Properties

Label 6525.2.a.bh.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} +0.369102 q^{4} -0.630898 q^{7} -2.51026 q^{8} +O(q^{10})\) \(q+1.53919 q^{2} +0.369102 q^{4} -0.630898 q^{7} -2.51026 q^{8} -0.290725 q^{11} +0.921622 q^{13} -0.971071 q^{14} -4.60197 q^{16} +4.97107 q^{17} -6.04945 q^{19} -0.447480 q^{22} +2.29072 q^{23} +1.41855 q^{26} -0.232866 q^{28} -1.00000 q^{29} +10.0494 q^{31} -2.06278 q^{32} +7.65142 q^{34} -1.55252 q^{37} -9.31124 q^{38} -0.340173 q^{41} +5.70928 q^{43} -0.107307 q^{44} +3.52586 q^{46} -1.12783 q^{47} -6.60197 q^{49} +0.340173 q^{52} -0.340173 q^{53} +1.58372 q^{56} -1.53919 q^{58} -9.75872 q^{59} +3.07838 q^{61} +15.4680 q^{62} +6.02893 q^{64} +5.70928 q^{67} +1.83483 q^{68} -9.07838 q^{71} +6.94441 q^{73} -2.38962 q^{74} -2.23287 q^{76} +0.183417 q^{77} +12.3896 q^{79} -0.523590 q^{82} +2.78765 q^{83} +8.78765 q^{86} +0.729794 q^{88} -4.73820 q^{89} -0.581449 q^{91} +0.845512 q^{92} -1.73594 q^{94} +15.8927 q^{97} -10.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 8 q^{11} + 6 q^{13} + 12 q^{14} + 5 q^{16} - 2 q^{22} + 14 q^{23} - 10 q^{26} + 22 q^{28} - 3 q^{29} + 12 q^{31} + 11 q^{32} - 14 q^{34} - 4 q^{37} - 2 q^{38} + 10 q^{41} + 10 q^{43} - 12 q^{44} + 8 q^{46} + 18 q^{47} - q^{49} - 10 q^{52} + 10 q^{53} + 32 q^{56} - 3 q^{58} - 4 q^{59} + 6 q^{61} + 14 q^{62} + 33 q^{64} + 10 q^{67} - 36 q^{68} - 24 q^{71} + 4 q^{73} + 22 q^{74} + 16 q^{76} - 4 q^{77} + 8 q^{79} + 14 q^{82} - 2 q^{83} + 16 q^{86} - 38 q^{88} - 22 q^{89} - 16 q^{91} + 22 q^{92} + 36 q^{97} + 23 q^{98}+O(q^{100}) \) 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 \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 0 0
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) 0 0
\(7\) −0.630898 −0.238457 −0.119228 0.992867i \(-0.538042\pi\)
−0.119228 + 0.992867i \(0.538042\pi\)
\(8\) −2.51026 −0.887511
\(9\) 0 0
\(10\) 0 0
\(11\) −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 \(-0.459207\pi\)
0.127806 + 0.991799i \(0.459207\pi\)
\(14\) −0.971071 −0.259530
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 4.97107 1.20566 0.602831 0.797869i \(-0.294039\pi\)
0.602831 + 0.797869i \(0.294039\pi\)
\(18\) 0 0
\(19\) −6.04945 −1.38784 −0.693919 0.720053i \(-0.744118\pi\)
−0.693919 + 0.720053i \(0.744118\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.447480 −0.0954031
\(23\) 2.29072 0.477649 0.238825 0.971063i \(-0.423238\pi\)
0.238825 + 0.971063i \(0.423238\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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 0
\(36\) 0 0
\(37\) −1.55252 −0.255233 −0.127616 0.991824i \(-0.540733\pi\)
−0.127616 + 0.991824i \(0.540733\pi\)
\(38\) −9.31124 −1.51048
\(39\) 0 0
\(40\) 0 0
\(41\) −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 \(-0.356632\pi\)
0.435328 + 0.900272i \(0.356632\pi\)
\(44\) −0.107307 −0.0161772
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) −1.12783 −0.164510 −0.0822552 0.996611i \(-0.526212\pi\)
−0.0822552 + 0.996611i \(0.526212\pi\)
\(48\) 0 0
\(49\) −6.60197 −0.943138
\(50\) 0 0
\(51\) 0 0
\(52\) 0.340173 0.0471735
