Embedded newform 805.2.bi.a.792.32

• Label: 805.2.bi.a.792.32
• Level: $805$
• Weight: $2$
• Character: 805.792
• Analytic conductor: $6.428$
• Analytic rank: $0$
• Dimension: $1440$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 805 = 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	805.bi (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.42795736271\)
Analytic rank:	\(0\)
Dimension:	\(1440\)
Relative dimension:	\(72\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			792.32
Character	\(\chi\)	\(=\)	805.792
Dual form			805.2.bi.a.743.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0369063 + 0.516018i) q^{2} +(-2.58727 + 0.562826i) q^{3} +(1.71473 + 0.246541i) q^{4} +(-2.23466 + 0.0791996i) q^{5} +(-0.194942 - 1.35585i) q^{6} +(0.877679 - 0.479249i) q^{7} +(-0.410439 + 1.88676i) q^{8} +(3.64829 - 1.66612i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 8 q^{3} + 16 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/805\mathbb{Z}\right)^\times\).

\(n\)	\(162\)	\(281\)	\(346\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0369063	+	0.516018i	−0.0260967	+	0.364880i	0.967577	+	0.252576i	\(0.0812776\pi\)
							−0.993674	+	0.112304i	\(0.964177\pi\)
\(3\)	−2.58727	+	0.562826i	−1.49376	+	0.324948i	−0.883943	−	0.467595i	\(-0.845121\pi\)
							−0.609817	+	0.792542i	\(0.708757\pi\)
\(5\)	−2.23466	+	0.0791996i	−0.999373	+	0.0354191i
							\(7\)	0.877679	−	0.479249i	0.331731	−	0.181139i
\(11\)	1.22435	+	1.06091i	0.369156	+	0.319875i								0.819607	−	0.572926i	\(-0.194192\pi\)
							−0.450452	+	0.892801i	\(0.648737\pi\)
\(13\)	−1.51501	+	2.77453i	−0.420188	+	0.769517i	−0.998942	−	0.0459853i	\(-0.985357\pi\)
							0.578754	+	0.815502i	\(0.303539\pi\)
\(17\)	−3.41153	−	2.55384i	−0.827417	−	0.619397i	0.0993415	−	0.995053i	\(-0.468326\pi\)
							−0.926759	+	0.375656i	\(0.877417\pi\)
\(19\)	−0.880782	+	6.12597i	−0.202065	+	1.40539i	0.596078	+	0.802927i	\(0.296725\pi\)
							−0.798143	+	0.602468i	\(0.794184\pi\)
\(23\)	−1.97108	−	4.37205i	−0.410998	−	0.911636i
							\(29\)	−6.58639	+	0.946980i	−1.22306	+	0.175850i	−0.723436	−	0.690391i	\(-0.757438\pi\)
−0.499626	+	0.866241i	\(0.666529\pi\)
\(31\)	−0.763381	+	0.490595i	−0.137107	+	0.0881135i	−0.607396	−	0.794399i	\(-0.707786\pi\)
							0.470289	+	0.882512i	\(0.344150\pi\)
\(37\)	−9.01650	−	3.36298i	−1.48230	−	0.552871i	−0.527356	−	0.849645i	\(-0.676816\pi\)
							−0.954948	+	0.296774i	\(0.904089\pi\)
\(41\)	2.62073	−	5.73859i	0.409289	−	0.896218i	−0.586955	−	0.809620i	\(-0.699673\pi\)
							0.996243	−	0.0865976i	\(-0.0275995\pi\)
\(43\)	0.0875894	+	0.402642i	0.0133573	+	0.0614023i	0.983343	−	0.181759i	\(-0.0581791\pi\)
							−0.969986	+	0.243161i	\(0.921816\pi\)
\(47\)	−7.40957	−	7.40957i	−1.08080	−	1.08080i	−0.996435	−	0.0843621i	\(-0.973115\pi\)
							−0.0843621	−	0.996435i	\(-0.526885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	805.2.bi.a.792.32	yes	1440
5.3	odd	4	inner	805.2.bi.a.148.41	✓	1440
23.7	odd	22	inner	805.2.bi.a.582.41	yes	1440
115.53	even	44	inner	805.2.bi.a.743.32	yes	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.bi.a.148.41	✓	1440	5.3	odd	4	inner
805.2.bi.a.582.41	yes	1440	23.7	odd	22	inner
805.2.bi.a.743.32	yes	1440	115.53	even	44	inner
805.2.bi.a.792.32	yes	1440	1.1	even	1	trivial