Properties

Label 4056.2.a.bi.1.3
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $6$
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] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.27700337.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.72245\) of defining polynomial
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.920510 q^{5} -4.87031 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.920510 q^{5} -4.87031 q^{7} +1.00000 q^{9} -0.964624 q^{11} -0.920510 q^{15} +2.47547 q^{17} +3.12841 q^{19} -4.87031 q^{21} -4.92788 q^{23} -4.15266 q^{25} +1.00000 q^{27} +6.90825 q^{29} -4.61253 q^{31} -0.964624 q^{33} +4.48317 q^{35} +4.81522 q^{37} -3.43483 q^{41} +10.0752 q^{43} -0.920510 q^{45} -8.64406 q^{47} +16.7199 q^{49} +2.47547 q^{51} +0.841166 q^{53} +0.887946 q^{55} +3.12841 q^{57} -9.35767 q^{59} +0.793428 q^{61} -4.87031 q^{63} +6.99350 q^{67} -4.92788 q^{69} +15.5279 q^{71} +5.72664 q^{73} -4.15266 q^{75} +4.69801 q^{77} -13.8726 q^{79} +1.00000 q^{81} +14.8552 q^{83} -2.27869 q^{85} +6.90825 q^{87} -10.6563 q^{89} -4.61253 q^{93} -2.87973 q^{95} +9.53602 q^{97} -0.964624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + q^{5} - 5 q^{7} + 6 q^{9} - 6 q^{11} + q^{15} + 9 q^{17} + 7 q^{19} - 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} - 11 q^{31} - 6 q^{33} + 6 q^{35} + 6 q^{37} + 13 q^{41} + 15 q^{43} + q^{45} - 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} + 7 q^{57} - 7 q^{59} + 25 q^{61} - 5 q^{63} + 5 q^{67} + 12 q^{69} + 8 q^{71} + 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} - 13 q^{83} - 35 q^{85} + 7 q^{87} + 33 q^{89} - 11 q^{93} + 47 q^{95} + 50 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.920510 −0.411665 −0.205832 0.978587i \(-0.565990\pi\)
−0.205832 + 0.978587i \(0.565990\pi\)
\(6\) 0 0
\(7\) −4.87031 −1.84080 −0.920401 0.390975i \(-0.872138\pi\)
−0.920401 + 0.390975i \(0.872138\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.964624 −0.290845 −0.145422 0.989370i \(-0.546454\pi\)
−0.145422 + 0.989370i \(0.546454\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −0.920510 −0.237675
\(16\) 0 0
\(17\) 2.47547 0.600389 0.300195 0.953878i \(-0.402948\pi\)
0.300195 + 0.953878i \(0.402948\pi\)
\(18\) 0 0
\(19\) 3.12841 0.717706 0.358853 0.933394i \(-0.383168\pi\)
0.358853 + 0.933394i \(0.383168\pi\)
\(20\) 0 0
\(21\) −4.87031 −1.06279
\(22\) 0 0
\(23\) −4.92788 −1.02753 −0.513767 0.857930i \(-0.671750\pi\)
−0.513767 + 0.857930i \(0.671750\pi\)
\(24\) 0 0
\(25\) −4.15266 −0.830532
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.90825 1.28283 0.641415 0.767194i \(-0.278348\pi\)
0.641415 + 0.767194i \(0.278348\pi\)
\(30\) 0 0
\(31\) −4.61253 −0.828435 −0.414218 0.910178i \(-0.635945\pi\)
−0.414218 + 0.910178i \(0.635945\pi\)
\(32\) 0 0
\(33\) −0.964624 −0.167919
\(34\) 0 0
\(35\) 4.48317 0.757793
\(36\) 0 0
\(37\) 4.81522 0.791617 0.395808 0.918333i \(-0.370464\pi\)
0.395808 + 0.918333i \(0.370464\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.43483 −0.536431 −0.268216 0.963359i \(-0.586434\pi\)
−0.268216 + 0.963359i \(0.586434\pi\)
\(42\) 0 0
\(43\) 10.0752 1.53646 0.768229 0.640175i \(-0.221138\pi\)
0.768229 + 0.640175i \(0.221138\pi\)
\(44\) 0 0
\(45\) −0.920510 −0.137222
\(46\) 0 0
\(47\) −8.64406 −1.26087 −0.630433 0.776244i \(-0.717122\pi\)
−0.630433 + 0.776244i \(0.717122\pi\)
\(48\) 0 0
\(49\) 16.7199 2.38855
\(50\) 0 0
\(51\) 2.47547 0.346635
\(52\) 0 0
\(53\) 0.841166 0.115543 0.0577715 0.998330i \(-0.481601\pi\)
0.0577715 + 0.998330i \(0.481601\pi\)
\(54\) 0 0
\(55\) 0.887946 0.119731
\(56\) 0 0
\(57\) 3.12841 0.414368
\(58\) 0 0
\(59\) −9.35767 −1.21826 −0.609132 0.793069i \(-0.708482\pi\)
−0.609132 + 0.793069i \(0.708482\pi\)
\(60\) 0 0
\(61\) 0.793428 0.101588 0.0507940 0.998709i \(-0.483825\pi\)
0.0507940 + 0.998709i \(0.483825\pi\)
\(62\) 0 0
\(63\) −4.87031 −0.613601
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.99350 0.854392 0.427196 0.904159i \(-0.359501\pi\)
0.427196 + 0.904159i \(0.359501\pi\)
\(68\) 0 0
\(69\) −4.92788 −0.593247
\(70\) 0 0
\(71\) 15.5279 1.84282 0.921411 0.388588i \(-0.127037\pi\)
