LMFDB - Newform orbit 290.2.k.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [290,2,Mod(81,290)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("290.81"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(290, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 290 = 2 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	290.k (of order $7$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.31566165862$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	12.0.4413675765625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 $ x^12 - x^11 + 2x^10 - 3x^9 + 5x^8 - 8x^7 + 13x^6 + 8x^5 + 5x^4 + 3x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{9} q^{2} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{3} + \beta_{3} q^{4} - \beta_{9} q^{5} + (\beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{6} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots + 6) q^{99}+O(q^{100}) $ q + b9 * q^2 + (-b11 + b10 - b7 + b5 - b2) * q^3 + b3 * q^4 - b9 * q^5 + (b11 - b8 + b7 - b4 - 1) * q^6 + (b11 + b10 + b9 + b8 + b6 - b4 - b3 - b2 - b1) * q^7 + (-b11 - b9 - b7 + b5 + b3 + 1) * q^8 + (b11 + b9 + b7 - b6 - b2 - 1) * q^9 - b3 * q^10 + (b8 - b6 + 3b5 - b4 - b2 - b1) q^11 + (-b10 - b9 - b8 - b7 + b6 - b4 + b2 + 1) * q^12 + (b7 - b5 - 2b4 - b3) q^13 + (b11 - 2b10 + b9 - 2b8 + b6 - b5 - 2b4 + 2b1 - 1) * q^14 + (-b11 + b8 - b7 + b4 + 1) * q^15 + b5 * q^16 + (2b11 - 2b8 + 2b6 - 2b3 + b2 - 1) * q^17 + (-b9 - b8 - b7 + b3 + b2 + 1) * q^18 + (-2b10 - 2b8 + 2b5 - 3b4 + b3 + 3b2 + b1 + 2) q^19 + (b11 + b9 + b7 - b5 - b3 - 1) * q^20 + (b11 + b9 + 3b7 - 3b6 - 2b5 - b4 - 2b3 - 3b2 + b1 - 3) q^21 + (3b11 - 2b10 - 2b8 + b6 - b4 + 2b2 + 2b1) q^22 + (3b8 + 2b7 - 2b6 + 3b4) * q^23 + (-b10 + b9 + b6 - b3 - 1) * q^24 + b3 * q^25 + (-2b10 + b9 - 2b8 + b7 + 2b6 - b5 - 2b4 - b3 + 2b2 + 2b1) * q^26 + (2b11 - 3b10 + 2b9 - 3b8 + 3b6 - 2b5 - 2b3 + b2 + b1 - 2) q^27 + (-b11 - b9 - 2b8 - b7 + 2b6 + b3 + b2) * q^28 + (-2b11 - b10 - 2b9 - 2b8 + b7 - 3b6 + b4 + 2b3 + b2 + 1) q^29 + (b10 + b9 + b8 + b7 - b6 + b4 - b2 - 1) * q^30 + (-2b11 + 2b10 + 6b8 - 2b6 + 2b5 + b3 - b2 - b1 + 1) q^31 + b11 * q^32 + (3b11 - 3b10 - 3b8 + 3b7 - 4b5 + b4 + b3 - b2 + 3b1 - 4) * q^33 + (2b11 + b9 + b8 - 2b5 - 2b3 - 2b2 - 2b1 - 2) q^34 + (-b11 + 2b10 - b9 + 2b8 - b6 + b5 + 2b4 - 2b1 + 1) * q^35 + (-b11 + b8 - b7 + b5 - b1) * q^36 + (-3b11 - 2b10 - 3b9 - 3b7 + b6 + b2 + 3) * q^37 + (b11 - 2b10 + b9 - b7 + 3b6 + b5 - b4 + b3 + 3b2 + b1 + 1) q^38 + (-b11 - b10 - b8 - b7 - b5 - 2b4 + b3 + 2b2 + 3b1 - 1) q^39 - b5 * q^40 + (-2b11 + 3b10 - 2b9 + 3b8 - 2b7 - 3b6 + 3b4 + 2b3 - 6b2 + 2) q^41 + (-b9 - 4b8 + b7 + b6 - 2b5 - b4 - b3 + 4b2 + b1 + 1) q^42 + (5b10 + 2b9 + 5b8 - b7 - 3b6 - 2b5 + 5b4 - 5b1) q^43 + (b10 + b8 - 3b7 + b6 + b4 - b1) q^44 + (b9 + b8 + b7 - b3 - b2 - 1) * q^45 + (3b10 + 3b8 - 3b6 + 3b4 - b2 + 2) * q^46 + (-3b7 + 2b6 + 2b5 - 2b4 + 3b3 + 2b1) * q^47 + (b11 + b7 - b5 + b4 - b2 - 1) * q^48 + (b11 + 3b10 + b9 + 3b7 + 2b6 + 2b2 - 3) * q^49 + (-b11 - b9 - b7 + b5 + b3 + 1) * q^50 + (-2b11 + 5b10 + b9 + 5b8 + 2b7 - 4b6 - 2b5 + 4b4 - b3 - 5b2 - 5b1) q^51 + (b9 + b7 + 2b6 - b5) q^52 + (-5b11 + 3b10 + 3b9 + 2b8 - 3b6 + 5b5 - 2b2 - 2b1) * q^53 + (b10 + b8 - 2b5 + 3b4 - 3b2 - 3b1 - 2) * q^54 + (-3b11 + 2b10 + 2b8 - b6 + b4 - 2b2 - 2b1) q^55 + (-b11 - b9 + b8 + b5 - 2b2 - 2b1) * q^56 + (-4b11 - b10 - 4b9 - 3b8 - 4b7 + 3b6 - b4 + 4b3 + 2b2 - 2) q^57 + (-2b11 + b10 - b9 + 2b8 - b6 + 2b5 + 2b4 + 2b2 - 3b1 + 3) * q^58 + (-b11 - 2b9 - 2b8 - 2b7 + 2b6 + b3 + b2 + 8) * q^59 + (b10 - b9 - b6 + b3 + 1) * q^60 + (-6b11 + 4b10 + 2b9 + 4b8 - 3b7 - 7b6 + 3b5 + 7b4 - 2b3 - 4b2 - 4b1) q^61 + (b11 - b10 - b8 + b7 + b5 - 2b4 + b3 + 2b2 + 6b1 + 1) q^62 + (2b11 - 2b9 - 3b8 - 2b5 + 2b3 + 2) q^63 - b7 * q^64 + (2b10 - b9 + 2b8 - b7 - 2b6 + b5 + 2b4 + b3 - 2b2 - 2b1) * q^65 + (-5b11 + 4b10 - 5b9 - 4b7 - b6 + b5 + 4b4 + b3 - b2 - 4b1 + 4) * q^66 + (5b11 - 4b10 + 5b9 + 2b7 + 3b6 - 2b5 - 2b4 - 2b3 + 3b2 + 2b1 - 2) * q^67 + (-2b10 - 2b8 - 2b5 - b3 + b1 - 2) q^68 + (-5b9 + b8 - 4b7 + 2b6 + 6b5 + 3b4 + 4b3 - b2 + 2b1 + 5) q^69 + (b11 + b9 + 2b8 + b7 - 2b6 - b3 - b2) * q^70 + (-b9 - 3b8 + 4b7 - 2b5 - 2b4 - 4b3 + 3b2 + 1) * q^71 + (b11 - b10 + b7 + b1 - 1) * q^72 + (-4b11 + b10 + 2b9 + 4b8 - 6b7 + 3b6 - 2b5 + 4b4 - b1 + 4) q^73 + (3b9 + b8 + 3b7 + 2b4 - 3b3 - b2 - 3) * q^74 + (-b10 - b9 - b8 - b7 + b6 - b4 + b2 + 1) * q^75 + (2b8 - 2b7 + b6 + b5 + b4 + 2b3 - 2b2 + b1) * q^76 + (8b11 - 3b10 - 3b8 + 8b7 - 3b5 - 2b3 - 2b1 - 3) q^77 + (-2b11 + b10 - 2b9 + 2b6 + b5 - b4 + b3 + 2b2 + b1) * q^78 + (4b11 - 2b10 + 4b9 + 7b6 - b5 - 3b4 - b3 + 7b2 + 3b1) q^79 - b11 * q^80 + (-b9 + b8 + 7b7 - b6 + b5 + b4) q^81 + (-2b11 + 3b10 - 3b8 - 3b6 + 2b5) q^82 + (5b11 - b10 - b8 + 5b7 - 7b5 - 2b4 - 8b3 + 2b2 - b1 - 7) * q^83 + (-b11 + 2b9 + 3b8 + b7 + b6 - b5 - b4 - 2b3 - 3b1) * q^84 + (-2b11 - b9 - b8 + 2b5 + 2b3 + 2b2 + 2b1 + 2) q^85 + (-2b11 + 5b8 - 5b6 + 2b3 - 2b2 - 1) q^86 + (-3b11 - b10 - b9 - 2b8 - 8b7 + 6b6 + 2b5 - b4 + 6b3 + 6b2 + 5) q^87 + (b8 - b6 - 2b2 - 3) q^88 + (2b11 - 8b9 - 10b8 - 2b5 - 3b3 + 3b2 + 3b1 - 3) q^89 + (b11 - b8 + b7 - b5 + b1) * q^90 + (4b11 + 4b10 + 4b8 + 4b7 - 3b5 - 3b4 - 2b3 + 3b2 - b1 - 3) * q^91 + (3b10 + 2b9 + 2b8 - 3b6) * q^92 + (5b11 - 7b10 - 2b9 - 2b8 + b7 + b6 + 2b5 - 2b4 + 7b1 - 5) q^93 + (-b11 - 3b9 - 2b8 - 3b7 + 2b6 + 3b5 - 2b4 + 3b3 + 2b1) * q^94 + (-b11 + 2b10 - b9 + b7 - 3b6 - b5 + b4 - b3 - 3b2 - b1 - 1) q^95 + (-b11 + b10 - b9 - b7 - b6 + b4 - b2 - b1 + 1) * q^96 + (-7b11 - 2b10 - 2b8 - 7b7 - 2b5 + 3b4 + 4b3 - 3b2 - 5b1 - 2) q^97 + (-3b9 + 2b8 - b7 - 3b4 + b3 - 2b2 + 3) * q^98 + (-2b11 + 2b10 - 6b9 - b8 - 6b7 + b6 + 2b4 + 2b3 + 2b2 + 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{5} - 6 q^{6} + 11 q^{7} + 2 q^{8} - q^{9} + 2 q^{10} - 2 q^{11} + 6 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 2 q^{16} - 6 q^{17} + q^{18} + 8 q^{19} - 2 q^{20}+ \cdots + 26 q^{99}+O(q^{100}) $ 12 * q + 2 * q^2 - q^3 - 2 * q^4 - 2 * q^5 - 6 * q^6 + 11 * q^7 + 2 * q^8 - q^9 + 2 * q^10 - 2 * q^11 + 6 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 2 * q^16 - 6 * q^17 + q^18 + 8 * q^19 - 2 * q^20 - q^21 + 2 * q^22 - 4 * q^23 - 6 * q^24 - 2 * q^25 + 6 * q^26 - 4 * q^27 - 10 * q^28 - 5 * q^29 - 6 * q^30 - 3 * q^31 + 2 * q^32 - 16 * q^33 - q^34 - 3 * q^35 - 8 * q^36 + 15 * q^37 - q^38 - 21 * q^39 + 2 * q^40 + 32 * q^41 + q^42 - 17 * q^43 - 9 * q^44 - q^45 + 18 * q^46 - 10 * q^47 - q^48 - 39 * q^49 + 2 * q^50 + 15 * q^51 + 8 * q^52 - 12 * q^53 - 10 * q^54 - 2 * q^55 + 3 * q^56 - 60 * q^57 + 5 * q^58 + 82 * q^59 + 6 * q^60 - 18 * q^61 + 10 * q^62 + 27 * q^63 - 2 * q^64 - 6 * q^65 + 9 * q^66 + q^67 - 13 * q^68 + 28 * q^69 + 10 * q^70 + 17 * q^71 - 6 * q^72 + 29 * q^73 - 15 * q^74 + 6 * q^75 + q^76 + 10 * q^77 - 21 * q^78 - 7 * q^79 - 2 * q^80 + 7 * q^81 - 11 * q^82 - 43 * q^83 + 6 * q^84 + q^85 - 18 * q^86 - 6 * q^87 - 26 * q^88 - 43 * q^89 + 8 * q^90 - 34 * q^91 - 4 * q^92 - 37 * q^93 - 18 * q^94 + q^95 + 8 * q^96 - 42 * q^97 + 39 * q^98 + 26 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 $ x^12 - x^11 + 2*x^10 - 3*x^9 + 5*x^8 - 8*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{7} + 8 ) / 13 $ (v^7 + 8) / 13
$\beta_{3}$	$=$	$ ( \nu^{8} + 21\nu ) / 13 $ (v^8 + 21*v) / 13
$\beta_{4}$	$=$	$ ( -\nu^{9} - 21\nu^{2} ) / 13 $ (-v^9 - 21*v^2) / 13
$\beta_{5}$	$=$	$ ( \nu^{9} + 34\nu^{2} ) / 13 $ (v^9 + 34*v^2) / 13
$\beta_{6}$	$=$	$ ( \nu^{10} + 34\nu^{3} ) / 13 $ (v^10 + 34*v^3) / 13
$\beta_{7}$	$=$	$ ( -2\nu^{10} - 55\nu^{3} ) / 13 $ (-2v^10 - 55v^3) / 13
$\beta_{8}$	$=$	$ ( -2\nu^{11} - 55\nu^{4} ) / 13 $ (-2v^11 - 55v^4) / 13
$\beta_{9}$	$=$	$ ( -3\nu^{11} - 89\nu^{4} ) / 13 $ (-3v^11 - 89v^4) / 13
$\beta_{10}$	$=$	$ ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 $ (-3v^11 + 6v^10 - 9v^9 + 15v^8 - 24v^7 + 39v^6 - 65v^5 + 15v^4 + 9v^3 + 6v^2 + 3*v + 3) / 13
$\beta_{11}$	$=$	$ ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 $ (8v^11 - 8v^10 + 16v^9 - 24v^8 + 40v^7 - 65v^6 + 104v^5 + 64v^4 + 40v^3 + 24v^2 + 16*v + 8) / 13

