LMFDB - Newform orbit 1140.2.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1140,2,Mod(569,1140)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1140.569"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1140, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1140.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$9.10294583043$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.11007531417600000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{6} $
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{3} + \beta_{5} q^{5} - \beta_{9} q^{7} + ( - \beta_{5} - \beta_{4} + 1) q^{9} - \beta_{3} q^{11} + (\beta_{7} - \beta_{6}) q^{13} - \beta_{14} q^{15} + (\beta_{8} + \beta_{5}) q^{17} + ( - \beta_{2} - 2) q^{19}+ \cdots + ( - \beta_{3} - 2 \beta_1 - 2) q^{99}+O(q^{100}) $ q + b6 * q^3 + b5 * q^5 - b9 * q^7 + (-b5 - b4 + 1) * q^9 - b3 * q^11 + (b7 - b6) * q^13 - b14 * q^15 + (b8 + b5) * q^17 + (-b2 - 2) * q^19 + (b11 - b2) * q^21 + (b8 - b5 + 2b4) q^23 + (2b12 - b5 - b4 + 1) q^25 + (b14 + b13 + b6) * q^27 + (-b15 - b14) * q^29 + (-b15 - b13 + b11 + b2) * q^31 + (b14 + b13 + b10 - b7) * q^33 + (-b8 + b4 - b3) * q^35 + (-b14 - b13 + b7 + 3b6) q^37 + (b5 + b4 - b3 + b1 - 3) * q^39 + (b15 + b14) * q^41 + (-2b12 + b9 + b5 + b4) q^43 + (-2b12 + 2b5 + b4 + 4) * q^45 + (b8 + b5) * q^47 + (b1 - 2) * q^49 + (b15 + b13 + b11 + b2) * q^51 + (-b10 + b6) * q^53 + (-2b12 - 3b9 + b5 + b4 + b1 + 1) * q^55 + (-b12 - 2b9 + b8 - 2b6 + b5 + b4) * q^57 + (-2b15 - b14 - b13 - 2b11) * q^59 + (b1 + 1) * q^61 + (-b9 + 2b8 + 2b5) * q^63 + (-b15 - b13 - 2b11 - b10 + b6) q^65 + (-2b7 + 2b6) * q^67 + (b15 + 2b14 - b13 + b11 + b2) q^69 + (-b14 + b13 + 2b11) q^71 + (2b12 + 2b9 - b5 - b4) * q^73 + (-2b15 - b14 - b13 + 2b11 + b6) * q^75 + (-b8 - 4b5 + 3b4) * q^77 + (-2b15 - 2b13 + 2b11) q^79 + (-2b5 - 2b4 - 7) * q^81 + (3b8 + 3b5) * q^83 + (-2b9 - b1 + 3) q^85 + (-2b12 - b9 - b8 + 2b5 - b4) * q^87 + (-b15 - 2b14 + b13 + 2b11) * q^89 + (3b15 + 3b13 - 3b11 + 3b2) * q^91 + (4b12 + 2b9 - b8 - b5 - 4b4) q^93 + (-b15 - b11 + b10 - 2b5) q^95 + (-b7 + b6) * q^97 + (-b3 - 2b1 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9} - 32 q^{19} + 16 q^{25} - 48 q^{39} + 64 q^{45} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 112 q^{81} + 48 q^{85} - 32 q^{99}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^19 + 16 * q^25 - 48 * q^39 + 64 * q^45 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 112 * q^81 + 48 * q^85 - 32 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7*x^12 + 48*x^8 - 7*x^4 + 1 :

