Properties

Label 775.2.a.i.1.3
Level $775$
Weight $2$
Character 775.1
Self dual yes
Analytic conductor $6.188$
Analytic rank $1$
Dimension $5$
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] = [775,2,Mod(1,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.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: \(6.18840615665\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.205225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.418933\) of defining polynomial
Character \(\chi\) \(=\) 775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.581067 q^{2} -2.46572 q^{3} -1.66236 q^{4} +1.43275 q^{6} -1.67419 q^{7} +2.12808 q^{8} +3.07975 q^{9} +O(q^{10})\) \(q-0.581067 q^{2} -2.46572 q^{3} -1.66236 q^{4} +1.43275 q^{6} -1.67419 q^{7} +2.12808 q^{8} +3.07975 q^{9} +5.38333 q^{11} +4.09891 q^{12} -0.325810 q^{13} +0.972816 q^{14} +2.08817 q^{16} -0.627408 q^{17} -1.78954 q^{18} -1.81068 q^{19} +4.12808 q^{21} -3.12808 q^{22} +0.558839 q^{23} -5.24723 q^{24} +0.189317 q^{26} -0.196646 q^{27} +2.78311 q^{28} +1.60221 q^{29} +1.00000 q^{31} -5.46952 q^{32} -13.2738 q^{33} +0.364566 q^{34} -5.11966 q^{36} -6.68929 q^{37} +1.05213 q^{38} +0.803354 q^{39} -8.14768 q^{41} -2.39869 q^{42} -10.8130 q^{43} -8.94905 q^{44} -0.324723 q^{46} +9.93459 q^{47} -5.14883 q^{48} -4.19709 q^{49} +1.54701 q^{51} +0.541613 q^{52} -3.51513 q^{53} +0.114264 q^{54} -3.56281 q^{56} +4.46463 q^{57} -0.930990 q^{58} +10.8460 q^{59} -8.69720 q^{61} -0.581067 q^{62} -5.15609 q^{63} -0.998180 q^{64} +7.71295 q^{66} +0.686019 q^{67} +1.04298 q^{68} -1.37794 q^{69} +1.00341 q^{71} +6.55395 q^{72} +5.03034 q^{73} +3.88692 q^{74} +3.01001 q^{76} -9.01273 q^{77} -0.466802 q^{78} -13.6275 q^{79} -8.75438 q^{81} +4.73434 q^{82} +4.77167 q^{83} -6.86236 q^{84} +6.28308 q^{86} -3.95059 q^{87} +11.4561 q^{88} -10.5620 q^{89} +0.545467 q^{91} -0.928992 q^{92} -2.46572 q^{93} -5.77266 q^{94} +13.4863 q^{96} +15.0252 q^{97} +2.43879 q^{98} +16.5793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{2} - q^{3} + 6 q^{4} - q^{6} - 6 q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{2} - q^{3} + 6 q^{4} - q^{6} - 6 q^{7} - 15 q^{8} + 2 q^{9} + 11 q^{12} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 11 q^{17} - 19 q^{18} - 4 q^{19} - 5 q^{21} + 10 q^{22} - 12 q^{23} - 26 q^{24} + 6 q^{26} + 2 q^{27} - 18 q^{28} - 6 q^{29} + 5 q^{31} - 29 q^{32} - 21 q^{33} - 5 q^{34} + 23 q^{36} + 2 q^{37} + 6 q^{38} + 7 q^{39} - 2 q^{41} + 24 q^{42} - 7 q^{43} - 28 q^{44} + 27 q^{46} - 8 q^{47} + 39 q^{48} - q^{49} - 19 q^{51} + 6 q^{52} - 25 q^{53} - 18 q^{54} + 35 q^{56} - 20 q^{57} + q^{58} + 4 q^{59} - 17 q^{61} - 4 q^{62} - 10 q^{63} + 27 q^{64} + 27 q^{66} + 13 q^{67} - 18 q^{68} - 10 q^{69} - 6 q^{71} - 26 q^{72} - 7 q^{73} + 6 q^{74} - 5 q^{76} - 7 q^{77} - 22 q^{78} + 12 q^{79} - 11 q^{81} + 21 q^{82} + 4 q^{83} - 63 q^{84} - 41 q^{86} - q^{87} + 49 q^{88} - 3 q^{89} - 22 q^{91} - 34 q^{92} - q^{93} - 34 q^{94} - 64 q^{96} - 25 q^{97} + 22 q^{98} + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.581067 −0.410876 −0.205438 0.978670i \(-0.565862\pi\)
−0.205438 + 0.978670i \(0.565862\pi\)
\(3\) −2.46572 −1.42358 −0.711791 0.702392i \(-0.752116\pi\)
−0.711791 + 0.702392i \(0.752116\pi\)
\(4\) −1.66236 −0.831181
\(5\) 0 0
\(6\) 1.43275 0.584916
\(7\) −1.67419 −0.632784 −0.316392 0.948628i \(-0.602472\pi\)
−0.316392 + 0.948628i \(0.602472\pi\)
\(8\) 2.12808 0.752389
\(9\) 3.07975 1.02658
\(10\) 0 0
\(11\) 5.38333 1.62314 0.811568 0.584258i \(-0.198614\pi\)
0.811568 + 0.584258i \(0.198614\pi\)
\(12\) 4.09891 1.18325
\(13\) −0.325810 −0.0903634 −0.0451817 0.998979i \(-0.514387\pi\)
−0.0451817 + 0.998979i \(0.514387\pi\)
\(14\) 0.972816 0.259996
\(15\) 0 0
\(16\) 2.08817 0.522042
\(17\) −0.627408 −0.152169 −0.0760844 0.997101i \(-0.524242\pi\)
−0.0760844 + 0.997101i \(0.524242\pi\)
\(18\) −1.78954 −0.421799
\(19\) −1.81068 −0.415399 −0.207700 0.978193i \(-0.566598\pi\)
−0.207700 + 0.978193i \(0.566598\pi\)
\(20\) 0 0
\(21\) 4.12808 0.900820
\(22\) −3.12808 −0.666908
\(23\) 0.558839 0.116526 0.0582630 0.998301i \(-0.481444\pi\)
0.0582630 + 0.998301i \(0.481444\pi\)
\(24\) −5.24723 −1.07109
\(25\) 0 0
\(26\) 0.189317 0.0371282
\(27\) −0.196646 −0.0378445
\(28\) 2.78311 0.525958
\(29\) 1.60221 0.297523 0.148761 0.988873i \(-0.452471\pi\)
0.148761 + 0.988873i \(0.452471\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −5.46952 −0.966883
\(33\) −13.2738 −2.31067
\(34\) 0.364566 0.0625226
\(35\) 0 0
\(36\) −5.11966 −0.853277
\(37\) −6.68929 −1.09971 −0.549856 0.835259i \(-0.685317\pi\)
−0.549856 + 0.835259i \(0.685317\pi\)
\(38\) 1.05213 0.170678
\(39\) 0.803354 0.128640
\(40\) 0 0
\(41\) −8.14768 −1.27245 −0.636227 0.771502i \(-0.719506\pi\)
−0.636227 + 0.771502i \(0.719506\pi\)
\(42\) −2.39869 −0.370126
\(43\) −10.8130 −1.64897 −0.824484 0.565885i \(-0.808535\pi\)
−0.824484 + 0.565885i \(0.808535\pi\)
