LMFDB - Newform orbit 392.2.m.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,2,Mod(19,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("392.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	392.m (of order $6$, 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:	$3.13013575923$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 4x^{14} + 6x^{12} + 8x^{10} + 20x^{8} + 32x^{6} + 96x^{4} + 256x^{2} + 256 $ x^16 + 4x^14 + 6x^12 + 8x^10 + 20x^8 + 32x^6 + 96x^4 + 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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_{14} + \beta_{5}) q^{2} - \beta_{15} q^{3} + (\beta_{11} - \beta_{8} - \beta_{7} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{14} + \beta_{11} + \beta_{5}) q^{9}+O(q^{10}) $ q + (b14 + b5) * q^2 - b15 * q^3 + (b11 - b8 - b7 + b6 + b5 + 1) * q^4 + (-2b15 - 2b12 - b10 - b3 - 2b2) q^5 + (b15 + b13 + b10 + b9) * q^6 + (b14 - b7 + b5 + 2b4 + 2) q^8 + (-b14 + b11 + b5) * q^9	$ q + (\beta_{14} + \beta_{5}) q^{2} - \beta_{15} q^{3} + (\beta_{11} - \beta_{8} - \beta_{7} + \cdots + 1) q^{4}+ \cdots + (3 \beta_{14} - 3 \beta_{7} + 3 \beta_{6} + \cdots + 9) q^{99}+O(q^{100}) $ q + (b14 + b5) * q^2 - b15 * q^3 + (b11 - b8 - b7 + b6 + b5 + 1) * q^4 + (-2b15 - 2b12 - b10 - b3 - 2b2) q^5 + (b15 + b13 + b10 + b9) * q^6 + (b14 - b7 + b5 + 2b4 + 2) q^8 + (-b14 + b11 + b5) * q^9 + (b15 + b13 + b12 - 2b9 - 3b3 - 2b1) q^10 + (-3b11 + 3b7 - 3b6 - 3b5 - 3) * q^11 + (-b15 - b12 - b10 - b3 - b2 + b1) * q^12 + (-b15 - b13 - b10 - 2b2) q^13 + (-4b4 - 2) q^15 + (-b14 + 3b11 + b8 + 2b5 - b4) * q^16 + (b15 - 2b13 + 2b3) * q^17 + (-b11 + b8 - b7 - b6 + b5 + 1) * q^18 + (-2b10 + b3) q^19 + (-4b15 + b13 - 4b10 + 4b9 - b2) q^20 + (3b6 - 3b4 - 3) * q^22 + (2b14 + 2b11 + 2b5) q^23 + (b15 - b13 + 2b12 - b9 - b1) q^24 + (2b11 - 2b7 + 2b6 - 5b5 - 5) * q^25 + (b15 + b12 - 3b3 + b2 - 2b1) * q^26 + (3b15 - b13 + 3b10) * q^27 + (2b14 - 2b7 - 2b6 + 2b5) * q^29 + (2b14 - 4b11 - 4b8 - 2b5 + 4b4) q^30 + (-2b15 - 4b12 - 4b9 - 4b3 - 4b1) q^31 + (-2b11 - b7 - 2b6 + 4b5 + 4) q^32 + (3b10 + 3b3) * q^33 + (b15 - b13 + b10 - 3b9 + 2b2) * q^34 + (-b14 + b7 - 2b6 - b5 - 4) q^36 + (4b8 + 2b5 - 4b4) q^37 + (-2b15 - 2b13 - b12 - 3b9 - b3 - 3b1) * q^38 + (-4b8 - 2b5 - 2) * q^39 + (-b15 - b12 + 6b10 - 3b3 - b2 - 2b1) q^40 + (2b15 - 3b13 + 2b10) q^41 + (-b14 + b7 - b6 - b5 + 3) * q^43 + (-3b11 - 3b8 - 9b5 + 3b4) * q^44 + (b15 + b13 + 2b12 + 4b9 + 3b3 + 4b1) * q^45 + (2b11 - 2b8 - 2b7 + 2b6 + 6b5 + 6) q^46 + (4b15 + 4b12 + 2b10 + 2b3 + 4b2) q^47 + (-b15 - b10 + 3b9 + b2) q^48 + (-7b14 + 7b7 - 2b6 - 7b5 + 2b4 + 2) q^50 + (-b14 + b11 + 2b5) q^51 + (-3b15 + b13 + b12 + 4b9 + 3b3 + 4b1) * q^52 + (2b11 + 4b8 + 2b7 + 2b6 + 2b5 + 2) q^53 + (b15 + b12 - 2b10 + b3 + b2 + 4b1) * q^54 + (-6b13 - 12b9) * q^55 + (b14 - b7 + b6 + b5 - 3) * q^57 + (-2b14 + 2b11 - 2b8 + 4b5 + 2b4) q^58 + (2b15 - 5b13 + 5b3) q^59 + (6b11 + 2b8 - 2b7 + 6b6 - 2b5 - 2) q^60 + (-2b15 - 2b12 - b10 - 3b3 - 2b2 - 4b1) q^61 + (-2b15 + 6b13 - 2b10 - 2b9) * q^62 + (3b14 - 3b7 - b6 + 3b5 + b4 - 5) q^64 + (-2b14 + 2b11 - 12b5) q^65 + (3b15 + 3b13 - 3b12 - 3b3) * q^66 + (2b5 + 2) q^67 + (b15 + b12 - b10 + 5b3 + b2 + b1) q^68 + (2b13 + 4b9) * q^69 + (-6b14 + 6b7 + 6b6 - 6b5 + 4b4 + 2) q^71 + (-3b14 - b11 + b8 - b4) q^72 + (-2b15 + b13 - b3) q^73 + (-4b11 - 4b8 + 2b7 - 4b6 - 4b5 - 4) q^74 + (-9b10 - 2b3) * q^75 + (b15 + 5b13 + b10 + b9 + 2b2) * q^76 + (2b14 - 2b7 + 4b6 + 2b5 + 4b4 + 4) q^78 + (-4b14 - 4b11 - 4b5) q^79 + (5b15 + 9b13 + b12 + 8b9 - b3 + 8b1) * q^80 + (-5b11 + 5b7 - 5b6 - 2b5 - 2) * q^81 + (3b15 + 3b12 + b10 + 3b3 + 3b2 + 5b1) q^82 + 5b13 q^83 + (4b14 - 4b7 - 4b6 + 4b5 + 4b4 + 2) q^85 + (4b14 - b11 + b8 + 5b5 - b4) * q^86 + (2b13 + 4b9 + 2b3 + 4b1) * q^87 + (3b11 + 3b8 + 6b7 + 3b6 - 3b5 - 3) q^88 + (4b10 + 3b3) * q^89 + (2b15 - 5b13 + 2b10 - b2) q^90 + (6b14 - 6b7 + 6b5 + 4b4 + 4) * q^92 + (-2b14 - 2b11 + 4b8 - 4b4) * q^93 + (-2b15 - 2b13 - 2b12 + 4b9 + 6b3 + 4b1) * q^94 + (-2b11 + 8b8 - 2b7 - 2b6 + 