
// Make newform 7488.2.a.by in Magma, downloaded from the LMFDB on 11 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7488_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_7488_2_a_by();".
// 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_7488_2_a_by();".
// 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_7488_a();"
function MakeCharacter_7488_a()
    N := 7488;
    order := 1;
    char_gens := [703, 6085, 5825, 5761];
    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_7488_a_Hecke( : Kf := Kf)
    return MakeCharacter_7488_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]];
    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], [2], [4], [-4], [-1], [-2], [0], [0], [-10], [-4], [2], [-6], [-12], [0], [6], [-12], [2], [-8], [0], [2], [-8], [-4], [2], [10], [-18], [0], [-12], [2], [6], [16], [-4], [-6], [12], [-6], [-4], [18], [8], [-8], [6], [-4], [10], [8], [18], [18], [-8], [-20], [-4], [20], [10], [14], [-24], [10], [12], [-26], [24], [22], [12], [10], [10], [12], [-6], [-16], [0], [-6], [26], [-16], [18], [12], [26], [2], [24], [-16], [26], [-24], [-16], [22], [-38], [-22], [34], [-4], [10], [0], [34], [-32], [4], [-22], [2], [-38], [-4], [-12], [-24], [-12], [12], [-24], [-8], [10], [-18], [44], [-30], [4], [18], [12], [-34], [-4], [-46], [-28], [26], [-40], [-38], [16], [2], [-22], [24], [-20], [-2], [-40], [-8], [6], [-28], [-30], [-14], [22], [12], [-24], [-34], [26], [-24], [-8], [-30], [32], [48], [-8], [-22], [10], [-30], [10], [-32], [46], [30], [8], [-22], [0], [-4], [34], [48], [-30], [46], [-44], [16], [10], [-58], [-44], [-48], [12], [-40], [48], [-30], [26], [-14], [60], [30], [52], [60], [42], [16], [-16], [26], [-6], [-18], [12], [-22], [-16], [18], [-24], [-42], [-24], [-6], [-16], [-30], [-20], [12], [58], [-30], [-16], [-42], [2], [-16], [-46], [48], [34], [-28], [52], [50], [-36], [30], [34], [2], [-38], [48], [-54], [16], [-22], [2], [36], [-34], [-52], [-4], [34], [28], [2], [54], [16], [44], [40], [-22], [-8], [-50], [0], [18], [-38], [20], [-22], [-52], [-12], [26], [54], [32], [-40], [-44], [18], [20], [-44], [-42], [-68], [24], [-70], [10], [52], [16], [-4], [28], [16], [-6], [-54], [-48], [-4], [-36], [56], [-24], [-62], [-62], [-48], [10], [6], [20], [10], [-40], [6], [-38], [48], [-28], [10], [2], [42], [28], [-22], [42], [56], [-2], [-62], [24], [-14], [-56], [-30], [-4], [44], [-22], [-38], [-60], [24], [4], [16], [82], [-40], [16], [34], [-6], [-20], [-74], [-18], [84], [66], [36], [2], [-50], [0], [-26], [28], [32], [-22], [58], [8], [4], [-4], [-6], [-44], [-46], [48], [-6], [-16], [66], [-10], [-20], [8], [26], [36], [16], [58], [78], [44], [-86], [-18], [36], [66], [-30], [-40], [-80], [64], [30], [-14], [-90], [-40], [-92], [40], [-28], [74], [-14], [2], [8], [-30], [62], [26], [-80], [26], [-36], [26], [-32], [80], [22], [-72], [-30], [66], [16], [58], [-2], [80], [12], [54], [-56], [-78], [54], [-48], [-84], [-20], [-38], [-6], [92], [-54], [-84], [-60], [64], [-42], [0], [-30], [-36], [48], [82], [62], [-54], [-30], [42], [4], [-54], [40], [16], [-60], [-54], [56], [-16], [-22], [-22], [-68], [-52], [-40], [10], [4], [-90], [52], [-30], [58], [30], [92], [-30], [50], [-32], [-46], [10], [0], [-12], [50], [38], [12], [-68], [-6], [30], [-40], [-32], [-6], [-64], [-26], [-78], [8], [4], [-70], [34], [36], [66], [-16], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7488_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7488_2_a_by();".
// 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_7488_2_a_by(:prec:=1)
    chi := MakeCharacter_7488_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_7488_2_a_by();".
// 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_7488_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7488_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<11,R![4, 1]>,<17,R![2, 1]>,<19,R![0, 1]>,<29,R![10, 1]>],Snew);
    return Vf;
end function;