LMFDB - Newform orbit 1890.2.i.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1890,2,Mod(991,1890)]

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([4, 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("1890.991");

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.i (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 + q^{2} + q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{10} - \beta_{4} - \beta_1) q^{7} + q^{8}+O(q^{10}) $ q + q^2 + q^4 + (b4 + 1) * q^5 + (-b10 - b4 - b1) * q^7 + q^8	$ q + q^{2} + q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{10} - \beta_{4} - \beta_1) q^{7} + q^{8} + (\beta_{4} + 1) q^{10} + ( - \beta_{5} - \beta_{4}) q^{11} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{11} - 2 \beta_{9} - 2 \beta_1 + 1) q^{98}+O(q^{100}) $ q + q^2 + q^4 + (b4 + 1) * q^5 + (-b10 - b4 - b1) * q^7 + q^8 + (b4 + 1) * q^10 + (-b5 - b4) * q^11 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^13 + (-b10 - b4 - b1) * q^14 + q^16 + (b8 - b7 + b6 - b5 + b2) * q^17 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^19 + (b4 + 1) * q^20 + (-b5 - b4) * q^22 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^23 + b4 * q^25 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^26 + (-b10 - b4 - b1) * q^28 + (b9 - b7 - b6 - b5 - b2 + b1) * q^29 + (-b7 - 2b2 + 1) q^31 + q^32 + (b8 - b7 + b6 - b5 + b2) * q^34 + (-b10 + 1) * q^35 + (-2b8 + b6 + 2b3) * q^37 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^38 + (b4 + 1) * q^40 + (-4b11 - 4b10 - 4b9 - 2b8 - b6 - b5 - 4b4 + 2b3 - 4b1) q^41 + (-b11 + b10 + b9 + b7 + b6 + b5 + 5b4 + b2 + 2b1 + 5) * q^43 + (-b5 - b4) * q^44 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^46 + (2b11 + b10 + b9 - b7 - b2 - b1 - 1) q^47 + (b11 - 2b9 - 2b1 + 1) * q^49 + b4 * q^50 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^52 + (-b9 - 3b8 + 3b6 + 2b4 + 3b2 - b1 + 2) * q^53 + (b7 + 1) * q^55 + (-b10 - b4 - b1) * q^56 + (b9 - b7 - b6 - b5 - b2 + b1) * q^58 + (b11 + b10 + b7 - 3b3 + 3b2 + 3) * q^59 + (-b7 + 2b3 - 2b2 + 3) * q^61 + (-b7 - 2b2 + 1) q^62 + q^64 + (b11 + b10 - b3 - b2 - 1) * q^65 + (-2b11 - b10 - b9 + 2b7 + b3 + 4b2 + b1 + 2) q^67 + (b8 - b7 + b6 - b5 + b2) * q^68 + (-b10 + 1) * q^70 + (-2b11 - 2b9 + b7 + 2b3 + 2b1 - 2) * q^71 + (2b11 - 2b10 - b6 + 3b4 - b2 - 2b1 + 3) * q^73 + (-2b8 + b6 + 2b3) * q^74 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^76 + (b11 + b9 + 2b8 - 2b7 - 3b6 - b5 - 2b4 - b3 - b2 - b1) * q^77 + (b7 - b3 - 4b2 - 1) q^79 + (b4 + 1) * q^80 + (-4b11 - 4b10 - 4b9 - 2b8 - b6 - b5 - 4b4 + 2b3 - 4b1) q^82 + (b11 - b10 + 3b9 + 2b8 - b6 - b2 + 2b1) q^83 + (b8 + b6 - b5 - b3) * q^85 + (-b11 + b10 + b9 + b7 + b6 + b5 + 5b4 + b2 + 2b1 + 5) * q^86 + (-b5 - b4) * q^88 + (-3b11 - 3b10 - 3b9 - 3b8 + 2b6 + 3b3 - 3b1) q^89 + (2b11 + 2b10 + 2b9 + 2b8 - b7 + b6 + 3b4 - 3b3 - 2b2 + b1) q^91 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^92 + (2b11 + b10 + b9 - b7 - b2 - b1 - 1) q^94 + (b11 + b10 - b3 - b2 - 1) * q^95 + (-b11 + b10 + b9 - 5b4 + 2b1 - 5) * q^97 + (b11 - 2b9 - 2b1 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 6 * q^5 + 4 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} + 12 q^{8} + 6 q^{10} + 7 q^{11} - 2 q^{13} + 4 q^{14} + 12 q^{16} - q^{17} - 2 q^{19} + 6 q^{20} + 7 q^{22} + 9 q^{23} - 6 q^{25} - 2 q^{26} + 4 q^{28} - 3 q^{29} + 18 q^{31} + 12 q^{32} - q^{34} + 8 q^{35} + 6 q^{37} - 2 q^{38} + 6 q^{40} + 11 q^{41} + 23 q^{43} + 7 q^{44} + 9 q^{46} + 2 q^{47} + 24 q^{49} - 6 q^{50} - 2 q^{52} + 4 q^{53} + 14 q^{55} + 4 q^{56} - 3 q^{58} + 22 q^{59} + 50 q^{61} + 18 q^{62} + 12 q^{64} - 4 q^{65} + 4 q^{67} - q^{68} + 8 q^{70} - 22 q^{71} + 24 q^{73} + 6 q^{74} - 2 q^{76} + 11 q^{77} + 2 q^{79} + 6 q^{80} + 11 q^{82} - 4 q^{83} + q^{85} + 23 q^{86} + 7 q^{88} - 2 q^{89} - 8 q^{91} + 9 q^{92} + 2 q^{94} - 4 q^{95} - 36 q^{97} + 24 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 6 * q^5 + 4 * q^7 + 12 * q^8 + 6 * q^10 + 7 * q^11 - 2 * q^13 + 4 * q^14 + 12 * q^16 - q^17 - 2 * q^19 + 6 * q^20 + 7 * q^22 + 9 * q^23 - 6 * q^25 - 2 * q^26 + 4 * q^28 - 3 * q^29 + 18 * q^31 + 12 * q^32 - q^34 + 8 * q^35 + 6 * q^37 - 2 * q^38 + 6 * q^40 + 11 * q^41 + 23 * q^43 + 7 * q^44 + 9 * q^46 + 2 * q^47 + 24 * q^49 - 6 * q^50 - 2 * q^52 + 4 * q^53 + 14 * q^55 + 4 * q^56 - 3 * q^58 + 22 * q^59 + 50 * q^61 + 18 * q^62 + 12 * q^64 - 4 * q^65 + 4 * q^67 - q^68 + 8 * q^70 - 22 * q^71 + 24 * q^73 + 6 * q^74 - 2 * q^76 + 11 * q^77 + 2 * q^79 + 6 * q^80 + 11 * q^82 - 4 * q^83 + q^85 + 23 * q^86 + 7 * q^88 - 2 * q^89 - 8 * q^91 + 9 * q^92 + 2 * q^94 - 4 * 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 :

