Properties

Label 108.3.k.a.65.6
Level $108$
Weight $3$
Character 108.65
Analytic conductor $2.943$
Analytic rank $0$
Dimension $36$
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] = [108,3,Mod(5,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.k (of order \(18\), degree \(6\), 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: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 65.6
Character \(\chi\) \(=\) 108.65
Dual form 108.3.k.a.5.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.99363 + 0.195330i) q^{3} +(-2.50118 + 6.87194i) q^{5} +(-1.62729 + 9.22884i) q^{7} +(8.92369 + 1.16949i) q^{9} +O(q^{10})\) \(q+(2.99363 + 0.195330i) q^{3} +(-2.50118 + 6.87194i) q^{5} +(-1.62729 + 9.22884i) q^{7} +(8.92369 + 1.16949i) q^{9} +(-4.02712 - 11.0644i) q^{11} +(17.4349 - 14.6296i) q^{13} +(-8.82992 + 20.0835i) q^{15} +(-13.5275 + 7.81011i) q^{17} +(9.08487 - 15.7354i) q^{19} +(-6.67419 + 27.3099i) q^{21} +(23.0574 - 4.06564i) q^{23} +(-21.8166 - 18.3063i) q^{25} +(26.4858 + 5.24410i) q^{27} +(-24.8898 + 29.6625i) q^{29} +(-4.47532 - 25.3808i) q^{31} +(-9.89451 - 33.9094i) q^{33} +(-59.3499 - 34.2657i) q^{35} +(1.45889 + 2.52687i) q^{37} +(55.0514 - 40.3903i) q^{39} +(-26.1577 - 31.1736i) q^{41} +(35.7017 - 12.9943i) q^{43} +(-30.3565 + 58.3980i) q^{45} +(-18.5209 - 3.26574i) q^{47} +(-36.4785 - 13.2771i) q^{49} +(-42.0220 + 20.7383i) q^{51} -12.3119i q^{53} +86.1066 q^{55} +(30.2704 - 45.3316i) q^{57} +(4.38973 - 12.0607i) q^{59} +(-15.6708 + 88.8733i) q^{61} +(-25.3145 + 80.4522i) q^{63} +(56.9261 + 156.403i) q^{65} +(-6.68767 + 5.61162i) q^{67} +(69.8195 - 7.66724i) q^{69} +(81.5508 - 47.0834i) q^{71} +(-25.3297 + 43.8724i) q^{73} +(-61.7350 - 59.0637i) q^{75} +(108.665 - 19.1606i) q^{77} +(-78.4939 - 65.8642i) q^{79} +(78.2646 + 20.8724i) q^{81} +(-12.1497 + 14.4795i) q^{83} +(-19.8359 - 112.495i) q^{85} +(-80.3049 + 83.9370i) q^{87} +(-52.8907 - 30.5364i) q^{89} +(106.643 + 184.711i) q^{91} +(-8.43984 - 76.8550i) q^{93} +(85.4102 + 101.788i) q^{95} +(-139.585 + 50.8047i) q^{97} +(-22.9970 - 103.445i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 9 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 9 q^{5} + 6 q^{9} + 36 q^{11} + 45 q^{15} + 42 q^{21} - 18 q^{23} - 9 q^{25} - 18 q^{29} + 45 q^{31} - 153 q^{33} - 243 q^{35} - 123 q^{39} - 198 q^{41} + 90 q^{43} - 333 q^{45} - 243 q^{47} + 72 q^{49} - 99 q^{51} + 243 q^{57} + 252 q^{59} - 144 q^{61} + 381 q^{63} + 747 q^{65} + 108 q^{67} + 585 q^{69} + 324 q^{71} - 63 q^{73} + 597 q^{75} + 495 q^{77} + 36 q^{79} - 54 q^{81} - 27 q^{83} - 180 q^{85} - 441 q^{87} - 567 q^{89} + 99 q^{91} - 699 q^{93} - 1044 q^{95} - 216 q^{97} - 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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
\(3\) 2.99363 + 0.195330i 0.997878 + 0.0651100i
\(4\) 0 0
\(5\) −2.50118 + 6.87194i −0.500236 + 1.37439i 0.390808 + 0.920472i \(0.372196\pi\)
−0.891044 + 0.453916i \(0.850027\pi\)
\(6\) 0 0
\(7\) −1.62729 + 9.22884i −0.232470 + 1.31841i 0.615405 + 0.788211i \(0.288992\pi\)
−0.847876 + 0.530195i \(0.822119\pi\)
\(8\) 0 0
\(9\) 8.92369 + 1.16949i 0.991521 + 0.129944i
\(10\) 0 0
\(11\) −4.02712 11.0644i −0.366102 1.00586i −0.976830 0.214016i \(-0.931345\pi\)
0.610728 0.791840i \(-0.290877\pi\)
\(12\) 0 0
\(13\) 17.4349 14.6296i 1.34115 1.12536i 0.359820 0.933022i \(-0.382838\pi\)
0.981329 0.192336i \(-0.0616063\pi\)
\(14\) 0 0
\(15\) −8.82992 + 20.0835i −0.588661 + 1.33890i
\(16\) 0 0
\(17\) −13.5275 + 7.81011i −0.795736 + 0.459418i −0.841978 0.539512i \(-0.818609\pi\)
0.0462422 + 0.998930i \(0.485275\pi\)
\(18\) 0 0
\(19\) 9.08487 15.7354i 0.478151 0.828182i −0.521535 0.853230i \(-0.674641\pi\)
0.999686 + 0.0250481i \(0.00797389\pi\)
\(20\) 0 0
\(21\) −6.67419 + 27.3099i −0.317819 + 1.30047i
\(22\) 0 0
\(23\) 23.0574 4.06564i 1.00250 0.176767i 0.351775 0.936085i \(-0.385578\pi\)
0.650720 + 0.759318i \(0.274467\pi\)
\(24\) 0 0
\(25\) −21.8166 18.3063i −0.872662 0.732251i
\(26\) 0 0
\(27\) 26.4858 + 5.24410i 0.980957 + 0.194226i
\(28\) 0 0
\(29\) −24.8898 + 29.6625i −0.858269 + 1.02284i 0.141191 + 0.989982i \(0.454907\pi\)
−0.999460 + 0.0328626i \(0.989538\pi\)
\(30\) 0 0
\(31\) −4.47532 25.3808i −0.144365 0.818735i −0.967875 0.251433i \(-0.919098\pi\)
0.823510 0.567302i \(-0.192013\pi\)
\(32\) 0 0
\(33\) −9.89451 33.9094i −0.299834 1.02756i
\(34\) 0 0
\(35\) −59.3499 34.2657i −1.69571 0.979019i
\(36\) 0 0
\(37\) 1.45889 + 2.52687i 0.0394294 + 0.0682937i 0.885067 0.465464i \(-0.154113\pi\)
−0.845637 + 0.533758i \(0.820779\pi\)
\(38\) 0 0
\(39\) 55.0514 40.3903i 1.41158 1.03565i
\(40\) 0 0
\(41\) −26.1577 31.1736i −0.637993 0.760331i 0.346058 0.938213i \(-0.387520\pi\)
−0.984052 + 0.177882i \(0.943075\pi\)
\(42\) 0 0
\(43\) 35.7017 12.9943i 0.830271 0.302194i 0.108301 0.994118i \(-0.465459\pi\)
0.721970 + 0.691924i \(0.243237\pi\)
\(44\) 0 0
\(45\) −30.3565 + 58.3980i −0.674588 + 1.29773i
\(46\) 0 0
\(47\) −18.5209 3.26574i −0.394063 0.0694839i −0.0268941 0.999638i \(-0.508562\pi\)
−0.367169 + 0.930154i \(0.619673\pi\)
\(48\) 0 0
\(49\) −36.4785 13.2771i −0.744458 0.270961i
\(50\) 0 0
\(51\) −42.0220 + 20.7383i −0.823960 + 0.406633i
\(52\) 0 0
\(53\) 12.3119i 0.232300i −0.993232 0.116150i \(-0.962945\pi\)
0.993232 0.116150i \(-0.0370554\pi\)
\(54\) 0 0
\(55\) 86.1066 1.56557
\(56\) 0 0
\(57\) 30.2704 45.3316i 0.531059 0.795292i
\(58\) 0 0
\(59\) 4.38973 12.0607i 0.0744022 0.204418i −0.896916 0.442200i \(-0.854198\pi\)
0.971319 + 0.237782i \(0.0764203\pi\)
\(60\) 0 0
\(61\) −15.6708 + 88.8733i −0.256898 + 1.45694i 0.534257 + 0.845322i \(0.320592\pi\)
