LMFDB - Newform orbit 37.2.f.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(7,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("37.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	37.f (of order $9$, degree $6$, 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:	$0.295446487479$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-\zeta_{18}^{5}$

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} $

7.1

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

1.26604

−

0.460802i

−1.76604

−

0.642788i

−0.141559

0.118782i

0.233956

1.32683i

−2.53209

−0.120615

−

0.684040i

−1.47178

2.54920i

0.407604

0.342020i

0.907604

1.57202i

9.1

−0.439693

−

2.49362i

−0.0603074

0.342020i

−4.14543

1.50881i

1.93969

1.62760i

0.879385

−2.34730

−

1.96962i

3.05303

5.28801i

2.70574

0.984808i

3.20574

−

5.55250i

12.1

0.673648

−

0.565258i

−1.17365

−

0.984808i

−0.213011

1.20805i

0.826352

−

0.300767i

−1.34730

−3.53209

1.28558i

1.41875

2.45734i

−0.113341

−

0.642788i

0.386659

−

0.669713i

16.1

1.26604

0.460802i

−1.76604

0.642788i

−0.141559

−

0.118782i

0.233956

−

1.32683i

−2.53209

−0.120615

0.684040i

−1.47178

−

2.54920i

0.407604

−

0.342020i

0.907604

−

1.57202i

33.1

−0.439693

2.49362i

−0.0603074

−

0.342020i

−4.14543

−

1.50881i

1.93969

−

1.62760i

0.879385

−2.34730

1.96962i

3.05303

−

5.28801i

2.70574

−

0.984808i

3.20574

5.55250i

34.1

0.673648

0.565258i

−1.17365

0.984808i

−0.213011

−

1.20805i

0.826352

0.300767i

−1.34730

−3.53209

−

1.28558i

1.41875

−

2.45734i

−0.113341

0.642788i

0.386659

0.669713i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.f	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.2.f.b	✓	6
3.b	odd	2	1		333.2.x.a		6
4.b	odd	2	1		592.2.bc.c		6
5.b	even	2	1		925.2.p.a		6
5.c	odd	4	2		925.2.bc.b		12
37.f	even	9	1	inner	37.2.f.b	✓	6
37.f	even	9	1		1369.2.a.i		3
37.h	even	18	1		1369.2.a.l		3
37.i	odd	36	2		1369.2.b.e		6
111.p	odd	18	1		333.2.x.a		6
148.p	odd	18	1		592.2.bc.c		6
185.x	even	18	1		925.2.p.a		6
185.bd	odd	36	2		925.2.bc.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.2.f.b	✓	6	1.a	even	1	1	trivial
37.2.f.b	✓	6	37.f	even	9	1	inner
333.2.x.a		6	3.b	odd	2	1
333.2.x.a		6	111.p	odd	18	1
592.2.bc.c		6	4.b	odd	2	1
592.2.bc.c		6	148.p	odd	18	1
925.2.p.a		6	5.b	even	2	1
925.2.p.a		6	185.x	even	18	1
925.2.bc.b		12	5.c	odd	4	2
925.2.bc.b		12	185.bd	odd	36	2
1369.2.a.i		3	37.f	even	9	1
1369.2.a.l		3	37.h	even	18	1
1369.2.b.e		6	37.i	odd	36	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 $ T2^6 - 3*T2^5 + 9*T2^4 - 24*T2^3 + 36*T2^2 - 27*T2 + 9 acting on $S_{2}^{\mathrm{new}}(37, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9 $ T^6 - 3T^5 + 9T^4 - 24T^3 + 36T^2 - 27*T + 9
