Properties

Label 325.6.b.h.274.15
Level $325$
Weight $6$
Character 325.274
Analytic conductor $52.125$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(274,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not 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: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.15
Root \(6.50844i\) of defining polynomial
Character \(\chi\) \(=\) 325.274
Dual form 325.6.b.h.274.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.50844i q^{2} -17.3551i q^{3} -24.3767 q^{4} +130.310 q^{6} -149.455i q^{7} +57.2388i q^{8} -58.2000 q^{9} +O(q^{10})\) \(q+7.50844i q^{2} -17.3551i q^{3} -24.3767 q^{4} +130.310 q^{6} -149.455i q^{7} +57.2388i q^{8} -58.2000 q^{9} -492.989 q^{11} +423.061i q^{12} +169.000i q^{13} +1122.18 q^{14} -1209.83 q^{16} +1824.24i q^{17} -436.991i q^{18} -424.035 q^{19} -2593.81 q^{21} -3701.58i q^{22} +3660.46i q^{23} +993.387 q^{24} -1268.93 q^{26} -3207.23i q^{27} +3643.23i q^{28} +8193.74 q^{29} +8113.24 q^{31} -7252.30i q^{32} +8555.87i q^{33} -13697.2 q^{34} +1418.73 q^{36} -6297.67i q^{37} -3183.84i q^{38} +2933.01 q^{39} +4298.36 q^{41} -19475.5i q^{42} -5822.12i q^{43} +12017.5 q^{44} -27484.4 q^{46} -8502.70i q^{47} +20996.7i q^{48} -5529.90 q^{49} +31660.0 q^{51} -4119.67i q^{52} -7049.74i q^{53} +24081.3 q^{54} +8554.65 q^{56} +7359.17i q^{57} +61522.2i q^{58} +21509.2 q^{59} +35743.1 q^{61} +60917.8i q^{62} +8698.30i q^{63} +15738.9 q^{64} -64241.3 q^{66} +15239.3i q^{67} -44469.1i q^{68} +63527.7 q^{69} +69104.1 q^{71} -3331.30i q^{72} +4134.60i q^{73} +47285.7 q^{74} +10336.6 q^{76} +73679.8i q^{77} +22022.4i q^{78} +16065.3 q^{79} -69804.4 q^{81} +32274.0i q^{82} +115739. i q^{83} +63228.7 q^{84} +43715.0 q^{86} -142203. i q^{87} -28218.1i q^{88} +35250.6 q^{89} +25258.0 q^{91} -89230.1i q^{92} -140806. i q^{93} +63842.1 q^{94} -125864. q^{96} +32169.5i q^{97} -41520.9i q^{98} +28691.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.50844i 1.32732i 0.748035 + 0.663659i \(0.230997\pi\)
−0.748035 + 0.663659i \(0.769003\pi\)
\(3\) − 17.3551i − 1.11333i −0.830737 0.556666i \(-0.812081\pi\)
0.830737 0.556666i \(-0.187919\pi\)
\(4\) −24.3767 −0.761773
\(5\) 0 0
\(6\) 130.310 1.47774
\(7\) − 149.455i − 1.15283i −0.817156 0.576416i \(-0.804451\pi\)
0.817156 0.576416i \(-0.195549\pi\)
\(8\) 57.2388i 0.316203i
\(9\) −58.2000 −0.239506
\(10\) 0 0
\(11\) −492.989 −1.22844 −0.614222 0.789133i \(-0.710530\pi\)
−0.614222 + 0.789133i \(0.710530\pi\)
\(12\) 423.061i 0.848106i
\(13\) 169.000i 0.277350i
\(14\) 1122.18 1.53018
\(15\) 0 0
\(16\) −1209.83 −1.18147
\(17\) 1824.24i 1.53095i 0.643466 + 0.765475i \(0.277496\pi\)
−0.643466 + 0.765475i \(0.722504\pi\)
\(18\) − 436.991i − 0.317901i
\(19\) −424.035 −0.269474 −0.134737 0.990881i \(-0.543019\pi\)
−0.134737 + 0.990881i \(0.543019\pi\)
\(20\) 0 0
\(21\) −2593.81 −1.28348
\(22\) − 3701.58i − 1.63054i
\(23\) 3660.46i 1.44283i 0.692501 + 0.721417i \(0.256509\pi\)
−0.692501 + 0.721417i \(0.743491\pi\)
\(24\) 993.387 0.352039
\(25\) 0 0
\(26\) −1268.93 −0.368132
\(27\) − 3207.23i − 0.846682i
\(28\) 3643.23i 0.878197i
\(29\) 8193.74 1.80920 0.904601 0.426259i \(-0.140169\pi\)
0.904601 + 0.426259i \(0.140169\pi\)
\(30\) 0 0
\(31\) 8113.24 1.51632 0.758158 0.652070i \(-0.226099\pi\)
0.758158 + 0.652070i \(0.226099\pi\)
\(32\) − 7252.30i − 1.25199i
\(33\) 8555.87i 1.36766i
\(34\) −13697.2 −2.03206
\(35\) 0 0
\(36\) 1418.73 0.182449
\(37\) − 6297.67i − 0.756268i −0.925751 0.378134i \(-0.876566\pi\)
0.925751 0.378134i \(-0.123434\pi\)
\(38\) − 3183.84i − 0.357678i
\(39\) 2933.01 0.308782
\(40\) 0 0
\(41\) 4298.36 0.399341 0.199670 0.979863i \(-0.436013\pi\)
0.199670 + 0.979863i \(0.436013\pi\)
\(42\) − 19475.5i − 1.70359i
\(43\) − 5822.12i − 0.480186i −0.970750 0.240093i \(-0.922822\pi\)
0.970750 0.240093i \(-0.0771780\pi\)
\(44\) 12017.5 0.935795
\(45\) 0 0
\(46\) −27484.4 −1.91510
\(47\) − 8502.70i − 0.561452i −0.959788 0.280726i \(-0.909425\pi\)
0.959788 0.280726i \(-0.0905751\pi\)
\(48\) 20996.7i 1.31537i
\(49\) −5529.90 −0.329023
\(50\) 0 0
\(51\) 31660.0 1.70445
\(52\) − 4119.67i − 0.211278i
\(53\) − 7049.74i − 0.344733i −0.985033 0.172367i \(-0.944859\pi\)
0.985033 0.172367i \(-0.0551414\pi\)
\(54\) 24081.3 1.12382
\(55\) 0 0
\(56\) 8554.65 0.364529
\(57\) 7359.17i 0.300014i
\(58\) 61522.2i 2.40139i
\(59\) 21509.2 0.804440 0.402220 0.915543i \(-0.368239\pi\)
0.402220 + 0.915543i \(0.368239\pi\)
\(60\) 0 0
\(61\) 35743.1 1.22989 0.614947 0.788568i \(-0.289177\pi\)
0.614947 + 0.788568i \(0.289177\pi\)
\(62\) 60917.8i 2.01263i
\(63\) 8698.30i 0.276110i
\(64\) 15738.9 0.480314
\(65\) 0 0
\(66\) −64241.3 −1.81533
\(67\) 15239.3i 0.414742i 0.978262 + 0.207371i \(0.0664907\pi\)
−0.978262 + 0.207371i \(0.933509\pi\)
\(68\) − 44469.1i − 1.16624i
\(69\) 63527.7 1.60635
\(70\) 0 0
\(71\) 69104.1 1.62689 0.813444 0.581643i \(-0.197590\pi\)
0.813444 + 0.581643i \(0.197590\pi\)
\(72\) − 3331.30i − 0.0757325i
\(73\) 4134.60i 0.0908085i 0.998969 + 0.0454043i \(0.0144576\pi\)
