Properties

Label 900.3.c.n.451.3
Level $900$
Weight $3$
Character 900.451
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
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] = [900,3,Mod(451,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.451");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4069419264.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 50x^{4} - 84x^{3} + 55x^{2} - 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 451.3
Root \(-2.65095 + 1.53053i\) of defining polynomial
Character \(\chi\) \(=\) 900.451
Dual form 900.3.c.n.451.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.534079 - 1.92737i) q^{2} +(-3.42952 + 2.05874i) q^{4} -11.9716i q^{7} +(5.79958 + 5.51043i) q^{8} +O(q^{10})\) \(q+(-0.534079 - 1.92737i) q^{2} +(-3.42952 + 2.05874i) q^{4} -11.9716i q^{7} +(5.79958 + 5.51043i) q^{8} +14.5382i q^{11} -22.4802 q^{13} +(-23.0738 + 6.39379i) q^{14} +(7.52322 - 14.1209i) q^{16} +12.6890 q^{17} +8.76336i q^{19} +(28.0205 - 7.76455i) q^{22} -4.99653i q^{23} +(12.0062 + 43.3278i) q^{26} +(24.6464 + 41.0570i) q^{28} -2.74712 q^{29} +16.3466i q^{31} +(-31.2343 - 6.95833i) q^{32} +(-6.77695 - 24.4565i) q^{34} -32.4872 q^{37} +(16.8902 - 4.68032i) q^{38} -42.7586 q^{41} +16.5435i q^{43} +(-29.9303 - 49.8591i) q^{44} +(-9.63018 + 2.66854i) q^{46} +48.5912i q^{47} -94.3200 q^{49} +(77.0964 - 46.2809i) q^{52} +94.1066 q^{53} +(65.9689 - 69.4305i) q^{56} +(1.46718 + 5.29471i) q^{58} +43.2650i q^{59} +56.7678 q^{61} +(31.5060 - 8.73038i) q^{62} +(3.27028 + 63.9164i) q^{64} +61.1106i q^{67} +(-43.5173 + 26.1234i) q^{68} +39.6643i q^{71} +99.5452 q^{73} +(17.3507 + 62.6149i) q^{74} +(-18.0414 - 30.0541i) q^{76} +174.046 q^{77} +10.7780i q^{79} +(22.8365 + 82.4118i) q^{82} +140.263i q^{83} +(31.8855 - 8.83554i) q^{86} +(-80.1118 + 84.3156i) q^{88} -54.8723 q^{89} +269.125i q^{91} +(10.2865 + 17.1357i) q^{92} +(93.6533 - 25.9515i) q^{94} -14.1601 q^{97} +(50.3743 + 181.790i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 8 q^{4} - 20 q^{8} + 8 q^{13} - 22 q^{14} + 40 q^{16} + 4 q^{22} + 66 q^{26} + 104 q^{28} + 32 q^{29} - 112 q^{32} + 124 q^{34} - 176 q^{37} + 170 q^{38} + 16 q^{41} - 40 q^{44} - 76 q^{46} + 16 q^{49} + 56 q^{52} + 304 q^{53} + 172 q^{56} - 12 q^{58} + 136 q^{61} + 238 q^{62} + 16 q^{64} - 88 q^{68} + 240 q^{73} + 108 q^{74} + 120 q^{76} + 384 q^{77} + 320 q^{82} - 214 q^{86} - 200 q^{88} - 128 q^{89} - 312 q^{92} + 12 q^{94} + 216 q^{97} - 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.534079 1.92737i −0.267039 0.963686i
\(3\) 0 0
\(4\) −3.42952 + 2.05874i −0.857380 + 0.514684i
\(5\) 0 0
\(6\) 0 0
\(7\) 11.9716i 1.71023i −0.518436 0.855117i \(-0.673485\pi\)
0.518436 0.855117i \(-0.326515\pi\)
\(8\) 5.79958 + 5.51043i 0.724948 + 0.688804i
\(9\) 0 0
\(10\) 0 0
\(11\) 14.5382i 1.32166i 0.750537 + 0.660828i \(0.229795\pi\)
−0.750537 + 0.660828i \(0.770205\pi\)
\(12\) 0 0
\(13\) −22.4802 −1.72925 −0.864625 0.502418i \(-0.832444\pi\)
−0.864625 + 0.502418i \(0.832444\pi\)
\(14\) −23.0738 + 6.39379i −1.64813 + 0.456699i
\(15\) 0 0
\(16\) 7.52322 14.1209i 0.470201 0.882559i
\(17\) 12.6890 0.746414 0.373207 0.927748i \(-0.378258\pi\)
0.373207 + 0.927748i \(0.378258\pi\)
\(18\) 0 0
\(19\) 8.76336i 0.461229i 0.973045 + 0.230615i \(0.0740737\pi\)
−0.973045 + 0.230615i \(0.925926\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 28.0205 7.76455i 1.27366 0.352934i
\(23\) 4.99653i 0.217241i −0.994083 0.108620i \(-0.965357\pi\)
0.994083 0.108620i \(-0.0346433\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 12.0062 + 43.3278i 0.461778 + 1.66645i
\(27\) 0 0
\(28\) 24.6464 + 41.0570i 0.880229 + 1.46632i
\(29\) −2.74712 −0.0947282 −0.0473641 0.998878i \(-0.515082\pi\)
−0.0473641 + 0.998878i \(0.515082\pi\)
\(30\) 0 0
\(31\) 16.3466i 0.527310i 0.964617 + 0.263655i \(0.0849281\pi\)
−0.964617 + 0.263655i \(0.915072\pi\)
\(32\) −31.2343 6.95833i −0.976072 0.217448i
\(33\) 0 0
\(34\) −6.77695 24.4565i −0.199322 0.719309i
\(35\) 0 0
\(36\) 0 0
\(37\) −32.4872 −0.878032 −0.439016 0.898479i \(-0.644673\pi\)
−0.439016 + 0.898479i \(0.644673\pi\)
\(38\) 16.8902 4.68032i 0.444480 0.123166i
\(39\) 0 0
\(40\) 0 0
\(41\) −42.7586 −1.04289 −0.521447 0.853284i \(-0.674607\pi\)
−0.521447 + 0.853284i \(0.674607\pi\)
\(42\) 0 0
\(43\) 16.5435i 0.384733i 0.981323 + 0.192367i \(0.0616163\pi\)
−0.981323 + 0.192367i \(0.938384\pi\)
\(44\) −29.9303 49.8591i −0.680235 1.13316i
\(45\) 0 0
\(46\) −9.63018 + 2.66854i −0.209352 + 0.0580118i
\(47\) 48.5912i 1.03386i 0.856029 + 0.516928i \(0.172924\pi\)
−0.856029 + 0.516928i \(0.827076\pi\)
\(48\) 0 0
\(49\) −94.3200 −1.92490
\(50\) 0 0
\(51\) 0 0
\(52\) 77.0964 46.2809i 1.48262 0.890017i
\(53\) 94.1066 1.77560 0.887798 0.460233i \(-0.152234\pi\)
0.887798 + 0.460233i \(0.152234\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 65.9689 69.4305i 1.17802 1.23983i
\(57\) 0 0
\(58\) 1.46718 + 5.29471i 0.0252961 + 0.0912882i
\(59\) 43.2650i 0.733305i 0.930358 + 0.366653i \(0.119496\pi\)
−0.930358 + 0.366653i \(0.880504\pi\)
\(60\) 0 0
\(61\) 56.7678 0.930620 0.465310 0.885148i \(-0.345943\pi\)
0.465310 + 0.885148i \(0.345943\pi\)
\(62\) 31.5060 8.73038i 0.508161 0.140813i
\(63\) 0 0
\(64\) 3.27028 + 63.9164i 0.0510981 + 0.998694i
