Properties

Label 5225.2.a.bc.1.19
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.661223 q^{2} -3.01131 q^{3} -1.56278 q^{4} -1.99115 q^{6} -0.592449 q^{7} -2.35579 q^{8} +6.06799 q^{9} +O(q^{10})\) \(q+0.661223 q^{2} -3.01131 q^{3} -1.56278 q^{4} -1.99115 q^{6} -0.592449 q^{7} -2.35579 q^{8} +6.06799 q^{9} +1.00000 q^{11} +4.70603 q^{12} -3.68157 q^{13} -0.391741 q^{14} +1.56786 q^{16} -0.251015 q^{17} +4.01230 q^{18} +1.00000 q^{19} +1.78405 q^{21} +0.661223 q^{22} -2.58194 q^{23} +7.09403 q^{24} -2.43434 q^{26} -9.23869 q^{27} +0.925870 q^{28} +2.22782 q^{29} -7.50257 q^{31} +5.74830 q^{32} -3.01131 q^{33} -0.165977 q^{34} -9.48297 q^{36} -9.64498 q^{37} +0.661223 q^{38} +11.0864 q^{39} +5.78336 q^{41} +1.17965 q^{42} +0.283462 q^{43} -1.56278 q^{44} -1.70724 q^{46} +3.78624 q^{47} -4.72132 q^{48} -6.64900 q^{49} +0.755885 q^{51} +5.75350 q^{52} -3.89730 q^{53} -6.10883 q^{54} +1.39569 q^{56} -3.01131 q^{57} +1.47309 q^{58} -2.98649 q^{59} -8.94657 q^{61} -4.96087 q^{62} -3.59498 q^{63} +0.665182 q^{64} -1.99115 q^{66} -3.70414 q^{67} +0.392283 q^{68} +7.77503 q^{69} +16.1865 q^{71} -14.2950 q^{72} -7.91613 q^{73} -6.37748 q^{74} -1.56278 q^{76} -0.592449 q^{77} +7.33056 q^{78} -15.4023 q^{79} +9.61657 q^{81} +3.82409 q^{82} -3.74222 q^{83} -2.78808 q^{84} +0.187432 q^{86} -6.70866 q^{87} -2.35579 q^{88} -3.78573 q^{89} +2.18114 q^{91} +4.03502 q^{92} +22.5926 q^{93} +2.50355 q^{94} -17.3099 q^{96} -9.18878 q^{97} -4.39647 q^{98} +6.06799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.661223 0.467555 0.233778 0.972290i \(-0.424891\pi\)
0.233778 + 0.972290i \(0.424891\pi\)
\(3\) −3.01131 −1.73858 −0.869291 0.494301i \(-0.835424\pi\)
−0.869291 + 0.494301i \(0.835424\pi\)
\(4\) −1.56278 −0.781392
\(5\) 0 0
\(6\) −1.99115 −0.812883
\(7\) −0.592449 −0.223925 −0.111962 0.993712i \(-0.535714\pi\)
−0.111962 + 0.993712i \(0.535714\pi\)
\(8\) −2.35579 −0.832899
\(9\) 6.06799 2.02266
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 4.70603 1.35851
\(13\) −3.68157 −1.02108 −0.510542 0.859853i \(-0.670555\pi\)
−0.510542 + 0.859853i \(0.670555\pi\)
\(14\) −0.391741 −0.104697
\(15\) 0 0
\(16\) 1.56786 0.391966
\(17\) −0.251015 −0.0608801 −0.0304401 0.999537i \(-0.509691\pi\)
−0.0304401 + 0.999537i \(0.509691\pi\)
\(18\) 4.01230 0.945708
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.78405 0.389311
\(22\) 0.661223 0.140973
\(23\) −2.58194 −0.538372 −0.269186 0.963088i \(-0.586755\pi\)
−0.269186 + 0.963088i \(0.586755\pi\)
\(24\) 7.09403 1.44806
\(25\) 0 0
\(26\) −2.43434 −0.477414
\(27\) −9.23869 −1.77799
\(28\) 0.925870 0.174973
\(29\) 2.22782 0.413696 0.206848 0.978373i \(-0.433679\pi\)
0.206848 + 0.978373i \(0.433679\pi\)
\(30\) 0 0
\(31\) −7.50257 −1.34750 −0.673751 0.738959i \(-0.735318\pi\)
−0.673751 + 0.738959i \(0.735318\pi\)
\(32\) 5.74830 1.01616
\(33\) −3.01131 −0.524202
\(34\) −0.165977 −0.0284648
\(35\) 0 0
\(36\) −9.48297 −1.58049
\(37\) −9.64498 −1.58562 −0.792812 0.609466i \(-0.791384\pi\)
−0.792812 + 0.609466i \(0.791384\pi\)
\(38\) 0.661223 0.107265
\(39\) 11.0864 1.77524
\(40\) 0 0
\(41\) 5.78336 0.903209 0.451605 0.892218i \(-0.350852\pi\)
0.451605 + 0.892218i \(0.350852\pi\)
\(42\) 1.17965 0.182025
\(43\) 0.283462 0.0432275 0.0216138 0.999766i \(-0.493120\pi\)
0.0216138 + 0.999766i \(0.493120\pi\)
\(44\) −1.56278 −0.235599
\(45\) 0 0
\(46\) −1.70724 −0.251719
\(47\) 3.78624 0.552279 0.276140 0.961118i \(-0.410945\pi\)
0.276140 + 0.961118i \(0.410945\pi\)
\(48\) −4.72132 −0.681464
\(49\) −6.64900 −0.949858
\(50\) 0 0
\(51\) 0.755885 0.105845
\(52\) 5.75350 0.797868
\(53\) −3.89730 −0.535336 −0.267668 0.963511i \(-0.586253\pi\)
−0.267668 + 0.963511i \(0.586253\pi\)
\(54\) −6.10883 −0.831307
\(55\) 0 0
\(56\) 1.39569 0.186507
\(57\) −3.01131 −0.398858
\(58\) 1.47309 0.193426
\(59\) −2.98649 −0.388808 −0.194404 0.980922i \(-0.562277\pi\)
−0.194404 + 0.980922i \(0.562277\pi\)
\(60\) 0 0
\(61\) −8.94657 −1.14549 −0.572746 0.819733i \(-0.694122\pi\)
−0.572746 + 0.819733i \(0.694122\pi\)
\(62\) −4.96087 −0.630031
\(63\) −3.59498 −0.452925
\(64\) 0.665182 0.0831477
\(65\) 0 0
\(66\) −1.99115 −0.245093
\(67\) −3.70414 −0.452533 −0.226266 0.974065i \(-0.572652\pi\)
−0.226266 + 0.974065i \(0.572652\pi\)
\(68\) 0.392283 0.0475713
\(69\) 7.77503 0.936003
\(70\) 0 0
\(71\) 16.1865 1.92099 0.960493 0.278304i \(-0.0897724\pi\)
0.960493 + 0.278304i \(0.0897724\pi\)
\(72\) −14.2950 −1.68468
\(73\) −7.91613 −0.926513 −0.463257 0.886224i \(-0.653319\pi\)
−0.463257 + 0.886224i \(0.653319\pi\)
\(74\) −6.37748 −0.741367
\(75\) 0 0
\(76\) −1.56278 −0.179264
\(77\) −0.592449 −0.0675158
\(78\) 7.33056 0.830022
\(79\) −15.4023 −1.73289 −0.866447 0.499270i \(-0.833602\pi\)
