Newform orbit 45.6.e.b

• Label: 45.6.e.b
• Level: $45$
• Weight: $6$
• Character orbit: 45.e
• Analytic conductor: $7.217$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	45.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.21727189158\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{2} - 11 q^{3} - 208 q^{4} - 275 q^{5} + 167 q^{6} - 225 q^{7} + 624 q^{8} + 473 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−5.51002	+	9.54364i	−8.26871	−	13.2147i	−44.7207	−	77.4586i	−12.5000	−	21.6506i	171.677	−	6.10027i	−76.9862	+	133.344i	633.008			−106.257	+	218.537i	275.501
16.2	−4.58829	+	7.94714i	−5.21295	+	14.6910i	−26.1047	−	45.2147i	−12.5000	−	21.6506i	−92.8329	−	108.835i	68.5076	−	118.659i	185.453			−188.650	−	153.167i	229.414
16.3	−3.86491	+	6.69422i	15.4976	+	1.68057i	−13.8751	−	24.0324i	−12.5000	−	21.6506i	−71.1470	+	97.2492i	−90.3140	+	156.428i	−32.8503			237.351	+	52.0898i	193.246
16.4	−2.71418	+	4.70110i	7.66499	−	13.5738i	1.26645	+	2.19356i	−12.5000	−	21.6506i	43.0076	+	72.8756i	50.4557	−	87.3918i	−187.457			−125.496	−	208.086i	135.709
16.5	−1.40485	+	2.43327i	−14.3473	+	6.09555i	12.0528	+	20.8761i	−12.5000	−	21.6506i	5.32364	−	43.4741i	−77.3368	+	133.951i	−157.640			168.689	−	174.909i	70.2424
16.6	−0.567163	+	0.982355i	−10.4949	−	11.5264i	15.3567	+	26.5985i	−12.5000	−	21.6506i	17.2753	−	3.77232i	6.11924	−	10.5988i	−71.1373			−22.7162	+	241.936i	28.3581
16.7	1.80440	−	3.12531i	6.16270	+	14.3186i	9.48831	+	16.4342i	−12.5000	−	21.6506i	55.8698	+	6.57604i	−83.4512	+	144.542i	183.964			−167.042	+	176.482i	−90.2198
16.8	2.36599	−	4.09802i	12.6470	−	9.11335i	4.80416	+	8.32105i	−12.5000	−	21.6506i	−7.42395	−	73.3898i	60.6499	−	105.049i	196.890			76.8937	−	230.513i	−118.300
16.9	2.73912	−	4.74429i	−11.2787	+	10.7607i	0.994478	+	1.72249i	−12.5000	−	21.6506i	20.1582	+	82.9839i	109.585	−	189.806i	186.199			11.4160	−	242.732i	−136.956
16.10	4.55512	−	7.88970i	−13.3725	−	8.01100i	−25.4983	−	44.1643i	−12.5000	−	21.6506i	−124.118	+	69.0140i	−11.7427	+	20.3390i	−173.063			114.648	+	214.254i	−227.756
16.11	5.18479	−	8.98031i	15.5026	−	1.63336i	−37.7640	−	65.4092i	−12.5000	−	21.6506i	65.7098	−	147.687i	−67.9859	+	117.755i	−451.367			237.664	−	50.6428i	−259.239
31.1	−5.51002	−	9.54364i	−8.26871	+	13.2147i	−44.7207	+	77.4586i	−12.5000	+	21.6506i	171.677	+	6.10027i	−76.9862	−	133.344i	633.008			−106.257	−	218.537i	275.501
31.2	−4.58829	−	7.94714i	−5.21295	−	14.6910i	−26.1047	+	45.2147i	−12.5000	+	21.6506i	−92.8329	+	108.835i	68.5076	+	118.659i	185.453			−188.650	+	153.167i	229.414
31.3	−3.86491	−	6.69422i	15.4976	−	1.68057i	−13.8751	+	24.0324i	−12.5000	+	21.6506i	−71.1470	−	97.2492i	−90.3140	−	156.428i	−32.8503			237.351	−	52.0898i	193.246
31.4	−2.71418	−	4.70110i	7.66499	+	13.5738i	1.26645	−	2.19356i	−12.5000	+	21.6506i	43.0076	−	72.8756i	50.4557	+	87.3918i	−187.457			−125.496	+	208.086i	135.709
31.5	−1.40485	−	2.43327i	−14.3473	−	6.09555i	12.0528	−	20.8761i	−12.5000	+	21.6506i	5.32364	+	43.4741i	−77.3368	−	133.951i	−157.640			168.689	+	174.909i	70.2424
31.6	−0.567163	−	0.982355i	−10.4949	+	11.5264i	15.3567	−	26.5985i	−12.5000	+	21.6506i	17.2753	+	3.77232i	6.11924	+	10.5988i	−71.1373			−22.7162	−	241.936i	28.3581
31.7	1.80440	+	3.12531i	6.16270	−	14.3186i	9.48831	−	16.4342i	−12.5000	+	21.6506i	55.8698	−	6.57604i	−83.4512	−	144.542i	183.964			−167.042	−	176.482i	−90.2198
31.8	2.36599	+	4.09802i	12.6470	+	9.11335i	4.80416	−	8.32105i	−12.5000	+	21.6506i	−7.42395	+	73.3898i	60.6499	+	105.049i	196.890			76.8937	+	230.513i	−118.300
31.9	2.73912	+	4.74429i	−11.2787	−	10.7607i	0.994478	−	1.72249i	−12.5000	+	21.6506i	20.1582	−	82.9839i	109.585	+	189.806i	186.199			11.4160	+	242.732i	−136.956

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.6.e.b	✓	22
3.b	odd	2	1		135.6.e.b		22
9.c	even	3	1	inner	45.6.e.b	✓	22
9.c	even	3	1		405.6.a.j		11
9.d	odd	6	1		135.6.e.b		22
9.d	odd	6	1		405.6.a.i		11

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.6.e.b	✓	22	1.a	even	1	1	trivial
45.6.e.b	✓	22	9.c	even	3	1	inner
135.6.e.b		22	3.b	odd	2	1
135.6.e.b		22	9.d	odd	6	1
405.6.a.i		11	9.d	odd	6	1
405.6.a.j		11	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^22 + 4*T2^21 + 288*T2^20 + 752*T2^19 + 51527*T2^18 + 111006*T2^17 + 5518288*T2^16 + 7631392*T2^15 + 419003217*T2^14 + 416902582*T2^13 + 20993003320*T2^12 + 6860487096*T2^11 + 741711291440*T2^10 + 174258683840*T2^9 + 17189695228608*T2^8 + 1655030452096*T2^7 + 272221657026304*T2^6 + 145211944946688*T2^5 + 2263207067606016*T2^4 + 2085645093894144*T2^3 + 13670297235136512*T2^2 + 12479488802611200*T2 + 14284755027247104