Properties

Label 7605.2.a.bw.1.1
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
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] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.26180\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} +O(q^{10})\) \(q-2.26180 q^{2} +3.11575 q^{4} +1.00000 q^{5} +1.26180 q^{7} -2.52360 q^{8} -2.26180 q^{10} +4.52360 q^{11} -2.85395 q^{14} -0.523604 q^{16} +4.49330 q^{17} -5.11575 q^{19} +3.11575 q^{20} -10.2315 q^{22} -2.23150 q^{23} +1.00000 q^{25} +3.93146 q^{28} +1.37755 q^{29} -8.87085 q^{31} +6.23150 q^{32} -10.1630 q^{34} +1.26180 q^{35} +0.231499 q^{37} +11.5708 q^{38} -2.52360 q^{40} +1.14605 q^{41} +6.37755 q^{43} +14.0944 q^{44} +5.04721 q^{46} -10.7854 q^{47} -5.40786 q^{49} -2.26180 q^{50} -4.52360 q^{53} +4.52360 q^{55} -3.18429 q^{56} -3.11575 q^{58} -0.853947 q^{59} +4.63935 q^{61} +20.0641 q^{62} -13.0472 q^{64} -13.1327 q^{67} +14.0000 q^{68} -2.85395 q^{70} -9.60905 q^{71} -13.7854 q^{73} -0.523604 q^{74} -15.9394 q^{76} +5.70789 q^{77} -8.87085 q^{79} -0.523604 q^{80} -2.59214 q^{82} -8.23150 q^{83} +4.49330 q^{85} -14.4248 q^{86} -11.4158 q^{88} +6.62245 q^{89} -6.95279 q^{92} +24.3945 q^{94} -5.11575 q^{95} -10.6697 q^{97} +12.2315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40} + 9 q^{43} - 12 q^{44} - 12 q^{46} - 12 q^{47} - 6 q^{49} - 30 q^{56} - 6 q^{58} - 6 q^{59} - 3 q^{61} + 6 q^{62} - 12 q^{64} - 9 q^{67} + 42 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 12 q^{74} - 48 q^{76} + 24 q^{77} - 3 q^{79} + 12 q^{80} - 18 q^{82} - 18 q^{83} - 6 q^{86} - 48 q^{88} + 30 q^{89} - 48 q^{92} + 36 q^{94} - 12 q^{95} - 15 q^{97} + 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.26180 −1.59934 −0.799668 0.600443i \(-0.794991\pi\)
−0.799668 + 0.600443i \(0.794991\pi\)
\(3\) 0 0
\(4\) 3.11575 1.55787
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.26180 0.476916 0.238458 0.971153i \(-0.423358\pi\)
0.238458 + 0.971153i \(0.423358\pi\)
\(8\) −2.52360 −0.892229
\(9\) 0 0
\(10\) −2.26180 −0.715245
\(11\) 4.52360 1.36392 0.681959 0.731390i \(-0.261128\pi\)
0.681959 + 0.731390i \(0.261128\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −2.85395 −0.762749
\(15\) 0 0
\(16\) −0.523604 −0.130901
\(17\) 4.49330 1.08979 0.544893 0.838506i \(-0.316570\pi\)
0.544893 + 0.838506i \(0.316570\pi\)
\(18\) 0 0
\(19\) −5.11575 −1.17363 −0.586817 0.809720i \(-0.699619\pi\)
−0.586817 + 0.809720i \(0.699619\pi\)
\(20\) 3.11575 0.696703
\(21\) 0 0
\(22\) −10.2315 −2.18136
\(23\) −2.23150 −0.465300 −0.232650 0.972561i \(-0.574740\pi\)
−0.232650 + 0.972561i \(0.574740\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 3.93146 0.742976
\(29\) 1.37755 0.255805 0.127902 0.991787i \(-0.459176\pi\)
0.127902 + 0.991787i \(0.459176\pi\)
\(30\) 0 0
\(31\) −8.87085 −1.59325 −0.796626 0.604472i \(-0.793384\pi\)
−0.796626 + 0.604472i \(0.793384\pi\)
\(32\) 6.23150 1.10158
\(33\) 0 0
\(34\) −10.1630 −1.74293
\(35\) 1.26180 0.213284
\(36\) 0 0
\(37\) 0.231499 0.0380582 0.0190291 0.999819i \(-0.493942\pi\)
0.0190291 + 0.999819i \(0.493942\pi\)
\(38\) 11.5708 1.87703
\(39\) 0 0
\(40\) −2.52360 −0.399017
\(41\) 1.14605 0.178983 0.0894917 0.995988i \(-0.471476\pi\)
0.0894917 + 0.995988i \(0.471476\pi\)
\(42\) 0 0
\(43\) 6.37755 0.972568 0.486284 0.873801i \(-0.338352\pi\)
0.486284 + 0.873801i \(0.338352\pi\)
\(44\) 14.0944 2.12481
\(45\) 0 0
\(46\) 5.04721 0.744170
\(47\) −10.7854 −1.57321 −0.786607 0.617454i \(-0.788164\pi\)
−0.786607 + 0.617454i \(0.788164\pi\)
\(48\) 0 0
\(49\) −5.40786 −0.772551
\(50\) −2.26180 −0.319867
\(51\) 0 0
\(52\) 0 0
\(53\) −4.52360 −0.621365 −0.310682 0.950514i \(-0.600558\pi\)
−0.310682 + 0.950514i \(0.600558\pi\)
\(54\) 0 0
\(55\) 4.52360 0.609963
\(56\) −3.18429 −0.425519
\(57\) 0 0
\(58\) −3.11575 −0.409118
\(59\) −0.853947 −0.111174 −0.0555872 0.998454i \(-0.517703\pi\)
−0.0555872 + 0.998454i \(0.517703\pi\)
\(60\) 0 0
\(61\) 4.63935 0.594008 0.297004 0.954876i \(-0.404012\pi\)
0.297004 + 0.954876i \(0.404012\pi\)
\(62\) 20.0641 2.54815
\(63\) 0 0
\(64\) −13.0472 −1.63090
\(65\) 0 0
\(66\) 0 0
\(67\) −13.1327 −1.60441 −0.802205 0.597049i \(-0.796340\pi\)
−0.802205 + 0.597049i \(0.796340\pi\)
\(68\) 14.0000 1.69775
\(69\) 0 0
\(70\) −2.85395 −0.341112
\(71\) −9.60905 −1.14038 −0.570192 0.821511i \(-0.693131\pi\)
−0.570192 + 0.821511i \(0.693131\pi\)
\(72\) 0 0
\(73\) −13.7854 −1.61346 −0.806730 0.590920i \(-0.798765\pi\)
−0.806730 + 0.590920i \(0.798765\pi\)
