LMFDB - Newform orbit 1890.2.l.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1890,2,Mod(361,1890)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1890, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0, 4]))

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

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

chi := DirichletCharacter("1890.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1890.l (of order $3$, degree $2$, not 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:	$15.0917259820$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	12.0.91830304992969.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{11} + x^{10} + 4x^{9} - 7x^{8} + x^{7} + 7x^{6} + 2x^{5} - 28x^{4} + 32x^{3} + 16x^{2} - 64x + 64 $ x^12 - 2x^11 + x^10 + 4x^9 - 7x^8 + x^7 + 7x^6 + 2x^5 - 28x^4 + 32x^3 + 16x^2 - 64*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 630)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} - q^{5} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + q^{8}+O(q^{10}) $ q + (-b3 - 1) * q^2 + b3 * q^4 - q^5 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^7 + q^8	$ q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} - q^{5} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 1) q^{98}+O(q^{100}) $ q + (-b3 - 1) * q^2 + b3 * q^4 - q^5 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^7 + q^8 + (b3 + 1) * q^10 + (-b4 - 1) * q^11 + (b8 + b5 - b3 + b1 - 1) * q^13 + (b11 + b10 - b9 - b8 - b5 + b3 + 1) * q^14 + (-b3 - 1) * q^16 + (-b10 + b9 - b7 - b1) * q^17 + (-b5 + b3 + b2) * q^19 - b3 * q^20 + (b7 + b3 + 1) * q^22 + (b10 - b8 + b4 - 1) * q^23 + q^25 + (-b5 + b3 + b2) * q^26 + (b6 - b3) * q^28 + (b9 + b7 + b5 - b4 + b2) * q^29 + (-b7 + b4 + b3 - 2b2) q^31 + b3 * q^32 + (-b9 + b7 - b4 - b2) * q^34 + (b11 + b10 - b9 - b8 + b6 - b5 + 1) * q^35 + (-2b9 + b2) q^37 + (-b8 - b2 - b1 + 1) * q^38 - q^40 + (2b10 - 2b9 - 4b8 + b7 - 4b5 + 4b3 - b1 + 4) q^41 + (2b11 + b10 - b8 - b7 + b6 + b4 - 5b3 - b2) * q^43 + (-b7 + b4 - b3) * q^44 + (-b10 + b9 + b8 - b7 + b5 + b3 + 1) * q^46 + (-b11 + b10 - b8 + b7 - 2b6 + b3 - b1 + 1) q^47 + (2b10 + b9 - 2b8 - b6 + b5 + b3) * q^49 + (-b3 - 1) * q^50 + (-b8 - b2 - b1 + 1) * q^52 + (2b10 - 2b9 + b8 + b5 + 2b3 - 3b1 + 2) * q^53 + (b4 + 1) * q^55 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^56 + (-b10 + b8 + b4 - b2 - b1) * q^58 + (-2b9 + b7 + b5 - b4 + 3b3 + 3b2) q^59 + (2b10 - 2b9 + b7 - 3b3 - 2b1 - 3) * q^61 + (-b4 + 2b2 + 2b1 + 1) * q^62 + q^64 + (-b8 - b5 + b3 - b1 + 1) * q^65 + (-2b11 - b10 + b9 + b8 + 2b7 - b6 - 2b4 + 2b3 + 4b2) q^67 + (b10 + b4 + b2 + b1) * q^68 + (-b11 - b10 + b9 + b8 + b5 - b3 - 1) * q^70 + (2b11 - 2b8 - 2b6 + b4 - 2) q^71 + (2b11 - 2b9 + 4b6 - 2b5 + 3b3 + b1 + 3) q^73 + (2b10 - b2 - b1) q^74 + (b8 + b5 - b3 + b1 - 1) * q^76 + (2b11 + 3b10 - 2b9 - b8 - b7 + b6 - b5 + 2b4 + 2b3 - b2 - 3b1 + 2) * q^77 + (-b10 + b9 - b7 + b3 - 4b1 + 1) q^79 + (b3 + 1) * q^80 + (2b9 - b7 + 4b5 + b4 - 4b3 - b2) q^82 + (-2b11 - b10 + 2b9 + b8 - b6 + 4b5 + b2) q^83 + (b10 - b9 + b7 + b1) * q^85 + (-b11 - 2b10 + 2b8 + b6 - b4 + b2 + b1 - 5) * q^86 + (-b4 - 1) * q^88 + (3b5 + 2b2) * q^89 + (-b11 + b9 + b8 - b5 - b4 + 3b3 + 3b2 + 2b1) q^91 + (-b9 + b7 - b5 - b4 - b3) * q^92 + (2b11 + b10 - b8 - b7 + b6 + b4 - b3 - b2) q^94 + (b5 - b3 - b2) * q^95 + (2b11 + b10 - b8 + b6 + 5b3) * q^97 + (b11 - 2b10 + b9 + 2b8 + b6 + b5 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 8 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 8 * q^7 + 12 * q^8	$ 12 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 8 q^{7} + 12 q^{8} + 6 q^{10} - 14 q^{11} - 2 q^{13} + 4 q^{14} - 6 q^{16} - q^{17} - 2 q^{19} + 6 q^{20} + 7 q^{22} - 18 q^{23} + 12 q^{25} - 2 q^{26} + 4 q^{28} - 3 q^{29} - 9 q^{31} - 6 q^{32} - q^{34} + 8 q^{35} + 6 q^{37} + 4 q^{38} - 12 q^{40} + 11 q^{41} + 23 q^{43} + 7 q^{44} + 9 q^{46} - q^{47} - 24 q^{49} - 6 q^{50} + 4 q^{52} + 4 q^{53} + 14 q^{55} - 8 q^{56} + 6 q^{58} - 11 q^{59} - 25 q^{61} + 18 q^{62} + 12 q^{64} + 2 q^{65} - 2 q^{67} + 2 q^{68} - 4 q^{70} - 22 q^{71} + 24 q^{73} - 12 q^{74} - 2 q^{76} - q^{77} - q^{79} + 6 q^{80} + 11 q^{82} - 4 q^{83} + q^{85} - 46 q^{86} - 14 q^{88} - 2 q^{89} - 8 q^{91} + 9 q^{92} - q^{94} + 2 q^{95} - 36 q^{97} + 24 q^{98}+O(q^{100}) $ 12 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 8 * q^7 + 12 * q^8 + 6 * q^10 - 14 * q^11 - 2 * q^13 + 4 * q^14 - 6 * q^16 - q^17 - 2 * q^19 + 6 * q^20 + 7 * q^22 - 18 * q^23 + 12 * q^25 - 2 * q^26 + 4 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - q^34 + 8 * q^35 + 6 * q^37 + 4 * q^38 - 12 * q^40 + 11 * q^41 + 23 * q^43 + 7 * q^44 + 9 * q^46 - q^47 - 24 * q^49 - 6 * q^50 + 4 * q^52 + 4 * q^53 + 14 * q^55 - 8 * q^56 + 6 * q^58 - 11 * q^59 - 25 * q^61 + 18 * q^62 + 12 * q^64 + 2 * q^65 - 2 * q^67 + 2 * q^68 - 4 * q^70 - 22 * q^71 + 24 * q^73 - 12 * q^74 - 2 * q^76 - q^77 - q^79 + 6 * q^80 + 11 * q^82 - 4 * q^83 + q^85 - 46 * q^86 - 14 * q^88 - 2 * q^89 - 8 * q^91 + 9 * q^92 - q^94 + 2 * q^95 - 36 * q^97 + 24 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 2x^{11} + x^{10} + 4x^{9} - 7x^{8} + x^{7} + 7x^{6} + 2x^{5} - 28x^{4} + 32x^{3} + 16x^{2} - 64x + 64 $ x^12 - 2*x^11 + x^10 + 4*x^9 - 7*x^8 + x^7 + 7*x^6 + 2*x^5 - 28*x^4 + 32*x^3 + 16*x^2 - 64*x + 64 :

