LMFDB - Newform orbit 693.2.i.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(100,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("693.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.i (of order $3$, degree $2$, 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:	$5.53363286007$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity $\zeta_{18}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (\zeta_{18}^{5} - \zeta_{18}^{4} - 1) q^{8}+O(q^{10}) $ q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (-z^4 + 2z^3 - z^2) q^5 + (2z^4 - 3z) * q^7 + (z^5 - z^4 - 1) * q^8	$ q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + \cdots + 5) q^{98}+O(q^{100}) $ q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (-z^4 + 2z^3 - z^2) q^5 + (2z^4 - 3z) * q^7 + (z^5 - z^4 - 1) * q^8 + (z^3 - 3z^2 + 3z - 1) * q^10 + (-z^3 + 1) * q^11 + (2z^5 - z^4 - z^2 - z + 1) q^13 + (3z^5 - 2z^3 - z^2 + 3) * q^14 + (-z^5 - 2z^4 + 2z^3 - 2z^2 - z) q^16 + (-4z^5 - 4z^4 + z^3 + z^2 + 3z - 1) q^17 + (-z^5 - z^4 - 3z^3 - z^2 - z) q^19 + (-2z^5 + z^4 + z^2 + z - 2) q^20 + (z^5 - z^4) * q^22 + (-5z^5 + 3z^4 + 3z^2 - 5z) * q^23 + (-3z^5 - 3z^4 + z^3 - z^2 + 4z - 1) q^25 + (2z^5 - 3z^4 + 3z^3 - 3z^2 + 2z) q^26 + (3z^5 + 3z^3 - z^2 - 1) * q^28 + (-z^5 + 4z^4 - 3z^2 - 3z + 1) q^29 + (z^5 + z^4 + 3z^3 - z - 3) q^31 + (z^5 + z^4 + 3z^3 - 3z^2 + 2z - 3) q^32 + (-4z^5 + 5z^4 - z^2 - z - 2) * q^34 + (z^5 - 2z^4 + z^3 + 2z^2 - 4z + 2) q^35 + (-4z^5 + 2z^4 + 2z^2 - 4z) * q^37 + (z^5 + z^4 + z^2 - 2z) q^38 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^40 + (2z^5 + z^4 - 3z^2 - 3z - 3) q^41 + (-2z^5 + z^4 + z^2 + z) q^43 + (-z^4 - z^2) * q^44 + (5z^5 + 5z^4 - 8z^3 - 2z^2 - 3z + 8) q^46 + (3z^5 + 3z^4 - z^3 + 3z^2 + 3z) * q^47 + (-8z^5 + 5z^2) * q^49 + (-5z^5 + 4z^4 + z^2 + z - 5) * q^50 + (3z^3 - 2z^2 + 2z - 3) q^52 + (-3z^5 - 3z^4 - 3z^3 - 3z^2 + 6z + 3) q^53 + (z^5 - z^2 - z + 2) * q^55 + (z^5 - 2z^4 - 3z^3 + 2z^2 + 3z + 1) * q^56 + (4z^5 - 5z^3 + 4z) q^58 + (5z^5 + 5z^4 + z^2 - 6z) q^59 + (6z^5 - 2z^4 - 4z^3 - 2z^2 + 6z) q^61 + (-2z^5 + 2z^4 + 1) * q^62 + (-z^5 + 4z^4 - 3z^2 - 3z - 1) q^64 + (4z^5 - 6z^4 + 5z^3 - 6z^2 + 4z) q^65 + (-2z^5 - 2z^4 + 2z^2) q^67 + (-3z^5 - 7z^3 - 3z) q^68 + (6z^5 - 3z^4 + 3z^3 - 9z^2 + z + 6) * q^70 + (-7z^5 + 5z^4 + 2z^2 + 2z + 3) * q^71 + (z^5 + z^4 + 2z^3 - z^2 - 2) q^73 + (4z^5 + 4z^4 - 6z^3 - 2z^2 - 2z + 6) q^74 + (4z^5 + z^4 - 5z^2 - 5z - 3) q^76 + (3z^4 - z) q^77 + (2z^5 - 7z^4 + z^3 - 7z^2 + 2z) * q^79 + (-5z^5 - 5z^4 + 9z^3 + 5z - 9) * q^80 + (z^4 + z^3 + z^2) * q^82 + (3z^5 - 5z^4 + 2z^2 + 2z + 5) * q^83 + (-6z^5 + z^4 + 5z^2 + 5z - 9) q^85 + (-z^5 + 2z^4 - 3z^3 + 2z^2 - z) q^86 + (z^3 + z^2 - z - 1) * q^88 + (-z^4 - 5z^3 - z^2) q^89 + (-z^5 + 2z^4 - 5z^3 + 5z^2 - 3z + 4) * q^91 + (5z^5 - 3z^4 - 2z^2 - 2z + 1) * q^92 + (-3z^5 - 3z^4 + 7z^2 - 4z) * q^94 + (z^5 + z^4 - 3z^3 + z^2 - 2z + 3) * q^95 + (-z^4 + z^2 + z + 15) * q^97 + (8z^4 - 8z^3 - 5z + 5) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^5 - 6 * q^8	$ 6 q + 6 q^{5} - 6 q^{8} - 3 q^{10} + 3 q^{11} + 6 q^{13} + 12 q^{14} + 6 q^{16} - 3 q^{17} - 9 q^{19} - 12 q^{20} - 3 q^{25} + 9 q^{26} + 3 q^{28} + 6 q^{29} - 9 q^{31} - 9 q^{32} - 12 q^{34} + 15 q^{35} - 3 q^{40} - 18 q^{41} + 24 q^{46} - 3 q^{47} - 30 q^{50} - 9 q^{52} + 9 q^{53} + 12 q^{55} - 3 q^{56} - 15 q^{58} - 12 q^{61} + 6 q^{62} - 6 q^{64} + 15 q^{65} - 21 q^{68} + 45 q^{70} + 18 q^{71} - 6 q^{73} + 18 q^{74} - 18 q^{76} + 3 q^{79} - 27 q^{80} + 3 q^{82} + 30 q^{83} - 54 q^{85} - 9 q^{86} - 3 q^{88} - 15 q^{89} + 9 q^{91} + 6 q^{92} + 9 q^{95} + 90 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^5 - 6 * q^8 - 3 * q^10 + 3 * q^11 + 6 * q^13 + 12 * q^14 + 6 * q^16 - 3 * q^17 - 9 * q^19 - 12 * q^20 - 3 * q^25 + 9 * q^26 + 3 * q^28 + 6 * q^29 - 9 * q^31 - 9 * q^32 - 12 * q^34 + 15 * q^35 - 3 * q^40 - 18 * q^41 + 24 * q^46 - 3 * q^47 - 30 * q^50 - 9 * q^52 + 9 * q^53 + 12 * q^55 - 3 * q^56 - 15 * q^58 - 12 * q^61 + 6 * q^62 - 6 * q^64 + 15 * q^65 - 21 * q^68 + 45 * q^70 + 18 * q^71 - 6 * q^73 + 18 * q^74 - 18 * q^76 + 3 * q^79 - 27 * q^80 + 3 * q^82 + 30 * q^83 - 54 * q^85 - 9 * q^86 - 3 * q^88 - 15 * q^89 + 9 * q^91 + 6 * q^92 + 9 * q^95 + 90 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/693\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$442$
$\chi(n)$	$1$	$-\zeta_{18}^{3}$	$1$

