Embedded newform 845.2.t.e.657.5

• Label: 845.2.t.e.657.5
• 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			657.5
Root			\(2.08794i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.657
Dual form			845.2.t.e.418.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.80821 + 1.04397i) q^{2} +(0.713171 + 2.66159i) q^{3} +(1.17974 + 2.04338i) q^{4} +(-2.22760 + 0.194361i) q^{5} +(-1.48906 + 5.55724i) q^{6} +(1.45563 + 2.52122i) q^{7} +0.750585i q^{8} +(-3.97738 + 2.29634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.80821	+	1.04397i	1.27860	+	0.738198i	0.976590	−	0.215108i	\(-0.0690105\pi\)
							0.302006	+	0.953306i	\(0.402344\pi\)
\(3\)	0.713171	+	2.66159i	0.411750	+	1.53667i	0.791258	+	0.611482i	\(0.209426\pi\)
							−0.379508	+	0.925188i	\(0.623907\pi\)
\(5\)	−2.22760	+	0.194361i	−0.996215	+	0.0869210i
							\(7\)	1.45563	+	2.52122i	0.550176	+	0.952933i	0.998261	+	0.0589424i	\(0.0187728\pi\)
−0.448085	+	0.893991i	\(0.647894\pi\)
\(11\)	−0.00681908	−	0.0254491i	−0.00205603	−	0.00767321i	0.964890	−	0.262654i	\(-0.0845976\pi\)
							−0.966946	+	0.254980i	\(0.917931\pi\)
\(13\)		0			0
							\(17\)	2.76664	+	0.741318i	0.671008	+	0.179796i	0.578209	−	0.815889i	\(-0.303752\pi\)
0.0927992	+	0.995685i	\(0.470419\pi\)
\(19\)	−4.62320	−	1.23878i	−1.06063	−	0.284196i	−0.313995	−	0.949425i	\(-0.601667\pi\)
							−0.746640	+	0.665229i	\(0.768334\pi\)
\(23\)	0.358680	−	0.0961080i	0.0747899	−	0.0200399i	−0.221230	−	0.975222i	\(-0.571007\pi\)
							0.296020	+	0.955182i	\(0.404340\pi\)
\(29\)	3.62262	+	2.09152i	0.672703	+	0.388386i	0.797100	−	0.603847i	\(-0.206366\pi\)
							−0.124397	+	0.992233i	\(0.539700\pi\)
\(31\)	0.835277	+	0.835277i	0.150020	+	0.150020i	0.778127	−	0.628107i	\(-0.216170\pi\)
							−0.628107	+	0.778127i	\(0.716170\pi\)
\(37\)	3.22588	−	5.58739i	0.530332	−	0.918561i	−0.469042	−	0.883176i	\(-0.655401\pi\)
							0.999374	−	0.0353856i	\(-0.0112659\pi\)
\(41\)	−7.57344	+	2.02930i	−1.18277	+	0.316923i	−0.796026	−	0.605263i	\(-0.793068\pi\)
							−0.386747	+	0.922186i	\(0.626401\pi\)
\(43\)	−1.79436	+	6.69664i	−0.273637	+	1.02123i	0.683112	+	0.730314i	\(0.260626\pi\)
							−0.956749	+	0.290915i	\(0.906040\pi\)
\(47\)	−0.833377			−0.121561			−0.0607803	−	0.998151i	\(-0.519359\pi\)
							−0.0607803	+	0.998151i	\(0.519359\pi\)