
// Make newform 2175.2.a.a in Magma, downloaded from the LMFDB on 17 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2175_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_2175_2_a_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_2175_2_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2175_a();"
function MakeCharacter_2175_a()
    N := 2175;
    order := 1;
    char_gens := [1451, 2002, 901];
    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_2175_a_Hecke( : Kf := Kf)
    return MakeCharacter_2175_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 := [[-2], [1], [0], [-2], [-3], [4], [8], [0], [-1], [1], [-8], [-7], [7], [9], [-12], [9], [10], [2], [8], [-8], [-1], [-10], [9], [10], [13], [17], [4], [-12], [-5], [14], [3], [12], [-2], [5], [0], [-3], [-2], [-1], [8], [9], [10], [17], [-3], [-6], [13], [-25], [12], [-6], [3], [20], [-1], [30], [-3], [12], [23], [4], [-10], [12], [28], [-18], [14], [-6], [-17], [17], [4], [-12], [32], [-22], [3], [-5], [-26], [-15], [3], [4], [-10], [39], [-25], [8], [-28], [-10], [0], [-28], [12], [19], [20], [-6], [5], [28], [-13], [-6], [-42], [0], [-22], [-28], [20], [4], [-40], [22], [4], [-8], [8], [-47], [-46], [-10], [-23], [18], [-12], [-6], [0], [-8], [8], [34], [18], [-30], [-8], [-43], [-16], [33], [-36], [25], [-13], [-6], [-22], [39], [-8], [-18], [-45], [30], [8], [34], [20], [44], [22], [3], [42], [-40], [24], [38], [8], [-45], [-13], [-48], [-16], [-22], [50], [-40], [19], [-7], [-30], [44], [-52], [57], [4], [-22], [-37], [27], [0], [20], [-22], [42], [-12], [-6], [3], [-3], [13], [-36], [-33], [-7], [-35], [-6], [50], [-23], [-18], [-51], [-15], [20], [17], [-3], [-11], [-5], [8], [-53], [44], [-17], [4], [20], [-22], [19], [-30], [52], [-6], [-16], [-28], [37], [-12], [14], [-18], [-36], [43], [39], [-45], [-33], [33], [-50], [45], [13], [0], [-36], [0], [42], [-17], [-28], [-71], [48], [-45], [7], [28], [-33], [-32], [14], [2], [35], [-65], [-31], [13], [40], [34], [-45], [8], [-8], [19], [60], [-8], [-38], [44], [38], [60], [-36], [-40], [-63], [54], [-13], [14], [-70], [39], [20], [-72], [12], [-5], [14], [-2], [2], [48], [70], [74], [30], [2], [-47], [38], [-2], [4], [-42], [-5], [-51], [48], [-70], [-45], [-33], [-31], [-11], [22], [48], [54], [-50], [-57], [-61], [-12], [30], [-28], [52], [-51], [-23], [33], [82], [73], [72], [34], [-32], [-20], [30], [72], [3], [-51], [-8], [-1], [0], [52], [39], [50], [53], [-41], [-42], [70], [-66], [-68], [8], [-72], [-50], [-10], [-26], [24], [-35], [72], [-36], [-12], [-25], [5], [-38], [-66], [-65], [-8], [58], [12], [59], [-6], [52], [-55], [-46], [48], [-21], [-28], [58], [75], [-36], [-8], [-67], [50], [54], [52], [-62], [-1], [13], [45], [-13], [-21], [-15], [-48], [28], [77], [-62], [52], [-2], [-18], [14], [30], [-6], [-15], [72], [48], [-21], [-12], [92], [-22], [-70], [88], [-41], [18], [74], [-58], [27], [-45], [19], [55], [-48], [78], [65], [92], [-51], [40], [-2], [-53], [39], [-47], [28], [-25], [-51], [22], [88], [-76], [-92], [0], [29], [65], [3], [87], [74], [75], [-55], [47], [-73], [45], [64], [48], [-17], [60], [-43], [-52], [-13], [59], [50], [-96], [-42], [-1], [57], [-77], [-23], [-55], [-102], [98], [-26], [-10], [-7], [-92], [35], [-16], [63], [-21], [-30], [-83], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2175_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2175_2_a_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_2175_2_a_a(:prec:=1)
    chi := MakeCharacter_2175_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_2175_2_a_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_2175_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2175_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;