Properties

Label 592.2.be.e.415.2
Level $592$
Weight $2$
Character 592.415
Analytic conductor $4.727$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [592,2,Mod(319,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.319");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.be (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 46 x^{18} + 885 x^{16} + 9292 x^{14} + 58264 x^{12} + 224256 x^{10} + 523884 x^{8} + 706272 x^{6} + 490896 x^{4} + 133632 x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 415.2
Root \(-2.56563i\) of defining polynomial
Character \(\chi\) \(=\) 592.415
Dual form 592.2.be.e.495.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.28281 + 2.22190i) q^{3} +(3.31311 - 0.887746i) q^{5} +(-0.921902 - 0.532260i) q^{7} +(-1.79122 - 3.10248i) q^{9} +O(q^{10})\) \(q+(-1.28281 + 2.22190i) q^{3} +(3.31311 - 0.887746i) q^{5} +(-0.921902 - 0.532260i) q^{7} +(-1.79122 - 3.10248i) q^{9} +4.10723 q^{11} +(4.46062 - 1.19522i) q^{13} +(-2.27762 + 8.50021i) q^{15} +(-0.143130 + 0.534169i) q^{17} +(-1.22535 - 4.57306i) q^{19} +(2.36526 - 1.36558i) q^{21} +(3.27882 - 3.27882i) q^{23} +(5.85850 - 3.38241i) q^{25} +1.49432 q^{27} +(-6.59744 + 6.59744i) q^{29} +(5.20966 + 5.20966i) q^{31} +(-5.26881 + 9.12584i) q^{33} +(-3.52688 - 0.945024i) q^{35} +(-4.97186 + 3.50437i) q^{37} +(-3.06649 + 11.4443i) q^{39} +(4.62353 + 2.66940i) q^{41} +(1.34410 - 1.34410i) q^{43} +(-8.68873 - 8.68873i) q^{45} +9.47900i q^{47} +(-2.93340 - 5.08079i) q^{49} +(-1.00326 - 1.00326i) q^{51} +(1.36189 + 2.35887i) q^{53} +(13.6077 - 3.64618i) q^{55} +(11.7328 + 3.14378i) q^{57} +(-4.84103 - 1.29715i) q^{59} +(-2.42351 - 9.04465i) q^{61} +3.81358i q^{63} +(13.7175 - 7.91979i) q^{65} +(-2.43414 + 4.21606i) q^{67} +(3.07909 + 11.4913i) q^{69} +(-2.19312 - 1.26620i) q^{71} +12.9196i q^{73} +17.3560i q^{75} +(-3.78646 - 2.18611i) q^{77} +(-0.448216 - 1.67276i) q^{79} +(3.45672 - 5.98722i) q^{81} +(12.9919 - 7.50085i) q^{83} +1.89683i q^{85} +(-6.19556 - 23.1221i) q^{87} +(-14.7714 - 3.95797i) q^{89} +(-4.74842 - 1.27234i) q^{91} +(-18.2584 + 4.89231i) q^{93} +(-8.11942 - 14.0633i) q^{95} +(9.15861 + 9.15861i) q^{97} +(-7.35695 - 12.7426i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} - 4 q^{5} - 6 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} - 4 q^{5} - 6 q^{7} - 16 q^{9} + 4 q^{11} + 4 q^{13} + 8 q^{15} - 4 q^{17} + 10 q^{19} - 18 q^{21} + 8 q^{23} + 42 q^{25} + 68 q^{27} - 8 q^{29} + 28 q^{31} - 20 q^{33} - 10 q^{35} - 24 q^{37} - 14 q^{39} - 6 q^{41} + 32 q^{43} + 8 q^{45} - 12 q^{49} - 58 q^{51} + 6 q^{53} + 26 q^{55} - 2 q^{57} - 56 q^{59} - 8 q^{61} + 6 q^{65} + 20 q^{67} + 26 q^{69} - 30 q^{71} + 60 q^{77} + 50 q^{79} - 22 q^{81} + 36 q^{83} - 32 q^{87} - 20 q^{89} - 50 q^{91} - 50 q^{93} - 72 q^{95} + 32 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(-1\)

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
\(3\) −1.28281 + 2.22190i −0.740633 + 1.28281i 0.211575 + 0.977362i \(0.432141\pi\)
−0.952208 + 0.305452i \(0.901193\pi\)
\(4\) 0 0
\(5\) 3.31311 0.887746i 1.48167 0.397012i 0.574754 0.818326i \(-0.305098\pi\)
0.906915 + 0.421314i \(0.138431\pi\)
\(6\) 0 0
\(7\) −0.921902 0.532260i −0.348446 0.201175i 0.315555 0.948907i \(-0.397809\pi\)
−0.664001 + 0.747732i \(0.731143\pi\)
\(8\) 0 0
\(9\) −1.79122 3.10248i −0.597073 1.03416i
\(10\) 0 0
\(11\) 4.10723 1.23838 0.619188 0.785243i \(-0.287462\pi\)
0.619188 + 0.785243i \(0.287462\pi\)
\(12\) 0 0
\(13\) 4.46062 1.19522i 1.23715 0.331494i 0.419792 0.907620i \(-0.362103\pi\)
0.817361 + 0.576126i \(0.195436\pi\)
\(14\) 0 0
\(15\) −2.27762 + 8.50021i −0.588080 + 2.19475i
\(16\) 0 0
\(17\) −0.143130 + 0.534169i −0.0347142 + 0.129555i −0.981108 0.193459i \(-0.938029\pi\)
0.946394 + 0.323014i \(0.104696\pi\)
\(18\) 0 0
\(19\) −1.22535 4.57306i −0.281114 1.04913i −0.951633 0.307238i \(-0.900595\pi\)
0.670519 0.741893i \(-0.266072\pi\)
\(20\) 0 0
\(21\) 2.36526 1.36558i 0.516141 0.297994i
\(22\) 0 0
\(23\) 3.27882 3.27882i 0.683681 0.683681i −0.277147 0.960828i \(-0.589389\pi\)
0.960828 + 0.277147i \(0.0893888\pi\)
\(24\) 0 0
\(25\) 5.85850 3.38241i 1.17170 0.676481i
\(26\) 0 0
\(27\) 1.49432 0.287583
\(28\) 0 0
\(29\) −6.59744 + 6.59744i −1.22511 + 1.22511i −0.259324 + 0.965790i \(0.583500\pi\)
−0.965790 + 0.259324i \(0.916500\pi\)
\(30\) 0 0
\(31\) 5.20966 + 5.20966i 0.935683 + 0.935683i 0.998053 0.0623700i \(-0.0198659\pi\)
−0.0623700 + 0.998053i \(0.519866\pi\)
\(32\) 0 0
\(33\) −5.26881 + 9.12584i −0.917182 + 1.58861i
\(34\) 0 0
\(35\) −3.52688 0.945024i −0.596151 0.159738i
\(36\) 0 0
\(37\) −4.97186 + 3.50437i −0.817369 + 0.576115i
\(38\) 0 0
\(39\) −3.06649 + 11.4443i −0.491031 + 1.83255i
\(40\) 0 0
\(41\) 4.62353 + 2.66940i 0.722075 + 0.416890i 0.815516 0.578735i \(-0.196453\pi\)
−0.0934410 + 0.995625i \(0.529787\pi\)
\(42\) 0 0
\(43\) 1.34410 1.34410i 0.204974 0.204974i −0.597153 0.802127i \(-0.703702\pi\)
0.802127 + 0.597153i \(0.203702\pi\)
\(44\) 0 0
\(45\) −8.68873 8.68873i −1.29524 1.29524i
\(46\) 0 0
\(47\) 9.47900i 1.38265i 0.722542 + 0.691327i \(0.242974\pi\)
−0.722542 + 0.691327i \(0.757026\pi\)
\(48\) 0 0
\(49\) −2.93340 5.08079i −0.419057 0.725828i
\(50\) 0 0
\(51\) −1.00326 1.00326i −0.140485 0.140485i
\(52\) 0 0
\(53\) 1.36189 + 2.35887i 0.187071 + 0.324016i 0.944272 0.329165i \(-0.106767\pi\)
−0.757202 + 0.653181i \(0.773434\pi\)
\(54\) 0 0
\(55\) 13.6077 3.64618i 1.83486 0.491650i
\(56\) 0 0
\(57\) 11.7328 + 3.14378i 1.55404 + 0.416404i
