Embedded newform 252.2.o.a.95.21

• Label: 252.2.o.a.95.21
• Level: $252$
• Weight: $2$
• Character: 252.95
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $88$
• 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			95.21
Character	\(\chi\)	\(=\)	252.95
Dual form			252.2.o.a.191.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.182752 - 1.40236i) q^{2} +(-0.773010 - 1.54999i) q^{3} +(-1.93320 + 0.512566i) q^{4} +2.45799i q^{5} +(-2.03236 + 1.36730i) q^{6} +(-2.25199 + 1.38871i) q^{7} +(1.07210 + 2.61737i) q^{8} +(-1.80491 + 2.39631i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 3 q^{2} + q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.182752	−	1.40236i	−0.129225	−	0.991615i
							\(3\)	−0.773010	−	1.54999i	−0.446298	−	0.894885i
\(5\)			2.45799i			1.09925i								0.835413	+	0.549623i	\(0.185229\pi\)
							−0.835413	+	0.549623i	\(0.814771\pi\)
\(7\)	−2.25199	+	1.38871i	−0.851174	+	0.524884i
							\(11\)	−3.73252			−1.12540			−0.562698	−	0.826663i	\(-0.690237\pi\)
−0.562698	+	0.826663i	\(0.690237\pi\)
\(13\)	−0.708051	+	1.22638i	−0.196378	+	0.340137i	−0.947351	−	0.320196i	\(-0.896251\pi\)
							0.750973	+	0.660332i	\(0.229585\pi\)
\(17\)	−3.50791	−	2.02529i	−0.850794	−	0.491206i	0.0101248	−	0.999949i	\(-0.496777\pi\)
							−0.860919	+	0.508743i	\(0.830110\pi\)
\(19\)	1.98185	−	1.14422i	0.454668	−	0.262503i	−0.255132	−	0.966906i	\(-0.582119\pi\)
							0.709800	+	0.704404i	\(0.248785\pi\)
\(23\)	−3.61698			−0.754192			−0.377096	−	0.926174i	\(-0.623077\pi\)
							−0.377096	+	0.926174i	\(0.623077\pi\)
\(29\)	−4.59226	+	2.65135i	−0.852762	+	0.492342i	−0.861582	−	0.507619i	\(-0.830526\pi\)
							0.00881973	+	0.999961i	\(0.497193\pi\)
\(31\)	5.38753	−	3.11049i	0.967629	−	0.558661i	0.0691164	−	0.997609i	\(-0.477982\pi\)
							0.898513	+	0.438948i	\(0.144649\pi\)
\(37\)	2.37386	+	4.11165i	0.390261	+	0.675952i	0.992484	−	0.122376i	\(-0.0390515\pi\)
							−0.602223	+	0.798328i	\(0.705718\pi\)
\(41\)	−7.67307	−	4.43005i	−1.19833	−	0.691857i	−0.238149	−	0.971229i	\(-0.576541\pi\)
							−0.960183	+	0.279372i	\(0.909874\pi\)
\(43\)	−10.5542	+	6.09346i	−1.60950	+	0.929244i	−0.620016	+	0.784590i	\(0.712874\pi\)
							−0.989482	+	0.144654i	\(0.953793\pi\)
\(47\)	−4.33970	+	7.51657i	−0.633010	+	1.09640i	0.353923	+	0.935274i	\(0.384847\pi\)
							−0.986933	+	0.161130i	\(0.948486\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.o.a.95.21	yes	88
3.2	odd	2		756.2.o.a.179.24		88
4.3	odd	2	inner	252.2.o.a.95.7	✓	88
7.2	even	3		252.2.bb.a.23.9	yes	88
9.2	odd	6		252.2.bb.a.11.36	yes	88
9.7	even	3		756.2.bb.a.683.9		88
12.11	even	2		756.2.o.a.179.38		88
21.2	odd	6		756.2.bb.a.611.36		88
28.23	odd	6		252.2.bb.a.23.36	yes	88
36.7	odd	6		756.2.bb.a.683.36		88
36.11	even	6		252.2.bb.a.11.9	yes	88
63.2	odd	6	inner	252.2.o.a.191.7	yes	88
63.16	even	3		756.2.o.a.359.38		88
84.23	even	6		756.2.bb.a.611.9		88
252.79	odd	6		756.2.o.a.359.24		88
252.191	even	6	inner	252.2.o.a.191.21	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.7	✓	88	4.3	odd	2	inner
252.2.o.a.95.21	yes	88	1.1	even	1	trivial
252.2.o.a.191.7	yes	88	63.2	odd	6	inner
252.2.o.a.191.21	yes	88	252.191	even	6	inner
252.2.bb.a.11.9	yes	88	36.11	even	6
252.2.bb.a.11.36	yes	88	9.2	odd	6
252.2.bb.a.23.9	yes	88	7.2	even	3
252.2.bb.a.23.36	yes	88	28.23	odd	6
756.2.o.a.179.24		88	3.2	odd	2
756.2.o.a.179.38		88	12.11	even	2
756.2.o.a.359.24		88	252.79	odd	6
756.2.o.a.359.38		88	63.16	even	3
756.2.bb.a.611.9		88	84.23	even	6
756.2.bb.a.611.36		88	21.2	odd	6
756.2.bb.a.683.9		88	9.7	even	3
756.2.bb.a.683.36		88	36.7	odd	6