Embedded newform 43.7.f.a.22.9

• Label: 43.7.f.a.22.9
• Level: $43$
• Weight: $7$
• Character: 43.22
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.9
Character	\(\chi\)	\(=\)	43.22
Dual form			43.7.f.a.2.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12993 + 2.34633i) q^{2} +(12.1637 + 25.2581i) q^{3} +(35.6748 + 44.7348i) q^{4} +(-135.516 + 30.9307i) q^{5} -73.0081 q^{6} +261.823i q^{7} +(-307.765 + 70.2454i) q^{8} +(-35.4943 + 44.5085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12993	+	2.34633i	−0.141242	+	0.293292i	−0.959575	−	0.281452i	\(-0.909184\pi\)
							0.818333	+	0.574744i	\(0.194898\pi\)
\(3\)	12.1637	+	25.2581i	0.450507	+	0.935487i	0.995293	+	0.0969117i	\(0.0308964\pi\)
							−0.544786	+	0.838575i	\(0.683389\pi\)
\(5\)	−135.516	+	30.9307i	−1.08413	+	0.247445i	−0.727019	−	0.686617i	\(-0.759095\pi\)
							−0.357109	+	0.934063i	\(0.616238\pi\)
\(7\)			261.823i			0.763332i	0.924300	+	0.381666i	\(0.124649\pi\)
							−0.924300	+	0.381666i	\(0.875351\pi\)
\(11\)	582.215	−	730.075i	0.437427	−	0.548516i	−0.513436	−	0.858128i	\(-0.671628\pi\)
							0.950863	+	0.309612i	\(0.100199\pi\)
\(13\)	−361.705	−	1584.73i	−0.164636	−	0.721316i	−0.988083	−	0.153923i	\(-0.950809\pi\)
							0.823447	−	0.567393i	\(-0.192048\pi\)
\(17\)	−322.665	+	1413.69i	−0.0656758	+	0.287745i	−0.997092	−	0.0762123i	\(-0.975717\pi\)
							0.931416	+	0.363957i	\(0.118574\pi\)
\(19\)	−10177.4	+	8116.18i	−1.48380	+	1.18329i	−0.545186	+	0.838315i	\(0.683541\pi\)
							−0.938612	+	0.344974i	\(0.887888\pi\)
\(23\)	4291.92	−	5381.90i	0.352751	−	0.442336i	−0.573521	−	0.819191i	\(-0.694423\pi\)
							0.926272	+	0.376855i	\(0.122994\pi\)
\(29\)	1702.21	−	3534.67i	0.0697940	−	0.144929i	−0.863153	−	0.504942i	\(-0.831514\pi\)
							0.932947	+	0.360014i	\(0.117228\pi\)
\(31\)	16673.2	+	8029.41i	0.559674	+	0.269525i	0.692262	−	0.721647i	\(-0.256614\pi\)
							−0.132588	+	0.991171i	\(0.542329\pi\)
\(37\)			89903.8i			1.77490i	0.460908	+	0.887448i	\(0.347524\pi\)
							−0.460908	+	0.887448i	\(0.652476\pi\)
\(41\)	112447.	+	54151.4i	1.63153	+	0.785702i	0.999946	+	0.0103934i	\(0.00330838\pi\)
							0.631582	+	0.775309i	\(0.282406\pi\)
\(43\)	−64029.3	−	47134.0i	−0.805329	−	0.592828i
							\(47\)	30487.4	+	38230.0i	0.293648	+	0.368222i	0.906668	−	0.421844i	\(-0.138617\pi\)
−0.613021	+	0.790067i	\(0.710046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.22.9	yes	126
43.2	odd	14	inner	43.7.f.a.2.9	✓	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.9	✓	126	43.2	odd	14	inner
43.7.f.a.22.9	yes	126	1.1	even	1	trivial