$3$	$ T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 $ T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1
$5$	$ T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 $ T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$7$	$ T^{6} + 12 T^{5} + 60 T^{4} + 152 T^{3} + \cdots + 64 $ T^6 + 12T^5 + 60T^4 + 152T^3 + 192T^2 + 96*T + 64
$11$	$ 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
$13$	$ T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 5329 $ T^6 + 12T^5 + 78T^4 + 476T^3 + 2226T^2 + 5037*T + 5329
$17$	$ T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9 $ T^6 - 3T^5 + 30T^3 + 36*T^2 + 9
$19$	$ T^{6} + 9 T^{5} + 18 T^{4} - 28 T^{3} + \cdots + 1 $ T^6 + 9T^5 + 18T^4 - 28T^3 + 18T^2 + 1
$23$	$ (T^{2} - 6 T + 36)^{3} $ (T^2 - 6*T + 36)^3
$29$	$ T^{6} - 9 T^{5} + 81 T^{4} - 54 T^{3} + \cdots + 729 $ T^6 - 9T^5 + 81T^4 - 54T^3 + 243T^2 + 729
$31$	$ (T^{3} + 9 T^{2} + 6 T - 53)^{2} $ (T^3 + 9T^2 + 6T - 53)^2
$37$	$ T^{6} + 12 T^{5} - 30 T^{4} + \cdots + 50653 $ T^6 + 12T^5 - 30T^4 - 803T^3 - 1110T^2 + 16428*T + 50653
$41$	$ T^{6} - 15 T^{5} + 72 T^{4} - 84 T^{3} + \cdots + 9 $ T^6 - 15T^5 + 72T^4 - 84T^3 + 63T^2 - 27*T + 9
$43$	$ (T^{3} + 9 T^{2} + 24 T + 19)^{2} $ (T^3 + 9T^2 + 24T + 19)^2
$47$	$ T^{6} - 9 T^{5} + 117 T^{4} + 342 T^{3} + \cdots + 81 $ T^6 - 9T^5 + 117T^4 + 342T^3 + 1215T^2 + 324*T + 81
$53$	$ T^{6} - 21 T^{5} + 180 T^{4} + \cdots + 2601 $ T^6 - 21T^5 + 180T^4 - 915T^3 + 3681T^2 - 4590*T + 2601
$59$	$ T^{6} + 6 T^{5} + 18 T^{4} + 3 T^{3} + \cdots + 9 $ T^6 + 6T^5 + 18T^4 + 3T^3 + 90T^2 - 54*T + 9
$61$	$ T^{6} + 9 T^{5} + 45 T^{4} + 152 T^{3} + \cdots + 289 $ T^6 + 9T^5 + 45T^4 + 152T^3 + 342T^2 + 459*T + 289
$67$	$ T^{6} - 24 T^{5} + 258 T^{4} + \cdots + 1369 $ T^6 - 24T^5 + 258T^4 - 1108T^3 + 2010T^2 - 1443*T + 1369
$71$	$ T^{6} + 12 T^{5} + 126 T^{4} + \cdots + 1418481 $ T^6 + 12T^5 + 126T^4 - 651T^3 - 1854T^2 - 171504*T + 1418481
$73$	$ (T^{3} - 39 T - 89)^{2} $ (T^3 - 39*T - 89)^2
$79$	$ T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 104329 $ T^6 + 3T^5 + 114T^4 + 656T^3 + 2631T^2 - 45543*T + 104329
$83$	$ T^{6} - 3 T^{5} - 117 T^{4} + \cdots + 751689 $ T^6 - 3T^5 - 117T^4 - 1104T^3 + 27306T^2 - 85833*T + 751689
$89$	$ T^{6} + 24 T^{5} + 207 T^{4} + \cdots + 3249 $ T^6 + 24T^5 + 207T^4 + 759T^3 + 2196T^2 + 3591*T + 3249
$97$	$ T^{6} + 27 T^{5} + 525 T^{4} + \cdots + 128881 $ T^6 + 27T^5 + 525T^4 + 4790T^3 + 31923T^2 + 73236*T + 128881
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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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

