Properties

Label 1305.2.a.o.1.2
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $3$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +0.630898 q^{7} +2.51026 q^{8} +O(q^{10})\) \(q-1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +0.630898 q^{7} +2.51026 q^{8} -1.53919 q^{10} -0.290725 q^{11} -0.921622 q^{13} -0.971071 q^{14} -4.60197 q^{16} -4.97107 q^{17} -6.04945 q^{19} +0.369102 q^{20} +0.447480 q^{22} -2.29072 q^{23} +1.00000 q^{25} +1.41855 q^{26} +0.232866 q^{28} -1.00000 q^{29} +10.0494 q^{31} +2.06278 q^{32} +7.65142 q^{34} +0.630898 q^{35} +1.55252 q^{37} +9.31124 q^{38} +2.51026 q^{40} -0.340173 q^{41} -5.70928 q^{43} -0.107307 q^{44} +3.52586 q^{46} +1.12783 q^{47} -6.60197 q^{49} -1.53919 q^{50} -0.340173 q^{52} +0.340173 q^{53} -0.290725 q^{55} +1.58372 q^{56} +1.53919 q^{58} -9.75872 q^{59} +3.07838 q^{61} -15.4680 q^{62} +6.02893 q^{64} -0.921622 q^{65} -5.70928 q^{67} -1.83483 q^{68} -0.971071 q^{70} -9.07838 q^{71} -6.94441 q^{73} -2.38962 q^{74} -2.23287 q^{76} -0.183417 q^{77} +12.3896 q^{79} -4.60197 q^{80} +0.523590 q^{82} -2.78765 q^{83} -4.97107 q^{85} +8.78765 q^{86} -0.729794 q^{88} -4.73820 q^{89} -0.581449 q^{91} -0.845512 q^{92} -1.73594 q^{94} -6.04945 q^{95} -15.8927 q^{97} +10.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} - 2 q^{7} - 9 q^{8} - 3 q^{10} - 8 q^{11} - 6 q^{13} + 12 q^{14} + 5 q^{16} + 5 q^{20} + 2 q^{22} - 14 q^{23} + 3 q^{25} - 10 q^{26} - 22 q^{28} - 3 q^{29} + 12 q^{31} - 11 q^{32} - 14 q^{34} - 2 q^{35} + 4 q^{37} + 2 q^{38} - 9 q^{40} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 8 q^{46} - 18 q^{47} - q^{49} - 3 q^{50} + 10 q^{52} - 10 q^{53} - 8 q^{55} + 32 q^{56} + 3 q^{58} - 4 q^{59} + 6 q^{61} - 14 q^{62} + 33 q^{64} - 6 q^{65} - 10 q^{67} + 36 q^{68} + 12 q^{70} - 24 q^{71} - 4 q^{73} + 22 q^{74} + 16 q^{76} + 4 q^{77} + 8 q^{79} + 5 q^{80} - 14 q^{82} + 2 q^{83} + 16 q^{86} + 38 q^{88} - 22 q^{89} - 16 q^{91} - 22 q^{92} - 36 q^{97} - 23 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53919 −1.08837 −0.544185 0.838965i \(-0.683161\pi\)
−0.544185 + 0.838965i \(0.683161\pi\)
\(3\) 0 0
\(4\) 0.369102 0.184551
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.630898 0.238457 0.119228 0.992867i \(-0.461958\pi\)
0.119228 + 0.992867i \(0.461958\pi\)
\(8\) 2.51026 0.887511
\(9\) 0 0
\(10\) −1.53919 −0.486734
\(11\) −0.290725 −0.0876568 −0.0438284 0.999039i \(-0.513955\pi\)
−0.0438284 + 0.999039i \(0.513955\pi\)
\(12\) 0 0
\(13\) −0.921622 −0.255612 −0.127806 0.991799i \(-0.540793\pi\)
−0.127806 + 0.991799i \(0.540793\pi\)
\(14\) −0.971071 −0.259530
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) −4.97107 −1.20566 −0.602831 0.797869i \(-0.705961\pi\)
−0.602831 + 0.797869i \(0.705961\pi\)
\(18\) 0 0
\(19\) −6.04945 −1.38784 −0.693919 0.720053i \(-0.744118\pi\)
−0.693919 + 0.720053i \(0.744118\pi\)
\(20\) 0.369102 0.0825338
\(21\) 0 0
\(22\) 0.447480 0.0954031
\(23\) −2.29072 −0.477649 −0.238825 0.971063i \(-0.576762\pi\)
−0.238825 + 0.971063i \(0.576762\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.41855 0.278201
\(27\) 0 0
\(28\) 0.232866 0.0440075
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 10.0494 1.80493 0.902467 0.430759i \(-0.141754\pi\)
0.902467 + 0.430759i \(0.141754\pi\)
\(32\) 2.06278 0.364651
\(33\) 0 0
\(34\) 7.65142 1.31221
\(35\) 0.630898 0.106641
\(36\) 0 0
\(37\) 1.55252 0.255233 0.127616 0.991824i \(-0.459267\pi\)
0.127616 + 0.991824i \(0.459267\pi\)
\(38\) 9.31124 1.51048
\(39\) 0 0
\(40\) 2.51026 0.396907
\(41\) −0.340173 −0.0531261 −0.0265630 0.999647i \(-0.508456\pi\)
−0.0265630 + 0.999647i \(0.508456\pi\)
\(42\) 0 0
\(43\) −5.70928 −0.870656 −0.435328 0.900272i \(-0.643368\pi\)
−0.435328 + 0.900272i \(0.643368\pi\)
\(44\) −0.107307 −0.0161772
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) 1.12783 0.164510 0.0822552 0.996611i \(-0.473788\pi\)
0.0822552 + 0.996611i \(0.473788\pi\)
\(48\) 0 0
\(49\) −6.60197 −0.943138
\(50\) −1.53919 −0.217674
\(51\) 0 0
\(52\) −0.340173 −0.0471735
\(53\) 0.340173 0.0467264 0.0233632 0.999727i \(-0.492563\pi\)
0.0233632 + 0.999727i \(0.492563\pi\)
\(54\) 0 0
\(55\) −0.290725 −0.0392013
\(56\) 1.58372 0.211633
\(57\) 0 0
\(58\) 1.53919 0.202105
\(59\) −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 \(0.436856\pi\)
\(62\) −15.4680 −1.96444
\(63\) 0 0
