LMFDB - Newform orbit 1152.2.l.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(287,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("1152.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, 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:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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_{3} q^{5} - \beta_{9} q^{7}+O(q^{10}) $ q - b3 * q^5 - b9 * q^7	$ q - \beta_{3} q^{5} - \beta_{9} q^{7} + (\beta_{14} + \beta_{2}) q^{11} - \beta_{4} q^{13} + ( - \beta_{14} - \beta_{10} + \cdots - \beta_{7}) q^{17}+ \cdots + (\beta_{12} + 2 \beta_{9} - \beta_{4}) q^{97}+O(q^{100}) $ q - b3 * q^5 - b9 * q^7 + (b14 + b2) * q^11 - b4 * q^13 + (-b14 - b10 + b8 - b7) * q^17 + (b12 - b6 - b5 + 1) * q^19 + (-b15 - b14 - b10 + b8 + b3 + b2) * q^23 + (-2b11 + b6 + b5 - b1) q^25 + (-b15 - b13 - b10 - b7 + b2) * q^29 + (b12 - b11 - 2b6 + b5 + b4 - b1) q^31 + (-b15 + b13 + b10 + b8 - 2b7 - b3) q^35 + (b12 - 2b11 + 2b9 + b5) * q^37 + (-b14 - 2b13 - b8) q^41 + (-b11 - b9 - 2b6 - 2b1 - 2) * q^43 + (b14 + b13 + 3b10 + b8 + b3 - b2) q^47 + (b12 + b5 - b4 + b1 + 1) * q^49 + (b15 - b13 + b10 - b7 - b3) * q^53 + (-2b9 - 2b5 - 2b1 + 4) q^55 + (-b15 - b13 + 2b10 + 2b7) * q^59 + (2b6 - b4 - b1 + 2) q^61 + (-2b15 - b14 - b10 + b8 - 2b7) * q^65 + (b12 + b11 - b9 + b6 + b5 - 1) * q^67 + (-b15 - 4b7) q^71 + (-b12 + b5 - b4 - b1) * q^73 + (-2b14 - 2b10 - 2b7) q^77 + (b12 - b11 + 2b6 - b5 + b4 + b1) q^79 + (-b15 + b13 - 3b10 + b8 + 2b7 + 3b3) q^83 + (-2b6 + b5 + 2) q^85 + (-b10 - 2b3 + 2b2) * q^89 + (-2b11 - 2b9 + 3b6 + b4 + b1 + 3) q^91 + (b14 - 5b10 + b8 - 3b3 + 3b2) q^95 + (b12 + 2b9 - b4) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{19} - 32 q^{43} + 16 q^{49} + 64 q^{55} + 32 q^{61} - 16 q^{67} + 32 q^{85} + 48 q^{91}+O(q^{100}) $ 16 * q + 16 * q^19 - 32 * q^43 + 16 * q^49 + 64 * q^55 + 32 * q^61 - 16 * q^67 + 32 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4*x^14 + 6*x^12 - 12*x^10 + 33*x^8 - 48*x^6 + 96*x^4 - 256*x^2 + 256 :

