Properties

Label 1344.4.a.be.1.2
Level $1344$
Weight $4$
Character 1344.1
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(1,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

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: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-5.35235\) of defining polynomial
Character \(\chi\) \(=\) 1344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +6.70470 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +6.70470 q^{5} +7.00000 q^{7} +9.00000 q^{9} +0.704700 q^{11} -35.4094 q^{13} -20.1141 q^{15} +26.7047 q^{17} +30.8188 q^{19} -21.0000 q^{21} +5.88590 q^{23} -80.0470 q^{25} -27.0000 q^{27} -252.094 q^{29} -37.6376 q^{31} -2.11410 q^{33} +46.9329 q^{35} +139.866 q^{37} +106.228 q^{39} -46.7047 q^{41} -254.094 q^{43} +60.3423 q^{45} +607.960 q^{47} +49.0000 q^{49} -80.1141 q^{51} -298.685 q^{53} +4.72480 q^{55} -92.4564 q^{57} -178.591 q^{59} -390.188 q^{61} +63.0000 q^{63} -237.409 q^{65} -349.141 q^{67} -17.6577 q^{69} +348.477 q^{71} -646.416 q^{73} +240.141 q^{75} +4.93290 q^{77} +1047.50 q^{79} +81.0000 q^{81} -278.819 q^{83} +179.047 q^{85} +756.282 q^{87} +439.309 q^{89} -247.866 q^{91} +112.913 q^{93} +206.631 q^{95} -154.309 q^{97} +6.34230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} - 22 q^{11} - 24 q^{13} + 30 q^{15} + 30 q^{17} - 32 q^{19} - 42 q^{21} + 82 q^{23} + 74 q^{25} - 54 q^{27} - 36 q^{29} + 112 q^{31} + 66 q^{33} - 70 q^{35} - 48 q^{37} + 72 q^{39} - 70 q^{41} - 40 q^{43} - 90 q^{45} + 420 q^{47} + 98 q^{49} - 90 q^{51} - 176 q^{53} + 384 q^{55} + 96 q^{57} - 404 q^{59} + 156 q^{61} + 126 q^{63} - 428 q^{65} + 4 q^{67} - 246 q^{69} + 814 q^{71} - 216 q^{73} - 222 q^{75} - 154 q^{77} + 1580 q^{79} + 162 q^{81} - 464 q^{83} + 124 q^{85} + 108 q^{87} - 1158 q^{89} - 168 q^{91} - 336 q^{93} + 1256 q^{95} - 1760 q^{97} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 6.70470 0.599687 0.299843 0.953988i \(-0.403066\pi\)
0.299843 + 0.953988i \(0.403066\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0.704700 0.0193159 0.00965796 0.999953i \(-0.496926\pi\)
0.00965796 + 0.999953i \(0.496926\pi\)
\(12\) 0 0
\(13\) −35.4094 −0.755446 −0.377723 0.925919i \(-0.623293\pi\)
−0.377723 + 0.925919i \(0.623293\pi\)
\(14\) 0 0
\(15\) −20.1141 −0.346229
\(16\) 0 0
\(17\) 26.7047 0.380991 0.190495 0.981688i \(-0.438991\pi\)
0.190495 + 0.981688i \(0.438991\pi\)
\(18\) 0 0
\(19\) 30.8188 0.372122 0.186061 0.982538i \(-0.440428\pi\)
0.186061 + 0.982538i \(0.440428\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 5.88590 0.0533607 0.0266803 0.999644i \(-0.491506\pi\)
0.0266803 + 0.999644i \(0.491506\pi\)
\(24\) 0 0
\(25\) −80.0470 −0.640376
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −252.094 −1.61423 −0.807115 0.590394i \(-0.798972\pi\)
−0.807115 + 0.590394i \(0.798972\pi\)
\(30\) 0 0
\(31\) −37.6376 −0.218062 −0.109031 0.994038i \(-0.534775\pi\)
−0.109031 + 0.994038i \(0.534775\pi\)
\(32\) 0 0
\(33\) −2.11410 −0.0111520
\(34\) 0 0
\(35\) 46.9329 0.226660
\(36\) 0 0
\(37\) 139.866 0.621454 0.310727 0.950499i \(-0.399428\pi\)
0.310727 + 0.950499i \(0.399428\pi\)
\(38\) 0 0
\(39\) 106.228 0.436157
\(40\) 0 0
\(41\) −46.7047 −0.177904 −0.0889518 0.996036i \(-0.528352\pi\)
−0.0889518 + 0.996036i \(0.528352\pi\)
\(42\) 0 0
\(43\) −254.094 −0.901139 −0.450569 0.892741i \(-0.648779\pi\)
−0.450569 + 0.892741i \(0.648779\pi\)
\(44\) 0 0
\(45\) 60.3423 0.199896
\(46\) 0 0
\(47\) 607.960 1.88681 0.943405 0.331643i \(-0.107603\pi\)
0.943405 + 0.331643i \(0.107603\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −80.1141 −0.219965
\(52\) 0 0
\(53\) −298.685 −0.774103 −0.387052 0.922058i \(-0.626507\pi\)
−0.387052 + 0.922058i \(0.626507\pi\)
\(54\) 0 0
\(55\) 4.72480 0.0115835
\(56\) 0 0
\(57\) −92.4564 −0.214845
\(58\) 0 0
\(59\) −178.591 −0.394077 −0.197038 0.980396i \(-0.563132\pi\)
−0.197038 + 0.980396i \(0.563132\pi\)
\(60\) 0 0
\(61\) −390.188 −0.818991 −0.409496 0.912312i \(-0.634295\pi\)
−0.409496 + 0.912312i \(0.634295\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −237.409 −0.453031
\(66\) 0 0
\(67\) −349.141 −0.636632 −0.318316 0.947985i \(-0.603117\pi\)
−0.318316 + 0.947985i \(0.603117\pi\)
\(68\) 0 0
