Properties

Label 8016.2.a.z.1.7
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $8$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.22210\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.56175 q^{5} -4.60397 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.56175 q^{5} -4.60397 q^{7} +1.00000 q^{9} +0.677161 q^{11} +2.42101 q^{13} +1.56175 q^{15} -0.541579 q^{17} -2.86493 q^{19} -4.60397 q^{21} +5.52905 q^{23} -2.56093 q^{25} +1.00000 q^{27} +4.48876 q^{29} -6.27589 q^{31} +0.677161 q^{33} -7.19026 q^{35} -4.48690 q^{37} +2.42101 q^{39} -2.64946 q^{41} -10.5513 q^{43} +1.56175 q^{45} +0.130851 q^{47} +14.1965 q^{49} -0.541579 q^{51} -1.86586 q^{53} +1.05756 q^{55} -2.86493 q^{57} +1.40860 q^{59} -4.55445 q^{61} -4.60397 q^{63} +3.78102 q^{65} +2.39601 q^{67} +5.52905 q^{69} -9.71089 q^{71} +0.237876 q^{73} -2.56093 q^{75} -3.11763 q^{77} -11.7562 q^{79} +1.00000 q^{81} +0.539478 q^{83} -0.845813 q^{85} +4.48876 q^{87} +5.24695 q^{89} -11.1462 q^{91} -6.27589 q^{93} -4.47431 q^{95} -18.9230 q^{97} +0.677161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.56175 0.698437 0.349218 0.937041i \(-0.386447\pi\)
0.349218 + 0.937041i \(0.386447\pi\)
\(6\) 0 0
\(7\) −4.60397 −1.74014 −0.870068 0.492932i \(-0.835925\pi\)
−0.870068 + 0.492932i \(0.835925\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.677161 0.204172 0.102086 0.994776i \(-0.467448\pi\)
0.102086 + 0.994776i \(0.467448\pi\)
\(12\) 0 0
\(13\) 2.42101 0.671467 0.335734 0.941957i \(-0.391016\pi\)
0.335734 + 0.941957i \(0.391016\pi\)
\(14\) 0 0
\(15\) 1.56175 0.403243
\(16\) 0 0
\(17\) −0.541579 −0.131352 −0.0656761 0.997841i \(-0.520920\pi\)
−0.0656761 + 0.997841i \(0.520920\pi\)
\(18\) 0 0
\(19\) −2.86493 −0.657260 −0.328630 0.944459i \(-0.606587\pi\)
−0.328630 + 0.944459i \(0.606587\pi\)
\(20\) 0 0
\(21\) −4.60397 −1.00467
\(22\) 0 0
\(23\) 5.52905 1.15289 0.576443 0.817137i \(-0.304440\pi\)
0.576443 + 0.817137i \(0.304440\pi\)
\(24\) 0 0
\(25\) −2.56093 −0.512186
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.48876 0.833543 0.416771 0.909011i \(-0.363162\pi\)
0.416771 + 0.909011i \(0.363162\pi\)
\(30\) 0 0
\(31\) −6.27589 −1.12718 −0.563591 0.826054i \(-0.690581\pi\)
−0.563591 + 0.826054i \(0.690581\pi\)
\(32\) 0 0
\(33\) 0.677161 0.117879
\(34\) 0 0
\(35\) −7.19026 −1.21538
\(36\) 0 0
\(37\) −4.48690 −0.737642 −0.368821 0.929500i \(-0.620239\pi\)
−0.368821 + 0.929500i \(0.620239\pi\)
\(38\) 0 0
\(39\) 2.42101 0.387672
\(40\) 0 0
\(41\) −2.64946 −0.413776 −0.206888 0.978365i \(-0.566334\pi\)
−0.206888 + 0.978365i \(0.566334\pi\)
\(42\) 0 0
\(43\) −10.5513 −1.60906 −0.804532 0.593909i \(-0.797584\pi\)
−0.804532 + 0.593909i \(0.797584\pi\)
\(44\) 0 0
\(45\) 1.56175 0.232812
\(46\) 0 0
\(47\) 0.130851 0.0190866 0.00954328 0.999954i \(-0.496962\pi\)
0.00954328 + 0.999954i \(0.496962\pi\)
\(48\) 0 0
\(49\) 14.1965 2.02807
\(50\) 0 0
\(51\) −0.541579 −0.0758362
\(52\) 0 0
\(53\) −1.86586 −0.256296 −0.128148 0.991755i \(-0.540903\pi\)
−0.128148 + 0.991755i \(0.540903\pi\)
\(54\) 0 0
\(55\) 1.05756 0.142601
\(56\) 0 0
\(57\) −2.86493 −0.379469
\(58\) 0 0
\(59\) 1.40860 0.183384 0.0916920 0.995787i \(-0.470772\pi\)
0.0916920 + 0.995787i \(0.470772\pi\)
\(60\) 0 0
\(61\) −4.55445 −0.583138 −0.291569 0.956550i \(-0.594177\pi\)
−0.291569 + 0.956550i \(0.594177\pi\)
\(62\) 0 0
\(63\) −4.60397 −0.580045
\(64\) 0 0
\(65\) 3.78102 0.468977
\(66\) 0 0
\(67\) 2.39601 0.292719 0.146359 0.989231i \(-0.453244\pi\)
0.146359 + 0.989231i \(0.453244\pi\)
\(68\) 0 0
\(69\) 5.52905 0.665619
\(70\) 0 0
\(71\) −9.71089 −1.15247 −0.576235 0.817284i \(-0.695479\pi\)
−0.576235 + 0.817284i \(0.695479\pi\)
\(72\) 0 0
\(73\) 0.237876 0.0278413 0.0139207 0.999903i \(-0.495569\pi\)
0.0139207 + 0.999903i \(0.495569\pi\)
\(74\) 0 0
\(75\) −2.56093 −0.295711
\(76\) 0 0
\(77\) −3.11763 −0.355286
\(78\) 0 0
\(79\) −11.7562 −1.32267 −0.661337 0.750089i \(-0.730011\pi\)
−0.661337 + 0.750089i \(0.730011\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.539478 0.0592154 0.0296077 0.999562i \(-0.490574\pi\)
0.0296077 + 0.999562i \(0.490574\pi\)