\(44\) −8.94905 −1.34912
\(45\) 0 0
\(46\) −0.324723 −0.0478777
\(47\) 9.93459 1.44911 0.724554 0.689218i \(-0.242046\pi\)
0.724554 + 0.689218i \(0.242046\pi\)
\(48\) −5.14883 −0.743169
\(49\) −4.19709 −0.599584
\(50\) 0 0
\(51\) 1.54701 0.216625
\(52\) 0.541613 0.0751083
\(53\) −3.51513 −0.482840 −0.241420 0.970421i \(-0.577613\pi\)
−0.241420 + 0.970421i \(0.577613\pi\)
\(54\) 0.114264 0.0155494
\(55\) 0 0
\(56\) −3.56281 −0.476100
\(57\) 4.46463 0.591354
\(58\) −0.930990 −0.122245
\(59\) 10.8460 1.41203 0.706013 0.708199i \(-0.250492\pi\)
0.706013 + 0.708199i \(0.250492\pi\)
\(60\) 0 0
\(61\) −8.69720 −1.11356 −0.556781 0.830659i \(-0.687964\pi\)
−0.556781 + 0.830659i \(0.687964\pi\)
\(62\) −0.581067 −0.0737956
\(63\) −5.15609 −0.649606
\(64\) −0.998180 −0.124773
\(65\) 0 0
\(66\) 7.71295 0.949398
\(67\) 0.686019 0.0838106 0.0419053 0.999122i \(-0.486657\pi\)
0.0419053 + 0.999122i \(0.486657\pi\)
\(68\) 1.04298 0.126480
\(69\) −1.37794 −0.165884
\(70\) 0 0
\(71\) 1.00341 0.119083 0.0595416 0.998226i \(-0.481036\pi\)
0.0595416 + 0.998226i \(0.481036\pi\)
\(72\) 6.55395 0.772390
\(73\) 5.03034 0.588757 0.294378 0.955689i \(-0.404887\pi\)
0.294378 + 0.955689i \(0.404887\pi\)
\(74\) 3.88692 0.451846
\(75\) 0 0
\(76\) 3.01001 0.345272
\(77\) −9.01273 −1.02710
\(78\) −0.466802 −0.0528550
\(79\) −13.6275 −1.53322 −0.766609 0.642114i \(-0.778058\pi\)
−0.766609 + 0.642114i \(0.778058\pi\)
\(80\) 0 0
\(81\) −8.75438 −0.972709
\(82\) 4.73434 0.522821
\(83\) 4.77167 0.523759 0.261879 0.965101i \(-0.415658\pi\)
0.261879 + 0.965101i \(0.415658\pi\)
\(84\) −6.86236 −0.748744
\(85\) 0 0
\(86\) 6.28308 0.677522
\(87\) −3.95059 −0.423548
\(88\) 11.4561 1.22123
\(89\) −10.5620 −1.11957 −0.559785 0.828638i \(-0.689116\pi\)
−0.559785 + 0.828638i \(0.689116\pi\)
\(90\) 0 0
\(91\) 0.545467 0.0571805
\(92\) −0.928992 −0.0968541
\(93\) −2.46572 −0.255683
\(94\) −5.77266 −0.595404
\(95\) 0 0
\(96\) 13.4863 1.37644
\(97\) 15.0252 1.52558 0.762789 0.646648i \(-0.223830\pi\)
0.762789 + 0.646648i \(0.223830\pi\)
\(98\) 2.43879 0.246355
\(99\) 16.5793 1.66629
\(100\) 0 0
\(101\) −18.6414 −1.85488 −0.927442 0.373966i \(-0.877998\pi\)
−0.927442 + 0.373966i \(0.877998\pi\)
\(102\) −0.898916 −0.0890060
\(103\) 16.0308 1.57956 0.789780 0.613390i \(-0.210195\pi\)
0.789780 + 0.613390i \(0.210195\pi\)
\(104\) −0.693348 −0.0679884
\(105\) 0 0
\(106\) 2.04252 0.198387
\(107\) −16.5838 −1.60322 −0.801610 0.597847i \(-0.796023\pi\)
−0.801610 + 0.597847i \(0.796023\pi\)
\(108\) 0.326897 0.0314557
\(109\) −3.87084 −0.370759 −0.185379 0.982667i \(-0.559351\pi\)
−0.185379 + 0.982667i \(0.559351\pi\)
\(110\) 0 0
\(111\) 16.4939 1.56553
\(112\) −3.49599 −0.330340
\(113\) −3.50164 −0.329407 −0.164703 0.986343i \(-0.552667\pi\)
−0.164703 + 0.986343i \(0.552667\pi\)
\(114\) −2.59425 −0.242974
\(115\) 0 0
\(116\) −2.66345 −0.247295
\(117\) −1.00341 −0.0927656
\(118\) −6.30224 −0.580168
\(119\) 1.05040 0.0962901
\(120\) 0 0
\(121\) 17.9803 1.63457
\(122\) 5.05366 0.457536
\(123\) 20.0898 1.81144
\(124\) −1.66236 −0.149284
\(125\) 0 0
\(126\) 2.99603 0.266908
\(127\) −17.1431 −1.52121 −0.760603 0.649218i \(-0.775096\pi\)
−0.760603 + 0.649218i \(0.775096\pi\)
\(128\) 11.5190 1.01815
\(129\) 26.6618 2.34744
\(130\) 0 0
\(131\) −16.9361 −1.47972 −0.739858 0.672763i \(-0.765107\pi\)
−0.739858 + 0.672763i \(0.765107\pi\)
\(132\) 22.0658 1.92058
\(133\) 3.03143 0.262858
\(134\) −0.398623 −0.0344358
\(135\) 0 0
\(136\) −1.33517 −0.114490
\(137\) −9.79885 −0.837173 −0.418586 0.908177i \(-0.637474\pi\)
−0.418586 + 0.908177i \(0.637474\pi\)
\(138\) 0.800674 0.0681579
\(139\) 16.4079 1.39170 0.695849 0.718188i \(-0.255028\pi\)
0.695849 + 0.718188i \(0.255028\pi\)
\(140\) 0 0
\(141\) −24.4959 −2.06292
\(142\) −0.583050 −0.0489285
\(143\) −1.75394 −0.146672
\(144\) 6.43104 0.535920
\(145\) 0 0
\(146\) −2.92296 −0.241906
\(147\) 10.3488 0.853556
\(148\) 11.1200 0.914060
\(149\) 1.86445 0.152742 0.0763709 0.997079i \(-0.475667\pi\)
0.0763709 + 0.997079i \(0.475667\pi\)
\(150\) 0 0
\(151\) 9.28758 0.755813 0.377906 0.925844i \(-0.376644\pi\)
0.377906 + 0.925844i \(0.376644\pi\)
\(152\) −3.85327 −0.312542
\(153\) −1.93226 −0.156214
\(154\) 5.23700 0.422009
\(155\) 0 0
\(156\) −1.33546 −0.106923
\(157\) 12.2141 0.974789 0.487394 0.873182i \(-0.337947\pi\)
0.487394 + 0.873182i \(0.337947\pi\)
\(158\) 7.91851 0.629963
\(159\) 8.66730 0.687362
\(160\) 0 0
\(161\) −0.935602 −0.0737358
\(162\) 5.08688 0.399663
\(163\) −12.4555 −0.975590 −0.487795 0.872958i \(-0.662199\pi\)
−0.487795 + 0.872958i \(0.662199\pi\)
\(164\) 13.5444 1.05764
\(165\) 0 0
\(166\) −2.77266 −0.215200
\(167\) −6.58092 −0.509247 −0.254624 0.967040i \(-0.581952\pi\)
−0.254624 + 0.967040i \(0.581952\pi\)
\(168\) 8.78486 0.677767
\(169\) −12.8938 −0.991834
\(170\) 0 0
\(171\) −5.57645 −0.426442
\(172\) 17.9751 1.37059
\(173\) 4.41674 0.335799 0.167899 0.985804i \(-0.446302\pi\)
