Embedded newform 950.2.r.a.11.2

• Label: 950.2.r.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(-2.02768 - 2.25196i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-0.209287 - 2.22625i) q^{5} +(2.02768 - 2.25196i) q^{6} +1.71696 q^{7} +(-0.309017 - 0.951057i) q^{8} +(-0.646281 + 6.14895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.02768	−	2.25196i	−1.17068	−	1.30017i	−0.945413	−	0.325876i	\(-0.894341\pi\)
−0.225268	−	0.974297i	\(-0.572326\pi\)
\(5\)	−0.209287	−	2.22625i	−0.0935962	−	0.995610i
							\(7\)	1.71696			0.648950			0.324475	−	0.945894i	\(-0.394812\pi\)
0.324475	+	0.945894i	\(0.394812\pi\)
\(11\)	−4.36820	+	3.17368i	−1.31706	+	0.956901i	−0.317097	+	0.948393i	\(0.602708\pi\)
							−0.999964	+	0.00850777i	\(0.997292\pi\)
\(13\)	−0.709391	+	6.74940i	−0.196750	+	1.87195i	0.237821	+	0.971309i	\(0.423567\pi\)
							−0.434571	+	0.900638i	\(0.643100\pi\)
\(17\)	4.32159	+	0.918583i	1.04814	+	0.222789i	0.699622	−	0.714513i	\(-0.253352\pi\)
							0.348518	+	0.937302i	\(0.386685\pi\)
\(19\)	2.14510	−	3.79454i	0.492120	−	0.870527i
							\(23\)	0.460190	+	0.204890i	0.0959563	+	0.0427225i	0.454153	−	0.890923i	\(-0.349942\pi\)
−0.358197	+	0.933646i	\(0.616608\pi\)
\(29\)	7.12799	−	1.51510i	1.32363	−	0.281347i	0.508722	−	0.860931i	\(-0.330118\pi\)
							0.814913	+	0.579584i	\(0.196785\pi\)
\(31\)	−0.472219	−	1.45334i	−0.0848131	−	0.261028i	0.899652	−	0.436607i	\(-0.143820\pi\)
							−0.984465	+	0.175579i	\(0.943820\pi\)
\(37\)	2.57517	+	1.87097i	0.423355	+	0.307586i	0.778986	−	0.627041i	\(-0.215734\pi\)
							−0.355631	+	0.934626i	\(0.615734\pi\)
\(41\)	8.17424	−	3.63941i	1.27660	−	0.568380i	0.347319	−	0.937747i	\(-0.387092\pi\)
							0.929283	+	0.369367i	\(0.120426\pi\)
\(43\)	−4.06126	+	7.03431i	−0.619337	+	1.07272i	0.370270	+	0.928924i	\(0.379265\pi\)
							−0.989607	+	0.143799i	\(0.954068\pi\)
\(47\)	6.39545	−	1.35940i	0.932873	−	0.198288i	0.283686	−	0.958917i	\(-0.408443\pi\)
							0.649187	+	0.760629i	\(0.275109\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.2	✓	200
19.7	even	3	inner	950.2.r.a.311.24	yes	200
25.16	even	5	inner	950.2.r.a.391.24	yes	200
475.216	even	15	inner	950.2.r.a.691.2	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.2	✓	200	1.1	even	1	trivial
950.2.r.a.311.24	yes	200	19.7	even	3	inner
950.2.r.a.391.24	yes	200	25.16	even	5	inner
950.2.r.a.691.2	yes	200	475.216	even	15	inner