LMFDB - Newform orbit 1512.2.c.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(757,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.757"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	20.0.1646001224014411784746814245175296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} - q^{7} + \beta_{3} q^{8} + (\beta_{17} - 1) q^{10} + ( - \beta_{14} - \beta_{12} + \cdots + \beta_{3}) q^{11} + ( - \beta_{13} - \beta_{2}) q^{13}+ \cdots + \beta_1 q^{98}+O(q^{100}) $ q + b1 * q^2 + b2 * q^4 + b14 * q^5 - q^7 + b3 * q^8 + (b17 - 1) * q^10 + (-b14 - b12 + b6 + b3) * q^11 + (-b13 - b2) * q^13 - b1 * q^14 + (b4 - 1) * q^16 + (b9 + b8 + b3) * q^17 + (b18 + b17 - b13 - b10) * q^19 + (-b14 - b12 + b8 + b6 + b3) * q^20 + (-b17 + 2b13 + b10 + b4) q^22 + (b12 + b3) * q^23 + (-b15 + b4 - 2) * q^25 + (-b16 - b3) * q^26 - b2 * q^28 + (-b16 + b14 - b6 - 2b1) q^29 + (-b13 + b2 + 2) * q^31 + (b7 + b6 - b1) * q^32 + (-b18 + b5 - b2) * q^34 - b14 * q^35 + (b17 + b15 - b11 + b5 + b4) * q^37 + (b19 + b14 + b9 + b8 - b6) * q^38 + (-b18 - b17 + 2b13 + b11 + b10 - b2 - 1) q^40 + (b19 + b9 + b8 - b7 + b3 + b1) * q^41 + (-b13 - b10 + b5 - b4 - b2 + 1) * q^43 + (-b19 + b16 + b14 - b9 - b8 + b7 - b3 - b1) * q^44 + (-2b13 + b4 - 1) q^46 + (b16 + b14 + b9 - b8 - b6 - b3 - 2b1) q^47 + q^49 + (2b12 + b7 + b6 - b1) q^50 + (-b4 + 5) * q^52 + (-b19 - b16 + 2b14 - b7 - 2b6 - b1) * q^53 + (b18 - b15 + 3b13 + b11 + 2b10 - 2b2 + 1) q^55 - b3 * q^56 + (b17 - b10 - 2b2 + 3) q^58 + (-b16 + b6 - 2b1) q^59 + (b18 + 2b17 - b15 - b13 - b11 - 1) q^61 + (-b16 + b3 + 2b1) q^62 + (b18 + b17 + b5 - b2 + 1) * q^64 + (b14 + b12 - 2b8 - b6 - b3) q^65 + (b15 + b4) * q^67 + (-3b14 + b12 - 2b9 - b8 + b7 + 2b6 - b3 + b1) q^68 + (-b17 + 1) * q^70 + (-b19 + b16 + b7 - 3b1) q^71 + (-b18 + b17 + b15 + 2b13 - b10 - b4 - 3b2) * q^73 + (-2b12 - 2b9 + b7 + 3b6) q^74 + (-b18 + b17 + 2b15 - 2b10 + b5 - b4 - b2 + 2) * q^76 + (b14 + b12 - b6 - b3) * q^77 + (b17 - b15 + b13 + b11 + b10 - b2 + 2) * q^79 + (-b19 + b16 - 3b14 - b9 - b8 + b7 - b3 - b1) q^80 + (-2b18 - b17 + 2b15 + 1) * q^82 + (-b19 + b16 - 2b12 - b7 - b6 + 2b3 + 3b1) q^83 + (-b18 - b17 + b15 - 2b13 + b5 - 3b2 + 1) * q^85 + (b19 - b14 + 2b12 - b9 - b8 - b6 - 2b3 + b1) * q^86 + (2b18 + 2b17 - 2b15 - 6) q^88 + (b19 + b16 - b7 - b1) * q^89 + (b13 + b2) * q^91 + (-2b16 + b7 + b6 - b1) q^92 + (b18 + b17 - 2b11 - b10 + b5 - b2 - 1) q^94 + (b19 + b16 + b14 + 2b12 + 2b9 - b7 - b6 + 2b3 - b1) q^95 + (-b17 + b13 - b11 - b10 + b4 - b2 + 3) * q^97 + b1 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{4} - 20 q^{7} - 12 q^{10} - 14 q^{16} + 8 q^{22} - 28 q^{25} + 2 q^{28} + 36 q^{31} - 6 q^{34} - 16 q^{40} - 18 q^{46} + 20 q^{49} + 94 q^{52} + 48 q^{55} + 66 q^{58} + 22 q^{64} + 12 q^{70}+ \cdots + 56 q^{97}+O(q^{100}) $ 20 * q - 2 * q^4 - 20 * q^7 - 12 * q^10 - 14 * q^16 + 8 * q^22 - 28 * q^25 + 2 * q^28 + 36 * q^31 - 6 * q^34 - 16 * q^40 - 18 * q^46 + 20 * q^49 + 94 * q^52 + 48 * q^55 + 66 * q^58 + 22 * q^64 + 12 * q^70 + 12 * q^76 + 64 * q^79 - 92 * q^88 - 24 * q^94 + 56 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4*x^16 + 8*x^12 + 4*x^10 + 32*x^8 + 256*x^4 + 256*x^2 + 1024 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} $ v^2
$\beta_{3}$	$=$	$ \nu^{3} $ v^3
$\beta_{4}$	$=$	$ \nu^{4} + 1 $ v^4 + 1
