
// Make newform 432.6.a.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_432_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_432_6_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_432_6_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_432_a();"
function MakeCharacter_432_a()
    N := 432;
    order := 1;
    char_gens := [271, 325, 353];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_432_a_Hecke(Kf)
    return MakeCharacter_432_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 6;
    raw_aps := [[0], [0], [0], [25], [0], [-427], [0], [1711], [0], [0], [10324], [-6661], [0], [3352], [0], [0], [0], [56927], [37939], [0], [79577], [-90857], [0], [0], [177725], [0], [211477], [0], [114482], [0], [267100], [0], [0], [454657], [0], [127627], [109214], [325873], [0], [0], [0], [619001], [0], [656375], [0], [102493], [-1236299], [-304052], [0], [1269818], [0], [0], [-1733551], [0], [0], [0], [0], [365851], [-2486614], [0], [-1952], [0], [2301136], [0], [-3300973], [0], [-3692999], [-4116811], [0], [-787957], [0], [0], [-5160539], [1983875], [-69893], [0], [0], [5789486], [0], [-5170207], [0], [7111901], [0], [1857602], [-3993332], [0], [0], [-8257450], [0], [9039073], [0], [0], [8305489], [0], [8361568], [0], [0], [0], [11728675], [13008077], [11304889], [0], [0], [0], [-9549977], [-12909925], [0], [0], [0], [6021626], [84361], [16390223], [0], [12226039], [1987651], [0], [-20670200], [0], [0], [0], [19792751], [-22418773], [0], [0], [-17363048], [0], [-935611], [0], [28451236], [-11302750], [29603536], [0], [-26183201], [-31143475], [0], [-30884593], [0], [25702375], [0], [0], [-22770776], [0], [38714323], [0], [-16780807], [0], [38219123], [0], [-35829707], [0], [-37391125], [0], [-16846775], [0], [-26211989], [0], [-46303868], [0], [5297075], [0], [0], [0], [32345311], [0], [0], [0], [-60552827], [-44815450], [-64590139], [0], [0], [59500574], [0], [26034725], [36018157], [0], [49647424], [0], [9167527], [-3474418], [52747711], [0], [-78373702], [0], [0], [0], [-82699039], [83819077], [67658189], [0], [90280025], [0], [90100477], [0], [0], [0], [78492299], [-18906127], [0], [0], [0], [-70390901], [91539689], [5539493], [0], [0], [-96112061], [0], [0], [-83236901], [80360711], [0], [96415423], [0], [0], [67061351], [34794964], [0], [0], [0], [-8052223], [80069932], [0], [-148772900], [0], [-117406393], [0], [0], [-46443611], [0], [-148717075], [134356264], [129116476], [0], [-168318227], [0], [-31759843], [0], [0], [0], [0], [103473199], [186824875], [64083707], [0], [0], [-146306825], [0], [-173769632], [0], [-48022750], [0], [0], [30695357], [0], [0], [-92127727], [-170531600], [0], [200077850], [-68102975], [0], [-94649107], [191178575], [0], [-229459136], [0], [0], [144785800], [0], [219533423], [232503589], [-144985450], [-194217593], [197198414], [-105699275], [0], [-87625282], [-229903126], [0], [0], [284820652], [0], [-36264949], [-298264739], [0], [253600577], [0], [223046593], [0], [0], [0], [0], [0], [-228995698], [0], [281230873], [0], [0], [80933839], [-110416477], [0], [13787332], [0], [-361677299], [-340663714], [0], [298672061], [0], [381782627], [0], [0], [0], [289789825], [0], [335242118], [0], [0], [287249198], [0], [133746451], [237111611], [0], [-359699852], [0], [-222800851], [-58180232], [-188521448], [0], [0], [361720901], [0], [-446735693], [0], [360308176], [0], [-25730461], [0], [-312675274], [-434756900], [405483275], [0], [0], [322131823], [0], [0], [-496436326], [-239827136], [0], [0], [494097727], [-280189711], [0], [550668748], [-453511957], [0], [0], [0], [0], [0], [521998175], [0], [0], [0], [-285806339], [41338775], [0], [-215227223], [-451057249], [0], [-393095339], [0], [0], [-654949751], [-174198475], [0], [0], [516401927], [0], [169584725], [0], [0], [445376539], [0], [-510534257], [0], [-596816924], [338517761], [-547383527], [0], [-46663618], [0], [0], [485419039], [0], [-172877575], [-517183757], [0], [-684858824], [0], [491811614], [0], [-628317764], [0], [0], [692522548], [193835225], [0], [270197977], [0], [-360091102], [0], [0], [863829601], [-109931575], [0], [0], [679383475], [0], [0], [0], [-520998586], [0], [0], [331621850], [0], [0], [0], [-413471576], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_432_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_432_6_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_432_6_a_e(:prec:=1)
    chi := MakeCharacter_432_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_432_6_a_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_432_6_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-25, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;