Properties

Label 384.6.c.d.383.4
Level $384$
Weight $6$
Character 384.383
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
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] = [384,6,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + 18093172325 x^{8} + 74958811500 x^{6} + 79355888475 x^{4} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.4
Root \(2.58939i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.d.383.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-12.3458 + 9.51744i) q^{3} +26.7752i q^{5} -70.6872i q^{7} +(61.8366 - 235.001i) q^{9} +O(q^{10})\) \(q+(-12.3458 + 9.51744i) q^{3} +26.7752i q^{5} -70.6872i q^{7} +(61.8366 - 235.001i) q^{9} -98.7796 q^{11} -210.273 q^{13} +(-254.832 - 330.561i) q^{15} -524.555i q^{17} +1114.76i q^{19} +(672.761 + 872.688i) q^{21} +866.788 q^{23} +2408.09 q^{25} +(1473.18 + 3489.79i) q^{27} -2162.01i q^{29} -1726.91i q^{31} +(1219.51 - 940.129i) q^{33} +1892.66 q^{35} +12357.1 q^{37} +(2595.99 - 2001.26i) q^{39} -1392.98i q^{41} +9131.49i q^{43} +(6292.19 + 1655.69i) q^{45} -21857.9 q^{47} +11810.3 q^{49} +(4992.42 + 6476.04i) q^{51} +7601.83i q^{53} -2644.85i q^{55} +(-10609.7 - 13762.6i) q^{57} -40981.9 q^{59} +24132.2 q^{61} +(-16611.5 - 4371.05i) q^{63} -5630.11i q^{65} +8969.02i q^{67} +(-10701.2 + 8249.60i) q^{69} +4758.61 q^{71} -65015.7 q^{73} +(-29729.7 + 22918.8i) q^{75} +6982.45i q^{77} +100325. i q^{79} +(-51401.5 - 29063.3i) q^{81} +29063.3 q^{83} +14045.1 q^{85} +(20576.8 + 26691.8i) q^{87} +619.603i q^{89} +14863.6i q^{91} +(16435.8 + 21320.1i) q^{93} -29848.1 q^{95} -74920.5 q^{97} +(-6108.19 + 23213.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 948 q^{11} + 852 q^{15} - 1640 q^{21} + 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} - 15056 q^{37} - 12980 q^{39} - 11800 q^{45} + 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} - 73264 q^{61} + 23608 q^{63} + 84024 q^{69} + 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} + 30112 q^{85} - 225700 q^{87} + 148016 q^{93} + 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −12.3458 + 9.51744i −0.791982 + 0.610544i
\(4\) 0 0
\(5\) 26.7752i 0.478970i 0.970900 + 0.239485i \(0.0769786\pi\)
−0.970900 + 0.239485i \(0.923021\pi\)
\(6\) 0 0
\(7\) 70.6872i 0.545250i −0.962120 0.272625i \(-0.912108\pi\)
0.962120 0.272625i \(-0.0878918\pi\)
\(8\) 0 0
\(9\) 61.8366 235.001i 0.254472 0.967080i
\(10\) 0 0
\(11\) −98.7796 −0.246142 −0.123071 0.992398i \(-0.539274\pi\)
−0.123071 + 0.992398i \(0.539274\pi\)
\(12\) 0 0
\(13\) −210.273 −0.345084 −0.172542 0.985002i \(-0.555198\pi\)
−0.172542 + 0.985002i \(0.555198\pi\)
\(14\) 0 0
\(15\) −254.832 330.561i −0.292432 0.379336i
\(16\) 0 0
\(17\) 524.555i 0.440219i −0.975475 0.220109i \(-0.929359\pi\)
0.975475 0.220109i \(-0.0706414\pi\)
\(18\) 0 0
\(19\) 1114.76i 0.708434i 0.935163 + 0.354217i \(0.115253\pi\)
−0.935163 + 0.354217i \(0.884747\pi\)
\(20\) 0 0
\(21\) 672.761 + 872.688i 0.332899 + 0.431828i
\(22\) 0 0
\(23\) 866.788 0.341659 0.170830 0.985301i \(-0.445355\pi\)
0.170830 + 0.985301i \(0.445355\pi\)
\(24\) 0 0
\(25\) 2408.09 0.770588
\(26\) 0 0
\(27\) 1473.18 + 3489.79i 0.388908 + 0.921276i
\(28\) 0 0
\(29\) 2162.01i 0.477379i −0.971096 0.238690i \(-0.923282\pi\)
0.971096 0.238690i \(-0.0767179\pi\)
\(30\) 0 0
\(31\) 1726.91i 0.322750i −0.986893 0.161375i \(-0.948407\pi\)
0.986893 0.161375i \(-0.0515928\pi\)
\(32\) 0 0
\(33\) 1219.51 940.129i 0.194940 0.150281i
\(34\) 0 0
\(35\) 1892.66 0.261158
\(36\) 0 0
\(37\) 12357.1 1.48393 0.741964 0.670440i \(-0.233895\pi\)
0.741964 + 0.670440i \(0.233895\pi\)
\(38\) 0 0
\(39\) 2595.99 2001.26i 0.273301 0.210689i
\(40\) 0 0
\(41\) 1392.98i 0.129415i −0.997904 0.0647075i \(-0.979389\pi\)
0.997904 0.0647075i \(-0.0206114\pi\)
\(42\) 0 0
\(43\) 9131.49i 0.753131i 0.926390 + 0.376566i \(0.122895\pi\)
−0.926390 + 0.376566i \(0.877105\pi\)
\(44\) 0 0
\(45\) 6292.19 + 1655.69i 0.463202 + 0.121884i
\(46\) 0 0
\(47\) −21857.9 −1.44332 −0.721662 0.692246i \(-0.756621\pi\)
−0.721662 + 0.692246i \(0.756621\pi\)
\(48\) 0 0
\(49\) 11810.3 0.702703
\(50\) 0 0
\(51\) 4992.42 + 6476.04i 0.268773 + 0.348645i
\(52\) 0 0
\(53\) 7601.83i 0.371731i 0.982575 + 0.185865i \(0.0595088\pi\)
−0.982575 + 0.185865i \(0.940491\pi\)
\(54\) 0 0
\(55\) 2644.85i 0.117895i
\(56\) 0 0
\(57\) −10609.7 13762.6i −0.432530 0.561067i
\(58\) 0 0
\(59\) −40981.9 −1.53272 −0.766359 0.642413i \(-0.777933\pi\)
−0.766359 + 0.642413i \(0.777933\pi\)
\(60\) 0 0
\(61\) 24132.2 0.830370 0.415185 0.909737i \(-0.363717\pi\)
0.415185 + 0.909737i \(0.363717\pi\)
\(62\) 0 0
\(63\) −16611.5 4371.05i −0.527300 0.138751i
\(64\) 0 0
\(65\) 5630.11i 0.165285i
\(66\) 0 0
\(67\) 8969.02i 0.244094i 0.992524 + 0.122047i \(0.0389459\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(68\) 0 0
\(69\) −10701.2 + 8249.60i −0.270588 + 0.208598i
