Embedded newform 1000.2.bd.a.109.138

• Label: 1000.2.bd.a.109.138
• Level: $1000$
• Weight: $2$
• Character: 1000.109
• Analytic conductor: $7.985$
• Analytic rank: $0$
• Dimension: $2960$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.bd (of order \(50\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.98504020213\)
Analytic rank:	\(0\)
Dimension:	\(2960\)
Relative dimension:	\(148\) over \(\Q(\zeta_{50})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{50}]$

Embedding invariants

Embedding label			109.138
Character	\(\chi\)	\(=\)	1000.109
Dual form			1000.2.bd.a.789.138

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35919 - 0.390640i) q^{2} +(0.290927 + 0.351670i) q^{3} +(1.69480 - 1.06191i) q^{4} +(1.00978 + 1.99508i) q^{5} +(0.532802 + 0.364339i) q^{6} +(3.27785 - 1.06504i) q^{7} +(1.88873 - 2.10539i) q^{8} +(0.523110 - 2.74224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 50 q^{7} - 35 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\)	\(377\)	\(501\)	\(751\)
\(\chi(n)\)	\(e\left(\frac{27}{50}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35919	−	0.390640i	0.961093	−	0.276224i
							\(3\)	0.290927	+	0.351670i	0.167967	+	0.203037i	0.847780	−	0.530349i	\(-0.177939\pi\)
−0.679813	+	0.733386i	\(0.737939\pi\)
\(5\)	1.00978	+	1.99508i	0.451586	+	0.892227i
							\(7\)	3.27785	−	1.06504i	1.23891	−	0.402547i	0.384977	−	0.922926i	\(-0.374209\pi\)
0.853935	+	0.520379i	\(0.174209\pi\)
\(11\)	−0.505692	+	4.00297i	−0.152472	+	1.20694i	0.711238	+	0.702951i	\(0.248135\pi\)
							−0.863710	+	0.503989i	\(0.831865\pi\)
\(13\)	−0.348849	+	1.82873i	−0.0967533	+	0.507199i	0.900610	+	0.434628i	\(0.143120\pi\)
							−0.997363	+	0.0725705i	\(0.976880\pi\)
\(17\)	−0.351077	−	1.36735i	−0.0851487	−	0.331632i	0.911710	−	0.410835i	\(-0.134763\pi\)
							−0.996859	+	0.0792024i	\(0.974763\pi\)
\(19\)	−4.46290	−	3.69203i	−1.02386	−	0.847010i	−0.0355335	−	0.999368i	\(-0.511313\pi\)
							−0.988325	+	0.152359i	\(0.951313\pi\)
\(23\)	−2.55506	+	2.72086i	−0.532766	+	0.567338i	−0.937894	−	0.346922i	\(-0.887227\pi\)
							0.405128	+	0.914260i	\(0.367227\pi\)
\(29\)	−8.15530	+	0.513088i	−1.51440	+	0.0952780i	−0.798105	−	0.602519i	\(-0.794164\pi\)
							−0.716296	+	0.697797i	\(0.754164\pi\)
\(31\)	5.40358	−	1.38740i	0.970511	−	0.249185i	0.270027	−	0.962853i	\(-0.412967\pi\)
							0.700484	+	0.713668i	\(0.252967\pi\)
\(37\)	4.18774	+	2.30223i	0.688460	+	0.378484i	0.787258	−	0.616624i	\(-0.211500\pi\)
							−0.0987985	+	0.995107i	\(0.531500\pi\)
\(41\)	−2.71681	+	2.55125i	−0.424294	+	0.398439i	−0.867064	−	0.498197i	\(-0.833996\pi\)
							0.442770	+	0.896635i	\(0.353996\pi\)
\(43\)	−0.804593	+	0.584571i	−0.122699	+	0.0891463i	−0.647442	−	0.762114i	\(-0.724161\pi\)
							0.524743	+	0.851261i	\(0.324161\pi\)
\(47\)	−2.83510	−	7.16065i	−0.413542	−	1.04449i	−0.976021	−	0.217676i	\(-0.930152\pi\)
							0.562479	−	0.826811i	\(-0.309848\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.bd.a.109.138	yes	2960
8.5	even	2	inner	1000.2.bd.a.109.25	✓	2960
125.39	even	50	inner	1000.2.bd.a.789.25	yes	2960
1000.789	even	50	inner	1000.2.bd.a.789.138	yes	2960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.bd.a.109.25	✓	2960	8.5	even	2	inner
1000.2.bd.a.109.138	yes	2960	1.1	even	1	trivial
1000.2.bd.a.789.25	yes	2960	125.39	even	50	inner
1000.2.bd.a.789.138	yes	2960	1000.789	even	50	inner