Properties

Label 4010.2.a.j.1.4
Level $4010$
Weight $2$
Character 4010.1
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.a (trivial)

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: yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 93 x^{8} - 162 x^{7} - 238 x^{6} + 391 x^{5} + 240 x^{4} - 408 x^{3} - 42 x^{2} + 120 x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.16326\) of defining polynomial
Character \(\chi\) \(=\) 4010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.16326 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.16326 q^{6} -3.35475 q^{7} +1.00000 q^{8} -1.64681 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.16326 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.16326 q^{6} -3.35475 q^{7} +1.00000 q^{8} -1.64681 q^{9} -1.00000 q^{10} +2.96520 q^{11} -1.16326 q^{12} +2.83453 q^{13} -3.35475 q^{14} +1.16326 q^{15} +1.00000 q^{16} +5.60775 q^{17} -1.64681 q^{18} -6.23686 q^{19} -1.00000 q^{20} +3.90247 q^{21} +2.96520 q^{22} -2.99565 q^{23} -1.16326 q^{24} +1.00000 q^{25} +2.83453 q^{26} +5.40548 q^{27} -3.35475 q^{28} +6.71384 q^{29} +1.16326 q^{30} +7.99833 q^{31} +1.00000 q^{32} -3.44932 q^{33} +5.60775 q^{34} +3.35475 q^{35} -1.64681 q^{36} -8.79576 q^{37} -6.23686 q^{38} -3.29731 q^{39} -1.00000 q^{40} -12.2963 q^{41} +3.90247 q^{42} -0.840974 q^{43} +2.96520 q^{44} +1.64681 q^{45} -2.99565 q^{46} -1.73855 q^{47} -1.16326 q^{48} +4.25438 q^{49} +1.00000 q^{50} -6.52330 q^{51} +2.83453 q^{52} -10.2164 q^{53} +5.40548 q^{54} -2.96520 q^{55} -3.35475 q^{56} +7.25512 q^{57} +6.71384 q^{58} +13.1910 q^{59} +1.16326 q^{60} -8.53190 q^{61} +7.99833 q^{62} +5.52466 q^{63} +1.00000 q^{64} -2.83453 q^{65} -3.44932 q^{66} +9.32681 q^{67} +5.60775 q^{68} +3.48473 q^{69} +3.35475 q^{70} -9.13221 q^{71} -1.64681 q^{72} -3.85112 q^{73} -8.79576 q^{74} -1.16326 q^{75} -6.23686 q^{76} -9.94753 q^{77} -3.29731 q^{78} +9.04751 q^{79} -1.00000 q^{80} -1.34756 q^{81} -12.2963 q^{82} -15.8755 q^{83} +3.90247 q^{84} -5.60775 q^{85} -0.840974 q^{86} -7.80997 q^{87} +2.96520 q^{88} -4.65541 q^{89} +1.64681 q^{90} -9.50914 q^{91} -2.99565 q^{92} -9.30417 q^{93} -1.73855 q^{94} +6.23686 q^{95} -1.16326 q^{96} -2.71538 q^{97} +4.25438 q^{98} -4.88314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 12 q^{5} - 2 q^{6} - 9 q^{7} + 12 q^{8} - 12 q^{10} + q^{11} - 2 q^{12} - 6 q^{13} - 9 q^{14} + 2 q^{15} + 12 q^{16} - 11 q^{17} - 13 q^{19} - 12 q^{20} - 14 q^{21} + q^{22} - 21 q^{23} - 2 q^{24} + 12 q^{25} - 6 q^{26} - 2 q^{27} - 9 q^{28} - 10 q^{29} + 2 q^{30} - 11 q^{31} + 12 q^{32} - 22 q^{33} - 11 q^{34} + 9 q^{35} - 29 q^{37} - 13 q^{38} - 2 q^{39} - 12 q^{40} - q^{41} - 14 q^{42} - 23 q^{43} + q^{44} - 21 q^{46} - 17 q^{47} - 2 q^{48} - 3 q^{49} + 12 q^{50} - 19 q^{51} - 6 q^{52} - 47 q^{53} - 2 q^{54} - q^{55} - 9 q^{56} - 11 q^{57} - 10 q^{58} + 14 q^{59} + 2 q^{60} - 22 q^{61} - 11 q^{62} - 28 q^{63} + 12 q^{64} + 6 q^{65} - 22 q^{66} - 28 q^{67} - 11 q^{68} - q^{69} + 9 q^{70} - 18 q^{71} - 2 q^{73} - 29 q^{74} - 2 q^{75} - 13 q^{76} - 11 q^{77} - 2 q^{78} - 39 q^{79} - 12 q^{80} - 44 q^{81} - q^{82} - 5 q^{83} - 14 q^{84} + 11 q^{85} - 23 q^{86} - 6 q^{87} + q^{88} - 8 q^{89} - 12 q^{91} - 21 q^{92} - 30 q^{93} - 17 q^{94} + 13 q^{95} - 2 q^{96} - 32 q^{97} - 3 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.16326 −0.671611 −0.335806 0.941931i \(-0.609009\pi\)
−0.335806 + 0.941931i \(0.609009\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.16326 −0.474901
\(7\) −3.35475 −1.26798 −0.633989 0.773342i \(-0.718584\pi\)
−0.633989 + 0.773342i \(0.718584\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.64681 −0.548938
