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Properties

Label 1008.3.dc.d
Level $1008$
Weight $3$
Character orbit 1008.dc
Analytic conductor $27.466$
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] = [1008,3,Mod(305,1008)]
 
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, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.92844527616.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 13x^{6} - 2x^{5} + 239x^{4} - 200x^{3} - 50x^{2} - 288x + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 252)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{5} + ( - \beta_{7} - \beta_{2} - 2) q^{7} + ( - \beta_{6} + 2 \beta_{4}) q^{11} + (\beta_1 - 14) q^{13} + (4 \beta_{6} + 3 \beta_{4}) q^{17} + ( - 3 \beta_{7} + 2 \beta_{2} + \cdots + 2) q^{19}+ \cdots + ( - 16 \beta_1 - 69) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{7} - 112 q^{13} + 8 q^{19} + 24 q^{25} - 20 q^{31} + 160 q^{37} + 80 q^{43} - 172 q^{49} + 40 q^{55} + 8 q^{61} + 144 q^{67} + 288 q^{73} - 140 q^{79} + 896 q^{85} + 352 q^{91} - 552 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 13x^{6} - 2x^{5} + 239x^{4} - 200x^{3} - 50x^{2} - 288x + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 51\nu^{7} + 184\nu^{6} - 705\nu^{5} - 834\nu^{4} + 1069\nu^{3} + 1164\nu^{2} + 756\nu + 369692 ) / 54766 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10084 \nu^{7} + 8183 \nu^{6} + 142618 \nu^{5} + 185843 \nu^{4} - 2214086 \nu^{3} + \cdots + 2726532 ) / 492894 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18952 \nu^{7} + 14847 \nu^{6} + 258762 \nu^{5} + 371130 \nu^{4} - 4017174 \nu^{3} + \cdots + 5841234 ) / 410745 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49527 \nu^{7} + 56486 \nu^{6} + 691081 \nu^{5} + 687494 \nu^{4} - 11140841 \nu^{3} + \cdots + 13632768 ) / 821490 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 164132 \nu^{7} - 129997 \nu^{6} - 2265662 \nu^{5} - 3155995 \nu^{4} + 35173474 \nu^{3} + \cdots - 51144534 ) / 2464470 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 218011 \nu^{7} - 243803 \nu^{6} - 3016903 \nu^{5} - 3093167 \nu^{4} + 48961283 \nu^{3} + \cdots - 59937804 ) / 2464470 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 68687 \nu^{7} + 53938 \nu^{6} + 975269 \nu^{5} + 1290754 \nu^{4} - 15051769 \nu^{3} + \cdots + 18545040 ) / 492894 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - \beta_{6} - \beta_{4} - 15\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{6} + 11\beta_{5} + 16\beta_{4} + 16\beta_{3} - 3\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28\beta_{7} + 23\beta_{5} + 33\beta_{3} - 189\beta_{2} + 28\beta _1 - 189 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 75\beta_{7} + 139\beta_{6} + 204\beta_{4} - 511\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 475\beta_{6} + 475\beta_{5} + 695\beta_{4} + 695\beta_{3} + 338\beta _1 - 2289 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1449\beta_{7} - 1583\beta_{5} - 2318\beta_{3} - 9827\beta_{2} + 1449\beta _1 - 9827 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
3.55140 − 1.47305i
1.10191 − 0.0588384i
−0.601911 + 0.924864i
−3.05140 + 2.33908i
3.55140 + 1.47305i
1.10191 + 0.0588384i
−0.601911 − 0.924864i
−3.05140 − 2.33908i
0 0 0 −6.60280 + 3.81213i 0 −4.89116 + 5.00764i 0 0 0
305.2 0 0 0 −1.70382 + 0.983702i 0 1.89116 − 6.73970i 0 0 0
305.3 0 0 0 1.70382 − 0.983702i 0 1.89116 − 6.73970i 0 0 0
305.4 0 0 0 6.60280 − 3.81213i 0 −4.89116 + 5.00764i 0 0 0
737.1 0 0 0 −6.60280 − 3.81213i 0 −4.89116 − 5.00764i 0 0 0
737.2 0 0 0 −1.70382 − 0.983702i 0 1.89116 + 6.73970i 0 0 0
737.3 0 0 0 1.70382 + 0.983702i 0 1.89116 + 6.73970i 0 0 0
737.4 0 0 0 6.60280 + 3.81213i 0 −4.89116 − 5.00764i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h 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.dc.d 8
3.b odd 2 1 inner 1008.3.dc.d 8
4.b odd 2 1 252.3.bk.b ✓ 8
7.c even 3 1 inner 1008.3.dc.d 8
12.b even 2 1 252.3.bk.b ✓ 8
21.h odd 6 1 inner 1008.3.dc.d 8
28.d even 2 1 1764.3.bk.f 8
28.f even 6 1 1764.3.c.g 4
28.f even 6 1 1764.3.bk.f 8
28.g odd 6 1 252.3.bk.b ✓ 8
28.g odd 6 1 1764.3.c.e 4
84.h odd 2 1 1764.3.bk.f 8
84.j odd 6 1 1764.3.c.g 4
84.j odd 6 1 1764.3.bk.f 8
84.n even 6 1 252.3.bk.b ✓ 8
84.n even 6 1 1764.3.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.bk.b ✓ 8 4.b odd 2 1
252.3.bk.b ✓ 8 12.b even 2 1
252.3.bk.b ✓ 8 28.g odd 6 1
252.3.bk.b ✓ 8 84.n even 6 1
1008.3.dc.d 8 1.a even 1 1 trivial
1008.3.dc.d 8 3.b odd 2 1 inner
1008.3.dc.d 8 7.c even 3 1 inner
1008.3.dc.d 8 21.h odd 6 1 inner
1764.3.c.e 4 28.g odd 6 1
1764.3.c.e 4 84.n even 6 1
1764.3.c.g 4 28.f even 6 1
1764.3.c.g 4 84.j odd 6 1
1764.3.bk.f 8 28.d even 2 1
1764.3.bk.f 8 28.f even 6 1
1764.3.bk.f 8 84.h odd 2 1
1764.3.bk.f 8 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 \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{8} - 62T_{5}^{6} + 3619T_{5}^{4} - 13950T_{5}^{2} + 50625 \) Copy content Toggle raw display
\( T_{11}^{8} - 242T_{11}^{6} + 52939T_{11}^{4} - 1361250T_{11}^{2} + 31640625 \) 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
$5$ \( T^{8} - 62 T^{6} + \cdots + 50625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 242 T^{6} + \cdots + 31640625 \) Copy content Toggle raw display
$13$ \( (T^{2} + 28 T + 150)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 10226063376 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 168100)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 4669488810000 \) Copy content Toggle raw display
$29$ \( (T^{4} + 62 T^{2} + 225)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + \cdots + 441)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 80 T^{3} + \cdots + 1406596)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2232 T^{2} + 291600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 20 T - 314)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1458 T^{2} + 2125764)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 558 T^{6} + \cdots + 332150625 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 587380254493041 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 5062500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 72 T^{3} + \cdots + 313600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8132 T^{2} + 11088900)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 144 T^{3} + \cdots + 341056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 70 T^{3} + \cdots + 1390041)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 21698 T^{2} + 9455625)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 138 T - 7015)^{4} \) Copy content Toggle raw display
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