Embedded newform 39.3.l.a.37.1

• Label: 39.3.l.a.37.1
• Level: $39$
• Weight: $3$
• Character: 39.37
• 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			37.1
Root			\(1.11361 - 1.42401i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.37
Dual form			39.3.l.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.247589 + 0.924013i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(2.67160 + 1.54245i) q^{4} +(5.21405 + 5.21405i) q^{5} +(1.60044 - 0.428836i) q^{6} +(-2.33731 - 8.72296i) q^{7} +(-4.79240 + 4.79240i) q^{8} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{4} + 16 q^{5} - 6 q^{6} + 14 q^{7} - 24 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.247589	+	0.924013i	−0.123794	+	0.462007i	−0.999794	−	0.0203054i	\(-0.993536\pi\)
							0.876000	+	0.482312i	\(0.160203\pi\)
\(3\)	−0.866025	−	1.50000i	−0.288675	−	0.500000i
							\(5\)	5.21405	+	5.21405i	1.04281	+	1.04281i	0.999042	+	0.0437686i	\(0.0139364\pi\)
0.0437686	+	0.999042i	\(0.486064\pi\)
\(7\)	−2.33731	−	8.72296i	−0.333902	−	1.24614i	−0.905055	−	0.425294i	\(-0.860171\pi\)
							0.571153	−	0.820843i	\(-0.306496\pi\)
\(11\)	−13.9926	−	3.74932i	−1.27206	−	0.340847i	−0.441239	−	0.897390i	\(-0.645461\pi\)
							−0.830819	+	0.556542i	\(0.812128\pi\)
\(13\)	6.62499	−	11.1852i	0.509614	−	0.860403i
							\(17\)	−1.28651	−	0.742766i	−0.0756769	−	0.0436921i	0.461684	−	0.887044i	\(-0.347245\pi\)
−0.537361	+	0.843352i	\(0.680579\pi\)
\(19\)	−23.0011	+	6.16313i	−1.21058	+	0.324375i	−0.806991	−	0.590564i	\(-0.798905\pi\)
							−0.403593	+	0.914939i	\(0.632239\pi\)
\(23\)	16.8580	−	9.73295i	0.732955	−	0.423172i	−0.0865474	−	0.996248i	\(-0.527583\pi\)
							0.819502	+	0.573076i	\(0.194250\pi\)
\(29\)	−10.4677	−	18.1306i	−0.360956	−	0.625195i	0.627162	−	0.778889i	\(-0.284216\pi\)
							−0.988118	+	0.153694i	\(0.950883\pi\)
\(31\)	27.4605	+	27.4605i	0.885823	+	0.885823i	0.994119	−	0.108296i	\(-0.0345395\pi\)
							−0.108296	+	0.994119i	\(0.534539\pi\)
\(37\)	4.23961	+	1.13600i	0.114584	+	0.0307027i	0.315655	−	0.948874i	\(-0.397776\pi\)
							−0.201071	+	0.979577i	\(0.564442\pi\)
\(41\)	0.229153	−	0.855210i	0.00558910	−	0.0208588i	−0.963075	−	0.269233i	\(-0.913230\pi\)
							0.968664	+	0.248375i	\(0.0798964\pi\)
\(43\)	32.9120	+	19.0018i	0.765395	+	0.441901i	0.831230	−	0.555929i	\(-0.187638\pi\)
							−0.0658342	+	0.997831i	\(0.520971\pi\)
\(47\)	−28.9494	+	28.9494i	−0.615944	+	0.615944i	−0.944489	−	0.328544i	\(-0.893442\pi\)
							0.328544	+	0.944489i	\(0.393442\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.l.a.19.1	✓	8	13.6	odd	12	inner
39.3.l.a.37.1	yes	8	1.1	even	1	trivial
117.3.bd.c.19.2		8	39.32	even	12
117.3.bd.c.37.2		8	3.2	odd	2