
// Make newform 350.4.c.e in Magma, downloaded from the LMFDB on 14 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_c_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_350_4_c_e();".
// 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_350_4_c_e();".
// 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_350_c();"
function MakeCharacter_350_c()
    N := 350;
    order := 2;
    char_gens := [127, 101];
    v := [1, 2];
    // 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_350_c_Hecke();"
function MakeCharacter_350_c_Hecke( : Kf := false)
    N := 350;
    order := 2;
    char_gens := [127, 101];
    char_values := [[-1, 0], [1, 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 := 4;
    raw_aps := [[0, 2], [0, 3], [0, 0], [0, 7], [37, 0], [0, 18], [0, 121], [45, 0], [0, -72], [-210, 0], [-148, 0], [0, 136], [227, 0], [0, -32], [0, 346], [0, -452], [140, 0], [-578, 0], [0, 801], [-478, 0], [0, -247], [-610, 0], [0, 653], [1115, 0], [0, -614], [-768, 0], [0, 148], [0, 531], [1650, 0], [0, 2063], [0, 1706], [1712, 0], [0, -2949], [1685, 0], [-420, 0], [3032, 0], [0, -1084], [0, -517], [0, -2844], [0, 818], [-3405, 0], [3442, 0], [2862, 0], [0, -2827], [0, -1634], [1270, 0], [-2393, 0], [0, 5338], [0, -3284], [5150, 0], [0, -3642], [-1540, 0], [3187, 0], [-2883, 0], [0, 5326], [0, -382], [-5250, 0], [-3378, 0], [0, -5994], [-7858, 0], [0, -7387], [0, -5102], [0, 9341], [8682, 0], [0, 998], [0, -4084], [11857, 0], [0, -1219], [0, -6589], [5240, 0], [0, -10482], [-11750, 0], [0, 56], [0, -6892], [6465, 0], [0, 11958], [-9920, 0], [0, 4886], [-1763, 0], [9645, 0], [-4215, 0], [5012, 0], [-7548, 0], [0, -7177], [16580, 0], [0, -557], [10935, 0], [0, -7489], [-13718, 0], [0, 12008], [0, -5324], [16290, 0], [0, -8954], [5892, 0], [16620, 0], [0, 8238], [20870, 0], [-1213, 0], [0, -16637], [7302, 0], [0, -5589], [0, 2276], [0, -8332], [21555, 0], [10332, 0], [0, -7549], [0, 15451], [0, 20073], [-1800, 0], [-17403, 0], [0, 21596], [0, 10938], [0, -11294], [-10700, 0], [17432, 0], [-8638, 0], [0, 25288], [0, 17576], [0, -11692], [11465, 0], [2182, 0], [0, 13858], [0, -7824], [0, -13267], [-1553, 0], [-9368, 0], [3740, 0], [-20640, 0], [0, 10686], [0, -35552], [5660, 0], [0, 18528], [11222, 0], [0, 22466], [-20973, 0], [-20155, 0], [0, 5388], [0, 32396], [0, -23394], [30570, 0], [-21868, 0], [-14448, 0], [0, 15958], [0, -17999], [32140, 0], [-23870, 0], [0, -24442], [0, 15381], [-46995, 0], [0, -16652], [0, -25424], [21242, 0], [0, -25887], [0, -24434], [0, -45164], [-43668, 0], [-27770, 0], [2930, 0], [0, -17849], [47552, 0], [0, 31876], [0, 26673], [0, -31714], [-26013, 0], [0, 29571], [0, -21052], [45772, 0], [0, -5894]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_4_c_e();".
// 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_350_4_c_e(:prec:=2)
    chi := MakeCharacter_350_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_350_4_c_e();".
// 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_350_4_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![9, 0, 1]>,<11,R![-37, 1]>],Snew);
    return Vf;
end function;