4b5 + 4) q^95 + (5b10 + 2b3 + 3b1) q^96 + (-9b15 - 9b10) * q^97 + (3b14 - 3b7 + 3b6 + 3b5 + 9) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 4 q^{4} + 8 q^{8} - 8 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 4 * q^4 + 8 * q^8 - 8 * q^9	$ 16 q - 4 q^{2} + 4 q^{4} + 8 q^{8} - 8 q^{9} - 4 q^{16} + 4 q^{18} - 48 q^{22} - 56 q^{25} - 8 q^{30} + 36 q^{32} - 40 q^{36} + 64 q^{43} + 48 q^{44} + 40 q^{46} + 88 q^{50} - 16 q^{51} - 64 q^{57} - 40 q^{58} - 56 q^{60} - 104 q^{64} + 96 q^{65} + 16 q^{67} - 12 q^{72} + 8 q^{74} - 16 q^{78} + 24 q^{81} - 24 q^{86} - 24 q^{88} - 16 q^{92} + 96 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 4 * q^4 + 8 * q^8 - 8 * q^9 - 4 * q^16 + 4 * q^18 - 48 * q^22 - 56 * q^25 - 8 * q^30 + 36 * q^32 - 40 * q^36 + 64 * q^43 + 48 * q^44 + 40 * q^46 + 88 * q^50 - 16 * q^51 - 64 * q^57 - 40 * q^58 - 56 * q^60 - 104 * q^64 + 96 * q^65 + 16 * q^67 - 12 * q^72 + 8 * q^74 - 16 * q^78 + 24 * q^81 - 24 * q^86 - 24 * q^88 - 16 * q^92 + 96 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 4x^{14} + 6x^{12} + 8x^{10} + 20x^{8} + 32x^{6} + 96x^{4} + 256x^{2} + 256 $ x^16 + 4*x^14 + 6*x^12 + 8*x^10 + 20*x^8 + 32*x^6 + 96*x^4 + 256*x^2 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{15} + 6\nu^{13} + 4\nu^{11} - 12\nu^{9} + 24\nu^{7} + 24\nu^{5} + 88\nu^{3} + 320\nu ) / 224 $ (v^15 + 6v^13 + 4v^11 - 12v^9 + 24v^7 + 24v^5 + 88v^3 + 320*v) / 224
$\beta_{3}$	$=$	$ ( -5\nu^{15} - 44\nu^{13} - 6\nu^{11} - 24\nu^{9} - 148\nu^{7} - 64\nu^{5} - 384\nu^{3} - 2944\nu ) / 896 $ (-5v^15 - 44v^13 - 6v^11 - 24v^9 - 148v^7 - 64v^5 - 384v^3 - 2944v) / 896
$\beta_{4}$	$=$	$ ( 5\nu^{14} - 12\nu^{12} + 6\nu^{10} + 24\nu^{8} - 76\nu^{6} + 64\nu^{4} + 384\nu^{2} - 1088 ) / 448 $ (5v^14 - 12v^12 + 6v^10 + 24v^8 - 76v^6 + 64v^4 + 384*v^2 - 1088) / 448
$\beta_{5}$	$=$	$ ( 3\nu^{14} - 3\nu^{12} - 2\nu^{10} + 6\nu^{8} - 12\nu^{6} - 12\nu^{4} + 96\nu^{2} - 384 ) / 224 $ (3v^14 - 3v^12 - 2v^10 + 6v^8 - 12v^6 - 12v^4 + 96*v^2 - 384) / 224
$\beta_{6}$	$=$	$ ( 5\nu^{14} + 16\nu^{12} + 6\nu^{10} + 24\nu^{8} + 36\nu^{6} + 64\nu^{4} + 384\nu^{2} + 704 ) / 224 $ (5v^14 + 16v^12 + 6v^10 + 24v^8 + 36v^6 + 64v^4 + 384*v^2 + 704) / 224
$\beta_{7}$	$=$	$ ( 5\nu^{14} + 9\nu^{12} + 6\nu^{10} + 38\nu^{8} + 36\nu^{6} + 36\nu^{4} + 272\nu^{2} + 480 ) / 224 $ (5v^14 + 9v^12 + 6v^10 + 38v^8 + 36v^6 + 36v^4 + 272*v^2 + 480) / 224
$\beta_{8}$	$=$	$ ( 5\nu^{14} + 9\nu^{12} + 6\nu^{10} + 38\nu^{8} + 36\nu^{6} + 36\nu^{4} + 496\nu^{2} + 480 ) / 224 $ (5v^14 + 9v^12 + 6v^10 + 38v^8 + 36v^6 + 36v^4 + 496*v^2 + 480) / 224
$\beta_{9}$	$=$	$ ( 3\nu^{15} + 18\nu^{13} + 12\nu^{11} + 20\nu^{9} + 72\nu^{7} + 72\nu^{5} + 264\nu^{3} + 960\nu ) / 224 $ (3v^15 + 18v^13 + 12v^11 + 20v^9 + 72v^7 + 72v^5 + 264v^3 + 960v) / 224
$\beta_{10}$	$=$	$ ( 15\nu^{15} + 20\nu^{13} + 18\nu^{11} + 72\nu^{9} - 4\nu^{7} + 192\nu^{5} + 1152\nu^{3} + 768\nu ) / 896 $ (15v^15 + 20v^13 + 18v^11 + 72v^9 - 4v^7 + 192v^5 + 1152v^3 + 768v) / 896
$\beta_{11}$	$=$	$ ( 9\nu^{14} + 26\nu^{12} + 22\nu^{10} + 60\nu^{8} + 132\nu^{6} + 104\nu^{4} + 736\nu^{2} + 1536 ) / 224 $ (9v^14 + 26v^12 + 22v^10 + 60v^8 + 132v^6 + 104v^4 + 736*v^2 + 1536) / 224
$\beta_{12}$	$=$	$ ( 11\nu^{15} + 24\nu^{13} + 2\nu^{11} + 64\nu^{9} + 124\nu^{7} + 208\nu^{5} + 800\nu^{3} + 1280\nu ) / 448 $ (11v^15 + 24v^13 + 2v^11 + 64v^9 + 124v^7 + 208v^5 + 800v^3 + 1280v) / 448
$\beta_{13}$	$=$	$ ( -6\nu^{15} - 15\nu^{13} - 10\nu^{11} - 26\nu^{9} - 60\nu^{7} - 60\nu^{5} - 360\nu^{3} - 800\nu ) / 224 $ (-6v^15 - 15v^13 - 10v^11 - 26v^9 - 60v^7 - 60v^5 - 360v^3 - 800v) / 224
$\beta_{14}$	$=$	$ ( 29\nu^{14} + 76\nu^{12} + 46\nu^{10} + 184\nu^{8} + 388\nu^{6} + 416\nu^{4} + 2272\nu^{2} + 4352 ) / 448 $ (29v^14 + 76v^12 + 46v^10 + 184v^8 + 388v^6 + 416v^4 + 2272*v^2 + 4352) / 448
$\beta_{15}$	$=$	$ ( 29\nu^{15} + 76\nu^{13} + 46\nu^{11} + 184\nu^{9} + 388\nu^{7} + 192\nu^{5} + 2272\nu^{3} + 4352\nu ) / 896 $ (29v^15 + 76v^13 + 46v^11 + 184v^9 + 388v^7 + 192v^5 + 2272v^3 + 4352v) / 896

