Properties

Label 6008.2.a.d.1.2
Level $6008$
Weight $2$
Character 6008.1
Self dual yes
Analytic conductor $47.974$
Analytic rank $0$
Dimension $49$
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] = [6008,2,Mod(1,6008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6008.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: \(47.9741215344\)
Analytic rank: \(0\)
Dimension: \(49\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 6008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.93504 q^{3} -2.06805 q^{5} +2.42601 q^{7} +5.61445 q^{9} +O(q^{10})\) \(q-2.93504 q^{3} -2.06805 q^{5} +2.42601 q^{7} +5.61445 q^{9} -2.57617 q^{11} -1.28339 q^{13} +6.06981 q^{15} +6.19094 q^{17} +1.69686 q^{19} -7.12044 q^{21} +6.28188 q^{23} -0.723164 q^{25} -7.67350 q^{27} +5.09394 q^{29} +5.11764 q^{31} +7.56115 q^{33} -5.01712 q^{35} -10.0401 q^{37} +3.76681 q^{39} -0.199780 q^{41} +8.26243 q^{43} -11.6110 q^{45} -5.60927 q^{47} -1.11445 q^{49} -18.1706 q^{51} -3.03567 q^{53} +5.32765 q^{55} -4.98033 q^{57} +3.92521 q^{59} +5.50032 q^{61} +13.6207 q^{63} +2.65412 q^{65} +5.34148 q^{67} -18.4376 q^{69} -4.73831 q^{71} +4.46890 q^{73} +2.12251 q^{75} -6.24982 q^{77} -0.736412 q^{79} +5.67866 q^{81} -7.29873 q^{83} -12.8032 q^{85} -14.9509 q^{87} +3.85260 q^{89} -3.11353 q^{91} -15.0205 q^{93} -3.50918 q^{95} -1.69150 q^{97} -14.4638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 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
\(3\) −2.93504 −1.69454 −0.847272 0.531159i \(-0.821757\pi\)
−0.847272 + 0.531159i \(0.821757\pi\)
\(4\) 0 0
\(5\) −2.06805 −0.924861 −0.462430 0.886656i \(-0.653022\pi\)
−0.462430 + 0.886656i \(0.653022\pi\)
\(6\) 0 0
\(7\) 2.42601 0.916947 0.458474 0.888708i \(-0.348396\pi\)
0.458474 + 0.888708i \(0.348396\pi\)
\(8\) 0 0
\(9\) 5.61445 1.87148
\(10\) 0 0
\(11\) −2.57617 −0.776744 −0.388372 0.921503i \(-0.626962\pi\)
−0.388372 + 0.921503i \(0.626962\pi\)
\(12\) 0 0
\(13\) −1.28339 −0.355949 −0.177975 0.984035i \(-0.556954\pi\)
−0.177975 + 0.984035i \(0.556954\pi\)
\(14\) 0 0
\(15\) 6.06981 1.56722
\(16\) 0 0
\(17\) 6.19094 1.50152 0.750762 0.660573i \(-0.229687\pi\)
0.750762 + 0.660573i \(0.229687\pi\)
\(18\) 0 0
\(19\) 1.69686 0.389285 0.194643 0.980874i \(-0.437645\pi\)
0.194643 + 0.980874i \(0.437645\pi\)
\(20\) 0 0
\(21\) −7.12044 −1.55381
\(22\) 0 0
\(23\) 6.28188 1.30986 0.654931 0.755689i \(-0.272698\pi\)
0.654931 + 0.755689i \(0.272698\pi\)
\(24\) 0 0
\(25\) −0.723164 −0.144633
\(26\) 0 0
\(27\) −7.67350 −1.47676
\(28\) 0 0
\(29\) 5.09394 0.945920 0.472960 0.881084i \(-0.343186\pi\)
0.472960 + 0.881084i \(0.343186\pi\)
\(30\) 0 0
\(31\) 5.11764 0.919156 0.459578 0.888137i \(-0.348001\pi\)
0.459578 + 0.888137i \(0.348001\pi\)
\(32\) 0 0
\(33\) 7.56115 1.31623
\(34\) 0 0
\(35\) −5.01712 −0.848048
\(36\) 0 0
\(37\) −10.0401 −1.65058 −0.825291 0.564707i \(-0.808989\pi\)
−0.825291 + 0.564707i \(0.808989\pi\)
\(38\) 0 0
\(39\) 3.76681 0.603172
\(40\) 0 0
\(41\) −0.199780 −0.0312003 −0.0156002 0.999878i \(-0.504966\pi\)
−0.0156002 + 0.999878i \(0.504966\pi\)
\(42\) 0 0
\(43\) 8.26243 1.26001 0.630005 0.776591i \(-0.283053\pi\)
0.630005 + 0.776591i \(0.283053\pi\)
\(44\) 0 0
\(45\) −11.6110 −1.73086
\(46\) 0 0
\(47\) −5.60927 −0.818197 −0.409098 0.912490i \(-0.634157\pi\)
−0.409098 + 0.912490i \(0.634157\pi\)
\(48\) 0 0
\(49\) −1.11445 −0.159208
\(50\) 0 0
\(51\) −18.1706 −2.54440
\(52\) 0 0
\(53\) −3.03567 −0.416982 −0.208491 0.978024i \(-0.566855\pi\)
−0.208491 + 0.978024i \(0.566855\pi\)
\(54\) 0 0
\(55\) 5.32765 0.718380
\(56\) 0 0
\(57\) −4.98033 −0.659661
\(58\) 0 0
\(59\) 3.92521 0.511019 0.255509 0.966807i \(-0.417757\pi\)
0.255509 + 0.966807i \(0.417757\pi\)
\(60\) 0 0
\(61\) 5.50032 0.704244 0.352122 0.935954i \(-0.385460\pi\)
0.352122 + 0.935954i \(0.385460\pi\)
\(62\) 0 0
\(63\) 13.6207 1.71605
\(64\) 0 0
\(65\) 2.65412 0.329203
\(66\) 0 0
\(67\) 5.34148 0.652565 0.326283 0.945272i \(-0.394204\pi\)
0.326283 + 0.945272i \(0.394204\pi\)
\(68\) 0 0
\(69\) −18.4376 −2.21962
\(70\) 0 0
\(71\) −4.73831 −0.562334 −0.281167 0.959659i \(-0.590722\pi\)
−0.281167 + 0.959659i \(0.590722\pi\)