\(69\) −17.6577 −0.0308078
\(70\) 0 0
\(71\) 348.477 0.582487 0.291243 0.956649i \(-0.405931\pi\)
0.291243 + 0.956649i \(0.405931\pi\)
\(72\) 0 0
\(73\) −646.416 −1.03640 −0.518201 0.855259i \(-0.673398\pi\)
−0.518201 + 0.855259i \(0.673398\pi\)
\(74\) 0 0
\(75\) 240.141 0.369721
\(76\) 0 0
\(77\) 4.93290 0.00730073
\(78\) 0 0
\(79\) 1047.50 1.49181 0.745907 0.666050i \(-0.232016\pi\)
0.745907 + 0.666050i \(0.232016\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −278.819 −0.368727 −0.184363 0.982858i \(-0.559022\pi\)
−0.184363 + 0.982858i \(0.559022\pi\)
\(84\) 0 0
\(85\) 179.047 0.228475
\(86\) 0 0
\(87\) 756.282 0.931976
\(88\) 0 0
\(89\) 439.309 0.523221 0.261610 0.965174i \(-0.415746\pi\)
0.261610 + 0.965174i \(0.415746\pi\)
\(90\) 0 0
\(91\) −247.866 −0.285532
\(92\) 0 0
\(93\) 112.913 0.125898
\(94\) 0 0
\(95\) 206.631 0.223157
\(96\) 0 0
\(97\) −154.309 −0.161522 −0.0807612 0.996733i \(-0.525735\pi\)
−0.0807612 + 0.996733i \(0.525735\pi\)
\(98\) 0 0
\(99\) 6.34230 0.00643864
\(100\) 0 0
\(101\) −166.060 −0.163600 −0.0818001 0.996649i \(-0.526067\pi\)
−0.0818001 + 0.996649i \(0.526067\pi\)
\(102\) 0 0
\(103\) 42.4428 0.0406021 0.0203010 0.999794i \(-0.493538\pi\)
0.0203010 + 0.999794i \(0.493538\pi\)
\(104\) 0 0
\(105\) −140.799 −0.130862
\(106\) 0 0
\(107\) −293.470 −0.265148 −0.132574 0.991173i \(-0.542324\pi\)
−0.132574 + 0.991173i \(0.542324\pi\)
\(108\) 0 0
\(109\) 1176.01 1.03341 0.516705 0.856164i \(-0.327158\pi\)
0.516705 + 0.856164i \(0.327158\pi\)
\(110\) 0 0
\(111\) −419.597 −0.358797
\(112\) 0 0
\(113\) −182.537 −0.151961 −0.0759806 0.997109i \(-0.524209\pi\)
−0.0759806 + 0.997109i \(0.524209\pi\)
\(114\) 0 0
\(115\) 39.4632 0.0319997
\(116\) 0 0
\(117\) −318.685 −0.251815
\(118\) 0 0
\(119\) 186.933 0.144001
\(120\) 0 0
\(121\) −1330.50 −0.999627
\(122\) 0 0
\(123\) 140.114 0.102713
\(124\) 0 0
\(125\) −1374.78 −0.983711
\(126\) 0 0
\(127\) 1374.59 0.960435 0.480217 0.877150i \(-0.340558\pi\)
0.480217 + 0.877150i \(0.340558\pi\)
\(128\) 0 0
\(129\) 762.282 0.520273
\(130\) 0 0
\(131\) −1316.91 −0.878315 −0.439157 0.898410i \(-0.644723\pi\)
−0.439157 + 0.898410i \(0.644723\pi\)
\(132\) 0 0
\(133\) 215.732 0.140649
\(134\) 0 0
\(135\) −181.027 −0.115410
\(136\) 0 0
\(137\) −2266.98 −1.41373 −0.706866 0.707348i \(-0.749891\pi\)
−0.706866 + 0.707348i \(0.749891\pi\)
\(138\) 0 0
\(139\) −1162.93 −0.709627 −0.354813 0.934937i \(-0.615456\pi\)
−0.354813 + 0.934937i \(0.615456\pi\)
\(140\) 0 0
\(141\) −1823.88 −1.08935
\(142\) 0 0
\(143\) −24.9530 −0.0145921
\(144\) 0 0
\(145\) −1690.21 −0.968032
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) 0 0
\(149\) 54.0674 0.0297273 0.0148637 0.999890i \(-0.495269\pi\)
0.0148637 + 0.999890i \(0.495269\pi\)
\(150\) 0 0
\(151\) −862.014 −0.464567 −0.232284 0.972648i \(-0.574620\pi\)
−0.232284 + 0.972648i \(0.574620\pi\)
\(152\) 0 0
\(153\) 240.342 0.126997
\(154\) 0 0
\(155\) −252.349 −0.130769
\(156\) 0 0
\(157\) −1459.75 −0.742040 −0.371020 0.928625i \(-0.620992\pi\)
−0.371020 + 0.928625i \(0.620992\pi\)
\(158\) 0 0
\(159\) 896.054 0.446929
\(160\) 0 0
\(161\) 41.2013 0.0201684
\(162\) 0 0
\(163\) −1067.77 −0.513094 −0.256547 0.966532i \(-0.582585\pi\)
−0.256547 + 0.966532i \(0.582585\pi\)
\(164\) 0 0
\(165\) −14.1744 −0.00668773
\(166\) 0 0
\(167\) −2082.86 −0.965129 −0.482564 0.875861i \(-0.660295\pi\)
−0.482564 + 0.875861i \(0.660295\pi\)
\(168\) 0 0
\(169\) −943.174 −0.429301
\(170\) 0 0
\(171\) 277.369 0.124041
\(172\) 0 0
\(173\) −2210.17 −0.971306 −0.485653 0.874152i \(-0.661418\pi\)
−0.485653 + 0.874152i \(0.661418\pi\)
\(174\) 0 0
\(175\) −560.329 −0.242039
\(176\) 0 0
\(177\) 535.772 0.227520
\(178\) 0 0
\(179\) −934.718 −0.390302 −0.195151 0.980773i \(-0.562520\pi\)
−0.195151 + 0.980773i \(0.562520\pi\)
\(180\) 0 0
\(181\) −5.85221 −0.00240327 −0.00120163 0.999999i \(-0.500382\pi\)
−0.00120163 + 0.999999i \(0.500382\pi\)
\(182\) 0 0
\(183\) 1170.56 0.472845
\(184\) 0 0
\(185\) 937.758 0.372678
\(186\) 0 0
\(187\) 18.8188 0.00735918
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 2377.34 0.900618 0.450309 0.892873i \(-0.351314\pi\)
0.450309 + 0.892873i \(0.351314\pi\)
\(192\) 0 0
\(193\) −2233.85 −0.833141 −0.416570 0.909103i \(-0.636768\pi\)
