Properties

Label 725.2.b.d.349.5
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.5
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.d.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919i q^{2} -1.70928i q^{3} -0.369102 q^{4} +2.63090 q^{6} +0.630898i q^{7} +2.51026i q^{8} +0.0783777 q^{9} +O(q^{10})\) \(q+1.53919i q^{2} -1.70928i q^{3} -0.369102 q^{4} +2.63090 q^{6} +0.630898i q^{7} +2.51026i q^{8} +0.0783777 q^{9} +0.290725 q^{11} +0.630898i q^{12} +0.921622i q^{13} -0.971071 q^{14} -4.60197 q^{16} +4.97107i q^{17} +0.120638i q^{18} +6.04945 q^{19} +1.07838 q^{21} +0.447480i q^{22} -2.29072i q^{23} +4.29072 q^{24} -1.41855 q^{26} -5.26180i q^{27} -0.232866i q^{28} -1.00000 q^{29} +10.0494 q^{31} -2.06278i q^{32} -0.496928i q^{33} -7.65142 q^{34} -0.0289294 q^{36} +1.55252i q^{37} +9.31124i q^{38} +1.57531 q^{39} +0.340173 q^{41} +1.65983i q^{42} +5.70928i q^{43} -0.107307 q^{44} +3.52586 q^{46} -1.12783i q^{47} +7.86603i q^{48} +6.60197 q^{49} +8.49693 q^{51} -0.340173i q^{52} +0.340173i q^{53} +8.09890 q^{54} -1.58372 q^{56} -10.3402i q^{57} -1.53919i q^{58} -9.75872 q^{59} +3.07838 q^{61} +15.4680i q^{62} +0.0494483i q^{63} -6.02893 q^{64} +0.764867 q^{66} -5.70928i q^{67} -1.83483i q^{68} -3.91548 q^{69} +9.07838 q^{71} +0.196748i q^{72} +6.94441i q^{73} -2.38962 q^{74} -2.23287 q^{76} +0.183417i q^{77} +2.42469i q^{78} -12.3896 q^{79} -8.75872 q^{81} +0.523590i q^{82} -2.78765i q^{83} -0.398032 q^{84} -8.78765 q^{86} +1.70928i q^{87} +0.729794i q^{88} -4.73820 q^{89} -0.581449 q^{91} +0.845512i q^{92} -17.1773i q^{93} +1.73594 q^{94} -3.52586 q^{96} -15.8927i q^{97} +10.1617i q^{98} +0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 8 q^{6} - 6 q^{9} + 16 q^{11} + 24 q^{14} + 10 q^{16} + 40 q^{24} + 20 q^{26} - 6 q^{29} + 24 q^{31} + 28 q^{34} - 30 q^{36} - 32 q^{39} - 20 q^{41} - 24 q^{44} + 16 q^{46} + 2 q^{49} + 16 q^{51} - 24 q^{54} - 64 q^{56} - 8 q^{59} + 12 q^{61} - 66 q^{64} + 24 q^{66} + 40 q^{69} + 48 q^{71} + 44 q^{74} + 32 q^{76} - 16 q^{79} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 44 q^{89} - 32 q^{91} - 16 q^{96} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919i 1.08837i 0.838965 + 0.544185i \(0.183161\pi\)
−0.838965 + 0.544185i \(0.816839\pi\)
\(3\) − 1.70928i − 0.986851i −0.869788 0.493425i \(-0.835745\pi\)
0.869788 0.493425i \(-0.164255\pi\)
\(4\) −0.369102 −0.184551
\(5\) 0 0
\(6\) 2.63090 1.07406
\(7\) 0.630898i 0.238457i 0.992867 + 0.119228i \(0.0380421\pi\)
−0.992867 + 0.119228i \(0.961958\pi\)
\(8\) 2.51026i 0.887511i
\(9\) 0.0783777 0.0261259
\(10\) 0 0
\(11\) 0.290725 0.0876568 0.0438284 0.999039i \(-0.486045\pi\)
0.0438284 + 0.999039i \(0.486045\pi\)
\(12\) 0.630898i 0.182124i
\(13\) 0.921622i 0.255612i 0.991799 + 0.127806i \(0.0407935\pi\)
−0.991799 + 0.127806i \(0.959207\pi\)
\(14\) −0.971071 −0.259530
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 4.97107i 1.20566i 0.797869 + 0.602831i \(0.205961\pi\)
−0.797869 + 0.602831i \(0.794039\pi\)
\(18\) 0.120638i 0.0284347i
\(19\) 6.04945 1.38784 0.693919 0.720053i \(-0.255882\pi\)
0.693919 + 0.720053i \(0.255882\pi\)
\(20\) 0 0
\(21\) 1.07838 0.235321
\(22\) 0.447480i 0.0954031i
\(23\) − 2.29072i − 0.477649i −0.971063 0.238825i \(-0.923238\pi\)
0.971063 0.238825i \(-0.0767621\pi\)
\(24\) 4.29072 0.875840
\(25\) 0 0
\(26\) −1.41855 −0.278201
\(27\) − 5.26180i − 1.01263i
\(28\) − 0.232866i − 0.0440075i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.0494 1.80493 0.902467 0.430759i \(-0.141754\pi\)
0.902467 + 0.430759i \(0.141754\pi\)
\(32\) − 2.06278i − 0.364651i
\(33\) − 0.496928i − 0.0865041i
\(34\) −7.65142 −1.31221
\(35\) 0 0
\(36\) −0.0289294 −0.00482157
\(37\) 1.55252i 0.255233i 0.991824 + 0.127616i \(0.0407326\pi\)
−0.991824 + 0.127616i \(0.959267\pi\)
\(38\) 9.31124i 1.51048i
\(39\) 1.57531 0.252251
\(40\) 0 0
\(41\) 0.340173 0.0531261 0.0265630 0.999647i \(-0.491544\pi\)
0.0265630 + 0.999647i \(0.491544\pi\)
\(42\) 1.65983i 0.256117i
\(43\) 5.70928i 0.870656i 0.900272 + 0.435328i \(0.143368\pi\)
−0.900272 + 0.435328i \(0.856632\pi\)
\(44\) −0.107307 −0.0161772
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) − 1.12783i − 0.164510i −0.996611 0.0822552i \(-0.973788\pi\)
0.996611 0.0822552i \(-0.0262122\pi\)
\(48\) 7.86603i 1.13536i
\(49\) 6.60197 0.943138
\(50\) 0 0
\(51\) 8.49693 1.18981
\(52\) − 0.340173i − 0.0471735i
\(53\) 0.340173i 0.0467264i 0.999727 + 0.0233632i \(0.00743741\pi\)
−0.999727 + 0.0233632i \(0.992563\pi\)
\(54\) 8.09890 1.10212
\(55\) 0 0
\(56\) −1.58372 −0.211633
\(57\) − 10.3402i − 1.36959i
\(58\) − 1.53919i − 0.202105i
\(59\) −9.75872 −1.27048 −0.635239 0.772316i \(-0.719098\pi\)
−0.635239 + 0.772316i \(0.719098\pi\)
\(60\) 0 0
\(61\) 3.07838 0.394146 0.197073 0.980389i \(-0.436856\pi\)
0.197073 + 0.980389i \(0.436856\pi\)
\(62\) 15.4680i 1.96444i
\(63\) 0.0494483i 0.00622990i
\(64\) −6.02893 −0.753616
\(65\) 0 0
\(66\) 0.764867 0.0941486
\(67\) − 5.70928i − 0.697499i −0.937216 0.348749i \(-0.886606\pi\)
