Properties

Label 840.2.k.b.209.1
Level $840$
Weight $2$
Character 840.209
Analytic conductor $6.707$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(209,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.k (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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.1
Character \(\chi\) \(=\) 840.209
Dual form 840.2.k.b.209.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.70556 - 0.301745i) q^{3} +(-2.04579 - 0.902636i) q^{5} +(1.30232 + 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} +O(q^{10})\) \(q+(-1.70556 - 0.301745i) q^{3} +(-2.04579 - 0.902636i) q^{5} +(1.30232 + 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} -5.71375i q^{11} -3.76758 q^{13} +(3.21686 + 2.15681i) q^{15} +3.22968i q^{17} -0.786936i q^{19} +(-1.52625 - 4.32094i) q^{21} -1.52714 q^{23} +(3.37050 + 3.69320i) q^{25} +(-4.49553 - 2.60581i) q^{27} +6.77221i q^{29} +1.56709i q^{31} +(-1.72410 + 9.74517i) q^{33} +(-0.585457 - 5.88704i) q^{35} +8.83080i q^{37} +(6.42585 + 1.13685i) q^{39} +6.65914 q^{41} +8.85072i q^{43} +(-4.83575 - 4.64925i) q^{45} +1.75054i q^{47} +(-3.60795 + 5.99856i) q^{49} +(0.974541 - 5.50843i) q^{51} -0.616602 q^{53} +(-5.15744 + 11.6891i) q^{55} +(-0.237454 + 1.34217i) q^{57} +6.38642 q^{59} +14.4554i q^{61} +(1.29930 + 7.83019i) q^{63} +(7.70767 + 3.40075i) q^{65} -6.26557i q^{67} +(2.60463 + 0.460806i) q^{69} -5.09956i q^{71} +12.5631 q^{73} +(-4.63419 - 7.31603i) q^{75} +(13.1590 - 7.44111i) q^{77} -12.8517 q^{79} +(6.88112 + 5.80088i) q^{81} +10.5003i q^{83} +(2.91523 - 6.60724i) q^{85} +(2.04348 - 11.5504i) q^{87} -15.6873 q^{89} +(-4.90658 - 8.67687i) q^{91} +(0.472863 - 2.67278i) q^{93} +(-0.710317 + 1.60990i) q^{95} -7.69604 q^{97} +(5.88112 - 16.1008i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{9} + 2 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} - 2 q^{39} + 6 q^{51} + 24 q^{53} + 8 q^{57} + 16 q^{63} + 16 q^{65} + 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} + 32 q^{93} - 24 q^{95} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.70556 0.301745i −0.984708 0.174213i
\(4\) 0 0
\(5\) −2.04579 0.902636i −0.914904 0.403671i
\(6\) 0 0
\(7\) 1.30232 + 2.30304i 0.492229 + 0.870466i
\(8\) 0 0
\(9\) 2.81790 + 1.02929i 0.939300 + 0.343097i
\(10\) 0 0
\(11\) 5.71375i 1.72276i −0.507960 0.861381i \(-0.669600\pi\)
0.507960 0.861381i \(-0.330400\pi\)
\(12\) 0 0
\(13\) −3.76758 −1.04494 −0.522469 0.852658i \(-0.674989\pi\)
−0.522469 + 0.852658i \(0.674989\pi\)
\(14\) 0 0
\(15\) 3.21686 + 2.15681i 0.830589 + 0.556886i
\(16\) 0 0
\(17\) 3.22968i 0.783313i 0.920112 + 0.391656i \(0.128098\pi\)
−0.920112 + 0.391656i \(0.871902\pi\)
\(18\) 0 0
\(19\) 0.786936i 0.180536i −0.995918 0.0902678i \(-0.971228\pi\)
0.995918 0.0902678i \(-0.0287723\pi\)
\(20\) 0 0
\(21\) −1.52625 4.32094i −0.333056 0.942907i
\(22\) 0 0
\(23\) −1.52714 −0.318430 −0.159215 0.987244i \(-0.550896\pi\)
−0.159215 + 0.987244i \(0.550896\pi\)
\(24\) 0 0
\(25\) 3.37050 + 3.69320i 0.674099 + 0.738641i
\(26\) 0 0
\(27\) −4.49553 2.60581i −0.865164 0.501489i
\(28\) 0 0
\(29\) 6.77221i 1.25757i 0.777580 + 0.628784i \(0.216447\pi\)
−0.777580 + 0.628784i \(0.783553\pi\)
\(30\) 0 0
\(31\) 1.56709i 0.281458i 0.990048 + 0.140729i \(0.0449447\pi\)
−0.990048 + 0.140729i \(0.955055\pi\)
\(32\) 0 0
\(33\) −1.72410 + 9.74517i −0.300127 + 1.69642i
\(34\) 0 0
\(35\) −0.585457 5.88704i −0.0989603 0.995091i
\(36\) 0 0
\(37\) 8.83080i 1.45177i 0.687814 + 0.725887i \(0.258570\pi\)
−0.687814 + 0.725887i \(0.741430\pi\)
\(38\) 0 0
\(39\) 6.42585 + 1.13685i 1.02896 + 0.182042i
\(40\) 0 0
\(41\) 6.65914 1.03998 0.519992 0.854171i \(-0.325935\pi\)
0.519992 + 0.854171i \(0.325935\pi\)
\(42\) 0 0
\(43\) 8.85072i 1.34972i 0.737945 + 0.674861i \(0.235796\pi\)
−0.737945 + 0.674861i \(0.764204\pi\)
\(44\) 0 0
\(45\) −4.83575 4.64925i −0.720871 0.693069i
\(46\) 0 0
\(47\) 1.75054i 0.255342i 0.991817 + 0.127671i \(0.0407502\pi\)
−0.991817 + 0.127671i \(0.959250\pi\)
\(48\) 0 0
\(49\) −3.60795 + 5.99856i −0.515421 + 0.856937i
\(50\) 0 0
\(51\) 0.974541 5.50843i 0.136463 0.771335i
\(52\) 0 0
\(53\) −0.616602 −0.0846967 −0.0423484 0.999103i \(-0.513484\pi\)
−0.0423484 + 0.999103i \(0.513484\pi\)
\(54\) 0 0
\(55\) −5.15744 + 11.6891i −0.695429 + 1.57616i
\(56\) 0 0
\(57\) −0.237454 + 1.34217i −0.0314516 + 0.177775i
\(58\) 0 0
\(59\) 6.38642 0.831441 0.415721 0.909492i \(-0.363529\pi\)
0.415721 + 0.909492i \(0.363529\pi\)
\(60\) 0 0
\(61\) 14.4554i 1.85082i 0.378964 + 0.925412i \(0.376281\pi\)
−0.378964 + 0.925412i \(0.623719\pi\)
\(62\) 0 0
\(63\) 1.29930 + 7.83019i 0.163696 + 0.986511i
\(64\) 0 0
\(65\) 7.70767 + 3.40075i 0.956019 + 0.421812i
\(66\) 0 0
\(67\) 6.26557i 0.765462i −0.923860 0.382731i \(-0.874984\pi\)
0.923860 0.382731i \(-0.125016\pi\)
\(68\) 0 0
\(69\) 2.60463 + 0.460806i 0.313561 + 0.0554746i
\(70\) 0 0
\(71\) 5.09956i 0.605206i −0.953117 0.302603i \(-0.902144\pi\)
0.953117 0.302603i \(-0.0978556\pi\)
\(72\) 0 0
\(73\) 12.5631 1.47039 0.735197 0.677853i \(-0.237089\pi\)
