LMFDB - Newform orbit 900.2.h.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(899,900)] 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("900.899"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	900.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
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_{9} q^{2} + \beta_{3} q^{4} + (\beta_{10} + \beta_{5}) q^{7} + (\beta_{14} + \beta_{9} - \beta_{7}) q^{8} + ( - \beta_{12} - \beta_{8}) q^{11} + (\beta_{10} - \beta_{5} + \cdots - \beta_{2}) q^{13}+ \cdots + (2 \beta_{11} + 2 \beta_{9} + 2 \beta_{7}) q^{98}+O(q^{100}) $ q - b9 * q^2 + b3 * q^4 + (b10 + b5) * q^7 + (b14 + b9 - b7) * q^8 + (-b12 - b8) * q^11 + (b10 - b5 + 2b4 - b2) q^13 + (b15 + b12 - b8 + 2b6) q^14 + (b13 - b3 + b1 - 2) * q^16 + (-b14 - b11 - 2b9 + b7) q^17 + (b13 + 2b3 + 1) q^19 + (b10 + 3b4 + b2) q^22 + (b14 - 2b11 + b9) q^23 + (-2b15 + b12 + 2b6) * q^26 + (-b10 - 2b5 + 3b4) * q^28 + (b15 + 2b12 - b8 + 3b6) * q^29 + (-b13 + 2b3 - 4b1 + 3) * q^31 + 4b7 q^32 + (2b3 - b1 + 4) q^34 + (b10 - b5 - b4 - b2) * q^37 + (3b14 + b11 + b9) q^38 + (3b15 + 6b12 - 3b8 + 2b6) * q^41 + (-b10 + 3b5 - 2b4 - 2b2) q^43 + (b15 + 3b12 - 2b8 - b6) * q^44 + (3b13 - b3 + 2) q^46 + (-2b14 - b11 + 3b9) * q^47 + (b13 - b3 - b1 - 2) * q^49 + (-b10 - 2b5 - 5b4 + 2b2) q^52 + (b14 + b11 + 2b9 - 6b7) * q^53 + (-2b15 + b12 + b8 - 3b6) * q^56 + (-2b10 - 3b5 + 2b4) q^58 + (-3b15 + 3b12) * q^59 + (-2b13 + 2b3 + 2b1 + 5) q^61 + (b14 + 3b11 + 3b9 - 8b7) q^62 - 4b1 q^64 + (2b10 + 4b5 - b4 - b2) * q^67 + (2b14 + b11 - b9 - 3b7) * q^68 + (b15 - 4b12 - 3b8) * q^71 + (b10 - b5 - 5b4 - b2) q^73 + (-2b15 - 2b12 + 2b6) q^74 + (2b13 - b3 - 8) q^76 + (-b14 - b11 - 2b9 + b7) q^77 + (-4b3 + 4b1 - 4) * q^79 + (-6b10 - 2b5 + 6b4) q^82 + (-2b14 - b11 + 3b9) * q^83 + (b15 + b12 + 3b8 + 6b6) * q^86 + (-3b10 + b5 + 3b4 + b2) * q^88 + (-2b15 - 4b12 + 2b8 - 4b6) * q^89 + (-b13 + 4b3 - 6b1 + 5) * q^91 + (2b14 + 3b11 - 3b9 + 7b7) * q^92 + (-b13 - 3b3 + 6) q^94 - 3b4 q^97 + (2b11 + 2b9 + 2b7) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 16 q^{16} + 40 q^{34} + 40 q^{46} - 32 q^{49} + 80 q^{61} - 32 q^{64} - 120 q^{76} + 120 q^{94}+O(q^{100}) $ 16 * q - 8 * q^4 - 16 * q^16 + 40 * q^34 + 40 * q^46 - 32 * q^49 + 80 * q^61 - 32 * q^64 - 120 * q^76 + 120 * q^94

