Embedded newform 429.2.t.a.329.9

• Label: 429.2.t.a.329.9
• Level: $429$
• Weight: $2$
• Character: 429.329
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(52\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			329.9
Character	\(\chi\)	\(=\)	429.329
Dual form			429.2.t.a.296.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63490 - 0.943908i) q^{2} +(0.664837 - 1.59937i) q^{3} +(0.781925 + 1.35433i) q^{4} +0.196251 q^{5} +(-2.59660 + 1.98726i) q^{6} +(1.83210 + 3.17329i) q^{7} +0.823370i q^{8} +(-2.11598 - 2.12665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63490	−	0.943908i	−1.15605	−	0.667444i	−0.205693	−	0.978617i	\(-0.565945\pi\)
							−0.950353	+	0.311173i	\(0.899278\pi\)
\(3\)	0.664837	−	1.59937i	0.383844	−	0.923398i
							\(5\)	0.196251			0.0877663			0.0438832	−	0.999037i	\(-0.486027\pi\)
0.0438832	+	0.999037i	\(0.486027\pi\)
\(7\)	1.83210	+	3.17329i	0.692468	+	1.19939i	0.971027	+	0.238971i	\(0.0768100\pi\)
							−0.278559	+	0.960419i	\(0.589857\pi\)
\(11\)	−0.100432	−	3.31510i	−0.0302814	−	0.999541i
							\(13\)	0.632817	−	3.54958i	0.175512	−	0.984477i
\(17\)	−2.64622	−	4.58339i	−0.641804	−	1.11164i								−0.985030	−	0.172384i	\(-0.944853\pi\)
							0.343226	−	0.939253i	\(-0.388480\pi\)
\(19\)	−2.23117	−	3.86451i	−0.511866	−	0.886579i	−0.999905	−	0.0137569i	\(-0.995621\pi\)
							0.488039	−	0.872822i	\(-0.337712\pi\)
\(23\)	3.35850	+	1.93903i	0.700295	+	0.404315i	0.807457	−	0.589926i	\(-0.200843\pi\)
							−0.107162	+	0.994242i	\(0.534177\pi\)
\(29\)	2.60331	−	4.50906i	0.483422	−	0.837311i	−0.516397	−	0.856349i	\(-0.672727\pi\)
							0.999819	+	0.0190384i	\(0.00606048\pi\)
\(31\)			0.506312i			0.0909363i	0.998966	+	0.0454682i	\(0.0144780\pi\)
							−0.998966	+	0.0454682i	\(0.985522\pi\)
\(37\)	2.99511	+	1.72923i	0.492393	+	0.284283i	0.725567	−	0.688152i	\(-0.241578\pi\)
							−0.233174	+	0.972435i	\(0.574911\pi\)
\(41\)	2.86810	+	1.65590i	0.447921	+	0.258607i	0.706952	−	0.707262i	\(-0.250070\pi\)
							−0.259031	+	0.965869i	\(0.583403\pi\)
\(43\)	0.184817	−	0.106704i	0.0281843	−	0.0162722i	−0.485842	−	0.874047i	\(-0.661487\pi\)
							0.514026	+	0.857775i	\(0.328153\pi\)
\(47\)	7.59321			1.10758			0.553791	−	0.832655i	\(-0.313181\pi\)
							0.553791	+	0.832655i	\(0.313181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.329.9	yes	104
3.2	odd	2	inner	429.2.t.a.329.44	yes	104
11.10	odd	2	inner	429.2.t.a.329.43	yes	104
13.10	even	6	inner	429.2.t.a.296.10	yes	104
33.32	even	2	inner	429.2.t.a.329.10	yes	104
39.23	odd	6	inner	429.2.t.a.296.43	yes	104
143.10	odd	6	inner	429.2.t.a.296.44	yes	104
429.296	even	6	inner	429.2.t.a.296.9	✓	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.9	✓	104	429.296	even	6	inner
429.2.t.a.296.10	yes	104	13.10	even	6	inner
429.2.t.a.296.43	yes	104	39.23	odd	6	inner
429.2.t.a.296.44	yes	104	143.10	odd	6	inner
429.2.t.a.329.9	yes	104	1.1	even	1	trivial
429.2.t.a.329.10	yes	104	33.32	even	2	inner
429.2.t.a.329.43	yes	104	11.10	odd	2	inner
429.2.t.a.329.44	yes	104	3.2	odd	2	inner