\(10\) −1.00000 −0.316228
\(11\) 2.96520 0.894042 0.447021 0.894523i \(-0.352485\pi\)
0.447021 + 0.894523i \(0.352485\pi\)
\(12\) −1.16326 −0.335806
\(13\) 2.83453 0.786156 0.393078 0.919505i \(-0.371410\pi\)
0.393078 + 0.919505i \(0.371410\pi\)
\(14\) −3.35475 −0.896596
\(15\) 1.16326 0.300354
\(16\) 1.00000 0.250000
\(17\) 5.60775 1.36008 0.680039 0.733176i \(-0.261963\pi\)
0.680039 + 0.733176i \(0.261963\pi\)
\(18\) −1.64681 −0.388158
\(19\) −6.23686 −1.43083 −0.715417 0.698698i \(-0.753763\pi\)
−0.715417 + 0.698698i \(0.753763\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.90247 0.851588
\(22\) 2.96520 0.632183
\(23\) −2.99565 −0.624636 −0.312318 0.949978i \(-0.601105\pi\)
−0.312318 + 0.949978i \(0.601105\pi\)
\(24\) −1.16326 −0.237450
\(25\) 1.00000 0.200000
\(26\) 2.83453 0.555896
\(27\) 5.40548 1.04028
\(28\) −3.35475 −0.633989
\(29\) 6.71384 1.24673 0.623364 0.781931i \(-0.285765\pi\)
0.623364 + 0.781931i \(0.285765\pi\)
\(30\) 1.16326 0.212382
\(31\) 7.99833 1.43654 0.718271 0.695764i \(-0.244934\pi\)
0.718271 + 0.695764i \(0.244934\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.44932 −0.600449
\(34\) 5.60775 0.961721
\(35\) 3.35475 0.567057
\(36\) −1.64681 −0.274469
\(37\) −8.79576 −1.44601 −0.723007 0.690841i \(-0.757241\pi\)
−0.723007 + 0.690841i \(0.757241\pi\)
\(38\) −6.23686 −1.01175
\(39\) −3.29731 −0.527991
\(40\) −1.00000 −0.158114
\(41\) −12.2963 −1.92035 −0.960177 0.279393i \(-0.909867\pi\)
−0.960177 + 0.279393i \(0.909867\pi\)
\(42\) 3.90247 0.602164
\(43\) −0.840974 −0.128247 −0.0641237 0.997942i \(-0.520425\pi\)
−0.0641237 + 0.997942i \(0.520425\pi\)
\(44\) 2.96520 0.447021
\(45\) 1.64681 0.245493
\(46\) −2.99565 −0.441684
\(47\) −1.73855 −0.253594 −0.126797 0.991929i \(-0.540470\pi\)
−0.126797 + 0.991929i \(0.540470\pi\)
\(48\) −1.16326 −0.167903
\(49\) 4.25438 0.607768
\(50\) 1.00000 0.141421
\(51\) −6.52330 −0.913444
\(52\) 2.83453 0.393078
\(53\) −10.2164 −1.40333 −0.701664 0.712508i \(-0.747559\pi\)
−0.701664 + 0.712508i \(0.747559\pi\)
\(54\) 5.40548 0.735592
\(55\) −2.96520 −0.399828
\(56\) −3.35475 −0.448298
\(57\) 7.25512 0.960964
\(58\) 6.71384 0.881570
\(59\) 13.1910 1.71733 0.858663 0.512541i \(-0.171296\pi\)
0.858663 + 0.512541i \(0.171296\pi\)
\(60\) 1.16326 0.150177
\(61\) −8.53190 −1.09240 −0.546199 0.837655i \(-0.683926\pi\)
−0.546199 + 0.837655i \(0.683926\pi\)
\(62\) 7.99833 1.01579
\(63\) 5.52466 0.696042
\(64\) 1.00000 0.125000
\(65\) −2.83453 −0.351580
\(66\) −3.44932 −0.424582
\(67\) 9.32681 1.13945 0.569726 0.821835i \(-0.307050\pi\)
0.569726 + 0.821835i \(0.307050\pi\)
\(68\) 5.60775 0.680039
\(69\) 3.48473 0.419512
\(70\) 3.35475 0.400970
\(71\) −9.13221 −1.08379 −0.541897 0.840445i \(-0.682294\pi\)
−0.541897 + 0.840445i \(0.682294\pi\)
\(72\) −1.64681 −0.194079
\(73\) −3.85112 −0.450740 −0.225370 0.974273i \(-0.572359\pi\)
−0.225370 + 0.974273i \(0.572359\pi\)
\(74\) −8.79576 −1.02249
\(75\) −1.16326 −0.134322
\(76\) −6.23686 −0.715417
\(77\) −9.94753 −1.13363
\(78\) −3.29731 −0.373346
\(79\) 9.04751 1.01792 0.508962 0.860789i \(-0.330029\pi\)
0.508962 + 0.860789i \(0.330029\pi\)
\(80\) −1.00000 −0.111803
\(81\) −1.34756 −0.149728
\(82\) −12.2963 −1.35790
\(83\) −15.8755 −1.74256 −0.871281 0.490785i \(-0.836710\pi\)
−0.871281 + 0.490785i \(0.836710\pi\)
\(84\) 3.90247 0.425794
\(85\) −5.60775 −0.608246
\(86\) −0.840974 −0.0906846
\(87\) −7.80997 −0.837317
\(88\) 2.96520 0.316092
\(89\) −4.65541 −0.493472 −0.246736 0.969083i \(-0.579358\pi\)
−0.246736 + 0.969083i \(0.579358\pi\)
\(90\) 1.64681 0.173590
