Properties

Label 1600.2.j.c.1007.4
Level $1600$
Weight $2$
Character 1600.1007
Analytic conductor $12.776$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(143,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (of order \(4\), degree \(2\), not 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1007.4
Root \(1.35949 - 0.389597i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1007
Dual form 1600.2.j.c.143.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.207468i q^{3} +(1.32185 + 1.32185i) q^{7} +2.95696 q^{9} +O(q^{10})\) \(q-0.207468i q^{3} +(1.32185 + 1.32185i) q^{7} +2.95696 q^{9} +(2.39286 + 2.39286i) q^{11} +4.20208 q^{13} +(-3.29071 - 3.29071i) q^{17} +(-0.838342 - 0.838342i) q^{19} +(0.274241 - 0.274241i) q^{21} +(-2.67277 + 2.67277i) q^{23} -1.23588i q^{27} +(2.55451 - 2.55451i) q^{29} +6.23723i q^{31} +(0.496441 - 0.496441i) q^{33} +4.29451 q^{37} -0.871798i q^{39} +7.06598i q^{41} -9.43258 q^{43} +(-8.31820 + 8.31820i) q^{47} -3.50544i q^{49} +(-0.682716 + 0.682716i) q^{51} -7.66672i q^{53} +(-0.173929 + 0.173929i) q^{57} +(7.07557 - 7.07557i) q^{59} +(8.74267 + 8.74267i) q^{61} +(3.90865 + 3.90865i) q^{63} +12.5494 q^{67} +(0.554514 + 0.554514i) q^{69} +10.4624 q^{71} +(4.79095 + 4.79095i) q^{73} +6.32598i q^{77} +8.69963 q^{79} +8.61447 q^{81} -4.14519i q^{83} +(-0.529980 - 0.529980i) q^{87} -0.548482 q^{89} +(5.55451 + 5.55451i) q^{91} +1.29403 q^{93} +(-12.2022 - 12.2022i) q^{97} +(7.07557 + 7.07557i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 8 q^{11} + 8 q^{19} + 16 q^{29} + 48 q^{51} + 8 q^{59} - 16 q^{61} - 16 q^{69} + 32 q^{71} - 80 q^{79} + 16 q^{81} + 64 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 0.207468i 0.119782i −0.998205 0.0598908i \(-0.980925\pi\)
0.998205 0.0598908i \(-0.0190753\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.32185 + 1.32185i 0.499611 + 0.499611i 0.911317 0.411706i \(-0.135067\pi\)
−0.411706 + 0.911317i \(0.635067\pi\)
\(8\) 0 0
\(9\) 2.95696 0.985652
\(10\) 0 0
\(11\) 2.39286 + 2.39286i 0.721473 + 0.721473i 0.968905 0.247432i \(-0.0795867\pi\)
−0.247432 + 0.968905i \(0.579587\pi\)
\(12\) 0 0
\(13\) 4.20208 1.16545 0.582724 0.812670i \(-0.301987\pi\)
0.582724 + 0.812670i \(0.301987\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.29071 3.29071i −0.798114 0.798114i 0.184684 0.982798i \(-0.440874\pi\)
−0.982798 + 0.184684i \(0.940874\pi\)
\(18\) 0 0
\(19\) −0.838342 0.838342i −0.192329 0.192329i 0.604373 0.796702i \(-0.293424\pi\)
−0.796702 + 0.604373i \(0.793424\pi\)
\(20\) 0 0
\(21\) 0.274241 0.274241i 0.0598443 0.0598443i
\(22\) 0 0
\(23\) −2.67277 + 2.67277i −0.557311 + 0.557311i −0.928541 0.371230i \(-0.878936\pi\)
0.371230 + 0.928541i \(0.378936\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.23588i 0.237845i
\(28\) 0 0
\(29\) 2.55451 2.55451i 0.474361 0.474361i −0.428961 0.903323i \(-0.641120\pi\)
0.903323 + 0.428961i \(0.141120\pi\)
\(30\) 0 0
\(31\) 6.23723i 1.12024i 0.828412 + 0.560120i \(0.189245\pi\)
−0.828412 + 0.560120i \(0.810755\pi\)
\(32\) 0 0
\(33\) 0.496441 0.496441i 0.0864192 0.0864192i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.29451 0.706013 0.353006 0.935621i \(-0.385159\pi\)
0.353006 + 0.935621i \(0.385159\pi\)
\(38\) 0 0
\(39\) 0.871798i 0.139599i
\(40\) 0 0
\(41\) 7.06598i 1.10352i 0.834002 + 0.551761i \(0.186044\pi\)
−0.834002 + 0.551761i \(0.813956\pi\)
\(42\) 0 0
\(43\) −9.43258 −1.43845 −0.719227 0.694775i \(-0.755504\pi\)
−0.719227 + 0.694775i \(0.755504\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.31820 + 8.31820i −1.21333 + 1.21333i −0.243410 + 0.969923i \(0.578266\pi\)
−0.969923 + 0.243410i \(0.921734\pi\)
\(48\) 0 0
\(49\) 3.50544i 0.500777i
\(50\) 0 0
\(51\) −0.682716 + 0.682716i −0.0955994 + 0.0955994i
\(52\) 0 0
\(53\) 7.66672i 1.05311i −0.850143 0.526553i \(-0.823484\pi\)
0.850143 0.526553i \(-0.176516\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.173929 + 0.173929i −0.0230375 + 0.0230375i
\(58\) 0 0
\(59\) 7.07557 7.07557i 0.921161 0.921161i −0.0759506 0.997112i \(-0.524199\pi\)
0.997112 + 0.0759506i \(0.0241991\pi\)
\(60\) 0 0
\(61\) 8.74267 + 8.74267i 1.11938 + 1.11938i 0.991832 + 0.127552i \(0.0407120\pi\)
0.127552 + 0.991832i \(0.459288\pi\)
\(62\) 0 0
\(63\) 3.90865 + 3.90865i 0.492443 + 0.492443i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.5494 1.53315 0.766574 0.642156i \(-0.221960\pi\)
0.766574 + 0.642156i \(0.221960\pi\)
\(68\) 0 0
\(69\) 0.554514 + 0.554514i 0.0667556 + 0.0667556i
\(70\) 0 0
\(71\) 10.4624 1.24166 0.620829 0.783946i \(-0.286796\pi\)
0.620829 + 0.783946i \(0.286796\pi\)
\(72\) 0 0
\(73\) 4.79095 + 4.79095i 0.560738 + 0.560738i 0.929517 0.368779i \(-0.120224\pi\)
−0.368779 + 0.929517i \(0.620224\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.32598i 0.720912i