\(84\) 0 0
\(85\) −0.845813 −0.0917412
\(86\) 0 0
\(87\) 4.48876 0.481246
\(88\) 0 0
\(89\) 5.24695 0.556175 0.278088 0.960556i \(-0.410299\pi\)
0.278088 + 0.960556i \(0.410299\pi\)
\(90\) 0 0
\(91\) −11.1462 −1.16844
\(92\) 0 0
\(93\) −6.27589 −0.650779
\(94\) 0 0
\(95\) −4.47431 −0.459054
\(96\) 0 0
\(97\) −18.9230 −1.92134 −0.960671 0.277690i \(-0.910431\pi\)
−0.960671 + 0.277690i \(0.910431\pi\)
\(98\) 0 0
\(99\) 0.677161 0.0680572
\(100\) 0 0
\(101\) 19.2071 1.91118 0.955591 0.294697i \(-0.0952187\pi\)
0.955591 + 0.294697i \(0.0952187\pi\)
\(102\) 0 0
\(103\) 14.1285 1.39212 0.696060 0.717983i \(-0.254935\pi\)
0.696060 + 0.717983i \(0.254935\pi\)
\(104\) 0 0
\(105\) −7.19026 −0.701697
\(106\) 0 0
\(107\) 2.40350 0.232355 0.116178 0.993228i \(-0.462936\pi\)
0.116178 + 0.993228i \(0.462936\pi\)
\(108\) 0 0
\(109\) −4.48595 −0.429676 −0.214838 0.976650i \(-0.568922\pi\)
−0.214838 + 0.976650i \(0.568922\pi\)
\(110\) 0 0
\(111\) −4.48690 −0.425878
\(112\) 0 0
\(113\) 10.2896 0.967961 0.483980 0.875079i \(-0.339191\pi\)
0.483980 + 0.875079i \(0.339191\pi\)
\(114\) 0 0
\(115\) 8.63500 0.805218
\(116\) 0 0
\(117\) 2.42101 0.223822
\(118\) 0 0
\(119\) 2.49341 0.228571
\(120\) 0 0
\(121\) −10.5415 −0.958314
\(122\) 0 0
\(123\) −2.64946 −0.238894
\(124\) 0 0
\(125\) −11.8083 −1.05617
\(126\) 0 0
\(127\) −14.3566 −1.27394 −0.636970 0.770889i \(-0.719812\pi\)
−0.636970 + 0.770889i \(0.719812\pi\)
\(128\) 0 0
\(129\) −10.5513 −0.928994
\(130\) 0 0
\(131\) 10.1829 0.889682 0.444841 0.895610i \(-0.353260\pi\)
0.444841 + 0.895610i \(0.353260\pi\)
\(132\) 0 0
\(133\) 13.1900 1.14372
\(134\) 0 0
\(135\) 1.56175 0.134414
\(136\) 0 0
\(137\) −8.65142 −0.739141 −0.369570 0.929203i \(-0.620495\pi\)
−0.369570 + 0.929203i \(0.620495\pi\)
\(138\) 0 0
\(139\) 17.8616 1.51500 0.757499 0.652836i \(-0.226421\pi\)
0.757499 + 0.652836i \(0.226421\pi\)
\(140\) 0 0
\(141\) 0.130851 0.0110196
\(142\) 0 0
\(143\) 1.63941 0.137095
\(144\) 0 0
\(145\) 7.01034 0.582177
\(146\) 0 0
\(147\) 14.1965 1.17091
\(148\) 0 0
\(149\) 12.3329 1.01035 0.505173 0.863018i \(-0.331429\pi\)
0.505173 + 0.863018i \(0.331429\pi\)
\(150\) 0 0
\(151\) −18.5880 −1.51267 −0.756336 0.654184i \(-0.773012\pi\)
−0.756336 + 0.654184i \(0.773012\pi\)
\(152\) 0 0
\(153\) −0.541579 −0.0437841
\(154\) 0 0
\(155\) −9.80138 −0.787266
\(156\) 0 0
\(157\) −11.5265 −0.919916 −0.459958 0.887941i \(-0.652136\pi\)
−0.459958 + 0.887941i \(0.652136\pi\)
\(158\) 0 0
\(159\) −1.86586 −0.147972
\(160\) 0 0
\(161\) −25.4555 −2.00618
\(162\) 0 0
\(163\) −0.881223 −0.0690227 −0.0345114 0.999404i \(-0.510987\pi\)
−0.0345114 + 0.999404i \(0.510987\pi\)
\(164\) 0 0
\(165\) 1.05756 0.0823308
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −7.13872 −0.549132
\(170\) 0 0
\(171\) −2.86493 −0.219087
\(172\) 0 0
\(173\) 19.5335 1.48510 0.742551 0.669790i \(-0.233616\pi\)
0.742551 + 0.669790i \(0.233616\pi\)
\(174\) 0 0
\(175\) 11.7904 0.891273
\(176\) 0 0
\(177\) 1.40860 0.105877
\(178\) 0 0
\(179\) −21.0890 −1.57627 −0.788134 0.615504i \(-0.788952\pi\)
−0.788134 + 0.615504i \(0.788952\pi\)
\(180\) 0 0
\(181\) 14.3226 1.06459 0.532295 0.846559i \(-0.321330\pi\)
0.532295 + 0.846559i \(0.321330\pi\)
\(182\) 0 0
\(183\) −4.55445 −0.336675
\(184\) 0 0
\(185\) −7.00743 −0.515197
\(186\) 0 0
\(187\) −0.366736 −0.0268184
\(188\) 0 0
\(189\) −4.60397 −0.334889
\(190\) 0 0
\(191\) −1.93785 −0.140218 −0.0701091 0.997539i \(-0.522335\pi\)
−0.0701091 + 0.997539i \(0.522335\pi\)
\(192\) 0 0
\(193\) −21.1251 −1.52062 −0.760310 0.649561i \(-0.774953\pi\)
−0.760310 + 0.649561i \(0.774953\pi\)
\(194\) 0 0
\(195\) 3.78102 0.270764
\(196\) 0 0
\(197\) 0.902280 0.0642848 0.0321424 0.999483i \(-0.489767\pi\)
0.0321424 + 0.999483i \(0.489767\pi\)
\(198\) 0 0
\(199\) −10.0097 −0.709568 −0.354784 0.934948i \(-0.615446\pi\)
−0.354784 + 0.934948i \(0.615446\pi\)
\(200\) 0 0
\(201\) 2.39601 0.169001
\(202\) 0 0
\(203\) −20.6661 −1.45048
\(204\) 0 0
\(205\) −4.13780 −0.288996
\(206\) 0 0
\(207\) 5.52905 0.384295
\(208\) 0 0
\(209\) −1.94002 −0.134194
\(210\) 0 0
