LMFDB - Newform orbit 162.2.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,2,Mod(19,162)] 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("162.19"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	162.e (of order $9$, degree $6$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.29357651274$
Analytic rank:	$0$
Dimension:	$6$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-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}^{4} + \zeta_{18}) q^{2} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} + \cdots + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} + \cdots + \zeta_{18}^{3}) q^{7} - \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} + \cdots + 1) q^{10} + \cdots + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \cdots + 1) q^{98} +O(q^{100}) $ q + (-z^4 + z) * q^2 - z^5 * q^4 + (-z^5 - z^4 + z^3 + z^2) * q^5 + (z^5 - 2z^4 + z^3) q^7 - z^3 * q^8 + (-z^3 - z^2 + z + 1) * q^10 + (-3z^5 + 3z^3 + 2z^2 + z - 1) q^11 + (-2z^4 - 3z^2 - 2) * q^13 + (z^3 - 2z^2 + z) q^14 - z * q^16 + (4z^5 + 4z^4 - 2z^3 - z^2 - 3z + 2) * q^17 + (-3z^5 + 4z^4 + 3z^3 + 4z^2 - 3z) q^19 + (-z^5 - z^4 + z^2 - 1) * q^20 + (-z^5 + z^4 - 3z^3 + z^2 + 2z + 2) * q^22 + (-z^5 - 2z^4 - 2z^3 + 2) * q^23 + (z^5 + 2z^4 - z^3 + z^2 - 2z - 1) * q^25 + (2z^4 - 2z^2 - 2z - 3) q^26 + (-z^5 + z^2 + z - 2) * q^28 + (3z^5 + z^4 + z^3 - z^2 - z - 3) q^29 + (4z^5 + 3z - 3) * q^31 + (z^5 - z^2) * q^32 + (3z^5 - 2z^4 + 4z^3 + z^2 - 1) q^34 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^35 + (3z^5 + 3z^4 - 5z^3 - z^2 - 2z + 5) * q^37 + (3z^5 - 3z^3 + z^2 + 3z + 4) q^38 + (z^4 - z^3 - z^2 - z + 1) * q^40 + (3z^4 - 5z^3 - z^2 - 5z + 3) q^41 + (-2z^5 + 2z^3 - 2z^2 + 4z - 4) * q^43 + (-2z^5 - 2z^4 - z^3 + 3z^2 - z + 1) q^44 + (-2z^4 - z^3 - 2z^2) * q^46 + (-2z^5 + 3z^4 - z^3 + z^2 - 1) * q^47 + (-z^5 - 4z^4 + z^3 + z^2 + 3z + 3) * q^49 + (2z^5 + z^4 + z^3 - 2z + 1) * q^50 + (2z^5 + 3z^4 - 3z - 2) q^52 + (-2z^5 - z^4 + 3z^2 + 3z + 2) q^53 + (-z^5 - 4z^4 + 5z^2 + 5z - 3) q^55 + (-z^5 + 2z^4 - z^3 + z^2 - 2z + 1) * q^56 + (z^5 + 3z^4 + 3z^3 - 2z - 1) q^58 + (z^5 - 3z^4 + 4z^3 - z^2 - z - 1) * q^59 + (z^5 + 2z^4 - 4z^3 - 5z^2 + 5) q^61 + (-3z^5 + 3z^4 + 4z^3 + 3z^2 - 3z) q^62 + (z^3 - 1) * q^64 + (z^5 - z^3 - 4z^2 - z - 3) q^65 + (-3z^3 - z^2 - 3z) * q^67 + (z^4 + 3z^3 - 2z^2 + 3z + 1) q^68 + (-3z^5 + 3z^3 + z^2 - z - 2) * q^70 + (-5z^5 - 5z^4 + 4z^3 - z^2 + 6z - 4) * q^71 + (-5z^5 - 2z^4 + z^3 - 2z^2 - 5z) * q^73 + (2z^5 - 5z^4 + 3z^3 + z^2 - 1) q^74 + (-3z^5 - 4z^4 + 3z^3 + 3z^2 + z + 1) * q^76 + (-z^5 - z^4 - z^3 + 7z - 6) q^77 + (-8z^5 - 5z^3 + 5z^2 + 8) q^79 + (z^5 - z^4 - 1) * q^80 + (5z^5 - 3z^4 - 2z^2 - 2z - 1) * q^82 + (-2z^5 + 2z^3 - 2z^2 + 2) q^83 + (5z^4 + 5z^3 - 7z + 2) q^85 + (-4z^5 + 4z^4 - 2z^3 + 4z^2 - 2z - 2) q^86 + (z^5 - z^4 - 2z^3 - 3z^2 + 3) * q^88 + (-z^5 - 3z^4 + 5z^3 - 3z^2 - z) q^89 + (-z^5 - z^4 + 4z^3 - 4z^2 + 5z - 4) q^91 + (-2z^2 - z - 2) q^92 + (z^4 - 2z^3 + 3z^2 - 2z + 1) q^94 + (-5z^4 + 3z^3 + 10z^2 + 3z - 5) * q^95 + (-2z^5 + 2z^3 - z^2 + 6z - 3) q^97 + (-3z^5 - 3z^4 - z^3 - z^2 + 4z + 1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 12 q^{13} + 3 q^{14} + 6 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{22} + 6 q^{23} - 9 q^{25} - 18 q^{26} - 12 q^{28} - 15 q^{29} - 18 q^{31} + 6 q^{34}+ \cdots + 3 q^{98}+O(q^{100}) $ 6 * q + 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 + 3 * q^11 - 12 * q^13 + 3 * q^14 + 6 * q^17 + 9 * q^19 - 6 * q^20 + 3 * q^22 + 6 * q^23 - 9 * q^25 - 18 * q^26 - 12 * q^28 - 15 * q^29 - 18 * q^31 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 18 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 21 * q^49 + 9 * q^50 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 3 * q^56 + 3 * q^58 + 6 * q^59 + 18 * q^61 + 12 * q^62 - 3 * q^64 - 21 * q^65 - 9 * q^67 + 15 * q^68 - 3 * q^70 - 12 * q^71 + 3 * q^73 + 3 * q^74 + 15 * q^76 - 39 * q^77 + 33 * q^79 - 6 * q^80 - 6 * q^82 + 18 * q^83 + 27 * q^85 - 18 * q^86 + 12 * q^88 + 15 * q^89 - 12 * q^91 - 12 * q^92 - 21 * q^95 - 12 * q^97 + 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-\zeta_{18}^{5}$

