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Properties

Label 1400.2.x.b
Level $1400$
Weight $2$
Character orbit 1400.x
Analytic conductor $11.179$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1400,2,Mod(657,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.657");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.x (of order \(4\), degree \(2\), 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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7} + 8 q^{11} + 16 q^{21} + 32 q^{23} + 8 q^{37} - 16 q^{43} - 24 q^{51} + 16 q^{53} - 20 q^{63} + 32 q^{67} - 32 q^{71} + 40 q^{77} - 72 q^{81} - 64 q^{91} - 72 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
657.1 0 −2.16993 − 2.16993i 0 0 0 −2.07939 + 1.63589i 0 6.41716i 0
657.2 0 −2.16341 − 2.16341i 0 0 0 2.61380 + 0.409966i 0 6.36066i 0
657.3 0 −1.41195 − 1.41195i 0 0 0 −2.64529 − 0.0496428i 0 0.987218i 0
657.4 0 −1.03893 − 1.03893i 0 0 0 −1.58883 − 2.11557i 0 − 0.841261i 0
657.5 0 −0.730185 − 0.730185i 0 0 0 1.08445 − 2.41329i 0 − 1.93366i 0
657.6 0 −0.0703127 − 0.0703127i 0 0 0 0.562657 + 2.58523i 0 − 2.99011i 0
657.7 0 0.0703127 + 0.0703127i 0 0 0 2.58523 + 0.562657i 0 − 2.99011i 0
657.8 0 0.730185 + 0.730185i 0 0 0 −2.41329 + 1.08445i 0 − 1.93366i 0
657.9 0 1.03893 + 1.03893i 0 0 0 −2.11557 − 1.58883i 0 − 0.841261i 0
657.10 0 1.41195 + 1.41195i 0 0 0 −0.0496428 − 2.64529i 0 0.987218i 0
657.11 0 2.16341 + 2.16341i 0 0 0 0.409966 + 2.61380i 0 6.36066i 0
657.12 0 2.16993 + 2.16993i 0 0 0 1.63589 − 2.07939i 0 6.41716i 0
993.1 0 −2.16993 + 2.16993i 0 0 0 −2.07939 − 1.63589i 0 − 6.41716i 0
993.2 0 −2.16341 + 2.16341i 0 0 0 2.61380 − 0.409966i 0 − 6.36066i 0
993.3 0 −1.41195 + 1.41195i 0 0 0 −2.64529 + 0.0496428i 0 − 0.987218i 0
993.4 0 −1.03893 + 1.03893i 0 0 0 −1.58883 + 2.11557i 0 0.841261i 0
993.5 0 −0.730185 + 0.730185i 0 0 0 1.08445 + 2.41329i 0 1.93366i 0
993.6 0 −0.0703127 + 0.0703127i 0 0 0 0.562657 − 2.58523i 0 2.99011i 0
993.7 0 0.0703127 − 0.0703127i 0 0 0 2.58523 − 0.562657i 0 2.99011i 0
993.8 0 0.730185 − 0.730185i 0 0 0 −2.41329 − 1.08445i 0 1.93366i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 657.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.x.b 24
5.b even 2 1 280.2.x.a ✓ 24
5.c odd 4 1 280.2.x.a ✓ 24
5.c odd 4 1 inner 1400.2.x.b 24
7.b odd 2 1 inner 1400.2.x.b 24
20.d odd 2 1 560.2.bj.d 24
20.e even 4 1 560.2.bj.d 24
35.c odd 2 1 280.2.x.a ✓ 24
35.f even 4 1 280.2.x.a ✓ 24
35.f even 4 1 inner 1400.2.x.b 24
140.c even 2 1 560.2.bj.d 24
140.j odd 4 1 560.2.bj.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.x.a ✓ 24 5.b even 2 1
280.2.x.a ✓ 24 5.c odd 4 1
280.2.x.a ✓ 24 35.c odd 2 1
280.2.x.a ✓ 24 35.f even 4 1
560.2.bj.d 24 20.d odd 2 1
560.2.bj.d 24 20.e even 4 1
560.2.bj.d 24 140.c even 2 1
560.2.bj.d 24 140.j odd 4 1
1400.2.x.b 24 1.a even 1 1 trivial
1400.2.x.b 24 5.c odd 4 1 inner
1400.2.x.b 24 7.b odd 2 1 inner
1400.2.x.b 24 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 198T_{3}^{20} + 11693T_{3}^{16} + 185852T_{3}^{12} + 772212T_{3}^{8} + 654688T_{3}^{4} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\). Copy content Toggle raw display