\(64\) 6.02893 0.753616
\(65\) −0.921622 −0.114313
\(66\) 0 0
\(67\) −5.70928 −0.697499 −0.348749 0.937216i \(-0.613394\pi\)
−0.348749 + 0.937216i \(0.613394\pi\)
\(68\) −1.83483 −0.222506
\(69\) 0 0
\(70\) −0.971071 −0.116065
\(71\) −9.07838 −1.07741 −0.538703 0.842496i \(-0.681085\pi\)
−0.538703 + 0.842496i \(0.681085\pi\)
\(72\) 0 0
\(73\) −6.94441 −0.812782 −0.406391 0.913699i \(-0.633213\pi\)
−0.406391 + 0.913699i \(0.633213\pi\)
\(74\) −2.38962 −0.277788
\(75\) 0 0
\(76\) −2.23287 −0.256127
\(77\) −0.183417 −0.0209024
\(78\) 0 0
\(79\) 12.3896 1.39394 0.696971 0.717100i \(-0.254531\pi\)
0.696971 + 0.717100i \(0.254531\pi\)
\(80\) −4.60197 −0.514516
\(81\) 0 0
\(82\) 0.523590 0.0578209
\(83\) −2.78765 −0.305985 −0.152992 0.988227i \(-0.548891\pi\)
−0.152992 + 0.988227i \(0.548891\pi\)
\(84\) 0 0
\(85\) −4.97107 −0.539188
\(86\) 8.78765 0.947597
\(87\) 0 0
\(88\) −0.729794 −0.0777963
\(89\) −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
\(92\) −0.845512 −0.0881507
\(93\) 0 0
\(94\) −1.73594 −0.179048
\(95\) −6.04945 −0.620660
\(96\) 0 0
\(97\) −15.8927 −1.61366 −0.806829 0.590785i \(-0.798818\pi\)
−0.806829 + 0.590785i \(0.798818\pi\)
\(98\) 10.1617 1.02648
\(99\) 0 0
\(100\) 0.369102 0.0369102
\(101\) 12.2557 1.21948 0.609741 0.792600i \(-0.291273\pi\)
0.609741 + 0.792600i \(0.291273\pi\)
\(102\) 0 0
\(103\) −7.86603 −0.775063 −0.387532 0.921856i \(-0.626672\pi\)
−0.387532 + 0.921856i \(0.626672\pi\)
\(104\) −2.31351 −0.226858
\(105\) 0 0
\(106\) −0.523590 −0.0508556
\(107\) −12.7298 −1.23064 −0.615318 0.788279i \(-0.710972\pi\)
−0.615318 + 0.788279i \(0.710972\pi\)
\(108\) 0 0
\(109\) 12.4391 1.19145 0.595723 0.803190i \(-0.296865\pi\)
0.595723 + 0.803190i \(0.296865\pi\)
\(110\) 0.447480 0.0426656
\(111\) 0 0
\(112\) −2.90337 −0.274343
\(113\) −12.5730 −1.18277 −0.591386 0.806389i \(-0.701419\pi\)
−0.591386 + 0.806389i \(0.701419\pi\)
\(114\) 0 0
\(115\) −2.29072 −0.213611
\(116\) −0.369102 −0.0342703
\(117\) 0 0
\(118\) 15.0205 1.38275
\(119\) −3.13624 −0.287498
\(120\) 0 0
\(121\) −10.9155 −0.992316
\(122\) −4.73820 −0.428977
\(123\) 0 0
\(124\) 3.70928 0.333103
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.9132 −1.85575 −0.927874 0.372895i \(-0.878365\pi\)
−0.927874 + 0.372895i \(0.878365\pi\)
\(128\) −13.4052 −1.18487
\(129\) 0 0
\(130\) 1.41855 0.124415
\(131\) 13.4680 1.17670 0.588352 0.808605i \(-0.299777\pi\)
0.588352 + 0.808605i \(0.299777\pi\)
\(132\) 0 0
\(133\) −3.81658 −0.330940
\(134\) 8.78765 0.759138
\(135\) 0 0
\(136\) −12.4787 −1.07004
\(137\) 13.5525 1.15787 0.578935 0.815374i \(-0.303469\pi\)
0.578935 + 0.815374i \(0.303469\pi\)
\(138\) 0 0
\(139\) −4.89496 −0.415185 −0.207593 0.978215i \(-0.566563\pi\)
−0.207593 + 0.978215i \(0.566563\pi\)
\(140\) 0.232866 0.0196808
\(141\) 0 0
\(142\) 13.9733 1.17262
\(143\) 0.267938 0.0224061
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 10.6888 0.884608
\(147\) 0 0
\(148\) 0.573039 0.0471035
\(149\) 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 \(0.399899\pi\)
\(152\) −15.1857 −1.23172
\(153\) 0 0
\(154\) 0.282314 0.0227495
\(155\) 10.0494 0.807191
\(156\) 0 0
\(157\) −24.8865 −1.98616 −0.993081 0.117428i \(-0.962535\pi\)
−0.993081 + 0.117428i \(0.962535\pi\)
\(158\) −19.0700 −1.51713
\(159\) 0 0
\(160\) 2.06278 0.163077
\(161\) −1.44521 −0.113899
\(162\) 0 0
\(163\) 0.447480 0.0350493 0.0175247 0.999846i \(-0.494421\pi\)
0.0175247 + 0.999846i \(0.494421\pi\)
\(164\) −0.125559 −0.00980448
\(165\) 0 0
\(166\) 4.29072 0.333025
\(167\) −19.8660 −1.53728 −0.768640 0.639682i \(-0.779066\pi\)
−0.768640 + 0.639682i \(0.779066\pi\)
\(168\) 0 0
\(169\) −12.1506 −0.934662
\(170\) 7.65142 0.586837
\(171\) 0 0
\(172\) −2.10731 −0.160681
\(173\) 25.4329 1.93363 0.966815 0.255478i \(-0.0822329\pi\)
0.966815 + 0.255478i \(0.0822329\pi\)
\(174\) 0 0
\(175\) 0.630898 0.0476914
\(176\) 1.33791 0.100848
\(177\) 0 0
\(178\) 7.29299 0.546633
\(179\) −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 \(0.570556\pi\)
\(182\) 0.894960 0.0663389
\(183\) 0 0
\(184\) −5.75031 −0.423919
\(185\) 1.55252 0.114144
\(186\) 0 0
\(187\) 1.44521 0.105684
\(188\) 0.416283 0.0303606
\(189\) 0 0
\(190\) 9.31124 0.675509
\(191\) −7.02893 −0.508595 −0.254298 0.967126i \(-0.581844\pi\)
−0.254298 + 0.967126i \(0.581844\pi\)
\(192\) 0 0