\(70\) 0 0
\(71\) 4758.61 0.112030 0.0560150 0.998430i \(-0.482161\pi\)
0.0560150 + 0.998430i \(0.482161\pi\)
\(72\) 0 0
\(73\) −65015.7 −1.42794 −0.713972 0.700174i \(-0.753106\pi\)
−0.713972 + 0.700174i \(0.753106\pi\)
\(74\) 0 0
\(75\) −29729.7 + 22918.8i −0.610292 + 0.470478i
\(76\) 0 0
\(77\) 6982.45i 0.134209i
\(78\) 0 0
\(79\) 100325.i 1.80860i 0.426902 + 0.904298i \(0.359605\pi\)
−0.426902 + 0.904298i \(0.640395\pi\)
\(80\) 0 0
\(81\) −51401.5 29063.3i −0.870488 0.492189i
\(82\) 0 0
\(83\) 29063.3 0.463073 0.231537 0.972826i \(-0.425625\pi\)
0.231537 + 0.972826i \(0.425625\pi\)
\(84\) 0 0
\(85\) 14045.1 0.210851
\(86\) 0 0
\(87\) 20576.8 + 26691.8i 0.291461 + 0.378076i
\(88\) 0 0
\(89\) 619.603i 0.00829160i 0.999991 + 0.00414580i \(0.00131965\pi\)
−0.999991 + 0.00414580i \(0.998680\pi\)
\(90\) 0 0
\(91\) 14863.6i 0.188157i
\(92\) 0 0
\(93\) 16435.8 + 21320.1i 0.197053 + 0.255612i
\(94\) 0 0
\(95\) −29848.1 −0.339318
\(96\) 0 0
\(97\) −74920.5 −0.808484 −0.404242 0.914652i \(-0.632465\pi\)
−0.404242 + 0.914652i \(0.632465\pi\)
\(98\) 0 0
\(99\) −6108.19 + 23213.3i −0.0626361 + 0.238039i
\(100\) 0 0
\(101\) 154823.i 1.51019i 0.655617 + 0.755094i \(0.272409\pi\)
−0.655617 + 0.755094i \(0.727591\pi\)
\(102\) 0 0
\(103\) 156724.i 1.45560i 0.685790 + 0.727799i \(0.259457\pi\)
−0.685790 + 0.727799i \(0.740543\pi\)
\(104\) 0 0
\(105\) −23366.4 + 18013.3i −0.206833 + 0.159449i
\(106\) 0 0
\(107\) −163885. −1.38382 −0.691910 0.721984i \(-0.743231\pi\)
−0.691910 + 0.721984i \(0.743231\pi\)
\(108\) 0 0
\(109\) 58919.6 0.475000 0.237500 0.971388i \(-0.423672\pi\)
0.237500 + 0.971388i \(0.423672\pi\)
\(110\) 0 0
\(111\) −152558. + 117608.i −1.17524 + 0.906003i
\(112\) 0 0
\(113\) 206289.i 1.51978i −0.650051 0.759890i \(-0.725253\pi\)
0.650051 0.759890i \(-0.274747\pi\)
\(114\) 0 0
\(115\) 23208.4i 0.163644i
\(116\) 0 0
\(117\) −13002.6 + 49414.3i −0.0878142 + 0.333724i
\(118\) 0 0
\(119\) −37079.3 −0.240029
\(120\) 0 0
\(121\) −151294. −0.939414
\(122\) 0 0
\(123\) 13257.6 + 17197.4i 0.0790135 + 0.102494i
\(124\) 0 0
\(125\) 148150.i 0.848058i
\(126\) 0 0
\(127\) 34658.0i 0.190675i 0.995445 + 0.0953375i \(0.0303930\pi\)
−0.995445 + 0.0953375i \(0.969607\pi\)
\(128\) 0 0
\(129\) −86908.4 112735.i −0.459820 0.596466i
\(130\) 0 0
\(131\) −146438. −0.745548 −0.372774 0.927922i \(-0.621593\pi\)
−0.372774 + 0.927922i \(0.621593\pi\)
\(132\) 0 0
\(133\) 78799.5 0.386273
\(134\) 0 0
\(135\) −93439.9 + 39444.8i −0.441264 + 0.186275i
\(136\) 0 0
\(137\) 78954.6i 0.359398i 0.983722 + 0.179699i \(0.0575124\pi\)
−0.983722 + 0.179699i \(0.942488\pi\)
\(138\) 0 0
\(139\) 284001.i 1.24676i 0.781919 + 0.623380i \(0.214241\pi\)
−0.781919 + 0.623380i \(0.785759\pi\)
\(140\) 0 0
\(141\) 269853. 208031.i 1.14309 0.881213i
\(142\) 0 0
\(143\) 20770.7 0.0849398
\(144\) 0 0
\(145\) 57888.4 0.228650
\(146\) 0 0
\(147\) −145808. + 112404.i −0.556528 + 0.429031i
\(148\) 0 0
\(149\) 174820.i 0.645097i 0.946553 + 0.322549i \(0.104540\pi\)
−0.946553 + 0.322549i \(0.895460\pi\)
\(150\) 0 0
\(151\) 90430.8i 0.322756i −0.986893 0.161378i \(-0.948406\pi\)
0.986893 0.161378i \(-0.0515938\pi\)
\(152\) 0 0
\(153\) −123271. 32436.7i −0.425727 0.112023i
\(154\) 0 0
\(155\) 46238.4 0.154587
\(156\) 0 0
\(157\) 435845. 1.41118 0.705591 0.708619i \(-0.250681\pi\)
0.705591 + 0.708619i \(0.250681\pi\)
\(158\) 0 0
\(159\) −72350.0 93850.5i −0.226958 0.294404i
\(160\) 0 0
\(161\) 61270.8i 0.186290i
\(162\) 0 0
\(163\) 357566.i 1.05411i 0.849830 + 0.527057i \(0.176704\pi\)
−0.849830 + 0.527057i \(0.823296\pi\)
\(164\) 0 0
\(165\) 25172.2 + 32652.7i 0.0719798 + 0.0933704i
\(166\) 0 0
\(167\) −206409. −0.572714 −0.286357 0.958123i \(-0.592444\pi\)
−0.286357 + 0.958123i \(0.592444\pi\)
\(168\) 0 0
\(169\) −327078. −0.880917
\(170\) 0 0
\(171\) 261970. + 68933.2i 0.685112 + 0.180276i
\(172\) 0 0
\(173\) 425865.i 1.08183i −0.841079 0.540913i \(-0.818079\pi\)
0.841079 0.540913i \(-0.181921\pi\)
\(174\) 0 0
\(175\) 170221.i 0.420163i
\(176\) 0 0
\(177\) 505954. 390043.i 1.21388 0.935792i
\(178\) 0 0
\(179\) −275598. −0.642900 −0.321450 0.946927i \(-0.604170\pi\)
−0.321450 + 0.946927i \(0.604170\pi\)
\(180\) 0 0
\(181\) −160334. −0.363773 −0.181886 0.983320i \(-0.558220\pi\)
−0.181886 + 0.983320i \(0.558220\pi\)
\(182\) 0 0
\(183\) −297930. + 229676.i −0.657638 + 0.506977i
\(184\) 0 0
\(185\) 330864.i 0.710756i
\(186\) 0 0
\(187\) 51815.3i 0.108356i
\(188\) 0 0
\(189\) 246683. 104135.i 0.502326 0.212052i
\(190\) 0 0
\(191\) 449025. 0.890608 0.445304 0.895379i \(-0.353096\pi\)
0.445304 + 0.895379i \(0.353096\pi\)
\(192\) 0 0
\(193\) 467575. 0.903562 0.451781 0.892129i \(-0.350789\pi\)
0.451781 + 0.892129i \(0.350789\pi\)
\(194\) 0 0
\(195\) 53584.2 + 69508.1i 0.100914 + 0.130903i
\(196\) 0 0
\(197\) 888657.i 1.63143i 0.578453 + 0.815715i \(0.303656\pi\)
