Properties

Label 325.6.b.i.274.22
Level $325$
Weight $6$
Character 325.274
Analytic conductor $52.125$
Analytic rank $0$
Dimension $22$
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: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.22
Character \(\chi\) \(=\) 325.274
Dual form 325.6.b.i.274.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.2252i q^{2} -19.1897i q^{3} -94.0044 q^{4} +215.408 q^{6} +70.9909i q^{7} -696.009i q^{8} -125.246 q^{9} -161.025 q^{11} +1803.92i q^{12} -169.000i q^{13} -796.885 q^{14} +4804.68 q^{16} -121.397i q^{17} -1405.90i q^{18} +3111.76 q^{19} +1362.30 q^{21} -1807.53i q^{22} +3100.00i q^{23} -13356.2 q^{24} +1897.05 q^{26} -2259.68i q^{27} -6673.46i q^{28} -3172.08 q^{29} -3515.11 q^{31} +31661.0i q^{32} +3090.02i q^{33} +1362.70 q^{34} +11773.6 q^{36} +6991.30i q^{37} +34930.1i q^{38} -3243.06 q^{39} -19551.6 q^{41} +15292.0i q^{42} -13398.3i q^{43} +15137.0 q^{44} -34798.0 q^{46} +7637.52i q^{47} -92200.5i q^{48} +11767.3 q^{49} -2329.57 q^{51} +15886.7i q^{52} -27750.4i q^{53} +25365.2 q^{54} +49410.3 q^{56} -59713.9i q^{57} -35607.1i q^{58} -33042.3 q^{59} -33195.2 q^{61} -39457.6i q^{62} -8891.30i q^{63} -201651. q^{64} -34686.0 q^{66} +24591.9i q^{67} +11411.8i q^{68} +59488.1 q^{69} +16641.4 q^{71} +87172.0i q^{72} +5252.97i q^{73} -78478.5 q^{74} -292519. q^{76} -11431.3i q^{77} -36403.9i q^{78} +3121.54 q^{79} -73797.2 q^{81} -219470. i q^{82} +25125.8i q^{83} -128062. q^{84} +150399. q^{86} +60871.3i q^{87} +112075. i q^{88} -634.739 q^{89} +11997.5 q^{91} -291413. i q^{92} +67453.9i q^{93} -85732.5 q^{94} +607566. q^{96} +156159. i q^{97} +132090. i q^{98} +20167.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36}+ \cdots - 853396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.2252i 1.98435i 0.124864 + 0.992174i \(0.460151\pi\)
−0.124864 + 0.992174i \(0.539849\pi\)
\(3\) − 19.1897i − 1.23102i −0.788129 0.615511i \(-0.788950\pi\)
0.788129 0.615511i \(-0.211050\pi\)
\(4\) −94.0044 −2.93764
\(5\) 0 0
\(6\) 215.408 2.44277
\(7\) 70.9909i 0.547593i 0.961788 + 0.273796i \(0.0882795\pi\)
−0.961788 + 0.273796i \(0.911721\pi\)
\(8\) − 696.009i − 3.84494i
\(9\) −125.246 −0.515414
\(10\) 0 0
\(11\) −161.025 −0.401246 −0.200623 0.979669i \(-0.564297\pi\)
−0.200623 + 0.979669i \(0.564297\pi\)
\(12\) 1803.92i 3.61629i
\(13\) − 169.000i − 0.277350i
\(14\) −796.885 −1.08661
\(15\) 0 0
\(16\) 4804.68 4.69207
\(17\) − 121.397i − 0.101879i −0.998702 0.0509396i \(-0.983778\pi\)
0.998702 0.0509396i \(-0.0162216\pi\)
\(18\) − 1405.90i − 1.02276i
\(19\) 3111.76 1.97753 0.988764 0.149485i \(-0.0477617\pi\)
0.988764 + 0.149485i \(0.0477617\pi\)
\(20\) 0 0
\(21\) 1362.30 0.674099
\(22\) − 1807.53i − 0.796211i
\(23\) 3100.00i 1.22192i 0.791662 + 0.610959i \(0.209216\pi\)
−0.791662 + 0.610959i \(0.790784\pi\)
\(24\) −13356.2 −4.73321
\(25\) 0 0
\(26\) 1897.05 0.550359
\(27\) − 2259.68i − 0.596536i
\(28\) − 6673.46i − 1.60863i
\(29\) −3172.08 −0.700404 −0.350202 0.936674i \(-0.613887\pi\)
−0.350202 + 0.936674i \(0.613887\pi\)
\(30\) 0 0
\(31\) −3515.11 −0.656953 −0.328476 0.944512i \(-0.606535\pi\)
−0.328476 + 0.944512i \(0.606535\pi\)
\(32\) 31661.0i 5.46575i
\(33\) 3090.02i 0.493942i
\(34\) 1362.70 0.202164
\(35\) 0 0
\(36\) 11773.6 1.51410
\(37\) 6991.30i 0.839563i 0.907625 + 0.419782i \(0.137893\pi\)
−0.907625 + 0.419782i \(0.862107\pi\)
\(38\) 34930.1i 3.92410i
\(39\) −3243.06 −0.341424
\(40\) 0 0
\(41\) −19551.6 −1.81644 −0.908222 0.418488i \(-0.862560\pi\)
−0.908222 + 0.418488i \(0.862560\pi\)
\(42\) 15292.0i 1.33765i
\(43\) − 13398.3i − 1.10504i −0.833498 0.552522i \(-0.813665\pi\)
0.833498 0.552522i \(-0.186335\pi\)
\(44\) 15137.0 1.17871
\(45\) 0 0
\(46\) −34798.0 −2.42471
\(47\) 7637.52i 0.504322i 0.967685 + 0.252161i \(0.0811412\pi\)
−0.967685 + 0.252161i \(0.918859\pi\)
\(48\) − 92200.5i − 5.77604i
\(49\) 11767.3 0.700142
\(50\) 0 0
\(51\) −2329.57 −0.125415
\(52\) 15886.7i 0.814754i
\(53\) − 27750.4i − 1.35700i −0.734600 0.678500i \(-0.762630\pi\)
0.734600 0.678500i \(-0.237370\pi\)
\(54\) 25365.2 1.18374
\(55\) 0 0
\(56\) 49410.3 2.10546
\(57\) − 59713.9i − 2.43438i
\(58\) − 35607.1i − 1.38985i
\(59\) −33042.3 −1.23578 −0.617889 0.786265i \(-0.712012\pi\)
−0.617889 + 0.786265i \(0.712012\pi\)
\(60\) 0 0
\(61\) −33195.2 −1.14222 −0.571110 0.820873i \(-0.693487\pi\)
−0.571110 + 0.820873i \(0.693487\pi\)
\(62\) − 39457.6i − 1.30362i
\(63\) − 8891.30i − 0.282237i
\(64\) −201651. −6.15389
\(65\) 0 0
\(66\) −34686.0 −0.980153
\(67\) 24591.9i 0.669275i 0.942347 + 0.334637i \(0.108614\pi\)
−0.942347 + 0.334637i \(0.891386\pi\)
\(68\) 11411.8i 0.299284i
\(69\) 59488.1 1.50421
\(70\) 0 0
\(71\) 16641.4 0.391781 0.195891 0.980626i \(-0.437240\pi\)
0.195891 + 0.980626i \(0.437240\pi\)
\(72\) 87172.0i 1.98174i
\(73\) 5252.97i 0.115371i 0.998335 + 0.0576856i \(0.0183721\pi\)
−0.998335 + 0.0576856i \(0.981628\pi\)
\(74\) −78478.5 −1.66599
\(75\) 0 0
\(76\) −292519. −5.80926
\(77\) − 11431.3i − 0.219719i
