Properties

Label 3150.2.bf.f.1601.8
Level $3150$
Weight $2$
Character 3150.1601
Analytic conductor $25.153$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1601.8
Character \(\chi\) \(=\) 3150.1601
Dual form 3150.2.bf.f.1151.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.29693 + 2.30608i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.29693 + 2.30608i) q^{7} -1.00000i q^{8} +(-1.11120 + 0.641550i) q^{11} +6.14864i q^{13} +(0.0298666 - 2.64558i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.26128 - 5.64871i) q^{17} +(-5.22868 - 3.01878i) q^{19} +1.28310 q^{22} +(2.49343 + 1.43958i) q^{23} +(3.07432 - 5.32488i) q^{26} +(-1.34866 + 2.27621i) q^{28} +1.35052i q^{29} +(-7.49558 + 4.32758i) q^{31} +(0.866025 - 0.500000i) q^{32} +6.52257i q^{34} +(4.76690 - 8.25652i) q^{37} +(3.01878 + 5.22868i) q^{38} -8.71759 q^{41} -5.35859 q^{43} +(-1.11120 - 0.641550i) q^{44} +(-1.43958 - 2.49343i) q^{46} +(-0.403086 + 0.698165i) q^{47} +(-3.63597 + 5.98162i) q^{49} +(-5.32488 + 3.07432i) q^{52} +(5.77488 - 3.33413i) q^{53} +(2.30608 - 1.29693i) q^{56} +(0.675260 - 1.16959i) q^{58} +(-0.798110 - 1.38237i) q^{59} +(-5.50239 - 3.17681i) q^{61} +8.65515 q^{62} -1.00000 q^{64} +(-2.69731 - 4.67188i) q^{67} +(3.26128 - 5.64871i) q^{68} -15.6787i q^{71} +(10.7532 - 6.20837i) q^{73} +(-8.25652 + 4.76690i) q^{74} -6.03756i q^{76} +(-2.92060 - 1.73046i) q^{77} +(5.59093 - 9.68377i) q^{79} +(7.54966 + 4.35880i) q^{82} -3.74493 q^{83} +(4.64067 + 2.67929i) q^{86} +(0.641550 + 1.11120i) q^{88} +(1.81971 - 3.15183i) q^{89} +(-14.1792 + 7.97433i) q^{91} +2.87917i q^{92} +(0.698165 - 0.403086i) q^{94} +8.76818i q^{97} +(6.13965 - 3.36225i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{4} - 16 q^{16} - 48 q^{19} + 24 q^{31} - 16 q^{46} + 56 q^{49} + 48 q^{61} - 32 q^{64} - 8 q^{79} - 56 q^{91} + 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 0.500000i −0.612372 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.29693 + 2.30608i 0.490192 + 0.871614i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.11120 + 0.641550i −0.335038 + 0.193435i −0.658076 0.752952i \(-0.728629\pi\)
0.323037 + 0.946386i \(0.395296\pi\)
\(12\) 0 0
\(13\) 6.14864i 1.70533i 0.522462 + 0.852663i \(0.325014\pi\)
−0.522462 + 0.852663i \(0.674986\pi\)
\(14\) 0.0298666 2.64558i 0.00798219 0.707062i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −3.26128 5.64871i −0.790977 1.37001i −0.925362 0.379084i \(-0.876239\pi\)
0.134385 0.990929i \(-0.457094\pi\)
\(18\) 0 0
\(19\) −5.22868 3.01878i −1.19954 0.692555i −0.239088 0.970998i \(-0.576848\pi\)
−0.960453 + 0.278443i \(0.910182\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.28310 0.273558
\(23\) 2.49343 + 1.43958i 0.519917 + 0.300174i 0.736901 0.676001i \(-0.236289\pi\)
−0.216984 + 0.976175i \(0.569622\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.07432 5.32488i 0.602923 1.04429i
\(27\) 0 0
\(28\) −1.34866 + 2.27621i −0.254872 + 0.430163i
\(29\) 1.35052i 0.250785i 0.992107 + 0.125393i \(0.0400191\pi\)
−0.992107 + 0.125393i \(0.959981\pi\)
\(30\) 0 0
\(31\) −7.49558 + 4.32758i −1.34625 + 0.777255i −0.987716 0.156263i \(-0.950055\pi\)
−0.358530 + 0.933518i \(0.616722\pi\)
\(32\) 0.866025 0.500000i 0.153093 0.0883883i
\(33\) 0 0
\(34\) 6.52257i 1.11861i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.76690 8.25652i 0.783674 1.35736i −0.146114 0.989268i \(-0.546677\pi\)
0.929788 0.368095i \(-0.119990\pi\)
\(38\) 3.01878 + 5.22868i 0.489710 + 0.848203i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.71759 −1.36146 −0.680730 0.732535i \(-0.738337\pi\)
−0.680730 + 0.732535i \(0.738337\pi\)
\(42\) 0 0
\(43\) −5.35859 −0.817177 −0.408588 0.912719i \(-0.633979\pi\)
−0.408588 + 0.912719i \(0.633979\pi\)
\(44\) −1.11120 0.641550i −0.167519 0.0967173i
\(45\) 0 0
\(46\) −1.43958 2.49343i −0.212255 0.367637i
\(47\) −0.403086 + 0.698165i −0.0587961 + 0.101838i −0.893925 0.448216i \(-0.852060\pi\)
0.835129 + 0.550054i \(0.185393\pi\)
\(48\) 0 0
\(49\) −3.63597 + 5.98162i −0.519424 + 0.854517i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.32488 + 3.07432i −0.738427 + 0.426331i
\(53\) 5.77488 3.33413i 0.793241 0.457978i −0.0478611 0.998854i \(-0.515240\pi\)
0.841102 + 0.540876i \(0.181907\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.30608 1.29693i 0.308162 0.173309i
\(57\) 0 0
\(58\) 0.675260 1.16959i 0.0886660 0.153574i
\(59\) −0.798110 1.38237i −0.103905 0.179969i 0.809385 0.587278i \(-0.199800\pi\)
−0.913290 + 0.407309i \(0.866467\pi\)
\(60\) 0 0
\(61\) −5.50239 3.17681i −0.704509 0.406748i 0.104516 0.994523i \(-0.466671\pi\)
−0.809025 + 0.587775i \(0.800004\pi\)
\(62\) 8.65515 1.09921
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.69731 4.67188i −0.329529 0.570761i 0.652889 0.757453i \(-0.273557\pi\)
−0.982418 + 0.186692i \(0.940223\pi\)
\(68\) 3.26128 5.64871i 0.395489 0.685006i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6787i 1.86072i −0.366650 0.930359i \(-0.619495\pi\)
0.366650 0.930359i \(-0.380505\pi\)
\(72\) 0 0
\(73\) 10.7532 6.20837i 1.25857 0.726635i 0.285772 0.958298i \(-0.407750\pi\)
0.972796 + 0.231663i \(0.0744166\pi\)
\(74\) −8.25652 + 4.76690i −0.959801 + 0.554141i
\(75\) 0 0
\(76\) 6.03756i 0.692555i
\(77\) −2.92060 1.73046i −0.332834 0.197204i
\(78\) 0 0
\(79\) 5.59093 9.68377i 0.629028 1.08951i −0.358719 0.933446i \(-0.616786\pi\)
0.987747 0.156064i \(-0.0498805\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.54966 + 4.35880i 0.833720 + 0.481349i
\(83\) −3.74493 −0.411059 −0.205530 0.978651i \(-0.565892\pi\)