\(65\) 0 0
\(66\) 0 0
\(67\) 61.1106i 0.912098i 0.889955 + 0.456049i \(0.150736\pi\)
−0.889955 + 0.456049i \(0.849264\pi\)
\(68\) −43.5173 + 26.1234i −0.639961 + 0.384167i
\(69\) 0 0
\(70\) 0 0
\(71\) 39.6643i 0.558652i 0.960196 + 0.279326i \(0.0901110\pi\)
−0.960196 + 0.279326i \(0.909889\pi\)
\(72\) 0 0
\(73\) 99.5452 1.36363 0.681817 0.731523i \(-0.261190\pi\)
0.681817 + 0.731523i \(0.261190\pi\)
\(74\) 17.3507 + 62.6149i 0.234469 + 0.846147i
\(75\) 0 0
\(76\) −18.0414 30.0541i −0.237387 0.395449i
\(77\) 174.046 2.26034
\(78\) 0 0
\(79\) 10.7780i 0.136430i 0.997671 + 0.0682151i \(0.0217304\pi\)
−0.997671 + 0.0682151i \(0.978270\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 22.8365 + 82.4118i 0.278494 + 1.00502i
\(83\) 140.263i 1.68991i 0.534837 + 0.844955i \(0.320373\pi\)
−0.534837 + 0.844955i \(0.679627\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 31.8855 8.83554i 0.370762 0.102739i
\(87\) 0 0
\(88\) −80.1118 + 84.3156i −0.910362 + 0.958131i
\(89\) −54.8723 −0.616543 −0.308271 0.951298i \(-0.599751\pi\)
−0.308271 + 0.951298i \(0.599751\pi\)
\(90\) 0 0
\(91\) 269.125i 2.95742i
\(92\) 10.2865 + 17.1357i 0.111810 + 0.186258i
\(93\) 0 0
\(94\) 93.6533 25.9515i 0.996312 0.276080i
\(95\) 0 0
\(96\) 0 0
\(97\) −14.1601 −0.145980 −0.0729902 0.997333i \(-0.523254\pi\)
−0.0729902 + 0.997333i \(0.523254\pi\)
\(98\) 50.3743 + 181.790i 0.514023 + 1.85500i
\(99\) 0 0
\(100\) 0 0
\(101\) 163.410 1.61792 0.808962 0.587861i \(-0.200030\pi\)
0.808962 + 0.587861i \(0.200030\pi\)
\(102\) 0 0
\(103\) 169.591i 1.64651i −0.567672 0.823255i \(-0.692156\pi\)
0.567672 0.823255i \(-0.307844\pi\)
\(104\) −130.376 123.876i −1.25362 1.19111i
\(105\) 0 0
\(106\) −50.2603 181.378i −0.474154 1.71112i
\(107\) 8.14840i 0.0761532i 0.999275 + 0.0380766i \(0.0121231\pi\)
−0.999275 + 0.0380766i \(0.987877\pi\)
\(108\) 0 0
\(109\) −25.2322 −0.231488 −0.115744 0.993279i \(-0.536925\pi\)
−0.115744 + 0.993279i \(0.536925\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −169.051 90.0652i −1.50938 0.804153i
\(113\) 97.8142 0.865613 0.432806 0.901487i \(-0.357523\pi\)
0.432806 + 0.901487i \(0.357523\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.42129 5.65559i 0.0812180 0.0487551i
\(117\) 0 0
\(118\) 83.3877 23.1069i 0.706676 0.195821i
\(119\) 151.909i 1.27654i
\(120\) 0 0
\(121\) −90.3597 −0.746774
\(122\) −30.3185 109.413i −0.248512 0.896825i
\(123\) 0 0
\(124\) −33.6534 56.0611i −0.271398 0.452105i
\(125\) 0 0
\(126\) 0 0
\(127\) 167.563i 1.31939i 0.751533 + 0.659695i \(0.229315\pi\)
−0.751533 + 0.659695i \(0.770685\pi\)
\(128\) 121.444 40.4394i 0.948782 0.315933i
\(129\) 0 0
\(130\) 0 0
\(131\) 82.0465i 0.626309i 0.949702 + 0.313155i \(0.101386\pi\)
−0.949702 + 0.313155i \(0.898614\pi\)
\(132\) 0 0
\(133\) 104.912 0.788810
\(134\) 117.783 32.6378i 0.878976 0.243566i
\(135\) 0 0
\(136\) 73.5911 + 69.9221i 0.541111 + 0.514133i
\(137\) −254.459 −1.85737 −0.928683 0.370874i \(-0.879058\pi\)
−0.928683 + 0.370874i \(0.879058\pi\)
\(138\) 0 0
\(139\) 78.9483i 0.567974i 0.958828 + 0.283987i \(0.0916572\pi\)
−0.958828 + 0.283987i \(0.908343\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 76.4478 21.1838i 0.538365 0.149182i
\(143\) 326.823i 2.28547i
\(144\) 0 0
\(145\) 0 0
\(146\) −53.1650 191.861i −0.364144 1.31411i
\(147\) 0 0
\(148\) 111.415 66.8825i 0.752807 0.451909i
\(149\) 32.3433 0.217069 0.108534 0.994093i \(-0.465384\pi\)
0.108534 + 0.994093i \(0.465384\pi\)
\(150\) 0 0
\(151\) 38.7953i 0.256922i −0.991715 0.128461i \(-0.958996\pi\)
0.991715 0.128461i \(-0.0410038\pi\)
\(152\) −48.2899 + 50.8238i −0.317697 + 0.334367i
\(153\) 0 0
\(154\) −92.9543 335.452i −0.603600 2.17826i
\(155\) 0 0
\(156\) 0 0
\(157\) −44.2021 −0.281542 −0.140771 0.990042i \(-0.544958\pi\)
−0.140771 + 0.990042i \(0.544958\pi\)
\(158\) 20.7732 5.75629i 0.131476 0.0364322i
\(159\) 0 0
\(160\) 0 0
\(161\) −59.8167 −0.371532
\(162\) 0 0
\(163\) 52.9366i 0.324764i −0.986728 0.162382i \(-0.948082\pi\)
0.986728 0.162382i \(-0.0519177\pi\)
\(164\) 146.642 88.0287i 0.894156 0.536760i
\(165\) 0 0
\(166\) 270.338 74.9112i 1.62854 0.451272i
\(167\) 179.273i 1.07349i 0.843744 + 0.536745i \(0.180346\pi\)
−0.843744 + 0.536745i \(0.819654\pi\)
\(168\) 0 0
\(169\) 336.361 1.99030
\(170\) 0 0
\(171\) 0 0
\(172\) −34.0587 56.7364i −0.198016 0.329863i
\(173\) −177.276 −1.02471 −0.512357 0.858772i \(-0.671228\pi\)
−0.512357 + 0.858772i \(0.671228\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 205.293 + 109.374i 1.16644 + 0.621444i
\(177\) 0 0
\(178\) 29.3061 + 105.759i 0.164641 + 0.594154i
\(179\) 102.849i 0.574573i 0.957845 + 0.287286i \(0.0927532\pi\)
−0.957845 + 0.287286i \(0.907247\pi\)
\(180\) 0 0
\(181\) −115.413 −0.637640 −0.318820 0.947815i \(-0.603286\pi\)
−0.318820 + 0.947815i \(0.603286\pi\)
\(182\) 518.704 143.734i 2.85002 0.789747i
\(183\) 0 0
\(184\) 27.5331 28.9778i 0.149636 0.157488i
\(185\) 0 0
\(186\) 0 0
\(187\) 184.476i 0.986503i
\(188\) −100.036 166.645i −0.532109 0.886407i
\(189\) 0 0
\(190\) 0 0
\(191\) 191.305i 1.00160i 0.865563 + 0.500799i \(0.166960\pi\)
−0.865563 + 0.500799i \(0.833040\pi\)
\(192\) 0 0
\(193\) −160.332 −0.830734 −0.415367 0.909654i \(-0.636347\pi\)
−0.415367 + 0.909654i \(0.636347\pi\)
