Properties

Label 1452.4.a.t.1.3
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
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] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-9.47700\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -0.981524 q^{5} +17.7745 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -0.981524 q^{5} +17.7745 q^{7} +9.00000 q^{9} +61.8967 q^{13} -2.94457 q^{15} -130.198 q^{17} -116.281 q^{19} +53.3235 q^{21} -26.4920 q^{23} -124.037 q^{25} +27.0000 q^{27} -179.702 q^{29} -319.935 q^{31} -17.4461 q^{35} -177.998 q^{37} +185.690 q^{39} +338.949 q^{41} -159.872 q^{43} -8.83372 q^{45} +98.9479 q^{47} -27.0675 q^{49} -390.593 q^{51} +562.690 q^{53} -348.844 q^{57} +443.734 q^{59} -551.701 q^{61} +159.970 q^{63} -60.7531 q^{65} +454.264 q^{67} -79.4761 q^{69} -214.257 q^{71} -162.610 q^{73} -372.110 q^{75} -167.620 q^{79} +81.0000 q^{81} -357.601 q^{83} +127.792 q^{85} -539.105 q^{87} +1215.19 q^{89} +1100.18 q^{91} -959.805 q^{93} +114.133 q^{95} -1117.50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) 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
\(5\) −0.981524 −0.0877902 −0.0438951 0.999036i \(-0.513977\pi\)
−0.0438951 + 0.999036i \(0.513977\pi\)
\(6\) 0 0
\(7\) 17.7745 0.959732 0.479866 0.877342i \(-0.340685\pi\)
0.479866 + 0.877342i \(0.340685\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 61.8967 1.32054 0.660272 0.751027i \(-0.270441\pi\)
0.660272 + 0.751027i \(0.270441\pi\)
\(14\) 0 0
\(15\) −2.94457 −0.0506857
\(16\) 0 0
\(17\) −130.198 −1.85751 −0.928753 0.370700i \(-0.879118\pi\)
−0.928753 + 0.370700i \(0.879118\pi\)
\(18\) 0 0
\(19\) −116.281 −1.40404 −0.702019 0.712158i \(-0.747718\pi\)
−0.702019 + 0.712158i \(0.747718\pi\)
\(20\) 0 0
\(21\) 53.3235 0.554102
\(22\) 0 0
\(23\) −26.4920 −0.240173 −0.120086 0.992763i \(-0.538317\pi\)
−0.120086 + 0.992763i \(0.538317\pi\)
\(24\) 0 0
\(25\) −124.037 −0.992293
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −179.702 −1.15068 −0.575341 0.817914i \(-0.695131\pi\)
−0.575341 + 0.817914i \(0.695131\pi\)
\(30\) 0 0
\(31\) −319.935 −1.85361 −0.926807 0.375538i \(-0.877458\pi\)
−0.926807 + 0.375538i \(0.877458\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −17.4461 −0.0842551
\(36\) 0 0
\(37\) −177.998 −0.790885 −0.395442 0.918491i \(-0.629409\pi\)
−0.395442 + 0.918491i \(0.629409\pi\)
\(38\) 0 0
\(39\) 185.690 0.762416
\(40\) 0 0
\(41\) 338.949 1.29110 0.645548 0.763720i \(-0.276629\pi\)
0.645548 + 0.763720i \(0.276629\pi\)
\(42\) 0 0
\(43\) −159.872 −0.566984 −0.283492 0.958975i \(-0.591493\pi\)
−0.283492 + 0.958975i \(0.591493\pi\)
\(44\) 0 0
\(45\) −8.83372 −0.0292634
\(46\) 0 0
\(47\) 98.9479 0.307086 0.153543 0.988142i \(-0.450932\pi\)
0.153543 + 0.988142i \(0.450932\pi\)
\(48\) 0 0
\(49\) −27.0675 −0.0789141
\(50\) 0 0
\(51\) −390.593 −1.07243
\(52\) 0 0
\(53\) 562.690 1.45833 0.729164 0.684339i \(-0.239909\pi\)
0.729164 + 0.684339i \(0.239909\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −348.844 −0.810622
\(58\) 0 0
\(59\) 443.734 0.979140 0.489570 0.871964i \(-0.337154\pi\)
0.489570 + 0.871964i \(0.337154\pi\)
\(60\) 0 0
\(61\) −551.701 −1.15800 −0.579001 0.815327i \(-0.696557\pi\)
−0.579001 + 0.815327i \(0.696557\pi\)
\(62\) 0 0
\(63\) 159.970 0.319911
\(64\) 0 0
\(65\) −60.7531 −0.115931
\(66\) 0 0
\(67\) 454.264 0.828316 0.414158 0.910205i \(-0.364076\pi\)
0.414158 + 0.910205i \(0.364076\pi\)
\(68\) 0 0
\(69\) −79.4761 −0.138664
\(70\) 0 0
\(71\) −214.257 −0.358136 −0.179068 0.983837i \(-0.557308\pi\)
−0.179068 + 0.983837i \(0.557308\pi\)
\(72\) 0 0
\(73\) −162.610 −0.260713 −0.130357 0.991467i \(-0.541612\pi\)
−0.130357 + 0.991467i \(0.541612\pi\)
\(74\) 0 0
\(75\) −372.110 −0.572901
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −167.620 −0.238719 −0.119359 0.992851i \(-0.538084\pi\)
−0.119359 + 0.992851i \(0.538084\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −357.601 −0.472913 −0.236457 0.971642i \(-0.575986\pi\)
−0.236457 + 0.971642i \(0.575986\pi\)
\(84\) 0 0
\(85\) 127.792 0.163071
\(86\) 0 0
