Embedded newform 950.2.bc.b.61.10

• Label: 950.2.bc.b.61.10
• 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			61.10
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(-0.902497 - 0.126838i) q^{3} +(0.848048 - 0.529919i) q^{4} +(1.77071 - 1.36550i) q^{5} +(0.902497 - 0.126838i) q^{6} +(-2.28164 + 3.95191i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-2.08537 - 0.597971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.902497	−	0.126838i	−0.521057	−	0.0732298i	−0.126258	−	0.991997i	\(-0.540297\pi\)
−0.394799	+	0.918768i	\(0.629186\pi\)
\(5\)	1.77071	−	1.36550i	0.791885	−	0.610670i
							\(7\)	−2.28164	+	3.95191i	−0.862378	+	1.49368i	0.00724986	+	0.999974i	\(0.497692\pi\)
−0.869628	+	0.493708i	\(0.835641\pi\)
\(11\)	−0.190929	−	0.0850073i	−0.0575674	−	0.0256307i	0.377751	−	0.925907i	\(-0.376698\pi\)
							−0.435318	+	0.900277i	\(0.643364\pi\)
\(13\)	0.0625282	−	0.0603828i	0.0173422	−	0.0167472i	−0.685937	−	0.727661i	\(-0.740607\pi\)
							0.703279	+	0.710914i	\(0.251718\pi\)
\(17\)	0.0266984	−	0.764543i	0.00647532	−	0.185429i	−0.991843	−	0.127463i	\(-0.959316\pi\)
							0.998319	−	0.0579655i	\(-0.0184613\pi\)
\(19\)	−0.563475	−	4.32233i	−0.129270	−	0.991609i
							\(23\)	4.85551	+	0.339530i	1.01244	+	0.0707970i	0.566329	−	0.824179i	\(-0.308363\pi\)
0.446114	+	0.894976i	\(0.352807\pi\)
\(29\)	−0.164284	−	4.70449i	−0.0305068	−	0.873602i	−0.914134	−	0.405411i	\(-0.867128\pi\)
							0.883628	−	0.468190i	\(-0.155094\pi\)
\(31\)	−2.98342	−	0.634146i	−0.535839	−	0.113896i	−0.0679588	−	0.997688i	\(-0.521649\pi\)
							−0.467880	+	0.883792i	\(0.654982\pi\)
\(37\)	−7.37016	−	5.35474i	−1.21165	−	0.880313i	−0.216268	−	0.976334i	\(-0.569388\pi\)
							−0.995379	+	0.0960210i	\(0.969388\pi\)
\(41\)	0.746511	−	1.53057i	0.116585	−	0.239035i	−0.832519	−	0.553996i	\(-0.813102\pi\)
							0.949105	+	0.314960i	\(0.101991\pi\)
\(43\)	3.13435	−	2.63003i	0.477984	−	0.401077i	−0.371713	−	0.928348i	\(-0.621229\pi\)
							0.849697	+	0.527271i	\(0.176785\pi\)
\(47\)	−0.0639678	−	1.83180i	−0.00933066	−	0.267195i	−0.995086	−	0.0990113i	\(-0.968432\pi\)
							0.985756	−	0.168184i	\(-0.0537902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.10	✓	600
19.5	even	9	inner	950.2.bc.b.461.16	yes	600
25.16	even	5	inner	950.2.bc.b.441.16	yes	600
475.366	even	45	inner	950.2.bc.b.841.10	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.10	✓	600	1.1	even	1	trivial
950.2.bc.b.441.16	yes	600	25.16	even	5	inner
950.2.bc.b.461.16	yes	600	19.5	even	9	inner
950.2.bc.b.841.10	yes	600	475.366	even	45	inner