LMFDB - Newform orbit 841.2.b.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [841,2,Mod(840,841)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("841.840"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
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 $1,\beta_1,\ldots,\beta_{5}$ 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_{3} q^{3} + \beta_{2} q^{4} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{5} + (\beta_{4} - 1) q^{6} + ( - \beta_{4} + \beta_{2} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{4} + 1) q^{9}+ \cdots + ( - 5 \beta_{5} - \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^2 + b3 * q^3 + b2 * q^4 + (-2b4 - b2 + 2) q^5 + (b4 - 1) * q^6 + (-b4 + b2 - 1) * q^7 + (b3 + b1) * q^8 + (b4 + 1) * q^9 + (-2b5 + b3 + b1) q^10 + (-2b5 - 2b3 - b1) * q^11 + (b5 + b3) * q^12 + (b4 + 4b2 - 2) q^13 + (-b5 + 2b3 - 3b1) * q^14 + (-3b5 + b3 + 2b1) * q^15 + (b4 + 3b2 - 3) q^16 + (-2b5 - 2b3) * q^17 + (b5 - b3 + 2b1) q^18 + (2b5 + b3 - 2b1) * q^19 + (-b4 - b2 + 1) * q^20 + b1 * q^21 + (-b2 + 4) * q^22 + (2b4 + 3b2 - 4) * q^23 + (2b4 - 3) q^24 + (-7b4 - 3b2 + 4) * q^25 + (b5 + 3b3 - 5b1) * q^26 + (b5 + 4b3 - b1) q^27 + (b4 - b2 + 2) * q^28 + (4b4 + 2b2 - 5) * q^30 + (5b5 + b3 + b1) q^31 + (b5 + 4b3 - 3b1) * q^32 + (-b4 + 2b2 + 3) q^33 + 2 * q^34 + (-b4 - b2 + 2) * q^35 + (2b2 - 1) q^36 + (-2b5 - 2b3 - b1) * q^37 + (-b4 - 2b2 + 3) q^38 + (5b5 + 2b3 - b1) * q^39 + (-5b5 + 2b3 + 3b1) q^40 + (4b5 - 2b1) * q^41 + (b2 - 2) * q^42 + (b5 + 3b3 - 3b1) * q^43 + (-4b5 - 5b3 + 3b1) q^44 - q^45 + (2b5 + b3 - 5b1) * q^46 + (-6b5 - b1) q^47 + (4b5 - b1) q^48 + (3b4 - 4b2 - 1) * q^49 + (-7b5 + 4b3) * q^50 + (2b2 + 2) q^51 + (4b4 + 3b2 + 3) * q^52 + (-2b4 - b2 - 2) q^53 + (3b4 - b2 - 2) q^54 + (6b5 - b3 - 3b1) * q^55 + (-b5 + 2b3 - 2b1) * q^56 + (-3b4 - 2b2 + 2) * q^57 + (-2b4 - 2b2 - 8) * q^59 + (-2b5 + b1) q^60 + (2b5 - b3 + 2b1) * q^61 + (-4b4 + b2 - 3) q^62 + (-2b4 + 3b2 - 4) * q^63 + (5b4 + 3b2 - 4) * q^64 + (2b4 - b2 - 3) q^65 + (-b5 + 3b3) q^66 + (7b4 + 4b2 - 8) * q^67 + (-4b5 - 4b3 + 2b1) q^68 + (5b5 - b3 - 2b1) * q^69 + (-b5 + 2b1) q^70 + (3b4 + 6) q^71 + (2b5 + b1) q^72 + (5b5 + 3b3 - 7b1) q^73 + (-b2 + 4) * q^74 + (-10b5 + b3 + 7b1) * q^75 + (3b5 + b3) q^76 + (b5 - 4b3 + 5b1) * q^77 + (-3b4 - b2) q^78 + (6b5 - 3b1) * q^79 + (5b4 + b2 - 6) q^80 + (5b4 - b2 - 3) q^81 + (-4b4 - 2b2 + 4) * q^82 + (b4 + 5b2 - 5) q^83 + (b3 - b1) * q^84 + (4b5 - 2b1) * q^85 + (2b4 - 3b2 + 3) * q^86 + (-b4 + b2 + 7) * q^88 + (-b5 + 5b3 + b1) q^89 - b1 * q^90 + (5b4 - 4b2 + 7) * q^91 + (3b4 + b2 + 1) q^92 + (-3b4 - 5b2 + 2) * q^93 + (6b4 - b2 + 2) q^94 + (3b5 - 3b3 - 2b1) q^95 + (-b2 - 4) * q^96 + (-3b3 + 8b1) * q^97 + (3b5 - 7b3 + 6b1) q^98 + (-5b5 - b3 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} + 8 q^{9} - 2 q^{13} - 10 q^{16} + 2 q^{20} + 22 q^{22} - 14 q^{23} - 14 q^{24} + 4 q^{25} + 12 q^{28} - 18 q^{30} + 20 q^{33} + 12 q^{34} + 8 q^{35} - 2 q^{36}+ \cdots - 26 q^{96}+O(q^{100}) $ 6 * q + 2 * q^4 + 6 * q^5 - 4 * q^6 - 6 * q^7 + 8 * q^9 - 2 * q^13 - 10 * q^16 + 2 * q^20 + 22 * q^22 - 14 * q^23 - 14 * q^24 + 4 * q^25 + 12 * q^28 - 18 * q^30 + 20 * q^33 + 12 * q^34 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 10 * q^42 - 6 * q^45 - 8 * q^49 + 16 * q^51 + 32 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^57 - 56 * q^59 - 24 * q^62 - 22 * q^63 - 8 * q^64 - 16 * q^65 - 26 * q^67 + 42 * q^71 + 22 * q^74 - 8 * q^78 - 24 * q^80 - 10 * q^81 + 12 * q^82 - 18 * q^83 + 16 * q^86 + 42 * q^88 + 44 * q^91 + 14 * q^92 - 4 * q^93 + 22 * q^94 - 26 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5*x^4 + 6*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 2 $ v^2 + 2
$\beta_{3}$	$=$	$ \nu^{3} + 3\nu $ v^3 + 3*v
$\beta_{4}$	$=$	$ \nu^{4} + 3\nu^{2} + 1 $ v^4 + 3*v^2 + 1
$\beta_{5}$	$=$	$ \nu^{5} + 4\nu^{3} + 3\nu $ v^5 + 4v^3 + 3v