\(87\) −539.105 −0.664347
\(88\) 0 0
\(89\) 1215.19 1.44730 0.723648 0.690169i \(-0.242464\pi\)
0.723648 + 0.690169i \(0.242464\pi\)
\(90\) 0 0
\(91\) 1100.18 1.26737
\(92\) 0 0
\(93\) −959.805 −1.07018
\(94\) 0 0
\(95\) 114.133 0.123261
\(96\) 0 0
\(97\) −1117.50 −1.16974 −0.584868 0.811128i \(-0.698854\pi\)
−0.584868 + 0.811128i \(0.698854\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −405.605 −0.399596 −0.199798 0.979837i \(-0.564029\pi\)
−0.199798 + 0.979837i \(0.564029\pi\)
\(102\) 0 0
\(103\) 444.313 0.425044 0.212522 0.977156i \(-0.431832\pi\)
0.212522 + 0.977156i \(0.431832\pi\)
\(104\) 0 0
\(105\) −52.3383 −0.0486447
\(106\) 0 0
\(107\) −1247.30 −1.12693 −0.563465 0.826140i \(-0.690532\pi\)
−0.563465 + 0.826140i \(0.690532\pi\)
\(108\) 0 0
\(109\) −1343.63 −1.18070 −0.590349 0.807148i \(-0.701010\pi\)
−0.590349 + 0.807148i \(0.701010\pi\)
\(110\) 0 0
\(111\) −533.995 −0.456618
\(112\) 0 0
\(113\) 1193.63 0.993693 0.496846 0.867838i \(-0.334491\pi\)
0.496846 + 0.867838i \(0.334491\pi\)
\(114\) 0 0
\(115\) 26.0026 0.0210848
\(116\) 0 0
\(117\) 557.071 0.440181
\(118\) 0 0
\(119\) −2314.20 −1.78271
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 1016.85 0.745414
\(124\) 0 0
\(125\) 244.435 0.174904
\(126\) 0 0
\(127\) −640.650 −0.447626 −0.223813 0.974632i \(-0.571851\pi\)
−0.223813 + 0.974632i \(0.571851\pi\)
\(128\) 0 0
\(129\) −479.617 −0.327348
\(130\) 0 0
\(131\) −379.349 −0.253007 −0.126503 0.991966i \(-0.540375\pi\)
−0.126503 + 0.991966i \(0.540375\pi\)
\(132\) 0 0
\(133\) −2066.84 −1.34750
\(134\) 0 0
\(135\) −26.5011 −0.0168952
\(136\) 0 0
\(137\) −1193.17 −0.744083 −0.372041 0.928216i \(-0.621342\pi\)
−0.372041 + 0.928216i \(0.621342\pi\)
\(138\) 0 0
\(139\) −2945.39 −1.79730 −0.898650 0.438666i \(-0.855451\pi\)
−0.898650 + 0.438666i \(0.855451\pi\)
\(140\) 0 0
\(141\) 296.844 0.177296
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 176.382 0.101019
\(146\) 0 0
\(147\) −81.2026 −0.0455611
\(148\) 0 0
\(149\) −3305.66 −1.81752 −0.908760 0.417320i \(-0.862969\pi\)
−0.908760 + 0.417320i \(0.862969\pi\)
\(150\) 0 0
\(151\) 1670.75 0.900421 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(152\) 0 0
\(153\) −1171.78 −0.619168
\(154\) 0 0
\(155\) 314.024 0.162729
\(156\) 0 0
\(157\) 561.400 0.285379 0.142690 0.989767i \(-0.454425\pi\)
0.142690 + 0.989767i \(0.454425\pi\)
\(158\) 0 0
\(159\) 1688.07 0.841966
\(160\) 0 0
\(161\) −470.883 −0.230502
\(162\) 0 0
\(163\) 3799.01 1.82553 0.912765 0.408485i \(-0.133943\pi\)
0.912765 + 0.408485i \(0.133943\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 693.606 0.321394 0.160697 0.987004i \(-0.448626\pi\)
0.160697 + 0.987004i \(0.448626\pi\)
\(168\) 0 0
\(169\) 1634.21 0.743835
\(170\) 0 0
\(171\) −1046.53 −0.468013
\(172\) 0 0
\(173\) 2766.95 1.21600 0.607998 0.793939i \(-0.291973\pi\)
0.607998 + 0.793939i \(0.291973\pi\)
\(174\) 0 0
\(175\) −2204.69 −0.952335
\(176\) 0 0
\(177\) 1331.20 0.565307
\(178\) 0 0
\(179\) 2640.76 1.10268 0.551340 0.834281i \(-0.314117\pi\)
0.551340 + 0.834281i \(0.314117\pi\)
\(180\) 0 0
\(181\) −4399.56 −1.80672 −0.903361 0.428882i \(-0.858908\pi\)
−0.903361 + 0.428882i \(0.858908\pi\)
\(182\) 0 0
\(183\) −1655.10 −0.668572
\(184\) 0 0
\(185\) 174.710 0.0694319
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 479.911 0.184701
\(190\) 0 0
\(191\) −1575.94 −0.597019 −0.298510 0.954407i \(-0.596490\pi\)
−0.298510 + 0.954407i \(0.596490\pi\)
\(192\) 0 0
\(193\) 187.213 0.0698232 0.0349116 0.999390i \(-0.488885\pi\)
0.0349116 + 0.999390i \(0.488885\pi\)
\(194\) 0 0
\(195\) −182.259 −0.0669326
\(196\) 0 0
\(197\) −1465.07 −0.529856 −0.264928 0.964268i \(-0.585348\pi\)
−0.264928 + 0.964268i \(0.585348\pi\)
\(198\) 0 0
\(199\) 3519.83 1.25384 0.626920 0.779083i \(-0.284315\pi\)
0.626920 + 0.779083i \(0.284315\pi\)
\(200\) 0 0
\(201\) 1362.79 0.478228
\(202\) 0 0
\(203\) −3194.11 −1.10435
\(204\) 0 0
\(205\) −332.686 −0.113345
\(206\) 0 0
\(207\) −238.428 −0.0800576
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 504.847 0.164716 0.0823580 0.996603i \(-0.473755\pi\)
