Newform orbit 120.4.w.c

• Label: 120.4.w.c
• Level: $120$
• Weight: $4$
• Character orbit: 120.w
• Analytic conductor: $7.080$
• Analytic rank: $0$
• Dimension: $128$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.08022920069\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 4 q^{6} - 144 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−2.82746	−	0.0740349i	−4.92707	−	1.65046i	7.98904	+	0.418661i	9.98780	−	5.02432i	13.8089	+	5.03137i	−17.9179	+	17.9179i	−22.5577	−	1.77521i	21.5520	+	16.2638i	−28.6121	+	13.4666i
53.2	−2.82670	−	0.0987097i	−0.743209	+	5.14273i	7.98051	+	0.558046i	10.6120	+	3.51928i	2.60847	−	14.4636i	22.4668	−	22.4668i	−22.5035	−	2.36519i	−25.8953	−	7.64424i	−29.6496	−	10.9955i
53.3	−2.79977	+	0.401606i	4.62659	−	2.36530i	7.67742	−	2.24881i	3.00421	−	10.7692i	−12.0035	+	8.48037i	13.4333	−	13.4333i	−20.5919	+	9.37945i	15.8107	−	21.8866i	−4.08613	+	31.3577i
53.4	−2.79310	+	0.445660i	−4.73057	−	2.14982i	7.60277	−	2.48954i	−4.08469	+	10.4075i	14.1710	+	3.89642i	13.5500	−	13.5500i	−20.1258	+	10.3418i	17.7566	+	20.3397i	6.77075	−	30.8894i
53.5	−2.75807	+	0.626941i	0.195243	−	5.19248i	7.21389	−	3.45829i	−10.6250	−	3.47979i	2.71688	+	14.4436i	−17.6061	+	17.6061i	−17.7283	+	14.0609i	−26.9238	−	2.02759i	31.4862	+	2.93625i
53.6	−2.75693	−	0.631946i	3.22170	+	4.07684i	7.20129	+	3.48446i	−8.53855	−	7.21756i	−6.30564	−	13.2755i	−9.44868	+	9.44868i	−17.6514	−	14.1572i	−6.24131	+	26.2687i	18.9790	+	25.2942i
53.7	−2.70249	+	0.834608i	4.97128	+	1.51207i	6.60686	−	4.51103i	2.32852	+	10.9352i	−14.6968	+	0.0627124i	−7.15720	+	7.15720i	−14.0900	+	17.7051i	22.4273	+	15.0339i	−15.4194	−	27.6087i
53.8	−2.68461	−	0.890420i	−4.50848	+	2.58334i	6.41430	+	4.78087i	−7.39959	−	8.38130i	14.4038	−	2.92083i	8.77577	−	8.77577i	−12.9629	−	18.5462i	13.6527	−	23.2938i	12.4022	+	29.0893i
53.9	−2.62963	−	1.04165i	0.323055	−	5.18610i	5.82991	+	5.47833i	4.01180	+	10.4358i	−6.25164	+	13.3010i	−2.52357	+	2.52357i	−9.62399	−	20.4787i	−26.7913	−	3.35079i	0.320933	−	31.6211i
53.10	−2.52995	−	1.26466i	4.80218	−	1.98471i	4.80127	+	6.39905i	−9.85625	+	5.27772i	−14.6592	−	1.05193i	6.52335	−	6.52335i	−4.05433	−	22.2612i	19.1219	−	19.0618i	31.6103	−	0.887536i
53.11	−2.49705	+	1.32844i	0.0934179	+	5.19531i	4.47048	−	6.63436i	5.77473	−	9.57353i	−7.13494	−	12.8488i	−15.4977	+	15.4977i	−2.34964	+	22.5051i	−26.9825	+	0.970671i	−1.70189	+	31.5769i
53.12	−2.43671	+	1.43613i	−3.22666	+	4.07292i	3.87507	−	6.99884i	−9.99090	+	5.01815i	2.01318	−	14.5584i	−4.98039	+	4.98039i	0.608813	+	22.6192i	−6.17736	−	26.2838i	17.1382	−	26.5760i
53.13	−2.38175	−	1.52553i	−3.32206	+	3.99548i	3.34551	+	7.26688i	1.66448	+	11.0557i	14.0076	−	4.44835i	−23.0437	+	23.0437i	3.11769	−	22.4116i	−4.92780	−	26.5465i	12.9015	−	28.8713i
53.14	−2.33291	−	1.59924i	−1.39809	−	5.00453i	2.88489	+	7.46173i	6.09990	−	9.36970i	−4.74181	+	13.9110i	16.1295	−	16.1295i	5.20287	−	22.0211i	−23.0907	+	13.9936i	−29.2148	+	12.1035i
53.15	−2.22682	−	1.74392i	4.46520	+	2.65743i	1.91749	+	7.76680i	10.9958	−	2.02288i	−5.30887	−	13.7046i	−3.45928	+	3.45928i	9.27479	−	20.6392i	12.8761	+	23.7320i	−28.0135	−	14.6712i
53.16	−2.22355	+	1.74810i	1.34563	−	5.01889i	1.88832	−	7.77395i	10.3996	+	4.10459i	5.78143	+	13.5120i	2.75574	−	2.75574i	9.39082	+	20.5867i	−23.3786	−	13.5072i	−30.2993	+	9.05281i
53.17	−1.74810	+	2.22355i	−5.01889	+	1.34563i	−1.88832	−	7.77395i	10.3996	+	4.10459i	5.78143	−	13.5120i	2.75574	−	2.75574i	20.5867	+	9.39082i	23.3786	−	13.5072i	−27.3063	+	15.9489i
53.18	−1.74392	−	2.22682i	−4.46520	−	2.65743i	−1.91749	+	7.76680i	−10.9958	+	2.02288i	1.86933	+	14.5776i	−3.45928	+	3.45928i	20.6392	−	9.27479i	12.8761	+	23.7320i	23.6804	+	20.9580i
53.19	−1.59924	−	2.33291i	1.39809	+	5.00453i	−2.88489	+	7.46173i	−6.09990	+	9.36970i	9.43922	−	11.2650i	16.1295	−	16.1295i	22.0211	−	5.20287i	−23.0907	+	13.9936i	31.6138	−	0.753871i
53.20	−1.52553	−	2.38175i	3.32206	−	3.99548i	−3.34551	+	7.26688i	−1.66448	−	11.0557i	−14.5842	−	1.81710i	−23.0437	+	23.0437i	22.4116	−	3.11769i	−4.92780	−	26.5465i	−23.7929	+	20.8303i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
8.b	even	2	1	inner
15.e	even	4	1	inner
24.h	odd	2	1	inner
40.i	odd	4	1	inner
120.w	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	120.4.w.c	✓	128
3.b	odd	2	1	inner	120.4.w.c	✓	128
5.c	odd	4	1	inner	120.4.w.c	✓	128
8.b	even	2	1	inner	120.4.w.c	✓	128
15.e	even	4	1	inner	120.4.w.c	✓	128
24.h	odd	2	1	inner	120.4.w.c	✓	128
40.i	odd	4	1	inner	120.4.w.c	✓	128
120.w	even	4	1	inner	120.4.w.c	✓	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.w.c	✓	128	1.a	even	1	1	trivial
120.4.w.c	✓	128	3.b	odd	2	1	inner
120.4.w.c	✓	128	5.c	odd	4	1	inner
120.4.w.c	✓	128	8.b	even	2	1	inner
120.4.w.c	✓	128	15.e	even	4	1	inner
120.4.w.c	✓	128	24.h	odd	2	1	inner
120.4.w.c	✓	128	40.i	odd	4	1	inner
120.4.w.c	✓	128	120.w	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\):