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} + \beta_{4} $ b5 + b4
$\nu^{3}$	$=$	$ \beta_{7} + 2\beta_{6} $ b7 + 2*b6
$\nu^{4}$	$=$	$ -2\beta_{9} + 3\beta_{8} $ -2b9 + 3b8
$\nu^{5}$	$=$	$ -3\beta_{11} - 5\beta_{10} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 $ -3b11 - 5b10 - 3b9 - 3b7 + 3b5 + 3b3 + 3
$\nu^{6}$	$=$	$ -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 $ -5b11 - 8b10 - 8b8 + 8b6 - 8b4 + 8b2 + 8*b1
$\nu^{7}$	$=$	$ 13\beta_{2} - 8 $ 13*b2 - 8
$\nu^{8}$	$=$	$ 13\beta_{3} - 21\beta_1 $ 13b3 - 21b1
$\nu^{9}$	$=$	$ -21\beta_{5} - 34\beta_{4} $ -21b5 - 34b4
$\nu^{10}$	$=$	$ -34\beta_{7} - 55\beta_{6} $ -34b7 - 55b6
$\nu^{11}$	$=$	$ 55\beta_{9} - 89\beta_{8} $ 55b9 - 89b8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/290\mathbb{Z}\right)^\times$.

$n$	$31$	$117$
$\chi(n)$	$\beta_{3}$	$1$

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} $

81.1

−0.137526	−	0.602539i
0.360046	+	1.57747i
−0.137526	+	0.602539i
0.360046	−	1.57747i
−0.556829	−	0.268155i
1.45780	+	0.702039i
0.385338	+	0.483198i
−1.00883	−	1.26503i
−0.556829	+	0.268155i
1.45780	−	0.702039i
0.385338	−	0.483198i
−1.00883	+	1.26503i