\(72\) 0 0
\(73\) 4.46890 0.523046 0.261523 0.965197i \(-0.415775\pi\)
0.261523 + 0.965197i \(0.415775\pi\)
\(74\) 0 0
\(75\) 2.12251 0.245087
\(76\) 0 0
\(77\) −6.24982 −0.712233
\(78\) 0 0
\(79\) −0.736412 −0.0828528 −0.0414264 0.999142i \(-0.513190\pi\)
−0.0414264 + 0.999142i \(0.513190\pi\)
\(80\) 0 0
\(81\) 5.67866 0.630962
\(82\) 0 0
\(83\) −7.29873 −0.801139 −0.400570 0.916266i \(-0.631188\pi\)
−0.400570 + 0.916266i \(0.631188\pi\)
\(84\) 0 0
\(85\) −12.8032 −1.38870
\(86\) 0 0
\(87\) −14.9509 −1.60290
\(88\) 0 0
\(89\) 3.85260 0.408375 0.204187 0.978932i \(-0.434545\pi\)
0.204187 + 0.978932i \(0.434545\pi\)
\(90\) 0 0
\(91\) −3.11353 −0.326387
\(92\) 0 0
\(93\) −15.0205 −1.55755
\(94\) 0 0
\(95\) −3.50918 −0.360035
\(96\) 0 0
\(97\) −1.69150 −0.171746 −0.0858728 0.996306i \(-0.527368\pi\)
−0.0858728 + 0.996306i \(0.527368\pi\)
\(98\) 0 0
\(99\) −14.4638 −1.45366
\(100\) 0 0
\(101\) −6.23235 −0.620142 −0.310071 0.950713i \(-0.600353\pi\)
−0.310071 + 0.950713i \(0.600353\pi\)
\(102\) 0 0
\(103\) 5.79770 0.571264 0.285632 0.958339i \(-0.407796\pi\)
0.285632 + 0.958339i \(0.407796\pi\)
\(104\) 0 0
\(105\) 14.7254 1.43706
\(106\) 0 0
\(107\) 2.30489 0.222822 0.111411 0.993774i \(-0.464463\pi\)
0.111411 + 0.993774i \(0.464463\pi\)
\(108\) 0 0
\(109\) −2.34406 −0.224520 −0.112260 0.993679i \(-0.535809\pi\)
−0.112260 + 0.993679i \(0.535809\pi\)
\(110\) 0 0
\(111\) 29.4681 2.79699
\(112\) 0 0
\(113\) −5.06566 −0.476537 −0.238269 0.971199i \(-0.576580\pi\)
−0.238269 + 0.971199i \(0.576580\pi\)
\(114\) 0 0
\(115\) −12.9912 −1.21144
\(116\) 0 0
\(117\) −7.20554 −0.666152
\(118\) 0 0
\(119\) 15.0193 1.37682
\(120\) 0 0
\(121\) −4.36335 −0.396668
\(122\) 0 0
\(123\) 0.586361 0.0528704
\(124\) 0 0
\(125\) 11.8358 1.05863
\(126\) 0 0
\(127\) 4.81718 0.427456 0.213728 0.976893i \(-0.431439\pi\)
0.213728 + 0.976893i \(0.431439\pi\)
\(128\) 0 0
\(129\) −24.2506 −2.13514
\(130\) 0 0
\(131\) −12.7616 −1.11499 −0.557494 0.830181i \(-0.688237\pi\)
−0.557494 + 0.830181i \(0.688237\pi\)
\(132\) 0 0
\(133\) 4.11659 0.356954
\(134\) 0 0
\(135\) 15.8692 1.36580
\(136\) 0 0
\(137\) −8.55453 −0.730863 −0.365431 0.930838i \(-0.619078\pi\)
−0.365431 + 0.930838i \(0.619078\pi\)
\(138\) 0 0
\(139\) 0.655924 0.0556348 0.0278174 0.999613i \(-0.491144\pi\)
0.0278174 + 0.999613i \(0.491144\pi\)
\(140\) 0 0
\(141\) 16.4634 1.38647
\(142\) 0 0
\(143\) 3.30624 0.276481
\(144\) 0 0
\(145\) −10.5345 −0.874844
\(146\) 0 0
\(147\) 3.27097 0.269785
\(148\) 0 0
\(149\) 4.95189 0.405675 0.202837 0.979212i \(-0.434984\pi\)
0.202837 + 0.979212i \(0.434984\pi\)
\(150\) 0 0
\(151\) −1.55315 −0.126393 −0.0631966 0.998001i \(-0.520130\pi\)
−0.0631966 + 0.998001i \(0.520130\pi\)
\(152\) 0 0
\(153\) 34.7587 2.81007
\(154\) 0 0
\(155\) −10.5836 −0.850091
\(156\) 0 0
\(157\) −4.22482 −0.337178 −0.168589 0.985686i \(-0.553921\pi\)
−0.168589 + 0.985686i \(0.553921\pi\)
\(158\) 0 0
\(159\) 8.90982 0.706595
\(160\) 0 0
\(161\) 15.2399 1.20107
\(162\) 0 0
\(163\) 19.5311 1.52979 0.764895 0.644155i \(-0.222791\pi\)
0.764895 + 0.644155i \(0.222791\pi\)
\(164\) 0 0
\(165\) −15.6369 −1.21733
\(166\) 0 0
\(167\) −20.6163 −1.59533 −0.797667 0.603098i \(-0.793933\pi\)
−0.797667 + 0.603098i \(0.793933\pi\)
\(168\) 0 0
\(169\) −11.3529 −0.873300
\(170\) 0 0
\(171\) 9.52690 0.728540
\(172\) 0 0
\(173\) −12.1882 −0.926649 −0.463325 0.886189i \(-0.653344\pi\)
−0.463325 + 0.886189i \(0.653344\pi\)
\(174\) 0 0
\(175\) −1.75441 −0.132621
\(176\) 0 0
\(177\) −11.5206 −0.865944
\(178\) 0 0
\(179\) 9.39333 0.702091 0.351045 0.936358i \(-0.385826\pi\)
0.351045 + 0.936358i \(0.385826\pi\)
\(180\) 0 0
\(181\) −8.09033 −0.601349 −0.300675 0.953727i \(-0.597212\pi\)
−0.300675 + 0.953727i \(0.597212\pi\)
\(182\) 0 0
\(183\) −16.1437 −1.19337
\(184\) 0 0
\(185\) 20.7634 1.52656
\(186\) 0 0
\(187\) −15.9489 −1.16630
\(188\) 0 0
\(189\) −18.6160 −1.35412
\(190\) 0 0
\(191\) 14.4868 1.04823 0.524114 0.851648i \(-0.324397\pi\)
0.524114 + 0.851648i \(0.324397\pi\)
\(192\) 0 0
\(193\) −2.99199 −0.215368 −0.107684 0.994185i \(-0.534344\pi\)
−0.107684 + 0.994185i \(0.534344\pi\)