$\beta_{5}$	$=$	$ ( - \nu^{18} + 11 \nu^{16} - 24 \nu^{14} - 48 \nu^{12} + 120 \nu^{10} - 100 \nu^{8} + 1168 \nu^{6} + \cdots + 512 ) / 1792 $ (-v^18 + 11v^16 - 24v^14 - 48v^12 + 120v^10 - 100v^8 + 1168v^6 - 128v^4 + 1280v^2 + 512) / 1792
$\beta_{6}$	$=$	$ ( - \nu^{19} - 3 \nu^{17} + 4 \nu^{15} - 6 \nu^{13} + 8 \nu^{11} - 100 \nu^{9} - 120 \nu^{7} + \cdots - 832 \nu ) / 896 $ (-v^19 - 3v^17 + 4v^15 - 6v^13 + 8v^11 - 100v^9 - 120v^7 + 40v^5 - 288v^3 - 832*v) / 896
$\beta_{7}$	$=$	$ ( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} + 120 \nu^{7} + \cdots + 1728 \nu ) / 896 $ (v^19 + 3v^17 - 4v^15 + 6v^13 - 8v^11 + 100v^9 + 120v^7 + 856v^5 + 288v^3 + 1728*v) / 896
$\beta_{8}$	$=$	$ ( - \nu^{19} - 10 \nu^{17} + 11 \nu^{15} - 20 \nu^{13} - 48 \nu^{11} - 44 \nu^{9} + 188 \nu^{7} + \cdots - 3072 \nu ) / 896 $ (-v^19 - 10v^17 + 11v^15 - 20v^13 - 48v^11 - 44v^9 + 188v^7 - 128v^5 - 288v^3 - 3072*v) / 896
$\beta_{9}$	$=$	$ ( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} - 328 \nu^{7} + \cdots + 1728 \nu ) / 896 $ (v^19 + 3v^17 - 4v^15 + 6v^13 - 8v^11 + 100v^9 - 328v^7 + 408v^5 - 608v^3 + 1728*v) / 896
$\beta_{10}$	$=$	$ ( -\nu^{18} + 4\nu^{16} - 3\nu^{14} + 8\nu^{12} - 48\nu^{10} - 44\nu^{8} + 20\nu^{6} - 16\nu^{4} - 288\nu^{2} + 512 ) / 448 $ (-v^18 + 4v^16 - 3v^14 + 8v^12 - 48v^10 - 44v^8 + 20v^6 - 16v^4 - 288v^2 + 512) / 448
$\beta_{11}$	$=$	$ ( - 5 \nu^{18} + 27 \nu^{16} - 36 \nu^{14} - 16 \nu^{12} - 72 \nu^{10} + 620 \nu^{8} + 352 \nu^{6} + \cdots + 6144 ) / 1792 $ (-5v^18 + 27v^16 - 36v^14 - 16v^12 - 72v^10 + 620v^8 + 352v^6 + 1600v^4 - 1664*v^2 + 6144) / 1792
$\beta_{12}$	$=$	$ ( \nu^{19} - 3\nu^{17} - 16\nu^{13} + 8\nu^{11} - 28\nu^{9} + 16\nu^{7} - 128\nu^{5} + 256\nu^{3} - 768\nu ) / 512 $ (v^19 - 3v^17 - 16v^13 + 8v^11 - 28v^9 + 16v^7 - 128v^5 + 256v^3 - 768v) / 512
$\beta_{13}$	$=$	$ ( \nu^{18} + \nu^{16} + 4\nu^{14} + 8\nu^{10} + 4\nu^{8} + 32\nu^{6} + 256\nu^{2} + 256 ) / 256 $ (v^18 + v^16 + 4v^14 + 8v^10 + 4v^8 + 32v^6 + 256*v^2 + 256) / 256
$\beta_{14}$	$=$	$ ( 9 \nu^{19} - 15 \nu^{17} + 20 \nu^{15} + 40 \nu^{13} + 40 \nu^{11} - 220 \nu^{9} + 128 \nu^{7} + \cdots - 1024 \nu ) / 3584 $ (9v^19 - 15v^17 + 20v^15 + 40v^13 + 40v^11 - 220v^9 + 128v^7 + 32v^5 + 2816v^3 - 1024v) / 3584
$\beta_{15}$	$=$	$ ( -\nu^{18} + 3\nu^{16} + 16\nu^{12} - 8\nu^{10} + 28\nu^{8} - 16\nu^{6} + 128\nu^{4} - 256\nu^{2} + 512 ) / 256 $ (-v^18 + 3v^16 + 16v^12 - 8v^10 + 28v^8 - 16v^6 + 128v^4 - 256*v^2 + 512) / 256
$\beta_{16}$	$=$	$ ( \nu^{19} + \nu^{17} + 4\nu^{15} + 8\nu^{11} + 4\nu^{9} + 32\nu^{7} + 256\nu^{3} + 256\nu ) / 256 $ (v^19 + v^17 + 4v^15 + 8v^11 + 4v^9 + 32v^7 + 256v^3 + 256v) / 256
$\beta_{17}$	$=$	$ ( -3\nu^{18} - 2\nu^{16} + 5\nu^{14} - 4\nu^{12} - 32\nu^{10} - 20\nu^{8} + 4\nu^{6} + 64\nu^{4} - 416\nu^{2} - 704 ) / 448 $ (-3v^18 - 2v^16 + 5v^14 - 4v^12 - 32v^10 - 20v^8 + 4v^6 + 64v^4 - 416*v^2 - 704) / 448
$\beta_{18}$	$=$	$ ( 13 \nu^{18} - 3 \nu^{16} + 4 \nu^{14} + 64 \nu^{12} + 8 \nu^{10} + 180 \nu^{8} + 608 \nu^{6} + \cdots + 512 ) / 1792 $ (13v^18 - 3v^16 + 4v^14 + 64v^12 + 8v^10 + 180v^8 + 608v^6 - 128v^4 + 2176*v^2 + 512) / 1792
$\beta_{19}$	$=$	$ ( - 17 \nu^{19} - 37 \nu^{17} - 16 \nu^{15} + 24 \nu^{13} + 248 \nu^{11} + 92 \nu^{9} + \cdots - 9216 \nu ) / 3584 $ (-17v^19 - 37v^17 - 16v^15 + 24v^13 + 248v^11 + 92v^9 + 592v^7 - 608v^5 - 640v^3 - 9216v) / 3584

