Properties

Label 387.5.b.c.343.5
Level $387$
Weight $5$
Character 387.343
Analytic conductor $40.004$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,5,Mod(343,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.343");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 387.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.0041757134\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 343.5
Root \(-2.75662i\) of defining polynomial
Character \(\chi\) \(=\) 387.343
Dual form 387.5.b.c.343.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75662i q^{2} +8.40105 q^{4} -21.9831i q^{5} -63.3272i q^{7} -67.2644i q^{8} +O(q^{10})\) \(q-2.75662i q^{2} +8.40105 q^{4} -21.9831i q^{5} -63.3272i q^{7} -67.2644i q^{8} -60.5989 q^{10} +1.17035 q^{11} -173.901 q^{13} -174.569 q^{14} -51.0054 q^{16} -469.315 q^{17} -27.8220i q^{19} -184.681i q^{20} -3.22622i q^{22} -79.3541 q^{23} +141.745 q^{25} +479.379i q^{26} -532.015i q^{28} +696.895i q^{29} +1190.63 q^{31} -935.628i q^{32} +1293.72i q^{34} -1392.12 q^{35} +1535.36i q^{37} -76.6946 q^{38} -1478.68 q^{40} +2455.37 q^{41} +(-1435.68 - 1165.17i) q^{43} +9.83220 q^{44} +218.749i q^{46} -1694.64 q^{47} -1609.33 q^{49} -390.737i q^{50} -1460.95 q^{52} +1990.15 q^{53} -25.7279i q^{55} -4259.66 q^{56} +1921.07 q^{58} -3563.55 q^{59} -5447.36i q^{61} -3282.13i q^{62} -3395.26 q^{64} +3822.88i q^{65} +4930.23 q^{67} -3942.74 q^{68} +3837.56i q^{70} +9660.68i q^{71} -5609.18i q^{73} +4232.39 q^{74} -233.734i q^{76} -74.1151i q^{77} -11423.8 q^{79} +1121.26i q^{80} -6768.51i q^{82} -5524.25 q^{83} +10317.0i q^{85} +(-3211.94 + 3957.62i) q^{86} -78.7231i q^{88} +2559.89i q^{89} +11012.7i q^{91} -666.658 q^{92} +4671.49i q^{94} -611.612 q^{95} +3659.79 q^{97} +4436.31i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 92 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75662i 0.689155i −0.938758 0.344577i \(-0.888022\pi\)
0.938758 0.344577i \(-0.111978\pi\)
\(3\) 0 0
\(4\) 8.40105 0.525066
\(5\) 21.9831i 0.879322i −0.898164 0.439661i \(-0.855099\pi\)
0.898164 0.439661i \(-0.144901\pi\)
\(6\) 0 0
\(7\) 63.3272i 1.29239i −0.763172 0.646196i \(-0.776359\pi\)
0.763172 0.646196i \(-0.223641\pi\)
\(8\) 67.2644i 1.05101i
\(9\) 0 0
\(10\) −60.5989 −0.605989
\(11\) 1.17035 0.00967234 0.00483617 0.999988i \(-0.498461\pi\)
0.00483617 + 0.999988i \(0.498461\pi\)
\(12\) 0 0
\(13\) −173.901 −1.02900 −0.514501 0.857490i \(-0.672023\pi\)
−0.514501 + 0.857490i \(0.672023\pi\)
\(14\) −174.569 −0.890657
\(15\) 0 0
\(16\) −51.0054 −0.199240
\(17\) −469.315 −1.62393 −0.811963 0.583709i \(-0.801601\pi\)
−0.811963 + 0.583709i \(0.801601\pi\)
\(18\) 0 0
\(19\) 27.8220i 0.0770692i −0.999257 0.0385346i \(-0.987731\pi\)
0.999257 0.0385346i \(-0.0122690\pi\)
\(20\) 184.681i 0.461702i
\(21\) 0 0
\(22\) 3.22622i 0.00666574i
\(23\) −79.3541 −0.150008 −0.0750039 0.997183i \(-0.523897\pi\)
−0.0750039 + 0.997183i \(0.523897\pi\)
\(24\) 0 0
\(25\) 141.745 0.226792
\(26\) 479.379i 0.709141i
\(27\) 0 0
\(28\) 532.015i 0.678590i
\(29\) 696.895i 0.828650i 0.910129 + 0.414325i \(0.135982\pi\)
−0.910129 + 0.414325i \(0.864018\pi\)
\(30\) 0 0
\(31\) 1190.63 1.23895 0.619477 0.785015i \(-0.287345\pi\)
0.619477 + 0.785015i \(0.287345\pi\)
\(32\) 935.628i 0.913699i
\(33\) 0 0
\(34\) 1293.72i 1.11914i
\(35\) −1392.12 −1.13643
\(36\) 0 0
\(37\) 1535.36i 1.12152i 0.827980 + 0.560758i \(0.189490\pi\)
−0.827980 + 0.560758i \(0.810510\pi\)
\(38\) −76.6946 −0.0531126
\(39\) 0 0
\(40\) −1478.68 −0.924173
\(41\) 2455.37 1.46066 0.730329 0.683095i \(-0.239367\pi\)
0.730329 + 0.683095i \(0.239367\pi\)
\(42\) 0 0
\(43\) −1435.68 1165.17i −0.776463 0.630163i
\(44\) 9.83220 0.00507861
\(45\) 0 0
\(46\) 218.749i 0.103379i
\(47\) −1694.64 −0.767154 −0.383577 0.923509i \(-0.625308\pi\)
−0.383577 + 0.923509i \(0.625308\pi\)
\(48\) 0 0
\(49\) −1609.33 −0.670275
\(50\) 390.737i 0.156295i
\(51\) 0 0
\(52\) −1460.95 −0.540294
\(53\) 1990.15 0.708489 0.354245 0.935153i \(-0.384738\pi\)
0.354245 + 0.935153i \(0.384738\pi\)
\(54\) 0 0
\(55\) 25.7279i 0.00850510i
\(56\) −4259.66 −1.35831
\(57\) 0 0
\(58\) 1921.07 0.571068
\(59\) −3563.55 −1.02371 −0.511857 0.859071i \(-0.671042\pi\)
−0.511857 + 0.859071i \(0.671042\pi\)
\(60\) 0 0
\(61\) 5447.36i 1.46395i −0.681331 0.731975i \(-0.738599\pi\)
0.681331 0.731975i \(-0.261401\pi\)
\(62\) 3282.13i 0.853831i
\(63\) 0 0
\(64\) −3395.26 −0.828920
\(65\) 3822.88i 0.904824i
\(66\) 0 0
\(67\) 4930.23 1.09829 0.549145 0.835727i \(-0.314953\pi\)
0.549145 + 0.835727i \(0.314953\pi\)
\(68\) −3942.74 −0.852668
\(69\) 0 0
\(70\) 3837.56i 0.783175i