−0.791154 + 0.611617i \(0.790520\pi\)
\(62\) 0 0
\(63\) −25.3145 + 80.4522i −0.401818 + 1.27702i
\(64\) 0 0
\(65\) 56.9261 + 156.403i 0.875787 + 2.40620i
\(66\) 0 0
\(67\) −6.68767 + 5.61162i −0.0998159 + 0.0837555i −0.691330 0.722539i \(-0.742975\pi\)
0.591514 + 0.806295i \(0.298530\pi\)
\(68\) 0 0
\(69\) 69.8195 7.66724i 1.01188 0.111119i
\(70\) 0 0
\(71\) 81.5508 47.0834i 1.14860 0.663146i 0.200057 0.979784i \(-0.435887\pi\)
0.948546 + 0.316638i \(0.102554\pi\)
\(72\) 0 0
\(73\) −25.3297 + 43.8724i −0.346983 + 0.600992i −0.985712 0.168440i \(-0.946127\pi\)
0.638729 + 0.769432i \(0.279460\pi\)
\(74\) 0 0
\(75\) −61.7350 59.0637i −0.823134 0.787516i
\(76\) 0 0
\(77\) 108.665 19.1606i 1.41123 0.248839i
\(78\) 0 0
\(79\) −78.4939 65.8642i −0.993593 0.833724i −0.00750946 0.999972i \(-0.502390\pi\)
−0.986084 + 0.166248i \(0.946835\pi\)
\(80\) 0 0
\(81\) 78.2646 + 20.8724i 0.966229 + 0.257684i
\(82\) 0 0
\(83\) −12.1497 + 14.4795i −0.146382 + 0.174451i −0.834253 0.551381i \(-0.814101\pi\)
0.687871 + 0.725833i \(0.258545\pi\)
\(84\) 0 0
\(85\) −19.8359 112.495i −0.233363 1.32347i
\(86\) 0 0
\(87\) −80.3049 + 83.9370i −0.923045 + 0.964793i
\(88\) 0 0
\(89\) −52.8907 30.5364i −0.594277 0.343106i 0.172510 0.985008i \(-0.444812\pi\)
−0.766787 + 0.641902i \(0.778146\pi\)
\(90\) 0 0
\(91\) 106.643 + 184.711i 1.17190 + 2.02979i
\(92\) 0 0
\(93\) −8.43984 76.8550i −0.0907510 0.826398i
\(94\) 0 0
\(95\) 85.4102 + 101.788i 0.899055 + 1.07145i
\(96\) 0 0
\(97\) −139.585 + 50.8047i −1.43902 + 0.523760i −0.939501 0.342545i \(-0.888711\pi\)
−0.499516 + 0.866305i \(0.666489\pi\)
\(98\) 0 0
\(99\) −22.9970 103.445i −0.232293 1.04490i
\(100\) 0 0
\(101\) 3.22458 + 0.568581i 0.0319266 + 0.00562952i 0.189589 0.981864i \(-0.439285\pi\)
−0.157662 + 0.987493i \(0.550396\pi\)
\(102\) 0 0
\(103\) 18.2867 + 6.65582i 0.177541 + 0.0646196i 0.429261 0.903180i \(-0.358774\pi\)
−0.251720 + 0.967800i \(0.580996\pi\)
\(104\) 0 0
\(105\) −170.979 114.172i −1.62837 1.08735i
\(106\) 0 0
\(107\) 24.7281i 0.231103i 0.993301 + 0.115552i \(0.0368636\pi\)
−0.993301 + 0.115552i \(0.963136\pi\)
\(108\) 0 0
\(109\) 1.01801 0.00933957 0.00466979 0.999989i \(-0.498514\pi\)
0.00466979 + 0.999989i \(0.498514\pi\)
\(110\) 0 0
\(111\) 3.87380 + 7.84947i 0.0348991 + 0.0707160i
\(112\) 0 0
\(113\) 53.0907 145.865i 0.469829 1.29084i −0.448059 0.894004i \(-0.647884\pi\)
0.917888 0.396840i \(-0.129893\pi\)
\(114\) 0 0
\(115\) −29.7319 + 168.618i −0.258538 + 1.46624i
\(116\) 0 0
\(117\) 172.693 110.160i 1.47601 0.941542i
\(118\) 0 0
\(119\) −50.0650 137.553i −0.420714 1.15590i
\(120\) 0 0
\(121\) −13.5123 + 11.3382i −0.111672 + 0.0937041i
\(122\) 0 0
\(123\) −72.2175 98.4316i −0.587134 0.800257i
\(124\) 0 0
\(125\) 22.0364 12.7227i 0.176291 0.101782i
\(126\) 0 0
\(127\) 35.9061 62.1913i 0.282726 0.489695i −0.689330 0.724448i \(-0.742095\pi\)
0.972055 + 0.234753i \(0.0754281\pi\)
\(128\) 0 0
\(129\) 109.416 31.9267i 0.848185 0.247494i
\(130\) 0 0
\(131\) −17.2894 + 3.04859i −0.131980 + 0.0232717i −0.239248 0.970958i \(-0.576901\pi\)
0.107268 + 0.994230i \(0.465790\pi\)
\(132\) 0 0
\(133\) 130.436 + 109.449i 0.980723 + 0.822924i
\(134\) 0 0
\(135\) −102.283 + 168.893i −0.757652 + 1.25106i
\(136\) 0 0
\(137\) −13.3559 + 15.9169i −0.0974881 + 0.116182i −0.812583 0.582846i \(-0.801939\pi\)
0.715095 + 0.699028i \(0.246384\pi\)
\(138\) 0 0
\(139\) −1.08034 6.12690i −0.00777221 0.0440784i 0.980675 0.195644i \(-0.0626797\pi\)
−0.988447 + 0.151565i \(0.951569\pi\)
\(140\) 0 0
\(141\) −54.8070 13.3941i −0.388702 0.0949938i
\(142\) 0 0
\(143\) −232.081 133.992i −1.62295 0.937008i
\(144\) 0 0
\(145\) −141.585 245.233i −0.976449 1.69126i
\(146\) 0 0
\(147\) −106.610 46.8720i −0.725236 0.318857i
\(148\) 0 0
\(149\) 144.811 + 172.579i 0.971888 + 1.15825i 0.987380 + 0.158371i \(0.0506242\pi\)
−0.0154918 + 0.999880i \(0.504931\pi\)
\(150\) 0 0
\(151\) −221.794 + 80.7264i −1.46883 + 0.534612i −0.947784 0.318914i \(-0.896682\pi\)
−0.521051 + 0.853526i \(0.674460\pi\)
\(152\) 0 0
\(153\) −129.849 + 53.8747i −0.848687 + 0.352122i
\(154\) 0 0
\(155\) 185.609 + 32.7279i 1.19748 + 0.211148i
\(156\) 0 0
\(157\) −34.1578 12.4324i −0.217566 0.0791874i 0.230938 0.972969i \(-0.425821\pi\)
−0.448503 + 0.893781i \(0.648043\pi\)
\(158\) 0 0
\(159\) 2.40488 36.8573i 0.0151251 0.231807i
\(160\) 0 0
\(161\) 219.409i 1.36279i
\(162\) 0 0
\(163\) −68.8785 −0.422568 −0.211284 0.977425i \(-0.567764\pi\)
−0.211284 + 0.977425i \(0.567764\pi\)
\(164\) 0 0
\(165\) 257.772 + 16.8192i 1.56225 + 0.101935i
\(166\) 0 0
\(167\) 38.6584 106.213i 0.231488 0.636007i −0.768505 0.639844i \(-0.778999\pi\)
0.999993 + 0.00383681i \(0.00122130\pi\)
\(168\) 0 0
\(169\) 60.6039 343.702i 0.358603 2.03374i
\(170\) 0 0
\(171\) 99.4730 129.794i 0.581714 0.759027i
\(172\) 0 0
\(173\) 46.9901 + 129.104i 0.271619 + 0.746267i 0.998244 + 0.0592329i \(0.0188655\pi\)
−0.726625 + 0.687034i \(0.758912\pi\)
\(174\) 0 0
\(175\) 204.448 171.552i 1.16827 0.980296i
\(176\) 0 0
\(177\) 15.4971 35.2478i 0.0875540 0.199140i
\(178\) 0 0
\(179\) 166.758 96.2777i 0.931608 0.537864i 0.0442884 0.999019i \(-0.485898\pi\)
0.887320 + 0.461155i \(0.152565\pi\)
\(180\) 0 0
\(181\) −38.0528 + 65.9094i −0.210237 + 0.364140i −0.951788 0.306755i \(-0.900757\pi\)
0.741552 + 0.670895i \(0.234090\pi\)
\(182\) 0 0
\(183\) −64.2721 + 262.993i −0.351214 + 1.43712i
\(184\) 0 0
\(185\) −21.0134 + 3.70523i −0.113586 + 0.0200283i
\(186\) 0 0
\(187\) 140.891 + 118.222i 0.753429 + 0.632202i
\(188\) 0 0
\(189\) −91.4971 + 235.900i −0.484112 + 1.24815i
\(190\) 0 0