Basis of coefficient ring

$\beta_{1}$	$=$	$ 2\zeta_{40}^{4} $ 2*v^4
$\beta_{2}$	$=$	$ 2\zeta_{40}^{6} + 2\zeta_{40}^{2} $ 2v^6 + 2v^2
$\beta_{3}$	$=$	$ 2\zeta_{40}^{8} $ 2*v^8
$\beta_{4}$	$=$	$ \zeta_{40}^{10} $ v^10
$\beta_{5}$	$=$	$ 2\zeta_{40}^{14} $ 2*v^14
$\beta_{6}$	$=$	$ \zeta_{40}^{15} + \zeta_{40}^{5} $ v^15 + v^5
$\beta_{7}$	$=$	$ -\zeta_{40}^{15} + \zeta_{40}^{5} $ -v^15 + v^5
$\beta_{8}$	$=$	$ -2\zeta_{40}^{13} + \zeta_{40}^{11} + \zeta_{40}^{9} + \zeta_{40}^{7} - \zeta_{40}^{5} + \zeta_{40}^{3} + 2\zeta_{40} $ -2v^13 + v^11 + v^9 + v^7 - v^5 + v^3 + 2v
$\beta_{9}$	$=$	$ -\zeta_{40}^{15} + \zeta_{40}^{11} + \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} $ -v^15 + v^11 + v^9 - v^7 + v^3
$\beta_{10}$	$=$	$ -2\zeta_{40}^{14} + \zeta_{40}^{10} - 2\zeta_{40}^{6} + 2\zeta_{40}^{2} $ -2v^14 + v^10 - 2v^6 + 2*v^2
$\beta_{11}$	$=$	$ 2\zeta_{40}^{13} - \zeta_{40}^{11} - \zeta_{40}^{9} + \zeta_{40}^{7} + \zeta_{40}^{5} + \zeta_{40}^{3} $ 2*v^13 - v^11 - v^9 + v^7 + v^5 + v^3
$\beta_{12}$	$=$	$ -\zeta_{40}^{15} + \zeta_{40}^{11} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} $ -v^15 + v^11 - v^9 - v^7 + v^3
$\beta_{13}$	$=$	$ 4\zeta_{40}^{12} - 2\zeta_{40}^{8} + 2\zeta_{40}^{4} - 2 $ 4v^12 - 2v^8 + 2*v^4 - 2
$\beta_{14}$	$=$	$ -2\zeta_{40}^{13} - \zeta_{40}^{11} + \zeta_{40}^{9} - \zeta_{40}^{7} - \zeta_{40}^{5} - \zeta_{40}^{3} + 2\zeta_{40} $ -2v^13 - v^11 + v^9 - v^7 - v^5 - v^3 + 2v
$\beta_{15}$	$=$	$ -2\zeta_{40}^{13} - \zeta_{40}^{11} + \zeta_{40}^{9} + \zeta_{40}^{7} - \zeta_{40}^{5} + \zeta_{40}^{3} $ -2*v^13 - v^11 + v^9 + v^7 - v^5 + v^3

Display $\zeta_{40}^j$ in terms of $\beta_i$

$\zeta_{40}$	$=$	$ ( -\beta_{15} + \beta_{14} + \beta_{11} + \beta_{8} ) / 4 $ (-b15 + b14 + b11 + b8) / 4
$\zeta_{40}^{2}$	$=$	$ ( \beta_{10} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 $ (b10 + b5 - b4 + b2) / 4
$\zeta_{40}^{3}$	$=$	$ ( \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} ) / 4 $ (b15 + b12 + b11 + b9 - b7 + b6) / 4
$\zeta_{40}^{4}$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\zeta_{40}^{5}$	$=$	$ ( \beta_{7} + \beta_{6} ) / 2 $ (b7 + b6) / 2
$\zeta_{40}^{6}$	$=$	$ ( -\beta_{10} - \beta_{5} + \beta_{4} + \beta_{2} ) / 4 $ (-b10 - b5 + b4 + b2) / 4
$\zeta_{40}^{7}$	$=$	$ ( -\beta_{14} - \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} ) / 4 $ (-b14 - b12 - b9 + b8 + b7 - b6) / 4
$\zeta_{40}^{8}$	$=$	$ ( \beta_{3} ) / 2 $ (b3) / 2
$\zeta_{40}^{9}$	$=$	$ ( -\beta_{12} + \beta_{9} ) / 2 $ (-b12 + b9) / 2
$\zeta_{40}^{10}$	$=$	$ \beta_{4} $ b4
$\zeta_{40}^{11}$	$=$	$ ( -\beta_{15} - \beta_{14} - \beta_{11} + \beta_{8} ) / 4 $ (-b15 - b14 - b11 + b8) / 4
$\zeta_{40}^{12}$	$=$	$ ( \beta_{13} + \beta_{3} - \beta _1 + 2 ) / 4 $ (b13 + b3 - b1 + 2) / 4
$\zeta_{40}^{13}$	$=$	$ ( -\beta_{15} - \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} ) / 4 $ (-b15 - b12 + b11 + b9 - b7 - b6) / 4
$\zeta_{40}^{14}$	$=$	$ ( \beta_{5} ) / 2 $ (b5) / 2
$\zeta_{40}^{15}$	$=$	$ ( -\beta_{7} + \beta_{6} ) / 2 $ (-b7 + b6) / 2