\(78\) − 36403.9i − 0.677504i
\(79\) 3121.54 0.0562731 0.0281366 0.999604i \(-0.491043\pi\)
0.0281366 + 0.999604i \(0.491043\pi\)
\(80\) 0 0
\(81\) −73797.2 −1.24976
\(82\) − 219470.i − 3.60446i
\(83\) 25125.8i 0.400336i 0.979762 + 0.200168i \(0.0641488\pi\)
−0.979762 + 0.200168i \(0.935851\pi\)
\(84\) −128062. −1.98026
\(85\) 0 0
\(86\) 150399. 2.19279
\(87\) 60871.3i 0.862213i
\(88\) 112075.i 1.54277i
\(89\) −634.739 −0.00849416 −0.00424708 0.999991i \(-0.501352\pi\)
−0.00424708 + 0.999991i \(0.501352\pi\)
\(90\) 0 0
\(91\) 11997.5 0.151875
\(92\) − 291413.i − 3.58955i
\(93\) 67453.9i 0.808723i
\(94\) −85732.5 −1.00075
\(95\) 0 0
\(96\) 607566. 6.72846
\(97\) 156159.i 1.68514i 0.538584 + 0.842572i \(0.318959\pi\)
−0.538584 + 0.842572i \(0.681041\pi\)
\(98\) 132090.i 1.38933i
\(99\) 20167.6 0.206808
\(100\) 0 0
\(101\) −165081. −1.61026 −0.805128 0.593101i \(-0.797903\pi\)
−0.805128 + 0.593101i \(0.797903\pi\)
\(102\) − 26149.9i − 0.248868i
\(103\) − 75861.6i − 0.704578i −0.935891 0.352289i \(-0.885403\pi\)
0.935891 0.352289i \(-0.114597\pi\)
\(104\) −117626. −1.06640
\(105\) 0 0
\(106\) 311503. 2.69276
\(107\) − 141728.i − 1.19673i −0.801223 0.598366i \(-0.795817\pi\)
0.801223 0.598366i \(-0.204183\pi\)
\(108\) 212419.i 1.75241i
\(109\) −11820.9 −0.0952981 −0.0476490 0.998864i \(-0.515173\pi\)
−0.0476490 + 0.998864i \(0.515173\pi\)
\(110\) 0 0
\(111\) 134161. 1.03352
\(112\) 341089.i 2.56934i
\(113\) 43552.7i 0.320862i 0.987047 + 0.160431i \(0.0512885\pi\)
−0.987047 + 0.160431i \(0.948712\pi\)
\(114\) 670298. 4.83065
\(115\) 0 0
\(116\) 298189. 2.05753
\(117\) 21166.5i 0.142950i
\(118\) − 370906.i − 2.45222i
\(119\) 8618.08 0.0557883
\(120\) 0 0
\(121\) −135122. −0.839002
\(122\) − 372621.i − 2.26656i
\(123\) 375189.i 2.23608i
\(124\) 330435. 1.92989
\(125\) 0 0
\(126\) 99806.3 0.560056
\(127\) 229412.i 1.26214i 0.775727 + 0.631068i \(0.217383\pi\)
−0.775727 + 0.631068i \(0.782617\pi\)
\(128\) − 1.25041e6i − 6.74570i
\(129\) −257110. −1.36033
\(130\) 0 0
\(131\) −161719. −0.823347 −0.411674 0.911331i \(-0.635056\pi\)
−0.411674 + 0.911331i \(0.635056\pi\)
\(132\) − 290475.i − 1.45102i
\(133\) 220907.i 1.08288i
\(134\) −276048. −1.32807
\(135\) 0 0
\(136\) −84493.4 −0.391720
\(137\) 233597.i 1.06332i 0.846956 + 0.531662i \(0.178433\pi\)
−0.846956 + 0.531662i \(0.821567\pi\)
\(138\) 667764.i 2.98487i
\(139\) −97992.9 −0.430187 −0.215094 0.976593i \(-0.569006\pi\)
−0.215094 + 0.976593i \(0.569006\pi\)
\(140\) 0 0
\(141\) 146562. 0.620831
\(142\) 186802.i 0.777430i
\(143\) 27213.1i 0.111286i
\(144\) −601765. −2.41836
\(145\) 0 0
\(146\) −58965.4 −0.228937
\(147\) − 225811.i − 0.861890i
\(148\) − 657212.i − 2.46633i
\(149\) −267805. −0.988219 −0.494110 0.869400i \(-0.664506\pi\)
−0.494110 + 0.869400i \(0.664506\pi\)
\(150\) 0 0
\(151\) −125041. −0.446283 −0.223141 0.974786i \(-0.571631\pi\)
−0.223141 + 0.974786i \(0.571631\pi\)
\(152\) − 2.16582e6i − 7.60348i
\(153\) 15204.4i 0.0525099i
\(154\) 128318. 0.436000
\(155\) 0 0
\(156\) 304862. 1.00298
\(157\) − 323793.i − 1.04838i −0.851602 0.524189i \(-0.824368\pi\)
0.851602 0.524189i \(-0.175632\pi\)
\(158\) 35039.8i 0.111665i
\(159\) −532523. −1.67050
\(160\) 0 0
\(161\) −220072. −0.669113
\(162\) − 828386.i − 2.47996i
\(163\) 483015.i 1.42394i 0.702210 + 0.711970i \(0.252197\pi\)
−0.702210 + 0.711970i \(0.747803\pi\)
\(164\) 1.83793e6 5.33605
\(165\) 0 0
\(166\) −282041. −0.794406
\(167\) 272660.i 0.756538i 0.925696 + 0.378269i \(0.123480\pi\)
−0.925696 + 0.378269i \(0.876520\pi\)
\(168\) − 948171.i − 2.59187i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) −389734. −1.01924
\(172\) 1.25950e6i 3.24622i
\(173\) − 248584.i − 0.631477i −0.948846 0.315738i \(-0.897748\pi\)
0.948846 0.315738i \(-0.102252\pi\)
\(174\) −683290. −1.71093
\(175\) 0 0
\(176\) −773671. −1.88267
\(177\) 634073.i 1.52127i
\(178\) − 7125.05i − 0.0168554i
\(179\) −29132.9 −0.0679597 −0.0339798 0.999423i \(-0.510818\pi\)
−0.0339798 + 0.999423i \(0.510818\pi\)
\(180\) 0 0
\(181\) 583931. 1.32485 0.662423 0.749130i \(-0.269528\pi\)
0.662423 + 0.749130i \(0.269528\pi\)
\(182\) 134674.i 0.301373i
\(183\) 637006.i 1.40610i
\(184\) 2.15763e6 4.69820
\(185\) 0 0
\(186\) −757181. −1.60479
\(187\) 19547.9i 0.0408786i
\(188\) − 717960.i − 1.48151i
\(189\) 160417. 0.326659
\(190\) 0 0
\(191\) 708228. 1.40472 0.702359 0.711822i \(-0.252130\pi\)
0.702359 + 0.711822i \(0.252130\pi\)
\(192\) 3.86962e6i 7.57557i
\(193\) 596277.i 1.15227i 0.817354 + 0.576136i \(0.195440\pi\)
−0.817354 + 0.576136i \(0.804560\pi\)
\(194\) −1.75291e6 −3.34391
\(195\) 0 0
\(196\) −1.10618e6 −2.05676
\(197\) − 23822.5i − 0.0437342i −0.999761 0.0218671i \(-0.993039\pi\)
0.999761 0.0218671i \(-0.00696107\pi\)
\(198\) 226385.i 0.410378i
\(199\) −470656. −0.842503 −0.421251 0.906944i \(-0.638409\pi\)
−0.421251 + 0.906944i \(0.638409\pi\)
\(200\) 0 0
\(201\) 471911. 0.823892
\(202\) − 1.85307e6i − 3.19531i
\(203\) − 225189.i − 0.383536i
\(204\) 218990. 0.368425
\(205\) 0 0