0.0823580 + 0.996603i \(0.473755\pi\)
\(212\) 0 0
\(213\) −642.772 −0.206770
\(214\) 0 0
\(215\) 156.919 0.0497756
\(216\) 0 0
\(217\) −5686.68 −1.77897
\(218\) 0 0
\(219\) −487.830 −0.150523
\(220\) 0 0
\(221\) −8058.81 −2.45292
\(222\) 0 0
\(223\) −599.729 −0.180093 −0.0900467 0.995938i \(-0.528702\pi\)
−0.0900467 + 0.995938i \(0.528702\pi\)
\(224\) 0 0
\(225\) −1116.33 −0.330764
\(226\) 0 0
\(227\) 2340.46 0.684326 0.342163 0.939641i \(-0.388840\pi\)
0.342163 + 0.939641i \(0.388840\pi\)
\(228\) 0 0
\(229\) −3465.87 −1.00014 −0.500069 0.865986i \(-0.666692\pi\)
−0.500069 + 0.865986i \(0.666692\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2976.18 0.836807 0.418404 0.908261i \(-0.362590\pi\)
0.418404 + 0.908261i \(0.362590\pi\)
\(234\) 0 0
\(235\) −97.1197 −0.0269591
\(236\) 0 0
\(237\) −502.861 −0.137824
\(238\) 0 0
\(239\) 4379.37 1.18526 0.592632 0.805473i \(-0.298089\pi\)
0.592632 + 0.805473i \(0.298089\pi\)
\(240\) 0 0
\(241\) −4045.04 −1.08118 −0.540590 0.841286i \(-0.681799\pi\)
−0.540590 + 0.841286i \(0.681799\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 26.5674 0.00692788
\(246\) 0 0
\(247\) −7197.42 −1.85409
\(248\) 0 0
\(249\) −1072.80 −0.273037
\(250\) 0 0
\(251\) −6862.99 −1.72585 −0.862925 0.505333i \(-0.831370\pi\)
−0.862925 + 0.505333i \(0.831370\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 383.377 0.0941489
\(256\) 0 0
\(257\) 6424.73 1.55939 0.779696 0.626158i \(-0.215373\pi\)
0.779696 + 0.626158i \(0.215373\pi\)
\(258\) 0 0
\(259\) −3163.83 −0.759038
\(260\) 0 0
\(261\) −1617.32 −0.383561
\(262\) 0 0
\(263\) −6510.99 −1.52656 −0.763280 0.646068i \(-0.776412\pi\)
−0.763280 + 0.646068i \(0.776412\pi\)
\(264\) 0 0
\(265\) −552.294 −0.128027
\(266\) 0 0
\(267\) 3645.56 0.835597
\(268\) 0 0
\(269\) 5029.52 1.13998 0.569992 0.821650i \(-0.306946\pi\)
0.569992 + 0.821650i \(0.306946\pi\)
\(270\) 0 0
\(271\) 900.678 0.201890 0.100945 0.994892i \(-0.467813\pi\)
0.100945 + 0.994892i \(0.467813\pi\)
\(272\) 0 0
\(273\) 3300.55 0.731715
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4634.96 1.00537 0.502686 0.864469i \(-0.332345\pi\)
0.502686 + 0.864469i \(0.332345\pi\)
\(278\) 0 0
\(279\) −2879.42 −0.617871
\(280\) 0 0
\(281\) −934.894 −0.198474 −0.0992368 0.995064i \(-0.531640\pi\)
−0.0992368 + 0.995064i \(0.531640\pi\)
\(282\) 0 0
\(283\) 1335.92 0.280608 0.140304 0.990108i \(-0.455192\pi\)
0.140304 + 0.990108i \(0.455192\pi\)
\(284\) 0 0
\(285\) 342.398 0.0711646
\(286\) 0 0
\(287\) 6024.64 1.23911
\(288\) 0 0
\(289\) 12038.5 2.45033
\(290\) 0 0
\(291\) −3352.49 −0.675348
\(292\) 0 0
\(293\) −2972.82 −0.592745 −0.296372 0.955072i \(-0.595777\pi\)
−0.296372 + 0.955072i \(0.595777\pi\)
\(294\) 0 0
\(295\) −435.536 −0.0859589
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1639.77 −0.317159
\(300\) 0 0
\(301\) −2841.65 −0.544153
\(302\) 0 0
\(303\) −1216.82 −0.230707
\(304\) 0 0
\(305\) 541.508 0.101661
\(306\) 0 0
\(307\) −644.628 −0.119840 −0.0599200 0.998203i \(-0.519085\pi\)
−0.0599200 + 0.998203i \(0.519085\pi\)
\(308\) 0 0
\(309\) 1332.94 0.245399
\(310\) 0 0
\(311\) 4531.91 0.826305 0.413152 0.910662i \(-0.364428\pi\)
0.413152 + 0.910662i \(0.364428\pi\)
\(312\) 0 0
\(313\) 4647.06 0.839193 0.419597 0.907711i \(-0.362172\pi\)
0.419597 + 0.907711i \(0.362172\pi\)
\(314\) 0 0
\(315\) −157.015 −0.0280850
\(316\) 0 0
\(317\) −628.109 −0.111287 −0.0556437 0.998451i \(-0.517721\pi\)
−0.0556437 + 0.998451i \(0.517721\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3741.91 −0.650633
\(322\) 0 0
\(323\) 15139.5 2.60801
\(324\) 0 0
\(325\) −7677.46 −1.31037
\(326\) 0 0
\(327\) −4030.88 −0.681676
\(328\) 0 0
\(329\) 1758.75 0.294720
\(330\) 0 0
\(331\) −1614.45 −0.268091 −0.134045 0.990975i \(-0.542797\pi\)
−0.134045 + 0.990975i \(0.542797\pi\)
\(332\) 0 0
\(333\) −1601.98 −0.263628
\(334\) 0 0
\(335\) −445.871 −0.0727180
\(336\) 0 0
\(337\) 359.175 0.0580579 0.0290290 0.999579i \(-0.490758\pi\)
0.0290290 + 0.999579i \(0.490758\pi\)
\(338\) 0 0
\(339\) 3580.89 0.573709
