LMFDB - Newform orbit 2016.2.b.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(1567,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2016.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.b (of order $2$, degree $1$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30} $
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_{2} q^{5} - \beta_{8} q^{7}+O(q^{10}) $ q - b2 * q^5 - b8 * q^7	\( q - \beta_{2} q^{5} - \beta_{8} q^{7} + (\beta_{11} + \beta_{7}) q^{11} + \beta_{15} q^{13} + ( - \beta_{6} + \beta_{2}) q^{17} - \beta_{4} q^{19} + \beta_{11} q^{23} + (\beta_{12} - 2) q^{25} + \beta_{14} q^{29} + ( - \beta_{8} - \beta_{5} - \beta_{4}) q^{31} + (\beta_{11} + \beta_{3}) q^{35} + ( - \beta_{12} + 1) q^{37} + ( - \beta_{6} + \beta_{2}) q^{41} + (\beta_{8} - \beta_{5} - \beta_1) q^{43} + (\beta_{13} + \beta_{3}) q^{47} + (\beta_{12} - \beta_{10} - 2) q^{49} + (\beta_{14} - \beta_{9}) q^{53} + (2 \beta_{8} + 2 \beta_{5} + 3 \beta_{4}) q^{55} + (\beta_{13} - \beta_{3}) q^{59} + ( - \beta_{15} + 2 \beta_{10}) q^{61} + (2 \beta_{14} + \beta_{9}) q^{65} + ( - \beta_{8} + \beta_{5} + \beta_1) q^{67} + (3 \beta_{11} + 2 \beta_{7}) q^{71} - 2 \beta_{15} q^{73} + (\beta_{14} + \beta_{9} - \beta_{2}) q^{77} + (\beta_{8} - \beta_{5}) q^{79} + (\beta_{13} + \beta_{3}) q^{83} + ( - 3 \beta_{12} + 5) q^{85} + ( - \beta_{6} - \beta_{2}) q^{89} + (\beta_{8} + \beta_{5} + 3 \beta_{4} + \beta_1) q^{91} + (2 \beta_{11} + 4 \beta_{7}) q^{95} - 2 \beta_{10} q^{97}+O(q^{100}) \) q - b2 * q^5 - b8 * q^7 + (b11 + b7) * q^11 + b15 * q^13 + (-b6 + b2) * q^17 - b4 * q^19 + b11 * q^23 + (b12 - 2) * q^25 + b14 * q^29 + (-b8 - b5 - b4) * q^31 + (b11 + b3) * q^35 + (-b12 + 1) * q^37 + (-b6 + b2) * q^41 + (b8 - b5 - b1) * q^43 + (b13 + b3) * q^47 + (b12 - b10 - 2) * q^49 + (b14 - b9) * q^53 + (2b8 + 2b5 + 3b4) q^55 + (b13 - b3) * q^59 + (-b15 + 2b10) q^61 + (2b14 + b9) q^65 + (-b8 + b5 + b1) * q^67 + (3b11 + 2b7) * q^71 - 2b15 q^73 + (b14 + b9 - b2) * q^77 + (b8 - b5) * q^79 + (b13 + b3) * q^83 + (-3b12 + 5) q^85 + (-b6 - b2) * q^89 + (b8 + b5 + 3b4 + b1) q^91 + (2b11 + 4b7) * q^95 - 2b10 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{25} + 16 q^{37} - 32 q^{49} + 80 q^{85}+O(q^{100}) $ 16 * q - 32 * q^25 + 16 * q^37 - 32 * q^49 + 80 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7*x^12 + 40*x^8 - 112*x^4 + 256 :

