LMFDB - Newform orbit 8028.2.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.a (trivial)

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:	yes
Analytic conductor:	$64.1039027427$
Analytic rank:	$1$
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} - x^{7} - 9x^{6} + 6x^{5} + 24x^{4} - 10x^{3} - 19x^{2} + 7x + 2 $ x^8 - x^7 - 9x^6 + 6x^5 + 24x^4 - 10x^3 - 19x^2 + 7x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 892)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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_{7} + \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{3}) q^{7}+O(q^{10}) $ q + (b7 + b6 + b3 - b2) * q^5 + (-b7 - b6 - b5 - b3) * q^7	$ q + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \cdots + 1) q^{97}+O(q^{100}) $ q + (b7 + b6 + b3 - b2) * q^5 + (-b7 - b6 - b5 - b3) * q^7 + (-b7 - b6 + b4 - b3 - b1 - 2) * q^11 + (b7 - b4 - b2 + 1) * q^13 + (b5 - b4 + 2b3 + 2b1) * q^17 + (-2b7 - b3 + b2 - 2b1 + 1) * q^19 + (-b6 - b5 + 2b4 - 2b3 + b2 - 4) * q^23 + (b7 + b6 + b5 - b4 + b3 + 2b1 + 2) q^25 + (b7 + b6 + 2b5 - b4 + 2b3 + 2b2 - 1) q^29 + (2b7 + b5 + 2) q^31 + (b6 - 2b3 + b2 - 3b1 - 3) * q^35 + (2b7 + 5b6 - b4 + 3b3 + b2 + b1 + 2) q^37 + (-3b7 + b6 - b5 + 2b4 + 2b2 + 2b1) * q^41 + (-2b6 + 3b4 + b2 + 1) * q^43 + (-3b6 - b5 + 3b4 - 3b3 + b2 - b1 - 5) q^47 + (-b7 - 3b6 - b5 - b4 + 2b3 - 2b2 + 3b1 - 1) * q^49 + (b7 + b5 - 2b4 - 2b3 + b2 - 1) * q^53 + (-b7 - b5 + 4b2) q^55 + (2b7 + 2b6 + 2b5 - 3b4 + 2b3 - 3b2 + 7b1 + 2) q^59 + (4b6 - b4 + b3 + b2 - 3b1) * q^61 + (b7 + b6 - b4 - 2b2 + 3b1 + 2) * q^65 + (-b7 - 4b6 + 2b5 - 2b2 + b1 + 1) q^67 + (-2b7 - b6 + 3b5 - 2b4 + b3 + b2 - 4) q^71 + (-3b7 - b6 + 2b4 - b3 + 5b2 - 3b1 - 3) * q^73 + (3b7 + 2b5 + b4 - b2 - 2b1 - 1) q^77 + (-b7 + b6 - b5 + 2b4 - 2b3 + 5b2 - 5b1 - 3) * q^79 + (-b7 + 4b6 + 2b5 - 4b4 - 2b1) * q^83 + (2b7 - 3b6 + 2b5 + 3b4 - b3 - 3) * q^85 + (2b7 + 2b6 + b5 + 3b3 + b2 - 4b1 + 2) * q^89 + (-b7 - 2b6 - 2b5 + 2b4 - 3b3 - 3b1 - 1) q^91 + (-b7 + 2b6 - 5b5 + 3b4 - 3b3 + 3b2 - 3b1 - 6) * q^95 + (2b7 + 2b6 + 3b5 - 5b4 + 4b3 - b2 + 2b1 + 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{5} + 5 q^{7}+O(q^{10}) $ 8 * q - 5 * q^5 + 5 * q^7	$ 8 q - 5 q^{5} + 5 q^{7} - 13 q^{11} + 3 q^{13} - q^{17} + 8 q^{19} - 21 q^{23} + 11 q^{25} - 11 q^{29} + 13 q^{31} - 29 