Properties

Label 4020.2.q.j.3781.3
Level $4020$
Weight $2$
Character 4020.3781
Analytic conductor $32.100$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 30 x^{10} - 53 x^{9} + 798 x^{8} - 1096 x^{7} + 4060 x^{6} - 915 x^{5} + 10392 x^{4} - 7038 x^{3} + 4869 x^{2} - 675 x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3781.3
Root \(0.307496 + 0.532598i\) of defining polynomial
Character \(\chi\) \(=\) 4020.3781
Dual form 4020.2.q.j.841.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +(-0.550200 - 0.952975i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +(-0.550200 - 0.952975i) q^{7} +1.00000 q^{9} +(-0.589441 - 1.02094i) q^{11} +(0.500000 - 0.866025i) q^{13} -1.00000 q^{15} +(-1.83022 + 3.17003i) q^{17} +(-3.26255 + 5.65090i) q^{19} +(-0.550200 - 0.952975i) q^{21} +(4.11217 - 7.12248i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(4.58485 + 7.94120i) q^{29} +(1.12291 + 1.94493i) q^{31} +(-0.589441 - 1.02094i) q^{33} +(0.550200 + 0.952975i) q^{35} +(-5.58203 + 9.66835i) q^{37} +(0.500000 - 0.866025i) q^{39} +(-4.73507 - 8.20139i) q^{41} +3.52510 q^{43} -1.00000 q^{45} +(5.16548 + 8.94687i) q^{47} +(2.89456 - 5.01352i) q^{49} +(-1.83022 + 3.17003i) q^{51} +1.24582 q^{53} +(0.589441 + 1.02094i) q^{55} +(-3.26255 + 5.65090i) q^{57} +8.73327 q^{59} +(7.40027 - 12.8176i) q^{61} +(-0.550200 - 0.952975i) q^{63} +(-0.500000 + 0.866025i) q^{65} +(8.12438 - 0.997229i) q^{67} +(4.11217 - 7.12248i) q^{69} +(3.07575 + 5.32736i) q^{71} +(-4.45329 + 7.71333i) q^{73} +1.00000 q^{75} +(-0.648621 + 1.12345i) q^{77} +(-1.22597 - 2.12344i) q^{79} +1.00000 q^{81} +(-5.48704 + 9.50384i) q^{83} +(1.83022 - 3.17003i) q^{85} +(4.58485 + 7.94120i) q^{87} +7.19709 q^{89} -1.10040 q^{91} +(1.12291 + 1.94493i) q^{93} +(3.26255 - 5.65090i) q^{95} +(3.18799 - 5.52175i) q^{97} +(-0.589441 - 1.02094i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 12 q^{5} + 2 q^{7} + 12 q^{9} - 5 q^{11} + 6 q^{13} - 12 q^{15} + 9 q^{17} - 11 q^{19} + 2 q^{21} + 19 q^{23} + 12 q^{25} + 12 q^{27} + 8 q^{29} - 4 q^{31} - 5 q^{33} - 2 q^{35} + 5 q^{37} + 6 q^{39} - 9 q^{41} - 14 q^{43} - 12 q^{45} - 6 q^{47} - 46 q^{49} + 9 q^{51} - 20 q^{53} + 5 q^{55} - 11 q^{57} + 6 q^{59} + 27 q^{61} + 2 q^{63} - 6 q^{65} + 3 q^{67} + 19 q^{69} + 25 q^{71} - 8 q^{73} + 12 q^{75} + 10 q^{77} - 2 q^{79} + 12 q^{81} + 20 q^{83} - 9 q^{85} + 8 q^{87} + 12 q^{89} + 4 q^{91} - 4 q^{93} + 11 q^{95} - q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(7\) −0.550200 0.952975i −0.207956 0.360191i 0.743114 0.669164i \(-0.233348\pi\)
−0.951071 + 0.308974i \(0.900014\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.589441 1.02094i −0.177723 0.307826i 0.763377 0.645953i \(-0.223540\pi\)
−0.941100 + 0.338127i \(0.890207\pi\)
\(12\) 0 0
\(13\) 0.500000 0.866025i 0.138675 0.240192i −0.788320 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.83022 + 3.17003i −0.443894 + 0.768846i −0.997974 0.0636166i \(-0.979737\pi\)
0.554081 + 0.832463i \(0.313070\pi\)
\(18\) 0 0
\(19\) −3.26255 + 5.65090i −0.748480 + 1.29641i 0.200071 + 0.979781i \(0.435883\pi\)
−0.948551 + 0.316624i \(0.897451\pi\)
\(20\) 0 0
\(21\) −0.550200 0.952975i −0.120064 0.207956i
\(22\) 0 0
\(23\) 4.11217 7.12248i 0.857446 1.48514i −0.0169115 0.999857i \(-0.505383\pi\)
0.874357 0.485283i \(-0.161283\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.58485 + 7.94120i 0.851386 + 1.47464i 0.879958 + 0.475052i \(0.157571\pi\)
−0.0285719 + 0.999592i \(0.509096\pi\)
\(30\) 0 0
\(31\) 1.12291 + 1.94493i 0.201680 + 0.349320i 0.949070 0.315066i \(-0.102027\pi\)
−0.747390 + 0.664386i \(0.768693\pi\)
\(32\) 0 0
\(33\) −0.589441 1.02094i −0.102609 0.177723i
\(34\) 0 0
\(35\) 0.550200 + 0.952975i 0.0930008 + 0.161082i
\(36\) 0 0
\(37\) −5.58203 + 9.66835i −0.917680 + 1.58947i −0.114750 + 0.993394i \(0.536607\pi\)
−0.802930 + 0.596073i \(0.796727\pi\)
\(38\) 0 0
\(39\) 0.500000 0.866025i 0.0800641 0.138675i
\(40\) 0 0
\(41\) −4.73507 8.20139i −0.739494 1.28084i −0.952723 0.303839i \(-0.901731\pi\)
0.213229 0.977002i \(-0.431602\pi\)
\(42\) 0 0
\(43\) 3.52510 0.537572 0.268786 0.963200i \(-0.413377\pi\)
0.268786 + 0.963200i \(0.413377\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 5.16548 + 8.94687i 0.753462 + 1.30503i 0.946135 + 0.323772i \(0.104951\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(48\) 0 0
\(49\) 2.89456 5.01352i 0.413508 0.716218i
\(50\) 0 0
\(51\) −1.83022 + 3.17003i −0.256282 + 0.443894i
\(52\) 0 0
\(53\) 1.24582 0.171126 0.0855629 0.996333i \(-0.472731\pi\)
0.0855629 + 0.996333i \(0.472731\pi\)
\(54\) 0 0
\(55\) 0.589441 + 1.02094i 0.0794802 + 0.137664i
\(56\) 0 0
\(57\) −3.26255 + 5.65090i −0.432135 + 0.748480i
\(58\) 0 0
\(59\) 8.73327 1.13697 0.568487 0.822692i \(-0.307529\pi\)
0.568487 + 0.822692i \(0.307529\pi\)
\(60\) 0 0
\(61\) 7.40027 12.8176i 0.947507 1.64113i 0.196854 0.980433i \(-0.436927\pi\)
0.750652 0.660697i \(-0.229739\pi\)
\(62\) 0 0
\(63\) −0.550200 0.952975i −0.0693187 0.120064i
\(64\) 0 0
\(65\) −0.500000 + 0.866025i −0.0620174 + 0.107417i
\(66\) 0 0
\(67\) 8.12438 0.997229i 0.992551 0.121831i
\(68\) 0 0