0.921411 + 0.388588i \(0.127037\pi\)
\(72\) 0 0
\(73\) 5.72664 0.670253 0.335126 0.942173i \(-0.391221\pi\)
0.335126 + 0.942173i \(0.391221\pi\)
\(74\) 0 0
\(75\) −4.15266 −0.479508
\(76\) 0 0
\(77\) 4.69801 0.535388
\(78\) 0 0
\(79\) −13.8726 −1.56079 −0.780395 0.625287i \(-0.784982\pi\)
−0.780395 + 0.625287i \(0.784982\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.8552 1.63057 0.815284 0.579061i \(-0.196581\pi\)
0.815284 + 0.579061i \(0.196581\pi\)
\(84\) 0 0
\(85\) −2.27869 −0.247159
\(86\) 0 0
\(87\) 6.90825 0.740642
\(88\) 0 0
\(89\) −10.6563 −1.12957 −0.564783 0.825239i \(-0.691040\pi\)
−0.564783 + 0.825239i \(0.691040\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.61253 −0.478297
\(94\) 0 0
\(95\) −2.87973 −0.295454
\(96\) 0 0
\(97\) 9.53602 0.968236 0.484118 0.875003i \(-0.339141\pi\)
0.484118 + 0.875003i \(0.339141\pi\)
\(98\) 0 0
\(99\) −0.964624 −0.0969483
\(100\) 0 0
\(101\) 19.3566 1.92605 0.963025 0.269413i \(-0.0868295\pi\)
0.963025 + 0.269413i \(0.0868295\pi\)
\(102\) 0 0
\(103\) −10.7878 −1.06295 −0.531477 0.847073i \(-0.678363\pi\)
−0.531477 + 0.847073i \(0.678363\pi\)
\(104\) 0 0
\(105\) 4.48317 0.437512
\(106\) 0 0
\(107\) −2.55352 −0.246859 −0.123429 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(108\) 0 0
\(109\) 11.1717 1.07005 0.535027 0.844835i \(-0.320301\pi\)
0.535027 + 0.844835i \(0.320301\pi\)
\(110\) 0 0
\(111\) 4.81522 0.457040
\(112\) 0 0
\(113\) 12.1234 1.14047 0.570236 0.821481i \(-0.306852\pi\)
0.570236 + 0.821481i \(0.306852\pi\)
\(114\) 0 0
\(115\) 4.53616 0.422999
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0563 −1.10520
\(120\) 0 0
\(121\) −10.0695 −0.915409
\(122\) 0 0
\(123\) −3.43483 −0.309709
\(124\) 0 0
\(125\) 8.42512 0.753565
\(126\) 0 0
\(127\) 19.8748 1.76360 0.881802 0.471619i \(-0.156330\pi\)
0.881802 + 0.471619i \(0.156330\pi\)
\(128\) 0 0
\(129\) 10.0752 0.887074
\(130\) 0 0
\(131\) 17.5906 1.53690 0.768450 0.639910i \(-0.221028\pi\)
0.768450 + 0.639910i \(0.221028\pi\)
\(132\) 0 0
\(133\) −15.2363 −1.32115
\(134\) 0 0
\(135\) −0.920510 −0.0792249
\(136\) 0 0
\(137\) 17.3054 1.47850 0.739251 0.673430i \(-0.235180\pi\)
0.739251 + 0.673430i \(0.235180\pi\)
\(138\) 0 0
\(139\) 12.2816 1.04171 0.520856 0.853644i \(-0.325613\pi\)
0.520856 + 0.853644i \(0.325613\pi\)
\(140\) 0 0
\(141\) −8.64406 −0.727961
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.35911 −0.528095
\(146\) 0 0
\(147\) 16.7199 1.37903
\(148\) 0 0
\(149\) 2.37832 0.194840 0.0974199 0.995243i \(-0.468941\pi\)
0.0974199 + 0.995243i \(0.468941\pi\)
\(150\) 0 0
\(151\) 6.80104 0.553460 0.276730 0.960948i \(-0.410749\pi\)
0.276730 + 0.960948i \(0.410749\pi\)
\(152\) 0 0
\(153\) 2.47547 0.200130
\(154\) 0 0
\(155\) 4.24588 0.341037
\(156\) 0 0
\(157\) −15.8025 −1.26118 −0.630588 0.776118i \(-0.717186\pi\)
−0.630588 + 0.776118i \(0.717186\pi\)
\(158\) 0 0
\(159\) 0.841166 0.0667088
\(160\) 0 0
\(161\) 24.0003 1.89149
\(162\) 0 0
\(163\) 1.66375 0.130315 0.0651575 0.997875i \(-0.479245\pi\)
0.0651575 + 0.997875i \(0.479245\pi\)
\(164\) 0 0
\(165\) 0.887946 0.0691265
\(166\) 0 0
\(167\) 6.04374 0.467679 0.233839 0.972275i \(-0.424871\pi\)
0.233839 + 0.972275i \(0.424871\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.12841 0.239235
\(172\) 0 0
\(173\) 10.9425 0.831946 0.415973 0.909377i \(-0.363441\pi\)
0.415973 + 0.909377i \(0.363441\pi\)
\(174\) 0 0
\(175\) 20.2247 1.52885
\(176\) 0 0
\(177\) −9.35767 −0.703365
\(178\) 0 0
\(179\) −2.06741 −0.154525 −0.0772627 0.997011i \(-0.524618\pi\)
−0.0772627 + 0.997011i \(0.524618\pi\)
\(180\) 0 0
\(181\) −5.10368 −0.379353 −0.189677 0.981847i \(-0.560744\pi\)
−0.189677 + 0.981847i \(0.560744\pi\)
\(182\) 0 0
\(183\) 0.793428 0.0586519
\(184\) 0 0
\(185\) −4.43245 −0.325881
\(186\) 0 0
\(187\) −2.38790 −0.174620
\(188\) 0 0
\(189\) −4.87031 −0.354263
\(190\) 0 0
\(191\) 0.862551 0.0624120 0.0312060 0.999513i \(-0.490065\pi\)
0.0312060 + 0.999513i \(0.490065\pi\)
\(192\) 0 0
\(193\) −5.55727 −0.400021 −0.200010 0.979794i \(-0.564098\pi\)
−0.200010 + 0.979794i \(0.564098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.78884 0.483685 0.241842 0.970316i \(-0.422248\pi\)