−0.578453 + 0.815715i \(0.696344\pi\)
\(198\) 0 0
\(199\) 90005.4i 0.161115i −0.996750 0.0805575i \(-0.974330\pi\)
0.996750 0.0805575i \(-0.0256701\pi\)
\(200\) 0 0
\(201\) −85362.1 110729.i −0.149030 0.193318i
\(202\) 0 0
\(203\) −152827. −0.260291
\(204\) 0 0
\(205\) 37297.3 0.0619858
\(206\) 0 0
\(207\) 53599.2 203696.i 0.0869426 0.330412i
\(208\) 0 0
\(209\) 110116.i 0.174375i
\(210\) 0 0
\(211\) 272052.i 0.420674i −0.977629 0.210337i \(-0.932544\pi\)
0.977629 0.210337i \(-0.0674562\pi\)
\(212\) 0 0
\(213\) −58748.8 + 45289.8i −0.0887258 + 0.0683993i
\(214\) 0 0
\(215\) −244498. −0.360727
\(216\) 0 0
\(217\) −122070. −0.175979
\(218\) 0 0
\(219\) 802670. 618783.i 1.13091 0.871823i
\(220\) 0 0
\(221\) 110300.i 0.151913i
\(222\) 0 0
\(223\) 400949.i 0.539917i 0.962872 + 0.269958i \(0.0870100\pi\)
−0.962872 + 0.269958i \(0.912990\pi\)
\(224\) 0 0
\(225\) 148908. 565902.i 0.196093 0.745220i
\(226\) 0 0
\(227\) −388147. −0.499955 −0.249978 0.968252i \(-0.580423\pi\)
−0.249978 + 0.968252i \(0.580423\pi\)
\(228\) 0 0
\(229\) −999721. −1.25977 −0.629883 0.776690i \(-0.716897\pi\)
−0.629883 + 0.776690i \(0.716897\pi\)
\(230\) 0 0
\(231\) −66455.1 86203.8i −0.0819404 0.106291i
\(232\) 0 0
\(233\) 1.53632e6i 1.85392i 0.375163 + 0.926959i \(0.377587\pi\)
−0.375163 + 0.926959i \(0.622413\pi\)
\(234\) 0 0
\(235\) 585250.i 0.691308i
\(236\) 0 0
\(237\) −954837. 1.23859e6i −1.10423 1.43238i
\(238\) 0 0
\(239\) 1.24345e6 1.40810 0.704051 0.710150i \(-0.251373\pi\)
0.704051 + 0.710150i \(0.251373\pi\)
\(240\) 0 0
\(241\) 227499. 0.252311 0.126156 0.992010i \(-0.459736\pi\)
0.126156 + 0.992010i \(0.459736\pi\)
\(242\) 0 0
\(243\) 911199. 130402.i 0.989914 0.141667i
\(244\) 0 0
\(245\) 316224.i 0.336573i
\(246\) 0 0
\(247\) 234405.i 0.244469i
\(248\) 0 0
\(249\) −358809. + 276608.i −0.366746 + 0.282727i
\(250\) 0 0
\(251\) −113966. −0.114180 −0.0570899 0.998369i \(-0.518182\pi\)
−0.0570899 + 0.998369i \(0.518182\pi\)
\(252\) 0 0
\(253\) −85621.0 −0.0840967
\(254\) 0 0
\(255\) −173397. + 133673.i −0.166991 + 0.128734i
\(256\) 0 0
\(257\) 114461.i 0.108100i 0.998538 + 0.0540501i \(0.0172131\pi\)
−0.998538 + 0.0540501i \(0.982787\pi\)
\(258\) 0 0
\(259\) 873489.i 0.809111i
\(260\) 0 0
\(261\) −508074. 133692.i −0.461664 0.121479i
\(262\) 0 0
\(263\) 1.92150e6 1.71297 0.856487 0.516168i \(-0.172642\pi\)
0.856487 + 0.516168i \(0.172642\pi\)
\(264\) 0 0
\(265\) −203541. −0.178048
\(266\) 0 0
\(267\) −5897.03 7649.48i −0.00506239 0.00656680i
\(268\) 0 0
\(269\) 2.09389e6i 1.76431i −0.470964 0.882153i \(-0.656094\pi\)
0.470964 0.882153i \(-0.343906\pi\)
\(270\) 0 0
\(271\) 536501.i 0.443759i −0.975074 0.221880i \(-0.928781\pi\)
0.975074 0.221880i \(-0.0712192\pi\)
\(272\) 0 0
\(273\) −141464. 183503.i −0.114878 0.149017i
\(274\) 0 0
\(275\) −237870. −0.189674
\(276\) 0 0
\(277\) −1.88269e6 −1.47428 −0.737141 0.675739i \(-0.763825\pi\)
−0.737141 + 0.675739i \(0.763825\pi\)
\(278\) 0 0
\(279\) −405825. 106786.i −0.312125 0.0821306i
\(280\) 0 0
\(281\) 2.07557e6i 1.56809i 0.620705 + 0.784045i \(0.286847\pi\)
−0.620705 + 0.784045i \(0.713153\pi\)
\(282\) 0 0
\(283\) 959147.i 0.711900i −0.934505 0.355950i \(-0.884157\pi\)
0.934505 0.355950i \(-0.115843\pi\)
\(284\) 0 0
\(285\) 368498. 284077.i 0.268734 0.207169i
\(286\) 0 0
\(287\) −98465.6 −0.0705635
\(288\) 0 0
\(289\) 1.14470e6 0.806207
\(290\) 0 0
\(291\) 924952. 713051.i 0.640305 0.493615i
\(292\) 0 0
\(293\) 1.40544e6i 0.956406i 0.878249 + 0.478203i \(0.158712\pi\)
−0.878249 + 0.478203i \(0.841288\pi\)
\(294\) 0 0
\(295\) 1.09730e6i 0.734125i
\(296\) 0 0
\(297\) −145520. 344720.i −0.0957267 0.226765i
\(298\) 0 0
\(299\) −182262. −0.117901
\(300\) 0 0
\(301\) 645479. 0.410645
\(302\) 0 0
\(303\) −1.47352e6 1.91141e6i −0.922036 1.19604i
\(304\) 0 0
\(305\) 646144.i 0.397722i
\(306\) 0 0
\(307\) 2.45801e6i 1.48846i −0.667921 0.744232i \(-0.732816\pi\)
0.667921 0.744232i \(-0.267184\pi\)
\(308\) 0 0
\(309\) −1.49161e6 1.93488e6i −0.888707 1.15281i
\(310\) 0 0
\(311\) 2.05824e6 1.20669 0.603344 0.797481i \(-0.293835\pi\)
0.603344 + 0.797481i \(0.293835\pi\)
\(312\) 0 0
\(313\) 1.08280e6 0.624722 0.312361 0.949964i \(-0.398880\pi\)
0.312361 + 0.949964i \(0.398880\pi\)
\(314\) 0 0
\(315\) 117036. 444777.i 0.0664573 0.252561i
\(316\) 0 0
\(317\) 2.77359e6i 1.55022i 0.631824 + 0.775112i \(0.282306\pi\)
−0.631824 + 0.775112i \(0.717694\pi\)
\(318\) 0 0
\(319\) 213563.i 0.117503i
\(320\) 0 0
\(321\) 2.02329e6 1.55977e6i 1.09596 0.844884i
\(322\) 0 0
\(323\) 584755. 0.311866
\(324\) 0 0
\(325\) −506356. −0.265918
\(326\) 0 0
\(327\) −727408. + 560764.i −0.376191 + 0.290008i
\(328\) 0 0
\(329\) 1.54507e6i 0.786972i
\(330\) 0 0
\(331\) 1.19074e6i 0.597377i 0.954351 + 0.298689i \(0.0965492\pi\)
−0.954351 + 0.298689i \(0.903451\pi\)
\(332\) 0 0
\(333\) 764122. 2.90393e6i 0.377617 1.43508i