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 2 $ b2 - 2
$\nu^{3}$	$=$	$ \beta_{3} - 3\beta_1 $ b3 - 3*b1
$\nu^{4}$	$=$	$ \beta_{4} - 3\beta_{2} + 5 $ b4 - 3*b2 + 5
$\nu^{5}$	$=$	$ \beta_{5} - 4\beta_{3} + 9\beta_1 $ b5 - 4b3 + 9b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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} $

840.1

	−	1.80194i
	−	1.24698i
	−	0.445042i
		0.445042i
		1.24698i
		1.80194i

−

1.80194i

0.445042i

−1.24698

−0.356896

0.801938

−4.04892

−

1.35690i

2.80194

0.643104i

840.2

−

1.24698i

−

1.80194i

0.445042

4.04892

−2.24698

0.692021

−

3.04892i

−0.246980

−

5.04892i

840.3

−

0.445042i

−

1.24698i

1.80194

−0.692021

−0.554958

0.356896

−

1.69202i

1.44504

0.307979i

840.4

0.445042i

1.24698i

1.80194

−0.692021

−0.554958

0.356896

1.69202i

1.44504

−

0.307979i

840.5

1.24698i

1.80194i

0.445042

4.04892

−2.24698

0.692021

3.04892i

−0.246980

5.04892i

840.6

1.80194i

−

0.445042i

−1.24698

−0.356896

0.801938

−4.04892

1.35690i

2.80194

−

0.643104i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	841.2.b.c		6
29.b	even	2	1	inner	841.2.b.c		6
29.c	odd	4	1		841.2.a.e		3
29.c	odd	4	1		841.2.a.f		3
29.d	even	7	2		841.2.e.b		12
29.d	even	7	2		841.2.e.c		12
29.d	even	7	2		841.2.e.d		12
29.e	even	14	2		841.2.e.b		12
29.e	even	14	2		841.2.e.c		12
29.e	even	14	2		841.2.e.d		12
29.f	odd	28	2		29.2.d.a	✓	6
29.f	odd	28	2		841.2.d.a		6
29.f	odd	28	2		841.2.d.b		6
29.f	odd	28	2		841.2.d.c		6
29.f	odd	28	2		841.2.d.d		6
29.f	odd	28	2		841.2.d.e		6
87.f	even	4	1		7569.2.a.p		3
87.f	even	4	1		7569.2.a.r		3
87.k	even	28	2		261.2.k.a		6
116.l	even	28	2		464.2.u.f		6
145.o	even	28	2		725.2.r.b		12
145.s	odd	28	2		725.2.l.b		6
145.t	even	28	2		725.2.r.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.2.d.a	✓	6	29.f	odd	28	2
261.2.k.a		6	87.k	even	28	2
464.2.u.f		6	116.l	even	28	2
725.2.l.b		6	145.s	odd	28	2
725.2.r.b		12	145.o	even	28	2
725.2.r.b		12	145.t	even	28	2
841.2.a.e		3	29.c	odd	4	1
841.2.a.f		3	29.c	odd	4	1
841.2.b.c		6	1.a	even	1	1	trivial
841.2.b.c		6	29.b	even	2	1	inner
841.2.d.a		6	29.f	odd	28	2
841.2.d.b		6	29.f	odd	28	2
841.2.d.c		6	29.f	odd	28	2
841.2.d.d		6	29.f	odd	28	2
841.2.d.e		6	29.f	odd	28	2
841.2.e.b		12	29.d	even	7	2
841.2.e.b		12	29.e	even	14	2
841.2.e.c		12	29.d	even	7	2
841.2.e.c		12	29.e	even	14	2
841.2.e.d		12	29.d	even	7	2
841.2.e.d		12	29.e	even	14	2
7569.2.a.p		3	87.f	even	4	1