0.937216 0.348749i \(-0.113394\pi\)
\(68\) − 1.83483i − 0.222506i
\(69\) −3.91548 −0.471368
\(70\) 0 0
\(71\) 9.07838 1.07741 0.538703 0.842496i \(-0.318915\pi\)
0.538703 + 0.842496i \(0.318915\pi\)
\(72\) 0.196748i 0.0231870i
\(73\) 6.94441i 0.812782i 0.913699 + 0.406391i \(0.133213\pi\)
−0.913699 + 0.406391i \(0.866787\pi\)
\(74\) −2.38962 −0.277788
\(75\) 0 0
\(76\) −2.23287 −0.256127
\(77\) 0.183417i 0.0209024i
\(78\) 2.42469i 0.274543i
\(79\) −12.3896 −1.39394 −0.696971 0.717100i \(-0.745469\pi\)
−0.696971 + 0.717100i \(0.745469\pi\)
\(80\) 0 0
\(81\) −8.75872 −0.973192
\(82\) 0.523590i 0.0578209i
\(83\) − 2.78765i − 0.305985i −0.988227 0.152992i \(-0.951109\pi\)
0.988227 0.152992i \(-0.0488910\pi\)
\(84\) −0.398032 −0.0434288
\(85\) 0 0
\(86\) −8.78765 −0.947597
\(87\) 1.70928i 0.183254i
\(88\) 0.729794i 0.0777963i
\(89\) −4.73820 −0.502249 −0.251124 0.967955i \(-0.580800\pi\)
−0.251124 + 0.967955i \(0.580800\pi\)
\(90\) 0 0
\(91\) −0.581449 −0.0609524
\(92\) 0.845512i 0.0881507i
\(93\) − 17.1773i − 1.78120i
\(94\) 1.73594 0.179048
\(95\) 0 0
\(96\) −3.52586 −0.359856
\(97\) − 15.8927i − 1.61366i −0.590785 0.806829i \(-0.701182\pi\)
0.590785 0.806829i \(-0.298818\pi\)
\(98\) 10.1617i 1.02648i
\(99\) 0.0227863 0.00229011
\(100\) 0 0
\(101\) −12.2557 −1.21948 −0.609741 0.792600i \(-0.708727\pi\)
−0.609741 + 0.792600i \(0.708727\pi\)
\(102\) 13.0784i 1.29495i
\(103\) 7.86603i 0.775063i 0.921856 + 0.387532i \(0.126672\pi\)
−0.921856 + 0.387532i \(0.873328\pi\)
\(104\) −2.31351 −0.226858
\(105\) 0 0
\(106\) −0.523590 −0.0508556
\(107\) 12.7298i 1.23064i 0.788279 + 0.615318i \(0.210972\pi\)
−0.788279 + 0.615318i \(0.789028\pi\)
\(108\) 1.94214i 0.186883i
\(109\) −12.4391 −1.19145 −0.595723 0.803190i \(-0.703135\pi\)
−0.595723 + 0.803190i \(0.703135\pi\)
\(110\) 0 0
\(111\) 2.65368 0.251877
\(112\) − 2.90337i − 0.274343i
\(113\) − 12.5730i − 1.18277i −0.806389 0.591386i \(-0.798581\pi\)
0.806389 0.591386i \(-0.201419\pi\)
\(114\) 15.9155 1.49062
\(115\) 0 0
\(116\) 0.369102 0.0342703
\(117\) 0.0722347i 0.00667810i
\(118\) − 15.0205i − 1.38275i
\(119\) −3.13624 −0.287498
\(120\) 0 0
\(121\) −10.9155 −0.992316
\(122\) 4.73820i 0.428977i
\(123\) − 0.581449i − 0.0524275i
\(124\) −3.70928 −0.333103
\(125\) 0 0
\(126\) −0.0761103 −0.00678045
\(127\) − 20.9132i − 1.85575i −0.372895 0.927874i \(-0.621635\pi\)
0.372895 0.927874i \(-0.378365\pi\)
\(128\) − 13.4052i − 1.18487i
\(129\) 9.75872 0.859208
\(130\) 0 0
\(131\) −13.4680 −1.17670 −0.588352 0.808605i \(-0.700223\pi\)
−0.588352 + 0.808605i \(0.700223\pi\)
\(132\) 0.183417i 0.0159644i
\(133\) 3.81658i 0.330940i
\(134\) 8.78765 0.759138
\(135\) 0 0
\(136\) −12.4787 −1.07004
\(137\) − 13.5525i − 1.15787i −0.815374 0.578935i \(-0.803469\pi\)
0.815374 0.578935i \(-0.196531\pi\)
\(138\) − 6.02666i − 0.513024i
\(139\) 4.89496 0.415185 0.207593 0.978215i \(-0.433437\pi\)
0.207593 + 0.978215i \(0.433437\pi\)
\(140\) 0 0
\(141\) −1.92777 −0.162347
\(142\) 13.9733i 1.17262i
\(143\) 0.267938i 0.0224061i
\(144\) −0.360692 −0.0300577
\(145\) 0 0
\(146\) −10.6888 −0.884608
\(147\) − 11.2846i − 0.930737i
\(148\) − 0.573039i − 0.0471035i
\(149\) 12.5236 1.02597 0.512986 0.858397i \(-0.328539\pi\)
0.512986 + 0.858397i \(0.328539\pi\)
\(150\) 0 0
\(151\) 7.60197 0.618639 0.309320 0.950958i \(-0.399899\pi\)
0.309320 + 0.950958i \(0.399899\pi\)
\(152\) 15.1857i 1.23172i
\(153\) 0.389621i 0.0314990i
\(154\) −0.282314 −0.0227495
\(155\) 0 0
\(156\) −0.581449 −0.0465532
\(157\) − 24.8865i − 1.98616i −0.117428 0.993081i \(-0.537465\pi\)
0.117428 0.993081i \(-0.462535\pi\)
\(158\) − 19.0700i − 1.51713i
\(159\) 0.581449 0.0461119
\(160\) 0 0
\(161\) 1.44521 0.113899
\(162\) − 13.4813i − 1.05919i
\(163\) − 0.447480i − 0.0350493i −0.999846 0.0175247i \(-0.994421\pi\)
0.999846 0.0175247i \(-0.00557856\pi\)
\(164\) −0.125559 −0.00980448
\(165\) 0 0
\(166\) 4.29072 0.333025
\(167\) 19.8660i 1.53728i 0.639682 + 0.768640i \(0.279066\pi\)
−0.639682 + 0.768640i \(0.720934\pi\)
\(168\) 2.70701i 0.208850i
\(169\) 12.1506 0.934662
\(170\) 0 0
\(171\) 0.474142 0.0362586
\(172\) − 2.10731i − 0.160681i
\(173\) 25.4329i 1.93363i 0.255478 + 0.966815i \(0.417767\pi\)
−0.255478 + 0.966815i \(0.582233\pi\)
\(174\) −2.63090 −0.199448
\(175\) 0 0
\(176\) −1.33791 −0.100848
\(177\) 16.6803i 1.25377i
\(178\) − 7.29299i − 0.546633i
\(179\) −14.8371 −1.10898 −0.554489 0.832191i \(-0.687086\pi\)
−0.554489 + 0.832191i \(0.687086\pi\)
\(180\) 0 0
\(181\) −5.91548 −0.439694 −0.219847 0.975534i \(-0.570556\pi\)
−0.219847 + 0.975534i \(0.570556\pi\)
\(182\) − 0.894960i − 0.0663389i
\(183\) − 5.26180i − 0.388963i
\(184\) 5.75031 0.423919
\(185\) 0 0
\(186\) 26.4391 1.93861
\(187\) 1.44521i 0.105684i
\(188\) 0.416283i 0.0303606i
\(189\) 3.31965 0.241469
\(190\) 0 0
\(191\) 7.02893 0.508595 0.254298 0.967126i \(-0.418156\pi\)
0.254298 + 0.967126i \(0.418156\pi\)
\(192\) 10.3051i 0.743707i
\(193\) − 17.8660i − 1.28603i −0.765856 0.643013i \(-0.777684\pi\)
0.765856 0.643013i \(-0.222316\pi\)
