Embedded newform 19.9.f.a.2.7

• Label: 19.9.f.a.2.7
• Level: $19$
• Weight: $9$
• Character: 19.2
• Analytic conductor: $7.740$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.74019359116\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.54293 + 0.624714i) q^{2} +(-19.5543 + 23.3039i) q^{3} +(-228.399 - 83.1305i) q^{4} +(773.341 - 281.473i) q^{5} +(-83.8379 + 70.3484i) q^{6} +(2158.31 + 3738.30i) q^{7} +(-1554.86 - 897.701i) q^{8} +(978.604 + 5549.94i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.54293	+	0.624714i	0.221433	+	0.0390446i	0.283264	−	0.959042i	\(-0.408583\pi\)
							−0.0618307	+	0.998087i	\(0.519694\pi\)
\(3\)	−19.5543	+	23.3039i	−0.241411	+	0.287703i	−0.873122	−	0.487501i	\(-0.837909\pi\)
							0.631711	+	0.775204i	\(0.282353\pi\)
\(5\)	773.341	−	281.473i	1.23734	−	0.450357i	0.361238	−	0.932474i	\(-0.382354\pi\)
							0.876107	+	0.482117i	\(0.160132\pi\)
\(7\)	2158.31	+	3738.30i	0.898921	+	1.55698i	0.828876	+	0.559433i	\(0.188981\pi\)
							0.0700454	+	0.997544i	\(0.477686\pi\)
\(11\)	888.052	−	1538.15i	0.0606552	−	0.105058i	−0.834103	−	0.551608i	\(-0.814014\pi\)
							0.894758	+	0.446550i	\(0.147348\pi\)
\(13\)	17733.1	+	21133.5i	0.620884	+	0.739941i	0.981222	−	0.192881i	\(-0.0617830\pi\)
							−0.360338	+	0.932822i	\(0.617339\pi\)
\(17\)	3717.98	−	21085.7i	0.0445156	−	0.252460i	−0.954427	−	0.298446i	\(-0.903532\pi\)
							0.998942	+	0.0459856i	\(0.0146428\pi\)
\(19\)	62279.8	−	114476.i	0.477895	−	0.878417i
							\(23\)	91077.8	+	33149.6i	0.325463	+	0.118459i	0.499584	−	0.866266i	\(-0.333486\pi\)
−0.174121	+	0.984724i	\(0.555708\pi\)
\(29\)	−1.33754e6	+	235844.i	−1.89110	+	0.333452i	−0.994090	−	0.108560i	\(-0.965376\pi\)
							−0.897009	+	0.442012i	\(0.854265\pi\)
\(31\)	299443.	−	172883.i	0.324240	−	0.187200i	−0.329041	−	0.944316i	\(-0.606725\pi\)
							0.653281	+	0.757115i	\(0.273392\pi\)
\(37\)			2.54697e6i			1.35899i	0.733680	+	0.679495i	\(0.237801\pi\)
							−0.733680	+	0.679495i	\(0.762199\pi\)
\(41\)	363039.	−	432653.i	0.128475	−	0.153110i	−0.697972	−	0.716125i	\(-0.745914\pi\)
							0.826447	+	0.563015i	\(0.190359\pi\)
\(43\)	3.59367e6	−	1.30799e6i	1.05115	−	0.382587i	0.242053	−	0.970263i	\(-0.422179\pi\)
							0.809096	+	0.587676i	\(0.199957\pi\)
\(47\)	−977948.	−	5.54622e6i	−0.200412	−	1.13659i	−0.904498	−	0.426479i	\(-0.859754\pi\)
							0.704085	−	0.710115i	\(-0.251357\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.7	✓	72
19.10	odd	18	inner	19.9.f.a.10.7	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.7	✓	72	1.1	even	1	trivial
19.9.f.a.10.7	yes	72	19.10	odd	18	inner