Embedded newform 804.2.u.a.43.10

• Label: 804.2.u.a.43.10
• 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			43.10
Character	\(\chi\)	\(=\)	804.43
Dual form			804.2.u.a.187.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.827747 + 1.14666i) q^{2} +(0.841254 - 0.540641i) q^{3} +(-0.629670 - 1.89829i) q^{4} +(2.62053 + 0.376775i) q^{5} +(-0.0764124 + 1.41215i) q^{6} +(0.207219 - 0.453747i) q^{7} +(2.69791 + 0.849287i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 34 q^{3} - 2 q^{4} + 11 q^{6} - 4 q^{7} + 39 q^{8} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.827747	+	1.14666i	−0.585305	+	0.810813i
							\(3\)	0.841254	−	0.540641i	0.485698	−	0.312139i
\(5\)	2.62053	+	0.376775i	1.17193	+	0.168499i								0.700665	−	0.713490i	\(-0.252887\pi\)
							0.471270	+	0.881989i	\(0.343796\pi\)
\(7\)	0.207219	−	0.453747i	0.0783215	−	0.171500i	−0.866420	−	0.499315i	\(-0.833585\pi\)
							0.944742	+	0.327815i	\(0.106312\pi\)
\(11\)	0.849844	−	5.91079i	0.256238	−	1.78217i	−0.302825	−	0.953046i	\(-0.597930\pi\)
							0.559063	−	0.829125i	\(-0.311161\pi\)
\(13\)	−0.295747	+	1.00722i	−0.0820254	+	0.279353i	−0.990285	−	0.139055i	\(-0.955594\pi\)
							0.908259	+	0.418408i	\(0.137412\pi\)
\(17\)	−0.925414	+	1.06798i	−0.224446	+	0.259024i	−0.856793	−	0.515661i	\(-0.827546\pi\)
							0.632347	+	0.774685i	\(0.282092\pi\)
\(19\)	5.74664	−	2.62440i	1.31837	−	0.602080i	0.372925	−	0.927861i	\(-0.378355\pi\)
							0.945445	+	0.325782i	\(0.105627\pi\)
\(23\)	−0.917694	−	1.42796i	−0.191352	−	0.297750i	0.732299	−	0.680983i	\(-0.238447\pi\)
							−0.923652	+	0.383233i	\(0.874811\pi\)
\(29\)	−7.59137			−1.40968			−0.704841	−	0.709365i	\(-0.748982\pi\)
							−0.704841	+	0.709365i	\(0.748982\pi\)
\(31\)	−5.70138	+	1.67408i	−1.02400	+	0.300673i	−0.750268	−	0.661134i	\(-0.770076\pi\)
							−0.273730	+	0.961807i	\(0.588257\pi\)
\(37\)	3.65143			0.600292			0.300146	−	0.953893i	\(-0.402965\pi\)
							0.300146	+	0.953893i	\(0.402965\pi\)
\(41\)	4.24875	+	3.68156i	0.663543	+	0.574963i	0.920157	−	0.391551i	\(-0.128061\pi\)
							−0.256613	+	0.966514i	\(0.582607\pi\)
\(43\)	3.55654	−	4.10446i	0.542367	−	0.625925i	−0.416721	−	0.909035i	\(-0.636821\pi\)
							0.959087	+	0.283110i	\(0.0913661\pi\)
\(47\)	−5.93753	−	9.23897i	−0.866077	−	1.34764i	−0.936698	−	0.350139i	\(-0.886134\pi\)
							0.0706207	−	0.997503i	\(-0.477502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.u.a.43.10	✓	340
4.3	odd	2		804.2.u.b.43.20	yes	340
67.53	odd	22		804.2.u.b.187.20	yes	340
268.187	even	22	inner	804.2.u.a.187.10	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.u.a.43.10	✓	340	1.1	even	1	trivial
804.2.u.a.187.10	yes	340	268.187	even	22	inner
804.2.u.b.43.20	yes	340	4.3	odd	2
804.2.u.b.187.20	yes	340	67.53	odd	22