Newform orbit 224.2.bd.a

• Label: 224.2.bd.a
• Level: $224$
• Weight: $2$
• Character orbit: 224.bd
• 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.bd (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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{8} - 4 q^{9}+O(q^{10}) \)

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	−1.41418	−	0.00960786i	−1.76562	+	2.30100i	1.99982	+	0.0271745i	2.61963	−	2.01011i	2.51901	−	3.23706i	−1.37753	−	2.25885i	−2.82784	−	0.0576436i	−1.40073	−	5.22759i	−3.72395	+	2.81750i
37.2	−1.41287	+	0.0616517i	−0.0537004	+	0.0699837i	1.99240	−	0.174212i	−0.644996	+	0.494923i	0.0715570	−	0.102189i	−2.22522	+	1.43123i	−2.80426	+	0.368973i	0.774443	+	2.89026i	0.880782	−	0.739026i
37.3	−1.32342	+	0.498548i	0.615442	−	0.802059i	1.50290	−	1.31958i	−1.98693	+	1.52462i	−0.414625	+	1.36829i	2.32280	−	1.26674i	−1.33110	+	2.49563i	0.511927	+	1.91054i	1.86945	−	3.00830i
37.4	−1.30182	−	0.552515i	1.55155	−	2.02202i	1.38945	+	1.43855i	0.189180	−	0.145163i	−3.13703	+	1.77505i	−1.52456	−	2.16234i	−1.01400	−	2.64042i	−0.904803	−	3.37677i	−0.326482	+	0.0844507i
37.5	−1.29489	−	0.568565i	−1.02607	+	1.33720i	1.35347	+	1.47246i	−1.35700	+	1.04126i	2.08893	−	1.14813i	2.49879	−	0.869520i	−0.915400	−	2.67620i	0.0411741	+	0.153664i	2.34919	−	0.576775i
37.6	−1.16516	+	0.801498i	1.71349	−	2.23306i	0.715201	−	1.86775i	1.90405	−	1.46103i	−0.206695	+	3.97523i	−0.262719	+	2.63268i	0.663674	+	2.74946i	−1.27406	−	4.75485i	−1.04751	+	3.22843i
37.7	−1.13266	−	0.846802i	0.0986144	−	0.128517i	0.565853	+	1.91828i	2.39507	−	1.83780i	−0.220525	+	0.0620594i	0.134369	+	2.64234i	0.983485	−	2.65193i	0.769665	+	2.87243i	−4.26906	+	0.0534610i
37.8	−0.940911	+	1.05579i	−1.04486	+	1.36169i	−0.229373	−	1.98680i	1.80014	−	1.38129i	−0.454533	−	2.38438i	2.29881	+	1.30976i	2.31346	+	1.62724i	0.0139922	+	0.0522197i	−0.235417	+	3.20024i
37.9	−0.760837	+	1.19211i	−1.44087	+	1.87778i	−0.842255	−	1.81400i	−2.75892	+	2.11700i	−1.14225	−	3.14636i	−1.14468	−	2.38531i	2.80331	+	0.376098i	−0.673491	−	2.51350i	−0.424604	−	4.89963i
37.10	−0.631870	−	1.26520i	0.678058	−	0.883663i	−1.20148	+	1.59889i	−2.77837	+	2.13192i	−1.54646	−	0.299522i	−2.64040	−	0.168196i	2.78210	+	0.509831i	0.455360	+	1.69943i	4.45288	+	2.16811i
37.11	−0.480321	−	1.33015i	−0.758838	+	0.988936i	−1.53858	+	1.27780i	0.171111	−	0.131298i	1.67992	+	0.534359i	1.01488	−	2.44336i	2.43867	+	1.43279i	0.374296	+	1.39689i	−0.256834	−	0.164537i
37.12	−0.460775	+	1.33704i	1.47397	−	1.92092i	−1.57537	−	1.23215i	0.169674	−	0.130196i	1.88918	+	2.85588i	−0.114544	−	2.64327i	2.37333	−	1.53860i	−0.740878	−	2.76499i	0.0958956	+	0.286853i
37.13	−0.436149	+	1.34528i	0.756979	−	0.986515i	−1.61955	−	1.17348i	−2.52010	+	1.93374i	0.996981	+	1.44862i	−0.259491	+	2.63300i	2.28503	−	1.66693i	0.376264	+	1.40424i	−1.50228	−	4.23364i
37.14	−0.399779	−	1.35653i	1.62045	−	2.11181i	−1.68035	+	1.08463i	1.05240	−	0.807534i	−3.51256	−	1.35393i	2.64452	−	0.0808662i	2.14310	+	1.84584i	−1.05744	−	3.94640i	−1.51617	−	1.10478i
37.15	0.0713999	+	1.41241i	0.118403	−	0.154306i	−1.98980	+	0.201692i	2.39583	−	1.83839i	0.226398	+	0.156217i	1.47323	−	2.19764i	−0.426943	−	2.79602i	0.766666	+	2.86124i	2.76762	+	3.25263i
37.16	0.131818	−	1.40806i	−0.751048	+	0.978785i	−1.96525	−	0.371216i	−1.09067	+	0.836901i	1.27918	+	1.18654i	0.448114	+	2.60753i	−0.781748	+	2.71825i	0.382510	+	1.42755i	1.03463	+	1.64604i
37.17	0.148173	+	1.40643i	−1.23048	+	1.60359i	−1.95609	+	0.416790i	0.591453	−	0.453838i	−2.43766	−	1.49297i	−2.18592	+	1.49056i	−0.876026	−	2.68935i	−0.280969	−	1.04859i	0.725929	+	0.764591i
37.18	0.361256	−	1.36729i	0.338207	−	0.440760i	−1.73899	−	0.987888i	3.17926	−	2.43953i	−0.480469	−	0.621656i	−2.54805	−	0.712336i	−1.97895	+	2.02083i	0.696572	+	2.59964i	−2.18703	−	5.22828i
37.19	0.674919	−	1.24277i	−1.68352	+	2.19400i	−1.08897	−	1.67754i	−1.03656	+	0.795384i	1.59041	+	3.57300i	−2.17678	−	1.50387i	−2.81977	+	0.221138i	−1.20296	−	4.48951i	0.288885	+	1.82503i
37.20	0.699836	+	1.22891i	0.666690	−	0.868847i	−1.02046	+	1.72008i	−0.367682	+	0.282133i	1.53431	+	0.211254i	1.90974	+	1.83109i	−2.82798	−	0.0502881i	0.466037	+	1.73928i	−0.604034	−	0.254404i

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	224.2.bd.a	✓	240
4.b	odd	2	1		896.2.bh.a		240
7.c	even	3	1	inner	224.2.bd.a	✓	240
28.g	odd	6	1		896.2.bh.a		240
32.g	even	8	1	inner	224.2.bd.a	✓	240
32.h	odd	8	1		896.2.bh.a		240
224.bd	even	24	1	inner	224.2.bd.a	✓	240
224.bf	odd	24	1		896.2.bh.a		240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.bd.a	✓	240	1.a	even	1	1	trivial
224.2.bd.a	✓	240	7.c	even	3	1	inner
224.2.bd.a	✓	240	32.g	even	8	1	inner
224.2.bd.a	✓	240	224.bd	even	24	1	inner
896.2.bh.a		240	4.b	odd	2	1
896.2.bh.a		240	28.g	odd	6	1
896.2.bh.a		240	32.h	odd	8	1
896.2.bh.a		240	224.bf	odd	24	1

Hecke kernels

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