$\beta_{1}$	$=$	$ ( - 5 \nu^{11} - 2 \nu^{10} + 11 \nu^{9} - 16 \nu^{8} - 5 \nu^{7} + 15 \nu^{6} - 7 \nu^{5} - 70 \nu^{4} + \cdots + 96 ) / 96 $ (-5v^11 - 2v^10 + 11v^9 - 16v^8 - 5v^7 + 15v^6 - 7v^5 - 70v^4 + 76v^3 + 16v^2 - 160*v + 96) / 96
$\beta_{2}$	$=$	$ ( \nu^{11} - 2 \nu^{10} + \nu^{9} + 4 \nu^{8} - 7 \nu^{7} + \nu^{6} + 7 \nu^{5} + 2 \nu^{4} - 28 \nu^{3} + \cdots - 64 ) / 32 $ (v^11 - 2v^10 + v^9 + 4v^8 - 7v^7 + v^6 + 7v^5 + 2v^4 - 28v^3 + 32v^2 - 16v - 64) / 32
$\beta_{3}$	$=$	$ ( - \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
$\beta_{4}$	$=$	$ ( 3 \nu^{11} - 11 \nu^{10} + 13 \nu^{9} - \nu^{8} - \nu^{7} - 2 \nu^{6} + 24 \nu^{5} + 11 \nu^{4} + \cdots - 16 ) / 144 $ (3v^11 - 11v^10 + 13v^9 - v^8 - v^7 - 2v^6 + 24v^5 + 11v^4 - 94v^3 + 172v^2 - 152*v - 16) / 144
$\beta_{5}$	$=$	$ ( 27 \nu^{11} - 26 \nu^{10} + 31 \nu^{9} + 32 \nu^{8} - \nu^{7} - 41 \nu^{6} - 27 \nu^{5} + 134 \nu^{4} + \cdots + 320 ) / 288 $ (27v^11 - 26v^10 + 31v^9 + 32v^8 - v^7 - 41v^6 - 27v^5 + 134v^4 - 184v^3 + 328v^2 - 80v + 320) / 288
$\beta_{6}$	$=$	$ ( 3 \nu^{11} - 10 \nu^{10} + 23 \nu^{9} - 32 \nu^{8} + 7 \nu^{7} - \nu^{6} - 3 \nu^{5} - 26 \nu^{4} + \cdots + 352 ) / 96 $ (3v^11 - 10v^10 + 23v^9 - 32v^8 + 7v^7 - v^6 - 3v^5 - 26v^4 - 56v^3 + 248v^2 - 352v + 352) / 96
$\beta_{7}$	$=$	$ ( \nu^{11} - 5 \nu^{10} - \nu^{9} + 5 \nu^{8} - 11 \nu^{7} - 6 \nu^{6} + 20 \nu^{5} + 41 \nu^{4} + \cdots - 96 ) / 48 $ (v^11 - 5v^10 - v^9 + 5v^8 - 11v^7 - 6v^6 + 20v^5 + 41v^4 - 62v^3 + 40v^2 + 80*v - 96) / 48
$\beta_{8}$	$=$	$ ( 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
$\beta_{9}$	$=$	$ ( - 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
$\beta_{10}$	$=$	$ ( - 9 \nu^{11} + 35 \nu^{10} - 49 \nu^{9} + 13 \nu^{8} + 37 \nu^{7} - 4 \nu^{6} - 36 \nu^{5} + \cdots - 32 ) / 144 $ (-9v^11 + 35v^10 - 49v^9 + 13v^8 + 37v^7 - 4v^6 - 36v^5 + 19v^4 + 328v^3 - 508v^2 + 368*v - 32) / 144
$\beta_{11}$	$=$	$ ( 21 \nu^{11} - 76 \nu^{10} + 77 \nu^{9} - 38 \nu^{8} - 71 \nu^{7} + 35 \nu^{6} + 69 \nu^{5} + \cdots + 352 ) / 288 $ (21v^11 - 76v^10 + 77v^9 - 38v^8 - 71v^7 + 35v^6 + 69v^5 - 32v^4 - 500v^3 + 1136v^2 - 928*v + 352) / 288