−0.416570 + 0.909103i \(0.636768\pi\)
\(194\) 0 0
\(195\) 712.228 0.261558
\(196\) 0 0
\(197\) −2358.95 −0.853139 −0.426570 0.904455i \(-0.640278\pi\)
−0.426570 + 0.904455i \(0.640278\pi\)
\(198\) 0 0
\(199\) 367.759 0.131004 0.0655018 0.997852i \(-0.479135\pi\)
0.0655018 + 0.997852i \(0.479135\pi\)
\(200\) 0 0
\(201\) 1047.42 0.367560
\(202\) 0 0
\(203\) −1764.66 −0.610122
\(204\) 0 0
\(205\) −313.141 −0.106686
\(206\) 0 0
\(207\) 52.9731 0.0177869
\(208\) 0 0
\(209\) 21.7180 0.00718787
\(210\) 0 0
\(211\) −1873.13 −0.611144 −0.305572 0.952169i \(-0.598848\pi\)
−0.305572 + 0.952169i \(0.598848\pi\)
\(212\) 0 0
\(213\) −1045.43 −0.336299
\(214\) 0 0
\(215\) −1703.62 −0.540401
\(216\) 0 0
\(217\) −263.463 −0.0824196
\(218\) 0 0
\(219\) 1939.25 0.598367
\(220\) 0 0
\(221\) −945.597 −0.287818
\(222\) 0 0
\(223\) −3352.83 −1.00683 −0.503413 0.864046i \(-0.667923\pi\)
−0.503413 + 0.864046i \(0.667923\pi\)
\(224\) 0 0
\(225\) −720.423 −0.213459
\(226\) 0 0
\(227\) 1006.08 0.294167 0.147084 0.989124i \(-0.453011\pi\)
0.147084 + 0.989124i \(0.453011\pi\)
\(228\) 0 0
\(229\) −1550.82 −0.447515 −0.223758 0.974645i \(-0.571832\pi\)
−0.223758 + 0.974645i \(0.571832\pi\)
\(230\) 0 0
\(231\) −14.7987 −0.00421508
\(232\) 0 0
\(233\) 592.215 0.166512 0.0832559 0.996528i \(-0.473468\pi\)
0.0832559 + 0.996528i \(0.473468\pi\)
\(234\) 0 0
\(235\) 4076.19 1.13149
\(236\) 0 0
\(237\) −3142.51 −0.861299
\(238\) 0 0
\(239\) −5096.56 −1.37937 −0.689684 0.724111i \(-0.742251\pi\)
−0.689684 + 0.724111i \(0.742251\pi\)
\(240\) 0 0
\(241\) −4499.44 −1.20263 −0.601316 0.799011i \(-0.705357\pi\)
−0.601316 + 0.799011i \(0.705357\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 328.530 0.0856695
\(246\) 0 0
\(247\) −1091.28 −0.281118
\(248\) 0 0
\(249\) 836.456 0.212885
\(250\) 0 0
\(251\) 1602.75 0.403047 0.201523 0.979484i \(-0.435411\pi\)
0.201523 + 0.979484i \(0.435411\pi\)
\(252\) 0 0
\(253\) 4.14779 0.00103071
\(254\) 0 0
\(255\) −537.141 −0.131910
\(256\) 0 0
\(257\) −4953.82 −1.20238 −0.601188 0.799107i \(-0.705306\pi\)
−0.601188 + 0.799107i \(0.705306\pi\)
\(258\) 0 0
\(259\) 979.061 0.234888
\(260\) 0 0
\(261\) −2268.85 −0.538077
\(262\) 0 0
\(263\) −3675.40 −0.861730 −0.430865 0.902416i \(-0.641792\pi\)
−0.430865 + 0.902416i \(0.641792\pi\)
\(264\) 0 0
\(265\) −2002.59 −0.464219
\(266\) 0 0
\(267\) −1317.93 −0.302082
\(268\) 0 0
\(269\) −1989.64 −0.450969 −0.225485 0.974247i \(-0.572397\pi\)
−0.225485 + 0.974247i \(0.572397\pi\)
\(270\) 0 0
\(271\) 1638.85 0.367354 0.183677 0.982987i \(-0.441200\pi\)
0.183677 + 0.982987i \(0.441200\pi\)
\(272\) 0 0
\(273\) 743.597 0.164852
\(274\) 0 0
\(275\) −56.4091 −0.0123694
\(276\) 0 0
\(277\) 7810.87 1.69426 0.847130 0.531386i \(-0.178329\pi\)
0.847130 + 0.531386i \(0.178329\pi\)
\(278\) 0 0
\(279\) −338.738 −0.0726872
\(280\) 0 0
\(281\) 3018.34 0.640779 0.320389 0.947286i \(-0.396186\pi\)
0.320389 + 0.947286i \(0.396186\pi\)
\(282\) 0 0
\(283\) 7348.13 1.54347 0.771734 0.635946i \(-0.219390\pi\)
0.771734 + 0.635946i \(0.219390\pi\)
\(284\) 0 0
\(285\) −619.892 −0.128839
\(286\) 0 0
\(287\) −326.933 −0.0672413
\(288\) 0 0
\(289\) −4199.86 −0.854846
\(290\) 0 0
\(291\) 462.926 0.0932550
\(292\) 0 0
\(293\) −4211.91 −0.839804 −0.419902 0.907569i \(-0.637936\pi\)
−0.419902 + 0.907569i \(0.637936\pi\)
\(294\) 0 0
\(295\) −1197.40 −0.236322
\(296\) 0 0
\(297\) −19.0269 −0.00371735
\(298\) 0 0
\(299\) −208.416 −0.0403111
\(300\) 0 0
\(301\) −1778.66 −0.340598
\(302\) 0 0
\(303\) 498.181 0.0944546
\(304\) 0 0
\(305\) −2616.09 −0.491138
\(306\) 0 0
\(307\) 7553.16 1.40417 0.702087 0.712091i \(-0.252252\pi\)
0.702087 + 0.712091i \(0.252252\pi\)
\(308\) 0 0
\(309\) −127.328 −0.0234416
\(310\) 0 0
\(311\) −2656.58 −0.484375 −0.242188 0.970229i \(-0.577865\pi\)
−0.242188 + 0.970229i \(0.577865\pi\)
\(312\) 0 0
\(313\) −10819.2 −1.95379 −0.976893 0.213731i \(-0.931439\pi\)
−0.976893 + 0.213731i \(0.931439\pi\)
\(314\) 0 0
\(315\) 422.396 0.0755534
\(316\) 0 0
\(317\) −1704.91 −0.302074 −0.151037 0.988528i \(-0.548261\pi\)
−0.151037 + 0.988528i \(0.548261\pi\)
\(318\) 0 0
\(319\) −177.651 −0.0311803
\(320\) 0 0
\(321\) 880.409 0.153083
\(322\) 0 0
\(323\) 823.007 0.141775
