Embedded newform 950.2.bg.a.9.7

• Label: 950.2.bg.a.9.7
• 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			9.7
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-0.838664 + 1.34214i) q^{3} +(0.615661 + 0.788011i) q^{4} +(1.22007 + 1.87388i) q^{5} +(1.34214 - 0.838664i) q^{6} +(0.0845996 + 0.0488436i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(0.217124 + 0.445170i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−0.838664	+	1.34214i	−0.484203	+	0.774886i	−0.996471	−	0.0839393i	\(-0.973250\pi\)
0.512268	+	0.858826i	\(0.328805\pi\)
\(5\)	1.22007	+	1.87388i	0.545632	+	0.838025i
							\(7\)	0.0845996	+	0.0488436i	0.0319756	+	0.0184611i	0.515903	−	0.856647i	\(-0.327457\pi\)
−0.483927	+	0.875108i	\(0.660790\pi\)
\(11\)	0.478690	−	4.55443i	0.144331	−	1.37321i	−0.647310	−	0.762227i	\(-0.724106\pi\)
							0.791641	−	0.610987i	\(-0.209227\pi\)
\(13\)	0.334526	−	4.78394i	0.0927808	−	1.32683i	−0.698604	−	0.715509i	\(-0.746195\pi\)
							0.791385	−	0.611318i	\(-0.209361\pi\)
\(17\)	0.944494	−	6.72042i	0.229073	−	1.62994i	−0.448005	−	0.894031i	\(-0.647865\pi\)
							0.677078	−	0.735911i	\(-0.263246\pi\)
\(19\)	2.99404	−	3.16792i	0.686880	−	0.726771i
							\(23\)	1.09901	−	3.83269i	0.229159	−	0.799171i	−0.759695	−	0.650280i	\(-0.774652\pi\)
0.988854	−	0.148891i	\(-0.0475705\pi\)
\(29\)	8.95331	−	1.25831i	1.66259	−	0.233662i	0.755665	−	0.654958i	\(-0.227314\pi\)
							0.906923	+	0.421297i	\(0.138425\pi\)
\(31\)	−3.02387	−	3.35835i	−0.543103	−	0.603177i	0.407645	−	0.913140i	\(-0.366350\pi\)
							−0.950749	+	0.309963i	\(0.899683\pi\)
\(37\)	1.72897	+	2.37973i	0.284241	+	0.391225i	0.927133	−	0.374732i	\(-0.122265\pi\)
							−0.642892	+	0.765957i	\(0.722265\pi\)
\(41\)	−2.84064	+	11.3932i	−0.443633	+	1.77931i	0.165462	+	0.986216i	\(0.447089\pi\)
							−0.609095	+	0.793098i	\(0.708467\pi\)
\(43\)	1.51136	−	4.15244i	0.230481	−	0.633241i	−0.769505	−	0.638641i	\(-0.779497\pi\)
							0.999985	+	0.00540059i	\(0.00171907\pi\)
\(47\)	−0.565139	−	4.02117i	−0.0824340	−	0.586548i	−0.987061	−	0.160344i	\(-0.948740\pi\)
							0.904627	−	0.426204i	\(-0.140149\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.7	✓	1200
19.17	even	9	inner	950.2.bg.a.359.32	yes	1200
25.14	even	10	inner	950.2.bg.a.389.32	yes	1200
475.264	even	90	inner	950.2.bg.a.739.7	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.7	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.32	yes	1200	19.17	even	9	inner
950.2.bg.a.389.32	yes	1200	25.14	even	10	inner
950.2.bg.a.739.7	yes	1200	475.264	even	90	inner