\(194\) 0 0
\(195\) −7.78995 −0.557850
\(196\) 0 0
\(197\) −5.65268 −0.402737 −0.201368 0.979516i \(-0.564539\pi\)
−0.201368 + 0.979516i \(0.564539\pi\)
\(198\) 0 0
\(199\) 19.1790 1.35956 0.679782 0.733414i \(-0.262075\pi\)
0.679782 + 0.733414i \(0.262075\pi\)
\(200\) 0 0
\(201\) −15.6774 −1.10580
\(202\) 0 0
\(203\) 12.3580 0.867359
\(204\) 0 0
\(205\) 0.413155 0.0288560
\(206\) 0 0
\(207\) 35.2693 2.45138
\(208\) 0 0
\(209\) −4.37139 −0.302375
\(210\) 0 0
\(211\) 7.73576 0.532551 0.266276 0.963897i \(-0.414207\pi\)
0.266276 + 0.963897i \(0.414207\pi\)
\(212\) 0 0
\(213\) 13.9071 0.952901
\(214\) 0 0
\(215\) −17.0871 −1.16533
\(216\) 0 0
\(217\) 12.4155 0.842818
\(218\) 0 0
\(219\) −13.1164 −0.886324
\(220\) 0 0
\(221\) −7.94541 −0.534466
\(222\) 0 0
\(223\) 4.46039 0.298690 0.149345 0.988785i \(-0.452284\pi\)
0.149345 + 0.988785i \(0.452284\pi\)
\(224\) 0 0
\(225\) −4.06016 −0.270678
\(226\) 0 0
\(227\) −0.693983 −0.0460613 −0.0230306 0.999735i \(-0.507332\pi\)
−0.0230306 + 0.999735i \(0.507332\pi\)
\(228\) 0 0
\(229\) −12.8794 −0.851092 −0.425546 0.904937i \(-0.639918\pi\)
−0.425546 + 0.904937i \(0.639918\pi\)
\(230\) 0 0
\(231\) 18.3435 1.20691
\(232\) 0 0
\(233\) −15.9830 −1.04708 −0.523540 0.852001i \(-0.675389\pi\)
−0.523540 + 0.852001i \(0.675389\pi\)
\(234\) 0 0
\(235\) 11.6003 0.756718
\(236\) 0 0
\(237\) 2.16140 0.140398
\(238\) 0 0
\(239\) 6.74647 0.436393 0.218196 0.975905i \(-0.429983\pi\)
0.218196 + 0.975905i \(0.429983\pi\)
\(240\) 0 0
\(241\) 16.2087 1.04409 0.522047 0.852917i \(-0.325169\pi\)
0.522047 + 0.852917i \(0.325169\pi\)
\(242\) 0 0
\(243\) 6.35340 0.407571
\(244\) 0 0
\(245\) 2.30475 0.147245
\(246\) 0 0
\(247\) −2.17773 −0.138566
\(248\) 0 0
\(249\) 21.4220 1.35757
\(250\) 0 0
\(251\) 3.62965 0.229101 0.114551 0.993417i \(-0.463457\pi\)
0.114551 + 0.993417i \(0.463457\pi\)
\(252\) 0 0
\(253\) −16.1832 −1.01743
\(254\) 0 0
\(255\) 37.5778 2.35322
\(256\) 0 0
\(257\) 21.4727 1.33943 0.669716 0.742618i \(-0.266416\pi\)
0.669716 + 0.742618i \(0.266416\pi\)
\(258\) 0 0
\(259\) −24.3574 −1.51350
\(260\) 0 0
\(261\) 28.5996 1.77027
\(262\) 0 0
\(263\) −0.124008 −0.00764668 −0.00382334 0.999993i \(-0.501217\pi\)
−0.00382334 + 0.999993i \(0.501217\pi\)
\(264\) 0 0
\(265\) 6.27793 0.385650
\(266\) 0 0
\(267\) −11.3075 −0.692009
\(268\) 0 0
\(269\) −8.24251 −0.502555 −0.251277 0.967915i \(-0.580851\pi\)
−0.251277 + 0.967915i \(0.580851\pi\)
\(270\) 0 0
\(271\) 25.9873 1.57861 0.789307 0.613998i \(-0.210440\pi\)
0.789307 + 0.613998i \(0.210440\pi\)
\(272\) 0 0
\(273\) 9.13833 0.553077
\(274\) 0 0
\(275\) 1.86299 0.112343
\(276\) 0 0
\(277\) 1.14163 0.0685941 0.0342971 0.999412i \(-0.489081\pi\)
0.0342971 + 0.999412i \(0.489081\pi\)
\(278\) 0 0
\(279\) 28.7327 1.72018
\(280\) 0 0
\(281\) 5.18883 0.309540 0.154770 0.987951i \(-0.450536\pi\)
0.154770 + 0.987951i \(0.450536\pi\)
\(282\) 0 0
\(283\) 18.4060 1.09412 0.547062 0.837092i \(-0.315746\pi\)
0.547062 + 0.837092i \(0.315746\pi\)
\(284\) 0 0
\(285\) 10.2996 0.610095
\(286\) 0 0
\(287\) −0.484668 −0.0286091
\(288\) 0 0
\(289\) 21.3278 1.25457
\(290\) 0 0
\(291\) 4.96461 0.291031
\(292\) 0 0
\(293\) 30.4604 1.77952 0.889758 0.456432i \(-0.150873\pi\)
0.889758 + 0.456432i \(0.150873\pi\)
\(294\) 0 0
\(295\) −8.11753 −0.472621
\(296\) 0 0
\(297\) 19.7682 1.14707
\(298\) 0 0
\(299\) −8.06212 −0.466244
\(300\) 0 0
\(301\) 20.0448 1.15536
\(302\) 0 0
\(303\) 18.2922 1.05086
\(304\) 0 0
\(305\) −11.3749 −0.651328
\(306\) 0 0
\(307\) 0.441213 0.0251814 0.0125907 0.999921i \(-0.495992\pi\)
0.0125907 + 0.999921i \(0.495992\pi\)
\(308\) 0 0
\(309\) −17.0165 −0.968033
\(310\) 0 0
\(311\) 0.426103 0.0241621 0.0120810 0.999927i \(-0.496154\pi\)
0.0120810 + 0.999927i \(0.496154\pi\)
\(312\) 0 0
\(313\) 10.6708 0.603149 0.301575 0.953443i \(-0.402488\pi\)
0.301575 + 0.953443i \(0.402488\pi\)
\(314\) 0 0
\(315\) −28.1684 −1.58711
\(316\) 0 0
\(317\) 25.3371 1.42308 0.711538 0.702648i \(-0.247999\pi\)
0.711538 + 0.702648i \(0.247999\pi\)
\(318\) 0 0
\(319\) −13.1228 −0.734738
\(320\) 0 0
\(321\) −6.76494 −0.377582
\(322\) 0 0
\(323\) 10.5051 0.584521