\(324\) 0 0
\(325\) 2834.42 0.483770
\(326\) 0 0
\(327\) −3528.04 −0.596639
\(328\) 0 0
\(329\) 4255.72 0.713147
\(330\) 0 0
\(331\) −75.1144 −0.0124733 −0.00623665 0.999981i \(-0.501985\pi\)
−0.00623665 + 0.999981i \(0.501985\pi\)
\(332\) 0 0
\(333\) 1258.79 0.207151
\(334\) 0 0
\(335\) −2340.89 −0.381780
\(336\) 0 0
\(337\) −1474.97 −0.238417 −0.119208 0.992869i \(-0.538036\pi\)
−0.119208 + 0.992869i \(0.538036\pi\)
\(338\) 0 0
\(339\) 547.610 0.0877349
\(340\) 0 0
\(341\) −26.5232 −0.00421206
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −118.390 −0.0184750
\(346\) 0 0
\(347\) −73.8327 −0.0114223 −0.00571116 0.999984i \(-0.501818\pi\)
−0.00571116 + 0.999984i \(0.501818\pi\)
\(348\) 0 0
\(349\) −3803.45 −0.583364 −0.291682 0.956515i \(-0.594215\pi\)
−0.291682 + 0.956515i \(0.594215\pi\)
\(350\) 0 0
\(351\) 956.054 0.145386
\(352\) 0 0
\(353\) 7569.15 1.14126 0.570631 0.821207i \(-0.306699\pi\)
0.570631 + 0.821207i \(0.306699\pi\)
\(354\) 0 0
\(355\) 2336.43 0.349309
\(356\) 0 0
\(357\) −560.799 −0.0831390
\(358\) 0 0
\(359\) −5094.05 −0.748896 −0.374448 0.927248i \(-0.622168\pi\)
−0.374448 + 0.927248i \(0.622168\pi\)
\(360\) 0 0
\(361\) −5909.20 −0.861525
\(362\) 0 0
\(363\) 3991.51 0.577135
\(364\) 0 0
\(365\) −4334.03 −0.621516
\(366\) 0 0
\(367\) 8647.95 1.23003 0.615013 0.788517i \(-0.289151\pi\)
0.615013 + 0.788517i \(0.289151\pi\)
\(368\) 0 0
\(369\) −420.342 −0.0593012
\(370\) 0 0
\(371\) −2090.79 −0.292584
\(372\) 0 0
\(373\) 13760.1 1.91011 0.955053 0.296436i \(-0.0957982\pi\)
0.955053 + 0.296436i \(0.0957982\pi\)
\(374\) 0 0
\(375\) 4124.34 0.567946
\(376\) 0 0
\(377\) 8926.50 1.21946
\(378\) 0 0
\(379\) 2462.66 0.333768 0.166884 0.985977i \(-0.446629\pi\)
0.166884 + 0.985977i \(0.446629\pi\)
\(380\) 0 0
\(381\) −4123.77 −0.554507
\(382\) 0 0
\(383\) −2553.40 −0.340659 −0.170330 0.985387i \(-0.554483\pi\)
−0.170330 + 0.985387i \(0.554483\pi\)
\(384\) 0 0
\(385\) 33.0736 0.00437815
\(386\) 0 0
\(387\) −2286.85 −0.300380
\(388\) 0 0
\(389\) 5247.13 0.683907 0.341954 0.939717i \(-0.388911\pi\)
0.341954 + 0.939717i \(0.388911\pi\)
\(390\) 0 0
\(391\) 157.181 0.0203299
\(392\) 0 0
\(393\) 3950.74 0.507095
\(394\) 0 0
\(395\) 7023.20 0.894621
\(396\) 0 0
\(397\) 11500.6 1.45390 0.726952 0.686688i \(-0.240936\pi\)
0.726952 + 0.686688i \(0.240936\pi\)
\(398\) 0 0
\(399\) −647.195 −0.0812037
\(400\) 0 0
\(401\) −288.322 −0.0359055 −0.0179528 0.999839i \(-0.505715\pi\)
−0.0179528 + 0.999839i \(0.505715\pi\)
\(402\) 0 0
\(403\) 1332.72 0.164734
\(404\) 0 0
\(405\) 543.081 0.0666318
\(406\) 0 0
\(407\) 98.5634 0.0120039
\(408\) 0 0
\(409\) 13567.8 1.64030 0.820151 0.572147i \(-0.193889\pi\)
0.820151 + 0.572147i \(0.193889\pi\)
\(410\) 0 0
\(411\) 6800.94 0.816218
\(412\) 0 0
\(413\) −1250.13 −0.148947
\(414\) 0 0
\(415\) −1869.40 −0.221121
\(416\) 0 0
\(417\) 3488.78 0.409703
\(418\) 0 0
\(419\) −12094.2 −1.41012 −0.705059 0.709148i \(-0.749080\pi\)
−0.705059 + 0.709148i \(0.749080\pi\)
\(420\) 0 0
\(421\) −2030.79 −0.235094 −0.117547 0.993067i \(-0.537503\pi\)
−0.117547 + 0.993067i \(0.537503\pi\)
\(422\) 0 0
\(423\) 5471.64 0.628937
\(424\) 0 0
\(425\) −2137.63 −0.243977
\(426\) 0 0
\(427\) −2731.32 −0.309550
\(428\) 0 0
\(429\) 74.8590 0.00842477
\(430\) 0 0
\(431\) −11236.7 −1.25580 −0.627902 0.778292i \(-0.716086\pi\)
−0.627902 + 0.778292i \(0.716086\pi\)
\(432\) 0 0
\(433\) −2847.80 −0.316066 −0.158033 0.987434i \(-0.550515\pi\)
−0.158033 + 0.987434i \(0.550515\pi\)
\(434\) 0 0
\(435\) 5070.64 0.558894
\(436\) 0 0
\(437\) 181.396 0.0198567
\(438\) 0 0
\(439\) 596.831 0.0648866 0.0324433 0.999474i \(-0.489671\pi\)
0.0324433 + 0.999474i \(0.489671\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −10883.5 −1.16725 −0.583624 0.812024i \(-0.698366\pi\)
−0.583624 + 0.812024i \(0.698366\pi\)
\(444\) 0 0
\(445\) 2945.43 0.313768
\(446\) 0 0
\(447\) −162.202 −0.0171631
\(448\) 0 0
\(449\) −6593.68 −0.693040 −0.346520 0.938043i \(-0.612637\pi\)
−0.346520 + 0.938043i \(0.612637\pi\)
\(450\) 0 0
\(451\) −32.9128 −0.00343637
\(452\) 0 0
\(453\) 2586.04 0.268218
\(454\) 0 0
\(455\) −1661.87 −0.171230
\(456\) 0 0
\(457\) 14848.9 1.51992 0.759960 0.649970i \(-0.225218\pi\)
