Newform orbit 72.4.n.a

• Label: 72.4.n.a
• Level: $72$
• Weight: $4$
• Character orbit: 72.n
• Analytic conductor: $4.248$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24813752041\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - q^{2} - q^{4} - 25 q^{6} - 2 q^{7} - 10 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−2.80501	−	0.363209i	−5.19539	−	0.0892698i	7.73616	+	2.03761i	10.5646	+	6.09948i	14.5407	+	2.13741i	−9.53003	−	16.5065i	−20.9599	−	8.52535i	26.9841	+	0.927582i	−27.4185	−	20.9463i
13.2	−2.79478	+	0.434983i	−3.80570	+	3.53789i	7.62158	−	2.43136i	−13.2040	−	7.62335i	9.09715	−	11.5430i	6.18711	+	10.7164i	−20.2430	+	10.1104i	1.96663	−	26.9283i	40.2184	+	15.5620i
13.3	−2.75744	−	0.629699i	3.11870	+	4.15616i	7.20696	+	3.47272i	2.76156	+	1.59439i	−5.98250	−	13.4242i	1.73091	+	2.99802i	−17.6860	−	14.1140i	−7.54740	+	25.9237i	−6.61085	−	6.13538i
13.4	−2.65450	+	0.976535i	4.63039	−	2.35786i	6.09276	−	5.18443i	7.42736	+	4.28819i	−9.98884	+	10.7807i	−10.3525	−	17.9310i	−11.1105	+	19.7119i	15.8810	−	21.8356i	−23.9035	−	4.12993i
13.5	−2.53942	+	1.24553i	−1.68289	−	4.91608i	4.89731	−	6.32585i	−6.66588	−	3.84855i	10.3967	+	10.3879i	4.44653	+	7.70162i	−4.55727	+	22.1637i	−21.3358	+	16.5464i	21.7209	+	1.47052i
13.6	−2.53536	−	1.25377i	0.137064	−	5.19434i	4.85613	+	6.35752i	13.2324	+	7.63974i	−6.86001	+	12.9977i	11.2586	+	19.5004i	−4.34120	−	22.2071i	−26.9624	−	1.42391i	−23.9705	−	35.9599i
13.7	−2.53312	−	1.25830i	4.57518	−	2.46327i	4.83336	+	6.37484i	−17.3643	−	10.0253i	−14.6890	+	0.482803i	1.69820	+	2.94137i	−4.22200	−	22.2300i	14.8646	−	22.5398i	31.3710	+	47.2447i
13.8	−2.14152	+	1.84767i	−0.844287	+	5.12710i	1.17220	−	7.91366i	14.6790	+	8.47493i	−7.66516	−	12.5397i	9.54171	+	16.5267i	12.1116	+	19.1131i	−25.5744	−	8.65749i	−47.0943	+	8.97282i
13.9	−1.80733	−	2.17568i	−1.25203	+	5.04306i	−1.46713	+	7.86432i	−0.772898	−	0.446233i	13.2349	−	6.39046i	−14.6308	−	25.3412i	19.7618	−	11.0214i	−23.8649	−	12.6281i	0.426023	+	2.48806i
13.10	−1.73446	−	2.23420i	−4.97517	−	1.49923i	−1.98329	+	7.75026i	−7.81354	−	4.51115i	5.27965	+	13.7159i	6.37921	+	11.0491i	20.7556	−	9.01146i	22.5046	+	14.9179i	3.47347	+	25.2814i
13.11	−1.70888	+	2.25383i	2.73187	+	4.42006i	−2.15947	−	7.70303i	−13.3655	−	7.71659i	−14.6305	−	1.39619i	−14.8241	−	25.6761i	21.0516	+	8.29647i	−12.0738	+	24.1500i	40.2319	−	16.9369i
13.12	−1.38757	+	2.46468i	−5.00405	−	1.39982i	−4.14930	−	6.83983i	4.46589	+	2.57838i	10.3936	−	10.3910i	−4.47414	−	7.74943i	22.6154	−	0.735958i	23.0810	+	14.0095i	−12.5516	+	7.42931i