Display $\beta_i$ in terms of $\zeta_{40}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

899.1

0.156434	−	0.987688i
0.987688	+	0.156434i
0.156434	+	0.987688i
0.987688	−	0.156434i
−0.453990	−	0.891007i
0.891007	−	0.453990i
−0.453990	+	0.891007i
0.891007	+	0.453990i
0.453990	−	0.891007i
−0.891007	−	0.453990i
0.453990	+	0.891007i
−0.891007	+	0.453990i
−0.156434	−	0.987688i
−0.987688	+	0.156434i
−0.156434	+	0.987688i
−0.987688	−	0.156434i

−1.14412

−

0.831254i

0.618034

1.90211i

−0.726543

0.874032

−

2.68999i

899.2

−1.14412

−

0.831254i

0.618034

1.90211i

0.726543

0.874032

−

2.68999i

899.3

−1.14412

0.831254i

0.618034

−

1.90211i

−0.726543

0.874032

2.68999i

899.4

−1.14412

0.831254i

0.618034

−

1.90211i

0.726543

0.874032

2.68999i

899.5

−0.437016

−

1.34500i

−1.61803

1.17557i

−3.07768

2.28825

1.66251i

899.6

−0.437016

−

1.34500i

−1.61803

1.17557i

3.07768

2.28825

1.66251i

899.7

−0.437016

1.34500i

−1.61803

−

1.17557i

−3.07768

2.28825

−

1.66251i

899.8

−0.437016

1.34500i

−1.61803

−

1.17557i

3.07768

2.28825

−

1.66251i

899.9

0.437016

−

1.34500i

−1.61803

−

1.17557i

−3.07768

−2.28825

1.66251i

899.10

0.437016

−

1.34500i

−1.61803

−

1.17557i

3.07768

−2.28825

1.66251i

899.11

0.437016

1.34500i

−1.61803

1.17557i

−3.07768

−2.28825

−

1.66251i

899.12

0.437016

1.34500i

−1.61803

1.17557i

3.07768

−2.28825

−

1.66251i

899.13

1.14412

−

0.831254i

0.618034

−

1.90211i

−0.726543

−0.874032

−

2.68999i

899.14

1.14412

−

0.831254i

0.618034

−

1.90211i

0.726543

−0.874032

−

2.68999i

899.15

1.14412

0.831254i

0.618034

1.90211i

−0.726543

−0.874032

2.68999i

899.16

1.14412

0.831254i

0.618034

1.90211i

0.726543

−0.874032

2.68999i

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
5.b	even	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.h.d		16
3.b	odd	2	1	inner	900.2.h.d		16
4.b	odd	2	1	inner	900.2.h.d		16
5.b	even	2	1	inner	900.2.h.d		16
5.c	odd	4	1		900.2.e.e	✓	8
5.c	odd	4	1		900.2.e.g	yes	8
12.b	even	2	1	inner	900.2.h.d		16
15.d	odd	2	1	inner	900.2.h.d		16
15.e	even	4	1		900.2.e.e	✓	8
15.e	even	4	1		900.2.e.g	yes	8
20.d	odd	2	1	inner	900.2.h.d		16
20.e	even	4	1		900.2.e.e	✓	8
20.e	even	4	1		900.2.e.g	yes	8
60.h	even	2	1	inner	900.2.h.d		16
60.l	odd	4	1		900.2.e.e	✓	8
60.l	odd	4	1		900.2.e.g	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.e.e	✓	8	5.c	odd	4	1
900.2.e.e	✓	8	15.e	even	4	1
900.2.e.e	✓	8	20.e	even	4	1
900.2.e.e	✓	8	60.l	odd	4	1
900.2.e.g	yes	8	5.c	odd	4	1
900.2.e.g	yes	8	15.e	even	4	1
900.2.e.g	yes	8	20.e	even	4	1
900.2.e.g	yes	8	60.l	odd	4	1
900.2.h.d		16	1.a	even	1	1	trivial
900.2.h.d		16	3.b	odd	2	1	inner
900.2.h.d		16	4.b	odd	2	1	inner
900.2.h.d		16	5.b	even	2	1	inner
900.2.h.d		16	12.b	even	2	1	inner
900.2.h.d		16	15.d	odd	2	1	inner
900.2.h.d		16	20.d	odd	2	1	inner
900.2.h.d		16	60.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(900, [\chi])$:

$ T_{7}^{4} - 10T_{7}^{2} + 5 $

T7^4 - 10*T7^2 + 5

$ T_{17}^{2} - 10 $

T17^2 - 10

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} $ (T^8 + 2T^6 + 4T^4 + 8*T^2 + 16)^2
$3$	$ T^{16} $ T^16
$5$	$ T^{16} $ T^16
$7$	$ (T^{4} - 10 T^{2} + 5)^{4} $ (T^4 - 10*T^2 + 5)^4
$11$	$ (T^{4} - 20 T^{2} + 20)^{4} $ (T^4 - 20*T^2 + 20)^4
$13$	$ (T^{4} + 42 T^{2} + 361)^{4} $ (T^4 + 42*T^2 + 361)^4
$17$	$ (T^{2} - 10)^{8} $ (T^2 - 10)^8
$19$	$ (T^{4} + 50 T^{2} + 125)^{4} $ (T^4 + 50*T^2 + 125)^4
$23$	$ (T^{4} + 100 T^{2} + 2420)^{4} $ (T^4 + 100*T^2 + 2420)^4
$29$	$ (T^{4} + 36 T^{2} + 4)^{4} $ (T^4 + 36*T^2 + 4)^4
$31$	$ (T^{4} + 130 T^{2} + 1805)^{4} $ (T^4 + 130*T^2 + 1805)^4
$37$	$ (T^{4} + 48 T^{2} + 256)^{4} $ (T^4 + 48*T^2 + 256)^4
$41$	$ (T^{4} + 184 T^{2} + 7744)^{4} $ (T^4 + 184*T^2 + 7744)^4
$43$	$ (T^{4} - 170 T^{2} + 4805)^{4} $ (T^4 - 170*T^2 + 4805)^4
$47$	$ (T^{4} + 100 T^{2} + 2420)^{4} $ (T^4 + 100*T^2 + 2420)^4
$53$	$ (T^{4} - 120 T^{2} + 1600)^{4} $ (T^4 - 120*T^2 + 1600)^4
$59$	$ (T^{4} - 180 T^{2} + 1620)^{4} $ (T^4 - 180*T^2 + 1620)^4
$61$	$ (T^{2} - 10 T - 55)^{8} $ (T^2 - 10*T - 55)^8
$67$	$ (T^{4} - 130 T^{2} + 1805)^{4} $ (T^4 - 130*T^2 + 1805)^4
$71$	$ (T^{4} - 200 T^{2} + 8000)^{4} $ (T^4 - 200*T^2 + 8000)^4
$73$	$ (T^{4} + 112 T^{2} + 256)^{4} $ (T^4 + 112*T^2 + 256)^4
$79$	$ (T^{4} + 160 T^{2} + 1280)^{4} $ (T^4 + 160*T^2 + 1280)^4
$83$	$ (T^{4} + 100 T^{2} + 2420)^{4} $ (T^4 + 100*T^2 + 2420)^4
$89$	$ (T^{4} + 96 T^{2} + 1024)^{4} $ (T^4 + 96*T^2 + 1024)^4
$97$	$ (T^{2} + 9)^{8} $ (T^2 + 9)^8