7569.2.a.r		3	87.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{2} + 1 $ T2^6 + 5*T2^4 + 6*T2^2 + 1 acting on $S_{2}^{\mathrm{new}}(841, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 5 T^{4} + \cdots + 1 $ T^6 + 5T^4 + 6T^2 + 1
$3$	$ T^{6} + 5 T^{4} + \cdots + 1 $ T^6 + 5T^4 + 6T^2 + 1
$5$	$ (T^{3} - 3 T^{2} - 4 T - 1)^{2} $ (T^3 - 3T^2 - 4T - 1)^2
$7$	$ (T^{3} + 3 T^{2} - 4 T + 1)^{2} $ (T^3 + 3T^2 - 4T + 1)^2
$11$	$ T^{6} + 41 T^{4} + \cdots + 1681 $ T^6 + 41T^4 + 474T^2 + 1681
$13$	$ (T^{3} + T^{2} - 30 T - 43)^{2} $ (T^3 + T^2 - 30*T - 43)^2
$17$	$ T^{6} + 24 T^{4} + \cdots + 64 $ T^6 + 24T^4 + 80T^2 + 64
$19$	$ T^{6} + 17 T^{4} + \cdots + 169 $ T^6 + 17T^4 + 94T^2 + 169
$23$	$ (T^{3} + 7 T^{2} - 49)^{2} $ (T^3 + 7*T^2 - 49)^2
$29$	$ T^{6} $ T^6
$31$	$ T^{6} + 89 T^{4} + \cdots + 6889 $ T^6 + 89T^4 + 2134T^2 + 6889
$37$	$ T^{6} + 41 T^{4} + \cdots + 1681 $ T^6 + 41T^4 + 474T^2 + 1681
$41$	$ T^{6} + 52 T^{4} + \cdots + 64 $ T^6 + 52T^4 + 416T^2 + 64
$43$	$ T^{6} + 45 T^{4} + \cdots + 169 $ T^6 + 45T^4 + 402T^2 + 169
$47$	$ T^{6} + 125 T^{4} + \cdots + 57121 $ T^6 + 125T^4 + 4842T^2 + 57121
$53$	$ (T^{3} + 9 T^{2} + 20 T - 1)^{2} $ (T^3 + 9T^2 + 20T - 1)^2
$59$	$ (T^{3} + 28 T^{2} + \cdots + 728)^{2} $ (T^3 + 28T^2 + 252T + 728)^2
$61$	$ T^{6} + 41 T^{4} + \cdots + 169 $ T^6 + 41T^4 + 166T^2 + 169
$67$	$ (T^{3} + 13 T^{2} + \cdots - 13)^{2} $ (T^3 + 13T^2 - 30T - 13)^2
$71$	$ (T^{3} - 21 T^{2} + \cdots - 189)^{2} $ (T^3 - 21T^2 + 126T - 189)^2
$73$	$ T^{6} + 181 T^{4} + \cdots + 175561 $ T^6 + 181T^4 + 10274T^2 + 175561
$79$	$ T^{6} + 117 T^{4} + \cdots + 729 $ T^6 + 117T^4 + 2106T^2 + 729
$83$	$ (T^{3} + 9 T^{2} + \cdots - 169)^{2} $ (T^3 + 9T^2 - 22T - 169)^2
$89$	$ T^{6} + 161 T^{4} + \cdots + 8281 $ T^6 + 161T^4 + 4410T^2 + 8281
$97$	$ T^{6} + 269 T^{4} + \cdots + 169 $ T^6 + 269T^4 + 5190T^2 + 169
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Level:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{5} + (\beta_{4} - 1) q^{6} + ( - \beta_{4} + \beta_{2} - 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{4} + 1) q^{9}+ \cdots + ( - 5 \beta_{5} - \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^2 + b3 * q^3 + b2 * q^4 + (-2b4 - b2 + 2) q^5 + (b4 - 1) * q^6 + (-b4 + b2 - 1) * q^7 + (b3 + b1) * q^8 + (b4 + 1) * q^9 + (-2b5 + b3 + b1) q^10 + (-2b5 - 2b3 - b1) * q^11 + (b5 + b3) * q^12 + (b4 + 4b2 - 2) q^13 + (-b5 + 2b3 - 3b1) * q^14 + (-3b5 + b3 + 2b1) * q^15 + (b4 + 3b2 - 3) q^16 + (-2b5 - 2b3) * q^17 + (b5 - b3 + 2b1) q^18 + (2b5 + b3 - 2b1) * q^19 + (-b4 - b2 + 1) * q^20 + b1 * q^21 + (-b2 + 