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{8} - \beta_{7} $ b8 - b7
$\nu^{3}$	$=$	$ \beta_{15} + 2\beta_{13} + \beta_{10} + \beta_{2} $ b15 + 2*b13 + b10 + b2
$\nu^{4}$	$=$	$ 2\beta_{14} - 2\beta_{11} - 2\beta_{8} - 2\beta_{5} + 2\beta_{4} $ 2b14 - 2b11 - 2b8 - 2b5 + 2*b4
$\nu^{5}$	$=$	$ -2\beta_{15} + 2\beta_{12} + 2\beta_{9} + 2\beta_{3} + 2\beta_1 $ -2b15 + 2b12 + 2b9 + 2b3 + 2*b1
$\nu^{6}$	$=$	$ 2\beta_{14} - 2\beta_{7} - 4\beta_{6} + 2\beta_{5} - 2\beta_{4} - 4 $ 2b14 - 2b7 - 4b6 + 2b5 - 2*b4 - 4
$\nu^{7}$	$=$	$ 2\beta_{15} + 2\beta_{12} - 6\beta_{10} + 4\beta_{3} + 2\beta_{2} $ 2b15 + 2b12 - 6b10 + 4b3 + 2*b2
$\nu^{8}$	$=$	$ -4\beta_{11} + 4\beta_{8} + 12\beta_{7} - 4\beta_{6} - 8\beta_{5} - 8 $ -4b11 + 4b8 + 12b7 - 4b6 - 8*b5 - 8
$\nu^{9}$	$=$	$ 4\beta_{9} - 12\beta_{2} $ 4b9 - 12b2
$\nu^{10}$	$=$	$ -12\beta_{14} + 24\beta_{11} - 12\beta_{8} - 4\beta_{5} + 12\beta_{4} $ -12b14 + 24b11 - 12b8 - 4b5 + 12*b4
$\nu^{11}$	$=$	$ -8\beta_{13} - 12\beta_{12} + 16\beta_{9} + 24\beta_{3} + 16\beta_1 $ -8b13 - 12b12 + 16b9 + 24b3 + 16*b1
$\nu^{12}$	$=$	$ -8\beta_{14} + 8\beta_{7} + 24\beta_{6} - 8\beta_{5} - 8\beta_{4} - 48 $ -8b14 + 8b7 + 24b6 - 8b5 - 8*b4 - 48
$\nu^{13}$	$=$	$ -8\beta_{15} - 8\beta_{12} + 16\beta_{10} - 40\beta_{3} - 8\beta_{2} - 72\beta_1 $ -8b15 - 8b12 + 16b10 - 40b3 - 8b2 - 72b1
$\nu^{14}$	$=$	$ 16\beta_{11} - 56\beta_{8} + 8\beta_{7} + 16\beta_{6} + 80\beta_{5} + 80 $ 16b11 - 56b8 + 8b7 + 16b6 + 80*b5 + 80
$\nu^{15}$	$=$	$ -40\beta_{15} - 144\beta_{13} - 40\beta_{10} - 64\beta_{9} - 8\beta_{2} $ -40b15 - 144b13 - 40b10 - 64b9 - 8*b2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$-1$	$-1$	$-\beta_{5}$