$\beta_{1}$	$=$	$ ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 1920 $ (17v^14 - 72v^12 + 6v^10 - 36v^8 + 513v^6 - 1332v^4 + 3696*v^2 - 4544) / 1920
$\beta_{2}$	$=$	$ ( 11\nu^{15} - 66\nu^{13} + 138\nu^{11} - 168\nu^{9} + 339\nu^{7} - 486\nu^{5} + 1008\nu^{3} - 1952\nu ) / 1920 $ (11v^15 - 66v^13 + 138v^11 - 168v^9 + 339v^7 - 486v^5 + 1008v^3 - 1952v) / 1920
$\beta_{3}$	$=$	$ ( 19\nu^{15} - 54\nu^{13} + 42\nu^{11} - 192\nu^{9} + 651\nu^{7} - 954\nu^{5} + 1872\nu^{3} - 1888\nu ) / 1920 $ (19v^15 - 54v^13 + 42v^11 - 192v^9 + 651v^7 - 954v^5 + 1872v^3 - 1888v) / 1920
$\beta_{4}$	$=$	$ ( 2\nu^{14} - 9\nu^{12} - 30\nu^{8} + 78\nu^{6} + 15\nu^{4} + 192\nu^{2} - 560 ) / 96 $ (2v^14 - 9v^12 - 30v^8 + 78v^6 + 15v^4 + 192v^2 - 560) / 96
$\beta_{5}$	$=$	$ ( 43\nu^{14} - 48\nu^{12} - 126\nu^{10} + 36\nu^{8} + 27\nu^{6} + 492\nu^{4} + 144\nu^{2} - 1216 ) / 1920 $ (43v^14 - 48v^12 - 126v^10 + 36v^8 + 27v^6 + 492v^4 + 144*v^2 - 1216) / 1920
$\beta_{6}$	$=$	$ ( 43\nu^{14} - 108\nu^{12} + 114\nu^{10} - 324\nu^{8} + 747\nu^{6} - 528\nu^{4} + 3024\nu^{2} - 6016 ) / 1920 $ (43v^14 - 108v^12 + 114v^10 - 324v^8 + 747v^6 - 528v^4 + 3024*v^2 - 6016) / 1920
$\beta_{7}$	$=$	$ ( -\nu^{15} + \nu^{13} - 2\nu^{11} + 10\nu^{9} - 13\nu^{7} + 13\nu^{5} - 56\nu^{3} + 80\nu ) / 64 $ (-v^15 + v^13 - 2v^11 + 10v^9 - 13v^7 + 13v^5 - 56v^3 + 80v) / 64
$\beta_{8}$	$=$	$ ( 61\nu^{15} - 96\nu^{13} + 78\nu^{11} - 228\nu^{9} + 909\nu^{7} - 156\nu^{5} + 5328\nu^{3} - 2752\nu ) / 3840 $ (61v^15 - 96v^13 + 78v^11 - 228v^9 + 909v^7 - 156v^5 + 5328v^3 - 2752v) / 3840
$\beta_{9}$	$=$	$ ( -\nu^{14} + 3\nu^{12} - 2\nu^{10} + 6\nu^{8} - 21\nu^{6} + 31\nu^{4} - 48\nu^{2} + 144 ) / 32 $ (-v^14 + 3v^12 - 2v^10 + 6v^8 - 21v^6 + 31v^4 - 48v^2 + 144) / 32
$\beta_{10}$	$=$	$ ( -89\nu^{15} + 204\nu^{13} - 102\nu^{11} + 732\nu^{9} - 1401\nu^{7} + 984\nu^{5} - 4752\nu^{3} + 9728\nu ) / 3840 $ (-89v^15 + 204v^13 - 102v^11 + 732v^9 - 1401v^7 + 984v^5 - 4752v^3 + 9728v) / 3840
$\beta_{11}$	$=$	$ ( 32\nu^{14} - 57\nu^{12} + 36\nu^{10} - 246\nu^{8} + 588\nu^{6} - 297\nu^{4} + 1776\nu^{2} - 3824 ) / 480 $ (32v^14 - 57v^12 + 36v^10 - 246v^8 + 588v^6 - 297v^4 + 1776*v^2 - 3824) / 480
$\beta_{12}$	$=$	$ ( -17\nu^{14} + 30\nu^{12} - 30\nu^{10} + 168\nu^{8} - 201\nu^{6} + 234\nu^{4} - 1296\nu^{2} + 1760 ) / 192 $ (-17v^14 + 30v^12 - 30v^10 + 168v^8 - 201v^6 + 234v^4 - 1296*v^2 + 1760) / 192
$\beta_{13}$	$=$	$ ( 199\nu^{15} - 504\nu^{13} + 522\nu^{11} - 2172\nu^{9} + 3351\nu^{7} - 3564\nu^{5} + 17232\nu^{3} - 33088\nu ) / 3840 $ (199v^15 - 504v^13 + 522v^11 - 2172v^9 + 3351v^7 - 3564v^5 + 17232v^3 - 33088v) / 3840
$\beta_{14}$	$=$	$ ( -7\nu^{15} + 18\nu^{13} - 18\nu^{11} + 56\nu^{9} - 143\nu^{7} + 134\nu^{5} - 528\nu^{3} + 1184\nu ) / 128 $ (-7v^15 + 18v^13 - 18v^11 + 56v^9 - 143v^7 + 134v^5 - 528v^3 + 1184v) / 128
$\beta_{15}$	$=$	$ ( 17\nu^{15} - 32\nu^{13} + 38\nu^{11} - 116\nu^{9} + 225\nu^{7} - 332\nu^{5} + 1136\nu^{3} - 1600\nu ) / 256 $ (17v^15 - 32v^13 + 38v^11 - 116v^9 + 225v^7 - 332v^5 + 1136v^3 - 1600v) / 256