−0.623490

0.781831i

−0.561205

2.45880i

−0.222521

−

0.974928i

0.623490

−

0.781831i

−1.57246

−

1.97180i

−0.658956

2.88708i

0.900969

0.433884i

−3.02783

−

1.45813i

0.222521

0.974928i

81.2

−0.623490

0.781831i

−0.339764

1.48860i

−0.222521

−

0.974928i

0.623490

−

0.781831i

−0.951998

−

1.19377i

1.07954

−

4.72977i

0.900969

0.433884i

0.602403

0.290102i

0.222521

0.974928i

111.1

−0.623490

−

0.781831i

−0.561205

−

2.45880i

−0.222521

0.974928i

0.623490

0.781831i

−1.57246

1.97180i

−0.658956

−

2.88708i

0.900969

−

0.433884i

−3.02783

1.45813i

0.222521

−

0.974928i

111.2

−0.623490

−

0.781831i

−0.339764

−

1.48860i

−0.222521

0.974928i

0.623490

0.781831i

−0.951998

1.19377i

1.07954

4.72977i

0.900969

−

0.433884i

0.602403

−

0.290102i

0.222521

−

0.974928i

141.1

0.222521

−

0.974928i

−1.50337

0.723986i

−0.900969

−

0.433884i

−0.222521

0.974928i

0.371302

1.62678i

1.96325

−

0.945453i

−0.623490

0.781831i

−0.134498

0.168655i

0.900969

0.433884i

141.2

0.222521

−

0.974928i

2.12686

−

1.02424i

−0.900969

−

0.433884i

−0.222521

0.974928i

−0.525291

−

2.30145i

2.18469

−

1.05209i

−0.623490

0.781831i

1.60400

−

2.01135i

0.900969

0.433884i

161.1

0.900969

0.433884i

−0.980508

1.22952i

0.623490

0.781831i

−0.900969

−

0.433884i

−1.41688

0.682331i

−1.34938

1.69207i

0.222521

0.974928i

0.117243

0.513677i

−0.623490

−

0.781831i

161.2

0.900969

0.433884i

0.757988

−

0.950486i

0.623490

0.781831i

−0.900969

−

0.433884i

1.09532

−

0.527480i

2.28085

−

2.86010i

0.222521

0.974928i

0.338684

1.48387i

−0.623490

−

0.781831i

181.1

0.222521

0.974928i

−1.50337

−

0.723986i

−0.900969

0.433884i

−0.222521

−

0.974928i

0.371302

−

1.62678i

1.96325

0.945453i

−0.623490

−

0.781831i

−0.134498

−

0.168655i

0.900969

−

0.433884i

181.2

0.222521

0.974928i

2.12686

1.02424i

−0.900969

0.433884i

−0.222521

−

0.974928i

−0.525291

2.30145i

2.18469

1.05209i

−0.623490

−

0.781831i

1.60400

2.01135i

0.900969

−

0.433884i

281.1

0.900969

−

0.433884i

−0.980508

−

1.22952i

0.623490

−

0.781831i

−0.900969

0.433884i

−1.41688

−

0.682331i

−1.34938

−

1.69207i

0.222521

−

0.974928i

0.117243

−

0.513677i

−0.623490

0.781831i

281.2

0.900969

−

0.433884i

0.757988

0.950486i

0.623490

−

0.781831i

−0.900969

0.433884i

1.09532

0.527480i

2.28085

2.86010i

0.222521

−

0.974928i

0.338684

−

1.48387i

−0.623490

0.781831i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	290.2.k.b	✓	12
29.d	even	7	1	inner	290.2.k.b	✓	12
29.d	even	7	1		8410.2.a.bh		6
29.e	even	14	1		8410.2.a.bi		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
290.2.k.b	✓	12	1.a	even	1	1	trivial
290.2.k.b	✓	12	29.d	even	7	1	inner
8410.2.a.bh		6	29.d	even	7	1
8410.2.a.bi		6	29.e	even	14	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} + T_{3}^{11} + 4 T_{3}^{10} - 7 T_{3}^{9} - 9 T_{3}^{8} + 12 T_{3}^{7} + 132 T_{3}^{6} + \cdots + 841 $ T3^12 + T3^11 + 4*T3^10 - 7*T3^9 - 9*T3^8 + 12*T3^7 + 132*T3^6 + 237*T3^5 + 360*T3^4 + 273*T3^3 + 492*T3^2 + 464*T3 + 841 acting on $S_{2}^{\mathrm{new}}(290, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{12} + T^{11} + \cdots + 841 $ T^12 + T^11 + 4T^10 - 7T^9 - 9T^8 + 12T^7 + 132T^6 + 237T^5 + 360T^4 + 273T^3 + 492T^2 + 464T + 841
$5$	$ (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} $ (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$7$	$ T^{12} - 11 T^{11} + \cdots + 361201 $ T^12 - 11T^11 + 87T^10 - 457T^9 + 1845T^8 - 5884T^7 + 16344T^6 - 39609T^5 + 79006T^4 - 140663T^3 + 282516T^2 - 460967*T + 361201
$11$	$ T^{12} + 2 T^{11} + \cdots + 32761 $ T^12 + 2T^11 + 2T^10 + 35T^9 + 206T^8 - 183T^7 + 3898T^6 - 14870T^5 + 34179T^4 - 80801T^3 + 183488T^2 - 99007*T + 32761
$13$	$ T^{12} - 8 T^{11} + \cdots + 5041 $ T^12 - 8T^11 + 53T^10 - 196T^9 + 369T^8 - 502T^7 + 2799T^6 - 4822T^5 + 7591T^4 + 4662T^3 + 4111T^2 + 7810*T + 5041
$17$	$ (T^{6} + 3 T^{5} + \cdots + 281)^{2} $ (T^6 + 3T^5 - 34T^4 - 17T^3 + 382T^2 - 645*T + 281)^2
$19$	$ T^{12} - 8 T^{11} + \cdots + 1413721 $ T^12 - 8T^11 + 5T^10 + 194T^9 - 461T^8 - 3740T^7 + 42573T^6 - 174712T^5 + 540696T^4 - 847950T^3 + 1350565T^2 - 1242505*T + 1413721
$23$	$ T^{12} + 4 T^{11} + \cdots + 5041 $ T^12 + 4T^11 + 8T^10 + 210T^9 + 251T^8 - 13066T^7 + 34852T^6 + 252640T^5 + 826769T^4 + 833028T^3 + 503987T^2 + 30104*T + 5041
$29$	$ T^{12} + \cdots + 594823321 $ T^12 + 5T^11 + 101T^10 + 493T^9 + 4618T^8 + 22324T^7 + 146601T^6 + 647396T^5 + 3883738T^4 + 12023777T^3 + 71435381T^2 + 102555745*T + 594823321
