LMFDB - Newform orbit 297.3.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,3,Mod(26,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("297.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	297.m (of order $10$, 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:	$8.09266385150$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.484000000.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 $ x^8 - x^6 + 16x^4 - 66x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-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_{4} + \beta_1) q^{2} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} - \beta_{7} q^{5} + (9 \beta_{5} - 9 \beta_{3} + 7 \beta_{2} + 7) q^{7} + (2 \beta_{7} - 3 \beta_{6} + \cdots + 3 \beta_1) q^{8}+ \cdots + ( - 36 \beta_{7} + 32 \beta_{4} - 36 \beta_1) q^{98}+O(q^{100}) $ q + (-b4 + b1) * q^2 + (b5 - b3 - b2 - 1) * q^4 - b7 * q^5 + (9b5 - 9b3 + 7b2 + 7) q^7 + (2b7 - 3b6 + 2b4 + 3b1) * q^8 + (-2b5 - 2b2 + 3) * q^10 + (-b7 + 2b6 - 6b4 + 2b1) q^11 + (-7b5 + 19b3 - 19b2) q^13 + (2b7 + 9b6 + 2b4 - 9b1) * q^14 + (-8b3 - 7b2 + 8) * q^16 + (2b7 + 15b6) * q^17 + (b5 - 3b3 + 1) q^19 + (2b7 - 2b6 - b4 + 3b1) q^20 + (22b5 - 22b3 + 22b2 + 11) q^22 + (3b7 + 6b4 + 3b1) q^23 + (-2b5 - 20b3 - 2) * q^25 + (-7b6 - 7b4 + 26b1) q^26 + (16b3 - 3b2 - 16) * q^28 + (20b6 + 20b4 - 7b1) q^29 + (-49b5 + 16b3 - 16b2) q^31 + (3b7 - 16b4 + 3b1) q^32 + (49b5 + 49b2 + 9) * q^34 + (2b7 - 2b6 - 9b4 + 11b1) * q^35 + (-24b5 + 24b3 - 11b2 - 11) q^37 + (3b7 + b6) q^38 + (-11b5 + 11b3 - 11b2 - 11) q^40 + (-10b7 + 20b6 - 10b4 - 20b1) * q^41 + (-3b5 - 3b2 - 43) * q^43 + (8b7 + 6b6 + 15b4 - 5b1) * q^44 + (3b5 + 12b3 - 12b2) q^46 + (-22b7 - 13b6 - 22b4 + 13b1) * q^47 + (32b3 + 4b2 - 32) * q^49 + (20b7 - 2b6) * q^50 + (-5b5 - 45b3 - 5) * q^52 + (-9b7 + 9b6 + 29b4 - 38b1) * q^53 + (-11b3 + 22) q^55 + (23b7 + 8b4 + 23b1) q^56 + (-b5 + 74b3 - 1) q^58 + (-13b6 - 13b4 - 22b1) q^59 + (-37b3 + 81b2 + 37) * q^61 + (-49b6 - 49b4 + 65b1) q^62 + (-13b5 - 22b3 + 22b2) q^64 + (-7b7 + 19b4 - 7b1) q^65 + (-48b5 - 48b2 - 61) * q^67 + (11b7 - 11b6 + 32b4 - 21b1) * q^68 + (49b5 - 49b3 + 25b2 + 25) q^70 + (-40b7 + 21b6) * q^71 + (-45b5 + 45b3 - 65b2 - 65) q^73 + (-13b7 - 24b6 - 13b4 + 24b1) * q^74 + (5b5 + 5b2) * q^76 + (-8b7 + 38b6 + 7b4 - 61b1) * q^77 + (-70b5 + 64b3 - 64b2) q^79 + (-8b7 - 15b6 - 8b4 + 15b1) * q^80 + (10b3 + 70b2 - 10) * q^82 + (20b7 - 24b6) * q^83 + (49b5 - 40b3 + 49) * q^85 + (3b7 - 3b6 + 40b4 - 37b1) * q^86 + (33b5 + 11b3 - 55b2 + 11) q^88 + (-12b7 + 12b4 - 12b1) q^89 + (147b5 - 101b3 + 147) * q^91 + (-9b6 - 9b4 - 3b1) q^92 + (-92b3 - 17b2 + 92) * q^94 + (-3b6 - 3b4 + 4b1) q^95 + (44b5 + 25b3 - 25b2) q^97 + (-36b7 + 32b4 - 36b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{4} + 6 q^{7} + 32 q^{10} + 90 q^{13} + 62 q^{16} - 44 q^{22} - 52 q^{25} - 90 q^{28} + 162 q^{31} - 124 q^{34} + 30 q^{37} - 22 q^{40} - 332 q^{43} + 42 q^{46} - 200 q^{49} - 120 q^{52} + 154 q^{55}+ \cdots + 12 q^{97}+O(q^{100}) $ 8 * q - 10 * q^4 + 6 * q^7 + 32 * q^10 + 90 * q^13 + 62 * q^16 - 44 * q^22 - 52 * q^25 - 90 * q^28 + 162 * q^31 - 124 * q^34 + 30 * q^37 - 22 * q^40 - 332 * q^43 + 42 * q^46 - 200 * q^49 - 120 * q^52 + 154 * q^55 + 142 * q^58 + 60 * q^61 - 62 * q^64 - 296 * q^67 - 46 * q^70 - 210 * q^73 - 20 * q^76 + 396 * q^79 - 200 * q^82 + 214 * q^85 + 154 * q^88 + 680 * q^91 + 586 * q^94 + 12 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 $ x^8 - x^6 + 16*x^4 - 66*x^2 + 121 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 $ (-7v^6 - 37v^4 - 629*v^2 - 363) / 1991
$\beta_{3}$	$=$	$ ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 $ (-28v^6 - 148v^4 - 525*v^2 + 539) / 1991
$\beta_{4}$	$=$	$ ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 $ (-28v^7 - 148v^5 - 525v^3 + 539v) / 1991
$\beta_{5}$	$=$	$ ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 $ (40v^6 - 73v^4 + 750*v^2 - 2761) / 1991
$\beta_{6}$	$=$	$ ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 $ (61v^7 + 38v^5 + 646v^3 - 1672v) / 1991
$\beta_{7}$	$=$	$ ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 $ (68v^7 + 75v^5 + 1275v^3 - 3300v) / 1991

