Embedded newform 605.2.s.a.16.15

• Label: 605.2.s.a.16.15
• Level: $605$
• Weight: $2$
• Character: 605.16
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $880$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.s (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(880\)
Relative dimension:	\(22\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			16.15
Character	\(\chi\)	\(=\)	605.16
Dual form			605.2.s.a.416.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.06134 + 0.121777i) q^{2} +(-0.605637 - 0.440021i) q^{3} +(-0.836420 - 0.194501i) q^{4} +(0.564443 - 0.825472i) q^{5} +(-0.589199 - 0.540762i) q^{6} +(-1.40253 - 1.44317i) q^{7} +(-2.87638 - 1.02629i) q^{8} +(-0.753874 - 2.32018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.06134	+	0.121777i	0.750478	+	0.0861093i	0.480787	−	0.876838i	\(-0.340351\pi\)
							0.269691	+	0.962947i	\(0.413078\pi\)
\(3\)	−0.605637	−	0.440021i	−0.349664	−	0.254046i	0.399064	−	0.916923i	\(-0.369335\pi\)
							−0.748728	+	0.662877i	\(0.769335\pi\)
\(5\)	0.564443	−	0.825472i	0.252427	−	0.369162i
							\(7\)	−1.40253	−	1.44317i	−0.530105	−	0.545465i	0.398193	−	0.917302i	\(-0.369637\pi\)
−0.928298	+	0.371836i	\(0.878728\pi\)
\(11\)	−0.348669	+	3.29825i	−0.105128	+	0.994459i
							\(13\)	−3.22678	+	5.35101i	−0.894946	+	1.48410i	−0.0200460	+	0.999799i	\(0.506381\pi\)
−0.874900	+	0.484303i	\(0.839073\pi\)
\(17\)	−5.60461	+	4.58287i	−1.35932	+	1.11151i	−0.376606	+	0.926374i	\(0.622909\pi\)
							−0.982713	+	0.185136i	\(0.940728\pi\)
\(19\)	2.33097	+	0.133289i	0.534760	+	0.0305787i	0.322383	−	0.946609i	\(-0.395516\pi\)
							0.212377	+	0.977188i	\(0.431880\pi\)
\(23\)	0.398935	−	2.77465i	0.0831837	−	0.578555i	−0.905015	−	0.425379i	\(-0.860141\pi\)
							0.988199	−	0.153176i	\(-0.0489502\pi\)
\(29\)	3.17780	−	8.16212i	0.590103	−	1.51567i	−0.247220	−	0.968959i	\(-0.579517\pi\)
							0.837323	−	0.546708i	\(-0.184119\pi\)
\(31\)	−8.74916	−	3.69742i	−1.57139	−	0.664077i	−0.582518	−	0.812817i	\(-0.697933\pi\)
							−0.988876	+	0.148741i	\(0.952478\pi\)
\(37\)	0.855084	−	9.95556i	0.140575	−	1.63668i	−0.498037	−	0.867156i	\(-0.665946\pi\)
							0.638612	−	0.769529i	\(-0.279509\pi\)
\(41\)	−6.10300	+	3.44652i	−0.953128	+	0.538256i	−0.888112	−	0.459628i	\(-0.847983\pi\)
							−0.0650166	+	0.997884i	\(0.520710\pi\)
\(43\)	−0.526247	+	0.154520i	−0.0802519	+	0.0235641i	−0.321612	−	0.946872i	\(-0.604225\pi\)
							0.241360	+	0.970436i	\(0.422406\pi\)
\(47\)	0.785990	−	3.87901i	0.114648	−	0.565812i	−0.881039	−	0.473043i	\(-0.843155\pi\)
							0.995688	−	0.0927689i	\(-0.0295718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.16.15	✓	880
121.53	even	55	inner	605.2.s.a.416.15	yes	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.15	✓	880	1.1	even	1	trivial
605.2.s.a.416.15	yes	880	121.53	even	55	inner