0.735197 + 0.677853i \(0.237089\pi\)
\(74\) 0 0
\(75\) −4.63419 7.31603i −0.535110 0.844782i
\(76\) 0 0
\(77\) 13.1590 7.44111i 1.49960 0.847993i
\(78\) 0 0
\(79\) −12.8517 −1.44593 −0.722965 0.690885i \(-0.757221\pi\)
−0.722965 + 0.690885i \(0.757221\pi\)
\(80\) 0 0
\(81\) 6.88112 + 5.80088i 0.764569 + 0.644542i
\(82\) 0 0
\(83\) 10.5003i 1.15256i 0.817254 + 0.576278i \(0.195495\pi\)
−0.817254 + 0.576278i \(0.804505\pi\)
\(84\) 0 0
\(85\) 2.91523 6.60724i 0.316201 0.716656i
\(86\) 0 0
\(87\) 2.04348 11.5504i 0.219084 1.23834i
\(88\) 0 0
\(89\) −15.6873 −1.66285 −0.831423 0.555640i \(-0.812473\pi\)
−0.831423 + 0.555640i \(0.812473\pi\)
\(90\) 0 0
\(91\) −4.90658 8.67687i −0.514349 0.909583i
\(92\) 0 0
\(93\) 0.472863 2.67278i 0.0490336 0.277154i
\(94\) 0 0
\(95\) −0.710317 + 1.60990i −0.0728770 + 0.165173i
\(96\) 0 0
\(97\) −7.69604 −0.781414 −0.390707 0.920515i \(-0.627769\pi\)
−0.390707 + 0.920515i \(0.627769\pi\)
\(98\) 0 0
\(99\) 5.88112 16.1008i 0.591075 1.61819i
\(100\) 0 0
\(101\) −10.2658 −1.02149 −0.510745 0.859732i \(-0.670630\pi\)
−0.510745 + 0.859732i \(0.670630\pi\)
\(102\) 0 0
\(103\) 3.62328 0.357013 0.178506 0.983939i \(-0.442873\pi\)
0.178506 + 0.983939i \(0.442873\pi\)
\(104\) 0 0
\(105\) −0.777851 + 10.2174i −0.0759105 + 0.997115i
\(106\) 0 0
\(107\) −8.65500 −0.836710 −0.418355 0.908284i \(-0.637393\pi\)
−0.418355 + 0.908284i \(0.637393\pi\)
\(108\) 0 0
\(109\) 8.32109 0.797016 0.398508 0.917165i \(-0.369528\pi\)
0.398508 + 0.917165i \(0.369528\pi\)
\(110\) 0 0
\(111\) 2.66465 15.0615i 0.252917 1.42957i
\(112\) 0 0
\(113\) −14.0986 −1.32628 −0.663142 0.748494i \(-0.730777\pi\)
−0.663142 + 0.748494i \(0.730777\pi\)
\(114\) 0 0
\(115\) 3.12420 + 1.37845i 0.291333 + 0.128541i
\(116\) 0 0
\(117\) −10.6167 3.87794i −0.981511 0.358516i
\(118\) 0 0
\(119\) −7.43807 + 4.20606i −0.681847 + 0.385569i
\(120\) 0 0
\(121\) −21.6470 −1.96791
\(122\) 0 0
\(123\) −11.3576 2.00936i −1.02408 0.181178i
\(124\) 0 0
\(125\) −3.56170 10.5978i −0.318568 0.947900i
\(126\) 0 0
\(127\) 12.7643i 1.13265i −0.824181 0.566326i \(-0.808364\pi\)
0.824181 0.566326i \(-0.191636\pi\)
\(128\) 0 0
\(129\) 2.67066 15.0955i 0.235139 1.32908i
\(130\) 0 0
\(131\) 16.9991 1.48522 0.742610 0.669724i \(-0.233588\pi\)
0.742610 + 0.669724i \(0.233588\pi\)
\(132\) 0 0
\(133\) 1.81234 1.02484i 0.157150 0.0888648i
\(134\) 0 0
\(135\) 6.84479 + 9.38876i 0.589106 + 0.808056i
\(136\) 0 0
\(137\) 12.3946 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(138\) 0 0
\(139\) 11.2465i 0.953912i 0.878927 + 0.476956i \(0.158260\pi\)
−0.878927 + 0.476956i \(0.841740\pi\)
\(140\) 0 0
\(141\) 0.528217 2.98566i 0.0444839 0.251438i
\(142\) 0 0
\(143\) 21.5270i 1.80018i
\(144\) 0 0
\(145\) 6.11284 13.8545i 0.507644 1.15055i
\(146\) 0 0
\(147\) 7.96363 9.14225i 0.656829 0.754040i
\(148\) 0 0
\(149\) 2.85057i 0.233528i 0.993160 + 0.116764i \(0.0372521\pi\)
−0.993160 + 0.116764i \(0.962748\pi\)
\(150\) 0 0
\(151\) 16.1515 1.31439 0.657197 0.753719i \(-0.271742\pi\)
0.657197 + 0.753719i \(0.271742\pi\)
\(152\) 0 0
\(153\) −3.32428 + 9.10092i −0.268752 + 0.735766i
\(154\) 0 0
\(155\) 1.41452 3.20594i 0.113617 0.257507i
\(156\) 0 0
\(157\) 7.21577 0.575881 0.287941 0.957648i \(-0.407029\pi\)
0.287941 + 0.957648i \(0.407029\pi\)
\(158\) 0 0
\(159\) 1.05165 + 0.186057i 0.0834016 + 0.0147552i
\(160\) 0 0
\(161\) −1.98881 3.51705i −0.156740 0.277182i
\(162\) 0 0
\(163\) 3.50516i 0.274545i 0.990533 + 0.137273i \(0.0438337\pi\)
−0.990533 + 0.137273i \(0.956166\pi\)
\(164\) 0 0
\(165\) 12.3235 18.3803i 0.959382 1.43091i
\(166\) 0 0
\(167\) 1.38365i 0.107070i 0.998566 + 0.0535349i \(0.0170489\pi\)
−0.998566 + 0.0535349i \(0.982951\pi\)
\(168\) 0 0
\(169\) 1.19466 0.0918967
\(170\) 0 0
\(171\) 0.809987 2.21751i 0.0619413 0.169577i
\(172\) 0 0
\(173\) 10.1282i 0.770032i 0.922910 + 0.385016i \(0.125804\pi\)
−0.922910 + 0.385016i \(0.874196\pi\)
\(174\) 0 0
\(175\) −4.11613 + 12.5721i −0.311151 + 0.950361i
\(176\) 0 0
\(177\) −10.8925 1.92707i −0.818727 0.144848i
\(178\) 0 0
\(179\) 0.542452i 0.0405447i 0.999794 + 0.0202724i \(0.00645334\pi\)
−0.999794 + 0.0202724i \(0.993547\pi\)
\(180\) 0 0
\(181\) 1.15745i 0.0860323i −0.999074 0.0430162i \(-0.986303\pi\)
0.999074 0.0430162i \(-0.0136967\pi\)
\(182\) 0 0
\(183\) 4.36184 24.6546i 0.322437 1.82252i
\(184\) 0 0
\(185\) 7.97100 18.0659i 0.586039 1.32823i
\(186\) 0 0
\(187\) 18.4536 1.34946
\(188\) 0 0
\(189\) 0.146685 13.7469i 0.0106697 0.999943i
\(190\) 0 0
\(191\) 10.8558i 0.785495i 0.919646 + 0.392748i \(0.128475\pi\)
−0.919646 + 0.392748i \(0.871525\pi\)
\(192\) 0 0
\(193\) 24.6409i 1.77369i −0.462067 0.886845i \(-0.652892\pi\)
0.462067 0.886845i \(-0.347108\pi\)
\(194\) 0 0
\(195\) −12.1198 8.12596i −0.867914 0.581912i
\(196\) 0 0
\(197\) −3.15816 −0.225009 −0.112505 0.993651i \(-0.535887\pi\)
−0.112505 + 0.993651i \(0.535887\pi\)
\(198\) 0 0
\(199\) 3.60377i 0.255464i −0.991809 0.127732i \(-0.959230\pi\)
0.991809 0.127732i \(-0.0407698\pi\)
\(200\) 0 0
\(201\) −1.89061 + 10.6863i −0.133353 + 0.753756i
\(202\) 0 0
