Embedded newform 950.2.bd.a.27.8

• Label: 950.2.bd.a.27.8
• 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			27.8
Character	\(\chi\)	\(=\)	950.27
Dual form			950.2.bd.a.563.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(-0.0775294 + 1.47935i) q^{3} +(0.207912 + 0.978148i) q^{4} +(-1.34338 - 1.78755i) q^{5} +(0.991237 - 1.10088i) q^{6} +(1.75517 + 1.75517i) q^{7} +(0.453990 - 0.891007i) q^{8} +(0.801101 + 0.0841991i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 800 q - 4 q^{5} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−0.0775294	+	1.47935i	−0.0447616	+	0.854103i	0.880331	+	0.474361i	\(0.157321\pi\)
−0.925092	+	0.379742i	\(0.876013\pi\)
\(5\)	−1.34338	−	1.78755i	−0.600779	−	0.799415i
							\(7\)	1.75517	+	1.75517i	0.663392	+	0.663392i	0.956178	−	0.292786i	\(-0.0945823\pi\)
−0.292786	+	0.956178i	\(0.594582\pi\)
\(11\)	0.349990	−	0.254283i	0.105526	−	0.0766691i	−0.533771	−	0.845629i	\(-0.679226\pi\)
							0.639297	+	0.768960i	\(0.279226\pi\)
\(13\)	2.18819	−	1.77196i	0.606894	−	0.491453i	−0.275917	−	0.961181i	\(-0.588982\pi\)
							0.882811	+	0.469728i	\(0.155648\pi\)
\(17\)	−1.38624	−	2.13463i	−0.336213	−	0.517723i	0.629615	−	0.776908i	\(-0.283213\pi\)
							−0.965828	+	0.259185i	\(0.916546\pi\)
\(19\)	−4.28319	−	0.808898i	−0.982630	−	0.185574i
							\(23\)	3.49493	−	1.34158i	0.728743	−	0.279738i	0.0344256	−	0.999407i	\(-0.489040\pi\)
0.694317	+	0.719669i	\(0.255707\pi\)
\(29\)	7.96758	−	1.69356i	1.47954	−	0.314487i	0.603750	−	0.797174i	\(-0.293672\pi\)
							0.875793	+	0.482687i	\(0.160339\pi\)
\(31\)	3.84330	−	1.24876i	0.690278	−	0.224285i	0.0571882	−	0.998363i	\(-0.481787\pi\)
							0.633089	+	0.774079i	\(0.281787\pi\)
\(37\)	−3.53966	+	0.560627i	−0.581916	+	0.0921665i	−0.440451	−	0.897777i	\(-0.645181\pi\)
							−0.141465	+	0.989943i	\(0.545181\pi\)
\(41\)	−2.26746	−	5.09280i	−0.354118	−	0.795362i	−0.999504	−	0.0315050i	\(-0.989970\pi\)
							0.645386	−	0.763857i	\(-0.276697\pi\)
\(43\)	0.256833	+	0.958515i	0.0391667	+	0.146172i	0.982740	−	0.184991i	\(-0.0592256\pi\)
							−0.943573	+	0.331163i	\(0.892559\pi\)
\(47\)	10.6670	+	6.92725i	1.55595	+	1.01044i	0.981401	+	0.191972i	\(0.0614881\pi\)
							0.574546	+	0.818472i	\(0.305179\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bd.a.27.8	✓	800
19.12	odd	6	inner	950.2.bd.a.677.18	yes	800
25.13	odd	20	inner	950.2.bd.a.863.18	yes	800
475.88	even	60	inner	950.2.bd.a.563.8	yes	800

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bd.a.27.8	✓	800	1.1	even	1	trivial
950.2.bd.a.563.8	yes	800	475.88	even	60	inner
950.2.bd.a.677.18	yes	800	19.12	odd	6	inner
950.2.bd.a.863.18	yes	800	25.13	odd	20	inner