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

$\nu$	$=$	$ ( \beta_{14} - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2} ) / 4 $ (b14 - b10 + b8 + b3 + b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{9} + \beta_{6} + \beta _1 + 1 ) / 2 $ (b9 + b6 + b1 + 1) / 2
$\nu^{3}$	$=$	$ ( \beta_{14} + 2\beta_{13} + 3\beta_{10} + 3\beta_{8} + 2\beta_{7} + \beta_{3} - \beta_{2} ) / 4 $ (b14 + 2b13 + 3b10 + 3b8 + 2b7 + b3 - b2) / 4
$\nu^{4}$	$=$	$ ( \beta_{12} + 4\beta_{9} + 8\beta_{6} + \beta_{5} + \beta_{4} - \beta _1 + 2 ) / 4 $ (b12 + 4b9 + 8b6 + b5 + b4 - b1 + 2) / 4
$\nu^{5}$	$=$	$ ( -4\beta_{15} - \beta_{14} - 3\beta_{10} + 7\beta_{8} - 4\beta_{7} - 5\beta_{3} + 3\beta_{2} ) / 4 $ (-4b15 - b14 - 3b10 + 7b8 - 4b7 - 5b3 + 3b2) / 4
$\nu^{6}$	$=$	$ ( 2\beta_{12} + 4\beta_{11} + \beta_{9} - \beta_{6} - 2\beta_{5} + \beta _1 + 7 ) / 2 $ (2b12 + 4b11 + b9 - b6 - 2*b5 + b1 + 7) / 2
$\nu^{7}$	$=$	$ ( -8\beta_{15} - 7\beta_{14} - 6\beta_{13} - \beta_{10} + 7\beta_{8} - 14\beta_{7} + 9\beta_{3} + 3\beta_{2} ) / 4 $ (-8b15 - 7b14 - 6b13 - b10 + 7b8 - 14b7 + 9b3 + 3*b2) / 4
$\nu^{8}$	$=$	$ ( 9\beta_{12} + 4\beta_{11} + 8\beta_{9} + 32\beta_{6} + 5\beta_{5} - 7\beta_{4} + 11\beta _1 + 2 ) / 4 $ (9b12 + 4b11 + 8b9 + 32b6 + 5b5 - 7b4 + 11*b1 + 2) / 4
$\nu^{9}$	$=$	$ ( 8\beta_{15} - 13\beta_{14} - 12\beta_{13} + 13\beta_{10} + 7\beta_{8} + 20\beta_{7} - 9\beta_{3} - 5\beta_{2} ) / 4 $ (8b15 - 13b14 - 12b13 + 13b10 + 7b8 + 20b7 - 9b3 - 5b2) / 4
$\nu^{10}$	$=$	$ ( 4\beta_{12} + 5\beta_{9} + 41\beta_{6} - 8\beta_{5} - 8\beta_{4} - 7\beta _1 + 1 ) / 2 $ (4b12 + 5b9 + 41b6 - 8b5 - 8b4 - 7b1 + 1) / 2
$\nu^{11}$	$=$	$ ( -15\beta_{14} - 14\beta_{13} + 43\beta_{10} + 11\beta_{8} - 46\beta_{7} - 31\beta_{3} + 55\beta_{2} ) / 4 $ (-15b14 - 14b13 + 43b10 + 11b8 - 46b7 - 31b3 + 55*b2) / 4
$\nu^{12}$	$=$	$ ( 9\beta_{12} + 72\beta_{11} + 44\beta_{9} - 56\beta_{6} - 31\beta_{5} - 39\beta_{4} + 15\beta _1 - 94 ) / 4 $ (9b12 + 72b11 + 44b9 - 56b6 - 31b5 - 39b4 + 15*b1 - 94) / 4
$\nu^{13}$	$=$	$ ( -12\beta_{15} - 49\beta_{14} - 24\beta_{13} + 133\beta_{10} - \beta_{8} - 180\beta_{7} - 5\beta_{3} - 53\beta_{2} ) / 4 $ (-12b15 - 49b14 - 24b13 + 133b10 - b8 - 180b7 - 5b3 - 53*b2) / 4
$\nu^{14}$	$=$	$ ( 6\beta_{12} + 36\beta_{11} + 9\beta_{9} + 27\beta_{6} + 42\beta_{5} - 48\beta_{4} - 15\beta _1 - 13 ) / 2 $ (6b12 + 36b11 + 9b9 + 27b6 + 42b5 - 48b4 - 15*b1 - 13) / 2
$\nu^{15}$	$=$	$ ( 120 \beta_{15} - 47 \beta_{14} - 150 \beta_{13} - 97 \beta_{10} - 41 \beta_{8} - 126 \beta_{7} + \cdots - 77 \beta_{2} ) / 4 $ (120b15 - 47b14 - 150b13 - 97b10 - 41b8 - 126b7 - 191b3 - 77b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-\beta_{6}$

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} $

287.1

−0.517174	−	1.31626i
0.944649	−	1.05244i
1.36166	−	0.381939i
1.40927	−	0.118126i
−1.40927	+	0.118126i
−1.36166	+	0.381939i
−0.944649	+	1.05244i
0.517174	+	1.31626i
−0.517174	+	1.31626i
0.944649	+	1.05244i
1.36166	+	0.381939i
1.40927	+	0.118126i
−1.40927	−	0.118126i
−1.36166	−	0.381939i
−0.944649	−	1.05244i
0.517174	−	1.31626i