$31$	$ T^{12} + \cdots + 147646801 $ T^12 + 3T^11 - 18T^10 + 61T^9 + 3511T^8 + 5484T^7 - 5797T^6 - 540662T^5 + 4602637T^4 + 18906561T^3 + 217703790T^2 + 90366987*T + 147646801
$37$	$ T^{12} - 15 T^{11} + \cdots + 1681 $ T^12 - 15T^11 + 75T^10 - 310T^9 + 4726T^8 - 34974T^7 + 111677T^6 - 305150T^5 + 1320979T^4 - 602642T^3 + 180585T^2 - 13284*T + 1681
$41$	$ (T^{6} - 16 T^{5} + \cdots + 349)^{2} $ (T^6 - 16T^5 - 21T^4 + 996T^3 - 1813T^2 + 475*T + 349)^2
$43$	$ T^{12} + \cdots + 10316261761 $ T^12 + 17T^11 + 143T^10 + 124T^9 + 1516T^8 + 43004T^7 + 291976T^6 - 1092675T^5 + 11075978T^4 - 79542216T^3 + 651931696T^2 - 68660644*T + 10316261761
$47$	$ T^{12} + 10 T^{11} + \cdots + 5041 $ T^12 + 10T^11 + 120T^10 + 560T^9 + 1826T^8 + 1474T^7 + 1567T^6 + 1950T^5 - 1296T^4 - 3948T^3 + 4005T^2 + 6674*T + 5041
$53$	$ T^{12} + \cdots + 594823321 $ T^12 + 12T^11 + 141T^10 - 822T^9 + 2866T^8 - 51146T^7 + 475509T^6 - 1294676T^5 + 17324600T^4 - 60533498T^3 + 270181342T^2 - 635845619*T + 594823321
$59$	$ (T^{6} - 41 T^{5} + \cdots + 58841)^{2} $ (T^6 - 41T^5 + 672T^4 - 5651T^3 + 25837T^2 - 61178*T + 58841)^2
$61$	$ T^{12} + \cdots + 382132912561 $ T^12 + 18T^11 + 486T^10 + 3544T^9 + 49200T^8 - 83513T^7 + 1987061T^6 - 10689690T^5 + 220287141T^4 + 212652648T^3 - 7403458745T^2 + 14502862909*T + 382132912561
$67$	$ T^{12} - T^{11} + \cdots + 19321 $ T^12 - T^11 + 27T^10 + 906T^9 + 7804T^8 + 35128T^7 + 104573T^6 + 234112T^5 + 465349T^4 + 607148T^3 + 353799T^2 - 7228T + 19321
$71$	$ T^{12} + \cdots + 594823321 $ T^12 - 17T^11 + 260T^10 - 1799T^9 + 12319T^8 - 14620T^7 + 28993T^6 + 708172T^5 + 738191T^4 - 15753815T^3 + 7047580T^2 + 166211035*T + 594823321
$73$	$ T^{12} - 29 T^{11} + \cdots + 96020401 $ T^12 - 29T^11 + 664T^10 - 9573T^9 + 135085T^8 - 1288096T^7 + 9189369T^6 - 47562840T^5 + 163231771T^4 - 330937779T^3 + 407448620T^2 - 288943113*T + 96020401
$79$	$ T^{12} + \cdots + 54596528281 $ T^12 + 7T^11 + 80T^10 + 1869T^9 + 29514T^8 + 228277T^7 + 2436314T^6 + 7026236T^5 + 77230409T^4 - 218643775T^3 + 1247286871T^2 - 14149688063*T + 54596528281
$83$	$ T^{12} + \cdots + 15136626961 $ T^12 + 43T^11 + 1012T^10 + 13405T^9 + 110311T^8 + 434838T^7 + 429813T^6 - 4520090T^5 - 14797171T^4 + 109069961T^3 + 1456316738T^2 + 4156848397*T + 15136626961
$89$	$ T^{12} + \cdots + 5855378804521 $ T^12 + 43T^11 + 998T^10 + 17451T^9 + 311351T^8 + 5423206T^7 + 84955471T^6 + 1031025820T^5 + 10102984083T^4 + 71391189821T^3 + 423394922876T^2 + 2065606903859*T + 5855378804521
$97$	$ T^{12} + \cdots + 5256294460921 $ T^12 + 42T^11 + 728T^10 + 10815T^9 + 282436T^8 + 5727855T^7 + 74183991T^6 + 741528956T^5 + 6719261269T^4 + 45797001222T^3 + 262703057218T^2 + 818399733865*T + 5256294460921
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 290 = 2 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	290.k (of order \(7\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{9} q^{2} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{3} + \beta_{3} q^{4} - \beta_{9} q^{5} + (\beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{6} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots + 6) q^{99}+O(q^{100}) \) q + b9 * q^2 + (-b11 + b10 - b7 + b5 - b2) * q^3 + b3 * q^4 - b9 * q^5 + (b11 - b8 + b7 - b4 - 1) * q^6 + (b11 + b10 + b9 + b8 + b6 - b4 - b3 - b2 - b1) * q^7 + (-b11 - b9 - b7 + b5 + b3 + 1) * q^8 + (b11 + b9 + b7 - b6 - b2 - 1) * q^9 - b3 * q^10 + (b8 - b6 + 3b5 - b4 - b2 - b1) q^11 + (-b10 - b9 - b8 - b7 + b6 - b4 + b2 + 1) * q^12 + (b7 - b5 - 2b4 - b3) q^13 + (b11 - 2b10 + b9 - 2b8 + b6 - b5 - 2b4 + 2b1 - 1) * q^14 + (-b11 + b8 - b7 + b4 + 1) * q^15 + b5 * q^16 + (2b11 - 2b8 + 2b6 - 2b3 + b2 - 1) * q^17 + (-b9 - b8 - b7 + b3 + b2 + 1) * q^18 + (-2b10 - 2b8 + 2b5 - 3b4 + b3 + 3b2 + b1 + 2) q^19 + (b11 + b9 + b7 - b5 - b3 - 1) * q^20 + (b11 + b9 + 3b7 - 3b6 - 2b5 - b4 - 2b3 - 3b2 + b1 - 3) q^21 + (3b11 - 2b10 - 2b8 + b6 - b4 + 2b2 + 2b1) q^22 + (3b8 + 2b7 - 2b6 + 3b4) * q^23 + (-b10 + b9 + b6 - b3 - 1) * q^24 + b3 * q^25 + (-2b10 + b9 - 2b8 + b7 + 2b6 - b5 - 2b4 - b3 + 2b2 + 2b1) * q^26 + (2b11 - 3b10 + 2b9 - 3b8 + 3b6 - 2b5 - 2b3 + b2 + b1 - 2) q^27 + (-b11 - b9 - 2b8 - b7 + 2b6 + b3 + b2) * q^28 + (-2b11 - b10 - 2b9 - 2b8 + b7 - 3b6 + b4 + 2b3 + b2 + 1) q^29 + (b10 + b9 + b8 + b7 - b6 + b4 - b2 - 1) * q^30 + (-2b11 + 2b10 + 6b8 - 2b6 + 2b5 + b3 - b2 - b1 + 1) q^31 + b11 * q^32 + (3b11 - 3b10 - 3b8 + 3b7 - 4b5 + b4 + b3 - b2 + 3b1 - 4) * q^33 + (2b11 + b9 + b8 - 2b5 - 2b3 - 2b2 - 2b1 - 2) q^34 + (-b11 + 2b10 - b9 + 2b8 - b6 + b5 + 2b4 - 2b1 + 1) * q^35 + (-b11 + b8 - b7 + b5 - b1) * q^36 + (-3b11 - 2b10 - 3b9 - 3b7 + b6 + b2 + 3) * q^37 + (b11 - 2b10 + b9 - b7 + 3b6 + b5 - b4 + b3 + 3b2 + b1 + 1) q^38 + (-b11 - b10 - b8 - b7 - b5 - 2b4 + b3 + 2b2 + 3b1 - 1) q^39 - b5 * q^40 + (-2b11 + 3b10 - 2b9 + 3b8 - 2b7 - 3b6 + 3b4 + 2b3 - 6b2 + 2) q^41 + (-b9 - 4b8 + b7 + b6 - 2b5 - b4 - b3 + 4b2 + b1 + 1) q^42 + (5b10 + 2b9 + 5b8 - b7 - 3b6 - 2b5 + 5b4 - 5b1) q^43 + (b10 + b8 - 3b7 + b6 + b4 - b1) q^44 + (b9 + b8 + b7 - b3 - b2 - 1) * q^45 + (3b10 + 3b8 - 3b6 + 3b4 - b2 + 2) * q^46 + (-3b7 + 2b6 + 2b5 - 2b4 + 3b3 + 2b1) * q^47 + (b11 + b7 - b5 + b4 - b2 - 1) * q^48 + (b11 + 3b10 + b9 + 3b7 + 2b6 + 2b2 - 3) * q^49 + (-b11 - b9 - b7 + b5 + b3 + 1) * q^50 + (-2b11 + 5b10 + b9 + 5b8 + 2b7 - 4b6 - 2b5 + 4b4 - b3 - 5b2 - 5b1) q^51 + (b9 + b7 + 2b6 - b5) q^52 + (-5b11 + 3b10 + 3b9 + 2b8 - 3b6 + 5b5 - 2b2 - 2b1) * q^53 + (b10 + b8 - 2b5 + 3b4 - 3b2 - 3b1 - 2) * q^54 + (-3b11 + 2b10 + 2b8 - b6 + b4 - 2b2 - 2b1) q^55 + (-b11 - b9 + b8 + b5 - 2b2 - 2b1) * q^56 + (-4b11 - b10 - 4b9 - 3b8 - 4b7 + 3b6 - b4 + 4b3 + 2b2 - 2) q^57 + (-2b11 + b10 - b9 + 2b8 - b6 + 2b5 + 2b4 + 2b2 - 3b1 + 3) * q^58 + (-b11 - 2b9 - 2b8 - 2b7 + 2b6 + b3 + b2 + 8) * q^59 + (b10 - b9 - b6 + b3 + 1) * q^60 + (-6b11 + 4b10 + 2b9 + 4b8 - 3b7 - 7b6 + 3b5 + 7b4 - 2b3 - 4b2 - 4b1) q^61 + (b11 - b10 - b8 + b7 + b5 - 2b4 + b3 + 2b2 + 6b1 + 1) q^62 + (2b11 - 2b9 - 3b8 - 2b5 + 2b3 + 2) q^63 - b7 * q^64 + (2b10 - b9 + 2b8 - b7 - 2b6 + b5 + 2b4 + b3 - 2b2 - 2b1) * q^65 + (-5b11 + 4b10 - 5b9 - 4b7 - b6 + b5 + 4b4 + b3 - b2 - 4b1 + 4) * q^66 + (5b11 - 4b10 + 5b9 + 2b7 + 3b6 - 2b5 - 2b4 - 2b3 + 3b2 + 2b1 - 2) * q^67 + (-2b10 - 2b8 - 2b5 - b3 + b1 - 2) q^68 + (-5b9 + b8 - 4b7 + 2b6 + 6b5 + 3b4 + 4b3 - b2 + 2b1 + 5) q^69 + (b11 + b9 + 2b8 + b7 - 2b6 - b3 - b2) * q^70 + (-b9 - 3b8 + 4b7 - 2b5 - 2b4 - 4b3 + 3b2 + 1) * q^71 + (b11 - b10 + b7 + b1 - 1) * q^72 + (-4b11 + b10 + 2b9 + 4b8 - 6b7 + 3b6 - 2b5 + 4b4 - b1 + 4) q^73 + (3b9 + b8 + 3b7 + 2b4 - 3b3 - b2 - 3) * q^74 + (-b10 - b9 - b8 - b7 + b6 - b4 + b2 + 1) * q^75 + (2b8 - 2b7 + b6 + b5 + b4 + 2b3 - 2b2 + b1) * q^76 + (8b11 - 3b10 - 3b8 + 8b7 - 3b5 - 2b3 - 2b1 - 3) q^77 + (-2b11 + b10 - 2b9 + 2b6 + b5 - b4 + b3 + 2b2 + b1) * q^78 + (4b11 - 2b10 + 4b9 + 7b6 - b5 - 3b4 - b3 + 7b2 + 3b1) q^79 - b11 * q^80 + (-b9 + b8 + 7b7 - b6 + b5 + b4) q^81 + (-2b11 + 3b10 - 3b8 - 3b6 + 2b5) q^82 + (5b11 - b10 - b8 + 5b7 - 7b5 - 2b4 - 8b3 + 2b2 - b1 - 7) * q^83 + (-b11 + 2b9 + 3b8 + b7 + b6 - b5 - b4 - 2b3 - 3b1) * q^84 + (-2b11 - b9 - b8 + 2b5 + 2b3 + 2b2 + 2b1 + 2) q^85 + (-2b11 + 5b8 - 5b6 + 2b3 - 2b2 - 1) q^86 + (-3b11 - b10 - b9 - 2b8 - 8b7 + 6b6 + 2b5 - b4 + 6b3 + 6b2 + 5) q^87 + (b8 - b6 - 2b2 - 3) q^88 + (2b11 - 8b9 - 10b8 - 2b5 - 3b3 + 3b2 + 3b1 - 3) q^89 + (b11 - b8 + b7 - b5 + b1) * q^90 + (4b11 + 4b10 + 4b8 + 4b7 - 3b5 - 3b4 - 2b3 + 3b2 - b1 - 3) * q^91 + (3b10 + 2b9 + 2b8 - 3b6) * q^92 + (5b11 - 7b10 - 2b9 - 2b8 + b7 + b6 + 2b5 - 2b4 + 7b1 - 5) q^93 + (-b11 - 3b9 - 2b8 - 3b7 + 2b6 + 3b5 - 2b4 + 3b3 + 2b1) * q^94 + (-b11 + 2b10 - b9 + b7 - 3b6 - b5 + b4 - b3 - 3b2 - b1 - 1) q^95 + (-b11 + b10 - b9 - b7 - b6 + b4 - b2 - b1 + 1) * q^96 + (-7b11 - 2b10 - 2b8 - 7b7 - 2b5 + 3b4 + 4b3 - 3b2 - 5b1 - 2) q^97 + (-3b9 + 2b8 - b7 - 3b4 + b3 - 2b2 + 3) * q^98 + (-2b11 + 2b10 - 6b9 - b8 - 6b7 + b6 + 2b4 + 2b3 + 2b2 + 6) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{5} - 6 q^{6} + 11 q^{7} + 2 q^{8} - q^{9} + 2 q^{10} - 2 q^{11} + 6 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 2 q^{16} - 6 q^{17} + q^{18} + 8 q^{19} - 2 q^{20}+ \cdots + 26 q^{99}+O(q^{100}) \) 12 * q + 2 * q^2 - q^3 - 2 * q^4 - 2 * q^5 - 6 * q^6 + 11 * q^7 + 2 * q^8 - q^9 + 2 * q^10 - 2 * q^11 + 6 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 2 * q^16 - 6 * q^17 + q^18 + 8 * q^19 - 2 * q^20 - q^21 + 2 * q^22 - 4 * q^23 - 6 * q^24 - 2 * q^25 + 6 * q^26 - 4 * q^27 - 10 * q^28 - 5 * q^29 - 6 * q^30 - 3 * q^31 + 2 * q^32 - 16 * q^33 - q^34 - 3 * q^35 - 8 * q^36 + 15 * q^37 - q^38 - 21 * q^39 + 2 * q^40 + 32 * q^41 + q^42 - 17 * q^43 - 9 * q^44 - q^45 + 18 * q^46 - 10 * q^47 - q^48 - 39 * q^49 + 2 * q^50 + 15 * q^51 + 8 * q^52 - 12 * q^53 - 10 * q^54 - 2 * q^55 + 3 * q^56 - 60 * q^57 + 5 * q^58 + 82 * q^59 + 6 * q^60 - 18 * q^61 + 10 * q^62 + 27 * q^63 - 2 * q^64 - 6 * q^65 + 9 * q^66 + q^67 - 13 * q^68 + 28 * q^69 + 10 * q^70 + 17 * q^71 - 6 * q^72 + 29 * q^73 - 15 * q^74 + 6 * q^75 + q^76 + 10 * q^77 - 21 * q^78 - 7 * q^79 - 2 * q^80 + 7 * q^81 - 11 * q^82 - 43 * q^83 + 6 * q^84 + q^85 - 18 * q^86 - 6 * q^87 - 26 * q^88 - 43 * q^89 + 8 * q^90 - 34 * q^91 - 4 * q^92 - 37 * q^93 - 18 * q^94 + q^95 + 8 * q^96 - 42 * q^97 + 39 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	290.2.k.b