\(74\) −0.523604 −0.0608678
\(75\) 0 0
\(76\) −15.9394 −1.82837
\(77\) 5.70789 0.650475
\(78\) 0 0
\(79\) −8.87085 −0.998049 −0.499024 0.866588i \(-0.666308\pi\)
−0.499024 + 0.866588i \(0.666308\pi\)
\(80\) −0.523604 −0.0585408
\(81\) 0 0
\(82\) −2.59214 −0.286255
\(83\) −8.23150 −0.903524 −0.451762 0.892138i \(-0.649204\pi\)
−0.451762 + 0.892138i \(0.649204\pi\)
\(84\) 0 0
\(85\) 4.49330 0.487367
\(86\) −14.4248 −1.55546
\(87\) 0 0
\(88\) −11.4158 −1.21693
\(89\) 6.62245 0.701978 0.350989 0.936380i \(-0.385845\pi\)
0.350989 + 0.936380i \(0.385845\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.95279 −0.724879
\(93\) 0 0
\(94\) 24.3945 2.51610
\(95\) −5.11575 −0.524865
\(96\) 0 0
\(97\) −10.6697 −1.08334 −0.541670 0.840591i \(-0.682208\pi\)
−0.541670 + 0.840591i \(0.682208\pi\)
\(98\) 12.2315 1.23557
\(99\) 0 0
\(100\) 3.11575 0.311575
\(101\) 17.0472 1.69626 0.848130 0.529788i \(-0.177728\pi\)
0.848130 + 0.529788i \(0.177728\pi\)
\(102\) 0 0
\(103\) −13.9484 −1.37437 −0.687187 0.726481i \(-0.741155\pi\)
−0.687187 + 0.726481i \(0.741155\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.2315 0.993771
\(107\) −12.4933 −1.20777 −0.603886 0.797070i \(-0.706382\pi\)
−0.603886 + 0.797070i \(0.706382\pi\)
\(108\) 0 0
\(109\) 2.27871 0.218261 0.109130 0.994027i \(-0.465193\pi\)
0.109130 + 0.994027i \(0.465193\pi\)
\(110\) −10.2315 −0.975535
\(111\) 0 0
\(112\) −0.660685 −0.0624289
\(113\) 1.47640 0.138888 0.0694438 0.997586i \(-0.477878\pi\)
0.0694438 + 0.997586i \(0.477878\pi\)
\(114\) 0 0
\(115\) −2.23150 −0.208088
\(116\) 4.29211 0.398512
\(117\) 0 0
\(118\) 1.93146 0.177805
\(119\) 5.66966 0.519737
\(120\) 0 0
\(121\) 9.46300 0.860273
\(122\) −10.4933 −0.950019
\(123\) 0 0
\(124\) −27.6394 −2.48209
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −17.7854 −1.57820 −0.789100 0.614265i \(-0.789453\pi\)
−0.789100 + 0.614265i \(0.789453\pi\)
\(128\) 17.0472 1.50677
\(129\) 0 0
\(130\) 0 0
\(131\) −11.6697 −1.01958 −0.509791 0.860298i \(-0.670277\pi\)
−0.509791 + 0.860298i \(0.670277\pi\)
\(132\) 0 0
\(133\) −6.45506 −0.559725
\(134\) 29.7035 2.56599
\(135\) 0 0
\(136\) −11.3393 −0.972338
\(137\) −20.0641 −1.71419 −0.857096 0.515156i \(-0.827734\pi\)
−0.857096 + 0.515156i \(0.827734\pi\)
\(138\) 0 0
\(139\) 16.3393 1.38588 0.692941 0.720994i \(-0.256314\pi\)
0.692941 + 0.720994i \(0.256314\pi\)
\(140\) 3.93146 0.332269
\(141\) 0 0
\(142\) 21.7338 1.82386
\(143\) 0 0
\(144\) 0 0
\(145\) 1.37755 0.114399
\(146\) 31.1799 2.58046
\(147\) 0 0
\(148\) 0.721292 0.0592899
\(149\) −3.24490 −0.265832 −0.132916 0.991127i \(-0.542434\pi\)
−0.132916 + 0.991127i \(0.542434\pi\)
\(150\) 0 0
\(151\) −5.69996 −0.463856 −0.231928 0.972733i \(-0.574503\pi\)
−0.231928 + 0.972733i \(0.574503\pi\)
\(152\) 12.9101 1.04715
\(153\) 0 0
\(154\) −12.9101 −1.04033
\(155\) −8.87085 −0.712524
\(156\) 0 0
\(157\) −3.85395 −0.307578 −0.153789 0.988104i \(-0.549148\pi\)
−0.153789 + 0.988104i \(0.549148\pi\)
\(158\) 20.0641 1.59622
\(159\) 0 0
\(160\) 6.23150 0.492643
\(161\) −2.81571 −0.221909
\(162\) 0 0
\(163\) 9.72480 0.761705 0.380853 0.924636i \(-0.375631\pi\)
0.380853 + 0.924636i \(0.375631\pi\)
\(164\) 3.57081 0.278834
\(165\) 0 0
\(166\) 18.6180 1.44504
\(167\) −19.2787 −1.49183 −0.745916 0.666040i \(-0.767988\pi\)
−0.745916 + 0.666040i \(0.767988\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −10.1630 −0.779463
\(171\) 0 0
\(172\) 19.8709 1.51514
\(173\) 3.01691 0.229371 0.114686 0.993402i \(-0.463414\pi\)
0.114686 + 0.993402i \(0.463414\pi\)
\(174\) 0 0
\(175\) 1.26180 0.0953833
\(176\) −2.36858 −0.178538
\(177\) 0 0
\(178\) −14.9787 −1.12270
\(179\) 5.31694 0.397407 0.198704 0.980060i \(-0.436327\pi\)
0.198704 + 0.980060i \(0.436327\pi\)
\(180\) 0 0
\(181\) 25.8709 1.92297 0.961483 0.274866i \(-0.0886333\pi\)
0.961483 + 0.274866i \(0.0886333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.63142 0.415154
\(185\) 0.231499 0.0170201
\(186\) 0 0
\(187\) 20.3259 1.48638
\(188\) −33.6046 −2.45087
\(189\) 0 0
\(190\) 11.5708 0.839435
\(191\) −10.6563 −0.771060 −0.385530 0.922695i \(-0.625981\pi\)
−0.385530 + 0.922695i \(0.625981\pi\)
\(192\) 0 0
\(193\) −7.98309 −0.574636 −0.287318 0.957835i \(-0.592764\pi\)
−0.287318 + 0.957835i \(0.592764\pi\)
\(194\) 24.1327 1.73262
\(195\) 0 0
\(196\) −16.8495 −1.20354
\(197\) 17.3393 1.23538 0.617688 0.786424i \(-0.288070\pi\)
0.617688 + 0.786424i \(0.288070\pi\)
\(198\) 0 0
\(199\) −1.23150 −0.0872986 −0.0436493 0.999047i \(-0.513898\pi\)
