Embedded newform 68.2.i.b.11.6

• Label: 68.2.i.b.11.6
• Level: $68$
• Weight: $2$
• Character: 68.11
• Analytic conductor: $0.543$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.542982733745\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	68.11
Dual form			68.2.i.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07774 - 0.915678i) q^{2} +(-0.935416 + 0.186066i) q^{3} +(0.323068 - 1.97373i) q^{4} +(0.763429 + 0.510107i) q^{5} +(-0.837764 + 1.05707i) q^{6} +(0.225807 + 0.337944i) q^{7} +(-1.45912 - 2.42301i) q^{8} +(-1.93126 + 0.799952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07774	−	0.915678i	0.762081	−	0.647482i
							\(3\)	−0.935416	+	0.186066i	−0.540063	+	0.107425i	−0.457582	−	0.889167i	\(-0.651284\pi\)
−0.0824809	+	0.996593i	\(0.526284\pi\)
\(5\)	0.763429	+	0.510107i	0.341416	+	0.228127i	0.714443	−	0.699694i	\(-0.246680\pi\)
							−0.373027	+	0.927820i	\(0.621680\pi\)
\(7\)	0.225807	+	0.337944i	0.0853469	+	0.127731i	0.871687	−	0.490064i	\(-0.163026\pi\)
							−0.786340	+	0.617794i	\(0.788026\pi\)
\(11\)	−0.892912	+	4.48897i	−0.269223	+	1.35348i	0.575294	+	0.817947i	\(0.304888\pi\)
							−0.844517	+	0.535529i	\(0.820112\pi\)
\(13\)	−1.21544	+	1.21544i	−0.337103	+	0.337103i	−0.855276	−	0.518173i	\(-0.826612\pi\)
							0.518173	+	0.855276i	\(0.326612\pi\)
\(17\)	2.80335	−	3.02345i	0.679912	−	0.733294i
							\(19\)	1.29031	−	3.11507i	0.296016	−	0.714647i	−0.703974	−	0.710226i	\(-0.748593\pi\)
0.999990	−	0.00442090i	\(-0.00140722\pi\)
\(23\)	1.22170	+	0.243011i	0.254742	+	0.0506713i	0.320809	−	0.947144i	\(-0.396045\pi\)
							−0.0660673	+	0.997815i	\(0.521045\pi\)
\(29\)	−3.14845	+	4.71199i	−0.584653	+	0.874995i	−0.999389	−	0.0349470i	\(-0.988874\pi\)
							0.414737	+	0.909942i	\(0.363874\pi\)
\(31\)	−1.67369	−	8.41419i	−0.300603	−	1.51123i	−0.775588	−	0.631239i	\(-0.782547\pi\)
							0.474985	−	0.879994i	\(-0.342453\pi\)
\(37\)	1.30555	+	6.56346i	0.214632	+	1.07903i	0.926380	+	0.376591i	\(0.122904\pi\)
							−0.711748	+	0.702435i	\(0.752096\pi\)
\(41\)	1.11203	−	0.743033i	0.173670	−	0.116042i	−0.465697	−	0.884944i	\(-0.654196\pi\)
							0.639367	+	0.768902i	\(0.279196\pi\)
\(43\)	−2.03481	−	4.91247i	−0.310306	−	0.749145i	−0.999694	−	0.0247525i	\(-0.992120\pi\)
							0.689387	−	0.724393i	\(-0.257880\pi\)
\(47\)	9.03666	+	9.03666i	1.31813	+	1.31813i	0.915253	+	0.402879i	\(0.131990\pi\)
							0.402879	+	0.915253i	\(0.368010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.i.b.11.6	yes	48
3.2	odd	2		612.2.bd.d.487.1		48
4.3	odd	2	inner	68.2.i.b.11.5	✓	48
12.11	even	2		612.2.bd.d.487.2		48
17.14	odd	16	inner	68.2.i.b.31.5	yes	48
51.14	even	16		612.2.bd.d.235.2		48
68.31	even	16	inner	68.2.i.b.31.6	yes	48
204.167	odd	16		612.2.bd.d.235.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.i.b.11.5	✓	48	4.3	odd	2	inner
68.2.i.b.11.6	yes	48	1.1	even	1	trivial
68.2.i.b.31.5	yes	48	17.14	odd	16	inner
68.2.i.b.31.6	yes	48	68.31	even	16	inner
612.2.bd.d.235.1		48	204.167	odd	16
612.2.bd.d.235.2		48	51.14	even	16
612.2.bd.d.487.1		48	3.2	odd	2
612.2.bd.d.487.2		48	12.11	even	2