\(211\) −23.8910 −1.64473 −0.822364 0.568962i \(-0.807345\pi\)
−0.822364 + 0.568962i \(0.807345\pi\)
\(212\) 0 0
\(213\) −9.71089 −0.665379
\(214\) 0 0
\(215\) −16.4786 −1.12383
\(216\) 0 0
\(217\) 28.8940 1.96145
\(218\) 0 0
\(219\) 0.237876 0.0160742
\(220\) 0 0
\(221\) −1.31117 −0.0881987
\(222\) 0 0
\(223\) 13.6415 0.913504 0.456752 0.889594i \(-0.349013\pi\)
0.456752 + 0.889594i \(0.349013\pi\)
\(224\) 0 0
\(225\) −2.56093 −0.170729
\(226\) 0 0
\(227\) −8.28044 −0.549592 −0.274796 0.961503i \(-0.588610\pi\)
−0.274796 + 0.961503i \(0.588610\pi\)
\(228\) 0 0
\(229\) −16.9527 −1.12026 −0.560132 0.828404i \(-0.689249\pi\)
−0.560132 + 0.828404i \(0.689249\pi\)
\(230\) 0 0
\(231\) −3.11763 −0.205125
\(232\) 0 0
\(233\) −14.9174 −0.977269 −0.488635 0.872489i \(-0.662505\pi\)
−0.488635 + 0.872489i \(0.662505\pi\)
\(234\) 0 0
\(235\) 0.204357 0.0133308
\(236\) 0 0
\(237\) −11.7562 −0.763647
\(238\) 0 0
\(239\) 16.2260 1.04958 0.524788 0.851233i \(-0.324145\pi\)
0.524788 + 0.851233i \(0.324145\pi\)
\(240\) 0 0
\(241\) −2.95693 −0.190473 −0.0952363 0.995455i \(-0.530361\pi\)
−0.0952363 + 0.995455i \(0.530361\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 22.1714 1.41648
\(246\) 0 0
\(247\) −6.93602 −0.441328
\(248\) 0 0
\(249\) 0.539478 0.0341880
\(250\) 0 0
\(251\) −26.2547 −1.65718 −0.828591 0.559854i \(-0.810857\pi\)
−0.828591 + 0.559854i \(0.810857\pi\)
\(252\) 0 0
\(253\) 3.74405 0.235387
\(254\) 0 0
\(255\) −0.845813 −0.0529668
\(256\) 0 0
\(257\) −25.8500 −1.61248 −0.806240 0.591588i \(-0.798501\pi\)
−0.806240 + 0.591588i \(0.798501\pi\)
\(258\) 0 0
\(259\) 20.6576 1.28360
\(260\) 0 0
\(261\) 4.48876 0.277848
\(262\) 0 0
\(263\) 9.05966 0.558642 0.279321 0.960198i \(-0.409891\pi\)
0.279321 + 0.960198i \(0.409891\pi\)
\(264\) 0 0
\(265\) −2.91401 −0.179007
\(266\) 0 0
\(267\) 5.24695 0.321108
\(268\) 0 0
\(269\) −22.5095 −1.37243 −0.686213 0.727401i \(-0.740728\pi\)
−0.686213 + 0.727401i \(0.740728\pi\)
\(270\) 0 0
\(271\) −9.12378 −0.554230 −0.277115 0.960837i \(-0.589378\pi\)
−0.277115 + 0.960837i \(0.589378\pi\)
\(272\) 0 0
\(273\) −11.1462 −0.674601
\(274\) 0 0
\(275\) −1.73416 −0.104574
\(276\) 0 0
\(277\) −14.2854 −0.858328 −0.429164 0.903227i \(-0.641192\pi\)
−0.429164 + 0.903227i \(0.641192\pi\)
\(278\) 0 0
\(279\) −6.27589 −0.375727
\(280\) 0 0
\(281\) −23.0548 −1.37533 −0.687666 0.726028i \(-0.741364\pi\)
−0.687666 + 0.726028i \(0.741364\pi\)
\(282\) 0 0
\(283\) 27.7777 1.65121 0.825607 0.564246i \(-0.190833\pi\)
0.825607 + 0.564246i \(0.190833\pi\)
\(284\) 0 0
\(285\) −4.47431 −0.265035
\(286\) 0 0
\(287\) 12.1980 0.720026
\(288\) 0 0
\(289\) −16.7067 −0.982747
\(290\) 0 0
\(291\) −18.9230 −1.10929
\(292\) 0 0
\(293\) 25.0738 1.46483 0.732414 0.680860i \(-0.238394\pi\)
0.732414 + 0.680860i \(0.238394\pi\)
\(294\) 0 0
\(295\) 2.19988 0.128082
\(296\) 0 0
\(297\) 0.677161 0.0392929
\(298\) 0 0
\(299\) 13.3859 0.774125
\(300\) 0 0
\(301\) 48.5780 2.79999
\(302\) 0 0
\(303\) 19.2071 1.10342
\(304\) 0 0
\(305\) −7.11293 −0.407285
\(306\) 0 0
\(307\) −11.1106 −0.634115 −0.317058 0.948406i \(-0.602695\pi\)
−0.317058 + 0.948406i \(0.602695\pi\)
\(308\) 0 0
\(309\) 14.1285 0.803741
\(310\) 0 0
\(311\) −0.770347 −0.0436824 −0.0218412 0.999761i \(-0.506953\pi\)
−0.0218412 + 0.999761i \(0.506953\pi\)
\(312\) 0 0
\(313\) −2.71480 −0.153450 −0.0767248 0.997052i \(-0.524446\pi\)
−0.0767248 + 0.997052i \(0.524446\pi\)
\(314\) 0 0
\(315\) −7.19026 −0.405125
\(316\) 0 0
\(317\) −20.2749 −1.13875 −0.569375 0.822078i \(-0.692815\pi\)
−0.569375 + 0.822078i \(0.692815\pi\)
\(318\) 0 0
\(319\) 3.03962 0.170186
\(320\) 0 0
\(321\) 2.40350 0.134150
\(322\) 0 0
\(323\) 1.55159 0.0863325
\(324\) 0 0
\(325\) −6.20003 −0.343916
\(326\) 0 0
\(327\) −4.48595 −0.248074
\(328\) 0 0
\(329\) −0.602433 −0.0332132
\(330\) 0 0
\(331\) −11.5475 −0.634706 −0.317353 0.948308i \(-0.602794\pi\)
−0.317353 + 0.948308i \(0.602794\pi\)
\(332\) 0 0
\(333\) −4.48690 −0.245881
\(334\) 0 0
\(335\) 3.74197 0.204446
\(336\) 0 0
\(337\) −28.5512 −1.55528 −0.777640 0.628709i \(-0.783584\pi\)
−0.777640 + 0.628709i \(0.783584\pi\)