−0.0436493 + 0.999047i \(0.513898\pi\)
\(200\) −2.52360 −0.178446
\(201\) 0 0
\(202\) −38.5574 −2.71289
\(203\) 1.73820 0.121998
\(204\) 0 0
\(205\) 1.14605 0.0800438
\(206\) 31.5484 2.19808
\(207\) 0 0
\(208\) 0 0
\(209\) −23.1416 −1.60074
\(210\) 0 0
\(211\) 10.3393 0.711788 0.355894 0.934526i \(-0.384176\pi\)
0.355894 + 0.934526i \(0.384176\pi\)
\(212\) −14.0944 −0.968009
\(213\) 0 0
\(214\) 28.2574 1.93163
\(215\) 6.37755 0.434945
\(216\) 0 0
\(217\) −11.1933 −0.759848
\(218\) −5.15399 −0.349072
\(219\) 0 0
\(220\) 14.0944 0.950245
\(221\) 0 0
\(222\) 0 0
\(223\) −21.5708 −1.44449 −0.722244 0.691638i \(-0.756889\pi\)
−0.722244 + 0.691638i \(0.756889\pi\)
\(224\) 7.86292 0.525363
\(225\) 0 0
\(226\) −3.33931 −0.222128
\(227\) 5.44609 0.361470 0.180735 0.983532i \(-0.442152\pi\)
0.180735 + 0.983532i \(0.442152\pi\)
\(228\) 0 0
\(229\) −21.9315 −1.44927 −0.724636 0.689132i \(-0.757992\pi\)
−0.724636 + 0.689132i \(0.757992\pi\)
\(230\) 5.04721 0.332803
\(231\) 0 0
\(232\) −3.47640 −0.228237
\(233\) −25.3125 −1.65828 −0.829139 0.559042i \(-0.811169\pi\)
−0.829139 + 0.559042i \(0.811169\pi\)
\(234\) 0 0
\(235\) −10.7854 −0.703562
\(236\) −2.66069 −0.173196
\(237\) 0 0
\(238\) −12.8236 −0.831233
\(239\) 22.5236 1.45693 0.728465 0.685083i \(-0.240234\pi\)
0.728465 + 0.685083i \(0.240234\pi\)
\(240\) 0 0
\(241\) −11.1157 −0.716028 −0.358014 0.933716i \(-0.616546\pi\)
−0.358014 + 0.933716i \(0.616546\pi\)
\(242\) −21.4034 −1.37586
\(243\) 0 0
\(244\) 14.4551 0.925391
\(245\) −5.40786 −0.345495
\(246\) 0 0
\(247\) 0 0
\(248\) 22.3865 1.42155
\(249\) 0 0
\(250\) −2.26180 −0.143049
\(251\) −11.0472 −0.697294 −0.348647 0.937254i \(-0.613359\pi\)
−0.348647 + 0.937254i \(0.613359\pi\)
\(252\) 0 0
\(253\) −10.0944 −0.634631
\(254\) 40.2271 2.52407
\(255\) 0 0
\(256\) −12.4630 −0.778937
\(257\) 2.03030 0.126647 0.0633234 0.997993i \(-0.479830\pi\)
0.0633234 + 0.997993i \(0.479830\pi\)
\(258\) 0 0
\(259\) 0.292106 0.0181506
\(260\) 0 0
\(261\) 0 0
\(262\) 26.3945 1.63066
\(263\) −3.24840 −0.200305 −0.100153 0.994972i \(-0.531933\pi\)
−0.100153 + 0.994972i \(0.531933\pi\)
\(264\) 0 0
\(265\) −4.52360 −0.277883
\(266\) 14.6001 0.895188
\(267\) 0 0
\(268\) −40.9181 −2.49947
\(269\) −26.9484 −1.64307 −0.821535 0.570157i \(-0.806882\pi\)
−0.821535 + 0.570157i \(0.806882\pi\)
\(270\) 0 0
\(271\) 19.8495 1.20577 0.602886 0.797827i \(-0.294017\pi\)
0.602886 + 0.797827i \(0.294017\pi\)
\(272\) −2.35271 −0.142654
\(273\) 0 0
\(274\) 45.3811 2.74157
\(275\) 4.52360 0.272784
\(276\) 0 0
\(277\) 16.0944 0.967020 0.483510 0.875339i \(-0.339362\pi\)
0.483510 + 0.875339i \(0.339362\pi\)
\(278\) −36.9563 −2.21649
\(279\) 0 0
\(280\) −3.18429 −0.190298
\(281\) −5.37755 −0.320798 −0.160399 0.987052i \(-0.551278\pi\)
−0.160399 + 0.987052i \(0.551278\pi\)
\(282\) 0 0
\(283\) 26.8664 1.59704 0.798522 0.601966i \(-0.205616\pi\)
0.798522 + 0.601966i \(0.205616\pi\)
\(284\) −29.9394 −1.77658
\(285\) 0 0
\(286\) 0 0
\(287\) 1.44609 0.0853601
\(288\) 0 0
\(289\) 3.18975 0.187633
\(290\) −3.11575 −0.182963
\(291\) 0 0
\(292\) −42.9519 −2.51357
\(293\) 21.4193 1.25133 0.625664 0.780092i \(-0.284828\pi\)
0.625664 + 0.780092i \(0.284828\pi\)
\(294\) 0 0
\(295\) −0.853947 −0.0497187
\(296\) −0.584211 −0.0339566
\(297\) 0 0
\(298\) 7.33931 0.425155
\(299\) 0 0
\(300\) 0 0
\(301\) 8.04721 0.463833
\(302\) 12.8922 0.741862
\(303\) 0 0
\(304\) 2.67863 0.153630
\(305\) 4.63935 0.265649
\(306\) 0 0
\(307\) 7.85395 0.448248 0.224124 0.974561i \(-0.428048\pi\)
0.224124 + 0.974561i \(0.428048\pi\)
\(308\) 17.7844 1.01336
\(309\) 0 0
\(310\) 20.0641 1.13957
\(311\) 27.8744 1.58061 0.790305 0.612714i \(-0.209922\pi\)
0.790305 + 0.612714i \(0.209922\pi\)
\(312\) 0 0
\(313\) 7.88776 0.445842 0.222921 0.974836i \(-0.428441\pi\)
0.222921 + 0.974836i \(0.428441\pi\)
\(314\) 8.71687 0.491921
\(315\) 0 0
\(316\) −27.6394 −1.55484
\(317\) 13.2181 0.742403 0.371201 0.928552i \(-0.378946\pi\)
0.371201 + 0.928552i \(0.378946\pi\)
\(318\) 0 0
\(319\) 6.23150 0.348897
\(320\) −13.0472 −0.729361
\(321\) 0 0
\(322\) 6.36858 0.354907
\(323\) −22.9866 −1.27901
\(324\) 0 0
\(325\) 0 0
\(326\) −21.9956 −1.21822
\(327\) 0 0
\(328\) −2.89218 −0.159694
\(329\) −13.6091 −0.750291
\(330\) 0 0
\(331\) 23.8495 1.31089 0.655444 0.755244i \(-0.272481\pi\)
0.655444 + 0.755244i \(0.272481\pi\)
\(332\) −25.6473 −1.40758
\(333\) 0 0
\(334\) 43.6046 2.38594
\(335\) −13.1327 −0.717514
\(336\) 0 0
\(337\) 5.68306 0.309576 0.154788 0.987948i \(-0.450531\pi\)