\(194\) 24.4619 1.75626
\(195\) 0 0
\(196\) −2.43680 −0.174057
\(197\) 6.09890i 0.434528i 0.976113 + 0.217264i \(0.0697133\pi\)
−0.976113 + 0.217264i \(0.930287\pi\)
\(198\) 0.0350725i 0.00249249i
\(199\) 9.75872 0.691778 0.345889 0.938276i \(-0.387577\pi\)
0.345889 + 0.938276i \(0.387577\pi\)
\(200\) 0 0
\(201\) −9.75872 −0.688327
\(202\) − 18.8638i − 1.32725i
\(203\) − 0.630898i − 0.0442803i
\(204\) −3.13624 −0.219580
\(205\) 0 0
\(206\) −12.1073 −0.843556
\(207\) − 0.179542i − 0.0124790i
\(208\) − 4.24128i − 0.294080i
\(209\) 1.75872 0.121653
\(210\) 0 0
\(211\) 9.86603 0.679206 0.339603 0.940569i \(-0.389707\pi\)
0.339603 + 0.940569i \(0.389707\pi\)
\(212\) − 0.125559i − 0.00862340i
\(213\) − 15.5174i − 1.06324i
\(214\) −19.5936 −1.33939
\(215\) 0 0
\(216\) 13.2085 0.898723
\(217\) 6.34017i 0.430399i
\(218\) − 19.1461i − 1.29674i
\(219\) 11.8699 0.802094
\(220\) 0 0
\(221\) −4.58145 −0.308182
\(222\) 4.08452i 0.274135i
\(223\) − 10.9711i − 0.734677i −0.930087 0.367339i \(-0.880269\pi\)
0.930087 0.367339i \(-0.119731\pi\)
\(224\) 1.30140 0.0869536
\(225\) 0 0
\(226\) 19.3523 1.28729
\(227\) − 12.5464i − 0.832732i −0.909197 0.416366i \(-0.863303\pi\)
0.909197 0.416366i \(-0.136697\pi\)
\(228\) 3.81658i 0.252759i
\(229\) −23.3607 −1.54372 −0.771859 0.635794i \(-0.780673\pi\)
−0.771859 + 0.635794i \(0.780673\pi\)
\(230\) 0 0
\(231\) 0.313511 0.0206275
\(232\) − 2.51026i − 0.164807i
\(233\) − 12.4703i − 0.816954i −0.912769 0.408477i \(-0.866060\pi\)
0.912769 0.408477i \(-0.133940\pi\)
\(234\) −0.111183 −0.00726825
\(235\) 0 0
\(236\) 3.60197 0.234468
\(237\) 21.1773i 1.37561i
\(238\) − 4.82726i − 0.312905i
\(239\) −13.7587 −0.889978 −0.444989 0.895536i \(-0.646792\pi\)
−0.444989 + 0.895536i \(0.646792\pi\)
\(240\) 0 0
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) − 16.8010i − 1.08001i
\(243\) − 0.814315i − 0.0522383i
\(244\) −1.13624 −0.0727401
\(245\) 0 0
\(246\) 0.894960 0.0570606
\(247\) 5.57531i 0.354748i
\(248\) 25.2267i 1.60190i
\(249\) −4.76487 −0.301961
\(250\) 0 0
\(251\) 15.4413 0.974649 0.487324 0.873221i \(-0.337973\pi\)
0.487324 + 0.873221i \(0.337973\pi\)
\(252\) − 0.0182515i − 0.00114974i
\(253\) − 0.665970i − 0.0418692i
\(254\) 32.1894 2.01974
\(255\) 0 0
\(256\) 8.57531 0.535957
\(257\) − 6.28231i − 0.391880i −0.980616 0.195940i \(-0.937224\pi\)
0.980616 0.195940i \(-0.0627758\pi\)
\(258\) 15.0205i 0.935137i
\(259\) −0.979481 −0.0608620
\(260\) 0 0
\(261\) −0.0783777 −0.00485146
\(262\) − 20.7298i − 1.28069i
\(263\) − 10.0761i − 0.621320i −0.950521 0.310660i \(-0.899450\pi\)
0.950521 0.310660i \(-0.100550\pi\)
\(264\) 1.24742 0.0767734
\(265\) 0 0
\(266\) −5.87444 −0.360185
\(267\) 8.09890i 0.495644i
\(268\) 2.10731i 0.128724i
\(269\) −28.1711 −1.71762 −0.858812 0.512291i \(-0.828797\pi\)
−0.858812 + 0.512291i \(0.828797\pi\)
\(270\) 0 0
\(271\) 28.8020 1.74960 0.874799 0.484485i \(-0.160993\pi\)
0.874799 + 0.484485i \(0.160993\pi\)
\(272\) − 22.8767i − 1.38710i
\(273\) 0.993857i 0.0601510i
\(274\) 20.8599 1.26019
\(275\) 0 0
\(276\) 1.44521 0.0869916
\(277\) 0.0266620i 0.00160196i 1.00000 0.000800982i \(0.000254960\pi\)
−1.00000 0.000800982i \(0.999745\pi\)
\(278\) 7.53427i 0.451875i
\(279\) 0.787653 0.0471556
\(280\) 0 0
\(281\) −28.0722 −1.67465 −0.837325 0.546706i \(-0.815881\pi\)
−0.837325 + 0.546706i \(0.815881\pi\)
\(282\) − 2.96719i − 0.176694i
\(283\) 20.8143i 1.23728i 0.785674 + 0.618641i \(0.212317\pi\)
−0.785674 + 0.618641i \(0.787683\pi\)
\(284\) −3.35085 −0.198836
\(285\) 0 0
\(286\) −0.412408 −0.0243862
\(287\) 0.214614i 0.0126683i
\(288\) − 0.161676i − 0.00952685i
\(289\) −7.71154 −0.453620
\(290\) 0 0
\(291\) −27.1650 −1.59244
\(292\) − 2.56320i − 0.150000i
\(293\) 15.4101i 0.900270i 0.892961 + 0.450135i \(0.148624\pi\)
−0.892961 + 0.450135i \(0.851376\pi\)
\(294\) 17.3691 1.01299
\(295\) 0 0
\(296\) −3.89723 −0.226522
\(297\) − 1.52973i − 0.0887641i
\(298\) 19.2762i 1.11664i
\(299\) 2.11118 0.122093
\(300\) 0 0
\(301\) −3.60197 −0.207614
\(302\) 11.7009i 0.673309i
\(303\) 20.9483i 1.20345i
\(304\) −27.8394 −1.59670
\(305\) 0 0
\(306\) −0.599701 −0.0342826
\(307\) − 28.4307i − 1.62262i −0.584614 0.811312i \(-0.698754\pi\)
0.584614 0.811312i \(-0.301246\pi\)
\(308\) − 0.0676998i − 0.00385756i
\(309\) 13.4452 0.764871
\(310\) 0 0
\(311\) −19.6248 −1.11282 −0.556409 0.830909i \(-0.687821\pi\)
−0.556409 + 0.830909i \(0.687821\pi\)
\(312\) 3.95443i 0.223875i
\(313\) − 22.9093i − 1.29491i −0.762103 0.647456i \(-0.775833\pi\)
0.762103 0.647456i \(-0.224167\pi\)
\(314\) 38.3051 2.16168
\(315\) 0 0
\(316\) 4.57304 0.257254
\(317\) 22.8599i 1.28394i 0.766730 + 0.641970i \(0.221882\pi\)
−0.766730 + 0.641970i \(0.778118\pi\)
\(318\) 0.894960i 0.0501869i
\(319\) −0.290725 −0.0162775
\(320\) 0 0
\(321\) 21.7587 1.21445
\(322\) 2.22446i 0.123964i
\(323\) 30.0722i 1.67326i
\(324\) 3.23287 0.179604
\(325\) 0 0
\(326\) 0.688756 0.0381467
\(327\) 21.2618i 1.17578i
\(328\) 0.853922i 0.0471500i
\(329\) 0.711543 0.0392286
\(330\) 0 0
