Embedded newform 950.2.r.b.11.10

• Label: 950.2.r.b.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.b.691.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-0.475190 - 0.527752i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(1.37456 - 1.76369i) q^{5} +(-0.475190 + 0.527752i) q^{6} -4.63378 q^{7} +(0.309017 + 0.951057i) q^{8} +(0.260869 - 2.48200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.475190	−	0.527752i	−0.274351	−	0.304698i	0.590185	−	0.807268i	\(-0.299055\pi\)
−0.864537	+	0.502570i	\(0.832388\pi\)
\(5\)	1.37456	−	1.76369i	0.614720	−	0.788745i
							\(7\)	−4.63378			−1.75140			−0.875702	−	0.482851i	\(-0.839601\pi\)
−0.875702	+	0.482851i	\(0.839601\pi\)
\(11\)	−4.16391	+	3.02526i	−1.25547	+	0.912149i	−0.998526	−	0.0542781i	\(-0.982714\pi\)
							−0.256940	+	0.966427i	\(0.582714\pi\)
\(13\)	−0.597998	+	5.68957i	−0.165855	+	1.57800i	0.522507	+	0.852635i	\(0.324997\pi\)
							−0.688362	+	0.725367i	\(0.741670\pi\)
\(17\)	0.705347	+	0.149926i	0.171072	+	0.0363624i	0.292651	−	0.956219i	\(-0.405463\pi\)
							−0.121579	+	0.992582i	\(0.538796\pi\)
\(19\)	4.27003	+	0.875705i	0.979612	+	0.200901i
							\(23\)	6.59092	+	2.93447i	1.37430	+	0.611879i	0.955173	−	0.296048i	\(-0.0956688\pi\)
0.419129	+	0.907927i	\(0.362336\pi\)
\(29\)	−5.33806	+	1.13464i	−0.991253	+	0.210697i	−0.674872	−	0.737935i	\(-0.735801\pi\)
							−0.316381	+	0.948632i	\(0.602468\pi\)
\(31\)	0.176718	+	0.543883i	0.0317395	+	0.0976842i	0.965671	−	0.259767i	\(-0.0836458\pi\)
							−0.933932	+	0.357451i	\(0.883646\pi\)
\(37\)	0.194032	+	0.140972i	0.0318986	+	0.0231757i	0.603620	−	0.797272i	\(-0.293724\pi\)
							−0.571722	+	0.820448i	\(0.693724\pi\)
\(41\)	5.72055	−	2.54695i	0.893400	−	0.397767i	0.0919040	−	0.995768i	\(-0.470705\pi\)
							0.801496	+	0.598001i	\(0.204038\pi\)
\(43\)	0.196940	−	0.341110i	0.0300330	−	0.0520187i	−0.850618	−	0.525784i	\(-0.823772\pi\)
							0.880651	+	0.473765i	\(0.157105\pi\)
\(47\)	−7.84079	+	1.66661i	−1.14370	+	0.243100i	−0.740528	−	0.672025i	\(-0.765425\pi\)
							−0.403169	+	0.915126i	\(0.632091\pi\)

(See \(a_n\) instead)

Twists

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

    

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