LMFDB - Newform orbit 480.2.bs.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(59,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 1, 4, 4]))

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

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

chi := DirichletCharacter("480.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.bs (of order $8$, degree $4$, 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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	16.0.721389578983833600000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 17x^{8} + 256 $ x^16 + 17*x^8 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

$q$-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_1 q^{2} + ( - \beta_{12} - \beta_{9}) q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{14}) q^{5} + (\beta_{13} - 2 \beta_{10}) q^{6} + (\beta_{8} + 2 \beta_{6}) q^{8} - 3 \beta_{5} q^{9}+O(q^{10}) $ q + b1 * q^2 + (-b12 - b9) * q^3 + b2 * q^4 + (b15 + b14) * q^5 + (b13 - 2b10) q^6 + (b8 + 2b6) q^8 - 3b5 q^9	$ q + \beta_1 q^{2} + ( - \beta_{12} - \beta_{9}) q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{14}) q^{5} + (\beta_{13} - 2 \beta_{10}) q^{6} + (\beta_{8} + 2 \beta_{6}) q^{8} - 3 \beta_{5} q^{9} + (\beta_{3} + 3) q^{10} + ( - 2 \beta_{15} - \beta_{14}) q^{12} + ( - 2 \beta_{11} - \beta_{7}) q^{15} + (\beta_{11} + 4 \beta_{7}) q^{16} + ( - \beta_{12} - 3 \beta_{9} + \cdots + 3 \beta_{6}) q^{17}+ \cdots + 7 \beta_{9} q^{98}+O(q^{100}) $ q + b1 * q^2 + (-b12 - b9) * q^3 + b2 * q^4 + (b15 + b14) * q^5 + (b13 - 2b10) q^6 + (b8 + 2b6) q^8 - 3b5 q^9 + (b3 + 3) * q^10 + (-2b15 - b14) q^12 + (-2b11 - b7) q^15 + (b11 + 4b7) q^16 + (-b12 - 3b9 + b8 + 3b6) * q^17 + 3b8 q^18 + (-2b13 - 2b11 - b10 - 3b7) q^19 + (2b4 + 3b1) * q^20 + (3b8 - b6 + 3b4 - b1) * q^23 + (-b3 - 5) * q^24 + 5b10 q^25 + (-3b15 + 3b14) * q^27 + (-4b12 - b9) q^30 + (-2b13 + 5b10 + 3b5 - 2b2) * q^31 + (2b12 - 3b9) * q^32 + (3b13 + b11 - 2b10 + 6b7) q^34 + 3b11 q^36 + (4b15 - 3b14 - 4b12 + b9) q^38 + (4b5 + 3b2) * q^40 + (-3b4 + 3b1) * q^45 + (3b11 - 2b7 + 6b5 - b2) q^46 + (b15 - 5b14 + 5b4 + b1) * q^47 + (-2b4 - 5b1) * q^48 - 7b7 q^49 + 5b14 q^50 + (-7b5 - 2b3 - 2b2 - 7) q^51 + (3b12 + 5b9 - 5b4 - 3b1) * q^53 + (3b3 - 3) q^54 + (-b8 + 5b6 - b4 + 5b1) * q^57 + (b13 - 8b10) q^60 + (-b5 - 4b3 - 4b2 - 1) * q^61 + (4b15 + 3b14 - 5b8 - 4b6) * q^62 + (3b13 + 4b10) * q^64 + (-6b15 + b14 + 2b12 - 5b9) q^68 + (-4b13 - b10 + 4b3 - 1) * q^69 + (6b12 + 3b9) * q^72 + (-5b8 - 5b6) * q^75 + (-b13 - 8b10 - 3b3 + 5) * q^76 + 16 * q^79 + (-b8 + 6b6) q^80 - 9b7 q^81 + (-5b15 - 3b14 + 5b8 - 3b6) * q^83 + (-4b11 + 3b7 + 7b5 + 4b2) * q^85 + (-6b5 + 3b2) * q^90 + (6b12 + 5b9 - 7b8 - 2b6) * q^92 + (7b15 - b14 - 7b8 - b6) * q^93 + (10b5 - 5b3 + b2 - 3) * q^94 + (3b12 + 7b9 - 3b8 - 7b6) * q^95 + (-4b5 - 5b2) * q^96 + 7b9 q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 40 q^{10} - 72 q^{24} - 96 q^{51} - 72 q^{54} + 16 q^{61} - 48 q^{69} + 104 q^{76} + 256 q^{79} - 8 q^{94}+O(q^{100}) $ 16 * q + 40 * q^10 - 72 * q^24 - 96 * q^51 - 72 * q^54 + 16 * q^61 - 48 * q^69 + 104 * q^76 + 256 * q^79 - 8 * q^94

