
# q-expansion of newform 1875.4.a.f, downloaded from the LMFDB on 24 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 1875
weight = 4
poly_data = [42064, 394896, 579848, -371750, -718713, 125779, 257291, -12987, -36970, 517, 2512, -7, -81, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-12, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-29392108806288, -107932745742176, -19532525709704, 128798752266598, 15486244836395, -43818200690808, -3872467992151, 6092827974316, 370272399598, -404069408515, -14915595999, 12768694032, 214678293, -154777191], 9573422384000], [[-61602820198807408, -85954039584517216, 80521422793290536, 41656166382191418, -22364379485677555, -11311893366723728, 2255658862839559, 1577495446704356, -86380604207582, -107039813741365, 617153824991, 3381917381312, 18793858963, -39842017881], 2039138967792000], [[-6510547762224, -31728784810848, 18185501095208, 77070946463954, -17558372991415, -34953965143184, 3995422441227, 5882559371668, -370715223046, -450292207345, 15269104723, 15999036336, -231741561, -214678293], 9573422384000], [[59456059266864, 141903211930528, 58597577129112, -376822834415794, -46458734509185, 131454602072424, 11617403976453, -18278483922948, -1110817198794, 1212208225545, 44746787997, -38306082096, -644034879, 464331573], 9573422384000], [[-24134699084790608, -16843498300706016, 52601762509172536, -7399996906970082, -20698866949726305, 7351605029930072, 3540851664674509, -1435508388868644, -284994761799082, 114153083031385, 10590731904141, -4038749648088, -146787187287, 52774069869], 679712989264000], [[-30074258777775312, 9290772958676576, 60972601507328504, -67233637278192898, -30340650774791145, 36372690966037208, 6864964367337301, -6126575299424516, -626111554905898, 453486628821265, 24130842178949, -15446080755832, -330612632143, 198268707541], 2039138967792000], [[-9141952716611952, -21750605688750304, 15485072554819544, 19640680670547962, -6141712440159795, -6436658577450832, 849773703313171, 907965942281524, -54212075098438, -61648333032485, 1654834981079, 1994905817228, -19698303673, -24723383429], 203913896779200], [[-212680826375024, -7322318299380448, -1764109520445592, 6973604604781954, 586318929748585, -2247892398399984, -117002587471373, 310790823668868, 10887849198954, -20765197179345, -440450615677, 663666222536, 6363639039, -8146109293], 53661551784000], [[-228794009350704, -1216927897475808, -151211736871832, 1483963872871634, 108613804503785, -516374760265864, -26188788364733, 74321331141828, 2389171674634, -5108559447745, -90007487317, 166990685056, 1186234119, -2089067853], 9573422384000], [[16216865165616, 70339095459552, -13288458871592, -145774591408826, 10543225997835, 61605242598856, -684436038823, -9449742817332, -31650082226, 