Embedded newform 889.2.bm.a.2.3

• Label: 889.2.bm.a.2.3
• Level: $889$
• Weight: $2$
• Character: 889.2
• Analytic conductor: $7.099$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 889 = 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	889.bm (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.09870073969\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(84\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	889.2
Dual form			889.2.bm.a.445.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.24110 + 1.52796i) q^{2} +(-2.03600 + 0.628021i) q^{3} +(1.95721 - 4.98688i) q^{4} +(0.0276021 + 0.0703289i) q^{5} +(3.60329 - 4.51838i) q^{6} +(2.62302 - 0.346077i) q^{7} +(2.02631 + 8.87783i) q^{8} +(1.27215 - 0.867338i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 7 q^{3} + 84 q^{4} - 9 q^{5} - 28 q^{6} - 19 q^{7} - 30 q^{8} + 67 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/889\mathbb{Z}\right)^\times\).

\(n\)	\(255\)	\(638\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.24110	+	1.52796i	−1.58470	+	1.08043i	−0.632672	+	0.774420i	\(0.718042\pi\)
							−0.952028	+	0.306010i	\(0.901006\pi\)
\(3\)	−2.03600	+	0.628021i	−1.17548	+	0.362588i	−0.820148	−	0.572151i	\(-0.806109\pi\)
							−0.355334	+	0.934739i	\(0.615633\pi\)
\(5\)	0.0276021	+	0.0703289i	0.0123440	+	0.0314521i	0.936913	−	0.349563i	\(-0.113670\pi\)
							−0.924569	+	0.381015i	\(0.875575\pi\)
\(7\)	2.62302	−	0.346077i	0.991408	−	0.130805i
							\(11\)	2.90182	+	1.97843i	0.874933	+	0.596519i	0.915348	−	0.402664i	\(-0.131916\pi\)
−0.0404149	+	0.999183i	\(0.512868\pi\)
\(13\)	0.864665	−	1.08426i	0.239815	−	0.300718i	−0.647330	−	0.762210i	\(-0.724114\pi\)
							0.887144	+	0.461492i	\(0.152686\pi\)
\(17\)	−2.09075	−	0.315130i	−0.507081	−	0.0764302i	−0.109481	−	0.993989i	\(-0.534919\pi\)
							−0.397600	+	0.917559i	\(0.630157\pi\)
\(19\)	1.40588	−	2.43506i	0.322532	−	0.558641i	−0.658478	−	0.752600i	\(-0.728799\pi\)
							0.981010	+	0.193959i	\(0.0621328\pi\)
\(23\)	−3.96672	+	2.70447i	−0.827119	+	0.563920i	−0.901236	−	0.433329i	\(-0.857339\pi\)
							0.0741167	+	0.997250i	\(0.476386\pi\)
\(29\)	0.753991	−	3.30345i	0.140013	−	0.613436i	−0.855417	−	0.517940i	\(-0.826699\pi\)
							0.995430	−	0.0954960i	\(-0.0304437\pi\)
\(31\)	2.45762	−	6.26192i	0.441402	−	1.12467i	−0.521123	−	0.853481i	\(-0.674487\pi\)
							0.962525	−	0.271192i	\(-0.0874179\pi\)
\(37\)	−4.43735	+	7.68571i	−0.729495	+	1.26352i	0.227601	+	0.973754i	\(0.426912\pi\)
							−0.957097	+	0.289769i	\(0.906422\pi\)
\(41\)	2.61489	−	3.27897i	0.408377	−	0.512089i	−0.534528	−	0.845151i	\(-0.679510\pi\)
							0.942905	+	0.333062i	\(0.108082\pi\)
\(43\)	0.0584156	−	0.255936i	0.00890830	−	0.0390298i	−0.970278	−	0.241991i	\(-0.922199\pi\)
							0.979187	+	0.202961i	\(0.0650566\pi\)
\(47\)	−3.50769	+	0.528700i	−0.511650	+	0.0771188i	−0.399793	−	0.916606i	\(-0.630918\pi\)
							−0.111857	+	0.993724i	\(0.535680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	889.2.bm.a.2.3	✓	1008
7.4	even	3	inner	889.2.bm.a.256.3	yes	1008
127.64	even	7	inner	889.2.bm.a.191.3	yes	1008
889.445	even	21	inner	889.2.bm.a.445.3	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.bm.a.2.3	✓	1008	1.1	even	1	trivial
889.2.bm.a.191.3	yes	1008	127.64	even	7	inner
889.2.bm.a.256.3	yes	1008	7.4	even	3	inner
889.2.bm.a.445.3	yes	1008	889.445	even	21	inner