$\beta_{1}$	$=$	$ ( -\nu^{10} + 3\nu^{6} - 12\nu^{2} ) / 4 $ (-v^10 + 3v^6 - 12v^2) / 4
$\beta_{2}$	$=$	$ ( \nu^{12} - 11\nu^{8} + 84\nu^{4} - 192 ) / 64 $ (v^12 - 11v^8 + 84v^4 - 192) / 64
$\beta_{3}$	$=$	$ ( \nu^{14} - 3\nu^{10} - 4\nu^{6} + 96\nu^{2} ) / 32 $ (v^14 - 3v^10 - 4v^6 + 96*v^2) / 32
$\beta_{4}$	$=$	$ ( -\nu^{15} - 2\nu^{13} + 11\nu^{11} - 10\nu^{9} - 20\nu^{7} + 56\nu^{5} - 384\nu ) / 128 $ (-v^15 - 2v^13 + 11v^11 - 10v^9 - 20v^7 + 56v^5 - 384v) / 128
$\beta_{5}$	$=$	$ ( - 3 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} - 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + \cdots + 256 \nu ) / 256 $ (-3v^15 + 4v^14 - 2v^13 + 9v^11 - 28v^10 + 22v^9 - 52v^7 + 160v^6 - 40v^5 + 32v^3 - 192v^2 + 256v) / 256
$\beta_{6}$	$=$	$ ( 3\nu^{12} - \nu^{8} + 28\nu^{4} + 64 ) / 32 $ (3v^12 - v^8 + 28v^4 + 64) / 32
$\beta_{7}$	$=$	$ ( \nu^{15} - 6\nu^{13} - 3\nu^{11} + 34\nu^{9} - 4\nu^{7} - 152\nu^{5} - 32\nu^{3} + 256\nu ) / 128 $ (v^15 - 6v^13 - 3v^11 + 34v^9 - 4v^7 - 152v^5 - 32v^3 + 256*v) / 128
$\beta_{8}$	$=$	$ ( - 3 \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} + 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + \cdots + 256 \nu ) / 256 $ (-3v^15 - 4v^14 - 2v^13 + 9v^11 + 28v^10 + 22v^9 - 52v^7 - 160v^6 - 40v^5 + 32v^3 + 192v^2 + 256v) / 256
$\beta_{9}$	$=$	$ ( \nu^{15} + 6\nu^{13} - 3\nu^{11} - 34\nu^{9} - 4\nu^{7} + 152\nu^{5} - 32\nu^{3} - 256\nu ) / 64 $ (v^15 + 6v^13 - 3v^11 - 34v^9 - 4v^7 + 152v^5 - 32v^3 - 256*v) / 64
$\beta_{10}$	$=$	$ ( \nu^{15} + \nu^{11} + 16\nu^{9} + 16\nu^{7} - 48\nu^{5} - 16\nu^{3} + 320\nu ) / 64 $ (v^15 + v^11 + 16v^9 + 16v^7 - 48v^5 - 16v^3 + 320*v) / 64
$\beta_{11}$	$=$	$ ( -3\nu^{15} + 6\nu^{13} + 33\nu^{11} - 2\nu^{9} - 124\nu^{7} + 56\nu^{5} + 448\nu^{3} - 128\nu ) / 256 $ (-3v^15 + 6v^13 + 33v^11 - 2v^9 - 124v^7 + 56v^5 + 448v^3 - 128v) / 256
$\beta_{12}$	$=$	$ ( -\nu^{12} + 7\nu^{8} - 24\nu^{4} + 56 ) / 8 $ (-v^12 + 7v^8 - 24v^4 + 56) / 8
$\beta_{13}$	$=$	$ ( -\nu^{14} + 5\nu^{10} - 18\nu^{6} + 40\nu^{2} ) / 8 $ (-v^14 + 5v^10 - 18v^6 + 40*v^2) / 8
$\beta_{14}$	$=$	$ ( -\nu^{15} + 15\nu^{11} - 16\nu^{9} - 64\nu^{7} + 48\nu^{5} + 208\nu^{3} - 64\nu ) / 64 $ (-v^15 + 15v^11 - 16v^9 - 64v^7 + 48v^5 + 208v^3 - 64v) / 64
$\beta_{15}$	$=$	$ ( \nu^{15} - \nu^{13} - 5\nu^{11} + 3\nu^{9} + 18\nu^{7} + 4\nu^{5} - 8\nu^{3} - 32\nu ) / 32 $ (v^15 - v^13 - 5v^11 + 3v^9 + 18v^7 + 4v^5 - 8v^3 - 32v) / 32