\(69\) 4.11217 7.12248i 0.495047 0.857446i
\(70\) 0 0
\(71\) 3.07575 + 5.32736i 0.365024 + 0.632241i 0.988780 0.149379i \(-0.0477273\pi\)
−0.623756 + 0.781619i \(0.714394\pi\)
\(72\) 0 0
\(73\) −4.45329 + 7.71333i −0.521218 + 0.902777i 0.478477 + 0.878100i \(0.341189\pi\)
−0.999695 + 0.0246766i \(0.992144\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.648621 + 1.12345i −0.0739173 + 0.128028i
\(78\) 0 0
\(79\) −1.22597 2.12344i −0.137932 0.238906i 0.788781 0.614674i \(-0.210712\pi\)
−0.926714 + 0.375768i \(0.877379\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.48704 + 9.50384i −0.602281 + 1.04318i 0.390194 + 0.920733i \(0.372408\pi\)
−0.992475 + 0.122449i \(0.960925\pi\)
\(84\) 0 0
\(85\) 1.83022 3.17003i 0.198515 0.343839i
\(86\) 0 0
\(87\) 4.58485 + 7.94120i 0.491548 + 0.851386i
\(88\) 0 0
\(89\) 7.19709 0.762890 0.381445 0.924392i \(-0.375427\pi\)
0.381445 + 0.924392i \(0.375427\pi\)
\(90\) 0 0
\(91\) −1.10040 −0.115353
\(92\) 0 0
\(93\) 1.12291 + 1.94493i 0.116440 + 0.201680i
\(94\) 0 0
\(95\) 3.26255 5.65090i 0.334730 0.579770i
\(96\) 0 0
\(97\) 3.18799 5.52175i 0.323691 0.560649i −0.657556 0.753406i \(-0.728410\pi\)
0.981247 + 0.192757i \(0.0617429\pi\)
\(98\) 0 0
\(99\) −0.589441 1.02094i −0.0592411 0.102609i
\(100\) 0 0
\(101\) 7.77764 + 13.4713i 0.773904 + 1.34044i 0.935408 + 0.353570i \(0.115032\pi\)
−0.161504 + 0.986872i \(0.551634\pi\)
\(102\) 0 0
\(103\) −2.21517 3.83680i −0.218268 0.378051i 0.736011 0.676970i \(-0.236707\pi\)
−0.954278 + 0.298919i \(0.903374\pi\)
\(104\) 0 0
\(105\) 0.550200 + 0.952975i 0.0536940 + 0.0930008i
\(106\) 0 0
\(107\) 14.6613 1.41736 0.708682 0.705528i \(-0.249290\pi\)
0.708682 + 0.705528i \(0.249290\pi\)
\(108\) 0 0
\(109\) −0.127209 −0.0121844 −0.00609222 0.999981i \(-0.501939\pi\)
−0.00609222 + 0.999981i \(0.501939\pi\)
\(110\) 0 0
\(111\) −5.58203 + 9.66835i −0.529823 + 0.917680i
\(112\) 0 0
\(113\) 2.12291 + 3.67698i 0.199706 + 0.345902i 0.948433 0.316977i \(-0.102668\pi\)
−0.748727 + 0.662879i \(0.769335\pi\)
\(114\) 0 0
\(115\) −4.11217 + 7.12248i −0.383461 + 0.664175i
\(116\) 0 0
\(117\) 0.500000 0.866025i 0.0462250 0.0800641i
\(118\) 0 0
\(119\) 4.02795 0.369242
\(120\) 0 0
\(121\) 4.80512 8.32271i 0.436829 0.756610i
\(122\) 0 0
\(123\) −4.73507 8.20139i −0.426947 0.739494i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.22138 9.04370i −0.463323 0.802499i 0.535801 0.844344i \(-0.320010\pi\)
−0.999124 + 0.0418455i \(0.986676\pi\)
\(128\) 0 0
\(129\) 3.52510 0.310368
\(130\) 0 0
\(131\) 4.81740 0.420899 0.210449 0.977605i \(-0.432507\pi\)
0.210449 + 0.977605i \(0.432507\pi\)
\(132\) 0 0
\(133\) 7.18022 0.622604
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −0.833998 −0.0712533 −0.0356266 0.999365i \(-0.511343\pi\)
−0.0356266 + 0.999365i \(0.511343\pi\)
\(138\) 0 0
\(139\) −10.8727 −0.922206 −0.461103 0.887347i \(-0.652546\pi\)
−0.461103 + 0.887347i \(0.652546\pi\)
\(140\) 0 0
\(141\) 5.16548 + 8.94687i 0.435012 + 0.753462i
\(142\) 0 0
\(143\) −1.17888 −0.0985831
\(144\) 0 0
\(145\) −4.58485 7.94120i −0.380751 0.659481i
\(146\) 0 0
\(147\) 2.89456 5.01352i 0.238739 0.413508i
\(148\) 0 0
\(149\) −3.32992 −0.272798 −0.136399 0.990654i \(-0.543553\pi\)
−0.136399 + 0.990654i \(0.543553\pi\)
\(150\) 0 0
\(151\) −0.834754 + 1.44584i −0.0679313 + 0.117661i −0.897991 0.440015i \(-0.854973\pi\)
0.830059 + 0.557675i \(0.188307\pi\)
\(152\) 0 0
\(153\) −1.83022 + 3.17003i −0.147965 + 0.256282i
\(154\) 0 0
\(155\) −1.12291 1.94493i −0.0901941 0.156221i
\(156\) 0 0
\(157\) 9.29941 16.1071i 0.742174 1.28548i −0.209330 0.977845i \(-0.567128\pi\)
0.951504 0.307638i \(-0.0995385\pi\)
\(158\) 0 0
\(159\) 1.24582 0.0987996
\(160\) 0 0
\(161\) −9.05006 −0.713245
\(162\) 0 0
\(163\) 4.29454 + 7.43836i 0.336374 + 0.582617i 0.983748 0.179555i \(-0.0574659\pi\)
−0.647374 + 0.762173i \(0.724133\pi\)
\(164\) 0 0
\(165\) 0.589441 + 1.02094i 0.0458879 + 0.0794802i
\(166\) 0 0
\(167\) −5.73292 9.92970i −0.443626 0.768384i 0.554329 0.832298i \(-0.312975\pi\)
−0.997955 + 0.0639141i \(0.979642\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) −3.26255 + 5.65090i −0.249493 + 0.432135i
\(172\) 0 0
\(173\) −5.77239 + 9.99807i −0.438867 + 0.760139i −0.997602 0.0692065i \(-0.977953\pi\)
0.558736 + 0.829346i \(0.311287\pi\)
\(174\) 0 0
\(175\) −0.550200 0.952975i −0.0415912 0.0720381i
\(176\) 0 0
\(177\) 8.73327 0.656433
\(178\) 0 0
\(179\) 23.1723 1.73198 0.865990 0.500061i \(-0.166689\pi\)
0.865990 + 0.500061i \(0.166689\pi\)
\(180\) 0 0
\(181\) 11.6327 + 20.1484i 0.864650 + 1.49762i 0.867394 + 0.497621i \(0.165793\pi\)
−0.00274431 + 0.999996i \(0.500874\pi\)
\(182\) 0 0
\(183\) 7.40027 12.8176i 0.547043 0.947507i
\(184\) 0 0
\(185\) 5.58203 9.66835i 0.410399 0.710832i
\(186\) 0 0
\(187\) 4.31523 0.315561
\(188\) 0 0
\(189\) −0.550200 0.952975i −0.0400212 0.0693187i
\(190\) 0 0
\(191\) −1.97540 + 3.42150i −0.142935 + 0.247571i −0.928601 0.371080i \(-0.878987\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(192\) 0 0
\(193\) −16.6912 −1.20146 −0.600729 0.799452i \(-0.705123\pi\)
−0.600729 + 0.799452i \(0.705123\pi\)
\(194\) 0 0