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

$\nu$	$=$	$ ( -\beta_{10} - \beta_{9} - \beta_{6} - 2\beta_{4} - 2\beta_{2} - \beta _1 - 1 ) / 3 $ (-b10 - b9 - b6 - 2b4 - 2b2 - b1 - 1) / 3
$\nu^{2}$	$=$	$ ( -\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{3} - 2\beta_1 ) / 3 $ (-b11 - b10 - 2b9 - b8 + 2b3 - 2*b1) / 3
$\nu^{3}$	$=$	$ ( 2\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{5} + 2\beta _1 - 3 ) / 3 $ (2b10 - b9 + b7 + 2b5 + 2*b1 - 3) / 3
$\nu^{4}$	$=$	$ ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 1 ) / 3 $ (-2b11 + b10 + 2b9 + b8 + 4b7 + b6 + 2b5 + 2b4 + b3 + 2b2 + 2*b1 + 1) / 3
$\nu^{5}$	$=$	$ ( 5\beta_{10} - \beta_{9} + 3\beta_{8} + \beta_{7} - \beta_{5} + 3\beta_{4} - 6\beta_{3} + 2\beta _1 + 3 ) / 3 $ (5b10 - b9 + 3b8 + b7 - b5 + 3b4 - 6b3 + 2*b1 + 3) / 3
$\nu^{6}$	$=$	$ ( 6 \beta_{11} + 3 \beta_{9} + 6 \beta_{8} - \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \cdots + 2 ) / 3 $ (6b11 + 3b9 + 6b8 - b7 - 10b6 - 2b5 - 5b4 - 3b3 - 5b2 - 3*b1 + 2) / 3
$\nu^{7}$	$=$	$ ( 3 \beta_{11} - \beta_{10} - \beta_{9} - 6 \beta_{7} - 7 \beta_{6} - 3 \beta_{5} + 19 \beta_{4} + \cdots - 7 ) / 3 $ (3b11 - b10 - b9 - 6b7 - 7b6 - 3b5 + 19b4 - 14b2 - 7*b1 - 7) / 3
$\nu^{8}$	$=$	$ ( - 7 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} + \cdots + 32 ) / 3 $ (-7b11 - 4b10 - 2b9 - 4b8 - 4b7 - 7b6 + 4b5 + 25b4 + 8b3 + 7b2 + b1 + 32) / 3
$\nu^{9}$	$=$	$ ( - 21 \beta_{11} - 13 \beta_{10} - \beta_{9} + 6 \beta_{8} + 6 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} + \cdots - 4 ) / 3 $ (-21b11 - 13b10 - b9 + 6b8 + 6b7 - 4b6 + 12b5 - 2b4 - 3b3 - 2b2 + 11b1 - 4) / 3
$\nu^{10}$	$=$	$ ( - 35 \beta_{11} - 14 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 14 \beta_{6} + \beta_{5} + \cdots + 14 ) / 3 $ (-35b11 - 14b10 - 10b9 + 4b8 + 2b7 + 14b6 + b5 - 29b4 + 4b3 + 28b2 - 16b1 + 14) / 3
$\nu^{11}$	$=$	$ ( 30 \beta_{11} + 29 \beta_{10} + 2 \beta_{9} + 21 \beta_{8} - 14 \beta_{7} + 3 \beta_{6} + 14 \beta_{5} + \cdots - 78 ) / 3 $ (30b11 + 29b10 + 2b9 + 21b8 - 14b7 + 3b6 + 14b5 - 75b4 - 42b3 - 3b2 - 7*b1 - 78) / 3

