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Properties

Label 125.4.e.a
Level $125$
Weight $4$
Character orbit 125.e
Analytic conductor $7.375$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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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: \(24\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{2} + 5 q^{3} + 13 q^{4} - 7 q^{6} + 110 q^{8} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 5 q^{2} + 5 q^{3} + 13 q^{4} - 7 q^{6} + 110 q^{8} + 47 q^{9} + 83 q^{11} + 165 q^{12} + 5 q^{13} + 31 q^{14} + 89 q^{16} + 165 q^{17} - 115 q^{19} + 138 q^{21} - 30 q^{22} - 75 q^{23} + 640 q^{24} - 822 q^{26} - 595 q^{27} - 675 q^{28} + 125 q^{29} + 633 q^{31} + 1285 q^{33} - 779 q^{34} - 531 q^{36} + 1510 q^{37} - 255 q^{38} - 1241 q^{39} - 117 q^{41} - 1745 q^{42} + 516 q^{44} + 1233 q^{46} + 95 q^{47} - 1410 q^{48} + 1148 q^{49} - 2022 q^{51} - 1740 q^{52} - 2580 q^{53} + 1745 q^{54} + 3160 q^{56} + 4230 q^{58} - 1905 q^{59} - 567 q^{61} + 6880 q^{62} + 1950 q^{63} - 3612 q^{64} - 2774 q^{66} - 4195 q^{67} + 539 q^{69} + 2473 q^{71} - 215 q^{72} + 845 q^{73} + 3596 q^{74} - 3280 q^{76} - 3870 q^{77} - 9295 q^{78} + 775 q^{79} + 3309 q^{81} + 4625 q^{83} - 5694 q^{84} - 3897 q^{86} + 8485 q^{87} + 1650 q^{88} - 2410 q^{89} - 382 q^{91} - 4090 q^{92} + 5401 q^{94} + 6488 q^{96} - 5185 q^{97} - 5180 q^{98} + 6024 q^{99}+O(q^{100}) \) 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 −3.47870 − 1.13030i −1.22604 + 1.68750i 4.35165 + 3.16166i 0 6.17240 − 4.48451i − 8.69522i 5.63517 + 7.75614i 6.99898 + 21.5406i 0
24.2 −3.33666 − 1.08415i 3.80499 − 5.23712i 3.48577 + 2.53256i 0 −18.3737 + 13.3493i 25.7483i 7.61219 + 10.4773i −4.60602 − 14.1759i 0
24.3 −0.331937 − 0.107853i −3.76234 + 5.17842i −6.37359 − 4.63068i 0 1.80737 − 1.31313i − 14.4107i 3.25738 + 4.48340i −4.31736 − 13.2875i 0
24.4 1.48841 + 0.483614i 4.16357 − 5.73067i −4.49065 − 3.26265i 0 8.96855 − 6.51603i − 1.18261i −12.4652 − 17.1569i −7.16175 − 22.0416i 0
24.5 2.51328 + 0.816614i −1.75639 + 2.41747i −0.822422 − 0.597525i 0 −6.38844 + 4.64147i 26.3705i −14.0054 − 19.2767i 5.58423 + 17.1865i 0
24.6 4.95462 + 1.60985i 1.70326 − 2.34434i 15.4845 + 11.2501i 0 12.2131 − 8.87330i − 24.5811i 34.1117 + 46.9507i 5.74864 + 17.6925i 0
49.1 −2.81526 + 3.87487i 3.96562 − 1.28851i −4.61680 − 14.2091i 0 −6.17144 + 18.9937i − 14.5499i 31.6143 + 10.2721i −7.77757 + 5.65074i 0
49.2 −1.81265 + 2.49489i −3.16804 + 1.02936i −0.466674 − 1.43628i 0 3.17440 − 9.76980i 5.10302i −19.0341 − 6.18456i −12.8665 + 9.34809i 0