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Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{2} + \beta_{3} q^{4} + (\beta_{10} + \beta_{5}) q^{7} + (\beta_{14} + \beta_{9} - \beta_{7}) q^{8} + ( - \beta_{12} - \beta_{8}) q^{11} + (\beta_{10} - \beta_{5} + \cdots - \beta_{2}) q^{13}+ \cdots + (2 \beta_{11} + 2 \beta_{9} + 2 \beta_{7}) q^{98}+O(q^{100}) \) q - b9 * q^2 + b3 * q^4 + (b10 + b5) * q^7 + (b14 + b9 - b7) * q^8 + (-b12 - b8) * q^11 + (b10 - b5 + 2b4 - b2) q^13 + (b15 + b12 - b8 + 2b6) q^14 + (b13 - b3 + b1 - 2) * q^16 + (-b14 - b11 - 2b9 + b7) q^17 + (b13 + 2b3 + 1) q^19 + (b10 + 3b4 + b2) q^22 + (b14 - 2b11 + b9) q^23 + (-2b15 + b12 + 2b6) * q^26 + (-b10 - 2b5 + 3b4) * q^28 + (b15 + 2b12 - b8 + 3b6) * q^29 + (-b13 + 2b3 - 4b1 + 3) * q^31 + 4b7 q^32 + (2b3 - b1 + 4) q^34 + (b10 - b5 - b4 - b2) * q^37 + (3b14 + b11 + b9) q^38 + (3b15 + 6b12 - 3b8 + 2b6) * q^41 + (-b10 + 3b5 - 2b4 - 2b2) q^43 + (b15 + 3b12 - 2b8 - b6) * q^44 + (3b13 - b3 + 2) q^46 + (-2b14 - b11 + 3b9) * q^47 + (b13 - b3 - b1 - 2) * q^49 + (-b10 - 2b5 - 5b4 + 2b2) q^52 + (b14 + b11 + 2b9 - 6b7) * q^53 + (-2b15 + b12 + b8 - 3b6) * q^56 + (-2b10 - 3b5 + 2b4) q^58 + (-3b15 + 3b12) * q^59 + (-2b13 + 2b3 + 2b1 + 5) q^61 + (b14 + 3b11 + 3b9 - 8b7) q^62 - 4b1 q^64 + (2b10 + 4b5 - b4 - b2) * q^67 + (2b14 + b11 - b9 - 3b7) * q^68 + (b15 - 4b12 - 3b8) * q^71 + (b10 - b5 - 5b4 - b2) q^73 + (-2b15 - 2b12 + 2b6) q^74 + (2b13 - b3 - 8) q^76 + (-b14 - b11 - 2b9 + b7) q^77 + (-4b3 + 4b1 - 4) * q^79 + (-6b10 - 2b5 + 6b4) q^82 + (-2b14 - b11 + 3b9) * q^83 + (b15 + b12 + 3b8 + 6b6) * q^86 + (-3b10 + b5 + 3b4 + b2) * q^88 + (-2b15 - 4b12 + 2b8 - 4b6) * q^89 + (-b13 + 4b3 - 6b1 + 5) * q^91 + (2b14 + 3b11 - 3b9 + 7b7) * q^92 + (-b13 - 3b3 + 6) q^94 - 3b4 q^97 + (2b11 + 2b9 + 2b7) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 16 q^{16} + 40 q^{34} + 40 q^{46} - 32 q^{49} + 80 q^{61} - 32 q^{64} - 120 q^{76} + 120 q^{94}+O(q^{100}) \) 16 * q - 8 * q^4 - 16 * q^16 + 40 * q^34 + 40 * q^46 - 32 * q^49 + 80 * q^61 - 32 * q^64 - 120 * q^76 + 120 * q^94

