Newform orbit 72.3.j.a

• Label: 72.3.j.a
• Level: $72$
• Weight: $3$
• Character orbit: 72.j
• Analytic conductor: $1.962$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.96185790339\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 3 q^{2} - q^{4} + 5 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.95473	−	0.423142i	2.84278	+	0.958433i	3.64190	+	1.65425i	−3.64648	+	6.31589i	−5.15130	−	3.07637i	−0.487126	−	0.843726i	−6.41894	−	4.77465i	7.16281	+	5.44923i	9.80039	−	10.8029i
5.2	−1.94520	+	0.464985i	−2.50633	+	1.64872i	3.56758	−	1.80898i	−1.64388	+	2.84729i	4.10868	−	4.37250i	−4.94431	−	8.56379i	−6.09849	+	5.17768i	3.56342	−	8.26451i	1.87373	−	6.30292i
5.3	−1.89895	−	0.627676i	0.340809	−	2.98058i	3.21205	+	2.38385i	1.53127	−	2.65223i	−2.51802	+	5.44606i	−0.720479	−	1.24791i	−4.60324	−	6.54295i	−8.76770	−	2.03162i	−4.57254	+	4.07532i
5.4	−1.87413	+	0.698310i	1.07504	+	2.80077i	3.02473	−	2.61745i	3.98823	−	6.90782i	−3.97056	−	4.49829i	5.64852	+	9.78353i	−3.84094	+	7.01763i	−6.68859	+	6.02186i	−2.65067	+	15.7312i
5.5	−1.55605	−	1.25647i	−2.97841	+	0.359265i	0.842586	+	3.91025i	0.661853	−	1.14636i	5.08596	+	3.18324i	4.89334	+	8.47551i	3.60199	−	7.14323i	8.74186	−	2.14008i	−2.47024	+	0.952203i
5.6	−1.54182	+	1.27389i	−1.07504	−	2.80077i	0.754413	−	3.92821i	−3.98823	+	6.90782i	5.22538	+	2.94880i	5.64852	+	9.78353i	3.84094	+	7.01763i	−6.68859	+	6.02186i	−2.65067	−	15.7312i
5.7	−1.37529	+	1.45210i	2.50633	−	1.64872i	−0.217170	−	3.99410i	1.64388	−	2.84729i	−1.05282	+	5.90691i	−4.94431	−	8.56379i	6.09849	+	5.17768i	3.56342	−	8.26451i	1.87373	+	6.30292i
5.8	−0.820362	−	1.82401i	2.99204	+	0.218346i	−2.65401	+	2.99269i	2.90774	−	5.03636i	−2.05629	−	5.63663i	−0.363382	−	0.629396i	7.63595	+	2.38585i	8.90465	+	1.30660i	−11.5718	−	1.17211i
5.9	−0.610911	+	1.90441i	−2.84278	−	0.958433i	−3.25357	−	2.32685i	3.64648	−	6.31589i	3.56194	−	4.82831i	−0.487126	−	0.843726i	6.41894	−	4.77465i	7.16281	+	5.44923i	9.80039	+	10.8029i
5.10	−0.467688	−	1.94455i	−1.79120	−	2.40657i	−3.56254	+	1.81888i	−1.89538	+	3.28290i	−3.84198	+	4.60860i	−5.70744	−	9.88558i	5.20306	+	6.07685i	−2.58320	+	8.62132i	7.27021	+	2.15029i
5.11	−0.405893	+	1.95838i	−0.340809	+	2.98058i	−3.67050	−	1.58979i	−1.53127	+	2.65223i	−5.69877	−	1.87723i	−0.720479	−	1.24791i	4.60324	−	6.54295i	−8.76770	−	2.03162i	−4.57254	−	4.07532i
5.12	−0.390748	−	1.96146i	−0.102534	+	2.99825i	−3.69463	+	1.53287i	−3.47699	+	6.02232i	5.92100	−	0.970445i	2.29534	+	3.97565i	4.45034	+	6.64790i	−8.97897	−	0.614843i	13.1711	+	4.46675i
5.13	0.310106	+	1.97581i	2.97841	−	0.359265i	−3.80767	+	1.22542i	−0.661853	+	1.14636i	1.63346	+	5.77337i	4.89334	+	8.47551i	−3.60199	−	7.14323i	8.74186	−	2.14008i	−2.47024	−	0.952203i
5.14	0.676143	−	1.88224i	−2.21491	+	2.02340i	−3.08566	−	2.54533i	4.28090	−	7.41474i	2.31093	+	5.53711i	−3.75800	−	6.50904i	−6.87727	+	4.08695i	0.811683	−	8.96332i	−11.0618	−	13.0711i
