LMFDB - Newform orbit 77.2.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(23,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("77.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.e (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:	$0.614848095564$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-\zeta_{18}^{3}$	$1$

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} $

23.1

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

−0.939693

1.62760i

−0.326352

−

0.565258i

−0.766044

−

1.32683i

−1.76604

3.05888i

1.22668

0.418748

2.61240i

−0.879385

1.28699

−

2.22913i

−3.31908

−

5.74881i

23.2

0.173648

−

0.300767i

0.266044

0.460802i

0.939693

1.62760i

−0.0603074

0.104455i

0.184793

−2.47178

−

0.943555i

1.34730

1.35844

−

2.35289i

0.0209445

0.0362770i

23.3

0.766044

−

1.32683i

−1.43969

−

2.49362i

−0.173648

−

0.300767i

−1.17365

2.03282i

−4.41147

2.05303

−

1.66885i

2.53209

−2.64543

4.58202i

1.79813

3.11446i

67.1

−0.939693

−

1.62760i

−0.326352

0.565258i

−0.766044

1.32683i

−1.76604

−

3.05888i

1.22668

0.418748

−

2.61240i

−0.879385

1.28699

2.22913i

−3.31908

5.74881i

67.2

0.173648

0.300767i

0.266044

−

0.460802i

0.939693

−

1.62760i

−0.0603074

−

0.104455i

0.184793

−2.47178

0.943555i

1.34730

1.35844

2.35289i

0.0209445

−

0.0362770i

67.3

0.766044

1.32683i

−1.43969

2.49362i

−0.173648

0.300767i

−1.17365

−

2.03282i

−4.41147

2.05303

1.66885i

2.53209

−2.64543

−

4.58202i

1.79813

−

3.11446i

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	77.2.e.a	✓	6
3.b	odd	2	1		693.2.i.h		6
4.b	odd	2	1		1232.2.q.m		6
7.b	odd	2	1		539.2.e.m		6
7.c	even	3	1	inner	77.2.e.a	✓	6
7.c	even	3	1		539.2.a.j		3
7.d	odd	6	1		539.2.a.g		3
7.d	odd	6	1		539.2.e.m		6
11.b	odd	2	1		847.2.e.c		6
11.c	even	5	4		847.2.n.g		24
11.d	odd	10	4		847.2.n.f		24
21.g	even	6	1		4851.2.a.bk		3
21.h	odd	6	1		693.2.i.h		6
21.h	odd	6	1		4851.2.a.bj		3
28.f	even	6	1		8624.2.a.co		3
28.g	odd	6	1		1232.2.q.m		6
28.g	odd	6	1		8624.2.a.ch		3
77.h	odd	6	1		847.2.e.c		6
77.h	odd	6	1		5929.2.a.x		3
77.i	even	6	1		5929.2.a.u		3
77.m	even	15	4		847.2.n.g		24
77.o	odd	30	4		847.2.n.f		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	1.a	even	1	1	trivial
77.2.e.a	✓	6	7.c	even	3	1	inner
539.2.a.g		3	7.d	odd	6	1
539.2.a.j		3	7.c	even	3	1
539.2.e.m		6	7.b	odd	2	1
539.2.e.m		6	7.d	odd	6	1
693.2.i.h		6	3.b	odd	2	1
693.2.i.h		6	21.h	odd	6	1
847.2.e.c		6	11.b	odd	2	1
847.2.e.c		6	77.h	odd	6	1
847.2.n.f		24	11.d	odd	10	4
847.2.n.f		24	77.o	odd	30	4
847.2.n.g		24	11.c	even	5	4
847.2.n.g		24	77.m	even	15	4
1232.2.q.m		6	4.b	odd	2	1
1232.2.q.m		6	28.g	odd	6	1
4851.2.a.bj		3	21.h	odd	6	1
4851.2.a.bk		3	21.g	even	6	1
5929.2.a.u		3	77.i	even	6	1
5929.2.a.x		3	77.h	odd	6	1
8624.2.a.ch		3	28.g	odd	6	1
8624.2.a.co		3	28.f	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} + 3T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1 $ T2^6 + 3*T2^4 - 2*T2^3 + 9*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(77, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$3$	$ T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1 $ T^6 + 3T^5 + 9T^4 + 2T^3 + 3T^2 + 1
$5$	$ T^{6} + 6 T^{5} + 27 T^{4} + 52 T^{3} + \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} + 45 T^{4} + \cdots + 16129 $ T^6 - 3T^5 + 45T^4 - 146T^3 + 1677T^2 - 4572*T + 16129
$19$	$ T^{6} + 9 T^{5} + 63 T^{4} + 144 T^{3} + \cdots + 81 $ T^6 + 9T^5 + 63T^4 + 144T^3 + 243T^2 + 162*T + 81
$23$	$ T^{6} + 57 T^{4} - 214 T^{3} + \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} + 57 T^{4} + 178 T^{3} + \cdots + 361 $ T^6 + 9T^5 + 57T^4 + 178T^3 + 405T^2 + 456*T + 361