\(191\) −132.171 + 157.515i −0.691992 + 0.824684i −0.991595 0.129380i \(-0.958701\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(192\) 0 0
\(193\) 50.6060 + 287.001i 0.262207 + 1.48705i 0.776872 + 0.629659i \(0.216805\pi\)
−0.514665 + 0.857391i \(0.672084\pi\)
\(194\) 0 0
\(195\) 139.866 + 479.334i 0.717261 + 2.45812i
\(196\) 0 0
\(197\) −263.147 151.928i −1.33577 0.771209i −0.349595 0.936901i \(-0.613681\pi\)
−0.986178 + 0.165692i \(0.947014\pi\)
\(198\) 0 0
\(199\) 133.234 + 230.767i 0.669515 + 1.15963i 0.978040 + 0.208418i \(0.0668314\pi\)
−0.308525 + 0.951216i \(0.599835\pi\)
\(200\) 0 0
\(201\) −21.1165 + 15.4928i −0.105057 + 0.0770788i
\(202\) 0 0
\(203\) −233.247 277.974i −1.14900 1.36933i
\(204\) 0 0
\(205\) 279.648 101.784i 1.36414 0.496505i
\(206\) 0 0
\(207\) 210.512 9.31507i 1.01697 0.0450004i
\(208\) 0 0
\(209\) −210.689 37.1502i −1.00808 0.177752i
\(210\) 0 0
\(211\) −83.5395 30.4059i −0.395922 0.144104i 0.136384 0.990656i \(-0.456452\pi\)
−0.532306 + 0.846552i \(0.678674\pi\)
\(212\) 0 0
\(213\) 253.330 125.021i 1.18934 0.586954i
\(214\) 0 0
\(215\) 277.841i 1.29228i
\(216\) 0 0
\(217\) 241.518 1.11299
\(218\) 0 0
\(219\) −84.3976 + 126.390i −0.385377 + 0.577124i
\(220\) 0 0
\(221\) −121.592 + 334.071i −0.550190 + 1.51164i
\(222\) 0 0
\(223\) 50.4092 285.885i 0.226050 1.28199i −0.634617 0.772827i \(-0.718842\pi\)
0.860667 0.509168i \(-0.170047\pi\)
\(224\) 0 0
\(225\) −173.275 188.874i −0.770112 0.839439i
\(226\) 0 0
\(227\) −63.1774 173.578i −0.278315 0.764663i −0.997554 0.0699006i \(-0.977732\pi\)
0.719239 0.694762i \(-0.244490\pi\)
\(228\) 0 0
\(229\) −261.099 + 219.088i −1.14017 + 0.956715i −0.999444 0.0333342i \(-0.989387\pi\)
−0.140724 + 0.990049i \(0.544943\pi\)
\(230\) 0 0
\(231\) 329.046 36.1342i 1.42444 0.156425i
\(232\) 0 0
\(233\) 262.837 151.749i 1.12806 0.651283i 0.184611 0.982812i \(-0.440898\pi\)
0.943445 + 0.331528i \(0.107564\pi\)
\(234\) 0 0
\(235\) 68.7662 119.107i 0.292622 0.506837i
\(236\) 0 0
\(237\) −222.117 212.505i −0.937201 0.896648i
\(238\) 0 0
\(239\) −431.057 + 76.0070i −1.80359 + 0.318021i −0.971574 0.236735i \(-0.923923\pi\)
−0.832013 + 0.554756i \(0.812812\pi\)
\(240\) 0 0
\(241\) −124.647 104.591i −0.517209 0.433990i 0.346449 0.938069i \(-0.387387\pi\)
−0.863657 + 0.504079i \(0.831832\pi\)
\(242\) 0 0
\(243\) 230.219 + 77.7717i 0.947401 + 0.320048i
\(244\) 0 0
\(245\) 182.479 217.469i 0.744810 0.887630i
\(246\) 0 0
\(247\) −71.8100 407.255i −0.290729 1.64881i
\(248\) 0 0
\(249\) −39.2001 + 40.9730i −0.157430 + 0.164550i
\(250\) 0 0
\(251\) 394.843 + 227.963i 1.57308 + 0.908219i 0.995788 + 0.0916811i \(0.0292240\pi\)
0.577292 + 0.816538i \(0.304109\pi\)
\(252\) 0 0
\(253\) −137.839 238.744i −0.544817 0.943652i
\(254\) 0 0
\(255\) −37.4077 340.643i −0.146697 1.33585i
\(256\) 0 0
\(257\) −266.990 318.187i −1.03887 1.23808i −0.970672 0.240408i \(-0.922719\pi\)
−0.0682006 0.997672i \(-0.521726\pi\)
\(258\) 0 0
\(259\) −25.6941 + 9.35188i −0.0992049 + 0.0361076i
\(260\) 0 0
\(261\) −256.799 + 235.591i −0.983904 + 0.902646i
\(262\) 0 0
\(263\) −420.070 74.0696i −1.59722 0.281633i −0.697003 0.717068i \(-0.745483\pi\)
−0.900220 + 0.435435i \(0.856595\pi\)
\(264\) 0 0
\(265\) 84.6067 + 30.7943i 0.319271 + 0.116205i
\(266\) 0 0
\(267\) −152.371 101.746i −0.570676 0.381071i
\(268\) 0 0
\(269\) 229.784i 0.854215i 0.904201 + 0.427108i \(0.140467\pi\)
−0.904201 + 0.427108i \(0.859533\pi\)
\(270\) 0 0
\(271\) −247.394 −0.912892 −0.456446 0.889751i \(-0.650878\pi\)
−0.456446 + 0.889751i \(0.650878\pi\)
\(272\) 0 0
\(273\) 283.170 + 573.788i 1.03725 + 2.10179i
\(274\) 0 0
\(275\) −114.690 + 315.109i −0.417056 + 1.14585i
\(276\) 0 0
\(277\) 25.5477 144.888i 0.0922299 0.523062i −0.903331 0.428944i \(-0.858886\pi\)
0.995561 0.0941179i \(-0.0300031\pi\)
\(278\) 0 0
\(279\) −10.2537 231.724i −0.0367517 0.830553i
\(280\) 0 0
\(281\) −15.8083 43.4331i −0.0562575 0.154566i 0.908381 0.418145i \(-0.137319\pi\)
−0.964638 + 0.263578i \(0.915097\pi\)
\(282\) 0 0
\(283\) 94.8323 79.5738i 0.335097 0.281179i −0.459676 0.888087i \(-0.652034\pi\)
0.794773 + 0.606907i \(0.207590\pi\)
\(284\) 0 0
\(285\) 235.805 + 321.399i 0.827385 + 1.12772i
\(286\) 0 0
\(287\) 330.262 190.677i 1.15074 0.664379i
\(288\) 0 0
\(289\) −22.5044 + 38.9787i −0.0778698 + 0.134874i
\(290\) 0 0
\(291\) −427.789 + 124.826i −1.47007 + 0.428954i
\(292\) 0 0
\(293\) 134.669 23.7457i 0.459621 0.0810435i 0.0609563 0.998140i \(-0.480585\pi\)
0.398664 + 0.917097i \(0.369474\pi\)
\(294\) 0 0
\(295\) 71.9008 + 60.3319i 0.243731 + 0.204515i
\(296\) 0 0
\(297\) −48.6387 314.169i −0.163767 1.05781i
\(298\) 0 0
\(299\) 342.525 408.206i 1.14557 1.36524i
\(300\) 0 0
\(301\) 61.8256 + 350.630i 0.205401 + 1.16489i
\(302\) 0 0
\(303\) 9.54217 + 2.33198i 0.0314923 + 0.00769631i
\(304\) 0 0
\(305\) −571.537 329.977i −1.87389 1.08189i
\(306\) 0 0
\(307\) −148.518 257.240i −0.483771 0.837915i 0.516056 0.856555i \(-0.327400\pi\)
−0.999826 + 0.0186398i \(0.994066\pi\)
\(308\) 0 0
\(309\) 53.4436 + 23.4970i 0.172957 + 0.0760422i
\(310\) 0 0
\(311\) 237.230 + 282.720i 0.762797 + 0.909067i 0.998021 0.0628746i \(-0.0200268\pi\)
−0.235224 + 0.971941i \(0.575582\pi\)
\(312\) 0 0
\(313\) 322.695 117.451i 1.03097 0.375244i 0.229522 0.973303i \(-0.426284\pi\)
0.801452 + 0.598059i \(0.204061\pi\)
\(314\) 0 0
\(315\) −489.547 375.186i −1.55412 1.19107i
\(316\) 0 0
\(317\) 537.404 + 94.7588i 1.69528 + 0.298924i 0.936042 0.351889i \(-0.114460\pi\)
0.759239 + 0.650812i \(0.225572\pi\)
\(318\) 0 0
\(319\) 428.433 + 155.937i 1.34305 + 0.488830i
\(320\) 0 0
\(321\) −4.83013 + 74.0268i −0.0150471 + 0.230613i