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Label	900.2.h.d

Level	$900$
Weight	$2$
Character orbit	900.h
Analytic conductor	$7.187$
Analytic rank	$0$
Dimension	$16$
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{16} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( 2\zeta_{40}^{4} \) 2*v^4
\(\beta_{2}\)	\(=\)	\( 2\zeta_{40}^{6} + 2\zeta_{40}^{2} \) 2v^6 + 2v^2
\(\beta_{3}\)	\(=\)	\( 2\zeta_{40}^{8} \) 2*v^8
\(\beta_{4}\)	\(=\)	\( \zeta_{40}^{10} \) v^10
\(\beta_{5}\)	\(=\)	\( 2\zeta_{40}^{14} \) 2*v^14
\(\beta_{6}\)	\(=\)	\( \zeta_{40}^{15} + \zeta_{40}^{5} \) v^15 + v^5
\(\beta_{7}\)	\(=\)	\( -\zeta_{40}^{15} + \zeta_{40}^{5} \) -v^15 + v^5
\(\beta_{8}\)	\(=\)	\( -2\zeta_{40}^{13} + \zeta_{40}^{11} + \zeta_{40}^{9} + \zeta_{40}^{7} - \zeta_{40}^{5} + \zeta_{40}^{3} + 2\zeta_{40} \) -2v^13 + v^11 + v^9 + v^7 - v^5 + v^3 + 2v
\(\beta_{9}\)	\(=\)	\( -\zeta_{40}^{15} + \zeta_{40}^{11} + \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} \) -v^15 + v^11 + v^9 - v^7 + v^3
\(\beta_{10}\)	\(=\)	\( -2\zeta_{40}^{14} + \zeta_{40}^{10} - 2\zeta_{40}^{6} + 2\zeta_{40}^{2} \) -2v^14 + v^10 - 2v^6 + 2*v^2
\(\beta_{11}\)	\(=\)	\( 2\zeta_{40}^{13} - \zeta_{40}^{11} - \zeta_{40}^{9} + \zeta_{40}^{7} + \zeta_{40}^{5} + \zeta_{40}^{3} \) 2*v^13 - v^11 - v^9 + v^7 + v^5 + v^3
\(\beta_{12}\)	\(=\)	\( -\zeta_{40}^{15} + \zeta_{40}^{11} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} \) -v^15 + v^11 - v^9 - v^7 + v^3
\(\beta_{13}\)	\(=\)	\( 4\zeta_{40}^{12} - 2\zeta_{40}^{8} + 2\zeta_{40}^{4} - 2 \) 4v^12 - 2v^8 + 2*v^4 - 2
\(\beta_{14}\)	\(=\)	\( -2\zeta_{40}^{13} - \zeta_{40}^{11} + \zeta_{40}^{9} - \zeta_{40}^{7} - \zeta_{40}^{5} - \zeta_{40}^{3} + 2\zeta_{40} \) -2v^13 - v^11 + v^9 - v^7 - v^5 - v^3 + 2v
\(\beta_{15}\)	\(=\)	\( -2\zeta_{40}^{13} - \zeta_{40}^{11} + \zeta_{40}^{9} + \zeta_{40}^{7} - \zeta_{40}^{5} + \zeta_{40}^{3} \) -2*v^13 - v^11 + v^9 + v^7 - v^5 + v^3

\(\zeta_{40}\)	\(=\)	\( ( -\beta_{15} + \beta_{14} + \beta_{11} + \beta_{8} ) / 4 \) (-b15 + b14 + b11 + b8) / 4
\(\zeta_{40}^{2}\)	\(=\)	\( ( \beta_{10} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 \) (b10 + b5 - b4 + b2) / 4
\(\zeta_{40}^{3}\)	\(=\)	\( ( \beta_{15} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} ) / 4 \) (b15 + b12 + b11 + b9 - b7 + b6) / 4
\(\zeta_{40}^{4}\)	\(=\)	\( ( \beta_1 ) / 2 \) (b1) / 2
\(\zeta_{40}^{5}\)	\(=\)	\( ( \beta_{7} + \beta_{6} ) / 2 \) (b7 + b6) / 2
\(\zeta_{40}^{6}\)	\(=\)	\( ( -\beta_{10} - \beta_{5} + \beta_{4} + \beta_{2} ) / 4 \) (-b10 - b5 + b4 + b2) / 4
\(\zeta_{40}^{7}\)	\(=\)	\( ( -\beta_{14} - \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} ) / 4 \) (-b14 - b12 - b9 + b8 + b7 - b6) / 4
\(\zeta_{40}^{8}\)	\(=\)	\( ( \beta_{3} ) / 2 \) (b3) / 2
\(\zeta_{40}^{9}\)	\(=\)	\( ( -\beta_{12} + \beta_{9} ) / 2 \) (-b12 + b9) / 2
\(\zeta_{40}^{10}\)	\(=\)	\( \beta_{4} \) b4
\(\zeta_{40}^{11}\)	\(=\)	\( ( -\beta_{15} - \beta_{14} - \beta_{11} + \beta_{8} ) / 4 \) (-b15 - b14 - b11 + b8) / 4
\(\zeta_{40}^{12}\)	\(=\)	\( ( \beta_{13} + \beta_{3} - \beta _1 + 2 ) / 4 \) (b13 + b3 - b1 + 2) / 4
\(\zeta_{40}^{13}\)	\(=\)	\( ( -\beta_{15} - \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} ) / 4 \) (-b15 - b12 + b11 + b9 - b7 - b6) / 4
\(\zeta_{40}^{14}\)	\(=\)	\( ( \beta_{5} ) / 2 \) (b5) / 2
\(\zeta_{40}^{15}\)	\(=\)	\( ( -\beta_{7} + \beta_{6} ) / 2 \) (-b7 + b6) / 2