0.167899 + 0.985804i \(0.446302\pi\)
\(174\) 2.29556 0.174026
\(175\) 0 0
\(176\) 11.2413 0.847345
\(177\) −26.7431 −2.01013
\(178\) 6.13723 0.460004
\(179\) 11.1637 0.834412 0.417206 0.908812i \(-0.363009\pi\)
0.417206 + 0.908812i \(0.363009\pi\)
\(180\) 0 0
\(181\) 17.8449 1.32640 0.663200 0.748443i \(-0.269198\pi\)
0.663200 + 0.748443i \(0.269198\pi\)
\(182\) −0.316953 −0.0234941
\(183\) 21.4448 1.58525
\(184\) 1.18925 0.0876728
\(185\) 0 0
\(186\) 1.43275 0.105054
\(187\) −3.37755 −0.246991
\(188\) −16.5149 −1.20447
\(189\) 0.329223 0.0239474
\(190\) 0 0
\(191\) −25.5220 −1.84671 −0.923354 0.383950i \(-0.874563\pi\)
−0.923354 + 0.383950i \(0.874563\pi\)
\(192\) 2.46123 0.177624
\(193\) −13.1431 −0.946062 −0.473031 0.881046i \(-0.656840\pi\)
−0.473031 + 0.881046i \(0.656840\pi\)
\(194\) −8.73064 −0.626824
\(195\) 0 0
\(196\) 6.97707 0.498362
\(197\) 0.217382 0.0154879 0.00774393 0.999970i \(-0.497535\pi\)
0.00774393 + 0.999970i \(0.497535\pi\)
\(198\) −9.63370 −0.684637
\(199\) −2.75888 −0.195572 −0.0977860 0.995207i \(-0.531176\pi\)
−0.0977860 + 0.995207i \(0.531176\pi\)
\(200\) 0 0
\(201\) −1.69153 −0.119311
\(202\) 10.8319 0.762128
\(203\) −2.68240 −0.188268
\(204\) −2.57169 −0.180054
\(205\) 0 0
\(206\) −9.31496 −0.649004
\(207\) 1.72108 0.119624
\(208\) −0.680345 −0.0471735
\(209\) −9.74751 −0.674249
\(210\) 0 0
\(211\) −17.3289 −1.19297 −0.596484 0.802625i \(-0.703436\pi\)
−0.596484 + 0.802625i \(0.703436\pi\)
\(212\) 5.84341 0.401327
\(213\) −2.47413 −0.169525
\(214\) 9.63632 0.658725
\(215\) 0 0
\(216\) −0.418478 −0.0284738
\(217\) −1.67419 −0.113651
\(218\) 2.24921 0.152336
\(219\) −12.4034 −0.838143
\(220\) 0 0
\(221\) 0.204416 0.0137505
\(222\) −9.58405 −0.643239
\(223\) 20.1921 1.35216 0.676081 0.736827i \(-0.263677\pi\)
0.676081 + 0.736827i \(0.263677\pi\)
\(224\) 9.15701 0.611829
\(225\) 0 0
\(226\) 2.03469 0.135345
\(227\) −6.31694 −0.419270 −0.209635 0.977780i \(-0.567228\pi\)
−0.209635 + 0.977780i \(0.567228\pi\)
\(228\) −7.42183 −0.491522
\(229\) −12.0127 −0.793822 −0.396911 0.917857i \(-0.629918\pi\)
−0.396911 + 0.917857i \(0.629918\pi\)
\(230\) 0 0
\(231\) 22.2228 1.46215
\(232\) 3.40962 0.223853
\(233\) −28.3769 −1.85903 −0.929516 0.368781i \(-0.879775\pi\)
−0.929516 + 0.368781i \(0.879775\pi\)
\(234\) 0.583050 0.0381152
\(235\) 0 0
\(236\) −18.0299 −1.17365
\(237\) 33.6016 2.18266
\(238\) −0.610353 −0.0395633
\(239\) 8.20713 0.530875 0.265438 0.964128i \(-0.414484\pi\)
0.265438 + 0.964128i \(0.414484\pi\)
\(240\) 0 0
\(241\) −11.5115 −0.741521 −0.370761 0.928728i \(-0.620903\pi\)
−0.370761 + 0.928728i \(0.620903\pi\)
\(242\) −10.4477 −0.671607
\(243\) 22.1758 1.42258
\(244\) 14.4579 0.925572
\(245\) 0 0
\(246\) −11.6735 −0.744278
\(247\) 0.589938 0.0375369
\(248\) 2.12808 0.135133
\(249\) −11.7656 −0.745613
\(250\) 0 0
\(251\) 10.4877 0.661977 0.330988 0.943635i \(-0.392618\pi\)
0.330988 + 0.943635i \(0.392618\pi\)
\(252\) 8.57129 0.539940
\(253\) 3.00842 0.189137
\(254\) 9.96129 0.625027
\(255\) 0 0
\(256\) −4.69698 −0.293561
\(257\) −19.3291 −1.20571 −0.602857 0.797849i \(-0.705971\pi\)
−0.602857 + 0.797849i \(0.705971\pi\)
\(258\) −15.4923 −0.964508
\(259\) 11.1991 0.695881
\(260\) 0 0
\(261\) 4.93440 0.305432
\(262\) 9.84102 0.607980
\(263\) −4.86634 −0.300071 −0.150036 0.988681i \(-0.547939\pi\)
−0.150036 + 0.988681i \(0.547939\pi\)
\(264\) −28.2476 −1.73852
\(265\) 0 0
\(266\) −1.76146 −0.108002
\(267\) 26.0429 1.59380
\(268\) −1.14041 −0.0696617
\(269\) 17.7368 1.08143 0.540717 0.841204i \(-0.318153\pi\)
0.540717 + 0.841204i \(0.318153\pi\)
\(270\) 0 0
\(271\) −23.2534 −1.41254 −0.706272 0.707941i \(-0.749624\pi\)
−0.706272 + 0.707941i \(0.749624\pi\)
\(272\) −1.31013 −0.0794385
\(273\) −1.34497 −0.0814011
\(274\) 5.69379 0.343974
\(275\) 0 0
\(276\) 2.29063 0.137880
\(277\) −6.12148 −0.367804 −0.183902 0.982945i \(-0.558873\pi\)
−0.183902 + 0.982945i \(0.558873\pi\)
\(278\) −9.53408 −0.571816
\(279\) 3.07975 0.184380
\(280\) 0 0
\(281\) −15.0759 −0.899354 −0.449677 0.893191i \(-0.648461\pi\)
−0.449677 + 0.893191i \(0.648461\pi\)
\(282\) 14.2337 0.847607
\(283\) 11.8948 0.707072 0.353536 0.935421i \(-0.384979\pi\)
0.353536 + 0.935421i \(0.384979\pi\)
\(284\) −1.66804 −0.0989797
\(285\) 0 0
\(286\) 1.01916 0.0602641
\(287\) 13.6408 0.805189
\(288\) −16.8448 −0.992587
\(289\) −16.6064 −0.976845
\(290\) 0 0
\(291\) −37.0479 −2.17178
\(292\) −8.36224 −0.489363
\(293\) −11.2627 −0.657974 −0.328987 0.944334i \(-0.606707\pi\)
−0.328987 + 0.944334i \(0.606707\pi\)
\(294\) −6.01336 −0.350706
\(295\) 0 0
\(296\) −14.2353 −0.827411
\(297\) −1.05861 −0.0614268
\(298\) −1.08337 −0.0627580
\(299\) −0.182075 −0.0105297
\(300\) 0 0
\(301\) 18.1030 1.04344
\(302\) −5.39671 −0.310545
\(303\) 45.9643 2.64058
\(304\) −3.78101 −0.216856
\(305\) 0 0
\(306\) 1.12277 0.0641847
\(307\) −15.1566 −0.865034 −0.432517 0.901626i \(-0.642374\pi\)
−0.432517 + 0.901626i \(0.642374\pi\)
