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Properties

Label 125.4.d.b
Level $125$
Weight $4$
Character orbit 125.d
Analytic conductor $7.375$
Analytic rank $0$
Dimension $48$
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(26,125)]
 
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([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.26");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 125.d (of order \(5\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 26 q^{4} - 14 q^{6} - 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 26 q^{4} - 14 q^{6} - 94 q^{9} + 166 q^{11} - 62 q^{14} + 178 q^{16} + 230 q^{19} + 276 q^{21} - 1280 q^{24} - 1644 q^{26} - 250 q^{29} + 1266 q^{31} + 1558 q^{34} - 1062 q^{36} + 2482 q^{39} - 234 q^{41} - 1032 q^{44} + 2466 q^{46} - 2296 q^{49} - 4044 q^{51} - 3490 q^{54} + 6320 q^{56} + 3810 q^{59} - 1134 q^{61} + 7224 q^{64} - 5548 q^{66} - 1078 q^{69} + 4946 q^{71} - 7192 q^{74} - 6560 q^{76} - 1550 q^{79} + 6618 q^{81} + 11388 q^{84} - 7794 q^{86} + 4820 q^{89} - 764 q^{91} - 10802 q^{94} + 12976 q^{96} - 12048 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.60985 − 4.95462i −2.34434 + 1.70326i −15.4845 + 11.2501i 0 12.2131 + 8.87330i 24.5811 46.9507 + 34.1117i −5.74864 + 17.6925i 0
26.2 −1.13030 − 3.47870i −1.68750 + 1.22604i −4.35165 + 3.16166i 0 6.17240 + 4.48451i −8.69522 −7.75614 − 5.63517i −6.99898 + 21.5406i 0
26.3 −1.08415 − 3.33666i 5.23712 − 3.80499i −3.48577 + 2.53256i 0 −18.3737 − 13.3493i 25.7483 −10.4773 − 7.61219i 4.60602 − 14.1759i 0
26.4 −0.816614 − 2.51328i 2.41747 − 1.75639i 0.822422 − 0.597525i 0 −6.38844 − 4.64147i −26.3705 −19.2767 − 14.0054i −5.58423 + 17.1865i 0
26.5 −0.483614 − 1.48841i −5.73067 + 4.16357i 4.49065 − 3.26265i 0 8.96855 + 6.51603i 1.18261 −17.1569 − 12.4652i 7.16175 − 22.0416i 0
26.6 −0.107853 − 0.331937i −5.17842 + 3.76234i 6.37359 − 4.63068i 0 1.80737 + 1.31313i −14.4107 −4.48340 − 3.25738i 4.31736 − 13.2875i 0
26.7 0.107853 + 0.331937i 5.17842 − 3.76234i 6.37359 − 4.63068i 0 1.80737 + 1.31313i 14.4107 4.48340 + 3.25738i 4.31736 − 13.2875i 0
26.8 0.483614 + 1.48841i 5.73067 − 4.16357i 4.49065 − 3.26265i 0 8.96855 + 6.51603i −1.18261 17.1569 + 12.4652i 7.16175 − 22.0416i 0
26.9 0.816614 + 2.51328i −2.41747 + 1.75639i 0.822422 − 0.597525i 0 −6.38844 − 4.64147i 26.3705 19.2767 + 14.0054i −5.58423 + 17.1865i 0
26.10 1.08415 + 3.33666i −5.23712 + 3.80499i −3.48577 + 2.53256i 0 −18.3737 − 13.3493i −25.7483 10.4773 + 7.61219i 4.60602 − 14.1759i 0
26.11 1.13030 + 3.47870i 1.68750 − 1.22604i −4.35165 + 3.16166i 0 6.17240 + 4.48451i 8.69522 7.75614 + 5.63517i −6.99898 + 21.5406i 0
26.12 1.60985 + 4.95462i 2.34434 − 1.70326i −15.4845 + 11.2501i 0 12.2131 + 8.87330i −24.5811 −46.9507 − 34.1117i −5.74864 + 17.6925i 0
51.1 −3.87487 + 2.81526i −1.28851 + 3.96562i 4.61680 − 14.2091i 0 −6.17144 − 18.9937i 14.5499 10.2721 + 31.6143i 7.77757 + 5.65074i 0
51.2 −3.33341 + 2.42187i 1.15776 − 3.56321i 2.77407 − 8.53772i 0 4.77034 + 14.6816i −26.4674 1.24408 + 3.82887i 10.4874 + 7.61952i 0
51.3 −3.07931 + 2.23725i −2.89162 + 8.89950i 2.00472 − 6.16988i 0 −11.0062 − 33.8735i 4.54748 −1.77911 − 5.47554i −48.9961 − 35.5978i 0
51.4 −2.49489 + 1.81265i 1.02936 − 3.16804i 0.466674 − 1.43628i 0 3.17440 + 9.76980i −5.10302 −6.18456 − 19.0341i 12.8665 + 9.34809i 0
51.5 −0.608718 + 0.442260i 2.19509 − 6.75579i −2.29719 + 7.07003i 0 1.65162 + 5.08317i 18.3105 −3.58852 − 11.0443i −18.9788 − 13.7889i 0
51.6 −0.299383 + 0.217515i −0.859131 + 2.64413i −2.42982 + 7.47821i 0 −0.317928 − 0.978483i −0.707538 −1.81401 − 5.58294i 15.5901 + 11.3269i 0
51.7 0.299383 − 0.217515i 0.859131 − 2.64413i −2.42982 + 7.47821i 0 −0.317928 − 0.978483i 0.707538 1.81401 + 5.58294i 15.5901 + 11.3269i 0
51.8 0.608718 − 0.442260i −2.19509 + 6.75579i −2.29719 + 7.07003i 0 1.65162 + 5.08317i −18.3105 3.58852 + 11.0443i −18.9788 − 13.7889i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.12
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.d.b 48
5.b even 2 1 inner 125.4.d.b 48
5.c odd 4 1 25.4.e.a ✓ 24
5.c odd 4 1 125.4.e.a 24
15.e even 4 1 225.4.m.a 24
25.d even 5 1 inner 125.4.d.b 48
25.d even 5 1 625.4.a.g 24
25.e even 10 1 inner 125.4.d.b 48
25.e even 10 1 625.4.a.g 24
25.f odd 20 1 25.4.e.a ✓ 24
25.f odd 20 1 125.4.e.a 24
75.l even 20 1 225.4.m.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.e.a ✓ 24 5.c odd 4 1
25.4.e.a ✓ 24 25.f odd 20 1
125.4.d.b 48 1.a even 1 1 trivial
125.4.d.b 48 5.b even 2 1 inner
125.4.d.b 48 25.d even 5 1 inner
125.4.d.b 48 25.e even 10 1 inner
125.4.e.a 24 5.c odd 4 1
125.4.e.a 24 25.f odd 20 1
225.4.m.a 24 15.e even 4 1
225.4.m.a 24 75.l even 20 1
625.4.a.g 24 25.d even 5 1
625.4.a.g 24 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 61 T_{2}^{46} + 2416 T_{2}^{44} + 77901 T_{2}^{42} + 2321001 T_{2}^{40} + \cdots + 15\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display