Embedded newform 950.2.r.a.11.12

• Label: 950.2.r.a.11.12
• Level: $950$
• Weight: $2$
• Character: 950.11
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.r (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(25\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			11.12
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(-0.261407 - 0.290322i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.28550 + 1.82962i) q^{5} +(0.261407 - 0.290322i) q^{6} -0.691099 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.297632 - 2.83178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.104528	+	0.994522i	0.0739128	+	0.703233i
							\(3\)	−0.261407	−	0.290322i	−0.150923	−	0.167617i	0.662942	−	0.748670i	\(-0.269307\pi\)
−0.813866	+	0.581053i	\(0.802641\pi\)
\(5\)	1.28550	+	1.82962i	0.574892	+	0.818230i
							\(7\)	−0.691099			−0.261211			−0.130605	−	0.991434i	\(-0.541692\pi\)
−0.130605	+	0.991434i	\(0.541692\pi\)
\(11\)	1.91826	−	1.39370i	0.578377	−	0.420215i	−0.259762	−	0.965673i	\(-0.583644\pi\)
							0.838139	+	0.545457i	\(0.183644\pi\)
\(13\)	−0.701321	+	6.67263i	−0.194512	+	1.85065i	0.267222	+	0.963635i	\(0.413894\pi\)
							−0.461733	+	0.887019i	\(0.652772\pi\)
\(17\)	5.68545	+	1.20848i	1.37892	+	0.293099i	0.836942	−	0.547291i	\(-0.184341\pi\)
							0.541981	+	0.840390i	\(0.317674\pi\)
\(19\)	1.80627	−	3.96704i	0.414388	−	0.910100i
							\(23\)	2.71006	+	1.20660i	0.565087	+	0.251593i	0.669338	−	0.742958i	\(-0.266578\pi\)
−0.104251	+	0.994551i	\(0.533244\pi\)
\(29\)	−9.62798	+	2.04649i	−1.78787	+	0.380024i	−0.978332	−	0.207043i	\(-0.933616\pi\)
							−0.809539	+	0.587066i	\(0.800283\pi\)
\(31\)	2.64086	+	8.12774i	0.474313	+	1.45979i	0.846882	+	0.531781i	\(0.178477\pi\)
							−0.372569	+	0.928004i	\(0.621523\pi\)
\(37\)	7.08276	+	5.14593i	1.16440	+	0.845985i	0.990328	−	0.138748i	\(-0.0443077\pi\)
							0.174071	+	0.984733i	\(0.444308\pi\)
\(41\)	−1.11122	+	0.494746i	−0.173543	+	0.0772664i	−0.491669	−	0.870782i	\(-0.663613\pi\)
							0.318126	+	0.948048i	\(0.396946\pi\)
\(43\)	4.52624	−	7.83968i	0.690246	−	1.19554i	−0.281512	−	0.959558i	\(-0.590836\pi\)
							0.971757	−	0.235983i	\(-0.0758308\pi\)
\(47\)	−2.53516	+	0.538864i	−0.369791	+	0.0786014i	−0.389059	−	0.921213i	\(-0.627200\pi\)
							0.0192682	+	0.999814i	\(0.493866\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.12	✓	200
19.7	even	3	inner	950.2.r.a.311.14	yes	200
25.16	even	5	inner	950.2.r.a.391.14	yes	200
475.216	even	15	inner	950.2.r.a.691.12	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.12	✓	200	1.1	even	1	trivial
950.2.r.a.311.14	yes	200	19.7	even	3	inner
950.2.r.a.391.14	yes	200	25.16	even	5	inner
950.2.r.a.691.12	yes	200	475.216	even	15	inner