−0.205530 + 0.978651i \(0.565892\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.64067 + 2.67929i 0.500417 + 0.288916i
\(87\) 0 0
\(88\) 0.641550 + 1.11120i 0.0683894 + 0.118454i
\(89\) 1.81971 3.15183i 0.192889 0.334093i −0.753318 0.657657i \(-0.771548\pi\)
0.946206 + 0.323564i \(0.104881\pi\)
\(90\) 0 0
\(91\) −14.1792 + 7.97433i −1.48639 + 0.835937i
\(92\) 2.87917i 0.300174i
\(93\) 0 0
\(94\) 0.698165 0.403086i 0.0720102 0.0415751i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.76818i 0.890274i 0.895463 + 0.445137i \(0.146845\pi\)
−0.895463 + 0.445137i \(0.853155\pi\)
\(98\) 6.13965 3.36225i 0.620198 0.339639i
\(99\) 0 0
\(100\) 0 0
\(101\) −0.853270 1.47791i −0.0849035 0.147057i 0.820447 0.571723i \(-0.193725\pi\)
−0.905350 + 0.424666i \(0.860392\pi\)
\(102\) 0 0
\(103\) 8.97583 + 5.18220i 0.884415 + 0.510617i 0.872112 0.489307i \(-0.162750\pi\)
0.0123035 + 0.999924i \(0.496084\pi\)
\(104\) 6.14864 0.602923
\(105\) 0 0
\(106\) −6.66826 −0.647679
\(107\) 9.97106 + 5.75680i 0.963939 + 0.556531i 0.897383 0.441252i \(-0.145466\pi\)
0.0665560 + 0.997783i \(0.478799\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.73205i 0.0957826 + 0.165900i 0.909935 0.414751i \(-0.136131\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.64558 0.0298666i −0.249984 0.00282213i
\(113\) 13.3875i 1.25939i −0.776843 0.629695i \(-0.783180\pi\)
0.776843 0.629695i \(-0.216820\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.16959 + 0.675260i −0.108593 + 0.0626963i
\(117\) 0 0
\(118\) 1.59622i 0.146944i
\(119\) 8.79670 14.8467i 0.806392 1.36100i
\(120\) 0 0
\(121\) −4.67683 + 8.10050i −0.425166 + 0.736409i
\(122\) 3.17681 + 5.50239i 0.287615 + 0.498163i
\(123\) 0 0
\(124\) −7.49558 4.32758i −0.673123 0.388628i
\(125\) 0 0
\(126\) 0 0
\(127\) −5.65313 −0.501634 −0.250817 0.968035i \(-0.580699\pi\)
−0.250817 + 0.968035i \(0.580699\pi\)
\(128\) 0.866025 + 0.500000i 0.0765466 + 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.23366 10.7970i 0.544638 0.943340i −0.453992 0.891006i \(-0.650001\pi\)
0.998630 0.0523344i \(-0.0166662\pi\)
\(132\) 0 0
\(133\) 0.180321 15.9729i 0.0156358 1.38502i
\(134\) 5.39463i 0.466025i
\(135\) 0 0
\(136\) −5.64871 + 3.26128i −0.484373 + 0.279653i
\(137\) −0.0441401 + 0.0254843i −0.00377114 + 0.00217727i −0.501884 0.864935i \(-0.667360\pi\)
0.498113 + 0.867112i \(0.334026\pi\)
\(138\) 0 0
\(139\) 1.05069i 0.0891184i 0.999007 + 0.0445592i \(0.0141883\pi\)
−0.999007 + 0.0445592i \(0.985812\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.83934 + 13.5781i −0.657863 + 1.13945i
\(143\) −3.94466 6.83235i −0.329869 0.571350i
\(144\) 0 0
\(145\) 0 0
\(146\) −12.4167 −1.02762
\(147\) 0 0
\(148\) 9.53381 0.783674
\(149\) −7.12137 4.11153i −0.583406 0.336829i 0.179080 0.983835i \(-0.442688\pi\)
−0.762486 + 0.647005i \(0.776021\pi\)
\(150\) 0 0
\(151\) 2.13357 + 3.69546i 0.173628 + 0.300732i 0.939686 0.342040i \(-0.111118\pi\)
−0.766058 + 0.642772i \(0.777784\pi\)
\(152\) −3.01878 + 5.22868i −0.244855 + 0.424102i
\(153\) 0 0
\(154\) 1.66409 + 2.95892i 0.134096 + 0.238437i
\(155\) 0 0
\(156\) 0 0
\(157\) 4.18466 2.41601i 0.333972 0.192819i −0.323631 0.946183i \(-0.604904\pi\)
0.657603 + 0.753364i \(0.271570\pi\)
\(158\) −9.68377 + 5.59093i −0.770399 + 0.444790i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.0859910 + 7.61708i −0.00677704 + 0.600310i
\(162\) 0 0
\(163\) −5.37661 + 9.31256i −0.421128 + 0.729416i −0.996050 0.0887927i \(-0.971699\pi\)
0.574922 + 0.818208i \(0.305032\pi\)
\(164\) −4.35880 7.54966i −0.340365 0.589529i
\(165\) 0 0
\(166\) 3.24320 + 1.87246i 0.251721 + 0.145331i
\(167\) −18.9267 −1.46459 −0.732295 0.680988i \(-0.761551\pi\)
−0.732295 + 0.680988i \(0.761551\pi\)
\(168\) 0 0
\(169\) −24.8057 −1.90813
\(170\) 0 0
\(171\) 0 0
\(172\) −2.67929 4.64067i −0.204294 0.353848i
\(173\) −5.92176 + 10.2568i −0.450223 + 0.779810i −0.998400 0.0565531i \(-0.981989\pi\)
0.548176 + 0.836363i \(0.315322\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.28310i 0.0967173i
\(177\) 0 0
\(178\) −3.15183 + 1.81971i −0.236239 + 0.136393i
\(179\) −3.27843 + 1.89280i −0.245041 + 0.141475i −0.617492 0.786577i \(-0.711851\pi\)
0.372450 + 0.928052i \(0.378518\pi\)
\(180\) 0 0
\(181\) 19.0033i 1.41251i −0.707960 0.706253i \(-0.750384\pi\)
0.707960 0.706253i \(-0.249616\pi\)
\(182\) 16.2667 + 0.183639i 1.20577 + 0.0136122i
\(183\) 0 0
\(184\) 1.43958 2.49343i 0.106128 0.183818i
\(185\) 0 0
\(186\) 0 0
\(187\) 7.24785 + 4.18455i 0.530016 + 0.306005i
\(188\) −0.806172 −0.0587961
\(189\) 0 0
\(190\) 0 0
\(191\) 2.44949 + 1.41421i 0.177239 + 0.102329i 0.585995 0.810315i \(-0.300704\pi\)
−0.408756 + 0.912644i \(0.634037\pi\)
\(192\) 0 0
\(193\) 1.76946 + 3.06479i 0.127368 + 0.220608i 0.922656 0.385623i \(-0.126014\pi\)
−0.795288 + 0.606232i \(0.792680\pi\)
\(194\) 4.38409 7.59347i 0.314759 0.545179i
\(195\) 0 0
\(196\) −6.99822 0.158029i −0.499873 0.0112878i
\(197\) 2.36728i 0.168662i −0.996438 0.0843308i \(-0.973125\pi\)
0.996438 0.0843308i \(-0.0268752\pi\)
\(198\) 0 0
\(199\) 3.00000 1.73205i 0.212664 0.122782i −0.389885 0.920864i \(-0.627485\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.70654i 0.120072i
\(203\) −3.11440 + 1.75153i −0.218588 + 0.122933i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.18220 8.97583i −0.361061 0.625376i
\(207\) 0 0
\(208\) −5.32488 3.07432i −0.369214 0.213166i
\(209\) 7.74679 0.535856
\(210\) 0 0
\(211\) −14.3620 −0.988721 −0.494361 0.869257i \(-0.664598\pi\)
−0.494361 + 0.869257i \(0.664598\pi\)
\(212\) 5.77488 + 3.33413i 0.396621 + 0.228989i
\(213\) 0 0