\(78\) 0 0
\(79\) 8.69963 0.978784 0.489392 0.872064i \(-0.337219\pi\)
0.489392 + 0.872064i \(0.337219\pi\)
\(80\) 0 0
\(81\) 8.61447 0.957163
\(82\) 0 0
\(83\) 4.14519i 0.454993i −0.973779 0.227497i \(-0.926946\pi\)
0.973779 0.227497i \(-0.0730541\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.529980 0.529980i −0.0568198 0.0568198i
\(88\) 0 0
\(89\) −0.548482 −0.0581390 −0.0290695 0.999577i \(-0.509254\pi\)
−0.0290695 + 0.999577i \(0.509254\pi\)
\(90\) 0 0
\(91\) 5.55451 + 5.55451i 0.582271 + 0.582271i
\(92\) 0 0
\(93\) 1.29403 0.134184
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.2022 12.2022i −1.23895 1.23895i −0.960431 0.278518i \(-0.910157\pi\)
−0.278518 0.960431i \(-0.589843\pi\)
\(98\) 0 0
\(99\) 7.07557 + 7.07557i 0.711122 + 0.711122i
\(100\) 0 0
\(101\) −6.46843 + 6.46843i −0.643633 + 0.643633i −0.951447 0.307814i \(-0.900403\pi\)
0.307814 + 0.951447i \(0.400403\pi\)
\(102\) 0 0
\(103\) 6.55234 6.55234i 0.645621 0.645621i −0.306310 0.951932i \(-0.599095\pi\)
0.951932 + 0.306310i \(0.0990945\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.3708i 1.58262i −0.611413 0.791312i \(-0.709399\pi\)
0.611413 0.791312i \(-0.290601\pi\)
\(108\) 0 0
\(109\) 2.00000 2.00000i 0.191565 0.191565i −0.604807 0.796372i \(-0.706750\pi\)
0.796372 + 0.604807i \(0.206750\pi\)
\(110\) 0 0
\(111\) 0.890973i 0.0845674i
\(112\) 0 0
\(113\) −11.8688 + 11.8688i −1.11652 + 1.11652i −0.124275 + 0.992248i \(0.539661\pi\)
−0.992248 + 0.124275i \(0.960339\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.4254 1.14873
\(118\) 0 0
\(119\) 8.69963i 0.797493i
\(120\) 0 0
\(121\) 0.451518i 0.0410471i
\(122\) 0 0
\(123\) 1.46597 0.132182
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.55946 + 5.55946i −0.493322 + 0.493322i −0.909351 0.416029i \(-0.863421\pi\)
0.416029 + 0.909351i \(0.363421\pi\)
\(128\) 0 0
\(129\) 1.95696i 0.172300i
\(130\) 0 0
\(131\) −3.16166 + 3.16166i −0.276235 + 0.276235i −0.831604 0.555369i \(-0.812577\pi\)
0.555369 + 0.831604i \(0.312577\pi\)
\(132\) 0 0
\(133\) 2.21632i 0.192179i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.507360 + 0.507360i −0.0433467 + 0.0433467i −0.728448 0.685101i \(-0.759758\pi\)
0.685101 + 0.728448i \(0.259758\pi\)
\(138\) 0 0
\(139\) −5.52106 + 5.52106i −0.468290 + 0.468290i −0.901360 0.433070i \(-0.857430\pi\)
0.433070 + 0.901360i \(0.357430\pi\)
\(140\) 0 0
\(141\) 1.72576 + 1.72576i 0.145335 + 0.145335i
\(142\) 0 0
\(143\) 10.0550 + 10.0550i 0.840840 + 0.840840i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.727266 −0.0599839
\(148\) 0 0
\(149\) −2.82875 2.82875i −0.231741 0.231741i 0.581678 0.813419i \(-0.302396\pi\)
−0.813419 + 0.581678i \(0.802396\pi\)
\(150\) 0 0
\(151\) −7.12820 −0.580085 −0.290042 0.957014i \(-0.593669\pi\)
−0.290042 + 0.957014i \(0.593669\pi\)
\(152\) 0 0
\(153\) −9.73048 9.73048i −0.786663 0.786663i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20.7371i 1.65500i 0.561464 + 0.827501i \(0.310238\pi\)
−0.561464 + 0.827501i \(0.689762\pi\)
\(158\) 0 0
\(159\) −1.59060 −0.126143
\(160\) 0 0
\(161\) −7.06598 −0.556878
\(162\) 0 0
\(163\) 10.8985i 0.853640i −0.904337 0.426820i \(-0.859634\pi\)
0.904337 0.426820i \(-0.140366\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.20680 + 1.20680i 0.0933853 + 0.0933853i 0.752256 0.658871i \(-0.228966\pi\)
−0.658871 + 0.752256i \(0.728966\pi\)
\(168\) 0 0
\(169\) 4.65751 0.358270
\(170\) 0 0
\(171\) −2.47894 2.47894i −0.189569 0.189569i
\(172\) 0 0
\(173\) 5.12438 0.389599 0.194800 0.980843i \(-0.437594\pi\)
0.194800 + 0.980843i \(0.437594\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.46795 1.46795i −0.110338 0.110338i
\(178\) 0 0
\(179\) −12.6301 12.6301i −0.944017 0.944017i 0.0544970 0.998514i \(-0.482644\pi\)
−0.998514 + 0.0544970i \(0.982644\pi\)
\(180\) 0 0
\(181\) 8.46843 8.46843i 0.629453 0.629453i −0.318477 0.947931i \(-0.603171\pi\)
0.947931 + 0.318477i \(0.103171\pi\)
\(182\) 0 0
\(183\) 1.81382 1.81382i 0.134082 0.134082i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.7484i 1.15164i
\(188\) 0 0
\(189\) 1.63364 1.63364i 0.118830 0.118830i
\(190\) 0 0
\(191\) 11.3534i 0.821501i 0.911748 + 0.410750i \(0.134733\pi\)
−0.911748 + 0.410750i \(0.865267\pi\)
\(192\) 0 0
\(193\) −11.0245 + 11.0245i −0.793561 + 0.793561i −0.982071 0.188510i \(-0.939634\pi\)
0.188510 + 0.982071i \(0.439634\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.1953 1.51010 0.755050 0.655667i \(-0.227612\pi\)
0.755050 + 0.655667i \(0.227612\pi\)
\(198\) 0 0
\(199\) 8.68045i 0.615341i 0.951493 + 0.307670i \(0.0995494\pi\)
−0.951493 + 0.307670i \(0.900451\pi\)
\(200\) 0 0
\(201\) 2.60359i 0.183643i
\(202\) 0 0
\(203\) 6.75335 0.473993