\(322\) 0 0
\(323\) 283.815i 0.878685i
\(324\) 0 0
\(325\) −648.185 −1.99441
\(326\) 0 0
\(327\) 3.04756 + 0.198848i 0.00931975 + 0.000608099i
\(328\) 0 0
\(329\) 60.2780 165.612i 0.183216 0.503381i
\(330\) 0 0
\(331\) −31.4888 + 178.582i −0.0951322 + 0.539522i 0.899575 + 0.436767i \(0.143877\pi\)
−0.994707 + 0.102754i \(0.967234\pi\)
\(332\) 0 0
\(333\) 10.0635 + 24.2551i 0.0302207 + 0.0728382i
\(334\) 0 0
\(335\) −21.8356 59.9929i −0.0651810 0.179083i
\(336\) 0 0
\(337\) −287.391 + 241.149i −0.852791 + 0.715577i −0.960403 0.278616i \(-0.910124\pi\)
0.107611 + 0.994193i \(0.465680\pi\)
\(338\) 0 0
\(339\) 187.426 426.298i 0.552879 1.25751i
\(340\) 0 0
\(341\) −262.801 + 151.728i −0.770678 + 0.444951i
\(342\) 0 0
\(343\) −47.7015 + 82.6215i −0.139072 + 0.240879i
\(344\) 0 0
\(345\) −121.943 + 498.973i −0.353457 + 1.44630i
\(346\) 0 0
\(347\) 167.538 29.5415i 0.482820 0.0851341i 0.0730598 0.997328i \(-0.476724\pi\)
0.409760 + 0.912193i \(0.365613\pi\)
\(348\) 0 0
\(349\) 126.546 + 106.185i 0.362596 + 0.304254i 0.805824 0.592155i \(-0.201723\pi\)
−0.443229 + 0.896409i \(0.646167\pi\)
\(350\) 0 0
\(351\) 538.498 296.048i 1.53418 0.843442i
\(352\) 0 0
\(353\) −54.0593 + 64.4254i −0.153143 + 0.182508i −0.837161 0.546956i \(-0.815786\pi\)
0.684018 + 0.729465i \(0.260231\pi\)
\(354\) 0 0
\(355\) 119.581 + 678.177i 0.336847 + 1.91036i
\(356\) 0 0
\(357\) −123.008 421.561i −0.344561 1.18084i
\(358\) 0 0
\(359\) 289.723 + 167.271i 0.807027 + 0.465937i 0.845922 0.533306i \(-0.179051\pi\)
−0.0388953 + 0.999243i \(0.512384\pi\)
\(360\) 0 0
\(361\) 15.4304 + 26.7263i 0.0427436 + 0.0740340i
\(362\) 0 0
\(363\) −42.6657 + 31.3031i −0.117536 + 0.0862343i
\(364\) 0 0
\(365\) −238.134 283.797i −0.652423 0.777527i
\(366\) 0 0
\(367\) 149.979 54.5880i 0.408663 0.148741i −0.129505 0.991579i \(-0.541339\pi\)
0.538168 + 0.842838i \(0.319117\pi\)
\(368\) 0 0
\(369\) −196.966 308.775i −0.533784 0.836787i
\(370\) 0 0
\(371\) 113.625 + 20.0351i 0.306266 + 0.0540029i
\(372\) 0 0
\(373\) 307.437 + 111.898i 0.824227 + 0.299994i 0.719487 0.694506i \(-0.244377\pi\)
0.104740 + 0.994500i \(0.466599\pi\)
\(374\) 0 0
\(375\) 68.4541 33.7828i 0.182544 0.0900875i
\(376\) 0 0
\(377\) 881.293i 2.33765i
\(378\) 0 0
\(379\) −244.316 −0.644634 −0.322317 0.946632i \(-0.604462\pi\)
−0.322317 + 0.946632i \(0.604462\pi\)
\(380\) 0 0
\(381\) 119.638 179.164i 0.314010 0.470248i
\(382\) 0 0
\(383\) −132.824 + 364.931i −0.346799 + 0.952822i 0.636573 + 0.771217i \(0.280351\pi\)
−0.983372 + 0.181605i \(0.941871\pi\)
\(384\) 0 0
\(385\) −140.121 + 794.664i −0.363950 + 2.06406i
\(386\) 0 0
\(387\) 333.787 74.2047i 0.862500 0.191743i
\(388\) 0 0
\(389\) −183.766 504.892i −0.472406 1.29792i −0.915813 0.401604i \(-0.868453\pi\)
0.443408 0.896320i \(-0.353769\pi\)
\(390\) 0 0
\(391\) −280.156 + 235.079i −0.716511 + 0.601224i
\(392\) 0 0
\(393\) −52.3536 + 5.74922i −0.133215 + 0.0146291i
\(394\) 0 0
\(395\) 648.942 374.667i 1.64289 0.948524i
\(396\) 0 0
\(397\) −8.55286 + 14.8140i −0.0215437 + 0.0373148i −0.876596 0.481227i \(-0.840191\pi\)
0.855053 + 0.518541i \(0.173525\pi\)
\(398\) 0 0
\(399\) 369.100 + 353.128i 0.925062 + 0.885033i
\(400\) 0 0
\(401\) 534.855 94.3095i 1.33380 0.235186i 0.539130 0.842222i \(-0.318753\pi\)
0.794674 + 0.607037i \(0.207642\pi\)
\(402\) 0 0
\(403\) −449.339 377.040i −1.11499 0.935584i
\(404\) 0 0
\(405\) −339.188 + 485.624i −0.837501 + 1.19907i
\(406\) 0 0
\(407\) 22.0832 26.3177i 0.0542585 0.0646627i
\(408\) 0 0
\(409\) 14.3282 + 81.2594i 0.0350323 + 0.198678i 0.997301 0.0734234i \(-0.0233924\pi\)
−0.962269 + 0.272102i \(0.912281\pi\)
\(410\) 0 0
\(411\) −43.0916 + 45.0406i −0.104846 + 0.109588i
\(412\) 0 0
\(413\) 104.163 + 60.1384i 0.252210 + 0.145613i
\(414\) 0 0
\(415\) −69.1134 119.708i −0.166538 0.288453i
\(416\) 0 0
\(417\) −2.03737 18.5527i −0.00488578 0.0444909i
\(418\) 0 0
\(419\) 478.636 + 570.416i 1.14233 + 1.36137i 0.922574 + 0.385819i \(0.126081\pi\)
0.219755 + 0.975555i \(0.429474\pi\)
\(420\) 0 0
\(421\) 670.000 243.860i 1.59145 0.579240i 0.613796 0.789465i \(-0.289642\pi\)
0.977653 + 0.210225i \(0.0674196\pi\)
\(422\) 0 0
\(423\) −161.456 50.8026i −0.381693 0.120101i
\(424\) 0 0
\(425\) 438.098 + 77.2484i 1.03082 + 0.181761i
\(426\) 0 0
\(427\) −794.696 289.246i −1.86112 0.677391i
\(428\) 0 0
\(429\) −668.593 446.456i −1.55849 1.04069i
\(430\) 0 0
\(431\) 60.8696i 0.141229i −0.997504 0.0706144i \(-0.977504\pi\)
0.997504 0.0706144i \(-0.0224960\pi\)
\(432\) 0 0
\(433\) 542.738 1.25344 0.626718 0.779246i \(-0.284398\pi\)
0.626718 + 0.779246i \(0.284398\pi\)
\(434\) 0 0
\(435\) −375.953 761.792i −0.864259 1.75125i
\(436\) 0 0
\(437\) 145.499 399.754i 0.332949 0.914769i
\(438\) 0 0
\(439\) −6.32635 + 35.8785i −0.0144108 + 0.0817278i −0.991165 0.132635i \(-0.957656\pi\)
0.976754 + 0.214363i \(0.0687674\pi\)
\(440\) 0 0
\(441\) −309.995 161.142i −0.702937 0.365401i
\(442\) 0 0
\(443\) −102.451 281.481i −0.231266 0.635398i 0.768725 0.639579i \(-0.220891\pi\)
−0.999991 + 0.00418102i \(0.998669\pi\)
\(444\) 0 0
\(445\) 342.134 287.084i 0.768840 0.645133i
\(446\) 0 0
\(447\) 399.802 + 544.926i 0.894412 + 1.21907i
\(448\) 0 0
\(449\) 569.528 328.817i 1.26844 0.732332i 0.293745 0.955884i \(-0.405098\pi\)
0.974692 + 0.223551i \(0.0717651\pi\)
\(450\) 0 0
\(451\) −239.577 + 414.960i −0.531213 + 0.920088i
\(452\) 0 0
\(453\) −679.738 + 198.342i −1.50053 + 0.437842i
\(454\) 0 0
\(455\) −1536.06 + 270.848i −3.37595 + 0.595271i
\(456\) 0 0
\(457\) −191.262 160.488i −0.418517 0.351177i 0.409082 0.912498i \(-0.365849\pi\)
−0.827598 + 0.561321i \(0.810293\pi\)
\(458\) 0 0
