
// Make newform 9075.2.a.r in Magma, downloaded from the LMFDB on 20 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9075_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_9075_2_a_r();".
// 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_9075_2_a_r();".
// 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_9075_a();"
function MakeCharacter_9075_a()
    N := 9075;
    order := 1;
    char_gens := [3026, 727, 5326];
    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_9075_a_Hecke( : Kf := Kf)
    return MakeCharacter_9075_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]];
    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 := 2;
    raw_aps := [[1], [1], [0], [4], [0], [2], [2], [-8], [-4], [-4], [-8], [-8], [-12], [-8], [4], [4], [-8], [0], [4], [-12], [2], [-8], [4], [6], [8], [4], [12], [12], [8], [-12], [-4], [-8], [4], [-16], [-20], [0], [-8], [20], [24], [14], [0], [-10], [12], [-6], [6], [-16], [16], [4], [-12], [-22], [6], [24], [-8], [0], [-4], [0], [-26], [-16], [10], [-12], [-16], [-6], [24], [28], [-16], [-20], [4], [26], [12], [-16], [-12], [-24], [-36], [22], [-4], [36], [18], [-8], [-2], [40], [8], [-10], [24], [-16], [0], [-12], [-30], [-2], [12], [-28], [-12], [24], [-20], [0], [-12], [24], [26], [22], [24], [-8], [-8], [14], [36], [-44], [40], [0], [-12], [-14], [-12], [32], [20], [-42], [28], [-36], [8], [-2], [-12], [28], [-28], [-24], [6], [-38], [-38], [-4], [-20], [-12], [26], [36], [-28], [14], [-16], [-24], [-16], [0], [36], [-16], [36], [32], [-12], [44], [8], [-20], [-20], [-20], [-14], [20], [6], [-38], [4], [36], [-2], [-30], [4], [-16], [52], [-36], [0], [-18], [14], [4], [-28], [54], [4], [16], [-36], [4], [0], [-42], [24], [-36], [24], [-14], [48], [-34], [-40], [-22], [-8], [-18], [28], [16], [20], [-48], [-48], [38], [12], [50], [30], [28], [40], [40], [-40], [-44], [52], [-42], [-28], [12], [-16], [-16], [30], [48], [20], [-40], [16], [-24], [-40], [-12], [-56], [-36], [60], [-28], [22], [50], [12], [-12], [-8], [-42], [44], [-20], [12], [36], [-8], [-24], [2], [44], [-36], [32], [-12], [-20], [-44], [40], [32], [-64], [8], [-4], [-20], [24], [-2], [42], [48], [-36], [52], [0], [60], [-2], [14], [8], [44], [-40], [16], [24], [18], [-4], [44], [-6], [-50], [-8], [-54], [8], [-52], [-34], [-52], [-28], [-40], [34], [60], [-68], [58], [54], [-16], [26], [-14], [-28], [-56], [-32], [-26], [20], [52], [-80], [-64], [-8], [40], [-32], [-48], [64], [0], [60], [-8], [-86], [24], [-36], [6], [-36], [70], [-72], [66], [-20], [-8], [12], [-8], [-16], [14], [78], [-64], [-20], [-28], [-88], [84], [30], [36], [-10], [24], [62], [12], [28], [-24], [56], [0], [-32], [40], [-12], [0], [32], [-12], [-44], [-42], [-34], [-36], [84], [-16], [6], [8], [12], [40], [-44], [-48], [-68], [-78], [78], [-26], [52], [40], [-12], [-36], [-64], [84], [-80], [-6], [28], [40], [-68], [-40], [88], [-38], [-12], [-48], [-22], [-20], [72], [-2], [-12], [26], [-36], [-60], [-80], [-68], [-16], [-82], [-44], [16], [-40], [100], [40], [4], [-40], [-64], [-24], [0], [38], [20], [-82], [74], [-52], [-44], [66], [56], [84], [-48], [16], [32], [28], [62], [-60], [-48], [-44], [60], [-18], [80], [-26], [-52], [-84], [-16], [20], [76], [92], [-36], [-64], [16], [60], [60], [-72], [-10], [-22], [4], [8], [98], [-38], [0], [4], [-84], [-64], [70], [-38], [60], [16], [32], [-4], [84], [12], [60], [40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9075_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9075_2_a_r();".
// 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_9075_2_a_r(:prec:=1)
    chi := MakeCharacter_9075_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_9075_2_a_r();".
// 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_9075_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9075_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![8, 1]>,<23,R![4, 1]>,<37,R![8, 1]>],Snew);
    return Vf;
end function;