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.90326 + 7.90326i −0.549315 + 0.549315i
\(208\) 0 0
\(209\) 4.01206i 0.277520i
\(210\) 0 0
\(211\) 6.28986 6.28986i 0.433012 0.433012i −0.456640 0.889652i \(-0.650947\pi\)
0.889652 + 0.456640i \(0.150947\pi\)
\(212\) 0 0
\(213\) 2.17061i 0.148728i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.24467 + 8.24467i −0.559684 + 0.559684i
\(218\) 0 0
\(219\) 0.993968 0.993968i 0.0671661 0.0671661i
\(220\) 0 0
\(221\) −13.8278 13.8278i −0.930160 0.930160i
\(222\) 0 0
\(223\) 3.50264 + 3.50264i 0.234554 + 0.234554i 0.814591 0.580036i \(-0.196962\pi\)
−0.580036 + 0.814591i \(0.696962\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.622404 0.0413104 0.0206552 0.999787i \(-0.493425\pi\)
0.0206552 + 0.999787i \(0.493425\pi\)
\(228\) 0 0
\(229\) 9.35940 + 9.35940i 0.618487 + 0.618487i 0.945143 0.326657i \(-0.105922\pi\)
−0.326657 + 0.945143i \(0.605922\pi\)
\(230\) 0 0
\(231\) 1.31244 0.0863521
\(232\) 0 0
\(233\) −1.79047 1.79047i −0.117297 0.117297i 0.646022 0.763319i \(-0.276432\pi\)
−0.763319 + 0.646022i \(0.776432\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.80489i 0.117240i
\(238\) 0 0
\(239\) −20.9056 −1.35227 −0.676136 0.736777i \(-0.736347\pi\)
−0.676136 + 0.736777i \(0.736347\pi\)
\(240\) 0 0
\(241\) −3.70055 −0.238374 −0.119187 0.992872i \(-0.538029\pi\)
−0.119187 + 0.992872i \(0.538029\pi\)
\(242\) 0 0
\(243\) 5.49486i 0.352495i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.52278 3.52278i −0.224149 0.224149i
\(248\) 0 0
\(249\) −0.859993 −0.0544999
\(250\) 0 0
\(251\) −0.630086 0.630086i −0.0397707 0.0397707i 0.686942 0.726712i \(-0.258953\pi\)
−0.726712 + 0.686942i \(0.758953\pi\)
\(252\) 0 0
\(253\) −12.7911 −0.804170
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.62414 3.62414i −0.226068 0.226068i 0.584980 0.811048i \(-0.301102\pi\)
−0.811048 + 0.584980i \(0.801102\pi\)
\(258\) 0 0
\(259\) 5.67668 + 5.67668i 0.352732 + 0.352732i
\(260\) 0 0
\(261\) 7.55359 7.55359i 0.467555 0.467555i
\(262\) 0 0
\(263\) −5.25957 + 5.25957i −0.324319 + 0.324319i −0.850421 0.526102i \(-0.823653\pi\)
0.526102 + 0.850421i \(0.323653\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.113792i 0.00696398i
\(268\) 0 0
\(269\) −8.82875 + 8.82875i −0.538299 + 0.538299i −0.923029 0.384730i \(-0.874294\pi\)
0.384730 + 0.923029i \(0.374294\pi\)
\(270\) 0 0
\(271\) 30.9177i 1.87812i −0.343760 0.939058i \(-0.611701\pi\)
0.343760 0.939058i \(-0.388299\pi\)
\(272\) 0 0
\(273\) 1.15238 1.15238i 0.0697454 0.0697454i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.9322 −1.13753 −0.568764 0.822501i \(-0.692578\pi\)
−0.568764 + 0.822501i \(0.692578\pi\)
\(278\) 0 0
\(279\) 18.4432i 1.10417i
\(280\) 0 0
\(281\) 6.51750i 0.388802i −0.980922 0.194401i \(-0.937724\pi\)
0.980922 0.194401i \(-0.0622762\pi\)
\(282\) 0 0
\(283\) −16.0721 −0.955390 −0.477695 0.878526i \(-0.658528\pi\)
−0.477695 + 0.878526i \(0.658528\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.34015 + 9.34015i −0.551332 + 0.551332i
\(288\) 0 0
\(289\) 4.65751i 0.273971i
\(290\) 0 0
\(291\) −2.53157 + 2.53157i −0.148403 + 0.148403i
\(292\) 0 0
\(293\) 0.829872i 0.0484816i −0.999706 0.0242408i \(-0.992283\pi\)
0.999706 0.0242408i \(-0.00771685\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.95728 2.95728i 0.171599 0.171599i
\(298\) 0 0
\(299\) −11.2312 + 11.2312i −0.649517 + 0.649517i
\(300\) 0 0
\(301\) −12.4684 12.4684i −0.718668 0.718668i
\(302\) 0 0
\(303\) 1.34199 + 1.34199i 0.0770954 + 0.0770954i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.4115 1.62153 0.810766 0.585371i \(-0.199051\pi\)
0.810766 + 0.585371i \(0.199051\pi\)
\(308\) 0 0
\(309\) −1.35940 1.35940i −0.0773336 0.0773336i
\(310\) 0 0
\(311\) −31.8087 −1.80370 −0.901852 0.432046i \(-0.857792\pi\)
−0.901852 + 0.432046i \(0.857792\pi\)
\(312\) 0 0
\(313\) 18.7727 + 18.7727i 1.06110 + 1.06110i 0.998008 + 0.0630896i \(0.0200954\pi\)
0.0630896 + 0.998008i \(0.479905\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.6205i 0.596506i −0.954487 0.298253i \(-0.903596\pi\)
0.954487 0.298253i \(-0.0964039\pi\)
\(318\) 0 0
\(319\) 12.2252 0.684478
\(320\) 0 0
\(321\) −3.39641 −0.189569
\(322\) 0 0
\(323\) 5.51748i 0.307001i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.414936 0.414936i −0.0229460 0.0229460i
\(328\) 0 0
\(329\) −21.9908 −1.21239
\(330\) 0 0
\(331\) −22.4098 22.4098i −1.23175 1.23175i −0.963290 0.268462i \(-0.913485\pi\)
−0.268462 0.963290i \(-0.586515\pi\)
\(332\) 0 0
\(333\) 12.6987 0.695883
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.95728 2.95728i −0.161093 0.161093i 0.621958 0.783051i \(-0.286338\pi\)
−0.783051 + 0.621958i \(0.786338\pi\)
\(338\) 0 0
\(339\) 2.46240 + 2.46240i 0.133739 + 0.133739i