\(338\) 0 0
\(339\) 10.2896 0.558852
\(340\) 0 0
\(341\) −4.24978 −0.230139
\(342\) 0 0
\(343\) −33.1325 −1.78898
\(344\) 0 0
\(345\) 8.63500 0.464893
\(346\) 0 0
\(347\) 30.1278 1.61735 0.808673 0.588258i \(-0.200186\pi\)
0.808673 + 0.588258i \(0.200186\pi\)
\(348\) 0 0
\(349\) −3.21506 −0.172098 −0.0860491 0.996291i \(-0.527424\pi\)
−0.0860491 + 0.996291i \(0.527424\pi\)
\(350\) 0 0
\(351\) 2.42101 0.129224
\(352\) 0 0
\(353\) 2.71593 0.144555 0.0722773 0.997385i \(-0.476973\pi\)
0.0722773 + 0.997385i \(0.476973\pi\)
\(354\) 0 0
\(355\) −15.1660 −0.804928
\(356\) 0 0
\(357\) 2.49341 0.131965
\(358\) 0 0
\(359\) 34.4845 1.82002 0.910010 0.414587i \(-0.136074\pi\)
0.910010 + 0.414587i \(0.136074\pi\)
\(360\) 0 0
\(361\) −10.7922 −0.568010
\(362\) 0 0
\(363\) −10.5415 −0.553283
\(364\) 0 0
\(365\) 0.371504 0.0194454
\(366\) 0 0
\(367\) −5.74370 −0.299819 −0.149909 0.988700i \(-0.547898\pi\)
−0.149909 + 0.988700i \(0.547898\pi\)
\(368\) 0 0
\(369\) −2.64946 −0.137925
\(370\) 0 0
\(371\) 8.59036 0.445990
\(372\) 0 0
\(373\) 21.4774 1.11206 0.556030 0.831162i \(-0.312324\pi\)
0.556030 + 0.831162i \(0.312324\pi\)
\(374\) 0 0
\(375\) −11.8083 −0.609778
\(376\) 0 0
\(377\) 10.8673 0.559696
\(378\) 0 0
\(379\) −0.430617 −0.0221193 −0.0110597 0.999939i \(-0.503520\pi\)
−0.0110597 + 0.999939i \(0.503520\pi\)
\(380\) 0 0
\(381\) −14.3566 −0.735509
\(382\) 0 0
\(383\) 2.88548 0.147441 0.0737207 0.997279i \(-0.476513\pi\)
0.0737207 + 0.997279i \(0.476513\pi\)
\(384\) 0 0
\(385\) −4.86896 −0.248145
\(386\) 0 0
\(387\) −10.5513 −0.536355
\(388\) 0 0
\(389\) −7.70045 −0.390428 −0.195214 0.980761i \(-0.562540\pi\)
−0.195214 + 0.980761i \(0.562540\pi\)
\(390\) 0 0
\(391\) −2.99442 −0.151434
\(392\) 0 0
\(393\) 10.1829 0.513658
\(394\) 0 0
\(395\) −18.3603 −0.923805
\(396\) 0 0
\(397\) −1.91661 −0.0961921 −0.0480961 0.998843i \(-0.515315\pi\)
−0.0480961 + 0.998843i \(0.515315\pi\)
\(398\) 0 0
\(399\) 13.1900 0.660328
\(400\) 0 0
\(401\) −38.3129 −1.91325 −0.956627 0.291316i \(-0.905907\pi\)
−0.956627 + 0.291316i \(0.905907\pi\)
\(402\) 0 0
\(403\) −15.1940 −0.756866
\(404\) 0 0
\(405\) 1.56175 0.0776041
\(406\) 0 0
\(407\) −3.03836 −0.150606
\(408\) 0 0
\(409\) −14.7420 −0.728946 −0.364473 0.931214i \(-0.618751\pi\)
−0.364473 + 0.931214i \(0.618751\pi\)
\(410\) 0 0
\(411\) −8.65142 −0.426743
\(412\) 0 0
\(413\) −6.48514 −0.319113
\(414\) 0 0
\(415\) 0.842532 0.0413582
\(416\) 0 0
\(417\) 17.8616 0.874685
\(418\) 0 0
\(419\) −25.7668 −1.25879 −0.629396 0.777085i \(-0.716698\pi\)
−0.629396 + 0.777085i \(0.716698\pi\)
\(420\) 0 0
\(421\) 4.26556 0.207891 0.103945 0.994583i \(-0.466853\pi\)
0.103945 + 0.994583i \(0.466853\pi\)
\(422\) 0 0
\(423\) 0.130851 0.00636219
\(424\) 0 0
\(425\) 1.38695 0.0672767
\(426\) 0 0
\(427\) 20.9685 1.01474
\(428\) 0 0
\(429\) 1.63941 0.0791516
\(430\) 0 0
\(431\) 17.3243 0.834482 0.417241 0.908796i \(-0.362997\pi\)
0.417241 + 0.908796i \(0.362997\pi\)
\(432\) 0 0
\(433\) 24.5917 1.18180 0.590901 0.806744i \(-0.298772\pi\)
0.590901 + 0.806744i \(0.298772\pi\)
\(434\) 0 0
\(435\) 7.01034 0.336120
\(436\) 0 0
\(437\) −15.8403 −0.757745
\(438\) 0 0
\(439\) 23.8929 1.14035 0.570173 0.821525i \(-0.306876\pi\)
0.570173 + 0.821525i \(0.306876\pi\)
\(440\) 0 0
\(441\) 14.1965 0.676024
\(442\) 0 0
\(443\) −8.36955 −0.397649 −0.198825 0.980035i \(-0.563712\pi\)
−0.198825 + 0.980035i \(0.563712\pi\)
\(444\) 0 0
\(445\) 8.19443 0.388453
\(446\) 0 0
\(447\) 12.3329 0.583324
\(448\) 0 0
\(449\) 19.5398 0.922142 0.461071 0.887363i \(-0.347465\pi\)
0.461071 + 0.887363i \(0.347465\pi\)
\(450\) 0 0
\(451\) −1.79411 −0.0844814
\(452\) 0 0
\(453\) −18.5880 −0.873341
\(454\) 0 0
\(455\) −17.4077 −0.816084
\(456\) 0 0
\(457\) −29.7136 −1.38994 −0.694972 0.719037i \(-0.744583\pi\)
−0.694972 + 0.719037i \(0.744583\pi\)
\(458\) 0 0
\(459\) −0.541579 −0.0252787
\(460\) 0 0
\(461\) −11.3628 −0.529216 −0.264608 0.964356i \(-0.585243\pi\)
−0.264608 + 0.964356i \(0.585243\pi\)
\(462\) 0 0
\(463\) −32.2498 −1.49878 −0.749388 0.662131i \(-0.769652\pi\)
−0.749388 + 0.662131i \(0.769652\pi\)
\(464\) 0 0