q^{35} + 6 q^{37} + 12 q^{41} + 23 q^{43} - 23 q^{47} + q^{49} - 14 q^{53} + 15 q^{55} - 2 q^{59} - 13 q^{61} + 8 q^{65} + 9 q^{67} - 38 q^{71} - 6 q^{73} - 17 q^{77} - 12 q^{79} - 28 q^{83} - 16 q^{85} + 9 q^{89} + 2 q^{91} - 30 q^{95} - 14 q^{97}+O(q^{100}) $ 8 * q - 5 * q^5 + 5 * q^7 - 13 * q^11 + 3 * q^13 - q^17 + 8 * q^19 - 21 * q^23 + 11 * q^25 - 11 * q^29 + 13 * q^31 - 29 * q^35 + 6 * q^37 + 12 * q^41 + 23 * q^43 - 23 * q^47 + q^49 - 14 * q^53 + 15 * q^55 - 2 * q^59 - 13 * q^61 + 8 * q^65 + 9 * q^67 - 38 * q^71 - 6 * q^73 - 17 * q^77 - 12 * q^79 - 28 * q^83 - 16 * q^85 + 9 * q^89 + 2 * q^91 - 30 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - x^{7} - 9x^{6} + 6x^{5} + 24x^{4} - 10x^{3} - 19x^{2} + 7x + 2 $ x^8 - x^7 - 9*x^6 + 6*x^5 + 24*x^4 - 10*x^3 - 19*x^2 + 7*x + 2 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 2 $ v^2 - 2
$\beta_{3}$	$=$	$ \nu^{7} - \nu^{6} - 8\nu^{5} + 4\nu^{4} + 17\nu^{3} + \nu^{2} - 7\nu - 3 $ v^7 - v^6 - 8v^5 + 4v^4 + 17v^3 + v^2 - 7v - 3
$\beta_{4}$	$=$	$ \nu^{7} - 2\nu^{6} - 6\nu^{5} + 11\nu^{4} + 6\nu^{3} - 12\nu^{2} + 4\nu + 1 $ v^7 - 2v^6 - 6v^5 + 11v^4 + 6v^3 - 12v^2 + 4v + 1
$\beta_{5}$	$=$	$ -\nu^{6} + 2\nu^{5} + 7\nu^{4} - 12\nu^{3} - 12\nu^{2} + 14\nu + 2 $ -v^6 + 2v^5 + 7v^4 - 12v^3 - 12v^2 + 14*v + 2
$\beta_{6}$	$=$	$ \nu^{7} - 3\nu^{6} - 5\nu^{5} + 19\nu^{4} + \nu^{3} - 28\nu^{2} + 8\nu + 5 $ v^7 - 3v^6 - 5v^5 + 19v^4 + v^3 - 28v^2 + 8*v + 5
$\beta_{7}$	$=$	$ 2\nu^{6} - 3\nu^{5} - 14\nu^{4} + 16\nu^{3} + 23\nu^{2} - 16\nu - 4 $ 2v^6 - 3v^5 - 14v^4 + 16v^3 + 23v^2 - 16v - 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 2 $ b2 + 2
$\nu^{3}$	$=$	$ -\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 3\beta_1 $ -b5 + b4 - b3 + b2 + 3*b1
$\nu^{4}$	$=$	$ \beta_{7} + \beta_{6} - \beta_{3} + 6\beta_{2} + \beta _1 + 8 $ b7 + b6 - b3 + 6*b2 + b1 + 8
$\nu^{5}$	$=$	$ \beta_{7} - 6\beta_{5} + 8\beta_{4} - 8\beta_{3} + 9\beta_{2} + 12\beta _1 + 2 $ b7 - 6b5 + 8b4 - 8b3 + 9b2 + 12*b1 + 2
$\nu^{6}$	$=$	$ 9\beta_{7} + 7\beta_{6} - \beta_{5} + 4\beta_{4} - 11\beta_{3} + 36\beta_{2} + 9\beta _1 + 38 $ 9b7 + 7b6 - b5 + 4b4 - 11b3 + 36b2 + 9b1 + 38
$\nu^{7}$	$=$	$ 13\beta_{7} + 3\beta_{6} - 32\beta_{5} + 51\beta_{4} - 53\beta_{3} + 66\beta_{2} + 57\beta _1 + 23 $ 13b7 + 3b6 - 32b5 + 51b4 - 53b3 + 66b2 + 57*b1 + 23

