Newform orbit 229.4.i.a

• Label: 229.4.i.a
• Level: $229$
• Weight: $4$
• Character orbit: 229.i
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $2052$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.i (of order \(57\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(2052\)
Relative dimension:	\(57\) over \(\Q(\zeta_{57})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{57}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2052 q - 30 q^{2} - 33 q^{3} - 514 q^{4} - 99 q^{5} - 33 q^{6} - 43 q^{7} + 46 q^{8} + 464 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−4.79781	−	2.59644i	−5.27411	+	5.13073i	11.9019	+	18.2172i	4.16414	+	13.3135i	38.6258	−	10.9223i	0.979568	−	2.61631i	−6.19906	−	74.8116i	0.747841	−	27.1303i	14.5890	−	74.6876i
3.2	−4.73477	−	2.56233i	−0.107397	+	0.104477i	11.4769	+	17.5668i	−3.86773	−	12.3658i	0.776202	−	0.219489i	−2.74867	+	7.34138i	−5.77221	−	69.6602i	−0.743349	+	26.9673i	−13.3725	+	68.4597i
3.3	−4.57408	−	2.47537i	3.93528	−	3.82830i	10.4192	+	15.9478i	5.15540	+	16.4827i	−27.4768	+	7.76969i	0.203543	−	0.543638i	−4.74572	−	57.2723i	0.0865862	−	3.14119i	17.2196	−	88.1548i
3.4	−4.49506	−	2.43260i	6.64898	−	6.46823i	9.91239	+	15.1720i	−4.70999	−	15.0587i	−45.6222	+	12.9007i	4.42313	−	11.8136i	−4.27264	−	51.5631i	1.62700	−	59.0244i	−15.4601	+	79.1471i
3.5	−4.29957	−	2.32681i	−1.37486	+	1.33748i	8.69669	+	13.3113i	−1.61874	−	5.17541i	9.02337	−	2.55157i	8.44801	−	22.5636i	−3.18950	−	38.4916i	−0.642592	+	23.3121i	−5.08231	+	26.0186i
3.6	−4.20663	−	2.27651i	4.80572	−	4.67507i	8.13761	+	12.4555i	−0.320933	−	1.02608i	−30.8587	+	8.72601i	−7.16052	+	19.1249i	−2.71680	−	32.7869i	0.494633	−	17.9444i	−0.985839	+	5.04694i
3.7	−3.99232	−	2.16053i	−5.95093	+	5.78915i	6.89511	+	10.5538i	−2.05374	−	6.56617i	36.2657	−	10.2549i	2.06026	−	5.50272i	−1.72682	−	20.8396i	1.15529	−	41.9118i	−5.98726	+	30.6514i
3.8	−3.94279	−	2.13373i	−1.06676	+	1.03776i	6.61718	+	10.1283i	2.59467	+	8.29564i	6.42030	−	1.81549i	−11.9853	+	32.0113i	−1.51727	−	18.3107i	−0.682937	+	24.7757i	7.47039	−	38.2443i
3.9	−3.93475	−	2.12938i	−5.06435	+	4.92668i	6.57241	+	10.0598i	−4.78240	−	15.2902i	30.4177	−	8.60131i	−8.13419	+	21.7255i	−1.48394	−	17.9085i	0.631539	−	22.9111i	−13.7411	+	70.3466i
3.10	−3.87415	−	2.09659i	3.33880	−	3.24803i	6.23780	+	9.54766i	2.16399	+	6.91865i	−19.7448	+	5.58329i	12.0737	−	32.2473i	−1.23853	−	14.9469i	−0.146113	+	5.30070i	6.12193	−	31.3409i
3.11	−3.50020	−	1.89421i	−2.79213	+	2.71623i	4.28777	+	6.56292i	2.73826	+	8.75469i	14.9181	−	4.21844i	5.11089	−	13.6506i	0.0527682	+	0.636818i	−0.325861	+	11.8216i	6.99881	−	35.8300i
3.12	−3.00437	−	1.62588i	3.04302	−	2.96030i	2.00714	+	3.07216i	−0.149107	−	0.476721i	−13.9555	+	3.94623i	−3.10771	+	8.30031i	1.22157	+	14.7421i	−0.247345	+	8.97323i	−0.327121	+	1.67468i
3.13	−2.99200	−	1.61919i	1.29863	−	1.26333i	1.95469	+	2.99188i	−5.16013	−	16.4979i	−5.93107	+	1.67715i	0.408789	−	1.09183i	1.24348	+	15.0066i	−0.653522	+	23.7086i	−11.2740	+	57.7168i
3.14	−2.79604	−	1.51314i	−1.94337	+	1.89054i	1.15265	+	1.76426i	4.73127	+	15.1267i	8.29438	−	2.34543i	−2.76899	+	7.39564i	1.54702	+	18.6698i	−0.541420	+	19.6417i	9.66001	−	49.4539i
3.15	−2.65021	−	1.43422i	−7.04533	+	6.85380i	0.591039	+	0.904653i	2.05030	+	6.55517i	28.5015	−	8.05945i	−2.59814	+	6.93933i	1.72186	+	20.7797i	1.91809	−	69.5848i	3.96785	−	20.3132i
3.16	−2.52316	−	1.36546i	5.99563	−	5.83264i	0.126242	+	0.193229i	−1.01221	−	3.23621i	−23.0922	+	6.52984i	−2.76192	+	7.37676i	1.84064	+	22.2132i	1.18393	−	42.9509i	−1.86497	+	9.54759i
3.17	−2.28317	−	1.23559i	6.68643	−	6.50466i	−0.689394	−	1.05520i	5.48079	+	17.5231i	−23.3034	+	6.58957i	−5.27938	+	14.1006i	1.98527	+	23.9586i	1.65378	−	59.9962i	9.13775	−	46.7802i
3.18	−2.22113	−	1.20201i	−4.70160	+	4.57378i	−0.887021	−	1.35769i	−4.21094	−	13.4631i	15.9406	−	4.50757i	11.3771	−	30.3869i	2.00667	+	24.2169i	0.441545	−	16.0184i	−6.82983	+	34.9649i
3.19	−2.06855	−	1.11945i	5.67973	−	5.52533i	−1.34983	−	2.06606i	−2.56564	−	8.20282i	−17.9341	+	5.07129i	10.3564	−	27.6608i	2.03318	+	24.5368i	0.986130	−	35.7750i	−3.87543	+	19.8401i
3.20	−1.73238	−	0.937518i	2.72537	−	2.65128i	−2.25338	−	3.44905i	5.31963	+	17.0078i	−7.20702	+	2.03795i	7.16729	−	19.1430i	1.97148	+	23.7922i	−0.345615	+	12.5383i	6.72950	−	34.4513i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
229.i	even	57	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	229.4.i.a	✓	2052
229.i	even	57	1	inner	229.4.i.a	✓	2052

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
229.4.i.a	✓	2052	1.a	even	1	1	trivial
229.4.i.a	✓	2052	229.i	even	57	1	inner

Hecke kernels

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