\(53\) −0.340173 −0.0467264 −0.0233632 0.999727i \(-0.507437\pi\)
−0.0233632 + 0.999727i \(0.507437\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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 0
\(66\) 0 0
\(67\) 5.70928 0.697499 0.348749 0.937216i \(-0.386606\pi\)
0.348749 + 0.937216i \(0.386606\pi\)
\(68\) 1.83483 0.222506
\(69\) 0 0
\(70\) 0 0
\(71\) −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 \(-0.366787\pi\)
0.406391 + 0.913699i \(0.366787\pi\)
\(74\) −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 \(0.254531\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.523590 −0.0578209
\(83\) 2.78765 0.305985 0.152992 0.988227i \(-0.451109\pi\)
0.152992 + 0.988227i \(0.451109\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 15.8927 1.61366 0.806829 0.590785i \(-0.201182\pi\)
0.806829 + 0.590785i \(0.201182\pi\)
\(98\) −10.1617 −1.02648
\(99\) 0 0
\(100\) 0 0
\(101\) 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 \(-0.373328\pi\)
0.387532 + 0.921856i \(0.373328\pi\)
\(104\) −2.31351 −0.226858
\(105\) 0 0
\(106\) −0.523590 −0.0508556
\(107\) 12.7298 1.23064 0.615318 0.788279i \(-0.289028\pi\)
0.615318 + 0.788279i \(0.289028\pi\)
\(108\) 0 0
\(109\) 12.4391 1.19145 0.595723 0.803190i \(-0.296865\pi\)
0.595723 + 0.803190i \(0.296865\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.90337 0.274343
\(113\) 12.5730 1.18277 0.591386 0.806389i \(-0.298581\pi\)
0.591386 + 0.806389i \(0.298581\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) 20.9132 1.85575 0.927874 0.372895i \(-0.121635\pi\)
0.927874 + 0.372895i \(0.121635\pi\)
\(128\) 13.4052 1.18487
\(129\) 0 0
\(130\) 0 0
\(131\) 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 \(-0.696531\pi\)
−0.578935 + 0.815374i \(0.696531\pi\)
\(138\) 0 0
\(139\) −4.89496 −0.415185 −0.207593 0.978215i \(-0.566563\pi\)
−0.207593 + 0.978215i \(0.566563\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −13.9733 −1.17262
\(143\) −0.267938 −0.0224061
\(144\) 0 0
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 24.8865 1.98616 0.993081 0.117428i \(-0.0374648\pi\)
0.993081 + 0.117428i \(0.0374648\pi\)
\(158\) 19.0700 1.51713
\(159\) 0 0
\(160\) 0 0
\(161\) −1.44521 −0.113899
\(162\) 0 0
\(163\) −0.447480 −0.0350493 −0.0175247 0.999846i \(-0.505579\pi\)
−0.0175247 + 0.999846i \(0.505579\pi\)
\(164\) −0.125559 −0.00980448
\(165\) 0 0
\(166\) 4.29072 0.333025
\(167\) 19.8660 1.53728 0.768640 0.639682i \(-0.220934\pi\)
0.768640 + 0.639682i \(0.220934\pi\)
\(168\) 0 0
\(169\) −12.1506 −0.934662
\(170\) 0 0
\(171\) 0 0
\(172\) 2.10731 0.160681
\(173\) −25.4329 −1.93363 −0.966815 0.255478i \(-0.917767\pi\)
−0.966815 + 0.255478i \(0.917767\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 0 0
\(187\) −1.44521 −0.105684
\(188\) −0.416283 −0.0303606
\(189\) 0 0
\(190\) 0 0
\(191\) −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 \(-0.722316\pi\)
−0.643013 + 0.765856i \(0.722316\pi\)
\(194\) 24.4619 1.75626
\(195\) 0 0
\(196\) −2.43680 −0.174057
\(197\) 6.09890 0.434528 0.217264 0.976113i \(-0.430287\pi\)
0.217264 + 0.976113i \(0.430287\pi\)
\(198\) 0 0
\(199\) −9.75872 −0.691778 −0.345889 0.938276i \(-0.612423\pi\)