\(58\) 0 0
\(59\) −4.84103 1.29715i −0.630248 0.168874i −0.0704657 0.997514i \(-0.522449\pi\)
−0.559782 + 0.828640i \(0.689115\pi\)
\(60\) 0 0
\(61\) −2.42351 9.04465i −0.310298 1.15805i −0.928288 0.371862i \(-0.878719\pi\)
0.617990 0.786186i \(-0.287947\pi\)
\(62\) 0 0
\(63\) 3.81358i 0.480466i
\(64\) 0 0
\(65\) 13.7175 7.91979i 1.70144 0.982329i
\(66\) 0 0
\(67\) −2.43414 + 4.21606i −0.297378 + 0.515074i −0.975535 0.219843i \(-0.929445\pi\)
0.678157 + 0.734917i \(0.262779\pi\)
\(68\) 0 0
\(69\) 3.07909 + 11.4913i 0.370679 + 1.38339i
\(70\) 0 0
\(71\) −2.19312 1.26620i −0.260276 0.150270i 0.364185 0.931327i \(-0.381348\pi\)
−0.624460 + 0.781057i \(0.714681\pi\)
\(72\) 0 0
\(73\) 12.9196i 1.51213i 0.654498 + 0.756064i \(0.272880\pi\)
−0.654498 + 0.756064i \(0.727120\pi\)
\(74\) 0 0
\(75\) 17.3560i 2.00410i
\(76\) 0 0
\(77\) −3.78646 2.18611i −0.431507 0.249131i
\(78\) 0 0
\(79\) −0.448216 1.67276i −0.0504282 0.188201i 0.936117 0.351688i \(-0.114392\pi\)
−0.986545 + 0.163488i \(0.947726\pi\)
\(80\) 0 0
\(81\) 3.45672 5.98722i 0.384080 0.665246i
\(82\) 0 0
\(83\) 12.9919 7.50085i 1.42604 0.823325i 0.429236 0.903192i \(-0.358783\pi\)
0.996806 + 0.0798671i \(0.0254496\pi\)
\(84\) 0 0
\(85\) 1.89683i 0.205740i
\(86\) 0 0
\(87\) −6.19556 23.1221i −0.664233 2.47895i
\(88\) 0 0
\(89\) −14.7714 3.95797i −1.56576 0.419544i −0.631279 0.775556i \(-0.717470\pi\)
−0.934482 + 0.356011i \(0.884136\pi\)
\(90\) 0 0
\(91\) −4.74842 1.27234i −0.497770 0.133377i
\(92\) 0 0
\(93\) −18.2584 + 4.89231i −1.89330 + 0.507309i
\(94\) 0 0
\(95\) −8.11942 14.0633i −0.833035 1.44286i
\(96\) 0 0
\(97\) 9.15861 + 9.15861i 0.929916 + 0.929916i 0.997700 0.0677837i \(-0.0215928\pi\)
−0.0677837 + 0.997700i \(0.521593\pi\)
\(98\) 0 0
\(99\) −7.35695 12.7426i −0.739401 1.28068i
\(100\) 0 0
\(101\) 9.03388i 0.898905i −0.893304 0.449452i \(-0.851619\pi\)
0.893304 0.449452i \(-0.148381\pi\)
\(102\) 0 0
\(103\) −4.83399 4.83399i −0.476307 0.476307i 0.427641 0.903949i \(-0.359345\pi\)
−0.903949 + 0.427641i \(0.859345\pi\)
\(104\) 0 0
\(105\) 6.62407 6.62407i 0.646443 0.646443i
\(106\) 0 0
\(107\) −5.01433 2.89502i −0.484753 0.279872i 0.237642 0.971353i \(-0.423626\pi\)
−0.722395 + 0.691480i \(0.756959\pi\)
\(108\) 0 0
\(109\) 3.33051 12.4296i 0.319005 1.19054i −0.601198 0.799100i \(-0.705310\pi\)
0.920203 0.391442i \(-0.128024\pi\)
\(110\) 0 0
\(111\) −1.40838 15.5424i −0.133677 1.47522i
\(112\) 0 0
\(113\) 0.775994 + 0.207927i 0.0729994 + 0.0195601i 0.295134 0.955456i \(-0.404636\pi\)
−0.222135 + 0.975016i \(0.571302\pi\)
\(114\) 0 0
\(115\) 7.95234 13.7739i 0.741559 1.28442i
\(116\) 0 0
\(117\) −11.6981 11.6981i −1.08149 1.08149i
\(118\) 0 0
\(119\) 0.416269 0.416269i 0.0381593 0.0381593i
\(120\) 0 0
\(121\) 5.86933 0.533576
\(122\) 0 0
\(123\) −11.8623 + 6.84868i −1.06958 + 0.617525i
\(124\) 0 0
\(125\) 4.28031 4.28031i 0.382843 0.382843i
\(126\) 0 0
\(127\) −16.5643 + 9.56340i −1.46984 + 0.848615i −0.999428 0.0338302i \(-0.989229\pi\)
−0.470416 + 0.882445i \(0.655896\pi\)
\(128\) 0 0
\(129\) 1.26223 + 4.71070i 0.111133 + 0.414754i
\(130\) 0 0
\(131\) 3.70654 13.8330i 0.323842 1.20859i −0.591629 0.806210i \(-0.701515\pi\)
0.915471 0.402384i \(-0.131818\pi\)
\(132\) 0 0
\(133\) −1.30441 + 4.86811i −0.113106 + 0.422119i
\(134\) 0 0
\(135\) 4.95087 1.32658i 0.426103 0.114174i
\(136\) 0 0
\(137\) 5.33351 0.455672 0.227836 0.973700i \(-0.426835\pi\)
0.227836 + 0.973700i \(0.426835\pi\)
\(138\) 0 0
\(139\) 0.464616 + 0.804738i 0.0394082 + 0.0682570i 0.885057 0.465483i \(-0.154119\pi\)
−0.845649 + 0.533740i \(0.820786\pi\)
\(140\) 0 0
\(141\) −21.0614 12.1598i −1.77369 1.02404i
\(142\) 0 0
\(143\) 18.3208 4.90904i 1.53206 0.410515i
\(144\) 0 0
\(145\) −16.0012 + 27.7149i −1.32883 + 2.30160i
\(146\) 0 0
\(147\) 15.0520 1.24147
\(148\) 0 0
\(149\) 7.86230 0.644105 0.322053 0.946722i \(-0.395627\pi\)
0.322053 + 0.946722i \(0.395627\pi\)
\(150\) 0 0
\(151\) −8.11982 + 14.0639i −0.660781 + 1.14451i 0.319629 + 0.947543i \(0.396442\pi\)
−0.980411 + 0.196964i \(0.936892\pi\)
\(152\) 0 0
\(153\) 1.91363 0.512755i 0.154708 0.0414538i
\(154\) 0 0
\(155\) 21.8851 + 12.6353i 1.75785 + 1.01490i
\(156\) 0 0
\(157\) −4.91552 8.51392i −0.392301 0.679485i 0.600452 0.799661i \(-0.294987\pi\)
−0.992753 + 0.120176i \(0.961654\pi\)
\(158\) 0 0
\(159\) −6.98823 −0.554202
\(160\) 0 0
\(161\) −4.76793 + 1.27756i −0.375766 + 0.100686i
\(162\) 0 0
\(163\) −0.732053 + 2.73206i −0.0573388 + 0.213991i −0.988651 0.150231i \(-0.951998\pi\)
0.931312 + 0.364222i \(0.118665\pi\)
\(164\) 0 0
\(165\) −9.35473 + 34.9123i −0.728264 + 2.71792i
\(166\) 0 0
\(167\) −2.95349 11.0226i −0.228548 0.852953i −0.980952 0.194251i \(-0.937772\pi\)
0.752404 0.658702i \(-0.228894\pi\)
\(168\) 0 0
\(169\) 7.21024 4.16284i 0.554634 0.320218i
\(170\) 0 0
\(171\) −11.9930 + 11.9930i −0.917125 + 0.917125i
\(172\) 0 0
\(173\) −9.41607 + 5.43637i −0.715890 + 0.413319i −0.813238 0.581931i \(-0.802297\pi\)
0.0973479 + 0.995250i \(0.468964\pi\)
\(174\) 0 0
\(175\) −7.20128 −0.544366
\(176\) 0 0
\(177\) 9.09226 9.09226i 0.683416 0.683416i
\(178\) 0 0
\(179\) −13.8879 13.8879i −1.03803 1.03803i −0.999248 0.0387800i \(-0.987653\pi\)
−0.0387800 0.999248i \(-0.512347\pi\)
\(180\) 0 0
\(181\) 3.21521 5.56891i 0.238985 0.413934i −0.721438 0.692479i \(-0.756519\pi\)
0.960423 + 0.278545i \(0.0898520\pi\)
\(182\) 0 0
\(183\) 23.2052 + 6.21781i 1.71538 + 0.459634i
\(184\) 0 0
\(185\) −13.3613 + 16.0241i −0.982346 + 1.17812i