\(331\) 24.0905 1.32413 0.662066 0.749445i \(-0.269680\pi\)
0.662066 + 0.749445i \(0.269680\pi\)
\(332\) 1.02893i 0.0564698i
\(333\) 0.121683i 0.00666819i
\(334\) −30.5776 −1.67313
\(335\) 0 0
\(336\) −4.96266 −0.270735
\(337\) 12.7877i 0.696588i 0.937385 + 0.348294i \(0.113239\pi\)
−0.937385 + 0.348294i \(0.886761\pi\)
\(338\) 18.7021i 1.01726i
\(339\) −21.4908 −1.16722
\(340\) 0 0
\(341\) 2.92162 0.158215
\(342\) 0.729794i 0.0394628i
\(343\) 8.58145i 0.463355i
\(344\) −14.3318 −0.772717
\(345\) 0 0
\(346\) −39.1461 −2.10451
\(347\) 8.41628i 0.451810i 0.974149 + 0.225905i \(0.0725339\pi\)
−0.974149 + 0.225905i \(0.927466\pi\)
\(348\) − 0.630898i − 0.0338197i
\(349\) −22.1978 −1.18822 −0.594110 0.804384i \(-0.702496\pi\)
−0.594110 + 0.804384i \(0.702496\pi\)
\(350\) 0 0
\(351\) 4.84939 0.258841
\(352\) − 0.599701i − 0.0319642i
\(353\) 6.18342i 0.329110i 0.986368 + 0.164555i \(0.0526188\pi\)
−0.986368 + 0.164555i \(0.947381\pi\)
\(354\) −25.6742 −1.36457
\(355\) 0 0
\(356\) 1.74888 0.0926906
\(357\) 5.36069i 0.283718i
\(358\) − 22.8371i − 1.20698i
\(359\) −5.05559 −0.266824 −0.133412 0.991061i \(-0.542593\pi\)
−0.133412 + 0.991061i \(0.542593\pi\)
\(360\) 0 0
\(361\) 17.5958 0.926096
\(362\) − 9.10504i − 0.478550i
\(363\) 18.6576i 0.979268i
\(364\) 0.214614 0.0112488
\(365\) 0 0
\(366\) 8.09890 0.423336
\(367\) 29.5402i 1.54199i 0.636843 + 0.770994i \(0.280240\pi\)
−0.636843 + 0.770994i \(0.719760\pi\)
\(368\) 10.5418i 0.549531i
\(369\) 0.0266620 0.00138797
\(370\) 0 0
\(371\) −0.214614 −0.0111422
\(372\) 6.34017i 0.328723i
\(373\) − 14.4124i − 0.746246i −0.927782 0.373123i \(-0.878287\pi\)
0.927782 0.373123i \(-0.121713\pi\)
\(374\) −2.22446 −0.115024
\(375\) 0 0
\(376\) 2.83114 0.146005
\(377\) − 0.921622i − 0.0474660i
\(378\) 5.10957i 0.262808i
\(379\) −14.1340 −0.726013 −0.363007 0.931787i \(-0.618250\pi\)
−0.363007 + 0.931787i \(0.618250\pi\)
\(380\) 0 0
\(381\) −35.7464 −1.83135
\(382\) 10.8188i 0.553541i
\(383\) 15.7815i 0.806397i 0.915112 + 0.403199i \(0.132102\pi\)
−0.915112 + 0.403199i \(0.867898\pi\)
\(384\) −22.9132 −1.16928
\(385\) 0 0
\(386\) 27.4992 1.39967
\(387\) 0.447480i 0.0227467i
\(388\) 5.86603i 0.297803i
\(389\) 13.8166 0.700529 0.350264 0.936651i \(-0.386092\pi\)
0.350264 + 0.936651i \(0.386092\pi\)
\(390\) 0 0
\(391\) 11.3874 0.575883
\(392\) 16.5727i 0.837045i
\(393\) 23.0205i 1.16123i
\(394\) −9.38735 −0.472928
\(395\) 0 0
\(396\) −0.00841049 −0.000422643 0
\(397\) 9.05172i 0.454293i 0.973861 + 0.227146i \(0.0729396\pi\)
−0.973861 + 0.227146i \(0.927060\pi\)
\(398\) 15.0205i 0.752911i
\(399\) 6.52359 0.326588
\(400\) 0 0
\(401\) 19.7587 0.986704 0.493352 0.869830i \(-0.335772\pi\)
0.493352 + 0.869830i \(0.335772\pi\)
\(402\) − 15.0205i − 0.749155i
\(403\) 9.26180i 0.461363i
\(404\) 4.52359 0.225057
\(405\) 0 0
\(406\) 0.971071 0.0481934
\(407\) 0.451356i 0.0223729i
\(408\) 21.3295i 1.05597i
\(409\) 1.71769 0.0849341 0.0424670 0.999098i \(-0.486478\pi\)
0.0424670 + 0.999098i \(0.486478\pi\)
\(410\) 0 0
\(411\) −23.1650 −1.14264
\(412\) − 2.90337i − 0.143039i
\(413\) − 6.15676i − 0.302954i
\(414\) 0.276349 0.0135818
\(415\) 0 0
\(416\) 1.90110 0.0932093
\(417\) − 8.36683i − 0.409726i
\(418\) 2.70701i 0.132404i
\(419\) 35.5318 1.73584 0.867922 0.496701i \(-0.165456\pi\)
0.867922 + 0.496701i \(0.165456\pi\)
\(420\) 0 0
\(421\) −12.0722 −0.588365 −0.294182 0.955749i \(-0.595047\pi\)
−0.294182 + 0.955749i \(0.595047\pi\)
\(422\) 15.1857i 0.739228i
\(423\) − 0.0883965i − 0.00429798i
\(424\) −0.853922 −0.0414701
\(425\) 0 0
\(426\) 23.8843 1.15720
\(427\) 1.94214i 0.0939868i
\(428\) − 4.69860i − 0.227115i
\(429\) 0.457980 0.0221115
\(430\) 0 0
\(431\) 19.8310 0.955224 0.477612 0.878571i \(-0.341503\pi\)
0.477612 + 0.878571i \(0.341503\pi\)
\(432\) 24.2146i 1.16503i
\(433\) − 14.8143i − 0.711931i −0.934499 0.355965i \(-0.884152\pi\)
0.934499 0.355965i \(-0.115848\pi\)
\(434\) −9.75872 −0.468434
\(435\) 0 0
\(436\) 4.59129 0.219883
\(437\) − 13.8576i − 0.662900i
\(438\) 18.2700i 0.872976i
\(439\) 17.8576 0.852298 0.426149 0.904653i \(-0.359870\pi\)
0.426149 + 0.904653i \(0.359870\pi\)
\(440\) 0 0
\(441\) 0.517447 0.0246404
\(442\) − 7.05172i − 0.335416i
\(443\) − 33.5936i − 1.59608i −0.602606 0.798039i \(-0.705871\pi\)
0.602606 0.798039i \(-0.294129\pi\)
\(444\) −0.979481 −0.0464841
\(445\) 0 0
\(446\) 16.8865 0.799601
\(447\) − 21.4063i − 1.01248i
\(448\) − 3.80364i − 0.179705i
\(449\) −7.07838 −0.334049 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(450\) 0 0
\(451\) 0.0988967 0.00465686
\(452\) 4.64074i 0.218282i
\(453\) − 12.9939i − 0.610505i
\(454\) 19.3112 0.906322
\(455\) 0 0
\(456\) 25.9565 1.21553
\(457\) − 5.81658i − 0.272088i −0.990703 0.136044i \(-0.956561\pi\)
0.990703 0.136044i \(-0.0434389\pi\)
\(458\) − 35.9565i − 1.68014i
\(459\) 26.1568 1.22089
\(460\) 0 0
\(461\) −32.3090 −1.50478 −0.752390 0.658718i \(-0.771099\pi\)
−0.752390 + 0.658718i \(0.771099\pi\)
\(462\) 0.482553i 0.0224504i
\(463\) − 1.44134i − 0.0669846i −0.999439 0.0334923i \(-0.989337\pi\)