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

Character values

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

$n$	$757$	$1081$	$1541$
$\chi(n)$	$1$	$\beta_{4}$	$\beta_{4}$

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} $

991.1

−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
1.04029	−	0.958022i
0.309529	−	1.37992i
0.683706	+	1.23796i
−1.41396	+	0.0268737i

1.00000

0.500000

−

0.866025i

−2.56238

0.658939i

1.00000

0.500000

−

0.866025i

991.2

1.00000

0.500000

−

0.866025i

−2.56238

0.658939i

1.00000

0.500000

−

0.866025i

991.3

1.00000

0.500000

−

0.866025i

1.23855

2.33795i

1.00000

0.500000

−

0.866025i

991.4

1.00000

0.500000

−

0.866025i

1.23855

2.33795i

1.00000

0.500000

−

0.866025i

991.5

1.00000

0.500000

−

0.866025i

2.32383

−

1.26483i

1.00000

0.500000

−

0.866025i

991.6

1.00000

0.500000

−

0.866025i

2.32383

−

1.26483i

1.00000

0.500000

−

0.866025i

1171.1

1.00000

0.500000

0.866025i

−2.56238

−

0.658939i

1.00000

0.500000

0.866025i

1171.2

1.00000

0.500000

0.866025i

−2.56238

−

0.658939i

1.00000

0.500000

0.866025i

1171.3

1.00000

0.500000

0.866025i

1.23855

−

2.33795i

1.00000

0.500000

0.866025i

1171.4

1.00000

0.500000

0.866025i

1.23855

−

2.33795i

1.00000

0.500000

0.866025i

1171.5

1.00000

0.500000

0.866025i

2.32383

1.26483i

1.00000

0.500000

0.866025i

1171.6

1.00000

0.500000

0.866025i

2.32383

1.26483i

1.00000

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	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.i.h		12
3.b	odd	2	1		630.2.i.f	✓	12
7.c	even	3	1		1890.2.l.f		12
9.c	even	3	1		1890.2.l.f		12
9.d	odd	6	1		630.2.l.h	yes	12
21.h	odd	6	1		630.2.l.h	yes	12
63.h	even	3	1	inner	1890.2.i.h		12
63.j	odd	6	1		630.2.i.f	✓	12

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

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}^{12} - 7 T_{11}^{11} + 57 T_{11}^{10} - 208 T_{11}^{9} + 1120 T_{11}^{8} - 3390 T_{11}^{7} + \cdots + 531441 $

$ 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 - 1)^{12} $ (T - 1)^12
$3$	$ T^{12} $ T^12
$5$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$7$	$ (T^{6} - 2 T^{5} + \cdots + 343)^{2} $ (T^6 - 2T^5 - 4T^4 + 31T^3 - 28T^2 - 98*T + 343)^2
$11$	$ T^{12} - 7 T^{11} + \cdots + 531441 $ T^12 - 7T^11 + 57T^10 - 208T^9 + 1120T^8 - 3390T^7 + 14424T^6 - 30303T^5 + 87408T^4 - 128304T^3 + 332424T^2 - 354294*T + 531441
$13$	$ 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^{12} - 9 T^{11} + \cdots + 59049 $ T^12 - 9T^11 + 90T^10 - 351T^9 + 2628T^8 - 12150T^7 + 46575T^6 - 116397T^5 + 221454T^4 - 285768T^3 + 273375T^2 - 157464*T + 59049
$29$	$ 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^{6} - 9 T^{5} + \cdots + 109)^{2} $ (T^6 - 9T^5 - 36T^4 + 308T^3 - 342T^2 - 324*T + 109)^2
$37$	$ 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^{6} - T^{5} - 62 T^{4} + \cdots + 81)^{2} $ (T^6 - T^5 - 62T^4 + 159T^3 - 15T^2 - 162T + 81)^2
$53$	$ 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^{6} - 11 T^{5} + \cdots + 276399)^{2} $ (T^6 - 11T^5 - 203T^4 + 2520T^3 + 6240T^2 - 129645*T + 276399)^2
$61$	$ (T^{6} - 25 T^{5} + \cdots - 3821)^{2} $ (T^6 - 25T^5 + 134T^4 + 674T^3 - 5734T^2 + 10100*T - 3821)^2
$67$	$ (T^{6} - 2 T^{5} + \cdots + 21339)^{2} $ (T^6 - 2T^5 - 225T^4 + 427T^3 + 11110T^2 - 36321*T + 21339)^2
$71$	$ (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^{6} - T^{5} + \cdots + 247267)^{2} $ (T^6 - T^5 - 295T^4 + 1310T^3 + 20885T^2 - 159430T + 247267)^2
$83$	$ 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
show more
show less