\(195\) −0.500000 + 0.866025i −0.0358057 + 0.0620174i
\(196\) 0 0
\(197\) 8.67221 + 15.0207i 0.617869 + 1.07018i 0.989874 + 0.141950i \(0.0453372\pi\)
−0.372005 + 0.928231i \(0.621330\pi\)
\(198\) 0 0
\(199\) −12.8250 + 22.2136i −0.909141 + 1.57468i −0.0938817 + 0.995583i \(0.529928\pi\)
−0.815260 + 0.579096i \(0.803406\pi\)
\(200\) 0 0
\(201\) 8.12438 0.997229i 0.573050 0.0703391i
\(202\) 0 0
\(203\) 5.04518 8.73850i 0.354102 0.613322i
\(204\) 0 0
\(205\) 4.73507 + 8.20139i 0.330712 + 0.572810i
\(206\) 0 0
\(207\) 4.11217 7.12248i 0.285815 0.495047i
\(208\) 0 0
\(209\) 7.69232 0.532089
\(210\) 0 0
\(211\) −5.91507 + 10.2452i −0.407210 + 0.705309i −0.994576 0.104013i \(-0.966832\pi\)
0.587366 + 0.809322i \(0.300165\pi\)
\(212\) 0 0
\(213\) 3.07575 + 5.32736i 0.210747 + 0.365024i
\(214\) 0 0
\(215\) −3.52510 −0.240410
\(216\) 0 0
\(217\) 1.23565 2.14021i 0.0838813 0.145287i
\(218\) 0 0
\(219\) −4.45329 + 7.71333i −0.300926 + 0.521218i
\(220\) 0 0
\(221\) 1.83022 + 3.17003i 0.123114 + 0.213240i
\(222\) 0 0
\(223\) 14.1821 0.949705 0.474853 0.880065i \(-0.342501\pi\)
0.474853 + 0.880065i \(0.342501\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 11.0669 + 19.1684i 0.734533 + 1.27225i 0.954928 + 0.296838i \(0.0959322\pi\)
−0.220395 + 0.975411i \(0.570734\pi\)
\(228\) 0 0
\(229\) 8.26322 14.3123i 0.546049 0.945784i −0.452491 0.891769i \(-0.649465\pi\)
0.998540 0.0540155i \(-0.0172020\pi\)
\(230\) 0 0
\(231\) −0.648621 + 1.12345i −0.0426762 + 0.0739173i
\(232\) 0 0
\(233\) 2.96682 + 5.13868i 0.194363 + 0.336646i 0.946691 0.322142i \(-0.104403\pi\)
−0.752329 + 0.658788i \(0.771069\pi\)
\(234\) 0 0
\(235\) −5.16548 8.94687i −0.336959 0.583629i
\(236\) 0 0
\(237\) −1.22597 2.12344i −0.0796353 0.137932i
\(238\) 0 0
\(239\) 12.6186 + 21.8561i 0.816230 + 1.41375i 0.908442 + 0.418012i \(0.137273\pi\)
−0.0922117 + 0.995739i \(0.529394\pi\)
\(240\) 0 0
\(241\) −16.6342 −1.07150 −0.535752 0.844376i \(-0.679972\pi\)
−0.535752 + 0.844376i \(0.679972\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.89456 + 5.01352i −0.184927 + 0.320302i
\(246\) 0 0
\(247\) 3.26255 + 5.65090i 0.207591 + 0.359558i
\(248\) 0 0
\(249\) −5.48704 + 9.50384i −0.347727 + 0.602281i
\(250\) 0 0
\(251\) 11.1153 19.2523i 0.701593 1.21520i −0.266313 0.963886i \(-0.585806\pi\)
0.967907 0.251309i \(-0.0808610\pi\)
\(252\) 0 0
\(253\) −9.69552 −0.609552
\(254\) 0 0
\(255\) 1.83022 3.17003i 0.114613 0.198515i
\(256\) 0 0
\(257\) −4.81593 8.34143i −0.300409 0.520324i 0.675819 0.737067i \(-0.263790\pi\)
−0.976229 + 0.216743i \(0.930457\pi\)
\(258\) 0 0
\(259\) 12.2849 0.763349
\(260\) 0 0
\(261\) 4.58485 + 7.94120i 0.283795 + 0.491548i
\(262\) 0 0
\(263\) −11.9838 −0.738955 −0.369477 0.929240i \(-0.620463\pi\)
−0.369477 + 0.929240i \(0.620463\pi\)
\(264\) 0 0
\(265\) −1.24582 −0.0765298
\(266\) 0 0
\(267\) 7.19709 0.440455
\(268\) 0 0
\(269\) −27.5058 −1.67706 −0.838531 0.544854i \(-0.816585\pi\)
−0.838531 + 0.544854i \(0.816585\pi\)
\(270\) 0 0
\(271\) −6.99762 −0.425075 −0.212538 0.977153i \(-0.568173\pi\)
−0.212538 + 0.977153i \(0.568173\pi\)
\(272\) 0 0
\(273\) −1.10040 −0.0665993
\(274\) 0 0
\(275\) −0.589441 1.02094i −0.0355446 0.0615651i
\(276\) 0 0
\(277\) −3.51028 −0.210912 −0.105456 0.994424i \(-0.533630\pi\)
−0.105456 + 0.994424i \(0.533630\pi\)
\(278\) 0 0
\(279\) 1.12291 + 1.94493i 0.0672267 + 0.116440i
\(280\) 0 0
\(281\) 1.60328 2.77696i 0.0956437 0.165660i −0.814233 0.580538i \(-0.802842\pi\)
0.909877 + 0.414878i \(0.136176\pi\)
\(282\) 0 0
\(283\) 0.280187 0.0166554 0.00832771 0.999965i \(-0.497349\pi\)
0.00832771 + 0.999965i \(0.497349\pi\)
\(284\) 0 0
\(285\) 3.26255 5.65090i 0.193257 0.334730i
\(286\) 0 0
\(287\) −5.21048 + 9.02481i −0.307565 + 0.532718i
\(288\) 0 0
\(289\) 1.80059 + 3.11871i 0.105917 + 0.183454i
\(290\) 0 0
\(291\) 3.18799 5.52175i 0.186883 0.323691i
\(292\) 0 0
\(293\) 7.10378 0.415007 0.207504 0.978234i \(-0.433466\pi\)
0.207504 + 0.978234i \(0.433466\pi\)
\(294\) 0 0
\(295\) −8.73327 −0.508471
\(296\) 0 0
\(297\) −0.589441 1.02094i −0.0342028 0.0592411i
\(298\) 0 0
\(299\) −4.11217 7.12248i −0.237813 0.411904i
\(300\) 0 0
\(301\) −1.93951 3.35933i −0.111791 0.193629i
\(302\) 0 0
\(303\) 7.77764 + 13.4713i 0.446814 + 0.773904i
\(304\) 0 0
\(305\) −7.40027 + 12.8176i −0.423738 + 0.733936i
\(306\) 0 0
\(307\) −5.76165 + 9.97946i −0.328835 + 0.569558i −0.982281 0.187415i \(-0.939989\pi\)
0.653446 + 0.756973i \(0.273323\pi\)
\(308\) 0 0
\(309\) −2.21517 3.83680i −0.126017 0.218268i
\(310\) 0 0
\(311\) 9.66677 0.548152 0.274076 0.961708i \(-0.411628\pi\)
0.274076 + 0.961708i \(0.411628\pi\)
\(312\) 0 0
\(313\) −19.8904 −1.12427 −0.562136 0.827045i \(-0.690020\pi\)
−0.562136 + 0.827045i \(0.690020\pi\)
\(314\) 0 0
\(315\) 0.550200 + 0.952975i 0.0310003 + 0.0536940i
\(316\) 0 0
\(317\) −2.59315 + 4.49147i −0.145646 + 0.252266i −0.929614 0.368535i \(-0.879859\pi\)
0.783968 + 0.620801i \(0.213193\pi\)
\(318\) 0 0
\(319\) 5.40500 9.36174i 0.302622 0.524157i
\(320\) 0 0
\(321\) 14.6613 0.818316
\(322\) 0 0
\(323\) −11.9424 20.6848i −0.664491 1.15093i
\(324\) 0 0
\(325\) 0.500000 0.866025i 0.0277350 0.0480384i
\(326\) 0 0
