Properties

Label 384.6.c.a.383.19
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 383.19
Root \(-0.852576i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.a.383.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(15.4777 - 1.85457i) q^{3} +55.8580i q^{5} +225.953i q^{7} +(236.121 - 57.4091i) q^{9} +O(q^{10})\) \(q+(15.4777 - 1.85457i) q^{3} +55.8580i q^{5} +225.953i q^{7} +(236.121 - 57.4091i) q^{9} -36.9229 q^{11} +795.597 q^{13} +(103.593 + 864.557i) q^{15} -129.541i q^{17} -998.033i q^{19} +(419.046 + 3497.25i) q^{21} +3748.76 q^{23} +4.87856 q^{25} +(3548.15 - 1326.47i) q^{27} +921.479i q^{29} +423.688i q^{31} +(-571.482 + 68.4759i) q^{33} -12621.3 q^{35} -10929.6 q^{37} +(12314.0 - 1475.49i) q^{39} +14465.5i q^{41} +20897.9i q^{43} +(3206.76 + 13189.3i) q^{45} -12002.2 q^{47} -34247.9 q^{49} +(-240.242 - 2005.00i) q^{51} -11365.4i q^{53} -2062.44i q^{55} +(-1850.92 - 15447.3i) q^{57} -36606.6 q^{59} +38251.8 q^{61} +(12971.8 + 53352.4i) q^{63} +44440.5i q^{65} -8975.18i q^{67} +(58022.3 - 6952.33i) q^{69} -61606.6 q^{71} +2569.04 q^{73} +(75.5091 - 9.04762i) q^{75} -8342.84i q^{77} +61551.9i q^{79} +(52457.4 - 27111.0i) q^{81} +113300. q^{83} +7235.89 q^{85} +(1708.95 + 14262.4i) q^{87} -125521. i q^{89} +179768. i q^{91} +(785.759 + 6557.74i) q^{93} +55748.2 q^{95} -156644. q^{97} +(-8718.27 + 2119.71i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) 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
\(3\) 15.4777 1.85457i 0.992898 0.118971i
\(4\) 0 0
\(5\) 55.8580i 0.999219i 0.866251 + 0.499610i \(0.166523\pi\)
−0.866251 + 0.499610i \(0.833477\pi\)
\(6\) 0 0
\(7\) 225.953i 1.74290i 0.490480 + 0.871452i \(0.336821\pi\)
−0.490480 + 0.871452i \(0.663179\pi\)
\(8\) 0 0
\(9\) 236.121 57.4091i 0.971692 0.236251i
\(10\) 0 0
\(11\) −36.9229 −0.0920054 −0.0460027 0.998941i \(-0.514648\pi\)
−0.0460027 + 0.998941i \(0.514648\pi\)
\(12\) 0 0
\(13\) 795.597 1.30567 0.652837 0.757499i \(-0.273579\pi\)
0.652837 + 0.757499i \(0.273579\pi\)
\(14\) 0 0
\(15\) 103.593 + 864.557i 0.118878 + 0.992122i
\(16\) 0 0
\(17\) 129.541i 0.108714i −0.998522 0.0543568i \(-0.982689\pi\)
0.998522 0.0543568i \(-0.0173108\pi\)
\(18\) 0 0
\(19\) 998.033i 0.634251i −0.948384 0.317125i \(-0.897282\pi\)
0.948384 0.317125i \(-0.102718\pi\)
\(20\) 0 0
\(21\) 419.046 + 3497.25i 0.207354 + 1.73053i
\(22\) 0 0
\(23\) 3748.76 1.47764 0.738819 0.673904i \(-0.235384\pi\)
0.738819 + 0.673904i \(0.235384\pi\)
\(24\) 0 0
\(25\) 4.87856 0.00156114
\(26\) 0 0
\(27\) 3548.15 1326.47i 0.936684 0.350176i
\(28\) 0 0
\(29\) 921.479i 0.203465i 0.994812 + 0.101733i \(0.0324386\pi\)
−0.994812 + 0.101733i \(0.967561\pi\)
\(30\) 0 0
\(31\) 423.688i 0.0791849i 0.999216 + 0.0395925i \(0.0126060\pi\)
−0.999216 + 0.0395925i \(0.987394\pi\)
\(32\) 0 0
\(33\) −571.482 + 68.4759i −0.0913520 + 0.0109459i
\(34\) 0 0
\(35\) −12621.3 −1.74154
\(36\) 0 0
\(37\) −10929.6 −1.31250 −0.656249 0.754544i \(-0.727858\pi\)
−0.656249 + 0.754544i \(0.727858\pi\)
\(38\) 0 0
\(39\) 12314.0 1475.49i 1.29640 0.155337i
\(40\) 0 0
\(41\) 14465.5i 1.34392i 0.740586 + 0.671962i \(0.234548\pi\)
−0.740586 + 0.671962i \(0.765452\pi\)
\(42\) 0 0
\(43\) 20897.9i 1.72358i 0.507268 + 0.861788i \(0.330655\pi\)
−0.507268 + 0.861788i \(0.669345\pi\)
\(44\) 0 0
\(45\) 3206.76 + 13189.3i 0.236067 + 0.970933i
\(46\) 0 0
\(47\) −12002.2 −0.792534 −0.396267 0.918135i \(-0.629695\pi\)
−0.396267 + 0.918135i \(0.629695\pi\)
\(48\) 0 0
\(49\) −34247.9 −2.03772
\(50\) 0 0
\(51\) −240.242 2005.00i −0.0129337 0.107942i
\(52\) 0 0
\(53\) 11365.4i 0.555769i −0.960615 0.277884i \(-0.910367\pi\)
0.960615 0.277884i \(-0.0896332\pi\)
\(54\) 0 0
\(55\) 2062.44i 0.0919336i
\(56\) 0 0
\(57\) −1850.92 15447.3i −0.0754572 0.629746i
\(58\) 0 0
\(59\) −36606.6 −1.36908 −0.684542 0.728974i \(-0.739998\pi\)
−0.684542 + 0.728974i \(0.739998\pi\)
\(60\) 0 0
\(61\) 38251.8 1.31622 0.658108 0.752923i \(-0.271357\pi\)
0.658108 + 0.752923i \(0.271357\pi\)
\(62\) 0 0
\(63\) 12971.8 + 53352.4i 0.411763 + 1.69357i
\(64\) 0 0
\(65\) 44440.5i 1.30465i
\(66\) 0 0
\(67\) 8975.18i 0.244262i −0.992514 0.122131i \(-0.961027\pi\)
0.992514 0.122131i \(-0.0389728\pi\)
\(68\) 0 0
\(69\) 58022.3 6952.33i 1.46714 0.175795i
\(70\) 0 0
\(71\) −61606.6 −1.45038 −0.725189 0.688549i \(-0.758248\pi\)
−0.725189 + 0.688549i \(0.758248\pi\)
\(72\) 0 0
\(73\) 2569.04 0.0564240 0.0282120 0.999602i \(-0.491019\pi\)
0.0282120 + 0.999602i \(0.491019\pi\)
\(74\) 0 0
\(75\) 75.5091 9.04762i 0.00155005 0.000185730i
\(76\) 0 0
\(77\) 8342.84i 0.160357i
\(78\) 0 0
\(79\) 61551.9i 1.10962i 0.831978 + 0.554809i \(0.187209\pi\)
−0.831978 + 0.554809i \(0.812791\pi\)
\(80\) 0 0
\(81\) 52457.4 27111.0i 0.888371 0.459127i
\(82\) 0 0
