Newform orbit 72.3.p.b

• Label: 72.3.p.b
• Level: $72$
• Weight: $3$
• Character orbit: 72.p
• Analytic conductor: $1.962$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96185790339\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 5 q^{2} - 6 q^{3} + 7 q^{4} + 3 q^{6} + 46 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.99054	+	0.194264i	−0.136174	+	2.99691i	3.92452	−	0.773382i	−3.84571	+	2.22032i	−0.311131	−	5.99193i	−0.704321	−	0.406640i	−7.66169	+	2.30184i	−8.96291	−	0.816203i	7.22371	−	5.16672i
43.2	−1.95218	−	0.434750i	2.76566	−	1.16239i	3.62198	+	1.69742i	−0.0166003	+	0.00958419i	−5.90440	+	1.06681i	4.07208	+	2.35102i	−6.33280	−	4.88832i	6.29772	−	6.42952i	0.0365734	−	0.0114930i
43.3	−1.75132	+	0.965848i	−2.96823	−	0.435462i	2.13428	−	3.38303i	1.50948	−	0.871501i	5.61892	−	2.10422i	7.93804	+	4.58303i	−0.470322	+	7.98616i	8.62075	+	2.58510i	−1.80186	+	2.98421i
43.4	−1.70852	−	1.03970i	−2.73922	+	1.22340i	1.83805	+	3.55269i	6.07342	−	3.50649i	5.95196	+	0.757771i	−8.07247	−	4.66064i	0.553388	−	7.98084i	6.00661	−	6.70229i	−14.0222	−	0.323638i
43.5	−1.58502	+	1.21972i	0.418145	−	2.97072i	1.02458	−	3.86655i	−4.40783	+	2.54486i	2.96067	+	5.21866i	−10.9609	−	6.32830i	3.09212	+	7.37826i	−8.65031	−	2.48438i	3.88249	−	9.40996i
43.6	−1.49437	−	1.32923i	−1.96889	−	2.26351i	0.466308	+	3.97273i	−5.84790	+	3.37629i	−0.0664586	+	5.99963i	3.50808	+	2.02539i	4.58382	−	6.55657i	−1.24694	+	8.91320i	13.2268	+	2.72775i
43.7	−1.45504	+	1.37217i	2.66358	+	1.38035i	0.234289	−	3.99313i	8.07964	−	4.66478i	−5.76969	+	1.64642i	−4.91220	−	2.83606i	5.13836	+	6.13166i	5.18928	+	7.35333i	−5.35532	+	17.8741i
43.8	−0.865989	−	1.80279i	1.33710	+	2.68555i	−2.50013	+	3.12240i	1.70411	−	0.983869i	3.68357	−	4.73617i	8.69613	+	5.02071i	7.79412	+	1.80325i	−5.42431	+	7.18171i	−3.24945	−	2.22014i
43.9	−0.564251	−	1.91875i	1.21551	−	2.74272i	−3.36324	+	2.16532i	5.15803	−	2.97799i	−5.94847	−	0.784682i	−4.09037	−	2.36158i	6.05243	+	5.23145i	−6.04507	−	6.66762i	−8.62446	−	8.21666i
43.10	−0.460815	+	1.94619i	2.66358	+	1.38035i	−3.57530	−	1.79367i	−8.07964	+	4.66478i	−3.91383	+	4.54774i	4.91220	+	2.83606i	5.13836	−	6.13166i	5.18928	+	7.35333i	−5.35532	−	17.8741i
43.11	−0.263796	+	1.98253i	0.418145	−	2.97072i	−3.86082	−	1.04596i	4.40783	−	2.54486i	5.77922	+	1.61265i	10.9609	+	6.32830i	3.09212	−	7.37826i	−8.65031	−	2.48438i	3.88249	+	9.40996i
43.12	0.0392136	+	1.99962i	−2.96823	−	0.435462i	−3.99692	+	0.156824i	−1.50948	+	0.871501i	0.754361	−	5.95239i	−7.93804	−	4.58303i	−0.470322	−	7.98616i	8.62075	+	2.58510i	−1.80186	−	2.98421i
43.13	0.104500	−	1.99727i	−2.08749	+	2.15462i	−3.97816	−	0.417428i	−5.42020	+	3.12935i	4.08522	+	4.39443i	−5.96345	−	3.44300i	−1.24943	+	7.90183i	−0.284811	−	8.99549i	5.68375	+	11.1526i
43.14	0.827034	+	1.82099i	−0.136174	+	2.99691i	−2.63203	+	3.01205i	3.84571	−	2.22032i	−5.56997	+	2.23057i	0.704321	+	0.406640i	−7.66169	−	2.30184i	−8.96291	−	0.816203i	7.22371	+	5.16672i
43.15	1.35259	+	1.47326i	2.76566	−	1.16239i	−0.340985	+	3.98544i	0.0166003	−	0.00958419i	5.45330	+	2.50229i	−4.07208	−	2.35102i	−6.33280	+	4.88832i	6.29772	−	6.42952i	0.0365734	+	0.0114930i