\(203\) −15.5966 + 8.81956i −1.09467 + 0.619011i
\(204\) 0 0
\(205\) −13.6232 6.01078i −0.951485 0.419811i
\(206\) 0 0
\(207\) −4.30332 1.57187i −0.299101 0.109252i
\(208\) 0 0
\(209\) −4.49636 −0.311020
\(210\) 0 0
\(211\) 0.159466 0.0109781 0.00548905 0.999985i \(-0.498253\pi\)
0.00548905 + 0.999985i \(0.498253\pi\)
\(212\) 0 0
\(213\) −1.53877 + 8.69762i −0.105435 + 0.595951i
\(214\) 0 0
\(215\) 7.98898 18.1067i 0.544844 1.23487i
\(216\) 0 0
\(217\) −3.60907 + 2.04085i −0.245000 + 0.138542i
\(218\) 0 0
\(219\) −21.4271 3.79084i −1.44791 0.256161i
\(220\) 0 0
\(221\) 12.1681i 0.818514i
\(222\) 0 0
\(223\) 4.39412 0.294252 0.147126 0.989118i \(-0.452998\pi\)
0.147126 + 0.989118i \(0.452998\pi\)
\(224\) 0 0
\(225\) 5.69633 + 13.8763i 0.379756 + 0.925087i
\(226\) 0 0
\(227\) 19.1642i 1.27197i −0.771700 0.635987i \(-0.780593\pi\)
0.771700 0.635987i \(-0.219407\pi\)
\(228\) 0 0
\(229\) 1.25651i 0.0830328i −0.999138 0.0415164i \(-0.986781\pi\)
0.999138 0.0415164i \(-0.0132189\pi\)
\(230\) 0 0
\(231\) −24.6888 + 8.72063i −1.62440 + 0.573775i
\(232\) 0 0
\(233\) −5.73752 −0.375878 −0.187939 0.982181i \(-0.560181\pi\)
−0.187939 + 0.982181i \(0.560181\pi\)
\(234\) 0 0
\(235\) 1.58010 3.58123i 0.103074 0.233614i
\(236\) 0 0
\(237\) 21.9194 + 3.87794i 1.42382 + 0.251899i
\(238\) 0 0
\(239\) 18.1836i 1.17620i 0.808789 + 0.588099i \(0.200123\pi\)
−0.808789 + 0.588099i \(0.799877\pi\)
\(240\) 0 0
\(241\) 18.1259i 1.16759i 0.811901 + 0.583795i \(0.198433\pi\)
−0.811901 + 0.583795i \(0.801567\pi\)
\(242\) 0 0
\(243\) −9.98580 11.9701i −0.640589 0.767884i
\(244\) 0 0
\(245\) 12.7956 9.01511i 0.817482 0.575954i
\(246\) 0 0
\(247\) 2.96485i 0.188649i
\(248\) 0 0
\(249\) 3.16841 17.9089i 0.200790 1.13493i
\(250\) 0 0
\(251\) −0.598823 −0.0377974 −0.0188987 0.999821i \(-0.506016\pi\)
−0.0188987 + 0.999821i \(0.506016\pi\)
\(252\) 0 0
\(253\) 8.72568i 0.548579i
\(254\) 0 0
\(255\) −6.96581 + 10.3894i −0.436216 + 0.650611i
\(256\) 0 0
\(257\) 15.4070i 0.961063i −0.876977 0.480532i \(-0.840444\pi\)
0.876977 0.480532i \(-0.159556\pi\)
\(258\) 0 0
\(259\) −20.3376 + 11.5005i −1.26372 + 0.714605i
\(260\) 0 0
\(261\) −6.97058 + 19.0834i −0.431468 + 1.18123i
\(262\) 0 0
\(263\) −9.66296 −0.595843 −0.297922 0.954590i \(-0.596293\pi\)
−0.297922 + 0.954590i \(0.596293\pi\)
\(264\) 0 0
\(265\) 1.26144 + 0.556567i 0.0774894 + 0.0341896i
\(266\) 0 0
\(267\) 26.7556 + 4.73355i 1.63742 + 0.289689i
\(268\) 0 0
\(269\) 13.2397 0.807241 0.403620 0.914927i \(-0.367752\pi\)
0.403620 + 0.914927i \(0.367752\pi\)
\(270\) 0 0
\(271\) 15.6372i 0.949890i −0.880016 0.474945i \(-0.842468\pi\)
0.880016 0.474945i \(-0.157532\pi\)
\(272\) 0 0
\(273\) 5.75028 + 16.2795i 0.348023 + 0.985280i
\(274\) 0 0
\(275\) 21.1021 19.2582i 1.27250 1.16131i
\(276\) 0 0
\(277\) 3.48524i 0.209408i −0.994503 0.104704i \(-0.966611\pi\)
0.994503 0.104704i \(-0.0333895\pi\)
\(278\) 0 0
\(279\) −1.61300 + 4.41591i −0.0965676 + 0.264374i
\(280\) 0 0
\(281\) 21.4311i 1.27847i 0.769011 + 0.639236i \(0.220749\pi\)
−0.769011 + 0.639236i \(0.779251\pi\)
\(282\) 0 0
\(283\) −7.97240 −0.473910 −0.236955 0.971521i \(-0.576149\pi\)
−0.236955 + 0.971521i \(0.576149\pi\)
\(284\) 0 0
\(285\) 1.69727 2.53146i 0.100538 0.149951i
\(286\) 0 0
\(287\) 8.67230 + 15.3362i 0.511910 + 0.905270i
\(288\) 0 0
\(289\) 6.56915 0.386421
\(290\) 0 0
\(291\) 13.1261 + 2.32224i 0.769465 + 0.136132i
\(292\) 0 0
\(293\) 12.8728i 0.752035i 0.926613 + 0.376018i \(0.122707\pi\)
−0.926613 + 0.376018i \(0.877293\pi\)
\(294\) 0 0
\(295\) −13.0653 5.76461i −0.760689 0.335629i
\(296\) 0 0
\(297\) −14.8890 + 25.6863i −0.863945 + 1.49047i
\(298\) 0 0
\(299\) 5.75361 0.332740
\(300\) 0 0
\(301\) −20.3835 + 11.5264i −1.17489 + 0.664372i
\(302\) 0 0
\(303\) 17.5091 + 3.09767i 1.00587 + 0.177956i
\(304\) 0 0
\(305\) 13.0480 29.5727i 0.747124 1.69333i
\(306\) 0 0
\(307\) −7.16507 −0.408932 −0.204466 0.978874i \(-0.565546\pi\)
−0.204466 + 0.978874i \(0.565546\pi\)
\(308\) 0 0
\(309\) −6.17974 1.09331i −0.351553 0.0621961i
\(310\) 0 0
\(311\) −34.4684 −1.95453 −0.977263 0.212030i \(-0.931992\pi\)
−0.977263 + 0.212030i \(0.931992\pi\)
\(312\) 0 0
\(313\) 28.5587 1.61423 0.807117 0.590392i \(-0.201027\pi\)
0.807117 + 0.590392i \(0.201027\pi\)
\(314\) 0 0
\(315\) 4.40972 17.1917i 0.248460 0.968642i
\(316\) 0 0
\(317\) −20.9809 −1.17840 −0.589202 0.807986i \(-0.700558\pi\)
−0.589202 + 0.807986i \(0.700558\pi\)
\(318\) 0 0
\(319\) 38.6947 2.16649
\(320\) 0 0
\(321\) 14.7617 + 2.61160i 0.823915 + 0.145766i
\(322\) 0 0
\(323\) 2.54155 0.141416
\(324\) 0 0
\(325\) −12.6986 13.9144i −0.704392 0.771834i
\(326\) 0 0
\(327\) −14.1922 2.51085i −0.784828 0.138850i
\(328\) 0 0
\(329\) −4.03156 + 2.27975i −0.222267 + 0.125687i
\(330\) 0 0
\(331\) 10.9404 0.601340 0.300670 0.953728i \(-0.402790\pi\)
0.300670 + 0.953728i \(0.402790\pi\)
\(332\) 0 0
\(333\) −9.08946 + 24.8843i −0.498100 + 1.36365i
\(334\) 0 0
\(335\) −5.65553 + 12.8180i −0.308995 + 0.700324i
\(336\) 0 0
\(337\) 5.98022i 0.325763i −0.986646 0.162882i \(-0.947921\pi\)
0.986646 0.162882i \(-0.0520789\pi\)
\(338\) 0 0
