Properties

Label 4020.2.q.i.841.2
Level $4020$
Weight $2$
Character 4020.841
Analytic conductor $32.100$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 841.2
Root \(1.68614 - 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 4020.841
Dual form 4020.2.q.i.3781.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +(1.18614 - 2.05446i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +(1.18614 - 2.05446i) q^{7} +1.00000 q^{9} +(2.18614 - 3.78651i) q^{11} +(0.500000 + 0.866025i) q^{13} -1.00000 q^{15} +(-0.813859 - 1.40965i) q^{17} +(-3.18614 - 5.51856i) q^{19} +(1.18614 - 2.05446i) q^{21} +(-0.813859 - 1.40965i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(-2.18614 + 3.78651i) q^{29} +(-3.87228 + 6.70699i) q^{31} +(2.18614 - 3.78651i) q^{33} +(-1.18614 + 2.05446i) q^{35} +(-4.55842 - 7.89542i) q^{37} +(0.500000 + 0.866025i) q^{39} +(5.18614 - 8.98266i) q^{41} +3.37228 q^{43} -1.00000 q^{45} +(0.813859 - 1.40965i) q^{47} +(0.686141 + 1.18843i) q^{49} +(-0.813859 - 1.40965i) q^{51} -8.74456 q^{53} +(-2.18614 + 3.78651i) q^{55} +(-3.18614 - 5.51856i) q^{57} -2.74456 q^{59} +(-5.50000 - 9.52628i) q^{61} +(1.18614 - 2.05446i) q^{63} +(-0.500000 - 0.866025i) q^{65} +(-1.05842 + 8.11663i) q^{67} +(-0.813859 - 1.40965i) q^{69} +(-6.55842 + 11.3595i) q^{71} +(6.50000 + 11.2583i) q^{73} +1.00000 q^{75} +(-5.18614 - 8.98266i) q^{77} +(2.12772 - 3.68532i) q^{79} +1.00000 q^{81} +(-3.55842 - 6.16337i) q^{83} +(0.813859 + 1.40965i) q^{85} +(-2.18614 + 3.78651i) q^{87} -11.4891 q^{89} +2.37228 q^{91} +(-3.87228 + 6.70699i) q^{93} +(3.18614 + 5.51856i) q^{95} +(0.500000 + 0.866025i) q^{97} +(2.18614 - 3.78651i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} + 3 q^{11} + 2 q^{13} - 4 q^{15} - 9 q^{17} - 7 q^{19} - q^{21} - 9 q^{23} + 4 q^{25} + 4 q^{27} - 3 q^{29} - 4 q^{31} + 3 q^{33} + q^{35} - q^{37} + 2 q^{39} + 15 q^{41} + 2 q^{43} - 4 q^{45} + 9 q^{47} - 3 q^{49} - 9 q^{51} - 12 q^{53} - 3 q^{55} - 7 q^{57} + 12 q^{59} - 22 q^{61} - q^{63} - 2 q^{65} + 13 q^{67} - 9 q^{69} - 9 q^{71} + 26 q^{73} + 4 q^{75} - 15 q^{77} + 20 q^{79} + 4 q^{81} + 3 q^{83} + 9 q^{85} - 3 q^{87} - 2 q^{91} - 4 q^{93} + 7 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.18614 2.05446i 0.448319 0.776511i −0.549958 0.835192i \(-0.685356\pi\)
0.998277 + 0.0586811i \(0.0186895\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.18614 3.78651i 0.659146 1.14167i −0.321691 0.946845i \(-0.604251\pi\)
0.980837 0.194830i \(-0.0624155\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.813859 1.40965i −0.197390 0.341889i 0.750291 0.661107i \(-0.229913\pi\)
−0.947681 + 0.319218i \(0.896580\pi\)
\(18\) 0 0
\(19\) −3.18614 5.51856i −0.730951 1.26604i −0.956477 0.291807i \(-0.905744\pi\)
0.225526 0.974237i \(-0.427590\pi\)
\(20\) 0 0
\(21\) 1.18614 2.05446i 0.258837 0.448319i
\(22\) 0 0
\(23\) −0.813859 1.40965i −0.169701 0.293931i 0.768613 0.639713i \(-0.220947\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.18614 + 3.78651i −0.405956 + 0.703137i −0.994432 0.105378i \(-0.966395\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(30\) 0 0
\(31\) −3.87228 + 6.70699i −0.695482 + 1.20461i 0.274536 + 0.961577i \(0.411476\pi\)
−0.970018 + 0.243034i \(0.921857\pi\)
\(32\) 0 0
\(33\) 2.18614 3.78651i 0.380558 0.659146i
\(34\) 0 0
\(35\) −1.18614 + 2.05446i −0.200494 + 0.347266i
\(36\) 0 0
\(37\) −4.55842 7.89542i −0.749400 1.29800i −0.948111 0.317941i \(-0.897009\pi\)
0.198711 0.980058i \(-0.436325\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.0800641 + 0.138675i
\(40\) 0 0
\(41\) 5.18614 8.98266i 0.809939 1.40286i −0.102966 0.994685i \(-0.532833\pi\)
0.912906 0.408171i \(-0.133833\pi\)
\(42\) 0 0
\(43\) 3.37228 0.514268 0.257134 0.966376i \(-0.417222\pi\)
0.257134 + 0.966376i \(0.417222\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0.813859 1.40965i 0.118714 0.205618i −0.800545 0.599273i \(-0.795456\pi\)
0.919258 + 0.393655i \(0.128790\pi\)
\(48\) 0 0
\(49\) 0.686141 + 1.18843i 0.0980201 + 0.169776i
\(50\) 0 0
\(51\) −0.813859 1.40965i −0.113963 0.197390i
\(52\) 0 0
\(53\) −8.74456 −1.20116 −0.600579 0.799565i \(-0.705063\pi\)
−0.600579 + 0.799565i \(0.705063\pi\)
\(54\) 0 0
\(55\) −2.18614 + 3.78651i −0.294779 + 0.510572i
\(56\) 0 0
\(57\) −3.18614 5.51856i −0.422015 0.730951i
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) −5.50000 9.52628i −0.704203 1.21972i −0.966978 0.254858i \(-0.917971\pi\)
0.262776 0.964857i \(-0.415362\pi\)
\(62\) 0 0
\(63\) 1.18614 2.05446i 0.149440 0.258837i
\(64\) 0 0
\(65\) −0.500000 0.866025i −0.0620174 0.107417i
\(66\) 0 0
\(67\) −1.05842 + 8.11663i −0.129307 + 0.991605i
\(68\) 0 0
\(69\) −0.813859 1.40965i −0.0979772 0.169701i
\(70\) 0 0
\(71\) −6.55842 + 11.3595i −0.778341 + 1.34813i 0.154556 + 0.987984i \(0.450605\pi\)
−0.932897 + 0.360143i \(0.882728\pi\)
\(72\) 0 0
\(73\) 6.50000 + 11.2583i 0.760767 + 1.31769i 0.942455 + 0.334332i \(0.108511\pi\)
−0.181688 + 0.983356i \(0.558156\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −5.18614 8.98266i −0.591016 1.02367i
