Embedded newform 39.3.l.a.28.1

• Label: 39.3.l.a.28.1
• Level: $39$
• Weight: $3$
• Character: 39.28
• 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			28.1
Root			\(2.22833 + 1.32913i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.28
Dual form			39.3.l.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.09436 - 0.829131i) q^{2} +(0.866025 + 1.50000i) q^{3} +(5.42349 + 3.13125i) q^{4} +(3.29224 - 3.29224i) q^{5} +(-1.43610 - 5.35958i) q^{6} +(9.85686 - 2.64114i) q^{7} +(-5.12509 - 5.12509i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{4} + 16 q^{5} - 6 q^{6} + 14 q^{7} - 24 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.09436	−	0.829131i	−1.54718	−	0.414565i	−0.618602	−	0.785705i	\(-0.712300\pi\)
							−0.928577	+	0.371140i	\(0.878967\pi\)
\(3\)	0.866025	+	1.50000i	0.288675	+	0.500000i
							\(5\)	3.29224	−	3.29224i	0.658447	−	0.658447i	−0.296565	−	0.955013i	\(-0.595841\pi\)
0.955013	+	0.296565i	\(0.0958413\pi\)
\(7\)	9.85686	−	2.64114i	1.40812	−	0.377305i	0.526868	−	0.849947i	\(-0.323366\pi\)
							0.881255	+	0.472642i	\(0.156699\pi\)
\(11\)	−0.184273	+	0.687718i	−0.0167521	+	0.0625198i	−0.973796	−	0.227424i	\(-0.926970\pi\)
							0.957044	+	0.289943i	\(0.0936365\pi\)
\(13\)	−9.55574	+	8.81407i	−0.735057	+	0.678006i
							\(17\)	16.0788	+	9.28307i	0.945809	+	0.546063i	0.891777	−	0.452476i	\(-0.149459\pi\)
0.0540324	+	0.998539i	\(0.482793\pi\)
\(19\)	−9.02474	−	33.6808i	−0.474986	−	1.77267i	−0.621446	−	0.783457i	\(-0.713455\pi\)
							0.146460	−	0.989217i	\(-0.453212\pi\)
\(23\)	−21.3355	+	12.3180i	−0.927629	+	0.535567i	−0.886061	−	0.463569i	\(-0.846569\pi\)
							−0.0415682	+	0.999136i	\(0.513235\pi\)
\(29\)	−9.06508	−	15.7012i	−0.312589	−	0.541420i	0.666333	−	0.745654i	\(-0.267863\pi\)
							−0.978922	+	0.204234i	\(0.934530\pi\)
\(31\)	−22.3611	+	22.3611i	−0.721326	+	0.721326i	−0.968875	−	0.247550i	\(-0.920375\pi\)
							0.247550	+	0.968875i	\(0.420375\pi\)
\(37\)	−5.44167	+	20.3086i	−0.147072	+	0.548881i	0.852582	+	0.522593i	\(0.175035\pi\)
							−0.999654	+	0.0262879i	\(0.991631\pi\)
\(41\)	−10.7053	−	2.86847i	−0.261105	−	0.0699628i	0.125892	−	0.992044i	\(-0.459821\pi\)
							−0.386996	+	0.922081i	\(0.626487\pi\)
\(43\)	19.9262	+	11.5044i	0.463401	+	0.267544i	0.713473	−	0.700683i	\(-0.247121\pi\)
							−0.250072	+	0.968227i	\(0.580454\pi\)
\(47\)	−41.1206	−	41.1206i	−0.874906	−	0.874906i	0.118096	−	0.993002i	\(-0.462321\pi\)
							−0.993002	+	0.118096i	\(0.962321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.3.l.a.28.1	yes	8
3.2	odd	2		117.3.bd.c.28.2		8
13.7	odd	12	inner	39.3.l.a.7.1	✓	8
39.20	even	12		117.3.bd.c.46.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.l.a.7.1	✓	8	13.7	odd	12	inner
39.3.l.a.28.1	yes	8	1.1	even	1	trivial
117.3.bd.c.28.2		8	3.2	odd	2
117.3.bd.c.46.2		8	39.20	even	12