\(340\) 0 0
\(341\) −14.9248 + 14.9248i −0.808223 + 0.808223i
\(342\) 0 0
\(343\) 13.8866 13.8866i 0.749805 0.749805i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.8901i 0.745659i 0.927900 + 0.372830i \(0.121612\pi\)
−0.927900 + 0.372830i \(0.878388\pi\)
\(348\) 0 0
\(349\) −13.4914 + 13.4914i −0.722176 + 0.722176i −0.969048 0.246872i \(-0.920597\pi\)
0.246872 + 0.969048i \(0.420597\pi\)
\(350\) 0 0
\(351\) 5.19326i 0.277196i
\(352\) 0 0
\(353\) 15.9785 15.9785i 0.850448 0.850448i −0.139740 0.990188i \(-0.544627\pi\)
0.990188 + 0.139740i \(0.0446268\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.80489 −0.0955251
\(358\) 0 0
\(359\) 28.2831i 1.49273i −0.665539 0.746363i \(-0.731798\pi\)
0.665539 0.746363i \(-0.268202\pi\)
\(360\) 0 0
\(361\) 17.5944i 0.926019i
\(362\) 0 0
\(363\) 0.0936755 0.00491669
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.8285 16.8285i 0.878439 0.878439i −0.114934 0.993373i \(-0.536666\pi\)
0.993373 + 0.114934i \(0.0366656\pi\)
\(368\) 0 0
\(369\) 20.8938i 1.08769i
\(370\) 0 0
\(371\) 10.1342 10.1342i 0.526143 0.526143i
\(372\) 0 0
\(373\) 8.03845i 0.416215i −0.978106 0.208108i \(-0.933270\pi\)
0.978106 0.208108i \(-0.0667304\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.7343 10.7343i 0.552844 0.552844i
\(378\) 0 0
\(379\) −2.30677 + 2.30677i −0.118491 + 0.118491i −0.763866 0.645375i \(-0.776701\pi\)
0.645375 + 0.763866i \(0.276701\pi\)
\(380\) 0 0
\(381\) 1.15341 + 1.15341i 0.0590910 + 0.0590910i
\(382\) 0 0
\(383\) −7.85530 7.85530i −0.401387 0.401387i 0.477335 0.878722i \(-0.341603\pi\)
−0.878722 + 0.477335i \(0.841603\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −27.8917 −1.41782
\(388\) 0 0
\(389\) −27.2171 27.2171i −1.37996 1.37996i −0.844660 0.535303i \(-0.820197\pi\)
−0.535303 0.844660i \(-0.679803\pi\)
\(390\) 0 0
\(391\) 17.5906 0.889595
\(392\) 0 0
\(393\) 0.655943 + 0.655943i 0.0330879 + 0.0330879i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.38693i 0.220174i 0.993922 + 0.110087i \(0.0351129\pi\)
−0.993922 + 0.110087i \(0.964887\pi\)
\(398\) 0 0
\(399\) −0.459815 −0.0230196
\(400\) 0 0
\(401\) −24.5944 −1.22818 −0.614092 0.789234i \(-0.710478\pi\)
−0.614092 + 0.789234i \(0.710478\pi\)
\(402\) 0 0
\(403\) 26.2094i 1.30558i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.2761 + 10.2761i 0.509369 + 0.509369i
\(408\) 0 0
\(409\) −7.06598 −0.349390 −0.174695 0.984623i \(-0.555894\pi\)
−0.174695 + 0.984623i \(0.555894\pi\)
\(410\) 0 0
\(411\) 0.105261 + 0.105261i 0.00519214 + 0.00519214i
\(412\) 0 0
\(413\) 18.7057 0.920445
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.14544 + 1.14544i 0.0560926 + 0.0560926i
\(418\) 0 0
\(419\) −4.54400 4.54400i −0.221989 0.221989i 0.587347 0.809336i \(-0.300173\pi\)
−0.809336 + 0.587347i \(0.800173\pi\)
\(420\) 0 0
\(421\) 5.47539 5.47539i 0.266854 0.266854i −0.560977 0.827831i \(-0.689574\pi\)
0.827831 + 0.560977i \(0.189574\pi\)
\(422\) 0 0
\(423\) −24.5966 + 24.5966i −1.19593 + 1.19593i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 23.1129i 1.11851i
\(428\) 0 0
\(429\) 2.08609 2.08609i 0.100717 0.100717i
\(430\) 0 0
\(431\) 12.6658i 0.610090i −0.952338 0.305045i \(-0.901328\pi\)
0.952338 0.305045i \(-0.0986716\pi\)
\(432\) 0 0
\(433\) 11.0499 11.0499i 0.531022 0.531022i −0.389854 0.920876i \(-0.627475\pi\)
0.920876 + 0.389854i \(0.127475\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.48139 0.214374
\(438\) 0 0
\(439\) 20.4936i 0.978108i 0.872254 + 0.489054i \(0.162658\pi\)
−0.872254 + 0.489054i \(0.837342\pi\)
\(440\) 0 0
\(441\) 10.3654i 0.493592i
\(442\) 0 0
\(443\) −11.0123 −0.523212 −0.261606 0.965175i \(-0.584252\pi\)
−0.261606 + 0.965175i \(0.584252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.586876 + 0.586876i −0.0277583 + 0.0277583i
\(448\) 0 0
\(449\) 15.4423i 0.728767i −0.931249 0.364383i \(-0.881280\pi\)
0.931249 0.364383i \(-0.118720\pi\)
\(450\) 0 0
\(451\) −16.9079 + 16.9079i −0.796161 + 0.796161i
\(452\) 0 0
\(453\) 1.47887i 0.0694835i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.14212 3.14212i 0.146982 0.146982i −0.629786 0.776769i \(-0.716857\pi\)
0.776769 + 0.629786i \(0.216857\pi\)
\(458\) 0 0
\(459\) −4.06691 + 4.06691i −0.189827 + 0.189827i
\(460\) 0 0
\(461\) 8.45636 + 8.45636i 0.393852 + 0.393852i 0.876058 0.482206i \(-0.160164\pi\)
−0.482206 + 0.876058i \(0.660164\pi\)
\(462\) 0 0
\(463\) −21.5859 21.5859i −1.00318 1.00318i −0.999995 0.00318643i \(-0.998986\pi\)
−0.00318643 0.999995i \(-0.501014\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.4880 −1.04062 −0.520311 0.853977i \(-0.674184\pi\)
−0.520311 + 0.853977i \(0.674184\pi\)
\(468\) 0 0
\(469\) 16.5883 + 16.5883i 0.765978 + 0.765978i
\(470\) 0 0
\(471\) 4.30229 0.198239
