LMFDB - Newform orbit 77.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(4,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([20, 6]))

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

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

chi := DirichletCharacter("77.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.m (of order $15$, degree $8$, 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:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

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

$n$	$45$	$57$
$\chi(n)$	$-1 - \zeta_{15}^{5}$	$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$

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} $

4.1

−0.978148	+	0.207912i
−0.104528	−	0.994522i
0.913545	−	0.406737i
0.669131	−	0.743145i
0.669131	+	0.743145i
0.913545	+	0.406737i
−0.978148	−	0.207912i
−0.104528	+	0.994522i

0.0646021

−

0.614648i

0.669131

0.743145i

1.58268

0.336408i

−2.04275

−

0.909491i

0.500000

−

0.363271i

−0.0510966

2.64526i

0.690983

−

2.12663i

0.209057

−

1.98904i

−0.690983

1.19682i

9.1

1.08268

1.20243i

0.913545

0.406737i

−0.0646021

0.614648i

−2.18720

−

0.464905i

0.500000

1.53884i

−2.53158

−

0.768834i

1.80902

−

1.31433i

−1.33826

−

1.48629i

−1.80902

−

3.13331i

16.1

−1.58268

−

0.336408i

−0.104528

0.994522i

0.564602

0.251377i

1.49622

1.66172i

0.500000

−

1.53884i

−1.51351

2.17009i

1.80902

1.31433i

1.95630

0.415823i

−1.80902

−

3.13331i

25.1

−0.564602

−

0.251377i

−0.978148

−

0.207912i

−1.08268

−

1.20243i

0.233733

−

2.22382i

0.500000

0.363271i

1.59618

−

2.11002i

0.690983

2.12663i

−1.82709

−

0.813473i

−0.690983

1.19682i

37.1

−0.564602

0.251377i

−0.978148

0.207912i

−1.08268

1.20243i

0.233733

2.22382i

0.500000

−

0.363271i

1.59618

2.11002i

0.690983

−

2.12663i

−1.82709

0.813473i

−0.690983

−

1.19682i

53.1

−1.58268

0.336408i

−0.104528

−

0.994522i

0.564602

−

0.251377i

1.49622

−

1.66172i

0.500000

1.53884i

−1.51351

−

2.17009i

1.80902

−

1.31433i

1.95630

−

0.415823i

−1.80902

3.13331i

58.1

0.0646021

0.614648i

0.669131

−

0.743145i

1.58268

−

0.336408i

−2.04275

0.909491i

0.500000

0.363271i

−0.0510966

−

2.64526i

0.690983

2.12663i

0.209057

1.98904i

−0.690983

−

1.19682i

60.1

1.08268

−

1.20243i

0.913545

−

0.406737i

−0.0646021

−

0.614648i

−2.18720

0.464905i

0.500000

−

1.53884i

−2.53158