−0.345889 + 0.938276i \(0.612423\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.8638 1.32725
\(203\) 0.630898 0.0442803
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 0 0
\(217\) −6.34017 −0.430399
\(218\) 19.1461 1.29674
\(219\) 0 0
\(220\) 0 0
\(221\) 4.58145 0.308182
\(222\) 0 0
\(223\) −10.9711 −0.734677 −0.367339 0.930087i \(-0.619731\pi\)
−0.367339 + 0.930087i \(0.619731\pi\)
\(224\) 1.30140 0.0869536
\(225\) 0 0
\(226\) 19.3523 1.28729
\(227\) −12.5464 −0.832732 −0.416366 0.909197i \(-0.636697\pi\)
−0.416366 + 0.909197i \(0.636697\pi\)
\(228\) 0 0
\(229\) 23.3607 1.54372 0.771859 0.635794i \(-0.219327\pi\)
0.771859 + 0.635794i \(0.219327\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.51026 0.164807
\(233\) 12.4703 0.816954 0.408477 0.912769i \(-0.366060\pi\)
0.408477 + 0.912769i \(0.366060\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −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
\(245\) 0 0
\(246\) 0 0
\(247\) −5.57531 −0.354748
\(248\) −25.2267 −1.60190
\(249\) 0 0
\(250\) 0 0
\(251\) −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 \(-0.562776\pi\)
−0.195940 + 0.980616i \(0.562776\pi\)
\(258\) 0 0
\(259\) 0.979481 0.0608620
\(260\) 0 0
\(261\) 0 0
\(262\) 20.7298 1.28069
\(263\) 10.0761 0.621320 0.310660 0.950521i \(-0.399450\pi\)
0.310660 + 0.950521i \(0.399450\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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 0
\(276\) 0 0
\(277\) −0.0266620 −0.00160196 −0.000800982 1.00000i \(-0.500255\pi\)
−0.000800982 1.00000i \(0.500255\pi\)
\(278\) −7.53427 −0.451875
\(279\) 0 0
\(280\) 0 0
\(281\) 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 \(-0.287683\pi\)
0.618641 + 0.785674i \(0.287683\pi\)
\(284\) −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
\(290\) 0 0
\(291\) 0 0
\(292\) 2.56320 0.150000
\(293\) −15.4101 −0.900270 −0.450135 0.892961i \(-0.648624\pi\)
−0.450135 + 0.892961i \(0.648624\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 28.4307 1.62262 0.811312 0.584614i \(-0.198754\pi\)
0.811312 + 0.584614i \(0.198754\pi\)
\(308\) 0.0676998 0.00385756
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(-0.724167\pi\)
−0.647456 + 0.762103i \(0.724167\pi\)
\(314\) 38.3051 2.16168
\(315\) 0 0
\(316\) 4.57304 0.257254
\(317\) 22.8599 1.28394 0.641970 0.766730i \(-0.278118\pi\)
0.641970 + 0.766730i \(0.278118\pi\)
\(318\) 0 0
\(319\) 0.290725 0.0162775
\(320\) 0 0
\(321\) 0 0
\(322\) −2.22446 −0.123964
\(323\) −30.0722 −1.67326
\(324\) 0 0
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) −12.7877 −0.696588 −0.348294 0.937385i \(-0.613239\pi\)
−0.348294 + 0.937385i \(0.613239\pi\)
\(338\) −18.7021 −1.01726
\(339\) 0 0
\(340\) 0 0
\(341\) −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 \(-0.427466\pi\)
0.225905 + 0.974149i \(0.427466\pi\)
\(348\) 0 0
\(349\) 22.1978 1.18822 0.594110 0.804384i \(-0.297504\pi\)
0.594110 + 0.804384i \(0.297504\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.599701 0.0319642
\(353\) −6.18342 −0.329110 −0.164555 0.986368i \(-0.552619\pi\)