\(78\) 0 0
\(79\) 2.12772 3.68532i 0.239387 0.414631i −0.721152 0.692777i \(-0.756387\pi\)
0.960539 + 0.278147i \(0.0897202\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.55842 6.16337i −0.390588 0.676517i 0.601940 0.798542i \(-0.294395\pi\)
−0.992527 + 0.122024i \(0.961061\pi\)
\(84\) 0 0
\(85\) 0.813859 + 1.40965i 0.0882754 + 0.152898i
\(86\) 0 0
\(87\) −2.18614 + 3.78651i −0.234379 + 0.405956i
\(88\) 0 0
\(89\) −11.4891 −1.21784 −0.608922 0.793230i \(-0.708398\pi\)
−0.608922 + 0.793230i \(0.708398\pi\)
\(90\) 0 0
\(91\) 2.37228 0.248683
\(92\) 0 0
\(93\) −3.87228 + 6.70699i −0.401537 + 0.695482i
\(94\) 0 0
\(95\) 3.18614 + 5.51856i 0.326891 + 0.566192i
\(96\) 0 0
\(97\) 0.500000 + 0.866025i 0.0507673 + 0.0879316i 0.890292 0.455389i \(-0.150500\pi\)
−0.839525 + 0.543321i \(0.817167\pi\)
\(98\) 0 0
\(99\) 2.18614 3.78651i 0.219715 0.380558i
\(100\) 0 0
\(101\) 7.93070 13.7364i 0.789134 1.36682i −0.137363 0.990521i \(-0.543863\pi\)
0.926498 0.376300i \(-0.122804\pi\)
\(102\) 0 0
\(103\) −7.55842 + 13.0916i −0.744753 + 1.28995i 0.205556 + 0.978645i \(0.434100\pi\)
−0.950310 + 0.311306i \(0.899234\pi\)
\(104\) 0 0
\(105\) −1.18614 + 2.05446i −0.115755 + 0.200494i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 12.1168 1.16058 0.580292 0.814409i \(-0.302939\pi\)
0.580292 + 0.814409i \(0.302939\pi\)
\(110\) 0 0
\(111\) −4.55842 7.89542i −0.432666 0.749400i
\(112\) 0 0
\(113\) 6.55842 11.3595i 0.616964 1.06861i −0.373072 0.927802i \(-0.621696\pi\)
0.990036 0.140811i \(-0.0449711\pi\)
\(114\) 0 0
\(115\) 0.813859 + 1.40965i 0.0758928 + 0.131450i
\(116\) 0 0
\(117\) 0.500000 + 0.866025i 0.0462250 + 0.0800641i
\(118\) 0 0
\(119\) −3.86141 −0.353975
\(120\) 0 0
\(121\) −4.05842 7.02939i −0.368947 0.639036i
\(122\) 0 0
\(123\) 5.18614 8.98266i 0.467619 0.809939i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.12772 3.68532i 0.188805 0.327019i −0.756047 0.654517i \(-0.772872\pi\)
0.944852 + 0.327498i \(0.106205\pi\)
\(128\) 0 0
\(129\) 3.37228 0.296913
\(130\) 0 0
\(131\) 17.4891 1.52803 0.764016 0.645197i \(-0.223225\pi\)
0.764016 + 0.645197i \(0.223225\pi\)
\(132\) 0 0
\(133\) −15.1168 −1.31080
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −20.7446 −1.77233 −0.886164 0.463372i \(-0.846639\pi\)
−0.886164 + 0.463372i \(0.846639\pi\)
\(138\) 0 0
\(139\) 0.627719 0.0532424 0.0266212 0.999646i \(-0.491525\pi\)
0.0266212 + 0.999646i \(0.491525\pi\)
\(140\) 0 0
\(141\) 0.813859 1.40965i 0.0685393 0.118714i
\(142\) 0 0
\(143\) 4.37228 0.365629
\(144\) 0 0
\(145\) 2.18614 3.78651i 0.181549 0.314452i
\(146\) 0 0
\(147\) 0.686141 + 1.18843i 0.0565919 + 0.0980201i
\(148\) 0 0
\(149\) −8.74456 −0.716382 −0.358191 0.933648i \(-0.616606\pi\)
−0.358191 + 0.933648i \(0.616606\pi\)
\(150\) 0 0
\(151\) 0.500000 + 0.866025i 0.0406894 + 0.0704761i 0.885653 0.464348i \(-0.153711\pi\)
−0.844963 + 0.534824i \(0.820378\pi\)
\(152\) 0 0
\(153\) −0.813859 1.40965i −0.0657966 0.113963i
\(154\) 0 0
\(155\) 3.87228 6.70699i 0.311029 0.538718i
\(156\) 0 0
\(157\) −1.81386 3.14170i −0.144762 0.250735i 0.784522 0.620101i \(-0.212908\pi\)
−0.929284 + 0.369366i \(0.879575\pi\)
\(158\) 0 0
\(159\) −8.74456 −0.693489
\(160\) 0 0
\(161\) −3.86141 −0.304321
\(162\) 0 0
\(163\) 3.50000 6.06218i 0.274141 0.474826i −0.695777 0.718258i \(-0.744940\pi\)
0.969918 + 0.243432i \(0.0782731\pi\)
\(164\) 0 0
\(165\) −2.18614 + 3.78651i −0.170191 + 0.294779i
\(166\) 0 0
\(167\) 6.81386 11.8020i 0.527272 0.913262i −0.472223 0.881479i \(-0.656548\pi\)
0.999495 0.0317830i \(-0.0101185\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) −3.18614 5.51856i −0.243650 0.422015i
\(172\) 0 0
\(173\) 0.558422 + 0.967215i 0.0424560 + 0.0735360i 0.886473 0.462781i \(-0.153148\pi\)
−0.844017 + 0.536317i \(0.819815\pi\)
\(174\) 0 0
\(175\) 1.18614 2.05446i 0.0896638 0.155302i
\(176\) 0 0
\(177\) −2.74456 −0.206294
\(178\) 0 0
\(179\) −14.7446 −1.10206 −0.551030 0.834485i \(-0.685765\pi\)
−0.551030 + 0.834485i \(0.685765\pi\)
\(180\) 0 0
\(181\) −6.87228 + 11.9031i −0.510813 + 0.884753i 0.489109 + 0.872223i \(0.337322\pi\)
−0.999921 + 0.0125307i \(0.996011\pi\)
\(182\) 0 0
\(183\) −5.50000 9.52628i −0.406572 0.704203i
\(184\) 0 0
\(185\) 4.55842 + 7.89542i 0.335142 + 0.580483i
\(186\) 0 0
\(187\) −7.11684 −0.520435
\(188\) 0 0
\(189\) 1.18614 2.05446i 0.0862790 0.149440i
\(190\) 0 0
\(191\) −9.55842 16.5557i −0.691623 1.19793i −0.971306 0.237834i \(-0.923563\pi\)
0.279683 0.960092i \(-0.409771\pi\)
\(192\) 0 0
\(193\) 0.116844 0.00841061 0.00420531 0.999991i \(-0.498661\pi\)
0.00420531 + 0.999991i \(0.498661\pi\)
\(194\) 0 0
\(195\) −0.500000 0.866025i −0.0358057 0.0620174i
\(196\) 0 0
\(197\) 5.18614 8.98266i 0.369497 0.639988i −0.619990 0.784610i \(-0.712863\pi\)
0.989487 + 0.144622i \(0.0461966\pi\)
\(198\) 0 0
\(199\) 1.87228 + 3.24289i 0.132723 + 0.229882i 0.924725 0.380636i \(-0.124295\pi\)
−0.792003 + 0.610518i \(0.790961\pi\)
\(200\) 0 0
\(201\) −1.05842 + 8.11663i −0.0746553 + 0.572503i
\(202\) 0 0
