Embedded newform 845.2.w.a.116.52

• Label: 845.2.w.a.116.52
• Level: $845$
• Weight: $2$
• Character: 845.116
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $744$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.w (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(744\)
Relative dimension:	\(62\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			116.52
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37796 + 1.55540i) q^{2} +(0.924057 + 2.43654i) q^{3} +(-0.279410 + 2.30115i) q^{4} +(0.239316 + 0.970942i) q^{5} +(-2.51647 + 4.79474i) q^{6} +(1.93354 - 1.33462i) q^{7} +(-0.543920 + 0.375441i) q^{8} +(-2.83731 + 2.51363i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{7}{26}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37796	+	1.55540i	0.974368	+	1.09983i	0.994983	+	0.100046i	\(0.0318991\pi\)
							−0.0206151	+	0.999787i	\(0.506562\pi\)
\(3\)	0.924057	+	2.43654i	0.533505	+	1.40674i	0.882811	+	0.469728i	\(0.155648\pi\)
							−0.349307	+	0.937008i	\(0.613583\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	1.93354	−	1.33462i	0.730808	−	0.504441i	−0.143597	−	0.989636i	\(-0.545867\pi\)
0.874406	+	0.485196i	\(0.161252\pi\)
\(11\)	−0.775813	+	0.875712i	−0.233916	+	0.264037i	−0.853735	−	0.520708i	\(-0.825668\pi\)
							0.619818	+	0.784745i	\(0.287206\pi\)
\(13\)	−2.45174	+	2.64366i	−0.679991	+	0.733221i
							\(17\)	−1.42823	−	2.06915i	−0.346398	−	0.501844i	0.610600	−	0.791939i	\(-0.290928\pi\)
−0.956998	+	0.290096i	\(0.906313\pi\)
\(19\)		−	3.47262i		−	0.796674i	−0.917239	−	0.398337i	\(-0.869587\pi\)
							0.917239	−	0.398337i	\(-0.130413\pi\)
\(23\)	−4.67002			−0.973766			−0.486883	−	0.873467i	\(-0.661866\pi\)
							−0.486883	+	0.873467i	\(0.661866\pi\)
\(29\)	5.48950	−	4.86327i	1.01937	−	0.903086i	0.0240529	−	0.999711i	\(-0.492343\pi\)
							0.995321	+	0.0966243i	\(0.0308045\pi\)
\(31\)	0.517203	−	0.985448i	0.0928924	−	0.176992i	−0.834576	−	0.550892i	\(-0.814288\pi\)
							0.927469	+	0.373901i	\(0.121980\pi\)
\(37\)	−1.30796	+	2.49211i	−0.215027	+	0.409700i	−0.968948	−	0.247266i	\(-0.920468\pi\)
							0.753921	+	0.656966i	\(0.228160\pi\)
\(41\)	−2.81170	+	1.06634i	−0.439114	+	0.166534i	−0.564247	−	0.825606i	\(-0.690833\pi\)
							0.125133	+	0.992140i	\(0.460064\pi\)
\(43\)	6.78237	−	3.55966i	1.03430	−	0.542844i	0.139927	−	0.990162i	\(-0.455313\pi\)
							0.894375	+	0.447318i	\(0.147621\pi\)
\(47\)	8.61518	−	1.04607i	1.25665	−	0.152585i	0.535007	−	0.844847i	\(-0.320309\pi\)
							0.721646	+	0.692262i	\(0.243386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.52	yes	744
169.51	even	26	inner	845.2.w.a.51.52	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.52	✓	744	169.51	even	26	inner
845.2.w.a.116.52	yes	744	1.1	even	1	trivial