LMFDB - Newform orbit 1120.2.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(111,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("1120.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.h (of order $2$, degree $1$, 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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} - q^{5} - \beta_{8} q^{7} + ( - \beta_{12} - 1) q^{9}+O(q^{10}) $ q + b2 * q^3 - q^5 - b8 * q^7 + (-b12 - 1) * q^9	$ q + \beta_{2} q^{3} - q^{5} - \beta_{8} q^{7} + ( - \beta_{12} - 1) q^{9} + \beta_{10} q^{11} - \beta_{5} q^{13} - \beta_{2} q^{15} + (\beta_{13} - \beta_{2}) q^{17} + (\beta_{13} + \beta_{11} - \beta_{7}) q^{19} + ( - \beta_{15} + \beta_{7}) q^{21} + ( - \beta_{14} - \beta_{2}) q^{23} + q^{25} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{2}) q^{27}+ \cdots + ( - \beta_{15} - \beta_{12} - \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) $ q + b2 * q^3 - q^5 - b8 * q^7 + (-b12 - 1) * q^9 + b10 * q^11 - b5 * q^13 - b2 * q^15 + (b13 - b2) * q^17 + (b13 + b11 - b7) * q^19 + (-b15 + b7) * q^21 + (-b14 - b2) * q^23 + q^25 + (-b15 - b14 + b13 + b9 + b7 + b4 - b2) * q^27 + (b11 - b2) * q^29 + (-b15 + b10 + b6 + b3 - b1 - 2) * q^31 + (b14 - b8 - b7 + b4 + b2 + b1) * q^33 + b8 * q^35 + (-b15 + b13 + b11 + b9) * q^37 + (-b15 + b13 + b9 + b8 + b7 - b4 - b1) * q^39 + (-b15 + b14 + b11 + b9 + b8 + b7 - b4 + 2b2 - b1) q^41 + (-b12 - b8 + b6 + b5 - b1) * q^43 + (b12 + 1) * q^45 + (-b12 + b10 + b8 - b6 - b5 + b1) * q^47 + (-b15 - b14 + b13 + b10 + b9 + b8 + b3 - 1) * q^49 + (-b15 + b12 + b5 + b3 - b1 + 2) * q^51 + (b15 + b14 - b13 - b9 - b8 - b7 - b4 + b1) * q^53 - b10 * q^55 + (-b15 - b12 - b9 - 2b8 + 2b6 + b5 - 2b1 - 2) q^57 + (b14 - b13 + b11 - b8 + b7 + b2 + b1) * q^59 + (-b15 + b12 + b9 + b5 + 2b3 - 2b1) * q^61 + (-b13 + 2b12 + b9 + b8 + b7 - b6 - b4 + b3 + b2 - b1 + 2) q^63 + b5 * q^65 + (-b15 - b12 + b10 + b8 - b5 + b3 - 2) * q^67 + (b12 + 2b10 - b5 + 2) q^69 + (-b15 + b11 + b9 + b4 + b2) * q^71 + (-b14 + b13 + b8 + b7 + b4 - b1) * q^73 + b2 * q^75 + (b13 - b12 + b10 - b9 - b4 - b3 + b2 - b1) * q^77 + (2b14 - b13 + b11 - b8 - b7 + b2 + b1) q^79 + (-b15 + 2b12 + b9 + 2b8 + 2b3 + 1) q^81 + (b15 - b14 + b13 - 2b11 - b9 - b7 + b4 - 2b2) * q^83 + (-b13 + b2) * q^85 + (b12 + b10 + b6 - b5 + 4) * q^87 + (-b14 - b11 + b8 - b7 + b4 - 2b2 - b1) q^89 + (-b15 + b14 + b12 + 2b9 + b8 - b5 + b3 + b2 + 2) q^91 + (b14 - 2b13 - b11 + b8 + b7 - b4 - b1) q^93 + (-b13 - b11 + b7) * q^95 + (-b13 + 2b11 - 2b8 + b2 + 2b1) q^97 + (-b15 - b12 - b10 - 2b9 - 2b8 + 2b6 + b5 - b3 - b1 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 - 16 * q^9	$ 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2*x^12 + 6*x^11 - 12*x^9 + 8*x^8 - 24*x^7 + 48*x^5 - 32*x^4 - 128*x + 256 :