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} $ b2
$\nu^{3}$	$=$	$ \beta_{3} $ b3
$\nu^{4}$	$=$	$ \beta_{4} - 1 $ b4 - 1
$\nu^{5}$	$=$	$ \beta_{7} + \beta_{6} - \beta_1 $ b7 + b6 - b1
$\nu^{6}$	$=$	$ \beta_{18} + \beta_{17} + \beta_{5} - \beta_{2} + 1 $ b18 + b17 + b5 - b2 + 1
$\nu^{7}$	$=$	$ -2\beta_{9} + \beta_{7} - \beta_{6} - 2\beta_{3} + \beta_1 $ -2b9 + b7 - b6 - 2b3 + b1
$\nu^{8}$	$=$	$ \beta_{18} + \beta_{17} + 2\beta_{11} - 2\beta_{10} - \beta_{5} - 2\beta_{4} + \beta_{2} - 1 $ b18 + b17 + 2b11 - 2b10 - b5 - 2*b4 + b2 - 1
$\nu^{9}$	$=$	$ 2\beta_{19} + 2\beta_{16} - 2\beta_{14} + 4\beta_{9} + 2\beta_{8} - \beta_{7} - 5\beta_{6} + 2\beta_{3} - \beta_1 $ 2b19 + 2b16 - 2b14 + 4b9 + 2b8 - b7 - 5b6 + 2*b3 - b1
$\nu^{10}$	$=$	$ -3\beta_{18} - 3\beta_{17} + 4\beta_{15} - 2\beta_{11} - 6\beta_{10} + 3\beta_{5} - 3\beta_{2} + 1 $ -3b18 - 3b17 + 4b15 - 2b11 - 6b10 + 3b5 - 3*b2 + 1
$\nu^{11}$	$=$	$ 6\beta_{19} + 6\beta_{16} - 2\beta_{14} + 4\beta_{12} - 6\beta_{8} + \beta_{7} + 5\beta_{6} - 6\beta_{3} - 3\beta_1 $ 6b19 + 6b16 - 2b14 + 4b12 - 6b8 + b7 + 5b6 - 6b3 - 3b1
$\nu^{12}$	$=$	$ 7\beta_{18} - \beta_{17} + 12\beta_{15} - 8\beta_{13} - 6\beta_{11} - 2\beta_{10} + \beta_{5} + 3\beta_{2} + 3 $ 7b18 - b17 + 12b15 - 8b13 - 6b11 - 2b10 + b5 + 3b2 + 3
$\nu^{13}$	$=$	$ 2 \beta_{19} - 6 \beta_{16} + 26 \beta_{14} - 20 \beta_{12} - 2 \beta_{8} - 5 \beta_{7} + \cdots - 17 \beta_1 $ 2b19 - 6b16 + 26b14 - 20b12 - 2b8 - 5b7 - 9b6 - 6b3 - 17*b1
$\nu^{14}$	$=$	$ - 3 \beta_{18} + 21 \beta_{17} + 4 \beta_{15} + 40 \beta_{13} - 2 \beta_{11} - 6 \beta_{10} - 5 \beta_{5} + \cdots + 5 $ -3b18 + 21b17 + 4b15 + 40b13 - 2b11 - 6b10 - 5b5 - 4b4 - 15*b2 + 5
$\nu^{15}$	$=$	$ 6 \beta_{19} + 46 \beta_{16} - 18 \beta_{14} - 28 \beta_{12} + 16 \beta_{9} + 26 \beta_{8} + \cdots + 25 \beta_1 $ 6b19 + 46b16 - 18b14 - 28b12 + 16b9 + 26b8 - 11b7 + 25b6 + 14b3 + 25b1
$\nu^{16}$	$=$	$ - 37 \beta_{18} - 29 \beta_{17} + 12 \beta_{15} + 56 \beta_{13} + 10 \beta_{11} + 30 \beta_{10} + \cdots - 173 $ -37b18 - 29b17 + 12b15 + 56b13 + 10b11 + 30b10 + 5b5 - 12b4 - b2 - 173
$\nu^{17}$	$=$	$ - 30 \beta_{19} + 26 \beta_{16} - 70 \beta_{14} - 20 \beta_{12} - 40 \beta_{9} - 34 \beta_{8} + \cdots - 177 \beta_1 $ -30b19 + 26b16 - 70b14 - 20b12 - 40b9 - 34b8 + 3b7 + 23b6 + 2b3 - 177b1
$\nu^{18}$	$=$	$ 37 \beta_{18} - 67 \beta_{17} - 60 \beta_{15} + 40 \beta_{13} + 6 \beta_{11} + 50 \beta_{10} + \cdots - 139 $ 37b18 - 67b17 - 60b15 + 40b13 + 6b11 + 50b10 - 37b5 + 36b4 - 143*b2 - 139
$\nu^{19}$	$=$	$ - 50 \beta_{19} - 10 \beta_{16} + 166 \beta_{14} + 100 \beta_{12} + 24 \beta_{9} - 30 \beta_{8} + \cdots - 183 \beta_1 $ -50b19 - 10b16 + 166b14 + 100b12 + 24b9 - 30b8 + 5b7 - 111b6 - 210b3 - 183b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$-1$	$1$	$1$	$1$

