Embedded newform 68.6.i.b.3.8

• Label: 68.6.i.b.3.8
• Level: $68$
• Weight: $6$
• Character: 68.3
• Analytic conductor: $10.906$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	68.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9060997473\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(42\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	68.3
Dual form			68.6.i.b.23.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.03616 + 2.57627i) q^{2} +(3.80789 - 19.1436i) q^{3} +(18.7257 - 25.9489i) q^{4} +(8.73287 + 13.0697i) q^{5} +(30.1418 + 106.220i) q^{6} +(126.682 + 84.6465i) q^{7} +(-27.4543 + 178.925i) q^{8} +(-127.474 - 52.8014i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.03616	+	2.57627i	−0.890275	+	0.455424i
							\(3\)	3.80789	−	19.1436i	0.244277	−	1.22806i	−0.642653	−	0.766157i	\(-0.722166\pi\)
0.886930	−	0.461904i	\(-0.152834\pi\)
\(5\)	8.73287	+	13.0697i	0.156218	+	0.233797i	0.901320	−	0.433153i	\(-0.142599\pi\)
							−0.745102	+	0.666951i	\(0.767599\pi\)
\(7\)	126.682	+	84.6465i	0.977172	+	0.652926i	0.938120	−	0.346311i	\(-0.112566\pi\)
							0.0390528	+	0.999237i	\(0.487566\pi\)
\(11\)	307.741	−	61.2134i	0.766838	−	0.152533i	0.203854	−	0.979001i	\(-0.434653\pi\)
							0.562984	+	0.826468i	\(0.309653\pi\)
\(13\)	−241.273	−	241.273i	−0.395958	−	0.395958i	0.480846	−	0.876805i	\(-0.340329\pi\)
							−0.876805	+	0.480846i	\(0.840329\pi\)
\(17\)	1046.25	−	570.278i	0.878038	−	0.478591i
							\(19\)	208.039	+	502.250i	0.132209	+	0.319180i	0.976096	−	0.217341i	\(-0.0697382\pi\)
−0.843887	+	0.536521i	\(0.819738\pi\)
\(23\)	185.511	+	932.627i	0.0731224	+	0.367611i	0.999969	−	0.00782835i	\(-0.00249187\pi\)
							−0.926847	+	0.375439i	\(0.877492\pi\)
\(29\)	4155.51	−	2776.63i	0.917550	−	0.613087i	−0.00457062	−	0.999990i	\(-0.501455\pi\)
							0.922121	+	0.386902i	\(0.126455\pi\)
\(31\)	−3729.27	−	741.797i	−0.696978	−	0.138638i	−0.166129	−	0.986104i	\(-0.553127\pi\)
							−0.530849	+	0.847466i	\(0.678127\pi\)
\(37\)	−9473.37	−	1884.37i	−1.13763	−	0.226288i	−0.409884	−	0.912138i	\(-0.634431\pi\)
							−0.727744	+	0.685849i	\(0.759431\pi\)
\(41\)	1377.47	−	2061.53i	0.127974	−	0.191527i	−0.761949	−	0.647636i	\(-0.775758\pi\)
							0.889924	+	0.456110i	\(0.150758\pi\)
\(43\)	1783.32	−	4305.32i	0.147082	−	0.355087i	−0.833119	−	0.553094i	\(-0.813447\pi\)
							0.980201	+	0.198007i	\(0.0634470\pi\)
\(47\)	−3562.33	+	3562.33i	−0.235228	+	0.235228i	−0.814871	−	0.579643i	\(-0.803192\pi\)
							0.579643	+	0.814871i	\(0.303192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.6.i.b.3.8	yes	336
4.3	odd	2	inner	68.6.i.b.3.4	✓	336
17.6	odd	16	inner	68.6.i.b.23.4	yes	336
68.23	even	16	inner	68.6.i.b.23.8	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.6.i.b.3.4	✓	336	4.3	odd	2	inner
68.6.i.b.3.8	yes	336	1.1	even	1	trivial
68.6.i.b.23.4	yes	336	17.6	odd	16	inner
68.6.i.b.23.8	yes	336	68.23	even	16	inner