Embedded newform 27.7.f.a.2.13

• Label: 27.7.f.a.2.13
• Level: $27$
• Weight: $7$
• Character: 27.2
• Analytic conductor: $6.211$
• Analytic rank: $0$
• Dimension: $102$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.21146025774\)
Analytic rank:	\(0\)
Dimension:	\(102\)
Relative dimension:	\(17\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.40620 + 1.30591i) q^{2} +(20.7548 - 17.2696i) q^{3} +(-6.99392 - 2.54558i) q^{4} +(93.4519 + 111.372i) q^{5} +(176.267 - 100.798i) q^{6} +(468.870 - 170.655i) q^{7} +(-465.300 - 268.641i) q^{8} +(132.523 - 716.853i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.40620	+	1.30591i	0.925775	+	0.163239i	0.616158	−	0.787622i	\(-0.288688\pi\)
							0.309617	+	0.950861i	\(0.399799\pi\)
\(3\)	20.7548	−	17.2696i	0.768696	−	0.639614i
							\(5\)	93.4519	+	111.372i	0.747615	+	0.890973i	0.996998	−	0.0774321i	\(-0.0246721\pi\)
−0.249383	+	0.968405i	\(0.580228\pi\)
\(7\)	468.870	−	170.655i	1.36697	−	0.497535i	0.448766	−	0.893649i	\(-0.351864\pi\)
							0.918201	+	0.396114i	\(0.129641\pi\)
\(11\)	−224.091	+	267.061i	−0.168363	+	0.200647i	−0.843628	−	0.536928i	\(-0.819585\pi\)
							0.675265	+	0.737575i	\(0.264029\pi\)
\(13\)	134.825	+	764.633i	0.0613679	+	0.348035i	0.999995	+	0.00310171i	\(0.000987307\pi\)
							−0.938627	+	0.344933i	\(0.887902\pi\)
\(17\)	−5550.68	+	3204.68i	−1.12979	+	0.652287i	−0.943883	−	0.330280i	\(-0.892857\pi\)
							−0.185911	+	0.982567i	\(0.559523\pi\)
\(19\)	−1943.92	+	3366.96i	−0.283411	+	0.490882i	−0.972223	−	0.234058i	\(-0.924799\pi\)
							0.688812	+	0.724940i	\(0.258133\pi\)
\(23\)	−5229.60	+	14368.2i	−0.429818	+	1.18092i	0.516105	+	0.856526i	\(0.327382\pi\)
							−0.945923	+	0.324391i	\(0.894841\pi\)
\(29\)	−1445.02	−	254.796i	−0.0592489	−	0.0104472i	0.143945	−	0.989586i	\(-0.454021\pi\)
							−0.203194	+	0.979138i	\(0.565132\pi\)
\(31\)	−31916.9	−	11616.8i	−1.07136	−	0.389944i	−0.254676	−	0.967027i	\(-0.581969\pi\)
							−0.816685	+	0.577083i	\(0.804191\pi\)
\(37\)	−45438.5	−	78701.8i	−0.897054	−	1.55374i	−0.831242	−	0.555911i	\(-0.812369\pi\)
							−0.0658126	−	0.997832i	\(-0.520964\pi\)
\(41\)	123662.	−	21805.0i	1.79426	−	0.316376i	0.825505	−	0.564395i	\(-0.190890\pi\)
							0.968755	+	0.248018i	\(0.0797794\pi\)
\(43\)	103394.	+	86758.2i	1.30044	+	1.09120i	0.990068	+	0.140587i	\(0.0448991\pi\)
							0.310375	+	0.950614i	\(0.399545\pi\)
\(47\)	−7944.94	−	21828.6i	−0.0765239	−	0.210248i	0.895532	−	0.444997i	\(-0.146795\pi\)
							−0.972056	+	0.234749i	\(0.924573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.13	✓	102
3.2	odd	2		81.7.f.a.8.5		102
27.13	even	9		81.7.f.a.71.5		102
27.14	odd	18	inner	27.7.f.a.14.13	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.13	✓	102	1.1	even	1	trivial
27.7.f.a.14.13	yes	102	27.14	odd	18	inner
81.7.f.a.8.5		102	3.2	odd	2
81.7.f.a.71.5		102	27.13	even	9