Embedded newform 845.2.t.e.427.5

• Label: 845.2.t.e.427.5
• Level: $845$
• Weight: $2$
• Character: 845.427
• 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			427.5
Root			\(-1.51805i\) of defining polynomial
Character	\(\chi\)	\(=\)	845.427
Dual form			845.2.t.e.188.5

$q$-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 \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

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