Newform orbit 153.4.s.a

• Label: 153.4.s.a
• Level: $153$
• Weight: $4$
• Character orbit: 153.s
• Analytic conductor: $9.027$
• Analytic rank: $0$
• Dimension: $832$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.s (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.02729223088\)
Analytic rank:	\(0\)
Dimension:	\(832\)
Relative dimension:	\(52\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 832 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−4.29555	−	3.29609i	−4.98529	+	1.46521i	5.51698	+	20.5896i	1.11170	−	1.26765i	26.2440	+	10.1381i	1.52138	−	1.33422i	27.5908	−	66.6101i	22.7063	−	14.6090i	−8.95366	+	1.78099i
5.2	−4.22921	−	3.24519i	4.65415	+	2.31060i	5.28444	+	19.7218i	12.5721	−	14.3357i	−12.1850	−	24.8756i	−9.68176	+	8.49068i	25.3319	−	61.1566i	16.3222	+	21.5078i	−99.6923	+	19.8300i
5.3	−4.17083	−	3.20039i	2.80611	+	4.37330i	5.08277	+	18.9691i	−10.9559	+	12.4928i	2.29245	−	27.2209i	19.4627	−	17.0683i	23.4145	−	56.5276i	−11.2515	+	24.5439i	85.6769	−	17.0422i
5.4	−4.04087	−	3.10067i	0.0205008	−	5.19611i	4.64393	+	17.3314i	−10.6221	+	12.1121i	−16.1943	+	20.9332i	−10.7139	+	9.39583i	19.3801	−	46.7876i	−26.9992	−	0.213049i	80.4781	−	16.0081i
5.5	−3.90532	−	2.99666i	−1.37623	−	5.01059i	4.20103	+	15.6784i	10.2749	−	11.7163i	−9.64042	+	23.6921i	0.545254	−	0.478175i	15.5064	−	37.4357i	−23.2120	+	13.7914i	−75.2365	+	14.9655i
5.6	−3.60914	−	2.76939i	4.64018	−	2.33853i	3.28580	+	12.2628i	−2.64521	+	3.01628i	−23.2233	−	4.41039i	−14.0566	+	12.3273i	8.17420	−	19.7343i	16.0626	−	21.7024i	17.9002	−	3.56057i
5.7	−3.49146	−	2.67909i	−0.171974	+	5.19331i	2.94222	+	10.9805i	−2.40454	+	2.74185i	14.5138	−	17.6715i	−24.8303	+	21.7756i	5.67200	−	13.6934i	−26.9408	−	1.78623i	15.7410	−	3.13108i
5.8	−3.31868	−	2.54651i	−4.14677	−	3.13118i	2.45835	+	9.17469i	−3.19087	+	3.63849i	5.78819	+	20.9512i	16.9114	−	14.8309i	2.39854	−	5.79059i	7.39137	+	25.9686i	19.8549	−	3.94939i
5.9	−3.30283	−	2.53435i	5.19373	+	0.158803i	2.41519	+	9.01363i	−3.84077	+	4.37956i	−16.7515	−	13.6872i	1.62349	−	1.42376i	2.12144	−	5.12160i	26.9496	+	1.64956i	23.7847	−	4.73107i
5.10	−3.27889	−	2.51598i	−2.21477	+	4.70051i	2.35040	+	8.77180i	2.81137	−	3.20575i	19.0884	−	9.84012i	9.26511	−	8.12529i	1.71009	−	4.12851i	−17.1896	−	20.8211i	−17.2838	+	3.43795i
5.11	−3.08693	−	2.36868i	3.50260	−	3.83820i	1.84791	+	6.89649i	6.36879	−	7.26221i	−19.9038	+	3.55168i	16.5745	−	14.5354i	−1.28090	+	3.09236i	−2.46353	−	26.8874i	−36.8619	+	7.33228i
5.12	−2.79393	−	2.14386i	−5.07664	+	1.10802i	1.13938	+	4.25221i	−13.5965	+	15.5038i	16.5592	+	7.78788i	−12.7877	+	11.2145i	−4.84869	+	11.7058i	24.5446	−	11.2500i	71.2257	−	14.1677i
5.13	−2.55395	−	1.95971i	2.93030	+	4.29108i	0.611619	+	2.28259i	9.34538	−	10.6564i	0.925436	−	16.7017i	23.0015	−	20.1718i	−6.94423	+	16.7649i	−9.82665	+	25.1483i	−44.7510	+	8.90153i
5.14	−2.49213	−	1.91228i	−4.75754	+	2.08946i	0.483354	+	1.80390i	14.3872	−	16.4054i	15.8520	+	3.89054i	−8.24283	+	7.22877i	−7.37189	+	17.7973i	18.2683	−	19.8813i	−67.2265	+	13.3722i
5.15	−2.28512	−	1.75343i	−4.63567	−	2.34747i	0.0766901	+	0.286211i	4.30203	−	4.90552i	6.47693	+	13.4926i	−8.75114	+	7.67454i	−8.49143	+	20.5001i	15.9788	+	21.7641i	−18.4322	+	3.66639i
5.16	−1.91748	−	1.47134i	1.58091	−	4.94982i	−0.558640	−	2.08487i	5.44240	−	6.20587i	−10.3142	+	7.16515i	−20.6388	+	18.0998i	−9.39574	+	22.6833i	−22.0014	−	15.6505i	−19.5667	+	3.89205i
5.17	−1.86507	−	1.43112i	1.64118	−	4.93016i	−0.640164	−	2.38912i	−10.8656	+	12.3898i	−10.1166	+	6.84638i	21.9057	−	19.2108i	−9.42229	+	22.7474i	−21.6130	−	16.1826i	37.9963	−	7.55794i
5.18	−1.71856	−	1.31870i	3.26991	+	4.03827i	−0.856066	−	3.19488i	2.51318	−	2.86574i	−0.294283	−	11.2520i	−13.6533	+	11.9736i	−9.37363	+	22.6299i	−5.61532	+	26.4096i	−8.09810	+	1.61081i
5.19	−1.66237	−	1.27558i	5.14095	+	0.755411i	−0.934186	−	3.48643i	−8.50256	+	9.69531i	−7.58257	−	7.81347i	3.43021	−	3.00821i	−9.30917	+	22.4743i	25.8587	+	7.76705i	26.5016	−	5.27149i
5.20	−1.43563	−	1.10160i	−2.32307	−	4.64794i	−1.22304	−	4.56444i	−5.77104	+	6.58061i	−1.78509	+	9.23180i	−9.10323	+	7.98332i	−8.81228	+	21.2747i	−16.2067	+	21.5950i	15.5342	−	3.08995i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
17.e	odd	16	1	inner
153.s	even	48	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	153.4.s.a	✓	832
9.d	odd	6	1	inner	153.4.s.a	✓	832
17.e	odd	16	1	inner	153.4.s.a	✓	832
153.s	even	48	1	inner	153.4.s.a	✓	832

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
153.4.s.a	✓	832	1.a	even	1	1	trivial
153.4.s.a	✓	832	9.d	odd	6	1	inner
153.4.s.a	✓	832	17.e	odd	16	1	inner
153.4.s.a	✓	832	153.s	even	48	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(153, [\chi])\).