\(214\) −5.75680 9.97106i −0.393527 0.681608i
\(215\) 0 0
\(216\) 0 0
\(217\) −19.7009 11.6728i −1.33739 0.792403i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) 34.7319 20.0524i 2.33632 1.34887i
\(222\) 0 0
\(223\) 13.5975i 0.910557i −0.890349 0.455279i \(-0.849540\pi\)
0.890349 0.455279i \(-0.150460\pi\)
\(224\) 2.27621 + 1.34866i 0.152086 + 0.0901109i
\(225\) 0 0
\(226\) −6.69375 + 11.5939i −0.445261 + 0.771215i
\(227\) −6.17797 10.7006i −0.410046 0.710221i 0.584848 0.811143i \(-0.301154\pi\)
−0.994894 + 0.100922i \(0.967821\pi\)
\(228\) 0 0
\(229\) 2.09008 + 1.20671i 0.138116 + 0.0797414i 0.567466 0.823397i \(-0.307924\pi\)
−0.429350 + 0.903138i \(0.641257\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.35052 0.0886660
\(233\) −11.1506 6.43780i −0.730500 0.421754i 0.0881051 0.996111i \(-0.471919\pi\)
−0.818605 + 0.574357i \(0.805252\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.798110 1.38237i 0.0519525 0.0899844i
\(237\) 0 0
\(238\) −15.0415 + 8.45929i −0.974997 + 0.548334i
\(239\) 27.2546i 1.76296i 0.472226 + 0.881478i \(0.343451\pi\)
−0.472226 + 0.881478i \(0.656549\pi\)
\(240\) 0 0
\(241\) 2.26690 1.30880i 0.146024 0.0843070i −0.425208 0.905096i \(-0.639799\pi\)
0.571232 + 0.820789i \(0.306466\pi\)
\(242\) 8.10050 4.67683i 0.520720 0.300638i
\(243\) 0 0
\(244\) 6.35361i 0.406748i
\(245\) 0 0
\(246\) 0 0
\(247\) 18.5614 32.1492i 1.18103 2.04561i
\(248\) 4.32758 + 7.49558i 0.274801 + 0.475970i
\(249\) 0 0
\(250\) 0 0
\(251\) −26.7173 −1.68638 −0.843190 0.537615i \(-0.819325\pi\)
−0.843190 + 0.537615i \(0.819325\pi\)
\(252\) 0 0
\(253\) −3.69426 −0.232256
\(254\) 4.89575 + 2.82656i 0.307187 + 0.177354i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) −5.28063 + 9.14632i −0.329397 + 0.570532i −0.982392 0.186830i \(-0.940179\pi\)
0.652996 + 0.757362i \(0.273512\pi\)
\(258\) 0 0
\(259\) 25.2225 + 0.284743i 1.56725 + 0.0176930i
\(260\) 0 0
\(261\) 0 0
\(262\) −10.7970 + 6.23366i −0.667042 + 0.385117i
\(263\) 7.21550 4.16587i 0.444927 0.256879i −0.260759 0.965404i \(-0.583973\pi\)
0.705685 + 0.708526i \(0.250639\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.14259 + 13.7427i −0.499254 + 0.842621i
\(267\) 0 0
\(268\) 2.69731 4.67188i 0.164765 0.285381i
\(269\) −7.14171 12.3698i −0.435438 0.754201i 0.561893 0.827210i \(-0.310073\pi\)
−0.997331 + 0.0730091i \(0.976740\pi\)
\(270\) 0 0
\(271\) −6.58566 3.80223i −0.400050 0.230969i 0.286456 0.958094i \(-0.407523\pi\)
−0.686506 + 0.727125i \(0.740856\pi\)
\(272\) 6.52257 0.395489
\(273\) 0 0
\(274\) 0.0509686 0.00307912
\(275\) 0 0
\(276\) 0 0
\(277\) 6.02002 + 10.4270i 0.361708 + 0.626497i 0.988242 0.152897i \(-0.0488604\pi\)
−0.626534 + 0.779394i \(0.715527\pi\)
\(278\) 0.525345 0.909924i 0.0315081 0.0545736i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7879i 0.762861i 0.924397 + 0.381431i \(0.124569\pi\)
−0.924397 + 0.381431i \(0.875431\pi\)
\(282\) 0 0
\(283\) −13.4685 + 7.77607i −0.800622 + 0.462239i −0.843689 0.536833i \(-0.819621\pi\)
0.0430666 + 0.999072i \(0.486287\pi\)
\(284\) 13.5781 7.83934i 0.805715 0.465180i
\(285\) 0 0
\(286\) 7.88931i 0.466505i
\(287\) −11.3061 20.1034i −0.667376 1.18667i
\(288\) 0 0
\(289\) −12.7719 + 22.1216i −0.751290 + 1.30127i
\(290\) 0 0
\(291\) 0 0
\(292\) 10.7532 + 6.20837i 0.629284 + 0.363317i
\(293\) −11.5536 −0.674967 −0.337483 0.941331i \(-0.609576\pi\)
−0.337483 + 0.941331i \(0.609576\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.25652 4.76690i −0.479900 0.277071i
\(297\) 0 0
\(298\) 4.11153 + 7.12137i 0.238174 + 0.412530i
\(299\) −8.85148 + 15.3312i −0.511894 + 0.886627i
\(300\) 0 0
\(301\) −6.94969 12.3573i −0.400573 0.712263i
\(302\) 4.26715i 0.245547i
\(303\) 0 0
\(304\) 5.22868 3.01878i 0.299885 0.173139i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.57617i 0.147030i 0.997294 + 0.0735150i \(0.0234217\pi\)
−0.997294 + 0.0735150i \(0.976578\pi\)
\(308\) 0.0383219 3.39455i 0.00218359 0.193422i
\(309\) 0 0
\(310\) 0 0
\(311\) −12.3570 21.4030i −0.700702 1.21365i −0.968220 0.250098i \(-0.919537\pi\)
0.267519 0.963553i \(-0.413796\pi\)
\(312\) 0 0
\(313\) 7.79573 + 4.50087i 0.440641 + 0.254404i 0.703869 0.710329i \(-0.251454\pi\)
−0.263229 + 0.964733i \(0.584787\pi\)
\(314\) −4.83203 −0.272687
\(315\) 0 0
\(316\) 11.1819 0.629028
\(317\) −20.7031 11.9529i −1.16280 0.671344i −0.210828 0.977523i \(-0.567616\pi\)
−0.951974 + 0.306180i \(0.900949\pi\)
\(318\) 0 0
\(319\) −0.866426 1.50069i −0.0485106 0.0840227i
\(320\) 0 0
\(321\) 0 0
\(322\) 3.88301 6.55359i 0.216392 0.365217i
\(323\) 39.3804i 2.19118i
\(324\) 0 0
\(325\) 0 0
\(326\) 9.31256 5.37661i 0.515775 0.297783i
\(327\) 0 0
\(328\) 8.71759i 0.481349i
\(329\) −2.13279 0.0240776i −0.117585 0.00132744i
\(330\) 0 0
\(331\) −2.18364 + 3.78217i −0.120024 + 0.207887i −0.919777 0.392442i \(-0.871630\pi\)
0.799753 + 0.600329i \(0.204964\pi\)
\(332\) −1.87246 3.24320i −0.102765 0.177994i
\(333\) 0 0
\(334\) 16.3910 + 9.46334i 0.896874 + 0.517811i
\(335\) 0 0
\(336\) 0 0
\(337\) 19.5159 1.06310 0.531549 0.847028i \(-0.321610\pi\)
0.531549 + 0.847028i \(0.321610\pi\)
\(338\) 21.4824 + 12.4029i 1.16849 + 0.674627i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.55271 9.61758i 0.300696 0.520821i
\(342\) 0 0
\(343\) −18.5096 0.627092i −0.999427 0.0338598i
\(344\) 5.35859i 0.288916i
\(345\) 0 0
\(346\) 10.2568 5.92176i 0.551409 0.318356i
\(347\) −23.3088 + 13.4574i −1.25128 + 0.722429i −0.971364 0.237595i \(-0.923641\pi\)
−0.279919 + 0.960024i \(0.590308\pi\)
\(348\) 0 0
\(349\) 10.3821i 0.555741i 0.960619 + 0.277870i \(0.0896286\pi\)