\(71\) 9660.68i 1.91642i 0.286065 + 0.958210i \(0.407653\pi\)
−0.286065 + 0.958210i \(0.592347\pi\)
\(72\) 0 0
\(73\) 5609.18i 1.05258i −0.850306 0.526288i \(-0.823583\pi\)
0.850306 0.526288i \(-0.176417\pi\)
\(74\) 4232.39 0.772898
\(75\) 0 0
\(76\) 233.734i 0.0404664i
\(77\) 74.1151i 0.0125004i
\(78\) 0 0
\(79\) −11423.8 −1.83044 −0.915218 0.402959i \(-0.867982\pi\)
−0.915218 + 0.402959i \(0.867982\pi\)
\(80\) 1121.26i 0.175196i
\(81\) 0 0
\(82\) 6768.51i 1.00662i
\(83\) −5524.25 −0.801894 −0.400947 0.916101i \(-0.631319\pi\)
−0.400947 + 0.916101i \(0.631319\pi\)
\(84\) 0 0
\(85\) 10317.0i 1.42795i
\(86\) −3211.94 + 3957.62i −0.434280 + 0.535103i
\(87\) 0 0
\(88\) 78.7231i 0.0101657i
\(89\) 2559.89i 0.323177i 0.986858 + 0.161589i \(0.0516618\pi\)
−0.986858 + 0.161589i \(0.948338\pi\)
\(90\) 0 0
\(91\) 11012.7i 1.32987i
\(92\) −666.658 −0.0787640
\(93\) 0 0
\(94\) 4671.49i 0.528688i
\(95\) −611.612 −0.0677686
\(96\) 0 0
\(97\) 3659.79 0.388967 0.194484 0.980906i \(-0.437697\pi\)
0.194484 + 0.980906i \(0.437697\pi\)
\(98\) 4436.31i 0.461923i
\(99\) 0 0
\(100\) 1190.81 0.119081
\(101\) 10505.1 1.02981 0.514907 0.857246i \(-0.327827\pi\)
0.514907 + 0.857246i \(0.327827\pi\)
\(102\) 0 0
\(103\) 2014.90 0.189924 0.0949619 0.995481i \(-0.469727\pi\)
0.0949619 + 0.995481i \(0.469727\pi\)
\(104\) 11697.4i 1.08149i
\(105\) 0 0
\(106\) 5486.08i 0.488259i
\(107\) −8728.48 −0.762379 −0.381190 0.924497i \(-0.624486\pi\)
−0.381190 + 0.924497i \(0.624486\pi\)
\(108\) 0 0
\(109\) −9286.44 −0.781621 −0.390811 0.920471i \(-0.627805\pi\)
−0.390811 + 0.920471i \(0.627805\pi\)
\(110\) −70.9221 −0.00586133
\(111\) 0 0
\(112\) 3230.03i 0.257496i
\(113\) 16690.4i 1.30711i −0.756881 0.653553i \(-0.773278\pi\)
0.756881 0.653553i \(-0.226722\pi\)
\(114\) 0 0
\(115\) 1744.45i 0.131905i
\(116\) 5854.65i 0.435096i
\(117\) 0 0
\(118\) 9823.34i 0.705497i
\(119\) 29720.4i 2.09875i
\(120\) 0 0
\(121\) −14639.6 −0.999906
\(122\) −15016.3 −1.00889
\(123\) 0 0
\(124\) 10002.6 0.650532
\(125\) 16855.4i 1.07875i
\(126\) 0 0
\(127\) −17016.4 −1.05502 −0.527510 0.849549i \(-0.676874\pi\)
−0.527510 + 0.849549i \(0.676874\pi\)
\(128\) 5610.62i 0.342445i
\(129\) 0 0
\(130\) 10538.2 0.623564
\(131\) 13378.5i 0.779590i −0.920902 0.389795i \(-0.872546\pi\)
0.920902 0.389795i \(-0.127454\pi\)
\(132\) 0 0
\(133\) −1761.89 −0.0996035
\(134\) 13590.8i 0.756892i
\(135\) 0 0
\(136\) 31568.2i 1.70676i
\(137\) 9129.47i 0.486412i 0.969975 + 0.243206i \(0.0781991\pi\)
−0.969975 + 0.243206i \(0.921801\pi\)
\(138\) 0 0
\(139\) 28220.0 1.46059 0.730293 0.683134i \(-0.239384\pi\)
0.730293 + 0.683134i \(0.239384\pi\)
\(140\) −11695.3 −0.596700
\(141\) 0 0
\(142\) 26630.8 1.32071
\(143\) −203.526 −0.00995285
\(144\) 0 0
\(145\) 15319.9 0.728651
\(146\) −15462.4 −0.725388
\(147\) 0 0
\(148\) 12898.6i 0.588870i
\(149\) 11221.8i 0.505464i 0.967536 + 0.252732i \(0.0813291\pi\)
−0.967536 + 0.252732i \(0.918671\pi\)
\(150\) 0 0
\(151\) 37652.1i 1.65133i −0.564158 0.825667i \(-0.690799\pi\)
0.564158 0.825667i \(-0.309201\pi\)
\(152\) −1871.43 −0.0810002
\(153\) 0 0
\(154\) −204.307 −0.00861474
\(155\) 26173.8i 1.08944i
\(156\) 0 0
\(157\) 9904.16i 0.401808i 0.979611 + 0.200904i \(0.0643879\pi\)
−0.979611 + 0.200904i \(0.935612\pi\)
\(158\) 31490.9i 1.26145i
\(159\) 0 0
\(160\) −20568.0 −0.803436
\(161\) 5025.27i 0.193869i
\(162\) 0 0
\(163\) 20673.8i 0.778117i −0.921213 0.389059i \(-0.872800\pi\)
0.921213 0.389059i \(-0.127200\pi\)
\(164\) 20627.7 0.766942
\(165\) 0 0
\(166\) 15228.2i 0.552629i
\(167\) 313.151 0.0112285 0.00561424 0.999984i \(-0.498213\pi\)
0.00561424 + 0.999984i \(0.498213\pi\)
\(168\) 0 0
\(169\) 1680.65 0.0588441
\(170\) 28439.9 0.984081
\(171\) 0 0
\(172\) −12061.2 9788.67i −0.407694 0.330877i
\(173\) 5952.11 0.198874 0.0994372 0.995044i \(-0.468296\pi\)
0.0994372 + 0.995044i \(0.468296\pi\)
\(174\) 0 0
\(175\) 8976.32i 0.293104i
\(176\) −59.6944 −0.00192712
\(177\) 0 0
\(178\) 7056.63 0.222719
\(179\) 39430.1i 1.23062i −0.788287 0.615308i \(-0.789032\pi\)
0.788287 0.615308i \(-0.210968\pi\)
\(180\) 0 0
\(181\) 29726.1 0.907361 0.453681 0.891164i \(-0.350111\pi\)
0.453681 + 0.891164i \(0.350111\pi\)
\(182\) 30357.7 0.916488
\(183\) 0 0
\(184\) 5337.71i 0.157659i
\(185\) 33751.8 0.986174
\(186\) 0 0
\(187\) −549.264 −0.0157072
\(188\) −14236.8 −0.402806
\(189\) 0 0
\(190\) 1685.98i 0.0467031i
\(191\) 35347.9i 0.968940i −0.874808 0.484470i \(-0.839012\pi\)
0.874808 0.484470i \(-0.160988\pi\)
\(192\) 0 0
\(193\) −74147.3 −1.99058 −0.995292 0.0969181i \(-0.969102\pi\)
−0.995292 + 0.0969181i \(0.969102\pi\)
\(194\) 10088.6i 0.268058i
\(195\) 0 0
\(196\) −13520.1 −0.351938
