LMFDB - Newform orbit 784.2.m.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(197,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("784.197");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.m (of order $4$, degree $2$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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^{2} + (\beta_{4} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{4} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2} + \cdots + 2) q^{6}+ \cdots + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{9}+O(q^{10}) $ q - b3 * q^2 + (b4 + b1) * q^3 + (b7 - b6 + b4 + 1) * q^4 + (b4 - b2 - b1 + 1) * q^5 + (b5 + b3 + b2 + b1 + 2) * q^6 + (b7 + b6 + b5 - b2 + b1 - 1) * q^8 + (-b7 - b5 + b2) * q^9	$ q - \beta_{3} q^{2} + (\beta_{4} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{4} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_{2} + \cdots + 2) q^{6}+ \cdots + ( - 2 \beta_{7} + 4 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) $ q - b3 * q^2 + (b4 + b1) * q^3 + (b7 - b6 + b4 + 1) * q^4 + (b4 - b2 - b1 + 1) * q^5 + (b5 + b3 + b2 + b1 + 2) * q^6 + (b7 + b6 + b5 - b2 + b1 - 1) * q^8 + (-b7 - b5 + b2) * q^9 + (b5 - b4 + b2 + b1 - 2) * q^10 + 2b5 q^11 + (-b7 + b6 + b5 - 2b3 - b2 - b1 + 1) q^12 + (2b6 - b4 - 2b3 - b2 - b1 - 1) * q^13 + (-b7 + b6 + b5 - b4 - b3 + b1 - 2) * q^15 + (b7 + b6 + b5 - b4 + b3 + b2 - b1 + 1) * q^16 + (b6 - b4 - b3 + b1 - 4) * q^17 + (-b7 - b6 - b5 + b4 + b2 + b1 + 1) * q^18 + (2b7 - b6 + 2b4 + b3 + 2b2 + 2b1 + 2) * q^19 + (b4 + b3 - 2b2 - 2b1 + 2) * q^20 + (2b5 - 2b2 - 2b1) q^22 + (-b7 + b6 - b5 + b4 + b3 + b1) * q^23 + (b7 - 3b6 + b5 + b4 - b3 + b2 - b1 + 1) q^24 + (-b7 + 2b6 - b5 + 2b4 - 2b3 - b2 - 2b1) * q^25 + (2b7 - 2b6 - b5 + b4 + 3b2 - b1) q^26 + (-b6 - 2b5 - b4 - b3 + 2b2 + b1 - 2) * q^27 + (2b7 - 2b6 + 2b4 + 2b3 - b2 + 2b1 - 1) q^29 + (-2b6 + b4 + b3 - 2b1 + 4) * q^30 + (-3b6 + 3b4 + b3 - b1) * q^31 + (2b6 - 2b3 - 2b1 - 4) q^32 + (-2b7 - 2b6 + 2b5 + 2b4 + 4) * q^33 + (b7 - b6 - b5 + b4 + 3b3 + b2 - b1 + 3) q^34 + (-b7 - b6 + b4 + 2b1 + 3) q^36 + (-2b6 - 4b5 + 2b4 - 2b3 - 3b2 - 2b1 + 3) * q^37 + (b7 + 3b6 + 2b5 + b4 + 2b1 + 1) q^38 + (3b7 - b6 + 3b5 - b4 + 3b3 + 3b1) * q^39 + (-b7 + b6 + b5 - 3b4 - b3 + b2 + b1 - 5) q^40 + (-4b7 - b6 - 4b5 - b4 - 3b3 - 3b1) * q^41 + (-2b6 - 4b5 - 2b3 + 4b2 - 4) * q^43 + (2b5 - 2b4 - 2b2 - 2b1 - 4) * q^44 + (b4 + b2 + b1 + 1) * q^45 + (-2b7 - b4 + b3 + 4b2 + 2b1 + 2) q^46 + (3b6 - 3b4 + b3 - b1 - 4) * q^47 + (2b7 + 2b5 + 2b4 - 6b2 - 2) * q^48 + (b7 - 3b6 + b5 + b4 + 2b3 + 7b2 + 3b1 - 1) * q^50 + (2b7 - 2b6 - 2b4 + 2b3 + 2b2 - 2b1 + 2) * q^51 + (2b7 + 2b6 + 3b4 + b3 - 2b2 + 2b1 - 4) q^52 + (-2b6 - 2b3 + b2 - 1) * q^53 + (b7 - b6 - 3b5 + 3b4 + b3 - b2 + b1 + 3) * q^54 + (-2b6 - 2b4) * q^55 + (-b7 + 2b6 - b5 + 2b4 - 2b3 + 2b2 - 2b1) q^57 + (4b6 + 2b5 - 3b4 + 3b3 - 2b2 + 2b1) * q^58 + (2b5 - 3b4 - 2b2 + 3b1 + 2) * q^59 + (-b7 + b6 + b5 - b4 - 3b3 - 3b2 + b1 - 5) * q^60 + (-b6 + b3 - 3b2 - 3) q^61 + (-b7 + b6 + 3b5 - b4 + 3b3 - 3b2 + 3b1 - 3) * q^62 + (2b7 - 2b6 + 2b4 + 4b3 + 4b2 - 2) q^64 + (-b7 + 2b6 + b5 - 2b4 - 2b3 + 2b1 - 8) * q^65 + (-2b7 - 2b6 + 4b5 - 2b3 - 4b2 + 2) q^66 + (4b7 - 4b6 + 4b4 + 4b3 + 2b2 + 4b1 + 2) * q^67 + (-2b7 + 4b6 - 2b4 - 2b3 + 2b1 - 6) q^68 + (b6 - 4b5 - b4 + b3 + 6b2 + b1 - 6) * q^69 + (-2b7 - 2b5 - 6b3 - 10b2 - 6b1) q^71 + (-b7 - b6 + b5 - 2b3 - b2 + b1 + 5) q^72 + (2b7 + 2b6 + 2b5 + 2b4 + 2b3 + 6b2 + 2b1) q^73 + (2b7 - 2b6 - 2b5 - b4 - b3 + 2b2 + 6b1 - 2) q^74 + (4b6 + 2b5 - 5b4 + 4b3 + 6b2 + 5b1 - 6) * q^75 + (b7 + b6 + 3b5 + 5b2 - b1 + 3) * q^76 + (6b6 + 2b5 - 3b4 - b3 - 6b2 - 4b1) q^78 + (2b7 + 2b6 - 2b5 - 2b4 - 4b3 + 4b1 - 6) * q^79 + (-2b6 - 2b5 + 2b4 + 2b3 - 2b2 - 4b1 + 4) * q^80 + (-b7 + 2b6 + b5 - 2b4 + 2b3 - 2b1 + 5) * q^81 + (-b7 - 7b6 - 5b5 + 3b4 - b3 + b2 + 3b1 + 1) * q^82 + (b4 + b1) * q^83 + (2b5 - 6b4 + 8b2 + 6b1 - 8) * q^85 + (2b7 - 2b6 - 4b5 + 6b4 + 4b3 + 4b1 + 2) * q^86 + (-2b7 - b6 - 2b5 - b4 - 7b3 - 7b1) * q^87 + (-2b4 + 2b3 - 4b2 - 4b1 - 4) * q^88 + (-4b7 - 4b5 - 6b3 - 6b2 - 6b1) q^89 + (b5 + b4 + b2 + b1 + 2) * q^90 + (-3b7 - b6 - b5 + 3b4 - 3b3 - b2 - b1 + 5) q^92 + (-6b7 + 4b6 - 4b4 - 4b3 - 4b1) q^93 + (-b7 + b6 - 3b5 - b4 + b3 + 3b2 - 3b1 - 3) q^94 + (-b7 - b6 + b5 + b4 + 3b3 - 3b1 + 4) * q^95 + (2b7 + 2b6 + 4b5 - 6b4 + 4b3 - 2) q^96 + (2b7 - 3b6 - 2b5 + 3b4 - 3b3 + 3b1 + 4) * q^97 + (-2b7 + 4b4 + 6b2 + 4b1 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 4 q^{4} + 4 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 + 4 * q^4 + 4 * q^5 + 16 * q^6 - 4 * q^8	$ 8 q + 2 q^{2} + 4 q^{4} + 4 q^{5} + 16 q^{6} - 4 q^{8} - 12 q^{10} + 12 q^{12} - 8 q^{15} + 8 q^{16} - 24 q^{17} + 6 q^{18} + 12 q^{19} + 8 q^{20} - 4 q^{22} - 8 q^{26} - 12 q^{27} - 16 q^{29} + 20 q^{30} - 16 q^{31} - 28 q^{32} + 24 q^{33} + 12 q^{34} + 24 q^{36} + 16 q^{37} + 16 q^{38} - 28 q^{40} - 32 q^{43} - 32 q^{44} + 8 q^{45} + 20 q^{46} - 24 q^{47} - 20 q^{48} - 14 q^{50} + 8 q^{51} - 32 q^{52} - 8 q^{53} + 16 q^{54} + 12 q^{58} + 28 q^{59} - 28 q^{60} - 28 q^{61} - 20 q^{62} - 32 q^{64} - 48 q^{65} + 16 q^{66} - 28 q^{68} - 44 q^{69} + 44 q^{72} - 4 q^{74} - 28 q^{75} + 24 q^{76} + 12 q^{78} - 24 q^{79} + 12 q^{80} + 40 q^{81} - 4 q^{82} - 40 q^{85} - 40 q^{88} + 16 q^{90} + 36 q^{92} + 16 q^{93} - 28 q^{94} + 16 q^{95} - 8 q^{96} + 32 q^{97} + 48 q^{99}+O(q^{100}) $ 8 * q + 2 * q^2 + 4 * q^4 + 4 * q^5 + 16 * q^6 - 4 * q^8 - 12 * q^10 + 12 * q^12 - 8 * q^15 + 8 * q^16 - 24 * q^17 + 6 * q^18 + 12 * q^19 + 8 * q^20 - 4 * q^22 - 8 * q^26 - 12 * q^27 - 16 * q^29 + 20 * q^30 - 16 * q^31 - 28 * q^32 + 24 * q^33 + 12 * q^34 + 24 * q^36 + 16 * q^37 + 16 * q^38 - 28 * q^40 - 32 * q^43 - 32 * q^44 + 8 * q^45 + 20 * q^46 - 24 * q^47 - 20 * q^48 - 14 * q^50 + 8 * q^51 - 32 * q^52 - 8 * q^53 + 16 * q^54 + 12 * q^58 + 28 * q^59 - 28 * q^60 - 28 * q^61 - 20 * q^62 - 32 * q^64 - 48 * q^65 + 16 * q^66 - 28 * q^68 - 44 * q^69 + 44 * q^72 - 4 * q^74 - 28 * q^75 + 24 * q^76 + 12 * q^78 - 24 * q^79 + 12 * q^80 + 40 * q^81 - 4 * q^82 - 40 * q^85 - 40 * q^88 + 16 * q^90 + 36 * q^92 + 16 * q^93 - 28 * q^94 + 16 * q^95 - 8 * q^96 + 32 * q^97 + 48 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2*x^7 - 2*x^5 + 9*x^4 - 4*x^3 - 16*x + 16 :

