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Properties

Label 125.4.e.b
Level $125$
Weight $4$
Character orbit 125.e
Analytic conductor $7.375$
Analytic rank $0$
Dimension $56$
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] = [125,4,Mod(24,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 125.e (of order \(10\), 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: \(7.37523875072\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 62 q^{4} + 2 q^{6} + 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 62 q^{4} + 2 q^{6} + 68 q^{9} - 178 q^{11} + 34 q^{14} - 414 q^{16} + 230 q^{19} - 288 q^{21} - 1560 q^{24} + 1172 q^{26} + 10 q^{29} - 1278 q^{31} + 1554 q^{34} + 1346 q^{36} + 2266 q^{39} + 682 q^{41} - 1096 q^{44} - 2478 q^{46} - 2688 q^{49} + 4012 q^{51} - 3230 q^{54} - 5820 q^{56} + 3810 q^{59} + 2782 q^{61} + 7192 q^{64} + 7264 q^{66} - 5374 q^{69} - 7438 q^{71} - 9696 q^{74} + 7040 q^{76} - 1550 q^{79} - 7424 q^{81} + 15224 q^{84} + 7782 q^{86} + 10150 q^{89} + 752 q^{91} - 7146 q^{94} - 15508 q^{96} - 13144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −4.85470 − 1.57739i 4.56378 − 6.28151i 14.6078 + 10.6132i 0 −32.0642 + 23.2960i − 5.45530i −30.1725 − 41.5290i −10.2858 − 31.6563i 0
24.2 −4.45449 − 1.44735i 0.861326 − 1.18551i 11.2755 + 8.19213i 0 −5.55262 + 4.03421i − 13.4350i −16.3455 − 22.4976i 7.67990 + 23.6363i 0
24.3 −4.36711 − 1.41896i −5.13823 + 7.07217i 10.5860 + 7.69120i 0 32.4743 − 23.5940i − 20.8866i −13.7247 − 18.8904i −15.2707 − 46.9983i 0
24.4 −2.78910 − 0.906232i −3.99049 + 5.49244i 0.485661 + 0.352853i 0 16.1073 − 11.7026i 23.0864i 12.7553 + 17.5561i −5.89942 − 18.1565i 0
24.5 −1.92350 − 0.624983i 1.23409 − 1.69858i −3.16288 − 2.29797i 0 −3.43535 + 2.49593i 24.6755i 14.1579 + 19.4867i 6.98127 + 21.4861i 0
24.6 −1.82048 − 0.591509i 5.51128 − 7.58563i −3.50788 − 2.54862i 0 −14.5201 + 10.5495i − 14.5331i 13.8794 + 19.1034i −18.8241 − 57.9345i 0
24.7 −0.408651 − 0.132779i 0.0448763 − 0.0617670i −6.32277 − 4.59376i 0 −0.0265401 + 0.0192825i − 16.1597i 3.99433 + 5.49773i 8.34166 + 25.6730i 0
24.8 0.408651 + 0.132779i −0.0448763 + 0.0617670i −6.32277 − 4.59376i 0 −0.0265401 + 0.0192825i 16.1597i −3.99433 − 5.49773i 8.34166 + 25.6730i 0
24.9 1.82048 + 0.591509i −5.51128 + 7.58563i −3.50788 − 2.54862i 0 −14.5201 + 10.5495i 14.5331i −13.8794 − 19.1034i −18.8241 − 57.9345i 0
24.10 1.92350 + 0.624983i −1.23409 + 1.69858i −3.16288 − 2.29797i 0 −3.43535 + 2.49593i − 24.6755i −14.1579 − 19.4867i 6.98127 + 21.4861i 0
24.11 2.78910 + 0.906232i 3.99049 − 5.49244i 0.485661 + 0.352853i 0 16.1073 − 11.7026i − 23.0864i −12.7553 − 17.5561i −5.89942 − 18.1565i 0
24.12 4.36711 + 1.41896i 5.13823 − 7.07217i 10.5860 + 7.69120i 0 32.4743 − 23.5940i 20.8866i 13.7247 + 18.8904i −15.2707 − 46.9983i 0
24.13 4.45449 + 1.44735i −0.861326 + 1.18551i 11.2755 + 8.19213i 0 −5.55262 + 4.03421i 13.4350i 16.3455 + 22.4976i 7.67990 + 23.6363i 0
24.14 4.85470 + 1.57739i −4.56378 + 6.28151i 14.6078 + 10.6132i 0 −32.0642 + 23.2960i 5.45530i 30.1725 + 41.5290i −10.2858 − 31.6563i 0
49.1 −3.12208 + 4.29718i −5.96393 + 1.93780i −6.24620 − 19.2238i 0 10.2928 − 31.6780i − 9.63602i 61.6963 + 20.0464i 9.96993 − 7.24358i 0
49.2 −2.99151 + 4.11746i 3.64082 − 1.18297i −5.53220 − 17.0263i 0 −6.02069 + 18.5298i 12.1888i 47.9320 + 15.5740i −9.98733 + 7.25622i 0
49.3 −2.04622 + 2.81638i −8.11420 + 2.63646i −1.27284 − 3.91740i 0 9.17815 − 28.2474i − 12.5082i −12.8494 − 4.17503i 37.0458 − 26.9153i 0
49.4 −1.65029 + 2.27143i 1.26614 − 0.411392i 0.0362106 + 0.111445i 0 −1.15504 + 3.55485i − 35.1773i −21.6747 − 7.04252i −20.4096 + 14.8284i 0
49.5 −1.39571 + 1.92104i 6.11509 − 1.98691i 0.729774 + 2.24601i 0 −4.71799 + 14.5205i 25.3460i −23.3997 − 7.60304i 11.6030 − 8.43010i 0
49.6 −1.10196 + 1.51671i 1.52739 − 0.496278i 1.38603 + 4.26575i 0 −0.930402 + 2.86348i 5.91678i −22.2613 − 7.23313i −19.7568 + 14.3542i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.4.e.b 56
5.b even 2 1 inner 125.4.e.b 56
5.c odd 4 1 25.4.d.a ✓ 28
5.c odd 4 1 125.4.d.a 28
15.e even 4 1 225.4.h.b 28
25.d even 5 1 inner 125.4.e.b 56
25.e even 10 1 inner 125.4.e.b 56
25.f odd 20 1 25.4.d.a ✓ 28
25.f odd 20 1 125.4.d.a 28
25.f odd 20 1 625.4.a.c 14
25.f odd 20 1 625.4.a.d 14
75.l even 20 1 225.4.h.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.d.a ✓ 28 5.c odd 4 1
25.4.d.a ✓ 28 25.f odd 20 1
125.4.d.a 28 5.c odd 4 1
125.4.d.a 28 25.f odd 20 1
125.4.e.b 56 1.a even 1 1 trivial
125.4.e.b 56 5.b even 2 1 inner
125.4.e.b 56 25.d even 5 1 inner
125.4.e.b 56 25.e even 10 1 inner
225.4.h.b 28 15.e even 4 1
225.4.h.b 28 75.l even 20 1
625.4.a.c 14 25.f odd 20 1
625.4.a.d 14 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 87 T_{2}^{54} + 4648 T_{2}^{52} - 199799 T_{2}^{50} + 7517125 T_{2}^{48} + \cdots + 19\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display