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} $

757.1

−1.37874	−	0.314750i
−1.37874	+	0.314750i
−1.19566	−	0.755240i
−1.19566	+	0.755240i
−0.885915	−	1.10234i
−0.885915	+	1.10234i
−0.725842	−	1.21374i
−0.725842	+	1.21374i
−0.328272	−	1.37559i
−0.328272	+	1.37559i
0.328272	−	1.37559i
0.328272	+	1.37559i
0.725842	−	1.21374i
0.725842	+	1.21374i
0.885915	−	1.10234i
0.885915	+	1.10234i
1.19566	−	0.755240i
1.19566	+	0.755240i
1.37874	−	0.314750i
1.37874	+	0.314750i

−1.37874

−

0.314750i

1.80186

0.867919i

−

0.114591i

−1.00000

−2.21113

−

1.76377i

−0.0360676

0.157992i

757.2

−1.37874

0.314750i

1.80186

−

0.867919i

0.114591i

−1.00000

−2.21113

1.76377i

−0.0360676

−

0.157992i

757.3

−1.19566

−

0.755240i

0.859226

1.80603i

−

3.16969i

−1.00000

0.336637

−

2.80832i

−2.39387

3.78988i

757.4

−1.19566

0.755240i

0.859226

−

1.80603i

3.16969i

−1.00000

0.336637

2.80832i

−2.39387

−

3.78988i

757.5

−0.885915

−

1.10234i

−0.430309

1.95316i

3.50133i

−1.00000

2.53426

−

1.25599i

3.85966

−

3.10188i

757.6

−0.885915

1.10234i

−0.430309

−

1.95316i

−

3.50133i

−1.00000

2.53426

1.25599i

3.85966

3.10188i

757.7

−0.725842

−

1.21374i

−0.946308

1.76196i

−

3.06888i

−1.00000

2.82542

−

0.130337i

−3.72481

2.22752i

757.8

−0.725842

1.21374i

−0.946308

−

1.76196i

3.06888i

−1.00000

2.82542

0.130337i

−3.72481

−

2.22752i

757.9

−0.328272

−

1.37559i

−1.78447

0.903134i

−

0.512447i

−1.00000

1.82813

2.15822i

−0.704915

0.168222i

757.10

−0.328272

1.37559i

−1.78447

−

0.903134i

0.512447i

−1.00000

1.82813

−

2.15822i

−0.704915

−

0.168222i

757.11

0.328272

−

1.37559i

−1.78447

−

0.903134i

−

0.512447i

−1.00000

−1.82813

2.15822i

−0.704915

−

0.168222i

757.12

0.328272

1.37559i

−1.78447

0.903134i

0.512447i

−1.00000

−1.82813

−

2.15822i

−0.704915

0.168222i

757.13

0.725842

−

1.21374i

−0.946308

−

1.76196i

−

3.06888i

−1.00000

−2.82542

−

0.130337i

−3.72481

−

2.22752i

757.14

0.725842

1.21374i

−0.946308

1.76196i

3.06888i

−1.00000

−2.82542

0.130337i

−3.72481

2.22752i

757.15

0.885915

−

1.10234i

−0.430309

−

1.95316i

3.50133i

−1.00000

−2.53426

−

1.25599i

3.85966

3.10188i

757.16

0.885915

1.10234i

−0.430309

1.95316i

−

3.50133i

−1.00000

−2.53426

1.25599i

3.85966

−

3.10188i

757.17

1.19566

−

0.755240i

0.859226

−

1.80603i

−

3.16969i

−1.00000

−0.336637

−

2.80832i

−2.39387

−

3.78988i

757.18

1.19566

0.755240i

0.859226

1.80603i

3.16969i

−1.00000

−0.336637

2.80832i

−2.39387

3.78988i

757.19

1.37874

−

0.314750i

1.80186

−

0.867919i

−

0.114591i

−1.00000

2.21113

−

1.76377i

−0.0360676

−

0.157992i

757.20

1.37874

0.314750i

1.80186

0.867919i

0.114591i

−1.00000

2.21113

1.76377i

−0.0360676

0.157992i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1512.2.c.e	✓	20
3.b	odd	2	1	inner	1512.2.c.e	✓	20
4.b	odd	2	1		6048.2.c.e		20
8.b	even	2	1	inner	1512.2.c.e	✓	20
8.d	odd	2	1		6048.2.c.e		20
12.b	even	2	1		6048.2.c.e		20
24.f	even	2	1		6048.2.c.e		20
24.h	odd	2	1	inner	1512.2.c.e	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1512.2.c.e	✓	20	1.a	even	1	1	trivial
1512.2.c.e	✓	20	3.b	odd	2	1	inner
1512.2.c.e	✓	20	8.b	even	2	1	inner
1512.2.c.e	✓	20	24.h	odd	2	1	inner
6048.2.c.e		20	4.b	odd	2	1
6048.2.c.e		20	8.d	odd	2	1
6048.2.c.e		20	12.b	even	2	1
6048.2.c.e		20	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1512, [\chi])$:

$ T_{5}^{10} + 32T_{5}^{8} + 342T_{5}^{6} + 1252T_{5}^{4} + 321T_{5}^{2} + 4 $

T5^10 + 32*T5^8 + 342*T5^6 + 1252*T5^4 + 321*T5^2 + 4

$ T_{17}^{10} - 131T_{17}^{8} + 5266T_{17}^{6} - 69022T_{17}^{4} + 319757T_{17}^{2} - 395839 $

T17^10 - 131*T17^8 + 5266*T17^6 - 69022*T17^4 + 319757*T17^2 - 395839

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} + T^{18} + \cdots + 1024 $ T^20 + T^18 + 4T^16 + 8T^12 + 4T^10 + 32T^8 + 256T^4 + 256T^2 + 1024
$3$	$ T^{20} $ T^20
$5$	$ (T^{10} + 32 T^{8} + \cdots + 4)^{2} $ (T^10 + 32T^8 + 342T^6 + 1252T^4 + 321T^2 + 4)^2
$7$	$ (T + 1)^{20} $ (T + 1)^20
$11$	$ (T^{10} + 76 T^{8} + \cdots + 5184)^{2} $ (T^10 + 76T^8 + 1832T^6 + 14672T^4 + 32848T^2 + 5184)^2
$13$	$ (T^{10} + 47 T^{8} + \cdots + 24304)^{2} $ (T^10 + 47T^8 + 816T^6 + 6328T^4 + 20928T^2 + 24304)^2
$17$	$ (T^{10} - 131 T^{8} + \cdots - 395839)^{2} $ (T^10 - 131T^8 + 5266T^6 - 69022T^4 + 319757T^2 - 395839)^2
$19$	$ (T^{10} + 120 T^{8} + \cdots + 3668416)^{2} $ (T^10 + 120T^8 + 5320T^6 + 109152T^4 + 1035920T^2 + 3668416)^2
$23$	$ (T^{10} - 91 T^{8} + \cdots - 97216)^{2} $ (T^10 - 91T^8 + 3020T^6 - 43424T^4 + 233536T^2 - 97216)^2
$29$	$ (T^{10} + 101 T^{8} + \cdots + 63504)^{2} $ (T^10 + 101T^8 + 2732T^6 + 24088T^4 + 72976T^2 + 63504)^2
$31$	$ (T^{5} - 9 T^{4} + 16 T^{3} + \cdots + 4)^{4} $ (T^5 - 9T^4 + 16T^3 + 24T^2 - 40T + 4)^4
$37$	$ (T^{10} + 348 T^{8} + \cdots + 425308096)^{2} $ (T^10 + 348T^8 + 43174T^6 + 2337372T^4 + 53431265T^2 + 425308096)^2
$41$	$ (T^{10} - 220 T^{8} + \cdots - 3928816)^{2} $ (T^10 - 220T^8 + 13166T^6 - 254444T^4 + 1751785T^2 - 3928816)^2
$43$	$ (T^{10} + 299 T^{8} + \cdots + 2771431)^{2} $ (T^10 + 299T^8 + 25266T^6 + 486478T^4 + 2192877T^2 + 2771431)^2
$47$	$ (T^{10} - 360 T^{8} + \cdots - 593088156)^{2} $ (T^10 - 360T^8 + 47142T^6 - 2743956T^4 + 69920577T^2 - 593088156)^2
$53$	$ (T^{10} + 337 T^{8} + \cdots + 20736)^{2} $ (T^10 + 337T^8 + 37040T^6 + 1320032T^4 + 355840T^2 + 20736)^2
$59$	$ (T^{10} + 141 T^{8} + \cdots + 169)^{2} $ (T^10 + 141T^8 + 6346T^6 + 95898T^4 + 384773T^2 + 169)^2
$61$	$ (T^{10} + 424 T^{8} + \cdots + 1063074816)^{2} $ (T^10 + 424T^8 + 62984T^6 + 3962000T^4 + 108214672T^2 + 1063074816)^2
$67$	$ (T^{10} + 175 T^{8} + \cdots + 31497984)^{2} $ (T^10 + 175T^8 + 11888T^6 + 386720T^4 + 5875456T^2 + 31497984)^2
$71$	$ (T^{10} - 367 T^{8} + \cdots - 1142784)^{2} $ (T^10 - 367T^8 + 34224T^6 - 925312T^4 + 3426304T^2 - 1142784)^2
$73$	$ (T^{5} - 216 T^{3} + \cdots - 11664)^{4} $ (T^5 - 216T^3 + 432T^2 + 8424*T - 11664)^4
$79$	$ (T^{5} - 16 T^{4} + \cdots + 9882)^{4} $ (T^5 - 16T^4 - 60T^3 + 1854T^2 - 8073T + 9882)^4
$83$	$ (T^{10} + 692 T^{8} + \cdots + 4368152464)^{2} $ (T^10 + 692T^8 + 163110T^6 + 15188164T^4 + 461644257T^2 + 4368152464)^2
$89$	$ (T^{10} - 343 T^{8} + \cdots - 21458944)^{2} $ (T^10 - 343T^8 + 40976T^6 - 1990784T^4 + 33353728T^2 - 21458944)^2
$97$	$ (T^{5} - 14 T^{4} + \cdots - 23296)^{4} $ (T^5 - 14T^4 - 144T^3 + 1952T^2 + 852T - 23296)^4
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} - q^{7} + \beta_{3} q^{8} + (\beta_{17} - 1) q^{10} + ( - \beta_{14} - \beta_{12} + \cdots + \beta_{3}) q^{11} + ( - \beta_{13} - \beta_{2}) q^{13}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) q + b1 * q^2 + b2 * q^4 + b14 * q^5 - q^7 + b3 * q^8 + (b17 - 1) * q^10 + (-b14 - b12 + b6 + b3) * q^11 + (-b13 - b2) * q^13 - b1 * q^14 + (b4 - 1) * q^16 + (b9 + b8 + b3) * q^17 + (b18 + b17 - b13 - b10) * q^19 + (-b14 - b12 + b8 + b6 + b3) * q^20 + (-b17 + 2b13 + b10 + b4) q^22 + (b12 + b3) * q^23 + (-b15 + b4 - 2) * q^25 + (-b16 - b3) * q^26 - b2 * q^28 + (-b16 + b14 - b6 - 2b1) q^29 + (-b13 + b2 + 2) * q^31 + (b7 + b6 - b1) * q^32 + (-b18 + b5 - b2) * q^34 - b14 * q^35 + (b17 + b15 - b11 + b5 + b4) * q^37 + (b19 + b14 + b9 + b8 - b6) * q^38 + (-b18 - b17 + 2b13 + b11 + b10 - b2 - 1) q^40 + (b19 + b9 + b8 - b7 + b3 + b1) * q^41 + (-b13 - b10 + b5 - b4 - b2 + 1) * q^43 + (-b19 + b16 + b14 - b9 - b8 + b7 - b3 - b1) * q^44 + (-2b13 + b4 - 1) q^46 + (b16 + b14 + b9 - b8 - b6 - b3 - 2b1) q^47 + q^49 + (2b12 + b7 + b6 - b1) q^50 + (-b4 + 5) * q^52 + (-b19 - b16 + 2b14 - b7 - 2b6 - b1) * q^53 + (b18 - b15 + 3b13 + b11 + 2b10 - 2b2 + 1) q^55 - b3 * q^56 + (b17 - b10 - 2b2 + 3) q^58 + (-b16 + b6 - 2b1) q^59 + (b18 + 2b17 - b15 - b13 - b11 - 1) q^61 + (-b16 + b3 + 2b1) q^62 + (b18 + b17 + b5 - b2 + 1) * q^64 + (b14 + b12 - 2b8 - b6 - b3) q^65 + (b15 + b4) * q^67 + (-3b14 + b12 - 2b9 - b8 + b7 + 2b6 - b3 + b1) q^68 + (-b17 + 1) * q^70 + (-b19 + b16 + b7 - 3b1) q^71 + (-b18 + b17 + b15 + 2b13 - b10 - b4 - 3b2) * q^73 + (-2b12 - 2b9 + b7 + 3b6) q^74 + (-b18 + b17 + 2b15 - 2b10 + b5 - b4 - b2 + 2) * q^76 + (b14 + b12 - b6 - b3) * q^77 + (b17 - b15 + b13 + b11 + b10 - b2 + 2) * q^79 + (-b19 + b16 - 3b14 - b9 - b8 + b7 - b3 - b1) q^80 + (-2b18 - b17 + 2b15 + 1) * q^82 + (-b19 + b16 - 2b12 - b7 - b6 + 2b3 + 3b1) q^83 + (-b18 - b17 + b15 - 2b13 + b5 - 3b2 + 1) * q^85 + (b19 - b14 + 2b12 - b9 - b8 - b6 - 2b3 + b1) * q^86 + (2b18 + 2b17 - 2b15 - 6) q^88 + (b19 + b16 - b7 - b1) * q^89 + (b13 + b2) * q^91 + (-2b16 + b7 + b6 - b1) q^92 + (b18 + b17 - 2b11 - b10 + b5 - b2 - 1) q^94 + (b19 + b16 + b14 + 2b12 + 2b9 - b7 - b6 + 2b3 - b1) q^95 + (-b17 + b13 - b11 - b10 + b4 - b2 + 3) * q^97 + b1 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{4} - 20 q^{7} - 12 q^{10} - 14 q^{16} + 8 q^{22} - 28 q^{25} + 2 q^{28} + 36 q^{31} - 6 q^{34} - 16 q^{40} - 18 q^{46} + 20 q^{49} + 94 q^{52} + 48 q^{55} + 66 q^{58} + 22 q^{64} + 12 q^{70}+ \cdots + 56 q^{97}+O(q^{100}) \) 20 * q - 2 * q^4 - 20 * q^7 - 12 * q^10 - 14 * q^16 + 8 * q^22 - 28 * q^25 + 2 * q^28 + 36 * q^31 - 6 * q^34 - 16 * q^40 - 18 * q^46 + 20 * q^49 + 94 * q^52 + 48 * q^55 + 66 * q^58 + 22 * q^64 + 12 * q^70 + 12 * q^76 + 64 * q^79 - 92 * q^88 - 24 * q^94 + 56 * q^97