\(83\) 113300. 1.80523 0.902617 0.430444i \(-0.141643\pi\)
0.902617 + 0.430444i \(0.141643\pi\)
\(84\) 0 0
\(85\) 7235.89 0.108629
\(86\) 0 0
\(87\) 1708.95 + 14262.4i 0.0242064 + 0.202020i
\(88\) 0 0
\(89\) 125521.i 1.67974i −0.542790 0.839868i \(-0.682632\pi\)
0.542790 0.839868i \(-0.317368\pi\)
\(90\) 0 0
\(91\) 179768.i 2.27566i
\(92\) 0 0
\(93\) 785.759 + 6557.74i 0.00942067 + 0.0786225i
\(94\) 0 0
\(95\) 55748.2 0.633756
\(96\) 0 0
\(97\) −156644. −1.69038 −0.845191 0.534465i \(-0.820513\pi\)
−0.845191 + 0.534465i \(0.820513\pi\)
\(98\) 0 0
\(99\) −8718.27 + 2119.71i −0.0894010 + 0.0217364i
\(100\) 0 0
\(101\) 34422.0i 0.335763i −0.985807 0.167881i \(-0.946307\pi\)
0.985807 0.167881i \(-0.0536926\pi\)
\(102\) 0 0
\(103\) 28205.4i 0.261962i −0.991385 0.130981i \(-0.958187\pi\)
0.991385 0.130981i \(-0.0418127\pi\)
\(104\) 0 0
\(105\) −195349. + 23407.1i −1.72917 + 0.207192i
\(106\) 0 0
\(107\) 42182.3 0.356181 0.178090 0.984014i \(-0.443008\pi\)
0.178090 + 0.984014i \(0.443008\pi\)
\(108\) 0 0
\(109\) 88788.0 0.715794 0.357897 0.933761i \(-0.383494\pi\)
0.357897 + 0.933761i \(0.383494\pi\)
\(110\) 0 0
\(111\) −169165. + 20269.6i −1.30318 + 0.156149i
\(112\) 0 0
\(113\) 238904.i 1.76006i −0.474921 0.880028i \(-0.657523\pi\)
0.474921 0.880028i \(-0.342477\pi\)
\(114\) 0 0
\(115\) 209398.i 1.47648i
\(116\) 0 0
\(117\) 187857. 45674.5i 1.26871 0.308467i
\(118\) 0 0
\(119\) 29270.2 0.189477
\(120\) 0 0
\(121\) −159688. −0.991535
\(122\) 0 0
\(123\) 26827.3 + 223894.i 0.159887 + 1.33438i
\(124\) 0 0
\(125\) 174829.i 1.00078i
\(126\) 0 0
\(127\) 258860.i 1.42415i 0.702105 + 0.712074i \(0.252244\pi\)
−0.702105 + 0.712074i \(0.747756\pi\)
\(128\) 0 0
\(129\) 38756.5 + 323452.i 0.205055 + 1.71133i
\(130\) 0 0
\(131\) 9462.28 0.0481745 0.0240873 0.999710i \(-0.492332\pi\)
0.0240873 + 0.999710i \(0.492332\pi\)
\(132\) 0 0
\(133\) 225509. 1.10544
\(134\) 0 0
\(135\) 74093.8 + 198193.i 0.349903 + 0.935952i
\(136\) 0 0
\(137\) 146703.i 0.667788i −0.942611 0.333894i \(-0.891637\pi\)
0.942611 0.333894i \(-0.108363\pi\)
\(138\) 0 0
\(139\) 150825.i 0.662119i −0.943610 0.331060i \(-0.892594\pi\)
0.943610 0.331060i \(-0.107406\pi\)
\(140\) 0 0
\(141\) −185768. + 22259.0i −0.786905 + 0.0942882i
\(142\) 0 0
\(143\) −29375.7 −0.120129
\(144\) 0 0
\(145\) −51472.0 −0.203306
\(146\) 0 0
\(147\) −530080. + 63515.1i −2.02324 + 0.242428i
\(148\) 0 0
\(149\) 223409.i 0.824396i −0.911094 0.412198i \(-0.864761\pi\)
0.911094 0.412198i \(-0.135239\pi\)
\(150\) 0 0
\(151\) 398109.i 1.42089i 0.703754 + 0.710443i \(0.251506\pi\)
−0.703754 + 0.710443i \(0.748494\pi\)
\(152\) 0 0
\(153\) −7436.81 30587.3i −0.0256837 0.105636i
\(154\) 0 0
\(155\) −23666.4 −0.0791231
\(156\) 0 0
\(157\) 457758. 1.48213 0.741066 0.671432i \(-0.234320\pi\)
0.741066 + 0.671432i \(0.234320\pi\)
\(158\) 0 0
\(159\) −21077.9 175910.i −0.0661202 0.551822i
\(160\) 0 0
\(161\) 847044.i 2.57538i
\(162\) 0 0
\(163\) 90580.0i 0.267032i −0.991047 0.133516i \(-0.957373\pi\)
0.991047 0.133516i \(-0.0426268\pi\)
\(164\) 0 0
\(165\) −3824.93 31921.9i −0.0109374 0.0912807i
\(166\) 0 0
\(167\) −160677. −0.445822 −0.222911 0.974839i \(-0.571556\pi\)
−0.222911 + 0.974839i \(0.571556\pi\)
\(168\) 0 0
\(169\) 261681. 0.704784
\(170\) 0 0
\(171\) −57296.2 235657.i −0.149843 0.616296i
\(172\) 0 0
\(173\) 425382.i 1.08060i 0.841473 + 0.540299i \(0.181689\pi\)
−0.841473 + 0.540299i \(0.818311\pi\)
\(174\) 0 0
\(175\) 1102.33i 0.00272092i
\(176\) 0 0
\(177\) −566588. + 67889.5i −1.35936 + 0.162881i
\(178\) 0 0
\(179\) 19674.5 0.0458956 0.0229478 0.999737i \(-0.492695\pi\)
0.0229478 + 0.999737i \(0.492695\pi\)
\(180\) 0 0
\(181\) −46347.1 −0.105154 −0.0525771 0.998617i \(-0.516744\pi\)
−0.0525771 + 0.998617i \(0.516744\pi\)
\(182\) 0 0
\(183\) 592052. 70940.6i 1.30687 0.156591i
\(184\) 0 0
\(185\) 610504.i 1.31147i
\(186\) 0 0
\(187\) 4783.01i 0.0100022i
\(188\) 0 0
\(189\) 299719. + 801717.i 0.610324 + 1.63255i
\(190\) 0 0
\(191\) 57120.9 0.113295 0.0566476 0.998394i \(-0.481959\pi\)
0.0566476 + 0.998394i \(0.481959\pi\)
\(192\) 0 0
\(193\) −257879. −0.498336 −0.249168 0.968460i \(-0.580157\pi\)
−0.249168 + 0.968460i \(0.580157\pi\)
\(194\) 0 0
\(195\) 82417.9 + 687838.i 0.155215 + 1.29539i
\(196\) 0 0
\(197\) 704799.i 1.29390i −0.762534 0.646948i \(-0.776045\pi\)
0.762534 0.646948i \(-0.223955\pi\)
\(198\) 0 0
\(199\) 259574.i 0.464654i 0.972638 + 0.232327i \(0.0746339\pi\)
−0.972638 + 0.232327i \(0.925366\pi\)
\(200\) 0 0
\(201\) −16645.1 138916.i −0.0290600 0.242527i
\(202\) 0 0
\(203\) −208211. −0.354621
\(204\) 0 0
\(205\) −808016. −1.34287
\(206\) 0 0
\(207\) 885161. 215213.i 1.43581 0.349094i
\(208\) 0 0
\(209\) 36850.2i 0.0583545i
\(210\) 0 0
\(211\) 534281.i 0.826159i 0.910695 + 0.413080i \(0.135547\pi\)
