Newform orbit 60.6.h.c

• Label: 60.6.h.c
• Level: $60$
• Weight: $6$
• Character orbit: 60.h
• Analytic conductor: $9.623$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.62302918878\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{4} + 420 q^{6} + 1440 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−5.44678	−	1.52727i	−11.2078	+	10.8345i	27.3349	+	16.6374i	42.6056	−	36.1906i	77.5935	−	41.8958i	−128.907			−123.478	−	132.368i	8.22823	−	242.861i	−287.336	+	132.052i
59.2	−5.44678	−	1.52727i	11.2078	+	10.8345i	27.3349	+	16.6374i	42.6056	+	36.1906i	−44.4991	−	76.1303i	128.907			−123.478	−	132.368i	8.22823	+	242.861i	−176.791	−	262.193i
59.3	−5.44678	+	1.52727i	−11.2078	−	10.8345i	27.3349	−	16.6374i	42.6056	+	36.1906i	77.5935	+	41.8958i	−128.907			−123.478	+	132.368i	8.22823	+	242.861i	−287.336	−	132.052i
59.4	−5.44678	+	1.52727i	11.2078	−	10.8345i	27.3349	−	16.6374i	42.6056	−	36.1906i	−44.4991	+	76.1303i	128.907			−123.478	+	132.368i	8.22823	−	242.861i	−176.791	+	262.193i
59.5	−5.20228	−	2.22177i	−15.5789	+	0.545517i	22.1275	+	23.1165i	−17.3193	+	53.1511i	82.2579	+	31.7747i	106.220			−63.7541	−	169.421i	242.405	−	16.9971i	208.189	−	238.028i
59.6	−5.20228	−	2.22177i	15.5789	+	0.545517i	22.1275	+	23.1165i	−17.3193	−	53.1511i	−79.8339	−	37.4506i	−106.220			−63.7541	−	169.421i	242.405	+	16.9971i	−27.9895	+	314.987i
59.7	−5.20228	+	2.22177i	−15.5789	−	0.545517i	22.1275	−	23.1165i	−17.3193	−	53.1511i	82.2579	−	31.7747i	106.220			−63.7541	+	169.421i	242.405	+	16.9971i	208.189	+	238.028i
59.8	−5.20228	+	2.22177i	15.5789	−	0.545517i	22.1275	−	23.1165i	−17.3193	+	53.1511i	−79.8339	+	37.4506i	−106.220			−63.7541	+	169.421i	242.405	−	16.9971i	−27.9895	−	314.987i
59.9	−4.16295	−	3.83012i	−7.28769	−	13.7800i	2.66035	+	31.8892i	45.7166	−	32.1713i	−22.4409	+	85.2784i	213.200			111.065	−	142.943i	−136.779	+	200.849i	−313.536	−	41.1725i
59.10	−4.16295	−	3.83012i	7.28769	−	13.7800i	2.66035	+	31.8892i	45.7166	+	32.1713i	−83.1175	+	29.4530i	−213.200			111.065	−	142.943i	−136.779	−	200.849i	−67.0961	−	309.028i
59.11	−4.16295	+	3.83012i	−7.28769	+	13.7800i	2.66035	−	31.8892i	45.7166	+	32.1713i	−22.4409	−	85.2784i	213.200			111.065	+	142.943i	−136.779	−	200.849i	−313.536	+	41.1725i
59.12	−4.16295	+	3.83012i	7.28769	+	13.7800i	2.66035	−	31.8892i	45.7166	−	32.1713i	−83.1175	−	29.4530i	−213.200			111.065	+	142.943i	−136.779	+	200.849i	−67.0961	+	309.028i
59.13	−3.28930	−	4.60223i	−3.33289	+	15.2280i	−10.3610	+	30.2762i	−17.3746	+	53.1331i	81.0456	−	34.7508i	−121.108			173.418	−	51.9042i	−220.784	−	101.506i	301.681	−	94.8087i
59.14	−3.28930	−	4.60223i	3.33289	+	15.2280i	−10.3610	+	30.2762i	−17.3746	−	53.1331i	59.1198	−	65.4282i	121.108			173.418	−	51.9042i	−220.784	+	101.506i	−187.380	+	254.733i
59.15	−3.28930	+	4.60223i	−3.33289	−	15.2280i	−10.3610	−	30.2762i	−17.3746	−	53.1331i	81.0456	+	34.7508i	−121.108			173.418	+	51.9042i	−220.784	+	101.506i	301.681	+	94.8087i
59.16	−3.28930	+	4.60223i	3.33289	−	15.2280i	−10.3610	−	30.2762i	−17.3746	+	53.1331i	59.1198	+	65.4282i	121.108			173.418	+	51.9042i	−220.784	−	101.506i	−187.380	−	254.733i
59.17	−3.08938	−	4.73875i	−14.4591	−	5.82522i	−12.9114	+	29.2796i	−43.0117	−	35.7070i	17.0656	+	86.5145i	−168.076			178.637	−	29.2717i	175.134	+	168.455i	−36.3270	+	314.134i
59.18	−3.08938	−	4.73875i	14.4591	−	5.82522i	−12.9114	+	29.2796i	−43.0117	+	35.7070i	−72.2740	−	50.5219i	168.076			178.637	−	29.2717i	175.134	−	168.455i	302.086	+	93.5090i
59.19	−3.08938	+	4.73875i	−14.4591	+	5.82522i	−12.9114	−	29.2796i	−43.0117	+	35.7070i	17.0656	−	86.5145i	−168.076			178.637	+	29.2717i	175.134	−	168.455i	−36.3270	−	314.134i
59.20	−3.08938	+	4.73875i	14.4591	+	5.82522i	−12.9114	−	29.2796i	−43.0117	−	35.7070i	−72.2740	+	50.5219i	168.076			178.637	+	29.2717i	175.134	+	168.455i	302.086	−	93.5090i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
5.b	even	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	60.6.h.c	✓	48
3.b	odd	2	1	inner	60.6.h.c	✓	48
4.b	odd	2	1	inner	60.6.h.c	✓	48
5.b	even	2	1	inner	60.6.h.c	✓	48
12.b	even	2	1	inner	60.6.h.c	✓	48
15.d	odd	2	1	inner	60.6.h.c	✓	48
20.d	odd	2	1	inner	60.6.h.c	✓	48
60.h	even	2	1	inner	60.6.h.c	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.6.h.c	✓	48	1.a	even	1	1	trivial
60.6.h.c	✓	48	3.b	odd	2	1	inner
60.6.h.c	✓	48	4.b	odd	2	1	inner
60.6.h.c	✓	48	5.b	even	2	1	inner
60.6.h.c	✓	48	12.b	even	2	1	inner
60.6.h.c	✓	48	15.d	odd	2	1	inner
60.6.h.c	✓	48	20.d	odd	2	1	inner
60.6.h.c	✓	48	60.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^12 - 124828*T7^10 + 6013060404*T7^8 - 144328380358144*T7^6 + 1836468169716195328*T7^4 - 11821917603469598392320*T7^2 + 30215962439806704471244800