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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{10} - \beta_{4} - \beta_1) q^{7} + q^{8}+O(q^{10}) \) q + q^2 + q^4 + (b4 + 1) * q^5 + (-b10 - b4 - b1) * q^7 + q^8	\( q + q^{2} + q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{10} - \beta_{4} - \beta_1) q^{7} + q^{8} + (\beta_{4} + 1) q^{10} + ( - \beta_{5} - \beta_{4}) q^{11} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{11} - 2 \beta_{9} - 2 \beta_1 + 1) q^{98}+O(q^{100}) \) q + q^2 + q^4 + (b4 + 1) * q^5 + (-b10 - b4 - b1) * q^7 + q^8 + (b4 + 1) * q^10 + (-b5 - b4) * q^11 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^13 + (-b10 - b4 - b1) * q^14 + q^16 + (b8 - b7 + b6 - b5 + b2) * q^17 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^19 + (b4 + 1) * q^20 + (-b5 - b4) * q^22 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^23 + b4 * q^25 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^26 + (-b10 - b4 - b1) * q^28 + (b9 - b7 - b6 - b5 - b2 + b1) * q^29 + (-b7 - 2b2 + 1) q^31 + q^32 + (b8 - b7 + b6 - b5 + b2) * q^34 + (-b10 + 1) * q^35 + (-2b8 + b6 + 2b3) * q^37 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^38 + (b4 + 1) * q^40 + (-4b11 - 4b10 - 4b9 - 2b8 - b6 - b5 - 4b4 + 2b3 - 4b1) q^41 + (-b11 + b10 + b9 + b7 + b6 + b5 + 5b4 + b2 + 2b1 + 5) * q^43 + (-b5 - b4) * q^44 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^46 + (2b11 + b10 + b9 - b7 - b2 - b1 - 1) q^47 + (b11 - 2b9 - 2b1 + 1) * q^49 + b4 * q^50 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^52 + (-b9 - 3b8 + 3b6 + 2b4 + 3b2 - b1 + 2) * q^53 + (b7 + 1) * q^55 + (-b10 - b4 - b1) * q^56 + (b9 - b7 - b6 - b5 - b2 + b1) * q^58 + (b11 + b10 + b7 - 3b3 + 3b2 + 3) * q^59 + (-b7 + 2b3 - 2b2 + 3) * q^61 + (-b7 - 2b2 + 1) q^62 + q^64 + (b11 + b10 - b3 - b2 - 1) * q^65 + (-2b11 - b10 - b9 + 2b7 + b3 + 4b2 + b1 + 2) q^67 + (b8 - b7 + b6 - b5 + b2) * q^68 + (-b10 + 1) * q^70 + (-2b11 - 2b9 + b7 + 2b3 + 2b1 - 2) * q^71 + (2b11 - 2b10 - b6 + 3b4 - b2 - 2b1 + 3) * q^73 + (-2b8 + b6 + 2b3) * q^74 + (b11 + b10 + b9 + b8 + b6 + b4 - b3 + b1) * q^76 + (b11 + b9 + 2b8 - 2b7 - 3b6 - b5 - 2b4 - b3 - b2 - b1) * q^77 + (b7 - b3 - 4b2 - 1) q^79 + (b4 + 1) * q^80 + (-4b11 - 4b10 - 4b9 - 2b8 - b6 - b5 - 4b4 + 2b3 - 4b1) q^82 + (b11 - b10 + 3b9 + 2b8 - b6 - b2 + 2b1) q^83 + (b8 + b6 - b5 - b3) * q^85 + (-b11 + b10 + b9 + b7 + b6 + b5 + 5b4 + b2 + 2b1 + 5) * q^86 + (-b5 - b4) * q^88 + (-3b11 - 3b10 - 3b9 - 3b8 + 2b6 + 3b3 - 3b1) q^89 + (2b11 + 2b10 + 2b9 + 2b8 - b7 + b6 + 3b4 - 3b3 - 2b2 + b1) q^91 + (-b9 - b7 - b5 + b4 - b1 + 1) * q^92 + (2b11 + b10 + b9 - b7 - b2 - b1 - 1) q^94 + (b11 + b10 - b3 - b2 - 1) * q^95 + (-b11 + b10 + b9 - 5b4 + 2b1 - 5) * q^97 + (b11 - 2b9 - 2b1 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 6 * q^5 + 4 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 6 q^{5} + 4 q^{7} + 12 q^{8} + 6 q^{10} + 7 q^{11} - 2 q^{13} + 4 q^{14} + 12 q^{16} - q^{17} - 2 q^{19} + 6 q^{20} + 7 q^{22} + 9 q^{23} - 6 q^{25} - 2 q^{26} + 4 q^{28} - 3 q^{29} + 18 q^{31} + 12 q^{32} - q^{34} + 8 q^{35} + 6 q^{37} - 2 q^{38} + 6 q^{40} + 11 q^{41} + 23 q^{43} + 7 q^{44} + 9 q^{46} + 2 q^{47} + 24 q^{49} - 6 q^{50} - 2 q^{52} + 4 q^{53} + 14 q^{55} + 4 q^{56} - 3 q^{58} + 22 q^{59} + 50 q^{61} + 18 q^{62} + 12 q^{64} - 4 q^{65} + 4 q^{67} - q^{68} + 8 q^{70} - 22 q^{71} + 24 q^{73} + 6 q^{74} - 2 q^{76} + 11 q^{77} + 2 q^{79} + 6 q^{80} + 11 q^{82} - 4 q^{83} + q^{85} + 23 q^{86} + 7 q^{88} - 2 q^{89} - 8 q^{91} + 9 q^{92} + 2 q^{94} - 4 q^{95} - 36 q^{97} + 24 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 6 * q^5 + 4 * q^7 + 12 * q^8 + 6 * q^10 + 7 * q^11 - 2 * q^13 + 4 * q^14 + 12 * q^16 - q^17 - 2 * q^19 + 6 * q^20 + 7 * q^22 + 9 * q^23 - 6 * q^25 - 2 * q^26 + 4 * q^28 - 3 * q^29 + 18 * q^31 + 12 * q^32 - q^34 + 8 * q^35 + 6 * q^37 - 2 * q^38 + 6 * q^40 + 11 * q^41 + 23 * q^43 + 7 * q^44 + 9 * q^46 + 2 * q^47 + 24 * q^49 - 6 * q^50 - 2 * q^52 + 4 * q^53 + 14 * q^55 + 4 * q^56 - 3 * q^58 + 22 * q^59 + 50 * q^61 + 18 * q^62 + 12 * q^64 - 4 * q^65 + 4 * q^67 - q^68 + 8 * q^70 - 22 * q^71 + 24 * q^73 + 6 * q^74 - 2 * q^76 + 11 * q^77 + 2 * q^79 + 6 * q^80 + 11 * q^82 - 4 * q^83 + q^85 + 23 * q^86 + 7 * q^88 - 2 * q^89 - 8 * q^91 + 9 * q^92 + 2 * q^94 - 4 * 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.i.h