\(203\) 5.18614 + 8.98266i 0.363996 + 0.630459i
\(204\) 0 0
\(205\) −5.18614 + 8.98266i −0.362216 + 0.627376i
\(206\) 0 0
\(207\) −0.813859 1.40965i −0.0565671 0.0979772i
\(208\) 0 0
\(209\) −27.8614 −1.92721
\(210\) 0 0
\(211\) 1.87228 + 3.24289i 0.128893 + 0.223250i 0.923248 0.384204i \(-0.125524\pi\)
−0.794355 + 0.607454i \(0.792191\pi\)
\(212\) 0 0
\(213\) −6.55842 + 11.3595i −0.449376 + 0.778341i
\(214\) 0 0
\(215\) −3.37228 −0.229988
\(216\) 0 0
\(217\) 9.18614 + 15.9109i 0.623596 + 1.08010i
\(218\) 0 0
\(219\) 6.50000 + 11.2583i 0.439229 + 0.760767i
\(220\) 0 0
\(221\) 0.813859 1.40965i 0.0547461 0.0948230i
\(222\) 0 0
\(223\) −5.88316 −0.393965 −0.196983 0.980407i \(-0.563114\pi\)
−0.196983 + 0.980407i \(0.563114\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.9307 18.9325i 0.725496 1.25660i −0.233273 0.972411i \(-0.574944\pi\)
0.958769 0.284185i \(-0.0917230\pi\)
\(228\) 0 0
\(229\) −9.61684 16.6569i −0.635499 1.10072i −0.986409 0.164307i \(-0.947461\pi\)
0.350910 0.936409i \(-0.385872\pi\)
\(230\) 0 0
\(231\) −5.18614 8.98266i −0.341223 0.591016i
\(232\) 0 0
\(233\) 3.81386 6.60580i 0.249854 0.432760i −0.713631 0.700522i \(-0.752951\pi\)
0.963485 + 0.267762i \(0.0862840\pi\)
\(234\) 0 0
\(235\) −0.813859 + 1.40965i −0.0530903 + 0.0919551i
\(236\) 0 0
\(237\) 2.12772 3.68532i 0.138210 0.239387i
\(238\) 0 0
\(239\) 5.44158 9.42509i 0.351986 0.609658i −0.634611 0.772832i \(-0.718840\pi\)
0.986597 + 0.163173i \(0.0521730\pi\)
\(240\) 0 0
\(241\) 24.1168 1.55350 0.776751 0.629808i \(-0.216866\pi\)
0.776751 + 0.629808i \(0.216866\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.686141 1.18843i −0.0438359 0.0759260i
\(246\) 0 0
\(247\) 3.18614 5.51856i 0.202729 0.351137i
\(248\) 0 0
\(249\) −3.55842 6.16337i −0.225506 0.390588i
\(250\) 0 0
\(251\) −9.55842 16.5557i −0.603322 1.04498i −0.992314 0.123744i \(-0.960510\pi\)
0.388992 0.921241i \(-0.372823\pi\)
\(252\) 0 0
\(253\) −7.11684 −0.447432
\(254\) 0 0
\(255\) 0.813859 + 1.40965i 0.0509658 + 0.0882754i
\(256\) 0 0
\(257\) 5.18614 8.98266i 0.323503 0.560323i −0.657706 0.753275i \(-0.728473\pi\)
0.981208 + 0.192952i \(0.0618062\pi\)
\(258\) 0 0
\(259\) −21.6277 −1.34388
\(260\) 0 0
\(261\) −2.18614 + 3.78651i −0.135319 + 0.234379i
\(262\) 0 0
\(263\) 5.48913 0.338474 0.169237 0.985575i \(-0.445870\pi\)
0.169237 + 0.985575i \(0.445870\pi\)
\(264\) 0 0
\(265\) 8.74456 0.537174
\(266\) 0 0
\(267\) −11.4891 −0.703123
\(268\) 0 0
\(269\) 20.7446 1.26482 0.632409 0.774635i \(-0.282066\pi\)
0.632409 + 0.774635i \(0.282066\pi\)
\(270\) 0 0
\(271\) −26.1168 −1.58649 −0.793243 0.608906i \(-0.791609\pi\)
−0.793243 + 0.608906i \(0.791609\pi\)
\(272\) 0 0
\(273\) 2.37228 0.143577
\(274\) 0 0
\(275\) 2.18614 3.78651i 0.131829 0.228335i
\(276\) 0 0
\(277\) 14.8614 0.892935 0.446468 0.894800i \(-0.352682\pi\)
0.446468 + 0.894800i \(0.352682\pi\)
\(278\) 0 0
\(279\) −3.87228 + 6.70699i −0.231827 + 0.401537i
\(280\) 0 0
\(281\) −6.81386 11.8020i −0.406481 0.704045i 0.588012 0.808852i \(-0.299911\pi\)
−0.994493 + 0.104807i \(0.966578\pi\)
\(282\) 0 0
\(283\) 13.4891 0.801845 0.400923 0.916112i \(-0.368690\pi\)
0.400923 + 0.916112i \(0.368690\pi\)
\(284\) 0 0
\(285\) 3.18614 + 5.51856i 0.188731 + 0.326891i
\(286\) 0 0
\(287\) −12.3030 21.3094i −0.726222 1.25785i
\(288\) 0 0
\(289\) 7.17527 12.4279i 0.422074 0.731054i
\(290\) 0 0
\(291\) 0.500000 + 0.866025i 0.0293105 + 0.0507673i
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 2.74456 0.159795
\(296\) 0 0
\(297\) 2.18614 3.78651i 0.126853 0.219715i
\(298\) 0 0
\(299\) 0.813859 1.40965i 0.0470667 0.0815219i
\(300\) 0 0
\(301\) 4.00000 6.92820i 0.230556 0.399335i
\(302\) 0 0
\(303\) 7.93070 13.7364i 0.455607 0.789134i
\(304\) 0 0
\(305\) 5.50000 + 9.52628i 0.314929 + 0.545473i
\(306\) 0 0
\(307\) 0.500000 + 0.866025i 0.0285365 + 0.0494267i 0.879941 0.475083i \(-0.157582\pi\)
−0.851404 + 0.524510i \(0.824249\pi\)
\(308\) 0 0
\(309\) −7.55842 + 13.0916i −0.429984 + 0.744753i
\(310\) 0 0
\(311\) 20.2337 1.14735 0.573674 0.819084i \(-0.305518\pi\)
0.573674 + 0.819084i \(0.305518\pi\)
\(312\) 0 0
\(313\) 19.4891 1.10159 0.550795 0.834640i \(-0.314325\pi\)
0.550795 + 0.834640i \(0.314325\pi\)
\(314\) 0 0
\(315\) −1.18614 + 2.05446i −0.0668315 + 0.115755i
\(316\) 0 0
\(317\) 0.558422 + 0.967215i 0.0313641 + 0.0543242i 0.881281 0.472592i \(-0.156682\pi\)
−0.849917 + 0.526916i \(0.823348\pi\)
\(318\) 0 0
\(319\) 9.55842 + 16.5557i 0.535169 + 0.926940i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.18614 + 8.98266i −0.288565 + 0.499809i
\(324\) 0 0
\(325\) 0.500000 + 0.866025i 0.0277350 + 0.0480384i
\(326\) 0 0
\(327\) 12.1168 0.670063
\(328\) 0 0
\(329\) −1.93070 3.34408i −0.106443 0.184365i
\(330\) 0 0
\(331\) −7.98913 + 13.8376i −0.439122 + 0.760582i −0.997622 0.0689229i \(-0.978044\pi\)
0.558500 + 0.829505i \(0.311377\pi\)
\(332\) 0 0
\(333\) −4.55842 7.89542i −0.249800 0.432666i
\(334\) 0 0
\(335\) 1.05842 8.11663i 0.0578278 0.443459i
\(336\) 0 0