\(308\) 14.9824 0.853702
\(309\) −39.5273 −2.24863
\(310\) 0 0
\(311\) 18.6418 1.05708 0.528539 0.848909i \(-0.322740\pi\)
0.528539 + 0.848909i \(0.322740\pi\)
\(312\) 1.70960 0.0967870
\(313\) 1.23926 0.0700469 0.0350235 0.999386i \(-0.488849\pi\)
0.0350235 + 0.999386i \(0.488849\pi\)
\(314\) −7.09719 −0.400518
\(315\) 0 0
\(316\) 22.6539 1.27438
\(317\) 9.07207 0.509538 0.254769 0.967002i \(-0.418001\pi\)
0.254769 + 0.967002i \(0.418001\pi\)
\(318\) −5.03628 −0.282421
\(319\) 8.62522 0.482920
\(320\) 0 0
\(321\) 40.8910 2.28231
\(322\) 0.543648 0.0302963
\(323\) 1.13604 0.0632108
\(324\) 14.5529 0.808497
\(325\) 0 0
\(326\) 7.23748 0.400847
\(327\) 9.54438 0.527805
\(328\) −17.3389 −0.957379
\(329\) −16.6324 −0.916973
\(330\) 0 0
\(331\) 30.2154 1.66079 0.830394 0.557176i \(-0.188115\pi\)
0.830394 + 0.557176i \(0.188115\pi\)
\(332\) −7.93224 −0.435338
\(333\) −20.6014 −1.12895
\(334\) 3.82396 0.209238
\(335\) 0 0
\(336\) 8.62012 0.470266
\(337\) 7.31819 0.398647 0.199324 0.979934i \(-0.436125\pi\)
0.199324 + 0.979934i \(0.436125\pi\)
\(338\) 7.49219 0.407521
\(339\) 8.63405 0.468937
\(340\) 0 0
\(341\) 5.38333 0.291524
\(342\) 3.24029 0.175215
\(343\) 18.7461 1.01219
\(344\) −23.0109 −1.24067
\(345\) 0 0
\(346\) −2.56642 −0.137972
\(347\) −18.8592 −1.01241 −0.506206 0.862412i \(-0.668953\pi\)
−0.506206 + 0.862412i \(0.668953\pi\)
\(348\) 6.56731 0.352045
\(349\) −3.95224 −0.211559 −0.105779 0.994390i \(-0.533734\pi\)
−0.105779 + 0.994390i \(0.533734\pi\)
\(350\) 0 0
\(351\) 0.0640692 0.00341976
\(352\) −29.4442 −1.56938
\(353\) −29.1807 −1.55313 −0.776565 0.630037i \(-0.783040\pi\)
−0.776565 + 0.630037i \(0.783040\pi\)
\(354\) 15.5395 0.825916
\(355\) 0 0
\(356\) 17.5579 0.930564
\(357\) −2.58999 −0.137077
\(358\) −6.48684 −0.342840
\(359\) −19.0423 −1.00502 −0.502508 0.864573i \(-0.667589\pi\)
−0.502508 + 0.864573i \(0.667589\pi\)
\(360\) 0 0
\(361\) −15.7214 −0.827444
\(362\) −10.3691 −0.544986
\(363\) −44.3343 −2.32695
\(364\) −0.906764 −0.0475273
\(365\) 0 0
\(366\) −12.4609 −0.651340
\(367\) −21.8273 −1.13938 −0.569689 0.821861i \(-0.692936\pi\)
−0.569689 + 0.821861i \(0.692936\pi\)
\(368\) 1.16695 0.0608314
\(369\) −25.0928 −1.30628
\(370\) 0 0
\(371\) 5.88499 0.305534
\(372\) 4.09891 0.212519
\(373\) 0.572836 0.0296603 0.0148302 0.999890i \(-0.495279\pi\)
0.0148302 + 0.999890i \(0.495279\pi\)
\(374\) 1.96258 0.101483
\(375\) 0 0
\(376\) 21.1416 1.09029
\(377\) −0.522015 −0.0268851
\(378\) −0.191300 −0.00983943
\(379\) 23.4864 1.20642 0.603208 0.797584i \(-0.293889\pi\)
0.603208 + 0.797584i \(0.293889\pi\)
\(380\) 0 0
\(381\) 42.2700 2.16556
\(382\) 14.8300 0.758768
\(383\) −24.4037 −1.24697 −0.623485 0.781835i \(-0.714284\pi\)
−0.623485 + 0.781835i \(0.714284\pi\)
\(384\) −28.4027 −1.44942
\(385\) 0 0
\(386\) 7.63703 0.388714
\(387\) −33.3014 −1.69280
\(388\) −24.9773 −1.26803
\(389\) 22.0685 1.11892 0.559458 0.828859i \(-0.311009\pi\)
0.559458 + 0.828859i \(0.311009\pi\)
\(390\) 0 0
\(391\) −0.350620 −0.0177316
\(392\) −8.93172 −0.451120
\(393\) 41.7597 2.10650
\(394\) −0.126314 −0.00636359
\(395\) 0 0
\(396\) −27.5608 −1.38498
\(397\) 12.7949 0.642156 0.321078 0.947053i \(-0.395955\pi\)
0.321078 + 0.947053i \(0.395955\pi\)
\(398\) 1.60310 0.0803559
\(399\) −7.47464 −0.374200
\(400\) 0 0
\(401\) 10.9209 0.545362 0.272681 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(402\) 0.982891 0.0490221
\(403\) −0.325810 −0.0162297
\(404\) 30.9887 1.54174
\(405\) 0 0
\(406\) 1.55865 0.0773547
\(407\) −36.0107 −1.78498
\(408\) 3.29216 0.162986
\(409\) 10.9359 0.540744 0.270372 0.962756i \(-0.412853\pi\)
0.270372 + 0.962756i \(0.412853\pi\)
\(410\) 0 0
\(411\) 24.1612 1.19178
\(412\) −26.6490 −1.31290
\(413\) −18.1582 −0.893508
\(414\) −1.00007 −0.0491505
\(415\) 0 0
\(416\) 1.78202 0.0873708
\(417\) −40.4572 −1.98120
\(418\) 5.66395 0.277033
\(419\) 10.6004 0.517863 0.258932 0.965896i \(-0.416630\pi\)
0.258932 + 0.965896i \(0.416630\pi\)
\(420\) 0 0
\(421\) −26.7943 −1.30587 −0.652936 0.757413i \(-0.726463\pi\)
−0.652936 + 0.757413i \(0.726463\pi\)
\(422\) 10.0692 0.490162
\(423\) 30.5961 1.48763
\(424\) −7.48046 −0.363283
\(425\) 0 0
\(426\) 1.43764 0.0696537
\(427\) 14.5608 0.704645
\(428\) 27.5683 1.33257
\(429\) 4.32472 0.208800
\(430\) 0 0
\(431\) 21.9475 1.05717 0.528586 0.848880i \(-0.322722\pi\)
0.528586 + 0.848880i \(0.322722\pi\)
\(432\) −0.410630 −0.0197564
\(433\) −20.8138 −1.00025 −0.500124 0.865954i \(-0.666712\pi\)
−0.500124 + 0.865954i \(0.666712\pi\)
\(434\) 0.972816 0.0466967
\(435\) 0 0
\(436\) 6.43473 0.308168
\(437\) −1.01188 −0.0484048
\(438\) 7.20720 0.344373
\(439\) −17.8641 −0.852608 −0.426304 0.904580i \(-0.640185\pi\)
−0.426304 + 0.904580i \(0.640185\pi\)
\(440\) 0 0
\(441\) −12.9260 −0.615523
\(442\) −0.118779 −0.00564975
\(443\) 0.492530 0.0234008 0.0117004 0.999932i \(-0.496276\pi\)
0.0117004 + 0.999932i \(0.496276\pi\)