\(339\) 24.0460 + 4.25418i 1.30600 + 0.231055i
\(340\) 0 0
\(341\) 8.95399 0.484886
\(342\) 0 0
\(343\) −18.5136 0.497225i −0.999640 0.0268476i
\(344\) 0 0
\(345\) −4.91258 3.29375i −0.264484 0.177329i
\(346\) 0 0
\(347\) −32.3200 −1.73503 −0.867514 0.497412i \(-0.834284\pi\)
−0.867514 + 0.497412i \(0.834284\pi\)
\(348\) 0 0
\(349\) 12.9630i 0.693896i −0.937884 0.346948i \(-0.887218\pi\)
0.937884 0.346948i \(-0.112782\pi\)
\(350\) 0 0
\(351\) 16.9373 + 9.81760i 0.904044 + 0.524025i
\(352\) 0 0
\(353\) 18.4705i 0.983083i −0.870854 0.491541i \(-0.836434\pi\)
0.870854 0.491541i \(-0.163566\pi\)
\(354\) 0 0
\(355\) −4.60304 + 10.4326i −0.244304 + 0.553705i
\(356\) 0 0
\(357\) 13.9553 4.92931i 0.738591 0.260887i
\(358\) 0 0
\(359\) 24.2434i 1.27952i 0.768576 + 0.639759i \(0.220966\pi\)
−0.768576 + 0.639759i \(0.779034\pi\)
\(360\) 0 0
\(361\) 18.3807 0.967407
\(362\) 0 0
\(363\) 36.9203 + 6.53187i 1.93781 + 0.342834i
\(364\) 0 0
\(365\) −25.7014 11.3399i −1.34527 0.593556i
\(366\) 0 0
\(367\) −1.55647 −0.0812468 −0.0406234 0.999175i \(-0.512934\pi\)
−0.0406234 + 0.999175i \(0.512934\pi\)
\(368\) 0 0
\(369\) 18.7648 + 6.85420i 0.976856 + 0.356815i
\(370\) 0 0
\(371\) −0.803010 1.42006i −0.0416902 0.0737256i
\(372\) 0 0
\(373\) 22.0453i 1.14146i 0.821137 + 0.570731i \(0.193340\pi\)
−0.821137 + 0.570731i \(0.806660\pi\)
\(374\) 0 0
\(375\) 2.87686 + 19.1500i 0.148560 + 0.988903i
\(376\) 0 0
\(377\) 25.5149i 1.31408i
\(378\) 0 0
\(379\) 9.42384 0.484070 0.242035 0.970268i \(-0.422185\pi\)
0.242035 + 0.970268i \(0.422185\pi\)
\(380\) 0 0
\(381\) −3.85158 + 21.7704i −0.197322 + 1.11533i
\(382\) 0 0
\(383\) 27.3334i 1.39667i 0.715771 + 0.698335i \(0.246075\pi\)
−0.715771 + 0.698335i \(0.753925\pi\)
\(384\) 0 0
\(385\) −33.6371 + 3.34516i −1.71430 + 0.170485i
\(386\) 0 0
\(387\) −9.10997 + 24.9404i −0.463086 + 1.26779i
\(388\) 0 0
\(389\) 13.3761i 0.678195i 0.940751 + 0.339097i \(0.110122\pi\)
−0.940751 + 0.339097i \(0.889878\pi\)
\(390\) 0 0
\(391\) 4.93217i 0.249430i
\(392\) 0 0
\(393\) −28.9931 5.12940i −1.46251 0.258744i
\(394\) 0 0
\(395\) 26.2919 + 11.6004i 1.32289 + 0.583680i
\(396\) 0 0
\(397\) −9.36101 −0.469815 −0.234908 0.972018i \(-0.575479\pi\)
−0.234908 + 0.972018i \(0.575479\pi\)
\(398\) 0 0
\(399\) −3.40031 + 1.20106i −0.170228 + 0.0601284i
\(400\) 0 0
\(401\) 24.5115i 1.22405i −0.790840 0.612024i \(-0.790356\pi\)
0.790840 0.612024i \(-0.209644\pi\)
\(402\) 0 0
\(403\) 5.90415i 0.294107i
\(404\) 0 0
\(405\) −8.84122 18.0785i −0.439324 0.898329i
\(406\) 0 0
\(407\) 50.4570 2.50106
\(408\) 0 0
\(409\) 0.0679660i 0.00336070i 0.999999 + 0.00168035i \(0.000534872\pi\)
−0.999999 + 0.00168035i \(0.999465\pi\)
\(410\) 0 0
\(411\) −21.1397 3.74000i −1.04275 0.184481i
\(412\) 0 0
\(413\) 8.31713 + 14.7082i 0.409259 + 0.723741i
\(414\) 0 0
\(415\) 9.47794 21.4814i 0.465254 1.05448i
\(416\) 0 0
\(417\) 3.39356 19.1816i 0.166184 0.939325i
\(418\) 0 0
\(419\) −32.7469 −1.59979 −0.799896 0.600138i \(-0.795112\pi\)
−0.799896 + 0.600138i \(0.795112\pi\)
\(420\) 0 0
\(421\) 16.8869 0.823017 0.411508 0.911406i \(-0.365002\pi\)
0.411508 + 0.911406i \(0.365002\pi\)
\(422\) 0 0
\(423\) −1.80182 + 4.93285i −0.0876073 + 0.239843i
\(424\) 0 0
\(425\) −11.9279 + 10.8856i −0.578587 + 0.528031i
\(426\) 0 0
\(427\) −33.2913 + 18.8255i −1.61108 + 0.911029i
\(428\) 0 0
\(429\) 6.49567 36.7157i 0.313614 1.77265i
\(430\) 0 0
\(431\) 29.7369i 1.43238i 0.697907 + 0.716188i \(0.254115\pi\)
−0.697907 + 0.716188i \(0.745885\pi\)
\(432\) 0 0
\(433\) −15.1503 −0.728076 −0.364038 0.931384i \(-0.618602\pi\)
−0.364038 + 0.931384i \(0.618602\pi\)
\(434\) 0 0
\(435\) −14.6064 + 21.7852i −0.700322 + 1.04452i
\(436\) 0 0
\(437\) 1.20176i 0.0574880i
\(438\) 0 0
\(439\) 9.26454i 0.442172i −0.975254 0.221086i \(-0.929040\pi\)
0.975254 0.221086i \(-0.0709602\pi\)
\(440\) 0 0
\(441\) −16.3411 + 13.1897i −0.778148 + 0.628081i
\(442\) 0 0
\(443\) 5.88820 0.279757 0.139878 0.990169i \(-0.455329\pi\)
0.139878 + 0.990169i \(0.455329\pi\)
\(444\) 0 0
\(445\) 32.0928 + 14.1599i 1.52134 + 0.671243i
\(446\) 0 0
\(447\) 0.860147 4.86184i 0.0406836 0.229957i
\(448\) 0 0
\(449\) 14.4606i 0.682439i −0.939984 0.341219i \(-0.889160\pi\)
0.939984 0.341219i \(-0.110840\pi\)
\(450\) 0 0
\(451\) 38.0487i 1.79164i
\(452\) 0 0
\(453\) −27.5475 4.87365i −1.29429 0.228984i
\(454\) 0 0
\(455\) 2.20576 + 22.1799i 0.103407 + 1.03981i
\(456\) 0 0
\(457\) 19.9157i 0.931617i 0.884885 + 0.465809i \(0.154236\pi\)
−0.884885 + 0.465809i \(0.845764\pi\)
\(458\) 0 0
\(459\) 8.41594 14.5191i 0.392822 0.677694i
\(460\) 0 0
\(461\) 42.4755 1.97828 0.989142 0.146966i \(-0.0469507\pi\)
0.989142 + 0.146966i \(0.0469507\pi\)
\(462\) 0 0
\(463\) 34.4952i 1.60313i 0.597910 + 0.801564i \(0.295998\pi\)
−0.597910 + 0.801564i \(0.704002\pi\)
\(464\) 0 0
\(465\) −3.37992 + 5.04112i −0.156740 + 0.233776i
\(466\) 0 0
\(467\) 5.47336i 0.253277i 0.991949 + 0.126639i \(0.0404188\pi\)
−0.991949 + 0.126639i \(0.959581\pi\)
\(468\) 0 0
\(469\) 14.4298 8.15975i 0.666308 0.376782i
\(470\) 0 0
\(471\) −12.3070 2.17732i −0.567075 0.100326i