$\beta_{1}$	$=$	$ ( \nu^{11} + \nu^{10} + 2\nu^{8} - 2\nu^{7} - 2\nu^{6} - 8\nu^{2} - 32\nu - 16 ) / 16 $ (v^11 + v^10 + 2v^8 - 2v^7 - 2v^6 - 8v^2 - 32*v - 16) / 16
$\beta_{2}$	$=$	$ ( \nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + \cdots + 128 \nu ) / 128 $ (v^15 + v^14 + 2v^13 + 4v^12 - 2v^11 - 6v^10 + 4v^9 - 4v^8 + 16v^7 - 8v^6 - 48v^5 - 16v^4 - 64v^3 - 64v^2 + 128*v) / 128
$\beta_{3}$	$=$	$ ( -\nu^{14} + 3\nu^{12} - 6\nu^{8} + 4\nu^{7} + 8\nu^{6} - 16\nu^{5} + 8\nu^{4} - 16\nu^{3} - 96\nu^{2} + 32 ) / 32 $ (-v^14 + 3v^12 - 6v^8 + 4v^7 + 8v^6 - 16v^5 + 8v^4 - 16v^3 - 96v^2 + 32) / 32
$\beta_{4}$	$=$	$ ( - \nu^{15} - \nu^{14} - 6 \nu^{13} + 2 \nu^{11} - 10 \nu^{10} + 20 \nu^{9} - 20 \nu^{8} + \cdots + 128 \nu^{3} ) / 128 $ (-v^15 - v^14 - 6v^13 + 2v^11 - 10v^10 + 20v^9 - 20v^8 - 16v^7 + 24v^6 - 16v^5 + 112v^4 + 128v^3) / 128
$\beta_{5}$	$=$	$ ( - \nu^{15} + 3 \nu^{14} - 2 \nu^{13} + 10 \nu^{11} - 2 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} + \cdots + 256 ) / 128 $ (-v^15 + 3v^14 - 2v^13 + 10v^11 - 2v^10 - 4v^9 + 12v^8 - 48v^7 - 8v^6 + 48v^5 - 80v^4 - 128v^2 - 384v + 256) / 128
$\beta_{6}$	$=$	$ ( - \nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + \cdots + 128 ) / 64 $ (-v^15 + v^14 + 2v^11 - 6v^10 + 12v^8 - 8v^7 + 24v^6 - 48v^4 + 32v^3 + 128v + 128) / 64
$\beta_{7}$	$=$	$ ( - \nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + \cdots + 128 ) / 64 $ (-v^15 + v^14 + 2v^11 - 6v^10 + 12v^8 - 8v^7 + 24v^6 - 48v^4 + 32v^3 - 128v + 128) / 64
$\beta_{8}$	$=$	$ ( - \nu^{15} + \nu^{14} + 6 \nu^{11} - 10 \nu^{10} - 8 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 32 \nu^{6} + \cdots + 320 ) / 64 $ (-v^15 + v^14 + 6v^11 - 10v^10 - 8v^9 + 4v^8 - 16v^7 + 32v^6 + 16v^5 - 80v^4 - 32*v^2 + 320) / 64
$\beta_{9}$	$=$	$ ( - 3 \nu^{15} + \nu^{14} + 2 \nu^{13} + 14 \nu^{11} + 10 \nu^{10} - 12 \nu^{9} + 4 \nu^{8} + \cdots - 128 \nu ) / 128 $ (-3v^15 + v^14 + 2v^13 + 14v^11 + 10v^10 - 12v^9 + 4v^8 - 16v^7 + 40v^6 + 112v^5 + 16v^4 - 64v^3 - 128v^2 - 128*v) / 128
$\beta_{10}$	$=$	$ ( - 3 \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 14 \nu^{11} - 6 \nu^{10} - 20 \nu^{9} + \cdots + 512 ) / 128 $ (-3v^15 + v^14 - 2v^13 + 4v^12 + 14v^11 - 6v^10 - 20v^9 + 12v^8 - 16v^7 + 56v^6 + 112v^5 - 80v^4 - 64v^2 - 256*v + 512) / 128
$\beta_{11}$	$=$	$ ( - \nu^{15} - \nu^{14} - \nu^{13} - \nu^{12} + 2 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + \cdots + 64 ) / 32 $ (-v^15 - v^14 - v^13 - v^12 + 2v^11 - 2v^10 - 2v^9 + 2v^8 + 12v^6 + 32v^5 + 8v^4 + 16v^3 - 16v^2 - 32v + 64) / 32
$\beta_{12}$	$=$	$ ( 3 \nu^{15} + 3 \nu^{14} - 6 \nu^{13} - 8 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 128 $ (3v^15 + 3v^14 - 6v^13 - 8v^12 - 6v^11 - 2v^10 + 20v^9 + 12v^8 - 16v^7 - 40v^6 - 80v^5 + 48v^4 + 256v^3 + 128v^2 - 256) / 128
$\beta_{13}$	$=$	$ ( \nu^{15} + \nu^{12} + 4 \nu^{10} + 8 \nu^{9} - 2 \nu^{8} - 12 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} + \cdots - 128 ) / 32 $ (v^15 + v^12 + 4v^10 + 8v^9 - 2v^8 - 12v^6 - 16v^5 + 8v^4 + 16v^3 - 48v^2 - 64*v - 128) / 32
$\beta_{14}$	$=$	$ ( 3 \nu^{15} - \nu^{14} - 6 \nu^{13} + 4 \nu^{12} + 2 \nu^{11} + 22 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} + \cdots - 512 ) / 128 $ (3v^15 - v^14 - 6v^13 + 4v^12 + 2v^11 + 22v^10 + 36v^9 - 28v^8 - 16v^7 + 8v^6 - 48v^5 + 144v^4 - 192v^2 - 256*v - 512) / 128
$\beta_{15}$	$=$	$ ( - 5 \nu^{15} - \nu^{14} - 2 \nu^{13} + 8 \nu^{12} + 18 \nu^{11} - 26 \nu^{10} - 20 \nu^{9} + \cdots + 1024 ) / 128 $ (-5v^15 - v^14 - 2v^13 + 8v^12 + 18v^11 - 26v^10 - 20v^9 + 12v^8 - 48v^7 + 152v^6 + 144v^5 - 80v^4 + 64v^3 - 384v^2 - 128v + 1024) / 128