\(472\) 0 0
\(473\) −22.5708 22.5708i −1.03781 1.03781i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 22.6702i 1.03800i
\(478\) 0 0
\(479\) −0.852623 −0.0389573 −0.0194787 0.999810i \(-0.506201\pi\)
−0.0194787 + 0.999810i \(0.506201\pi\)
\(480\) 0 0
\(481\) 18.0459 0.822821
\(482\) 0 0
\(483\) 1.46597i 0.0667037i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.49958 2.49958i −0.113267 0.113267i 0.648202 0.761469i \(-0.275521\pi\)
−0.761469 + 0.648202i \(0.775521\pi\)
\(488\) 0 0
\(489\) −2.26110 −0.102250
\(490\) 0 0
\(491\) −22.1177 22.1177i −0.998157 0.998157i 0.00184099 0.999998i \(-0.499414\pi\)
−0.999998 + 0.00184099i \(0.999414\pi\)
\(492\) 0 0
\(493\) −16.8123 −0.757189
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.8297 + 13.8297i 0.620346 + 0.620346i
\(498\) 0 0
\(499\) 7.73911 + 7.73911i 0.346450 + 0.346450i 0.858786 0.512335i \(-0.171219\pi\)
−0.512335 + 0.858786i \(0.671219\pi\)
\(500\) 0 0
\(501\) 0.250373 0.250373i 0.0111858 0.0111858i
\(502\) 0 0
\(503\) −13.9144 + 13.9144i −0.620413 + 0.620413i −0.945637 0.325224i \(-0.894560\pi\)
0.325224 + 0.945637i \(0.394560\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.966284i 0.0429142i
\(508\) 0 0
\(509\) 4.48049 4.48049i 0.198594 0.198594i −0.600803 0.799397i \(-0.705152\pi\)
0.799397 + 0.600803i \(0.205152\pi\)
\(510\) 0 0
\(511\) 12.6658i 0.560302i
\(512\) 0 0
\(513\) −1.03609 + 1.03609i −0.0457444 + 0.0457444i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −39.8085 −1.75078
\(518\) 0 0
\(519\) 1.06314i 0.0466669i
\(520\) 0 0
\(521\) 3.26728i 0.143142i −0.997435 0.0715711i \(-0.977199\pi\)
0.997435 0.0715711i \(-0.0228013\pi\)
\(522\) 0 0
\(523\) 10.1372 0.443270 0.221635 0.975130i \(-0.428861\pi\)
0.221635 + 0.975130i \(0.428861\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.5249 20.5249i 0.894079 0.894079i
\(528\) 0 0
\(529\) 8.71262i 0.378809i
\(530\) 0 0
\(531\) 20.9222 20.9222i 0.907945 0.907945i
\(532\) 0 0
\(533\) 29.6919i 1.28610i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.62034 + 2.62034i −0.113076 + 0.113076i
\(538\) 0 0
\(539\) 8.38801 8.38801i 0.361297 0.361297i
\(540\) 0 0
\(541\) −18.3823 18.3823i −0.790319 0.790319i 0.191227 0.981546i \(-0.438753\pi\)
−0.981546 + 0.191227i \(0.938753\pi\)
\(542\) 0 0
\(543\) −1.75693 1.75693i −0.0753970 0.0753970i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.60503 0.154140 0.0770699 0.997026i \(-0.475444\pi\)
0.0770699 + 0.997026i \(0.475444\pi\)
\(548\) 0 0
\(549\) 25.8517 + 25.8517i 1.10332 + 1.10332i
\(550\) 0 0
\(551\) −4.28311 −0.182467
\(552\) 0 0
\(553\) 11.4996 + 11.4996i 0.489012 + 0.489012i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.5670i 0.913823i −0.889512 0.456911i \(-0.848956\pi\)
0.889512 0.456911i \(-0.151044\pi\)
\(558\) 0 0
\(559\) −39.6365 −1.67644
\(560\) 0 0
\(561\) −3.26728 −0.137945
\(562\) 0 0
\(563\) 26.5306i 1.11813i −0.829123 0.559066i \(-0.811160\pi\)
0.829123 0.559066i \(-0.188840\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 11.3870 + 11.3870i 0.478209 + 0.478209i
\(568\) 0 0
\(569\) 9.23630 0.387206 0.193603 0.981080i \(-0.437983\pi\)
0.193603 + 0.981080i \(0.437983\pi\)
\(570\) 0 0
\(571\) 31.6530 + 31.6530i 1.32464 + 1.32464i 0.909975 + 0.414663i \(0.136100\pi\)
0.414663 + 0.909975i \(0.363900\pi\)
\(572\) 0 0
\(573\) 2.35546 0.0984007
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.8539 + 23.8539i 0.993051 + 0.993051i 0.999976 0.00692502i \(-0.00220432\pi\)
−0.00692502 + 0.999976i \(0.502204\pi\)
\(578\) 0 0
\(579\) 2.28723 + 2.28723i 0.0950541 + 0.0950541i
\(580\) 0 0
\(581\) 5.47931 5.47931i 0.227320 0.227320i
\(582\) 0 0
\(583\) 18.3454 18.3454i 0.759787 0.759787i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.2348i 1.00028i 0.865946 + 0.500138i \(0.166718\pi\)
−0.865946 + 0.500138i \(0.833282\pi\)
\(588\) 0 0
\(589\) 5.22893 5.22893i 0.215454 0.215454i
\(590\) 0 0
\(591\) 4.39734i 0.180882i
\(592\) 0 0
\(593\) 15.4929 15.4929i 0.636219 0.636219i −0.313402 0.949621i \(-0.601469\pi\)
0.949621 + 0.313402i \(0.101469\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.80091 0.0737065
\(598\) 0 0
\(599\) 19.1812i 0.783722i 0.920024 + 0.391861i \(0.128169\pi\)
−0.920024 + 0.391861i \(0.871831\pi\)
\(600\) 0 0
\(601\) 21.7464i 0.887056i 0.896261 + 0.443528i \(0.146273\pi\)
−0.896261 + 0.443528i \(0.853727\pi\)
\(602\) 0 0
\(603\) 37.1079 1.51115
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.2541 20.2541i 0.822088 0.822088i −0.164319 0.986407i \(-0.552543\pi\)
0.986407 + 0.164319i \(0.0525428\pi\)
\(608\) 0 0
\(609\) 1.40110i 0.0567756i
\(610\) 0 0
\(611\) −34.9538 + 34.9538i −1.41408 + 1.41408i
\(612\) 0 0
\(613\) 6.00698i 0.242620i 0.992615 + 0.121310i \(0.0387094\pi\)
