Embedded newform 74.4.f.a.49.1

• Label: 74.4.f.a.49.1
• Level: $74$
• Weight: $4$
• Character: 74.49
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $24$
• 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:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.1
Character	\(\chi\)	\(=\)	74.49
Dual form			74.4.f.a.71.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53209 + 1.28558i) q^{2} +(-6.58177 - 5.52276i) q^{3} +(0.694593 - 3.93923i) q^{4} +(-2.85390 + 1.03873i) q^{5} +17.1838 q^{6} +(-0.561982 + 0.204545i) q^{7} +(4.00000 + 6.92820i) q^{8} +(8.13033 + 46.1094i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.58177	−	5.52276i	−1.26666	−	1.06286i	−0.994939	−	0.100481i	\(-0.967962\pi\)
−0.271724	−	0.962375i	\(-0.587594\pi\)
\(5\)	−2.85390	+	1.03873i	−0.255261	+	0.0929073i	−0.466481	−	0.884531i	\(-0.654478\pi\)
							0.211220	+	0.977438i	\(0.432256\pi\)
\(7\)	−0.561982	+	0.204545i	−0.0303442	+	0.0110444i	−0.357148	−	0.934048i	\(-0.616251\pi\)
							0.326803	+	0.945092i	\(0.394028\pi\)
\(11\)	18.8860	+	32.7116i	0.517668	+	0.896628i	0.999789	+	0.0205233i	\(0.00653324\pi\)
							−0.482121	+	0.876105i	\(0.660133\pi\)
\(13\)	−5.40683	+	30.6637i	−0.115353	+	0.654198i	0.871222	+	0.490889i	\(0.163328\pi\)
							−0.986575	+	0.163309i	\(0.947783\pi\)
\(17\)	9.73221	+	55.1941i	0.138847	+	0.787443i	0.972103	+	0.234555i	\(0.0753632\pi\)
							−0.833255	+	0.552888i	\(0.813526\pi\)
\(19\)	−27.5057	−	23.0800i	−0.332118	−	0.278680i	0.461444	−	0.887169i	\(-0.347332\pi\)
							−0.793562	+	0.608489i	\(0.791776\pi\)
\(23\)	44.7504	−	77.5099i	0.405700	−	0.702693i	−0.588703	−	0.808350i	\(-0.700361\pi\)
							0.994403	+	0.105657i	\(0.0336944\pi\)
\(29\)	84.1091	+	145.681i	0.538575	+	0.932839i	0.998981	+	0.0451309i	\(0.0143705\pi\)
							−0.460406	+	0.887708i	\(0.652296\pi\)
\(31\)	248.887			1.44198			0.720990	−	0.692946i	\(-0.243688\pi\)
							0.720990	+	0.692946i	\(0.243688\pi\)
\(37\)	−223.583	−	25.7593i	−0.993429	−	0.114454i
							\(41\)	−57.4574	+	325.857i	−0.218862	+	1.24123i	0.655216	+	0.755442i	\(0.272578\pi\)
−0.874078	+	0.485786i	\(0.838534\pi\)
\(43\)	−526.034			−1.86557			−0.932783	−	0.360437i	\(-0.882627\pi\)
							−0.932783	+	0.360437i	\(0.882627\pi\)
\(47\)	−45.4760	+	78.7667i	−0.141135	+	0.244453i	−0.927924	−	0.372769i	\(-0.878409\pi\)
							0.786789	+	0.617222i	\(0.211742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.49.1	✓	24
37.34	even	9	inner	74.4.f.a.71.1	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.49.1	✓	24	1.1	even	1	trivial
74.4.f.a.71.1	yes	24	37.34	even	9	inner