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} $

19.1

−0.349313	+	1.37039i
0.349313	−	1.37039i
−1.33546	+	0.465333i
1.33546	−	0.465333i
1.01214	−	0.987711i
−1.01214	+	0.987711i
−0.264742	−	1.38921i
0.264742	+	1.38921i
−0.349313	−	1.37039i
0.349313	+	1.37039i
−1.33546	−	0.465333i
1.33546	+	0.465333i
1.01214	+	0.987711i
−1.01214	−	0.987711i
−0.264742	+	1.38921i
0.264742	−	1.38921i

−1.24165

0.676979i

−1.60021

0.923880i

1.08340

−

1.68114i

−1.92538

3.33486i

1.36145

−

2.23044i

−0.207107

2.82083i

0.207107

−

0.358719i

0.133026

−

5.44419i

19.2

−1.24165

0.676979i

1.60021

−

0.923880i

1.08340

−

1.68114i

1.92538

−

3.33486i

−1.36145

2.23044i

−0.207107

2.82083i

0.207107

−

0.358719i

−0.133026

5.44419i

19.3

−1.10799

−

0.878843i

−0.662827

0.382683i

0.455270

1.94749i

1.51423

−

2.62272i

1.07072

0.158513i

1.20711

−

2.55791i

−1.20711

2.09077i

−3.98271

1.57517i

19.4

−1.10799

−

0.878843i

0.662827

−

0.382683i

0.455270

1.94749i

−1.51423

2.62272i

−1.07072

−

0.158513i

1.20711

−

2.55791i

−1.20711

2.09077i

3.98271

−

1.57517i

19.5

0.0345453

1.41379i

−1.60021

0.923880i

−1.99761

0.0976797i

1.92538

−

3.33486i

−1.36145

−

2.23044i

−0.207107

−

2.82083i

0.207107

−

0.358719i

4.78132

2.60689i

19.6

0.0345453

1.41379i

1.60021

−

0.923880i

−1.99761

0.0976797i

−1.92538

3.33486i

1.36145

2.23044i

−0.207107

−

2.82083i

0.207107

−

0.358719i

−4.78132

−

2.60689i

19.7

1.31509

0.520123i

−0.662827

0.382683i

1.45894

1.36802i

−1.51423

2.62272i

−1.07072

0.158513i

1.20711

2.55791i

−1.20711

2.09077i

−3.35549

2.66154i

19.8

1.31509

0.520123i

0.662827

−

0.382683i

1.45894

1.36802i

1.51423

−

2.62272i

1.07072

−

0.158513i

1.20711

2.55791i

−1.20711

2.09077i

3.35549

−

2.66154i

227.1

−1.24165

−

0.676979i

−1.60021

−

0.923880i

1.08340

1.68114i

−1.92538

−

3.33486i

1.36145

2.23044i

−0.207107

−

2.82083i

0.207107

0.358719i

0.133026

5.44419i

227.2

−1.24165

−

0.676979i

1.60021

0.923880i

1.08340

1.68114i

1.92538

3.33486i

−1.36145

−

2.23044i

−0.207107

−

2.82083i

0.207107

0.358719i

−0.133026

−

5.44419i

227.3

−1.10799

0.878843i

−0.662827

−

0.382683i

0.455270

−

1.94749i

1.51423

2.62272i

1.07072

−

0.158513i

1.20711

2.55791i

−1.20711

−

2.09077i

−3.98271

−

1.57517i

227.4

−1.10799

0.878843i

0.662827

0.382683i

0.455270

−

1.94749i

−1.51423

−

2.62272i

−1.07072

0.158513i

1.20711

2.55791i

−1.20711

−

2.09077i

3.98271

1.57517i

227.5

0.0345453

−

1.41379i

−1.60021

−

0.923880i

−1.99761

−

0.0976797i

1.92538

3.33486i

−1.36145

2.23044i

−0.207107

2.82083i

0.207107

0.358719i

4.78132

−

2.60689i

227.6

0.0345453

−

1.41379i

1.60021

0.923880i

−1.99761

−

0.0976797i

−1.92538

−

3.33486i

1.36145

−

2.23044i

−0.207107

2.82083i

0.207107

0.358719i

−4.78132

2.60689i

227.7

1.31509

−

0.520123i

−0.662827

−

0.382683i

1.45894

−

1.36802i

−1.51423

−

2.62272i

−1.07072

−

0.158513i

1.20711

−

2.55791i

−1.20711

−

2.09077i

−3.35549

−

2.66154i

227.8

1.31509

−

0.520123i

0.662827

0.382683i

1.45894

−

1.36802i

1.51423

2.62272i

1.07072

0.158513i

1.20711

−

2.55791i

−1.20711

−

2.09077i

3.35549

2.66154i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
8.d	odd	2	1	inner
56.e	even	2	1	inner
56.k	odd	6	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.2.m.h		16
4.b	odd	2	1		1568.2.q.h		16
7.b	odd	2	1	inner	392.2.m.h		16
7.c	even	3	1		392.2.e.d	✓	8
7.c	even	3	1	inner	392.2.m.h		16
7.d	odd	6	1		392.2.e.d	✓	8
7.d	odd	6	1	inner	392.2.m.h		16
8.b	even	2	1		1568.2.q.h		16
8.d	odd	2	1	inner	392.2.m.h		16
28.d	even	2	1		1568.2.q.h		16
28.f	even	6	1		1568.2.e.d		8
28.f	even	6	1		1568.2.q.h		16
28.g	odd	6	1		1568.2.e.d		8
28.g	odd	6	1		1568.2.q.h		16
56.e	even	2	1	inner	392.2.m.h		16
56.h	odd	2	1		1568.2.q.h		16
56.j	odd	6	1		1568.2.e.d		8
56.j	odd	6	1		1568.2.q.h		16
56.k	odd	6	1		392.2.e.d	✓	8
56.k	odd	6	1	inner	392.2.m.h		16
56.m	even	6	1		392.2.e.d	✓	8
56.m	even	6	1	inner	392.2.m.h		16
56.p	even	6	1		1568.2.e.d		8
56.p	even	6	1		1568.2.q.h		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.2.e.d	✓	8	7.c	even	3	1
392.2.e.d	✓	8	7.d	odd	6	1
392.2.e.d	✓	8	56.k	odd	6	1
392.2.e.d	✓	8	56.m	even	6	1
392.2.m.h		16	1.a	even	1	1	trivial
392.2.m.h		16	7.b	odd	2	1	inner
392.2.m.h		16	7.c	even	3	1	inner
392.2.m.h		16	7.d	odd	6	1	inner
392.2.m.h		16	8.d	odd	2	1	inner
392.2.m.h		16	56.e	even	2	1	inner
392.2.m.h		16	56.k	odd	6	1	inner
392.2.m.h		16	56.m	even	6	1	inner
1568.2.e.d		8	28.f	even	6	1
1568.2.e.d		8	28.g	odd	6	1
1568.2.e.d		8	56.j	odd	6	1
1568.2.e.d		8	56.p	even	6	1
1568.2.q.h		16	4.b	odd	2	1
1568.2.q.h		16	8.b	even	2	1
1568.2.q.h		16	28.d	even	2	1
1568.2.q.h		16	28.f	even	6	1
1568.2.q.h		16	28.g	odd	6	1
1568.2.q.h		16	56.h	odd	2	1
1568.2.q.h		16	56.j	odd	6	1
1568.2.q.h		16	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(392, [\chi])$:

$ T_{3}^{8} - 4T_{3}^{6} + 14T_{3}^{4} - 8T_{3}^{2} + 4 $

$ T_{5}^{8} + 24T_{5}^{6} + 440T_{5}^{4} + 3264T_{5}^{2} + 18496 $

$ T_{23}^{8} - 40T_{23}^{6} + 1328T_{23}^{4} - 10880T_{23}^{2} + 73984 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 2 T^{7} + T^{6} + \cdots + 16)^{2} $ (T^8 + 2T^7 + T^6 - 2T^5 - 3T^4 - 4T^3 + 4T^2 + 16T + 16)^2
$3$	$ (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$5$	$ (T^{8} + 24 T^{6} + \cdots + 18496)^{2} $ (T^8 + 24T^6 + 440T^4 + 3264*T^2 + 18496)^2
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} + 18 T^{2} + 324)^{4} $ (T^4 + 18*T^2 + 324)^4
$13$	$ (T^{4} - 24 T^{2} + 136)^{4} $ (T^4 - 24*T^2 + 136)^4
$17$	$ (T^{8} - 20 T^{6} + 398 T^{4} + \cdots + 4)^{2} $ (T^8 - 20T^6 + 398T^4 - 40*T^2 + 4)^2
$19$	$ (T^{8} - 20 T^{6} + \cdots + 9604)^{2} $ (T^8 - 20T^6 + 302T^4 - 1960*T^2 + 9604)^2
$23$	$ (T^{8} - 40 T^{6} + \cdots + 73984)^{2} $ (T^8 - 40T^6 + 1328T^4 - 10880*T^2 + 73984)^2
$29$	$ (T^{4} + 40 T^{2} + 272)^{4} $ (T^4 + 40*T^2 + 272)^4
$31$	$ (T^{8} + 112 T^{6} + \cdots + 295936)^{2} $ (T^8 + 112T^6 + 12000T^4 + 60928*T^2 + 295936)^2
$37$	$ (T^{8} - 56 T^{6} + \cdots + 73984)^{2} $ (T^8 - 56T^6 + 2864T^4 - 15232*T^2 + 73984)^2
$41$	$ (T^{4} + 52 T^{2} + 98)^{4} $ (T^4 + 52*T^2 + 98)^4
$43$	$ (T^{2} - 8 T + 14)^{8} $ (T^2 - 8*T + 14)^8
$47$	$ (T^{8} + 96 T^{6} + \cdots + 4734976)^{2} $ (T^8 + 96T^6 + 7040T^4 + 208896*T^2 + 4734976)^2
$53$	$ (T^{8} - 112 T^{6} + \cdots + 1183744)^{2} $ (T^8 - 112T^6 + 11456T^4 - 121856*T^2 + 1183744)^2
$59$	$ (T^{8} - 116 T^{6} + \cdots + 4)^{2} $ (T^8 - 116T^6 + 13454T^4 - 232*T^2 + 4)^2
$61$	$ (T^{8} + 56 T^{6} + \cdots + 18496)^{2} $ (T^8 + 56T^6 + 3000T^4 + 7616*T^2 + 18496)^2
$67$	$ (T^{2} - 2 T + 4)^{8} $ (T^2 - 2*T + 4)^8
$71$	$ (T^{4} + 464 T^{2} + 53312)^{4} $ (T^4 + 464*T^2 + 53312)^4
$73$	$ (T^{8} - 20 T^{6} + \cdots + 9604)^{2} $ (T^8 - 20T^6 + 302T^4 - 1960*T^2 + 9604)^2
$79$	$ (T^{8} - 160 T^{6} + \cdots + 18939904)^{2} $ (T^8 - 160T^6 + 21248T^4 - 696320*T^2 + 18939904)^2
$83$	$ (T^{4} + 100 T^{2} + 1250)^{4} $ (T^4 + 100*T^2 + 1250)^4
$89$	$ (T^{8} - 100 T^{6} + \cdots + 334084)^{2} $ (T^8 - 100T^6 + 9422T^4 - 57800*T^2 + 334084)^2
$97$	$ (T^{4} + 324 T^{2} + 13122)^{4} $ (T^4 + 324*T^2 + 13122)^4
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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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{14} + \beta_{5}) q^{2} - \beta_{15} q^{3} + (\beta_{11} - \beta_{8} - \beta_{7} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{14} + \beta_{11} + \beta_{5}) q^{9}+O(q^{10}) \) q + (b14 + b5) * q^2 - b15 * q^3 + (b11 - b8 - b7 + b6 + b5 + 1) * q^4 + (-2b15 - 2b12 - b10 - b3 - 2b2) q^5 + (b15 + b13 + b10 + b9) * q^6 + (b14 - b7 + b5 + 2b4 + 2) q^8 + (-b14 + b11 + b5) * q^9	\( q + (\beta_{14} + \beta_{5}) q^{2} - \beta_{15} q^{3} + (\beta_{11} - \beta_{8} - \beta_{7} + \cdots + 1) q^{4}+ \cdots + (3 \beta_{14} - 3 \beta_{7} + 3 \beta_{6} + \cdots + 9) q^{99}+O(q^{100}) \) q + (b14 + b5) * q^2 - b15 * q^3 + (b11 - b8 - b7 + b6 + b5 + 1) * q^4 + (-2b15 - 2b12 - b10 - b3 - 2b2) q^5 + (b15 + b13 + b10 + b9) * q^6 + (b14 - b7 + b5 + 2b4 + 2) q^8 + (-b14 + b11 + b5) * q^9 + (b15 + b13 + b12 - 2b9 - 3b3 - 2b1) q^10 + (-3b11 + 3b7 - 3b6 - 3b5 - 3) * q^11 + (-b15 - b12 - b10 - b3 - b2 + b1) * q^12 + (-b15 - b13 - b10 - 2b2) q^13 + (-4b4 - 2) q^15 + (-b14 + 3b11 + b8 + 2b5 - b4) * q^16 + (b15 - 2b13 + 2b3) * q^17 + (-b11 + b8 - b7 - b6 + b5 + 1) * q^18 + (-2b10 + b3) q^19 + (-4b15 + b13 - 4b10 + 4b9 - b2) q^20 + (3b6 - 3b4 - 3) * q^22 + (2b14 + 2b11 + 2b5) q^23 + (b15 - b13 + 2b12 - b9 - b1) q^24 + (2b11 - 2b7 + 2b6 - 5b5 - 5) * q^25 + (b15 + b12 - 3b3 + b2 - 2b1) * q^26 + (3b15 - b13 + 3b10) * q^27 + (2b14 - 2b7 - 2b6 + 2b5) * q^29 + (2b14 - 4b11 - 4b8 - 2b5 + 4b4) q^30 + (-2b15 - 4b12 - 4b9 - 4b3 - 4b1) q^31 + (-2b11 - b7 - 2b6 + 4b5 + 4) q^32 + (3b10 + 3b3) * q^33 + (b15 - b13 + b10 - 3b9 + 2b2) * q^34 + (-b14 + b7 - 2b6 - b5 - 4) q^36 + (4b8 + 2b5 - 4b4) q^37 + (-2b15 - 2b13 - b12 - 3b9 - b3 - 3b1) * q^38 + (-4b8 - 2b5 - 2) * q^39 + (-b15 - b12 + 6b10 - 3b3 - b2 - 2b1) q^40 + (2b15 - 3b13 + 2b10) q^41 + (-b14 + b7 - b6 - b5 + 3) * q^43 + (-3b11 - 3b8 - 9b5 + 3b4) * q^44 + (b15 + b13 + 2b12 + 4b9 + 3b3 + 4b1) * q^45 + (2b11 - 2b8 - 2b7 + 2b6 + 6b5 + 6) q^46 + (4b15 + 4b12 + 2b10 + 2b3 + 4b2) q^47 + (-b15 - b10 + 3b9 + b2) q^48 + (-7b14 + 7b7 - 2b6 - 7b5 + 2b4 + 2) q^50 + (-b14 + b11 + 2b5) q^51 + (-3b15 + b13 + b12 + 4b9 + 3b3 + 4b1) * q^52 + (2b11 + 4b8 + 2b7 + 2b6 + 2b5 + 2) q^53 + (b15 + b12 - 2b10 + b3 + b2 + 4b1) * q^54 + (-6b13 - 12b9) * q^55 + (b14 - b7 + b6 + b5 - 3) * q^57 + (-2b14 + 2b11 - 2b8 + 4b5 + 2b4) q^58 + (2b15 - 5b13 + 5b3) q^59 + (6b11 + 2b8 - 2b7 + 6b6 - 2b5 - 2) q^60 + (-2b15 - 2b12 - b10 - 3b3 - 2b2 - 4b1) q^61 + (-2b15 + 6b13 - 2b10 - 2b9) * q^62 + (3b14 - 3b7 - b6 + 3b5 + b4 - 5) q^64 + (-2b14 + 2b11 - 12b5) q^65 + (3b15 + 3b13 - 3b12 - 3b3) * q^66 + (2b5 + 2) q^67 + (b15 + b12 - b10 + 5b3 + b2 + b1) q^68 + (2b13 + 4b9) * q^69 + (-6b14 + 6b7 + 6b6 - 6b5 + 4b4 + 2) q^71 + (-3b14 - b11 + b8 - b4) q^72 + (-2b15 + b13 - b3) q^73 + (-4b11 - 4b8 + 2b7 - 4b6 - 4b5 - 4) q^74 + (-9b10 - 2b3) * q^75 + (b15 + 5b13 + b10 + b9 + 2b2) * q^76 + (2b14 - 2b7 + 4b6 + 2b5 + 4b4 + 4) q^78 + (-4b14 - 4b11 - 4b5) q^79 + (5b15 + 9b13 + b12 + 8b9 - b3 + 8b1) * q^80 + (-5b11 + 5b7 - 5b6 - 2b5 - 2) * q^81 + (3b15 + 3b12 + b10 + 3b3 + 3b2 + 5b1) q^82 + 5b13 q^83 + (4b14 - 4b7 - 4b6 + 4b5 + 4b4 + 2) q^85 + (4b14 - b11 + b8 + 5b5 - b4) * q^86 + (2b13 + 4b9 + 2b3 + 4b1) * q^87 + (3b11 + 3b8 + 6b7 + 3b6 - 3b5 - 3) q^88 + (4b10 + 3b3) * q^89 + (2b15 - 5b13 + 2b10 - b2) q^90 + (6b14 - 6b7 + 6b5 + 4b4 + 4) * q^92 + (-2b14 - 2b11 + 4b8 - 4b4) * q^93 + (-2b15 - 2b13 - 2b12 + 4b9 + 6b3 + 4b1) * q^94 + (-2b11 + 8b8 - 2b7 - 2b6 + 4b5 + 4) q^95 + (5b10 + 2b3 + 3b1) q^96 + (-9b15 - 9b10) * q^97 + (3b14 - 3b7 + 3b6 + 3b5 + 9) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 4 q^{4} + 8 q^{8} - 8 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 4 * q^4 + 8 * q^8 - 8 * q^9	\( 16 q - 4 q^{2} + 4 q^{4} + 8 q^{8} - 8 q^{9} - 4 q^{16} + 4 q^{18} - 48 q^{22} - 56 q^{25} - 8 q^{30} + 36 q^{32} - 40 q^{36} + 64 q^{43} + 48 q^{44} + 40 q^{46} + 88 q^{50} - 16 q^{51} - 64 q^{57} - 40 q^{58} - 56 q^{60} - 104 q^{64} + 96 q^{65} + 16 q^{67} - 12 q^{72} + 8 q^{74} - 16 q^{78} + 24 q^{81} - 24 q^{86} - 24 q^{88} - 16 q^{92} + 96 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 4 * q^4 + 8 * q^8 - 8 * q^9 - 4 * q^16 + 4 * q^18 - 48 * q^22 - 56 * q^25 - 8 * q^30 + 36 * q^32 - 40 * q^36 + 64 * q^43 + 48 * q^44 + 40 * q^46 + 88 * q^50 - 16 * q^51 - 64 * q^57 - 40 * q^58 - 56 * q^60 - 104 * q^64 + 96 * q^65 + 16 * q^67 - 12 * q^72 + 8 * q^74 - 16 * q^78 + 24 * q^81 - 24 * q^86 - 24 * q^88 - 16 * q^92 + 96 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	392.2.m.h

