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Properties

Label 1008.4.bt.a
Level $1008$
Weight $4$
Character orbit 1008.bt
Analytic conductor $59.474$
Analytic rank $0$
Dimension $16$
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(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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(59.4739252858\)
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: \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{5} + (\beta_{4} - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{5} + (\beta_{4} - 3) q^{7} + (\beta_{14} - 3 \beta_{6} - \beta_1) q^{11} + ( - \beta_{10} + \beta_{9} + 4 \beta_{5} + \cdots + 9) q^{13}+ \cdots + ( - 7 \beta_{10} - 36 \beta_{9} + \cdots + 365) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 56 q^{7} + 612 q^{19} - 20 q^{25} - 1128 q^{31} - 1196 q^{37} - 328 q^{43} + 784 q^{49} - 1632 q^{61} - 308 q^{67} + 4068 q^{73} + 2176 q^{79} - 4608 q^{85} - 924 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2582250337 \nu^{14} + 122400945426 \nu^{12} - 4183683881139 \nu^{10} + \cdots + 43\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50455191 \nu^{14} + 2307248098 \nu^{12} - 75951189685 \nu^{10} + 1149235164582 \nu^{8} + \cdots + 266811223623624 ) / 36533932100064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3065808613 \nu^{14} - 119299855274 \nu^{12} + 3755758223311 \nu^{10} + \cdots + 37\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14645418817 \nu^{14} + 682142342066 \nu^{12} - 23315734296099 \nu^{10} + \cdots + 38\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 937699527 \nu^{15} + 42257710318 \nu^{13} - 1411537509445 \nu^{11} + \cdots - 41\!\cdots\!12 \nu ) / 730678642001280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 128271858533 \nu^{14} - 6298544893834 \nu^{12} + 217538780633951 \nu^{10} + \cdots - 31\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + \cdots + 699777739755000 \nu ) / 913348302501600 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 101634687971 \nu^{14} - 4897105890358 \nu^{12} + 168224430945137 \nu^{10} + \cdots - 14\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 113844178871 \nu^{14} + 5451040376758 \nu^{12} - 186603617326637 \nu^{10} + \cdots + 54\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 253797 \nu^{15} - 12085406 \nu^{13} + 413081109 \nu^{11} - 6837155440 \nu^{9} + \cdots - 69093405000 \nu ) / 49371246000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 360099673999 \nu^{15} - 17075216598302 \nu^{13} + 583205171768653 \nu^{11} + \cdots - 70\!\cdots\!00 \nu ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 383305735249 \nu^{15} + 18114452956802 \nu^{13} - 618137711412403 \nu^{11} + \cdots - 36\!\cdots\!00 \nu ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1280411313 \nu^{15} + 57618445232 \nu^{13} - 1927428289955 \nu^{11} + \cdots - 74\!\cdots\!48 \nu ) / 91334830250160 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64647369298 \nu^{15} + 3114318637179 \nu^{13} - 106972651093931 \nu^{11} + \cdots + 11\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{4} - 25\beta_{2} + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{13} - \beta_{12} + \beta_{11} - 5\beta_{8} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{10} - 12\beta_{9} - 12\beta_{7} + 12\beta_{5} - 13\beta_{4} + 12\beta_{3} - 259\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{15} + 27\beta_{13} - 54\beta_{12} + 13\beta_{11} - 110\beta_{8} - 41\beta_{6} - 110\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -283\beta_{10} - 314\beta_{9} + 31\beta_{7} - 31\beta_{4} + 259\beta_{3} - 5751 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86\beta_{14} - 659\beta_{13} - 659\beta_{12} - 722\beta_{6} - 2457\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 831\beta_{10} - 831\beta_{9} + 7510\beta_{7} - 5527\beta_{5} + 6679\beta_{4} + 127972\beta_{2} - 121293 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2814 \beta_{15} + 2814 \beta_{14} - 31702 \beta_{13} + 15851 \beta_{12} + 9607 \beta_{11} + \cdots + 15851 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 179186 \beta_{10} + 