Embedded newform 667.2.x.a.10.1

• Label: 667.2.x.a.10.1
• Level: $667$
• Weight: $2$
• Character: 667.10
• Analytic conductor: $5.326$
• Analytic rank: $0$
• Dimension: $6960$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.x (of order \(308\), degree \(120\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.32602181482\)
Analytic rank:	\(0\)
Dimension:	\(6960\)
Relative dimension:	\(58\) over \(\Q(\zeta_{308})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{308}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	667.10
Dual form			667.2.x.a.467.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.65729 - 0.809781i) q^{2} +(-0.342570 - 0.438674i) q^{3} +(4.74534 + 3.18827i) q^{4} +(1.70314 + 1.30227i) q^{5} +(0.555077 + 1.44309i) q^{6} +(-0.549665 - 3.33810i) q^{7} +(-6.51978 - 8.00663i) q^{8} +(0.652004 - 2.61000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6960 q - 112 q^{2} - 108 q^{3} - 126 q^{4} - 154 q^{5} - 126 q^{6} - 110 q^{7} - 102 q^{8} - 126 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\)	\(465\)	\(553\)
\(\chi(n)\)	\(e\left(\frac{3}{22}\right)\)	\(e\left(\frac{23}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.65729	−	0.809781i	−1.87899	−	0.572602i	−0.994327	−	0.106364i	\(-0.966079\pi\)
							−0.884659	−	0.466238i	\(-0.845609\pi\)
\(3\)	−0.342570	−	0.438674i	−0.197783	−	0.253269i	0.679019	−	0.734121i	\(-0.262405\pi\)
							−0.876802	+	0.480852i	\(0.840327\pi\)
\(5\)	1.70314	+	1.30227i	0.761669	+	0.582395i	0.913895	−	0.405951i	\(-0.133060\pi\)
							−0.152226	+	0.988346i	\(0.548644\pi\)
\(7\)	−0.549665	−	3.33810i	−0.207754	−	1.26168i	−0.862587	−	0.505909i	\(-0.831157\pi\)
							0.654833	−	0.755774i	\(-0.272739\pi\)
\(11\)	1.90935	−	1.75958i	0.575692	−	0.530533i	−0.335756	−	0.941949i	\(-0.608992\pi\)
							0.911448	+	0.411416i	\(0.134966\pi\)
\(13\)	−3.82375	+	5.44773i	−1.06052	+	1.51093i	−0.215270	+	0.976555i	\(0.569063\pi\)
							−0.845248	+	0.534374i	\(0.820547\pi\)
\(17\)	−5.48781	−	4.10812i	−1.33099	−	0.996366i	−0.998582	−	0.0532303i	\(-0.983048\pi\)
							−0.332407	−	0.943136i	\(-0.607861\pi\)
\(19\)	−0.336582	−	1.71495i	−0.0772172	−	0.393438i	−0.999968	−	0.00803978i	\(-0.997441\pi\)
							0.922750	−	0.385398i	\(-0.125936\pi\)
\(23\)	4.68500	−	1.02505i	0.976891	−	0.213739i
							\(29\)	−3.49009	−	4.10113i	−0.648094	−	0.761561i
\(31\)	−2.42645	+	5.17290i	−0.435804	+	0.929080i								0.559180	+	0.829046i	\(0.311116\pi\)
							−0.994984	+	0.100034i	\(0.968105\pi\)
\(37\)	1.07351	+	1.78845i	0.176483	+	0.294019i	0.932945	−	0.360018i	\(-0.117230\pi\)
							−0.756462	+	0.654038i	\(0.773074\pi\)
\(41\)	−1.08203	−	2.90102i	−0.168984	−	0.453064i	0.824660	−	0.565628i	\(-0.191366\pi\)
							−0.993644	+	0.112564i	\(0.964094\pi\)
\(43\)	−0.532821	−	1.13591i	−0.0812544	−	0.173224i	0.860568	−	0.509336i	\(-0.170109\pi\)
							−0.941822	+	0.336111i	\(0.890888\pi\)
\(47\)	−1.64784	−	0.185667i	−0.240362	−	0.0270823i	−0.00903828	−	0.999959i	\(-0.502877\pi\)
							−0.231324	+	0.972877i	\(0.574306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.x.a.10.1	✓	6960
23.7	odd	22	inner	667.2.x.a.329.1	yes	6960
29.3	odd	28	inner	667.2.x.a.148.1	yes	6960
667.467	even	308	inner	667.2.x.a.467.1	yes	6960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.x.a.10.1	✓	6960	1.1	even	1	trivial
667.2.x.a.148.1	yes	6960	29.3	odd	28	inner
667.2.x.a.329.1	yes	6960	23.7	odd	22	inner
667.2.x.a.467.1	yes	6960	667.467	even	308	inner