\(337\) 0.500000 + 0.866025i 0.0272367 + 0.0471754i 0.879322 0.476227i \(-0.157996\pi\)
−0.852086 + 0.523402i \(0.824663\pi\)
\(338\) 0 0
\(339\) 6.55842 11.3595i 0.356205 0.616964i
\(340\) 0 0
\(341\) 16.9307 + 29.3248i 0.916849 + 1.58803i
\(342\) 0 0
\(343\) 19.8614 1.07242
\(344\) 0 0
\(345\) 0.813859 + 1.40965i 0.0438167 + 0.0758928i
\(346\) 0 0
\(347\) −0.558422 + 0.967215i −0.0299777 + 0.0519228i −0.880625 0.473814i \(-0.842877\pi\)
0.850647 + 0.525737i \(0.176210\pi\)
\(348\) 0 0
\(349\) −2.62772 −0.140659 −0.0703293 0.997524i \(-0.522405\pi\)
−0.0703293 + 0.997524i \(0.522405\pi\)
\(350\) 0 0
\(351\) 0.500000 + 0.866025i 0.0266880 + 0.0462250i
\(352\) 0 0
\(353\) 16.6753 + 28.8824i 0.887535 + 1.53726i 0.842780 + 0.538257i \(0.180917\pi\)
0.0447544 + 0.998998i \(0.485749\pi\)
\(354\) 0 0
\(355\) 6.55842 11.3595i 0.348085 0.602901i
\(356\) 0 0
\(357\) −3.86141 −0.204367
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −10.8030 + 18.7113i −0.568578 + 0.984806i
\(362\) 0 0
\(363\) −4.05842 7.02939i −0.213012 0.368947i
\(364\) 0 0
\(365\) −6.50000 11.2583i −0.340226 0.589288i
\(366\) 0 0
\(367\) −7.55842 + 13.0916i −0.394546 + 0.683374i −0.993043 0.117751i \(-0.962432\pi\)
0.598497 + 0.801125i \(0.295765\pi\)
\(368\) 0 0
\(369\) 5.18614 8.98266i 0.269980 0.467619i
\(370\) 0 0
\(371\) −10.3723 + 17.9653i −0.538502 + 0.932713i
\(372\) 0 0
\(373\) 10.6168 18.3889i 0.549719 0.952142i −0.448574 0.893746i \(-0.648068\pi\)
0.998293 0.0583962i \(-0.0185987\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −4.37228 −0.225184
\(378\) 0 0
\(379\) 15.2446 + 26.4044i 0.783061 + 1.35630i 0.930151 + 0.367178i \(0.119676\pi\)
−0.147090 + 0.989123i \(0.546991\pi\)
\(380\) 0 0
\(381\) 2.12772 3.68532i 0.109006 0.188805i
\(382\) 0 0
\(383\) 8.44158 + 14.6212i 0.431344 + 0.747111i 0.996989 0.0775384i \(-0.0247060\pi\)
−0.565645 + 0.824649i \(0.691373\pi\)
\(384\) 0 0
\(385\) 5.18614 + 8.98266i 0.264310 + 0.457799i
\(386\) 0 0
\(387\) 3.37228 0.171423
\(388\) 0 0
\(389\) 1.93070 + 3.34408i 0.0978905 + 0.169551i 0.910811 0.412823i \(-0.135457\pi\)
−0.812921 + 0.582374i \(0.802124\pi\)
\(390\) 0 0
\(391\) −1.32473 + 2.29451i −0.0669947 + 0.116038i
\(392\) 0 0
\(393\) 17.4891 0.882210
\(394\) 0 0
\(395\) −2.12772 + 3.68532i −0.107057 + 0.185428i
\(396\) 0 0
\(397\) −11.8832 −0.596399 −0.298199 0.954504i \(-0.596386\pi\)
−0.298199 + 0.954504i \(0.596386\pi\)
\(398\) 0 0
\(399\) −15.1168 −0.756789
\(400\) 0 0
\(401\) −31.7228 −1.58416 −0.792081 0.610416i \(-0.791002\pi\)
−0.792081 + 0.610416i \(0.791002\pi\)
\(402\) 0 0
\(403\) −7.74456 −0.385784
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −39.8614 −1.97586
\(408\) 0 0
\(409\) −13.3030 + 23.0414i −0.657790 + 1.13933i 0.323396 + 0.946264i \(0.395175\pi\)
−0.981186 + 0.193062i \(0.938158\pi\)
\(410\) 0 0
\(411\) −20.7446 −1.02325
\(412\) 0 0
\(413\) −3.25544 + 5.63858i −0.160190 + 0.277457i
\(414\) 0 0
\(415\) 3.55842 + 6.16337i 0.174676 + 0.302548i
\(416\) 0 0
\(417\) 0.627719 0.0307395
\(418\) 0 0
\(419\) −7.67527 13.2940i −0.374961 0.649452i 0.615360 0.788246i \(-0.289011\pi\)
−0.990321 + 0.138794i \(0.955677\pi\)
\(420\) 0 0
\(421\) 13.8723 + 24.0275i 0.676094 + 1.17103i 0.976148 + 0.217106i \(0.0696619\pi\)
−0.300054 + 0.953922i \(0.597005\pi\)
\(422\) 0 0
\(423\) 0.813859 1.40965i 0.0395712 0.0685393i
\(424\) 0 0
\(425\) −0.813859 1.40965i −0.0394780 0.0683779i
\(426\) 0 0
\(427\) −26.0951 −1.26283
\(428\) 0 0
\(429\) 4.37228 0.211096
\(430\) 0 0
\(431\) −18.0475 + 31.2593i −0.869320 + 1.50571i −0.00662675 + 0.999978i \(0.502109\pi\)
−0.862693 + 0.505728i \(0.831224\pi\)
\(432\) 0 0
\(433\) −7.81386 + 13.5340i −0.375510 + 0.650403i −0.990403 0.138208i \(-0.955866\pi\)
0.614893 + 0.788610i \(0.289199\pi\)
\(434\) 0 0
\(435\) 2.18614 3.78651i 0.104817 0.181549i
\(436\) 0 0
\(437\) −5.18614 + 8.98266i −0.248087 + 0.429699i
\(438\) 0 0
\(439\) 0.500000 + 0.866025i 0.0238637 + 0.0413331i 0.877711 0.479191i \(-0.159070\pi\)
−0.853847 + 0.520524i \(0.825737\pi\)
\(440\) 0 0
\(441\) 0.686141 + 1.18843i 0.0326734 + 0.0565919i
\(442\) 0 0
\(443\) −7.93070 + 13.7364i −0.376799 + 0.652635i −0.990595 0.136830i \(-0.956309\pi\)
0.613795 + 0.789465i \(0.289642\pi\)
\(444\) 0 0
\(445\) 11.4891 0.544637
\(446\) 0 0
\(447\) −8.74456 −0.413604
\(448\) 0 0
\(449\) 19.9307 34.5210i 0.940588 1.62915i 0.176235 0.984348i \(-0.443608\pi\)
0.764353 0.644798i \(-0.223059\pi\)
\(450\) 0 0
\(451\) −22.6753 39.2747i −1.06774 1.84937i
\(452\) 0 0
\(453\) 0.500000 + 0.866025i 0.0234920 + 0.0406894i
\(454\) 0 0
\(455\) −2.37228 −0.111214
\(456\) 0 0
\(457\) 17.9891 31.1581i 0.841496 1.45751i −0.0471342 0.998889i \(-0.515009\pi\)
0.888630 0.458625i \(-0.151658\pi\)
\(458\) 0 0
\(459\) −0.813859 1.40965i −0.0379877 0.0657966i
\(460\) 0 0
\(461\) 0.510875 0.0237938 0.0118969 0.999929i \(-0.496213\pi\)
0.0118969 + 0.999929i \(0.496213\pi\)
\(462\) 0 0
\(463\) 5.55842 + 9.62747i 0.258322 + 0.447426i 0.965792 0.259316i \(-0.0834972\pi\)
−0.707471 + 0.706743i \(0.750164\pi\)
\(464\) 0 0
