Embedded newform 950.2.bc.b.461.12

• Label: 950.2.bc.b.461.12
• Level: $950$
• Weight: $2$
• Character: 950.461
• 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			461.12
Character	\(\chi\)	\(=\)	950.461
Dual form			950.2.bc.b.441.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.559193 + 0.829038i) q^{2} +(-0.711435 - 0.378277i) q^{3} +(-0.374607 - 0.927184i) q^{4} +(-0.561422 - 2.16444i) q^{5} +(0.711435 - 0.378277i) q^{6} +(1.52948 + 2.64913i) q^{7} +(0.978148 + 0.207912i) q^{8} +(-1.31453 - 1.94887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.559193	+	0.829038i	−0.395409	+	0.586218i
							\(3\)	−0.711435	−	0.378277i	−0.410747	−	0.218398i	0.251222	−	0.967930i	\(-0.419168\pi\)
−0.661969	+	0.749531i	\(0.730279\pi\)
\(5\)	−0.561422	−	2.16444i	−0.251076	−	0.967967i
							\(7\)	1.52948	+	2.64913i	0.578088	+	1.00128i	0.995699	+	0.0926520i	\(0.0295344\pi\)
−0.417610	+	0.908626i	\(0.637132\pi\)
\(11\)	−0.186913	−	1.77835i	−0.0563563	−	0.536194i	−0.985883	−	0.167438i	\(-0.946450\pi\)
							0.929526	−	0.368756i	\(-0.120216\pi\)
\(13\)	0.668452	−	1.37053i	0.185395	−	0.380116i	−0.785772	−	0.618516i	\(-0.787734\pi\)
							0.971167	+	0.238400i	\(0.0766229\pi\)
\(17\)	0.996010	+	1.27484i	0.241568	+	0.309193i	0.893481	−	0.449101i	\(-0.148256\pi\)
							−0.651913	+	0.758294i	\(0.726033\pi\)
\(19\)	−1.39251	−	4.13049i	−0.319463	−	0.947599i
							\(23\)	−0.798720	+	3.20349i	−0.166545	+	0.667974i	0.827731	+	0.561125i	\(0.189631\pi\)
−0.994275	+	0.106848i	\(0.965924\pi\)
\(29\)	−5.12685	+	6.56207i	−0.952032	+	1.21855i	0.0237314	+	0.999718i	\(0.492445\pi\)
							−0.975764	+	0.218827i	\(0.929777\pi\)
\(31\)	6.80804	−	7.56109i	1.22276	−	1.35801i	0.309362	−	0.950944i	\(-0.399884\pi\)
							0.913398	−	0.407068i	\(-0.133449\pi\)
\(37\)	−8.41905	−	6.11680i	−1.38408	−	1.00560i	−0.996485	−	0.0837678i	\(-0.973305\pi\)
							−0.387599	−	0.921828i	\(-0.626695\pi\)
\(41\)	−2.95037	+	2.84914i	−0.460770	+	0.444961i	−0.888453	−	0.458969i	\(-0.848219\pi\)
							0.427682	+	0.903929i	\(0.359330\pi\)
\(43\)	−4.77999	−	4.01088i	−0.728941	−	0.611654i	0.200902	−	0.979611i	\(-0.435613\pi\)
							−0.929843	+	0.367957i	\(0.880057\pi\)
\(47\)	2.05069	−	2.62476i	0.299124	−	0.382861i	−0.614843	−	0.788649i	\(-0.710781\pi\)
							0.913967	+	0.405788i	\(0.133003\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.461.12	yes	600
19.4	even	9	inner	950.2.bc.b.61.14	✓	600
25.16	even	5	inner	950.2.bc.b.841.14	yes	600
475.441	even	45	inner	950.2.bc.b.441.12	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.61.14	✓	600	19.4	even	9	inner
950.2.bc.b.441.12	yes	600	475.441	even	45	inner
950.2.bc.b.461.12	yes	600	1.1	even	1	trivial
950.2.bc.b.841.14	yes	600	25.16	even	5	inner