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{3} - 4\beta_{2} - 1 $ b3 - 4*b2 - 1
$\nu^{3}$	$=$	$ 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 $ 4b7 - 4b6 + b4 + 3*b1
$\nu^{4}$	$=$	$ -7\beta_{5} - 10\beta_{3} - 7 $ -7b5 - 10b3 - 7
$\nu^{5}$	$=$	$ -7\beta_{7} - 17\beta_{4} - 7\beta_1 $ -7b7 - 17b4 - 7*b1
$\nu^{6}$	$=$	$ 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 $ 37b5 - 37b3 + 75*b2 + 75
$\nu^{7}$	$=$	$ -38\beta_{7} + 75\beta_{6} $ -38b7 + 75b6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$-1$	$-1 - \beta_{2} + \beta_{3} - \beta_{5}$

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} $

26.1

−1.46782	+	0.476925i
1.46782	−	0.476925i
−1.26313	+	1.73855i
1.26313	−	1.73855i
−1.46782	−	0.476925i
1.46782	+	0.476925i
−1.26313	−	1.73855i
1.26313	+	1.73855i

−1.46782

2.02029i

−0.690983

−

2.12663i

−1.46782

−

2.02029i

−3.16312

−

9.73508i

−4.18932

−

1.36119i

6.23607

26.2

1.46782

−

2.02029i

−0.690983

−

2.12663i

1.46782

2.02029i

−3.16312

−

9.73508i

4.18932

1.36119i

6.23607

53.1

−1.26313

−

0.410415i

−1.80902

−

1.31433i

−1.26313

0.410415i

4.66312

3.38795i

4.86822

6.70053i

1.76393

53.2

1.26313

0.410415i

−1.80902

−

1.31433i

1.26313

−

0.410415i

4.66312

3.38795i

−4.86822

−

6.70053i

1.76393

80.1

−1.46782

−

2.02029i

−0.690983

2.12663i

−1.46782

2.02029i

−3.16312

9.73508i

−4.18932

1.36119i

6.23607

80.2

1.46782

2.02029i

−0.690983

2.12663i

1.46782

−

2.02029i

−3.16312

9.73508i

4.18932

−

1.36119i

6.23607

269.1

−1.26313

0.410415i

−1.80902

1.31433i

−1.26313

−

0.410415i

4.66312

−

3.38795i

4.86822

−

6.70053i

1.76393

269.2

1.26313

−

0.410415i

−1.80902

1.31433i

1.26313

0.410415i

4.66312

−

3.38795i

−4.86822

6.70053i

1.76393

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
11.c	even	5	1	inner
33.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.3.m.a	✓	8
3.b	odd	2	1	inner	297.3.m.a	✓	8
11.c	even	5	1	inner	297.3.m.a	✓	8
33.h	odd	10	1	inner	297.3.m.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
297.3.m.a	✓	8	1.a	even	1	1	trivial
297.3.m.a	✓	8	3.b	odd	2	1	inner
297.3.m.a	✓	8	11.c	even	5	1	inner
297.3.m.a	✓	8	33.h	odd	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + T_{2}^{6} + 31T_{2}^{4} - 99T_{2}^{2} + 121 $ T2^8 + T2^6 + 31*T2^4 - 99*T2^2 + 121 acting on $S_{3}^{\mathrm{new}}(297, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + T^{6} + \cdots + 121 $ T^8 + T^6 + 31T^4 - 99T^2 + 121