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

$\nu$	$=$	$ ( -\beta_{7} + \beta_{6} ) / 4 $ (-b7 + b6) / 4
$\nu^{2}$	$=$	$ ( -\beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 4 $ (-b12 - b11 + b10 - b9 - b5 + b4 - b3 - b2 + b1) / 4
$\nu^{3}$	$=$	$ ( - 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + \cdots - \beta_1 ) / 4 $ (-2b14 + 2b13 + b12 - b11 + b10 + b9 - b5 + b4 + b3 - 3*b2 - b1) / 4
$\nu^{4}$	$=$	$ ( 2 \beta_{15} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + \cdots + 4 ) / 4 $ (2b15 - 2b13 + b12 - b11 - b10 + b9 - 2b8 - 2b6 - b5 + b4 - b3 + 3b2 + 3b1 + 4) / 4
$\nu^{5}$	$=$	$ ( 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots - 4 ) / 4 $ (2b15 + 2b13 - b12 + b11 + b10 - b9 - 2b8 - 2b7 + b5 - b4 - 3b3 + b2 - 3b1 - 4) / 4
$\nu^{6}$	$=$	$ ( 2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots - 4 ) / 4 $ (2b15 + 4b14 - 2b13 + b12 - b11 - b10 + b9 + 2b8 + 4b7 - 2b6 - 5b5 - 3b4 - b3 - b2 - b1 - 4) / 4
$\nu^{7}$	$=$	$ ( - 6 \beta_{15} + 2 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + \cdots + 12 ) / 4 $ (-6b15 + 2b13 + 5b12 + 7b11 + 7b10 + 5b9 + 2b8 + 4b7 - 2b6 - 5b5 - 3b4 + 3b3 + 11b2 - 5b1 + 12) / 4
$\nu^{8}$	$=$	$ ( 6 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} - 7 \beta_{9} + \cdots + 4 ) / 4 $ (6b15 + 4b14 - 6b13 + b12 + 3b11 - b10 - 7b9 - 10b8 + 6b6 - b5 - 7b4 - 5b3 + 11b2 + 15*b1 + 4) / 4
$\nu^{9}$	$=$	$ ( - 2 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + \cdots + 36 ) / 4 $ (-2b15 + 4b14 + 10b13 - 11b12 + 7b11 - 5b10 - 3b9 - 2b8 + 4b7 + 10b6 + 3b5 + 5b4 - 9b3 + 7b2 - 5*b1 + 36) / 4
$\nu^{10}$	$=$	$ ( - 10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + \beta_{12} - 5 \beta_{11} + 7 \beta_{10} + 9 \beta_{9} + \cdots + 52 ) / 4 $ (-10b15 + 4b14 + 2b13 + b12 - 5b11 + 7b10 + 9b9 - 2b8 + 14b6 - b5 - 7b4 + 11b3 - 29b2 - 9b1 + 52) / 4
$\nu^{11}$	$=$	$ ( - 10 \beta_{15} - 4 \beta_{14} + 10 \beta_{13} + \beta_{12} + 3 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + \cdots + 20 ) / 4 $ (-10b15 - 4b14 + 10b13 + b12 + 3b11 + 15b10 + 9b9 + 30b8 - 16b7 - 2b6 - 25b5 + 17b4 - 5b3 + 19b2 + 39b1 + 20) / 4
$\nu^{12}$	$=$	$ ( 14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} + 47 \beta_{10} + \cdots - 60 ) / 4 $ (14b15 - 12b14 + 10b13 - 23b12 - 37b11 + 47b10 - 31b9 - 42b8 - 16b7 + 22b6 - b5 + 25b4 - 5b3 - 13b2 + 15b1 - 60) / 4
$\nu^{13}$	$=$	$ ( 30 \beta_{15} - 44 \beta_{14} + 42 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} - 49 \beta_{10} + \cdots + 68 ) / 4 $ (30b15 - 44b14 + 42b13 - 23b12 - 37b11 - 49b10 + 33b9 + 6b8 + 24b7 - 34b6 - 33b5 + 25b4 - 21b3 - 29b2 + 15*b1 + 68) / 4
$\nu^{14}$	$=$	$ ( - 18 \beta_{15} + 4 \beta_{14} - 22 \beta_{13} + 57 \beta_{12} - 21 \beta_{11} + 31 \beta_{10} + \cdots + 36 ) / 4 $ (-18b15 + 4b14 - 22b13 + 57b12 - 21b11 + 31b10 + 81b9 - 26b8 + 32b7 - 10b6 + 31b5 - 7b4 + 11b3 + 83b2 - 81*b1 + 36) / 4
$\nu^{15}$	$=$	$ ( 94 \beta_{15} + 52 \beta_{14} + 10 \beta_{13} + 33 \beta_{12} - 13 \beta_{11} + 7 \beta_{10} - 71 \beta_{9} + \cdots - 60 ) / 4 $ (94b15 + 52b14 + 10b13 + 33b12 - 13b11 + 7b10 - 71b9 + 54b8 - 64b7 - 90b6 - 89b5 - 79b4 - 93b3 + 75b2 + 71*b1 - 60) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

111.1

−0.275585	−	1.38710i
1.07046	+	0.924187i
1.41214	−	0.0765298i
0.244064	+	1.39299i
−1.24098	+	0.678208i
−1.38133	−	0.303194i
1.14218	−	0.833926i
−0.470943	−	1.33350i
−0.470943	+	1.33350i
1.14218	+	0.833926i
−1.38133	+	0.303194i
−1.24098	−	0.678208i
0.244064	−	1.39299i
1.41214	+	0.0765298i
1.07046	−	0.924187i
−0.275585	+	1.38710i

