Embedded newform 43.9.f.a.22.18

• Label: 43.9.f.a.22.18
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.18
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.66053 - 7.60117i) q^{2} +(-21.3391 - 44.3111i) q^{3} +(115.235 + 144.500i) q^{4} +(-661.934 + 151.082i) q^{5} -414.929 q^{6} +1496.55i q^{7} +(3625.83 - 827.571i) q^{8} +(2582.60 - 3238.48i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.66053	−	7.60117i	0.228783	−	0.475073i	−0.754701	−	0.656069i	\(-0.772218\pi\)
							0.983484	+	0.180996i	\(0.0579322\pi\)
\(3\)	−21.3391	−	44.3111i	−0.263446	−	0.547051i	0.726723	−	0.686931i	\(-0.241043\pi\)
							−0.990168	+	0.139880i	\(0.955328\pi\)
\(5\)	−661.934	+	151.082i	−1.05909	+	0.241731i	−0.716382	−	0.697709i	\(-0.754203\pi\)
							−0.342713	+	0.939440i	\(0.611346\pi\)
\(7\)			1496.55i			0.623301i	0.950197	+	0.311651i	\(0.100882\pi\)
							−0.950197	+	0.311651i	\(0.899118\pi\)
\(11\)	11842.3	−	14849.8i	0.808844	−	1.01426i	−0.190625	−	0.981663i	\(-0.561051\pi\)
							0.999469	−	0.0325951i	\(-0.0103772\pi\)
\(13\)	886.406	+	3883.60i	0.0310355	+	0.135976i	0.988072	−	0.153991i	\(-0.0492128\pi\)
							−0.957037	+	0.289967i	\(0.906356\pi\)
\(17\)	13735.0	−	60176.9i	0.164449	−	0.720500i	−0.823703	−	0.567022i	\(-0.808095\pi\)
							0.988152	−	0.153478i	\(-0.0490474\pi\)
\(19\)	4230.57	−	3373.77i	0.0324627	−	0.0258881i	−0.607124	−	0.794607i	\(-0.707677\pi\)
							0.639587	+	0.768719i	\(0.279105\pi\)
\(23\)	304616.	−	381976.i	1.08853	−	1.36497i	0.162858	−	0.986649i	\(-0.447929\pi\)
							0.925673	−	0.378325i	\(-0.123500\pi\)
\(29\)	297850.	−	618493.i	0.421120	−	0.874465i	−0.577204	−	0.816600i	\(-0.695856\pi\)
							0.998324	−	0.0578654i	\(-0.0184294\pi\)
\(31\)	978786.	+	471359.i	1.05984	+	0.510393i	0.880820	−	0.473452i	\(-0.156992\pi\)
							0.179022	+	0.983845i	\(0.442707\pi\)
\(37\)		−	1.33452e6i		−	0.712065i	−0.934474	−	0.356032i	\(-0.884129\pi\)
							0.934474	−	0.356032i	\(-0.115871\pi\)
\(41\)	−1.03070e6	−	496358.i	−0.364751	−	0.175655i	0.242526	−	0.970145i	\(-0.422024\pi\)
							−0.607277	+	0.794490i	\(0.707738\pi\)
\(43\)	741059.	−	3.33752e6i	0.216760	−	0.976225i
							\(47\)	4.02723e6	+	5.04999e6i	0.825306	+	1.03490i	0.998746	+	0.0500568i	\(0.0159402\pi\)
−0.173440	+	0.984844i	\(0.555488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.18	yes	174
43.2	odd	14	inner	43.9.f.a.2.18	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.18	✓	174	43.2	odd	14	inner
43.9.f.a.22.18	yes	174	1.1	even	1	trivial