$\beta_{1}$	$=$	$ ( \nu^{7} - 2\nu^{4} + 5\nu^{3} - 2\nu^{2} + 4\nu - 8 ) / 8 $ (v^7 - 2v^4 + 5v^3 - 2v^2 + 4v - 8) / 8
$\beta_{2}$	$=$	$ ( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8 $ (-5v^7 + 2v^6 + 4v^5 + 18v^4 - 21v^3 - 12v^2 - 20*v + 56) / 8
$\beta_{3}$	$=$	$ ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} - 18\nu^{4} + 25\nu^{3} + 10\nu^{2} + 24\nu - 64 ) / 8 $ (5v^7 - 4v^6 - 4v^5 - 18v^4 + 25v^3 + 10v^2 + 24*v - 64) / 8
$\beta_{4}$	$=$	$ ( -3\nu^{7} + 2\nu^{6} + 2\nu^{5} + 10\nu^{4} - 15\nu^{3} - 8\nu^{2} - 10\nu + 36 ) / 4 $ (-3v^7 + 2v^6 + 2v^5 + 10v^4 - 15v^3 - 8v^2 - 10*v + 36) / 4
$\beta_{5}$	$=$	$ ( -7\nu^{7} + 4\nu^{6} + 4\nu^{5} + 22\nu^{4} - 27\nu^{3} - 14\nu^{2} - 32\nu + 72 ) / 8 $ (-7v^7 + 4v^6 + 4v^5 + 22v^4 - 27v^3 - 14v^2 - 32*v + 72) / 8
$\beta_{6}$	$=$	$ ( -7\nu^{7} + 4\nu^{6} + 8\nu^{5} + 22\nu^{4} - 35\nu^{3} - 22\nu^{2} - 20\nu + 88 ) / 8 $ (-7v^7 + 4v^6 + 8v^5 + 22v^4 - 35v^3 - 22v^2 - 20*v + 88) / 8
$\beta_{7}$	$=$	$ ( -9\nu^{7} + 6\nu^{6} + 8\nu^{5} + 26\nu^{4} - 41\nu^{3} - 16\nu^{2} - 24\nu + 96 ) / 8 $ (-9v^7 + 6v^6 + 8v^5 + 26v^4 - 41v^3 - 16v^2 - 24*v + 96) / 8