\(193\) 17.8660 1.28603 0.643013 0.765856i \(-0.277684\pi\)
0.643013 + 0.765856i \(0.277684\pi\)
\(194\) 24.4619 1.75626
\(195\) 0 0
\(196\) −2.43680 −0.174057
\(197\) −6.09890 −0.434528 −0.217264 0.976113i \(-0.569713\pi\)
−0.217264 + 0.976113i \(0.569713\pi\)
\(198\) 0 0
\(199\) −9.75872 −0.691778 −0.345889 0.938276i \(-0.612423\pi\)
−0.345889 + 0.938276i \(0.612423\pi\)
\(200\) 2.51026 0.177502
\(201\) 0 0
\(202\) −18.8638 −1.32725
\(203\) −0.630898 −0.0442803
\(204\) 0 0
\(205\) −0.340173 −0.0237587
\(206\) 12.1073 0.843556
\(207\) 0 0
\(208\) 4.24128 0.294080
\(209\) 1.75872 0.121653
\(210\) 0 0
\(211\) 9.86603 0.679206 0.339603 0.940569i \(-0.389707\pi\)
0.339603 + 0.940569i \(0.389707\pi\)
\(212\) 0.125559 0.00862340
\(213\) 0 0
\(214\) 19.5936 1.33939
\(215\) −5.70928 −0.389369
\(216\) 0 0
\(217\) 6.34017 0.430399
\(218\) −19.1461 −1.29674
\(219\) 0 0
\(220\) −0.107307 −0.00723465
\(221\) 4.58145 0.308182
\(222\) 0 0
\(223\) 10.9711 0.734677 0.367339 0.930087i \(-0.380269\pi\)
0.367339 + 0.930087i \(0.380269\pi\)
\(224\) 1.30140 0.0869536
\(225\) 0 0
\(226\) 19.3523 1.28729
\(227\) 12.5464 0.832732 0.416366 0.909197i \(-0.363303\pi\)
0.416366 + 0.909197i \(0.363303\pi\)
\(228\) 0 0
\(229\) 23.3607 1.54372 0.771859 0.635794i \(-0.219327\pi\)
0.771859 + 0.635794i \(0.219327\pi\)
\(230\) 3.52586 0.232488
\(231\) 0 0
\(232\) −2.51026 −0.164807
\(233\) −12.4703 −0.816954 −0.408477 0.912769i \(-0.633940\pi\)
−0.408477 + 0.912769i \(0.633940\pi\)
\(234\) 0 0
\(235\) 1.12783 0.0735713
\(236\) −3.60197 −0.234468
\(237\) 0 0
\(238\) 4.82726 0.312905
\(239\) −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 \(0.656765\pi\)
\(242\) 16.8010 1.08001
\(243\) 0 0
\(244\) 1.13624 0.0727401
\(245\) −6.60197 −0.421784
\(246\) 0 0
\(247\) 5.57531 0.354748
\(248\) 25.2267 1.60190
\(249\) 0 0
\(250\) −1.53919 −0.0973469
\(251\) −15.4413 −0.974649 −0.487324 0.873221i \(-0.662027\pi\)
−0.487324 + 0.873221i \(0.662027\pi\)
\(252\) 0 0
\(253\) 0.665970 0.0418692
\(254\) 32.1894 2.01974
\(255\) 0 0
\(256\) 8.57531 0.535957
\(257\) 6.28231 0.391880 0.195940 0.980616i \(-0.437224\pi\)
0.195940 + 0.980616i \(0.437224\pi\)
\(258\) 0 0
\(259\) 0.979481 0.0608620
\(260\) −0.340173 −0.0210966
\(261\) 0 0
\(262\) −20.7298 −1.28069
\(263\) −10.0761 −0.621320 −0.310660 0.950521i \(-0.600550\pi\)
−0.310660 + 0.950521i \(0.600550\pi\)
\(264\) 0 0
\(265\) 0.340173 0.0208967
\(266\) 5.87444 0.360185
\(267\) 0 0
\(268\) −2.10731 −0.128724
\(269\) −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 \(0.160993\pi\)
\(272\) 22.8767 1.38710
\(273\) 0 0
\(274\) −20.8599 −1.26019
\(275\) −0.290725 −0.0175314
\(276\) 0 0
\(277\) 0.0266620 0.00160196 0.000800982 1.00000i \(-0.499745\pi\)
0.000800982 1.00000i \(0.499745\pi\)
\(278\) 7.53427 0.451875
\(279\) 0 0
\(280\) 1.58372 0.0946452
\(281\) 28.0722 1.67465 0.837325 0.546706i \(-0.184119\pi\)
0.837325 + 0.546706i \(0.184119\pi\)
\(282\) 0 0
\(283\) −20.8143 −1.23728 −0.618641 0.785674i \(-0.712317\pi\)
−0.618641 + 0.785674i \(0.712317\pi\)
\(284\) −3.35085 −0.198836
\(285\) 0 0
\(286\) −0.412408 −0.0243862
\(287\) −0.214614 −0.0126683
\(288\) 0 0
\(289\) 7.71154 0.453620
\(290\) 1.53919 0.0903843
\(291\) 0 0
\(292\) −2.56320 −0.150000
\(293\) 15.4101 0.900270 0.450135 0.892961i \(-0.351376\pi\)
0.450135 + 0.892961i \(0.351376\pi\)
\(294\) 0 0
\(295\) −9.75872 −0.568175
\(296\) 3.89723 0.226522
\(297\) 0 0
\(298\) −19.2762 −1.11664
\(299\) 2.11118 0.122093
\(300\) 0 0
\(301\) −3.60197 −0.207614
\(302\) −11.7009 −0.673309
\(303\) 0 0
\(304\) 27.8394 1.59670
\(305\) 3.07838 0.176267
\(306\) 0 0
\(307\) −28.4307 −1.62262 −0.811312 0.584614i \(-0.801246\pi\)
−0.811312 + 0.584614i \(0.801246\pi\)
\(308\) −0.0676998 −0.00385756
\(309\) 0 0
\(310\) −15.4680 −0.878523
\(311\) 19.6248 1.11282 0.556409 0.830909i \(-0.312179\pi\)
0.556409 + 0.830909i \(0.312179\pi\)
\(312\) 0 0
\(313\) 22.9093 1.29491 0.647456 0.762103i \(-0.275833\pi\)
0.647456 + 0.762103i \(0.275833\pi\)
\(314\) 38.3051 2.16168
\(315\) 0 0
\(316\) 4.57304 0.257254
\(317\) −22.8599 −1.28394 −0.641970 0.766730i \(-0.721882\pi\)
−0.641970 + 0.766730i \(0.721882\pi\)
\(318\) 0 0
\(319\) 0.290725 0.0162775
\(320\) 6.02893 0.337027
\(321\) 0 0
\(322\) 2.22446 0.123964
\(323\) 30.0722 1.67326
\(324\) 0 0
\(325\) −0.921622 −0.0511224
\(326\) −0.688756 −0.0381467
\(327\) 0 0