\(186\) 0 0
\(187\) −0.587869 + 2.19396i −0.0429892 + 0.160438i
\(188\) 0 0
\(189\) −1.37762 0.795369i −0.100207 0.0578546i
\(190\) 0 0
\(191\) 12.1844 12.1844i 0.881632 0.881632i −0.112069 0.993700i \(-0.535748\pi\)
0.993700 + 0.112069i \(0.0357476\pi\)
\(192\) 0 0
\(193\) 15.4383 + 15.4383i 1.11127 + 1.11127i 0.992978 + 0.118296i \(0.0377433\pi\)
0.118296 + 0.992978i \(0.462257\pi\)
\(194\) 0 0
\(195\) 40.6385i 2.91018i
\(196\) 0 0
\(197\) 4.95390 + 8.58040i 0.352951 + 0.611328i 0.986765 0.162157i \(-0.0518450\pi\)
−0.633814 + 0.773485i \(0.718512\pi\)
\(198\) 0 0
\(199\) −8.52004 8.52004i −0.603970 0.603970i 0.337394 0.941364i \(-0.390455\pi\)
−0.941364 + 0.337394i \(0.890455\pi\)
\(200\) 0 0
\(201\) −6.24510 10.8168i −0.440496 0.762961i
\(202\) 0 0
\(203\) 9.59375 2.57064i 0.673349 0.180423i
\(204\) 0 0
\(205\) 17.6880 + 4.73950i 1.23539 + 0.331021i
\(206\) 0 0
\(207\) −16.0456 4.29940i −1.11524 0.298829i
\(208\) 0 0
\(209\) −5.03278 18.7826i −0.348125 1.29922i
\(210\) 0 0
\(211\) 4.16265i 0.286569i −0.989682 0.143284i \(-0.954234\pi\)
0.989682 0.143284i \(-0.0457664\pi\)
\(212\) 0 0
\(213\) 5.62673 3.24860i 0.385537 0.222590i
\(214\) 0 0
\(215\) 3.25995 5.64639i 0.222327 0.385081i
\(216\) 0 0
\(217\) −2.02990 7.57569i −0.137799 0.514272i
\(218\) 0 0
\(219\) −28.7061 16.5735i −1.93978 1.11993i
\(220\) 0 0
\(221\) 2.55380i 0.171787i
\(222\) 0 0
\(223\) 22.1491i 1.48321i 0.670834 + 0.741607i \(0.265936\pi\)
−0.670834 + 0.741607i \(0.734064\pi\)
\(224\) 0 0
\(225\) −20.9877 12.1173i −1.39918 0.807818i
\(226\) 0 0
\(227\) 3.17314 + 11.8423i 0.210609 + 0.786003i 0.987666 + 0.156573i \(0.0500448\pi\)
−0.777057 + 0.629430i \(0.783289\pi\)
\(228\) 0 0
\(229\) −12.8225 + 22.2091i −0.847332 + 1.46762i 0.0362489 + 0.999343i \(0.488459\pi\)
−0.883581 + 0.468279i \(0.844874\pi\)
\(230\) 0 0
\(231\) 9.71465 5.60875i 0.639177 0.369029i
\(232\) 0 0
\(233\) 11.3416i 0.743011i −0.928431 0.371506i \(-0.878842\pi\)
0.928431 0.371506i \(-0.121158\pi\)
\(234\) 0 0
\(235\) 8.41495 + 31.4050i 0.548931 + 2.04864i
\(236\) 0 0
\(237\) 4.29169 + 1.14995i 0.278775 + 0.0746976i
\(238\) 0 0
\(239\) 25.8335 + 6.92207i 1.67103 + 0.447751i 0.965388 0.260817i \(-0.0839919\pi\)
0.705642 + 0.708568i \(0.250659\pi\)
\(240\) 0 0
\(241\) −8.07195 + 2.16287i −0.519960 + 0.139323i −0.509248 0.860620i \(-0.670076\pi\)
−0.0107119 + 0.999943i \(0.503410\pi\)
\(242\) 0 0
\(243\) 11.1101 + 19.2433i 0.712716 + 1.23446i
\(244\) 0 0
\(245\) −14.2291 14.2291i −0.909066 0.909066i
\(246\) 0 0
\(247\) −10.9316 18.9341i −0.695562 1.20475i
\(248\) 0 0
\(249\) 38.4888i 2.43913i
\(250\) 0 0
\(251\) −8.80235 8.80235i −0.555600 0.555600i 0.372452 0.928051i \(-0.378517\pi\)
−0.928051 + 0.372452i \(0.878517\pi\)
\(252\) 0 0
\(253\) 13.4669 13.4669i 0.846654 0.846654i
\(254\) 0 0
\(255\) −4.21455 2.43327i −0.263926 0.152378i
\(256\) 0 0
\(257\) 8.15736 30.4437i 0.508842 1.89902i 0.0771108 0.997023i \(-0.475430\pi\)
0.431731 0.902002i \(-0.357903\pi\)
\(258\) 0 0
\(259\) 6.44880 0.584359i 0.400709 0.0363103i
\(260\) 0 0
\(261\) 32.2859 + 8.65099i 1.99845 + 0.535483i
\(262\) 0 0
\(263\) −4.20493 + 7.28315i −0.259287 + 0.449098i −0.966051 0.258351i \(-0.916821\pi\)
0.706764 + 0.707449i \(0.250154\pi\)
\(264\) 0 0
\(265\) 6.60619 + 6.60619i 0.405815 + 0.405815i
\(266\) 0 0
\(267\) 27.7431 27.7431i 1.69785 1.69785i
\(268\) 0 0
\(269\) −24.5521 −1.49697 −0.748483 0.663154i \(-0.769217\pi\)
−0.748483 + 0.663154i \(0.769217\pi\)
\(270\) 0 0
\(271\) −12.8491 + 7.41843i −0.780527 + 0.450637i −0.836617 0.547788i \(-0.815470\pi\)
0.0560902 + 0.998426i \(0.482137\pi\)
\(272\) 0 0
\(273\) 8.91834 8.91834i 0.539762 0.539762i
\(274\) 0 0
\(275\) 24.0622 13.8923i 1.45100 0.837738i
\(276\) 0 0
\(277\) −2.96327 11.0591i −0.178046 0.664475i −0.996013 0.0892104i \(-0.971566\pi\)
0.817967 0.575265i \(-0.195101\pi\)
\(278\) 0 0
\(279\) 6.83124 25.4946i 0.408976 1.52632i
\(280\) 0 0
\(281\) 5.56147 20.7557i 0.331770 1.23818i −0.575559 0.817760i \(-0.695216\pi\)
0.907329 0.420421i \(-0.138118\pi\)
\(282\) 0 0
\(283\) −16.6287 + 4.45566i −0.988476 + 0.264861i −0.716610 0.697474i \(-0.754307\pi\)
−0.271866 + 0.962335i \(0.587641\pi\)
\(284\) 0 0
\(285\) 41.6628 2.46789
\(286\) 0 0
\(287\) −2.84163 4.92185i −0.167736 0.290527i
\(288\) 0 0
\(289\) 14.4576 + 8.34709i 0.850446 + 0.491005i
\(290\) 0 0
\(291\) −32.0983 + 8.60071i −1.88164 + 0.504183i
\(292\) 0 0
\(293\) −1.77150 + 3.06833i −0.103492 + 0.179254i −0.913121 0.407688i \(-0.866335\pi\)
0.809629 + 0.586942i \(0.199668\pi\)
\(294\) 0 0
\(295\) −17.1904 −1.00086
\(296\) 0 0
\(297\) 6.13753 0.356136
\(298\) 0 0
\(299\) 10.7067 18.5445i 0.619182 1.07245i
\(300\) 0 0
\(301\) −1.95455 + 0.523719i −0.112658 + 0.0301867i
\(302\) 0 0
\(303\) 20.0724 + 11.5888i 1.15313 + 0.665758i
\(304\) 0 0
\(305\) −16.0587 27.8145i −0.919518 1.59265i
\(306\) 0 0
\(307\) −12.7996 −0.730511 −0.365256 0.930907i \(-0.619018\pi\)
−0.365256 + 0.930907i \(0.619018\pi\)
\(308\) 0 0
\(309\) 16.9417 4.53953i 0.963782 0.258245i
\(310\) 0 0
\(311\) −8.68446 + 32.4109i −0.492451 + 1.83785i 0.0514129 + 0.998677i \(0.483628\pi\)
−0.543864 + 0.839174i \(0.683039\pi\)
\(312\) 0 0
\(313\) −2.95371 + 11.0234i −0.166954 + 0.623079i 0.830829 + 0.556527i \(0.187867\pi\)
−0.997783 + 0.0665521i \(0.978800\pi\)
\(314\) 0 0
\(315\) 3.38549 + 12.6348i 0.190751 + 0.711892i
\(316\) 0 0
\(317\) −6.60059 + 3.81086i −0.370726 + 0.214039i −0.673776 0.738936i \(-0.735329\pi\)
