Newform orbit 27.8.e.a

• Label: 27.8.e.a
• Level: $27$
• Weight: $8$
• Character orbit: 27.e
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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	−20.1276	−	7.32584i	46.7116	−	2.24141i	253.398	+	212.626i	26.6624	−	151.210i	−956.612	−	297.088i	−1123.74	+	942.932i	−2171.79	−	3761.65i	2176.95	−	209.400i	−1644.39	+	2848.17i
4.2	−19.1819	−	6.98163i	−45.3621	−	11.3703i	221.148	+	185.565i	73.2555	−	415.452i	790.746	+	534.805i	1090.38	−	914.934i	−1640.05	−	2840.66i	1928.43	+	1031.56i	−4305.71	+	7457.72i
4.3	−16.4594	−	5.99073i	7.02376	+	46.2349i	136.969	+	114.931i	−40.0449	+	227.106i	161.374	−	803.077i	882.649	−	740.631i	−444.907	−	770.602i	−2088.33	+	649.486i	2019.65	−	3498.13i
4.4	−15.5171	−	5.64776i	−4.90318	−	46.5076i	110.829	+	92.9968i	−48.1497	+	273.071i	−186.581	+	749.355i	−92.9707	+	78.0117i	−137.696	−	238.496i	−2138.92	+	456.070i	2289.38	−	3965.33i
4.5	−13.9790	−	5.08793i	−44.6310	+	13.9670i	71.4707	+	59.9711i	−48.7897	+	276.700i	694.958	+	31.8356i	−1113.53	+	934.359i	258.112	+	447.063i	1796.85	−	1246.72i	2089.86	−	3619.74i
4.6	−9.62864	−	3.50454i	31.7200	−	34.3634i	−17.6248	−	14.7890i	40.7285	−	230.983i	−425.848	+	219.708i	500.665	−	420.108i	773.656	+	1340.01i	−174.680	−	2180.01i	−1201.65	+	2081.32i
4.7	−9.32840	−	3.39526i	39.8042	+	24.5484i	−22.5624	−	18.9321i	12.2615	−	69.5386i	−287.961	−	364.143i	48.0590	−	40.3263i	781.525	+	1353.64i	981.750	+	1954.26i	−350.482	+	607.053i
4.8	−9.04432	−	3.29186i	−20.3312	+	42.1146i	−27.0904	−	22.7315i	82.6399	−	468.674i	322.518	−	313.970i	−558.283	+	468.455i	786.170	+	1361.69i	−1360.28	−	1712.48i	−2290.23	+	3966.80i
4.9	−2.79327	−	1.01667i	−34.1687	−	31.9296i	−91.2849	−	76.5971i	17.7337	−	100.573i	62.9806	+	123.926i	−71.4548	+	59.9577i	367.352	+	636.273i	147.997	+	2181.99i	−151.785	+	262.899i
4.10	−2.03404	−	0.740329i	45.5454	−	10.6122i	−94.4645	−	79.2651i	−90.6912	+	514.336i	−100.497	−	12.1330i	−162.436	+	136.300i	271.995	+	471.109i	1961.76	−	966.672i	565.246	−	979.036i
4.11	−0.493529	−	0.179630i	−38.9037	+	25.9520i	−97.8424	−	82.0995i	−47.2754	+	268.112i	23.8618	−	5.81978i	932.688	−	782.618i	67.1535	+	116.313i	839.991	−	2019.25i	71.4927	−	123.829i
4.12	4.29826	+	1.56444i	11.1293	+	45.4218i	−82.0261	−	68.8281i	−20.3422	+	115.366i	−23.2230	+	212.646i	−610.300	+	512.103i	−537.635	−	931.211i	−1939.28	+	1011.02i	−267.920	+	464.051i
4.13	6.42719	+	2.33931i	21.8338	−	41.3556i	−62.2172	−	52.2064i	40.2664	−	228.362i	237.073	−	214.725i	−1346.32	+	1129.70i	−715.495	−	1239.27i	−1233.57	−	1805.90i	793.009	−	1373.53i
4.14	7.46251	+	2.71613i	42.7022	+	19.0664i	−49.7420	−	41.7385i	71.4552	−	405.243i	266.878	+	258.268i	898.235	−	753.709i	−766.085	−	1326.90i	1459.95	+	1628.35i	1633.93	−	2830.04i
4.15	9.90100	+	3.60367i	7.82827	−	46.1055i	−13.0104	−	10.9170i	−29.9915	+	170.090i	243.657	−	428.280i	1223.81	−	1026.90i	−763.805	−	1322.95i	−2064.44	−	721.852i	−909.893	+	1575.98i
4.16	11.7328	+	4.27037i	−44.8117	+	13.3756i	21.3678	+	17.9297i	58.4278	−	331.360i	−582.884	−	34.4297i	−23.9260	+	20.0763i	−624.951	−	1082.45i	1829.18	−	1198.77i	2100.55	−	3638.26i
4.17	14.3397	+	5.21922i	−40.6699	−	23.0860i	80.3327	+	67.4072i	−74.8052	+	424.241i	−462.703	−	543.310i	−616.615	+	517.401i	−176.505	−	305.716i	1121.08	+	1877.81i	−3286.89	+	5693.06i
4.18	16.7921	+	6.11184i	46.7511	−	1.15397i	146.567	+	122.985i	−21.3907	+	121.313i	792.104	+	266.358i	−285.632	+	239.674i	565.850	+	980.080i	2184.34	−	107.899i	−1100.64	+	1906.36i
4.19	17.6042	+	6.40739i	−7.16273	+	46.2136i	170.798	+	143.317i	−14.0863	+	79.8875i	−422.203	+	767.657i	245.838	−	206.282i	889.502	+	1540.66i	−2084.39	−	662.031i	−759.849	+	1316.10i
4.20	20.7458	+	7.55086i	−11.1469	−	45.4175i	275.320	+	231.021i	74.3769	−	421.812i	111.689	−	1026.39i	181.714	−	152.476i	2554.39	+	4424.34i	−1938.49	+	1012.53i	4728.06	−	8189.24i

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	27.8.e.a	✓	120
3.b	odd	2	1		81.8.e.a		120
27.e	even	9	1	inner	27.8.e.a	✓	120
27.f	odd	18	1		81.8.e.a		120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.8.e.a	✓	120	1.a	even	1	1	trivial
27.8.e.a	✓	120	27.e	even	9	1	inner
81.8.e.a		120	3.b	odd	2	1
81.8.e.a		120	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(27, [\chi])\).