Embedded newform 43.8.g.a.10.6

• Label: 43.8.g.a.10.6
• Level: $43$
• Weight: $8$
• Character: 43.10
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.6
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.26039 - 14.2847i) q^{2} +(-61.7305 - 57.2775i) q^{3} +(-78.0983 + 37.6102i) q^{4} +(148.612 - 22.3997i) q^{5} +(-616.926 + 1068.55i) q^{6} +(860.703 + 1490.78i) q^{7} +(-377.452 - 473.309i) q^{8} +(366.504 + 4890.65i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.26039	−	14.2847i	−0.288180	−	1.26260i	−0.887020	−	0.461731i	\(-0.847228\pi\)
							0.598840	−	0.800869i	\(-0.295629\pi\)
\(3\)	−61.7305	−	57.2775i	−1.32000	−	1.22478i	−0.955993	−	0.293390i	\(-0.905216\pi\)
							−0.364011	−	0.931394i	\(-0.618593\pi\)
\(5\)	148.612	−	22.3997i	0.531692	−	0.0801396i	0.122291	−	0.992494i	\(-0.460976\pi\)
							0.409401	+	0.912355i	\(0.365738\pi\)
\(7\)	860.703	+	1490.78i	0.948441	+	1.64275i	0.748711	+	0.662897i	\(0.230673\pi\)
							0.199730	+	0.979851i	\(0.435993\pi\)
\(11\)	−530.996	−	255.714i	−0.120286	−	0.0579269i	0.372773	−	0.927923i	\(-0.378407\pi\)
							−0.493059	+	0.869996i	\(0.664121\pi\)
\(13\)	−1774.50	+	4521.36i	−0.224014	+	0.570779i	−0.998054	−	0.0623584i	\(-0.980138\pi\)
							0.774040	+	0.633137i	\(0.218233\pi\)
\(17\)	8354.71	+	1259.27i	0.412439	+	0.0621652i	0.351985	−	0.936006i	\(-0.385507\pi\)
							0.0604546	+	0.998171i	\(0.480745\pi\)
\(19\)	−891.449	+	11895.6i	−0.0298167	+	0.397876i	0.962250	+	0.272166i	\(0.0877399\pi\)
							−0.992067	+	0.125710i	\(0.959879\pi\)
\(23\)	−64495.3	+	43972.1i	−1.10530	+	0.753581i	−0.971408	−	0.237415i	\(-0.923700\pi\)
							−0.133892	+	0.990996i	\(0.542748\pi\)
\(29\)	1567.26	−	1454.20i	0.0119330	−	0.0110722i	−0.674184	−	0.738564i	\(-0.735504\pi\)
							0.686117	+	0.727492i	\(0.259314\pi\)
\(31\)	216083.	+	66652.8i	1.30273	+	0.401839i	0.867032	−	0.498253i	\(-0.166025\pi\)
							0.435700	+	0.900092i	\(0.356501\pi\)
\(37\)	−152161.	+	263551.i	−0.493854	+	0.855380i	−0.999975	−	0.00708257i	\(-0.997746\pi\)
							0.506121	+	0.862462i	\(0.331079\pi\)
\(41\)	−123875.	−	542733.i	−0.280699	−	1.22982i	−0.896899	−	0.442234i	\(-0.854186\pi\)
							0.616200	−	0.787590i	\(-0.288671\pi\)
\(43\)	−148226.	+	499848.i	−0.284305	+	0.958734i
							\(47\)	−44796.8	+	21573.0i	−0.0629367	+	0.0303087i	−0.465087	−	0.885265i	\(-0.653977\pi\)
0.402151	+	0.915574i	\(0.368263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.6	✓	300
43.13	even	21	inner	43.8.g.a.13.6	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.6	✓	300	1.1	even	1	trivial
43.8.g.a.13.6	yes	300	43.13	even	21	inner