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

$\nu$	$=$	$ ( \beta_{14} - 2\beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} ) / 8 $ (b14 - 2*b11 + b10 + b8 - b7 + b5 - b4) / 8
$\nu^{2}$	$=$	$ ( \beta_{13} - 2\beta_{8} + 2\beta_{5} + 2\beta_{3} ) / 8 $ (b13 - 2b8 + 2b5 + 2*b3) / 8
$\nu^{3}$	$=$	$ ( 2\beta_{15} + \beta_{14} + 2\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - 3\beta_{7} - \beta_{5} - 3\beta_{4} ) / 8 $ (2b15 + b14 + 2b11 - b10 - 2b9 - b8 - 3b7 - b5 - 3*b4) / 8
$\nu^{4}$	$=$	$ ( 2\beta_{12} + \beta_{6} + 10\beta_{2} + 14 ) / 8 $ (2b12 + b6 + 10b2 + 14) / 8
$\nu^{5}$	$=$	$ ( 6 \beta_{15} + \beta_{14} - 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 9 \beta_{8} - 5 \beta_{7} + \cdots + 3 \beta_{4} ) / 8 $ (6b15 + b14 - 2b11 + 3b10 + 2b9 + 9b8 - 5b7 + 9b5 + 3b4) / 8
$\nu^{6}$	$=$	$ ( 3\beta_{13} - 14\beta_{8} + 14\beta_{5} - 2\beta_{3} - 4\beta_1 ) / 8 $ (3b13 - 14b8 + 14b5 - 2b3 - 4*b1) / 8
$\nu^{7}$	$=$	$ ( 2 \beta_{15} - \beta_{14} - 2 \beta_{11} + 5 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} + \cdots + 3 \beta_{4} ) / 8 $ (2b15 - b14 - 2b11 + 5b10 - 10b9 - 7b8 - 21b7 - 7b5 + 3b4) / 8
$\nu^{8}$	$=$	$ ( 14\beta_{12} + 15\beta_{6} + 22\beta_{2} - 62 ) / 8 $ (14b12 + 15b6 + 22*b2 - 62) / 8
$\nu^{9}$	$=$	$ ( 18 \beta_{15} - 17 \beta_{14} + 34 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 23 \beta_{8} + \cdots + 13 \beta_{4} ) / 8 $ (18b15 - 17b14 + 34b11 + 5b10 + 6b9 + 23b8 + 5b7 + 23b5 + 13*b4) / 8
$\nu^{10}$	$=$	$ ( -3\beta_{13} - 18\beta_{8} + 18\beta_{5} - 30\beta_{3} - 44\beta_1 ) / 8 $ (-3b13 - 18b8 + 18b5 - 30b3 - 44*b1) / 8
$\nu^{11}$	$=$	$ ( - 18 \beta_{15} + \beta_{14} + 2 \beta_{11} + 43 \beta_{10} - 6 \beta_{9} - 25 \beta_{8} + \cdots + 61 \beta_{4} ) / 8 $ (-18b15 + b14 + 2b11 + 43b10 - 6b9 - 25b8 - 11b7 - 25b5 + 61b4) / 8
$\nu^{12}$	$=$	$ ( -14\beta_{12} + 81\beta_{6} - 86\beta_{2} - 322 ) / 8 $ (-14b12 + 81b6 - 86*b2 - 322) / 8
$\nu^{13}$	$=$	$ ( - 50 \beta_{15} - 79 \beta_{14} + 158 \beta_{11} - 5 \beta_{10} + 26 \beta_{9} - 55 \beta_{8} + \cdots - 45 \beta_{4} ) / 8 $ (-50b15 - 79b14 + 158b11 - 5b10 + 26b9 - 55b8 + 27b7 - 55b5 - 45*b4) / 8
$\nu^{14}$	$=$	$ ( -93\beta_{13} + 82\beta_{8} - 82\beta_{5} - 34\beta_{3} - 148\beta_1 ) / 8 $ (-93b13 + 82b8 - 82b5 - 34b3 - 148*b1) / 8
$\nu^{15}$	$=$	$ ( 18 \beta_{15} + 31 \beta_{14} + 62 \beta_{11} + 117 \beta_{10} + 134 \beta_{9} - 135 \beta_{8} + \cdots + 99 \beta_{4} ) / 8 $ (18b15 + 31b14 + 62b11 + 117b10 + 134b9 - 135b8 + 299b7 - 135b5 + 99*b4) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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} $

1567.1

0.481610	−	1.32968i
−1.32968	−	0.481610i
−0.481610	+	1.32968i
1.32968	+	0.481610i
0.281691	−	1.38588i
1.38588	+	0.281691i
−0.281691	+	1.38588i
−1.38588	−	0.281691i
1.38588	−	0.281691i
0.281691	+	1.38588i
−1.38588	+	0.281691i
−0.281691	−	1.38588i
−1.32968	+	0.481610i
0.481610	+	1.32968i
1.32968	−	0.481610i
−0.481610	−	1.32968i

