Embedded newform 74.4.f.a.49.4

• Label: 74.4.f.a.49.4
• 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			49.4
Character	\(\chi\)	\(=\)	74.49
Dual form			74.4.f.a.71.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53209 + 1.28558i) q^{2} +(5.19166 + 4.35632i) q^{3} +(0.694593 - 3.93923i) q^{4} +(-18.6628 + 6.79271i) q^{5} -13.5545 q^{6} +(-8.13840 + 2.96214i) q^{7} +(4.00000 + 6.92820i) q^{8} +(3.28731 + 18.6432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

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
							\(3\)	5.19166	+	4.35632i	0.999135	+	0.838374i	0.986864	−	0.161551i	\(-0.0516495\pi\)
0.0122709	+	0.999925i	\(0.496094\pi\)
\(5\)	−18.6628	+	6.79271i	−1.66925	+	0.607559i	−0.991776	−	0.127985i	\(-0.959149\pi\)
							−0.677477	+	0.735544i	\(0.736927\pi\)
\(7\)	−8.13840	+	2.96214i	−0.439432	+	0.159940i	−0.552256	−	0.833675i	\(-0.686233\pi\)
							0.112823	+	0.993615i	\(0.464011\pi\)
\(11\)	−1.63132	−	2.82553i	−0.0447148	−	0.0774482i	0.842802	−	0.538224i	\(-0.180905\pi\)
							−0.887517	+	0.460776i	\(0.847571\pi\)
\(13\)	−3.84611	+	21.8123i	−0.0820552	+	0.465358i	0.915898	+	0.401411i	\(0.131480\pi\)
							−0.997953	+	0.0639473i	\(0.979631\pi\)
\(17\)	13.0124	+	73.7972i	0.185646	+	1.05285i	0.925123	+	0.379668i	\(0.123962\pi\)
							−0.739477	+	0.673182i	\(0.764927\pi\)
\(19\)	−51.6291	−	43.3219i	−0.623396	−	0.523091i	0.275473	−	0.961309i	\(-0.411165\pi\)
							−0.898869	+	0.438218i	\(0.855610\pi\)
\(23\)	−81.8962	+	141.848i	−0.742458	+	1.28597i	0.208915	+	0.977934i	\(0.433007\pi\)
							−0.951373	+	0.308041i	\(0.900327\pi\)
\(29\)	110.651	+	191.652i	0.708527	+	1.22721i	0.965403	+	0.260761i	\(0.0839735\pi\)
							−0.256876	+	0.966444i	\(0.582693\pi\)
\(31\)	255.761			1.48180			0.740902	−	0.671613i	\(-0.234398\pi\)
							0.740902	+	0.671613i	\(0.234398\pi\)
\(37\)	−64.2330	+	215.701i	−0.285401	+	0.958408i
							\(41\)	52.1826	−	295.942i	0.198770	−	1.12728i	−0.708177	−	0.706034i	\(-0.750482\pi\)
0.906947	−	0.421244i	\(-0.138407\pi\)
\(43\)	129.080			0.457780			0.228890	−	0.973452i	\(-0.426490\pi\)
							0.228890	+	0.973452i	\(0.426490\pi\)
\(47\)	−210.514	+	364.621i	−0.653333	+	1.13161i	0.328976	+	0.944338i	\(0.393297\pi\)
							−0.982309	+	0.187268i	\(0.940037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.49.4	✓	24
37.34	even	9	inner	74.4.f.a.71.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.49.4	✓	24	1.1	even	1	trivial
74.4.f.a.71.4	yes	24	37.34	even	9	inner