−0.164555 + 0.986368i \(0.552619\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −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
\(365\) 0 0
\(366\) 0 0
\(367\) −29.5402 −1.54199 −0.770994 0.636843i \(-0.780240\pi\)
−0.770994 + 0.636843i \(0.780240\pi\)
\(368\) −10.5418 −0.549531
\(369\) 0 0
\(370\) 0 0
\(371\) 0.214614 0.0111422
\(372\) 0 0
\(373\) −14.4124 −0.746246 −0.373123 0.927782i \(-0.621713\pi\)
−0.373123 + 0.927782i \(0.621713\pi\)
\(374\) −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 \(0.381750\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −10.8188 −0.553541
\(383\) −15.7815 −0.806397 −0.403199 0.915112i \(-0.632102\pi\)
−0.403199 + 0.915112i \(0.632102\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −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
\(395\) 0 0
\(396\) 0 0
\(397\) −9.05172 −0.454293 −0.227146 0.973861i \(-0.572940\pi\)
−0.227146 + 0.973861i \(0.572940\pi\)
\(398\) −15.0205 −0.752911
\(399\) 0 0
\(400\) 0 0
\(401\) −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 0
\(411\) 0 0
\(412\) 2.90337 0.143039
\(413\) 6.15676 0.302954
\(414\) 0 0
\(415\) 0 0
\(416\) −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
\(425\) 0 0
\(426\) 0 0
\(427\) −1.94214 −0.0939868
\(428\) 4.69860 0.227115
\(429\) 0 0
\(430\) 0 0
\(431\) −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 \(-0.615848\pi\)
−0.355965 + 0.934499i \(0.615848\pi\)
\(434\) −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 0
\(441\) 0 0
\(442\) 7.05172 0.335416
\(443\) 33.5936 1.59608 0.798039 0.602606i \(-0.205871\pi\)
0.798039 + 0.602606i \(0.205871\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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 0
\(456\) 0 0
\(457\) 5.81658 0.272088 0.136044 0.990703i \(-0.456561\pi\)
0.136044 + 0.990703i \(0.456561\pi\)
\(458\) 35.9565 1.68014
\(459\) 0 0
\(460\) 0 0
\(461\) 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 \(-0.510663\pi\)
−0.0334923 + 0.999439i \(0.510663\pi\)
\(464\) 4.60197 0.213641
\(465\) 0 0
\(466\) 19.1941 0.889149
\(467\) −11.7503 −0.543740 −0.271870 0.962334i \(-0.587642\pi\)
−0.271870 + 0.962334i \(0.587642\pi\)
\(468\) 0 0
\(469\) −3.60197 −0.166323
\(470\) 0 0
\(471\) 0 0
\(472\) 24.4969 1.12756
\(473\) −1.65983 −0.0763189
\(474\) 0 0
\(475\) 0 0
\(476\) −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
\(485\) 0 0
\(486\) 0 0
\(487\) 4.10277 0.185914 0.0929572 0.995670i \(-0.470368\pi\)
0.0929572 + 0.995670i \(0.470368\pi\)
\(488\) −7.72753 −0.349809
\(489\) 0 0
\(490\) 0 0
\(491\) −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 0
\(501\) 0 0
\(502\) −23.7671 −1.06078
\(503\) −21.4947 −0.958400 −0.479200 0.877706i \(-0.659073\pi\)
−0.479200 + 0.877706i \(0.659073\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −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
\(515\) 0 0
\(516\) 0 0
\(517\) 0.327887 0.0144204
\(518\) 1.50761 0.0662404
\(519\) 0 0
\(520\) 0 0
\(521\) −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 \(-0.347158\pi\)
0.461928 + 0.886918i \(0.347158\pi\)
\(524\) 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 0
\(531\) 0 0
\(532\) 1.40871 0.0610753
