Embedded newform 845.2.w.a.116.27

• Label: 845.2.w.a.116.27
• 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			116.27
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.373593 - 0.421699i) q^{2} +(-0.840433 - 2.21604i) q^{3} +(0.202815 - 1.67033i) q^{4} +(-0.239316 - 0.970942i) q^{5} +(-0.620523 + 1.18231i) q^{6} +(2.92476 - 2.01881i) q^{7} +(-1.70746 + 1.17857i) q^{8} +(-1.95897 + 1.73550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.373593	−	0.421699i	−0.264170	−	0.298187i	0.601427	−	0.798928i	\(-0.294599\pi\)
							−0.865597	+	0.500741i	\(0.833061\pi\)
\(3\)	−0.840433	−	2.21604i	−0.485224	−	1.27943i	−0.924535	−	0.381096i	\(-0.875547\pi\)
							0.439311	−	0.898335i	\(-0.355223\pi\)
\(5\)	−0.239316	−	0.970942i	−0.107025	−	0.434218i
							\(7\)	2.92476	−	2.01881i	1.10545	−	0.763040i	0.131847	−	0.991270i	\(-0.457909\pi\)
0.973607	+	0.228231i	\(0.0732940\pi\)
\(11\)	−2.20150	+	2.48498i	−0.663777	+	0.749249i	−0.979904	−	0.199468i	\(-0.936079\pi\)
							0.316128	+	0.948717i	\(0.397617\pi\)
\(13\)	−1.74138	−	3.15715i	−0.482972	−	0.875636i
							\(17\)	−1.08258	−	1.56839i	−0.262565	−	0.380391i	0.669422	−	0.742882i	\(-0.266542\pi\)
−0.931987	+	0.362491i	\(0.881926\pi\)
\(19\)			1.97974i			0.454183i	0.973873	+	0.227092i	\(0.0729217\pi\)
							−0.973873	+	0.227092i	\(0.927078\pi\)
\(23\)	2.00806			0.418710			0.209355	−	0.977840i	\(-0.432864\pi\)
							0.209355	+	0.977840i	\(0.432864\pi\)
\(29\)	4.81255	−	4.26355i	0.893668	−	0.791721i	−0.0851682	−	0.996367i	\(-0.527143\pi\)
							0.978836	+	0.204646i	\(0.0656043\pi\)
\(31\)	0.483793	−	0.921790i	0.0868917	−	0.165558i	−0.838132	−	0.545467i	\(-0.816352\pi\)
							0.925024	+	0.379909i	\(0.124045\pi\)
\(37\)	−0.155435	+	0.296156i	−0.0255533	+	0.0486877i	−0.897898	−	0.440204i	\(-0.854906\pi\)
							0.872345	+	0.488891i	\(0.162599\pi\)
\(41\)	−6.23682	+	2.36531i	−0.974027	+	0.369400i	−0.789719	−	0.613469i	\(-0.789774\pi\)
							−0.184308	+	0.982868i	\(0.559004\pi\)
\(43\)	9.42829	−	4.94835i	1.43780	−	0.754616i	0.447596	−	0.894236i	\(-0.352280\pi\)
							0.990205	+	0.139620i	\(0.0445880\pi\)
\(47\)	−5.41546	+	0.657555i	−0.789925	+	0.0959143i	−0.505548	−	0.862798i	\(-0.668710\pi\)
							−0.284377	+	0.958713i	\(0.591787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.27	yes	744
169.51	even	26	inner	845.2.w.a.51.27	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.27	✓	744	169.51	even	26	inner
845.2.w.a.116.27	yes	744	1.1	even	1	trivial