4) * q^22 + (2b4 + 3b2 - 4) * q^23 + (2b4 - 3) q^24 + (-7b4 - 3b2 + 4) * q^25 + (b5 + 3b3 - 5b1) * q^26 + (b5 + 4b3 - b1) q^27 + (b4 - b2 + 2) * q^28 + (4b4 + 2b2 - 5) * q^30 + (5b5 + b3 + b1) q^31 + (b5 + 4b3 - 3b1) * q^32 + (-b4 + 2b2 + 3) q^33 + 2 * q^34 + (-b4 - b2 + 2) * q^35 + (2b2 - 1) q^36 + (-2b5 - 2b3 - b1) * q^37 + (-b4 - 2b2 + 3) q^38 + (5b5 + 2b3 - b1) * q^39 + (-5b5 + 2b3 + 3b1) q^40 + (4b5 - 2b1) * q^41 + (b2 - 2) * q^42 + (b5 + 3b3 - 3b1) * q^43 + (-4b5 - 5b3 + 3b1) q^44 - q^45 + (2b5 + b3 - 5b1) * q^46 + (-6b5 - b1) q^47 + (4b5 - b1) q^48 + (3b4 - 4b2 - 1) * q^49 + (-7b5 + 4b3) * q^50 + (2b2 + 2) q^51 + (4b4 + 3b2 + 3) * q^52 + (-2b4 - b2 - 2) q^53 + (3b4 - b2 - 2) q^54 + (6b5 - b3 - 3b1) * q^55 + (-b5 + 2b3 - 2b1) * q^56 + (-3b4 - 2b2 + 2) * q^57 + (-2b4 - 2b2 - 8) * q^59 + (-2b5 + b1) q^60 + (2b5 - b3 + 2b1) * q^61 + (-4b4 + b2 - 3) q^62 + (-2b4 + 3b2 - 4) * q^63 + (5b4 + 3b2 - 4) * q^64 + (2b4 - b2 - 3) q^65 + (-b5 + 3b3) q^66 + (7b4 + 4b2 - 8) * q^67 + (-4b5 - 4b3 + 2b1) q^68 + (5b5 - b3 - 2b1) * q^69 + (-b5 + 2b1) q^70 + (3b4 + 6) q^71 + (2b5 + b1) q^72 + (5b5 + 3b3 - 7b1) q^73 + (-b2 + 4) * q^74 + (-10b5 + b3 + 7b1) * q^75 + (3b5 + b3) q^76 + (b5 - 4b3 + 5b1) * q^77 + (-3b4 - b2) q^78 + (6b5 - 3b1) * q^79 + (5b4 + b2 - 6) q^80 + (5b4 - b2 - 3) q^81 + (-4b4 - 2b2 + 4) * q^82 + (b4 + 5b2 - 5) q^83 + (b3 - b1) * q^84 + (4b5 - 2b1) * q^85 + (2b4 - 3b2 + 3) * q^86 + (-b4 + b2 + 7) * q^88 + (-b5 + 5b3 + b1) q^89 - b1 * q^90 + (5b4 - 4b2 + 7) * q^91 + (3b4 + b2 + 1) q^92 + (-3b4 - 5b2 + 2) * q^93 + (6b4 - b2 + 2) q^94 + (3b5 - 3b3 - 2b1) q^95 + (-b2 - 4) * q^96 + (-3b3 + 8b1) * q^97 + (3b5 - 7b3 + 6b1) q^98 + (-5b5 - b3 - 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{7} + 8 q^{9} - 2 q^{13} - 10 q^{16} + 2 q^{20} + 22 q^{22} - 14 q^{23} - 14 q^{24} + 4 q^{25} + 12 q^{28} - 18 q^{30} + 20 q^{33} + 12 q^{34} + 8 q^{35} - 2 q^{36}+ \cdots - 26 q^{96}+O(q^{100}) \) 6 * q + 2 * q^4 + 6 * q^5 - 4 * q^6 - 6 * q^7 + 8 * q^9 - 2 * q^13 - 10 * q^16 + 2 * q^20 + 22 * q^22 - 14 * q^23 - 14 * q^24 + 4 * q^25 + 12 * q^28 - 18 * q^30 + 20 * q^33 + 12 * q^34 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 10 * q^42 - 6 * q^45 - 8 * q^49 + 16 * q^51 + 32 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^57 - 56 * q^59 - 24 * q^62 - 22 * q^63 - 8 * q^64 - 16 * q^65 - 26 * q^67 + 42 * q^71 + 22 * q^74 - 8 * q^78 - 24 * q^80 - 10 * q^81 + 12 * q^82 - 18 * q^83 + 16 * q^86 + 42 * q^88 + 44 * q^91 + 14 * q^92 - 4 * q^93 + 22 * q^94 - 26 * q^96

