Newform orbit 468.2.k.a

• Label: 468.2.k.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.k
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$468 = 2^{2} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	468.k (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.73699881460$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 28 * q - 4 * q^7 - 2 * q^9 - 4 * q^11 + q^13 - 4 * q^15 - 8 * q^17 - q^19 + 14 * q^21 + 8 * q^23 - 14 * q^25 - 13 * q^29 + 2 * q^31 - 25 * q^33 + 3 * q^35 - q^37 - 3 * q^39 - 8 * q^41 - 4 * q^43 - 38 * q^45 + 11 * q^47 + 24 * q^49 + 5 * q^51 + 52 * q^53 - q^57 - 8 * q^59 + 14 * q^61 - 21 * q^63 + 38 * q^65 + 14 * q^67 - 21 * q^69 - 12 * q^71 + 14 * q^73 - 14 * q^75 - 28 * q^77 + 5 * q^79 + 10 * q^81 + 9 * q^83 + 18 * q^85 + 18 * q^87 - 11 * q^89 - q^91 - 9 * q^93 + 28 * q^95 + 50 * q^97 - 29 * q^99

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
61.1		0		−1.72988	−	0.0866924i		0		0.826802	+	1.43206i		0		−3.50986				0		2.98497	+	0.299935i		0
61.2		0		−1.72274	+	0.179356i		0		−1.94663	−	3.37166i		0		−2.50913				0		2.93566	−	0.617967i		0
61.3		0		−1.43490	−	0.970080i		0		−0.883304	−	1.52993i		0		3.13090				0		1.11789	+	2.78394i		0
61.4		0		−0.958062	−	1.44295i		0		1.10190	+	1.90855i		0		0.307428				0		−1.16424	+	2.76488i		0
61.5		0		−0.824641	+	1.52314i		0		−0.913993	−	1.58308i		0		3.54787				0		−1.63994	−	2.51209i		0
61.6		0		−0.674997	+	1.59511i		0		−0.0842838	−	0.145984i		0		−2.38663				0		−2.08876	−	2.15339i		0
61.7		0		−0.328747	+	1.70057i		0		2.03195	+	3.51944i		0		−1.23400				0		−2.78385	−	1.11811i		0
61.8		0		0.180043	−	1.72267i		0		0.0380222	+	0.0658564i		0		−3.72476				0		−2.93517	−	0.620309i		0
61.9		0		0.583701	−	1.63073i		0		−1.31191	−	2.27229i		0		3.59326				0		−2.31859	−	1.90372i		0
61.10		0		0.964854	+	1.43842i		0		−0.468893	−	0.812147i		0		−0.897167				0		−1.13811	+	2.77573i		0
61.11		0		1.20815	−	1.24112i		0		1.66845	+	2.88984i		0		−0.0102323				0		−0.0807596	−	2.99891i		0
61.12		0		1.36018	+	1.07234i		0		1.62465	+	2.81398i		0		4.02812				0		0.700171	+	2.91715i		0
61.13		0		1.64886	−	0.530344i		0		−1.46582	−	2.53887i		0		−3.91907				0		2.43747	−	1.74892i		0
61.14		0		1.72819	+	0.115654i		0		−0.216950	−	0.375768i		0		1.58327				0		2.97325	+	0.399741i		0
445.1		0		−1.72988	+	0.0866924i		0		0.826802	−	1.43206i		0		−3.50986				0		2.98497	−	0.299935i		0
445.2		0		−1.72274	−	0.179356i		0		−1.94663	+	3.37166i		0		−2.50913				0		2.93566	+	0.617967i		0
445.3		0		−1.43490	+	0.970080i		0		−0.883304	+	1.52993i		0		3.13090				0		1.11789	−	2.78394i		0
445.4		0		−0.958062	+	1.44295i		0		1.10190	−	1.90855i		0		0.307428				0		−1.16424	−	2.76488i		0
445.5		0		−0.824641	−	1.52314i		0		−0.913993	+	1.58308i		0		3.54787				0		−1.63994	+	2.51209i		0
445.6		0		−0.674997	−	1.59511i		0		−0.0842838	+	0.145984i		0		−2.38663				0		−2.08876	+	2.15339i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.f	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.k.a	yes	28
3.b	odd	2	1		1404.2.k.a		28
9.c	even	3	1		468.2.j.a	✓	28
9.d	odd	6	1		1404.2.j.a		28
13.c	even	3	1		468.2.j.a	✓	28
39.i	odd	6	1		1404.2.j.a		28
117.f	even	3	1	inner	468.2.k.a	yes	28
117.u	odd	6	1		1404.2.k.a		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.j.a	✓	28	9.c	even	3	1
468.2.j.a	✓	28	13.c	even	3	1
468.2.k.a	yes	28	1.a	even	1	1	trivial
468.2.k.a	yes	28	117.f	even	3	1	inner
1404.2.j.a		28	9.d	odd	6	1
1404.2.j.a		28	39.i	odd	6	1
1404.2.k.a		28	3.b	odd	2	1
1404.2.k.a		28	117.u	odd	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(468, [\chi])$.