\(194\) 7.56261 + 27.2918i 0.0389825 + 0.140679i
\(195\) 0 0
\(196\) 323.472 194.180i 1.65037 0.990714i
\(197\) −355.081 −1.80244 −0.901220 0.433362i \(-0.857327\pi\)
−0.901220 + 0.433362i \(0.857327\pi\)
\(198\) 0 0
\(199\) 88.2032i 0.443232i −0.975134 0.221616i \(-0.928867\pi\)
0.975134 0.221616i \(-0.0711332\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −87.2740 314.952i −0.432049 1.55917i
\(203\) 32.8875i 0.162007i
\(204\) 0 0
\(205\) 0 0
\(206\) −326.864 + 90.5747i −1.58672 + 0.439683i
\(207\) 0 0
\(208\) −169.124 + 317.442i −0.813095 + 1.52617i
\(209\) −127.404 −0.609586
\(210\) 0 0
\(211\) 190.584i 0.903243i 0.892210 + 0.451622i \(0.149154\pi\)
−0.892210 + 0.451622i \(0.850846\pi\)
\(212\) −322.741 + 193.741i −1.52236 + 0.913871i
\(213\) 0 0
\(214\) 15.7050 4.35188i 0.0733878 0.0203359i
\(215\) 0 0
\(216\) 0 0
\(217\) 195.696 0.901824
\(218\) 13.4760 + 48.6318i 0.0618164 + 0.223082i
\(219\) 0 0
\(220\) 0 0
\(221\) −285.253 −1.29074
\(222\) 0 0
\(223\) 79.2869i 0.355547i −0.984071 0.177773i \(-0.943111\pi\)
0.984071 0.177773i \(-0.0568894\pi\)
\(224\) −83.3026 + 373.926i −0.371887 + 1.66931i
\(225\) 0 0
\(226\) −52.2405 188.524i −0.231153 0.834178i
\(227\) 353.645i 1.55791i −0.627082 0.778953i \(-0.715751\pi\)
0.627082 0.778953i \(-0.284249\pi\)
\(228\) 0 0
\(229\) −22.7911 −0.0995244 −0.0497622 0.998761i \(-0.515846\pi\)
−0.0497622 + 0.998761i \(0.515846\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −15.9321 15.1378i −0.0686730 0.0652491i
\(233\) −189.710 −0.814205 −0.407103 0.913382i \(-0.633461\pi\)
−0.407103 + 0.913382i \(0.633461\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −89.0712 148.378i −0.377420 0.628721i
\(237\) 0 0
\(238\) −292.784 + 81.1311i −1.23019 + 0.340887i
\(239\) 267.778i 1.12041i 0.828355 + 0.560204i \(0.189277\pi\)
−0.828355 + 0.560204i \(0.810723\pi\)
\(240\) 0 0
\(241\) −301.663 −1.25171 −0.625857 0.779938i \(-0.715251\pi\)
−0.625857 + 0.779938i \(0.715251\pi\)
\(242\) 48.2592 + 174.157i 0.199418 + 0.719656i
\(243\) 0 0
\(244\) −194.686 + 116.870i −0.797895 + 0.478975i
\(245\) 0 0
\(246\) 0 0
\(247\) 197.002i 0.797580i
\(248\) −90.0769 + 94.8035i −0.363213 + 0.382272i
\(249\) 0 0
\(250\) 0 0
\(251\) 63.1891i 0.251749i 0.992046 + 0.125875i \(0.0401737\pi\)
−0.992046 + 0.125875i \(0.959826\pi\)
\(252\) 0 0
\(253\) 72.6407 0.287117
\(254\) 322.955 89.4916i 1.27148 0.352329i
\(255\) 0 0
\(256\) −142.802 212.470i −0.557822 0.829961i
\(257\) 150.719 0.586456 0.293228 0.956043i \(-0.405271\pi\)
0.293228 + 0.956043i \(0.405271\pi\)
\(258\) 0 0
\(259\) 388.925i 1.50164i
\(260\) 0 0
\(261\) 0 0
\(262\) 158.134 43.8193i 0.603565 0.167249i
\(263\) 203.755i 0.774735i −0.921925 0.387368i \(-0.873384\pi\)
0.921925 0.387368i \(-0.126616\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −56.0311 202.204i −0.210643 0.760165i
\(267\) 0 0
\(268\) −125.810 209.580i −0.469442 0.782014i
\(269\) −76.3986 −0.284010 −0.142005 0.989866i \(-0.545355\pi\)
−0.142005 + 0.989866i \(0.545355\pi\)
\(270\) 0 0
\(271\) 169.216i 0.624414i −0.950014 0.312207i \(-0.898932\pi\)
0.950014 0.312207i \(-0.101068\pi\)
\(272\) 95.4624 179.181i 0.350965 0.658755i
\(273\) 0 0
\(274\) 135.901 + 490.437i 0.495990 + 1.78992i
\(275\) 0 0
\(276\) 0 0
\(277\) 273.891 0.988774 0.494387 0.869242i \(-0.335393\pi\)
0.494387 + 0.869242i \(0.335393\pi\)
\(278\) 152.163 42.1646i 0.547348 0.151671i
\(279\) 0 0
\(280\) 0 0
\(281\) 311.672 1.10915 0.554577 0.832133i \(-0.312880\pi\)
0.554577 + 0.832133i \(0.312880\pi\)
\(282\) 0 0
\(283\) 264.566i 0.934861i −0.884030 0.467431i \(-0.845180\pi\)
0.884030 0.467431i \(-0.154820\pi\)
\(284\) −81.6583 136.029i −0.287529 0.478977i
\(285\) 0 0
\(286\) −629.909 + 174.549i −2.20248 + 0.610311i
\(287\) 511.891i 1.78359i
\(288\) 0 0
\(289\) −127.988 −0.442866
\(290\) 0 0
\(291\) 0 0
\(292\) −341.392 + 204.937i −1.16915 + 0.701840i
\(293\) 121.281 0.413927 0.206964 0.978349i \(-0.433642\pi\)
0.206964 + 0.978349i \(0.433642\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −188.412 179.018i −0.636527 0.604792i
\(297\) 0 0
\(298\) −17.2739 62.3375i −0.0579659 0.209186i
\(299\) 112.323i 0.375663i
\(300\) 0 0
\(301\) 198.053 0.657984
\(302\) −74.7729 + 20.7197i −0.247592 + 0.0686083i
\(303\) 0 0
\(304\) 123.747 + 65.9286i 0.407062 + 0.216870i
\(305\) 0 0
\(306\) 0 0
\(307\) 161.768i 0.526932i 0.964669 + 0.263466i \(0.0848656\pi\)
−0.964669 + 0.263466i \(0.915134\pi\)
\(308\) −596.895 + 358.315i −1.93797 + 1.16336i
\(309\) 0 0
\(310\) 0 0
\(311\) 26.3813i 0.0848273i −0.999100 0.0424137i \(-0.986495\pi\)
0.999100 0.0424137i \(-0.0135047\pi\)
\(312\) 0 0
\(313\) −5.39902 −0.0172493 −0.00862463 0.999963i \(-0.502745\pi\)
−0.00862463 + 0.999963i \(0.502745\pi\)
\(314\) 23.6074 + 85.1938i 0.0751827 + 0.271318i
\(315\) 0 0
\(316\) −22.1890 36.9633i −0.0702184 0.116972i
\(317\) 270.157 0.852231 0.426116 0.904669i \(-0.359882\pi\)
0.426116 + 0.904669i \(0.359882\pi\)
\(318\) 0 0
\(319\) 39.9382i 0.125198i
\(320\) 0 0
\(321\) 0 0
\(322\) 31.9468 + 115.289i 0.0992137 + 0.358040i
\(323\) 111.199i 0.344268i
\(324\) 0 0
\(325\) 0 0
\(326\) −102.028 + 28.2723i −0.312971 + 0.0867248i
\(327\) 0 0
\(328\) −247.982 235.619i −0.756043 0.718349i
\(329\) 581.716 1.76813