0.154788 + 0.987948i \(0.450531\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 14.0000 0.759257
\(341\) −40.1282 −2.17307
\(342\) 0 0
\(343\) −15.6563 −0.845359
\(344\) −16.0944 −0.867753
\(345\) 0 0
\(346\) −6.82364 −0.366841
\(347\) 21.4496 1.15147 0.575737 0.817635i \(-0.304715\pi\)
0.575737 + 0.817635i \(0.304715\pi\)
\(348\) 0 0
\(349\) −6.87085 −0.367788 −0.183894 0.982946i \(-0.558870\pi\)
−0.183894 + 0.982946i \(0.558870\pi\)
\(350\) −2.85395 −0.152550
\(351\) 0 0
\(352\) 28.1888 1.50247
\(353\) 34.4968 1.83608 0.918040 0.396488i \(-0.129771\pi\)
0.918040 + 0.396488i \(0.129771\pi\)
\(354\) 0 0
\(355\) −9.60905 −0.509995
\(356\) 20.6339 1.09359
\(357\) 0 0
\(358\) −12.0259 −0.635587
\(359\) −8.15945 −0.430639 −0.215320 0.976544i \(-0.569079\pi\)
−0.215320 + 0.976544i \(0.569079\pi\)
\(360\) 0 0
\(361\) 7.17089 0.377415
\(362\) −58.5148 −3.07547
\(363\) 0 0
\(364\) 0 0
\(365\) −13.7854 −0.721561
\(366\) 0 0
\(367\) −6.84055 −0.357074 −0.178537 0.983933i \(-0.557136\pi\)
−0.178537 + 0.983933i \(0.557136\pi\)
\(368\) 1.16842 0.0609082
\(369\) 0 0
\(370\) −0.523604 −0.0272209
\(371\) −5.70789 −0.296339
\(372\) 0 0
\(373\) 13.8192 0.715532 0.357766 0.933811i \(-0.383539\pi\)
0.357766 + 0.933811i \(0.383539\pi\)
\(374\) −45.9732 −2.37722
\(375\) 0 0
\(376\) 27.2181 1.40367
\(377\) 0 0
\(378\) 0 0
\(379\) −11.5157 −0.591520 −0.295760 0.955262i \(-0.595573\pi\)
−0.295760 + 0.955262i \(0.595573\pi\)
\(380\) −15.9394 −0.817674
\(381\) 0 0
\(382\) 24.1024 1.23318
\(383\) 22.8798 1.16910 0.584552 0.811356i \(-0.301270\pi\)
0.584552 + 0.811356i \(0.301270\pi\)
\(384\) 0 0
\(385\) 5.70789 0.290901
\(386\) 18.0562 0.919035
\(387\) 0 0
\(388\) −33.2440 −1.68771
\(389\) 4.35271 0.220691 0.110346 0.993893i \(-0.464804\pi\)
0.110346 + 0.993893i \(0.464804\pi\)
\(390\) 0 0
\(391\) −10.0268 −0.507077
\(392\) 13.6473 0.689292
\(393\) 0 0
\(394\) −39.2181 −1.97578
\(395\) −8.87085 −0.446341
\(396\) 0 0
\(397\) −7.62245 −0.382560 −0.191280 0.981536i \(-0.561264\pi\)
−0.191280 + 0.981536i \(0.561264\pi\)
\(398\) 2.78541 0.139620
\(399\) 0 0
\(400\) −0.523604 −0.0261802
\(401\) 20.8495 1.04118 0.520588 0.853808i \(-0.325713\pi\)
0.520588 + 0.853808i \(0.325713\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 53.1148 2.64256
\(405\) 0 0
\(406\) −3.93146 −0.195115
\(407\) 1.04721 0.0519082
\(408\) 0 0
\(409\) 16.9653 0.838879 0.419439 0.907783i \(-0.362227\pi\)
0.419439 + 0.907783i \(0.362227\pi\)
\(410\) −2.59214 −0.128017
\(411\) 0 0
\(412\) −43.4596 −2.14110
\(413\) −1.07751 −0.0530209
\(414\) 0 0
\(415\) −8.23150 −0.404068
\(416\) 0 0
\(417\) 0 0
\(418\) 52.3418 2.56012
\(419\) −2.09884 −0.102535 −0.0512676 0.998685i \(-0.516326\pi\)
−0.0512676 + 0.998685i \(0.516326\pi\)
\(420\) 0 0
\(421\) 39.6598 1.93290 0.966449 0.256857i \(-0.0826870\pi\)
0.966449 + 0.256857i \(0.0826870\pi\)
\(422\) −23.3855 −1.13839
\(423\) 0 0
\(424\) 11.4158 0.554400
\(425\) 4.49330 0.217957
\(426\) 0 0
\(427\) 5.85395 0.283292
\(428\) −38.9260 −1.88156
\(429\) 0 0
\(430\) −14.4248 −0.695624
\(431\) −22.1282 −1.06588 −0.532940 0.846153i \(-0.678913\pi\)
−0.532940 + 0.846153i \(0.678913\pi\)
\(432\) 0 0
\(433\) −31.3562 −1.50688 −0.753442 0.657515i \(-0.771608\pi\)
−0.753442 + 0.657515i \(0.771608\pi\)
\(434\) 25.3169 1.21525
\(435\) 0 0
\(436\) 7.09988 0.340023
\(437\) 11.4158 0.546091
\(438\) 0 0
\(439\) 4.31344 0.205869 0.102935 0.994688i \(-0.467177\pi\)
0.102935 + 0.994688i \(0.467177\pi\)
\(440\) −11.4158 −0.544226
\(441\) 0 0
\(442\) 0 0
\(443\) −0.493301 −0.0234374 −0.0117187 0.999931i \(-0.503730\pi\)
−0.0117187 + 0.999931i \(0.503730\pi\)
\(444\) 0 0
\(445\) 6.62245 0.313934
\(446\) 48.7889 2.31022
\(447\) 0 0
\(448\) −16.4630 −0.777804
\(449\) −23.3125 −1.10019 −0.550093 0.835103i \(-0.685408\pi\)
−0.550093 + 0.835103i \(0.685408\pi\)
\(450\) 0 0
\(451\) 5.18429 0.244119
\(452\) 4.60008 0.216369
\(453\) 0 0
\(454\) −12.3180 −0.578112
\(455\) 0 0
\(456\) 0 0
\(457\) −6.66966 −0.311993 −0.155997 0.987758i \(-0.549859\pi\)
−0.155997 + 0.987758i \(0.549859\pi\)
\(458\) 49.6046 2.31787
\(459\) 0 0
\(460\) −6.95279 −0.324176
\(461\) 29.1193 1.35622 0.678110 0.734961i \(-0.262800\pi\)
0.678110 + 0.734961i \(0.262800\pi\)
\(462\) 0 0
\(463\) 1.79334 0.0833436 0.0416718 0.999131i \(-0.486732\pi\)
0.0416718 + 0.999131i \(0.486732\pi\)
\(464\) −0.721292 −0.0334852
\(465\) 0 0
\(466\) 57.2519 2.65214
\(467\) −16.8192 −0.778301 −0.389150 0.921174i \(-0.627231\pi\)