0.759960 + 0.649970i \(0.225218\pi\)
\(458\) 0 0
\(459\) −721.027 −0.0733217
\(460\) 0 0
\(461\) −1723.59 −0.174134 −0.0870668 0.996202i \(-0.527749\pi\)
−0.0870668 + 0.996202i \(0.527749\pi\)
\(462\) 0 0
\(463\) 17899.2 1.79664 0.898321 0.439339i \(-0.144787\pi\)
0.898321 + 0.439339i \(0.144787\pi\)
\(464\) 0 0
\(465\) 757.046 0.0754993
\(466\) 0 0
\(467\) −17921.1 −1.77579 −0.887893 0.460051i \(-0.847831\pi\)
−0.887893 + 0.460051i \(0.847831\pi\)
\(468\) 0 0
\(469\) −2443.99 −0.240624
\(470\) 0 0
\(471\) 4379.24 0.428417
\(472\) 0 0
\(473\) −179.060 −0.0174063
\(474\) 0 0
\(475\) −2466.95 −0.238298
\(476\) 0 0
\(477\) −2688.16 −0.258034
\(478\) 0 0
\(479\) 1507.10 0.143760 0.0718801 0.997413i \(-0.477100\pi\)
0.0718801 + 0.997413i \(0.477100\pi\)
\(480\) 0 0
\(481\) −4952.56 −0.469475
\(482\) 0 0
\(483\) −123.604 −0.0116442
\(484\) 0 0
\(485\) −1034.59 −0.0968628
\(486\) 0 0
\(487\) 7618.50 0.708885 0.354443 0.935078i \(-0.384671\pi\)
0.354443 + 0.935078i \(0.384671\pi\)
\(488\) 0 0
\(489\) 3203.32 0.296235
\(490\) 0 0
\(491\) 7346.20 0.675212 0.337606 0.941287i \(-0.390383\pi\)
0.337606 + 0.941287i \(0.390383\pi\)
\(492\) 0 0
\(493\) −6732.09 −0.615007
\(494\) 0 0
\(495\) 42.5232 0.00386116
\(496\) 0 0
\(497\) 2439.34 0.220159
\(498\) 0 0
\(499\) 18775.3 1.68436 0.842182 0.539193i \(-0.181271\pi\)
0.842182 + 0.539193i \(0.181271\pi\)
\(500\) 0 0
\(501\) 6248.58 0.557217
\(502\) 0 0
\(503\) −2556.27 −0.226597 −0.113299 0.993561i \(-0.536142\pi\)
−0.113299 + 0.993561i \(0.536142\pi\)
\(504\) 0 0
\(505\) −1113.38 −0.0981088
\(506\) 0 0
\(507\) 2829.52 0.247857
\(508\) 0 0
\(509\) −1902.18 −0.165644 −0.0828219 0.996564i \(-0.526393\pi\)
−0.0828219 + 0.996564i \(0.526393\pi\)
\(510\) 0 0
\(511\) −4524.91 −0.391723
\(512\) 0 0
\(513\) −832.108 −0.0716149
\(514\) 0 0
\(515\) 284.566 0.0243485
\(516\) 0 0
\(517\) 428.429 0.0364454
\(518\) 0 0
\(519\) 6630.50 0.560784
\(520\) 0 0
\(521\) 14253.1 1.19854 0.599272 0.800545i \(-0.295457\pi\)
0.599272 + 0.800545i \(0.295457\pi\)
\(522\) 0 0
\(523\) −10544.3 −0.881591 −0.440796 0.897608i \(-0.645304\pi\)
−0.440796 + 0.897608i \(0.645304\pi\)
\(524\) 0 0
\(525\) 1680.99 0.139741
\(526\) 0 0
\(527\) −1005.10 −0.0830795
\(528\) 0 0
\(529\) −12132.4 −0.997153
\(530\) 0 0
\(531\) −1607.32 −0.131359
\(532\) 0 0
\(533\) 1653.79 0.134397
\(534\) 0 0
\(535\) −1967.63 −0.159005
\(536\) 0 0
\(537\) 2804.15 0.225341
\(538\) 0 0
\(539\) 34.5303 0.00275942
\(540\) 0 0
\(541\) 7563.14 0.601044 0.300522 0.953775i \(-0.402839\pi\)
0.300522 + 0.953775i \(0.402839\pi\)
\(542\) 0 0
\(543\) 17.5566 0.00138753
\(544\) 0 0
\(545\) 7884.82 0.619722
\(546\) 0 0
\(547\) 9059.07 0.708113 0.354057 0.935224i \(-0.384802\pi\)
0.354057 + 0.935224i \(0.384802\pi\)
\(548\) 0 0
\(549\) −3511.69 −0.272997
\(550\) 0 0
\(551\) −7769.23 −0.600691
\(552\) 0 0
\(553\) 7332.52 0.563853
\(554\) 0 0
\(555\) −2813.27 −0.215166
\(556\) 0 0
\(557\) 23719.2 1.80434 0.902169 0.431384i \(-0.141974\pi\)
0.902169 + 0.431384i \(0.141974\pi\)
\(558\) 0 0
\(559\) 8997.32 0.680762
\(560\) 0 0
\(561\) −56.4564 −0.00424883
\(562\) 0 0
\(563\) −14719.8 −1.10189 −0.550947 0.834540i \(-0.685733\pi\)
−0.550947 + 0.834540i \(0.685733\pi\)
\(564\) 0 0
\(565\) −1223.85 −0.0911291
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −5184.65 −0.381989 −0.190994 0.981591i \(-0.561171\pi\)
−0.190994 + 0.981591i \(0.561171\pi\)
\(570\) 0 0
\(571\) −13173.6 −0.965496 −0.482748 0.875759i \(-0.660361\pi\)
−0.482748 + 0.875759i \(0.660361\pi\)
\(572\) 0 0
\(573\) −7132.01 −0.519972
\(574\) 0 0
\(575\) −471.149 −0.0341709
\(576\) 0 0
\(577\) 2354.29 0.169862 0.0849311 0.996387i \(-0.472933\pi\)
0.0849311 + 0.996387i \(0.472933\pi\)
\(578\) 0 0
\(579\) 6701.56 0.481014
\(580\) 0 0
\(581\) −1951.73 −0.139366
\(582\) 0 0
\(583\) −210.483 −0.0149525
\(584\) 0 0
\(585\) −2136.68 −0.151010
\(586\) 0 0
\(587\) −17138.5 −1.20508 −0.602541 0.798088i \(-0.705845\pi\)
−0.602541 + 0.798088i \(0.705845\pi\)
\(588\) 0 0
\(589\) −1159.95 −0.0811455
\(590\) 0 0
\(591\) 7076.86 0.492560
\(592\) 0 0
\(593\) −20553.0 −1.42329 −0.711645 0.702540i \(-0.752049\pi\)
−0.711645 + 0.702540i \(0.752049\pi\)
\(594\) 0 0
\(595\) 1253.33 0.0863554
\(596\) 0 0
\(597\) −1103.28 −0.0756350