\(472\) 0 0
\(473\) 50.5708 2.32525
\(474\) 0 0
\(475\) 2.90632 2.65237i 0.133351 0.121699i
\(476\) 0 0
\(477\) −1.73752 0.634663i −0.0795556 0.0290592i
\(478\) 0 0
\(479\) −25.9132 −1.18400 −0.592001 0.805937i \(-0.701662\pi\)
−0.592001 + 0.805937i \(0.701662\pi\)
\(480\) 0 0
\(481\) 33.2707i 1.51701i
\(482\) 0 0
\(483\) 2.33080 + 6.59867i 0.106055 + 0.300250i
\(484\) 0 0
\(485\) 15.7445 + 6.94672i 0.714919 + 0.315434i
\(486\) 0 0
\(487\) 8.92087i 0.404243i 0.979360 + 0.202122i \(0.0647836\pi\)
−0.979360 + 0.202122i \(0.935216\pi\)
\(488\) 0 0
\(489\) 1.05767 5.97828i 0.0478293 0.270347i
\(490\) 0 0
\(491\) 36.1241i 1.63026i −0.579278 0.815130i \(-0.696666\pi\)
0.579278 0.815130i \(-0.303334\pi\)
\(492\) 0 0
\(493\) −21.8721 −0.985070
\(494\) 0 0
\(495\) −26.5647 + 27.6303i −1.19399 + 1.24189i
\(496\) 0 0
\(497\) 11.7445 6.64123i 0.526811 0.297900i
\(498\) 0 0
\(499\) −24.8816 −1.11385 −0.556927 0.830562i \(-0.688020\pi\)
−0.556927 + 0.830562i \(0.688020\pi\)
\(500\) 0 0
\(501\) 0.417509 2.35990i 0.0186529 0.105433i
\(502\) 0 0
\(503\) 35.3029i 1.57408i −0.616902 0.787040i \(-0.711612\pi\)
0.616902 0.787040i \(-0.288388\pi\)
\(504\) 0 0
\(505\) 21.0017 + 9.26632i 0.934565 + 0.412346i
\(506\) 0 0
\(507\) −2.03756 0.360482i −0.0904914 0.0160096i
\(508\) 0 0
\(509\) −13.6651 −0.605696 −0.302848 0.953039i \(-0.597938\pi\)
−0.302848 + 0.953039i \(0.597938\pi\)
\(510\) 0 0
\(511\) 16.3611 + 28.9332i 0.723771 + 1.27993i
\(512\) 0 0
\(513\) −2.05061 + 3.53769i −0.0905365 + 0.156193i
\(514\) 0 0
\(515\) −7.41247 3.27051i −0.326632 0.144116i
\(516\) 0 0
\(517\) 10.0022 0.439894
\(518\) 0 0
\(519\) 3.05613 17.2743i 0.134149 0.758257i
\(520\) 0 0
\(521\) −9.66702 −0.423520 −0.211760 0.977322i \(-0.567919\pi\)
−0.211760 + 0.977322i \(0.567919\pi\)
\(522\) 0 0
\(523\) 3.51766 0.153817 0.0769083 0.997038i \(-0.475495\pi\)
0.0769083 + 0.997038i \(0.475495\pi\)
\(524\) 0 0
\(525\) 10.8139 20.2005i 0.471957 0.881621i
\(526\) 0 0
\(527\) −5.06121 −0.220470
\(528\) 0 0
\(529\) −20.6679 −0.898602
\(530\) 0 0
\(531\) 17.9963 + 6.57349i 0.780973 + 0.285265i
\(532\) 0 0
\(533\) −25.0888 −1.08672
\(534\) 0 0
\(535\) 17.7063 + 7.81231i 0.765510 + 0.337756i
\(536\) 0 0
\(537\) 0.163682 0.925186i 0.00706341 0.0399247i
\(538\) 0 0
\(539\) 34.2743 + 20.6149i 1.47630 + 0.887948i
\(540\) 0 0
\(541\) −5.55020 −0.238622 −0.119311 0.992857i \(-0.538069\pi\)
−0.119311 + 0.992857i \(0.538069\pi\)
\(542\) 0 0
\(543\) −0.349254 + 1.97410i −0.0149879 + 0.0847167i
\(544\) 0 0
\(545\) −17.0232 7.51092i −0.729193 0.321732i
\(546\) 0 0
\(547\) 16.5496i 0.707609i −0.935319 0.353805i \(-0.884888\pi\)
0.935319 0.353805i \(-0.115112\pi\)
\(548\) 0 0
\(549\) −14.8788 + 40.7338i −0.635012 + 1.73848i
\(550\) 0 0
\(551\) 5.32930 0.227036
\(552\) 0 0
\(553\) −16.7370 29.5979i −0.711728 1.25863i
\(554\) 0 0
\(555\) −19.0464 + 28.4074i −0.808473 + 1.20583i
\(556\) 0 0
\(557\) −25.7014 −1.08900 −0.544501 0.838760i \(-0.683281\pi\)
−0.544501 + 0.838760i \(0.683281\pi\)
\(558\) 0 0
\(559\) 33.3458i 1.41038i
\(560\) 0 0
\(561\) −31.4738 5.56829i −1.32883 0.235093i
\(562\) 0 0
\(563\) 1.70532i 0.0718707i −0.999354 0.0359354i \(-0.988559\pi\)
0.999354 0.0359354i \(-0.0114410\pi\)
\(564\) 0 0
\(565\) 28.8427 + 12.7259i 1.21342 + 0.535382i
\(566\) 0 0
\(567\) −4.39825 + 23.4020i −0.184709 + 0.982793i
\(568\) 0 0
\(569\) 36.2249i 1.51863i 0.650725 + 0.759314i \(0.274465\pi\)
−0.650725 + 0.759314i \(0.725535\pi\)
\(570\) 0 0
\(571\) 12.2875 0.514215 0.257107 0.966383i \(-0.417231\pi\)
0.257107 + 0.966383i \(0.417231\pi\)
\(572\) 0 0
\(573\) 3.27567 18.5152i 0.136843 0.773483i
\(574\) 0 0
\(575\) −5.14721 5.64003i −0.214653 0.235205i
\(576\) 0 0
\(577\) −19.9917 −0.832265 −0.416132 0.909304i \(-0.636615\pi\)
−0.416132 + 0.909304i \(0.636615\pi\)
\(578\) 0 0
\(579\) −7.43527 + 42.0266i −0.308999 + 1.74657i
\(580\) 0 0
\(581\) −24.1825 + 13.6747i −1.00326 + 0.567321i
\(582\) 0 0
\(583\) 3.52311i 0.145912i
\(584\) 0 0
\(585\) 18.2191 + 17.5164i 0.753266 + 0.724215i
\(586\) 0 0
\(587\) 12.6341i 0.521465i 0.965411 + 0.260733i \(0.0839641\pi\)
−0.965411 + 0.260733i \(0.916036\pi\)
\(588\) 0 0
\(589\) 1.23320 0.0508132
\(590\) 0 0
\(591\) 5.38644 + 0.952958i 0.221568 + 0.0391995i
\(592\) 0 0
\(593\) 25.1383i 1.03231i −0.856496 0.516153i \(-0.827364\pi\)
0.856496 0.516153i \(-0.172636\pi\)
\(594\) 0 0
\(595\) 19.0133 1.89084i 0.779468 0.0775169i
\(596\) 0 0
\(597\) −1.08742 + 6.14645i −0.0445051 + 0.251558i
\(598\) 0 0
\(599\) 7.03306i 0.287363i 0.989624 + 0.143682i \(0.0458941\pi\)
−0.989624 + 0.143682i \(0.954106\pi\)
\(600\) 0 0
\(601\) 41.7407i 1.70264i 0.524646 + 0.851320i \(0.324198\pi\)
−0.524646 + 0.851320i \(0.675802\pi\)
\(602\) 0 0
\(603\) 6.44910 17.6558i 0.262628 0.718998i
\(604\) 0 0
\(605\) 44.2851 + 19.5393i 1.80045 + 0.794387i
\(606\) 0 0
\(607\) 43.5247 1.76661 0.883306 0.468797i \(-0.155312\pi\)
0.883306 + 0.468797i \(0.155312\pi\)
\(608\) 0 0
\(609\) 29.2623 10.3361i 1.18577 0.418840i
\(610\) 0 0
\(611\) 6.59530i 0.266817i
\(612\) 0 0
