Newform orbit 160.4.z.a

• Label: 160.4.z.a
• Level: $160$
• Weight: $4$
• Character orbit: 160.z
• Analytic conductor: $9.440$
• Analytic rank: $0$
• Dimension: $280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.z (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.44030560092\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 280 * q - 8 * q^4 - 4 * q^5 - 8 * q^6 - 8 * q^9 - 4 * q^10 - 8 * q^11 + 200 * q^14 - 8 * q^16 - 8 * q^19 - 4 * q^20 - 8 * q^21 - 464 * q^24 - 4 * q^25 - 48 * q^26 - 8 * q^29 + 1124 * q^30 + 1472 * q^31 - 72 * q^34 - 460 * q^35 - 888 * q^36 - 8 * q^39 - 1112 * q^40 - 8 * q^41 - 8 * q^44 - 112 * q^45 - 8 * q^46 + 3556 * q^50 - 1712 * q^51 + 856 * q^54 + 284 * q^55 + 328 * q^56 - 1384 * q^59 - 5248 * q^60 - 1832 * q^61 + 832 * q^64 - 8 * q^65 + 6088 * q^66 + 1048 * q^69 - 616 * q^70 + 216 * q^71 - 2064 * q^74 + 2204 * q^75 - 5128 * q^76 - 656 * q^80 - 6904 * q^84 - 504 * q^85 - 1192 * q^86 - 8 * q^89 + 2720 * q^90 - 8 * q^91 - 920 * q^94 - 8 * q^95 - 4152 * q^96 + 5520 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−2.82582	−	0.121466i	1.13930	−	2.75051i	7.97049	+	0.686483i	−11.1458	+	0.878057i	−3.55354	+	7.63405i	2.46301	+	2.46301i	−22.4398	−	2.90802i	12.8246	+	12.8246i	31.6027	−	1.12739i
29.2	−2.82422	−	0.154155i	3.56147	−	8.59816i	7.95247	+	0.870736i	9.02558	−	6.59840i	−11.3838	+	23.7341i	−18.6905	−	18.6905i	−22.3253	−	3.68507i	−42.1524	−	42.1524i	−26.5074	+	17.2440i
29.3	−2.82041	+	0.212828i	−2.00307	+	4.83583i	7.90941	−	1.20053i	3.32107	+	10.6757i	4.62026	−	14.0653i	18.8414	+	18.8414i	−22.0523	+	5.06932i	−0.281096	−	0.281096i	−11.6389	−	29.4030i
29.4	−2.76649	+	0.588676i	−3.34280	+	8.07023i	7.30692	−	3.25713i	8.17012	−	7.63212i	4.49707	−	24.2940i	5.36942	+	5.36942i	−18.2971	+	13.3122i	−34.8625	−	34.8625i	−18.1097	+	25.9237i
29.5	−2.76626	−	0.589757i	−2.89779	+	6.99588i	7.30437	+	3.26284i	−10.6728	+	3.33034i	12.1419	−	17.6434i	−7.51720	−	7.51720i	−18.2815	−	13.3337i	−21.4532	−	21.4532i	31.4878	−	2.91824i
29.6	−2.71925	−	0.778249i	0.155312	−	0.374956i	6.78866	+	4.23251i	−1.64450	−	11.0587i	−0.714142	+	0.898729i	−8.18573	−	8.18573i	−15.1661	−	16.7925i	18.9754	+	18.9754i	−4.13465	+	31.3513i
29.7	−2.71650	+	0.787786i	−0.748614	+	1.80731i	6.75878	−	4.28005i	4.89753	+	10.0506i	0.609835	−	5.49932i	−19.1083	−	19.1083i	−14.9885	+	16.9513i	16.3859	+	16.3859i	−21.2219	−	23.4442i
29.8	−2.71274	+	0.800660i	2.64369	−	6.38243i	6.71789	−	4.34396i	−5.38122	+	9.80012i	−2.06148	+	19.4306i	−1.79865	−	1.79865i	−14.7458	+	17.1628i	−14.6545	−	14.6545i	6.75128	−	30.8937i