−0.910695 + 0.413080i \(0.864453\pi\)
\(212\) 0 0
\(213\) −953531. + 114254.i −1.44008 + 0.172552i
\(214\) 0 0
\(215\) −1.16731e6 −1.72223
\(216\) 0 0
\(217\) −95733.8 −0.138012
\(218\) 0 0
\(219\) 39762.9 4764.46i 0.0560232 0.00671279i
\(220\) 0 0
\(221\) 103062.i 0.141945i
\(222\) 0 0
\(223\) 864006.i 1.16347i −0.813379 0.581734i \(-0.802374\pi\)
0.813379 0.581734i \(-0.197626\pi\)
\(224\) 0 0
\(225\) 1151.93 280.074i 0.00151695 0.000368821i
\(226\) 0 0
\(227\) 889864. 1.14620 0.573098 0.819487i \(-0.305741\pi\)
0.573098 + 0.819487i \(0.305741\pi\)
\(228\) 0 0
\(229\) 296508. 0.373635 0.186818 0.982395i \(-0.440183\pi\)
0.186818 + 0.982395i \(0.440183\pi\)
\(230\) 0 0
\(231\) −15472.4 129128.i −0.0190777 0.159218i
\(232\) 0 0
\(233\) 763288.i 0.921083i −0.887638 0.460541i \(-0.847655\pi\)
0.887638 0.460541i \(-0.152345\pi\)
\(234\) 0 0
\(235\) 670422.i 0.791915i
\(236\) 0 0
\(237\) 114152. + 952684.i 0.132012 + 1.10174i
\(238\) 0 0
\(239\) 36450.5 0.0412771 0.0206385 0.999787i \(-0.493430\pi\)
0.0206385 + 0.999787i \(0.493430\pi\)
\(240\) 0 0
\(241\) 1.29506e6 1.43631 0.718155 0.695884i \(-0.244987\pi\)
0.718155 + 0.695884i \(0.244987\pi\)
\(242\) 0 0
\(243\) 761643. 516903.i 0.827439 0.561556i
\(244\) 0 0
\(245\) 1.91302e6i 2.03613i
\(246\) 0 0
\(247\) 794032.i 0.828125i
\(248\) 0 0
\(249\) 1.75362e6 210122.i 1.79241 0.214770i
\(250\) 0 0
\(251\) 466045. 0.466921 0.233460 0.972366i \(-0.424995\pi\)
0.233460 + 0.972366i \(0.424995\pi\)
\(252\) 0 0
\(253\) −138415. −0.135951
\(254\) 0 0
\(255\) 111995. 13419.5i 0.107857 0.0129236i
\(256\) 0 0
\(257\) 571612.i 0.539844i −0.962882 0.269922i \(-0.913002\pi\)
0.962882 0.269922i \(-0.0869979\pi\)
\(258\) 0 0
\(259\) 2.46957e6i 2.28756i
\(260\) 0 0
\(261\) 52901.2 + 217581.i 0.0480689 + 0.197706i
\(262\) 0 0
\(263\) −435018. −0.387809 −0.193905 0.981020i \(-0.562115\pi\)
−0.193905 + 0.981020i \(0.562115\pi\)
\(264\) 0 0
\(265\) 634848. 0.555335
\(266\) 0 0
\(267\) −232787. 1.94278e6i −0.199839 1.66781i
\(268\) 0 0
\(269\) 1.11982e6i 0.943556i −0.881717 0.471778i \(-0.843612\pi\)
0.881717 0.471778i \(-0.156388\pi\)
\(270\) 0 0
\(271\) 201205.i 0.166424i −0.996532 0.0832118i \(-0.973482\pi\)
0.996532 0.0832118i \(-0.0265178\pi\)
\(272\) 0 0
\(273\) 333392. + 2.78240e6i 0.270737 + 2.25950i
\(274\) 0 0
\(275\) −180.130 −0.000143633
\(276\) 0 0
\(277\) −302438. −0.236830 −0.118415 0.992964i \(-0.537781\pi\)
−0.118415 + 0.992964i \(0.537781\pi\)
\(278\) 0 0
\(279\) 24323.6 + 100042.i 0.0187075 + 0.0769433i
\(280\) 0 0
\(281\) 1.77044e6i 1.33757i −0.743457 0.668784i \(-0.766815\pi\)
0.743457 0.668784i \(-0.233185\pi\)
\(282\) 0 0
\(283\) 809916.i 0.601138i 0.953760 + 0.300569i \(0.0971765\pi\)
−0.953760 + 0.300569i \(0.902824\pi\)
\(284\) 0 0
\(285\) 862856. 103389.i 0.629255 0.0753983i
\(286\) 0 0
\(287\) −3.26853e6 −2.34233
\(288\) 0 0
\(289\) 1.40308e6 0.988181
\(290\) 0 0
\(291\) −2.42450e6 + 290507.i −1.67838 + 0.201106i
\(292\) 0 0
\(293\) 1.73657e6i 1.18174i −0.806767 0.590870i \(-0.798784\pi\)
0.806767 0.590870i \(-0.201216\pi\)
\(294\) 0 0
\(295\) 2.04478e6i 1.36801i
\(296\) 0 0
\(297\) −131008. + 48976.9i −0.0861800 + 0.0322181i
\(298\) 0 0
\(299\) 2.98250e6 1.92931
\(300\) 0 0
\(301\) −4.72194e6 −3.00403
\(302\) 0 0
\(303\) −63837.9 532775.i −0.0399459 0.333378i
\(304\) 0 0
\(305\) 2.13667e6i 1.31519i
\(306\) 0 0
\(307\) 596701.i 0.361335i 0.983544 + 0.180668i \(0.0578259\pi\)
−0.983544 + 0.180668i \(0.942174\pi\)
\(308\) 0 0
\(309\) −52308.8 436556.i −0.0311658 0.260102i
\(310\) 0 0
\(311\) 2.86268e6 1.67831 0.839154 0.543894i \(-0.183051\pi\)
0.839154 + 0.543894i \(0.183051\pi\)
\(312\) 0 0
\(313\) −1.53442e6 −0.885287 −0.442644 0.896698i \(-0.645959\pi\)
−0.442644 + 0.896698i \(0.645959\pi\)
\(314\) 0 0
\(315\) −2.98016e6 + 724578.i −1.69224 + 0.411442i
\(316\) 0 0
\(317\) 3.33250e6i 1.86261i 0.364241 + 0.931305i \(0.381329\pi\)
−0.364241 + 0.931305i \(0.618671\pi\)
\(318\) 0 0
\(319\) 34023.6i 0.0187199i
\(320\) 0 0
\(321\) 652886. 78229.9i 0.353651 0.0423750i
\(322\) 0 0
\(323\) −129286. −0.0689517
\(324\) 0 0
\(325\) 3881.37 0.00203834
\(326\) 0 0
\(327\) 1.37424e6 164663.i 0.710711 0.0851585i
\(328\) 0 0
\(329\) 2.71195e6i 1.38131i
\(330\) 0 0
\(331\) 2.02136e6i 1.01408i 0.861922 + 0.507041i \(0.169261\pi\)
−0.861922 + 0.507041i \(0.830739\pi\)
\(332\) 0 0
\(333\) −2.58070e6 + 627456.i −1.27534 + 0.310079i
\(334\) 0 0
\(335\) 501336. 0.244071
\(336\) 0 0
\(337\) −1.12008e6 −0.537249 −0.268625 0.963245i \(-0.586569\pi\)
−0.268625 + 0.963245i \(0.586569\pi\)
\(338\) 0 0
\(339\) −443063. 3.69769e6i −0.209395 1.74756i
\(340\) 0 0
\(341\) 15643.8i 0.00728544i
\(342\) 0 0
\(343\) 3.94083e6i 1.80864i
\(344\) 0 0
\(345\) 388343. + 3.24101e6i 0.175658 + 1.46600i