$\beta_{1}$	$=$	$ ( - \nu^{11} + 2 \nu^{10} - 17 \nu^{9} + 76 \nu^{8} - 73 \nu^{7} + 15 \nu^{6} + 73 \nu^{5} - 2 \nu^{4} + \cdots - 960 ) / 288 $ (-v^11 + 2v^10 - 17v^9 + 76v^8 - 73v^7 + 15v^6 + 73v^5 - 2v^4 - 100v^3 - 112v^2 + 784v - 960) / 288
$\beta_{2}$	$=$	$ ( - \nu^{11} - 10 \nu^{10} + 43 \nu^{9} - 56 \nu^{8} + 11 \nu^{7} + 3 \nu^{6} + \nu^{5} - 98 \nu^{4} + \cdots + 672 ) / 288 $ (-v^11 - 10v^10 + 43v^9 - 56v^8 + 11v^7 + 3v^6 + v^5 - 98v^4 - 16v^3 + 344v^2 - 992*v + 672) / 288
$\beta_{3}$	$=$	$ ( - 3 \nu^{11} + 11 \nu^{10} - 13 \nu^{9} + \nu^{8} + \nu^{7} + 2 \nu^{6} - 24 \nu^{5} - 11 \nu^{4} + \cdots - 128 ) / 144 $ (-3v^11 + 11v^10 - 13v^9 + v^8 + v^7 + 2v^6 - 24v^5 - 11v^4 + 94v^3 - 172v^2 + 152*v - 128) / 144
$\beta_{4}$	$=$	$ ( 5 \nu^{11} + 20 \nu^{10} - 11 \nu^{9} - 14 \nu^{8} + 65 \nu^{7} + 27 \nu^{6} - 131 \nu^{5} + \cdots + 576 ) / 288 $ (5v^11 + 20v^10 - 11v^9 - 14v^8 + 65v^7 + 27v^6 - 131v^5 - 128v^4 + 236v^3 - 184v^2 - 416*v + 576) / 288
$\beta_{5}$	$=$	$ ( - 3 \nu^{11} + 38 \nu^{10} - 139 \nu^{9} + 172 \nu^{8} - 35 \nu^{7} - 187 \nu^{6} + 3 \nu^{5} + \cdots - 1568 ) / 288 $ (-3v^11 + 38v^10 - 139v^9 + 172v^8 - 35v^7 - 187v^6 + 3v^5 + 250v^4 + 4v^3 - 1432v^2 + 2240*v - 1568) / 288
$\beta_{6}$	$=$	$ ( - 3 \nu^{11} + 23 \nu^{10} - 37 \nu^{9} + 25 \nu^{8} + 25 \nu^{7} - 22 \nu^{6} - 60 \nu^{5} + \cdots - 176 ) / 144 $ (-3v^11 + 23v^10 - 37v^9 + 25v^8 + 25v^7 - 22v^6 - 60v^5 + 85v^4 + 82v^3 - 520v^2 + 416*v - 176) / 144
$\beta_{7}$	$=$	$ ( 13 \nu^{11} + 24 \nu^{10} - 47 \nu^{9} + 54 \nu^{8} + 53 \nu^{7} - 49 \nu^{6} - 139 \nu^{5} + \cdots + 256 ) / 288 $ (13v^11 + 24v^10 - 47v^9 + 54v^8 + 53v^7 - 49v^6 - 139v^5 + 192v^4 + 272v^3 - 648v^2 + 496*v + 256) / 288
$\beta_{8}$	$=$	$ ( - \nu^{11} + \nu^{10} + \nu^{9} - 5 \nu^{8} + 3 \nu^{7} + 6 \nu^{6} - 8 \nu^{5} - 9 \nu^{4} + \cdots + 48 ) / 16 $ (-v^11 + v^10 + v^9 - 5v^8 + 3v^7 + 6v^6 - 8v^5 - 9v^4 + 26v^3 + 12v^2 - 48v + 48) / 16
$\beta_{9}$	$=$	$ ( - 9 \nu^{11} + \nu^{10} + 49 \nu^{9} - 85 \nu^{8} + 35 \nu^{7} + 94 \nu^{6} - 72 \nu^{5} - 145 \nu^{4} + \cdots + 752 ) / 144 $ (-9v^11 + v^10 + 49v^9 - 85v^8 + 35v^7 + 94v^6 - 72v^5 - 145v^4 + 194v^3 + 220v^2 - 944v + 752) / 144
$\beta_{10}$	$=$	$ ( - 7 \nu^{11} + 8 \nu^{10} + 13 \nu^{9} - 26 \nu^{8} + 17 \nu^{7} + 27 \nu^{6} - 47 \nu^{5} + \cdots + 192 ) / 96 $ (-7v^11 + 8v^10 + 13v^9 - 26v^8 + 17v^7 + 27v^6 - 47v^5 - 56v^4 + 104v^3 + 32v^2 - 224*v + 192) / 96
$\beta_{11}$	$=$	$ ( - 33 \nu^{11} + 64 \nu^{10} - 65 \nu^{9} - 22 \nu^{8} + 59 \nu^{7} + 37 \nu^{6} - 93 \nu^{5} + \cdots + 224 ) / 288 $ (-33v^11 + 64v^10 - 65v^9 - 22v^8 + 59v^7 + 37v^6 - 93v^5 - 172v^4 + 884v^3 - 968v^2 + 256*v + 224) / 288