0.768834i

1.80902

1.31433i

−1.33826

1.48629i

−1.80902

3.13331i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.c	even	5	1	inner
77.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.2.m.a	✓	8
3.b	odd	2	1		693.2.by.a		8
7.b	odd	2	1		539.2.q.a		8
7.c	even	3	1	inner	77.2.m.a	✓	8
7.c	even	3	1		539.2.f.a		4
7.d	odd	6	1		539.2.f.b		4
7.d	odd	6	1		539.2.q.a		8
11.b	odd	2	1		847.2.n.b		8
11.c	even	5	1	inner	77.2.m.a	✓	8
11.c	even	5	1		847.2.e.a		4
11.c	even	5	2		847.2.n.c		8
11.d	odd	10	1		847.2.e.b		4
11.d	odd	10	2		847.2.n.a		8
11.d	odd	10	1		847.2.n.b		8
21.h	odd	6	1		693.2.by.a		8
33.h	odd	10	1		693.2.by.a		8
77.h	odd	6	1		847.2.n.b		8
77.j	odd	10	1		539.2.q.a		8
77.m	even	15	1	inner	77.2.m.a	✓	8
77.m	even	15	1		539.2.f.a		4
77.m	even	15	1		847.2.e.a		4
77.m	even	15	2		847.2.n.c		8
77.m	even	15	1		5929.2.a.q		2
77.n	even	30	1		5929.2.a.j		2
77.o	odd	30	1		847.2.e.b		4
77.o	odd	30	2		847.2.n.a		8
77.o	odd	30	1		847.2.n.b		8
77.o	odd	30	1		5929.2.a.l		2
77.p	odd	30	1		539.2.f.b		4
77.p	odd	30	1		539.2.q.a		8
77.p	odd	30	1		5929.2.a.o		2
231.z	odd	30	1		693.2.by.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.m.a	✓	8	1.a	even	1	1	trivial
77.2.m.a	✓	8	7.c	even	3	1	inner
77.2.m.a	✓	8	11.c	even	5	1	inner
77.2.m.a	✓	8	77.m	even	15	1	inner
539.2.f.a		4	7.c	even	3	1
539.2.f.a		4	77.m	even	15	1
539.2.f.b		4	7.d	odd	6	1
539.2.f.b		4	77.p	odd	30	1
539.2.q.a		8	7.b	odd	2	1
539.2.q.a		8	7.d	odd	6	1
539.2.q.a		8	77.j	odd	10	1
539.2.q.a		8	77.p	odd	30	1
693.2.by.a		8	3.b	odd	2	1
693.2.by.a		8	21.h	odd	6	1
693.2.by.a		8	33.h	odd	10	1
693.2.by.a		8	231.z	odd	30	1
847.2.e.a		4	11.c	even	5	1
847.2.e.a		4	77.m	even	15	1
847.2.e.b		4	11.d	odd	10	1
847.2.e.b		4	77.o	odd	30	1
847.2.n.a		8	11.d	odd	10	2
847.2.n.a		8	77.o	odd	30	2
847.2.n.b		8	11.b	odd	2	1
847.2.n.b		8	11.d	odd	10	1
847.2.n.b		8	77.h	odd	6	1
847.2.n.b		8	77.o	odd	30	1
847.2.n.c		8	11.c	even	5	2
847.2.n.c		8	77.m	even	15	2
5929.2.a.j		2	77.n	even	30	1
5929.2.a.l		2	77.o	odd	30	1
5929.2.a.o		2	77.p	odd	30	1
5929.2.a.q		2	77.m	even	15	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} + 2T_{2}^{7} + 2T_{2}^{5} + 9T_{2}^{4} + 8T_{2}^{3} + 5T_{2}^{2} + 3T_{2} + 1 $ T2^8 + 2*T2^7 + 2*T2^5 + 9*T2^4 + 8*T2^3 + 5*T2^2 + 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(77, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 2T^5 + 9T^4 + 8T^3 + 5T^2 + 3T + 1