\(206\) 851560. 1.39813
\(207\) − 388261.i − 0.629793i
\(208\) − 811991.i − 1.30135i
\(209\) −501070. −0.793475
\(210\) 0 0
\(211\) −619805. −0.958405 −0.479202 0.877704i \(-0.659074\pi\)
−0.479202 + 0.877704i \(0.659074\pi\)
\(212\) 2.60866e6i 3.98637i
\(213\) − 319344.i − 0.482291i
\(214\) 1.59092e6 2.37473
\(215\) 0 0
\(216\) −1.57276e6 −2.29365
\(217\) − 249541.i − 0.359743i
\(218\) − 132692.i − 0.189105i
\(219\) 100803. 0.142024
\(220\) 0 0
\(221\) −20516.1 −0.0282562
\(222\) 1.50598e6i 2.05086i
\(223\) 310997.i 0.418787i 0.977831 + 0.209394i \(0.0671490\pi\)
−0.977831 + 0.209394i \(0.932851\pi\)
\(224\) −2.24765e6 −2.99301
\(225\) 0 0
\(226\) −488886. −0.636702
\(227\) − 450336.i − 0.580059i −0.957018 0.290030i \(-0.906335\pi\)
0.957018 0.290030i \(-0.0936651\pi\)
\(228\) 5.61337e6i 7.15132i
\(229\) −855126. −1.07756 −0.538780 0.842447i \(-0.681114\pi\)
−0.538780 + 0.842447i \(0.681114\pi\)
\(230\) 0 0
\(231\) −219363. −0.270479
\(232\) 2.20779e6i 2.69301i
\(233\) 1.09514e6i 1.32153i 0.750591 + 0.660767i \(0.229769\pi\)
−0.750591 + 0.660767i \(0.770231\pi\)
\(234\) −237597. −0.283663
\(235\) 0 0
\(236\) 3.10612e6 3.63027
\(237\) − 59901.4i − 0.0692734i
\(238\) 96739.4i 0.110703i
\(239\) −917921. −1.03947 −0.519733 0.854329i \(-0.673969\pi\)
−0.519733 + 0.854329i \(0.673969\pi\)
\(240\) 0 0
\(241\) −642632. −0.712721 −0.356361 0.934349i \(-0.615982\pi\)
−0.356361 + 0.934349i \(0.615982\pi\)
\(242\) − 1.51677e6i − 1.66487i
\(243\) 867047.i 0.941948i
\(244\) 3.12049e6 3.35543
\(245\) 0 0
\(246\) −4.21156e6 −4.43717
\(247\) − 525888.i − 0.548468i
\(248\) 2.44655e6i 2.52595i
\(249\) 482157. 0.492822
\(250\) 0 0
\(251\) −96744.1 −0.0969260 −0.0484630 0.998825i \(-0.515432\pi\)
−0.0484630 + 0.998825i \(0.515432\pi\)
\(252\) 835821.i 0.829109i
\(253\) − 499176.i − 0.490289i
\(254\) −2.57518e6 −2.50452
\(255\) 0 0
\(256\) 7.58322e6 7.23192
\(257\) 1.06681e6i 1.00752i 0.863844 + 0.503759i \(0.168050\pi\)
−0.863844 + 0.503759i \(0.831950\pi\)
\(258\) − 2.88611e6i − 2.69937i
\(259\) −496319. −0.459739
\(260\) 0 0
\(261\) 397288. 0.360998
\(262\) − 1.81532e6i − 1.63381i
\(263\) − 503116.i − 0.448517i −0.974530 0.224258i \(-0.928004\pi\)
0.974530 0.224258i \(-0.0719960\pi\)
\(264\) 2.15068e6 1.89918
\(265\) 0 0
\(266\) −2.47972e6 −2.14881
\(267\) 12180.5i 0.0104565i
\(268\) − 2.31174e6i − 1.96609i
\(269\) 104820. 0.0883212 0.0441606 0.999024i \(-0.485939\pi\)
0.0441606 + 0.999024i \(0.485939\pi\)
\(270\) 0 0
\(271\) −168792. −0.139614 −0.0698069 0.997561i \(-0.522238\pi\)
−0.0698069 + 0.997561i \(0.522238\pi\)
\(272\) − 583273.i − 0.478024i
\(273\) − 230228.i − 0.186961i
\(274\) −2.62217e6 −2.11001
\(275\) 0 0
\(276\) −5.59214e6 −4.41881
\(277\) − 863313.i − 0.676034i −0.941140 0.338017i \(-0.890244\pi\)
0.941140 0.338017i \(-0.109756\pi\)
\(278\) − 1.09999e6i − 0.853641i
\(279\) 440251. 0.338602
\(280\) 0 0
\(281\) −1.56951e6 −1.18577 −0.592883 0.805289i \(-0.702010\pi\)
−0.592883 + 0.805289i \(0.702010\pi\)
\(282\) 1.64518e6i 1.23194i
\(283\) − 1.10553e6i − 0.820546i −0.911963 0.410273i \(-0.865433\pi\)
0.911963 0.410273i \(-0.134567\pi\)
\(284\) −1.56436e6 −1.15091
\(285\) 0 0
\(286\) −305472. −0.220829
\(287\) − 1.38799e6i − 0.994672i
\(288\) − 3.96540e6i − 2.81712i
\(289\) 1.40512e6 0.989621
\(290\) 0 0
\(291\) 2.99664e6 2.07445
\(292\) − 493802.i − 0.338919i
\(293\) 116407.i 0.0792155i 0.999215 + 0.0396077i \(0.0126108\pi\)
−0.999215 + 0.0396077i \(0.987389\pi\)
\(294\) 2.53477e6 1.71029
\(295\) 0 0
\(296\) 4.86601e6 3.22807
\(297\) 363863.i 0.239358i
\(298\) − 3.00616e6i − 1.96097i
\(299\) 523900. 0.338899
\(300\) 0 0
\(301\) 951161. 0.605115
\(302\) − 1.40361e6i − 0.885580i
\(303\) 3.16787e6i 1.98226i
\(304\) 1.49510e7 9.27870
\(305\) 0 0
\(306\) −170672. −0.104198
\(307\) − 830234.i − 0.502753i −0.967889 0.251376i \(-0.919117\pi\)
0.967889 0.251376i \(-0.0808832\pi\)
\(308\) 1.07459e6i 0.645455i
\(309\) −1.45576e6 −0.867351
\(310\) 0 0
\(311\) 204863. 0.120105 0.0600526 0.998195i \(-0.480873\pi\)
0.0600526 + 0.998195i \(0.480873\pi\)
\(312\) 2.25720e6i 1.31276i
\(313\) 1.96760e6i 1.13521i 0.823302 + 0.567604i \(0.192129\pi\)
−0.823302 + 0.567604i \(0.807871\pi\)
\(314\) 3.63463e6 2.08035
\(315\) 0 0
\(316\) −293438. −0.165310
\(317\) − 2.47859e6i − 1.38534i −0.721255 0.692670i \(-0.756434\pi\)
0.721255 0.692670i \(-0.243566\pi\)
\(318\) − 5.97766e6i − 3.31485i
\(319\) 510782. 0.281034
\(320\) 0 0
\(321\) −2.71973e6 −1.47320
\(322\) − 2.47034e6i − 1.32775i
\(323\) − 377759.i − 0.201469i
\(324\) 6.93726e6 3.67135
\(325\) 0 0
\(326\) −5.42193e6 −2.82559
\(327\) 226840.i 0.117314i
\(328\) 1.36081e7i 6.98413i
\(329\) −542195. −0.276163
\(330\) 0 0
\(331\) −3.17720e6 −1.59395 −0.796974 0.604013i \(-0.793567\pi\)
−0.796974 + 0.604013i \(0.793567\pi\)
\(332\) − 2.36193e6i − 1.17604i
\(333\) − 875629.i − 0.432722i
\(334\) −3.06066e6 −1.50123
\(335\) 0 0
\(336\) 6.54540e6 3.16292
\(337\) 1.04972e6i 0.503500i 0.967792 + 0.251750i \(0.0810061\pi\)