\(327\) −0.127209 −0.00703469
\(328\) 0 0
\(329\) 5.68409 9.84514i 0.313374 0.542780i
\(330\) 0 0
\(331\) −3.39598 5.88201i −0.186660 0.323305i 0.757475 0.652865i \(-0.226433\pi\)
−0.944135 + 0.329560i \(0.893100\pi\)
\(332\) 0 0
\(333\) −5.58203 + 9.66835i −0.305893 + 0.529823i
\(334\) 0 0
\(335\) −8.12438 + 0.997229i −0.443882 + 0.0544845i
\(336\) 0 0
\(337\) 6.38886 11.0658i 0.348023 0.602794i −0.637875 0.770140i \(-0.720186\pi\)
0.985898 + 0.167346i \(0.0535197\pi\)
\(338\) 0 0
\(339\) 2.12291 + 3.67698i 0.115301 + 0.199706i
\(340\) 0 0
\(341\) 1.32378 2.29285i 0.0716865 0.124165i
\(342\) 0 0
\(343\) −14.0732 −0.759879
\(344\) 0 0
\(345\) −4.11217 + 7.12248i −0.221392 + 0.383461i
\(346\) 0 0
\(347\) −6.55027 11.3454i −0.351637 0.609053i 0.634900 0.772595i \(-0.281041\pi\)
−0.986536 + 0.163542i \(0.947708\pi\)
\(348\) 0 0
\(349\) −11.1701 −0.597921 −0.298961 0.954265i \(-0.596640\pi\)
−0.298961 + 0.954265i \(0.596640\pi\)
\(350\) 0 0
\(351\) 0.500000 0.866025i 0.0266880 0.0462250i
\(352\) 0 0
\(353\) 12.0197 20.8187i 0.639744 1.10807i −0.345745 0.938329i \(-0.612374\pi\)
0.985489 0.169741i \(-0.0542930\pi\)
\(354\) 0 0
\(355\) −3.07575 5.32736i −0.163244 0.282747i
\(356\) 0 0
\(357\) 4.02795 0.213182
\(358\) 0 0
\(359\) 2.57364 0.135831 0.0679157 0.997691i \(-0.478365\pi\)
0.0679157 + 0.997691i \(0.478365\pi\)
\(360\) 0 0
\(361\) −11.7885 20.4182i −0.620445 1.07464i
\(362\) 0 0
\(363\) 4.80512 8.32271i 0.252203 0.436829i
\(364\) 0 0
\(365\) 4.45329 7.71333i 0.233096 0.403734i
\(366\) 0 0
\(367\) −4.74432 8.21741i −0.247652 0.428945i 0.715222 0.698897i \(-0.246326\pi\)
−0.962874 + 0.269952i \(0.912992\pi\)
\(368\) 0 0
\(369\) −4.73507 8.20139i −0.246498 0.426947i
\(370\) 0 0
\(371\) −0.685448 1.18723i −0.0355867 0.0616379i
\(372\) 0 0
\(373\) −16.2664 28.1742i −0.842241 1.45880i −0.887996 0.459852i \(-0.847903\pi\)
0.0457545 0.998953i \(-0.485431\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 9.16971 0.472264
\(378\) 0 0
\(379\) 19.4141 33.6262i 0.997235 1.72726i 0.434265 0.900785i \(-0.357008\pi\)
0.562970 0.826477i \(-0.309659\pi\)
\(380\) 0 0
\(381\) −5.22138 9.04370i −0.267500 0.463323i
\(382\) 0 0
\(383\) −19.3203 + 33.4638i −0.987223 + 1.70992i −0.355615 + 0.934633i \(0.615728\pi\)
−0.631608 + 0.775288i \(0.717605\pi\)
\(384\) 0 0
\(385\) 0.648621 1.12345i 0.0330568 0.0572561i
\(386\) 0 0
\(387\) 3.52510 0.179191
\(388\) 0 0
\(389\) 15.5748 26.9763i 0.789673 1.36775i −0.136495 0.990641i \(-0.543584\pi\)
0.926168 0.377112i \(-0.123083\pi\)
\(390\) 0 0
\(391\) 15.0523 + 26.0714i 0.761229 + 1.31849i
\(392\) 0 0
\(393\) 4.81740 0.243006
\(394\) 0 0
\(395\) 1.22597 + 2.12344i 0.0616853 + 0.106842i
\(396\) 0 0
\(397\) 7.53785 0.378314 0.189157 0.981947i \(-0.439425\pi\)
0.189157 + 0.981947i \(0.439425\pi\)
\(398\) 0 0
\(399\) 7.18022 0.359461
\(400\) 0 0
\(401\) 21.2999 1.06366 0.531832 0.846850i \(-0.321504\pi\)
0.531832 + 0.846850i \(0.321504\pi\)
\(402\) 0 0
\(403\) 2.24582 0.111872
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 13.1611 0.652372
\(408\) 0 0
\(409\) −4.93011 8.53921i −0.243778 0.422237i 0.718009 0.696034i \(-0.245054\pi\)
−0.961787 + 0.273797i \(0.911720\pi\)
\(410\) 0 0
\(411\) −0.833998 −0.0411381
\(412\) 0 0
\(413\) −4.80505 8.32259i −0.236441 0.409528i
\(414\) 0 0
\(415\) 5.48704 9.50384i 0.269348 0.466525i
\(416\) 0 0
\(417\) −10.8727 −0.532436
\(418\) 0 0
\(419\) 0.661016 1.14491i 0.0322927 0.0559327i −0.849427 0.527706i \(-0.823052\pi\)
0.881720 + 0.471773i \(0.156386\pi\)
\(420\) 0 0
\(421\) −8.94550 + 15.4941i −0.435977 + 0.755134i −0.997375 0.0724119i \(-0.976930\pi\)
0.561398 + 0.827546i \(0.310264\pi\)
\(422\) 0 0
\(423\) 5.16548 + 8.94687i 0.251154 + 0.435012i
\(424\) 0 0
\(425\) −1.83022 + 3.17003i −0.0887787 + 0.153769i
\(426\) 0 0
\(427\) −16.2865 −0.788160
\(428\) 0 0
\(429\) −1.17888 −0.0569170
\(430\) 0 0
\(431\) 1.59453 + 2.76181i 0.0768059 + 0.133032i 0.901870 0.432007i \(-0.142194\pi\)
−0.825064 + 0.565039i \(0.808861\pi\)
\(432\) 0 0
\(433\) −4.21629 7.30283i −0.202622 0.350952i 0.746750 0.665104i \(-0.231613\pi\)
−0.949372 + 0.314153i \(0.898280\pi\)
\(434\) 0 0
\(435\) −4.58485 7.94120i −0.219827 0.380751i
\(436\) 0 0
\(437\) 26.8323 + 46.4749i 1.28356 + 2.22320i
\(438\) 0 0
\(439\) −15.7918 + 27.3521i −0.753700 + 1.30545i 0.192318 + 0.981333i \(0.438399\pi\)
−0.946018 + 0.324114i \(0.894934\pi\)
\(440\) 0 0
\(441\) 2.89456 5.01352i 0.137836 0.238739i
\(442\) 0 0
\(443\) −15.7479 27.2761i −0.748203 1.29593i −0.948683 0.316228i \(-0.897583\pi\)
0.200480 0.979698i \(-0.435750\pi\)
\(444\) 0 0
\(445\) −7.19709 −0.341175
\(446\) 0 0
\(447\) −3.32992 −0.157500
\(448\) 0 0
\(449\) 17.1863 + 29.7675i 0.811070 + 1.40482i 0.912116 + 0.409933i \(0.134448\pi\)
−0.101045 + 0.994882i \(0.532219\pi\)
\(450\) 0 0
\(451\) −5.58209 + 9.66847i −0.262851 + 0.455271i
\(452\) 0 0
\(453\) −0.834754 + 1.44584i −0.0392202 + 0.0679313i
\(454\) 0 0
\(455\) 1.10040 0.0515876
\(456\) 0 0
\(457\) 16.3788 + 28.3689i 0.766167 + 1.32704i 0.939627 + 0.342199i \(0.111172\pi\)
−0.173461 + 0.984841i \(0.555495\pi\)
\(458\) 0 0
\(459\) −1.83022 + 3.17003i −0.0854274 + 0.147965i