−

3.33513i

−0.662153

−

2.56155i

1567.2

−

3.33513i

−0.662153

2.56155i

1567.3

−

3.33513i

0.662153

−

2.56155i

1567.4

−

3.33513i

0.662153

2.56155i

1567.5

−

1.69614i

−2.13578

−

1.56155i

1567.6

−

1.69614i

−2.13578

1.56155i

1567.7

−

1.69614i

2.13578

−

1.56155i

1567.8

−

1.69614i

2.13578

1.56155i

1567.9

1.69614i

−2.13578

−

1.56155i

1567.10

1.69614i

−2.13578

1.56155i

1567.11

1.69614i

2.13578

−

1.56155i

1567.12

1.69614i

2.13578

1.56155i

1567.13

3.33513i

−0.662153

−

2.56155i

1567.14

3.33513i

−0.662153

2.56155i

1567.15

3.33513i

0.662153

−

2.56155i

1567.16

3.33513i

0.662153

2.56155i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.b	odd	2	1	inner
12.b	even	2	1	inner
21.c	even	2	1	inner
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2016.2.b.d	✓	16
3.b	odd	2	1	inner	2016.2.b.d	✓	16
4.b	odd	2	1	inner	2016.2.b.d	✓	16
7.b	odd	2	1	inner	2016.2.b.d	✓	16
8.b	even	2	1		4032.2.b.r		16
8.d	odd	2	1		4032.2.b.r		16
12.b	even	2	1	inner	2016.2.b.d	✓	16
21.c	even	2	1	inner	2016.2.b.d	✓	16
24.f	even	2	1		4032.2.b.r		16
24.h	odd	2	1		4032.2.b.r		16
28.d	even	2	1	inner	2016.2.b.d	✓	16
56.e	even	2	1		4032.2.b.r		16
56.h	odd	2	1		4032.2.b.r		16
84.h	odd	2	1	inner	2016.2.b.d	✓	16
168.e	odd	2	1		4032.2.b.r		16
168.i	even	2	1		4032.2.b.r		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2016.2.b.d	✓	16	1.a	even	1	1	trivial
2016.2.b.d	✓	16	3.b	odd	2	1	inner
2016.2.b.d	✓	16	4.b	odd	2	1	inner
2016.2.b.d	✓	16	7.b	odd	2	1	inner
2016.2.b.d	✓	16	12.b	even	2	1	inner
2016.2.b.d	✓	16	21.c	even	2	1	inner
2016.2.b.d	✓	16	28.d	even	2	1	inner
2016.2.b.d	✓	16	84.h	odd	2	1	inner
4032.2.b.r		16	8.b	even	2	1
4032.2.b.r		16	8.d	odd	2	1
4032.2.b.r		16	24.f	even	2	1
4032.2.b.r		16	24.h	odd	2	1
4032.2.b.r		16	56.e	even	2	1
4032.2.b.r		16	56.h	odd	2	1
4032.2.b.r		16	168.e	odd	2	1
4032.2.b.r		16	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2016, [\chi])$:

$ T_{5}^{4} + 14T_{5}^{2} + 32 $

$ T_{19}^{4} - 28T_{19}^{2} + 128 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} + 14 T^{2} + 32)^{4} $ (T^4 + 14*T^2 + 32)^4
$7$	$ (T^{8} + 8 T^{6} + \cdots + 2401)^{2} $ (T^8 + 8T^6 + 46T^4 + 392*T^2 + 2401)^2
$11$	$ (T^{4} + 26 T^{2} + 16)^{4} $ (T^4 + 26*T^2 + 16)^4
$13$	$ (T^{4} + 40 T^{2} + 128)^{4} $ (T^4 + 40*T^2 + 128)^4
$17$	$ (T^{4} + 46 T^{2} + 512)^{4} $ (T^4 + 46*T^2 + 512)^4
$19$	$ (T^{4} - 28 T^{2} + 128)^{4} $ (T^4 - 28*T^2 + 128)^4
$23$	$ (T^{4} + 18 T^{2} + 64)^{4} $ (T^4 + 18*T^2 + 64)^4
$29$	$ (T^{4} - 72 T^{2} + 1024)^{4} $ (T^4 - 72*T^2 + 1024)^4
$31$	$ (T^{4} - 40 T^{2} + 128)^{4} $ (T^4 - 40*T^2 + 128)^4
$37$	$ (T^{2} - 2 T - 16)^{8} $ (T^2 - 2*T - 16)^8
$41$	$ (T^{4} + 46 T^{2} + 512)^{4} $ (T^4 + 46*T^2 + 512)^4
$43$	$ (T^{4} + 84 T^{2} + 64)^{4} $ (T^4 + 84*T^2 + 64)^4
$47$	$ (T^{4} - 160 T^{2} + 2048)^{4} $ (T^4 - 160*T^2 + 2048)^4
$53$	$ (T^{4} - 104 T^{2} + 256)^{4} $ (T^4 - 104*T^2 + 256)^4
$59$	$ (T^{4} - 224 T^{2} + 8192)^{4} $ (T^4 - 224*T^2 + 8192)^4
$61$	$ (T^{4} + 296 T^{2} + 21632)^{4} $ (T^4 + 296*T^2 + 21632)^4
$67$	$ (T^{4} + 84 T^{2} + 64)^{4} $ (T^4 + 84*T^2 + 64)^4
$71$	$ (T^{4} + 178 T^{2} + 4096)^{4} $ (T^4 + 178*T^2 + 4096)^4
$73$	$ (T^{4} + 160 T^{2} + 2048)^{4} $ (T^4 + 160*T^2 + 2048)^4
$79$	$ (T^{4} + 36 T^{2} + 256)^{4} $ (T^4 + 36*T^2 + 256)^4
$83$	$ (T^{4} - 160 T^{2} + 2048)^{4} $ (T^4 - 160*T^2 + 2048)^4
$89$	$ (T^{4} + 62 T^{2} + 128)^{4} $ (T^4 + 62*T^2 + 128)^4
$97$	$ (T^{4} + 224 T^{2} + 8192)^{4} $ (T^4 + 224*T^2 + 8192)^4
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} - \beta_{8} q^{7}+O(q^{10}) \) q - b2 * q^5 - b8 * q^7	\( q - \beta_{2} q^{5} - \beta_{8} q^{7} + (\beta_{11} + \beta_{7}) q^{11} + \beta_{15} q^{13} + ( - \beta_{6} + \beta_{2}) q^{17} - \beta_{4} q^{19} + \beta_{11} q^{23} + (\beta_{12} - 2) q^{25} + \beta_{14} q^{29} + ( - \beta_{8} - \beta_{5} - \beta_{4}) q^{31} + (\beta_{11} + \beta_{3}) q^{35} + ( - \beta_{12} + 1) q^{37} + ( - \beta_{6} + \beta_{2}) q^{41} + (\beta_{8} - \beta_{5} - \beta_1) q^{43} + (\beta_{13} + \beta_{3}) q^{47} + (\beta_{12} - \beta_{10} - 2) q^{49} + (\beta_{14} - \beta_{9}) q^{53} + (2 \beta_{8} + 2 \beta_{5} + 3 \beta_{4}) q^{55} + (\beta_{13} - \beta_{3}) q^{59} + ( - \beta_{15} + 2 \beta_{10}) q^{61} + (2 \beta_{14} + \beta_{9}) q^{65} + ( - \beta_{8} + \beta_{5} + \beta_1) q^{67} + (3 \beta_{11} + 2 \beta_{7}) q^{71} - 2 \beta_{15} q^{73} + (\beta_{14} + \beta_{9} - \beta_{2}) q^{77} + (\beta_{8} - \beta_{5}) q^{79} + (\beta_{13} + \beta_{3}) q^{83} + ( - 3 \beta_{12} + 5) q^{85} + ( - \beta_{6} - \beta_{2}) q^{89} + (\beta_{8} + \beta_{5} + 3 \beta_{4} + \beta_1) q^{91} + (2 \beta_{11} + 4 \beta_{7}) q^{95} - 2 \beta_{10} q^{97}+O(q^{100}) \) q - b2 * q^5 - b8 * q^7 + (b11 + b7) * q^11 + b15 * q^13 + (-b6 + b2) * q^17 - b4 * q^19 + b11 * q^23 + (b12 - 2) * q^25 + b14 * q^29 + (-b8 - b5 - b4) * q^31 + (b11 + b3) * q^35 + (-b12 + 1) * q^37 + (-b6 + b2) * q^41 + (b8 - b5 - b1) * q^43 + (b13 + b3) * q^47 + (b12 - b10 - 2) * q^49 + (b14 - b9) * q^53 + (2b8 + 2b5 + 3b4) q^55 + (b13 - b3) * q^59 + (-b15 + 2b10) q^61 + (2b14 + b9) q^65 + (-b8 + b5 + b1) * q^67 + (3b11 + 2b7) * q^71 - 2b15 q^73 + (b14 + b9 - b2) * q^77 + (b8 - b5) * q^79 + (b13 + b3) * q^83 + (-3b12 + 5) q^85 + (-b6 - b2) * q^89 + (b8 + b5 + 3b4 + b1) q^91 + (2b11 + 4b7) * q^95 - 2b10 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{25} + 16 q^{37} - 32 q^{49} + 80 q^{85}+O(q^{100}) \) 16 * q - 32 * q^25 + 16 * q^37 - 32 * q^49 + 80 * q^85

