
// Make newform 7350.2.a.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_k();".
// 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_7350_2_a_k();".
// 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_7350_a();"
function MakeCharacter_7350_a()
    N := 7350;
    order := 1;
    char_gens := [4901, 1177, 2551];
    v := [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_7350_a_Hecke(Kf)
    return MakeCharacter_7350_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+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 := 2;
    raw_aps := [[-1], [-1], [0], [0], [0], [-5], [6], [7], [-6], [0], [-8], [-1], [0], [8], [-6], [-6], [6], [1], [-13], [12], [-5], [-7], [18], [6], [7], [0], [13], [-18], [-7], [-6], [11], [-6], [-18], [-11], [12], [-1], [1], [-1], [6], [-18], [12], [10], [24], [-10], [-12], [-11], [-13], [-17], [-12], [-5], [24], [-12], [-5], [24], [24], [0], [24], [-20], [-19], [-18], [-5], [24], [28], [-18], [-14], [-18], [-19], [-34], [18], [-26], [24], [30], [-8], [-13], [11], [6], [-12], [-14], [-36], [-5], [0], [5], [-18], [34], [-5], [-42], [6], [29], [-6], [-31], [-30], [-18], [-40], [-6], [5], [0], [0], [-42], [-32], [-43], [-28], [-12], [12], [-36], [5], [-14], [30], [6], [-30], [-35], [43], [38], [48], [-20], [-19], [-24], [-11], [12], [-36], [0], [-35], [-1], [-42], [24], [7], [-30], [-7], [36], [37], [-41], [-25], [30], [-25], [29], [-18], [10], [0], [-23], [-54], [-30], [-11], [0], [5], [-24], [7], [18], [-26], [-18], [4], [30], [47], [12], [-25], [6], [-7], [30], [8], [36], [10], [6], [42], [-12], [41], [0], [0], [30], [32], [13], [-22], [-12], [24], [-11], [-6], [-19], [-5], [12], [20], [-60], [-29], [-26], [32], [30], [-34], [18], [42], [-6], [5], [4], [-22], [-24], [-29], [-36], [53], [-6], [12], [-12], [-13], [-10], [-42], [-6], [-42], [-65], [-14], [-26], [-30], [48], [-23], [42], [48], [7], [11], [-30], [29], [-72], [-24], [-35], [53], [-6], [12], [18], [-22], [31], [6], [-16], [18], [35], [6], [66], [40], [30], [41], [52], [-40], [60], [49], [-6], [67], [-30], [18], [-48], [-18], [52], [67], [-49], [36], [-6], [-56], [-66], [-31], [48], [-43], [36], [0], [19], [-30], [18], [-58], [55], [12], [34], [35], [12], [-41], [-62], [-24], [37], [-60], [66], [68], [6], [13], [-52], [55], [23], [-5], [7], [-54], [2], [11], [18], [-24], [41], [-30], [73], [-71], [36], [29], [-6], [37], [36], [24], [-42], [30], [-18], [35], [-30], [-8], [78], [-36], [43], [22], [60], [41], [-24], [83], [-7], [18], [31], [54], [47], [-36], [-48], [-12], [-19], [-42], [-38], [24], [-54], [19], [42], [-47], [-10], [-54], [-61], [30], [-17], [44], [-17], [-18], [0], [53], [30], [40], [-18], [8], [48], [47], [72], [22], [1], [-46], [48], [-54], [89], [42], [-18], [7], [11], [12], [36], [-83], [2], [6], [85], [-85], [-30], [12], [90], [-6], [18], [11], [-78], [-48], [66], [-23], [-49], [-18], [-13], [83], [-12], [-44], [-6], [30], [67], [38], [-24], [-60], [-83], [84], [19], [18], [6], [-16], [-18], [40], [0], [-1], [22], [-43], [-84], [-55], [78], [54], [-35], [-30], [-31], [91], [-84], [23], [0], [22], [-30], [23], [-48], [66], [85], [-43], [-24], [-77], [30], [-71], [-72], [-78], [47], [38], [-36], [-54], [-17], [6], [66], [24], [-74], [-12], [-12], [70], [30], [48], [-12], [-17], [24], [58], [24], [95], [-90], [-23], [-102], [101], [-85], [17], [31], [-36], [66], [-73], [84], [43], [60], [-17], [-72], [37], [34], [-13], [-18], [-54], [-30], [89], [-42], [-7], [42], [22], [-48], [-7], [-40], [-78], [-109], [-17], [-43], [65], [-54], [108], [-65], [-52], [72], [-36], [41]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7350_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_k();".
// 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_7350_2_a_k(:prec:=1)
    chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_k();".
// 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_7350_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![5, 1]>,<17,R![-6, 1]>,<19,R![-7, 1]>,<23,R![6, 1]>,<31,R![8, 1]>],Snew);
    return Vf;
end function;