$\beta_{1}$	$=$	$ ( \nu^{12} + 161 ) / 24 $ (v^12 + 161) / 24
$\beta_{2}$	$=$	$ ( -2\nu^{12} + 14\nu^{8} - 94\nu^{4} + 7 ) / 3 $ (-2v^12 + 14v^8 - 94*v^4 + 7) / 3
$\beta_{3}$	$=$	$ ( -7\nu^{15} - 131\nu^{13} + 912\nu^{9} - 6288\nu^{5} - 2351\nu^{3} + 917\nu ) / 144 $ (-7v^15 - 131v^13 + 912v^9 - 6288v^5 - 2351v^3 + 917v) / 144
$\beta_{4}$	$=$	$ ( - \nu^{15} - 27 \nu^{14} - 17 \nu^{13} + 192 \nu^{10} + 120 \nu^{9} - 1320 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 $ (-v^15 - 27v^14 - 17v^13 + 192v^10 + 120v^9 - 1320v^6 - 816v^5 - 305v^3 + 381v^2 + 119*v) / 72
$\beta_{5}$	$=$	$ ( - \nu^{15} + 27 \nu^{14} - 17 \nu^{13} - 192 \nu^{10} + 120 \nu^{9} + 1320 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 $ (-v^15 + 27v^14 - 17v^13 - 192v^10 + 120v^9 + 1320v^6 - 816v^5 - 305v^3 - 381v^2 + 119*v) / 72
$\beta_{6}$	$=$	$ ( - \nu^{15} - 28 \nu^{14} + 17 \nu^{13} + 192 \nu^{10} - 120 \nu^{9} - 1320 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 $ (-v^15 - 28v^14 + 17v^13 + 192v^10 - 120v^9 - 1320v^6 + 816v^5 - 305v^3 + 4v^2 - 119*v) / 72
$\beta_{7}$	$=$	$ ( - 5 \nu^{15} - 56 \nu^{14} + 97 \nu^{13} + 384 \nu^{10} - 672 \nu^{9} - 2640 \nu^{6} + 4656 \nu^{5} + \cdots - 679 \nu ) / 144 $ (-5v^15 - 56v^14 + 97v^13 + 384v^10 - 672v^9 - 2640v^6 + 4656v^5 - 1741v^3 + 8v^2 - 679v) / 144
$\beta_{8}$	$=$	$ ( \nu^{15} - 60 \nu^{14} + 17 \nu^{13} + 432 \nu^{10} - 120 \nu^{9} - 2952 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 $ (v^15 - 60v^14 + 17v^13 + 432v^10 - 120v^9 - 2952v^6 + 816v^5 + 305v^3 + 852v^2 - 119*v) / 72
$\beta_{9}$	$=$	$ ( 37\nu^{15} + 7\nu^{13} - 256\nu^{11} - 48\nu^{9} + 1760\nu^{7} + 336\nu^{5} - 131\nu^{3} + 47\nu ) / 48 $ (37v^15 + 7v^13 - 256v^11 - 48v^9 + 1760v^7 + 336v^5 - 131v^3 + 47v) / 48
$\beta_{10}$	$=$	$ ( - \nu^{15} - 133 \nu^{14} + 17 \nu^{13} + 912 \nu^{10} - 120 \nu^{9} - 6216 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 $ (-v^15 - 133v^14 + 17v^13 + 912v^10 - 120v^9 - 6216v^6 + 816v^5 - 305v^3 + 19v^2 - 119*v) / 72
$\beta_{11}$	$=$	$ ( 37 \nu^{15} - 7 \nu^{13} - 14 \nu^{12} - 256 \nu^{11} + 48 \nu^{9} + 96 \nu^{8} + 1760 \nu^{7} + \cdots + 50 ) / 48 $ (37v^15 - 7v^13 - 14v^12 - 256v^11 + 48v^9 + 96v^8 + 1760v^7 - 336v^5 - 672v^4 - 131v^3 - 47*v + 50) / 48
$\beta_{12}$	$=$	$ ( 89\nu^{15} + \nu^{13} - 624\nu^{11} + 4272\nu^{7} - 623\nu^{3} + 233\nu ) / 72 $ (89v^15 + v^13 - 624v^11 + 4272v^7 - 623v^3 + 233*v) / 72
$\beta_{13}$	$=$	$ ( - 89 \nu^{15} - 56 \nu^{14} + \nu^{13} + 21 \nu^{12} + 624 \nu^{11} + 384 \nu^{10} - 144 \nu^{8} + \cdots - 75 ) / 72 $ (-89v^15 - 56v^14 + v^13 + 21v^12 + 624v^11 + 384v^10 - 144v^8 - 4272v^7 - 2640v^6 + 1008v^4 + 623v^3 + 8v^2 + 233v - 75) / 72
$\beta_{14}$	$=$	$ ( 91 \nu^{15} - 56 \nu^{14} - 35 \nu^{13} - 21 \nu^{12} - 624 \nu^{11} + 384 \nu^{10} + 240 \nu^{9} + \cdots + 75 ) / 72 $ (91v^15 - 56v^14 - 35v^13 - 21v^12 - 624v^11 + 384v^10 + 240v^9 + 144v^8 + 4272v^7 - 2640v^6 - 1632v^5 - 1008v^4 - 13v^3 + 8v^2 + 5*v + 75) / 72
$\beta_{15}$	$=$	$ ( 289 \nu^{15} + 112 \nu^{14} - 23 \nu^{13} + 42 \nu^{12} - 2016 \nu^{11} - 768 \nu^{10} + 144 \nu^{9} + \cdots - 150 ) / 144 $ (289v^15 + 112v^14 - 23v^13 + 42v^12 - 2016v^11 - 768v^10 + 144v^9 - 288v^8 + 13824v^7 + 5280v^6 - 1008v^5 + 2016v^4 - 1639v^3 - 16v^2 - 607*v - 150) / 144

