Embedded newform 804.2.u.b.43.19

• Label: 804.2.u.b.43.19
• Level: $804$
• Weight: $2$
• Character: 804.43
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $340$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(340\)
Relative dimension:	\(34\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			43.19
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.b.187.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.302140 + 1.38156i) q^{2} +(-0.841254 + 0.540641i) q^{3} +(-1.81742 + 0.834849i) q^{4} +(-1.61949 - 0.232847i) q^{5} +(-1.00110 - 0.998894i) q^{6} +(1.26738 - 2.77517i) q^{7} +(-1.70251 - 2.25864i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q + 34 q^{3} - 2 q^{4} - 11 q^{6} + 4 q^{7} + 27 q^{8} - 34 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.302140	+	1.38156i	0.213645	+	0.976911i
							\(3\)	−0.841254	+	0.540641i	−0.485698	+	0.312139i
\(5\)	−1.61949	−	0.232847i	−0.724258	−	0.104133i								−0.229680	−	0.973266i	\(-0.573768\pi\)
							−0.494577	+	0.869134i	\(0.664677\pi\)
\(7\)	1.26738	−	2.77517i	0.479023	−	1.04891i	−0.503708	−	0.863874i	\(-0.668031\pi\)
							0.982731	−	0.185040i	\(-0.0592416\pi\)
\(11\)	−0.679795	+	4.72808i	−0.204966	+	1.42557i	0.584310	+	0.811531i	\(0.301365\pi\)
							−0.789276	+	0.614039i	\(0.789544\pi\)
\(13\)	0.879715	−	2.99604i	0.243989	−	0.830951i	−0.742879	−	0.669425i	\(-0.766540\pi\)
							0.986869	−	0.161525i	\(-0.0516414\pi\)
\(17\)	3.25140	−	3.75232i	0.788581	−	0.910071i	−0.209117	−	0.977891i	\(-0.567059\pi\)
							0.997698	+	0.0678198i	\(0.0216043\pi\)
\(19\)	0.0643062	−	0.0293676i	0.0147528	−	0.00673740i	−0.408025	−	0.912971i	\(-0.633782\pi\)
							0.422778	+	0.906233i	\(0.361055\pi\)
\(23\)	0.765241	+	1.19074i	0.159564	+	0.248286i	0.911825	−	0.410579i	\(-0.134673\pi\)
							−0.752261	+	0.658865i	\(0.771037\pi\)
\(29\)	7.41428			1.37680			0.688399	−	0.725333i	\(-0.258314\pi\)
							0.688399	+	0.725333i	\(0.258314\pi\)
\(31\)	5.34758	−	1.57019i	0.960454	−	0.282015i	0.236321	−	0.971675i	\(-0.424058\pi\)
							0.724133	+	0.689660i	\(0.242240\pi\)
\(37\)	2.22904			0.366452			0.183226	−	0.983071i	\(-0.441346\pi\)
							0.183226	+	0.983071i	\(0.441346\pi\)
\(41\)	8.23114	+	7.13232i	1.28549	+	1.11388i	0.987214	+	0.159401i	\(0.0509563\pi\)
							0.298274	+	0.954480i	\(0.403589\pi\)
\(43\)	2.29234	−	2.64551i	0.349579	−	0.403436i	−0.553542	−	0.832821i	\(-0.686724\pi\)
							0.903121	+	0.429385i	\(0.141270\pi\)
\(47\)	−5.85032	−	9.10327i	−0.853356	−	1.32785i	−0.943314	−	0.331903i	\(-0.892309\pi\)
							0.0899574	−	0.995946i	\(-0.471327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.b.43.19	yes	340
4.3	odd	2		804.2.u.a.43.29	✓	340
67.53	odd	22		804.2.u.a.187.29	yes	340
268.187	even	22	inner	804.2.u.b.187.19	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.29	✓	340	4.3	odd	2
804.2.u.a.187.29	yes	340	67.53	odd	22
804.2.u.b.43.19	yes	340	1.1	even	1	trivial
804.2.u.b.187.19	yes	340	268.187	even	22	inner