Embedded newform 845.2.w.a.116.49

• Label: 845.2.w.a.116.49
• 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			116.49
Character	\(\chi\)	\(=\)	845.116
Dual form			845.2.w.a.51.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16765 + 1.31801i) q^{2} +(0.0897217 + 0.236577i) q^{3} +(-0.132657 + 1.09253i) q^{4} +(0.239316 + 0.970942i) q^{5} +(-0.207046 + 0.394493i) q^{6} +(1.17298 - 0.809651i) q^{7} +(1.30342 - 0.899687i) q^{8} +(2.19761 - 1.94692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 744 q + 4 q^{3} + 66 q^{4} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16765	+	1.31801i	0.825655	+	0.931972i	0.998597	−	0.0529601i	\(-0.0168656\pi\)
							−0.172941	+	0.984932i	\(0.555327\pi\)
\(3\)	0.0897217	+	0.236577i	0.0518008	+	0.136588i	0.958419	−	0.285364i	\(-0.0921145\pi\)
							−0.906618	+	0.421952i	\(0.861345\pi\)
\(5\)	0.239316	+	0.970942i	0.107025	+	0.434218i
							\(7\)	1.17298	−	0.809651i	0.443346	−	0.306019i	−0.325397	−	0.945577i	\(-0.605498\pi\)
0.768743	+	0.639558i	\(0.220883\pi\)
\(11\)	3.41945	−	3.85976i	1.03100	−	1.16376i	0.0446682	−	0.999002i	\(-0.485777\pi\)
							0.986334	−	0.164759i	\(-0.0526846\pi\)
\(13\)	−3.48500	−	0.924555i	−0.966564	−	0.256426i
							\(17\)	−1.40214	−	2.03135i	−0.340069	−	0.492675i	0.615217	−	0.788358i	\(-0.289068\pi\)
−0.955286	+	0.295683i	\(0.904453\pi\)
\(19\)			3.97782i			0.912574i	0.889833	+	0.456287i	\(0.150821\pi\)
							−0.889833	+	0.456287i	\(0.849179\pi\)
\(23\)	0.222928			0.0464837			0.0232419	−	0.999730i	\(-0.492601\pi\)
							0.0232419	+	0.999730i	\(0.492601\pi\)
\(29\)	−0.224845	+	0.199196i	−0.0417528	+	0.0369897i	−0.683741	−	0.729725i	\(-0.739648\pi\)
							0.641988	+	0.766714i	\(0.278110\pi\)
\(31\)	−1.08666	+	2.07045i	−0.195170	+	0.371864i	−0.963299	−	0.268430i	\(-0.913495\pi\)
							0.768130	+	0.640294i	\(0.221188\pi\)
\(37\)	−1.03458	+	1.97123i	−0.170084	+	0.324068i	−0.955546	−	0.294842i	\(-0.904733\pi\)
							0.785462	+	0.618910i	\(0.212425\pi\)
\(41\)	−0.717700	+	0.272188i	−0.112086	+	0.0425086i	−0.410010	−	0.912081i	\(-0.634475\pi\)
							0.297924	+	0.954589i	\(0.403706\pi\)
\(43\)	−3.94465	+	2.07031i	−0.601553	+	0.315719i	−0.737867	−	0.674946i	\(-0.764167\pi\)
							0.136314	+	0.990666i	\(0.456474\pi\)
\(47\)	−10.6240	+	1.28998i	−1.54966	+	0.188163i	−0.850182	−	0.526488i	\(-0.823508\pi\)
							−0.699481	+	0.714651i	\(0.746585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.w.a.116.49	yes	744
169.51	even	26	inner	845.2.w.a.51.49	✓	744

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.w.a.51.49	✓	744	169.51	even	26	inner
845.2.w.a.116.49	yes	744	1.1	even	1	trivial