LMFDB - Newform orbit 240.2.o.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,2,Mod(239,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	240.o (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:	$1.91640964851$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) $ q + b2 * q^3 - b1 * q^5 + (b4 - b1 + 1) * q^9	$ q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{11} + \beta_{6} q^{13} + ( - \beta_{7} - \beta_{3}) q^{15} + ( - \beta_{4} - \beta_1) q^{17} - \beta_{3} q^{19} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{23} + ( - \beta_{6} + 1) q^{25} + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{27} + ( - \beta_{4} + \beta_1) q^{29} + \beta_{3} q^{31} + (\beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{33} - \beta_{6} q^{37} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{39} + ( - 2 \beta_{4} + 2 \beta_1) q^{41} + (\beta_{7} + \beta_{5} + 4 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_1 - 4) q^{45} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{47} - 7 q^{49} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{51} + (3 \beta_{4} + 3 \beta_1) q^{53} + ( - \beta_{7} - \beta_{5} + \cdots - 4 \beta_{2}) q^{55}+ \cdots + (\beta_{7} - \beta_{5} + 5 \beta_{3}) q^{99}+O(q^{100}) $ q + b2 * q^3 - b1 * q^5 + (b4 - b1 + 1) * q^9 + (b7 - b5 + b3) * q^11 + b6 * q^13 + (-b7 - b3) * q^15 + (-b4 - b1) * q^17 - b3 * q^19 + (b7 + b5 - 2b2) q^23 + (-b6 + 1) * q^25 + (-b7 - b5 + b2) * q^27 + (-b4 + b1) * q^29 + b3 * q^31 + (b6 + 2b4 + 2b1) * q^33 - b6 * q^37 + (-b7 + b5 + b3) * q^39 + (-2b4 + 2b1) * q^41 + (b7 + b5 + 4b2) q^43 + (-b6 - b1 - 4) * q^45 + (b7 + b5 - 2b2) q^47 - 7 * q^49 + (-b7 + b5 - 2b3) q^51 + (3b4 + 3b1) * q^53 + (-b7 - b5 + 2b3 - 4b2) * q^55 + (-b6 + b4 + b1) * q^57 + (b7 - b5 + b3) * q^59 - 2 * q^61 + (-5b4 + b1) q^65 + (-b7 - b5 - 4b2) q^67 + (-3b4 + 3b1 + 6) * q^69 + (-2b7 + 2b5 - 2b3) q^71 + 2b6 q^73 + (b7 - b5 - b3 + b2) * q^75 - 3b3 q^79 + (2b4 - 2b1 - 7) * q^81 + (b7 + b5 - 2b2) q^83 + (-b6 + 6) * q^85 + (b7 + b5) * q^87 + (2b4 - 2b1) * q^89 + (b6 - b4 - b1) * q^93 + (-2b5 + b3 + 2b2) * q^95 + (b7 - b5 + 5b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{25} - 32 q^{45} - 56 q^{49} - 16 q^{61} + 48 q^{69} - 56 q^{81} + 48 q^{85}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^25 - 32 * q^45 - 56 * q^49 - 16 * q^61 + 48 * q^69 - 56 * q^81 + 48 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} $ -v^6 - v^5 - v^3 + 2*v^2 + v
$\beta_{2}$	$=$	$ \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} $ v^6 - v^5 + v^3 + v
$\beta_{3}$	$=$	$ 4\zeta_{24}^{4} - 2 $ 4*v^4 - 2
$\beta_{4}$	$=$	$ -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} $ -v^6 + v^5 + v^3 + 2*v^2 - v
$\beta_{5}$	$=$	$ 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1 $ 2v^7 - 2v^6 - 2v^5 + 2v^4 - 1
$\beta_{6}$	$=$	$ 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} $ 4v^7 + 2v^5 - 2v^3 + 2v
$\beta_{7}$	$=$	$ -2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1 $ -2v^7 - 2v^6 - 2v^4 + 2v^3 + 2*v + 1