\(613\) 22.7849i 0.920272i −0.887849 0.460136i \(-0.847801\pi\)
0.887849 0.460136i \(-0.152199\pi\)
\(614\) 0 0
\(615\) 21.4215 + 14.3625i 0.863798 + 0.579152i
\(616\) 0 0
\(617\) 8.44461 0.339967 0.169984 0.985447i \(-0.445629\pi\)
0.169984 + 0.985447i \(0.445629\pi\)
\(618\) 0 0
\(619\) 28.1187i 1.13018i 0.825028 + 0.565092i \(0.191159\pi\)
−0.825028 + 0.565092i \(0.808841\pi\)
\(620\) 0 0
\(621\) 6.86528 + 3.97943i 0.275494 + 0.159689i
\(622\) 0 0
\(623\) −20.4298 36.1283i −0.818501 1.44745i
\(624\) 0 0
\(625\) −2.27952 + 24.8959i −0.0911807 + 0.995834i
\(626\) 0 0
\(627\) 7.66883 + 1.35675i 0.306264 + 0.0541836i
\(628\) 0 0
\(629\) −28.5207 −1.13719
\(630\) 0 0
\(631\) 0.462386 0.0184073 0.00920365 0.999958i \(-0.497070\pi\)
0.00920365 + 0.999958i \(0.497070\pi\)
\(632\) 0 0
\(633\) −0.271980 0.0481181i −0.0108102 0.00191252i
\(634\) 0 0
\(635\) −11.5216 + 26.1131i −0.457219 + 1.03627i
\(636\) 0 0
\(637\) 13.5932 22.6000i 0.538584 0.895446i
\(638\) 0 0
\(639\) 5.24893 14.3700i 0.207644 0.568470i
\(640\) 0 0
\(641\) 38.3510i 1.51477i −0.652967 0.757387i \(-0.726476\pi\)
0.652967 0.757387i \(-0.273524\pi\)
\(642\) 0 0
\(643\) −23.0759 −0.910024 −0.455012 0.890485i \(-0.650365\pi\)
−0.455012 + 0.890485i \(0.650365\pi\)
\(644\) 0 0
\(645\) −19.0893 + 28.4715i −0.751642 + 1.12106i
\(646\) 0 0
\(647\) 28.2861i 1.11204i 0.831169 + 0.556020i \(0.187672\pi\)
−0.831169 + 0.556020i \(0.812328\pi\)
\(648\) 0 0
\(649\) 36.4904i 1.43237i
\(650\) 0 0
\(651\) 6.77132 2.39178i 0.265389 0.0937413i
\(652\) 0 0
\(653\) 18.6348 0.729238 0.364619 0.931157i \(-0.381199\pi\)
0.364619 + 0.931157i \(0.381199\pi\)
\(654\) 0 0
\(655\) −34.7766 15.3440i −1.35883 0.599540i
\(656\) 0 0
\(657\) 35.4014 + 12.9311i 1.38114 + 0.504488i
\(658\) 0 0
\(659\) 5.86589i 0.228503i −0.993452 0.114251i \(-0.963553\pi\)
0.993452 0.114251i \(-0.0364469\pi\)
\(660\) 0 0
\(661\) 6.13161i 0.238492i −0.992865 0.119246i \(-0.961952\pi\)
0.992865 0.119246i \(-0.0380477\pi\)
\(662\) 0 0
\(663\) −3.67166 + 20.7535i −0.142595 + 0.805997i
\(664\) 0 0
\(665\) −4.63273 + 0.460717i −0.179649 + 0.0178659i
\(666\) 0 0
\(667\) 10.3421i 0.400448i
\(668\) 0 0
\(669\) −7.49445 1.32590i −0.289752 0.0512624i
\(670\) 0 0
\(671\) 82.5945 3.18853
\(672\) 0 0
\(673\) 8.78377i 0.338589i −0.985565 0.169295i \(-0.945851\pi\)
0.985565 0.169295i \(-0.0541490\pi\)
\(674\) 0 0
\(675\) −5.52836 25.3858i −0.212787 0.977099i
\(676\) 0 0
\(677\) 38.7594i 1.48964i 0.667264 + 0.744822i \(0.267465\pi\)
−0.667264 + 0.744822i \(0.732535\pi\)
\(678\) 0 0
\(679\) −10.0227 17.7243i −0.384635 0.680194i
\(680\) 0 0
\(681\) −5.78271 + 32.6858i −0.221594 + 1.25252i
\(682\) 0 0
\(683\) 4.80345 0.183799 0.0918994 0.995768i \(-0.470706\pi\)
0.0918994 + 0.995768i \(0.470706\pi\)
\(684\) 0 0
\(685\) −25.3566 11.1878i −0.968828 0.427463i
\(686\) 0 0
\(687\) −0.379147 + 2.14307i −0.0144654 + 0.0817631i
\(688\) 0 0
\(689\) 2.32310 0.0885029
\(690\) 0 0
\(691\) 49.9522i 1.90027i −0.311836 0.950136i \(-0.600944\pi\)
0.311836 0.950136i \(-0.399056\pi\)
\(692\) 0 0
\(693\) 44.7397 7.42387i 1.69952 0.282009i
\(694\) 0 0
\(695\) 10.1515 23.0079i 0.385067 0.872738i
\(696\) 0 0
\(697\) 21.5069i 0.814632i
\(698\) 0 0
\(699\) 9.78571 + 1.73127i 0.370130 + 0.0654826i
\(700\) 0 0
\(701\) 2.56829i 0.0970031i 0.998823 + 0.0485016i \(0.0154446\pi\)
−0.998823 + 0.0485016i \(0.984555\pi\)
\(702\) 0 0
\(703\) 6.94927 0.262097
\(704\) 0 0
\(705\) −3.77558 + 5.63124i −0.142197 + 0.212085i
\(706\) 0 0
\(707\) −13.3694 23.6426i −0.502807 0.889172i
\(708\) 0 0
\(709\) 2.70387 0.101546 0.0507730 0.998710i \(-0.483832\pi\)
0.0507730 + 0.998710i \(0.483832\pi\)
\(710\) 0 0
\(711\) −36.2148 13.2281i −1.35816 0.496094i
\(712\) 0 0
\(713\) 2.39317i 0.0896248i
\(714\) 0 0
\(715\) 19.4311 44.0397i 0.726681 1.64699i
\(716\) 0 0
\(717\) 5.48681 31.0133i 0.204909 1.15821i
\(718\) 0 0
\(719\) −21.3073 −0.794627 −0.397313 0.917683i \(-0.630057\pi\)
−0.397313 + 0.917683i \(0.630057\pi\)
\(720\) 0 0
\(721\) 4.71866 + 8.34455i 0.175732 + 0.310767i
\(722\) 0 0
\(723\) 5.46939 30.9148i 0.203409 1.14973i
\(724\) 0 0
\(725\) −25.0112 + 22.8257i −0.928891 + 0.847726i
\(726\) 0 0
\(727\) −36.9056 −1.36875 −0.684377 0.729129i \(-0.739926\pi\)
−0.684377 + 0.729129i \(0.739926\pi\)
\(728\) 0 0
\(729\) 13.4195 + 23.4290i 0.497019 + 0.867740i
\(730\) 0 0
\(731\) −28.5850 −1.05725
\(732\) 0 0
\(733\) 31.8436 1.17617 0.588086 0.808798i \(-0.299882\pi\)
0.588086 + 0.808798i \(0.299882\pi\)
\(734\) 0 0
\(735\) −24.5440 + 11.5148i −0.905320 + 0.424731i
\(736\) 0 0
\(737\) −35.7999 −1.31871
\(738\) 0 0
\(739\) −2.76708 −0.101789 −0.0508944 0.998704i \(-0.516207\pi\)
−0.0508944 + 0.998704i \(0.516207\pi\)
\(740\) 0 0
\(741\) 0.894628 5.05673i 0.0328650 0.185764i
\(742\) 0 0
\(743\) 37.7284 1.38412 0.692060 0.721840i \(-0.256703\pi\)
0.692060 + 0.721840i \(0.256703\pi\)
\(744\) 0 0
\(745\) 2.57303 5.83167i 0.0942686 0.213656i
\(746\) 0 0
\(747\) −10.8079 + 29.5887i −0.395439 + 1.08260i
\(748\) 0 0
\(749\) −11.2715 19.9328i −0.411853 0.728328i
\(750\) 0 0
\(751\) −10.7810 −0.393403 −0.196701 0.980463i \(-0.563023\pi\)