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + T^{6} + \cdots + 121 $ T^8 + T^6 + 31T^4 - 99T^2 + 121
$7$	$ (T^{4} - 3 T^{3} + \cdots + 3481)^{2} $ (T^4 - 3T^3 + 79T^2 - 767*T + 3481)^2
$11$	$ T^{8} + 209 T^{6} + \cdots + 214358881 $ T^8 + 209T^6 + 27951T^4 + 3059969*T^2 + 214358881
$13$	$ (T^{4} - 45 T^{3} + \cdots + 198025)^{2} $ (T^4 - 45T^3 + 1360T^2 - 22250*T + 198025)^2
$17$	$ T^{8} + \cdots + 480264246121 $ T^8 - 926T^6 + 699856T^4 - 513521151*T^2 + 480264246121
$19$	$ (T^{4} + 10 T^{2} + \cdots + 25)^{2} $ (T^4 + 10T^2 + 25T + 25)^2
$23$	$ (T^{4} + 288 T^{2} + 891)^{2} $ (T^4 + 288*T^2 + 891)^2
$29$	$ T^{8} + \cdots + 7032525875881 $ T^8 + 391T^6 + 6967096T^4 - 11281144314*T^2 + 7032525875881
$31$	$ (T^{4} - 81 T^{3} + \cdots + 1852321)^{2} $ (T^4 - 81T^3 + 2641T^2 - 1361*T + 1852321)^2
$37$	$ (T^{4} - 15 T^{3} + \cdots + 9025)^{2} $ (T^4 - 15T^3 + 940T^2 + 4750*T + 9025)^2
$41$	$ T^{8} + \cdots + 7562500000000 $ T^8 - 500T^6 + 4000000T^4 - 8250000000*T^2 + 7562500000000
$43$	$ (T^{2} + 83 T + 1711)^{4} $ (T^2 + 83*T + 1711)^4
$47$	$ T^{8} + \cdots + 15786405650521 $ T^8 - 6581T^6 + 16297711T^4 + 5042004759*T^2 + 15786405650521
$53$	$ T^{8} + \cdots + 7502181258121 $ T^8 + 4009T^6 + 52888456T^4 - 32391544086*T^2 + 7502181258121
$59$	$ T^{8} + \cdots + 15786405650521 $ T^8 + 4534T^6 + 69386656T^4 - 53840982261*T^2 + 15786405650521
$61$	$ (T^{4} - 30 T^{3} + \cdots + 4818025)^{2} $ (T^4 - 30T^3 + 9040T^2 + 340225*T + 4818025)^2
$67$	$ (T^{2} + 74 T - 1511)^{4} $ (T^2 + 74*T - 1511)^4
$71$	$ T^{8} + \cdots + 19\!\cdots\!21 $ T^8 - 12806T^6 + 69970816T^4 - 61582371411*T^2 + 1932130903032121
$73$	$ (T^{4} + 105 T^{3} + \cdots + 6375625)^{2} $ (T^4 + 105T^3 + 4525T^2 + 12625*T + 6375625)^2
$79$	$ (T^{4} - 198 T^{3} + \cdots + 13512976)^{2} $ (T^4 - 198T^3 + 16804T^2 - 448472*T + 13512976)^2
$83$	$ T^{8} + \cdots + 1033426763776 $ T^8 - 9776T^6 + 36288256T^4 + 3562082304*T^2 + 1033426763776
$89$	$ (T^{4} + 2448 T^{2} + 228096)^{2} $ (T^4 + 2448*T^2 + 228096)^2
$97$	$ (T^{4} - 6 T^{3} + \cdots + 5812921)^{2} $ (T^4 - 6T^3 + 5536T^2 + 286909*T + 5812921)^2
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Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	297.m (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} - \beta_{7} q^{5} + (9 \beta_{5} - 9 \beta_{3} + 7 \beta_{2} + 7) q^{7} + (2 \beta_{7} - 3 \beta_{6} + \cdots + 3 \beta_1) q^{8}+ \cdots + ( - 36 \beta_{7} + 32 \beta_{4} - 36 \beta_1) q^{98}+O(q^{100}) \) q + (-b4 + b1) * q^2 + (b5 - b3 - b2 - 1) * q^4 - b7 * q^5 + (9b5 - 9b3 + 7b2 + 7) q^7 + (2b7 - 3b6 + 2b4 + 3b1) * q^8 + (-2b5 - 2b2 + 3) * q^10 + (-b7 + 2b6 - 6b4 + 2b1) q^11 + (-7b5 + 19b3 - 19b2) q^13 + (2b7 + 9b6 + 2b4 - 9b1) * q^14 + (-8b3 - 7b2 + 8) * q^16 + (2b7 + 15b6) * q^17 + (b5 - 3b3 + 1) q^19 + (2b7 - 2b6 - b4 + 3b1) q^20 + (22b5 - 22b3 + 22b2 + 11) q^22 + (3b7 + 6b4 + 3b1) q^23 + (-2b5 - 20b3 - 2) * q^25 + (-7b6 - 7b4 + 26b1) q^26 + (16b3 - 3b2 - 16) * q^28 + (20b6 + 20b4 - 7b1) q^29 + (-49b5 + 16b3 - 16b2) q^31 + (3b7 - 16b4 + 3b1) q^32 + (49b5 + 49b2 + 9) * q^34 + (2b7 - 2b6 - 9b4 + 11b1) * q^35 + (-24b5 + 24b3 - 11b2 - 11) q^37 + (3b7 + b6) q^38 + (-11b5 + 11b3 - 11b2 - 11) q^40 + (-10b7 + 20b6 - 10b4 - 20b1) * q^41 + (-3b5 - 3b2 - 43) * q^43 + (8b7 + 6b6 + 15b4 - 5b1) * q^44 + (3b5 + 12b3 - 12b2) q^46 + (-22b7 - 13b6 - 22b4 + 13b1) * q^47 + (32b3 + 4b2 - 32) * q^49 + (20b7 - 2b6) * q^50 + (-5b5 - 45b3 - 5) * q^52 + (-9b7 + 9b6 + 29b4 - 38b1) * q^53 + (-11b3 + 22) q^55 + (23b7 + 8b4 + 23b1) q^56 + (-b5 + 74b3 - 1) q^58 + (-13b6 - 13b4 - 22b1) q^59 + (-37b3 + 81b2 + 37) * q^61 + (-49b6 - 49b4 + 65b1) q^62 + (-13b5 - 22b3 + 22b2) q^64 + (-7b7 + 19b4 - 7b1) q^65 + (-48b5 - 48b2 - 61) * q^67 + (11b7 - 11b6 + 32b4 - 21b1) * q^68 + (49b5 - 49b3 + 25b2 + 25) q^70 + (-40b7 + 21b6) * q^71 + (-45b5 + 45b3 - 65b2 - 65) q^73 + (-13b7 - 24b6 - 13b4 + 24b1) * q^74 + (5b5 + 5b2) * q^76 + (-8b7 + 38b6 + 7b4 - 61b1) * q^77 + (-70b5 + 64b3 - 64b2) q^79 + (-8b7 - 15b6 - 8b4 + 15b1) * q^80 + (10b3 + 70b2 - 10) * q^82 + (20b7 - 24b6) * q^83 + (49b5 - 40b3 + 49) * q^85 + (3b7 - 3b6 + 40b4 - 37b1) * q^86 + (33b5 + 11b3 - 55b2 + 11) q^88 + (-12b7 + 12b4 - 12b1) q^89 + (147b5 - 101b3 + 147) * q^91 + (-9b6 - 9b4 - 3b1) q^92 + (-92b3 - 17b2 + 92) * q^94 + (-3b6 - 3b4 + 4b1) q^95 + (44b5 + 25b3 - 25b2) q^97 + (-36b7 + 32b4 - 36b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4} + 6 q^{7} + 32 q^{10} + 90 q^{13} + 62 q^{16} - 44 q^{22} - 52 q^{25} - 90 q^{28} + 162 q^{31} - 124 q^{34} + 30 q^{37} - 22 q^{40} - 332 q^{43} + 42 q^{46} - 200 q^{49} - 120 q^{52} + 154 q^{55}+ \cdots + 12 q^{97}+O(q^{100}) \) 8 * q - 10 * q^4 + 6 * q^7 + 32 * q^10 + 90 * q^13 + 62 * q^16 - 44 * q^22 - 52 * q^25 - 90 * q^28 + 162 * q^31 - 124 * q^34 + 30 * q^37 - 22 * q^40 - 332 * q^43 + 42 * q^46 - 200 * q^49 - 120 * q^52 + 154 * q^55 + 142 * q^58 + 60 * q^61 - 62 * q^64 - 296 * q^67 - 46 * q^70 - 210 * q^73 - 20 * q^76 + 396 * q^79 - 200 * q^82 + 214 * q^85 + 154 * q^88 + 680 * q^91 + 586 * q^94 + 12 * q^97