0.241842 + 0.970316i \(0.422248\pi\)
\(198\) 0 0
\(199\) 7.51699 0.532865 0.266433 0.963854i \(-0.414155\pi\)
0.266433 + 0.963854i \(0.414155\pi\)
\(200\) 0 0
\(201\) 6.99350 0.493283
\(202\) 0 0
\(203\) −33.6453 −2.36143
\(204\) 0 0
\(205\) 3.16180 0.220830
\(206\) 0 0
\(207\) −4.92788 −0.342511
\(208\) 0 0
\(209\) −3.01774 −0.208741
\(210\) 0 0
\(211\) 15.9780 1.09997 0.549984 0.835175i \(-0.314634\pi\)
0.549984 + 0.835175i \(0.314634\pi\)
\(212\) 0 0
\(213\) 15.5279 1.06395
\(214\) 0 0
\(215\) −9.27435 −0.632505
\(216\) 0 0
\(217\) 22.4644 1.52499
\(218\) 0 0
\(219\) 5.72664 0.386971
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −26.5641 −1.77887 −0.889433 0.457066i \(-0.848900\pi\)
−0.889433 + 0.457066i \(0.848900\pi\)
\(224\) 0 0
\(225\) −4.15266 −0.276844
\(226\) 0 0
\(227\) −9.03932 −0.599961 −0.299980 0.953945i \(-0.596980\pi\)
−0.299980 + 0.953945i \(0.596980\pi\)
\(228\) 0 0
\(229\) −10.5057 −0.694235 −0.347118 0.937822i \(-0.612840\pi\)
−0.347118 + 0.937822i \(0.612840\pi\)
\(230\) 0 0
\(231\) 4.69801 0.309106
\(232\) 0 0
\(233\) −15.7349 −1.03083 −0.515415 0.856941i \(-0.672362\pi\)
−0.515415 + 0.856941i \(0.672362\pi\)
\(234\) 0 0
\(235\) 7.95694 0.519054
\(236\) 0 0
\(237\) −13.8726 −0.901122
\(238\) 0 0
\(239\) −18.3360 −1.18606 −0.593028 0.805182i \(-0.702068\pi\)
−0.593028 + 0.805182i \(0.702068\pi\)
\(240\) 0 0
\(241\) 3.01936 0.194494 0.0972471 0.995260i \(-0.468996\pi\)
0.0972471 + 0.995260i \(0.468996\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −15.3908 −0.983283
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 14.8552 0.941409
\(250\) 0 0
\(251\) 17.3054 1.09231 0.546154 0.837685i \(-0.316091\pi\)
0.546154 + 0.837685i \(0.316091\pi\)
\(252\) 0 0
\(253\) 4.75355 0.298853
\(254\) 0 0
\(255\) −2.27869 −0.142697
\(256\) 0 0
\(257\) −6.83635 −0.426440 −0.213220 0.977004i \(-0.568395\pi\)
−0.213220 + 0.977004i \(0.568395\pi\)
\(258\) 0 0
\(259\) −23.4516 −1.45721
\(260\) 0 0
\(261\) 6.90825 0.427610
\(262\) 0 0
\(263\) 0.720029 0.0443989 0.0221995 0.999754i \(-0.492933\pi\)
0.0221995 + 0.999754i \(0.492933\pi\)
\(264\) 0 0
\(265\) −0.774302 −0.0475650
\(266\) 0 0
\(267\) −10.6563 −0.652156
\(268\) 0 0
\(269\) 11.1138 0.677622 0.338811 0.940854i \(-0.389975\pi\)
0.338811 + 0.940854i \(0.389975\pi\)
\(270\) 0 0
\(271\) 21.3436 1.29653 0.648266 0.761414i \(-0.275494\pi\)
0.648266 + 0.761414i \(0.275494\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00576 0.241556
\(276\) 0 0
\(277\) −16.9693 −1.01959 −0.509794 0.860296i \(-0.670279\pi\)
−0.509794 + 0.860296i \(0.670279\pi\)
\(278\) 0 0
\(279\) −4.61253 −0.276145
\(280\) 0 0
\(281\) 31.9782 1.90766 0.953829 0.300349i \(-0.0971031\pi\)
0.953829 + 0.300349i \(0.0971031\pi\)
\(282\) 0 0
\(283\) −12.7438 −0.757541 −0.378770 0.925491i \(-0.623653\pi\)
−0.378770 + 0.925491i \(0.623653\pi\)
\(284\) 0 0
\(285\) −2.87973 −0.170581
\(286\) 0 0
\(287\) 16.7287 0.987464
\(288\) 0 0
\(289\) −10.8721 −0.639533
\(290\) 0 0
\(291\) 9.53602 0.559011
\(292\) 0 0
\(293\) −28.9771 −1.69286 −0.846431 0.532498i \(-0.821253\pi\)
−0.846431 + 0.532498i \(0.821253\pi\)
\(294\) 0 0
\(295\) 8.61383 0.501516
\(296\) 0 0
\(297\) −0.964624 −0.0559731
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −49.0694 −2.82832
\(302\) 0 0
\(303\) 19.3566 1.11201
\(304\) 0 0
\(305\) −0.730358 −0.0418202
\(306\) 0 0
\(307\) −22.8992 −1.30693 −0.653464 0.756957i \(-0.726685\pi\)
−0.653464 + 0.756957i \(0.726685\pi\)
\(308\) 0 0
\(309\) −10.7878 −0.613696
\(310\) 0 0
\(311\) −33.8141 −1.91742 −0.958710 0.284385i \(-0.908211\pi\)
−0.958710 + 0.284385i \(0.908211\pi\)
\(312\) 0 0
\(313\) −11.1846 −0.632190 −0.316095 0.948728i \(-0.602372\pi\)
−0.316095 + 0.948728i \(0.602372\pi\)
\(314\) 0 0
\(315\) 4.48317 0.252598
\(316\) 0 0
\(317\) 22.4998 1.26372 0.631858 0.775084i \(-0.282292\pi\)
0.631858 + 0.775084i \(0.282292\pi\)
\(318\) 0 0
\(319\) −6.66386 −0.373104
\(320\) 0 0
\(321\) −2.55352 −0.142524
\(322\) 0 0
\(323\) 7.74427 0.430903
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.1717 0.617796
\(328\) 0 0
\(329\) 42.0992 2.32100
\(330\) 0 0