−0.866447 + 0.499270i \(0.833602\pi\)
\(80\) 0 0
\(81\) 9.61657 1.06851
\(82\) 3.82409 0.422300
\(83\) −3.74222 −0.410762 −0.205381 0.978682i \(-0.565843\pi\)
−0.205381 + 0.978682i \(0.565843\pi\)
\(84\) −2.78808 −0.304205
\(85\) 0 0
\(86\) 0.187432 0.0202113
\(87\) −6.70866 −0.719244
\(88\) −2.35579 −0.251129
\(89\) −3.78573 −0.401286 −0.200643 0.979664i \(-0.564303\pi\)
−0.200643 + 0.979664i \(0.564303\pi\)
\(90\) 0 0
\(91\) 2.18114 0.228646
\(92\) 4.03502 0.420680
\(93\) 22.5926 2.34274
\(94\) 2.50355 0.258221
\(95\) 0 0
\(96\) −17.3099 −1.76669
\(97\) −9.18878 −0.932979 −0.466489 0.884527i \(-0.654481\pi\)
−0.466489 + 0.884527i \(0.654481\pi\)
\(98\) −4.39647 −0.444111
\(99\) 6.06799 0.609856
\(100\) 0 0
\(101\) −14.1183 −1.40482 −0.702411 0.711772i \(-0.747893\pi\)
−0.702411 + 0.711772i \(0.747893\pi\)
\(102\) 0.499809 0.0494884
\(103\) −5.20319 −0.512686 −0.256343 0.966586i \(-0.582518\pi\)
−0.256343 + 0.966586i \(0.582518\pi\)
\(104\) 8.67303 0.850461
\(105\) 0 0
\(106\) −2.57699 −0.250299
\(107\) 12.8863 1.24576 0.622881 0.782316i \(-0.285962\pi\)
0.622881 + 0.782316i \(0.285962\pi\)
\(108\) 14.4381 1.38930
\(109\) 3.10981 0.297866 0.148933 0.988847i \(-0.452416\pi\)
0.148933 + 0.988847i \(0.452416\pi\)
\(110\) 0 0
\(111\) 29.0440 2.75674
\(112\) −0.928879 −0.0877708
\(113\) −7.27809 −0.684665 −0.342333 0.939579i \(-0.611217\pi\)
−0.342333 + 0.939579i \(0.611217\pi\)
\(114\) −1.99115 −0.186488
\(115\) 0 0
\(116\) −3.48160 −0.323259
\(117\) −22.3398 −2.06531
\(118\) −1.97474 −0.181789
\(119\) 0.148714 0.0136326
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.91568 −0.535581
\(123\) −17.4155 −1.57030
\(124\) 11.7249 1.05293
\(125\) 0 0
\(126\) −2.37708 −0.211767
\(127\) −16.8696 −1.49694 −0.748468 0.663171i \(-0.769210\pi\)
−0.748468 + 0.663171i \(0.769210\pi\)
\(128\) −11.0568 −0.977289
\(129\) −0.853592 −0.0751546
\(130\) 0 0
\(131\) −11.4376 −0.999309 −0.499655 0.866225i \(-0.666540\pi\)
−0.499655 + 0.866225i \(0.666540\pi\)
\(132\) 4.70603 0.409607
\(133\) −0.592449 −0.0513718
\(134\) −2.44926 −0.211584
\(135\) 0 0
\(136\) 0.591340 0.0507070
\(137\) 16.7276 1.42914 0.714569 0.699565i \(-0.246623\pi\)
0.714569 + 0.699565i \(0.246623\pi\)
\(138\) 5.14103 0.437633
\(139\) 16.6330 1.41079 0.705397 0.708813i \(-0.250769\pi\)
0.705397 + 0.708813i \(0.250769\pi\)
\(140\) 0 0
\(141\) −11.4015 −0.960182
\(142\) 10.7029 0.898167
\(143\) −3.68157 −0.307869
\(144\) 9.51378 0.792815
\(145\) 0 0
\(146\) −5.23433 −0.433196
\(147\) 20.0222 1.65140
\(148\) 15.0730 1.23899
\(149\) 14.3567 1.17614 0.588072 0.808808i \(-0.299887\pi\)
0.588072 + 0.808808i \(0.299887\pi\)
\(150\) 0 0
\(151\) 16.0364 1.30502 0.652511 0.757779i \(-0.273716\pi\)
0.652511 + 0.757779i \(0.273716\pi\)
\(152\) −2.35579 −0.191080
\(153\) −1.52316 −0.123140
\(154\) −0.391741 −0.0315674
\(155\) 0 0
\(156\) −17.3256 −1.38716
\(157\) 24.1525 1.92758 0.963788 0.266670i \(-0.0859234\pi\)
0.963788 + 0.266670i \(0.0859234\pi\)
\(158\) −10.1844 −0.810223
\(159\) 11.7360 0.930725
\(160\) 0 0
\(161\) 1.52967 0.120555
\(162\) 6.35870 0.499587
\(163\) 16.3016 1.27684 0.638421 0.769687i \(-0.279588\pi\)
0.638421 + 0.769687i \(0.279588\pi\)
\(164\) −9.03814 −0.705760
\(165\) 0 0
\(166\) −2.47444 −0.192054
\(167\) 16.9851 1.31435 0.657173 0.753740i \(-0.271752\pi\)
0.657173 + 0.753740i \(0.271752\pi\)
\(168\) −4.20285 −0.324257
\(169\) 0.553985 0.0426142
\(170\) 0 0
\(171\) 6.06799 0.464031
\(172\) −0.442990 −0.0337776
\(173\) 6.48231 0.492841 0.246420 0.969163i \(-0.420746\pi\)
0.246420 + 0.969163i \(0.420746\pi\)
\(174\) −4.43592 −0.336286
\(175\) 0 0
\(176\) 1.56786 0.118182
\(177\) 8.99326 0.675975
\(178\) −2.50321 −0.187624
\(179\) 2.50231 0.187031 0.0935157 0.995618i \(-0.470189\pi\)
0.0935157 + 0.995618i \(0.470189\pi\)
\(180\) 0 0
\(181\) 3.08136 0.229036 0.114518 0.993421i \(-0.463468\pi\)
0.114518 + 0.993421i \(0.463468\pi\)
\(182\) 1.44222 0.106905
\(183\) 26.9409 1.99153
\(184\) 6.08252 0.448410
\(185\) 0 0
\(186\) 14.9387 1.09536
\(187\) −0.251015 −0.0183561
\(188\) −5.91707 −0.431547
\(189\) 5.47345 0.398135
\(190\) 0 0
\(191\) 6.02462 0.435926 0.217963 0.975957i \(-0.430059\pi\)
0.217963 + 0.975957i \(0.430059\pi\)
\(192\) −2.00307 −0.144559
\(193\) −9.85410 −0.709314 −0.354657 0.934996i \(-0.615402\pi\)
−0.354657 + 0.934996i \(0.615402\pi\)
\(194\) −6.07583 −0.436219
\(195\) 0 0
\(196\) 10.3910 0.742211
\(197\) −10.3472 −0.737208 −0.368604 0.929587i \(-0.620164\pi\)
−0.368604 + 0.929587i \(0.620164\pi\)
\(198\) 4.01230 0.285142
\(199\) 9.67189 0.685622 0.342811 0.939404i \(-0.388621\pi\)
0.342811 + 0.939404i \(0.388621\pi\)
\(200\) 0 0
\(201\) 11.1543 0.786765