−0.992615 + 0.121310i \(0.961291\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.9289 + 20.9289i −0.842566 + 0.842566i −0.989192 0.146626i \(-0.953159\pi\)
0.146626 + 0.989192i \(0.453159\pi\)
\(618\) 0 0
\(619\) 3.30074 3.30074i 0.132668 0.132668i −0.637655 0.770322i \(-0.720095\pi\)
0.770322 + 0.637655i \(0.220095\pi\)
\(620\) 0 0
\(621\) 3.30321 + 3.30321i 0.132553 + 0.132553i
\(622\) 0 0
\(623\) −0.725009 0.725009i −0.0290469 0.0290469i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.832374 −0.0332418
\(628\) 0 0
\(629\) −14.1320 14.1320i −0.563479 0.563479i
\(630\) 0 0
\(631\) −4.39734 −0.175055 −0.0875276 0.996162i \(-0.527897\pi\)
−0.0875276 + 0.996162i \(0.527897\pi\)
\(632\) 0 0
\(633\) −1.30494 1.30494i −0.0518669 0.0518669i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.7301i 0.583630i
\(638\) 0 0
\(639\) 30.9369 1.22384
\(640\) 0 0
\(641\) 8.74978 0.345596 0.172798 0.984957i \(-0.444719\pi\)
0.172798 + 0.984957i \(0.444719\pi\)
\(642\) 0 0
\(643\) 26.4968i 1.04493i 0.852660 + 0.522466i \(0.174988\pi\)
−0.852660 + 0.522466i \(0.825012\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.8343 15.8343i −0.622512 0.622512i 0.323661 0.946173i \(-0.395086\pi\)
−0.946173 + 0.323661i \(0.895086\pi\)
\(648\) 0 0
\(649\) 33.8616 1.32919
\(650\) 0 0
\(651\) 1.71050 + 1.71050i 0.0670399 + 0.0670399i
\(652\) 0 0
\(653\) −26.4608 −1.03549 −0.517746 0.855534i \(-0.673229\pi\)
−0.517746 + 0.855534i \(0.673229\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.1666 + 14.1666i 0.552693 + 0.552693i
\(658\) 0 0
\(659\) −30.5128 30.5128i −1.18861 1.18861i −0.977452 0.211156i \(-0.932277\pi\)
−0.211156 0.977452i \(-0.567723\pi\)
\(660\) 0 0
\(661\) −4.81695 + 4.81695i −0.187358 + 0.187358i −0.794553 0.607195i \(-0.792295\pi\)
0.607195 + 0.794553i \(0.292295\pi\)
\(662\) 0 0
\(663\) −2.86883 + 2.86883i −0.111416 + 0.111416i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.6552i 0.528733i
\(668\) 0 0
\(669\) 0.726685 0.726685i 0.0280953 0.0280953i
\(670\) 0 0
\(671\) 41.8399i 1.61521i
\(672\) 0 0
\(673\) 2.80519 2.80519i 0.108132 0.108132i −0.650971 0.759103i \(-0.725638\pi\)
0.759103 + 0.650971i \(0.225638\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.5498 −0.674492 −0.337246 0.941417i \(-0.609495\pi\)
−0.337246 + 0.941417i \(0.609495\pi\)
\(678\) 0 0
\(679\) 32.2590i 1.23799i
\(680\) 0 0
\(681\) 0.129129i 0.00494823i
\(682\) 0 0
\(683\) 21.7834 0.833518 0.416759 0.909017i \(-0.363166\pi\)
0.416759 + 0.909017i \(0.363166\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.94178 1.94178i 0.0740833 0.0740833i
\(688\) 0 0
\(689\) 32.2162i 1.22734i
\(690\) 0 0
\(691\) 31.2147 31.2147i 1.18746 1.18746i 0.209694 0.977767i \(-0.432753\pi\)
0.977767 0.209694i \(-0.0672469\pi\)
\(692\) 0 0
\(693\) 18.7057i 0.710569i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 23.2521 23.2521i 0.880736 0.880736i
\(698\) 0 0
\(699\) −0.371464 + 0.371464i −0.0140501 + 0.0140501i
\(700\) 0 0
\(701\) 20.9248 + 20.9248i 0.790318 + 0.790318i 0.981546 0.191227i \(-0.0612468\pi\)
−0.191227 + 0.981546i \(0.561247\pi\)
\(702\) 0 0
\(703\) −3.60027 3.60027i −0.135787 0.135787i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.1005 −0.643132
\(708\) 0 0
\(709\) −26.3703 26.3703i −0.990357 0.990357i 0.00959736 0.999954i \(-0.496945\pi\)
−0.999954 + 0.00959736i \(0.996945\pi\)
\(710\) 0 0
\(711\) 25.7244 0.964741
\(712\) 0 0
\(713\) −16.6707 16.6707i −0.624322 0.624322i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.33724i 0.161977i
\(718\) 0 0
\(719\) −2.65374 −0.0989679 −0.0494840 0.998775i \(-0.515758\pi\)
−0.0494840 + 0.998775i \(0.515758\pi\)
\(720\) 0 0
\(721\) 17.3224 0.645119
\(722\) 0 0
\(723\) 0.767746i 0.0285528i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −30.8999 30.8999i −1.14601 1.14601i −0.987329 0.158685i \(-0.949275\pi\)
−0.158685 0.987329i \(-0.550725\pi\)
\(728\) 0 0
\(729\) 24.7034 0.914940
\(730\) 0 0
\(731\) 31.0399 + 31.0399i 1.14805 + 1.14805i
\(732\) 0 0
\(733\) −46.2726 −1.70912 −0.854559 0.519354i \(-0.826172\pi\)
−0.854559 + 0.519354i \(0.826172\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0288 + 30.0288i 1.10613 + 1.10613i
\(738\) 0 0
\(739\) 12.0504 + 12.0504i 0.443280 + 0.443280i 0.893113 0.449833i \(-0.148516\pi\)
−0.449833 + 0.893113i \(0.648516\pi\)
\(740\) 0 0
\(741\) −0.730865 + 0.730865i −0.0268490 + 0.0268490i
\(742\) 0 0
\(743\) 21.4100 21.4100i 0.785456 0.785456i −0.195290 0.980746i \(-0.562565\pi\)
0.980746 + 0.195290i \(0.0625648\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.2571i 0.448465i
\(748\) 0 0
\(749\) 21.6397 21.6397i 0.790696 0.790696i
\(750\) 0 0
\(751\) 8.50828i 0.310472i 0.987877 + 0.155236i \(0.0496137\pi\)
−0.987877 + 0.155236i \(0.950386\pi\)