−0.967792 + 0.251750i \(0.918994\pi\)
\(338\) − 320602.i − 0.152642i
\(339\) 835764. 0.394988
\(340\) 0 0
\(341\) 566018. 0.263599
\(342\) − 4.37483e6i − 2.02254i
\(343\) 2.02852e6i 0.930986i
\(344\) −9.32537e6 −4.24883
\(345\) 0 0
\(346\) 2.79039e6 1.25307
\(347\) 502238.i 0.223916i 0.993713 + 0.111958i \(0.0357123\pi\)
−0.993713 + 0.111958i \(0.964288\pi\)
\(348\) − 5.72217e6i − 2.53287i
\(349\) 1.28824e6 0.566152 0.283076 0.959098i \(-0.408645\pi\)
0.283076 + 0.959098i \(0.408645\pi\)
\(350\) 0 0
\(351\) −381885. −0.165449
\(352\) − 5.09820e6i − 2.19311i
\(353\) − 3.80443e6i − 1.62500i −0.582962 0.812499i \(-0.698107\pi\)
0.582962 0.812499i \(-0.301893\pi\)
\(354\) −7.11758e6 −3.01873
\(355\) 0 0
\(356\) 59668.2 0.0249527
\(357\) − 165379.i − 0.0686766i
\(358\) − 327022.i − 0.134856i
\(359\) −3.18303e6 −1.30348 −0.651740 0.758442i \(-0.725961\pi\)
−0.651740 + 0.758442i \(0.725961\pi\)
\(360\) 0 0
\(361\) 7.20697e6 2.91062
\(362\) 6.55472e6i 2.62895i
\(363\) 2.59296e6i 1.03283i
\(364\) −1.12781e6 −0.446153
\(365\) 0 0
\(366\) −7.15050e6 −2.79019
\(367\) 1.51403e6i 0.586770i 0.955994 + 0.293385i \(0.0947818\pi\)
−0.955994 + 0.293385i \(0.905218\pi\)
\(368\) 1.48945e7i 5.73332i
\(369\) 2.44875e6 0.936221
\(370\) 0 0
\(371\) 1.97003e6 0.743084
\(372\) − 6.34096e6i − 2.37573i
\(373\) − 1.59920e6i − 0.595156i −0.954698 0.297578i \(-0.903821\pi\)
0.954698 0.297578i \(-0.0961788\pi\)
\(374\) −219428. −0.0811173
\(375\) 0 0
\(376\) 5.31579e6 1.93909
\(377\) 536081.i 0.194257i
\(378\) 1.80070e6i 0.648205i
\(379\) −2.52547e6 −0.903116 −0.451558 0.892242i \(-0.649132\pi\)
−0.451558 + 0.892242i \(0.649132\pi\)
\(380\) 0 0
\(381\) 4.40235e6 1.55372
\(382\) 7.94997e6i 2.78745i
\(383\) 3.16883e6i 1.10383i 0.833900 + 0.551915i \(0.186103\pi\)
−0.833900 + 0.551915i \(0.813897\pi\)
\(384\) −2.39950e7 −8.30410
\(385\) 0 0
\(386\) −6.69331e6 −2.28651
\(387\) 1.67808e6i 0.569555i
\(388\) − 1.46796e7i − 4.95034i
\(389\) −1.03129e6 −0.345545 −0.172773 0.984962i \(-0.555273\pi\)
−0.172773 + 0.984962i \(0.555273\pi\)
\(390\) 0 0
\(391\) 376330. 0.124488
\(392\) − 8.19014e6i − 2.69201i
\(393\) 3.10334e6i 1.01356i
\(394\) 267411. 0.0867838
\(395\) 0 0
\(396\) −1.89584e6 −0.607525
\(397\) 634862.i 0.202164i 0.994878 + 0.101082i \(0.0322304\pi\)
−0.994878 + 0.101082i \(0.967770\pi\)
\(398\) − 5.28320e6i − 1.67182i
\(399\) 4.23915e6 1.33305
\(400\) 0 0
\(401\) 2.19913e6 0.682952 0.341476 0.939890i \(-0.389073\pi\)
0.341476 + 0.939890i \(0.389073\pi\)
\(402\) 5.29728e6i 1.63489i
\(403\) 594053.i 0.182206i
\(404\) 1.55184e7 4.73035
\(405\) 0 0
\(406\) 2.52778e6 0.761070
\(407\) − 1.12577e6i − 0.336871i
\(408\) 1.62140e6i 0.482215i
\(409\) 4.25223e6 1.25692 0.628461 0.777841i \(-0.283685\pi\)
0.628461 + 0.777841i \(0.283685\pi\)
\(410\) 0 0
\(411\) 4.48266e6 1.30898
\(412\) 7.13132e6i 2.06979i
\(413\) − 2.34571e6i − 0.676704i
\(414\) 4.35829e6 1.24973
\(415\) 0 0
\(416\) 5.35071e6 1.51593
\(417\) 1.88046e6i 0.529570i
\(418\) − 5.62460e6i − 1.57453i
\(419\) −3.54751e6 −0.987161 −0.493581 0.869700i \(-0.664312\pi\)
−0.493581 + 0.869700i \(0.664312\pi\)
\(420\) 0 0
\(421\) 1.40038e6 0.385071 0.192535 0.981290i \(-0.438329\pi\)
0.192535 + 0.981290i \(0.438329\pi\)
\(422\) − 6.95742e6i − 1.90181i
\(423\) − 956565.i − 0.259934i
\(424\) −1.93146e7 −5.21759
\(425\) 0 0
\(426\) 3.58469e6 0.957033
\(427\) − 2.35656e6i − 0.625472i
\(428\) 1.33231e7i 3.51556i
\(429\) 522213. 0.136995
\(430\) 0 0
\(431\) 5.41665e6 1.40455 0.702276 0.711905i \(-0.252167\pi\)
0.702276 + 0.711905i \(0.252167\pi\)
\(432\) − 1.08570e7i − 2.79899i
\(433\) − 5.58069e6i − 1.43043i −0.698902 0.715217i \(-0.746328\pi\)
0.698902 0.715217i \(-0.253672\pi\)
\(434\) 2.80113e6 0.713854
\(435\) 0 0
\(436\) 1.11122e6 0.279951
\(437\) 9.64646e6i 2.41638i
\(438\) 1.13153e6i 0.281826i
\(439\) 1.76748e6 0.437716 0.218858 0.975757i \(-0.429767\pi\)
0.218858 + 0.975757i \(0.429767\pi\)
\(440\) 0 0
\(441\) −1.47380e6 −0.360863
\(442\) − 230296.i − 0.0560701i
\(443\) 2.11511e6i 0.512062i 0.966669 + 0.256031i \(0.0824150\pi\)
−0.966669 + 0.256031i \(0.917585\pi\)
\(444\) −1.26117e7 −3.03611
\(445\) 0 0
\(446\) −3.49099e6 −0.831019
\(447\) 5.13911e6i 1.21652i
\(448\) − 1.43154e7i − 3.36982i
\(449\) −2.74804e6 −0.643290 −0.321645 0.946860i \(-0.604236\pi\)
−0.321645 + 0.946860i \(0.604236\pi\)
\(450\) 0 0
\(451\) 3.14828e6 0.728841
\(452\) − 4.09414e6i − 0.942577i
\(453\) 2.39950e6i 0.549384i
\(454\) 5.05510e6 1.15104
\(455\) 0 0
\(456\) −4.15614e7 −9.36005
\(457\) 3.46825e6i 0.776820i 0.921487 + 0.388410i \(0.126976\pi\)
−0.921487 + 0.388410i \(0.873024\pi\)
\(458\) − 9.59893e6i − 2.13825i
\(459\) −274318. −0.0607746
\(460\) 0 0
\(461\) 5.16521e6 1.13197 0.565986 0.824415i \(-0.308496\pi\)
0.565986 + 0.824415i \(0.308496\pi\)
\(462\) − 2.46239e6i − 0.536725i
\(463\) − 6.87934e6i − 1.49140i −0.666282 0.745700i \(-0.732115\pi\)
0.666282 0.745700i \(-0.267885\pi\)
\(464\) −1.52408e7 −3.28634
\(465\) 0 0
\(466\) −1.22931e7 −2.62238
