
// Make newform 6300.2.v.d in Magma, downloaded from the LMFDB on 03 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6300_v();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_6300_v_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_6300_2_v_d();".
// 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_6300_2_v_d();".
// 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, 0, 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_6300_v();"
function MakeCharacter_6300_v()
    N := 6300;
    order := 4;
    char_gens := [3151, 2801, 3277, 3601];
    v := [4, 2, 1, 4];
    // 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_6300_v_Hecke();"
function MakeCharacter_6300_v_Hecke( : Kf := false)
    N := 6300;
    order := 4;
    char_gens := [3151, 2801, 3277, 3601];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, -1, 4, -1], [2, 0, -2, 2], [0, 0, 0, -4], [0, 2, 0, 2], [1, 4, 1, 0], [0, 1, 0, -1], [0, 2, 0, -2], [-4, 2, -4, 0], [0, -2, -4, -2], [4, 0, -4, 4], [4, 0, -4, 4], [5, -4, 5, 0], [4, 0, 0, 0], [2, -8, 0, 8], [4, 2, 4, 0], [0, 3, -4, 3], [-6, 0, 6, 6], [0, -4, 4, -4], [4, 4, 4, 0], [-8, 2, 0, -2], [-2, 6, -2, 0], [0, -2, 4, -2], [-8, 0, 8, 4], [7, 0, -7, -4], [0, -8, 0, -8], [11, -4, 11, 0], [8, 2, 8, 0], [0, 6, -12, 6], [3, 0, -3, 4], [0, 6, -8, 6], [0, 1, 0, -1], [-2, 8, 0, -8], [2, -2, 2, 0], [8, 0, -8, 10], [12, 0, -12, 8], [10, 0, 10, 0], [-4, -5, 0, 5], [14, -4, 0, 4], [0, 5, 12, 5], [0, 0, 0, 18], [-3, 0, 3, -20], [0, -2, 0, -2], [-6, -12, 0, 12], [-4, 0, 4, 4], [10, 0, -10, 4], [0, 8, -6, 8], [13, 4, 13, 0], [-4, 3, 0, -3], [-2, 4, 0, -4], [0, -12, 4, -12], [6, 0, -6, 4], [5, 12, 5, 0], [4, -8, 0, 8], [4, -10, 0, 10], [-4, 8, -4, 0], [0, 7, -8, 7], [12, 0, -12, 12], [10, 12, 10, 0], [-8, 12, -8, 0], [0, -6, -4, -6], [14, 0, -14, 14], [1, 0, -1, 12], [-10, -4, 0, 4], [-8, -6, -8, 0], [9, 0, -9, -20], [0, -12, -2, -12], [12, 8, 12, 0], [20, 9, 0, -9], [4, 24, 4, 0], [-8, 0, 8, 12], [0, -8, 6, -8], [2, 0, 2, 0], [24, -5, 0, 5], [6, -10, 6, 0], [0, 11, 0, 11], [0, -4, -2, -4], [-16, -12, 0, 12], [-10, 8, 0, -8], [0, -9, -4, -9], [-2, 0, 2, 18], [0, -2, 12, -2], [5, -12, 5, 0], [-8, 15, 0, -15], [20, 6, 20, 0], [0, -22, -8, -22], [-8, 0, 8, 6], [10, 0, -10, -4], [-28, 10, 0, -10], [-4, 0, -4, 0], [0, 15, 12, 15], [0, 0, 18, 0], [2, -40, 2, 0], [12, 4, 0, -4], [0, 0, 4, 0], [-8, 0, 8, -4], [-18, 0, 0, 0], [-4, 30, -4, 0], [-3, 0, 3, 20], [-6, 32, -6, 0], [-32, 11, 0, -11], [16, 0, 0, 0], [-2, -10, -2, 0], [2, 0, -2, -24], [16, 16, 16, 0], [-12, -5, 0, 5], [30, -12, 0, 12], [0, -4, 0, 0], [4, 0, -4, 8], [1, 0, -1, 20], [0, 2, -4, 2], [-12, 12, 0, -12], [0, -7, -32, -7], [4, 0, -4, 8], [12, 0, -12, 0], [-1, -20, -1, 0], [28, -5, 0, 5], [10, -24, 0, 24], [-12, 0, 12, -14], [-22, 0, 22, -12], [-21, 20, -21, 0], [-20, 18, 0, -18], [0, 9, 24, 9], [0, 16, 0, 16], [20, 0, 0, 0], [0, 24, 0, 0], [14, 0, -14, 6], [0, 8, -32, 8], [-11, 4, -11, 0], [-2, -12, 0, 