−0.389150 + 0.921174i \(0.627231\pi\)
\(468\) 0 0
\(469\) −16.5708 −0.765169
\(470\) 24.3945 1.12523
\(471\) 0 0
\(472\) 2.15502 0.0991931
\(473\) 28.8495 1.32650
\(474\) 0 0
\(475\) −5.11575 −0.234727
\(476\) 17.6652 0.809685
\(477\) 0 0
\(478\) −50.9439 −2.33012
\(479\) 35.1531 1.60618 0.803092 0.595855i \(-0.203187\pi\)
0.803092 + 0.595855i \(0.203187\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 25.1416 1.14517
\(483\) 0 0
\(484\) 29.4843 1.34020
\(485\) −10.6697 −0.484484
\(486\) 0 0
\(487\) 8.62596 0.390879 0.195440 0.980716i \(-0.437387\pi\)
0.195440 + 0.980716i \(0.437387\pi\)
\(488\) −11.7079 −0.529991
\(489\) 0 0
\(490\) 12.2315 0.552563
\(491\) 23.2137 1.04762 0.523809 0.851836i \(-0.324510\pi\)
0.523809 + 0.851836i \(0.324510\pi\)
\(492\) 0 0
\(493\) 6.18975 0.278773
\(494\) 0 0
\(495\) 0 0
\(496\) 4.64482 0.208558
\(497\) −12.1247 −0.543868
\(498\) 0 0
\(499\) −29.0393 −1.29998 −0.649988 0.759944i \(-0.725226\pi\)
−0.649988 + 0.759944i \(0.725226\pi\)
\(500\) 3.11575 0.139341
\(501\) 0 0
\(502\) 24.9866 1.11521
\(503\) 17.3999 0.775824 0.387912 0.921696i \(-0.373196\pi\)
0.387912 + 0.921696i \(0.373196\pi\)
\(504\) 0 0
\(505\) 17.0472 0.758591
\(506\) 22.8316 1.01499
\(507\) 0 0
\(508\) −55.4149 −2.45864
\(509\) −22.8763 −1.01397 −0.506987 0.861953i \(-0.669241\pi\)
−0.506987 + 0.861953i \(0.669241\pi\)
\(510\) 0 0
\(511\) −17.3945 −0.769485
\(512\) −5.90558 −0.260992
\(513\) 0 0
\(514\) −4.59214 −0.202551
\(515\) −13.9484 −0.614638
\(516\) 0 0
\(517\) −48.7889 −2.14573
\(518\) −0.660685 −0.0290288
\(519\) 0 0
\(520\) 0 0
\(521\) 26.3642 1.15503 0.577517 0.816378i \(-0.304022\pi\)
0.577517 + 0.816378i \(0.304022\pi\)
\(522\) 0 0
\(523\) −26.2921 −1.14967 −0.574837 0.818268i \(-0.694934\pi\)
−0.574837 + 0.818268i \(0.694934\pi\)
\(524\) −36.3597 −1.58838
\(525\) 0 0
\(526\) 7.34725 0.320355
\(527\) −39.8594 −1.73630
\(528\) 0 0
\(529\) −18.0204 −0.783496
\(530\) 10.2315 0.444428
\(531\) 0 0
\(532\) −20.1124 −0.871981
\(533\) 0 0
\(534\) 0 0
\(535\) −12.4933 −0.540133
\(536\) 33.1416 1.43150
\(537\) 0 0
\(538\) 60.9519 2.62782
\(539\) −24.4630 −1.05370
\(540\) 0 0
\(541\) 0.107816 0.00463537 0.00231768 0.999997i \(-0.499262\pi\)
0.00231768 + 0.999997i \(0.499262\pi\)
\(542\) −44.8957 −1.92844
\(543\) 0 0
\(544\) 28.0000 1.20049
\(545\) 2.27871 0.0976091
\(546\) 0 0
\(547\) 12.1193 0.518182 0.259091 0.965853i \(-0.416577\pi\)
0.259091 + 0.965853i \(0.416577\pi\)
\(548\) −62.5148 −2.67050
\(549\) 0 0
\(550\) −10.2315 −0.436273
\(551\) −7.04721 −0.300221
\(552\) 0 0
\(553\) −11.1933 −0.475986
\(554\) −36.4024 −1.54659
\(555\) 0 0
\(556\) 50.9092 2.15903
\(557\) 38.0070 1.61041 0.805204 0.592997i \(-0.202056\pi\)
0.805204 + 0.592997i \(0.202056\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.660685 −0.0279191
\(561\) 0 0
\(562\) 12.1630 0.513063
\(563\) −15.5102 −0.653677 −0.326839 0.945080i \(-0.605983\pi\)
−0.326839 + 0.945080i \(0.605983\pi\)
\(564\) 0 0
\(565\) 1.47640 0.0621124
\(566\) −60.7665 −2.55421
\(567\) 0 0
\(568\) 24.2494 1.01748
\(569\) −19.2405 −0.806602 −0.403301 0.915067i \(-0.632137\pi\)
−0.403301 + 0.915067i \(0.632137\pi\)
\(570\) 0 0
\(571\) 27.7338 1.16062 0.580311 0.814395i \(-0.302931\pi\)
0.580311 + 0.814395i \(0.302931\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.27077 −0.136519
\(575\) −2.23150 −0.0930599
\(576\) 0 0
\(577\) −26.9866 −1.12347 −0.561733 0.827318i \(-0.689865\pi\)
−0.561733 + 0.827318i \(0.689865\pi\)
\(578\) −7.21459 −0.300088
\(579\) 0 0
\(580\) 4.29211 0.178220
\(581\) −10.3865 −0.430906
\(582\) 0 0
\(583\) −20.4630 −0.847491
\(584\) 34.7889 1.43958
\(585\) 0 0
\(586\) −48.4462 −2.00129
\(587\) −30.8530 −1.27344 −0.636720 0.771095i \(-0.719709\pi\)
−0.636720 + 0.771095i \(0.719709\pi\)
\(588\) 0 0
\(589\) 45.3811 1.86989
\(590\) 1.93146 0.0795169
\(591\) 0 0
\(592\) −0.121214 −0.00498186
\(593\) 4.29211 0.176256 0.0881278 0.996109i \(-0.471912\pi\)
0.0881278 + 0.996109i \(0.471912\pi\)
\(594\) 0 0
\(595\) 5.66966 0.232433
\(596\) −10.1103 −0.414133
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0224 0.981527 0.490764 0.871293i \(-0.336718\pi\)
0.490764 + 0.871293i \(0.336718\pi\)
\(600\) 0 0
\(601\) 2.53154 0.103264 0.0516318 0.998666i \(-0.483558\pi\)
0.0516318 + 0.998666i \(0.483558\pi\)
\(602\) −18.2012 −0.741825
\(603\) 0 0
\(604\) −17.7596 −0.722630
\(605\) 9.46300 0.384726
\(606\) 0 0
\(607\) 11.4417 0.464403 0.232201 0.972668i \(-0.425407\pi\)
0.232201 + 0.972668i \(0.425407\pi\)
