Embedded newform 950.2.bc.b.111.1

• Label: 950.2.bc.b.111.1
• 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			111.1
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.b.291.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.997564 + 0.0697565i) q^{2} +(-0.111264 - 3.18617i) q^{3} +(0.990268 + 0.139173i) q^{4} +(-1.21162 - 1.87935i) q^{5} +(0.111264 - 3.18617i) q^{6} +(0.0756516 + 0.131032i) q^{7} +(0.978148 + 0.207912i) q^{8} +(-7.14661 + 0.499740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.111264	−	3.18617i	−0.0642380	−	1.83954i	−0.423026	−	0.906118i	\(-0.639032\pi\)
0.358788	−	0.933419i	\(-0.383190\pi\)
\(5\)	−1.21162	−	1.87935i	−0.541855	−	0.840472i
							\(7\)	0.0756516	+	0.131032i	0.0285936	+	0.0495256i	0.879968	−	0.475033i	\(-0.157564\pi\)
−0.851374	+	0.524558i	\(0.824230\pi\)
\(11\)	−0.142475	−	1.35556i	−0.0429579	−	0.408717i	−0.994778	−	0.102061i	\(-0.967456\pi\)
							0.951820	−	0.306656i	\(-0.0992103\pi\)
\(13\)	−0.154013	−	0.228334i	−0.0427156	−	0.0633284i	0.807075	−	0.590449i	\(-0.201049\pi\)
							−0.849790	+	0.527121i	\(0.823272\pi\)
\(17\)	0.0911971	−	0.225721i	0.0221185	−	0.0547453i	−0.915720	−	0.401816i	\(-0.868379\pi\)
							0.937839	+	0.347071i	\(0.112824\pi\)
\(19\)	−4.15462	−	1.31875i	−0.953136	−	0.302543i
							\(23\)	2.39605	+	2.31384i	0.499611	+	0.482469i	0.901232	−	0.433337i	\(-0.142664\pi\)
−0.401621	+	0.915806i	\(0.631553\pi\)
\(29\)	−0.615529	−	1.52349i	−0.114301	−	0.282904i	0.859180	−	0.511674i	\(-0.170974\pi\)
							−0.973481	+	0.228769i	\(0.926530\pi\)
\(31\)	5.99298	−	6.65588i	1.07637	−	1.19543i	0.0965966	−	0.995324i	\(-0.469204\pi\)
							0.979774	−	0.200107i	\(-0.0641290\pi\)
\(37\)	−2.95488	−	2.14684i	−0.485779	−	0.352939i	0.317780	−	0.948165i	\(-0.397063\pi\)
							−0.803559	+	0.595226i	\(0.797063\pi\)
\(41\)	6.94744	+	1.99215i	1.08501	+	0.311121i	0.770049	−	0.637985i	\(-0.220232\pi\)
							0.314958	+	0.949105i	\(0.398009\pi\)
\(43\)	0.136849	−	0.776107i	0.0208692	−	0.118355i	−0.972594	−	0.232512i	\(-0.925306\pi\)
							0.993463	+	0.114157i	\(0.0364166\pi\)
\(47\)	1.35871	+	3.36293i	0.198188	+	0.490534i	0.993250	−	0.115995i	\(-0.0370058\pi\)
							−0.795061	+	0.606529i	\(0.792561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.111.1	✓	600
19.6	even	9	inner	950.2.bc.b.861.25	yes	600
25.16	even	5	inner	950.2.bc.b.491.25	yes	600
475.291	even	45	inner	950.2.bc.b.291.1	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.111.1	✓	600	1.1	even	1	trivial
950.2.bc.b.291.1	yes	600	475.291	even	45	inner
950.2.bc.b.491.25	yes	600	25.16	even	5	inner
950.2.bc.b.861.25	yes	600	19.6	even	9	inner