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

2.53950
1.84978
−2.07466
0.662444
−1.29396
−1.37353
−0.196521
0.886950

−3.24778

1.01176

1.2

−2.71287

5.19804

1.3

−2.65914

−1.17810

1.4

−2.61196

0.134673

1.5

−0.888594

3.74843

1.6

1.44274

−0.978552

1.7

2.51463

0.560605

1.8

3.16298

−3.49686

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$223$	$1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8028.2.a.k		8
3.b	odd	2	1		892.2.a.e	✓	8
12.b	even	2	1		3568.2.a.p		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
892.2.a.e	✓	8	3.b	odd	2	1
3568.2.a.p		8	12.b	even	2	1
8028.2.a.k		8	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8028))$:

$ T_{5}^{8} + 5T_{5}^{7} - 13T_{5}^{6} - 93T_{5}^{5} - T_{5}^{4} + 502T_{5}^{3} + 372T_{5}^{2} - 720T_{5} - 624 $

$ T_{7}^{8} - 5T_{7}^{7} - 16T_{7}^{6} + 72T_{7}^{5} + 54T_{7}^{4} - 119T_{7}^{3} - 33T_{7}^{2} + 51T_{7} - 6 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + 5 T^{7} + \cdots - 624 $ T^8 + 5T^7 - 13T^6 - 93T^5 - T^4 + 502T^3 + 372T^2 - 720T - 624
$7$	$ T^{8} - 5 T^{7} + \cdots - 6 $ T^8 - 5T^7 - 16T^6 + 72T^5 + 54T^4 - 119T^3 - 33T^2 + 51*T - 6
$11$	$ T^{8} + 13 T^{7} + \cdots - 536 $ T^8 + 13T^7 + 46T^6 - 49T^5 - 546T^4 - 631T^3 + 862T^2 + 1196*T - 536
$13$	$ T^{8} - 3 T^{7} + \cdots + 208 $ T^8 - 3T^7 - 19T^6 + 61T^5 + 75T^4 - 246T^3 - 164T^2 + 288*T + 208
$17$	$ T^{8} + T^{7} + \cdots - 1583 $ T^8 + T^7 - 57T^6 - 130T^5 + 667T^4 + 1993T^3 - 65T^2 - 2925T - 1583
$19$	$ T^{8} - 8 T^{7} + \cdots - 1448 $ T^8 - 8T^7 - 51T^6 + 523T^5 - 60T^4 - 6575T^3 + 10770T^2 - 2791*T - 1448
$23$	$ T^{8} + 21 T^{7} + \cdots + 17624 $ T^8 + 21T^7 + 114T^6 - 458T^5 - 7363T^4 - 30439T^3 - 51864T^2 - 22900*T + 17624
$29$	$ T^{8} + 11 T^{7} + \cdots - 26139 $ T^8 + 11T^7 - 65T^6 - 1136T^5 - 3065T^4 + 5809T^3 + 26919T^2 + 8121*T - 26139
$31$	$ T^{8} - 13 T^{7} + \cdots + 2456 $ T^8 - 13T^7 + 2T^6 + 578T^5 - 2284T^4 + 313T^3 + 8755T^2 - 9099*T + 2456
$37$	$ T^{8} - 6 T^{7} + \cdots + 29397 $ T^8 - 6T^7 - 151T^6 + 1210T^5 + 3252T^4 - 49361T^3 + 132384T^2 - 117264*T + 29397
