Embedded newform 950.2.bg.a.9.2

• Label: 950.2.bg.a.9.2
• Level: $950$
• Weight: $2$
• Character: 950.9
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $1200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bg (of order \(90\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(1200\)
Relative dimension:	\(50\) over \(\Q(\zeta_{90})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{90}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-1.54722 + 2.47606i) q^{3} +(0.615661 + 0.788011i) q^{4} +(1.80261 + 1.32311i) q^{5} +(2.47606 - 1.54722i) q^{6} +(0.146440 + 0.0845473i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(-2.42190 - 4.96562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.898794	−	0.438371i	−0.635543	−	0.309975i
							\(3\)	−1.54722	+	2.47606i	−0.893285	+	1.42956i	0.00973265	+	0.999953i	\(0.496902\pi\)
−0.903018	+	0.429603i	\(0.858654\pi\)
\(5\)	1.80261	+	1.32311i	0.806150	+	0.591711i
							\(7\)	0.146440	+	0.0845473i	0.0553492	+	0.0319559i	0.527419	−	0.849605i	\(-0.323160\pi\)
−0.472070	+	0.881561i	\(0.656493\pi\)
\(11\)	0.142965	−	1.36023i	0.0431057	−	0.410123i	−0.951598	−	0.307344i	\(-0.900560\pi\)
							0.994704	−	0.102779i	\(-0.0327736\pi\)
\(13\)	−0.0157003	+	0.224525i	−0.00435449	+	0.0622722i	−0.999255	−	0.0385864i	\(-0.987715\pi\)
							0.994901	+	0.100859i	\(0.0321590\pi\)
\(17\)	−0.407143	+	2.89697i	−0.0987466	+	0.702619i	0.876484	+	0.481430i	\(0.159882\pi\)
							−0.975231	+	0.221188i	\(0.929006\pi\)
\(19\)	−3.21431	+	2.94418i	−0.737413	+	0.675442i
							\(23\)	−1.39729	+	4.87294i	−0.291356	+	1.01608i	0.670361	+	0.742035i	\(0.266139\pi\)
−0.961717	+	0.274044i	\(0.911639\pi\)
\(29\)	−6.18444	+	0.869166i	−1.14842	+	0.161400i	−0.687598	−	0.726092i	\(-0.741335\pi\)
							−0.460824	+	0.887492i	\(0.652446\pi\)
\(31\)	−3.47668	−	3.86124i	−0.624429	−	0.693499i	0.345074	−	0.938575i	\(-0.387854\pi\)
							−0.969504	+	0.245076i	\(0.921187\pi\)
\(37\)	−0.264602	−	0.364193i	−0.0435002	−	0.0598729i	0.786713	−	0.617319i	\(-0.211781\pi\)
							−0.830213	+	0.557446i	\(0.811781\pi\)
\(41\)	1.18211	−	4.74118i	0.184615	−	0.740448i	−0.804581	−	0.593843i	\(-0.797610\pi\)
							0.989195	−	0.146605i	\(-0.0468346\pi\)
\(43\)	−3.50029	+	9.61698i	−0.533790	+	1.46658i	0.320737	+	0.947168i	\(0.396069\pi\)
							−0.854527	+	0.519407i	\(0.826153\pi\)
\(47\)	−0.465265	−	3.31053i	−0.0678658	−	0.482890i	−0.994316	−	0.106472i	\(-0.966044\pi\)
							0.926450	−	0.376418i	\(-0.122844\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.2	✓	1200
19.17	even	9	inner	950.2.bg.a.359.27	yes	1200
25.14	even	10	inner	950.2.bg.a.389.27	yes	1200
475.264	even	90	inner	950.2.bg.a.739.2	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.2	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.27	yes	1200	19.17	even	9	inner
950.2.bg.a.389.27	yes	1200	25.14	even	10	inner
950.2.bg.a.739.2	yes	1200	475.264	even	90	inner