Embedded newform 605.2.r.a.32.11

• Label: 605.2.r.a.32.11
• 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			32.11
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29475 - 1.72958i) q^{2} +(-2.08349 + 2.08349i) q^{3} +(-0.751609 + 2.55974i) q^{4} +(-2.21633 - 0.296468i) q^{5} +(6.30115 + 0.905969i) q^{6} +(-0.317966 + 1.46166i) q^{7} +(1.35184 - 0.504211i) q^{8} -5.68187i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29475	−	1.72958i	−0.915524	−	1.22300i	−0.974420	−	0.224733i	\(-0.927849\pi\)
							0.0588966	−	0.998264i	\(-0.481242\pi\)
\(3\)	−2.08349	+	2.08349i	−1.20290	+	1.20290i	−0.229625	+	0.973279i	\(0.573750\pi\)
							−0.973279	+	0.229625i	\(0.926250\pi\)
\(5\)	−2.21633	−	0.296468i	−0.991172	−	0.132585i
							\(7\)	−0.317966	+	1.46166i	−0.120180	+	0.552457i	0.877201	+	0.480124i	\(0.159408\pi\)
−0.997380	+	0.0723336i	\(0.976955\pi\)
\(11\)	−1.05526	+	3.14427i	−0.318172	+	0.948033i
							\(13\)	2.54464	+	4.66017i	0.705757	+	1.29250i	0.946750	+	0.321970i	\(0.104345\pi\)
−0.240993	+	0.970527i	\(0.577473\pi\)
\(17\)	0.475695	+	6.65109i	0.115373	+	1.61313i	0.644514	+	0.764592i	\(0.277060\pi\)
							−0.529141	+	0.848534i	\(0.677486\pi\)
\(19\)	−2.14643	−	2.47711i	−0.492425	−	0.568288i	0.454087	−	0.890957i	\(-0.349965\pi\)
							−0.946512	+	0.322669i	\(0.895420\pi\)
\(23\)	−3.32896	+	0.724172i	−0.694137	+	0.151000i	−0.545774	−	0.837933i	\(-0.683764\pi\)
							−0.148363	+	0.988933i	\(0.547401\pi\)
\(29\)	−5.89930	−	6.80816i	−1.09547	−	1.26424i	−0.961958	−	0.273196i	\(-0.911919\pi\)
							−0.133514	−	0.991047i	\(-0.542626\pi\)
\(31\)	−2.62113	+	0.769633i	−0.470768	+	0.138230i	−0.508508	−	0.861058i	\(-0.669803\pi\)
							0.0377391	+	0.999288i	\(0.487984\pi\)
\(37\)	−6.72846	−	3.67402i	−1.10615	−	0.604005i	−0.180955	−	0.983491i	\(-0.557919\pi\)
							−0.925197	+	0.379487i	\(0.876101\pi\)
\(41\)	1.64893	+	0.237081i	0.257520	+	0.0370258i	0.269866	−	0.962898i	\(-0.413021\pi\)
							−0.0123454	+	0.999924i	\(0.503930\pi\)
\(43\)	−0.723449	+	0.269833i	−0.110325	+	0.0411491i	−0.404020	−	0.914750i	\(-0.632388\pi\)
							0.293695	+	0.955899i	\(0.405115\pi\)
\(47\)	−1.87182	+	2.50046i	−0.273033	+	0.364730i	−0.916010	−	0.401156i	\(-0.868608\pi\)
							0.642976	+	0.765886i	\(0.277699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.11	✓	1280
5.3	odd	4	inner	605.2.r.a.153.11	yes	1280
121.87	odd	22	inner	605.2.r.a.87.11	yes	1280
605.208	even	44	inner	605.2.r.a.208.11	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.11	✓	1280	1.1	even	1	trivial
605.2.r.a.87.11	yes	1280	121.87	odd	22	inner
605.2.r.a.153.11	yes	1280	5.3	odd	4	inner
605.2.r.a.208.11	yes	1280	605.208	even	44	inner