\(331\) 8.39583 0.461477 0.230738 0.973016i \(-0.425886\pi\)
0.230738 + 0.973016i \(0.425886\pi\)
\(332\) 0 0
\(333\) 4.81522 0.263872
\(334\) 0 0
\(335\) −6.43759 −0.351723
\(336\) 0 0
\(337\) 3.76684 0.205193 0.102596 0.994723i \(-0.467285\pi\)
0.102596 + 0.994723i \(0.467285\pi\)
\(338\) 0 0
\(339\) 12.1234 0.658451
\(340\) 0 0
\(341\) 4.44936 0.240946
\(342\) 0 0
\(343\) −47.3387 −2.55605
\(344\) 0 0
\(345\) 4.53616 0.244219
\(346\) 0 0
\(347\) 10.3473 0.555470 0.277735 0.960658i \(-0.410416\pi\)
0.277735 + 0.960658i \(0.410416\pi\)
\(348\) 0 0
\(349\) −6.75407 −0.361537 −0.180769 0.983526i \(-0.557859\pi\)
−0.180769 + 0.983526i \(0.557859\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.6346 1.25794 0.628972 0.777428i \(-0.283476\pi\)
0.628972 + 0.777428i \(0.283476\pi\)
\(354\) 0 0
\(355\) −14.2936 −0.758625
\(356\) 0 0
\(357\) −12.0563 −0.638086
\(358\) 0 0
\(359\) −8.87652 −0.468485 −0.234242 0.972178i \(-0.575261\pi\)
−0.234242 + 0.972178i \(0.575261\pi\)
\(360\) 0 0
\(361\) −9.21307 −0.484898
\(362\) 0 0
\(363\) −10.0695 −0.528512
\(364\) 0 0
\(365\) −5.27143 −0.275919
\(366\) 0 0
\(367\) −1.13397 −0.0591929 −0.0295965 0.999562i \(-0.509422\pi\)
−0.0295965 + 0.999562i \(0.509422\pi\)
\(368\) 0 0
\(369\) −3.43483 −0.178810
\(370\) 0 0
\(371\) −4.09674 −0.212692
\(372\) 0 0
\(373\) −14.0604 −0.728020 −0.364010 0.931395i \(-0.618593\pi\)
−0.364010 + 0.931395i \(0.618593\pi\)
\(374\) 0 0
\(375\) 8.42512 0.435071
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −24.1182 −1.23887 −0.619435 0.785048i \(-0.712638\pi\)
−0.619435 + 0.785048i \(0.712638\pi\)
\(380\) 0 0
\(381\) 19.8748 1.01822
\(382\) 0 0
\(383\) −7.39256 −0.377742 −0.188871 0.982002i \(-0.560483\pi\)
−0.188871 + 0.982002i \(0.560483\pi\)
\(384\) 0 0
\(385\) −4.32457 −0.220400
\(386\) 0 0
\(387\) 10.0752 0.512153
\(388\) 0 0
\(389\) 35.4230 1.79602 0.898010 0.439975i \(-0.145013\pi\)
0.898010 + 0.439975i \(0.145013\pi\)
\(390\) 0 0
\(391\) −12.1988 −0.616920
\(392\) 0 0
\(393\) 17.5906 0.887330
\(394\) 0 0
\(395\) 12.7699 0.642522
\(396\) 0 0
\(397\) 18.4553 0.926244 0.463122 0.886294i \(-0.346729\pi\)
0.463122 + 0.886294i \(0.346729\pi\)
\(398\) 0 0
\(399\) −15.2363 −0.762769
\(400\) 0 0
\(401\) 21.9812 1.09769 0.548845 0.835924i \(-0.315068\pi\)
0.548845 + 0.835924i \(0.315068\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.920510 −0.0457405
\(406\) 0 0
\(407\) −4.64487 −0.230238
\(408\) 0 0
\(409\) 11.1327 0.550479 0.275239 0.961376i \(-0.411243\pi\)
0.275239 + 0.961376i \(0.411243\pi\)
\(410\) 0 0
\(411\) 17.3054 0.853613
\(412\) 0 0
\(413\) 45.5747 2.24258
\(414\) 0 0
\(415\) −13.6743 −0.671247
\(416\) 0 0
\(417\) 12.2816 0.601433
\(418\) 0 0
\(419\) 7.42461 0.362716 0.181358 0.983417i \(-0.441951\pi\)
0.181358 + 0.983417i \(0.441951\pi\)
\(420\) 0 0
\(421\) 14.3283 0.698318 0.349159 0.937064i \(-0.386467\pi\)
0.349159 + 0.937064i \(0.386467\pi\)
\(422\) 0 0
\(423\) −8.64406 −0.420288
\(424\) 0 0
\(425\) −10.2798 −0.498643
\(426\) 0 0
\(427\) −3.86424 −0.187003
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −22.3212 −1.07517 −0.537586 0.843209i \(-0.680664\pi\)
−0.537586 + 0.843209i \(0.680664\pi\)
\(432\) 0 0
\(433\) −37.0848 −1.78218 −0.891092 0.453823i \(-0.850060\pi\)
−0.891092 + 0.453823i \(0.850060\pi\)
\(434\) 0 0
\(435\) −6.35911 −0.304896
\(436\) 0 0
\(437\) −15.4164 −0.737467
\(438\) 0 0
\(439\) −0.458627 −0.0218891 −0.0109445 0.999940i \(-0.503484\pi\)
−0.0109445 + 0.999940i \(0.503484\pi\)
\(440\) 0 0
\(441\) 16.7199 0.796184
\(442\) 0 0
\(443\) −14.0996 −0.669893 −0.334947 0.942237i \(-0.608718\pi\)
−0.334947 + 0.942237i \(0.608718\pi\)
\(444\) 0 0
\(445\) 9.80924 0.465003
\(446\) 0 0
\(447\) 2.37832 0.112491
\(448\) 0 0
\(449\) 18.6386 0.879608 0.439804 0.898094i \(-0.355048\pi\)
0.439804 + 0.898094i \(0.355048\pi\)
\(450\) 0 0
\(451\) 3.31332 0.156018
\(452\) 0 0
\(453\) 6.80104 0.319541
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.6100 −0.589869 −0.294934 0.955518i \(-0.595298\pi\)
−0.294934 + 0.955518i \(0.595298\pi\)
\(458\) 0 0
\(459\) 2.47547 0.115545
\(460\) 0 0
\(461\) 5.05767 0.235559 0.117780 0.993040i \(-0.462422\pi\)
