Properties

Label 6032.2.a.ba.1.7
Level $6032$
Weight $2$
Character 6032.1
Self dual yes
Analytic conductor $48.166$
Analytic rank $0$
Dimension $10$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.866798\) of defining polynomial
Character \(\chi\) \(=\) 6032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.866798 q^{3} +2.19814 q^{5} +4.35947 q^{7} -2.24866 q^{9} +O(q^{10})\) \(q+0.866798 q^{3} +2.19814 q^{5} +4.35947 q^{7} -2.24866 q^{9} +3.62729 q^{11} +1.00000 q^{13} +1.90534 q^{15} +0.809117 q^{17} +5.53409 q^{19} +3.77878 q^{21} -3.43856 q^{23} -0.168191 q^{25} -4.54953 q^{27} -1.00000 q^{29} -0.227533 q^{31} +3.14413 q^{33} +9.58271 q^{35} +4.90883 q^{37} +0.866798 q^{39} +6.88191 q^{41} -2.53289 q^{43} -4.94287 q^{45} +2.69537 q^{47} +12.0049 q^{49} +0.701341 q^{51} -3.14083 q^{53} +7.97328 q^{55} +4.79694 q^{57} -12.9502 q^{59} +7.54000 q^{61} -9.80296 q^{63} +2.19814 q^{65} -8.95057 q^{67} -2.98054 q^{69} +7.11287 q^{71} +15.6224 q^{73} -0.145788 q^{75} +15.8130 q^{77} -1.98066 q^{79} +2.80246 q^{81} +1.59895 q^{83} +1.77855 q^{85} -0.866798 q^{87} -8.71529 q^{89} +4.35947 q^{91} -0.197225 q^{93} +12.1647 q^{95} -15.3323 q^{97} -8.15655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.866798 0.500446 0.250223 0.968188i \(-0.419496\pi\)
0.250223 + 0.968188i \(0.419496\pi\)
\(4\) 0 0
\(5\) 2.19814 0.983037 0.491519 0.870867i \(-0.336442\pi\)
0.491519 + 0.870867i \(0.336442\pi\)
\(6\) 0 0
\(7\) 4.35947 1.64772 0.823862 0.566791i \(-0.191815\pi\)
0.823862 + 0.566791i \(0.191815\pi\)
\(8\) 0 0
\(9\) −2.24866 −0.749554
\(10\) 0 0
\(11\) 3.62729 1.09367 0.546835 0.837241i \(-0.315833\pi\)
0.546835 + 0.837241i \(0.315833\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.90534 0.491957
\(16\) 0 0
\(17\) 0.809117 0.196240 0.0981198 0.995175i \(-0.468717\pi\)
0.0981198 + 0.995175i \(0.468717\pi\)
\(18\) 0 0
\(19\) 5.53409 1.26961 0.634803 0.772674i \(-0.281081\pi\)
0.634803 + 0.772674i \(0.281081\pi\)
\(20\) 0 0
\(21\) 3.77878 0.824597
\(22\) 0 0
\(23\) −3.43856 −0.716989 −0.358495 0.933532i \(-0.616710\pi\)
−0.358495 + 0.933532i \(0.616710\pi\)
\(24\) 0 0
\(25\) −0.168191 −0.0336382
\(26\) 0 0
\(27\) −4.54953 −0.875557
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −0.227533 −0.0408662 −0.0204331 0.999791i \(-0.506505\pi\)
−0.0204331 + 0.999791i \(0.506505\pi\)
\(32\) 0 0
\(33\) 3.14413 0.547322
\(34\) 0 0
\(35\) 9.58271 1.61977
\(36\) 0 0
\(37\) 4.90883 0.807007 0.403504 0.914978i \(-0.367792\pi\)
0.403504 + 0.914978i \(0.367792\pi\)
\(38\) 0 0
\(39\) 0.866798 0.138799
\(40\) 0 0
\(41\) 6.88191 1.07477 0.537387 0.843336i \(-0.319411\pi\)
0.537387 + 0.843336i \(0.319411\pi\)
\(42\) 0 0
\(43\) −2.53289 −0.386262 −0.193131 0.981173i \(-0.561864\pi\)
−0.193131 + 0.981173i \(0.561864\pi\)
\(44\) 0 0
\(45\) −4.94287 −0.736839
\(46\) 0 0
\(47\) 2.69537 0.393160 0.196580 0.980488i \(-0.437016\pi\)
0.196580 + 0.980488i \(0.437016\pi\)
\(48\) 0 0
\(49\) 12.0049 1.71499
\(50\) 0 0
\(51\) 0.701341 0.0982074
\(52\) 0 0
\(53\) −3.14083 −0.431427 −0.215713 0.976457i \(-0.569208\pi\)
−0.215713 + 0.976457i \(0.569208\pi\)
\(54\) 0 0
\(55\) 7.97328 1.07512
\(56\) 0 0
\(57\) 4.79694 0.635370
\(58\) 0 0
\(59\) −12.9502 −1.68598 −0.842989 0.537930i \(-0.819206\pi\)
−0.842989 + 0.537930i \(0.819206\pi\)
\(60\) 0 0
\(61\) 7.54000 0.965398 0.482699 0.875786i \(-0.339656\pi\)
0.482699 + 0.875786i \(0.339656\pi\)
\(62\) 0 0
\(63\) −9.80296 −1.23506
\(64\) 0 0
\(65\) 2.19814 0.272645
\(66\) 0 0
\(67\) −8.95057 −1.09349 −0.546743 0.837300i \(-0.684133\pi\)
−0.546743 + 0.837300i \(0.684133\pi\)
\(68\) 0 0
\(69\) −2.98054 −0.358815
\(70\) 0 0
\(71\) 7.11287 0.844143 0.422071 0.906563i \(-0.361303\pi\)
0.422071 + 0.906563i \(0.361303\pi\)
\(72\) 0 0
\(73\) 15.6224 1.82847 0.914235 0.405185i \(-0.132793\pi\)
0.914235 + 0.405185i \(0.132793\pi\)
\(74\) 0 0
\(75\) −0.145788 −0.0168341
\(76\) 0 0
\(77\) 15.8130 1.80206
\(78\) 0 0
\(79\) −1.98066 −0.222841 −0.111421 0.993773i \(-0.535540\pi\)
−0.111421 + 0.993773i \(0.535540\pi\)
\(80\) 0 0
\(81\) 2.80246 0.311384
\(82\) 0 0