5.15	0.848214	−	1.81122i	1.69228	−	2.47713i	−2.56107	−	3.07261i	−0.344546	+	0.596772i	−3.05123	−	5.16624i	3.20652	+	5.55385i	−7.73752	+	2.03243i	−3.27238	−	8.38400i	0.788638	+	1.13024i
5.16	1.16946	+	1.62246i	−2.99204	−	0.218346i	−1.26474	+	3.79479i	−2.90774	+	5.03636i	−3.14481	−	5.10981i	−0.363382	−	0.629396i	−7.63595	+	2.38585i	8.90465	+	1.30660i	−11.5718	+	1.17211i
5.17	1.45018	+	1.37730i	1.79120	+	2.40657i	0.206068	+	3.99469i	1.89538	−	3.28290i	−0.717012	+	5.95700i	−5.70744	−	9.88558i	−5.20306	+	6.07685i	−2.58320	+	8.62132i	7.27021	−	2.15029i
5.18	1.50330	+	1.31913i	0.102534	−	2.99825i	0.519809	+	3.96608i	3.47699	−	6.02232i	4.10921	−	4.37200i	2.29534	+	3.97565i	−4.45034	+	6.64790i	−8.97897	−	0.614843i	13.1711	−	4.46675i
5.19	1.55921	−	1.25254i	2.13383	+	2.10874i	0.862276	−	3.90595i	−0.693019	+	1.20034i	5.96837	+	0.615265i	−0.562989	−	0.975125i	−3.54790	−	7.17024i	0.106423	+	8.99937i	0.422919	+	2.73963i
5.20	1.86434	−	0.724045i	−2.13383	−	2.10874i	2.95152	−	2.69973i	0.693019	−	1.20034i	−5.50500	−	2.38642i	−0.562989	−	0.975125i	3.54790	−	7.17024i	0.106423	+	8.99937i	0.422919	−	2.73963i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
9.d	odd	6	1	inner
72.j	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.j.a	✓	44
3.b	odd	2	1		216.3.j.a		44
4.b	odd	2	1		288.3.n.a		44
8.b	even	2	1	inner	72.3.j.a	✓	44
8.d	odd	2	1		288.3.n.a		44
9.c	even	3	1		216.3.j.a		44
9.c	even	3	1		648.3.h.a		44
9.d	odd	6	1	inner	72.3.j.a	✓	44
9.d	odd	6	1		648.3.h.a		44
12.b	even	2	1		864.3.n.a		44
24.f	even	2	1		864.3.n.a		44
24.h	odd	2	1		216.3.j.a		44
36.f	odd	6	1		864.3.n.a		44
36.f	odd	6	1		2592.3.h.a		44
36.h	even	6	1		288.3.n.a		44
36.h	even	6	1		2592.3.h.a		44
72.j	odd	6	1	inner	72.3.j.a	✓	44
72.j	odd	6	1		648.3.h.a		44
72.l	even	6	1		288.3.n.a		44
72.l	even	6	1		2592.3.h.a		44
72.n	even	6	1		216.3.j.a		44
72.n	even	6	1		648.3.h.a		44
72.p	odd	6	1		864.3.n.a		44
72.p	odd	6	1		2592.3.h.a		44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.3.j.a	✓	44	1.a	even	1	1	trivial
72.3.j.a	✓	44	8.b	even	2	1	inner
72.3.j.a	✓	44	9.d	odd	6	1	inner
72.3.j.a	✓	44	72.j	odd	6	1	inner
216.3.j.a		44	3.b	odd	2	1
216.3.j.a		44	9.c	even	3	1
216.3.j.a		44	24.h	odd	2	1
216.3.j.a		44	72.n	even	6	1
288.3.n.a		44	4.b	odd	2	1
288.3.n.a		44	8.d	odd	2	1
288.3.n.a		44	36.h	even	6	1
288.3.n.a		44	72.l	even	6	1
648.3.h.a		44	9.c	even	3	1
648.3.h.a		44	9.d	odd	6	1
648.3.h.a		44	72.j	odd	6	1
648.3.h.a		44	72.n	even	6	1
864.3.n.a		44	12.b	even	2	1
864.3.n.a		44	24.f	even	2	1
864.3.n.a		44	36.f	odd	6	1
864.3.n.a		44	72.p	odd	6	1
2592.3.h.a		44	36.f	odd	6	1
2592.3.h.a		44	36.h	even	6	1
2592.3.h.a		44	72.l	even	6	1
2592.3.h.a		44	72.p	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(72, [\chi])\).