Embedded newform 950.2.bi.a.117.41

• Label: 950.2.bi.a.117.41
• Level: $950$
• Weight: $2$
• Character: 950.117
• Analytic conductor: $7.586$
• Analytic rank: $0$
• Dimension: $2400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.bi (of order \(180\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.58578819202\)
Analytic rank:	\(0\)
Dimension:	\(2400\)
Relative dimension:	\(50\) over \(\Q(\zeta_{180})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{180}]$

Embedding invariants

Embedding label			117.41
Character	\(\chi\)	\(=\)	950.117
Dual form			950.2.bi.a.203.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.390731 + 0.920505i) q^{2} +(0.783123 - 0.152224i) q^{3} +(-0.694658 + 0.719340i) q^{4} +(0.394023 + 2.20108i) q^{5} +(0.446113 + 0.661390i) q^{6} +(0.314348 - 1.17316i) q^{7} +(-0.933580 - 0.358368i) q^{8} +(-2.19144 + 0.885400i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2400 q + 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{13}{20}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.390731	+	0.920505i	0.276289	+	0.650895i
							\(3\)	0.783123	−	0.152224i	0.452136	−	0.0878863i	0.0403880	−	0.999184i	\(-0.487141\pi\)
0.411748	+	0.911298i	\(0.364918\pi\)
\(5\)	0.394023	+	2.20108i	0.176212	+	0.984352i
							\(7\)	0.314348	−	1.17316i	0.118812	−	0.443413i	−0.880732	−	0.473616i	\(-0.842949\pi\)
0.999544	+	0.0302025i	\(0.00961520\pi\)
\(11\)	−4.55956	−	0.969164i	−1.37476	−	0.292214i	−0.539460	−	0.842011i	\(-0.681372\pi\)
							−0.835298	+	0.549798i	\(0.814705\pi\)
\(13\)	−1.74798	+	0.214625i	−0.484802	+	0.0595262i	−0.360580	−	0.932728i	\(-0.617421\pi\)
							−0.124222	+	0.992254i	\(0.539643\pi\)
\(17\)	−2.38158	+	3.96361i	−0.577617	+	0.961317i	0.420912	+	0.907102i	\(0.361710\pi\)
							−0.998529	+	0.0542152i	\(0.982734\pi\)
\(19\)	2.35207	+	3.66984i	0.539602	+	0.841920i
							\(23\)	−0.680612	+	2.22618i	−0.141917	+	0.464191i	−0.998671	−	0.0515347i	\(-0.983589\pi\)
0.856754	+	0.515726i	\(0.172478\pi\)
\(29\)	0.663078	−	2.65946i	0.123130	−	0.493849i	−0.876785	−	0.480882i	\(-0.840317\pi\)
							0.999916	−	0.0129676i	\(-0.00412784\pi\)
\(31\)	−4.97220	−	0.522599i	−0.893034	−	0.0938616i	−0.353116	−	0.935580i	\(-0.614878\pi\)
							−0.539918	+	0.841718i	\(0.681545\pi\)
\(37\)	2.41307	+	4.73593i	0.396707	+	0.778581i	0.999818	−	0.0190759i	\(-0.00607243\pi\)
							−0.603111	+	0.797657i	\(0.706072\pi\)
\(41\)	10.8629	−	0.379341i	1.69650	−	0.0592431i	0.829797	−	0.558066i	\(-0.188456\pi\)
							0.866704	+	0.498822i	\(0.166234\pi\)
\(43\)	−2.15868	−	3.08291i	−0.329195	−	0.470139i	0.620101	−	0.784522i	\(-0.287091\pi\)
							−0.949296	+	0.314383i	\(0.898203\pi\)
\(47\)	5.85467	−	3.51784i	0.853992	−	0.513130i	−0.0195894	−	0.999808i	\(-0.506236\pi\)
							0.873581	+	0.486678i	\(0.161791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bi.a.117.41	yes	2400
19.13	odd	18	inner	950.2.bi.a.317.10	yes	2400
25.3	odd	20	inner	950.2.bi.a.3.10	✓	2400
475.203	even	180	inner	950.2.bi.a.203.41	yes	2400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bi.a.3.10	✓	2400	25.3	odd	20	inner
950.2.bi.a.117.41	yes	2400	1.1	even	1	trivial
950.2.bi.a.203.41	yes	2400	475.203	even	180	inner
950.2.bi.a.317.10	yes	2400	19.13	odd	18	inner