Introduction

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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	297.3.m.a

Level	$297$
Weight	$3$
Character orbit	297.m
Analytic conductor	$8.093$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(8.09266385150\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	8.0.484000000.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) x^8 - x^6 + 16x^4 - 66x^2 + 121
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 \) (-7v^6 - 37v^4 - 629*v^2 - 363) / 1991
\(\beta_{3}\)	\(=\)	\( ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 \) (-28v^6 - 148v^4 - 525*v^2 + 539) / 1991
\(\beta_{4}\)	\(=\)	\( ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 \) (-28v^7 - 148v^5 - 525v^3 + 539v) / 1991
\(\beta_{5}\)	\(=\)	\( ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 \) (40v^6 - 73v^4 + 750*v^2 - 2761) / 1991
\(\beta_{6}\)	\(=\)	\( ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 \) (61v^7 + 38v^5 + 646v^3 - 1672v) / 1991
\(\beta_{7}\)	\(=\)	\( ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 \) (68v^7 + 75v^5 + 1275v^3 - 3300v) / 1991

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{3} - 4\beta_{2} - 1 \) b3 - 4*b2 - 1
\(\nu^{3}\)	\(=\)	\( 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 \) 4b7 - 4b6 + b4 + 3*b1
\(\nu^{4}\)	\(=\)	\( -7\beta_{5} - 10\beta_{3} - 7 \) -7b5 - 10b3 - 7
\(\nu^{5}\)	\(=\)	\( -7\beta_{7} - 17\beta_{4} - 7\beta_1 \) -7b7 - 17b4 - 7*b1
\(\nu^{6}\)	\(=\)	\( 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 \) 37b5 - 37b3 + 75*b2 + 75
\(\nu^{7}\)	\(=\)	\( -38\beta_{7} + 75\beta_{6} \) -38b7 + 75b6