\(598\) 0 0
\(599\) 19280.3 1.31514 0.657571 0.753392i \(-0.271584\pi\)
0.657571 + 0.753392i \(0.271584\pi\)
\(600\) 0 0
\(601\) 24736.4 1.67890 0.839451 0.543436i \(-0.182877\pi\)
0.839451 + 0.543436i \(0.182877\pi\)
\(602\) 0 0
\(603\) −3142.27 −0.212211
\(604\) 0 0
\(605\) −8920.63 −0.599463
\(606\) 0 0
\(607\) 4926.07 0.329395 0.164698 0.986344i \(-0.447335\pi\)
0.164698 + 0.986344i \(0.447335\pi\)
\(608\) 0 0
\(609\) 5293.97 0.352254
\(610\) 0 0
\(611\) −21527.5 −1.42538
\(612\) 0 0
\(613\) 28344.4 1.86757 0.933784 0.357837i \(-0.116486\pi\)
0.933784 + 0.357837i \(0.116486\pi\)
\(614\) 0 0
\(615\) 939.423 0.0615954
\(616\) 0 0
\(617\) 21881.8 1.42776 0.713879 0.700269i \(-0.246937\pi\)
0.713879 + 0.700269i \(0.246937\pi\)
\(618\) 0 0
\(619\) −8355.39 −0.542538 −0.271269 0.962504i \(-0.587443\pi\)
−0.271269 + 0.962504i \(0.587443\pi\)
\(620\) 0 0
\(621\) −158.919 −0.0102693
\(622\) 0 0
\(623\) 3075.16 0.197759
\(624\) 0 0
\(625\) 788.397 0.0504574
\(626\) 0 0
\(627\) −65.1540 −0.00414992
\(628\) 0 0
\(629\) 3735.07 0.236768
\(630\) 0 0
\(631\) 10780.0 0.680100 0.340050 0.940407i \(-0.389556\pi\)
0.340050 + 0.940407i \(0.389556\pi\)
\(632\) 0 0
\(633\) 5619.38 0.352844
\(634\) 0 0
\(635\) 9216.22 0.575960
\(636\) 0 0
\(637\) −1735.06 −0.107921
\(638\) 0 0
\(639\) 3136.29 0.194162
\(640\) 0 0
\(641\) −20161.7 −1.24234 −0.621171 0.783675i \(-0.713343\pi\)
−0.621171 + 0.783675i \(0.713343\pi\)
\(642\) 0 0
\(643\) −18277.8 −1.12100 −0.560502 0.828153i \(-0.689392\pi\)
−0.560502 + 0.828153i \(0.689392\pi\)
\(644\) 0 0
\(645\) 5110.87 0.312001
\(646\) 0 0
\(647\) −2534.86 −0.154027 −0.0770136 0.997030i \(-0.524538\pi\)
−0.0770136 + 0.997030i \(0.524538\pi\)
\(648\) 0 0
\(649\) −125.853 −0.00761195
\(650\) 0 0
\(651\) 790.390 0.0475850
\(652\) 0 0
\(653\) 16869.8 1.01098 0.505488 0.862833i \(-0.331312\pi\)
0.505488 + 0.862833i \(0.331312\pi\)
\(654\) 0 0
\(655\) −8829.51 −0.526713
\(656\) 0 0
\(657\) −5817.75 −0.345467
\(658\) 0 0
\(659\) 7724.65 0.456616 0.228308 0.973589i \(-0.426681\pi\)
0.228308 + 0.973589i \(0.426681\pi\)
\(660\) 0 0
\(661\) −6387.26 −0.375848 −0.187924 0.982184i \(-0.560176\pi\)
−0.187924 + 0.982184i \(0.560176\pi\)
\(662\) 0 0
\(663\) 2836.79 0.166172
\(664\) 0 0
\(665\) 1446.42 0.0843453
\(666\) 0 0
\(667\) −1483.80 −0.0861364
\(668\) 0 0
\(669\) 10058.5 0.581291
\(670\) 0 0
\(671\) −274.965 −0.0158196
\(672\) 0 0
\(673\) 14558.1 0.833837 0.416919 0.908944i \(-0.363110\pi\)
0.416919 + 0.908944i \(0.363110\pi\)
\(674\) 0 0
\(675\) 2161.27 0.123240
\(676\) 0 0
\(677\) 32914.0 1.86852 0.934259 0.356595i \(-0.116063\pi\)
0.934259 + 0.356595i \(0.116063\pi\)
\(678\) 0 0
\(679\) −1080.16 −0.0610497
\(680\) 0 0
\(681\) −3018.24 −0.169837
\(682\) 0 0
\(683\) −27334.4 −1.53137 −0.765683 0.643218i \(-0.777599\pi\)
−0.765683 + 0.643218i \(0.777599\pi\)
\(684\) 0 0
\(685\) −15199.4 −0.847796
\(686\) 0 0
\(687\) 4652.46 0.258373
\(688\) 0 0
\(689\) 10576.2 0.584794
\(690\) 0 0
\(691\) −25255.2 −1.39038 −0.695190 0.718826i \(-0.744680\pi\)
−0.695190 + 0.718826i \(0.744680\pi\)
\(692\) 0 0
\(693\) 44.3961 0.00243358
\(694\) 0 0
\(695\) −7797.07 −0.425554
\(696\) 0 0
\(697\) −1247.23 −0.0677796
\(698\) 0 0
\(699\) −1776.64 −0.0961357
\(700\) 0 0
\(701\) −16418.5 −0.884620 −0.442310 0.896862i \(-0.645841\pi\)
−0.442310 + 0.896862i \(0.645841\pi\)
\(702\) 0 0
\(703\) 4310.50 0.231257
\(704\) 0 0
\(705\) −12228.6 −0.653269
\(706\) 0 0
\(707\) −1162.42 −0.0618351
\(708\) 0 0
\(709\) 13186.2 0.698475 0.349238 0.937034i \(-0.386441\pi\)
0.349238 + 0.937034i \(0.386441\pi\)
\(710\) 0 0
\(711\) 9427.53 0.497271
\(712\) 0 0
\(713\) −221.531 −0.0116359
\(714\) 0 0
\(715\) −167.302 −0.00875071
\(716\) 0 0
\(717\) 15289.7 0.796378
\(718\) 0 0
\(719\) −17244.2 −0.894438 −0.447219 0.894425i \(-0.647585\pi\)
−0.447219 + 0.894425i \(0.647585\pi\)
\(720\) 0 0
\(721\) 297.100 0.0153461
\(722\) 0 0
\(723\) 13498.3 0.694340
\(724\) 0 0
\(725\) 20179.4 1.03371
\(726\) 0 0
\(727\) −3297.69 −0.168232 −0.0841160 0.996456i \(-0.526807\pi\)
−0.0841160 + 0.996456i \(0.526807\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −6785.50 −0.343325
\(732\) 0 0
\(733\) −14191.5 −0.715112 −0.357556 0.933892i \(-0.616390\pi\)
