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Properties

Label 100.4.e.f
Level $100$
Weight $4$
Character orbit 100.e
Analytic conductor $5.900$
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] = [100,4,Mod(7,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.e (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: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 36 q^{6} - 676 q^{16} + 512 q^{21} + 2072 q^{26} - 4600 q^{36} - 392 q^{41} + 5016 q^{46} - 8224 q^{56} + 1088 q^{61} + 11140 q^{66} - 6700 q^{76} - 2424 q^{81} + 5216 q^{86} + 796 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.66990 − 0.933623i −3.69377 − 3.69377i 6.25670 + 4.98535i 0 6.41339 + 13.3106i 19.3489 − 19.3489i −12.0503 − 19.1518i 0.287847i 0
7.2 −2.38148 + 1.52596i −0.243692 − 0.243692i 3.34287 − 7.26809i 0 0.952213 + 0.208482i −9.53656 + 9.53656i 3.12986 + 22.4099i − 26.8812i 0
7.3 −2.21943 − 1.75332i 6.14790 + 6.14790i 1.85174 + 7.78274i 0 −2.86560 − 24.4241i 16.4522 − 16.4522i 9.53582 − 20.5199i 48.5934i 0
7.4 −1.75332 − 2.21943i −6.14790 − 6.14790i −1.85174 + 7.78274i 0 −2.86560 + 24.4241i −16.4522 + 16.4522i 20.5199 − 9.53582i 48.5934i 0
7.5 −1.52596 + 2.38148i −0.243692 − 0.243692i −3.34287 − 7.26809i 0 0.952213 − 0.208482i −9.53656 + 9.53656i 22.4099 + 3.12986i − 26.8812i 0
7.6 −0.933623 − 2.66990i 3.69377 + 3.69377i −6.25670 + 4.98535i 0 6.41339 − 13.3106i −19.3489 + 19.3489i 19.1518 + 12.0503i 0.287847i 0
7.7 0.933623 + 2.66990i −3.69377 − 3.69377i −6.25670 + 4.98535i 0 6.41339 − 13.3106i 19.3489 − 19.3489i −19.1518 − 12.0503i 0.287847i 0
7.8 1.52596 − 2.38148i 0.243692 + 0.243692i −3.34287 − 7.26809i 0 0.952213 − 0.208482i 9.53656 − 9.53656i −22.4099 − 3.12986i − 26.8812i 0
7.9 1.75332 + 2.21943i 6.14790 + 6.14790i −1.85174 + 7.78274i 0 −2.86560 + 24.4241i 16.4522 − 16.4522i −20.5199 + 9.53582i 48.5934i 0
7.10 2.21943 + 1.75332i −6.14790 − 6.14790i 1.85174 + 7.78274i 0 −2.86560 − 24.4241i −16.4522 + 16.4522i −9.53582 + 20.5199i 48.5934i 0
7.11 2.38148 − 1.52596i 0.243692 + 0.243692i 3.34287 − 7.26809i 0 0.952213 + 0.208482i 9.53656 − 9.53656i −3.12986 − 22.4099i − 26.8812i 0
7.12 2.66990 + 0.933623i 3.69377 + 3.69377i 6.25670 + 4.98535i 0 6.41339 + 13.3106i −19.3489 + 19.3489i 12.0503 + 19.1518i 0.287847i 0
43.1 −2.66990 + 0.933623i −3.69377 + 3.69377i 6.25670 − 4.98535i 0 6.41339 − 13.3106i 19.3489 + 19.3489i −12.0503 + 19.1518i − 0.287847i 0
43.2 −2.38148 − 1.52596i −0.243692 + 0.243692i 3.34287 + 7.26809i 0 0.952213 − 0.208482i −9.53656 − 9.53656i 3.12986 − 22.4099i 26.8812i 0
43.3 −2.21943 + 1.75332i 6.14790 − 6.14790i 1.85174 − 7.78274i 0 −2.86560 + 24.4241i 16.4522 + 16.4522i 9.53582 + 20.5199i − 48.5934i 0
43.4 −1.75332 + 2.21943i −6.14790 + 6.14790i −1.85174 − 7.78274i 0 −2.86560 − 24.4241i −16.4522 − 16.4522i 20.5199 + 9.53582i − 48.5934i 0
43.5 −1.52596 − 2.38148i −0.243692 + 0.243692i −3.34287 + 7.26809i 0 0.952213 + 0.208482i −9.53656 − 9.53656i 22.4099 − 3.12986i 26.8812i 0
43.6 −0.933623 + 2.66990i 3.69377 − 3.69377i −6.25670 − 4.98535i 0 6.41339 + 13.3106i −19.3489 − 19.3489i 19.1518 − 12.0503i − 0.287847i 0
43.7 0.933623 − 2.66990i −3.69377 + 3.69377i −6.25670 − 4.98535i 0 6.41339 + 13.3106i 19.3489 + 19.3489i −19.1518 + 12.0503i − 0.287847i 0
43.8 1.52596 + 2.38148i 0.243692 − 0.243692i −3.34287 + 7.26809i 0 0.952213 + 0.208482i 9.53656 + 9.53656i −22.4099 + 3.12986i 26.8812i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.f ✓ 24
4.b odd 2 1 inner 100.4.e.f ✓ 24
5.b even 2 1 inner 100.4.e.f ✓ 24
5.c odd 4 2 inner 100.4.e.f ✓ 24
20.d odd 2 1 inner 100.4.e.f ✓ 24
20.e even 4 2 inner 100.4.e.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.e.f ✓ 24 1.a even 1 1 trivial
100.4.e.f ✓ 24 4.b odd 2 1 inner
100.4.e.f ✓ 24 5.b even 2 1 inner
100.4.e.f ✓ 24 5.c odd 4 2 inner
100.4.e.f ✓ 24 20.d odd 2 1 inner
100.4.e.f ✓ 24 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 6459T_{3}^{8} + 4255155T_{3}^{4} + 60025 \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display