Embedded newform 605.2.r.a.32.4

• Label: 605.2.r.a.32.4
• Level: $605$
• Weight: $2$
• Character: 605.32
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			32.4
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55810 - 2.08138i) q^{2} +(-0.762046 + 0.762046i) q^{3} +(-1.34100 + 4.56704i) q^{4} +(-2.23099 + 0.150681i) q^{5} +(2.77346 + 0.398763i) q^{6} +(0.945323 - 4.34558i) q^{7} +(6.72311 - 2.50759i) q^{8} +1.83857i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55810	−	2.08138i	−1.10175	−	1.47176i	−0.866066	−	0.499930i	\(-0.833359\pi\)
							−0.235681	−	0.971831i	\(-0.575732\pi\)
\(3\)	−0.762046	+	0.762046i	−0.439968	+	0.439968i	−0.892001	−	0.452033i	\(-0.850699\pi\)
							0.452033	+	0.892001i	\(0.350699\pi\)
\(5\)	−2.23099	+	0.150681i	−0.997727	+	0.0673864i
							\(7\)	0.945323	−	4.34558i	0.357299	−	1.64248i	−0.350864	−	0.936426i	\(-0.614112\pi\)
0.708163	−	0.706049i	\(-0.249524\pi\)
\(11\)	3.26140	−	0.602727i	0.983349	−	0.181729i
							\(13\)	−1.25119	−	2.29139i	−0.347018	−	0.635516i	0.644629	−	0.764495i	\(-0.277012\pi\)
−0.991647	+	0.128979i	\(0.958830\pi\)
\(17\)	0.0449579	+	0.628594i	0.0109039	+	0.152456i	0.999987	+	0.00509762i	\(0.00162263\pi\)
							−0.989083	+	0.147359i	\(0.952923\pi\)
\(19\)	−4.18834	−	4.83360i	−0.960870	−	1.10890i	−0.993992	−	0.109452i	\(-0.965090\pi\)
							0.0331221	−	0.999451i	\(-0.489455\pi\)
\(23\)	−3.86595	+	0.840987i	−0.806107	+	0.175358i	−0.596693	−	0.802469i	\(-0.703519\pi\)
							−0.209414	+	0.977827i	\(0.567155\pi\)
\(29\)	5.58183	+	6.44178i	1.03652	+	1.19621i	0.980244	+	0.197793i	\(0.0633775\pi\)
							0.0562769	+	0.998415i	\(0.482077\pi\)
\(31\)	−6.90902	+	2.02867i	−1.24090	+	0.364360i	−0.835352	−	0.549715i	\(-0.814736\pi\)
							−0.405544	+	0.914075i	\(0.632918\pi\)
\(37\)	3.24799	+	1.77354i	0.533966	+	0.291568i	0.723521	−	0.690303i	\(-0.242523\pi\)
							−0.189555	+	0.981870i	\(0.560704\pi\)
\(41\)	−3.24251	−	0.466203i	−0.506396	−	0.0728087i	−0.115618	−	0.993294i	\(-0.536885\pi\)
							−0.390778	+	0.920485i	\(0.627794\pi\)
\(43\)	−2.98815	+	1.11452i	−0.455689	+	0.169963i	−0.566818	−	0.823843i	\(-0.691826\pi\)
							0.111130	+	0.993806i	\(0.464553\pi\)
\(47\)	−4.97512	+	6.64598i	−0.725696	+	0.969416i	0.274274	+	0.961652i	\(0.411562\pi\)
							−0.999970	+	0.00776443i	\(0.997528\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.4	✓	1280
5.3	odd	4	inner	605.2.r.a.153.4	yes	1280
121.87	odd	22	inner	605.2.r.a.87.4	yes	1280
605.208	even	44	inner	605.2.r.a.208.4	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.4	✓	1280	1.1	even	1	trivial
605.2.r.a.87.4	yes	1280	121.87	odd	22	inner
605.2.r.a.153.4	yes	1280	5.3	odd	4	inner
605.2.r.a.208.4	yes	1280	605.208	even	44	inner