Embedded newform 804.2.u.a.43.3

• Label: 804.2.u.a.43.3
• 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			43.3
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.a.187.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37601 + 0.326492i) q^{2} +(0.841254 - 0.540641i) q^{3} +(1.78681 - 0.898513i) q^{4} +(0.977951 + 0.140608i) q^{5} +(-0.981058 + 1.01859i) q^{6} +(2.04888 - 4.48642i) q^{7} +(-2.16530 + 1.81974i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 34 q^{3} - 2 q^{4} + 11 q^{6} - 4 q^{7} + 39 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37601	+	0.326492i	−0.972986	+	0.230865i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	0.977951	+	0.140608i	0.437353	+	0.0628818i								0.357475	−	0.933923i	\(-0.383638\pi\)
							0.0798780	+	0.996805i	\(0.474547\pi\)
\(7\)	2.04888	−	4.48642i	0.774404	−	1.69571i	0.0576982	−	0.998334i	\(-0.481624\pi\)
							0.716706	−	0.697375i	\(-0.245649\pi\)
\(11\)	0.00935924	−	0.0650950i	0.00282192	−	0.0196269i	−0.988362	−	0.152121i	\(-0.951390\pi\)
							0.991184	+	0.132494i	\(0.0422987\pi\)
\(13\)	−0.139566	+	0.475317i	−0.0387086	+	0.131829i	−0.976581	−	0.215152i	\(-0.930975\pi\)
							0.937872	+	0.346982i	\(0.112793\pi\)
\(17\)	1.62327	−	1.87335i	0.393700	−	0.454354i	−0.523947	−	0.851751i	\(-0.675541\pi\)
							0.917647	+	0.397397i	\(0.130086\pi\)
\(19\)	2.43244	−	1.11086i	0.558041	−	0.254849i	−0.116369	−	0.993206i	\(-0.537125\pi\)
							0.674410	+	0.738357i	\(0.264398\pi\)
\(23\)	0.199194	+	0.309952i	0.0415349	+	0.0646295i	0.861400	−	0.507927i	\(-0.169588\pi\)
							−0.819865	+	0.572557i	\(0.805952\pi\)
\(29\)	−8.19081			−1.52100			−0.760498	−	0.649340i	\(-0.775045\pi\)
							−0.760498	+	0.649340i	\(0.775045\pi\)
\(31\)	1.91796	−	0.563165i	0.344476	−	0.101147i	−0.104918	−	0.994481i	\(-0.533458\pi\)
							0.449394	+	0.893334i	\(0.351640\pi\)
\(37\)	0.326272			0.0536388			0.0268194	−	0.999640i	\(-0.491462\pi\)
							0.0268194	+	0.999640i	\(0.491462\pi\)
\(41\)	4.16505	+	3.60903i	0.650471	+	0.563636i	0.916354	−	0.400369i	\(-0.131118\pi\)
							−0.265883	+	0.964005i	\(0.585664\pi\)
\(43\)	−3.90036	+	4.50126i	−0.594799	+	0.686435i	−0.970719	−	0.240217i	\(-0.922781\pi\)
							0.375920	+	0.926652i	\(0.377327\pi\)
\(47\)	7.02541	+	10.9317i	1.02476	+	1.59456i	0.780807	+	0.624772i	\(0.214808\pi\)
							0.243954	+	0.969787i	\(0.421555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.a.43.3	✓	340
4.3	odd	2		804.2.u.b.43.11	yes	340
67.53	odd	22		804.2.u.b.187.11	yes	340
268.187	even	22	inner	804.2.u.a.187.3	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.3	✓	340	1.1	even	1	trivial
804.2.u.a.187.3	yes	340	268.187	even	22	inner
804.2.u.b.43.11	yes	340	4.3	odd	2
804.2.u.b.187.11	yes	340	67.53	odd	22