Embedded newform 950.2.r.a.11.10

• Label: 950.2.r.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.a.691.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.104528 + 0.994522i) q^{2} +(-0.286801 - 0.318525i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.03010 + 1.98466i) q^{5} +(0.286801 - 0.318525i) q^{6} +3.29125 q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.294382 - 2.80086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 25 q^{2} + 25 q^{4} + q^{5} - 36 q^{7} + 50 q^{8} + 37 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.286801	−	0.318525i	−0.165585	−	0.183900i	0.654642	−	0.755939i	\(-0.272819\pi\)
−0.820227	+	0.572039i	\(0.806153\pi\)
\(5\)	−1.03010	+	1.98466i	−0.460676	+	0.887568i
							\(7\)	3.29125			1.24398			0.621988	−	0.783027i	\(-0.286325\pi\)
0.621988	+	0.783027i	\(0.286325\pi\)
\(11\)	−5.28781	+	3.84182i	−1.59433	+	1.15835i	−0.696933	+	0.717137i	\(0.745452\pi\)
							−0.897402	+	0.441215i	\(0.854548\pi\)
\(13\)	−0.203736	+	1.93842i	−0.0565063	+	0.537622i	0.929251	+	0.369449i	\(0.120453\pi\)
							−0.985757	+	0.168173i	\(0.946213\pi\)
\(17\)	2.77118	+	0.589032i	0.672110	+	0.142861i	0.531311	−	0.847177i	\(-0.321700\pi\)
							0.140799	+	0.990038i	\(0.455033\pi\)
\(19\)	−0.257477	+	4.35129i	−0.0590694	+	0.998254i
							\(23\)	−3.82900	−	1.70478i	−0.798402	−	0.355472i	−0.0333534	−	0.999444i	\(-0.510619\pi\)
−0.765049	+	0.643972i	\(0.777285\pi\)
\(29\)	1.59236	−	0.338467i	0.295694	−	0.0628518i	−0.0576758	−	0.998335i	\(-0.518369\pi\)
							0.353370	+	0.935484i	\(0.385036\pi\)
\(31\)	1.94103	+	5.97386i	0.348618	+	1.07294i	0.959618	+	0.281306i	\(0.0907677\pi\)
							−0.611000	+	0.791631i	\(0.709232\pi\)
\(37\)	−0.242164	−	0.175942i	−0.0398114	−	0.0289247i	0.567702	−	0.823234i	\(-0.307833\pi\)
							−0.607513	+	0.794310i	\(0.707833\pi\)
\(41\)	−11.3170	+	5.03867i	−1.76743	+	0.786908i	−0.780618	+	0.625009i	\(0.785095\pi\)
							−0.986807	+	0.161900i	\(0.948238\pi\)
\(43\)	−1.17009	+	2.02666i	−0.178437	+	0.309063i	−0.941346	−	0.337444i	\(-0.890438\pi\)
							0.762908	+	0.646507i	\(0.223771\pi\)
\(47\)	−2.99839	+	0.637327i	−0.437360	+	0.0929637i	−0.421329	−	0.906908i	\(-0.638436\pi\)
							−0.0160306	+	0.999872i	\(0.505103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.a.11.10	✓	200
19.7	even	3	inner	950.2.r.a.311.16	yes	200
25.16	even	5	inner	950.2.r.a.391.16	yes	200
475.216	even	15	inner	950.2.r.a.691.10	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.a.11.10	✓	200	1.1	even	1	trivial
950.2.r.a.311.16	yes	200	19.7	even	3	inner
950.2.r.a.391.16	yes	200	25.16	even	5	inner
950.2.r.a.691.10	yes	200	475.216	even	15	inner