−0.960619 + 0.277870i \(0.910371\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.641550 + 1.11120i −0.0341947 + 0.0592270i
\(353\) 0.0844756 + 0.146316i 0.00449618 + 0.00778761i 0.868265 0.496101i \(-0.165235\pi\)
−0.863769 + 0.503889i \(0.831902\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.63941 0.192889
\(357\) 0 0
\(358\) 3.78561 0.200075
\(359\) −32.6198 18.8330i −1.72161 0.993970i −0.915631 0.402021i \(-0.868308\pi\)
−0.805975 0.591949i \(-0.798359\pi\)
\(360\) 0 0
\(361\) 8.72604 + 15.1139i 0.459265 + 0.795471i
\(362\) −9.50166 + 16.4574i −0.499396 + 0.864979i
\(363\) 0 0
\(364\) −13.9956 8.29240i −0.733568 0.434640i
\(365\) 0 0
\(366\) 0 0
\(367\) −13.1253 + 7.57787i −0.685133 + 0.395562i −0.801786 0.597611i \(-0.796117\pi\)
0.116653 + 0.993173i \(0.462783\pi\)
\(368\) −2.49343 + 1.43958i −0.129979 + 0.0750435i
\(369\) 0 0
\(370\) 0 0
\(371\) 15.1784 + 8.99319i 0.788021 + 0.466903i
\(372\) 0 0
\(373\) 6.02002 10.4270i 0.311705 0.539889i −0.667027 0.745034i \(-0.732433\pi\)
0.978732 + 0.205145i \(0.0657667\pi\)
\(374\) −4.18455 7.24785i −0.216378 0.374778i
\(375\) 0 0
\(376\) 0.698165 + 0.403086i 0.0360051 + 0.0207876i
\(377\) −8.30386 −0.427671
\(378\) 0 0
\(379\) −18.0918 −0.929312 −0.464656 0.885491i \(-0.653822\pi\)
−0.464656 + 0.885491i \(0.653822\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.41421 2.44949i −0.0723575 0.125327i
\(383\) −14.5769 + 25.2480i −0.744846 + 1.29011i 0.205421 + 0.978674i \(0.434144\pi\)
−0.950267 + 0.311437i \(0.899190\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.53892i 0.180126i
\(387\) 0 0
\(388\) −7.59347 + 4.38409i −0.385500 + 0.222568i
\(389\) 17.0297 9.83207i 0.863438 0.498506i −0.00172432 0.999999i \(-0.500549\pi\)
0.865162 + 0.501493i \(0.167216\pi\)
\(390\) 0 0
\(391\) 18.7796i 0.949723i
\(392\) 5.98162 + 3.63597i 0.302117 + 0.183644i
\(393\) 0 0
\(394\) −1.18364 + 2.05012i −0.0596309 + 0.103284i
\(395\) 0 0
\(396\) 0 0
\(397\) −4.92295 2.84227i −0.247076 0.142649i 0.371349 0.928493i \(-0.378895\pi\)
−0.618425 + 0.785844i \(0.712229\pi\)
\(398\) −3.46410 −0.173640
\(399\) 0 0
\(400\) 0 0
\(401\) 8.84787 + 5.10832i 0.441842 + 0.255097i 0.704378 0.709825i \(-0.251226\pi\)
−0.262537 + 0.964922i \(0.584559\pi\)
\(402\) 0 0
\(403\) −26.6087 46.0876i −1.32547 2.29579i
\(404\) 0.853270 1.47791i 0.0424518 0.0735286i
\(405\) 0 0
\(406\) 3.57291 + 0.0403355i 0.177321 + 0.00200182i
\(407\) 12.2328i 0.606359i
\(408\) 0 0
\(409\) −12.0969 + 6.98414i −0.598153 + 0.345344i −0.768314 0.640073i \(-0.778904\pi\)
0.170162 + 0.985416i \(0.445571\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.3644i 0.510617i
\(413\) 2.15275 3.63333i 0.105930 0.178784i
\(414\) 0 0
\(415\) 0 0
\(416\) 3.07432 + 5.32488i 0.150731 + 0.261074i
\(417\) 0 0
\(418\) −6.70891 3.87339i −0.328144 0.189454i
\(419\) 11.3181 0.552924 0.276462 0.961025i \(-0.410838\pi\)
0.276462 + 0.961025i \(0.410838\pi\)
\(420\) 0 0
\(421\) −13.9099 −0.677928 −0.338964 0.940799i \(-0.610077\pi\)
−0.338964 + 0.940799i \(0.610077\pi\)
\(422\) 12.4379 + 7.18100i 0.605466 + 0.349566i
\(423\) 0 0
\(424\) −3.33413 5.77488i −0.161920 0.280453i
\(425\) 0 0
\(426\) 0 0
\(427\) 0.189761 16.8090i 0.00918318 0.813445i
\(428\) 11.5136i 0.556531i
\(429\) 0 0
\(430\) 0 0
\(431\) −22.5947 + 13.0451i −1.08835 + 0.628359i −0.933137 0.359522i \(-0.882940\pi\)
−0.155213 + 0.987881i \(0.549606\pi\)
\(432\) 0 0
\(433\) 8.42614i 0.404935i 0.979289 + 0.202467i \(0.0648960\pi\)
−0.979289 + 0.202467i \(0.935104\pi\)
\(434\) 11.2251 + 19.9594i 0.538822 + 0.958083i
\(435\) 0 0
\(436\) −1.00000 + 1.73205i −0.0478913 + 0.0829502i
\(437\) −8.69157 15.0542i −0.415774 0.720142i
\(438\) 0 0
\(439\) 23.9529 + 13.8292i 1.14321 + 0.660033i 0.947224 0.320573i \(-0.103875\pi\)
0.195987 + 0.980606i \(0.437209\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −40.1049 −1.90759
\(443\) 2.04142 + 1.17861i 0.0969906 + 0.0559975i 0.547711 0.836668i \(-0.315499\pi\)
−0.450720 + 0.892665i \(0.648833\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −6.79876 + 11.7758i −0.321931 + 0.557600i
\(447\) 0 0
\(448\) −1.29693 2.30608i −0.0612740 0.108952i
\(449\) 12.8021i 0.604169i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976824\pi\)
\(450\) 0 0
\(451\) 9.68696 5.59277i 0.456141 0.263353i
\(452\) 11.5939 6.69375i 0.545332 0.314847i
\(453\) 0 0
\(454\) 12.3559i 0.579893i
\(455\) 0 0
\(456\) 0 0
\(457\) 17.2621 29.8989i 0.807488 1.39861i −0.107111 0.994247i \(-0.534160\pi\)
0.914599 0.404363i \(-0.132507\pi\)
\(458\) −1.20671 2.09008i −0.0563857 0.0976628i
\(459\) 0 0
\(460\) 0 0
\(461\) −20.2692 −0.944032 −0.472016 0.881590i \(-0.656474\pi\)
−0.472016 + 0.881590i \(0.656474\pi\)
\(462\) 0 0
\(463\) 13.1246 0.609952 0.304976 0.952360i \(-0.401352\pi\)
0.304976 + 0.952360i \(0.401352\pi\)
\(464\) −1.16959 0.675260i −0.0542966 0.0313482i
\(465\) 0 0
\(466\) 6.43780 + 11.1506i 0.298225 + 0.516541i
\(467\) −0.765836 + 1.32647i −0.0354387 + 0.0613816i −0.883201 0.468995i \(-0.844616\pi\)
0.847762 + 0.530377i \(0.177950\pi\)
\(468\) 0 0
\(469\) 7.27550 12.2793i 0.335951 0.567005i
\(470\) 0 0
\(471\) 0 0
\(472\) −1.38237 + 0.798110i −0.0636286 + 0.0367360i
\(473\) 5.95444 3.43780i 0.273786 0.158070i
\(474\) 0 0
\(475\) 0 0
\(476\) 17.2560 + 0.194807i 0.790927 + 0.00892896i
\(477\) 0 0
\(478\) 13.6273 23.6032i 0.623299 1.07959i
\(479\) 14.6585 + 25.3893i 0.669764 + 1.16007i 0.977970 + 0.208746i \(0.0669382\pi\)
−0.308205 + 0.951320i \(0.599728\pi\)
\(480\) 0 0
\(481\) 50.7663 + 29.3100i 2.31475 + 1.33642i
\(482\) −2.61759 −0.119228
\(483\) 0 0