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1512.2.c.e

Level	$1512$
Weight	$2$
Character orbit	1512.c
Analytic conductor	$12.073$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(12.0733807856\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	20.0.1646001224014411784746814245175296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 \) x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{8}\cdot 3^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} \) v^2
\(\beta_{3}\)	\(=\)	\( \nu^{3} \) v^3
\(\beta_{4}\)	\(=\)	\( \nu^{4} + 1 \) v^4 + 1
\(\beta_{5}\)	\(=\)	\( ( - \nu^{18} + 11 \nu^{16} - 24 \nu^{14} - 48 \nu^{12} + 120 \nu^{10} - 100 \nu^{8} + 1168 \nu^{6} + \cdots + 512 ) / 1792 \) (-v^18 + 11v^16 - 24v^14 - 48v^12 + 120v^10 - 100v^8 + 1168v^6 - 128v^4 + 1280v^2 + 512) / 1792
\(\beta_{6}\)	\(=\)	\( ( - \nu^{19} - 3 \nu^{17} + 4 \nu^{15} - 6 \nu^{13} + 8 \nu^{11} - 100 \nu^{9} - 120 \nu^{7} + \cdots - 832 \nu ) / 896 \) (-v^19 - 3v^17 + 4v^15 - 6v^13 + 8v^11 - 100v^9 - 120v^7 + 40v^5 - 288v^3 - 832*v) / 896
\(\beta_{7}\)	\(=\)	\( ( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} + 120 \nu^{7} + \cdots + 1728 \nu ) / 896 \) (v^19 + 3v^17 - 4v^15 + 6v^13 - 8v^11 + 100v^9 + 120v^7 + 856v^5 + 288v^3 + 1728*v) / 896
\(\beta_{8}\)	\(=\)	\( ( - \nu^{19} - 10 \nu^{17} + 11 \nu^{15} - 20 \nu^{13} - 48 \nu^{11} - 44 \nu^{9} + 188 \nu^{7} + \cdots - 3072 \nu ) / 896 \) (-v^19 - 10v^17 + 11v^15 - 20v^13 - 48v^11 - 44v^9 + 188v^7 - 128v^5 - 288v^3 - 3072*v) / 896
\(\beta_{9}\)	\(=\)	\( ( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} - 328 \nu^{7} + \cdots + 1728 \nu ) / 896 \) (v^19 + 3v^17 - 4v^15 + 6v^13 - 8v^11 + 100v^9 - 328v^7 + 408v^5 - 608v^3 + 1728*v) / 896
\(\beta_{10}\)	\(=\)	\( ( -\nu^{18} + 4\nu^{16} - 3\nu^{14} + 8\nu^{12} - 48\nu^{10} - 44\nu^{8} + 20\nu^{6} - 16\nu^{4} - 288\nu^{2} + 512 ) / 448 \) (-v^18 + 4v^16 - 3v^14 + 8v^12 - 48v^10 - 44v^8 + 20v^6 - 16v^4 - 288v^2 + 512) / 448
\(\beta_{11}\)	\(=\)	\( ( - 5 \nu^{18} + 27 \nu^{16} - 36 \nu^{14} - 16 \nu^{12} - 72 \nu^{10} + 620 \nu^{8} + 352 \nu^{6} + \cdots + 6144 ) / 1792 \) (-5v^18 + 27v^16 - 36v^14 - 16v^12 - 72v^10 + 620v^8 + 352v^6 + 1600v^4 - 1664*v^2 + 6144) / 1792
\(\beta_{12}\)	\(=\)	\( ( \nu^{19} - 3\nu^{17} - 16\nu^{13} + 8\nu^{11} - 28\nu^{9} + 16\nu^{7} - 128\nu^{5} + 256\nu^{3} - 768\nu ) / 512 \) (v^19 - 3v^17 - 16v^13 + 8v^11 - 28v^9 + 16v^7 - 128v^5 + 256v^3 - 768v) / 512
\(\beta_{13}\)	\(=\)	\( ( \nu^{18} + \nu^{16} + 4\nu^{14} + 8\nu^{10} + 4\nu^{8} + 32\nu^{6} + 256\nu^{2} + 256 ) / 256 \) (v^18 + v^16 + 4v^14 + 8v^10 + 4v^8 + 32v^6 + 256*v^2 + 256) / 256
\(\beta_{14}\)	\(=\)	\( ( 9 \nu^{19} - 15 \nu^{17} + 20 \nu^{15} + 40 \nu^{13} + 40 \nu^{11} - 220 \nu^{9} + 128 \nu^{7} + \cdots - 1024 \nu ) / 3584 \) (9v^19 - 15v^17 + 20v^15 + 40v^13 + 40v^11 - 220v^9 + 128v^7 + 32v^5 + 2816v^3 - 1024v) / 3584
\(\beta_{15}\)	\(=\)	\( ( -\nu^{18} + 3\nu^{16} + 16\nu^{12} - 8\nu^{10} + 28\nu^{8} - 16\nu^{6} + 128\nu^{4} - 256\nu^{2} + 512 ) / 256 \) (-v^18 + 3v^16 + 16v^12 - 8v^10 + 28v^8 - 16v^6 + 128v^4 - 256*v^2 + 512) / 256
\(\beta_{16}\)	\(=\)	\( ( \nu^{19} + \nu^{17} + 4\nu^{15} + 8\nu^{11} + 4\nu^{9} + 32\nu^{7} + 256\nu^{3} + 256\nu ) / 256 \) (v^19 + v^17 + 4v^15 + 8v^11 + 4v^9 + 32v^7 + 256v^3 + 256v) / 256
\(\beta_{17}\)	\(=\)	\( ( -3\nu^{18} - 2\nu^{16} + 5\nu^{14} - 4\nu^{12} - 32\nu^{10} - 20\nu^{8} + 4\nu^{6} + 64\nu^{4} - 416\nu^{2} - 704 ) / 448 \) (-3v^18 - 2v^16 + 5v^14 - 4v^12 - 32v^10 - 20v^8 + 4v^6 + 64v^4 - 416*v^2 - 704) / 448
\(\beta_{18}\)	\(=\)	\( ( 13 \nu^{18} - 3 \nu^{16} + 4 \nu^{14} + 64 \nu^{12} + 8 \nu^{10} + 180 \nu^{8} + 608 \nu^{6} + \cdots + 512 ) / 1792 \) (13v^18 - 3v^16 + 4v^14 + 64v^12 + 8v^10 + 180v^8 + 608v^6 - 128v^4 + 2176*v^2 + 512) / 1792
\(\beta_{19}\)	\(=\)	\( ( - 17 \nu^{19} - 37 \nu^{17} - 16 \nu^{15} + 24 \nu^{13} + 248 \nu^{11} + 92 \nu^{9} + \cdots - 9216 \nu ) / 3584 \) (-17v^19 - 37v^17 - 16v^15 + 24v^13 + 248v^11 + 92v^9 + 592v^7 - 608v^5 - 640v^3 - 9216v) / 3584