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 17x^{8} + 256 $ x^16 + 17*x^8 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} $ v^2
$\beta_{3}$	$=$	$ ( \nu^{8} + 5 ) / 7 $ (v^8 + 5) / 7
$\beta_{4}$	$=$	$ ( \nu^{9} + 5\nu ) / 14 $ (v^9 + 5*v) / 14
$\beta_{5}$	$=$	$ ( \nu^{10} + 5\nu^{2} ) / 28 $ (v^10 + 5*v^2) / 28
$\beta_{6}$	$=$	$ ( \nu^{11} + 33\nu^{3} ) / 56 $ (v^11 + 33*v^3) / 56
$\beta_{7}$	$=$	$ ( \nu^{12} + 33\nu^{4} ) / 112 $ (v^12 + 33*v^4) / 112
$\beta_{8}$	$=$	$ ( -\nu^{11} - 5\nu^{3} ) / 28 $ (-v^11 - 5*v^3) / 28
$\beta_{9}$	$=$	$ ( -\nu^{13} - 33\nu^{5} ) / 112 $ (-v^13 - 33*v^5) / 112
$\beta_{10}$	$=$	$ ( -3\nu^{14} + 13\nu^{6} ) / 448 $ (-3v^14 + 13v^6) / 448
$\beta_{11}$	$=$	$ ( -\nu^{12} - 5\nu^{4} ) / 28 $ (-v^12 - 5*v^4) / 28
$\beta_{12}$	$=$	$ ( -3\nu^{13} + 13\nu^{5} ) / 224 $ (-3v^13 + 13v^5) / 224
$\beta_{13}$	$=$	$ ( \nu^{14} + 33\nu^{6} ) / 112 $ (v^14 + 33*v^6) / 112
$\beta_{14}$	$=$	$ ( -3\nu^{15} + 13\nu^{7} ) / 448 $ (-3v^15 + 13v^7) / 448
$\beta_{15}$	$=$	$ ( -\nu^{15} - 17\nu^{7} ) / 128 $ (-v^15 - 17*v^7) / 128

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} $ b2
$\nu^{3}$	$=$	$ \beta_{8} + 2\beta_{6} $ b8 + 2*b6
$\nu^{4}$	$=$	$ \beta_{11} + 4\beta_{7} $ b11 + 4*b7
$\nu^{5}$	$=$	$ 2\beta_{12} - 3\beta_{9} $ 2b12 - 3b9
$\nu^{6}$	$=$	$ 3\beta_{13} + 4\beta_{10} $ 3b13 + 4b10
$\nu^{7}$	$=$	$ -6\beta_{15} + 7\beta_{14} $ -6b15 + 7b14
$\nu^{8}$	$=$	$ 7\beta_{3} - 5 $ 7*b3 - 5
$\nu^{9}$	$=$	$ 14\beta_{4} - 5\beta_1 $ 14b4 - 5b1
$\nu^{10}$	$=$	$ 28\beta_{5} - 5\beta_{2} $ 28b5 - 5b2
$\nu^{11}$	$=$	$ -33\beta_{8} - 10\beta_{6} $ -33b8 - 10b6
$\nu^{12}$	$=$	$ -33\beta_{11} - 20\beta_{7} $ -33b11 - 20b7
$\nu^{13}$	$=$	$ -66\beta_{12} - 13\beta_{9} $ -66b12 - 13b9
$\nu^{14}$	$=$	$ 13\beta_{13} - 132\beta_{10} $ 13b13 - 132b10
$\nu^{15}$	$=$	$ -26\beta_{15} - 119\beta_{14} $ -26b15 - 119b14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$-1$	$-1$	$\beta_{10}$

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} $

59.1

−1.36434	−	0.372250i
−0.701515	+	1.22796i
0.701515	−	1.22796i
1.36434	+	0.372250i
−1.36434	+	0.372250i
−0.701515	−	1.22796i
0.701515	+	1.22796i
1.36434	−	0.372250i
−1.22796	−	0.701515i
−0.372250	+	1.36434i
0.372250	−	1.36434i
1.22796	+	0.701515i
−1.22796	+	0.701515i
−0.372250	−	1.36434i
0.372250	+	1.36434i
1.22796	−	0.701515i