662126520605, 3229526593, -21662421664, -60162011, 268898217], 957342238400], [[-108688968627954224, -351247214799828448, 70922058848184008, 469758003849131554, -27127244705617415, -166468628891531184, 951266497072627, 23644160022864468, 171646433410554, -1593816456351345, -12148429797277, 51141259110536, 211929067839, -629711814493], 2039138967792000]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0], [-4, 0, 0, -1, 1, -1, -1, 1, -1, -2, 0, 0, 0, -1], [15, -1, 0, 5, 1, 0, -1, -2, 0, 0, 0, 0, 2, -1], [15, 0, 1, 7, -2, -2, 0, -2, 0, 2, -2, 2, 2, -1], [-14, -7, -3, 2, 0, -9, -2, -2, -1, 1, -1, 4, 1, -2], [12, 9, 0, 1, 3, -6, -1, 1, 2, -4, -1, 1, -3, -1], [35, -4, 0, 35, -2, -7, -1, -4, 2, -2, 0, 0, 2, -1], [-71, -11, -9, -8, -1, -5, 0, -6, -1, 0, -5, -4, 6, 4], [44, 6, -4, 7, 0, 13, -11, 8, -1, -12, 5, -4, -6, -2], [-23, 16, -17, 1, -1, -17, -13, 4, -1, -1, 7, -6, -1, -7], [52, -14, 9, -8, 6, -7, -1, 6, 6, -8, -6, 4, -2, -2], [-23, 17, 0, -25, 1, -19, 9, 4, 4, -13, 3, -13, 0, -3], [102, 50, 16, 61, -3, -34, -13, -3, -5, 3, 5, 15, 10, -7], [37, -39, 20, 55, -19, 0, 14, -20, -14, 30, -11, -4, -2, 17], [-107, 30, 28, -181, -7, -29, 2, -23, 5, 9, -5, 35, 6, 2], [201, -28, 2, 104, -2, -47, 18, 1, 13, 2, -19, -15, 1, 6], [179, -66, 40, -33, 19, 1, 2, 4, 4, -7, -11, 2, 15, -3], [-36, 0, 22, 25, -25, -4, 6, -10, 14, 1, 6, 48, 7, 6], [7, -28, 6, 122, 20, -10, 3, 15, 3, -15, -5, -40, -21, -2], [422, -156, 37, 60, 7, 58, 3, 11, -23, -7, 7, 11, -12, 18], [16, -91, 26, 196, -1, 76, 18, -24, -13, 16, -12, 52, 30, 16], [-151, -233, 19, -356, 10, -2, -1, 3, 8, 4, -11, -22, 16, -13], [-93, 43, 27, -80, -27, -61, 8, -21, -22, 23, 5, -32, 7, 8], [124, -29, 26, 294, -1, -149, -47, 9, -24, 26, 14, -15, -7, 4], [-50, -24, -33, -277, 3, 149, -22, -28, -20, 11, -19, -8, 25, 10], [144, 17, -19, 82, -20, 36, -26, 38, 18, -15, 19, 0, 21, -42], [423, 156, 13, -136, 3, -45, 0, 2, 20, -44, 11, 23, -3, 5], [145, -85, -29, -306, -30, -11, 13, -7, -34, 0, 16, 14, -28, 4], [-723, 44, 6, -91, 49, -205, 34, 14, 9, -36, 0, -65, 17, -11], [538, -48, 38, 141, 27, 62, -38, 56, -63, 8, 11, -95, -9, -38], [-367, -64, 45, 208, -26, 116, -55, 39, -17, -38, 83, 9, -33, -50], [-358, -59, 30, -15, 17, 34, 30, 5, 80, -44, -28, 91, 3, 23], [-718, 216, 13, -518, 41, -229, 7, 64, 33, -65, -15, 20, 41, -23], [542, 51, -36, -83, 8, 186, 42, 39, 12, -45, 18, -77, -32, 65], [187, 89, -23, 123, 17, -72, 14, 42, 20, -5, -59, -23, 52, -31], [-349, -92, -59, -199, -43, -91, 5, -67, -76, 39, -9, 83, 60, 42], [694, -108, -234, 723, -17, 1, 35, 62, -40, -45, 48, -146, -55, 45], [-549, -46, 91, -828, -35, 41, -7, -10, -69, 80, -3, 107, -47, 34], [-1107, -213, 153, -558, -27, -56, 67, -11, -58, 64, -47, -51, 17, 56], [1638, 274, 17, 604, 12, -132, 19, -25, 