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Properties

Label 735.2.y.j
Level $735$
Weight $2$
Character orbit 735.y
Analytic conductor $5.869$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,2,Mod(128,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.128"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 4])) N = Newforms(chi, 2, names="a")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [48,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} + 16 q^{10} - 16 q^{12} - 16 q^{13} - 32 q^{15} + 16 q^{16} + 20 q^{18} + 16 q^{22} + 16 q^{25} - 32 q^{27} - 20 q^{30} - 28 q^{33} + 32 q^{36} + 16 q^{37} - 64 q^{40} - 80 q^{43} - 20 q^{45}+ \cdots + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 80-90
Embeddings: e.g. 1-3 or 128.12
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 \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{48} - 76 T_{2}^{44} + 4074 T_{2}^{40} - 105632 T_{2}^{36} + 1979523 T_{2}^{32} - 17766888 T_{2}^{28} + \cdots + 256 \) Copy content Toggle raw display
\( T_{11}^{24} - 62 T_{11}^{22} + 2451 T_{11}^{20} - 57038 T_{11}^{18} + 955849 T_{11}^{16} + \cdots + 22663495936 \) Copy content Toggle raw display
\( T_{13}^{12} + 4 T_{13}^{11} + 8 T_{13}^{10} - 124 T_{13}^{9} + 625 T_{13}^{8} + 160 T_{13}^{7} + \cdots + 33856 \) Copy content Toggle raw display
\( T_{17}^{48} - 1218 T_{17}^{44} + 1078419 T_{17}^{40} - 440592322 T_{17}^{36} + 131887163297 T_{17}^{32} + \cdots + 65536 \) Copy content Toggle raw display