Level	$392$
Weight	$2$
Character orbit	392.m
Analytic conductor	$3.130$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 4x^{14} + 6x^{12} + 8x^{10} + 20x^{8} + 32x^{6} + 96x^{4} + 256x^{2} + 256 \) x^16 + 4x^14 + 6x^12 + 8x^10 + 20x^8 + 32x^6 + 96x^4 + 256*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{15} + 6\nu^{13} + 4\nu^{11} - 12\nu^{9} + 24\nu^{7} + 24\nu^{5} + 88\nu^{3} + 320\nu ) / 224 \) (v^15 + 6v^13 + 4v^11 - 12v^9 + 24v^7 + 24v^5 + 88v^3 + 320*v) / 224
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{15} - 44\nu^{13} - 6\nu^{11} - 24\nu^{9} - 148\nu^{7} - 64\nu^{5} - 384\nu^{3} - 2944\nu ) / 896 \) (-5v^15 - 44v^13 - 6v^11 - 24v^9 - 148v^7 - 64v^5 - 384v^3 - 2944v) / 896
\(\beta_{4}\)	\(=\)	\( ( 5\nu^{14} - 12\nu^{12} + 6\nu^{10} + 24\nu^{8} - 76\nu^{6} + 64\nu^{4} + 384\nu^{2} - 1088 ) / 448 \) (5v^14 - 12v^12 + 6v^10 + 24v^8 - 76v^6 + 64v^4 + 384*v^2 - 1088) / 448
\(\beta_{5}\)	\(=\)	\( ( 3\nu^{14} - 3\nu^{12} - 2\nu^{10} + 6\nu^{8} - 12\nu^{6} - 12\nu^{4} + 96\nu^{2} - 384 ) / 224 \) (3v^14 - 3v^12 - 2v^10 + 6v^8 - 12v^6 - 12v^4 + 96*v^2 - 384) / 224
\(\beta_{6}\)	\(=\)	\( ( 5\nu^{14} + 16\nu^{12} + 6\nu^{10} + 24\nu^{8} + 36\nu^{6} + 64\nu^{4} + 384\nu^{2} + 704 ) / 224 \) (5v^14 + 16v^12 + 6v^10 + 24v^8 + 36v^6 + 64v^4 + 384*v^2 + 704) / 224
\(\beta_{7}\)	\(=\)	\( ( 5\nu^{14} + 9\nu^{12} + 6\nu^{10} + 38\nu^{8} + 36\nu^{6} + 36\nu^{4} + 272\nu^{2} + 480 ) / 224 \) (5v^14 + 9v^12 + 6v^10 + 38v^8 + 36v^6 + 36v^4 + 272*v^2 + 480) / 224
\(\beta_{8}\)	\(=\)	\( ( 5\nu^{14} + 9\nu^{12} + 6\nu^{10} + 38\nu^{8} + 36\nu^{6} + 36\nu^{4} + 496\nu^{2} + 480 ) / 224 \) (5v^14 + 9v^12 + 6v^10 + 38v^8 + 36v^6 + 36v^4 + 496*v^2 + 480) / 224
\(\beta_{9}\)	\(=\)	\( ( 3\nu^{15} + 18\nu^{13} + 12\nu^{11} + 20\nu^{9} + 72\nu^{7} + 72\nu^{5} + 264\nu^{3} + 960\nu ) / 224 \) (3v^15 + 18v^13 + 12v^11 + 20v^9 + 72v^7 + 72v^5 + 264v^3 + 960v) / 224
\(\beta_{10}\)	\(=\)	\( ( 15\nu^{15} + 20\nu^{13} + 18\nu^{11} + 72\nu^{9} - 4\nu^{7} + 192\nu^{5} + 1152\nu^{3} + 768\nu ) / 896 \) (15v^15 + 20v^13 + 18v^11 + 72v^9 - 4v^7 + 192v^5 + 1152v^3 + 768v) / 896
\(\beta_{11}\)	\(=\)	\( ( 9\nu^{14} + 26\nu^{12} + 22\nu^{10} + 60\nu^{8} + 132\nu^{6} + 104\nu^{4} + 736\nu^{2} + 1536 ) / 224 \) (9v^14 + 26v^12 + 22v^10 + 60v^8 + 132v^6 + 104v^4 + 736*v^2 + 1536) / 224
\(\beta_{12}\)	\(=\)	\( ( 11\nu^{15} + 24\nu^{13} + 2\nu^{11} + 64\nu^{9} + 124\nu^{7} + 208\nu^{5} + 800\nu^{3} + 1280\nu ) / 448 \) (11v^15 + 24v^13 + 2v^11 + 64v^9 + 124v^7 + 208v^5 + 800v^3 + 1280v) / 448
\(\beta_{13}\)	\(=\)	\( ( -6\nu^{15} - 15\nu^{13} - 10\nu^{11} - 26\nu^{9} - 60\nu^{7} - 60\nu^{5} - 360\nu^{3} - 800\nu ) / 224 \) (-6v^15 - 15v^13 - 10v^11 - 26v^9 - 60v^7 - 60v^5 - 360v^3 - 800v) / 224
\(\beta_{14}\)	\(=\)	\( ( 29\nu^{14} + 76\nu^{12} + 46\nu^{10} + 184\nu^{8} + 388\nu^{6} + 416\nu^{4} + 2272\nu^{2} + 4352 ) / 448 \) (29v^14 + 76v^12 + 46v^10 + 184v^8 + 388v^6 + 416v^4 + 2272*v^2 + 4352) / 448
\(\beta_{15}\)	\(=\)	\( ( 29\nu^{15} + 76\nu^{13} + 46\nu^{11} + 184\nu^{9} + 388\nu^{7} + 192\nu^{5} + 2272\nu^{3} + 4352\nu ) / 896 \) (29v^15 + 76v^13 + 46v^11 + 184v^9 + 388v^7 + 192v^5 + 2272v^3 + 4352v) / 896