\(533\) −0.313511 −0.0135797
\(534\) 0 0
\(535\) 0 0
\(536\) −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
\(545\) 0 0
\(546\) 0 0
\(547\) 22.1073 0.945240 0.472620 0.881266i \(-0.343308\pi\)
0.472620 + 0.881266i \(0.343308\pi\)
\(548\) −5.00227 −0.213686
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.820230\pi\)
−0.844715 + 0.535216i \(0.820230\pi\)
\(558\) 0 0
\(559\) 5.26180 0.222550
\(560\) 0 0
\(561\) 0 0
\(562\) 43.2085 1.82264
\(563\) 10.1217 0.426578 0.213289 0.976989i \(-0.431582\pi\)
0.213289 + 0.976989i \(0.431582\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −46.1171 −1.91988 −0.959941 0.280202i \(-0.909598\pi\)
−0.959941 + 0.280202i \(0.909598\pi\)
\(578\) 11.8695 0.493707
\(579\) 0 0
\(580\) 0 0
\(581\) −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 \(-0.504700\pi\)
−0.0147642 + 0.999891i \(0.504700\pi\)
\(588\) 0 0
\(589\) −60.7936 −2.50496
\(590\) 0 0
\(591\) 0 0
\(592\) 7.14465 0.293643
\(593\) 15.5441 0.638320 0.319160 0.947701i \(-0.396599\pi\)
0.319160 + 0.947701i \(0.396599\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) −31.6970 −1.28654 −0.643271 0.765639i \(-0.722423\pi\)
−0.643271 + 0.765639i \(0.722423\pi\)
\(608\) 12.4787 0.506077
\(609\) 0 0
\(610\) 0 0
\(611\) −1.03943 −0.0420508
\(612\) 0 0
\(613\) −1.20394 −0.0486265 −0.0243133 0.999704i \(-0.507740\pi\)
−0.0243133 + 0.999704i \(0.507740\pi\)
\(614\) 43.7602 1.76602
\(615\) 0 0
\(616\) −0.460425 −0.0185511
\(617\) −37.9337 −1.52715 −0.763577 0.645716i \(-0.776559\pi\)
−0.763577 + 0.645716i \(0.776559\pi\)
\(618\) 0 0
\(619\) −4.60424 −0.185060 −0.0925299 0.995710i \(-0.529495\pi\)
−0.0925299 + 0.995710i \(0.529495\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.2062 1.21116
\(623\) 2.98932 0.119765
\(624\) 0 0
\(625\) 0 0
\(626\) −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
\(635\) 0 0
\(636\) 0 0
\(637\) −6.08452 −0.241077
\(638\) 0.447480 0.0177159
\(639\) 0 0
\(640\) 0 0
\(641\) −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 \(-0.486860\pi\)
0.0412686 + 0.999148i \(0.486860\pi\)
\(644\) −0.533431 −0.0210201
\(645\) 0 0
\(646\) −46.2868 −1.82113
\(647\) 45.1955 1.77682 0.888410 0.459051i \(-0.151811\pi\)
0.888410 + 0.459051i \(0.151811\pi\)
\(648\) 0 0
\(649\) 2.83710 0.111366
\(650\) 0 0
\(651\) 0 0
\(652\) −0.165166 −0.00646840
\(653\) −2.14834 −0.0840712 −0.0420356 0.999116i \(-0.513384\pi\)
−0.0420356 + 0.999116i \(0.513384\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 0 0
\(667\) −2.29072 −0.0886972
\(668\) 7.33260 0.283707
\(669\) 0 0
\(670\) 0 0
\(671\) −0.894960 −0.0345496
\(672\) 0 0
\(673\) 17.4719 0.673491 0.336746 0.941596i \(-0.390674\pi\)
0.336746 + 0.941596i \(0.390674\pi\)
\(674\) −19.6826 −0.758146
\(675\) 0 0
\(676\) −4.48482 −0.172493
\(677\) −40.0372 −1.53875 −0.769377 0.638796i \(-0.779433\pi\)
−0.769377 + 0.638796i \(0.779433\pi\)
\(678\) 0 0
\(679\) −10.0267 −0.384788
\(680\) 0 0
\(681\) 0 0
\(682\) −4.49693 −0.172196
\(683\) 2.07611 0.0794402 0.0397201 0.999211i \(-0.487353\pi\)
0.0397201 + 0.999211i \(0.487353\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 0 0
\(697\) −1.69102 −0.0640521