0.117780 + 0.993040i \(0.462422\pi\)
\(462\) 0 0
\(463\) 3.32946 0.154733 0.0773665 0.997003i \(-0.475349\pi\)
0.0773665 + 0.997003i \(0.475349\pi\)
\(464\) 0 0
\(465\) 4.24588 0.196898
\(466\) 0 0
\(467\) 18.4337 0.853009 0.426504 0.904486i \(-0.359745\pi\)
0.426504 + 0.904486i \(0.359745\pi\)
\(468\) 0 0
\(469\) −34.0605 −1.57277
\(470\) 0 0
\(471\) −15.8025 −0.728140
\(472\) 0 0
\(473\) −9.71880 −0.446871
\(474\) 0 0
\(475\) −12.9912 −0.596078
\(476\) 0 0
\(477\) 0.841166 0.0385144
\(478\) 0 0
\(479\) −30.8641 −1.41022 −0.705108 0.709100i \(-0.749102\pi\)
−0.705108 + 0.709100i \(0.749102\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.0003 1.09205
\(484\) 0 0
\(485\) −8.77800 −0.398588
\(486\) 0 0
\(487\) −1.52808 −0.0692438 −0.0346219 0.999400i \(-0.511023\pi\)
−0.0346219 + 0.999400i \(0.511023\pi\)
\(488\) 0 0
\(489\) 1.66375 0.0752374
\(490\) 0 0
\(491\) −42.2011 −1.90451 −0.952254 0.305308i \(-0.901241\pi\)
−0.952254 + 0.305308i \(0.901241\pi\)
\(492\) 0 0
\(493\) 17.1011 0.770197
\(494\) 0 0
\(495\) 0.887946 0.0399102
\(496\) 0 0
\(497\) −75.6256 −3.39227
\(498\) 0 0
\(499\) −4.73294 −0.211875 −0.105938 0.994373i \(-0.533784\pi\)
−0.105938 + 0.994373i \(0.533784\pi\)
\(500\) 0 0
\(501\) 6.04374 0.270014
\(502\) 0 0
\(503\) 1.83801 0.0819529 0.0409765 0.999160i \(-0.486953\pi\)
0.0409765 + 0.999160i \(0.486953\pi\)
\(504\) 0 0
\(505\) −17.8179 −0.792887
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 39.7800 1.76322 0.881608 0.471982i \(-0.156461\pi\)
0.881608 + 0.471982i \(0.156461\pi\)
\(510\) 0 0
\(511\) −27.8905 −1.23380
\(512\) 0 0
\(513\) 3.12841 0.138123
\(514\) 0 0
\(515\) 9.93028 0.437580
\(516\) 0 0
\(517\) 8.33826 0.366716
\(518\) 0 0
\(519\) 10.9425 0.480324
\(520\) 0 0
\(521\) 18.3812 0.805295 0.402648 0.915355i \(-0.368090\pi\)
0.402648 + 0.915355i \(0.368090\pi\)
\(522\) 0 0
\(523\) 11.9577 0.522874 0.261437 0.965221i \(-0.415804\pi\)
0.261437 + 0.965221i \(0.415804\pi\)
\(524\) 0 0
\(525\) 20.2247 0.882679
\(526\) 0 0
\(527\) −11.4182 −0.497384
\(528\) 0 0
\(529\) 1.28399 0.0558256
\(530\) 0 0
\(531\) −9.35767 −0.406088
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 2.35055 0.101623
\(536\) 0 0
\(537\) −2.06741 −0.0892153
\(538\) 0 0
\(539\) −16.1284 −0.694699
\(540\) 0 0
\(541\) 39.9455 1.71739 0.858696 0.512485i \(-0.171275\pi\)
0.858696 + 0.512485i \(0.171275\pi\)
\(542\) 0 0
\(543\) −5.10368 −0.219020
\(544\) 0 0
\(545\) −10.2837 −0.440504
\(546\) 0 0
\(547\) −36.0434 −1.54110 −0.770551 0.637378i \(-0.780019\pi\)
−0.770551 + 0.637378i \(0.780019\pi\)
\(548\) 0 0
\(549\) 0.793428 0.0338627
\(550\) 0 0
\(551\) 21.6118 0.920694
\(552\) 0 0
\(553\) 67.5638 2.87310
\(554\) 0 0
\(555\) −4.43245 −0.188147
\(556\) 0 0
\(557\) 14.2929 0.605611 0.302806 0.953052i \(-0.402077\pi\)
0.302806 + 0.953052i \(0.402077\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −2.38790 −0.100817
\(562\) 0 0
\(563\) −1.32405 −0.0558021 −0.0279010 0.999611i \(-0.508882\pi\)
−0.0279010 + 0.999611i \(0.508882\pi\)
\(564\) 0 0
\(565\) −11.1597 −0.469492
\(566\) 0 0
\(567\) −4.87031 −0.204534
\(568\) 0 0
\(569\) 17.3321 0.726598 0.363299 0.931673i \(-0.381650\pi\)
0.363299 + 0.931673i \(0.381650\pi\)
\(570\) 0 0
\(571\) −12.6753 −0.530446 −0.265223 0.964187i \(-0.585446\pi\)
−0.265223 + 0.964187i \(0.585446\pi\)
\(572\) 0 0
\(573\) 0.862551 0.0360336
\(574\) 0 0
\(575\) 20.4638 0.853400
\(576\) 0 0
\(577\) −5.62372 −0.234119 −0.117059 0.993125i \(-0.537347\pi\)
−0.117059 + 0.993125i \(0.537347\pi\)
\(578\) 0 0
\(579\) −5.55727 −0.230952
\(580\) 0 0
\(581\) −72.3493 −3.00155
\(582\) 0 0
\(583\) −0.811409 −0.0336051
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.3576 0.798973 0.399487 0.916739i \(-0.369188\pi\)
0.399487 + 0.916739i \(0.369188\pi\)
\(588\) 0 0
\(589\) −14.4299 −0.594573
\(590\) 0 0
\(591\) 6.78884 0.279255
\(592\) 0 0
\(593\) −3.31980 −0.136328 −0.0681638 0.997674i \(-0.521714\pi\)
−0.0681638 + 0.997674i \(0.521714\pi\)
\(594\) 0 0
\(595\) 11.0979 0.454971
\(596\) 0 0
\(597\) 7.51699 0.307650
\(598\) 0 0
\(599\) 12.9233 0.528032 0.264016 0.964518i \(-0.414953\pi\)
0.264016 + 0.964518i \(0.414953\pi\)
\(600\) 0 0