157707 \beta_{9} + 157707 \beta_{7} - 118179 \beta_{5} + 179186 \beta_{4} + \cdots + 179186 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 82486 \beta_{15} - 379851 \beta_{13} + 759702 \beta_{12} + 395191 \beta_{11} + 1247719 \beta_{8} + \cdots + 1247719 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3725147\beta_{10} + 4269970\beta_{9} - 544823\beta_{7} + 544823\beta_{4} - 2537675\beta_{3} + 65697303 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -2277118\beta_{14} + 9084763\beta_{13} + 9084763\beta_{12} + 21877186\beta_{6} + 28315551\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 13638999 \beta_{10} + 13638999 \beta_{9} - 101655338 \beta_{7} + 54753563 \beta_{5} + \cdots + 1392243261 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 60540774 \beta_{15} - 60540774 \beta_{14} + 433899350 \beta_{13} - 216949675 \beta_{12} + \cdots - 216949675 \beta_{6} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.57646 − 0.910170i
4.21355 + 2.43270i
−0.648633 − 0.374489i
−3.91663 − 2.26127i
3.91663 + 2.26127i
0.648633 + 0.374489i
−4.21355 − 2.43270i
1.57646 + 0.910170i
−1.57646 + 0.910170i
4.21355 − 2.43270i
−0.648633 + 0.374489i
−3.91663 + 2.26127i
3.91663 − 2.26127i
0.648633 − 0.374489i
−4.21355 + 2.43270i
1.57646 − 0.910170i
0 0 0 −7.54372 − 13.0661i 0 −16.2919 + 8.80760i 0 0 0
17.2 0 0 0 −6.38217 − 11.0542i 0 −2.53897 − 18.3454i 0 0 0
17.3 0 0 0 −5.42768 − 9.40102i 0 18.2341 − 3.24321i 0 0 0
17.4 0 0 0 −0.632851 − 1.09613i 0 −13.4032 + 12.7810i 0 0 0
17.5 0 0 0 0.632851 + 1.09613i 0 −13.4032 + 12.7810i 0 0 0
17.6 0 0 0 5.42768 + 9.40102i 0 18.2341 − 3.24321i 0 0 0
17.7 0 0 0 6.38217 + 11.0542i 0 −2.53897 − 18.3454i 0 0 0
17.8 0 0 0 7.54372 + 13.0661i 0 −16.2919 + 8.80760i 0 0 0
593.1 0 0 0 −7.54372 + 13.0661i 0 −16.2919 − 8.80760i 0 0 0
593.2 0 0 0 −6.38217 + 11.0542i 0 −2.53897 + 18.3454i 0 0 0
593.3 0 0 0 −5.42768 + 9.40102i 0 18.2341 + 3.24321i 0 0 0
593.4 0 0 0 −0.632851 + 1.09613i 0 −13.4032 − 12.7810i 0 0 0
593.5 0 0 0 0.632851 − 1.09613i 0 −13.4032 − 12.7810i 0 0 0
593.6 0 0 0 5.42768 − 9.40102i 0 18.2341 + 3.24321i 0 0 0
593.7 0 0 0 6.38217 − 11.0542i 0 −2.53897 + 18.3454i 0 0 0
593.8 0 0 0 7.54372 − 13.0661i 0 −16.2919 − 8.80760i 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 1008.4.bt.a 16
3.b odd 2 1 inner 1008.4.bt.a 16
4.b odd 2 1 63.4.p.a ✓ 16
7.d odd 6 1 inner 1008.4.bt.a 16
12.b even 2 1 63.4.p.a ✓ 16
21.g even 6 1 inner 1008.4.bt.a 16
28.d even 2 1 441.4.p.c 16
28.f even 6 1 63.4.p.a ✓ 16
28.f even 6 1 441.4.c.a 16
28.g odd 6 1 441.4.c.a 16
28.g odd 6 1 441.4.p.c 16
84.h odd 2 1 441.4.p.c 16
84.j odd 6 1 63.4.p.a ✓ 16
84.j odd 6 1 441.4.c.a 16
84.n even 6 1 441.4.c.a 16
84.n even 6 1 441.4.p.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.p.a ✓ 16 4.b odd 2 1
63.4.p.a ✓ 16 12.b even 2 1
63.4.p.a ✓ 16 28.f even 6 1
63.4.p.a ✓ 16 84.j odd 6 1
441.4.c.a 16 28.f even 6 1
441.4.c.a 16 28.g odd 6 1
441.4.c.a 16 84.j odd 6 1
441.4.c.a 16 84.n even 6 1
441.4.p.c 16 28.d even 2 1
441.4.p.c 16 28.g odd 6 1
441.4.p.c 16 84.h odd 2 1
441.4.p.c 16 84.n even 6 1
1008.4.bt.a 16 1.a even 1 1 trivial
1008.4.bt.a 16 3.b odd 2 1 inner
1008.4.bt.a 16 7.d odd 6 1 inner
1008.4.bt.a 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 510 T_{5}^{14} + 176175 T_{5}^{12} + 33794766 T_{5}^{10} + 4739623389 T_{5}^{8} + \cdots + 49018425731856 \) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\). 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
$5$ \( T^{16} + \cdots + 49018425731856 \) Copy content Toggle raw display
$7$ \( (T^{8} + 28 T^{7} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 2752420357764)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} - 306 T^{7} + \cdots + 567106596)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 20\!\cdots\!41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 32\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 82 T^{3} + \cdots + 144774182)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 84\!\cdots\!49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 75\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
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