\(197\) −33924.2 −0.874133 −0.437066 0.899429i \(-0.643983\pi\)
−0.437066 + 0.899429i \(0.643983\pi\)
\(198\) 0 0
\(199\) 20327.0i 0.513296i −0.966505 0.256648i \(-0.917382\pi\)
0.966505 0.256648i \(-0.0826182\pi\)
\(200\) 9534.40i 0.238360i
\(201\) 0 0
\(202\) 28958.6i 0.709701i
\(203\) 44132.4 1.07094
\(204\) 0 0
\(205\) 53976.5i 1.28439i
\(206\) 5554.31i 0.130887i
\(207\) 0 0
\(208\) 8869.91 0.205018
\(209\) 32.5615i 0.000745439i
\(210\) 0 0
\(211\) 6957.00i 0.156263i −0.996943 0.0781316i \(-0.975105\pi\)
0.996943 0.0781316i \(-0.0248954\pi\)
\(212\) 16719.3 0.372004
\(213\) 0 0
\(214\) 24061.1i 0.525397i
\(215\) −25614.1 + 31560.6i −0.554117 + 0.682761i
\(216\) 0 0
\(217\) 75399.5i 1.60121i
\(218\) 25599.2i 0.538658i
\(219\) 0 0
\(220\) 216.142i 0.00446574i
\(221\) 81614.4 1.67102
\(222\) 0 0
\(223\) 2228.35i 0.0448099i −0.999749 0.0224050i \(-0.992868\pi\)
0.999749 0.0224050i \(-0.00713232\pi\)
\(224\) −59250.7 −1.18086
\(225\) 0 0
\(226\) −46009.1 −0.900798
\(227\) 61573.4i 1.19493i −0.801896 0.597464i \(-0.796175\pi\)
0.801896 0.597464i \(-0.203825\pi\)
\(228\) 0 0
\(229\) −40330.0 −0.769054 −0.384527 0.923114i \(-0.625635\pi\)
−0.384527 + 0.923114i \(0.625635\pi\)
\(230\) 4808.77 0.0909031
\(231\) 0 0
\(232\) 46876.2 0.870917
\(233\) 74382.7i 1.37013i −0.728484 0.685063i \(-0.759775\pi\)
0.728484 0.685063i \(-0.240225\pi\)
\(234\) 0 0
\(235\) 37253.4i 0.674576i
\(236\) −29937.5 −0.537517
\(237\) 0 0
\(238\) 81927.7 1.44636
\(239\) −11874.5 −0.207883 −0.103942 0.994583i \(-0.533146\pi\)
−0.103942 + 0.994583i \(0.533146\pi\)
\(240\) 0 0
\(241\) 23536.3i 0.405233i 0.979258 + 0.202616i \(0.0649444\pi\)
−0.979258 + 0.202616i \(0.935056\pi\)
\(242\) 40355.9i 0.689090i
\(243\) 0 0
\(244\) 45763.6i 0.768670i
\(245\) 35378.0i 0.589388i
\(246\) 0 0
\(247\) 4838.28i 0.0793043i
\(248\) 80087.3i 1.30215i
\(249\) 0 0
\(250\) −46463.9 −0.743423
\(251\) 54757.4 0.869151 0.434575 0.900635i \(-0.356898\pi\)
0.434575 + 0.900635i \(0.356898\pi\)
\(252\) 0 0
\(253\) −92.8723 −0.00145093
\(254\) 46907.8i 0.727072i
\(255\) 0 0
\(256\) −69790.4 −1.06492
\(257\) 57759.9i 0.874501i 0.899340 + 0.437250i \(0.144048\pi\)
−0.899340 + 0.437250i \(0.855952\pi\)
\(258\) 0 0
\(259\) 97229.7 1.44944
\(260\) 32116.2i 0.475092i
\(261\) 0 0
\(262\) −36879.5 −0.537258
\(263\) 7567.24i 0.109402i −0.998503 0.0547011i \(-0.982579\pi\)
0.998503 0.0547011i \(-0.0174206\pi\)
\(264\) 0 0
\(265\) 43749.5i 0.622991i
\(266\) 4856.85i 0.0686422i
\(267\) 0 0
\(268\) 41419.1 0.576675
\(269\) 39923.2 0.551723 0.275861 0.961197i \(-0.411037\pi\)
0.275861 + 0.961197i \(0.411037\pi\)
\(270\) 0 0
\(271\) 119787. 1.63107 0.815533 0.578710i \(-0.196444\pi\)
0.815533 + 0.578710i \(0.196444\pi\)
\(272\) 23937.6 0.323551
\(273\) 0 0
\(274\) 25166.5 0.335213
\(275\) 165.892 0.00219361
\(276\) 0 0
\(277\) 37526.7i 0.489081i 0.969639 + 0.244540i \(0.0786371\pi\)
−0.969639 + 0.244540i \(0.921363\pi\)
\(278\) 77791.7i 1.00657i
\(279\) 0 0
\(280\) 93640.4i 1.19439i
\(281\) 29935.1 0.379113 0.189556 0.981870i \(-0.439295\pi\)
0.189556 + 0.981870i \(0.439295\pi\)
\(282\) 0 0
\(283\) −54918.3 −0.685716 −0.342858 0.939387i \(-0.611395\pi\)
−0.342858 + 0.939387i \(0.611395\pi\)
\(284\) 81159.9i 1.00625i
\(285\) 0 0
\(286\) 561.043i 0.00685905i
\(287\) 155491.i 1.88774i
\(288\) 0 0
\(289\) 136735. 1.63713
\(290\) 42231.1i 0.502153i
\(291\) 0 0
\(292\) 47123.0i 0.552672i
\(293\) 155740. 1.81412 0.907060 0.421002i \(-0.138322\pi\)
0.907060 + 0.421002i \(0.138322\pi\)
\(294\) 0 0
\(295\) 78337.6i 0.900174i
\(296\) 103275. 1.17872
\(297\) 0 0
\(298\) 30934.2 0.348343
\(299\) 13799.8 0.154358
\(300\) 0 0
\(301\) −73787.1 + 90917.5i −0.814418 + 1.00349i
\(302\) −103792. −1.13802
\(303\) 0 0
\(304\) 1419.07i 0.0153553i
\(305\) −119750. −1.28728
\(306\) 0 0
\(307\) 70887.6 0.752131 0.376065 0.926593i \(-0.377277\pi\)
0.376065 + 0.926593i \(0.377277\pi\)
\(308\) 622.645i 0.00656356i
\(309\) 0 0
\(310\) −72151.2 −0.750793
\(311\) 27909.5 0.288557 0.144278 0.989537i \(-0.453914\pi\)
0.144278 + 0.989537i \(0.453914\pi\)
\(312\) 0 0
\(313\) 50613.7i 0.516630i −0.966061 0.258315i \(-0.916833\pi\)
0.966061 0.258315i \(-0.0831672\pi\)
\(314\) 27302.0 0.276908
\(315\) 0 0
\(316\) −95971.5 −0.961099
\(317\) −111161. −1.10620 −0.553099 0.833116i \(-0.686555\pi\)
−0.553099 + 0.833116i \(0.686555\pi\)
\(318\) 0 0
\(319\) 815.613i 0.00801498i
\(320\) 74638.1i 0.728888i
\(321\) 0 0
\(322\) 13852.8 0.133606
\(323\) 13057.3i 0.125155i
\(324\) 0 0
\(325\) −24649.7 −0.233369
\(326\) −56989.8 −0.536243
\(327\) 0 0
\(328\) 165159.i 1.53516i
\(329\) 107317.i 0.991463i
\(330\) 0 0