\(330\) 0 0
\(331\) 480.728i 1.45235i 0.687510 + 0.726174i \(0.258704\pi\)
−0.687510 + 0.726174i \(0.741296\pi\)
\(332\) −288.763 481.033i −0.869770 1.44890i
\(333\) 0 0
\(334\) 345.526 95.7459i 1.03451 0.286664i
\(335\) 0 0
\(336\) 0 0
\(337\) −568.382 −1.68659 −0.843297 0.537448i \(-0.819388\pi\)
−0.843297 + 0.537448i \(0.819388\pi\)
\(338\) −179.643 648.293i −0.531489 1.91803i
\(339\) 0 0
\(340\) 0 0
\(341\) −237.651 −0.696923
\(342\) 0 0
\(343\) 542.554i 1.58179i
\(344\) −91.1620 + 95.9455i −0.265006 + 0.278911i
\(345\) 0 0
\(346\) 94.6791 + 341.676i 0.273639 + 0.987503i
\(347\) 370.184i 1.06681i −0.845859 0.533406i \(-0.820912\pi\)
0.845859 0.533406i \(-0.179088\pi\)
\(348\) 0 0
\(349\) −488.570 −1.39991 −0.699957 0.714185i \(-0.746797\pi\)
−0.699957 + 0.714185i \(0.746797\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 101.162 454.091i 0.287391 1.29003i
\(353\) −649.728 −1.84059 −0.920295 0.391226i \(-0.872051\pi\)
−0.920295 + 0.391226i \(0.872051\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 188.186 112.968i 0.528612 0.317325i
\(357\) 0 0
\(358\) 198.227 54.9292i 0.553708 0.153434i
\(359\) 405.910i 1.13067i −0.824862 0.565334i \(-0.808747\pi\)
0.824862 0.565334i \(-0.191253\pi\)
\(360\) 0 0
\(361\) 284.204 0.787268
\(362\) 61.6395 + 222.443i 0.170275 + 0.614484i
\(363\) 0 0
\(364\) −554.058 922.970i −1.52214 2.53563i
\(365\) 0 0
\(366\) 0 0
\(367\) 46.2347i 0.125980i 0.998014 + 0.0629900i \(0.0200636\pi\)
−0.998014 + 0.0629900i \(0.979936\pi\)
\(368\) −70.5558 37.5900i −0.191728 0.102147i
\(369\) 0 0
\(370\) 0 0
\(371\) 1126.61i 3.03668i
\(372\) 0 0
\(373\) −138.262 −0.370676 −0.185338 0.982675i \(-0.559338\pi\)
−0.185338 + 0.982675i \(0.559338\pi\)
\(374\) 355.554 98.5247i 0.950679 0.263435i
\(375\) 0 0
\(376\) −267.759 + 281.809i −0.712124 + 0.749491i
\(377\) 61.7559 0.163809
\(378\) 0 0
\(379\) 254.516i 0.671546i 0.941943 + 0.335773i \(0.108998\pi\)
−0.941943 + 0.335773i \(0.891002\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 368.716 102.172i 0.965226 0.267466i
\(383\) 62.7205i 0.163761i 0.996642 + 0.0818805i \(0.0260926\pi\)
−0.996642 + 0.0818805i \(0.973907\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 85.6298 + 309.019i 0.221839 + 0.800567i
\(387\) 0 0
\(388\) 48.5623 29.1519i 0.125161 0.0751338i
\(389\) −110.130 −0.283112 −0.141556 0.989930i \(-0.545210\pi\)
−0.141556 + 0.989930i \(0.545210\pi\)
\(390\) 0 0
\(391\) 63.4012i 0.162152i
\(392\) −547.016 519.744i −1.39545 1.32588i
\(393\) 0 0
\(394\) 189.641 + 684.372i 0.481322 + 1.73698i
\(395\) 0 0
\(396\) 0 0
\(397\) −292.953 −0.737916 −0.368958 0.929446i \(-0.620285\pi\)
−0.368958 + 0.929446i \(0.620285\pi\)
\(398\) −170.000 + 47.1074i −0.427136 + 0.118360i
\(399\) 0 0
\(400\) 0 0
\(401\) −518.103 −1.29203 −0.646014 0.763325i \(-0.723565\pi\)
−0.646014 + 0.763325i \(0.723565\pi\)
\(402\) 0 0
\(403\) 367.476i 0.911851i
\(404\) −560.419 + 336.419i −1.38718 + 0.832720i
\(405\) 0 0
\(406\) 63.3864 17.5645i 0.156124 0.0432623i
\(407\) 472.306i 1.16046i
\(408\) 0 0
\(409\) −181.984 −0.444948 −0.222474 0.974939i \(-0.571413\pi\)
−0.222474 + 0.974939i \(0.571413\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 349.142 + 581.614i 0.847432 + 1.41168i
\(413\) 517.953 1.25412
\(414\) 0 0
\(415\) 0 0
\(416\) 702.155 + 156.425i 1.68787 + 0.376022i
\(417\) 0 0
\(418\) 68.0435 + 245.554i 0.162784 + 0.587450i
\(419\) 163.347i 0.389849i 0.980818 + 0.194925i \(0.0624462\pi\)
−0.980818 + 0.194925i \(0.937554\pi\)
\(420\) 0 0
\(421\) −467.206 −1.10975 −0.554876 0.831933i \(-0.687234\pi\)
−0.554876 + 0.831933i \(0.687234\pi\)
\(422\) 367.327 101.787i 0.870442 0.241201i
\(423\) 0 0
\(424\) 545.779 + 518.568i 1.28721 + 1.22304i
\(425\) 0 0
\(426\) 0 0
\(427\) 679.603i 1.59158i
\(428\) −16.7754 27.9451i −0.0391948 0.0652923i
\(429\) 0 0
\(430\) 0 0
\(431\) 685.527i 1.59055i 0.606248 + 0.795275i \(0.292674\pi\)
−0.606248 + 0.795275i \(0.707326\pi\)
\(432\) 0 0
\(433\) −592.777 −1.36900 −0.684500 0.729013i \(-0.739979\pi\)
−0.684500 + 0.729013i \(0.739979\pi\)
\(434\) −104.517 377.178i −0.240822 0.869074i
\(435\) 0 0
\(436\) 86.5343 51.9464i 0.198473 0.119143i
\(437\) 43.7864 0.100198
\(438\) 0 0
\(439\) 464.439i 1.05795i −0.848638 0.528974i \(-0.822577\pi\)
0.848638 0.528974i \(-0.177423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 152.347 + 549.788i 0.344677 + 1.24386i
\(443\) 54.2868i 0.122544i 0.998121 + 0.0612718i \(0.0195156\pi\)
−0.998121 + 0.0612718i \(0.980484\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −152.815 + 42.3454i −0.342635 + 0.0949449i
\(447\) 0 0
\(448\) 765.184 39.1506i 1.70800 0.0873897i
\(449\) 428.051 0.953343 0.476671 0.879082i \(-0.341843\pi\)
0.476671 + 0.879082i \(0.341843\pi\)
\(450\) 0 0
\(451\) 621.634i 1.37835i
\(452\) −335.456 + 201.374i −0.742159 + 0.445517i
\(453\) 0 0
\(454\) −681.605 + 188.874i −1.50133 + 0.416022i
\(455\) 0 0
\(456\) 0 0
\(457\) 66.5848 0.145700 0.0728498 0.997343i \(-0.476791\pi\)
0.0728498 + 0.997343i \(0.476791\pi\)
\(458\) 12.1722 + 43.9269i 0.0265769 + 0.0959103i
\(459\) 0 0
\(460\) 0 0
\(461\) 238.626 0.517627 0.258814 0.965927i \(-0.416668\pi\)
0.258814 + 0.965927i \(0.416668\pi\)
\(462\) 0 0