\(91\) −9.50914 −0.996829
\(92\) −2.99565 −0.312318
\(93\) −9.30417 −0.964798
\(94\) −1.73855 −0.179318
\(95\) 6.23686 0.639888
\(96\) −1.16326 −0.118725
\(97\) −2.71538 −0.275705 −0.137852 0.990453i \(-0.544020\pi\)
−0.137852 + 0.990453i \(0.544020\pi\)
\(98\) 4.25438 0.429757
\(99\) −4.88314 −0.490774
\(100\) 1.00000 0.100000
\(101\) −13.7228 −1.36547 −0.682736 0.730665i \(-0.739210\pi\)
−0.682736 + 0.730665i \(0.739210\pi\)
\(102\) −6.52330 −0.645903
\(103\) −15.4538 −1.52271 −0.761354 0.648336i \(-0.775465\pi\)
−0.761354 + 0.648336i \(0.775465\pi\)
\(104\) 2.83453 0.277948
\(105\) −3.90247 −0.380842
\(106\) −10.2164 −0.992303
\(107\) −2.41946 −0.233898 −0.116949 0.993138i \(-0.537311\pi\)
−0.116949 + 0.993138i \(0.537311\pi\)
\(108\) 5.40548 0.520142
\(109\) −5.07441 −0.486040 −0.243020 0.970021i \(-0.578138\pi\)
−0.243020 + 0.970021i \(0.578138\pi\)
\(110\) −2.96520 −0.282721
\(111\) 10.2318 0.971159
\(112\) −3.35475 −0.316994
\(113\) −16.5001 −1.55220 −0.776101 0.630609i \(-0.782805\pi\)
−0.776101 + 0.630609i \(0.782805\pi\)
\(114\) 7.25512 0.679504
\(115\) 2.99565 0.279346
\(116\) 6.71384 0.623364
\(117\) −4.66794 −0.431551
\(118\) 13.1910 1.21433
\(119\) −18.8126 −1.72455
\(120\) 1.16326 0.106191
\(121\) −2.20757 −0.200688
\(122\) −8.53190 −0.772442
\(123\) 14.3038 1.28973
\(124\) 7.99833 0.718271
\(125\) −1.00000 −0.0894427
\(126\) 5.52466 0.492176
\(127\) 12.7407 1.13055 0.565277 0.824901i \(-0.308769\pi\)
0.565277 + 0.824901i \(0.308769\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.978276 0.0861324
\(130\) −2.83453 −0.248604
\(131\) 4.48683 0.392016 0.196008 0.980602i \(-0.437202\pi\)
0.196008 + 0.980602i \(0.437202\pi\)
\(132\) −3.44932 −0.300224
\(133\) 20.9231 1.81427
\(134\) 9.32681 0.805714
\(135\) −5.40548 −0.465229
\(136\) 5.60775 0.480860
\(137\) 13.7647 1.17599 0.587997 0.808863i \(-0.299917\pi\)
0.587997 + 0.808863i \(0.299917\pi\)
\(138\) 3.48473 0.296640
\(139\) 15.8688 1.34598 0.672989 0.739653i \(-0.265010\pi\)
0.672989 + 0.739653i \(0.265010\pi\)
\(140\) 3.35475 0.283528
\(141\) 2.02240 0.170316
\(142\) −9.13221 −0.766358
\(143\) 8.40495 0.702857
\(144\) −1.64681 −0.137235
\(145\) −6.71384 −0.557554
\(146\) −3.85112 −0.318721
\(147\) −4.94897 −0.408184
\(148\) −8.79576 −0.723007
\(149\) −22.9427 −1.87954 −0.939769 0.341811i \(-0.888960\pi\)
−0.939769 + 0.341811i \(0.888960\pi\)
\(150\) −1.16326 −0.0949802
\(151\) −12.5535 −1.02159 −0.510793 0.859704i \(-0.670648\pi\)
−0.510793 + 0.859704i \(0.670648\pi\)
\(152\) −6.23686 −0.505876
\(153\) −9.23492 −0.746599
\(154\) −9.94753 −0.801595
\(155\) −7.99833 −0.642441
\(156\) −3.29731 −0.263996
\(157\) −7.44522 −0.594194 −0.297097 0.954847i \(-0.596018\pi\)
−0.297097 + 0.954847i \(0.596018\pi\)
\(158\) 9.04751 0.719781
\(159\) 11.8844 0.942491
\(160\) −1.00000 −0.0790569
\(161\) 10.0497 0.792025
\(162\) −1.34756 −0.105874
\(163\) −12.7158 −0.995980 −0.497990 0.867183i \(-0.665928\pi\)
−0.497990 + 0.867183i \(0.665928\pi\)
\(164\) −12.2963 −0.960177
\(165\) 3.44932 0.268529
\(166\) −15.8755 −1.23218
\(167\) 16.6894 1.29147 0.645734 0.763563i \(-0.276552\pi\)
0.645734 + 0.763563i \(0.276552\pi\)
\(168\) 3.90247 0.301082
\(169\) −4.96546 −0.381958
\(170\) −5.60775 −0.430095
\(171\) 10.2710 0.785440
\(172\) −0.840974 −0.0641237
\(173\) −13.7598 −1.04614 −0.523070 0.852290i \(-0.675213\pi\)
−0.523070 + 0.852290i \(0.675213\pi\)
\(174\) −7.80997 −0.592073
\(175\) −3.35475 −0.253596
\(176\) 2.96520 0.223511
\(177\) −15.3447 −1.15338
\(178\) −4.65541 −0.348938
\(179\) −2.09821 −0.156827 −0.0784137 0.996921i \(-0.524986\pi\)
−0.0784137 + 0.996921i \(0.524986\pi\)
