LMFDB - Newform orbit 441.2.f.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(148,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("441.148");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$441 = 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	441.f (of order $3$ , 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:	$3.52140272914$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.991381711347.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$ x^10 - 2x^9 + 9x^8 - 8x^7 + 40x^6 - 36x^5 + 90x^4 - 3x^3 + 36x^2 - 9*x + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{8} + \beta_{6} + \beta_{4} + \cdots - 1) q^{4} + ( - \beta_{8} + \beta_{7} + \beta_{5}) q^{5} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{6} + (\beta_{9} - \beta_{3} - 1) q^{8}+ \cdots + ( - 3 \beta_{9} + \beta_{7} + \beta_{5} + \cdots + 5) q^{99}+O(q^{100})$ q + b1 * q^2 + b4 * q^3 + (b8 + b6 + b4 - b2 - 1) * q^4 + (-b8 + b7 + b5) * q^5 + (b9 - b8 + b7 - b6 + b5 - b4 - b3 - b1 + 1) * q^6 + (b9 - b3 - 1) * q^8 + (-b9 - b7 - 2b6 + b4 + b2 + b1) q^9 + (b9 - b8 + b7 + b5 - b4 - b3 - b2 - b1 - 1) * q^10 + (-b8 - b7 + b6 + b3 + b2) * q^11 + (-b7 - 3b6 - b5 - b3 + b1 + 1) q^12 + (-b8 + b7 - 2b6 + 2) q^13 + (-b9 - b8 + b7 - 2b6 + b5 - b4 - b1) q^15 + (b8 + b7 + b6 - b4 - b3 - 2b2 - b1) q^16 + (b7 + b5 - b4 - b1 + 3) * q^17 + (-b9 + b8 + b6 - 2b5 + b4 + b3 + b2 - 1) q^18 + (b9 + b8 + b5 - b3 + b2 - b1) * q^19 + (-b8 - b7 - 2b6 - 2b3 + b2) * q^20 + (b7 + b4 - b2) * q^22 + (2b9 - 3b8 + b7 - b6 + b5 - 2b4 + 2b2 + 1) * q^23 + (-b7 - 2b5 + 2b2 + b1) * q^24 + (b8 + b7 + 2b6 - 2b4 - 3b2 - 3b1) * q^25 + (b9 - b8 + 2b7 - b5 - 2b4 - b3 - b2 + b1 + 2) * q^26 + (2b7 + 2b5 - b4 - b3 + b1 + 1) * q^27 + (-2b8 - 2b7 + b6 + b3 + 2b2 + b1) q^29 + (-b9 - 2b6 - 2b5 - b4 + b3 + b2 + 2) * q^30 + (-b9 + 2b8 - b7 - b5 + b4 - b2) q^31 + (b9 + b8 - 2b7 + b6 + b5 - b4 + b2 - 1) q^32 + (-b9 - b8 - b7 + b6 + 2b5 + b3 - 2b1 + 2) * q^33 + (-b8 - b7 - 2b6 + b3 + b2 + 4b1) * q^34 + (b9 + b8 + b7 - b6 + b4 + 2b2 + 2b1 + 3) * q^36 + (2b9 + 2b5 - 2b3 - 2b1) * q^37 + (-3b8 - 3b7 - 5b6 + b3 + 3b2 + 2b1) q^38 + (-b9 + b8 - 2b6 + b5 + 2b4 + b3 + b2 - 1) * q^39 + (-b9 - 2b8 - b6 - b5 - 2b4 + 2b2 + 1) q^40 + (-b9 + 3b8 - 2b7 + 2b6 - b5 + b4 - b2 - 2) q^41 + (b8 + b7 - 3b4 - b3 - 4b2 - b1) * q^43 + (-b8 + 2b5 - b2 - 2b1 + 3) * q^44 + (-b9 + b8 + 3b7 + b6 + b5 - 2b4 + b3 - 2b2 - 1) q^45 + (b9 - 3b8 + b7 - 2b5 - b4 - b3 - 3b2 + 2b1 - 1) * q^46 + (2b8 + 2b7 - 5b6 + b4 + b3 - b2) q^47 + (b9 + b7 - b6 - 3b5 - b2 + 2b1) * q^48 + (-b9 + b8 - 2b7 - 4b6 + b5 - b4 + b2 + 4) * q^50 + (b8 + b7 + 3b6 - b5 + 2b4 - 2b2 - b1 - 3) q^51 + (b8 + b7 + b6 - b3 - b2 + 2b1) q^52 + (2b8 - 2b7 - b5 + 2b4 + 2b2 + b1 + 3) * q^53 + (-2b7 + 3b6 + b5 + b4 + b3 + 2b1 - 7) q^54 + (-b9 - b8 + 2b5 + b3 - b2 - 2b1 + 1) * q^55 + (2b8 - b7 + 6b6 - 4b5 + b4 - b3 + b1 - 5) q^57 + (-b9 + b8 + b7 + 2b6 - b5 + 2b4 - 2b2 - 2) q^58 + (b9 + 2b8 - b7 + 7b6 + b4 - b2 - 7) * q^59 + (-b9 - b8 + 2b7 + b6 - b5 + b3 + 3b2 + 4b1 + 2) q^60 + (-b8 - b7 + b6 + b4 - b3 + 2b2 + 2b1) * q^61 + (-b9 + 2b8 - b7 + b4 + b3 + 2b2 + 1) * q^62 + (-b8 - b7 + b4 - b2 - 5) * q^64 + (b8 + b7 - 2b6 + b4 + b3) q^65 + (-b9 - b7 - 5b6 + b4 + b2 + b1) q^66 + (-b9 + b8 + b7 - 2b6 + 5b5 + 2b4 - 2b2 + 2) * q^67 + (2b8 + b7 + 6b6 - b5 + 3b4 - 3b2 - 6) * q^68 + (-b9 - b7 + 4b6 + b5 + 2b3 + 2b2 - 5b1 - 2) * q^69 + (-3b9 + 2b8 - b7 + b4 + 3b3 + 2b2) * q^71 + (b9 - b8 - b7 + 5b6 + b5 + b4 + b3 + 2b2 + b1 - 4) * q^72 + (-3b9 - b7 + b4 + 3b3 + 4) * q^73 + (-4b8 - 4b7 - 8b6 - 2b4 - 2b3 + 2b2) * q^74 + (2b8 - b7 - 4b5 + b4 + 2b3 - 3b2 + b1 + 1) * q^75 + (2b8 - b7 + 4b6 - 6b5 + b4 - b2 - 4) q^76 + (2b9 - 2b8 + b7 - 2b6 + b4 + b3 + 2b2 - b1 - 1) * q^78 + (-2b8 - 2b7 - 2b6 - b4 + b2 + 3b1) * q^79 + (b9 + b8 - b7 - 3b5 + b4 - b3 + b2 + 3b1 + 2) * q^80 + (2b9 - 2b8 + 5b7 + b6 - 3b4 - 3b3 - 2b2 - 2b1 - 3) q^81 + (-2b9 + 3b8 - 3b7 + b5 + 3b4 + 2b3 + 3b2 - b1 - 1) * q^82 + (-2b8 - 2b7 + b6 - 2b4 + 4b3 - 2b1) q^83 + (b9 + b7 + b6 + b5 + b4 - b2 - 1) * q^85 + (-3b9 + 5b8 - 4b7 + 3b6 - 2b5 + b4 - b2 - 3) q^86 + (-b9 - 3b8 + b6 + 4b5 - b4 + b3 + b2 - 3b1 + 5) q^87 + (b8 + b7 - 5b6 + b4 - 2b3 + b1) * q^88 + (-2b9 - b8 + b7 + 3b5 - b4 + 2b3 - b2 - 3b1 + 8) * q^89 + (b9 + 2b8 + 5b6 + 2b5 + 2b4 + 2b3 - b2 - 5) q^90 + (7b6 - 2b4 - 2b3 - 2b2 - 4b1) q^92 + (b9 + b8 - b6 - b5 + b4 - b3 - b2 + 3b1 + 1) q^93 + (4b9 - 2b8 + 2b7 + b6 - 2b5 - 1) * q^94 + (-b9 - 2b8 + 2b6 - 2b4 + 2b2 - 2) * q^95 + (2b9 + b8 + 7b6 - 2b5 - 2b4 - 2b3 - 2b2 - 3b1 + 2) q^96 + (4b8 + 4b7 + 4b6 - 2b4 - b3 - 6b2 - b1) q^97 + (-3b9 + b7 + b5 + b4 + b3 - 3b2 - b1 + 5) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 2 q^{2} + q^{3} - 4 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{8} - 7 q^{9} - 14 q^{10} + 4 q^{11} + 2 q^{12} + 8 q^{13} - 19 q^{15} + 2 q^{16} + 24 q^{17} - 2 q^{18} + 2 q^{19} - 5 q^{20} - q^{22} + 3 q^{23}+ \cdots + 35 q^{99}+O(q^{100})$ 10 * q + 2 * q^2 + q^3 - 4 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^8 - 7 * q^9 - 14 * q^10 + 4 * q^11 + 2 * q^12 + 8 * q^13 - 19 * q^15 + 2 * q^16 + 24 * q^17 - 2 * q^18 + 2 * q^19 - 5 * q^20 - q^22 + 3 * q^23 + 9 * q^24 - q^25 + 22 * q^26 + 7 * q^27 + 7 * q^29 + 10 * q^30 + 3 * q^31 - 2 * q^32 + 13 * q^33 - 3 * q^34 + 34 * q^36 - 20 * q^38 - 22 * q^39 + 3 * q^40 - 5 * q^41 - 7 * q^43 + 20 * q^44 - 17 * q^45 - 6 * q^46 - 27 * q^47 + 5 * q^48 + 19 * q^50 - 15 * q^51 + 10 * q^52 + 42 * q^53 - 52 * q^54 - 4 * q^55 - 4 * q^57 - 10 * q^58 - 30 * q^59 + 31 * q^60 + 14 * q^61 + 12 * q^62 - 50 * q^64 - 11 * q^65 - 22 * q^66 - 2 * q^67 - 27 * q^68 - 15 * q^69 - 6 * q^71 - 12 * q^72 + 30 * q^73 - 36 * q^74 + 17 * q^75 - 5 * q^76 - 20 * q^78 - 4 * q^79 + 40 * q^80 - 31 * q^81 - 10 * q^82 - 9 * q^83 - 6 * q^85 - 8 * q^86 + 34 * q^87 - 18 * q^88 + 56 * q^89 - 28 * q^90 + 27 * q^92 + 18 * q^93 + 3 * q^94 - 14 * q^95 + 58 * q^96 + 12 * q^97 + 35 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$ x^10 - 2*x^9 + 9*x^8 - 8*x^7 + 40*x^6 - 36*x^5 + 90*x^4 - 3*x^3 + 36*x^2 - 9*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 2\nu^{9} + 9\nu^{8} - 3\nu^{7} + 95\nu^{6} + 18\nu^{5} + 402\nu^{4} - 87\nu^{3} + 936\nu^{2} + 342\nu + 72 ) / 189$ (2v^9 + 9v^8 - 3v^7 + 95v^6 + 18v^5 + 402v^4 - 87v^3 + 936v^2 + 342*v + 72) / 189
$\beta_{3}$	$=$	$( -2\nu^{9} + \nu^{8} - 12\nu^{7} - 8\nu^{6} - 68\nu^{5} - 30\nu^{4} - 123\nu^{3} - 204\nu^{2} - 270\nu - 63 ) / 63$ (-2v^9 + v^8 - 12v^7 - 8v^6 - 68v^5 - 30v^4 - 123v^3 - 204v^2 - 270v - 63) / 63
$\beta_{4}$	$=$	$( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} + \cdots + 639 ) / 567$ (17v^9 - 24v^8 + 159v^7 - 106v^6 + 786v^5 - 417v^4 + 1893v^3 - 27v^2 + 1395*v + 639) / 567
$\beta_{5}$	$=$	$( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} + \cdots - 180 ) / 567$ (16v^9 - 39v^8 + 156v^7 - 176v^6 + 663v^5 - 780v^4 + 1680v^3 - 351v^2 + 684*v - 180) / 567
$\beta_{6}$	$=$	$( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + \cdots + 504 ) / 567$ (20v^9 - 24v^8 + 141v^7 - 4v^6 + 624v^5 - 57v^4 + 1020v^3 + 1620v^2 + 369*v + 504) / 567
$\beta_{7}$	$=$	$( 8\nu^{9} - 12\nu^{8} + 69\nu^{7} - 43\nu^{6} + 330\nu^{5} - 219\nu^{4} + 732\nu^{3} - 45\nu^{2} + 477\nu - 306 ) / 189$ (8v^9 - 12v^8 + 69v^7 - 43v^6 + 330v^5 - 219v^4 + 732v^3 - 45v^2 + 477*v - 306) / 189
$\beta_{8}$	$=$	$( - 71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} + \cdots - 234 ) / 567$ (-71v^9 + 123v^8 - 591v^7 + 403v^6 - 2604v^5 + 1794v^4 - 5214v^3 - 1458v^2 - 1476*v - 234) / 567
$\beta_{9}$	$=$	$( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} + \cdots + 720 ) / 567$ (-82v^9 + 165v^8 - 732v^7 + 632v^6 - 3264v^5 + 2850v^4 - 7260v^3 - 432v^2 - 2898*v + 720) / 567