\(328\) −0.853922 −0.0471500
\(329\) 0.711543 0.0392286
\(330\) 0 0
\(331\) 24.0905 1.32413 0.662066 0.749445i \(-0.269680\pi\)
0.662066 + 0.749445i \(0.269680\pi\)
\(332\) −1.02893 −0.0564698
\(333\) 0 0
\(334\) 30.5776 1.67313
\(335\) −5.70928 −0.311931
\(336\) 0 0
\(337\) 12.7877 0.696588 0.348294 0.937385i \(-0.386761\pi\)
0.348294 + 0.937385i \(0.386761\pi\)
\(338\) 18.7021 1.01726
\(339\) 0 0
\(340\) −1.83483 −0.0995078
\(341\) −2.92162 −0.158215
\(342\) 0 0
\(343\) −8.58145 −0.463355
\(344\) −14.3318 −0.772717
\(345\) 0 0
\(346\) −39.1461 −2.10451
\(347\) −8.41628 −0.451810 −0.225905 0.974149i \(-0.572534\pi\)
−0.225905 + 0.974149i \(0.572534\pi\)
\(348\) 0 0
\(349\) 22.1978 1.18822 0.594110 0.804384i \(-0.297504\pi\)
0.594110 + 0.804384i \(0.297504\pi\)
\(350\) −0.971071 −0.0519059
\(351\) 0 0
\(352\) −0.599701 −0.0319642
\(353\) 6.18342 0.329110 0.164555 0.986368i \(-0.447381\pi\)
0.164555 + 0.986368i \(0.447381\pi\)
\(354\) 0 0
\(355\) −9.07838 −0.481830
\(356\) −1.74888 −0.0926906
\(357\) 0 0
\(358\) 22.8371 1.20698
\(359\) −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
\(362\) 9.10504 0.478550
\(363\) 0 0
\(364\) −0.214614 −0.0112488
\(365\) −6.94441 −0.363487
\(366\) 0 0
\(367\) 29.5402 1.54199 0.770994 0.636843i \(-0.219760\pi\)
0.770994 + 0.636843i \(0.219760\pi\)
\(368\) 10.5418 0.549531
\(369\) 0 0
\(370\) −2.38962 −0.124230
\(371\) 0.214614 0.0111422
\(372\) 0 0
\(373\) 14.4124 0.746246 0.373123 0.927782i \(-0.378287\pi\)
0.373123 + 0.927782i \(0.378287\pi\)
\(374\) −2.22446 −0.115024
\(375\) 0 0
\(376\) 2.83114 0.146005
\(377\) 0.921622 0.0474660
\(378\) 0 0
\(379\) 14.1340 0.726013 0.363007 0.931787i \(-0.381750\pi\)
0.363007 + 0.931787i \(0.381750\pi\)
\(380\) −2.23287 −0.114544
\(381\) 0 0
\(382\) 10.8188 0.553541
\(383\) 15.7815 0.806397 0.403199 0.915112i \(-0.367898\pi\)
0.403199 + 0.915112i \(0.367898\pi\)
\(384\) 0 0
\(385\) −0.183417 −0.00934782
\(386\) −27.4992 −1.39967
\(387\) 0 0
\(388\) −5.86603 −0.297803
\(389\) 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
\(392\) −16.5727 −0.837045
\(393\) 0 0
\(394\) 9.38735 0.472928
\(395\) 12.3896 0.623390
\(396\) 0 0
\(397\) 9.05172 0.454293 0.227146 0.973861i \(-0.427060\pi\)
0.227146 + 0.973861i \(0.427060\pi\)
\(398\) 15.0205 0.752911
\(399\) 0 0
\(400\) −4.60197 −0.230098
\(401\) −19.7587 −0.986704 −0.493352 0.869830i \(-0.664228\pi\)
−0.493352 + 0.869830i \(0.664228\pi\)
\(402\) 0 0
\(403\) −9.26180 −0.461363
\(404\) 4.52359 0.225057
\(405\) 0 0
\(406\) 0.971071 0.0481934
\(407\) −0.451356 −0.0223729
\(408\) 0 0
\(409\) −1.71769 −0.0849341 −0.0424670 0.999098i \(-0.513522\pi\)
−0.0424670 + 0.999098i \(0.513522\pi\)
\(410\) 0.523590 0.0258583
\(411\) 0 0
\(412\) −2.90337 −0.143039
\(413\) −6.15676 −0.302954
\(414\) 0 0
\(415\) −2.78765 −0.136841
\(416\) −1.90110 −0.0932093
\(417\) 0 0
\(418\) −2.70701 −0.132404
\(419\) 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 \(0.595047\pi\)
\(422\) −15.1857 −0.739228
\(423\) 0 0
\(424\) 0.853922 0.0414701
\(425\) −4.97107 −0.241132
\(426\) 0 0
\(427\) 1.94214 0.0939868
\(428\) −4.69860 −0.227115
\(429\) 0 0
\(430\) 8.78765 0.423778
\(431\) −19.8310 −0.955224 −0.477612 0.878571i \(-0.658497\pi\)
−0.477612 + 0.878571i \(0.658497\pi\)
\(432\) 0 0
\(433\) 14.8143 0.711931 0.355965 0.934499i \(-0.384152\pi\)
0.355965 + 0.934499i \(0.384152\pi\)
\(434\) −9.75872 −0.468434
\(435\) 0 0
\(436\) 4.59129 0.219883
\(437\) 13.8576 0.662900
\(438\) 0 0
\(439\) −17.8576 −0.852298 −0.426149 0.904653i \(-0.640130\pi\)
−0.426149 + 0.904653i \(0.640130\pi\)
\(440\) −0.729794 −0.0347916
\(441\) 0 0
\(442\) −7.05172 −0.335416
\(443\) −33.5936 −1.59608 −0.798039 0.602606i \(-0.794129\pi\)
−0.798039 + 0.602606i \(0.794129\pi\)
\(444\) 0 0
\(445\) −4.73820 −0.224612
\(446\) −16.8865 −0.799601
\(447\) 0 0
\(448\) 3.80364 0.179705
\(449\) −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
\(452\) −4.64074 −0.218282
\(453\) 0 0
\(454\) −19.3112 −0.906322
\(455\) −0.581449 −0.0272588
\(456\) 0 0
\(457\) −5.81658 −0.272088 −0.136044 0.990703i \(-0.543439\pi\)
−0.136044 + 0.990703i \(0.543439\pi\)
\(458\) −35.9565 −1.68014
\(459\) 0 0
\(460\) −0.845512 −0.0394222
\(461\) 32.3090 1.50478 0.752390 0.658718i \(-0.228901\pi\)
0.752390 + 0.658718i \(0.228901\pi\)
\(462\) 0 0
\(463\) 1.44134 0.0669846 0.0334923 0.999439i \(-0.489337\pi\)