−1.36434

−

0.372250i

1.60021

−

0.662827i

1.72286

1.01575i

−2.06586

−

0.855706i

−2.42997

0.308646i

−1.97246

−

2.02717i

2.12132

−

2.12132i

2.50000

1.93649i

59.2

−0.701515

1.22796i

−1.60021

0.662827i

−1.01575

−

1.72286i

−2.06586

−

0.855706i

0.308646

−

2.42997i

2.82816

−

0.0386880i

2.12132

−

2.12132i

2.50000

−

1.93649i

59.3

0.701515

−

1.22796i

1.60021

−

0.662827i

−1.01575

−

1.72286i

2.06586

0.855706i

0.308646

−

2.42997i

−2.82816

0.0386880i

2.12132

−

2.12132i

2.50000

−

1.93649i

59.4

1.36434

0.372250i

−1.60021

0.662827i

1.72286

1.01575i

2.06586

0.855706i

−2.42997

0.308646i

1.97246

2.02717i

2.12132

−

2.12132i

2.50000

1.93649i

179.1

−1.36434

0.372250i

1.60021

0.662827i

1.72286

−

1.01575i

−2.06586

0.855706i

−2.42997

−

0.308646i

−1.97246

2.02717i

2.12132

2.12132i

2.50000

−

1.93649i

179.2

−0.701515

−

1.22796i

−1.60021

−

0.662827i

−1.01575

1.72286i

−2.06586

0.855706i

0.308646

2.42997i

2.82816

0.0386880i

2.12132

2.12132i

2.50000

1.93649i

179.3

0.701515

1.22796i

1.60021

0.662827i

−1.01575

1.72286i

2.06586

−

0.855706i

0.308646

2.42997i

−2.82816

−

0.0386880i

2.12132

2.12132i

2.50000

1.93649i

179.4

1.36434

−

0.372250i

−1.60021

−

0.662827i

1.72286

−

1.01575i

2.06586

−

0.855706i

−2.42997

−

0.308646i

1.97246

−

2.02717i

2.12132

2.12132i

2.50000

−

1.93649i

299.1

−1.22796

−

0.701515i

−0.662827

−

1.60021i

1.01575

1.72286i

−0.855706

2.06586i

−0.308646

2.42997i

−0.0386880

−

2.82816i

−2.12132

2.12132i

2.50000

−

1.93649i

299.2

−0.372250

1.36434i

−0.662827

−

1.60021i

−1.72286

−

1.01575i

0.855706

−

2.06586i

2.42997

−

0.308646i

2.02717

−

1.97246i

−2.12132

2.12132i

2.50000

1.93649i

299.3

0.372250

−

1.36434i

0.662827

1.60021i

−1.72286

−

1.01575i

−0.855706

2.06586i

2.42997

−

0.308646i

−2.02717

1.97246i

−2.12132

2.12132i

2.50000

1.93649i

299.4

1.22796

0.701515i

0.662827

1.60021i

1.01575

1.72286i

0.855706

−

2.06586i

−0.308646

2.42997i

0.0386880

2.82816i

−2.12132

2.12132i

2.50000

−

1.93649i

419.1

−1.22796

0.701515i

−0.662827

1.60021i

1.01575

−

1.72286i

−0.855706

−

2.06586i

−0.308646

−

2.42997i

−0.0386880

2.82816i

−2.12132

−

2.12132i

2.50000

1.93649i

419.2

−0.372250

−

1.36434i

−0.662827

1.60021i

−1.72286

1.01575i

0.855706

2.06586i

2.42997

0.308646i

2.02717

1.97246i

−2.12132

−

2.12132i

2.50000

−

1.93649i

419.3

0.372250

1.36434i

0.662827

−

1.60021i

−1.72286

1.01575i

−0.855706

−

2.06586i

2.42997

0.308646i

−2.02717

−

1.97246i

−2.12132

−

2.12132i

2.50000

−

1.93649i

419.4

1.22796

−

0.701515i

0.662827

−

1.60021i

1.01575

−

1.72286i

0.855706

2.06586i

−0.308646

−

2.42997i

0.0386880

−

2.82816i

−2.12132

−

2.12132i

2.50000

1.93649i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
32.h	odd	8	1	inner
96.o	even	8	1	inner
160.y	odd	8	1	inner
480.bs	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	480.2.bs.a	✓	16
3.b	odd	2	1	inner	480.2.bs.a	✓	16
5.b	even	2	1	inner	480.2.bs.a	✓	16
15.d	odd	2	1	CM	480.2.bs.a	✓	16
32.h	odd	8	1	inner	480.2.bs.a	✓	16
96.o	even	8	1	inner	480.2.bs.a	✓	16
160.y	odd	8	1	inner	480.2.bs.a	✓	16
480.bs	even	8	1	inner	480.2.bs.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
480.2.bs.a	✓	16	1.a	even	1	1	trivial
480.2.bs.a	✓	16	3.b	odd	2	1	inner
480.2.bs.a	✓	16	5.b	even	2	1	inner
480.2.bs.a	✓	16	15.d	odd	2	1	CM
480.2.bs.a	✓	16	32.h	odd	8	1	inner
480.2.bs.a	✓	16	96.o	even	8	1	inner
480.2.bs.a	✓	16	160.y	odd	8	1	inner
480.2.bs.a	✓	16	480.bs	even	8	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} $ T7 acting on $S_{2}^{\mathrm{new}}(480, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 17T^{8} + 256 $ T^16 + 17*T^8 + 256