\(83\) 1.59895 0.175508 0.0877538 0.996142i \(-0.472031\pi\)
0.0877538 + 0.996142i \(0.472031\pi\)
\(84\) 0 0
\(85\) 1.77855 0.192911
\(86\) 0 0
\(87\) −0.866798 −0.0929305
\(88\) 0 0
\(89\) −8.71529 −0.923818 −0.461909 0.886927i \(-0.652835\pi\)
−0.461909 + 0.886927i \(0.652835\pi\)
\(90\) 0 0
\(91\) 4.35947 0.456996
\(92\) 0 0
\(93\) −0.197225 −0.0204513
\(94\) 0 0
\(95\) 12.1647 1.24807
\(96\) 0 0
\(97\) −15.3323 −1.55676 −0.778378 0.627796i \(-0.783957\pi\)
−0.778378 + 0.627796i \(0.783957\pi\)
\(98\) 0 0
\(99\) −8.15655 −0.819764
\(100\) 0 0
\(101\) 1.74668 0.173801 0.0869005 0.996217i \(-0.472304\pi\)
0.0869005 + 0.996217i \(0.472304\pi\)
\(102\) 0 0
\(103\) 3.96219 0.390406 0.195203 0.980763i \(-0.437463\pi\)
0.195203 + 0.980763i \(0.437463\pi\)
\(104\) 0 0
\(105\) 8.30627 0.810609
\(106\) 0 0
\(107\) −10.9317 −1.05681 −0.528406 0.848992i \(-0.677210\pi\)
−0.528406 + 0.848992i \(0.677210\pi\)
\(108\) 0 0
\(109\) −3.21484 −0.307926 −0.153963 0.988077i \(-0.549204\pi\)
−0.153963 + 0.988077i \(0.549204\pi\)
\(110\) 0 0
\(111\) 4.25497 0.403864
\(112\) 0 0
\(113\) 13.0885 1.23127 0.615633 0.788033i \(-0.288900\pi\)
0.615633 + 0.788033i \(0.288900\pi\)
\(114\) 0 0
\(115\) −7.55843 −0.704827
\(116\) 0 0
\(117\) −2.24866 −0.207889
\(118\) 0 0
\(119\) 3.52732 0.323349
\(120\) 0 0
\(121\) 2.15723 0.196112
\(122\) 0 0
\(123\) 5.96523 0.537867
\(124\) 0 0
\(125\) −11.3604 −1.01610
\(126\) 0 0
\(127\) 1.79424 0.159213 0.0796066 0.996826i \(-0.474634\pi\)
0.0796066 + 0.996826i \(0.474634\pi\)
\(128\) 0 0
\(129\) −2.19551 −0.193304
\(130\) 0 0
\(131\) 2.50063 0.218481 0.109241 0.994015i \(-0.465158\pi\)
0.109241 + 0.994015i \(0.465158\pi\)
\(132\) 0 0
\(133\) 24.1257 2.09196
\(134\) 0 0
\(135\) −10.0005 −0.860705
\(136\) 0 0
\(137\) −11.2810 −0.963798 −0.481899 0.876227i \(-0.660053\pi\)
−0.481899 + 0.876227i \(0.660053\pi\)
\(138\) 0 0
\(139\) −18.1314 −1.53789 −0.768945 0.639315i \(-0.779218\pi\)
−0.768945 + 0.639315i \(0.779218\pi\)
\(140\) 0 0
\(141\) 2.33634 0.196756
\(142\) 0 0
\(143\) 3.62729 0.303329
\(144\) 0 0
\(145\) −2.19814 −0.182545
\(146\) 0 0
\(147\) 10.4059 0.858261
\(148\) 0 0
\(149\) −6.47830 −0.530723 −0.265361 0.964149i \(-0.585491\pi\)
−0.265361 + 0.964149i \(0.585491\pi\)
\(150\) 0 0
\(151\) 0.449097 0.0365470 0.0182735 0.999833i \(-0.494183\pi\)
0.0182735 + 0.999833i \(0.494183\pi\)
\(152\) 0 0
\(153\) −1.81943 −0.147092
\(154\) 0 0
\(155\) −0.500149 −0.0401729
\(156\) 0 0
\(157\) 2.05118 0.163702 0.0818508 0.996645i \(-0.473917\pi\)
0.0818508 + 0.996645i \(0.473917\pi\)
\(158\) 0 0
\(159\) −2.72247 −0.215906
\(160\) 0 0
\(161\) −14.9903 −1.18140
\(162\) 0 0
\(163\) 13.0998 1.02606 0.513029 0.858371i \(-0.328523\pi\)
0.513029 + 0.858371i \(0.328523\pi\)
\(164\) 0 0
\(165\) 6.91123 0.538038
\(166\) 0 0
\(167\) 3.02735 0.234264 0.117132 0.993116i \(-0.462630\pi\)
0.117132 + 0.993116i \(0.462630\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −12.4443 −0.951638
\(172\) 0 0
\(173\) 2.87327 0.218451 0.109225 0.994017i \(-0.465163\pi\)
0.109225 + 0.994017i \(0.465163\pi\)
\(174\) 0 0
\(175\) −0.733223 −0.0554264
\(176\) 0 0
\(177\) −11.2252 −0.843741
\(178\) 0 0
\(179\) −1.33055 −0.0994501 −0.0497250 0.998763i \(-0.515834\pi\)
−0.0497250 + 0.998763i \(0.515834\pi\)
\(180\) 0 0
\(181\) 15.5300 1.15433 0.577167 0.816626i \(-0.304158\pi\)
0.577167 + 0.816626i \(0.304158\pi\)
\(182\) 0 0
\(183\) 6.53566 0.483130
\(184\) 0 0
\(185\) 10.7903 0.793318
\(186\) 0 0
\(187\) 2.93490 0.214621
\(188\) 0 0
\(189\) −19.8335 −1.44268
\(190\) 0 0
\(191\) −8.87561 −0.642216 −0.321108 0.947043i \(-0.604055\pi\)
−0.321108 + 0.947043i \(0.604055\pi\)
\(192\) 0 0
\(193\) 14.7284 1.06018 0.530088 0.847943i \(-0.322159\pi\)
0.530088 + 0.847943i \(0.322159\pi\)
\(194\) 0 0
\(195\) 1.90534 0.136444
\(196\) 0 0
\(197\) 0.728224 0.0518838 0.0259419 0.999663i \(-0.491742\pi\)
0.0259419 + 0.999663i \(0.491742\pi\)
\(198\) 0 0
\(199\) 9.75714 0.691665 0.345833 0.938296i \(-0.387597\pi\)
0.345833 + 0.938296i \(0.387597\pi\)
\(200\) 0 0
\(201\) −7.75834 −0.547231
\(202\) 0 0
\(203\) −4.35947 −0.305975
\(204\) 0 0
