Embedded newform 43.6.g.a.10.5

• Label: 43.6.g.a.10.5
• Level: $43$
• Weight: $6$
• Character: 43.10
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.5
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.14814 - 5.03031i) q^{2} +(6.48344 + 6.01575i) q^{3} +(4.84518 - 2.33332i) q^{4} +(-81.5186 + 12.2870i) q^{5} +(22.8172 - 39.5206i) q^{6} +(-25.0456 - 43.3802i) q^{7} +(-120.244 - 150.782i) q^{8} +(-12.3137 - 164.315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.14814	−	5.03031i	−0.202964	−	0.889242i	−0.969120	−	0.246588i	\(-0.920691\pi\)
							0.766157	−	0.642654i	\(-0.222167\pi\)
\(3\)	6.48344	+	6.01575i	0.415913	+	0.385910i	0.860186	−	0.509981i	\(-0.170348\pi\)
							−0.444273	+	0.895891i	\(0.646538\pi\)
\(5\)	−81.5186	+	12.2870i	−1.45825	+	0.219796i	−0.829803	−	0.558057i	\(-0.811547\pi\)
							−0.628447	+	0.777853i	\(0.716309\pi\)
\(7\)	−25.0456	−	43.3802i	−0.193191	−	0.334616i	0.753115	−	0.657889i	\(-0.228550\pi\)
							−0.946306	+	0.323273i	\(0.895217\pi\)
\(11\)	−332.738	−	160.238i	−0.829127	−	0.399287i	−0.0293394	−	0.999570i	\(-0.509340\pi\)
							−0.799788	+	0.600283i	\(0.795055\pi\)
\(13\)	−104.144	+	265.354i	−0.170913	+	0.435479i	−0.990506	−	0.137466i	\(-0.956104\pi\)
							0.819593	+	0.572946i	\(0.194199\pi\)
\(17\)	−649.469	−	97.8916i	−0.545049	−	0.0821530i	−0.129255	−	0.991611i	\(-0.541258\pi\)
							−0.415795	+	0.909458i	\(0.636497\pi\)
\(19\)	−67.2940	+	897.976i	−0.0427654	+	0.570664i	0.934128	+	0.356938i	\(0.116179\pi\)
							−0.976894	+	0.213727i	\(0.931440\pi\)
\(23\)	1651.81	−	1126.19i	0.651090	−	0.443905i	−0.192206	−	0.981355i	\(-0.561564\pi\)
							0.843296	+	0.537449i	\(0.180612\pi\)
\(29\)	3647.34	−	3384.24i	0.805344	−	0.747250i	−0.165694	−	0.986177i	\(-0.552986\pi\)
							0.971037	+	0.238927i	\(0.0767958\pi\)
\(31\)	−1020.81	−	314.877i	−0.190783	−	0.0588487i	0.197891	−	0.980224i	\(-0.436591\pi\)
							−0.388674	+	0.921375i	\(0.627067\pi\)
\(37\)	3858.20	−	6682.59i	0.463319	−	0.802492i	−0.535805	−	0.844342i	\(-0.679992\pi\)
							0.999124	+	0.0418497i	\(0.0133250\pi\)
\(41\)	104.669	+	458.584i	0.00972427	+	0.0426048i	0.979557	−	0.201165i	\(-0.0644729\pi\)
							−0.969833	+	0.243770i	\(0.921616\pi\)
\(43\)	1162.35	+	12068.9i	0.0958663	+	0.995394i
							\(47\)	−20142.6	+	9700.18i	−1.33006	+	0.640524i	−0.957755	−	0.287586i	\(-0.907147\pi\)
−0.372306	+	0.928110i	\(0.621433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.5	✓	204
43.13	even	21	inner	43.6.g.a.13.5	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.5	✓	204	1.1	even	1	trivial
43.6.g.a.13.5	yes	204	43.13	even	21	inner