Embedded newform 950.2.bc.a.111.13

• Label: 950.2.bc.a.111.13
• 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			111.13
Character	\(\chi\)	\(=\)	950.111
Dual form			950.2.bc.a.291.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.997564 - 0.0697565i) q^{2} +(-0.00161455 - 0.0462348i) q^{3} +(0.990268 + 0.139173i) q^{4} +(-1.32595 - 1.80052i) q^{5} +(-0.00161455 + 0.0462348i) q^{6} +(-2.07837 - 3.59984i) q^{7} +(-0.978148 - 0.207912i) q^{8} +(2.99056 - 0.209120i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

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
							\(3\)	−0.00161455	−	0.0462348i	−0.000932164	−	0.0266937i	0.998836	−	0.0482432i	\(-0.0153623\pi\)
−0.999768	+	0.0215495i	\(0.993140\pi\)
\(5\)	−1.32595	−	1.80052i	−0.592981	−	0.805216i
							\(7\)	−2.07837	−	3.59984i	−0.785549	−	1.36061i	−0.928671	−	0.370905i	\(-0.879048\pi\)
0.143122	−	0.989705i	\(-0.454286\pi\)
\(11\)	0.430255	+	4.09360i	0.129727	+	1.23427i	0.844748	+	0.535165i	\(0.179751\pi\)
							−0.715021	+	0.699103i	\(0.753583\pi\)
\(13\)	1.13626	+	1.68458i	0.315142	+	0.467217i	0.953015	−	0.302925i	\(-0.0979631\pi\)
							−0.637872	+	0.770142i	\(0.720185\pi\)
\(17\)	2.98702	−	7.39313i	0.724459	−	1.79310i	0.125861	−	0.992048i	\(-0.459831\pi\)
							0.598597	−	0.801050i	\(-0.295725\pi\)
\(19\)	4.08227	−	1.52808i	0.936538	−	0.350566i
							\(23\)	−0.935123	−	0.903038i	−0.194987	−	0.188296i	0.590755	−	0.806851i	\(-0.298830\pi\)
−0.785742	+	0.618554i	\(0.787719\pi\)
\(29\)	−2.93254	−	7.25829i	−0.544559	−	1.34783i	−0.909074	−	0.416634i	\(-0.863210\pi\)
							0.364515	−	0.931197i	\(-0.381235\pi\)
\(31\)	−5.05063	+	5.60929i	−0.907120	+	1.00746i	0.0928107	+	0.995684i	\(0.470415\pi\)
							−0.999931	+	0.0117750i	\(0.996252\pi\)
\(37\)	−7.44656	−	5.41024i	−1.22421	−	0.889439i	−0.227765	−	0.973716i	\(-0.573142\pi\)
							−0.996442	+	0.0842777i	\(0.973142\pi\)
\(41\)	−7.23292	−	2.07401i	−1.12959	−	0.323905i	−0.341755	−	0.939789i	\(-0.611021\pi\)
							−0.787837	+	0.615884i	\(0.788799\pi\)
\(43\)	1.29213	−	7.32804i	0.197048	−	1.11752i	−0.712423	−	0.701750i	\(-0.752402\pi\)
							0.909472	−	0.415766i	\(-0.136487\pi\)
\(47\)	4.08017	+	10.0988i	0.595153	+	1.47306i	0.860778	+	0.508980i	\(0.169977\pi\)
							−0.265625	+	0.964076i	\(0.585578\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.111.13	✓	600
19.6	even	9	inner	950.2.bc.a.861.13	yes	600
25.16	even	5	inner	950.2.bc.a.491.13	yes	600
475.291	even	45	inner	950.2.bc.a.291.13	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.111.13	✓	600	1.1	even	1	trivial
950.2.bc.a.291.13	yes	600	475.291	even	45	inner
950.2.bc.a.491.13	yes	600	25.16	even	5	inner
950.2.bc.a.861.13	yes	600	19.6	even	9	inner