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{8} - \beta_{6} + \beta_{3} - 2\beta_{2} - \beta _1 + 2 ) / 3 $ (b10 - b8 - b6 + b3 - 2*b2 - b1 + 2) / 3
$\nu^{2}$	$=$	$ ( \beta_{10} - 2\beta_{9} + \beta_{8} - \beta_{5} ) / 3 $ (b10 - 2*b9 + b8 - b5) / 3
$\nu^{3}$	$=$	$ ( \beta_{11} + 2\beta_{7} - 2\beta_{6} - \beta_{4} - 3 ) / 3 $ (b11 + 2b7 - 2b6 - b4 - 3) / 3
$\nu^{4}$	$=$	$ ( - \beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + \cdots - 2 ) / 3 $ (-b10 - b9 + 2b8 + 2b7 + 3b6 - 2b5 - 4b4 - b3 + 2b2 + b1 - 2) / 3
$\nu^{5}$	$=$	$ ( 5\beta_{11} - \beta_{10} - 3\beta_{9} - 5\beta_{8} - \beta_{7} + 3\beta_{6} - 6\beta_{5} - \beta_{4} - 3\beta_{3} ) / 3 $ (5b11 - b10 - 3b9 - 5b8 - b7 + 3b6 - 6b5 - b4 - 3b3) / 3
$\nu^{6}$	$=$	$ ( - 6 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 $ (-6b10 + 3b9 + 3b8 - 2b7 + 6b6 - 3b5 + b4 - 5b3 - 5b2 + 5*b1 + 2) / 3
$\nu^{7}$	$=$	$ ( 6 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots - 19 ) / 3 $ (6b11 + 7b10 - 3b9 - 7b8 - 3b7 + 2b6 - 3b5 + 6b4 - 26b3 - 14b2 - 7*b1 - 19) / 3
$\nu^{8}$	$=$	$ ( 3 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \cdots - 7 ) / 3 $ (3b11 + 4b10 - 2b9 + 4b8 + 4b7 + 3b6 + 2b5 + 4b4 - 32b3 + 7b2 + 14*b1 - 7) / 3
$\nu^{9}$	$=$	$ ( 4 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} - 8 \beta_{6} - 3 \beta_{5} + \cdots - 4 ) / 3 $ (4b11 + 21b10 + 3b9 - 24b8 + 12b7 - 8b6 - 3b5 - 6b4 - 2b3 - 2b2 + 2*b1 - 4) / 3
$\nu^{10}$	$=$	$ ( 6 \beta_{11} + 20 \beta_{10} - 37 \beta_{9} - 16 \beta_{8} + \beta_{7} + 27 \beta_{6} - 41 \beta_{5} + \cdots + 29 ) / 3 $ (6b11 + 20b10 - 37b9 - 16b8 + b7 + 27b6 - 41b5 - 2b4 + 43b3 + 28b2 + 14b1 + 29) / 3
$\nu^{11}$	$=$	$ ( - \beta_{11} - 13 \beta_{10} - 6 \beta_{9} - 29 \beta_{8} + 14 \beta_{7} - 9 \beta_{6} - 27 \beta_{5} + \cdots + 3 ) / 3 $ (-b11 - 13b10 - 6b9 - 29b8 + 14b7 - 9b6 - 27b5 + 14b4 + 78b3 - 3b2 - 6b1 + 3) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times$.

$n$	$757$	$1081$	$1541$
$\chi(n)$	$1$	$\beta_{3}$	$-1 - \beta_{3}$

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} $

361.1

1.04029	−	0.958022i
0.309529	−	1.37992i
0.683706	+	1.23796i
−1.41396	+	0.0268737i
−0.989378	−	1.01051i
1.36982	+	0.351572i
1.04029	+	0.958022i
0.309529	+	1.37992i
0.683706	−	1.23796i
−1.41396	−	0.0268737i
−0.989378	+	1.01051i
1.36982	−	0.351572i

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

−2.64400

−

0.0963576i

1.00000

0.500000

−

0.866025i

361.2

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

−2.64400

−

0.0963576i

1.00000

0.500000

−

0.866025i

361.3

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

−0.0665372

2.64491i

1.00000

0.500000

−

0.866025i

361.4

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

−0.0665372

2.64491i

1.00000

0.500000

−

0.866025i

361.5

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

0.710533

−

2.54856i

1.00000

0.500000

−

0.866025i

361.6

−0.500000

0.866025i

−0.500000

−

0.866025i

−1.00000

0.710533

−

2.54856i

1.00000

0.500000

−

0.866025i

1801.1

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

−2.64400

0.0963576i

1.00000

0.500000

0.866025i

1801.2

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

−2.64400

0.0963576i

1.00000

0.500000

0.866025i

1801.3

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

−0.0665372

−

2.64491i

1.00000

0.500000

0.866025i

1801.4

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

−0.0665372

−

2.64491i

1.00000

0.500000

0.866025i

1801.5

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

0.710533

2.54856i

1.00000

0.500000

0.866025i

1801.6

−0.500000

−

0.866025i

−0.500000

0.866025i

−1.00000

0.710533

2.54856i

1.00000

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1890.2.l.f		12
3.b	odd	2	1		630.2.l.h	yes	12
7.c	even	3	1		1890.2.i.h		12
9.c	even	3	1		1890.2.i.h		12
9.d	odd	6	1		630.2.i.f	✓	12
21.h	odd	6	1		630.2.i.f	✓	12
63.g	even	3	1	inner	1890.2.l.f		12
63.n	odd	6	1		630.2.l.h	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.i.f	✓	12	9.d	odd	6	1
630.2.i.f	✓	12	21.h	odd	6	1
630.2.l.h	yes	12	3.b	odd	2	1
630.2.l.h	yes	12	63.n	odd	6	1
1890.2.i.h		12	7.c	even	3	1
1890.2.i.h		12	9.c	even	3	1
1890.2.l.f		12	1.a	even	1	1	trivial
1890.2.l.f		12	63.g	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1890, [\chi])$:

$ T_{11}^{6} + 7T_{11}^{5} - 8T_{11}^{4} - 132T_{11}^{3} - 132T_{11}^{2} + 486T_{11} + 729 $

$ T_{13}^{12} + 2 T_{13}^{11} + 34 T_{13}^{10} - 28 T_{13}^{9} + 808 T_{13}^{8} - 67 T_{13}^{7} + 4072 T_{13}^{6} + \cdots + 9 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$3$	$ T^{12} $ T^12
$5$	$ (T + 1)^{12} $ (T + 1)^12
$7$	$ (T^{6} + 4 T^{5} + \cdots + 343)^{2} $ (T^6 + 4T^5 + 14T^4 + 55T^3 + 98T^2 + 196*T + 343)^2
$11$	$ (T^{6} + 7 T^{5} + \cdots + 729)^{2} $ (T^6 + 7T^5 - 8T^4 - 132T^3 - 132T^2 + 486*T + 729)^2
$13$	$ T^{12} + 2 T^{11} + \cdots + 9 $ T^12 + 2T^11 + 34T^10 - 28T^9 + 808T^8 - 67T^7 + 4072T^6 - 5050T^5 + 14470T^4 - 6420T^3 + 2973T^2 + 153*T + 9
$17$	$ T^{12} + T^{11} + \cdots + 49660209 $ T^12 + T^11 + 72T^10 - 47T^9 + 3667T^8 - 2001T^7 + 84537T^6 - 35181T^5 + 1419687T^4 - 281394T^3 + 9773703T^2 + 570807T + 49660209
$19$	$ T^{12} + 2 T^{11} + \cdots + 9 $ T^12 + 2T^11 + 34T^10 - 28T^9 + 808T^8 - 67T^7 + 4072T^6 - 5050T^5 + 14470T^4 - 6420T^3 + 2973T^2 + 153*T + 9
$23$	$ (T^{6} + 9 T^{5} + \cdots - 243)^{2} $ (T^6 + 9T^5 - 9T^4 - 216T^3 - 603T^2 - 648*T - 243)^2
$29$	$ T^{12} + 3 T^{11} + \cdots + 531441 $ T^12 + 3T^11 + 87T^10 + 720T^9 + 8217T^8 + 41661T^7 + 173502T^6 + 374949T^5 + 665577T^4 + 524880T^3 + 570807T^2 + 177147*T + 531441
$31$	$ T^{12} + 9 T^{11} + \cdots + 11881 $ T^12 + 9T^11 + 117T^10 + 292T^9 + 4410T^8 + 17568T^7 + 79854T^6 + 129645T^5 + 220680T^4 - 43664T^3 + 142254T^2 + 35316*T + 11881
$37$	$ T^{12} - 6 T^{11} + \cdots + 32137561 $ T^12 - 6T^11 + 111T^10 - 86T^9 + 5958T^8 - 6507T^7 + 145869T^6 + 119664T^5 + 1657926T^4 + 862159T^3 + 10141824T^2 + 9676983*T + 32137561
$41$	$ T^{12} + \cdots + 44545901481 $ T^12 - 11T^11 + 333T^10 - 1898T^9 + 55570T^8 - 263886T^7 + 5701371T^6 - 10618002T^5 + 312887673T^4 - 289775826T^3 + 11421796797T^2 + 19747524276*T + 44545901481
$43$	$ T^{12} - 23 T^{11} + \cdots + 11771761 $ T^12 - 23T^11 + 371T^10 - 3060T^9 + 19222T^8 - 81202T^7 + 295363T^6 - 830482T^5 + 2329825T^4 - 5111616T^3 + 10433735T^2 - 12550598*T + 11771761
$47$	$ T^{12} + T^{11} + \cdots + 6561 $ T^12 + T^11 + 63T^10 + 256T^9 + 4018T^8 + 10050T^7 + 24351T^6 + 22554T^5 + 31005T^4 + 23328T^3 + 27459T^2 + 13122T + 6561
$53$	$ T^{12} + \cdots + 503418969 $ T^12 - 4T^11 + 249T^10 - 1456T^9 + 47860T^8 - 255171T^7 + 3814839T^6 - 17750772T^5 + 210310992T^4 - 799933401T^3 + 4185352296T^2 - 1494506133*T + 503418969
$59$	$ T^{12} + \cdots + 76396407201 $ T^12 + 11T^11 + 324T^10 + 2807T^9 + 62689T^8 + 503925T^7 + 6743823T^6 + 39951459T^5 + 421751997T^4 + 2202035760T^3 + 15083096265T^2 + 35833748355*T + 76396407201
$61$	$ T^{12} + 25 T^{11} + \cdots + 14600041 $ T^12 + 25T^11 + 491T^10 + 4698T^9 + 40540T^8 + 186284T^7 + 1467490T^6 + 6475991T^5 + 26583370T^4 + 52762692T^3 + 80100386T^2 + 38592100*T + 14600041
$67$	$ T^{12} + \cdots + 455352921 $ T^12 + 2T^11 + 229T^10 + 404T^9 + 40369T^8 + 87956T^7 + 2652115T^6 + 11643158T^5 + 143742442T^4 + 421749816T^3 + 1082138751T^2 + 775053819*T + 455352921
$71$	$ (T^{6} + 11 T^{5} + \cdots + 27)^{2} $ (T^6 + 11T^5 - 182T^4 - 1860T^3 - 5397T^2 - 5031*T + 27)^2
$73$	$ T^{12} - 24 T^{11} + \cdots + 1 $ T^12 - 24T^11 + 480T^10 - 3596T^9 + 26046T^8 - 1269T^7 + 553326T^6 - 926316T^5 + 1523682T^4 - 482630T^3 + 133095T^2 + 363*T + 1
$79$	$ T^{12} + \cdots + 61140969289 $ T^12 + T^11 + 296T^10 + 2325T^9 + 67450T^8 + 504110T^7 + 8212279T^6 + 66951617T^5 + 717980290T^4 + 3977535090T^3 + 20253753605T^2 + 39421777810T + 61140969289
$83$	$ T^{12} + \cdots + 130439241 $ T^12 + 4T^11 + 204T^10 - 620T^9 + 31780T^8 - 15111T^7 + 688758T^6 + 869148T^5 + 12711366T^4 + 10378368T^3 + 53360613T^2 - 35462205*T + 130439241
$89$	$ T^{12} + 2 T^{11} + \cdots + 4549689 $ T^12 + 2T^11 + 279T^10 + 386T^9 + 64354T^8 + 68361T^7 + 3603237T^6 - 12039660T^5 + 144210276T^4 - 138518289T^3 + 106465590T^2 - 24552963*T + 4549689
$97$	$ (T^{6} + 18 T^{5} + \cdots + 49)^{2} $ (T^6 + 18T^5 + 255T^4 + 1256T^3 + 4887T^2 - 483*T + 49)^2
