# Properties

 Label 25.4.e.a Level $25$ Weight $4$ Character orbit 25.e Analytic conductor $1.475$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,4,Mod(4,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, 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("25.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.e (of order $$10$$, degree $$4$$, 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: $$1.47504775014$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 5 q^{2} - 5 q^{3} + 13 q^{4} + 15 q^{5} - 7 q^{6} - 110 q^{8} + 47 q^{9}+O(q^{10})$$ 24 * q - 5 * q^2 - 5 * q^3 + 13 * q^4 + 15 * q^5 - 7 * q^6 - 110 * q^8 + 47 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 5 q^{2} - 5 q^{3} + 13 q^{4} + 15 q^{5} - 7 q^{6} - 110 q^{8} + 47 q^{9} - 5 q^{10} + 83 q^{11} - 165 q^{12} - 5 q^{13} + 31 q^{14} - 225 q^{15} + 89 q^{16} - 165 q^{17} - 115 q^{19} + 395 q^{20} + 138 q^{21} + 30 q^{22} + 75 q^{23} + 640 q^{24} + 595 q^{25} - 822 q^{26} + 595 q^{27} + 675 q^{28} + 125 q^{29} - 965 q^{30} + 633 q^{31} - 1285 q^{33} - 779 q^{34} + 190 q^{35} - 531 q^{36} - 1510 q^{37} + 255 q^{38} - 1241 q^{39} - 2470 q^{40} - 117 q^{41} + 1745 q^{42} + 516 q^{44} + 1725 q^{45} + 1233 q^{46} - 95 q^{47} + 1410 q^{48} + 1148 q^{49} + 2165 q^{50} - 2022 q^{51} + 1740 q^{52} + 2580 q^{53} + 1745 q^{54} - 315 q^{55} + 3160 q^{56} - 4230 q^{58} - 1905 q^{59} + 560 q^{60} - 567 q^{61} - 6880 q^{62} - 1950 q^{63} - 3612 q^{64} - 3170 q^{65} - 2774 q^{66} + 4195 q^{67} + 539 q^{69} + 3630 q^{70} + 2473 q^{71} + 215 q^{72} - 845 q^{73} + 3596 q^{74} + 2045 q^{75} - 3280 q^{76} + 3870 q^{77} + 9295 q^{78} + 775 q^{79} - 3670 q^{80} + 3309 q^{81} - 4625 q^{83} - 5694 q^{84} - 1460 q^{85} - 3897 q^{86} - 8485 q^{87} - 1650 q^{88} - 2410 q^{89} - 8845 q^{90} - 382 q^{91} + 4090 q^{92} + 5401 q^{94} + 3545 q^{95} + 6488 q^{96} + 5185 q^{97} + 5180 q^{98} + 6024 q^{99}+O(q^{100})$$ 24 * q - 5 * q^2 - 5 * q^3 + 13 * q^4 + 15 * q^5 - 7 * q^6 - 110 * q^8 + 47 * q^9 - 5 * q^10 + 83 * q^11 - 165 * q^12 - 5 * q^13 + 31 * q^14 - 225 * q^15 + 89 * q^16 - 165 * q^17 - 115 * q^19 + 395 * q^20 + 138 * q^21 + 30 * q^22 + 75 * q^23 + 640 * q^24 + 595 * q^25 - 822 * q^26 + 595 * q^27 + 675 * q^28 + 125 * q^29 - 965 * q^30 + 633 * q^31 - 1285 * q^33 - 779 * q^34 + 190 * q^35 - 531 * q^36 - 1510 * q^37 + 255 * q^38 - 1241 * q^39 - 2470 * q^40 - 117 * q^41 + 1745 * q^42 + 516 * q^44 + 1725 * q^45 + 1233 * q^46 - 95 * q^47 + 1410 * q^48 + 1148 * q^49 + 2165 * q^50 - 2022 * q^51 + 1740 * q^52 + 2580 * q^53 + 1745 * q^54 - 315 * q^55 + 3160 * q^56 - 4230 * q^58 - 1905 * q^59 + 560 * q^60 - 567 * q^61 - 6880 * q^62 - 1950 * q^63 - 3612 * q^64 - 3170 * q^65 - 2774 * q^66 + 4195 * q^67 + 539 * q^69 + 3630 * q^70 + 2473 * q^71 + 215 * q^72 - 845 * q^73 + 3596 * q^74 + 2045 * q^75 - 3280 * q^76 + 3870 * q^77 + 9295 * q^78 + 775 * q^79 - 3670 * q^80 + 3309 * q^81 - 4625 * q^83 - 5694 * q^84 - 1460 * q^85 - 3897 * q^86 - 8485 * q^87 - 1650 * q^88 - 2410 * q^89 - 8845 * q^90 - 382 * q^91 + 4090 * q^92 + 5401 * q^94 + 3545 * q^95 + 6488 * q^96 + 5185 * q^97 + 5180 * q^98 + 6024 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −4.95462 1.60985i −1.70326 + 2.34434i 15.4845 + 11.2501i 11.0057 1.96864i 12.2131 8.87330i 24.5811i −34.1117 46.9507i 5.74864 + 17.6925i −57.6981 7.96365i
4.2 −2.51328 0.816614i 1.75639 2.41747i −0.822422 0.597525i −5.31827 9.83443i −6.38844 + 4.64147i 26.3705i 14.0054 + 19.2767i 5.58423 + 17.1865i 5.33537 + 29.0596i