\(n\)	\(56\)	\(244\)
\(\chi(n)\)	\(-1\)	\(-1 - \beta_{2} + \beta_{3} - \beta_{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 26.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + T^{6} + \cdots + 121 \) T^8 + T^6 + 31T^4 - 99T^2 + 121
$3$	\( T^{8} \) T^8
$5$	\( T^{8} + T^{6} + \cdots + 121 \) T^8 + T^6 + 31T^4 - 99T^2 + 121
$7$	\( (T^{4} - 3 T^{3} + \cdots + 3481)^{2} \) (T^4 - 3T^3 + 79T^2 - 767*T + 3481)^2
$11$	\( T^{8} + 209 T^{6} + \cdots + 214358881 \) T^8 + 209T^6 + 27951T^4 + 3059969*T^2 + 214358881
$13$	\( (T^{4} - 45 T^{3} + \cdots + 198025)^{2} \) (T^4 - 45T^3 + 1360T^2 - 22250*T + 198025)^2
$17$	\( T^{8} + \cdots + 480264246121 \) T^8 - 926T^6 + 699856T^4 - 513521151*T^2 + 480264246121
$19$	\( (T^{4} + 10 T^{2} + \cdots + 25)^{2} \) (T^4 + 10T^2 + 25T + 25)^2
$23$	\( (T^{4} + 288 T^{2} + 891)^{2} \) (T^4 + 288*T^2 + 891)^2
$29$	\( T^{8} + \cdots + 7032525875881 \) T^8 + 391T^6 + 6967096T^4 - 11281144314*T^2 + 7032525875881
$31$	\( (T^{4} - 81 T^{3} + \cdots + 1852321)^{2} \) (T^4 - 81T^3 + 2641T^2 - 1361*T + 1852321)^2
$37$	\( (T^{4} - 15 T^{3} + \cdots + 9025)^{2} \) (T^4 - 15T^3 + 940T^2 + 4750*T + 9025)^2
$41$	\( T^{8} + \cdots + 7562500000000 \) T^8 - 500T^6 + 4000000T^4 - 8250000000*T^2 + 7562500000000
$43$	\( (T^{2} + 83 T + 1711)^{4} \) (T^2 + 83*T + 1711)^4
$47$	\( T^{8} + \cdots + 15786405650521 \) T^8 - 6581T^6 + 16297711T^4 + 5042004759*T^2 + 15786405650521
$53$	\( T^{8} + \cdots + 7502181258121 \) T^8 + 4009T^6 + 52888456T^4 - 32391544086*T^2 + 7502181258121
$59$	\( T^{8} + \cdots + 15786405650521 \) T^8 + 4534T^6 + 69386656T^4 - 53840982261*T^2 + 15786405650521
$61$	\( (T^{4} - 30 T^{3} + \cdots + 4818025)^{2} \) (T^4 - 30T^3 + 9040T^2 + 340225*T + 4818025)^2
$67$	\( (T^{2} + 74 T - 1511)^{4} \) (T^2 + 74*T - 1511)^4
$71$	\( T^{8} + \cdots + 19\!\cdots\!21 \) T^8 - 12806T^6 + 69970816T^4 - 61582371411*T^2 + 1932130903032121
$73$	\( (T^{4} + 105 T^{3} + \cdots + 6375625)^{2} \) (T^4 + 105T^3 + 4525T^2 + 12625*T + 6375625)^2
$79$	\( (T^{4} - 198 T^{3} + \cdots + 13512976)^{2} \) (T^4 - 198T^3 + 16804T^2 - 448472*T + 13512976)^2
$83$	\( T^{8} + \cdots + 1033426763776 \) T^8 - 9776T^6 + 36288256T^4 + 3562082304*T^2 + 1033426763776
$89$	\( (T^{4} + 2448 T^{2} + 228096)^{2} \) (T^4 + 2448*T^2 + 228096)^2
$97$	\( (T^{4} - 6 T^{3} + \cdots + 5812921)^{2} \) (T^4 - 6T^3 + 5536T^2 + 286909*T + 5812921)^2
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