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} $

19.1

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

−0.939693

0.342020i

0.766044

−

0.642788i

−0.439693

−

2.49362i

−1.79813

−

1.50881i

−0.500000

0.866025i

1.26604

2.19285i

37.1

0.766044

−

0.642788i

0.173648

−

0.984808i

1.26604

−

0.460802i

−0.0209445

−

0.118782i

−0.500000

−

0.866025i

0.673648

−

1.16679i

73.1

0.173648

−

0.984808i

−0.939693

−

0.342020i

0.673648

−

0.565258i

3.31908

−

1.20805i

−0.500000

0.866025i

−0.439693

−

0.761570i

91.1

0.173648

0.984808i

−0.939693

0.342020i

0.673648

0.565258i

3.31908

1.20805i

−0.500000

−

0.866025i

−0.439693

0.761570i

127.1

0.766044

0.642788i

0.173648

0.984808i

1.26604

0.460802i

−0.0209445

0.118782i

−0.500000

0.866025i

0.673648

1.16679i

145.1

−0.939693

−

0.342020i

0.766044

0.642788i

−0.439693

2.49362i

−1.79813

1.50881i

−0.500000

−

0.866025i

1.26604

−

2.19285i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.2.e.a		6
3.b	odd	2	1		54.2.e.a	✓	6
9.c	even	3	1		486.2.e.a		6
9.c	even	3	1		486.2.e.c		6
9.d	odd	6	1		486.2.e.b		6
9.d	odd	6	1		486.2.e.d		6
12.b	even	2	1		432.2.u.a		6
27.e	even	9	1	inner	162.2.e.a		6
27.e	even	9	1		486.2.e.a		6
27.e	even	9	1		486.2.e.c		6
27.e	even	9	1		1458.2.a.d		3
27.e	even	9	2		1458.2.c.a		6
27.f	odd	18	1		54.2.e.a	✓	6
27.f	odd	18	1		486.2.e.b		6
27.f	odd	18	1		486.2.e.d		6
27.f	odd	18	1		1458.2.a.a		3
27.f	odd	18	2		1458.2.c.d		6
108.l	even	18	1		432.2.u.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.2.e.a	✓	6	3.b	odd	2	1
54.2.e.a	✓	6	27.f	odd	18	1
162.2.e.a		6	1.a	even	1	1	trivial
162.2.e.a		6	27.e	even	9	1	inner
432.2.u.a		6	12.b	even	2	1
432.2.u.a		6	108.l	even	18	1
486.2.e.a		6	9.c	even	3	1
486.2.e.a		6	27.e	even	9	1
486.2.e.b		6	9.d	odd	6	1
486.2.e.b		6	27.f	odd	18	1
486.2.e.c		6	9.c	even	3	1
486.2.e.c		6	27.e	even	9	1
486.2.e.d		6	9.d	odd	6	1
486.2.e.d		6	27.f	odd	18	1
1458.2.a.a		3	27.f	odd	18	1
1458.2.a.d		3	27.e	even	9	1
1458.2.c.a		6	27.e	even	9	2
1458.2.c.d		6	27.f	odd	18	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - 3T_{5}^{5} + 9T_{5}^{4} - 24T_{5}^{3} + 36T_{5}^{2} - 27T_{5} + 9 $ T5^6 - 3*T5^5 + 9*T5^4 - 24*T5^3 + 36*T5^2 - 27*T5 + 9 acting on $S_{2}^{\mathrm{new}}(162, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 3 T^{5} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 24T^3 + 36T^2 - 27*T + 9
$7$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 - 6T^4 + 8T^3 + 69T^2 + 3*T + 1
$11$	$ T^{6} - 3 T^{5} + \cdots + 3249 $ T^6 - 3T^5 + 9T^4 - 24T^3 - 18T^2 - 513*T + 3249
$13$	$ T^{6} + 12 T^{5} + \cdots + 289 $ T^6 + 12T^5 + 78T^4 + 269T^3 + 786T^2 + 816*T + 289