\(205\) 15.1274 1.05654
\(206\) 0 0
\(207\) 7.73216 0.537422
\(208\) 0 0
\(209\) 20.0737 1.38853
\(210\) 0 0
\(211\) −5.49906 −0.378571 −0.189285 0.981922i \(-0.560617\pi\)
−0.189285 + 0.981922i \(0.560617\pi\)
\(212\) 0 0
\(213\) 6.16543 0.422448
\(214\) 0 0
\(215\) −5.56764 −0.379710
\(216\) 0 0
\(217\) −0.991923 −0.0673361
\(218\) 0 0
\(219\) 13.5415 0.915050
\(220\) 0 0
\(221\) 0.809117 0.0544271
\(222\) 0 0
\(223\) −17.6940 −1.18488 −0.592440 0.805615i \(-0.701835\pi\)
−0.592440 + 0.805615i \(0.701835\pi\)
\(224\) 0 0
\(225\) 0.378205 0.0252136
\(226\) 0 0
\(227\) 5.78798 0.384162 0.192081 0.981379i \(-0.438476\pi\)
0.192081 + 0.981379i \(0.438476\pi\)
\(228\) 0 0
\(229\) 22.8632 1.51084 0.755422 0.655239i \(-0.227432\pi\)
0.755422 + 0.655239i \(0.227432\pi\)
\(230\) 0 0
\(231\) 13.7067 0.901836
\(232\) 0 0
\(233\) −13.8442 −0.906963 −0.453482 0.891266i \(-0.649818\pi\)
−0.453482 + 0.891266i \(0.649818\pi\)
\(234\) 0 0
\(235\) 5.92480 0.386491
\(236\) 0 0
\(237\) −1.71683 −0.111520
\(238\) 0 0
\(239\) −8.74900 −0.565926 −0.282963 0.959131i \(-0.591317\pi\)
−0.282963 + 0.959131i \(0.591317\pi\)
\(240\) 0 0
\(241\) −18.5363 −1.19403 −0.597014 0.802231i \(-0.703646\pi\)
−0.597014 + 0.802231i \(0.703646\pi\)
\(242\) 0 0
\(243\) 16.0778 1.03139
\(244\) 0 0
\(245\) 26.3885 1.68590
\(246\) 0 0
\(247\) 5.53409 0.352126
\(248\) 0 0
\(249\) 1.38597 0.0878321
\(250\) 0 0
\(251\) 6.68464 0.421931 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(252\) 0 0
\(253\) −12.4727 −0.784149
\(254\) 0 0
\(255\) 1.54164 0.0965415
\(256\) 0 0
\(257\) 20.9053 1.30404 0.652019 0.758203i \(-0.273922\pi\)
0.652019 + 0.758203i \(0.273922\pi\)
\(258\) 0 0
\(259\) 21.3999 1.32972
\(260\) 0 0
\(261\) 2.24866 0.139189
\(262\) 0 0
\(263\) −23.7095 −1.46199 −0.730995 0.682383i \(-0.760944\pi\)
−0.730995 + 0.682383i \(0.760944\pi\)
\(264\) 0 0
\(265\) −6.90398 −0.424108
\(266\) 0 0
\(267\) −7.55439 −0.462321
\(268\) 0 0
\(269\) −10.7389 −0.654763 −0.327381 0.944892i \(-0.606166\pi\)
−0.327381 + 0.944892i \(0.606166\pi\)
\(270\) 0 0
\(271\) −27.5160 −1.67148 −0.835739 0.549127i \(-0.814960\pi\)
−0.835739 + 0.549127i \(0.814960\pi\)
\(272\) 0 0
\(273\) 3.77878 0.228702
\(274\) 0 0
\(275\) −0.610077 −0.0367891
\(276\) 0 0
\(277\) −18.4171 −1.10658 −0.553289 0.832989i \(-0.686628\pi\)
−0.553289 + 0.832989i \(0.686628\pi\)
\(278\) 0 0
\(279\) 0.511645 0.0306314
\(280\) 0 0
\(281\) 3.07889 0.183671 0.0918356 0.995774i \(-0.470727\pi\)
0.0918356 + 0.995774i \(0.470727\pi\)
\(282\) 0 0
\(283\) 5.30920 0.315599 0.157800 0.987471i \(-0.449560\pi\)
0.157800 + 0.987471i \(0.449560\pi\)
\(284\) 0 0
\(285\) 10.5443 0.624592
\(286\) 0 0
\(287\) 30.0015 1.77093
\(288\) 0 0
\(289\) −16.3453 −0.961490
\(290\) 0 0
\(291\) −13.2900 −0.779072
\(292\) 0 0
\(293\) 16.8332 0.983405 0.491702 0.870763i \(-0.336375\pi\)
0.491702 + 0.870763i \(0.336375\pi\)
\(294\) 0 0
\(295\) −28.4664 −1.65738
\(296\) 0 0
\(297\) −16.5025 −0.957570
\(298\) 0 0
\(299\) −3.43856 −0.198857
\(300\) 0 0
\(301\) −11.0421 −0.636454
\(302\) 0 0
\(303\) 1.51402 0.0869780
\(304\) 0 0
\(305\) 16.5740 0.949022
\(306\) 0 0
\(307\) −25.9689 −1.48213 −0.741063 0.671436i \(-0.765678\pi\)
−0.741063 + 0.671436i \(0.765678\pi\)
\(308\) 0 0
\(309\) 3.43442 0.195377
\(310\) 0 0
\(311\) −10.3043 −0.584304 −0.292152 0.956372i \(-0.594371\pi\)
−0.292152 + 0.956372i \(0.594371\pi\)
\(312\) 0 0
\(313\) 30.2526 1.70998 0.854989 0.518646i \(-0.173564\pi\)
0.854989 + 0.518646i \(0.173564\pi\)
\(314\) 0 0
\(315\) −21.5483 −1.21411
\(316\) 0 0
\(317\) −18.4428 −1.03585 −0.517926 0.855425i \(-0.673296\pi\)
−0.517926 + 0.855425i \(0.673296\pi\)
\(318\) 0 0
\(319\) −3.62729 −0.203089
\(320\) 0 0
\(321\) −9.47561 −0.528877
\(322\) 0 0
\(323\) 4.47772 0.249147
\(324\) 0 0
\(325\) −0.168191 −0.00932956
\(326\) 0 0
\(327\) −2.78662 −0.154100
\(328\) 0 0
\(329\) 11.7504 0.647820
\(330\) 0 0
\(331\) 5.65331 0.310734 0.155367 0.987857i \(-0.450344\pi\)
0.155367 + 0.987857i \(0.450344\pi\)
\(332\) 0 0
\(333\) −11.0383 −0.604895
\(334\) 0 0
\(335\) −19.6746 −1.07494
\(336\) 0 0