\(331\) 199722.i 1.82293i 0.411376 + 0.911466i \(0.365048\pi\)
−0.411376 + 0.911466i \(0.634952\pi\)
\(332\) −46409.5 −0.421047
\(333\) 0 0
\(334\) 863.238i 0.00773816i
\(335\) 108381.i 0.965752i
\(336\) 0 0
\(337\) −19032.7 −0.167588 −0.0837938 0.996483i \(-0.526704\pi\)
−0.0837938 + 0.996483i \(0.526704\pi\)
\(338\) 4632.90i 0.0405527i
\(339\) 0 0
\(340\) 86673.4i 0.749770i
\(341\) 1393.46 0.0119836
\(342\) 0 0
\(343\) 50134.2i 0.426134i
\(344\) −78374.6 + 96570.1i −0.662306 + 0.816067i
\(345\) 0 0
\(346\) 16407.7i 0.137055i
\(347\) 13299.5i 0.110453i −0.998474 0.0552263i \(-0.982412\pi\)
0.998474 0.0552263i \(-0.0175880\pi\)
\(348\) 0 0
\(349\) 207713.i 1.70535i −0.522442 0.852675i \(-0.674979\pi\)
0.522442 0.852675i \(-0.325021\pi\)
\(350\) −24744.3 −0.201994
\(351\) 0 0
\(352\) 1095.01i 0.00883760i
\(353\) −69576.5 −0.558358 −0.279179 0.960239i \(-0.590062\pi\)
−0.279179 + 0.960239i \(0.590062\pi\)
\(354\) 0 0
\(355\) 212371. 1.68515
\(356\) 21505.7i 0.169689i
\(357\) 0 0
\(358\) −108694. −0.848084
\(359\) 165095. 1.28099 0.640495 0.767963i \(-0.278729\pi\)
0.640495 + 0.767963i \(0.278729\pi\)
\(360\) 0 0
\(361\) 129547. 0.994060
\(362\) 81943.4i 0.625312i
\(363\) 0 0
\(364\) 92518.1i 0.698271i
\(365\) −123307. −0.925554
\(366\) 0 0
\(367\) −58660.7 −0.435527 −0.217764 0.976002i \(-0.569876\pi\)
−0.217764 + 0.976002i \(0.569876\pi\)
\(368\) 4047.49 0.0298876
\(369\) 0 0
\(370\) 93040.8i 0.679626i
\(371\) 126030.i 0.915645i
\(372\) 0 0
\(373\) 179031.i 1.28680i 0.765530 + 0.643401i \(0.222477\pi\)
−0.765530 + 0.643401i \(0.777523\pi\)
\(374\) 1514.11i 0.0108247i
\(375\) 0 0
\(376\) 113989.i 0.806284i
\(377\) 121191.i 0.852682i
\(378\) 0 0
\(379\) −47513.9 −0.330783 −0.165391 0.986228i \(-0.552889\pi\)
−0.165391 + 0.986228i \(0.552889\pi\)
\(380\) −5138.19 −0.0355830
\(381\) 0 0
\(382\) −97440.7 −0.667749
\(383\) 59722.7i 0.407138i 0.979061 + 0.203569i \(0.0652541\pi\)
−0.979061 + 0.203569i \(0.934746\pi\)
\(384\) 0 0
\(385\) −1629.28 −0.0109919
\(386\) 204396.i 1.37182i
\(387\) 0 0
\(388\) 30746.1 0.204233
\(389\) 254851.i 1.68417i −0.539343 0.842086i \(-0.681327\pi\)
0.539343 0.842086i \(-0.318673\pi\)
\(390\) 0 0
\(391\) 37242.0 0.243601
\(392\) 108251.i 0.704463i
\(393\) 0 0
\(394\) 93516.1i 0.602413i
\(395\) 251129.i 1.60954i
\(396\) 0 0
\(397\) −79062.9 −0.501639 −0.250820 0.968034i \(-0.580700\pi\)
−0.250820 + 0.968034i \(0.580700\pi\)
\(398\) −56033.9 −0.353741
\(399\) 0 0
\(400\) −7229.77 −0.0451861
\(401\) 196389. 1.22132 0.610658 0.791894i \(-0.290905\pi\)
0.610658 + 0.791894i \(0.290905\pi\)
\(402\) 0 0
\(403\) −207053. −1.27489
\(404\) 88254.1 0.540720
\(405\) 0 0
\(406\) 121656.i 0.738043i
\(407\) 1796.91i 0.0108477i
\(408\) 0 0
\(409\) 115044.i 0.687729i 0.939019 + 0.343864i \(0.111736\pi\)
−0.939019 + 0.343864i \(0.888264\pi\)
\(410\) −148793. −0.885143
\(411\) 0 0
\(412\) 16927.3 0.0997225
\(413\) 225669.i 1.32304i
\(414\) 0 0
\(415\) 121440.i 0.705123i
\(416\) 162707.i 0.940198i
\(417\) 0 0
\(418\) −89.7597 −0.000513723
\(419\) 281568.i 1.60382i 0.597446 + 0.801909i \(0.296182\pi\)
−0.597446 + 0.801909i \(0.703818\pi\)
\(420\) 0 0
\(421\) 182603.i 1.03025i −0.857115 0.515125i \(-0.827745\pi\)
0.857115 0.515125i \(-0.172255\pi\)
\(422\) −19177.8 −0.107690
\(423\) 0 0
\(424\) 133866.i 0.744627i
\(425\) −66523.0 −0.368294
\(426\) 0 0
\(427\) −344966. −1.89200
\(428\) −73328.4 −0.400299
\(429\) 0 0
\(430\) 87000.6 + 70608.2i 0.470528 + 0.381872i
\(431\) −295412. −1.59028 −0.795139 0.606427i \(-0.792602\pi\)
−0.795139 + 0.606427i \(0.792602\pi\)
\(432\) 0 0
\(433\) 315628.i 1.68345i 0.539908 + 0.841724i \(0.318459\pi\)
−0.539908 + 0.841724i \(0.681541\pi\)
\(434\) −207848. −1.10348
\(435\) 0 0
\(436\) −78015.9 −0.410403
\(437\) 2207.79i 0.0115610i
\(438\) 0 0
\(439\) 289252. 1.50088 0.750442 0.660937i \(-0.229841\pi\)
0.750442 + 0.660937i \(0.229841\pi\)
\(440\) −1730.57 −0.00893892
\(441\) 0 0
\(442\) 224980.i 1.15159i
\(443\) 13612.9 0.0693655 0.0346827 0.999398i \(-0.488958\pi\)
0.0346827 + 0.999398i \(0.488958\pi\)
\(444\) 0 0
\(445\) 56274.1 0.284177
\(446\) −6142.72 −0.0308810
\(447\) 0 0
\(448\) 215012.i 1.07129i
\(449\) 164810.i 0.817506i 0.912645 + 0.408753i \(0.134036\pi\)
−0.912645 + 0.408753i \(0.865964\pi\)
\(450\) 0 0
\(451\) 2873.65 0.0141280
\(452\) 140217.i 0.686316i
\(453\) 0 0
\(454\) −169734. −0.823490
\(455\) 242092. 1.16939
\(456\) 0 0
\(457\) 157048.i 0.751971i −0.926626 0.375985i \(-0.877304\pi\)
0.926626 0.375985i \(-0.122696\pi\)
\(458\) 111174.i 0.529997i
\(459\) 0 0
\(460\) 14655.2i 0.0692589i
\(461\) −332439. −1.56426 −0.782131 0.623114i \(-0.785867\pi\)