$17$	$ T^{6} - 6 T^{5} + \cdots + 25281 $ T^6 - 6T^5 + 63T^4 - 156T^3 + 1683T^2 - 4293*T + 25281
$19$	$ T^{6} - 9 T^{5} + \cdots + 32041 $ T^6 - 9T^5 + 93T^4 - 250T^3 + 1755T^2 - 2148*T + 32041
$23$	$ T^{6} - 6 T^{5} + \cdots + 9 $ T^6 - 6T^5 + 36T^4 - 111T^3 + 144T^2 - 54*T + 9
$29$	$ T^{6} + 15 T^{5} + \cdots + 9 $ T^6 + 15T^5 + 99T^4 + 300T^3 + 360T^2 + 27*T + 9
$31$	$ T^{6} + 18 T^{5} + \cdots + 5041 $ T^6 + 18T^5 + 171T^4 + 881T^3 + 3636T^2 + 7029*T + 5041
$37$	$ T^{6} - 15 T^{5} + \cdots + 289 $ T^6 - 15T^5 + 171T^4 - 844T^3 + 3171T^2 + 918*T + 289
$41$	$ T^{6} - 3 T^{5} + \cdots + 47961 $ T^6 - 3T^5 + 36T^4 - 51T^3 - 369T^2 - 7884*T + 47961
$43$	$ T^{6} + 18 T^{5} + \cdots + 64 $ T^6 + 18T^5 + 144T^4 + 224T^3 + 432T^2 + 288*T + 64
$47$	$ T^{6} + 9 T^{5} + \cdots + 81 $ T^6 + 9T^5 + 9T^4 - 72T^3 + 162T^2 - 81*T + 81
$53$	$ (T^{3} - 6 T^{2} - 9 T - 3)^{2} $ (T^3 - 6T^2 - 9T - 3)^2
$59$	$ T^{6} - 6 T^{5} + \cdots + 3249 $ T^6 - 6T^5 + 36T^4 - 84T^3 + 954T^2 + 3591*T + 3249
$61$	$ T^{6} - 18 T^{5} + \cdots + 2809 $ T^6 - 18T^5 + 153T^4 - 379T^3 + 2124T^2 - 4293*T + 2809
$67$	$ T^{6} + 9 T^{5} + \cdots + 1 $ T^6 + 9T^5 + 45T^4 + 80T^3 + 576T^2 + 45*T + 1
$71$	$ T^{6} + 12 T^{5} + \cdots + 106929 $ T^6 + 12T^5 + 189T^4 + 114T^3 + 5949T^2 + 14715*T + 106929
$73$	$ T^{6} - 3 T^{5} + \cdots + 72361 $ T^6 - 3T^5 + 123T^4 - 196T^3 + 13803T^2 - 30666*T + 72361
$79$	$ T^{6} - 33 T^{5} + \cdots + 466489 $ T^6 - 33T^5 + 510T^4 - 4168T^3 + 25707T^2 - 145479*T + 466489
$83$	$ T^{6} - 18 T^{5} + \cdots + 5184 $ T^6 - 18T^5 + 144T^4 - 720T^3 + 2592T^2 - 5184*T + 5184
$89$	$ T^{6} - 15 T^{5} + \cdots + 25281 $ T^6 - 15T^5 + 189T^4 - 858T^3 + 3681T^2 + 5724*T + 25281
$97$	$ T^{6} + 12 T^{5} + \cdots + 16129 $ T^6 + 12T^5 + 51T^4 - 469T^3 + 1092T^2 - 4953*T + 16129
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.e (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{4} + \zeta_{18}) q^{2} - \zeta_{18}^{5} q^{4} + ( - \zeta_{18}^{5} + \cdots + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} + \cdots + \zeta_{18}^{3}) q^{7} - \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} + \cdots + 1) q^{10} + \cdots + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \cdots + 1) q^{98} +O(q^{100}) \) q + (-z^4 + z) * q^2 - z^5 * q^4 + (-z^5 - z^4 + z^3 + z^2) * q^5 + (z^5 - 2z^4 + z^3) q^7 - z^3 * q^8 + (-z^3 - z^2 + z + 1) * q^10 + (-3z^5 + 3z^3 + 2z^2 + z - 1) q^11 + (-2z^4 - 3z^2 - 2) * q^13 + (z^3 - 2z^2 + z) q^14 - z * q^16 + (4z^5 + 4z^4 - 2z^3 - z^2 - 3z + 2) * q^17 + (-3z^5 + 4z^4 + 3z^3 + 4z^2 - 3z) q^19 + (-z^5 - z^4 + z^2 - 1) * q^20 + (-z^5 + z^4 - 3z^3 + z^2 + 2z + 2) * q^22 + (-z^5 - 2z^4 - 2z^3 + 2) * q^23 + (z^5 + 2z^4 - z^3 + z^2 - 2z - 1) * q^25 + (2z^4 - 2z^2 - 2z - 3) q^26 + (-z^5 + z^2 + z - 2) * q^28 + (3z^5 + z^4 + z^3 - z^2 - z - 3) q^29 + (4z^5 + 3z - 3) * q^31 + (z^5 - z^2) * q^32 + (3z^5 - 2z^4 + 4z^3 + z^2 - 1) q^34 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^35 + (3z^5 + 3z^4 - 5z^3 - z^2 - 2z + 5) * q^37 + (3z^5 - 3z^3 + z^2 + 3z + 4) q^38 + (z^4 - z^3 - z^2 - z + 1) * q^40 + (3z^4 - 5z^3 - z^2 - 5z + 3) q^41 + (-2z^5 + 2z^3 - 2z^2 + 4z - 4) * q^43 + (-2z^5 - 2z^4 - z^3 + 3z^2 - z + 1) q^44 + (-2z^4 - z^3 - 2z^2) * q^46 + (-2z^5 + 3z^4 - z^3 + z^2 - 1) * q^47 + (-z^5 - 4z^4 + z^3 + z^2 + 3z + 3) * q^49 + (2z^5 + z^4 + z^3 - 2z + 1) * q^50 + (2z^5 + 3z^4 - 3z - 2) q^52 + (-2z^5 - z^4 + 3z^2 + 3z + 2) q^53 + (-z^5 - 4z^4 + 5z^2 + 5z - 3) q^55 + (-z^5 + 2z^4 - z^3 + z^2 - 2z + 1) * q^56 + (z^5 + 3z^4 + 3z^3 - 2z - 1) q^58 + (z^5 - 3z^4 + 4z^3 - z^2 - z - 1) * q^59 + (z^5 + 2z^4 - 4z^3 - 5z^2 + 5) q^61 + (-3z^5 + 3z^4 + 4z^3 + 3z^2 - 3z) q^62 + (z^3 - 1) * q^64 + (z^5 - z^3 - 4z^2 - z - 3) q^65 + (-3z^3 - z^2 - 3z) * q^67 + (z^4 + 3z^3 - 2z^2 + 3z + 1) q^68 + (-3z^5 + 3z^3 + z^2 - z - 2) * q^70 + (-5z^5 - 5z^4 + 4z^3 - z^2 + 6z - 4) * q^71 + (-5z^5 - 2z^4 + z^3 - 2z^2 - 5z) * q^73 + (2z^5 - 5z^4 + 3z^3 + z^2 - 1) q^74 + (-3z^5 - 4z^4 + 3z^3 + 3z^2 + z + 1) * q^76 + (-z^5 - z^4 - z^3 + 7z - 6) q^77 + (-8z^5 - 5z^3 + 5z^2 + 8) q^79 + (z^5 - z^4 - 1) * q^80 + (5z^5 - 3z^4 - 2z^2 - 2z - 1) * q^82 + (-2z^5 + 2z^3 - 2z^2 + 2) q^83 + (5z^4 + 5z^3 - 7z + 2) q^85 + (-4z^5 + 4z^4 - 2z^3 + 4z^2 - 2z - 2) q^86 + (z^5 - z^4 - 2z^3 - 3z^2 + 3) * q^88 + (-z^5 - 3z^4 + 5z^3 - 3z^2 - z) q^89 + (-z^5 - z^4 + 4z^3 - 4z^2 + 5z - 4) q^91 + (-2z^2 - z - 2) q^92 + (z^4 - 2z^3 + 3z^2 - 2z + 1) q^94 + (-5z^4 + 3z^3 + 10z^2 + 3z - 5) * q^95 + (-2z^5 + 2z^3 - z^2 + 6z - 3) q^97 + (-3z^5 - 3z^4 - z^3 - z^2 + 4z + 1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} + 3 q^{7} - 3 q^{8} + 3 q^{10} + 3 q^{11} - 12 q^{13} + 3 q^{14} + 6 q^{17} + 9 q^{19} - 6 q^{20} + 3 q^{22} + 6 q^{23} - 9 q^{25} - 18 q^{26} - 12 q^{28} - 15 q^{29} - 18 q^{31} + 6 q^{34}+ \cdots + 3 q^{98}+O(q^{100}) \) 6 * q + 3 * q^5 + 3 * q^7 - 3 * q^8 + 3 * q^10 + 3 * q^11 - 12 * q^13 + 3 * q^14 + 6 * q^17 + 9 * q^19 - 6 * q^20 + 3 * q^22 + 6 * q^23 - 9 * q^25 - 18 * q^26 - 12 * q^28 - 15 * q^29 - 18 * q^31 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 18 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 21 * q^49 + 9 * q^50 - 12 * q^52 + 12 * q^53 - 18 * q^55 + 3 * q^56 + 3 * q^58 + 6 * q^59 + 18 * q^61 + 12 * q^62 - 3 * q^64 - 21 * q^65 - 9 * q^67 + 15 * q^68 - 3 * q^70 - 12 * q^71 + 3 * q^73 + 3 * q^74 + 15 * q^76 - 39 * q^77 + 33 * q^79 - 6 * q^80 - 6 * q^82 + 18 * q^83 + 27 * q^85 - 18 * q^86 + 12 * q^88 + 15 * q^89 - 12 * q^91 - 12 * q^92 - 21 * q^95 - 12 * q^97 + 3 * q^98