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6577.76 −1.03547
\(344\) 0 0
\(345\) 78.0077 0.0121733
\(346\) 0 0
\(347\) −1980.58 −0.306407 −0.153203 0.988195i \(-0.548959\pi\)
−0.153203 + 0.988195i \(0.548959\pi\)
\(348\) 0 0
\(349\) −1550.65 −0.237835 −0.118918 0.992904i \(-0.537942\pi\)
−0.118918 + 0.992904i \(0.537942\pi\)
\(350\) 0 0
\(351\) 1671.21 0.254139
\(352\) 0 0
\(353\) 261.089 0.0393664 0.0196832 0.999806i \(-0.493734\pi\)
0.0196832 + 0.999806i \(0.493734\pi\)
\(354\) 0 0
\(355\) 210.299 0.0314408
\(356\) 0 0
\(357\) −6942.59 −1.02925
\(358\) 0 0
\(359\) −3.02135 −0.000444181 0 −0.000222090 1.00000i \(-0.500071\pi\)
−0.000222090 1.00000i \(0.500071\pi\)
\(360\) 0 0
\(361\) 6662.31 0.971324
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 159.606 0.0228881
\(366\) 0 0
\(367\) −10582.1 −1.50513 −0.752563 0.658520i \(-0.771183\pi\)
−0.752563 + 0.658520i \(0.771183\pi\)
\(368\) 0 0
\(369\) 3050.54 0.430365
\(370\) 0 0
\(371\) 10001.5 1.39960
\(372\) 0 0
\(373\) 0.122254 1.69707e−5 0 8.48537e−6 1.00000i \(-0.499997\pi\)
8.48537e−6 1.00000i \(0.499997\pi\)
\(374\) 0 0
\(375\) 733.306 0.100981
\(376\) 0 0
\(377\) −11123.0 −1.51953
\(378\) 0 0
\(379\) 6236.92 0.845300 0.422650 0.906293i \(-0.361100\pi\)
0.422650 + 0.906293i \(0.361100\pi\)
\(380\) 0 0
\(381\) −1921.95 −0.258437
\(382\) 0 0
\(383\) −8554.88 −1.14134 −0.570671 0.821179i \(-0.693317\pi\)
−0.570671 + 0.821179i \(0.693317\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1438.85 −0.188995
\(388\) 0 0
\(389\) −3604.00 −0.469743 −0.234871 0.972026i \(-0.575467\pi\)
−0.234871 + 0.972026i \(0.575467\pi\)
\(390\) 0 0
\(391\) 3449.20 0.446122
\(392\) 0 0
\(393\) −1138.05 −0.146074
\(394\) 0 0
\(395\) 164.523 0.0209572
\(396\) 0 0
\(397\) 10515.4 1.32935 0.664674 0.747133i \(-0.268570\pi\)
0.664674 + 0.747133i \(0.268570\pi\)
\(398\) 0 0
\(399\) −6200.52 −0.777980
\(400\) 0 0
\(401\) −10070.2 −1.25407 −0.627035 0.778991i \(-0.715732\pi\)
−0.627035 + 0.778991i \(0.715732\pi\)
\(402\) 0 0
\(403\) −19802.9 −2.44778
\(404\) 0 0
\(405\) −79.5034 −0.00975446
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 146.224 0.0176780 0.00883900 0.999961i \(-0.497186\pi\)
0.00883900 + 0.999961i \(0.497186\pi\)
\(410\) 0 0
\(411\) −3579.51 −0.429596
\(412\) 0 0
\(413\) 7887.15 0.939712
\(414\) 0 0
\(415\) 350.994 0.0415171
\(416\) 0 0
\(417\) −8836.17 −1.03767
\(418\) 0 0
\(419\) −12686.3 −1.47915 −0.739577 0.673072i \(-0.764974\pi\)
−0.739577 + 0.673072i \(0.764974\pi\)
\(420\) 0 0
\(421\) −8158.85 −0.944508 −0.472254 0.881462i \(-0.656560\pi\)
−0.472254 + 0.881462i \(0.656560\pi\)
\(422\) 0 0
\(423\) 890.531 0.102362
\(424\) 0 0
\(425\) 16149.3 1.84319
\(426\) 0 0
\(427\) −9806.20 −1.11137
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6289.60 −0.702922 −0.351461 0.936202i \(-0.614315\pi\)
−0.351461 + 0.936202i \(0.614315\pi\)
\(432\) 0 0
\(433\) 5741.35 0.637210 0.318605 0.947888i \(-0.396786\pi\)
0.318605 + 0.947888i \(0.396786\pi\)
\(434\) 0 0
\(435\) 529.145 0.0583231
\(436\) 0 0
\(437\) 3080.53 0.337212
\(438\) 0 0
\(439\) −3601.82 −0.391584 −0.195792 0.980645i \(-0.562728\pi\)
−0.195792 + 0.980645i \(0.562728\pi\)
\(440\) 0 0
\(441\) −243.608 −0.0263047
\(442\) 0 0
\(443\) −3487.54 −0.374036 −0.187018 0.982356i \(-0.559882\pi\)
−0.187018 + 0.982356i \(0.559882\pi\)
\(444\) 0 0
\(445\) −1192.73 −0.127058
\(446\) 0 0
\(447\) −9916.99 −1.04935
\(448\) 0 0
\(449\) −9135.55 −0.960208 −0.480104 0.877212i \(-0.659401\pi\)
−0.480104 + 0.877212i \(0.659401\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5012.24 0.519858
\(454\) 0 0
\(455\) −1079.86 −0.111262
\(456\) 0 0
\(457\) −858.984 −0.0879247 −0.0439623 0.999033i \(-0.513998\pi\)
−0.0439623 + 0.999033i \(0.513998\pi\)
\(458\) 0 0
\(459\) −3515.34 −0.357477
\(460\) 0 0
\(461\) 15172.7 1.53289 0.766447 0.642308i \(-0.222023\pi\)
0.766447 + 0.642308i \(0.222023\pi\)
\(462\) 0 0
\(463\) 808.597 0.0811634 0.0405817 0.999176i \(-0.487079\pi\)
0.0405817 + 0.999176i \(0.487079\pi\)
\(464\) 0 0
\(465\) 942.072 0.0939517
\(466\) 0 0
\(467\) 16387.7 1.62384 0.811921 0.583767i \(-0.198422\pi\)