T7^32 + 36*T7^31 + 648*T7^30 - 1740*T7^29 + 1620908*T7^28 + 57297240*T7^27 + 1013866056*T7^26 - 2375638224*T7^25 + 731492240080*T7^24 + 24975324911744*T7^23 + 427341590470656*T7^22 - 916889336020992*T7^21 + 136362456134883008*T7^20 + 4315208692905337600*T7^19 + 68102487674579414016*T7^18 - 88093098750273088768*T7^17 + 11035430906434512293632*T7^16 + 305244604205728966097408*T7^15 + 4064111248461101988756992*T7^14 + 5601538285028432192453632*T7^13 + 275745745575331800248317952*T7^12 + 6643609941217291837808877568*T7^11 + 77983517072420744184138649600*T7^10 + 256879469016815578419839479808*T7^9 + 2004409640147234123809173790720*T7^8 + 35158196508260351509745237172224*T7^7 + 380691662233556470529572242071552*T7^6 + 1643684282790463848659418114883584*T7^5 + 3869664891891946360826385278173184*T7^4 + 7394596565021238896235576855592960*T7^3 + 85030076063358823138565829886771200*T7^2 + 363890970536958764384762921902080000*T7 + 778645889600649169822644929536000000

T11^32 - 20810*T11^30 + 191023336*T11^28 - 1022421127568*T11^26 + 3553679389874704*T11^24 - 8455616526471386528*T11^22 + 14149986466063530383104*T11^20 - 16851663245440505546524160*T11^18 + 14295854934310340645923047424*T11^16 - 8562990004332032042895064784896*T11^14 + 3549263536738485403506845718216704*T11^12 - 981419971433314609353852726225141760*T11^10 + 169712355429910390012673341377309310976*T11^8 - 16297216053046428707932165915559034093568*T11^6 + 692535105121800147978155110333348239638528*T11^4 - 11609291066388285165420992959155272687812608*T11^2 + 58439477872525209909029762811185622889267200