Embedded newform 950.2.bg.a.9.1

• Label: 950.2.bg.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-1.80064 + 2.88162i) q^{3} +(0.615661 + 0.788011i) q^{4} +(-1.98988 - 1.01998i) q^{5} +(2.88162 - 1.80064i) q^{6} +(-1.84599 - 1.06578i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(-3.74633 - 7.68112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−1.80064	+	2.88162i	−1.03960	+	1.66370i	−0.354853	+	0.934922i	\(0.615469\pi\)
−0.684745	+	0.728783i	\(0.740086\pi\)
\(5\)	−1.98988	−	1.01998i	−0.889903	−	0.456151i
							\(7\)	−1.84599	−	1.06578i	−0.697719	−	0.402828i	0.108778	−	0.994066i	\(-0.465306\pi\)
−0.806497	+	0.591238i	\(0.798640\pi\)
\(11\)	−0.100624	+	0.957375i	−0.0303393	+	0.288659i	0.968824	+	0.247752i	\(0.0796917\pi\)
							−0.999163	+	0.0409079i	\(0.986975\pi\)
\(13\)	−0.0544791	+	0.779087i	−0.0151098	+	0.216080i	0.984141	+	0.177386i	\(0.0567641\pi\)
							−0.999251	+	0.0386938i	\(0.987680\pi\)
\(17\)	0.423306	−	3.01198i	0.102667	−	0.730512i	−0.869112	−	0.494616i	\(-0.835309\pi\)
							0.971778	−	0.235896i	\(-0.0758025\pi\)
\(19\)	1.68596	−	4.01964i	0.386786	−	0.922169i
							\(23\)	−2.30598	+	8.04191i	−0.480830	+	1.67685i	0.229760	+	0.973247i	\(0.426206\pi\)
−0.710590	+	0.703606i	\(0.751572\pi\)
\(29\)	4.18712	−	0.588461i	0.777528	−	0.109274i	0.260764	−	0.965402i	\(-0.416025\pi\)
							0.516764	+	0.856128i	\(0.327137\pi\)
\(31\)	−3.98710	−	4.42812i	−0.716104	−	0.795314i	0.269748	−	0.962931i	\(-0.413060\pi\)
							−0.985852	+	0.167617i	\(0.946393\pi\)
\(37\)	−6.41084	−	8.82376i	−1.05394	−	1.45062i	−0.885347	−	0.464931i	\(-0.846079\pi\)
							−0.168588	−	0.985687i	\(-0.553921\pi\)
\(41\)	−0.814208	+	3.26561i	−0.127158	+	0.510002i	0.872609	+	0.488419i	\(0.162426\pi\)
							−0.999767	+	0.0215833i	\(0.993129\pi\)
\(43\)	−0.131505	+	0.361308i	−0.0200544	+	0.0550989i	−0.949316	−	0.314324i	\(-0.898222\pi\)
							0.929262	+	0.369423i	\(0.120445\pi\)
\(47\)	1.75416	+	12.4815i	0.255871	+	1.82062i	0.517769	+	0.855521i	\(0.326763\pi\)
							−0.261898	+	0.965096i	\(0.584348\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.1	✓	1200
19.17	even	9	inner	950.2.bg.a.359.26	yes	1200
25.14	even	10	inner	950.2.bg.a.389.26	yes	1200
475.264	even	90	inner	950.2.bg.a.739.1	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.1	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.26	yes	1200	19.17	even	9	inner
950.2.bg.a.389.26	yes	1200	25.14	even	10	inner
950.2.bg.a.739.1	yes	1200	475.264	even	90	inner