\(459\) −399.244 + 135.918i −0.869813 + 0.296117i
\(460\) 0 0
\(461\) −100.503 + 119.775i −0.218012 + 0.259816i −0.863955 0.503569i \(-0.832020\pi\)
0.645943 + 0.763385i \(0.276464\pi\)
\(462\) 0 0
\(463\) −112.206 636.349i −0.242345 1.37440i −0.826580 0.562820i \(-0.809716\pi\)
0.584235 0.811585i \(-0.301395\pi\)
\(464\) 0 0
\(465\) 549.253 + 134.230i 1.18119 + 0.288667i
\(466\) 0 0
\(467\) −142.134 82.0612i −0.304356 0.175720i 0.340042 0.940410i \(-0.389559\pi\)
−0.644398 + 0.764690i \(0.722892\pi\)
\(468\) 0 0
\(469\) −40.9059 70.8512i −0.0872195 0.151069i
\(470\) 0 0
\(471\) −99.8275 43.8902i −0.211948 0.0931851i
\(472\) 0 0
\(473\) −287.550 342.688i −0.607927 0.724500i
\(474\) 0 0
\(475\) −486.258 + 176.983i −1.02370 + 0.372597i
\(476\) 0 0
\(477\) 14.3987 109.868i 0.0301859 0.230331i
\(478\) 0 0
\(479\) −192.731 33.9838i −0.402362 0.0709473i −0.0311949 0.999513i \(-0.509931\pi\)
−0.371167 + 0.928566i \(0.621042\pi\)
\(480\) 0 0
\(481\) 62.4027 + 22.7127i 0.129735 + 0.0472198i
\(482\) 0 0
\(483\) −42.8571 + 656.830i −0.0887311 + 1.35990i
\(484\) 0 0
\(485\) 1086.29i 2.23977i
\(486\) 0 0
\(487\) 459.602 0.943741 0.471870 0.881668i \(-0.343579\pi\)
0.471870 + 0.881668i \(0.343579\pi\)
\(488\) 0 0
\(489\) −206.197 13.4540i −0.421671 0.0275134i
\(490\) 0 0
\(491\) 157.868 433.740i 0.321524 0.883381i −0.668654 0.743573i \(-0.733129\pi\)
0.990179 0.139807i \(-0.0446483\pi\)
\(492\) 0 0
\(493\) 105.029 595.652i 0.213042 1.20822i
\(494\) 0 0
\(495\) 768.389 + 100.701i 1.55230 + 0.203436i
\(496\) 0 0
\(497\) 301.818 + 829.238i 0.607279 + 1.66849i
\(498\) 0 0
\(499\) −55.9965 + 46.9867i −0.112218 + 0.0941617i −0.697170 0.716906i \(-0.745558\pi\)
0.584952 + 0.811068i \(0.301113\pi\)
\(500\) 0 0
\(501\) 136.476 310.412i 0.272407 0.619585i
\(502\) 0 0
\(503\) −667.951 + 385.641i −1.32793 + 0.766683i −0.984980 0.172670i \(-0.944761\pi\)
−0.342954 + 0.939352i \(0.611427\pi\)
\(504\) 0 0
\(505\) −11.9725 + 20.7370i −0.0237080 + 0.0410634i
\(506\) 0 0
\(507\) 248.561 1017.08i 0.490258 2.00607i
\(508\) 0 0
\(509\) −141.711 + 24.9874i −0.278410 + 0.0490912i −0.311110 0.950374i \(-0.600701\pi\)
0.0326996 + 0.999465i \(0.489590\pi\)
\(510\) 0 0
\(511\) −363.672 305.157i −0.711687 0.597177i
\(512\) 0 0
\(513\) 323.138 369.125i 0.629900 0.719541i
\(514\) 0 0
\(515\) −91.4768 + 109.018i −0.177625 + 0.211685i
\(516\) 0 0
\(517\) 38.4525 + 218.075i 0.0743762 + 0.421809i
\(518\) 0 0
\(519\) 115.453 + 395.669i 0.222453 + 0.762369i
\(520\) 0 0
\(521\) 19.8121 + 11.4385i 0.0380271 + 0.0219550i 0.518893 0.854839i \(-0.326344\pi\)
−0.480866 + 0.876794i \(0.659678\pi\)
\(522\) 0 0
\(523\) 504.779 + 874.303i 0.965160 + 1.67171i 0.709183 + 0.705024i \(0.249064\pi\)
0.255977 + 0.966683i \(0.417603\pi\)
\(524\) 0 0
\(525\) 645.550 473.629i 1.22962 0.902150i
\(526\) 0 0
\(527\) 258.767 + 308.386i 0.491018 + 0.585173i
\(528\) 0 0
\(529\) 18.0163 6.55741i 0.0340574 0.0123959i
\(530\) 0 0
\(531\) 53.2775 102.492i 0.100334 0.193017i
\(532\) 0 0
\(533\) −912.117 160.831i −1.71129 0.301746i
\(534\) 0 0
\(535\) −169.930 61.8494i −0.317626 0.115606i
\(536\) 0 0
\(537\) 518.018 255.647i 0.964652 0.476066i
\(538\) 0 0
\(539\) 457.081i 0.848017i
\(540\) 0 0
\(541\) −593.097 −1.09630 −0.548149 0.836381i \(-0.684667\pi\)
−0.548149 + 0.836381i \(0.684667\pi\)
\(542\) 0 0
\(543\) −126.790 + 189.876i −0.233500 + 0.349679i
\(544\) 0 0
\(545\) −2.54624 + 6.99573i −0.00467199 + 0.0128362i
\(546\) 0 0
\(547\) −31.2102 + 177.002i −0.0570570 + 0.323586i −0.999955 0.00949559i \(-0.996977\pi\)
0.942898 + 0.333082i \(0.108089\pi\)
\(548\) 0 0
\(549\) −243.778 + 774.751i −0.444039 + 1.41120i
\(550\) 0 0
\(551\) 240.632 + 661.132i 0.436719 + 1.19988i
\(552\) 0 0
\(553\) 735.583 617.227i 1.33017 1.11614i
\(554\) 0 0
\(555\) −63.6302 + 6.98756i −0.114649 + 0.0125902i
\(556\) 0 0
\(557\) −312.376 + 180.351i −0.560819 + 0.323789i −0.753474 0.657477i \(-0.771624\pi\)
0.192655 + 0.981267i \(0.438290\pi\)
\(558\) 0 0
\(559\) 432.353 748.858i 0.773441 1.33964i
\(560\) 0 0
\(561\) 398.685 + 381.433i 0.710668 + 0.679916i
\(562\) 0 0
\(563\) 571.890 100.840i 1.01579 0.179111i 0.359122 0.933291i \(-0.383076\pi\)
0.656669 + 0.754179i \(0.271965\pi\)
\(564\) 0 0
\(565\) 869.589 + 729.672i 1.53910 + 1.29145i
\(566\) 0 0
\(567\) −319.987 + 688.326i −0.564351 + 1.21398i
\(568\) 0 0
\(569\) −356.121 + 424.408i −0.625871 + 0.745884i −0.982068 0.188527i \(-0.939629\pi\)
0.356197 + 0.934411i \(0.384073\pi\)
\(570\) 0 0
\(571\) −41.9016 237.636i −0.0733828 0.416175i −0.999264 0.0383598i \(-0.987787\pi\)
0.925881 0.377815i \(-0.123324\pi\)
\(572\) 0 0
\(573\) −426.438 + 445.725i −0.744219 + 0.777879i
\(574\) 0 0
\(575\) −577.460 333.396i −1.00428 0.579820i
\(576\) 0 0
\(577\) 9.74580 + 16.8802i 0.0168905 + 0.0292552i 0.874347 0.485301i \(-0.161290\pi\)
−0.857457 + 0.514556i \(0.827957\pi\)
\(578\) 0 0
\(579\) 95.4359 + 869.060i 0.164829 + 1.50097i
\(580\) 0 0
\(581\) −113.858 135.690i −0.195968 0.233546i
\(582\) 0 0
\(583\) −136.224 + 49.5815i −0.233661 + 0.0850455i
\(584\) 0 0
\(585\) 325.079 + 1462.27i 0.555690 + 2.49961i
\(586\) 0 0
\(587\) −489.063 86.2350i −0.833156 0.146908i −0.259231 0.965815i \(-0.583469\pi\)
−0.573926 + 0.818907i \(0.694580\pi\)
\(588\) 0 0
\(589\) −440.036 160.160i −0.747090 0.271918i
\(590\) 0 0
\(591\) −758.091 506.218i −1.28273 0.856545i
\(592\) 0 0
\(593\) 629.836i 1.06212i −0.847335 0.531059i \(-0.821794\pi\)
0.847335 0.531059i \(-0.178206\pi\)
\(594\) 0 0
\(595\) 1070.47 1.79912
\(596\) 0 0
\(597\) 353.777 + 716.857i 0.592591 + 1.20077i
\(598\) 0 0