\(698\) 34.1666 1.29322
\(699\) 0 0
\(700\) 0 0
\(701\) 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 \(0.527578\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 11.8941 0.445751
\(713\) 23.0205 0.862125
\(714\) 0 0
\(715\) 0 0
\(716\) −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
\(725\) 0 0
\(726\) 0 0
\(727\) 26.9711 1.00030 0.500151 0.865938i \(-0.333278\pi\)
0.500151 + 0.865938i \(0.333278\pi\)
\(728\) 1.45959 0.0540960
\(729\) 0 0
\(730\) 0 0
\(731\) 28.3812 1.04972
\(732\) 0 0
\(733\) 30.0638 1.11043 0.555216 0.831706i \(-0.312635\pi\)
0.555216 + 0.831706i \(0.312635\pi\)
\(734\) −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 0
\(741\) 0 0
\(742\) 0.330332 0.0121269
\(743\) 11.1857 0.410363 0.205181 0.978724i \(-0.434222\pi\)
0.205181 + 0.978724i \(0.434222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −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
\(755\) 0 0
\(756\) 0 0
\(757\) −15.8927 −0.577630 −0.288815 0.957385i \(-0.593261\pi\)
−0.288815 + 0.957385i \(0.593261\pi\)
\(758\) 21.7548 0.790172
\(759\) 0 0
\(760\) 0 0
\(761\) −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 0
\(771\) 0 0
\(772\) −6.59439 −0.237337
\(773\) −0.488518 −0.0175708 −0.00878539 0.999961i \(-0.502797\pi\)
−0.00878539 + 0.999961i \(0.502797\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 0 0
\(787\) 1.99159 0.0709925 0.0354962 0.999370i \(-0.488699\pi\)
0.0354962 + 0.999370i \(0.488699\pi\)
\(788\) 2.25112 0.0801927
\(789\) 0 0
\(790\) 0 0
\(791\) −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 \(-0.598948\pi\)
−0.305871 + 0.952073i \(0.598948\pi\)
\(798\) 0 0
\(799\) −5.60650 −0.198344
\(800\) 0 0
\(801\) 0 0
\(802\) −30.4124 −1.07390
\(803\) −2.01891 −0.0712458
\(804\) 0 0
\(805\) 0 0
\(806\) 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 0
\(816\) 0 0
\(817\) −34.5380 −1.20833
\(818\) −2.64384 −0.0924398
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.617752\pi\)
−0.361548 + 0.932353i \(0.617752\pi\)
\(824\) −19.7458 −0.687877
\(825\) 0 0
\(826\) 9.47641 0.329726
\(827\) −9.12783 −0.317406 −0.158703 0.987326i \(-0.550731\pi\)
−0.158703 + 0.987326i \(0.550731\pi\)
\(828\) 0 0
\(829\) 31.8576 1.10646 0.553230 0.833028i \(-0.313395\pi\)
0.553230 + 0.833028i \(0.313395\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.55640 0.192633
\(833\) −32.8188 −1.13711
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 0 0
\(847\) 6.88655 0.236625
\(848\) 1.56547 0.0537583
\(849\) 0 0
\(850\) 0 0
\(851\) −3.55640 −0.121912
\(852\) 0 0
\(853\) 56.0515 1.91917 0.959584 0.281422i \(-0.0908061\pi\)
0.959584 + 0.281422i \(0.0908061\pi\)
\(854\) −2.98932 −0.102292
\(855\) 0 0
\(856\) −31.9551 −1.09220
\(857\) −6.08452 −0.207843 −0.103922 0.994585i \(-0.533139\pi\)
−0.103922 + 0.994585i \(0.533139\pi\)
\(858\) 0 0
\(859\) 35.5936 1.21444 0.607218 0.794535i \(-0.292285\pi\)
0.607218 + 0.794535i \(0.292285\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −30.5236 −1.03964
\(863\) −12.8287 −0.436694 −0.218347 0.975871i \(-0.570066\pi\)