0.0334923 + 0.999439i \(0.489337\pi\)
\(464\) 4.60197 0.213641
\(465\) 0 0
\(466\) 19.1941 0.889149
\(467\) 11.7503 0.543740 0.271870 0.962334i \(-0.412358\pi\)
0.271870 + 0.962334i \(0.412358\pi\)
\(468\) 0 0
\(469\) −3.60197 −0.166323
\(470\) −1.73594 −0.0800728
\(471\) 0 0
\(472\) −24.4969 −1.12756
\(473\) 1.65983 0.0763189
\(474\) 0 0
\(475\) −6.04945 −0.277568
\(476\) −1.15759 −0.0530582
\(477\) 0 0
\(478\) 21.1773 0.968626
\(479\) −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
\(482\) 22.5958 1.02921
\(483\) 0 0
\(484\) −4.02893 −0.183133
\(485\) −15.8927 −0.721650
\(486\) 0 0
\(487\) −4.10277 −0.185914 −0.0929572 0.995670i \(-0.529632\pi\)
−0.0929572 + 0.995670i \(0.529632\pi\)
\(488\) 7.72753 0.349809
\(489\) 0 0
\(490\) 10.1617 0.459058
\(491\) −40.7708 −1.83996 −0.919981 0.391963i \(-0.871796\pi\)
−0.919981 + 0.391963i \(0.871796\pi\)
\(492\) 0 0
\(493\) 4.97107 0.223886
\(494\) −8.58145 −0.386098
\(495\) 0 0
\(496\) −46.2472 −2.07656
\(497\) −5.72753 −0.256915
\(498\) 0 0
\(499\) −18.4703 −0.826843 −0.413421 0.910540i \(-0.635666\pi\)
−0.413421 + 0.910540i \(0.635666\pi\)
\(500\) 0.369102 0.0165068
\(501\) 0 0
\(502\) 23.7671 1.06078
\(503\) 21.4947 0.958400 0.479200 0.877706i \(-0.340927\pi\)
0.479200 + 0.877706i \(0.340927\pi\)
\(504\) 0 0
\(505\) 12.2557 0.545369
\(506\) −1.02505 −0.0455692
\(507\) 0 0
\(508\) −7.71912 −0.342480
\(509\) −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
\(512\) 13.6114 0.601546
\(513\) 0 0
\(514\) −9.66967 −0.426511
\(515\) −7.86603 −0.346619
\(516\) 0 0
\(517\) −0.327887 −0.0144204
\(518\) −1.50761 −0.0662404
\(519\) 0 0
\(520\) −2.31351 −0.101454
\(521\) −12.8059 −0.561037 −0.280518 0.959849i \(-0.590506\pi\)
−0.280518 + 0.959849i \(0.590506\pi\)
\(522\) 0 0
\(523\) −21.1278 −0.923855 −0.461928 0.886918i \(-0.652842\pi\)
−0.461928 + 0.886918i \(0.652842\pi\)
\(524\) 4.97107 0.217162
\(525\) 0 0
\(526\) 15.5090 0.676226
\(527\) −49.9565 −2.17614
\(528\) 0 0
\(529\) −17.7526 −0.771851
\(530\) −0.523590 −0.0227433
\(531\) 0 0
\(532\) −1.40871 −0.0610753
\(533\) 0.313511 0.0135797
\(534\) 0 0
\(535\) −12.7298 −0.550357
\(536\) −14.3318 −0.619038
\(537\) 0 0
\(538\) 43.3607 1.86941
\(539\) 1.91935 0.0826725
\(540\) 0 0
\(541\) 32.7382 1.40753 0.703763 0.710435i \(-0.251502\pi\)
0.703763 + 0.710435i \(0.251502\pi\)
\(542\) −44.3318 −1.90421
\(543\) 0 0
\(544\) −10.2542 −0.439646
\(545\) 12.4391 0.532831
\(546\) 0 0
\(547\) −22.1073 −0.945240 −0.472620 0.881266i \(-0.656692\pi\)
−0.472620 + 0.881266i \(0.656692\pi\)
\(548\) 5.00227 0.213686
\(549\) 0 0
\(550\) 0.447480 0.0190806
\(551\) 6.04945 0.257715
\(552\) 0 0
\(553\) 7.81658 0.332395
\(554\) −0.0410378 −0.00174353
\(555\) 0 0
\(556\) −1.80674 −0.0766229
\(557\) 39.8720 1.68943 0.844715 0.535216i \(-0.179770\pi\)
0.844715 + 0.535216i \(0.179770\pi\)
\(558\) 0 0
\(559\) 5.26180 0.222550
\(560\) −2.90337 −0.122690
\(561\) 0 0
\(562\) −43.2085 −1.82264
\(563\) −10.1217 −0.426578 −0.213289 0.976989i \(-0.568418\pi\)
−0.213289 + 0.976989i \(0.568418\pi\)
\(564\) 0 0
\(565\) −12.5730 −0.528952
\(566\) 32.0372 1.34662
\(567\) 0 0
\(568\) −22.7891 −0.956209
\(569\) 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 \(0.701320\pi\)
\(572\) 0.0988967 0.00413508
\(573\) 0 0
\(574\) 0.330332 0.0137878
\(575\) −2.29072 −0.0955298
\(576\) 0 0
\(577\) 46.1171 1.91988 0.959941 0.280202i \(-0.0904015\pi\)
0.959941 + 0.280202i \(0.0904015\pi\)
\(578\) −11.8695 −0.493707
\(579\) 0 0
\(580\) −0.369102 −0.0153261
\(581\) −1.75872 −0.0729642
\(582\) 0 0
\(583\) −0.0988967 −0.00409588
\(584\) −17.4323 −0.721352
\(585\) 0 0
\(586\) −23.7191 −0.979828
\(587\) 0.715418 0.0295285 0.0147642 0.999891i \(-0.495300\pi\)
0.0147642 + 0.999891i \(0.495300\pi\)
\(588\) 0 0
\(589\) −60.7936 −2.50496
\(590\) 15.0205 0.618385
\(591\) 0 0
\(592\) −7.14465 −0.293643
\(593\) −15.5441 −0.638320 −0.319160 0.947701i \(-0.603401\pi\)
−0.319160 + 0.947701i \(0.603401\pi\)
\(594\) 0 0
\(595\) −3.13624 −0.128573
\(596\) 4.62249 0.189344
\(597\) 0 0
\(598\) −3.24951 −0.132882
\(599\) 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 \(0.455643\pi\)
\(602\) 5.54411 0.225961
\(603\) 0 0
\(604\) 2.80590 0.114171
\(605\) −10.9155 −0.443777
\(606\) 0 0
\(607\) 31.6970 1.28654 0.643271 0.765639i \(-0.277577\pi\)
0.643271 + 0.765639i \(0.277577\pi\)