13.13	−1.20810	+	2.55744i	5.11365	−	0.922265i	−5.08101	−	6.17927i	−2.06876	−	1.19440i	−3.81914	+	14.1920i	17.9335	+	31.0618i	21.9415	−	5.52925i	25.2989	−	9.43228i	5.55387	−	3.84779i
13.14	−1.06764	−	2.61919i	4.97517	+	1.49923i	−5.72028	+	5.59271i	7.81354	+	4.51115i	−1.38493	−	14.6315i	6.37921	+	11.0491i	20.7556	+	9.01146i	22.5046	+	14.9179i	3.47347	−	25.2814i
13.15	−0.980526	−	2.65303i	1.25203	−	5.04306i	−6.07714	+	5.20273i	0.772898	+	0.446233i	−14.6070	+	1.62318i	−14.6308	−	25.3412i	19.7618	+	11.0214i	−23.8649	−	12.6281i	0.426023	−	2.48806i
13.16	−0.0804569	+	2.82728i	1.39010	−	5.00676i	−7.98705	−	0.454949i	−6.59200	−	3.80589i	14.0437	+	4.33302i	−7.67810	−	13.2989i	1.92888	−	22.5451i	−23.1353	−	13.9197i	11.2907	−	18.3312i
13.17	0.176838	−	2.82289i	−4.57518	+	2.46327i	−7.93746	−	0.998392i	17.3643	+	10.0253i	6.14449	+	13.3509i	1.69820	+	2.94137i	−4.22200	+	22.2300i	14.8646	−	22.5398i	31.3710	−	47.2447i
13.18	0.181886	−	2.82257i	−0.137064	+	5.19434i	−7.93383	−	1.02677i	−13.2324	−	7.63974i	14.6365	+	1.33165i	11.2586	+	19.5004i	−4.34120	+	22.2071i	−26.9624	−	1.42391i	−23.9705	+	35.9599i
13.19	0.262215	+	2.81625i	−3.60475	+	3.74243i	−7.86249	+	1.47692i	−3.38868	−	1.95645i	−11.4848	−	9.17054i	1.48218	+	2.56721i	−6.22103	−	21.7554i	−1.01155	−	26.9810i	4.62129	−	10.0564i
13.20	0.487912	+	2.78603i	4.64700	+	2.32494i	−7.52388	+	2.71867i	17.5797	+	10.1497i	−4.21002	+	14.0810i	−10.6896	−	18.5150i	−11.2453	−	19.6353i	16.1893	+	21.6080i	−19.6999	+	53.9298i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
9.c	even	3	1	inner
72.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.4.n.a	✓	68
3.b	odd	2	1		216.4.n.a		68
4.b	odd	2	1		288.4.r.a		68
8.b	even	2	1	inner	72.4.n.a	✓	68
8.d	odd	2	1		288.4.r.a		68
9.c	even	3	1	inner	72.4.n.a	✓	68
9.d	odd	6	1		216.4.n.a		68
12.b	even	2	1		864.4.r.a		68
24.f	even	2	1		864.4.r.a		68
24.h	odd	2	1		216.4.n.a		68
36.f	odd	6	1		288.4.r.a		68
36.h	even	6	1		864.4.r.a		68
72.j	odd	6	1		216.4.n.a		68
72.l	even	6	1		864.4.r.a		68
72.n	even	6	1	inner	72.4.n.a	✓	68
72.p	odd	6	1		288.4.r.a		68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.4.n.a	✓	68	1.a	even	1	1	trivial
72.4.n.a	✓	68	8.b	even	2	1	inner
72.4.n.a	✓	68	9.c	even	3	1	inner
72.4.n.a	✓	68	72.n	even	6	1	inner
216.4.n.a		68	3.b	odd	2	1
216.4.n.a		68	9.d	odd	6	1
216.4.n.a		68	24.h	odd	2	1
216.4.n.a		68	72.j	odd	6	1
288.4.r.a		68	4.b	odd	2	1
288.4.r.a		68	8.d	odd	2	1
288.4.r.a		68	36.f	odd	6	1
288.4.r.a		68	72.p	odd	6	1
864.4.r.a		68	12.b	even	2	1
864.4.r.a		68	24.f	even	2	1
864.4.r.a		68	36.h	even	6	1
864.4.r.a		68	72.l	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(72, [\chi])\).