\(608\) −31.8788 −1.29286
\(609\) 0 0
\(610\) −10.4933 −0.424861
\(611\) 0 0
\(612\) 0 0
\(613\) 7.19326 0.290533 0.145267 0.989393i \(-0.453596\pi\)
0.145267 + 0.989393i \(0.453596\pi\)
\(614\) −17.7641 −0.716900
\(615\) 0 0
\(616\) −14.4045 −0.580373
\(617\) 25.5708 1.02944 0.514721 0.857358i \(-0.327895\pi\)
0.514721 + 0.857358i \(0.327895\pi\)
\(618\) 0 0
\(619\) −5.98660 −0.240622 −0.120311 0.992736i \(-0.538389\pi\)
−0.120311 + 0.992736i \(0.538389\pi\)
\(620\) −27.6394 −1.11002
\(621\) 0 0
\(622\) −63.0463 −2.52793
\(623\) 8.35622 0.334785
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −17.8405 −0.713052
\(627\) 0 0
\(628\) −12.0079 −0.479169
\(629\) 1.04019 0.0414752
\(630\) 0 0
\(631\) 43.0259 1.71283 0.856417 0.516285i \(-0.172686\pi\)
0.856417 + 0.516285i \(0.172686\pi\)
\(632\) 22.3865 0.890488
\(633\) 0 0
\(634\) −29.8967 −1.18735
\(635\) −17.7854 −0.705792
\(636\) 0 0
\(637\) 0 0
\(638\) −14.0944 −0.558003
\(639\) 0 0
\(640\) 17.0472 0.673850
\(641\) 6.69450 0.264417 0.132208 0.991222i \(-0.457793\pi\)
0.132208 + 0.991222i \(0.457793\pi\)
\(642\) 0 0
\(643\) −8.18078 −0.322619 −0.161309 0.986904i \(-0.551572\pi\)
−0.161309 + 0.986904i \(0.551572\pi\)
\(644\) −8.77305 −0.345706
\(645\) 0 0
\(646\) 51.9911 2.04556
\(647\) 11.2787 0.443412 0.221706 0.975114i \(-0.428838\pi\)
0.221706 + 0.975114i \(0.428838\pi\)
\(648\) 0 0
\(649\) −3.86292 −0.151633
\(650\) 0 0
\(651\) 0 0
\(652\) 30.3000 1.18664
\(653\) −27.2484 −1.06631 −0.533156 0.846017i \(-0.678994\pi\)
−0.533156 + 0.846017i \(0.678994\pi\)
\(654\) 0 0
\(655\) −11.6697 −0.455971
\(656\) −0.600078 −0.0234291
\(657\) 0 0
\(658\) 30.7810 1.19997
\(659\) −6.75510 −0.263141 −0.131571 0.991307i \(-0.542002\pi\)
−0.131571 + 0.991307i \(0.542002\pi\)
\(660\) 0 0
\(661\) 26.5102 1.03113 0.515564 0.856851i \(-0.327583\pi\)
0.515564 + 0.856851i \(0.327583\pi\)
\(662\) −53.9429 −2.09655
\(663\) 0 0
\(664\) 20.7730 0.806151
\(665\) −6.45506 −0.250317
\(666\) 0 0
\(667\) −3.07400 −0.119026
\(668\) −60.0676 −2.32409
\(669\) 0 0
\(670\) 29.7035 1.14755
\(671\) 20.9866 0.810179
\(672\) 0 0
\(673\) −0.677591 −0.0261192 −0.0130596 0.999915i \(-0.504157\pi\)
−0.0130596 + 0.999915i \(0.504157\pi\)
\(674\) −12.8539 −0.495116
\(675\) 0 0
\(676\) 0 0
\(677\) 10.4630 0.402126 0.201063 0.979578i \(-0.435560\pi\)
0.201063 + 0.979578i \(0.435560\pi\)
\(678\) 0 0
\(679\) −13.4630 −0.516662
\(680\) −11.3393 −0.434843
\(681\) 0 0
\(682\) 90.7621 3.47546
\(683\) −18.7516 −0.717510 −0.358755 0.933432i \(-0.616799\pi\)
−0.358755 + 0.933432i \(0.616799\pi\)
\(684\) 0 0
\(685\) −20.0641 −0.766610
\(686\) 35.4114 1.35201
\(687\) 0 0
\(688\) −3.33931 −0.127310
\(689\) 0 0
\(690\) 0 0
\(691\) −43.5653 −1.65730 −0.828652 0.559764i \(-0.810892\pi\)
−0.828652 + 0.559764i \(0.810892\pi\)
\(692\) 9.39992 0.357331
\(693\) 0 0
\(694\) −48.5148 −1.84159
\(695\) 16.3393 0.619786
\(696\) 0 0
\(697\) 5.14956 0.195054
\(698\) 15.5405 0.588217
\(699\) 0 0
\(700\) 3.93146 0.148595
\(701\) 19.0810 0.720680 0.360340 0.932821i \(-0.382661\pi\)
0.360340 + 0.932821i \(0.382661\pi\)
\(702\) 0 0
\(703\) −1.18429 −0.0446663
\(704\) −59.0204 −2.22442
\(705\) 0 0
\(706\) −78.0250 −2.93651
\(707\) 21.5102 0.808975
\(708\) 0 0
\(709\) −44.6046 −1.67516 −0.837581 0.546313i \(-0.816031\pi\)
−0.837581 + 0.546313i \(0.816031\pi\)
\(710\) 21.7338 0.815654
\(711\) 0 0
\(712\) −16.7124 −0.626325
\(713\) 19.7953 0.741340
\(714\) 0 0
\(715\) 0 0
\(716\) 16.5663 0.619110
\(717\) 0 0
\(718\) 18.4551 0.688737
\(719\) 12.9822 0.484153 0.242077 0.970257i \(-0.422171\pi\)
0.242077 + 0.970257i \(0.422171\pi\)
\(720\) 0 0
\(721\) −17.6001 −0.655461
\(722\) −16.2191 −0.603614
\(723\) 0 0
\(724\) 80.6071 2.99574
\(725\) 1.37755 0.0511610
\(726\) 0 0
\(727\) 17.3980 0.645255 0.322627 0.946526i \(-0.395434\pi\)
0.322627 + 0.946526i \(0.395434\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 31.1799 1.15402
\(731\) 28.6563 1.05989
\(732\) 0 0
\(733\) −45.8272 −1.69266 −0.846332 0.532655i \(-0.821194\pi\)
−0.846332 + 0.532655i \(0.821194\pi\)
\(734\) 15.4720 0.571081
\(735\) 0 0
\(736\) −13.9056 −0.512567
\(737\) −59.4069 −2.18828
\(738\) 0 0
\(739\) −17.5708 −0.646353 −0.323176 0.946339i \(-0.604751\pi\)
−0.323176 + 0.946339i \(0.604751\pi\)
\(740\) 0.721292 0.0265152
\(741\) 0 0
\(742\) 12.9101 0.473946
\(743\) −30.3562 −1.11366 −0.556831 0.830626i \(-0.687983\pi\)
−0.556831 + 0.830626i \(0.687983\pi\)
\(744\) 0 0
\(745\) −3.24490 −0.118884
\(746\) −31.2563 −1.14438