−0.357556 + 0.933892i \(0.616390\pi\)
\(734\) 0 0
\(735\) −985.591 −0.0494613
\(736\) 0 0
\(737\) −246.040 −0.0122971
\(738\) 0 0
\(739\) 31437.7 1.56489 0.782446 0.622718i \(-0.213972\pi\)
0.782446 + 0.622718i \(0.213972\pi\)
\(740\) 0 0
\(741\) 3273.83 0.162304
\(742\) 0 0
\(743\) −9095.98 −0.449124 −0.224562 0.974460i \(-0.572095\pi\)
−0.224562 + 0.974460i \(0.572095\pi\)
\(744\) 0 0
\(745\) 362.506 0.0178271
\(746\) 0 0
\(747\) −2509.37 −0.122909
\(748\) 0 0
\(749\) −2054.29 −0.100216
\(750\) 0 0
\(751\) −19791.9 −0.961676 −0.480838 0.876810i \(-0.659667\pi\)
−0.480838 + 0.876810i \(0.659667\pi\)
\(752\) 0 0
\(753\) −4808.25 −0.232699
\(754\) 0 0
\(755\) −5779.54 −0.278595
\(756\) 0 0
\(757\) −11194.5 −0.537479 −0.268739 0.963213i \(-0.586607\pi\)
−0.268739 + 0.963213i \(0.586607\pi\)
\(758\) 0 0
\(759\) −12.4434 −0.000595080 0
\(760\) 0 0
\(761\) 24445.6 1.16446 0.582230 0.813024i \(-0.302180\pi\)
0.582230 + 0.813024i \(0.302180\pi\)
\(762\) 0 0
\(763\) 8232.10 0.390592
\(764\) 0 0
\(765\) 1611.42 0.0761583
\(766\) 0 0
\(767\) 6323.79 0.297704
\(768\) 0 0
\(769\) 33296.7 1.56139 0.780695 0.624912i \(-0.214865\pi\)
0.780695 + 0.624912i \(0.214865\pi\)
\(770\) 0 0
\(771\) 14861.5 0.694192
\(772\) 0 0
\(773\) 31129.8 1.44846 0.724230 0.689559i \(-0.242195\pi\)
0.724230 + 0.689559i \(0.242195\pi\)
\(774\) 0 0
\(775\) 3012.78 0.139641
\(776\) 0 0
\(777\) −2937.18 −0.135612
\(778\) 0 0
\(779\) −1439.38 −0.0662018
\(780\) 0 0
\(781\) 245.571 0.0112513
\(782\) 0 0
\(783\) 6806.54 0.310659
\(784\) 0 0
\(785\) −9787.15 −0.444992
\(786\) 0 0
\(787\) 22918.3 1.03805 0.519027 0.854758i \(-0.326294\pi\)
0.519027 + 0.854758i \(0.326294\pi\)
\(788\) 0 0
\(789\) 11026.2 0.497520
\(790\) 0 0
\(791\) −1277.76 −0.0574359
\(792\) 0 0
\(793\) 13816.3 0.618704
\(794\) 0 0
\(795\) 6007.77 0.268017
\(796\) 0 0
\(797\) −40462.6 −1.79832 −0.899158 0.437624i \(-0.855820\pi\)
−0.899158 + 0.437624i \(0.855820\pi\)
\(798\) 0 0
\(799\) 16235.4 0.718857
\(800\) 0 0
\(801\) 3953.78 0.174407
\(802\) 0 0
\(803\) −455.529 −0.0200190
\(804\) 0 0
\(805\) 276.242 0.0120947
\(806\) 0 0
\(807\) 5968.93 0.260367
\(808\) 0 0
\(809\) −8286.03 −0.360100 −0.180050 0.983657i \(-0.557626\pi\)
−0.180050 + 0.983657i \(0.557626\pi\)
\(810\) 0 0
\(811\) −15527.8 −0.672323 −0.336162 0.941804i \(-0.609129\pi\)
−0.336162 + 0.941804i \(0.609129\pi\)
\(812\) 0 0
\(813\) −4916.54 −0.212092
\(814\) 0 0
\(815\) −7159.09 −0.307696
\(816\) 0 0
\(817\) −7830.87 −0.335334
\(818\) 0 0
\(819\) −2230.79 −0.0951773
\(820\) 0 0
\(821\) 13165.6 0.559663 0.279831 0.960049i \(-0.409721\pi\)
0.279831 + 0.960049i \(0.409721\pi\)
\(822\) 0 0
\(823\) 9659.67 0.409131 0.204565 0.978853i \(-0.434422\pi\)
0.204565 + 0.978853i \(0.434422\pi\)
\(824\) 0 0
\(825\) 169.227 0.00714150
\(826\) 0 0
\(827\) 31937.7 1.34290 0.671452 0.741048i \(-0.265671\pi\)
0.671452 + 0.741048i \(0.265671\pi\)
\(828\) 0 0
\(829\) −29987.5 −1.25635 −0.628173 0.778074i \(-0.716197\pi\)
−0.628173 + 0.778074i \(0.716197\pi\)
\(830\) 0 0
\(831\) −23432.6 −0.978181
\(832\) 0 0
\(833\) 1308.53 0.0544272
\(834\) 0 0
\(835\) −13964.9 −0.578775
\(836\) 0 0
\(837\) 1016.22 0.0419660
\(838\) 0 0
\(839\) −25864.3 −1.06428 −0.532141 0.846656i \(-0.678613\pi\)
−0.532141 + 0.846656i \(0.678613\pi\)
\(840\) 0 0
\(841\) 39162.4 1.60574
\(842\) 0 0
\(843\) −9055.01 −0.369954
\(844\) 0 0
\(845\) −6323.70 −0.257446
\(846\) 0 0
\(847\) −9313.52 −0.377823
\(848\) 0 0
\(849\) −22044.4 −0.891121
\(850\) 0 0
\(851\) 823.236 0.0331612
\(852\) 0 0
\(853\) 9652.50 0.387451 0.193725 0.981056i \(-0.437943\pi\)
0.193725 + 0.981056i \(0.437943\pi\)
\(854\) 0 0
\(855\) 1859.68 0.0743855
\(856\) 0 0
\(857\) 7058.37 0.281341 0.140671 0.990056i \(-0.455074\pi\)
0.140671 + 0.990056i \(0.455074\pi\)
\(858\) 0 0
\(859\) 5146.18 0.204407 0.102203 0.994764i \(-0.467411\pi\)
0.102203 + 0.994764i \(0.467411\pi\)
\(860\) 0 0
\(861\) 980.799 0.0388218
\(862\) 0 0
\(863\) −23233.4 −0.916426 −0.458213 0.888842i \(-0.651510\pi\)
−0.458213 + 0.888842i \(0.651510\pi\)
\(864\) 0 0
\(865\) −14818.5 −0.582479
\(866\) 0 0
\(867\) 12599.6 0.493546
\(868\) 0 0
\(869\) 738.176 0.0288158
\(870\) 0 0
\(871\) 12362.9 0.480941
\(872\) 0 0