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1890.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} - q^{5} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + q^{8}+O(q^{10}) \) q + (-b3 - 1) * q^2 + b3 * q^4 - q^5 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^7 + q^8	\( q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} - q^{5} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 1) q^{98}+O(q^{100}) \) q + (-b3 - 1) * q^2 + b3 * q^4 - q^5 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^7 + q^8 + (b3 + 1) * q^10 + (-b4 - 1) * q^11 + (b8 + b5 - b3 + b1 - 1) * q^13 + (b11 + b10 - b9 - b8 - b5 + b3 + 1) * q^14 + (-b3 - 1) * q^16 + (-b10 + b9 - b7 - b1) * q^17 + (-b5 + b3 + b2) * q^19 - b3 * q^20 + (b7 + b3 + 1) * q^22 + (b10 - b8 + b4 - 1) * q^23 + q^25 + (-b5 + b3 + b2) * q^26 + (b6 - b3) * q^28 + (b9 + b7 + b5 - b4 + b2) * q^29 + (-b7 + b4 + b3 - 2b2) q^31 + b3 * q^32 + (-b9 + b7 - b4 - b2) * q^34 + (b11 + b10 - b9 - b8 + b6 - b5 + 1) * q^35 + (-2b9 + b2) q^37 + (-b8 - b2 - b1 + 1) * q^38 - q^40 + (2b10 - 2b9 - 4b8 + b7 - 4b5 + 4b3 - b1 + 4) q^41 + (2b11 + b10 - b8 - b7 + b6 + b4 - 5b3 - b2) * q^43 + (-b7 + b4 - b3) * q^44 + (-b10 + b9 + b8 - b7 + b5 + b3 + 1) * q^46 + (-b11 + b10 - b8 + b7 - 2b6 + b3 - b1 + 1) q^47 + (2b10 + b9 - 2b8 - b6 + b5 + b3) * q^49 + (-b3 - 1) * q^50 + (-b8 - b2 - b1 + 1) * q^52 + (2b10 - 2b9 + b8 + b5 + 2b3 - 3b1 + 2) * q^53 + (b4 + 1) * q^55 + (-b11 - b10 + b9 + b8 - b6 + b5 - 1) * q^56 + (-b10 + b8 + b4 - b2 - b1) * q^58 + (-2b9 + b7 + b5 - b4 + 3b3 + 3b2) q^59 + (2b10 - 2b9 + b7 - 3b3 - 2b1 - 3) * q^61 + (-b4 + 2b2 + 2b1 + 1) * q^62 + q^64 + (-b8 - b5 + b3 - b1 + 1) * q^65 + (-2b11 - b10 + b9 + b8 + 2b7 - b6 - 2b4 + 2b3 + 4b2) q^67 + (b10 + b4 + b2 + b1) * q^68 + (-b11 - b10 + b9 + b8 + b5 - b3 - 1) * q^70 + (2b11 - 2b8 - 2b6 + b4 - 2) q^71 + (2b11 - 2b9 + 4b6 - 2b5 + 3b3 + b1 + 3) q^73 + (2b10 - b2 - b1) q^74 + (b8 + b5 - b3 + b1 - 1) * q^76 + (2b11 + 3b10 - 2b9 - b8 - b7 + b6 - b5 + 2b4 + 2b3 - b2 - 3b1 + 2) * q^77 + (-b10 + b9 - b7 + b3 - 4b1 + 1) q^79 + (b3 + 1) * q^80 + (2b9 - b7 + 4b5 + b4 - 4b3 - b2) q^82 + (-2b11 - b10 + 2b9 + b8 - b6 + 4b5 + b2) q^83 + (b10 - b9 + b7 + b1) * q^85 + (-b11 - 2b10 + 2b8 + b6 - b4 + b2 + b1 - 5) * q^86 + (-b4 - 1) * q^88 + (3b5 + 2b2) * q^89 + (-b11 + b9 + b8 - b5 - b4 + 3b3 + 3b2 + 2b1) q^91 + (-b9 + b7 - b5 - b4 - b3) * q^92 + (2b11 + b10 - b8 - b7 + b6 + b4 - b3 - b2) q^94 + (b5 - b3 - b2) * q^95 + (2b11 + b10 - b8 + b6 + 5b3) * q^97 + (b11 - 2b10 + b9 + 2b8 + b6 + b5 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 8 * q^7 + 12 * q^8	\( 12 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 8 q^{7} + 12 q^{8} + 6 q^{10} - 14 q^{11} - 2 q^{13} + 4 q^{14} - 6 q^{16} - q^{17} - 2 q^{19} + 6 q^{20} + 7 q^{22} - 18 q^{23} + 12 q^{25} - 2 q^{26} + 4 q^{28} - 3 q^{29} - 9 q^{31} - 6 q^{32} - q^{34} + 8 q^{35} + 6 q^{37} + 4 q^{38} - 12 q^{40} + 11 q^{41} + 23 q^{43} + 7 q^{44} + 9 q^{46} - q^{47} - 24 q^{49} - 6 q^{50} + 4 q^{52} + 4 q^{53} + 14 q^{55} - 8 q^{56} + 6 q^{58} - 11 q^{59} - 25 q^{61} + 18 q^{62} + 12 q^{64} + 2 q^{65} - 2 q^{67} + 2 q^{68} - 4 q^{70} - 22 q^{71} + 24 q^{73} - 12 q^{74} - 2 q^{76} - q^{77} - q^{79} + 6 q^{80} + 11 q^{82} - 4 q^{83} + q^{85} - 46 q^{86} - 14 q^{88} - 2 q^{89} - 8 q^{91} + 9 q^{92} - q^{94} + 2 q^{95} - 36 q^{97} + 24 q^{98}+O(q^{100}) \) 12 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 8 * q^7 + 12 * q^8 + 6 * q^10 - 14 * q^11 - 2 * q^13 + 4 * q^14 - 6 * q^16 - q^17 - 2 * q^19 + 6 * q^20 + 7 * q^22 - 18 * q^23 + 12 * q^25 - 2 * q^26 + 4 * q^28 - 3 * q^29 - 9 * q^31 - 6 * q^32 - q^34 + 8 * q^35 + 6 * q^37 + 4 * q^38 - 12 * q^40 + 11 * q^41 + 23 * q^43 + 7 * q^44 + 9 * q^46 - q^47 - 24 * q^49 - 6 * q^50 + 4 * q^52 + 4 * q^53 + 14 * q^55 - 8 * q^56 + 6 * q^58 - 11 * q^59 - 25 * q^61 + 18 * q^62 + 12 * q^64 + 2 * q^65 - 2 * q^67 + 2 * q^68 - 4 * q^70 - 22 * q^71 + 24 * q^73 - 12 * q^74 - 2 * q^76 - q^77 - q^79 + 6 * q^80 + 11 * q^82 - 4 * q^83 + q^85 - 46 * q^86 - 14 * q^88 - 2 * q^89 - 8 * q^91 + 9 * q^92 - q^94 + 2 * q^95 - 36 * q^97 + 24 * q^98

