Embedded newform 43.4.e.a.21.7

• Label: 43.4.e.a.21.7
• Level: $43$
• Weight: $4$
• Character: 43.21
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			21.7
Character	\(\chi\)	\(=\)	43.21
Dual form			43.4.e.a.41.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.748275 + 0.360350i) q^{2} +(-6.31340 + 3.04037i) q^{3} +(-4.55785 - 5.71537i) q^{4} +(-1.01538 - 4.44868i) q^{5} -5.81976 q^{6} -20.3840 q^{7} +(-2.82946 - 12.3967i) q^{8} +(13.7810 - 17.2808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.748275	+	0.360350i	0.264555	+	0.127403i	0.561460	−	0.827504i	\(-0.310240\pi\)
							−0.296904	+	0.954907i	\(0.595954\pi\)
\(3\)	−6.31340	+	3.04037i	−1.21502	+	0.585120i	−0.927919	−	0.372781i	\(-0.878404\pi\)
							−0.287096	+	0.957902i	\(0.592690\pi\)
\(5\)	−1.01538	−	4.44868i	−0.0908185	−	0.397902i	0.909003	−	0.416789i	\(-0.136845\pi\)
							−0.999822	+	0.0188874i	\(0.993988\pi\)
\(7\)	−20.3840			−1.10063			−0.550317	−	0.834956i	\(-0.685493\pi\)
							−0.550317	+	0.834956i	\(0.685493\pi\)
\(11\)	−9.32270	+	11.6903i	−0.255536	+	0.320432i	−0.893007	−	0.450042i	\(-0.851409\pi\)
							0.637471	+	0.770474i	\(0.279980\pi\)
\(13\)	8.56214	+	37.5132i	0.182670	+	0.800330i	0.980353	+	0.197252i	\(0.0632019\pi\)
							−0.797683	+	0.603077i	\(0.793941\pi\)
\(17\)	12.5218	−	54.8614i	0.178645	−	0.782697i	−0.803611	−	0.595155i	\(-0.797091\pi\)
							0.982257	−	0.187542i	\(-0.0600521\pi\)
\(19\)	−51.4867	−	64.5623i	−0.621677	−	0.779558i	0.366903	−	0.930259i	\(-0.380418\pi\)
							−0.988579	+	0.150701i	\(0.951847\pi\)
\(23\)	−52.4564	+	65.7782i	−0.475561	+	0.596335i	−0.960523	−	0.278200i	\(-0.910262\pi\)
							0.484962	+	0.874535i	\(0.338834\pi\)
\(29\)	−130.210	−	62.7060i	−0.833775	−	0.401525i	−0.0322453	−	0.999480i	\(-0.510266\pi\)
							−0.801530	+	0.597955i	\(0.795980\pi\)
\(31\)	−123.399	−	59.4259i	−0.714940	−	0.344297i	0.0408010	−	0.999167i	\(-0.487009\pi\)
							−0.755741	+	0.654870i	\(0.772723\pi\)
\(37\)	250.262			1.11197			0.555984	−	0.831193i	\(-0.312342\pi\)
							0.555984	+	0.831193i	\(0.312342\pi\)
\(41\)	−402.991	−	194.070i	−1.53504	−	0.739236i	−0.540281	−	0.841485i	\(-0.681682\pi\)
							−0.994759	+	0.102248i	\(0.967396\pi\)
\(43\)	−273.272	−	69.4956i	−0.969152	−	0.246465i
							\(47\)	211.037	+	264.632i	0.654954	+	0.821287i	0.992783	−	0.119921i	\(-0.0382640\pi\)
−0.337829	+	0.941207i	\(0.609693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.21.7	✓	60
43.16	even	7		1849.4.a.h.1.14		30
43.27	odd	14		1849.4.a.g.1.17		30
43.41	even	7	inner	43.4.e.a.41.7	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.21.7	✓	60	1.1	even	1	trivial
43.4.e.a.41.7	yes	60	43.41	even	7	inner
1849.4.a.g.1.17		30	43.27	odd	14
1849.4.a.h.1.14		30	43.16	even	7