
// Make newform 6525.2.a.k in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_6525_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_6525_2_a_k();".
// 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_6525_2_a_k();".
// 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_6525_a();"
function MakeCharacter_6525_a()
    N := 6525;
    order := 1;
    char_gens := [1451, 4177, 901];
    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_6525_a_Hecke( : Kf := Kf)
    return MakeCharacter_6525_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], [0], [0], [2], [0], [-4], [2], [0], [2], [-1], [4], [-2], [-10], [0], [12], [-12], [-4], [2], [-2], [8], [-14], [8], [-6], [-10], [-10], [-2], [2], [6], [-14], [14], [-12], [12], [-14], [-4], [6], [-24], [-22], [4], [2], [0], [20], [2], [-12], [-6], [-8], [-16], [-12], [18], [-6], [-22], [20], [-24], [30], [0], [8], [16], [-26], [-24], [8], [6], [10], [6], [8], [-8], [-28], [-6], [8], [-2], [-6], [-2], [28], [0], [24], [-4], [20], [18], [-2], [-20], [34], [-22], [-12], [-22], [0], [14], [-16], [-24], [34], [8], [-26], [30], [0], [-36], [-14], [4], [-4], [4], [-26], [-10], [-34], [-2], [46], [-20], [-28], [-14], [4], [-6], [42], [36], [36], [-2], [28], [-40], [18], [36], [-8], [-50], [10], [-18], [-6], [-28], [26], [12], [14], [-18], [-44], [30], [-6], [0], [40], [50], [-40], [20], [-20], [-42], [42], [38], [-30], [34], [-22], [30], [44], [42], [40], [8], [-34], [-20], [26], [32], [36], [50], [-8], [-42], [38], [44], [-8], [48], [24], [-14], [16], [42], [48], [12], [48], [-60], [-32], [-12], [0], [-26], [34], [-18], [-20], [-2], [-24], [-42], [-48], [-38], [-28], [54], [32], [46], [28], [20], [52], [16], [10], [-26], [-50], [-64], [-30], [8], [36], [56], [56], [26], [-12], [56], [6], [-60], [-8], [-54], [-6], [48], [42], [-2], [36], [16], [60], [-42], [-54], [60], [-10], [10], [56], [12], [60], [10], [14], [18], [-2], [-46], [-22], [-64], [-34], [40], [-14], [22], [-14], [60], [-44], [-28], [-34], [-60], [-8], [-10], [-50], [-64], [-54], [-18], [-20], [-24], [-72], [-52], [-14], [50], [-36], [64], [-54], [12], [-20], [32], [50], [-14], [-72], [10], [-34], [36], [14], [20], [-28], [-52], [8], [12], [58], [-18], [18], [20], [-54], [-6], [-8], [28], [-14], [66], [-60], [40], [-6], [16], [-12], [-42], [38], [-52], [-42], [-8], [-54], [-26], [8], [-72], [-10], [-50], [-56], [54], [-18], [-78], [-48], [-40], [34], [66], [4], [-36], [76], [64], [-10], [36], [4], [-48], [-8], [28], [-24], [-26], [16], [68], [6], [50], [18], [-78], [-42], [-46], [-56], [56], [-36], [2], [-44], [20], [-78], [-8], [60], [-2], [-4], [70], [36], [-12], [-46], [40], [-24], [24], [-32], [-46], [-34], [54], [-26], [50], [-38], [-32], [78], [32], [0], [0], [-6], [8], [28], [22], [-92], [-16], [90], [22], [42], [30], [-24], [60], [54], [54], [-36], [-50], [32], [-92], [20], [86], [-54], [70], [-82], [-60], [-12], [-38], [-82], [-48], [0], [4], [76], [90], [-34], [-28], [-34], [48], [-64], [-26], [-52], [42], [16], [-76], [28], [-20], [-78], [44], [-20], [-84], [60], [-20], [24], [46], [-100], [-14], [-30], [34], [-48], [88], [42], [32], [-56], [10], [16], [-20], [-36], [-60], [50], [84], [50], [38], [4], [54], [-70], [52], [94], [-2], [-92], [16], [-56], [-84], [-42], [-6], [-20], [32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6525_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6525_2_a_k();".
// 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_6525_2_a_k(:prec:=1)
    chi := MakeCharacter_6525_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_6525_2_a_k();".
// 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_6525_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;