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	1890.2.l.f

Level	$1890$
Weight	$2$
Character orbit	1890.l
Analytic conductor	$15.092$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(15.0917259820\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	12.0.91830304992969.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 2x^{11} + x^{10} + 4x^{9} - 7x^{8} + x^{7} + 7x^{6} + 2x^{5} - 28x^{4} + 32x^{3} + 16x^{2} - 64x + 64 \) x^12 - 2x^11 + x^10 + 4x^9 - 7x^8 + x^7 + 7x^6 + 2x^5 - 28x^4 + 32x^3 + 16x^2 - 64*x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{6} \)
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - \nu^{11} + 2 \nu^{10} - 17 \nu^{9} + 76 \nu^{8} - 73 \nu^{7} + 15 \nu^{6} + 73 \nu^{5} - 2 \nu^{4} + \cdots - 960 ) / 288 \) (-v^11 + 2v^10 - 17v^9 + 76v^8 - 73v^7 + 15v^6 + 73v^5 - 2v^4 - 100v^3 - 112v^2 + 784v - 960) / 288
\(\beta_{2}\)	\(=\)	\( ( - \nu^{11} - 10 \nu^{10} + 43 \nu^{9} - 56 \nu^{8} + 11 \nu^{7} + 3 \nu^{6} + \nu^{5} - 98 \nu^{4} + \cdots + 672 ) / 288 \) (-v^11 - 10v^10 + 43v^9 - 56v^8 + 11v^7 + 3v^6 + v^5 - 98v^4 - 16v^3 + 344v^2 - 992*v + 672) / 288
\(\beta_{3}\)	\(=\)	\( ( - 3 \nu^{11} + 11 \nu^{10} - 13 \nu^{9} + \nu^{8} + \nu^{7} + 2 \nu^{6} - 24 \nu^{5} - 11 \nu^{4} + \cdots - 128 ) / 144 \) (-3v^11 + 11v^10 - 13v^9 + v^8 + v^7 + 2v^6 - 24v^5 - 11v^4 + 94v^3 - 172v^2 + 152*v - 128) / 144
\(\beta_{4}\)	\(=\)	\( ( 5 \nu^{11} + 20 \nu^{10} - 11 \nu^{9} - 14 \nu^{8} + 65 \nu^{7} + 27 \nu^{6} - 131 \nu^{5} + \cdots + 576 ) / 288 \) (5v^11 + 20v^10 - 11v^9 - 14v^8 + 65v^7 + 27v^6 - 131v^5 - 128v^4 + 236v^3 - 184v^2 - 416*v + 576) / 288
\(\beta_{5}\)	\(=\)	\( ( - 3 \nu^{11} + 38 \nu^{10} - 139 \nu^{9} + 172 \nu^{8} - 35 \nu^{7} - 187 \nu^{6} + 3 \nu^{5} + \cdots - 1568 ) / 288 \) (-3v^11 + 38v^10 - 139v^9 + 172v^8 - 35v^7 - 187v^6 + 3v^5 + 250v^4 + 4v^3 - 1432v^2 + 2240*v - 1568) / 288
\(\beta_{6}\)	\(=\)	\( ( - 3 \nu^{11} + 23 \nu^{10} - 37 \nu^{9} + 25 \nu^{8} + 25 \nu^{7} - 22 \nu^{6} - 60 \nu^{5} + \cdots - 176 ) / 144 \) (-3v^11 + 23v^10 - 37v^9 + 25v^8 + 25v^7 - 22v^6 - 60v^5 + 85v^4 + 82v^3 - 520v^2 + 416*v - 176) / 144
\(\beta_{7}\)	\(=\)	\( ( 13 \nu^{11} + 24 \nu^{10} - 47 \nu^{9} + 54 \nu^{8} + 53 \nu^{7} - 49 \nu^{6} - 139 \nu^{5} + \cdots + 256 ) / 288 \) (13v^11 + 24v^10 - 47v^9 + 54v^8 + 53v^7 - 49v^6 - 139v^5 + 192v^4 + 272v^3 - 648v^2 + 496*v + 256) / 288
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} + \nu^{9} - 5 \nu^{8} + 3 \nu^{7} + 6 \nu^{6} - 8 \nu^{5} - 9 \nu^{4} + \cdots + 48 ) / 16 \) (-v^11 + v^10 + v^9 - 5v^8 + 3v^7 + 6v^6 - 8v^5 - 9v^4 + 26v^3 + 12v^2 - 48v + 48) / 16
\(\beta_{9}\)	\(=\)	\( ( - 9 \nu^{11} + \nu^{10} + 49 \nu^{9} - 85 \nu^{8} + 35 \nu^{7} + 94 \nu^{6} - 72 \nu^{5} - 145 \nu^{4} + \cdots + 752 ) / 144 \) (-9v^11 + v^10 + 49v^9 - 85v^8 + 35v^7 + 94v^6 - 72v^5 - 145v^4 + 194v^3 + 220v^2 - 944v + 752) / 144
\(\beta_{10}\)	\(=\)	\( ( - 7 \nu^{11} + 8 \nu^{10} + 13 \nu^{9} - 26 \nu^{8} + 17 \nu^{7} + 27 \nu^{6} - 47 \nu^{5} + \cdots + 192 ) / 96 \) (-7v^11 + 8v^10 + 13v^9 - 26v^8 + 17v^7 + 27v^6 - 47v^5 - 56v^4 + 104v^3 + 32v^2 - 224*v + 192) / 96
\(\beta_{11}\)	\(=\)	\( ( - 33 \nu^{11} + 64 \nu^{10} - 65 \nu^{9} - 22 \nu^{8} + 59 \nu^{7} + 37 \nu^{6} - 93 \nu^{5} + \cdots + 224 ) / 288 \) (-33v^11 + 64v^10 - 65v^9 - 22v^8 + 59v^7 + 37v^6 - 93v^5 - 172v^4 + 884v^3 - 968v^2 + 256*v + 224) / 288