$41$	$ T^{8} - 12 T^{7} + \cdots + 2655469 $ T^8 - 12T^7 - 216T^6 + 2607T^5 + 13183T^4 - 161405T^3 - 219004T^2 + 2463601*T + 2655469
$43$	$ T^{8} - 23 T^{7} + \cdots + 2748 $ T^8 - 23T^7 + 117T^6 + 820T^5 - 8305T^4 + 8816T^3 + 63864T^2 - 83919*T + 2748
$47$	$ T^{8} + 23 T^{7} + \cdots + 712344 $ T^8 + 23T^7 + 89T^6 - 1406T^5 - 11601T^4 + 698T^3 + 238188T^2 + 753357*T + 712344
$53$	$ T^{8} + 14 T^{7} + \cdots - 22767 $ T^8 + 14T^7 - 103T^6 - 1648T^5 + 1032T^4 + 46321T^3 + 62466T^2 - 93594*T - 22767
$59$	$ T^{8} + 2 T^{7} + \cdots - 13465784 $ T^8 + 2T^7 - 363T^6 + 430T^5 + 41195T^4 - 172099T^3 - 1197992T^2 + 8621348*T - 13465784
$61$	$ T^{8} + 13 T^{7} + \cdots - 2868496 $ T^8 + 13T^7 - 112T^6 - 1846T^5 + 2303T^4 + 84704T^3 + 95784T^2 - 1260192*T - 2868496
$67$	$ T^{8} - 9 T^{7} + \cdots - 669384 $ T^8 - 9T^7 - 204T^6 + 1797T^5 + 9920T^4 - 82043T^3 - 98460T^2 + 860436*T - 669384
$71$	$ T^{8} + 38 T^{7} + \cdots + 7248896 $ T^8 + 38T^7 + 366T^6 - 2339T^5 - 54245T^4 - 170356T^3 + 1112160T^2 + 6673472*T + 7248896
$73$	$ T^{8} + 6 T^{7} + \cdots + 58077 $ T^8 + 6T^7 - 237T^6 - 520T^5 + 17328T^4 - 21609T^3 - 177984T^2 + 103680*T + 58077
$79$	$ T^{8} + 12 T^{7} + \cdots - 5329152 $ T^8 + 12T^7 - 268T^6 - 2633T^5 + 21987T^4 + 194248T^3 - 476784T^2 - 4916160*T - 5329152
$83$	$ T^{8} + 28 T^{7} + \cdots - 99670376 $ T^8 + 28T^7 - 160T^6 - 8995T^5 - 13379T^4 + 909312T^3 + 2772141T^2 - 29054501*T - 99670376
$89$	$ T^{8} - 9 T^{7} + \cdots + 478339 $ T^8 - 9T^7 - 218T^6 + 1225T^5 + 15520T^4 - 32593T^3 - 259611T^2 + 342922*T + 478339
$97$	$ T^{8} + 14 T^{7} + \cdots - 1579184 $ T^8 + 14T^7 - 190T^6 - 3501T^5 - 4363T^4 + 105474T^3 + 219476T^2 - 858144*T - 1579184
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{3}) q^{7}+O(q^{10}) \) q + (b7 + b6 + b3 - b2) * q^5 + (-b7 - b6 - b5 - b3) * q^7	\( q + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \cdots + 1) q^{97}+O(q^{100}) \) q + (b7 + b6 + b3 - b2) * q^5 + (-b7 - b6 - b5 - b3) * q^7 + (-b7 - b6 + b4 - b3 - b1 - 2) * q^11 + (b7 - b4 - b2 + 1) * q^13 + (b5 - b4 + 