Embedded newform 950.2.r.b.11.9

• Label: 950.2.r.b.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-0.809690 - 0.899251i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.75482 - 1.38587i) q^{5} +(-0.809690 + 0.899251i) q^{6} +2.11907 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.160530 - 1.52734i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.809690	−	0.899251i	−0.467474	−	0.519183i	0.462593	−	0.886571i	\(-0.346919\pi\)
−0.930068	+	0.367388i	\(0.880252\pi\)
\(5\)	−1.75482	−	1.38587i	−0.784777	−	0.619778i
							\(7\)	2.11907			0.800935			0.400467	−	0.916311i	\(-0.368848\pi\)
0.400467	+	0.916311i	\(0.368848\pi\)
\(11\)	4.47327	−	3.25002i	1.34874	−	0.979919i	0.349669	−	0.936873i	\(-0.386294\pi\)
							0.999073	−	0.0430453i	\(-0.0137060\pi\)
\(13\)	0.509276	−	4.84543i	0.141248	−	1.34388i	−0.662567	−	0.749002i	\(-0.730533\pi\)
							0.803815	−	0.594879i	\(-0.202800\pi\)
\(17\)	1.71402	+	0.364327i	0.415711	+	0.0883622i	0.411019	−	0.911627i	\(-0.365173\pi\)
							0.00469249	+	0.999989i	\(0.498506\pi\)
\(19\)	2.33338	+	3.68176i	0.535313	+	0.844654i
							\(23\)	4.56756	+	2.03361i	0.952403	+	0.424037i	0.823308	−	0.567595i	\(-0.192126\pi\)
0.129095	+	0.991632i	\(0.458793\pi\)
\(29\)	−5.21602	+	1.10870i	−0.968591	+	0.205880i	−0.664935	−	0.746902i	\(-0.731541\pi\)
							−0.303656	+	0.952782i	\(0.598207\pi\)
\(31\)	−3.19820	−	9.84304i	−0.574413	−	1.76786i	−0.638168	−	0.769897i	\(-0.720308\pi\)
							0.0637549	−	0.997966i	\(-0.479692\pi\)
\(37\)	−7.01299	−	5.09523i	−1.15293	−	0.837651i	−0.164060	−	0.986450i	\(-0.552459\pi\)
							−0.988867	+	0.148799i	\(0.952459\pi\)
\(41\)	8.64991	−	3.85119i	1.35089	−	0.601455i	0.401588	−	0.915821i	\(-0.368459\pi\)
							0.949302	+	0.314366i	\(0.101792\pi\)
\(43\)	−0.846313	+	1.46586i	−0.129061	+	0.223541i	−0.923313	−	0.384048i	\(-0.874530\pi\)
							0.794252	+	0.607589i	\(0.207863\pi\)
\(47\)	2.29088	−	0.486941i	0.334159	−	0.0710277i	−0.0377766	−	0.999286i	\(-0.512028\pi\)
							0.371936	+	0.928259i	\(0.378694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.9	✓	200
19.7	even	3	inner	950.2.r.b.311.17	yes	200
25.16	even	5	inner	950.2.r.b.391.17	yes	200
475.216	even	15	inner	950.2.r.b.691.9	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.9	✓	200	1.1	even	1	trivial
950.2.r.b.311.17	yes	200	19.7	even	3	inner
950.2.r.b.391.17	yes	200	25.16	even	5	inner
950.2.r.b.691.9	yes	200	475.216	even	15	inner