\(202\) −9.33533 −0.656832
\(203\) −1.31987 −0.0926367
\(204\) −1.18128 −0.0827065
\(205\) 0 0
\(206\) −3.44047 −0.239709
\(207\) −15.6672 −1.08895
\(208\) −5.77220 −0.400230
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −5.34846 −0.368203 −0.184102 0.982907i \(-0.558938\pi\)
−0.184102 + 0.982907i \(0.558938\pi\)
\(212\) 6.09065 0.418307
\(213\) −48.7426 −3.33979
\(214\) 8.52070 0.582463
\(215\) 0 0
\(216\) 21.7644 1.48088
\(217\) 4.44489 0.301739
\(218\) 2.05628 0.139269
\(219\) 23.8379 1.61082
\(220\) 0 0
\(221\) 0.924131 0.0621638
\(222\) 19.2046 1.28893
\(223\) −3.76862 −0.252365 −0.126183 0.992007i \(-0.540273\pi\)
−0.126183 + 0.992007i \(0.540273\pi\)
\(224\) −3.40557 −0.227544
\(225\) 0 0
\(226\) −4.81244 −0.320119
\(227\) 3.70626 0.245993 0.122997 0.992407i \(-0.460750\pi\)
0.122997 + 0.992407i \(0.460750\pi\)
\(228\) 4.70603 0.311664
\(229\) −7.17137 −0.473897 −0.236949 0.971522i \(-0.576147\pi\)
−0.236949 + 0.971522i \(0.576147\pi\)
\(230\) 0 0
\(231\) 1.78405 0.117382
\(232\) −5.24829 −0.344567
\(233\) 6.50149 0.425927 0.212964 0.977060i \(-0.431688\pi\)
0.212964 + 0.977060i \(0.431688\pi\)
\(234\) −14.7716 −0.965648
\(235\) 0 0
\(236\) 4.66724 0.303812
\(237\) 46.3811 3.01278
\(238\) 0.0983329 0.00637398
\(239\) 23.9030 1.54616 0.773079 0.634310i \(-0.218716\pi\)
0.773079 + 0.634310i \(0.218716\pi\)
\(240\) 0 0
\(241\) 15.7035 1.01155 0.505777 0.862664i \(-0.331206\pi\)
0.505777 + 0.862664i \(0.331206\pi\)
\(242\) 0.661223 0.0425050
\(243\) −1.24243 −0.0797022
\(244\) 13.9816 0.895078
\(245\) 0 0
\(246\) −11.5155 −0.734203
\(247\) −3.68157 −0.234253
\(248\) 17.6745 1.12233
\(249\) 11.2690 0.714144
\(250\) 0 0
\(251\) 8.82062 0.556753 0.278376 0.960472i \(-0.410204\pi\)
0.278376 + 0.960472i \(0.410204\pi\)
\(252\) 5.61817 0.353912
\(253\) −2.58194 −0.162325
\(254\) −11.1546 −0.699901
\(255\) 0 0
\(256\) −8.64135 −0.540084
\(257\) −9.32567 −0.581719 −0.290860 0.956766i \(-0.593941\pi\)
−0.290860 + 0.956766i \(0.593941\pi\)
\(258\) −0.564415 −0.0351389
\(259\) 5.71416 0.355060
\(260\) 0 0
\(261\) 13.5184 0.836768
\(262\) −7.56282 −0.467232
\(263\) −23.7768 −1.46614 −0.733070 0.680153i \(-0.761913\pi\)
−0.733070 + 0.680153i \(0.761913\pi\)
\(264\) 7.09403 0.436607
\(265\) 0 0
\(266\) −0.391741 −0.0240192
\(267\) 11.4000 0.697669
\(268\) 5.78877 0.353605
\(269\) −14.8523 −0.905562 −0.452781 0.891622i \(-0.649568\pi\)
−0.452781 + 0.891622i \(0.649568\pi\)
\(270\) 0 0
\(271\) 13.9947 0.850117 0.425059 0.905166i \(-0.360253\pi\)
0.425059 + 0.905166i \(0.360253\pi\)
\(272\) −0.393557 −0.0238629
\(273\) −6.56811 −0.397520
\(274\) 11.0607 0.668201
\(275\) 0 0
\(276\) −12.1507 −0.731386
\(277\) 3.84886 0.231256 0.115628 0.993293i \(-0.463112\pi\)
0.115628 + 0.993293i \(0.463112\pi\)
\(278\) 10.9981 0.659624
\(279\) −45.5255 −2.72554
\(280\) 0 0
\(281\) 17.9323 1.06975 0.534877 0.844930i \(-0.320358\pi\)
0.534877 + 0.844930i \(0.320358\pi\)
\(282\) −7.53896 −0.448938
\(283\) −15.7342 −0.935304 −0.467652 0.883913i \(-0.654900\pi\)
−0.467652 + 0.883913i \(0.654900\pi\)
\(284\) −25.2960 −1.50104
\(285\) 0 0
\(286\) −2.43434 −0.143946
\(287\) −3.42635 −0.202251
\(288\) 34.8806 2.05536
\(289\) −16.9370 −0.996294
\(290\) 0 0
\(291\) 27.6703 1.62206
\(292\) 12.3712 0.723970
\(293\) −20.1233 −1.17562 −0.587808 0.809000i \(-0.700009\pi\)
−0.587808 + 0.809000i \(0.700009\pi\)
\(294\) 13.2392 0.772123
\(295\) 0 0
\(296\) 22.7216 1.32067
\(297\) −9.23869 −0.536083
\(298\) 9.49297 0.549913
\(299\) 9.50561 0.549723
\(300\) 0 0
\(301\) −0.167937 −0.00967971
\(302\) 10.6036 0.610170
\(303\) 42.5145 2.44240
\(304\) 1.56786 0.0899231
\(305\) 0 0
\(306\) −1.00715 −0.0575748
\(307\) −13.0048 −0.742226 −0.371113 0.928588i \(-0.621024\pi\)
−0.371113 + 0.928588i \(0.621024\pi\)
\(308\) 0.925870 0.0527563
\(309\) 15.6684 0.891346
\(310\) 0 0
\(311\) 23.0064 1.30457 0.652287 0.757972i \(-0.273810\pi\)
0.652287 + 0.757972i \(0.273810\pi\)
\(312\) −26.1172 −1.47860
\(313\) 10.8793 0.614936 0.307468 0.951558i \(-0.400518\pi\)
0.307468 + 0.951558i \(0.400518\pi\)
\(314\) 15.9702 0.901248
\(315\) 0 0
\(316\) 24.0705 1.35407
\(317\) 13.6489 0.766599 0.383300 0.923624i \(-0.374788\pi\)
0.383300 + 0.923624i \(0.374788\pi\)
\(318\) 7.76011 0.435165
\(319\) 2.22782 0.124734
\(320\) 0 0
\(321\) −38.8046 −2.16586
\(322\) 1.01145 0.0563660
\(323\) −0.251015 −0.0139669
\(324\) −15.0286 −0.834924
\(325\) 0 0
\(326\) 10.7790 0.596994
\(327\) −9.36461 −0.517864
\(328\) −13.6244 −0.752282
\(329\) −2.24315 −0.123669
\(330\) 0 0
\(331\) 8.45886 0.464941 0.232471 0.972603i \(-0.425319\pi\)
0.232471 + 0.972603i \(0.425319\pi\)
\(332\) 5.84829 0.320966
\(333\) −58.5257 −3.20719
\(334\) 11.2309 0.614529