\(180\) 1.64681 0.122746
\(181\) 16.1159 1.19788 0.598942 0.800792i \(-0.295588\pi\)
0.598942 + 0.800792i \(0.295588\pi\)
\(182\) −9.50914 −0.704864
\(183\) 9.92486 0.733667
\(184\) −2.99565 −0.220842
\(185\) 8.79576 0.646677
\(186\) −9.30417 −0.682215
\(187\) 16.6281 1.21597
\(188\) −1.73855 −0.126797
\(189\) −18.1340 −1.31906
\(190\) 6.23686 0.452469
\(191\) 6.77893 0.490506 0.245253 0.969459i \(-0.421129\pi\)
0.245253 + 0.969459i \(0.421129\pi\)
\(192\) −1.16326 −0.0839514
\(193\) 20.3721 1.46641 0.733207 0.680006i \(-0.238023\pi\)
0.733207 + 0.680006i \(0.238023\pi\)
\(194\) −2.71538 −0.194953
\(195\) 3.29731 0.236125
\(196\) 4.25438 0.303884
\(197\) −18.4708 −1.31599 −0.657994 0.753023i \(-0.728595\pi\)
−0.657994 + 0.753023i \(0.728595\pi\)
\(198\) −4.88314 −0.347030
\(199\) −24.5092 −1.73741 −0.868704 0.495331i \(-0.835047\pi\)
−0.868704 + 0.495331i \(0.835047\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.8495 −0.765268
\(202\) −13.7228 −0.965534
\(203\) −22.5233 −1.58082
\(204\) −6.52330 −0.456722
\(205\) 12.2963 0.858808
\(206\) −15.4538 −1.07672
\(207\) 4.93328 0.342887
\(208\) 2.83453 0.196539
\(209\) −18.4936 −1.27923
\(210\) −3.90247 −0.269296
\(211\) 0.410059 0.0282296 0.0141148 0.999900i \(-0.495507\pi\)
0.0141148 + 0.999900i \(0.495507\pi\)
\(212\) −10.2164 −0.701664
\(213\) 10.6232 0.727888
\(214\) −2.41946 −0.165391
\(215\) 0.840974 0.0573540
\(216\) 5.40548 0.367796
\(217\) −26.8324 −1.82150
\(218\) −5.07441 −0.343682
\(219\) 4.47987 0.302722
\(220\) −2.96520 −0.199914
\(221\) 15.8953 1.06923
\(222\) 10.2318 0.686713
\(223\) −1.72577 −0.115566 −0.0577830 0.998329i \(-0.518403\pi\)
−0.0577830 + 0.998329i \(0.518403\pi\)
\(224\) −3.35475 −0.224149
\(225\) −1.64681 −0.109788
\(226\) −16.5001 −1.09757
\(227\) −8.01949 −0.532273 −0.266136 0.963935i \(-0.585747\pi\)
−0.266136 + 0.963935i \(0.585747\pi\)
\(228\) 7.25512 0.480482
\(229\) 12.8189 0.847098 0.423549 0.905873i \(-0.360784\pi\)
0.423549 + 0.905873i \(0.360784\pi\)
\(230\) 2.99565 0.197527
\(231\) 11.5716 0.761356
\(232\) 6.71384 0.440785
\(233\) −1.42208 −0.0931633 −0.0465817 0.998914i \(-0.514833\pi\)
−0.0465817 + 0.998914i \(0.514833\pi\)
\(234\) −4.66794 −0.305153
\(235\) 1.73855 0.113411
\(236\) 13.1910 0.858663
\(237\) −10.5246 −0.683649
\(238\) −18.8126 −1.21944
\(239\) −9.78933 −0.633219 −0.316609 0.948556i \(-0.602544\pi\)
−0.316609 + 0.948556i \(0.602544\pi\)
\(240\) 1.16326 0.0750884
\(241\) 0.658045 0.0423884 0.0211942 0.999775i \(-0.493253\pi\)
0.0211942 + 0.999775i \(0.493253\pi\)
\(242\) −2.20757 −0.141908
\(243\) −14.6489 −0.939725
\(244\) −8.53190 −0.546199
\(245\) −4.25438 −0.271802
\(246\) 14.3038 0.911978
\(247\) −17.6785 −1.12486
\(248\) 7.99833 0.507894
\(249\) 18.4674 1.17032
\(250\) −1.00000 −0.0632456
\(251\) 14.2838 0.901587 0.450793 0.892628i \(-0.351141\pi\)
0.450793 + 0.892628i \(0.351141\pi\)
\(252\) 5.52466 0.348021
\(253\) −8.88271 −0.558451
\(254\) 12.7407 0.799423
\(255\) 6.52330 0.408505
\(256\) 1.00000 0.0625000
\(257\) −0.865966 −0.0540175 −0.0270087 0.999635i \(-0.508598\pi\)
−0.0270087 + 0.999635i \(0.508598\pi\)
\(258\) 0.978276 0.0609048
\(259\) 29.5076 1.83351
\(260\) −2.83453 −0.175790
\(261\) −11.0565 −0.684377
\(262\) 4.48683 0.277197
\(263\) −13.8708 −0.855312 −0.427656 0.903942i \(-0.640661\pi\)
−0.427656 + 0.903942i \(0.640661\pi\)
\(264\) −3.44932 −0.212291
\(265\) 10.2164 0.627587
\(266\) 20.9231 1.28288
\(267\) 5.41547 0.331422
\(268\) 9.32681 0.569726
\(269\) 27.4725 1.67503 0.837513 0.546418i \(-0.184009\pi\)
0.837513 + 0.546418i \(0.184009\pi\)
\(270\) −5.40548 −0.328967