\(484\) −9.35366 −0.425166
\(485\) 0 0
\(486\) 0 0
\(487\) 4.09770 + 7.09743i 0.185685 + 0.321615i 0.943807 0.330497i \(-0.107216\pi\)
−0.758122 + 0.652112i \(0.773883\pi\)
\(488\) −3.17681 + 5.50239i −0.143807 + 0.249082i
\(489\) 0 0
\(490\) 0 0
\(491\) 40.4383i 1.82495i 0.409129 + 0.912477i \(0.365833\pi\)
−0.409129 + 0.912477i \(0.634167\pi\)
\(492\) 0 0
\(493\) 7.62869 4.40443i 0.343579 0.198366i
\(494\) −32.1492 + 18.5614i −1.44646 + 0.835116i
\(495\) 0 0
\(496\) 8.65515i 0.388628i
\(497\) 36.1562 20.3341i 1.62183 0.912109i
\(498\) 0 0
\(499\) 8.72450 15.1113i 0.390562 0.676474i −0.601961 0.798525i \(-0.705614\pi\)
0.992524 + 0.122051i \(0.0389473\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23.1379 + 13.3586i 1.03269 + 0.596226i
\(503\) −1.91949 −0.0855860 −0.0427930 0.999084i \(-0.513626\pi\)
−0.0427930 + 0.999084i \(0.513626\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.19932 + 1.84713i 0.142227 + 0.0821149i
\(507\) 0 0
\(508\) −2.82656 4.89575i −0.125408 0.217214i
\(509\) −5.54549 + 9.60508i −0.245800 + 0.425737i −0.962356 0.271792i \(-0.912384\pi\)
0.716556 + 0.697529i \(0.245717\pi\)
\(510\) 0 0
\(511\) 28.2631 + 16.7459i 1.25029 + 0.740796i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 9.14632 5.28063i 0.403427 0.232919i
\(515\) 0 0
\(516\) 0 0
\(517\) 1.03440i 0.0454928i
\(518\) −21.7009 12.8578i −0.953484 0.564941i
\(519\) 0 0
\(520\) 0 0
\(521\) 19.8706 + 34.4168i 0.870546 + 1.50783i 0.861433 + 0.507871i \(0.169567\pi\)
0.00911254 + 0.999958i \(0.497099\pi\)
\(522\) 0 0
\(523\) 27.0900 + 15.6404i 1.18456 + 0.683907i 0.957065 0.289872i \(-0.0936129\pi\)
0.227496 + 0.973779i \(0.426946\pi\)
\(524\) 12.4673 0.544638
\(525\) 0 0
\(526\) −8.33174 −0.363281
\(527\) 48.8904 + 28.2269i 2.12970 + 1.22958i
\(528\) 0 0
\(529\) −7.35519 12.7396i −0.319791 0.553894i
\(530\) 0 0
\(531\) 0 0
\(532\) 13.9231 7.83026i 0.603641 0.339485i
\(533\) 53.6013i 2.32173i
\(534\) 0 0
\(535\) 0 0
\(536\) −4.67188 + 2.69731i −0.201795 + 0.116506i
\(537\) 0 0
\(538\) 14.2834i 0.615802i
\(539\) 0.202767 8.97941i 0.00873380 0.386771i
\(540\) 0 0
\(541\) 12.3987 21.4752i 0.533061 0.923290i −0.466193 0.884683i \(-0.654375\pi\)
0.999254 0.0386065i \(-0.0122919\pi\)
\(542\) 3.80223 + 6.58566i 0.163320 + 0.282878i
\(543\) 0 0
\(544\) −5.64871 3.26128i −0.242186 0.139826i
\(545\) 0 0
\(546\) 0 0
\(547\) −40.5380 −1.73328 −0.866640 0.498934i \(-0.833725\pi\)
−0.866640 + 0.498934i \(0.833725\pi\)
\(548\) −0.0441401 0.0254843i −0.00188557 0.00108863i
\(549\) 0 0
\(550\) 0 0
\(551\) 4.07692 7.06144i 0.173683 0.300827i
\(552\) 0 0
\(553\) 29.5825 + 0.333964i 1.25798 + 0.0142016i
\(554\) 12.0400i 0.511532i
\(555\) 0 0
\(556\) −0.909924 + 0.525345i −0.0385894 + 0.0222796i
\(557\) −20.5650 + 11.8732i −0.871367 + 0.503084i −0.867802 0.496910i \(-0.834468\pi\)
−0.00356477 + 0.999994i \(0.501135\pi\)
\(558\) 0 0
\(559\) 32.9480i 1.39355i
\(560\) 0 0
\(561\) 0 0
\(562\) 6.39394 11.0746i 0.269712 0.467155i
\(563\) −4.21434 7.29944i −0.177613 0.307635i 0.763449 0.645868i \(-0.223504\pi\)
−0.941062 + 0.338233i \(0.890171\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 15.5521 0.653705
\(567\) 0 0
\(568\) −15.6787 −0.657863
\(569\) 20.3139 + 11.7283i 0.851605 + 0.491674i 0.861192 0.508280i \(-0.169718\pi\)
−0.00958727 + 0.999954i \(0.503052\pi\)
\(570\) 0 0
\(571\) −7.67946 13.3012i −0.321376 0.556639i 0.659397 0.751795i \(-0.270812\pi\)
−0.980772 + 0.195157i \(0.937479\pi\)
\(572\) 3.94466 6.83235i 0.164934 0.285675i
\(573\) 0 0
\(574\) −0.260365 + 23.0631i −0.0108674 + 0.962636i
\(575\) 0 0
\(576\) 0 0
\(577\) −12.3329 + 7.12041i −0.513426 + 0.296426i −0.734241 0.678889i \(-0.762462\pi\)
0.220815 + 0.975316i \(0.429128\pi\)
\(578\) 22.1216 12.7719i 0.920139 0.531242i
\(579\) 0 0
\(580\) 0 0
\(581\) −4.85690 8.63609i −0.201498 0.358285i
\(582\) 0 0
\(583\) −4.27802 + 7.40975i −0.177178 + 0.306881i
\(584\) −6.20837 10.7532i −0.256904 0.444971i
\(585\) 0 0
\(586\) 10.0057 + 5.77679i 0.413331 + 0.238637i
\(587\) −12.5249 −0.516958 −0.258479 0.966017i \(-0.583221\pi\)
−0.258479 + 0.966017i \(0.583221\pi\)
\(588\) 0 0
\(589\) 52.2560 2.15317
\(590\) 0 0
\(591\) 0 0
\(592\) 4.76690 + 8.25652i 0.195919 + 0.339341i
\(593\) 7.35750 12.7436i 0.302136 0.523316i −0.674483 0.738290i \(-0.735634\pi\)
0.976620 + 0.214975i \(0.0689669\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.22305i 0.336829i
\(597\) 0 0
\(598\) 15.3312 8.85148i 0.626940 0.361964i
\(599\) 23.9304 13.8162i 0.977769 0.564515i 0.0761732 0.997095i \(-0.475730\pi\)
0.901596 + 0.432579i \(0.142396\pi\)
\(600\) 0 0
\(601\) 8.82137i 0.359831i −0.983682 0.179916i \(-0.942418\pi\)
0.983682 0.179916i \(-0.0575825\pi\)
\(602\) −0.160043 + 14.1766i −0.00652286 + 0.577794i
\(603\) 0 0
\(604\) −2.13357 + 3.69546i −0.0868139 + 0.150366i
\(605\) 0 0
\(606\) 0 0
\(607\) 22.9803 + 13.2677i 0.932743 + 0.538519i 0.887678 0.460465i \(-0.152317\pi\)
0.0450649 + 0.998984i \(0.485651\pi\)
\(608\) −6.03756 −0.244855
\(609\) 0 0
\(610\) 0 0
\(611\) −4.29276 2.47843i −0.173667 0.100266i
\(612\) 0 0
\(613\) 0.456842 + 0.791273i 0.0184517 + 0.0319592i 0.875104 0.483935i \(-0.160793\pi\)
−0.856652 + 0.515894i \(0.827460\pi\)
\(614\) 1.28809 2.23103i 0.0519830 0.0900371i
\(615\) 0 0
\(616\) −1.73046 + 2.92060i −0.0697223 + 0.117674i
\(617\) 35.0558i 1.41129i −0.708565 0.705646i \(-0.750657\pi\)
0.708565 0.705646i \(-0.249343\pi\)
\(618\) 0 0
\(619\) −16.5382 + 9.54835i −0.664727 + 0.383781i −0.794076 0.607819i \(-0.792045\pi\)
0.129348 + 0.991599i \(0.458711\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.7140i 0.990942i