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Newspace parameters

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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	2016.2.b.d

Level	$2016$
Weight	$2$
Character orbit	2016.b
Analytic conductor	$16.098$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(16.0978410475\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{30} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{10} + 3\nu^{6} - 12\nu^{2} ) / 4 \) (-v^10 + 3v^6 - 12v^2) / 4
\(\beta_{2}\)	\(=\)	\( ( \nu^{12} - 11\nu^{8} + 84\nu^{4} - 192 ) / 64 \) (v^12 - 11v^8 + 84v^4 - 192) / 64
\(\beta_{3}\)	\(=\)	\( ( \nu^{14} - 3\nu^{10} - 4\nu^{6} + 96\nu^{2} ) / 32 \) (v^14 - 3v^10 - 4v^6 + 96*v^2) / 32
\(\beta_{4}\)	\(=\)	\( ( -\nu^{15} - 2\nu^{13} + 11\nu^{11} - 10\nu^{9} - 20\nu^{7} + 56\nu^{5} - 384\nu ) / 128 \) (-v^15 - 2v^13 + 11v^11 - 10v^9 - 20v^7 + 56v^5 - 384v) / 128
\(\beta_{5}\)	\(=\)	\( ( - 3 \nu^{15} + 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} - 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + \cdots + 256 \nu ) / 256 \) (-3v^15 + 4v^14 - 2v^13 + 9v^11 - 28v^10 + 22v^9 - 52v^7 + 160v^6 - 40v^5 + 32v^3 - 192v^2 + 256v) / 256
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{12} - \nu^{8} + 28\nu^{4} + 64 ) / 32 \) (3v^12 - v^8 + 28v^4 + 64) / 32
\(\beta_{7}\)	\(=\)	\( ( \nu^{15} - 6\nu^{13} - 3\nu^{11} + 34\nu^{9} - 4\nu^{7} - 152\nu^{5} - 32\nu^{3} + 256\nu ) / 128 \) (v^15 - 6v^13 - 3v^11 + 34v^9 - 4v^7 - 152v^5 - 32v^3 + 256*v) / 128
\(\beta_{8}\)	\(=\)	\( ( - 3 \nu^{15} - 4 \nu^{14} - 2 \nu^{13} + 9 \nu^{11} + 28 \nu^{10} + 22 \nu^{9} - 52 \nu^{7} + \cdots + 256 \nu ) / 256 \) (-3v^15 - 4v^14 - 2v^13 + 9v^11 + 28v^10 + 22v^9 - 52v^7 - 160v^6 - 40v^5 + 32v^3 + 192v^2 + 256v) / 256
\(\beta_{9}\)	\(=\)	\( ( \nu^{15} + 6\nu^{13} - 3\nu^{11} - 34\nu^{9} - 4\nu^{7} + 152\nu^{5} - 32\nu^{3} - 256\nu ) / 64 \) (v^15 + 6v^13 - 3v^11 - 34v^9 - 4v^7 + 152v^5 - 32v^3 - 256*v) / 64
\(\beta_{10}\)	\(=\)	\( ( \nu^{15} + \nu^{11} + 16\nu^{9} + 16\nu^{7} - 48\nu^{5} - 16\nu^{3} + 320\nu ) / 64 \) (v^15 + v^11 + 16v^9 + 16v^7 - 48v^5 - 16v^3 + 320*v) / 64
\(\beta_{11}\)	\(=\)	\( ( -3\nu^{15} + 6\nu^{13} + 33\nu^{11} - 2\nu^{9} - 124\nu^{7} + 56\nu^{5} + 448\nu^{3} - 128\nu ) / 256 \) (-3v^15 + 6v^13 + 33v^11 - 2v^9 - 124v^7 + 56v^5 + 448v^3 - 128v) / 256
\(\beta_{12}\)	\(=\)	\( ( -\nu^{12} + 7\nu^{8} - 24\nu^{4} + 56 ) / 8 \) (-v^12 + 7v^8 - 24v^4 + 56) / 8
\(\beta_{13}\)	\(=\)	\( ( -\nu^{14} + 5\nu^{10} - 18\nu^{6} + 40\nu^{2} ) / 8 \) (-v^14 + 5v^10 - 18v^6 + 40*v^2) / 8
\(\beta_{14}\)	\(=\)	\( ( -\nu^{15} + 15\nu^{11} - 16\nu^{9} - 64\nu^{7} + 48\nu^{5} + 208\nu^{3} - 64\nu ) / 64 \) (-v^15 + 15v^11 - 16v^9 - 64v^7 + 48v^5 + 208v^3 - 64v) / 64
\(\beta_{15}\)	\(=\)	\( ( \nu^{15} - \nu^{13} - 5\nu^{11} + 3\nu^{9} + 18\nu^{7} + 4\nu^{5} - 8\nu^{3} - 32\nu ) / 32 \) (v^15 - v^13 - 5v^11 + 3v^9 + 18v^7 + 4v^5 - 8v^3 - 32v) / 32

