Newform orbit 228.3.l.a

• Label: 228.3.l.a
• Level: $228$
• Weight: $3$
• Character orbit: 228.l
• Analytic conductor: $6.213$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$228 = 2^{2} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	228.l (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.21255002741$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (z + 1) * q^3 + (6*z - 6) * q^5 - 5 * q^7 + 3*z * q^9 + (11*z - 22) * q^13 + (6*z - 12) * q^15 + (6*z - 6) * q^17 + (-16*z - 5) * q^19 + (-5*z - 5) * q^21 + 24*z * q^23 - 11*z * q^25 + (6*z - 3) * q^27 + (-18*z + 36) * q^29 + (34*z - 17) * q^31 + (-30*z + 30) * q^35 + (70*z - 35) * q^37 - 33 * q^39 + (-24*z - 24) * q^41 + (-25*z + 25) * q^43 - 18 * q^45 + 42*z * q^47 - 24 * q^49 + (6*z - 12) * q^51 + (-36*z + 72) * q^53 + (-37*z + 11) * q^57 + (42*z + 42) * q^59 - 43*z * q^61 - 15*z * q^63 + (-132*z + 66) * q^65 + (-33*z + 66) * q^67 + (48*z - 24) * q^69 + (36*z + 36) * q^71 + (11*z - 11) * q^73 + (-22*z + 11) * q^75 + (z + 1) * q^79 + (9*z - 9) * q^81 + 126 * q^83 - 36*z * q^85 + 54 * q^87 + (6*z - 12) * q^89 + (-55*z + 110) * q^91 + (51*z - 51) * q^93 + (-30*z + 126) * q^95 + (-76*z - 76) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 3 * q^3 - 6 * q^5 - 10 * q^7 + 3 * q^9 - 33 * q^13 - 18 * q^15 - 6 * q^17 - 26 * q^19 - 15 * q^21 + 24 * q^23 - 11 * q^25 + 54 * q^29 + 30 * q^35 - 66 * q^39 - 72 * q^41 + 25 * q^43 - 36 * q^45 + 42 * q^47 - 48 * q^49 - 18 * q^51 + 108 * q^53 - 15 * q^57 + 126 * q^59 - 43 * q^61 - 15 * q^63 + 99 * q^67 + 108 * q^71 - 11 * q^73 + 3 * q^79 - 9 * q^81 + 252 * q^83 - 36 * q^85 + 108 * q^87 - 18 * q^89 + 165 * q^91 - 51 * q^93 + 222 * q^95 - 228 * q^97

Character values

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

$n$	$77$	$97$	$115$
$\chi(n)$	$1$	$\zeta_{6}$	$1$

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.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

145.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

1.50000

−

0.866025i

0

−3.00000

−

5.19615i

0

−5.00000

0

1.50000

−

2.59808i

0

217.1

0

1.50000

+

0.866025i

0

−3.00000

+

5.19615i

0

−5.00000

0

1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.3.l.a	✓	2
3.b	odd	2	1		684.3.y.e		2
4.b	odd	2	1		912.3.be.a		2
19.d	odd	6	1	inner	228.3.l.a	✓	2
57.f	even	6	1		684.3.y.e		2
76.f	even	6	1		912.3.be.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.3.l.a	✓	2	1.a	even	1	1	trivial
228.3.l.a	✓	2	19.d	odd	6	1	inner
684.3.y.e		2	3.b	odd	2	1
684.3.y.e		2	57.f	even	6	1
912.3.be.a		2	4.b	odd	2	1
912.3.be.a		2	76.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(228, [\chi])$:

$T_{5}^{2} + 6T_{5} + 36$

$T_{7} + 5$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 3T + 3$
$5$	$T^{2} + 6T + 36$
$7$	$(T + 5)^{2}$
$11$	$T^{2}$