Properties

Label 735.2.g.b
Level $735$
Weight $2$
Character orbit 735.g
Analytic conductor $5.869$
Analytic rank $0$
Dimension $24$
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(734,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.734");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.g (of order \(2\), degree \(1\), 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: \(24\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 12 q^{9} - 24 q^{15} + 24 q^{16} + 24 q^{25} - 36 q^{30} + 84 q^{36} + 24 q^{39} - 72 q^{46} + 24 q^{51} - 24 q^{60} + 24 q^{64} - 96 q^{79} + 12 q^{81} + 48 q^{85} - 48 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
734.1 −2.50798 −1.47954 0.900534i 4.28995 −1.94085 1.11045i 3.71065 + 2.25852i 0 −5.74313 1.37808 + 2.66475i 4.86760 + 2.78499i
734.2 −2.50798 −1.47954 + 0.900534i 4.28995 −1.94085 + 1.11045i 3.71065 2.25852i 0 −5.74313 1.37808 2.66475i 4.86760 2.78499i
734.3 −2.50798 1.47954 0.900534i 4.28995 1.94085 1.11045i −3.71065 + 2.25852i 0 −5.74313 1.37808 2.66475i −4.86760 + 2.78499i
734.4 −2.50798 1.47954 + 0.900534i 4.28995 1.94085 + 1.11045i −3.71065 2.25852i 0 −5.74313 1.37808 + 2.66475i −4.86760 2.78499i
734.5 −1.51469 −1.66512 0.476833i 0.294280 0.775809 2.09717i 2.52214 + 0.722254i 0 2.58363 2.54526 + 1.58797i −1.17511 + 3.17656i
734.6 −1.51469 −1.66512 + 0.476833i 0.294280 0.775809 + 2.09717i 2.52214 0.722254i 0 2.58363 2.54526 1.58797i −1.17511 3.17656i
734.7 −1.51469 1.66512 0.476833i 0.294280 −0.775809 2.09717i −2.52214 + 0.722254i 0 2.58363 2.54526 1.58797i 1.17511 + 3.17656i
734.8 −1.51469 1.66512 + 0.476833i 0.294280 −0.775809 + 2.09717i −2.52214 0.722254i 0 2.58363 2.54526 + 1.58797i 1.17511 3.17656i
734.9 −0.644806 −0.536966 1.64671i −1.58423 2.15203 0.607270i 0.346239 + 1.06181i 0 2.31113 −2.42334 + 1.76846i −1.38764 + 0.391571i
734.10 −0.644806 −0.536966 + 1.64671i −1.58423 2.15203 + 0.607270i 0.346239 1.06181i 0 2.31113 −2.42334 1.76846i −1.38764 0.391571i
734.11 −0.644806 0.536966 1.64671i −1.58423 −2.15203 0.607270i −0.346239 + 1.06181i 0 2.31113 −2.42334 1.76846i 1.38764 + 0.391571i
734.12 −0.644806 0.536966 + 1.64671i −1.58423 −2.15203 + 0.607270i −0.346239 1.06181i 0 2.31113 −2.42334 + 1.76846i 1.38764 0.391571i
734.13 0.644806 −0.536966 1.64671i −1.58423 −2.15203 0.607270i −0.346239 1.06181i 0 −2.31113 −2.42334 + 1.76846i −1.38764 0.391571i
734.14 0.644806 −0.536966 + 1.64671i −1.58423 −2.15203 + 0.607270i −0.346239 + 1.06181i 0 −2.31113 −2.42334 1.76846i −1.38764 + 0.391571i
734.15 0.644806 0.536966 1.64671i −1.58423 2.15203 0.607270i 0.346239 1.06181i 0 −2.31113 −2.42334 1.76846i 1.38764 0.391571i
734.16 0.644806 0.536966 + 1.64671i −1.58423 2.15203 + 0.607270i 0.346239 + 1.06181i 0 −2.31113 −2.42334 + 1.76846i 1.38764 + 0.391571i
734.17 1.51469 −1.66512 0.476833i 0.294280 −0.775809 2.09717i −2.52214 0.722254i 0 −2.58363 2.54526 + 1.58797i −1.17511 3.17656i
734.18 1.51469 −1.66512 + 0.476833i 0.294280 −0.775809 + 2.09717i −2.52214 + 0.722254i 0 −2.58363 2.54526 1.58797i −1.17511 + 3.17656i
734.19 1.51469 1.66512 0.476833i 0.294280 0.775809 2.09717i 2.52214 0.722254i 0 −2.58363 2.54526 1.58797i 1.17511 3.17656i
734.20 1.51469 1.66512 + 0.476833i 0.294280 0.775809 + 2.09717i 2.52214 + 0.722254i 0 −2.58363 2.54526 + 1.58797i 1.17511 + 3.17656i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 734.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9T_{2}^{4} + 18T_{2}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display