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{8} - \beta_{6} + \beta_{3} - 2\beta_{2} - \beta _1 + 2 ) / 3 \) (b10 - b8 - b6 + b3 - 2*b2 - b1 + 2) / 3
\(\nu^{2}\)	\(=\)	\( ( \beta_{10} - 2\beta_{9} + \beta_{8} - \beta_{5} ) / 3 \) (b10 - 2*b9 + b8 - b5) / 3
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + 2\beta_{7} - 2\beta_{6} - \beta_{4} - 3 ) / 3 \) (b11 + 2b7 - 2b6 - b4 - 3) / 3
\(\nu^{4}\)	\(=\)	\( ( - \beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + \cdots - 2 ) / 3 \) (-b10 - b9 + 2b8 + 2b7 + 3b6 - 2b5 - 4b4 - b3 + 2b2 + b1 - 2) / 3
\(\nu^{5}\)	\(=\)	\( ( 5\beta_{11} - \beta_{10} - 3\beta_{9} - 5\beta_{8} - \beta_{7} + 3\beta_{6} - 6\beta_{5} - \beta_{4} - 3\beta_{3} ) / 3 \) (5b11 - b10 - 3b9 - 5b8 - b7 + 3b6 - 6b5 - b4 - 3b3) / 3
\(\nu^{6}\)	\(=\)	\( ( - 6 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 \) (-6b10 + 3b9 + 3b8 - 2b7 + 6b6 - 3b5 + b4 - 5b3 - 5b2 + 5*b1 + 2) / 3
\(\nu^{7}\)	\(=\)	\( ( 6 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots - 19 ) / 3 \) (6b11 + 7b10 - 3b9 - 7b8 - 3b7 + 2b6 - 3b5 + 6b4 - 26b3 - 14b2 - 7*b1 - 19) / 3
\(\nu^{8}\)	\(=\)	\( ( 3 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \cdots - 7 ) / 3 \) (3b11 + 4b10 - 2b9 + 4b8 + 4b7 + 3b6 + 2b5 + 4b4 - 32b3 + 7b2 + 14*b1 - 7) / 3
\(\nu^{9}\)	\(=\)	\( ( 4 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} - 24 \beta_{8} + 12 \beta_{7} - 8 \beta_{6} - 3 \beta_{5} + \cdots - 4 ) / 3 \) (4b11 + 21b10 + 3b9 - 24b8 + 12b7 - 8b6 - 3b5 - 6b4 - 2b3 - 2b2 + 2*b1 - 4) / 3
\(\nu^{10}\)	\(=\)	\( ( 6 \beta_{11} + 20 \beta_{10} - 37 \beta_{9} - 16 \beta_{8} + \beta_{7} + 27 \beta_{6} - 41 \beta_{5} + \cdots + 29 ) / 3 \) (6b11 + 20b10 - 37b9 - 16b8 + b7 + 27b6 - 41b5 - 2b4 + 43b3 + 28b2 + 14b1 + 29) / 3
\(\nu^{11}\)	\(=\)	\( ( - \beta_{11} - 13 \beta_{10} - 6 \beta_{9} - 29 \beta_{8} + 14 \beta_{7} - 9 \beta_{6} - 27 \beta_{5} + \cdots + 3 ) / 3 \) (-b11 - 13b10 - 6b9 - 29b8 + 14b7 - 9b6 - 27b5 + 14b4 + 78b3 - 3b2 - 6b1 + 3) / 3