\(335\) 0 0
\(336\) 2.79714 0.152597
\(337\) −5.76453 −0.314014 −0.157007 0.987598i \(-0.550184\pi\)
−0.157007 + 0.987598i \(0.550184\pi\)
\(338\) 0.366308 0.0199245
\(339\) 21.9166 1.19035
\(340\) 0 0
\(341\) −7.50257 −0.406287
\(342\) 4.01230 0.216960
\(343\) 8.08634 0.436621
\(344\) −0.667778 −0.0360042
\(345\) 0 0
\(346\) 4.28625 0.230430
\(347\) −20.2740 −1.08837 −0.544183 0.838966i \(-0.683160\pi\)
−0.544183 + 0.838966i \(0.683160\pi\)
\(348\) 10.4842 0.562011
\(349\) −4.09845 −0.219385 −0.109693 0.993966i \(-0.534987\pi\)
−0.109693 + 0.993966i \(0.534987\pi\)
\(350\) 0 0
\(351\) 34.0129 1.81547
\(352\) 5.74830 0.306385
\(353\) 26.6199 1.41684 0.708418 0.705793i \(-0.249409\pi\)
0.708418 + 0.705793i \(0.249409\pi\)
\(354\) 5.94655 0.316056
\(355\) 0 0
\(356\) 5.91628 0.313562
\(357\) −0.447823 −0.0237013
\(358\) 1.65459 0.0874476
\(359\) 22.2264 1.17307 0.586534 0.809925i \(-0.300492\pi\)
0.586534 + 0.809925i \(0.300492\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.03747 0.107087
\(363\) −3.01131 −0.158053
\(364\) −3.40866 −0.178662
\(365\) 0 0
\(366\) 17.8140 0.931150
\(367\) 24.0546 1.25564 0.627820 0.778358i \(-0.283947\pi\)
0.627820 + 0.778358i \(0.283947\pi\)
\(368\) −4.04813 −0.211023
\(369\) 35.0934 1.82689
\(370\) 0 0
\(371\) 2.30895 0.119875
\(372\) −35.3073 −1.83060
\(373\) 0.342698 0.0177442 0.00887211 0.999961i \(-0.497176\pi\)
0.00887211 + 0.999961i \(0.497176\pi\)
\(374\) −0.165977 −0.00858247
\(375\) 0 0
\(376\) −8.91959 −0.459993
\(377\) −8.20188 −0.422418
\(378\) 3.61917 0.186150
\(379\) −30.4031 −1.56170 −0.780851 0.624717i \(-0.785214\pi\)
−0.780851 + 0.624717i \(0.785214\pi\)
\(380\) 0 0
\(381\) 50.7997 2.60255
\(382\) 3.98361 0.203819
\(383\) 26.5955 1.35897 0.679483 0.733692i \(-0.262204\pi\)
0.679483 + 0.733692i \(0.262204\pi\)
\(384\) 33.2953 1.69910
\(385\) 0 0
\(386\) −6.51576 −0.331643
\(387\) 1.72005 0.0874348
\(388\) 14.3601 0.729022
\(389\) 4.46007 0.226134 0.113067 0.993587i \(-0.463932\pi\)
0.113067 + 0.993587i \(0.463932\pi\)
\(390\) 0 0
\(391\) 0.648107 0.0327762
\(392\) 15.6637 0.791136
\(393\) 34.4422 1.73738
\(394\) −6.84181 −0.344686
\(395\) 0 0
\(396\) −9.48297 −0.476537
\(397\) −6.23802 −0.313077 −0.156539 0.987672i \(-0.550034\pi\)
−0.156539 + 0.987672i \(0.550034\pi\)
\(398\) 6.39528 0.320566
\(399\) 1.78405 0.0893141
\(400\) 0 0
\(401\) 18.8302 0.940336 0.470168 0.882577i \(-0.344193\pi\)
0.470168 + 0.882577i \(0.344193\pi\)
\(402\) 7.37549 0.367856
\(403\) 27.6213 1.37591
\(404\) 22.0638 1.09772
\(405\) 0 0
\(406\) −0.872728 −0.0433128
\(407\) −9.64498 −0.478084
\(408\) −1.78071 −0.0881583
\(409\) −29.9488 −1.48087 −0.740437 0.672126i \(-0.765381\pi\)
−0.740437 + 0.672126i \(0.765381\pi\)
\(410\) 0 0
\(411\) −50.3721 −2.48467
\(412\) 8.13146 0.400609
\(413\) 1.76935 0.0870638
\(414\) −10.3595 −0.509142
\(415\) 0 0
\(416\) −21.1628 −1.03759
\(417\) −50.0872 −2.45278
\(418\) 0.661223 0.0323415
\(419\) −27.8201 −1.35910 −0.679551 0.733628i \(-0.737826\pi\)
−0.679551 + 0.733628i \(0.737826\pi\)
\(420\) 0 0
\(421\) −28.7544 −1.40140 −0.700701 0.713455i \(-0.747129\pi\)
−0.700701 + 0.713455i \(0.747129\pi\)
\(422\) −3.53652 −0.172155
\(423\) 22.9749 1.11708
\(424\) 9.18125 0.445881
\(425\) 0 0
\(426\) −32.2298 −1.56154
\(427\) 5.30039 0.256504
\(428\) −20.1385 −0.973429
\(429\) 11.0864 0.535255
\(430\) 0 0
\(431\) 6.72669 0.324013 0.162007 0.986790i \(-0.448203\pi\)
0.162007 + 0.986790i \(0.448203\pi\)
\(432\) −14.4850 −0.696909
\(433\) −32.1724 −1.54611 −0.773053 0.634341i \(-0.781272\pi\)
−0.773053 + 0.634341i \(0.781272\pi\)
\(434\) 2.93906 0.141080
\(435\) 0 0
\(436\) −4.85997 −0.232750
\(437\) −2.58194 −0.123511
\(438\) 15.7622 0.753147
\(439\) 31.4984 1.50334 0.751668 0.659541i \(-0.229249\pi\)
0.751668 + 0.659541i \(0.229249\pi\)
\(440\) 0 0
\(441\) −40.3461 −1.92124
\(442\) 0.611057 0.0290650
\(443\) 18.9443 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(444\) −45.3895 −2.15409
\(445\) 0 0
\(446\) −2.49190 −0.117995
\(447\) −43.2324 −2.04482
\(448\) −0.394086 −0.0186188
\(449\) 31.1737 1.47118 0.735590 0.677427i \(-0.236905\pi\)
0.735590 + 0.677427i \(0.236905\pi\)
\(450\) 0 0
\(451\) 5.78336 0.272328
\(452\) 11.3741 0.534992
\(453\) −48.2905 −2.26889
\(454\) 2.45067 0.115016
\(455\) 0 0
\(456\) 7.09403 0.332208
\(457\) 34.4215 1.61017 0.805085 0.593159i \(-0.202120\pi\)
0.805085 + 0.593159i \(0.202120\pi\)
\(458\) −4.74187 −0.221573
\(459\) 2.31905 0.108244
\(460\) 0 0
\(461\) −24.8901 −1.15925 −0.579624 0.814884i \(-0.696801\pi\)
−0.579624 + 0.814884i \(0.696801\pi\)
\(462\) 1.17965 0.0548825
\(463\) 26.3939 1.22663 0.613314 0.789839i \(-0.289836\pi\)
0.613314 + 0.789839i \(0.289836\pi\)