\(601\) 46.5474 1.89871 0.949354 0.314208i \(-0.101739\pi\)
0.949354 + 0.314208i \(0.101739\pi\)
\(602\) 0 0
\(603\) 6.99350 0.284797
\(604\) 0 0
\(605\) 9.26908 0.376842
\(606\) 0 0
\(607\) −5.28623 −0.214562 −0.107281 0.994229i \(-0.534214\pi\)
−0.107281 + 0.994229i \(0.534214\pi\)
\(608\) 0 0
\(609\) −33.6453 −1.36338
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −19.9724 −0.806678 −0.403339 0.915051i \(-0.632150\pi\)
−0.403339 + 0.915051i \(0.632150\pi\)
\(614\) 0 0
\(615\) 3.16180 0.127496
\(616\) 0 0
\(617\) −35.8208 −1.44209 −0.721045 0.692888i \(-0.756338\pi\)
−0.721045 + 0.692888i \(0.756338\pi\)
\(618\) 0 0
\(619\) 38.1115 1.53183 0.765915 0.642942i \(-0.222287\pi\)
0.765915 + 0.642942i \(0.222287\pi\)
\(620\) 0 0
\(621\) −4.92788 −0.197749
\(622\) 0 0
\(623\) 51.8995 2.07931
\(624\) 0 0
\(625\) 13.0079 0.520316
\(626\) 0 0
\(627\) −3.01774 −0.120517
\(628\) 0 0
\(629\) 11.9199 0.475278
\(630\) 0 0
\(631\) −11.5675 −0.460494 −0.230247 0.973132i \(-0.573953\pi\)
−0.230247 + 0.973132i \(0.573953\pi\)
\(632\) 0 0
\(633\) 15.9780 0.635067
\(634\) 0 0
\(635\) −18.2950 −0.726014
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 15.5279 0.614274
\(640\) 0 0
\(641\) −30.9802 −1.22364 −0.611822 0.790995i \(-0.709563\pi\)
−0.611822 + 0.790995i \(0.709563\pi\)
\(642\) 0 0
\(643\) 7.04913 0.277991 0.138995 0.990293i \(-0.455613\pi\)
0.138995 + 0.990293i \(0.455613\pi\)
\(644\) 0 0
\(645\) −9.27435 −0.365177
\(646\) 0 0
\(647\) −15.7204 −0.618033 −0.309016 0.951057i \(-0.600000\pi\)
−0.309016 + 0.951057i \(0.600000\pi\)
\(648\) 0 0
\(649\) 9.02663 0.354326
\(650\) 0 0
\(651\) 22.4644 0.880451
\(652\) 0 0
\(653\) 44.6261 1.74635 0.873177 0.487403i \(-0.162056\pi\)
0.873177 + 0.487403i \(0.162056\pi\)
\(654\) 0 0
\(655\) −16.1924 −0.632688
\(656\) 0 0
\(657\) 5.72664 0.223418
\(658\) 0 0
\(659\) 32.3994 1.26210 0.631050 0.775742i \(-0.282624\pi\)
0.631050 + 0.775742i \(0.282624\pi\)
\(660\) 0 0
\(661\) 42.3068 1.64555 0.822773 0.568371i \(-0.192426\pi\)
0.822773 + 0.568371i \(0.192426\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 14.0252 0.543873
\(666\) 0 0
\(667\) −34.0430 −1.31815
\(668\) 0 0
\(669\) −26.5641 −1.02703
\(670\) 0 0
\(671\) −0.765359 −0.0295464
\(672\) 0 0
\(673\) 23.3519 0.900150 0.450075 0.892991i \(-0.351397\pi\)
0.450075 + 0.892991i \(0.351397\pi\)
\(674\) 0 0
\(675\) −4.15266 −0.159836
\(676\) 0 0
\(677\) −5.86331 −0.225345 −0.112673 0.993632i \(-0.535941\pi\)
−0.112673 + 0.993632i \(0.535941\pi\)
\(678\) 0 0
\(679\) −46.4433 −1.78233
\(680\) 0 0
\(681\) −9.03932 −0.346387
\(682\) 0 0
\(683\) 25.7385 0.984857 0.492428 0.870353i \(-0.336109\pi\)
0.492428 + 0.870353i \(0.336109\pi\)
\(684\) 0 0
\(685\) −15.9298 −0.608647
\(686\) 0 0
\(687\) −10.5057 −0.400817
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 32.7218 1.24479 0.622397 0.782701i \(-0.286159\pi\)
0.622397 + 0.782701i \(0.286159\pi\)
\(692\) 0 0
\(693\) 4.69801 0.178463
\(694\) 0 0
\(695\) −11.3053 −0.428836
\(696\) 0 0
\(697\) −8.50282 −0.322067
\(698\) 0 0
\(699\) −15.7349 −0.595150
\(700\) 0 0
\(701\) 37.3888 1.41216 0.706078 0.708134i \(-0.250463\pi\)
0.706078 + 0.708134i \(0.250463\pi\)
\(702\) 0 0
\(703\) 15.0640 0.568148
\(704\) 0 0
\(705\) 7.95694 0.299676
\(706\) 0 0
\(707\) −94.2724 −3.54548
\(708\) 0 0
\(709\) −20.5069 −0.770153 −0.385076 0.922885i \(-0.625825\pi\)
−0.385076 + 0.922885i \(0.625825\pi\)
\(710\) 0 0
\(711\) −13.8726 −0.520263
\(712\) 0 0
\(713\) 22.7300 0.851245
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −18.3360 −0.684770
\(718\) 0 0
\(719\) 23.8237 0.888476 0.444238 0.895909i \(-0.353475\pi\)
0.444238 + 0.895909i \(0.353475\pi\)
\(720\) 0 0
\(721\) 52.5399 1.95669
\(722\) 0 0
\(723\) 3.01936 0.112291
\(724\) 0 0
\(725\) −28.6876 −1.06543
\(726\) 0 0
\(727\) 1.65273 0.0612962 0.0306481 0.999530i \(-0.490243\pi\)
0.0306481 + 0.999530i \(0.490243\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.9409 0.922473
\(732\) 0 0
\(733\) −42.0736 −1.55402 −0.777011 0.629486i \(-0.783265\pi\)
−0.777011 + 0.629486i \(0.783265\pi\)
\(734\) 0 0
\(735\) −15.3908 −0.567699
\(736\) 0 0
\(737\) −6.74609 −0.248496
