Embedded newform 605.2.r.a.32.17

• Label: 605.2.r.a.32.17
• 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			32.17
Character	\(\chi\)	\(=\)	605.32
Dual form			605.2.r.a.208.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.989614 - 1.32197i) q^{2} +(-0.0346243 + 0.0346243i) q^{3} +(-0.204802 + 0.697492i) q^{4} +(0.766483 + 2.10060i) q^{5} +(0.0800369 + 0.0115076i) q^{6} +(0.0122172 - 0.0561614i) q^{7} +(-1.96972 + 0.734666i) q^{8} +2.99760i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.989614	−	1.32197i	−0.699763	−	0.934773i	0.300054	−	0.953922i	\(-0.402995\pi\)
							−0.999817	+	0.0191488i	\(0.993904\pi\)
\(3\)	−0.0346243	+	0.0346243i	−0.0199903	+	0.0199903i	−0.717031	−	0.697041i	\(-0.754500\pi\)
							0.697041	+	0.717031i	\(0.254500\pi\)
\(5\)	0.766483	+	2.10060i	0.342781	+	0.939415i
							\(7\)	0.0122172	−	0.0561614i	0.00461765	−	0.0212270i	−0.974780	−	0.223166i	\(-0.928361\pi\)
0.979398	+	0.201939i	\(0.0647244\pi\)
\(11\)	−1.67011	−	2.86544i	−0.503556	−	0.863963i
							\(13\)	−1.81934	−	3.33187i	−0.504594	−	0.924095i	−0.998509	−	0.0545846i	\(-0.982617\pi\)
0.493915	−	0.869510i	\(-0.335565\pi\)
\(17\)	0.501152	+	7.00703i	0.121547	+	1.69945i	0.583148	+	0.812366i	\(0.301821\pi\)
							−0.461601	+	0.887088i	\(0.652725\pi\)
\(19\)	2.65760	+	3.06704i	0.609696	+	0.703627i	0.973716	−	0.227764i	\(-0.0731415\pi\)
							−0.364020	+	0.931391i	\(0.618596\pi\)
\(23\)	−8.96791	+	1.95085i	−1.86994	+	0.406780i	−0.996747	−	0.0805952i	\(-0.974318\pi\)
							−0.873192	+	0.487376i	\(0.837954\pi\)
\(29\)	−0.0531117	−	0.0612942i	−0.00986259	−	0.0113820i	0.750797	−	0.660533i	\(-0.229670\pi\)
							−0.760659	+	0.649151i	\(0.775124\pi\)
\(31\)	−8.96080	+	2.63113i	−1.60941	+	0.472564i	−0.958144	−	0.286288i	\(-0.907579\pi\)
							−0.651263	+	0.758852i	\(0.725761\pi\)
\(37\)	3.05698	+	1.66924i	0.502565	+	0.274421i	0.710470	−	0.703728i	\(-0.248483\pi\)
							−0.207905	+	0.978149i	\(0.566664\pi\)
\(41\)	11.1170	+	1.59839i	1.73619	+	0.249626i	0.936469	−	0.350751i	\(-0.114074\pi\)
							0.799717	+	0.600377i	\(0.204983\pi\)
\(43\)	3.23296	−	1.20583i	0.493021	−	0.183887i	−0.0906416	−	0.995884i	\(-0.528892\pi\)
							0.583663	+	0.811996i	\(0.301619\pi\)
\(47\)	−5.19687	+	6.94220i	−0.758041	+	1.01262i	0.241071	+	0.970507i	\(0.422501\pi\)
							−0.999113	+	0.0421172i	\(0.986590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.32.17	✓	1280
5.3	odd	4	inner	605.2.r.a.153.17	yes	1280
121.87	odd	22	inner	605.2.r.a.87.17	yes	1280
605.208	even	44	inner	605.2.r.a.208.17	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.17	✓	1280	1.1	even	1	trivial
605.2.r.a.87.17	yes	1280	121.87	odd	22	inner
605.2.r.a.153.17	yes	1280	5.3	odd	4	inner
605.2.r.a.208.17	yes	1280	605.208	even	44	inner