Embedded newform 950.2.bc.a.61.7

• Label: 950.2.bc.a.61.7
• Level: $950$
• Weight: $2$
• Character: 950.61
• 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			61.7
Character	\(\chi\)	\(=\)	950.61
Dual form			950.2.bc.a.841.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961262 - 0.275637i) q^{2} +(-1.97569 - 0.277665i) q^{3} +(0.848048 - 0.529919i) q^{4} +(-0.0284411 + 2.23589i) q^{5} +(-1.97569 + 0.277665i) q^{6} +(-1.68657 + 2.92123i) q^{7} +(0.669131 - 0.743145i) q^{8} +(0.942462 + 0.270247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 600 q - 42 q^{7} + 75 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961262	−	0.275637i	0.679715	−	0.194905i
							\(3\)	−1.97569	−	0.277665i	−1.14066	−	0.160310i	−0.456559	−	0.889693i	\(-0.650918\pi\)
−0.684105	+	0.729383i	\(0.739807\pi\)
\(5\)	−0.0284411	+	2.23589i	−0.0127192	+	0.999919i
							\(7\)	−1.68657	+	2.92123i	−0.637464	+	1.10412i	0.348524	+	0.937300i	\(0.386683\pi\)
−0.985987	+	0.166820i	\(0.946650\pi\)
\(11\)	−0.713674	−	0.317748i	−0.215181	−	0.0958046i	0.296316	−	0.955090i	\(-0.404242\pi\)
							−0.511496	+	0.859285i	\(0.670909\pi\)
\(13\)	−0.697761	+	0.673820i	−0.193524	+	0.186884i	−0.785107	−	0.619360i	\(-0.787392\pi\)
							0.591583	+	0.806244i	\(0.298503\pi\)
\(17\)	0.248960	−	7.12929i	0.0603817	−	1.72911i	−0.471137	−	0.882060i	\(-0.656156\pi\)
							0.531519	−	0.847046i	\(-0.321621\pi\)
\(19\)	−1.49877	−	4.09313i	−0.343842	−	0.939028i
							\(23\)	−5.04252	−	0.352607i	−1.05144	−	0.0735237i	−0.466436	−	0.884555i	\(-0.654462\pi\)
−0.585003	+	0.811031i	\(0.698907\pi\)
\(29\)	−0.191150	−	5.47381i	−0.0354956	−	1.01646i	−0.878261	−	0.478181i	\(-0.841296\pi\)
							0.842766	−	0.538281i	\(-0.180926\pi\)
\(31\)	−4.78053	−	1.01613i	−0.858609	−	0.182503i	−0.242488	−	0.970154i	\(-0.577963\pi\)
							−0.616121	+	0.787652i	\(0.711297\pi\)
\(37\)	0.952710	+	0.692185i	0.156625	+	0.113794i	0.663337	−	0.748321i	\(-0.269139\pi\)
							−0.506713	+	0.862115i	\(0.669139\pi\)
\(41\)	2.89690	−	5.93953i	0.452420	−	0.927599i	−0.543674	−	0.839297i	\(-0.682967\pi\)
							0.996094	−	0.0883020i	\(-0.0281440\pi\)
\(43\)	−4.08349	+	3.42645i	−0.622726	+	0.522529i	−0.898659	−	0.438648i	\(-0.855458\pi\)
							0.275933	+	0.961177i	\(0.411013\pi\)
\(47\)	0.249656	+	7.14921i	0.0364161	+	1.04282i	0.870800	+	0.491637i	\(0.163601\pi\)
							−0.834384	+	0.551183i	\(0.814177\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bc.a.61.7	✓	600
19.5	even	9	inner	950.2.bc.a.461.19	yes	600
25.16	even	5	inner	950.2.bc.a.441.19	yes	600
475.366	even	45	inner	950.2.bc.a.841.7	yes	600

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bc.a.61.7	✓	600	1.1	even	1	trivial
950.2.bc.a.441.19	yes	600	25.16	even	5	inner
950.2.bc.a.461.19	yes	600	19.5	even	9	inner
950.2.bc.a.841.7	yes	600	475.366	even	45	inner