$37$	$ T^{6} + 36 T^{4} + 144 T^{3} + \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} + 87 T^{4} + \cdots + 104329 $ T^6 - 3T^5 + 87T^4 - 412T^3 + 7053T^2 - 25194*T + 104329
$53$	$ T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 210681 $ T^6 + 9T^5 + 135T^4 + 432T^3 + 7047T^2 + 24786*T + 210681
$59$	$ T^{6} + 93 T^{4} + 38 T^{3} + \cdots + 361 $ T^6 + 93T^4 + 38T^3 + 8649T^2 + 1767T + 361
$61$	$ T^{6} + 12 T^{5} + 180 T^{4} + \cdots + 576 $ T^6 + 12T^5 + 180T^4 - 480T^3 + 1008T^2 - 864*T + 576
$67$	$ T^{6} + 12 T^{4} - 16 T^{3} + 144 T^{2} + \cdots + 64 $ T^6 + 12T^4 - 16T^3 + 144T^2 - 96T + 64
$71$	$ (T^{3} + 9 T^{2} - 90 T - 801)^{2} $ (T^3 + 9T^2 - 90T - 801)^2
$73$	$ T^{6} + 6 T^{5} + 27 T^{4} + 52 T^{3} + \cdots + 1 $ T^6 + 6T^5 + 27T^4 + 52T^3 + 75T^2 + 9*T + 1
$79$	$ T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369 $ T^6 - 3T^5 + 123T^4 + 416T^3 + 12885T^2 + 4218*T + 1369
$83$	$ (T^{3} + 15 T^{2} + 18 T - 267)^{2} $ (T^3 + 15T^2 + 18T - 267)^2
$89$	$ T^{6} - 15 T^{5} + 153 T^{4} + \cdots + 12321 $ T^6 - 15T^5 + 153T^4 - 858T^3 + 3519T^2 - 7992*T + 12321
$97$	$ (T^{3} - 45 T^{2} + 672 T - 3329)^{2} $ (T^3 - 45T^2 + 672T - 3329)^2
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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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - \zeta_{18}) q^{2} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{4} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{6} + (2 \zeta_{18}^{4} - 3 \zeta_{18}) q^{7} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{8} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{9}+O(q^{10}) \) q + (-z^5 + z^4 + z^2 - z) * q^2 + (-z^5 - z^4 + z^3 + z^2 - 1) * q^3 + (-z^5 - z^4 + z) * q^4 + (z^4 - 2z^3 + z^2) q^5 + (z^5 - 2z^4 + z^2 + z - 1) q^6 + (2z^4 - 3z) * q^7 + (-z^5 + z^4 + 1) * q^8 + (3z^5 - z^4 - z^2 + 3z) * q^9	\( q + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - \zeta_{18}) q^{2} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{4} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{5} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{6} + (2 \zeta_{18}^{4} - 3 \zeta_{18}) q^{7} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{8} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{9} + (\zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{10} + (\zeta_{18}^{3} - 1) q^{11} - \zeta_{18}^{3} q^{12} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 1) q^{13} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 3) q^{14} + (2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{15} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18}) q^{16} + (4 \zeta_{18}^{5} + 4 \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} - 3 \zeta_{18} + 1) q^{17} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - \zeta_{18} + 4) q^{18} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{19} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 2) q^{20} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} - 3) q^{21} + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{22} + (5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 5 \zeta_{18}) q^{23} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - \zeta_{18} - 2) q^{24} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 4 \zeta_{18} - 1) q^{25} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18}) q^{26} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2) q^{27} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} - 