Level	$1890$
Weight	$2$
Character orbit	1890.i
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:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( - 5 \nu^{11} - 2 \nu^{10} + 11 \nu^{9} - 16 \nu^{8} - 5 \nu^{7} + 15 \nu^{6} - 7 \nu^{5} - 70 \nu^{4} + \cdots + 96 ) / 96 \) (-5v^11 - 2v^10 + 11v^9 - 16v^8 - 5v^7 + 15v^6 - 7v^5 - 70v^4 + 76v^3 + 16v^2 - 160*v + 96) / 96
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} - 2 \nu^{10} + \nu^{9} + 4 \nu^{8} - 7 \nu^{7} + \nu^{6} + 7 \nu^{5} + 2 \nu^{4} - 28 \nu^{3} + \cdots - 64 ) / 32 \) (v^11 - 2v^10 + v^9 + 4v^8 - 7v^7 + v^6 + 7v^5 + 2v^4 - 28v^3 + 32v^2 - 16v - 64) / 32
\(\beta_{3}\)	\(=\)	\( ( - \nu^{11} + \nu^{10} + \nu^{9} - 5 \nu^{8} + 3 \nu^{7} + 6 \nu^{6} - 8 \nu^{5} - 9 \nu^{4} + \cdots + 48 ) / 16 \) (-v^11 + v^10 + v^9 - 5v^8 + 3v^7 + 6v^6 - 8v^5 - 9v^4 + 26v^3 + 12v^2 - 48v + 48) / 16
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{11} - 11 \nu^{10} + 13 \nu^{9} - \nu^{8} - \nu^{7} - 2 \nu^{6} + 24 \nu^{5} + 11 \nu^{4} + \cdots - 16 ) / 144 \) (3v^11 - 11v^10 + 13v^9 - v^8 - v^7 - 2v^6 + 24v^5 + 11v^4 - 94v^3 + 172v^2 - 152*v - 16) / 144
\(\beta_{5}\)	\(=\)	\( ( 27 \nu^{11} - 26 \nu^{10} + 31 \nu^{9} + 32 \nu^{8} - \nu^{7} - 41 \nu^{6} - 27 \nu^{5} + 134 \nu^{4} + \cdots + 320 ) / 288 \) (27v^11 - 26v^10 + 31v^9 + 32v^8 - v^7 - 41v^6 - 27v^5 + 134v^4 - 184v^3 + 328v^2 - 80v + 320) / 288
\(\beta_{6}\)	\(=\)	\( ( 3 \nu^{11} - 10 \nu^{10} + 23 \nu^{9} - 32 \nu^{8} + 7 \nu^{7} - \nu^{6} - 3 \nu^{5} - 26 \nu^{4} + \cdots + 352 ) / 96 \) (3v^11 - 10v^10 + 23v^9 - 32v^8 + 7v^7 - v^6 - 3v^5 - 26v^4 - 56v^3 + 248v^2 - 352v + 352) / 96
\(\beta_{7}\)	\(=\)	\( ( \nu^{11} - 5 \nu^{10} - \nu^{9} + 5 \nu^{8} - 11 \nu^{7} - 6 \nu^{6} + 20 \nu^{5} + 41 \nu^{4} + \cdots - 96 ) / 48 \) (v^11 - 5v^10 - v^9 + 5v^8 - 11v^7 - 6v^6 + 20v^5 + 41v^4 - 62v^3 + 40v^2 + 80*v - 96) / 48
\(\beta_{8}\)	\(=\)	\( ( 3 \nu^{11} - 38 \nu^{10} + 139 \nu^{9} - 172 \nu^{8} + 35 \nu^{7} + 187 \nu^{6} - 3 \nu^{5} + \cdots + 1568 ) / 288 \) (3v^11 - 38v^10 + 139v^9 - 172v^8 + 35v^7 + 187v^6 - 3v^5 - 250v^4 - 4v^3 + 1432v^2 - 2240*v + 1568) / 288
\(\beta_{9}\)	\(=\)	\( ( - 3 \nu^{11} + 23 \nu^{10} - 37 \nu^{9} + 25 \nu^{8} + 25 \nu^{7} - 22 \nu^{6} - 60 \nu^{5} + \cdots - 176 ) / 144 \) (-3v^11 + 23v^10 - 37v^9 + 25v^8 + 25v^7 - 22v^6 - 60v^5 + 85v^4 + 82v^3 - 520v^2 + 416*v - 176) / 144
\(\beta_{10}\)	\(=\)	\( ( - 9 \nu^{11} + 35 \nu^{10} - 49 \nu^{9} + 13 \nu^{8} + 37 \nu^{7} - 4 \nu^{6} - 36 \nu^{5} + \cdots - 32 ) / 144 \) (-9v^11 + 35v^10 - 49v^9 + 13v^8 + 37v^7 - 4v^6 - 36v^5 + 19v^4 + 328v^3 - 508v^2 + 368*v - 32) / 144
\(\beta_{11}\)	\(=\)	\( ( 21 \nu^{11} - 76 \nu^{10} + 77 \nu^{9} - 38 \nu^{8} - 71 \nu^{7} + 35 \nu^{6} + 69 \nu^{5} + \cdots + 352 ) / 288 \) (21v^11 - 76v^10 + 77v^9 - 38v^8 - 71v^7 + 35v^6 + 69v^5 - 32v^4 - 500v^3 + 1136v^2 - 928*v + 352) / 288

