Embedded newform 19.7.f.a.2.3

• Label: 19.7.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	19.2
Dual form			19.7.f.a.10.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.05321 - 1.42000i) q^{2} +(3.00004 - 3.57530i) q^{3} +(2.69755 + 0.981826i) q^{4} +(-7.95224 + 2.89438i) q^{5} +(-29.2369 + 24.5326i) q^{6} +(213.780 + 370.279i) q^{7} +(432.910 + 249.941i) q^{8} +(122.807 + 696.473i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 6 q^{2} - 36 q^{3} + 192 q^{4} - 6 q^{5} + 720 q^{6} - 219 q^{7} - 9 q^{8} - 2076 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.05321	−	1.42000i	−1.00665	−	0.177500i	−0.354071	−	0.935219i	\(-0.615203\pi\)
							−0.652581	+	0.757719i	\(0.726314\pi\)
\(3\)	3.00004	−	3.57530i	0.111112	−	0.132419i	−0.707622	−	0.706592i	\(-0.750232\pi\)
							0.818734	+	0.574173i	\(0.194676\pi\)
\(5\)	−7.95224	+	2.89438i	−0.0636179	+	0.0231550i	−0.373633	−	0.927577i	\(-0.621888\pi\)
							0.310015	+	0.950732i	\(0.399666\pi\)
\(7\)	213.780	+	370.279i	0.623267	+	1.07953i	0.988873	+	0.148760i	\(0.0475282\pi\)
							−0.365607	+	0.930769i	\(0.619138\pi\)
\(11\)	−121.706	+	210.800i	−0.0914392	+	0.158377i	−0.908117	−	0.418717i	\(-0.862480\pi\)
							0.816678	+	0.577094i	\(0.195813\pi\)
\(13\)	31.3654	+	37.3798i	0.0142765	+	0.0170140i	0.773135	−	0.634241i	\(-0.218687\pi\)
							−0.758859	+	0.651255i	\(0.774243\pi\)
\(17\)	215.800	−	1223.86i	0.0439243	−	0.249107i	−0.954937	−	0.296807i	\(-0.904078\pi\)
							0.998862	+	0.0477002i	\(0.0151892\pi\)
\(19\)	−1993.76	+	6562.83i	−0.290679	+	0.956821i
							\(23\)	9048.75	+	3293.47i	0.743712	+	0.270689i	0.685957	−	0.727642i	\(-0.259384\pi\)
0.0577549	+	0.998331i	\(0.481606\pi\)
\(29\)	10048.6	−	1771.83i	0.412012	−	0.0726488i	0.0361995	−	0.999345i	\(-0.488475\pi\)
							0.375812	+	0.926696i	\(0.377364\pi\)
\(31\)	−8314.75	+	4800.52i	−0.279103	+	0.161140i	−0.633017	−	0.774138i	\(-0.718184\pi\)
							0.353914	+	0.935278i	\(0.384850\pi\)
\(37\)			52836.3i			1.04310i	0.853220	+	0.521551i	\(0.174646\pi\)
							−0.853220	+	0.521551i	\(0.825354\pi\)
\(41\)	42407.9	−	50539.8i	0.615312	−	0.733300i	−0.364945	−	0.931029i	\(-0.618912\pi\)
							0.980257	+	0.197729i	\(0.0633565\pi\)
\(43\)	−56771.6	+	20663.2i	−0.714045	+	0.259891i	−0.673396	−	0.739282i	\(-0.735165\pi\)
							−0.0406494	+	0.999173i	\(0.512943\pi\)
\(47\)	−24165.5	−	137049.i	−0.232757	−	1.32003i	−0.847286	−	0.531136i	\(-0.821765\pi\)
							0.614530	−	0.788894i	\(-0.289346\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.7.f.a.2.3	✓	54
19.10	odd	18	inner	19.7.f.a.10.3	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.f.a.2.3	✓	54	1.1	even	1	trivial
19.7.f.a.10.3	yes	54	19.10	odd	18	inner