\(324\) 0 0
\(325\) 0.928103 0.0514819
\(326\) 0 0
\(327\) 6.87991 0.380460
\(328\) 0 0
\(329\) −13.6082 −0.750243
\(330\) 0 0
\(331\) 27.9722 1.53749 0.768745 0.639555i \(-0.220881\pi\)
0.768745 + 0.639555i \(0.220881\pi\)
\(332\) 0 0
\(333\) −56.3696 −3.08904
\(334\) 0 0
\(335\) −11.0465 −0.603532
\(336\) 0 0
\(337\) −0.747034 −0.0406935 −0.0203468 0.999793i \(-0.506477\pi\)
−0.0203468 + 0.999793i \(0.506477\pi\)
\(338\) 0 0
\(339\) 14.8679 0.807514
\(340\) 0 0
\(341\) −13.1839 −0.713949
\(342\) 0 0
\(343\) −19.6858 −1.06293
\(344\) 0 0
\(345\) 38.1298 2.05284
\(346\) 0 0
\(347\) −17.3371 −0.930703 −0.465351 0.885126i \(-0.654072\pi\)
−0.465351 + 0.885126i \(0.654072\pi\)
\(348\) 0 0
\(349\) −2.97310 −0.159147 −0.0795733 0.996829i \(-0.525356\pi\)
−0.0795733 + 0.996829i \(0.525356\pi\)
\(350\) 0 0
\(351\) 9.84811 0.525653
\(352\) 0 0
\(353\) 22.8009 1.21357 0.606784 0.794867i \(-0.292459\pi\)
0.606784 + 0.794867i \(0.292459\pi\)
\(354\) 0 0
\(355\) 9.79907 0.520081
\(356\) 0 0
\(357\) −44.0823 −2.33308
\(358\) 0 0
\(359\) −19.2242 −1.01461 −0.507306 0.861766i \(-0.669359\pi\)
−0.507306 + 0.861766i \(0.669359\pi\)
\(360\) 0 0
\(361\) −16.1207 −0.848457
\(362\) 0 0
\(363\) 12.8066 0.672172
\(364\) 0 0
\(365\) −9.24192 −0.483744
\(366\) 0 0
\(367\) 1.91709 0.100071 0.0500357 0.998747i \(-0.484066\pi\)
0.0500357 + 0.998747i \(0.484066\pi\)
\(368\) 0 0
\(369\) −1.12165 −0.0583909
\(370\) 0 0
\(371\) −7.36459 −0.382350
\(372\) 0 0
\(373\) 25.6220 1.32666 0.663328 0.748328i \(-0.269143\pi\)
0.663328 + 0.748328i \(0.269143\pi\)
\(374\) 0 0
\(375\) −34.7385 −1.79389
\(376\) 0 0
\(377\) −6.53752 −0.336699
\(378\) 0 0
\(379\) 36.1324 1.85600 0.927998 0.372585i \(-0.121528\pi\)
0.927998 + 0.372585i \(0.121528\pi\)
\(380\) 0 0
\(381\) −14.1386 −0.724343
\(382\) 0 0
\(383\) −8.79073 −0.449186 −0.224593 0.974453i \(-0.572105\pi\)
−0.224593 + 0.974453i \(0.572105\pi\)
\(384\) 0 0
\(385\) 12.9250 0.658717
\(386\) 0 0
\(387\) 46.3890 2.35808
\(388\) 0 0
\(389\) −35.0473 −1.77697 −0.888484 0.458907i \(-0.848241\pi\)
−0.888484 + 0.458907i \(0.848241\pi\)
\(390\) 0 0
\(391\) 38.8907 1.96679
\(392\) 0 0
\(393\) 37.4558 1.88940
\(394\) 0 0
\(395\) 1.52294 0.0766273
\(396\) 0 0
\(397\) 23.9137 1.20020 0.600098 0.799927i \(-0.295128\pi\)
0.600098 + 0.799927i \(0.295128\pi\)
\(398\) 0 0
\(399\) −12.0824 −0.604875
\(400\) 0 0
\(401\) −1.03703 −0.0517867 −0.0258933 0.999665i \(-0.508243\pi\)
−0.0258933 + 0.999665i \(0.508243\pi\)
\(402\) 0 0
\(403\) −6.56795 −0.327173
\(404\) 0 0
\(405\) −11.7438 −0.583552
\(406\) 0 0
\(407\) 25.8650 1.28208
\(408\) 0 0
\(409\) 8.86456 0.438325 0.219162 0.975688i \(-0.429668\pi\)
0.219162 + 0.975688i \(0.429668\pi\)
\(410\) 0 0
\(411\) 25.1079 1.23848
\(412\) 0 0
\(413\) 9.52261 0.468577
\(414\) 0 0
\(415\) 15.0941 0.740942
\(416\) 0 0
\(417\) −1.92516 −0.0942756
\(418\) 0 0
\(419\) 19.2682 0.941311 0.470656 0.882317i \(-0.344017\pi\)
0.470656 + 0.882317i \(0.344017\pi\)
\(420\) 0 0
\(421\) 16.3278 0.795768 0.397884 0.917436i \(-0.369745\pi\)
0.397884 + 0.917436i \(0.369745\pi\)
\(422\) 0 0
\(423\) −31.4930 −1.53124
\(424\) 0 0
\(425\) −4.47706 −0.217170
\(426\) 0 0
\(427\) 13.3439 0.645755
\(428\) 0 0
\(429\) −9.70393 −0.468510
\(430\) 0 0
\(431\) −8.37338 −0.403331 −0.201666 0.979454i \(-0.564635\pi\)
−0.201666 + 0.979454i \(0.564635\pi\)
\(432\) 0 0
\(433\) 0.553303 0.0265901 0.0132950 0.999912i \(-0.495768\pi\)
0.0132950 + 0.999912i \(0.495768\pi\)
\(434\) 0 0
\(435\) 30.9192 1.48246
\(436\) 0 0
\(437\) 10.6594 0.509910
\(438\) 0 0
\(439\) 23.9717 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(440\) 0 0
\(441\) −6.25704 −0.297954
\(442\) 0 0
\(443\) 33.9876 1.61480 0.807399 0.590005i \(-0.200874\pi\)
0.807399 + 0.590005i \(0.200874\pi\)
\(444\) 0 0
\(445\) −7.96737 −0.377690
\(446\) 0 0
\(447\) −14.5340 −0.687434
\(448\) 0 0
\(449\) 3.87173 0.182718 0.0913590 0.995818i \(-0.470879\pi\)
0.0913590 + 0.995818i \(0.470879\pi\)
\(450\) 0 0
\(451\) 0.514666 0.0242347
\(452\) 0 0
\(453\) 4.55854 0.214179
\(454\) 0 0
\(455\) 6.43894 0.301862
\(456\) 0 0
\(457\) 2.35380 0.110106 0.0550531 0.998483i \(-0.482467\pi\)