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{13} - 2\beta_{11} + 3\beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} ) / 12 $ (-b15 - b13 - 2b11 + 3b9 - b7 + b6 - b5 - b4 + b3) / 12
$\nu^{2}$	$=$	$ ( -\beta_{14} - \beta_{13} + \beta_{10} - 3\beta_{8} - 3\beta_{6} - 6\beta_{5} + 3\beta_{4} ) / 12 $ (-b14 - b13 + b10 - 3b8 - 3b6 - 6b5 + 3b4) / 12
$\nu^{3}$	$=$	$ ( -\beta_{14} - \beta_{13} - 4\beta_{7} + 8\beta_{6} + 7\beta_{5} + 7\beta_{4} - 4\beta_{3} ) / 12 $ (-b14 - b13 - 4b7 + 8b6 + 7b5 + 7b4 - 4*b3) / 12
$\nu^{4}$	$=$	$ ( 7\beta_{15} + 7\beta_{13} - 7\beta_{11} + 9\beta_{2} - 3\beta _1 + 21 ) / 12 $ (7b15 + 7b13 - 7b11 + 9b2 - 3*b1 + 21) / 12
$\nu^{5}$	$=$	$ ( - 2 \beta_{15} + 6 \beta_{14} - 5 \beta_{13} - 9 \beta_{12} - 13 \beta_{11} + 15 \beta_{9} + \cdots - 5 \beta_{3} ) / 12 $ (-2b15 + 6b14 - 5b13 - 9b12 - 13b11 + 15b9 + 5b7 - 11b6 + 14b5 + 14b4 - 5*b3) / 12
$\nu^{6}$	$=$	$ ( -5\beta_{14} - 5\beta_{13} + 8\beta_{10} - 18\beta_{6} ) / 6 $ (-5b14 - 5b13 + 8b10 - 18b6) / 6
$\nu^{7}$	$=$	$ ( 5 \beta_{15} - 16 \beta_{14} + 13 \beta_{13} - 24 \beta_{12} + 34 \beta_{11} + 39 \beta_{9} + \cdots - 13 \beta_{3} ) / 12 $ (5b15 - 16b14 + 13b13 - 24b12 + 34b11 + 39b9 - 13b7 + 29b6 + 37b5 + 37b4 - 13*b3) / 12
$\nu^{8}$	$=$	$ ( 47\beta_{15} + 47\beta_{13} - 47\beta_{11} + 63\beta_{2} + 21\beta _1 - 141 ) / 12 $ (47b15 + 47b13 - 47b11 + 63b2 + 21*b1 - 141) / 12
$\nu^{9}$	$=$	$ ( 21\beta_{14} + 21\beta_{13} + 68\beta_{7} - 152\beta_{6} + 131\beta_{5} + 131\beta_{4} - 68\beta_{3} ) / 12 $ (21b14 + 21b13 + 68b7 - 152b6 + 131b5 + 131b4 - 68*b3) / 12
$\nu^{10}$	$=$	$ ( -34\beta_{14} - 34\beta_{13} + 55\beta_{10} + 165\beta_{8} - 123\beta_{6} + 267\beta_{5} - 102\beta_{4} ) / 12 $ (-34b14 - 34b13 + 55b10 + 165b8 - 123b6 + 267b5 - 102*b4) / 12
$\nu^{11}$	$=$	$ ( 34 \beta_{15} - 55 \beta_{14} + 144 \beta_{13} - 165 \beta_{12} + 233 \beta_{11} + 267 \beta_{9} + \cdots + 89 \beta_{3} ) / 12 $ (34b15 - 55b14 + 144b13 - 165b12 + 233b11 + 267b9 + 89b7 - 199b6 - 89b5 - 89b4 + 89*b3) / 12
$\nu^{12}$	$=$	$ 24\beta _1 - 161 $ 24*b1 - 161
$\nu^{13}$	$=$	$ ( 89 \beta_{15} - 144 \beta_{14} + 377 \beta_{13} + 432 \beta_{12} + 610 \beta_{11} + \cdots - 233 \beta_{3} ) / 12 $ (89b15 - 144b14 + 377b13 + 432b12 + 610b11 - 699b9 + 233b7 - 521b6 + 233b5 + 233b4 - 233*b3) / 12
$\nu^{14}$	$=$	$ ( 233\beta_{14} + 233\beta_{13} - 377\beta_{10} + 1131\beta_{8} + 843\beta_{6} + 1830\beta_{5} - 699\beta_{4} ) / 12 $ (233b14 + 233b13 - 377b10 + 1131b8 + 843b6 + 1830b5 - 699*b4) / 12
$\nu^{15}$	$=$	$ ( 377\beta_{14} + 377\beta_{13} + 1220\beta_{7} - 2728\beta_{6} - 2351\beta_{5} - 2351\beta_{4} + 1220\beta_{3} ) / 12 $ (377b14 + 377b13 + 1220b7 - 2728b6 - 2351b5 - 2351b4 + 1220*b3) / 12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times$.