−0.196701 + 0.980463i \(0.563023\pi\)
\(752\) 0 0
\(753\) 1.02133 + 0.180692i 0.0372194 + 0.00658478i
\(754\) 0 0
\(755\) −33.0426 14.5790i −1.20254 0.530583i
\(756\) 0 0
\(757\) 22.0222i 0.800409i −0.916426 0.400205i \(-0.868939\pi\)
0.916426 0.400205i \(-0.131061\pi\)
\(758\) 0 0
\(759\) 2.63293 14.8822i 0.0955694 0.540190i
\(760\) 0 0
\(761\) −0.133605 −0.00484319 −0.00242160 0.999997i \(-0.500771\pi\)
−0.00242160 + 0.999997i \(0.500771\pi\)
\(762\) 0 0
\(763\) 10.8367 + 19.1638i 0.392314 + 0.693775i
\(764\) 0 0
\(765\) 15.0156 15.6179i 0.542890 0.564668i
\(766\) 0 0
\(767\) −24.0613 −0.868805
\(768\) 0 0
\(769\) 26.2686i 0.947269i 0.880722 + 0.473634i \(0.157058\pi\)
−0.880722 + 0.473634i \(0.842942\pi\)
\(770\) 0 0
\(771\) −4.64899 + 26.2777i −0.167429 + 0.946367i
\(772\) 0 0
\(773\) 4.91789i 0.176884i 0.996081 + 0.0884422i \(0.0281889\pi\)
−0.996081 + 0.0884422i \(0.971811\pi\)
\(774\) 0 0
\(775\) −5.78760 + 5.28188i −0.207897 + 0.189731i
\(776\) 0 0
\(777\) 38.1574 13.4780i 1.36889 0.483521i
\(778\) 0 0
\(779\) 5.24032i 0.187754i
\(780\) 0 0
\(781\) −29.1376 −1.04263
\(782\) 0 0
\(783\) 17.6471 30.4447i 0.630656 1.08800i
\(784\) 0 0
\(785\) −14.7619 6.51322i −0.526876 0.232467i
\(786\) 0 0
\(787\) 21.2704 0.758207 0.379104 0.925354i \(-0.376232\pi\)
0.379104 + 0.925354i \(0.376232\pi\)
\(788\) 0 0
\(789\) 16.4808 + 2.91575i 0.586732 + 0.103803i
\(790\) 0 0
\(791\) −18.3608 32.4695i −0.652835 1.15448i
\(792\) 0 0
\(793\) 54.4618i 1.93400i
\(794\) 0 0
\(795\) −1.98352 1.32989i −0.0703482 0.0471664i
\(796\) 0 0
\(797\) 27.3045i 0.967175i 0.875296 + 0.483588i \(0.160667\pi\)
−0.875296 + 0.483588i \(0.839333\pi\)
\(798\) 0 0
\(799\) −5.65369 −0.200013
\(800\) 0 0
\(801\) −44.2051 16.1468i −1.56191 0.570518i
\(802\) 0 0
\(803\) 71.7822i 2.53314i
\(804\) 0 0
\(805\) 0.894073 + 8.99032i 0.0315119 + 0.316867i
\(806\) 0 0
\(807\) −22.5812 3.99503i −0.794896 0.140632i
\(808\) 0 0
\(809\) 37.9979i 1.33593i 0.744191 + 0.667967i \(0.232835\pi\)
−0.744191 + 0.667967i \(0.767165\pi\)
\(810\) 0 0
\(811\) 46.6063i 1.63657i −0.574813 0.818285i \(-0.694925\pi\)
0.574813 0.818285i \(-0.305075\pi\)
\(812\) 0 0
\(813\) −4.71844 + 26.6702i −0.165483 + 0.935364i
\(814\) 0 0
\(815\) 3.16389 7.17082i 0.110826 0.251183i
\(816\) 0 0
\(817\) 6.96495 0.243673
\(818\) 0 0
\(819\) −4.89521 29.5009i −0.171052 1.03084i
\(820\) 0 0
\(821\) 13.3057i 0.464371i −0.972672 0.232185i \(-0.925412\pi\)
0.972672 0.232185i \(-0.0745876\pi\)
\(822\) 0 0
\(823\) 6.85875i 0.239081i −0.992829 0.119541i \(-0.961858\pi\)
0.992829 0.119541i \(-0.0381421\pi\)
\(824\) 0 0
\(825\) −41.8020 + 26.4786i −1.45536 + 0.921867i
\(826\) 0 0
\(827\) 17.9117 0.622852 0.311426 0.950270i \(-0.399193\pi\)
0.311426 + 0.950270i \(0.399193\pi\)
\(828\) 0 0
\(829\) 45.4320i 1.57792i −0.614445 0.788960i \(-0.710620\pi\)
0.614445 0.788960i \(-0.289380\pi\)
\(830\) 0 0
\(831\) −1.05165 + 5.94430i −0.0364815 + 0.206205i
\(832\) 0 0
\(833\) −19.3734 11.6525i −0.671250 0.403736i
\(834\) 0 0
\(835\) 1.24893 2.83065i 0.0432210 0.0979587i
\(836\) 0 0
\(837\) 4.08355 7.04491i 0.141148 0.243508i
\(838\) 0 0
\(839\) 8.18091 0.282436 0.141218 0.989979i \(-0.454898\pi\)
0.141218 + 0.989979i \(0.454898\pi\)
\(840\) 0 0
\(841\) −16.8629 −0.581478
\(842\) 0 0
\(843\) 6.46673 36.5521i 0.222726 1.25892i
\(844\) 0 0
\(845\) −2.44401 1.07834i −0.0840766 0.0370960i
\(846\) 0 0
\(847\) −28.1912 49.8537i −0.968660 1.71299i
\(848\) 0 0
\(849\) 13.5974 + 2.40563i 0.466663 + 0.0825611i
\(850\) 0 0
\(851\) 13.4858i 0.462288i
\(852\) 0 0
\(853\) −1.25962 −0.0431287 −0.0215644 0.999767i \(-0.506865\pi\)
−0.0215644 + 0.999767i \(0.506865\pi\)
\(854\) 0 0
\(855\) −3.65866 + 3.80543i −0.125124 + 0.130143i
\(856\) 0 0
\(857\) 0.221305i 0.00755962i −0.999993 0.00377981i \(-0.998797\pi\)
0.999993 0.00377981i \(-0.00120315\pi\)
\(858\) 0 0
\(859\) 24.2694i 0.828062i 0.910263 + 0.414031i \(0.135880\pi\)
−0.910263 + 0.414031i \(0.864120\pi\)
\(860\) 0 0
\(861\) −10.1635 28.7738i −0.346372 0.980607i
\(862\) 0 0
\(863\) −0.515972 −0.0175639 −0.00878194 0.999961i \(-0.502795\pi\)
−0.00878194 + 0.999961i \(0.502795\pi\)
\(864\) 0 0
\(865\) 9.14207 20.7201i 0.310840 0.704506i
\(866\) 0 0
\(867\) −11.2041 1.98221i −0.380512 0.0673194i
\(868\) 0 0
\(869\) 73.4314i 2.49099i
\(870\) 0 0
\(871\) 23.6060i 0.799860i
\(872\) 0 0
\(873\) −21.6867 7.92147i −0.733982 0.268101i
\(874\) 0 0
\(875\) 19.7688 22.0045i 0.668306 0.743886i
\(876\) 0 0
\(877\) 45.4080i 1.53332i −0.642053 0.766660i \(-0.721917\pi\)
0.642053 0.766660i \(-0.278083\pi\)
\(878\) 0 0
\(879\) 3.88430 21.9553i 0.131014 0.740535i
\(880\) 0 0
\(881\) 30.6497 1.03262 0.516308 0.856403i \(-0.327306\pi\)
0.516308 + 0.856403i \(0.327306\pi\)
\(882\) 0 0
\(883\) 20.1004i 0.676433i −0.941068 0.338217i \(-0.890176\pi\)
0.941068 0.338217i \(-0.109824\pi\)
\(884\) 0 0
\(885\) 20.5442 + 13.7743i 0.690586 + 0.463018i
\(886\) 0 0
\(887\) 38.4432i 1.29080i −0.763846 0.645398i \(-0.776691\pi\)
0.763846 0.645398i \(-0.223309\pi\)
\(888\) 0 0