\(444\) −27.4188 −1.30124
\(445\) 0 0
\(446\) −11.7330 −0.555571
\(447\) −4.59721 −0.217440
\(448\) 1.67114 0.0789541
\(449\) 22.5350 1.06349 0.531746 0.846904i \(-0.321536\pi\)
0.531746 + 0.846904i \(0.321536\pi\)
\(450\) 0 0
\(451\) −43.8617 −2.06536
\(452\) 5.82100 0.273797
\(453\) −22.9005 −1.07596
\(454\) 3.67056 0.172268
\(455\) 0 0
\(456\) 9.50107 0.444928
\(457\) 5.72325 0.267722 0.133861 0.991000i \(-0.457262\pi\)
0.133861 + 0.991000i \(0.457262\pi\)
\(458\) 6.98019 0.326163
\(459\) 0.123377 0.00575876
\(460\) 0 0
\(461\) −2.00807 −0.0935249 −0.0467625 0.998906i \(-0.514890\pi\)
−0.0467625 + 0.998906i \(0.514890\pi\)
\(462\) −12.9129 −0.600764
\(463\) 20.5390 0.954528 0.477264 0.878760i \(-0.341629\pi\)
0.477264 + 0.878760i \(0.341629\pi\)
\(464\) 3.34568 0.155319
\(465\) 0 0
\(466\) 16.4889 0.763832
\(467\) 5.95062 0.275362 0.137681 0.990477i \(-0.456035\pi\)
0.137681 + 0.990477i \(0.456035\pi\)
\(468\) 1.66804 0.0771050
\(469\) −1.14853 −0.0530340
\(470\) 0 0
\(471\) −30.1164 −1.38769
\(472\) 23.0811 1.06239
\(473\) −58.2100 −2.67650
\(474\) −19.5248 −0.896804
\(475\) 0 0
\(476\) −1.74615 −0.0800344
\(477\) −10.8257 −0.495676
\(478\) −4.76889 −0.218124
\(479\) −22.2840 −1.01818 −0.509090 0.860713i \(-0.670018\pi\)
−0.509090 + 0.860713i \(0.670018\pi\)
\(480\) 0 0
\(481\) 2.17944 0.0993737
\(482\) 6.68896 0.304674
\(483\) 2.30693 0.104969
\(484\) −29.8897 −1.35862
\(485\) 0 0
\(486\) −12.8856 −0.584502
\(487\) 38.0611 1.72472 0.862358 0.506300i \(-0.168987\pi\)
0.862358 + 0.506300i \(0.168987\pi\)
\(488\) −18.5083 −0.837832
\(489\) 30.7117 1.38883
\(490\) 0 0
\(491\) 17.5955 0.794073 0.397037 0.917803i \(-0.370039\pi\)
0.397037 + 0.917803i \(0.370039\pi\)
\(492\) −33.3966 −1.50563
\(493\) −1.00524 −0.0452737
\(494\) −0.342793 −0.0154230
\(495\) 0 0
\(496\) 2.08817 0.0937615
\(497\) −1.67990 −0.0753540
\(498\) 6.83659 0.306355
\(499\) −7.07301 −0.316632 −0.158316 0.987389i \(-0.550606\pi\)
−0.158316 + 0.987389i \(0.550606\pi\)
\(500\) 0 0
\(501\) 16.2267 0.724955
\(502\) −6.09405 −0.271991
\(503\) −4.43425 −0.197713 −0.0988567 0.995102i \(-0.531519\pi\)
−0.0988567 + 0.995102i \(0.531519\pi\)
\(504\) −10.9726 −0.488757
\(505\) 0 0
\(506\) −1.74809 −0.0777121
\(507\) 31.7926 1.41196
\(508\) 28.4980 1.26440
\(509\) 15.4718 0.685775 0.342888 0.939376i \(-0.388595\pi\)
0.342888 + 0.939376i \(0.388595\pi\)
\(510\) 0 0
\(511\) −8.42175 −0.372556
\(512\) −20.3088 −0.897532
\(513\) 0.356064 0.0157206
\(514\) 11.2315 0.495400
\(515\) 0 0
\(516\) −44.3215 −1.95115
\(517\) 53.4812 2.35210
\(518\) −6.50745 −0.285921
\(519\) −10.8904 −0.478037
\(520\) 0 0
\(521\) 8.92019 0.390800 0.195400 0.980724i \(-0.437399\pi\)
0.195400 + 0.980724i \(0.437399\pi\)
\(522\) −2.86722 −0.125495
\(523\) −37.7360 −1.65008 −0.825039 0.565075i \(-0.808847\pi\)
−0.825039 + 0.565075i \(0.808847\pi\)
\(524\) 28.1540 1.22991
\(525\) 0 0
\(526\) 2.82767 0.123292
\(527\) −0.627408 −0.0273303
\(528\) −27.7179 −1.20626
\(529\) −22.6877 −0.986422
\(530\) 0 0
\(531\) 33.4029 1.44956
\(532\) −5.03933 −0.218483
\(533\) 2.65459 0.114983
\(534\) −15.1327 −0.654854
\(535\) 0 0
\(536\) 1.45990 0.0630581
\(537\) −27.5264 −1.18785
\(538\) −10.3063 −0.444336
\(539\) −22.5943 −0.973206
\(540\) 0 0
\(541\) 29.8561 1.28361 0.641806 0.766867i \(-0.278185\pi\)
0.641806 + 0.766867i \(0.278185\pi\)
\(542\) 13.5118 0.580381
\(543\) −44.0004 −1.88824
\(544\) 3.43162 0.147130
\(545\) 0 0
\(546\) 0.781516 0.0334458
\(547\) 42.0150 1.79643 0.898215 0.439556i \(-0.144864\pi\)
0.898215 + 0.439556i \(0.144864\pi\)
\(548\) 16.2892 0.695842
\(549\) −26.7852 −1.14317
\(550\) 0 0
\(551\) −2.90109 −0.123591
\(552\) −2.93236 −0.124809
\(553\) 22.8151 0.970197
\(554\) 3.55699 0.151122
\(555\) 0 0
\(556\) −27.2758 −1.15675
\(557\) 24.1764 1.02439 0.512194 0.858870i \(-0.328833\pi\)
0.512194 + 0.858870i \(0.328833\pi\)
\(558\) −1.78954 −0.0757573
\(559\) 3.52298 0.149006
\(560\) 0 0
\(561\) 8.32807 0.351611
\(562\) 8.76012 0.369523
\(563\) 2.85978 0.120525 0.0602627 0.998183i \(-0.480806\pi\)
0.0602627 + 0.998183i \(0.480806\pi\)
\(564\) 40.7210 1.71466
\(565\) 0 0
\(566\) −6.91167 −0.290519
\(567\) 14.6565 0.615515
\(568\) 2.13534 0.0895969
\(569\) 9.28905 0.389417 0.194709 0.980861i \(-0.437624\pi\)
0.194709 + 0.980861i \(0.437624\pi\)
\(570\) 0 0
\(571\) −19.1986 −0.803437 −0.401719 0.915763i \(-0.631587\pi\)
−0.401719 + 0.915763i \(0.631587\pi\)
\(572\) 2.91569 0.121911
\(573\) 62.9300 2.62894
\(574\) −7.92619 −0.330833
\(575\) 0 0
\(576\) −3.07415 −0.128089
\(577\) 34.8805 1.45209 0.726047 0.687645i \(-0.241355\pi\)
0.726047 + 0.687645i \(0.241355\pi\)
\(578\) 9.64940 0.401362
\(579\) 32.4072 1.34680
\(580\) 0 0
\(581\) −7.98868 −0.331426
\(582\) 21.5273 0.892334
\(583\) −18.9231 −0.783715
\(584\) 10.7050 0.442974
\(585\) 0 0
\(586\) 6.54438 0.270346
\(587\) 44.2409 1.82602 0.913009 0.407938i \(-0.133752\pi\)
0.913009 + 0.407938i \(0.133752\pi\)