0.999439 0.0334923i \(-0.0106629\pi\)
\(464\) 4.60197 0.213641
\(465\) 0 0
\(466\) 19.1941 0.889149
\(467\) − 11.7503i − 0.543740i −0.962334 0.271870i \(-0.912358\pi\)
0.962334 0.271870i \(-0.0876420\pi\)
\(468\) − 0.0266620i − 0.00123245i
\(469\) 3.60197 0.166323
\(470\) 0 0
\(471\) −42.5380 −1.96005
\(472\) − 24.4969i − 1.12756i
\(473\) 1.65983i 0.0763189i
\(474\) −32.5958 −1.49718
\(475\) 0 0
\(476\) 1.15759 0.0530582
\(477\) 0.0266620i 0.00122077i
\(478\) − 21.1773i − 0.968626i
\(479\) −17.1689 −0.784465 −0.392233 0.919866i \(-0.628297\pi\)
−0.392233 + 0.919866i \(0.628297\pi\)
\(480\) 0 0
\(481\) −1.43084 −0.0652405
\(482\) − 22.5958i − 1.02921i
\(483\) − 2.47027i − 0.112401i
\(484\) 4.02893 0.183133
\(485\) 0 0
\(486\) 1.25338 0.0568547
\(487\) − 4.10277i − 0.185914i −0.995670 0.0929572i \(-0.970368\pi\)
0.995670 0.0929572i \(-0.0296320\pi\)
\(488\) 7.72753i 0.349809i
\(489\) −0.764867 −0.0345885
\(490\) 0 0
\(491\) 40.7708 1.83996 0.919981 0.391963i \(-0.128204\pi\)
0.919981 + 0.391963i \(0.128204\pi\)
\(492\) 0.214614i 0.00967556i
\(493\) − 4.97107i − 0.223886i
\(494\) −8.58145 −0.386098
\(495\) 0 0
\(496\) −46.2472 −2.07656
\(497\) 5.72753i 0.256915i
\(498\) − 7.33403i − 0.328646i
\(499\) 18.4703 0.826843 0.413421 0.910540i \(-0.364334\pi\)
0.413421 + 0.910540i \(0.364334\pi\)
\(500\) 0 0
\(501\) 33.9565 1.51707
\(502\) 23.7671i 1.06078i
\(503\) 21.4947i 0.958400i 0.877706 + 0.479200i \(0.159073\pi\)
−0.877706 + 0.479200i \(0.840927\pi\)
\(504\) −0.124128 −0.00552911
\(505\) 0 0
\(506\) 1.02505 0.0455692
\(507\) − 20.7687i − 0.922372i
\(508\) 7.71912i 0.342480i
\(509\) −3.75872 −0.166602 −0.0833012 0.996524i \(-0.526546\pi\)
−0.0833012 + 0.996524i \(0.526546\pi\)
\(510\) 0 0
\(511\) −4.38121 −0.193813
\(512\) − 13.6114i − 0.601546i
\(513\) − 31.8310i − 1.40537i
\(514\) 9.66967 0.426511
\(515\) 0 0
\(516\) −3.60197 −0.158568
\(517\) − 0.327887i − 0.0144204i
\(518\) − 1.50761i − 0.0662404i
\(519\) 43.4719 1.90820
\(520\) 0 0
\(521\) 12.8059 0.561037 0.280518 0.959849i \(-0.409494\pi\)
0.280518 + 0.959849i \(0.409494\pi\)
\(522\) − 0.120638i − 0.00528019i
\(523\) 21.1278i 0.923855i 0.886918 + 0.461928i \(0.152842\pi\)
−0.886918 + 0.461928i \(0.847158\pi\)
\(524\) 4.97107 0.217162
\(525\) 0 0
\(526\) 15.5090 0.676226
\(527\) 49.9565i 2.17614i
\(528\) 2.28685i 0.0995223i
\(529\) 17.7526 0.771851
\(530\) 0 0
\(531\) −0.764867 −0.0331924
\(532\) − 1.40871i − 0.0610753i
\(533\) 0.313511i 0.0135797i
\(534\) −12.4657 −0.539445
\(535\) 0 0
\(536\) 14.3318 0.619038
\(537\) 25.3607i 1.09439i
\(538\) − 43.3607i − 1.86941i
\(539\) 1.91935 0.0826725
\(540\) 0 0
\(541\) 32.7382 1.40753 0.703763 0.710435i \(-0.251502\pi\)
0.703763 + 0.710435i \(0.251502\pi\)
\(542\) 44.3318i 1.90421i
\(543\) 10.1112i 0.433912i
\(544\) 10.2542 0.439646
\(545\) 0 0
\(546\) −1.52973 −0.0654666
\(547\) − 22.1073i − 0.945240i −0.881266 0.472620i \(-0.843308\pi\)
0.881266 0.472620i \(-0.156692\pi\)
\(548\) 5.00227i 0.213686i
\(549\) 0.241276 0.0102974
\(550\) 0 0
\(551\) −6.04945 −0.257715
\(552\) − 9.82887i − 0.418344i
\(553\) − 7.81658i − 0.332395i
\(554\) −0.0410378 −0.00174353
\(555\) 0 0
\(556\) −1.80674 −0.0766229
\(557\) − 39.8720i − 1.68943i −0.535216 0.844715i \(-0.679770\pi\)
0.535216 0.844715i \(-0.320230\pi\)
\(558\) 1.21235i 0.0513227i
\(559\) −5.26180 −0.222550
\(560\) 0 0
\(561\) 2.47027 0.104295
\(562\) − 43.2085i − 1.82264i
\(563\) − 10.1217i − 0.426578i −0.976989 0.213289i \(-0.931582\pi\)
0.976989 0.213289i \(-0.0684176\pi\)
\(564\) 0.711543 0.0299614
\(565\) 0 0
\(566\) −32.0372 −1.34662
\(567\) − 5.52586i − 0.232064i
\(568\) 22.7891i 0.956209i
\(569\) 24.4391 1.02454 0.512270 0.858825i \(-0.328805\pi\)
0.512270 + 0.858825i \(0.328805\pi\)
\(570\) 0 0
\(571\) −28.2511 −1.18227 −0.591136 0.806572i \(-0.701320\pi\)
−0.591136 + 0.806572i \(0.701320\pi\)
\(572\) − 0.0988967i − 0.00413508i
\(573\) − 12.0144i − 0.501908i
\(574\) −0.330332 −0.0137878
\(575\) 0 0
\(576\) −0.472534 −0.0196889
\(577\) 46.1171i 1.91988i 0.280202 + 0.959941i \(0.409598\pi\)
−0.280202 + 0.959941i \(0.590402\pi\)
\(578\) − 11.8695i − 0.493707i
\(579\) −30.5380 −1.26911
\(580\) 0 0
\(581\) 1.75872 0.0729642
\(582\) − 41.8120i − 1.73317i
\(583\) 0.0988967i 0.00409588i
\(584\) −17.4323 −0.721352
\(585\) 0 0
\(586\) −23.7191 −0.979828
\(587\) − 0.715418i − 0.0295285i −0.999891 0.0147642i \(-0.995300\pi\)
0.999891 0.0147642i \(-0.00469977\pi\)
\(588\) 4.16517i 0.171769i
\(589\) 60.7936 2.50496
\(590\) 0 0
\(591\) 10.4247 0.428815
\(592\) − 7.14465i − 0.293643i
\(593\) − 15.5441i − 0.638320i −0.947701 0.319160i \(-0.896599\pi\)
0.947701 0.319160i \(-0.103401\pi\)
\(594\) 2.35455 0.0966083
\(595\) 0 0
\(596\) −4.62249 −0.189344
\(597\) − 16.6803i − 0.682681i
\(598\) 3.24951i 0.132882i
\(599\) 9.59809 0.392167 0.196084 0.980587i \(-0.437178\pi\)
0.196084 + 0.980587i \(0.437178\pi\)
\(600\) 0 0
\(601\) 6.81044 0.277804 0.138902 0.990306i \(-0.455643\pi\)
0.138902 + 0.990306i \(0.455643\pi\)
\(602\) − 5.54411i − 0.225961i
