Newform orbit 29.5.f.a

• Label: 29.5.f.a
• Level: $29$
• Weight: $5$
• Character orbit: 29.f
• Analytic conductor: $2.998$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99772892943\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(9\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} - 176 q^{8} - 14 q^{9}+O(q^{10}) \)

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	−6.73327	−	0.758657i	3.70958	+	2.33088i	29.1625	+	6.65615i	−12.0904	+	9.64178i	−23.2093	−	18.5088i	−19.9638	−	87.4670i	−88.9790	−	31.1351i	−26.8166	−	55.6853i	88.7228	−	55.7482i
2.2	−5.52783	−	0.622837i	−9.47794	−	5.95539i	14.5702	+	3.32554i	1.89549	−	1.51160i	48.6833	+	38.8236i	12.4209	+	54.4193i	5.54018	+	1.93859i	19.2202	+	39.9112i	−11.4194	+	7.17529i
2.3	−3.44049	−	0.387651i	12.4787	+	7.84089i	−3.91212	−	0.892916i	−6.64868	+	5.30215i	−39.8934	−	31.8139i	15.0152	+	65.7861i	65.4011	+	22.8848i	59.0939	+	122.710i	24.9301	−	15.6646i
2.4	−1.95865	−	0.220687i	1.47124	+	0.924442i	−11.8112	−	2.69584i	33.2452	−	26.5122i	−2.67763	−	2.13534i	−9.18650	−	40.2487i	52.3061	+	18.3027i	−33.8346	−	70.2583i	−70.9665	+	44.5912i
2.5	0.272341	+	0.0306854i	−4.11089	−	2.58304i	−15.5256	−	3.54362i	−22.7686	+	18.1573i	−1.04030	−	0.829612i	1.28534	+	5.63143i	−8.25847	−	2.88976i	−24.9173	−	51.7413i	−6.75797	+	4.24631i
2.6	3.63289	+	0.409328i	−14.5226	−	9.12514i	−2.56850	−	0.586244i	18.8745	−	15.0519i	−49.0237	−	39.0951i	−8.09747	−	35.4773i	−64.3027	−	22.5005i	92.4924	+	192.062i	74.7302	−	46.9561i
2.7	4.11921	+	0.464124i	10.8013	+	6.78691i	1.15364	+	0.263311i	−9.08180	+	7.24250i	41.3429	+	32.9698i	−10.6222	−	46.5391i	−57.9726	−	20.2855i	35.4614	+	73.6364i	−40.7713	+	25.6183i
2.8	5.03785	+	0.567629i	2.02228	+	1.27068i	9.45888	+	2.15893i	20.0579	−	15.9956i	9.46665	+	7.54940i	16.1221	+	70.6355i	−30.1367	−	10.5453i	−32.6696	−	67.8391i	110.128	−	69.1981i
2.9	7.67142	+	0.864362i	−3.80559	−	2.39121i	42.5048	+	9.70144i	−20.0125	+	15.9594i	−27.1274	−	21.6334i	−10.9905	−	48.1524i	201.099	+	70.3674i	−26.3800	−	54.7786i	−167.319	+	105.133i
3.1	−6.34646	−	3.98774i	13.1212	−	4.59131i	17.4333	+	36.2006i	33.6799	−	7.68723i	−101.582	−	23.1855i	−16.2405	−	7.82103i	20.2917	−	180.094i	87.7577	−	69.9844i	−244.403	−	85.5203i
3.2	−4.59434	−	2.88681i	−1.03420	+	0.361882i	5.83212	+	12.1105i	−26.4459	+	6.03611i	5.79614	+	1.32293i	87.2324	+	42.0089i	−1.55425	+	13.7943i	−62.3897	+	49.7542i	138.927	+	48.6126i
3.3	−4.55042	−	2.85922i	−14.5612	+	5.09520i	5.58904	+	11.6058i	31.9975	−	7.30323i	80.8280	+	18.4485i	−8.96993	−	4.31969i	−1.87649	+	16.6543i	122.741	−	97.8823i	−166.484	−	58.2552i
3.4	−3.22425	−	2.02593i	2.10719	−	0.737336i	−0.650743	−	1.35128i	−7.14131	+	1.62996i	−8.28789	−	1.89166i	−67.7403	−	32.6220i	−7.46106	+	66.2187i	−59.4318	+	47.3953i	26.3276	+	9.21241i
3.5	−0.159758	−	0.100383i	14.7543	−	5.16274i	−6.92669	−	14.3834i	−13.1719	+	3.00639i	−2.87536	−	0.656282i	17.6244	+	8.48744i	−0.675254	+	5.99305i	127.706	−	101.842i	2.40610	+	0.841931i
3.6	0.850861	+	0.534631i	0.0346598	−	0.0121280i	−6.50401	−	13.5057i	43.9571	−	10.0329i	0.0359746	+	0.00821098i	39.2865	+	18.9194i	3.48675	−	30.9458i	−63.3273	+	50.5018i	42.7653	+	14.9642i
3.7	1.13138	+	0.710891i	−9.69350	+	3.39190i	−6.16749	−	12.8069i	−21.3012	+	4.86187i	−13.3783	−	3.05351i	−18.6509	−	8.98181i	4.52025	−	40.1184i	19.1306	−	15.2561i	−27.5560	−	9.64226i
3.8	4.83745	+	3.03957i	6.19652	−	2.16826i	7.21978	+	14.9920i	−2.78296	+	0.635192i	36.5659	+	8.34593i	−22.5098	−	10.8401i	−0.409326	+	3.63287i	−29.6329	+	23.6314i	−15.3931	−	5.38629i
3.9	5.82640	+	3.66097i	−12.7067	+	4.44627i	13.6021	+	28.2451i	1.33657	−	0.305063i	−90.3121	−	20.6132i	58.6150	+	28.2275i	−11.8260	+	104.958i	78.3631	−	62.4925i	8.90421	+	3.11572i
8.1	−7.20083	−	2.51968i	−0.768193	+	6.81790i	32.9939	+	26.3118i	−6.41700	−	13.3250i	22.7106	−	47.1589i	−26.2859	−	32.9614i	−106.345	−	169.247i	33.0755	+	7.54928i	12.6329	+	112.120i
8.2	−5.35553	−	1.87398i	1.86096	−	16.5165i	12.6606	+	10.0965i	−0.579591	−	1.20353i	−40.9179	+	84.9670i	−13.0418	−	16.3539i	−0.584210	−	0.929766i	−190.361	−	43.4486i	0.848619	+	7.53170i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.f	odd	28	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	29.5.f.a	✓	108
29.f	odd	28	1	inner	29.5.f.a	✓	108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.5.f.a	✓	108	1.a	even	1	1	trivial
29.5.f.a	✓	108	29.f	odd	28	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(29, [\chi])\).