\(337\) −19.5324 −1.06400 −0.532000 0.846744i \(-0.678559\pi\)
−0.532000 + 0.846744i \(0.678559\pi\)
\(338\) 0 0
\(339\) 11.3451 0.616183
\(340\) 0 0
\(341\) −0.825329 −0.0446941
\(342\) 0 0
\(343\) 21.8189 1.17811
\(344\) 0 0
\(345\) −6.55163 −0.352728
\(346\) 0 0
\(347\) 6.63857 0.356377 0.178189 0.983996i \(-0.442976\pi\)
0.178189 + 0.983996i \(0.442976\pi\)
\(348\) 0 0
\(349\) −4.45417 −0.238426 −0.119213 0.992869i \(-0.538037\pi\)
−0.119213 + 0.992869i \(0.538037\pi\)
\(350\) 0 0
\(351\) −4.54953 −0.242836
\(352\) 0 0
\(353\) 7.34190 0.390770 0.195385 0.980727i \(-0.437404\pi\)
0.195385 + 0.980727i \(0.437404\pi\)
\(354\) 0 0
\(355\) 15.6351 0.829824
\(356\) 0 0
\(357\) 3.05747 0.161819
\(358\) 0 0
\(359\) −23.0325 −1.21561 −0.607804 0.794087i \(-0.707949\pi\)
−0.607804 + 0.794087i \(0.707949\pi\)
\(360\) 0 0
\(361\) 11.6261 0.611901
\(362\) 0 0
\(363\) 1.86988 0.0981435
\(364\) 0 0
\(365\) 34.3403 1.79745
\(366\) 0 0
\(367\) −26.9306 −1.40576 −0.702882 0.711306i \(-0.748104\pi\)
−0.702882 + 0.711306i \(0.748104\pi\)
\(368\) 0 0
\(369\) −15.4751 −0.805601
\(370\) 0 0
\(371\) −13.6924 −0.710872
\(372\) 0 0
\(373\) 31.9916 1.65646 0.828230 0.560388i \(-0.189348\pi\)
0.828230 + 0.560388i \(0.189348\pi\)
\(374\) 0 0
\(375\) −9.84717 −0.508506
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) 11.4552 0.588413 0.294207 0.955742i \(-0.404945\pi\)
0.294207 + 0.955742i \(0.404945\pi\)
\(380\) 0 0
\(381\) 1.55525 0.0796777
\(382\) 0 0
\(383\) −0.901149 −0.0460466 −0.0230233 0.999735i \(-0.507329\pi\)
−0.0230233 + 0.999735i \(0.507329\pi\)
\(384\) 0 0
\(385\) 34.7593 1.77150
\(386\) 0 0
\(387\) 5.69561 0.289524
\(388\) 0 0
\(389\) 0.198850 0.0100821 0.00504104 0.999987i \(-0.498395\pi\)
0.00504104 + 0.999987i \(0.498395\pi\)
\(390\) 0 0
\(391\) −2.78220 −0.140702
\(392\) 0 0
\(393\) 2.16754 0.109338
\(394\) 0 0
\(395\) −4.35376 −0.219061
\(396\) 0 0
\(397\) 22.6657 1.13756 0.568780 0.822490i \(-0.307416\pi\)
0.568780 + 0.822490i \(0.307416\pi\)
\(398\) 0 0
\(399\) 20.9121 1.04691
\(400\) 0 0
\(401\) −3.32964 −0.166274 −0.0831372 0.996538i \(-0.526494\pi\)
−0.0831372 + 0.996538i \(0.526494\pi\)
\(402\) 0 0
\(403\) −0.227533 −0.0113342
\(404\) 0 0
\(405\) 6.16019 0.306102
\(406\) 0 0
\(407\) 17.8058 0.882599
\(408\) 0 0
\(409\) −5.47625 −0.270783 −0.135391 0.990792i \(-0.543229\pi\)
−0.135391 + 0.990792i \(0.543229\pi\)
\(410\) 0 0
\(411\) −9.77833 −0.482329
\(412\) 0 0
\(413\) −56.4562 −2.77803
\(414\) 0 0
\(415\) 3.51471 0.172531
\(416\) 0 0
\(417\) −15.7163 −0.769631
\(418\) 0 0
\(419\) −10.9483 −0.534860 −0.267430 0.963577i \(-0.586174\pi\)
−0.267430 + 0.963577i \(0.586174\pi\)
\(420\) 0 0
\(421\) 0.726095 0.0353877 0.0176939 0.999843i \(-0.494368\pi\)
0.0176939 + 0.999843i \(0.494368\pi\)
\(422\) 0 0
\(423\) −6.06098 −0.294695
\(424\) 0 0
\(425\) −0.136086 −0.00660115
\(426\) 0 0
\(427\) 32.8704 1.59071
\(428\) 0 0
\(429\) 3.14413 0.151800
\(430\) 0 0
\(431\) 14.7541 0.710680 0.355340 0.934737i \(-0.384365\pi\)
0.355340 + 0.934737i \(0.384365\pi\)
\(432\) 0 0
\(433\) 29.8348 1.43377 0.716884 0.697193i \(-0.245568\pi\)
0.716884 + 0.697193i \(0.245568\pi\)
\(434\) 0 0
\(435\) −1.90534 −0.0913541
\(436\) 0 0
\(437\) −19.0293 −0.910294
\(438\) 0 0
\(439\) 37.5016 1.78985 0.894927 0.446212i \(-0.147227\pi\)
0.894927 + 0.446212i \(0.147227\pi\)
\(440\) 0 0
\(441\) −26.9951 −1.28548
\(442\) 0 0
\(443\) 34.3849 1.63368 0.816839 0.576866i \(-0.195725\pi\)
0.816839 + 0.576866i \(0.195725\pi\)
\(444\) 0 0
\(445\) −19.1574 −0.908148
\(446\) 0 0
\(447\) −5.61538 −0.265598
\(448\) 0 0
\(449\) 26.6260 1.25656 0.628280 0.777988i \(-0.283759\pi\)
0.628280 + 0.777988i \(0.283759\pi\)
\(450\) 0 0
\(451\) 24.9627 1.17545
\(452\) 0 0
\(453\) 0.389276 0.0182898
\(454\) 0 0
\(455\) 9.58271 0.449244
\(456\) 0 0
\(457\) 1.49872 0.0701071 0.0350535 0.999385i \(-0.488840\pi\)
0.0350535 + 0.999385i \(0.488840\pi\)
\(458\) 0 0
\(459\) −3.68110 −0.171819
\(460\) 0 0
\(461\) −11.9893 −0.558397 −0.279198 0.960233i \(-0.590069\pi\)
−0.279198 + 0.960233i \(0.590069\pi\)
\(462\) 0 0
\(463\) 9.20541 0.427812 0.213906 0.976854i \(-0.431381\pi\)
