Embedded newform 950.2.bc.b.111.6

• Label: 950.2.bc.b.111.6
• 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			111.6
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.b.291.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.997564 + 0.0697565i) q^{2} +(-0.0568237 - 1.62722i) q^{3} +(0.990268 + 0.139173i) q^{4} +(0.280090 - 2.21846i) q^{5} +(0.0568237 - 1.62722i) q^{6} +(-1.45652 - 2.52277i) q^{7} +(0.978148 + 0.207912i) q^{8} +(0.348082 - 0.0243402i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q + 54 q^{7} - 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.0568237	−	1.62722i	−0.0328072	−	0.939475i	−0.898520	−	0.438933i	\(-0.855356\pi\)
0.865712	−	0.500542i	\(-0.166866\pi\)
\(5\)	0.280090	−	2.21846i	0.125260	−	0.992124i
							\(7\)	−1.45652	−	2.52277i	−0.550514	−	0.953519i	−0.998237	−	0.0593464i	\(-0.981098\pi\)
0.447723	−	0.894172i	\(-0.352235\pi\)
\(11\)	−0.0569275	−	0.541629i	−0.0171643	−	0.163307i	0.982582	−	0.185829i	\(-0.0594971\pi\)
							−0.999746	+	0.0225219i	\(0.992830\pi\)
\(13\)	−1.45693	−	2.15999i	−0.404080	−	0.599074i	0.570993	−	0.820955i	\(-0.306558\pi\)
							−0.975074	+	0.221881i	\(0.928780\pi\)
\(17\)	−1.13512	+	2.80952i	−0.275307	+	0.681408i	−0.999970	−	0.00769459i	\(-0.997551\pi\)
							0.724664	+	0.689103i	\(0.241995\pi\)
\(19\)	2.00626	+	3.86974i	0.460269	+	0.887780i
							\(23\)	1.42943	+	1.38038i	0.298057	+	0.287830i	0.828616	−	0.559818i	\(-0.189129\pi\)
−0.530559	+	0.847648i	\(0.678018\pi\)
\(29\)	−1.43855	−	3.56053i	−0.267132	−	0.661175i	0.732680	−	0.680573i	\(-0.238269\pi\)
							−0.999812	+	0.0193988i	\(0.993825\pi\)
\(31\)	−4.35806	+	4.84011i	−0.782730	+	0.869310i	−0.994141	−	0.108087i	\(-0.965527\pi\)
							0.211411	+	0.977397i	\(0.432194\pi\)
\(37\)	3.83234	+	2.78436i	0.630034	+	0.457746i	0.856412	−	0.516293i	\(-0.172689\pi\)
							−0.226378	+	0.974039i	\(0.572689\pi\)
\(41\)	2.12321	+	0.608821i	0.331590	+	0.0950818i	0.437301	−	0.899315i	\(-0.355934\pi\)
							−0.105712	+	0.994397i	\(0.533712\pi\)
\(43\)	1.93089	−	10.9506i	0.294458	−	1.66995i	−0.374939	−	0.927050i	\(-0.622336\pi\)
							0.669397	−	0.742905i	\(-0.266553\pi\)
\(47\)	0.875973	+	2.16811i	0.127774	+	0.316251i	0.977488	−	0.210993i	\(-0.0676698\pi\)
							−0.849714	+	0.527244i	\(0.823225\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.b.111.6	✓	600
19.6	even	9	inner	950.2.bc.b.861.20	yes	600
25.16	even	5	inner	950.2.bc.b.491.20	yes	600
475.291	even	45	inner	950.2.bc.b.291.6	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.b.111.6	✓	600	1.1	even	1	trivial
950.2.bc.b.291.6	yes	600	475.291	even	45	inner
950.2.bc.b.491.20	yes	600	25.16	even	5	inner
950.2.bc.b.861.20	yes	600	19.6	even	9	inner