Newform orbit 735.2.j.h

• Label: 735.2.j.h
• Level: $735$
• Weight: $2$
• Character orbit: 735.j
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$735 = 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	735.j (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 24 * q + 4 * q^3 + 16 * q^10 - 16 * q^12 + 8 * q^13 - 16 * q^15 - 16 * q^16 - 20 * q^18 + 8 * q^22 - 16 * q^25 + 16 * q^27 + 20 * q^30 - 28 * q^33 + 16 * q^36 - 16 * q^37 - 64 * q^40 - 40 * q^43 - 20 * q^45 - 64 * q^46 - 16 * q^48 - 20 * q^51 - 40 * q^55 + 4 * q^57 + 40 * q^58 + 32 * q^60 - 32 * q^61 + 16 * q^66 + 24 * q^67 - 8 * q^72 - 32 * q^73 + 60 * q^75 - 32 * q^76 + 60 * q^78 + 52 * q^81 + 80 * q^82 + 24 * q^85 - 4 * q^87 + 96 * q^88 + 24 * q^90 - 76 * q^93 + 96 * q^96 - 24 * q^97

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
197.1	−1.79963	+	1.79963i	1.66094	+	0.491204i		−	4.47734i	−1.87996	+	1.21069i	−3.87306	+	2.10509i		0		4.45829	+	4.45829i	2.51744	+	1.63172i	1.20443	−	5.56202i
197.2	−1.54414	+	1.54414i	0.00622252	−	1.73204i		−	2.76875i	−0.252500	−	2.22177i	2.66491	+	2.68412i		0		1.18705	+	1.18705i	−2.99992	−	0.0215553i	3.82062	+	3.04083i
197.3	−1.24414	+	1.24414i	−1.66575	+	0.474620i		−	1.09578i	−1.67522	+	1.48109i	1.48194	−	2.66293i		0		−1.12498	−	1.12498i	2.54947	−	1.58120i	0.241524	−	3.92690i
197.4	−0.800553	+	0.800553i	1.34285	−	1.09397i			0.718229i	2.10480	+	0.754855i	−0.199242	+	1.95080i		0		−2.17609	−	2.17609i	0.606476	−	2.93806i	−2.28931	+	1.08070i
197.5	−0.347054	+	0.347054i	−1.72305	+	0.176396i			1.75911i	1.16790	−	1.90683i	0.536770	−	0.659208i		0		−1.30461	−	1.30461i	2.93777	−	0.607876i	0.256447	+	1.06710i
197.6	−0.260263	+	0.260263i	1.52191	+	0.826909i			1.86453i	−0.895238	−	2.04904i	−0.611312	+	0.180884i		0		−1.00579	−	1.00579i	1.63244	+	2.51697i	0.766286	+	0.300291i
197.7	0.260263	−	0.260263i	−0.826909	−	1.52191i			1.86453i	0.895238	+	2.04904i	−0.611312	−	0.180884i		0		1.00579	+	1.00579i	−1.63244	+	2.51697i	0.766286	+	0.300291i
197.8	0.347054	−	0.347054i	−0.176396	+	1.72305i			1.75911i	−1.16790	+	1.90683i	0.536770	+	0.659208i		0		1.30461	+	1.30461i	−2.93777	−	0.607876i	0.256447	+	1.06710i
197.9	0.800553	−	0.800553i	1.09397	−	1.34285i			0.718229i	−2.10480	−	0.754855i	−0.199242	−	1.95080i		0		2.17609	+	2.17609i	−0.606476	−	2.93806i	−2.28931	+	1.08070i
197.10	1.24414	−	1.24414i	−0.474620	+	1.66575i		−	1.09578i	1.67522	−	1.48109i	1.48194	+	2.66293i		0		1.12498	+	1.12498i	−2.54947	−	1.58120i	0.241524	−	3.92690i
197.11	1.54414	−	1.54414i	1.73204	−	0.00622252i		−	2.76875i	0.252500	+	2.22177i	2.66491	−	2.68412i		0		−1.18705	−	1.18705i	2.99992	−	0.0215553i	3.82062	+	3.04083i
197.12	1.79963	−	1.79963i	−0.491204	−	1.66094i		−	4.47734i	1.87996	−	1.21069i	−3.87306	−	2.10509i		0		−4.45829	−	4.45829i	−2.51744	+	1.63172i	1.20443	−	5.56202i
638.1	−1.79963	−	1.79963i	1.66094	−	0.491204i			4.47734i	−1.87996	−	1.21069i	−3.87306	−	2.10509i		0		4.45829	−	4.45829i	2.51744	−	1.63172i	1.20443	+	5.56202i