\(\nu\)	\(=\)	\( ( \beta_{14} - 2\beta_{11} + \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} ) / 8 \) (b14 - 2*b11 + b10 + b8 - b7 + b5 - b4) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{13} - 2\beta_{8} + 2\beta_{5} + 2\beta_{3} ) / 8 \) (b13 - 2b8 + 2b5 + 2*b3) / 8
\(\nu^{3}\)	\(=\)	\( ( 2\beta_{15} + \beta_{14} + 2\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - 3\beta_{7} - \beta_{5} - 3\beta_{4} ) / 8 \) (2b15 + b14 + 2b11 - b10 - 2b9 - b8 - 3b7 - b5 - 3*b4) / 8
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{12} + \beta_{6} + 10\beta_{2} + 14 ) / 8 \) (2b12 + b6 + 10b2 + 14) / 8
\(\nu^{5}\)	\(=\)	\( ( 6 \beta_{15} + \beta_{14} - 2 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 9 \beta_{8} - 5 \beta_{7} + \cdots + 3 \beta_{4} ) / 8 \) (6b15 + b14 - 2b11 + 3b10 + 2b9 + 9b8 - 5b7 + 9b5 + 3b4) / 8
\(\nu^{6}\)	\(=\)	\( ( 3\beta_{13} - 14\beta_{8} + 14\beta_{5} - 2\beta_{3} - 4\beta_1 ) / 8 \) (3b13 - 14b8 + 14b5 - 2b3 - 4*b1) / 8
\(\nu^{7}\)	\(=\)	\( ( 2 \beta_{15} - \beta_{14} - 2 \beta_{11} + 5 \beta_{10} - 10 \beta_{9} - 7 \beta_{8} + \cdots + 3 \beta_{4} ) / 8 \) (2b15 - b14 - 2b11 + 5b10 - 10b9 - 7b8 - 21b7 - 7b5 + 3b4) / 8
\(\nu^{8}\)	\(=\)	\( ( 14\beta_{12} + 15\beta_{6} + 22\beta_{2} - 62 ) / 8 \) (14b12 + 15b6 + 22*b2 - 62) / 8
\(\nu^{9}\)	\(=\)	\( ( 18 \beta_{15} - 17 \beta_{14} + 34 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 23 \beta_{8} + \cdots + 13 \beta_{4} ) / 8 \) (18b15 - 17b14 + 34b11 + 5b10 + 6b9 + 23b8 + 5b7 + 23b5 + 13*b4) / 8
\(\nu^{10}\)	\(=\)	\( ( -3\beta_{13} - 18\beta_{8} + 18\beta_{5} - 30\beta_{3} - 44\beta_1 ) / 8 \) (-3b13 - 18b8 + 18b5 - 30b3 - 44*b1) / 8
\(\nu^{11}\)	\(=\)	\( ( - 18 \beta_{15} + \beta_{14} + 2 \beta_{11} + 43 \beta_{10} - 6 \beta_{9} - 25 \beta_{8} + \cdots + 61 \beta_{4} ) / 8 \) (-18b15 + b14 + 2b11 + 43b10 - 6b9 - 25b8 - 11b7 - 25b5 + 61b4) / 8
\(\nu^{12}\)	\(=\)	\( ( -14\beta_{12} + 81\beta_{6} - 86\beta_{2} - 322 ) / 8 \) (-14b12 + 81b6 - 86*b2 - 322) / 8
\(\nu^{13}\)	\(=\)	\( ( - 50 \beta_{15} - 79 \beta_{14} + 158 \beta_{11} - 5 \beta_{10} + 26 \beta_{9} - 55 \beta_{8} + \cdots - 45 \beta_{4} ) / 8 \) (-50b15 - 79b14 + 158b11 - 5b10 + 26b9 - 55b8 + 27b7 - 55b5 - 45*b4) / 8
\(\nu^{14}\)	\(=\)	\( ( -93\beta_{13} + 82\beta_{8} - 82\beta_{5} - 34\beta_{3} - 148\beta_1 ) / 8 \) (-93b13 + 82b8 - 82b5 - 34b3 - 148*b1) / 8
\(\nu^{15}\)	\(=\)	\( ( 18 \beta_{15} + 31 \beta_{14} + 62 \beta_{11} + 117 \beta_{10} + 134 \beta_{9} - 135 \beta_{8} + \cdots + 99 \beta_{4} ) / 8 \) (18b15 + 31b14 + 62b11 + 117b10 + 134b9 - 135b8 + 299b7 - 135b5 + 99*b4) / 8

