Properties

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

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

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