\(271\) 31.6496 1.92258 0.961289 0.275542i \(-0.0888574\pi\)
0.961289 + 0.275542i \(0.0888574\pi\)
\(272\) 5.60775 0.340020
\(273\) 11.0616 0.669481
\(274\) 13.7647 0.831553
\(275\) 2.96520 0.178808
\(276\) 3.48473 0.209756
\(277\) −2.48866 −0.149529 −0.0747644 0.997201i \(-0.523820\pi\)
−0.0747644 + 0.997201i \(0.523820\pi\)
\(278\) 15.8688 0.951750
\(279\) −13.1718 −0.788573
\(280\) 3.35475 0.200485
\(281\) 7.69352 0.458957 0.229478 0.973314i \(-0.426298\pi\)
0.229478 + 0.973314i \(0.426298\pi\)
\(282\) 2.02240 0.120432
\(283\) −10.9512 −0.650981 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(284\) −9.13221 −0.541897
\(285\) −7.25512 −0.429756
\(286\) 8.40495 0.496995
\(287\) 41.2509 2.43497
\(288\) −1.64681 −0.0970395
\(289\) 14.4468 0.849814
\(290\) −6.71384 −0.394250
\(291\) 3.15870 0.185167
\(292\) −3.85112 −0.225370
\(293\) 16.5844 0.968869 0.484435 0.874828i \(-0.339025\pi\)
0.484435 + 0.874828i \(0.339025\pi\)
\(294\) −4.94897 −0.288630
\(295\) −13.1910 −0.768011
\(296\) −8.79576 −0.511243
\(297\) 16.0283 0.930058
\(298\) −22.9427 −1.32903
\(299\) −8.49125 −0.491061
\(300\) −1.16326 −0.0671611
\(301\) 2.82126 0.162615
\(302\) −12.5535 −0.722370
\(303\) 15.9633 0.917066
\(304\) −6.23686 −0.357709
\(305\) 8.53190 0.488535
\(306\) −9.23492 −0.527925
\(307\) −17.5543 −1.00188 −0.500938 0.865483i \(-0.667011\pi\)
−0.500938 + 0.865483i \(0.667011\pi\)
\(308\) −9.94753 −0.566813
\(309\) 17.9769 1.02267
\(310\) −7.99833 −0.454274
\(311\) 14.0724 0.797973 0.398986 0.916957i \(-0.369362\pi\)
0.398986 + 0.916957i \(0.369362\pi\)
\(312\) −3.29731 −0.186673
\(313\) −19.1958 −1.08501 −0.542507 0.840052i \(-0.682525\pi\)
−0.542507 + 0.840052i \(0.682525\pi\)
\(314\) −7.44522 −0.420158
\(315\) −5.52466 −0.311279
\(316\) 9.04751 0.508962
\(317\) −25.6040 −1.43807 −0.719033 0.694976i \(-0.755415\pi\)
−0.719033 + 0.694976i \(0.755415\pi\)
\(318\) 11.8844 0.666442
\(319\) 19.9079 1.11463
\(320\) −1.00000 −0.0559017
\(321\) 2.81447 0.157088
\(322\) 10.0497 0.560046
\(323\) −34.9747 −1.94605
\(324\) −1.34756 −0.0748642
\(325\) 2.83453 0.157231
\(326\) −12.7158 −0.704264
\(327\) 5.90288 0.326430
\(328\) −12.2963 −0.678948
\(329\) 5.83241 0.321551
\(330\) 3.44932 0.189879
\(331\) −13.4718 −0.740480 −0.370240 0.928936i \(-0.620725\pi\)
−0.370240 + 0.928936i \(0.620725\pi\)
\(332\) −15.8755 −0.871281
\(333\) 14.4850 0.793772
\(334\) 16.6894 0.913206
\(335\) −9.32681 −0.509578
\(336\) 3.90247 0.212897
\(337\) 12.1885 0.663952 0.331976 0.943288i \(-0.392285\pi\)
0.331976 + 0.943288i \(0.392285\pi\)
\(338\) −4.96546 −0.270085
\(339\) 19.1940 1.04248
\(340\) −5.60775 −0.304123
\(341\) 23.7167 1.28433
\(342\) 10.2710 0.555390
\(343\) 9.21089 0.497341
\(344\) −0.840974 −0.0453423
\(345\) −3.48473 −0.187612
\(346\) −13.7598 −0.739733
\(347\) 1.94541 0.104435 0.0522176 0.998636i \(-0.483371\pi\)
0.0522176 + 0.998636i \(0.483371\pi\)
\(348\) −7.80997 −0.418659
\(349\) −10.2515 −0.548752 −0.274376 0.961623i \(-0.588471\pi\)
−0.274376 + 0.961623i \(0.588471\pi\)
\(350\) −3.35475 −0.179319
\(351\) 15.3220 0.817826
\(352\) 2.96520 0.158046
\(353\) −32.3292 −1.72071 −0.860355 0.509696i \(-0.829758\pi\)
−0.860355 + 0.509696i \(0.829758\pi\)
\(354\) −15.3447 −0.815559
\(355\) 9.13221 0.484687
\(356\) −4.65541 −0.246736
\(357\) 21.8841 1.15823
\(358\) −2.09821 −0.110894
\(359\) 4.50747 0.237895 0.118948 0.992901i \(-0.462048\pi\)
0.118948 + 0.992901i \(0.462048\pi\)
\(360\) 1.64681 0.0867948
\(361\) 19.8984 1.04729
\(362\) 16.1159 0.847032
\(363\) 2.56799 0.134784
\(364\) −9.50914 −0.498414
\(365\) 3.85112 0.201577
\(366\) 9.92486 0.518781
