Embedded newform 950.2.bd.a.27.3

• Label: 950.2.bd.a.27.3
• 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			27.3
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(-0.146876 + 2.80257i) q^{3} +(0.207912 + 0.978148i) q^{4} +(0.359551 + 2.20697i) q^{5} +(1.87786 - 2.08557i) q^{6} +(-2.28843 - 2.28843i) q^{7} +(0.453990 - 0.891007i) q^{8} +(-4.84924 - 0.509675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−0.146876	+	2.80257i	−0.0847990	+	1.61806i	0.543032	+	0.839712i	\(0.317276\pi\)
−0.627831	+	0.778350i	\(0.716057\pi\)
\(5\)	0.359551	+	2.20697i	0.160796	+	0.986988i
							\(7\)	−2.28843	−	2.28843i	−0.864947	−	0.864947i	0.126961	−	0.991908i	\(-0.459478\pi\)
−0.991908	+	0.126961i	\(0.959478\pi\)
\(11\)	2.17374	−	1.57932i	0.655408	−	0.476182i	−0.209701	−	0.977766i	\(-0.567249\pi\)
							0.865109	+	0.501584i	\(0.167249\pi\)
\(13\)	0.360601	−	0.292009i	0.100013	−	0.0809886i	−0.578010	−	0.816030i	\(-0.696171\pi\)
							0.678023	+	0.735041i	\(0.262837\pi\)
\(17\)	−3.53500	−	5.44342i	−0.857364	−	1.32022i	−0.946002	−	0.324160i	\(-0.894918\pi\)
							0.0886384	−	0.996064i	\(-0.471748\pi\)
\(19\)	−2.55110	−	3.53439i	−0.585262	−	0.810844i
							\(23\)	−4.12529	+	1.58355i	−0.860182	+	0.330193i	−0.748164	−	0.663513i	\(-0.769065\pi\)
−0.112017	+	0.993706i	\(0.535731\pi\)
\(29\)	0.794532	−	0.168883i	0.147541	−	0.0313608i	−0.133549	−	0.991042i	\(-0.542637\pi\)
							0.281090	+	0.959681i	\(0.409304\pi\)
\(31\)	−5.58867	+	1.81587i	−1.00375	+	0.326140i	−0.764365	−	0.644784i	\(-0.776947\pi\)
							−0.239390	+	0.970924i	\(0.576947\pi\)
\(37\)	−1.70869	+	0.270630i	−0.280907	+	0.0444914i	−0.295298	−	0.955405i	\(-0.595419\pi\)
							0.0143910	+	0.999896i	\(0.495419\pi\)
\(41\)	1.11029	+	2.49375i	0.173398	+	0.389458i	0.979249	−	0.202660i	\(-0.0649586\pi\)
							−0.805851	+	0.592118i	\(0.798292\pi\)
\(43\)	−2.59751	−	9.69402i	−0.396116	−	1.47832i	−0.819871	−	0.572548i	\(-0.805955\pi\)
							0.423756	−	0.905777i	\(-0.360712\pi\)
\(47\)	−3.13231	−	2.03415i	−0.456895	−	0.296711i	0.295597	−	0.955313i	\(-0.404481\pi\)
							−0.752492	+	0.658602i	\(0.771148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.3	✓	800
19.12	odd	6	inner	950.2.bd.a.677.23	yes	800
25.13	odd	20	inner	950.2.bd.a.863.23	yes	800
475.88	even	60	inner	950.2.bd.a.563.3	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.3	✓	800	1.1	even	1	trivial
950.2.bd.a.563.3	yes	800	475.88	even	60	inner
950.2.bd.a.677.23	yes	800	19.12	odd	6	inner
950.2.bd.a.863.23	yes	800	25.13	odd	20	inner