71, -69, -9, 114, -31, 21], [1610, 84, -29, 727, -132, -249, 49, -14, 80, 55, -30, 39, -46, 67], [1197, -158, 13, 351, 8, -78, -5, -19, 12, 128, -38, -69, 3, 39], [-589, -521, -31, 399, -75, 186, 1, -51, -64, 86, 3, -19, -85, 98], [-40, 289, -208, 480, -30, -22, 7, 12, -19, -19, 47, -66, -43, 100], [449, -38, -75, -325, -21, -480, -86, -137, 2, 64, -29, 198, 60, -92], [75, -379, -28, -1164, -18, 187, -62, -56, -84, 124, -38, -73, 71, -11], [-638, -228, -37, -129, 3, 216, 65, 51, -55, 76, -1, -82, 28, 57], [1545, -18, 79, 1000, -60, -16, 87, -150, 133, 111, -46, 153, 58, 47], [317, -22, 233, 1329, 19, 176, 17, 134, -106, -97, 29, 38, -67, 37], [1965, -7, 179, 807, 61, -47, 1, 67, -46, -62, -18, 157, 87, 27], [-1160, 171, 37, -1007, -16, 41, -68, 109, -102, 48, 81, -142, -102, -20], [295, 315, -183, -1707, -171, -465, 46, -176, 42, 232, -20, 56, -2, 87], [248, -25, 92, 584, 8, 426, -155, -64, -38, 64, 60, 217, -9, -108], [355, 162, -53, -2081, -94, -165, 113, -118, -12, 96, -18, 20, 60, 64], [-77, 333, 9, 553, 70, 425, -86, 117, -130, -73, 7, 57, 0, -55], [1218, -614, 116, -956, -11, 791, 61, 5, 11, 96, 22, -18, -72, -9], [-900, 822, -176, -427, 91, -624, -56, 28, 57, -154, -19, -67, -51, -48], [902, 198, -103, 1250, 20, -566, 99, -63, -179, 111, -61, -90, -21, 215], [528, 171, 10, -123, 98, 488, 45, -14, 2, 76, -130, 67, 163, -32], [880, -170, -13, 563, 67, -140, -189, -42, -106, 150, 25, -15, 43, -162], [2691, 340, 132, -222, -10, -125, -44, -63, -53, 64, 13, -11, -139, 42], [-286, 149, 25, -425, 55, 296, 139, -23, 120, -89, 47, 88, -37, -111], [-397, 1021, -72, 1334, 122, -175, 192, 38, 252, -208, -72, 57, -15, 81], [-1856, 363, 188, 369, 7, 149, 3, -70, 38, 190, -135, 24, -106, -70], [2113, -898, -132, 1861, -64, 390, 118, -3, -44, 29, 93, -250, -7, 131], [1670, -94, 88, 398, -4, 448, -273, 135, -73, -192, 117, -79, -37, -195], [-2756, -466, 30, -351, -202, 379, 391, -84, 259, -40, -71, -134, -4, 96], [-1624, -522, 112, -2559, 47, 56, 178, -101, -46, 90, -35, 110, 40, 145], [1117, 9, -23, -616, 172, 831, 18, 58, -18, 123, -108, -188, 29, 189], [2187, -530, 94, 704, -145, 599, -214, -85, -133, 266, 49, 2, 62, -157], [-643, 407, 145, -2487, -31, -549, 44, 165, 67, -122, -12, 181, -67, 36], [-2718, 416, 36, -359, -143, 304, 22, -217, 166, -30, -129, 16, 140, 9], [1067, 1016, 217, -252, 68, -827, -161, -22, -343, 162, -83, 164, 28, -101], [-424, -384, -68, 637, 37, 479, -167, -40, 227, -21, 61, 406, 113, -66], [2030, 487, -278, 972, 240, 443, 87, 67, 73, -85, -25, -439, -80, 207], [1755, 147, 113, 89, -49, 255, 2, -234, -12, 280, -208, -224, 278, 151], [337, 728, 121, -2380, -9, -557, -261, 55, -118, -109, 9, 11, 10, -164], [2379, 1569, -164, 623, -120, 82, 71, 259, 93, -76, -14, -38, -148, 85], [-1723, 1272, 253, 