\(603\) − 0.447480i − 0.0182228i
\(604\) −2.80590 −0.114171
\(605\) 0 0
\(606\) −32.2434 −1.30980
\(607\) 31.6970i 1.28654i 0.765639 + 0.643271i \(0.222423\pi\)
−0.765639 + 0.643271i \(0.777577\pi\)
\(608\) − 12.4787i − 0.506077i
\(609\) −1.07838 −0.0436981
\(610\) 0 0
\(611\) 1.03943 0.0420508
\(612\) − 0.143810i − 0.00581318i
\(613\) − 1.20394i − 0.0486265i −0.999704 0.0243133i \(-0.992260\pi\)
0.999704 0.0243133i \(-0.00773992\pi\)
\(614\) 43.7602 1.76602
\(615\) 0 0
\(616\) −0.460425 −0.0185511
\(617\) − 37.9337i − 1.52715i −0.645716 0.763577i \(-0.723441\pi\)
0.645716 0.763577i \(-0.276559\pi\)
\(618\) 20.6947i 0.832464i
\(619\) 4.60424 0.185060 0.0925299 0.995710i \(-0.470505\pi\)
0.0925299 + 0.995710i \(0.470505\pi\)
\(620\) 0 0
\(621\) −12.0533 −0.483683
\(622\) − 30.2062i − 1.21116i
\(623\) − 2.98932i − 0.119765i
\(624\) −7.24951 −0.290213
\(625\) 0 0
\(626\) 35.2618 1.40934
\(627\) − 3.00614i − 0.120054i
\(628\) 9.18568i 0.366549i
\(629\) −7.71769 −0.307724
\(630\) 0 0
\(631\) −8.41241 −0.334893 −0.167446 0.985881i \(-0.553552\pi\)
−0.167446 + 0.985881i \(0.553552\pi\)
\(632\) − 31.1012i − 1.23714i
\(633\) − 16.8638i − 0.670274i
\(634\) −35.1857 −1.39740
\(635\) 0 0
\(636\) −0.214614 −0.00851001
\(637\) 6.08452i 0.241077i
\(638\) − 0.447480i − 0.0177159i
\(639\) 0.711543 0.0281482
\(640\) 0 0
\(641\) 32.5380 1.28517 0.642586 0.766213i \(-0.277861\pi\)
0.642586 + 0.766213i \(0.277861\pi\)
\(642\) 33.4908i 1.32178i
\(643\) 2.09293i 0.0825372i 0.999148 + 0.0412686i \(0.0131399\pi\)
−0.999148 + 0.0412686i \(0.986860\pi\)
\(644\) −0.533431 −0.0210201
\(645\) 0 0
\(646\) −46.2868 −1.82113
\(647\) 45.1955i 1.77682i 0.459051 + 0.888410i \(0.348189\pi\)
−0.459051 + 0.888410i \(0.651811\pi\)
\(648\) − 21.9867i − 0.863718i
\(649\) −2.83710 −0.111366
\(650\) 0 0
\(651\) 10.8371 0.424739
\(652\) 0.165166i 0.00646840i
\(653\) 2.14834i 0.0840712i 0.999116 + 0.0420356i \(0.0133843\pi\)
−0.999116 + 0.0420356i \(0.986616\pi\)
\(654\) −32.7259 −1.27968
\(655\) 0 0
\(656\) −1.56547 −0.0611211
\(657\) 0.544287i 0.0212347i
\(658\) 1.09520i 0.0426953i
\(659\) 45.0843 1.75624 0.878118 0.478444i \(-0.158799\pi\)
0.878118 + 0.478444i \(0.158799\pi\)
\(660\) 0 0
\(661\) −36.3234 −1.41281 −0.706407 0.707806i \(-0.749685\pi\)
−0.706407 + 0.707806i \(0.749685\pi\)
\(662\) 37.0798i 1.44115i
\(663\) 7.83096i 0.304129i
\(664\) 6.99773 0.271565
\(665\) 0 0
\(666\) −0.187293 −0.00725746
\(667\) 2.29072i 0.0886972i
\(668\) − 7.33260i − 0.283707i
\(669\) −18.7526 −0.725017
\(670\) 0 0
\(671\) 0.894960 0.0345496
\(672\) − 2.22446i − 0.0858102i
\(673\) 17.4719i 0.673491i 0.941596 + 0.336746i \(0.109326\pi\)
−0.941596 + 0.336746i \(0.890674\pi\)
\(674\) −19.6826 −0.758146
\(675\) 0 0
\(676\) −4.48482 −0.172493
\(677\) − 40.0372i − 1.53875i −0.638796 0.769377i \(-0.720567\pi\)
0.638796 0.769377i \(-0.279433\pi\)
\(678\) − 33.0784i − 1.27037i
\(679\) 10.0267 0.384788
\(680\) 0 0
\(681\) −21.4452 −0.821782
\(682\) 4.49693i 0.172196i
\(683\) − 2.07611i − 0.0794402i −0.999211 0.0397201i \(-0.987353\pi\)
0.999211 0.0397201i \(-0.0126466\pi\)
\(684\) −0.175007 −0.00669156
\(685\) 0 0
\(686\) −13.2085 −0.504302
\(687\) 39.9299i 1.52342i
\(688\) − 26.2739i − 1.00168i
\(689\) −0.313511 −0.0119438
\(690\) 0 0
\(691\) 26.7070 1.01598 0.507991 0.861362i \(-0.330388\pi\)
0.507991 + 0.861362i \(0.330388\pi\)
\(692\) − 9.38735i − 0.356854i
\(693\) 0.0143758i 0 0.000546093i
\(694\) −12.9542 −0.491737
\(695\) 0 0
\(696\) −4.29072 −0.162639
\(697\) 1.69102i 0.0640521i
\(698\) − 34.1666i − 1.29322i
\(699\) −21.3151 −0.806212
\(700\) 0 0
\(701\) −21.9155 −0.827736 −0.413868 0.910337i \(-0.635823\pi\)
−0.413868 + 0.910337i \(0.635823\pi\)
\(702\) 7.46412i 0.281715i
\(703\) 9.39189i 0.354222i
\(704\) −1.75276 −0.0660596
\(705\) 0 0
\(706\) −9.51745 −0.358194
\(707\) − 7.73206i − 0.290794i
\(708\) − 6.15676i − 0.231385i
\(709\) 4.60811 0.173061 0.0865306 0.996249i \(-0.472422\pi\)
0.0865306 + 0.996249i \(0.472422\pi\)
\(710\) 0 0
\(711\) −0.971071 −0.0364180
\(712\) − 11.8941i − 0.445751i
\(713\) − 23.0205i − 0.862125i
\(714\) −8.25112 −0.308790
\(715\) 0 0
\(716\) 5.47641 0.204663
\(717\) 23.5174i 0.878275i
\(718\) − 7.78151i − 0.290403i
\(719\) −6.80590 −0.253817 −0.126909 0.991914i \(-0.540506\pi\)
−0.126909 + 0.991914i \(0.540506\pi\)
\(720\) 0 0
\(721\) −4.96266 −0.184819
\(722\) 27.0833i 1.00794i
\(723\) 25.0928i 0.933210i
\(724\) 2.18342 0.0811461
\(725\) 0 0
\(726\) −28.7175 −1.06581
\(727\) − 26.9711i − 1.00030i −0.865938 0.500151i \(-0.833278\pi\)
0.865938 0.500151i \(-0.166722\pi\)
\(728\) − 1.45959i − 0.0540960i
\(729\) −27.6681 −1.02474
\(730\) 0 0
\(731\) −28.3812 −1.04972
\(732\) 1.94214i 0.0717836i
\(733\) 30.0638i 1.11043i 0.831706 + 0.555216i \(0.187365\pi\)
−0.831706 + 0.555216i \(0.812635\pi\)
\(734\) −45.4680 −1.67825
\(735\) 0 0
\(736\) −4.72526 −0.174175
\(737\) − 1.65983i − 0.0611405i
\(738\) 0.0410378i 0.00151062i
\(739\) −51.1422 −1.88130 −0.940648 0.339383i \(-0.889782\pi\)
−0.940648 + 0.339383i \(0.889782\pi\)
