Embedded newform 74.6.h.a.3.16

• Label: 74.6.h.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(21.8303 - 18.3178i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(-10.1389 + 27.8564i) q^{5} -113.990i q^{6} +(-22.2808 - 8.10956i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(98.8244 - 560.461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

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
							\(3\)	21.8303	−	18.3178i	1.40042	−	1.17509i	0.439504	−	0.898241i	\(-0.355154\pi\)
0.960913	−	0.276849i	\(-0.0892902\pi\)
\(5\)	−10.1389	+	27.8564i	−0.181370	+	0.498311i	−0.996745	−	0.0806233i	\(-0.974309\pi\)
							0.815374	+	0.578934i	\(0.196531\pi\)
\(7\)	−22.2808	−	8.10956i	−0.171865	−	0.0625536i	0.254655	−	0.967032i	\(-0.418038\pi\)
							−0.426519	+	0.904478i	\(0.640260\pi\)
\(11\)	248.670	−	430.709i	0.619643	−	1.07325i	−0.369908	−	0.929069i	\(-0.620611\pi\)
							0.989551	−	0.144185i	\(-0.0460560\pi\)
\(13\)	−177.375	+	31.2759i	−0.291094	+	0.0513277i	−0.317288	−	0.948329i	\(-0.602772\pi\)
							0.0261943	+	0.999657i	\(0.491661\pi\)
\(17\)	−733.299	−	129.300i	−0.615402	−	0.108512i	−0.142747	−	0.989759i	\(-0.545593\pi\)
							−0.472656	+	0.881247i	\(0.656705\pi\)
\(19\)	1413.03	+	1683.98i	0.897979	+	1.07017i	0.997176	+	0.0751014i	\(0.0239280\pi\)
							−0.0991974	+	0.995068i	\(0.531628\pi\)
\(23\)	−1234.02	+	712.464i	−0.486412	+	0.280830i	−0.723085	−	0.690759i	\(-0.757276\pi\)
							0.236673	+	0.971589i	\(0.423943\pi\)
\(29\)	4108.38	+	2371.98i	0.907144	+	0.523740i	0.879511	−	0.475878i	\(-0.157870\pi\)
							0.0276327	+	0.999618i	\(0.491203\pi\)
\(31\)		−	1791.28i		−	0.334780i	−0.985891	−	0.167390i	\(-0.946466\pi\)
							0.985891	−	0.167390i	\(-0.0535339\pi\)
\(37\)	4112.11	−	7241.17i	0.493810	−	0.869570i
							\(41\)	780.392	+	4425.82i	0.0725026	+	0.411182i	0.999360	+	0.0357692i	\(0.0113881\pi\)
−0.926858	+	0.375413i	\(0.877501\pi\)
\(43\)			8705.51i			0.717998i	0.933338	+	0.358999i	\(0.116882\pi\)
							−0.933338	+	0.358999i	\(0.883118\pi\)
\(47\)	13231.3	+	22917.3i	0.873691	+	1.51328i	0.858150	+	0.513398i	\(0.171614\pi\)
							0.0155407	+	0.999879i	\(0.495053\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.16	✓	96
37.25	even	18	inner	74.6.h.a.25.16	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.16	✓	96	1.1	even	1	trivial
74.6.h.a.25.16	yes	96	37.25	even	18	inner