\(346\) 0 0
\(347\) 2.39720e6 1.06876 0.534380 0.845244i \(-0.320545\pi\)
0.534380 + 0.845244i \(0.320545\pi\)
\(348\) 0 0
\(349\) 1.86085e6 0.817801 0.408900 0.912579i \(-0.365912\pi\)
0.408900 + 0.912579i \(0.365912\pi\)
\(350\) 0 0
\(351\) 2.82290e6 1.05533e6i 1.22300 0.457216i
\(352\) 0 0
\(353\) 2.14786e6i 0.917423i −0.888585 0.458711i \(-0.848311\pi\)
0.888585 0.458711i \(-0.151689\pi\)
\(354\) 0 0
\(355\) 3.44122e6i 1.44925i
\(356\) 0 0
\(357\) 453036. 54283.5i 0.188132 0.0225423i
\(358\) 0 0
\(359\) −1.48571e6 −0.608412 −0.304206 0.952606i \(-0.598391\pi\)
−0.304206 + 0.952606i \(0.598391\pi\)
\(360\) 0 0
\(361\) 1.48003e6 0.597726
\(362\) 0 0
\(363\) −2.47161e6 + 296152.i −0.984493 + 0.117964i
\(364\) 0 0
\(365\) 143501.i 0.0563799i
\(366\) 0 0
\(367\) 2.58403e6i 1.00146i 0.865604 + 0.500729i \(0.166935\pi\)
−0.865604 + 0.500729i \(0.833065\pi\)
\(368\) 0 0
\(369\) 830452. + 3.41562e6i 0.317504 + 1.30588i
\(370\) 0 0
\(371\) 2.56804e6 0.968652
\(372\) 0 0
\(373\) −3.75492e6 −1.39743 −0.698713 0.715402i \(-0.746244\pi\)
−0.698713 + 0.715402i \(0.746244\pi\)
\(374\) 0 0
\(375\) 324232. + 2.70596e6i 0.119063 + 0.993671i
\(376\) 0 0
\(377\) 733126.i 0.265659i
\(378\) 0 0
\(379\) 4.37130e6i 1.56319i −0.623784 0.781597i \(-0.714405\pi\)
0.623784 0.781597i \(-0.285595\pi\)
\(380\) 0 0
\(381\) 480073. + 4.00656e6i 0.169432 + 1.41403i
\(382\) 0 0
\(383\) 1.26259e6 0.439812 0.219906 0.975521i \(-0.429425\pi\)
0.219906 + 0.975521i \(0.429425\pi\)
\(384\) 0 0
\(385\) 466015. 0.160231
\(386\) 0 0
\(387\) 1.19973e6 + 4.93442e6i 0.407197 + 1.67479i
\(388\) 0 0
\(389\) 2.13040e6i 0.713817i 0.934139 + 0.356908i \(0.116169\pi\)
−0.934139 + 0.356908i \(0.883831\pi\)
\(390\) 0 0
\(391\) 485617.i 0.160639i
\(392\) 0 0
\(393\) 146455. 17548.4i 0.0478324 0.00573135i
\(394\) 0 0
\(395\) −3.43817e6 −1.10875
\(396\) 0 0
\(397\) 3.68644e6 1.17390 0.586950 0.809623i \(-0.300328\pi\)
0.586950 + 0.809623i \(0.300328\pi\)
\(398\) 0 0
\(399\) 3.49037e6 418222.i 1.09759 0.131515i
\(400\) 0 0
\(401\) 3.06338e6i 0.951350i −0.879621 0.475675i \(-0.842204\pi\)
0.879621 0.475675i \(-0.157796\pi\)
\(402\) 0 0
\(403\) 337085.i 0.103390i
\(404\) 0 0
\(405\) 1.51437e6 + 2.93017e6i 0.458768 + 0.887677i
\(406\) 0 0
\(407\) 403551. 0.120757
\(408\) 0 0
\(409\) 4.44429e6 1.31369 0.656847 0.754024i \(-0.271890\pi\)
0.656847 + 0.754024i \(0.271890\pi\)
\(410\) 0 0
\(411\) −272071. 2.27064e6i −0.0794472 0.663046i
\(412\) 0 0
\(413\) 8.27139e6i 2.38618i
\(414\) 0 0
\(415\) 6.32870e6i 1.80382i
\(416\) 0 0
\(417\) −279715. 2.33443e6i −0.0787727 0.657417i
\(418\) 0 0
\(419\) 1.62681e6 0.452691 0.226346 0.974047i \(-0.427322\pi\)
0.226346 + 0.974047i \(0.427322\pi\)
\(420\) 0 0
\(421\) 1.43223e6 0.393828 0.196914 0.980421i \(-0.436908\pi\)
0.196914 + 0.980421i \(0.436908\pi\)
\(422\) 0 0
\(423\) −2.83398e6 + 689038.i −0.770099 + 0.187237i
\(424\) 0 0
\(425\) 631.972i 0.000169717i
\(426\) 0 0
\(427\) 8.64312e6i 2.29404i
\(428\) 0 0
\(429\) −454670. + 54479.2i −0.119276 + 0.0142918i
\(430\) 0 0
\(431\) −5.60361e6 −1.45303 −0.726515 0.687151i \(-0.758861\pi\)
−0.726515 + 0.687151i \(0.758861\pi\)
\(432\) 0 0
\(433\) −4.49780e6 −1.15287 −0.576435 0.817143i \(-0.695557\pi\)
−0.576435 + 0.817143i \(0.695557\pi\)
\(434\) 0 0
\(435\) −796671. + 95458.3i −0.201862 + 0.0241875i
\(436\) 0 0
\(437\) 3.74139e6i 0.937193i
\(438\) 0 0
\(439\) 5.39182e6i 1.33529i 0.744481 + 0.667643i \(0.232697\pi\)
−0.744481 + 0.667643i \(0.767303\pi\)
\(440\) 0 0
\(441\) −8.08665e6 + 1.96614e6i −1.98003 + 0.481413i
\(442\) 0 0
\(443\) 697144. 0.168777 0.0843885 0.996433i \(-0.473106\pi\)
0.0843885 + 0.996433i \(0.473106\pi\)
\(444\) 0 0
\(445\) 7.01136e6 1.67843
\(446\) 0 0
\(447\) −414328. 3.45787e6i −0.0980788 0.818541i
\(448\) 0 0
\(449\) 2.66314e6i 0.623416i 0.950178 + 0.311708i \(0.100901\pi\)
−0.950178 + 0.311708i \(0.899099\pi\)
\(450\) 0 0
\(451\) 534108.i 0.123648i
\(452\) 0 0
\(453\) 738320. + 6.16183e6i 0.169044 + 1.41080i
\(454\) 0 0
\(455\) −1.00415e7 −2.27389
\(456\) 0 0
\(457\) −2.12657e6 −0.476310 −0.238155 0.971227i \(-0.576543\pi\)
−0.238155 + 0.971227i \(0.576543\pi\)
\(458\) 0 0
\(459\) −171831. 459630.i −0.0380689 0.101830i
\(460\) 0 0
\(461\) 4.19441e6i 0.919218i −0.888121 0.459609i \(-0.847989\pi\)
0.888121 0.459609i \(-0.152011\pi\)
\(462\) 0 0
\(463\) 4.23099e6i 0.917254i −0.888629 0.458627i \(-0.848341\pi\)
0.888629 0.458627i \(-0.151659\pi\)
\(464\) 0 0
\(465\) −366303. + 43891.0i −0.0785611 + 0.00941332i
\(466\) 0 0
\(467\) −4.90063e6 −1.03982 −0.519912 0.854220i \(-0.674035\pi\)
−0.519912 + 0.854220i \(0.674035\pi\)
\(468\) 0 0
\(469\) 2.02797e6 0.425726
\(470\) 0 0
\(471\) 7.08506e6 848944.i 1.47161 0.176330i
\(472\) 0 0
\(473\) 771608.i 0.158578i
\(474\) 0 0