0.303049 + 0.952975i \(0.401995\pi\)
\(318\) 0 0
\(319\) −27.0972 + 27.0972i −1.51715 + 1.51715i
\(320\) 0 0
\(321\) 12.8649 7.42755i 0.718048 0.414565i
\(322\) 0 0
\(323\) 2.61817 0.145679
\(324\) 0 0
\(325\) 22.0898 22.0898i 1.22532 1.22532i
\(326\) 0 0
\(327\) 23.3449 + 23.3449i 1.29098 + 1.29098i
\(328\) 0 0
\(329\) 5.04530 8.73871i 0.278156 0.481781i
\(330\) 0 0
\(331\) −16.3553 4.38239i −0.898969 0.240878i −0.220396 0.975411i \(-0.570735\pi\)
−0.678574 + 0.734532i \(0.737402\pi\)
\(332\) 0 0
\(333\) 19.7779 + 9.14803i 1.08382 + 0.501309i
\(334\) 0 0
\(335\) −4.32180 + 16.1292i −0.236125 + 0.881231i
\(336\) 0 0
\(337\) −20.4415 11.8019i −1.11352 0.642892i −0.173783 0.984784i \(-0.555599\pi\)
−0.939739 + 0.341892i \(0.888932\pi\)
\(338\) 0 0
\(339\) −1.45745 + 1.45745i −0.0791577 + 0.0791577i
\(340\) 0 0
\(341\) 21.3973 + 21.3973i 1.15873 + 1.15873i
\(342\) 0 0
\(343\) 13.6970i 0.739567i
\(344\) 0 0
\(345\) 20.4027 + 35.3386i 1.09845 + 1.90256i
\(346\) 0 0
\(347\) 9.54031 + 9.54031i 0.512150 + 0.512150i 0.915185 0.403034i \(-0.132045\pi\)
−0.403034 + 0.915185i \(0.632045\pi\)
\(348\) 0 0
\(349\) −4.94731 8.56899i −0.264823 0.458687i 0.702694 0.711492i \(-0.251980\pi\)
−0.967517 + 0.252805i \(0.918647\pi\)
\(350\) 0 0
\(351\) 6.66561 1.78605i 0.355784 0.0953320i
\(352\) 0 0
\(353\) −29.2935 7.84916i −1.55913 0.417769i −0.626744 0.779225i \(-0.715613\pi\)
−0.932389 + 0.361456i \(0.882280\pi\)
\(354\) 0 0
\(355\) −8.39012 2.24813i −0.445302 0.119318i
\(356\) 0 0
\(357\) 0.390912 + 1.45890i 0.0206892 + 0.0772133i
\(358\) 0 0
\(359\) 4.63014i 0.244370i −0.992507 0.122185i \(-0.961010\pi\)
0.992507 0.122185i \(-0.0389901\pi\)
\(360\) 0 0
\(361\) −2.95688 + 1.70716i −0.155626 + 0.0898504i
\(362\) 0 0
\(363\) −7.52926 + 13.0411i −0.395183 + 0.684478i
\(364\) 0 0
\(365\) 11.4693 + 42.8042i 0.600333 + 2.24047i
\(366\) 0 0
\(367\) 27.3558 + 15.7939i 1.42796 + 0.824433i 0.996960 0.0779192i \(-0.0248276\pi\)
0.431000 + 0.902352i \(0.358161\pi\)
\(368\) 0 0
\(369\) 19.1259i 0.995656i
\(370\) 0 0
\(371\) 2.89953i 0.150536i
\(372\) 0 0
\(373\) −17.3245 10.0023i −0.897030 0.517901i −0.0207947 0.999784i \(-0.506620\pi\)
−0.876235 + 0.481883i \(0.839953\pi\)
\(374\) 0 0
\(375\) 4.01958 + 15.0013i 0.207570 + 0.774662i
\(376\) 0 0
\(377\) −21.5433 + 37.3141i −1.10954 + 1.92177i
\(378\) 0 0
\(379\) 30.7317 17.7429i 1.57858 0.911393i 0.583522 0.812098i \(-0.301674\pi\)
0.995058 0.0992957i \(-0.0316590\pi\)
\(380\) 0 0
\(381\) 49.0722i 2.51405i
\(382\) 0 0
\(383\) −0.0671562 0.250630i −0.00343152 0.0128066i 0.964189 0.265217i \(-0.0854435\pi\)
−0.967620 + 0.252410i \(0.918777\pi\)
\(384\) 0 0
\(385\) −14.4857 3.88143i −0.738259 0.197816i
\(386\) 0 0
\(387\) −6.57765 1.76248i −0.334361 0.0895917i
\(388\) 0 0
\(389\) −13.1144 + 3.51399i −0.664925 + 0.178166i −0.575468 0.817824i \(-0.695180\pi\)
−0.0894574 + 0.995991i \(0.528513\pi\)
\(390\) 0 0
\(391\) 1.28215 + 2.22074i 0.0648409 + 0.112308i
\(392\) 0 0
\(393\) 25.9807 + 25.9807i 1.31055 + 1.31055i
\(394\) 0 0
\(395\) −2.96998 5.14415i −0.149436 0.258830i
\(396\) 0 0
\(397\) 2.67669i 0.134339i 0.997742 + 0.0671695i \(0.0213968\pi\)
−0.997742 + 0.0671695i \(0.978603\pi\)
\(398\) 0 0
\(399\) −9.14314 9.14314i −0.457729 0.457729i
\(400\) 0 0
\(401\) 13.6170 13.6170i 0.679998 0.679998i −0.280001 0.960000i \(-0.590335\pi\)
0.960000 + 0.280001i \(0.0903349\pi\)
\(402\) 0 0
\(403\) 29.4650 + 17.0116i 1.46776 + 0.847410i
\(404\) 0 0
\(405\) 6.13738 22.9050i 0.304969 1.13816i
\(406\) 0 0
\(407\) −20.4206 + 14.3932i −1.01221 + 0.713447i
\(408\) 0 0
\(409\) 28.5669 + 7.65448i 1.41254 + 0.378490i 0.882832 0.469689i \(-0.155634\pi\)
0.529711 + 0.848178i \(0.322301\pi\)
\(410\) 0 0
\(411\) −6.84189 + 11.8505i −0.337486 + 0.584542i
\(412\) 0 0
\(413\) 3.77253 + 3.77253i 0.185634 + 0.185634i
\(414\) 0 0
\(415\) 36.3846 36.3846i 1.78605 1.78605i
\(416\) 0 0
\(417\) −2.38406 −0.116748
\(418\) 0 0
\(419\) −8.45985 + 4.88430i −0.413291 + 0.238613i −0.692203 0.721703i \(-0.743360\pi\)
0.278912 + 0.960317i \(0.410026\pi\)
\(420\) 0 0
\(421\) −5.78874 + 5.78874i −0.282126 + 0.282126i −0.833956 0.551830i \(-0.813930\pi\)
0.551830 + 0.833956i \(0.313930\pi\)
\(422\) 0 0
\(423\) 29.4085 16.9790i 1.42989 0.825546i
\(424\) 0 0
\(425\) 0.968249 + 3.61355i 0.0469670 + 0.175283i
\(426\) 0 0
\(427\) −2.57987 + 9.62821i −0.124849 + 0.465942i
\(428\) 0 0
\(429\) −12.5948 + 47.0043i −0.608081 + 2.26939i
\(430\) 0 0
\(431\) −8.65634 + 2.31946i −0.416961 + 0.111724i −0.461199 0.887297i \(-0.652581\pi\)
0.0442379 + 0.999021i \(0.485914\pi\)
\(432\) 0 0
\(433\) −22.4023 −1.07658 −0.538292 0.842759i \(-0.680930\pi\)
−0.538292 + 0.842759i \(0.680930\pi\)
\(434\) 0 0
\(435\) −41.0532 71.1061i −1.96835 3.40928i
\(436\) 0 0
\(437\) −19.0119 10.9765i −0.909463 0.525079i
\(438\) 0 0
\(439\) −10.5938 + 2.83859i −0.505612 + 0.135478i −0.502604 0.864517i \(-0.667624\pi\)
−0.00300894 + 0.999995i \(0.500958\pi\)
\(440\) 0 0
\(441\) −10.5087 + 18.2016i −0.500415 + 0.866745i
\(442\) 0 0
\(443\) 18.4761 0.877827 0.438914 0.898529i \(-0.355363\pi\)
0.438914 + 0.898529i \(0.355363\pi\)
\(444\) 0 0
\(445\) −52.4529 −2.48650
\(446\) 0 0
\(447\) −10.0859 + 17.4692i −0.477045 + 0.826267i
\(448\) 0 0
\(449\) −9.34310 + 2.50348i −0.440928 + 0.118146i −0.472451 0.881357i \(-0.656631\pi\)
0.0315234 + 0.999503i \(0.489964\pi\)
\(450\) 0 0
\(451\) 18.9899 + 10.9638i 0.894200 + 0.516267i
\(452\) 0 0
\(453\) −20.8324 36.0828i −0.978793 1.69532i
