6 1.a even 1 1 trivial
1008.3.cd.m 6 28.g odd 6 1 inner
2352.3.m.m 6 21.g even 6 1
2352.3.m.m 6 84.j odd 6 1
2352.3.m.p 6 21.h odd 6 1
2352.3.m.p 6 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{6} - 2T_{5}^{5} + 69T_{5}^{4} - 74T_{5}^{3} + 4429T_{5}^{2} - 6630T_{5} + 10404 \) Copy content Toggle raw display
\( T_{11}^{6} - 12T_{11}^{5} - 25T_{11}^{4} + 876T_{11}^{3} + 4681T_{11}^{2} - 11826T_{11} + 8748 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} - 172T_{13} + 272 \) Copy content Toggle raw display
\( T_{19}^{6} - 51T_{19}^{5} + 623T_{19}^{4} + 12444T_{19}^{3} + 17920T_{19}^{2} - 597312T_{19} + 1997568 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 69 T^{4} + \cdots + 10404 \) Copy content Toggle raw display
$7$ \( T^{6} - 11 T^{5} + 30 T^{4} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{6} - 12 T^{5} - 25 T^{4} + \cdots + 8748 \) Copy content Toggle raw display
$13$ \( (T^{3} - 5 T^{2} - 172 T + 272)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 28 T^{5} + \cdots + 115605504 \) Copy content Toggle raw display
$19$ \( T^{6} - 51 T^{5} + 623 T^{4} + \cdots + 1997568 \) Copy content Toggle raw display
$23$ \( T^{6} - 36 T^{5} + 128 T^{4} + \cdots + 442368 \) Copy content Toggle raw display
$29$ \( (T^{3} - 16 T^{2} - 881 T - 2436)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 21 T^{5} + \cdots + 217311363 \) Copy content Toggle raw display
$37$ \( T^{6} - 33 T^{5} + \cdots + 972441856 \) Copy content Toggle raw display
$41$ \( (T^{3} + 94 T^{2} + 2224 T + 8352)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 6227 T^{4} + \cdots + 1237244592 \) Copy content Toggle raw display
$47$ \( T^{6} + 102 T^{5} + \cdots + 77964057792 \) Copy content Toggle raw display
$53$ \( T^{6} - 56 T^{5} + \cdots + 359936784 \) Copy content Toggle raw display
$59$ \( T^{6} + 90 T^{5} + \cdots + 42160884912 \) Copy content Toggle raw display
$61$ \( T^{6} + 62 T^{5} + 4600 T^{4} + \cdots + 1382976 \) Copy content Toggle raw display
$67$ \( T^{6} + 369 T^{5} + \cdots + 3610604592 \) Copy content Toggle raw display
$71$ \( T^{6} + 13892 T^{4} + \cdots + 63605068992 \) Copy content Toggle raw display
$73$ \( T^{6} + 33 T^{5} + \cdots + 177689227024 \) Copy content Toggle raw display
$79$ \( T^{6} - 105 T^{5} + \cdots + 24831387 \) Copy content Toggle raw display
$83$ \( T^{6} + 18002 T^{4} + \cdots + 67710764268 \) Copy content Toggle raw display
$89$ \( T^{6} - 256 T^{5} + \cdots + 357929385984 \) Copy content Toggle raw display
$97$ \( (T^{3} - 176 T^{2} + 1109 T + 489542)^{2} \) Copy content Toggle raw display
show more
show less