Embedded newform 950.2.r.b.11.1

• Label: 950.2.r.b.11.1
• 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			11.1
Character	\(\chi\)	\(=\)	950.11
Dual form			950.2.r.b.691.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104528 - 0.994522i) q^{2} +(-1.98778 - 2.20765i) q^{3} +(-0.978148 + 0.207912i) q^{4} +(-1.48151 + 1.67485i) q^{5} +(-1.98778 + 2.20765i) q^{6} +5.05644 q^{7} +(0.309017 + 0.951057i) q^{8} +(-0.608880 + 5.79310i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q + 25 q^{2} + 25 q^{4} + q^{5} + 28 q^{7} - 50 q^{8} + 13 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.98778	−	2.20765i	−1.14765	−	1.27459i	−0.956079	−	0.293111i	\(-0.905310\pi\)
−0.191568	−	0.981479i	\(-0.561357\pi\)
\(5\)	−1.48151	+	1.67485i	−0.662553	+	0.749015i
							\(7\)	5.05644			1.91115			0.955577	−	0.294742i	\(-0.0952338\pi\)
0.955577	+	0.294742i	\(0.0952338\pi\)
\(11\)	−2.01448	+	1.46361i	−0.607390	+	0.441294i	−0.848494	−	0.529205i	\(-0.822490\pi\)
							0.241105	+	0.970499i	\(0.422490\pi\)
\(13\)	0.637273	−	6.06325i	0.176748	−	1.68164i	−0.442750	−	0.896645i	\(-0.645997\pi\)
							0.619498	−	0.784998i	\(-0.287336\pi\)
\(17\)	5.27781	+	1.12183i	1.28006	+	0.272085i	0.797254	−	0.603644i	\(-0.206285\pi\)
							0.482804	+	0.875729i	\(0.339618\pi\)
\(19\)	−3.14982	−	3.01308i	−0.722617	−	0.691248i
							\(23\)	−3.21723	−	1.43240i	−0.670838	−	0.298677i	0.0429097	−	0.999079i	\(-0.486337\pi\)
−0.713748	+	0.700402i	\(0.753004\pi\)
\(29\)	6.00346	−	1.27608i	1.11481	−	0.236961i	0.386558	−	0.922265i	\(-0.373664\pi\)
							0.728257	+	0.685304i	\(0.240331\pi\)
\(31\)	−0.800574	−	2.46391i	−0.143787	−	0.442532i	0.853066	−	0.521803i	\(-0.174741\pi\)
							−0.996853	+	0.0792716i	\(0.974741\pi\)
\(37\)	−2.75306	−	2.00021i	−0.452600	−	0.328833i	0.338022	−	0.941138i	\(-0.390242\pi\)
							−0.790621	+	0.612305i	\(0.790242\pi\)
\(41\)	4.16268	−	1.85335i	0.650102	−	0.289444i	−0.0550679	−	0.998483i	\(-0.517538\pi\)
							0.705170	+	0.709039i	\(0.250871\pi\)
\(43\)	−3.74809	+	6.49189i	−0.571579	+	0.990004i	0.424825	+	0.905275i	\(0.360336\pi\)
							−0.996404	+	0.0847285i	\(0.972998\pi\)
\(47\)	−0.0832821	+	0.0177022i	−0.0121479	+	0.00258213i	−0.213982	−	0.976838i	\(-0.568643\pi\)
							0.201834	+	0.979420i	\(0.435310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.r.b.11.1	✓	200
19.7	even	3	inner	950.2.r.b.311.25	yes	200
25.16	even	5	inner	950.2.r.b.391.25	yes	200
475.216	even	15	inner	950.2.r.b.691.1	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.r.b.11.1	✓	200	1.1	even	1	trivial
950.2.r.b.311.25	yes	200	19.7	even	3	inner
950.2.r.b.391.25	yes	200	25.16	even	5	inner
950.2.r.b.691.1	yes	200	475.216	even	15	inner