1) q^{28} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{29} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18}) q^{30} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18} - 3) q^{31} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{32} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}) q^{33} + ( - 4 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} - 2) q^{34} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 4 \zeta_{18} - 2) q^{35} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{36} + ( - 4 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 4 \zeta_{18}) q^{37} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{38} + (\zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18} - 1) q^{39} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{40} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} + 3) q^{41} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{42} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}) q^{43} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{44} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 5 \zeta_{18} - 1) q^{45} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} + 8) q^{46} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{47} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 6) q^{48} + ( - 8 \zeta_{18}^{5} + 5 \zeta_{18}^{2}) q^{49} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 5) q^{50} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 6 \zeta_{18}^{3} + \zeta_{18}^{2} - 3 \zeta_{18}) q^{51} + (3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{52} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 6 \zeta_{18} - 3) q^{53} + ( - 2 \zeta_{18}^{5} - \zeta_{18}^{4} + 6 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18}) q^{54} + (\zeta_{18}^{5} - \zeta_{18}^{2} - \zeta_{18} + 2) q^{55} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{56} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{57} + (4 \zeta_{18}^{5} - 5 \zeta_{18}^{3} + 4 \zeta_{18}) q^{58} + ( - 5 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - \zeta_{18}^{2} + 6 \zeta_{18}) q^{59} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18} - 2) q^{60} + (6 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 6 \zeta_{18}) q^{61} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 1) q^{62} + (7 \zeta_{18}^{5} - 8 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 5) q^{63} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{2} - 3 \zeta_{18} - 1) q^{64} + ( - 4 \zeta_{18}^{5} + 6 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 4 \zeta_{18}) q^{65} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{66} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2}) q^{67} + (3 \zeta_{18}^{5} + 7 \zeta_{18}^{3} + 3 \zeta_{18}) q^{68} + ( - 5 \zeta_{18}^{5} + 10 \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} + 7) q^{69} + (6 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 9 \zeta_{18}^{2} + \zeta_{18} + 6) q^{70} + (7 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 3) q^{71} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{3} + 5 \zeta_{18}) q^{72} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{73} + ( - 4 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 6) q^{74} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18}) q^{75} + (4 \zeta_{18}^{5} + \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} - 3) q^{76} + ( - 3 \zeta_{18}^{4} + \zeta_{18}) q^{77} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 2) q^{78} + (2 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 