Embedded newform 252.2.bb.a.11.3

• Label: 252.2.bb.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	252.11
Dual form			252.2.bb.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40608 - 0.151434i) q^{2} +(0.377313 - 1.69045i) q^{3} +(1.95414 + 0.425857i) q^{4} +(-0.941391 - 0.543513i) q^{5} +(-0.786525 + 2.31978i) q^{6} +(-2.38059 - 1.15446i) q^{7} +(-2.68319 - 0.894712i) q^{8} +(-2.71527 - 1.27566i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40608	−	0.151434i	−0.994250	−	0.107080i
							\(3\)	0.377313	−	1.69045i	0.217842	−	0.975984i
\(5\)	−0.941391	−	0.543513i	−0.421003	−	0.243066i								0.274503	−	0.961586i	\(-0.411487\pi\)
							−0.695506	+	0.718520i	\(0.744820\pi\)
\(7\)	−2.38059	−	1.15446i	−0.899780	−	0.436343i
							\(11\)	−0.751874	−	1.30228i	−0.226699	−	0.392653i	0.730129	−	0.683309i	\(-0.239460\pi\)
−0.956828	+	0.290656i	\(0.906127\pi\)
\(13\)	1.59238	+	2.75809i	0.441647	+	0.764955i	0.997812	−	0.0661167i	\(-0.0210610\pi\)
							−0.556165	+	0.831072i	\(0.687728\pi\)
\(17\)	−5.40337	−	3.11964i	−1.31051	−	0.756624i	−0.328330	−	0.944563i	\(-0.606486\pi\)
							−0.982181	+	0.187939i	\(0.939819\pi\)
\(19\)	2.29977	−	1.32777i	0.527604	−	0.304612i	−0.212436	−	0.977175i	\(-0.568140\pi\)
							0.740040	+	0.672563i	\(0.234806\pi\)
\(23\)	−1.62579	+	2.81595i	−0.339000	+	0.587165i	−0.984245	−	0.176811i	\(-0.943422\pi\)
							0.645245	+	0.763976i	\(0.276755\pi\)
\(29\)	−1.78455	−	1.03031i	−0.331382	−	0.191324i	0.325072	−	0.945689i	\(-0.394611\pi\)
							−0.656455	+	0.754365i	\(0.727945\pi\)
\(31\)			0.818407i			0.146990i	0.997296	+	0.0734951i	\(0.0234153\pi\)
							−0.997296	+	0.0734951i	\(0.976585\pi\)
\(37\)	−4.35941	−	7.55073i	−0.716683	−	1.24133i	−0.962307	−	0.271967i	\(-0.912326\pi\)
							0.245623	−	0.969365i	\(-0.421007\pi\)
\(41\)	6.57171	−	3.79418i	1.02633	−	0.592552i	0.110398	−	0.993887i	\(-0.464787\pi\)
							0.915931	+	0.401336i	\(0.131454\pi\)
\(43\)	−3.95088	−	2.28104i	−0.602504	−	0.347856i	0.167522	−	0.985868i	\(-0.446423\pi\)
							−0.770026	+	0.638012i	\(0.779757\pi\)
\(47\)	13.6215			1.98690			0.993451	−	0.114259i	\(-0.0364492\pi\)
							0.993451	+	0.114259i	\(0.0364492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.11.3	yes	88
3.2	odd	2		756.2.bb.a.683.42		88
4.3	odd	2	inner	252.2.bb.a.11.42	yes	88
7.2	even	3		252.2.o.a.191.28	yes	88
9.4	even	3		756.2.o.a.179.31		88
9.5	odd	6		252.2.o.a.95.14	✓	88
12.11	even	2		756.2.bb.a.683.3		88
21.2	odd	6		756.2.o.a.359.17		88
28.23	odd	6		252.2.o.a.191.14	yes	88
36.23	even	6		252.2.o.a.95.28	yes	88
36.31	odd	6		756.2.o.a.179.17		88
63.23	odd	6	inner	252.2.bb.a.23.42	yes	88
63.58	even	3		756.2.bb.a.611.3		88
84.23	even	6		756.2.o.a.359.31		88
252.23	even	6	inner	252.2.bb.a.23.3	yes	88
252.247	odd	6		756.2.bb.a.611.42		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.14	✓	88	9.5	odd	6
252.2.o.a.95.28	yes	88	36.23	even	6
252.2.o.a.191.14	yes	88	28.23	odd	6
252.2.o.a.191.28	yes	88	7.2	even	3
252.2.bb.a.11.3	yes	88	1.1	even	1	trivial
252.2.bb.a.11.42	yes	88	4.3	odd	2	inner
252.2.bb.a.23.3	yes	88	252.23	even	6	inner
252.2.bb.a.23.42	yes	88	63.23	odd	6	inner
756.2.o.a.179.17		88	36.31	odd	6
756.2.o.a.179.31		88	9.4	even	3
756.2.o.a.359.17		88	21.2	odd	6
756.2.o.a.359.31		88	84.23	even	6
756.2.bb.a.611.3		88	63.58	even	3
756.2.bb.a.611.42		88	252.247	odd	6
756.2.bb.a.683.3		88	12.11	even	2
756.2.bb.a.683.42		88	3.2	odd	2