\(873\) −1388.78 −0.0538408
\(874\) 0 0
\(875\) −9623.45 −0.371808
\(876\) 0 0
\(877\) −17998.3 −0.692997 −0.346498 0.938051i \(-0.612629\pi\)
−0.346498 + 0.938051i \(0.612629\pi\)
\(878\) 0 0
\(879\) 12635.7 0.484861
\(880\) 0 0
\(881\) 10400.3 0.397723 0.198861 0.980028i \(-0.436276\pi\)
0.198861 + 0.980028i \(0.436276\pi\)
\(882\) 0 0
\(883\) 35365.0 1.34782 0.673912 0.738812i \(-0.264613\pi\)
0.673912 + 0.738812i \(0.264613\pi\)
\(884\) 0 0
\(885\) 3592.19 0.136441
\(886\) 0 0
\(887\) −39216.8 −1.48452 −0.742261 0.670111i \(-0.766246\pi\)
−0.742261 + 0.670111i \(0.766246\pi\)
\(888\) 0 0
\(889\) 9622.13 0.363010
\(890\) 0 0
\(891\) 57.0807 0.00214621
\(892\) 0 0
\(893\) 18736.6 0.702123
\(894\) 0 0
\(895\) −6267.01 −0.234059
\(896\) 0 0
\(897\) 625.249 0.0232736
\(898\) 0 0
\(899\) 9488.21 0.352002
\(900\) 0 0
\(901\) −7976.28 −0.294926
\(902\) 0 0
\(903\) 5335.97 0.196645
\(904\) 0 0
\(905\) −39.2373 −0.00144121
\(906\) 0 0
\(907\) 42804.2 1.56702 0.783511 0.621378i \(-0.213427\pi\)
0.783511 + 0.621378i \(0.213427\pi\)
\(908\) 0 0
\(909\) −1494.54 −0.0545334
\(910\) 0 0
\(911\) 32448.5 1.18010 0.590048 0.807368i \(-0.299109\pi\)
0.590048 + 0.807368i \(0.299109\pi\)
\(912\) 0 0
\(913\) −196.484 −0.00712230
\(914\) 0 0
\(915\) 7848.28 0.283559
\(916\) 0 0
\(917\) −9218.39 −0.331972
\(918\) 0 0
\(919\) −21991.6 −0.789375 −0.394688 0.918815i \(-0.629147\pi\)
−0.394688 + 0.918815i \(0.629147\pi\)
\(920\) 0 0
\(921\) −22659.5 −0.810700
\(922\) 0 0
\(923\) −12339.3 −0.440037
\(924\) 0 0
\(925\) −11195.8 −0.397964
\(926\) 0 0
\(927\) 381.985 0.0135340
\(928\) 0 0
\(929\) −1281.57 −0.0452606 −0.0226303 0.999744i \(-0.507204\pi\)
−0.0226303 + 0.999744i \(0.507204\pi\)
\(930\) 0 0
\(931\) 1510.12 0.0531603
\(932\) 0 0
\(933\) 7969.73 0.279654
\(934\) 0 0
\(935\) 126.174 0.00441320
\(936\) 0 0
\(937\) 23582.2 0.822194 0.411097 0.911592i \(-0.365146\pi\)
0.411097 + 0.911592i \(0.365146\pi\)
\(938\) 0 0
\(939\) 32457.5 1.12802
\(940\) 0 0
\(941\) −27439.5 −0.950586 −0.475293 0.879828i \(-0.657658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(942\) 0 0
\(943\) −274.899 −0.00949305
\(944\) 0 0
\(945\) −1267.19 −0.0436208
\(946\) 0 0
\(947\) 17543.6 0.601996 0.300998 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(948\) 0 0
\(949\) 22889.2 0.782945
\(950\) 0 0
\(951\) 5114.74 0.174403
\(952\) 0 0
\(953\) −13884.1 −0.471930 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(954\) 0 0
\(955\) 15939.3 0.540088
\(956\) 0 0
\(957\) 532.952 0.0180020
\(958\) 0 0
\(959\) −15868.9 −0.534340
\(960\) 0 0
\(961\) −28374.4 −0.952449
\(962\) 0 0
\(963\) −2641.23 −0.0883825
\(964\) 0 0
\(965\) −14977.3 −0.499623
\(966\) 0 0
\(967\) 39807.7 1.32382 0.661908 0.749585i \(-0.269747\pi\)
0.661908 + 0.749585i \(0.269747\pi\)
\(968\) 0 0
\(969\) −2469.02 −0.0818538
\(970\) 0 0
\(971\) −20963.7 −0.692851 −0.346425 0.938077i \(-0.612605\pi\)
−0.346425 + 0.938077i \(0.612605\pi\)
\(972\) 0 0
\(973\) −8140.48 −0.268214
\(974\) 0 0
\(975\) −8503.25 −0.279305
\(976\) 0 0
\(977\) −38199.7 −1.25089 −0.625443 0.780270i \(-0.715082\pi\)
−0.625443 + 0.780270i \(0.715082\pi\)
\(978\) 0 0
\(979\) 309.581 0.0101065
\(980\) 0 0
\(981\) 10584.1 0.344470
\(982\) 0 0
\(983\) 18877.5 0.612510 0.306255 0.951950i \(-0.400924\pi\)
0.306255 + 0.951950i \(0.400924\pi\)
\(984\) 0 0
\(985\) −15816.1 −0.511616
\(986\) 0 0
\(987\) −12767.2 −0.411736
\(988\) 0 0
\(989\) −1495.57 −0.0480854
\(990\) 0 0
\(991\) −1894.95 −0.0607418 −0.0303709 0.999539i \(-0.509669\pi\)
−0.0303709 + 0.999539i \(0.509669\pi\)
\(992\) 0 0
\(993\) 225.343 0.00720146
\(994\) 0 0
\(995\) 2465.71 0.0785611
\(996\) 0 0
\(997\) −60576.0 −1.92424 −0.962118 0.272635i \(-0.912105\pi\)
−0.962118 + 0.272635i \(0.912105\pi\)
\(998\) 0 0
\(999\) −3776.38 −0.119599
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 1344.4.a.be.1.2 2
4.3 odd 2 1344.4.a.bm.1.2 2
8.3 odd 2 672.4.a.h.1.1 2
8.5 even 2 672.4.a.m.1.1 yes 2
24.5 odd 2 2016.4.a.j.1.2 2
24.11 even 2 2016.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.a.h.1.1 2 8.3 odd 2
672.4.a.m.1.1 yes 2 8.5 even 2
1344.4.a.be.1.2 2 1.1 even 1 trivial
1344.4.a.bm.1.2 2 4.3 odd 2
2016.4.a.i.1.2 2 24.11 even 2
2016.4.a.j.1.2 2 24.5 odd 2