↑

Properties

Label 1008.3.d.e
Level $1008$
Weight $3$
Character orbit 1008.d
Analytic conductor $27.466$
Analytic rank $0$
Dimension $8$
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(449,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 46x^{6} - 56x^{5} + 789x^{4} + 1736x^{3} - 1668x^{2} - 336x + 18468 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} + \beta_{2}) q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} + \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - \beta_{6} - \beta_{5} + \cdots + \beta_{2}) q^{11}+ \cdots + (4 \beta_{7} - 23 \beta_{3} + \cdots + 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{13} + 24 q^{19} - 32 q^{25} + 104 q^{31} + 24 q^{37} - 96 q^{43} + 56 q^{49} - 104 q^{55} - 136 q^{61} - 328 q^{67} - 464 q^{73} + 200 q^{79} + 440 q^{85} + 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 46x^{6} - 56x^{5} + 789x^{4} + 1736x^{3} - 1668x^{2} - 336x + 18468 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 77003 \nu^{7} - 350722 \nu^{6} + 5286660 \nu^{5} + 14470168 \nu^{4} - 111325075 \nu^{3} + \cdots + 41659956 ) / 349131960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40511 \nu^{7} - 52766 \nu^{6} - 1592250 \nu^{5} - 375676 \nu^{4} + 26275405 \nu^{3} + \cdots + 130497948 ) / 174565980 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 33\nu^{7} - 218\nu^{6} - 420\nu^{5} + 5032\nu^{4} - 2415\nu^{3} - 49922\nu^{2} + 128658\nu + 218724 ) / 135480 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26383 \nu^{7} + 135628 \nu^{6} + 946470 \nu^{5} - 2843887 \nu^{4} - 19313945 \nu^{3} + \cdots - 199512594 ) / 87282990 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 55\nu^{6} - 708\nu^{5} + 2182\nu^{4} + 11888\nu^{3} - 23397\nu^{2} - 60132\nu + 145278 ) / 30924 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 138158 \nu^{7} - 116393 \nu^{6} - 6312350 \nu^{5} - 795228 \nu^{4} + 94004700 \nu^{3} + \cdots + 511517934 ) / 58188660 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18651 \nu^{7} - 90366 \nu^{6} - 550212 \nu^{5} + 1842264 \nu^{4} + 8787875 \nu^{3} + \cdots - 58330812 ) / 7758488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 4\beta_{4} - 4\beta_{3} + \beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 6\beta_{6} + 6\beta_{5} + 6\beta_{4} - 31\beta_{3} + 80\beta_{2} + 13\beta _1 + 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{7} - 16\beta_{6} + 64\beta_{5} + 136\beta_{4} - 88\beta_{3} + 176\beta_{2} + 43\beta _1 + 269 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 42\beta_{7} - 230\beta_{6} + 230\beta_{5} + 510\beta_{4} - 319\beta_{3} + 2508\beta_{2} + 109\beta _1 + 1050 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -37\beta_{7} - 824\beta_{6} + 2288\beta_{5} + 4156\beta_{4} - 580\beta_{3} + 10072\beta_{2} + 355\beta _1 - 919 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 338 \beta_{7} - 6370 \beta_{6} + 8722 \beta_{5} + 19698 \beta_{4} + 9317 \beta_{3} + 69676 \beta_{2} + \cdots - 9114 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−2.80526 − 1.41421i
1.21848 + 1.41421i
−3.86423 − 1.41421i
5.45101 + 1.41421i
5.45101 − 1.41421i
−3.86423 + 1.41421i
1.21848 − 1.41421i
−2.80526 + 1.41421i
0 0 0 − 9.12311i 0 2.64575 0 0 0
449.2 0 0 0 − 4.05064i 0 −2.64575 0 0 0
449.3 0 0 0 − 3.13741i 0 −2.64575 0 0 0
449.4 0 0 0 − 2.55302i 0 2.64575 0 0 0
449.5 0 0 0 2.55302i 0 2.64575 0 0 0
449.6 0 0 0 3.13741i 0 −2.64575 0 0 0
449.7 0 0 0 4.05064i 0 −2.64575 0 0 0
449.8 0 0 0 9.12311i 0 2.64575 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.d.e 8
3.b odd 2 1 inner 1008.3.d.e 8
4.b odd 2 1 504.3.d.b ✓ 8
8.b even 2 1 4032.3.d.l 8
8.d odd 2 1 4032.3.d.m 8
12.b even 2 1 504.3.d.b ✓ 8
24.f even 2 1 4032.3.d.m 8
24.h odd 2 1 4032.3.d.l 8
28.d even 2 1 3528.3.d.e 8
84.h odd 2 1 3528.3.d.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.d.b ✓ 8 4.b odd 2 1
504.3.d.b ✓ 8 12.b even 2 1
1008.3.d.e 8 1.a even 1 1 trivial
1008.3.d.e 8 3.b odd 2 1 inner
3528.3.d.e 8 28.d even 2 1
3528.3.d.e 8 84.h odd 2 1
4032.3.d.l 8 8.b even 2 1
4032.3.d.l 8 24.h odd 2 1
4032.3.d.m 8 8.d odd 2 1
4032.3.d.m 8 24.f even 2 1

Hecke kernels

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

\( T_{5}^{8} + 116T_{5}^{6} + 3060T_{5}^{4} + 28736T_{5}^{2} + 87616 \) Copy content Toggle raw display
\( T_{11}^{8} + 644T_{11}^{6} + 95076T_{11}^{4} + 3157056T_{11}^{2} + 1679616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 116 T^{6} + \cdots + 87616 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 644 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$13$ \( (T^{4} - 12 T^{3} + \cdots + 80448)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1204 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{3} + \cdots - 28544)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 72744562944 \) Copy content Toggle raw display
$29$ \( T^{8} + 3544 T^{6} + \cdots + 681210000 \) Copy content Toggle raw display
$31$ \( (T^{4} - 52 T^{3} + \cdots - 995200)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + \cdots + 946752)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 2043596047936 \) Copy content Toggle raw display
$43$ \( (T^{4} + 48 T^{3} + \cdots + 815616)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15423653871616 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2276 T^{2} + 427716)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 40179400704 \) Copy content Toggle raw display
$61$ \( (T^{4} + 68 T^{3} + \cdots + 161232)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 164 T^{3} + \cdots - 23731200)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 62261189611776 \) Copy content Toggle raw display
$73$ \( (T^{4} + 232 T^{3} + \cdots + 4802624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 100 T^{3} + \cdots + 28736)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 76868845830144 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 614668544064 \) Copy content Toggle raw display
$97$ \( (T^{4} - 96 T^{3} + \cdots - 91278528)^{2} \) Copy content Toggle raw display
show more
show less