LMFDB - Newform orbit 213.2.b.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [213,2,Mod(212,213)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(213, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("213.212");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 213 = 3 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	213.b (of order $2$, degree $1$, 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:	$1.70081356305$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 21x^{7} + 128 $ x^14 - 21*x^7 + 128
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{2} - \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} - \beta_{9} q^{5} + \beta_{8} q^{6} + (\beta_{10} + \beta_{5} + \cdots - \beta_{2}) q^{8}+ \cdots + \beta_1 q^{9}+O(q^{10}) $ q - b4 * q^2 - b2 * q^3 + (b3 - 2) * q^4 - b9 * q^5 + b8 * q^6 + (b10 + b5 + 2b4 - b2) q^8 + b1 * q^9	$ q - \beta_{4} q^{2} - \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} - \beta_{9} q^{5} + \beta_{8} q^{6} + (\beta_{10} + \beta_{5} + \cdots - \beta_{2}) q^{8}+ \cdots - 7 \beta_{4} q^{98}+O(q^{100}) $ q - b4 * q^2 - b2 * q^3 + (b3 - 2) * q^4 - b9 * q^5 + b8 * q^6 + (b10 + b5 + 2b4 - b2) q^8 + b1 * q^9 + (b13 - b10 - b6 + b5 - b2 - b1) * q^10 + (-b13 + b9 - b4 - b3 + 2b2) q^12 + (-b13 + b8 - b7 + b6 - b5 + b4 + b3) * q^15 + (b11 + b5 - 2b3 + 2b2 + 4) * q^16 + (-b12 - b9 + b8 + b7 - b5) * q^18 + (b13 + b6 + b4 - b1) * q^19 + (2b12 - b11 + 2b9 + b5 + 2b2 + 1) q^20 + (-b12 - b8 + b6 - b5 - b4 + b1 - 3) * q^24 + (-b11 + b10 - b8 - b5 - b2 - 5) * q^25 + (b9 + b7 + b3) * q^27 + (b11 - b10 - b8 - b5 - b2) * q^29 + (-b12 + b11 + b10 - b9 - b8 - b7 - b6 + 2b5 + b4 - b3 + b1 + 3) q^30 + (-2b10 + b9 - b8 + 2b7 - b5 - 4b4 - b3 + 2b2) * q^32 + (2b13 - b11 - b10 - b6 + b5 - 2b1) * q^36 + (b13 - 2b8 - 3b6 + 2b5 - b4 - b1) q^37 + (2b12 - 2b7 + b3 + 1) * q^38 + (-4b13 + 2b10 + b9 - b8 - 2b7 + 2b6 - b5 - b4 - b3 + 2b2 + 4b1) * q^40 + (-b9 + 2b8 + 2b7 - 2b5) q^43 + (-b13 - b11 - b10 - b6 - 2b5 - 3b4) * q^45 + (2b13 - b12 - 2b9 - b6 + 3b4 + 2b3 - 4b2 - 2b1 - 6) * q^48 + 7 * q^49 + (b11 + 2b10 - 3b9 - 2b7 + b5 + 5b4 + b3 - 4b2) q^50 + (-b13 - b11 + 2b10 + b8 + 2b6 - b5 + 3b4 - b2) q^54 + (b11 + b10 - b9 - b8 - b7 - b5 - b3) * q^57 + (-b11 - 2b10 - b9 + 4b8 + 2b7 - 3b5 + 3b3 - 4b2) * q^58 + (2b13 - b12 + 2b11 - 4b10 + 2b7 - b6 - 3b4 - 2b3 + b2 - 2b1 + 6) q^60 + (b13 - 2b11 + b8 + 3b6 - 3b5 + 2b4 + 4b3 - 4b2 - b1 - 8) * q^64 + (-2b12 + b8 - b5 - 1) q^71 + (2b12 - b11 + 2b10 + 3b9 - b8 - b7 + b5 + b3 + 2b2) * q^72 + (b9 - 2b8 - 2b7 + 2b5 + 2b3) * q^73 + (2b12 + 4b9 - 2b8 + 2b7 + 2b5 - b3 + 1) q^74 + (2b12 + 2b6 - 2b4 + 5b2 + b1 + 3) * q^75 + (-4b13 + 2b11 + b10 + 2b8 + 2b6 - b5 - b4 + 5b2 + 4b1) * q^76 + (b9 - 2b8 - 2b7 + 2b5 - 4b3) * q^79 + (3b13 - 4b12 + 2b11 - 6b9 - 3b8 + 2b7 - b6 + b5 - b3 - 4b2 + 3b1 - 2) * q^80 + (2b9 - b8 + b7 + b5 - 3b3) * q^81 + (-3b13 + 2b8 + b6 - 2b5 + b4 - 3b1) * q^83 + (-3b13 - 2b11 - b10 + 4b8 + b6 - 3b5 - 2b4 + 5b2 - 3b1) q^86 + (2b12 - 2b8 - 2b6 + 2b5 + 2b4 + b1 - 3) q^87 + (-b11 - 3b10 + b8 - 3b5 + 5b2) q^89 + (-b12 - b11 + 2b10 - b8 - b7 + b5 + 4b3 - 4b2 - 9) q^90 + (3b13 - b11 + 3b10 - b8 - b6 + 5b5 + 3b4 - b2 + 3b1) q^95 + (2b13 + 2b12 + 3b9 - b7 - b6 + 4b5 + 6b4 - 2b3 - 2b1 + 6) q^96 - 7b4 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 28 q^{4}+O(q^{10}) $ 14 * q - 28 * q^4	$ 14 q - 28 q^{4} + 56 q^{16} + 7 q^{18} - 35 q^{24} - 70 q^{25} + 49 q^{30} - 77 q^{48} + 98 q^{49} + 91 q^{60} - 112 q^{64} - 14 q^{72} + 28 q^{75} - 56 q^{87} - 119 q^{90} + 70 q^{96}+O(q^{100}) $ 14 * q - 28 * q^4 + 56 * q^16 + 7 * q^18 - 35 * q^24 - 70 * q^25 + 49 * q^30 - 77 * q^48 + 98 * q^49 + 91 * q^60 - 112 * q^64 - 14 * q^72 + 28 * q^75 - 56 * q^87 - 119 * q^90 + 70 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} - 21x^{7} + 128 $ x^14 - 21*x^7 + 128 :

