Embedded newform 605.2.s.a.416.8

• Label: 605.2.s.a.416.8
• Level: $605$
• Weight: $2$
• Character: 605.416
• 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			416.8
Character	\(\chi\)	\(=\)	605.416
Dual form			605.2.s.a.16.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07976 + 0.123891i) q^{2} +(0.718084 - 0.521719i) q^{3} +(-0.797485 + 0.185447i) q^{4} +(0.564443 + 0.825472i) q^{5} +(-0.710724 + 0.652297i) q^{6} +(-0.657495 + 0.676547i) q^{7} +(2.88540 - 1.02951i) q^{8} +(-0.683596 + 2.10389i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{53}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07976	+	0.123891i	−0.763508	+	0.0876043i	−0.486994	−	0.873406i	\(-0.661907\pi\)
							−0.276514	+	0.961010i	\(0.589179\pi\)
\(3\)	0.718084	−	0.521719i	0.414586	−	0.301214i	−0.360870	−	0.932616i	\(-0.617520\pi\)
							0.775456	+	0.631402i	\(0.217520\pi\)
\(5\)	0.564443	+	0.825472i	0.252427	+	0.369162i
							\(7\)	−0.657495	+	0.676547i	−0.248510	+	0.255711i	−0.829897	−	0.557916i	\(-0.811601\pi\)
0.581387	+	0.813627i	\(0.302510\pi\)
\(11\)	2.57672	−	2.08819i	0.776909	−	0.629613i
							\(13\)	−0.244974	−	0.406244i	−0.0679436	−	0.112672i	0.820522	−	0.571615i	\(-0.193683\pi\)
−0.888466	+	0.458943i	\(0.848228\pi\)
\(17\)	0.621524	+	0.508218i	0.150742	+	0.123261i	0.705376	−	0.708833i	\(-0.250778\pi\)
							−0.554634	+	0.832094i	\(0.687142\pi\)
\(19\)	−4.96081	+	0.283669i	−1.13809	+	0.0650782i	−0.615951	−	0.787784i	\(-0.711228\pi\)
							−0.522136	+	0.852862i	\(0.674865\pi\)
\(23\)	1.12065	+	7.79427i	0.233671	+	1.62522i	0.682003	+	0.731350i	\(0.261109\pi\)
							−0.448332	+	0.893867i	\(0.647982\pi\)
\(29\)	1.48912	+	3.82478i	0.276523	+	0.710244i	0.999806	+	0.0196921i	\(0.00626860\pi\)
							−0.723283	+	0.690552i	\(0.757368\pi\)
\(31\)	1.90365	−	0.804489i	0.341906	−	0.144490i	−0.211555	−	0.977366i	\(-0.567853\pi\)
							0.553460	+	0.832876i	\(0.313307\pi\)
\(37\)	0.211744	+	2.46529i	0.0348104	+	0.405291i	0.992911	+	0.118859i	\(0.0379237\pi\)
							−0.958101	+	0.286432i	\(0.907531\pi\)
\(41\)	4.88501	+	2.75869i	0.762910	+	0.430835i	0.823492	−	0.567328i	\(-0.192023\pi\)
							−0.0605820	+	0.998163i	\(0.519296\pi\)
\(43\)	10.5763	+	3.10549i	1.61287	+	0.473583i	0.959091	−	0.283099i	\(-0.0913623\pi\)
							0.653784	+	0.756681i	\(0.273181\pi\)
\(47\)	0.315449	+	1.55680i	0.0460130	+	0.227083i	0.996490	−	0.0837117i	\(-0.0266775\pi\)
							−0.950477	+	0.310795i	\(0.899405\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.s.a.416.8	yes	880
121.16	even	55	inner	605.2.s.a.16.8	✓	880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.s.a.16.8	✓	880	121.16	even	55	inner
605.2.s.a.416.8	yes	880	1.1	even	1	trivial