−0.998969 + 0.0454043i \(0.985542\pi\)
\(74\) 47285.7 1.00381
\(75\) 0 0
\(76\) 10336.6 0.205278
\(77\) 73679.8i 1.41619i
\(78\) 22022.4i 0.409853i
\(79\) 16065.3 0.289615 0.144808 0.989460i \(-0.453744\pi\)
0.144808 + 0.989460i \(0.453744\pi\)
\(80\) 0 0
\(81\) −69804.4 −1.18214
\(82\) 32274.0i 0.530052i
\(83\) 115739.i 1.84409i 0.387079 + 0.922047i \(0.373484\pi\)
−0.387079 + 0.922047i \(0.626516\pi\)
\(84\) 63228.7 0.977724
\(85\) 0 0
\(86\) 43715.0 0.637360
\(87\) − 142203.i − 2.01424i
\(88\) − 28218.1i − 0.388438i
\(89\) 35250.6 0.471728 0.235864 0.971786i \(-0.424208\pi\)
0.235864 + 0.971786i \(0.424208\pi\)
\(90\) 0 0
\(91\) 25258.0 0.319738
\(92\) − 89230.1i − 1.09911i
\(93\) − 140806.i − 1.68816i
\(94\) 63842.1 0.745225
\(95\) 0 0
\(96\) −125864. −1.39388
\(97\) 32169.5i 0.347148i 0.984821 + 0.173574i \(0.0555316\pi\)
−0.984821 + 0.173574i \(0.944468\pi\)
\(98\) − 41520.9i − 0.436719i
\(99\) 28691.9 0.294220
\(100\) 0 0
\(101\) −38266.9 −0.373267 −0.186634 0.982430i \(-0.559758\pi\)
−0.186634 + 0.982430i \(0.559758\pi\)
\(102\) 237717.i 2.26235i
\(103\) − 52058.1i − 0.483499i −0.970339 0.241750i \(-0.922279\pi\)
0.970339 0.241750i \(-0.0777212\pi\)
\(104\) −9673.37 −0.0876989
\(105\) 0 0
\(106\) 52932.6 0.457571
\(107\) 121233.i 1.02367i 0.859083 + 0.511836i \(0.171034\pi\)
−0.859083 + 0.511836i \(0.828966\pi\)
\(108\) 78181.7i 0.644979i
\(109\) −140412. −1.13198 −0.565989 0.824413i \(-0.691506\pi\)
−0.565989 + 0.824413i \(0.691506\pi\)
\(110\) 0 0
\(111\) −109297. −0.841977
\(112\) 180816.i 1.36204i
\(113\) − 8725.69i − 0.0642841i −0.999483 0.0321421i \(-0.989767\pi\)
0.999483 0.0321421i \(-0.0102329\pi\)
\(114\) −55255.9 −0.398214
\(115\) 0 0
\(116\) −199737. −1.37820
\(117\) − 9835.80i − 0.0664270i
\(118\) 161500.i 1.06775i
\(119\) 272643. 1.76493
\(120\) 0 0
\(121\) 81986.8 0.509074
\(122\) 268375.i 1.63246i
\(123\) − 74598.6i − 0.444598i
\(124\) −197774. −1.15509
\(125\) 0 0
\(126\) −65310.7 −0.366486
\(127\) − 125413.i − 0.689974i −0.938607 0.344987i \(-0.887883\pi\)
0.938607 0.344987i \(-0.112117\pi\)
\(128\) − 113899.i − 0.614461i
\(129\) −101044. −0.534606
\(130\) 0 0
\(131\) −392187. −1.99671 −0.998354 0.0573546i \(-0.981733\pi\)
−0.998354 + 0.0573546i \(0.981733\pi\)
\(132\) − 208564.i − 1.04185i
\(133\) 63374.3i 0.310659i
\(134\) −114423. −0.550494
\(135\) 0 0
\(136\) −104418. −0.484091
\(137\) 10040.7i 0.0457051i 0.999739 + 0.0228525i \(0.00727482\pi\)
−0.999739 + 0.0228525i \(0.992725\pi\)
\(138\) 476994.i 2.13214i
\(139\) 413226. 1.81406 0.907028 0.421071i \(-0.138346\pi\)
0.907028 + 0.421071i \(0.138346\pi\)
\(140\) 0 0
\(141\) −147565. −0.625081
\(142\) 518864.i 2.15940i
\(143\) − 83315.1i − 0.340709i
\(144\) 70412.1 0.282970
\(145\) 0 0
\(146\) −31044.4 −0.120532
\(147\) 95972.0i 0.366312i
\(148\) 153517.i 0.576105i
\(149\) −111303. −0.410716 −0.205358 0.978687i \(-0.565836\pi\)
−0.205358 + 0.978687i \(0.565836\pi\)
\(150\) 0 0
\(151\) 329006. 1.17425 0.587127 0.809495i \(-0.300259\pi\)
0.587127 + 0.809495i \(0.300259\pi\)
\(152\) − 24271.3i − 0.0852086i
\(153\) − 106171.i − 0.366672i
\(154\) −553221. −1.87973
\(155\) 0 0
\(156\) −71497.3 −0.235222
\(157\) 324164.i 1.04958i 0.851232 + 0.524790i \(0.175856\pi\)
−0.851232 + 0.524790i \(0.824144\pi\)
\(158\) 120626.i 0.384412i
\(159\) −122349. −0.383803
\(160\) 0 0
\(161\) 547076. 1.66335
\(162\) − 524122.i − 1.56908i
\(163\) 471803.i 1.39089i 0.718581 + 0.695443i \(0.244792\pi\)
−0.718581 + 0.695443i \(0.755208\pi\)
\(164\) −104780. −0.304207
\(165\) 0 0
\(166\) −869017. −2.44770
\(167\) 318786.i 0.884522i 0.896886 + 0.442261i \(0.145823\pi\)
−0.896886 + 0.442261i \(0.854177\pi\)
\(168\) − 148467.i − 0.405842i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) 24678.8 0.0645407
\(172\) 141924.i 0.365793i
\(173\) 340132.i 0.864036i 0.901865 + 0.432018i \(0.142198\pi\)
−0.901865 + 0.432018i \(0.857802\pi\)
\(174\) 1.06773e6 2.67354
\(175\) 0 0
\(176\) 596433. 1.45138
\(177\) − 373294.i − 0.895608i
\(178\) 264677.i 0.626134i
\(179\) 749176. 1.74764 0.873819 0.486251i \(-0.161636\pi\)
0.873819 + 0.486251i \(0.161636\pi\)
\(180\) 0 0
\(181\) −496170. −1.12573 −0.562865 0.826549i \(-0.690301\pi\)
−0.562865 + 0.826549i \(0.690301\pi\)
\(182\) 189648.i 0.424394i
\(183\) − 620326.i − 1.36928i
\(184\) −209521. −0.456228
\(185\) 0 0
\(186\) 1.05724e6 2.24073
\(187\) − 899332.i − 1.88068i
\(188\) 207268.i 0.427699i
\(189\) −479337. −0.976082
\(190\) 0 0
\(191\) 784516. 1.55603 0.778015 0.628245i \(-0.216227\pi\)
0.778015 + 0.628245i \(0.216227\pi\)
\(192\) − 273151.i − 0.534748i
\(193\) − 394155.i − 0.761683i −0.924640 0.380842i \(-0.875634\pi\)
0.924640 0.380842i \(-0.124366\pi\)
\(194\) −241543. −0.460776
\(195\) 0 0
\(196\) 134801. 0.250641
\(197\) − 913374.i − 1.67681i −0.545051 0.838403i \(-0.683490\pi\)
0.545051 0.838403i \(-0.316510\pi\)
\(198\) 215432.i 0.390523i
\(199\) 340138. 0.608866 0.304433 0.952534i \(-0.401533\pi\)
0.304433 + 0.952534i \(0.401533\pi\)
\(200\) 0 0
\(201\) 264480. 0.461745