\(467\) 569700.i 0.120880i 0.998172 + 0.0604400i \(0.0192504\pi\)
−0.998172 + 0.0604400i \(0.980750\pi\)
\(468\) − 1.98974e6i − 0.419935i
\(469\) −1.74580e6 −0.366490
\(470\) 0 0
\(471\) −6.21350e6 −1.29058
\(472\) 2.29978e7i 4.75150i
\(473\) 2.15746e6i 0.443394i
\(474\) 672404. 0.137463
\(475\) 0 0
\(476\) −810137. −0.163886
\(477\) 3.47562e6i 0.699417i
\(478\) − 1.03038e7i − 2.06266i
\(479\) 3.33256e6 0.663649 0.331825 0.943341i \(-0.392336\pi\)
0.331825 + 0.943341i \(0.392336\pi\)
\(480\) 0 0
\(481\) 1.18153e6 0.232853
\(482\) − 7.21365e6i − 1.41429i
\(483\) 4.22312e6i 0.823693i
\(484\) 1.27021e7 2.46468
\(485\) 0 0
\(486\) −9.73275e6 −1.86915
\(487\) 5.24387e6i 1.00191i 0.865473 + 0.500956i \(0.167018\pi\)
−0.865473 + 0.500956i \(0.832982\pi\)
\(488\) 2.31041e7i 4.39177i
\(489\) 9.26893e6 1.75290
\(490\) 0 0
\(491\) −4.52706e6 −0.847447 −0.423724 0.905792i \(-0.639277\pi\)
−0.423724 + 0.905792i \(0.639277\pi\)
\(492\) − 3.52694e7i − 6.56880i
\(493\) 385080.i 0.0713566i
\(494\) 5.90318e6 1.08835
\(495\) 0 0
\(496\) −1.68890e7 −3.08247
\(497\) 1.18139e6i 0.214537i
\(498\) 5.41229e6i 0.977930i
\(499\) −8.06232e6 −1.44947 −0.724734 0.689029i \(-0.758037\pi\)
−0.724734 + 0.689029i \(0.758037\pi\)
\(500\) 0 0
\(501\) 5.23227e6 0.931314
\(502\) − 1.08597e6i − 0.192335i
\(503\) 3.91235e6i 0.689474i 0.938699 + 0.344737i \(0.112032\pi\)
−0.938699 + 0.344737i \(0.887968\pi\)
\(504\) −6.18842e6 −1.08518
\(505\) 0 0
\(506\) 5.60333e6 0.972904
\(507\) 548078.i 0.0946940i
\(508\) − 2.15657e7i − 3.70770i
\(509\) 1.01584e7 1.73792 0.868959 0.494884i \(-0.164790\pi\)
0.868959 + 0.494884i \(0.164790\pi\)
\(510\) 0 0
\(511\) −372913. −0.0631765
\(512\) 4.51098e7i 7.60495i
\(513\) − 7.03158e6i − 1.17967i
\(514\) −1.19751e7 −1.99927
\(515\) 0 0
\(516\) 2.41695e7 3.99616
\(517\) − 1.22983e6i − 0.202357i
\(518\) − 5.57126e6i − 0.912282i
\(519\) −4.77025e6 −0.777361
\(520\) 0 0
\(521\) 1.44957e6 0.233962 0.116981 0.993134i \(-0.462678\pi\)
0.116981 + 0.993134i \(0.462678\pi\)
\(522\) 4.45963e6i 0.716345i
\(523\) 203871.i 0.0325912i 0.999867 + 0.0162956i \(0.00518728\pi\)
−0.999867 + 0.0162956i \(0.994813\pi\)
\(524\) 1.52023e7 2.41869
\(525\) 0 0
\(526\) 5.64756e6 0.890013
\(527\) 426723.i 0.0669298i
\(528\) 1.48465e7i 2.31761i
\(529\) −3.17365e6 −0.493082
\(530\) 0 0
\(531\) 4.13841e6 0.636937
\(532\) − 2.07662e7i − 3.18111i
\(533\) 3.30422e6i 0.503791i
\(534\) −136728. −0.0207493
\(535\) 0 0
\(536\) 1.71162e7 2.57332
\(537\) 559052.i 0.0836598i
\(538\) 1.17663e6i 0.175260i
\(539\) −1.89482e6 −0.280929
\(540\) 0 0
\(541\) 7.28883e6 1.07069 0.535346 0.844633i \(-0.320181\pi\)
0.535346 + 0.844633i \(0.320181\pi\)
\(542\) − 1.89472e6i − 0.277042i
\(543\) − 1.12055e7i − 1.63091i
\(544\) 3.84355e6 0.556846
\(545\) 0 0
\(546\) 2.58435e6 0.370996
\(547\) − 5.64096e6i − 0.806092i −0.915180 0.403046i \(-0.867951\pi\)
0.915180 0.403046i \(-0.132049\pi\)
\(548\) − 2.19591e7i − 3.12366i
\(549\) 4.15754e6 0.588716
\(550\) 0 0
\(551\) −9.87075e6 −1.38507
\(552\) − 4.14043e7i − 5.78359i
\(553\) 221601.i 0.0308148i
\(554\) 9.69083e6 1.34149
\(555\) 0 0
\(556\) 9.21176e6 1.26373
\(557\) 910831.i 0.124394i 0.998064 + 0.0621971i \(0.0198107\pi\)
−0.998064 + 0.0621971i \(0.980189\pi\)
\(558\) 4.94189e6i 0.671905i
\(559\) −2.26432e6 −0.306484
\(560\) 0 0
\(561\) 375119. 0.0503224
\(562\) − 1.76180e7i − 2.35297i
\(563\) − 7.96586e6i − 1.05916i −0.848260 0.529580i \(-0.822349\pi\)
0.848260 0.529580i \(-0.177651\pi\)
\(564\) −1.37775e7 −1.82378
\(565\) 0 0
\(566\) 1.24097e7 1.62825
\(567\) − 5.23893e6i − 0.684361i
\(568\) − 1.15826e7i − 1.50638i
\(569\) 3.26286e6 0.422492 0.211246 0.977433i \(-0.432248\pi\)
0.211246 + 0.977433i \(0.432248\pi\)
\(570\) 0 0
\(571\) −7.89949e6 −1.01393 −0.506966 0.861966i \(-0.669233\pi\)
−0.506966 + 0.861966i \(0.669233\pi\)
\(572\) − 2.55815e6i − 0.326916i
\(573\) − 1.35907e7i − 1.72924i
\(574\) 1.55804e7 1.97378
\(575\) 0 0
\(576\) 2.52558e7 3.17180
\(577\) − 1.91866e6i − 0.239916i −0.992779 0.119958i \(-0.961724\pi\)
0.992779 0.119958i \(-0.0382760\pi\)
\(578\) 1.57727e7i 1.96375i
\(579\) 1.14424e7 1.41847
\(580\) 0 0
\(581\) −1.78370e6 −0.219221
\(582\) 3.36378e7i 4.11642i
\(583\) 4.46850e6i 0.544491i
\(584\) 3.65611e6 0.443596
\(585\) 0 0
\(586\) −1.30669e6 −0.157191
\(587\) 1.98302e6i 0.237537i 0.992922 + 0.118769i \(0.0378947\pi\)
−0.992922 + 0.118769i \(0.962105\pi\)
\(588\) 2.12272e7i 2.53192i
\(589\) −1.09382e7 −1.29914
\(590\) 0 0
\(591\) −457146. −0.0538377
\(592\) 3.35909e7i 3.93929i
\(593\) − 9.43563e6i − 1.10188i −0.834545 0.550940i \(-0.814269\pi\)
0.834545 0.550940i \(-0.185731\pi\)
\(594\) −4.08443e6 −0.474969
\(595\) 0 0
\(596\) 2.51749e7 2.90303
\(597\) 9.03177e6i 1.03714i
\(598\) 5.88086e6i 0.672493i
\(599\) −6.57359e6 −0.748576 −0.374288 0.927313i \(-0.622113\pi\)
−0.374288 + 0.927313i \(0.622113\pi\)
\(600\) 0 0
\(601\) 9.04817e6 1.02182 0.510910 0.859634i \(-0.329308\pi\)
0.510910 + 0.859634i \(0.329308\pi\)
\(602\) 1.06769e7i 1.20076i
\(603\) − 3.08002e6i − 0.344953i