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} $

100.1

−0.173648	+	0.984808i
0.939693	−	0.342020i
−0.766044	−	0.642788i
−0.173648	−	0.984808i
0.939693	+	0.342020i
−0.766044	+	0.642788i

−0.766044

1.32683i

−0.173648

−

0.300767i

1.17365

−

2.03282i

2.05303

−

1.66885i

−2.53209

1.79813

3.11446i

100.2

−0.173648

0.300767i

0.939693

1.62760i

0.0603074

−

0.104455i

−2.47178

−

0.943555i

−1.34730

0.0209445

0.0362770i

100.3

0.939693

−

1.62760i

−0.766044

−

1.32683i

1.76604

−

3.05888i

0.418748

2.61240i

0.879385

−3.31908

−

5.74881i

298.1

−0.766044

−

1.32683i

−0.173648

0.300767i

1.17365

2.03282i

2.05303

1.66885i

−2.53209

1.79813

−

3.11446i

298.2

−0.173648

−

0.300767i

0.939693

−

1.62760i

0.0603074

0.104455i

−2.47178

0.943555i

−1.34730

0.0209445

−

0.0362770i

298.3

0.939693

1.62760i

−0.766044

1.32683i

1.76604

3.05888i

0.418748

−

2.61240i

0.879385

−3.31908

5.74881i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.i.h		6
3.b	odd	2	1		77.2.e.a	✓	6
7.c	even	3	1	inner	693.2.i.h		6
7.c	even	3	1		4851.2.a.bj		3
7.d	odd	6	1		4851.2.a.bk		3
12.b	even	2	1		1232.2.q.m		6
21.c	even	2	1		539.2.e.m		6
21.g	even	6	1		539.2.a.g		3
21.g	even	6	1		539.2.e.m		6
21.h	odd	6	1		77.2.e.a	✓	6
21.h	odd	6	1		539.2.a.j		3
33.d	even	2	1		847.2.e.c		6
33.f	even	10	4		847.2.n.f		24
33.h	odd	10	4		847.2.n.g		24
84.j	odd	6	1		8624.2.a.co		3
84.n	even	6	1		1232.2.q.m		6
84.n	even	6	1		8624.2.a.ch		3
231.k	odd	6	1		5929.2.a.u		3
231.l	even	6	1		847.2.e.c		6
231.l	even	6	1		5929.2.a.x		3
231.z	odd	30	4		847.2.n.g		24
231.be	even	30	4		847.2.n.f		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	3.b	odd	2	1
77.2.e.a	✓	6	21.h	odd	6	1
539.2.a.g		3	21.g	even	6	1
539.2.a.j		3	21.h	odd	6	1
539.2.e.m		6	21.c	even	2	1
539.2.e.m		6	21.g	even	6	1
693.2.i.h		6	1.a	even	1	1	trivial
693.2.i.h		6	7.c	even	3	1	inner
847.2.e.c		6	33.d	even	2	1
847.2.e.c		6	231.l	even	6	1
847.2.n.f		24	33.f	even	10	4
847.2.n.f		24	231.be	even	30	4
847.2.n.g		24	33.h	odd	10	4
847.2.n.g		24	231.z	odd	30	4
1232.2.q.m		6	12.b	even	2	1
1232.2.q.m		6	84.n	even	6	1
4851.2.a.bj		3	7.c	even	3	1
4851.2.a.bk		3	7.d	odd	6	1
5929.2.a.u		3	231.k	odd	6	1
5929.2.a.x		3	231.l	even	6	1
8624.2.a.ch		3	84.n	even	6	1
8624.2.a.co		3	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(693, [\chi])$:

$ T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1 $

$ T_{5}^{6} - 6T_{5}^{5} + 27T_{5}^{4} - 52T_{5}^{3} + 75T_{5}^{2} - 9T_{5} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 27T^4 - 52T^3 + 75T^2 - 9*T + 1
$7$	$ T^{6} + 17T^{3} + 343 $ T^6 + 17*T^3 + 343
$11$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$13$	$ (T^{3} - 3 T^{2} - 6 T - 1)^{2} $ (T^3 - 3T^2 - 6T - 1)^2
$17$	$ T^{6} + 3 T^{5} + \cdots + 16129 $ T^6 + 3T^5 + 45T^4 + 146T^3 + 1677T^2 + 4572*T + 16129
$19$	$ T^{6} + 9 T^{5} + \cdots + 81 $ T^6 + 9T^5 + 63T^4 + 144T^3 + 243T^2 + 162*T + 81
$23$	$ T^{6} + 57 T^{4} + \cdots + 11449 $ T^6 + 57T^4 + 214T^3 + 3249T^2 + 6099T + 11449
$29$	$ (T^{3} - 3 T^{2} - 36 T - 51)^{2} $ (T^3 - 3T^2 - 36T - 51)^2
$31$	$ T^{6} + 9 T^{5} + \cdots + 361 $ T^6 + 9T^5 + 57T^4 + 178T^3 + 405T^2 + 456*T + 361
$37$	$ T^{6} + 36 T^{4} + \cdots + 5184 $ T^6 + 36T^4 + 144T^3 + 1296T^2 + 2592T + 5184
$41$	$ (T^{3} + 9 T^{2} + 6 T + 1)^{2} $ (T^3 + 9T^2 + 6T + 1)^2
$43$	$ (T^{3} - 9 T + 9)^{2} $ (T^3 - 9*T + 9)^2
$47$	$ T^{6} + 3 T^{5} + \cdots + 104329 $ T^6 + 3T^5 + 87T^4 + 412T^3 + 7053T^2 + 25194*T + 104329
$53$	$ T^{6} - 9 T^{5} + \cdots + 210681 $ T^6 - 9T^5 + 135T^4 - 432T^3 + 7047T^2 - 24786*T + 210681
$59$	$ T^{6} + 93 T^{4} + \cdots + 361 $ T^6 + 93T^4 - 38T^3 + 8649T^2 - 1767T + 361
$61$	$ T^{6} + 12 T^{5} + \cdots + 576 $ T^6 + 12T^5 + 180T^4 - 480T^3 + 1008T^2 - 864*T + 576
$67$	$ T^{6} + 12 T^{4} + \cdots + 64 $ T^6 + 12T^4 - 16T^3 + 144T^2 - 96T + 64
$71$	$ (T^{3} - 9 T^{2} + \cdots + 801)^{2} $ (T^3 - 9T^2 - 90T + 801)^2
$73$	$ T^{6} + 6 T^{5} + \cdots + 1 $ T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$79$	$ T^{6} - 3 T^{5} + \cdots + 1369 $ T^6 - 3T^5 + 123T^4 + 416T^3 + 12885T^2 + 4218*T + 1369
$83$	$ (T^{3} - 15 T^{2} + \cdots + 267)^{2} $ (T^3 - 15T^2 + 18T + 267)^2
$89$	$ T^{6} + 15 T^{5} + \cdots + 12321 $ T^6 + 15T^5 + 153T^4 + 858T^3 + 3519T^2 + 7992*T + 12321
$97$	$ (T^{3} - 45 T^{2} + \cdots - 3329)^{2} $ (T^3 - 45T^2 + 672T - 3329)^2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (\zeta_{18}^{5} - \zeta_{18}^{4} - 1) q^{8}+O(q^{10}) \) q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (-z^4 + 2z^3 - z^2) q^5 + (2z^4 - 3z) * q^7 + (z^5 - z^4 - 1) * q^8	\( q + (\zeta_{18}^{5} - \zeta_{18}^{4} + \cdots + \zeta_{18}) q^{2}+ \cdots + (8 \zeta_{18}^{4} - 8 \zeta_{18}^{3} + \cdots + 5) q^{98}+O(q^{100}) \) q + (z^5 - z^4 - z^2 + z) * q^2 + (-z^5 - z^4 + z) * q^4 + (-z^4 + 2z^3 - z^2) q^5 + (2z^4 - 3z) * q^7 + (z^5 - z^4 - 1) * q^8 + (z^3 - 3z^2 + 3z - 1) * q^10 + (-z^3 + 1) * q^11 + (2z^5 - z^4 - z^2 - z + 1) q^13 + (3z^5 - 2z^3 - z^2 + 3) * q^14 + (-z^5 - 2z^4 + 2z^3 - 2z^2 - z) q^16 + (-4z^5 - 4z^4 + z^3 + z^2 + 3z - 1) q^17 + (-z^5 - z^4 - 3z^3 - z^2 - z) q^19 + (-2z^5 + z^4 + z^2 + z - 2) q^20 + (z^5 - z^4) * q^22 + (-5z^5 + 3z^4 + 3z^2 - 5z) * q^23 + (-3z^5 - 3z^4 + z^3 - z^2 + 4z - 1) q^25 + (2z^5 - 3z^4 + 3z^3 - 3z^2 + 2z) q^26 + (3z^5 + 3z^3 - z^2 - 1) * q^28 + (-z^5 + 4z^4 - 3z^2 - 3z + 1) q^29 + (z^5 + z^4 + 3z^3 - z - 3) q^31 + (z^5 + z^4 + 3z^3 - 3z^2 + 2z - 3) q^32 + (-4z^5 + 5z^4 - z^2 - z - 2) * q^34 + (z^5 - 2z^4 + z^3 + 2z^2 - 4z + 2) q^35 + (-4z^5 + 2z^4 + 2z^2 - 4z) * q^37 + (z^5 + z^4 + z^2 - 2z) q^38 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^40 + (2z^5 + z^4 - 3z^2 - 3z - 3) q^41 + (-2z^5 + z^4 + z^2 + z) q^43 + (-z^4 - z^2) * q^44 + (5z^5 + 5z^4 - 8z^3 - 2z^2 - 3z + 8) q^46 + (3z^5 + 3z^4 - z^3 + 3z^2 + 3z) * q^47 + (-8z^5 + 5z^2) * q^49 + (-5z^5 + 4z^4 + z^2 + z - 5) * q^50 + (3z^3 - 2z^2 + 2z - 3) q^52 + (-3z^5 - 3z^4 - 3z^3 - 3z^2 + 6z + 3) q^53 + (z^5 - z^2 - z + 2) * q^55 + (z^5 - 2z^4 - 3z^3 + 2z^2 + 3z + 1) * q^56 + (4z^5 - 5z^3 + 4z) q^58 + (5z^5 + 5z^4 + z^2 - 6z) q^59 + (6z^5 - 2z^4 - 4z^3 - 2z^2 + 6z) q^61 + (-2z^5 + 2z^4 + 1) * q^62 + (-z^5 + 4z^4 - 3z^2 - 3z - 1) q^64 + (4z^5 - 6z^4 + 5z^3 - 6z^2 + 4z) q^65 + (-2z^5 - 2z^4 + 2z^2) q^67 + (-3z^5 - 7z^3 - 3z) q^68 + (6z^5 - 3z^4 + 3z^3 - 9z^2 + z + 6) * q^70 + (-7z^5 + 5z^4 + 2z^2 + 2z + 3) * q^71 + (z^5 + z^4 + 2z^3 - z^2 - 2) q^73 + (4z^5 + 4z^4 - 6z^3 - 2z^2 - 2z + 6) q^74 + (4z^5 + z^4 - 5z^2 - 5z - 3) q^76 + (3z^4 - z) q^77 + (2z^5 - 7z^4 + z^3 - 7z^2 + 2z) * q^79 + (-5z^5 - 5z^4 + 9z^3 + 5z - 9) * q^80 + (z^4 + z^3 + z^2) * q^82 + (3z^5 - 5z^4 + 2z^2 + 2z + 5) * q^83 + (-6z^5 + z^4 + 5z^2 + 5z - 9) q^85 + (-z^5 + 2z^4 - 3z^3 + 2z^2 - z) q^86 + (z^3 + z^2 - z - 1) * q^88 + (-z^4 - 5z^3 - z^2) q^89 + (-z^5 + 2z^4 - 5z^3 + 5z^2 - 3z + 4) * q^91 + (5z^5 - 3z^4 - 2z^2 - 2z + 1) * q^92 + (-3z^5 - 3z^4 + 7z^2 - 4z) * q^94 + (z^5 + z^4 - 3z^3 + z^2 - 2z + 3) * q^95 + (-z^4 + z^2 + z + 15) * q^97 + (8z^4 - 8z^3 - 5z + 5) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^5 - 6 * q^8	\( 6 q + 6 q^{5} - 6 q^{8} - 3 q^{10} + 3 q^{11} + 6 q^{13} + 12 q^{14} + 6 q^{16} - 3 q^{17} - 9 q^{19} - 12 q^{20} - 3 q^{25} + 9 q^{26} + 3 q^{28} + 6 q^{29} - 9 q^{31} - 9 q^{32} - 12 q^{34} + 15 q^{35} - 3 q^{40} - 18 q^{41} + 24 q^{46} - 3 q^{47} - 30 q^{50} - 9 q^{52} + 9 q^{53} + 12 q^{55} - 3 q^{56} - 15 q^{58} - 12 q^{61} + 6 q^{62} - 6 q^{64} + 15 q^{65} - 21 q^{68} + 45 q^{70} + 18 q^{71} - 6 q^{73} + 18 q^{74} - 18 q^{76} + 3 q^{79} - 27 q^{80} + 3 q^{82} + 30 q^{83} - 54 q^{85} - 9 q^{86} - 3 q^{88} - 15 q^{89} + 9 q^{91} + 6 q^{92} + 9 q^{95} + 90 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^5 - 6 * q^8 - 3 * q^10 + 3 * q^11 + 6 * q^13 + 12 * q^14 + 6 * q^16 - 3 * q^17 - 9 * q^19 - 12 * q^20 - 3 * q^25 + 9 * q^26 + 3 * q^28 + 6 * q^29 - 9 * q^31 - 9 * q^32 - 12 * q^34 + 15 * q^35 - 3 * q^40 - 18 * q^41 + 24 * q^46 - 3 * q^47 - 30 * q^50 - 9 * q^52 + 9 * q^53 + 12 * q^55 - 3 * q^56 - 15 * q^58 - 12 * q^61 + 6 * q^62 - 6 * q^64 + 15 * q^65 - 21 * q^68 + 45 * q^70 + 18 * q^71 - 6 * q^73 + 18 * q^74 - 18 * q^76 + 3 * q^79 - 27 * q^80 + 3 * q^82 + 30 * q^83 - 54 * q^85 - 9 * q^86 - 3 * q^88 - 15 * q^89 + 9 * q^91 + 6 * q^92 + 9 * q^95 + 90 * q^97 + 6 * q^98