0.213906 + 0.976854i \(0.431381\pi\)
\(464\) 0 0
\(465\) −0.433528 −0.0201044
\(466\) 0 0
\(467\) −19.9910 −0.925072 −0.462536 0.886601i \(-0.653060\pi\)
−0.462536 + 0.886601i \(0.653060\pi\)
\(468\) 0 0
\(469\) −39.0197 −1.80176
\(470\) 0 0
\(471\) 1.77796 0.0819238
\(472\) 0 0
\(473\) −9.18753 −0.422443
\(474\) 0 0
\(475\) −0.930784 −0.0427073
\(476\) 0 0
\(477\) 7.06267 0.323377
\(478\) 0 0
\(479\) 35.9438 1.64231 0.821157 0.570702i \(-0.193329\pi\)
0.821157 + 0.570702i \(0.193329\pi\)
\(480\) 0 0
\(481\) 4.90883 0.223823
\(482\) 0 0
\(483\) −12.9936 −0.591227
\(484\) 0 0
\(485\) −33.7024 −1.53035
\(486\) 0 0
\(487\) 15.9734 0.723822 0.361911 0.932213i \(-0.382124\pi\)
0.361911 + 0.932213i \(0.382124\pi\)
\(488\) 0 0
\(489\) 11.3549 0.513486
\(490\) 0 0
\(491\) −9.28873 −0.419194 −0.209597 0.977788i \(-0.567215\pi\)
−0.209597 + 0.977788i \(0.567215\pi\)
\(492\) 0 0
\(493\) −0.809117 −0.0364408
\(494\) 0 0
\(495\) −17.9292 −0.805858
\(496\) 0 0
\(497\) 31.0083 1.39091
\(498\) 0 0
\(499\) −14.2188 −0.636519 −0.318259 0.948004i \(-0.603098\pi\)
−0.318259 + 0.948004i \(0.603098\pi\)
\(500\) 0 0
\(501\) 2.62411 0.117236
\(502\) 0 0
\(503\) −22.0456 −0.982963 −0.491482 0.870888i \(-0.663545\pi\)
−0.491482 + 0.870888i \(0.663545\pi\)
\(504\) 0 0
\(505\) 3.83944 0.170853
\(506\) 0 0
\(507\) 0.866798 0.0384959
\(508\) 0 0
\(509\) 21.1535 0.937614 0.468807 0.883301i \(-0.344684\pi\)
0.468807 + 0.883301i \(0.344684\pi\)
\(510\) 0 0
\(511\) 68.1055 3.01281
\(512\) 0 0
\(513\) −25.1775 −1.11161
\(514\) 0 0
\(515\) 8.70943 0.383783
\(516\) 0 0
\(517\) 9.77690 0.429987
\(518\) 0 0
\(519\) 2.49055 0.109323
\(520\) 0 0
\(521\) 38.5905 1.69068 0.845341 0.534227i \(-0.179397\pi\)
0.845341 + 0.534227i \(0.179397\pi\)
\(522\) 0 0
\(523\) 24.8645 1.08725 0.543625 0.839328i \(-0.317051\pi\)
0.543625 + 0.839328i \(0.317051\pi\)
\(524\) 0 0
\(525\) −0.635556 −0.0277380
\(526\) 0 0
\(527\) −0.184101 −0.00801956
\(528\) 0 0
\(529\) −11.1763 −0.485926
\(530\) 0 0
\(531\) 29.1207 1.26373
\(532\) 0 0
\(533\) 6.88191 0.298089
\(534\) 0 0
\(535\) −24.0295 −1.03888
\(536\) 0 0
\(537\) −1.15332 −0.0497694
\(538\) 0 0
\(539\) 43.5454 1.87563
\(540\) 0 0
\(541\) 11.6371 0.500317 0.250158 0.968205i \(-0.419517\pi\)
0.250158 + 0.968205i \(0.419517\pi\)
\(542\) 0 0
\(543\) 13.4614 0.577682
\(544\) 0 0
\(545\) −7.06666 −0.302703
\(546\) 0 0
\(547\) 0.690340 0.0295168 0.0147584 0.999891i \(-0.495302\pi\)
0.0147584 + 0.999891i \(0.495302\pi\)
\(548\) 0 0
\(549\) −16.9549 −0.723618
\(550\) 0 0
\(551\) −5.53409 −0.235760
\(552\) 0 0
\(553\) −8.63461 −0.367181
\(554\) 0 0
\(555\) 9.35300 0.397013
\(556\) 0 0
\(557\) −40.1097 −1.69950 −0.849751 0.527184i \(-0.823248\pi\)
−0.849751 + 0.527184i \(0.823248\pi\)
\(558\) 0 0
\(559\) −2.53289 −0.107130
\(560\) 0 0
\(561\) 2.54397 0.107406
\(562\) 0 0
\(563\) 32.1905 1.35667 0.678333 0.734754i \(-0.262703\pi\)
0.678333 + 0.734754i \(0.262703\pi\)
\(564\) 0 0
\(565\) 28.7704 1.21038
\(566\) 0 0
\(567\) 12.2172 0.513075
\(568\) 0 0
\(569\) 20.1691 0.845531 0.422766 0.906239i \(-0.361059\pi\)
0.422766 + 0.906239i \(0.361059\pi\)
\(570\) 0 0
\(571\) 6.43260 0.269196 0.134598 0.990900i \(-0.457026\pi\)
0.134598 + 0.990900i \(0.457026\pi\)
\(572\) 0 0
\(573\) −7.69336 −0.321395
\(574\) 0 0
\(575\) 0.578335 0.0241182
\(576\) 0 0
\(577\) −27.4692 −1.14356 −0.571779 0.820408i \(-0.693747\pi\)
−0.571779 + 0.820408i \(0.693747\pi\)
\(578\) 0 0
\(579\) 12.7666 0.530561
\(580\) 0 0
\(581\) 6.97057 0.289188
\(582\) 0 0
\(583\) −11.3927 −0.471838
\(584\) 0 0
\(585\) −4.94287 −0.204362
\(586\) 0 0
\(587\) 32.2797 1.33233 0.666163 0.745806i \(-0.267935\pi\)
0.666163 + 0.745806i \(0.267935\pi\)
\(588\) 0 0
\(589\) −1.25919 −0.0518839
\(590\) 0 0
\(591\) 0.631223 0.0259650
\(592\) 0 0
\(593\) −29.9968 −1.23182 −0.615911 0.787815i \(-0.711212\pi\)
−0.615911 + 0.787815i \(0.711212\pi\)
\(594\) 0 0
\(595\) 7.75353 0.317864
\(596\) 0 0
\(597\) 8.45747 0.346141
\(598\) 0 0
\(599\) −38.5665 −1.57579 −0.787893 0.615812i \(-0.788828\pi\)
−0.787893 + 0.615812i \(0.788828\pi\)
\(600\) 0 0