$n$	$457$	$571$	$761$	$781$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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} $

569.1

0.596975	+	0.159959i
−1.56290	−	0.418778i
0.418778	+	1.56290i
−0.159959	−	0.596975i
−1.56290	+	0.418778i
0.596975	−	0.159959i
−0.159959	+	0.596975i
0.418778	−	1.56290i
1.56290	+	0.418778i
−0.596975	−	0.159959i
0.159959	+	0.596975i
−0.418778	−	1.56290i
−0.596975	+	0.159959i
1.56290	−	0.418778i
−0.418778	+	1.56290i
0.159959	−	0.596975i

−1.41421

−

1.00000i

−1.73205

−

1.41421i

−

1.51387i

1.00000

2.82843i

569.2

−1.41421

−

1.00000i

−1.73205

−

1.41421i

3.96336i

1.00000

2.82843i

569.3

−1.41421

−

1.00000i

1.73205

−

1.41421i

−

3.96336i

1.00000

2.82843i

569.4

−1.41421

−

1.00000i

1.73205

−

1.41421i

1.51387i

1.00000

2.82843i

569.5

−1.41421

1.00000i

−1.73205

1.41421i

−

3.96336i

1.00000

−

2.82843i

569.6

−1.41421

1.00000i

−1.73205

1.41421i

1.51387i

1.00000

−

2.82843i

569.7

−1.41421

1.00000i

1.73205

1.41421i

−

1.51387i

1.00000

−

2.82843i

569.8

−1.41421

1.00000i

1.73205

1.41421i

3.96336i

1.00000

−

2.82843i

569.9

1.41421

−

1.00000i

−1.73205

1.41421i

−

3.96336i

1.00000

−

2.82843i

569.10

1.41421

−

1.00000i

−1.73205

1.41421i

1.51387i

1.00000

−

2.82843i

569.11

1.41421

−

1.00000i

1.73205

1.41421i

−

1.51387i

1.00000

−

2.82843i

569.12

1.41421

−

1.00000i

1.73205

1.41421i

3.96336i

1.00000

−

2.82843i

569.13

1.41421

1.00000i

−1.73205

−

1.41421i

−

1.51387i

1.00000

2.82843i

569.14

1.41421

1.00000i

−1.73205

−

1.41421i

3.96336i

1.00000

2.82843i

569.15

1.41421

1.00000i

1.73205

−

1.41421i

−

3.96336i

1.00000

2.82843i

569.16

1.41421

1.00000i

1.73205

−

1.41421i

1.51387i

1.00000

2.82843i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
19.b	odd	2	1	inner
57.d	even	2	1	inner
95.d	odd	2	1	inner
285.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1140.2.e.a	✓	16
3.b	odd	2	1	inner	1140.2.e.a	✓	16
5.b	even	2	1	inner	1140.2.e.a	✓	16
15.d	odd	2	1	inner	1140.2.e.a	✓	16
19.b	odd	2	1	inner	1140.2.e.a	✓	16
57.d	even	2	1	inner	1140.2.e.a	✓	16
95.d	odd	2	1	inner	1140.2.e.a	✓	16
285.b	even	2	1	inner	1140.2.e.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1140.2.e.a	✓	16	1.a	even	1	1	trivial
1140.2.e.a	✓	16	3.b	odd	2	1	inner
1140.2.e.a	✓	16	5.b	even	2	1	inner
1140.2.e.a	✓	16	15.d	odd	2	1	inner
1140.2.e.a	✓	16	19.b	odd	2	1	inner
1140.2.e.a	✓	16	57.d	even	2	1	inner
1140.2.e.a	✓	16	95.d	odd	2	1	inner
1140.2.e.a	✓	16	285.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{4} + 18T_{7}^{2} + 36 $ T7^4 + 18*T7^2 + 36 acting on $S_{2}^{\mathrm{new}}(1140, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} - 2 T^{2} + 9)^{4} $ (T^4 - 2*T^2 + 9)^4
$5$	$ (T^{4} - 2 T^{2} + 25)^{4} $ (T^4 - 2*T^2 + 25)^4
$7$	$ (T^{4} + 18 T^{2} + 36)^{4} $ (T^4 + 18*T^2 + 36)^4
$11$	$ (T^{4} + 46 T^{2} + 484)^{4} $ (T^4 + 46*T^2 + 484)^4
$13$	$ (T^{4} - 54 T^{2} + 324)^{4} $ (T^4 - 54*T^2 + 324)^4
$17$	$ (T^{4} - 36 T^{2} + 144)^{4} $ (T^4 - 36*T^2 + 144)^4
$19$	$ (T^{2} + 4 T + 19)^{8} $ (T^2 + 4*T + 19)^8
$23$	$ (T^{4} - 84 T^{2} + 144)^{4} $ (T^4 - 84*T^2 + 144)^4