Level	$290$
Weight	$2$
Character orbit	290.k
Analytic conductor	$2.316$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.31566165862\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{7})\)
Coefficient field:	12.0.4413675765625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) x^12 - x^11 + 2x^10 - 3x^9 + 5x^8 - 8x^7 + 13x^6 + 8x^5 + 5x^4 + 3x^3 + 2*x^2 + x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} + 8 ) / 13 \) (v^7 + 8) / 13
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} + 21\nu ) / 13 \) (v^8 + 21*v) / 13
\(\beta_{4}\)	\(=\)	\( ( -\nu^{9} - 21\nu^{2} ) / 13 \) (-v^9 - 21*v^2) / 13
\(\beta_{5}\)	\(=\)	\( ( \nu^{9} + 34\nu^{2} ) / 13 \) (v^9 + 34*v^2) / 13
\(\beta_{6}\)	\(=\)	\( ( \nu^{10} + 34\nu^{3} ) / 13 \) (v^10 + 34*v^3) / 13
\(\beta_{7}\)	\(=\)	\( ( -2\nu^{10} - 55\nu^{3} ) / 13 \) (-2v^10 - 55v^3) / 13
\(\beta_{8}\)	\(=\)	\( ( -2\nu^{11} - 55\nu^{4} ) / 13 \) (-2v^11 - 55v^4) / 13
\(\beta_{9}\)	\(=\)	\( ( -3\nu^{11} - 89\nu^{4} ) / 13 \) (-3v^11 - 89v^4) / 13
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 \) (-3v^11 + 6v^10 - 9v^9 + 15v^8 - 24v^7 + 39v^6 - 65v^5 + 15v^4 + 9v^3 + 6v^2 + 3*v + 3) / 13
\(\beta_{11}\)	\(=\)	\( ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 \) (8v^11 - 8v^10 + 16v^9 - 24v^8 + 40v^7 - 65v^6 + 104v^5 + 64v^4 + 40v^3 + 24v^2 + 16*v + 8) / 13

