
// Make newform 7360.2.a.p in Magma, downloaded from the LMFDB on 20 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_7360_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_7360_2_a_p();".
// 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_7360_2_a_p();".
// 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_7360_a();"
function MakeCharacter_7360_a()
    N := 7360;
    order := 1;
    char_gens := [1151, 5061, 4417, 6721];
    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_7360_a_Hecke( : Kf := Kf)
    return MakeCharacter_7360_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], [1], [0], [4], [2], [2], [-4], [-1], [2], [-8], [-10], [-6], [-12], [0], [6], [0], [-10], [-4], [16], [-6], [8], [-4], [10], [2], [18], [8], [-4], [-10], [10], [-16], [-16], [-6], [-16], [6], [8], [14], [-16], [-16], [-6], [-24], [14], [-8], [2], [18], [-8], [0], [8], [-28], [-2], [26], [-24], [10], [-4], [18], [8], [-14], [16], [-30], [-6], [4], [6], [-32], [8], [10], [10], [-16], [-22], [32], [-14], [-14], [-8], [-16], [-26], [-36], [-24], [-18], [34], [-38], [10], [-20], [22], [24], [2], [16], [0], [34], [-22], [-30], [16], [28], [-32], [32], [40], [-8], [-16], [34], [18], [44], [2], [24], [6], [4], [18], [-20], [2], [24], [34], [24], [-6], [8], [-2], [18], [-12], [-32], [10], [20], [-24], [2], [-36], [-18], [-14], [6], [24], [-16], [-26], [6], [40], [-32], [-34], [-40], [-24], [-32], [14], [42], [10], [14], [4], [-26], [-6], [0], [2], [40], [28], [34], [-32], [-38], [10], [-8], [-24], [-22], [-30], [-8], [-32], [20], [40], [40], [-30], [34], [-18], [48], [-38], [-40], [4], [-46], [48], [-8], [-46], [-22], [26], [-52], [2], [48], [34], [0], [-6], [8], [2], [40], [14], [-24], [12], [-14], [-22], [16], [-18], [2], [-44], [10], [-64], [18], [40], [52], [-6], [28], [50], [50], [-34], [58], [-24], [6], [0], [50], [50], [-12], [26], [-24], [24], [-54], [-8], [-62], [18], [56], [-12], [0], [-46], [40], [50], [72], [-30], [-62], [0], [34], [24], [-48], [2], [-30], [16], [72], [24], [-6], [-44], [32], [74], [-12], [32], [10], [30], [-48], [0], [20], [-24], [-24], [-70], [-14], [-24], [-16], [12], [-44], [40], [22], [-78], [-24], [-54], [34], [-8], [62], [28], [18], [-22], [24], [12], [58], [54], [66], [-60], [6], [-6], [4], [-30], [50], [-76], [18], [16], [2], [80], [-8], [10], [58], [-44], [-24], [32], [48], [-34], [32], [8], [-30], [14], [-64], [50], [-18], [-52], [10], [-28], [-86], [-18], [-72], [-86], [40], [-8], [42], [6], [24], [24], [12], [-62], [0], [86], [48], [-22], [-8], [-10], [-54], [-40], [-48], [26], [-24], [48], [42], [-30], [20], [26], [26], [40], [-38], [42], [-60], [72], [24], [46], [18], [-14], [16], [56], [20], [-48], [-66], [42], [-38], [88], [10], [42], [82], [72], [70], [-72], [-78], [-80], [-24], [22], [56], [10], [-78], [-64], [-34], [90], [0], [-4], [50], [-16], [62], [10], [-48], [4], [-40], [26], [18], [16], [-62], [8], [-8], [16], [46], [32], [-62], [80], [-40], [-54], [-22], [-22], [38], [-6], [64], [-46], [-20], [-8], [0], [50], [-52], [16], [50], [-78], [16], [88], [8], [-54], [48], [-30], [-28], [2], [-38], [18], [64], [90], [-30], [-56], [-74], [34], [-12], [-56], [-62], [-54], [-84], [60], [82], [-30], [-96], [-56], [-30], [0], [54], [-10], [24], [80], [-70], [2], [4], [-54], [-80], [-48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7360_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7360_2_a_p();".
// 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_7360_2_a_p(:prec:=1)
    chi := MakeCharacter_7360_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_7360_2_a_p();".
// 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_7360_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;