\(738\) 0 0
\(739\) 42.0726 1.54766 0.773832 0.633391i \(-0.218337\pi\)
0.773832 + 0.633391i \(0.218337\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.78602 0.359014 0.179507 0.983757i \(-0.442550\pi\)
0.179507 + 0.983757i \(0.442550\pi\)
\(744\) 0 0
\(745\) −2.18927 −0.0802086
\(746\) 0 0
\(747\) 14.8552 0.543523
\(748\) 0 0
\(749\) 12.4364 0.454418
\(750\) 0 0
\(751\) −25.7737 −0.940495 −0.470247 0.882535i \(-0.655835\pi\)
−0.470247 + 0.882535i \(0.655835\pi\)
\(752\) 0 0
\(753\) 17.3054 0.630644
\(754\) 0 0
\(755\) −6.26042 −0.227840
\(756\) 0 0
\(757\) −20.6200 −0.749446 −0.374723 0.927137i \(-0.622262\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(758\) 0 0
\(759\) 4.75355 0.172543
\(760\) 0 0
\(761\) 5.18524 0.187965 0.0939824 0.995574i \(-0.470040\pi\)
0.0939824 + 0.995574i \(0.470040\pi\)
\(762\) 0 0
\(763\) −54.4096 −1.96976
\(764\) 0 0
\(765\) −2.27869 −0.0823863
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 45.1989 1.62992 0.814958 0.579521i \(-0.196760\pi\)
0.814958 + 0.579521i \(0.196760\pi\)
\(770\) 0 0
\(771\) −6.83635 −0.246205
\(772\) 0 0
\(773\) 12.0422 0.433130 0.216565 0.976268i \(-0.430515\pi\)
0.216565 + 0.976268i \(0.430515\pi\)
\(774\) 0 0
\(775\) 19.1543 0.688042
\(776\) 0 0
\(777\) −23.4516 −0.841320
\(778\) 0 0
\(779\) −10.7456 −0.385000
\(780\) 0 0
\(781\) −14.9786 −0.535976
\(782\) 0 0
\(783\) 6.90825 0.246881
\(784\) 0 0
\(785\) 14.5463 0.519181
\(786\) 0 0
\(787\) −40.1313 −1.43053 −0.715264 0.698855i \(-0.753693\pi\)
−0.715264 + 0.698855i \(0.753693\pi\)
\(788\) 0 0
\(789\) 0.720029 0.0256337
\(790\) 0 0
\(791\) −59.0445 −2.09938
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.774302 −0.0274617
\(796\) 0 0
\(797\) 8.86591 0.314047 0.157023 0.987595i \(-0.449810\pi\)
0.157023 + 0.987595i \(0.449810\pi\)
\(798\) 0 0
\(799\) −21.3981 −0.757010
\(800\) 0 0
\(801\) −10.6563 −0.376522
\(802\) 0 0
\(803\) −5.52405 −0.194940
\(804\) 0 0
\(805\) −22.0925 −0.778658
\(806\) 0 0
\(807\) 11.1138 0.391225
\(808\) 0 0
\(809\) −28.3977 −0.998411 −0.499206 0.866484i \(-0.666375\pi\)
−0.499206 + 0.866484i \(0.666375\pi\)
\(810\) 0 0
\(811\) −24.6373 −0.865132 −0.432566 0.901602i \(-0.642392\pi\)
−0.432566 + 0.901602i \(0.642392\pi\)
\(812\) 0 0
\(813\) 21.3436 0.748554
\(814\) 0 0
\(815\) −1.53150 −0.0536461
\(816\) 0 0
\(817\) 31.5194 1.10272
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.2417 0.462139 0.231070 0.972937i \(-0.425777\pi\)
0.231070 + 0.972937i \(0.425777\pi\)
\(822\) 0 0
\(823\) 25.4249 0.886257 0.443129 0.896458i \(-0.353868\pi\)
0.443129 + 0.896458i \(0.353868\pi\)
\(824\) 0 0
\(825\) 4.00576 0.139462
\(826\) 0 0
\(827\) −44.5293 −1.54844 −0.774218 0.632920i \(-0.781856\pi\)
−0.774218 + 0.632920i \(0.781856\pi\)
\(828\) 0 0
\(829\) −31.7843 −1.10392 −0.551958 0.833872i \(-0.686119\pi\)
−0.551958 + 0.833872i \(0.686119\pi\)
\(830\) 0 0
\(831\) −16.9693 −0.588660
\(832\) 0 0
\(833\) 41.3895 1.43406
\(834\) 0 0
\(835\) −5.56332 −0.192527
\(836\) 0 0
\(837\) −4.61253 −0.159432
\(838\) 0 0
\(839\) 51.0143 1.76121 0.880604 0.473853i \(-0.157137\pi\)
0.880604 + 0.473853i \(0.157137\pi\)
\(840\) 0 0
\(841\) 18.7239 0.645651
\(842\) 0 0
\(843\) 31.9782 1.10139
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 49.0415 1.68509
\(848\) 0 0
\(849\) −12.7438 −0.437366
\(850\) 0 0
\(851\) −23.7288 −0.813413
\(852\) 0 0
\(853\) −1.34869 −0.0461784 −0.0230892 0.999733i \(-0.507350\pi\)
−0.0230892 + 0.999733i \(0.507350\pi\)
\(854\) 0 0
\(855\) −2.87973 −0.0984847
\(856\) 0 0
\(857\) −24.7695 −0.846112 −0.423056 0.906104i \(-0.639043\pi\)
−0.423056 + 0.906104i \(0.639043\pi\)
\(858\) 0 0
\(859\) −39.2069 −1.33772 −0.668861 0.743387i \(-0.733218\pi\)
−0.668861 + 0.743387i \(0.733218\pi\)
\(860\) 0 0
\(861\) 16.7287 0.570112
\(862\) 0 0
\(863\) 9.70416 0.330333 0.165167 0.986266i \(-0.447184\pi\)
0.165167 + 0.986266i \(0.447184\pi\)
\(864\) 0 0
\(865\) −10.0727 −0.342483
\(866\) 0 0
\(867\) −10.8721 −0.369234
\(868\) 0 0
\(869\) 13.3818 0.453948
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9.53602 0.322745
\(874\) 0 0
\(875\) −41.0329 −1.38716
\(876\) 0 0
