Embedded newform 950.2.bc.a.61.5

• Label: 950.2.bc.a.61.5
• 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			61.5
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(-2.26586 - 0.318446i) q^{3} +(0.848048 - 0.529919i) q^{4} +(0.868613 - 2.06046i) q^{5} +(-2.26586 + 0.318446i) q^{6} +(-1.77884 + 3.08104i) q^{7} +(0.669131 - 0.743145i) q^{8} +(2.14894 + 0.616198i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−2.26586	−	0.318446i	−1.30820	−	0.183855i	−0.549606	−	0.835424i	\(-0.685222\pi\)
−0.758590	+	0.651569i	\(0.774111\pi\)
\(5\)	0.868613	−	2.06046i	0.388456	−	0.921467i
							\(7\)	−1.77884	+	3.08104i	−0.672338	+	1.16452i	0.304901	+	0.952384i	\(0.401377\pi\)
−0.977239	+	0.212140i	\(0.931957\pi\)
\(11\)	3.16496	+	1.40913i	0.954270	+	0.424868i	0.823987	−	0.566608i	\(-0.191745\pi\)
							0.130283	+	0.991477i	\(0.458411\pi\)
\(13\)	−4.60089	+	4.44303i	−1.27606	+	1.23227i	−0.317208	+	0.948356i	\(0.602745\pi\)
							−0.958849	+	0.283917i	\(0.908366\pi\)
\(17\)	−0.0655201	+	1.87625i	−0.0158910	+	0.455057i	0.965200	+	0.261512i	\(0.0842210\pi\)
							−0.981091	+	0.193546i	\(0.938001\pi\)
\(19\)	4.31607	+	0.609544i	0.990174	+	0.139839i
							\(23\)	−3.56293	−	0.249145i	−0.742923	−	0.0519502i	−0.306722	−	0.951799i	\(-0.599232\pi\)
−0.436201	+	0.899849i	\(0.643676\pi\)
\(29\)	0.0569207	+	1.63000i	0.0105699	+	0.302683i	0.993150	+	0.116844i	\(0.0372778\pi\)
							−0.982580	+	0.185838i	\(0.940500\pi\)
\(31\)	8.13847	+	1.72988i	1.46171	+	0.310696i	0.869038	−	0.494746i	\(-0.164739\pi\)
							0.592674	+	0.805442i	\(0.298072\pi\)
\(37\)	2.67742	+	1.94526i	0.440164	+	0.319798i	0.785700	−	0.618607i	\(-0.212303\pi\)
							−0.345536	+	0.938406i	\(0.612303\pi\)
\(41\)	−3.37779	+	6.92550i	−0.527522	+	1.08158i	0.454136	+	0.890932i	\(0.349948\pi\)
							−0.981658	+	0.190649i	\(0.938941\pi\)
\(43\)	9.81168	−	8.23298i	1.49627	−	1.25552i	0.609949	−	0.792441i	\(-0.291190\pi\)
							0.886318	−	0.463076i	\(-0.153254\pi\)
\(47\)	0.285964	+	8.18894i	0.0417121	+	1.19448i	0.822159	+	0.569258i	\(0.192769\pi\)
							−0.780447	+	0.625222i	\(0.785008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.5	✓	600
19.5	even	9	inner	950.2.bc.a.461.21	yes	600
25.16	even	5	inner	950.2.bc.a.441.21	yes	600
475.366	even	45	inner	950.2.bc.a.841.5	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.5	✓	600	1.1	even	1	trivial
950.2.bc.a.441.21	yes	600	25.16	even	5	inner
950.2.bc.a.461.21	yes	600	19.5	even	9	inner
950.2.bc.a.841.5	yes	600	475.366	even	45	inner