\(608\) −12.4787 −0.506077
\(609\) 0 0
\(610\) −4.73820 −0.191844
\(611\) −1.03943 −0.0420508
\(612\) 0 0
\(613\) 1.20394 0.0486265 0.0243133 0.999704i \(-0.492260\pi\)
0.0243133 + 0.999704i \(0.492260\pi\)
\(614\) 43.7602 1.76602
\(615\) 0 0
\(616\) −0.460425 −0.0185511
\(617\) 37.9337 1.52715 0.763577 0.645716i \(-0.223441\pi\)
0.763577 + 0.645716i \(0.223441\pi\)
\(618\) 0 0
\(619\) −4.60424 −0.185060 −0.0925299 0.995710i \(-0.529495\pi\)
−0.0925299 + 0.995710i \(0.529495\pi\)
\(620\) 3.70928 0.148968
\(621\) 0 0
\(622\) −30.2062 −1.21116
\(623\) −2.98932 −0.119765
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −35.2618 −1.40934
\(627\) 0 0
\(628\) −9.18568 −0.366549
\(629\) −7.71769 −0.307724
\(630\) 0 0
\(631\) −8.41241 −0.334893 −0.167446 0.985881i \(-0.553552\pi\)
−0.167446 + 0.985881i \(0.553552\pi\)
\(632\) 31.1012 1.23714
\(633\) 0 0
\(634\) 35.1857 1.39740
\(635\) −20.9132 −0.829915
\(636\) 0 0
\(637\) 6.08452 0.241077
\(638\) −0.447480 −0.0177159
\(639\) 0 0
\(640\) −13.4052 −0.529888
\(641\) −32.5380 −1.28517 −0.642586 0.766213i \(-0.722139\pi\)
−0.642586 + 0.766213i \(0.722139\pi\)
\(642\) 0 0
\(643\) −2.09293 −0.0825372 −0.0412686 0.999148i \(-0.513140\pi\)
−0.0412686 + 0.999148i \(0.513140\pi\)
\(644\) −0.533431 −0.0210201
\(645\) 0 0
\(646\) −46.2868 −1.82113
\(647\) −45.1955 −1.77682 −0.888410 0.459051i \(-0.848189\pi\)
−0.888410 + 0.459051i \(0.848189\pi\)
\(648\) 0 0
\(649\) 2.83710 0.111366
\(650\) 1.41855 0.0556401
\(651\) 0 0
\(652\) 0.165166 0.00646840
\(653\) 2.14834 0.0840712 0.0420356 0.999116i \(-0.486616\pi\)
0.0420356 + 0.999116i \(0.486616\pi\)
\(654\) 0 0
\(655\) 13.4680 0.526238
\(656\) 1.56547 0.0611211
\(657\) 0 0
\(658\) −1.09520 −0.0426953
\(659\) 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 \(0.749685\pi\)
\(662\) −37.0798 −1.44115
\(663\) 0 0
\(664\) −6.99773 −0.271565
\(665\) −3.81658 −0.148001
\(666\) 0 0
\(667\) 2.29072 0.0886972
\(668\) −7.33260 −0.283707
\(669\) 0 0
\(670\) 8.78765 0.339497
\(671\) −0.894960 −0.0345496
\(672\) 0 0
\(673\) −17.4719 −0.673491 −0.336746 0.941596i \(-0.609326\pi\)
−0.336746 + 0.941596i \(0.609326\pi\)
\(674\) −19.6826 −0.758146
\(675\) 0 0
\(676\) −4.48482 −0.172493
\(677\) 40.0372 1.53875 0.769377 0.638796i \(-0.220567\pi\)
0.769377 + 0.638796i \(0.220567\pi\)
\(678\) 0 0
\(679\) −10.0267 −0.384788
\(680\) −12.4787 −0.478535
\(681\) 0 0
\(682\) 4.49693 0.172196
\(683\) −2.07611 −0.0794402 −0.0397201 0.999211i \(-0.512647\pi\)
−0.0397201 + 0.999211i \(0.512647\pi\)
\(684\) 0 0
\(685\) 13.5525 0.517815
\(686\) 13.2085 0.504302
\(687\) 0 0
\(688\) 26.2739 1.00168
\(689\) −0.313511 −0.0119438
\(690\) 0 0
\(691\) 26.7070 1.01598 0.507991 0.861362i \(-0.330388\pi\)
0.507991 + 0.861362i \(0.330388\pi\)
\(692\) 9.38735 0.356854
\(693\) 0 0
\(694\) 12.9542 0.491737
\(695\) −4.89496 −0.185676
\(696\) 0 0
\(697\) 1.69102 0.0640521
\(698\) −34.1666 −1.29322
\(699\) 0 0
\(700\) 0.232866 0.00880150
\(701\) 21.9155 0.827736 0.413868 0.910337i \(-0.364177\pi\)
0.413868 + 0.910337i \(0.364177\pi\)
\(702\) 0 0
\(703\) −9.39189 −0.354222
\(704\) −1.75276 −0.0660596
\(705\) 0 0
\(706\) −9.51745 −0.358194
\(707\) 7.73206 0.290794
\(708\) 0 0
\(709\) −4.60811 −0.173061 −0.0865306 0.996249i \(-0.527578\pi\)
−0.0865306 + 0.996249i \(0.527578\pi\)
\(710\) 13.9733 0.524410
\(711\) 0 0
\(712\) −11.8941 −0.445751
\(713\) −23.0205 −0.862125
\(714\) 0 0
\(715\) 0.267938 0.0100203
\(716\) −5.47641 −0.204663
\(717\) 0 0
\(718\) 7.78151 0.290403
\(719\) −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
\(722\) −27.0833 −1.00794
\(723\) 0 0
\(724\) −2.18342 −0.0811461
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −26.9711 −1.00030 −0.500151 0.865938i \(-0.666722\pi\)
−0.500151 + 0.865938i \(0.666722\pi\)
\(728\) −1.45959 −0.0540960
\(729\) 0 0
\(730\) 10.6888 0.395609
\(731\) 28.3812 1.04972
\(732\) 0 0
\(733\) −30.0638 −1.11043 −0.555216 0.831706i \(-0.687365\pi\)
−0.555216 + 0.831706i \(0.687365\pi\)
\(734\) −45.4680 −1.67825
\(735\) 0 0
\(736\) −4.72526 −0.174175
\(737\) 1.65983 0.0611405
\(738\) 0 0
\(739\) 51.1422 1.88130 0.940648 0.339383i \(-0.110218\pi\)
0.940648 + 0.339383i \(0.110218\pi\)
\(740\) 0.573039 0.0210653
\(741\) 0 0
\(742\) −0.330332 −0.0121269
\(743\) −11.1857 −0.410363 −0.205181 0.978724i \(-0.565778\pi\)
−0.205181 + 0.978724i \(0.565778\pi\)
