Embedded newform 252.2.bb.a.11.5

• Label: 252.2.bb.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	252.11
Dual form			252.2.bb.a.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30394 - 0.547479i) q^{2} +(-1.06571 + 1.36538i) q^{3} +(1.40053 + 1.42776i) q^{4} +(-1.40549 - 0.811459i) q^{5} +(2.13714 - 1.19692i) q^{6} +(1.53742 - 2.15322i) q^{7} +(-1.04455 - 2.62848i) q^{8} +(-0.728526 - 2.91020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 2 q^{4} - 6 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30394	−	0.547479i	−0.922027	−	0.387126i
							\(3\)	−1.06571	+	1.36538i	−0.615288	+	0.788303i
\(5\)	−1.40549	−	0.811459i	−0.628554	−	0.362896i								0.151638	−	0.988436i	\(-0.451545\pi\)
							−0.780192	+	0.625540i	\(0.784879\pi\)
\(7\)	1.53742	−	2.15322i	0.581089	−	0.813840i
							\(11\)	0.553960	+	0.959487i	0.167025	+	0.289296i	0.937373	−	0.348328i	\(-0.113251\pi\)
−0.770347	+	0.637624i	\(0.779917\pi\)
\(13\)	2.95905	+	5.12523i	0.820694	+	1.42148i	0.905166	+	0.425058i	\(0.139746\pi\)
							−0.0844726	+	0.996426i	\(0.526921\pi\)
\(17\)	6.47792	+	3.74003i	1.57113	+	0.907090i	0.996031	+	0.0890039i	\(0.0283683\pi\)
							0.575095	+	0.818086i	\(0.304965\pi\)
\(19\)	3.12725	−	1.80552i	0.717440	−	0.414214i	−0.0963698	−	0.995346i	\(-0.530723\pi\)
							0.813810	+	0.581131i	\(0.197390\pi\)
\(23\)	−0.263598	+	0.456565i	−0.0549640	+	0.0952005i	−0.892198	−	0.451644i	\(-0.850838\pi\)
							0.837234	+	0.546844i	\(0.184171\pi\)
\(29\)	−1.30418	−	0.752968i	−0.242180	−	0.139823i	0.373998	−	0.927429i	\(-0.377987\pi\)
							−0.616178	+	0.787607i	\(0.711320\pi\)
\(31\)		−	1.85308i		−	0.332824i	−0.986056	−	0.166412i	\(-0.946782\pi\)
							0.986056	−	0.166412i	\(-0.0532181\pi\)
\(37\)	3.78593	+	6.55743i	0.622403	+	1.07803i	0.989037	+	0.147669i	\(0.0471769\pi\)
							−0.366634	+	0.930365i	\(0.619490\pi\)
\(41\)	5.31155	−	3.06663i	0.829525	−	0.478926i	−0.0241650	−	0.999708i	\(-0.507693\pi\)
							0.853690	+	0.520782i	\(0.174359\pi\)
\(43\)	3.91977	+	2.26308i	0.597759	+	0.345116i	0.768160	−	0.640258i	\(-0.221173\pi\)
							−0.170400	+	0.985375i	\(0.554506\pi\)
\(47\)	3.96658			0.578585			0.289293	−	0.957241i	\(-0.406580\pi\)
							0.289293	+	0.957241i	\(0.406580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bb.a.11.5	yes	88
3.2	odd	2		756.2.bb.a.683.40		88
4.3	odd	2	inner	252.2.bb.a.11.40	yes	88
7.2	even	3		252.2.o.a.191.25	yes	88
9.4	even	3		756.2.o.a.179.35		88
9.5	odd	6		252.2.o.a.95.10	✓	88
12.11	even	2		756.2.bb.a.683.5		88
21.2	odd	6		756.2.o.a.359.20		88
28.23	odd	6		252.2.o.a.191.10	yes	88
36.23	even	6		252.2.o.a.95.25	yes	88
36.31	odd	6		756.2.o.a.179.20		88
63.23	odd	6	inner	252.2.bb.a.23.40	yes	88
63.58	even	3		756.2.bb.a.611.5		88
84.23	even	6		756.2.o.a.359.35		88
252.23	even	6	inner	252.2.bb.a.23.5	yes	88
252.247	odd	6		756.2.bb.a.611.40		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.o.a.95.10	✓	88	9.5	odd	6
252.2.o.a.95.25	yes	88	36.23	even	6
252.2.o.a.191.10	yes	88	28.23	odd	6
252.2.o.a.191.25	yes	88	7.2	even	3
252.2.bb.a.11.5	yes	88	1.1	even	1	trivial
252.2.bb.a.11.40	yes	88	4.3	odd	2	inner
252.2.bb.a.23.5	yes	88	252.23	even	6	inner
252.2.bb.a.23.40	yes	88	63.23	odd	6	inner
756.2.o.a.179.20		88	36.31	odd	6
756.2.o.a.179.35		88	9.4	even	3
756.2.o.a.359.20		88	21.2	odd	6
756.2.o.a.359.35		88	84.23	even	6
756.2.bb.a.611.5		88	63.58	even	3
756.2.bb.a.611.40		88	252.247	odd	6
756.2.bb.a.683.5		88	12.11	even	2
756.2.bb.a.683.40		88	3.2	odd	2