12], [-16, -10, -16, 0], [0, 10, 32, 10], [0, -28, -14, -28], [-10, 0, -10, 0], [-12, -8, -12, 0], [-22, 0, 22, 16], [-16, -3, 0, 3], [20, 18, 0, -18], [0, 29, -8, 29], [16, 0, -16, -2], [11, 0, -11, 20], [0, 8, -26, 8], [0, -26, 0, 26], [-6, 0, 6, -38], [-6, 0, 6, -16], [0, -30, 4, -30], [-17, 20, -17, 0], [-4, -48, -4, 0], [0, -36, -8, -36], [16, 0, -16, -20], [-8, 0, 8, -44], [8, 36, 8, 0], [0, 23, -20, 23], [0, -4, 10, -4], [0, 2, 0, -2], [-6, -26, -6, 0], [0, 8, -40, 8], [-7, 0, 7, -20], [-5, 28, -5, 0], [12, -6, 12, 0], [0, 14, 24, 14], [-19, 0, 19, -28], [-22, -28, -22, 0], [6, -20, 0, 20], [-22, 2, -22, 0], [0, 8, 26, 8], [-10, -36, -10, 0], [4, -21, 0, 21], [34, 16, 0, -16], [0, 25, -12, 25], [0, 0, 0, -16], [0, 18, 20, 18], [20, -16, 0, 16], [-32, 4, 0, -4], [0, -21, 24, -21], [12, 0, -12, -12], [0, 4, 10, 4], [-12, 26, -12, 0], [0, -30, -8, -30], [-8, 0, 8, -4], [-12, 0, 12, 36], [-29, 4, -29, 0], [-60, -2, 0, 2], [12, -26, 12, 0], [-4, 0, 4, -28], [0, 0, 14, 0], [0, -20, -28, -20], [-30, 0, 30, -2], [-19, -12, -19, 0], [-18, -24, 0, 24], [0, 10, 16, 10], [-3, 0, 3, 52], [12, -32, 12, 0], [40, -16, 0, 16], [8, 0, -8, -52], [-10, 0, 10, -28], [-20, 16, -20, 0], [-32, -3, 0, 3], [48, 6, 0, -6], [-6, -54, -6, 0], [0, -16, -22, -16], [-16, -26, 0, 26], [-34, 0, 34, 0], [0, -6, 32, -6], [-23, -12, -23, 0], [40, 19, 0, -19], [32, 2, 0, -2], [-28, -4, -28, 0], [0, -44, -4, -44], [8, 0, -8, -40], [20, 0, -20, 36], [48, 2, 0, -2], [-10, 0, 0, 0], [4, -32, 4, 0], [0, -12, 48, -12], [19, 0, -19, 20], [-21, 12, -21, 0], [-24, 0, 0, 0], [0, 2, -32, 2], [40, 3, 0, -3], [32, 0, -32, 0], [-4, 0, 4, -8], [0, -24, 28, -24], [-10, 4, -10, 0], [-12, -11, 0, 11], [8, 40, 8, 0], [0, 3, 28, 3], [8, 0, -8, -18], [0, 2, 12, 2], [-18, -16, 0, 16], [0, -29, -8, -29], [12, 0, -12, -48], [-26, 0, 26, -20], [0, -4, -30, -4], [-15, 44, -15, 0], [-20, 21, 0, -21], [0, -40, 12, -40], [-3, -44, -3, 0], [-8, -30, 0, 30], [-4, 0, 4, 16], [0, 24, -8, 24], [-2, 0, -2, 0], [-32, 14, 0, -14], [-12, -8, -12, 0], [0, -16, -8, -16], [0, -4, 0, -4], [-15, -12, -15, 0], [-4, -54, -4, 0], [0, 20, -48, 20], [5, 0, -5, -60], [0, 32, -18, 32], [-18, 32, -18, 0], [-20, 15, 0, -15], [2, 40, 0, -40], [8, 52, 8, 0], [-34, 0, 34, -8], [2, 6, 2, 0], [32, 0, -32, -6], [-13, 0, 13, -60], [0, 0, -30, 0], [26, 0, -26, 22], [-4, 0, 4, 8], [0, 30, 20, 30], [-24, 25, 0, -25], [0, -48, 4, -48], [12, 0, -12, 18], [-33, -4, -33, 0], [-30, -24, 0, 24], [4, -28, 4, 0], [38, 0, -38, -10], [0, -20, 54, -20], [-46, -18, -46, 0], [28, 0, -28, -32], [-9, 0, 9, 20], [0, -24, -8, -24], [-52, 16, 0, -16], [0, 32, -12, 32], [6, 12, 6, 0], [12, 12, 0, -12], [22, 0, -22, 44], [30, 24, 0, -24], [-32, 24, -32, 0], [0, 33, -4, 33], [12, 0, -12, 50], [-23, 0, 23, 28], [0, -10, 12, -10], [-56, -18, 0, 18], [0, -29, 8, -29], [8, 0, -8, -28], [23, 20, 23, 0], [0, -32, -16, -32], [8, 0, -8, -52], [-8, -46, 0, 46], [28, -26, 0, 26], [-8, -16, -8, 0], [28, 24, 0, -24], [20, 