\(475\) 4868.97i 0.000990154i
\(476\) 0 0
\(477\) −652476. 2.68361e6i −0.131301 0.540036i
\(478\) 0 0
\(479\) −3.45294e6 −0.687622 −0.343811 0.939039i \(-0.611718\pi\)
−0.343811 + 0.939039i \(0.611718\pi\)
\(480\) 0 0
\(481\) −8.69553e6 −1.71369
\(482\) 0 0
\(483\) 1.57090e6 + 1.31103e7i 0.306395 + 2.55709i
\(484\) 0 0
\(485\) 8.74983e6i 1.68906i
\(486\) 0 0
\(487\) 1.62872e6i 0.311188i 0.987821 + 0.155594i \(0.0497292\pi\)
−0.987821 + 0.155594i \(0.950271\pi\)
\(488\) 0 0
\(489\) −167987. 1.40197e6i −0.0317690 0.265136i
\(490\) 0 0
\(491\) 5.79077e6 1.08401 0.542004 0.840376i \(-0.317666\pi\)
0.542004 + 0.840376i \(0.317666\pi\)
\(492\) 0 0
\(493\) 119369. 0.0221195
\(494\) 0 0
\(495\) −118403. 486985.i −0.0217194 0.0893311i
\(496\) 0 0
\(497\) 1.39202e7i 2.52787i
\(498\) 0 0
\(499\) 4.75964e6i 0.855702i 0.903849 + 0.427851i \(0.140729\pi\)
−0.903849 + 0.427851i \(0.859271\pi\)
\(500\) 0 0
\(501\) −2.48691e6 + 297986.i −0.442656 + 0.0530397i
\(502\) 0 0
\(503\) 55691.2 0.00981446 0.00490723 0.999988i \(-0.498438\pi\)
0.00490723 + 0.999988i \(0.498438\pi\)
\(504\) 0 0
\(505\) 1.92275e6 0.335501
\(506\) 0 0
\(507\) 4.05024e6 485306.i 0.699778 0.0838486i
\(508\) 0 0
\(509\) 2.84383e6i 0.486530i 0.969960 + 0.243265i \(0.0782185\pi\)
−0.969960 + 0.243265i \(0.921781\pi\)
\(510\) 0 0
\(511\) 580483.i 0.0983416i
\(512\) 0 0
\(513\) −1.32386e6 3.54118e6i −0.222100 0.594093i
\(514\) 0 0
\(515\) 1.57550e6 0.261758
\(516\) 0 0
\(517\) 443157. 0.0729174
\(518\) 0 0
\(519\) 788900. + 6.58396e6i 0.128559 + 1.07292i
\(520\) 0 0
\(521\) 1.07097e7i 1.72855i 0.503019 + 0.864275i \(0.332223\pi\)
−0.503019 + 0.864275i \(0.667777\pi\)
\(522\) 0 0
\(523\) 4.05988e6i 0.649022i 0.945882 + 0.324511i \(0.105200\pi\)
−0.945882 + 0.324511i \(0.894800\pi\)
\(524\) 0 0
\(525\) 2044.34 + 17061.5i 0.000323709 + 0.00270159i
\(526\) 0 0
\(527\) 54884.9 0.00860848
\(528\) 0 0
\(529\) 7.61684e6 1.18341
\(530\) 0 0
\(531\) −8.64360e6 + 2.10155e6i −1.33033 + 0.323448i
\(532\) 0 0
\(533\) 1.15087e7i 1.75473i
\(534\) 0 0
\(535\) 2.35622e6i 0.355903i
\(536\) 0 0
\(537\) 304517. 36487.7i 0.0455696 0.00546022i
\(538\) 0 0
\(539\) 1.26453e6 0.187481
\(540\) 0 0
\(541\) 3.26069e6 0.478979 0.239490 0.970899i \(-0.423020\pi\)
0.239490 + 0.970899i \(0.423020\pi\)
\(542\) 0 0
\(543\) −717349. + 85953.9i −0.104407 + 0.0125102i
\(544\) 0 0
\(545\) 4.95953e6i 0.715235i
\(546\) 0 0
\(547\) 5.81195e6i 0.830526i −0.909701 0.415263i \(-0.863690\pi\)
0.909701 0.415263i \(-0.136310\pi\)
\(548\) 0 0
\(549\) 9.03206e6 2.19600e6i 1.27896 0.310958i
\(550\) 0 0
\(551\) 919667. 0.129048
\(552\) 0 0
\(553\) −1.39078e7 −1.93396
\(554\) 0 0
\(555\) −1.13222e6 9.44923e6i −0.156027 1.30216i
\(556\) 0 0
\(557\) 5.08607e6i 0.694615i −0.937751 0.347308i \(-0.887096\pi\)
0.937751 0.347308i \(-0.112904\pi\)
\(558\) 0 0
\(559\) 1.66263e7i 2.25043i
\(560\) 0 0
\(561\) 8870.42 + 74030.3i 0.00118997 + 0.00993121i
\(562\) 0 0
\(563\) 9.79359e6 1.30218 0.651090 0.759001i \(-0.274312\pi\)
0.651090 + 0.759001i \(0.274312\pi\)
\(564\) 0 0
\(565\) 1.33447e7 1.75868
\(566\) 0 0
\(567\) 6.12582e6 + 1.18529e7i 0.800214 + 1.54835i
\(568\) 0 0
\(569\) 8.50753e6i 1.10160i −0.834638 0.550799i \(-0.814323\pi\)
0.834638 0.550799i \(-0.185677\pi\)
\(570\) 0 0
\(571\) 4.37026e6i 0.560941i 0.959863 + 0.280470i \(0.0904905\pi\)
−0.959863 + 0.280470i \(0.909510\pi\)
\(572\) 0 0
\(573\) 884103. 105935.i 0.112491 0.0134788i
\(574\) 0 0
\(575\) 18288.5 0.00230680
\(576\) 0 0
\(577\) 8.87042e6 1.10919 0.554594 0.832121i \(-0.312874\pi\)
0.554594 + 0.832121i \(0.312874\pi\)
\(578\) 0 0
\(579\) −3.99138e6 + 478254.i −0.494797 + 0.0592874i
\(580\) 0 0
\(581\) 2.56004e7i 3.14635i
\(582\) 0 0
\(583\) 419642.i 0.0511338i
\(584\) 0 0
\(585\) 2.55129e6 + 1.04933e7i 0.308226 + 1.26772i
\(586\) 0 0
\(587\) −1.40119e6 −0.167842 −0.0839209 0.996472i \(-0.526744\pi\)
−0.0839209 + 0.996472i \(0.526744\pi\)
\(588\) 0 0
\(589\) 422855. 0.0502231
\(590\) 0 0
\(591\) −1.30710e6 1.09087e7i −0.153936 1.28471i
\(592\) 0 0
\(593\) 1.59083e7i 1.85774i −0.370401 0.928872i \(-0.620780\pi\)
0.370401 0.928872i \(-0.379220\pi\)
\(594\) 0 0
\(595\) 1.63497e6i 0.189330i
\(596\) 0 0
\(597\) 481398. + 4.01763e6i 0.0552801 + 0.461353i
\(598\) 0 0
\(599\) 1.27346e7 1.45017 0.725085 0.688660i \(-0.241801\pi\)
0.725085 + 0.688660i \(0.241801\pi\)
\(600\) 0 0
\(601\) −8.53611e6 −0.963992 −0.481996 0.876173i \(-0.660088\pi\)
−0.481996 + 0.876173i \(0.660088\pi\)
\(602\) 0 0
\(603\) −515257. 2.11923e6i −0.0577072 0.237348i
\(604\) 0 0
\(605\) 8.91984e6i 0.990761i
\(606\) 0 0
\(607\) 3.68744e6i 0.406212i −0.979157 0.203106i \(-0.934896\pi\)
0.979157 0.203106i \(-0.0651036\pi\)
\(608\) 0 0
\(609\) −3.22264e6 + 386142.i −0.352102 + 0.0421894i
\(610\) 0 0