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 899.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) (T^8 + 2T^6 + 4T^4 + 8*T^2 + 16)^2
$3$	\( T^{16} \) T^16
$5$	\( T^{16} \) T^16
$7$	\( (T^{4} - 10 T^{2} + 5)^{4} \) (T^4 - 10*T^2 + 5)^4
$11$	\( (T^{4} - 20 T^{2} + 20)^{4} \) (T^4 - 20*T^2 + 20)^4
$13$	\( (T^{4} + 42 T^{2} + 361)^{4} \) (T^4 + 42*T^2 + 361)^4
$17$	\( (T^{2} - 10)^{8} \) (T^2 - 10)^8
$19$	\( (T^{4} + 50 T^{2} + 125)^{4} \) (T^4 + 50*T^2 + 125)^4
$23$	\( (T^{4} + 100 T^{2} + 2420)^{4} \) (T^4 + 100*T^2 + 2420)^4
$29$	\( (T^{4} + 36 T^{2} + 4)^{4} \) (T^4 + 36*T^2 + 4)^4
$31$	\( (T^{4} + 130 T^{2} + 1805)^{4} \) (T^4 + 130*T^2 + 1805)^4
$37$	\( (T^{4} + 48 T^{2} + 256)^{4} \) (T^4 + 48*T^2 + 256)^4
$41$	\( (T^{4} + 184 T^{2} + 7744)^{4} \) (T^4 + 184*T^2 + 7744)^4
$43$	\( (T^{4} - 170 T^{2} + 4805)^{4} \) (T^4 - 170*T^2 + 4805)^4
$47$	\( (T^{4} + 100 T^{2} + 2420)^{4} \) (T^4 + 100*T^2 + 2420)^4
$53$	\( (T^{4} - 120 T^{2} + 1600)^{4} \) (T^4 - 120*T^2 + 1600)^4
$59$	\( (T^{4} - 180 T^{2} + 1620)^{4} \) (T^4 - 180*T^2 + 1620)^4
$61$	\( (T^{2} - 10 T - 55)^{8} \) (T^2 - 10*T - 55)^8
$67$	\( (T^{4} - 130 T^{2} + 1805)^{4} \) (T^4 - 130*T^2 + 1805)^4
$71$	\( (T^{4} - 200 T^{2} + 8000)^{4} \) (T^4 - 200*T^2 + 8000)^4
$73$	\( (T^{4} + 112 T^{2} + 256)^{4} \) (T^4 + 112*T^2 + 256)^4
$79$	\( (T^{4} + 160 T^{2} + 1280)^{4} \) (T^4 + 160*T^2 + 1280)^4
$83$	\( (T^{4} + 100 T^{2} + 2420)^{4} \) (T^4 + 100*T^2 + 2420)^4
$89$	\( (T^{4} + 96 T^{2} + 1024)^{4} \) (T^4 + 96*T^2 + 1024)^4
$97$	\( (T^{2} + 9)^{8} \) (T^2 + 9)^8
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\(n\)	\(101\)	\(451\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)