\(202\) − 287325.i − 0.495444i
\(203\) − 1.22460e6i − 2.08571i
\(204\) −771767. −1.29841
\(205\) 0 0
\(206\) 390876. 0.641757
\(207\) − 213039.i − 0.345567i
\(208\) − 204461.i − 0.327682i
\(209\) 209044. 0.331034
\(210\) 0 0
\(211\) −582499. −0.900719 −0.450360 0.892847i \(-0.648704\pi\)
−0.450360 + 0.892847i \(0.648704\pi\)
\(212\) 171850.i 0.262609i
\(213\) − 1.19931e6i − 1.81127i
\(214\) −910269. −1.35874
\(215\) 0 0
\(216\) 183578. 0.267723
\(217\) − 1.21257e6i − 1.74806i
\(218\) − 1.05428e6i − 1.50249i
\(219\) 71756.5 0.101100
\(220\) 0 0
\(221\) −308297. −0.424609
\(222\) − 820649.i − 1.11757i
\(223\) − 580068.i − 0.781119i −0.920578 0.390559i \(-0.872282\pi\)
0.920578 0.390559i \(-0.127718\pi\)
\(224\) −1.08389e6 −1.44333
\(225\) 0 0
\(226\) 65516.4 0.0853255
\(227\) 833624.i 1.07376i 0.843660 + 0.536878i \(0.180396\pi\)
−0.843660 + 0.536878i \(0.819604\pi\)
\(228\) − 179393.i − 0.228543i
\(229\) 10222.2 0.0128811 0.00644057 0.999979i \(-0.497950\pi\)
0.00644057 + 0.999979i \(0.497950\pi\)
\(230\) 0 0
\(231\) 1.27872e6 1.57669
\(232\) 469000.i 0.572075i
\(233\) 932235.i 1.12496i 0.826812 + 0.562478i \(0.190152\pi\)
−0.826812 + 0.562478i \(0.809848\pi\)
\(234\) 73851.5 0.0881698
\(235\) 0 0
\(236\) −524324. −0.612801
\(237\) − 278815.i − 0.322438i
\(238\) 2.04713e6i 2.34262i
\(239\) 1.35733e6 1.53706 0.768530 0.639814i \(-0.220988\pi\)
0.768530 + 0.639814i \(0.220988\pi\)
\(240\) 0 0
\(241\) 59546.1 0.0660406 0.0330203 0.999455i \(-0.489487\pi\)
0.0330203 + 0.999455i \(0.489487\pi\)
\(242\) 615594.i 0.675703i
\(243\) 432107.i 0.469435i
\(244\) −871301. −0.936901
\(245\) 0 0
\(246\) 560119. 0.590123
\(247\) − 71661.9i − 0.0747387i
\(248\) 464392.i 0.479464i
\(249\) 2.00866e6 2.05309
\(250\) 0 0
\(251\) −1.20903e6 −1.21130 −0.605651 0.795731i \(-0.707087\pi\)
−0.605651 + 0.795731i \(0.707087\pi\)
\(252\) − 212036.i − 0.210333i
\(253\) − 1.80457e6i − 1.77244i
\(254\) 941656. 0.915815
\(255\) 0 0
\(256\) 1.35885e6 1.29590
\(257\) 1.91933e6i 1.81266i 0.422573 + 0.906329i \(0.361127\pi\)
−0.422573 + 0.906329i \(0.638873\pi\)
\(258\) − 758679.i − 0.709593i
\(259\) −941221. −0.871851
\(260\) 0 0
\(261\) −476875. −0.433315
\(262\) − 2.94471e6i − 2.65027i
\(263\) − 637187.i − 0.568038i −0.958819 0.284019i \(-0.908332\pi\)
0.958819 0.284019i \(-0.0916679\pi\)
\(264\) −489728. −0.432460
\(265\) 0 0
\(266\) −475842. −0.412343
\(267\) − 611779.i − 0.525190i
\(268\) − 371484.i − 0.315939i
\(269\) −250073. −0.210711 −0.105355 0.994435i \(-0.533598\pi\)
−0.105355 + 0.994435i \(0.533598\pi\)
\(270\) 0 0
\(271\) 994132. 0.822282 0.411141 0.911572i \(-0.365130\pi\)
0.411141 + 0.911572i \(0.365130\pi\)
\(272\) − 2.20703e6i − 1.80878i
\(273\) − 438355.i − 0.355975i
\(274\) −75390.4 −0.0606652
\(275\) 0 0
\(276\) −1.54860e6 −1.22368
\(277\) 1.80516e6i 1.41356i 0.707432 + 0.706782i \(0.249854\pi\)
−0.707432 + 0.706782i \(0.750146\pi\)
\(278\) 3.10268e6i 2.40783i
\(279\) −472190. −0.363167
\(280\) 0 0
\(281\) −349545. −0.264081 −0.132040 0.991244i \(-0.542153\pi\)
−0.132040 + 0.991244i \(0.542153\pi\)
\(282\) − 1.10799e6i − 0.829682i
\(283\) 204139.i 0.151516i 0.997126 + 0.0757582i \(0.0241377\pi\)
−0.997126 + 0.0757582i \(0.975862\pi\)
\(284\) −1.68453e6 −1.23932
\(285\) 0 0
\(286\) 625567. 0.452229
\(287\) − 642413.i − 0.460373i
\(288\) 422084.i 0.299859i
\(289\) −1.90801e6 −1.34381
\(290\) 0 0
\(291\) 558305. 0.386491
\(292\) − 100788.i − 0.0691755i
\(293\) − 78795.6i − 0.0536207i −0.999641 0.0268104i \(-0.991465\pi\)
0.999641 0.0268104i \(-0.00853503\pi\)
\(294\) −720600. −0.486212
\(295\) 0 0
\(296\) 360471. 0.239134
\(297\) 1.58113e6i 1.04010i
\(298\) − 835713.i − 0.545150i
\(299\) −618618. −0.400170
\(300\) 0 0
\(301\) −870146. −0.553575
\(302\) 2.47033e6i 1.55861i
\(303\) 664126.i 0.415570i
\(304\) 513010. 0.318377
\(305\) 0 0
\(306\) 797179. 0.486690
\(307\) − 2.05653e6i − 1.24534i −0.782483 0.622672i \(-0.786047\pi\)
0.782483 0.622672i \(-0.213953\pi\)
\(308\) − 1.79607e6i − 1.07882i
\(309\) −903475. −0.538295
\(310\) 0 0
\(311\) −3119.84 −0.00182908 −0.000914538 1.00000i \(-0.500291\pi\)
−0.000914538 1.00000i \(0.500291\pi\)
\(312\) 167882.i 0.0976379i
\(313\) 984315.i 0.567902i 0.958839 + 0.283951i \(0.0916453\pi\)
−0.958839 + 0.283951i \(0.908355\pi\)
\(314\) −2.43397e6 −1.39313
\(315\) 0 0
\(316\) −391620. −0.220621
\(317\) 1.90110e6i 1.06257i 0.847194 + 0.531283i \(0.178290\pi\)
−0.847194 + 0.531283i \(0.821710\pi\)
\(318\) − 918651.i − 0.509428i
\(319\) −4.03942e6 −2.22250
\(320\) 0 0
\(321\) 2.10401e6 1.13969
\(322\) 4.10769e6i 2.20779i
\(323\) − 773543.i − 0.412552i
\(324\) 1.70160e6 0.900525
\(325\) 0 0
\(326\) −3.54251e6 −1.84615
\(327\) 2.43687e6i 1.26027i
\(328\) 246033.i 0.126273i
\(329\) −1.27077e6 −0.647260
\(330\) 0 0
\(331\) −1.74888e6 −0.877386 −0.438693 0.898637i \(-0.644558\pi\)
−0.438693 + 0.898637i \(0.644558\pi\)
\(332\) − 2.82133e6i − 1.40478i
\(333\) 366524.i 0.181131i
\(334\) −2.39359e6 −1.17404
\(335\) 0 0
\(336\) 3.13808e6 1.51640