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{8} + 3\beta_{6} + \beta_{4} - \beta_{2} - 3$ b8 + 3*b6 + b4 - b2 - 3
$\nu^{3}$	$=$	$\beta_{9} + 4\beta_{5} - \beta_{3} - 4\beta _1 - 1$ b9 + 4b5 - b3 - 4b1 - 1
$\nu^{4}$	$=$	$-5\beta_{8} - 5\beta_{7} - 13\beta_{6} - \beta_{4} - \beta_{3} + 4\beta_{2} - \beta_1$ -5b8 - 5b7 - 13b6 - b4 - b3 + 4b2 - b1
$\nu^{5}$	$=$	$-7\beta_{9} + \beta_{8} - 2\beta_{7} - 7\beta_{6} - 19\beta_{5} - \beta_{4} + \beta_{2} + 7$ -7b9 + b8 - 2b7 - 7b6 - 19b5 - b4 + b2 + 7
$\nu^{6}$	$=$	$-10\beta_{9} + 9\beta_{8} + 15\beta_{7} - 10\beta_{5} - 15\beta_{4} + 10\beta_{3} + 9\beta_{2} + 10\beta _1 + 61$ -10b9 + 9b8 + 15b7 - 10b5 - 15b4 + 10b3 + 9b2 + 10b1 + 61
$\nu^{7}$	$=$	$11\beta_{8} + 11\beta_{7} + 46\beta_{6} + 19\beta_{4} + 43\beta_{3} + 8\beta_{2} + 94\beta_1$ 11b8 + 11b7 + 46b6 + 19b4 + 43b3 + 8b2 + 94*b1
$\nu^{8}$	$=$	$73\beta_{9} + 56\beta_{8} + 62\beta_{7} + 298\beta_{6} + 76\beta_{5} + 118\beta_{4} - 118\beta_{2} - 298$ 73b9 + 56b8 + 62b7 + 298b6 + 76b5 + 118b4 - 118*b2 - 298
$\nu^{9}$	$=$	$253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} + \cdots - 295$ 253b9 - 135b8 + 48b7 + 478b5 - 48b4 - 253b3 - 135b2 - 478b1 - 295