\(465\) −9.80138 −0.454528
\(466\) 0 0
\(467\) 3.07502 0.142295 0.0711475 0.997466i \(-0.477334\pi\)
0.0711475 + 0.997466i \(0.477334\pi\)
\(468\) 0 0
\(469\) −11.0311 −0.509370
\(470\) 0 0
\(471\) −11.5265 −0.531114
\(472\) 0 0
\(473\) −7.14496 −0.328525
\(474\) 0 0
\(475\) 7.33688 0.336639
\(476\) 0 0
\(477\) −1.86586 −0.0854320
\(478\) 0 0
\(479\) 27.9296 1.27614 0.638069 0.769980i \(-0.279734\pi\)
0.638069 + 0.769980i \(0.279734\pi\)
\(480\) 0 0
\(481\) −10.8628 −0.495303
\(482\) 0 0
\(483\) −25.4555 −1.15827
\(484\) 0 0
\(485\) −29.5531 −1.34194
\(486\) 0 0
\(487\) 11.7804 0.533823 0.266912 0.963721i \(-0.413997\pi\)
0.266912 + 0.963721i \(0.413997\pi\)
\(488\) 0 0
\(489\) −0.881223 −0.0398503
\(490\) 0 0
\(491\) 35.8557 1.61814 0.809072 0.587710i \(-0.199970\pi\)
0.809072 + 0.587710i \(0.199970\pi\)
\(492\) 0 0
\(493\) −2.43102 −0.109488
\(494\) 0 0
\(495\) 1.05756 0.0475337
\(496\) 0 0
\(497\) 44.7086 2.00545
\(498\) 0 0
\(499\) 19.2035 0.859666 0.429833 0.902908i \(-0.358572\pi\)
0.429833 + 0.902908i \(0.358572\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −30.8353 −1.37488 −0.687440 0.726241i \(-0.741265\pi\)
−0.687440 + 0.726241i \(0.741265\pi\)
\(504\) 0 0
\(505\) 29.9968 1.33484
\(506\) 0 0
\(507\) −7.13872 −0.317041
\(508\) 0 0
\(509\) 3.93321 0.174336 0.0871682 0.996194i \(-0.472218\pi\)
0.0871682 + 0.996194i \(0.472218\pi\)
\(510\) 0 0
\(511\) −1.09517 −0.0484477
\(512\) 0 0
\(513\) −2.86493 −0.126490
\(514\) 0 0
\(515\) 22.0652 0.972309
\(516\) 0 0
\(517\) 0.0886071 0.00389694
\(518\) 0 0
\(519\) 19.5335 0.857424
\(520\) 0 0
\(521\) −7.97954 −0.349590 −0.174795 0.984605i \(-0.555926\pi\)
−0.174795 + 0.984605i \(0.555926\pi\)
\(522\) 0 0
\(523\) −24.7612 −1.08273 −0.541365 0.840788i \(-0.682092\pi\)
−0.541365 + 0.840788i \(0.682092\pi\)
\(524\) 0 0
\(525\) 11.7904 0.514577
\(526\) 0 0
\(527\) 3.39889 0.148058
\(528\) 0 0
\(529\) 7.57036 0.329146
\(530\) 0 0
\(531\) 1.40860 0.0611280
\(532\) 0 0
\(533\) −6.41436 −0.277837
\(534\) 0 0
\(535\) 3.75368 0.162286
\(536\) 0 0
\(537\) −21.0890 −0.910058
\(538\) 0 0
\(539\) 9.61332 0.414075
\(540\) 0 0
\(541\) 34.1179 1.46685 0.733423 0.679773i \(-0.237922\pi\)
0.733423 + 0.679773i \(0.237922\pi\)
\(542\) 0 0
\(543\) 14.3226 0.614641
\(544\) 0 0
\(545\) −7.00594 −0.300102
\(546\) 0 0
\(547\) 3.75529 0.160565 0.0802823 0.996772i \(-0.474418\pi\)
0.0802823 + 0.996772i \(0.474418\pi\)
\(548\) 0 0
\(549\) −4.55445 −0.194379
\(550\) 0 0
\(551\) −12.8600 −0.547854
\(552\) 0 0
\(553\) 54.1251 2.30163
\(554\) 0 0
\(555\) −7.00743 −0.297449
\(556\) 0 0
\(557\) 29.3144 1.24209 0.621045 0.783775i \(-0.286708\pi\)
0.621045 + 0.783775i \(0.286708\pi\)
\(558\) 0 0
\(559\) −25.5449 −1.08043
\(560\) 0 0
\(561\) −0.366736 −0.0154836
\(562\) 0 0
\(563\) 11.8655 0.500070 0.250035 0.968237i \(-0.419558\pi\)
0.250035 + 0.968237i \(0.419558\pi\)
\(564\) 0 0
\(565\) 16.0698 0.676060
\(566\) 0 0
\(567\) −4.60397 −0.193348
\(568\) 0 0
\(569\) −18.9575 −0.794739 −0.397370 0.917659i \(-0.630077\pi\)
−0.397370 + 0.917659i \(0.630077\pi\)
\(570\) 0 0
\(571\) 16.8768 0.706270 0.353135 0.935572i \(-0.385116\pi\)
0.353135 + 0.935572i \(0.385116\pi\)
\(572\) 0 0
\(573\) −1.93785 −0.0809550
\(574\) 0 0
\(575\) −14.1595 −0.590492
\(576\) 0 0
\(577\) 23.2738 0.968901 0.484451 0.874819i \(-0.339019\pi\)
0.484451 + 0.874819i \(0.339019\pi\)
\(578\) 0 0
\(579\) −21.1251 −0.877930
\(580\) 0 0
\(581\) −2.48374 −0.103043
\(582\) 0 0
\(583\) −1.26349 −0.0523284
\(584\) 0 0
\(585\) 3.78102 0.156326
\(586\) 0 0
\(587\) −14.9457 −0.616873 −0.308437 0.951245i \(-0.599806\pi\)
−0.308437 + 0.951245i \(0.599806\pi\)
\(588\) 0 0
\(589\) 17.9800 0.740851
\(590\) 0 0
\(591\) 0.902280 0.0371149
\(592\) 0 0
\(593\) 20.4973 0.841724 0.420862 0.907125i \(-0.361728\pi\)
0.420862 + 0.907125i \(0.361728\pi\)
\(594\) 0 0
\(595\) 3.89409 0.159642
\(596\) 0 0
\(597\) −10.0097 −0.409669
\(598\) 0 0
\(599\) −12.8824 −0.526361 −0.263180 0.964747i \(-0.584771\pi\)
−0.263180 + 0.964747i \(0.584771\pi\)
\(600\) 0 0
\(601\) 18.2054 0.742613 0.371307 0.928510i \(-0.378910\pi\)