\(334\) 0 0
\(335\) −240147. −0.116914
\(336\) 0 0
\(337\) 2.18912e6 1.05001 0.525006 0.851099i \(-0.324063\pi\)
0.525006 + 0.851099i \(0.324063\pi\)
\(338\) 0 0
\(339\) 1.96335e6 + 2.54680e6i 0.927893 + 1.20364i
\(340\) 0 0
\(341\) 170584.i 0.0794422i
\(342\) 0 0
\(343\) 2.02288e6i 0.928398i
\(344\) 0 0
\(345\) −220885. 286526.i −0.0999122 0.129604i
\(346\) 0 0
\(347\) −806476. −0.359557 −0.179778 0.983707i \(-0.557538\pi\)
−0.179778 + 0.983707i \(0.557538\pi\)
\(348\) 0 0
\(349\) 1.13913e6 0.500624 0.250312 0.968165i \(-0.419467\pi\)
0.250312 + 0.968165i \(0.419467\pi\)
\(350\) 0 0
\(351\) −309771. 733809.i −0.134206 0.317918i
\(352\) 0 0
\(353\) 405481.i 0.173195i 0.996243 + 0.0865973i \(0.0275993\pi\)
−0.996243 + 0.0865973i \(0.972401\pi\)
\(354\) 0 0
\(355\) 127413.i 0.0536590i
\(356\) 0 0
\(357\) 457773. 352900.i 0.190099 0.146548i
\(358\) 0 0
\(359\) 3.48388e6 1.42668 0.713341 0.700817i \(-0.247181\pi\)
0.713341 + 0.700817i \(0.247181\pi\)
\(360\) 0 0
\(361\) 1.23340e6 0.498122
\(362\) 0 0
\(363\) 1.86784e6 1.43993e6i 0.743999 0.573554i
\(364\) 0 0
\(365\) 1.74081e6i 0.683942i
\(366\) 0 0
\(367\) 430810.i 0.166963i 0.996509 + 0.0834816i \(0.0266040\pi\)
−0.996509 + 0.0834816i \(0.973396\pi\)
\(368\) 0 0
\(369\) −327350. 86137.0i −0.125155 0.0329324i
\(370\) 0 0
\(371\) 537352. 0.202686
\(372\) 0 0
\(373\) 1.72685e6 0.642661 0.321330 0.946967i \(-0.395870\pi\)
0.321330 + 0.946967i \(0.395870\pi\)
\(374\) 0 0
\(375\) −1.41001e6 1.82902e6i −0.517777 0.671647i
\(376\) 0 0
\(377\) 454614.i 0.164736i
\(378\) 0 0
\(379\) 4.14141e6i 1.48098i 0.672066 + 0.740491i \(0.265407\pi\)
−0.672066 + 0.740491i \(0.734593\pi\)
\(380\) 0 0
\(381\) −329855. 427880.i −0.116416 0.151011i
\(382\) 0 0
\(383\) 1.77835e6 0.619469 0.309734 0.950823i \(-0.399760\pi\)
0.309734 + 0.950823i \(0.399760\pi\)
\(384\) 0 0
\(385\) −186957. −0.0642820
\(386\) 0 0
\(387\) 2.14591e6 + 564660.i 0.728338 + 0.191650i
\(388\) 0 0
\(389\) 2.42987e6i 0.814159i 0.913393 + 0.407079i \(0.133453\pi\)
−0.913393 + 0.407079i \(0.866547\pi\)
\(390\) 0 0
\(391\) 454678.i 0.150405i
\(392\) 0 0
\(393\) 1.80789e6 1.39372e6i 0.590461 0.455190i
\(394\) 0 0
\(395\) −2.68622e6 −0.866263
\(396\) 0 0
\(397\) −2.87276e6 −0.914794 −0.457397 0.889263i \(-0.651218\pi\)
−0.457397 + 0.889263i \(0.651218\pi\)
\(398\) 0 0
\(399\) −972842. + 749970.i −0.305921 + 0.235837i
\(400\) 0 0
\(401\) 622456.i 0.193307i 0.995318 + 0.0966535i \(0.0308139\pi\)
−0.995318 + 0.0966535i \(0.969186\pi\)
\(402\) 0 0
\(403\) 363123.i 0.111376i
\(404\) 0 0
\(405\) 778175. 1.37629e6i 0.235744 0.416938i
\(406\) 0 0
\(407\) −1.22063e6 −0.365257
\(408\) 0 0
\(409\) −2.35930e6 −0.697389 −0.348694 0.937236i \(-0.613375\pi\)
−0.348694 + 0.937236i \(0.613375\pi\)
\(410\) 0 0
\(411\) −751446. 974756.i −0.219429 0.284637i
\(412\) 0 0
\(413\) 2.89689e6i 0.835714i
\(414\) 0 0
\(415\) 778177.i 0.221798i
\(416\) 0 0
\(417\) −2.70296e6 3.50622e6i −0.761202 0.987412i
\(418\) 0 0
\(419\) −2.85277e6 −0.793839 −0.396919 0.917854i \(-0.629921\pi\)
−0.396919 + 0.917854i \(0.629921\pi\)
\(420\) 0 0
\(421\) 2.61879e6 0.720105 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(422\) 0 0
\(423\) −1.35162e6 + 5.13662e6i −0.367285 + 1.39581i
\(424\) 0 0
\(425\) 1.26317e6i 0.339227i
\(426\) 0 0
\(427\) 1.70583e6i 0.452759i
\(428\) 0 0
\(429\) −256430. + 197684.i −0.0672708 + 0.0518595i
\(430\) 0 0
\(431\) −1.48960e6 −0.386257 −0.193128 0.981173i \(-0.561863\pi\)
−0.193128 + 0.981173i \(0.561863\pi\)
\(432\) 0 0
\(433\) −3.69553e6 −0.947233 −0.473617 0.880731i \(-0.657052\pi\)
−0.473617 + 0.880731i \(0.657052\pi\)
\(434\) 0 0
\(435\) −714678. + 550950.i −0.181087 + 0.139601i
\(436\) 0 0
\(437\) 966264.i 0.242043i
\(438\) 0 0
\(439\) 1.97023e6i 0.487927i −0.969784 0.243963i \(-0.921552\pi\)
0.969784 0.243963i \(-0.0784477\pi\)
\(440\) 0 0
\(441\) 730310. 2.77543e6i 0.178818 0.679570i
\(442\) 0 0
\(443\) −3.26291e6 −0.789943 −0.394971 0.918693i \(-0.629245\pi\)
−0.394971 + 0.918693i \(0.629245\pi\)
\(444\) 0 0
\(445\) −16590.0 −0.00397142
\(446\) 0 0
\(447\) −1.66384e6 2.15829e6i −0.393860 0.510906i
\(448\) 0 0
\(449\) 5.83012e6i 1.36478i 0.730989 + 0.682389i \(0.239059\pi\)
−0.730989 + 0.682389i \(0.760941\pi\)
\(450\) 0 0
\(451\) 137598.i 0.0318544i
\(452\) 0 0
\(453\) 860669. + 1.11644e6i 0.197057 + 0.255617i
\(454\) 0 0
\(455\) −397976. −0.0901216
\(456\) 0 0
\(457\) −3.58653e6 −0.803310 −0.401655 0.915791i \(-0.631565\pi\)
−0.401655 + 0.915791i \(0.631565\pi\)
\(458\) 0 0
\(459\) 1.83059e6 772765.i 0.405563 0.171205i
\(460\) 0 0
\(461\) 3.80727e6i 0.834376i 0.908820 + 0.417188i \(0.136984\pi\)
−0.908820 + 0.417188i \(0.863016\pi\)
\(462\) 0 0
\(463\) 7.35067e6i 1.59358i 0.604255 + 0.796791i \(0.293471\pi\)
−0.604255 + 0.796791i \(0.706529\pi\)
\(464\) 0 0
\(465\) −570850. + 440072.i −0.122430 + 0.0943824i
