Newform orbit 440.2.z.a

• Label: 440.2.z.a
• Level: $440$
• Weight: $2$
• Character orbit: 440.z
• Analytic conductor: $3.513$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.z (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.51341768894\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 192 * q - 4 * q^4 - 48 * q^9 + 8 * q^11 - 10 * q^14 - 16 * q^16 - 50 * q^18 + 4 * q^20 - 56 * q^22 + 60 * q^24 + 48 * q^25 + 20 * q^26 + 50 * q^28 + 8 * q^33 - 12 * q^34 - 64 * q^36 - 50 * q^38 - 110 * q^42 + 34 * q^44 - 10 * q^46 - 14 * q^48 - 48 * q^49 - 50 * q^52 - 120 * q^56 + 62 * q^58 - 16 * q^59 - 42 * q^60 - 64 * q^64 + 54 * q^66 + 80 * q^67 - 130 * q^68 + 40 * q^70 + 50 * q^72 + 120 * q^74 + 92 * q^78 - 24 * q^80 - 64 * q^81 + 12 * q^82 + 120 * q^84 + 108 * q^86 - 8 * q^88 - 128 * q^91 + 72 * q^92 + 120 * q^94 + 170 * q^96 - 192 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.41067	+	0.100069i	−0.666731	+	2.05199i	1.97997	−	0.282327i	0.587785	+	0.809017i	0.735197	−	2.96139i	0.587432	+	1.80793i	−2.76483	+	0.596404i	−1.33907	−	0.972894i	−0.910128	−	1.08244i
51.2	−1.40817	+	0.130636i	−0.594187	+	1.82872i	1.96587	−	0.367913i	−0.587785	−	0.809017i	0.597819	−	2.65277i	−0.938517	−	2.88846i	−2.72021	+	0.774896i	−0.564110	−	0.409850i	0.933386	+	1.06245i
51.3	−1.39580	+	0.227480i	0.306155	−	0.942248i	1.89651	−	0.635032i	0.587785	+	0.809017i	−0.212988	+	1.38483i	−0.491825	−	1.51368i	−2.50268	+	1.31779i	1.63295	+	1.18641i	−1.00446	−	0.995515i
51.4	−1.39567	+	0.228239i	0.998789	−	3.07396i	1.89581	−	0.637095i	−0.587785	−	0.809017i	−0.692387	+	4.51821i	0.520561	+	1.60212i	−2.50053	+	1.32188i	−6.02458	−	4.37711i	1.00501	+	0.994969i
51.5	−1.35051	−	0.419669i	0.109136	−	0.335887i	1.64776	+	1.13353i	−0.587785	−	0.809017i	−0.288351	+	0.407818i	−0.624312	−	1.92144i	−1.74960	−	2.22236i	2.32614	+	1.69004i	0.454291	+	1.33926i
51.6	−1.30140	−	0.553504i	−1.02351	+	3.15004i	1.38727	+	1.44066i	0.587785	+	0.809017i	3.07555	−	3.53293i	−0.961524	−	2.95927i	−1.00797	−	2.64272i	−6.44810	−	4.68482i	−0.317148	−	1.37819i
51.7	−1.29442	+	0.569632i	0.168837	−	0.519626i	1.35104	−	1.47468i	−0.587785	−	0.809017i	0.0774504	+	0.768789i	0.996469	+	3.06682i	−0.908781	+	2.67845i	2.18555	+	1.58789i	1.22168	+	0.712385i
51.8	−1.16000	−	0.808956i	0.734943	−	2.26192i	0.691182	+	1.87677i	0.587785	+	0.809017i	−2.68233	+	2.02929i	1.41521	+	4.35558i	0.716456	−	2.73618i	−2.14910	−	1.56142i	−0.0273697	−	1.41395i
51.9	−1.10740	+	0.879582i	−0.399394	+	1.22921i	0.452671	−	1.94810i	0.587785	+	0.809017i	−0.638901	−	1.71253i	−1.45788	−	4.48690i	1.21222	+	2.55549i	1.07561	+	0.781477i	−1.36251	−	0.378900i
