Embedded newform 950.2.bg.a.9.10

• Label: 950.2.bg.a.9.10
• 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			9.10
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-0.443586 + 0.709885i) q^{3} +(0.615661 + 0.788011i) q^{4} +(0.161877 - 2.23020i) q^{5} +(0.709885 - 0.443586i) q^{6} +(-3.38490 - 1.95427i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(1.00794 + 2.06659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−0.443586	+	0.709885i	−0.256104	+	0.409853i	−0.950968	−	0.309289i	\(-0.899909\pi\)
0.694864	+	0.719141i	\(0.255465\pi\)
\(5\)	0.161877	−	2.23020i	0.0723936	−	0.997376i
							\(7\)	−3.38490	−	1.95427i	−1.27937	−	0.738645i	−0.302638	−	0.953106i	\(-0.597867\pi\)
−0.976733	+	0.214461i	\(0.931201\pi\)
\(11\)	0.250145	−	2.37997i	0.0754216	−	0.717589i	−0.889834	−	0.456283i	\(-0.849180\pi\)
							0.965256	−	0.261306i	\(-0.0841530\pi\)
\(13\)	0.268648	−	3.84184i	0.0745094	−	1.06553i	−0.806127	−	0.591742i	\(-0.798440\pi\)
							0.880637	−	0.473792i	\(-0.157115\pi\)
\(17\)	0.330267	−	2.34997i	0.0801016	−	0.569953i	−0.908326	−	0.418262i	\(-0.862639\pi\)
							0.988428	−	0.151691i	\(-0.0484717\pi\)
\(19\)	2.45784	+	3.59986i	0.563867	+	0.825866i
							\(23\)	−2.43367	+	8.48720i	−0.507455	+	1.76970i	0.121853	+	0.992548i	\(0.461116\pi\)
−0.629308	+	0.777156i	\(0.716662\pi\)
\(29\)	−8.72393	+	1.22607i	−1.61999	+	0.227675i	−0.890060	−	0.455844i	\(-0.849338\pi\)
							−0.729933	+	0.683519i	\(0.760449\pi\)
\(31\)	1.60295	+	1.78026i	0.287898	+	0.319743i	0.869694	−	0.493592i	\(-0.164316\pi\)
							−0.581795	+	0.813335i	\(0.697650\pi\)
\(37\)	−1.93589	−	2.66453i	−0.318259	−	0.438046i	0.619676	−	0.784858i	\(-0.287264\pi\)
							−0.937935	+	0.346812i	\(0.887264\pi\)
\(41\)	−0.0656189	+	0.263183i	−0.0102479	+	0.0411023i	−0.975202	−	0.221316i	\(-0.928965\pi\)
							0.964954	+	0.262419i	\(0.0845202\pi\)
\(43\)	2.25930	−	6.20737i	0.344540	−	0.946616i	−0.639519	−	0.768775i	\(-0.720867\pi\)
							0.984059	−	0.177841i	\(-0.0569111\pi\)
\(47\)	0.0381979	+	0.271792i	0.00557174	+	0.0396450i	0.992856	−	0.119323i	\(-0.0380723\pi\)
							−0.987284	+	0.158968i	\(0.949183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.10	✓	1200
19.17	even	9	inner	950.2.bg.a.359.35	yes	1200
25.14	even	10	inner	950.2.bg.a.389.35	yes	1200
475.264	even	90	inner	950.2.bg.a.739.10	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.10	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.35	yes	1200	19.17	even	9	inner
950.2.bg.a.389.35	yes	1200	25.14	even	10	inner
950.2.bg.a.739.10	yes	1200	475.264	even	90	inner