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} \) b2
\(\nu^{3}\)	\(=\)	\( \beta_{3} \) b3
\(\nu^{4}\)	\(=\)	\( \beta_{4} - 1 \) b4 - 1
\(\nu^{5}\)	\(=\)	\( \beta_{7} + \beta_{6} - \beta_1 \) b7 + b6 - b1
\(\nu^{6}\)	\(=\)	\( \beta_{18} + \beta_{17} + \beta_{5} - \beta_{2} + 1 \) b18 + b17 + b5 - b2 + 1
\(\nu^{7}\)	\(=\)	\( -2\beta_{9} + \beta_{7} - \beta_{6} - 2\beta_{3} + \beta_1 \) -2b9 + b7 - b6 - 2b3 + b1
\(\nu^{8}\)	\(=\)	\( \beta_{18} + \beta_{17} + 2\beta_{11} - 2\beta_{10} - \beta_{5} - 2\beta_{4} + \beta_{2} - 1 \) b18 + b17 + 2b11 - 2b10 - b5 - 2*b4 + b2 - 1
\(\nu^{9}\)	\(=\)	\( 2\beta_{19} + 2\beta_{16} - 2\beta_{14} + 4\beta_{9} + 2\beta_{8} - \beta_{7} - 5\beta_{6} + 2\beta_{3} - \beta_1 \) 2b19 + 2b16 - 2b14 + 4b9 + 2b8 - b7 - 5b6 + 2*b3 - b1
\(\nu^{10}\)	\(=\)	\( -3\beta_{18} - 3\beta_{17} + 4\beta_{15} - 2\beta_{11} - 6\beta_{10} + 3\beta_{5} - 3\beta_{2} + 1 \) -3b18 - 3b17 + 4b15 - 2b11 - 6b10 + 3b5 - 3*b2 + 1
\(\nu^{11}\)	\(=\)	\( 6\beta_{19} + 6\beta_{16} - 2\beta_{14} + 4\beta_{12} - 6\beta_{8} + \beta_{7} + 5\beta_{6} - 6\beta_{3} - 3\beta_1 \) 6b19 + 6b16 - 2b14 + 4b12 - 6b8 + b7 + 5b6 - 6b3 - 3b1
\(\nu^{12}\)	\(=\)	\( 7\beta_{18} - \beta_{17} + 12\beta_{15} - 8\beta_{13} - 6\beta_{11} - 2\beta_{10} + \beta_{5} + 3\beta_{2} + 3 \) 7b18 - b17 + 12b15 - 8b13 - 6b11 - 2b10 + b5 + 3b2 + 3
\(\nu^{13}\)	\(=\)	\( 2 \beta_{19} - 6 \beta_{16} + 26 \beta_{14} - 20 \beta_{12} - 2 \beta_{8} - 5 \beta_{7} + \cdots - 17 \beta_1 \) 2b19 - 6b16 + 26b14 - 20b12 - 2b8 - 5b7 - 9b6 - 6b3 - 17*b1
\(\nu^{14}\)	\(=\)	\( - 3 \beta_{18} + 21 \beta_{17} + 4 \beta_{15} + 40 \beta_{13} - 2 \beta_{11} - 6 \beta_{10} - 5 \beta_{5} + \cdots + 5 \) -3b18 + 21b17 + 4b15 + 40b13 - 2b11 - 6b10 - 5b5 - 4b4 - 15*b2 + 5
\(\nu^{15}\)	\(=\)	\( 6 \beta_{19} + 46 \beta_{16} - 18 \beta_{14} - 28 \beta_{12} + 16 \beta_{9} + 26 \beta_{8} + \cdots + 25 \beta_1 \) 6b19 + 46b16 - 18b14 - 28b12 + 16b9 + 26b8 - 11b7 + 25b6 + 14b3 + 25b1
\(\nu^{16}\)	\(=\)	\( - 37 \beta_{18} - 29 \beta_{17} + 12 \beta_{15} + 56 \beta_{13} + 10 \beta_{11} + 30 \beta_{10} + \cdots - 173 \) -37b18 - 29b17 + 12b15 + 56b13 + 10b11 + 30b10 + 5b5 - 12b4 - b2 - 173
\(\nu^{17}\)	\(=\)	\( - 30 \beta_{19} + 26 \beta_{16} - 70 \beta_{14} - 20 \beta_{12} - 40 \beta_{9} - 34 \beta_{8} + \cdots - 177 \beta_1 \) -30b19 + 26b16 - 70b14 - 20b12 - 40b9 - 34b8 + 3b7 + 23b6 + 2b3 - 177b1
\(\nu^{18}\)	\(=\)	\( 37 \beta_{18} - 67 \beta_{17} - 60 \beta_{15} + 40 \beta_{13} + 6 \beta_{11} + 50 \beta_{10} + \cdots - 139 \) 37b18 - 67b17 - 60b15 + 40b13 + 6b11 + 50b10 - 37b5 + 36b4 - 143*b2 - 139
\(\nu^{19}\)	\(=\)	\( - 50 \beta_{19} - 10 \beta_{16} + 166 \beta_{14} + 100 \beta_{12} + 24 \beta_{9} - 30 \beta_{8} + \cdots - 183 \beta_1 \) -50b19 - 10b16 + 166b14 + 100b12 + 24b9 - 30b8 + 5b7 - 111b6 - 210b3 - 183b1