2b3 + 2b1) * q^17 + (-2b7 - b3 + b2 - 2b1 + 1) * q^19 + (-b6 - b5 + 2b4 - 2b3 + b2 - 4) * q^23 + (b7 + b6 + b5 - b4 + b3 + 2b1 + 2) q^25 + (b7 + b6 + 2b5 - b4 + 2b3 + 2b2 - 1) q^29 + (2b7 + b5 + 2) q^31 + (b6 - 2b3 + b2 - 3b1 - 3) * q^35 + (2b7 + 5b6 - b4 + 3b3 + b2 + b1 + 2) q^37 + (-3b7 + b6 - b5 + 2b4 + 2b2 + 2b1) * q^41 + (-2b6 + 3b4 + b2 + 1) * q^43 + (-3b6 - b5 + 3b4 - 3b3 + b2 - b1 - 5) q^47 + (-b7 - 3b6 - b5 - b4 + 2b3 - 2b2 + 3b1 - 1) * q^49 + (b7 + b5 - 2b4 - 2b3 + b2 - 1) * q^53 + (-b7 - b5 + 4b2) q^55 + (2b7 + 2b6 + 2b5 - 3b4 + 2b3 - 3b2 + 7b1 + 2) q^59 + (4b6 - b4 + b3 + b2 - 3b1) * q^61 + (b7 + b6 - b4 - 2b2 + 3b1 + 2) * q^65 + (-b7 - 4b6 + 2b5 - 2b2 + b1 + 1) q^67 + (-2b7 - b6 + 3b5 - 2b4 + b3 + b2 - 4) q^71 + (-3b7 - b6 + 2b4 - b3 + 5b2 - 3b1 - 3) * q^73 + (3b7 + 2b5 + b4 - b2 - 2b1 - 1) q^77 + (-b7 + b6 - b5 + 2b4 - 2b3 + 5b2 - 5b1 - 3) * q^79 + (-b7 + 4b6 + 2b5 - 4b4 - 2b1) * q^83 + (2b7 - 3b6 + 2b5 + 3b4 - b3 - 3) * q^85 + (2b7 + 2b6 + b5 + 3b3 + b2 - 4b1 + 2) * q^89 + (-b7 - 2b6 - 2b5 + 2b4 - 3b3 - 3b1 - 1) q^91 + (-b7 + 2b6 - 5b5 + 3b4 - 3b3 + 3b2 - 3b1 - 6) * q^95 + (2b7 + 2b6 + 3b5 - 5b4 + 4b3 - b2 + 2b1 + 1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{5} + 5 q^{7}+O(q^{10}) \) 8 * q - 5 * q^5 + 5 * q^7	\( 8 q - 5 q^{5} + 5 q^{7} - 13 q^{11} + 3 q^{13} - q^{17} + 8 q^{19} - 21 q^{23} + 11 q^{25} - 11 q^{29} + 13 q^{31} - 29 q^{35} + 6 q^{37} + 12 q^{41} + 23 q^{43} - 23 q^{47} + q^{49} - 14 q^{53} + 15 q^{55} - 2 q^{59} - 13 q^{61} + 8 q^{65} + 9 q^{67} - 38 q^{71} - 6 q^{73} - 17 q^{77} - 12 q^{79} - 28 q^{83} - 16 q^{85} + 9 q^{89} + 2 q^{91} - 30 q^{95} - 14 q^{97}+O(q^{100}) \) 8 * q - 5 * q^5 + 5 * q^7 - 13 * q^11 + 3 * q^13 - q^17 + 8 * q^19 - 21 * q^23 + 11 * q^25 - 11 * q^29 + 13 * q^31 - 29 * q^35 + 6 * q^37 + 12 * q^41 + 23 * q^43 - 23 * q^47 + q^49 - 14 * q^53 + 15 * q^55 - 2 * q^59 - 13 * q^61 + 8 * q^65 + 9 * q^67 - 38 * q^71 - 6 * q^73 - 17 * q^77 - 12 * q^79 - 28 * q^83 - 16 * q^85 + 9 * q^89 + 2 * q^91 - 30 * q^95 - 14 * q^97

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