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{5} + \beta_{4} \) b5 + b4
\(\nu^{3}\)	\(=\)	\( \beta_{7} + 2\beta_{6} \) b7 + 2*b6
\(\nu^{4}\)	\(=\)	\( -2\beta_{9} + 3\beta_{8} \) -2b9 + 3b8
\(\nu^{5}\)	\(=\)	\( -3\beta_{11} - 5\beta_{10} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 \) -3b11 - 5b10 - 3b9 - 3b7 + 3b5 + 3b3 + 3
\(\nu^{6}\)	\(=\)	\( -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 \) -5b11 - 8b10 - 8b8 + 8b6 - 8b4 + 8b2 + 8*b1
\(\nu^{7}\)	\(=\)	\( 13\beta_{2} - 8 \) 13*b2 - 8
\(\nu^{8}\)	\(=\)	\( 13\beta_{3} - 21\beta_1 \) 13b3 - 21b1
\(\nu^{9}\)	\(=\)	\( -21\beta_{5} - 34\beta_{4} \) -21b5 - 34b4
\(\nu^{10}\)	\(=\)	\( -34\beta_{7} - 55\beta_{6} \) -34b7 - 55b6
\(\nu^{11}\)	\(=\)	\( 55\beta_{9} - 89\beta_{8} \) 55b9 - 89b8

