Newform orbit 39.5.i.b

• Label: 39.5.i.b
• Level: $39$
• Weight: $5$
• Character orbit: 39.i
• Analytic conductor: $4.031$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	39.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.03142856027\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} + 142 q^{4} + 122 q^{7} + 114 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−6.33131	+	3.65538i	7.39769	+	5.12584i	18.7236	−	32.4303i		−	21.2308i	−65.5739	−	5.41191i	35.0835	−	60.7664i			156.796i	28.4515	+	75.8387i	77.6067	+	134.419i
29.2	−5.96411	+	3.44338i	−8.99846	−	0.166218i	15.7137	−	27.2170i		−	27.0884i	54.2402	−	29.9938i	−39.3076	+	68.0828i			106.245i	80.9447	+	2.99141i	93.2757	+	161.558i
29.3	−5.70393	+	3.29317i	2.61666	−	8.61122i	13.6899	−	23.7116i			26.9859i	13.4329	+	57.7349i	−6.28578	+	10.8873i			74.9512i	−67.3062	−	45.0653i	−88.8692	−	153.926i
29.4	−4.18280	+	2.41494i	−2.70769	+	8.58303i	3.66386	−	6.34600i			24.8103i	−9.40176	−	42.4400i	−4.36411	+	7.55885i		−	41.8860i	−66.3368	−	46.4804i	−59.9153	−	103.776i
29.5	−3.22377	+	1.86124i	−7.97580	−	4.16973i	−1.07155	+	1.85597i			15.5623i	33.4730	−	1.40267i	40.6154	−	70.3479i		−	67.5374i	46.2268	+	66.5138i	−28.9652	−	50.1691i
29.6	−2.83347	+	1.63590i	6.59382	−	6.12548i	−2.64764	+	4.58585i		−	28.1535i	−8.66269	+	28.1432i	−1.35822	+	2.35251i		−	69.6740i	5.95697	−	80.7807i	46.0564	+	79.7720i
29.7	−2.18006	+	1.25866i	7.31254	+	5.24659i	−4.83157	+	8.36852i		−	1.90367i	−22.5454	−	2.23390i	−15.0949	+	26.1452i		−	64.6021i	25.9465	+	76.7319i	2.39607	+	4.15011i
29.8	−0.624020	+	0.360278i	−6.61180	+	6.10607i	−7.74040	+	13.4068i		−	46.9495i	1.92602	−	6.19239i	21.2118	−	36.7399i		−	22.6837i	6.43193	−	80.7442i	16.9149	+	29.2974i
29.9	0.624020	−	0.360278i	8.59391	−	2.67296i	−7.74040	+	13.4068i			46.9495i	4.39976	−	4.76417i	21.2118	−	36.7399i			22.6837i	66.7106	−	45.9423i	16.9149	+	29.2974i
29.10	2.18006	−	1.25866i	0.887413	+	8.95614i	−4.83157	+	8.36852i			1.90367i	13.2073	+	18.4080i	−15.0949	+	26.1452i			64.6021i	−79.4250	+	15.8956i	2.39607	+	4.15011i
29.11	2.83347	−	1.63590i	−8.60173	+	2.64768i	−2.64764	+	4.58585i			28.1535i	−20.0414	+	21.5737i	−1.35822	+	2.35251i			69.6740i	66.9796	−	45.5492i	46.0564	+	79.7720i
29.12	3.22377	−	1.86124i	0.376810	−	8.99211i	−1.07155	+	1.85597i		−	15.5623i	−15.5218	−	29.6898i	40.6154	−	70.3479i			67.5374i	−80.7160	−	6.77664i	−28.9652	−	50.1691i
29.13	4.18280	−	2.41494i	8.78697	+	1.94658i	3.66386	−	6.34600i		−	24.8103i	41.4550	−	13.0778i	−4.36411	+	7.55885i			41.8860i	73.4216	+	34.2091i	−59.9153	−	103.776i
29.14	5.70393	−	3.29317i	−8.76586	−	2.03952i	13.6899	−	23.7116i		−	26.9859i	−56.7164	+	17.2342i	−6.28578	+	10.8873i		−	74.9512i	72.6808	+	35.7562i	−88.8692	−	153.926i
29.15	5.96411	−	3.44338i	4.35528	−	7.87601i	15.7137	−	27.2170i			27.0884i	−1.14470	−	61.9703i	−39.3076	+	68.0828i		−	106.245i	−43.0630	−	68.6045i	93.2757	+	161.558i
29.16	6.33131	−	3.65538i	0.740266	+	8.96950i	18.7236	−	32.4303i			21.2308i	37.4738	+	54.0827i	35.0835	−	60.7664i		−	156.796i	−79.9040	+	13.2796i	77.6067	+	134.419i
35.1	−6.33131	−	3.65538i	7.39769	−	5.12584i	18.7236	+	32.4303i			21.2308i	−65.5739	+	5.41191i	35.0835	+	60.7664i		−	156.796i	28.4515	−	75.8387i	77.6067	−	134.419i
35.2	−5.96411	−	3.44338i	−8.99846	+	0.166218i	15.7137	+	27.2170i			27.0884i	54.2402	+	29.9938i	−39.3076	−	68.0828i		−	106.245i	80.9447	−	2.99141i	93.2757	−	161.558i
35.3	−5.70393	−	3.29317i	2.61666	+	8.61122i	13.6899	+	23.7116i		−	26.9859i	13.4329	−	57.7349i	−6.28578	−	10.8873i		−	74.9512i	−67.3062	+	45.0653i	−88.8692	+	153.926i
35.4	−4.18280	−	2.41494i	−2.70769	−	8.58303i	3.66386	+	6.34600i		−	24.8103i	−9.40176	+	42.4400i	−4.36411	−	7.55885i			41.8860i	−66.3368	+	46.4804i	−59.9153	+	103.776i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.c	even	3	1	inner
39.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	39.5.i.b	✓	32
3.b	odd	2	1	inner	39.5.i.b	✓	32
13.c	even	3	1	inner	39.5.i.b	✓	32
39.i	odd	6	1	inner	39.5.i.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.5.i.b	✓	32	1.a	even	1	1	trivial
39.5.i.b	✓	32	3.b	odd	2	1	inner
39.5.i.b	✓	32	13.c	even	3	1	inner
39.5.i.b	✓	32	39.i	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 199*T2^30 + 23740*T2^28 - 1858841*T2^26 + 107912428*T2^24 - 4674522829*T2^22 + 156652093415*T2^20 - 4001068589866*T2^18 + 79326429910348*T2^16 - 1191132918164960*T2^14 + 13716078999111280*T2^12 - 115798942388292448*T2^10 + 717309417968812096*T2^8 - 2865340414885318656*T2^6 + 7286321910069146880*T2^4 - 3691060352857612800*T2^2 + 1567234861601670144