$29$	$ (T^{4} - 42 T^{2} + 36)^{4} $ (T^4 - 42*T^2 + 36)^4
$31$	$ (T^{4} + 84 T^{2} + 144)^{4} $ (T^4 + 84*T^2 + 144)^4
$37$	$ (T^{4} - 126 T^{2} + 324)^{4} $ (T^4 - 126*T^2 + 324)^4
$41$	$ (T^{4} - 42 T^{2} + 36)^{4} $ (T^4 - 42*T^2 + 36)^4
$43$	$ (T^{4} + 90 T^{2} + 900)^{4} $ (T^4 + 90*T^2 + 900)^4
$47$	$ (T^{4} - 36 T^{2} + 144)^{4} $ (T^4 - 36*T^2 + 144)^4
$53$	$ (T^{4} + 108 T^{2} + 1296)^{4} $ (T^4 + 108*T^2 + 1296)^4
$59$	$ (T^{2} - 120)^{8} $ (T^2 - 120)^8
$61$	$ (T^{2} - 2 T - 44)^{8} $ (T^2 - 2*T - 44)^8
$67$	$ (T^{4} - 216 T^{2} + 5184)^{4} $ (T^4 - 216*T^2 + 5184)^4
$71$	$ (T^{4} - 168 T^{2} + 576)^{4} $ (T^4 - 168*T^2 + 576)^4
$73$	$ (T^{4} + 72 T^{2} + 576)^{4} $ (T^4 + 72*T^2 + 576)^4
$79$	$ (T^{2} + 108)^{8} $ (T^2 + 108)^8
$83$	$ (T^{4} - 324 T^{2} + 11664)^{4} $ (T^4 - 324*T^2 + 11664)^4
$89$	$ (T^{4} - 258 T^{2} + 12996)^{4} $ (T^4 - 258*T^2 + 12996)^4
$97$	$ (T^{4} - 54 T^{2} + 324)^{4} $ (T^4 - 54*T^2 + 324)^4
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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:	\( N \)	\(=\)	\( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1140.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{3} + \beta_{5} q^{5} - \beta_{9} q^{7} + ( - \beta_{5} - \beta_{4} + 1) q^{9} - \beta_{3} q^{11} + (\beta_{7} - \beta_{6}) q^{13} - \beta_{14} q^{15} + (\beta_{8} + \beta_{5}) q^{17} + ( - \beta_{2} - 2) q^{19}+ \cdots + ( - \beta_{3} - 2 \beta_1 - 2) q^{99}+O(q^{100}) \) q + b6 * q^3 + b5 * q^5 - b9 * q^7 + (-b5 - b4 + 1) * q^9 - b3 * q^11 + (b7 - b6) * q^13 - b14 * q^15 + (b8 + b5) * q^17 + (-b2 - 2) * q^19 + (b11 - b2) * q^21 + (b8 - b5 + 2b4) q^23 + (2b12 - b5 - b4 + 1) q^25 + (b14 + b13 + b6) * q^27 + (-b15 - b14) * q^29 + (-b15 - b13 + b11 + b2) * q^31 + (b14 + b13 + b10 - b7) * q^33 + (-b8 + b4 - b3) * q^35 + (-b14 - b13 + b7 + 3b6) q^37 + (b5 + b4 - b3 + b1 - 3) * q^39 + (b15 + b14) * q^41 + (-2b12 + b9 + b5 + b4) q^43 + (-2b12 + 2b5 + b4 + 4) * q^45 + (b8 + b5) * q^47 + (b1 - 2) * q^49 + (b15 + b13 + b11 + b2) * q^51 + (-b10 + b6) * q^53 + (-2b12 - 3b9 + b5 + b4 + b1 + 1) * q^55 + (-b12 - 2b9 + b8 - 2b6 + b5 + b4) * q^57 + (-2b15 - b14 - b13 - 2b11) * q^59 + (b1 + 1) * q^61 + (-b9 + 2b8 + 2b5) * q^63 + (-b15 - b13 - 2b11 - b10 + b6) q^65 + (-2b7 + 2b6) * q^67 + (b15 + 2b14 - b13 + b11 + b2) q^69 + (-b14 + b13 + 2b11) q^71 + (2b12 + 2b9 - b5 - b4) * q^73 + (-2b15 - b14 - b13 + 2b11 + b6) * q^75 + (-b8 - 4b5 + 3b4) * q^77 + (-2b15 - 2b13 + 2b11) q^79 + (-2b5 - 2b4 - 7) * q^81 + (3b8 + 3b5) * q^83 + (-2b9 - b1 + 3) q^85 + (-2b12 - b9 - b8 + 2b5 - b4) * q^87 + (-b15 - 2b14 + b13 + 2b11) * q^89 + (3b15 + 3b13 - 3b11 + 3b2) * q^91 + (4b12 + 2b9 - b8 - b5 - 4b4) q^93 + (-b15 - b11 + b10 - 2b5) q^95 + (-b7 + b6) * q^97 + (-b3 - 2b1 - 2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9} - 32 q^{19} + 16 q^{25} - 48 q^{39} + 64 q^{45} - 32 q^{49} + 16 q^{55} + 16 q^{61} - 112 q^{81} + 48 q^{85} - 32 q^{99}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^19 + 16 * q^25 - 48 * q^39 + 64 * q^45 - 32 * q^49 + 16 * q^55 + 16 * q^61 - 112 * q^81 + 48 * q^85 - 32 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	1140.2.e.a

