Embedded newform 950.2.bd.a.27.13

• Label: 950.2.bd.a.27.13
• 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			27.13
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(0.00693568 - 0.132341i) q^{3} +(0.207912 + 0.978148i) q^{4} +(1.23454 - 1.86438i) q^{5} +(-0.0886748 + 0.0984833i) q^{6} +(-2.47768 - 2.47768i) q^{7} +(0.453990 - 0.891007i) q^{8} +(2.96610 + 0.311750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	0.00693568	−	0.132341i	0.00400432	−	0.0764070i	−0.995896	−	0.0905011i	\(-0.971153\pi\)
0.999901	+	0.0140942i	\(0.00448646\pi\)
\(5\)	1.23454	−	1.86438i	0.552103	−	0.833776i
							\(7\)	−2.47768	−	2.47768i	−0.936476	−	0.936476i	0.0616237	−	0.998099i	\(-0.480372\pi\)
−0.998099	+	0.0616237i	\(0.980372\pi\)
\(11\)	1.66959	−	1.21303i	0.503400	−	0.365741i	−0.306914	−	0.951737i	\(-0.599296\pi\)
							0.810314	+	0.585996i	\(0.199296\pi\)
\(13\)	3.92630	−	3.17946i	1.08896	−	0.881823i	0.0954351	−	0.995436i	\(-0.469576\pi\)
							0.993525	+	0.113613i	\(0.0362424\pi\)
\(17\)	3.55815	+	5.47908i	0.862979	+	1.32887i	0.943245	+	0.332098i	\(0.107756\pi\)
							−0.0802660	+	0.996773i	\(0.525577\pi\)
\(19\)	−4.18951	+	1.20334i	−0.961139	+	0.276066i
							\(23\)	5.13722	−	1.97199i	1.07118	−	0.411189i	0.242065	−	0.970260i	\(-0.422175\pi\)
0.829120	+	0.559071i	\(0.188842\pi\)
\(29\)	−5.21369	+	1.10820i	−0.968158	+	0.205788i	−0.664745	−	0.747071i	\(-0.731460\pi\)
							−0.303413	+	0.952859i	\(0.598126\pi\)
\(31\)	−9.90209	+	3.21738i	−1.77847	+	0.577859i	−0.998829	−	0.0483701i	\(-0.984597\pi\)
							−0.779639	+	0.626229i	\(0.784597\pi\)
\(37\)	10.2260	−	1.61964i	1.68115	−	0.266268i	0.758433	−	0.651751i	\(-0.225965\pi\)
							0.922715	+	0.385484i	\(0.125965\pi\)
\(41\)	−3.84619	−	8.63869i	−0.600674	−	1.34914i	−0.916391	−	0.400283i	\(-0.868912\pi\)
							0.315717	−	0.948853i	\(-0.397755\pi\)
\(43\)	−0.814145	−	3.03843i	−0.124156	−	0.463356i	0.875652	−	0.482942i	\(-0.160432\pi\)
							−0.999808	+	0.0195860i	\(0.993765\pi\)
\(47\)	0.509228	+	0.330696i	0.0742785	+	0.0482370i	0.581244	−	0.813729i	\(-0.302566\pi\)
							−0.506966	+	0.861966i	\(0.669233\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.13	✓	800
19.12	odd	6	inner	950.2.bd.a.677.13	yes	800
25.13	odd	20	inner	950.2.bd.a.863.13	yes	800
475.88	even	60	inner	950.2.bd.a.563.13	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.13	✓	800	1.1	even	1	trivial
950.2.bd.a.563.13	yes	800	475.88	even	60	inner
950.2.bd.a.677.13	yes	800	19.12	odd	6	inner
950.2.bd.a.863.13	yes	800	25.13	odd	20	inner