Newform orbit 72.5.j.a

• Label: 72.5.j.a
• Level: $72$
• Weight: $5$
• Character orbit: 72.j
• Analytic conductor: $7.443$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	72.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.44263734204\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 3 q^{2} - q^{4} - 19 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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	−3.99936	−	0.0715673i	−8.43375	−	3.14194i	15.9898	+	0.572447i	−5.94208	+	10.2920i	33.5048	+	13.1693i	21.5744	+	37.3679i	−63.9078	−	3.43376i	61.2564	+	52.9967i	24.5011	−	40.7361i
5.2	−3.96631	−	0.518044i	−4.87156	+	7.56756i	15.4633	+	4.10945i	9.54377	−	16.5303i	23.2425	−	27.4916i	−6.73634	−	11.6677i	−59.2032	−	24.3100i	−33.5358	−	73.7316i	−46.4170	+	60.6202i
5.3	−3.95490	−	0.598961i	0.933380	−	8.95147i	15.2825	+	4.73766i	−12.1310	+	21.0114i	−9.05300	+	34.8431i	−22.4004	−	38.7986i	−57.6031	−	27.8906i	−79.2576	−	16.7102i	60.5617	−	75.8321i
5.4	−3.93003	+	0.744918i	8.05757	+	4.00943i	14.8902	−	5.85509i	13.0982	−	22.6868i	−34.6532	−	9.75495i	−43.7982	−	75.8606i	−54.1573	+	34.1026i	48.8489	+	64.6126i	−34.5765	+	98.9167i
5.5	−3.75717	+	1.37247i	8.39079	−	3.25495i	12.2327	−	10.3132i	−10.8201	+	18.7410i	−27.0583	+	23.7455i	18.2392	+	31.5913i	−31.8056	+	55.5374i	59.8107	−	54.6231i	14.9316	−	85.2634i
5.6	−3.63985	−	1.65877i	4.81678	+	7.60254i	10.4970	+	12.0753i	−5.01037	+	8.67822i	−4.92146	−	35.6620i	23.4688	+	40.6492i	−18.1772	−	61.3644i	−34.5973	+	73.2395i	32.6322	−	23.2763i
5.7	−3.58461	−	1.77498i	7.77894	−	4.52637i	9.69890	+	12.7252i	19.2792	−	33.3925i	−35.9187	+	2.41784i	36.2264	+	62.7460i	−12.1798	−	62.8303i	40.0239	−	70.4208i	−128.379	+	85.4792i
5.8	−3.48346	+	1.96609i	−0.573879	−	8.98168i	8.26898	−	13.6976i	16.4840	−	28.5511i	19.6579	+	30.1590i	2.21331	+	3.83357i	−1.87399	+	63.9726i	−80.3413	+	10.3088i	−1.28728	+	131.866i
5.9	−3.44441	+	2.03372i	0.573879	+	8.98168i	7.72797	−	14.0099i	−16.4840	+	28.5511i	−20.2429	−	29.7695i	2.21331	+	3.83357i	1.87399	+	63.9726i	−80.3413	+	10.3088i	−1.28728	−	131.866i
5.10	−3.09807	−	2.53021i	−7.96182	−	4.19635i	3.19606	+	15.6775i	20.1379	−	34.8798i	14.0486	+	33.1457i	−36.9510	−	64.0010i	29.7659	−	56.6568i	45.7812	+	66.8213i	−150.642	+	57.1069i
5.11	−3.06718	+	2.56757i	−8.39079	+	3.25495i	2.81517	−	15.7504i	10.8201	−	18.7410i	17.3787	−	31.5274i	18.2392	+	31.5913i	31.8056	+	55.5374i	59.8107	−	54.6231i	14.9316	+	85.2634i
5.12	−2.83732	−	2.81950i	−7.20486	+	5.39351i	0.100792	+	15.9997i	−21.1127	+	36.5683i	35.6495	+	5.01101i	−16.9047	−	29.2798i	44.8252	−	45.6804i	22.8201	−	77.7190i	163.008	−	44.2286i
5.13	−2.81177	−	2.84499i	8.34560	−	3.36912i	−0.187929	+	15.9989i	−8.74932	+	15.1543i	−33.0510	−	14.2700i	−30.7371	−	53.2382i	46.0451	−	44.4505i	58.2981	−	56.2346i	67.7148	−	17.7186i
5.14	−2.61013	+	3.03104i	−8.05757	−	4.00943i	−2.37444	−	15.8228i	−13.0982	+	22.6868i	33.1841	−	13.9577i	−43.7982	−	75.8606i	54.1573	+	34.1026i	48.8489	+	64.6126i	−34.5765	−	98.9167i
5.15	−2.21293	−	3.33211i	−3.11153	−	8.44502i	−6.20585	+	14.7475i	−5.93817	+	10.2852i	−21.2541	+	29.0562i	29.0886	+	50.3830i	62.8732	−	11.9566i	−61.6367	+	52.5539i	47.4122	−	2.97388i
5.16	−1.93770	+	3.49933i	8.43375	+	3.14194i	−8.49063	−	13.5613i	5.94208	−	10.2920i	−27.3368	+	23.4244i	21.5744	+	37.3679i	63.9078	−	3.43376i	61.2564	+	52.9967i	24.5011	+	40.7361i
5.17	−1.67951	−	3.63032i	5.83929	+	6.84855i	−10.3585	+	12.1943i	3.78105	−	6.54897i	15.0553	−	32.7007i	−14.0251	−	24.2922i	61.6666	+	17.1240i	−12.8053	+	79.9814i	−30.1252	−	2.72734i
5.18	−1.53452	+	3.69395i	4.87156	−	7.56756i	−11.2905	−	11.3369i	−9.54377	+	16.5303i	20.4787	+	29.6078i	−6.73634	−	11.6677i	59.2032	−	24.3100i	−33.5358	−	73.7316i	−46.4170	−	60.6202i
5.19	−1.45874	+	3.72453i	−0.933380	+	8.95147i	−11.7442	−	10.8662i	12.1310	−	21.0114i	−31.9784	−	16.5342i	−22.4004	−	38.7986i	57.6031	−	27.8906i	−79.2576	−	16.7102i	60.5617	+	75.8321i
5.20	−1.38904	−	3.75108i	−4.45965	+	7.81739i	−12.1411	+	10.4208i	15.7860	−	27.3421i	35.5182	+	5.86981i	22.5676	+	39.0883i	55.9537	+	31.0674i	−41.2231	−	69.7256i	−124.490	−	21.2351i