\(588\) −17.2035 −0.709460
\(589\) −1.81068 −0.0746079
\(590\) 0 0
\(591\) −0.536003 −0.0220482
\(592\) −13.9684 −0.574096
\(593\) −23.2861 −0.956247 −0.478124 0.878293i \(-0.658683\pi\)
−0.478124 + 0.878293i \(0.658683\pi\)
\(594\) 0.615124 0.0252388
\(595\) 0 0
\(596\) −3.09939 −0.126956
\(597\) 6.80262 0.278413
\(598\) 0.105798 0.00432639
\(599\) 24.4588 0.999359 0.499679 0.866210i \(-0.333451\pi\)
0.499679 + 0.866210i \(0.333451\pi\)
\(600\) 0 0
\(601\) 20.7169 0.845061 0.422531 0.906349i \(-0.361142\pi\)
0.422531 + 0.906349i \(0.361142\pi\)
\(602\) −10.5191 −0.428725
\(603\) 2.11277 0.0860386
\(604\) −15.4393 −0.628217
\(605\) 0 0
\(606\) −26.7083 −1.08495
\(607\) −29.4411 −1.19498 −0.597488 0.801878i \(-0.703835\pi\)
−0.597488 + 0.801878i \(0.703835\pi\)
\(608\) 9.90356 0.401643
\(609\) 6.61404 0.268014
\(610\) 0 0
\(611\) −3.23679 −0.130946
\(612\) 3.21212 0.129842
\(613\) 21.8105 0.880916 0.440458 0.897773i \(-0.354816\pi\)
0.440458 + 0.897773i \(0.354816\pi\)
\(614\) 8.80701 0.355422
\(615\) 0 0
\(616\) −19.1798 −0.772775
\(617\) −23.2460 −0.935848 −0.467924 0.883769i \(-0.654998\pi\)
−0.467924 + 0.883769i \(0.654998\pi\)
\(618\) 22.9680 0.923910
\(619\) −2.69894 −0.108480 −0.0542398 0.998528i \(-0.517274\pi\)
−0.0542398 + 0.998528i \(0.517274\pi\)
\(620\) 0 0
\(621\) −0.109893 −0.00440987
\(622\) −10.8321 −0.434328
\(623\) 17.6828 0.708446
\(624\) 1.67754 0.0671553
\(625\) 0 0
\(626\) −0.720091 −0.0287806
\(627\) 24.0346 0.959849
\(628\) −20.3042 −0.810226
\(629\) 4.19691 0.167342
\(630\) 0 0
\(631\) 3.16304 0.125918 0.0629592 0.998016i \(-0.479946\pi\)
0.0629592 + 0.998016i \(0.479946\pi\)
\(632\) −29.0005 −1.15358
\(633\) 42.7281 1.69829
\(634\) −5.27148 −0.209357
\(635\) 0 0
\(636\) −14.4082 −0.571322
\(637\) 1.36745 0.0541804
\(638\) −5.01183 −0.198420
\(639\) 3.09026 0.122249
\(640\) 0 0
\(641\) −24.1094 −0.952264 −0.476132 0.879374i \(-0.657962\pi\)
−0.476132 + 0.879374i \(0.657962\pi\)
\(642\) −23.7604 −0.937749
\(643\) −6.01512 −0.237213 −0.118607 0.992941i \(-0.537843\pi\)
−0.118607 + 0.992941i \(0.537843\pi\)
\(644\) 1.55531 0.0612878
\(645\) 0 0
\(646\) −0.660113 −0.0259718
\(647\) −2.25081 −0.0884884 −0.0442442 0.999021i \(-0.514088\pi\)
−0.0442442 + 0.999021i \(0.514088\pi\)
\(648\) −18.6300 −0.731855
\(649\) 58.3875 2.29191
\(650\) 0 0
\(651\) 4.12808 0.161792
\(652\) 20.7055 0.810892
\(653\) 24.7232 0.967495 0.483747 0.875208i \(-0.339275\pi\)
0.483747 + 0.875208i \(0.339275\pi\)
\(654\) −5.54592 −0.216863
\(655\) 0 0
\(656\) −17.0137 −0.664274
\(657\) 15.4922 0.604408
\(658\) 9.66453 0.376763
\(659\) −21.5721 −0.840330 −0.420165 0.907448i \(-0.638028\pi\)
−0.420165 + 0.907448i \(0.638028\pi\)
\(660\) 0 0
\(661\) 28.9338 1.12540 0.562698 0.826663i \(-0.309763\pi\)
0.562698 + 0.826663i \(0.309763\pi\)
\(662\) −17.5572 −0.682379
\(663\) −0.504031 −0.0195749
\(664\) 10.1545 0.394070
\(665\) 0 0
\(666\) 11.9708 0.463858
\(667\) 0.895376 0.0346691
\(668\) 10.9399 0.423276
\(669\) −49.7879 −1.92491
\(670\) 0 0
\(671\) −46.8199 −1.80746
\(672\) −22.5786 −0.870988
\(673\) −11.0864 −0.427350 −0.213675 0.976905i \(-0.568543\pi\)
−0.213675 + 0.976905i \(0.568543\pi\)
\(674\) −4.25236 −0.163795
\(675\) 0 0
\(676\) 21.4342 0.824394
\(677\) −31.0895 −1.19487 −0.597434 0.801918i \(-0.703813\pi\)
−0.597434 + 0.801918i \(0.703813\pi\)
\(678\) −5.01696 −0.192675
\(679\) −25.1550 −0.965362
\(680\) 0 0
\(681\) 15.5758 0.596865
\(682\) −3.12808 −0.119780
\(683\) 4.07538 0.155940 0.0779701 0.996956i \(-0.475156\pi\)
0.0779701 + 0.996956i \(0.475156\pi\)
\(684\) 9.27008 0.354450
\(685\) 0 0
\(686\) −10.8927 −0.415886
\(687\) 29.6199 1.13007
\(688\) −22.5794 −0.860831
\(689\) 1.14526 0.0436310
\(690\) 0 0
\(691\) 7.04037 0.267828 0.133914 0.990993i \(-0.457245\pi\)
0.133914 + 0.990993i \(0.457245\pi\)
\(692\) −7.34223 −0.279110
\(693\) −27.7570 −1.05440
\(694\) 10.9584 0.415976
\(695\) 0 0
\(696\) −8.40716 −0.318672
\(697\) 5.11192 0.193628
\(698\) 2.29652 0.0869245
\(699\) 69.9694 2.64648
\(700\) 0 0
\(701\) 22.8283 0.862211 0.431106 0.902301i \(-0.358124\pi\)
0.431106 + 0.902301i \(0.358124\pi\)
\(702\) −0.0372285 −0.00140510
\(703\) 12.1122 0.456820
\(704\) −5.37354 −0.202523
\(705\) 0 0
\(706\) 16.9559 0.638144
\(707\) 31.2092 1.17374
\(708\) 44.4567 1.67078
\(709\) −19.3406 −0.726352 −0.363176 0.931721i \(-0.618308\pi\)
−0.363176 + 0.931721i \(0.618308\pi\)
\(710\) 0 0
\(711\) −41.9695 −1.57398
\(712\) −22.4767 −0.842351
\(713\) 0.558839 0.0209287
\(714\) 1.50496 0.0563216
\(715\) 0 0
\(716\) −18.5581 −0.693547
\(717\) −20.2365 −0.755744
\(718\) 11.0649 0.412937
\(719\) −42.3226 −1.57837 −0.789184 0.614157i \(-0.789496\pi\)
−0.789184 + 0.614157i \(0.789496\pi\)
\(720\) 0 0
\(721\) −26.8386 −0.999521
\(722\) 9.13520 0.339977
\(723\) 28.3841 1.05562
\(724\) −29.6646 −1.10248
\(725\) 0 0
\(726\) 25.7612 0.956087
\(727\) 25.0220 0.928015 0.464008 0.885831i \(-0.346411\pi\)