−0.782131 + 0.623114i \(0.785867\pi\)
\(462\) 0 0
\(463\) 197697.i 0.922226i −0.887341 0.461113i \(-0.847450\pi\)
0.887341 0.461113i \(-0.152550\pi\)
\(464\) 35545.4i 0.165100i
\(465\) 0 0
\(466\) −205045. −0.944228
\(467\) 121639.i 0.557749i −0.960328 0.278874i \(-0.910039\pi\)
0.960328 0.278874i \(-0.0899613\pi\)
\(468\) 0 0
\(469\) 312217.i 1.41942i
\(470\) 102694. 0.464887
\(471\) 0 0
\(472\) 239700.i 1.07593i
\(473\) −1680.25 1363.66i −0.00751021 0.00609515i
\(474\) 0 0
\(475\) 3943.63i 0.0174787i
\(476\) 249682.i 1.10198i
\(477\) 0 0
\(478\) 32733.5i 0.143264i
\(479\) 141700. 0.617586 0.308793 0.951129i \(-0.400075\pi\)
0.308793 + 0.951129i \(0.400075\pi\)
\(480\) 0 0
\(481\) 267000.i 1.15404i
\(482\) 64880.6 0.279268
\(483\) 0 0
\(484\) −122988. −0.525017
\(485\) 80453.4i 0.342027i
\(486\) 0 0
\(487\) 194637. 0.820668 0.410334 0.911935i \(-0.365412\pi\)
0.410334 + 0.911935i \(0.365412\pi\)
\(488\) −366413. −1.53862
\(489\) 0 0
\(490\) 97523.6 0.406179
\(491\) 143694.i 0.596040i −0.954560 0.298020i \(-0.903674\pi\)
0.954560 0.298020i \(-0.0963263\pi\)
\(492\) 0 0
\(493\) 327063.i 1.34567i
\(494\) 13337.3 0.0546529
\(495\) 0 0
\(496\) −60728.9 −0.246849
\(497\) 611783. 2.47676
\(498\) 0 0
\(499\) 328853.i 1.32069i 0.750962 + 0.660345i \(0.229590\pi\)
−0.750962 + 0.660345i \(0.770410\pi\)
\(500\) 141603.i 0.566413i
\(501\) 0 0
\(502\) 150945.i 0.598979i
\(503\) 278129.i 1.09929i −0.835399 0.549643i \(-0.814764\pi\)
0.835399 0.549643i \(-0.185236\pi\)
\(504\) 0 0
\(505\) 230935.i 0.905538i
\(506\) 256.014i 0.000999912i
\(507\) 0 0
\(508\) −142956. −0.553955
\(509\) −53485.5 −0.206443 −0.103222 0.994658i \(-0.532915\pi\)
−0.103222 + 0.994658i \(0.532915\pi\)
\(510\) 0 0
\(511\) −355213. −1.36034
\(512\) 102616.i 0.391448i
\(513\) 0 0
\(514\) 159222. 0.602666
\(515\) 44293.7i 0.167004i
\(516\) 0 0
\(517\) −1983.33 −0.00742017
\(518\) 268025.i 0.998886i
\(519\) 0 0
\(520\) 257144. 0.950976
\(521\) 43791.8i 0.161331i 0.996741 + 0.0806655i \(0.0257045\pi\)
−0.996741 + 0.0806655i \(0.974295\pi\)
\(522\) 0 0
\(523\) 104920.i 0.383578i 0.981436 + 0.191789i \(0.0614290\pi\)
−0.981436 + 0.191789i \(0.938571\pi\)
\(524\) 112394.i 0.409336i
\(525\) 0 0
\(526\) −20860.0 −0.0753950
\(527\) −558782. −2.01197
\(528\) 0 0
\(529\) −273544. −0.977498
\(530\) −120601. −0.429337
\(531\) 0 0
\(532\) −14801.7 −0.0522984
\(533\) −426991. −1.50302
\(534\) 0 0
\(535\) 191879.i 0.670377i
\(536\) 331629.i 1.15431i
\(537\) 0 0
\(538\) 110053.i 0.380222i
\(539\) −1883.48 −0.00648312
\(540\) 0 0
\(541\) −60893.0 −0.208052 −0.104026 0.994575i \(-0.533173\pi\)
−0.104026 + 0.994575i \(0.533173\pi\)
\(542\) 330208.i 1.12406i
\(543\) 0 0
\(544\) 439104.i 1.48378i
\(545\) 204144.i 0.687297i
\(546\) 0 0
\(547\) 77111.4 0.257717 0.128859 0.991663i \(-0.458869\pi\)
0.128859 + 0.991663i \(0.458869\pi\)
\(548\) 76697.1i 0.255398i
\(549\) 0 0
\(550\) 457.300i 0.00151174i
\(551\) 19389.0 0.0638634
\(552\) 0 0
\(553\) 723434.i 2.36564i
\(554\) 103447. 0.337052
\(555\) 0 0
\(556\) 237078. 0.766904
\(557\) 496754. 1.60115 0.800573 0.599235i \(-0.204529\pi\)
0.800573 + 0.599235i \(0.204529\pi\)
\(558\) 0 0
\(559\) 249666. + 202625.i 0.798981 + 0.648439i
\(560\) 71006.0 0.226422
\(561\) 0 0
\(562\) 82519.7i 0.261267i
\(563\) 510115. 1.60935 0.804676 0.593714i \(-0.202339\pi\)
0.804676 + 0.593714i \(0.202339\pi\)
\(564\) 0 0
\(565\) −366907. −1.14937
\(566\) 151389.i 0.472564i
\(567\) 0 0
\(568\) 649820. 2.01417
\(569\) 18540.2 0.0572650 0.0286325 0.999590i \(-0.490885\pi\)
0.0286325 + 0.999590i \(0.490885\pi\)
\(570\) 0 0
\(571\) 145098.i 0.445029i −0.974929 0.222515i \(-0.928573\pi\)
0.974929 0.222515i \(-0.0714266\pi\)
\(572\) −1709.83 −0.00522590
\(573\) 0 0
\(574\) −428631. −1.30095
\(575\) −11248.1 −0.0340206
\(576\) 0 0
\(577\) 492406.i 1.47901i −0.673150 0.739506i \(-0.735059\pi\)
0.673150 0.739506i \(-0.264941\pi\)
\(578\) 376927.i 1.12824i
\(579\) 0 0
\(580\) 128703. 0.382590
\(581\) 349835.i 1.03636i
\(582\) 0 0
\(583\) 2329.17 0.00685275
\(584\) −377298. −1.10626
\(585\) 0 0
\(586\) 429317.i 1.25021i
\(587\) 113204.i 0.328539i 0.986416 + 0.164269i \(0.0525266\pi\)
−0.986416 + 0.164269i \(0.947473\pi\)
\(588\) 0 0
\(589\) 33125.8i 0.0954852i
\(590\) 215947. 0.620359
\(591\) 0 0
\(592\) 78311.5i 0.223451i
\(593\) 378754.i 1.07708i −0.842600 0.538540i \(-0.818976\pi\)
0.842600 0.538540i \(-0.181024\pi\)
\(594\) 0 0
\(595\) 653344. 1.84548
\(596\) 94275.0i 0.265402i
\(597\) 0 0
\(598\) 38040.7i 0.106377i
\(599\) 70600.5 0.196768 0.0983840 0.995149i \(-0.468633\pi\)
0.0983840 + 0.995149i \(0.468633\pi\)
\(600\) 0 0
