
// Make newform 400.3.p.g in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_p_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_400_3_p_g();".
// 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_400_3_p_g();".
// 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
        poly := [1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_400_p();"
function MakeCharacter_400_p()
    N := 400;
    order := 4;
    char_gens := [351, 101, 177];
    v := [4, 4, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_p_Hecke();"
function MakeCharacter_400_p_Hecke( : Kf := false)
    N := 400;
    order := 4;
    char_gens := [351, 101, 177];
    char_values := [[1, 0], [1, 0], [0, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[0, 0], [3, -3], [0, 0], [-3, -3], [-12, 0], [12, -12], [-12, -12], [0, -20], [3, -3], [0, -30], [8, 0], [48, 48], [-48, 0], [-27, 27], [27, 27], [12, -12], [0, 60], [32, 0], [-3, -3], [48, 0], [12, -12], [0, -40], [93, -93], [0, 30], [-12, -12], [-78, 0], [93, -93], [27, 27], [0, 160], [72, -72], [117, 117], [-132, 0], [168, 168], [0, 100], [0, 0], [248, 0], [-72, -72], [93, -93], [-3, -3], [-168, 168], [0, 300], [142, 0], [-192, 0], [132, -132], [-132, -132], [0, -160], [28, 0], [-117, 117], [-93, -93], [0, -370], [252, -252], [0, -360], [32, 0], [-252, 0], [-192, -192], [333, -333], [0, 480], [88, 0], [288, 288], [-288, 0], [-117, 117], [-168, 168], [-243, -243], [-552, 0], [-48, 48], [228, 228], [148, 0], [-192, -192], [117, 117], [0, -130], [-288, 288], [0, 120], [-213, -213], [-168, 168], [0, -20], [123, -123], [0, 0], [108, 108], [-18, 0], [0, 80], [0, 540], [-608, 0], [-312, 0], [252, -252], [0, -40], [213, -213], [0, 480], [-432, -432], [222, 0], [213, -213], [-3, -3], [0, 240], [627, 627], [588, 0], [0, -460], [-627, 627], [0, 450], [-558, 0], [123, -123], [542, 0], [147, 147], [288, 288], [-477, 477], [0, -240], [-692, 0], [168, 168], [-213, -213], [312, -312], [0, -240], [-608, 0], [267, 267], [-228, 228], [348, 348], [0, -940], [808, 0], [-768, 0], [-477, 477], [627, 627], [12, -12], [0, 540], [352, 0], [732, -732], [108, 108], [933, -933], [68, 0], [192, 0], [0, 50], [0, -840], [-963, -963], [72, -72], [0, -20], [243, -243], [-1072, 0], [408, 408], [1362, 0], [0, 370], [132, -132], [-93, -93], [228, 228], [0, -750], [-412, 0], [672, 0], [-717, 717], [-123, -123], [0, 1280], [0, -1560], [372, -372], [-552, -552], [0, 620], [123, -123], [1128, 1128], [912, 0], [-957, 957], [-483, -483], [1077, 1077], [1128, 0], [0, 1600], [0, 960], [-492, -492], [-738, 0], [237, 237], [-648, 648], [627, 627], [708, 0], [-612, -612], [-627, 627], [1168, 0], [108, 108]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_p_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_3_p_g();".
// 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_400_3_p_g(:prec:=2)
    chi := MakeCharacter_400_p();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_400_3_p_g();".
// 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_400_3_p_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![18, -6, 1]>,<7,R![18, 6, 1]>],Snew);
    return Vf;
end function;