$3$	$ (T^{8} + 81)^{2} $ (T^8 + 81)^2
$5$	$ (T^{8} + 625)^{2} $ (T^8 + 625)^2
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} $ T^16
$17$	$ (T^{8} + 136 T^{6} + \cdots + 145924)^{2} $ (T^8 + 136T^6 + 5780T^4 + 78608*T^2 + 145924)^2
$19$	$ (T^{8} - 44 T^{6} + \cdots + 56644)^{2} $ (T^8 - 44T^6 - 304T^5 + 968T^4 + 24928T^3 + 120216T^2 + 72352T + 56644)^2
$23$	$ T^{16} + 12696 T^{12} + \cdots + 92236816 $ T^16 + 12696T^12 + 40316312T^8 + 37693966944*T^4 + 92236816
$29$	$ T^{16} $ T^16
$31$	$ (T^{4} + 124 T^{2} + 4)^{4} $ (T^4 + 124*T^2 + 4)^4
$37$	$ T^{16} $ T^16
$41$	$ T^{16} $ T^16
$43$	$ T^{16} $ T^16
$47$	$ (T^{8} - 376 T^{6} + \cdots + 17825284)^{2} $ (T^8 - 376T^6 + 44180T^4 - 1661168*T^2 + 17825284)^2
$53$	$ T^{16} + \cdots + 29\!\cdots\!56 $ T^16 + 799741472*T^8 + 2992179271065856
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} - 8 T^{7} + \cdots + 42016324)^{2} $ (T^8 - 8T^7 + 268T^6 - 1496T^5 + 31752T^4 - 230464T^3 - 957560T^2 + 9282224*T + 42016324)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} $ T^16
$79$	$ (T - 16)^{16} $ (T - 16)^16
$83$	$ T^{16} + \cdots + 31\!\cdots\!36 $ T^16 + 2817937952*T^8 + 316348490636206336
$89$	$ T^{16} $ T^16
$97$	$ T^{16} $ T^16
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.bs (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + ( - \beta_{12} - \beta_{9}) q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{14}) q^{5} + (\beta_{13} - 2 \beta_{10}) q^{6} + (\beta_{8} + 2 \beta_{6}) q^{8} - 3 \beta_{5} q^{9}+O(q^{10}) \) q + b1 * q^2 + (-b12 - b9) * q^3 + b2 * q^4 + (b15 + b14) * q^5 + (b13 - 2b10) q^6 + (b8 + 2b6) q^8 - 3b5 q^9	\( q + \beta_1 q^{2} + ( - \beta_{12} - \beta_{9}) q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{14}) q^{5} + (\beta_{13} - 2 \beta_{10}) q^{6} + (\beta_{8} + 2 \beta_{6}) q^{8} - 3 \beta_{5} q^{9} + (\beta_{3} + 3) q^{10} + ( - 2 \beta_{15} - \beta_{14}) q^{12} + ( - 2 \beta_{11} - \beta_{7}) q^{15} + (\beta_{11} + 4 \beta_{7}) q^{16} + ( - \beta_{12} - 3 \beta_{9} + \cdots + 3 \beta_{6}) q^{17}+ \cdots + 7 \beta_{9} q^{98}+O(q^{100}) \) q + b1 * q^2 + (-b12 - b9) * q^3 + b2 * q^4 + (b15 + b14) * q^5 + (b13 - 2b10) q^6 + (b8 + 2b6) q^8 - 3b5 q^9 + (b3 + 3) * q^10 + (-2b15 - b14) q^12 + (-2b11 - b7) q^15 + (b11 + 4b7) q^16 + (-b12 - 3b9 + b8 + 3b6) * q^17 + 3b8 q^18 + (-2b13 - 2b11 - b10 - 3b7) q^19 + (2b4 + 3b1) * q^20 + (3b8 - b6 + 3b4 - b1) * q^23 + (-b3 - 5) * q^24 + 5b10 q^25 + (-3b15 + 3b14) * q^27 + (-4b12 - b9) q^30 + (-2b13 + 5b10 + 3b5 - 2b2) * q^31 + (2b12 - 3b9) * q^32 + (3b13 + b11 - 2b10 + 6b7) q^34 + 3b11 q^36 + (4b15 - 3b14 - 4b12 + b9) q^38 + (4b5 + 3b2) * q^40 + (-3b4 + 3b1) * q^45 + (3b11 - 2b7 + 6b5 - b2) q^46 + (b15 - 5b14 + 5b4 + b1) * q^47 + (-2b4 - 5b1) * q^48 - 7b7 q^49 + 5b14 q^50 + (-7b5 - 2b3 - 2b2 - 7) q^51 + (3b12 + 5b9 - 5b4 - 3b1) * q^53 + (3b3 - 3) q^54 + (-b8 + 5b6 - b4 + 5b1) * q^57 + (b13 - 8b10) q^60 + (-b5 - 4b3 - 4b2 - 1) * q^61 + (4b15 + 3b14 - 5b8 - 4b6) * q^62 + (3b13 + 4b10) * q^64 + (-6b15 + b14 + 2b12 - 5b9) q^68 + (-4b13 - b10 + 4b3 - 1) * q^69 + (6b12 + 3b9) * q^72 + (-5b8 - 5b6) * q^75 + (-b13 - 8b10 - 3b3 + 5) * q^76 + 16 * q^79 + (-b8 + 6b6) q^80 - 9b7 q^81 + (-5b15 - 3b14 + 5b8 - 3b6) * q^83 + (-4b11 + 3b7 + 7b5 + 4b2) * q^85 + (-6b5 + 3b2) * q^90 + (6b12 + 5b9 - 7b8 - 2b6) * q^92 + (7b15 - b14 - 7b8 - b6) * q^93 + (10b5 - 5b3 + b2 - 3) * q^94 + (3b12 + 7b9 - 3b8 - 7b6) * q^95 + (-4b5 - 5b2) * q^96 + 7b9 q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 40 q^{10} - 72 q^{24} - 96 q^{51} - 72 q^{54} + 16 q^{61} - 48 q^{69} + 104 q^{76} + 256 q^{79} - 8 q^{94}+O(q^{100}) \) 16 * q + 40 * q^10 - 72 * q^24 - 96 * q^51 - 72 * q^54 + 16 * q^61 - 48 * q^69 + 104 * q^76 + 256 * q^79 - 8 * q^94