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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	693.2.i.h

Level	$693$
Weight	$2$
Character orbit	693.i
Analytic conductor	$5.534$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(n\)	\(155\)	\(199\)	\(442\)
\(\chi(n)\)	\(1\)	\(-\zeta_{18}^{3}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 100.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$3$	\( T^{6} \) T^6
$5$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 27T^4 - 52T^3 + 75T^2 - 9*T + 1
$7$	\( T^{6} + 17T^{3} + 343 \) T^6 + 17*T^3 + 343
$11$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$13$	\( (T^{3} - 3 T^{2} - 6 T - 1)^{2} \) (T^3 - 3T^2 - 6T - 1)^2
$17$	\( T^{6} + 3 T^{5} + \cdots + 16129 \) T^6 + 3T^5 + 45T^4 + 146T^3 + 1677T^2 + 4572*T + 16129
$19$	\( T^{6} + 9 T^{5} + \cdots + 81 \) T^6 + 9T^5 + 63T^4 + 144T^3 + 243T^2 + 162*T + 81
$23$	\( T^{6} + 57 T^{4} + \cdots + 11449 \) T^6 + 57T^4 + 214T^3 + 3249T^2 + 6099T + 11449
$29$	\( (T^{3} - 3 T^{2} - 36 T - 51)^{2} \) (T^3 - 3T^2 - 36T - 51)^2
$31$	\( T^{6} + 9 T^{5} + \cdots + 361 \) T^6 + 9T^5 + 57T^4 + 178T^3 + 405T^2 + 456*T + 361
$37$	\( T^{6} + 36 T^{4} + \cdots + 5184 \) T^6 + 36T^4 + 144T^3 + 1296T^2 + 2592T + 5184
$41$	\( (T^{3} + 9 T^{2} + 6 T + 1)^{2} \) (T^3 + 9T^2 + 6T + 1)^2
$43$	\( (T^{3} - 9 T + 9)^{2} \) (T^3 - 9*T + 9)^2
$47$	\( T^{6} + 3 T^{5} + \cdots + 104329 \) T^6 + 3T^5 + 87T^4 + 412T^3 + 7053T^2 + 25194*T + 104329
$53$	\( T^{6} - 9 T^{5} + \cdots + 210681 \) T^6 - 9T^5 + 135T^4 - 432T^3 + 7047T^2 - 24786*T + 210681
$59$	\( T^{6} + 93 T^{4} + \cdots + 361 \) T^6 + 93T^4 - 38T^3 + 8649T^2 - 1767T + 361
$61$	\( T^{6} + 12 T^{5} + \cdots + 576 \) T^6 + 12T^5 + 180T^4 - 480T^3 + 1008T^2 - 864*T + 576
$67$	\( T^{6} + 12 T^{4} + \cdots + 64 \) T^6 + 12T^4 - 16T^3 + 144T^2 - 96T + 64
$71$	\( (T^{3} - 9 T^{2} + \cdots + 801)^{2} \) (T^3 - 9T^2 - 90T + 801)^2
$73$	\( T^{6} + 6 T^{5} + \cdots + 1 \) T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$79$	\( T^{6} - 3 T^{5} + \cdots + 1369 \) T^6 - 3T^5 + 123T^4 + 416T^3 + 12885T^2 + 4218*T + 1369
$83$	\( (T^{3} - 15 T^{2} + \cdots + 267)^{2} \) (T^3 - 15T^2 + 18T + 267)^2
$89$	\( T^{6} + 15 T^{5} + \cdots + 12321 \) T^6 + 15T^5 + 153T^4 + 858T^3 + 3519T^2 + 7992*T + 12321
$97$	\( (T^{3} - 45 T^{2} + \cdots - 3329)^{2} \) (T^3 - 45T^2 + 672T - 3329)^2
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