−2.63251

2.63251i

0.207188

287.2

−2.10489

2.10489i

4.40731

287.3

−0.763878

0.763878i

−1.33620

287.4

−0.236253

0.236253i

−3.27830

287.5

0.236253

−

0.236253i

−3.27830

287.6

0.763878

−

0.763878i

−1.33620

287.7

2.10489

−

2.10489i

4.40731

287.8

2.63251

−

2.63251i

0.207188

863.1

−2.63251

−

2.63251i

0.207188

863.2

−2.10489

−

2.10489i

4.40731

863.3

−0.763878

−

0.763878i

−1.33620

863.4

−0.236253

−

0.236253i

−3.27830

863.5

0.236253

0.236253i

−3.27830

863.6

0.763878

0.763878i

−1.33620

863.7

2.10489

2.10489i

4.40731

863.8

2.63251

2.63251i

0.207188

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
16.f	odd	4	1	inner
48.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1152.2.l.b		16
3.b	odd	2	1	inner	1152.2.l.b		16
4.b	odd	2	1		1152.2.l.a		16
8.b	even	2	1		576.2.l.a		16
8.d	odd	2	1		144.2.l.a	✓	16
12.b	even	2	1		1152.2.l.a		16
16.e	even	4	1		144.2.l.a	✓	16
16.e	even	4	1		1152.2.l.a		16
16.f	odd	4	1		576.2.l.a		16
16.f	odd	4	1	inner	1152.2.l.b		16
24.f	even	2	1		144.2.l.a	✓	16
24.h	odd	2	1		576.2.l.a		16
48.i	odd	4	1		144.2.l.a	✓	16
48.i	odd	4	1		1152.2.l.a		16
48.k	even	4	1		576.2.l.a		16
48.k	even	4	1	inner	1152.2.l.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
144.2.l.a	✓	16	8.d	odd	2	1
144.2.l.a	✓	16	16.e	even	4	1
144.2.l.a	✓	16	24.f	even	2	1
144.2.l.a	✓	16	48.i	odd	4	1
576.2.l.a		16	8.b	even	2	1
576.2.l.a		16	16.f	odd	4	1
576.2.l.a		16	24.h	odd	2	1
576.2.l.a		16	48.k	even	4	1
1152.2.l.a		16	4.b	odd	2	1
1152.2.l.a		16	12.b	even	2	1
1152.2.l.a		16	16.e	even	4	1
1152.2.l.a		16	48.i	odd	4	1
1152.2.l.b		16	1.a	even	1	1	trivial
1152.2.l.b		16	3.b	odd	2	1	inner
1152.2.l.b		16	16.f	odd	4	1	inner
1152.2.l.b		16	48.k	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} - 16T_{7}^{2} - 16T_{7} + 4 $ T7^4 - 16*T7^2 - 16*T7 + 4 acting on $S_{2}^{\mathrm{new}}(1152, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 272 T^{12} + \cdots + 256 $ T^16 + 272T^12 + 15456T^8 + 20736*T^4 + 256
$7$	$ (T^{4} - 16 T^{2} - 16 T + 4)^{4} $ (T^4 - 16T^2 - 16T + 4)^4
$11$	$ T^{16} + 960 T^{12} + \cdots + 65536 $ T^16 + 960T^12 + 181760T^8 + 7585792*T^4 + 65536
$13$	$ (T^{8} - 64 T^{5} + \cdots + 400)^{2} $ (T^8 - 64T^5 + 744T^4 - 1792T^3 + 2048T^2 + 1280*T + 400)^2
$17$	$ (T^{8} + 72 T^{6} + \cdots + 1936)^{2} $ (T^8 + 72T^6 + 1208T^4 + 5024*T^2 + 1936)^2
$19$	$ (T^{8} - 8 T^{7} + \cdots + 30976)^{2} $ (T^8 - 8T^7 + 32T^6 - 32T^5 + 1376T^4 - 10624T^3 + 41472T^2 - 50688*T + 30976)^2
$23$	$ (T^{8} + 96 T^{6} + \cdots + 6400)^{2} $ (T^8 + 96T^6 + 1440T^4 + 5632*T^2 + 6400)^2
$29$	$ T^{16} + \cdots + 16062013696 $ T^16 + 5520T^12 + 7056992T^8 + 1768163584*T^4 + 16062013696
$31$	$ (T^{8} + 192 T^{6} + \cdots + 1648656)^{2} $ (T^8 + 192T^6 + 12296T^4 + 288256*T^2 + 1648656)^2
$37$	$ (T^{8} + 512 T^{5} + \cdots + 35344)^{2} $ (T^8 + 512T^5 + 8840T^4 + 47104T^3 + 131072T^2 + 96256*T + 35344)^2