\(465\) 3.87228 6.70699i 0.179573 0.311029i
\(466\) 0 0
\(467\) 1.93070 + 3.34408i 0.0893423 + 0.154745i 0.907233 0.420628i \(-0.138190\pi\)
−0.817891 + 0.575373i \(0.804857\pi\)
\(468\) 0 0
\(469\) 15.4198 + 11.8020i 0.712022 + 0.544963i
\(470\) 0 0
\(471\) −1.81386 3.14170i −0.0835782 0.144762i
\(472\) 0 0
\(473\) 7.37228 12.7692i 0.338978 0.587127i
\(474\) 0 0
\(475\) −3.18614 5.51856i −0.146190 0.253209i
\(476\) 0 0
\(477\) −8.74456 −0.400386
\(478\) 0 0
\(479\) 4.67527 + 8.09780i 0.213618 + 0.369998i 0.952844 0.303460i \(-0.0981418\pi\)
−0.739226 + 0.673458i \(0.764808\pi\)
\(480\) 0 0
\(481\) 4.55842 7.89542i 0.207846 0.360000i
\(482\) 0 0
\(483\) −3.86141 −0.175700
\(484\) 0 0
\(485\) −0.500000 0.866025i −0.0227038 0.0393242i
\(486\) 0 0
\(487\) 3.75544 + 6.50461i 0.170175 + 0.294752i 0.938481 0.345331i \(-0.112233\pi\)
−0.768306 + 0.640083i \(0.778900\pi\)
\(488\) 0 0
\(489\) 3.50000 6.06218i 0.158275 0.274141i
\(490\) 0 0
\(491\) 2.74456 0.123860 0.0619302 0.998080i \(-0.480274\pi\)
0.0619302 + 0.998080i \(0.480274\pi\)
\(492\) 0 0
\(493\) 7.11684 0.320527
\(494\) 0 0
\(495\) −2.18614 + 3.78651i −0.0982597 + 0.170191i
\(496\) 0 0
\(497\) 15.5584 + 26.9480i 0.697891 + 1.20878i
\(498\) 0 0
\(499\) −6.44158 11.1571i −0.288365 0.499462i 0.685055 0.728491i \(-0.259778\pi\)
−0.973420 + 0.229029i \(0.926445\pi\)
\(500\) 0 0
\(501\) 6.81386 11.8020i 0.304421 0.527272i
\(502\) 0 0
\(503\) −2.44158 + 4.22894i −0.108865 + 0.188559i −0.915311 0.402749i \(-0.868055\pi\)
0.806446 + 0.591308i \(0.201388\pi\)
\(504\) 0 0
\(505\) −7.93070 + 13.7364i −0.352912 + 0.611261i
\(506\) 0 0
\(507\) 6.00000 10.3923i 0.266469 0.461538i
\(508\) 0 0
\(509\) 20.7446 0.919487 0.459743 0.888052i \(-0.347941\pi\)
0.459743 + 0.888052i \(0.347941\pi\)
\(510\) 0 0
\(511\) 30.8397 1.36427
\(512\) 0 0
\(513\) −3.18614 5.51856i −0.140672 0.243650i
\(514\) 0 0
\(515\) 7.55842 13.0916i 0.333064 0.576884i
\(516\) 0 0
\(517\) −3.55842 6.16337i −0.156499 0.271065i
\(518\) 0 0
\(519\) 0.558422 + 0.967215i 0.0245120 + 0.0424560i
\(520\) 0 0
\(521\) −31.7228 −1.38980 −0.694901 0.719106i \(-0.744552\pi\)
−0.694901 + 0.719106i \(0.744552\pi\)
\(522\) 0 0
\(523\) 11.5584 + 20.0198i 0.505415 + 0.875404i 0.999980 + 0.00626346i \(0.00199373\pi\)
−0.494566 + 0.869140i \(0.664673\pi\)
\(524\) 0 0
\(525\) 1.18614 2.05446i 0.0517674 0.0896638i
\(526\) 0 0
\(527\) 12.6060 0.549125
\(528\) 0 0
\(529\) 10.1753 17.6241i 0.442403 0.766264i
\(530\) 0 0
\(531\) −2.74456 −0.119104
\(532\) 0 0
\(533\) 10.3723 0.449273
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14.7446 −0.636275
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 29.6060 1.27286 0.636430 0.771334i \(-0.280410\pi\)
0.636430 + 0.771334i \(0.280410\pi\)
\(542\) 0 0
\(543\) −6.87228 + 11.9031i −0.294918 + 0.510813i
\(544\) 0 0
\(545\) −12.1168 −0.519029
\(546\) 0 0
\(547\) 21.5000 37.2391i 0.919274 1.59223i 0.118753 0.992924i \(-0.462110\pi\)
0.800521 0.599305i \(-0.204556\pi\)
\(548\) 0 0
\(549\) −5.50000 9.52628i −0.234734 0.406572i
\(550\) 0 0
\(551\) 27.8614 1.18694
\(552\) 0 0
\(553\) −5.04755 8.74261i −0.214644 0.371774i
\(554\) 0 0
\(555\) 4.55842 + 7.89542i 0.193494 + 0.335142i
\(556\) 0 0
\(557\) 3.81386 6.60580i 0.161598 0.279897i −0.773844 0.633377i \(-0.781668\pi\)
0.935442 + 0.353480i \(0.115002\pi\)
\(558\) 0 0
\(559\) 1.68614 + 2.92048i 0.0713162 + 0.123523i
\(560\) 0 0
\(561\) −7.11684 −0.300473
\(562\) 0 0
\(563\) −25.7228 −1.08409 −0.542044 0.840350i \(-0.682349\pi\)
−0.542044 + 0.840350i \(0.682349\pi\)
\(564\) 0 0
\(565\) −6.55842 + 11.3595i −0.275915 + 0.477899i
\(566\) 0 0
\(567\) 1.18614 2.05446i 0.0498132 0.0862790i
\(568\) 0 0
\(569\) −4.06930 + 7.04823i −0.170594 + 0.295477i −0.938628 0.344932i \(-0.887902\pi\)
0.768034 + 0.640409i \(0.221235\pi\)
\(570\) 0 0
\(571\) 6.75544 11.7008i 0.282706 0.489662i −0.689344 0.724434i \(-0.742101\pi\)
0.972050 + 0.234772i \(0.0754345\pi\)
\(572\) 0 0
\(573\) −9.55842 16.5557i −0.399309 0.691623i
\(574\) 0 0
\(575\) −0.813859 1.40965i −0.0339403 0.0587863i
\(576\) 0 0
\(577\) −21.6168 + 37.4415i −0.899921 + 1.55871i −0.0723275 + 0.997381i \(0.523043\pi\)
−0.827593 + 0.561328i \(0.810291\pi\)
\(578\) 0 0
\(579\) 0.116844 0.00485587
\(580\) 0 0
\(581\) −16.8832 −0.700431
\(582\) 0 0
\(583\) −19.1168 + 33.1113i −0.791739 + 1.37133i
\(584\) 0 0
\(585\) −0.500000 0.866025i −0.0206725 0.0358057i
\(586\) 0 0
\(587\) −0.302985 0.524785i −0.0125055 0.0216602i 0.859705 0.510791i \(-0.170647\pi\)
−0.872210 + 0.489131i \(0.837314\pi\)
\(588\) 0 0
\(589\) 49.3505 2.03345
\(590\) 0 0
\(591\) 5.18614 8.98266i 0.213329 0.369497i
\(592\) 0 0
\(593\) 11.1861 + 19.3750i 0.459360 + 0.795634i 0.998927 0.0463080i \(-0.0147456\pi\)
−0.539567 + 0.841942i \(0.681412\pi\)
\(594\) 0 0
\(595\) 3.86141 0.158302
\(596\) 0 0
\(597\) 1.87228 + 3.24289i 0.0766274 + 0.132723i
\(598\) 0 0
\(599\) 3.55842 6.16337i 0.145393 0.251828i −0.784126 0.620601i \(-0.786889\pi\)
0.929520 + 0.368773i \(0.120222\pi\)
\(600\) 0 0