Level	$1140$
Weight	$2$
Character orbit	1140.e
Analytic conductor	$9.103$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(9.10294583043\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.11007531417600000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{6} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{12} + 161 ) / 24 \) (v^12 + 161) / 24
\(\beta_{2}\)	\(=\)	\( ( -2\nu^{12} + 14\nu^{8} - 94\nu^{4} + 7 ) / 3 \) (-2v^12 + 14v^8 - 94*v^4 + 7) / 3
\(\beta_{3}\)	\(=\)	\( ( -7\nu^{15} - 131\nu^{13} + 912\nu^{9} - 6288\nu^{5} - 2351\nu^{3} + 917\nu ) / 144 \) (-7v^15 - 131v^13 + 912v^9 - 6288v^5 - 2351v^3 + 917v) / 144
\(\beta_{4}\)	\(=\)	\( ( - \nu^{15} - 27 \nu^{14} - 17 \nu^{13} + 192 \nu^{10} + 120 \nu^{9} - 1320 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 \) (-v^15 - 27v^14 - 17v^13 + 192v^10 + 120v^9 - 1320v^6 - 816v^5 - 305v^3 + 381v^2 + 119*v) / 72
\(\beta_{5}\)	\(=\)	\( ( - \nu^{15} + 27 \nu^{14} - 17 \nu^{13} - 192 \nu^{10} + 120 \nu^{9} + 1320 \nu^{6} - 816 \nu^{5} + \cdots + 119 \nu ) / 72 \) (-v^15 + 27v^14 - 17v^13 - 192v^10 + 120v^9 + 1320v^6 - 816v^5 - 305v^3 - 381v^2 + 119*v) / 72
\(\beta_{6}\)	\(=\)	\( ( - \nu^{15} - 28 \nu^{14} + 17 \nu^{13} + 192 \nu^{10} - 120 \nu^{9} - 1320 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 \) (-v^15 - 28v^14 + 17v^13 + 192v^10 - 120v^9 - 1320v^6 + 816v^5 - 305v^3 + 4v^2 - 119*v) / 72
\(\beta_{7}\)	\(=\)	\( ( - 5 \nu^{15} - 56 \nu^{14} + 97 \nu^{13} + 384 \nu^{10} - 672 \nu^{9} - 2640 \nu^{6} + 4656 \nu^{5} + \cdots - 679 \nu ) / 144 \) (-5v^15 - 56v^14 + 97v^13 + 384v^10 - 672v^9 - 2640v^6 + 4656v^5 - 1741v^3 + 8v^2 - 679v) / 144
\(\beta_{8}\)	\(=\)	\( ( \nu^{15} - 60 \nu^{14} + 17 \nu^{13} + 432 \nu^{10} - 120 \nu^{9} - 2952 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 \) (v^15 - 60v^14 + 17v^13 + 432v^10 - 120v^9 - 2952v^6 + 816v^5 + 305v^3 + 852v^2 - 119*v) / 72
\(\beta_{9}\)	\(=\)	\( ( 37\nu^{15} + 7\nu^{13} - 256\nu^{11} - 48\nu^{9} + 1760\nu^{7} + 336\nu^{5} - 131\nu^{3} + 47\nu ) / 48 \) (37v^15 + 7v^13 - 256v^11 - 48v^9 + 1760v^7 + 336v^5 - 131v^3 + 47v) / 48
\(\beta_{10}\)	\(=\)	\( ( - \nu^{15} - 133 \nu^{14} + 17 \nu^{13} + 912 \nu^{10} - 120 \nu^{9} - 6216 \nu^{6} + 816 \nu^{5} + \cdots - 119 \nu ) / 72 \) (-v^15 - 133v^14 + 17v^13 + 912v^10 - 120v^9 - 6216v^6 + 816v^5 - 305v^3 + 19v^2 - 119*v) / 72
\(\beta_{11}\)	\(=\)	\( ( 37 \nu^{15} - 7 \nu^{13} - 14 \nu^{12} - 256 \nu^{11} + 48 \nu^{9} + 96 \nu^{8} + 1760 \nu^{7} + \cdots + 50 ) / 48 \) (37v^15 - 7v^13 - 14v^12 - 256v^11 + 48v^9 + 96v^8 + 1760v^7 - 336v^5 - 672v^4 - 131v^3 - 47*v + 50) / 48
\(\beta_{12}\)	\(=\)	\( ( 89\nu^{15} + \nu^{13} - 624\nu^{11} + 4272\nu^{7} - 623\nu^{3} + 233\nu ) / 72 \) (89v^15 + v^13 - 624v^11 + 4272v^7 - 623v^3 + 233*v) / 72
\(\beta_{13}\)	\(=\)	\( ( - 89 \nu^{15} - 56 \nu^{14} + \nu^{13} + 21 \nu^{12} + 624 \nu^{11} + 384 \nu^{10} - 144 \nu^{8} + \cdots - 75 ) / 72 \) (-89v^15 - 56v^14 + v^13 + 21v^12 + 624v^11 + 384v^10 - 144v^8 - 4272v^7 - 2640v^6 + 1008v^4 + 623v^3 + 8v^2 + 233v - 75) / 72
\(\beta_{14}\)	\(=\)	\( ( 91 \nu^{15} - 56 \nu^{14} - 35 \nu^{13} - 21 \nu^{12} - 624 \nu^{11} + 384 \nu^{10} + 240 \nu^{9} + \cdots + 75 ) / 72 \) (91v^15 - 56v^14 - 35v^13 - 21v^12 - 624v^11 + 384v^10 + 240v^9 + 144v^8 + 4272v^7 - 2640v^6 - 1632v^5 - 1008v^4 - 13v^3 + 8v^2 + 5*v + 75) / 72
\(\beta_{15}\)	\(=\)	\( ( 289 \nu^{15} + 112 \nu^{14} - 23 \nu^{13} + 42 \nu^{12} - 2016 \nu^{11} - 768 \nu^{10} + 144 \nu^{9} + \cdots - 150 ) / 144 \) (289v^15 + 112v^14 - 23v^13 + 42v^12 - 2016v^11 - 768v^10 + 144v^9 - 288v^8 + 13824v^7 + 5280v^6 - 1008v^5 + 2016v^4 - 1639v^3 - 16v^2 - 607*v - 150) / 144