0.371307 + 0.928510i \(0.378910\pi\)
\(602\) 0 0
\(603\) 2.39601 0.0975729
\(604\) 0 0
\(605\) −16.4631 −0.669322
\(606\) 0 0
\(607\) 21.2271 0.861583 0.430791 0.902452i \(-0.358234\pi\)
0.430791 + 0.902452i \(0.358234\pi\)
\(608\) 0 0
\(609\) −20.6661 −0.837433
\(610\) 0 0
\(611\) 0.316791 0.0128160
\(612\) 0 0
\(613\) −4.85113 −0.195935 −0.0979676 0.995190i \(-0.531234\pi\)
−0.0979676 + 0.995190i \(0.531234\pi\)
\(614\) 0 0
\(615\) −4.13780 −0.166852
\(616\) 0 0
\(617\) −33.5898 −1.35228 −0.676138 0.736775i \(-0.736348\pi\)
−0.676138 + 0.736775i \(0.736348\pi\)
\(618\) 0 0
\(619\) 0.864216 0.0347358 0.0173679 0.999849i \(-0.494471\pi\)
0.0173679 + 0.999849i \(0.494471\pi\)
\(620\) 0 0
\(621\) 5.52905 0.221873
\(622\) 0 0
\(623\) −24.1568 −0.967820
\(624\) 0 0
\(625\) −5.63700 −0.225480
\(626\) 0 0
\(627\) −1.94002 −0.0774769
\(628\) 0 0
\(629\) 2.43001 0.0968910
\(630\) 0 0
\(631\) 17.6907 0.704254 0.352127 0.935952i \(-0.385458\pi\)
0.352127 + 0.935952i \(0.385458\pi\)
\(632\) 0 0
\(633\) −23.8910 −0.949584
\(634\) 0 0
\(635\) −22.4214 −0.889767
\(636\) 0 0
\(637\) 34.3699 1.36178
\(638\) 0 0
\(639\) −9.71089 −0.384157
\(640\) 0 0
\(641\) 34.8330 1.37582 0.687910 0.725796i \(-0.258528\pi\)
0.687910 + 0.725796i \(0.258528\pi\)
\(642\) 0 0
\(643\) 32.7079 1.28987 0.644937 0.764236i \(-0.276884\pi\)
0.644937 + 0.764236i \(0.276884\pi\)
\(644\) 0 0
\(645\) −16.4786 −0.648844
\(646\) 0 0
\(647\) 45.0734 1.77202 0.886008 0.463669i \(-0.153467\pi\)
0.886008 + 0.463669i \(0.153467\pi\)
\(648\) 0 0
\(649\) 0.953849 0.0374418
\(650\) 0 0
\(651\) 28.8940 1.13244
\(652\) 0 0
\(653\) 37.5770 1.47050 0.735252 0.677794i \(-0.237064\pi\)
0.735252 + 0.677794i \(0.237064\pi\)
\(654\) 0 0
\(655\) 15.9031 0.621387
\(656\) 0 0
\(657\) 0.237876 0.00928044
\(658\) 0 0
\(659\) 32.1271 1.25149 0.625747 0.780026i \(-0.284794\pi\)
0.625747 + 0.780026i \(0.284794\pi\)
\(660\) 0 0
\(661\) 19.2467 0.748610 0.374305 0.927306i \(-0.377881\pi\)
0.374305 + 0.927306i \(0.377881\pi\)
\(662\) 0 0
\(663\) −1.31117 −0.0509215
\(664\) 0 0
\(665\) 20.5996 0.798817
\(666\) 0 0
\(667\) 24.8186 0.960980
\(668\) 0 0
\(669\) 13.6415 0.527412
\(670\) 0 0
\(671\) −3.08410 −0.119060
\(672\) 0 0
\(673\) 37.4707 1.44439 0.722195 0.691690i \(-0.243133\pi\)
0.722195 + 0.691690i \(0.243133\pi\)
\(674\) 0 0
\(675\) −2.56093 −0.0985702
\(676\) 0 0
\(677\) 45.5562 1.75087 0.875433 0.483339i \(-0.160576\pi\)
0.875433 + 0.483339i \(0.160576\pi\)
\(678\) 0 0
\(679\) 87.1209 3.34339
\(680\) 0 0
\(681\) −8.28044 −0.317307
\(682\) 0 0
\(683\) −8.30725 −0.317868 −0.158934 0.987289i \(-0.550806\pi\)
−0.158934 + 0.987289i \(0.550806\pi\)
\(684\) 0 0
\(685\) −13.5114 −0.516243
\(686\) 0 0
\(687\) −16.9527 −0.646784
\(688\) 0 0
\(689\) −4.51727 −0.172094
\(690\) 0 0
\(691\) −51.1454 −1.94566 −0.972831 0.231516i \(-0.925631\pi\)
−0.972831 + 0.231516i \(0.925631\pi\)
\(692\) 0 0
\(693\) −3.11763 −0.118429
\(694\) 0 0
\(695\) 27.8954 1.05813
\(696\) 0 0
\(697\) 1.43489 0.0543504
\(698\) 0 0
\(699\) −14.9174 −0.564227
\(700\) 0 0
\(701\) 33.8929 1.28012 0.640059 0.768325i \(-0.278910\pi\)
0.640059 + 0.768325i \(0.278910\pi\)
\(702\) 0 0
\(703\) 12.8547 0.484823
\(704\) 0 0
\(705\) 0.204357 0.00769652
\(706\) 0 0
\(707\) −88.4290 −3.32571
\(708\) 0 0
\(709\) 7.85245 0.294905 0.147452 0.989069i \(-0.452893\pi\)
0.147452 + 0.989069i \(0.452893\pi\)
\(710\) 0 0
\(711\) −11.7562 −0.440892
\(712\) 0 0
\(713\) −34.6997 −1.29951
\(714\) 0 0
\(715\) 2.56036 0.0957519
\(716\) 0 0
\(717\) 16.2260 0.605973
\(718\) 0 0
\(719\) 32.9194 1.22768 0.613842 0.789429i \(-0.289623\pi\)
0.613842 + 0.789429i \(0.289623\pi\)
\(720\) 0 0
\(721\) −65.0471 −2.42248
\(722\) 0 0
\(723\) −2.95693 −0.109969
\(724\) 0 0
\(725\) −11.4954 −0.426929
\(726\) 0 0
\(727\) −14.2103 −0.527029 −0.263515 0.964655i \(-0.584882\pi\)
−0.263515 + 0.964655i \(0.584882\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.71439 0.211354
\(732\) 0 0
\(733\) −32.0616 −1.18422 −0.592112 0.805856i \(-0.701706\pi\)
−0.592112 + 0.805856i \(0.701706\pi\)
\(734\) 0 0
\(735\) 22.1714 0.817805
\(736\) 0 0