0.464008 + 0.885831i \(0.346411\pi\)
\(728\) 1.16080 0.0430220
\(729\) −28.4159 −1.05244
\(730\) 0 0
\(731\) 6.78417 0.250922
\(732\) −35.6490 −1.31763
\(733\) 38.1240 1.40814 0.704071 0.710130i \(-0.251364\pi\)
0.704071 + 0.710130i \(0.251364\pi\)
\(734\) 12.6831 0.468143
\(735\) 0 0
\(736\) −3.05658 −0.112667
\(737\) 3.69307 0.136036
\(738\) 14.5806 0.536719
\(739\) 47.7727 1.75735 0.878673 0.477424i \(-0.158429\pi\)
0.878673 + 0.477424i \(0.158429\pi\)
\(740\) 0 0
\(741\) −1.45462 −0.0534368
\(742\) −3.41957 −0.125536
\(743\) 53.5451 1.96438 0.982189 0.187896i \(-0.0601667\pi\)
0.982189 + 0.187896i \(0.0601667\pi\)
\(744\) −5.24723 −0.192373
\(745\) 0 0
\(746\) −0.332856 −0.0121867
\(747\) 14.6956 0.537682
\(748\) 5.61470 0.205294
\(749\) 27.7645 1.01449
\(750\) 0 0
\(751\) −37.9303 −1.38410 −0.692048 0.721852i \(-0.743291\pi\)
−0.692048 + 0.721852i \(0.743291\pi\)
\(752\) 20.7451 0.756495
\(753\) −25.8597 −0.942378
\(754\) 0.303326 0.0110465
\(755\) 0 0
\(756\) −0.547287 −0.0199046
\(757\) −30.4110 −1.10531 −0.552653 0.833412i \(-0.686384\pi\)
−0.552653 + 0.833412i \(0.686384\pi\)
\(758\) −13.6472 −0.495688
\(759\) −7.41790 −0.269253
\(760\) 0 0
\(761\) 21.8253 0.791167 0.395583 0.918430i \(-0.370542\pi\)
0.395583 + 0.918430i \(0.370542\pi\)
\(762\) −24.5617 −0.889777
\(763\) 6.48052 0.234610
\(764\) 42.4268 1.53495
\(765\) 0 0
\(766\) 14.1802 0.512350
\(767\) −3.53373 −0.127595
\(768\) 11.5814 0.417908
\(769\) 37.4313 1.34981 0.674904 0.737905i \(-0.264185\pi\)
0.674904 + 0.737905i \(0.264185\pi\)
\(770\) 0 0
\(771\) 47.6600 1.71643
\(772\) 21.8486 0.786348
\(773\) −22.0976 −0.794797 −0.397398 0.917646i \(-0.630087\pi\)
−0.397398 + 0.917646i \(0.630087\pi\)
\(774\) 19.3503 0.695533
\(775\) 0 0
\(776\) 31.9748 1.14783
\(777\) −27.6139 −0.990643
\(778\) −12.8233 −0.459736
\(779\) 14.7529 0.528576
\(780\) 0 0
\(781\) 5.40171 0.193288
\(782\) 0.203734 0.00728550
\(783\) −0.315068 −0.0112596
\(784\) −8.76422 −0.313008
\(785\) 0 0
\(786\) −24.2652 −0.865510
\(787\) 20.3038 0.723753 0.361877 0.932226i \(-0.382136\pi\)
0.361877 + 0.932226i \(0.382136\pi\)
\(788\) −0.361368 −0.0128732
\(789\) 11.9990 0.427176
\(790\) 0 0
\(791\) 5.86242 0.208444
\(792\) 35.2821 1.25369
\(793\) 2.83363 0.100625
\(794\) −7.43468 −0.263847
\(795\) 0 0
\(796\) 4.58626 0.162556
\(797\) −15.5996 −0.552565 −0.276282 0.961077i \(-0.589102\pi\)
−0.276282 + 0.961077i \(0.589102\pi\)
\(798\) 4.34326 0.153750
\(799\) −6.23304 −0.220509
\(800\) 0 0
\(801\) −32.5283 −1.14933
\(802\) −6.34576 −0.224076
\(803\) 27.0800 0.955633
\(804\) 2.81193 0.0991692
\(805\) 0 0
\(806\) 0.189317 0.00666841
\(807\) −43.7340 −1.53951
\(808\) −39.6702 −1.39559
\(809\) 18.5700 0.652887 0.326443 0.945217i \(-0.394150\pi\)
0.326443 + 0.945217i \(0.394150\pi\)
\(810\) 0 0
\(811\) −1.53602 −0.0539369 −0.0269684 0.999636i \(-0.508585\pi\)
−0.0269684 + 0.999636i \(0.508585\pi\)
\(812\) 4.45912 0.156484
\(813\) 57.3363 2.01087
\(814\) 20.9246 0.733407
\(815\) 0 0
\(816\) 3.23042 0.113087
\(817\) 19.5789 0.684980
\(818\) −6.35447 −0.222179
\(819\) 1.67990 0.0587006
\(820\) 0 0
\(821\) 21.9559 0.766267 0.383133 0.923693i \(-0.374845\pi\)
0.383133 + 0.923693i \(0.374845\pi\)
\(822\) −14.0393 −0.489676
\(823\) 50.5105 1.76069 0.880343 0.474338i \(-0.157313\pi\)
0.880343 + 0.474338i \(0.157313\pi\)
\(824\) 34.1147 1.18844
\(825\) 0 0
\(826\) 10.5511 0.367121
\(827\) 20.7591 0.721864 0.360932 0.932592i \(-0.382459\pi\)
0.360932 + 0.932592i \(0.382459\pi\)
\(828\) −2.86107 −0.0994289
\(829\) 14.7970 0.513920 0.256960 0.966422i \(-0.417279\pi\)
0.256960 + 0.966422i \(0.417279\pi\)
\(830\) 0 0
\(831\) 15.0938 0.523599
\(832\) 0.325217 0.0112749
\(833\) 2.63329 0.0912380
\(834\) 23.5083 0.814027
\(835\) 0 0
\(836\) 16.2039 0.560423
\(837\) −0.196646 −0.00679708
\(838\) −6.15954 −0.212778
\(839\) −19.1059 −0.659608 −0.329804 0.944049i \(-0.606983\pi\)
−0.329804 + 0.944049i \(0.606983\pi\)
\(840\) 0 0
\(841\) −26.4329 −0.911480
\(842\) 15.5693 0.536552
\(843\) 37.1729 1.28030
\(844\) 28.8068 0.991572
\(845\) 0 0
\(846\) −17.7784 −0.611233
\(847\) −30.1024 −1.03433
\(848\) −7.34018 −0.252063
\(849\) −29.3292 −1.00657
\(850\) 0 0
\(851\) −3.73823 −0.128145
\(852\) 4.11290 0.140906
\(853\) −36.3723 −1.24536 −0.622681 0.782476i \(-0.713957\pi\)
−0.622681 + 0.782476i \(0.713957\pi\)
\(854\) −8.46078 −0.289522
\(855\) 0 0
\(856\) −35.2917 −1.20624
\(857\) 13.6923 0.467719 0.233859 0.972270i \(-0.424864\pi\)
0.233859 + 0.972270i \(0.424864\pi\)
\(858\) −2.51295 −0.0857908
\(859\) −45.9661 −1.56834 −0.784171 0.620545i \(-0.786912\pi\)
−0.784171 + 0.620545i \(0.786912\pi\)
\(860\) 0 0
\(861\) −33.6342 −1.14625
\(862\) −12.7529 −0.434367
\(863\) 52.8198 1.79801 0.899004 0.437941i \(-0.144292\pi\)
0.899004 + 0.437941i \(0.144292\pi\)
\(864\) 1.07556 0.0365913
\(865\) 0 0
\(866\) 12.0942 0.410978
\(867\) 40.9466 1.39062
\(868\) 2.78311 0.0944649