\(744\) 0 0
\(745\) 12.5236 0.458829
\(746\) −22.1834 −0.812193
\(747\) 0 0
\(748\) 0.533431 0.0195042
\(749\) −8.03120 −0.293454
\(750\) 0 0
\(751\) −18.3630 −0.670074 −0.335037 0.942205i \(-0.608749\pi\)
−0.335037 + 0.942205i \(0.608749\pi\)
\(752\) −5.19022 −0.189268
\(753\) 0 0
\(754\) −1.41855 −0.0516606
\(755\) 7.60197 0.276664
\(756\) 0 0
\(757\) 15.8927 0.577630 0.288815 0.957385i \(-0.406739\pi\)
0.288815 + 0.957385i \(0.406739\pi\)
\(758\) −21.7548 −0.790172
\(759\) 0 0
\(760\) −15.1857 −0.550843
\(761\) −13.8843 −0.503305 −0.251652 0.967818i \(-0.580974\pi\)
−0.251652 + 0.967818i \(0.580974\pi\)
\(762\) 0 0
\(763\) 7.84778 0.284109
\(764\) −2.59439 −0.0938619
\(765\) 0 0
\(766\) −24.2907 −0.877660
\(767\) 8.99386 0.324749
\(768\) 0 0
\(769\) −35.4063 −1.27678 −0.638391 0.769712i \(-0.720400\pi\)
−0.638391 + 0.769712i \(0.720400\pi\)
\(770\) 0.282314 0.0101739
\(771\) 0 0
\(772\) 6.59439 0.237337
\(773\) 0.488518 0.0175708 0.00878539 0.999961i \(-0.497203\pi\)
0.00878539 + 0.999961i \(0.497203\pi\)
\(774\) 0 0
\(775\) 10.0494 0.360987
\(776\) −39.8948 −1.43214
\(777\) 0 0
\(778\) −21.2663 −0.762435
\(779\) 2.05786 0.0737304
\(780\) 0 0
\(781\) 2.63931 0.0944419
\(782\) −17.5273 −0.626775
\(783\) 0 0
\(784\) 30.3820 1.08507
\(785\) −24.8865 −0.888239
\(786\) 0 0
\(787\) −1.99159 −0.0709925 −0.0354962 0.999370i \(-0.511301\pi\)
−0.0354962 + 0.999370i \(0.511301\pi\)
\(788\) −2.25112 −0.0801927
\(789\) 0 0
\(790\) −19.0700 −0.678479
\(791\) −7.93230 −0.282040
\(792\) 0 0
\(793\) −2.83710 −0.100748
\(794\) −13.9323 −0.494439
\(795\) 0 0
\(796\) −3.60197 −0.127668
\(797\) 17.2702 0.611742 0.305871 0.952073i \(-0.401052\pi\)
0.305871 + 0.952073i \(0.401052\pi\)
\(798\) 0 0
\(799\) −5.60650 −0.198344
\(800\) 2.06278 0.0729303
\(801\) 0 0
\(802\) 30.4124 1.07390
\(803\) 2.01891 0.0712458
\(804\) 0 0
\(805\) −1.44521 −0.0509371
\(806\) 14.2557 0.502134
\(807\) 0 0
\(808\) 30.7649 1.08230
\(809\) 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 \(0.549261\pi\)
\(812\) −0.232866 −0.00817199
\(813\) 0 0
\(814\) 0.694722 0.0243500
\(815\) 0.447480 0.0156745
\(816\) 0 0
\(817\) 34.5380 1.20833
\(818\) 2.64384 0.0924398
\(819\) 0 0
\(820\) −0.125559 −0.00438470
\(821\) 28.1568 0.982678 0.491339 0.870969i \(-0.336508\pi\)
0.491339 + 0.870969i \(0.336508\pi\)
\(822\) 0 0
\(823\) 20.7442 0.723096 0.361548 0.932353i \(-0.382248\pi\)
0.361548 + 0.932353i \(0.382248\pi\)
\(824\) −19.7458 −0.687877
\(825\) 0 0
\(826\) 9.47641 0.329726
\(827\) 9.12783 0.317406 0.158703 0.987326i \(-0.449269\pi\)
0.158703 + 0.987326i \(0.449269\pi\)
\(828\) 0 0
\(829\) 31.8576 1.10646 0.553230 0.833028i \(-0.313395\pi\)
0.553230 + 0.833028i \(0.313395\pi\)
\(830\) 4.29072 0.148933
\(831\) 0 0
\(832\) −5.55640 −0.192633
\(833\) 32.8188 1.13711
\(834\) 0 0
\(835\) −19.8660 −0.687492
\(836\) 0.649149 0.0224513
\(837\) 0 0
\(838\) −54.6902 −1.88924
\(839\) −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
\(842\) 18.5814 0.640359
\(843\) 0 0
\(844\) 3.64158 0.125348
\(845\) −12.1506 −0.417994
\(846\) 0 0
\(847\) −6.88655 −0.236625
\(848\) −1.56547 −0.0537583
\(849\) 0 0
\(850\) 7.65142 0.262441
\(851\) −3.55640 −0.121912
\(852\) 0 0
\(853\) −56.0515 −1.91917 −0.959584 0.281422i \(-0.909194\pi\)
−0.959584 + 0.281422i \(0.909194\pi\)
\(854\) −2.98932 −0.102292
\(855\) 0 0
\(856\) −31.9551 −1.09220
\(857\) 6.08452 0.207843 0.103922 0.994585i \(-0.466861\pi\)
0.103922 + 0.994585i \(0.466861\pi\)
\(858\) 0 0
\(859\) 35.5936 1.21444 0.607218 0.794535i \(-0.292285\pi\)
0.607218 + 0.794535i \(0.292285\pi\)
\(860\) −2.10731 −0.0718586
\(861\) 0 0
\(862\) 30.5236 1.03964
\(863\) 12.8287 0.436694 0.218347 0.975871i \(-0.429934\pi\)
0.218347 + 0.975871i \(0.429934\pi\)
\(864\) 0 0
\(865\) 25.4329 0.864745
\(866\) −22.8020 −0.774844
\(867\) 0 0
\(868\) 2.34017 0.0794306
\(869\) −3.60197 −0.122188
\(870\) 0 0
\(871\) 5.26180 0.178289
\(872\) 31.2253 1.05742
\(873\) 0 0
\(874\) −21.3295 −0.721481
\(875\) 0.630898 0.0213282
\(876\) 0 0
\(877\) −1.50307 −0.0507551 −0.0253776 0.999678i \(-0.508079\pi\)
−0.0253776 + 0.999678i \(0.508079\pi\)
\(878\) 27.4863 0.927616
\(879\) 0 0
\(880\) 1.33791 0.0451008
\(881\) −23.4908 −0.791425 −0.395712 0.918375i \(-0.629502\pi\)
−0.395712 + 0.918375i \(0.629502\pi\)
\(882\) 0 0
\(883\) −29.0433 −0.977385 −0.488693 0.872456i \(-0.662526\pi\)
