Embedded newform 605.2.r.a.32.10

• Label: 605.2.r.a.32.10
• 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			32.10
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29735 - 1.73306i) q^{2} +(2.35487 - 2.35487i) q^{3} +(-0.756913 + 2.57781i) q^{4} +(0.140101 + 2.23167i) q^{5} +(-7.13625 - 1.02604i) q^{6} +(-0.147963 + 0.680177i) q^{7} +(1.39275 - 0.519471i) q^{8} -8.09087i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29735	−	1.73306i	−0.917368	−	1.22546i	−0.973881	−	0.227059i	\(-0.927089\pi\)
							0.0565126	−	0.998402i	\(-0.482002\pi\)
\(3\)	2.35487	−	2.35487i	1.35959	−	1.35959i	0.485165	−	0.874423i	\(-0.338760\pi\)
							0.874423	−	0.485165i	\(-0.161240\pi\)
\(5\)	0.140101	+	2.23167i	0.0626551	+	0.998035i
							\(7\)	−0.147963	+	0.680177i	−0.0559249	+	0.257083i	−0.996618	−	0.0821767i	\(-0.973813\pi\)
0.940693	+	0.339259i	\(0.110176\pi\)
\(11\)	3.27114	−	0.547411i	0.986285	−	0.165051i
							\(13\)	−2.28251	−	4.18010i	−0.633053	−	1.15935i	−0.975539	−	0.219828i	\(-0.929450\pi\)
0.342485	−	0.939523i	\(-0.388731\pi\)
\(17\)	−0.489819	−	6.84857i	−0.118799	−	1.66102i	−0.611815	−	0.791001i	\(-0.709560\pi\)
							0.493016	−	0.870020i	\(-0.335894\pi\)
\(19\)	−0.232011	−	0.267755i	−0.0532270	−	0.0614272i	0.728512	−	0.685033i	\(-0.240212\pi\)
							−0.781739	+	0.623606i	\(0.785667\pi\)
\(23\)	−5.39568	+	1.17376i	−1.12508	+	0.244745i	−0.736345	−	0.676606i	\(-0.763450\pi\)
							−0.388731	+	0.921351i	\(0.627086\pi\)
\(29\)	0.248398	+	0.286666i	0.0461263	+	0.0532326i	0.778344	−	0.627838i	\(-0.216060\pi\)
							−0.732218	+	0.681071i	\(0.761515\pi\)
\(31\)	3.30732	−	0.971117i	0.594012	−	0.174418i	0.0291108	−	0.999576i	\(-0.490732\pi\)
							0.564901	+	0.825158i	\(0.308914\pi\)
\(37\)	7.15027	+	3.90434i	1.17550	+	0.641870i	0.943670	−	0.330887i	\(-0.107348\pi\)
							0.231827	+	0.972757i	\(0.425530\pi\)
\(41\)	0.229083	+	0.0329371i	0.0357767	+	0.00514391i	0.160180	−	0.987088i	\(-0.448793\pi\)
							−0.124403	+	0.992232i	\(0.539702\pi\)
\(43\)	−2.34086	+	0.873096i	−0.356978	+	0.133146i	−0.521554	−	0.853219i	\(-0.674647\pi\)
							0.164575	+	0.986364i	\(0.447375\pi\)
\(47\)	−0.177884	+	0.237625i	−0.0259470	+	0.0346612i	−0.813327	−	0.581807i	\(-0.802346\pi\)
							0.787380	+	0.616468i	\(0.211437\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.10	✓	1280
5.3	odd	4	inner	605.2.r.a.153.10	yes	1280
121.87	odd	22	inner	605.2.r.a.87.10	yes	1280
605.208	even	44	inner	605.2.r.a.208.10	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.10	✓	1280	1.1	even	1	trivial
605.2.r.a.87.10	yes	1280	121.87	odd	22	inner
605.2.r.a.153.10	yes	1280	5.3	odd	4	inner
605.2.r.a.208.10	yes	1280	605.208	even	44	inner