↑

Properties

Label 1008.2.bt.d
Level $1008$
Weight $2$
Character orbit 1008.bt
Analytic conductor $8.049$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(17,1008)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 504)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{4}) q^{5} + (\beta_{9} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{4}) q^{5} + (\beta_{9} + \beta_{3}) q^{7} + ( - \beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{11}+ \cdots + (4 \beta_{9} + 4 \beta_{6} + 6 \beta_{5} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} - 12 q^{19} + 12 q^{25} - 24 q^{31} + 4 q^{37} - 8 q^{43} + 32 q^{49} + 28 q^{67} - 60 q^{73} + 32 q^{79} - 32 q^{85} + 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 34158 \nu^{14} + 852503 \nu^{12} + 4914956 \nu^{10} + 18664744 \nu^{8} + 59406470 \nu^{6} + \cdots + 139803027 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 66287 \nu^{15} - 2125573 \nu^{13} - 20719966 \nu^{11} - 144889584 \nu^{9} + \cdots - 6339564777 \nu ) / 2750894300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 382234 \nu^{14} - 3667549 \nu^{12} - 21832848 \nu^{10} - 94498292 \nu^{8} - 297554910 \nu^{6} + \cdots - 1050178241 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 88213 \nu^{15} - 1057753 \nu^{13} - 8202336 \nu^{11} - 41107094 \nu^{9} + \cdots - 905297057 \nu ) / 785969800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 623003 \nu^{14} + 5366258 \nu^{12} + 30938216 \nu^{10} + 125675214 \nu^{8} + 373946420 \nu^{6} + \cdots + 1273004222 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 83051 \nu^{14} + 542488 \nu^{12} + 2797264 \nu^{10} + 9444519 \nu^{8} + 21581337 \nu^{6} + \cdots + 3256785 ) / 39298490 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17628 \nu^{14} + 144183 \nu^{12} + 817216 \nu^{10} + 3201064 \nu^{8} + 9190570 \nu^{6} + \cdots + 14486947 ) / 8020100 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3705 \nu^{14} + 31546 \nu^{12} + 171760 \nu^{10} + 672790 \nu^{8} + 1902192 \nu^{6} + \cdots + 7562162 ) / 1604020 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1744691 \nu^{14} + 15899931 \nu^{12} + 90773542 \nu^{10} + 373226028 \nu^{8} + 1092704010 \nu^{6} + \cdots + 3124455159 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} + \cdots - 238292243 \nu ) / 239208200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13169 \nu^{15} - 130759 \nu^{13} - 753868 \nu^{11} - 3141922 \nu^{9} - 9111910 \nu^{7} + \cdots - 21443331 \nu ) / 17355800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7738601 \nu^{15} + 58163261 \nu^{13} + 313306372 \nu^{11} + 1167320938 \nu^{9} + \cdots + 9719272549 \nu ) / 5501788600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9919057 \nu^{15} - 78396587 \nu^{13} - 451981724 \nu^{11} - 1839736626 \nu^{9} + \cdots - 7354565183 \nu ) / 5501788600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 438892 \nu^{15} - 3881427 \nu^{13} - 22712419 \nu^{11} - 93951256 \nu^{9} + \cdots - 752948308 \nu ) / 125040650 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 965736 \nu^{15} + 8386545 \nu^{13} + 48089618 \nu^{11} + 197972096 \nu^{9} + \cdots + 1747120823 \nu ) / 275089430 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{13} + 3\beta_{10} - 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 2\beta_{5} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} + 5\beta_{14} - \beta_{13} + 2\beta_{12} - 4\beta_{11} - 3\beta_{10} - 8\beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} + 2\beta_{6} - 3\beta_{5} - 5\beta_{3} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{15} + 3\beta_{14} - 3\beta_{13} + 28\beta_{12} - 41\beta_{10} + 34\beta_{4} - 28\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{9} + 19\beta_{8} + 24\beta_{7} - 14\beta_{6} - 7\beta_{5} - 7\beta_{3} - 7\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 98 \beta_{15} - 83 \beta_{14} + 91 \beta_{13} - 158 \beta_{12} + 256 \beta_{11} + 69 \beta_{10} + \cdots + 90 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 62\beta_{9} - 31\beta_{8} - 37\beta_{7} - 31\beta_{5} + 68\beta_{3} - 81\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 112 \beta_{15} + 235 \beta_{14} - 535 \beta_{13} - 112 \beta_{12} - 1092 \beta_{11} + \cdots - 112 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -404\beta_{8} - 99\beta_{7} + 198\beta_{6} + 454\beta_{5} + 32\beta_{3} + 503\beta _1 + 503 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1526 \beta_{15} - 759 \beta_{14} + 759 \beta_{13} + 1942 \beta_{12} - 1671 \beta_{10} + \cdots + 614 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -332\beta_{9} + 1742\beta_{8} - 788\beta_{7} - 332\beta_{6} - 166\beta_{5} - 166\beta_{3} - 166\beta _1 - 3969 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7424 \beta_{15} + 7163 \beta_{14} + 3621 \beta_{13} + 1616 \beta_{12} + 5808 \beta_{11} + \cdots - 3360 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -1244\beta_{9} + 622\beta_{8} + 6831\beta_{7} - 10592\beta_{5} - 7453\beta_{3} - 4438\beta _1 + 10592 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 15030 \beta_{15} - 29095 \beta_{14} - 13493 \beta_{13} - 15030 \beta_{12} + 22012 \beta_{11} + \cdots - 15030 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.45333 + 1.51725i