\(454\) 0 0
\(455\) −16.8616 −0.790482
\(456\) 0 0
\(457\) 2.74958 0.736748i 0.128620 0.0344636i −0.193935 0.981014i \(-0.562125\pi\)
0.322555 + 0.946551i \(0.395458\pi\)
\(458\) 0 0
\(459\) −0.213883 + 0.798222i −0.00998320 + 0.0372578i
\(460\) 0 0
\(461\) 2.59232 9.67468i 0.120737 0.450595i −0.878915 0.476978i \(-0.841732\pi\)
0.999652 + 0.0263829i \(0.00839890\pi\)
\(462\) 0 0
\(463\) 4.84681 + 18.0885i 0.225250 + 0.840646i 0.982304 + 0.187293i \(0.0599714\pi\)
−0.757054 + 0.653353i \(0.773362\pi\)
\(464\) 0 0
\(465\) −56.1489 + 32.4176i −2.60384 + 1.50333i
\(466\) 0 0
\(467\) 6.00990 6.00990i 0.278105 0.278105i −0.554247 0.832352i \(-0.686994\pi\)
0.832352 + 0.554247i \(0.186994\pi\)
\(468\) 0 0
\(469\) 4.48808 2.59119i 0.207240 0.119650i
\(470\) 0 0
\(471\) 25.2228 1.16220
\(472\) 0 0
\(473\) 5.52055 5.52055i 0.253835 0.253835i
\(474\) 0 0
\(475\) −22.6466 22.6466i −1.03910 1.03910i
\(476\) 0 0
\(477\) 4.87891 8.45051i 0.223390 0.386922i
\(478\) 0 0
\(479\) 10.7260 + 2.87403i 0.490084 + 0.131318i 0.495394 0.868668i \(-0.335024\pi\)
−0.00530995 + 0.999986i \(0.501690\pi\)
\(480\) 0 0
\(481\) −17.9891 + 21.5741i −0.820232 + 0.983695i
\(482\) 0 0
\(483\) 3.27775 12.2327i 0.149143 0.556609i
\(484\) 0 0
\(485\) 38.4740 + 22.2130i 1.74702 + 1.00864i
\(486\) 0 0
\(487\) 4.74492 4.74492i 0.215013 0.215013i −0.591380 0.806393i \(-0.701417\pi\)
0.806393 + 0.591380i \(0.201417\pi\)
\(488\) 0 0
\(489\) −5.13127 5.13127i −0.232044 0.232044i
\(490\) 0 0
\(491\) 16.6226i 0.750168i −0.926991 0.375084i \(-0.877614\pi\)
0.926991 0.375084i \(-0.122386\pi\)
\(492\) 0 0
\(493\) −2.57986 4.46844i −0.116191 0.201249i
\(494\) 0 0
\(495\) −35.6866 35.6866i −1.60399 1.60399i
\(496\) 0 0
\(497\) 1.34790 + 2.33462i 0.0604614 + 0.104722i
\(498\) 0 0
\(499\) −6.12818 + 1.64204i −0.274335 + 0.0735079i −0.393364 0.919383i \(-0.628689\pi\)
0.119029 + 0.992891i \(0.462022\pi\)
\(500\) 0 0
\(501\) 28.2798 + 7.57756i 1.26345 + 0.338540i
\(502\) 0 0
\(503\) −23.9100 6.40667i −1.06609 0.285659i −0.317207 0.948356i \(-0.602745\pi\)
−0.748888 + 0.662697i \(0.769412\pi\)
\(504\) 0 0
\(505\) −8.01979 29.9303i −0.356876 1.33188i
\(506\) 0 0
\(507\) 21.3606i 0.948656i
\(508\) 0 0
\(509\) −9.01265 + 5.20346i −0.399479 + 0.230639i −0.686259 0.727357i \(-0.740748\pi\)
0.286780 + 0.957996i \(0.407415\pi\)
\(510\) 0 0
\(511\) 6.87660 11.9106i 0.304203 0.526895i
\(512\) 0 0
\(513\) −1.83107 6.83363i −0.0808435 0.301712i
\(514\) 0 0
\(515\) −20.3069 11.7242i −0.894830 0.516630i
\(516\) 0 0
\(517\) 38.9324i 1.71225i
\(518\) 0 0
\(519\) 27.8954i 1.22447i
\(520\) 0 0
\(521\) −16.9426 9.78182i −0.742269 0.428549i 0.0806248 0.996745i \(-0.474308\pi\)
−0.822894 + 0.568195i \(0.807642\pi\)
\(522\) 0 0
\(523\) −4.19342 15.6501i −0.183365 0.684329i −0.994975 0.100128i \(-0.968075\pi\)
0.811609 0.584201i \(-0.198592\pi\)
\(524\) 0 0
\(525\) 9.23790 16.0005i 0.403175 0.698319i
\(526\) 0 0
\(527\) −3.52850 + 2.03718i −0.153704 + 0.0887410i
\(528\) 0 0
\(529\) 1.49870i 0.0651611i
\(530\) 0 0
\(531\) 4.64696 + 17.3427i 0.201661 + 0.752608i
\(532\) 0 0
\(533\) 23.8143 + 6.38103i 1.03151 + 0.276393i
\(534\) 0 0
\(535\) −19.1831 5.14009i −0.829357 0.222226i
\(536\) 0 0
\(537\) 48.6730 13.0419i 2.10039 0.562799i
\(538\) 0 0
\(539\) −12.0481 20.8680i −0.518950 0.898848i
\(540\) 0 0
\(541\) 7.57423 + 7.57423i 0.325641 + 0.325641i 0.850926 0.525285i \(-0.176041\pi\)
−0.525285 + 0.850926i \(0.676041\pi\)
\(542\) 0 0
\(543\) 8.24904 + 14.2878i 0.354000 + 0.613146i
\(544\) 0 0
\(545\) 44.1374i 1.89064i
\(546\) 0 0
\(547\) 15.4542 + 15.4542i 0.660776 + 0.660776i 0.955563 0.294787i \(-0.0952488\pi\)
−0.294787 + 0.955563i \(0.595249\pi\)
\(548\) 0 0
\(549\) −23.7198 + 23.7198i −1.01234 + 1.01234i
\(550\) 0 0
\(551\) 38.2546 + 22.0863i 1.62970 + 0.940909i
\(552\) 0 0
\(553\) −0.477135 + 1.78069i −0.0202898 + 0.0757227i
\(554\) 0 0
\(555\) −18.4638 50.2435i −0.783746 2.13272i
\(556\) 0 0
\(557\) −4.59438 1.23106i −0.194670 0.0521617i 0.160167 0.987090i \(-0.448797\pi\)
−0.354837 + 0.934928i \(0.615464\pi\)
\(558\) 0 0
\(559\) 4.38904 7.60204i 0.185637 0.321532i
\(560\) 0 0
\(561\) −4.12062 4.12062i −0.173973 0.173973i
\(562\) 0 0
\(563\) 6.34335 6.34335i 0.267340 0.267340i −0.560687 0.828028i \(-0.689463\pi\)
0.828028 + 0.560687i \(0.189463\pi\)
\(564\) 0 0
\(565\) 2.75554 0.115927
\(566\) 0 0
\(567\) −6.37351 + 3.67975i −0.267662 + 0.154535i
\(568\) 0 0
\(569\) 13.3609 13.3609i 0.560117 0.560117i −0.369224 0.929341i \(-0.620376\pi\)
0.929341 + 0.369224i \(0.120376\pi\)
\(570\) 0 0
\(571\) 28.0106 16.1719i 1.17221 0.676774i 0.218008 0.975947i \(-0.430044\pi\)
0.954199 + 0.299173i \(0.0967109\pi\)
\(572\) 0 0
\(573\) 11.4422 + 42.7028i 0.478004 + 1.78393i
\(574\) 0 0
\(575\) 8.11866 30.2992i 0.338571 1.26357i
\(576\) 0 0
\(577\) 10.2424 38.2253i 0.426398 1.59134i −0.334452 0.942413i \(-0.608551\pi\)
0.760850 0.648928i \(-0.224782\pi\)
\(578\) 0 0
\(579\) −54.1069 + 14.4979i −2.24860 + 0.602512i
\(580\) 0 0
\(581\) −15.9696 −0.662531
\(582\) 0 0
\(583\) 5.59361 + 9.68842i 0.231664 + 0.401253i
\(584\) 0 0
\(585\) −49.1421 28.3722i −2.03177 1.17305i
\(586\) 0 0
\(587\) 40.2799 10.7930i 1.66253 0.445474i 0.699449 0.714682i \(-0.253429\pi\)
0.963082 + 0.269208i \(0.0867621\pi\)
\(588\) 0 0
\(589\) 17.4404 30.2077i 0.718621 1.24469i
\(590\) 0 0
\(591\) −25.4197 −1.04563
\(592\) 0 0
\(593\) −6.05194 −0.248523 −0.124262 0.992249i \(-0.539656\pi\)