\(877\) −49.7250 −1.67910 −0.839548 0.543286i \(-0.817180\pi\)
−0.839548 + 0.543286i \(0.817180\pi\)
\(878\) 0 0
\(879\) −28.9771 −0.977375
\(880\) 0 0
\(881\) −22.1672 −0.746831 −0.373416 0.927664i \(-0.621813\pi\)
−0.373416 + 0.927664i \(0.621813\pi\)
\(882\) 0 0
\(883\) −45.2133 −1.52155 −0.760774 0.649016i \(-0.775181\pi\)
−0.760774 + 0.649016i \(0.775181\pi\)
\(884\) 0 0
\(885\) 8.61383 0.289551
\(886\) 0 0
\(887\) 45.5752 1.53027 0.765133 0.643872i \(-0.222673\pi\)
0.765133 + 0.643872i \(0.222673\pi\)
\(888\) 0 0
\(889\) −96.7964 −3.24645
\(890\) 0 0
\(891\) −0.964624 −0.0323161
\(892\) 0 0
\(893\) −27.0421 −0.904930
\(894\) 0 0
\(895\) 1.90307 0.0636126
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −31.8645 −1.06274
\(900\) 0 0
\(901\) 2.08228 0.0693708
\(902\) 0 0
\(903\) −49.0694 −1.63293
\(904\) 0 0
\(905\) 4.69799 0.156166
\(906\) 0 0
\(907\) 31.0483 1.03094 0.515471 0.856907i \(-0.327617\pi\)
0.515471 + 0.856907i \(0.327617\pi\)
\(908\) 0 0
\(909\) 19.3566 0.642017
\(910\) 0 0
\(911\) 48.2719 1.59932 0.799660 0.600454i \(-0.205013\pi\)
0.799660 + 0.600454i \(0.205013\pi\)
\(912\) 0 0
\(913\) −14.3297 −0.474243
\(914\) 0 0
\(915\) −0.730358 −0.0241449
\(916\) 0 0
\(917\) −85.6717 −2.82913
\(918\) 0 0
\(919\) −30.4700 −1.00511 −0.502557 0.864544i \(-0.667607\pi\)
−0.502557 + 0.864544i \(0.667607\pi\)
\(920\) 0 0
\(921\) −22.8992 −0.754555
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −19.9960 −0.657463
\(926\) 0 0
\(927\) −10.7878 −0.354318
\(928\) 0 0
\(929\) −0.335144 −0.0109957 −0.00549787 0.999985i \(-0.501750\pi\)
−0.00549787 + 0.999985i \(0.501750\pi\)
\(930\) 0 0
\(931\) 52.3066 1.71428
\(932\) 0 0
\(933\) −33.8141 −1.10702
\(934\) 0 0
\(935\) 2.19808 0.0718850
\(936\) 0 0
\(937\) −40.6902 −1.32929 −0.664646 0.747158i \(-0.731418\pi\)
−0.664646 + 0.747158i \(0.731418\pi\)
\(938\) 0 0
\(939\) −11.1846 −0.364995
\(940\) 0 0
\(941\) −45.6192 −1.48714 −0.743572 0.668655i \(-0.766870\pi\)
−0.743572 + 0.668655i \(0.766870\pi\)
\(942\) 0 0
\(943\) 16.9264 0.551201
\(944\) 0 0
\(945\) 4.48317 0.145837
\(946\) 0 0
\(947\) 9.60704 0.312187 0.156093 0.987742i \(-0.450110\pi\)
0.156093 + 0.987742i \(0.450110\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 22.4998 0.729607
\(952\) 0 0
\(953\) −2.36871 −0.0767299 −0.0383650 0.999264i \(-0.512215\pi\)
−0.0383650 + 0.999264i \(0.512215\pi\)
\(954\) 0 0
\(955\) −0.793986 −0.0256928
\(956\) 0 0
\(957\) −6.66386 −0.215412
\(958\) 0 0
\(959\) −84.2827 −2.72163
\(960\) 0 0
\(961\) −9.72455 −0.313695
\(962\) 0 0
\(963\) −2.55352 −0.0822862
\(964\) 0 0
\(965\) 5.11552 0.164674
\(966\) 0 0
\(967\) −1.08026 −0.0347388 −0.0173694 0.999849i \(-0.505529\pi\)
−0.0173694 + 0.999849i \(0.505529\pi\)
\(968\) 0 0
\(969\) 7.74427 0.248782
\(970\) 0 0
\(971\) 27.3056 0.876278 0.438139 0.898907i \(-0.355638\pi\)
0.438139 + 0.898907i \(0.355638\pi\)
\(972\) 0 0
\(973\) −59.8152 −1.91759
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.9136 0.765064 0.382532 0.923942i \(-0.375052\pi\)
0.382532 + 0.923942i \(0.375052\pi\)
\(978\) 0 0
\(979\) 10.2793 0.328529
\(980\) 0 0
\(981\) 11.1717 0.356685
\(982\) 0 0
\(983\) 8.89131 0.283589 0.141794 0.989896i \(-0.454713\pi\)
0.141794 + 0.989896i \(0.454713\pi\)
\(984\) 0 0
\(985\) −6.24919 −0.199116
\(986\) 0 0
\(987\) 42.0992 1.34003
\(988\) 0 0
\(989\) −49.6495 −1.57876
\(990\) 0 0
\(991\) 21.8568 0.694303 0.347152 0.937809i \(-0.387149\pi\)
0.347152 + 0.937809i \(0.387149\pi\)
\(992\) 0 0
\(993\) 8.39583 0.266434
\(994\) 0 0
\(995\) −6.91947 −0.219362
\(996\) 0 0
\(997\) −38.9135 −1.23240 −0.616202 0.787589i \(-0.711329\pi\)
−0.616202 + 0.787589i \(0.711329\pi\)
\(998\) 0 0
\(999\) 4.81522 0.152347
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 4056.2.a.bi.1.3 yes 6
4.3 odd 2 8112.2.a.cu.1.3 6
13.5 odd 4 4056.2.c.r.337.7 12
13.8 odd 4 4056.2.c.r.337.6 12
13.12 even 2 4056.2.a.bh.1.4 6
52.51 odd 2 8112.2.a.ct.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4056.2.a.bh.1.4 6 13.12 even 2
4056.2.a.bi.1.3 yes 6 1.1 even 1 trivial
4056.2.c.r.337.6 12 13.8 odd 4
4056.2.c.r.337.7 12 13.5 odd 4
8112.2.a.ct.1.4 6 52.51 odd 2
8112.2.a.cu.1.3 6 4.3 odd 2