Embedded newform 252.2.bb.a.11.1

• Label: 252.2.bb.a.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	252.11
Dual form			252.2.bb.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41415 + 0.0135638i) q^{2} +(1.41411 + 1.00015i) q^{3} +(1.99963 - 0.0383626i) q^{4} +(-2.92354 - 1.68790i) q^{5} +(-2.01332 - 1.39518i) q^{6} +(-0.479205 + 2.60199i) q^{7} +(-2.82726 + 0.0813731i) q^{8} +(0.999400 + 2.82864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41415	+	0.0135638i	−0.999954	+	0.00959108i
							\(3\)	1.41411	+	1.00015i	0.816435	+	0.577437i
\(5\)	−2.92354	−	1.68790i	−1.30745	−	0.754854i								−0.325776	−	0.945447i	\(-0.605626\pi\)
							−0.981669	+	0.190593i	\(0.938959\pi\)
\(7\)	−0.479205	+	2.60199i	−0.181123	+	0.983461i
							\(11\)	2.63661	+	4.56674i	0.794968	+	1.37692i	0.922860	+	0.385136i	\(0.125846\pi\)
−0.127892	+	0.991788i	\(0.540821\pi\)
\(13\)	1.01362	+	1.75565i	0.281128	+	0.486928i	0.971663	−	0.236371i	\(-0.0759580\pi\)
							−0.690535	+	0.723299i	\(0.742625\pi\)
\(17\)	0.756255	+	0.436624i	0.183419	+	0.105897i	0.588898	−	0.808207i	\(-0.299562\pi\)
							−0.405479	+	0.914104i	\(0.632895\pi\)
\(19\)	−2.93018	+	1.69174i	−0.672230	+	0.388112i	−0.796921	−	0.604084i	\(-0.793539\pi\)
							0.124691	+	0.992196i	\(0.460206\pi\)
\(23\)	0.671428	−	1.16295i	0.140002	−	0.242491i	−0.787495	−	0.616321i	\(-0.788622\pi\)
							0.927497	+	0.373830i	\(0.121956\pi\)
\(29\)	5.64970	+	3.26186i	1.04912	+	0.605712i	0.922404	−	0.386227i	\(-0.126222\pi\)
							0.126720	+	0.991939i	\(0.459555\pi\)
\(31\)		−	6.89210i		−	1.23786i	−0.785447	−	0.618929i	\(-0.787567\pi\)
							0.785447	−	0.618929i	\(-0.212433\pi\)
\(37\)	−1.11577	−	1.93257i	−0.183431	−	0.317712i	0.759616	−	0.650372i	\(-0.225387\pi\)
							−0.943047	+	0.332660i	\(0.892054\pi\)
\(41\)	−0.715268	+	0.412960i	−0.111706	+	0.0644935i	−0.554812	−	0.831976i	\(-0.687210\pi\)
							0.443106	+	0.896469i	\(0.353877\pi\)
\(43\)	−0.269989	−	0.155878i	−0.0411730	−	0.0237712i	0.479272	−	0.877666i	\(-0.340901\pi\)
							−0.520445	+	0.853895i	\(0.674234\pi\)
\(47\)	−9.12157			−1.33052			−0.665259	−	0.746612i	\(-0.731679\pi\)
							−0.665259	+	0.746612i	\(0.731679\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.11.1	yes	88
3.2	odd	2		756.2.bb.a.683.44		88
4.3	odd	2	inner	252.2.bb.a.11.44	yes	88
7.2	even	3		252.2.o.a.191.31	yes	88
9.4	even	3		756.2.o.a.179.29		88
9.5	odd	6		252.2.o.a.95.16	✓	88
12.11	even	2		756.2.bb.a.683.1		88
21.2	odd	6		756.2.o.a.359.14		88
28.23	odd	6		252.2.o.a.191.16	yes	88
36.23	even	6		252.2.o.a.95.31	yes	88
36.31	odd	6		756.2.o.a.179.14		88
63.23	odd	6	inner	252.2.bb.a.23.44	yes	88
63.58	even	3		756.2.bb.a.611.1		88
84.23	even	6		756.2.o.a.359.29		88
252.23	even	6	inner	252.2.bb.a.23.1	yes	88
252.247	odd	6		756.2.bb.a.611.44		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.16	✓	88	9.5	odd	6
252.2.o.a.95.31	yes	88	36.23	even	6
252.2.o.a.191.16	yes	88	28.23	odd	6
252.2.o.a.191.31	yes	88	7.2	even	3
252.2.bb.a.11.1	yes	88	1.1	even	1	trivial
252.2.bb.a.11.44	yes	88	4.3	odd	2	inner
252.2.bb.a.23.1	yes	88	252.23	even	6	inner
252.2.bb.a.23.44	yes	88	63.23	odd	6	inner
756.2.o.a.179.14		88	36.31	odd	6
756.2.o.a.179.29		88	9.4	even	3
756.2.o.a.359.14		88	21.2	odd	6
756.2.o.a.359.29		88	84.23	even	6
756.2.bb.a.611.1		88	63.58	even	3
756.2.bb.a.611.44		88	252.247	odd	6
756.2.bb.a.683.1		88	12.11	even	2
756.2.bb.a.683.44		88	3.2	odd	2