Newform orbit 224.4.bd.a

• Label: 224.4.bd.a
• Level: $224$
• Weight: $4$
• Character orbit: 224.bd
• Analytic conductor: $13.216$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	224.bd (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2164278413\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(94\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−2.82840	−	0.0123104i	−1.74699	+	2.27673i	7.99970	+	0.0696373i	1.25375	−	0.962039i	4.96923	−	6.41799i	−4.99803	−	17.8331i	−22.6255	−	0.295441i	4.85662	+	18.1251i	−3.55796	+	2.70560i
37.2	−2.82634	−	0.108734i	−5.15263	+	6.71503i	7.97635	+	0.614638i	−13.1063	+	10.0568i	15.2932	−	18.4187i	17.8518	−	4.93088i	−22.4770	−	2.60448i	−11.5540	−	43.1201i	38.1364	−	26.9989i
37.3	−2.80896	+	0.331234i	3.53338	−	4.60480i	7.78057	−	1.86085i	−8.59530	+	6.59541i	−8.39989	+	14.1051i	−17.0302	+	7.27828i	−21.2390	+	7.80425i	−1.73123	−	6.46104i	21.9593	−	21.3733i
37.4	−2.78871	+	0.472327i	−1.56649	+	2.04149i	7.55381	−	2.63437i	6.35230	−	4.87429i	3.40424	−	6.43303i	14.5574	+	11.4491i	−19.8211	+	10.9144i	5.27432	+	19.6840i	−15.4125	+	16.5933i
37.5	−2.78180	−	0.511440i	−5.18785	+	6.76093i	7.47686	+	2.84545i	5.70746	−	4.37949i	17.8894	−	16.1543i	−3.88508	+	18.1082i	−19.3439	−	11.7395i	−11.8084	−	44.0694i	−18.1169	+	9.26385i
37.6	−2.75723	−	0.630617i	3.30408	−	4.30596i	7.20465	+	3.47751i	11.9404	−	9.16218i	−11.8255	+	9.78892i	−17.9334	−	4.62528i	−17.6719	−	14.1317i	−0.636234	−	2.37446i	−38.7002	+	17.7324i
37.7	−2.74124	+	0.696867i	3.09604	−	4.03484i	7.02875	−	3.82055i	−5.04365	+	3.87013i	−5.67523	+	13.2180i	15.3437	+	10.3716i	−16.6051	+	15.3711i	0.293667	+	1.09598i	11.1289	−	14.1237i
37.8	−2.73501	−	0.720921i	4.76254	−	6.20666i	6.96055	+	3.94345i	11.1293	−	8.53978i	−17.5001	+	13.5419i	6.18647	+	17.4564i	−16.1942	−	15.8034i	−8.85277	−	33.0390i	−36.5951	+	15.3331i
37.9	−2.72441	−	0.759977i	3.73117	−	4.86256i	6.84487	+	4.14098i	−15.1236	+	11.6048i	−13.8607	+	10.4120i	2.66113	−	18.3281i	−15.5012	−	16.4837i	−2.73470	−	10.2060i	50.0224	−	20.1226i
37.10	−2.70573	+	0.824025i	−3.12290	+	4.06985i	6.64196	−	4.45918i	−12.4343	+	9.54116i	5.09608	−	13.5853i	−18.4815	−	1.19809i	−14.2969	+	17.5385i	0.176982	+	0.660506i	25.7817	−	36.0620i
37.11	−2.67901	+	0.907130i	6.22563	−	8.11340i	6.35423	−	4.86043i	4.48711	−	3.44308i	−9.31864	+	27.3834i	5.02571	−	17.8253i	−12.6140	+	18.7853i	−20.0807	−	74.9422i	−8.89771	+	13.2945i
37.12	−2.66137	−	0.957645i	0.959860	−	1.25091i	6.16583	+	5.09731i	3.35831	−	2.57692i	−3.75248	+	2.40995i	17.5759	−	5.83861i	−11.5282	−	19.4705i	6.34466	+	23.6786i	−11.4055	+	3.64208i
37.13	−2.64687	+	0.997026i	−0.539436	+	0.703006i	6.01188	−	5.27800i	16.9486	−	13.0052i	0.726902	−	2.39860i	−12.6736	+	13.5048i	−10.6504	+	19.9642i	6.78489	+	25.3215i	−31.8944	+	51.3213i
37.14	−2.56251	−	1.19732i	−0.552446	+	0.719961i	5.13287	+	6.13626i	−12.3126	+	9.44779i	2.27767	−	1.18345i	0.932925	+	18.4967i	−5.80597	−	21.8699i	6.77497	+	25.2845i	42.8631	−	9.46794i
37.15	−2.45155	+	1.41064i	1.56476	−	2.03924i	4.02018	−	6.91651i	7.84875	−	6.02255i	−0.959459	+	7.20662i	−1.66873	−	18.4449i	−0.0989617	+	22.6272i	5.27810	+	19.6981i	−10.7459	+	25.8364i
37.16	−2.44633	−	1.41967i	−2.36263	+	3.07904i	3.96909	+	6.94596i	3.84043	−	2.94686i	10.1510	−	4.17821i	−17.8684	+	4.87037i	0.151232	−	22.6269i	3.08964	+	11.5307i	−13.5785	+	1.75688i
37.17	−2.42639	+	1.45349i	−4.84581	+	6.31519i	3.77472	−	7.05348i	7.93134	−	6.08593i	2.57874	−	22.3664i	16.8521	−	7.68169i	1.09325	+	22.6010i	−9.41156	−	35.1244i	−10.3987	+	26.2950i
37.18	−2.28328	−	1.66932i	−3.35733	+	4.37536i	2.42677	+	7.62304i	17.2678	−	13.2501i	14.9696	−	4.38574i	11.8856	−	14.2033i	7.18427	−	21.4566i	−0.883991	−	3.29910i	−61.5459	+	1.42821i
37.19	−2.19776	−	1.78040i	−5.30524	+	6.91392i	1.66033	+	7.82581i	−6.69215	+	5.13506i	23.9692	−	5.74970i	−9.30836	−	16.0111i	10.2841	−	20.1553i	−12.6686	−	47.2799i	23.8502	+	0.629061i
37.20	−2.19140	+	1.78823i	−0.444503	+	0.579288i	1.60447	−	7.83745i	−13.5084	+	10.3654i	−0.0618156	−	2.06433i	11.7085	−	14.3496i	10.4991	+	20.0442i	6.85012	+	25.5650i	11.0667	−	46.8709i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
32.g	even	8	1	inner
224.bd	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.4.bd.a	✓	752
7.c	even	3	1	inner	224.4.bd.a	✓	752
32.g	even	8	1	inner	224.4.bd.a	✓	752
224.bd	even	24	1	inner	224.4.bd.a	✓	752

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.4.bd.a	✓	752	1.a	even	1	1	trivial
224.4.bd.a	✓	752	7.c	even	3	1	inner
224.4.bd.a	✓	752	32.g	even	8	1	inner
224.4.bd.a	✓	752	224.bd	even	24	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(224, [\chi])\).