−0.124262 + 0.992249i \(0.539656\pi\)
\(594\) 0 0
\(595\) 1.00961 1.74869i 0.0413898 0.0716892i
\(596\) 0 0
\(597\) 29.8603 8.00104i 1.22210 0.327461i
\(598\) 0 0
\(599\) −23.5639 13.6046i −0.962795 0.555870i −0.0657630 0.997835i \(-0.520948\pi\)
−0.897032 + 0.441965i \(0.854281\pi\)
\(600\) 0 0
\(601\) 13.9110 + 24.0946i 0.567443 + 0.982840i 0.996818 + 0.0797133i \(0.0254005\pi\)
−0.429375 + 0.903126i \(0.641266\pi\)
\(602\) 0 0
\(603\) 17.4403 0.710226
\(604\) 0 0
\(605\) 19.4458 5.21047i 0.790582 0.211836i
\(606\) 0 0
\(607\) 9.42849 35.1876i 0.382691 1.42822i −0.459084 0.888393i \(-0.651822\pi\)
0.841775 0.539829i \(-0.181511\pi\)
\(608\) 0 0
\(609\) −6.59530 + 24.6140i −0.267255 + 0.997409i
\(610\) 0 0
\(611\) 11.3295 + 42.2822i 0.458342 + 1.71056i
\(612\) 0 0
\(613\) 32.0996 18.5327i 1.29649 0.748529i 0.316693 0.948528i \(-0.397427\pi\)
0.979796 + 0.199999i \(0.0640940\pi\)
\(614\) 0 0
\(615\) −33.2211 + 33.2211i −1.33961 + 1.33961i
\(616\) 0 0
\(617\) −22.9221 + 13.2341i −0.922809 + 0.532784i −0.884530 0.466483i \(-0.845521\pi\)
−0.0382787 + 0.999267i \(0.512187\pi\)
\(618\) 0 0
\(619\) −28.2202 −1.13427 −0.567133 0.823626i \(-0.691947\pi\)
−0.567133 + 0.823626i \(0.691947\pi\)
\(620\) 0 0
\(621\) 4.89962 4.89962i 0.196615 0.196615i
\(622\) 0 0
\(623\) 11.5111 + 11.5111i 0.461181 + 0.461181i
\(624\) 0 0
\(625\) −6.53069 + 11.3115i −0.261228 + 0.452460i
\(626\) 0 0
\(627\) 48.1891 + 12.9122i 1.92449 + 0.515665i
\(628\) 0 0
\(629\) −1.16030 3.15740i −0.0462643 0.125894i
\(630\) 0 0
\(631\) −4.44449 + 16.5871i −0.176932 + 0.660321i 0.819282 + 0.573391i \(0.194372\pi\)
−0.996214 + 0.0869300i \(0.972294\pi\)
\(632\) 0 0
\(633\) 9.24899 + 5.33991i 0.367614 + 0.212242i
\(634\) 0 0
\(635\) −46.3895 + 46.3895i −1.84091 + 1.84091i
\(636\) 0 0
\(637\) −19.1574 19.1574i −0.759045 0.759045i
\(638\) 0 0
\(639\) 9.07217i 0.358889i
\(640\) 0 0
\(641\) −0.689148 1.19364i −0.0272197 0.0471459i 0.852095 0.523388i \(-0.175332\pi\)
−0.879314 + 0.476242i \(0.841999\pi\)
\(642\) 0 0
\(643\) −22.0565 22.0565i −0.869824 0.869824i 0.122629 0.992453i \(-0.460868\pi\)
−0.992453 + 0.122629i \(0.960868\pi\)
\(644\) 0 0
\(645\) 8.36381 + 14.4865i 0.329325 + 0.570407i
\(646\) 0 0
\(647\) −18.7398 + 5.02131i −0.736736 + 0.197408i −0.607627 0.794222i \(-0.707878\pi\)
−0.129109 + 0.991630i \(0.541212\pi\)
\(648\) 0 0
\(649\) −19.8832 5.32769i −0.780484 0.209130i
\(650\) 0 0
\(651\) 19.4364 + 5.20797i 0.761773 + 0.204116i
\(652\) 0 0
\(653\) −2.36083 8.81075i −0.0923866 0.344791i 0.904224 0.427059i \(-0.140450\pi\)
−0.996610 + 0.0822678i \(0.973784\pi\)
\(654\) 0 0
\(655\) 49.1207i 1.91931i
\(656\) 0 0
\(657\) 40.0829 23.1419i 1.56378 0.902851i
\(658\) 0 0
\(659\) 9.90050 17.1482i 0.385669 0.667997i −0.606193 0.795317i \(-0.707304\pi\)
0.991862 + 0.127320i \(0.0406375\pi\)
\(660\) 0 0
\(661\) 1.01710 + 3.79588i 0.0395607 + 0.147642i 0.982881 0.184241i \(-0.0589826\pi\)
−0.943320 + 0.331883i \(0.892316\pi\)
\(662\) 0 0
\(663\) −5.67428 3.27605i −0.220371 0.127231i
\(664\) 0 0
\(665\) 17.2866i 0.670345i
\(666\) 0 0
\(667\) 43.2636i 1.67517i
\(668\) 0 0
\(669\) −49.2131 28.4132i −1.90269 1.09852i
\(670\) 0 0
\(671\) −9.95389 37.1484i −0.384266 1.43410i
\(672\) 0 0
\(673\) −3.68093 + 6.37555i −0.141889 + 0.245760i −0.928208 0.372062i \(-0.878651\pi\)
0.786319 + 0.617821i \(0.211984\pi\)
\(674\) 0 0
\(675\) 8.75450 5.05441i 0.336961 0.194544i
\(676\) 0 0
\(677\) 7.16592i 0.275409i −0.990473 0.137704i \(-0.956028\pi\)
0.990473 0.137704i \(-0.0439724\pi\)
\(678\) 0 0
\(679\) −3.56858 13.3181i −0.136949 0.511102i
\(680\) 0 0
\(681\) −30.3830 8.14110i −1.16428 0.311968i
\(682\) 0 0
\(683\) −23.0903 6.18702i −0.883525 0.236740i −0.211598 0.977357i \(-0.567867\pi\)
−0.671927 + 0.740617i \(0.734533\pi\)
\(684\) 0 0
\(685\) 17.6705 4.73480i 0.675155 0.180907i
\(686\) 0 0
\(687\) −32.8976 56.9804i −1.25512 2.17394i
\(688\) 0 0
\(689\) 8.89426 + 8.89426i 0.338844 + 0.338844i
\(690\) 0 0
\(691\) −16.1197 27.9201i −0.613221 1.06213i −0.990694 0.136109i \(-0.956540\pi\)
0.377473 0.926021i \(-0.376793\pi\)
\(692\) 0 0
\(693\) 15.6632i 0.594998i
\(694\) 0 0
\(695\) 2.25373 + 2.25373i 0.0854888 + 0.0854888i
\(696\) 0 0
\(697\) −2.08768 + 2.08768i −0.0790765 + 0.0790765i
\(698\) 0 0
\(699\) 25.1998 + 14.5491i 0.953145 + 0.550298i
\(700\) 0 0
\(701\) 1.94640 7.26405i 0.0735144 0.274360i −0.919378 0.393375i \(-0.871307\pi\)
0.992892 + 0.119016i \(0.0379739\pi\)
\(702\) 0 0
\(703\) 22.1179 + 18.4425i 0.834193 + 0.695573i
\(704\) 0 0
\(705\) −80.5735 21.5896i −3.03457 0.813112i
\(706\) 0 0
\(707\) −4.80837 + 8.32835i −0.180838 + 0.313220i
\(708\) 0 0
\(709\) 14.6322 + 14.6322i 0.549524 + 0.549524i 0.926303 0.376779i \(-0.122968\pi\)
−0.376779 + 0.926303i \(0.622968\pi\)
\(710\) 0 0
\(711\) −4.38687 + 4.38687i −0.164521 + 0.164521i
\(712\) 0 0
\(713\) 34.1631 1.27942
\(714\) 0 0
\(715\) 56.3409 32.5284i 2.10703 1.21649i
\(716\) 0 0
\(717\) −48.5197 + 48.5197i −1.81200 + 1.81200i
\(718\) 0 0
\(719\) 7.82407 4.51723i 0.291789 0.168464i −0.346960 0.937880i \(-0.612786\pi\)
0.638748 + 0.769416i \(0.279453\pi\)
\(720\) 0 0
\(721\) 1.88352 + 7.02941i 0.0701461 + 0.261789i
\(722\) 0 0
\(723\) 5.54912 20.7096i 0.206374 0.770199i
\(724\) 0 0
\(725\) −16.3359 + 60.9663i −0.606699 + 2.26423i
\(726\) 0 0
\(727\) −18.8017 + 5.03789i −0.697315 + 0.186845i −0.590028 0.807383i \(-0.700883\pi\)
−0.107287 + 0.994228i \(0.534216\pi\)
\(728\) 0 0
\(729\) −36.2686 −1.34328
\(730\) 0 0
