Embedded newform 845.2.t.e.657.3

• Label: 845.2.t.e.657.3
• Level: $845$
• Weight: $2$
• Character: 845.657
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.t (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 26*x^18 + 279*x^16 + 1604*x^14 + 5353*x^12 + 10466*x^10 + 11441*x^8 + 6176*x^6 + 1263*x^4 + 78*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			657.3
Root			\(-0.131303i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.657
Dual form			845.2.t.e.418.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.113711 - 0.0656513i) q^{2} +(-0.0890070 - 0.332179i) q^{3} +(-0.991380 - 1.71712i) q^{4} +(-2.08297 - 0.813169i) q^{5} +(-0.0116869 + 0.0436159i) q^{6} +(-1.39069 - 2.40874i) q^{7} +0.522947i q^{8} +(2.49566 - 1.44087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.113711	−	0.0656513i	−0.0804061	−	0.0464225i	0.459258	−	0.888303i	\(-0.348115\pi\)
							−0.539664	+	0.841881i	\(0.681449\pi\)
\(3\)	−0.0890070	−	0.332179i	−0.0513882	−	0.191783i	0.935460	−	0.353432i	\(-0.114985\pi\)
							−0.986848	+	0.161649i	\(0.948319\pi\)
\(5\)	−2.08297	−	0.813169i	−0.931532	−	0.363660i
							\(7\)	−1.39069	−	2.40874i	−0.525630	−	0.910418i	−0.999554	−	0.0298522i	\(-0.990496\pi\)
0.473924	−	0.880566i	\(-0.342837\pi\)
\(11\)	−1.04957	−	3.91706i	−0.316459	−	1.18104i	−0.922624	−	0.385701i	\(-0.873960\pi\)
							0.606165	−	0.795339i	\(-0.292707\pi\)
\(13\)		0			0
							\(17\)	2.34186	+	0.627499i	0.567984	+	0.152191i	0.531372	−	0.847139i	\(-0.321677\pi\)
0.0366120	+	0.999330i	\(0.488343\pi\)
\(19\)	1.83459	+	0.491577i	0.420883	+	0.112775i	0.463044	−	0.886335i	\(-0.346757\pi\)
							−0.0421602	+	0.999111i	\(0.513424\pi\)
\(23\)	−7.70544	+	2.06467i	−1.60670	+	0.430513i	−0.947056	−	0.321069i	\(-0.895958\pi\)
							−0.659640	+	0.751582i	\(0.729291\pi\)
\(29\)	3.96565	+	2.28957i	0.736403	+	0.425162i	0.820760	−	0.571273i	\(-0.193550\pi\)
							−0.0843571	+	0.996436i	\(0.526884\pi\)
\(31\)	−3.87352	−	3.87352i	−0.695704	−	0.695704i	0.267777	−	0.963481i	\(-0.413711\pi\)
							−0.963481	+	0.267777i	\(0.913711\pi\)
\(37\)	−3.50510	+	6.07101i	−0.576234	+	0.998067i	0.419672	+	0.907676i	\(0.362145\pi\)
							−0.995906	+	0.0903914i	\(0.971188\pi\)
\(41\)	−6.20184	+	1.66178i	−0.968565	+	0.259526i	−0.708222	−	0.705990i	\(-0.750502\pi\)
							−0.260343	+	0.965516i	\(0.583836\pi\)
\(43\)	1.67299	−	6.24368i	0.255128	−	0.952152i	−0.712891	−	0.701275i	\(-0.752614\pi\)
							0.968019	−	0.250877i	\(-0.0807189\pi\)
\(47\)	0.512375			0.0747376			0.0373688	−	0.999302i	\(-0.488102\pi\)
							0.0373688	+	0.999302i	\(0.488102\pi\)