0.0550531 + 0.998483i \(0.482467\pi\)
\(458\) 0 0
\(459\) −47.5062 −2.21740
\(460\) 0 0
\(461\) 16.7486 0.780059 0.390030 0.920802i \(-0.372465\pi\)
0.390030 + 0.920802i \(0.372465\pi\)
\(462\) 0 0
\(463\) 22.2328 1.03324 0.516622 0.856213i \(-0.327189\pi\)
0.516622 + 0.856213i \(0.327189\pi\)
\(464\) 0 0
\(465\) 31.0631 1.44052
\(466\) 0 0
\(467\) 5.49920 0.254472 0.127236 0.991872i \(-0.459389\pi\)
0.127236 + 0.991872i \(0.459389\pi\)
\(468\) 0 0
\(469\) 12.9585 0.598368
\(470\) 0 0
\(471\) 12.4000 0.571363
\(472\) 0 0
\(473\) −21.2854 −0.978705
\(474\) 0 0
\(475\) −1.22710 −0.0563034
\(476\) 0 0
\(477\) −17.0436 −0.780374
\(478\) 0 0
\(479\) 2.98726 0.136491 0.0682456 0.997669i \(-0.478260\pi\)
0.0682456 + 0.997669i \(0.478260\pi\)
\(480\) 0 0
\(481\) 12.8854 0.587524
\(482\) 0 0
\(483\) −44.7298 −2.03527
\(484\) 0 0
\(485\) 3.49810 0.158841
\(486\) 0 0
\(487\) 35.0775 1.58951 0.794757 0.606928i \(-0.207598\pi\)
0.794757 + 0.606928i \(0.207598\pi\)
\(488\) 0 0
\(489\) −57.3244 −2.59230
\(490\) 0 0
\(491\) −34.1593 −1.54159 −0.770793 0.637086i \(-0.780140\pi\)
−0.770793 + 0.637086i \(0.780140\pi\)
\(492\) 0 0
\(493\) 31.5363 1.42032
\(494\) 0 0
\(495\) 29.9118 1.34444
\(496\) 0 0
\(497\) −11.4952 −0.515631
\(498\) 0 0
\(499\) −22.2523 −0.996150 −0.498075 0.867134i \(-0.665960\pi\)
−0.498075 + 0.867134i \(0.665960\pi\)
\(500\) 0 0
\(501\) 60.5095 2.70337
\(502\) 0 0
\(503\) −14.2409 −0.634971 −0.317486 0.948263i \(-0.602839\pi\)
−0.317486 + 0.948263i \(0.602839\pi\)
\(504\) 0 0
\(505\) 12.8888 0.573545
\(506\) 0 0
\(507\) 33.3212 1.47985
\(508\) 0 0
\(509\) 11.1870 0.495855 0.247928 0.968779i \(-0.420251\pi\)
0.247928 + 0.968779i \(0.420251\pi\)
\(510\) 0 0
\(511\) 10.8416 0.479605
\(512\) 0 0
\(513\) −13.0208 −0.574883
\(514\) 0 0
\(515\) −11.9899 −0.528340
\(516\) 0 0
\(517\) 14.4504 0.635530
\(518\) 0 0
\(519\) 35.7727 1.57025
\(520\) 0 0
\(521\) −31.2133 −1.36748 −0.683739 0.729727i \(-0.739647\pi\)
−0.683739 + 0.729727i \(0.739647\pi\)
\(522\) 0 0
\(523\) 5.71810 0.250035 0.125017 0.992155i \(-0.460101\pi\)
0.125017 + 0.992155i \(0.460101\pi\)
\(524\) 0 0
\(525\) 5.14925 0.224732
\(526\) 0 0
\(527\) 31.6830 1.38013
\(528\) 0 0
\(529\) 16.4620 0.715739
\(530\) 0 0
\(531\) 22.0379 0.956362
\(532\) 0 0
\(533\) 0.256396 0.0111057
\(534\) 0 0
\(535\) −4.76663 −0.206079
\(536\) 0 0
\(537\) −27.5698 −1.18972
\(538\) 0 0
\(539\) 2.87102 0.123664
\(540\) 0 0
\(541\) −12.4407 −0.534868 −0.267434 0.963576i \(-0.586176\pi\)
−0.267434 + 0.963576i \(0.586176\pi\)
\(542\) 0 0
\(543\) 23.7454 1.01901
\(544\) 0 0
\(545\) 4.84764 0.207650
\(546\) 0 0
\(547\) 25.6933 1.09857 0.549284 0.835636i \(-0.314901\pi\)
0.549284 + 0.835636i \(0.314901\pi\)
\(548\) 0 0
\(549\) 30.8813 1.31798
\(550\) 0 0
\(551\) 8.64367 0.368233
\(552\) 0 0
\(553\) −1.78655 −0.0759717
\(554\) 0 0
\(555\) −60.9415 −2.58682
\(556\) 0 0
\(557\) −16.5549 −0.701453 −0.350726 0.936478i \(-0.614065\pi\)
−0.350726 + 0.936478i \(0.614065\pi\)
\(558\) 0 0
\(559\) −10.6039 −0.448499
\(560\) 0 0
\(561\) 46.8107 1.97635
\(562\) 0 0
\(563\) 27.5804 1.16237 0.581187 0.813770i \(-0.302589\pi\)
0.581187 + 0.813770i \(0.302589\pi\)
\(564\) 0 0
\(565\) 10.4760 0.440731
\(566\) 0 0
\(567\) 13.7765 0.578559
\(568\) 0 0
\(569\) −19.7709 −0.828839 −0.414420 0.910086i \(-0.636015\pi\)
−0.414420 + 0.910086i \(0.636015\pi\)
\(570\) 0 0
\(571\) 8.71600 0.364753 0.182377 0.983229i \(-0.441621\pi\)
0.182377 + 0.983229i \(0.441621\pi\)
\(572\) 0 0
\(573\) −42.5193 −1.77627
\(574\) 0 0
\(575\) −4.54283 −0.189449
\(576\) 0 0
\(577\) −6.83706 −0.284631 −0.142315 0.989821i \(-0.545455\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(578\) 0 0
\(579\) 8.78161 0.364951
\(580\) 0 0
\(581\) −17.7068 −0.734603
\(582\) 0 0
\(583\) 7.82041 0.323888
\(584\) 0 0
\(585\) 14.9014 0.616098
\(586\) 0 0
\(587\) 12.2754 0.506660 0.253330 0.967380i \(-0.418474\pi\)
0.253330 + 0.967380i \(0.418474\pi\)
\(588\) 0 0
\(589\) 8.68390 0.357814
\(590\) 0 0
\(591\) 16.5908 0.682455
\(592\) 0 0
\(593\) −37.8978 −1.55627 −0.778137 0.628094i \(-0.783835\pi\)
−0.778137 + 0.628094i \(0.783835\pi\)