$41$	$ (T^{8} - 168 T^{6} + \cdots + 144)^{2} $ (T^8 - 168T^6 + 7032T^4 - 30752*T^2 + 144)^2
$43$	$ (T^{8} + 16 T^{7} + \cdots + 4129024)^{2} $ (T^8 + 16T^7 + 128T^6 + 512T^5 + 5088T^4 + 65280T^3 + 524288T^2 + 2080768*T + 4129024)^2
$47$	$ (T^{8} - 160 T^{6} + \cdots + 665856)^{2} $ (T^8 - 160T^6 + 7840T^4 - 129536*T^2 + 665856)^2
$53$	$ T^{16} + \cdots + 22663495936 $ T^16 + 5904T^12 + 6358112T^8 + 1216180480*T^4 + 22663495936
$59$	$ T^{16} + \cdots + 2186423566336 $ T^16 + 15104T^12 + 34824192T^8 + 18575523840*T^4 + 2186423566336
$61$	$ (T^{8} - 16 T^{7} + \cdots + 258064)^{2} $ (T^8 - 16T^7 + 128T^6 - 416T^5 + 1032T^4 - 6720T^3 + 61952T^2 - 178816*T + 258064)^2
$67$	$ (T^{8} + 8 T^{7} + \cdots + 7573504)^{2} $ (T^8 + 8T^7 + 32T^6 + 704T^5 + 21888T^4 + 243200T^3 + 1492992T^2 + 4755456*T + 7573504)^2
$71$	$ (T^{8} + 192 T^{6} + \cdots + 73984)^{2} $ (T^8 + 192T^6 + 7712T^4 + 78848*T^2 + 73984)^2
$73$	$ (T^{8} + 176 T^{6} + \cdots + 20736)^{2} $ (T^8 + 176T^6 + 8032T^4 + 90880*T^2 + 20736)^2
$79$	$ (T^{8} + 224 T^{6} + \cdots + 3825936)^{2} $ (T^8 + 224T^6 + 15432T^4 + 415360*T^2 + 3825936)^2
$83$	$ T^{16} + \cdots + 102776124276736 $ T^16 + 40384T^12 + 167437824T^8 + 231440760832*T^4 + 102776124276736
$89$	$ (T^{8} - 200 T^{6} + \cdots + 104976)^{2} $ (T^8 - 200T^6 + 8600T^4 - 60704*T^2 + 104976)^2
$97$	$ (T^{4} - 72 T^{2} + \cdots - 176)^{4} $ (T^4 - 72T^2 - 256T - 176)^4
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) q - b3 * q^5 - b9 * q^7	\( q - \beta_{3} q^{5} - \beta_{9} q^{7} + (\beta_{14} + \beta_{2}) q^{11} - \beta_{4} q^{13} + ( - \beta_{14} - \beta_{10} + \cdots - \beta_{7}) q^{17}+ \cdots + (\beta_{12} + 2 \beta_{9} - \beta_{4}) q^{97}+O(q^{100}) \) q - b3 * q^5 - b9 * q^7 + (b14 + b2) * q^11 - b4 * q^13 + (-b14 - b10 + b8 - b7) * q^17 + (b12 - b6 - b5 + 1) * q^19 + (-b15 - b14 - b10 + b8 + b3 + b2) * q^23 + (-2b11 + b6 + b5 - b1) q^25 + (-b15 - b13 - b10 - b7 + b2) * q^29 + (b12 - b11 - 2b6 + b5 + b4 - b1) q^31 + (-b15 + b13 + b10 + b8 - 2b7 - b3) q^35 + (b12 - 2b11 + 2b9 + b5) * q^37 + (-b14 - 2b13 - b8) q^41 + (-b11 - b9 - 2b6 - 2b1 - 2) * q^43 + (b14 + b13 + 3b10 + b8 + b3 - b2) q^47 + (b12 + b5 - b4 + b1 + 1) * q^49 + (b15 - b13 + b10 - b7 - b3) * q^53 + (-2b9 - 2b5 - 2b1 + 4) q^55 + (-b15 - b13 + 2b10 + 2b7) * q^59 + (2b6 - b4 - b1 + 2) q^61 + (-2b15 - b14 - b10 + b8 - 2b7) * q^65 + (b12 + b11 - b9 + b6 + b5 - 1) * q^67 + (-b15 - 4b7) q^71 + (-b12 + b5 - b4 - b1) * q^73 + (-2b14 - 2b10 - 2b7) q^77 + (b12 - b11 + 2b6 - b5 + b4 + b1) q^79 + (-b15 + b13 - 3b10 + b8 + 2b7 + 3b3) q^83 + (-2b6 + b5 + 2) q^85 + (-b10 - 2b3 + 2b2) * q^89 + (-2b11 - 2b9 + 3b6 + b4 + b1 + 3) q^91 + (b14 - 5b10 + b8 - 3b3 + 3b2) q^95 + (b12 + 2b9 - b4) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{19} - 32 q^{43} + 16 q^{49} + 64 q^{55} + 32 q^{61} - 16 q^{67} + 32 q^{85} + 48 q^{91}+O(q^{100}) \) 16 * q + 16 * q^19 - 32 * q^43 + 16 * q^49 + 64 * q^55 + 32 * q^61 - 16 * q^67 + 32 * q^85 + 48 * q^91

