Newform orbit 243.2.g.a

• Label: 243.2.g.a
• Level: $243$
• Weight: $2$
• Character orbit: 243.g
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.g (of order \(27\), degree \(18\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	no (minimal twist has level 81)
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−1.94437	−	0.976500i		0		1.63271	+	2.19311i	0.185416	+	0.619334i		0		−0.724564	−	1.67973i	−0.277373	−	1.57306i		0		0.244261	−	1.38527i
10.2	−1.38037	−	0.693250i		0		0.230520	+	0.309643i	−0.145961	−	0.487543i		0		1.57975	+	3.66226i	0.432916	+	2.45519i		0		−0.136509	+	0.774179i
10.3	−0.744407	−	0.373855i		0		−0.779943	−	1.04765i	−0.345929	−	1.15548i		0		−0.520803	−	1.20736i	0.478230	+	2.71217i		0		−0.174471	+	0.989477i
10.4	0.507233	+	0.254742i		0		−1.00193	−	1.34582i	0.832798	+	2.78174i		0		1.23966	+	2.87385i	−0.362501	−	2.05585i		0		−0.286203	+	1.62314i
10.5	0.698417	+	0.350758i		0		−0.829563	−	1.11430i	−0.424677	−	1.41852i		0		−1.50560	−	3.49038i	−0.459961	−	2.60857i		0		0.200956	−	1.13968i
10.6	1.07782	+	0.541304i		0		−0.325622	−	0.437386i	−1.12732	−	3.76550i		0		0.736444	+	1.70727i	−0.533084	−	3.02327i		0		0.823230	−	4.66877i
10.7	1.89071	+	0.949549i		0		1.47882	+	1.98639i	1.11651	+	3.72940i		0		−1.32100	−	3.06243i	0.175036	+	0.992677i		0		−1.43025	+	8.11138i
10.8	2.38576	+	1.19817i		0		3.06192	+	4.11287i	−0.727126	−	2.42877i		0		0.202369	+	0.469143i	1.44988	+	8.22270i		0		1.17534	−	6.66569i
19.1	−2.43604	+	0.284732i		0		3.90713	−	0.926007i	3.57352	+	1.79469i		0		0.377600	+	1.26127i	−4.64483	+	1.69058i		0		−9.21624	−	3.35444i
19.2	−1.69853	+	0.198530i		0		0.899496	−	0.213184i	−1.22732	−	0.616384i		0		0.0889729	+	0.297190i	1.72842	−	0.629095i		0		2.20701	+	0.803287i
19.3	−1.04137	+	0.121718i		0		−0.876460	+	0.207725i	0.681964	+	0.342495i		0		−0.831257	−	2.77659i	2.85789	−	1.04019i		0		−0.751863	−	0.273656i
19.4	0.186105	−	0.0217526i		0		−1.91193	+	0.453135i	−1.41557	−	0.710926i		0		1.16062	+	3.87673i	−0.698107	+	0.254090i		0		−0.278909	−	0.101515i
19.5	0.702679	−	0.0821314i		0		−1.45908	+	0.345808i	−3.06981	−	1.54172i		0		−0.775884	−	2.59163i	−2.32646	+	0.846761i		0		−2.28372	−	0.831205i
19.6	0.907715	−	0.106097i		0		−1.13340	+	0.268621i	3.40320	+	1.70915i		0		0.208293	+	0.695746i	−2.71786	+	0.989221i		0		3.27047	+	1.19035i
19.7	2.10031	−	0.245492i		0		2.40496	−	0.569987i	0.401399	+	0.201590i		0		0.699754	+	2.33734i	0.937080	−	0.341069i		0		0.892552	+	0.324862i
19.8	2.25893	−	0.264031i		0		3.08696	−	0.731623i	0.524452	+	0.263390i		0		−1.09261	−	3.64956i	2.50575	−	0.912019i		0		1.25424	+	0.456507i
37.1	−0.761699	+	2.54425i		0		−4.22205	−	2.77689i	1.12810	−	2.61524i		0		−0.111224	−	1.90965i	6.21207	−	5.21255i		0		5.79455	+	4.86220i
37.2	−0.587043	+	1.96086i		0		−1.82938	−	1.20320i	−0.00823679	+	0.0190950i		0		0.0874756	+	1.50190i	0.297287	−	0.249453i		0		−0.0326074	−	0.0273608i
37.3	−0.423731	+	1.41536i		0		−0.152717	−	0.100444i	−1.03062	+	2.38925i		0		0.0288073	+	0.494602i	−2.05667	+	1.72575i		0		−2.94494	−	2.47110i
37.4	−0.0876154	+	0.292656i		0		1.59300	+	1.04774i	1.04144	−	2.41432i		0		−0.211661	−	3.63407i	−0.914234	+	0.767134i		0		0.615319	+	0.516314i

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	243.2.g.a		144
3.b	odd	2	1		81.2.g.a	✓	144
9.c	even	3	1		729.2.g.a		144
9.c	even	3	1		729.2.g.b		144
9.d	odd	6	1		729.2.g.c		144
9.d	odd	6	1		729.2.g.d		144
81.g	even	27	1	inner	243.2.g.a		144
81.g	even	27	1		729.2.g.a		144
81.g	even	27	1		729.2.g.b		144
81.g	even	27	1		6561.2.a.d		72
81.h	odd	54	1		81.2.g.a	✓	144
81.h	odd	54	1		729.2.g.c		144
81.h	odd	54	1		729.2.g.d		144
81.h	odd	54	1		6561.2.a.c		72

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(243, [\chi])\).