0.811921 + 0.583767i \(0.198422\pi\)
\(468\) 0 0
\(469\) 8074.31 0.794962
\(470\) 0 0
\(471\) 1684.20 0.164764
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 14423.1 1.39322
\(476\) 0 0
\(477\) 5064.21 0.486110
\(478\) 0 0
\(479\) −18147.3 −1.73105 −0.865523 0.500870i \(-0.833014\pi\)
−0.865523 + 0.500870i \(0.833014\pi\)
\(480\) 0 0
\(481\) −11017.5 −1.04440
\(482\) 0 0
\(483\) −1412.65 −0.133080
\(484\) 0 0
\(485\) 1096.85 0.102691
\(486\) 0 0
\(487\) −6192.06 −0.576158 −0.288079 0.957607i \(-0.593017\pi\)
−0.288079 + 0.957607i \(0.593017\pi\)
\(488\) 0 0
\(489\) 11397.0 1.05397
\(490\) 0 0
\(491\) 691.676 0.0635741 0.0317871 0.999495i \(-0.489880\pi\)
0.0317871 + 0.999495i \(0.489880\pi\)
\(492\) 0 0
\(493\) 23396.8 2.13740
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3808.31 −0.343715
\(498\) 0 0
\(499\) −6553.93 −0.587964 −0.293982 0.955811i \(-0.594981\pi\)
−0.293982 + 0.955811i \(0.594981\pi\)
\(500\) 0 0
\(501\) 2080.82 0.185557
\(502\) 0 0
\(503\) 7930.22 0.702964 0.351482 0.936195i \(-0.385678\pi\)
0.351482 + 0.936195i \(0.385678\pi\)
\(504\) 0 0
\(505\) 398.111 0.0350806
\(506\) 0 0
\(507\) 4902.62 0.429453
\(508\) 0 0
\(509\) 8186.17 0.712860 0.356430 0.934322i \(-0.383994\pi\)
0.356430 + 0.934322i \(0.383994\pi\)
\(510\) 0 0
\(511\) −2890.31 −0.250215
\(512\) 0 0
\(513\) −3139.59 −0.270207
\(514\) 0 0
\(515\) −436.104 −0.0373147
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 8300.85 0.702056
\(520\) 0 0
\(521\) −6788.92 −0.570879 −0.285440 0.958397i \(-0.592140\pi\)
−0.285440 + 0.958397i \(0.592140\pi\)
\(522\) 0 0
\(523\) 12912.6 1.07959 0.539797 0.841795i \(-0.318501\pi\)
0.539797 + 0.841795i \(0.318501\pi\)
\(524\) 0 0
\(525\) −6614.06 −0.549831
\(526\) 0 0
\(527\) 41654.8 3.44310
\(528\) 0 0
\(529\) −11465.2 −0.942317
\(530\) 0 0
\(531\) 3993.61 0.326380
\(532\) 0 0
\(533\) 20979.8 1.70495
\(534\) 0 0
\(535\) 1224.26 0.0989333
\(536\) 0 0
\(537\) 7922.28 0.636633
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8014.83 −0.636940 −0.318470 0.947933i \(-0.603169\pi\)
−0.318470 + 0.947933i \(0.603169\pi\)
\(542\) 0 0
\(543\) −13198.7 −1.04311
\(544\) 0 0
\(545\) 1318.80 0.103654
\(546\) 0 0
\(547\) −9922.03 −0.775567 −0.387784 0.921750i \(-0.626759\pi\)
−0.387784 + 0.921750i \(0.626759\pi\)
\(548\) 0 0
\(549\) −4965.31 −0.386000
\(550\) 0 0
\(551\) 20895.9 1.61560
\(552\) 0 0
\(553\) −2979.37 −0.229106
\(554\) 0 0
\(555\) 524.129 0.0400865
\(556\) 0 0
\(557\) 22180.3 1.68727 0.843636 0.536916i \(-0.180411\pi\)
0.843636 + 0.536916i \(0.180411\pi\)
\(558\) 0 0
\(559\) −9895.58 −0.748727
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16205.7 −1.21312 −0.606562 0.795036i \(-0.707452\pi\)
−0.606562 + 0.795036i \(0.707452\pi\)
\(564\) 0 0
\(565\) −1171.58 −0.0872365
\(566\) 0 0
\(567\) 1439.73 0.106637
\(568\) 0 0
\(569\) 17252.2 1.27109 0.635543 0.772065i \(-0.280776\pi\)
0.635543 + 0.772065i \(0.280776\pi\)
\(570\) 0 0
\(571\) 3002.90 0.220083 0.110042 0.993927i \(-0.464902\pi\)
0.110042 + 0.993927i \(0.464902\pi\)
\(572\) 0 0
\(573\) −4727.81 −0.344689
\(574\) 0 0
\(575\) 3285.98 0.238322
\(576\) 0 0
\(577\) −11249.6 −0.811659 −0.405830 0.913949i \(-0.633017\pi\)
−0.405830 + 0.913949i \(0.633017\pi\)
\(578\) 0 0
\(579\) 561.639 0.0403124
\(580\) 0 0
\(581\) −6356.17 −0.453870
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −546.778 −0.0386436
\(586\) 0 0
\(587\) 12915.4 0.908139 0.454070 0.890966i \(-0.349972\pi\)
0.454070 + 0.890966i \(0.349972\pi\)
\(588\) 0 0
\(589\) 37202.4 2.60255
\(590\) 0 0
\(591\) −4395.20 −0.305913
\(592\) 0 0
\(593\) 8524.23 0.590300 0.295150 0.955451i \(-0.404630\pi\)
0.295150 + 0.955451i \(0.404630\pi\)
\(594\) 0 0
\(595\) 2271.44 0.156504
\(596\) 0 0
\(597\) 10559.5 0.723905
\(598\) 0 0
\(599\) 9761.84 0.665873 0.332936 0.942949i \(-0.391961\pi\)
0.332936 + 0.942949i \(0.391961\pi\)
\(600\) 0 0
\(601\) −3962.71 −0.268956 −0.134478 0.990917i \(-0.542936\pi\)
−0.134478 + 0.990917i \(0.542936\pi\)
\(602\) 0 0
\(603\) 4088.38 0.276105