\(869\) −73.3616 −2.48862
\(870\) 0 0
\(871\) −0.223512 −0.00757341
\(872\) −8.23744 −0.278955
\(873\) 46.2739 1.56613
\(874\) 0.587970 0.0198884
\(875\) 0 0
\(876\) 20.6189 0.696648
\(877\) −31.8767 −1.07640 −0.538201 0.842817i \(-0.680896\pi\)
−0.538201 + 0.842817i \(0.680896\pi\)
\(878\) 10.3802 0.350316
\(879\) 27.7706 0.936679
\(880\) 0 0
\(881\) −11.3499 −0.382387 −0.191194 0.981552i \(-0.561236\pi\)
−0.191194 + 0.981552i \(0.561236\pi\)
\(882\) 7.51086 0.252904
\(883\) −40.3011 −1.35624 −0.678120 0.734951i \(-0.737205\pi\)
−0.678120 + 0.734951i \(0.737205\pi\)
\(884\) −0.339813 −0.0114291
\(885\) 0 0
\(886\) −0.286193 −0.00961484
\(887\) −48.6547 −1.63367 −0.816833 0.576875i \(-0.804272\pi\)
−0.816833 + 0.576875i \(0.804272\pi\)
\(888\) 35.1002 1.17789
\(889\) 28.7008 0.962595
\(890\) 0 0
\(891\) −47.1278 −1.57884
\(892\) −33.5665 −1.12389
\(893\) −17.9884 −0.601958
\(894\) 2.67128 0.0893411
\(895\) 0 0
\(896\) −19.2851 −0.644269
\(897\) 0.448945 0.0149899
\(898\) −13.0943 −0.436963
\(899\) 1.60221 0.0534366
\(900\) 0 0
\(901\) 2.20542 0.0734732
\(902\) 25.4866 0.848609
\(903\) −44.6369 −1.48542
\(904\) −7.45176 −0.247842
\(905\) 0 0
\(906\) 13.3067 0.442087
\(907\) −40.5901 −1.34777 −0.673886 0.738835i \(-0.735376\pi\)
−0.673886 + 0.738835i \(0.735376\pi\)
\(908\) 10.5010 0.348489
\(909\) −57.4108 −1.90419
\(910\) 0 0
\(911\) 7.30922 0.242165 0.121083 0.992642i \(-0.461363\pi\)
0.121083 + 0.992642i \(0.461363\pi\)
\(912\) 9.32289 0.308712
\(913\) 25.6875 0.850132
\(914\) −3.32559 −0.110001
\(915\) 0 0
\(916\) 19.9695 0.659810
\(917\) 28.3543 0.936342
\(918\) −0.0716905 −0.00236614
\(919\) −14.5908 −0.481305 −0.240652 0.970611i \(-0.577361\pi\)
−0.240652 + 0.970611i \(0.577361\pi\)
\(920\) 0 0
\(921\) 37.3719 1.23145
\(922\) 1.16682 0.0384272
\(923\) −0.326922 −0.0107608
\(924\) −36.9423 −1.21531
\(925\) 0 0
\(926\) −11.9345 −0.392193
\(927\) 49.3708 1.62155
\(928\) −8.76331 −0.287670
\(929\) −21.7265 −0.712824 −0.356412 0.934329i \(-0.616000\pi\)
−0.356412 + 0.934329i \(0.616000\pi\)
\(930\) 0 0
\(931\) 7.59959 0.249067
\(932\) 47.1727 1.54519
\(933\) −45.9653 −1.50484
\(934\) −3.45771 −0.113140
\(935\) 0 0
\(936\) −2.13534 −0.0697958
\(937\) −12.8632 −0.420221 −0.210111 0.977678i \(-0.567382\pi\)
−0.210111 + 0.977678i \(0.567382\pi\)
\(938\) 0.667371 0.0217904
\(939\) −3.05565 −0.0997175
\(940\) 0 0
\(941\) 41.0452 1.33804 0.669018 0.743246i \(-0.266715\pi\)
0.669018 + 0.743246i \(0.266715\pi\)
\(942\) 17.4997 0.570170
\(943\) −4.55324 −0.148274
\(944\) 22.6482 0.737137
\(945\) 0 0
\(946\) 33.8239 1.09971
\(947\) −10.4562 −0.339782 −0.169891 0.985463i \(-0.554342\pi\)
−0.169891 + 0.985463i \(0.554342\pi\)
\(948\) −55.8581 −1.81419
\(949\) −1.63893 −0.0532020
\(950\) 0 0
\(951\) −22.3691 −0.725369
\(952\) 2.23533 0.0724476
\(953\) 35.9951 1.16600 0.582998 0.812474i \(-0.301880\pi\)
0.582998 + 0.812474i \(0.301880\pi\)
\(954\) 6.29047 0.203661
\(955\) 0 0
\(956\) −13.6432 −0.441253
\(957\) −21.2673 −0.687475
\(958\) 12.9485 0.418346
\(959\) 16.4051 0.529750
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −1.26640 −0.0408303
\(963\) −51.0741 −1.64584
\(964\) 19.1363 0.616338
\(965\) 0 0
\(966\) −1.34048 −0.0431292
\(967\) 34.1188 1.09719 0.548593 0.836089i \(-0.315164\pi\)
0.548593 + 0.836089i \(0.315164\pi\)
\(968\) 38.2634 1.22983
\(969\) −2.80114 −0.0899857
\(970\) 0 0
\(971\) −40.8349 −1.31046 −0.655228 0.755431i \(-0.727427\pi\)
−0.655228 + 0.755431i \(0.727427\pi\)
\(972\) −36.8641 −1.18242
\(973\) −27.4699 −0.880645
\(974\) −22.1161 −0.708645
\(975\) 0 0
\(976\) −18.1612 −0.581326
\(977\) −7.47323 −0.239090 −0.119545 0.992829i \(-0.538144\pi\)
−0.119545 + 0.992829i \(0.538144\pi\)
\(978\) −17.8456 −0.570638
\(979\) −56.8587 −1.81721
\(980\) 0 0
\(981\) −11.9212 −0.380615
\(982\) −10.2242 −0.326266
\(983\) −27.7520 −0.885152 −0.442576 0.896731i \(-0.645935\pi\)
−0.442576 + 0.896731i \(0.645935\pi\)
\(984\) 42.7527 1.36291
\(985\) 0 0
\(986\) 0.584111 0.0186019
\(987\) 41.0107 1.30539
\(988\) −0.980690 −0.0311999
\(989\) −6.04273 −0.192148
\(990\) 0 0
\(991\) −49.0311 −1.55752 −0.778762 0.627320i \(-0.784152\pi\)
−0.778762 + 0.627320i \(0.784152\pi\)
\(992\) −5.46952 −0.173657
\(993\) −74.5026 −2.36427
\(994\) 0.976137 0.0309612
\(995\) 0 0
\(996\) 19.5586 0.619739
\(997\) 12.6691 0.401235 0.200617 0.979670i \(-0.435705\pi\)
0.200617 + 0.979670i \(0.435705\pi\)
\(998\) 4.10989 0.130096
\(999\) 1.31542 0.0416181
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 775.2.a.i.1.3 5
3.2 odd 2 6975.2.a.bx.1.3 5
5.2 odd 4 775.2.b.h.249.4 10
5.3 odd 4 775.2.b.h.249.7 10
5.4 even 2 775.2.a.j.1.3 yes 5
15.14 odd 2 6975.2.a.bq.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
775.2.a.i.1.3 5 1.1 even 1 trivial
775.2.a.j.1.3 yes 5 5.4 even 2
775.2.b.h.249.4 10 5.2 odd 4
775.2.b.h.249.7 10 5.3 odd 4
6975.2.a.bq.1.3 5 15.14 odd 2
6975.2.a.bx.1.3 5 3.2 odd 2