\(752\) 0 0
\(753\) −0.130723 + 0.130723i −0.00476379 + 0.00476379i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −29.2794 −1.06418 −0.532089 0.846688i \(-0.678593\pi\)
−0.532089 + 0.846688i \(0.678593\pi\)
\(758\) 0 0
\(759\) 2.65374i 0.0963248i
\(760\) 0 0
\(761\) 39.3132i 1.42510i 0.701621 + 0.712551i \(0.252460\pi\)
−0.701621 + 0.712551i \(0.747540\pi\)
\(762\) 0 0
\(763\) 5.28739 0.191416
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 29.7321 29.7321i 1.07357 1.07357i
\(768\) 0 0
\(769\) 26.4222i 0.952809i 0.879226 + 0.476404i \(0.158060\pi\)
−0.879226 + 0.476404i \(0.841940\pi\)
\(770\) 0 0
\(771\) −0.751892 + 0.751892i −0.0270787 + 0.0270787i
\(772\) 0 0
\(773\) 8.08416i 0.290767i 0.989375 + 0.145383i \(0.0464416\pi\)
−0.989375 + 0.145383i \(0.953558\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.17773 1.17773i 0.0422508 0.0422508i
\(778\) 0 0
\(779\) 5.92371 5.92371i 0.212239 0.212239i
\(780\) 0 0
\(781\) 25.0350 + 25.0350i 0.895823 + 0.895823i
\(782\) 0 0
\(783\) −3.15707 3.15707i −0.112824 0.112824i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −40.5596 −1.44579 −0.722897 0.690955i \(-0.757190\pi\)
−0.722897 + 0.690955i \(0.757190\pi\)
\(788\) 0 0
\(789\) 1.09119 + 1.09119i 0.0388474 + 0.0388474i
\(790\) 0 0
\(791\) −31.3775 −1.11566
\(792\) 0 0
\(793\) 36.7374 + 36.7374i 1.30458 + 1.30458i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.85060i 0.0655517i −0.999463 0.0327759i \(-0.989565\pi\)
0.999463 0.0327759i \(-0.0104347\pi\)
\(798\) 0 0
\(799\) 54.7455 1.93676
\(800\) 0 0
\(801\) −1.62184 −0.0573048
\(802\) 0 0
\(803\) 22.9281i 0.809115i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.83168 + 1.83168i 0.0644783 + 0.0644783i
\(808\) 0 0
\(809\) −26.5823 −0.934584 −0.467292 0.884103i \(-0.654770\pi\)
−0.467292 + 0.884103i \(0.654770\pi\)
\(810\) 0 0
\(811\) −6.49812 6.49812i −0.228180 0.228180i 0.583752 0.811932i \(-0.301584\pi\)
−0.811932 + 0.583752i \(0.801584\pi\)
\(812\) 0 0
\(813\) −6.41443 −0.224964
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.90773 + 7.90773i 0.276656 + 0.276656i
\(818\) 0 0
\(819\) 16.4245 + 16.4245i 0.573917 + 0.573917i
\(820\) 0 0
\(821\) 6.00000 6.00000i 0.209401 0.209401i −0.594612 0.804013i \(-0.702694\pi\)
0.804013 + 0.594612i \(0.202694\pi\)
\(822\) 0 0
\(823\) 15.1106 15.1106i 0.526722 0.526722i −0.392871 0.919593i \(-0.628518\pi\)
0.919593 + 0.392871i \(0.128518\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.3792i 0.465241i 0.972568 + 0.232621i \(0.0747301\pi\)
−0.972568 + 0.232621i \(0.925270\pi\)
\(828\) 0 0
\(829\) −5.81788 + 5.81788i −0.202063 + 0.202063i −0.800883 0.598820i \(-0.795636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(830\) 0 0
\(831\) 3.92783i 0.136255i
\(832\) 0 0
\(833\) −11.5354 + 11.5354i −0.399677 + 0.399677i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.70845 0.266443
\(838\) 0 0
\(839\) 27.4305i 0.947006i −0.880792 0.473503i \(-0.842989\pi\)
0.880792 0.473503i \(-0.157011\pi\)
\(840\) 0 0
\(841\) 15.9489i 0.549963i
\(842\) 0 0
\(843\) −1.35217 −0.0465713
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.596838 + 0.596838i −0.0205076 + 0.0205076i
\(848\) 0 0
\(849\) 3.33445i 0.114438i
\(850\) 0 0
\(851\) −11.4782 + 11.4782i −0.393469 + 0.393469i
\(852\) 0 0
\(853\) 16.6692i 0.570742i −0.958417 0.285371i \(-0.907883\pi\)
0.958417 0.285371i \(-0.0921169\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.73331 + 4.73331i −0.161687 + 0.161687i −0.783314 0.621627i \(-0.786472\pi\)
0.621627 + 0.783314i \(0.286472\pi\)
\(858\) 0 0
\(859\) 26.3237 26.3237i 0.898152 0.898152i −0.0971203 0.995273i \(-0.530963\pi\)
0.995273 + 0.0971203i \(0.0309632\pi\)
\(860\) 0 0
\(861\) 1.93778 + 1.93778i 0.0660394 + 0.0660394i
\(862\) 0 0
\(863\) −8.71130 8.71130i −0.296536 0.296536i 0.543119 0.839655i \(-0.317243\pi\)
−0.839655 + 0.543119i \(0.817243\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.966284 0.0328167
\(868\) 0 0
\(869\) 20.8170 + 20.8170i 0.706167 + 0.706167i
\(870\) 0 0
\(871\) 52.7334 1.78680
\(872\) 0 0
\(873\) −36.0815 36.0815i −1.22117 1.22117i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 33.6674i 1.13687i 0.822730 + 0.568433i \(0.192450\pi\)
−0.822730 + 0.568433i \(0.807550\pi\)
\(878\) 0 0
\(879\) −0.172172 −0.00580721
\(880\) 0 0
\(881\) 27.3058 0.919956 0.459978 0.887930i \(-0.347857\pi\)
0.459978 + 0.887930i \(0.347857\pi\)
\(882\) 0 0
\(883\) 27.9628i 0.941023i 0.882394 + 0.470511i \(0.155930\pi\)
−0.882394 + 0.470511i \(0.844070\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.4497 + 12.4497i 0.418020 + 0.418020i 0.884521 0.466501i \(-0.154486\pi\)
−0.466501 + 0.884521i \(0.654486\pi\)
\(888\) 0 0
\(889\) −14.6975 −0.492939
\(890\) 0 0
