Embedded newform 950.2.bg.a.9.20

• Label: 950.2.bg.a.9.20
• 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			9.20
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(1.25757 - 2.01253i) q^{3} +(0.615661 + 0.788011i) q^{4} +(-0.556445 + 2.16573i) q^{5} +(-2.01253 + 1.25757i) q^{6} +(-1.28276 - 0.740599i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(-1.15369 - 2.36542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	1.25757	−	2.01253i	0.726058	−	1.16194i	−0.254759	−	0.967005i	\(-0.581996\pi\)
0.980817	−	0.194931i	\(-0.0624484\pi\)
\(5\)	−0.556445	+	2.16573i	−0.248850	+	0.968542i
							\(7\)	−1.28276	−	0.740599i	−0.484836	−	0.279920i	0.237594	−	0.971365i	\(-0.423641\pi\)
−0.722430	+	0.691444i	\(0.756975\pi\)
\(11\)	0.568513	−	5.40904i	0.171413	−	1.63089i	−0.483613	−	0.875282i	\(-0.660676\pi\)
							0.655027	−	0.755606i	\(-0.272657\pi\)
\(13\)	−0.105241	+	1.50501i	−0.0291885	+	0.417415i	0.960893	+	0.276920i	\(0.0893138\pi\)
							−0.990081	+	0.140495i	\(0.955131\pi\)
\(17\)	0.217640	−	1.54859i	0.0527856	−	0.375589i	−0.945906	−	0.324442i	\(-0.894823\pi\)
							0.998691	−	0.0511468i	\(-0.0162876\pi\)
\(19\)	3.87092	−	2.00399i	0.888050	−	0.459748i
							\(23\)	0.304537	−	1.06205i	0.0635004	−	0.221452i	−0.923113	−	0.384529i	\(-0.874364\pi\)
0.986613	+	0.163076i	\(0.0521417\pi\)
\(29\)	−7.16469	+	1.00693i	−1.33045	+	0.186983i	−0.768320	−	0.640066i	\(-0.778907\pi\)
							−0.562130	+	0.827049i	\(0.690018\pi\)
\(31\)	−4.31595	−	4.79335i	−0.775167	−	0.860910i	0.218199	−	0.975904i	\(-0.429982\pi\)
							−0.993366	+	0.114994i	\(0.963315\pi\)
\(37\)	1.29986	+	1.78911i	0.213696	+	0.294128i	0.902386	−	0.430928i	\(-0.141814\pi\)
							−0.688690	+	0.725056i	\(0.741814\pi\)
\(41\)	2.74665	−	11.0162i	0.428954	−	1.72044i	−0.234403	−	0.972140i	\(-0.575314\pi\)
							0.663357	−	0.748303i	\(-0.269131\pi\)
\(43\)	1.67190	−	4.59351i	0.254962	−	0.700503i	−0.744497	−	0.667626i	\(-0.767311\pi\)
							0.999459	−	0.0328775i	\(-0.0104671\pi\)
\(47\)	1.08155	+	7.69565i	0.157761	+	1.12253i	0.894732	+	0.446603i	\(0.147366\pi\)
							−0.736972	+	0.675924i	\(0.763745\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.20	✓	1200
19.17	even	9	inner	950.2.bg.a.359.45	yes	1200
25.14	even	10	inner	950.2.bg.a.389.45	yes	1200
475.264	even	90	inner	950.2.bg.a.739.20	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.20	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.45	yes	1200	19.17	even	9	inner
950.2.bg.a.389.45	yes	1200	25.14	even	10	inner
950.2.bg.a.739.20	yes	1200	475.264	even	90	inner