Embedded newform 252.2.bb.a.11.16

• Label: 252.2.bb.a.11.16
• Level: $252$
• Weight: $2$
• Character: 252.11
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.16
Character	\(\chi\)	\(=\)	252.11
Dual form			252.2.bb.a.23.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.692480 - 1.23307i) q^{2} +(-1.73156 + 0.0411748i) q^{3} +(-1.04094 + 1.70776i) q^{4} +(1.47575 + 0.852024i) q^{5} +(1.24984 + 2.10663i) q^{6} +(-0.540398 - 2.58997i) q^{7} +(2.82662 + 0.100974i) q^{8} +(2.99661 - 0.142593i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.692480	−	1.23307i	−0.489657	−	0.871915i
							\(3\)	−1.73156	+	0.0411748i	−0.999717	+	0.0237723i
\(5\)	1.47575	+	0.852024i	0.659975	+	0.381037i								0.792267	−	0.610174i	\(-0.208901\pi\)
							−0.132293	+	0.991211i	\(0.542234\pi\)
\(7\)	−0.540398	−	2.58997i	−0.204251	−	0.978918i
							\(11\)	−0.0990477	−	0.171556i	−0.0298640	−	0.0517260i	0.850707	−	0.525640i	\(-0.176174\pi\)
−0.880571	+	0.473914i	\(0.842841\pi\)
\(13\)	−2.11577	−	3.66462i	−0.586808	−	1.01638i	−0.994647	−	0.103328i	\(-0.967051\pi\)
							0.407839	−	0.913054i	\(-0.366282\pi\)
\(17\)	−4.85542	−	2.80328i	−1.17761	−	0.679895i	−0.222152	−	0.975012i	\(-0.571308\pi\)
							−0.955461	+	0.295117i	\(0.904641\pi\)
\(19\)	6.97331	−	4.02604i	1.59979	−	0.923637i	0.608259	−	0.793739i	\(-0.291868\pi\)
							0.991527	−	0.129898i	\(-0.0414650\pi\)
\(23\)	−0.838905	+	1.45303i	−0.174924	+	0.302977i	−0.940135	−	0.340803i	\(-0.889301\pi\)
							0.765211	+	0.643779i	\(0.222635\pi\)
\(29\)	4.04957	+	2.33802i	0.751986	+	0.434160i	0.826411	−	0.563067i	\(-0.190379\pi\)
							−0.0744248	+	0.997227i	\(0.523712\pi\)
\(31\)		−	5.66572i		−	1.01759i	−0.860887	−	0.508797i	\(-0.830090\pi\)
							0.860887	−	0.508797i	\(-0.169910\pi\)
\(37\)	−2.01511	−	3.49027i	−0.331282	−	0.573797i	0.651481	−	0.758665i	\(-0.274148\pi\)
							−0.982764	+	0.184867i	\(0.940815\pi\)
\(41\)	7.25892	−	4.19094i	1.13365	−	0.654514i	0.188801	−	0.982015i	\(-0.439540\pi\)
							0.944851	+	0.327501i	\(0.106206\pi\)
\(43\)	3.72962	+	2.15330i	0.568762	+	0.328375i	0.756655	−	0.653815i	\(-0.226832\pi\)
							−0.187893	+	0.982190i	\(0.560166\pi\)
\(47\)	−3.31254			−0.483183			−0.241592	−	0.970378i	\(-0.577669\pi\)
							−0.241592	+	0.970378i	\(0.577669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.11.16	yes	88
3.2	odd	2		756.2.bb.a.683.29		88
4.3	odd	2	inner	252.2.bb.a.11.29	yes	88
7.2	even	3		252.2.o.a.191.15	yes	88
9.4	even	3		756.2.o.a.179.44		88
9.5	odd	6		252.2.o.a.95.1	✓	88
12.11	even	2		756.2.bb.a.683.16		88
21.2	odd	6		756.2.o.a.359.30		88
28.23	odd	6		252.2.o.a.191.1	yes	88
36.23	even	6		252.2.o.a.95.15	yes	88
36.31	odd	6		756.2.o.a.179.30		88
63.23	odd	6	inner	252.2.bb.a.23.29	yes	88
63.58	even	3		756.2.bb.a.611.16		88
84.23	even	6		756.2.o.a.359.44		88
252.23	even	6	inner	252.2.bb.a.23.16	yes	88
252.247	odd	6		756.2.bb.a.611.29		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.1	✓	88	9.5	odd	6
252.2.o.a.95.15	yes	88	36.23	even	6
252.2.o.a.191.1	yes	88	28.23	odd	6
252.2.o.a.191.15	yes	88	7.2	even	3
252.2.bb.a.11.16	yes	88	1.1	even	1	trivial
252.2.bb.a.11.29	yes	88	4.3	odd	2	inner
252.2.bb.a.23.16	yes	88	252.23	even	6	inner
252.2.bb.a.23.29	yes	88	63.23	odd	6	inner
756.2.o.a.179.30		88	36.31	odd	6
756.2.o.a.179.44		88	9.4	even	3
756.2.o.a.359.30		88	21.2	odd	6
756.2.o.a.359.44		88	84.23	even	6
756.2.bb.a.611.16		88	63.58	even	3
756.2.bb.a.611.29		88	252.247	odd	6
756.2.bb.a.683.16		88	12.11	even	2
756.2.bb.a.683.29		88	3.2	odd	2