\(601\) 576111.i 1.59499i 0.603327 + 0.797494i \(0.293841\pi\)
−0.603327 + 0.797494i \(0.706159\pi\)
\(602\) 250625. + 203403.i 0.691562 + 0.561260i
\(603\) 0 0
\(604\) 316317.i 0.867059i
\(605\) 321824.i 0.879240i
\(606\) 0 0
\(607\) 179183.i 0.486317i 0.969987 + 0.243159i \(0.0781835\pi\)
−0.969987 + 0.243159i \(0.921816\pi\)
\(608\) −26031.0 −0.0704180
\(609\) 0 0
\(610\) 330104.i 0.887138i
\(611\) 294701. 0.789403
\(612\) 0 0
\(613\) 527970. 1.40504 0.702519 0.711665i \(-0.252059\pi\)
0.702519 + 0.711665i \(0.252059\pi\)
\(614\) 195410.i 0.518334i
\(615\) 0 0
\(616\) −4985.31 −0.0131380
\(617\) −21720.1 −0.0570547 −0.0285274 0.999593i \(-0.509082\pi\)
−0.0285274 + 0.999593i \(0.509082\pi\)
\(618\) 0 0
\(619\) −64840.3 −0.169225 −0.0846124 0.996414i \(-0.526965\pi\)
−0.0846124 + 0.996414i \(0.526965\pi\)
\(620\) 219887.i 0.572028i
\(621\) 0 0
\(622\) 76935.9i 0.198860i
\(623\) 162110. 0.417671
\(624\) 0 0
\(625\) −281943. −0.721773
\(626\) −139523. −0.356038
\(627\) 0 0
\(628\) 83205.4i 0.210976i
\(629\) 720564.i 1.82126i
\(630\) 0 0
\(631\) 639737.i 1.60673i −0.595488 0.803364i \(-0.703041\pi\)
0.595488 0.803364i \(-0.296959\pi\)
\(632\) 768412.i 1.92380i
\(633\) 0 0
\(634\) 306428.i 0.762341i
\(635\) 374073.i 0.927702i
\(636\) 0 0
\(637\) 279865. 0.689714
\(638\) 2248.33 0.00552356
\(639\) 0 0
\(640\) −123339. −0.301120
\(641\) 92614.4i 0.225404i 0.993629 + 0.112702i \(0.0359506\pi\)
−0.993629 + 0.112702i \(0.964049\pi\)
\(642\) 0 0
\(643\) 517514. 1.25170 0.625850 0.779943i \(-0.284752\pi\)
0.625850 + 0.779943i \(0.284752\pi\)
\(644\) 42217.6i 0.101794i
\(645\) 0 0
\(646\) 35993.9 0.0862509
\(647\) 320760.i 0.766253i −0.923696 0.383126i \(-0.874847\pi\)
0.923696 0.383126i \(-0.125153\pi\)
\(648\) 0 0
\(649\) −4170.61 −0.00990170
\(650\) 67949.7i 0.160828i
\(651\) 0 0
\(652\) 173682.i 0.408563i
\(653\) 271078.i 0.635724i 0.948137 + 0.317862i \(0.102965\pi\)
−0.948137 + 0.317862i \(0.897035\pi\)
\(654\) 0 0
\(655\) −294101. −0.685511
\(656\) −125237. −0.291022
\(657\) 0 0
\(658\) 295832. 0.683271
\(659\) 297067. 0.684044 0.342022 0.939692i \(-0.388888\pi\)
0.342022 + 0.939692i \(0.388888\pi\)
\(660\) 0 0
\(661\) 91628.4 0.209714 0.104857 0.994487i \(-0.466562\pi\)
0.104857 + 0.994487i \(0.466562\pi\)
\(662\) 550558. 1.25628
\(663\) 0 0
\(664\) 371585.i 0.842796i
\(665\) 38731.7i 0.0875836i
\(666\) 0 0
\(667\) 55301.5i 0.124304i
\(668\) 2630.80 0.00589569
\(669\) 0 0
\(670\) −298766. −0.665552
\(671\) 6375.33i 0.0141598i
\(672\) 0 0
\(673\) 601565.i 1.32817i 0.747658 + 0.664084i \(0.231178\pi\)
−0.747658 + 0.664084i \(0.768822\pi\)
\(674\) 52466.0i 0.115494i
\(675\) 0 0
\(676\) 14119.2 0.0308970
\(677\) 211198.i 0.460800i 0.973096 + 0.230400i \(0.0740034\pi\)
−0.973096 + 0.230400i \(0.925997\pi\)
\(678\) 0 0
\(679\) 231764.i 0.502698i
\(680\) 693965. 1.50079
\(681\) 0 0
\(682\) 3841.25i 0.00825854i
\(683\) 260659. 0.558768 0.279384 0.960179i \(-0.409870\pi\)
0.279384 + 0.960179i \(0.409870\pi\)
\(684\) 0 0
\(685\) 200694. 0.427713
\(686\) −138201. −0.293672
\(687\) 0 0
\(688\) 73227.5 + 59430.1i 0.154702 + 0.125554i
\(689\) −346089. −0.729037
\(690\) 0 0
\(691\) 670752.i 1.40477i −0.711797 0.702386i \(-0.752118\pi\)
0.711797 0.702386i \(-0.247882\pi\)
\(692\) 50004.0 0.104422
\(693\) 0 0
\(694\) −36661.6 −0.0761190
\(695\) 620361.i 1.28433i
\(696\) 0 0
\(697\) −1.15234e6 −2.37200
\(698\) −572586. −1.17525
\(699\) 0 0
\(700\) 75410.5i 0.153899i
\(701\) −104357. −0.212366 −0.106183 0.994347i \(-0.533863\pi\)
−0.106183 + 0.994347i \(0.533863\pi\)
\(702\) 0 0
\(703\) 42716.6 0.0864343
\(704\) −3973.65 −0.00801759
\(705\) 0 0
\(706\) 191796.i 0.384795i
\(707\) 665260.i 1.33092i
\(708\) 0 0
\(709\) 135186. 0.268930 0.134465 0.990918i \(-0.457068\pi\)
0.134465 + 0.990918i \(0.457068\pi\)
\(710\) 585426.i 1.16133i
\(711\) 0 0
\(712\) 172189. 0.339661
\(713\) −94481.8 −0.185853
\(714\) 0 0
\(715\) 4474.12i 0.00875176i
\(716\) 331255.i 0.646154i
\(717\) 0 0
\(718\) 455105.i 0.882800i
\(719\) −99461.8 −0.192397 −0.0961985 0.995362i \(-0.530668\pi\)
−0.0961985 + 0.995362i \(0.530668\pi\)
\(720\) 0 0
\(721\) 127598.i 0.245456i
\(722\) 357112.i 0.685061i
\(723\) 0 0
\(724\) 249730. 0.476424
\(725\) 98781.4i 0.187931i
\(726\) 0 0
\(727\) 440731.i 0.833882i −0.908933 0.416941i \(-0.863102\pi\)
0.908933 0.416941i \(-0.136898\pi\)
\(728\) 740761. 1.39770
\(729\) 0 0
\(730\) 339910.i 0.637850i
\(731\) 673785. + 546832.i 1.26092 + 1.02334i
\(732\) 0 0
\(733\) 353122.i 0.657229i 0.944464 + 0.328615i \(0.106582\pi\)
−0.944464 + 0.328615i \(0.893418\pi\)
\(734\) 161705.i 0.300146i
\(735\) 0 0
\(736\) 74245.9i 0.137062i