\(601\) −14.2446 24.6723i −0.581048 1.00640i −0.995356 0.0962675i \(-0.969310\pi\)
0.414308 0.910137i \(-0.364024\pi\)
\(602\) 0 0
\(603\) −1.05842 + 8.11663i −0.0431023 + 0.330535i
\(604\) 0 0
\(605\) 4.05842 + 7.02939i 0.164998 + 0.285785i
\(606\) 0 0
\(607\) −19.9891 + 34.6222i −0.811334 + 1.40527i 0.100597 + 0.994927i \(0.467925\pi\)
−0.911931 + 0.410344i \(0.865409\pi\)
\(608\) 0 0
\(609\) 5.18614 + 8.98266i 0.210153 + 0.363996i
\(610\) 0 0
\(611\) 1.62772 0.0658504
\(612\) 0 0
\(613\) 16.6168 + 28.7812i 0.671148 + 1.16246i 0.977579 + 0.210570i \(0.0675318\pi\)
−0.306431 + 0.951893i \(0.599135\pi\)
\(614\) 0 0
\(615\) −5.18614 + 8.98266i −0.209125 + 0.362216i
\(616\) 0 0
\(617\) −4.97825 −0.200417 −0.100208 0.994966i \(-0.531951\pi\)
−0.100208 + 0.994966i \(0.531951\pi\)
\(618\) 0 0
\(619\) −7.38316 12.7880i −0.296754 0.513993i 0.678637 0.734474i \(-0.262571\pi\)
−0.975391 + 0.220480i \(0.929238\pi\)
\(620\) 0 0
\(621\) −0.813859 1.40965i −0.0326591 0.0565671i
\(622\) 0 0
\(623\) −13.6277 + 23.6039i −0.545983 + 0.945670i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −27.8614 −1.11268
\(628\) 0 0
\(629\) −7.41983 + 12.8515i −0.295848 + 0.512424i
\(630\) 0 0
\(631\) −23.4198 40.5643i −0.932329 1.61484i −0.779330 0.626614i \(-0.784440\pi\)
−0.152999 0.988226i \(-0.548893\pi\)
\(632\) 0 0
\(633\) 1.87228 + 3.24289i 0.0744165 + 0.128893i
\(634\) 0 0
\(635\) −2.12772 + 3.68532i −0.0844359 + 0.146247i
\(636\) 0 0
\(637\) −0.686141 + 1.18843i −0.0271859 + 0.0470873i
\(638\) 0 0
\(639\) −6.55842 + 11.3595i −0.259447 + 0.449376i
\(640\) 0 0
\(641\) 22.6753 39.2747i 0.895619 1.55126i 0.0625827 0.998040i \(-0.480066\pi\)
0.833036 0.553218i \(-0.186600\pi\)
\(642\) 0 0
\(643\) 36.4674 1.43813 0.719066 0.694941i \(-0.244570\pi\)
0.719066 + 0.694941i \(0.244570\pi\)
\(644\) 0 0
\(645\) −3.37228 −0.132783
\(646\) 0 0
\(647\) −12.3030 21.3094i −0.483680 0.837759i 0.516144 0.856502i \(-0.327367\pi\)
−0.999824 + 0.0187430i \(0.994034\pi\)
\(648\) 0 0
\(649\) −6.00000 + 10.3923i −0.235521 + 0.407934i
\(650\) 0 0
\(651\) 9.18614 + 15.9109i 0.360033 + 0.623596i
\(652\) 0 0
\(653\) 14.4416 + 25.0135i 0.565143 + 0.978856i 0.997036 + 0.0769313i \(0.0245122\pi\)
−0.431894 + 0.901924i \(0.642154\pi\)
\(654\) 0 0
\(655\) −17.4891 −0.683357
\(656\) 0 0
\(657\) 6.50000 + 11.2583i 0.253589 + 0.439229i
\(658\) 0 0
\(659\) 6.81386 11.8020i 0.265430 0.459739i −0.702246 0.711934i \(-0.747819\pi\)
0.967676 + 0.252196i \(0.0811527\pi\)
\(660\) 0 0
\(661\) 15.8832 0.617783 0.308892 0.951097i \(-0.400042\pi\)
0.308892 + 0.951097i \(0.400042\pi\)
\(662\) 0 0
\(663\) 0.813859 1.40965i 0.0316077 0.0547461i
\(664\) 0 0
\(665\) 15.1168 0.586206
\(666\) 0 0
\(667\) 7.11684 0.275565
\(668\) 0 0
\(669\) −5.88316 −0.227456
\(670\) 0 0
\(671\) −48.0951 −1.85669
\(672\) 0 0
\(673\) 29.6060 1.14123 0.570613 0.821219i \(-0.306706\pi\)
0.570613 + 0.821219i \(0.306706\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 22.6753 39.2747i 0.871481 1.50945i 0.0110168 0.999939i \(-0.496493\pi\)
0.860464 0.509511i \(-0.170174\pi\)
\(678\) 0 0
\(679\) 2.37228 0.0910398
\(680\) 0 0
\(681\) 10.9307 18.9325i 0.418865 0.725496i
\(682\) 0 0
\(683\) 14.4416 + 25.0135i 0.552592 + 0.957117i 0.998087 + 0.0618323i \(0.0196944\pi\)
−0.445495 + 0.895284i \(0.646972\pi\)
\(684\) 0 0
\(685\) 20.7446 0.792609
\(686\) 0 0
\(687\) −9.61684 16.6569i −0.366905 0.635499i
\(688\) 0 0
\(689\) −4.37228 7.57301i −0.166571 0.288509i
\(690\) 0 0
\(691\) 18.6753 32.3465i 0.710441 1.23052i −0.254251 0.967138i \(-0.581829\pi\)
0.964692 0.263381i \(-0.0848376\pi\)
\(692\) 0 0
\(693\) −5.18614 8.98266i −0.197005 0.341223i
\(694\) 0 0
\(695\) −0.627719 −0.0238107
\(696\) 0 0
\(697\) −16.8832 −0.639495
\(698\) 0 0
\(699\) 3.81386 6.60580i 0.144253 0.249854i
\(700\) 0 0
\(701\) −6.81386 + 11.8020i −0.257356 + 0.445754i −0.965533 0.260282i \(-0.916185\pi\)
0.708177 + 0.706035i \(0.249518\pi\)
\(702\) 0 0
\(703\) −29.0475 + 50.3118i −1.09555 + 1.89755i
\(704\) 0 0
\(705\) −0.813859 + 1.40965i −0.0306517 + 0.0530903i
\(706\) 0 0
\(707\) −18.8139 32.5866i −0.707568 1.22554i
\(708\) 0 0
\(709\) 4.69702 + 8.13547i 0.176400 + 0.305534i 0.940645 0.339392i \(-0.110221\pi\)
−0.764245 + 0.644926i \(0.776888\pi\)
\(710\) 0 0
\(711\) 2.12772 3.68532i 0.0797957 0.138210i
\(712\) 0 0
\(713\) 12.6060 0.472097
\(714\) 0 0
\(715\) −4.37228 −0.163514
\(716\) 0 0
\(717\) 5.44158 9.42509i 0.203219 0.351986i
\(718\) 0 0
\(719\) −17.4416 30.2097i −0.650461 1.12663i −0.983011 0.183546i \(-0.941242\pi\)
0.332550 0.943086i \(-0.392091\pi\)
\(720\) 0 0
\(721\) 17.9307 + 31.0569i 0.667774 + 1.15662i
\(722\) 0 0
\(723\) 24.1168 0.896915
\(724\) 0 0
\(725\) −2.18614 + 3.78651i −0.0811912 + 0.140627i
\(726\) 0 0
\(727\) 10.6168 + 18.3889i 0.393757 + 0.682007i 0.992942 0.118604i \(-0.0378418\pi\)
−0.599185 + 0.800611i \(0.704508\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.74456 4.75372i −0.101511 0.175823i
\(732\) 0 0
\(733\) −22.5584 + 39.0723i −0.833214 + 1.44317i 0.0622617 + 0.998060i \(0.480169\pi\)