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

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	37.2.f.b

Level	$37$
Weight	$2$
Character orbit	37.f
Analytic conductor	$0.295$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.295446487479\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 34.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9 \) T^6 - 3T^5 + 9T^4 - 24T^3 + 36T^2 - 27*T + 9
$3$	\( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1
$5$	\( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$7$	\( T^{6} + 12 T^{5} + 60 T^{4} + 152 T^{3} + \cdots + 64 \) T^6 + 12T^5 + 60T^4 + 152T^3 + 192T^2 + 96*T + 64
$11$	\( T^{6} - 9 T^{5} + 63 T^{4} - 144 T^{3} + \cdots + 81 \) T^6 - 9T^5 + 63T^4 - 144T^3 + 243T^2 - 162*T + 81
$13$	\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 5329 \) T^6 + 12T^5 + 78T^4 + 476T^3 + 2226T^2 + 5037*T + 5329
$17$	\( T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9 \) T^6 - 3T^5 + 30T^3 + 36*T^2 + 9
$19$	\( T^{6} + 9 T^{5} + 18 T^{4} - 28 T^{3} + \cdots + 1 \) T^6 + 9T^5 + 18T^4 - 28T^3 + 18T^2 + 1
$23$	\( (T^{2} - 6 T + 36)^{3} \) (T^2 - 6*T + 36)^3
$29$	\( T^{6} - 9 T^{5} + 81 T^{4} - 54 T^{3} + \cdots + 729 \) T^6 - 9T^5 + 81T^4 - 54T^3 + 243T^2 + 729
$31$	\( (T^{3} + 9 T^{2} + 6 T - 53)^{2} \) (T^3 + 9T^2 + 6T - 53)^2
$37$	\( T^{6} + 12 T^{5} - 30 T^{4} + \cdots + 50653 \) T^6 + 12T^5 - 30T^4 - 803T^3 - 1110T^2 + 16428*T + 50653
$41$	\( T^{6} - 15 T^{5} + 72 T^{4} - 84 T^{3} + \cdots + 9 \) T^6 - 15T^5 + 72T^4 - 84T^3 + 63T^2 - 27*T + 9
$43$	\( (T^{3} + 9 T^{2} + 24 T + 19)^{2} \) (T^3 + 9T^2 + 24T + 19)^2
$47$	\( T^{6} - 9 T^{5} + 117 T^{4} + 342 T^{3} + \cdots + 81 \) T^6 - 9T^5 + 117T^4 + 342T^3 + 1215T^2 + 324*T + 81
$53$	\( T^{6} - 21 T^{5} + 180 T^{4} + \cdots + 2601 \) T^6 - 21T^5 + 180T^4 - 915T^3 + 3681T^2 - 4590*T + 2601
$59$	\( T^{6} + 6 T^{5} + 18 T^{4} + 3 T^{3} + \cdots + 9 \) T^6 + 6T^5 + 18T^4 + 3T^3 + 90T^2 - 54*T + 9
$61$	\( T^{6} + 9 T^{5} + 45 T^{4} + 152 T^{3} + \cdots + 289 \) T^6 + 9T^5 + 45T^4 + 152T^3 + 342T^2 + 459*T + 289
$67$	\( T^{6} - 24 T^{5} + 258 T^{4} + \cdots + 1369 \) T^6 - 24T^5 + 258T^4 - 1108T^3 + 2010T^2 - 1443*T + 1369
$71$	\( T^{6} + 12 T^{5} + 126 T^{4} + \cdots + 1418481 \) T^6 + 12T^5 + 126T^4 - 651T^3 - 1854T^2 - 171504*T + 1418481
$73$	\( (T^{3} - 39 T - 89)^{2} \) (T^3 - 39*T - 89)^2
$79$	\( T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 104329 \) T^6 + 3T^5 + 114T^4 + 656T^3 + 2631T^2 - 45543*T + 104329
$83$	\( T^{6} - 3 T^{5} - 117 T^{4} + \cdots + 751689 \) T^6 - 3T^5 - 117T^4 - 1104T^3 + 27306T^2 - 85833*T + 751689
$89$	\( T^{6} + 24 T^{5} + 207 T^{4} + \cdots + 3249 \) T^6 + 24T^5 + 207T^4 + 759T^3 + 2196T^2 + 3591*T + 3249
$97$	\( T^{6} + 27 T^{5} + 525 T^{4} + \cdots + 128881 \) T^6 + 27T^5 + 525T^4 + 4790T^3 + 31923T^2 + 73236*T + 128881
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\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{18}^{5}\)