
// Make newform 8400.2.a.ci in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_8400_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_8400_2_a_ci();".
// 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_8400_2_a_ci();".
// 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_8400_a();"
function MakeCharacter_8400_a()
    N := 8400;
    order := 1;
    char_gens := [3151, 2101, 2801, 5377, 3601];
    v := [1, 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_8400_a_Hecke( : Kf := Kf)
    return MakeCharacter_8400_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], [1], [0], [1], [-6], [1], [-3], [4], [-3], [3], [-5], [10], [9], [-1], [0], [-9], [-9], [11], [-4], [12], [10], [10], [9], [-6], [-14], [-12], [17], [18], [8], [6], [-10], [0], [0], [16], [-9], [-2], [10], [5], [18], [12], [-18], [2], [15], [22], [15], [-8], [25], [-19], [3], [-22], [-12], [0], [2], [27], [-15], [9], [-18], [-20], [10], [-24], [-4], [0], [2], [18], [10], [27], [19], [13], [18], [17], [30], [3], [-37], [-8], [-35], [-24], [-6], [-29], [-12], [38], [21], [-34], [15], [-38], [-41], [24], [24], [25], [12], [32], [-3], [-24], [2], [-42], [19], [6], [-24], [-21], [20], [20], [17], [18], [21], [-6], [13], [34], [33], [18], [-9], [-4], [-28], [-2], [18], [-26], [22], [-18], [14], [6], [-18], [-42], [38], [37], [12], [18], [34], [3], [-40], [-30], [-7], [-29], [37], [-9], [-8], [-14], [18], [-4], [48], [26], [6], [0], [34], [-42], [-34], [6], [29], [-36], [19], [-18], [40], [-48], [-32], [33], [5], [54], [53], [-15], [-20], [-27], [-2], [24], [-24], [-24], [-28], [12], [-42], [24], [16], [-26], [-25], [-60], [-18], [-43], [39], [1], [28], [-3], [-44], [33], [-55], [35], [-10], [-21], [22], [15], [-24], [30], [-20], [26], [41], [-12], [4], [-12], [-56], [-42], [60], [-57], [5], [10], [30], [-36], [-6], [37], [31], [-10], [27], [48], [-17], [42], [-54], [-62], [10], [30], [26], [-39], [-6], [56], [-28], [-18], [-21], [-33], [32], [55], [-30], [-16], [60], [-22], [-30], [24], [-28], [-30], [40], [-20], [4], [66], [-4], [-12], [-22], [6], [6], [30], [12], [-2], [-61], [2], [-15], [-36], [-7], [12], [4], [72], [-32], [21], [-21], [-10], [-12], [-30], [2], [-58], [-42], [34], [56], [-36], [-19], [34], [-63], [22], [75], [-63], [-13], [-69], [41], [-7], [4], [4], [76], [-76], [-48], [-46], [-25], [51], [0], [16], [-12], [-37], [32], [39], [13], [-42], [-11], [-30], [-30], [60], [-36], [24], [34], [-24], [-35], [30], [-33], [26], [-68], [63], [-14], [24], [52], [-53], [-18], [65], [-63], [-38], [18], [-33], [54], [56], [-15], [38], [-12], [-69], [-8], [-12], [-14], [58], [54], [-16], [-30], [-28], [25], [-34], [0], [27], [-46], [-69], [-35], [39], [-11], [12], [68], [69], [14], [29], [10], [-66], [72], [-8], [-15], [-48], [-94], [29], [48], [-18], [-32], [-11], [-15], [-13], [-58], [-69], [-6], [-63], [-66], [-63], [4], [-42], [-48], [84], [-70], [-17], [-6], [-16], [17], [-72], [16], [-3], [-15], [-5], [-92], [-45], [-24], [64], [27], [4], [30], [24], [20], [0], [-68], [96], [-2], [-86], [-52], [66], [-73], [-36], [6], [-58], [15], [-26], [-23], [51], [-92], [30], [-13], [90], [32], [39], [-48], [13], [-50], [-102], [86], [24], [-8], [90], [-84], [-71], [13], [36], [-57], [-25], [105], [78], [-99], [-11], [-9], [21], [16], [0], [-30], [78], [-8], [18], [50], [6], [43], [54], [49], [42], [59], [32], [-76], [-56], [81], [-12], [92], [-93], [104], [96], [-10], [-66], [56], [77], [17], [-108], [42], [-45], [7], [-30], [110], [33], [-5], [-66], [-20], [58], [-60], [110], [38], [10], [-86], [3], [-54], [52], [86], [-54], [72], [-7], [36], [-14], [54], [-35], [108], [-18], [40], [-75], [-8], [60], [47], [96], [-40], [-63], [28], [-92], [-83], [18], [-1], [-6], [48], [89], [2], [51], [31], [-89], [-105], [56], [84], [44], [76], [-30], [-33], [64], [70], [-4], [84], [84], [-56], [15], [25], [82], [120], [83], [15], [-1], [-26], [-41], [72], [27], [80], [-9], [-14], [54], [48], [-30], [-46], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8400_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8400_2_a_ci();".
// 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_8400_2_a_ci(:prec:=1)
    chi := MakeCharacter_8400_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(3833) - 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_8400_2_a_ci();".
// 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_8400_2_a_ci( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![6, 1]>,<13,R![-1, 1]>,<17,R![3, 1]>,<19,R![-4, 1]>,<23,R![3, 1]>],Snew);
    return Vf;
end function;