Introduction

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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	162.2.e.a

Level	$162$
Weight	$2$
Character orbit	162.e
Analytic conductor	$1.294$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(6\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$3$	\( T^{6} \) T^6
$5$	\( T^{6} - 3 T^{5} + \cdots + 9 \) T^6 - 3T^5 + 9T^4 - 24T^3 + 36T^2 - 27*T + 9
$7$	\( T^{6} - 3 T^{5} + \cdots + 1 \) T^6 - 3T^5 - 6T^4 + 8T^3 + 69T^2 + 3*T + 1
$11$	\( T^{6} - 3 T^{5} + \cdots + 3249 \) T^6 - 3T^5 + 9T^4 - 24T^3 - 18T^2 - 513*T + 3249
$13$	\( T^{6} + 12 T^{5} + \cdots + 289 \) T^6 + 12T^5 + 78T^4 + 269T^3 + 786T^2 + 816*T + 289
$17$	\( T^{6} - 6 T^{5} + \cdots + 25281 \) T^6 - 6T^5 + 63T^4 - 156T^3 + 1683T^2 - 4293*T + 25281
$19$	\( T^{6} - 9 T^{5} + \cdots + 32041 \) T^6 - 9T^5 + 93T^4 - 250T^3 + 1755T^2 - 2148*T + 32041
$23$	\( T^{6} - 6 T^{5} + \cdots + 9 \) T^6 - 6T^5 + 36T^4 - 111T^3 + 144T^2 - 54*T + 9
$29$	\( T^{6} + 15 T^{5} + \cdots + 9 \) T^6 + 15T^5 + 99T^4 + 300T^3 + 360T^2 + 27*T + 9
$31$	\( T^{6} + 18 T^{5} + \cdots + 5041 \) T^6 + 18T^5 + 171T^4 + 881T^3 + 3636T^2 + 7029*T + 5041
$37$	\( T^{6} - 15 T^{5} + \cdots + 289 \) T^6 - 15T^5 + 171T^4 - 844T^3 + 3171T^2 + 918*T + 289
$41$	\( T^{6} - 3 T^{5} + \cdots + 47961 \) T^6 - 3T^5 + 36T^4 - 51T^3 - 369T^2 - 7884*T + 47961
$43$	\( T^{6} + 18 T^{5} + \cdots + 64 \) T^6 + 18T^5 + 144T^4 + 224T^3 + 432T^2 + 288*T + 64
$47$	\( T^{6} + 9 T^{5} + \cdots + 81 \) T^6 + 9T^5 + 9T^4 - 72T^3 + 162T^2 - 81*T + 81
$53$	\( (T^{3} - 6 T^{2} - 9 T - 3)^{2} \) (T^3 - 6T^2 - 9T - 3)^2
$59$	\( T^{6} - 6 T^{5} + \cdots + 3249 \) T^6 - 6T^5 + 36T^4 - 84T^3 + 954T^2 + 3591*T + 3249
$61$	\( T^{6} - 18 T^{5} + \cdots + 2809 \) T^6 - 18T^5 + 153T^4 - 379T^3 + 2124T^2 - 4293*T + 2809
$67$	\( T^{6} + 9 T^{5} + \cdots + 1 \) T^6 + 9T^5 + 45T^4 + 80T^3 + 576T^2 + 45*T + 1
$71$	\( T^{6} + 12 T^{5} + \cdots + 106929 \) T^6 + 12T^5 + 189T^4 + 114T^3 + 5949T^2 + 14715*T + 106929
$73$	\( T^{6} - 3 T^{5} + \cdots + 72361 \) T^6 - 3T^5 + 123T^4 - 196T^3 + 13803T^2 - 30666*T + 72361
$79$	\( T^{6} - 33 T^{5} + \cdots + 466489 \) T^6 - 33T^5 + 510T^4 - 4168T^3 + 25707T^2 - 145479*T + 466489
$83$	\( T^{6} - 18 T^{5} + \cdots + 5184 \) T^6 - 18T^5 + 144T^4 - 720T^3 + 2592T^2 - 5184*T + 5184
$89$	\( T^{6} - 15 T^{5} + \cdots + 25281 \) T^6 - 15T^5 + 189T^4 - 858T^3 + 3681T^2 + 5724*T + 25281
$97$	\( T^{6} + 12 T^{5} + \cdots + 16129 \) T^6 + 12T^5 + 51T^4 - 469T^3 + 1092T^2 - 4953*T + 16129
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\(n\)	\(83\)
\(\chi(n)\)	\(-\zeta_{18}^{5}\)