Embedded newform 950.2.r.b.11.14

• Label: 950.2.r.b.11.14
• 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			11.14
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(0.343520 + 0.381518i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.76429 + 1.37378i) q^{5} +(0.343520 - 0.381518i) q^{6} -3.26090 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.286036 - 2.72145i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	0.343520	+	0.381518i	0.198332	+	0.220270i	0.834104	−	0.551607i	\(-0.185985\pi\)
−0.635773	+	0.771876i	\(0.719318\pi\)
\(5\)	1.76429	+	1.37378i	0.789015	+	0.614374i
							\(7\)	−3.26090			−1.23250			−0.616252	−	0.787549i	\(-0.711350\pi\)
−0.616252	+	0.787549i	\(0.711350\pi\)
\(11\)	4.07390	−	2.95986i	1.22833	−	0.892432i	0.231564	−	0.972820i	\(-0.425616\pi\)
							0.996764	+	0.0803872i	\(0.0256157\pi\)
\(13\)	−0.0864972	+	0.822966i	−0.0239900	+	0.228250i	0.975957	+	0.217963i	\(0.0699411\pi\)
							−0.999947	+	0.0102869i	\(0.996726\pi\)
\(17\)	−4.74964	−	1.00957i	−1.15196	−	0.244856i	−0.407934	−	0.913011i	\(-0.633751\pi\)
							−0.744022	+	0.668155i	\(0.767084\pi\)
\(19\)	2.86587	−	3.28432i	0.657476	−	0.753476i
							\(23\)	4.70129	+	2.09315i	0.980287	+	0.436452i	0.833381	−	0.552699i	\(-0.186402\pi\)
0.146906	+	0.989150i	\(0.453069\pi\)
\(29\)	6.57295	−	1.39712i	1.22057	−	0.259439i	0.447802	−	0.894133i	\(-0.352207\pi\)
							0.772764	+	0.634694i	\(0.218874\pi\)
\(31\)	−0.568204	−	1.74875i	−0.102053	−	0.314085i	0.886975	−	0.461818i	\(-0.152803\pi\)
							−0.989027	+	0.147732i	\(0.952803\pi\)
\(37\)	−3.46766	−	2.51940i	−0.570080	−	0.414187i	0.265055	−	0.964233i	\(-0.414610\pi\)
							−0.835134	+	0.550046i	\(0.814610\pi\)
\(41\)	7.00447	−	3.11859i	1.09391	−	0.487042i	0.221178	−	0.975234i	\(-0.429010\pi\)
							0.872736	+	0.488192i	\(0.162343\pi\)
\(43\)	−0.0448371	+	0.0776602i	−0.00683760	+	0.0118431i	−0.869424	−	0.494067i	\(-0.835510\pi\)
							0.862586	+	0.505910i	\(0.168843\pi\)
\(47\)	10.7748	−	2.29026i	1.57167	−	0.334069i	0.662035	−	0.749473i	\(-0.269693\pi\)
							0.909637	+	0.415404i	\(0.136360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.14	✓	200
19.7	even	3	inner	950.2.r.b.311.12	yes	200
25.16	even	5	inner	950.2.r.b.391.12	yes	200
475.216	even	15	inner	950.2.r.b.691.14	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.14	✓	200	1.1	even	1	trivial
950.2.r.b.311.12	yes	200	19.7	even	3	inner
950.2.r.b.391.12	yes	200	25.16	even	5	inner
950.2.r.b.691.14	yes	200	475.216	even	15	inner