−

3.19977i

−1.00000

−2.59303

−

0.525543i

−7.23851

111.2

−

2.99734i

−1.00000

0.183359

2.63939i

−5.98405

111.3

−

2.21915i

−1.00000

1.20923

−

2.35325i

−1.92464

111.4

−

1.68420i

−1.00000

0.695780

−

2.55262i

0.163484

111.5

−

1.61069i

−1.00000

2.13463

1.56312i

0.405694

111.6

−

1.34113i

−1.00000

−1.28003

2.31550i

1.20136

111.7

−

0.586834i

−1.00000

−2.52442

0.792014i

2.65563

111.8

−

0.528177i

−1.00000

2.17448

−

1.50719i

2.72103

111.9

0.528177i

−1.00000

2.17448

1.50719i

2.72103

111.10

0.586834i

−1.00000

−2.52442

−

0.792014i

2.65563

111.11

1.34113i

−1.00000

−1.28003

−

2.31550i

1.20136

111.12

1.61069i

−1.00000

2.13463

−

1.56312i

0.405694

111.13

1.68420i

−1.00000

0.695780

2.55262i

0.163484

111.14

2.21915i

−1.00000

1.20923

2.35325i

−1.92464

111.15

2.99734i

−1.00000

0.183359

−

2.63939i

−5.98405

111.16

3.19977i

−1.00000

−2.59303

0.525543i

−7.23851

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1120.2.h.a		16
4.b	odd	2	1		280.2.h.a	✓	16
7.b	odd	2	1		1120.2.h.b		16
8.b	even	2	1		280.2.h.b	yes	16
8.d	odd	2	1		1120.2.h.b		16
28.d	even	2	1		280.2.h.b	yes	16
56.e	even	2	1	inner	1120.2.h.a		16
56.h	odd	2	1		280.2.h.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.2.h.a	✓	16	4.b	odd	2	1
280.2.h.a	✓	16	56.h	odd	2	1
280.2.h.b	yes	16	8.b	even	2	1
280.2.h.b	yes	16	28.d	even	2	1
1120.2.h.a		16	1.a	even	1	1	trivial
1120.2.h.a		16	56.e	even	2	1	inner
1120.2.h.b		16	7.b	odd	2	1
1120.2.h.b		16	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{13}^{8} - 62T_{13}^{6} - 12T_{13}^{5} + 1213T_{13}^{4} + 228T_{13}^{3} - 7792T_{13}^{2} + 1232T_{13} + 10032 $ T13^8 - 62*T13^6 - 12*T13^5 + 1213*T13^4 + 228*T13^3 - 7792*T13^2 + 1232*T13 + 10032 acting on $S_{2}^{\mathrm{new}}(1120, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} + 32 T^{14} + \cdots + 576 $ T^16 + 32T^14 + 398T^12 + 2468T^10 + 8217T^8 + 14676T^6 + 13024T^4 + 4720*T^2 + 576
$5$	$ (T + 1)^{16} $ (T + 1)^16
$7$	$ T^{16} + 4 T^{14} + \cdots + 5764801 $ T^16 + 4T^14 + 36T^13 + 4T^12 + 204T^11 + 700T^10 + 904T^9 + 5814T^8 + 6328T^7 + 34300T^6 + 69972T^5 + 9604T^4 + 605052T^3 + 470596*T^2 + 5764801
$11$	$ (T^{8} - 2 T^{7} + \cdots + 2400)^{2} $ (T^8 - 2T^7 - 50T^6 + 52T^5 + 761T^4 - 138T^3 - 3932T^2 - 1960*T + 2400)^2
$13$	$ (T^{8} - 62 T^{6} + \cdots + 10032)^{2} $ (T^8 - 62T^6 - 12T^5 + 1213T^4 + 228T^3 - 7792T^2 + 1232T + 10032)^2
$17$	$ T^{16} + 136 T^{14} + \cdots + 2359296 $ T^16 + 136T^14 + 7086T^12 + 178476T^10 + 2255465T^8 + 13596164T^6 + 34001456T^4 + 27167680*T^2 + 2359296
$19$	$ T^{16} + 184 T^{14} + \cdots + 1327104 $ T^16 + 184T^14 + 12736T^12 + 423168T^10 + 6949696T^8 + 49375232T^6 + 81127424T^4 + 26099712*T^2 + 1327104
$23$	$ T^{16} + 160 T^{14} + \cdots + 7573504 $ T^16 + 160T^14 + 10096T^12 + 325872T^10 + 5807360T^8 + 57190400T^6 + 289346560T^4 + 585268224*T^2 + 7573504
$29$	$ T^{16} + 172 T^{14} + \cdots + 65536 $ T^16 + 172T^14 + 9718T^12 + 217548T^10 + 2200529T^8 + 9795008T^6 + 15058432T^4 + 2408448*T^2 + 65536
$31$	$ (T^{8} + 8 T^{7} + \cdots + 98304)^{2} $ (T^8 + 8T^7 - 92T^6 - 440T^5 + 3496T^4 + 3040T^3 - 33920T^2 - 2816*T + 98304)^2
$37$	$ T^{16} + \cdots + 4063297536 $ T^16 + 252T^14 + 26192T^12 + 1448768T^10 + 45832768T^8 + 825741568T^6 + 7832304640T^4 + 30390333440*T^2 + 4063297536
$41$	$ T^{16} + \cdots + 6423183360000 $ T^16 + 424T^14 + 73760T^12 + 6823040T^10 + 364052032T^8 + 11370832896T^6 + 201056874496T^4 + 1819708211200*T^2 + 6423183360000
$43$	$ (T^{8} - 2 T^{7} + \cdots - 69504)^{2} $ (T^8 - 2T^7 - 128T^6 + 332T^5 + 4232T^4 - 14912T^3 - 17664T^2 + 91360*T - 69504)^2
$47$	$ (T^{8} - 166 T^{6} + \cdots - 10368)^{2} $ (T^8 - 166T^6 + 424T^5 + 5461T^4 - 28292T^3 + 36188T^2 + 5820T - 10368)^2
$53$	$ T^{16} + 404 T^{14} + \cdots + 262144 $ T^16 + 404T^14 + 58128T^12 + 3445312T^10 + 69431040T^8 + 251149312T^6 + 158445568T^4 + 27312128*T^2 + 262144
$59$	$ T^{16} + \cdots + 1090584576 $ T^16 + 400T^14 + 55552T^12 + 3287040T^10 + 82300416T^8 + 734793728T^6 + 2362048512T^4 + 2931687424*T^2 + 1090584576
$61$	$ (T^{8} - 4 T^{7} + \cdots + 10368)^{2} $ (T^8 - 4T^7 - 268T^6 + 1256T^5 + 15176T^4 - 85216T^3 + 26432T^2 + 51712*T + 10368)^2