29.9	−2.68188	+	0.898609i	1.66254	−	4.01373i	6.38500	−	4.81993i	2.22134	−	10.9574i	−0.851969	+	12.2583i	24.4939	+	24.4939i	−12.7926	+	18.6641i	5.74587	+	5.74587i	3.88908	+	31.3827i
29.10	−2.62962	−	1.04167i	0.240464	−	0.580532i	5.82985	+	5.47840i	10.8412	+	2.73281i	−1.23705	+	1.27610i	8.45542	+	8.45542i	−9.62365	−	20.4789i	18.8127	+	18.8127i	−25.6616	−	18.4792i
29.11	−2.44333	+	1.42483i	−1.70942	+	4.12691i	3.93973	−	6.96265i	−6.60470	−	9.02097i	−1.70346	−	12.5190i	−11.0262	−	11.0262i	0.294527	+	22.6255i	4.98262	+	4.98262i	28.9908	+	12.6307i
29.12	−2.42938	+	1.44848i	0.822121	−	1.98478i	3.80379	−	7.03784i	11.1418	−	0.927768i	0.877668	+	6.01261i	−8.30658	−	8.30658i	0.953323	+	22.6073i	15.8284	+	15.8284i	−25.7238	+	18.3926i
29.13	−2.42251	−	1.45995i	2.54494	−	6.14403i	3.73710	+	7.07348i	−0.364663	+	11.1744i	−15.1351	+	11.1685i	−1.31712	−	1.31712i	1.27377	−	22.5915i	−12.1805	−	12.1805i	17.1974	−	26.5377i
29.14	−2.31596	−	1.62368i	−2.87385	+	6.93809i	2.72732	+	7.52075i	11.1803	+	0.0240294i	17.9210	−	11.4021i	−23.8772	−	23.8772i	5.89493	−	21.8460i	−20.7862	−	20.7862i	−25.8541	−	18.2089i
29.15	−2.30569	−	1.63823i	−2.74110	+	6.61760i	2.63243	+	7.55449i	−5.49720	−	9.73554i	17.1613	−	10.7676i	20.0512	+	20.0512i	6.30639	−	21.7308i	−17.1871	−	17.1871i	−3.27418	+	31.4528i
29.16	−2.22148	−	1.75073i	3.65386	−	8.82120i	1.86992	+	7.77839i	−8.53482	−	7.22197i	−23.5605	+	13.1992i	17.6199	+	17.6199i	9.46386	−	20.5532i	−45.3710	−	45.3710i	6.31621	+	30.9856i
29.17	−2.08004	+	1.91662i	−1.01808	+	2.45786i	0.653158	−	7.97329i	−10.3533	+	4.22013i	−2.59313	−	7.06372i	18.2161	+	18.2161i	13.9231	+	17.8366i	14.0873	+	14.0873i	13.4469	−	28.6213i
29.18	−1.89989	+	2.09533i	3.43485	−	8.29247i	−0.780855	−	7.96180i	−9.81187	−	5.35978i	10.8497	+	22.9519i	−3.51241	−	3.51241i	18.1662	+	13.4904i	−37.8750	−	37.8750i	29.8720	−	10.3762i
29.19	−1.81015	−	2.17332i	−1.45135	+	3.50386i	−1.44668	+	7.86811i	−3.62694	+	10.5757i	10.2422	−	3.18828i	0.932083	+	0.932083i	19.7187	−	11.0984i	8.92125	+	8.92125i	29.5497	−	11.2611i
29.20	−1.80049	+	2.18134i	2.73103	−	6.59330i	−1.51645	−	7.85496i	7.36406	+	8.41253i	9.46500	+	17.8285i	8.76485	+	8.76485i	19.8647	+	10.8349i	−16.9212	−	16.9212i	−31.6095	+	0.916789i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
32.g	even	8	1	inner
160.z	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	160.4.z.a	✓	280
5.b	even	2	1	inner	160.4.z.a	✓	280
32.g	even	8	1	inner	160.4.z.a	✓	280
160.z	even	8	1	inner	160.4.z.a	✓	280

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.4.z.a	✓	280	1.a	even	1	1	trivial
160.4.z.a	✓	280	5.b	even	2	1	inner
160.4.z.a	✓	280	32.g	even	8	1	inner
160.4.z.a	✓	280	160.z	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(160, [\chi])\).