\(623\) 9.62837 + 0.108697i 0.385753 + 0.00435485i
\(624\) 0 0
\(625\) 0 0
\(626\) −4.50087 7.79573i −0.179891 0.311580i
\(627\) 0 0
\(628\) 4.18466 + 2.41601i 0.166986 + 0.0964095i
\(629\) −62.1849 −2.47947
\(630\) 0 0
\(631\) 21.5350 0.857296 0.428648 0.903472i \(-0.358990\pi\)
0.428648 + 0.903472i \(0.358990\pi\)
\(632\) −9.68377 5.59093i −0.385200 0.222395i
\(633\) 0 0
\(634\) 11.9529 + 20.7031i 0.474712 + 0.822225i
\(635\) 0 0
\(636\) 0 0
\(637\) −36.7788 22.3562i −1.45723 0.885786i
\(638\) 1.73285i 0.0686043i
\(639\) 0 0
\(640\) 0 0
\(641\) −28.6767 + 16.5565i −1.13266 + 0.653943i −0.944603 0.328214i \(-0.893553\pi\)
−0.188060 + 0.982158i \(0.560220\pi\)
\(642\) 0 0
\(643\) 3.31191i 0.130609i −0.997865 0.0653044i \(-0.979198\pi\)
0.997865 0.0653044i \(-0.0208018\pi\)
\(644\) −6.63958 + 3.73407i −0.261636 + 0.147143i
\(645\) 0 0
\(646\) 19.6902 34.1044i 0.774700 1.34182i
\(647\) 11.6669 + 20.2077i 0.458674 + 0.794447i 0.998891 0.0470789i \(-0.0149912\pi\)
−0.540217 + 0.841526i \(0.681658\pi\)
\(648\) 0 0
\(649\) 1.77371 + 1.02405i 0.0696244 + 0.0401977i
\(650\) 0 0
\(651\) 0 0
\(652\) −10.7532 −0.421128
\(653\) −15.1211 8.73019i −0.591735 0.341639i 0.174048 0.984737i \(-0.444315\pi\)
−0.765783 + 0.643099i \(0.777649\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.35880 7.54966i 0.170182 0.294765i
\(657\) 0 0
\(658\) 1.83501 + 1.08725i 0.0715363 + 0.0423854i
\(659\) 1.29864i 0.0505877i 0.999680 + 0.0252939i \(0.00805215\pi\)
−0.999680 + 0.0252939i \(0.991948\pi\)
\(660\) 0 0
\(661\) 4.39869 2.53959i 0.171089 0.0987785i −0.412010 0.911179i \(-0.635173\pi\)
0.583099 + 0.812401i \(0.301840\pi\)
\(662\) 3.78217 2.18364i 0.146998 0.0848695i
\(663\) 0 0
\(664\) 3.74493i 0.145331i
\(665\) 0 0
\(666\) 0 0
\(667\) −1.94419 + 3.36743i −0.0752793 + 0.130387i
\(668\) −9.46334 16.3910i −0.366147 0.634186i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.15232 0.314717
\(672\) 0 0
\(673\) −18.4124 −0.709747 −0.354873 0.934914i \(-0.615476\pi\)
−0.354873 + 0.934914i \(0.615476\pi\)
\(674\) −16.9012 9.75794i −0.651012 0.375862i
\(675\) 0 0
\(676\) −12.4029 21.4824i −0.477033 0.826246i
\(677\) −20.6397 + 35.7491i −0.793250 + 1.37395i 0.130695 + 0.991423i \(0.458279\pi\)
−0.923945 + 0.382526i \(0.875054\pi\)
\(678\) 0 0
\(679\) −20.2201 + 11.3717i −0.775976 + 0.436405i
\(680\) 0 0
\(681\) 0 0
\(682\) −9.61758 + 5.55271i −0.368276 + 0.212624i
\(683\) −19.9007 + 11.4896i −0.761477 + 0.439639i −0.829826 0.558022i \(-0.811560\pi\)
0.0683485 + 0.997662i \(0.478227\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.7163 + 9.79790i 0.600050 + 0.374085i
\(687\) 0 0
\(688\) 2.67929 4.64067i 0.102147 0.176924i
\(689\) 20.5004 + 35.5077i 0.781001 + 1.35273i
\(690\) 0 0
\(691\) 11.0048 + 6.35361i 0.418642 + 0.241703i 0.694496 0.719497i \(-0.255627\pi\)
−0.275854 + 0.961199i \(0.588961\pi\)
\(692\) −11.8435 −0.450223
\(693\) 0 0
\(694\) 26.9147 1.02167
\(695\) 0 0
\(696\) 0 0
\(697\) 28.4305 + 49.2431i 1.07688 + 1.86522i
\(698\) 5.19105 8.99116i 0.196484 0.340320i
\(699\) 0 0
\(700\) 0 0
\(701\) 26.5441i 1.00256i 0.865286 + 0.501279i \(0.167137\pi\)
−0.865286 + 0.501279i \(0.832863\pi\)
\(702\) 0 0
\(703\) −49.8492 + 28.7804i −1.88010 + 1.08548i
\(704\) 1.11120 0.641550i 0.0418798 0.0241793i
\(705\) 0 0
\(706\) 0.168951i 0.00635856i
\(707\) 2.30154 3.88444i 0.0865582 0.146089i
\(708\) 0 0
\(709\) 21.1766 36.6789i 0.795303 1.37751i −0.127343 0.991859i \(-0.540645\pi\)
0.922646 0.385647i \(-0.126022\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.15183 1.81971i −0.118120 0.0681964i
\(713\) −24.9196 −0.933248
\(714\) 0 0
\(715\) 0 0
\(716\) −3.27843 1.89280i −0.122521 0.0707374i
\(717\) 0 0
\(718\) 18.8330 + 32.6198i 0.702843 + 1.21736i
\(719\) 11.9507 20.6992i 0.445686 0.771951i −0.552413 0.833570i \(-0.686293\pi\)
0.998100 + 0.0616189i \(0.0196263\pi\)
\(720\) 0 0
\(721\) −0.309550 + 27.4199i −0.0115282 + 1.02117i
\(722\) 17.4521i 0.649499i
\(723\) 0 0
\(724\) 16.4574 9.50166i 0.611633 0.353126i
\(725\) 0 0
\(726\) 0 0
\(727\) 22.4529i 0.832731i 0.909197 + 0.416365i \(0.136696\pi\)
−0.909197 + 0.416365i \(0.863304\pi\)
\(728\) 7.97433 + 14.1792i 0.295548 + 0.525517i
\(729\) 0 0
\(730\) 0 0
\(731\) 17.4759 + 30.2691i 0.646368 + 1.11954i
\(732\) 0 0
\(733\) 23.8547 + 13.7725i 0.881094 + 0.508700i 0.871019 0.491249i \(-0.163460\pi\)
0.0100750 + 0.999949i \(0.496793\pi\)
\(734\) 15.1557 0.559409
\(735\) 0 0
\(736\) 2.87917 0.106128
\(737\) 5.99449 + 3.46092i 0.220810 + 0.127485i
\(738\) 0 0
\(739\) 15.5456 + 26.9258i 0.571856 + 0.990483i 0.996375 + 0.0850646i \(0.0271097\pi\)
−0.424520 + 0.905419i \(0.639557\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.64824 15.3775i −0.317487 0.564526i
\(743\) 1.12610i 0.0413127i −0.999787 0.0206564i \(-0.993424\pi\)
0.999787 0.0206564i \(-0.00657559\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.4270 + 6.02002i −0.381759 + 0.220409i
\(747\) 0 0
\(748\) 8.36910i 0.306005i
\(749\) −0.343872 + 30.4602i −0.0125648 + 1.11299i
\(750\) 0 0
\(751\) 4.77108 8.26375i 0.174099 0.301549i −0.765750 0.643138i \(-0.777632\pi\)
0.939849 + 0.341590i \(0.110965\pi\)
\(752\) −0.403086 0.698165i −0.0146990 0.0254595i
\(753\) 0 0
\(754\) 7.19135 + 4.15193i 0.261894 + 0.151204i
\(755\) 0 0
\(756\) 0 0
\(757\) 33.1687 1.20554 0.602768 0.797917i \(-0.294065\pi\)
0.602768 + 0.797917i \(0.294065\pi\)
\(758\) 15.6679 + 9.04589i 0.569085 + 0.328562i
\(759\) 0 0
\(760\) 0 0
\(761\) 2.17122 3.76067i 0.0787068 0.136324i −0.823985 0.566611i \(-0.808254\pi\)