\(737\) 5770.11 0.0106230
\(738\) 0 0
\(739\) 511160.i 0.935983i −0.883733 0.467992i \(-0.844978\pi\)
0.883733 0.467992i \(-0.155022\pi\)
\(740\) 283551. 0.517806
\(741\) 0 0
\(742\) −347418. −0.631021
\(743\) 162933.i 0.295141i −0.989052 0.147571i \(-0.952855\pi\)
0.989052 0.147571i \(-0.0471454\pi\)
\(744\) 0 0
\(745\) 246690. 0.444466
\(746\) 493521. 0.886805
\(747\) 0 0
\(748\) −4614.39 −0.00824729
\(749\) 552750.i 0.985292i
\(750\) 0 0
\(751\) 476066.i 0.844087i −0.906576 0.422044i \(-0.861313\pi\)
0.906576 0.422044i \(-0.138687\pi\)
\(752\) 86436.0 0.152848
\(753\) 0 0
\(754\) −334077. −0.587630
\(755\) −827708. −1.45205
\(756\) 0 0
\(757\) 513071.i 0.895335i −0.894200 0.447667i \(-0.852255\pi\)
0.894200 0.447667i \(-0.147745\pi\)
\(758\) 130978.i 0.227960i
\(759\) 0 0
\(760\) 41139.7i 0.0712253i
\(761\) 132174.i 0.228233i 0.993467 + 0.114116i \(0.0364036\pi\)
−0.993467 + 0.114116i \(0.963596\pi\)
\(762\) 0 0
\(763\) 588084.i 1.01016i
\(764\) 296960.i 0.508757i
\(765\) 0 0
\(766\) 164633. 0.280581
\(767\) 619705. 1.05340
\(768\) 0 0
\(769\) −549403. −0.929048 −0.464524 0.885561i \(-0.653775\pi\)
−0.464524 + 0.885561i \(0.653775\pi\)
\(770\) 4491.30i 0.00757513i
\(771\) 0 0
\(772\) −622915. −1.04519
\(773\) 357526.i 0.598340i −0.954200 0.299170i \(-0.903290\pi\)
0.954200 0.299170i \(-0.0967098\pi\)
\(774\) 0 0
\(775\) 168767. 0.280985
\(776\) 246174.i 0.408807i
\(777\) 0 0
\(778\) −702526. −1.16066
\(779\) 68313.2i 0.112572i
\(780\) 0 0
\(781\) 11306.4i 0.0185363i
\(782\) 102662.i 0.167879i
\(783\) 0 0
\(784\) 82084.6 0.133546
\(785\) 217724. 0.353319
\(786\) 0 0
\(787\) 11905.5 0.0192221 0.00961103 0.999954i \(-0.496941\pi\)
0.00961103 + 0.999954i \(0.496941\pi\)
\(788\) −284999. −0.458977
\(789\) 0 0
\(790\) 692267. 1.10922
\(791\) −1.05696e6 −1.68929
\(792\) 0 0
\(793\) 947303.i 1.50641i
\(794\) 217946.i 0.345707i
\(795\) 0 0
\(796\) 170769.i 0.269514i
\(797\) 54534.3 0.0858526 0.0429263 0.999078i \(-0.486332\pi\)
0.0429263 + 0.999078i \(0.486332\pi\)
\(798\) 0 0
\(799\) 795321. 1.24580
\(800\) 132621.i 0.207220i
\(801\) 0 0
\(802\) 541370.i 0.841676i
\(803\) 6564.72i 0.0101809i
\(804\) 0 0
\(805\) 110471. 0.170473
\(806\) 570766.i 0.878593i
\(807\) 0 0
\(808\) 706621.i 1.08234i
\(809\) −1.14580e6 −1.75070 −0.875350 0.483489i \(-0.839369\pi\)
−0.875350 + 0.483489i \(0.839369\pi\)
\(810\) 0 0
\(811\) 56880.8i 0.0864816i −0.999065 0.0432408i \(-0.986232\pi\)
0.999065 0.0432408i \(-0.0137683\pi\)
\(812\) 370758. 0.562314
\(813\) 0 0
\(814\) 4953.39 0.00747573
\(815\) −454473. −0.684216
\(816\) 0 0
\(817\) −32417.4 + 39943.4i −0.0485662 + 0.0598413i
\(818\) 317132. 0.473952
\(819\) 0 0
\(820\) 453459.i 0.674389i
\(821\) −200304. −0.297169 −0.148584 0.988900i \(-0.547472\pi\)
−0.148584 + 0.988900i \(0.547472\pi\)
\(822\) 0 0
\(823\) −1.07152e6 −1.58198 −0.790989 0.611830i \(-0.790434\pi\)
−0.790989 + 0.611830i \(0.790434\pi\)
\(824\) 135531.i 0.199611i
\(825\) 0 0
\(826\) 622084. 0.911778
\(827\) −767100. −1.12161 −0.560803 0.827949i \(-0.689508\pi\)
−0.560803 + 0.827949i \(0.689508\pi\)
\(828\) 0 0
\(829\) 358430.i 0.521549i −0.965400 0.260774i \(-0.916022\pi\)
0.965400 0.260774i \(-0.0839778\pi\)
\(830\) 334763. 0.485939
\(831\) 0 0
\(832\) 590439. 0.852960
\(833\) 755282. 1.08848
\(834\) 0 0
\(835\) 6884.02i 0.00987345i
\(836\) 273.551i 0.000391405i
\(837\) 0 0
\(838\) 776176. 1.10528
\(839\) 93647.6i 0.133037i −0.997785 0.0665185i \(-0.978811\pi\)
0.997785 0.0665185i \(-0.0211891\pi\)
\(840\) 0 0
\(841\) 221619. 0.313339
\(842\) −503366. −0.710002
\(843\) 0 0
\(844\) 58446.1i 0.0820485i
\(845\) 36945.8i 0.0517430i
\(846\) 0 0
\(847\) 927086.i 1.29227i
\(848\) −101508. −0.141159
\(849\) 0 0
\(850\) 183379.i 0.253811i
\(851\) 121837.i 0.168236i
\(852\) 0 0
\(853\) 368391. 0.506304 0.253152 0.967427i \(-0.418533\pi\)
0.253152 + 0.967427i \(0.418533\pi\)
\(854\) 950939.i 1.30388i
\(855\) 0 0
\(856\) 587116.i 0.801266i
\(857\) −851457. −1.15931 −0.579657 0.814860i \(-0.696814\pi\)
−0.579657 + 0.814860i \(0.696814\pi\)
\(858\) 0 0
\(859\) 236916.i 0.321076i 0.987030 + 0.160538i \(0.0513229\pi\)
−0.987030 + 0.160538i \(0.948677\pi\)
\(860\) −215185. + 265142.i −0.290948 + 0.358494i
\(861\) 0 0
\(862\) 814337.i 1.09595i
\(863\) 22247.8i 0.0298721i 0.999888 + 0.0149361i \(0.00475447\pi\)
−0.999888 + 0.0149361i \(0.995246\pi\)
\(864\) 0 0
\(865\) 130846.i 0.174875i
\(866\) 870066. 1.16016
\(867\) 0 0
\(868\) 633435.i 0.840742i
\(869\) −13369.8 −0.0177046
\(870\) 0 0
\(871\) −857373. −1.13014
\(872\) 624647.i 0.821489i
\(873\) 0 0
\(874\) 6086.03 0.00796730
\(875\) −1.06740e6 −1.39416
