Newform orbit 229.4.h.a

• Label: 229.4.h.a
• Level: $229$
• Weight: $4$
• Character orbit: 229.h
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.h (of order \(38\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(56\) over \(\Q(\zeta_{38})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{38}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 19 q^{2} - 13 q^{3} + 187 q^{4} - 101 q^{5} - 19 q^{6} - 19 q^{7} - 19 q^{8} - 481 q^{9}+O(q^{10}) \)

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	−5.12121	+	2.24637i	4.75145	+	7.27264i	15.7623	−	17.1225i	−15.4638	−	2.58046i	−40.6702	−	26.5712i	0.441155	−	0.566795i	−27.7325	+	80.7820i	−19.4692	+	44.3853i	84.9902	−	21.5224i
4.2	−4.94289	+	2.16815i	−0.404374	−	0.618941i	14.3130	−	15.5481i	−0.992626	−	0.165640i	3.34074	+	2.18261i	20.5920	−	26.4567i	−23.0166	+	67.0449i	10.6262	−	24.2253i	5.26557	−	1.33342i
4.3	−4.77482	+	2.09443i	0.636128	+	0.973666i	12.9940	−	14.1153i	13.6738	+	2.28175i	−5.07667	−	3.31675i	−8.12755	+	10.4423i	−18.9368	+	55.1611i	10.3024	−	23.4871i	−70.0687	+	17.7438i
4.4	−4.75481	+	2.08565i	−2.16015	−	3.30636i	12.8400	−	13.9480i	−13.1543	−	2.19506i	17.1670	+	11.2158i	−15.5734	+	20.0087i	−18.4742	+	53.8136i	4.58002	−	10.4414i	67.1243	−	16.9982i
4.5	−4.46356	+	1.95790i	−4.85394	−	7.42951i	10.6717	−	11.5926i	12.4944	+	2.08495i	36.2121	+	23.6585i	−0.625298	+	0.803383i	−12.2759	+	35.7585i	−20.7911	+	47.3989i	−59.8518	+	15.1565i
4.6	−4.46249	+	1.95743i	−3.74700	−	5.73522i	10.6640	−	11.5842i	−12.8757	−	2.14858i	27.9473	+	18.2589i	0.906430	−	1.16458i	−12.2550	+	35.6975i	−8.00689	+	18.2539i	61.6635	−	15.6153i
4.7	−4.27022	+	1.87309i	3.02027	+	4.62287i	9.30807	−	10.1113i	3.05575	+	0.509915i	−21.5563	−	14.0834i	−8.59538	+	11.0433i	−8.69570	+	25.3297i	−1.40310	+	3.19874i	−14.0039	+	3.54626i
4.8	−3.99106	+	1.75064i	4.62399	+	7.07754i	7.44556	−	8.08804i	10.7050	+	1.78635i	−30.8448	−	20.1519i	10.5575	−	13.5642i	−4.23578	+	12.3384i	−17.8645	+	40.7270i	−45.8517	+	11.6112i
4.9	−3.72170	+	1.63249i	−0.949850	−	1.45385i	5.76779	−	6.26549i	2.39551	+	0.399741i	5.90846	+	3.86019i	16.7801	−	21.5591i	−0.681028	+	1.98377i	9.63430	−	21.9640i	−9.56796	+	2.42294i
4.10	−3.70152	+	1.62364i	1.86455	+	2.85391i	5.64678	−	6.13404i	−15.3518	−	2.56176i	−11.5354	−	7.53643i	3.97073	−	5.10159i	−0.442832	+	1.28993i	6.17754	−	14.0834i	60.9842	−	15.4433i
4.11	−3.49464	+	1.53289i	1.45636	+	2.22913i	4.44449	−	4.82801i	−10.8716	−	1.81414i	−8.50646	−	5.55755i	−4.44030	+	5.70490i	1.78147	−	5.18925i	7.99776	−	18.2331i	40.7731	−	10.3252i
4.12	−3.27229	+	1.43536i	−1.63642	−	2.50473i	3.22938	−	3.50804i	2.30468	+	0.384584i	8.95003	+	5.84735i	−17.4344	+	22.3997i	3.74971	−	10.9225i	7.24998	−	16.5283i	−8.09362	+	2.04958i
4.13	−3.21576	+	1.41056i	−4.71456	−	7.21617i	2.93316	−	3.18626i	−6.36016	−	1.06132i	25.3397	+	16.5553i	16.5645	−	21.2821i	4.18360	−	12.1864i	−19.0003	+	43.3162i	21.9498	−	5.55845i
4.14	−3.20398	+	1.40540i	−1.32003	−	2.02045i	2.87210	−	3.11994i	20.2335	+	3.37637i	7.06887	+	4.61832i	8.79307	−	11.2973i	4.27069	−	12.4401i	8.50603	−	19.3918i	−69.5728	+	17.6182i
4.15	−2.71813	+	1.19228i	−3.52790	−	5.39985i	0.548450	−	0.595775i	6.73660	+	1.12414i	16.0274	+	10.4713i	−10.8753	+	13.9726i	6.92957	−	20.1851i	−5.86655	+	13.3744i	−19.6513	+	4.97638i
4.16	−2.58107	+	1.13216i	5.17494	+	7.92083i	−0.0381227	+	0.0414123i	−10.5413	−	1.75904i	−22.3245	−	14.5853i	−21.8412	+	28.0616i	7.37273	−	21.4760i	−25.1138	+	57.2537i	29.1994	−	7.39429i
4.17	−2.38852	+	1.04770i	3.73392	+	5.71519i	−0.810907	+	0.880880i	3.16529	+	0.528194i	−14.9064	−	9.73882i	9.90162	−	12.7216i	7.78901	−	22.6886i	−7.87549	+	17.9543i	−8.11376	+	2.05468i
4.18	−2.17224	+	0.952832i	3.81878	+	5.84508i	−1.60752	+	1.74623i	21.7216	+	3.62470i	−13.8647	−	9.05825i	−15.6717	+	20.1349i	7.98962	−	23.2730i	−8.73607	+	19.9162i	−50.6383	+	12.8234i
4.19	−2.15189	+	0.943907i	−2.10958	−	3.22895i	−1.67857	+	1.82342i	−16.1857	−	2.70092i	7.58740	+	4.95710i	6.65490	−	8.55021i	7.99483	−	23.2881i	4.87000	−	11.1025i	37.3793	−	9.46573i
4.20	−1.79452	+	0.787150i	4.64909	+	7.11596i	−2.81755	+	3.06068i	−16.0048	−	2.67073i	−13.9442	−	9.11020i	19.5607	−	25.1316i	7.73711	−	22.5374i	−18.1771	+	41.4395i	30.8233	−	7.80551i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
229.h	even	38	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	229.4.h.a	✓	1008
229.h	even	38	1	inner	229.4.h.a	✓	1008

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
229.4.h.a	✓	1008	1.a	even	1	1	trivial
229.4.h.a	✓	1008	229.h	even	38	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(229, [\chi])\).