# Properties

 Label 225.4.u.a Level $225$ Weight $4$ Character orbit 225.u Analytic conductor $13.275$ Analytic rank $0$ Dimension $704$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([10, 3]))

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

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

chi := DirichletCharacter("225.4");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.u (of order $$30$$, degree $$8$$, 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: $$704$$ Relative dimension: $$88$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9}+O(q^{10})$$ 704 * q - 5 * q^2 - 10 * q^3 - 347 * q^4 + 12 * q^5 + 10 * q^6 - 20 * q^8 - 38 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$704 q - 5 q^{2} - 10 q^{3} - 347 q^{4} + 12 q^{5} + 10 q^{6} - 20 q^{8} - 38 q^{9} - 91 q^{11} + 150 q^{12} - 5 q^{13} + 61 q^{14} - 363 q^{15} + 1293 q^{16} - 20 q^{17} - 12 q^{19} + q^{20} - 135 q^{21} - 5 q^{22} - 5 q^{23} - 250 q^{24} + 284 q^{25} - 2496 q^{26} - 340 q^{27} - 660 q^{28} + 345 q^{29} + 56 q^{30} + 33 q^{31} + 790 q^{33} - 19 q^{34} - 736 q^{35} - 852 q^{36} - 20 q^{37} - 3015 q^{38} + 500 q^{39} - 49 q^{40} - 659 q^{41} - 1790 q^{42} - 1996 q^{44} - 1083 q^{45} + 20 q^{46} - 955 q^{47} - 6225 q^{48} + 14888 q^{49} - 563 q^{50} + 204 q^{51} - 45 q^{52} - 20 q^{53} - 17 q^{54} - 50 q^{55} - 590 q^{56} - 5 q^{58} + 915 q^{59} - 2153 q^{60} - 3 q^{61} + 4900 q^{62} + 2385 q^{63} + 9156 q^{64} + 456 q^{65} - 3514 q^{66} + 1525 q^{67} - 476 q^{69} + 1254 q^{70} + 2432 q^{71} - 5090 q^{72} - 20 q^{73} - 3830 q^{74} - 4343 q^{75} + 152 q^{76} - 715 q^{77} - 1330 q^{78} - 255 q^{79} + 4778 q^{80} + 786 q^{81} - 145 q^{83} - 5595 q^{84} + 699 q^{85} - 4551 q^{86} - 6260 q^{87} - 5 q^{88} + 116 q^{89} + 3097 q^{90} - 2070 q^{91} + 12395 q^{92} - 2455 q^{94} + 1687 q^{95} - 5225 q^{96} - 5 q^{97} - 3370 q^{98} - 30 q^{99}+O(q^{100})$$ 704 * q - 5 * q^2 - 10 * q^3 - 347 * q^4 + 12 * q^5 + 10 * q^6 - 20 * q^8 - 38 * q^9 - 91 * q^11 + 150 * q^12 - 5 * q^13 + 61 * q^14 - 363 * q^15 + 1293 * q^16 - 20 * q^17 - 12 * q^19 + q^20 - 135 * q^21 - 5 * q^22 - 5 * q^23 - 250 * q^24 + 284 * q^25 - 2496 * q^26 - 340 * q^27 - 660 * q^28 + 345 * q^29 + 56 * q^30 + 33 * q^31 + 790 * q^33 - 19 * q^34 - 736 * q^35 - 852 * q^36 - 20 * q^37 - 3015 * q^38 + 500 * q^39 - 49 * q^40 - 659 * q^41 - 1790 * q^42 - 1996 * q^44 - 1083 * q^45 + 20 * q^46 - 955 * q^47 - 6225 * q^48 + 14888 * q^49 - 563 * q^50 + 204 * q^51 - 45 * q^52 - 20 * q^53 - 17 * q^54 - 50 * q^55 - 590 * q^56 - 5 * q^58 + 915 * q^59 - 2153 * q^60 - 3 * q^61 + 4900 * q^62 + 2385 * q^63 + 9156 * q^64 + 456 * q^65 - 3514 * q^66 + 1525 * q^67 - 476 * q^69 + 1254 * q^70 + 2432 * q^71 - 5090 * q^72 - 20 * q^73 - 3830 * q^74 - 4343 * q^75 + 152 * q^76 - 715 * q^77 - 1330 * q^78 - 255 * q^79 + 4778 * q^80 + 786 * q^81 - 145 * q^83 - 5595 * q^84 + 699 * q^85 - 4551 * q^86 - 6260 * q^87 - 5 * q^88 + 116 * q^89 + 3097 * q^90 - 2070 * q^91 + 12395 * q^92 - 2455 * q^94 + 1687 * q^95 - 5225 * q^96 - 5 * q^97 - 3370 * q^98 - 30 * 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 −1.13719 5.35004i −4.50950 + 2.58155i −20.0214 + 8.91409i 2.25903 10.9497i 18.9395 + 21.1903i 24.9371 14.3974i 44.7393 + 61.5784i 13.6712 23.2830i −61.1505 + 0.366009i
4.2 −1.12585 5.29669i −5.17983 + 0.411540i −19.4791 + 8.67264i 6.78358 + 8.88724i 8.01150 + 26.9726i −21.8175 + 12.5963i 42.4038 + 58.3638i 26.6613 4.26341i 39.4357 45.9362i
4.3 −1.11854 5.26231i 4.92317 + 1.66204i −19.1324 + 8.51831i 11.0988 1.34760i 3.23940 27.7663i −14.9230 + 8.61579i 40.9287 + 56.3335i 21.4753 + 16.3650i −19.5060 56.8982i