1.01089 + 0.750919i
0.587308 + 2.01725i
0.144868 + 1.25092i
−0.144868 − 1.25092i
−0.587308 − 2.01725i
−1.01089 − 0.750919i
−1.45333 − 1.51725i
1.45333 − 1.51725i
1.01089 − 0.750919i
0.587308 − 2.01725i
0.144868 − 1.25092i
−0.144868 + 1.25092i
−0.587308 + 2.01725i
−1.01089 + 0.750919i
−1.45333 + 1.51725i
0 0 0 −1.45333 − 2.51725i 0 2.64571 + 0.0146827i 0 0 0
17.2 0 0 0 −1.01089 − 1.75092i 0 −0.561961 − 2.58538i 0 0 0
17.3 0 0 0 −0.587308 − 1.01725i 0 −2.35282 + 1.21006i 0 0 0
17.4 0 0 0 −0.144868 − 0.250919i 0 2.26907 + 1.36064i 0 0 0
17.5 0 0 0 0.144868 + 0.250919i 0 2.26907 + 1.36064i 0 0 0
17.6 0 0 0 0.587308 + 1.01725i 0 −2.35282 + 1.21006i 0 0 0
17.7 0 0 0 1.01089 + 1.75092i 0 −0.561961 − 2.58538i 0 0 0
17.8 0 0 0 1.45333 + 2.51725i 0 2.64571 + 0.0146827i 0 0 0
593.1 0 0 0 −1.45333 + 2.51725i 0 2.64571 − 0.0146827i 0 0 0
593.2 0 0 0 −1.01089 + 1.75092i 0 −0.561961 + 2.58538i 0 0 0
593.3 0 0 0 −0.587308 + 1.01725i 0 −2.35282 − 1.21006i 0 0 0
593.4 0 0 0 −0.144868 + 0.250919i 0 2.26907 − 1.36064i 0 0 0
593.5 0 0 0 0.144868 − 0.250919i 0 2.26907 − 1.36064i 0 0 0
593.6 0 0 0 0.587308 − 1.01725i 0 −2.35282 − 1.21006i 0 0 0
593.7 0 0 0 1.01089 − 1.75092i 0 −0.561961 + 2.58538i 0 0 0
593.8 0 0 0 1.45333 − 2.51725i 0 2.64571 − 0.0146827i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bt.d 16
3.b odd 2 1 inner 1008.2.bt.d 16
4.b odd 2 1 504.2.bl.a ✓ 16
7.c even 3 1 7056.2.k.h 16
7.d odd 6 1 inner 1008.2.bt.d 16
7.d odd 6 1 7056.2.k.h 16
12.b even 2 1 504.2.bl.a ✓ 16
21.g even 6 1 inner 1008.2.bt.d 16
21.g even 6 1 7056.2.k.h 16
21.h odd 6 1 7056.2.k.h 16
28.d even 2 1 3528.2.bl.a 16
28.f even 6 1 504.2.bl.a ✓ 16
28.f even 6 1 3528.2.k.b 16
28.g odd 6 1 3528.2.k.b 16
28.g odd 6 1 3528.2.bl.a 16
84.h odd 2 1 3528.2.bl.a 16
84.j odd 6 1 504.2.bl.a ✓ 16
84.j odd 6 1 3528.2.k.b 16
84.n even 6 1 3528.2.k.b 16
84.n even 6 1 3528.2.bl.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bl.a ✓ 16 4.b odd 2 1
504.2.bl.a ✓ 16 12.b even 2 1
504.2.bl.a ✓ 16 28.f even 6 1
504.2.bl.a ✓ 16 84.j odd 6 1
1008.2.bt.d 16 1.a even 1 1 trivial
1008.2.bt.d 16 3.b odd 2 1 inner
1008.2.bt.d 16 7.d odd 6 1 inner
1008.2.bt.d 16 21.g even 6 1 inner
3528.2.k.b 16 28.f even 6 1
3528.2.k.b 16 28.g odd 6 1
3528.2.k.b 16 84.j odd 6 1
3528.2.k.b 16 84.n even 6 1
3528.2.bl.a 16 28.d even 2 1
3528.2.bl.a 16 28.g odd 6 1
3528.2.bl.a 16 84.h odd 2 1
3528.2.bl.a 16 84.n even 6 1
7056.2.k.h 16 7.c even 3 1
7056.2.k.h 16 7.d odd 6 1
7056.2.k.h 16 21.g even 6 1
7056.2.k.h 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} + 14T_{5}^{14} + 143T_{5}^{12} + 638T_{5}^{10} + 2077T_{5}^{8} + 2644T_{5}^{6} + 2492T_{5}^{4} + 208T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{16} - 54 T_{11}^{14} + 2215 T_{11}^{12} - 33222 T_{11}^{10} + 364221 T_{11}^{8} - 1394988 T_{11}^{6} + \cdots + 4477456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 14 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 54 T^{14} + \cdots + 4477456 \) Copy content Toggle raw display
$13$ \( (T^{8} + 82 T^{6} + \cdots + 24964)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$19$ \( (T^{8} + 6 T^{7} + \cdots + 111556)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2517630976 \) Copy content Toggle raw display
$29$ \( (T^{8} + 150 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 529)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + \cdots + 586756)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 124 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} - 33 T^{2} + \cdots - 98)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 100000000 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 9082363580416 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 64524128256 \) Copy content Toggle raw display
$61$ \( (T^{8} - 174 T^{6} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 14 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 176 T^{6} + \cdots + 26896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 30 T^{7} + \cdots + 446224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 16 T^{7} + \cdots + 11242609)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 226 T^{6} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 704536369954816 \) Copy content Toggle raw display
$97$ \( (T^{8} + 542 T^{6} + \cdots + 65480464)^{2} \) Copy content Toggle raw display
show more
show less