
// Make newform 2275.2.a.c in Magma, downloaded from the LMFDB on 07 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2275_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_2275_2_a_c();".
// 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_2275_2_a_c();".
// 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_2275_a();"
function MakeCharacter_2275_a()
    N := 2275;
    order := 1;
    char_gens := [1002, 976, 1926];
    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_2275_a_Hecke( : Kf := Kf)
    return MakeCharacter_2275_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 := [[-1], [0], [0], [1], [0], [1], [2], [-4], [0], [-2], [0], [-2], [6], [4], [8], [-6], [-4], [-10], [-12], [4], [10], [0], [-12], [-18], [2], [-2], [4], [4], [-2], [6], [-8], [-4], [-22], [-12], [-18], [-12], [14], [-20], [-8], [-10], [-4], [-10], [-8], [26], [-2], [16], [-20], [0], [4], [-22], [6], [20], [-26], [4], [2], [-24], [6], [8], [2], [-6], [8], [-26], [12], [-24], [2], [-18], [16], [22], [-4], [26], [-14], [-4], [4], [18], [24], [-24], [6], [-2], [10], [14], [-4], [6], [36], [-6], [8], [-28], [18], [18], [26], [-16], [-8], [40], [32], [-36], [32], [-20], [2], [-14], [-16], [38], [28], [6], [-40], [-38], [4], [-14], [36], [18], [-16], [-22], [4], [-26], [18], [20], [-44], [-30], [28], [36], [-30], [44], [18], [38], [38], [12], [20], [-18], [14], [-24], [44], [30], [16], [48], [40], [18], [-42], [-50], [6], [-4], [-42], [26], [-4], [6], [32], [-4], [-10], [-32], [-42], [-30], [-44], [8], [6], [-30], [4], [-12], [-52], [48], [56], [-34], [-22], [-38], [-12], [30], [-40], [-28], [2], [40], [-8], [-50], [58], [-50], [-24], [-30], [-32], [34], [8], [-30], [0], [-42], [44], [22], [-8], [-12], [18], [-22], [16], [30], [2], [-4], [-30], [-48], [18], [36], [44], [-30], [60], [18], [66], [10], [-62], [-28], [-34], [-8], [-26], [-46], [28], [-42], [8], [36], [-30], [-60], [-50], [14], [-64], [12], [-40], [22], [32], [-30], [48], [-26], [-10], [8], [-6], [64], [24], [22], [-54], [24], [36], [48], [-38], [-20], [-4], [42], [32], [-32], [6], [38], [60], [0], [-36], [76], [4], [-2], [-14], [24], [-56], [-12], [-56], [-16], [-58], [54], [48], [42], [30], [24], [22], [76], [-42], [42], [-8], [-20], [-14], [62], [2], [4], [-34], [78], [-12], [58], [6], [76], [10], [32], [-14], [-48], [12], [14], [2], [-12], [-12], [-4], [-28], [50], [60], [48], [-2], [30], [-80], [58], [78], [-72], [26], [-28], [-38], [30], [80], [-42], [-4], [-44], [74], [30], [72], [-28], [-28], [2], [84], [-74], [-60], [-22], [4], [30], [50], [-44], [-32], [-54], [12], [-76], [66], [-46], [-44], [-6], [-42], [-56], [26], [-6], [64], [-4], [48], [26], [-34], [-14], [-64], [60], [-8], [44], [-34], [34], [22], [92], [-74], [38], [66], [-80], [-18], [-60], [-90], [12], [24], [-18], [-44], [90], [-2], [-44], [-74], [-6], [80], [20], [-58], [56], [6], [-54], [-40], [80], [48], [86], [-82], [0], [90], [36], [92], [56], [-42], [-8], [-46], [84], [48], [2], [18], [-78], [66], [26], [-48], [2], [4], [-64], [52], [-10], [20], [68], [-38], [-42], [-80], [-40], [76], [14], [-96], [-30], [28], [-34], [98], [-18], [-80], [-38], [-6], [-24], [62], [-46], [-52], [-20], [34], [66], [4], [28], [-42], [-2], [-12], [-84], [98], [28], [-66], [-74], [0], [-52], [-86], [-18], [-76], [-30], [100], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2275_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2275_2_a_c();".
// 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_2275_2_a_c(:prec:=1)
    chi := MakeCharacter_2275_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_2275_2_a_c();".
// 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_2275_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2275_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<3,R![0, 1]>],Snew);
    return Vf;
end function;