\(367\) −4.62156 −0.241244 −0.120622 0.992699i \(-0.538489\pi\)
−0.120622 + 0.992699i \(0.538489\pi\)
\(368\) −2.99565 −0.156159
\(369\) 20.2497 1.05416
\(370\) 8.79576 0.457270
\(371\) 34.2735 1.77939
\(372\) −9.30417 −0.482399
\(373\) −23.2544 −1.20407 −0.602035 0.798470i \(-0.705643\pi\)
−0.602035 + 0.798470i \(0.705643\pi\)
\(374\) 16.6281 0.859819
\(375\) 1.16326 0.0600707
\(376\) −1.73855 −0.0896589
\(377\) 19.0306 0.980124
\(378\) −18.1340 −0.932715
\(379\) −18.0541 −0.927374 −0.463687 0.885999i \(-0.653474\pi\)
−0.463687 + 0.885999i \(0.653474\pi\)
\(380\) 6.23686 0.319944
\(381\) −14.8208 −0.759293
\(382\) 6.77893 0.346840
\(383\) 5.71349 0.291946 0.145973 0.989289i \(-0.453369\pi\)
0.145973 + 0.989289i \(0.453369\pi\)
\(384\) −1.16326 −0.0593626
\(385\) 9.94753 0.506973
\(386\) 20.3721 1.03691
\(387\) 1.38493 0.0703999
\(388\) −2.71538 −0.137852
\(389\) 26.0785 1.32223 0.661117 0.750283i \(-0.270083\pi\)
0.661117 + 0.750283i \(0.270083\pi\)
\(390\) 3.29731 0.166966
\(391\) −16.7988 −0.849554
\(392\) 4.25438 0.214878
\(393\) −5.21937 −0.263282
\(394\) −18.4708 −0.930545
\(395\) −9.04751 −0.455230
\(396\) −4.88314 −0.245387
\(397\) 4.13836 0.207698 0.103849 0.994593i \(-0.466884\pi\)
0.103849 + 0.994593i \(0.466884\pi\)
\(398\) −24.5092 −1.22853
\(399\) −24.3391 −1.21848
\(400\) 1.00000 0.0500000
\(401\) 1.00000 0.0499376
\(402\) −10.8495 −0.541126
\(403\) 22.6715 1.12935
\(404\) −13.7228 −0.682736
\(405\) 1.34756 0.0669606
\(406\) −22.5233 −1.11781
\(407\) −26.0812 −1.29280
\(408\) −6.52330 −0.322951
\(409\) 10.4102 0.514752 0.257376 0.966311i \(-0.417142\pi\)
0.257376 + 0.966311i \(0.417142\pi\)
\(410\) 12.2963 0.607269
\(411\) −16.0119 −0.789811
\(412\) −15.4538 −0.761354
\(413\) −44.2527 −2.17753
\(414\) 4.93328 0.242457
\(415\) 15.8755 0.779297
\(416\) 2.83453 0.138974
\(417\) −18.4597 −0.903974
\(418\) −18.4936 −0.904550
\(419\) 2.10590 0.102880 0.0514400 0.998676i \(-0.483619\pi\)
0.0514400 + 0.998676i \(0.483619\pi\)
\(420\) −3.90247 −0.190421
\(421\) 4.89803 0.238716 0.119358 0.992851i \(-0.461916\pi\)
0.119358 + 0.992851i \(0.461916\pi\)
\(422\) 0.410059 0.0199613
\(423\) 2.86307 0.139207
\(424\) −10.2164 −0.496151
\(425\) 5.60775 0.272016
\(426\) 10.6232 0.514695
\(427\) 28.6224 1.38514
\(428\) −2.41946 −0.116949
\(429\) −9.77718 −0.472047
\(430\) 0.840974 0.0405554
\(431\) 5.86490 0.282502 0.141251 0.989974i \(-0.454887\pi\)
0.141251 + 0.989974i \(0.454887\pi\)
\(432\) 5.40548 0.260071
\(433\) 7.60619 0.365530 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(434\) −26.8324 −1.28800
\(435\) 7.80997 0.374460
\(436\) −5.07441 −0.243020
\(437\) 18.6834 0.893750
\(438\) 4.47987 0.214057
\(439\) −2.73186 −0.130384 −0.0651922 0.997873i \(-0.520766\pi\)
−0.0651922 + 0.997873i \(0.520766\pi\)
\(440\) −2.96520 −0.141361
\(441\) −7.00617 −0.333627
\(442\) 15.8953 0.756063
\(443\) −21.0264 −0.998996 −0.499498 0.866315i \(-0.666482\pi\)
−0.499498 + 0.866315i \(0.666482\pi\)
\(444\) 10.2318 0.485580
\(445\) 4.65541 0.220688
\(446\) −1.72577 −0.0817176
\(447\) 26.6884 1.26232
\(448\) −3.35475 −0.158497
\(449\) 26.2757 1.24003 0.620013 0.784592i \(-0.287127\pi\)
0.620013 + 0.784592i \(0.287127\pi\)
\(450\) −1.64681 −0.0776316
\(451\) −36.4609 −1.71688
\(452\) −16.5001 −0.776101
\(453\) 14.6030 0.686108
\(454\) −8.01949 −0.376374
\(455\) 9.50914 0.445795
\(456\) 7.25512 0.339752
\(457\) −20.4868 −0.958332 −0.479166 0.877724i \(-0.659061\pi\)
−0.479166 + 0.877724i \(0.659061\pi\)
\(458\) 12.8189 0.598989
\(459\) 30.3126 1.41487
\(460\) 2.99565 0.139673
\(461\) −6.83475 −0.318326 −0.159163 0.987252i \(-0.550880\pi\)
−0.159163 + 0.987252i \(0.550880\pi\)