\(604\) 1.17544e7 1.31102
\(605\) 0 0
\(606\) −3.55598e7 −3.93349
\(607\) 2.54414e6i 0.280266i 0.990133 + 0.140133i \(0.0447530\pi\)
−0.990133 + 0.140133i \(0.955247\pi\)
\(608\) 9.85216e7i 10.8087i
\(609\) −4.32131e6 −0.472141
\(610\) 0 0
\(611\) 1.29074e6 0.139874
\(612\) − 1.42928e6i − 0.154255i
\(613\) − 7.53921e6i − 0.810354i −0.914238 0.405177i \(-0.867210\pi\)
0.914238 0.405177i \(-0.132790\pi\)
\(614\) 9.31951e6 0.997636
\(615\) 0 0
\(616\) −7.95628e6 −0.844808
\(617\) 1.44546e6i 0.152860i 0.997075 + 0.0764300i \(0.0243522\pi\)
−0.997075 + 0.0764300i \(0.975648\pi\)
\(618\) − 1.63412e7i − 1.72113i
\(619\) −832032. −0.0872797 −0.0436399 0.999047i \(-0.513895\pi\)
−0.0436399 + 0.999047i \(0.513895\pi\)
\(620\) 0 0
\(621\) 7.00499e6 0.728918
\(622\) 2.29962e6i 0.238330i
\(623\) − 45060.7i − 0.00465134i
\(624\) −1.55819e7 −1.60198
\(625\) 0 0
\(626\) −2.20866e7 −2.25265
\(627\) 9.61540e6i 0.976784i
\(628\) 3.04379e7i 3.07975i
\(629\) 848722. 0.0855340
\(630\) 0 0
\(631\) 1.42941e6 0.142916 0.0714582 0.997444i \(-0.477235\pi\)
0.0714582 + 0.997444i \(0.477235\pi\)
\(632\) − 2.17262e6i − 0.216367i
\(633\) 1.18939e7i 1.17982i
\(634\) 2.78226e7 2.74900
\(635\) 0 0
\(636\) 5.00595e7 4.90731
\(637\) − 1.98867e6i − 0.194184i
\(638\) 5.73362e6i 0.557669i
\(639\) −2.08426e6 −0.201929
\(640\) 0 0
\(641\) −1.54071e7 −1.48107 −0.740537 0.672015i \(-0.765429\pi\)
−0.740537 + 0.672015i \(0.765429\pi\)
\(642\) − 3.05294e7i − 2.92335i
\(643\) 3.71125e6i 0.353992i 0.984212 + 0.176996i \(0.0566379\pi\)
−0.984212 + 0.176996i \(0.943362\pi\)
\(644\) 2.06877e7 1.96561
\(645\) 0 0
\(646\) 4.24040e6 0.399784
\(647\) − 8.31438e6i − 0.780853i −0.920634 0.390426i \(-0.872328\pi\)
0.920634 0.390426i \(-0.127672\pi\)
\(648\) 5.13635e7i 4.80527i
\(649\) 5.32063e6 0.495851
\(650\) 0 0
\(651\) −4.78862e6 −0.442851
\(652\) − 4.54055e7i − 4.18302i
\(653\) − 2.58516e6i − 0.237249i −0.992939 0.118624i \(-0.962152\pi\)
0.992939 0.118624i \(-0.0378484\pi\)
\(654\) −2.54631e6 −0.232792
\(655\) 0 0
\(656\) −9.39391e7 −8.52289
\(657\) − 657911.i − 0.0594639i
\(658\) − 6.08623e6i − 0.548004i
\(659\) −1.77052e7 −1.58813 −0.794065 0.607833i \(-0.792039\pi\)
−0.794065 + 0.607833i \(0.792039\pi\)
\(660\) 0 0
\(661\) 2.01104e6 0.179027 0.0895134 0.995986i \(-0.471469\pi\)
0.0895134 + 0.995986i \(0.471469\pi\)
\(662\) − 3.56646e7i − 3.16295i
\(663\) 393698.i 0.0347840i
\(664\) 1.74878e7 1.53927
\(665\) 0 0
\(666\) 9.82908e6 0.858672
\(667\) − 9.83343e6i − 0.855836i
\(668\) − 2.56312e7i − 2.22243i
\(669\) 5.96794e6 0.515536
\(670\) 0 0
\(671\) 5.34523e6 0.458311
\(672\) 4.31317e7i 3.68446i
\(673\) 5.48300e6i 0.466639i 0.972400 + 0.233319i \(0.0749587\pi\)
−0.972400 + 0.233319i \(0.925041\pi\)
\(674\) −1.17833e7 −0.999119
\(675\) 0 0
\(676\) 2.68486e6 0.225972
\(677\) − 1.58127e7i − 1.32597i −0.748631 0.662987i \(-0.769288\pi\)
0.748631 0.662987i \(-0.230712\pi\)
\(678\) 9.38159e6i 0.783794i
\(679\) −1.10859e7 −0.922772
\(680\) 0 0
\(681\) −8.64183e6 −0.714065
\(682\) 6.35365e6i 0.523073i
\(683\) − 2.24599e7i − 1.84229i −0.389225 0.921143i \(-0.627257\pi\)
0.389225 0.921143i \(-0.372743\pi\)
\(684\) 3.66367e7 2.99417
\(685\) 0 0
\(686\) −2.27704e7 −1.84740
\(687\) 1.64096e7i 1.32650i
\(688\) − 6.43747e7i − 5.18495i
\(689\) −4.68982e6 −0.376364
\(690\) 0 0
\(691\) −8.90456e6 −0.709443 −0.354721 0.934972i \(-0.615424\pi\)
−0.354721 + 0.934972i \(0.615424\pi\)
\(692\) 2.33679e7i 1.85505i
\(693\) 1.43172e6i 0.113246i
\(694\) −5.63770e6 −0.444328
\(695\) 0 0
\(696\) 4.23670e7 3.31516
\(697\) 2.37350e6i 0.185058i
\(698\) 1.44607e7i 1.12344i
\(699\) 2.10154e7 1.62684
\(700\) 0 0
\(701\) −1.83550e7 −1.41078 −0.705392 0.708818i \(-0.749229\pi\)
−0.705392 + 0.708818i \(0.749229\pi\)
\(702\) − 4.28673e6i − 0.328309i
\(703\) 2.17553e7i 1.66026i
\(704\) 3.24707e7 2.46922
\(705\) 0 0
\(706\) 4.27054e7 3.22456
\(707\) − 1.17193e7i − 0.881765i
\(708\) − 5.96057e7i − 4.46894i
\(709\) 2.57581e7 1.92441 0.962207 0.272318i \(-0.0877902\pi\)
0.962207 + 0.272318i \(0.0877902\pi\)
\(710\) 0 0
\(711\) −390959. −0.0290039
\(712\) 441784.i 0.0326596i
\(713\) − 1.08968e7i − 0.802742i
\(714\) 1.85640e6 0.136278
\(715\) 0 0
\(716\) 2.73862e6 0.199641
\(717\) 1.76146e7i 1.27961i
\(718\) − 3.57300e7i − 2.58656i
\(719\) −1.69710e6 −0.122429 −0.0612146 0.998125i \(-0.519497\pi\)
−0.0612146 + 0.998125i \(0.519497\pi\)
\(720\) 0 0
\(721\) 5.38549e6 0.385822
\(722\) 8.08995e7i 5.77568i
\(723\) 1.23319e7i 0.877375i
\(724\) −5.48921e7 −3.89191
\(725\) 0 0
\(726\) −2.91064e7 −2.04949
\(727\) − 1.84315e7i − 1.29338i −0.762754 0.646689i \(-0.776153\pi\)
0.762754 0.646689i \(-0.223847\pi\)
\(728\) − 8.35035e6i − 0.583951i
\(729\) −1.29433e6 −0.0902041
\(730\) 0 0
\(731\) −1.62652e6 −0.112581
\(732\) − 5.98813e7i − 4.13060i
\(733\) 1.79395e7i 1.23325i 0.787258 + 0.616624i \(0.211500\pi\)
−0.787258 + 0.616624i \(0.788500\pi\)
\(734\) −1.69952e7 −1.16436
\(735\) 0 0
\(736\) −9.81491e7 −6.67870
\(737\) − 3.95989e6i − 0.268544i
