Embedded newform 270.2.r.a.113.12

• Label: 270.2.r.a.113.12
• Level: $270$
• Weight: $2$
• Character: 270.113
• Analytic conductor: $2.156$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.r (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.15596085457\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.12
Character	\(\chi\)	\(=\)	270.113
Dual form			270.2.r.a.227.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.996195 + 0.0871557i) q^{2} +(-0.967424 - 1.43669i) q^{3} +(0.984808 + 0.173648i) q^{4} +(1.28326 + 1.83119i) q^{5} +(-0.838527 - 1.51554i) q^{6} +(2.33095 + 1.63215i) q^{7} +(0.965926 + 0.258819i) q^{8} +(-1.12818 + 2.77979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q + 12 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(217\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.996195	+	0.0871557i	0.704416	+	0.0616284i
							\(3\)	−0.967424	−	1.43669i	−0.558543	−	0.829476i
\(5\)	1.28326	+	1.83119i	0.573891	+	0.818931i
							\(7\)	2.33095	+	1.63215i	0.881017	+	0.616895i	0.924089	−	0.382177i	\(-0.124826\pi\)
−0.0430721	+	0.999072i	\(0.513715\pi\)
\(11\)	0.754583	−	2.07320i	0.227515	−	0.625093i	−0.772435	−	0.635094i	\(-0.780961\pi\)
							0.999950	+	0.0100010i	\(0.00318346\pi\)
\(13\)	−0.310645	−	3.55068i	−0.0861573	−	0.984782i	−0.909078	−	0.416626i	\(-0.863212\pi\)
							0.822921	−	0.568156i	\(-0.192343\pi\)
\(17\)	−1.90217	+	0.509686i	−0.461345	+	0.123617i	−0.482003	−	0.876170i	\(-0.660091\pi\)
							0.0206580	+	0.999787i	\(0.493424\pi\)
\(19\)	3.47249	−	2.00484i	0.796644	−	0.459942i	−0.0456525	−	0.998957i	\(-0.514537\pi\)
							0.842296	+	0.539015i	\(0.181203\pi\)
\(23\)	−2.00124	−	2.85807i	−0.417287	−	0.595948i	0.554391	−	0.832256i	\(-0.312951\pi\)
							−0.971678	+	0.236309i	\(0.924062\pi\)
\(29\)	−5.57992	+	4.68211i	−1.03616	+	0.869445i	−0.991572	−	0.129560i	\(-0.958644\pi\)
							−0.0445931	+	0.999005i	\(0.514199\pi\)
\(31\)	−1.26447	+	7.17115i	−0.227105	+	1.28798i	0.631515	+	0.775364i	\(0.282433\pi\)
							−0.858620	+	0.512613i	\(0.828678\pi\)
\(37\)	−2.37558	−	8.86580i	−0.390544	−	1.45753i	−0.829240	−	0.558893i	\(-0.811226\pi\)
							0.438696	−	0.898636i	\(-0.355441\pi\)
\(41\)	−3.83276	+	4.56771i	−0.598577	+	0.713357i	−0.977230	−	0.212182i	\(-0.931943\pi\)
							0.378653	+	0.925539i	\(0.376387\pi\)
\(43\)	−2.90599	−	6.23192i	−0.443159	−	0.950358i	−0.993330	−	0.115306i	\(-0.963215\pi\)
							0.550171	−	0.835052i	\(-0.314563\pi\)
\(47\)	0.930319	−	1.32863i	0.135701	−	0.193801i	−0.745550	−	0.666450i	\(-0.767813\pi\)
							0.881251	+	0.472649i	\(0.156702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.r.a.113.12	✓	216
3.2	odd	2		810.2.s.a.773.3		216
5.2	odd	4	inner	270.2.r.a.167.2	yes	216
15.2	even	4		810.2.s.a.287.14		216
27.11	odd	18	inner	270.2.r.a.173.2	yes	216
27.16	even	9		810.2.s.a.683.14		216
135.92	even	36	inner	270.2.r.a.227.12	yes	216
135.97	odd	36		810.2.s.a.197.3		216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.r.a.113.12	✓	216	1.1	even	1	trivial
270.2.r.a.167.2	yes	216	5.2	odd	4	inner
270.2.r.a.173.2	yes	216	27.11	odd	18	inner
270.2.r.a.227.12	yes	216	135.92	even	36	inner
810.2.s.a.197.3		216	135.97	odd	36
810.2.s.a.287.14		216	15.2	even	4
810.2.s.a.683.14		216	27.16	even	9
810.2.s.a.773.3		216	3.2	odd	2