Embedded newform 19.7.f.a.2.6

• Label: 19.7.f.a.2.6
• Level: $19$
• Weight: $7$
• Character: 19.2
• Analytic conductor: $4.371$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.37102758878\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.6
Character	\(\chi\)	\(=\)	19.2
Dual form			19.7.f.a.10.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23760 + 0.570877i) q^{2} +(-11.8162 + 14.0820i) q^{3} +(-49.9842 - 18.1927i) q^{4} +(-103.262 + 37.5842i) q^{5} +(-46.2952 + 38.8462i) q^{6} +(-64.9288 - 112.460i) q^{7} +(-333.657 - 192.637i) q^{8} +(67.9097 + 385.135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.23760	+	0.570877i	0.404700	+	0.0713596i	0.372293	−	0.928115i	\(-0.378572\pi\)
							0.0324072	+	0.999475i	\(0.489683\pi\)
\(3\)	−11.8162	+	14.0820i	−0.437636	+	0.521554i	−0.939109	−	0.343619i	\(-0.888347\pi\)
							0.501473	+	0.865173i	\(0.332792\pi\)
\(5\)	−103.262	+	37.5842i	−0.826094	+	0.300674i	−0.720255	−	0.693709i	\(-0.755975\pi\)
							−0.105839	+	0.994383i	\(0.533753\pi\)
\(7\)	−64.9288	−	112.460i	−0.189297	−	0.327872i	0.755719	−	0.654896i	\(-0.227288\pi\)
							−0.945016	+	0.327024i	\(0.893954\pi\)
\(11\)	−461.406	+	799.178i	−0.346661	+	0.600434i	−0.985654	−	0.168778i	\(-0.946018\pi\)
							0.638993	+	0.769212i	\(0.279351\pi\)
\(13\)	−310.293	−	369.793i	−0.141235	−	0.168317i	0.690790	−	0.723055i	\(-0.257263\pi\)
							−0.832025	+	0.554738i	\(0.812818\pi\)
\(17\)	52.9698	−	300.406i	0.0107816	−	0.0611452i	−0.978942	−	0.204137i	\(-0.934561\pi\)
							0.989724	+	0.142992i	\(0.0456723\pi\)
\(19\)	6850.25	−	346.384i	0.998724	−	0.0505006i
							\(23\)	−9738.27	−	3544.44i	−0.800384	−	0.291316i	−0.0907382	−	0.995875i	\(-0.528923\pi\)
−0.709645	+	0.704559i	\(0.751145\pi\)
\(29\)	−23967.8	+	4226.17i	−0.982731	+	0.173282i	−0.641855	−	0.766826i	\(-0.721835\pi\)
							−0.340876	+	0.940108i	\(0.610724\pi\)
\(31\)	−35135.5	+	20285.5i	−1.17940	+	0.680926i	−0.955875	−	0.293773i	\(-0.905089\pi\)
							−0.223523	+	0.974699i	\(0.571756\pi\)
\(37\)		−	23682.3i		−	0.467540i	−0.972292	−	0.233770i	\(-0.924894\pi\)
							0.972292	−	0.233770i	\(-0.0751063\pi\)
\(41\)	3098.75	−	3692.95i	0.0449609	−	0.0535823i	−0.743095	−	0.669186i	\(-0.766643\pi\)
							0.788056	+	0.615603i	\(0.211088\pi\)
\(43\)	39742.6	−	14465.1i	0.499862	−	0.181935i	−0.0797698	−	0.996813i	\(-0.525419\pi\)
							0.579632	+	0.814878i	\(0.303196\pi\)
\(47\)	32279.6	+	183067.i	0.310910	+	1.76326i	0.594291	+	0.804250i	\(0.297433\pi\)
							−0.283381	+	0.959007i	\(0.591456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.2.6	✓	54
19.10	odd	18	inner	19.7.f.a.10.6	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.2.6	✓	54	1.1	even	1	trivial
19.7.f.a.10.6	yes	54	19.10	odd	18	inner