\(740\) 0 0
\(741\) 9.52973 0.350084
\(742\) − 0.330332i − 0.0121269i
\(743\) − 11.1857i − 0.410363i −0.978724 0.205181i \(-0.934222\pi\)
0.978724 0.205181i \(-0.0657785\pi\)
\(744\) 43.1194 1.58083
\(745\) 0 0
\(746\) 22.1834 0.812193
\(747\) − 0.218490i − 0.00799413i
\(748\) − 0.533431i − 0.0195042i
\(749\) −8.03120 −0.293454
\(750\) 0 0
\(751\) −18.3630 −0.670074 −0.335037 0.942205i \(-0.608749\pi\)
−0.335037 + 0.942205i \(0.608749\pi\)
\(752\) 5.19022i 0.189268i
\(753\) − 26.3935i − 0.961832i
\(754\) 1.41855 0.0516606
\(755\) 0 0
\(756\) −1.22529 −0.0445634
\(757\) 15.8927i 0.577630i 0.957385 + 0.288815i \(0.0932612\pi\)
−0.957385 + 0.288815i \(0.906739\pi\)
\(758\) − 21.7548i − 0.790172i
\(759\) −1.13833 −0.0413186
\(760\) 0 0
\(761\) 13.8843 0.503305 0.251652 0.967818i \(-0.419026\pi\)
0.251652 + 0.967818i \(0.419026\pi\)
\(762\) − 55.0205i − 1.99318i
\(763\) − 7.84778i − 0.284109i
\(764\) −2.59439 −0.0938619
\(765\) 0 0
\(766\) −24.2907 −0.877660
\(767\) − 8.99386i − 0.324749i
\(768\) − 14.6576i − 0.528909i
\(769\) 35.4063 1.27678 0.638391 0.769712i \(-0.279600\pi\)
0.638391 + 0.769712i \(0.279600\pi\)
\(770\) 0 0
\(771\) −10.7382 −0.386727
\(772\) 6.59439i 0.237337i
\(773\) 0.488518i 0.0175708i 0.999961 + 0.00878539i \(0.00279651\pi\)
−0.999961 + 0.00878539i \(0.997203\pi\)
\(774\) −0.688756 −0.0247568
\(775\) 0 0
\(776\) 39.8948 1.43214
\(777\) 1.67420i 0.0600617i
\(778\) 21.2663i 0.762435i
\(779\) 2.05786 0.0737304
\(780\) 0 0
\(781\) 2.63931 0.0944419
\(782\) 17.5273i 0.626775i
\(783\) 5.26180i 0.188041i
\(784\) −30.3820 −1.08507
\(785\) 0 0
\(786\) −35.4329 −1.26385
\(787\) − 1.99159i − 0.0709925i −0.999370 0.0354962i \(-0.988699\pi\)
0.999370 0.0354962i \(-0.0113012\pi\)
\(788\) − 2.25112i − 0.0801927i
\(789\) −17.2228 −0.613150
\(790\) 0 0
\(791\) 7.93230 0.282040
\(792\) 0.0571996i 0.00203250i
\(793\) 2.83710i 0.100748i
\(794\) −13.9323 −0.494439
\(795\) 0 0
\(796\) −3.60197 −0.127668
\(797\) − 17.2702i − 0.611742i −0.952073 0.305871i \(-0.901052\pi\)
0.952073 0.305871i \(-0.0989476\pi\)
\(798\) 10.0410i 0.355449i
\(799\) 5.60650 0.198344
\(800\) 0 0
\(801\) −0.371370 −0.0131217
\(802\) 30.4124i 1.07390i
\(803\) 2.01891i 0.0712458i
\(804\) 3.60197 0.127032
\(805\) 0 0
\(806\) −14.2557 −0.502134
\(807\) 48.1522i 1.69504i
\(808\) − 30.7649i − 1.08230i
\(809\) 56.5068 1.98667 0.993336 0.115254i \(-0.0367680\pi\)
0.993336 + 0.115254i \(0.0367680\pi\)
\(810\) 0 0
\(811\) −8.77924 −0.308281 −0.154140 0.988049i \(-0.549261\pi\)
−0.154140 + 0.988049i \(0.549261\pi\)
\(812\) 0.232866i 0.00817199i
\(813\) − 49.2306i − 1.72659i
\(814\) −0.694722 −0.0243500
\(815\) 0 0
\(816\) −39.1026 −1.36886
\(817\) 34.5380i 1.20833i
\(818\) 2.64384i 0.0924398i
\(819\) −0.0455727 −0.00159244
\(820\) 0 0
\(821\) −28.1568 −0.982678 −0.491339 0.870969i \(-0.663492\pi\)
−0.491339 + 0.870969i \(0.663492\pi\)
\(822\) − 35.6553i − 1.24362i
\(823\) − 20.7442i − 0.723096i −0.932353 0.361548i \(-0.882248\pi\)
0.932353 0.361548i \(-0.117752\pi\)
\(824\) −19.7458 −0.687877
\(825\) 0 0
\(826\) 9.47641 0.329726
\(827\) − 9.12783i − 0.317406i −0.987326 0.158703i \(-0.949269\pi\)
0.987326 0.158703i \(-0.0507312\pi\)
\(828\) 0.0662693i 0.00230302i
\(829\) −31.8576 −1.10646 −0.553230 0.833028i \(-0.686605\pi\)
−0.553230 + 0.833028i \(0.686605\pi\)
\(830\) 0 0
\(831\) 0.0455727 0.00158090
\(832\) − 5.55640i − 0.192633i
\(833\) 32.8188i 1.13711i
\(834\) 12.8781 0.445933
\(835\) 0 0
\(836\) −0.649149 −0.0224513
\(837\) − 52.8781i − 1.82774i
\(838\) 54.6902i 1.88924i
\(839\) −27.4413 −0.947380 −0.473690 0.880692i \(-0.657078\pi\)
−0.473690 + 0.880692i \(0.657078\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 18.5814i − 0.640359i
\(843\) 47.9832i 1.65263i
\(844\) −3.64158 −0.125348
\(845\) 0 0
\(846\) 0.136059 0.00467780
\(847\) − 6.88655i − 0.236625i
\(848\) − 1.56547i − 0.0537583i
\(849\) 35.5774 1.22101
\(850\) 0 0
\(851\) 3.55640 0.121912
\(852\) 5.72753i 0.196222i
\(853\) 56.0515i 1.91917i 0.281422 + 0.959584i \(0.409194\pi\)
−0.281422 + 0.959584i \(0.590806\pi\)
\(854\) −2.98932 −0.102292
\(855\) 0 0
\(856\) −31.9551 −1.09220
\(857\) − 6.08452i − 0.207843i −0.994585 0.103922i \(-0.966861\pi\)
0.994585 0.103922i \(-0.0331391\pi\)
\(858\) 0.704918i 0.0240655i
\(859\) −35.5936 −1.21444 −0.607218 0.794535i \(-0.707715\pi\)
−0.607218 + 0.794535i \(0.707715\pi\)
\(860\) 0 0
\(861\) 0.366835 0.0125017
\(862\) 30.5236i 1.03964i
\(863\) 12.8287i 0.436694i 0.975871 + 0.218347i \(0.0700664\pi\)
−0.975871 + 0.218347i \(0.929934\pi\)
\(864\) −10.8539 −0.369258
\(865\) 0 0
\(866\) 22.8020 0.774844
\(867\) 13.1812i 0.447655i
\(868\) − 2.34017i − 0.0794306i
\(869\) −3.60197 −0.122188
\(870\) 0 0
\(871\) 5.26180 0.178289
\(872\) − 31.2253i − 1.05742i
\(873\) − 1.24563i − 0.0421583i
\(874\) 21.3295 0.721481
\(875\) 0 0
\(876\) −4.38121 −0.148027
\(877\) − 1.50307i − 0.0507551i −0.999678 0.0253776i \(-0.991921\pi\)
0.999678 0.0253776i \(-0.00807880\pi\)
\(878\) 27.4863i 0.927616i
\(879\) 26.3402 0.888432