\(466\) 0 0
\(467\) −8.39614e6 −1.78151 −0.890753 0.454488i \(-0.849822\pi\)
−0.890753 + 0.454488i \(0.849822\pi\)
\(468\) 0 0
\(469\) 633994. 0.133092
\(470\) 0 0
\(471\) −5.38085e6 + 4.14813e6i −1.11763 + 0.861589i
\(472\) 0 0
\(473\) 902005.i 0.185377i
\(474\) 0 0
\(475\) 2.68445e6i 0.545910i
\(476\) 0 0
\(477\) 1.78643e6 + 470071.i 0.359493 + 0.0945949i
\(478\) 0 0
\(479\) 4.74882e6 0.945686 0.472843 0.881147i \(-0.343228\pi\)
0.472843 + 0.881147i \(0.343228\pi\)
\(480\) 0 0
\(481\) −2.59837e6 −0.512080
\(482\) 0 0
\(483\) 583141. + 756436.i 0.113738 + 0.147538i
\(484\) 0 0
\(485\) 2.00601e6i 0.387239i
\(486\) 0 0
\(487\) 5.94838e6i 1.13652i 0.822850 + 0.568259i \(0.192383\pi\)
−0.822850 + 0.568259i \(0.807617\pi\)
\(488\) 0 0
\(489\) −3.40312e6 4.41443e6i −0.643583 0.834839i
\(490\) 0 0
\(491\) 3.20829e6 0.600579 0.300289 0.953848i \(-0.402917\pi\)
0.300289 + 0.953848i \(0.402917\pi\)
\(492\) 0 0
\(493\) −1.13410e6 −0.210151
\(494\) 0 0
\(495\) −621540. 163548.i −0.114013 0.0300008i
\(496\) 0 0
\(497\) 336373.i 0.0610843i
\(498\) 0 0
\(499\) 3.23102e6i 0.580882i −0.956893 0.290441i \(-0.906198\pi\)
0.956893 0.290441i \(-0.0938021\pi\)
\(500\) 0 0
\(501\) 2.54828e6 1.96449e6i 0.453579 0.349667i
\(502\) 0 0
\(503\) −7.43429e6 −1.31014 −0.655072 0.755566i \(-0.727362\pi\)
−0.655072 + 0.755566i \(0.727362\pi\)
\(504\) 0 0
\(505\) −4.14541e6 −0.723334
\(506\) 0 0
\(507\) 4.03804e6 3.11295e6i 0.697670 0.537839i
\(508\) 0 0
\(509\) 8.05706e6i 1.37842i 0.724560 + 0.689211i \(0.242043\pi\)
−0.724560 + 0.689211i \(0.757957\pi\)
\(510\) 0 0
\(511\) 4.59578e6i 0.778586i
\(512\) 0 0
\(513\) −3.89029e6 + 1.64225e6i −0.652663 + 0.275516i
\(514\) 0 0
\(515\) −4.19631e6 −0.697188
\(516\) 0 0
\(517\) 2.15911e6 0.355262
\(518\) 0 0
\(519\) 4.05315e6 + 5.25764e6i 0.660502 + 0.856786i
\(520\) 0 0
\(521\) 1.22668e6i 0.197988i −0.995088 0.0989939i \(-0.968438\pi\)
0.995088 0.0989939i \(-0.0315624\pi\)
\(522\) 0 0
\(523\) 6.04029e6i 0.965614i 0.875727 + 0.482807i \(0.160383\pi\)
−0.875727 + 0.482807i \(0.839617\pi\)
\(524\) 0 0
\(525\) 1.62007e6 + 2.10151e6i 0.256528 + 0.332762i
\(526\) 0 0
\(527\) −905860. −0.142080
\(528\) 0 0
\(529\) −5.68502e6 −0.883269
\(530\) 0 0
\(531\) −2.53418e6 + 9.63077e6i −0.390033 + 1.48226i
\(532\) 0 0
\(533\) 292906.i 0.0446591i
\(534\) 0 0
\(535\) 4.38806e6i 0.662808i
\(536\) 0 0
\(537\) 3.40247e6 2.62299e6i 0.509165 0.392519i
\(538\) 0 0
\(539\) −1.16662e6 −0.172965
\(540\) 0 0
\(541\) 585938. 0.0860714 0.0430357 0.999074i \(-0.486297\pi\)
0.0430357 + 0.999074i \(0.486297\pi\)
\(542\) 0 0
\(543\) 1.97945e6 1.52597e6i 0.288102 0.222099i
\(544\) 0 0
\(545\) 1.57758e6i 0.227511i
\(546\) 0 0
\(547\) 1.72451e6i 0.246432i 0.992380 + 0.123216i \(0.0393208\pi\)
−0.992380 + 0.123216i \(0.960679\pi\)
\(548\) 0 0
\(549\) 1.49225e6 5.67107e6i 0.211305 0.803034i
\(550\) 0 0
\(551\) 2.41014e6 0.338192
\(552\) 0 0
\(553\) 7.09169e6 0.986136
\(554\) 0 0
\(555\) −3.14898e6 4.08478e6i −0.433948 0.562906i
\(556\) 0 0
\(557\) 7.06092e6i 0.964325i 0.876082 + 0.482163i \(0.160149\pi\)
−0.876082 + 0.482163i \(0.839851\pi\)
\(558\) 0 0
\(559\) 1.92011e6i 0.259894i
\(560\) 0 0
\(561\) −493149. 639701.i −0.0661563 0.0858163i
\(562\) 0 0
\(563\) 1.79606e6 0.238809 0.119404 0.992846i \(-0.461901\pi\)
0.119404 + 0.992846i \(0.461901\pi\)
\(564\) 0 0
\(565\) 5.52345e6 0.727929
\(566\) 0 0
\(567\) −2.05440e6 + 3.63342e6i −0.268366 + 0.474634i
\(568\) 0 0
\(569\) 1.28895e7i 1.66899i −0.551012 0.834497i \(-0.685758\pi\)
0.551012 0.834497i \(-0.314242\pi\)
\(570\) 0 0
\(571\) 9.88344e6i 1.26858i 0.773095 + 0.634290i \(0.218707\pi\)
−0.773095 + 0.634290i \(0.781293\pi\)
\(572\) 0 0
\(573\) −5.54356e6 + 4.27357e6i −0.705346 + 0.543756i
\(574\) 0 0
\(575\) 2.08730e6 0.263279
\(576\) 0 0
\(577\) −7.24804e6 −0.906319 −0.453160 0.891429i \(-0.649703\pi\)
−0.453160 + 0.891429i \(0.649703\pi\)
\(578\) 0 0
\(579\) −5.77258e6 + 4.45012e6i −0.715605 + 0.551665i
\(580\) 0 0
\(581\) 2.05440e6i 0.252491i
\(582\) 0 0
\(583\) 750906.i 0.0914985i
\(584\) 0 0
\(585\) −1.32308e6 348147.i −0.159844 0.0420603i
\(586\) 0 0
\(587\) −1.17837e6 −0.141152 −0.0705758 0.997506i \(-0.522484\pi\)
−0.0705758 + 0.997506i \(0.522484\pi\)
\(588\) 0 0
\(589\) 1.92510e6 0.228647
\(590\) 0 0
\(591\) −8.45774e6 1.09712e7i −0.996061 1.29206i
\(592\) 0 0
\(593\) 3.11009e6i 0.363191i −0.983373 0.181596i \(-0.941874\pi\)
0.983373 0.181596i \(-0.0581262\pi\)
\(594\) 0 0
\(595\) 992806.i 0.114967i
\(596\) 0 0
\(597\) 856621. + 1.11119e6i 0.0983678 + 0.127600i
\(598\) 0 0
\(599\) −8.95969e6 −1.02030 −0.510148 0.860087i \(-0.670409\pi\)
−0.510148 + 0.860087i \(0.670409\pi\)
\(600\) 0 0
\(601\) 1.54483e7 1.74459 0.872294 0.488981i \(-0.162631\pi\)
0.872294 + 0.488981i \(0.162631\pi\)
\(602\) 0 0
\(603\) 2.10772e6 + 554613.i 0.236059 + 0.0621151i
\(604\) 0 0
