Embedded newform 117.4.t.a.103.15

• Label: 117.4.t.a.103.15
• Level: $117$
• Weight: $4$
• Character: 117.103
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.90322347067\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.15
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55219 + 0.896159i) q^{2} +(-5.10957 - 0.944628i) q^{3} +(-2.39380 + 4.14618i) q^{4} +(1.81395 + 1.04728i) q^{5} +(8.77757 - 3.11274i) q^{6} +(-22.1814 + 12.8064i) q^{7} -22.9194i q^{8} +(25.2154 + 9.65328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55219	+	0.896159i	−0.548783	+	0.316840i	−0.748631	−	0.662987i	\(-0.769288\pi\)
							0.199848	+	0.979827i	\(0.435955\pi\)
\(3\)	−5.10957	−	0.944628i	−0.983337	−	0.181794i
							\(5\)	1.81395	+	1.04728i	0.162244	+	0.0936719i	0.578924	−	0.815382i	\(-0.303473\pi\)
−0.416679	+	0.909054i	\(0.636806\pi\)
\(7\)	−22.1814	+	12.8064i	−1.19768	+	0.691482i	−0.960038	−	0.279871i	\(-0.909708\pi\)
							−0.237643	+	0.971352i	\(0.576375\pi\)
\(11\)	−32.5884	+	18.8149i	−0.893253	+	0.515720i	−0.875005	−	0.484114i	\(-0.839142\pi\)
							−0.0182476	+	0.999833i	\(0.505809\pi\)
\(13\)	33.6108	−	32.6698i	0.717073	−	0.696998i
							\(17\)	86.9156			1.24001			0.620004	−	0.784599i	\(-0.287131\pi\)
0.620004	+	0.784599i	\(0.287131\pi\)
\(19\)		−	132.035i		−	1.59425i	−0.603812	−	0.797127i	\(-0.706352\pi\)
							0.603812	−	0.797127i	\(-0.293648\pi\)
\(23\)	35.1515	−	60.8841i	0.318678	−	0.551966i	−0.661535	−	0.749915i	\(-0.730095\pi\)
							0.980212	+	0.197949i	\(0.0634279\pi\)
\(29\)	116.674	+	202.084i	0.747094	+	1.29400i	0.949210	+	0.314643i	\(0.101885\pi\)
							−0.202116	+	0.979362i	\(0.564782\pi\)
\(31\)	−57.7500	−	33.3420i	−0.334587	−	0.193174i	0.323289	−	0.946300i	\(-0.395212\pi\)
							−0.657876	+	0.753126i	\(0.728545\pi\)
\(37\)			196.349i			0.872420i	0.899845	+	0.436210i	\(0.143679\pi\)
							−0.899845	+	0.436210i	\(0.856321\pi\)
\(41\)	−258.552	−	149.275i	−0.984853	−	0.568605i	−0.0811214	−	0.996704i	\(-0.525850\pi\)
							−0.903732	+	0.428099i	\(0.859183\pi\)
\(43\)	−65.3625	−	113.211i	−0.231807	−	0.401501i	0.726533	−	0.687132i	\(-0.241130\pi\)
							−0.958340	+	0.285630i	\(0.907797\pi\)
\(47\)	182.692	−	105.477i	0.566987	−	0.327350i	−0.188958	−	0.981985i	\(-0.560511\pi\)
							0.755945	+	0.654635i	\(0.227178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.15	yes	80
3.2	odd	2		351.4.t.a.64.26		80
9.2	odd	6		351.4.t.a.181.15		80
9.7	even	3	inner	117.4.t.a.25.26	yes	80
13.12	even	2	inner	117.4.t.a.103.26	yes	80
39.38	odd	2		351.4.t.a.64.15		80
117.25	even	6	inner	117.4.t.a.25.15	✓	80
117.38	odd	6		351.4.t.a.181.26		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.15	✓	80	117.25	even	6	inner
117.4.t.a.25.26	yes	80	9.7	even	3	inner
117.4.t.a.103.15	yes	80	1.1	even	1	trivial
117.4.t.a.103.26	yes	80	13.12	even	2	inner
351.4.t.a.64.15		80	39.38	odd	2
351.4.t.a.64.26		80	3.2	odd	2
351.4.t.a.181.15		80	9.2	odd	6
351.4.t.a.181.26		80	117.38	odd	6