Display $\zeta_{24}^j$ in terms of $\beta_i$

$\zeta_{24}$	$=$	$ ( 4\beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 3\beta_1 ) / 24 $ (4b7 + 3b6 - 2b5 - 3b4 + 3b3 + 4b2 + 3*b1) / 24
$\zeta_{24}^{2}$	$=$	$ ( -\beta_{7} - \beta_{5} + 3\beta_{4} + 2\beta_{2} + 3\beta_1 ) / 12 $ (-b7 - b5 + 3b4 + 2b2 + 3*b1) / 12
$\zeta_{24}^{3}$	$=$	$ ( \beta_{7} + \beta_{5} + 3\beta_{4} + 4\beta_{2} - 3\beta_1 ) / 12 $ (b7 + b5 + 3b4 + 4b2 - 3*b1) / 12
$\zeta_{24}^{4}$	$=$	$ ( \beta_{3} + 2 ) / 4 $ (b3 + 2) / 4
$\zeta_{24}^{5}$	$=$	$ ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} + 3\beta_{4} + 3\beta_{3} - 4\beta_{2} - 3\beta_1 ) / 24 $ (2b7 + 3b6 - 4b5 + 3b4 + 3b3 - 4b2 - 3*b1) / 24
$\zeta_{24}^{6}$	$=$	$ ( -\beta_{7} - \beta_{5} + 2\beta_{2} ) / 6 $ (-b7 - b5 + 2*b2) / 6
$\zeta_{24}^{7}$	$=$	$ ( -2\beta_{7} + 3\beta_{6} + 4\beta_{5} + 3\beta_{4} - 3\beta_{3} + 4\beta_{2} - 3\beta_1 ) / 24 $ (-2b7 + 3b6 + 4b5 + 3b4 - 3b3 + 4b2 - 3*b1) / 24

Display $\beta_i$ in terms of $\zeta_{24}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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} $

239.1

−0.965926	+	0.258819i
0.258819	−	0.965926i
−0.965926	−	0.258819i
0.258819	+	0.965926i
0.965926	−	0.258819i
−0.258819	+	0.965926i
0.965926	+	0.258819i
−0.258819	−	0.965926i