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{8} - \beta_{7} \) b8 - b7
\(\nu^{3}\)	\(=\)	\( \beta_{15} + 2\beta_{13} + \beta_{10} + \beta_{2} \) b15 + 2*b13 + b10 + b2
\(\nu^{4}\)	\(=\)	\( 2\beta_{14} - 2\beta_{11} - 2\beta_{8} - 2\beta_{5} + 2\beta_{4} \) 2b14 - 2b11 - 2b8 - 2b5 + 2*b4
\(\nu^{5}\)	\(=\)	\( -2\beta_{15} + 2\beta_{12} + 2\beta_{9} + 2\beta_{3} + 2\beta_1 \) -2b15 + 2b12 + 2b9 + 2b3 + 2*b1
\(\nu^{6}\)	\(=\)	\( 2\beta_{14} - 2\beta_{7} - 4\beta_{6} + 2\beta_{5} - 2\beta_{4} - 4 \) 2b14 - 2b7 - 4b6 + 2b5 - 2*b4 - 4
\(\nu^{7}\)	\(=\)	\( 2\beta_{15} + 2\beta_{12} - 6\beta_{10} + 4\beta_{3} + 2\beta_{2} \) 2b15 + 2b12 - 6b10 + 4b3 + 2*b2
\(\nu^{8}\)	\(=\)	\( -4\beta_{11} + 4\beta_{8} + 12\beta_{7} - 4\beta_{6} - 8\beta_{5} - 8 \) -4b11 + 4b8 + 12b7 - 4b6 - 8*b5 - 8
\(\nu^{9}\)	\(=\)	\( 4\beta_{9} - 12\beta_{2} \) 4b9 - 12b2
\(\nu^{10}\)	\(=\)	\( -12\beta_{14} + 24\beta_{11} - 12\beta_{8} - 4\beta_{5} + 12\beta_{4} \) -12b14 + 24b11 - 12b8 - 4b5 + 12*b4
\(\nu^{11}\)	\(=\)	\( -8\beta_{13} - 12\beta_{12} + 16\beta_{9} + 24\beta_{3} + 16\beta_1 \) -8b13 - 12b12 + 16b9 + 24b3 + 16*b1
\(\nu^{12}\)	\(=\)	\( -8\beta_{14} + 8\beta_{7} + 24\beta_{6} - 8\beta_{5} - 8\beta_{4} - 48 \) -8b14 + 8b7 + 24b6 - 8b5 - 8*b4 - 48
\(\nu^{13}\)	\(=\)	\( -8\beta_{15} - 8\beta_{12} + 16\beta_{10} - 40\beta_{3} - 8\beta_{2} - 72\beta_1 \) -8b15 - 8b12 + 16b10 - 40b3 - 8b2 - 72b1
\(\nu^{14}\)	\(=\)	\( 16\beta_{11} - 56\beta_{8} + 8\beta_{7} + 16\beta_{6} + 80\beta_{5} + 80 \) 16b11 - 56b8 + 8b7 + 16b6 + 80*b5 + 80
\(\nu^{15}\)	\(=\)	\( -40\beta_{15} - 144\beta_{13} - 40\beta_{10} - 64\beta_{9} - 8\beta_{2} \) -40b15 - 144b13 - 40b10 - 64b9 - 8*b2