\(876\) 0 0
\(877\) 2045.84 0.00265995 0.00132998 0.999999i \(-0.499577\pi\)
0.00132998 + 0.999999i \(0.499577\pi\)
\(878\) 797357.i 1.03434i
\(879\) 0 0
\(880\) 1312.26i 0.00169456i
\(881\) 196655. 0.253369 0.126684 0.991943i \(-0.459566\pi\)
0.126684 + 0.991943i \(0.459566\pi\)
\(882\) 0 0
\(883\) 59837.7 0.0767456 0.0383728 0.999263i \(-0.487783\pi\)
0.0383728 + 0.999263i \(0.487783\pi\)
\(884\) 685647. 0.877397
\(885\) 0 0
\(886\) 37525.6i 0.0478035i
\(887\) 577234.i 0.733676i 0.930285 + 0.366838i \(0.119560\pi\)
−0.930285 + 0.366838i \(0.880440\pi\)
\(888\) 0 0
\(889\) 1.07760e6i 1.36350i
\(890\) 155126.i 0.195842i
\(891\) 0 0
\(892\) 18720.5i 0.0235282i
\(893\) 47148.3i 0.0591239i
\(894\) 0 0
\(895\) −866795. −1.08211
\(896\) −355305. −0.442573
\(897\) 0 0
\(898\) 454318. 0.563388
\(899\) 829747.i 1.02666i
\(900\) 0 0
\(901\) −934005. −1.15053
\(902\) 7921.55i 0.00973637i
\(903\) 0 0
\(904\) −1.12267e6 −1.37378
\(905\) 653470.i 0.797863i
\(906\) 0 0
\(907\) −543357. −0.660496 −0.330248 0.943894i \(-0.607132\pi\)
−0.330248 + 0.943894i \(0.607132\pi\)
\(908\) 517282.i 0.627416i
\(909\) 0 0
\(910\) 667356.i 0.805888i
\(911\) 119998.i 0.144590i 0.997383 + 0.0722948i \(0.0230322\pi\)
−0.997383 + 0.0722948i \(0.976968\pi\)
\(912\) 0 0
\(913\) −6465.32 −0.00775619
\(914\) −432922. −0.518224
\(915\) 0 0
\(916\) −338814. −0.403804
\(917\) −847225. −1.00753
\(918\) 0 0
\(919\) 198081. 0.234537 0.117268 0.993100i \(-0.462586\pi\)
0.117268 + 0.993100i \(0.462586\pi\)
\(920\) 117339. 0.138633
\(921\) 0 0
\(922\) 916406.i 1.07802i
\(923\) 1.68000e6i 1.97200i
\(924\) 0 0
\(925\) 217629.i 0.254351i
\(926\) −544974. −0.635556
\(927\) 0 0
\(928\) 652034. 0.757137
\(929\) 1.28224e6i 1.48573i 0.669443 + 0.742864i \(0.266533\pi\)
−0.669443 + 0.742864i \(0.733467\pi\)
\(930\) 0 0
\(931\) 44774.7i 0.0516575i
\(932\) 624893.i 0.719406i
\(933\) 0 0
\(934\) −335312. −0.384375
\(935\) 12074.5i 0.0138117i
\(936\) 0 0
\(937\) 1.00382e6i 1.14334i 0.820483 + 0.571671i \(0.193705\pi\)
−0.820483 + 0.571671i \(0.806295\pi\)
\(938\) −860664. −0.978201
\(939\) 0 0
\(940\) 312968.i 0.354197i
\(941\) −893241. −1.00876 −0.504382 0.863481i \(-0.668280\pi\)
−0.504382 + 0.863481i \(0.668280\pi\)
\(942\) 0 0
\(943\) −194844. −0.219110
\(944\) 181760. 0.203965
\(945\) 0 0
\(946\) −3759.10 + 4631.81i −0.00420050 + 0.00517569i
\(947\) 66944.6 0.0746475 0.0373238 0.999303i \(-0.488117\pi\)
0.0373238 + 0.999303i \(0.488117\pi\)
\(948\) 0 0
\(949\) 975443.i 1.08310i
\(950\) −10871.1 −0.0120455
\(951\) 0 0
\(952\) 1.99912e6 2.20580
\(953\) 863195.i 0.950436i 0.879868 + 0.475218i \(0.157631\pi\)
−0.879868 + 0.475218i \(0.842369\pi\)
\(954\) 0 0
\(955\) −777055. −0.852011
\(956\) −99758.3 −0.109152
\(957\) 0 0
\(958\) 390612.i 0.425612i
\(959\) 578143. 0.628635
\(960\) 0 0
\(961\) 494090. 0.535007
\(962\) −736018. −0.795313
\(963\) 0 0
\(964\) 197730.i 0.212774i
\(965\) 1.62998e6i 1.75037i
\(966\) 0 0
\(967\) −1.55611e6 −1.66413 −0.832065 0.554677i \(-0.812842\pi\)
−0.832065 + 0.554677i \(0.812842\pi\)
\(968\) 984726.i 1.05091i
\(969\) 0 0
\(970\) −221779. −0.235710
\(971\) 734703. 0.779243 0.389622 0.920975i \(-0.372606\pi\)
0.389622 + 0.920975i \(0.372606\pi\)
\(972\) 0 0
\(973\) 1.78709e6i 1.88765i
\(974\) 536540.i 0.565567i
\(975\) 0 0
\(976\) 277845.i 0.291677i
\(977\) 857114. 0.897944 0.448972 0.893546i \(-0.351790\pi\)
0.448972 + 0.893546i \(0.351790\pi\)
\(978\) 0 0
\(979\) 2995.97i 0.00312588i
\(980\) 297212.i 0.309467i
\(981\) 0 0
\(982\) −396109. −0.410764
\(983\) 116039.i 0.120087i 0.998196 + 0.0600437i \(0.0191240\pi\)
−0.998196 + 0.0600437i \(0.980876\pi\)
\(984\) 0 0
\(985\) 745758.i 0.768645i
\(986\) −901588. −0.927372
\(987\) 0 0
\(988\) 40646.6i 0.0416400i
\(989\) 113927. + 92461.2i 0.116475 + 0.0945294i
\(990\) 0 0
\(991\) 1.12664e6i 1.14720i −0.819136 0.573600i \(-0.805546\pi\)
0.819136 0.573600i \(-0.194454\pi\)
\(992\) 1.11399e6i 1.13203i
\(993\) 0 0
\(994\) 1.68645e6i 1.70687i
\(995\) −446851. −0.451353
\(996\) 0 0
\(997\) 1.20881e6i 1.21610i −0.793900 0.608048i \(-0.791953\pi\)
0.793900 0.608048i \(-0.208047\pi\)
\(998\) 906523. 0.910160
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 387.5.b.c.343.5 12
3.2 odd 2 43.5.b.b.42.8 yes 12
12.11 even 2 688.5.b.d.257.3 12
43.42 odd 2 inner 387.5.b.c.343.8 12
129.128 even 2 43.5.b.b.42.5 12
516.515 odd 2 688.5.b.d.257.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.5.b.b.42.5 12 129.128 even 2
43.5.b.b.42.8 yes 12 3.2 odd 2
387.5.b.c.343.5 12 1.1 even 1 trivial
387.5.b.c.343.8 12 43.42 odd 2 inner
688.5.b.d.257.3 12 12.11 even 2
688.5.b.d.257.10 12 516.515 odd 2