\(337\) 1.09525e6i 0.525338i 0.964886 + 0.262669i \(0.0846027\pi\)
−0.964886 + 0.262669i \(0.915397\pi\)
\(338\) − 214449.i − 0.102101i
\(339\) −151435. −0.0715695
\(340\) 0 0
\(341\) −3.99973e6 −1.86271
\(342\) 185300.i 0.0856661i
\(343\) − 1.68542e6i − 0.773524i
\(344\) 333251. 0.151836
\(345\) 0 0
\(346\) −2.55386e6 −1.14685
\(347\) 1.99055e6i 0.887461i 0.896160 + 0.443730i \(0.146345\pi\)
−0.896160 + 0.443730i \(0.853655\pi\)
\(348\) 3.46645e6i 1.53439i
\(349\) 846086. 0.371835 0.185918 0.982565i \(-0.440474\pi\)
0.185918 + 0.982565i \(0.440474\pi\)
\(350\) 0 0
\(351\) 542021. 0.234827
\(352\) 3.57530e6i 1.53800i
\(353\) − 2.61806e6i − 1.11826i −0.829079 0.559131i \(-0.811135\pi\)
0.829079 0.559131i \(-0.188865\pi\)
\(354\) 2.80286e6 1.18876
\(355\) 0 0
\(356\) −859296. −0.359350
\(357\) − 4.73175e6i − 1.96495i
\(358\) 5.62515e6i 2.31967i
\(359\) 1.63224e6 0.668418 0.334209 0.942499i \(-0.391531\pi\)
0.334209 + 0.942499i \(0.391531\pi\)
\(360\) 0 0
\(361\) −2.29629e6 −0.927384
\(362\) − 3.72546e6i − 1.49420i
\(363\) − 1.42289e6i − 0.566768i
\(364\) −615706. −0.243568
\(365\) 0 0
\(366\) 4.65768e6 1.81747
\(367\) − 3.09644e6i − 1.20005i −0.799983 0.600023i \(-0.795158\pi\)
0.799983 0.600023i \(-0.204842\pi\)
\(368\) − 4.42854e6i − 1.70467i
\(369\) −250165. −0.0956445
\(370\) 0 0
\(371\) −1.05362e6 −0.397420
\(372\) 3.43239e6i 1.28600i
\(373\) 2.44663e6i 0.910533i 0.890355 + 0.455266i \(0.150456\pi\)
−0.890355 + 0.455266i \(0.849544\pi\)
\(374\) 6.75258e6 2.49627
\(375\) 0 0
\(376\) 486685. 0.177533
\(377\) 1.38474e6i 0.501782i
\(378\) − 3.59907e6i − 1.29557i
\(379\) 4.31307e6 1.54237 0.771184 0.636612i \(-0.219665\pi\)
0.771184 + 0.636612i \(0.219665\pi\)
\(380\) 0 0
\(381\) −2.17656e6 −0.768170
\(382\) 5.89049e6i 2.06535i
\(383\) − 4.18340e6i − 1.45724i −0.684916 0.728622i \(-0.740161\pi\)
0.684916 0.728622i \(-0.259839\pi\)
\(384\) −1.97673e6 −0.684098
\(385\) 0 0
\(386\) 2.95949e6 1.01100
\(387\) 338847.i 0.115008i
\(388\) − 784187.i − 0.264448i
\(389\) 3.01512e6 1.01025 0.505126 0.863045i \(-0.331446\pi\)
0.505126 + 0.863045i \(0.331446\pi\)
\(390\) 0 0
\(391\) −6.67758e6 −2.20891
\(392\) − 316525.i − 0.104038i
\(393\) 6.80644e6i 2.22300i
\(394\) 6.85801e6 2.22566
\(395\) 0 0
\(396\) −699416. −0.224129
\(397\) − 183544.i − 0.0584471i −0.999573 0.0292236i \(-0.990697\pi\)
0.999573 0.0292236i \(-0.00930347\pi\)
\(398\) 2.55390e6i 0.808159i
\(399\) 1.09987e6 0.345866
\(400\) 0 0
\(401\) −3.10417e6 −0.964015 −0.482008 0.876167i \(-0.660092\pi\)
−0.482008 + 0.876167i \(0.660092\pi\)
\(402\) 1.98583e6i 0.612882i
\(403\) 1.37114e6i 0.420551i
\(404\) 932822. 0.284345
\(405\) 0 0
\(406\) 9.19483e6 2.76840
\(407\) 3.10468e6i 0.929033i
\(408\) 1.81218e6i 0.538953i
\(409\) −2.45114e6 −0.724535 −0.362268 0.932074i \(-0.617997\pi\)
−0.362268 + 0.932074i \(0.617997\pi\)
\(410\) 0 0
\(411\) 174258. 0.0508849
\(412\) 1.26901e6i 0.368317i
\(413\) − 3.21466e6i − 0.927385i
\(414\) 1.59959e6 0.458678
\(415\) 0 0
\(416\) 1.22564e6 0.347240
\(417\) − 7.17158e6i − 2.01964i
\(418\) 1.56960e6i 0.439387i
\(419\) 1.38177e6 0.384503 0.192251 0.981346i \(-0.438421\pi\)
0.192251 + 0.981346i \(0.438421\pi\)
\(420\) 0 0
\(421\) −4.38636e6 −1.20614 −0.603071 0.797687i \(-0.706057\pi\)
−0.603071 + 0.797687i \(0.706057\pi\)
\(422\) − 4.37366e6i − 1.19554i
\(423\) 494857.i 0.134471i
\(424\) 403519. 0.109006
\(425\) 0 0
\(426\) 9.00495e6 2.40413
\(427\) − 5.34200e6i − 1.41786i
\(428\) − 2.95526e6i − 0.779805i
\(429\) −1.44594e6 −0.379322
\(430\) 0 0
\(431\) 1.36275e6 0.353365 0.176682 0.984268i \(-0.443463\pi\)
0.176682 + 0.984268i \(0.443463\pi\)
\(432\) 3.88020e6i 1.00033i
\(433\) 5.59344e6i 1.43370i 0.697226 + 0.716851i \(0.254417\pi\)
−0.697226 + 0.716851i \(0.745583\pi\)
\(434\) 9.10449e6 2.32023
\(435\) 0 0
\(436\) 3.42279e6 0.862310
\(437\) − 1.55216e6i − 0.388807i
\(438\) 538780.i 0.134192i
\(439\) 4.86469e6 1.20474 0.602371 0.798216i \(-0.294223\pi\)
0.602371 + 0.798216i \(0.294223\pi\)
\(440\) 0 0
\(441\) 321840. 0.0788031
\(442\) − 2.31483e6i − 0.563591i
\(443\) 4.59200e6i 1.11171i 0.831278 + 0.555857i \(0.187610\pi\)
−0.831278 + 0.555857i \(0.812390\pi\)
\(444\) 2.66430e6 0.641395
\(445\) 0 0
\(446\) 4.35541e6 1.03679
\(447\) 1.93168e6i 0.457263i
\(448\) − 2.35227e6i − 0.553722i
\(449\) 351998. 0.0823994 0.0411997 0.999151i \(-0.486882\pi\)
0.0411997 + 0.999151i \(0.486882\pi\)
\(450\) 0 0
\(451\) −2.11904e6 −0.490567
\(452\) 212704.i 0.0489699i
\(453\) − 5.70994e6i − 1.30733i
\(454\) −6.25922e6 −1.42521
\(455\) 0 0
\(456\) −421231. −0.0948654
\(457\) 546819.i 0.122477i 0.998123 + 0.0612383i \(0.0195050\pi\)
−0.998123 + 0.0612383i \(0.980495\pi\)
\(458\) 76752.6i 0.0170974i
\(459\) 5.85076e6 1.29623
\(460\) 0 0
\(461\) −6.35474e6 −1.39266 −0.696331 0.717721i \(-0.745185\pi\)
−0.696331 + 0.717721i \(0.745185\pi\)
\(462\) 9.60121e6i 2.09277i
\(463\) − 2.90818e6i − 0.630476i −0.949013 0.315238i \(-0.897916\pi\)
0.949013 0.315238i \(-0.102084\pi\)
\(464\) −9.91303e6 −2.13753
\(465\) 0 0