\(601\) 12.5601 0.512337 0.256169 0.966632i \(-0.417540\pi\)
0.256169 + 0.966632i \(0.417540\pi\)
\(602\) 0 0
\(603\) 20.1268 0.819627
\(604\) 0 0
\(605\) 4.74189 0.192785
\(606\) 0 0
\(607\) 36.0082 1.46153 0.730764 0.682630i \(-0.239164\pi\)
0.730764 + 0.682630i \(0.239164\pi\)
\(608\) 0 0
\(609\) −3.77878 −0.153124
\(610\) 0 0
\(611\) 2.69537 0.109043
\(612\) 0 0
\(613\) −34.4848 −1.39283 −0.696414 0.717640i \(-0.745222\pi\)
−0.696414 + 0.717640i \(0.745222\pi\)
\(614\) 0 0
\(615\) 13.1124 0.528743
\(616\) 0 0
\(617\) 1.81984 0.0732639 0.0366319 0.999329i \(-0.488337\pi\)
0.0366319 + 0.999329i \(0.488337\pi\)
\(618\) 0 0
\(619\) −0.718328 −0.0288720 −0.0144360 0.999896i \(-0.504595\pi\)
−0.0144360 + 0.999896i \(0.504595\pi\)
\(620\) 0 0
\(621\) 15.6438 0.627765
\(622\) 0 0
\(623\) −37.9940 −1.52220
\(624\) 0 0
\(625\) −24.1308 −0.965230
\(626\) 0 0
\(627\) 17.3999 0.694884
\(628\) 0 0
\(629\) 3.97182 0.158367
\(630\) 0 0
\(631\) −18.1962 −0.724378 −0.362189 0.932105i \(-0.617970\pi\)
−0.362189 + 0.932105i \(0.617970\pi\)
\(632\) 0 0
\(633\) −4.76658 −0.189454
\(634\) 0 0
\(635\) 3.94399 0.156513
\(636\) 0 0
\(637\) 12.0049 0.475653
\(638\) 0 0
\(639\) −15.9944 −0.632730
\(640\) 0 0
\(641\) −0.321087 −0.0126822 −0.00634109 0.999980i \(-0.502018\pi\)
−0.00634109 + 0.999980i \(0.502018\pi\)
\(642\) 0 0
\(643\) 19.4353 0.766455 0.383227 0.923654i \(-0.374813\pi\)
0.383227 + 0.923654i \(0.374813\pi\)
\(644\) 0 0
\(645\) −4.82602 −0.190025
\(646\) 0 0
\(647\) −30.5532 −1.20117 −0.600585 0.799561i \(-0.705065\pi\)
−0.600585 + 0.799561i \(0.705065\pi\)
\(648\) 0 0
\(649\) −46.9743 −1.84390
\(650\) 0 0
\(651\) −0.859797 −0.0336981
\(652\) 0 0
\(653\) −26.0930 −1.02110 −0.510549 0.859848i \(-0.670558\pi\)
−0.510549 + 0.859848i \(0.670558\pi\)
\(654\) 0 0
\(655\) 5.49674 0.214775
\(656\) 0 0
\(657\) −35.1296 −1.37054
\(658\) 0 0
\(659\) 2.22719 0.0867591 0.0433796 0.999059i \(-0.486188\pi\)
0.0433796 + 0.999059i \(0.486188\pi\)
\(660\) 0 0
\(661\) 35.6483 1.38656 0.693279 0.720669i \(-0.256165\pi\)
0.693279 + 0.720669i \(0.256165\pi\)
\(662\) 0 0
\(663\) 0.701341 0.0272378
\(664\) 0 0
\(665\) 53.0315 2.05647
\(666\) 0 0
\(667\) 3.43856 0.133142
\(668\) 0 0
\(669\) −15.3371 −0.592968
\(670\) 0 0
\(671\) 27.3498 1.05583
\(672\) 0 0
\(673\) −16.4224 −0.633039 −0.316519 0.948586i \(-0.602514\pi\)
−0.316519 + 0.948586i \(0.602514\pi\)
\(674\) 0 0
\(675\) 0.765190 0.0294522
\(676\) 0 0
\(677\) −22.9119 −0.880576 −0.440288 0.897857i \(-0.645124\pi\)
−0.440288 + 0.897857i \(0.645124\pi\)
\(678\) 0 0
\(679\) −66.8405 −2.56510
\(680\) 0 0
\(681\) 5.01701 0.192252
\(682\) 0 0
\(683\) −41.4800 −1.58719 −0.793594 0.608447i \(-0.791793\pi\)
−0.793594 + 0.608447i \(0.791793\pi\)
\(684\) 0 0
\(685\) −24.7971 −0.947450
\(686\) 0 0
\(687\) 19.8178 0.756096
\(688\) 0 0
\(689\) −3.14083 −0.119656
\(690\) 0 0
\(691\) 25.4177 0.966933 0.483467 0.875363i \(-0.339377\pi\)
0.483467 + 0.875363i \(0.339377\pi\)
\(692\) 0 0
\(693\) −35.5582 −1.35074
\(694\) 0 0
\(695\) −39.8554 −1.51180
\(696\) 0 0
\(697\) 5.56827 0.210913
\(698\) 0 0
\(699\) −12.0001 −0.453886
\(700\) 0 0
\(701\) −1.16013 −0.0438175 −0.0219087 0.999760i \(-0.506974\pi\)
−0.0219087 + 0.999760i \(0.506974\pi\)
\(702\) 0 0
\(703\) 27.1659 1.02458
\(704\) 0 0
\(705\) 5.13561 0.193418
\(706\) 0 0
\(707\) 7.61459 0.286376
\(708\) 0 0
\(709\) 7.32344 0.275038 0.137519 0.990499i \(-0.456087\pi\)
0.137519 + 0.990499i \(0.456087\pi\)
\(710\) 0 0
\(711\) 4.45383 0.167032
\(712\) 0 0
\(713\) 0.782386 0.0293006
\(714\) 0 0
\(715\) 7.97328 0.298184
\(716\) 0 0
\(717\) −7.58362 −0.283215
\(718\) 0 0
\(719\) 14.4802 0.540019 0.270010 0.962858i \(-0.412973\pi\)
0.270010 + 0.962858i \(0.412973\pi\)
\(720\) 0 0
\(721\) 17.2730 0.643281
\(722\) 0 0
\(723\) −16.0672 −0.597546
\(724\) 0 0
\(725\) 0.168191 0.00624646
\(726\) 0 0
\(727\) −43.5365 −1.61468 −0.807340 0.590087i \(-0.799094\pi\)
−0.807340 + 0.590087i \(0.799094\pi\)
\(728\) 0 0
\(729\) 5.52879 0.204770
\(730\) 0 0
\(731\) −2.04940 −0.0758000
\(732\) 0 0
\(733\) −3.72585 −0.137617 −0.0688086 0.997630i \(-0.521920\pi\)
