Embedded newform 950.2.bc.a.111.1

• Label: 950.2.bc.a.111.1
• Level: $950$
• Weight: $2$
• Character: 950.111
• 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			111.1
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.a.291.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997564 - 0.0697565i) q^{2} +(-0.118534 - 3.39437i) q^{3} +(0.990268 + 0.139173i) q^{4} +(-2.19232 + 0.440137i) q^{5} +(-0.118534 + 3.39437i) q^{6} +(-1.36724 - 2.36813i) q^{7} +(-0.978148 - 0.207912i) q^{8} +(-8.51502 + 0.595428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.997564	−	0.0697565i	−0.705384	−	0.0493253i
							\(3\)	−0.118534	−	3.39437i	−0.0684357	−	1.95974i	−0.230777	−	0.973007i	\(-0.574127\pi\)
0.162341	−	0.986735i	\(-0.448096\pi\)
\(5\)	−2.19232	+	0.440137i	−0.980437	+	0.196835i
							\(7\)	−1.36724	−	2.36813i	−0.516769	−	0.895070i	−0.999810	−	0.0194728i	\(-0.993801\pi\)
0.483041	−	0.875598i	\(-0.339532\pi\)
\(11\)	0.292074	+	2.77890i	0.0880636	+	0.837870i	0.946012	+	0.324133i	\(0.105073\pi\)
							−0.857948	+	0.513737i	\(0.828261\pi\)
\(13\)	−2.22943	−	3.30526i	−0.618332	−	0.916715i	0.381639	−	0.924311i	\(-0.375360\pi\)
							−0.999971	+	0.00759677i	\(0.997582\pi\)
\(17\)	−0.119273	+	0.295212i	−0.0289280	+	0.0715993i	−0.940956	−	0.338528i	\(-0.890071\pi\)
							0.912028	+	0.410127i	\(0.134516\pi\)
\(19\)	0.392781	−	4.34117i	0.0901102	−	0.995932i
							\(23\)	3.03558	+	2.93142i	0.632961	+	0.611244i	0.940296	−	0.340358i	\(-0.110548\pi\)
−0.307335	+	0.951602i	\(0.599437\pi\)
\(29\)	−1.66682	−	4.12553i	−0.309521	−	0.766092i	−0.999122	−	0.0419018i	\(-0.986658\pi\)
							0.689601	−	0.724190i	\(-0.257786\pi\)
\(31\)	−6.27301	+	6.96688i	−1.12667	+	1.25129i	−0.162293	+	0.986743i	\(0.551889\pi\)
							−0.964373	+	0.264546i	\(0.914778\pi\)
\(37\)	8.72515	+	6.33919i	1.43441	+	1.04216i	0.989175	+	0.146742i	\(0.0468786\pi\)
							0.445231	+	0.895416i	\(0.353121\pi\)
\(41\)	−2.68539	−	0.770022i	−0.419387	−	0.120257i	0.0593056	−	0.998240i	\(-0.481111\pi\)
							−0.478693	+	0.877983i	\(0.658889\pi\)
\(43\)	1.11007	−	6.29551i	0.169284	−	0.960056i	−0.775253	−	0.631651i	\(-0.782378\pi\)
							0.944537	−	0.328405i	\(-0.106511\pi\)
\(47\)	−2.29004	−	5.66805i	−0.334037	−	0.826770i	−0.996948	−	0.0780719i	\(-0.975124\pi\)
							0.662911	−	0.748698i	\(-0.269321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.111.1	✓	600
19.6	even	9	inner	950.2.bc.a.861.25	yes	600
25.16	even	5	inner	950.2.bc.a.491.25	yes	600
475.291	even	45	inner	950.2.bc.a.291.1	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.111.1	✓	600	1.1	even	1	trivial
950.2.bc.a.291.1	yes	600	475.291	even	45	inner
950.2.bc.a.491.25	yes	600	25.16	even	5	inner
950.2.bc.a.861.25	yes	600	19.6	even	9	inner