\(n\)	\(757\)	\(785\)	\(1081\)	\(1135\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 757.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} + T^{18} + \cdots + 1024 \) T^20 + T^18 + 4T^16 + 8T^12 + 4T^10 + 32T^8 + 256T^4 + 256T^2 + 1024
$3$	\( T^{20} \) T^20
$5$	\( (T^{10} + 32 T^{8} + \cdots + 4)^{2} \) (T^10 + 32T^8 + 342T^6 + 1252T^4 + 321T^2 + 4)^2
$7$	\( (T + 1)^{20} \) (T + 1)^20
$11$	\( (T^{10} + 76 T^{8} + \cdots + 5184)^{2} \) (T^10 + 76T^8 + 1832T^6 + 14672T^4 + 32848T^2 + 5184)^2
$13$	\( (T^{10} + 47 T^{8} + \cdots + 24304)^{2} \) (T^10 + 47T^8 + 816T^6 + 6328T^4 + 20928T^2 + 24304)^2
$17$	\( (T^{10} - 131 T^{8} + \cdots - 395839)^{2} \) (T^10 - 131T^8 + 5266T^6 - 69022T^4 + 319757T^2 - 395839)^2
$19$	\( (T^{10} + 120 T^{8} + \cdots + 3668416)^{2} \) (T^10 + 120T^8 + 5320T^6 + 109152T^4 + 1035920T^2 + 3668416)^2
$23$	\( (T^{10} - 91 T^{8} + \cdots - 97216)^{2} \) (T^10 - 91T^8 + 3020T^6 - 43424T^4 + 233536T^2 - 97216)^2
$29$	\( (T^{10} + 101 T^{8} + \cdots + 63504)^{2} \) (T^10 + 101T^8 + 2732T^6 + 24088T^4 + 72976T^2 + 63504)^2
$31$	\( (T^{5} - 9 T^{4} + 16 T^{3} + \cdots + 4)^{4} \) (T^5 - 9T^4 + 16T^3 + 24T^2 - 40T + 4)^4
$37$	\( (T^{10} + 348 T^{8} + \cdots + 425308096)^{2} \) (T^10 + 348T^8 + 43174T^6 + 2337372T^4 + 53431265T^2 + 425308096)^2
$41$	\( (T^{10} - 220 T^{8} + \cdots - 3928816)^{2} \) (T^10 - 220T^8 + 13166T^6 - 254444T^4 + 1751785T^2 - 3928816)^2
$43$	\( (T^{10} + 299 T^{8} + \cdots + 2771431)^{2} \) (T^10 + 299T^8 + 25266T^6 + 486478T^4 + 2192877T^2 + 2771431)^2
$47$	\( (T^{10} - 360 T^{8} + \cdots - 593088156)^{2} \) (T^10 - 360T^8 + 47142T^6 - 2743956T^4 + 69920577T^2 - 593088156)^2
$53$	\( (T^{10} + 337 T^{8} + \cdots + 20736)^{2} \) (T^10 + 337T^8 + 37040T^6 + 1320032T^4 + 355840T^2 + 20736)^2
$59$	\( (T^{10} + 141 T^{8} + \cdots + 169)^{2} \) (T^10 + 141T^8 + 6346T^6 + 95898T^4 + 384773T^2 + 169)^2
$61$	\( (T^{10} + 424 T^{8} + \cdots + 1063074816)^{2} \) (T^10 + 424T^8 + 62984T^6 + 3962000T^4 + 108214672T^2 + 1063074816)^2
$67$	\( (T^{10} + 175 T^{8} + \cdots + 31497984)^{2} \) (T^10 + 175T^8 + 11888T^6 + 386720T^4 + 5875456T^2 + 31497984)^2
$71$	\( (T^{10} - 367 T^{8} + \cdots - 1142784)^{2} \) (T^10 - 367T^8 + 34224T^6 - 925312T^4 + 3426304T^2 - 1142784)^2
$73$	\( (T^{5} - 216 T^{3} + \cdots - 11664)^{4} \) (T^5 - 216T^3 + 432T^2 + 8424*T - 11664)^4
$79$	\( (T^{5} - 16 T^{4} + \cdots + 9882)^{4} \) (T^5 - 16T^4 - 60T^3 + 1854T^2 - 8073T + 9882)^4
$83$	\( (T^{10} + 692 T^{8} + \cdots + 4368152464)^{2} \) (T^10 + 692T^8 + 163110T^6 + 15188164T^4 + 461644257T^2 + 4368152464)^2
$89$	\( (T^{10} - 343 T^{8} + \cdots - 21458944)^{2} \) (T^10 - 343T^8 + 40976T^6 - 1990784T^4 + 33353728T^2 - 21458944)^2
$97$	\( (T^{5} - 14 T^{4} + \cdots - 23296)^{4} \) (T^5 - 14T^4 - 144T^3 + 1952T^2 + 852T - 23296)^4
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