Embedded newform 43.8.e.a.4.20

• Label: 43.8.e.a.4.20
• Level: $43$
• Weight: $8$
• Character: 43.4
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.20
Character	\(\chi\)	\(=\)	43.4
Dual form			43.8.e.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.0321 - 12.5798i) q^{2} +(30.9538 + 38.8148i) q^{3} +(-29.1265 - 127.612i) q^{4} +(-433.070 - 208.556i) q^{5} +798.812 q^{6} -1469.95 q^{7} +(-41.9462 - 20.2002i) q^{8} +(-61.7996 + 270.762i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.0321	−	12.5798i	0.886717	−	1.11191i	−0.106347	−	0.994329i	\(-0.533915\pi\)
							0.993064	−	0.117579i	\(-0.0375132\pi\)
\(3\)	30.9538	+	38.8148i	0.661895	+	0.829990i	0.993548	−	0.113411i	\(-0.0361777\pi\)
							−0.331653	+	0.943401i	\(0.607606\pi\)
\(5\)	−433.070	−	208.556i	−1.54940	−	0.746151i	−0.553183	−	0.833060i	\(-0.686587\pi\)
							−0.996216	+	0.0869088i	\(0.972301\pi\)
\(7\)	−1469.95			−1.61979			−0.809896	−	0.586574i	\(-0.800476\pi\)
							−0.809896	+	0.586574i	\(0.800476\pi\)
\(11\)	392.276	−	1718.67i	0.0888623	−	0.389331i	−0.910864	−	0.412706i	\(-0.864584\pi\)
							0.999727	+	0.0233744i	\(0.00744099\pi\)
\(13\)	−10449.4	−	5032.18i	−1.31914	−	0.635265i	−0.363995	−	0.931401i	\(-0.618587\pi\)
							−0.955146	+	0.296136i	\(0.904302\pi\)
\(17\)	15471.7	−	7450.76i	0.763775	−	0.367815i	−0.0110922	−	0.999938i	\(-0.503531\pi\)
							0.774868	+	0.632124i	\(0.217817\pi\)
\(19\)	−3413.53	−	14955.7i	−0.114174	−	0.500228i	−0.999386	−	0.0350404i	\(-0.988844\pi\)
							0.885212	−	0.465188i	\(-0.154013\pi\)
\(23\)	−24683.2	+	108144.i	−0.423014	+	1.85334i	0.0914175	+	0.995813i	\(0.470860\pi\)
							−0.514431	+	0.857532i	\(0.671997\pi\)
\(29\)	−3173.33	+	3979.23i	−0.0241614	+	0.0302974i	−0.793765	−	0.608224i	\(-0.791882\pi\)
							0.769604	+	0.638521i	\(0.220454\pi\)
\(31\)	−17017.6	+	21339.4i	−0.102596	+	0.128652i	−0.830479	−	0.557050i	\(-0.811933\pi\)
							0.727883	+	0.685702i	\(0.240505\pi\)
\(37\)	−188747.			−0.612595			−0.306297	−	0.951936i	\(-0.599090\pi\)
							−0.306297	+	0.951936i	\(0.599090\pi\)
\(41\)	138389.	−	173534.i	0.313586	−	0.393225i	−0.599913	−	0.800065i	\(-0.704798\pi\)
							0.913499	+	0.406840i	\(0.133369\pi\)
\(43\)	−490102.	−	177816.i	−0.940041	−	0.341060i
							\(47\)	−226649.	−	993013.i	−0.318428	−	1.39512i	−0.840310	−	0.542106i	\(-0.817627\pi\)
0.521883	−	0.853017i	\(-0.325230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.e.a.4.20	✓	144
43.11	even	7	inner	43.8.e.a.11.20	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.e.a.4.20	✓	144	1.1	even	1	trivial
43.8.e.a.11.20	yes	144	43.11	even	7	inner