Newform orbit 224.2.be.a

• Label: 224.2.be.a
• Level: $224$
• Weight: $2$
• Character orbit: 224.be
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{2} - 12 q^{3} - 4 q^{4} - 12 q^{5} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.40664	−	0.146150i	−0.173475	+	0.133112i	1.95728	+	0.411162i	1.47800	+	1.13411i	0.263472	−	0.161888i	1.10840	−	2.40238i	−2.69310	−	0.864414i	−0.764082	+	2.85159i	−1.91327	−	1.81130i
3.2	−1.37290	+	0.339345i	−1.61210	+	1.23700i	1.76969	−	0.931772i	0.108918	+	0.0835753i	1.79347	−	2.24534i	−2.62908	+	0.296550i	−2.11341	+	1.87976i	0.292215	−	1.09056i	−0.177893	−	0.0777796i
3.3	−1.32601	+	0.491631i	1.63883	−	1.25752i	1.51660	−	1.30381i	2.28738	+	1.75517i	−1.55487	+	2.47319i	1.14998	+	2.38276i	−1.37003	+	2.47448i	0.327959	−	1.22396i	−3.89598	−	1.20282i
3.4	−1.28037	−	0.600540i	−2.17370	+	1.66793i	1.27870	+	1.53783i	0.725858	+	0.556970i	3.78480	−	0.830186i	1.28317	+	2.31376i	−0.713688	−	2.73691i	1.16649	−	4.35339i	−0.594885	−	1.14904i
3.5	−1.26996	+	0.622259i	−0.630556	+	0.483843i	1.22559	−	1.58049i	−3.20409	−	2.45858i	0.499704	−	1.00683i	2.00959	+	1.72092i	−0.572972	+	2.76978i	−0.612960	+	2.28760i	5.59893	+	1.12852i
3.6	−1.12720	+	0.854063i	2.03554	−	1.56193i	0.541153	−	1.92540i	−1.56026	−	1.19723i	−0.960477	+	3.49909i	−1.31328	−	2.29680i	1.03442	+	2.63248i	0.927364	−	3.46097i	2.78123	+	0.0169550i
3.7	−1.08124	−	0.911550i	0.158962	−	0.121976i	0.338152	+	1.97121i	−2.84975	−	2.18669i	−0.283063	−	0.0130170i	1.92735	−	1.81254i	1.43123	−	2.43959i	−0.766066	+	2.85900i	1.08798	+	4.96203i
3.8	−1.06453	−	0.931006i	2.25308	−	1.72885i	0.266455	+	1.98217i	1.40553	+	1.07850i	−4.00805	−	0.257217i	−2.00939	−	1.72115i	1.56176	−	2.35816i	1.31100	−	4.89272i	−0.492141	−	2.45666i
3.9	−0.886934	−	1.10152i	0.450846	−	0.345946i	−0.426696	+	1.95395i	0.813478	+	0.624204i	−0.780937	−	0.189784i	0.0551167	+	2.64518i	2.53077	−	1.26301i	−0.692874	+	2.58584i	−0.0339283	−	1.44969i
3.10	−0.810400	+	1.15899i	−2.13924	+	1.64150i	−0.686504	−	1.87849i	−0.625729	−	0.480139i	−0.168834	−	3.80962i	0.349257	−	2.62260i	2.73349	+	0.726676i	1.10538	−	4.12534i	1.06357	−	0.336108i
3.11	−0.670369	+	1.24523i	0.102906	−	0.0789622i	−1.10121	−	1.66953i	2.38833	+	1.83263i	0.0293417	+	0.181075i	−2.62469	−	0.333180i	2.81717	−	0.252064i	−0.772103	+	2.88153i	−3.88312	+	1.74549i
3.12	−0.229686	+	1.39544i	1.43074	−	1.09785i	−1.89449	−	0.641024i	−0.628945	−	0.482607i	1.20336	+	2.24867i	2.64136	−	0.152394i	1.32965	−	2.49641i	0.0652997	−	0.243702i	0.817907	−	0.766806i
3.13	−0.210188	−	1.39851i	−2.33910	+	1.79485i	−1.91164	+	0.587899i	−1.19071	−	0.913663i	3.00177	+	2.89399i	2.51525	−	0.820673i	1.22399	+	2.54987i	1.47343	−	5.49891i	−1.02749	+	1.85726i
3.14	−0.201712	−	1.39975i	−0.700660	+	0.537636i	−1.91862	+	0.564694i	−0.634957	−	0.487220i	0.893889	+	0.872305i	−2.64410	−	0.0934084i	1.17744	+	2.57170i	−0.574584	+	2.14438i	−0.553910	+	0.987063i
3.15	−0.124966	−	1.40868i	2.48500	−	1.90681i	−1.96877	+	0.352074i	−2.71407	−	2.08258i	−2.99663	−	3.26229i	1.12555	+	2.39440i	0.741988	+	2.72937i	1.76286	−	6.57908i	−2.59452	+	4.08351i
3.16	0.157697	−	1.40539i	1.49249	−	1.14523i	−1.95026	−	0.443252i	2.44964	+	1.87968i	−1.37414	−	2.27814i	1.65828	−	2.06158i	−0.930493	+	2.67099i	0.139527	−	0.520721i	3.02799	−	3.14630i
3.17	0.316867	+	1.37826i	−1.12518	+	0.863381i	−1.79919	+	0.873450i	2.90549	+	2.22946i	−1.54649	−	1.27721i	2.61828	−	0.380261i	−1.77394	−	2.20298i	−0.255854	+	0.954861i	−2.15212	+	4.71095i
3.18	0.337816	+	1.37327i	−0.983428	+	0.754611i	−1.77176	+	0.927827i	−2.45933	−	1.88711i	−1.36851	−	1.09560i	−0.990618	−	2.45330i	−1.87269	−	2.11968i	−0.378764	+	1.41357i	1.76072	−	4.01483i
3.19	0.510534	+	1.31885i	2.43405	−	1.86771i	−1.47871	+	1.34663i	1.80920	+	1.38825i	3.70589	+	2.25660i	−1.87347	+	1.86819i	−2.53093	−	1.26269i	1.65979	−	6.19441i	−0.907226	+	3.09480i
3.20	0.612401	−	1.27474i	−1.50340	+	1.15360i	−1.24993	−	1.56131i	3.01130	+	2.31065i	0.549854	+	2.62290i	−0.677804	+	2.55746i	−2.75572	+	0.637190i	0.152957	−	0.570843i	4.78960	−	2.42358i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
32.h	odd	8	1	inner
224.be	even	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.2.be.a	✓	240
4.b	odd	2	1		896.2.bi.a		240
7.d	odd	6	1	inner	224.2.be.a	✓	240
28.f	even	6	1		896.2.bi.a		240
32.g	even	8	1		896.2.bi.a		240
32.h	odd	8	1	inner	224.2.be.a	✓	240
224.bc	odd	24	1		896.2.bi.a		240
224.be	even	24	1	inner	224.2.be.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.be.a	✓	240	1.a	even	1	1	trivial
224.2.be.a	✓	240	7.d	odd	6	1	inner
224.2.be.a	✓	240	32.h	odd	8	1	inner
224.2.be.a	✓	240	224.be	even	24	1	inner
896.2.bi.a		240	4.b	odd	2	1
896.2.bi.a		240	28.f	even	6	1
896.2.bi.a		240	32.g	even	8	1
896.2.bi.a		240	224.bc	odd	24	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(224, [\chi])\).