$67$	$ (T^{8} + 10 T^{7} + \cdots + 165632)^{2} $ (T^8 + 10T^7 - 180T^6 - 1204T^5 + 7712T^4 + 42240T^3 - 55616T^2 - 197184*T + 165632)^2
$71$	$ T^{16} + \cdots + 2621440000 $ T^16 + 492T^14 + 82752T^12 + 6017280T^10 + 180376576T^8 + 1494106112T^6 + 4596301824T^4 + 5845811200*T^2 + 2621440000
$73$	$ T^{16} + \cdots + 28179280429056 $ T^16 + 612T^14 + 151344T^12 + 19483584T^10 + 1406521344T^8 + 57276760064T^6 + 1248988233728T^4 + 12310919774208*T^2 + 28179280429056
$79$	$ T^{16} + \cdots + 39806206534656 $ T^16 + 592T^14 + 145478T^12 + 19139020T^10 + 1444670401T^8 + 61858049132T^6 + 1373008608512T^4 + 12666930453504*T^2 + 39806206534656
$83$	$ T^{16} + \cdots + 88794464256 $ T^16 + 564T^14 + 109632T^12 + 8985456T^10 + 337953344T^8 + 6096578560T^6 + 51045990400T^4 + 165451399168*T^2 + 88794464256
$89$	$ T^{16} + \cdots + 347892350976 $ T^16 + 520T^14 + 95808T^12 + 8073856T^10 + 331338304T^8 + 6628880384T^6 + 61230546944T^4 + 245081571328*T^2 + 347892350976
$97$	$ T^{16} + \cdots + 17227945230336 $ T^16 + 888T^14 + 307550T^12 + 52726332T^10 + 4708474873T^8 + 212781536324T^6 + 4314469148336T^4 + 27670537460160*T^2 + 17227945230336
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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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - q^{5} - \beta_{8} q^{7} + ( - \beta_{12} - 1) q^{9}+O(q^{10}) \) q + b2 * q^3 - q^5 - b8 * q^7 + (-b12 - 1) * q^9	\( q + \beta_{2} q^{3} - q^{5} - \beta_{8} q^{7} + ( - \beta_{12} - 1) q^{9} + \beta_{10} q^{11} - \beta_{5} q^{13} - \beta_{2} q^{15} + (\beta_{13} - \beta_{2}) q^{17} + (\beta_{13} + \beta_{11} - \beta_{7}) q^{19} + ( - \beta_{15} + \beta_{7}) q^{21} + ( - \beta_{14} - \beta_{2}) q^{23} + q^{25} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{2}) q^{27}+ \cdots + ( - \beta_{15} - \beta_{12} - \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) q + b2 * q^3 - q^5 - b8 * q^7 + (-b12 - 1) * q^9 + b10 * q^11 - b5 * q^13 - b2 * q^15 + (b13 - b2) * q^17 + (b13 + b11 - b7) * q^19 + (-b15 + b7) * q^21 + (-b14 - b2) * q^23 + q^25 + (-b15 - b14 + b13 + b9 + b7 + b4 - b2) * q^27 + (b11 - b2) * q^29 + (-b15 + b10 + b6 + b3 - b1 - 2) * q^31 + (b14 - b8 - b7 + b4 + b2 + b1) * q^33 + b8 * q^35 + (-b15 + b13 + b11 + b9) * q^37 + (-b15 + b13 + b9 + b8 + b7 - b4 - b1) * q^39 + (-b15 + b14 + b11 + b9 + b8 + b7 - b4 + 2b2 - b1) q^41 + (-b12 - b8 + b6 + b5 - b1) * q^43 + (b12 + 1) * q^45 + (-b12 + b10 + b8 - b6 - b5 + b1) * q^47 + (-b15 - b14 + b13 + b10 + b9 + b8 + b3 - 1) * q^49 + (-b15 + b12 + b5 + b3 - b1 + 2) * q^51 + (b15 + b14 - b13 - b9 - b8 - b7 - b4 + b1) * q^53 - b10 * q^55 + (-b15 - b12 - b9 - 2b8 + 2b6 + b5 - 2b1 - 2) q^57 + (b14 - b13 + b11 - b8 + b7 + b2 + b1) * q^59 + (-b15 + b12 + b9 + b5 + 2b3 - 2b1) * q^61 + (-b13 + 2b12 + b9 + b8 + b7 - b6 - b4 + b3 + b2 - b1 + 2) q^63 + b5 * q^65 + (-b15 - b12 + b10 + b8 - b5 + b3 - 2) * q^67 + (b12 + 2b10 - b5 + 2) q^69 + (-b15 + b11 + b9 + b4 + b2) * q^71 + (-b14 + b13 + b8 + b7 + b4 - b1) * q^73 + b2 * q^75 + (b13 - b12 + b10 - b9 - b4 - b3 + b2 - b1) * q^77 + (2b14 - b13 + b11 - b8 - b7 + b2 + b1) q^79 + (-b15 + 2b12 + b9 + 2b8 + 2b3 + 1) q^81 + (b15 - b14 + b13 - 2b11 - b9 - b7 + b4 - 2b2) * q^83 + (-b13 + b2) * q^85 + (b12 + b10 + b6 - b5 + 4) * q^87 + (-b14 - b11 + b8 - b7 + b4 - 2b2 - b1) q^89 + (-b15 + b14 + b12 + 2b9 + b8 - b5 + b3 + b2 + 2) q^91 + (b14 - 2b13 - b11 + b8 + b7 - b4 - b1) q^93 + (-b13 - b11 + b7) * q^95 + (-b13 + 2b11 - 2b8 + b2 + 2b1) q^97 + (-b15 - b12 - b10 - 2b9 - 2b8 + 2b6 + b5 - b3 - b1 - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 - 16 * q^9	\( 16 q - 16 q^{5} - 16 q^{9} + 4 q^{11} + 4 q^{21} + 16 q^{25} - 16 q^{31} + 4 q^{43} + 16 q^{45} - 8 q^{49} + 40 q^{51} - 4 q^{55} - 16 q^{57} + 8 q^{61} + 28 q^{63} - 20 q^{67} + 40 q^{69} + 4 q^{77} + 24 q^{81} + 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 - 16 * q^9 + 4 * q^11 + 4 * q^21 + 16 * q^25 - 16 * q^31 + 4 * q^43 + 16 * q^45 - 8 * q^49 + 40 * q^51 - 4 * q^55 - 16 * q^57 + 8 * q^61 + 28 * q^63 - 20 * q^67 + 40 * q^69 + 4 * q^77 + 24 * q^81 + 72 * q^87 + 32 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1120.2.h.a

