Newform orbit 160.4.ba.a

• Label: 160.4.ba.a
• Level: $160$
• Weight: $4$
• Character orbit: 160.ba
• Analytic conductor: $9.440$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.ba (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 8 q^{7} - 88 q^{8}+O(q^{10}) \)

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	−2.82620	−	0.112193i	−5.02918	+	2.08316i	7.97483	+	0.634162i	−1.84839	−	11.0265i	14.4472	−	5.32318i	13.3189			−22.4673	−	2.68699i	1.86127	−	1.86127i	3.98683	+	31.3705i
3.2	−2.82295	−	0.175972i	4.02581	−	1.66754i	7.93807	+	0.993518i	7.26259	−	8.50028i	−11.6581	+	3.99896i	−35.0398			−22.2339	−	4.20152i	−5.66547	+	5.66547i	−21.9977	+	22.7178i
3.3	−2.82246	−	0.183639i	−5.27637	+	2.18554i	7.93255	+	1.03663i	−10.4064	+	4.08736i	15.2937	−	5.19966i	−19.5423			−22.1989	−	4.38256i	3.97156	−	3.97156i	30.1223	−	9.62538i
3.4	−2.82209	+	0.189157i	7.87428	−	3.26164i	7.92844	−	1.06764i	−5.83539	−	9.53668i	−21.6050	+	10.6941i	21.2036			−22.1729	+	4.51268i	32.2742	−	32.2742i	18.2720	+	25.8096i
3.5	−2.79487	+	0.434422i	−0.175127	+	0.0725400i	7.62256	−	2.42830i	7.70747	+	8.09907i	0.457944	−	0.278819i	−10.5403			−20.2491	+	10.0982i	−19.0665	+	19.0665i	−25.0598	−	19.2875i
3.6	−2.77152	−	0.564505i	−8.63229	+	3.57561i	7.36267	+	3.12907i	10.3503	+	4.22739i	25.9430	−	5.03692i	−6.56256			−18.6394	−	12.8286i	42.6395	−	42.6395i	−26.2998	−	17.5591i
3.7	−2.76499	+	0.595663i	4.38683	−	1.81709i	7.29037	−	3.29401i	−4.29208	+	10.3237i	−11.0472	+	7.63730i	17.8539			−18.1957	+	13.4505i	−3.14939	+	3.14939i	5.71813	−	31.1015i
3.8	−2.64909	−	0.991132i	1.67307	−	0.693006i	6.03532	+	5.25119i	−9.03485	+	6.58570i	−5.11896	+	0.177605i	3.19432			−10.7835	−	19.8926i	−16.7730	+	16.7730i	30.4614	−	8.49135i
3.9	−2.56752	−	1.18652i	1.46930	−	0.608604i	5.18433	+	6.09285i	11.1472	+	0.860428i	−4.49458	−	0.180755i	28.1375			−6.08156	−	21.7948i	−17.3034	+	17.3034i	−27.5997	−	15.4356i
3.10	−2.56368	−	1.19479i	9.19328	−	3.80798i	5.14496	+	6.12612i	6.22729	+	9.28552i	−28.1184	−	1.22156i	−11.4587			−5.87063	−	21.8526i	50.9238	−	50.9238i	−4.87056	−	31.2454i
3.11	−2.54654	+	1.23091i	−2.32747	+	0.964068i	4.96971	−	6.26913i	8.90423	−	6.76127i	4.74030	−	5.31995i	16.0647			−4.93878	+	22.0819i	−14.6042	+	14.6042i	−14.3524	+	28.1782i
3.12	−2.48776	+	1.34575i	1.30507	−	0.540577i	4.37789	−	6.69582i	−8.39625	−	7.38262i	−2.51921	+	3.10112i	−15.7029			−1.88021	+	22.5492i	−17.6809	+	17.6809i	30.8230	+	7.06690i
3.13	−2.47251	+	1.37357i	−7.53300	+	3.12027i	4.22658	−	6.79235i	−9.56055	+	5.79620i	14.3395	−	18.0620i	34.8226			−1.12046	+	22.5997i	27.9182	−	27.9182i	15.6770	−	27.4633i
3.14	−2.28693	−	1.66432i	−3.78770	+	1.56892i	2.46006	+	7.61236i	−3.88271	−	10.4845i	11.2734	+	2.71596i	6.46889			7.04344	−	21.5033i	−7.20671	+	7.20671i	−8.57010	+	30.4393i
3.15	−2.21144	−	1.76338i	4.81089	−	1.99274i	1.78097	+	7.79924i	−9.99813	−	5.00374i	−14.1530	−	4.07661i	−11.6957			9.81451	−	20.3881i	0.0818204	−	0.0818204i	13.2868	+	28.6960i
3.16	−2.18557	+	1.79535i	7.10263	−	2.94201i	1.55346	−	7.84772i	10.5754	−	3.62777i	−10.2414	+	19.1817i	2.95853			10.6942	+	19.9408i	22.7001	−	22.7001i	−16.6002	+	26.9153i
3.17	−2.11511	+	1.87785i	−4.19826	+	1.73898i	0.947364	−	7.94371i	4.19567	+	10.3632i	5.61424	−	11.5618i	−11.4589			12.9133	+	18.5808i	−4.49050	+	4.49050i	−28.3349	−	14.0405i
3.18	−2.05859	+	1.93965i	7.12247	−	2.95022i	0.475554	−	7.98585i	−7.88219	+	7.92912i	−8.93983	+	19.8884i	−27.8505			14.5108	+	17.3620i	22.9339	−	22.9339i	0.846492	−	31.6114i
3.19	−2.03915	+	1.96007i	−8.88596	+	3.68068i	0.316262	−	7.99375i	−1.79401	−	11.0355i	10.9054	−	24.9225i	−30.1289			15.0234	+	16.9203i	46.3209	−	46.3209i	25.2885	+	18.9866i
3.20	−1.95861	−	2.04055i	−5.60840	+	2.32308i	−0.327729	+	7.99328i	−1.27934	+	11.1069i	15.7250	+	6.89426i	12.5758			16.9526	−	14.9869i	6.96561	−	6.96561i	25.1700	−	19.1435i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
160.ba	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.ba.a	yes	280
5.c	odd	4	1		160.4.u.a	✓	280
32.h	odd	8	1		160.4.u.a	✓	280
160.ba	even	8	1	inner	160.4.ba.a	yes	280

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.4.u.a	✓	280	5.c	odd	4	1
160.4.u.a	✓	280	32.h	odd	8	1
160.4.ba.a	yes	280	1.a	even	1	1	trivial
160.4.ba.a	yes	280	160.ba	even	8	1	inner

Hecke kernels

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