Embedded newform 43.3.f.a.27.2

• Label: 43.3.f.a.27.2
• Level: $43$
• Weight: $3$
• Character: 43.27
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			27.2
Character	\(\chi\)	\(=\)	43.27
Dual form			43.3.f.a.8.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.63550 + 0.601536i) q^{2} +(-1.75981 - 0.401664i) q^{3} +(2.98015 - 1.43516i) q^{4} +(7.00305 - 5.58475i) q^{5} +4.87959 q^{6} -3.35098i q^{7} +(1.46315 - 1.16682i) q^{8} +(-5.17313 - 2.49125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.63550	+	0.601536i	−1.31775	+	0.300768i	−0.822888	−	0.568203i	\(-0.807639\pi\)
							−0.494863	+	0.868971i	\(0.664782\pi\)
\(3\)	−1.75981	−	0.401664i	−0.586602	−	0.133888i	−0.0810888	−	0.996707i	\(-0.525840\pi\)
							−0.505513	+	0.862819i	\(0.668697\pi\)
\(5\)	7.00305	−	5.58475i	1.40061	−	1.11695i	0.423094	−	0.906086i	\(-0.360944\pi\)
							0.977516	−	0.210864i	\(-0.0676276\pi\)
\(7\)		−	3.35098i		−	0.478712i	−0.970932	−	0.239356i	\(-0.923064\pi\)
							0.970932	−	0.239356i	\(-0.0769362\pi\)
\(11\)	11.3734	+	5.47712i	1.03394	+	0.497920i	0.872321	−	0.488933i	\(-0.162614\pi\)
							0.161620	+	0.986853i	\(0.448328\pi\)
\(13\)	−10.6319	−	13.3320i	−0.817841	−	1.02554i	−0.999113	−	0.0421074i	\(-0.986593\pi\)
							0.181272	−	0.983433i	\(-0.441979\pi\)
\(17\)	7.32636	−	9.18697i	0.430963	−	0.540410i	−0.518174	−	0.855275i	\(-0.673388\pi\)
							0.949136	+	0.314865i	\(0.101959\pi\)
\(19\)	7.01479	+	14.5664i	0.369200	+	0.766651i	0.999956	−	0.00932970i	\(-0.00296978\pi\)
							−0.630757	+	0.775980i	\(0.717255\pi\)
\(23\)	−0.485747	−	0.233923i	−0.0211194	−	0.0101706i	0.423294	−	0.905992i	\(-0.360874\pi\)
							−0.444414	+	0.895822i	\(0.646588\pi\)
\(29\)	−7.64377	+	1.74464i	−0.263578	+	0.0601600i	−0.352268	−	0.935899i	\(-0.614589\pi\)
							0.0886893	+	0.996059i	\(0.471732\pi\)
\(31\)	3.50953	+	15.3762i	0.113211	+	0.496008i	0.999462	+	0.0328030i	\(0.0104434\pi\)
							−0.886251	+	0.463205i	\(0.846699\pi\)
\(37\)			67.6217i			1.82761i	0.406151	+	0.913806i	\(0.366871\pi\)
							−0.406151	+	0.913806i	\(0.633129\pi\)
\(41\)	6.74717	+	29.5613i	0.164565	+	0.721006i	0.988109	+	0.153754i	\(0.0491364\pi\)
							−0.823544	+	0.567252i	\(0.808006\pi\)
\(43\)	−42.4996	−	6.54103i	−0.988362	−	0.152117i
							\(47\)	−26.6468	+	12.8324i	−0.566954	+	0.273031i	−0.695325	−	0.718696i	\(-0.744739\pi\)
0.128371	+	0.991726i	\(0.459025\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.27.2	yes	42
3.2	odd	2		387.3.w.b.199.6		42
43.8	odd	14	inner	43.3.f.a.8.2	✓	42
129.8	even	14		387.3.w.b.352.6		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.8.2	✓	42	43.8	odd	14	inner
43.3.f.a.27.2	yes	42	1.1	even	1	trivial
387.3.w.b.199.6		42	3.2	odd	2
387.3.w.b.352.6		42	129.8	even	14