\(611\) −9.54895e6 −1.03479
\(612\) 0 0
\(613\) −5.99314e6 −0.644174 −0.322087 0.946710i \(-0.604384\pi\)
−0.322087 + 0.946710i \(0.604384\pi\)
\(614\) 0 0
\(615\) −1.25063e7 + 1.49852e6i −1.33334 + 0.159762i
\(616\) 0 0
\(617\) 1.12913e7i 1.19407i 0.802215 + 0.597035i \(0.203655\pi\)
−0.802215 + 0.597035i \(0.796345\pi\)
\(618\) 0 0
\(619\) 7.53091e6i 0.789989i 0.918684 + 0.394994i \(0.129253\pi\)
−0.918684 + 0.394994i \(0.870747\pi\)
\(620\) 0 0
\(621\) 1.33012e7 4.97260e6i 1.38408 0.517433i
\(622\) 0 0
\(623\) 2.83619e7 2.92762
\(624\) 0 0
\(625\) −9.75036e6 −0.998436
\(626\) 0 0
\(627\) 68341.3 + 570359.i 0.00694247 + 0.0579401i
\(628\) 0 0
\(629\) 1.41582e6i 0.142686i
\(630\) 0 0
\(631\) 1.39881e7i 1.39858i −0.714840 0.699289i \(-0.753500\pi\)
0.714840 0.699289i \(-0.246500\pi\)
\(632\) 0 0
\(633\) 990861. + 8.26947e6i 0.0982887 + 0.820292i
\(634\) 0 0
\(635\) −1.44594e7 −1.42304
\(636\) 0 0
\(637\) −2.72475e7 −2.66059
\(638\) 0 0
\(639\) −1.45466e7 + 3.53678e6i −1.40932 + 0.342654i
\(640\) 0 0
\(641\) 9.54988e6i 0.918021i 0.888431 + 0.459011i \(0.151796\pi\)
−0.888431 + 0.459011i \(0.848204\pi\)
\(642\) 0 0
\(643\) 1.51800e6i 0.144792i −0.997376 0.0723959i \(-0.976935\pi\)
0.997376 0.0723959i \(-0.0230645\pi\)
\(644\) 0 0
\(645\) −1.80674e7 + 2.16486e6i −1.71000 + 0.204895i
\(646\) 0 0
\(647\) −1.58617e7 −1.48967 −0.744834 0.667250i \(-0.767471\pi\)
−0.744834 + 0.667250i \(0.767471\pi\)
\(648\) 0 0
\(649\) 1.35162e6 0.125963
\(650\) 0 0
\(651\) −1.48174e6 + 177545.i −0.137032 + 0.0164193i
\(652\) 0 0
\(653\) 543838.i 0.0499099i −0.999689 0.0249549i \(-0.992056\pi\)
0.999689 0.0249549i \(-0.00794423\pi\)
\(654\) 0 0
\(655\) 528544.i 0.0481369i
\(656\) 0 0
\(657\) 606604. 147486.i 0.0548267 0.0133302i
\(658\) 0 0
\(659\) 1.06321e7 0.953685 0.476843 0.878989i \(-0.341781\pi\)
0.476843 + 0.878989i \(0.341781\pi\)
\(660\) 0 0
\(661\) −1.47871e7 −1.31638 −0.658189 0.752853i \(-0.728677\pi\)
−0.658189 + 0.752853i \(0.728677\pi\)
\(662\) 0 0
\(663\) −191136. 1.59517e6i −0.0168872 0.140936i
\(664\) 0 0
\(665\) 1.25965e7i 1.10458i
\(666\) 0 0
\(667\) 3.45440e6i 0.300648i
\(668\) 0 0
\(669\) −1.60236e6 1.33729e7i −0.138419 1.15521i
\(670\) 0 0
\(671\) −1.41237e6 −0.121099
\(672\) 0 0
\(673\) 1.01183e7 0.861129 0.430565 0.902560i \(-0.358314\pi\)
0.430565 + 0.902560i \(0.358314\pi\)
\(674\) 0 0
\(675\) 17309.9 6471.24i 0.00146229 0.000546674i
\(676\) 0 0
\(677\) 9.55270e6i 0.801040i −0.916288 0.400520i \(-0.868829\pi\)
0.916288 0.400520i \(-0.131171\pi\)
\(678\) 0 0
\(679\) 3.53942e7i 2.94617i
\(680\) 0 0
\(681\) 1.37731e7 1.65031e6i 1.13806 0.136364i
\(682\) 0 0
\(683\) −1.31661e7 −1.07995 −0.539975 0.841681i \(-0.681566\pi\)
−0.539975 + 0.841681i \(0.681566\pi\)
\(684\) 0 0
\(685\) 8.19457e6 0.667267
\(686\) 0 0
\(687\) 4.58928e6 549894.i 0.370982 0.0444516i
\(688\) 0 0
\(689\) 9.04226e6i 0.725653i
\(690\) 0 0
\(691\) 1.90008e7i 1.51383i −0.653516 0.756913i \(-0.726707\pi\)
0.653516 0.756913i \(-0.273293\pi\)
\(692\) 0 0
\(693\) −478955. 1.96992e6i −0.0378845 0.155817i
\(694\) 0 0
\(695\) 8.42479e6 0.661602
\(696\) 0 0
\(697\) 1.87387e6 0.146103
\(698\) 0 0
\(699\) −1.41557e6 1.18140e7i −0.109582 0.914541i
\(700\) 0 0
\(701\) 7.43183e6i 0.571216i 0.958347 + 0.285608i \(0.0921956\pi\)
−0.958347 + 0.285608i \(0.907804\pi\)
\(702\) 0 0
\(703\) 1.09081e7i 0.832453i
\(704\) 0 0
\(705\) −1.24334e6 1.03766e7i −0.0942146 0.786291i
\(706\) 0 0
\(707\) 7.77776e6 0.585202
\(708\) 0 0
\(709\) 9.49641e6 0.709486 0.354743 0.934964i \(-0.384568\pi\)
0.354743 + 0.934964i \(0.384568\pi\)
\(710\) 0 0
\(711\) 3.53364e6 + 1.45337e7i 0.262149 + 1.07821i
\(712\) 0 0
\(713\) 1.58831e6i 0.117007i
\(714\) 0 0
\(715\) 1.64087e6i 0.120035i
\(716\) 0 0
\(717\) 564172. 67600.0i 0.0409839 0.00491076i
\(718\) 0 0
\(719\) −2.25267e6 −0.162508 −0.0812541 0.996693i \(-0.525893\pi\)
−0.0812541 + 0.996693i \(0.525893\pi\)
\(720\) 0 0
\(721\) 6.37310e6 0.456575
\(722\) 0 0
\(723\) 2.00446e7 2.40178e6i 1.42611 0.170879i
\(724\) 0 0
\(725\) 4495.49i 0.000317638i
\(726\) 0 0
\(727\) 9.23196e6i 0.647825i −0.946087 0.323913i \(-0.895002\pi\)
0.946087 0.323913i \(-0.104998\pi\)
\(728\) 0 0
\(729\) 1.08299e7 9.41301e6i 0.754753 0.656009i
\(730\) 0 0
\(731\) 2.70712e6 0.187376
\(732\) 0 0
\(733\) −3.66782e6 −0.252143 −0.126072 0.992021i \(-0.540237\pi\)
−0.126072 + 0.992021i \(0.540237\pi\)
\(734\) 0 0
\(735\) −3.54783e6 2.96092e7i −0.242239 2.02166i
\(736\) 0 0
\(737\) 331389.i 0.0224734i
\(738\) 0 0
\(739\) 2.25692e7i 1.52022i −0.649795 0.760109i \(-0.725145\pi\)
0.649795 0.760109i \(-0.274855\pi\)
\(740\) 0 0
\(741\) −1.47259e6 1.22898e7i −0.0985225 0.822243i
\(742\) 0 0
\(743\) −1.79409e7 −1.19226 −0.596130 0.802888i \(-0.703296\pi\)
−0.596130 + 0.802888i \(0.703296\pi\)