\(891\) 20.6132 + 20.6132i 0.690567 + 0.690567i
\(892\) 0 0
\(893\) 13.9470 0.466718
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.33011 + 2.33011i 0.0778002 + 0.0778002i
\(898\) 0 0
\(899\) 15.9331 + 15.9331i 0.531398 + 0.531398i
\(900\) 0 0
\(901\) −25.2289 + 25.2289i −0.840498 + 0.840498i
\(902\) 0 0
\(903\) −2.58680 + 2.58680i −0.0860833 + 0.0860833i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.3527i 1.27348i 0.771078 + 0.636740i \(0.219718\pi\)
−0.771078 + 0.636740i \(0.780282\pi\)
\(908\) 0 0
\(909\) −19.1269 + 19.1269i −0.634398 + 0.634398i
\(910\) 0 0
\(911\) 24.0192i 0.795791i −0.917431 0.397895i \(-0.869741\pi\)
0.917431 0.397895i \(-0.130259\pi\)
\(912\) 0 0
\(913\) 9.91884 9.91884i 0.328266 0.328266i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.35846 −0.276021
\(918\) 0 0
\(919\) 25.1211i 0.828668i 0.910125 + 0.414334i \(0.135986\pi\)
−0.910125 + 0.414334i \(0.864014\pi\)
\(920\) 0 0
\(921\) 5.89448i 0.194230i
\(922\) 0 0
\(923\) 43.9639 1.44709
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 19.3750 19.3750i 0.636358 0.636358i
\(928\) 0 0
\(929\) 39.8307i 1.30680i −0.757012 0.653401i \(-0.773342\pi\)
0.757012 0.653401i \(-0.226658\pi\)
\(930\) 0 0
\(931\) −2.93876 + 2.93876i −0.0963139 + 0.0963139i
\(932\) 0 0
\(933\) 6.59927i 0.216051i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.2763 18.2763i 0.597060 0.597060i −0.342469 0.939529i \(-0.611263\pi\)
0.939529 + 0.342469i \(0.111263\pi\)
\(938\) 0 0
\(939\) 3.89474 3.89474i 0.127100 0.127100i
\(940\) 0 0
\(941\) −2.34852 2.34852i −0.0765597 0.0765597i 0.667790 0.744350i \(-0.267240\pi\)
−0.744350 + 0.667790i \(0.767240\pi\)
\(942\) 0 0
\(943\) −18.8857 18.8857i −0.615004 0.615004i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.7829 −0.610363 −0.305181 0.952294i \(-0.598717\pi\)
−0.305181 + 0.952294i \(0.598717\pi\)
\(948\) 0 0
\(949\) 20.1320 + 20.1320i 0.653511 + 0.653511i
\(950\) 0 0
\(951\) −2.20341 −0.0714505
\(952\) 0 0
\(953\) −9.07454 9.07454i −0.293953 0.293953i 0.544687 0.838640i \(-0.316649\pi\)
−0.838640 + 0.544687i \(0.816649\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.53633i 0.0819879i
\(958\) 0 0
\(959\) −1.34130 −0.0433130
\(960\) 0 0
\(961\) −7.90304 −0.254937
\(962\) 0 0
\(963\) 48.4077i 1.55992i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.3057 + 13.3057i 0.427882 + 0.427882i 0.887906 0.460024i \(-0.152159\pi\)
−0.460024 + 0.887906i \(0.652159\pi\)
\(968\) 0 0
\(969\) 1.14470 0.0367730
\(970\) 0 0
\(971\) −18.6349 18.6349i −0.598023 0.598023i 0.341763 0.939786i \(-0.388976\pi\)
−0.939786 + 0.341763i \(0.888976\pi\)
\(972\) 0 0
\(973\) −14.5960 −0.467926
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.28407 7.28407i −0.233038 0.233038i 0.580921 0.813960i \(-0.302692\pi\)
−0.813960 + 0.580921i \(0.802692\pi\)
\(978\) 0 0
\(979\) −1.31244 1.31244i −0.0419457 0.0419457i
\(980\) 0 0
\(981\) 5.91391 5.91391i 0.188817 0.188817i
\(982\) 0 0
\(983\) −31.0322 + 31.0322i −0.989772 + 0.989772i −0.999948 0.0101761i \(-0.996761\pi\)
0.0101761 + 0.999948i \(0.496761\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.56238i 0.145222i
\(988\) 0 0
\(989\) 25.2111 25.2111i 0.801666 0.801666i
\(990\) 0 0
\(991\) 40.1060i 1.27401i 0.770860 + 0.637004i \(0.219827\pi\)
−0.770860 + 0.637004i \(0.780173\pi\)
\(992\) 0 0
\(993\) −4.64931 + 4.64931i −0.147541 + 0.147541i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −2.58204 −0.0817740 −0.0408870 0.999164i \(-0.513018\pi\)
−0.0408870 + 0.999164i \(0.513018\pi\)
\(998\) 0 0
\(999\) 5.30749i 0.167921i
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 1600.2.j.c.1007.4 16
4.3 odd 2 400.2.j.c.307.1 yes 16
5.2 odd 4 1600.2.s.c.943.4 16
5.3 odd 4 1600.2.s.c.943.5 16
5.4 even 2 inner 1600.2.j.c.1007.5 16
16.5 even 4 400.2.s.c.107.5 yes 16
16.11 odd 4 1600.2.s.c.207.5 16
20.3 even 4 400.2.s.c.243.5 yes 16
20.7 even 4 400.2.s.c.243.4 yes 16
20.19 odd 2 400.2.j.c.307.8 yes 16
80.27 even 4 inner 1600.2.j.c.143.4 16
80.37 odd 4 400.2.j.c.43.8 yes 16
80.43 even 4 inner 1600.2.j.c.143.5 16
80.53 odd 4 400.2.j.c.43.1 16
80.59 odd 4 1600.2.s.c.207.4 16
80.69 even 4 400.2.s.c.107.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.c.43.1 16 80.53 odd 4
400.2.j.c.43.8 yes 16 80.37 odd 4
400.2.j.c.307.1 yes 16 4.3 odd 2
400.2.j.c.307.8 yes 16 20.19 odd 2
400.2.s.c.107.4 yes 16 80.69 even 4
400.2.s.c.107.5 yes 16 16.5 even 4
400.2.s.c.243.4 yes 16 20.7 even 4
400.2.s.c.243.5 yes 16 20.3 even 4
1600.2.j.c.143.4 16 80.27 even 4 inner
1600.2.j.c.143.5 16 80.43 even 4 inner
1600.2.j.c.1007.4 16 1.1 even 1 trivial
1600.2.j.c.1007.5 16 5.4 even 2 inner
1600.2.s.c.207.4 16 80.59 odd 4
1600.2.s.c.207.5 16 16.11 odd 4
1600.2.s.c.943.4 16 5.2 odd 4
1600.2.s.c.943.5 16 5.3 odd 4