\(738\) 2.74876e7i 1.85779i
\(739\) −1.84597e7 −1.24341 −0.621703 0.783253i \(-0.713559\pi\)
−0.621703 + 0.783253i \(0.713559\pi\)
\(740\) 0 0
\(741\) −1.00916e7 −0.675175
\(742\) 2.21139e7i 1.47454i
\(743\) 3.97606e6i 0.264229i 0.991234 + 0.132115i \(0.0421767\pi\)
−0.991234 + 0.132115i \(0.957823\pi\)
\(744\) 4.69485e7 3.10949
\(745\) 0 0
\(746\) 1.79513e7 1.18100
\(747\) − 3.14689e6i − 0.206339i
\(748\) − 1.83759e6i − 0.120086i
\(749\) 1.00614e7 0.655322
\(750\) 0 0
\(751\) 2.25945e7 1.46185 0.730925 0.682458i \(-0.239089\pi\)
0.730925 + 0.682458i \(0.239089\pi\)
\(752\) 3.66958e7i 2.36631i
\(753\) 1.85649e6i 0.119318i
\(754\) −6.01760e6 −0.385474
\(755\) 0 0
\(756\) −1.50799e7 −0.959605
\(757\) − 1.40367e7i − 0.890279i −0.895461 0.445139i \(-0.853154\pi\)
0.895461 0.445139i \(-0.146846\pi\)
\(758\) − 2.83488e7i − 1.79210i
\(759\) −9.57905e6 −0.603556
\(760\) 0 0
\(761\) −6.05735e6 −0.379159 −0.189579 0.981865i \(-0.560712\pi\)
−0.189579 + 0.981865i \(0.560712\pi\)
\(762\) 4.94171e7i 3.08311i
\(763\) − 839177.i − 0.0521846i
\(764\) −6.65765e7 −4.12655
\(765\) 0 0
\(766\) −3.55707e7 −2.19038
\(767\) 5.58416e6i 0.342743i
\(768\) − 1.45520e8i − 8.90265i
\(769\) −1.07705e7 −0.656778 −0.328389 0.944543i \(-0.606506\pi\)
−0.328389 + 0.944543i \(0.606506\pi\)
\(770\) 0 0
\(771\) 2.04717e7 1.24028
\(772\) − 5.60527e7i − 3.38496i
\(773\) 8.92164e6i 0.537027i 0.963276 + 0.268514i \(0.0865324\pi\)
−0.963276 + 0.268514i \(0.913468\pi\)
\(774\) −1.88367e7 −1.13020
\(775\) 0 0
\(776\) 1.08688e8 6.47928
\(777\) 9.52422e6i 0.565949i
\(778\) − 1.15764e7i − 0.685682i
\(779\) −6.08399e7 −3.59207
\(780\) 0 0
\(781\) −2.67967e6 −0.157201
\(782\) 4.22437e6i 0.247027i
\(783\) 7.16787e6i 0.417816i
\(784\) 5.65380e7 3.28511
\(785\) 0 0
\(786\) −3.48356e7 −2.01125
\(787\) − 542225.i − 0.0312063i −0.999878 0.0156032i \(-0.995033\pi\)
0.999878 0.0156032i \(-0.00496684\pi\)
\(788\) 2.23941e6i 0.128475i
\(789\) −9.65466e6 −0.552134
\(790\) 0 0
\(791\) −3.09185e6 −0.175702
\(792\) − 1.40368e7i − 0.795163i
\(793\) 5.60998e6i 0.316795i
\(794\) −7.12643e6 −0.401163
\(795\) 0 0
\(796\) 4.42437e7 2.47497
\(797\) 1.82250e7i 1.01630i 0.861269 + 0.508149i \(0.169670\pi\)
−0.861269 + 0.508149i \(0.830330\pi\)
\(798\) 4.75851e7i 2.64523i
\(799\) 927172. 0.0513799
\(800\) 0 0
\(801\) 79498.2 0.00437800
\(802\) 2.46856e7i 1.35522i
\(803\) − 845857.i − 0.0462922i
\(804\) −4.43617e7 −2.42029
\(805\) 0 0
\(806\) −6.66834e6 −0.361560
\(807\) − 2.01147e6i − 0.108725i
\(808\) 1.14898e8i 6.19134i
\(809\) 3.56238e7 1.91368 0.956840 0.290615i \(-0.0938599\pi\)
0.956840 + 0.290615i \(0.0938599\pi\)
\(810\) 0 0
\(811\) 9.68050e6 0.516828 0.258414 0.966034i \(-0.416800\pi\)
0.258414 + 0.966034i \(0.416800\pi\)
\(812\) 2.11687e7i 1.12669i
\(813\) 3.23907e6i 0.171868i
\(814\) 1.26370e7 0.668470
\(815\) 0 0
\(816\) −1.11929e7 −0.588458
\(817\) − 4.16925e7i − 2.18526i
\(818\) 4.77320e7i 2.49417i
\(819\) −1.50263e6 −0.0782784
\(820\) 0 0
\(821\) 2.15879e7 1.11777 0.558885 0.829245i \(-0.311229\pi\)
0.558885 + 0.829245i \(0.311229\pi\)
\(822\) 5.03186e7i 2.59746i
\(823\) 2.95621e7i 1.52137i 0.649120 + 0.760686i \(0.275137\pi\)
−0.649120 + 0.760686i \(0.724863\pi\)
\(824\) −5.28004e7 −2.70906
\(825\) 0 0
\(826\) 2.63309e7 1.34282
\(827\) − 2.85332e7i − 1.45073i −0.688365 0.725364i \(-0.741671\pi\)
0.688365 0.725364i \(-0.258329\pi\)
\(828\) 3.64982e7i 1.85010i
\(829\) 2.02156e7 1.02165 0.510823 0.859686i \(-0.329341\pi\)
0.510823 + 0.859686i \(0.329341\pi\)
\(830\) 0 0
\(831\) −1.65667e7 −0.832213
\(832\) 3.40789e7i 1.70678i
\(833\) − 1.42851e6i − 0.0713299i
\(834\) −2.11084e7 −1.05085
\(835\) 0 0
\(836\) 4.71028e7 2.33094
\(837\) 7.94300e6i 0.391896i
\(838\) − 3.98214e7i − 1.95887i
\(839\) 6.64057e6 0.325687 0.162843 0.986652i \(-0.447933\pi\)
0.162843 + 0.986652i \(0.447933\pi\)
\(840\) 0 0
\(841\) −1.04491e7 −0.509434
\(842\) 1.57195e7i 0.764114i
\(843\) 3.01185e7i 1.45970i
\(844\) 5.82644e7 2.81544
\(845\) 0 0
\(846\) 1.07376e7 0.515800
\(847\) − 9.59244e6i − 0.459432i
\(848\) − 1.33332e8i − 6.36714i
\(849\) −2.12148e7 −1.01011
\(850\) 0 0
\(851\) −2.16730e7 −1.02588
\(852\) 3.00197e7i 1.41680i
\(853\) − 1.92258e7i − 0.904716i −0.891836 0.452358i \(-0.850583\pi\)
0.891836 0.452358i \(-0.149417\pi\)
\(854\) 2.64527e7 1.24115
\(855\) 0 0
\(856\) −9.86442e7 −4.60137
\(857\) 1.98896e6i 0.0925068i 0.998930 + 0.0462534i \(0.0147282\pi\)
−0.998930 + 0.0462534i \(0.985272\pi\)
\(858\) 5.86193e6i 0.271845i
\(859\) 7.32029e6 0.338490 0.169245 0.985574i \(-0.445867\pi\)
0.169245 + 0.985574i \(0.445867\pi\)
\(860\) 0 0
\(861\) −2.66351e7 −1.22446
\(862\) 6.08028e7i 2.78712i
\(863\) 3.13431e7i 1.43257i 0.697809 + 0.716283i \(0.254158\pi\)
−0.697809 + 0.716283i \(0.745842\pi\)
\(864\) 7.15437e7 3.26052
\(865\) 0 0
\(866\) 6.26441e7 2.83848
\(867\) − 2.69639e7i − 1.21824i
\(868\) 2.34579e7i 1.05679i
\(869\) −502644. −0.0225793
\(870\) 0 0
\(871\) 4.15603e6 0.185623
\(872\) 8.22745e6i 0.366416i
\(873\) − 1.95582e7i − 0.868546i
