Embedded newform 950.2.bc.b.61.16

• Label: 950.2.bc.b.61.16
• 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			61.16
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(1.00717 + 0.141549i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-1.45407 - 1.69873i) q^{5} +(-1.00717 + 0.141549i) q^{6} +(-1.19727 + 2.07373i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-1.88942 - 0.541783i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	1.00717	+	0.141549i	0.581492	+	0.0817233i	0.423785	−	0.905763i	\(-0.360701\pi\)
0.157707	+	0.987486i	\(0.449590\pi\)
\(5\)	−1.45407	−	1.69873i	−0.650280	−	0.759694i
							\(7\)	−1.19727	+	2.07373i	−0.452524	+	0.783795i	−0.998542	−	0.0539786i	\(-0.982810\pi\)
0.546018	+	0.837774i	\(0.316143\pi\)
\(11\)	3.05563	+	1.36045i	0.921306	+	0.410192i	0.811894	−	0.583805i	\(-0.198437\pi\)
							0.109412	+	0.993997i	\(0.465103\pi\)
\(13\)	0.685629	−	0.662104i	0.190159	−	0.183635i	−0.593484	−	0.804846i	\(-0.702248\pi\)
							0.783643	+	0.621211i	\(0.213359\pi\)
\(17\)	−0.0928348	+	2.65844i	−0.0225158	+	0.644767i	0.934708	+	0.355417i	\(0.115661\pi\)
							−0.957224	+	0.289350i	\(0.906561\pi\)
\(19\)	3.68749	−	2.32431i	0.845968	−	0.533233i
							\(23\)	3.65422	+	0.255528i	0.761958	+	0.0532813i	0.445448	−	0.895308i	\(-0.353044\pi\)
0.316510	+	0.948589i	\(0.397489\pi\)
\(29\)	0.0942772	+	2.69975i	0.0175068	+	0.501330i	0.976198	+	0.216881i	\(0.0695884\pi\)
							−0.958691	+	0.284449i	\(0.908189\pi\)
\(31\)	9.58610	+	2.03759i	1.72171	+	0.365962i	0.959575	−	0.281452i	\(-0.0908161\pi\)
							0.762139	+	0.647414i	\(0.224149\pi\)
\(37\)	6.91134	+	5.02138i	1.13622	+	0.825510i	0.986588	−	0.163232i	\(-0.0521919\pi\)
							0.149630	+	0.988742i	\(0.452192\pi\)
\(41\)	3.06191	−	6.27784i	0.478189	−	0.980434i	−0.514243	−	0.857644i	\(-0.671927\pi\)
							0.992433	−	0.122789i	\(-0.0391840\pi\)
\(43\)	0.232054	−	0.194716i	0.0353879	−	0.0296939i	−0.624922	−	0.780687i	\(-0.714869\pi\)
							0.660310	+	0.750993i	\(0.270425\pi\)
\(47\)	0.0170501	+	0.488251i	0.00248701	+	0.0712187i	1.00000		0.000732017i	\(0.000233008\pi\)
							−0.997513	+	0.0704867i	\(0.977545\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.16	✓	600
19.5	even	9	inner	950.2.bc.b.461.10	yes	600
25.16	even	5	inner	950.2.bc.b.441.10	yes	600
475.366	even	45	inner	950.2.bc.b.841.16	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.16	✓	600	1.1	even	1	trivial
950.2.bc.b.441.10	yes	600	25.16	even	5	inner
950.2.bc.b.461.10	yes	600	19.5	even	9	inner
950.2.bc.b.841.16	yes	600	475.366	even	45	inner