\(460\) 0 0
\(461\) −1.30790 −0.0609148 −0.0304574 0.999536i \(-0.509696\pi\)
−0.0304574 + 0.999536i \(0.509696\pi\)
\(462\) 0 0
\(463\) −4.87930 + 8.45120i −0.226760 + 0.392761i −0.956846 0.290595i \(-0.906147\pi\)
0.730086 + 0.683356i \(0.239480\pi\)
\(464\) 0 0
\(465\) −1.12291 1.94493i −0.0520736 0.0901941i
\(466\) 0 0
\(467\) −15.2035 + 26.3332i −0.703532 + 1.21855i 0.263687 + 0.964608i \(0.415062\pi\)
−0.967219 + 0.253945i \(0.918272\pi\)
\(468\) 0 0
\(469\) −5.42037 7.19365i −0.250289 0.332172i
\(470\) 0 0
\(471\) 9.29941 16.1071i 0.428494 0.742174i
\(472\) 0 0
\(473\) −2.07784 3.59892i −0.0955391 0.165479i
\(474\) 0 0
\(475\) −3.26255 + 5.65090i −0.149696 + 0.259281i
\(476\) 0 0
\(477\) 1.24582 0.0570420
\(478\) 0 0
\(479\) 1.11692 1.93456i 0.0510332 0.0883922i −0.839380 0.543545i \(-0.817082\pi\)
0.890414 + 0.455152i \(0.150415\pi\)
\(480\) 0 0
\(481\) 5.58203 + 9.66835i 0.254519 + 0.440839i
\(482\) 0 0
\(483\) −9.05006 −0.411792
\(484\) 0 0
\(485\) −3.18799 + 5.52175i −0.144759 + 0.250730i
\(486\) 0 0
\(487\) −15.2557 + 26.4237i −0.691304 + 1.19737i 0.280107 + 0.959969i \(0.409630\pi\)
−0.971411 + 0.237405i \(0.923703\pi\)
\(488\) 0 0
\(489\) 4.29454 + 7.43836i 0.194206 + 0.336374i
\(490\) 0 0
\(491\) 10.5642 0.476756 0.238378 0.971172i \(-0.423384\pi\)
0.238378 + 0.971172i \(0.423384\pi\)
\(492\) 0 0
\(493\) −33.5652 −1.51170
\(494\) 0 0
\(495\) 0.589441 + 1.02094i 0.0264934 + 0.0458879i
\(496\) 0 0
\(497\) 3.38456 5.86223i 0.151818 0.262957i
\(498\) 0 0
\(499\) −12.1578 + 21.0579i −0.544257 + 0.942680i 0.454397 + 0.890800i \(0.349855\pi\)
−0.998653 + 0.0518808i \(0.983478\pi\)
\(500\) 0 0
\(501\) −5.73292 9.92970i −0.256128 0.443626i
\(502\) 0 0
\(503\) 0.535297 + 0.927162i 0.0238677 + 0.0413401i 0.877713 0.479188i \(-0.159069\pi\)
−0.853845 + 0.520528i \(0.825735\pi\)
\(504\) 0 0
\(505\) −7.77764 13.4713i −0.346101 0.599464i
\(506\) 0 0
\(507\) 6.00000 + 10.3923i 0.266469 + 0.461538i
\(508\) 0 0
\(509\) −7.89902 −0.350118 −0.175059 0.984558i \(-0.556012\pi\)
−0.175059 + 0.984558i \(0.556012\pi\)
\(510\) 0 0
\(511\) 9.80081 0.433562
\(512\) 0 0
\(513\) −3.26255 + 5.65090i −0.144045 + 0.249493i
\(514\) 0 0
\(515\) 2.21517 + 3.83680i 0.0976123 + 0.169069i
\(516\) 0 0
\(517\) 6.08949 10.5473i 0.267815 0.463870i
\(518\) 0 0
\(519\) −5.77239 + 9.99807i −0.253380 + 0.438867i
\(520\) 0 0
\(521\) 31.4657 1.37854 0.689268 0.724506i \(-0.257932\pi\)
0.689268 + 0.724506i \(0.257932\pi\)
\(522\) 0 0
\(523\) 10.6369 18.4237i 0.465119 0.805610i −0.534088 0.845429i \(-0.679345\pi\)
0.999207 + 0.0398188i \(0.0126781\pi\)
\(524\) 0 0
\(525\) −0.550200 0.952975i −0.0240127 0.0415912i
\(526\) 0 0
\(527\) −8.22067 −0.358098
\(528\) 0 0
\(529\) −22.3198 38.6590i −0.970427 1.68083i
\(530\) 0 0
\(531\) 8.73327 0.378992
\(532\) 0 0
\(533\) −9.47015 −0.410198
\(534\) 0 0
\(535\) −14.6613 −0.633865
\(536\) 0 0
\(537\) 23.1723 0.999960
\(538\) 0 0
\(539\) −6.82469 −0.293960
\(540\) 0 0
\(541\) 0.429557 0.0184681 0.00923405 0.999957i \(-0.497061\pi\)
0.00923405 + 0.999957i \(0.497061\pi\)
\(542\) 0 0
\(543\) 11.6327 + 20.1484i 0.499206 + 0.864650i
\(544\) 0 0
\(545\) 0.127209 0.00544905
\(546\) 0 0
\(547\) −9.68237 16.7704i −0.413988 0.717049i 0.581333 0.813666i \(-0.302531\pi\)
−0.995322 + 0.0966166i \(0.969198\pi\)
\(548\) 0 0
\(549\) 7.40027 12.8176i 0.315836 0.547043i
\(550\) 0 0
\(551\) −59.8332 −2.54898
\(552\) 0 0
\(553\) −1.34906 + 2.33664i −0.0573678 + 0.0993640i
\(554\) 0 0
\(555\) 5.58203 9.66835i 0.236944 0.410399i
\(556\) 0 0
\(557\) 1.71292 + 2.96686i 0.0725787 + 0.125710i 0.900031 0.435826i \(-0.143544\pi\)
−0.827452 + 0.561536i \(0.810210\pi\)
\(558\) 0 0
\(559\) 1.76255 3.05282i 0.0745479 0.129121i
\(560\) 0 0
\(561\) 4.31523 0.182189
\(562\) 0 0
\(563\) −38.9250 −1.64049 −0.820247 0.572009i \(-0.806164\pi\)
−0.820247 + 0.572009i \(0.806164\pi\)
\(564\) 0 0
\(565\) −2.12291 3.67698i −0.0893114 0.154692i
\(566\) 0 0
\(567\) −0.550200 0.952975i −0.0231062 0.0400212i
\(568\) 0 0
\(569\) 16.3476 + 28.3149i 0.685328 + 1.18702i 0.973334 + 0.229394i \(0.0736744\pi\)
−0.288006 + 0.957629i \(0.592992\pi\)
\(570\) 0 0
\(571\) −4.36307 7.55706i −0.182589 0.316253i 0.760173 0.649721i \(-0.225114\pi\)
−0.942761 + 0.333468i \(0.891781\pi\)
\(572\) 0 0
\(573\) −1.97540 + 3.42150i −0.0825237 + 0.142935i
\(574\) 0 0
\(575\) 4.11217 7.12248i 0.171489 0.297028i
\(576\) 0 0
\(577\) 2.51029 + 4.34795i 0.104505 + 0.181008i 0.913536 0.406758i \(-0.133341\pi\)
−0.809031 + 0.587766i \(0.800008\pi\)
\(578\) 0 0
\(579\) −16.6912 −0.693663
\(580\) 0 0
\(581\) 12.0759 0.500992
\(582\) 0 0
\(583\) −0.734335 1.27191i −0.0304130 0.0526769i
\(584\) 0 0
\(585\) −0.500000 + 0.866025i −0.0206725 + 0.0358057i
\(586\) 0 0
\(587\) −1.17510 + 2.03533i −0.0485016 + 0.0840072i −0.889257 0.457408i \(-0.848778\pi\)
0.840755 + 0.541415i \(0.182111\pi\)
\(588\) 0 0
\(589\) −14.6542 −0.603814
\(590\) 0 0
\(591\) 8.67221 + 15.0207i 0.356727 + 0.617869i
\(592\) 0 0
\(593\) 19.1748 33.2117i 0.787414 1.36384i −0.140132 0.990133i \(-0.544753\pi\)
0.927546 0.373709i \(-0.121914\pi\)
\(594\) 0 0
\(595\) −4.02795 −0.165130
