Embedded newform 74.5.i.b.15.2

• Label: 74.5.i.b.15.2
• Level: $74$
• Weight: $5$
• Character: 74.15
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			15.2
Character	\(\chi\)	\(=\)	74.15
Dual form			74.5.i.b.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19534 + 2.56343i) q^{2} +(-3.78011 - 10.3858i) q^{3} +(-5.14230 + 6.12836i) q^{4} +(17.8200 + 25.4496i) q^{5} +(22.1046 - 22.1046i) q^{6} +(6.77159 - 38.4036i) q^{7} +(-21.8564 - 5.85641i) q^{8} +(-31.5252 + 26.4528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 114 q^{5} - 576 q^{8} - 210 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{13}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19534	+	2.56343i	0.298836	+	0.640856i
							\(3\)	−3.78011	−	10.3858i	−0.420012	−	1.15397i	−0.951699	−	0.307033i	\(-0.900664\pi\)
0.531687	−	0.846941i	\(-0.321558\pi\)
\(5\)	17.8200	+	25.4496i	0.712799	+	1.01798i	0.998223	+	0.0595814i	\(0.0189766\pi\)
							−0.285424	+	0.958401i	\(0.592135\pi\)
\(7\)	6.77159	−	38.4036i	0.138196	−	0.783747i	−0.834385	−	0.551182i	\(-0.814177\pi\)
							0.972581	−	0.232565i	\(-0.0747118\pi\)
\(11\)	119.184	−	68.8108i	0.984989	−	0.568684i	0.0812166	−	0.996696i	\(-0.474119\pi\)
							0.903773	+	0.428013i	\(0.140786\pi\)
\(13\)	258.515	−	22.6171i	1.52968	−	0.133829i	0.708994	−	0.705215i	\(-0.249149\pi\)
							0.820682	+	0.571385i	\(0.193594\pi\)
\(17\)	−9.01084	+	102.994i	−0.0311794	+	0.356382i	0.964539	+	0.263941i	\(0.0850225\pi\)
							−0.995718	+	0.0924409i	\(0.970533\pi\)
\(19\)	182.586	−	391.558i	0.505779	−	1.08465i	−0.473250	−	0.880928i	\(-0.656919\pi\)
							0.979029	−	0.203719i	\(-0.0653029\pi\)
\(23\)	161.869	+	604.103i	0.305990	+	1.14197i	0.932090	+	0.362228i	\(0.117984\pi\)
							−0.626099	+	0.779743i	\(0.715350\pi\)
\(29\)	−193.063	+	720.522i	−0.229564	+	0.856744i	0.750960	+	0.660347i	\(0.229591\pi\)
							−0.980524	+	0.196397i	\(0.937076\pi\)
\(31\)	−516.720	−	516.720i	−0.537690	−	0.537690i	0.385160	−	0.922850i	\(-0.374146\pi\)
							−0.922850	+	0.385160i	\(0.874146\pi\)
\(37\)	−1358.43	+	169.829i	−0.992276	+	0.124053i
							\(41\)	−424.758	+	506.207i	−0.252682	+	0.301134i	−0.877443	−	0.479682i	\(-0.840752\pi\)
0.624761	+	0.780816i	\(0.285197\pi\)
\(43\)	1672.73	−	1672.73i	0.904666	−	0.904666i	−0.0911695	−	0.995835i	\(-0.529061\pi\)
							0.995835	+	0.0911695i	\(0.0290605\pi\)
\(47\)	−1501.28	+	2600.30i	−0.679621	+	1.17714i	0.295474	+	0.955351i	\(0.404523\pi\)
							−0.975095	+	0.221788i	\(0.928811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.i.b.15.2	yes	72
37.5	odd	36	inner	74.5.i.b.5.2	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.i.b.5.2	✓	72	37.5	odd	36	inner
74.5.i.b.15.2	yes	72	1.1	even	1	trivial