Embedded newform 845.2.t.e.188.4

• Label: 845.2.t.e.188.4
• Level: $845$
• Weight: $2$
• Character: 845.188
• 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			188.4
Root			\(-0.274809i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.188
Dual form			845.2.t.e.427.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.237991 - 0.137404i) q^{2} +(-2.28256 + 0.611610i) q^{3} +(-0.962240 - 1.66665i) q^{4} +(-1.45395 + 1.69883i) q^{5} +(0.627267 + 0.168076i) q^{6} +(0.193052 + 0.334376i) q^{7} +1.07848i q^{8} +(2.23793 - 1.29207i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.237991	−	0.137404i	−0.168285	−	0.0971595i	0.413492	−	0.910508i	\(-0.364309\pi\)
							−0.581777	+	0.813348i	\(0.697642\pi\)
\(3\)	−2.28256	+	0.611610i	−1.31784	+	0.353113i	−0.848167	−	0.529729i	\(-0.822293\pi\)
							−0.469669	+	0.882842i	\(0.655627\pi\)
\(5\)	−1.45395	+	1.69883i	−0.650226	+	0.759741i
							\(7\)	0.193052	+	0.334376i	0.0729667	+	0.126382i	0.900200	−	0.435476i	\(-0.143420\pi\)
−0.827234	+	0.561858i	\(0.810087\pi\)
\(11\)	−4.21249	+	1.12873i	−1.27011	+	0.340326i	−0.830074	−	0.557653i	\(-0.811702\pi\)
							−0.440039	+	0.897979i	\(0.645035\pi\)
\(13\)		0			0
							\(17\)	−0.510514	+	1.90527i	−0.123818	+	0.462095i	−0.999795	−	0.0202583i	\(-0.993551\pi\)
0.875977	+	0.482353i	\(0.160218\pi\)
\(19\)	−1.29673	+	4.83947i	−0.297491	+	1.11025i	0.641728	+	0.766932i	\(0.278218\pi\)
							−0.939219	+	0.343318i	\(0.888449\pi\)
\(23\)	−0.0863441	−	0.322241i	−0.0180040	−	0.0671918i	0.956339	−	0.292258i	\(-0.0944067\pi\)
							−0.974343	+	0.225067i	\(0.927740\pi\)
\(29\)	−7.07031	−	4.08205i	−1.31292	−	0.758017i	−0.330345	−	0.943860i	\(-0.607165\pi\)
							−0.982579	+	0.185843i	\(0.940498\pi\)
\(31\)	2.54187	−	2.54187i	0.456534	−	0.456534i	−0.440982	−	0.897516i	\(-0.645370\pi\)
							0.897516	+	0.440982i	\(0.145370\pi\)
\(37\)	2.41251	−	4.17859i	0.396614	−	0.686956i	−0.596691	−	0.802471i	\(-0.703518\pi\)
							0.993306	+	0.115514i	\(0.0368517\pi\)
\(41\)	1.20515	+	4.49768i	0.188213	+	0.702419i	0.993920	+	0.110105i	\(0.0351188\pi\)
							−0.805707	+	0.592314i	\(0.798214\pi\)
\(43\)	6.58600	+	1.76471i	1.00436	+	0.269116i	0.723269	−	0.690566i	\(-0.242638\pi\)
							0.281087	+	0.959682i	\(0.409305\pi\)
\(47\)	9.83310			1.43430			0.717152	−	0.696917i	\(-0.245445\pi\)
							0.717152	+	0.696917i	\(0.245445\pi\)