Introduction

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Newspace parameters

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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	1152.2.l.b

Level	$1152$
Weight	$2$
Character orbit	1152.l
Analytic conductor	$9.199$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(9.19876631285\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{18} \)
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

\(\beta_{1}\)	\(=\)	\( ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 1920 \) (17v^14 - 72v^12 + 6v^10 - 36v^8 + 513v^6 - 1332v^4 + 3696*v^2 - 4544) / 1920
\(\beta_{2}\)	\(=\)	\( ( 11\nu^{15} - 66\nu^{13} + 138\nu^{11} - 168\nu^{9} + 339\nu^{7} - 486\nu^{5} + 1008\nu^{3} - 1952\nu ) / 1920 \) (11v^15 - 66v^13 + 138v^11 - 168v^9 + 339v^7 - 486v^5 + 1008v^3 - 1952v) / 1920
\(\beta_{3}\)	\(=\)	\( ( 19\nu^{15} - 54\nu^{13} + 42\nu^{11} - 192\nu^{9} + 651\nu^{7} - 954\nu^{5} + 1872\nu^{3} - 1888\nu ) / 1920 \) (19v^15 - 54v^13 + 42v^11 - 192v^9 + 651v^7 - 954v^5 + 1872v^3 - 1888v) / 1920
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{14} - 9\nu^{12} - 30\nu^{8} + 78\nu^{6} + 15\nu^{4} + 192\nu^{2} - 560 ) / 96 \) (2v^14 - 9v^12 - 30v^8 + 78v^6 + 15v^4 + 192v^2 - 560) / 96
\(\beta_{5}\)	\(=\)	\( ( 43\nu^{14} - 48\nu^{12} - 126\nu^{10} + 36\nu^{8} + 27\nu^{6} + 492\nu^{4} + 144\nu^{2} - 1216 ) / 1920 \) (43v^14 - 48v^12 - 126v^10 + 36v^8 + 27v^6 + 492v^4 + 144*v^2 - 1216) / 1920
\(\beta_{6}\)	\(=\)	\( ( 43\nu^{14} - 108\nu^{12} + 114\nu^{10} - 324\nu^{8} + 747\nu^{6} - 528\nu^{4} + 3024\nu^{2} - 6016 ) / 1920 \) (43v^14 - 108v^12 + 114v^10 - 324v^8 + 747v^6 - 528v^4 + 3024*v^2 - 6016) / 1920
\(\beta_{7}\)	\(=\)	\( ( -\nu^{15} + \nu^{13} - 2\nu^{11} + 10\nu^{9} - 13\nu^{7} + 13\nu^{5} - 56\nu^{3} + 80\nu ) / 64 \) (-v^15 + v^13 - 2v^11 + 10v^9 - 13v^7 + 13v^5 - 56v^3 + 80v) / 64
\(\beta_{8}\)	\(=\)	\( ( 61\nu^{15} - 96\nu^{13} + 78\nu^{11} - 228\nu^{9} + 909\nu^{7} - 156\nu^{5} + 5328\nu^{3} - 2752\nu ) / 3840 \) (61v^15 - 96v^13 + 78v^11 - 228v^9 + 909v^7 - 156v^5 + 5328v^3 - 2752v) / 3840
\(\beta_{9}\)	\(=\)	\( ( -\nu^{14} + 3\nu^{12} - 2\nu^{10} + 6\nu^{8} - 21\nu^{6} + 31\nu^{4} - 48\nu^{2} + 144 ) / 32 \) (-v^14 + 3v^12 - 2v^10 + 6v^8 - 21v^6 + 31v^4 - 48v^2 + 144) / 32
\(\beta_{10}\)	\(=\)	\( ( -89\nu^{15} + 204\nu^{13} - 102\nu^{11} + 732\nu^{9} - 1401\nu^{7} + 984\nu^{5} - 4752\nu^{3} + 9728\nu ) / 3840 \) (-89v^15 + 204v^13 - 102v^11 + 732v^9 - 1401v^7 + 984v^5 - 4752v^3 + 9728v) / 3840
\(\beta_{11}\)	\(=\)	\( ( 32\nu^{14} - 57\nu^{12} + 36\nu^{10} - 246\nu^{8} + 588\nu^{6} - 297\nu^{4} + 1776\nu^{2} - 3824 ) / 480 \) (32v^14 - 57v^12 + 36v^10 - 246v^8 + 588v^6 - 297v^4 + 1776*v^2 - 3824) / 480
\(\beta_{12}\)	\(=\)	\( ( -17\nu^{14} + 30\nu^{12} - 30\nu^{10} + 168\nu^{8} - 201\nu^{6} + 234\nu^{4} - 1296\nu^{2} + 1760 ) / 192 \) (-17v^14 + 30v^12 - 30v^10 + 168v^8 - 201v^6 + 234v^4 - 1296*v^2 + 1760) / 192
\(\beta_{13}\)	\(=\)	\( ( 199\nu^{15} - 504\nu^{13} + 522\nu^{11} - 2172\nu^{9} + 3351\nu^{7} - 3564\nu^{5} + 17232\nu^{3} - 33088\nu ) / 3840 \) (199v^15 - 504v^13 + 522v^11 - 2172v^9 + 3351v^7 - 3564v^5 + 17232v^3 - 33088v) / 3840
\(\beta_{14}\)	\(=\)	\( ( -7\nu^{15} + 18\nu^{13} - 18\nu^{11} + 56\nu^{9} - 143\nu^{7} + 134\nu^{5} - 528\nu^{3} + 1184\nu ) / 128 \) (-7v^15 + 18v^13 - 18v^11 + 56v^9 - 143v^7 + 134v^5 - 528v^3 + 1184v) / 128
\(\beta_{15}\)	\(=\)	\( ( 17\nu^{15} - 32\nu^{13} + 38\nu^{11} - 116\nu^{9} + 225\nu^{7} - 332\nu^{5} + 1136\nu^{3} - 1600\nu ) / 256 \) (17v^15 - 32v^13 + 38v^11 - 116v^9 + 225v^7 - 332v^5 + 1136v^3 - 1600v) / 256

