Embedded newform 43.8.g.a.9.20

• Label: 43.8.g.a.9.20
• Level: $43$
• Weight: $8$
• Character: 43.9
• 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			9.20
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.65759 + 10.8563i) q^{2} +(35.8811 + 5.40821i) q^{3} +(-14.4222 + 63.1876i) q^{4} +(369.067 + 251.626i) q^{5} +(251.931 + 436.357i) q^{6} +(471.340 - 816.384i) q^{7} +(790.511 - 380.690i) q^{8} +(-831.633 - 256.525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.65759	+	10.8563i	0.765230	+	0.959568i	0.999922	−	0.0125016i	\(-0.00397949\pi\)
							−0.234692	+	0.972070i	\(0.575408\pi\)
\(3\)	35.8811	+	5.40821i	0.767258	+	0.115646i	0.520998	−	0.853558i	\(-0.325560\pi\)
							0.246260	+	0.969204i	\(0.420798\pi\)
\(5\)	369.067	+	251.626i	1.32041	+	0.900243i	0.998911	−	0.0466648i	\(-0.0148593\pi\)
							0.321504	+	0.946908i	\(0.395812\pi\)
\(7\)	471.340	−	816.384i	0.519387	−	0.899604i	−0.480359	−	0.877072i	\(-0.659494\pi\)
							0.999746	−	0.0225325i	\(-0.00717293\pi\)
\(11\)	355.960	+	1559.56i	0.0806355	+	0.353287i	0.999109	−	0.0421943i	\(-0.0134348\pi\)
							−0.918474	+	0.395482i	\(0.870578\pi\)
\(13\)	300.304	−	4007.28i	0.0379105	−	0.505881i	−0.945636	−	0.325227i	\(-0.894559\pi\)
							0.983547	−	0.180654i	\(-0.0578215\pi\)
\(17\)	−4287.33	+	2923.05i	−0.211649	+	0.144300i	−0.664505	−	0.747284i	\(-0.731357\pi\)
							0.452856	+	0.891584i	\(0.350405\pi\)
\(19\)	−50959.8	+	15719.0i	−1.70447	+	0.525761i	−0.985558	−	0.169337i	\(-0.945838\pi\)
							−0.718916	+	0.695097i	\(0.755361\pi\)
\(23\)	−31263.1	+	29007.9i	−0.535777	+	0.497129i	−0.901016	−	0.433785i	\(-0.857178\pi\)
							0.365239	+	0.930914i	\(0.380987\pi\)
\(29\)	−67427.5	+	10163.1i	−0.513386	+	0.0773805i	−0.400626	−	0.916242i	\(-0.631207\pi\)
							−0.112760	+	0.993622i	\(0.535969\pi\)
\(31\)	49677.7	−	126577.i	0.299499	−	0.763111i	−0.699333	−	0.714796i	\(-0.746520\pi\)
							0.998832	−	0.0483149i	\(-0.0153851\pi\)
\(37\)	83601.8	+	144803.i	0.271337	+	0.469970i	0.969205	−	0.246257i	\(-0.0792008\pi\)
							−0.697867	+	0.716227i	\(0.745867\pi\)
\(41\)	−341934.	−	428772.i	−0.774816	−	0.971589i	0.225180	−	0.974317i	\(-0.427703\pi\)
							−0.999996	+	0.00272829i	\(0.999132\pi\)
\(43\)	508207.	+	116379.i	0.974768	+	0.223221i
							\(47\)	80755.5	−	353813.i	0.113457	−	0.497085i	−0.885986	−	0.463711i	\(-0.846518\pi\)
0.999443	−	0.0333742i	\(-0.0106253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.20	✓	300
43.24	even	21	inner	43.8.g.a.24.20	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.20	✓	300	1.1	even	1	trivial
43.8.g.a.24.20	yes	300	43.24	even	21	inner