Newform orbit 76.5.l.a

• Label: 76.5.l.a
• Level: $76$
• Weight: $5$
• Character orbit: 76.l
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−3.99987	−	0.0325330i	−1.55169	+	0.273604i	15.9979	+	0.260256i	3.23088	−	2.71103i	6.21545	−	1.04390i	4.97171	+	2.87042i	−63.9809	−	1.56145i	−73.7822	+	26.8545i	−13.0113	+	10.7387i
23.2	−3.99725	−	0.148413i	8.88596	−	1.56683i	15.9559	+	1.18649i	−25.4406	+	21.3472i	−35.7519	+	4.94423i	−18.4296	−	10.6403i	−63.6038	−	7.11074i	0.390232	−	0.142033i	104.860	−	81.5542i
23.3	−3.94825	+	0.641375i	−13.6721	+	2.41076i	15.1773	−	5.06461i	3.65652	−	3.06818i	52.4346	−	18.2872i	76.9302	+	44.4157i	−56.6753	+	29.7306i	105.000	−	38.2168i	−12.4690	+	14.4591i
23.4	−3.86299	−	1.03792i	13.6381	−	2.40477i	13.8454	+	8.01895i	29.8632	−	25.0582i	−55.1799	−	4.86566i	54.4581	+	31.4414i	−45.1618	−	45.3476i	104.100	−	37.8894i	−141.370	+	65.8041i
23.5	−3.62639	−	1.68799i	−13.6381	+	2.40477i	10.3014	+	12.2426i	29.8632	−	25.0582i	53.5163	+	14.3004i	−54.4581	−	31.4414i	−16.6913	−	61.7851i	104.100	−	37.8894i	−150.594	+	40.4619i
23.6	−3.55482	+	1.83391i	−12.3954	+	2.18564i	9.27354	−	13.0385i	−26.3005	+	22.0687i	40.0551	−	30.5016i	−82.9404	−	47.8857i	−9.05443	+	63.3563i	72.7534	−	26.4801i	53.0215	−	126.683i
23.7	−3.53287	+	1.87585i	3.32006	−	0.585416i	8.96237	−	13.2543i	31.7438	−	26.6362i	−10.6312	+	8.29614i	−51.6354	−	29.8117i	−6.79985	+	63.6377i	−65.4350	+	23.8164i	−62.1812	+	153.649i
23.8	−3.15747	−	2.45569i	−8.88596	+	1.56683i	3.93918	+	15.5075i	−25.4406	+	21.3472i	31.9048	+	16.8739i	18.4296	+	10.6403i	25.6438	−	58.6378i	0.390232	−	0.142033i	132.750	−	4.92883i
23.9	−3.08499	−	2.54614i	1.55169	−	0.273604i	3.03431	+	15.7096i	3.23088	−	2.71103i	−5.48357	−	3.10675i	−4.97171	−	2.87042i	30.6382	−	56.1899i	−73.7822	+	26.8545i	−16.8699	+	0.137212i
23.10	−3.06815	+	2.56640i	14.2932	−	2.52028i	2.82714	−	15.7482i	−7.84979	+	6.58676i	−37.3857	+	44.4148i	31.3486	+	18.0991i	31.7423	+	55.5736i	121.829	−	44.3421i	7.18009	−	40.3549i
23.11	−2.96694	+	2.68277i	−1.05794	+	0.186543i	1.60551	−	15.9192i	1.47493	−	1.23761i	2.63839	−	3.39166i	26.6122	+	15.3646i	37.9442	+	51.5387i	−75.0307	+	27.3089i	−1.05580	+	7.62880i
23.12	−2.61226	−	3.02920i	13.6721	−	2.41076i	−2.35216	+	15.8262i	3.65652	−	3.06818i	−43.0179	−	35.1181i	−76.9302	−	44.4157i	54.0851	−	34.2169i	105.000	−	38.2168i	−18.8459	−	3.06144i
23.13	−1.54434	−	3.68985i	12.3954	−	2.18564i	−11.2300	+	11.3968i	−26.3005	+	22.0687i	−27.2074	−	42.3618i	82.9404	+	47.8857i	59.3954	+	23.8368i	72.7534	−	26.4801i	122.047	+	62.9634i
23.14	−1.50056	−	3.70787i	−3.32006	+	0.585416i	−11.4966	+	11.1278i	31.7438	−	26.6362i	7.15261	+	11.4319i	51.6354	+	29.8117i	58.5118	+	25.9300i	−65.4350	+	23.8164i	−146.397	−	77.7326i
23.15	−1.47024	+	3.72000i	−11.9688	+	2.11041i	−11.6768	−	10.9386i	19.6524	−	16.4903i	9.74622	−	47.6266i	0.422999	+	0.244219i	57.8592	−	27.3552i	62.6822	−	22.8144i	32.4501	+	97.3515i
23.16	−1.33273	+	3.77145i	−4.88717	+	0.861739i	−12.4477	−	10.0526i	−29.7827	+	24.9906i	3.26326	−	19.5802i	20.5177	+	11.8459i	54.5024	−	33.5483i	−52.9733	+	19.2807i	−54.5587	−	145.630i
23.17	−1.14588	+	3.83236i	9.44677	−	1.66572i	−13.3739	−	8.78286i	−12.8360	+	10.7707i	−4.44126	+	38.1121i	−66.0612	−	38.1405i	48.9840	−	41.1894i	10.3517	−	3.76773i	−26.5686	−	61.5342i
23.18	−0.700689	−	3.93815i	−14.2932	+	2.52028i	−15.0181	+	5.51884i	−7.84979	+	6.58676i	19.9403	+	54.5229i	−31.3486	−	18.0991i	32.2570	+	55.2764i	121.829	−	44.3421i	31.4399	+	26.2984i
23.19	−0.548361	−	3.96223i	1.05794	−	0.186543i	−15.3986	+	4.34547i	1.47493	−	1.23761i	−1.31926	−	4.08950i	−26.6122	−	15.3646i	25.6618	+	58.6300i	−75.0307	+	27.3089i	−5.71249	−	5.16535i
23.20	−0.324370	+	3.98683i	10.8623	−	1.91531i	−15.7896	−	2.58641i	23.2093	−	19.4749i	4.11262	+	43.9272i	10.4065	+	6.00818i	15.4332	−	62.1113i	38.2053	−	13.9056i	70.1147	+	98.8485i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.e	even	9	1	inner
76.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	76.5.l.a	✓	228
4.b	odd	2	1	inner	76.5.l.a	✓	228
19.e	even	9	1	inner	76.5.l.a	✓	228
76.l	odd	18	1	inner	76.5.l.a	✓	228

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.5.l.a	✓	228	1.a	even	1	1	trivial
76.5.l.a	✓	228	4.b	odd	2	1	inner
76.5.l.a	✓	228	19.e	even	9	1	inner
76.5.l.a	✓	228	76.l	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(76, [\chi])\).