\(462\) 11.5716 0.538360
\(463\) −13.1377 −0.610560 −0.305280 0.952263i \(-0.598750\pi\)
−0.305280 + 0.952263i \(0.598750\pi\)
\(464\) 6.71384 0.311682
\(465\) 9.30417 0.431471
\(466\) −1.42208 −0.0658764
\(467\) −9.73693 −0.450571 −0.225286 0.974293i \(-0.572332\pi\)
−0.225286 + 0.974293i \(0.572332\pi\)
\(468\) −4.66794 −0.215776
\(469\) −31.2892 −1.44480
\(470\) 1.73855 0.0801934
\(471\) 8.66077 0.399067
\(472\) 13.1910 0.607166
\(473\) −2.49366 −0.114659
\(474\) −10.5246 −0.483413
\(475\) −6.23686 −0.286167
\(476\) −18.8126 −0.862275
\(477\) 16.8245 0.770341
\(478\) −9.78933 −0.447753
\(479\) −32.0677 −1.46521 −0.732604 0.680655i \(-0.761695\pi\)
−0.732604 + 0.680655i \(0.761695\pi\)
\(480\) 1.16326 0.0530955
\(481\) −24.9318 −1.13679
\(482\) 0.658045 0.0299731
\(483\) −11.6904 −0.531933
\(484\) −2.20757 −0.100344
\(485\) 2.71538 0.123299
\(486\) −14.6489 −0.664486
\(487\) 1.95740 0.0886984 0.0443492 0.999016i \(-0.485879\pi\)
0.0443492 + 0.999016i \(0.485879\pi\)
\(488\) −8.53190 −0.386221
\(489\) 14.7919 0.668911
\(490\) −4.25438 −0.192193
\(491\) 30.0347 1.35545 0.677725 0.735316i \(-0.262966\pi\)
0.677725 + 0.735316i \(0.262966\pi\)
\(492\) 14.3038 0.644866
\(493\) 37.6495 1.69565
\(494\) −17.6785 −0.795396
\(495\) 4.88314 0.219481
\(496\) 7.99833 0.359135
\(497\) 30.6363 1.37423
\(498\) 18.4674 0.827544
\(499\) −12.0810 −0.540821 −0.270410 0.962745i \(-0.587159\pi\)
−0.270410 + 0.962745i \(0.587159\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −19.4142 −0.867364
\(502\) 14.2838 0.637518
\(503\) −4.48425 −0.199943 −0.0999714 0.994990i \(-0.531875\pi\)
−0.0999714 + 0.994990i \(0.531875\pi\)
\(504\) 5.52466 0.246088
\(505\) 13.7228 0.610658
\(506\) −8.88271 −0.394884
\(507\) 5.77614 0.256528
\(508\) 12.7407 0.565277
\(509\) 0.883360 0.0391543 0.0195771 0.999808i \(-0.493768\pi\)
0.0195771 + 0.999808i \(0.493768\pi\)
\(510\) 6.52330 0.288856
\(511\) 12.9196 0.571528
\(512\) 1.00000 0.0441942
\(513\) −33.7132 −1.48847
\(514\) −0.865966 −0.0381961
\(515\) 15.4538 0.680976
\(516\) 0.978276 0.0430662
\(517\) −5.15516 −0.226724
\(518\) 29.5076 1.29649
\(519\) 16.0063 0.702600
\(520\) −2.83453 −0.124302
\(521\) 22.3938 0.981090 0.490545 0.871416i \(-0.336798\pi\)
0.490545 + 0.871416i \(0.336798\pi\)
\(522\) −11.0565 −0.483928
\(523\) −7.86014 −0.343700 −0.171850 0.985123i \(-0.554974\pi\)
−0.171850 + 0.985123i \(0.554974\pi\)
\(524\) 4.48683 0.196008
\(525\) 3.90247 0.170318
\(526\) −13.8708 −0.604797
\(527\) 44.8526 1.95381
\(528\) −3.44932 −0.150112
\(529\) −14.0261 −0.609830
\(530\) 10.2164 0.443771
\(531\) −21.7232 −0.942706
\(532\) 20.9231 0.907133
\(533\) −34.8541 −1.50970
\(534\) 5.41547 0.234351
\(535\) 2.41946 0.104602
\(536\) 9.32681 0.402857
\(537\) 2.44077 0.105327
\(538\) 27.4725 1.18442
\(539\) 12.6151 0.543370
\(540\) −5.40548 −0.232615
\(541\) −23.4352 −1.00756 −0.503779 0.863833i \(-0.668057\pi\)
−0.503779 + 0.863833i \(0.668057\pi\)
\(542\) 31.6496 1.35947
\(543\) −18.7470 −0.804512
\(544\) 5.60775 0.240430
\(545\) 5.07441 0.217364
\(546\) 11.0616 0.473395
\(547\) 12.4018 0.530262 0.265131 0.964212i \(-0.414585\pi\)
0.265131 + 0.964212i \(0.414585\pi\)
\(548\) 13.7647 0.587997
\(549\) 14.0505 0.599659
\(550\) 2.96520 0.126437
\(551\) −41.8733 −1.78386
\(552\) 3.48473 0.148320
\(553\) −30.3522 −1.29071
\(554\) −2.48866 −0.105733
\(555\) −10.2318 −0.434316
\(556\) 15.8688 0.672989
\(557\) 31.8490 1.34949 0.674743 0.738053i \(-0.264255\pi\)
0.674743 + 0.738053i \(0.264255\pi\)
\(558\) −13.1718 −0.557605
\(559\) −2.38376 −0.100822
\(560\) 3.35475 0.141764
\(561\) −19.3429 −0.816658
\(562\) 7.69352 0.324532
