Newform orbit 45.6.j.a

• Label: 45.6.j.a
• Level: $45$
• Weight: $6$
• Character orbit: 45.j
• Analytic conductor: $7.217$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.21727189158\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 414 q^{4} - 30 q^{5} + 12 q^{6} + 468 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−9.49551	+	5.48224i	9.75163	+	12.1617i	44.1098	−	76.4004i	−16.8805	+	53.2921i	−159.270	−	62.0204i	−90.9354	+	52.5016i			616.418i	−52.8116	+	237.192i	−131.871	−	598.579i
4.2	−8.77494	+	5.06621i	−9.60411	−	12.2785i	35.3330	−	61.1986i	−47.0619	−	30.1691i	146.481	+	59.0864i	115.876	−	66.9012i			391.781i	−58.5220	+	235.848i	565.809	+	26.3064i
4.3	−8.11521	+	4.68532i	9.54849	−	12.3218i	27.9045	−	48.3320i	44.0318	−	34.4412i	−19.7566	+	144.732i	−174.545	+	100.773i			223.105i	−60.6526	−	235.309i	−195.960	+	485.801i
4.4	−7.97990	+	4.60720i	−14.8858	+	4.62756i	26.4525	−	45.8171i	53.0966	+	17.4857i	97.4667	−	105.509i	68.5705	−	39.5892i			192.627i	200.171	−	137.769i	−504.266	+	105.092i
4.5	−6.83819	+	3.94803i	−3.71153	+	15.1402i	15.1739	−	26.2820i	−18.8028	−	52.6446i	−34.3937	−	118.185i	−2.59838	+	1.50018i		−	13.0460i	−215.449	−	112.386i	336.420	+	285.760i
4.6	−6.46655	+	3.73347i	11.5319	−	10.4888i	11.8775	−	20.5725i	−43.2756	+	35.3867i	−35.4118	+	110.881i	94.9826	−	54.8383i		−	61.5642i	22.9684	−	241.912i	147.729	−	390.398i
4.7	−5.78828	+	3.34187i	14.2219	+	6.38253i	6.33614	−	10.9745i	55.4347	−	7.21038i	−103.650	+	10.5839i	134.358	−	77.5715i		−	129.181i	161.527	+	181.544i	−296.776	+	226.991i
4.8	−5.40193	+	3.11881i	−15.5676	−	0.805240i	3.45391	−	5.98234i	−38.3848	+	40.6400i	86.6067	−	44.2026i	−214.711	+	123.964i		−	156.515i	241.703	+	25.0714i	80.6040	−	339.249i
4.9	−3.97109	+	2.29271i	−3.68862	−	15.1458i	−5.48699	+	9.50374i	33.0400	+	45.0928i	49.3726	+	51.6882i	50.5331	−	29.1753i		−	197.053i	−215.788	+	111.734i	−234.589	−	103.316i
4.10	−3.47377	+	2.00558i	14.5667	+	5.55075i	−7.95526	+	13.7789i	−49.2039	−	26.5325i	−61.7340	+	9.93271i	−86.2220	+	49.7803i		−	192.177i	181.378	+	161.712i	224.136	−	6.51460i
4.11	−3.02650	+	1.74735i	−9.72592	−	12.1822i	−9.89354	+	17.1361i	14.1623	−	54.0780i	50.7221	+	19.8749i	−44.9809	+	25.9697i		−	180.980i	−53.8131	+	236.967i	51.6311	+	188.413i
4.12	−2.41982	+	1.39709i	0.263169	+	15.5862i	−12.0963	+	20.9514i	4.74243	+	55.7002i	−22.4121	−	37.3483i	50.0291	−	28.8843i		−	157.012i	−242.861	+	8.20363i	−89.2938	−	128.159i
4.13	−0.620078	+	0.358002i	−15.0828	+	3.93798i	−15.7437	+	27.2688i	−55.4907	−	6.76608i	7.94274	−	7.84155i	187.320	−	108.150i		−	45.4572i	211.985	−	118.792i	36.8309	−	15.6703i
4.14	−0.227356	+	0.131264i	−10.9582	+	11.0868i	−15.9655	+	27.6531i	48.5617	−	27.6904i	1.03612	−	3.95907i	−109.161	+	63.0243i		−	16.7837i	−2.83439	−	242.983i	−7.40604	+	12.6700i
4.15	0.227356	−	0.131264i	10.9582	−	11.0868i	−15.9655	+	27.6531i	−0.300290	−	55.9009i	1.03612	−	3.95907i	109.161	−	63.0243i			16.7837i	−2.83439	−	242.983i	−7.40604	−	12.6700i
4.16	0.620078	−	0.358002i	15.0828	−	3.93798i	−15.7437	+	27.2688i	33.6050	+	44.6733i	7.94274	−	7.84155i	−187.320	+	108.150i			45.4572i	211.985	−	118.792i	36.8309	+	15.6703i
4.17	2.41982	−	1.39709i	−0.263169	−	15.5862i	−12.0963	+	20.9514i	−50.6090	+	23.7430i	−22.4121	−	37.3483i	−50.0291	+	28.8843i			157.012i	−242.861	+	8.20363i	−89.2938	+	128.159i
4.18	3.02650	−	1.74735i	9.72592	+	12.1822i	−9.89354	+	17.1361i	39.7518	−	39.3039i	50.7221	+	19.8749i	44.9809	−	25.9697i			180.980i	−53.8131	+	236.967i	51.6311	−	188.413i
4.19	3.47377	−	2.00558i	−14.5667	−	5.55075i	−7.95526	+	13.7789i	47.5798	+	29.3456i	−61.7340	+	9.93271i	86.2220	−	49.7803i			192.177i	181.378	+	161.712i	224.136	+	6.51460i
4.20	3.97109	−	2.29271i	3.68862	+	15.1458i	−5.48699	+	9.50374i	−55.5715	−	6.06703i	49.3726	+	51.6882i	−50.5331	+	29.1753i			197.053i	−215.788	+	111.734i	−234.589	+	103.316i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.6.j.a	✓	56
3.b	odd	2	1		135.6.j.a		56
5.b	even	2	1	inner	45.6.j.a	✓	56
9.c	even	3	1	inner	45.6.j.a	✓	56
9.d	odd	6	1		135.6.j.a		56
15.d	odd	2	1		135.6.j.a		56
45.h	odd	6	1		135.6.j.a		56
45.j	even	6	1	inner	45.6.j.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.6.j.a	✓	56	1.a	even	1	1	trivial
45.6.j.a	✓	56	5.b	even	2	1	inner
45.6.j.a	✓	56	9.c	even	3	1	inner
45.6.j.a	✓	56	45.j	even	6	1	inner
135.6.j.a		56	3.b	odd	2	1
135.6.j.a		56	9.d	odd	6	1
135.6.j.a		56	15.d	odd	2	1
135.6.j.a		56	45.h	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(45, [\chi])\).