Newform orbit 81.4.e.a

• Label: 81.4.e.a
• Level: $81$
• Weight: $4$
• Character orbit: 81.e
• Analytic conductor: $4.779$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.77915471046\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{2} - 6 q^{4} - 6 q^{5} - 6 q^{7} + 75 q^{8}+O(q^{10}) \)

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	−0.904728	−	5.13097i		0		−17.9907	+	6.54810i	−10.7601	−	9.02881i		0		9.67890	+	3.52283i	29.0343	+	50.2889i		0		−36.5916	+	63.3784i
10.2	−0.640527	−	3.63261i		0		−5.26804	+	1.91741i	1.23763	+	1.03850i		0		−31.9169	−	11.6168i	−4.41508	−	7.64714i		0		2.97972	−	5.16102i
10.3	−0.553775	−	3.14062i		0		−2.03925	+	0.742228i	10.9819	+	9.21490i		0		30.2596	+	11.0136i	−9.29592	−	16.1010i		0		22.8590	−	39.5929i
10.4	−0.111911	−	0.634678i		0		7.12725	−	2.59411i	0.393954	+	0.330567i		0		9.02395	+	3.28445i	−5.02191	−	8.69820i		0		0.165716	−	0.287028i
10.5	0.0518956	+	0.294314i		0		7.43361	−	2.70561i	−15.3604	−	12.8889i		0		−20.1945	−	7.35018i	2.37749	+	4.11794i		0		2.99626	−	5.18968i
10.6	0.404735	+	2.29536i		0		2.41266	−	0.878135i	4.78113	+	4.01185i		0		2.53990	+	0.924448i	12.3152	+	21.3306i		0		−7.27356	+	12.5982i
10.7	0.592608	+	3.36085i		0		−3.42657	+	1.24717i	8.41296	+	7.05931i		0		4.14024	+	1.50692i	7.42861	+	12.8667i		0		−18.7397	+	32.4581i
10.8	0.874714	+	4.96075i		0		−16.3263	+	5.94230i	−11.9326	−	10.0126i		0		−5.29732	−	1.92807i	−23.6100	−	40.8938i		0		39.2325	−	67.9527i
19.1	−4.06758	+	1.48048i		0		8.22505	−	6.90164i	0.745703	+	4.22909i		0		−15.6540	−	13.1353i	−5.92381	+	10.2603i		0		−9.29428	−	16.0982i
19.2	−3.53135	+	1.28531i		0		4.69008	−	3.93544i	−1.06026	−	6.01302i		0		13.2194	+	11.0924i	3.52788	−	6.11047i		0		11.4727	+	19.8713i
19.3	−1.82340	+	0.663664i		0		−3.24401	+	2.72205i	0.470388	+	2.66770i		0		−9.12942	−	7.66050i	11.8703	−	20.5600i		0		−2.62817	−	4.55212i
19.4	0.932125	−	0.339266i		0		−5.37460	+	4.50982i	−0.0250883	−	0.142283i		0		19.4381	+	16.3105i	−7.44756	+	12.8996i		0		−0.0716572	−	0.124114i
19.5	0.982993	−	0.357780i		0		−5.29009	+	4.43891i	−2.68017	−	15.2000i		0		−18.0349	−	15.1331i	−7.79628	+	13.5036i		0		−8.07284	−	13.9826i
19.6	2.12171	−	0.772239i		0		−2.22306	+	1.86537i	3.33920	+	18.9376i		0		0.500915	+	0.420318i	−12.3077	+	21.3175i		0		21.7091	+	37.6013i
19.7	4.05233	−	1.47493i		0		8.11761	−	6.81148i	−3.22691	−	18.3007i		0		14.4087	+	12.0903i	5.59920	−	9.69809i		0		−40.0687	−	69.4011i
19.8	4.97861	−	1.81207i		0		15.3746	−	12.9008i	1.82513	+	10.3508i		0		−5.92244	−	4.96952i	31.9745	−	55.3815i		0		27.8430	+	48.2255i
37.1	−4.06576	+	3.41158i		0		3.50237	−	19.8629i	5.18200	−	1.88609i		0		4.27574	+	24.2490i	32.2942	+	55.9352i		0		−14.6342	+	25.3472i
37.2	−2.44194	+	2.04903i		0		0.375355	−	2.12875i	−6.73772	+	2.45233i		0		0.843459	+	4.78350i	−9.30563	−	16.1178i		0		11.4282	−	19.7942i
37.3	−2.26133	+	1.89748i		0		0.123997	−	0.703222i	16.0156	−	5.82920i		0		−2.22794	−	12.6353i	−10.7539	−	18.6263i		0		−25.1558	+	43.5711i
37.4	−0.644303	+	0.540635i		0		−1.26634	+	7.18180i	−10.1642	+	3.69948i		0		−4.63497	−	26.2862i	−6.43113	−	11.1390i		0		4.54878	−	7.87873i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.4.e.a		48
3.b	odd	2	1		27.4.e.a	✓	48
9.c	even	3	1		243.4.e.a		48
9.c	even	3	1		243.4.e.b		48
9.d	odd	6	1		243.4.e.c		48
9.d	odd	6	1		243.4.e.d		48
27.e	even	9	1	inner	81.4.e.a		48
27.e	even	9	1		243.4.e.a		48
27.e	even	9	1		243.4.e.b		48
27.e	even	9	1		729.4.a.c		24
27.f	odd	18	1		27.4.e.a	✓	48
27.f	odd	18	1		243.4.e.c		48
27.f	odd	18	1		243.4.e.d		48
27.f	odd	18	1		729.4.a.d		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.4.e.a	✓	48	3.b	odd	2	1
27.4.e.a	✓	48	27.f	odd	18	1
81.4.e.a		48	1.a	even	1	1	trivial
81.4.e.a		48	27.e	even	9	1	inner
243.4.e.a		48	9.c	even	3	1
243.4.e.a		48	27.e	even	9	1
243.4.e.b		48	9.c	even	3	1
243.4.e.b		48	27.e	even	9	1
243.4.e.c		48	9.d	odd	6	1
243.4.e.c		48	27.f	odd	18	1
243.4.e.d		48	9.d	odd	6	1
243.4.e.d		48	27.f	odd	18	1
729.4.a.c		24	27.e	even	9	1
729.4.a.d		24	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(81, [\chi])\).