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\beta_{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 2 T^{7} + T^{6} + \cdots + 16)^{2} \) (T^8 + 2T^7 + T^6 - 2T^5 - 3T^4 - 4T^3 + 4T^2 + 16T + 16)^2
$3$	\( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 - 4T^6 + 14T^4 - 8*T^2 + 4)^2
$5$	\( (T^{8} + 24 T^{6} + \cdots + 18496)^{2} \) (T^8 + 24T^6 + 440T^4 + 3264*T^2 + 18496)^2
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} + 18 T^{2} + 324)^{4} \) (T^4 + 18*T^2 + 324)^4
$13$	\( (T^{4} - 24 T^{2} + 136)^{4} \) (T^4 - 24*T^2 + 136)^4
$17$	\( (T^{8} - 20 T^{6} + 398 T^{4} + \cdots + 4)^{2} \) (T^8 - 20T^6 + 398T^4 - 40*T^2 + 4)^2
$19$	\( (T^{8} - 20 T^{6} + \cdots + 9604)^{2} \) (T^8 - 20T^6 + 302T^4 - 1960*T^2 + 9604)^2
$23$	\( (T^{8} - 40 T^{6} + \cdots + 73984)^{2} \) (T^8 - 40T^6 + 1328T^4 - 10880*T^2 + 73984)^2
$29$	\( (T^{4} + 40 T^{2} + 272)^{4} \) (T^4 + 40*T^2 + 272)^4
$31$	\( (T^{8} + 112 T^{6} + \cdots + 295936)^{2} \) (T^8 + 112T^6 + 12000T^4 + 60928*T^2 + 295936)^2
$37$	\( (T^{8} - 56 T^{6} + \cdots + 73984)^{2} \) (T^8 - 56T^6 + 2864T^4 - 15232*T^2 + 73984)^2
$41$	\( (T^{4} + 52 T^{2} + 98)^{4} \) (T^4 + 52*T^2 + 98)^4
$43$	\( (T^{2} - 8 T + 14)^{8} \) (T^2 - 8*T + 14)^8
$47$	\( (T^{8} + 96 T^{6} + \cdots + 4734976)^{2} \) (T^8 + 96T^6 + 7040T^4 + 208896*T^2 + 4734976)^2
$53$	\( (T^{8} - 112 T^{6} + \cdots + 1183744)^{2} \) (T^8 - 112T^6 + 11456T^4 - 121856*T^2 + 1183744)^2
$59$	\( (T^{8} - 116 T^{6} + \cdots + 4)^{2} \) (T^8 - 116T^6 + 13454T^4 - 232*T^2 + 4)^2
$61$	\( (T^{8} + 56 T^{6} + \cdots + 18496)^{2} \) (T^8 + 56T^6 + 3000T^4 + 7616*T^2 + 18496)^2
$67$	\( (T^{2} - 2 T + 4)^{8} \) (T^2 - 2*T + 4)^8
$71$	\( (T^{4} + 464 T^{2} + 53312)^{4} \) (T^4 + 464*T^2 + 53312)^4
$73$	\( (T^{8} - 20 T^{6} + \cdots + 9604)^{2} \) (T^8 - 20T^6 + 302T^4 - 1960*T^2 + 9604)^2
$79$	\( (T^{8} - 160 T^{6} + \cdots + 18939904)^{2} \) (T^8 - 160T^6 + 21248T^4 - 696320*T^2 + 18939904)^2
$83$	\( (T^{4} + 100 T^{2} + 1250)^{4} \) (T^4 + 100*T^2 + 1250)^4
$89$	\( (T^{8} - 100 T^{6} + \cdots + 334084)^{2} \) (T^8 - 100T^6 + 9422T^4 - 57800*T^2 + 334084)^2
$97$	\( (T^{4} + 324 T^{2} + 13122)^{4} \) (T^4 + 324*T^2 + 13122)^4
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