Embedded newform 950.2.bd.a.27.18

• Label: 950.2.bd.a.27.18
• Level: $950$
• Weight: $2$
• Character: 950.27
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $800$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(800\)
Relative dimension:	\(50\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			27.18
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(0.0828442 - 1.58076i) q^{3} +(0.207912 + 0.978148i) q^{4} +(-2.09573 - 0.779702i) q^{5} +(-1.05919 + 1.17635i) q^{6} +(0.596283 + 0.596283i) q^{7} +(0.453990 - 0.891007i) q^{8} +(0.491621 + 0.0516714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777146	−	0.629320i	−0.549525	−	0.444997i
							\(3\)	0.0828442	−	1.58076i	0.0478301	−	0.912653i	−0.864357	−	0.502879i	\(-0.832274\pi\)
0.912187	−	0.409774i	\(-0.134393\pi\)
\(5\)	−2.09573	−	0.779702i	−0.937237	−	0.348694i
							\(7\)	0.596283	+	0.596283i	0.225374	+	0.225374i	0.810757	−	0.585383i	\(-0.199056\pi\)
−0.585383	+	0.810757i	\(0.699056\pi\)
\(11\)	3.30853	−	2.40379i	0.997558	−	0.724768i	0.0359950	−	0.999352i	\(-0.488540\pi\)
							0.961563	+	0.274583i	\(0.0885400\pi\)
\(13\)	3.36680	−	2.72638i	0.933781	−	0.756161i	−0.0361989	−	0.999345i	\(-0.511525\pi\)
							0.969980	+	0.243183i	\(0.0781917\pi\)
\(17\)	2.23605	+	3.44321i	0.542321	+	0.835101i	0.998498	−	0.0547881i	\(-0.0174483\pi\)
							−0.456177	+	0.889889i	\(0.650782\pi\)
\(19\)	4.21473	+	1.11179i	0.966925	+	0.255062i
							\(23\)	−8.15780	+	3.13149i	−1.70102	+	0.652960i	−0.998288	−	0.0584959i	\(-0.981370\pi\)
−0.702731	+	0.711456i	\(0.748036\pi\)
\(29\)	3.83001	−	0.814094i	0.711215	−	0.151174i	0.161923	−	0.986803i	\(-0.448231\pi\)
							0.549293	+	0.835630i	\(0.314897\pi\)
\(31\)	5.45474	−	1.77235i	0.979700	−	0.318324i	0.224975	−	0.974365i	\(-0.427770\pi\)
							0.754726	+	0.656041i	\(0.227770\pi\)
\(37\)	−4.91037	+	0.777726i	−0.807260	+	0.127857i	−0.546405	−	0.837521i	\(-0.684004\pi\)
							−0.260854	+	0.965378i	\(0.584004\pi\)
\(41\)	−4.61460	−	10.3646i	−0.720679	−	1.61867i	−0.784062	−	0.620683i	\(-0.786855\pi\)
							0.0633823	−	0.997989i	\(-0.479811\pi\)
\(43\)	−1.19620	−	4.46428i	−0.182419	−	0.680796i	−0.995168	−	0.0981830i	\(-0.968697\pi\)
							0.812750	−	0.582613i	\(-0.197970\pi\)
\(47\)	4.71891	+	3.06450i	0.688324	+	0.447003i	0.840830	−	0.541300i	\(-0.182068\pi\)
							−0.152506	+	0.988303i	\(0.548734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.18	✓	800
19.12	odd	6	inner	950.2.bd.a.677.8	yes	800
25.13	odd	20	inner	950.2.bd.a.863.8	yes	800
475.88	even	60	inner	950.2.bd.a.563.18	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.18	✓	800	1.1	even	1	trivial
950.2.bd.a.563.18	yes	800	475.88	even	60	inner
950.2.bd.a.677.8	yes	800	19.12	odd	6	inner
950.2.bd.a.863.8	yes	800	25.13	odd	20	inner