\(\nu\)	\(=\)	\( ( -\beta_{10} - \beta_{9} - \beta_{6} - 2\beta_{4} - 2\beta_{2} - \beta _1 - 1 ) / 3 \) (-b10 - b9 - b6 - 2b4 - 2b2 - b1 - 1) / 3
\(\nu^{2}\)	\(=\)	\( ( -\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{3} - 2\beta_1 ) / 3 \) (-b11 - b10 - 2b9 - b8 + 2b3 - 2*b1) / 3
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{5} + 2\beta _1 - 3 ) / 3 \) (2b10 - b9 + b7 + 2b5 + 2*b1 - 3) / 3
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 1 ) / 3 \) (-2b11 + b10 + 2b9 + b8 + 4b7 + b6 + 2b5 + 2b4 + b3 + 2b2 + 2*b1 + 1) / 3
\(\nu^{5}\)	\(=\)	\( ( 5\beta_{10} - \beta_{9} + 3\beta_{8} + \beta_{7} - \beta_{5} + 3\beta_{4} - 6\beta_{3} + 2\beta _1 + 3 ) / 3 \) (5b10 - b9 + 3b8 + b7 - b5 + 3b4 - 6b3 + 2*b1 + 3) / 3
\(\nu^{6}\)	\(=\)	\( ( 6 \beta_{11} + 3 \beta_{9} + 6 \beta_{8} - \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \cdots + 2 ) / 3 \) (6b11 + 3b9 + 6b8 - b7 - 10b6 - 2b5 - 5b4 - 3b3 - 5b2 - 3*b1 + 2) / 3
\(\nu^{7}\)	\(=\)	\( ( 3 \beta_{11} - \beta_{10} - \beta_{9} - 6 \beta_{7} - 7 \beta_{6} - 3 \beta_{5} + 19 \beta_{4} + \cdots - 7 ) / 3 \) (3b11 - b10 - b9 - 6b7 - 7b6 - 3b5 + 19b4 - 14b2 - 7*b1 - 7) / 3
\(\nu^{8}\)	\(=\)	\( ( - 7 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} + \cdots + 32 ) / 3 \) (-7b11 - 4b10 - 2b9 - 4b8 - 4b7 - 7b6 + 4b5 + 25b4 + 8b3 + 7b2 + b1 + 32) / 3
\(\nu^{9}\)	\(=\)	\( ( - 21 \beta_{11} - 13 \beta_{10} - \beta_{9} + 6 \beta_{8} + 6 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} + \cdots - 4 ) / 3 \) (-21b11 - 13b10 - b9 + 6b8 + 6b7 - 4b6 + 12b5 - 2b4 - 3b3 - 2b2 + 11b1 - 4) / 3
\(\nu^{10}\)	\(=\)	\( ( - 35 \beta_{11} - 14 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 14 \beta_{6} + \beta_{5} + \cdots + 14 ) / 3 \) (-35b11 - 14b10 - 10b9 + 4b8 + 2b7 + 14b6 + b5 - 29b4 + 4b3 + 28b2 - 16b1 + 14) / 3
\(\nu^{11}\)	\(=\)	\( ( 30 \beta_{11} + 29 \beta_{10} + 2 \beta_{9} + 21 \beta_{8} - 14 \beta_{7} + 3 \beta_{6} + 14 \beta_{5} + \cdots - 78 ) / 3 \) (30b11 + 29b10 + 2b9 + 21b8 - 14b7 + 3b6 + 14b5 - 75b4 - 42b3 - 3b2 - 7*b1 - 78) / 3

