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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(128,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
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 990-1000
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.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 \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{48} - 76 T_{2}^{44} + 4074 T_{2}^{40} - 105632 T_{2}^{36} + 1979523 T_{2}^{32} - 17766888 T_{2}^{28} + \cdots + 256 \) 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