Level	$1120$
Weight	$2$
Character orbit	1120.h
Analytic conductor	$8.943$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(8.94324502638\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 \) x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{12} \)
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} + \nu^{10} + 2\nu^{8} - 2\nu^{7} - 2\nu^{6} - 8\nu^{2} - 32\nu - 16 ) / 16 \) (v^11 + v^10 + 2v^8 - 2v^7 - 2v^6 - 8v^2 - 32*v - 16) / 16
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + \cdots + 128 \nu ) / 128 \) (v^15 + v^14 + 2v^13 + 4v^12 - 2v^11 - 6v^10 + 4v^9 - 4v^8 + 16v^7 - 8v^6 - 48v^5 - 16v^4 - 64v^3 - 64v^2 + 128*v) / 128
\(\beta_{3}\)	\(=\)	\( ( -\nu^{14} + 3\nu^{12} - 6\nu^{8} + 4\nu^{7} + 8\nu^{6} - 16\nu^{5} + 8\nu^{4} - 16\nu^{3} - 96\nu^{2} + 32 ) / 32 \) (-v^14 + 3v^12 - 6v^8 + 4v^7 + 8v^6 - 16v^5 + 8v^4 - 16v^3 - 96v^2 + 32) / 32
\(\beta_{4}\)	\(=\)	\( ( - \nu^{15} - \nu^{14} - 6 \nu^{13} + 2 \nu^{11} - 10 \nu^{10} + 20 \nu^{9} - 20 \nu^{8} + \cdots + 128 \nu^{3} ) / 128 \) (-v^15 - v^14 - 6v^13 + 2v^11 - 10v^10 + 20v^9 - 20v^8 - 16v^7 + 24v^6 - 16v^5 + 112v^4 + 128v^3) / 128
\(\beta_{5}\)	\(=\)	\( ( - \nu^{15} + 3 \nu^{14} - 2 \nu^{13} + 10 \nu^{11} - 2 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} + \cdots + 256 ) / 128 \) (-v^15 + 3v^14 - 2v^13 + 10v^11 - 2v^10 - 4v^9 + 12v^8 - 48v^7 - 8v^6 + 48v^5 - 80v^4 - 128v^2 - 384v + 256) / 128
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + \cdots + 128 ) / 64 \) (-v^15 + v^14 + 2v^11 - 6v^10 + 12v^8 - 8v^7 + 24v^6 - 48v^4 + 32v^3 + 128v + 128) / 64
\(\beta_{7}\)	\(=\)	\( ( - \nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + \cdots + 128 ) / 64 \) (-v^15 + v^14 + 2v^11 - 6v^10 + 12v^8 - 8v^7 + 24v^6 - 48v^4 + 32v^3 - 128v + 128) / 64
\(\beta_{8}\)	\(=\)	\( ( - \nu^{15} + \nu^{14} + 6 \nu^{11} - 10 \nu^{10} - 8 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 32 \nu^{6} + \cdots + 320 ) / 64 \) (-v^15 + v^14 + 6v^11 - 10v^10 - 8v^9 + 4v^8 - 16v^7 + 32v^6 + 16v^5 - 80v^4 - 32*v^2 + 320) / 64
\(\beta_{9}\)	\(=\)	\( ( - 3 \nu^{15} + \nu^{14} + 2 \nu^{13} + 14 \nu^{11} + 10 \nu^{10} - 12 \nu^{9} + 4 \nu^{8} + \cdots - 128 \nu ) / 128 \) (-3v^15 + v^14 + 2v^13 + 14v^11 + 10v^10 - 12v^9 + 4v^8 - 16v^7 + 40v^6 + 112v^5 + 16v^4 - 64v^3 - 128v^2 - 128*v) / 128
\(\beta_{10}\)	\(=\)	\( ( - 3 \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 14 \nu^{11} - 6 \nu^{10} - 20 \nu^{9} + \cdots + 512 ) / 128 \) (-3v^15 + v^14 - 2v^13 + 4v^12 + 14v^11 - 6v^10 - 20v^9 + 12v^8 - 16v^7 + 56v^6 + 112v^5 - 80v^4 - 64v^2 - 256*v + 512) / 128
\(\beta_{11}\)	\(=\)	\( ( - \nu^{15} - \nu^{14} - \nu^{13} - \nu^{12} + 2 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + \cdots + 64 ) / 32 \) (-v^15 - v^14 - v^13 - v^12 + 2v^11 - 2v^10 - 2v^9 + 2v^8 + 12v^6 + 32v^5 + 8v^4 + 16v^3 - 16v^2 - 32v + 64) / 32
\(\beta_{12}\)	\(=\)	\( ( 3 \nu^{15} + 3 \nu^{14} - 6 \nu^{13} - 8 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 128 \) (3v^15 + 3v^14 - 6v^13 - 8v^12 - 6v^11 - 2v^10 + 20v^9 + 12v^8 - 16v^7 - 40v^6 - 80v^5 + 48v^4 + 256v^3 + 128v^2 - 256) / 128
\(\beta_{13}\)	\(=\)	\( ( \nu^{15} + \nu^{12} + 4 \nu^{10} + 8 \nu^{9} - 2 \nu^{8} - 12 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} + \cdots - 128 ) / 32 \) (v^15 + v^12 + 4v^10 + 8v^9 - 2v^8 - 12v^6 - 16v^5 + 8v^4 + 16v^3 - 48v^2 - 64*v - 128) / 32
\(\beta_{14}\)	\(=\)	\( ( 3 \nu^{15} - \nu^{14} - 6 \nu^{13} + 4 \nu^{12} + 2 \nu^{11} + 22 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} + \cdots - 512 ) / 128 \) (3v^15 - v^14 - 6v^13 + 4v^12 + 2v^11 + 22v^10 + 36v^9 - 28v^8 - 16v^7 + 8v^6 - 48v^5 + 144v^4 - 192v^2 - 256*v - 512) / 128
\(\beta_{15}\)	\(=\)	\( ( - 5 \nu^{15} - \nu^{14} - 2 \nu^{13} + 8 \nu^{12} + 18 \nu^{11} - 26 \nu^{10} - 20 \nu^{9} + \cdots + 1024 ) / 128 \) (-5v^15 - v^14 - 2v^13 + 8v^12 + 18v^11 - 26v^10 - 20v^9 + 12v^8 - 48v^7 + 152v^6 + 144v^5 - 80v^4 + 64v^3 - 384v^2 - 128v + 1024) / 128