\(605\) 4.05092e6i 0.449951i
\(606\) 0 0
\(607\) 1.11892e7i 1.23261i 0.787507 + 0.616306i \(0.211371\pi\)
−0.787507 + 0.616306i \(0.788629\pi\)
\(608\) 0 0
\(609\) 1.88676e6 1.45452e6i 0.206146 0.158919i
\(610\) 0 0
\(611\) 4.59613e6 0.498068
\(612\) 0 0
\(613\) 7.89608e6 0.848713 0.424356 0.905495i \(-0.360500\pi\)
0.424356 + 0.905495i \(0.360500\pi\)
\(614\) 0 0
\(615\) −460464. + 354975.i −0.0490917 + 0.0378451i
\(616\) 0 0
\(617\) 1.91077e6i 0.202067i −0.994883 0.101033i \(-0.967785\pi\)
0.994883 0.101033i \(-0.0322149\pi\)
\(618\) 0 0
\(619\) 2.94989e6i 0.309441i 0.987958 + 0.154721i \(0.0494478\pi\)
−0.987958 + 0.154721i \(0.950552\pi\)
\(620\) 0 0
\(621\) 1.27694e6 + 3.02491e6i 0.132874 + 0.314763i
\(622\) 0 0
\(623\) 43798.0 0.00452099
\(624\) 0 0
\(625\) 3.55853e6 0.364394
\(626\) 0 0
\(627\) 1.04802e6 + 1.35947e6i 0.106464 + 0.138102i
\(628\) 0 0
\(629\) 6.48198e6i 0.653253i
\(630\) 0 0
\(631\) 1.88150e7i 1.88118i −0.339546 0.940589i \(-0.610274\pi\)
0.339546 0.940589i \(-0.389726\pi\)
\(632\) 0 0
\(633\) 2.58924e6 + 3.35870e6i 0.256840 + 0.333167i
\(634\) 0 0
\(635\) −927975. −0.0913276
\(636\) 0 0
\(637\) −2.48339e6 −0.242492
\(638\) 0 0
\(639\) 294256. 1.11828e6i 0.0285085 0.108342i
\(640\) 0 0
\(641\) 1.09172e7i 1.04946i 0.851267 + 0.524732i \(0.175835\pi\)
−0.851267 + 0.524732i \(0.824165\pi\)
\(642\) 0 0
\(643\) 1.00191e7i 0.955659i −0.878453 0.477829i \(-0.841424\pi\)
0.878453 0.477829i \(-0.158576\pi\)
\(644\) 0 0
\(645\) 3.01852e6 2.32699e6i 0.285689 0.220240i
\(646\) 0 0
\(647\) 1.37444e7 1.29082 0.645411 0.763836i \(-0.276686\pi\)
0.645411 + 0.763836i \(0.276686\pi\)
\(648\) 0 0
\(649\) 4.04818e6 0.377266
\(650\) 0 0
\(651\) 1.50706e6 1.16180e6i 0.139372 0.107443i
\(652\) 0 0
\(653\) 6.03271e6i 0.553643i 0.960921 + 0.276821i \(0.0892811\pi\)
−0.960921 + 0.276821i \(0.910719\pi\)
\(654\) 0 0
\(655\) 3.92091e6i 0.357095i
\(656\) 0 0
\(657\) −4.02035e6 + 1.52787e7i −0.363371 + 1.38094i
\(658\) 0 0
\(659\) 1.47900e7 1.32664 0.663320 0.748336i \(-0.269147\pi\)
0.663320 + 0.748336i \(0.269147\pi\)
\(660\) 0 0
\(661\) 1.07054e7 0.953010 0.476505 0.879172i \(-0.341903\pi\)
0.476505 + 0.879172i \(0.341903\pi\)
\(662\) 0 0
\(663\) −1.04977e6 1.36174e6i −0.0927494 0.120312i
\(664\) 0 0
\(665\) 2.10988e6i 0.185013i
\(666\) 0 0
\(667\) 1.87401e6i 0.163101i
\(668\) 0 0
\(669\) −3.81601e6 4.95003e6i −0.329643 0.427605i
\(670\) 0 0
\(671\) −2.38377e6 −0.204389
\(672\) 0 0
\(673\) −4.90789e6 −0.417693 −0.208847 0.977948i \(-0.566971\pi\)
−0.208847 + 0.977948i \(0.566971\pi\)
\(674\) 0 0
\(675\) 3.54755e6 + 8.40372e6i 0.299688 + 0.709925i
\(676\) 0 0
\(677\) 1.28439e7i 1.07702i −0.842618 0.538512i \(-0.818987\pi\)
0.842618 0.538512i \(-0.181013\pi\)
\(678\) 0 0
\(679\) 5.29592e6i 0.440825i
\(680\) 0 0
\(681\) 4.79197e6 3.69416e6i 0.395956 0.305245i
\(682\) 0 0
\(683\) −2.31400e6 −0.189807 −0.0949033 0.995486i \(-0.530254\pi\)
−0.0949033 + 0.995486i \(0.530254\pi\)
\(684\) 0 0
\(685\) −2.11403e6 −0.172141
\(686\) 0 0
\(687\) 1.23423e7 9.51478e6i 0.997713 0.769143i
\(688\) 0 0
\(689\) 1.59846e6i 0.128279i
\(690\) 0 0
\(691\) 1.02912e7i 0.819923i −0.912103 0.409961i \(-0.865542\pi\)
0.912103 0.409961i \(-0.134458\pi\)
\(692\) 0 0
\(693\) 1.64088e6 + 431771.i 0.129791 + 0.0341523i
\(694\) 0 0
\(695\) −7.60419e6 −0.597161
\(696\) 0 0
\(697\) −730693. −0.0569709
\(698\) 0 0
\(699\) −1.46218e7 1.89670e7i −1.13190 1.46827i
\(700\) 0 0
\(701\) 1.72471e7i 1.32562i −0.748786 0.662811i \(-0.769363\pi\)
0.748786 0.662811i \(-0.230637\pi\)
\(702\) 0 0
\(703\) 1.37753e7i 1.05126i
\(704\) 0 0
\(705\) 5.57008e6 + 7.22537e6i 0.422074 + 0.547504i
\(706\) 0 0
\(707\) 1.09440e7 0.823429
\(708\) 0 0
\(709\) −2.52733e7 −1.88819 −0.944095 0.329673i \(-0.893061\pi\)
−0.944095 + 0.329673i \(0.893061\pi\)
\(710\) 0 0
\(711\) 2.35764e7 + 6.20376e6i 1.74906 + 0.460236i
\(712\) 0 0
\(713\) 1.49687e6i 0.110270i
\(714\) 0 0
\(715\) 556140.i 0.0406836i
\(716\) 0 0
\(717\) −1.53514e7 + 1.18345e7i −1.11519 + 0.859708i
\(718\) 0 0
\(719\) −1.71570e7 −1.23771 −0.618857 0.785504i \(-0.712404\pi\)
−0.618857 + 0.785504i \(0.712404\pi\)
\(720\) 0 0
\(721\) 1.10784e7 0.793665
\(722\) 0 0
\(723\) −2.80865e6 + 2.16521e6i −0.199826 + 0.154047i
\(724\) 0 0
\(725\) 5.20632e6i 0.367863i
\(726\) 0 0
\(727\) 1.67975e7i 1.17871i −0.807873 0.589357i \(-0.799381\pi\)
0.807873 0.589357i \(-0.200619\pi\)
\(728\) 0 0
\(729\) −1.00084e7 + 1.02822e7i −0.697501 + 0.716584i
\(730\) 0 0
\(731\) 4.78997e6 0.331542
\(732\) 0 0
\(733\) −1.60515e7 −1.10346 −0.551730 0.834023i \(-0.686032\pi\)
−0.551730 + 0.834023i \(0.686032\pi\)
\(734\) 0 0
\(735\) −3.00964e6 3.90403e6i −0.205493 0.266560i
\(736\) 0 0
\(737\) 885956.i 0.0600819i
\(738\) 0 0
\(739\) 1.81409e7i 1.22194i −0.791655 0.610968i \(-0.790780\pi\)
0.791655 0.610968i \(-0.209220\pi\)
\(740\) 0 0