1299, -116, 396, 20, -302, 91, 297, -229, 171, 26, 241], [-2969, 616, -326, 450, -324, -154, 236, 4, 38, 119, 136, -389, -238, 213], [2674, 481, 187, -599, -261, -707, 59, -137, 140, 210, 26, 11, 189, 123], [-1712, 950, 282, 1377, -107, -927, -339, 39, -146, 201, -27, -59, -60, -117], [740, 440, 454, 70, -107, 513, 54, -139, -107, 53, 197, -205, -56, 292], [-990, 561, 70, -213, -8, -807, -342, 30, -1, 184, -53, -39, -45, 32], [-182, 68, 45, 117, 73, -528, 474, -175, -31, -266, -210, -92, -56, 177], [1148, 535, -72, -760, 250, -127, 18, 41, 211, 60, -188, 244, 104, -138], [-83, -685, 184, -2206, -314, 1067, -7, -53, -113, 270, 159, -62, -104, 234], [612, 1109, 510, 3548, 88, -272, -98, 189, -179, -146, -109, -186, -56, -189], [532, 673, -212, 3320, 214, 18, 333, 86, -226, -10, -44, -523, -323, 112], [-1508, -2553, 402, 923, 39, 870, 350, -52, -183, 362, -220, -280, -10, 228], [-2149, 90, 50, -4567, -14, 1538, -128, 137, 366, -237, -105, 160, 73, -337], [-2097, -1018, 323, -1294, 184, 826, -269, -65, -493, 70, -15, -39, 63, -168], [372, -1790, 772, -3042, -2, 963, 98, -263, -12, 218, -390, 5, 57, -29], [3116, 72, -402, 4889, -408, 70, 44, -110, 373, -99, 279, 27, -88, -230], [-1261, -52, -453, -762, -112, 550, -53, -374, 303, 179, -104, 219, 256, 259], [-5530, -369, -53, 797, -153, -1103, 508, -135, 154, -124, -55, -645, 41, 403], [-407, 412, 445, 3158, 55, -2624, 208, -41, -379, 301, -172, -481, -202, 192], [1714, 1415, 334, -2318, -236, 48, 93, -534, -26, 528, -186, 485, 15, 450], [1164, 2018, -120, -2938, 288, -685, -87, 33, -488, 133, -407, -442, 271, -26], [2466, 999, -1059, 2294, 109, 442, -20, 167, 15, -495, 69, -698, -247, 94], [1153, 255, -371, 2238, 67, 335, -135, -254, 151, -281, -6, -87, 246, -147], [-758, 1802, -287, 2984, 44, -408, -47, -321, -421, 155, 19, -768, -143, 259], [2382, -1204, 169, 2538, 479, 1206, 97, 119, 714, -396, -242, 838, 254, -368], [-2359, -3, -662, 2895, -133, 657, 338, -323, 545, -147, -152, 188, -219, 118], [-6152, 1527, 194, -6976, 19, 933, -112, 67, 89, -518, 162, -99, -267, 33], [-179, 645, -647, 523, 271, 754, -332, 346, 58, -317, 385, -333, -8, -293], [1484, 1279, -139, -738, 84, -1689, 263, -80, 726, -678, 251, 319, -415, -260], [1662, 1567, 438, 4195, 213, 333, 4, 105, 488, -661, 3, 642, -115, -331], [-1550, 300, 24, -2814, 333, -178, -325, 180, -475, -164, -56, -513, 283, -160], [4080, 289, 882, 8680, -400, -403, -11, -200, -378, 519, 39, 248, -15, 386], [-955, 1647, 529, 4234, 16, -1856, 448, 291, -167, -208, 65, 72, 10, 230], [-1069, -684, 372, 4260, -166, 591, 396, -81, 478, -525, 34, 202, 171, 172], [-5442, -1663, 84, -5959, -61, -1459, -237, -56, -615, 369, 130, -253, -348, 474], [-2819, -228, -549, -707, 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