−0.218347 + 0.975871i \(0.570066\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −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 0
\(876\) 0 0
\(877\) 1.50307 0.0507551 0.0253776 0.999678i \(-0.491921\pi\)
0.0253776 + 0.999678i \(0.491921\pi\)
\(878\) −27.4863 −0.927616
\(879\) 0 0
\(880\) 0 0
\(881\) −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 \(-0.337474\pi\)
0.488693 + 0.872456i \(0.337474\pi\)
\(884\) 1.69102 0.0568753
\(885\) 0 0
\(886\) 51.7068 1.73712
\(887\) −19.0700 −0.640307 −0.320153 0.947366i \(-0.603734\pi\)
−0.320153 + 0.947366i \(0.603734\pi\)
\(888\) 0 0
\(889\) −13.1941 −0.442516
\(890\) 0 0
\(891\) 0 0
\(892\) −4.04945 −0.135586
\(893\) 6.82273 0.228314
\(894\) 0 0
\(895\) 0 0
\(896\) −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
\(905\) 0 0
\(906\) 0 0
\(907\) −5.54023 −0.183960 −0.0919802 0.995761i \(-0.529320\pi\)
−0.0919802 + 0.995761i \(0.529320\pi\)
\(908\) −4.63090 −0.153682
\(909\) 0 0
\(910\) 0 0
\(911\) −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 \(0.287139\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 49.7296 1.63776
\(923\) −8.36683 −0.275398
\(924\) 0 0
\(925\) 0 0
\(926\) −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
\(935\) 0 0
\(936\) 0 0
\(937\) 22.8638 0.746927 0.373463 0.927645i \(-0.378170\pi\)
0.373463 + 0.927645i \(0.378170\pi\)
\(938\) −5.54411 −0.181022
\(939\) 0 0
\(940\) 0 0
\(941\) −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 \(-0.552068\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(948\) 0 0
\(949\) 6.40012 0.207757
\(950\) 0 0
\(951\) 0 0
\(952\) 7.87277 0.255158
\(953\) −8.15676 −0.264223 −0.132112 0.991235i \(-0.542176\pi\)
−0.132112 + 0.991235i \(0.542176\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −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
\(965\) 0 0
\(966\) 0 0
\(967\) −15.7671 −0.507037 −0.253518 0.967331i \(-0.581588\pi\)
−0.253518 + 0.967331i \(0.581588\pi\)
\(968\) 27.4007 0.880691
\(969\) 0 0
\(970\) 0 0
\(971\) 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 \(-0.843477\pi\)
−0.881517 + 0.472153i \(0.843477\pi\)
\(978\) 0 0
\(979\) 1.37751 0.0440255
\(980\) 0 0
\(981\) 0 0
\(982\) −62.7540 −2.00256
\(983\) −1.29687 −0.0413637 −0.0206818 0.999786i \(-0.506584\pi\)
−0.0206818 + 0.999786i \(0.506584\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) 0 0
\(997\) −30.2472 −0.957940 −0.478970 0.877831i \(-0.658990\pi\)
−0.478970 + 0.877831i \(0.658990\pi\)
\(998\) −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 6525.2.a.bh.1.2 3
3.2 odd 2 725.2.a.d.1.2 3
5.4 even 2 1305.2.a.o.1.2 3
15.2 even 4 725.2.b.d.349.2 6
15.8 even 4 725.2.b.d.349.5 6
15.14 odd 2 145.2.a.d.1.2 3
60.59 even 2 2320.2.a.s.1.1 3
105.104 even 2 7105.2.a.p.1.2 3
120.29 odd 2 9280.2.a.bu.1.1 3
120.59 even 2 9280.2.a.bm.1.3 3
435.434 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 15.14 odd 2
725.2.a.d.1.2 3 3.2 odd 2
725.2.b.d.349.2 6 15.2 even 4
725.2.b.d.349.5 6 15.8 even 4
1305.2.a.o.1.2 3 5.4 even 2
2320.2.a.s.1.1 3 60.59 even 2
4205.2.a.e.1.2 3 435.434 odd 2
6525.2.a.bh.1.2 3 1.1 even 1 trivial
7105.2.a.p.1.2 3 105.104 even 2
9280.2.a.bm.1.3 3 120.59 even 2
9280.2.a.bu.1.1 3 120.29 odd 2