2 \zeta_{18}) q^{79} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 9 \zeta_{18}^{3} - 5 \zeta_{18} + 9) q^{80} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} - 5) q^{81} + (\zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2}) q^{82} + ( - 3 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 5) q^{83} + (\zeta_{18}^{4} + 2 \zeta_{18}) q^{84} + ( - 6 \zeta_{18}^{5} + \zeta_{18}^{4} + 5 \zeta_{18}^{2} + 5 \zeta_{18} - 9) q^{85} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{86} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - 9 \zeta_{18}^{2} + 4 \zeta_{18} + 8) q^{87} + (\zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{88} + (\zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{89} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 7 \zeta_{18}^{2} + 7 \zeta_{18} - 12) q^{90} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 3 \zeta_{18} + 4) q^{91} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} - 1) q^{92} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 3 \zeta_{18}) q^{93} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 7 \zeta_{18}^{2} - 4 \zeta_{18}) q^{94} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{95} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{96} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} + 15) q^{97} + ( - 8 \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 5 \zeta_{18} - 5) q^{98} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{99} +O(q^{100}) \) q + (-z^5 + z^4 + z^2 - z) * q^2 + (-z^5 - z^4 + z^3 + z^2 - 1) * q^3 + (-z^5 - z^4 + z) * q^4 + (z^4 - 2z^3 + z^2) q^5 + (z^5 - 2z^4 + z^2 + z - 1) q^6 + (2z^4 - 3z) * q^7 + (-z^5 + z^4 + 1) * q^8 + (3z^5 - z^4 - z^2 + 3z) * q^9 + (z^3 - 3z^2 + 3z - 1) * q^10 + (z^3 - 1) * q^11 - z^3 * q^12 + (2z^5 - z^4 - z^2 - z + 1) q^13 + (-3z^5 + 2z^3 + z^2 - 3) * q^14 + (2z^4 - 2z^2 - 2z + 3) q^15 + (-z^5 - 2z^4 + 2z^3 - 2z^2 - z) q^16 + (4z^5 + 4z^4 - z^3 - z^2 - 3z + 1) q^17 + (3z^5 + 3z^4 - 4z^3 - 2z^2 - z + 4) * q^18 + (-z^5 - z^4 - 3z^3 - z^2 - z) q^19 + (2z^5 - z^4 - z^2 - z + 2) q^20 + (z^5 - 3z^4 + 2z^3 + 2z^2 + z - 3) q^21 + (z^5 - z^4) * q^22 + (5z^5 - 3z^4 - 3z^2 + 5z) * q^23 + (-2z^5 - 2z^4 + 2z^3 + 3z^2 - z - 2) * q^24 + (-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^4 + 2) * q^27 + (3z^5 + 3z^3 - z^2 - 1) * q^28 + (z^5 - 4z^4 + 3z^2 + 3z - 1) q^29 + (-5z^5 + 3z^4 + 2z^3 + 3z^2 - 5z) q^30 + (z^5 + z^4 + 3z^3 - z - 3) q^31 + (-z^5 - z^4 - 3z^3 + 3z^2 - 2z + 3) q^32 + (z^5 - z^3 + z) * q^33 + (-4z^5 + 5z^4 - z^2 - z - 2) * q^34 + (-z^5 + 2z^4 - z^3 - 2z^2 + 4z - 2) q^35 + (-3z^5 + z^4 + 2z^2 + 2z + 1) q^36 + (-4z^5 + 2z^4 + 2z^2 - 4z) * q^37 + (-z^5 - z^4 - z^2 + 2z) q^38 + (z^3 - z^2 + z - 1) * q^39 + (3z^5 - 2z^4 - z^3 - 2z^2 + 3z) * q^40 + (-2z^5 - z^4 + 3z^2 + 3z + 3) q^41 + (4z^5 - 2z^4 - 4z^3 + z^2 + 3z - 1) * q^42 + (-2z^5 + z^4 + z^2 + z) q^43 + (z^4 + z^2) * q^44 + (-2z^5 - 2z^4 + z^3 + 7z^2 - 5z - 1) * q^45 + (5z^5 + 5z^4 - 8z^3 - 2z^2 - 3z + 8) q^46 + (-3z^5 - 3z^4 + z^3 - 3z^2 - 3z) * q^47 + (z^5 - 4z^4 + 3z^2 + 3z - 6) q^48 + (-8z^5 + 5z^2) * q^49 + (5z^5 - 4z^4 - z^2 - z + 5) * q^50 + (-3z^5 + z^4 + 6z^3 + z^2 - 3z) q^51 + (3z^3 - 2z^2 + 2z - 3) q^52 + (3z^5 + 3z^4 + 3z^3 + 3z^2 - 6z - 3) q^53 + (-2z^5 - z^4 + 6z^3 - z^2 - 2z) q^54 + (z^5 - z^2 - z + 2) * q^55 + (-z^5 + 2z^4 + 3z^3 - 2z^2 - 3z - 1) * q^56 + (z^5 + z^4 - 2z^2 - 2z) * q^57 + (4z^5 - 5z^3 + 4z) q^58 + (-5z^5 - 5z^4 - z^2 + 6z) q^59 + (-z^5 - z^4 + 2z^3 + z - 2) q^60 + (6z^5 - 2z^4 - 4z^3 - 2z^2 + 6z) q^61 + (2z^5 - 2z^4 - 1) * q^62 + (7z^5 - 8z^3 - 7z^2 + 5) q^63 + (-z^5 + 4z^4 - 3z^2 - 3z - 1) q^64 + (-4z^5 + 6z^4 - 5z^3 + 6z^2 - 4z) q^65 + (z^5 + z^4 - z^3 - 2z^2 + z + 1) q^66 + (-2z^5 - 2z^4 + 2z^2) q^67 + (3z^5 + 7z^3 + 3z) q^68 + (-5z^5 + 10z^4 - 5z^2 - 5z + 7) * q^69 + (6z^5 - 3z^4 + 3z^3 - 9z^2 + z + 6) * q^70 + (7z^5 - 5z^4 - 2z^2 - 2z - 3) * q^71 + (5z^5 - 4z^3 + 5z) q^72 + (z^5 + z^4 + 2z^3 - z^2 - 2) q^73 + (-4z^5 - 4z^4 + 6z^3 + 2z^2 + 2z - 6) q^74 + (-z^5 + z^4 - 3z^3 + z^2 - z) q^75 + (4z^5 + z^4 - 5z^2 - 5z - 3) q^76 + (-3z^4 + z) q^77 + (2z^5 - z^4 - z^2 - z + 2) q^78 + (2z^5 - 7z^4 + z^3 - 7z^2 + 2z) * q^79 + (5z^5 + 5z^4 - 9z^3 - 5z + 9) * q^80 + (z^5 + z^4 + 5z^3 - z^2 - 5) q^81 + (z^4 + z^3 + z^2) * q^82 + (-3z^5 + 5z^4 - 2z^2 - 2z - 5) * q^83 + (z^4 + 2z) q^84 + (-6z^5 + z^4 + 5z^2 + 5z - 9) q^85 + (z^5 - 2z^4 + 3z^3 - 2z^2 + z) q^86 + (5z^5 + 5z^4 - 8z^3 - 9z^2 + 4z + 8) q^87 + (z^3 + z^2 - z - 1) * q^88 + (z^4 + 5z^3 + z^2) q^89 + (-4z^5 - 3z^4 + 7z^2 + 7z - 12) * q^90 + (-z^5 + 2z^4 - 5z^3 + 5z^2 - 3z + 4) * q^91 + (-5z^5 + 3z^4 + 2z^2 + 2z - 1) * q^92 + (3z^5 - 2z^3 + 3z) q^93 + (-3z^5 - 3z^4 + 7z^2 - 4z) * q^94 + (-z^5 - z^4 + 3z^3 - z^2 + 2z - 3) * q^95 + (3z^5 - 3z^4 - z^3 - 3z^2 + 3z) * q^96 + (-z^4 + z^2 + z + 15) * q^97 + (-8z^4 + 8z^3 + 5z - 5) q^98 + (-z^5 + 3z^4 - 2z^2 - 2z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} - 6 q^{6} + 6 q^{8}+O(q^{10}) \) 6 * q - 3 * q^3 - 6 * q^5 - 6 * q^6 + 6 * q^8	\( 6 q - 3 q^{3} - 6 q^{5} - 6 q^{6} + 6 q^{8} - 3 q^{10} - 3 q^{11} - 3 q^{12} + 6 q^{13} - 12 q^{14} + 18 q^{15} + 6 q^{16} + 3 q^{17} + 12 q^{18} - 9 q^{19} + 12 q^{20} - 12 q^{21} - 6 q^{24} - 3 q^{25} - 9 q^{26} + 12 q^{27} + 3 q^{28} - 6 q^{29} + 6 q^{30} - 9 q^{31} + 9 q^{32} - 3 q^{33} - 12 q^{34} - 15 q^{35} + 6 q^{36} - 3 q^{39} - 3 q^{40} + 18 q^{41} - 18 q^{42} - 3 q^{45} + 24 q^{46} + 3 q^{47} - 36 q^{48} + 30 q^{50} + 18 q^{51} - 9 q^{52} - 9 q^{53} + 18 q^{54} + 12 q^{55} + 3 q^{56} - 15 q^{58} - 6 q^{60} - 12 q^{61} - 6 q^{62} + 6 q^{63} - 6 q^{64} - 15 q^{65} + 3 q^{66} + 21 q^{68} + 42 q^{69} + 45 q^{70} - 18 q^{71} - 12 q^{72} - 6 q^{73} - 18 q^{74} - 9 q^{75} - 18 q^{76} + 12 q^{78} + 3 q^{79} + 27 q^{80} - 15 q^{81} + 3 q^{82} - 30 q^{83} - 54 q^{85} + 9 q^{86} + 24 q^{87} - 3 q^{88} + 15 q^{89} - 72 q^{90} + 9 q^{91} - 6 q^{92} - 6 q^{93} - 9 q^{95} - 3 q^{96} + 90 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 - 6 * q^6 + 6 * q^8 - 3 * q^10 - 3 * q^11 - 3 * q^12 + 6 * q^13 - 12 * q^14 + 18 * q^15 + 6 * q^16 + 3 * q^17 + 12 * q^18 - 9 * q^19 + 12 * q^20 - 12 * q^21 - 6 * q^24 - 3 * q^25 - 9 * q^26 + 12 * q^27 + 3 * q^28 - 6 * q^29 + 6 * q^30 - 9 * q^31 + 9 * q^32 - 3 * q^33 - 12 * q^34 - 15 * q^35 + 6 * q^36 - 3 * q^39 - 3 * q^40 + 18 * q^41 - 18 * q^42 - 3 * q^45 + 24 * q^46 + 3 * q^47 - 36 * q^48 + 30 * q^50 + 18 * q^51 - 9 * q^52 - 9 * q^53 + 18 * q^54 + 12 * q^55 + 3 * q^56 - 15 * q^58 - 6 * q^60 - 12 * q^61 - 6 * q^62 + 6 * q^63 - 6 * q^64 - 15 * q^65 + 3 * q^66 + 21 * q^68 + 42 * q^69 + 45 * q^70 - 18 * q^71 - 12 * q^72 - 6 * q^73 - 18 * q^74 - 9 * q^75 - 18 * q^76 + 12 * q^78 + 3 * q^79 + 27 * q^80 - 15 * q^81 + 3 * q^82 - 30 * q^83 - 54 * q^85 + 9 * q^86 + 24 * q^87 - 3 * q^88 + 15 * q^89 - 72 * q^90 + 9 * q^91 - 6 * q^92 - 6 * q^93 - 9 * q^95 - 3 * q^96 + 90 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	77.2.e.a

Level	$77$
Weight	$2$
Character orbit	77.e
Analytic conductor	$0.615$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.614848095564\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 67.3
Significant digits:
Format:

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