Embedded newform 252.2.bb.a.11.12

• Label: 252.2.bb.a.11.12
• 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			11.12
Character	\(\chi\)	\(=\)	252.11
Dual form			252.2.bb.a.23.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.966497 + 1.03242i) q^{2} +(-1.48378 - 0.893531i) q^{3} +(-0.131768 - 1.99565i) q^{4} +(-1.42201 - 0.820999i) q^{5} +(2.35656 - 0.668284i) q^{6} +(-2.57988 - 0.586703i) q^{7} +(2.18770 + 1.79275i) q^{8} +(1.40320 + 2.65161i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.966497	+	1.03242i	−0.683416	+	0.730029i
							\(3\)	−1.48378	−	0.893531i	−0.856661	−	0.515880i
\(5\)	−1.42201	−	0.820999i	−0.635943	−	0.367162i								0.147107	−	0.989121i	\(-0.453004\pi\)
							−0.783050	+	0.621959i	\(0.786337\pi\)
\(7\)	−2.57988	−	0.586703i	−0.975103	−	0.221753i
							\(11\)	3.06085	+	5.30154i	0.922880	+	1.59848i	0.794935	+	0.606694i	\(0.207505\pi\)
0.127945	+	0.991781i	\(0.459162\pi\)
\(13\)	0.599586	+	1.03851i	0.166295	+	0.288032i	0.937114	−	0.349022i	\(-0.113486\pi\)
							−0.770819	+	0.637054i	\(0.780153\pi\)
\(17\)	2.89068	+	1.66893i	0.701092	+	0.404776i	0.807754	−	0.589519i	\(-0.200683\pi\)
							−0.106662	+	0.994295i	\(0.534016\pi\)
\(19\)	3.61550	−	2.08741i	0.829452	−	0.478884i	−0.0242130	−	0.999707i	\(-0.507708\pi\)
							0.853665	+	0.520823i	\(0.174375\pi\)
\(23\)	−4.21762	+	7.30513i	−0.879434	+	1.52323i	−0.0274717	+	0.999623i	\(0.508746\pi\)
							−0.851963	+	0.523603i	\(0.824588\pi\)
\(29\)	−0.398864	−	0.230284i	−0.0740671	−	0.0427627i	0.462509	−	0.886615i	\(-0.346949\pi\)
							−0.536576	+	0.843852i	\(0.680283\pi\)
\(31\)			4.85676i			0.872299i	0.899874	+	0.436150i	\(0.143658\pi\)
							−0.899874	+	0.436150i	\(0.856342\pi\)
\(37\)	2.65132	+	4.59222i	0.435874	+	0.754956i	0.997367	−	0.0725258i	\(-0.0231060\pi\)
							−0.561492	+	0.827482i	\(0.689773\pi\)
\(41\)	−4.25113	+	2.45439i	−0.663914	+	0.383311i	−0.793767	−	0.608222i	\(-0.791883\pi\)
							0.129852	+	0.991533i	\(0.458550\pi\)
\(43\)	0.733230	+	0.423331i	0.111817	+	0.0645573i	0.554865	−	0.831940i	\(-0.312770\pi\)
							−0.443049	+	0.896498i	\(0.646103\pi\)
\(47\)	−8.42989			−1.22963			−0.614813	−	0.788673i	\(-0.710768\pi\)
							−0.614813	+	0.788673i	\(0.710768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.11.12	yes	88
3.2	odd	2		756.2.bb.a.683.33		88
4.3	odd	2	inner	252.2.bb.a.11.33	yes	88
7.2	even	3		252.2.o.a.191.41	yes	88
9.4	even	3		756.2.o.a.179.18		88
9.5	odd	6		252.2.o.a.95.27	✓	88
12.11	even	2		756.2.bb.a.683.12		88
21.2	odd	6		756.2.o.a.359.4		88
28.23	odd	6		252.2.o.a.191.27	yes	88
36.23	even	6		252.2.o.a.95.41	yes	88
36.31	odd	6		756.2.o.a.179.4		88
63.23	odd	6	inner	252.2.bb.a.23.33	yes	88
63.58	even	3		756.2.bb.a.611.12		88
84.23	even	6		756.2.o.a.359.18		88
252.23	even	6	inner	252.2.bb.a.23.12	yes	88
252.247	odd	6		756.2.bb.a.611.33		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.27	✓	88	9.5	odd	6
252.2.o.a.95.41	yes	88	36.23	even	6
252.2.o.a.191.27	yes	88	28.23	odd	6
252.2.o.a.191.41	yes	88	7.2	even	3
252.2.bb.a.11.12	yes	88	1.1	even	1	trivial
252.2.bb.a.11.33	yes	88	4.3	odd	2	inner
252.2.bb.a.23.12	yes	88	252.23	even	6	inner
252.2.bb.a.23.33	yes	88	63.23	odd	6	inner
756.2.o.a.179.4		88	36.31	odd	6
756.2.o.a.179.18		88	9.4	even	3
756.2.o.a.359.4		88	21.2	odd	6
756.2.o.a.359.18		88	84.23	even	6
756.2.bb.a.611.12		88	63.58	even	3
756.2.bb.a.611.33		88	252.247	odd	6
756.2.bb.a.683.12		88	12.11	even	2
756.2.bb.a.683.33		88	3.2	odd	2