\(596\) 0 0
\(597\) −12.8250 + 22.2136i −0.524893 + 0.909141i
\(598\) 0 0
\(599\) 15.1881 + 26.3066i 0.620570 + 1.07486i 0.989380 + 0.145354i \(0.0464321\pi\)
−0.368810 + 0.929505i \(0.620235\pi\)
\(600\) 0 0
\(601\) 18.8525 32.6535i 0.769009 1.33196i −0.169092 0.985600i \(-0.554083\pi\)
0.938101 0.346362i \(-0.112583\pi\)
\(602\) 0 0
\(603\) 8.12438 0.997229i 0.330850 0.0406103i
\(604\) 0 0
\(605\) −4.80512 + 8.32271i −0.195356 + 0.338366i
\(606\) 0 0
\(607\) 1.71292 + 2.96687i 0.0695254 + 0.120422i 0.898692 0.438579i \(-0.144518\pi\)
−0.829167 + 0.559001i \(0.811185\pi\)
\(608\) 0 0
\(609\) 5.04518 8.73850i 0.204441 0.354102i
\(610\) 0 0
\(611\) 10.3310 0.417946
\(612\) 0 0
\(613\) 22.3393 38.6929i 0.902277 1.56279i 0.0777468 0.996973i \(-0.475227\pi\)
0.824531 0.565817i \(-0.191439\pi\)
\(614\) 0 0
\(615\) 4.73507 + 8.20139i 0.190937 + 0.330712i
\(616\) 0 0
\(617\) −0.269780 −0.0108609 −0.00543047 0.999985i \(-0.501729\pi\)
−0.00543047 + 0.999985i \(0.501729\pi\)
\(618\) 0 0
\(619\) 2.26633 3.92540i 0.0910915 0.157775i −0.816879 0.576809i \(-0.804298\pi\)
0.907971 + 0.419034i \(0.137631\pi\)
\(620\) 0 0
\(621\) 4.11217 7.12248i 0.165016 0.285815i
\(622\) 0 0
\(623\) −3.95984 6.85864i −0.158648 0.274786i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.69232 0.307202
\(628\) 0 0
\(629\) −20.4327 35.3904i −0.814704 1.41111i
\(630\) 0 0
\(631\) 2.96580 5.13691i 0.118067 0.204497i −0.800935 0.598751i \(-0.795664\pi\)
0.919001 + 0.394254i \(0.128997\pi\)
\(632\) 0 0
\(633\) −5.91507 + 10.2452i −0.235103 + 0.407210i
\(634\) 0 0
\(635\) 5.22138 + 9.04370i 0.207204 + 0.358888i
\(636\) 0 0
\(637\) −2.89456 5.01352i −0.114687 0.198643i
\(638\) 0 0
\(639\) 3.07575 + 5.32736i 0.121675 + 0.210747i
\(640\) 0 0
\(641\) −1.10945 1.92162i −0.0438206 0.0758995i 0.843283 0.537469i \(-0.180620\pi\)
−0.887104 + 0.461570i \(0.847286\pi\)
\(642\) 0 0
\(643\) 35.6290 1.40507 0.702535 0.711649i \(-0.252051\pi\)
0.702535 + 0.711649i \(0.252051\pi\)
\(644\) 0 0
\(645\) −3.52510 −0.138801
\(646\) 0 0
\(647\) 21.2079 36.7331i 0.833767 1.44413i −0.0612630 0.998122i \(-0.519513\pi\)
0.895030 0.446005i \(-0.147154\pi\)
\(648\) 0 0
\(649\) −5.14775 8.91616i −0.202067 0.349990i
\(650\) 0 0
\(651\) 1.23565 2.14021i 0.0484289 0.0838813i
\(652\) 0 0
\(653\) 5.50356 9.53244i 0.215371 0.373033i −0.738016 0.674783i \(-0.764237\pi\)
0.953387 + 0.301749i \(0.0975706\pi\)
\(654\) 0 0
\(655\) −4.81740 −0.188232
\(656\) 0 0
\(657\) −4.45329 + 7.71333i −0.173739 + 0.300926i
\(658\) 0 0
\(659\) −1.49622 2.59154i −0.0582846 0.100952i 0.835411 0.549626i \(-0.185230\pi\)
−0.893695 + 0.448674i \(0.851896\pi\)
\(660\) 0 0
\(661\) −34.9573 −1.35968 −0.679842 0.733359i \(-0.737951\pi\)
−0.679842 + 0.733359i \(0.737951\pi\)
\(662\) 0 0
\(663\) 1.83022 + 3.17003i 0.0710799 + 0.123114i
\(664\) 0 0
\(665\) −7.18022 −0.278437
\(666\) 0 0
\(667\) 75.4147 2.92007
\(668\) 0 0
\(669\) 14.1821 0.548313
\(670\) 0 0
\(671\) −17.4481 −0.673576
\(672\) 0 0
\(673\) −18.1603 −0.700029 −0.350014 0.936744i \(-0.613823\pi\)
−0.350014 + 0.936744i \(0.613823\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −23.2194 40.2172i −0.892393 1.54567i −0.836998 0.547206i \(-0.815691\pi\)
−0.0553957 0.998464i \(-0.517642\pi\)
\(678\) 0 0
\(679\) −7.01612 −0.269254
\(680\) 0 0
\(681\) 11.0669 + 19.1684i 0.424083 + 0.734533i
\(682\) 0 0
\(683\) −13.4869 + 23.3600i −0.516062 + 0.893845i 0.483764 + 0.875198i \(0.339269\pi\)
−0.999826 + 0.0186468i \(0.994064\pi\)
\(684\) 0 0
\(685\) 0.833998 0.0318654
\(686\) 0 0
\(687\) 8.26322 14.3123i 0.315261 0.546049i
\(688\) 0 0
\(689\) 0.622908 1.07891i 0.0237309 0.0411031i
\(690\) 0 0
\(691\) −13.2380 22.9288i −0.503596 0.872254i −0.999991 0.00415771i \(-0.998677\pi\)
0.496395 0.868097i \(-0.334657\pi\)
\(692\) 0 0
\(693\) −0.648621 + 1.12345i −0.0246391 + 0.0426762i
\(694\) 0 0
\(695\) 10.8727 0.412423
\(696\) 0 0
\(697\) 34.6649 1.31303
\(698\) 0 0
\(699\) 2.96682 + 5.13868i 0.112215 + 0.194363i
\(700\) 0 0
\(701\) 18.9887 + 32.8894i 0.717193 + 1.24221i 0.962108 + 0.272669i \(0.0879065\pi\)
−0.244915 + 0.969544i \(0.578760\pi\)
\(702\) 0 0
\(703\) −36.4233 63.0870i −1.37373 2.37937i
\(704\) 0 0
\(705\) −5.16548 8.94687i −0.194543 0.336959i
\(706\) 0 0
\(707\) 8.55852 14.8238i 0.321876 0.557506i
\(708\) 0 0
\(709\) −16.8930 + 29.2596i −0.634431 + 1.09887i 0.352204 + 0.935923i \(0.385432\pi\)
−0.986635 + 0.162944i \(0.947901\pi\)
\(710\) 0 0
\(711\) −1.22597 2.12344i −0.0459775 0.0796353i
\(712\) 0 0
\(713\) 18.4703 0.691719
\(714\) 0 0
\(715\) 1.17888 0.0440877
\(716\) 0 0
\(717\) 12.6186 + 21.8561i 0.471250 + 0.816230i
\(718\) 0 0
\(719\) −24.8064 + 42.9659i −0.925121 + 1.60236i −0.133755 + 0.991014i \(0.542703\pi\)
−0.791366 + 0.611342i \(0.790630\pi\)
\(720\) 0 0
\(721\) −2.43758 + 4.22201i −0.0907802 + 0.157236i
\(722\) 0 0
\(723\) −16.6342 −0.618633
\(724\) 0 0
\(725\) 4.58485 + 7.94120i 0.170277 + 0.294929i
\(726\) 0 0
\(727\) −15.7465 + 27.2737i −0.584005 + 1.01153i 0.410994 + 0.911638i \(0.365182\pi\)
−0.994999 + 0.0998877i \(0.968152\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.45171 + 11.1747i −0.238625 + 0.413311i