\(\nu\)	\(=\)	\( ( -\beta_{7} + \beta_{6} ) / 4 \) (-b7 + b6) / 4
\(\nu^{2}\)	\(=\)	\( ( -\beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) (-b12 - b11 + b10 - b9 - b5 + b4 - b3 - b2 + b1) / 4
\(\nu^{3}\)	\(=\)	\( ( - 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + \cdots - \beta_1 ) / 4 \) (-2b14 + 2b13 + b12 - b11 + b10 + b9 - b5 + b4 + b3 - 3*b2 - b1) / 4
\(\nu^{4}\)	\(=\)	\( ( 2 \beta_{15} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + \cdots + 4 ) / 4 \) (2b15 - 2b13 + b12 - b11 - b10 + b9 - 2b8 - 2b6 - b5 + b4 - b3 + 3b2 + 3b1 + 4) / 4
\(\nu^{5}\)	\(=\)	\( ( 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots - 4 ) / 4 \) (2b15 + 2b13 - b12 + b11 + b10 - b9 - 2b8 - 2b7 + b5 - b4 - 3b3 + b2 - 3b1 - 4) / 4
\(\nu^{6}\)	\(=\)	\( ( 2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots - 4 ) / 4 \) (2b15 + 4b14 - 2b13 + b12 - b11 - b10 + b9 + 2b8 + 4b7 - 2b6 - 5b5 - 3b4 - b3 - b2 - b1 - 4) / 4
\(\nu^{7}\)	\(=\)	\( ( - 6 \beta_{15} + 2 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + \cdots + 12 ) / 4 \) (-6b15 + 2b13 + 5b12 + 7b11 + 7b10 + 5b9 + 2b8 + 4b7 - 2b6 - 5b5 - 3b4 + 3b3 + 11b2 - 5b1 + 12) / 4
\(\nu^{8}\)	\(=\)	\( ( 6 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} - 7 \beta_{9} + \cdots + 4 ) / 4 \) (6b15 + 4b14 - 6b13 + b12 + 3b11 - b10 - 7b9 - 10b8 + 6b6 - b5 - 7b4 - 5b3 + 11b2 + 15*b1 + 4) / 4
\(\nu^{9}\)	\(=\)	\( ( - 2 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + \cdots + 36 ) / 4 \) (-2b15 + 4b14 + 10b13 - 11b12 + 7b11 - 5b10 - 3b9 - 2b8 + 4b7 + 10b6 + 3b5 + 5b4 - 9b3 + 7b2 - 5*b1 + 36) / 4
\(\nu^{10}\)	\(=\)	\( ( - 10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + \beta_{12} - 5 \beta_{11} + 7 \beta_{10} + 9 \beta_{9} + \cdots + 52 ) / 4 \) (-10b15 + 4b14 + 2b13 + b12 - 5b11 + 7b10 + 9b9 - 2b8 + 14b6 - b5 - 7b4 + 11b3 - 29b2 - 9b1 + 52) / 4
\(\nu^{11}\)	\(=\)	\( ( - 10 \beta_{15} - 4 \beta_{14} + 10 \beta_{13} + \beta_{12} + 3 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + \cdots + 20 ) / 4 \) (-10b15 - 4b14 + 10b13 + b12 + 3b11 + 15b10 + 9b9 + 30b8 - 16b7 - 2b6 - 25b5 + 17b4 - 5b3 + 19b2 + 39b1 + 20) / 4
\(\nu^{12}\)	\(=\)	\( ( 14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} + 47 \beta_{10} + \cdots - 60 ) / 4 \) (14b15 - 12b14 + 10b13 - 23b12 - 37b11 + 47b10 - 31b9 - 42b8 - 16b7 + 22b6 - b5 + 25b4 - 5b3 - 13b2 + 15b1 - 60) / 4
\(\nu^{13}\)	\(=\)	\( ( 30 \beta_{15} - 44 \beta_{14} + 42 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} - 49 \beta_{10} + \cdots + 68 ) / 4 \) (30b15 - 44b14 + 42b13 - 23b12 - 37b11 - 49b10 + 33b9 + 6b8 + 24b7 - 34b6 - 33b5 + 25b4 - 21b3 - 29b2 + 15*b1 + 68) / 4
\(\nu^{14}\)	\(=\)	\( ( - 18 \beta_{15} + 4 \beta_{14} - 22 \beta_{13} + 57 \beta_{12} - 21 \beta_{11} + 31 \beta_{10} + \cdots + 36 ) / 4 \) (-18b15 + 4b14 - 22b13 + 57b12 - 21b11 + 31b10 + 81b9 - 26b8 + 32b7 - 10b6 + 31b5 - 7b4 + 11b3 + 83b2 - 81*b1 + 36) / 4
\(\nu^{15}\)	\(=\)	\( ( 94 \beta_{15} + 52 \beta_{14} + 10 \beta_{13} + 33 \beta_{12} - 13 \beta_{11} + 7 \beta_{10} - 71 \beta_{9} + \cdots - 60 ) / 4 \) (94b15 + 52b14 + 10b13 + 33b12 - 13b11 + 7b10 - 71b9 + 54b8 - 64b7 - 90b6 - 89b5 - 79b4 - 93b3 + 75b2 + 71*b1 - 60) / 4