$\beta_{1}$	$=$	$ ( \nu^{8} - 11\nu ) / 2 $ (v^8 - 11*v) / 2
$\beta_{2}$	$=$	$ ( \nu^{11} - 5\nu^{4} ) / 16 $ (v^11 - 5*v^4) / 16
$\beta_{3}$	$=$	$ ( -\nu^{12} + 21\nu^{5} + 32\nu^{2} ) / 32 $ (-v^12 + 21v^5 + 32v^2) / 32
$\beta_{4}$	$=$	$ ( -\nu^{13} + 21\nu^{6} - 64\nu ) / 64 $ (-v^13 + 21v^6 - 64v) / 64
$\beta_{5}$	$=$	$ ( \nu^{12} + 2\nu^{10} - 5\nu^{5} - 10\nu^{3} ) / 16 $ (v^12 + 2v^10 - 5v^5 - 10*v^3) / 16
$\beta_{6}$	$=$	$ ( -\nu^{13} + 4\nu^{12} + 21\nu^{6} - 20\nu^{5} + 128\nu ) / 64 $ (-v^13 + 4v^12 + 21v^6 - 20v^5 + 128v) / 64
$\beta_{7}$	$=$	$ ( -\nu^{12} + 8\nu^{9} + 21\nu^{5} - 104\nu^{2} ) / 32 $ (-v^12 + 8v^9 + 21v^5 - 104*v^2) / 32
$\beta_{8}$	$=$	$ ( -\nu^{12} + 2\nu^{10} + 5\nu^{5} - 10\nu^{3} ) / 16 $ (-v^12 + 2v^10 + 5v^5 - 10*v^3) / 16
$\beta_{9}$	$=$	$ ( -3\nu^{12} - 8\nu^{9} + 31\nu^{5} + 72\nu^{2} ) / 32 $ (-3v^12 - 8v^9 + 31v^5 + 72v^2) / 32
$\beta_{10}$	$=$	$ ( -\nu^{12} + 2\nu^{11} - 2\nu^{10} + 5\nu^{5} - 26\nu^{4} + 26\nu^{3} ) / 16 $ (-v^12 + 2v^11 - 2v^10 + 5v^5 - 26v^4 + 26*v^3) / 16
$\beta_{11}$	$=$	$ ( -\nu^{12} - 2\nu^{11} - 4\nu^{10} + 5\nu^{5} + 26\nu^{4} + 52\nu^{3} ) / 16 $ (-v^12 - 2v^11 - 4v^10 + 5v^5 + 26v^4 + 52*v^3) / 16
$\beta_{12}$	$=$	$ ( -\nu^{12} + 16\nu^{7} + 5\nu^{5} - 176 ) / 16 $ (-v^12 + 16v^7 + 5v^5 - 176) / 16
$\beta_{13}$	$=$	$ ( 5\nu^{13} - 4\nu^{12} - 41\nu^{6} + 20\nu^{5} + 64\nu ) / 64 $ (5v^13 - 4v^12 - 41v^6 + 20v^5 + 64*v) / 64