\(463\) 386.958i 0.835762i 0.908502 + 0.417881i \(0.137227\pi\)
−0.908502 + 0.417881i \(0.862773\pi\)
\(464\) −20.6672 + 38.7919i −0.0445413 + 0.0836032i
\(465\) 0 0
\(466\) 101.320 + 365.641i 0.217425 + 0.784638i
\(467\) 235.964i 0.505276i 0.967561 + 0.252638i \(0.0812981\pi\)
−0.967561 + 0.252638i \(0.918702\pi\)
\(468\) 0 0
\(469\) 731.593 1.55990
\(470\) 0 0
\(471\) 0 0
\(472\) −238.409 + 250.919i −0.505104 + 0.531608i
\(473\) −240.513 −0.508485
\(474\) 0 0
\(475\) 0 0
\(476\) 312.740 + 520.974i 0.657016 + 1.09448i
\(477\) 0 0
\(478\) 516.107 143.014i 1.07972 0.299193i
\(479\) 529.496i 1.10542i 0.833374 + 0.552710i \(0.186406\pi\)
−0.833374 + 0.552710i \(0.813594\pi\)
\(480\) 0 0
\(481\) 730.320 1.51834
\(482\) 161.112 + 581.417i 0.334257 + 1.20626i
\(483\) 0 0
\(484\) 309.890 186.027i 0.640270 0.384353i
\(485\) 0 0
\(486\) 0 0
\(487\) 880.801i 1.80863i −0.426869 0.904314i \(-0.640383\pi\)
0.426869 0.904314i \(-0.359617\pi\)
\(488\) 329.230 + 312.815i 0.674651 + 0.641015i
\(489\) 0 0
\(490\) 0 0
\(491\) 86.4466i 0.176062i 0.996118 + 0.0880312i \(0.0280575\pi\)
−0.996118 + 0.0880312i \(0.971942\pi\)
\(492\) 0 0
\(493\) −34.8583 −0.0707065
\(494\) −379.697 + 105.215i −0.768617 + 0.212985i
\(495\) 0 0
\(496\) 230.830 + 122.979i 0.465383 + 0.247942i
\(497\) 474.846 0.955425
\(498\) 0 0
\(499\) 874.536i 1.75258i −0.481786 0.876289i \(-0.660012\pi\)
0.481786 0.876289i \(-0.339988\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 121.789 33.7479i 0.242607 0.0672269i
\(503\) 17.5479i 0.0348865i −0.999848 0.0174433i \(-0.994447\pi\)
0.999848 0.0174433i \(-0.00555265\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −38.7958 140.006i −0.0766716 0.276691i
\(507\) 0 0
\(508\) −344.967 574.659i −0.679069 1.13122i
\(509\) −609.132 −1.19672 −0.598362 0.801226i \(-0.704181\pi\)
−0.598362 + 0.801226i \(0.704181\pi\)
\(510\) 0 0
\(511\) 1191.72i 2.33213i
\(512\) −333.241 + 388.709i −0.650861 + 0.759197i
\(513\) 0 0
\(514\) −80.4959 290.492i −0.156607 0.565159i
\(515\) 0 0
\(516\) 0 0
\(517\) −706.430 −1.36640
\(518\) 749.602 207.716i 1.44711 0.400997i
\(519\) 0 0
\(520\) 0 0
\(521\) −433.724 −0.832484 −0.416242 0.909254i \(-0.636653\pi\)
−0.416242 + 0.909254i \(0.636653\pi\)
\(522\) 0 0
\(523\) 473.223i 0.904823i −0.891809 0.452412i \(-0.850564\pi\)
0.891809 0.452412i \(-0.149436\pi\)
\(524\) −168.912 281.380i −0.322351 0.536985i
\(525\) 0 0
\(526\) −392.712 + 108.821i −0.746601 + 0.206885i
\(527\) 207.423i 0.393592i
\(528\) 0 0
\(529\) 504.035 0.952807
\(530\) 0 0
\(531\) 0 0
\(532\) −359.797 + 215.985i −0.676310 + 0.405988i
\(533\) 961.224 1.80342
\(534\) 0 0
\(535\) 0 0
\(536\) −336.746 + 354.416i −0.628257 + 0.661223i
\(537\) 0 0
\(538\) 40.8029 + 147.249i 0.0758418 + 0.273696i
\(539\) 1371.24i 2.54405i
\(540\) 0 0
\(541\) 294.889 0.545081 0.272540 0.962144i \(-0.412136\pi\)
0.272540 + 0.962144i \(0.412136\pi\)
\(542\) −326.142 + 90.3747i −0.601739 + 0.166743i
\(543\) 0 0
\(544\) −396.333 88.2946i −0.728554 0.162306i
\(545\) 0 0
\(546\) 0 0
\(547\) 966.695i 1.76727i 0.468179 + 0.883634i \(0.344910\pi\)
−0.468179 + 0.883634i \(0.655090\pi\)
\(548\) 872.673 523.864i 1.59247 0.955956i
\(549\) 0 0
\(550\) 0 0
\(551\) 24.0740i 0.0436914i
\(552\) 0 0
\(553\) 129.030 0.233327
\(554\) −146.279 527.889i −0.264042 0.952868i
\(555\) 0 0
\(556\) −162.534 270.755i −0.292327 0.486969i
\(557\) 74.2603 0.133322 0.0666609 0.997776i \(-0.478765\pi\)
0.0666609 + 0.997776i \(0.478765\pi\)
\(558\) 0 0
\(559\) 371.903i 0.665300i
\(560\) 0 0
\(561\) 0 0
\(562\) −166.457 600.708i −0.296187 1.06887i
\(563\) 663.688i 1.17884i 0.807826 + 0.589421i \(0.200644\pi\)
−0.807826 + 0.589421i \(0.799356\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −509.916 + 141.299i −0.900912 + 0.249645i
\(567\) 0 0
\(568\) −218.567 + 230.036i −0.384802 + 0.404993i
\(569\) 667.450 1.17302 0.586511 0.809941i \(-0.300501\pi\)
0.586511 + 0.809941i \(0.300501\pi\)
\(570\) 0 0
\(571\) 185.898i 0.325565i −0.986662 0.162782i \(-0.947953\pi\)
0.986662 0.162782i \(-0.0520469\pi\)
\(572\) 672.841 + 1120.84i 1.17630 + 1.95952i
\(573\) 0 0
\(574\) 986.603 273.390i 1.71882 0.476289i
\(575\) 0 0
\(576\) 0 0
\(577\) 664.331 1.15135 0.575676 0.817678i \(-0.304739\pi\)
0.575676 + 0.817678i \(0.304739\pi\)
\(578\) 68.3557 + 246.681i 0.118263 + 0.426783i
\(579\) 0 0
\(580\) 0 0
\(581\) 1679.17 2.89014
\(582\) 0 0
\(583\) 1368.14i 2.34673i
\(584\) 577.321 + 548.537i 0.988563 + 0.939276i
\(585\) 0 0
\(586\) −64.7734 233.753i −0.110535 0.398896i
\(587\) 763.083i 1.29997i 0.759946 + 0.649986i \(0.225225\pi\)
−0.759946 + 0.649986i \(0.774775\pi\)
\(588\) 0 0
\(589\) −143.251 −0.243211
\(590\) 0 0
\(591\) 0 0
\(592\) −244.408 + 458.750i −0.412852 + 0.774915i
\(593\) 286.193 0.482618 0.241309 0.970448i \(-0.422423\pi\)
0.241309 + 0.970448i \(0.422423\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −110.922 + 66.5862i −0.186111 + 0.111722i
\(597\) 0 0
\(598\) 216.489 59.9895i 0.362021 0.100317i
\(599\) 604.151i 1.00860i 0.863529 + 0.504300i \(0.168249\pi\)
−0.863529 + 0.504300i \(0.831751\pi\)
\(600\) 0 0
\(601\) 275.562 0.458505 0.229253 0.973367i \(-0.426372\pi\)
0.229253 + 0.973367i \(0.426372\pi\)
\(602\) −105.776 381.722i −0.175707 0.634089i
\(603\) 0 0