\(744\) 0 0
\(745\) 1.24792e7 0.823752
\(746\) 0 0
\(747\) 2.67525e7 6.50443e6i 1.75413 0.426489i
\(748\) 0 0
\(749\) 9.53122e6i 0.620789i
\(750\) 0 0
\(751\) 1.43657e7i 0.929451i 0.885455 + 0.464726i \(0.153847\pi\)
−0.885455 + 0.464726i \(0.846153\pi\)
\(752\) 0 0
\(753\) 7.21332e6 864311.i 0.463605 0.0555498i
\(754\) 0 0
\(755\) −2.22376e7 −1.41978
\(756\) 0 0
\(757\) 5.10716e6 0.323922 0.161961 0.986797i \(-0.448218\pi\)
0.161961 + 0.986797i \(0.448218\pi\)
\(758\) 0 0
\(759\) −2.14235e6 + 256700.i −0.134985 + 0.0161741i
\(760\) 0 0
\(761\) 8.66301e6i 0.542260i 0.962543 + 0.271130i \(0.0873973\pi\)
−0.962543 + 0.271130i \(0.912603\pi\)
\(762\) 0 0
\(763\) 2.00619e7i 1.24756i
\(764\) 0 0
\(765\) 1.70855e6 415406.i 0.105554 0.0256637i
\(766\) 0 0
\(767\) −2.91241e7 −1.78758
\(768\) 0 0
\(769\) 1.21621e6 0.0741639 0.0370819 0.999312i \(-0.488194\pi\)
0.0370819 + 0.999312i \(0.488194\pi\)
\(770\) 0 0
\(771\) −1.06009e6 8.84726e6i −0.0642256 0.536010i
\(772\) 0 0
\(773\) 1.17432e6i 0.0706867i −0.999375 0.0353434i \(-0.988748\pi\)
0.999375 0.0353434i \(-0.0112525\pi\)
\(774\) 0 0
\(775\) 2066.99i 0.000123619i
\(776\) 0 0
\(777\) −4.57999e6 3.82234e7i −0.272152 2.27131i
\(778\) 0 0
\(779\) 1.44371e7 0.852384
\(780\) 0 0
\(781\) 2.27469e6 0.133443
\(782\) 0 0
\(783\) 1.22231e6 + 3.26955e6i 0.0712487 + 0.190583i
\(784\) 0 0
\(785\) 2.55695e7i 1.48098i
\(786\) 0 0
\(787\) 1.61589e7i 0.929983i −0.885315 0.464991i \(-0.846057\pi\)
0.885315 0.464991i \(-0.153943\pi\)
\(788\) 0 0
\(789\) −6.73310e6 + 806771.i −0.385055 + 0.0461379i
\(790\) 0 0
\(791\) 5.39811e7 3.06761
\(792\) 0 0
\(793\) 3.04330e7 1.71855
\(794\) 0 0
\(795\) 9.82601e6 1.17737e6i 0.551391 0.0660685i
\(796\) 0 0
\(797\) 1.59354e7i 0.888622i −0.895873 0.444311i \(-0.853448\pi\)
0.895873 0.444311i \(-0.146552\pi\)
\(798\) 0 0
\(799\) 1.55478e6i 0.0861592i
\(800\) 0 0
\(801\) −7.20604e6 2.96382e7i −0.396840 1.63219i
\(802\) 0 0
\(803\) −94856.2 −0.00519131
\(804\) 0 0
\(805\) −4.73142e7 −2.57337
\(806\) 0 0
\(807\) −2.07678e6 1.73323e7i −0.112255 0.936855i
\(808\) 0 0
\(809\) 8.21438e6i 0.441269i −0.975357 0.220635i \(-0.929187\pi\)
0.975357 0.220635i \(-0.0708128\pi\)
\(810\) 0 0
\(811\) 2.18134e7i 1.16459i −0.812979 0.582293i \(-0.802156\pi\)
0.812979 0.582293i \(-0.197844\pi\)
\(812\) 0 0
\(813\) −373148. 3.11419e6i −0.0197995 0.165242i
\(814\) 0 0
\(815\) 5.05962e6 0.266824
\(816\) 0 0
\(817\) 2.08568e7 1.09318
\(818\) 0 0
\(819\) 1.03203e7 + 4.24470e7i 0.537629 + 2.21125i
\(820\) 0 0
\(821\) 8.73324e6i 0.452186i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(822\) 0 0
\(823\) 5.88986e6i 0.303114i −0.988449 0.151557i \(-0.951571\pi\)
0.988449 0.151557i \(-0.0484287\pi\)
\(824\) 0 0
\(825\) −2788.01 + 334.064i −0.000142613 + 1.70881e-5i
\(826\) 0 0
\(827\) 2.33435e7 1.18687 0.593435 0.804882i \(-0.297771\pi\)
0.593435 + 0.804882i \(0.297771\pi\)
\(828\) 0 0
\(829\) −1.06227e7 −0.536846 −0.268423 0.963301i \(-0.586502\pi\)
−0.268423 + 0.963301i \(0.586502\pi\)
\(830\) 0 0
\(831\) −4.68106e6 + 560892.i −0.235148 + 0.0281758i
\(832\) 0 0
\(833\) 4.43650e6i 0.221528i
\(834\) 0 0
\(835\) 8.97508e6i 0.445474i
\(836\) 0 0
\(837\) 562008. + 1.50331e6i 0.0277287 + 0.0741712i
\(838\) 0 0
\(839\) −2.24424e7 −1.10069 −0.550345 0.834937i \(-0.685504\pi\)
−0.550345 + 0.834937i \(0.685504\pi\)
\(840\) 0 0
\(841\) 1.96620e7 0.958602
\(842\) 0 0
\(843\) −3.28341e6 2.74025e7i −0.159131 1.32807i
\(844\) 0 0
\(845\) 1.46170e7i 0.704234i
\(846\) 0 0
\(847\) 3.60820e7i 1.72815i
\(848\) 0 0
\(849\) 1.50204e6 + 1.25357e7i 0.0715177 + 0.596868i
\(850\) 0 0
\(851\) −4.09723e7 −1.93940
\(852\) 0 0
\(853\) 2.61531e7 1.23070 0.615348 0.788256i \(-0.289015\pi\)
0.615348 + 0.788256i \(0.289015\pi\)
\(854\) 0 0
\(855\) 1.31633e7 3.20045e6i 0.615815 0.149726i
\(856\) 0 0
\(857\) 1.20270e7i 0.559380i 0.960090 + 0.279690i \(0.0902316\pi\)
−0.960090 + 0.279690i \(0.909768\pi\)
\(858\) 0 0
\(859\) 3.12884e7i 1.44677i 0.690443 + 0.723387i \(0.257416\pi\)
−0.690443 + 0.723387i \(0.742584\pi\)
\(860\) 0 0
\(861\) −5.05895e7 + 6.06172e6i −2.32569 + 0.278668i
\(862\) 0 0
\(863\) 1.08600e7 0.496368 0.248184 0.968713i \(-0.420166\pi\)
0.248184 + 0.968713i \(0.420166\pi\)
\(864\) 0 0
\(865\) −2.37610e7 −1.07975
\(866\) 0 0
\(867\) 2.17165e7 2.60210e6i 0.981163 0.117565i
\(868\) 0 0
\(869\) 2.27267e6i 0.102091i
\(870\) 0 0
\(871\) 7.14063e6i 0.318927i
\(872\) 0 0
\(873\) −3.69870e7 + 8.99279e6i −1.64253 + 0.399355i
\(874\) 0 0
\(875\) −3.95032e7 −1.74426
\(876\) 0 0
\(877\) 1.30114e7 0.571248 0.285624 0.958342i \(-0.407799\pi\)
0.285624 + 0.958342i \(0.407799\pi\)
\(878\) 0 0
\(879\) −3.22058e6 2.68781e7i −0.140592 1.17335i
\(880\) 0 0
\(881\) 4.19604e7i 1.82138i 0.413092 + 0.910689i \(0.364449\pi\)