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 81.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{12} + T^{11} + \cdots + 841 \) T^12 + T^11 + 4T^10 - 7T^9 - 9T^8 + 12T^7 + 132T^6 + 237T^5 + 360T^4 + 273T^3 + 492T^2 + 464T + 841
$5$	\( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) (T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$7$	\( T^{12} - 11 T^{11} + \cdots + 361201 \) T^12 - 11T^11 + 87T^10 - 457T^9 + 1845T^8 - 5884T^7 + 16344T^6 - 39609T^5 + 79006T^4 - 140663T^3 + 282516T^2 - 460967*T + 361201
$11$	\( T^{12} + 2 T^{11} + \cdots + 32761 \) T^12 + 2T^11 + 2T^10 + 35T^9 + 206T^8 - 183T^7 + 3898T^6 - 14870T^5 + 34179T^4 - 80801T^3 + 183488T^2 - 99007*T + 32761
$13$	\( T^{12} - 8 T^{11} + \cdots + 5041 \) T^12 - 8T^11 + 53T^10 - 196T^9 + 369T^8 - 502T^7 + 2799T^6 - 4822T^5 + 7591T^4 + 4662T^3 + 4111T^2 + 7810*T + 5041
$17$	\( (T^{6} + 3 T^{5} + \cdots + 281)^{2} \) (T^6 + 3T^5 - 34T^4 - 17T^3 + 382T^2 - 645*T + 281)^2
$19$	\( T^{12} - 8 T^{11} + \cdots + 1413721 \) T^12 - 8T^11 + 5T^10 + 194T^9 - 461T^8 - 3740T^7 + 42573T^6 - 174712T^5 + 540696T^4 - 847950T^3 + 1350565T^2 - 1242505*T + 1413721
$23$	\( T^{12} + 4 T^{11} + \cdots + 5041 \) T^12 + 4T^11 + 8T^10 + 210T^9 + 251T^8 - 13066T^7 + 34852T^6 + 252640T^5 + 826769T^4 + 833028T^3 + 503987T^2 + 30104*T + 5041
$29$	\( T^{12} + \cdots + 594823321 \) T^12 + 5T^11 + 101T^10 + 493T^9 + 4618T^8 + 22324T^7 + 146601T^6 + 647396T^5 + 3883738T^4 + 12023777T^3 + 71435381T^2 + 102555745*T + 594823321
$31$	\( T^{12} + \cdots + 147646801 \) T^12 + 3T^11 - 18T^10 + 61T^9 + 3511T^8 + 5484T^7 - 5797T^6 - 540662T^5 + 4602637T^4 + 18906561T^3 + 217703790T^2 + 90366987*T + 147646801
$37$	\( T^{12} - 15 T^{11} + \cdots + 1681 \) T^12 - 15T^11 + 75T^10 - 310T^9 + 4726T^8 - 34974T^7 + 111677T^6 - 305150T^5 + 1320979T^4 - 602642T^3 + 180585T^2 - 13284*T + 1681
$41$	\( (T^{6} - 16 T^{5} + \cdots + 349)^{2} \) (T^6 - 16T^5 - 21T^4 + 996T^3 - 1813T^2 + 475*T + 349)^2
$43$	\( T^{12} + \cdots + 10316261761 \) T^12 + 17T^11 + 143T^10 + 124T^9 + 1516T^8 + 43004T^7 + 291976T^6 - 1092675T^5 + 11075978T^4 - 79542216T^3 + 651931696T^2 - 68660644*T + 10316261761
$47$	\( T^{12} + 10 T^{11} + \cdots + 5041 \) T^12 + 10T^11 + 120T^10 + 560T^9 + 1826T^8 + 1474T^7 + 1567T^6 + 1950T^5 - 1296T^4 - 3948T^3 + 4005T^2 + 6674*T + 5041
$53$	\( T^{12} + \cdots + 594823321 \) T^12 + 12T^11 + 141T^10 - 822T^9 + 2866T^8 - 51146T^7 + 475509T^6 - 1294676T^5 + 17324600T^4 - 60533498T^3 + 270181342T^2 - 635845619*T + 594823321
$59$	\( (T^{6} - 41 T^{5} + \cdots + 58841)^{2} \) (T^6 - 41T^5 + 672T^4 - 5651T^3 + 25837T^2 - 61178*T + 58841)^2
$61$	\( T^{12} + \cdots + 382132912561 \) T^12 + 18T^11 + 486T^10 + 3544T^9 + 49200T^8 - 83513T^7 + 1987061T^6 - 10689690T^5 + 220287141T^4 + 212652648T^3 - 7403458745T^2 + 14502862909*T + 382132912561
$67$	\( T^{12} - T^{11} + \cdots + 19321 \) T^12 - T^11 + 27T^10 + 906T^9 + 7804T^8 + 35128T^7 + 104573T^6 + 234112T^5 + 465349T^4 + 607148T^3 + 353799T^2 - 7228T + 19321
$71$	\( T^{12} + \cdots + 594823321 \) T^12 - 17T^11 + 260T^10 - 1799T^9 + 12319T^8 - 14620T^7 + 28993T^6 + 708172T^5 + 738191T^4 - 15753815T^3 + 7047580T^2 + 166211035*T + 594823321
$73$	\( T^{12} - 29 T^{11} + \cdots + 96020401 \) T^12 - 29T^11 + 664T^10 - 9573T^9 + 135085T^8 - 1288096T^7 + 9189369T^6 - 47562840T^5 + 163231771T^4 - 330937779T^3 + 407448620T^2 - 288943113*T + 96020401
$79$	\( T^{12} + \cdots + 54596528281 \) T^12 + 7T^11 + 80T^10 + 1869T^9 + 29514T^8 + 228277T^7 + 2436314T^6 + 7026236T^5 + 77230409T^4 - 218643775T^3 + 1247286871T^2 - 14149688063*T + 54596528281
$83$	\( T^{12} + \cdots + 15136626961 \) T^12 + 43T^11 + 1012T^10 + 13405T^9 + 110311T^8 + 434838T^7 + 429813T^6 - 4520090T^5 - 14797171T^4 + 109069961T^3 + 1456316738T^2 + 4156848397*T + 15136626961
$89$	\( T^{12} + \cdots + 5855378804521 \) T^12 + 43T^11 + 998T^10 + 17451T^9 + 311351T^8 + 5423206T^7 + 84955471T^6 + 1031025820T^5 + 10102984083T^4 + 71391189821T^3 + 423394922876T^2 + 2065606903859*T + 5855378804521
$97$	\( T^{12} + \cdots + 5256294460921 \) T^12 + 42T^11 + 728T^10 + 10815T^9 + 282436T^8 + 5727855T^7 + 74183991T^6 + 741528956T^5 + 6719261269T^4 + 45797001222T^3 + 262703057218T^2 + 818399733865*T + 5256294460921
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\(n\)	\(31\)	\(117\)
\(\chi(n)\)	\(\beta_{3}\)	\(1\)