\(466\) −6.99963e6 −1.49317
\(467\) − 3.33677e6i − 0.708002i −0.935245 0.354001i \(-0.884821\pi\)
0.935245 0.354001i \(-0.115179\pi\)
\(468\) 239765.i 0.0506023i
\(469\) 2.27759e6 0.478128
\(470\) 0 0
\(471\) 5.62590e6 1.16853
\(472\) 1.23116e6i 0.254366i
\(473\) 2.87024e6i 0.589882i
\(474\) 2.09347e6 0.427978
\(475\) 0 0
\(476\) −6.64615e6 −1.34447
\(477\) 410295.i 0.0825658i
\(478\) 1.01914e7i 2.04017i
\(479\) −1.93937e6 −0.386208 −0.193104 0.981178i \(-0.561856\pi\)
−0.193104 + 0.981178i \(0.561856\pi\)
\(480\) 0 0
\(481\) 1.06431e6 0.209751
\(482\) 447099.i 0.0876568i
\(483\) − 9.49456e6i − 1.85185i
\(484\) −1.99857e6 −0.387799
\(485\) 0 0
\(486\) −3.24445e6 −0.623089
\(487\) − 5.88271e6i − 1.12397i −0.827147 0.561985i \(-0.810038\pi\)
0.827147 0.561985i \(-0.189962\pi\)
\(488\) 2.04590e6i 0.388896i
\(489\) 8.18820e6 1.54852
\(490\) 0 0
\(491\) 8.39552e6 1.57161 0.785804 0.618476i \(-0.212250\pi\)
0.785804 + 0.618476i \(0.212250\pi\)
\(492\) 1.81847e6i 0.338683i
\(493\) 1.49474e7i 2.76980i
\(494\) 538069. 0.0992021
\(495\) 0 0
\(496\) −9.81564e6 −1.79149
\(497\) − 1.03280e7i − 1.87553i
\(498\) 1.50819e7i 2.72510i
\(499\) −8.24513e6 −1.48234 −0.741168 0.671320i \(-0.765728\pi\)
−0.741168 + 0.671320i \(0.765728\pi\)
\(500\) 0 0
\(501\) 5.53257e6 0.984766
\(502\) − 9.07792e6i − 1.60778i
\(503\) − 8.99706e6i − 1.58555i −0.609513 0.792776i \(-0.708635\pi\)
0.609513 0.792776i \(-0.291365\pi\)
\(504\) −497880. −0.0873069
\(505\) 0 0
\(506\) 1.35495e7 2.35259
\(507\) 495679.i 0.0856409i
\(508\) 3.05716e6i 0.525604i
\(509\) −6.27444e6 −1.07345 −0.536723 0.843758i \(-0.680338\pi\)
−0.536723 + 0.843758i \(0.680338\pi\)
\(510\) 0 0
\(511\) 617938. 0.104687
\(512\) 6.55808e6i 1.10561i
\(513\) 1.35998e6i 0.228159i
\(514\) −1.44111e7 −2.40597
\(515\) 0 0
\(516\) 2.46311e6 0.407249
\(517\) 4.19174e6i 0.689712i
\(518\) − 7.06710e6i − 1.15722i
\(519\) 5.90302e6 0.961958
\(520\) 0 0
\(521\) −8.31964e6 −1.34280 −0.671398 0.741097i \(-0.734306\pi\)
−0.671398 + 0.741097i \(0.734306\pi\)
\(522\) − 3.58059e6i − 0.575147i
\(523\) 8.50917e6i 1.36029i 0.733076 + 0.680147i \(0.238084\pi\)
−0.733076 + 0.680147i \(0.761916\pi\)
\(524\) 9.56023e6 1.52104
\(525\) 0 0
\(526\) 4.78428e6 0.753967
\(527\) 1.48005e7i 2.32140i
\(528\) − 1.03512e7i − 1.61586i
\(529\) −6.96264e6 −1.08177
\(530\) 0 0
\(531\) −1.25183e6 −0.192668
\(532\) − 1.54486e6i − 0.236652i
\(533\) 726423.i 0.110757i
\(534\) 4.59351e6 0.697094
\(535\) 0 0
\(536\) −872280. −0.131143
\(537\) − 1.30020e7i − 1.94570i
\(538\) − 1.87766e6i − 0.279680i
\(539\) 2.72618e6 0.404187
\(540\) 0 0
\(541\) −1.15207e6 −0.169233 −0.0846164 0.996414i \(-0.526966\pi\)
−0.0846164 + 0.996414i \(0.526966\pi\)
\(542\) 7.46438e6i 1.09143i
\(543\) 8.61109e6i 1.25331i
\(544\) 1.32300e7 1.91673
\(545\) 0 0
\(546\) 3.29136e6 0.472491
\(547\) − 943671.i − 0.134850i −0.997724 0.0674252i \(-0.978522\pi\)
0.997724 0.0674252i \(-0.0214784\pi\)
\(548\) − 244761.i − 0.0348169i
\(549\) −2.08025e6 −0.294567
\(550\) 0 0
\(551\) −3.47443e6 −0.487534
\(552\) 3.63625e6i 0.507933i
\(553\) − 2.40105e6i − 0.333878i
\(554\) −1.35539e7 −1.87625
\(555\) 0 0
\(556\) −1.00731e7 −1.38190
\(557\) − 1.19731e7i − 1.63519i −0.575791 0.817597i \(-0.695306\pi\)
0.575791 0.817597i \(-0.304694\pi\)
\(558\) − 3.54541e6i − 0.482038i
\(559\) 983938. 0.133180
\(560\) 0 0
\(561\) −1.56080e7 −2.09382
\(562\) − 2.62454e6i − 0.350519i
\(563\) 3.19566e6i 0.424903i 0.977172 + 0.212452i \(0.0681448\pi\)
−0.977172 + 0.212452i \(0.931855\pi\)
\(564\) 3.59716e6 0.476170
\(565\) 0 0
\(566\) −1.53276e6 −0.201110
\(567\) 1.04326e7i 1.36281i
\(568\) 3.95544e6i 0.514427i
\(569\) −3.18417e6 −0.412303 −0.206151 0.978520i \(-0.566094\pi\)
−0.206151 + 0.978520i \(0.566094\pi\)
\(570\) 0 0
\(571\) −6.71439e6 −0.861820 −0.430910 0.902395i \(-0.641807\pi\)
−0.430910 + 0.902395i \(0.641807\pi\)
\(572\) 2.03095e6i 0.259543i
\(573\) − 1.36154e7i − 1.73238i
\(574\) 4.82352e6 0.611061
\(575\) 0 0
\(576\) −916005. −0.115038
\(577\) 4.75161e6i 0.594157i 0.954853 + 0.297078i \(0.0960122\pi\)
−0.954853 + 0.297078i \(0.903988\pi\)
\(578\) − 1.43262e7i − 1.78366i
\(579\) −6.84061e6 −0.848005
\(580\) 0 0
\(581\) 1.72977e7 2.12593
\(582\) 4.19200e6i 0.512996i
\(583\) 3.47544e6i 0.423486i
\(584\) −236660. −0.0287139
\(585\) 0 0
\(586\) 591632. 0.0711718
\(587\) − 639190.i − 0.0765658i −0.999267 0.0382829i \(-0.987811\pi\)
0.999267 0.0382829i \(-0.0121888\pi\)
\(588\) − 2.33948e6i − 0.279047i
\(589\) −3.44030e6 −0.408609
\(590\) 0 0
\(591\) −1.58517e7 −1.86684
\(592\) 7.61911e6i 0.893512i
\(593\) − 1.37403e6i − 0.160458i −0.996776 0.0802288i \(-0.974435\pi\)
0.996776 0.0802288i \(-0.0255651\pi\)
\(594\) −1.18718e7 −1.38054
\(595\) 0 0
\(596\) 2.71320e6 0.312872
\(597\) − 5.90313e6i − 0.677870i
\(598\) − 4.64486e6i − 0.531153i
\(599\) 1.28982e7 1.46880 0.734401 0.678716i \(-0.237463\pi\)
0.734401 + 0.678716i \(0.237463\pi\)
\(600\) 0 0
\(601\) 6.50916e6 0.735087 0.367543 0.930006i \(-0.380199\pi\)
0.367543 + 0.930006i \(0.380199\pi\)
