Embedded newform 252.2.bb.a.23.25

• Label: 252.2.bb.a.23.25
• Level: $252$
• Weight: $2$
• Character: 252.23
• 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			23.25
Character	\(\chi\)	\(=\)	252.23
Dual form			252.2.bb.a.11.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.237258 - 1.39417i) q^{2} +(-1.62755 - 0.592510i) q^{3} +(-1.88742 - 0.661555i) q^{4} +(-0.722678 + 0.417239i) q^{5} +(-1.21221 + 2.12851i) q^{6} +(-2.04221 + 1.68208i) q^{7} +(-1.37012 + 2.47442i) q^{8} +(2.29786 + 1.92868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.237258	−	1.39417i	0.167767	−	0.985827i
							\(3\)	−1.62755	−	0.592510i	−0.939669	−	0.342086i
\(5\)	−0.722678	+	0.417239i	−0.323192	+	0.186595i								−0.652814	−	0.757518i	\(-0.726412\pi\)
							0.329623	+	0.944113i	\(0.393078\pi\)
\(7\)	−2.04221	+	1.68208i	−0.771883	+	0.635765i
							\(11\)	−1.61094	+	2.79023i	−0.485716	+	0.841285i	−0.999865	−	0.0164158i	\(-0.994774\pi\)
0.514149	+	0.857701i	\(0.328108\pi\)
\(13\)	2.03757	−	3.52917i	0.565120	−	0.978817i	−0.431919	−	0.901913i	\(-0.642163\pi\)
							0.997039	−	0.0769039i	\(-0.0245035\pi\)
\(17\)	−3.20251	+	1.84897i	−0.776724	+	0.448442i	−0.835268	−	0.549843i	\(-0.814688\pi\)
							0.0585440	+	0.998285i	\(0.481354\pi\)
\(19\)	−4.26091	−	2.46004i	−0.977520	−	0.564371i	−0.0759994	−	0.997108i	\(-0.524215\pi\)
							−0.901520	+	0.432737i	\(0.857548\pi\)
\(23\)	0.567320	+	0.982627i	0.118294	+	0.204892i	0.919092	−	0.394043i	\(-0.128924\pi\)
							−0.800797	+	0.598935i	\(0.795591\pi\)
\(29\)	−8.69963	+	5.02273i	−1.61548	+	0.932698i	−0.627411	+	0.778688i	\(0.715886\pi\)
							−0.988069	+	0.154010i	\(0.950781\pi\)
\(31\)			2.10663i			0.378362i	0.981942	+	0.189181i	\(0.0605833\pi\)
							−0.981942	+	0.189181i	\(0.939417\pi\)
\(37\)	0.248579	−	0.430551i	0.0408661	−	0.0707822i	−0.844869	−	0.534973i	\(-0.820322\pi\)
							0.885735	+	0.464191i	\(0.153655\pi\)
\(41\)	−6.77313	−	3.91047i	−1.05778	−	0.610712i	−0.132966	−	0.991121i	\(-0.542450\pi\)
							−0.924819	+	0.380408i	\(0.875783\pi\)
\(43\)	7.81676	−	4.51301i	1.19204	−	0.688227i	0.233275	−	0.972411i	\(-0.425056\pi\)
							0.958770	+	0.284184i	\(0.0917225\pi\)
\(47\)	−3.97056			−0.579166			−0.289583	−	0.957153i	\(-0.593517\pi\)
							−0.289583	+	0.957153i	\(0.593517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.23.25	yes	88
3.2	odd	2		756.2.bb.a.611.20		88
4.3	odd	2	inner	252.2.bb.a.23.20	yes	88
7.4	even	3		252.2.o.a.95.35	✓	88
9.2	odd	6		252.2.o.a.191.39	yes	88
9.7	even	3		756.2.o.a.359.6		88
12.11	even	2		756.2.bb.a.611.25		88
21.11	odd	6		756.2.o.a.179.10		88
28.11	odd	6		252.2.o.a.95.39	yes	88
36.7	odd	6		756.2.o.a.359.10		88
36.11	even	6		252.2.o.a.191.35	yes	88
63.11	odd	6	inner	252.2.bb.a.11.20	yes	88
63.25	even	3		756.2.bb.a.683.25		88
84.11	even	6		756.2.o.a.179.6		88
252.11	even	6	inner	252.2.bb.a.11.25	yes	88
252.151	odd	6		756.2.bb.a.683.20		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.35	✓	88	7.4	even	3
252.2.o.a.95.39	yes	88	28.11	odd	6
252.2.o.a.191.35	yes	88	36.11	even	6
252.2.o.a.191.39	yes	88	9.2	odd	6
252.2.bb.a.11.20	yes	88	63.11	odd	6	inner
252.2.bb.a.11.25	yes	88	252.11	even	6	inner
252.2.bb.a.23.20	yes	88	4.3	odd	2	inner
252.2.bb.a.23.25	yes	88	1.1	even	1	trivial
756.2.o.a.179.6		88	84.11	even	6
756.2.o.a.179.10		88	21.11	odd	6
756.2.o.a.359.6		88	9.7	even	3
756.2.o.a.359.10		88	36.7	odd	6
756.2.bb.a.611.20		88	3.2	odd	2
756.2.bb.a.611.25		88	12.11	even	2
756.2.bb.a.683.20		88	252.151	odd	6
756.2.bb.a.683.25		88	63.25	even	3