Embedded newform 43.10.e.a.4.12

• Label: 43.10.e.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	43.4
Dual form			43.10.e.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.0261 + 13.8263i) q^{2} +(166.166 + 208.366i) q^{3} +(44.3393 + 194.263i) q^{4} +(324.564 + 156.302i) q^{5} -4713.08 q^{6} +9013.73 q^{7} +(-11332.6 - 5457.50i) q^{8} +(-11425.2 + 50057.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 23 q^{2} + 141 q^{3} - 7771 q^{4} + 677 q^{5} - 6572 q^{6} - 23872 q^{7} + 40269 q^{8} - 322935 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.0261	+	13.8263i	−0.487289	+	0.611041i	−0.963309	−	0.268393i	\(-0.913507\pi\)
							0.476020	+	0.879434i	\(0.342079\pi\)
\(3\)	166.166	+	208.366i	1.18439	+	1.48518i	0.836774	+	0.547548i	\(0.184439\pi\)
							0.347621	+	0.937635i	\(0.386990\pi\)
\(5\)	324.564	+	156.302i	0.232239	+	0.111840i	0.546385	−	0.837534i	\(-0.316004\pi\)
							−0.314145	+	0.949375i	\(0.601718\pi\)
\(7\)	9013.73			1.41894			0.709469	−	0.704736i	\(-0.248935\pi\)
							0.709469	+	0.704736i	\(0.248935\pi\)
\(11\)	−7447.70	+	32630.5i	−0.153375	+	0.671980i	0.838515	+	0.544879i	\(0.183424\pi\)
							−0.991890	+	0.127101i	\(0.959433\pi\)
\(13\)	−25401.5	−	12232.7i	−0.246669	−	0.118789i	0.306465	−	0.951882i	\(-0.400854\pi\)
							−0.553134	+	0.833093i	\(0.686568\pi\)
\(17\)	419743.	−	202138.i	1.21889	−	0.586985i	0.289886	−	0.957061i	\(-0.406383\pi\)
							0.929001	+	0.370076i	\(0.120668\pi\)
\(19\)	174363.	+	763933.i	0.306946	+	1.34482i	0.859412	+	0.511284i	\(0.170830\pi\)
							−0.552466	+	0.833536i	\(0.686313\pi\)
\(23\)	282769.	−	1.23889e6i	0.210696	−	0.923120i	−0.753399	−	0.657563i	\(-0.771587\pi\)
							0.964095	−	0.265556i	\(-0.0855557\pi\)
\(29\)	829224.	−	1.03981e6i	0.217711	−	0.273001i	−0.660968	−	0.750414i	\(-0.729854\pi\)
							0.878679	+	0.477413i	\(0.158425\pi\)
\(31\)	92860.4	−	116443.i	0.0180594	−	0.0226458i	−0.772720	−	0.634747i	\(-0.781104\pi\)
							0.790779	+	0.612102i	\(0.209676\pi\)
\(37\)	−1.07692e7			−0.944661			−0.472331	−	0.881421i	\(-0.656587\pi\)
							−0.472331	+	0.881421i	\(0.656587\pi\)
\(41\)	1.11170e7	−	1.39403e7i	0.614412	−	0.770448i	−0.373134	−	0.927777i	\(-0.621717\pi\)
							0.987546	+	0.157329i	\(0.0502884\pi\)
\(43\)	2.23040e7	+	2.26330e6i	0.994891	+	0.100957i
							\(47\)	−6.32532e6	−	2.77130e7i	−0.189079	−	0.828407i	−0.977104	−	0.212764i	\(-0.931753\pi\)
0.788025	−	0.615643i	\(-0.211104\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.e.a.4.12	✓	192
43.11	even	7	inner	43.10.e.a.11.12	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.e.a.4.12	✓	192	1.1	even	1	trivial
43.10.e.a.11.12	yes	192	43.11	even	7	inner