Embedded newform 605.2.o.a.34.5

• Label: 605.2.o.a.34.5
• Level: $605$
• Weight: $2$
• Character: 605.34
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			34.5
Character	\(\chi\)	\(=\)	605.34
Dual form			605.2.o.a.89.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33583 + 0.335842i) q^{2} +0.0506944i q^{3} +(3.42433 - 1.00547i) q^{4} +(-0.467375 + 2.18668i) q^{5} +(-0.0170253 - 0.118413i) q^{6} +(-0.376776 + 0.586275i) q^{7} +(-3.36779 + 1.53802i) q^{8} +2.99743 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 44 q^{4} - 7 q^{5} + 14 q^{6} - 644 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33583	+	0.335842i	−1.65168	+	0.237476i	−0.904293	−	0.426911i	\(-0.859602\pi\)
							−0.747388	+	0.664387i	\(0.768693\pi\)
\(3\)			0.0506944i			0.0292684i	0.999893	+	0.0146342i	\(0.00465838\pi\)
							−0.999893	+	0.0146342i	\(0.995342\pi\)
\(5\)	−0.467375	+	2.18668i	−0.209016	+	0.977912i
							\(7\)	−0.376776	+	0.586275i	−0.142408	+	0.221591i	−0.905130	−	0.425136i	\(-0.860226\pi\)
0.762722	+	0.646727i	\(0.223863\pi\)
\(11\)	2.67114	−	1.96597i	0.805379	−	0.592761i
							\(13\)	−1.29288	−	4.40314i	−0.358580	−	1.22121i	−0.919417	−	0.393284i	\(-0.871339\pi\)
0.560837	−	0.827926i	\(-0.310479\pi\)
\(17\)	1.17793	−	1.02069i	0.285691	−	0.247553i	−0.500211	−	0.865903i	\(-0.666744\pi\)
							0.785902	+	0.618351i	\(0.212199\pi\)
\(19\)	1.70434	−	1.96691i	0.391002	−	0.451241i	−0.525784	−	0.850618i	\(-0.676228\pi\)
							0.916787	+	0.399377i	\(0.130774\pi\)
\(23\)	3.68461	+	5.73336i	0.768294	+	1.19549i	0.976101	+	0.217318i	\(0.0697310\pi\)
							−0.207807	+	0.978170i	\(0.566633\pi\)
\(29\)	3.17034	−	3.65877i	0.588718	−	0.679417i	−0.380738	−	0.924683i	\(-0.624330\pi\)
							0.969456	+	0.245266i	\(0.0788754\pi\)
\(31\)	−7.33804	−	2.15464i	−1.31795	−	0.386986i	−0.454198	−	0.890901i	\(-0.650074\pi\)
							−0.863753	+	0.503915i	\(0.831892\pi\)
\(37\)	2.02218	−	6.88690i	0.332444	−	1.13220i	−0.608477	−	0.793572i	\(-0.708219\pi\)
							0.940921	−	0.338627i	\(-0.109963\pi\)
\(41\)	0.566707	+	3.94154i	0.0885048	+	0.615565i	0.985006	+	0.172522i	\(0.0551917\pi\)
							−0.896501	+	0.443042i	\(0.853899\pi\)
\(43\)	3.42820	−	1.56561i	0.522795	−	0.238753i	−0.136496	−	0.990641i	\(-0.543584\pi\)
							0.659291	+	0.751888i	\(0.270857\pi\)
\(47\)	7.96098	+	1.14462i	1.16123	+	0.166960i	0.695874	−	0.718164i	\(-0.255017\pi\)
							0.465355	+	0.885124i	\(0.345927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.o.a.34.5	✓	640
5.4	even	2	inner	605.2.o.a.34.60	yes	640
121.89	even	11	inner	605.2.o.a.89.60	yes	640
605.89	even	22	inner	605.2.o.a.89.5	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.o.a.34.5	✓	640	1.1	even	1	trivial
605.2.o.a.34.60	yes	640	5.4	even	2	inner
605.2.o.a.89.5	yes	640	605.89	even	22	inner
605.2.o.a.89.60	yes	640	121.89	even	11	inner