\(\nu\)	\(=\)	\( ( \beta_{14} - \beta_{10} + \beta_{8} + \beta_{3} + \beta_{2} ) / 4 \) (b14 - b10 + b8 + b3 + b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{9} + \beta_{6} + \beta _1 + 1 ) / 2 \) (b9 + b6 + b1 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{14} + 2\beta_{13} + 3\beta_{10} + 3\beta_{8} + 2\beta_{7} + \beta_{3} - \beta_{2} ) / 4 \) (b14 + 2b13 + 3b10 + 3b8 + 2b7 + b3 - b2) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{12} + 4\beta_{9} + 8\beta_{6} + \beta_{5} + \beta_{4} - \beta _1 + 2 ) / 4 \) (b12 + 4b9 + 8b6 + b5 + b4 - b1 + 2) / 4
\(\nu^{5}\)	\(=\)	\( ( -4\beta_{15} - \beta_{14} - 3\beta_{10} + 7\beta_{8} - 4\beta_{7} - 5\beta_{3} + 3\beta_{2} ) / 4 \) (-4b15 - b14 - 3b10 + 7b8 - 4b7 - 5b3 + 3b2) / 4
\(\nu^{6}\)	\(=\)	\( ( 2\beta_{12} + 4\beta_{11} + \beta_{9} - \beta_{6} - 2\beta_{5} + \beta _1 + 7 ) / 2 \) (2b12 + 4b11 + b9 - b6 - 2*b5 + b1 + 7) / 2
\(\nu^{7}\)	\(=\)	\( ( -8\beta_{15} - 7\beta_{14} - 6\beta_{13} - \beta_{10} + 7\beta_{8} - 14\beta_{7} + 9\beta_{3} + 3\beta_{2} ) / 4 \) (-8b15 - 7b14 - 6b13 - b10 + 7b8 - 14b7 + 9b3 + 3*b2) / 4
\(\nu^{8}\)	\(=\)	\( ( 9\beta_{12} + 4\beta_{11} + 8\beta_{9} + 32\beta_{6} + 5\beta_{5} - 7\beta_{4} + 11\beta _1 + 2 ) / 4 \) (9b12 + 4b11 + 8b9 + 32b6 + 5b5 - 7b4 + 11*b1 + 2) / 4
\(\nu^{9}\)	\(=\)	\( ( 8\beta_{15} - 13\beta_{14} - 12\beta_{13} + 13\beta_{10} + 7\beta_{8} + 20\beta_{7} - 9\beta_{3} - 5\beta_{2} ) / 4 \) (8b15 - 13b14 - 12b13 + 13b10 + 7b8 + 20b7 - 9b3 - 5b2) / 4
\(\nu^{10}\)	\(=\)	\( ( 4\beta_{12} + 5\beta_{9} + 41\beta_{6} - 8\beta_{5} - 8\beta_{4} - 7\beta _1 + 1 ) / 2 \) (4b12 + 5b9 + 41b6 - 8b5 - 8b4 - 7b1 + 1) / 2
\(\nu^{11}\)	\(=\)	\( ( -15\beta_{14} - 14\beta_{13} + 43\beta_{10} + 11\beta_{8} - 46\beta_{7} - 31\beta_{3} + 55\beta_{2} ) / 4 \) (-15b14 - 14b13 + 43b10 + 11b8 - 46b7 - 31b3 + 55*b2) / 4
\(\nu^{12}\)	\(=\)	\( ( 9\beta_{12} + 72\beta_{11} + 44\beta_{9} - 56\beta_{6} - 31\beta_{5} - 39\beta_{4} + 15\beta _1 - 94 ) / 4 \) (9b12 + 72b11 + 44b9 - 56b6 - 31b5 - 39b4 + 15*b1 - 94) / 4
\(\nu^{13}\)	\(=\)	\( ( -12\beta_{15} - 49\beta_{14} - 24\beta_{13} + 133\beta_{10} - \beta_{8} - 180\beta_{7} - 5\beta_{3} - 53\beta_{2} ) / 4 \) (-12b15 - 49b14 - 24b13 + 133b10 - b8 - 180b7 - 5b3 - 53*b2) / 4
\(\nu^{14}\)	\(=\)	\( ( 6\beta_{12} + 36\beta_{11} + 9\beta_{9} + 27\beta_{6} + 42\beta_{5} - 48\beta_{4} - 15\beta _1 - 13 ) / 2 \) (6b12 + 36b11 + 9b9 + 27b6 + 42b5 - 48b4 - 15*b1 - 13) / 2
\(\nu^{15}\)	\(=\)	\( ( 120 \beta_{15} - 47 \beta_{14} - 150 \beta_{13} - 97 \beta_{10} - 41 \beta_{8} - 126 \beta_{7} + \cdots - 77 \beta_{2} ) / 4 \) (120b15 - 47b14 - 150b13 - 97b10 - 41b8 - 126b7 - 191b3 - 77b2) / 4