−0.895476 + 0.445110i \(0.853165\pi\)
\(734\) 0 0
\(735\) −0.686141 1.18843i −0.0253087 0.0438359i
\(736\) 0 0
\(737\) 28.4198 + 21.7518i 1.04686 + 0.801239i
\(738\) 0 0
\(739\) −2.24456 3.88770i −0.0825676 0.143011i 0.821785 0.569798i \(-0.192979\pi\)
−0.904352 + 0.426787i \(0.859645\pi\)
\(740\) 0 0
\(741\) 3.18614 5.51856i 0.117046 0.202729i
\(742\) 0 0
\(743\) 9.30298 + 16.1132i 0.341293 + 0.591138i 0.984673 0.174410i \(-0.0558017\pi\)
−0.643380 + 0.765547i \(0.722468\pi\)
\(744\) 0 0
\(745\) 8.74456 0.320376
\(746\) 0 0
\(747\) −3.55842 6.16337i −0.130196 0.225506i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.4674 −0.746865 −0.373433 0.927657i \(-0.621819\pi\)
−0.373433 + 0.927657i \(0.621819\pi\)
\(752\) 0 0
\(753\) −9.55842 16.5557i −0.348328 0.603322i
\(754\) 0 0
\(755\) −0.500000 0.866025i −0.0181969 0.0315179i
\(756\) 0 0
\(757\) −13.7337 + 23.7874i −0.499159 + 0.864569i −1.00000 0.000970394i \(-0.999691\pi\)
0.500840 + 0.865540i \(0.333024\pi\)
\(758\) 0 0
\(759\) −7.11684 −0.258325
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 14.3723 24.8935i 0.520312 0.901206i
\(764\) 0 0
\(765\) 0.813859 + 1.40965i 0.0294251 + 0.0509658i
\(766\) 0 0
\(767\) −1.37228 2.37686i −0.0495502 0.0858235i
\(768\) 0 0
\(769\) 11.9891 20.7658i 0.432339 0.748833i −0.564735 0.825272i \(-0.691022\pi\)
0.997074 + 0.0764391i \(0.0243551\pi\)
\(770\) 0 0
\(771\) 5.18614 8.98266i 0.186774 0.323503i
\(772\) 0 0
\(773\) −14.1861 + 24.5711i −0.510240 + 0.883762i 0.489690 + 0.871897i \(0.337110\pi\)
−0.999930 + 0.0118648i \(0.996223\pi\)
\(774\) 0 0
\(775\) −3.87228 + 6.70699i −0.139096 + 0.240922i
\(776\) 0 0
\(777\) −21.6277 −0.775890
\(778\) 0 0
\(779\) −66.0951 −2.36810
\(780\) 0 0
\(781\) 28.6753 + 49.6670i 1.02608 + 1.77723i
\(782\) 0 0
\(783\) −2.18614 + 3.78651i −0.0781263 + 0.135319i
\(784\) 0 0
\(785\) 1.81386 + 3.14170i 0.0647394 + 0.112132i
\(786\) 0 0
\(787\) 5.55842 + 9.62747i 0.198136 + 0.343182i 0.947924 0.318496i \(-0.103178\pi\)
−0.749788 + 0.661678i \(0.769844\pi\)
\(788\) 0 0
\(789\) 5.48913 0.195418
\(790\) 0 0
\(791\) −15.5584 26.9480i −0.553194 0.958160i
\(792\) 0 0
\(793\) 5.50000 9.52628i 0.195311 0.338288i
\(794\) 0 0
\(795\) 8.74456 0.310138
\(796\) 0 0
\(797\) 1.06930 1.85208i 0.0378764 0.0656039i −0.846466 0.532443i \(-0.821274\pi\)
0.884342 + 0.466839i \(0.154607\pi\)
\(798\) 0 0
\(799\) −2.64947 −0.0937314
\(800\) 0 0
\(801\) −11.4891 −0.405948
\(802\) 0 0
\(803\) 56.8397 2.00583
\(804\) 0 0
\(805\) 3.86141 0.136097
\(806\) 0 0
\(807\) 20.7446 0.730243
\(808\) 0 0
\(809\) −23.4891 −0.825834 −0.412917 0.910769i \(-0.635490\pi\)
−0.412917 + 0.910769i \(0.635490\pi\)
\(810\) 0 0
\(811\) −19.9891 + 34.6222i −0.701913 + 1.21575i 0.265881 + 0.964006i \(0.414337\pi\)
−0.967794 + 0.251743i \(0.918996\pi\)
\(812\) 0 0
\(813\) −26.1168 −0.915958
\(814\) 0 0
\(815\) −3.50000 + 6.06218i −0.122600 + 0.212349i
\(816\) 0 0
\(817\) −10.7446 18.6101i −0.375905 0.651086i
\(818\) 0 0
\(819\) 2.37228 0.0828942
\(820\) 0 0
\(821\) 6.55842 + 11.3595i 0.228891 + 0.396450i 0.957480 0.288501i \(-0.0931569\pi\)
−0.728589 + 0.684951i \(0.759824\pi\)
\(822\) 0 0
\(823\) 7.87228 + 13.6352i 0.274410 + 0.475293i 0.969986 0.243160i \(-0.0781840\pi\)
−0.695576 + 0.718453i \(0.744851\pi\)
\(824\) 0 0
\(825\) 2.18614 3.78651i 0.0761116 0.131829i
\(826\) 0 0
\(827\) 9.30298 + 16.1132i 0.323496 + 0.560312i 0.981207 0.192959i \(-0.0618083\pi\)
−0.657710 + 0.753271i \(0.728475\pi\)
\(828\) 0 0
\(829\) 41.6060 1.44504 0.722518 0.691353i \(-0.242985\pi\)
0.722518 + 0.691353i \(0.242985\pi\)
\(830\) 0 0
\(831\) 14.8614 0.515536
\(832\) 0 0
\(833\) 1.11684 1.93443i 0.0386964 0.0670240i
\(834\) 0 0
\(835\) −6.81386 + 11.8020i −0.235803 + 0.408423i
\(836\) 0 0
\(837\) −3.87228 + 6.70699i −0.133846 + 0.231827i
\(838\) 0 0
\(839\) −9.30298 + 16.1132i −0.321175 + 0.556291i −0.980731 0.195365i \(-0.937411\pi\)
0.659556 + 0.751655i \(0.270744\pi\)
\(840\) 0 0
\(841\) 4.94158 + 8.55906i 0.170399 + 0.295140i
\(842\) 0 0
\(843\) −6.81386 11.8020i −0.234682 0.406481i
\(844\) 0 0
\(845\) −6.00000 + 10.3923i −0.206406 + 0.357506i
\(846\) 0 0
\(847\) −19.2554 −0.661625
\(848\) 0 0
\(849\) 13.4891 0.462946
\(850\) 0 0
\(851\) −7.41983 + 12.8515i −0.254348 + 0.440544i
\(852\) 0 0
\(853\) 2.38316 + 4.12775i 0.0815977 + 0.141331i 0.903936 0.427667i \(-0.140664\pi\)
−0.822339 + 0.568998i \(0.807331\pi\)
\(854\) 0 0
\(855\) 3.18614 + 5.51856i 0.108964 + 0.188731i
\(856\) 0 0
\(857\) 14.2337 0.486214 0.243107 0.970000i \(-0.421833\pi\)
0.243107 + 0.970000i \(0.421833\pi\)
\(858\) 0 0
\(859\) −9.87228 + 17.0993i −0.336838 + 0.583420i −0.983836 0.179071i \(-0.942691\pi\)
0.646998 + 0.762491i \(0.276024\pi\)
\(860\) 0 0
\(861\) −12.3030 21.3094i −0.419285 0.726222i
\(862\) 0 0
\(863\) 31.2119 1.06247 0.531233 0.847226i \(-0.321729\pi\)
0.531233 + 0.847226i \(0.321729\pi\)
\(864\) 0 0
\(865\) −0.558422 0.967215i −0.0189869 0.0328863i
\(866\) 0 0
\(867\) 7.17527 12.4279i 0.243685 0.422074i
\(868\) 0 0
\(869\) −9.30298 16.1132i −0.315582 0.546604i