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$ .

$n$	$199$	$344$
$\chi(n)$	$1$	$-1 + \beta_{6}$

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}

148.1

−1.02682	−	1.77851i
−0.335166	−	0.580525i
0.247934	+	0.429435i
0.920620	+	1.59456i
1.19343	+	2.06709i
−1.02682	+	1.77851i
−0.335166	+	0.580525i
0.247934	−	0.429435i
0.920620	−	1.59456i
1.19343	−	2.06709i

−1.02682

−

1.77851i

−0.608729

1.62156i

−1.10873

1.92038i

−0.0731228

0.126652i

3.50901

−

0.582422i

0.446582

−2.25890

−

1.97418i

0.300337

148.2

−0.335166

−

0.580525i

1.27533

−

1.17198i

0.775327

−

1.34291i

0.712469

−

1.23403i

−1.10781

−

0.347551i

−2.38012

0.252918

−

2.98932i

−0.955182

148.3

0.247934

0.429435i

1.37706

1.05058i

0.877057

−

1.51911i

−1.84629

3.19787i

−0.109735

0.851830i

1.86155

0.792574

2.89341i

−1.83103

148.4

0.920620

1.59456i

−0.195084

−

1.72103i

−0.695084

1.20392i

0.667377

−

1.15593i

2.56469

−

1.89549i

1.12285

−2.92388

0.671489i

2.45760

148.5

1.19343

2.06709i

−1.34857

1.08690i

−1.84857

3.20182i

−1.46043

2.52954i

−3.85615

−

1.49047i

−4.05086

0.637290

−

2.93153i

−6.97172

295.1