Embedded newform 39.3.l.a.19.2

• Label: 39.3.l.a.19.2
• Level: $39$
• Weight: $3$
• Character: 39.19
• Analytic conductor: $1.063$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	39.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.06267303101\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.1579585536.10
Defining polynomial:	\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			19.2
Root			\(0.252411 - 1.79004i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.19
Dual form			39.3.l.a.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.613614 + 2.29004i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(-1.40365 + 0.810399i) q^{4} +(-1.21405 + 1.21405i) q^{5} +(-3.96646 - 1.06281i) q^{6} +(1.50718 - 5.62489i) q^{7} +(3.98855 + 3.98855i) q^{8} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{4} + 16 q^{5} - 6 q^{6} + 14 q^{7} - 24 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.613614	+	2.29004i	0.306807	+	1.14502i	0.931379	+	0.364052i	\(0.118607\pi\)
							−0.624572	+	0.780967i	\(0.714726\pi\)
\(3\)	−0.866025	+	1.50000i	−0.288675	+	0.500000i
							\(5\)	−1.21405	+	1.21405i	−0.242810	+	0.242810i	−0.818012	−	0.575201i	\(-0.804924\pi\)
0.575201	+	0.818012i	\(0.304924\pi\)
\(7\)	1.50718	−	5.62489i	0.215312	−	0.803555i	−0.770744	−	0.637144i	\(-0.780115\pi\)
							0.986056	−	0.166411i	\(-0.0532179\pi\)
\(11\)	4.43046	−	1.18714i	0.402769	−	0.107922i	−0.0517467	−	0.998660i	\(-0.516479\pi\)
							0.454516	+	0.890738i	\(0.349812\pi\)
\(13\)	5.13334	−	11.9436i	0.394872	−	0.918736i
							\(17\)	3.18843	−	1.84084i	0.187555	−	0.108285i	−0.403283	−	0.915076i	\(-0.632131\pi\)
0.590837	+	0.806791i	\(0.298797\pi\)
\(19\)	−20.4175	−	5.47085i	−1.07460	−	0.287939i	−0.322221	−	0.946665i	\(-0.604429\pi\)
							−0.752384	+	0.658725i	\(0.771096\pi\)
\(23\)	−32.1522	−	18.5631i	−1.39792	−	0.807090i	−0.403746	−	0.914871i	\(-0.632292\pi\)
							−0.994175	+	0.107781i	\(0.965626\pi\)
\(29\)	−24.5848	+	42.5822i	−0.847753	+	1.46835i	0.0354565	+	0.999371i	\(0.488711\pi\)
							−0.883209	+	0.468979i	\(0.844622\pi\)
\(31\)	16.7882	−	16.7882i	0.541555	−	0.541555i	−0.382429	−	0.923985i	\(-0.624912\pi\)
							0.923985	+	0.382429i	\(0.124912\pi\)
\(37\)	48.3751	−	12.9621i	1.30744	−	0.350326i	0.463179	−	0.886265i	\(-0.346709\pi\)
							0.844257	+	0.535938i	\(0.180042\pi\)
\(41\)	−2.52338	−	9.41739i	−0.0615459	−	0.229692i	0.928301	−	0.371829i	\(-0.121269\pi\)
							−0.989847	+	0.142137i	\(0.954603\pi\)
\(43\)	−7.44086	+	4.29598i	−0.173043	+	0.0999065i	−0.584020	−	0.811739i	\(-0.698521\pi\)
							0.410977	+	0.911646i	\(0.365188\pi\)
\(47\)	−50.5865	−	50.5865i	−1.07631	−	1.07631i	−0.996837	−	0.0794720i	\(-0.974677\pi\)
							−0.0794720	−	0.996837i	\(-0.525323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.3.l.a.19.2	✓	8
3.2	odd	2		117.3.bd.c.19.1		8
13.11	odd	12	inner	39.3.l.a.37.2	yes	8
39.11	even	12		117.3.bd.c.37.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.l.a.19.2	✓	8	1.1	even	1	trivial
39.3.l.a.37.2	yes	8	13.11	odd	12	inner
117.3.bd.c.19.1		8	3.2	odd	2
117.3.bd.c.37.1		8	39.11	even	12