Embedded newform 19.9.f.a.2.11

• Label: 19.9.f.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	19.2
Dual form			19.9.f.a.10.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.4759 + 3.78678i) q^{2} +(79.9115 - 95.2348i) q^{3} +(206.314 + 75.0920i) q^{4} +(64.9468 - 23.6387i) q^{5} +(2076.80 - 1742.65i) q^{6} +(256.522 + 444.310i) q^{7} +(-688.296 - 397.388i) q^{8} +(-1544.51 - 8759.37i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 6 q^{2} + 162 q^{3} - 720 q^{4} - 6 q^{5} - 720 q^{6} + 3282 q^{7} - 9 q^{8} + 8562 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.4759	+	3.78678i	1.34224	+	0.236674i	0.798205	−	0.602386i	\(-0.205783\pi\)
							0.544039	+	0.839060i	\(0.316894\pi\)
\(3\)	79.9115	−	95.2348i	0.986561	−	1.17574i	0.00212455	−	0.999998i	\(-0.499324\pi\)
							0.984437	−	0.175740i	\(-0.0562318\pi\)
\(5\)	64.9468	−	23.6387i	0.103915	−	0.0378219i	−0.289540	−	0.957166i	\(-0.593502\pi\)
							0.393454	+	0.919344i	\(0.371280\pi\)
\(7\)	256.522	+	444.310i	0.106840	+	0.185052i	0.914488	−	0.404612i	\(-0.132593\pi\)
							−0.807649	+	0.589664i	\(0.799260\pi\)
\(11\)	−2530.10	+	4382.26i	−0.172809	+	0.299314i	−0.939401	−	0.342821i	\(-0.888618\pi\)
							0.766592	+	0.642135i	\(0.221951\pi\)
\(13\)	28601.5	+	34085.9i	1.00142	+	1.19344i	0.981071	+	0.193650i	\(0.0620325\pi\)
							0.0203466	+	0.999793i	\(0.493523\pi\)
\(17\)	−5508.13	+	31238.2i	−0.0659491	+	0.374016i	0.933914	+	0.357497i	\(0.116370\pi\)
							−0.999864	+	0.0165193i	\(0.994741\pi\)
\(19\)	118789.	+	53598.5i	0.911509	+	0.411281i
							\(23\)	−272946.	−	99344.3i	−0.975362	−	0.355003i	−0.195327	−	0.980738i	\(-0.562577\pi\)
−0.780035	+	0.625736i	\(0.784799\pi\)
\(29\)	−847367.	+	149414.i	−1.19806	+	0.211251i	−0.736861	−	0.676044i	\(-0.763693\pi\)
							−0.461202	+	0.887295i	\(0.652582\pi\)
\(31\)	209837.	−	121150.i	0.227214	−	0.131182i	−0.382072	−	0.924133i	\(-0.624789\pi\)
							0.609286	+	0.792950i	\(0.291456\pi\)
\(37\)		−	1.96708e6i		−	1.04958i	−0.851232	−	0.524790i	\(-0.824144\pi\)
							0.851232	−	0.524790i	\(-0.175856\pi\)
\(41\)	3.06042e6	−	3.64727e6i	1.08304	−	1.29072i	0.128806	−	0.991670i	\(-0.458885\pi\)
							0.954237	−	0.299051i	\(-0.0966701\pi\)
\(43\)	−3.17681e6	+	1.15627e6i	−0.929219	+	0.338208i	−0.761900	−	0.647695i	\(-0.775733\pi\)
							−0.167319	+	0.985903i	\(0.553511\pi\)
\(47\)	887261.	+	5.03191e6i	0.181828	+	1.03120i	0.929964	+	0.367651i	\(0.119838\pi\)
							−0.748136	+	0.663545i	\(0.769051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.9.f.a.2.11	✓	72
19.10	odd	18	inner	19.9.f.a.10.11	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.9.f.a.2.11	✓	72	1.1	even	1	trivial
19.9.f.a.10.11	yes	72	19.10	odd	18	inner