Newform orbit 54.3.f.a

• Label: 54.3.f.a
• Level: $54$
• Weight: $3$
• Character orbit: 54.f
• Analytic conductor: $1.471$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	54.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47139342755\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{5} + 12 q^{6} - 12 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−0.909039	−	1.08335i	−1.44503	+	2.62905i	−0.347296	+	1.96962i	0.696976	+	1.91493i	4.16177	−	0.824439i	2.23222	+	12.6596i	2.44949	−	1.41421i	−4.82380	−	7.59809i	1.44096	−	2.49581i
5.2	−0.909039	−	1.08335i	1.30968	−	2.69902i	−0.347296	+	1.96962i	−1.54033	−	4.23203i	−4.11454	+	1.03467i	0.223815	+	1.26932i	2.44949	−	1.41421i	−5.56946	−	7.06974i	−3.18455	+	5.51580i
5.3	−0.909039	−	1.08335i	2.80846	+	1.05478i	−0.347296	+	1.96962i	2.86430	+	7.86960i	−1.41030	−	4.00138i	−1.95208	−	11.0708i	2.44949	−	1.41421i	6.77487	+	5.92462i	5.92177	−	10.2568i
5.4	0.909039	+	1.08335i	−2.95590	+	0.512490i	−0.347296	+	1.96962i	2.71293	+	7.45370i	−3.24224	−	2.73640i	0.0787775	+	0.446769i	−2.44949	+	1.41421i	8.47471	−	3.02974i	−5.60882	+	9.71475i
5.5	0.909039	+	1.08335i	1.54884	+	2.56926i	−0.347296	+	1.96962i	−1.07911	−	2.96482i	−1.37545	+	4.01349i	−0.250410	−	1.42015i	−2.44949	+	1.41421i	−4.20220	+	7.95874i	2.23099	−	3.86419i
5.6	0.909039	+	1.08335i	2.14542	−	2.09694i	−0.347296	+	1.96962i	0.387127	+	1.06362i	4.22200	+	0.418040i	−0.332318	−	1.88467i	−2.44949	+	1.41421i	0.205664	−	8.99765i	−0.800362	+	1.38627i
11.1	−0.909039	+	1.08335i	−1.44503	−	2.62905i	−0.347296	−	1.96962i	0.696976	−	1.91493i	4.16177	+	0.824439i	2.23222	−	12.6596i	2.44949	+	1.41421i	−4.82380	+	7.59809i	1.44096	+	2.49581i
11.2	−0.909039	+	1.08335i	1.30968	+	2.69902i	−0.347296	−	1.96962i	−1.54033	+	4.23203i	−4.11454	−	1.03467i	0.223815	−	1.26932i	2.44949	+	1.41421i	−5.56946	+	7.06974i	−3.18455	−	5.51580i
11.3	−0.909039	+	1.08335i	2.80846	−	1.05478i	−0.347296	−	1.96962i	2.86430	−	7.86960i	−1.41030	+	4.00138i	−1.95208	+	11.0708i	2.44949	+	1.41421i	6.77487	−	5.92462i	5.92177	+	10.2568i
11.4	0.909039	−	1.08335i	−2.95590	−	0.512490i	−0.347296	−	1.96962i	2.71293	−	7.45370i	−3.24224	+	2.73640i	0.0787775	−	0.446769i	−2.44949	−	1.41421i	8.47471	+	3.02974i	−5.60882	−	9.71475i
11.5	0.909039	−	1.08335i	1.54884	−	2.56926i	−0.347296	−	1.96962i	−1.07911	+	2.96482i	−1.37545	−	4.01349i	−0.250410	+	1.42015i	−2.44949	−	1.41421i	−4.20220	−	7.95874i	2.23099	+	3.86419i
11.6	0.909039	−	1.08335i	2.14542	+	2.09694i	−0.347296	−	1.96962i	0.387127	−	1.06362i	4.22200	−	0.418040i	−0.332318	+	1.88467i	−2.44949	−	1.41421i	0.205664	+	8.99765i	−0.800362	−	1.38627i
23.1	−0.483690	+	1.32893i	−2.93655	−	0.613727i	−1.53209	−	1.28558i	−7.71206	−	1.35984i	2.23598	−	3.60561i	−0.690206	+	0.579152i	2.44949	−	1.41421i	8.24668	+	3.60448i	5.53738	−	9.59102i
23.2	−0.483690	+	1.32893i	−0.570511	+	2.94525i	−1.53209	−	1.28558i	3.98669	+	0.702961i	−3.63807	−	2.18276i	−10.1193	+	8.49107i	2.44949	−	1.41421i	−8.34903	−	3.36060i	−2.86250	+	4.95800i
23.3	−0.483690	+	1.32893i	−0.391734	−	2.97431i	−1.53209	−	1.28558i	7.52350	+	1.32660i	4.14212	+	0.918059i	4.40811	−	3.69885i	2.44949	−	1.41421i	−8.69309	+	2.33028i	−5.40199	+	9.35652i
23.4	0.483690	−	1.32893i	−1.47299	−	2.61348i	−1.53209	−	1.28558i	−2.99623	−	0.528316i	−4.18560	+	0.693383i	6.30359	−	5.28934i	−2.44949	+	1.41421i	−4.66059	+	7.69928i	−2.15134	+	3.72623i
23.5	0.483690	−	1.32893i	0.320982	+	2.98278i	−1.53209	−	1.28558i	5.90332	+	1.04091i	4.11915	+	1.01618i	5.59840	−	4.69762i	−2.44949	+	1.41421i	−8.79394	+	1.91484i	4.23867	−	7.34160i
23.6	0.483690	−	1.32893i	2.82413	−	1.01208i	−1.53209	−	1.28558i	0.891042	+	0.157115i	0.0210163	−	4.24259i	−5.50064	+	4.61559i	−2.44949	+	1.41421i	6.95137	−	5.71650i	0.639781	−	1.10813i
29.1	−1.39273	−	0.245576i	−2.97067	−	0.418457i	1.87939	+	0.684040i	5.54906	+	6.61311i	4.03458	+	1.31232i	7.83131	−	2.85036i	−2.44949	−	1.41421i	8.64979	+	2.48620i	−6.10431	−	10.5730i
29.2	−1.39273	−	0.245576i	−2.10518	+	2.13733i	1.87939	+	0.684040i	−5.37469	−	6.40531i	3.45683	−	2.45974i	−8.61655	+	3.13617i	−2.44949	−	1.41421i	−0.136395	−	8.99897i	5.91250	+	10.2407i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	54.3.f.a	✓	36
3.b	odd	2	1		162.3.f.a		36
4.b	odd	2	1		432.3.bc.c		36
27.e	even	9	1		162.3.f.a		36
27.e	even	9	1		1458.3.b.c		36
27.f	odd	18	1	inner	54.3.f.a	✓	36
27.f	odd	18	1		1458.3.b.c		36
108.l	even	18	1		432.3.bc.c		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.3.f.a	✓	36	1.a	even	1	1	trivial
54.3.f.a	✓	36	27.f	odd	18	1	inner
162.3.f.a		36	3.b	odd	2	1
162.3.f.a		36	27.e	even	9	1
432.3.bc.c		36	4.b	odd	2	1
432.3.bc.c		36	108.l	even	18	1
1458.3.b.c		36	27.e	even	9	1
1458.3.b.c		36	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(54, [\chi])\).