\(599\) 222.199 610.488i 0.370951 1.01918i −0.604044 0.796951i \(-0.706445\pi\)
0.974995 0.222228i \(-0.0713329\pi\)
\(600\) 0 0
\(601\) 69.7993 395.852i 0.116139 0.658655i −0.870042 0.492978i \(-0.835908\pi\)
0.986180 0.165677i \(-0.0529808\pi\)
\(602\) 0 0
\(603\) −66.2414 + 42.2552i −0.109853 + 0.0700749i
\(604\) 0 0
\(605\) −44.1186 121.215i −0.0729233 0.200355i
\(606\) 0 0
\(607\) −166.213 + 139.470i −0.273828 + 0.229769i −0.769352 0.638826i \(-0.779421\pi\)
0.495524 + 0.868594i \(0.334976\pi\)
\(608\) 0 0
\(609\) −643.961 877.711i −1.05741 1.44123i
\(610\) 0 0
\(611\) −370.688 + 214.017i −0.606691 + 0.350273i
\(612\) 0 0
\(613\) 469.156 812.602i 0.765344 1.32562i −0.174720 0.984618i \(-0.555902\pi\)
0.940064 0.340997i \(-0.110765\pi\)
\(614\) 0 0
\(615\) 857.046 250.079i 1.39357 0.406633i
\(616\) 0 0
\(617\) 474.755 83.7120i 0.769456 0.135676i 0.224879 0.974387i \(-0.427801\pi\)
0.544577 + 0.838711i \(0.316690\pi\)
\(618\) 0 0
\(619\) 71.9917 + 60.4082i 0.116303 + 0.0975900i 0.699084 0.715039i \(-0.253591\pi\)
−0.582781 + 0.812629i \(0.698036\pi\)
\(620\) 0 0
\(621\) 632.015 + 13.2333i 1.01774 + 0.0213097i
\(622\) 0 0
\(623\) 367.885 438.428i 0.590505 0.703736i
\(624\) 0 0
\(625\) −91.3226 517.916i −0.146116 0.828666i
\(626\) 0 0
\(627\) −623.471 152.368i −0.994371 0.243011i
\(628\) 0 0
\(629\) −39.4702 22.7881i −0.0627507 0.0362291i
\(630\) 0 0
\(631\) −266.288 461.224i −0.422010 0.730942i 0.574126 0.818767i \(-0.305342\pi\)
−0.996136 + 0.0878246i \(0.972008\pi\)
\(632\) 0 0
\(633\) −244.148 107.342i −0.385699 0.169576i
\(634\) 0 0
\(635\) 337.567 + 402.297i 0.531601 + 0.633538i
\(636\) 0 0
\(637\) −830.238 + 302.182i −1.30336 + 0.474383i
\(638\) 0 0
\(639\) 782.798 324.785i 1.22504 0.508270i
\(640\) 0 0
\(641\) 811.167 + 143.031i 1.26547 + 0.223137i 0.765800 0.643079i \(-0.222343\pi\)
0.499671 + 0.866215i \(0.333454\pi\)
\(642\) 0 0
\(643\) 875.130 + 318.521i 1.36101 + 0.495367i 0.916367 0.400340i \(-0.131108\pi\)
0.444644 + 0.895707i \(0.353330\pi\)
\(644\) 0 0
\(645\) −54.2706 + 831.754i −0.0841405 + 1.28954i
\(646\) 0 0
\(647\) 255.455i 0.394830i 0.980320 + 0.197415i \(0.0632546\pi\)
−0.980320 + 0.197415i \(0.936745\pi\)
\(648\) 0 0
\(649\) −151.122 −0.232854
\(650\) 0 0
\(651\) 723.016 + 47.1757i 1.11062 + 0.0724665i
\(652\) 0 0
\(653\) −277.404 + 762.162i −0.424815 + 1.16717i 0.524105 + 0.851653i \(0.324400\pi\)
−0.948920 + 0.315516i \(0.897822\pi\)
\(654\) 0 0
\(655\) 22.2942 126.437i 0.0340370 0.193033i
\(656\) 0 0
\(657\) −277.343 + 361.881i −0.422136 + 0.550808i
\(658\) 0 0
\(659\) 118.588 + 325.819i 0.179952 + 0.494414i 0.996569 0.0827659i \(-0.0263754\pi\)
−0.816617 + 0.577180i \(0.804153\pi\)
\(660\) 0 0
\(661\) −639.014 + 536.197i −0.966739 + 0.811190i −0.982036 0.188693i \(-0.939575\pi\)
0.0152975 + 0.999883i \(0.495130\pi\)
\(662\) 0 0
\(663\) −429.256 + 976.337i −0.647445 + 1.47261i
\(664\) 0 0
\(665\) −1078.37 + 622.598i −1.62161 + 0.936238i
\(666\) 0 0
\(667\) −453.297 + 785.133i −0.679605 + 1.17711i
\(668\) 0 0
\(669\) 206.749 845.988i 0.309041 1.26456i
\(670\) 0 0
\(671\) 1046.44 184.515i 1.55952 0.274986i
\(672\) 0 0
\(673\) 118.178 + 99.1631i 0.175599 + 0.147345i 0.726351 0.687324i \(-0.241215\pi\)
−0.550752 + 0.834669i \(0.685659\pi\)
\(674\) 0 0
\(675\) −481.830 599.265i −0.713822 0.887800i
\(676\) 0 0
\(677\) −576.975 + 687.612i −0.852252 + 1.01567i 0.147394 + 0.989078i \(0.452912\pi\)
−0.999646 + 0.0265971i \(0.991533\pi\)
\(678\) 0 0
\(679\) −241.723 1370.88i −0.355998 2.01897i
\(680\) 0 0
\(681\) −155.225 531.971i −0.227937 0.781161i
\(682\) 0 0
\(683\) −666.874 385.020i −0.976390 0.563719i −0.0752113 0.997168i \(-0.523963\pi\)
−0.901178 + 0.433449i \(0.857296\pi\)
\(684\) 0 0
\(685\) −75.9746 131.592i −0.110912 0.192105i
\(686\) 0 0
\(687\) −824.428 + 604.868i −1.20004 + 0.880448i
\(688\) 0 0
\(689\) −180.119 214.657i −0.261421 0.311549i
\(690\) 0 0
\(691\) −831.656 + 302.698i −1.20355 + 0.438058i −0.864463 0.502696i \(-0.832342\pi\)
−0.339091 + 0.940754i \(0.610119\pi\)
\(692\) 0 0
\(693\) 992.102 43.9002i 1.43160 0.0633480i
\(694\) 0 0
\(695\) 44.8058 + 7.90048i 0.0644688 + 0.0113676i
\(696\) 0 0
\(697\) 597.318 + 217.406i 0.856984 + 0.311917i
\(698\) 0 0
\(699\) 816.479 402.941i 1.16807 0.576454i
\(700\) 0 0
\(701\) 924.832i 1.31930i −0.751571 0.659652i \(-0.770704\pi\)
0.751571 0.659652i \(-0.229296\pi\)
\(702\) 0 0
\(703\) 53.0151 0.0754127
\(704\) 0 0
\(705\) 229.126 343.130i 0.325002 0.486709i
\(706\) 0 0
\(707\) −10.4947 + 28.8339i −0.0148440 + 0.0407835i
\(708\) 0 0
\(709\) −173.003 + 981.150i −0.244010 + 1.38385i 0.578770 + 0.815491i \(0.303533\pi\)
−0.822780 + 0.568360i \(0.807578\pi\)
\(710\) 0 0
\(711\) −623.428 679.550i −0.876832 0.955766i
\(712\) 0 0
\(713\) −206.378 567.020i −0.289451 0.795259i
\(714\) 0 0
\(715\) 1501.26 1259.71i 2.09967 1.76183i
\(716\) 0 0
\(717\) −1305.27 + 143.339i −1.82047 + 0.199915i
\(718\) 0 0
\(719\) −103.540 + 59.7791i −0.144006 + 0.0831420i −0.570272 0.821456i \(-0.693162\pi\)
0.426266 + 0.904598i \(0.359829\pi\)
\(720\) 0 0
\(721\) −91.1833 + 157.934i −0.126468 + 0.219049i
\(722\) 0 0
\(723\) −352.719 337.456i −0.487854 0.466744i
\(724\) 0 0
\(725\) 1086.02 191.495i 1.49796 0.264130i
\(726\) 0 0
\(727\) 391.062 + 328.140i 0.537911 + 0.451361i 0.870823 0.491597i \(-0.163587\pi\)
−0.332912 + 0.942958i \(0.608031\pi\)
\(728\) 0 0
\(729\) 673.999 + 277.789i 0.924553 + 0.381054i
\(730\) 0 0
\(731\) −381.467 + 454.615i −0.521843 + 0.621908i
\(732\) 0 0
\(733\) −61.0496 346.229i −0.0832872 0.472345i −0.997713 0.0675919i \(-0.978468\pi\)
0.914426 0.404754i \(-0.132643\pi\)
\(734\) 0 0