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −27291.7 −1.82493 −0.912467 0.409151i \(-0.865825\pi\)
−0.912467 + 0.409151i \(0.865825\pi\)
\(608\) 0 0
\(609\) −9582.32 −0.637595
\(610\) 0 0
\(611\) 6124.55 0.405520
\(612\) 0 0
\(613\) 9202.01 0.606306 0.303153 0.952942i \(-0.401961\pi\)
0.303153 + 0.952942i \(0.401961\pi\)
\(614\) 0 0
\(615\) −998.059 −0.0654400
\(616\) 0 0
\(617\) −21019.6 −1.37150 −0.685752 0.727835i \(-0.740526\pi\)
−0.685752 + 0.727835i \(0.740526\pi\)
\(618\) 0 0
\(619\) −28268.6 −1.83556 −0.917778 0.397094i \(-0.870019\pi\)
−0.917778 + 0.397094i \(0.870019\pi\)
\(620\) 0 0
\(621\) −715.285 −0.0462213
\(622\) 0 0
\(623\) 21599.3 1.38902
\(624\) 0 0
\(625\) 15264.7 0.976938
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23175.0 1.46907
\(630\) 0 0
\(631\) 17270.9 1.08961 0.544806 0.838562i \(-0.316603\pi\)
0.544806 + 0.838562i \(0.316603\pi\)
\(632\) 0 0
\(633\) 1514.54 0.0950988
\(634\) 0 0
\(635\) 628.814 0.0392972
\(636\) 0 0
\(637\) −1675.39 −0.104210
\(638\) 0 0
\(639\) −1928.31 −0.119379
\(640\) 0 0
\(641\) −23785.2 −1.46562 −0.732809 0.680435i \(-0.761791\pi\)
−0.732809 + 0.680435i \(0.761791\pi\)
\(642\) 0 0
\(643\) 5723.75 0.351046 0.175523 0.984475i \(-0.443838\pi\)
0.175523 + 0.984475i \(0.443838\pi\)
\(644\) 0 0
\(645\) 470.756 0.0287380
\(646\) 0 0
\(647\) 11170.0 0.678731 0.339366 0.940655i \(-0.389788\pi\)
0.339366 + 0.940655i \(0.389788\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −17060.0 −1.02709
\(652\) 0 0
\(653\) 31473.7 1.88615 0.943077 0.332573i \(-0.107917\pi\)
0.943077 + 0.332573i \(0.107917\pi\)
\(654\) 0 0
\(655\) 372.341 0.0222115
\(656\) 0 0
\(657\) −1463.49 −0.0869044
\(658\) 0 0
\(659\) −20262.0 −1.19772 −0.598858 0.800855i \(-0.704379\pi\)
−0.598858 + 0.800855i \(0.704379\pi\)
\(660\) 0 0
\(661\) −6117.04 −0.359948 −0.179974 0.983671i \(-0.557601\pi\)
−0.179974 + 0.983671i \(0.557601\pi\)
\(662\) 0 0
\(663\) −24176.4 −1.41619
\(664\) 0 0
\(665\) 2028.65 0.118297
\(666\) 0 0
\(667\) 4760.67 0.276362
\(668\) 0 0
\(669\) −1799.19 −0.103977
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 15872.7 0.909137 0.454569 0.890712i \(-0.349793\pi\)
0.454569 + 0.890712i \(0.349793\pi\)
\(674\) 0 0
\(675\) −3348.99 −0.190967
\(676\) 0 0
\(677\) 18848.6 1.07003 0.535016 0.844842i \(-0.320305\pi\)
0.535016 + 0.844842i \(0.320305\pi\)
\(678\) 0 0
\(679\) −19862.9 −1.12263
\(680\) 0 0
\(681\) 7021.39 0.395096
\(682\) 0 0
\(683\) 2430.48 0.136164 0.0680819 0.997680i \(-0.478312\pi\)
0.0680819 + 0.997680i \(0.478312\pi\)
\(684\) 0 0
\(685\) 1171.12 0.0653232
\(686\) 0 0
\(687\) −10397.6 −0.577430
\(688\) 0 0
\(689\) 34828.7 1.92579
\(690\) 0 0
\(691\) 28647.5 1.57714 0.788568 0.614947i \(-0.210823\pi\)
0.788568 + 0.614947i \(0.210823\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2890.97 0.157785
\(696\) 0 0
\(697\) −44130.4 −2.39822
\(698\) 0 0
\(699\) 8928.54 0.483131
\(700\) 0 0
\(701\) 3160.02 0.170260 0.0851300 0.996370i \(-0.472869\pi\)
0.0851300 + 0.996370i \(0.472869\pi\)
\(702\) 0 0
\(703\) 20697.9 1.11043
\(704\) 0 0
\(705\) −291.359 −0.0155649
\(706\) 0 0
\(707\) −7209.43 −0.383506
\(708\) 0 0
\(709\) 1450.00 0.0768066 0.0384033 0.999262i \(-0.487773\pi\)
0.0384033 + 0.999262i \(0.487773\pi\)
\(710\) 0 0
\(711\) −1508.58 −0.0795729
\(712\) 0 0
\(713\) 8475.74 0.445188
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13138.1 0.684313
\(718\) 0 0
\(719\) 8831.15 0.458062 0.229031 0.973419i \(-0.426444\pi\)
0.229031 + 0.973419i \(0.426444\pi\)
\(720\) 0 0
\(721\) 7897.44 0.407928
\(722\) 0 0
\(723\) −12135.1 −0.624219
\(724\) 0 0
\(725\) 22289.6 1.14181
\(726\) 0 0
\(727\) 20055.4 1.02313 0.511565 0.859245i \(-0.329066\pi\)
0.511565 + 0.859245i \(0.329066\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 20815.0 1.05318
\(732\) 0 0
\(733\) 8074.12 0.406854 0.203427 0.979090i \(-0.434792\pi\)
0.203427 + 0.979090i \(0.434792\pi\)
\(734\) 0 0
\(735\) 79.7023 0.00399982
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 3891.01 0.193685 0.0968425 0.995300i \(-0.469126\pi\)
0.0968425 + 0.995300i \(0.469126\pi\)