\(594\) 0 0
\(595\) −31.0607 −1.27337
\(596\) 0 0
\(597\) −56.2911 −2.30384
\(598\) 0 0
\(599\) 11.7004 0.478066 0.239033 0.971011i \(-0.423170\pi\)
0.239033 + 0.971011i \(0.423170\pi\)
\(600\) 0 0
\(601\) 3.21560 0.131167 0.0655835 0.997847i \(-0.479109\pi\)
0.0655835 + 0.997847i \(0.479109\pi\)
\(602\) 0 0
\(603\) 29.9894 1.22126
\(604\) 0 0
\(605\) 9.02364 0.366863
\(606\) 0 0
\(607\) 12.2564 0.497471 0.248736 0.968571i \(-0.419985\pi\)
0.248736 + 0.968571i \(0.419985\pi\)
\(608\) 0 0
\(609\) −36.2711 −1.46978
\(610\) 0 0
\(611\) 7.19890 0.291236
\(612\) 0 0
\(613\) 8.70143 0.351447 0.175724 0.984440i \(-0.443773\pi\)
0.175724 + 0.984440i \(0.443773\pi\)
\(614\) 0 0
\(615\) −1.21262 −0.0488977
\(616\) 0 0
\(617\) 19.7520 0.795185 0.397593 0.917562i \(-0.369846\pi\)
0.397593 + 0.917562i \(0.369846\pi\)
\(618\) 0 0
\(619\) −6.72356 −0.270243 −0.135121 0.990829i \(-0.543142\pi\)
−0.135121 + 0.990829i \(0.543142\pi\)
\(620\) 0 0
\(621\) −48.2040 −1.93436
\(622\) 0 0
\(623\) 9.34646 0.374458
\(624\) 0 0
\(625\) −20.8612 −0.834449
\(626\) 0 0
\(627\) 12.8302 0.512388
\(628\) 0 0
\(629\) −62.1577 −2.47839
\(630\) 0 0
\(631\) −13.6725 −0.544292 −0.272146 0.962256i \(-0.587733\pi\)
−0.272146 + 0.962256i \(0.587733\pi\)
\(632\) 0 0
\(633\) −22.7047 −0.902432
\(634\) 0 0
\(635\) −9.96218 −0.395337
\(636\) 0 0
\(637\) 1.43028 0.0566699
\(638\) 0 0
\(639\) −26.6030 −1.05240
\(640\) 0 0
\(641\) −23.7988 −0.939994 −0.469997 0.882668i \(-0.655745\pi\)
−0.469997 + 0.882668i \(0.655745\pi\)
\(642\) 0 0
\(643\) −48.6919 −1.92022 −0.960110 0.279624i \(-0.909790\pi\)
−0.960110 + 0.279624i \(0.909790\pi\)
\(644\) 0 0
\(645\) 50.1514 1.97471
\(646\) 0 0
\(647\) 12.5731 0.494298 0.247149 0.968977i \(-0.420506\pi\)
0.247149 + 0.968977i \(0.420506\pi\)
\(648\) 0 0
\(649\) −10.1120 −0.396931
\(650\) 0 0
\(651\) −36.4399 −1.42819
\(652\) 0 0
\(653\) −4.85923 −0.190156 −0.0950782 0.995470i \(-0.530310\pi\)
−0.0950782 + 0.995470i \(0.530310\pi\)
\(654\) 0 0
\(655\) 26.3917 1.03121
\(656\) 0 0
\(657\) 25.0904 0.978870
\(658\) 0 0
\(659\) −39.1649 −1.52565 −0.762824 0.646607i \(-0.776188\pi\)
−0.762824 + 0.646607i \(0.776188\pi\)
\(660\) 0 0
\(661\) 19.8068 0.770395 0.385197 0.922834i \(-0.374133\pi\)
0.385197 + 0.922834i \(0.374133\pi\)
\(662\) 0 0
\(663\) 23.3201 0.905677
\(664\) 0 0
\(665\) −8.51333 −0.330133
\(666\) 0 0
\(667\) 31.9995 1.23902
\(668\) 0 0
\(669\) −13.0914 −0.506143
\(670\) 0 0
\(671\) −14.1698 −0.547017
\(672\) 0 0
\(673\) −16.3830 −0.631517 −0.315759 0.948840i \(-0.602259\pi\)
−0.315759 + 0.948840i \(0.602259\pi\)
\(674\) 0 0
\(675\) 5.54919 0.213589
\(676\) 0 0
\(677\) −12.6860 −0.487564 −0.243782 0.969830i \(-0.578388\pi\)
−0.243782 + 0.969830i \(0.578388\pi\)
\(678\) 0 0
\(679\) −4.10360 −0.157482
\(680\) 0 0
\(681\) 2.03687 0.0780529
\(682\) 0 0
\(683\) −12.1906 −0.466460 −0.233230 0.972422i \(-0.574930\pi\)
−0.233230 + 0.972422i \(0.574930\pi\)
\(684\) 0 0
\(685\) 17.6912 0.675946
\(686\) 0 0
\(687\) 37.8014 1.44221
\(688\) 0 0
\(689\) 3.89596 0.148424
\(690\) 0 0
\(691\) 26.2060 0.996925 0.498463 0.866911i \(-0.333898\pi\)
0.498463 + 0.866911i \(0.333898\pi\)
\(692\) 0 0
\(693\) −35.0893 −1.33293
\(694\) 0 0
\(695\) −1.35649 −0.0514544
\(696\) 0 0
\(697\) −1.23682 −0.0468481
\(698\) 0 0
\(699\) 46.9107 1.77432
\(700\) 0 0
\(701\) −51.3595 −1.93982 −0.969911 0.243461i \(-0.921717\pi\)
−0.969911 + 0.243461i \(0.921717\pi\)
\(702\) 0 0
\(703\) −17.0366 −0.642547
\(704\) 0 0
\(705\) −34.0472 −1.28229
\(706\) 0 0
\(707\) −15.1198 −0.568637
\(708\) 0 0
\(709\) −26.8713 −1.00917 −0.504586 0.863361i \(-0.668355\pi\)
−0.504586 + 0.863361i \(0.668355\pi\)
\(710\) 0 0
\(711\) −4.13454 −0.155058
\(712\) 0 0
\(713\) 32.1484 1.20397
\(714\) 0 0
\(715\) −6.83747 −0.255707
\(716\) 0 0
\(717\) −19.8011 −0.739487
\(718\) 0 0
\(719\) 1.70033 0.0634115 0.0317058 0.999497i \(-0.489906\pi\)
0.0317058 + 0.999497i \(0.489906\pi\)
\(720\) 0 0
\(721\) 14.0653 0.523819
\(722\) 0 0
\(723\) −47.5731 −1.76926
\(724\) 0 0
\(725\) −3.68375 −0.136811
\(726\) 0 0
\(727\) −11.8931 −0.441089 −0.220545 0.975377i \(-0.570783\pi\)