−0.488693 + 0.872456i \(0.662526\pi\)
\(884\) 1.69102 0.0568753
\(885\) 0 0
\(886\) 51.7068 1.73712
\(887\) 19.0700 0.640307 0.320153 0.947366i \(-0.396266\pi\)
0.320153 + 0.947366i \(0.396266\pi\)
\(888\) 0 0
\(889\) −13.1941 −0.442516
\(890\) 7.29299 0.244462
\(891\) 0 0
\(892\) 4.04945 0.135586
\(893\) −6.82273 −0.228314
\(894\) 0 0
\(895\) −14.8371 −0.495950
\(896\) −8.45732 −0.282539
\(897\) 0 0
\(898\) 10.8950 0.363570
\(899\) −10.0494 −0.335168
\(900\) 0 0
\(901\) −1.69102 −0.0563362
\(902\) −0.152221 −0.00506839
\(903\) 0 0
\(904\) −31.5616 −1.04972
\(905\) −5.91548 −0.196637
\(906\) 0 0
\(907\) 5.54023 0.183960 0.0919802 0.995761i \(-0.470680\pi\)
0.0919802 + 0.995761i \(0.470680\pi\)
\(908\) 4.63090 0.153682
\(909\) 0 0
\(910\) 0.894960 0.0296676
\(911\) −53.2990 −1.76587 −0.882937 0.469492i \(-0.844437\pi\)
−0.882937 + 0.469492i \(0.844437\pi\)
\(912\) 0 0
\(913\) 0.810439 0.0268216
\(914\) 8.95282 0.296133
\(915\) 0 0
\(916\) 8.62249 0.284895
\(917\) 8.49693 0.280593
\(918\) 0 0
\(919\) 37.5897 1.23997 0.619985 0.784614i \(-0.287139\pi\)
0.619985 + 0.784614i \(0.287139\pi\)
\(920\) −5.75031 −0.189582
\(921\) 0 0
\(922\) −49.7296 −1.63776
\(923\) 8.36683 0.275398
\(924\) 0 0
\(925\) 1.55252 0.0510465
\(926\) −2.21849 −0.0729041
\(927\) 0 0
\(928\) −2.06278 −0.0677140
\(929\) −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
\(932\) −4.60281 −0.150770
\(933\) 0 0
\(934\) −18.0860 −0.591790
\(935\) 1.44521 0.0472635
\(936\) 0 0
\(937\) −22.8638 −0.746927 −0.373463 0.927645i \(-0.621830\pi\)
−0.373463 + 0.927645i \(0.621830\pi\)
\(938\) 5.54411 0.181022
\(939\) 0 0
\(940\) 0.416283 0.0135777
\(941\) −0.523590 −0.0170686 −0.00853428 0.999964i \(-0.502717\pi\)
−0.00853428 + 0.999964i \(0.502717\pi\)
\(942\) 0 0
\(943\) 0.779243 0.0253756
\(944\) 44.9093 1.46167
\(945\) 0 0
\(946\) −2.55479 −0.0830633
\(947\) 10.0228 0.325697 0.162848 0.986651i \(-0.447932\pi\)
0.162848 + 0.986651i \(0.447932\pi\)
\(948\) 0 0
\(949\) 6.40012 0.207757
\(950\) 9.31124 0.302097
\(951\) 0 0
\(952\) −7.87277 −0.255158
\(953\) 8.15676 0.264223 0.132112 0.991235i \(-0.457824\pi\)
0.132112 + 0.991235i \(0.457824\pi\)
\(954\) 0 0
\(955\) −7.02893 −0.227451
\(956\) −5.07838 −0.164246
\(957\) 0 0
\(958\) 26.4261 0.853789
\(959\) 8.55025 0.276102
\(960\) 0 0
\(961\) 69.9914 2.25779
\(962\) 2.20233 0.0710059
\(963\) 0 0
\(964\) −5.41855 −0.174520
\(965\) 17.8660 0.575128
\(966\) 0 0
\(967\) 15.7671 0.507037 0.253518 0.967331i \(-0.418412\pi\)
0.253518 + 0.967331i \(0.418412\pi\)
\(968\) −27.4007 −0.880691
\(969\) 0 0
\(970\) 24.4619 0.785423
\(971\) 17.8804 0.573810 0.286905 0.957959i \(-0.407374\pi\)
0.286905 + 0.957959i \(0.407374\pi\)
\(972\) 0 0
\(973\) −3.08822 −0.0990037
\(974\) 6.31494 0.202344
\(975\) 0 0
\(976\) −14.1666 −0.453462
\(977\) 55.1071 1.76303 0.881517 0.472153i \(-0.156523\pi\)
0.881517 + 0.472153i \(0.156523\pi\)
\(978\) 0 0
\(979\) 1.37751 0.0440255
\(980\) −2.43680 −0.0778408
\(981\) 0 0
\(982\) 62.7540 2.00256
\(983\) 1.29687 0.0413637 0.0206818 0.999786i \(-0.493416\pi\)
0.0206818 + 0.999786i \(0.493416\pi\)
\(984\) 0 0
\(985\) −6.09890 −0.194327
\(986\) −7.65142 −0.243671
\(987\) 0 0
\(988\) 2.05786 0.0654692
\(989\) 13.0784 0.415868
\(990\) 0 0
\(991\) −3.11942 −0.0990915 −0.0495458 0.998772i \(-0.515777\pi\)
−0.0495458 + 0.998772i \(0.515777\pi\)
\(992\) 20.7298 0.658172
\(993\) 0 0
\(994\) 8.81575 0.279618
\(995\) −9.75872 −0.309372
\(996\) 0 0
\(997\) 30.2472 0.957940 0.478970 0.877831i \(-0.341010\pi\)
0.478970 + 0.877831i \(0.341010\pi\)
\(998\) 28.4292 0.899912
\(999\) 0 0
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 1305.2.a.o.1.2 3
3.2 odd 2 145.2.a.d.1.2 3
5.4 even 2 6525.2.a.bh.1.2 3
12.11 even 2 2320.2.a.s.1.1 3
15.2 even 4 725.2.b.d.349.5 6
15.8 even 4 725.2.b.d.349.2 6
15.14 odd 2 725.2.a.d.1.2 3
21.20 even 2 7105.2.a.p.1.2 3
24.5 odd 2 9280.2.a.bu.1.1 3
24.11 even 2 9280.2.a.bm.1.3 3
87.86 odd 2 4205.2.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.2 3 3.2 odd 2
725.2.a.d.1.2 3 15.14 odd 2
725.2.b.d.349.2 6 15.8 even 4
725.2.b.d.349.5 6 15.2 even 4
1305.2.a.o.1.2 3 1.1 even 1 trivial
2320.2.a.s.1.1 3 12.11 even 2
4205.2.a.e.1.2 3 87.86 odd 2
6525.2.a.bh.1.2 3 5.4 even 2
7105.2.a.p.1.2 3 21.20 even 2
9280.2.a.bm.1.3 3 24.11 even 2
9280.2.a.bu.1.1 3 24.5 odd 2