\(464\) 3.49291 0.162154
\(465\) 0 0
\(466\) 4.29894 0.199144
\(467\) −26.6487 −1.23315 −0.616577 0.787295i \(-0.711481\pi\)
−0.616577 + 0.787295i \(0.711481\pi\)
\(468\) 34.9122 1.61382
\(469\) 2.19451 0.101333
\(470\) 0 0
\(471\) −72.7306 −3.35125
\(472\) 7.03557 0.323838
\(473\) 0.283462 0.0130336
\(474\) 30.6682 1.40864
\(475\) 0 0
\(476\) −0.232407 −0.0106524
\(477\) −23.6488 −1.08281
\(478\) 15.8052 0.722914
\(479\) −24.1076 −1.10150 −0.550751 0.834669i \(-0.685659\pi\)
−0.550751 + 0.834669i \(0.685659\pi\)
\(480\) 0 0
\(481\) 35.5087 1.61906
\(482\) 10.3835 0.472958
\(483\) −4.60631 −0.209594
\(484\) −1.56278 −0.0710356
\(485\) 0 0
\(486\) −0.821527 −0.0372652
\(487\) 11.9390 0.541006 0.270503 0.962719i \(-0.412810\pi\)
0.270503 + 0.962719i \(0.412810\pi\)
\(488\) 21.0763 0.954079
\(489\) −49.0893 −2.21989
\(490\) 0 0
\(491\) 10.8304 0.488768 0.244384 0.969678i \(-0.421414\pi\)
0.244384 + 0.969678i \(0.421414\pi\)
\(492\) 27.2167 1.22702
\(493\) −0.559217 −0.0251858
\(494\) −2.43434 −0.109526
\(495\) 0 0
\(496\) −11.7630 −0.528174
\(497\) −9.58969 −0.430156
\(498\) 7.45132 0.333902
\(499\) 14.6568 0.656130 0.328065 0.944655i \(-0.393604\pi\)
0.328065 + 0.944655i \(0.393604\pi\)
\(500\) 0 0
\(501\) −51.1474 −2.28510
\(502\) 5.83240 0.260313
\(503\) 21.7295 0.968872 0.484436 0.874827i \(-0.339025\pi\)
0.484436 + 0.874827i \(0.339025\pi\)
\(504\) 8.46903 0.377241
\(505\) 0 0
\(506\) −1.70724 −0.0758960
\(507\) −1.66822 −0.0740883
\(508\) 26.3636 1.16969
\(509\) −22.4914 −0.996914 −0.498457 0.866914i \(-0.666100\pi\)
−0.498457 + 0.866914i \(0.666100\pi\)
\(510\) 0 0
\(511\) 4.68990 0.207469
\(512\) 16.3997 0.724769
\(513\) −9.23869 −0.407898
\(514\) −6.16635 −0.271986
\(515\) 0 0
\(516\) 1.33398 0.0587252
\(517\) 3.78624 0.166518
\(518\) 3.77833 0.166010
\(519\) −19.5202 −0.856843
\(520\) 0 0
\(521\) −14.1999 −0.622109 −0.311055 0.950392i \(-0.600682\pi\)
−0.311055 + 0.950392i \(0.600682\pi\)
\(522\) 8.93868 0.391235
\(523\) 12.2976 0.537735 0.268867 0.963177i \(-0.413351\pi\)
0.268867 + 0.963177i \(0.413351\pi\)
\(524\) 17.8745 0.780852
\(525\) 0 0
\(526\) −15.7218 −0.685502
\(527\) 1.88326 0.0820360
\(528\) −4.72132 −0.205469
\(529\) −16.3336 −0.710156
\(530\) 0 0
\(531\) −18.1220 −0.786429
\(532\) 0.925870 0.0401416
\(533\) −21.2919 −0.922253
\(534\) 7.53795 0.326199
\(535\) 0 0
\(536\) 8.72619 0.376914
\(537\) −7.53524 −0.325169
\(538\) −9.82069 −0.423400
\(539\) −6.64900 −0.286393
\(540\) 0 0
\(541\) 31.2468 1.34341 0.671703 0.740821i \(-0.265563\pi\)
0.671703 + 0.740821i \(0.265563\pi\)
\(542\) 9.25362 0.397477
\(543\) −9.27894 −0.398197
\(544\) −1.44291 −0.0618643
\(545\) 0 0
\(546\) −4.34298 −0.185863
\(547\) −39.7365 −1.69901 −0.849505 0.527581i \(-0.823099\pi\)
−0.849505 + 0.527581i \(0.823099\pi\)
\(548\) −26.1417 −1.11672
\(549\) −54.2878 −2.31695
\(550\) 0 0
\(551\) 2.22782 0.0949083
\(552\) −18.3164 −0.779597
\(553\) 9.12507 0.388038
\(554\) 2.54496 0.108125
\(555\) 0 0
\(556\) −25.9938 −1.10238
\(557\) 14.7208 0.623740 0.311870 0.950125i \(-0.399045\pi\)
0.311870 + 0.950125i \(0.399045\pi\)
\(558\) −30.1025 −1.27434
\(559\) −1.04359 −0.0441390
\(560\) 0 0
\(561\) 0.755885 0.0319135
\(562\) 11.8573 0.500169
\(563\) 29.4737 1.24217 0.621083 0.783745i \(-0.286693\pi\)
0.621083 + 0.783745i \(0.286693\pi\)
\(564\) 17.8181 0.750279
\(565\) 0 0
\(566\) −10.4038 −0.437306
\(567\) −5.69733 −0.239265
\(568\) −38.1321 −1.59999
\(569\) −42.0413 −1.76246 −0.881231 0.472685i \(-0.843285\pi\)
−0.881231 + 0.472685i \(0.843285\pi\)
\(570\) 0 0
\(571\) 13.1875 0.551878 0.275939 0.961175i \(-0.411011\pi\)
0.275939 + 0.961175i \(0.411011\pi\)
\(572\) 5.75350 0.240566
\(573\) −18.1420 −0.757893
\(574\) −2.26558 −0.0945634
\(575\) 0 0
\(576\) 4.03632 0.168180
\(577\) 28.1175 1.17055 0.585273 0.810836i \(-0.300987\pi\)
0.585273 + 0.810836i \(0.300987\pi\)
\(578\) −11.1991 −0.465822
\(579\) 29.6738 1.23320
\(580\) 0 0
\(581\) 2.21708 0.0919798
\(582\) 18.2962 0.758403
\(583\) −3.89730 −0.161410
\(584\) 18.6488 0.771692
\(585\) 0 0
\(586\) −13.3060 −0.549665
\(587\) 0.510507 0.0210709 0.0105354 0.999945i \(-0.496646\pi\)
0.0105354 + 0.999945i \(0.496646\pi\)
\(588\) −31.2904 −1.29039
\(589\) −7.50257 −0.309138
\(590\) 0 0
\(591\) 31.1587 1.28170
\(592\) −15.1220 −0.621510
\(593\) 17.7779 0.730052 0.365026 0.930997i \(-0.381060\pi\)
0.365026 + 0.930997i \(0.381060\pi\)
\(594\) −6.10883 −0.250648
\(595\) 0 0
\(596\) −22.4364 −0.919030
\(597\) −29.1251 −1.19201
\(598\) 6.28533 0.257026
\(599\) 31.6622 1.29368 0.646842 0.762624i \(-0.276089\pi\)
0.646842 + 0.762624i \(0.276089\pi\)
\(600\) 0 0
\(601\) −35.7075 −1.45654 −0.728270 0.685290i \(-0.759675\pi\)