−0.413092 + 0.910689i \(0.635551\pi\)
\(882\) 0 0
\(883\) 2.89203e7i 1.24825i −0.781325 0.624124i \(-0.785456\pi\)
0.781325 0.624124i \(-0.214544\pi\)
\(884\) 0 0
\(885\) −3.79218e6 3.16485e7i −0.162753 1.35830i
\(886\) 0 0
\(887\) 3.04342e7 1.29883 0.649415 0.760434i \(-0.275014\pi\)
0.649415 + 0.760434i \(0.275014\pi\)
\(888\) 0 0
\(889\) −5.84902e7 −2.48215
\(890\) 0 0
\(891\) −1.93688e6 + 1.00102e6i −0.0817349 + 0.0422422i
\(892\) 0 0
\(893\) 1.19786e7i 0.502665i
\(894\) 0 0
\(895\) 1.09898e6i 0.0458597i
\(896\) 0 0
\(897\) 4.61624e7 5.53125e6i 1.91561 0.229531i
\(898\) 0 0
\(899\) −390420. −0.0161114
\(900\) 0 0
\(901\) −1.47228e6 −0.0604197
\(902\) 0 0
\(903\) −7.30850e7 + 8.75716e6i −2.98269 + 0.357391i
\(904\) 0 0
\(905\) 2.58886e6i 0.105072i
\(906\) 0 0
\(907\) 2.69233e7i 1.08670i −0.839506 0.543350i \(-0.817156\pi\)
0.839506 0.543350i \(-0.182844\pi\)
\(908\) 0 0
\(909\) −1.97613e6 8.12776e6i −0.0793244 0.326258i
\(910\) 0 0
\(911\) 3.27099e7 1.30582 0.652910 0.757435i \(-0.273548\pi\)
0.652910 + 0.757435i \(0.273548\pi\)
\(912\) 0 0
\(913\) −4.18335e6 −0.166091
\(914\) 0 0
\(915\) 3.96260e6 + 3.30709e7i 0.156469 + 1.30585i
\(916\) 0 0
\(917\) 2.13803e6i 0.0839636i
\(918\) 0 0
\(919\) 1.99356e7i 0.778648i 0.921101 + 0.389324i \(0.127291\pi\)
−0.921101 + 0.389324i \(0.872709\pi\)
\(920\) 0 0
\(921\) 1.10662e6 + 9.23558e6i 0.0429883 + 0.358769i
\(922\) 0 0
\(923\) −4.90140e7 −1.89372
\(924\) 0 0
\(925\) −53320.6 −0.00204899
\(926\) 0 0
\(927\) −1.61924e6 6.65989e6i −0.0618889 0.254547i
\(928\) 0 0
\(929\) 6.13766e6i 0.233326i −0.993172 0.116663i \(-0.962780\pi\)
0.993172 0.116663i \(-0.0372198\pi\)
\(930\) 0 0
\(931\) 3.41805e7i 1.29242i
\(932\) 0 0
\(933\) 4.43078e7 5.30903e6i 1.66639 0.199669i
\(934\) 0 0
\(935\) −267170. −0.00999444
\(936\) 0 0
\(937\) 2.01876e7 0.751166 0.375583 0.926789i \(-0.377443\pi\)
0.375583 + 0.926789i \(0.377443\pi\)
\(938\) 0 0
\(939\) −2.37494e7 + 2.84569e6i −0.879000 + 0.105323i
\(940\) 0 0
\(941\) 4.23966e7i 1.56083i 0.625259 + 0.780417i \(0.284993\pi\)
−0.625259 + 0.780417i \(0.715007\pi\)
\(942\) 0 0
\(943\) 5.42277e7i 1.98583i
\(944\) 0 0
\(945\) −4.47824e7 + 1.67417e7i −1.63128 + 0.609847i
\(946\) 0 0
\(947\) −4.58224e6 −0.166036 −0.0830181 0.996548i \(-0.526456\pi\)
−0.0830181 + 0.996548i \(0.526456\pi\)
\(948\) 0 0
\(949\) 2.04392e6 0.0736713
\(950\) 0 0
\(951\) 6.18035e6 + 5.15796e7i 0.221596 + 1.84938i
\(952\) 0 0
\(953\) 2.70977e7i 0.966498i −0.875483 0.483249i \(-0.839457\pi\)
0.875483 0.483249i \(-0.160543\pi\)
\(954\) 0 0
\(955\) 3.19066e6i 0.113207i
\(956\) 0 0
\(957\) −63099.1 526609.i −0.00222712 0.0185870i
\(958\) 0 0
\(959\) 3.31481e7 1.16389
\(960\) 0 0
\(961\) 2.84496e7 0.993730
\(962\) 0 0
\(963\) 9.96012e6 2.42164e6i 0.346098 0.0841481i
\(964\) 0 0
\(965\) 1.44046e7i 0.497947i
\(966\) 0 0
\(967\) 2.22367e7i 0.764724i 0.924013 + 0.382362i \(0.124889\pi\)
−0.924013 + 0.382362i \(0.875111\pi\)
\(968\) 0 0
\(969\) −2.00106e6 + 239770.i −0.0684620 + 0.00820323i
\(970\) 0 0
\(971\) 3.88599e7 1.32267 0.661337 0.750089i \(-0.269989\pi\)
0.661337 + 0.750089i \(0.269989\pi\)
\(972\) 0 0
\(973\) 3.40794e7 1.15401
\(974\) 0 0
\(975\) 60074.8 7198.26i 0.00202386 0.000242502i
\(976\) 0 0
\(977\) 3.97266e7i 1.33151i 0.746169 + 0.665756i \(0.231891\pi\)
−0.746169 + 0.665756i \(0.768109\pi\)
\(978\) 0 0
\(979\) 4.63459e6i 0.154545i
\(980\) 0 0
\(981\) 2.09647e7 5.09724e6i 0.695532 0.169107i
\(982\) 0 0
\(983\) 7.21692e6 0.238215 0.119107 0.992881i \(-0.461997\pi\)
0.119107 + 0.992881i \(0.461997\pi\)
\(984\) 0 0
\(985\) 3.93687e7 1.29289
\(986\) 0 0
\(987\) −5.02949e6 4.19748e7i −0.164335 1.37150i
\(988\) 0 0
\(989\) 7.83410e7i 2.54682i
\(990\) 0 0
\(991\) 2.46570e7i 0.797545i 0.917050 + 0.398773i \(0.130564\pi\)
−0.917050 + 0.398773i \(0.869436\pi\)
\(992\) 0 0
\(993\) 3.74875e6 + 3.12861e7i 0.120646 + 1.00688i
\(994\) 0 0
\(995\) −1.44993e7 −0.464291
\(996\) 0 0
\(997\) 1.96664e7 0.626596 0.313298 0.949655i \(-0.398566\pi\)
0.313298 + 0.949655i \(0.398566\pi\)
\(998\) 0 0
\(999\) −3.87798e7 + 1.44977e7i −1.22940 + 0.459606i
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.a.383.19 20
3.2 odd 2 384.6.c.d.383.1 yes 20
4.3 odd 2 384.6.c.d.383.2 yes 20
8.3 odd 2 384.6.c.b.383.19 yes 20
8.5 even 2 384.6.c.c.383.2 yes 20
12.11 even 2 inner 384.6.c.a.383.20 yes 20
24.5 odd 2 384.6.c.b.383.20 yes 20
24.11 even 2 384.6.c.c.383.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.19 20 1.1 even 1 trivial
384.6.c.a.383.20 yes 20 12.11 even 2 inner
384.6.c.b.383.19 yes 20 8.3 odd 2
384.6.c.b.383.20 yes 20 24.5 odd 2
384.6.c.c.383.1 yes 20 24.11 even 2
384.6.c.c.383.2 yes 20 8.5 even 2
384.6.c.d.383.1 yes 20 3.2 odd 2
384.6.c.d.383.2 yes 20 4.3 odd 2