Newform orbit 81.2.g.a

• Label: 81.2.g.a
• Level: $81$
• Weight: $2$
• Character orbit: 81.g
• Analytic conductor: $0.647$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.25893	−	0.264031i	−1.57391	−	0.723054i	3.08696	+	0.731623i	−0.524452	+	0.263390i	3.36444	+	2.04889i	−1.09261	+	3.64956i	−2.50575	−	0.912019i	1.95439	+	2.27604i	1.25424	−	0.456507i
4.2	−2.10031	−	0.245492i	1.49640	−	0.872226i	2.40496	+	0.569987i	−0.401399	+	0.201590i	−3.35704	+	1.46459i	0.699754	−	2.33734i	−0.937080	−	0.341069i	1.47844	−	2.61040i	0.892552	−	0.324862i
4.3	−0.907715	−	0.106097i	−0.350962	+	1.69612i	−1.13340	−	0.268621i	−3.40320	+	1.70915i	0.498526	−	1.50236i	0.208293	−	0.695746i	2.71786	+	0.989221i	−2.75365	−	1.19055i	3.27047	−	1.19035i
4.4	−0.702679	−	0.0821314i	1.28612	+	1.16012i	−1.45908	−	0.345808i	3.06981	−	1.54172i	−0.808449	−	0.920826i	−0.775884	+	2.59163i	2.32646	+	0.846761i	0.308226	+	2.98412i	−2.28372	+	0.831205i
4.5	−0.186105	−	0.0217526i	−0.680105	−	1.59294i	−1.91193	−	0.453135i	1.41557	−	0.710926i	0.0919205	+	0.311248i	1.16062	−	3.87673i	0.698107	+	0.254090i	−2.07491	+	2.16673i	−0.278909	+	0.101515i
4.6	1.04137	+	0.121718i	1.32397	−	1.11673i	−0.876460	−	0.207725i	−0.681964	+	0.342495i	1.51467	−	1.00178i	−0.831257	+	2.77659i	−2.85789	−	1.04019i	0.505808	−	2.95705i	−0.751863	+	0.273656i
4.7	1.69853	+	0.198530i	−1.07544	+	1.35773i	0.899496	+	0.213184i	1.22732	−	0.616384i	−2.09621	+	2.09264i	0.0889729	−	0.297190i	−1.72842	−	0.629095i	−0.686860	−	2.92031i	2.20701	−	0.803287i
4.8	2.43604	+	0.284732i	−1.18775	−	1.26065i	3.90713	+	0.926007i	−3.57352	+	1.79469i	−2.53446	−	3.40919i	0.377600	−	1.26127i	4.64483	+	1.69058i	−0.178488	+	2.99469i	−9.21624	+	3.35444i
7.1	−1.45464	−	1.95391i	−0.684440	+	1.59108i	−1.12821	+	3.76849i	−2.46571	+	1.62172i	4.10445	−	0.977107i	−1.57141	+	1.66559i	4.42639	−	1.61107i	−2.06308	−	2.17800i	6.75541	+	2.45877i
7.2	−1.32716	−	1.78269i	1.62274	−	0.605570i	−0.843016	+	2.81587i	2.46097	−	1.61861i	−3.23319	−	2.08915i	−2.02832	+	2.14989i	1.96178	−	0.714030i	2.26657	−	1.96536i	−6.15160	−	2.23900i
7.3	−0.886934	−	1.19136i	−0.340526	−	1.69825i	−0.0590788	+	0.197337i	−1.33926	+	0.880846i	−1.72120	+	1.91192i	1.30515	−	1.38337i	−2.50387	+	0.911335i	−2.76808	+	1.15659i	2.23724	+	0.814290i
7.4	−0.474186	−	0.636942i	−1.48279	+	0.895173i	0.392763	−	1.31192i	1.83182	−	1.20481i	1.27329	+	0.519973i	2.43989	−	2.58613i	−2.51422	+	0.915103i	1.39733	−	2.65471i	−1.63602	−	0.595462i