\(602\) − 6.53345e6i − 0.734769i
\(603\) − 886926.i − 0.0993332i
\(604\) −8.02010e6 −0.894515
\(605\) 0 0
\(606\) −4.98655e6 −0.551593
\(607\) 6.58784e6i 0.725724i 0.931843 + 0.362862i \(0.118200\pi\)
−0.931843 + 0.362862i \(0.881800\pi\)
\(608\) 3.07523e6i 0.337379i
\(609\) −2.12530e7 −2.32208
\(610\) 0 0
\(611\) 1.43696e6 0.155719
\(612\) 2.58810e6i 0.279321i
\(613\) 6.83056e6i 0.734184i 0.930185 + 0.367092i \(0.119647\pi\)
−0.930185 + 0.367092i \(0.880353\pi\)
\(614\) 1.54413e7 1.65297
\(615\) 0 0
\(616\) −4.21735e6 −0.447803
\(617\) − 5.07265e6i − 0.536441i −0.963357 0.268221i \(-0.913564\pi\)
0.963357 0.268221i \(-0.0864356\pi\)
\(618\) − 6.78369e6i − 0.714488i
\(619\) −2.32710e6 −0.244112 −0.122056 0.992523i \(-0.538949\pi\)
−0.122056 + 0.992523i \(0.538949\pi\)
\(620\) 0 0
\(621\) 1.17399e7 1.22162
\(622\) − 23425.2i − 0.00242776i
\(623\) − 5.26840e6i − 0.543824i
\(624\) −3.54845e6 −0.364819
\(625\) 0 0
\(626\) −7.39067e6 −0.753786
\(627\) − 3.62799e6i − 0.368551i
\(628\) − 7.90206e6i − 0.799542i
\(629\) 1.14885e7 1.15781
\(630\) 0 0
\(631\) 1.62914e7 1.62886 0.814432 0.580259i \(-0.197049\pi\)
0.814432 + 0.580259i \(0.197049\pi\)
\(632\) 919560.i 0.0915772i
\(633\) 1.01093e7i 1.00280i
\(634\) −1.42743e7 −1.41036
\(635\) 0 0
\(636\) 2.98247e6 0.292370
\(637\) − 934552.i − 0.0912547i
\(638\) − 3.03298e7i − 2.94997i
\(639\) −4.02186e6 −0.389650
\(640\) 0 0
\(641\) 9.20916e6 0.885268 0.442634 0.896702i \(-0.354044\pi\)
0.442634 + 0.896702i \(0.354044\pi\)
\(642\) 1.57978e7i 1.51272i
\(643\) − 1.30502e7i − 1.24477i −0.782710 0.622387i \(-0.786163\pi\)
0.782710 0.622387i \(-0.213837\pi\)
\(644\) −1.33359e7 −1.26709
\(645\) 0 0
\(646\) 5.80811e6 0.547587
\(647\) − 4.14869e6i − 0.389628i −0.980840 0.194814i \(-0.937590\pi\)
0.980840 0.194814i \(-0.0624103\pi\)
\(648\) − 3.99552e6i − 0.373797i
\(649\) −1.06038e7 −0.988210
\(650\) 0 0
\(651\) −2.10442e7 −1.94617
\(652\) − 1.15010e7i − 1.05954i
\(653\) − 1.41931e7i − 1.30255i −0.758844 0.651273i \(-0.774235\pi\)
0.758844 0.651273i \(-0.225765\pi\)
\(654\) −1.82971e7 −1.67277
\(655\) 0 0
\(656\) −5.20029e6 −0.471811
\(657\) − 240634.i − 0.0217492i
\(658\) − 9.54154e6i − 0.859120i
\(659\) −7.37184e6 −0.661245 −0.330622 0.943763i \(-0.607259\pi\)
−0.330622 + 0.943763i \(0.607259\pi\)
\(660\) 0 0
\(661\) 1.77538e7 1.58047 0.790235 0.612803i \(-0.209958\pi\)
0.790235 + 0.612803i \(0.209958\pi\)
\(662\) − 1.31314e7i − 1.16457i
\(663\) 5.35054e6i 0.472730i
\(664\) −6.62474e6 −0.583108
\(665\) 0 0
\(666\) −2.75203e6 −0.240418
\(667\) 2.99929e7i 2.61038i
\(668\) − 7.77097e6i − 0.673805i
\(669\) −1.00672e7 −0.869644
\(670\) 0 0
\(671\) −1.76210e7 −1.51086
\(672\) 1.88111e7i 1.60691i
\(673\) 274222.i 0.0233381i 0.999932 + 0.0116690i \(0.00371446\pi\)
−0.999932 + 0.0116690i \(0.996286\pi\)
\(674\) −8.22363e6 −0.697290
\(675\) 0 0
\(676\) 696224. 0.0585979
\(677\) − 3.03927e6i − 0.254857i −0.991848 0.127429i \(-0.959328\pi\)
0.991848 0.127429i \(-0.0406724\pi\)
\(678\) − 1.13704e6i − 0.0949955i
\(679\) 4.80790e6 0.400204
\(680\) 0 0
\(681\) 1.44676e7 1.19545
\(682\) − 3.00318e7i − 2.47241i
\(683\) − 1.26346e7i − 1.03636i −0.855273 0.518178i \(-0.826610\pi\)
0.855273 0.518178i \(-0.173390\pi\)
\(684\) −601589. −0.0491654
\(685\) 0 0
\(686\) 1.26549e7 1.02671
\(687\) − 177407.i − 0.0143410i
\(688\) 7.04377e6i 0.567328i
\(689\) 1.19141e6 0.0956119
\(690\) 0 0
\(691\) 1.53558e7 1.22343 0.611713 0.791080i \(-0.290481\pi\)
0.611713 + 0.791080i \(0.290481\pi\)
\(692\) − 8.29130e6i − 0.658199i
\(693\) − 4.28816e6i − 0.339186i
\(694\) −1.49459e7 −1.17794
\(695\) 0 0
\(696\) 8.13955e6 0.636909
\(697\) 7.84127e6i 0.611370i
\(698\) 6.35279e6i 0.493544i
\(699\) 1.61790e7 1.25245
\(700\) 0 0
\(701\) −2.13816e7 −1.64341 −0.821703 0.569916i \(-0.806976\pi\)
−0.821703 + 0.569916i \(0.806976\pi\)
\(702\) 4.06974e6i 0.311690i
\(703\) 2.67043e6i 0.203795i
\(704\) −7.75911e6 −0.590038
\(705\) 0 0
\(706\) 1.96576e7 1.48429
\(707\) 5.71919e6i 0.430315i
\(708\) 9.09970e6i 0.682250i
\(709\) 2.39380e7 1.78843 0.894214 0.447639i \(-0.147735\pi\)
0.894214 + 0.447639i \(0.147735\pi\)
\(710\) 0 0
\(711\) −935001. −0.0693646
\(712\) 2.01771e6i 0.149162i
\(713\) 2.96982e7i 2.18779i
\(714\) 3.55281e7 2.60811
\(715\) 0 0
\(716\) −1.82625e7 −1.33130
\(717\) − 2.35566e7i − 1.71126i
\(718\) 1.22556e7i 0.887203i
\(719\) 2.50209e6 0.180501 0.0902507 0.995919i \(-0.471233\pi\)
0.0902507 + 0.995919i \(0.471233\pi\)
\(720\) 0 0
\(721\) −7.78037e6 −0.557394
\(722\) − 1.72416e7i − 1.23093i
\(723\) − 1.03343e6i − 0.0735250i
\(724\) 1.20950e7 0.857550
\(725\) 0 0
\(726\) 1.06837e7 0.752281
\(727\) − 2.78011e7i − 1.95086i −0.220315 0.975429i \(-0.570709\pi\)
0.220315 0.975429i \(-0.429291\pi\)
\(728\) 1.44574e6i 0.101102i
\(729\) −9.46320e6 −0.659506
\(730\) 0 0
\(731\) 1.06210e7 0.735141
\(732\) 1.51215e7i 1.04308i
\(733\) 4.46741e6i 0.307111i 0.988140 + 0.153556i \(0.0490724\pi\)
−0.988140 + 0.153556i \(0.950928\pi\)
\(734\) 2.32495e7 1.59284
\(735\) 0 0
\(736\) 2.65468e7 1.80641