\(732\) 0 0
\(733\) −19.4492 33.6871i −0.718374 1.24426i −0.961644 0.274301i \(-0.911553\pi\)
0.243270 0.969959i \(-0.421780\pi\)
\(734\) 0 0
\(735\) −2.89456 + 5.01352i −0.106767 + 0.184927i
\(736\) 0 0
\(737\) −5.80696 7.70671i −0.213902 0.283880i
\(738\) 0 0
\(739\) −13.6016 + 23.5587i −0.500344 + 0.866622i 0.499656 + 0.866224i \(0.333460\pi\)
−1.00000 0.000397539i \(0.999873\pi\)
\(740\) 0 0
\(741\) 3.26255 + 5.65090i 0.119853 + 0.207591i
\(742\) 0 0
\(743\) −0.431423 + 0.747246i −0.0158274 + 0.0274138i −0.873831 0.486230i \(-0.838372\pi\)
0.858003 + 0.513644i \(0.171705\pi\)
\(744\) 0 0
\(745\) 3.32992 0.121999
\(746\) 0 0
\(747\) −5.48704 + 9.50384i −0.200760 + 0.347727i
\(748\) 0 0
\(749\) −8.06667 13.9719i −0.294750 0.510522i
\(750\) 0 0
\(751\) 51.9831 1.89689 0.948444 0.316944i \(-0.102657\pi\)
0.948444 + 0.316944i \(0.102657\pi\)
\(752\) 0 0
\(753\) 11.1153 19.2523i 0.405065 0.701593i
\(754\) 0 0
\(755\) 0.834754 1.44584i 0.0303798 0.0526194i
\(756\) 0 0
\(757\) −12.4768 21.6104i −0.453475 0.785442i 0.545124 0.838355i \(-0.316483\pi\)
−0.998599 + 0.0529134i \(0.983149\pi\)
\(758\) 0 0
\(759\) −9.69552 −0.351925
\(760\) 0 0
\(761\) −30.7143 −1.11339 −0.556697 0.830716i \(-0.687932\pi\)
−0.556697 + 0.830716i \(0.687932\pi\)
\(762\) 0 0
\(763\) 0.0699906 + 0.121227i 0.00253383 + 0.00438872i
\(764\) 0 0
\(765\) 1.83022 3.17003i 0.0661718 0.114613i
\(766\) 0 0
\(767\) 4.36664 7.56323i 0.157670 0.273093i
\(768\) 0 0
\(769\) −22.1945 38.4420i −0.800353 1.38625i −0.919384 0.393362i \(-0.871312\pi\)
0.119031 0.992891i \(-0.462021\pi\)
\(770\) 0 0
\(771\) −4.81593 8.34143i −0.173441 0.300409i
\(772\) 0 0
\(773\) −24.2720 42.0404i −0.873004 1.51209i −0.858874 0.512188i \(-0.828835\pi\)
−0.0141307 0.999900i \(-0.504498\pi\)
\(774\) 0 0
\(775\) 1.12291 + 1.94493i 0.0403360 + 0.0698641i
\(776\) 0 0
\(777\) 12.2849 0.440719
\(778\) 0 0
\(779\) 61.7936 2.21399
\(780\) 0 0
\(781\) 3.62595 6.28033i 0.129747 0.224728i
\(782\) 0 0
\(783\) 4.58485 + 7.94120i 0.163849 + 0.283795i
\(784\) 0 0
\(785\) −9.29941 + 16.1071i −0.331910 + 0.574885i
\(786\) 0 0
\(787\) −16.9491 + 29.3567i −0.604171 + 1.04645i 0.388011 + 0.921655i \(0.373162\pi\)
−0.992182 + 0.124800i \(0.960171\pi\)
\(788\) 0 0
\(789\) −11.9838 −0.426636
\(790\) 0 0
\(791\) 2.33605 4.04615i 0.0830603 0.143865i
\(792\) 0 0
\(793\) −7.40027 12.8176i −0.262791 0.455168i
\(794\) 0 0
\(795\) −1.24582 −0.0441845
\(796\) 0 0
\(797\) −21.4290 37.1161i −0.759053 1.31472i −0.943334 0.331845i \(-0.892329\pi\)
0.184281 0.982874i \(-0.441005\pi\)
\(798\) 0 0
\(799\) −37.8158 −1.33783
\(800\) 0 0
\(801\) 7.19709 0.254297
\(802\) 0 0
\(803\) 10.4998 0.370530
\(804\) 0 0
\(805\) 9.05006 0.318973
\(806\) 0 0
\(807\) −27.5058 −0.968252
\(808\) 0 0
\(809\) 42.4308 1.49179 0.745893 0.666065i \(-0.232023\pi\)
0.745893 + 0.666065i \(0.232023\pi\)
\(810\) 0 0
\(811\) 11.8418 + 20.5106i 0.415822 + 0.720224i 0.995514 0.0946111i \(-0.0301608\pi\)
−0.579693 + 0.814835i \(0.696827\pi\)
\(812\) 0 0
\(813\) −6.99762 −0.245417
\(814\) 0 0
\(815\) −4.29454 7.43836i −0.150431 0.260554i
\(816\) 0 0
\(817\) −11.5008 + 19.9200i −0.402362 + 0.696912i
\(818\) 0 0
\(819\) −1.10040 −0.0384511
\(820\) 0 0
\(821\) 4.69814 8.13742i 0.163966 0.283998i −0.772321 0.635232i \(-0.780904\pi\)
0.936288 + 0.351234i \(0.114238\pi\)
\(822\) 0 0
\(823\) −20.2804 + 35.1267i −0.706930 + 1.22444i 0.259061 + 0.965861i \(0.416587\pi\)
−0.965990 + 0.258578i \(0.916746\pi\)
\(824\) 0 0
\(825\) −0.589441 1.02094i −0.0205217 0.0355446i
\(826\) 0 0
\(827\) 15.1528 26.2454i 0.526915 0.912643i −0.472593 0.881281i \(-0.656682\pi\)
0.999508 0.0313624i \(-0.00998459\pi\)
\(828\) 0 0
\(829\) −22.3326 −0.775642 −0.387821 0.921735i \(-0.626772\pi\)
−0.387821 + 0.921735i \(0.626772\pi\)
\(830\) 0 0
\(831\) −3.51028 −0.121770
\(832\) 0 0
\(833\) 10.5954 + 18.3517i 0.367108 + 0.635849i
\(834\) 0 0
\(835\) 5.73292 + 9.92970i 0.198396 + 0.343632i
\(836\) 0 0
\(837\) 1.12291 + 1.94493i 0.0388134 + 0.0672267i
\(838\) 0 0
\(839\) 4.34310 + 7.52248i 0.149941 + 0.259705i 0.931205 0.364495i \(-0.118758\pi\)
−0.781265 + 0.624200i \(0.785425\pi\)
\(840\) 0 0
\(841\) −27.5418 + 47.7037i −0.949716 + 1.64496i
\(842\) 0 0
\(843\) 1.60328 2.77696i 0.0552199 0.0956437i
\(844\) 0 0
\(845\) −6.00000 10.3923i −0.206406 0.357506i
\(846\) 0 0
\(847\) −10.5751 −0.363365
\(848\) 0 0
\(849\) 0.280187 0.00961601
\(850\) 0 0
\(851\) 45.9084 + 79.5158i 1.57372 + 2.72576i
\(852\) 0 0
\(853\) 8.24107 14.2739i 0.282169 0.488731i −0.689750 0.724048i \(-0.742280\pi\)
0.971919 + 0.235317i \(0.0756129\pi\)
\(854\) 0 0
\(855\) 3.26255 5.65090i 0.111577 0.193257i
\(856\) 0 0
\(857\) −24.0863 −0.822772 −0.411386 0.911461i \(-0.634955\pi\)
−0.411386 + 0.911461i \(0.634955\pi\)
\(858\) 0 0
\(859\) 19.9354 + 34.5292i 0.680188 + 1.17812i 0.974923 + 0.222541i \(0.0714353\pi\)
−0.294735 + 0.955579i \(0.595231\pi\)
\(860\) 0 0
\(861\) −5.21048 + 9.02481i −0.177573 + 0.307565i
\(862\) 0 0
\(863\) −45.2344 −1.53980 −0.769898 0.638167i \(-0.779693\pi\)
−0.769898 + 0.638167i \(0.779693\pi\)
\(864\) 0 0
\(865\) 5.77239 9.99807i 0.196267 0.339945i
