Embedded newform 605.2.r.a.87.13

• Label: 605.2.r.a.87.13
• Level: $605$
• Weight: $2$
• Character: 605.87
• 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			87.13
Character	\(\chi\)	\(=\)	605.87
Dual form			605.2.r.a.153.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58098 - 1.18351i) q^{2} +(2.15121 - 2.15121i) q^{3} +(0.535343 + 1.82321i) q^{4} +(-2.10204 - 0.762514i) q^{5} +(-5.94699 + 0.855048i) q^{6} +(-2.51157 + 0.546359i) q^{7} +(-0.0688894 + 0.184700i) q^{8} -6.25540i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(486\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58098	−	1.18351i	−1.11792	−	0.836865i	−0.129953	−	0.991520i	\(-0.541483\pi\)
							−0.987969	+	0.154655i	\(0.950573\pi\)
\(3\)	2.15121	−	2.15121i	1.24200	−	1.24200i	0.282832	−	0.959170i	\(-0.408726\pi\)
							0.959170	−	0.282832i	\(-0.0912738\pi\)
\(5\)	−2.10204	−	0.762514i	−0.940061	−	0.341007i
							\(7\)	−2.51157	+	0.546359i	−0.949284	+	0.206504i	−0.660446	−	0.750874i	\(-0.729633\pi\)
−0.288838	+	0.957378i	\(0.593269\pi\)
\(11\)	−2.00171	+	2.64446i	−0.603539	+	0.797333i
							\(13\)	−4.23972	−	2.31506i	−1.17589	−	0.642083i	−0.232118	−	0.972688i	\(-0.574566\pi\)
−0.943769	+	0.330604i	\(0.892747\pi\)
\(17\)	5.73131	+	0.409911i	1.39005	+	0.0994180i	0.746192	−	0.665731i	\(-0.231880\pi\)
							0.643854	+	0.765149i	\(0.277335\pi\)
\(19\)	−0.804651	+	0.928616i	−0.184599	+	0.213039i	−0.840505	−	0.541804i	\(-0.817742\pi\)
							0.655905	+	0.754843i	\(0.272287\pi\)
\(23\)	−0.149196	+	0.685842i	−0.0311095	+	0.143008i	−0.990082	−	0.140491i	\(-0.955132\pi\)
							0.958972	+	0.283499i	\(0.0914954\pi\)
\(29\)	−5.08028	+	5.86296i	−0.943385	+	1.08872i	0.0525473	+	0.998618i	\(0.483266\pi\)
							−0.995932	+	0.0901058i	\(0.971279\pi\)
\(31\)	1.71177	+	0.502621i	0.307443	+	0.0902734i	0.431815	−	0.901962i	\(-0.357873\pi\)
							−0.124372	+	0.992236i	\(0.539692\pi\)
\(37\)	−2.48809	−	4.55659i	−0.409039	−	0.749099i	0.589227	−	0.807968i	\(-0.299433\pi\)
							−0.998266	+	0.0588686i	\(0.981251\pi\)
\(41\)	−8.36415	+	1.20258i	−1.30626	+	0.187812i	−0.760066	−	0.649846i	\(-0.774833\pi\)
							−0.546195	+	0.837658i	\(0.683924\pi\)
\(43\)	3.10411	−	8.32245i	0.473373	−	1.26916i	−0.452648	−	0.891689i	\(-0.649521\pi\)
							0.926021	−	0.377472i	\(-0.123207\pi\)
\(47\)	6.47832	−	4.84961i	0.944960	−	0.707388i	−0.0112906	−	0.999936i	\(-0.503594\pi\)
							0.956251	+	0.292548i	\(0.0945031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.r.a.87.13	yes	1280
5.3	odd	4	inner	605.2.r.a.208.13	yes	1280
121.32	odd	22	inner	605.2.r.a.32.13	✓	1280
605.153	even	44	inner	605.2.r.a.153.13	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.r.a.32.13	✓	1280	121.32	odd	22	inner
605.2.r.a.87.13	yes	1280	1.1	even	1	trivial
605.2.r.a.153.13	yes	1280	605.153	even	44	inner
605.2.r.a.208.13	yes	1280	5.3	odd	4	inner