Properties

Label 735.2.y.j
Level 735
Weight 2
Character orbit 735.y
Analytic conductor 5.869
Analytic rank 0
Dimension 48
CM no
Inner twists 8

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.y (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{3} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 4q^{3} + 16q^{10} - 16q^{12} - 16q^{13} - 32q^{15} + 16q^{16} + 20q^{18} + 16q^{22} + 16q^{25} - 32q^{27} - 20q^{30} - 28q^{33} + 32q^{36} + 16q^{37} - 64q^{40} - 80q^{43} - 20q^{45} + 64q^{46} + 32q^{48} + 20q^{51} + 80q^{55} + 8q^{57} - 40q^{58} - 32q^{60} - 32q^{61} + 16q^{66} - 24q^{67} + 8q^{72} - 32q^{73} + 60q^{75} + 64q^{76} + 120q^{78} - 52q^{81} + 80q^{82} + 48q^{85} - 4q^{87} - 96q^{88} - 48q^{90} + 76q^{93} + 96q^{96} + 48q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1 −2.45834 + 0.658710i −1.68402 + 0.405074i 3.87749 2.23867i 1.98846 1.02275i 3.87306 2.10509i 0 −4.45829 + 4.45829i 2.67183 1.36430i −4.21463 + 3.82408i
128.2 −2.10934 + 0.565195i 0.860631 + 1.50310i 2.39780 1.38437i −1.79786 1.32955i −2.66491 2.68412i 0 −1.18705 + 1.18705i −1.51863 + 2.58723i 4.54374 + 1.78834i
128.3 −1.69953 + 0.455388i 1.20528 1.24391i 0.948973 0.547890i 2.12027 0.710238i −1.48194 + 2.66293i 0 1.12498 1.12498i −0.0946229 2.99851i −3.28004 + 2.17262i
128.4 −1.09358 + 0.293023i −0.615956 + 1.61883i −0.622004 + 0.359114i −0.398678 + 2.20024i 0.199242 1.95080i 0 2.17609 2.17609i −2.24120 1.99425i −0.208736 2.52295i
128.5 −0.474084 + 0.127030i 1.40400 1.01429i −1.52343 + 0.879554i −2.23532 + 0.0580193i −0.536770 + 0.659208i 0 1.30461 1.30461i 0.942448 2.84812i 1.05236 0.311459i
128.6 −0.355526 + 0.0952630i −1.73147 + 0.0448327i −1.61473 + 0.932263i −1.32690 1.79982i 0.611312 0.180884i 0 1.00579 1.00579i 2.99598 0.155253i 0.643203 + 0.513478i
128.7 0.355526 0.0952630i 1.47708 + 0.904561i −1.61473 + 0.932263i 1.32690 + 1.79982i 0.611312 + 0.180884i 0 −1.00579 + 1.00579i 1.36354 + 2.67222i 0.643203 + 0.513478i
128.8 0.474084 0.127030i −0.708759 1.58040i −1.52343 + 0.879554i 2.23532 0.0580193i −0.536770 0.659208i 0 −1.30461 + 1.30461i −1.99532 + 2.24024i 1.05236 0.311459i
128.9 1.09358 0.293023i −0.275979 + 1.70992i −0.622004 + 0.359114i 0.398678 2.20024i 0.199242 + 1.95080i 0 −2.17609 + 2.17609i −2.84767 0.943806i −0.208736 2.52295i
128.10 1.69953 0.455388i −0.421844 1.67990i 0.948973 0.547890i −2.12027 + 0.710238i −1.48194 2.66293i 0 −1.12498 + 1.12498i −2.64410 + 1.41731i −3.28004 + 2.17262i
128.11 2.10934 0.565195i −1.49688 + 0.871409i 2.39780 1.38437i 1.79786 + 1.32955i −2.66491 + 2.68412i 0 1.18705 1.18705i 1.48129 2.60879i 4.54374 + 1.78834i
128.12 2.45834 0.658710i 1.25586 + 1.19281i 3.87749 2.23867i −1.98846 + 1.02275i 3.87306 + 2.10509i 0 4.45829 4.45829i 0.154393 + 2.99602i −4.21463 + 3.82408i
263.1 −0.658710 + 2.45834i 0.405074 1.68402i −3.87749 2.23867i 0.108509 2.23343i 3.87306 + 2.10509i 0 4.45829 4.45829i −2.67183 1.36430i 5.41906 + 1.73794i
263.2 −0.565195 + 2.10934i 1.50310 + 0.860631i −2.39780 1.38437i −2.05036 + 0.892212i −2.66491 + 2.68412i 0 1.18705 1.18705i 1.51863 + 2.58723i −0.723123 4.82916i
263.3 −0.455388 + 1.69953i −1.24391 + 1.20528i −0.948973 0.547890i 0.445053 2.19133i −1.48194 2.66293i 0 −1.12498 + 1.12498i 0.0946229 2.99851i 3.52156 + 1.75429i
263.4 −0.293023 + 1.09358i 1.61883 0.615956i 0.622004 + 0.359114i 1.70612 + 1.44538i 0.199242 + 1.95080i 0 −2.17609 + 2.17609i 2.24120 1.99425i −2.08057 + 1.44225i
263.5 −0.127030 + 0.474084i −1.01429 + 1.40400i 1.52343 + 0.879554i −1.06741 + 1.96485i −0.536770 0.659208i 0 −1.30461 + 1.30461i −0.942448 2.84812i −0.795910 0.755638i
263.6 −0.0952630 + 0.355526i 0.0448327 1.73147i 1.61473 + 0.932263i −2.22214 + 0.249219i 0.611312 + 0.180884i 0 −1.00579 + 1.00579i −2.99598 0.155253i 0.123083 0.813769i
263.7 0.0952630 0.355526i 0.904561 + 1.47708i 1.61473 + 0.932263i 2.22214 0.249219i 0.611312 0.180884i 0 1.00579 1.00579i −1.36354 + 2.67222i 0.123083 0.813769i
263.8 0.127030 0.474084i −1.58040 0.708759i 1.52343 + 0.879554i 1.06741 1.96485i −0.536770 + 0.659208i 0 1.30461 1.30461i 1.99532 + 2.24024i −0.795910 0.755638i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.12
Significant digits:
Format:

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
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

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

Hecke kernels

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

\(T_{2}^{48} - \cdots\)
\(T_{11}^{24} - \cdots\)
\(T_{13}^{12} + \cdots\)
\(T_{17}^{48} - \cdots\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database