\(866\) 0 0
\(867\) 1.80059 + 3.11871i 0.0611512 + 0.105917i
\(868\) 0 0
\(869\) −1.44528 + 2.50329i −0.0490276 + 0.0849183i
\(870\) 0 0
\(871\) 3.19856 7.53453i 0.108379 0.255298i
\(872\) 0 0
\(873\) 3.18799 5.52175i 0.107897 0.186883i
\(874\) 0 0
\(875\) 0.550200 + 0.952975i 0.0186002 + 0.0322164i
\(876\) 0 0
\(877\) 4.06823 7.04638i 0.137374 0.237939i −0.789128 0.614229i \(-0.789467\pi\)
0.926502 + 0.376290i \(0.122800\pi\)
\(878\) 0 0
\(879\) 7.10378 0.239604
\(880\) 0 0
\(881\) 13.8350 23.9630i 0.466114 0.807333i −0.533137 0.846029i \(-0.678987\pi\)
0.999251 + 0.0386961i \(0.0123204\pi\)
\(882\) 0 0
\(883\) 22.4030 + 38.8032i 0.753922 + 1.30583i 0.945908 + 0.324434i \(0.105174\pi\)
−0.191986 + 0.981398i \(0.561493\pi\)
\(884\) 0 0
\(885\) −8.73327 −0.293566
\(886\) 0 0
\(887\) 4.70319 8.14617i 0.157918 0.273522i −0.776200 0.630487i \(-0.782855\pi\)
0.934118 + 0.356965i \(0.116189\pi\)
\(888\) 0 0
\(889\) −5.74561 + 9.95169i −0.192702 + 0.333769i
\(890\) 0 0
\(891\) −0.589441 1.02094i −0.0197470 0.0342028i
\(892\) 0 0
\(893\) −67.4105 −2.25581
\(894\) 0 0
\(895\) −23.1723 −0.774565
\(896\) 0 0
\(897\) −4.11217 7.12248i −0.137301 0.237813i
\(898\) 0 0
\(899\) −10.2967 + 17.8345i −0.343415 + 0.594813i
\(900\) 0 0
\(901\) −2.28012 + 3.94928i −0.0759617 + 0.131569i
\(902\) 0 0
\(903\) −1.93951 3.35933i −0.0645429 0.111791i
\(904\) 0 0
\(905\) −11.6327 20.1484i −0.386683 0.669755i
\(906\) 0 0
\(907\) 6.19999 + 10.7387i 0.205867 + 0.356573i 0.950409 0.311004i \(-0.100665\pi\)
−0.744541 + 0.667576i \(0.767332\pi\)
\(908\) 0 0
\(909\) 7.77764 + 13.4713i 0.257968 + 0.446814i
\(910\) 0 0
\(911\) −32.3176 −1.07073 −0.535365 0.844621i \(-0.679826\pi\)
−0.535365 + 0.844621i \(0.679826\pi\)
\(912\) 0 0
\(913\) 12.9372 0.428157
\(914\) 0 0
\(915\) −7.40027 + 12.8176i −0.244645 + 0.423738i
\(916\) 0 0
\(917\) −2.65054 4.59086i −0.0875284 0.151604i
\(918\) 0 0
\(919\) 24.8807 43.0946i 0.820737 1.42156i −0.0843968 0.996432i \(-0.526896\pi\)
0.905134 0.425126i \(-0.139770\pi\)
\(920\) 0 0
\(921\) −5.76165 + 9.97946i −0.189853 + 0.328835i
\(922\) 0 0
\(923\) 6.15150 0.202479
\(924\) 0 0
\(925\) −5.58203 + 9.66835i −0.183536 + 0.317894i
\(926\) 0 0
\(927\) −2.21517 3.83680i −0.0727559 0.126017i
\(928\) 0 0
\(929\) 8.38322 0.275044 0.137522 0.990499i \(-0.456086\pi\)
0.137522 + 0.990499i \(0.456086\pi\)
\(930\) 0 0
\(931\) 18.8873 + 32.7137i 0.619006 + 1.07215i
\(932\) 0 0
\(933\) 9.66677 0.316476
\(934\) 0 0
\(935\) −4.31523 −0.141123
\(936\) 0 0
\(937\) −51.9476 −1.69706 −0.848528 0.529151i \(-0.822511\pi\)
−0.848528 + 0.529151i \(0.822511\pi\)
\(938\) 0 0
\(939\) −19.8904 −0.649099
\(940\) 0 0
\(941\) 16.2896 0.531026 0.265513 0.964107i \(-0.414459\pi\)
0.265513 + 0.964107i \(0.414459\pi\)
\(942\) 0 0
\(943\) −77.8856 −2.53630
\(944\) 0 0
\(945\) 0.550200 + 0.952975i 0.0178980 + 0.0310003i
\(946\) 0 0
\(947\) −2.49425 −0.0810522 −0.0405261 0.999178i \(-0.512903\pi\)
−0.0405261 + 0.999178i \(0.512903\pi\)
\(948\) 0 0
\(949\) 4.45329 + 7.71333i 0.144560 + 0.250385i
\(950\) 0 0
\(951\) −2.59315 + 4.49147i −0.0840887 + 0.145646i
\(952\) 0 0
\(953\) 11.9196 0.386113 0.193056 0.981188i \(-0.438160\pi\)
0.193056 + 0.981188i \(0.438160\pi\)
\(954\) 0 0
\(955\) 1.97540 3.42150i 0.0639226 0.110717i
\(956\) 0 0
\(957\) 5.40500 9.36174i 0.174719 0.302622i
\(958\) 0 0
\(959\) 0.458866 + 0.794779i 0.0148176 + 0.0256648i
\(960\) 0 0
\(961\) 12.9782 22.4788i 0.418650 0.725123i
\(962\) 0 0
\(963\) 14.6613 0.472455
\(964\) 0 0
\(965\) 16.6912 0.537309
\(966\) 0 0
\(967\) −6.64293 11.5059i −0.213622 0.370005i 0.739223 0.673460i \(-0.235193\pi\)
−0.952845 + 0.303456i \(0.901859\pi\)
\(968\) 0 0
\(969\) −11.9424 20.6848i −0.383644 0.664491i
\(970\) 0 0
\(971\) 9.04198 + 15.6612i 0.290171 + 0.502591i 0.973850 0.227192i \(-0.0729545\pi\)
−0.683679 + 0.729783i \(0.739621\pi\)
\(972\) 0 0
\(973\) 5.98214 + 10.3614i 0.191778 + 0.332170i
\(974\) 0 0
\(975\) 0.500000 0.866025i 0.0160128 0.0277350i
\(976\) 0 0
\(977\) 7.76151 13.4433i 0.248313 0.430090i −0.714745 0.699385i \(-0.753457\pi\)
0.963058 + 0.269295i \(0.0867906\pi\)
\(978\) 0 0
\(979\) −4.24226 7.34781i −0.135583 0.234837i
\(980\) 0 0
\(981\) −0.127209 −0.00406148
\(982\) 0 0
\(983\) −4.34963 −0.138732 −0.0693658 0.997591i \(-0.522098\pi\)
−0.0693658 + 0.997591i \(0.522098\pi\)
\(984\) 0 0
\(985\) −8.67221 15.0207i −0.276319 0.478599i
\(986\) 0 0
\(987\) 5.68409 9.84514i 0.180927 0.313374i
\(988\) 0 0
\(989\) 14.4958 25.1074i 0.460939 0.798370i
\(990\) 0 0
\(991\) −27.7433 −0.881296 −0.440648 0.897680i \(-0.645251\pi\)
−0.440648 + 0.897680i \(0.645251\pi\)
\(992\) 0 0
\(993\) −3.39598 5.88201i −0.107768 0.186660i
\(994\) 0 0
\(995\) 12.8250 22.2136i 0.406580 0.704218i
\(996\) 0 0
\(997\) −18.2526 −0.578065 −0.289033 0.957319i \(-0.593334\pi\)
−0.289033 + 0.957319i \(0.593334\pi\)
\(998\) 0 0
\(999\) −5.58203 + 9.66835i −0.176608 + 0.305893i
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.j.3781.3 yes 12
67.37 even 3 inner 4020.2.q.j.841.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.j.841.3 12 67.37 even 3 inner
4020.2.q.j.3781.3 yes 12 1.1 even 1 trivial