638.2	−1.54414	−	1.54414i	0.00622252	+	1.73204i			2.76875i	−0.252500	+	2.22177i	2.66491	−	2.68412i		0		1.18705	−	1.18705i	−2.99992	+	0.0215553i	3.82062	−	3.04083i
638.3	−1.24414	−	1.24414i	−1.66575	−	0.474620i			1.09578i	−1.67522	−	1.48109i	1.48194	+	2.66293i		0		−1.12498	+	1.12498i	2.54947	+	1.58120i	0.241524	+	3.92690i
638.4	−0.800553	−	0.800553i	1.34285	+	1.09397i		−	0.718229i	2.10480	−	0.754855i	−0.199242	−	1.95080i		0		−2.17609	+	2.17609i	0.606476	+	2.93806i	−2.28931	−	1.08070i
638.5	−0.347054	−	0.347054i	−1.72305	−	0.176396i		−	1.75911i	1.16790	+	1.90683i	0.536770	+	0.659208i		0		−1.30461	+	1.30461i	2.93777	+	0.607876i	0.256447	−	1.06710i
638.6	−0.260263	−	0.260263i	1.52191	−	0.826909i		−	1.86453i	−0.895238	+	2.04904i	−0.611312	−	0.180884i		0		−1.00579	+	1.00579i	1.63244	−	2.51697i	0.766286	−	0.300291i
638.7	0.260263	+	0.260263i	−0.826909	+	1.52191i		−	1.86453i	0.895238	−	2.04904i	−0.611312	+	0.180884i		0		1.00579	−	1.00579i	−1.63244	−	2.51697i	0.766286	−	0.300291i
638.8	0.347054	+	0.347054i	−0.176396	−	1.72305i		−	1.75911i	−1.16790	−	1.90683i	0.536770	−	0.659208i		0		1.30461	−	1.30461i	−2.93777	+	0.607876i	0.256447	−	1.06710i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.j.h		24
3.b	odd	2	1	inner	735.2.j.h		24
5.c	odd	4	1	inner	735.2.j.h		24
7.b	odd	2	1		105.2.j.a	✓	24
7.c	even	3	2		735.2.y.g		48
7.d	odd	6	2		735.2.y.j		48
15.e	even	4	1	inner	735.2.j.h		24
21.c	even	2	1		105.2.j.a	✓	24
21.g	even	6	2		735.2.y.j		48
21.h	odd	6	2		735.2.y.g		48
35.c	odd	2	1		525.2.j.b		24
35.f	even	4	1		105.2.j.a	✓	24
35.f	even	4	1		525.2.j.b		24
35.k	even	12	2		735.2.y.j		48
35.l	odd	12	2		735.2.y.g		48
105.g	even	2	1		525.2.j.b		24
105.k	odd	4	1		105.2.j.a	✓	24
105.k	odd	4	1		525.2.j.b		24
105.w	odd	12	2		735.2.y.j		48
105.x	even	12	2		735.2.y.g		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.j.a	✓	24	7.b	odd	2	1
105.2.j.a	✓	24	21.c	even	2	1
105.2.j.a	✓	24	35.f	even	4	1
105.2.j.a	✓	24	105.k	odd	4	1
525.2.j.b		24	35.c	odd	2	1
525.2.j.b		24	35.f	even	4	1
525.2.j.b		24	105.g	even	2	1
525.2.j.b		24	105.k	odd	4	1
735.2.j.h		24	1.a	even	1	1	trivial
735.2.j.h		24	3.b	odd	2	1	inner
735.2.j.h		24	5.c	odd	4	1	inner
735.2.j.h		24	15.e	even	4	1	inner
735.2.y.g		48	7.c	even	3	2
735.2.y.g		48	21.h	odd	6	2
735.2.y.g		48	35.l	odd	12	2
735.2.y.g		48	105.x	even	12	2
735.2.y.j		48	7.d	odd	6	2
735.2.y.j		48	21.g	even	6	2
735.2.y.j		48	35.k	even	12	2
735.2.y.j		48	105.w	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(735, [\chi])$:

$T_{2}^{24} + 76T_{2}^{20} + 1702T_{2}^{16} + 11860T_{2}^{12} + 15921T_{2}^{8} + 1160T_{2}^{4} + 16$

T13^12 - 4*T13^11 + 8*T13^10 + 124*T13^9 + 625*T13^8 - 160*T13^7 + 3328*T13^6 + 41056*T13^5 + 186352*T13^4 + 321280*T13^3 + 247808*T13^2 - 129536*T13 + 33856

$T_{17}^{24} + 1218T_{17}^{20} + 405105T_{17}^{16} + 26412784T_{17}^{12} + 52126816T_{17}^{8} + 244992T_{17}^{4} + 256$