51.10	−1.08101	+	0.911823i	−0.438021	+	1.34809i	0.337159	−	1.97138i	0.587785	+	0.809017i	−0.755715	−	1.85670i	0.797526	+	2.45453i	1.43307	+	2.43850i	0.801564	+	0.582370i	−1.37308	−	0.338598i
51.11	−1.07561	−	0.918190i	−0.214814	+	0.661130i	0.313853	+	1.97522i	0.587785	+	0.809017i	0.838099	−	0.513875i	0.110386	+	0.339734i	1.47605	−	2.41273i	2.03610	+	1.47932i	0.110607	−	1.40988i
51.12	−0.971538	−	1.02767i	−0.610310	+	1.87834i	−0.112226	+	1.99685i	−0.587785	−	0.809017i	2.52326	−	1.19768i	−0.0910362	−	0.280181i	2.16114	−	1.82468i	−0.728640	−	0.529388i	−0.260349	+	1.39004i
51.13	−0.925734	−	1.06912i	0.319915	−	0.984597i	−0.286035	+	1.97944i	−0.587785	−	0.809017i	−1.34881	+	0.569447i	0.427803	+	1.31664i	2.38105	−	1.52663i	1.55997	+	1.13338i	−0.320804	+	1.37735i
51.14	−0.916258	+	1.07725i	0.732229	−	2.25357i	−0.320942	−	1.97408i	0.587785	+	0.809017i	1.75675	+	2.85365i	1.04806	+	3.22559i	2.42065	+	1.46303i	−2.11536	−	1.53690i	−1.41008	−	0.108076i
51.15	−0.741388	+	1.20430i	0.732229	−	2.25357i	−0.900688	−	1.78571i	−0.587785	−	0.809017i	2.17111	+	2.55259i	−1.04806	−	3.22559i	2.81829	+	0.239202i	−2.11536	−	1.53690i	1.41008	−	0.108076i
51.16	−0.697229	−	1.23040i	0.475935	−	1.46478i	−1.02774	+	1.71573i	0.587785	+	0.809017i	−2.13409	+	0.435697i	−1.24708	−	3.83812i	2.82760	+	0.0682723i	0.507994	+	0.369080i	0.585590	−	1.28728i
51.17	−0.605678	−	1.27795i	0.877118	−	2.69949i	−1.26631	+	1.54805i	−0.587785	−	0.809017i	−3.98106	+	0.514110i	0.316509	+	0.974113i	2.74531	+	0.680657i	−4.09087	−	2.97219i	−0.677874	+	1.24116i
51.18	−0.533145	+	1.30987i	−0.438021	+	1.34809i	−1.43151	−	1.39670i	−0.587785	−	0.809017i	−1.53229	−	1.29248i	−0.797526	−	2.45453i	2.59270	−	1.13045i	0.801564	+	0.582370i	1.37308	−	0.338598i
51.19	−0.494327	+	1.32501i	−0.399394	+	1.22921i	−1.51128	−	1.30997i	−0.587785	−	0.809017i	−1.43128	−	1.13683i	1.45788	+	4.48690i	2.48279	−	1.35490i	1.07561	+	0.781477i	1.36251	−	0.378900i
51.20	−0.241174	−	1.39350i	−0.827456	+	2.54665i	−1.88367	+	0.672150i	−0.587785	−	0.809017i	3.74831	+	0.538873i	0.147083	+	0.452674i	1.39093	+	2.46279i	−3.37367	−	2.45112i	−0.985605	+	1.01419i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
11.d	odd	10	1	inner
88.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	440.2.z.a	✓	192
8.d	odd	2	1	inner	440.2.z.a	✓	192
11.d	odd	10	1	inner	440.2.z.a	✓	192
88.k	even	10	1	inner	440.2.z.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.2.z.a	✓	192	1.a	even	1	1	trivial
440.2.z.a	✓	192	8.d	odd	2	1	inner
440.2.z.a	✓	192	11.d	odd	10	1	inner
440.2.z.a	✓	192	88.k	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(440, [\chi])\).