Embedded newform 950.2.r.b.11.7

• Label: 950.2.r.b.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.b.691.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-1.08024 - 1.19973i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(0.757710 + 2.10378i) q^{5} +(-1.08024 + 1.19973i) q^{6} +2.77680 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.0411561 - 0.391575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.08024	−	1.19973i	−0.623677	−	0.692664i	0.345671	−	0.938356i	\(-0.387651\pi\)
−0.969348	+	0.245692i	\(0.920985\pi\)
\(5\)	0.757710	+	2.10378i	0.338858	+	0.940837i
							\(7\)	2.77680			1.04953			0.524766	−	0.851247i	\(-0.324153\pi\)
0.524766	+	0.851247i	\(0.324153\pi\)
\(11\)	−1.53557	+	1.11565i	−0.462991	+	0.336382i	−0.794703	−	0.606998i	\(-0.792374\pi\)
							0.331713	+	0.943380i	\(0.392374\pi\)
\(13\)	−0.595239	+	5.66332i	−0.165090	+	1.57072i	0.527604	+	0.849490i	\(0.323090\pi\)
							−0.692694	+	0.721232i	\(0.743576\pi\)
\(17\)	−5.75389	−	1.22303i	−1.39552	−	0.296628i	−0.552055	−	0.833807i	\(-0.686156\pi\)
							−0.843468	+	0.537180i	\(0.819490\pi\)
\(19\)	−2.57179	+	3.51936i	−0.590009	+	0.807397i
							\(23\)	−0.120083	−	0.0534642i	−0.0250390	−	0.0111481i	0.394179	−	0.919034i	\(-0.371029\pi\)
−0.419218	+	0.907886i	\(0.637696\pi\)
\(29\)	−4.40743	+	0.936827i	−0.818438	+	0.173964i	−0.598061	−	0.801450i	\(-0.704062\pi\)
							−0.220377	+	0.975415i	\(0.570729\pi\)
\(31\)	2.58101	+	7.94353i	0.463563	+	1.42670i	0.860781	+	0.508976i	\(0.169976\pi\)
							−0.397218	+	0.917724i	\(0.630024\pi\)
\(37\)	−6.61066	−	4.80293i	−1.08679	−	0.789596i	−0.107932	−	0.994158i	\(-0.534423\pi\)
							−0.978854	+	0.204562i	\(0.934423\pi\)
\(41\)	7.48566	−	3.33283i	1.16906	−	0.520501i	0.271955	−	0.962310i	\(-0.412330\pi\)
							0.897109	+	0.441809i	\(0.145663\pi\)
\(43\)	−3.72542	+	6.45262i	−0.568121	+	0.984015i	0.428631	+	0.903480i	\(0.358996\pi\)
							−0.996752	+	0.0805349i	\(0.974337\pi\)
\(47\)	8.31328	−	1.76704i	1.21262	−	0.257750i	0.443156	−	0.896444i	\(-0.353859\pi\)
							0.769460	+	0.638695i	\(0.220525\pi\)

(See \(a_n\) instead)

Twists

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

    

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