Embedded newform 74.4.f.b.53.3

• Label: 74.4.f.b.53.3
• Level: $74$
• Weight: $4$
• Character: 74.53
• 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			53.3
Character	\(\chi\)	\(=\)	74.53
Dual form			74.4.f.b.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87939 - 0.684040i) q^{2} +(-1.18036 + 0.429617i) q^{3} +(3.06418 + 2.57115i) q^{4} +(0.00632031 - 0.0358443i) q^{5} +2.51223 q^{6} +(-0.240077 + 1.36154i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(-19.4745 + 16.3411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87939	−	0.684040i	−0.664463	−	0.241845i
							\(3\)	−1.18036	+	0.429617i	−0.227161	+	0.0826797i	−0.453093	−	0.891463i	\(-0.649679\pi\)
0.225932	+	0.974143i	\(0.427457\pi\)
\(5\)	0.00632031	−	0.0358443i	0.000565306	−	0.00320601i	−0.984524	−	0.175251i	\(-0.943926\pi\)
							0.985089	+	0.172045i	\(0.0550374\pi\)
\(7\)	−0.240077	+	1.36154i	−0.0129629	+	0.0735164i	−0.990603	−	0.136769i	\(-0.956328\pi\)
							0.977640	+	0.210285i	\(0.0674393\pi\)
\(11\)	26.7562	+	46.3430i	0.733390	+	1.27027i	0.955426	+	0.295230i	\(0.0953962\pi\)
							−0.222037	+	0.975038i	\(0.571270\pi\)
\(13\)	58.2766	+	48.8999i	1.24331	+	1.04326i	0.997258	+	0.0740039i	\(0.0235777\pi\)
							0.246052	+	0.969257i	\(0.420867\pi\)
\(17\)	−18.5267	+	15.5457i	−0.264316	+	0.221788i	−0.765308	−	0.643664i	\(-0.777413\pi\)
							0.500991	+	0.865452i	\(0.332969\pi\)
\(19\)	−10.5616	+	3.84410i	−0.127526	+	0.0464157i	−0.404995	−	0.914319i	\(-0.632727\pi\)
							0.277469	+	0.960735i	\(0.410504\pi\)
\(23\)	−71.2311	+	123.376i	−0.645770	+	1.11851i	0.338353	+	0.941019i	\(0.390130\pi\)
							−0.984123	+	0.177487i	\(0.943203\pi\)
\(29\)	−145.354	−	251.761i	−0.930746	−	1.61210i	−0.782050	−	0.623216i	\(-0.785826\pi\)
							−0.148696	−	0.988883i	\(-0.547508\pi\)
\(31\)	−150.585			−0.872446			−0.436223	−	0.899839i	\(-0.643684\pi\)
							−0.436223	+	0.899839i	\(0.643684\pi\)
\(37\)	203.303	−	96.5441i	0.903320	−	0.428966i
							\(41\)	−77.5070	−	65.0361i	−0.295233	−	0.247730i	0.483124	−	0.875552i	\(-0.339502\pi\)
−0.778357	+	0.627822i	\(0.783947\pi\)
\(43\)	52.0294			0.184521			0.0922606	−	0.995735i	\(-0.470591\pi\)
							0.0922606	+	0.995735i	\(0.470591\pi\)
\(47\)	−100.246	+	173.632i	−0.311115	+	0.538867i	−0.978604	−	0.205752i	\(-0.934036\pi\)
							0.667489	+	0.744620i	\(0.267369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.53.3	yes	30
37.7	even	9	inner	74.4.f.b.7.3	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.7.3	✓	30	37.7	even	9	inner
74.4.f.b.53.3	yes	30	1.1	even	1	trivial