Properties

Label 115.4.e.a
Level $115$
Weight $4$
Character orbit 115.e
Analytic conductor $6.785$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,4,Mod(22,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} + 4 q^{3} + 32 q^{6} + 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} + 4 q^{3} + 32 q^{6} + 28 q^{8} + 200 q^{12} - 100 q^{13} - 832 q^{16} - 112 q^{18} - 46 q^{23} - 372 q^{25} - 312 q^{26} - 704 q^{27} - 1056 q^{31} + 560 q^{32} - 348 q^{35} + 2424 q^{36} + 2248 q^{41} - 96 q^{46} - 1412 q^{47} + 1532 q^{48} - 620 q^{50} + 112 q^{52} + 2688 q^{55} + 2044 q^{58} + 3948 q^{62} - 1836 q^{70} - 2272 q^{71} + 1128 q^{72} + 1100 q^{73} + 2828 q^{75} + 4216 q^{77} - 9180 q^{78} - 3580 q^{81} - 6516 q^{82} - 2684 q^{85} - 6304 q^{87} - 688 q^{92} - 1608 q^{93} - 8616 q^{95} - 544 q^{96} - 3612 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −3.71645 3.71645i 3.49413 3.49413i 19.6239i −10.7999 + 2.89187i −25.9715 2.69551 2.69551i 43.1997 43.1997i 2.58205i 50.8846 + 29.3896i
22.2 −3.71645 3.71645i 3.49413 3.49413i 19.6239i 10.7999 2.89187i −25.9715 −2.69551 + 2.69551i 43.1997 43.1997i 2.58205i −50.8846 29.3896i
22.3 −3.67169 3.67169i −4.81114 + 4.81114i 18.9626i −0.398194 11.1732i 35.3300 20.1910 20.1910i 40.2512 40.2512i 19.2941i −39.5626 + 42.4867i
22.4 −3.67169 3.67169i −4.81114 + 4.81114i 18.9626i 0.398194 + 11.1732i 35.3300 −20.1910 + 20.1910i 40.2512 40.2512i 19.2941i 39.5626 42.4867i
22.5 −2.78118 2.78118i 1.69288 1.69288i 7.46996i −9.46680 5.94809i −9.41642 1.06728 1.06728i −1.47414 + 1.47414i 21.2683i 9.78619 + 42.8716i
22.6 −2.78118 2.78118i 1.69288 1.69288i 7.46996i 9.46680 + 5.94809i −9.41642 −1.06728 + 1.06728i −1.47414 + 1.47414i 21.2683i −9.78619 42.8716i
22.7 −2.66509 2.66509i −2.23283 + 2.23283i 6.20541i −7.41137 + 8.37088i 11.9014 10.9048 10.9048i −4.78275 + 4.78275i 17.0290i 42.0611 2.55718i
22.8 −2.66509 2.66509i −2.23283 + 2.23283i 6.20541i 7.41137 8.37088i 11.9014 −10.9048 + 10.9048i −4.78275 + 4.78275i 17.0290i −42.0611 + 2.55718i
22.9 −2.37561 2.37561i 6.86048 6.86048i 3.28704i −2.55138 10.8853i −32.5957 −19.7347 + 19.7347i −11.1962 + 11.1962i 67.1325i −19.7982 + 31.9204i
22.10 −2.37561 2.37561i 6.86048 6.86048i 3.28704i 2.55138 + 10.8853i −32.5957 19.7347 19.7347i −11.1962 + 11.1962i 67.1325i 19.7982 31.9204i
22.11 −1.84990 1.84990i −5.85544 + 5.85544i 1.15573i −10.6129 3.51653i 21.6640 −7.25929 + 7.25929i −16.9372 + 16.9372i 41.5723i 13.1276 + 26.1381i
22.12 −1.84990 1.84990i −5.85544 + 5.85544i 1.15573i 10.6129 + 3.51653i 21.6640 7.25929 7.25929i −16.9372 + 16.9372i 41.5723i −13.1276 26.1381i
22.13 −1.32617 1.32617i 2.49169 2.49169i 4.48253i −3.09011 + 10.7448i −6.60883 −19.9254 + 19.9254i −16.5540 + 16.5540i 14.5829i 18.3475 10.1515i
22.14 −1.32617 1.32617i 2.49169 2.49169i 4.48253i 3.09011 10.7448i −6.60883 19.9254 19.9254i −16.5540 + 16.5540i 14.5829i −18.3475 + 10.1515i
22.15 −0.386420 0.386420i −0.970983 + 0.970983i 7.70136i −8.97655 6.66495i 0.750415 −6.23903 + 6.23903i −6.06732 + 6.06732i 25.1144i 0.893251 + 6.04419i
22.16 −0.386420 0.386420i −0.970983 + 0.970983i 7.70136i 8.97655 + 6.66495i 0.750415 6.23903 6.23903i −6.06732 + 6.06732i 25.1144i −0.893251 6.04419i
22.17 −0.152517 0.152517i −2.82764 + 2.82764i 7.95348i −5.56672 + 9.69596i 0.862527 20.1728 20.1728i −2.43318 + 2.43318i 11.0089i 2.32782 0.629778i
22.18 −0.152517 0.152517i −2.82764 + 2.82764i 7.95348i 5.56672 9.69596i 0.862527 −20.1728 + 20.1728i −2.43318 + 2.43318i 11.0089i −2.32782 + 0.629778i
22.19 0.304528 + 0.304528i 5.13977 5.13977i 7.81453i −11.1201 + 1.15901i 3.13040 2.31330 2.31330i 4.81596 4.81596i 25.8345i −3.73933 3.03343i
22.20 0.304528 + 0.304528i 5.13977 5.13977i 7.81453i 11.1201 1.15901i 3.13040 −2.31330 + 2.31330i 4.81596 4.81596i 25.8345i 3.73933 + 3.03343i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.34
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.4.e.a 68
5.c odd 4 1 inner 115.4.e.a 68
23.b odd 2 1 inner 115.4.e.a 68
115.e even 4 1 inner 115.4.e.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.e.a 68 1.a even 1 1 trivial
115.4.e.a 68 5.c odd 4 1 inner
115.4.e.a 68 23.b odd 2 1 inner
115.4.e.a 68 115.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(115, [\chi])\).