\(737\) 1.62248 0.0597649
\(738\) 0 0
\(739\) −4.29038 −0.157824 −0.0789121 0.996882i \(-0.525145\pi\)
−0.0789121 + 0.996882i \(0.525145\pi\)
\(740\) 0 0
\(741\) −6.93602 −0.254801
\(742\) 0 0
\(743\) −43.8196 −1.60758 −0.803792 0.594910i \(-0.797187\pi\)
−0.803792 + 0.594910i \(0.797187\pi\)
\(744\) 0 0
\(745\) 19.2609 0.705664
\(746\) 0 0
\(747\) 0.539478 0.0197385
\(748\) 0 0
\(749\) −11.0656 −0.404330
\(750\) 0 0
\(751\) 15.5132 0.566086 0.283043 0.959107i \(-0.408656\pi\)
0.283043 + 0.959107i \(0.408656\pi\)
\(752\) 0 0
\(753\) −26.2547 −0.956775
\(754\) 0 0
\(755\) −29.0299 −1.05651
\(756\) 0 0
\(757\) 20.1852 0.733644 0.366822 0.930291i \(-0.380446\pi\)
0.366822 + 0.930291i \(0.380446\pi\)
\(758\) 0 0
\(759\) 3.74405 0.135901
\(760\) 0 0
\(761\) 5.09852 0.184821 0.0924106 0.995721i \(-0.470543\pi\)
0.0924106 + 0.995721i \(0.470543\pi\)
\(762\) 0 0
\(763\) 20.6532 0.747695
\(764\) 0 0
\(765\) −0.845813 −0.0305804
\(766\) 0 0
\(767\) 3.41023 0.123136
\(768\) 0 0
\(769\) 34.3364 1.23820 0.619101 0.785311i \(-0.287497\pi\)
0.619101 + 0.785311i \(0.287497\pi\)
\(770\) 0 0
\(771\) −25.8500 −0.930966
\(772\) 0 0
\(773\) −37.6942 −1.35577 −0.677883 0.735169i \(-0.737103\pi\)
−0.677883 + 0.735169i \(0.737103\pi\)
\(774\) 0 0
\(775\) 16.0721 0.577327
\(776\) 0 0
\(777\) 20.6576 0.741086
\(778\) 0 0
\(779\) 7.59051 0.271958
\(780\) 0 0
\(781\) −6.57583 −0.235302
\(782\) 0 0
\(783\) 4.48876 0.160415
\(784\) 0 0
\(785\) −18.0016 −0.642504
\(786\) 0 0
\(787\) 14.9530 0.533016 0.266508 0.963833i \(-0.414130\pi\)
0.266508 + 0.963833i \(0.414130\pi\)
\(788\) 0 0
\(789\) 9.05966 0.322532
\(790\) 0 0
\(791\) −47.3728 −1.68438
\(792\) 0 0
\(793\) −11.0264 −0.391558
\(794\) 0 0
\(795\) −2.91401 −0.103349
\(796\) 0 0
\(797\) −2.47658 −0.0877250 −0.0438625 0.999038i \(-0.513966\pi\)
−0.0438625 + 0.999038i \(0.513966\pi\)
\(798\) 0 0
\(799\) −0.0708661 −0.00250706
\(800\) 0 0
\(801\) 5.24695 0.185392
\(802\) 0 0
\(803\) 0.161081 0.00568441
\(804\) 0 0
\(805\) −39.7553 −1.40119
\(806\) 0 0
\(807\) −22.5095 −0.792370
\(808\) 0 0
\(809\) −4.33836 −0.152528 −0.0762642 0.997088i \(-0.524299\pi\)
−0.0762642 + 0.997088i \(0.524299\pi\)
\(810\) 0 0
\(811\) −16.1575 −0.567368 −0.283684 0.958918i \(-0.591557\pi\)
−0.283684 + 0.958918i \(0.591557\pi\)
\(812\) 0 0
\(813\) −9.12378 −0.319985
\(814\) 0 0
\(815\) −1.37625 −0.0482080
\(816\) 0 0
\(817\) 30.2288 1.05757
\(818\) 0 0
\(819\) −11.1462 −0.389481
\(820\) 0 0
\(821\) −46.1374 −1.61021 −0.805104 0.593134i \(-0.797890\pi\)
−0.805104 + 0.593134i \(0.797890\pi\)
\(822\) 0 0
\(823\) −8.60911 −0.300095 −0.150047 0.988679i \(-0.547943\pi\)
−0.150047 + 0.988679i \(0.547943\pi\)
\(824\) 0 0
\(825\) −1.73416 −0.0603757
\(826\) 0 0
\(827\) 5.69291 0.197962 0.0989810 0.995089i \(-0.468442\pi\)
0.0989810 + 0.995089i \(0.468442\pi\)
\(828\) 0 0
\(829\) −29.2285 −1.01515 −0.507573 0.861609i \(-0.669457\pi\)
−0.507573 + 0.861609i \(0.669457\pi\)
\(830\) 0 0
\(831\) −14.2854 −0.495556
\(832\) 0 0
\(833\) −7.68853 −0.266392
\(834\) 0 0
\(835\) −1.56175 −0.0540467
\(836\) 0 0
\(837\) −6.27589 −0.216926
\(838\) 0 0
\(839\) −38.4973 −1.32907 −0.664537 0.747255i \(-0.731371\pi\)
−0.664537 + 0.747255i \(0.731371\pi\)
\(840\) 0 0
\(841\) −8.85099 −0.305207
\(842\) 0 0
\(843\) −23.0548 −0.794048
\(844\) 0 0
\(845\) −11.1489 −0.383534
\(846\) 0 0
\(847\) 48.5325 1.66760
\(848\) 0 0
\(849\) 27.7777 0.953328
\(850\) 0 0
\(851\) −24.8083 −0.850418
\(852\) 0 0
\(853\) −23.7885 −0.814503 −0.407252 0.913316i \(-0.633513\pi\)
−0.407252 + 0.913316i \(0.633513\pi\)
\(854\) 0 0
\(855\) −4.47431 −0.153018
\(856\) 0 0
\(857\) −41.9478 −1.43291 −0.716454 0.697634i \(-0.754236\pi\)
−0.716454 + 0.697634i \(0.754236\pi\)
\(858\) 0 0
\(859\) −7.74487 −0.264252 −0.132126 0.991233i \(-0.542180\pi\)
−0.132126 + 0.991233i \(0.542180\pi\)
\(860\) 0 0
\(861\) 12.1980 0.415707
\(862\) 0 0
\(863\) −12.3871 −0.421663 −0.210831 0.977522i \(-0.567617\pi\)
−0.210831 + 0.977522i \(0.567617\pi\)
\(864\) 0 0
\(865\) 30.5064 1.03725
\(866\) 0 0
\(867\) −16.7067 −0.567389
\(868\) 0 0
\(869\) −7.96083 −0.270053
\(870\) 0 0