\(747\) 0 0
\(748\) 63.3305 2.31559
\(749\) −15.7641 −0.576007
\(750\) 0 0
\(751\) 3.80323 0.138782 0.0693909 0.997590i \(-0.477894\pi\)
0.0693909 + 0.997590i \(0.477894\pi\)
\(752\) 5.64729 0.205935
\(753\) 0 0
\(754\) 0 0
\(755\) −5.69996 −0.207443
\(756\) 0 0
\(757\) −3.40786 −0.123861 −0.0619303 0.998080i \(-0.519726\pi\)
−0.0619303 + 0.998080i \(0.519726\pi\)
\(758\) 26.0462 0.946040
\(759\) 0 0
\(760\) 12.9101 0.468300
\(761\) 7.24490 0.262627 0.131314 0.991341i \(-0.458081\pi\)
0.131314 + 0.991341i \(0.458081\pi\)
\(762\) 0 0
\(763\) 2.87528 0.104092
\(764\) −33.2022 −1.20121
\(765\) 0 0
\(766\) −51.7496 −1.86979
\(767\) 0 0
\(768\) 0 0
\(769\) −31.1764 −1.12425 −0.562124 0.827053i \(-0.690016\pi\)
−0.562124 + 0.827053i \(0.690016\pi\)
\(770\) −12.9101 −0.465249
\(771\) 0 0
\(772\) −24.8733 −0.895210
\(773\) 19.4605 0.699947 0.349973 0.936760i \(-0.386191\pi\)
0.349973 + 0.936760i \(0.386191\pi\)
\(774\) 0 0
\(775\) −8.87085 −0.318650
\(776\) 26.9260 0.966587
\(777\) 0 0
\(778\) −9.84498 −0.352959
\(779\) −5.86292 −0.210061
\(780\) 0 0
\(781\) −43.4675 −1.55539
\(782\) 22.6786 0.810986
\(783\) 0 0
\(784\) 2.83158 0.101128
\(785\) −3.85395 −0.137553
\(786\) 0 0
\(787\) −14.4034 −0.513427 −0.256713 0.966488i \(-0.582640\pi\)
−0.256713 + 0.966488i \(0.582640\pi\)
\(788\) 54.0250 1.92456
\(789\) 0 0
\(790\) 20.0641 0.713849
\(791\) 1.86292 0.0662378
\(792\) 0 0
\(793\) 0 0
\(794\) 17.2405 0.611841
\(795\) 0 0
\(796\) −3.83704 −0.136000
\(797\) 23.6011 0.835994 0.417997 0.908448i \(-0.362732\pi\)
0.417997 + 0.908448i \(0.362732\pi\)
\(798\) 0 0
\(799\) −48.4621 −1.71447
\(800\) 6.23150 0.220317
\(801\) 0 0
\(802\) −47.1575 −1.66519
\(803\) −62.3597 −2.20063
\(804\) 0 0
\(805\) −2.81571 −0.0992407
\(806\) 0 0
\(807\) 0 0
\(808\) −43.0204 −1.51345
\(809\) −26.7551 −0.940659 −0.470330 0.882491i \(-0.655865\pi\)
−0.470330 + 0.882491i \(0.655865\pi\)
\(810\) 0 0
\(811\) 3.58421 0.125859 0.0629293 0.998018i \(-0.479956\pi\)
0.0629293 + 0.998018i \(0.479956\pi\)
\(812\) 5.41579 0.190057
\(813\) 0 0
\(814\) −2.36858 −0.0830187
\(815\) 9.72480 0.340645
\(816\) 0 0
\(817\) −32.6260 −1.14144
\(818\) −38.3721 −1.34165
\(819\) 0 0
\(820\) 3.57081 0.124698
\(821\) 15.8361 0.552685 0.276342 0.961059i \(-0.410878\pi\)
0.276342 + 0.961059i \(0.410878\pi\)
\(822\) 0 0
\(823\) 7.41579 0.258498 0.129249 0.991612i \(-0.458743\pi\)
0.129249 + 0.991612i \(0.458743\pi\)
\(824\) 35.2002 1.22626
\(825\) 0 0
\(826\) 2.43712 0.0847983
\(827\) −16.6036 −0.577363 −0.288682 0.957425i \(-0.593217\pi\)
−0.288682 + 0.957425i \(0.593217\pi\)
\(828\) 0 0
\(829\) −29.2842 −1.01708 −0.508541 0.861038i \(-0.669815\pi\)
−0.508541 + 0.861038i \(0.669815\pi\)
\(830\) 18.6180 0.646241
\(831\) 0 0
\(832\) 0 0
\(833\) −24.2991 −0.841915
\(834\) 0 0
\(835\) −19.2787 −0.667167
\(836\) −72.1035 −2.49375
\(837\) 0 0
\(838\) 4.74717 0.163988
\(839\) −30.1620 −1.04131 −0.520655 0.853767i \(-0.674312\pi\)
−0.520655 + 0.853767i \(0.674312\pi\)
\(840\) 0 0
\(841\) −27.1024 −0.934564
\(842\) −89.7026 −3.09135
\(843\) 0 0
\(844\) 32.2147 1.10888
\(845\) 0 0
\(846\) 0 0
\(847\) 11.9404 0.410278
\(848\) 2.36858 0.0813374
\(849\) 0 0
\(850\) −10.1630 −0.348587
\(851\) −0.516589 −0.0177085
\(852\) 0 0
\(853\) 2.50670 0.0858277 0.0429139 0.999079i \(-0.486336\pi\)
0.0429139 + 0.999079i \(0.486336\pi\)
\(854\) −13.2405 −0.453080
\(855\) 0 0
\(856\) 31.5282 1.07761
\(857\) 24.4327 0.834605 0.417302 0.908768i \(-0.362976\pi\)
0.417302 + 0.908768i \(0.362976\pi\)
\(858\) 0 0
\(859\) −7.10235 −0.242329 −0.121165 0.992632i \(-0.538663\pi\)
−0.121165 + 0.992632i \(0.538663\pi\)
\(860\) 19.8709 0.677590
\(861\) 0 0
\(862\) 50.0497 1.70470
\(863\) −3.24490 −0.110458 −0.0552288 0.998474i \(-0.517589\pi\)
−0.0552288 + 0.998474i \(0.517589\pi\)
\(864\) 0 0
\(865\) 3.01691 0.102578
\(866\) 70.9216 2.41001
\(867\) 0 0
\(868\) −34.8754 −1.18375
\(869\) −40.1282 −1.36126
\(870\) 0 0
\(871\) 0 0
\(872\) −5.75056 −0.194738
\(873\) 0 0
\(874\) −25.8203 −0.873383
\(875\) 1.26180 0.0426567
\(876\) 0 0
\(877\) −18.9866 −0.641132 −0.320566 0.947226i \(-0.603873\pi\)
−0.320566 + 0.947226i \(0.603873\pi\)
\(878\) −9.75614 −0.329254
\(879\) 0 0
\(880\) −2.36858 −0.0798448
\(881\) 19.4764 0.656176 0.328088 0.944647i \(-0.393596\pi\)
0.328088 + 0.944647i \(0.393596\pi\)
\(882\) 0 0
\(883\) −50.5664 −1.70169 −0.850847 0.525413i \(-0.823911\pi\)
−0.850847 + 0.525413i \(0.823911\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.11575 0.0374843
