Newform orbit 79.6.g.a

• Label: 79.6.g.a
• Level: $79$
• Weight: $6$
• Character orbit: 79.g
• Analytic conductor: $12.670$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 79 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	79.g (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.6703217652\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(32\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 26 q^{2} - 34 q^{3} + 424 q^{4} - 25 q^{5} + 19 q^{6} + 10 q^{7} - 82 q^{8} + 2434 q^{9}+O(q^{10}) \)

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	−8.92087	−	6.70360i	−17.9786	−	2.92181i	25.7407	+	88.8673i	63.2029	+	39.9671i	140.798	+	146.586i	−141.032	+	172.733i	239.477	−	631.450i	84.1988	+	28.1097i	−295.901	−	780.228i
2.2	−8.82910	−	6.63464i	10.0653	+	1.63577i	25.0316	+	86.4191i	−63.6477	−	40.2484i	−78.0145	−	81.2217i	31.1732	−	38.1803i	227.032	−	598.634i	−131.860	−	44.0215i	294.918	+	777.636i
2.3	−7.49802	−	5.63440i	25.5851	+	4.15798i	15.5709	+	53.7570i	64.4806	+	40.7750i	−168.410	−	175.333i	71.4168	−	87.4697i	79.7098	−	210.177i	406.814	+	135.814i	−253.734	−	669.041i
2.4	−7.41087	−	5.56891i	−26.9260	−	4.37589i	15.0053	+	51.8042i	−37.5289	−	23.7318i	175.176	+	182.377i	131.496	−	161.053i	72.1003	−	190.113i	475.364	+	158.700i	145.961	+	384.868i
2.5	−7.00659	−	5.26511i	14.2530	+	2.31634i	12.4680	+	43.0444i	3.19641	+	2.02129i	−87.6693	−	91.2735i	−60.5441	+	74.1530i	39.8235	−	105.006i	−32.7110	−	10.9205i	−11.7536	−	30.9918i
2.6	−6.85566	−	5.15169i	−6.68315	−	1.08612i	11.5571	+	39.8999i	44.5016	+	28.1411i	40.2221	+	41.8756i	66.5692	−	81.5324i	29.0104	−	76.4942i	−187.009	−	62.4329i	−160.114	−	422.185i
2.7	−6.34152	−	4.76534i	−11.6509	−	1.89346i	8.60342	+	29.7024i	−46.3799	−	29.3289i	64.8616	+	67.5281i	−47.7074	+	58.4309i	−3.02846	+	7.98538i	−98.3352	−	32.8291i	154.357	+	407.006i
2.8	−4.35704	−	3.27410i	26.5430	+	4.31366i	−0.638917	−	2.20580i	−48.2595	−	30.5174i	−101.526	−	105.699i	−121.859	+	149.250i	−66.2824	+	174.772i	455.429	+	152.045i	110.351	+	290.972i
2.9	−4.25686	−	3.19882i	12.2025	+	1.98309i	−1.01457	−	3.50271i	−72.9390	−	46.1238i	−45.6006	−	47.4752i	108.954	−	133.444i	−67.3079	+	177.476i	−85.5272	−	28.5532i	162.949	+	429.662i
2.10	−3.73467	−	2.80642i	−27.6593	−	4.49507i	−2.83121	−	9.77449i	28.1077	+	17.7743i	90.6832	+	94.4112i	−63.7016	+	78.0203i	−69.8678	+	184.226i	514.336	+	171.711i	−55.0910	−	145.263i
2.11	−3.65126	−	2.74375i	−3.89898	−	0.633647i	−3.09937	−	10.7003i	51.1074	+	32.3184i	12.4977	+	13.0114i	44.9082	−	55.0026i	−69.8686	+	184.228i	−215.694	−	72.0092i	−97.9333	−	258.229i
2.12	−3.01262	−	2.26383i	9.69231	+	1.57515i	−4.95205	−	17.0964i	81.4590	+	51.5115i	−25.6333	−	26.6871i	−146.846	+	179.854i	−66.5462	+	175.468i	−139.035	−	46.4166i	−128.791	−	339.594i
2.13	−2.93449	−	2.20513i	18.6087	+	3.02421i	−5.15430	−	17.7947i	9.57623	+	6.05564i	−47.9384	−	49.9091i	100.774	−	123.426i	−65.7668	+	173.413i	106.644	+	35.6030i	−14.7479	−	38.8871i
2.14	−2.43118	−	1.82692i	−9.71073	−	1.57815i	−6.32992	−	21.8534i	−46.9499	−	29.6893i	20.7254	+	21.5775i	−78.7139	+	96.4070i	−59.0436	+	155.685i	−138.687	−	46.3004i	59.9040	+	157.954i
2.15	−0.697404	−	0.524065i	−17.9020	−	2.90936i	−8.69123	−	30.0056i	48.6695	+	30.7767i	10.9602	+	11.4108i	111.900	−	137.053i	−19.5626	+	51.5823i	81.5234	+	27.2165i	−17.8133	−	46.9698i
2.16	−0.00388293	−	0.00291784i	−17.4230	−	2.83152i	−8.90295	−	30.7366i	−70.4050	−	44.5214i	0.0593906	+	0.0618321i	123.249	−	150.952i	−0.110229	+	0.290651i	65.0500	+	21.7169i	0.143472	+	0.378304i
2.17	0.754931	+	0.567294i	5.62672	+	0.914432i	−8.65486	−	29.8800i	3.36047	+	2.12503i	3.72904	+	3.88234i	3.76376	−	4.60977i	21.1325	−	55.7218i	−199.671	−	66.6598i	1.33140	+	3.51062i
2.18	0.816620	+	0.613650i	27.6787	+	4.49822i	−8.61266	−	29.7343i	25.0839	+	15.8621i	19.8426	+	20.6584i	−1.19880	+	1.46827i	22.8044	−	60.1303i	515.381	+	172.059i	10.7502	+	28.3461i
2.19	0.974354	+	0.732179i	6.05228	+	0.983591i	−8.48968	−	29.3098i	−20.8009	−	13.1537i	5.17689	+	5.38972i	−65.8662	+	80.6714i	27.0181	−	71.2409i	−194.832	−	65.0444i	−10.6366	−	28.0463i
2.20	2.85800	+	2.14765i	−14.5044	−	2.35720i	−5.34719	−	18.4606i	64.4858	+	40.7784i	−36.3912	−	37.8872i	−75.2131	+	92.1193i	64.9313	−	171.210i	−25.6725	−	8.57073i	96.7228	+	255.037i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
79.g	even	39	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	79.6.g.a	✓	768
79.g	even	39	1	inner	79.6.g.a	✓	768

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
79.6.g.a	✓	768	1.a	even	1	1	trivial
79.6.g.a	✓	768	79.g	even	39	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(79, [\chi])\).