
// Make newform 7245.2.a.m in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7245_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_7245_2_a_m();".
// 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_7245_2_a_m();".
// 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_7245_a();"
function MakeCharacter_7245_a()
    N := 7245;
    order := 1;
    char_gens := [5636, 5797, 5176, 1891];
    v := [1, 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_7245_a_Hecke(Kf)
    return MakeCharacter_7245_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 := [[1], [0], [-1], [1], [2], [-4], [2], [0], [-1], [6], [-2], [4], [2], [-8], [8], [0], [-2], [-14], [-12], [0], [8], [10], [-12], [-6], [-14], [-18], [8], [-4], [2], [8], [-16], [6], [20], [-10], [-2], [-20], [10], [-20], [16], [4], [-4], [-26], [18], [22], [-18], [-20], [8], [16], [-4], [-14], [-6], [-8], [-6], [-16], [0], [-20], [-10], [-10], [10], [18], [4], [10], [-20], [6], [-26], [-10], [-4], [0], [-36], [-26], [4], [-10], [32], [28], [26], [-24], [6], [0], [14], [34], [12], [-10], [-10], [2], [-14], [-4], [10], [-20], [-18], [-8], [-20], [-28], [-8], [12], [-24], [0], [-6], [2], [44], [2], [28], [-40], [-4], [-18], [10], [-8], [28], [-44], [12], [6], [36], [16], [8], [24], [10], [18], [12], [28], [-18], [30], [-26], [-10], [14], [-44], [14], [-6], [6], [-30], [-40], [-42], [0], [48], [2], [24], [6], [10], [-42], [12], [2], [30], [30], [42], [-32], [36], [-30], [4], [16], [-20], [14], [48], [46], [-18], [36], [28], [-24], [38], [38], [6], [-14], [30], [28], [44], [-8], [-56], [-48], [-16], [4], [12], [-26], [-24], [-30], [26], [-30], [-52], [30], [30], [40], [2], [-24], [34], [56], [-40], [14], [0], [-24], [-2], [38], [-4], [-6], [-50], [-8], [60], [-30], [42], [36], [-14], [-50], [24], [22], [-56], [50], [-62], [-12], [-14], [-48], [-12], [-52], [44], [54], [-38], [-2], [42], [-24], [-60], [30], [-26], [24], [-50], [20], [70], [38], [40], [66], [20], [24], [-22], [-54], [8], [56], [-24], [-18], [-52], [-6], [-18], [60], [32], [-14], [-28], [32], [-34], [0], [2], [-8], [-38], [-12], [-38], [40], [40], [-70], [-20], [-28], [18], [-12], [46], [44], [-48], [10], [12], [48], [-32], [8], [-60], [-18], [-14], [-12], [4], [-54], [-54], [-36], [66], [10], [-20], [-2], [-58], [-68], [36], [-28], [10], [6], [-52], [-56], [-10], [-16], [-6], [-72], [24], [56], [-32], [-46], [54], [26], [60], [-66], [-64], [42], [74], [-32], [36], [-62], [-56], [-38], [36], [58], [-28], [-54], [38], [52], [34], [-54], [-74], [-84], [18], [14], [-52], [24], [2], [2], [-60], [62], [82], [-84], [-32], [50], [48], [32], [42], [30], [-52], [60], [-40], [2], [-2], [-30], [-24], [-26], [0], [-22], [42], [62], [72], [22], [76], [22], [54], [64], [44], [-50], [-52], [-84], [50], [-70], [-18], [-18], [32], [86], [36], [-36], [-84], [74], [-40], [72], [6], [-72], [78], [4], [26], [72], [-4], [-34], [-32], [-30], [48], [-2], [80], [-50], [-38], [-66], [-94], [-54], [-28], [-30], [72], [-32], [54], [-16], [84], [8], [32], [88], [-24], [-70], [-52], [60], [-80], [-18], [28], [60], [22], [38], [58], [50], [10], [8], [-58], [14], [18], [-20], [66], [52], [-6], [-56], [6], [-36], [-94], [24], [-2], [88], [16], [30], [-8], [14], [78], [48], [-30], [-48], [24], [64], [-54], [-38], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7245_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7245_2_a_m();".
// 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_7245_2_a_m(:prec:=1)
    chi := MakeCharacter_7245_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_7245_2_a_m();".
// 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_7245_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7245_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;