−0.0688086 + 0.997630i \(0.521920\pi\)
\(734\) 0 0
\(735\) 22.8735 0.843703
\(736\) 0 0
\(737\) −32.4663 −1.19591
\(738\) 0 0
\(739\) −8.10272 −0.298063 −0.149032 0.988832i \(-0.547616\pi\)
−0.149032 + 0.988832i \(0.547616\pi\)
\(740\) 0 0
\(741\) 4.79694 0.176220
\(742\) 0 0
\(743\) −0.480988 −0.0176457 −0.00882287 0.999961i \(-0.502808\pi\)
−0.00882287 + 0.999961i \(0.502808\pi\)
\(744\) 0 0
\(745\) −14.2402 −0.521720
\(746\) 0 0
\(747\) −3.59550 −0.131552
\(748\) 0 0
\(749\) −47.6565 −1.74133
\(750\) 0 0
\(751\) −48.5082 −1.77009 −0.885044 0.465507i \(-0.845872\pi\)
−0.885044 + 0.465507i \(0.845872\pi\)
\(752\) 0 0
\(753\) 5.79423 0.211154
\(754\) 0 0
\(755\) 0.987177 0.0359270
\(756\) 0 0
\(757\) −42.0009 −1.52655 −0.763274 0.646075i \(-0.776410\pi\)
−0.763274 + 0.646075i \(0.776410\pi\)
\(758\) 0 0
\(759\) −10.8113 −0.392424
\(760\) 0 0
\(761\) −24.9513 −0.904484 −0.452242 0.891895i \(-0.649376\pi\)
−0.452242 + 0.891895i \(0.649376\pi\)
\(762\) 0 0
\(763\) −14.0150 −0.507377
\(764\) 0 0
\(765\) −3.99936 −0.144597
\(766\) 0 0
\(767\) −12.9502 −0.467606
\(768\) 0 0
\(769\) 42.0942 1.51796 0.758978 0.651116i \(-0.225699\pi\)
0.758978 + 0.651116i \(0.225699\pi\)
\(770\) 0 0
\(771\) 18.1207 0.652601
\(772\) 0 0
\(773\) 8.24607 0.296590 0.148295 0.988943i \(-0.452621\pi\)
0.148295 + 0.988943i \(0.452621\pi\)
\(774\) 0 0
\(775\) 0.0382690 0.00137466
\(776\) 0 0
\(777\) 18.5494 0.665456
\(778\) 0 0
\(779\) 38.0851 1.36454
\(780\) 0 0
\(781\) 25.8005 0.923213
\(782\) 0 0
\(783\) 4.54953 0.162587
\(784\) 0 0
\(785\) 4.50877 0.160925
\(786\) 0 0
\(787\) 40.8478 1.45607 0.728033 0.685542i \(-0.240435\pi\)
0.728033 + 0.685542i \(0.240435\pi\)
\(788\) 0 0
\(789\) −20.5514 −0.731647
\(790\) 0 0
\(791\) 57.0591 2.02879
\(792\) 0 0
\(793\) 7.54000 0.267753
\(794\) 0 0
\(795\) −5.98436 −0.212243
\(796\) 0 0
\(797\) 38.3461 1.35829 0.679144 0.734005i \(-0.262351\pi\)
0.679144 + 0.734005i \(0.262351\pi\)
\(798\) 0 0
\(799\) 2.18087 0.0771537
\(800\) 0 0
\(801\) 19.5977 0.692452
\(802\) 0 0
\(803\) 56.6671 1.99974
\(804\) 0 0
\(805\) −32.9507 −1.16136
\(806\) 0 0
\(807\) −9.30847 −0.327674
\(808\) 0 0
\(809\) −1.31360 −0.0461837 −0.0230919 0.999733i \(-0.507351\pi\)
−0.0230919 + 0.999733i \(0.507351\pi\)
\(810\) 0 0
\(811\) −55.3838 −1.94479 −0.972395 0.233342i \(-0.925034\pi\)
−0.972395 + 0.233342i \(0.925034\pi\)
\(812\) 0 0
\(813\) −23.8508 −0.836485
\(814\) 0 0
\(815\) 28.7952 1.00865
\(816\) 0 0
\(817\) −14.0172 −0.490401
\(818\) 0 0
\(819\) −9.80296 −0.342543
\(820\) 0 0
\(821\) 36.8984 1.28776 0.643882 0.765125i \(-0.277323\pi\)
0.643882 + 0.765125i \(0.277323\pi\)
\(822\) 0 0
\(823\) 41.9170 1.46113 0.730567 0.682841i \(-0.239256\pi\)
0.730567 + 0.682841i \(0.239256\pi\)
\(824\) 0 0
\(825\) −0.528814 −0.0184109
\(826\) 0 0
\(827\) −4.12791 −0.143542 −0.0717708 0.997421i \(-0.522865\pi\)
−0.0717708 + 0.997421i \(0.522865\pi\)
\(828\) 0 0
\(829\) 43.6665 1.51660 0.758300 0.651905i \(-0.226030\pi\)
0.758300 + 0.651905i \(0.226030\pi\)
\(830\) 0 0
\(831\) −15.9639 −0.553783
\(832\) 0 0
\(833\) 9.71340 0.336549
\(834\) 0 0
\(835\) 6.65454 0.230290
\(836\) 0 0
\(837\) 1.03517 0.0357807
\(838\) 0 0
\(839\) −52.3291 −1.80660 −0.903300 0.429008i \(-0.858863\pi\)
−0.903300 + 0.429008i \(0.858863\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 2.66878 0.0919175
\(844\) 0 0
\(845\) 2.19814 0.0756182
\(846\) 0 0
\(847\) 9.40438 0.323138
\(848\) 0 0
\(849\) 4.60201 0.157941
\(850\) 0 0
\(851\) −16.8793 −0.578616
\(852\) 0 0
\(853\) −16.7742 −0.574339 −0.287169 0.957880i \(-0.592714\pi\)
−0.287169 + 0.957880i \(0.592714\pi\)
\(854\) 0 0
\(855\) −27.3543 −0.935496
\(856\) 0 0
\(857\) 13.2516 0.452667 0.226334 0.974050i \(-0.427326\pi\)
0.226334 + 0.974050i \(0.427326\pi\)
\(858\) 0 0
\(859\) 20.3091 0.692936 0.346468 0.938062i \(-0.387381\pi\)
0.346468 + 0.938062i \(0.387381\pi\)
\(860\) 0 0
\(861\) 26.0052 0.886256
\(862\) 0 0
\(863\) 12.0234 0.409280 0.204640 0.978837i \(-0.434398\pi\)
0.204640 + 0.978837i \(0.434398\pi\)
\(864\) 0 0
\(865\) 6.31584 0.214745
\(866\) 0 0
\(867\) −14.1681 −0.481174