7.5	−0.120683	−	0.162105i	1.25647	+	1.19218i	0.561893	−	1.87685i	−0.761396	+	0.500778i	0.0416250	−	0.347556i	−0.112573	+	0.119321i	−0.751874	+	0.273660i	0.157410	+	2.99587i	0.173067	+	0.0629911i
7.6	0.470263	+	0.631673i	−1.10956	−	1.32999i	0.395743	−	1.32187i	2.32750	−	1.53082i	0.318331	−	1.32633i	−3.41908	+	3.62402i	2.50111	−	0.910331i	−0.537739	+	2.95141i	2.06152	+	0.750330i
7.7	0.925932	+	1.24374i	1.03494	−	1.38884i	−0.115939	+	0.387262i	−2.65494	+	1.74618i	2.68565	+	0.00122935i	−0.200969	+	0.213014i	2.32510	−	0.846267i	−0.857778	−	2.87476i	−4.63010	−	1.68522i
7.8	1.55705	+	2.09149i	−1.63477	−	0.572310i	−1.37629	+	4.59713i	−0.270787	+	0.178099i	−1.34844	−	4.31021i	2.64546	−	2.80403i	−6.85740	+	2.49589i	2.34492	+	1.87119i	−0.794122	−	0.289037i
13.1	−2.38576	−	1.19817i	1.71560	−	0.238151i	3.06192	+	4.11287i	0.727126	+	2.42877i	−4.37836	−	1.48742i	0.202369	+	0.469143i	−1.44988	−	8.22270i	2.88657	−	0.817145i	1.17534	−	6.66569i
13.2	−1.89071	−	0.949549i	−0.540889	+	1.64543i	1.47882	+	1.98639i	−1.11651	−	3.72940i	2.58508	−	2.59743i	−1.32100	−	3.06243i	−0.175036	−	0.992677i	−2.41488	−	1.77999i	−1.43025	+	8.11138i
13.3	−1.07782	−	0.541304i	−1.45923	+	0.933081i	−0.325622	−	0.437386i	1.12732	+	3.76550i	2.07788	−	0.215810i	0.736444	+	1.70727i	0.533084	+	3.02327i	1.25872	−	2.72317i	0.823230	−	4.66877i
13.4	−0.698417	−	0.350758i	0.295453	−	1.70667i	−0.829563	−	1.11430i	0.424677	+	1.41852i	−0.804976	+	1.08833i	−1.50560	−	3.49038i	0.459961	+	2.60857i	−2.82541	−	1.00848i	0.200956	−	1.13968i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
81.g	even	27	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.2.g.a	✓	144
3.b	odd	2	1		243.2.g.a		144
9.c	even	3	1		729.2.g.c		144
9.c	even	3	1		729.2.g.d		144
9.d	odd	6	1		729.2.g.a		144
9.d	odd	6	1		729.2.g.b		144
81.g	even	27	1	inner	81.2.g.a	✓	144
81.g	even	27	1		729.2.g.c		144
81.g	even	27	1		729.2.g.d		144
81.g	even	27	1		6561.2.a.c		72
81.h	odd	54	1		243.2.g.a		144
81.h	odd	54	1		729.2.g.a		144
81.h	odd	54	1		729.2.g.b		144
81.h	odd	54	1		6561.2.a.d		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
81.2.g.a	✓	144	1.a	even	1	1	trivial
81.2.g.a	✓	144	81.g	even	27	1	inner
243.2.g.a		144	3.b	odd	2	1
243.2.g.a		144	81.h	odd	54	1
729.2.g.a		144	9.d	odd	6	1
729.2.g.a		144	81.h	odd	54	1
729.2.g.b		144	9.d	odd	6	1
729.2.g.b		144	81.h	odd	54	1
729.2.g.c		144	9.c	even	3	1
729.2.g.c		144	81.g	even	27	1
729.2.g.d		144	9.c	even	3	1
729.2.g.d		144	81.g	even	27	1
6561.2.a.c		72	81.g	even	27	1
6561.2.a.d		72	81.h	odd	54	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(81, [\chi])\).