−0.220545 + 0.975377i \(0.570783\pi\)
\(728\) 0 0
\(729\) −35.6835 −1.32161
\(730\) 0 0
\(731\) 51.1522 1.89193
\(732\) 0 0
\(733\) 23.7838 0.878475 0.439237 0.898371i \(-0.355249\pi\)
0.439237 + 0.898371i \(0.355249\pi\)
\(734\) 0 0
\(735\) −6.76453 −0.249513
\(736\) 0 0
\(737\) −13.7606 −0.506876
\(738\) 0 0
\(739\) −44.2182 −1.62659 −0.813296 0.581851i \(-0.802329\pi\)
−0.813296 + 0.581851i \(0.802329\pi\)
\(740\) 0 0
\(741\) 6.39172 0.234806
\(742\) 0 0
\(743\) −47.7735 −1.75264 −0.876320 0.481729i \(-0.840009\pi\)
−0.876320 + 0.481729i \(0.840009\pi\)
\(744\) 0 0
\(745\) −10.2408 −0.375192
\(746\) 0 0
\(747\) −40.9783 −1.49932
\(748\) 0 0
\(749\) 5.59170 0.204316
\(750\) 0 0
\(751\) 1.00000 0.0364905
\(752\) 0 0
\(753\) −10.6531 −0.388222
\(754\) 0 0
\(755\) 3.21199 0.116896
\(756\) 0 0
\(757\) 22.4871 0.817308 0.408654 0.912689i \(-0.365998\pi\)
0.408654 + 0.912689i \(0.365998\pi\)
\(758\) 0 0
\(759\) 47.4982 1.72408
\(760\) 0 0
\(761\) −20.4890 −0.742727 −0.371364 0.928488i \(-0.621110\pi\)
−0.371364 + 0.928488i \(0.621110\pi\)
\(762\) 0 0
\(763\) −5.68673 −0.205873
\(764\) 0 0
\(765\) −71.8828 −2.59893
\(766\) 0 0
\(767\) −5.03758 −0.181897
\(768\) 0 0
\(769\) 36.1392 1.30321 0.651607 0.758557i \(-0.274095\pi\)
0.651607 + 0.758557i \(0.274095\pi\)
\(770\) 0 0
\(771\) −63.0232 −2.26973
\(772\) 0 0
\(773\) 38.1243 1.37124 0.685618 0.727961i \(-0.259532\pi\)
0.685618 + 0.727961i \(0.259532\pi\)
\(774\) 0 0
\(775\) −3.70089 −0.132940
\(776\) 0 0
\(777\) 71.4900 2.56469
\(778\) 0 0
\(779\) −0.338997 −0.0121458
\(780\) 0 0
\(781\) 12.2067 0.436790
\(782\) 0 0
\(783\) −39.0883 −1.39690
\(784\) 0 0
\(785\) 8.73715 0.311842
\(786\) 0 0
\(787\) −3.91552 −0.139573 −0.0697867 0.997562i \(-0.522232\pi\)
−0.0697867 + 0.997562i \(0.522232\pi\)
\(788\) 0 0
\(789\) 0.363969 0.0129576
\(790\) 0 0
\(791\) −12.2894 −0.436959
\(792\) 0 0
\(793\) −7.05907 −0.250675
\(794\) 0 0
\(795\) −18.4260 −0.653502
\(796\) 0 0
\(797\) 30.5703 1.08286 0.541429 0.840747i \(-0.317884\pi\)
0.541429 + 0.840747i \(0.317884\pi\)
\(798\) 0 0
\(799\) −34.7267 −1.22854
\(800\) 0 0
\(801\) 21.6302 0.764266
\(802\) 0 0
\(803\) −11.5126 −0.406273
\(804\) 0 0
\(805\) −31.5170 −1.11083
\(806\) 0 0
\(807\) 24.1921 0.851601
\(808\) 0 0
\(809\) 5.92027 0.208145 0.104073 0.994570i \(-0.466813\pi\)
0.104073 + 0.994570i \(0.466813\pi\)
\(810\) 0 0
\(811\) 53.6537 1.88404 0.942019 0.335561i \(-0.108926\pi\)
0.942019 + 0.335561i \(0.108926\pi\)
\(812\) 0 0
\(813\) −76.2736 −2.67503
\(814\) 0 0
\(815\) −40.3912 −1.41484
\(816\) 0 0
\(817\) 14.0202 0.490503
\(818\) 0 0
\(819\) −17.4807 −0.610827
\(820\) 0 0
\(821\) −29.5637 −1.03178 −0.515891 0.856654i \(-0.672539\pi\)
−0.515891 + 0.856654i \(0.672539\pi\)
\(822\) 0 0
\(823\) −44.3337 −1.54538 −0.772688 0.634786i \(-0.781088\pi\)
−0.772688 + 0.634786i \(0.781088\pi\)
\(824\) 0 0
\(825\) −5.46795 −0.190370
\(826\) 0 0
\(827\) 39.9154 1.38800 0.693998 0.719977i \(-0.255848\pi\)
0.693998 + 0.719977i \(0.255848\pi\)
\(828\) 0 0
\(829\) −2.35858 −0.0819169 −0.0409584 0.999161i \(-0.513041\pi\)
−0.0409584 + 0.999161i \(0.513041\pi\)
\(830\) 0 0
\(831\) −3.35074 −0.116236
\(832\) 0 0
\(833\) −6.89952 −0.239054
\(834\) 0 0
\(835\) 42.6355 1.47546
\(836\) 0 0
\(837\) −39.2702 −1.35738
\(838\) 0 0
\(839\) −24.9596 −0.861702 −0.430851 0.902423i \(-0.641787\pi\)
−0.430851 + 0.902423i \(0.641787\pi\)
\(840\) 0 0
\(841\) −3.05182 −0.105235
\(842\) 0 0
\(843\) −15.2294 −0.524529
\(844\) 0 0
\(845\) 23.4784 0.807681
\(846\) 0 0
\(847\) −10.5856 −0.363724
\(848\) 0 0
\(849\) −54.0224 −1.85404
\(850\) 0 0
\(851\) −63.0707 −2.16204
\(852\) 0 0
\(853\) −10.5205 −0.360216 −0.180108 0.983647i \(-0.557645\pi\)
−0.180108 + 0.983647i \(0.557645\pi\)
\(854\) 0 0
\(855\) −19.7021 −0.673798
\(856\) 0 0
\(857\) 23.7306 0.810621 0.405311 0.914179i \(-0.367163\pi\)
0.405311 + 0.914179i \(0.367163\pi\)
\(858\) 0 0
\(859\) −52.2305 −1.78208 −0.891042 0.453922i \(-0.850025\pi\)
−0.891042 + 0.453922i \(0.850025\pi\)
\(860\) 0 0
\(861\) 1.42252 0.0484793
\(862\) 0 0
\(863\) −41.2277 −1.40341 −0.701703 0.712469i \(-0.747577\pi\)