\(880\) 0 0
\(881\) 23.4908 0.791425 0.395712 0.918375i \(-0.370498\pi\)
0.395712 + 0.918375i \(0.370498\pi\)
\(882\) 0.796449i 0.0268178i
\(883\) 29.0433i 0.977385i 0.872456 + 0.488693i \(0.162526\pi\)
−0.872456 + 0.488693i \(0.837474\pi\)
\(884\) 1.69102 0.0568753
\(885\) 0 0
\(886\) 51.7068 1.73712
\(887\) − 19.0700i − 0.640307i −0.947366 0.320153i \(-0.896266\pi\)
0.947366 0.320153i \(-0.103734\pi\)
\(888\) 6.66144i 0.223543i
\(889\) 13.1941 0.442516
\(890\) 0 0
\(891\) −2.54638 −0.0853068
\(892\) 4.04945i 0.135586i
\(893\) − 6.82273i − 0.228314i
\(894\) 32.9483 1.10196
\(895\) 0 0
\(896\) 8.45732 0.282539
\(897\) − 3.60859i − 0.120487i
\(898\) − 10.8950i − 0.363570i
\(899\) −10.0494 −0.335168
\(900\) 0 0
\(901\) −1.69102 −0.0563362
\(902\) 0.152221i 0.00506839i
\(903\) 6.15676i 0.204884i
\(904\) 31.5616 1.04972
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) 5.54023i 0.183960i 0.995761 + 0.0919802i \(0.0293197\pi\)
−0.995761 + 0.0919802i \(0.970680\pi\)
\(908\) 4.63090i 0.153682i
\(909\) −0.960570 −0.0318601
\(910\) 0 0
\(911\) 53.2990 1.76587 0.882937 0.469492i \(-0.155563\pi\)
0.882937 + 0.469492i \(0.155563\pi\)
\(912\) 47.5851i 1.57570i
\(913\) − 0.810439i − 0.0268216i
\(914\) 8.95282 0.296133
\(915\) 0 0
\(916\) 8.62249 0.284895
\(917\) − 8.49693i − 0.280593i
\(918\) 40.2602i 1.32878i
\(919\) −37.5897 −1.23997 −0.619985 0.784614i \(-0.712861\pi\)
−0.619985 + 0.784614i \(0.712861\pi\)
\(920\) 0 0
\(921\) −48.5958 −1.60129
\(922\) − 49.7296i − 1.63776i
\(923\) 8.36683i 0.275398i
\(924\) −0.115718 −0.00380683
\(925\) 0 0
\(926\) 2.21849 0.0729041
\(927\) 0.616522i 0.0202492i
\(928\) 2.06278i 0.0677140i
\(929\) −37.3197 −1.22442 −0.612209 0.790696i \(-0.709719\pi\)
−0.612209 + 0.790696i \(0.709719\pi\)
\(930\) 0 0
\(931\) 39.9383 1.30892
\(932\) 4.60281i 0.150770i
\(933\) 33.5441i 1.09818i
\(934\) 18.0860 0.591790
\(935\) 0 0
\(936\) −0.181328 −0.00592688
\(937\) − 22.8638i − 0.746927i −0.927645 0.373463i \(-0.878170\pi\)
0.927645 0.373463i \(-0.121830\pi\)
\(938\) 5.54411i 0.181022i
\(939\) −39.1584 −1.27788
\(940\) 0 0
\(941\) 0.523590 0.0170686 0.00853428 0.999964i \(-0.497283\pi\)
0.00853428 + 0.999964i \(0.497283\pi\)
\(942\) − 65.4740i − 2.13326i
\(943\) − 0.779243i − 0.0253756i
\(944\) 44.9093 1.46167
\(945\) 0 0
\(946\) −2.55479 −0.0830633
\(947\) − 10.0228i − 0.325697i −0.986651 0.162848i \(-0.947932\pi\)
0.986651 0.162848i \(-0.0520681\pi\)
\(948\) − 7.81658i − 0.253871i
\(949\) −6.40012 −0.207757
\(950\) 0 0
\(951\) 39.0738 1.26706
\(952\) − 7.87277i − 0.255158i
\(953\) 8.15676i 0.264223i 0.991235 + 0.132112i \(0.0421757\pi\)
−0.991235 + 0.132112i \(0.957824\pi\)
\(954\) −0.0410378 −0.00132865
\(955\) 0 0
\(956\) 5.07838 0.164246
\(957\) 0.496928i 0.0160634i
\(958\) − 26.4261i − 0.853789i
\(959\) 8.55025 0.276102
\(960\) 0 0
\(961\) 69.9914 2.25779
\(962\) − 2.20233i − 0.0710059i
\(963\) 0.997733i 0.0321515i
\(964\) 5.41855 0.174520
\(965\) 0 0
\(966\) 3.80221 0.122334
\(967\) 15.7671i 0.507037i 0.967331 + 0.253518i \(0.0815878\pi\)
−0.967331 + 0.253518i \(0.918412\pi\)
\(968\) − 27.4007i − 0.880691i
\(969\) 51.4017 1.65126
\(970\) 0 0
\(971\) −17.8804 −0.573810 −0.286905 0.957959i \(-0.592626\pi\)
−0.286905 + 0.957959i \(0.592626\pi\)
\(972\) 0.300566i 0.00964065i
\(973\) 3.08822i 0.0990037i
\(974\) 6.31494 0.202344
\(975\) 0 0
\(976\) −14.1666 −0.453462
\(977\) − 55.1071i − 1.76303i −0.472153 0.881517i \(-0.656523\pi\)
0.472153 0.881517i \(-0.343477\pi\)
\(978\) − 1.17727i − 0.0376451i
\(979\) −1.37751 −0.0440255
\(980\) 0 0
\(981\) −0.974946 −0.0311276
\(982\) 62.7540i 2.00256i
\(983\) 1.29687i 0.0413637i 0.999786 + 0.0206818i \(0.00658370\pi\)
−0.999786 + 0.0206818i \(0.993416\pi\)
\(984\) 1.45959 0.0465300
\(985\) 0 0
\(986\) 7.65142 0.243671
\(987\) − 1.21622i − 0.0387128i
\(988\) − 2.05786i − 0.0654692i
\(989\) 13.0784 0.415868
\(990\) 0 0
\(991\) −3.11942 −0.0990915 −0.0495458 0.998772i \(-0.515777\pi\)
−0.0495458 + 0.998772i \(0.515777\pi\)
\(992\) − 20.7298i − 0.658172i
\(993\) − 41.1773i − 1.30672i
\(994\) −8.81575 −0.279618
\(995\) 0 0
\(996\) 1.75872 0.0557273
\(997\) 30.2472i 0.957940i 0.877831 + 0.478970i \(0.158990\pi\)
−0.877831 + 0.478970i \(0.841010\pi\)
\(998\) 28.4292i 0.899912i
\(999\) 8.16904 0.258457
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 725.2.b.d.349.5 6
5.2 odd 4 725.2.a.d.1.2 3
5.3 odd 4 145.2.a.d.1.2 3
5.4 even 2 inner 725.2.b.d.349.2 6
15.2 even 4 6525.2.a.bh.1.2 3
15.8 even 4 1305.2.a.o.1.2 3
20.3 even 4 2320.2.a.s.1.1 3
35.13 even 4 7105.2.a.p.1.2 3
40.3 even 4 9280.2.a.bm.1.3 3
40.13 odd 4 9280.2.a.bu.1.1 3
145.28 odd 4 4205.2.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.2 3 5.3 odd 4
725.2.a.d.1.2 3 5.2 odd 4
725.2.b.d.349.2 6 5.4 even 2 inner
725.2.b.d.349.5 6 1.1 even 1 trivial
1305.2.a.o.1.2 3 15.8 even 4
2320.2.a.s.1.1 3 20.3 even 4
4205.2.a.e.1.2 3 145.28 odd 4
6525.2.a.bh.1.2 3 15.2 even 4
7105.2.a.p.1.2 3 35.13 even 4
9280.2.a.bm.1.3 3 40.3 even 4
9280.2.a.bu.1.1 3 40.13 odd 4