\(741\) 2.23094e6 + 2.89391e6i 0.149259 + 0.193615i
\(742\) 0 0
\(743\) 1.71782e7 1.14158 0.570790 0.821096i \(-0.306637\pi\)
0.570790 + 0.821096i \(0.306637\pi\)
\(744\) 0 0
\(745\) −4.68084e6 −0.308982
\(746\) 0 0
\(747\) 1.79718e6 6.82989e6i 0.117839 0.447829i
\(748\) 0 0
\(749\) 1.15846e7i 0.754528i
\(750\) 0 0
\(751\) 9.57900e6i 0.619755i −0.950776 0.309878i \(-0.899712\pi\)
0.950776 0.309878i \(-0.100288\pi\)
\(752\) 0 0
\(753\) 1.40699e6 1.08466e6i 0.0904284 0.0697119i
\(754\) 0 0
\(755\) 2.42130e6 0.154590
\(756\) 0 0
\(757\) −1.72136e7 −1.09177 −0.545886 0.837860i \(-0.683807\pi\)
−0.545886 + 0.837860i \(0.683807\pi\)
\(758\) 0 0
\(759\) 1.05706e6 814893.i 0.0666031 0.0513448i
\(760\) 0 0
\(761\) 1.18209e7i 0.739925i −0.929047 0.369962i \(-0.879371\pi\)
0.929047 0.369962i \(-0.120629\pi\)
\(762\) 0 0
\(763\) 4.16486e6i 0.258993i
\(764\) 0 0
\(765\) 868499. 3.30060e6i 0.0536557 0.203910i
\(766\) 0 0
\(767\) 8.61739e6 0.528917
\(768\) 0 0
\(769\) 1.28407e7 0.783021 0.391511 0.920174i \(-0.371953\pi\)
0.391511 + 0.920174i \(0.371953\pi\)
\(770\) 0 0
\(771\) −1.08938e6 1.41312e6i −0.0659999 0.0856134i
\(772\) 0 0
\(773\) 8.84680e6i 0.532522i −0.963901 0.266261i \(-0.914212\pi\)
0.963901 0.266261i \(-0.0857883\pi\)
\(774\) 0 0
\(775\) 4.15855e6i 0.248707i
\(776\) 0 0
\(777\) 8.31338e6 + 1.07839e7i 0.493998 + 0.640802i
\(778\) 0 0
\(779\) 1.55284e6 0.0916819
\(780\) 0 0
\(781\) −470054. −0.0275753
\(782\) 0 0
\(783\) 7.54498e6 3.18504e6i 0.439798 0.185657i
\(784\) 0 0
\(785\) 1.16699e7i 0.675914i
\(786\) 0 0
\(787\) 2.54331e7i 1.46373i −0.681447 0.731867i \(-0.738649\pi\)
0.681447 0.731867i \(-0.261351\pi\)
\(788\) 0 0
\(789\) −2.37224e7 + 1.82878e7i −1.35665 + 1.04585i
\(790\) 0 0
\(791\) −1.45820e7 −0.828660
\(792\) 0 0
\(793\) −5.07434e6 −0.286548
\(794\) 0 0
\(795\) 2.51287e6 1.93719e6i 0.141011 0.108706i
\(796\) 0 0
\(797\) 3.09111e7i 1.72373i −0.507138 0.861865i \(-0.669297\pi\)
0.507138 0.861865i \(-0.330703\pi\)
\(798\) 0 0
\(799\) 1.14657e7i 0.635378i
\(800\) 0 0
\(801\) 145607. + 38314.1i 0.00801864 + 0.00210998i
\(802\) 0 0
\(803\) 6.42223e6 0.351477
\(804\) 0 0
\(805\) 1.64054e6 0.0892271
\(806\) 0 0
\(807\) 1.99285e7 + 2.58507e7i 1.07719 + 1.39730i
\(808\) 0 0
\(809\) 797413.i 0.0428363i −0.999771 0.0214181i \(-0.993182\pi\)
0.999771 0.0214181i \(-0.00681813\pi\)
\(810\) 0 0
\(811\) 2.10694e7i 1.12487i 0.826843 + 0.562433i \(0.190134\pi\)
−0.826843 + 0.562433i \(0.809866\pi\)
\(812\) 0 0
\(813\) 5.10612e6 + 6.62353e6i 0.270935 + 0.351450i
\(814\) 0 0
\(815\) −9.57391e6 −0.504889
\(816\) 0 0
\(817\) −1.01795e7 −0.533543
\(818\) 0 0
\(819\) 3.49296e6 + 919115.i 0.181963 + 0.0478807i
\(820\) 0 0
\(821\) 1.22210e6i 0.0632772i −0.999499 0.0316386i \(-0.989927\pi\)
0.999499 0.0316386i \(-0.0100726\pi\)
\(822\) 0 0
\(823\) 2.41506e7i 1.24288i −0.783463 0.621438i \(-0.786549\pi\)
0.783463 0.621438i \(-0.213451\pi\)
\(824\) 0 0
\(825\) 2.93669e6 2.26391e6i 0.150218 0.115804i
\(826\) 0 0
\(827\) 3.38595e7 1.72154 0.860769 0.508995i \(-0.169983\pi\)
0.860769 + 0.508995i \(0.169983\pi\)
\(828\) 0 0
\(829\) −1.08299e7 −0.547318 −0.273659 0.961827i \(-0.588234\pi\)
−0.273659 + 0.961827i \(0.588234\pi\)
\(830\) 0 0
\(831\) 2.32433e7 1.79184e7i 1.16760 0.900114i
\(832\) 0 0
\(833\) 6.19516e6i 0.309343i
\(834\) 0 0
\(835\) 5.52665e6i 0.274313i
\(836\) 0 0
\(837\) 6.02656e6 2.54406e6i 0.297342 0.125520i
\(838\) 0 0
\(839\) 1.78766e7 0.876760 0.438380 0.898790i \(-0.355552\pi\)
0.438380 + 0.898790i \(0.355552\pi\)
\(840\) 0 0
\(841\) 1.58368e7 0.772109
\(842\) 0 0
\(843\) −1.97541e7 2.56245e7i −0.957388 1.24190i
\(844\) 0 0
\(845\) 8.75759e6i 0.421932i
\(846\) 0 0
\(847\) 1.06945e7i 0.512215i
\(848\) 0 0
\(849\) 9.12862e6 + 1.18414e7i 0.434646 + 0.563812i
\(850\) 0 0
\(851\) 1.07110e7 0.506998
\(852\) 0 0
\(853\) 2.17436e6 0.102320 0.0511599 0.998690i \(-0.483708\pi\)
0.0511599 + 0.998690i \(0.483708\pi\)
\(854\) 0 0
\(855\) −1.84570e6 + 7.01431e6i −0.0863468 + 0.328148i
\(856\) 0 0
\(857\) 3.34641e7i 1.55642i 0.628004 + 0.778210i \(0.283872\pi\)
−0.628004 + 0.778210i \(0.716128\pi\)
\(858\) 0 0
\(859\) 5.77403e6i 0.266991i −0.991049 0.133495i \(-0.957380\pi\)
0.991049 0.133495i \(-0.0426202\pi\)
\(860\) 0 0
\(861\) 1.21563e6 937141.i 0.0558850 0.0430821i
\(862\) 0 0
\(863\) 3.35740e6 0.153453 0.0767267 0.997052i \(-0.475553\pi\)
0.0767267 + 0.997052i \(0.475553\pi\)
\(864\) 0 0
\(865\) 1.14026e7 0.518162
\(866\) 0 0
\(867\) −1.41322e7 + 1.08946e7i −0.638502 + 0.492225i
\(868\) 0 0
\(869\) 9.91007e6i 0.445171i
\(870\) 0 0
\(871\) 1.88594e6i 0.0842332i
\(872\) 0 0
\(873\) −4.63283e6 + 1.76064e7i −0.205736 + 0.781869i
\(874\) 0 0
\(875\) 1.04723e7 0.462403
\(876\) 0 0
\(877\) 12595.7 0.000552997 0.000276498 1.00000i \(-0.499912\pi\)
0.000276498 1.00000i \(0.499912\pi\)
\(878\) 0 0
\(879\) −1.33762e7 1.73512e7i −0.583928 0.757457i
