Embedded newform 950.2.bg.a.9.5

• Label: 950.2.bg.a.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	950.9
Dual form			950.2.bg.a.739.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.898794 - 0.438371i) q^{2} +(-1.21964 + 1.95183i) q^{3} +(0.615661 + 0.788011i) q^{4} +(0.512699 - 2.17650i) q^{5} +(1.95183 - 1.21964i) q^{6} +(1.49771 + 0.864704i) q^{7} +(-0.207912 - 0.978148i) q^{8} +(-1.00701 - 2.06468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1200 q+O(q^{10}) \)

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
							\(3\)	−1.21964	+	1.95183i	−0.704159	+	1.12689i	0.281730	+	0.959494i	\(0.409092\pi\)
−0.985890	+	0.167397i	\(0.946464\pi\)
\(5\)	0.512699	−	2.17650i	0.229286	−	0.973359i
							\(7\)	1.49771	+	0.864704i	0.566081	+	0.326827i	0.755583	−	0.655053i	\(-0.227354\pi\)
−0.189501	+	0.981880i	\(0.560687\pi\)
\(11\)	0.419665	−	3.99285i	0.126534	−	1.20389i	−0.728400	−	0.685152i	\(-0.759736\pi\)
							0.854934	−	0.518737i	\(-0.173597\pi\)
\(13\)	−0.244583	+	3.49770i	−0.0678352	+	0.970088i	0.837785	+	0.546000i	\(0.183850\pi\)
							−0.905621	+	0.424089i	\(0.860594\pi\)
\(17\)	0.424331	−	3.01927i	0.102915	−	0.732282i	−0.868636	−	0.495452i	\(-0.835003\pi\)
							0.971551	−	0.236830i	\(-0.0761085\pi\)
\(19\)	−0.634210	−	4.31251i	−0.145498	−	0.989359i
							\(23\)	1.12868	−	3.93618i	0.235347	−	0.820751i	−0.751588	−	0.659632i	\(-0.770712\pi\)
0.986935	−	0.161119i	\(-0.0515102\pi\)
\(29\)	−9.80016	+	1.37732i	−1.81984	+	0.255763i	−0.964836	−	0.262852i	\(-0.915337\pi\)
							−0.855008	+	0.518614i	\(0.826448\pi\)
\(31\)	0.390567	+	0.433768i	0.0701478	+	0.0779071i	0.777197	−	0.629257i	\(-0.216641\pi\)
							−0.707049	+	0.707164i	\(0.749974\pi\)
\(37\)	4.55283	+	6.26644i	0.748481	+	1.03020i	0.998086	+	0.0618481i	\(0.0196994\pi\)
							−0.249604	+	0.968348i	\(0.580301\pi\)
\(41\)	2.44957	−	9.82468i	0.382558	−	1.53436i	−0.402783	−	0.915295i	\(-0.631957\pi\)
							0.785342	−	0.619063i	\(-0.212487\pi\)
\(43\)	0.999744	−	2.74677i	0.152460	−	0.418879i	−0.839826	−	0.542856i	\(-0.817343\pi\)
							0.992285	+	0.123977i	\(0.0395650\pi\)
\(47\)	0.0195878	+	0.139374i	0.00285717	+	0.0203298i	0.991644	−	0.129001i	\(-0.0411770\pi\)
							−0.988787	+	0.149331i	\(0.952288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bg.a.9.5	✓	1200
19.17	even	9	inner	950.2.bg.a.359.30	yes	1200
25.14	even	10	inner	950.2.bg.a.389.30	yes	1200
475.264	even	90	inner	950.2.bg.a.739.5	yes	1200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bg.a.9.5	✓	1200	1.1	even	1	trivial
950.2.bg.a.359.30	yes	1200	19.17	even	9	inner
950.2.bg.a.389.30	yes	1200	25.14	even	10	inner
950.2.bg.a.739.5	yes	1200	475.264	even	90	inner