Embedded newform 950.2.r.b.11.6

• Label: 950.2.r.b.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-1.20234 - 1.33534i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-0.901641 - 2.04623i) q^{5} +(-1.20234 + 1.33534i) q^{6} +1.04913 q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.0239117 + 0.227505i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.20234	−	1.33534i	−0.694173	−	0.770958i	0.288264	−	0.957551i	\(-0.406922\pi\)
−0.982437	+	0.186593i	\(0.940255\pi\)
\(5\)	−0.901641	−	2.04623i	−0.403226	−	0.915100i
							\(7\)	1.04913			0.396535			0.198268	−	0.980148i	\(-0.436469\pi\)
0.198268	+	0.980148i	\(0.436469\pi\)
\(11\)	−3.65214	+	2.65344i	−1.10116	+	0.800042i	−0.981249	−	0.192743i	\(-0.938262\pi\)
							−0.119914	+	0.992784i	\(0.538262\pi\)
\(13\)	−0.0130470	+	0.124134i	−0.00361858	+	0.0344285i	−0.996182	−	0.0872986i	\(-0.972177\pi\)
							0.992564	+	0.121727i	\(0.0388433\pi\)
\(17\)	−1.67657	−	0.356366i	−0.406628	−	0.0864315i	5.67238e−5	−	1.00000i	\(-0.499982\pi\)
							−0.406685	+	0.913569i	\(0.633315\pi\)
\(19\)	−3.02001	+	3.14317i	−0.692838	+	0.721093i
							\(23\)	−1.40913	−	0.627385i	−0.293824	−	0.130819i	0.254531	−	0.967065i	\(-0.418079\pi\)
−0.548355	+	0.836246i	\(0.684746\pi\)
\(29\)	5.79200	−	1.23113i	1.07555	−	0.228615i	0.364104	−	0.931358i	\(-0.381375\pi\)
							0.711443	+	0.702744i	\(0.248042\pi\)
\(31\)	−2.01642	−	6.20591i	−0.362160	−	1.11461i	−0.951740	−	0.306904i	\(-0.900707\pi\)
							0.589580	−	0.807710i	\(-0.299293\pi\)
\(37\)	7.15255	+	5.19663i	1.17587	+	0.854321i	0.991700	−	0.128573i	\(-0.0410397\pi\)
							0.184172	+	0.982894i	\(0.441040\pi\)
\(41\)	−9.58483	+	4.26744i	−1.49690	+	0.666462i	−0.981670	−	0.190590i	\(-0.938960\pi\)
							−0.515229	+	0.857052i	\(0.672293\pi\)
\(43\)	−2.17567	+	3.76836i	−0.331786	+	0.574670i	−0.982862	−	0.184342i	\(-0.940985\pi\)
							0.651076	+	0.759012i	\(0.274318\pi\)
\(47\)	9.16219	−	1.94748i	1.33644	−	0.284070i	0.516388	−	0.856355i	\(-0.327276\pi\)
							0.820055	+	0.572285i	\(0.193943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.6	✓	200
19.7	even	3	inner	950.2.r.b.311.20	yes	200
25.16	even	5	inner	950.2.r.b.391.20	yes	200
475.216	even	15	inner	950.2.r.b.691.6	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.6	✓	200	1.1	even	1	trivial
950.2.r.b.311.20	yes	200	19.7	even	3	inner
950.2.r.b.391.20	yes	200	25.16	even	5	inner
950.2.r.b.691.6	yes	200	475.216	even	15	inner