−1.41421

−

1.00000i

−1.73205

1.41421i

1.00000

2.82843i

239.2

−1.41421

−

1.00000i

1.73205

1.41421i

1.00000

2.82843i

239.3

−1.41421

1.00000i

−1.73205

−

1.41421i

1.00000

−

2.82843i

239.4

−1.41421

1.00000i

1.73205

−

1.41421i

1.00000

−

2.82843i

239.5

1.41421

−

1.00000i

−1.73205

−

1.41421i

1.00000

−

2.82843i

239.6

1.41421

−

1.00000i

1.73205

−

1.41421i

1.00000

−

2.82843i

239.7

1.41421

1.00000i

−1.73205

1.41421i

1.00000

2.82843i

239.8

1.41421

1.00000i

1.73205

1.41421i

1.00000

2.82843i

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	240.2.o.b	✓	8
3.b	odd	2	1	inner	240.2.o.b	✓	8
4.b	odd	2	1	inner	240.2.o.b	✓	8
5.b	even	2	1	inner	240.2.o.b	✓	8
5.c	odd	4	1		1200.2.h.j		4
5.c	odd	4	1		1200.2.h.n		4
8.b	even	2	1		960.2.o.d		8
8.d	odd	2	1		960.2.o.d		8
12.b	even	2	1	inner	240.2.o.b	✓	8
15.d	odd	2	1	inner	240.2.o.b	✓	8
15.e	even	4	1		1200.2.h.j		4
15.e	even	4	1		1200.2.h.n		4
20.d	odd	2	1	inner	240.2.o.b	✓	8
20.e	even	4	1		1200.2.h.j		4
20.e	even	4	1		1200.2.h.n		4
24.f	even	2	1		960.2.o.d		8
24.h	odd	2	1		960.2.o.d		8
40.e	odd	2	1		960.2.o.d		8
40.f	even	2	1		960.2.o.d		8
60.h	even	2	1	inner	240.2.o.b	✓	8
60.l	odd	4	1		1200.2.h.j		4
60.l	odd	4	1		1200.2.h.n		4
120.i	odd	2	1		960.2.o.d		8
120.m	even	2	1		960.2.o.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.o.b	✓	8	1.a	even	1	1	trivial
240.2.o.b	✓	8	3.b	odd	2	1	inner
240.2.o.b	✓	8	4.b	odd	2	1	inner
240.2.o.b	✓	8	5.b	even	2	1	inner
240.2.o.b	✓	8	12.b	even	2	1	inner
240.2.o.b	✓	8	15.d	odd	2	1	inner
240.2.o.b	✓	8	20.d	odd	2	1	inner
240.2.o.b	✓	8	60.h	even	2	1	inner
960.2.o.d		8	8.b	even	2	1
960.2.o.d		8	8.d	odd	2	1
960.2.o.d		8	24.f	even	2	1
960.2.o.d		8	24.h	odd	2	1
960.2.o.d		8	40.e	odd	2	1
960.2.o.d		8	40.f	even	2	1
960.2.o.d		8	120.i	odd	2	1
960.2.o.d		8	120.m	even	2	1
1200.2.h.j		4	5.c	odd	4	1
1200.2.h.j		4	15.e	even	4	1
1200.2.h.j		4	20.e	even	4	1
1200.2.h.j		4	60.l	odd	4	1
1200.2.h.n		4	5.c	odd	4	1
1200.2.h.n		4	15.e	even	4	1
1200.2.h.n		4	20.e	even	4	1
1200.2.h.n		4	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} $ T7 acting on $S_{2}^{\mathrm{new}}(240, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} - 2 T^{2} + 9)^{2} $ (T^4 - 2*T^2 + 9)^2
$5$	$ (T^{4} - 2 T^{2} + 25)^{2} $ (T^4 - 2*T^2 + 25)^2
$7$	$ T^{8} $ T^8
$11$	$ (T^{2} - 24)^{4} $ (T^2 - 24)^4
$13$	$ (T^{2} + 24)^{4} $ (T^2 + 24)^4
$17$	$ (T^{2} - 12)^{4} $ (T^2 - 12)^4
$19$	$ (T^{2} + 12)^{4} $ (T^2 + 12)^4
$23$	$ (T^{2} + 36)^{4} $ (T^2 + 36)^4
$29$	$ (T^{2} + 8)^{4} $ (T^2 + 8)^4
$31$	$ (T^{2} + 12)^{4} $ (T^2 + 12)^4
$37$	$ (T^{2} + 24)^{4} $ (T^2 + 24)^4
$41$	$ (T^{2} + 32)^{4} $ (T^2 + 32)^4
$43$	$ (T^{2} - 72)^{4} $ (T^2 - 72)^4