\(740\) 0 0
\(741\) −21592.3 −1.07046
\(742\) 0 0
\(743\) 20048.6 0.989921 0.494960 0.868916i \(-0.335183\pi\)
0.494960 + 0.868916i \(0.335183\pi\)
\(744\) 0 0
\(745\) 3244.59 0.159560
\(746\) 0 0
\(747\) −3218.41 −0.157638
\(748\) 0 0
\(749\) −22170.2 −1.08155
\(750\) 0 0
\(751\) 23590.4 1.14624 0.573120 0.819472i \(-0.305733\pi\)
0.573120 + 0.819472i \(0.305733\pi\)
\(752\) 0 0
\(753\) −20589.0 −0.996419
\(754\) 0 0
\(755\) −1639.88 −0.0790481
\(756\) 0 0
\(757\) 13443.2 0.645446 0.322723 0.946493i \(-0.395402\pi\)
0.322723 + 0.946493i \(0.395402\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22458.3 1.06980 0.534898 0.844917i \(-0.320350\pi\)
0.534898 + 0.844917i \(0.320350\pi\)
\(762\) 0 0
\(763\) −23882.3 −1.13315
\(764\) 0 0
\(765\) 1150.13 0.0543569
\(766\) 0 0
\(767\) 27465.7 1.29300
\(768\) 0 0
\(769\) 32749.5 1.53573 0.767865 0.640612i \(-0.221319\pi\)
0.767865 + 0.640612i \(0.221319\pi\)
\(770\) 0 0
\(771\) 19274.2 0.900316
\(772\) 0 0
\(773\) −1660.59 −0.0772671 −0.0386335 0.999253i \(-0.512300\pi\)
−0.0386335 + 0.999253i \(0.512300\pi\)
\(774\) 0 0
\(775\) 39683.7 1.83933
\(776\) 0 0
\(777\) −9491.49 −0.438231
\(778\) 0 0
\(779\) −39413.4 −1.81275
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −4851.95 −0.221449
\(784\) 0 0
\(785\) −551.027 −0.0250535
\(786\) 0 0
\(787\) −41436.3 −1.87680 −0.938402 0.345545i \(-0.887694\pi\)
−0.938402 + 0.345545i \(0.887694\pi\)
\(788\) 0 0
\(789\) −19533.0 −0.881359
\(790\) 0 0
\(791\) 21216.2 0.953679
\(792\) 0 0
\(793\) −34148.5 −1.52919
\(794\) 0 0
\(795\) −1656.88 −0.0739164
\(796\) 0 0
\(797\) 23056.3 1.02471 0.512356 0.858773i \(-0.328773\pi\)
0.512356 + 0.858773i \(0.328773\pi\)
\(798\) 0 0
\(799\) −12882.8 −0.570414
\(800\) 0 0
\(801\) 10936.7 0.482432
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 462.183 0.0202358
\(806\) 0 0
\(807\) 15088.6 0.658170
\(808\) 0 0
\(809\) −15626.1 −0.679091 −0.339546 0.940590i \(-0.610273\pi\)
−0.339546 + 0.940590i \(0.610273\pi\)
\(810\) 0 0
\(811\) −19198.1 −0.831241 −0.415620 0.909538i \(-0.636436\pi\)
−0.415620 + 0.909538i \(0.636436\pi\)
\(812\) 0 0
\(813\) 2702.03 0.116561
\(814\) 0 0
\(815\) −3728.82 −0.160264
\(816\) 0 0
\(817\) 18590.2 0.796068
\(818\) 0 0
\(819\) 9901.64 0.422456
\(820\) 0 0
\(821\) 16425.9 0.698257 0.349129 0.937075i \(-0.386478\pi\)
0.349129 + 0.937075i \(0.386478\pi\)
\(822\) 0 0
\(823\) −11524.3 −0.488107 −0.244054 0.969762i \(-0.578477\pi\)
−0.244054 + 0.969762i \(0.578477\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32655.7 −1.37310 −0.686548 0.727084i \(-0.740875\pi\)
−0.686548 + 0.727084i \(0.740875\pi\)
\(828\) 0 0
\(829\) −15368.2 −0.643861 −0.321930 0.946763i \(-0.604332\pi\)
−0.321930 + 0.946763i \(0.604332\pi\)
\(830\) 0 0
\(831\) 13904.9 0.580451
\(832\) 0 0
\(833\) 3524.13 0.146583
\(834\) 0 0
\(835\) −680.791 −0.0282153
\(836\) 0 0
\(837\) −8638.25 −0.356728
\(838\) 0 0
\(839\) −40339.9 −1.65994 −0.829969 0.557809i \(-0.811642\pi\)
−0.829969 + 0.557809i \(0.811642\pi\)
\(840\) 0 0
\(841\) 7903.72 0.324069
\(842\) 0 0
\(843\) −2804.68 −0.114589
\(844\) 0 0
\(845\) −1604.01 −0.0653014
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4007.75 0.162009
\(850\) 0 0
\(851\) 4715.54 0.189949
\(852\) 0 0
\(853\) 12664.0 0.508331 0.254166 0.967161i \(-0.418199\pi\)
0.254166 + 0.967161i \(0.418199\pi\)
\(854\) 0 0
\(855\) 1027.19 0.0410869
\(856\) 0 0
\(857\) 16651.8 0.663729 0.331865 0.943327i \(-0.392322\pi\)
0.331865 + 0.943327i \(0.392322\pi\)
\(858\) 0 0
\(859\) 23620.4 0.938205 0.469103 0.883144i \(-0.344577\pi\)
0.469103 + 0.883144i \(0.344577\pi\)
\(860\) 0 0
\(861\) 18073.9 0.715398
\(862\) 0 0
\(863\) −19172.7 −0.756252 −0.378126 0.925754i \(-0.623431\pi\)
−0.378126 + 0.925754i \(0.623431\pi\)
\(864\) 0 0
\(865\) −2715.83 −0.106752
\(866\) 0 0
\(867\) 36115.4 1.41470
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 28117.5 1.09383
\(872\) 0 0
\(873\) −10057.5 −0.389912
\(874\) 0 0
\(875\) 4344.71 0.167861
\(876\) 0 0
\(877\) −6727.62 −0.259037 −0.129519 0.991577i \(-0.541343\pi\)