\(870\) 0 0
\(871\) −7.55842 + 3.14170i −0.256107 + 0.106452i
\(872\) 0 0
\(873\) 0.500000 + 0.866025i 0.0169224 + 0.0293105i
\(874\) 0 0
\(875\) −1.18614 + 2.05446i −0.0400989 + 0.0694533i
\(876\) 0 0
\(877\) −18.3614 31.8029i −0.620021 1.07391i −0.989481 0.144661i \(-0.953791\pi\)
0.369461 0.929246i \(-0.379542\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −14.1861 24.5711i −0.477943 0.827822i 0.521737 0.853106i \(-0.325284\pi\)
−0.999680 + 0.0252844i \(0.991951\pi\)
\(882\) 0 0
\(883\) −16.3030 + 28.2376i −0.548639 + 0.950271i 0.449729 + 0.893165i \(0.351521\pi\)
−0.998368 + 0.0571058i \(0.981813\pi\)
\(884\) 0 0
\(885\) 2.74456 0.0922575
\(886\) 0 0
\(887\) 18.0475 + 31.2593i 0.605977 + 1.04958i 0.991896 + 0.127051i \(0.0405511\pi\)
−0.385919 + 0.922533i \(0.626116\pi\)
\(888\) 0 0
\(889\) −5.04755 8.74261i −0.169289 0.293218i
\(890\) 0 0
\(891\) 2.18614 3.78651i 0.0732385 0.126853i
\(892\) 0 0
\(893\) −10.3723 −0.347095
\(894\) 0 0
\(895\) 14.7446 0.492856
\(896\) 0 0
\(897\) 0.813859 1.40965i 0.0271740 0.0470667i
\(898\) 0 0
\(899\) −16.9307 29.3248i −0.564671 0.978038i
\(900\) 0 0
\(901\) 7.11684 + 12.3267i 0.237096 + 0.410663i
\(902\) 0 0
\(903\) 4.00000 6.92820i 0.133112 0.230556i
\(904\) 0 0
\(905\) 6.87228 11.9031i 0.228442 0.395674i
\(906\) 0 0
\(907\) 12.6753 21.9542i 0.420875 0.728977i −0.575150 0.818048i \(-0.695056\pi\)
0.996025 + 0.0890706i \(0.0283897\pi\)
\(908\) 0 0
\(909\) 7.93070 13.7364i 0.263045 0.455607i
\(910\) 0 0
\(911\) 58.9783 1.95404 0.977018 0.213155i \(-0.0683741\pi\)
0.977018 + 0.213155i \(0.0683741\pi\)
\(912\) 0 0
\(913\) −31.1168 −1.02982
\(914\) 0 0
\(915\) 5.50000 + 9.52628i 0.181824 + 0.314929i
\(916\) 0 0
\(917\) 20.7446 35.9306i 0.685046 1.18653i
\(918\) 0 0
\(919\) −0.441578 0.764836i −0.0145663 0.0252296i 0.858650 0.512562i \(-0.171303\pi\)
−0.873217 + 0.487332i \(0.837970\pi\)
\(920\) 0 0
\(921\) 0.500000 + 0.866025i 0.0164756 + 0.0285365i
\(922\) 0 0
\(923\) −13.1168 −0.431746
\(924\) 0 0
\(925\) −4.55842 7.89542i −0.149880 0.259600i
\(926\) 0 0
\(927\) −7.55842 + 13.0916i −0.248251 + 0.429984i
\(928\) 0 0
\(929\) 50.2337 1.64811 0.824057 0.566507i \(-0.191706\pi\)
0.824057 + 0.566507i \(0.191706\pi\)
\(930\) 0 0
\(931\) 4.37228 7.57301i 0.143296 0.248195i
\(932\) 0 0
\(933\) 20.2337 0.662421
\(934\) 0 0
\(935\) 7.11684 0.232746
\(936\) 0 0
\(937\) 24.1168 0.787863 0.393931 0.919140i \(-0.371115\pi\)
0.393931 + 0.919140i \(0.371115\pi\)
\(938\) 0 0
\(939\) 19.4891 0.636004
\(940\) 0 0
\(941\) −12.5109 −0.407843 −0.203921 0.978987i \(-0.565369\pi\)
−0.203921 + 0.978987i \(0.565369\pi\)
\(942\) 0 0
\(943\) −16.8832 −0.549791
\(944\) 0 0
\(945\) −1.18614 + 2.05446i −0.0385852 + 0.0668315i
\(946\) 0 0
\(947\) 57.9565 1.88333 0.941667 0.336547i \(-0.109259\pi\)
0.941667 + 0.336547i \(0.109259\pi\)
\(948\) 0 0
\(949\) −6.50000 + 11.2583i −0.210999 + 0.365461i
\(950\) 0 0
\(951\) 0.558422 + 0.967215i 0.0181081 + 0.0313641i
\(952\) 0 0
\(953\) −3.25544 −0.105454 −0.0527270 0.998609i \(-0.516791\pi\)
−0.0527270 + 0.998609i \(0.516791\pi\)
\(954\) 0 0
\(955\) 9.55842 + 16.5557i 0.309303 + 0.535729i
\(956\) 0 0
\(957\) 9.55842 + 16.5557i 0.308980 + 0.535169i
\(958\) 0 0
\(959\) −24.6060 + 42.6188i −0.794568 + 1.37623i
\(960\) 0 0
\(961\) −14.4891 25.0959i −0.467391 0.809545i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.116844 −0.00376134
\(966\) 0 0
\(967\) −22.7337 + 39.3759i −0.731066 + 1.26624i 0.225362 + 0.974275i \(0.427644\pi\)
−0.956428 + 0.291969i \(0.905690\pi\)
\(968\) 0 0
\(969\) −5.18614 + 8.98266i −0.166603 + 0.288565i
\(970\) 0 0
\(971\) 2.18614 3.78651i 0.0701566 0.121515i −0.828813 0.559525i \(-0.810983\pi\)
0.898970 + 0.438011i \(0.144317\pi\)
\(972\) 0 0
\(973\) 0.744563 1.28962i 0.0238696 0.0413433i
\(974\) 0 0
\(975\) 0.500000 + 0.866025i 0.0160128 + 0.0277350i
\(976\) 0 0
\(977\) −10.9307 18.9325i −0.349704 0.605705i 0.636493 0.771283i \(-0.280385\pi\)
−0.986197 + 0.165577i \(0.947051\pi\)
\(978\) 0 0
\(979\) −25.1168 + 43.5036i −0.802738 + 1.39038i
\(980\) 0 0
\(981\) 12.1168 0.386861
\(982\) 0 0
\(983\) −13.0217 −0.415329 −0.207665 0.978200i \(-0.566586\pi\)
−0.207665 + 0.978200i \(0.566586\pi\)
\(984\) 0 0
\(985\) −5.18614 + 8.98266i −0.165244 + 0.286211i
\(986\) 0 0
\(987\) −1.93070 3.34408i −0.0614550 0.106443i
\(988\) 0 0
\(989\) −2.74456 4.75372i −0.0872720 0.151160i
\(990\) 0 0
\(991\) −56.4674 −1.79375 −0.896873 0.442289i \(-0.854167\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(992\) 0 0
\(993\) −7.98913 + 13.8376i −0.253527 + 0.439122i
\(994\) 0 0
\(995\) −1.87228 3.24289i −0.0593553 0.102806i
\(996\) 0 0
\(997\) −38.4674 −1.21827 −0.609137 0.793065i \(-0.708484\pi\)
−0.609137 + 0.793065i \(0.708484\pi\)
\(998\) 0 0
\(999\) −4.55842 7.89542i −0.144222 0.249800i
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 4020.2.q.i.841.2 4
67.29 even 3 inner 4020.2.q.i.3781.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.i.841.2 4 1.1 even 1 trivial
4020.2.q.i.3781.2 yes 4 67.29 even 3 inner