$47$	$ (T^{2} + 36)^{4} $ (T^2 + 36)^4
$53$	$ (T^{2} - 108)^{4} $ (T^2 - 108)^4
$59$	$ (T^{2} - 24)^{4} $ (T^2 - 24)^4
$61$	$ (T + 2)^{8} $ (T + 2)^8
$67$	$ (T^{2} - 72)^{4} $ (T^2 - 72)^4
$71$	$ (T^{2} - 96)^{4} $ (T^2 - 96)^4
$73$	$ (T^{2} + 96)^{4} $ (T^2 + 96)^4
$79$	$ (T^{2} + 108)^{4} $ (T^2 + 108)^4
$83$	$ (T^{2} + 36)^{4} $ (T^2 + 36)^4
$89$	$ (T^{2} + 32)^{4} $ (T^2 + 32)^4
$97$	$ T^{8} $ T^8
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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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \) q + b2 * q^3 - b1 * q^5 + (b4 - b1 + 1) * q^9	\( q + \beta_{2} q^{3} - \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{11} + \beta_{6} q^{13} + ( - \beta_{7} - \beta_{3}) q^{15} + ( - \beta_{4} - \beta_1) q^{17} - \beta_{3} q^{19} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{23} + ( - \beta_{6} + 1) q^{25} + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{27} + ( - \beta_{4} + \beta_1) q^{29} + \beta_{3} q^{31} + (\beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{33} - \beta_{6} q^{37} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{39} + ( - 2 \beta_{4} + 2 \beta_1) q^{41} + (\beta_{7} + \beta_{5} + 4 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_1 - 4) q^{45} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{47} - 7 q^{49} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{51} + (3 \beta_{4} + 3 \beta_1) q^{53} + ( - \beta_{7} - \beta_{5} + \cdots - 4 \beta_{2}) q^{55}+ \cdots + (\beta_{7} - \beta_{5} + 5 \beta_{3}) q^{99}+O(q^{100}) \) q + b2 * q^3 - b1 * q^5 + (b4 - b1 + 1) * q^9 + (b7 - b5 + b3) * q^11 + b6 * q^13 + (-b7 - b3) * q^15 + (-b4 - b1) * q^17 - b3 * q^19 + (b7 + b5 - 2b2) q^23 + (-b6 + 1) * q^25 + (-b7 - b5 + b2) * q^27 + (-b4 + b1) * q^29 + b3 * q^31 + (b6 + 2b4 + 2b1) * q^33 - b6 * q^37 + (-b7 + b5 + b3) * q^39 + (-2b4 + 2b1) * q^41 + (b7 + b5 + 4b2) q^43 + (-b6 - b1 - 4) * q^45 + (b7 + b5 - 2b2) q^47 - 7 * q^49 + (-b7 + b5 - 2b3) q^51 + (3b4 + 3b1) * q^53 + (-b7 - b5 + 2b3 - 4b2) * q^55 + (-b6 + b4 + b1) * q^57 + (b7 - b5 + b3) * q^59 - 2 * q^61 + (-5b4 + b1) q^65 + (-b7 - b5 - 4b2) q^67 + (-3b4 + 3b1 + 6) * q^69 + (-2b7 + 2b5 - 2b3) q^71 + 2b6 q^73 + (b7 - b5 - b3 + b2) * q^75 - 3b3 q^79 + (2b4 - 2b1 - 7) * q^81 + (b7 + b5 - 2b2) q^83 + (-b6 + 6) * q^85 + (b7 + b5) * q^87 + (2b4 - 2b1) * q^89 + (b6 - b4 - b1) * q^93 + (-2b5 + b3 + 2b2) * q^95 + (b7 - b5 + 5b3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{25} - 32 q^{45} - 56 q^{49} - 16 q^{61} + 48 q^{69} - 56 q^{81} + 48 q^{85}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^25 - 32 * q^45 - 56 * q^49 - 16 * q^61 + 48 * q^69 - 56 * q^81 + 48 * q^85