\(889\) 29.3967 16.6232i 0.985934 0.557524i
\(890\) 0 0
\(891\) 33.1448 39.3170i 1.11039 1.31717i
\(892\) 0 0
\(893\) 1.37756 0.0460984
\(894\) 0 0
\(895\) 0.489636 1.10974i 0.0163667 0.0370945i
\(896\) 0 0
\(897\) −9.81315 1.73612i −0.327652 0.0579675i
\(898\) 0 0
\(899\) −10.6127 −0.353953
\(900\) 0 0
\(901\) 1.99143i 0.0663440i
\(902\) 0 0
\(903\) 38.2435 13.5084i 1.27266 0.449533i
\(904\) 0 0
\(905\) −1.04475 + 2.36789i −0.0347288 + 0.0787113i
\(906\) 0 0
\(907\) 23.7121i 0.787346i 0.919251 + 0.393673i \(0.128796\pi\)
−0.919251 + 0.393673i \(0.871204\pi\)
\(908\) 0 0
\(909\) −28.9281 10.5665i −0.959485 0.350470i
\(910\) 0 0
\(911\) 35.4931i 1.17594i −0.808883 0.587970i \(-0.799927\pi\)
0.808883 0.587970i \(-0.200073\pi\)
\(912\) 0 0
\(913\) 59.9960 1.98558
\(914\) 0 0
\(915\) −31.1775 + 46.5009i −1.03070 + 1.53727i
\(916\) 0 0
\(917\) 22.1382 + 39.1496i 0.731068 + 1.29283i
\(918\) 0 0
\(919\) 41.0530 1.35421 0.677106 0.735885i \(-0.263234\pi\)
0.677106 + 0.735885i \(0.263234\pi\)
\(920\) 0 0
\(921\) 12.2205 + 2.16203i 0.402679 + 0.0712412i
\(922\) 0 0
\(923\) 19.2130i 0.632403i
\(924\) 0 0
\(925\) −32.6139 + 29.7642i −1.07234 + 0.978640i
\(926\) 0 0
\(927\) 10.2101 + 3.72942i 0.335342 + 0.122490i
\(928\) 0 0
\(929\) 17.1710 0.563362 0.281681 0.959508i \(-0.409108\pi\)
0.281681 + 0.959508i \(0.409108\pi\)
\(930\) 0 0
\(931\) 4.72048 + 2.83923i 0.154708 + 0.0930519i
\(932\) 0 0
\(933\) 58.7882 + 10.4007i 1.92464 + 0.340503i
\(934\) 0 0
\(935\) −37.7522 16.6569i −1.23463 0.544739i
\(936\) 0 0
\(937\) −21.1075 −0.689550 −0.344775 0.938685i \(-0.612045\pi\)
−0.344775 + 0.938685i \(0.612045\pi\)
\(938\) 0 0
\(939\) −48.7087 8.61745i −1.58955 0.281220i
\(940\) 0 0
\(941\) 29.7526 0.969908 0.484954 0.874540i \(-0.338836\pi\)
0.484954 + 0.874540i \(0.338836\pi\)
\(942\) 0 0
\(943\) −10.1694 −0.331162
\(944\) 0 0
\(945\) −12.7086 + 27.9909i −0.413410 + 0.910545i
\(946\) 0 0
\(947\) 15.4259 0.501274 0.250637 0.968081i \(-0.419360\pi\)
0.250637 + 0.968081i \(0.419360\pi\)
\(948\) 0 0
\(949\) −47.3323 −1.53647
\(950\) 0 0
\(951\) 35.7842 + 6.33088i 1.16038 + 0.205293i
\(952\) 0 0
\(953\) 43.6225 1.41307 0.706535 0.707678i \(-0.250257\pi\)
0.706535 + 0.707678i \(0.250257\pi\)
\(954\) 0 0
\(955\) 9.79880 22.2086i 0.317082 0.718653i
\(956\) 0 0
\(957\) −65.9964 11.6760i −2.13336 0.377430i
\(958\) 0 0
\(959\) 16.1416 + 28.5451i 0.521240 + 0.921770i
\(960\) 0 0
\(961\) 28.5442 0.920781
\(962\) 0 0
\(963\) −24.3889 8.90852i −0.785922 0.287073i
\(964\) 0 0
\(965\) −22.2418 + 50.4100i −0.715988 + 1.62276i
\(966\) 0 0
\(967\) 33.6963i 1.08360i 0.840508 + 0.541799i \(0.182257\pi\)
−0.840508 + 0.541799i \(0.817743\pi\)
\(968\) 0 0
\(969\) −4.33478 0.766902i −0.139253 0.0246364i
\(970\) 0 0
\(971\) 8.74825 0.280745 0.140372 0.990099i \(-0.455170\pi\)
0.140372 + 0.990099i \(0.455170\pi\)
\(972\) 0 0
\(973\) −25.9010 + 14.6464i −0.830348 + 0.469543i
\(974\) 0 0
\(975\) 17.4597 + 27.5637i 0.559157 + 0.882746i
\(976\) 0 0
\(977\) 44.2690 1.41629 0.708145 0.706067i \(-0.249532\pi\)
0.708145 + 0.706067i \(0.249532\pi\)
\(978\) 0 0
\(979\) 89.6331i 2.86469i
\(980\) 0 0
\(981\) 23.4480 + 8.56483i 0.748637 + 0.273454i
\(982\) 0 0
\(983\) 37.9368i 1.21000i −0.796227 0.604998i \(-0.793174\pi\)
0.796227 0.604998i \(-0.206826\pi\)
\(984\) 0 0
\(985\) 6.46092 + 2.85067i 0.205862 + 0.0908298i
\(986\) 0 0
\(987\) 7.56398 2.67177i 0.240764 0.0850432i
\(988\) 0 0
\(989\) 13.5163i 0.429792i
\(990\) 0 0
\(991\) −45.7568 −1.45351 −0.726757 0.686895i \(-0.758973\pi\)
−0.726757 + 0.686895i \(0.758973\pi\)
\(992\) 0 0
\(993\) −18.6596 3.30122i −0.592145 0.104761i
\(994\) 0 0
\(995\) −3.25289 + 7.37254i −0.103124 + 0.233725i
\(996\) 0 0
\(997\) 35.7188 1.13122 0.565612 0.824671i \(-0.308640\pi\)
0.565612 + 0.824671i \(0.308640\pi\)
\(998\) 0 0
\(999\) 23.0114 39.6991i 0.728048 1.25602i
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 840.2.k.b.209.1 yes 24
3.2 odd 2 840.2.k.a.209.2 yes 24
4.3 odd 2 1680.2.k.h.209.24 24
5.4 even 2 840.2.k.a.209.24 yes 24
7.6 odd 2 inner 840.2.k.b.209.24 yes 24
12.11 even 2 1680.2.k.i.209.23 24
15.14 odd 2 inner 840.2.k.b.209.23 yes 24
20.19 odd 2 1680.2.k.i.209.1 24
21.20 even 2 840.2.k.a.209.23 yes 24
28.27 even 2 1680.2.k.h.209.1 24
35.34 odd 2 840.2.k.a.209.1 24
60.59 even 2 1680.2.k.h.209.2 24
84.83 odd 2 1680.2.k.i.209.2 24
105.104 even 2 inner 840.2.k.b.209.2 yes 24
140.139 even 2 1680.2.k.i.209.24 24
420.419 odd 2 1680.2.k.h.209.23 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.k.a.209.1 24 35.34 odd 2
840.2.k.a.209.2 yes 24 3.2 odd 2
840.2.k.a.209.23 yes 24 21.20 even 2
840.2.k.a.209.24 yes 24 5.4 even 2
840.2.k.b.209.1 yes 24 1.1 even 1 trivial
840.2.k.b.209.2 yes 24 105.104 even 2 inner
840.2.k.b.209.23 yes 24 15.14 odd 2 inner
840.2.k.b.209.24 yes 24 7.6 odd 2 inner
1680.2.k.h.209.1 24 28.27 even 2
1680.2.k.h.209.2 24 60.59 even 2
1680.2.k.h.209.23 24 420.419 odd 2
1680.2.k.h.209.24 24 4.3 odd 2
1680.2.k.i.209.1 24 20.19 odd 2
1680.2.k.i.209.2 24 84.83 odd 2
1680.2.k.i.209.23 24 12.11 even 2
1680.2.k.i.209.24 24 140.139 even 2