
// Make newform 3525.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_c();".
// 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_3525_2_a_c();".
// 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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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 := 2;
    raw_aps := [[-2], [-1], [0], [3], [1], [2], [-2], [6], [-3], [3], [2], [7], [10], [10], [1], [-4], [8], [-10], [-10], [-14], [10], [17], [-8], [6], [-1], [-16], [-11], [-5], [6], [10], [8], [-6], [22], [-2], [-12], [-14], [3], [6], [-21], [0], [5], [12], [-18], [-4], [-8], [26], [16], [12], [-7], [4], [-23], [24], [11], [18], [31], [12], [-12], [-15], [-26], [27], [21], [-27], [-23], [-23], [-6], [3], [4], [13], [12], [16], [0], [-15], [8], [16], [35], [-12], [6], [2], [4], [-32], [25], [-20], [10], [0], [-15], [12], [7], [13], [31], [28], [21], [12], [8], [36], [-22], [21], [30], [-10], [13], [34], [8], [-3], [0], [15], [-20], [18], [12], [26], [-39], [7], [8], [-27], [-36], [-19], [-2], [17], [-28], [18], [-6], [-10], [-22], [-26], [30], [-4], [34], [-1], [-35], [2], [-10], [-41], [17], [-8], [2], [18], [-42], [-14], [36], [-32], [-29], [53], [-29], [6], [39], [24], [6], [-31], [34], [-3], [-18], [12], [-50], [-53], [44], [32], [-37], [50], [20], [-50], [14], [-14], [38], [-31], [13], [49], [-28], [-60], [-57], [-40], [-20], [6], [-16], [29], [0], [48], [38], [5], [47], [-20], [42], [0], [17], [-21], [-3], [-22], [-16], [-24], [-2], [44], [-51], [39], [14], [52], [34], [40], [-8], [12], [-56], [-44], [-16], [-32], [34], [-8], [18], [-31], [66], [52], [38], [26], [15], [-56], [-9], [6], [16], [7], [10], [34], [-68], [63], [-18], [45], [-23], [8], [-18], [-10], [12], [54], [-18], [27], [-23], [33], [36], [0], [-49], [28], [-34], [45], [-59], [2], [-36], [76], [-49], [-11], [-40], [2], [-30], [-18], [-57], [-23], [9], [-34], [8], [22], [6], [54], [-9], [46], [2], [-52], [29], [-61], [-33], [79], [-11], [-37], [63], [-51], [-30], [9], [-4], [11], [-29], [49], [-63], [44], [-40], [-14], [-38], [51], [-64], [34], [-66], [-24], [-6], [-85], [76], [-8], [-12], [-57], [38], [-60], [16], [62], [-69], [16], [5], [-65], [-40], [-39], [15], [16], [8], [-77], [-4], [-60], [-79], [52], [38], [22], [20], [35], [16], [-6], [-62], [50], [12], [-79], [-41], [-80], [18], [-62], [20], [-74], [-14], [-11], [79], [-76], [-17], [6], [-20], [58], [-53], [-34], [64], [-54], [-76], [68], [-30], [-86], [7], [22], [-2], [-87], [44], [-29], [-18], [60], [80], [18], [36], [56], [37], [37], [-77], [-44], [26], [-93], [54], [62], [43], [-21], [-62], [33], [-78], [-57], [16], [-56], [-91], [68], [-4], [3], [76], [-75], [83], [92], [32], [-3], [-58], [-61], [20], [47], [64], [28], [28], [-60], [39], [-71], [-2], [-28], [11], [20], [2], [-6], [-21], [-85], [42], [-2], [74], [8], [68], [-42], [38], [-36], [-36], [42], [34], [-29], [29], [-90], [-40], [7], [32], [-54], [69], [-94], [78], [-78], [-18], [10], [65], [34], [76], [-58], [-75], [72], [34], [6], [42], [60], [20], [-93]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_c();".
// 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_3525_2_a_c(:prec:=1)
    chi := MakeCharacter_3525_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_3525_2_a_c();".
// 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_3525_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![-3, 1]>,<11,R![-1, 1]>],Snew);
    return Vf;
end function;