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{13} - 2\beta_{11} + 3\beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} ) / 12 \) (-b15 - b13 - 2b11 + 3b9 - b7 + b6 - b5 - b4 + b3) / 12
\(\nu^{2}\)	\(=\)	\( ( -\beta_{14} - \beta_{13} + \beta_{10} - 3\beta_{8} - 3\beta_{6} - 6\beta_{5} + 3\beta_{4} ) / 12 \) (-b14 - b13 + b10 - 3b8 - 3b6 - 6b5 + 3b4) / 12
\(\nu^{3}\)	\(=\)	\( ( -\beta_{14} - \beta_{13} - 4\beta_{7} + 8\beta_{6} + 7\beta_{5} + 7\beta_{4} - 4\beta_{3} ) / 12 \) (-b14 - b13 - 4b7 + 8b6 + 7b5 + 7b4 - 4*b3) / 12
\(\nu^{4}\)	\(=\)	\( ( 7\beta_{15} + 7\beta_{13} - 7\beta_{11} + 9\beta_{2} - 3\beta _1 + 21 ) / 12 \) (7b15 + 7b13 - 7b11 + 9b2 - 3*b1 + 21) / 12
\(\nu^{5}\)	\(=\)	\( ( - 2 \beta_{15} + 6 \beta_{14} - 5 \beta_{13} - 9 \beta_{12} - 13 \beta_{11} + 15 \beta_{9} + \cdots - 5 \beta_{3} ) / 12 \) (-2b15 + 6b14 - 5b13 - 9b12 - 13b11 + 15b9 + 5b7 - 11b6 + 14b5 + 14b4 - 5*b3) / 12
\(\nu^{6}\)	\(=\)	\( ( -5\beta_{14} - 5\beta_{13} + 8\beta_{10} - 18\beta_{6} ) / 6 \) (-5b14 - 5b13 + 8b10 - 18b6) / 6
\(\nu^{7}\)	\(=\)	\( ( 5 \beta_{15} - 16 \beta_{14} + 13 \beta_{13} - 24 \beta_{12} + 34 \beta_{11} + 39 \beta_{9} + \cdots - 13 \beta_{3} ) / 12 \) (5b15 - 16b14 + 13b13 - 24b12 + 34b11 + 39b9 - 13b7 + 29b6 + 37b5 + 37b4 - 13*b3) / 12
\(\nu^{8}\)	\(=\)	\( ( 47\beta_{15} + 47\beta_{13} - 47\beta_{11} + 63\beta_{2} + 21\beta _1 - 141 ) / 12 \) (47b15 + 47b13 - 47b11 + 63b2 + 21*b1 - 141) / 12
\(\nu^{9}\)	\(=\)	\( ( 21\beta_{14} + 21\beta_{13} + 68\beta_{7} - 152\beta_{6} + 131\beta_{5} + 131\beta_{4} - 68\beta_{3} ) / 12 \) (21b14 + 21b13 + 68b7 - 152b6 + 131b5 + 131b4 - 68*b3) / 12
\(\nu^{10}\)	\(=\)	\( ( -34\beta_{14} - 34\beta_{13} + 55\beta_{10} + 165\beta_{8} - 123\beta_{6} + 267\beta_{5} - 102\beta_{4} ) / 12 \) (-34b14 - 34b13 + 55b10 + 165b8 - 123b6 + 267b5 - 102*b4) / 12
\(\nu^{11}\)	\(=\)	\( ( 34 \beta_{15} - 55 \beta_{14} + 144 \beta_{13} - 165 \beta_{12} + 233 \beta_{11} + 267 \beta_{9} + \cdots + 89 \beta_{3} ) / 12 \) (34b15 - 55b14 + 144b13 - 165b12 + 233b11 + 267b9 + 89b7 - 199b6 - 89b5 - 89b4 + 89*b3) / 12
\(\nu^{12}\)	\(=\)	\( 24\beta _1 - 161 \) 24*b1 - 161
\(\nu^{13}\)	\(=\)	\( ( 89 \beta_{15} - 144 \beta_{14} + 377 \beta_{13} + 432 \beta_{12} + 610 \beta_{11} + \cdots - 233 \beta_{3} ) / 12 \) (89b15 - 144b14 + 377b13 + 432b12 + 610b11 - 699b9 + 233b7 - 521b6 + 233b5 + 233b4 - 233*b3) / 12
\(\nu^{14}\)	\(=\)	\( ( 233\beta_{14} + 233\beta_{13} - 377\beta_{10} + 1131\beta_{8} + 843\beta_{6} + 1830\beta_{5} - 699\beta_{4} ) / 12 \) (233b14 + 233b13 - 377b10 + 1131b8 + 843b6 + 1830b5 - 699*b4) / 12
\(\nu^{15}\)	\(=\)	\( ( 377\beta_{14} + 377\beta_{13} + 1220\beta_{7} - 2728\beta_{6} - 2351\beta_{5} - 2351\beta_{4} + 1220\beta_{3} ) / 12 \) (377b14 + 377b13 + 1220b7 - 2728b6 - 2351b5 - 2351b4 + 1220*b3) / 12