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	72.5.j.a	✓	92
3.b	odd	2	1		216.5.j.a		92
4.b	odd	2	1		288.5.n.a		92
8.b	even	2	1	inner	72.5.j.a	✓	92
8.d	odd	2	1		288.5.n.a		92
9.c	even	3	1		216.5.j.a		92
9.d	odd	6	1	inner	72.5.j.a	✓	92
12.b	even	2	1		864.5.n.a		92
24.f	even	2	1		864.5.n.a		92
24.h	odd	2	1		216.5.j.a		92
36.f	odd	6	1		864.5.n.a		92
36.h	even	6	1		288.5.n.a		92
72.j	odd	6	1	inner	72.5.j.a	✓	92
72.l	even	6	1		288.5.n.a		92
72.n	even	6	1		216.5.j.a		92
72.p	odd	6	1		864.5.n.a		92

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.5.j.a	✓	92	1.a	even	1	1	trivial
72.5.j.a	✓	92	8.b	even	2	1	inner
72.5.j.a	✓	92	9.d	odd	6	1	inner
72.5.j.a	✓	92	72.j	odd	6	1	inner
216.5.j.a		92	3.b	odd	2	1
216.5.j.a		92	9.c	even	3	1
216.5.j.a		92	24.h	odd	2	1
216.5.j.a		92	72.n	even	6	1
288.5.n.a		92	4.b	odd	2	1
288.5.n.a		92	8.d	odd	2	1
288.5.n.a		92	36.h	even	6	1
288.5.n.a		92	72.l	even	6	1
864.5.n.a		92	12.b	even	2	1
864.5.n.a		92	24.f	even	2	1
864.5.n.a		92	36.f	odd	6	1
864.5.n.a		92	72.p	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(72, [\chi])\).