\(887\) 12.5271 0.420619 0.210310 0.977635i \(-0.432553\pi\)
0.210310 + 0.977635i \(0.432553\pi\)
\(888\) 0 0
\(889\) −22.4417 −0.752669
\(890\) −14.9787 −0.502086
\(891\) 0 0
\(892\) −67.2092 −2.25033
\(893\) 55.1754 1.84638
\(894\) 0 0
\(895\) 5.31694 0.177726
\(896\) 21.5102 0.718606
\(897\) 0 0
\(898\) 52.7283 1.75957
\(899\) −12.2201 −0.407562
\(900\) 0 0
\(901\) −20.3259 −0.677154
\(902\) −11.7258 −0.390428
\(903\) 0 0
\(904\) −3.72584 −0.123920
\(905\) 25.8709 0.859976
\(906\) 0 0
\(907\) 28.0417 0.931111 0.465555 0.885019i \(-0.345855\pi\)
0.465555 + 0.885019i \(0.345855\pi\)
\(908\) 16.9687 0.563125
\(909\) 0 0
\(910\) 0 0
\(911\) −16.1977 −0.536653 −0.268327 0.963328i \(-0.586471\pi\)
−0.268327 + 0.963328i \(0.586471\pi\)
\(912\) 0 0
\(913\) −37.2360 −1.23233
\(914\) 15.0854 0.498982
\(915\) 0 0
\(916\) −68.3329 −2.25778
\(917\) −14.7248 −0.486256
\(918\) 0 0
\(919\) −43.2778 −1.42760 −0.713801 0.700348i \(-0.753028\pi\)
−0.713801 + 0.700348i \(0.753028\pi\)
\(920\) 5.63142 0.185662
\(921\) 0 0
\(922\) −65.8620 −2.16905
\(923\) 0 0
\(924\) 0 0
\(925\) 0.231499 0.00761163
\(926\) −4.05618 −0.133294
\(927\) 0 0
\(928\) 8.58421 0.281791
\(929\) 13.7685 0.451730 0.225865 0.974159i \(-0.427479\pi\)
0.225865 + 0.974159i \(0.427479\pi\)
\(930\) 0 0
\(931\) 27.6652 0.906691
\(932\) −78.8675 −2.58339
\(933\) 0 0
\(934\) 38.0417 1.24476
\(935\) 20.3259 0.664729
\(936\) 0 0
\(937\) −38.1888 −1.24757 −0.623787 0.781594i \(-0.714407\pi\)
−0.623787 + 0.781594i \(0.714407\pi\)
\(938\) 37.4799 1.22376
\(939\) 0 0
\(940\) −33.6046 −1.09606
\(941\) −13.9618 −0.455140 −0.227570 0.973762i \(-0.573078\pi\)
−0.227570 + 0.973762i \(0.573078\pi\)
\(942\) 0 0
\(943\) −2.55742 −0.0832809
\(944\) 0.447131 0.0145529
\(945\) 0 0
\(946\) −65.2519 −2.12152
\(947\) 7.36962 0.239480 0.119740 0.992805i \(-0.461794\pi\)
0.119740 + 0.992805i \(0.461794\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 11.5708 0.375407
\(951\) 0 0
\(952\) −14.3080 −0.463724
\(953\) −23.2181 −0.752108 −0.376054 0.926598i \(-0.622719\pi\)
−0.376054 + 0.926598i \(0.622719\pi\)
\(954\) 0 0
\(955\) −10.6563 −0.344828
\(956\) 70.1779 2.26972
\(957\) 0 0
\(958\) −79.5093 −2.56883
\(959\) −25.3169 −0.817527
\(960\) 0 0
\(961\) 47.6920 1.53845
\(962\) 0 0
\(963\) 0 0
\(964\) −34.6339 −1.11548
\(965\) −7.98309 −0.256985
\(966\) 0 0
\(967\) 7.54402 0.242599 0.121300 0.992616i \(-0.461294\pi\)
0.121300 + 0.992616i \(0.461294\pi\)
\(968\) −23.8809 −0.767560
\(969\) 0 0
\(970\) 24.1327 0.774853
\(971\) 30.5842 0.981494 0.490747 0.871302i \(-0.336724\pi\)
0.490747 + 0.871302i \(0.336724\pi\)
\(972\) 0 0
\(973\) 20.6170 0.660950
\(974\) −19.5102 −0.625147
\(975\) 0 0
\(976\) −2.42919 −0.0777564
\(977\) −35.6811 −1.14154 −0.570770 0.821110i \(-0.693355\pi\)
−0.570770 + 0.821110i \(0.693355\pi\)
\(978\) 0 0
\(979\) 29.9573 0.957441
\(980\) −16.8495 −0.538238
\(981\) 0 0
\(982\) −52.5047 −1.67549
\(983\) 12.9866 0.414208 0.207104 0.978319i \(-0.433596\pi\)
0.207104 + 0.978319i \(0.433596\pi\)
\(984\) 0 0
\(985\) 17.3393 0.552477
\(986\) −14.0000 −0.445851
\(987\) 0 0
\(988\) 0 0
\(989\) −14.2315 −0.452535
\(990\) 0 0
\(991\) −22.7472 −0.722588 −0.361294 0.932452i \(-0.617665\pi\)
−0.361294 + 0.932452i \(0.617665\pi\)
\(992\) −55.2787 −1.75510
\(993\) 0 0
\(994\) 27.4237 0.869828
\(995\) −1.23150 −0.0390411
\(996\) 0 0
\(997\) −13.8460 −0.438508 −0.219254 0.975668i \(-0.570362\pi\)
−0.219254 + 0.975668i \(0.570362\pi\)
\(998\) 65.6811 2.07910
\(999\) 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 7605.2.a.bw.1.1 3
3.2 odd 2 2535.2.a.ba.1.3 3
13.4 even 6 585.2.j.f.406.1 6
13.10 even 6 585.2.j.f.451.1 6
13.12 even 2 7605.2.a.bv.1.3 3
39.17 odd 6 195.2.i.d.16.3 6
39.23 odd 6 195.2.i.d.61.3 yes 6
39.38 odd 2 2535.2.a.bb.1.1 3
195.17 even 12 975.2.bb.k.874.2 12
195.23 even 12 975.2.bb.k.724.2 12
195.62 even 12 975.2.bb.k.724.5 12
195.134 odd 6 975.2.i.l.601.1 6
195.173 even 12 975.2.bb.k.874.5 12
195.179 odd 6 975.2.i.l.451.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.3 6 39.17 odd 6
195.2.i.d.61.3 yes 6 39.23 odd 6
585.2.j.f.406.1 6 13.4 even 6
585.2.j.f.451.1 6 13.10 even 6
975.2.i.l.451.1 6 195.179 odd 6
975.2.i.l.601.1 6 195.134 odd 6
975.2.bb.k.724.2 12 195.23 even 12
975.2.bb.k.724.5 12 195.62 even 12
975.2.bb.k.874.2 12 195.17 even 12
975.2.bb.k.874.5 12 195.173 even 12
2535.2.a.ba.1.3 3 3.2 odd 2
2535.2.a.bb.1.1 3 39.38 odd 2
7605.2.a.bv.1.3 3 13.12 even 2
7605.2.a.bw.1.1 3 1.1 even 1 trivial