\(n\)	\(757\)	\(1081\)	\(1541\)
\(\chi(n)\)	\(1\)	\(\beta_{3}\)	\(-1 - \beta_{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$3$	\( T^{12} \) T^12
$5$	\( (T + 1)^{12} \) (T + 1)^12
$7$	\( (T^{6} + 4 T^{5} + \cdots + 343)^{2} \) (T^6 + 4T^5 + 14T^4 + 55T^3 + 98T^2 + 196*T + 343)^2
$11$	\( (T^{6} + 7 T^{5} + \cdots + 729)^{2} \) (T^6 + 7T^5 - 8T^4 - 132T^3 - 132T^2 + 486*T + 729)^2
$13$	\( T^{12} + 2 T^{11} + \cdots + 9 \) T^12 + 2T^11 + 34T^10 - 28T^9 + 808T^8 - 67T^7 + 4072T^6 - 5050T^5 + 14470T^4 - 6420T^3 + 2973T^2 + 153*T + 9
$17$	\( T^{12} + T^{11} + \cdots + 49660209 \) T^12 + T^11 + 72T^10 - 47T^9 + 3667T^8 - 2001T^7 + 84537T^6 - 35181T^5 + 1419687T^4 - 281394T^3 + 9773703T^2 + 570807T + 49660209
$19$	\( T^{12} + 2 T^{11} + \cdots + 9 \) T^12 + 2T^11 + 34T^10 - 28T^9 + 808T^8 - 67T^7 + 4072T^6 - 5050T^5 + 14470T^4 - 6420T^3 + 2973T^2 + 153*T + 9
$23$	\( (T^{6} + 9 T^{5} + \cdots - 243)^{2} \) (T^6 + 9T^5 - 9T^4 - 216T^3 - 603T^2 - 648*T - 243)^2
$29$	\( T^{12} + 3 T^{11} + \cdots + 531441 \) T^12 + 3T^11 + 87T^10 + 720T^9 + 8217T^8 + 41661T^7 + 173502T^6 + 374949T^5 + 665577T^4 + 524880T^3 + 570807T^2 + 177147*T + 531441
$31$	\( T^{12} + 9 T^{11} + \cdots + 11881 \) T^12 + 9T^11 + 117T^10 + 292T^9 + 4410T^8 + 17568T^7 + 79854T^6 + 129645T^5 + 220680T^4 - 43664T^3 + 142254T^2 + 35316*T + 11881
$37$	\( T^{12} - 6 T^{11} + \cdots + 32137561 \) T^12 - 6T^11 + 111T^10 - 86T^9 + 5958T^8 - 6507T^7 + 145869T^6 + 119664T^5 + 1657926T^4 + 862159T^3 + 10141824T^2 + 9676983*T + 32137561
$41$	\( T^{12} + \cdots + 44545901481 \) T^12 - 11T^11 + 333T^10 - 1898T^9 + 55570T^8 - 263886T^7 + 5701371T^6 - 10618002T^5 + 312887673T^4 - 289775826T^3 + 11421796797T^2 + 19747524276*T + 44545901481
$43$	\( T^{12} - 23 T^{11} + \cdots + 11771761 \) T^12 - 23T^11 + 371T^10 - 3060T^9 + 19222T^8 - 81202T^7 + 295363T^6 - 830482T^5 + 2329825T^4 - 5111616T^3 + 10433735T^2 - 12550598*T + 11771761
$47$	\( T^{12} + T^{11} + \cdots + 6561 \) T^12 + T^11 + 63T^10 + 256T^9 + 4018T^8 + 10050T^7 + 24351T^6 + 22554T^5 + 31005T^4 + 23328T^3 + 27459T^2 + 13122T + 6561
$53$	\( T^{12} + \cdots + 503418969 \) T^12 - 4T^11 + 249T^10 - 1456T^9 + 47860T^8 - 255171T^7 + 3814839T^6 - 17750772T^5 + 210310992T^4 - 799933401T^3 + 4185352296T^2 - 1494506133*T + 503418969
$59$	\( T^{12} + \cdots + 76396407201 \) T^12 + 11T^11 + 324T^10 + 2807T^9 + 62689T^8 + 503925T^7 + 6743823T^6 + 39951459T^5 + 421751997T^4 + 2202035760T^3 + 15083096265T^2 + 35833748355*T + 76396407201
$61$	\( T^{12} + 25 T^{11} + \cdots + 14600041 \) T^12 + 25T^11 + 491T^10 + 4698T^9 + 40540T^8 + 186284T^7 + 1467490T^6 + 6475991T^5 + 26583370T^4 + 52762692T^3 + 80100386T^2 + 38592100*T + 14600041
$67$	\( T^{12} + \cdots + 455352921 \) T^12 + 2T^11 + 229T^10 + 404T^9 + 40369T^8 + 87956T^7 + 2652115T^6 + 11643158T^5 + 143742442T^4 + 421749816T^3 + 1082138751T^2 + 775053819*T + 455352921
$71$	\( (T^{6} + 11 T^{5} + \cdots + 27)^{2} \) (T^6 + 11T^5 - 182T^4 - 1860T^3 - 5397T^2 - 5031*T + 27)^2
$73$	\( T^{12} - 24 T^{11} + \cdots + 1 \) T^12 - 24T^11 + 480T^10 - 3596T^9 + 26046T^8 - 1269T^7 + 553326T^6 - 926316T^5 + 1523682T^4 - 482630T^3 + 133095T^2 + 363*T + 1
$79$	\( T^{12} + \cdots + 61140969289 \) T^12 + T^11 + 296T^10 + 2325T^9 + 67450T^8 + 504110T^7 + 8212279T^6 + 66951617T^5 + 717980290T^4 + 3977535090T^3 + 20253753605T^2 + 39421777810T + 61140969289
$83$	\( T^{12} + \cdots + 130439241 \) T^12 + 4T^11 + 204T^10 - 620T^9 + 31780T^8 - 15111T^7 + 688758T^6 + 869148T^5 + 12711366T^4 + 10378368T^3 + 53360613T^2 - 35462205*T + 130439241
$89$	\( T^{12} + 2 T^{11} + \cdots + 4549689 \) T^12 + 2T^11 + 279T^10 + 386T^9 + 64354T^8 + 68361T^7 + 3603237T^6 - 12039660T^5 + 144210276T^4 - 138518289T^3 + 106465590T^2 - 24552963*T + 4549689
$97$	\( (T^{6} + 18 T^{5} + \cdots + 49)^{2} \) (T^6 + 18T^5 + 255T^4 + 1256T^3 + 4887T^2 - 483*T + 49)^2
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