\(n\)	\(457\)	\(571\)	\(761\)	\(781\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 569.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} - 2 T^{2} + 9)^{4} \) (T^4 - 2*T^2 + 9)^4
$5$	\( (T^{4} - 2 T^{2} + 25)^{4} \) (T^4 - 2*T^2 + 25)^4
$7$	\( (T^{4} + 18 T^{2} + 36)^{4} \) (T^4 + 18*T^2 + 36)^4
$11$	\( (T^{4} + 46 T^{2} + 484)^{4} \) (T^4 + 46*T^2 + 484)^4
$13$	\( (T^{4} - 54 T^{2} + 324)^{4} \) (T^4 - 54*T^2 + 324)^4
$17$	\( (T^{4} - 36 T^{2} + 144)^{4} \) (T^4 - 36*T^2 + 144)^4
$19$	\( (T^{2} + 4 T + 19)^{8} \) (T^2 + 4*T + 19)^8
$23$	\( (T^{4} - 84 T^{2} + 144)^{4} \) (T^4 - 84*T^2 + 144)^4
$29$	\( (T^{4} - 42 T^{2} + 36)^{4} \) (T^4 - 42*T^2 + 36)^4
$31$	\( (T^{4} + 84 T^{2} + 144)^{4} \) (T^4 + 84*T^2 + 144)^4
$37$	\( (T^{4} - 126 T^{2} + 324)^{4} \) (T^4 - 126*T^2 + 324)^4
$41$	\( (T^{4} - 42 T^{2} + 36)^{4} \) (T^4 - 42*T^2 + 36)^4
$43$	\( (T^{4} + 90 T^{2} + 900)^{4} \) (T^4 + 90*T^2 + 900)^4
$47$	\( (T^{4} - 36 T^{2} + 144)^{4} \) (T^4 - 36*T^2 + 144)^4
$53$	\( (T^{4} + 108 T^{2} + 1296)^{4} \) (T^4 + 108*T^2 + 1296)^4
$59$	\( (T^{2} - 120)^{8} \) (T^2 - 120)^8
$61$	\( (T^{2} - 2 T - 44)^{8} \) (T^2 - 2*T - 44)^8
$67$	\( (T^{4} - 216 T^{2} + 5184)^{4} \) (T^4 - 216*T^2 + 5184)^4
$71$	\( (T^{4} - 168 T^{2} + 576)^{4} \) (T^4 - 168*T^2 + 576)^4
$73$	\( (T^{4} + 72 T^{2} + 576)^{4} \) (T^4 + 72*T^2 + 576)^4
$79$	\( (T^{2} + 108)^{8} \) (T^2 + 108)^8
$83$	\( (T^{4} - 324 T^{2} + 11664)^{4} \) (T^4 - 324*T^2 + 11664)^4
$89$	\( (T^{4} - 258 T^{2} + 12996)^{4} \) (T^4 - 258*T^2 + 12996)^4
$97$	\( (T^{4} - 54 T^{2} + 324)^{4} \) (T^4 - 54*T^2 + 324)^4
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