Properties

 Label 225.4.h.d Level $225$ Weight $4$ Character orbit 225.h Analytic conductor $13.275$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(46,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 6]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.46");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.h (of order $$5$$, 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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 72 q^{4} + 20 q^{7}+O(q^{10})$$ 64 * q - 72 * q^4 + 20 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 72 q^{4} + 20 q^{7} + 50 q^{10} + 180 q^{13} - 244 q^{16} - 116 q^{19} - 210 q^{22} + 440 q^{25} + 980 q^{28} + 42 q^{31} + 40 q^{34} - 1170 q^{37} - 3040 q^{40} + 1040 q^{43} + 700 q^{46} + 4188 q^{49} + 3280 q^{52} + 2640 q^{55} - 4530 q^{58} + 88 q^{61} - 3018 q^{64} - 2860 q^{67} - 3720 q^{70} - 5280 q^{73} + 9288 q^{76} - 1144 q^{79} + 2840 q^{82} + 470 q^{85} + 7610 q^{88} - 1180 q^{91} + 2170 q^{94} + 2050 q^{97}+O(q^{100})$$ 64 * q - 72 * q^4 + 20 * q^7 + 50 * q^10 + 180 * q^13 - 244 * q^16 - 116 * q^19 - 210 * q^22 + 440 * q^25 + 980 * q^28 + 42 * q^31 + 40 * q^34 - 1170 * q^37 - 3040 * q^40 + 1040 * q^43 + 700 * q^46 + 4188 * q^49 + 3280 * q^52 + 2640 * q^55 - 4530 * q^58 + 88 * q^61 - 3018 * q^64 - 2860 * q^67 - 3720 * q^70 - 5280 * q^73 + 9288 * q^76 - 1144 * q^79 + 2840 * q^82 + 470 * q^85 + 7610 * q^88 - 1180 * q^91 + 2170 * q^94 + 2050 * q^97

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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1 −1.71061 + 5.26471i 0 −18.3188 13.3094i −11.1283 1.07706i 0 −30.2200 65.5791 47.6460i 0 24.7066 56.7450i
46.2 −1.45629 + 4.48200i 0 −11.4954 8.35189i 10.8155 + 2.83289i 0 2.19013 23.6729 17.1993i 0 −28.4475 + 44.3495i
46.3 −1.31991 + 4.06225i 0 −8.28762 6.02131i −4.79821 + 10.0984i 0 33.6094 7.75447 5.63395i 0 −34.6890 32.8205i
46.4 −1.20400 + 3.70554i 0 −5.80928 4.22069i 3.34114 10.6694i 0 6.27975 −2.58263 + 1.87639i 0 35.5133 + 25.2268i
46.5 −0.856692 + 2.63663i 0 0.254251 + 0.184724i −7.75979 8.04895i 0 −7.15708 −18.6477 + 13.5483i 0 27.8698 13.5642i
46.6 −0.626289 + 1.92752i 0 3.14904 + 2.28791i −3.12321 + 10.7352i 0 −16.4425 −19.4994 + 14.1671i 0 −18.7364 12.7434i
46.7 −0.426412 + 1.31236i 0 4.93167 + 3.58307i 10.9704 2.15653i 0 −13.9864 −15.7361 + 11.4329i 0 −1.84776 + 15.3167i
46.8 −0.0492832 + 0.151678i 0 6.45156 + 4.68733i −9.69768 5.56372i 0 29.3448 −2.06112 + 1.49749i 0 1.32183 1.19673i
46.9 0.0492832 0.151678i 0 6.45156 + 4.68733i 9.69768 + 5.56372i 0 29.3448 2.06112 1.49749i 0 1.32183 1.19673i
46.10 0.426412 1.31236i 0 4.93167 + 3.58307i −10.9704 + 2.15653i 0 −13.9864 15.7361 11.4329i 0 −1.84776 + 15.3167i
46.11 0.626289 1.92752i 0 3.14904 + 2.28791i 3.12321 10.7352i 0 −16.4425 19.4994 14.1671i 0 −18.7364 12.7434i
46.12 0.856692 2.63663i 0 0.254251 + 0.184724i 7.75979 + 8.04895i 0 −7.15708 18.6477 13.5483i 0 27.8698 13.5642i
46.13 1.20400 3.70554i 0 −5.80928 4.22069i −3.34114 + 10.6694i 0 6.27975 2.58263 1.87639i 0 35.5133 + 25.2268i
46.14 1.31991 4.06225i 0 −8.28762 6.02131i 4.79821 10.0984i 0 33.6094 −7.75447 + 5.63395i 0 −34.6890 32.8205i
46.15 1.45629 4.48200i 0 −11.4954 8.35189i −10.8155 2.83289i 0 2.19013 −23.6729 + 17.1993i 0 −28.4475 + 44.3495i
46.16 1.71061 5.26471i 0 −18.3188 13.3094i 11.1283 + 1.07706i 0 −30.2200 −65.5791 + 47.6460i 0 24.7066 56.7450i
91.1 −4.45962 3.24011i 0 6.91782 + 21.2908i −7.22670 8.53082i 0 −10.2696 24.5064 75.4228i 0 4.58760 + 61.4595i
91.2 −3.81348 2.77066i 0 4.39397 + 13.5233i −0.520226 + 11.1682i 0 30.0229 9.05902 27.8808i 0 32.9272 41.1485i
91.3 −3.54222 2.57358i 0 3.45191 + 10.6239i 10.9424 + 2.29411i 0 −21.8567 4.28988 13.2029i 0 −32.8565 36.2875i
91.4 −2.68030 1.94735i 0 0.919680 + 2.83048i −2.31093 10.9389i 0 10.7752 −5.14333 + 15.8295i 0 −15.1079 + 33.8197i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.h.d 64
3.b odd 2 1 inner 225.4.h.d 64
25.d even 5 1 inner 225.4.h.d 64
75.j odd 10 1 inner 225.4.h.d 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.h.d 64 1.a even 1 1 trivial
225.4.h.d 64 3.b odd 2 1 inner
225.4.h.d 64 25.d even 5 1 inner
225.4.h.d 64 75.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} + 100 T_{2}^{62} + 6005 T_{2}^{60} + 283035 T_{2}^{58} + 11612555 T_{2}^{56} + 401758275 T_{2}^{54} + 12005034550 T_{2}^{52} + 318342285625 T_{2}^{50} + \cdots + 15\!\cdots\!00$$ acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$.