Newform orbit 720.2.cu.e

• Label: 720.2.cu.e
• Level: $720$
• Weight: $2$
• Character orbit: 720.cu
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.cu (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
113.1		0		−1.73052	−	0.0727431i		0		0.484586	+	2.18293i		0		1.02200	+	0.273845i		0		2.98942	+	0.251767i		0
113.2		0		−1.60816	−	0.643288i		0		−0.153910	−	2.23076i		0		1.54994	+	0.415306i		0		2.17236	+	2.06902i		0
113.3		0		−1.60229	+	0.657781i		0		2.22211	−	0.249468i		0		−2.85561	−	0.765158i		0		2.13465	−	2.10791i		0
113.4		0		−1.54161	+	0.789587i		0		−1.90652	−	1.16840i		0		−3.69137	−	0.989099i		0		1.75310	−	2.43447i		0
113.5		0		−0.925409	+	1.46411i		0		1.76908	−	1.36761i		0		3.63616	+	0.974307i		0		−1.28724	−	2.70980i		0
113.6		0		−0.836139	−	1.51686i		0		−2.22494	−	0.222822i		0		3.21063	+	0.860287i		0		−1.60174	+	2.53662i		0
113.7		0		−0.771894	−	1.55054i		0		0.743858	+	2.10871i		0		−0.110362	−	0.0295714i		0		−1.80836	+	2.39371i		0
113.8		0		−0.684109	−	1.59122i		0		1.43117	−	1.71807i		0		−2.16574	−	0.580309i		0		−2.06399	+	2.17714i		0
113.9		0		−0.416476	+	1.68123i		0		−1.03051	+	1.98445i		0		−3.03419	−	0.813008i		0		−2.65309	−	1.40039i		0
113.10		0		−0.254049	+	1.71332i		0		−0.831407	−	2.07576i		0		1.77070	+	0.474457i		0		−2.87092	−	0.870532i		0
113.11		0		0.699583	−	1.58448i		0		−2.21911	−	0.274850i		0		1.41974	+	0.380419i		0		−2.02117	−	2.21695i		0
113.12		0		0.985769	+	1.42417i		0		2.06252	+	0.863710i		0		−2.64986	−	0.710029i		0		−1.05652	+	2.80780i		0
113.13		0		1.03585	−	1.38817i		0		−0.403585	+	2.19935i		0		−4.55631	−	1.22086i		0		−0.854037	−	2.87587i		0
113.14		0		1.16547	+	1.28128i		0		1.29543	+	1.82260i		0		4.79192	+	1.28399i		0		−0.283342	+	2.98659i		0
113.15		0		1.37710	−	1.05053i		0		2.15536	−	0.595338i		0		−0.205487	−	0.0550600i		0		0.792791	−	2.89335i		0
113.16		0		1.66953	+	0.461153i		0		−2.18181	−	0.489584i		0		−1.65362	−	0.443086i		0		2.57468	+	1.53982i		0
113.17		0		1.71407	−	0.248907i		0		0.528278	−	2.17277i		0		0.980572	+	0.262743i		0		2.87609	−	0.853289i		0
113.18		0		1.72328	+	0.174114i		0		−1.74059	+	1.40368i		0		2.54088	+	0.680826i		0		2.93937	+	0.600095i		0
257.1		0		−1.59122	+	0.684109i		0		−0.772302	+	2.09846i		0		0.580309	−	2.16574i		0		2.06399	−	2.17714i		0
257.2		0		−1.58448	−	0.699583i		0		−1.34758	−	1.78438i		0		−0.380419	+	1.41974i		0		2.02117	+	2.21695i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
9.d	odd	6	1	inner
45.l	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.cu.e		72
4.b	odd	2	1		360.2.bs.a	✓	72
5.c	odd	4	1	inner	720.2.cu.e		72
9.d	odd	6	1	inner	720.2.cu.e		72
12.b	even	2	1		1080.2.bt.a		72
20.e	even	4	1		360.2.bs.a	✓	72
36.f	odd	6	1		1080.2.bt.a		72
36.h	even	6	1		360.2.bs.a	✓	72
45.l	even	12	1	inner	720.2.cu.e		72
60.l	odd	4	1		1080.2.bt.a		72
180.v	odd	12	1		360.2.bs.a	✓	72
180.x	even	12	1		1080.2.bt.a		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.bs.a	✓	72	4.b	odd	2	1
360.2.bs.a	✓	72	20.e	even	4	1
360.2.bs.a	✓	72	36.h	even	6	1
360.2.bs.a	✓	72	180.v	odd	12	1
720.2.cu.e		72	1.a	even	1	1	trivial
720.2.cu.e		72	5.c	odd	4	1	inner
720.2.cu.e		72	9.d	odd	6	1	inner
720.2.cu.e		72	45.l	even	12	1	inner
1080.2.bt.a		72	12.b	even	2	1
1080.2.bt.a		72	36.f	odd	6	1
1080.2.bt.a		72	60.l	odd	4	1
1080.2.bt.a		72	180.x	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^72 + 8*T7^69 - 987*T7^68 + 120*T7^67 + 32*T7^66 - 11892*T7^65 + 671652*T7^64 + 82888*T7^63 - 56352*T7^62 + 11724288*T7^61 - 219665593*T7^60 - 61184472*T7^59 + 97338408*T7^58 - 4120818548*T7^57 + 50100320133*T7^56 + 20797122264*T7^55 - 35347466472*T7^54 + 989013994512*T7^53 - 7089555929226*T7^52 - 4177048489872*T7^51 + 8823255280056*T7^50 - 140757823600992*T7^49 + 715631133332041*T7^48 + 546356247804048*T7^47 - 1224956948512272*T7^46 + 14222319849994464*T7^45 - 46252934628904089*T7^44 - 46955578159140840*T7^43 + 121694865999945000*T7^42 - 862961598669627636*T7^41 + 2119702903956837849*T7^40 + 2415796853817027880*T7^39 - 6549600864207109176*T7^38 + 36161278167915544560*T7^37 - 67598232678394655026*T7^36 - 80520299368875709248*T7^35 + 248352974661978753576*T7^34 - 933420231302059346288*T7^33 + 1580190886310902412601*T7^32 + 1229181275059882451232*T7^31 - 4533268632725704778704*T7^30 + 15391507571517375876912*T7^29 - 28787321022957707983353*T7^28 - 4472792796368732670520*T7^27 + 55017977542776356620920*T7^26 - 171936685495939363214244*T7^25 + 372990497938513836266080*T7^24 - 247865265607820306035320*T7^23 - 64581241527760849976232*T7^22 + 740009793317551859455800*T7^21 - 2254846971044274585227535*T7^20 + 2313056472080647712919336*T7^19 - 895012496561685420011736*T7^18 - 1491680697686205045702300*T7^17 + 9060375083688175606037181*T7^16 - 16730450492264166046572504*T7^15 + 19898679405096960375933792*T7^14 - 24024524839751630277746640*T7^13 + 22518245025704668653001512*T7^12 - 13808393292276887790015792*T7^11 + 8799787659236422589586720*T7^10 - 4466397260040754233042720*T7^9 - 410782397104480618789956*T7^8 + 680090407795792771255584*T7^7 + 136925818503450931971072*T7^6 + 35435313691117439295456*T7^5 + 11903334693264827498304*T7^4 + 1519990464987382135680*T7^3 + 136068948011299276800*T7^2 + 17413815311996256000*T7 + 1114291571120010000