Embedded newform 68.6.i.b.23.8

• Label: 68.6.i.b.23.8
• Level: $68$
• Weight: $6$
• Character: 68.23
• 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			23.8
Character	\(\chi\)	\(=\)	68.23
Dual form			68.6.i.b.3.8

$q$-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 \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.6.i.b.3.4	✓	336	17.3	odd	16	inner
68.6.i.b.3.8	yes	336	68.3	even	16	inner
68.6.i.b.23.4	yes	336	4.3	odd	2	inner
68.6.i.b.23.8	yes	336	1.1	even	1	trivial