\(868\) 0 0
\(869\) −7.18442 −0.243715
\(870\) 0 0
\(871\) −8.95057 −0.303279
\(872\) 0 0
\(873\) 34.4771 1.16687
\(874\) 0 0
\(875\) −49.5253 −1.67426
\(876\) 0 0
\(877\) −34.5918 −1.16808 −0.584041 0.811724i \(-0.698529\pi\)
−0.584041 + 0.811724i \(0.698529\pi\)
\(878\) 0 0
\(879\) 14.5910 0.492141
\(880\) 0 0
\(881\) 33.5740 1.13114 0.565569 0.824701i \(-0.308657\pi\)
0.565569 + 0.824701i \(0.308657\pi\)
\(882\) 0 0
\(883\) 13.7440 0.462522 0.231261 0.972892i \(-0.425715\pi\)
0.231261 + 0.972892i \(0.425715\pi\)
\(884\) 0 0
\(885\) −24.6746 −0.829429
\(886\) 0 0
\(887\) −13.4578 −0.451868 −0.225934 0.974143i \(-0.572543\pi\)
−0.225934 + 0.974143i \(0.572543\pi\)
\(888\) 0 0
\(889\) 7.82194 0.262339
\(890\) 0 0
\(891\) 10.1653 0.340551
\(892\) 0 0
\(893\) 14.9164 0.499159
\(894\) 0 0
\(895\) −2.92473 −0.0977631
\(896\) 0 0
\(897\) −2.98054 −0.0995173
\(898\) 0 0
\(899\) 0.227533 0.00758865
\(900\) 0 0
\(901\) −2.54130 −0.0846630
\(902\) 0 0
\(903\) −9.57123 −0.318511
\(904\) 0 0
\(905\) 34.1370 1.13475
\(906\) 0 0
\(907\) 21.3789 0.709875 0.354937 0.934890i \(-0.384502\pi\)
0.354937 + 0.934890i \(0.384502\pi\)
\(908\) 0 0
\(909\) −3.92769 −0.130273
\(910\) 0 0
\(911\) −46.4415 −1.53868 −0.769338 0.638841i \(-0.779414\pi\)
−0.769338 + 0.638841i \(0.779414\pi\)
\(912\) 0 0
\(913\) 5.79986 0.191947
\(914\) 0 0
\(915\) 14.3663 0.474935
\(916\) 0 0
\(917\) 10.9014 0.359997
\(918\) 0 0
\(919\) −27.1716 −0.896307 −0.448154 0.893957i \(-0.647918\pi\)
−0.448154 + 0.893957i \(0.647918\pi\)
\(920\) 0 0
\(921\) −22.5098 −0.741724
\(922\) 0 0
\(923\) 7.11287 0.234123
\(924\) 0 0
\(925\) −0.825621 −0.0271463
\(926\) 0 0
\(927\) −8.90962 −0.292630
\(928\) 0 0
\(929\) 28.7343 0.942743 0.471372 0.881935i \(-0.343759\pi\)
0.471372 + 0.881935i \(0.343759\pi\)
\(930\) 0 0
\(931\) 66.4364 2.17737
\(932\) 0 0
\(933\) −8.93176 −0.292413
\(934\) 0 0
\(935\) 6.45132 0.210981
\(936\) 0 0
\(937\) −42.0740 −1.37450 −0.687249 0.726422i \(-0.741182\pi\)
−0.687249 + 0.726422i \(0.741182\pi\)
\(938\) 0 0
\(939\) 26.2229 0.855752
\(940\) 0 0
\(941\) −37.8989 −1.23547 −0.617734 0.786387i \(-0.711949\pi\)
−0.617734 + 0.786387i \(0.711949\pi\)
\(942\) 0 0
\(943\) −23.6639 −0.770602
\(944\) 0 0
\(945\) −43.5968 −1.41820
\(946\) 0 0
\(947\) −29.7705 −0.967411 −0.483706 0.875231i \(-0.660709\pi\)
−0.483706 + 0.875231i \(0.660709\pi\)
\(948\) 0 0
\(949\) 15.6224 0.507126
\(950\) 0 0
\(951\) −15.9862 −0.518388
\(952\) 0 0
\(953\) 47.3380 1.53343 0.766714 0.641989i \(-0.221890\pi\)
0.766714 + 0.641989i \(0.221890\pi\)
\(954\) 0 0
\(955\) −19.5098 −0.631322
\(956\) 0 0
\(957\) −3.14413 −0.101635
\(958\) 0 0
\(959\) −49.1790 −1.58807
\(960\) 0 0
\(961\) −30.9482 −0.998330
\(962\) 0 0
\(963\) 24.5818 0.792137
\(964\) 0 0
\(965\) 32.3751 1.04219
\(966\) 0 0
\(967\) 33.6621 1.08250 0.541249 0.840862i \(-0.317952\pi\)
0.541249 + 0.840862i \(0.317952\pi\)
\(968\) 0 0
\(969\) 3.88128 0.124685
\(970\) 0 0
\(971\) −51.1600 −1.64180 −0.820901 0.571070i \(-0.806528\pi\)
−0.820901 + 0.571070i \(0.806528\pi\)
\(972\) 0 0
\(973\) −79.0434 −2.53402
\(974\) 0 0
\(975\) −0.145788 −0.00466894
\(976\) 0 0
\(977\) 50.3817 1.61185 0.805926 0.592016i \(-0.201668\pi\)
0.805926 + 0.592016i \(0.201668\pi\)
\(978\) 0 0
\(979\) −31.6129 −1.01035
\(980\) 0 0
\(981\) 7.22909 0.230807
\(982\) 0 0
\(983\) −16.0901 −0.513194 −0.256597 0.966518i \(-0.582601\pi\)
−0.256597 + 0.966518i \(0.582601\pi\)
\(984\) 0 0
\(985\) 1.60074 0.0510037
\(986\) 0 0
\(987\) 10.1852 0.324199
\(988\) 0 0
\(989\) 8.70950 0.276946
\(990\) 0 0
\(991\) −31.2136 −0.991533 −0.495766 0.868456i \(-0.665113\pi\)
−0.495766 + 0.868456i \(0.665113\pi\)
\(992\) 0 0
\(993\) 4.90028 0.155506
\(994\) 0 0
\(995\) 21.4475 0.679932
\(996\) 0 0
\(997\) −61.7232 −1.95479 −0.977397 0.211414i \(-0.932193\pi\)
−0.977397 + 0.211414i \(0.932193\pi\)
\(998\) 0 0
\(999\) −22.3329 −0.706581
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 6032.2.a.ba.1.7 10
4.3 odd 2 3016.2.a.g.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3016.2.a.g.1.4 10 4.3 odd 2
6032.2.a.ba.1.7 10 1.1 even 1 trivial