Embedded newform 950.2.bc.a.61.20

• Label: 950.2.bc.a.61.20
• Level: $950$
• Weight: $2$
• Character: 950.61
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $600$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(600\)
Relative dimension:	\(25\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			61.20
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(2.20161 + 0.309416i) q^{3} +(0.848048 - 0.529919i) q^{4} +(1.57433 + 1.58792i) q^{5} +(2.20161 - 0.309416i) q^{6} +(1.15550 - 2.00139i) q^{7} +(0.669131 - 0.743145i) q^{8} +(1.86756 + 0.535515i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961262	−	0.275637i	0.679715	−	0.194905i
							\(3\)	2.20161	+	0.309416i	1.27110	+	0.178641i	0.742279	−	0.670091i	\(-0.233745\pi\)
0.528822	+	0.848733i	\(0.322634\pi\)
\(5\)	1.57433	+	1.58792i	0.704062	+	0.710139i
							\(7\)	1.15550	−	2.00139i	0.436739	−	0.756453i	−0.560697	−	0.828021i	\(-0.689467\pi\)
0.997436	+	0.0715676i	\(0.0228002\pi\)
\(11\)	−0.472319	−	0.210290i	−0.142409	−	0.0634048i	0.334295	−	0.942469i	\(-0.391502\pi\)
							−0.476704	+	0.879064i	\(0.658169\pi\)
\(13\)	−1.17440	+	1.13410i	−0.325719	+	0.314544i	−0.839481	−	0.543390i	\(-0.817141\pi\)
							0.513761	+	0.857933i	\(0.328252\pi\)
\(17\)	−0.000802464		0.0229796i	−0.000194626		0.00557336i	−0.999484	−	0.0321127i	\(-0.989776\pi\)
							0.999290	+	0.0376860i	\(0.0119987\pi\)
\(19\)	−4.30776	−	0.665720i	−0.988268	−	0.152727i
							\(23\)	4.51971	+	0.316049i	0.942425	+	0.0659007i	0.532696	−	0.846307i	\(-0.321179\pi\)
0.409729	+	0.912207i	\(0.365623\pi\)
\(29\)	−0.0309358	−	0.885885i	−0.00574463	−	0.164505i	−0.998873	−	0.0474712i	\(-0.984884\pi\)
							0.993128	−	0.117033i	\(-0.0373385\pi\)
\(31\)	−6.03068	−	1.28186i	−1.08314	−	0.230229i	−0.368435	−	0.929653i	\(-0.620106\pi\)
							−0.714706	+	0.699425i	\(0.753440\pi\)
\(37\)	−1.87634	−	1.36324i	−0.308469	−	0.224116i	0.422770	−	0.906237i	\(-0.361058\pi\)
							−0.731239	+	0.682121i	\(0.761058\pi\)
\(41\)	−0.507385	+	1.04029i	−0.0792402	+	0.162466i	−0.934736	−	0.355344i	\(-0.884364\pi\)
							0.855495	+	0.517810i	\(0.173253\pi\)
\(43\)	−3.71199	+	3.11473i	−0.566073	+	0.474992i	−0.880340	−	0.474343i	\(-0.842686\pi\)
							0.314267	+	0.949335i	\(0.398241\pi\)
\(47\)	−0.0537949	−	1.54049i	−0.00784680	−	0.224703i	−0.996979	−	0.0776732i	\(-0.975251\pi\)
							0.989132	−	0.147030i	\(-0.0469713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.20	✓	600
19.5	even	9	inner	950.2.bc.a.461.6	yes	600
25.16	even	5	inner	950.2.bc.a.441.6	yes	600
475.366	even	45	inner	950.2.bc.a.841.20	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.20	✓	600	1.1	even	1	trivial
950.2.bc.a.441.6	yes	600	25.16	even	5	inner
950.2.bc.a.461.6	yes	600	19.5	even	9	inner
950.2.bc.a.841.20	yes	600	475.366	even	45	inner