Embedded newform 950.2.r.a.11.7

• Label: 950.2.r.a.11.7
• Level: $950$
• Weight: $2$
• Character: 950.11
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.r (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.7
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(-1.32901 - 1.47601i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.09596 + 1.94907i) q^{5} +(1.32901 - 1.47601i) q^{6} -0.0211674 q^{7} +(-0.309017 - 0.951057i) q^{8} +(-0.0987671 + 0.939706i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.104528	+	0.994522i	0.0739128	+	0.703233i
							\(3\)	−1.32901	−	1.47601i	−0.767304	−	0.852177i	0.225210	−	0.974310i	\(-0.427693\pi\)
−0.992514	+	0.122133i	\(0.961026\pi\)
\(5\)	1.09596	+	1.94907i	0.490128	+	0.871650i
							\(7\)	−0.0211674			−0.00800051			−0.00400026	−	0.999992i	\(-0.501273\pi\)
−0.00400026	+	0.999992i	\(0.501273\pi\)
\(11\)	−0.534437	+	0.388292i	−0.161139	+	0.117074i	−0.665433	−	0.746458i	\(-0.731753\pi\)
							0.504294	+	0.863532i	\(0.331753\pi\)
\(13\)	0.501749	−	4.77383i	0.139160	−	1.32402i	−0.672590	−	0.740016i	\(-0.734818\pi\)
							0.811750	−	0.584005i	\(-0.198515\pi\)
\(17\)	0.491997	+	0.104577i	0.119327	+	0.0253637i	0.267188	−	0.963644i	\(-0.413906\pi\)
							−0.147861	+	0.989008i	\(0.547239\pi\)
\(19\)	4.00161	+	1.72832i	0.918033	+	0.396504i
							\(23\)	−4.14216	−	1.84421i	−0.863701	−	0.384544i	−0.0734286	−	0.997300i	\(-0.523394\pi\)
−0.790272	+	0.612756i	\(0.790061\pi\)
\(29\)	7.04588	−	1.49765i	1.30839	−	0.278106i	0.499625	−	0.866242i	\(-0.333471\pi\)
							0.808763	+	0.588135i	\(0.200138\pi\)
\(31\)	−0.850569	−	2.61778i	−0.152767	−	0.470168i	0.845161	−	0.534512i	\(-0.179504\pi\)
							−0.997928	+	0.0643441i	\(0.979504\pi\)
\(37\)	5.78108	+	4.20020i	0.950404	+	0.690509i	0.950902	−	0.309491i	\(-0.100159\pi\)
							−0.000498363		1.00000i	\(0.500159\pi\)
\(41\)	10.8027	−	4.80969i	1.68710	−	0.751147i	0.687413	−	0.726266i	\(-0.258746\pi\)
							0.999690	−	0.0248808i	\(-0.00792061\pi\)
\(43\)	4.34561	−	7.52682i	0.662700	−	1.14783i	−0.317204	−	0.948357i	\(-0.602744\pi\)
							0.979903	−	0.199472i	\(-0.0639228\pi\)
\(47\)	11.1037	−	2.36017i	1.61965	−	0.344267i	0.693216	−	0.720730i	\(-0.256193\pi\)
							0.926431	+	0.376464i	\(0.122860\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.7	✓	200
19.7	even	3	inner	950.2.r.a.311.19	yes	200
25.16	even	5	inner	950.2.r.a.391.19	yes	200
475.216	even	15	inner	950.2.r.a.691.7	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.7	✓	200	1.1	even	1	trivial
950.2.r.a.311.19	yes	200	19.7	even	3	inner
950.2.r.a.391.19	yes	200	25.16	even	5	inner
950.2.r.a.691.7	yes	200	475.216	even	15	inner