Embedded newform 252.2.o.a.191.44

• Label: 252.2.o.a.191.44
• Level: $252$
• Weight: $2$
• Character: 252.191
• 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.o (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			191.44
Character	\(\chi\)	\(=\)	252.191
Dual form			252.2.o.a.95.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41419 - 0.00770275i) q^{2} +(-1.69023 + 0.378305i) q^{3} +(1.99988 - 0.0217863i) q^{4} -2.17323i q^{5} +(-2.38740 + 0.548016i) q^{6} +(-0.191082 - 2.63884i) q^{7} +(2.82805 - 0.0462147i) q^{8} +(2.71377 - 1.27885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 3 q^{2} + q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41419	−	0.00770275i	0.999985	−	0.00544667i
							\(3\)	−1.69023	+	0.378305i	−0.975856	+	0.218415i
\(5\)		−	2.17323i		−	0.971896i								−0.873988	−	0.485948i	\(-0.838474\pi\)
							0.873988	−	0.485948i	\(-0.161526\pi\)
\(7\)	−0.191082	−	2.63884i	−0.0722223	−	0.997389i
							\(11\)	1.58540			0.478016			0.239008	−	0.971018i	\(-0.423178\pi\)
0.239008	+	0.971018i	\(0.423178\pi\)
\(13\)	1.44655	+	2.50550i	0.401201	+	0.694900i	0.993871	−	0.110545i	\(-0.0352597\pi\)
							−0.592671	+	0.805445i	\(0.701926\pi\)
\(17\)	−2.58107	+	1.49018i	−0.626002	+	0.361423i	−0.779202	−	0.626773i	\(-0.784376\pi\)
							0.153200	+	0.988195i	\(0.451042\pi\)
\(19\)	1.40642	+	0.811997i	0.322655	+	0.186285i	0.652575	−	0.757724i	\(-0.273689\pi\)
							−0.329920	+	0.944009i	\(0.607022\pi\)
\(23\)	−5.43086			−1.13241			−0.566206	−	0.824264i	\(-0.691589\pi\)
							−0.566206	+	0.824264i	\(0.691589\pi\)
\(29\)	−3.61383	−	2.08645i	−0.671072	−	0.387443i	0.125411	−	0.992105i	\(-0.459975\pi\)
							−0.796483	+	0.604661i	\(0.793308\pi\)
\(31\)	−2.97782	−	1.71924i	−0.534832	−	0.308785i	0.208150	−	0.978097i	\(-0.433256\pi\)
							−0.742982	+	0.669312i	\(0.766589\pi\)
\(37\)	−4.84724	+	8.39566i	−0.796881	+	1.38024i	0.124756	+	0.992187i	\(0.460185\pi\)
							−0.921638	+	0.388051i	\(0.873148\pi\)
\(41\)	8.56561	−	4.94536i	1.33772	−	0.772335i	0.351254	−	0.936280i	\(-0.385755\pi\)
							0.986469	+	0.163945i	\(0.0524221\pi\)
\(43\)	8.83531	+	5.10107i	1.34737	+	0.777906i	0.987877	−	0.155241i	\(-0.0496154\pi\)
							0.359496	+	0.933147i	\(0.382949\pi\)
\(47\)	−0.953235	−	1.65105i	−0.139044	−	0.240831i	0.788091	−	0.615558i	\(-0.211069\pi\)
							−0.927135	+	0.374728i	\(0.877736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.o.a.191.44	yes	88
3.2	odd	2		756.2.o.a.359.1		88
4.3	odd	2	inner	252.2.o.a.191.30	yes	88
7.4	even	3		252.2.bb.a.11.15	yes	88
9.4	even	3		756.2.bb.a.611.15		88
9.5	odd	6		252.2.bb.a.23.30	yes	88
12.11	even	2		756.2.o.a.359.15		88
21.11	odd	6		756.2.bb.a.683.30		88
28.11	odd	6		252.2.bb.a.11.30	yes	88
36.23	even	6		252.2.bb.a.23.15	yes	88
36.31	odd	6		756.2.bb.a.611.30		88
63.4	even	3		756.2.o.a.179.15		88
63.32	odd	6	inner	252.2.o.a.95.30	✓	88
84.11	even	6		756.2.bb.a.683.15		88
252.67	odd	6		756.2.o.a.179.1		88
252.95	even	6	inner	252.2.o.a.95.44	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.30	✓	88	63.32	odd	6	inner
252.2.o.a.95.44	yes	88	252.95	even	6	inner
252.2.o.a.191.30	yes	88	4.3	odd	2	inner
252.2.o.a.191.44	yes	88	1.1	even	1	trivial
252.2.bb.a.11.15	yes	88	7.4	even	3
252.2.bb.a.11.30	yes	88	28.11	odd	6
252.2.bb.a.23.15	yes	88	36.23	even	6
252.2.bb.a.23.30	yes	88	9.5	odd	6
756.2.o.a.179.1		88	252.67	odd	6
756.2.o.a.179.15		88	63.4	even	3
756.2.o.a.359.1		88	3.2	odd	2
756.2.o.a.359.15		88	12.11	even	2
756.2.bb.a.611.15		88	9.4	even	3
756.2.bb.a.611.30		88	36.31	odd	6
756.2.bb.a.683.15		88	84.11	even	6
756.2.bb.a.683.30		88	21.11	odd	6