
// Make newform 76.7.c.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_76_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_76_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_76_7_c_a();".
// 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_76_7_c_a();".
// 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 := [-14, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-4, 8]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_76_c();"
function MakeCharacter_76_c()
    N := 76;
    order := 2;
    char_gens := [39, 21];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_76_c_Hecke();"
function MakeCharacter_76_c_Hecke(Kf)
    N := 76;
    order := 2;
    char_gens := [39, 21];
    char_values := [[1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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 := 7;
    raw_aps := [[0, 0], [0, 0], [27, 7], [-305, 9], [531, -70], [0, 0], [4815, 56], [-6859, 0], [20610, 0], [0, 0], [0, 0], [0, 0], [0, 0], [71315, -2016], [37575, 5551], [0, 0], [0, 0], [28531, -12915], [0, 0], [0, 0], [-192025, -19404], [0, 0], [-1131030, 0], [0, 0], [0, 0], [2060298, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-651141, -123410], [-1111545, -132986], [-1886669, 109620], [-3121749, -62755], [0, 0], [-738470, 0], [-4301750, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-4686921, -295715], [0, 0], [-9749430, 0], [7635599, -111825], [0, 0], [0, 0], [0, 0], [-1334789, -684495], [12405735, -141694], [7666119, -647885], [0, 0], [2541699, 895160], [0, 0], [-8805105, 912989], [0, 0], [28422578, 0], [-14990165, 864171], [0, 0], [16637795, 882756], [0, 0], [0, 0], [-22143969, -1167635], [-31973870, 0], [0, 0], [0, 0], [0, 0], [-31859325, 1550374], [23102099, 2046555], [58000770, 0], [39802671, 1352995], [-20078350, 0], [0, 0], [0, 0], [0, 0], [-32937381, 2798075], [30618755, -3129651], [0, 0], [0, 0], [41118282, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-13201245, 4928434], [0, 0], [-73725625, 3476466], [97838019, -293965], [27588455, -5468211], [-83703285, -3327856], [-187496622, 0], [0, 0], [-109683558, 0], [123813451, -597870], [98461890, 0], [0, 0], [0, 0], [0, 0], [-152910829, -2357775], [0, 0], [-146882205, -5221559], [0, 0], [0, 0], [-351908422, 0], [178756255, 4034646], [130453875, 8865164], [31621410, 0], [0, 0], [0, 0], [0, 0], [153441595, -9853389], [-4175865, -13469456], [470840218, 0], [-61760609, 13967415], [0, 0], [-1604485, -15247134], [-123515505, -13824349], [213487875, 10252501], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-110511829, 17830260], [-294921702, 0], [-301928758, 0], [300221991, 12568745], [-67080385, -21699279], [-1311910, 0], [366258619, 9722160], [0, 0], [0, 0], [-432896885, 1602909], [-392016969, -11549230], [342346591, -17168130], [0, 0], [0, 0], [0, 0], [457355079, 15300740], [0, 0], [385246539, 22784965], [326488735, 25913979], [0, 0], [0, 0], [0, 0], [1155584810, 0], [0, 0], [11500579, -36347220], [0, 0], [0, 0], [-547770609, -23474990], [685469035, -3679956], [0, 0], [0, 0], [0, 0], [1006033282, 0], [-1397607678, 0], [422381735, 40488714], [0, 0], [-1205005590, 0], [0, 0], [-713911070, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-907115525, -22889601]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_76_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_76_7_c_a();".
// 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_76_7_c_a(:prec:=2)
    chi := MakeCharacter_76_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    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_76_7_c_a();".
// 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_76_7_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_76_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;