\(731\) 0.525597 + 0.910361i 0.0194399 + 0.0336709i
\(732\) 0 0
\(733\) 14.5379 + 8.39344i 0.536968 + 0.310019i 0.743849 0.668347i \(-0.232998\pi\)
−0.206881 + 0.978366i \(0.566331\pi\)
\(734\) 0 0
\(735\) 49.8690 13.3624i 1.83945 0.492878i
\(736\) 0 0
\(737\) −9.99758 + 17.3163i −0.368266 + 0.637855i
\(738\) 0 0
\(739\) −3.86556 −0.142197 −0.0710985 0.997469i \(-0.522650\pi\)
−0.0710985 + 0.997469i \(0.522650\pi\)
\(740\) 0 0
\(741\) 56.0929 2.06062
\(742\) 0 0
\(743\) 9.77216 16.9259i 0.358506 0.620950i −0.629206 0.777239i \(-0.716620\pi\)
0.987711 + 0.156289i \(0.0499530\pi\)
\(744\) 0 0
\(745\) 26.0487 6.97973i 0.954351 0.255717i
\(746\) 0 0
\(747\) −46.5425 26.8714i −1.70290 0.983171i
\(748\) 0 0
\(749\) 3.08181 + 5.33785i 0.112607 + 0.195041i
\(750\) 0 0
\(751\) 22.4625 0.819668 0.409834 0.912160i \(-0.365587\pi\)
0.409834 + 0.912160i \(0.365587\pi\)
\(752\) 0 0
\(753\) 30.8497 8.26615i 1.12423 0.301235i
\(754\) 0 0
\(755\) −14.4167 + 53.8037i −0.524676 + 1.95812i
\(756\) 0 0
\(757\) −0.398995 + 1.48907i −0.0145017 + 0.0541211i −0.972798 0.231657i \(-0.925585\pi\)
0.958296 + 0.285778i \(0.0922520\pi\)
\(758\) 0 0
\(759\) 12.6465 + 47.1974i 0.459039 + 1.71316i
\(760\) 0 0
\(761\) −12.5172 + 7.22682i −0.453749 + 0.261972i −0.709412 0.704794i \(-0.751040\pi\)
0.255663 + 0.966766i \(0.417706\pi\)
\(762\) 0 0
\(763\) −9.68620 + 9.68620i −0.350664 + 0.350664i
\(764\) 0 0
\(765\) 5.88487 3.39763i 0.212768 0.122842i
\(766\) 0 0
\(767\) −23.1443 −0.835694
\(768\) 0 0
\(769\) −13.1848 + 13.1848i −0.475456 + 0.475456i −0.903675 0.428219i \(-0.859141\pi\)
0.428219 + 0.903675i \(0.359141\pi\)
\(770\) 0 0
\(771\) 57.1784 + 57.1784i 2.05923 + 2.05923i
\(772\) 0 0
\(773\) 3.45856 5.99040i 0.124396 0.215460i −0.797101 0.603846i \(-0.793634\pi\)
0.921497 + 0.388386i \(0.126967\pi\)
\(774\) 0 0
\(775\) 48.1420 + 12.8996i 1.72931 + 0.463368i
\(776\) 0 0
\(777\) −6.97422 + 15.0782i −0.250199 + 0.540928i
\(778\) 0 0
\(779\) 6.54188 24.4146i 0.234387 0.874745i
\(780\) 0 0
\(781\) −9.00765 5.20057i −0.322319 0.186091i
\(782\) 0 0
\(783\) −9.85872 + 9.85872i −0.352322 + 0.352322i
\(784\) 0 0
\(785\) −23.8439 23.8439i −0.851024 0.851024i
\(786\) 0 0
\(787\) 50.2352i 1.79069i 0.445372 + 0.895346i \(0.353071\pi\)
−0.445372 + 0.895346i \(0.646929\pi\)
\(788\) 0 0
\(789\) −10.7883 18.6858i −0.384073 0.665234i
\(790\) 0 0
\(791\) −0.604719 0.604719i −0.0215013 0.0215013i
\(792\) 0 0
\(793\) −21.6207 37.4481i −0.767773 1.32982i
\(794\) 0 0
\(795\) −23.1528 + 6.20377i −0.821145 + 0.220025i
\(796\) 0 0
\(797\) 18.6254 + 4.99067i 0.659747 + 0.176779i 0.573132 0.819463i \(-0.305728\pi\)
0.0866151 + 0.996242i \(0.472395\pi\)
\(798\) 0 0
\(799\) −5.06339 1.35673i −0.179130 0.0479977i
\(800\) 0 0
\(801\) 14.1792 + 52.9175i 0.500997 + 1.86975i
\(802\) 0 0
\(803\) 53.0639i 1.87258i
\(804\) 0 0
\(805\) −14.6625 + 8.46543i −0.516787 + 0.298367i
\(806\) 0 0
\(807\) 31.4957 54.5522i 1.10870 1.92033i
\(808\) 0 0
\(809\) 4.11579 + 15.3603i 0.144703 + 0.540040i 0.999768 + 0.0215196i \(0.00685042\pi\)
−0.855065 + 0.518521i \(0.826483\pi\)
\(810\) 0 0
\(811\) 38.0696 + 21.9795i 1.33680 + 0.771804i 0.986332 0.164770i \(-0.0526882\pi\)
0.350471 + 0.936574i \(0.386022\pi\)
\(812\) 0 0
\(813\) 38.0658i 1.33503i
\(814\) 0 0
\(815\) 9.70149i 0.339828i
\(816\) 0 0
\(817\) −7.79366 4.49967i −0.272666 0.157424i
\(818\) 0 0
\(819\) 4.55807 + 17.0109i 0.159272 + 0.594410i
\(820\) 0 0
\(821\) −0.142888 + 0.247489i −0.00498683 + 0.00863744i −0.868508 0.495675i \(-0.834921\pi\)
0.863521 + 0.504313i \(0.168254\pi\)
\(822\) 0 0
\(823\) −11.3034 + 6.52601i −0.394011 + 0.227483i −0.683897 0.729579i \(-0.739716\pi\)
0.289885 + 0.957061i \(0.406383\pi\)
\(824\) 0 0
\(825\) 71.2850i 2.48182i
\(826\) 0 0
\(827\) −1.28254 4.78649i −0.0445981 0.166442i 0.940035 0.341078i \(-0.110792\pi\)
−0.984633 + 0.174635i \(0.944125\pi\)
\(828\) 0 0
\(829\) −3.11272 0.834050i −0.108109 0.0289677i 0.204359 0.978896i \(-0.434489\pi\)
−0.312468 + 0.949928i \(0.601156\pi\)
\(830\) 0 0
\(831\) 28.3734 + 7.60264i 0.984264 + 0.263733i
\(832\) 0 0
\(833\) 3.13386 0.839716i 0.108582 0.0290944i
\(834\) 0 0
\(835\) −19.5705 33.8971i −0.677265 1.17306i
\(836\) 0 0
\(837\) 7.78493 + 7.78493i 0.269086 + 0.269086i
\(838\) 0 0
\(839\) 15.6429 + 27.0943i 0.540053 + 0.935399i 0.998900 + 0.0468840i \(0.0149291\pi\)
−0.458847 + 0.888515i \(0.651738\pi\)
\(840\) 0 0
\(841\) 58.0525i 2.00181i
\(842\) 0 0
\(843\) 38.9827 + 38.9827i 1.34264 + 1.34264i
\(844\) 0 0
\(845\) 20.1928 20.1928i 0.694654 0.694654i
\(846\) 0 0
\(847\) −5.41095 3.12401i −0.185922 0.107342i
\(848\) 0 0
\(849\) 11.4316 42.6632i 0.392330 1.46420i
\(850\) 0 0
\(851\) −4.81164 + 27.7920i −0.164941 + 0.952698i
\(852\) 0 0
\(853\) 4.80192 + 1.28667i 0.164414 + 0.0440547i 0.340087 0.940394i \(-0.389543\pi\)
−0.175673 + 0.984449i \(0.556210\pi\)
\(854\) 0 0
\(855\) −29.0874 + 50.3808i −0.994766 + 1.72299i
\(856\) 0 0
\(857\) 17.0097 + 17.0097i 0.581040 + 0.581040i 0.935189 0.354149i \(-0.115230\pi\)
−0.354149 + 0.935189i \(0.615230\pi\)
\(858\) 0 0
\(859\) −21.2998 + 21.2998i −0.726741 + 0.726741i −0.969969 0.243228i \(-0.921794\pi\)
0.243228 + 0.969969i \(0.421794\pi\)
\(860\) 0 0
\(861\) 14.5811 0.496923
\(862\) 0 0
\(863\) −18.5645 + 10.7182i −0.631945 + 0.364853i −0.781505 0.623899i \(-0.785548\pi\)
0.149560 + 0.988753i \(0.452214\pi\)
\(864\) 0 0
\(865\) −26.3704 + 26.3704i −0.896620 + 0.896620i
\(866\) 0 0
\(867\) −37.0928 + 21.4155i −1.25974 + 0.727309i