−0.728270 + 0.685290i \(0.759675\pi\)
\(602\) −0.111044 −0.00452580
\(603\) −22.4767 −0.915322
\(604\) −25.0614 −1.01973
\(605\) 0 0
\(606\) 28.1116 1.14196
\(607\) −26.6864 −1.08317 −0.541584 0.840647i \(-0.682175\pi\)
−0.541584 + 0.840647i \(0.682175\pi\)
\(608\) 5.74830 0.233124
\(609\) 3.97454 0.161056
\(610\) 0 0
\(611\) −13.9393 −0.563924
\(612\) 2.38037 0.0962207
\(613\) −37.6483 −1.52060 −0.760301 0.649571i \(-0.774949\pi\)
−0.760301 + 0.649571i \(0.774949\pi\)
\(614\) −8.59910 −0.347032
\(615\) 0 0
\(616\) 1.39569 0.0562339
\(617\) 38.9725 1.56897 0.784486 0.620146i \(-0.212927\pi\)
0.784486 + 0.620146i \(0.212927\pi\)
\(618\) 10.3603 0.416753
\(619\) −18.4615 −0.742029 −0.371015 0.928627i \(-0.620990\pi\)
−0.371015 + 0.928627i \(0.620990\pi\)
\(620\) 0 0
\(621\) 23.8537 0.957218
\(622\) 15.2124 0.609960
\(623\) 2.24285 0.0898579
\(624\) 17.3819 0.695833
\(625\) 0 0
\(626\) 7.19366 0.287517
\(627\) −3.01131 −0.120260
\(628\) −37.7451 −1.50619
\(629\) 2.42104 0.0965330
\(630\) 0 0
\(631\) −13.9700 −0.556137 −0.278069 0.960561i \(-0.589694\pi\)
−0.278069 + 0.960561i \(0.589694\pi\)
\(632\) 36.2846 1.44333
\(633\) 16.1059 0.640151
\(634\) 9.02497 0.358427
\(635\) 0 0
\(636\) −18.3408 −0.727261
\(637\) 24.4788 0.969885
\(638\) 1.47309 0.0583200
\(639\) 98.2197 3.88551
\(640\) 0 0
\(641\) −15.9666 −0.630642 −0.315321 0.948985i \(-0.602112\pi\)
−0.315321 + 0.948985i \(0.602112\pi\)
\(642\) −25.6585 −1.01266
\(643\) 24.3556 0.960492 0.480246 0.877134i \(-0.340547\pi\)
0.480246 + 0.877134i \(0.340547\pi\)
\(644\) −2.39054 −0.0942006
\(645\) 0 0
\(646\) −0.165977 −0.00653028
\(647\) −0.437751 −0.0172098 −0.00860488 0.999963i \(-0.502739\pi\)
−0.00860488 + 0.999963i \(0.502739\pi\)
\(648\) −22.6547 −0.889960
\(649\) −2.98649 −0.117230
\(650\) 0 0
\(651\) −13.3849 −0.524597
\(652\) −25.4759 −0.997714
\(653\) 33.2515 1.30123 0.650616 0.759407i \(-0.274511\pi\)
0.650616 + 0.759407i \(0.274511\pi\)
\(654\) −6.19210 −0.242130
\(655\) 0 0
\(656\) 9.06751 0.354027
\(657\) −48.0350 −1.87403
\(658\) −1.48322 −0.0578221
\(659\) 2.76267 0.107618 0.0538092 0.998551i \(-0.482864\pi\)
0.0538092 + 0.998551i \(0.482864\pi\)
\(660\) 0 0
\(661\) 20.4256 0.794466 0.397233 0.917718i \(-0.369971\pi\)
0.397233 + 0.917718i \(0.369971\pi\)
\(662\) 5.59319 0.217386
\(663\) −2.78285 −0.108077
\(664\) 8.81591 0.342124
\(665\) 0 0
\(666\) −38.6985 −1.49954
\(667\) −5.75210 −0.222722
\(668\) −26.5440 −1.02702
\(669\) 11.3485 0.438758
\(670\) 0 0
\(671\) −8.94657 −0.345379
\(672\) 10.2552 0.395604
\(673\) 40.3538 1.55553 0.777763 0.628558i \(-0.216354\pi\)
0.777763 + 0.628558i \(0.216354\pi\)
\(674\) −3.81164 −0.146819
\(675\) 0 0
\(676\) −0.865759 −0.0332984
\(677\) 0.670468 0.0257682 0.0128841 0.999917i \(-0.495899\pi\)
0.0128841 + 0.999917i \(0.495899\pi\)
\(678\) 14.4918 0.556553
\(679\) 5.44388 0.208917
\(680\) 0 0
\(681\) −11.1607 −0.427680
\(682\) −4.96087 −0.189962
\(683\) 23.2360 0.889101 0.444551 0.895754i \(-0.353363\pi\)
0.444551 + 0.895754i \(0.353363\pi\)
\(684\) −9.48297 −0.362590
\(685\) 0 0
\(686\) 5.34687 0.204145
\(687\) 21.5952 0.823909
\(688\) 0.444429 0.0169437
\(689\) 14.3482 0.546623
\(690\) 0 0
\(691\) 38.1311 1.45058 0.725288 0.688446i \(-0.241707\pi\)
0.725288 + 0.688446i \(0.241707\pi\)
\(692\) −10.1304 −0.385102
\(693\) −3.59498 −0.136562
\(694\) −13.4056 −0.508871
\(695\) 0 0
\(696\) 15.8042 0.599057
\(697\) −1.45171 −0.0549875
\(698\) −2.70999 −0.102575
\(699\) −19.5780 −0.740509
\(700\) 0 0
\(701\) −10.9408 −0.413228 −0.206614 0.978423i \(-0.566244\pi\)
−0.206614 + 0.978423i \(0.566244\pi\)
\(702\) 22.4901 0.848835
\(703\) −9.64498 −0.363767
\(704\) 0.665182 0.0250700
\(705\) 0 0
\(706\) 17.6017 0.662449
\(707\) 8.36436 0.314574
\(708\) −14.0545 −0.528201
\(709\) 47.7317 1.79260 0.896300 0.443448i \(-0.146245\pi\)
0.896300 + 0.443448i \(0.146245\pi\)
\(710\) 0 0
\(711\) −93.4610 −3.50506
\(712\) 8.91840 0.334231
\(713\) 19.3712 0.725457
\(714\) −0.296111 −0.0110817
\(715\) 0 0
\(716\) −3.91057 −0.146145
\(717\) −71.9794 −2.68812
\(718\) 14.6966 0.548474
\(719\) −43.5137 −1.62279 −0.811394 0.584499i \(-0.801291\pi\)
−0.811394 + 0.584499i \(0.801291\pi\)
\(720\) 0 0
\(721\) 3.08263 0.114803
\(722\) 0.661223 0.0246082
\(723\) −47.2883 −1.75867
\(724\) −4.81550 −0.178967
\(725\) 0 0
\(726\) −1.99115 −0.0738984
\(727\) −6.09893 −0.226197 −0.113098 0.993584i \(-0.536078\pi\)
−0.113098 + 0.993584i \(0.536078\pi\)
\(728\) −5.13833 −0.190439
\(729\) −25.1084 −0.929939
\(730\) 0 0
\(731\) −0.0711532 −0.00263170
\(732\) −42.1028 −1.55617
\(733\) −18.9127 −0.698557 −0.349279 0.937019i \(-0.613573\pi\)
−0.349279 + 0.937019i \(0.613573\pi\)
\(734\) 15.9055 0.587081
\(735\) 0 0