43.16	1.67744	−	1.08913i	−2.08749	+	2.15462i	1.62758	−	3.65390i	5.42020	−	3.12935i	−1.15495	+	5.88779i	5.96345	+	3.44300i	−1.24943	−	7.90183i	−0.284811	−	8.99549i	5.68375	−	11.1526i
43.17	1.75466	+	0.959768i	−2.73922	+	1.22340i	2.15769	+	3.36814i	−6.07342	+	3.50649i	−5.98058	−	0.482363i	8.07247	+	4.66064i	0.553388	+	7.98084i	6.00661	−	6.70229i	−14.0222	+	0.323638i
43.18	1.89833	+	0.629552i	−1.96889	−	2.26351i	3.20733	+	2.39020i	5.84790	−	3.37629i	−2.31261	−	5.53641i	−3.50808	−	2.02539i	4.58382	+	6.55657i	−1.24694	+	8.91320i	13.2268	−	2.72775i
43.19	1.94382	−	0.470721i	1.21551	−	2.74272i	3.55684	−	1.82999i	−5.15803	+	2.97799i	1.07167	−	5.90352i	4.09037	+	2.36158i	6.05243	−	5.23145i	−6.04507	−	6.66762i	−8.62446	+	8.21666i
43.20	1.99426	−	0.151428i	1.33710	+	2.68555i	3.95414	−	0.603974i	−1.70411	+	0.983869i	3.07320	+	5.15320i	−8.69613	−	5.02071i	7.79412	−	1.80325i	−5.42431	+	7.18171i	−3.24945	+	2.22014i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
9.c	even	3	1	inner
72.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.3.p.b	✓	40
3.b	odd	2	1		216.3.p.b		40
4.b	odd	2	1		288.3.t.b		40
8.b	even	2	1		288.3.t.b		40
8.d	odd	2	1	inner	72.3.p.b	✓	40
9.c	even	3	1	inner	72.3.p.b	✓	40
9.c	even	3	1		648.3.b.f		20
9.d	odd	6	1		216.3.p.b		40
9.d	odd	6	1		648.3.b.e		20
12.b	even	2	1		864.3.t.b		40
24.f	even	2	1		216.3.p.b		40
24.h	odd	2	1		864.3.t.b		40
36.f	odd	6	1		288.3.t.b		40
36.f	odd	6	1		2592.3.b.e		20
36.h	even	6	1		864.3.t.b		40
36.h	even	6	1		2592.3.b.f		20
72.j	odd	6	1		864.3.t.b		40
72.j	odd	6	1		2592.3.b.f		20
72.l	even	6	1		216.3.p.b		40
72.l	even	6	1		648.3.b.e		20
72.n	even	6	1		288.3.t.b		40
72.n	even	6	1		2592.3.b.e		20
72.p	odd	6	1	inner	72.3.p.b	✓	40
72.p	odd	6	1		648.3.b.f		20

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.3.p.b	✓	40	1.a	even	1	1	trivial
72.3.p.b	✓	40	8.d	odd	2	1	inner
72.3.p.b	✓	40	9.c	even	3	1	inner
72.3.p.b	✓	40	72.p	odd	6	1	inner
216.3.p.b		40	3.b	odd	2	1
216.3.p.b		40	9.d	odd	6	1
216.3.p.b		40	24.f	even	2	1
216.3.p.b		40	72.l	even	6	1
288.3.t.b		40	4.b	odd	2	1
288.3.t.b		40	8.b	even	2	1
288.3.t.b		40	36.f	odd	6	1
288.3.t.b		40	72.n	even	6	1
648.3.b.e		20	9.d	odd	6	1
648.3.b.e		20	72.l	even	6	1
648.3.b.f		20	9.c	even	3	1
648.3.b.f		20	72.p	odd	6	1
864.3.t.b		40	12.b	even	2	1
864.3.t.b		40	24.h	odd	2	1
864.3.t.b		40	36.h	even	6	1
864.3.t.b		40	72.j	odd	6	1
2592.3.b.e		20	36.f	odd	6	1
2592.3.b.e		20	72.n	even	6	1
2592.3.b.f		20	36.h	even	6	1
2592.3.b.f		20	72.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^40 - 309*T5^38 + 55716*T5^36 - 6712983*T5^34 + 603811641*T5^32 - 41961529638*T5^30 + 2327328612189*T5^28 - 104016298280787*T5^26 + 3784106595265245*T5^24 - 111618882769275558*T5^22 + 2661925178189571849*T5^20 - 50373543727284752943*T5^18 + 743736619369179988308*T5^16 - 8157519137669082514317*T5^14 + 63934766382808694810673*T5^12 - 308579748941050774004376*T5^10 + 1058132405272496522900400*T5^8 - 2083606733974200670567296*T5^6 + 2657225112979175083194624*T5^4 - 976335082686666178560*T5^2 + 358628125468262400