Embedded newform 845.2.t.e.657.4

• Label: 845.2.t.e.657.4
• 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			657.4
Root			\(1.58474i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.657
Dual form			845.2.t.e.418.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37242 + 0.792369i) q^{2} +(-0.0510678 - 0.190588i) q^{3} +(0.255697 + 0.442881i) q^{4} +(2.23506 - 0.0672627i) q^{5} +(0.0809291 - 0.302032i) q^{6} +(-0.274164 - 0.474866i) q^{7} -2.35905i q^{8} +(2.56436 - 1.48053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37242	+	0.792369i	0.970450	+	0.560290i	0.899373	−	0.437181i	\(-0.144023\pi\)
							0.0710765	+	0.997471i	\(0.477357\pi\)
\(3\)	−0.0510678	−	0.190588i	−0.0294840	−	0.110036i	0.949616	−	0.313417i	\(-0.101474\pi\)
							−0.979100	+	0.203381i	\(0.934807\pi\)
\(5\)	2.23506	−	0.0672627i	0.999547	−	0.0300808i
							\(7\)	−0.274164	−	0.474866i	−0.103624	−	0.179482i	0.809551	−	0.587049i	\(-0.199711\pi\)
−0.913175	+	0.407567i	\(0.866377\pi\)
\(11\)	0.0396372	+	0.147928i	0.0119511	+	0.0446020i	0.971644	−	0.236449i	\(-0.0759836\pi\)
							−0.959693	+	0.281051i	\(0.909317\pi\)
\(13\)		0			0
							\(17\)	−3.03602	−	0.813499i	−0.736343	−	0.197303i	−0.128891	−	0.991659i	\(-0.541142\pi\)
−0.607452	+	0.794356i	\(0.707808\pi\)
\(19\)	−4.40678	−	1.18079i	−1.01098	−	0.270893i	−0.284944	−	0.958544i	\(-0.591975\pi\)
							−0.726041	+	0.687652i	\(0.758642\pi\)
\(23\)	3.41860	−	0.916011i	0.712828	−	0.191002i	0.115858	−	0.993266i	\(-0.463038\pi\)
							0.596969	+	0.802264i	\(0.296371\pi\)
\(29\)	−2.02878	−	1.17132i	−0.376735	−	0.217508i	0.299662	−	0.954045i	\(-0.403126\pi\)
							−0.676397	+	0.736538i	\(0.736459\pi\)
\(31\)	6.61000	+	6.61000i	1.18719	+	1.18719i	0.977841	+	0.209350i	\(0.0671347\pi\)
							0.209350	+	0.977841i	\(0.432865\pi\)
\(37\)	−3.40317	+	5.89447i	−0.559478	+	0.969045i	0.438062	+	0.898945i	\(0.355665\pi\)
							−0.997540	+	0.0700997i	\(0.977668\pi\)
\(41\)	3.45612	−	0.926064i	0.539755	−	0.144627i	0.0213659	−	0.999772i	\(-0.493199\pi\)
							0.518389	+	0.855145i	\(0.326532\pi\)
\(43\)	−1.84023	+	6.86784i	−0.280633	+	1.04734i	0.671339	+	0.741150i	\(0.265719\pi\)
							−0.951972	+	0.306185i	\(0.900947\pi\)
\(47\)	9.13956			1.33314			0.666571	−	0.745442i	\(-0.267761\pi\)
							0.666571	+	0.745442i	\(0.267761\pi\)