\(n\)	\(757\)	\(1081\)	\(1541\)
\(\chi(n)\)	\(1\)	\(\beta_{4}\)	\(\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 991.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 1)^{12} \) (T - 1)^12
$3$	\( T^{12} \) T^12
$5$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$7$	\( (T^{6} - 2 T^{5} + \cdots + 343)^{2} \) (T^6 - 2T^5 - 4T^4 + 31T^3 - 28T^2 - 98*T + 343)^2
$11$	\( T^{12} - 7 T^{11} + \cdots + 531441 \) T^12 - 7T^11 + 57T^10 - 208T^9 + 1120T^8 - 3390T^7 + 14424T^6 - 30303T^5 + 87408T^4 - 128304T^3 + 332424T^2 - 354294*T + 531441
$13$	\( T^{12} + 2 T^{11} + \cdots + 9 \) 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
$23$	\( T^{12} - 9 T^{11} + \cdots + 59049 \) T^12 - 9T^11 + 90T^10 - 351T^9 + 2628T^8 - 12150T^7 + 46575T^6 - 116397T^5 + 221454T^4 - 285768T^3 + 273375T^2 - 157464*T + 59049
$29$	\( T^{12} + 3 T^{11} + \cdots + 531441 \) 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
$31$	\( (T^{6} - 9 T^{5} + \cdots + 109)^{2} \) (T^6 - 9T^5 - 36T^4 + 308T^3 - 342T^2 - 324*T + 109)^2
$37$	\( T^{12} - 6 T^{11} + \cdots + 32137561 \) 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
$47$	\( (T^{6} - T^{5} - 62 T^{4} + \cdots + 81)^{2} \) (T^6 - T^5 - 62T^4 + 159T^3 - 15T^2 - 162T + 81)^2
$53$	\( T^{12} + \cdots + 503418969 \) 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
$59$	\( (T^{6} - 11 T^{5} + \cdots + 276399)^{2} \) (T^6 - 11T^5 - 203T^4 + 2520T^3 + 6240T^2 - 129645*T + 276399)^2
$61$	\( (T^{6} - 25 T^{5} + \cdots - 3821)^{2} \) (T^6 - 25T^5 + 134T^4 + 674T^3 - 5734T^2 + 10100*T - 3821)^2
$67$	\( (T^{6} - 2 T^{5} + \cdots + 21339)^{2} \) (T^6 - 2T^5 - 225T^4 + 427T^3 + 11110T^2 - 36321*T + 21339)^2
$71$	\( (T^{6} + 11 T^{5} + \cdots + 27)^{2} \) (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
$79$	\( (T^{6} - T^{5} + \cdots + 247267)^{2} \) (T^6 - T^5 - 295T^4 + 1310T^3 + 20885T^2 - 159430T + 247267)^2
$83$	\( T^{12} + \cdots + 130439241 \) 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
show more
show less