\(737\) − 7.51280e6i − 0.509487i
\(738\) − 1.87835e6i − 0.126951i
\(739\) 1.23320e7 0.830656 0.415328 0.909672i \(-0.363667\pi\)
0.415328 + 0.909672i \(0.363667\pi\)
\(740\) 0 0
\(741\) −1.24370e6 −0.0832090
\(742\) − 7.91106e6i − 0.527503i
\(743\) − 9.81721e6i − 0.652403i −0.945300 0.326202i \(-0.894231\pi\)
0.945300 0.326202i \(-0.105769\pi\)
\(744\) 8.05958e6 0.533802
\(745\) 0 0
\(746\) −1.83704e7 −1.20857
\(747\) − 6.73598e6i − 0.441672i
\(748\) 2.19228e7i 1.43265i
\(749\) 1.81189e7 1.18012
\(750\) 0 0
\(751\) 4.20655e6 0.272161 0.136081 0.990698i \(-0.456549\pi\)
0.136081 + 0.990698i \(0.456549\pi\)
\(752\) 1.02868e7i 0.663341i
\(753\) 2.09828e7i 1.34858i
\(754\) −1.03973e7 −0.666025
\(755\) 0 0
\(756\) 1.16847e7 0.743553
\(757\) 2.89772e7i 1.83788i 0.394400 + 0.918939i \(0.370952\pi\)
−0.394400 + 0.918939i \(0.629048\pi\)
\(758\) 3.23844e7i 2.04721i
\(759\) −3.13185e7 −1.97331
\(760\) 0 0
\(761\) −1.57354e6 −0.0984957 −0.0492478 0.998787i \(-0.515682\pi\)
−0.0492478 + 0.998787i \(0.515682\pi\)
\(762\) − 1.63425e7i − 1.01961i
\(763\) 2.09853e7i 1.30498i
\(764\) −1.91239e7 −1.18534
\(765\) 0 0
\(766\) 3.14108e7 1.93423
\(767\) 3.63505e6i 0.223112i
\(768\) − 2.35830e7i − 1.44276i
\(769\) 1.98322e7 1.20936 0.604680 0.796468i \(-0.293301\pi\)
0.604680 + 0.796468i \(0.293301\pi\)
\(770\) 0 0
\(771\) 3.33101e7 2.01809
\(772\) 9.60822e6i 0.580230i
\(773\) − 2.42031e7i − 1.45687i −0.685112 0.728437i \(-0.740247\pi\)
0.685112 0.728437i \(-0.259753\pi\)
\(774\) −2.54421e6 −0.152652
\(775\) 0 0
\(776\) −1.84134e6 −0.109769
\(777\) 1.63350e7i 0.970658i
\(778\) 2.26388e7i 1.34093i
\(779\) −1.82266e6 −0.107612
\(780\) 0 0
\(781\) −3.40675e7 −1.99854
\(782\) − 5.01382e7i − 2.93192i
\(783\) − 2.62792e7i − 1.53182i
\(784\) 6.69024e6 0.388733
\(785\) 0 0
\(786\) −5.11058e7 −2.95062
\(787\) 1.03509e6i 0.0595721i 0.999556 + 0.0297860i \(0.00948259\pi\)
−0.999556 + 0.0297860i \(0.990517\pi\)
\(788\) 2.22651e7i 1.27735i
\(789\) −1.10584e7 −0.632414
\(790\) 0 0
\(791\) −1.30410e6 −0.0741089
\(792\) 1.64229e6i 0.0930331i
\(793\) 6.04059e6i 0.341111i
\(794\) 1.37813e6 0.0775779
\(795\) 0 0
\(796\) −8.29144e6 −0.463818
\(797\) − 2.57309e7i − 1.43486i −0.696632 0.717429i \(-0.745319\pi\)
0.696632 0.717429i \(-0.254681\pi\)
\(798\) 8.25829e6i 0.459074i
\(799\) 1.55110e7 0.859554
\(800\) 0 0
\(801\) −2.05159e6 −0.112982
\(802\) − 2.33075e7i − 1.27955i
\(803\) − 2.03831e6i − 0.111553i
\(804\) −6.44715e6 −0.351745
\(805\) 0 0
\(806\) −1.02951e7 −0.558204
\(807\) 4.34005e6i 0.234591i
\(808\) − 2.19035e6i − 0.118028i
\(809\) −3.62907e7 −1.94950 −0.974751 0.223297i \(-0.928318\pi\)
−0.974751 + 0.223297i \(0.928318\pi\)
\(810\) 0 0
\(811\) −2.80415e7 −1.49709 −0.748546 0.663083i \(-0.769248\pi\)
−0.748546 + 0.663083i \(0.769248\pi\)
\(812\) 2.98517e7i 1.58884i
\(813\) − 1.72533e7i − 0.915472i
\(814\) −2.33113e7 −1.23312
\(815\) 0 0
\(816\) −3.83032e7 −2.01377
\(817\) 2.46878e6i 0.129398i
\(818\) − 1.84042e7i − 0.961688i
\(819\) −1.47001e6 −0.0765793
\(820\) 0 0
\(821\) −1.73781e7 −0.899795 −0.449897 0.893080i \(-0.648539\pi\)
−0.449897 + 0.893080i \(0.648539\pi\)
\(822\) 1.30841e6i 0.0675404i
\(823\) − 8.26775e6i − 0.425488i −0.977108 0.212744i \(-0.931760\pi\)
0.977108 0.212744i \(-0.0682401\pi\)
\(824\) 2.97975e6 0.152884
\(825\) 0 0
\(826\) 2.41371e7 1.23094
\(827\) − 9.80834e6i − 0.498691i −0.968415 0.249346i \(-0.919784\pi\)
0.968415 0.249346i \(-0.0802155\pi\)
\(828\) 5.19319e6i 0.263244i
\(829\) −2.34318e7 −1.18419 −0.592093 0.805870i \(-0.701698\pi\)
−0.592093 + 0.805870i \(0.701698\pi\)
\(830\) 0 0
\(831\) 3.13287e7 1.57376
\(832\) 2.65988e6i 0.133215i
\(833\) − 1.00879e7i − 0.503718i
\(834\) 5.38474e7 2.68071
\(835\) 0 0
\(836\) −5.09582e6 −0.252173
\(837\) − 2.60210e7i − 1.28384i
\(838\) 1.03749e7i 0.510357i
\(839\) 1.14704e7 0.562568 0.281284 0.959625i \(-0.409240\pi\)
0.281284 + 0.959625i \(0.409240\pi\)
\(840\) 0 0
\(841\) 4.66262e7 2.27321
\(842\) − 3.29347e7i − 1.60093i
\(843\) 6.06639e6i 0.294010i
\(844\) 1.41994e7 0.686144
\(845\) 0 0
\(846\) −3.71561e6 −0.178486
\(847\) − 1.22534e7i − 0.586877i
\(848\) 8.52899e6i 0.407294i
\(849\) 3.54285e6 0.168688
\(850\) 0 0
\(851\) 2.30524e7 1.09117
\(852\) 2.92352e7i 1.37977i
\(853\) − 2.31885e7i − 1.09119i −0.838049 0.545594i \(-0.816304\pi\)
0.838049 0.545594i \(-0.183696\pi\)
\(854\) 4.01101e7 1.88196
\(855\) 0 0
\(856\) −6.93922e6 −0.323688
\(857\) 3.35616e7i 1.56096i 0.625184 + 0.780478i \(0.285024\pi\)
−0.625184 + 0.780478i \(0.714976\pi\)
\(858\) − 1.08568e7i − 0.503481i
\(859\) 2.84860e7 1.31719 0.658594 0.752498i \(-0.271151\pi\)
0.658594 + 0.752498i \(0.271151\pi\)
\(860\) 0 0
\(861\) −1.11492e7 −0.512547
\(862\) 1.02321e7i 0.469028i
\(863\) 3.40787e7i 1.55760i 0.627273 + 0.778799i \(0.284171\pi\)
−0.627273 + 0.778799i \(0.715829\pi\)
\(864\) −2.32598e7 −1.06004
\(865\) 0 0
\(866\) −4.19980e7 −1.90298
\(867\) 3.31138e7i 1.49610i
\(868\) 2.95584e7i 1.33162i
\(869\) −7.92002e6 −0.355776
\(870\) 0 0
\(871\) −2.57544e6 −0.115029