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

$\nu$	$=$	$ ( \beta_{8} + 2\beta_{6} - \beta_{5} - 2\beta_{4} ) / 6 $ (b8 + 2b6 - b5 - 2b4) / 6
$\nu^{2}$	$=$	$ ( -2\beta_{9} + \beta_{8} - 2\beta_{7} - \beta_{5} + 4\beta_{3} ) / 6 $ (-2b9 + b8 - 2b7 - b5 + 4*b3) / 6
$\nu^{3}$	$=$	$ ( 2\beta_{11} + 2\beta_{10} + \beta_{8} + 5\beta_{5} ) / 6 $ (2b11 + 2b10 + b8 + 5*b5) / 6
$\nu^{4}$	$=$	$ ( 2\beta_{11} - 4\beta_{10} + \beta_{8} - \beta_{5} + 12\beta_{2} ) / 6 $ (2b11 - 4b10 + b8 - b5 + 12*b2) / 6
$\nu^{5}$	$=$	$ ( 4\beta_{9} - 5\beta_{8} + 4\beta_{7} + 5\beta_{5} + 4\beta_{3} ) / 6 $ (4b9 - 5b8 + 4b7 + 5b5 + 4*b3) / 6
$\nu^{6}$	$=$	$ ( 6\beta_{13} + \beta_{8} + 8\beta_{6} - \beta_{5} + 22\beta_{4} ) / 6 $ (6b13 + b8 + 8b6 - b5 + 22*b4) / 6
$\nu^{7}$	$=$	$ ( 2\beta_{12} - \beta_{8} + \beta_{5} + 22 ) / 2 $ (2*b12 - b8 + b5 + 22) / 2
$\nu^{8}$	$=$	$ ( 11\beta_{8} + 22\beta_{6} - 11\beta_{5} - 22\beta_{4} + 12\beta_1 ) / 6 $ (11b8 + 22b6 - 11b5 - 22b4 + 12*b1) / 6
$\nu^{9}$	$=$	$ ( -34\beta_{9} + 17\beta_{8} - 10\beta_{7} - 17\beta_{5} + 44\beta_{3} ) / 6 $ (-34b9 + 17b8 - 10b7 - 17b5 + 44*b3) / 6
$\nu^{10}$	$=$	$ ( 10\beta_{11} + 10\beta_{10} + 29\beta_{8} + 49\beta_{5} ) / 6 $ (10b11 + 10b10 + 29b8 + 49b5) / 6
$\nu^{11}$	$=$	$ ( 10\beta_{11} - 20\beta_{10} + 5\beta_{8} - 5\beta_{5} + 156\beta_{2} ) / 6 $ (10b11 - 20b10 + 5b8 - 5b5 + 156*b2) / 6
$\nu^{12}$	$=$	$ ( 20\beta_{9} - 73\beta_{8} + 20\beta_{7} + 73\beta_{5} + 20\beta_{3} ) / 6 $ (20b9 - 73b8 + 20b7 + 73b5 + 20*b3) / 6
$\nu^{13}$	$=$	$ ( 126\beta_{13} - 43\beta_{8} + 40\beta_{6} + 43\beta_{5} + 206\beta_{4} ) / 6 $ (126b13 - 43b8 + 40b6 + 43b5 + 206*b4) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/213\mathbb{Z}\right)^\times$.

$n$	$7$	$143$
$\chi(n)$	$-1$	$-1$

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} $

212.1

−0.239104	−	1.39385i
−0.389345	−	1.35956i
0.820196	−	1.15208i
0.940680	−	1.05599i
−1.23884	−	0.682115i
−1.30570	−	0.543271i
1.41211	−	0.0770525i
1.41211	+	0.0770525i
−1.30570	+	0.543271i
−1.23884	+	0.682115i
0.940680	+	1.05599i
0.820196	+	1.15208i
−0.389345	+	1.35956i
−0.239104	+	1.39385i

−

2.78771i

−1.73147

0.0448226i

−5.77132

−

3.47230i

0.124952

4.82684i

10.5133i

2.99598

−

0.155218i

−9.67975

212.2

−

2.71913i

0.341590

1.69803i

−5.39364

4.36842i

4.61716

−

0.928825i

9.22773i

−2.76663

1.16006i

11.8783

212.3

−

2.30415i

0.428987

−

1.67809i

−3.30911

−

0.0385424i

−3.86656

−

0.988452i

3.01640i

−2.63194

−

1.43975i

−0.0888076

212.4

−

2.11199i

1.57945

0.710873i

−2.46048

−

1.97504i

1.50135

−

3.33577i

0.972536i

1.98932

2.24558i

−4.17126

212.5

−

1.36423i

1.54055

−

0.791641i

0.138878

3.52036i

−1.07998

−

2.10167i

−

2.91792i

1.74661

−

2.43913i

4.80258

212.6

−

1.08654i

−1.04451

−

1.38166i

0.819427

−

1.90559i

−1.50124

1.13490i

−

3.06343i

−0.817995

2.88633i

−2.07050

212.7

−

0.154105i

−1.11460

1.32577i

1.97625

−

4.35127i

0.204308

0.171765i

−

0.612760i

−0.515342

−

2.95541i

−0.670553

212.8