\(880\) 0 0
\(881\) 3.16885e7i 1.37551i −0.725945 0.687753i \(-0.758597\pi\)
0.725945 0.687753i \(-0.241403\pi\)
\(882\) 0 0
\(883\) 2.11571e7i 0.913175i 0.889678 + 0.456588i \(0.150929\pi\)
−0.889678 + 0.456588i \(0.849071\pi\)
\(884\) 0 0
\(885\) 1.04435e7 + 1.35470e7i 0.448216 + 0.581414i
\(886\) 0 0
\(887\) −4.13156e7 −1.76321 −0.881607 0.471985i \(-0.843538\pi\)
−0.881607 + 0.471985i \(0.843538\pi\)
\(888\) 0 0
\(889\) 2.44987e6 0.103966
\(890\) 0 0
\(891\) 5.07742e6 + 2.87086e6i 0.214264 + 0.121148i
\(892\) 0 0
\(893\) 2.43664e7i 1.02250i
\(894\) 0 0
\(895\) 7.37920e6i 0.307930i
\(896\) 0 0
\(897\) 2.25017e6 1.73467e6i 0.0933757 0.0719840i
\(898\) 0 0
\(899\) −3.73361e6 −0.154074
\(900\) 0 0
\(901\) 3.98758e6 0.163643
\(902\) 0 0
\(903\) −7.96894e6 + 6.14331e6i −0.325223 + 0.250717i
\(904\) 0 0
\(905\) 4.29299e6i 0.174236i
\(906\) 0 0
\(907\) 6.87161e6i 0.277358i 0.990337 + 0.138679i \(0.0442856\pi\)
−0.990337 + 0.138679i \(0.955714\pi\)
\(908\) 0 0
\(909\) 3.63834e7 + 9.57370e6i 1.46047 + 0.384300i
\(910\) 0 0
\(911\) −1.68881e7 −0.674195 −0.337097 0.941470i \(-0.609445\pi\)
−0.337097 + 0.941470i \(0.609445\pi\)
\(912\) 0 0
\(913\) −2.87086e6 −0.113982
\(914\) 0 0
\(915\) −6.14964e6 7.97715e6i −0.242827 0.314989i
\(916\) 0 0
\(917\) 1.03513e7i 0.406510i
\(918\) 0 0
\(919\) 2.85463e7i 1.11497i −0.830189 0.557483i \(-0.811767\pi\)
0.830189 0.557483i \(-0.188233\pi\)
\(920\) 0 0
\(921\) 2.33940e7 + 3.03461e7i 0.908773 + 1.17884i
\(922\) 0 0
\(923\) −1.00061e6 −0.0386598
\(924\) 0 0
\(925\) 2.97570e7 1.14350
\(926\) 0 0
\(927\) 3.68301e7 + 9.69126e6i 1.40768 + 0.370408i
\(928\) 0 0
\(929\) 4.81809e7i 1.83162i −0.401610 0.915811i \(-0.631549\pi\)
0.401610 0.915811i \(-0.368451\pi\)
\(930\) 0 0
\(931\) 1.31657e7i 0.497818i
\(932\) 0 0
\(933\) −2.54106e7 + 1.95892e7i −0.955676 + 0.736737i
\(934\) 0 0
\(935\) −1.38737e6 −0.0518994
\(936\) 0 0
\(937\) −2.81871e7 −1.04882 −0.524411 0.851465i \(-0.675715\pi\)
−0.524411 + 0.851465i \(0.675715\pi\)
\(938\) 0 0
\(939\) −1.33680e7 + 1.03055e7i −0.494769 + 0.381420i
\(940\) 0 0
\(941\) 1.40620e7i 0.517693i −0.965918 0.258847i \(-0.916658\pi\)
0.965918 0.258847i \(-0.0833424\pi\)
\(942\) 0 0
\(943\) 1.20742e6i 0.0442158i
\(944\) 0 0
\(945\) 2.78824e6 + 6.60500e6i 0.101567 + 0.240599i
\(946\) 0 0
\(947\) −2.59975e7 −0.942013 −0.471006 0.882130i \(-0.656109\pi\)
−0.471006 + 0.882130i \(0.656109\pi\)
\(948\) 0 0
\(949\) 1.36711e7 0.492761
\(950\) 0 0
\(951\) −2.63975e7 3.42422e7i −0.946480 1.22775i
\(952\) 0 0
\(953\) 3.29770e7i 1.17619i −0.808791 0.588096i \(-0.799878\pi\)
0.808791 0.588096i \(-0.200122\pi\)
\(954\) 0 0
\(955\) 1.20227e7i 0.426574i
\(956\) 0 0
\(957\) −2.03257e6 2.63660e6i −0.0717408 0.0930603i
\(958\) 0 0
\(959\) 5.58108e6 0.195962
\(960\) 0 0
\(961\) 2.56469e7 0.895833
\(962\) 0 0
\(963\) −1.01341e7 + 3.85131e7i −0.352143 + 1.33827i
\(964\) 0 0
\(965\) 1.25194e7i 0.432779i
\(966\) 0 0
\(967\) 3.05302e7i 1.04994i 0.851122 + 0.524968i \(0.175923\pi\)
−0.851122 + 0.524968i \(0.824077\pi\)
\(968\) 0 0
\(969\) −7.21926e6 + 5.56537e6i −0.246992 + 0.190408i
\(970\) 0 0
\(971\) 2.86270e7 0.974377 0.487189 0.873297i \(-0.338022\pi\)
0.487189 + 0.873297i \(0.338022\pi\)
\(972\) 0 0
\(973\) 2.00752e7 0.679796
\(974\) 0 0
\(975\) 6.25136e6 4.81921e6i 0.210602 0.162355i
\(976\) 0 0
\(977\) 1.38661e7i 0.464749i −0.972626 0.232374i \(-0.925351\pi\)
0.972626 0.232374i \(-0.0746495\pi\)
\(978\) 0 0
\(979\) 61204.1i 0.00204091i
\(980\) 0 0
\(981\) 3.64339e6 1.38461e7i 0.120874 0.459363i
\(982\) 0 0
\(983\) −4.90160e7 −1.61791 −0.808954 0.587872i \(-0.799966\pi\)
−0.808954 + 0.587872i \(0.799966\pi\)
\(984\) 0 0
\(985\) −2.37940e7 −0.781406
\(986\) 0 0
\(987\) −1.47051e7 1.90751e7i −0.480481 0.623268i
\(988\) 0 0
\(989\) 7.91507e6i 0.257314i
\(990\) 0 0
\(991\) 9.29631e6i 0.300695i −0.988633 0.150348i \(-0.951961\pi\)
0.988633 0.150348i \(-0.0480393\pi\)
\(992\) 0 0
\(993\) −1.13328e7 1.47007e7i −0.364725 0.473112i
\(994\) 0 0
\(995\) 2.40991e6 0.0771692
\(996\) 0 0
\(997\) −3.47793e7 −1.10811 −0.554055 0.832480i \(-0.686920\pi\)
−0.554055 + 0.832480i \(0.686920\pi\)
\(998\) 0 0
\(999\) 1.82043e7 + 4.31237e7i 0.577112 + 1.36711i
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 384.6.c.d.383.4 yes 20
3.2 odd 2 384.6.c.a.383.18 yes 20
4.3 odd 2 384.6.c.a.383.17 20
8.3 odd 2 384.6.c.c.383.4 yes 20
8.5 even 2 384.6.c.b.383.17 yes 20
12.11 even 2 inner 384.6.c.d.383.3 yes 20
24.5 odd 2 384.6.c.c.383.3 yes 20
24.11 even 2 384.6.c.b.383.18 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.17 20 4.3 odd 2
384.6.c.a.383.18 yes 20 3.2 odd 2
384.6.c.b.383.17 yes 20 8.5 even 2
384.6.c.b.383.18 yes 20 24.11 even 2
384.6.c.c.383.3 yes 20 24.5 odd 2
384.6.c.c.383.4 yes 20 8.3 odd 2
384.6.c.d.383.3 yes 20 12.11 even 2 inner
384.6.c.d.383.4 yes 20 1.1 even 1 trivial