Embedded newform 950.2.bd.a.27.6

• Label: 950.2.bd.a.27.6
• Level: $950$
• Weight: $2$
• Character: 950.27
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $800$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(800\)
Relative dimension:	\(50\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			27.6
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(-0.0968046 + 1.84714i) q^{3} +(0.207912 + 0.978148i) q^{4} +(1.09978 + 1.94692i) q^{5} +(1.23768 - 1.37458i) q^{6} +(-0.632973 - 0.632973i) q^{7} +(0.453990 - 0.891007i) q^{8} +(-0.418995 - 0.0440382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777146	−	0.629320i	−0.549525	−	0.444997i
							\(3\)	−0.0968046	+	1.84714i	−0.0558902	+	1.06645i	0.816409	+	0.577475i	\(0.195962\pi\)
−0.872299	+	0.488973i	\(0.837372\pi\)
\(5\)	1.09978	+	1.94692i	0.491835	+	0.870689i
							\(7\)	−0.632973	−	0.632973i	−0.239241	−	0.239241i	0.577295	−	0.816536i	\(-0.304108\pi\)
−0.816536	+	0.577295i	\(0.804108\pi\)
\(11\)	−4.15603	+	3.01954i	−1.25309	+	0.910424i	−0.998397	−	0.0565973i	\(-0.981975\pi\)
							−0.254694	+	0.967022i	\(0.581975\pi\)
\(13\)	1.80537	−	1.46196i	0.500719	−	0.405474i	−0.345410	−	0.938452i	\(-0.612260\pi\)
							0.846130	+	0.532977i	\(0.178927\pi\)
\(17\)	1.56820	+	2.41481i	0.380343	+	0.585677i	0.976056	−	0.217521i	\(-0.0697969\pi\)
							−0.595713	+	0.803198i	\(0.703130\pi\)
\(19\)	3.40545	+	2.72083i	0.781263	+	0.624202i
							\(23\)	0.451587	−	0.173348i	0.0941624	−	0.0361456i	−0.310831	−	0.950465i	\(-0.600607\pi\)
0.404993	+	0.914320i	\(0.367274\pi\)
\(29\)	−7.30178	+	1.55204i	−1.35591	+	0.288207i	−0.827825	−	0.560987i	\(-0.810422\pi\)
							−0.528081	+	0.849194i	\(0.677088\pi\)
\(31\)	0.364621	−	0.118472i	0.0654878	−	0.0212783i	−0.276090	−	0.961132i	\(-0.589039\pi\)
							0.341578	+	0.939854i	\(0.389039\pi\)
\(37\)	2.85426	−	0.452070i	0.469237	−	0.0743199i	0.0826619	−	0.996578i	\(-0.473658\pi\)
							0.386576	+	0.922258i	\(0.373658\pi\)
\(41\)	−4.78953	−	10.7575i	−0.747999	−	1.68003i	−0.733078	−	0.680145i	\(-0.761917\pi\)
							−0.0149212	−	0.999889i	\(-0.504750\pi\)
\(43\)	2.02040	+	7.54023i	0.308108	+	1.14987i	0.930237	+	0.366960i	\(0.119601\pi\)
							−0.622129	+	0.782915i	\(0.713732\pi\)
\(47\)	4.35518	+	2.82829i	0.635268	+	0.412548i	0.821700	−	0.569920i	\(-0.193026\pi\)
							−0.186432	+	0.982468i	\(0.559692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.6	✓	800
19.12	odd	6	inner	950.2.bd.a.677.20	yes	800
25.13	odd	20	inner	950.2.bd.a.863.20	yes	800
475.88	even	60	inner	950.2.bd.a.563.6	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.6	✓	800	1.1	even	1	trivial
950.2.bd.a.563.6	yes	800	475.88	even	60	inner
950.2.bd.a.677.20	yes	800	19.12	odd	6	inner
950.2.bd.a.863.20	yes	800	25.13	odd	20	inner