\(604\) 79.8692 + 133.049i 0.132234 + 0.220280i
\(605\) 0 0
\(606\) 0 0
\(607\) 52.1487i 0.0859121i −0.999077 0.0429561i \(-0.986322\pi\)
0.999077 0.0429561i \(-0.0136775\pi\)
\(608\) 60.9784 273.717i 0.100293 0.450193i
\(609\) 0 0
\(610\) 0 0
\(611\) 1092.34i 1.78779i
\(612\) 0 0
\(613\) 898.128 1.46513 0.732567 0.680695i \(-0.238322\pi\)
0.732567 + 0.680695i \(0.238322\pi\)
\(614\) 311.787 86.3968i 0.507796 0.140711i
\(615\) 0 0
\(616\) 1009.39 + 959.070i 1.63863 + 1.55693i
\(617\) −636.868 −1.03220 −0.516101 0.856528i \(-0.672617\pi\)
−0.516101 + 0.856528i \(0.672617\pi\)
\(618\) 0 0
\(619\) 190.559i 0.307849i 0.988083 + 0.153925i \(0.0491913\pi\)
−0.988083 + 0.153925i \(0.950809\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −50.8466 + 14.0897i −0.0817469 + 0.0226522i
\(623\) 656.911i 1.05443i
\(624\) 0 0
\(625\) 0 0
\(626\) 2.88350 + 10.4059i 0.00460623 + 0.0166229i
\(627\) 0 0
\(628\) 151.592 91.0004i 0.241388 0.144905i
\(629\) −412.231 −0.655376
\(630\) 0 0
\(631\) 578.160i 0.916261i −0.888885 0.458130i \(-0.848519\pi\)
0.888885 0.458130i \(-0.151481\pi\)
\(632\) −59.3913 + 62.5078i −0.0939736 + 0.0989047i
\(633\) 0 0
\(634\) −144.285 520.693i −0.227579 0.821283i
\(635\) 0 0
\(636\) 0 0
\(637\) 2120.34 3.32863
\(638\) −76.9757 + 21.3301i −0.120652 + 0.0334328i
\(639\) 0 0
\(640\) 0 0
\(641\) 35.3085 0.0550834 0.0275417 0.999621i \(-0.491232\pi\)
0.0275417 + 0.999621i \(0.491232\pi\)
\(642\) 0 0
\(643\) 1045.67i 1.62623i 0.582100 + 0.813117i \(0.302231\pi\)
−0.582100 + 0.813117i \(0.697769\pi\)
\(644\) 205.143 123.147i 0.318544 0.191222i
\(645\) 0 0
\(646\) 214.321 59.3888i 0.331766 0.0919331i
\(647\) 2.71164i 0.00419110i −0.999998 0.00209555i \(-0.999333\pi\)
0.999998 0.00209555i \(-0.000667035\pi\)
\(648\) 0 0
\(649\) −628.996 −0.969177
\(650\) 0 0
\(651\) 0 0
\(652\) 108.982 + 181.547i 0.167151 + 0.278446i
\(653\) −206.765 −0.316639 −0.158319 0.987388i \(-0.550608\pi\)
−0.158319 + 0.987388i \(0.550608\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −321.682 + 603.792i −0.490370 + 0.920415i
\(657\) 0 0
\(658\) −310.682 1121.18i −0.472161 1.70393i
\(659\) 708.330i 1.07486i −0.843309 0.537428i \(-0.819396\pi\)
0.843309 0.537428i \(-0.180604\pi\)
\(660\) 0 0
\(661\) 1229.66 1.86031 0.930155 0.367167i \(-0.119672\pi\)
0.930155 + 0.367167i \(0.119672\pi\)
\(662\) 926.540 256.746i 1.39961 0.387834i
\(663\) 0 0
\(664\) −772.907 + 813.464i −1.16402 + 1.22510i
\(665\) 0 0
\(666\) 0 0
\(667\) 13.7261i 0.0205788i
\(668\) −369.076 614.820i −0.552508 0.920390i
\(669\) 0 0
\(670\) 0 0
\(671\) 825.303i 1.22996i
\(672\) 0 0
\(673\) 753.492 1.11960 0.559801 0.828627i \(-0.310878\pi\)
0.559801 + 0.828627i \(0.310878\pi\)
\(674\) 303.561 + 1095.48i 0.450387 + 1.62535i
\(675\) 0 0
\(676\) −1153.56 + 692.479i −1.70645 + 1.02438i
\(677\) 332.246 0.490762 0.245381 0.969427i \(-0.421087\pi\)
0.245381 + 0.969427i \(0.421087\pi\)
\(678\) 0 0
\(679\) 169.520i 0.249661i
\(680\) 0 0
\(681\) 0 0
\(682\) 126.924 + 458.041i 0.186106 + 0.671615i
\(683\) 1120.62i 1.64074i −0.571835 0.820368i \(-0.693768\pi\)
0.571835 0.820368i \(-0.306232\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1045.70 289.767i 1.52435 0.422400i
\(687\) 0 0
\(688\) 233.610 + 124.461i 0.339550 + 0.180902i
\(689\) −2115.54 −3.07045
\(690\) 0 0
\(691\) 331.115i 0.479182i −0.970874 0.239591i \(-0.922987\pi\)
0.970874 0.239591i \(-0.0770134\pi\)
\(692\) 607.970 364.964i 0.878570 0.527404i
\(693\) 0 0
\(694\) −713.482 + 197.707i −1.02807 + 0.284881i
\(695\) 0 0
\(696\) 0 0
\(697\) −542.566 −0.778431
\(698\) 260.935 + 941.655i 0.373832 + 1.34908i
\(699\) 0 0
\(700\) 0 0
\(701\) 564.971 0.805949 0.402975 0.915211i \(-0.367976\pi\)
0.402975 + 0.915211i \(0.367976\pi\)
\(702\) 0 0
\(703\) 284.697i 0.404974i
\(704\) −929.230 + 47.5440i −1.31993 + 0.0675341i
\(705\) 0 0
\(706\) 347.006 + 1252.27i 0.491510 + 1.77375i
\(707\) 1956.29i 2.76703i
\(708\) 0 0
\(709\) 1.56083 0.00220146 0.00110073 0.999999i \(-0.499650\pi\)
0.00110073 + 0.999999i \(0.499650\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −318.236 302.370i −0.446961 0.424677i
\(713\) 81.6765 0.114553
\(714\) 0 0
\(715\) 0 0
\(716\) −211.738 352.721i −0.295723 0.492627i
\(717\) 0 0
\(718\) −782.339 + 216.788i −1.08961 + 0.301933i
\(719\) 75.0325i 0.104357i 0.998638 + 0.0521784i \(0.0166164\pi\)
−0.998638 + 0.0521784i \(0.983384\pi\)
\(720\) 0 0
\(721\) −2030.28 −2.81592
\(722\) −151.787 547.766i −0.210231 0.758678i
\(723\) 0 0
\(724\) 395.810 237.604i 0.546699 0.328183i
\(725\) 0 0
\(726\) 0 0
\(727\) 1229.26i 1.69087i −0.534082 0.845433i \(-0.679343\pi\)
0.534082 0.845433i \(-0.320657\pi\)
\(728\) −1483.00 + 1560.81i −2.03708 + 2.14397i
\(729\) 0 0
\(730\) 0 0
\(731\) 209.922i 0.287170i
\(732\) 0 0
\(733\) 691.736 0.943705 0.471852 0.881678i \(-0.343586\pi\)
0.471852 + 0.881678i \(0.343586\pi\)
\(734\) 89.1114 24.6930i 0.121405 0.0336416i
\(735\) 0 0
\(736\) −34.7676 + 156.063i −0.0472385 + 0.212042i
\(737\) −888.438 −1.20548
\(738\) 0 0
\(739\) 71.4311i 0.0966591i −0.998831 0.0483296i \(-0.984610\pi\)
0.998831 0.0483296i \(-0.0153898\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2171.40 + 601.698i −2.92641 + 0.810914i
\(743\) 1006.92i 1.35521i 0.735426 + 0.677605i \(0.236982\pi\)