−0.701703 + 0.712469i \(0.747577\pi\)
\(864\) 0 0
\(865\) 25.2058 0.857021
\(866\) 0 0
\(867\) −62.5978 −2.12593
\(868\) 0 0
\(869\) 1.89712 0.0643554
\(870\) 0 0
\(871\) −6.85522 −0.232280
\(872\) 0 0
\(873\) −9.49682 −0.321419
\(874\) 0 0
\(875\) 28.7138 0.970704
\(876\) 0 0
\(877\) 20.3463 0.687047 0.343524 0.939144i \(-0.388379\pi\)
0.343524 + 0.939144i \(0.388379\pi\)
\(878\) 0 0
\(879\) −89.4025 −3.01547
\(880\) 0 0
\(881\) −14.3084 −0.482062 −0.241031 0.970517i \(-0.577486\pi\)
−0.241031 + 0.970517i \(0.577486\pi\)
\(882\) 0 0
\(883\) −22.3486 −0.752090 −0.376045 0.926601i \(-0.622716\pi\)
−0.376045 + 0.926601i \(0.622716\pi\)
\(884\) 0 0
\(885\) 23.8253 0.800877
\(886\) 0 0
\(887\) 41.3194 1.38737 0.693685 0.720278i \(-0.255986\pi\)
0.693685 + 0.720278i \(0.255986\pi\)
\(888\) 0 0
\(889\) 11.6865 0.391954
\(890\) 0 0
\(891\) −14.6292 −0.490096
\(892\) 0 0
\(893\) −9.51813 −0.318512
\(894\) 0 0
\(895\) −19.4259 −0.649336
\(896\) 0 0
\(897\) 23.6626 0.790072
\(898\) 0 0
\(899\) 26.0689 0.869448
\(900\) 0 0
\(901\) −18.7937 −0.626108
\(902\) 0 0
\(903\) −58.8322 −1.95781
\(904\) 0 0
\(905\) 16.7312 0.556164
\(906\) 0 0
\(907\) −26.8439 −0.891338 −0.445669 0.895198i \(-0.647034\pi\)
−0.445669 + 0.895198i \(0.647034\pi\)
\(908\) 0 0
\(909\) −34.9912 −1.16058
\(910\) 0 0
\(911\) 11.5951 0.384163 0.192081 0.981379i \(-0.438476\pi\)
0.192081 + 0.981379i \(0.438476\pi\)
\(912\) 0 0
\(913\) 18.8028 0.622280
\(914\) 0 0
\(915\) 33.3859 1.10370
\(916\) 0 0
\(917\) −30.9599 −1.02238
\(918\) 0 0
\(919\) −23.7360 −0.782980 −0.391490 0.920182i \(-0.628040\pi\)
−0.391490 + 0.920182i \(0.628040\pi\)
\(920\) 0 0
\(921\) −1.29498 −0.0426709
\(922\) 0 0
\(923\) 6.08112 0.200162
\(924\) 0 0
\(925\) 7.26064 0.238728
\(926\) 0 0
\(927\) 32.5509 1.06911
\(928\) 0 0
\(929\) 14.8191 0.486200 0.243100 0.970001i \(-0.421836\pi\)
0.243100 + 0.970001i \(0.421836\pi\)
\(930\) 0 0
\(931\) −1.89107 −0.0619772
\(932\) 0 0
\(933\) −1.25063 −0.0409437
\(934\) 0 0
\(935\) 32.9832 1.07867
\(936\) 0 0
\(937\) 31.9277 1.04303 0.521517 0.853241i \(-0.325366\pi\)
0.521517 + 0.853241i \(0.325366\pi\)
\(938\) 0 0
\(939\) −31.3192 −1.02206
\(940\) 0 0
\(941\) 22.5536 0.735225 0.367613 0.929979i \(-0.380175\pi\)
0.367613 + 0.929979i \(0.380175\pi\)
\(942\) 0 0
\(943\) −1.25499 −0.0408682
\(944\) 0 0
\(945\) 38.4989 1.25237
\(946\) 0 0
\(947\) 11.7270 0.381076 0.190538 0.981680i \(-0.438977\pi\)
0.190538 + 0.981680i \(0.438977\pi\)
\(948\) 0 0
\(949\) −5.73536 −0.186178
\(950\) 0 0
\(951\) −74.3654 −2.41146
\(952\) 0 0
\(953\) 26.9121 0.871770 0.435885 0.900002i \(-0.356435\pi\)
0.435885 + 0.900002i \(0.356435\pi\)
\(954\) 0 0
\(955\) −29.9594 −0.969464
\(956\) 0 0
\(957\) 38.5160 1.24505
\(958\) 0 0
\(959\) −20.7534 −0.670162
\(960\) 0 0
\(961\) −4.80972 −0.155152
\(962\) 0 0
\(963\) 12.9407 0.417008
\(964\) 0 0
\(965\) 6.18760 0.199186
\(966\) 0 0
\(967\) 53.2753 1.71322 0.856609 0.515966i \(-0.172567\pi\)
0.856609 + 0.515966i \(0.172567\pi\)
\(968\) 0 0
\(969\) −30.8330 −0.990497
\(970\) 0 0
\(971\) 34.3348 1.10186 0.550929 0.834552i \(-0.314274\pi\)
0.550929 + 0.834552i \(0.314274\pi\)
\(972\) 0 0
\(973\) 1.59128 0.0510142
\(974\) 0 0
\(975\) −2.72402 −0.0872384
\(976\) 0 0
\(977\) 30.7667 0.984315 0.492158 0.870506i \(-0.336208\pi\)
0.492158 + 0.870506i \(0.336208\pi\)
\(978\) 0 0
\(979\) −9.92494 −0.317203
\(980\) 0 0
\(981\) −13.1606 −0.420186
\(982\) 0 0
\(983\) −51.2741 −1.63539 −0.817695 0.575652i \(-0.804748\pi\)
−0.817695 + 0.575652i \(0.804748\pi\)
\(984\) 0 0
\(985\) 11.6900 0.372475
\(986\) 0 0
\(987\) 39.9405 1.27132
\(988\) 0 0
\(989\) 51.9036 1.65044
\(990\) 0 0
\(991\) −5.79576 −0.184108 −0.0920542 0.995754i \(-0.529343\pi\)
−0.0920542 + 0.995754i \(0.529343\pi\)
\(992\) 0 0
\(993\) −82.0994 −2.60535
\(994\) 0 0
\(995\) −39.6632 −1.25741
\(996\) 0 0
\(997\) 35.5585 1.12615 0.563075 0.826406i \(-0.309618\pi\)
0.563075 + 0.826406i \(0.309618\pi\)
\(998\) 0 0
\(999\) 77.0427 2.43752
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 6008.2.a.d.1.2 49
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6008.2.a.d.1.2 49 1.1 even 1 trivial