\(n\)	\(127\)	\(641\)	\(901\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\beta_{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 287.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 272 T^{12} + \cdots + 256 \) T^16 + 272T^12 + 15456T^8 + 20736*T^4 + 256
$7$	\( (T^{4} - 16 T^{2} - 16 T + 4)^{4} \) (T^4 - 16T^2 - 16T + 4)^4
$11$	\( T^{16} + 960 T^{12} + \cdots + 65536 \) T^16 + 960T^12 + 181760T^8 + 7585792*T^4 + 65536
$13$	\( (T^{8} - 64 T^{5} + \cdots + 400)^{2} \) (T^8 - 64T^5 + 744T^4 - 1792T^3 + 2048T^2 + 1280*T + 400)^2
$17$	\( (T^{8} + 72 T^{6} + \cdots + 1936)^{2} \) (T^8 + 72T^6 + 1208T^4 + 5024*T^2 + 1936)^2
$19$	\( (T^{8} - 8 T^{7} + \cdots + 30976)^{2} \) (T^8 - 8T^7 + 32T^6 - 32T^5 + 1376T^4 - 10624T^3 + 41472T^2 - 50688*T + 30976)^2
$23$	\( (T^{8} + 96 T^{6} + \cdots + 6400)^{2} \) (T^8 + 96T^6 + 1440T^4 + 5632*T^2 + 6400)^2
$29$	\( T^{16} + \cdots + 16062013696 \) T^16 + 5520T^12 + 7056992T^8 + 1768163584*T^4 + 16062013696
$31$	\( (T^{8} + 192 T^{6} + \cdots + 1648656)^{2} \) (T^8 + 192T^6 + 12296T^4 + 288256*T^2 + 1648656)^2
$37$	\( (T^{8} + 512 T^{5} + \cdots + 35344)^{2} \) (T^8 + 512T^5 + 8840T^4 + 47104T^3 + 131072T^2 + 96256*T + 35344)^2
$41$	\( (T^{8} - 168 T^{6} + \cdots + 144)^{2} \) (T^8 - 168T^6 + 7032T^4 - 30752*T^2 + 144)^2
$43$	\( (T^{8} + 16 T^{7} + \cdots + 4129024)^{2} \) (T^8 + 16T^7 + 128T^6 + 512T^5 + 5088T^4 + 65280T^3 + 524288T^2 + 2080768*T + 4129024)^2
$47$	\( (T^{8} - 160 T^{6} + \cdots + 665856)^{2} \) (T^8 - 160T^6 + 7840T^4 - 129536*T^2 + 665856)^2
$53$	\( T^{16} + \cdots + 22663495936 \) T^16 + 5904T^12 + 6358112T^8 + 1216180480*T^4 + 22663495936
$59$	\( T^{16} + \cdots + 2186423566336 \) T^16 + 15104T^12 + 34824192T^8 + 18575523840*T^4 + 2186423566336
$61$	\( (T^{8} - 16 T^{7} + \cdots + 258064)^{2} \) (T^8 - 16T^7 + 128T^6 - 416T^5 + 1032T^4 - 6720T^3 + 61952T^2 - 178816*T + 258064)^2
$67$	\( (T^{8} + 8 T^{7} + \cdots + 7573504)^{2} \) (T^8 + 8T^7 + 32T^6 + 704T^5 + 21888T^4 + 243200T^3 + 1492992T^2 + 4755456*T + 7573504)^2
$71$	\( (T^{8} + 192 T^{6} + \cdots + 73984)^{2} \) (T^8 + 192T^6 + 7712T^4 + 78848*T^2 + 73984)^2
$73$	\( (T^{8} + 176 T^{6} + \cdots + 20736)^{2} \) (T^8 + 176T^6 + 8032T^4 + 90880*T^2 + 20736)^2
$79$	\( (T^{8} + 224 T^{6} + \cdots + 3825936)^{2} \) (T^8 + 224T^6 + 15432T^4 + 415360*T^2 + 3825936)^2
$83$	\( T^{16} + \cdots + 102776124276736 \) T^16 + 40384T^12 + 167437824T^8 + 231440760832*T^4 + 102776124276736
$89$	\( (T^{8} - 200 T^{6} + \cdots + 104976)^{2} \) (T^8 - 200T^6 + 8600T^4 - 60704*T^2 + 104976)^2
$97$	\( (T^{4} - 72 T^{2} + \cdots - 176)^{4} \) (T^4 - 72T^2 - 256T - 176)^4
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