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Properties

Label 1008.3.cd.g
Level $1008$
Weight $3$
Character orbit 1008.cd
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
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] = [1008,3,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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{7} + (7 \beta_{2} + 5 \beta_1 + 2) q^{11} + (\beta_{3} + \beta_{2} + \beta_1 + 6) q^{13} + ( - 8 \beta_{3} + 8 \beta_{2} + \cdots - 12) q^{17}+ \cdots + (31 \beta_{3} + 31 \beta_{2} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} + 12 q^{7} + 27 q^{11} + 26 q^{13} - 20 q^{17} - 9 q^{19} - 12 q^{23} + 21 q^{25} + 2 q^{29} - 90 q^{31} - 36 q^{35} - 19 q^{37} + 4 q^{41} - 84 q^{47} - 10 q^{49} - 169 q^{53} - 27 q^{59} + 96 q^{61} + 22 q^{65} + 9 q^{67} - 191 q^{73} + 169 q^{77} - 168 q^{79} - 208 q^{85} - 122 q^{89} + 135 q^{91} - 366 q^{95} + 50 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display

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

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

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(1\) \(1\)

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
−1.63746 + 1.52274i
2.13746 − 0.656712i
−1.63746 − 1.52274i
2.13746 + 0.656712i
0 0 0 −2.13746 + 3.70219i 0 −0.774917 + 6.95698i 0 0 0
415.2 0 0 0 1.63746 − 2.83616i 0 6.77492 − 1.76082i 0 0 0
991.1 0 0 0 −2.13746 − 3.70219i 0 −0.774917 − 6.95698i 0 0 0
991.2 0 0 0 1.63746 + 2.83616i 0 6.77492 + 1.76082i 0 0 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
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.g 4
3.b odd 2 1 336.3.be.a ✓ 4
4.b odd 2 1 1008.3.cd.f 4
7.c even 3 1 1008.3.cd.f 4
12.b even 2 1 336.3.be.b yes 4
21.g even 6 1 2352.3.m.h 4
21.h odd 6 1 336.3.be.b yes 4
21.h odd 6 1 2352.3.m.g 4
28.g odd 6 1 inner 1008.3.cd.g 4
84.j odd 6 1 2352.3.m.h 4
84.n even 6 1 336.3.be.a ✓ 4
84.n even 6 1 2352.3.m.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.a ✓ 4 3.b odd 2 1
336.3.be.a ✓ 4 84.n even 6 1
336.3.be.b yes 4 12.b even 2 1
336.3.be.b yes 4 21.h odd 6 1
1008.3.cd.f 4 4.b odd 2 1
1008.3.cd.f 4 7.c even 3 1
1008.3.cd.g 4 1.a even 1 1 trivial
1008.3.cd.g 4 28.g odd 6 1 inner
2352.3.m.g 4 21.h odd 6 1
2352.3.m.g 4 84.n even 6 1
2352.3.m.h 4 21.g even 6 1
2352.3.m.h 4 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{4} + T_{5}^{3} + 15T_{5}^{2} - 14T_{5} + 196 \) Copy content Toggle raw display
\( T_{11}^{4} - 27T_{11}^{3} + 185T_{11}^{2} + 1566T_{11} + 3364 \) Copy content Toggle raw display
\( T_{13}^{2} - 13T_{13} + 28 \) Copy content Toggle raw display
\( T_{19}^{4} + 9T_{19}^{3} - 769T_{19}^{2} - 7164T_{19} + 633616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{4} - 12 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} - 27 T^{3} + \cdots + 3364 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13 T + 28)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 20 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + \cdots + 633616 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 1724)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 90 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$37$ \( T^{4} + 19 T^{3} + \cdots + 369664 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 1424)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 3839 T^{2} + 2829124 \) Copy content Toggle raw display
$47$ \( (T^{2} + 42 T + 588)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 169 T^{3} + \cdots + 46022656 \) Copy content Toggle raw display
$59$ \( T^{4} + 27 T^{3} + \cdots + 1721344 \) Copy content Toggle raw display
$61$ \( T^{4} - 96 T^{3} + \cdots + 4309776 \) Copy content Toggle raw display
$67$ \( T^{4} - 9 T^{3} + \cdots + 142884 \) Copy content Toggle raw display
$71$ \( T^{4} + 15416 T^{2} + 56130064 \) Copy content Toggle raw display
$73$ \( T^{4} + 191 T^{3} + \cdots + 82919236 \) Copy content Toggle raw display
$79$ \( T^{4} + 168 T^{3} + \cdots + 660969 \) Copy content Toggle raw display
$83$ \( T^{4} + 25559 T^{2} + 60124516 \) Copy content Toggle raw display
$89$ \( T^{4} + 122 T^{3} + \cdots + 13424896 \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T - 13538)^{2} \) Copy content Toggle raw display
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