\(n\)	\(351\)	\(421\)	\(801\)	\(897\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 111.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} + 32 T^{14} + \cdots + 576 \) T^16 + 32T^14 + 398T^12 + 2468T^10 + 8217T^8 + 14676T^6 + 13024T^4 + 4720*T^2 + 576
$5$	\( (T + 1)^{16} \) (T + 1)^16
$7$	\( T^{16} + 4 T^{14} + \cdots + 5764801 \) T^16 + 4T^14 + 36T^13 + 4T^12 + 204T^11 + 700T^10 + 904T^9 + 5814T^8 + 6328T^7 + 34300T^6 + 69972T^5 + 9604T^4 + 605052T^3 + 470596*T^2 + 5764801
$11$	\( (T^{8} - 2 T^{7} + \cdots + 2400)^{2} \) (T^8 - 2T^7 - 50T^6 + 52T^5 + 761T^4 - 138T^3 - 3932T^2 - 1960*T + 2400)^2
$13$	\( (T^{8} - 62 T^{6} + \cdots + 10032)^{2} \) (T^8 - 62T^6 - 12T^5 + 1213T^4 + 228T^3 - 7792T^2 + 1232T + 10032)^2
$17$	\( T^{16} + 136 T^{14} + \cdots + 2359296 \) T^16 + 136T^14 + 7086T^12 + 178476T^10 + 2255465T^8 + 13596164T^6 + 34001456T^4 + 27167680*T^2 + 2359296
$19$	\( T^{16} + 184 T^{14} + \cdots + 1327104 \) T^16 + 184T^14 + 12736T^12 + 423168T^10 + 6949696T^8 + 49375232T^6 + 81127424T^4 + 26099712*T^2 + 1327104
$23$	\( T^{16} + 160 T^{14} + \cdots + 7573504 \) T^16 + 160T^14 + 10096T^12 + 325872T^10 + 5807360T^8 + 57190400T^6 + 289346560T^4 + 585268224*T^2 + 7573504
$29$	\( T^{16} + 172 T^{14} + \cdots + 65536 \) T^16 + 172T^14 + 9718T^12 + 217548T^10 + 2200529T^8 + 9795008T^6 + 15058432T^4 + 2408448*T^2 + 65536
$31$	\( (T^{8} + 8 T^{7} + \cdots + 98304)^{2} \) (T^8 + 8T^7 - 92T^6 - 440T^5 + 3496T^4 + 3040T^3 - 33920T^2 - 2816*T + 98304)^2
$37$	\( T^{16} + \cdots + 4063297536 \) T^16 + 252T^14 + 26192T^12 + 1448768T^10 + 45832768T^8 + 825741568T^6 + 7832304640T^4 + 30390333440*T^2 + 4063297536
$41$	\( T^{16} + \cdots + 6423183360000 \) T^16 + 424T^14 + 73760T^12 + 6823040T^10 + 364052032T^8 + 11370832896T^6 + 201056874496T^4 + 1819708211200*T^2 + 6423183360000
$43$	\( (T^{8} - 2 T^{7} + \cdots - 69504)^{2} \) (T^8 - 2T^7 - 128T^6 + 332T^5 + 4232T^4 - 14912T^3 - 17664T^2 + 91360*T - 69504)^2
$47$	\( (T^{8} - 166 T^{6} + \cdots - 10368)^{2} \) (T^8 - 166T^6 + 424T^5 + 5461T^4 - 28292T^3 + 36188T^2 + 5820T - 10368)^2
$53$	\( T^{16} + 404 T^{14} + \cdots + 262144 \) T^16 + 404T^14 + 58128T^12 + 3445312T^10 + 69431040T^8 + 251149312T^6 + 158445568T^4 + 27312128*T^2 + 262144
$59$	\( T^{16} + \cdots + 1090584576 \) T^16 + 400T^14 + 55552T^12 + 3287040T^10 + 82300416T^8 + 734793728T^6 + 2362048512T^4 + 2931687424*T^2 + 1090584576
$61$	\( (T^{8} - 4 T^{7} + \cdots + 10368)^{2} \) (T^8 - 4T^7 - 268T^6 + 1256T^5 + 15176T^4 - 85216T^3 + 26432T^2 + 51712*T + 10368)^2
$67$	\( (T^{8} + 10 T^{7} + \cdots + 165632)^{2} \) (T^8 + 10T^7 - 180T^6 - 1204T^5 + 7712T^4 + 42240T^3 - 55616T^2 - 197184*T + 165632)^2
$71$	\( T^{16} + \cdots + 2621440000 \) T^16 + 492T^14 + 82752T^12 + 6017280T^10 + 180376576T^8 + 1494106112T^6 + 4596301824T^4 + 5845811200*T^2 + 2621440000
$73$	\( T^{16} + \cdots + 28179280429056 \) T^16 + 612T^14 + 151344T^12 + 19483584T^10 + 1406521344T^8 + 57276760064T^6 + 1248988233728T^4 + 12310919774208*T^2 + 28179280429056
$79$	\( T^{16} + \cdots + 39806206534656 \) T^16 + 592T^14 + 145478T^12 + 19139020T^10 + 1444670401T^8 + 61858049132T^6 + 1373008608512T^4 + 12666930453504*T^2 + 39806206534656
$83$	\( T^{16} + \cdots + 88794464256 \) T^16 + 564T^14 + 109632T^12 + 8985456T^10 + 337953344T^8 + 6096578560T^6 + 51045990400T^4 + 165451399168*T^2 + 88794464256
$89$	\( T^{16} + \cdots + 347892350976 \) T^16 + 520T^14 + 95808T^12 + 8073856T^10 + 331338304T^8 + 6628880384T^6 + 61230546944T^4 + 245081571328*T^2 + 347892350976
$97$	\( T^{16} + \cdots + 17227945230336 \) T^16 + 888T^14 + 307550T^12 + 52726332T^10 + 4708474873T^8 + 212781536324T^6 + 4314469148336T^4 + 27670537460160*T^2 + 17227945230336
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