12, 20, 0], [14, 0, -14, 50], [-19, 0, 19, -12], [0, 8, -20, 8], [-9, -4, -9, 0], [0, 32, 0, -32], [32, -2, 32, 0], [-1, 0, 1, 12], [0, 12, -50, 12], [4, 29, 0, -29], [-16, 0, 16, 2], [6, 12, 6, 0], [56, -1, 0, 1], [0, 19, 32, 19], [-32, 0, 32, 28], [-7, 0, 7, -36], [0, -20, -18, -20], [36, -30, 0, 30], [0, -3, 52, -3], [-38, 0, 38, -14], [-24, 23, 0, -23], [12, -6, 0, 6], [36, -26, 36, 0], [0, -38, -8, -38], [-40, 0, 40, 16], [1, 60, 1, 0], [18, -24, 0, 24], [0, -16, 16, -16], [-8, 0, 8, -24], [27, 0, -27, 28], [-1, -52, -1, 0], [-12, -32, 0, 32], [-17, 0, 17, 28], [0, -34, -40, -34], [30, -24, 30, 0], [0, 4, 0, -4], [14, 0, -14, -24], [0, -16, 66, -16], [34, -4, 34, 0], [58, -8, 0, 8], [-20, -48, -20, 0], [0, 0, 0, -44], [9, 0, -9, 12], [-44, 0, 0, 0], [-22, -24, 0, 24], [-21, -52, -21, 0], [-44, -1, 0, 1], [2, -36, 0, 36], [40, 36, 40, 0], [0, -20, -52, -20], [2, 0, -2, -36], [-40, -10, 0, 10], [12, -12, 12, 0], [0, -11, 24, -11], [8, 0, -8, 8], [0, -16, 38, -16], [56, -16, 56, 0], [-12, 0, 0, 0], [0, -38, -4, -38], [27, 0, -27, -12], [7, 52, 7, 0], [32, 12, 32, 0], [0, 18, 20, 18], [-5, 0, 5, -52], [12, 33, 0, -33], [-52, 12, -52, 0], [-20, 0, 20, -6], [0, 0, 0, 40], [36, 0, -36, -22], [20, -48, 0, 48], [0, -19, -20, -19], [0, 18, 60, 18], [33, -28, 33, 0], [32, 27, 0, -27], [44, -10, 0, 10], [-20, 16, -20, 0], [36, 18, 0, -18], [0, 5, 52, 5], [6, 0, -6, 86], [4, 8, 0, -8], [10, 54, 10, 0], [0, 18, -28, 18], [13, 60, 13, 0], [-36, 16, -36, 0], [35, 0, -35, -4], [0, 22, -40, 22], [14, -24, 14, 0], [-6, 12, 0, -12], [-46, -18, -46, 0], [-4, 0, 4, 22], [-36, 0, 36, -20], [0, 40, 30, 40], [0, 52, 0, 0], [-36, -21, 0, 21], [0, 56, 0, 0], [0, 11, -28, 11], [-24, 0, 24, 22], [0, -18, 0, -18], [-48, 8, 0, -8], [-48, 32, 0, -32], [0, -23, -8, -23], [0, -4, -54, -4], [21, -20, 21, 0], [-20, -6, -20, 0], [6, 0, -6, -72], [16, -24, 0, 24], [24, 38, 0, -38], [20, 16, 20, 0], [0, 23, -24, 23], [32, 0, -32, 36], [-44, 40, 0, -40], [14, 0, -14, -34], [-9, 0, 9, 20], [25, 20, 25, 0], [2, 4, 0, -4], [-24, 64, -24, 0], [0, 12, -52, 12], [-28, 23, 0, -23], [12, 12, 12, 0], [20, 0, -20, -52], [16, -20, 16, 0], [-40, 9, 0, -9], [-42, 22, -42, 0], [7, 0, -7, 84], [48, -38, 0, 38], [14, 0, -14, 78], [34, 0, -34, -28], [53, 28, 53, 0], [8, 49, 0, -49], [28, 2, 0, -2], [-24, 18, 0, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6300_v_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6300_2_v_d();".
// 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_6300_2_v_d(:prec:=4)
    chi := MakeCharacter_6300_v();
    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_6300_2_v_d();".
// 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_6300_2_v_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6300_v();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![196, 0, 36, 0, 1]>,<13,R![16, -32, 32, -8, 1]>,<17,R![256, 0, 0, 0, 1]>,<23,R![196, 56, 8, -4, 1]>],Snew);
    return Vf;
end function;