Embedded newform 43.10.e.a.4.13

• Label: 43.10.e.a.4.13
• Level: $43$
• Weight: $10$
• Character: 43.4
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(32\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.8792 + 13.6421i) q^{2} +(99.6054 + 124.901i) q^{3} +(46.1810 + 202.332i) q^{4} +(2039.80 + 982.318i) q^{5} -2787.54 q^{6} -10145.3 q^{7} +(-11311.8 - 5447.46i) q^{8} +(-1299.20 + 5692.15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.8792	+	13.6421i	−0.480798	+	0.602902i	−0.961778	−	0.273830i	\(-0.911709\pi\)
							0.480980	+	0.876731i	\(0.340281\pi\)
\(3\)	99.6054	+	124.901i	0.709965	+	0.890268i	0.997724	−	0.0674347i	\(-0.0214814\pi\)
							−0.287758	+	0.957703i	\(0.592910\pi\)
\(5\)	2039.80	+	982.318i	1.45957	+	0.702890i	0.984226	−	0.176913i	\(-0.0566112\pi\)
							0.475339	+	0.879803i	\(0.342325\pi\)
\(7\)	−10145.3			−1.59707			−0.798535	−	0.601948i	\(-0.794391\pi\)
							−0.798535	+	0.601948i	\(0.794391\pi\)
\(11\)	−3758.73	+	16468.1i	−0.0774058	+	0.339137i	−0.998771	−	0.0495608i	\(-0.984218\pi\)
							0.921365	+	0.388698i	\(0.127075\pi\)
\(13\)	23226.1	+	11185.1i	0.225544	+	0.108616i	0.543243	−	0.839575i	\(-0.317196\pi\)
							−0.317700	+	0.948191i	\(0.602910\pi\)
\(17\)	−269567.	+	129817.i	−0.782792	+	0.376973i	−0.782200	−	0.623027i	\(-0.785903\pi\)
							−0.000591491		1.00000i	\(0.500188\pi\)
\(19\)	21011.1	+	92055.8i	0.0369878	+	0.162054i	0.990049	−	0.140724i	\(-0.0449430\pi\)
							−0.953061	+	0.302778i	\(0.902086\pi\)
\(23\)	4304.64	−	18859.9i	0.00320746	−	0.0140528i	−0.973299	−	0.229542i	\(-0.926277\pi\)
							0.976506	+	0.215489i	\(0.0691345\pi\)
\(29\)	−2.95930e6	+	3.71084e6i	−0.776959	+	0.974275i	−1.00000		0.000533834i	\(-0.999830\pi\)
							0.223041	+	0.974809i	\(0.428402\pi\)
\(31\)	1.91462e6	−	2.40086e6i	0.372354	−	0.466917i	−0.559985	−	0.828503i	\(-0.689193\pi\)
							0.932339	+	0.361586i	\(0.117764\pi\)
\(37\)	−1.11343e7			−0.976683			−0.488342	−	0.872653i	\(-0.662398\pi\)
							−0.488342	+	0.872653i	\(0.662398\pi\)
\(41\)	−8.68820e6	+	1.08947e7i	−0.480178	+	0.602125i	−0.961631	−	0.274348i	\(-0.911538\pi\)
							0.481452	+	0.876472i	\(0.340110\pi\)
\(43\)	2.00905e7	−	9.94815e6i	0.896153	−	0.443746i
							\(47\)	8.09793e6	+	3.54793e7i	0.242066	+	1.06056i	0.939133	+	0.343554i	\(0.111631\pi\)
−0.697067	+	0.717006i	\(0.745512\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.13	✓	192
43.11	even	7	inner	43.10.e.a.11.13	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.13	✓	192	1.1	even	1	trivial
43.10.e.a.11.13	yes	192	43.11	even	7	inner