Embedded newform 950.2.bc.a.111.16

• Label: 950.2.bc.a.111.16
• Level: $950$
• Weight: $2$
• Character: 950.111
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $600$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(600\)
Relative dimension:	\(25\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			111.16
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.a.291.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997564 - 0.0697565i) q^{2} +(0.0163606 + 0.468506i) q^{3} +(0.990268 + 0.139173i) q^{4} +(-1.72130 + 1.42728i) q^{5} +(0.0163606 - 0.468506i) q^{6} +(1.96840 + 3.40937i) q^{7} +(-0.978148 - 0.207912i) q^{8} +(2.77346 - 0.193939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.997564	−	0.0697565i	−0.705384	−	0.0493253i
							\(3\)	0.0163606	+	0.468506i	0.00944579	+	0.270492i	0.994920	+	0.100668i	\(0.0320978\pi\)
−0.985474	+	0.169824i	\(0.945680\pi\)
\(5\)	−1.72130	+	1.42728i	−0.769789	+	0.638298i
							\(7\)	1.96840	+	3.40937i	0.743985	+	1.28862i	0.950667	+	0.310212i	\(0.100400\pi\)
−0.206682	+	0.978408i	\(0.566267\pi\)
\(11\)	0.484306	+	4.60786i	0.146024	+	1.38932i	0.784713	+	0.619859i	\(0.212810\pi\)
							−0.638690	+	0.769464i	\(0.720523\pi\)
\(13\)	0.00514262	+	0.00762424i	0.00142631	+	0.00211458i	0.829750	−	0.558135i	\(-0.188483\pi\)
							−0.828324	+	0.560250i	\(0.810705\pi\)
\(17\)	−0.904290	+	2.23820i	−0.219322	+	0.542842i	−0.996228	−	0.0867751i	\(-0.972344\pi\)
							0.776905	+	0.629617i	\(0.216788\pi\)
\(19\)	−0.681753	−	4.30525i	−0.156405	−	0.987693i
							\(23\)	1.98496	+	1.91685i	0.413892	+	0.399691i	0.872237	−	0.489083i	\(-0.162669\pi\)
−0.458345	+	0.888775i	\(0.651557\pi\)
\(29\)	−0.886024	−	2.19299i	−0.164531	−	0.407227i	0.822285	−	0.569076i	\(-0.192699\pi\)
							−0.986816	+	0.161848i	\(0.948254\pi\)
\(31\)	1.75083	−	1.94449i	0.314458	−	0.349241i	−0.565109	−	0.825017i	\(-0.691166\pi\)
							0.879567	+	0.475775i	\(0.157832\pi\)
\(37\)	−2.58762	−	1.88002i	−0.425402	−	0.309073i	0.354406	−	0.935092i	\(-0.384683\pi\)
							−0.779808	+	0.626019i	\(0.784683\pi\)
\(41\)	−3.43483	−	0.984921i	−0.536430	−	0.153819i	−0.00352483	−	0.999994i	\(-0.501122\pi\)
							−0.532905	+	0.846175i	\(0.678900\pi\)
\(43\)	−0.892319	+	5.06059i	−0.136077	+	0.771733i	0.838026	+	0.545630i	\(0.183710\pi\)
							−0.974103	+	0.226103i	\(0.927401\pi\)
\(47\)	2.66966	+	6.60763i	0.389410	+	0.963823i	0.986810	+	0.161880i	\(0.0517557\pi\)
							−0.597401	+	0.801943i	\(0.703800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.111.16	✓	600
19.6	even	9	inner	950.2.bc.a.861.10	yes	600
25.16	even	5	inner	950.2.bc.a.491.10	yes	600
475.291	even	45	inner	950.2.bc.a.291.16	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.111.16	✓	600	1.1	even	1	trivial
950.2.bc.a.291.16	yes	600	475.291	even	45	inner
950.2.bc.a.491.10	yes	600	25.16	even	5	inner
950.2.bc.a.861.10	yes	600	19.6	even	9	inner