−0.735426 + 0.677605i \(0.763018\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 73.8428 + 266.482i 0.0989850 + 0.357215i
\(747\) 0 0
\(748\) −379.787 632.664i −0.507737 0.845808i
\(749\) 97.5496 0.130240
\(750\) 0 0
\(751\) 1110.14i 1.47822i 0.673587 + 0.739108i \(0.264753\pi\)
−0.673587 + 0.739108i \(0.735247\pi\)
\(752\) 686.154 + 365.562i 0.912439 + 0.486120i
\(753\) 0 0
\(754\) −32.9825 119.026i −0.0437433 0.157860i
\(755\) 0 0
\(756\) 0 0
\(757\) 326.752 0.431641 0.215821 0.976433i \(-0.430757\pi\)
0.215821 + 0.976433i \(0.430757\pi\)
\(758\) 490.547 135.932i 0.647160 0.179329i
\(759\) 0 0
\(760\) 0 0
\(761\) −162.162 −0.213091 −0.106546 0.994308i \(-0.533979\pi\)
−0.106546 + 0.994308i \(0.533979\pi\)
\(762\) 0 0
\(763\) 302.070i 0.395898i
\(764\) −393.847 656.085i −0.515507 0.858750i
\(765\) 0 0
\(766\) 120.886 33.4977i 0.157814 0.0437306i
\(767\) 972.608i 1.26807i
\(768\) 0 0
\(769\) −154.694 −0.201162 −0.100581 0.994929i \(-0.532070\pi\)
−0.100581 + 0.994929i \(0.532070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 549.861 330.081i 0.712255 0.427566i
\(773\) 208.302 0.269472 0.134736 0.990882i \(-0.456981\pi\)
0.134736 + 0.990882i \(0.456981\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −82.1226 78.0283i −0.105828 0.100552i
\(777\) 0 0
\(778\) 58.8183 + 212.262i 0.0756019 + 0.272831i
\(779\) 374.709i 0.481013i
\(780\) 0 0
\(781\) −576.648 −0.738346
\(782\) −122.198 + 33.8612i −0.156263 + 0.0433008i
\(783\) 0 0
\(784\) −709.590 + 1331.89i −0.905089 + 1.69884i
\(785\) 0 0
\(786\) 0 0
\(787\) 377.158i 0.479235i −0.970867 0.239617i \(-0.922978\pi\)
0.970867 0.239617i \(-0.0770220\pi\)
\(788\) 1217.76 731.017i 1.54538 0.927687i
\(789\) 0 0
\(790\) 0 0
\(791\) 1171.00i 1.48040i
\(792\) 0 0
\(793\) −1276.15 −1.60927
\(794\) 156.460 + 564.629i 0.197053 + 0.711119i
\(795\) 0 0
\(796\) 181.587 + 302.495i 0.228124 + 0.380018i
\(797\) −321.141 −0.402937 −0.201468 0.979495i \(-0.564571\pi\)
−0.201468 + 0.979495i \(0.564571\pi\)
\(798\) 0 0
\(799\) 616.576i 0.771685i
\(800\) 0 0
\(801\) 0 0
\(802\) 276.708 + 998.577i 0.345022 + 1.24511i
\(803\) 1447.21i 1.80225i
\(804\) 0 0
\(805\) 0 0
\(806\) −708.263 + 196.261i −0.878738 + 0.243500i
\(807\) 0 0
\(808\) 947.712 + 900.462i 1.17291 + 1.11443i
\(809\) 861.938 1.06544 0.532718 0.846293i \(-0.321171\pi\)
0.532718 + 0.846293i \(0.321171\pi\)
\(810\) 0 0
\(811\) 1011.44i 1.24715i 0.781765 + 0.623574i \(0.214320\pi\)
−0.781765 + 0.623574i \(0.785680\pi\)
\(812\) −67.7066 112.788i −0.0833825 0.138902i
\(813\) 0 0
\(814\) −910.308 + 252.248i −1.11832 + 0.309887i
\(815\) 0 0
\(816\) 0 0
\(817\) −144.977 −0.177450
\(818\) 97.1935 + 350.750i 0.118818 + 0.428790i
\(819\) 0 0
\(820\) 0 0
\(821\) −68.0368 −0.0828707 −0.0414353 0.999141i \(-0.513193\pi\)
−0.0414353 + 0.999141i \(0.513193\pi\)
\(822\) 0 0
\(823\) 980.340i 1.19118i −0.803289 0.595589i \(-0.796919\pi\)
0.803289 0.595589i \(-0.203081\pi\)
\(824\) 934.517 983.554i 1.13412 1.19363i
\(825\) 0 0
\(826\) −276.628 998.288i −0.334900 1.20858i
\(827\) 1183.88i 1.43153i 0.698341 + 0.715766i \(0.253922\pi\)
−0.698341 + 0.715766i \(0.746078\pi\)
\(828\) 0 0
\(829\) 98.7892 0.119167 0.0595833 0.998223i \(-0.481023\pi\)
0.0595833 + 0.998223i \(0.481023\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −73.5167 1436.86i −0.0883614 1.72699i
\(833\) −1196.83 −1.43677
\(834\) 0 0
\(835\) 0 0
\(836\) 436.933 262.290i 0.522647 0.313744i
\(837\) 0 0
\(838\) 314.830 87.2401i 0.375692 0.104105i
\(839\) 407.965i 0.486251i −0.969995 0.243125i \(-0.921827\pi\)
0.969995 0.243125i \(-0.0781727\pi\)
\(840\) 0 0
\(841\) −833.453 −0.991027
\(842\) 249.525 + 900.479i 0.296347 + 1.06945i
\(843\) 0 0
\(844\) −392.363 653.613i −0.464885 0.774423i
\(845\) 0 0
\(846\) 0 0
\(847\) 1081.75i 1.27716i
\(848\) 707.984 1328.87i 0.834887 1.56707i
\(849\) 0 0
\(850\) 0 0
\(851\) 162.323i 0.190744i
\(852\) 0 0
\(853\) −907.020 −1.06333 −0.531665 0.846955i \(-0.678433\pi\)
−0.531665 + 0.846955i \(0.678433\pi\)
\(854\) −1309.85 + 362.962i −1.53378 + 0.425014i
\(855\) 0 0
\(856\) −44.9012 + 47.2573i −0.0524546 + 0.0552071i
\(857\) 693.549 0.809275 0.404638 0.914477i \(-0.367398\pi\)
0.404638 + 0.914477i \(0.367398\pi\)
\(858\) 0 0
\(859\) 397.401i 0.462632i −0.972879 0.231316i \(-0.925697\pi\)
0.972879 0.231316i \(-0.0743031\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1321.27 366.126i 1.53279 0.424740i
\(863\) 1193.06i 1.38246i −0.722636 0.691229i \(-0.757070\pi\)
0.722636 0.691229i \(-0.242930\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 316.590 + 1142.50i 0.365577 + 1.31929i
\(867\) 0 0
\(868\) −671.142 + 402.886i −0.773206 + 0.464154i
\(869\) −156.693 −0.180314
\(870\) 0 0
\(871\) 1373.78i 1.57724i
\(872\) −146.336 139.040i −0.167817 0.159450i
\(873\) 0 0
\(874\) −23.3854 84.3927i −0.0267567 0.0965591i
\(875\) 0 0
\(876\) 0 0
\(877\) −136.545 −0.155695 −0.0778477 0.996965i \(-0.524805\pi\)
−0.0778477 + 0.996965i \(0.524805\pi\)
\(878\) −895.146 + 248.047i −1.01953 + 0.282514i
\(879\) 0 0
\(880\) 0 0
\(881\) 836.578 0.949578 0.474789 0.880100i \(-0.342524\pi\)
0.474789 + 0.880100i \(0.342524\pi\)
\(882\) 0 0
\(883\) 632.625i 0.716450i −0.933635 0.358225i \(-0.883382\pi\)
0.933635 0.358225i \(-0.116618\pi\)
