Embedded newform 950.2.bc.a.111.9

• Label: 950.2.bc.a.111.9
• 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			111.9
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.a.291.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997564 - 0.0697565i) q^{2} +(-0.0352288 - 1.00882i) q^{3} +(0.990268 + 0.139173i) q^{4} +(-0.727261 + 2.11450i) q^{5} +(-0.0352288 + 1.00882i) q^{6} +(-1.27394 - 2.20653i) q^{7} +(-0.978148 - 0.207912i) q^{8} +(1.97621 - 0.138190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.0352288	−	1.00882i	−0.0203393	−	0.582443i	−0.966189	−	0.257836i	\(-0.916990\pi\)
0.945849	−	0.324606i	\(-0.105232\pi\)
\(5\)	−0.727261	+	2.11450i	−0.325241	+	0.945631i
							\(7\)	−1.27394	−	2.20653i	−0.481504	−	0.833989i	0.518271	−	0.855216i	\(-0.326576\pi\)
−0.999775	+	0.0212276i	\(0.993243\pi\)
\(11\)	0.431397	+	4.10447i	0.130071	+	1.23754i	0.843621	+	0.536939i	\(0.180419\pi\)
							−0.713550	+	0.700604i	\(0.752914\pi\)
\(13\)	−1.81248	−	2.68711i	−0.502690	−	0.745269i	0.489227	−	0.872157i	\(-0.337279\pi\)
							−0.991917	+	0.126887i	\(0.959501\pi\)
\(17\)	0.925273	−	2.29013i	0.224412	−	0.555438i	−0.772408	−	0.635126i	\(-0.780948\pi\)
							0.996820	+	0.0796882i	\(0.0253925\pi\)
\(19\)	0.663286	+	4.30814i	0.152168	+	0.988355i
							\(23\)	1.92231	+	1.85635i	0.400828	+	0.387075i	0.867567	−	0.497320i	\(-0.165682\pi\)
−0.466739	+	0.884395i	\(0.654571\pi\)
\(29\)	2.43978	+	6.03867i	0.453056	+	1.12135i	0.964980	+	0.262322i	\(0.0844882\pi\)
							−0.511925	+	0.859030i	\(0.671067\pi\)
\(31\)	4.46115	−	4.95461i	0.801246	−	0.889874i	−0.194604	−	0.980882i	\(-0.562342\pi\)
							0.995850	+	0.0910079i	\(0.0290089\pi\)
\(37\)	8.23068	+	5.97994i	1.35312	+	0.983096i	0.998850	+	0.0479518i	\(0.0152694\pi\)
							0.354266	+	0.935145i	\(0.384731\pi\)
\(41\)	−1.18287	−	0.339183i	−0.184733	−	0.0529714i	0.181995	−	0.983299i	\(-0.441744\pi\)
							−0.366729	+	0.930328i	\(0.619522\pi\)
\(43\)	−0.951349	+	5.39537i	−0.145079	+	0.822786i	0.822224	+	0.569164i	\(0.192733\pi\)
							−0.967303	+	0.253622i	\(0.918378\pi\)
\(47\)	3.92870	+	9.72387i	0.573059	+	1.41837i	0.883794	+	0.467875i	\(0.154980\pi\)
							−0.310735	+	0.950497i	\(0.600575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.111.9	✓	600
19.6	even	9	inner	950.2.bc.a.861.17	yes	600
25.16	even	5	inner	950.2.bc.a.491.17	yes	600
475.291	even	45	inner	950.2.bc.a.291.9	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.111.9	✓	600	1.1	even	1	trivial
950.2.bc.a.291.9	yes	600	475.291	even	45	inner
950.2.bc.a.491.17	yes	600	25.16	even	5	inner
950.2.bc.a.861.17	yes	600	19.6	even	9	inner