Embedded newform 74.6.h.a.3.11

• Label: 74.6.h.a.3.11
• 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			3.11
Character	\(\chi\)	\(=\)	74.3
Dual form			74.6.h.a.25.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57115 - 3.06418i) q^{2} +(-9.90239 + 8.30909i) q^{3} +(-2.77837 - 15.7569i) q^{4} +(16.4126 - 45.0933i) q^{5} +51.7066i q^{6} +(-20.3554 - 7.40876i) q^{7} +(-55.4256 - 32.0000i) q^{8} +(-13.1802 + 74.7486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 6 q^{3} + 90 q^{5} - 228 q^{7} - 186 q^{9}+O(q^{10}) \)

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
							\(3\)	−9.90239	+	8.30909i	−0.635239	+	0.533028i	−0.902552	−	0.430581i	\(-0.858309\pi\)
0.267313	+	0.963610i	\(0.413864\pi\)
\(5\)	16.4126	−	45.0933i	0.293598	−	0.806653i	−0.701936	−	0.712240i	\(-0.747681\pi\)
							0.995533	−	0.0944123i	\(-0.0300972\pi\)
\(7\)	−20.3554	−	7.40876i	−0.157013	−	0.0571480i	0.262318	−	0.964981i	\(-0.415513\pi\)
							−0.419331	+	0.907834i	\(0.637735\pi\)
\(11\)	24.0642	−	41.6805i	0.0599640	−	0.103861i	−0.834485	−	0.551031i	\(-0.814235\pi\)
							0.894449	+	0.447170i	\(0.147568\pi\)
\(13\)	−751.866	+	132.574i	−1.23391	+	0.217571i	−0.752302	−	0.658818i	\(-0.771057\pi\)
							−0.481604	+	0.876389i	\(0.659946\pi\)
\(17\)	−2195.01	−	387.039i	−1.84210	−	0.324812i	−0.859586	−	0.510991i	\(-0.829278\pi\)
							−0.982516	+	0.186179i	\(0.940390\pi\)
\(19\)	−1271.12	−	1514.87i	−0.807800	−	0.962698i	0.192026	−	0.981390i	\(-0.438494\pi\)
							−0.999825	+	0.0186916i	\(0.994050\pi\)
\(23\)	−2185.41	+	1261.74i	−0.861415	+	0.497338i	−0.864486	−	0.502657i	\(-0.832356\pi\)
							0.00307070	+	0.999995i	\(0.499023\pi\)
\(29\)	6420.04	+	3706.61i	1.41756	+	0.818431i	0.996085	−	0.0884062i	\(-0.0281773\pi\)
							0.421480	+	0.906838i	\(0.361511\pi\)
\(31\)			779.942i			0.145767i	0.997340	+	0.0728833i	\(0.0232201\pi\)
							−0.997340	+	0.0728833i	\(0.976780\pi\)
\(37\)	8143.82	+	1738.45i	0.977966	+	0.208765i
							\(41\)	−466.438	−	2645.30i	−0.0433345	−	0.245762i	0.955444	−	0.295172i	\(-0.0953771\pi\)
−0.998779	+	0.0494099i	\(0.984266\pi\)
\(43\)		−	10687.9i		−	0.881499i	−0.897630	−	0.440750i	\(-0.854713\pi\)
							0.897630	−	0.440750i	\(-0.145287\pi\)
\(47\)	−10512.2	−	18207.7i	−0.694143	−	1.20229i	−0.970469	−	0.241227i	\(-0.922450\pi\)
							0.276325	−	0.961064i	\(-0.410883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.h.a.3.11	✓	96
37.25	even	18	inner	74.6.h.a.25.11	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.h.a.3.11	✓	96	1.1	even	1	trivial
74.6.h.a.25.11	yes	96	37.25	even	18	inner