\(874\) −1.08283e8 −4.79493
\(875\) 0 0
\(876\) −9.47592e6 −0.417216
\(877\) 2.47672e7i 1.08737i 0.839289 + 0.543686i \(0.182972\pi\)
−0.839289 + 0.543686i \(0.817028\pi\)
\(878\) 1.98402e7i 0.868581i
\(879\) 2.23382e6 0.0975160
\(880\) 0 0
\(881\) 1.22752e7 0.532831 0.266415 0.963858i \(-0.414161\pi\)
0.266415 + 0.963858i \(0.414161\pi\)
\(882\) − 1.65436e7i − 0.716077i
\(883\) 2.56013e7i 1.10500i 0.833514 + 0.552498i \(0.186325\pi\)
−0.833514 + 0.552498i \(0.813675\pi\)
\(884\) 1.92860e6 0.0830064
\(885\) 0 0
\(886\) −2.37424e7 −1.01611
\(887\) 5.01026e6i 0.213821i 0.994269 + 0.106911i \(0.0340959\pi\)
−0.994269 + 0.106911i \(0.965904\pi\)
\(888\) − 9.33773e7i − 3.97383i
\(889\) −1.62862e7 −0.691137
\(890\) 0 0
\(891\) 1.18832e7 0.501462
\(892\) − 2.92350e7i − 1.23024i
\(893\) 2.37662e7i 0.997311i
\(894\) −5.76873e7 −2.41400
\(895\) 0 0
\(896\) 8.87676e7 3.69390
\(897\) − 1.00535e7i − 0.417192i
\(898\) − 3.08472e7i − 1.27651i
\(899\) 1.11502e7 0.460132
\(900\) 0 0
\(901\) −3.36882e6 −0.138250
\(902\) 3.53400e7i 1.44627i
\(903\) − 1.82525e7i − 0.744909i
\(904\) 3.03131e7 1.23370
\(905\) 0 0
\(906\) −2.69348e7 −1.09017
\(907\) 4.85441e7i 1.95938i 0.200527 + 0.979688i \(0.435734\pi\)
−0.200527 + 0.979688i \(0.564266\pi\)
\(908\) 4.23336e7i 1.70400i
\(909\) 2.06757e7 0.829948
\(910\) 0 0
\(911\) −3.03805e7 −1.21283 −0.606413 0.795150i \(-0.707392\pi\)
−0.606413 + 0.795150i \(0.707392\pi\)
\(912\) − 2.86906e8i − 11.4223i
\(913\) − 4.04587e6i − 0.160633i
\(914\) −3.89317e7 −1.54148
\(915\) 0 0
\(916\) 8.03855e7 3.16548
\(917\) − 1.14806e7i − 0.450859i
\(918\) − 3.07926e6i − 0.120598i
\(919\) 3.70802e7 1.44828 0.724141 0.689652i \(-0.242236\pi\)
0.724141 + 0.689652i \(0.242236\pi\)
\(920\) 0 0
\(921\) −1.59320e7 −0.618899
\(922\) 5.79803e7i 2.24623i
\(923\) − 2.81239e6i − 0.108661i
\(924\) 2.06211e7 0.794569
\(925\) 0 0
\(926\) 7.72217e7 2.95946
\(927\) 9.50133e6i 0.363149i
\(928\) − 1.00431e8i − 3.82824i
\(929\) −2.82561e6 −0.107417 −0.0537084 0.998557i \(-0.517104\pi\)
−0.0537084 + 0.998557i \(0.517104\pi\)
\(930\) 0 0
\(931\) 3.66170e7 1.38455
\(932\) − 1.02948e8i − 3.88219i
\(933\) − 3.93126e6i − 0.147852i
\(934\) −6.39498e6 −0.239868
\(935\) 0 0
\(936\) 1.47321e7 0.549635
\(937\) − 3.19406e6i − 0.118849i −0.998233 0.0594243i \(-0.981074\pi\)
0.998233 0.0594243i \(-0.0189265\pi\)
\(938\) − 1.95969e7i − 0.727244i
\(939\) 3.77576e7 1.39746
\(940\) 0 0
\(941\) −2.22120e7 −0.817738 −0.408869 0.912593i \(-0.634077\pi\)
−0.408869 + 0.912593i \(0.634077\pi\)
\(942\) − 6.97475e7i − 2.56095i
\(943\) − 6.06099e7i − 2.21955i
\(944\) −1.58758e8 −5.79836
\(945\) 0 0
\(946\) −2.42179e7 −0.879849
\(947\) − 2.24064e7i − 0.811889i −0.913898 0.405945i \(-0.866943\pi\)
0.913898 0.405945i \(-0.133057\pi\)
\(948\) 5.63100e6i 0.203500i
\(949\) 887751. 0.0319982
\(950\) 0 0
\(951\) −4.75634e7 −1.70538
\(952\) − 5.99826e6i − 0.214503i
\(953\) 1.48416e7i 0.529358i 0.964337 + 0.264679i \(0.0852660\pi\)
−0.964337 + 0.264679i \(0.914734\pi\)
\(954\) −3.90144e7 −1.38789
\(955\) 0 0
\(956\) 8.62886e7 3.05357
\(957\) − 9.80177e6i − 0.345959i
\(958\) 3.74085e7i 1.31691i
\(959\) −1.65833e7 −0.582269
\(960\) 0 0
\(961\) −1.62732e7 −0.568413
\(962\) 1.32629e7i 0.462061i
\(963\) 1.77508e7i 0.616812i
\(964\) 6.04102e7 2.09371
\(965\) 0 0
\(966\) −4.74052e7 −1.63449
\(967\) − 3.06919e7i − 1.05550i −0.849401 0.527749i \(-0.823036\pi\)
0.849401 0.527749i \(-0.176964\pi\)
\(968\) 9.40462e7i 3.22591i
\(969\) −7.24908e6 −0.248013
\(970\) 0 0
\(971\) −4.18932e7 −1.42592 −0.712961 0.701203i \(-0.752646\pi\)
−0.712961 + 0.701203i \(0.752646\pi\)
\(972\) − 8.15062e7i − 2.76710i
\(973\) − 6.95661e6i − 0.235568i
\(974\) −5.88633e7 −1.98814
\(975\) 0 0
\(976\) −1.59492e8 −5.35938
\(977\) 2.54822e7i 0.854084i 0.904232 + 0.427042i \(0.140444\pi\)
−0.904232 + 0.427042i \(0.859556\pi\)
\(978\) 1.04045e8i 3.47837i
\(979\) 102209. 0.00340824
\(980\) 0 0
\(981\) 1.48051e6 0.0491179
\(982\) − 5.08170e7i − 1.68163i
\(983\) − 5.91830e7i − 1.95350i −0.214388 0.976749i \(-0.568776\pi\)
0.214388 0.976749i \(-0.431224\pi\)
\(984\) 2.61135e8 8.59761
\(985\) 0 0
\(986\) −4.32259e6 −0.141596
\(987\) 1.04046e7i 0.339963i
\(988\) 4.94358e7i 1.61120i
\(989\) 4.15348e7 1.35027
\(990\) 0 0
\(991\) −625464. −0.0202310 −0.0101155 0.999949i \(-0.503220\pi\)
−0.0101155 + 0.999949i \(0.503220\pi\)
\(992\) − 1.11292e8i − 3.59074i
\(993\) 6.09695e7i 1.96218i
\(994\) −1.32613e7 −0.425715
\(995\) 0 0
\(996\) −4.53249e7 −1.44773
\(997\) − 5.76822e7i − 1.83783i −0.394461 0.918913i \(-0.629069\pi\)
0.394461 0.918913i \(-0.370931\pi\)
\(998\) − 9.05009e7i − 2.87625i
\(999\) 1.57981e7 0.500830
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.i.274.22 22
5.2 odd 4 325.6.a.j.1.1 11
5.3 odd 4 325.6.a.k.1.11 yes 11
5.4 even 2 inner 325.6.b.i.274.1 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.1 11 5.2 odd 4
325.6.a.k.1.11 yes 11 5.3 odd 4
325.6.b.i.274.1 22 5.4 even 2 inner
325.6.b.i.274.22 22 1.1 even 1 trivial