Embedded newform 74.6.h.a.3.9

• Label: 74.6.h.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(-20.6482 + 17.3259i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(-24.4685 + 67.2268i) q^{5} +107.817i q^{6} +(-39.0875 - 14.2267i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(83.9648 - 476.188i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

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
							\(3\)	−20.6482	+	17.3259i	−1.32458	+	1.11146i	−0.339271	+	0.940689i	\(0.610180\pi\)
−0.985311	+	0.170768i	\(0.945375\pi\)
\(5\)	−24.4685	+	67.2268i	−0.437707	+	1.20259i	0.503274	+	0.864127i	\(0.332129\pi\)
							−0.940980	+	0.338462i	\(0.890093\pi\)
\(7\)	−39.0875	−	14.2267i	−0.301504	−	0.109738i	0.186838	−	0.982391i	\(-0.440176\pi\)
							−0.488342	+	0.872652i	\(0.662398\pi\)
\(11\)	195.425	−	338.486i	0.486966	−	0.843450i	−0.512922	−	0.858435i	\(-0.671437\pi\)
							0.999888	+	0.0149858i	\(0.00477029\pi\)
\(13\)	234.895	−	41.4184i	0.385492	−	0.0679727i	0.0224563	−	0.999748i	\(-0.492851\pi\)
							0.363036	+	0.931775i	\(0.381740\pi\)
\(17\)	−675.405	−	119.092i	−0.566816	−	0.0999449i	−0.117106	−	0.993119i	\(-0.537362\pi\)
							−0.449710	+	0.893175i	\(0.648473\pi\)
\(19\)	−644.550	−	768.144i	−0.409612	−	0.488156i	0.521314	−	0.853365i	\(-0.325442\pi\)
							−0.930926	+	0.365209i	\(0.880998\pi\)
\(23\)	2240.77	−	1293.71i	0.883237	−	0.509937i	0.0115126	−	0.999934i	\(-0.496335\pi\)
							0.871724	+	0.489997i	\(0.163002\pi\)
\(29\)	−5672.97	−	3275.29i	−1.25261	−	0.723194i	−0.280982	−	0.959713i	\(-0.590660\pi\)
							−0.971627	+	0.236519i	\(0.923993\pi\)
\(31\)			1696.60i			0.317084i	0.987352	+	0.158542i	\(0.0506793\pi\)
							−0.987352	+	0.158542i	\(0.949321\pi\)
\(37\)	1820.80	−	8125.80i	0.218654	−	0.975803i
							\(41\)	3687.41	+	20912.4i	0.342580	+	1.94287i	0.333036	+	0.942914i	\(0.391927\pi\)
0.00954408	+	0.999954i	\(0.496962\pi\)
\(43\)		−	23059.7i		−	1.90188i	−0.309374	−	0.950940i	\(-0.600120\pi\)
							0.309374	−	0.950940i	\(-0.399880\pi\)
\(47\)	3749.54	+	6494.39i	0.247590	+	0.428838i	0.962857	−	0.270013i	\(-0.0870281\pi\)
							−0.715267	+	0.698852i	\(0.753695\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.9	✓	96
37.25	even	18	inner	74.6.h.a.25.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.9	✓	96	1.1	even	1	trivial
74.6.h.a.25.9	yes	96	37.25	even	18	inner