−0.129519 + 0.991577i \(0.541343\pi\)
\(878\) 0 0
\(879\) −8918.47 −0.342221
\(880\) 0 0
\(881\) −38006.5 −1.45343 −0.726714 0.686940i \(-0.758953\pi\)
−0.726714 + 0.686940i \(0.758953\pi\)
\(882\) 0 0
\(883\) −34461.2 −1.31338 −0.656688 0.754163i \(-0.728043\pi\)
−0.656688 + 0.754163i \(0.728043\pi\)
\(884\) 0 0
\(885\) −1306.61 −0.0496284
\(886\) 0 0
\(887\) −48302.1 −1.82844 −0.914220 0.405218i \(-0.867196\pi\)
−0.914220 + 0.405218i \(0.867196\pi\)
\(888\) 0 0
\(889\) −11387.2 −0.429601
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11505.8 −0.431160
\(894\) 0 0
\(895\) −2591.97 −0.0968045
\(896\) 0 0
\(897\) −4919.31 −0.183112
\(898\) 0 0
\(899\) 57492.9 2.13292
\(900\) 0 0
\(901\) −73261.0 −2.70885
\(902\) 0 0
\(903\) −8524.95 −0.314167
\(904\) 0 0
\(905\) 4318.27 0.158612
\(906\) 0 0
\(907\) 14658.4 0.536631 0.268316 0.963331i \(-0.413533\pi\)
0.268316 + 0.963331i \(0.413533\pi\)
\(908\) 0 0
\(909\) −3650.45 −0.133199
\(910\) 0 0
\(911\) −47552.5 −1.72940 −0.864701 0.502286i \(-0.832492\pi\)
−0.864701 + 0.502286i \(0.832492\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 1624.52 0.0586941
\(916\) 0 0
\(917\) −6742.74 −0.242819
\(918\) 0 0
\(919\) 8911.32 0.319867 0.159933 0.987128i \(-0.448872\pi\)
0.159933 + 0.987128i \(0.448872\pi\)
\(920\) 0 0
\(921\) −1933.88 −0.0691897
\(922\) 0 0
\(923\) −13261.8 −0.472934
\(924\) 0 0
\(925\) 22078.3 0.784789
\(926\) 0 0
\(927\) 3998.82 0.141681
\(928\) 0 0
\(929\) −8023.11 −0.283347 −0.141674 0.989913i \(-0.545248\pi\)
−0.141674 + 0.989913i \(0.545248\pi\)
\(930\) 0 0
\(931\) 3147.45 0.110798
\(932\) 0 0
\(933\) 13595.7 0.477067
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20712.2 0.722131 0.361065 0.932541i \(-0.382413\pi\)
0.361065 + 0.932541i \(0.382413\pi\)
\(938\) 0 0
\(939\) 13941.2 0.484509
\(940\) 0 0
\(941\) 39715.8 1.37588 0.687938 0.725770i \(-0.258516\pi\)
0.687938 + 0.725770i \(0.258516\pi\)
\(942\) 0 0
\(943\) −8979.45 −0.310086
\(944\) 0 0
\(945\) −471.044 −0.0162149
\(946\) 0 0
\(947\) −5331.12 −0.182934 −0.0914668 0.995808i \(-0.529156\pi\)
−0.0914668 + 0.995808i \(0.529156\pi\)
\(948\) 0 0
\(949\) −10065.0 −0.344283
\(950\) 0 0
\(951\) −1884.33 −0.0642518
\(952\) 0 0
\(953\) 44970.8 1.52859 0.764295 0.644866i \(-0.223087\pi\)
0.764295 + 0.644866i \(0.223087\pi\)
\(954\) 0 0
\(955\) 1546.82 0.0524124
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −21208.0 −0.714120
\(960\) 0 0
\(961\) 72567.5 2.43589
\(962\) 0 0
\(963\) −11225.7 −0.375643
\(964\) 0 0
\(965\) −183.754 −0.00612979
\(966\) 0 0
\(967\) 15679.1 0.521412 0.260706 0.965418i \(-0.416045\pi\)
0.260706 + 0.965418i \(0.416045\pi\)
\(968\) 0 0
\(969\) 45418.6 1.50573
\(970\) 0 0
\(971\) −2532.55 −0.0837008 −0.0418504 0.999124i \(-0.513325\pi\)
−0.0418504 + 0.999124i \(0.513325\pi\)
\(972\) 0 0
\(973\) −52352.8 −1.72493
\(974\) 0 0
\(975\) −23032.4 −0.756540
\(976\) 0 0
\(977\) 30233.1 0.990014 0.495007 0.868889i \(-0.335165\pi\)
0.495007 + 0.868889i \(0.335165\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −12092.6 −0.393566
\(982\) 0 0
\(983\) 16703.1 0.541960 0.270980 0.962585i \(-0.412652\pi\)
0.270980 + 0.962585i \(0.412652\pi\)
\(984\) 0 0
\(985\) 1438.00 0.0465162
\(986\) 0 0
\(987\) 5276.25 0.170157
\(988\) 0 0
\(989\) 4235.35 0.136174
\(990\) 0 0
\(991\) 13840.2 0.443640 0.221820 0.975088i \(-0.428800\pi\)
0.221820 + 0.975088i \(0.428800\pi\)
\(992\) 0 0
\(993\) −4843.34 −0.154782
\(994\) 0 0
\(995\) −3454.80 −0.110075
\(996\) 0 0
\(997\) 41518.7 1.31887 0.659434 0.751763i \(-0.270796\pi\)
0.659434 + 0.751763i \(0.270796\pi\)
\(998\) 0 0
\(999\) −4805.95 −0.152206
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 1452.4.a.t.1.3 6
11.7 odd 10 132.4.i.c.49.2 12
11.8 odd 10 132.4.i.c.97.2 yes 12
11.10 odd 2 1452.4.a.u.1.3 6
33.8 even 10 396.4.j.c.361.2 12
33.29 even 10 396.4.j.c.181.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.49.2 12 11.7 odd 10
132.4.i.c.97.2 yes 12 11.8 odd 10
396.4.j.c.181.2 12 33.29 even 10
396.4.j.c.361.2 12 33.8 even 10
1452.4.a.t.1.3 6 1.1 even 1 trivial
1452.4.a.u.1.3 6 11.10 odd 2