Newform orbit 180.3.s.a

• Label: 180.3.s.a
• Level: $180$
• Weight: $3$
• Character orbit: 180.s
• Analytic conductor: $4.905$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	180.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.90464475849\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{6} + 36 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.99526	−	0.137619i	1.64701	−	2.50746i	3.96212	+	0.549172i	−1.11803	+	1.93649i	−3.63128	+	4.77638i	−8.84742	+	5.10806i	−7.82989	−	1.64101i	−3.57473	−	8.25962i	2.49727	−	3.70994i
31.2	−1.97844	+	0.292901i	2.89213	+	0.797230i	3.82842	−	1.15897i	−1.11803	+	1.93649i	−5.95541	−	0.730161i	7.40061	−	4.27274i	−7.23482	+	3.41429i	7.72885	+	4.61139i	1.64476	−	4.15870i
31.3	−1.97026	−	0.343617i	−0.467822	+	2.96330i	3.76385	+	1.35403i	1.11803	−	1.93649i	1.93997	−	5.67772i	9.05772	−	5.22947i	−6.95050	−	3.96112i	−8.56229	−	2.77259i	−2.86823	+	3.43122i
31.4	−1.94447	+	0.468025i	1.81669	+	2.38739i	3.56191	−	1.82012i	1.11803	−	1.93649i	−4.64985	−	3.79195i	−10.3645	+	5.98396i	−6.07415	+	5.20622i	−2.39928	+	8.67430i	−1.26765	+	4.28871i
31.5	−1.94281	+	0.474867i	−2.93854	−	0.604145i	3.54900	−	1.84515i	1.11803	−	1.93649i	5.99590	−	0.221675i	1.14723	−	0.662352i	−6.01883	+	5.27007i	8.27002	+	3.55061i	−1.25255	+	4.29315i
31.6	−1.88646	−	0.664282i	2.19456	−	2.04546i	3.11746	+	2.50628i	1.11803	−	1.93649i	−5.49871	+	2.40087i	−0.307743	+	0.177676i	−4.21608	−	6.79887i	0.632177	−	8.97777i	−3.39550	+	2.91042i
31.7	−1.81768	−	0.834292i	−1.56419	−	2.55994i	2.60792	+	3.03295i	−1.11803	+	1.93649i	0.707458	+	5.95815i	8.61063	−	4.97135i	−2.20999	−	7.68869i	−4.10662	+	8.00848i	3.64783	−	2.58715i
31.8	−1.76950	+	0.932124i	−0.887178	+	2.86582i	2.26229	−	3.29879i	−1.11803	+	1.93649i	−1.10143	−	5.89804i	−1.24262	+	0.717427i	−0.928246	+	7.94596i	−7.42583	−	5.08498i	0.173316	−	4.46878i
31.9	−1.66115	+	1.11381i	−1.95671	−	2.27405i	1.51886	−	3.70042i	−1.11803	+	1.93649i	5.78325	+	1.59813i	−9.60564	+	5.54582i	1.59851	+	7.83867i	−1.34256	+	8.89930i	−0.299658	−	4.46209i
31.10	−1.57433	−	1.23348i	−1.93895	−	2.28921i	0.957049	+	3.88382i	1.11803	−	1.93649i	0.228853	+	5.99563i	−5.49529	+	3.17271i	3.28390	−	7.29493i	−1.48096	+	8.87732i	−4.14878	+	1.66961i
31.11	−1.53883	+	1.27750i	2.82321	−	1.01465i	0.735977	−	3.93171i	1.11803	−	1.93649i	−3.04821	+	5.16802i	−0.888940	+	0.513230i	3.89022	+	6.99043i	6.94098	−	5.72912i	0.753411	+	4.40822i
31.12	−1.29222	−	1.52649i	−0.0901554	+	2.99865i	−0.660317	+	3.94512i	−1.11803	+	1.93649i	4.69389	−	3.73730i	4.58579	−	2.64761i	6.87545	−	4.09001i	−8.98374	−	0.540688i	4.40078	−	0.795717i
31.13	−1.19930	−	1.60053i	2.96053	−	0.485064i	−1.12337	+	3.83902i	−1.11803	+	1.93649i	−4.32691	−	4.15666i	−2.72569	+	1.57368i	7.49170	−	2.80615i	8.52943	−	2.87209i	4.44026	−	0.532989i
31.14	−1.15123	+	1.63544i	−2.23178	+	2.00478i	−1.34933	−	3.76554i	1.11803	−	1.93649i	−0.709399	−	5.95792i	2.48029	−	1.43200i	7.71171	+	2.12827i	0.961702	−	8.94847i	1.87990	+	4.05783i
31.15	−1.14894	−	1.63705i	−1.46600	+	2.61741i	−1.35987	+	3.76175i	1.11803	−	1.93649i	5.96918	−	0.607325i	−9.52170	+	5.49736i	7.72059	−	2.09584i	−4.70167	−	7.67426i	−4.45469	+	0.394634i
31.16	−1.11110	+	1.66297i	−2.91067	−	0.726651i	−1.53092	−	3.69544i	−1.11803	+	1.93649i	4.44243	−	4.03296i	8.43435	−	4.86958i	7.84640	+	1.56012i	7.94396	+	4.23008i	−1.97808	−	4.01089i
31.17	−0.843258	−	1.81354i	1.46600	−	2.61741i	−2.57783	+	3.05856i	1.11803	−	1.93649i	−5.98299	−	0.451499i	9.52170	−	5.49736i	7.72059	+	2.09584i	−4.70167	−	7.67426i	−4.45469	−	0.394634i
31.18	−0.825531	+	1.82167i	2.96748	+	0.440495i	−2.63700	−	3.00770i	−1.11803	+	1.93649i	−3.25219	+	5.04215i	−6.98712	+	4.03402i	7.65597	−	2.32081i	8.61193	+	2.61433i	−2.60469	−	3.63533i
31.19	−0.801335	+	1.83245i	−0.547895	−	2.94954i	−2.71572	−	2.93681i	1.11803	−	1.93649i	5.84393	+	1.35959i	−4.64513	+	2.68186i	7.55775	−	2.62305i	−8.39962	+	3.23208i	2.65260	+	3.60052i
31.20	−0.786447	−	1.83889i	−2.96053	+	0.485064i	−2.76300	+	2.89237i	−1.11803	+	1.93649i	3.22027	+	5.06259i	2.72569	−	1.57368i	7.49170	+	2.80615i	8.52943	−	2.87209i	4.44026	+	0.532989i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	180.3.s.a	✓	96
3.b	odd	2	1		540.3.s.a		96
4.b	odd	2	1	inner	180.3.s.a	✓	96
9.c	even	3	1	inner	180.3.s.a	✓	96
9.d	odd	6	1		540.3.s.a		96
12.b	even	2	1		540.3.s.a		96
36.f	odd	6	1	inner	180.3.s.a	✓	96
36.h	even	6	1		540.3.s.a		96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.3.s.a	✓	96	1.a	even	1	1	trivial
180.3.s.a	✓	96	4.b	odd	2	1	inner
180.3.s.a	✓	96	9.c	even	3	1	inner
180.3.s.a	✓	96	36.f	odd	6	1	inner
540.3.s.a		96	3.b	odd	2	1
540.3.s.a		96	9.d	odd	6	1
540.3.s.a		96	12.b	even	2	1
540.3.s.a		96	36.h	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(180, [\chi])\).