Introduction

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Varieties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

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	240.2.o.b

Level	$240$
Weight	$2$
Character orbit	240.o
Analytic conductor	$1.916$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{10}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( -\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) -v^6 - v^5 - v^3 + 2*v^2 + v
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) v^6 - v^5 + v^3 + v
\(\beta_{3}\)	\(=\)	\( 4\zeta_{24}^{4} - 2 \) 4*v^4 - 2
\(\beta_{4}\)	\(=\)	\( -\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) -v^6 + v^5 + v^3 + 2*v^2 - v
\(\beta_{5}\)	\(=\)	\( 2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1 \) 2v^7 - 2v^6 - 2v^5 + 2v^4 - 1
\(\beta_{6}\)	\(=\)	\( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) 4v^7 + 2v^5 - 2v^3 + 2v
\(\beta_{7}\)	\(=\)	\( -2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1 \) -2v^7 - 2v^6 - 2v^4 + 2v^3 + 2*v + 1

\(\zeta_{24}\)	\(=\)	\( ( 4\beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 3\beta_1 ) / 24 \) (4b7 + 3b6 - 2b5 - 3b4 + 3b3 + 4b2 + 3*b1) / 24
\(\zeta_{24}^{2}\)	\(=\)	\( ( -\beta_{7} - \beta_{5} + 3\beta_{4} + 2\beta_{2} + 3\beta_1 ) / 12 \) (-b7 - b5 + 3b4 + 2b2 + 3*b1) / 12
\(\zeta_{24}^{3}\)	\(=\)	\( ( \beta_{7} + \beta_{5} + 3\beta_{4} + 4\beta_{2} - 3\beta_1 ) / 12 \) (b7 + b5 + 3b4 + 4b2 - 3*b1) / 12
\(\zeta_{24}^{4}\)	\(=\)	\( ( \beta_{3} + 2 ) / 4 \) (b3 + 2) / 4
\(\zeta_{24}^{5}\)	\(=\)	\( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} + 3\beta_{4} + 3\beta_{3} - 4\beta_{2} - 3\beta_1 ) / 24 \) (2b7 + 3b6 - 4b5 + 3b4 + 3b3 - 4b2 - 3*b1) / 24
\(\zeta_{24}^{6}\)	\(=\)	\( ( -\beta_{7} - \beta_{5} + 2\beta_{2} ) / 6 \) (-b7 - b5 + 2*b2) / 6
\(\zeta_{24}^{7}\)	\(=\)	\( ( -2\beta_{7} + 3\beta_{6} + 4\beta_{5} + 3\beta_{4} - 3\beta_{3} + 4\beta_{2} - 3\beta_1 ) / 24 \) (-2b7 + 3b6 + 4b5 + 3b4 - 3b3 + 4b2 - 3*b1) / 24

\(n\)	\(31\)	\(97\)	\(161\)	\(181\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 239.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} - 2 T^{2} + 9)^{2} \) (T^4 - 2*T^2 + 9)^2
$5$	\( (T^{4} - 2 T^{2} + 25)^{2} \) (T^4 - 2*T^2 + 25)^2
$7$	\( T^{8} \) T^8
$11$	\( (T^{2} - 24)^{4} \) (T^2 - 24)^4
$13$	\( (T^{2} + 24)^{4} \) (T^2 + 24)^4
$17$	\( (T^{2} - 12)^{4} \) (T^2 - 12)^4
$19$	\( (T^{2} + 12)^{4} \) (T^2 + 12)^4
$23$	\( (T^{2} + 36)^{4} \) (T^2 + 36)^4
$29$	\( (T^{2} + 8)^{4} \) (T^2 + 8)^4
$31$	\( (T^{2} + 12)^{4} \) (T^2 + 12)^4
$37$	\( (T^{2} + 24)^{4} \) (T^2 + 24)^4
$41$	\( (T^{2} + 32)^{4} \) (T^2 + 32)^4
$43$	\( (T^{2} - 72)^{4} \) (T^2 - 72)^4
$47$	\( (T^{2} + 36)^{4} \) (T^2 + 36)^4
$53$	\( (T^{2} - 108)^{4} \) (T^2 - 108)^4
$59$	\( (T^{2} - 24)^{4} \) (T^2 - 24)^4
$61$	\( (T + 2)^{8} \) (T + 2)^8
$67$	\( (T^{2} - 72)^{4} \) (T^2 - 72)^4
$71$	\( (T^{2} - 96)^{4} \) (T^2 - 96)^4
$73$	\( (T^{2} + 96)^{4} \) (T^2 + 96)^4
$79$	\( (T^{2} + 108)^{4} \) (T^2 + 108)^4
$83$	\( (T^{2} + 36)^{4} \) (T^2 + 36)^4
$89$	\( (T^{2} + 32)^{4} \) (T^2 + 32)^4
$97$	\( T^{8} \) T^8
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