Embedded newform 74.6.h.a.3.10

• Label: 74.6.h.a.3.10
• Level: $74$
• Weight: $6$
• Character: 74.3
• Analytic conductor: $11.868$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8684026662\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(-16.4794 + 13.8279i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(14.6671 - 40.2975i) q^{5} +86.0494i q^{6} +(36.2322 + 13.1874i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(38.1647 - 216.443i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.57115	−	3.06418i	0.454519	−	0.541675i
							\(3\)	−16.4794	+	13.8279i	−1.05716	+	0.887059i	−0.993828	−	0.110934i	\(-0.964616\pi\)
−0.0633276	+	0.997993i	\(0.520171\pi\)
\(5\)	14.6671	−	40.2975i	0.262373	−	0.720864i	−0.736633	−	0.676292i	\(-0.763586\pi\)
							0.999006	−	0.0445713i	\(-0.0141922\pi\)
\(7\)	36.2322	+	13.1874i	0.279479	+	0.101722i	0.477957	−	0.878383i	\(-0.341377\pi\)
							−0.198478	+	0.980105i	\(0.563600\pi\)
\(11\)	−365.427	+	632.937i	−0.910581	+	1.57717i	−0.0973347	+	0.995252i	\(0.531032\pi\)
							−0.813246	+	0.581920i	\(0.802302\pi\)
\(13\)	834.236	−	147.098i	1.36909	−	0.241407i	0.559705	−	0.828692i	\(-0.310914\pi\)
							0.809380	+	0.587285i	\(0.199803\pi\)
\(17\)	1984.62	+	349.942i	1.66554	+	0.293680i	0.925462	−	0.378840i	\(-0.123677\pi\)
							0.740079	+	0.672520i	\(0.234788\pi\)
\(19\)	1661.85	+	1980.51i	1.05610	+	1.25862i	0.964854	+	0.262788i	\(0.0846420\pi\)
							0.0912507	+	0.995828i	\(0.470914\pi\)
\(23\)	−3121.52	+	1802.21i	−1.23040	+	0.710372i	−0.967114	−	0.254345i	\(-0.918140\pi\)
							−0.263288	+	0.964717i	\(0.584807\pi\)
\(29\)	2043.87	+	1180.03i	0.451293	+	0.260554i	0.708376	−	0.705835i	\(-0.249428\pi\)
							−0.257083	+	0.966389i	\(0.582762\pi\)
\(31\)		−	6515.07i		−	1.21763i	−0.793313	−	0.608814i	\(-0.791646\pi\)
							0.793313	−	0.608814i	\(-0.208354\pi\)
\(37\)	−6582.30	+	5100.71i	−0.790449	+	0.612528i
							\(41\)	2455.14	+	13923.8i	0.228095	+	1.29359i	0.856679	+	0.515850i	\(0.172524\pi\)
−0.628583	+	0.777742i	\(0.716365\pi\)
\(43\)		−	7149.72i		−	0.589682i	−0.955546	−	0.294841i	\(-0.904733\pi\)
							0.955546	−	0.294841i	\(-0.0952667\pi\)
\(47\)	−1235.72	−	2140.32i	−0.0815970	−	0.141330i	0.822339	−	0.568998i	\(-0.192669\pi\)
							−0.903936	+	0.427668i	\(0.859335\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.10	✓	96
37.25	even	18	inner	74.6.h.a.25.10	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.10	✓	96	1.1	even	1	trivial
74.6.h.a.25.10	yes	96	37.25	even	18	inner