49.3 0.217515 − 0.299383i −2.64413 + 0.859131i 2.42982 + 7.47821i 0 −0.317928 + 0.978483i − 0.707538i 5.58294 + 1.81401i −15.5901 + 11.3269i 0
49.4 0.442260 − 0.608718i 6.75579 − 2.19509i 2.29719 + 7.07003i 0 1.65162 − 5.08317i 18.3105i 11.0443 + 3.58852i 18.9788 − 13.7889i 0
49.5 2.23725 − 3.07931i −8.89950 + 2.89162i −2.00472 − 6.16988i 0 −11.0062 + 33.8735i 4.54748i 5.47554 + 1.77911i 48.9961 − 35.5978i 0
49.6 2.42187 − 3.33341i 3.56321 − 1.15776i −2.77407 − 8.53772i 0 4.77034 − 14.6816i − 26.4674i −3.82887 − 1.24408i −10.4874 + 7.61952i 0
74.1 −2.81526 − 3.87487i 3.96562 + 1.28851i −4.61680 + 14.2091i 0 −6.17144 − 18.9937i 14.5499i 31.6143 − 10.2721i −7.77757 − 5.65074i 0
74.2 −1.81265 − 2.49489i −3.16804 − 1.02936i −0.466674 + 1.43628i 0 3.17440 + 9.76980i − 5.10302i −19.0341 + 6.18456i −12.8665 − 9.34809i 0
74.3 0.217515 + 0.299383i −2.64413 − 0.859131i 2.42982 − 7.47821i 0 −0.317928 − 0.978483i 0.707538i 5.58294 − 1.81401i −15.5901 − 11.3269i 0
74.4 0.442260 + 0.608718i 6.75579 + 2.19509i 2.29719 − 7.07003i 0 1.65162 + 5.08317i − 18.3105i 11.0443 − 3.58852i 18.9788 + 13.7889i 0
74.5 2.23725 + 3.07931i −8.89950 − 2.89162i −2.00472 + 6.16988i 0 −11.0062 − 33.8735i − 4.54748i 5.47554 − 1.77911i 48.9961 + 35.5978i 0
74.6 2.42187 + 3.33341i 3.56321 + 1.15776i −2.77407 + 8.53772i 0 4.77034 + 14.6816i 26.4674i −3.82887 + 1.24408i −10.4874 − 7.61952i 0
99.1 −3.47870 + 1.13030i −1.22604 − 1.68750i 4.35165 − 3.16166i 0 6.17240 + 4.48451i 8.69522i 5.63517 − 7.75614i 6.99898 − 21.5406i 0
99.2 −3.33666 + 1.08415i 3.80499 + 5.23712i 3.48577 − 2.53256i 0 −18.3737 − 13.3493i − 25.7483i 7.61219 − 10.4773i −4.60602 + 14.1759i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a 24
5.b even 2 1 25.4.e.a ✓ 24
5.c odd 4 2 125.4.d.b 48
15.d odd 2 1 225.4.m.a 24
25.d even 5 1 25.4.e.a ✓ 24
25.e even 10 1 inner 125.4.e.a 24
25.f odd 20 2 125.4.d.b 48
25.f odd 20 2 625.4.a.g 24
75.j odd 10 1 225.4.m.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.e.a ✓ 24 5.b even 2 1
25.4.e.a ✓ 24 25.d even 5 1
125.4.d.b 48 5.c odd 4 2
125.4.d.b 48 25.f odd 20 2
125.4.e.a 24 1.a even 1 1 trivial
125.4.e.a 24 25.e even 10 1 inner
225.4.m.a 24 15.d odd 2 1
225.4.m.a 24 75.j odd 10 1
625.4.a.g 24 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 5 T_{2}^{23} - 18 T_{2}^{22} + 95 T_{2}^{21} + 571 T_{2}^{20} - 2445 T_{2}^{19} + \cdots + 38738176 \) acting on \(S_{4}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display