Introduction

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Newspace parameters

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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	480.2.bs.a

Level	$480$
Weight	$2$
Character orbit	480.bs
Analytic conductor	$3.833$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(3.83281929702\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{8})\)
Coefficient field:	16.0.721389578983833600000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 17x^{8} + 256 \) x^16 + 17*x^8 + 256
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} \) v^2
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} + 5 ) / 7 \) (v^8 + 5) / 7
\(\beta_{4}\)	\(=\)	\( ( \nu^{9} + 5\nu ) / 14 \) (v^9 + 5*v) / 14
\(\beta_{5}\)	\(=\)	\( ( \nu^{10} + 5\nu^{2} ) / 28 \) (v^10 + 5*v^2) / 28
\(\beta_{6}\)	\(=\)	\( ( \nu^{11} + 33\nu^{3} ) / 56 \) (v^11 + 33*v^3) / 56
\(\beta_{7}\)	\(=\)	\( ( \nu^{12} + 33\nu^{4} ) / 112 \) (v^12 + 33*v^4) / 112
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} - 5\nu^{3} ) / 28 \) (-v^11 - 5*v^3) / 28
\(\beta_{9}\)	\(=\)	\( ( -\nu^{13} - 33\nu^{5} ) / 112 \) (-v^13 - 33*v^5) / 112
\(\beta_{10}\)	\(=\)	\( ( -3\nu^{14} + 13\nu^{6} ) / 448 \) (-3v^14 + 13v^6) / 448
\(\beta_{11}\)	\(=\)	\( ( -\nu^{12} - 5\nu^{4} ) / 28 \) (-v^12 - 5*v^4) / 28
\(\beta_{12}\)	\(=\)	\( ( -3\nu^{13} + 13\nu^{5} ) / 224 \) (-3v^13 + 13v^5) / 224
\(\beta_{13}\)	\(=\)	\( ( \nu^{14} + 33\nu^{6} ) / 112 \) (v^14 + 33*v^6) / 112
\(\beta_{14}\)	\(=\)	\( ( -3\nu^{15} + 13\nu^{7} ) / 448 \) (-3v^15 + 13v^7) / 448
\(\beta_{15}\)	\(=\)	\( ( -\nu^{15} - 17\nu^{7} ) / 128 \) (-v^15 - 17*v^7) / 128

