Embedded newform 74.4.f.b.49.4

• Label: 74.4.f.b.49.4
• Level: $74$
• Weight: $4$
• Character: 74.49
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.4
Character	\(\chi\)	\(=\)	74.49
Dual form			74.4.f.b.71.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53209 - 1.28558i) q^{2} +(4.80507 + 4.03193i) q^{3} +(0.694593 - 3.93923i) q^{4} +(14.5989 - 5.31355i) q^{5} +12.5451 q^{6} +(-23.2945 + 8.47851i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(2.14372 + 12.1576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53209	−	1.28558i	0.541675	−	0.454519i
							\(3\)	4.80507	+	4.03193i	0.924736	+	0.775946i	0.974865	−	0.222797i	\(-0.0715187\pi\)
−0.0501286	+	0.998743i	\(0.515963\pi\)
\(5\)	14.5989	−	5.31355i	1.30576	−	0.475258i	0.406892	−	0.913476i	\(-0.366612\pi\)
							0.898869	+	0.438218i	\(0.144390\pi\)
\(7\)	−23.2945	+	8.47851i	−1.25779	+	0.457797i	−0.883026	−	0.469325i	\(-0.844497\pi\)
							−0.374761	+	0.927122i	\(0.622275\pi\)
\(11\)	23.4595	+	40.6330i	0.643027	+	1.11376i	0.984753	+	0.173957i	\(0.0556552\pi\)
							−0.341726	+	0.939800i	\(0.611011\pi\)
\(13\)	6.16912	−	34.9868i	0.131616	−	0.746431i	−0.845541	−	0.533911i	\(-0.820722\pi\)
							0.977157	−	0.212520i	\(-0.0681670\pi\)
\(17\)	−5.54874	−	31.4685i	−0.0791628	−	0.448955i	−0.998464	−	0.0554004i	\(-0.982356\pi\)
							0.919301	−	0.393554i	\(-0.128755\pi\)
\(19\)	−82.3028	−	69.0603i	−0.993766	−	0.833869i	−0.00765777	−	0.999971i	\(-0.502438\pi\)
							−0.986109	+	0.166102i	\(0.946882\pi\)
\(23\)	−90.0798	+	156.023i	−0.816650	+	1.41448i	0.0914878	+	0.995806i	\(0.470838\pi\)
							−0.908137	+	0.418672i	\(0.862496\pi\)
\(29\)	50.8534	+	88.0807i	0.325629	+	0.564006i	0.981640	−	0.190746i	\(-0.0610906\pi\)
							−0.656010	+	0.754752i	\(0.727757\pi\)
\(31\)	−138.150			−0.800401			−0.400200	−	0.916428i	\(-0.631060\pi\)
							−0.400200	+	0.916428i	\(0.631060\pi\)
\(37\)	−62.5151	+	216.206i	−0.277768	+	0.960648i
							\(41\)	61.4092	−	348.269i	0.233915	−	1.32660i	−0.610972	−	0.791652i	\(-0.709221\pi\)
0.844887	−	0.534945i	\(-0.179668\pi\)
\(43\)	−342.309			−1.21399			−0.606995	−	0.794706i	\(-0.707625\pi\)
							−0.606995	+	0.794706i	\(0.707625\pi\)
\(47\)	177.261	−	307.025i	0.550132	−	0.952856i	−0.448133	−	0.893967i	\(-0.647911\pi\)
							0.998265	−	0.0588892i	\(-0.0187559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.49.4	✓	30
37.34	even	9	inner	74.4.f.b.71.4	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.49.4	✓	30	1.1	even	1	trivial
74.4.f.b.71.4	yes	30	37.34	even	9	inner