4.3 −1.48841 0.483614i −4.16357 + 5.73067i −4.49065 3.26265i −5.33345 + 9.82621i 8.96855 6.51603i 1.18261i 12.4652 + 17.1569i −7.16175 22.0416i 12.6905 12.0461i
4.4 0.331937 + 0.107853i 3.76234 5.17842i −6.37359 4.63068i 10.0178 + 4.96433i 1.80737 1.31313i 14.4107i −3.25738 4.48340i −4.31736 13.2875i 2.78985 + 2.72829i
4.5 3.33666 + 1.08415i −3.80499 + 5.23712i 3.48577 + 2.53256i 10.7075 3.21691i −18.3737 + 13.3493i 25.7483i −7.61219 10.4773i −4.60602 14.1759i 39.2150 + 0.874818i
4.6 3.47870 + 1.13030i 1.22604 1.68750i 4.35165 + 3.16166i −11.1800 0.0808329i 6.17240 4.48451i 8.69522i −5.63517 7.75614i 6.99898 + 21.5406i −38.8007 12.9180i
9.1 −2.42187 + 3.33341i −3.56321 + 1.15776i −2.77407 8.53772i −8.01402 7.79587i 4.77034 14.6816i 26.4674i 3.82887 + 1.24408i −10.4874 + 7.61952i 45.3957 7.83348i
9.2 −2.23725 + 3.07931i 8.89950 2.89162i −2.00472 6.16988i −3.48187 + 10.6243i −11.0062 + 33.8735i 4.54748i −5.47554 1.77911i 48.9961 35.5978i −24.9258 34.4910i
9.3 −0.442260 + 0.608718i −6.75579 + 2.19509i 2.29719 + 7.07003i 1.00498 + 11.1351i 1.65162 5.08317i 18.3105i −11.0443 3.58852i 18.9788 13.7889i −7.22259 4.31284i
9.4 −0.217515 + 0.299383i 2.64413 0.859131i 2.42982 + 7.47821i 7.78705 8.02258i −0.317928 + 0.978483i 0.707538i −5.58294 1.81401i −15.5901 + 11.3269i 0.708030 + 4.07634i
9.5 1.81265 2.49489i 3.16804 1.02936i −0.466674 1.43628i −10.8670 + 2.62823i 3.17440 9.76980i 5.10302i 19.0341 + 6.18456i −12.8665 + 9.34809i −13.1409 + 31.8762i
9.6 2.81526 3.87487i −3.96562 + 1.28851i −4.61680 14.2091i 11.1717 + 0.439337i −6.17144 + 18.9937i 14.5499i −31.6143 10.2721i −7.77757 + 5.65074i 33.1536 42.0521i
14.1 −2.42187 3.33341i −3.56321 1.15776i −2.77407 + 8.53772i −8.01402 + 7.79587i 4.77034 + 14.6816i 26.4674i 3.82887 1.24408i −10.4874 7.61952i 45.3957 + 7.83348i
14.2 −2.23725 3.07931i 8.89950 + 2.89162i −2.00472 + 6.16988i −3.48187 10.6243i −11.0062 33.8735i 4.54748i −5.47554 + 1.77911i 48.9961 + 35.5978i −24.9258 + 34.4910i
14.3 −0.442260 0.608718i −6.75579 2.19509i 2.29719 7.07003i 1.00498 11.1351i 1.65162 + 5.08317i 18.3105i −11.0443 + 3.58852i 18.9788 + 13.7889i −7.22259 + 4.31284i
14.4 −0.217515 0.299383i 2.64413 + 0.859131i 2.42982 7.47821i 7.78705 + 8.02258i −0.317928 0.978483i 0.707538i −5.58294 + 1.81401i −15.5901 11.3269i 0.708030 4.07634i
14.5 1.81265 + 2.49489i 3.16804 + 1.02936i −0.466674 + 1.43628i −10.8670 2.62823i 3.17440 + 9.76980i 5.10302i 19.0341 6.18456i −12.8665 9.34809i −13.1409 31.8762i
14.6 2.81526 + 3.87487i −3.96562 1.28851i −4.61680 + 14.2091i 11.1717 0.439337i −6.17144 18.9937i 14.5499i −31.6143 + 10.2721i −7.77757 5.65074i 33.1536 + 42.0521i
19.1 −4.95462 + 1.60985i −1.70326 2.34434i 15.4845 11.2501i 11.0057 + 1.96864i 12.2131 + 8.87330i 24.5811i −34.1117 + 46.9507i 5.74864 17.6925i −57.6981 + 7.96365i
19.2 −2.51328 + 0.816614i 1.75639 + 2.41747i −0.822422 + 0.597525i −5.31827 + 9.83443i −6.38844 4.64147i 26.3705i 14.0054 19.2767i 5.58423 17.1865i 5.33537 29.0596i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 25.4.e.a 24
3.b odd 2 1 225.4.m.a 24
5.b even 2 1 125.4.e.a 24
5.c odd 4 2 125.4.d.b 48
25.d even 5 1 125.4.e.a 24
25.e even 10 1 inner 25.4.e.a 24
25.f odd 20 2 125.4.d.b 48
25.f odd 20 2 625.4.a.g 24
75.h 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 1.a even 1 1 trivial
25.4.e.a 24 25.e even 10 1 inner
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 5.b even 2 1
125.4.e.a 24 25.d even 5 1
225.4.m.a 24 3.b odd 2 1
225.4.m.a 24 75.h odd 10 1
625.4.a.g 24 25.f odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(25, [\chi])$$.