$3$	$ T^{8} - T^{7} + T^{5} + \cdots + 1 $ T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	$ T^{8} + 5 T^{7} + \cdots + 625 $ T^8 + 5T^7 + 10T^6 + 25T^5 + 75T^4 + 125T^3 + 250T^2 + 625*T + 625
$7$	$ T^{8} + 5 T^{7} + \cdots + 2401 $ T^8 + 5T^7 + 18T^6 + 55T^5 + 149T^4 + 385T^3 + 882T^2 + 1715*T + 2401
$11$	$ T^{8} - 4 T^{7} + \cdots + 14641 $ T^8 - 4T^7 + 10T^6 + 64T^5 - 261T^4 + 704T^3 + 1210T^2 - 5324*T + 14641
$13$	$ (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} $ (T^4 + 5T^3 + 40T^2 + 50*T + 25)^2
$17$	$ T^{8} + 4 T^{7} + \cdots + 256 $ T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$19$	$ T^{8} + 3 T^{7} + \cdots + 6561 $ T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$23$	$ (T^{4} - 8 T^{3} + \cdots + 121)^{2} $ (T^4 - 8T^3 + 53T^2 - 88*T + 121)^2
$29$	$ (T^{4} - 12 T^{3} + \cdots + 81)^{2} $ (T^4 - 12T^3 + 54T^2 + 27*T + 81)^2
$31$	$ T^{8} - 8 T^{7} + \cdots + 707281 $ T^8 - 8T^7 + 30T^6 - 98T^5 - 381T^4 + 10498T^3 - 21025T^2 - 73167*T + 707281
$37$	$ T^{8} + 13 T^{7} + \cdots + 923521 $ T^8 + 13T^7 + 75T^6 + 478T^5 + 3039T^4 + 9982T^3 + 48050T^2 + 357492*T + 923521
$41$	$ (T^{4} - T^{3} + 16 T^{2} + \cdots + 121)^{2} $ (T^4 - T^3 + 16T^2 - 66T + 121)^2
$43$	$ (T^{2} - 7 T + 1)^{4} $ (T^2 - 7*T + 1)^4
$47$	$ T^{8} - 6 T^{7} + \cdots + 256 $ T^8 - 6T^7 - 40T^6 - 344T^5 + 5424T^4 - 4064T^3 + 1920T^2 - 896*T + 256
$53$	$ T^{8} + 12 T^{7} + \cdots + 923521 $ T^8 + 12T^7 + 50T^6 + 322T^5 + 3039T^4 + 14818T^3 + 72075T^2 + 387283*T + 923521
$59$	$ T^{8} + 18 T^{7} + \cdots + 1679616 $ T^8 + 18T^7 + 180T^6 + 1728T^5 + 11664T^4 + 15552T^3 + 559872T + 1679616
$61$	$ T^{8} - 18 T^{7} + \cdots + 33362176 $ T^8 - 18T^7 + 80T^6 - 1048T^5 + 23664T^4 - 200032T^3 + 1386240T^2 - 9657472*T + 33362176
$67$	$ (T^{4} + 19 T^{3} + \cdots + 7921)^{2} $ (T^4 + 19T^3 + 272T^2 + 1691*T + 7921)^2
$71$	$ (T^{4} + 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} $ (T^4 + 8T^3 + 24T^2 - 8*T + 16)^2
$73$	$ T^{8} - 15 T^{7} + \cdots + 4100625 $ T^8 - 15T^7 + 135T^6 - 1350T^5 + 6075T^4 + 60750T^3 - 182250T^2 + 4100625
$79$	$ T^{8} - 9 T^{7} + \cdots + 6561 $ T^8 - 9T^7 - 90T^6 - 1161T^5 + 27459T^4 - 30861T^3 + 21870T^2 - 15309*T + 6561
$83$	$ (T^{4} - 9 T^{3} + \cdots + 6561)^{2} $ (T^4 - 9T^3 + 81T^2 - 729*T + 6561)^2
$89$	$ (T^{4} + 12 T^{3} + \cdots + 256)^{2} $ (T^4 + 12T^3 + 128T^2 + 192*T + 256)^2
$97$	$ (T^{4} + 7 T^{3} + \cdots + 121)^{2} $ (T^4 + 7T^3 + 34T^2 + 88*T + 121)^2
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Classical	Maass
Hilbert	Bianchi