\(872\) − 8.03702e6i − 0.357935i
\(873\) − 1.87226e6i − 0.0831441i
\(874\) 1.16543e7 0.516070
\(875\) 0 0
\(876\) −1.74919e6 −0.0770152
\(877\) 1.29489e7i 0.568505i 0.958750 + 0.284252i \(0.0917453\pi\)
−0.958750 + 0.284252i \(0.908255\pi\)
\(878\) 3.65263e7i 1.59908i
\(879\) −1.36751e6 −0.0596976
\(880\) 0 0
\(881\) 8.39894e6 0.364573 0.182287 0.983245i \(-0.441650\pi\)
0.182287 + 0.983245i \(0.441650\pi\)
\(882\) 2.41652e6i 0.104597i
\(883\) − 2.38721e7i − 1.03036i −0.857083 0.515179i \(-0.827725\pi\)
0.857083 0.515179i \(-0.172275\pi\)
\(884\) 7.51528e6 0.323456
\(885\) 0 0
\(886\) −3.44788e7 −1.47560
\(887\) − 8.56803e6i − 0.365655i −0.983145 0.182828i \(-0.941475\pi\)
0.983145 0.182828i \(-0.0585250\pi\)
\(888\) − 6.25602e6i − 0.266236i
\(889\) −1.87436e7 −0.795425
\(890\) 0 0
\(891\) 3.44128e7 1.45220
\(892\) 1.41402e7i 0.595035i
\(893\) 3.60544e6i 0.151297i
\(894\) −1.45039e7 −0.606933
\(895\) 0 0
\(896\) −1.70228e7 −0.708370
\(897\) 1.07362e7i 0.445522i
\(898\) 2.64295e6i 0.109370i
\(899\) 6.64777e7 2.74332
\(900\) 0 0
\(901\) 1.28605e7 0.527769
\(902\) − 1.59107e7i − 0.651139i
\(903\) 1.51015e7i 0.616312i
\(904\) 499449. 0.0203268
\(905\) 0 0
\(906\) 4.28728e7 1.73525
\(907\) 2.84173e7i 1.14700i 0.819204 + 0.573502i \(0.194416\pi\)
−0.819204 + 0.573502i \(0.805584\pi\)
\(908\) − 2.03210e7i − 0.817958i
\(909\) 2.22713e6 0.0893997
\(910\) 0 0
\(911\) −1.90880e7 −0.762015 −0.381008 0.924572i \(-0.624423\pi\)
−0.381008 + 0.924572i \(0.624423\pi\)
\(912\) − 8.90335e6i − 0.354459i
\(913\) − 5.70578e7i − 2.26536i
\(914\) −4.10576e6 −0.162565
\(915\) 0 0
\(916\) −249183. −0.00981251
\(917\) 5.86144e7i 2.30187i
\(918\) 4.39301e7i 1.72050i
\(919\) 1.95252e7 0.762619 0.381310 0.924447i \(-0.375473\pi\)
0.381310 + 0.924447i \(0.375473\pi\)
\(920\) 0 0
\(921\) −3.56913e7 −1.38648
\(922\) − 4.77142e7i − 1.84850i
\(923\) 1.16786e7i 0.451218i
\(924\) −3.11710e7 −1.20108
\(925\) 0 0
\(926\) 2.18359e7 0.836843
\(927\) 3.02978e6i 0.115801i
\(928\) − 5.94235e7i − 2.26510i
\(929\) 1.62698e7 0.618504 0.309252 0.950980i \(-0.399921\pi\)
0.309252 + 0.950980i \(0.399921\pi\)
\(930\) 0 0
\(931\) 2.34487e6 0.0886634
\(932\) − 2.27248e7i − 0.856961i
\(933\) 54145.2i 0.00203637i
\(934\) 2.50540e7 0.939744
\(935\) 0 0
\(936\) 562990. 0.0210044
\(937\) 3.49494e7i 1.30044i 0.759745 + 0.650221i \(0.225324\pi\)
−0.759745 + 0.650221i \(0.774676\pi\)
\(938\) 1.71012e7i 0.634628i
\(939\) 1.70829e7 0.632263
\(940\) 0 0
\(941\) 2.04858e7 0.754187 0.377094 0.926175i \(-0.376923\pi\)
0.377094 + 0.926175i \(0.376923\pi\)
\(942\) 4.22418e7i 1.55101i
\(943\) 1.57340e7i 0.576182i
\(944\) −2.60225e7 −0.950426
\(945\) 0 0
\(946\) −2.15510e7 −0.782961
\(947\) 3.59599e7i 1.30300i 0.758650 + 0.651498i \(0.225859\pi\)
−0.758650 + 0.651498i \(0.774141\pi\)
\(948\) 6.79661e6i 0.245624i
\(949\) −698748. −0.0251858
\(950\) 0 0
\(951\) 3.29938e7 1.18299
\(952\) 1.56058e7i 0.558076i
\(953\) 2.09887e7i 0.748606i 0.927306 + 0.374303i \(0.122118\pi\)
−0.927306 + 0.374303i \(0.877882\pi\)
\(954\) −3.08068e6 −0.109591
\(955\) 0 0
\(956\) −3.30873e7 −1.17089
\(957\) 7.01046e7i 2.47438i
\(958\) − 1.45616e7i − 0.512621i
\(959\) 1.50064e6 0.0526903
\(960\) 0 0
\(961\) 3.71955e7 1.29922
\(962\) 7.99129e6i 0.278406i
\(963\) − 7.05574e6i − 0.245176i
\(964\) −1.45154e6 −0.0503079
\(965\) 0 0
\(966\) 7.12894e7 2.45800
\(967\) 5.30179e6i 0.182329i 0.995836 + 0.0911646i \(0.0290589\pi\)
−0.995836 + 0.0911646i \(0.970941\pi\)
\(968\) 4.69283e6i 0.160971i
\(969\) −1.34249e7 −0.459306
\(970\) 0 0
\(971\) −5.19084e7 −1.76681 −0.883405 0.468611i \(-0.844755\pi\)
−0.883405 + 0.468611i \(0.844755\pi\)
\(972\) − 1.05334e7i − 0.357603i
\(973\) − 6.17588e7i − 2.09130i
\(974\) 4.41700e7 1.49187
\(975\) 0 0
\(976\) −4.32431e7 −1.45309
\(977\) − 5.09506e7i − 1.70771i −0.520515 0.853853i \(-0.674260\pi\)
0.520515 0.853853i \(-0.325740\pi\)
\(978\) 6.14806e7i 2.05537i
\(979\) −1.73782e7 −0.579492
\(980\) 0 0
\(981\) 8.17198e6 0.271116
\(982\) 6.30373e7i 2.08602i
\(983\) − 4.03520e7i − 1.33193i −0.745984 0.665964i \(-0.768020\pi\)
0.745984 0.665964i \(-0.231980\pi\)
\(984\) 4.26994e6 0.140583
\(985\) 0 0
\(986\) −1.12232e8 −3.67640
\(987\) 2.20544e7i 0.720614i
\(988\) 1.74688e6i 0.0569340i
\(989\) 2.13116e7 0.692829
\(990\) 0 0
\(991\) −2.26833e7 −0.733705 −0.366852 0.930279i \(-0.619565\pi\)
−0.366852 + 0.930279i \(0.619565\pi\)
\(992\) − 5.88396e7i − 1.89841i
\(993\) 3.03521e7i 0.976821i
\(994\) 7.75470e7 2.48943
\(995\) 0 0
\(996\) −4.89645e7 −1.56399
\(997\) 840910.i 0.0267924i 0.999910 + 0.0133962i \(0.00426427\pi\)
−0.999910 + 0.0133962i \(0.995736\pi\)
\(998\) − 6.19081e7i − 1.96753i
\(999\) −2.01981e7 −0.640318
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 325.6.b.h.274.15 18
5.2 odd 4 325.6.a.h.1.3 9
5.3 odd 4 325.6.a.i.1.7 yes 9
5.4 even 2 inner 325.6.b.h.274.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.h.1.3 9 5.2 odd 4
325.6.a.i.1.7 yes 9 5.3 odd 4
325.6.b.h.274.4 18 5.4 even 2 inner
325.6.b.h.274.15 18 1.1 even 1 trivial