0.902692 + 0.430287i \(0.141588\pi\)
\(762\) 0 0
\(763\) −2.69731 + 4.55242i −0.0976493 + 0.164809i
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 25.2480 14.5769i 0.912246 0.526685i
\(767\) 8.49967 4.90729i 0.306905 0.177192i
\(768\) 0 0
\(769\) 46.1795i 1.66528i −0.553818 0.832638i \(-0.686830\pi\)
0.553818 0.832638i \(-0.313170\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.76946 + 3.06479i −0.0636842 + 0.110304i
\(773\) −8.98723 15.5663i −0.323248 0.559883i 0.657908 0.753098i \(-0.271442\pi\)
−0.981156 + 0.193216i \(0.938108\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.76818 0.314759
\(777\) 0 0
\(778\) −19.6641 −0.704994
\(779\) 45.5815 + 26.3165i 1.63313 + 0.942886i
\(780\) 0 0
\(781\) 10.0587 + 17.4221i 0.359927 + 0.623412i
\(782\) −9.38978 + 16.2636i −0.335778 + 0.581584i
\(783\) 0 0
\(784\) −3.36225 6.13965i −0.120080 0.219273i
\(785\) 0 0
\(786\) 0 0
\(787\) −11.9873 + 6.92087i −0.427301 + 0.246702i −0.698196 0.715907i \(-0.746014\pi\)
0.270895 + 0.962609i \(0.412680\pi\)
\(788\) 2.05012 1.18364i 0.0730326 0.0421654i
\(789\) 0 0
\(790\) 0 0
\(791\) 30.8726 17.3626i 1.09770 0.617342i
\(792\) 0 0
\(793\) 19.5330 33.8322i 0.693638 1.20142i
\(794\) 2.84227 + 4.92295i 0.100868 + 0.174709i
\(795\) 0 0
\(796\) 3.00000 + 1.73205i 0.106332 + 0.0613909i
\(797\) 7.46138 0.264296 0.132148 0.991230i \(-0.457813\pi\)
0.132148 + 0.991230i \(0.457813\pi\)
\(798\) 0 0
\(799\) 5.25831 0.186026
\(800\) 0 0
\(801\) 0 0
\(802\) −5.10832 8.84787i −0.180381 0.312429i
\(803\) −7.96596 + 13.7974i −0.281113 + 0.486901i
\(804\) 0 0
\(805\) 0 0
\(806\) 53.2174i 1.87450i
\(807\) 0 0
\(808\) −1.47791 + 0.853270i −0.0519926 + 0.0300179i
\(809\) 1.28750 0.743340i 0.0452662 0.0261344i −0.477196 0.878797i \(-0.658347\pi\)
0.522462 + 0.852662i \(0.325014\pi\)
\(810\) 0 0
\(811\) 31.5494i 1.10785i 0.832567 + 0.553925i \(0.186871\pi\)
−0.832567 + 0.553925i \(0.813129\pi\)
\(812\) −3.07407 1.82139i −0.107879 0.0639182i
\(813\) 0 0
\(814\) 6.11641 10.5939i 0.214380 0.371317i
\(815\) 0 0
\(816\) 0 0
\(817\) 28.0183 + 16.1764i 0.980237 + 0.565940i
\(818\) 13.9683 0.488390
\(819\) 0 0
\(820\) 0 0
\(821\) −27.5430 15.9020i −0.961257 0.554982i −0.0646976 0.997905i \(-0.520608\pi\)
−0.896560 + 0.442923i \(0.853942\pi\)
\(822\) 0 0
\(823\) 18.3186 + 31.7287i 0.638545 + 1.10599i 0.985752 + 0.168204i \(0.0537967\pi\)
−0.347207 + 0.937788i \(0.612870\pi\)
\(824\) 5.18220 8.97583i 0.180530 0.312688i
\(825\) 0 0
\(826\) −3.68100 + 2.07018i −0.128078 + 0.0720308i
\(827\) 21.9864i 0.764541i −0.924050 0.382271i \(-0.875142\pi\)
0.924050 0.382271i \(-0.124858\pi\)
\(828\) 0 0
\(829\) 21.0498 12.1531i 0.731090 0.422095i −0.0877305 0.996144i \(-0.527961\pi\)
0.818821 + 0.574049i \(0.194628\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.14864i 0.213166i
\(833\) 45.6463 + 1.03076i 1.58155 + 0.0357136i
\(834\) 0 0
\(835\) 0 0
\(836\) 3.87339 + 6.70891i 0.133964 + 0.232033i
\(837\) 0 0
\(838\) −9.80175 5.65904i −0.338596 0.195488i
\(839\) −27.3914 −0.945655 −0.472827 0.881155i \(-0.656767\pi\)
−0.472827 + 0.881155i \(0.656767\pi\)
\(840\) 0 0
\(841\) 27.1761 0.937107
\(842\) 12.0463 + 6.95496i 0.415145 + 0.239684i
\(843\) 0 0
\(844\) −7.18100 12.4379i −0.247180 0.428129i
\(845\) 0 0
\(846\) 0 0
\(847\) −24.7459 0.279362i −0.850278 0.00959899i
\(848\) 6.66826i 0.228989i
\(849\) 0 0
\(850\) 0 0
\(851\) 23.7719 13.7247i 0.814890 0.470477i
\(852\) 0 0
\(853\) 22.6073i 0.774060i 0.922067 + 0.387030i \(0.126499\pi\)
−0.922067 + 0.387030i \(0.873501\pi\)
\(854\) −8.56884 + 14.4622i −0.293220 + 0.494885i
\(855\) 0 0
\(856\) 5.75680 9.97106i 0.196763 0.340804i
\(857\) −23.9289 41.4461i −0.817396 1.41577i −0.907595 0.419848i \(-0.862083\pi\)
0.0901984 0.995924i \(-0.471250\pi\)
\(858\) 0 0
\(859\) −41.5770 24.0045i −1.41859 0.819022i −0.422413 0.906403i \(-0.638817\pi\)
−0.996175 + 0.0873810i \(0.972150\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 26.0901 0.888634
\(863\) 41.1855 + 23.7785i 1.40197 + 0.809428i 0.994595 0.103832i \(-0.0331104\pi\)
0.407376 + 0.913260i \(0.366444\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 4.21307 7.29726i 0.143166 0.247971i
\(867\) 0 0
\(868\) 0.258500 22.8979i 0.00877406 0.777206i
\(869\) 14.3474i 0.486703i
\(870\) 0 0
\(871\) 28.7257 16.5848i 0.973334 0.561955i
\(872\) 1.73205 1.00000i 0.0586546 0.0338643i
\(873\) 0 0
\(874\) 17.3831i 0.587993i
\(875\) 0 0
\(876\) 0 0
\(877\) 18.5002 32.0433i 0.624707 1.08202i −0.363890 0.931442i \(-0.618552\pi\)
0.988597 0.150583i \(-0.0481151\pi\)
\(878\) −13.8292 23.9529i −0.466714 0.808372i
\(879\) 0 0
\(880\) 0 0
\(881\) −5.97213 −0.201206 −0.100603 0.994927i \(-0.532077\pi\)
−0.100603 + 0.994927i \(0.532077\pi\)
\(882\) 0 0
\(883\) 11.0589 0.372163 0.186081 0.982534i \(-0.440421\pi\)
0.186081 + 0.982534i \(0.440421\pi\)
\(884\) 34.7319 + 20.0524i 1.16816 + 0.674437i
\(885\) 0 0
\(886\) −1.17861 2.04142i −0.0395962 0.0685827i
\(887\) 2.97535 5.15346i 0.0999026 0.173036i −0.811742 0.584017i \(-0.801480\pi\)
0.911644 + 0.410980i \(0.134814\pi\)
\(888\) 0 0
\(889\) −7.33169 13.0365i −0.245897 0.437231i
\(890\) 0 0
\(891\) 0 0
\(892\) 11.7758 6.79876i 0.394283 0.227639i
\(893\) 4.21521 2.43365i 0.141057 0.0814391i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.0298666 + 2.64558i −0.000997774 + 0.0883827i
\(897\) 0 0
\(898\) −6.40105 + 11.0869i −0.213606 + 0.369976i
\(899\) −5.84448 10.1229i −0.194924 0.337619i
\(900\) 0 0
\(901\) −37.6671 21.7471i −1.25487 0.724500i
\(902\) −11.1855 −0.372438
\(903\) 0 0
\(904\) −13.3875 −0.445261
\(905\) 0 0
\(906\) 0 0
