Embedded newform 468.2.b.c.181.2

• Label: 468.2.b.c.181.2
• Level: $468$
• Weight: $2$
• Character: 468.181
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$468 = 2^{2} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	468.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.73699881460$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	468.181
Dual form			468.2.b.c.181.1

$q$-expansion

$f(q)$	$=$	$q+2.00000i q^{5} -4.00000i q^{7} -6.00000i q^{11} +(3.00000 + 2.00000i) q^{13} +2.00000 q^{17} +8.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} +8.00000i q^{31} +8.00000 q^{35} -8.00000i q^{37} -2.00000i q^{41} -8.00000 q^{43} -6.00000i q^{47} -9.00000 q^{49} -6.00000 q^{53} +12.0000 q^{55} -2.00000i q^{59} +2.00000 q^{61} +(-4.00000 + 6.00000i) q^{65} -4.00000i q^{67} +6.00000i q^{71} +4.00000i q^{73} -24.0000 q^{77} +14.0000i q^{83} +4.00000i q^{85} +6.00000i q^{89} +(8.00000 - 12.0000i) q^{91} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{13} + 4 q^{17} + 16 q^{23} + 2 q^{25} - 4 q^{29} + 16 q^{35} - 16 q^{43} - 18 q^{49} - 12 q^{53} + 24 q^{55} + 4 q^{61} - 8 q^{65} - 48 q^{77} + 16 q^{91}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$
$2$		0			0
							$3$		0			0
$5$			2.00000i			0.894427i								0.894427	+	0.447214i	$0.147584\pi$
							−0.894427	+	0.447214i	$0.852416\pi$
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
							0.654654	−	0.755929i	$-0.272814\pi$
$11$		−	6.00000i		−	1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
							0.426401	−	0.904534i	$-0.359781\pi$
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
							$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
0.242536	+	0.970143i	$0.422021\pi$
$19$		0			0		1.00000			$0$
							−1.00000			$\pi$
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
							0.834058	+	0.551677i	$0.186012\pi$
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
							−0.185695	+	0.982607i	$0.559454\pi$
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
							−0.695608	+	0.718421i	$0.744865\pi$
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
							0.753371	−	0.657596i	$-0.228427\pi$
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
							0.987730	−	0.156174i	$-0.0499160\pi$
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
							−0.609994	+	0.792406i	$0.708828\pi$
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
							0.899172	−	0.437595i	$-0.144170\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.2.b.c.181.2		2
3.2	odd	2		156.2.b.b.25.1	✓	2
4.3	odd	2		1872.2.c.h.1585.2		2
12.11	even	2		624.2.c.d.337.1		2
13.5	odd	4		6084.2.a.n.1.1		1
13.8	odd	4		6084.2.a.d.1.1		1
13.12	even	2	inner	468.2.b.c.181.1		2
15.2	even	4		3900.2.j.e.649.1		2
15.8	even	4		3900.2.j.b.649.2		2
15.14	odd	2		3900.2.c.a.3301.2		2
21.20	even	2		7644.2.e.b.4705.2		2
24.5	odd	2		2496.2.c.b.961.2		2
24.11	even	2		2496.2.c.i.961.2		2
39.2	even	12		2028.2.i.a.529.1		2
39.5	even	4		2028.2.a.d.1.1		1
39.8	even	4		2028.2.a.f.1.1		1
39.11	even	12		2028.2.i.d.529.1		2
39.17	odd	6		2028.2.q.e.361.2		4
39.20	even	12		2028.2.i.d.2005.1		2
39.23	odd	6		2028.2.q.e.1837.2		4
39.29	odd	6		2028.2.q.e.1837.1		4
39.32	even	12		2028.2.i.a.2005.1		2
39.35	odd	6		2028.2.q.e.361.1		4
39.38	odd	2		156.2.b.b.25.2	yes	2
52.51	odd	2		1872.2.c.h.1585.1		2
156.47	odd	4		8112.2.a.l.1.1		1
156.83	odd	4		8112.2.a.d.1.1		1
156.155	even	2		624.2.c.d.337.2		2
195.38	even	4		3900.2.j.e.649.2		2
195.77	even	4		3900.2.j.b.649.1		2
195.194	odd	2		3900.2.c.a.3301.1		2
273.272	even	2		7644.2.e.b.4705.1		2
312.77	odd	2		2496.2.c.b.961.1		2
312.155	even	2		2496.2.c.i.961.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.b.25.1	✓	2	3.2	odd	2
156.2.b.b.25.2	yes	2	39.38	odd	2
468.2.b.c.181.1		2	13.12	even	2	inner
468.2.b.c.181.2		2	1.1	even	1	trivial
624.2.c.d.337.1		2	12.11	even	2
624.2.c.d.337.2		2	156.155	even	2
1872.2.c.h.1585.1		2	52.51	odd	2
1872.2.c.h.1585.2		2	4.3	odd	2
2028.2.a.d.1.1		1	39.5	even	4
2028.2.a.f.1.1		1	39.8	even	4
2028.2.i.a.529.1		2	39.2	even	12
2028.2.i.a.2005.1		2	39.32	even	12
2028.2.i.d.529.1		2	39.11	even	12
2028.2.i.d.2005.1		2	39.20	even	12
2028.2.q.e.361.1		4	39.35	odd	6
2028.2.q.e.361.2		4	39.17	odd	6
2028.2.q.e.1837.1		4	39.29	odd	6
2028.2.q.e.1837.2		4	39.23	odd	6
2496.2.c.b.961.1		2	312.77	odd	2
2496.2.c.b.961.2		2	24.5	odd	2
2496.2.c.i.961.1		2	312.155	even	2
2496.2.c.i.961.2		2	24.11	even	2
3900.2.c.a.3301.1		2	195.194	odd	2
3900.2.c.a.3301.2		2	15.14	odd	2
3900.2.j.b.649.1		2	195.77	even	4
3900.2.j.b.649.2		2	15.8	even	4
3900.2.j.e.649.1		2	15.2	even	4
3900.2.j.e.649.2		2	195.38	even	4
6084.2.a.d.1.1		1	13.8	odd	4
6084.2.a.n.1.1		1	13.5	odd	4
7644.2.e.b.4705.1		2	273.272	even	2
7644.2.e.b.4705.2		2	21.20	even	2
8112.2.a.d.1.1		1	156.83	odd	4
8112.2.a.l.1.1		1	156.47	odd	4