4.4 −1.10981 5.22123i −1.99679 4.79717i −18.7212 + 8.33520i −7.33503 + 8.43785i −22.8311 + 15.7496i 18.8450 10.8801i 39.1967 + 53.9496i −19.0257 + 19.1578i 52.1964 + 28.9335i
4.5 −1.07574 5.06097i 0.736845 5.14364i −17.1478 + 7.63470i −5.20342 9.89568i −26.8245 + 1.80408i −29.1252 + 16.8154i 32.7558 + 45.0845i −25.9141 7.58014i −44.4842 + 36.9795i
4.6 −1.06743 5.02188i −0.0785966 + 5.19556i −16.7715 + 7.46716i −8.45497 + 7.31529i 26.1754 5.15121i −9.64326 + 5.56754i 31.2598 + 43.0254i −26.9876 0.816706i 45.7616 + 34.6513i
4.7 −1.06231 4.99779i 5.10707 0.958021i −16.5410 + 7.36455i −10.9332 + 2.33771i −10.2133 24.5064i −5.12017 + 2.95613i 30.3522 + 41.7762i 25.1644 9.78537i 23.2979 + 52.1585i
4.8 −1.04294 4.90666i 2.73456 4.41839i −15.6792 + 6.98082i 10.5280 3.76300i −24.5315 8.80939i 16.9358 9.77791i 27.0170 + 37.1857i −12.0444 24.1647i −29.4439 47.7329i
4.9 −1.01603 4.78006i 3.24528 + 4.05810i −14.5083 + 6.45951i −5.06856 9.96543i 16.1007 19.6358i 11.0356 6.37141i 22.6385 + 31.1592i −5.93634 + 26.3393i −42.4855 + 34.3532i
4.10 −0.943007 4.43650i −5.11723 0.902178i −11.4849 + 5.11340i −11.1329 1.02923i 0.823077 + 23.5534i −6.46246 + 3.73110i 12.1882 + 16.7756i 25.3722 + 9.23331i 5.93219 + 50.3615i
4.11 −0.939504 4.42002i −1.03009 + 5.09303i −11.3455 + 5.05136i 9.01568 + 6.61193i 23.4791 0.231910i 13.1856 7.61268i 11.7378 + 16.1557i −24.8778 10.4925i 20.7546 46.0614i
4.12 −0.922831 4.34158i 4.70291 2.20967i −10.6893 + 4.75919i 1.62060 + 11.0623i −13.9335 18.3789i 8.42574 4.86461i 9.65541 + 13.2895i 17.2347 20.7838i 46.5321 17.2446i
4.13 −0.912762 4.29421i −3.93731 3.39081i −10.2987 + 4.58529i 5.02582 9.98705i −10.9670 + 20.0026i −4.48355 + 2.58858i 8.44680 + 11.6260i 4.00486 + 26.7013i −47.4739 12.4661i
4.14 −0.849046 3.99445i 0.343429 5.18479i −7.92637 + 3.52905i 5.82974 + 9.54013i −21.0020 + 3.03032i −17.1647 + 9.91002i 1.62379 + 2.23496i −26.7641 3.56121i 33.1578 31.3866i
4.15 −0.822434 3.86925i −4.35066 2.84108i −6.98632 + 3.11051i 9.64418 + 5.65595i −7.41470 + 19.1704i 13.6923 7.90527i −0.819642 1.12814i 10.8566 + 24.7211i 13.9526 41.9674i
4.16 −0.814957 3.83407i −3.33680 + 3.98318i −6.72759 + 2.99532i −9.34277 6.14107i 17.9912 + 9.54742i −7.03825 + 4.06354i −1.46471 2.01600i −4.73148 26.5822i −15.9313 + 40.8255i
4.17 −0.801643 3.77143i 5.10602 0.963611i −6.27271 + 2.79279i −2.58409 10.8776i −7.72740 18.4845i 11.3837 6.57239i −2.56924 3.53626i 25.1429 9.84043i −38.9527 + 18.4657i
4.18 −0.792549 3.72865i 2.16415 + 4.72403i −5.96634 + 2.65639i 9.75102 5.46969i 15.8991 11.8134i −4.93704 + 2.85040i −3.29151 4.53038i −17.6329 + 20.4470i −28.1227 32.0232i
4.19 −0.770399 3.62444i −4.10935 + 3.18013i −5.23470 + 2.33064i 9.67594 5.60144i 14.6920 + 12.4441i −20.0051 + 11.5499i −4.94384 6.80461i 6.77352 26.1366i −27.7564 30.7545i
4.20 −0.728639 3.42798i −4.82414 + 1.93071i −3.91175 + 1.74162i −5.26207 + 9.86462i 10.1335 + 15.1303i 23.0240 13.2929i −7.65893 10.5416i 19.5447 18.6280i 37.6498 + 10.8505i
See next 80 embeddings (of 704 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.88 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.e even 10 1 inner
225.u even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.u.a 704
9.c even 3 1 inner 225.4.u.a 704
25.e even 10 1 inner 225.4.u.a 704
225.u even 30 1 inner 225.4.u.a 704

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.u.a 704 1.a even 1 1 trivial
225.4.u.a 704 9.c even 3 1 inner
225.4.u.a 704 25.e even 10 1 inner
225.4.u.a 704 225.u even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(225, [\chi])$$.