Embedded newform 27.7.f.a.2.10

• Label: 27.7.f.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	27.2
Dual form			27.7.f.a.14.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.51454 + 0.267054i) q^{2} +(26.8676 - 2.67034i) q^{3} +(-57.9178 - 21.0804i) q^{4} +(-139.142 - 165.822i) q^{5} +(41.4052 + 3.13078i) q^{6} +(-39.0374 + 14.2084i) q^{7} +(-167.329 - 96.6072i) q^{8} +(714.739 - 143.491i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.51454	+	0.267054i	0.189318	+	0.0333818i	0.267503	−	0.963557i	\(-0.413801\pi\)
							−0.0781853	+	0.996939i	\(0.524913\pi\)
\(3\)	26.8676	−	2.67034i	0.995097	−	0.0989015i
							\(5\)	−139.142	−	165.822i	−1.11313	−	1.32658i	−0.939803	−	0.341716i	\(-0.888992\pi\)
−0.173329	−	0.984864i	\(-0.555452\pi\)
\(7\)	−39.0374	+	14.2084i	−0.113812	+	0.0414240i	−0.398298	−	0.917256i	\(-0.630399\pi\)
							0.284487	+	0.958680i	\(0.408177\pi\)
\(11\)	−170.419	+	203.097i	−0.128038	+	0.152590i	−0.826255	−	0.563297i	\(-0.809533\pi\)
							0.698217	+	0.715887i	\(0.253977\pi\)
\(13\)	−454.269	−	2576.29i	−0.206768	−	1.17264i	−0.894634	−	0.446800i	\(-0.852563\pi\)
							0.687866	−	0.725838i	\(-0.258548\pi\)
\(17\)	4585.03	−	2647.17i	0.933244	−	0.538809i	0.0454080	−	0.998969i	\(-0.485541\pi\)
							0.887836	+	0.460160i	\(0.152208\pi\)
\(19\)	−3572.08	+	6187.02i	−0.520787	+	0.902029i	0.478921	+	0.877858i	\(0.341028\pi\)
							−0.999708	+	0.0241713i	\(0.992305\pi\)
\(23\)	4130.38	−	11348.1i	0.339474	−	0.932698i	−0.646070	−	0.763278i	\(-0.723588\pi\)
							0.985544	−	0.169420i	\(-0.0541893\pi\)
\(29\)	14932.5	+	2633.00i	0.612263	+	0.107959i	0.471177	−	0.882039i	\(-0.343829\pi\)
							0.141086	+	0.989997i	\(0.454940\pi\)
\(31\)	−25151.2	−	9154.28i	−0.844254	−	0.307283i	−0.116559	−	0.993184i	\(-0.537186\pi\)
							−0.727695	+	0.685901i	\(0.759408\pi\)
\(37\)	7676.10	+	13295.4i	0.151543	+	0.262480i	0.931795	−	0.362985i	\(-0.118242\pi\)
							−0.780252	+	0.625465i	\(0.784909\pi\)
\(41\)	97509.7	−	17193.6i	1.41480	−	0.249468i	0.586591	−	0.809883i	\(-0.300470\pi\)
							0.828212	+	0.560415i	\(0.189358\pi\)
\(43\)	−67116.1	−	56317.1i	−0.844154	−	0.708329i	0.114340	−	0.993442i	\(-0.463525\pi\)
							−0.958494	+	0.285113i	\(0.907969\pi\)
\(47\)	−58261.5	−	160072.i	−0.561162	−	1.54178i	−0.817937	−	0.575308i	\(-0.804882\pi\)
							0.256775	−	0.966471i	\(-0.417340\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.f.a.2.10	✓	102
3.2	odd	2		81.7.f.a.8.8		102
27.13	even	9		81.7.f.a.71.8		102
27.14	odd	18	inner	27.7.f.a.14.10	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.f.a.2.10	✓	102	1.1	even	1	trivial
27.7.f.a.14.10	yes	102	27.14	odd	18	inner
81.7.f.a.8.8		102	3.2	odd	2
81.7.f.a.71.8		102	27.13	even	9