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.m (of order \(15\), degree \(8\), minimal)

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

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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	77.2.m.a

Level	$77$
Weight	$2$
Character orbit	77.m
Analytic conductor	$0.615$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.614848095564\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(-1 - \zeta_{15}^{5}\)	\(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 4.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 2T^5 + 9T^4 + 8T^3 + 5T^2 + 3T + 1
$3$	\( T^{8} - T^{7} + T^{5} + \cdots + 1 \) T^8 - T^7 + T^5 - T^4 + T^3 - T + 1
$5$	\( T^{8} + 5 T^{7} + \cdots + 625 \) T^8 + 5T^7 + 10T^6 + 25T^5 + 75T^4 + 125T^3 + 250T^2 + 625*T + 625
$7$	\( T^{8} + 5 T^{7} + \cdots + 2401 \) T^8 + 5T^7 + 18T^6 + 55T^5 + 149T^4 + 385T^3 + 882T^2 + 1715*T + 2401
$11$	\( T^{8} - 4 T^{7} + \cdots + 14641 \) T^8 - 4T^7 + 10T^6 + 64T^5 - 261T^4 + 704T^3 + 1210T^2 - 5324*T + 14641
$13$	\( (T^{4} + 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \) (T^4 + 5T^3 + 40T^2 + 50*T + 25)^2
$17$	\( T^{8} + 4 T^{7} + \cdots + 256 \) T^8 + 4T^7 + 16T^5 + 144T^4 + 256T^3 + 320T^2 + 384T + 256
$19$	\( T^{8} + 3 T^{7} + \cdots + 6561 \) T^8 + 3T^7 - 45T^6 + 378T^5 + 3159T^4 - 6318T^3 + 7290T^2 - 8748*T + 6561
$23$	\( (T^{4} - 8 T^{3} + \cdots + 121)^{2} \) (T^4 - 8T^3 + 53T^2 - 88*T + 121)^2
$29$	\( (T^{4} - 12 T^{3} + \cdots + 81)^{2} \) (T^4 - 12T^3 + 54T^2 + 27*T + 81)^2
$31$	\( T^{8} - 8 T^{7} + \cdots + 707281 \) T^8 - 8T^7 + 30T^6 - 98T^5 - 381T^4 + 10498T^3 - 21025T^2 - 73167*T + 707281
$37$	\( T^{8} + 13 T^{7} + \cdots + 923521 \) T^8 + 13T^7 + 75T^6 + 478T^5 + 3039T^4 + 9982T^3 + 48050T^2 + 357492*T + 923521
$41$	\( (T^{4} - T^{3} + 16 T^{2} + \cdots + 121)^{2} \) (T^4 - T^3 + 16T^2 - 66T + 121)^2
$43$	\( (T^{2} - 7 T + 1)^{4} \) (T^2 - 7*T + 1)^4
$47$	\( T^{8} - 6 T^{7} + \cdots + 256 \) T^8 - 6T^7 - 40T^6 - 344T^5 + 5424T^4 - 4064T^3 + 1920T^2 - 896*T + 256
$53$	\( T^{8} + 12 T^{7} + \cdots + 923521 \) T^8 + 12T^7 + 50T^6 + 322T^5 + 3039T^4 + 14818T^3 + 72075T^2 + 387283*T + 923521
$59$	\( T^{8} + 18 T^{7} + \cdots + 1679616 \) T^8 + 18T^7 + 180T^6 + 1728T^5 + 11664T^4 + 15552T^3 + 559872T + 1679616
$61$	\( T^{8} - 18 T^{7} + \cdots + 33362176 \) T^8 - 18T^7 + 80T^6 - 1048T^5 + 23664T^4 - 200032T^3 + 1386240T^2 - 9657472*T + 33362176
$67$	\( (T^{4} + 19 T^{3} + \cdots + 7921)^{2} \) (T^4 + 19T^3 + 272T^2 + 1691*T + 7921)^2
$71$	\( (T^{4} + 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) (T^4 + 8T^3 + 24T^2 - 8*T + 16)^2
$73$	\( T^{8} - 15 T^{7} + \cdots + 4100625 \) T^8 - 15T^7 + 135T^6 - 1350T^5 + 6075T^4 + 60750T^3 - 182250T^2 + 4100625
$79$	\( T^{8} - 9 T^{7} + \cdots + 6561 \) T^8 - 9T^7 - 90T^6 - 1161T^5 + 27459T^4 - 30861T^3 + 21870T^2 - 15309*T + 6561
$83$	\( (T^{4} - 9 T^{3} + \cdots + 6561)^{2} \) (T^4 - 9T^3 + 81T^2 - 729*T + 6561)^2
$89$	\( (T^{4} + 12 T^{3} + \cdots + 256)^{2} \) (T^4 + 12T^3 + 128T^2 + 192*T + 256)^2
$97$	\( (T^{4} + 7 T^{3} + \cdots + 121)^{2} \) (T^4 + 7T^3 + 34T^2 + 88*T + 121)^2
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