Embedded newform 1000.2.bd.a.109.41

• Label: 1000.2.bd.a.109.41
• 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			109.41
Character	\(\chi\)	\(=\)	1000.109
Dual form			1000.2.bd.a.789.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.894681 + 1.09524i) q^{2} +(0.859509 + 1.03897i) q^{3} +(-0.399093 - 1.95978i) q^{4} +(0.254836 - 2.22150i) q^{5} +(-1.90690 + 0.0118226i) q^{6} +(4.65603 - 1.51284i) q^{7} +(2.50348 + 1.31627i) q^{8} +(0.221445 - 1.16086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 50 q^{7} - 35 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.894681	+	1.09524i	−0.632635	+	0.774450i
							\(3\)	0.859509	+	1.03897i	0.496238	+	0.599849i	0.957863	−	0.287227i	\(-0.0927333\pi\)
−0.461625	+	0.887075i	\(0.652733\pi\)
\(5\)	0.254836	−	2.22150i	0.113966	−	0.993485i
							\(7\)	4.65603	−	1.51284i	1.75981	−	0.571799i	0.762634	−	0.646830i	\(-0.223906\pi\)
0.997181	+	0.0750320i	\(0.0239059\pi\)
\(11\)	−0.350697	+	2.77605i	−0.105739	+	0.837012i	0.846655	+	0.532142i	\(0.178613\pi\)
							−0.952394	+	0.304869i	\(0.901387\pi\)
\(13\)	−0.189832	+	0.995136i	−0.0526500	+	0.276001i	−0.998870	−	0.0475223i	\(-0.984867\pi\)
							0.946220	+	0.323524i	\(0.104867\pi\)
\(17\)	−1.66441	−	6.48244i	−0.403678	−	1.57222i	−0.767686	−	0.640826i	\(-0.778592\pi\)
							0.364008	−	0.931396i	\(-0.381408\pi\)
\(19\)	−1.29260	−	1.06933i	−0.296543	−	0.245322i	0.477239	−	0.878773i	\(-0.341637\pi\)
							−0.773782	+	0.633452i	\(0.781637\pi\)
\(23\)	0.102920	−	0.109598i	0.0214602	−	0.0228528i	−0.718149	−	0.695889i	\(-0.755010\pi\)
							0.739609	+	0.673037i	\(0.235010\pi\)
\(29\)	−2.65084	+	0.166777i	−0.492249	+	0.0309697i	−0.306976	−	0.951717i	\(-0.599317\pi\)
							−0.185273	+	0.982687i	\(0.559317\pi\)
\(31\)	4.75898	−	1.22190i	0.854738	−	0.219459i	0.204196	−	0.978930i	\(-0.434542\pi\)
							0.650542	+	0.759471i	\(0.274542\pi\)
\(37\)	−4.28495	−	2.35567i	−0.704442	−	0.387270i	0.0889052	−	0.996040i	\(-0.471663\pi\)
							−0.793347	+	0.608770i	\(0.791663\pi\)
\(41\)	2.05023	−	1.92529i	0.320192	−	0.300680i	−0.507738	−	0.861511i	\(-0.669518\pi\)
							0.827930	+	0.560831i	\(0.189518\pi\)
\(43\)	0.729437	−	0.529967i	0.111238	−	0.0808192i	−0.530776	−	0.847512i	\(-0.678099\pi\)
							0.642014	+	0.766693i	\(0.278099\pi\)
\(47\)	2.32115	+	5.86254i	0.338574	+	0.855140i	0.994919	+	0.100679i	\(0.0321015\pi\)
							−0.656345	+	0.754461i	\(0.727898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.bd.a.109.41	✓	2960
8.5	even	2	inner	1000.2.bd.a.109.95	yes	2960
125.39	even	50	inner	1000.2.bd.a.789.95	yes	2960
1000.789	even	50	inner	1000.2.bd.a.789.41	yes	2960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.bd.a.109.41	✓	2960	1.1	even	1	trivial
1000.2.bd.a.109.95	yes	2960	8.5	even	2	inner
1000.2.bd.a.789.41	yes	2960	1000.789	even	50	inner
1000.2.bd.a.789.95	yes	2960	125.39	even	50	inner