Newform orbit 81.5.h.a

• Label: 81.5.h.a
• Level: $81$
• Weight: $5$
• Character orbit: 81.h
• Analytic conductor: $8.373$
• Analytic rank: $0$
• Dimension: $630$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.h (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.37296700979\)
Analytic rank:	\(0\)
Dimension:	\(630\)
Relative dimension:	\(35\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 630 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−7.71559	−	0.449382i	−6.97310	−	5.68998i	43.4365	+	5.07700i	−7.64503	−	32.2569i	51.2446	+	47.0351i	24.7073	+	33.1877i	−211.077	−	37.2185i	16.2484	+	79.3536i	44.4902	+	252.317i
2.2	−7.55849	−	0.440232i	8.99915	−	0.123655i	41.0451	+	4.79749i	7.82671	+	33.0235i	−68.0744	−	3.02707i	52.3049	+	70.2576i	−188.827	−	33.2952i	80.9694	−	2.22557i	−44.6201	−	253.053i
2.3	−7.03180	−	0.409556i	−2.90231	+	8.51919i	33.3867	+	3.90235i	−1.13208	−	4.77662i	23.8976	−	58.7166i	−2.18531	−	2.93538i	−122.183	−	21.5442i	−64.1532	−	49.4507i	6.00427	+	34.0519i
2.4	−6.58789	−	0.383701i	0.257657	−	8.99631i	27.3612	+	3.19807i	5.40031	+	22.7857i	−5.14931	+	59.1678i	−40.4897	−	54.3871i	−75.0447	−	13.2324i	−80.8672	−	4.63593i	−26.8337	−	152.182i
2.5	−6.17799	−	0.359827i	6.86632	+	5.81839i	22.1462	+	2.58852i	−0.318134	−	1.34231i	−40.3264	−	38.4166i	−47.5806	−	63.9118i	−38.3766	−	6.76683i	13.2926	+	79.9019i	1.48243	+	8.40726i
2.6	−6.17723	−	0.359783i	7.90516	−	4.30214i	22.1369	+	2.58744i	−8.40125	−	35.4477i	−50.3799	+	23.7312i	−10.3710	−	13.9306i	−38.3150	−	6.75597i	43.9832	−	68.0182i	39.1430	+	221.991i
2.7	−5.63502	−	0.328203i	−8.39408	+	3.24646i	15.7540	+	1.84138i	5.01896	+	21.1767i	48.3663	−	15.5389i	4.54962	+	6.11120i	0.771436	+	0.136025i	59.9210	−	54.5020i	−21.3317	−	120.978i
2.8	−4.59881	−	0.267850i	−6.44405	−	6.28285i	5.18553	+	0.606102i	2.47764	+	10.4540i	27.9521	+	30.6197i	25.9842	+	34.9029i	48.9010	+	8.62257i	2.05158	+	80.9740i	−8.59410	−	48.7396i
2.9	−4.33543	−	0.252510i	−8.98142	+	0.578015i	2.84035	+	0.331989i	−9.04763	−	38.1749i	39.0842	−	0.238046i	−39.6102	−	53.2058i	56.1985	+	9.90930i	80.3318	−	10.3828i	29.5858	+	167.789i
2.10	−4.32349	−	0.251815i	2.76872	+	8.56354i	2.73735	+	0.319951i	−8.66399	−	36.5563i	−9.81412	−	37.7216i	53.9680	+	72.4916i	56.4860	+	9.96001i	−65.6684	+	47.4201i	28.2533	+	160.232i
2.11	−4.06669	−	0.236858i	4.66984	−	7.69367i	0.590049	+	0.0689669i	1.70338	+	7.18713i	−20.8131	+	30.1817i	32.5652	+	43.7427i	61.8039	+	10.8977i	−37.3853	−	71.8564i	−5.22479	−	29.6313i
2.12	−3.72738	−	0.217095i	3.31955	+	8.36544i	−2.04558	−	0.239094i	10.7778	+	45.4751i	−10.5571	−	31.9018i	0.513035	+	0.689125i	66.4043	+	11.7089i	−58.9612	+	55.5390i	−30.3005	−	171.843i
2.13	−2.98852	−	0.174062i	8.74786	+	2.11541i	−6.99084	−	0.817112i	−0.605697	−	2.55564i	−25.7750	−	7.84462i	2.63068	+	3.53362i	67.9198	+	11.9761i	72.0501	+	37.0106i	1.36530	+	7.74301i
2.14	−1.65883	−	0.0966160i	−1.26330	−	8.91090i	−13.1494	−	1.53695i	−6.75457	−	28.4998i	1.23467	+	14.9037i	−9.50523	−	12.7677i	47.8466	+	8.43664i	−77.8081	+	22.5143i	8.45117	+	47.9289i
2.15	−1.48523	−	0.0865049i	−3.70919	+	8.20012i	−13.6934	−	1.60053i	−3.01331	−	12.7141i	6.21835	−	11.8582i	−15.8634	−	21.3082i	43.6417	+	7.69521i	−53.4838	−	60.8316i	3.37562	+	19.1441i
2.16	−0.946526	−	0.0551289i	7.65218	−	4.73752i	−14.9989	−	1.75312i	4.59657	+	19.3945i	−7.50416	+	4.06233i	−32.6045	−	43.7955i	29.0399	+	5.12051i	36.1118	−	72.5048i	−3.28158	−	18.6108i
2.17	−0.836587	−	0.0487256i	−8.22208	−	3.66025i	−15.1943	−	1.77596i	7.80182	+	32.9185i	6.70014	+	3.46274i	−49.0013	−	65.8201i	25.8292	+	4.55439i	54.2052	+	60.1897i	−4.92293	−	27.9193i
2.18	−0.404846	−	0.0235796i	−6.78418	+	5.91396i	−15.7285	−	1.83839i	5.00281	+	21.1085i	2.88600	−	2.23427i	35.4210	+	47.5787i	12.7142	+	2.24185i	11.0502	−	80.2427i	−1.52764	−	8.66366i
2.19	1.01942	+	0.0593744i	−8.86585	−	1.54815i	−14.8561	−	1.73643i	−4.45851	−	18.8119i	−8.94610	−	2.10462i	32.3739	+	43.4857i	−31.1317	−	5.48935i	76.2065	+	27.4513i	−3.42814	−	19.4420i
2.20	1.34714	+	0.0784618i	5.56351	+	7.07442i	−14.0832	−	1.64609i	−6.74962	−	28.4789i	6.93973	+	9.96674i	−26.4108	−	35.4759i	−40.1056	−	7.07169i	−19.0948	+	78.7171i	−6.85816	−	38.8946i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
81.h	odd	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	81.5.h.a	✓	630
81.h	odd	54	1	inner	81.5.h.a	✓	630

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
81.5.h.a	✓	630	1.a	even	1	1	trivial
81.5.h.a	✓	630	81.h	odd	54	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(81, [\chi])\).