Embedded newform 845.2.t.e.188.5

• Label: 845.2.t.e.188.5
• 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			188.5
Root			\(1.51805i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.188
Dual form			845.2.t.e.427.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31467 + 0.759023i) q^{2} +(-0.653367 + 0.175069i) q^{3} +(0.152233 + 0.263675i) q^{4} +(2.15400 + 0.600231i) q^{5} +(-0.991842 - 0.265763i) q^{6} +(1.29744 + 2.24723i) q^{7} -2.57390i q^{8} +(-2.20184 + 1.27123i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - 2 q^{3} + 6 q^{4} - 4 q^{6} + 2 q^{7} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31467	+	0.759023i	0.929610	+	0.536710i	0.886688	−	0.462368i	\(-0.153000\pi\)
							0.0429217	+	0.999078i	\(0.486333\pi\)
\(3\)	−0.653367	+	0.175069i	−0.377222	+	0.101076i	−0.442448	−	0.896794i	\(-0.645890\pi\)
							0.0652261	+	0.997871i	\(0.479223\pi\)
\(5\)	2.15400	+	0.600231i	0.963299	+	0.268431i
							\(7\)	1.29744	+	2.24723i	0.490387	+	0.849375i	0.999939	−	0.0110652i	\(-0.00352222\pi\)
−0.509552	+	0.860440i	\(0.670189\pi\)
\(11\)	4.82908	−	1.29395i	1.45602	−	0.390140i	0.557909	−	0.829902i	\(-0.311604\pi\)
							0.898114	+	0.439762i	\(0.144937\pi\)
\(13\)		0			0
							\(17\)	−0.0211881	+	0.0790751i	−0.00513887	+	0.0191785i	−0.968448	−	0.249217i	\(-0.919827\pi\)
0.963309	+	0.268396i	\(0.0864934\pi\)
\(19\)	−0.726525	+	2.71143i	−0.166676	+	0.622045i	0.831144	+	0.556057i	\(0.187686\pi\)
							−0.997820	+	0.0659876i	\(0.978980\pi\)
\(23\)	1.05016	+	3.91925i	0.218973	+	0.817220i	0.984730	+	0.174089i	\(0.0556982\pi\)
							−0.765756	+	0.643131i	\(0.777635\pi\)
\(29\)	−4.31701	−	2.49243i	−0.801649	−	0.462833i	0.0423981	−	0.999101i	\(-0.486500\pi\)
							−0.844048	+	0.536268i	\(0.819834\pi\)
\(31\)	2.32124	−	2.32124i	0.416906	−	0.416906i	−0.467230	−	0.884136i	\(-0.654748\pi\)
							0.884136	+	0.467230i	\(0.154748\pi\)
\(37\)	0.285750	−	0.494934i	0.0469771	−	0.0813667i	−0.841581	−	0.540131i	\(-0.818375\pi\)
							0.888558	+	0.458765i	\(0.151708\pi\)
\(41\)	2.69458	+	10.0563i	0.420823	+	1.57053i	0.772879	+	0.634554i	\(0.218816\pi\)
							−0.352056	+	0.935979i	\(0.614517\pi\)
\(43\)	−0.132121	−	0.0354017i	−0.0201483	−	0.00539871i	0.248731	−	0.968573i	\(-0.419987\pi\)
							−0.268879	+	0.963174i	\(0.586653\pi\)
\(47\)	2.30053			0.335567			0.167784	−	0.985824i	\(-0.446339\pi\)
							0.167784	+	0.985824i	\(0.446339\pi\)