Embedded newform 950.2.bc.b.61.18

• Label: 950.2.bc.b.61.18
• 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			61.18
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.b.841.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.961262 + 0.275637i) q^{2} +(1.19617 + 0.168111i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-1.05511 + 1.97148i) q^{5} +(-1.19617 + 0.168111i) q^{6} +(-2.18297 + 3.78102i) q^{7} +(-0.669131 + 0.743145i) q^{8} +(-1.48123 - 0.424734i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	1.19617	+	0.168111i	0.690609	+	0.0970587i	0.475742	−	0.879585i	\(-0.342179\pi\)
0.214866	+	0.976643i	\(0.431068\pi\)
\(5\)	−1.05511	+	1.97148i	−0.471859	+	0.881674i
							\(7\)	−2.18297	+	3.78102i	−0.825086	+	1.42909i	0.0767681	+	0.997049i	\(0.475540\pi\)
−0.901854	+	0.432041i	\(0.857793\pi\)
\(11\)	3.51929	+	1.56689i	1.06111	+	0.472435i	0.861667	−	0.507475i	\(-0.169421\pi\)
							0.199440	+	0.979910i	\(0.436088\pi\)
\(13\)	−0.631710	+	0.610035i	−0.175205	+	0.169193i	−0.777090	−	0.629389i	\(-0.783305\pi\)
							0.601886	+	0.798582i	\(0.294416\pi\)
\(17\)	0.0861799	−	2.46787i	0.0209017	−	0.598546i	−0.943078	−	0.332571i	\(-0.892084\pi\)
							0.963980	−	0.265975i	\(-0.0856939\pi\)
\(19\)	3.38287	+	2.74885i	0.776083	+	0.630630i
							\(23\)	−3.82653	−	0.267577i	−0.797886	−	0.0557936i	−0.335008	−	0.942215i	\(-0.608739\pi\)
−0.462878	+	0.886422i	\(0.653183\pi\)
\(29\)	−0.158198	−	4.53021i	−0.0293767	−	0.841239i	−0.921254	−	0.388962i	\(-0.872834\pi\)
							0.891877	−	0.452278i	\(-0.149388\pi\)
\(31\)	−2.42021	−	0.514431i	−0.434682	−	0.0923945i	−0.0146268	−	0.999893i	\(-0.504656\pi\)
							−0.420055	+	0.907498i	\(0.637989\pi\)
\(37\)	−8.57707	−	6.23160i	−1.41006	−	1.02447i	−0.993314	−	0.115447i	\(-0.963170\pi\)
							−0.416747	−	0.909022i	\(-0.636830\pi\)
\(41\)	−1.94148	+	3.98063i	−0.303209	+	0.621670i	−0.994938	−	0.100486i	\(-0.967960\pi\)
							0.691730	+	0.722157i	\(0.256849\pi\)
\(43\)	−2.98911	+	2.50816i	−0.455834	+	0.382490i	−0.841596	−	0.540108i	\(-0.818383\pi\)
							0.385761	+	0.922599i	\(0.373939\pi\)
\(47\)	0.161224	+	4.61686i	0.0235170	+	0.673438i	0.952717	+	0.303860i	\(0.0982755\pi\)
							−0.929200	+	0.369578i	\(0.879502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.61.18	✓	600
19.5	even	9	inner	950.2.bc.b.461.8	yes	600
25.16	even	5	inner	950.2.bc.b.441.8	yes	600
475.366	even	45	inner	950.2.bc.b.841.18	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.18	✓	600	1.1	even	1	trivial
950.2.bc.b.441.8	yes	600	25.16	even	5	inner
950.2.bc.b.461.8	yes	600	19.5	even	9	inner
950.2.bc.b.841.18	yes	600	475.366	even	45	inner