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Properties

Label 1008.4.k.d
Level $1008$
Weight $4$
Character orbit 1008.k
Analytic conductor $59.474$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,4,Mod(881,1008)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.881");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.k (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: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.95072796278784.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 44x^{6} + 8x^{5} + 738x^{4} + 2416x^{3} + 3652x^{2} + 2824x + 946 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + (\beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_{5} q^{11} + (\beta_{7} + \beta_{6} + \beta_{2}) q^{13} + ( - 2 \beta_{4} - \beta_{3}) q^{17} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{2}) q^{19} + (5 \beta_{5} - 2 \beta_1) q^{23} + (4 \beta_{6} - 4 \beta_{2} - 5) q^{25} + ( - 10 \beta_{5} + \beta_1) q^{29} + ( - 6 \beta_{6} - 6 \beta_{2}) q^{31} + (4 \beta_{5} + \beta_{4} + \cdots - 4 \beta_1) q^{35}+ \cdots + (15 \beta_{7} - \beta_{6} - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{7} - 40 q^{25} + 512 q^{37} - 32 q^{43} + 632 q^{49} + 2336 q^{67} - 1792 q^{79} - 1344 q^{85} - 2496 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 44x^{6} + 8x^{5} + 738x^{4} + 2416x^{3} + 3652x^{2} + 2824x + 946 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 92664 \nu^{7} - 501309 \nu^{6} - 3333924 \nu^{5} + 5212827 \nu^{4} + 60114114 \nu^{3} + \cdots + 73262754 ) / 662137 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2138 \nu^{7} - 8330 \nu^{6} - 98982 \nu^{5} + 29213 \nu^{4} + 1739456 \nu^{3} + 4979240 \nu^{2} + \cdots + 2310265 ) / 13513 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 131700 \nu^{7} + 608310 \nu^{6} + 5519368 \nu^{5} - 5089780 \nu^{4} - 97299380 \nu^{3} + \cdots - 108572368 ) / 662137 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 237192 \nu^{7} + 713670 \nu^{6} + 12744356 \nu^{5} - 518804 \nu^{4} - 216361900 \nu^{3} + \cdots - 325962032 ) / 662137 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{7} - 111\nu^{6} - 576\nu^{5} + 1539\nu^{4} + 10758\nu^{3} + 18072\nu^{2} + 11880\nu + 1572 ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7186 \nu^{7} + 35734 \nu^{6} + 284298 \nu^{5} - 353329 \nu^{4} - 5035528 \nu^{3} - 12209692 \nu^{2} + \cdots - 5414873 ) / 13513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10160 \nu^{7} + 61580 \nu^{6} + 335976 \nu^{5} - 876944 \nu^{4} - 6170336 \nu^{3} - 10332920 \nu^{2} + \cdots - 751720 ) / 13513 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{6} - 9\beta_{3} - 3\beta_{2} + 2\beta _1 + 18 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 15\beta_{6} - 6\beta_{5} + 3\beta_{4} - 51\beta_{3} - 9\beta_{2} + 2\beta _1 + 468 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -45\beta_{7} + 264\beta_{6} - 72\beta_{5} + 18\beta_{4} - 684\beta_{3} - 120\beta_{2} + 160\beta _1 + 5112 ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -111\beta_{7} + 435\beta_{6} - 198\beta_{5} + 42\beta_{4} - 1110\beta_{3} - 159\beta_{2} + 286\beta _1 + 8694 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1785 \beta_{7} + 6312 \beta_{6} - 3090 \beta_{5} + 495 \beta_{4} - 14247 \beta_{3} - 1392 \beta_{2} + \cdots + 110628 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 14373 \beta_{7} + 43440 \beta_{6} - 25104 \beta_{5} + 3168 \beta_{4} - 90090 \beta_{3} + \cdots + 699660 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 110313 \beta_{7} + 299547 \beta_{6} - 192192 \beta_{5} + 19803 \beta_{4} - 560103 \beta_{3} + \cdots + 4355460 ) / 36 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
6.73116 − 0.707107i
6.73116 + 0.707107i
−1.34451 + 0.707107i
−1.34451 − 0.707107i
−2.45132 − 0.707107i
−2.45132 + 0.707107i
−0.935327 + 0.707107i
−0.935327 − 0.707107i
0 0 0 −15.3330 0 −16.3875 − 8.62844i 0 0 0
881.2 0 0 0 −15.3330 0 −16.3875 + 8.62844i 0 0 0
881.3 0 0 0 −2.21360 0 12.3875 − 13.7677i 0 0 0
881.4 0 0 0 −2.21360 0 12.3875 + 13.7677i 0 0 0
881.5 0 0 0 2.21360 0 12.3875 − 13.7677i 0 0 0
881.6 0 0 0 2.21360 0 12.3875 + 13.7677i 0 0 0
881.7 0 0 0 15.3330 0 −16.3875 − 8.62844i 0 0 0
881.8 0 0 0 15.3330 0 −16.3875 + 8.62844i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.k.d 8
3.b odd 2 1 inner 1008.4.k.d 8
4.b odd 2 1 252.4.f.a ✓ 8
7.b odd 2 1 inner 1008.4.k.d 8
12.b even 2 1 252.4.f.a ✓ 8
21.c even 2 1 inner 1008.4.k.d 8
28.d even 2 1 252.4.f.a ✓ 8
28.f even 6 2 1764.4.t.a 16
28.g odd 6 2 1764.4.t.a 16
84.h odd 2 1 252.4.f.a ✓ 8
84.j odd 6 2 1764.4.t.a 16
84.n even 6 2 1764.4.t.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.f.a ✓ 8 4.b odd 2 1
252.4.f.a ✓ 8 12.b even 2 1
252.4.f.a ✓ 8 28.d even 2 1
252.4.f.a ✓ 8 84.h odd 2 1
1008.4.k.d 8 1.a even 1 1 trivial
1008.4.k.d 8 3.b odd 2 1 inner
1008.4.k.d 8 7.b odd 2 1 inner
1008.4.k.d 8 21.c even 2 1 inner
1764.4.t.a 16 28.f even 6 2
1764.4.t.a 16 28.g odd 6 2
1764.4.t.a 16 84.j odd 6 2
1764.4.t.a 16 84.n even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 240T_{5}^{2} + 1152 \) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\). 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
$5$ \( (T^{4} - 240 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 414)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5088 T^{2} + 3875328)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16368 T^{2} + 194688)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 21792 T^{2} + 1663488)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 21996 T^{2} + 94128804)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 83124 T^{2} + 1700572644)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 38016 T^{2} + 292626432)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 128 T - 25712)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 328560 T^{2} + 22577275008)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 20684)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 407616 T^{2} + 41527314432)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 323028 T^{2} + 11397270564)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 102912 T^{2} + 1982988288)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 710400 T^{2} + 22302720000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 584 T + 72016)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 331308 T^{2} + 17756628516)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 463584 T^{2} + 53676605952)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 448 T - 69056)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1430592 T^{2} + 328711292928)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 503280 T^{2} + 24175366272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 816096 T^{2} + 149346427392)^{2} \) Copy content Toggle raw display
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