\(735\) 588.752 615.380i 0.801023 0.837252i
\(736\) 0 0
\(737\) 89.0213 + 51.3965i 0.120789 + 0.0697374i
\(738\) 0 0
\(739\) 43.2066 + 74.8360i 0.0584663 + 0.101267i 0.893777 0.448511i \(-0.148046\pi\)
−0.835311 + 0.549778i \(0.814712\pi\)
\(740\) 0 0
\(741\) −135.424 1233.20i −0.182758 1.66424i
\(742\) 0 0
\(743\) −435.424 518.918i −0.586035 0.698409i 0.388804 0.921320i \(-0.372888\pi\)
−0.974839 + 0.222912i \(0.928444\pi\)
\(744\) 0 0
\(745\) −1548.15 + 563.482i −2.07806 + 0.756352i
\(746\) 0 0
\(747\) −125.354 + 115.001i −0.167810 + 0.153951i
\(748\) 0 0
\(749\) −228.211 40.2398i −0.304688 0.0537247i
\(750\) 0 0
\(751\) −378.743 137.851i −0.504318 0.183557i 0.0773173 0.997007i \(-0.475365\pi\)
−0.581635 + 0.813450i \(0.697587\pi\)
\(752\) 0 0
\(753\) 1137.49 + 759.562i 1.51061 + 1.00871i
\(754\) 0 0
\(755\) 1726.07i 2.28618i
\(756\) 0 0
\(757\) 463.787 0.612665 0.306332 0.951925i \(-0.400898\pi\)
0.306332 + 0.951925i \(0.400898\pi\)
\(758\) 0 0
\(759\) −366.005 741.636i −0.482220 0.977122i
\(760\) 0 0
\(761\) −23.5534 + 64.7125i −0.0309506 + 0.0850361i −0.954206 0.299152i \(-0.903296\pi\)
0.923255 + 0.384188i \(0.125519\pi\)
\(762\) 0 0
\(763\) −1.65661 + 9.39508i −0.00217117 + 0.0123133i
\(764\) 0 0
\(765\) −45.4473 1027.07i −0.0594083 1.34257i
\(766\) 0 0
\(767\) −99.9089 274.497i −0.130259 0.357884i
\(768\) 0 0
\(769\) −587.623 + 493.074i −0.764139 + 0.641189i −0.939201 0.343369i \(-0.888432\pi\)
0.175062 + 0.984557i \(0.443988\pi\)
\(770\) 0 0
\(771\) −737.120 1004.69i −0.956057 1.30309i
\(772\) 0 0
\(773\) 215.221 124.258i 0.278423 0.160747i −0.354286 0.935137i \(-0.615276\pi\)
0.632709 + 0.774390i \(0.281943\pi\)
\(774\) 0 0
\(775\) −366.992 + 635.648i −0.473538 + 0.820191i
\(776\) 0 0
\(777\) −78.7454 + 22.9773i −0.101345 + 0.0295718i
\(778\) 0 0
\(779\) −728.169 + 128.396i −0.934749 + 0.164821i
\(780\) 0 0
\(781\) −849.365 712.702i −1.08754 0.912551i
\(782\) 0 0
\(783\) −814.780 + 655.112i −1.04059 + 0.836669i
\(784\) 0 0
\(785\) 170.870 203.635i 0.217668 0.259407i
\(786\) 0 0
\(787\) 226.645 + 1285.37i 0.287986 + 1.63325i 0.694419 + 0.719571i \(0.255662\pi\)
−0.406432 + 0.913681i \(0.633227\pi\)
\(788\) 0 0
\(789\) −1243.07 303.789i −1.57550 0.385031i
\(790\) 0 0
\(791\) 1259.77 + 727.331i 1.59264 + 0.919508i
\(792\) 0 0
\(793\) 1026.97 + 1778.76i 1.29504 + 2.24307i
\(794\) 0 0
\(795\) 247.267 + 108.713i 0.311027 + 0.136746i
\(796\) 0 0
\(797\) 385.545 + 459.474i 0.483745 + 0.576505i 0.951615 0.307292i \(-0.0994230\pi\)
−0.467870 + 0.883797i \(0.654979\pi\)
\(798\) 0 0
\(799\) 276.048 100.473i 0.345492 0.125749i
\(800\) 0 0
\(801\) −436.268 334.353i −0.544654 0.417420i
\(802\) 0 0
\(803\) 587.428 + 103.579i 0.731542 + 0.128991i
\(804\) 0 0
\(805\) −1507.77 548.782i −1.87300 0.681716i
\(806\) 0 0
\(807\) −44.8837 + 687.889i −0.0556179 + 0.852402i
\(808\) 0 0
\(809\) 574.876i 0.710601i 0.934752 + 0.355300i \(0.115621\pi\)
−0.934752 + 0.355300i \(0.884379\pi\)
\(810\) 0 0
\(811\) 942.729 1.16243 0.581214 0.813751i \(-0.302578\pi\)
0.581214 + 0.813751i \(0.302578\pi\)
\(812\) 0 0
\(813\) −740.607 48.3234i −0.910955 0.0594384i
\(814\) 0 0
\(815\) 172.278 473.329i 0.211384 0.580772i
\(816\) 0 0
\(817\) 119.873 679.833i 0.146723 0.832109i
\(818\) 0 0
\(819\) 735.631 + 1773.02i 0.898206 + 2.16486i
\(820\) 0 0
\(821\) 267.373 + 734.600i 0.325667 + 0.894763i 0.989194 + 0.146611i \(0.0468364\pi\)
−0.663527 + 0.748152i \(0.730941\pi\)
\(822\) 0 0
\(823\) −417.236 + 350.103i −0.506970 + 0.425398i −0.860061 0.510190i \(-0.829575\pi\)
0.353091 + 0.935589i \(0.385130\pi\)
\(824\) 0 0
\(825\) −404.891 + 920.919i −0.490777 + 1.11627i
\(826\) 0 0
\(827\) 813.492 469.670i 0.983666 0.567920i 0.0802912 0.996771i \(-0.474415\pi\)
0.903375 + 0.428851i \(0.141082\pi\)
\(828\) 0 0
\(829\) 340.470 589.711i 0.410699 0.711352i −0.584267 0.811562i \(-0.698618\pi\)
0.994966 + 0.100209i \(0.0319512\pi\)
\(830\) 0 0
\(831\) 104.781 428.752i 0.126091 0.515947i
\(832\) 0 0
\(833\) 597.158 105.295i 0.716876 0.126405i
\(834\) 0 0
\(835\) 633.199 + 531.317i 0.758322 + 0.636308i
\(836\) 0 0
\(837\) 14.5668 695.701i 0.0174036 0.831183i
\(838\) 0 0
\(839\) −24.0697 + 28.6852i −0.0286886 + 0.0341897i −0.780198 0.625533i \(-0.784882\pi\)
0.751509 + 0.659722i \(0.229326\pi\)
\(840\) 0 0
\(841\) −114.324 648.364i −0.135938 0.770944i
\(842\) 0 0
\(843\) −38.8406 133.111i −0.0460743 0.157901i
\(844\) 0 0
\(845\) 2210.32 + 1276.13i 2.61576 + 1.51021i
\(846\) 0 0
\(847\) −82.6499 143.154i −0.0975795 0.169013i
\(848\) 0 0
\(849\) 299.437 219.691i 0.352693 0.258765i
\(850\) 0 0
\(851\) 43.9114 + 52.3316i 0.0515998 + 0.0614943i
\(852\) 0 0
\(853\) 593.167 215.895i 0.695389 0.253101i 0.0299483 0.999551i \(-0.490466\pi\)
0.665441 + 0.746450i \(0.268244\pi\)
\(854\) 0 0
\(855\) 643.134 + 1008.21i 0.752204 + 1.17919i
\(856\) 0 0
\(857\) 1234.24 + 217.630i 1.44019 + 0.253944i 0.838550 0.544825i \(-0.183404\pi\)
0.601638 + 0.798769i \(0.294515\pi\)
\(858\) 0 0
\(859\) −753.601 274.288i −0.877300 0.319311i −0.136180 0.990684i \(-0.543483\pi\)
−0.741120 + 0.671373i \(0.765705\pi\)
\(860\) 0 0
\(861\) 1025.93 506.307i 1.19155 0.588045i
\(862\) 0 0
\(863\) 1108.04i 1.28394i −0.766731 0.641969i \(-0.778118\pi\)
0.766731 0.641969i \(-0.221882\pi\)
\(864\) 0 0
\(865\) −1004.73 −1.16153
\(866\) 0 0
\(867\) −74.9836 + 112.292i −0.0864862 + 0.129518i
\(868\) 0 0
\(869\) −412.645 + 1133.73i −0.474850 + 1.30464i
\(870\) 0 0
\(871\) −34.5030 + 195.676i −0.0396131 + 0.224657i
\(872\) 0 0
\(873\) −1305.03 + 290.122i −1.49488 + 0.332328i
\(874\) 0 0
\(875\) 81.5563 + 224.074i 0.0932072 + 0.256085i