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} \) b2
\(\nu^{3}\)	\(=\)	\( \beta_{8} + 2\beta_{6} \) b8 + 2*b6
\(\nu^{4}\)	\(=\)	\( \beta_{11} + 4\beta_{7} \) b11 + 4*b7
\(\nu^{5}\)	\(=\)	\( 2\beta_{12} - 3\beta_{9} \) 2b12 - 3b9
\(\nu^{6}\)	\(=\)	\( 3\beta_{13} + 4\beta_{10} \) 3b13 + 4b10
\(\nu^{7}\)	\(=\)	\( -6\beta_{15} + 7\beta_{14} \) -6b15 + 7b14
\(\nu^{8}\)	\(=\)	\( 7\beta_{3} - 5 \) 7*b3 - 5
\(\nu^{9}\)	\(=\)	\( 14\beta_{4} - 5\beta_1 \) 14b4 - 5b1
\(\nu^{10}\)	\(=\)	\( 28\beta_{5} - 5\beta_{2} \) 28b5 - 5b2
\(\nu^{11}\)	\(=\)	\( -33\beta_{8} - 10\beta_{6} \) -33b8 - 10b6
\(\nu^{12}\)	\(=\)	\( -33\beta_{11} - 20\beta_{7} \) -33b11 - 20b7
\(\nu^{13}\)	\(=\)	\( -66\beta_{12} - 13\beta_{9} \) -66b12 - 13b9
\(\nu^{14}\)	\(=\)	\( 13\beta_{13} - 132\beta_{10} \) 13b13 - 132b10
\(\nu^{15}\)	\(=\)	\( -26\beta_{15} - 119\beta_{14} \) -26b15 - 119b14

\(n\)	\(31\)	\(97\)	\(161\)	\(421\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(\beta_{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 59.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} + 17T^{8} + 256 \) T^16 + 17*T^8 + 256
$3$	\( (T^{8} + 81)^{2} \) (T^8 + 81)^2
$5$	\( (T^{8} + 625)^{2} \) (T^8 + 625)^2
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} \) T^16
$17$	\( (T^{8} + 136 T^{6} + \cdots + 145924)^{2} \) (T^8 + 136T^6 + 5780T^4 + 78608*T^2 + 145924)^2
$19$	\( (T^{8} - 44 T^{6} + \cdots + 56644)^{2} \) (T^8 - 44T^6 - 304T^5 + 968T^4 + 24928T^3 + 120216T^2 + 72352T + 56644)^2
$23$	\( T^{16} + 12696 T^{12} + \cdots + 92236816 \) T^16 + 12696T^12 + 40316312T^8 + 37693966944*T^4 + 92236816
$29$	\( T^{16} \) T^16
$31$	\( (T^{4} + 124 T^{2} + 4)^{4} \) (T^4 + 124*T^2 + 4)^4
$37$	\( T^{16} \) T^16
$41$	\( T^{16} \) T^16
$43$	\( T^{16} \) T^16
$47$	\( (T^{8} - 376 T^{6} + \cdots + 17825284)^{2} \) (T^8 - 376T^6 + 44180T^4 - 1661168*T^2 + 17825284)^2
$53$	\( T^{16} + \cdots + 29\!\cdots\!56 \) T^16 + 799741472*T^8 + 2992179271065856
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} - 8 T^{7} + \cdots + 42016324)^{2} \) (T^8 - 8T^7 + 268T^6 - 1496T^5 + 31752T^4 - 230464T^3 - 957560T^2 + 9282224*T + 42016324)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} \) T^16
$79$	\( (T - 16)^{16} \) (T - 16)^16
$83$	\( T^{16} + \cdots + 31\!\cdots\!36 \) T^16 + 2817937952*T^8 + 316348490636206336
$89$	\( T^{16} \) T^16
$97$	\( T^{16} \) T^16
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