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

$\nu$	$=$	$ ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 $ (b7 - b6 - b5 + b4 + b3 + b2 + b1 + 1) / 2
$\nu^{2}$	$=$	$ ( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 $ (b7 - b6 - b5 - b4 - b3 + b2 - b1 + 1) / 2
$\nu^{3}$	$=$	$ ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 3 ) / 2 $ (b7 - b6 + b5 - b4 + b3 + b2 + 3*b1 + 3) / 2
$\nu^{4}$	$=$	$ ( \beta_{7} - 3\beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + 5\beta_{2} + 3\beta _1 - 1 ) / 2 $ (b7 - 3b6 - 3b5 + b4 - b3 + 5b2 + 3b1 - 1) / 2
$\nu^{5}$	$=$	$ ( \beta_{7} + 3\beta_{6} - \beta_{5} - 7\beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 - 3 ) / 2 $ (b7 + 3b6 - b5 - 7b4 - 3*b3 + b2 + b1 - 3) / 2
$\nu^{6}$	$=$	$ ( 3\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} - 3\beta_{3} - 5\beta_{2} + 9\beta _1 - 1 ) / 2 $ (3b7 - 3b6 + b5 + b4 - 3b3 - 5b2 + 9*b1 - 1) / 2
$\nu^{7}$	$=$	$ ( -5\beta_{7} + \beta_{6} - 9\beta_{5} + \beta_{4} - 13\beta_{3} + 3\beta_{2} + \beta _1 - 3 ) / 2 $ (-5b7 + b6 - 9b5 + b4 - 13b3 + 3b2 + b1 - 3) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$-\beta_{2}$	$1$	$1$

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} $

197.1

1.23291	−	0.692769i
−1.08003	+	0.912978i
−0.565036	−	1.29643i
1.41216	+	0.0762223i
1.23291	+	0.692769i
−1.08003	−	0.912978i
−0.565036	+	1.29643i
1.41216	−	0.0762223i

−1.40101

0.192769i

−1.20825

1.20825i

1.92568

−

0.540143i

2.59378

2.59378i

1.45986

−

1.92568i

−2.59378

1.12796i

0.0802864i

−4.13393

−

3.13393i

197.2

−0.0591148

−

1.41298i

−1.47209

1.47209i

−1.99301

0.167056i

−0.353863

−

0.353863i

2.16706

1.99301i

0.353863

2.80620i

−

1.33411i

−0.479081

0.520919i

197.3

1.16863

0.796431i

1.96506

−

1.96506i

0.731395

1.86147i

0.627801

0.627801i

3.86147

−

0.731395i

−0.627801

2.75787i

−

4.72294i

0.233667

1.23367i

197.4

1.29150

−

0.576222i

0.715276

−

0.715276i

1.33594

−

1.48838i

−0.867721

−

0.867721i

0.511620

−

1.33594i

0.867721

−

2.69204i

1.97676i

−1.62066

−

0.620660i

589.1