Embedded newform 950.2.bi.a.117.2

• Label: 950.2.bi.a.117.2
• 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			117.2
Character	\(\chi\)	\(=\)	950.117
Dual form			950.2.bi.a.203.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.390731 - 0.920505i) q^{2} +(-2.94119 + 0.571709i) q^{3} +(-0.694658 + 0.719340i) q^{4} +(1.38921 + 1.75217i) q^{5} +(1.67547 + 2.48399i) q^{6} +(-1.10969 + 4.14141i) q^{7} +(0.933580 + 0.358368i) q^{8} +(5.54218 - 2.23918i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2400 q + 12 q^{7}+O(q^{10}) \)

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
							\(3\)	−2.94119	+	0.571709i	−1.69809	+	0.330076i	−0.944810	−	0.327618i	\(-0.893754\pi\)
−0.753285	+	0.657694i	\(0.771532\pi\)
\(5\)	1.38921	+	1.75217i	0.621275	+	0.783592i
							\(7\)	−1.10969	+	4.14141i	−0.419423	+	1.56531i	0.356386	+	0.934339i	\(0.384009\pi\)
−0.775809	+	0.630968i	\(0.782658\pi\)
\(11\)	0.841728	+	0.178915i	0.253790	+	0.0539448i	0.333050	−	0.942909i	\(-0.391922\pi\)
							−0.0792595	+	0.996854i	\(0.525256\pi\)
\(13\)	4.42518	−	0.543344i	1.22733	−	0.150697i	0.517878	−	0.855455i	\(-0.326722\pi\)
							0.709448	+	0.704758i	\(0.248944\pi\)
\(17\)	−0.0586886	+	0.0976743i	−0.0142341	+	0.0236895i	−0.864203	−	0.503144i	\(-0.832176\pi\)
							0.849968	+	0.526834i	\(0.176621\pi\)
\(19\)	4.23582	+	1.02850i	0.971764	+	0.235955i
							\(23\)	−1.53937	+	5.03504i	−0.320980	+	1.04988i	0.638858	+	0.769325i	\(0.279407\pi\)
−0.959838	+	0.280554i	\(0.909482\pi\)
\(29\)	0.513807	−	2.06077i	0.0954116	−	0.382675i	−0.903537	−	0.428509i	\(-0.859039\pi\)
							0.998949	+	0.0458340i	\(0.0145945\pi\)
\(31\)	1.23245	+	0.129536i	0.221355	+	0.0232653i	0.214557	−	0.976712i	\(-0.431169\pi\)
							0.00679826	+	0.999977i	\(0.497836\pi\)
\(37\)	−2.64727	−	5.19557i	−0.435209	−	0.854146i	−0.999590	−	0.0286317i	\(-0.990885\pi\)
							0.564381	−	0.825515i	\(-0.309115\pi\)
\(41\)	5.05679	−	0.176587i	0.789738	−	0.0275783i	0.362809	−	0.931863i	\(-0.381818\pi\)
							0.426929	+	0.904285i	\(0.359595\pi\)
\(43\)	5.46023	+	7.79801i	0.832677	+	1.18919i	0.979783	+	0.200065i	\(0.0641153\pi\)
							−0.147106	+	0.989121i	\(0.546996\pi\)
\(47\)	3.71415	−	2.23169i	0.541765	−	0.325525i	−0.217761	−	0.976002i	\(-0.569875\pi\)
							0.759526	+	0.650477i	\(0.225431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.bi.a.117.2	yes	2400
19.13	odd	18	inner	950.2.bi.a.317.49	yes	2400
25.3	odd	20	inner	950.2.bi.a.3.49	✓	2400
475.203	even	180	inner	950.2.bi.a.203.2	yes	2400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.bi.a.3.49	✓	2400	25.3	odd	20	inner
950.2.bi.a.117.2	yes	2400	1.1	even	1	trivial
950.2.bi.a.203.2	yes	2400	475.203	even	180	inner
950.2.bi.a.317.49	yes	2400	19.13	odd	18	inner