\(868\) 0 0
\(869\) −1.84092 6.87042i −0.0624491 0.233063i
\(870\) 0 0
\(871\) −5.81867 + 21.7156i −0.197158 + 0.735804i
\(872\) 0 0
\(873\) 12.0094 44.8196i 0.406455 1.51691i
\(874\) 0 0
\(875\) −6.22427 + 1.66779i −0.210419 + 0.0563815i
\(876\) 0 0
\(877\) 27.1912 0.918180 0.459090 0.888390i \(-0.348176\pi\)
0.459090 + 0.888390i \(0.348176\pi\)
\(878\) 0 0
\(879\) −4.54501 7.87219i −0.153299 0.265522i
\(880\) 0 0
\(881\) 12.9718 + 7.48930i 0.437032 + 0.252321i 0.702338 0.711844i \(-0.252140\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(882\) 0 0
\(883\) −6.29427 + 1.68655i −0.211819 + 0.0567568i −0.363168 0.931724i \(-0.618305\pi\)
0.151349 + 0.988480i \(0.451638\pi\)
\(884\) 0 0
\(885\) 22.0521 38.1953i 0.741272 1.28392i
\(886\) 0 0
\(887\) 44.7384 1.50217 0.751084 0.660206i \(-0.229531\pi\)
0.751084 + 0.660206i \(0.229531\pi\)
\(888\) 0 0
\(889\) 20.3609 0.682882
\(890\) 0 0
\(891\) 14.1975 24.5909i 0.475636 0.823825i
\(892\) 0 0
\(893\) 43.3480 11.6151i 1.45059 0.388683i
\(894\) 0 0
\(895\) −58.3410 33.6832i −1.95012 1.12590i
\(896\) 0 0
\(897\) 27.4693 + 47.5782i 0.917172 + 1.58859i
\(898\) 0 0
\(899\) −68.7409 −2.29264
\(900\) 0 0
\(901\) −1.45496 + 0.389857i −0.0484719 + 0.0129880i
\(902\) 0 0
\(903\) 1.34367 5.01463i 0.0447144 0.166877i
\(904\) 0 0
\(905\) 5.70859 21.3047i 0.189760 0.708193i
\(906\) 0 0
\(907\) −1.45454 5.42842i −0.0482972 0.180248i 0.937564 0.347814i \(-0.113076\pi\)
−0.985861 + 0.167566i \(0.946409\pi\)
\(908\) 0 0
\(909\) −28.0275 + 16.1817i −0.929613 + 0.536712i
\(910\) 0 0
\(911\) −13.6126 + 13.6126i −0.451007 + 0.451007i −0.895689 0.444682i \(-0.853317\pi\)
0.444682 + 0.895689i \(0.353317\pi\)
\(912\) 0 0
\(913\) 53.3605 30.8077i 1.76598 1.01959i
\(914\) 0 0
\(915\) 82.4012 2.72410
\(916\) 0 0
\(917\) −10.7798 + 10.7798i −0.355981 + 0.355981i
\(918\) 0 0
\(919\) −18.3650 18.3650i −0.605806 0.605806i 0.336041 0.941847i \(-0.390912\pi\)
−0.941847 + 0.336041i \(0.890912\pi\)
\(920\) 0 0
\(921\) 16.4195 28.4394i 0.541041 0.937110i
\(922\) 0 0
\(923\) −11.2961 3.02677i −0.371815 0.0996274i
\(924\) 0 0
\(925\) −17.2744 + 37.3472i −0.567980 + 1.22797i
\(926\) 0 0
\(927\) −6.33864 + 23.6561i −0.208188 + 0.776969i
\(928\) 0 0
\(929\) 45.3737 + 26.1965i 1.48866 + 0.859479i 0.999916 0.0129476i \(-0.00412147\pi\)
0.488745 + 0.872427i \(0.337455\pi\)
\(930\) 0 0
\(931\) −19.6403 + 19.6403i −0.643686 + 0.643686i
\(932\) 0 0
\(933\) −60.8731 60.8731i −1.99289 1.99289i
\(934\) 0 0
\(935\) 7.79070i 0.254783i
\(936\) 0 0
\(937\) −6.95084 12.0392i −0.227074 0.393304i 0.729866 0.683591i \(-0.239583\pi\)
−0.956940 + 0.290287i \(0.906249\pi\)
\(938\) 0 0
\(939\) −20.7038 20.7038i −0.675643 0.675643i
\(940\) 0 0
\(941\) −3.73019 6.46087i −0.121601 0.210618i 0.798798 0.601599i \(-0.205469\pi\)
−0.920399 + 0.390980i \(0.872136\pi\)
\(942\) 0 0
\(943\) 23.9122 6.40726i 0.778688 0.208649i
\(944\) 0 0
\(945\) −5.27030 1.41217i −0.171443 0.0459380i
\(946\) 0 0
\(947\) 25.5560 + 6.84770i 0.830457 + 0.222520i 0.648913 0.760863i \(-0.275224\pi\)
0.181544 + 0.983383i \(0.441891\pi\)
\(948\) 0 0
\(949\) 15.4418 + 57.6295i 0.501262 + 1.87073i
\(950\) 0 0
\(951\) 19.5545i 0.634097i
\(952\) 0 0
\(953\) −1.06205 + 0.613178i −0.0344033 + 0.0198628i −0.517103 0.855923i \(-0.672990\pi\)
0.482700 + 0.875786i \(0.339656\pi\)
\(954\) 0 0
\(955\) 29.5516 51.1849i 0.956268 1.65631i
\(956\) 0 0
\(957\) −25.4466 94.9679i −0.822571 3.06988i
\(958\) 0 0
\(959\) −4.91697 2.83881i −0.158777 0.0916700i
\(960\) 0 0
\(961\) 23.2812i 0.751006i
\(962\) 0 0
\(963\) 20.7425i 0.668418i
\(964\) 0 0
\(965\) 64.8542 + 37.4436i 2.08773 + 1.20535i
\(966\) 0 0
\(967\) 12.0837 + 45.0968i 0.388584 + 1.45022i 0.832439 + 0.554117i \(0.186944\pi\)
−0.443854 + 0.896099i \(0.646389\pi\)
\(968\) 0 0
\(969\) −3.35862 + 5.81731i −0.107895 + 0.186879i
\(970\) 0 0
\(971\) 48.7436 28.1421i 1.56426 0.903123i 0.567437 0.823417i \(-0.307935\pi\)
0.996818 0.0797061i \(-0.0253982\pi\)
\(972\) 0 0
\(973\) 0.989186i 0.0317119i
\(974\) 0 0
\(975\) 20.7442 + 77.4184i 0.664346 + 2.47937i
\(976\) 0 0
\(977\) 26.3628 + 7.06389i 0.843421 + 0.225994i 0.654560 0.756010i \(-0.272854\pi\)
0.188861 + 0.982004i \(0.439521\pi\)
\(978\) 0 0
\(979\) −60.6693 16.2563i −1.93900 0.519554i
\(980\) 0 0
\(981\) −44.5284 + 11.9313i −1.42168 + 0.380939i
\(982\) 0 0
\(983\) −18.6792 32.3533i −0.595774 1.03191i −0.993437 0.114379i \(-0.963512\pi\)
0.397663 0.917531i \(-0.369821\pi\)
\(984\) 0 0
\(985\) 24.0300 + 24.0300i 0.765661 + 0.765661i
\(986\) 0 0
\(987\) 12.9443 + 22.4203i 0.412023 + 0.713645i
\(988\) 0 0
\(989\) 8.81415i 0.280274i
\(990\) 0 0
\(991\) 31.7969 + 31.7969i 1.01006 + 1.01006i 0.999949 + 0.0101114i \(0.00321861\pi\)
0.0101114 + 0.999949i \(0.496781\pi\)
\(992\) 0 0
\(993\) 30.7180 30.7180i 0.974808 0.974808i
\(994\) 0 0
\(995\) −35.7915 20.6642i −1.13467 0.655100i
\(996\) 0 0
\(997\) −11.4993 + 42.9161i −0.364187 + 1.35917i 0.504332 + 0.863510i \(0.331739\pi\)
−0.868519 + 0.495655i \(0.834928\pi\)
\(998\) 0 0
\(999\) −7.42957 + 5.23666i −0.235061 + 0.165681i
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 592.2.be.e.415.2 20
4.3 odd 2 592.2.be.f.415.4 yes 20
37.14 odd 12 592.2.be.f.495.4 yes 20
148.51 even 12 inner 592.2.be.e.495.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
592.2.be.e.415.2 20 1.1 even 1 trivial
592.2.be.e.495.2 yes 20 148.51 even 12 inner
592.2.be.f.415.4 yes 20 4.3 odd 2
592.2.be.f.495.4 yes 20 37.14 odd 12