\(n\)	\(127\)	\(577\)	\(1765\)	\(1793\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1567.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} + 14 T^{2} + 32)^{4} \) (T^4 + 14*T^2 + 32)^4
$7$	\( (T^{8} + 8 T^{6} + \cdots + 2401)^{2} \) (T^8 + 8T^6 + 46T^4 + 392*T^2 + 2401)^2
$11$	\( (T^{4} + 26 T^{2} + 16)^{4} \) (T^4 + 26*T^2 + 16)^4
$13$	\( (T^{4} + 40 T^{2} + 128)^{4} \) (T^4 + 40*T^2 + 128)^4
$17$	\( (T^{4} + 46 T^{2} + 512)^{4} \) (T^4 + 46*T^2 + 512)^4
$19$	\( (T^{4} - 28 T^{2} + 128)^{4} \) (T^4 - 28*T^2 + 128)^4
$23$	\( (T^{4} + 18 T^{2} + 64)^{4} \) (T^4 + 18*T^2 + 64)^4
$29$	\( (T^{4} - 72 T^{2} + 1024)^{4} \) (T^4 - 72*T^2 + 1024)^4
$31$	\( (T^{4} - 40 T^{2} + 128)^{4} \) (T^4 - 40*T^2 + 128)^4
$37$	\( (T^{2} - 2 T - 16)^{8} \) (T^2 - 2*T - 16)^8
$41$	\( (T^{4} + 46 T^{2} + 512)^{4} \) (T^4 + 46*T^2 + 512)^4
$43$	\( (T^{4} + 84 T^{2} + 64)^{4} \) (T^4 + 84*T^2 + 64)^4
$47$	\( (T^{4} - 160 T^{2} + 2048)^{4} \) (T^4 - 160*T^2 + 2048)^4
$53$	\( (T^{4} - 104 T^{2} + 256)^{4} \) (T^4 - 104*T^2 + 256)^4
$59$	\( (T^{4} - 224 T^{2} + 8192)^{4} \) (T^4 - 224*T^2 + 8192)^4
$61$	\( (T^{4} + 296 T^{2} + 21632)^{4} \) (T^4 + 296*T^2 + 21632)^4
$67$	\( (T^{4} + 84 T^{2} + 64)^{4} \) (T^4 + 84*T^2 + 64)^4
$71$	\( (T^{4} + 178 T^{2} + 4096)^{4} \) (T^4 + 178*T^2 + 4096)^4
$73$	\( (T^{4} + 160 T^{2} + 2048)^{4} \) (T^4 + 160*T^2 + 2048)^4
$79$	\( (T^{4} + 36 T^{2} + 256)^{4} \) (T^4 + 36*T^2 + 256)^4
$83$	\( (T^{4} - 160 T^{2} + 2048)^{4} \) (T^4 - 160*T^2 + 2048)^4
$89$	\( (T^{4} + 62 T^{2} + 128)^{4} \) (T^4 + 62*T^2 + 128)^4
$97$	\( (T^{4} + 224 T^{2} + 8192)^{4} \) (T^4 + 224*T^2 + 8192)^4
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