\(871\) 5.80075 0.196551
\(872\) 0 0
\(873\) −18.9230 −0.640447
\(874\) 0 0
\(875\) 54.3650 1.83787
\(876\) 0 0
\(877\) −16.3076 −0.550667 −0.275334 0.961349i \(-0.588788\pi\)
−0.275334 + 0.961349i \(0.588788\pi\)
\(878\) 0 0
\(879\) 25.0738 0.845719
\(880\) 0 0
\(881\) −57.6688 −1.94291 −0.971455 0.237224i \(-0.923763\pi\)
−0.971455 + 0.237224i \(0.923763\pi\)
\(882\) 0 0
\(883\) 46.7514 1.57331 0.786654 0.617394i \(-0.211811\pi\)
0.786654 + 0.617394i \(0.211811\pi\)
\(884\) 0 0
\(885\) 2.19988 0.0739483
\(886\) 0 0
\(887\) 23.2845 0.781817 0.390908 0.920430i \(-0.372161\pi\)
0.390908 + 0.920430i \(0.372161\pi\)
\(888\) 0 0
\(889\) 66.0972 2.21683
\(890\) 0 0
\(891\) 0.677161 0.0226857
\(892\) 0 0
\(893\) −0.374879 −0.0125448
\(894\) 0 0
\(895\) −32.9358 −1.10092
\(896\) 0 0
\(897\) 13.3859 0.446941
\(898\) 0 0
\(899\) −28.1710 −0.939555
\(900\) 0 0
\(901\) 1.01051 0.0336650
\(902\) 0 0
\(903\) 48.5780 1.61658
\(904\) 0 0
\(905\) 22.3683 0.743549
\(906\) 0 0
\(907\) 28.2329 0.937457 0.468729 0.883342i \(-0.344712\pi\)
0.468729 + 0.883342i \(0.344712\pi\)
\(908\) 0 0
\(909\) 19.2071 0.637061
\(910\) 0 0
\(911\) −10.9302 −0.362134 −0.181067 0.983471i \(-0.557955\pi\)
−0.181067 + 0.983471i \(0.557955\pi\)
\(912\) 0 0
\(913\) 0.365314 0.0120901
\(914\) 0 0
\(915\) −7.11293 −0.235146
\(916\) 0 0
\(917\) −46.8816 −1.54817
\(918\) 0 0
\(919\) −11.5647 −0.381483 −0.190742 0.981640i \(-0.561089\pi\)
−0.190742 + 0.981640i \(0.561089\pi\)
\(920\) 0 0
\(921\) −11.1106 −0.366107
\(922\) 0 0
\(923\) −23.5101 −0.773846
\(924\) 0 0
\(925\) 11.4906 0.377810
\(926\) 0 0
\(927\) 14.1285 0.464040
\(928\) 0 0
\(929\) 47.7565 1.56684 0.783420 0.621492i \(-0.213473\pi\)
0.783420 + 0.621492i \(0.213473\pi\)
\(930\) 0 0
\(931\) −40.6720 −1.33297
\(932\) 0 0
\(933\) −0.770347 −0.0252200
\(934\) 0 0
\(935\) −0.572751 −0.0187310
\(936\) 0 0
\(937\) 32.0882 1.04828 0.524138 0.851634i \(-0.324388\pi\)
0.524138 + 0.851634i \(0.324388\pi\)
\(938\) 0 0
\(939\) −2.71480 −0.0885942
\(940\) 0 0
\(941\) 40.1773 1.30974 0.654872 0.755740i \(-0.272723\pi\)
0.654872 + 0.755740i \(0.272723\pi\)
\(942\) 0 0
\(943\) −14.6490 −0.477037
\(944\) 0 0
\(945\) −7.19026 −0.233899
\(946\) 0 0
\(947\) −15.1399 −0.491980 −0.245990 0.969272i \(-0.579113\pi\)
−0.245990 + 0.969272i \(0.579113\pi\)
\(948\) 0 0
\(949\) 0.575901 0.0186945
\(950\) 0 0
\(951\) −20.2749 −0.657457
\(952\) 0 0
\(953\) −48.8590 −1.58270 −0.791350 0.611364i \(-0.790621\pi\)
−0.791350 + 0.611364i \(0.790621\pi\)
\(954\) 0 0
\(955\) −3.02645 −0.0979336
\(956\) 0 0
\(957\) 3.03962 0.0982568
\(958\) 0 0
\(959\) 39.8309 1.28621
\(960\) 0 0
\(961\) 8.38674 0.270540
\(962\) 0 0
\(963\) 2.40350 0.0774518
\(964\) 0 0
\(965\) −32.9922 −1.06206
\(966\) 0 0
\(967\) 17.6147 0.566452 0.283226 0.959053i \(-0.408595\pi\)
0.283226 + 0.959053i \(0.408595\pi\)
\(968\) 0 0
\(969\) 1.55159 0.0498441
\(970\) 0 0
\(971\) 1.82455 0.0585526 0.0292763 0.999571i \(-0.490680\pi\)
0.0292763 + 0.999571i \(0.490680\pi\)
\(972\) 0 0
\(973\) −82.2341 −2.63630
\(974\) 0 0
\(975\) −6.20003 −0.198560
\(976\) 0 0
\(977\) 13.4709 0.430972 0.215486 0.976507i \(-0.430866\pi\)
0.215486 + 0.976507i \(0.430866\pi\)
\(978\) 0 0
\(979\) 3.55303 0.113555
\(980\) 0 0
\(981\) −4.48595 −0.143225
\(982\) 0 0
\(983\) −27.6193 −0.880920 −0.440460 0.897772i \(-0.645185\pi\)
−0.440460 + 0.897772i \(0.645185\pi\)
\(984\) 0 0
\(985\) 1.40914 0.0448989
\(986\) 0 0
\(987\) −0.602433 −0.0191757
\(988\) 0 0
\(989\) −58.3389 −1.85507
\(990\) 0 0
\(991\) 4.62752 0.146998 0.0734990 0.997295i \(-0.476583\pi\)
0.0734990 + 0.997295i \(0.476583\pi\)
\(992\) 0 0
\(993\) −11.5475 −0.366448
\(994\) 0 0
\(995\) −15.6327 −0.495589
\(996\) 0 0
\(997\) 60.6827 1.92184 0.960921 0.276824i \(-0.0892818\pi\)
0.960921 + 0.276824i \(0.0892818\pi\)
\(998\) 0 0
\(999\) −4.48690 −0.141959
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 8016.2.a.z.1.7 8
4.3 odd 2 501.2.a.d.1.3 8
12.11 even 2 1503.2.a.f.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.d.1.3 8 4.3 odd 2
1503.2.a.f.1.6 8 12.11 even 2
8016.2.a.z.1.7 8 1.1 even 1 trivial