\(907\) −27.6572 47.9037i −0.918343 1.59062i −0.801932 0.597415i \(-0.796195\pi\)
−0.116411 0.993201i \(-0.537139\pi\)
\(908\) 6.17797 10.7006i 0.205023 0.355110i
\(909\) 0 0
\(910\) 0 0
\(911\) 35.2080i 1.16649i 0.812295 + 0.583246i \(0.198218\pi\)
−0.812295 + 0.583246i \(0.801782\pi\)
\(912\) 0 0
\(913\) 4.16135 2.40256i 0.137721 0.0795131i
\(914\) −29.8989 + 17.2621i −0.988966 + 0.570980i
\(915\) 0 0
\(916\) 2.41341i 0.0797414i
\(917\) 32.9833 + 0.372357i 1.08921 + 0.0122963i
\(918\) 0 0
\(919\) −25.4437 + 44.0698i −0.839311 + 1.45373i 0.0511601 + 0.998690i \(0.483708\pi\)
−0.890471 + 0.455039i \(0.849625\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.5537 + 10.1346i 0.578099 + 0.333766i
\(923\) 96.4026 3.17313
\(924\) 0 0
\(925\) 0 0
\(926\) −11.3662 6.56230i −0.373518 0.215651i
\(927\) 0 0
\(928\) 0.675260 + 1.16959i 0.0221665 + 0.0383935i
\(929\) 29.8405 51.6853i 0.979036 1.69574i 0.313118 0.949714i \(-0.398627\pi\)
0.665918 0.746025i \(-0.268040\pi\)
\(930\) 0 0
\(931\) 37.0685 20.2998i 1.21487 0.665298i
\(932\) 12.8756i 0.421754i
\(933\) 0 0
\(934\) 1.32647 0.765836i 0.0434033 0.0250589i
\(935\) 0 0
\(936\) 0 0
\(937\) 24.2579i 0.792471i −0.918149 0.396235i \(-0.870316\pi\)
0.918149 0.396235i \(-0.129684\pi\)
\(938\) −12.4404 + 6.99643i −0.406194 + 0.228442i
\(939\) 0 0
\(940\) 0 0
\(941\) −15.0539 26.0742i −0.490744 0.849994i 0.509199 0.860649i \(-0.329942\pi\)
−0.999943 + 0.0106549i \(0.996608\pi\)
\(942\) 0 0
\(943\) −21.7367 12.5497i −0.707845 0.408675i
\(944\) 1.59622 0.0519525
\(945\) 0 0
\(946\) −6.87560 −0.223545
\(947\) 39.9779 + 23.0812i 1.29911 + 0.750039i 0.980250 0.197764i \(-0.0633678\pi\)
0.318857 + 0.947803i \(0.396701\pi\)
\(948\) 0 0
\(949\) 38.1730 + 66.1176i 1.23915 + 2.14627i
\(950\) 0 0
\(951\) 0 0
\(952\) −14.8467 8.79670i −0.481185 0.285103i
\(953\) 23.7121i 0.768110i −0.923310 0.384055i \(-0.874527\pi\)
0.923310 0.384055i \(-0.125473\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −23.6032 + 13.6273i −0.763382 + 0.440739i
\(957\) 0 0
\(958\) 29.3170i 0.947190i
\(959\) −0.116015 0.0687391i −0.00374632 0.00221970i
\(960\) 0 0
\(961\) 21.9558 38.0286i 0.708252 1.22673i
\(962\) −29.3100 50.7663i −0.944991 1.63677i
\(963\) 0 0
\(964\) 2.26690 + 1.30880i 0.0730120 + 0.0421535i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.0922780 0.00296746 0.00148373 0.999999i \(-0.499528\pi\)
0.00148373 + 0.999999i \(0.499528\pi\)
\(968\) 8.10050 + 4.67683i 0.260360 + 0.150319i
\(969\) 0 0
\(970\) 0 0
\(971\) −19.8502 + 34.3816i −0.637024 + 1.10336i 0.349058 + 0.937101i \(0.386502\pi\)
−0.986082 + 0.166257i \(0.946832\pi\)
\(972\) 0 0
\(973\) −2.42297 + 1.36267i −0.0776769 + 0.0436851i
\(974\) 8.19540i 0.262598i
\(975\) 0 0
\(976\) 5.50239 3.17681i 0.176127 0.101687i
\(977\) 31.8758 18.4035i 1.01980 0.588779i 0.105751 0.994393i \(-0.466275\pi\)
0.914045 + 0.405613i \(0.132942\pi\)
\(978\) 0 0
\(979\) 4.66973i 0.149245i
\(980\) 0 0
\(981\) 0 0
\(982\) 20.2191 35.0206i 0.645218 1.11755i
\(983\) 31.2865 + 54.1899i 0.997885 + 1.72839i 0.555235 + 0.831693i \(0.312628\pi\)
0.442650 + 0.896694i \(0.354038\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.80886 −0.280531
\(987\) 0 0
\(988\) 37.1227 1.18103
\(989\) −13.3613 7.71414i −0.424864 0.245295i
\(990\) 0 0
\(991\) 17.4019 + 30.1410i 0.552791 + 0.957462i 0.998072 + 0.0620708i \(0.0197704\pi\)
−0.445281 + 0.895391i \(0.646896\pi\)
\(992\) −4.32758 + 7.49558i −0.137401 + 0.237985i
\(993\) 0 0
\(994\) −41.4793 0.468269i −1.31564 0.0148526i
\(995\) 0 0
\(996\) 0 0
\(997\) −42.3167 + 24.4316i −1.34018 + 0.773755i −0.986834 0.161736i \(-0.948291\pi\)
−0.353349 + 0.935492i \(0.614957\pi\)
\(998\) −15.1113 + 8.72450i −0.478339 + 0.276169i
\(999\) 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 3150.2.bf.f.1601.8 32
3.2 odd 2 inner 3150.2.bf.f.1601.10 32
5.2 odd 4 630.2.bo.b.89.6 yes 16
5.3 odd 4 630.2.bo.a.89.1 16
5.4 even 2 inner 3150.2.bf.f.1601.9 32
7.3 odd 6 inner 3150.2.bf.f.1151.10 32
15.2 even 4 630.2.bo.a.89.3 yes 16
15.8 even 4 630.2.bo.b.89.8 yes 16
15.14 odd 2 inner 3150.2.bf.f.1601.7 32
21.17 even 6 inner 3150.2.bf.f.1151.8 32
35.2 odd 12 4410.2.d.a.4409.9 16
35.3 even 12 630.2.bo.a.269.3 yes 16
35.12 even 12 4410.2.d.a.4409.8 16
35.17 even 12 630.2.bo.b.269.8 yes 16
35.23 odd 12 4410.2.d.b.4409.10 16
35.24 odd 6 inner 3150.2.bf.f.1151.7 32
35.33 even 12 4410.2.d.b.4409.7 16
105.2 even 12 4410.2.d.b.4409.8 16
105.17 odd 12 630.2.bo.a.269.1 yes 16
105.23 even 12 4410.2.d.a.4409.7 16
105.38 odd 12 630.2.bo.b.269.6 yes 16
105.47 odd 12 4410.2.d.b.4409.9 16
105.59 even 6 inner 3150.2.bf.f.1151.9 32
105.68 odd 12 4410.2.d.a.4409.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.1 16 5.3 odd 4
630.2.bo.a.89.3 yes 16 15.2 even 4
630.2.bo.a.269.1 yes 16 105.17 odd 12
630.2.bo.a.269.3 yes 16 35.3 even 12
630.2.bo.b.89.6 yes 16 5.2 odd 4
630.2.bo.b.89.8 yes 16 15.8 even 4
630.2.bo.b.269.6 yes 16 105.38 odd 12
630.2.bo.b.269.8 yes 16 35.17 even 12
3150.2.bf.f.1151.7 32 35.24 odd 6 inner
3150.2.bf.f.1151.8 32 21.17 even 6 inner
3150.2.bf.f.1151.9 32 105.59 even 6 inner
3150.2.bf.f.1151.10 32 7.3 odd 6 inner
3150.2.bf.f.1601.7 32 15.14 odd 2 inner
3150.2.bf.f.1601.8 32 1.1 even 1 trivial
3150.2.bf.f.1601.9 32 5.4 even 2 inner
3150.2.bf.f.1601.10 32 3.2 odd 2 inner
4410.2.d.a.4409.7 16 105.23 even 12
4410.2.d.a.4409.8 16 35.12 even 12
4410.2.d.a.4409.9 16 35.2 odd 12
4410.2.d.a.4409.10 16 105.68 odd 12
4410.2.d.b.4409.7 16 35.33 even 12
4410.2.d.b.4409.8 16 105.2 even 12
4410.2.d.b.4409.9 16 105.47 odd 12
4410.2.d.b.4409.10 16 35.23 odd 12