Embedded newform 950.2.r.b.11.12

• Label: 950.2.r.b.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.b.691.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-0.302553 - 0.336019i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(0.565124 - 2.16348i) q^{5} +(-0.302553 + 0.336019i) q^{6} +3.20863 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.292215 - 2.78024i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.302553	−	0.336019i	−0.174679	−	0.194001i	0.649448	−	0.760406i	\(-0.275000\pi\)
−0.824127	+	0.566406i	\(0.808334\pi\)
\(5\)	0.565124	−	2.16348i	0.252731	−	0.967537i
							\(7\)	3.20863			1.21275			0.606374	−	0.795180i	\(-0.292624\pi\)
0.606374	+	0.795180i	\(0.292624\pi\)
\(11\)	1.84927	−	1.34357i	0.557574	−	0.405102i	−0.272996	−	0.962015i	\(-0.588015\pi\)
							0.830570	+	0.556914i	\(0.188015\pi\)
\(13\)	−0.311794	+	2.96652i	−0.0864761	+	0.822765i	0.862211	+	0.506549i	\(0.169079\pi\)
							−0.948687	+	0.316216i	\(0.897588\pi\)
\(17\)	5.38136	+	1.14384i	1.30517	+	0.277423i	0.807461	−	0.589921i	\(-0.200841\pi\)
							0.497710	+	0.867343i	\(0.334174\pi\)
\(19\)	−2.78166	−	3.35595i	−0.638156	−	0.769907i
							\(23\)	3.46764	+	1.54389i	0.723052	+	0.321924i	0.735058	−	0.678004i	\(-0.237155\pi\)
−0.0120057	+	0.999928i	\(0.503822\pi\)
\(29\)	6.95157	−	1.47760i	1.29087	−	0.274384i	0.489214	−	0.872164i	\(-0.337284\pi\)
							0.801660	+	0.597780i	\(0.203950\pi\)
\(31\)	2.76351	+	8.50520i	0.496341	+	1.52758i	0.814857	+	0.579662i	\(0.196815\pi\)
							−0.318517	+	0.947917i	\(0.603185\pi\)
\(37\)	−6.67642	−	4.85070i	−1.09760	−	0.797450i	−0.116931	−	0.993140i	\(-0.537305\pi\)
							−0.980666	+	0.195690i	\(0.937305\pi\)
\(41\)	−6.57711	+	2.92832i	−1.02717	+	0.457326i	−0.849960	−	0.526847i	\(-0.823374\pi\)
							−0.177211	+	0.984173i	\(0.556708\pi\)
\(43\)	5.56803	−	9.64412i	0.849117	−	1.47071i	−0.0328802	−	0.999459i	\(-0.510468\pi\)
							0.881997	−	0.471255i	\(-0.156199\pi\)
\(47\)	−2.19704	+	0.466996i	−0.320471	+	0.0681183i	−0.365339	−	0.930875i	\(-0.619047\pi\)
							0.0448673	+	0.998993i	\(0.485714\pi\)

(See \(a_n\) instead)

Twists

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

    

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