\(736\) −14.8418 −0.547075
\(737\) −3.70414 −0.136444
\(738\) 23.2046 0.854172
\(739\) −37.3950 −1.37560 −0.687799 0.725901i \(-0.741423\pi\)
−0.687799 + 0.725901i \(0.741423\pi\)
\(740\) 0 0
\(741\) 11.0864 0.407268
\(742\) 1.52673 0.0560482
\(743\) 47.1521 1.72984 0.864921 0.501908i \(-0.167369\pi\)
0.864921 + 0.501908i \(0.167369\pi\)
\(744\) −53.2235 −1.95127
\(745\) 0 0
\(746\) 0.226600 0.00829640
\(747\) −22.7078 −0.830834
\(748\) 0.392283 0.0143433
\(749\) −7.63446 −0.278957
\(750\) 0 0
\(751\) 41.7913 1.52499 0.762493 0.646996i \(-0.223975\pi\)
0.762493 + 0.646996i \(0.223975\pi\)
\(752\) 5.93630 0.216474
\(753\) −26.5616 −0.967960
\(754\) −5.42327 −0.197504
\(755\) 0 0
\(756\) −8.55382 −0.311099
\(757\) −46.1909 −1.67884 −0.839419 0.543485i \(-0.817105\pi\)
−0.839419 + 0.543485i \(0.817105\pi\)
\(758\) −20.1032 −0.730182
\(759\) 7.77503 0.282216
\(760\) 0 0
\(761\) 29.8786 1.08310 0.541550 0.840669i \(-0.317838\pi\)
0.541550 + 0.840669i \(0.317838\pi\)
\(762\) 33.5899 1.21683
\(763\) −1.84241 −0.0666996
\(764\) −9.41517 −0.340629
\(765\) 0 0
\(766\) 17.5855 0.635391
\(767\) 10.9950 0.397006
\(768\) 26.0218 0.938980
\(769\) −26.3059 −0.948615 −0.474307 0.880359i \(-0.657301\pi\)
−0.474307 + 0.880359i \(0.657301\pi\)
\(770\) 0 0
\(771\) 28.0825 1.01137
\(772\) 15.3998 0.554252
\(773\) 11.6861 0.420318 0.210159 0.977667i \(-0.432602\pi\)
0.210159 + 0.977667i \(0.432602\pi\)
\(774\) 1.13733 0.0408806
\(775\) 0 0
\(776\) 21.6469 0.777077
\(777\) −17.2071 −0.617301
\(778\) 2.94910 0.105730
\(779\) 5.78336 0.207210
\(780\) 0 0
\(781\) 16.1865 0.579199
\(782\) 0.428543 0.0153247
\(783\) −20.5821 −0.735545
\(784\) −10.4247 −0.372312
\(785\) 0 0
\(786\) 22.7740 0.812321
\(787\) −26.6657 −0.950529 −0.475265 0.879843i \(-0.657648\pi\)
−0.475265 + 0.879843i \(0.657648\pi\)
\(788\) 16.1705 0.576049
\(789\) 71.5994 2.54900
\(790\) 0 0
\(791\) 4.31190 0.153313
\(792\) −14.2950 −0.507949
\(793\) 32.9375 1.16964
\(794\) −4.12472 −0.146381
\(795\) 0 0
\(796\) −15.1151 −0.535740
\(797\) 4.54909 0.161137 0.0805686 0.996749i \(-0.474326\pi\)
0.0805686 + 0.996749i \(0.474326\pi\)
\(798\) 1.17965 0.0417593
\(799\) −0.950403 −0.0336228
\(800\) 0 0
\(801\) −22.9718 −0.811668
\(802\) 12.4510 0.439659
\(803\) −7.91613 −0.279354
\(804\) −17.4318 −0.614772
\(805\) 0 0
\(806\) 18.2638 0.643315
\(807\) 44.7249 1.57439
\(808\) 33.2598 1.17007
\(809\) −0.704828 −0.0247804 −0.0123902 0.999923i \(-0.503944\pi\)
−0.0123902 + 0.999923i \(0.503944\pi\)
\(810\) 0 0
\(811\) −1.69947 −0.0596763 −0.0298382 0.999555i \(-0.509499\pi\)
−0.0298382 + 0.999555i \(0.509499\pi\)
\(812\) 2.06267 0.0723856
\(813\) −42.1424 −1.47800
\(814\) −6.37748 −0.223531
\(815\) 0 0
\(816\) 1.18512 0.0414876
\(817\) 0.283462 0.00991708
\(818\) −19.8028 −0.692390
\(819\) 13.2352 0.462474
\(820\) 0 0
\(821\) 0.312402 0.0109029 0.00545146 0.999985i \(-0.498265\pi\)
0.00545146 + 0.999985i \(0.498265\pi\)
\(822\) −33.3072 −1.16172
\(823\) −37.7601 −1.31623 −0.658116 0.752916i \(-0.728647\pi\)
−0.658116 + 0.752916i \(0.728647\pi\)
\(824\) 12.2577 0.427016
\(825\) 0 0
\(826\) 1.16993 0.0407071
\(827\) −47.5469 −1.65337 −0.826684 0.562667i \(-0.809775\pi\)
−0.826684 + 0.562667i \(0.809775\pi\)
\(828\) 24.4845 0.850894
\(829\) −19.6186 −0.681381 −0.340690 0.940176i \(-0.610661\pi\)
−0.340690 + 0.940176i \(0.610661\pi\)
\(830\) 0 0
\(831\) −11.5901 −0.402057
\(832\) −2.44891 −0.0849008
\(833\) 1.66900 0.0578275
\(834\) −33.1188 −1.14681
\(835\) 0 0
\(836\) −1.56278 −0.0540500
\(837\) 69.3139 2.39584
\(838\) −18.3953 −0.635456
\(839\) −31.4596 −1.08611 −0.543053 0.839698i \(-0.682732\pi\)
−0.543053 + 0.839698i \(0.682732\pi\)
\(840\) 0 0
\(841\) −24.0368 −0.828856
\(842\) −19.0130 −0.655233
\(843\) −53.9999 −1.85985
\(844\) 8.35849 0.287711
\(845\) 0 0
\(846\) 15.1915 0.522295
\(847\) −0.592449 −0.0203568
\(848\) −6.11044 −0.209833
\(849\) 47.3807 1.62610
\(850\) 0 0
\(851\) 24.9028 0.853656
\(852\) 76.1742 2.60969
\(853\) 51.7295 1.77119 0.885593 0.464463i \(-0.153753\pi\)
0.885593 + 0.464463i \(0.153753\pi\)
\(854\) 3.50474 0.119930
\(855\) 0 0
\(856\) −30.3574 −1.03759
\(857\) −23.5776 −0.805394 −0.402697 0.915333i \(-0.631927\pi\)
−0.402697 + 0.915333i \(0.631927\pi\)
\(858\) 7.33056 0.250261
\(859\) 7.75928 0.264743 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(860\) 0 0
\(861\) 10.3178 0.351629
\(862\) 4.44784 0.151494
\(863\) 46.2297 1.57368 0.786839 0.617158i \(-0.211716\pi\)
0.786839 + 0.617158i \(0.211716\pi\)
\(864\) −53.1067 −1.80673
\(865\) 0 0
\(866\) −21.2731 −0.722890
\(867\) 51.0025 1.73214
\(868\) −6.94640 −0.235776
\(869\) −15.4023 −0.522487
\(870\) 0 0
\(871\) 13.6371 0.462074
\(872\) −7.32608 −0.248092
