Embedded newform 43.7.f.a.2.8

• Label: 43.7.f.a.2.8
• Level: $43$
• Weight: $7$
• Character: 43.2
• 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			2.8
Character	\(\chi\)	\(=\)	43.2
Dual form			43.7.f.a.22.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.72088 - 5.64997i) q^{2} +(-8.85233 + 18.3821i) q^{3} +(15.3844 - 19.2915i) q^{4} +(-52.8314 - 12.0584i) q^{5} +127.944 q^{6} -8.48034i q^{7} +(-542.136 - 123.739i) q^{8} +(194.988 + 244.507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.72088	−	5.64997i	−0.340110	−	0.706246i	0.658829	−	0.752292i	\(-0.271052\pi\)
							−0.998939	+	0.0460468i	\(0.985338\pi\)
\(3\)	−8.85233	+	18.3821i	−0.327864	+	0.680817i	−0.998118	−	0.0613147i	\(-0.980471\pi\)
							0.670254	+	0.742131i	\(0.266185\pi\)
\(5\)	−52.8314	−	12.0584i	−0.422651	−	0.0964674i	0.00590491	−	0.999983i	\(-0.498120\pi\)
							−0.428556	+	0.903515i	\(0.640978\pi\)
\(7\)		−	8.48034i		−	0.0247240i	−0.999924	−	0.0123620i	\(-0.996065\pi\)
							0.999924	−	0.0123620i	\(-0.00393505\pi\)
\(11\)	−101.284	−	127.006i	−0.0760963	−	0.0954217i	0.742326	−	0.670039i	\(-0.233722\pi\)
							−0.818422	+	0.574617i	\(0.805151\pi\)
\(13\)	−890.251	+	3900.44i	−0.405212	+	1.77535i	0.200529	+	0.979688i	\(0.435734\pi\)
							−0.605741	+	0.795662i	\(0.707123\pi\)
\(17\)	271.872	+	1191.15i	0.0553372	+	0.242448i	0.995031	−	0.0995700i	\(-0.0317467\pi\)
							−0.939693	+	0.342018i	\(0.888890\pi\)
\(19\)	9494.57	+	7571.67i	1.38425	+	1.10390i	0.982107	+	0.188326i	\(0.0603061\pi\)
							0.402143	+	0.915577i	\(0.368265\pi\)
\(23\)	13139.6	+	16476.5i	1.07994	+	1.35420i	0.930861	+	0.365372i	\(0.119058\pi\)
							0.149076	+	0.988826i	\(0.452370\pi\)
\(29\)	−14193.9	−	29473.9i	−0.581979	−	1.20849i	−0.959294	−	0.282408i	\(-0.908867\pi\)
							0.377315	−	0.926085i	\(-0.376847\pi\)
\(31\)	−33403.3	+	16086.2i	−1.12125	+	0.539968i	−0.900280	−	0.435312i	\(-0.856638\pi\)
							−0.220975	+	0.975280i	\(0.570924\pi\)
\(37\)			37913.0i			0.748484i	0.927331	+	0.374242i	\(0.122097\pi\)
							−0.927331	+	0.374242i	\(0.877903\pi\)
\(41\)	−23754.8	+	11439.7i	−0.344667	+	0.165983i	−0.598206	−	0.801342i	\(-0.704120\pi\)
							0.253539	+	0.967325i	\(0.418405\pi\)
\(43\)	−12762.6	−	78476.0i	−0.160521	−	0.987032i
							\(47\)	50642.7	−	63504.0i	0.487779	−	0.611656i	−0.475645	−	0.879638i	\(-0.657785\pi\)
0.963424	+	0.267981i	\(0.0863566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.2.8	✓	126
43.22	odd	14	inner	43.7.f.a.22.8	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.8	✓	126	1.1	even	1	trivial
43.7.f.a.22.8	yes	126	43.22	odd	14	inner