\(876\) 0 0
\(877\) −521.708 + 437.765i −0.594878 + 0.499162i −0.889795 0.456361i \(-0.849152\pi\)
0.294917 + 0.955523i \(0.404708\pi\)
\(878\) 0 0
\(879\) 407.787 44.7812i 0.463922 0.0509457i
\(880\) 0 0
\(881\) −1449.13 + 836.658i −1.64487 + 0.949669i −0.665810 + 0.746122i \(0.731914\pi\)
−0.979065 + 0.203547i \(0.934753\pi\)
\(882\) 0 0
\(883\) −155.212 + 268.834i −0.175778 + 0.304456i −0.940430 0.339987i \(-0.889577\pi\)
0.764653 + 0.644443i \(0.222911\pi\)
\(884\) 0 0
\(885\) 203.460 + 194.656i 0.229898 + 0.219950i
\(886\) 0 0
\(887\) −160.295 + 28.2644i −0.180716 + 0.0318652i −0.263274 0.964721i \(-0.584802\pi\)
0.0825576 + 0.996586i \(0.473691\pi\)
\(888\) 0 0
\(889\) 515.523 + 432.575i 0.579891 + 0.486587i
\(890\) 0 0
\(891\) −84.2400 950.008i −0.0945454 1.06623i
\(892\) 0 0
\(893\) −219.648 + 261.767i −0.245967 + 0.293132i
\(894\) 0 0
\(895\) 244.523 + 1386.76i 0.273210 + 1.54945i
\(896\) 0 0
\(897\) 1105.13 1155.11i 1.23203 1.28775i
\(898\) 0 0
\(899\) 864.248 + 498.974i 0.961343 + 0.555032i
\(900\) 0 0
\(901\) 96.1574 + 166.549i 0.106723 + 0.184850i
\(902\) 0 0
\(903\) 116.595 + 1061.74i 0.129119 + 1.17579i
\(904\) 0 0
\(905\) −357.749 426.348i −0.395302 0.471103i
\(906\) 0 0
\(907\) 797.885 290.406i 0.879697 0.320183i 0.137609 0.990487i \(-0.456058\pi\)
0.742087 + 0.670303i \(0.233836\pi\)
\(908\) 0 0
\(909\) 28.1103 + 8.84497i 0.0309244 + 0.00973044i
\(910\) 0 0
\(911\) −2.80554 0.494692i −0.00307962 0.000543021i 0.172108 0.985078i \(-0.444942\pi\)
−0.175188 + 0.984535i \(0.556053\pi\)
\(912\) 0 0
\(913\) 209.135 + 76.1191i 0.229064 + 0.0833725i
\(914\) 0 0
\(915\) −1646.52 1099.47i −1.79947 1.20160i
\(916\) 0 0
\(917\) 164.522i 0.179413i
\(918\) 0 0
\(919\) −188.322 −0.204920 −0.102460 0.994737i \(-0.532671\pi\)
−0.102460 + 0.994737i \(0.532671\pi\)
\(920\) 0 0
\(921\) −394.361 799.092i −0.428187 0.867636i
\(922\) 0 0
\(923\) 733.020 2013.96i 0.794171 2.18197i
\(924\) 0 0
\(925\) 14.4296 81.8343i 0.0155996 0.0884695i
\(926\) 0 0
\(927\) 155.401 + 80.7806i 0.167639 + 0.0871420i
\(928\) 0 0
\(929\) 270.413 + 742.955i 0.291080 + 0.799736i 0.995909 + 0.0903589i \(0.0288014\pi\)
−0.704829 + 0.709377i \(0.748976\pi\)
\(930\) 0 0
\(931\) −540.322 + 453.384i −0.580368 + 0.486986i
\(932\) 0 0
\(933\) 654.956 + 892.698i 0.701990 + 0.956803i
\(934\) 0 0
\(935\) −1164.81 + 672.502i −1.24578 + 0.719254i
\(936\) 0 0
\(937\) −630.512 + 1092.08i −0.672905 + 1.16551i 0.304172 + 0.952617i \(0.401620\pi\)
−0.977077 + 0.212888i \(0.931713\pi\)
\(938\) 0 0
\(939\) 988.972 288.574i 1.05322 0.307321i
\(940\) 0 0
\(941\) 534.133 94.1821i 0.567623 0.100087i 0.117531 0.993069i \(-0.462502\pi\)
0.450092 + 0.892982i \(0.351391\pi\)
\(942\) 0 0
\(943\) −729.869 612.433i −0.773987 0.649452i
\(944\) 0 0
\(945\) −1392.24 1218.79i −1.47327 1.28973i
\(946\) 0 0
\(947\) −104.599 + 124.656i −0.110453 + 0.131632i −0.818438 0.574595i \(-0.805160\pi\)
0.707985 + 0.706227i \(0.249604\pi\)
\(948\) 0 0
\(949\) 200.215 + 1135.48i 0.210975 + 1.19650i
\(950\) 0 0
\(951\) 1590.28 + 388.644i 1.67222 + 0.408669i
\(952\) 0 0
\(953\) −1529.94 883.311i −1.60539 0.926874i −0.990383 0.138354i \(-0.955819\pi\)
−0.615010 0.788519i \(-0.710848\pi\)
\(954\) 0 0
\(955\) −751.849 1302.24i −0.787277 1.36360i
\(956\) 0 0
\(957\) 1252.11 + 550.503i 1.30837 + 0.575238i
\(958\) 0 0
\(959\) −125.161 149.161i −0.130512 0.155538i
\(960\) 0 0
\(961\) 278.888 101.507i 0.290206 0.105626i
\(962\) 0 0
\(963\) −28.9193 + 220.666i −0.0300304 + 0.229144i
\(964\) 0 0
\(965\) −2098.83 370.080i −2.17495 0.383502i
\(966\) 0 0
\(967\) 1526.43 + 555.575i 1.57852 + 0.574535i 0.974882 0.222722i \(-0.0714943\pi\)
0.603640 + 0.797257i \(0.293717\pi\)
\(968\) 0 0
\(969\) −55.4376 + 849.639i −0.0572111 + 0.876820i
\(970\) 0 0
\(971\) 1343.18i 1.38330i −0.722235 0.691648i \(-0.756885\pi\)
0.722235 0.691648i \(-0.243115\pi\)
\(972\) 0 0
\(973\) 58.3022 0.0599200
\(974\) 0 0
\(975\) −1940.43 126.610i −1.99018 0.129856i
\(976\) 0 0
\(977\) −405.483 + 1114.06i −0.415029 + 1.14028i 0.539453 + 0.842016i \(0.318631\pi\)
−0.954482 + 0.298268i \(0.903591\pi\)
\(978\) 0 0
\(979\) −124.871 + 708.178i −0.127550 + 0.723369i
\(980\) 0 0
\(981\) 9.08444 + 1.19056i 0.00926038 + 0.00121362i
\(982\) 0 0
\(983\) 138.272 + 379.900i 0.140664 + 0.386470i 0.989942 0.141475i \(-0.0451846\pi\)
−0.849278 + 0.527946i \(0.822962\pi\)
\(984\) 0 0
\(985\) 1702.22 1428.33i 1.72814 1.45008i
\(986\) 0 0
\(987\) 212.799 484.009i 0.215602 0.490384i
\(988\) 0 0
\(989\) 770.357 444.766i 0.778925 0.449712i
\(990\) 0 0
\(991\) 500.942 867.656i 0.505491 0.875536i −0.494489 0.869184i \(-0.664645\pi\)
0.999980 0.00635207i \(-0.00202194\pi\)
\(992\) 0 0
\(993\) −129.148 + 528.458i −0.130059 + 0.532183i
\(994\) 0 0
\(995\) −1919.06 + 338.382i −1.92870 + 0.340083i
\(996\) 0 0
\(997\) −1154.65 968.864i −1.15812 0.971779i −0.158243 0.987400i \(-0.550583\pi\)
−0.999878 + 0.0156209i \(0.995027\pi\)
\(998\) 0 0
\(999\) 25.3887 + 74.5767i 0.0254141 + 0.0746513i
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 108.3.k.a.65.6 yes 36
3.2 odd 2 324.3.k.a.197.5 36
4.3 odd 2 432.3.bc.b.65.1 36
27.5 odd 18 inner 108.3.k.a.5.6 36
27.7 even 9 2916.3.c.b.1457.5 36
27.20 odd 18 2916.3.c.b.1457.32 36
27.22 even 9 324.3.k.a.125.5 36
108.59 even 18 432.3.bc.b.113.1 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.5.6 36 27.5 odd 18 inner
108.3.k.a.65.6 yes 36 1.1 even 1 trivial
324.3.k.a.125.5 36 27.22 even 9
324.3.k.a.197.5 36 3.2 odd 2
432.3.bc.b.65.1 36 4.3 odd 2
432.3.bc.b.113.1 36 108.59 even 18
2916.3.c.b.1457.5 36 27.7 even 9
2916.3.c.b.1457.32 36 27.20 odd 18