\(884\) 978.280 587.260i 1.10665 0.664321i
\(885\) 0 0
\(886\) 104.631 28.9934i 0.118093 0.0327239i
\(887\) 290.957i 0.328024i −0.986458 0.164012i \(-0.947556\pi\)
0.986458 0.164012i \(-0.0524435\pi\)
\(888\) 0 0
\(889\) 2006.00 2.25646
\(890\) 0 0
\(891\) 0 0
\(892\) 163.231 + 271.916i 0.182994 + 0.304838i
\(893\) −425.822 −0.476844
\(894\) 0 0
\(895\) 0 0
\(896\) −484.126 1453.88i −0.540319 1.62264i
\(897\) 0 0
\(898\) −228.613 825.013i −0.254580 0.918723i
\(899\) 44.9061i 0.0499511i
\(900\) 0 0
\(901\) 1194.12 1.32533
\(902\) −1198.12 + 332.001i −1.32829 + 0.368073i
\(903\) 0 0
\(904\) 567.282 + 538.999i 0.627524 + 0.596237i
\(905\) 0 0
\(906\) 0 0
\(907\) 473.341i 0.521875i 0.965356 + 0.260938i \(0.0840317\pi\)
−0.965356 + 0.260938i \(0.915968\pi\)
\(908\) 728.061 + 1212.83i 0.801830 + 1.33572i
\(909\) 0 0
\(910\) 0 0
\(911\) 1176.67i 1.29163i 0.763495 + 0.645813i \(0.223482\pi\)
−0.763495 + 0.645813i \(0.776518\pi\)
\(912\) 0 0
\(913\) −2039.17 −2.23348
\(914\) −35.5615 128.334i −0.0389075 0.140409i
\(915\) 0 0
\(916\) 78.1625 46.9208i 0.0853303 0.0512236i
\(917\) 982.230 1.07113
\(918\) 0 0
\(919\) 1491.24i 1.62267i 0.584580 + 0.811336i \(0.301259\pi\)
−0.584580 + 0.811336i \(0.698741\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −127.445 459.921i −0.138227 0.498830i
\(923\) 891.663i 0.966048i
\(924\) 0 0
\(925\) 0 0
\(926\) 745.812 206.666i 0.805412 0.223181i
\(927\) 0 0
\(928\) 85.8043 + 19.1154i 0.0924615 + 0.0205984i
\(929\) 1217.24 1.31027 0.655134 0.755513i \(-0.272612\pi\)
0.655134 + 0.755513i \(0.272612\pi\)
\(930\) 0 0
\(931\) 826.560i 0.887819i
\(932\) 650.613 390.562i 0.698083 0.419058i
\(933\) 0 0
\(934\) 454.790 126.023i 0.486927 0.134928i
\(935\) 0 0
\(936\) 0 0
\(937\) −468.840 −0.500363 −0.250182 0.968199i \(-0.580490\pi\)
−0.250182 + 0.968199i \(0.580490\pi\)
\(938\) −390.728 1410.05i −0.416555 1.50325i
\(939\) 0 0
\(940\) 0 0
\(941\) −358.033 −0.380481 −0.190241 0.981737i \(-0.560927\pi\)
−0.190241 + 0.981737i \(0.560927\pi\)
\(942\) 0 0
\(943\) 213.645i 0.226559i
\(944\) 610.943 + 325.492i 0.647185 + 0.344801i
\(945\) 0 0
\(946\) 128.453 + 463.559i 0.135785 + 0.490020i
\(947\) 1148.77i 1.21306i 0.795059 + 0.606532i \(0.207440\pi\)
−0.795059 + 0.606532i \(0.792560\pi\)
\(948\) 0 0
\(949\) −2237.80 −2.35806
\(950\) 0 0
\(951\) 0 0
\(952\) 837.082 881.006i 0.879288 0.925427i
\(953\) −911.785 −0.956753 −0.478376 0.878155i \(-0.658774\pi\)
−0.478376 + 0.878155i \(0.658774\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −551.283 918.348i −0.576656 0.960615i
\(957\) 0 0
\(958\) 1020.54 282.792i 1.06528 0.295190i
\(959\) 3046.29i 3.17653i
\(960\) 0 0
\(961\) 693.788 0.721944
\(962\) −390.048 1407.60i −0.405455 1.46320i
\(963\) 0 0
\(964\) 1034.56 621.045i 1.07320 0.644237i
\(965\) 0 0
\(966\) 0 0
\(967\) 311.565i 0.322197i −0.986938 0.161099i \(-0.948496\pi\)
0.986938 0.161099i \(-0.0515037\pi\)
\(968\) −524.048 497.921i −0.541372 0.514381i
\(969\) 0 0
\(970\) 0 0
\(971\) 370.530i 0.381596i −0.981629 0.190798i \(-0.938892\pi\)
0.981629 0.190798i \(-0.0611075\pi\)
\(972\) 0 0
\(973\) 945.141 0.971368
\(974\) −1697.63 + 470.417i −1.74295 + 0.482974i
\(975\) 0 0
\(976\) 427.077 801.615i 0.437578 0.821327i
\(977\) −1402.53 −1.43555 −0.717773 0.696278i \(-0.754838\pi\)
−0.717773 + 0.696278i \(0.754838\pi\)
\(978\) 0 0
\(979\) 797.746i 0.814858i
\(980\) 0 0
\(981\) 0 0
\(982\) 166.615 46.1693i 0.169669 0.0470156i
\(983\) 988.944i 1.00605i −0.864273 0.503023i \(-0.832221\pi\)
0.864273 0.503023i \(-0.167779\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.6171 + 67.1849i 0.0188814 + 0.0681388i
\(987\) 0 0
\(988\) 405.576 + 675.624i 0.410502 + 0.683830i
\(989\) 82.6603 0.0835797
\(990\) 0 0
\(991\) 90.1483i 0.0909670i −0.998965 0.0454835i \(-0.985517\pi\)
0.998965 0.0454835i \(-0.0144828\pi\)
\(992\) 113.745 510.575i 0.114663 0.514693i
\(993\) 0 0
\(994\) −253.605 915.205i −0.255136 0.920729i
\(995\) 0 0
\(996\) 0 0
\(997\) −655.605 −0.657578 −0.328789 0.944403i \(-0.606640\pi\)
−0.328789 + 0.944403i \(0.606640\pi\)
\(998\) −1685.56 + 467.071i −1.68893 + 0.468007i
\(999\) 0 0
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 900.3.c.n.451.3 8
3.2 odd 2 300.3.c.g.151.6 yes 8
4.3 odd 2 inner 900.3.c.n.451.4 8
5.2 odd 4 900.3.f.h.199.13 16
5.3 odd 4 900.3.f.h.199.4 16
5.4 even 2 900.3.c.t.451.6 8
12.11 even 2 300.3.c.g.151.5 yes 8
15.2 even 4 300.3.f.c.199.4 16
15.8 even 4 300.3.f.c.199.13 16
15.14 odd 2 300.3.c.e.151.3 8
20.3 even 4 900.3.f.h.199.14 16
20.7 even 4 900.3.f.h.199.3 16
20.19 odd 2 900.3.c.t.451.5 8
60.23 odd 4 300.3.f.c.199.3 16
60.47 odd 4 300.3.f.c.199.14 16
60.59 even 2 300.3.c.e.151.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.c.e.151.3 8 15.14 odd 2
300.3.c.e.151.4 yes 8 60.59 even 2
300.3.c.g.151.5 yes 8 12.11 even 2
300.3.c.g.151.6 yes 8 3.2 odd 2
300.3.f.c.199.3 16 60.23 odd 4
300.3.f.c.199.4 16 15.2 even 4
300.3.f.c.199.13 16 15.8 even 4
300.3.f.c.199.14 16 60.47 odd 4
900.3.c.n.451.3 8 1.1 even 1 trivial
900.3.c.n.451.4 8 4.3 odd 2 inner
900.3.c.t.451.5 8 20.19 odd 2
900.3.c.t.451.6 8 5.4 even 2
900.3.f.h.199.3 16 20.7 even 4
900.3.f.h.199.4 16 5.3 odd 4
900.3.f.h.199.13 16 5.2 odd 4
900.3.f.h.199.14 16 20.3 even 4