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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	841.2.b.c

Level	$841$
Weight	$2$
Character orbit	841.b
Analytic conductor	$6.715$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.71541880999\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} + 5x^{4} + 6x^{2} + 1 \) x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 2 \) v^2 + 2
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 3\nu \) v^3 + 3*v
\(\beta_{4}\)	\(=\)	\( \nu^{4} + 3\nu^{2} + 1 \) v^4 + 3*v^2 + 1
\(\beta_{5}\)	\(=\)	\( \nu^{5} + 4\nu^{3} + 3\nu \) v^5 + 4v^3 + 3v

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} - 2 \) b2 - 2
\(\nu^{3}\)	\(=\)	\( \beta_{3} - 3\beta_1 \) b3 - 3*b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} - 3\beta_{2} + 5 \) b4 - 3*b2 + 5
\(\nu^{5}\)	\(=\)	\( \beta_{5} - 4\beta_{3} + 9\beta_1 \) b5 - 4b3 + 9b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 840.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + 5 T^{4} + \cdots + 1 \) T^6 + 5T^4 + 6T^2 + 1
$3$	\( T^{6} + 5 T^{4} + \cdots + 1 \) T^6 + 5T^4 + 6T^2 + 1
$5$	\( (T^{3} - 3 T^{2} - 4 T - 1)^{2} \) (T^3 - 3T^2 - 4T - 1)^2
$7$	\( (T^{3} + 3 T^{2} - 4 T + 1)^{2} \) (T^3 + 3T^2 - 4T + 1)^2
$11$	\( T^{6} + 41 T^{4} + \cdots + 1681 \) T^6 + 41T^4 + 474T^2 + 1681
$13$	\( (T^{3} + T^{2} - 30 T - 43)^{2} \) (T^3 + T^2 - 30*T - 43)^2
$17$	\( T^{6} + 24 T^{4} + \cdots + 64 \) T^6 + 24T^4 + 80T^2 + 64
$19$	\( T^{6} + 17 T^{4} + \cdots + 169 \) T^6 + 17T^4 + 94T^2 + 169
$23$	\( (T^{3} + 7 T^{2} - 49)^{2} \) (T^3 + 7*T^2 - 49)^2
$29$	\( T^{6} \) T^6
$31$	\( T^{6} + 89 T^{4} + \cdots + 6889 \) T^6 + 89T^4 + 2134T^2 + 6889
$37$	\( T^{6} + 41 T^{4} + \cdots + 1681 \) T^6 + 41T^4 + 474T^2 + 1681
$41$	\( T^{6} + 52 T^{4} + \cdots + 64 \) T^6 + 52T^4 + 416T^2 + 64
$43$	\( T^{6} + 45 T^{4} + \cdots + 169 \) T^6 + 45T^4 + 402T^2 + 169
$47$	\( T^{6} + 125 T^{4} + \cdots + 57121 \) T^6 + 125T^4 + 4842T^2 + 57121
$53$	\( (T^{3} + 9 T^{2} + 20 T - 1)^{2} \) (T^3 + 9T^2 + 20T - 1)^2
$59$	\( (T^{3} + 28 T^{2} + \cdots + 728)^{2} \) (T^3 + 28T^2 + 252T + 728)^2
$61$	\( T^{6} + 41 T^{4} + \cdots + 169 \) T^6 + 41T^4 + 166T^2 + 169
$67$	\( (T^{3} + 13 T^{2} + \cdots - 13)^{2} \) (T^3 + 13T^2 - 30T - 13)^2
$71$	\( (T^{3} - 21 T^{2} + \cdots - 189)^{2} \) (T^3 - 21T^2 + 126T - 189)^2
$73$	\( T^{6} + 181 T^{4} + \cdots + 175561 \) T^6 + 181T^4 + 10274T^2 + 175561
$79$	\( T^{6} + 117 T^{4} + \cdots + 729 \) T^6 + 117T^4 + 2106T^2 + 729
$83$	\( (T^{3} + 9 T^{2} + \cdots - 169)^{2} \) (T^3 + 9T^2 - 22T - 169)^2
$89$	\( T^{6} + 161 T^{4} + \cdots + 8281 \) T^6 + 161T^4 + 4410T^2 + 8281
$97$	\( T^{6} + 269 T^{4} + \cdots + 169 \) T^6 + 269T^4 + 5190T^2 + 169
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\(n\)	\(2\)
\(\chi(n)\)	\(-1\)