LMFDB - Newform orbit 228.3.s.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,3,Mod(5,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 16]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("228.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	228.s (of order $18$, 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:	$6.21255002741$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$-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 - 3 \zeta_{18}^{4} q^{3} + (8 \zeta_{18}^{5} + \cdots - 5 \zeta_{18}) q^{7} + \cdots + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{9} +O(q^{10}) $ q - 3z^4 q^3 + (8z^5 + 8z^4 - 3z^2 - 5z) * q^7 + (9z^5 - 9z^2) * q^9	$ q - 3 \zeta_{18}^{4} q^{3} + (8 \zeta_{18}^{5} + \cdots - 5 \zeta_{18}) q^{7} + \cdots + 169 \zeta_{18} q^{97} +O(q^{100}) $ q - 3z^4 q^3 + (8z^5 + 8z^4 - 3z^2 - 5z) * q^7 + (9z^5 - 9z^2) * q^9 + (7z^4 + 7z^3 - 15z + 8) q^13 + (21z^3 - 5) q^19 + (-9z^5 + 9z^3 + 24z^2 + 15) q^21 - 25z^5 q^25 + 27z^3 q^27 + (35z^5 + 35z^4 - 24z^2 - 11z) * q^31 + (-7z^5 + 40z^4 - 33z^2 - 33z) * q^37 + (24z^5 - 45z^4 + 21z^2 + 21z) * q^39 + (13z^5 - 35z^3 + 35z^2 - 13) q^43 + (-16z^5 - 39z^4 - 49z^3 - 39z^2 - 16z) q^49 + (-48z^4 + 63z) * q^57 + (9z^4 + 56z^3 + 56z + 9) q^61 + (-72z^4 - 72z^3 + 27z + 45) q^63 + (77z^4 - 45z^3 - 32z - 32) q^67 + (-80z^5 - 63z^3 + 17z^2 - 17) q^73 - 75 * q^75 + (-51z^5 + 40z^3 + 91z^2 - 91) q^79 + (-81z^4 + 81z) * q^81 + (85z^4 - 96z^3 - 61z^2 - 96z + 85) * q^91 + (-72z^5 + 72z^3 + 105z^2 + 33) q^93 + 169z q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) $ 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$\zeta_{18}^{2}$	$1$

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} $

5.1

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

−0.520945

2.95442i

−6.99660

−

12.1185i

−8.45723

−

3.07818i

17.1

−2.29813

−

1.92836i

3.68732

6.38662i

1.56283

8.86327i

101.1

2.81908

1.02606i

3.30928

−

5.73184i

6.89440

5.78509i

137.1

−0.520945

−

2.95442i

−6.99660

12.1185i

−8.45723

3.07818i

149.1

2.81908

−

1.02606i

3.30928

5.73184i

6.89440

−

5.78509i

161.1

−2.29813

1.92836i

3.68732

−

6.38662i

1.56283

−

8.86327i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
19.e	even	9	1	inner
57.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.3.s.a	✓	6
3.b	odd	2	1	CM	228.3.s.a	✓	6
19.e	even	9	1	inner	228.3.s.a	✓	6
57.l	odd	18	1	inner	228.3.s.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.3.s.a	✓	6	1.a	even	1	1	trivial
228.3.s.a	✓	6	3.b	odd	2	1	CM
228.3.s.a	✓	6	19.e	even	9	1	inner
228.3.s.a	✓	6	57.l	odd	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{3}^{\mathrm{new}}(228, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 27T^{3} + 729 $ T^6 - 27*T^3 + 729
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 147 T^{4} + \cdots + 466489 $ T^6 + 147T^4 - 1366T^3 + 21609T^2 - 100401T + 466489
$11$	$ T^{6} $ T^6
$13$	$ T^{6} - 69 T^{5} + \cdots + 12439729 $ T^6 - 69T^5 + 2094T^4 - 31456T^3 + 246468T^2 - 1544826*T + 12439729
$17$	$ T^{6} $ T^6
$19$	$ (T^{2} - 11 T + 361)^{3} $ (T^2 - 11*T + 361)^3
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ T^{6} + 2883 T^{4} + \cdots + 573075721 $ T^6 + 2883T^4 - 47878T^3 + 8311689T^2 - 69016137T + 573075721
$37$	$ (T^{3} - 4107 T - 86183)^{2} $ (T^3 - 4107*T - 86183)^2
$41$	$ T^{6} $ T^6
$43$	$ T^{6} + \cdots + 1813652569 $ T^6 + 183T^5 + 16710T^4 + 1057688T^3 + 41373372T^2 + 410623854*T + 1813652569
$47$	$ T^{6} $ T^6
$53$	$ T^{6} $ T^6
$59$	$ T^{6} $ T^6
$61$	$ T^{6} + \cdots + 128048749921 $ T^6 - 222T^5 + 27591T^4 - 2120347T^3 + 175067868T^2 - 7728248883*T + 128048749921
$67$	$ T^{6} + \cdots + 342689647609 $ T^6 + 327T^5 + 49110T^4 + 3818312T^3 + 204190572T^2 + 10372064046*T + 342689647609
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + \cdots + 4425841729 $ T^6 + 291T^5 + 44214T^4 + 4718744T^3 + 284124252T^2 + 1704820902*T + 4425841729
$79$	$ T^{6} + \cdots + 537390158761 $ T^6 + 426T^5 + 79215T^4 + 7242929T^3 + 326359452T^2 + 10290089553*T + 537390158761
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ T^{6} + \cdots + 23298085122481 $ T^6 - 4826809*T^3 + 23298085122481
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Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	228.s (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - 3 \zeta_{18}^{4} q^{3} + (8 \zeta_{18}^{5} + \cdots - 5 \zeta_{18}) q^{7} + \cdots + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{9} +O(q^{10}) \) q - 3z^4 q^3 + (8z^5 + 8z^4 - 3z^2 - 5z) * q^7 + (9z^5 - 9z^2) * q^9	\( q - 3 \zeta_{18}^{4} q^{3} + (8 \zeta_{18}^{5} + \cdots - 5 \zeta_{18}) q^{7} + \cdots + 169 \zeta_{18} q^{97} +O(q^{100}) \) q - 3z^4 q^3 + (8z^5 + 8z^4 - 3z^2 - 5z) * q^7 + (9z^5 - 9z^2) * q^9 + (7z^4 + 7z^3 - 15z + 8) q^13 + (21z^3 - 5) q^19 + (-9z^5 + 9z^3 + 24z^2 + 15) q^21 - 25z^5 q^25 + 27z^3 q^27 + (35z^5 + 35z^4 - 24z^2 - 11z) * q^31 + (-7z^5 + 40z^4 - 33z^2 - 33z) * q^37 + (24z^5 - 45z^4 + 21z^2 + 21z) * q^39 + (13z^5 - 35z^3 + 35z^2 - 13) q^43 + (-16z^5 - 39z^4 - 49z^3 - 39z^2 - 16z) q^49 + (-48z^4 + 63z) * q^57 + (9z^4 + 56z^3 + 56z + 9) q^61 + (-72z^4 - 72z^3 + 27z + 45) q^63 + (77z^4 - 45z^3 - 32z - 32) q^67 + (-80z^5 - 63z^3 + 17z^2 - 17) q^73 - 75 * q^75 + (-51z^5 + 40z^3 + 91z^2 - 91) q^79 + (-81z^4 + 81z) * q^81 + (85z^4 - 96z^3 - 61z^2 - 96z + 85) * q^91 + (-72z^5 + 72z^3 + 105z^2 + 33) q^93 + 169z q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) \) 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

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