\(873\) −55.7574 −1.88710
\(874\) −1.70724 −0.0577482
\(875\) 0 0
\(876\) −37.2535 −1.25868
\(877\) 39.3550 1.32892 0.664461 0.747323i \(-0.268661\pi\)
0.664461 + 0.747323i \(0.268661\pi\)
\(878\) 20.8275 0.702893
\(879\) 60.5975 2.04390
\(880\) 0 0
\(881\) −0.350505 −0.0118088 −0.00590441 0.999983i \(-0.501879\pi\)
−0.00590441 + 0.999983i \(0.501879\pi\)
\(882\) −26.6778 −0.898288
\(883\) −15.3175 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(884\) −1.44422 −0.0485743
\(885\) 0 0
\(886\) 12.5264 0.420833
\(887\) 55.8068 1.87381 0.936904 0.349586i \(-0.113678\pi\)
0.936904 + 0.349586i \(0.113678\pi\)
\(888\) −68.4218 −2.29608
\(889\) 9.99439 0.335201
\(890\) 0 0
\(891\) 9.61657 0.322167
\(892\) 5.88954 0.197196
\(893\) 3.78624 0.126702
\(894\) −28.5863 −0.956068
\(895\) 0 0
\(896\) 6.55057 0.218839
\(897\) −28.6243 −0.955739
\(898\) 20.6128 0.687858
\(899\) −16.7144 −0.557455
\(900\) 0 0
\(901\) 0.978283 0.0325913
\(902\) 3.82409 0.127328
\(903\) 0.505710 0.0168290
\(904\) 17.1457 0.570257
\(905\) 0 0
\(906\) −31.9308 −1.06083
\(907\) −35.6759 −1.18460 −0.592300 0.805718i \(-0.701780\pi\)
−0.592300 + 0.805718i \(0.701780\pi\)
\(908\) −5.79209 −0.192217
\(909\) −85.6696 −2.84148
\(910\) 0 0
\(911\) 48.6660 1.61238 0.806189 0.591658i \(-0.201526\pi\)
0.806189 + 0.591658i \(0.201526\pi\)
\(912\) −4.72132 −0.156339
\(913\) −3.74222 −0.123849
\(914\) 22.7603 0.752844
\(915\) 0 0
\(916\) 11.2073 0.370300
\(917\) 6.77621 0.223770
\(918\) 1.53341 0.0506101
\(919\) 44.9545 1.48291 0.741456 0.671002i \(-0.234136\pi\)
0.741456 + 0.671002i \(0.234136\pi\)
\(920\) 0 0
\(921\) 39.1616 1.29042
\(922\) −16.4579 −0.542013
\(923\) −59.5918 −1.96149
\(924\) −2.78808 −0.0917212
\(925\) 0 0
\(926\) 17.4523 0.573517
\(927\) −31.5729 −1.03699
\(928\) 12.8062 0.420383
\(929\) −32.8433 −1.07756 −0.538778 0.842448i \(-0.681114\pi\)
−0.538778 + 0.842448i \(0.681114\pi\)
\(930\) 0 0
\(931\) −6.64900 −0.217912
\(932\) −10.1604 −0.332816
\(933\) −69.2794 −2.26811
\(934\) −17.6207 −0.576567
\(935\) 0 0
\(936\) 52.6279 1.72020
\(937\) 32.8654 1.07367 0.536833 0.843688i \(-0.319620\pi\)
0.536833 + 0.843688i \(0.319620\pi\)
\(938\) 1.45106 0.0473789
\(939\) −32.7610 −1.06912
\(940\) 0 0
\(941\) 42.2849 1.37845 0.689225 0.724548i \(-0.257951\pi\)
0.689225 + 0.724548i \(0.257951\pi\)
\(942\) −48.0911 −1.56689
\(943\) −14.9323 −0.486262
\(944\) −4.68241 −0.152399
\(945\) 0 0
\(946\) 0.187432 0.00609392
\(947\) −0.322414 −0.0104770 −0.00523852 0.999986i \(-0.501667\pi\)
−0.00523852 + 0.999986i \(0.501667\pi\)
\(948\) −72.4836 −2.35416
\(949\) 29.1438 0.946048
\(950\) 0 0
\(951\) −41.1011 −1.33279
\(952\) −0.350339 −0.0113546
\(953\) −36.2919 −1.17561 −0.587806 0.809002i \(-0.700008\pi\)
−0.587806 + 0.809002i \(0.700008\pi\)
\(954\) −15.6371 −0.506271
\(955\) 0 0
\(956\) −37.3552 −1.20816
\(957\) −6.70866 −0.216860
\(958\) −15.9405 −0.515013
\(959\) −9.91027 −0.320019
\(960\) 0 0
\(961\) 25.2885 0.815759
\(962\) 23.4792 0.756999
\(963\) 78.1938 2.51976
\(964\) −24.5413 −0.790421
\(965\) 0 0
\(966\) −3.04580 −0.0979969
\(967\) −3.37752 −0.108614 −0.0543069 0.998524i \(-0.517295\pi\)
−0.0543069 + 0.998524i \(0.517295\pi\)
\(968\) −2.35579 −0.0757181
\(969\) 0.755885 0.0242825
\(970\) 0 0
\(971\) −37.0852 −1.19012 −0.595061 0.803681i \(-0.702872\pi\)
−0.595061 + 0.803681i \(0.702872\pi\)
\(972\) 1.94166 0.0622787
\(973\) −9.85421 −0.315911
\(974\) 7.89432 0.252950
\(975\) 0 0
\(976\) −14.0270 −0.448993
\(977\) −15.5780 −0.498385 −0.249192 0.968454i \(-0.580165\pi\)
−0.249192 + 0.968454i \(0.580165\pi\)
\(978\) −32.4590 −1.03792
\(979\) −3.78573 −0.120992
\(980\) 0 0
\(981\) 18.8703 0.602483
\(982\) 7.16130 0.228526
\(983\) −17.9951 −0.573956 −0.286978 0.957937i \(-0.592651\pi\)
−0.286978 + 0.957937i \(0.592651\pi\)
\(984\) 41.0273 1.30790
\(985\) 0 0
\(986\) −0.369767 −0.0117758
\(987\) 6.75483 0.215009
\(988\) 5.75350 0.183043
\(989\) −0.731882 −0.0232725
\(990\) 0 0
\(991\) 5.95229 0.189081 0.0945403 0.995521i \(-0.469862\pi\)
0.0945403 + 0.995521i \(0.469862\pi\)
\(992\) −43.1270 −1.36928
\(993\) −25.4723 −0.808338
\(994\) −6.34092 −0.201122
\(995\) 0 0
\(996\) −17.6110 −0.558026
\(997\) 4.21169 0.133386 0.0666928 0.997774i \(-0.478755\pi\)
0.0666928 + 0.997774i \(0.478755\pi\)
\(998\) 9.69143 0.306777
\(999\) 89.1069 2.81922
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 5225.2.a.bc.1.19 30
5.2 odd 4 1045.2.b.e.419.19 yes 30
5.3 odd 4 1045.2.b.e.419.12 30
5.4 even 2 inner 5225.2.a.bc.1.12 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.e.419.12 30 5.3 odd 4
1045.2.b.e.419.19 yes 30 5.2 odd 4
5225.2.a.bc.1.12 30 5.4 even 2 inner
5225.2.a.bc.1.19 30 1.1 even 1 trivial