\(563\) 30.9414 1.30402 0.652012 0.758208i \(-0.273925\pi\)
0.652012 + 0.758208i \(0.273925\pi\)
\(564\) 2.02240 0.0851582
\(565\) 16.5001 0.694166
\(566\) −10.9512 −0.460313
\(567\) 4.52072 0.189852
\(568\) −9.13221 −0.383179
\(569\) −11.6170 −0.487011 −0.243506 0.969899i \(-0.578297\pi\)
−0.243506 + 0.969899i \(0.578297\pi\)
\(570\) −7.25512 −0.303884
\(571\) −12.6405 −0.528987 −0.264494 0.964387i \(-0.585205\pi\)
−0.264494 + 0.964387i \(0.585205\pi\)
\(572\) 8.40495 0.351429
\(573\) −7.88569 −0.329429
\(574\) 41.2509 1.72178
\(575\) −2.99565 −0.124927
\(576\) −1.64681 −0.0686173
\(577\) −21.8733 −0.910598 −0.455299 0.890339i \(-0.650468\pi\)
−0.455299 + 0.890339i \(0.650468\pi\)
\(578\) 14.4468 0.600909
\(579\) −23.6981 −0.984860
\(580\) −6.71384 −0.278777
\(581\) 53.2584 2.20953
\(582\) 3.15870 0.130933
\(583\) −30.2937 −1.25463
\(584\) −3.85112 −0.159361
\(585\) 4.66794 0.192996
\(586\) 16.5844 0.685094
\(587\) 15.2251 0.628406 0.314203 0.949356i \(-0.398263\pi\)
0.314203 + 0.949356i \(0.398263\pi\)
\(588\) −4.94897 −0.204092
\(589\) −49.8844 −2.05545
\(590\) −13.1910 −0.543066
\(591\) 21.4864 0.883833
\(592\) −8.79576 −0.361503
\(593\) −29.4662 −1.21003 −0.605015 0.796214i \(-0.706833\pi\)
−0.605015 + 0.796214i \(0.706833\pi\)
\(594\) 16.0283 0.657651
\(595\) 18.8126 0.771242
\(596\) −22.9427 −0.939769
\(597\) 28.5107 1.16686
\(598\) −8.49125 −0.347233
\(599\) 28.8464 1.17863 0.589315 0.807903i \(-0.299398\pi\)
0.589315 + 0.807903i \(0.299398\pi\)
\(600\) −1.16326 −0.0474901
\(601\) 48.2291 1.96731 0.983653 0.180073i \(-0.0576335\pi\)
0.983653 + 0.180073i \(0.0576335\pi\)
\(602\) 2.82126 0.114986
\(603\) −15.3595 −0.625488
\(604\) −12.5535 −0.510793
\(605\) 2.20757 0.0897505
\(606\) 15.9633 0.648464
\(607\) −13.9833 −0.567566 −0.283783 0.958889i \(-0.591589\pi\)
−0.283783 + 0.958889i \(0.591589\pi\)
\(608\) −6.23686 −0.252938
\(609\) 26.2005 1.06170
\(610\) 8.53190 0.345447
\(611\) −4.92797 −0.199364
\(612\) −9.23492 −0.373300
\(613\) −8.71593 −0.352033 −0.176017 0.984387i \(-0.556321\pi\)
−0.176017 + 0.984387i \(0.556321\pi\)
\(614\) −17.5543 −0.708433
\(615\) −14.3038 −0.576785
\(616\) −9.94753 −0.400797
\(617\) −10.5090 −0.423075 −0.211537 0.977370i \(-0.567847\pi\)
−0.211537 + 0.977370i \(0.567847\pi\)
\(618\) 17.9769 0.723136
\(619\) 13.2480 0.532484 0.266242 0.963906i \(-0.414218\pi\)
0.266242 + 0.963906i \(0.414218\pi\)
\(620\) −7.99833 −0.321220
\(621\) −16.1929 −0.649799
\(622\) 14.0724 0.564252
\(623\) 15.6178 0.625712
\(624\) −3.29731 −0.131998
\(625\) 1.00000 0.0400000
\(626\) −19.1958 −0.767220
\(627\) 21.5129 0.859143
\(628\) −7.44522 −0.297097
\(629\) −49.3244 −1.96669
\(630\) −5.52466 −0.220108
\(631\) 7.60870 0.302898 0.151449 0.988465i \(-0.451606\pi\)
0.151449 + 0.988465i \(0.451606\pi\)
\(632\) 9.04751 0.359891
\(633\) −0.477007 −0.0189593
\(634\) −25.6040 −1.01687
\(635\) −12.7407 −0.505600
\(636\) 11.8844 0.471246
\(637\) 12.0591 0.477801
\(638\) 19.9079 0.788161
\(639\) 15.0391 0.594936
\(640\) −1.00000 −0.0395285
\(641\) −39.9166 −1.57661 −0.788305 0.615285i \(-0.789041\pi\)
−0.788305 + 0.615285i \(0.789041\pi\)
\(642\) 2.81447 0.111078
\(643\) −4.98995 −0.196784 −0.0983922 0.995148i \(-0.531370\pi\)
−0.0983922 + 0.995148i \(0.531370\pi\)
\(644\) 10.0497 0.396012
\(645\) −0.978276 −0.0385196
\(646\) −34.9747 −1.37606
\(647\) 5.05124 0.198585 0.0992924 0.995058i \(-0.468342\pi\)
0.0992924 + 0.995058i \(0.468342\pi\)
\(648\) −1.34756 −0.0529370
\(649\) 39.1141 1.53536
\(650\) 2.83453 0.111179
\(651\) 31.2132 1.22334
\(652\) −12.7158 −0.497990
\(653\) −44.8420 −1.75480 −0.877402 0.479756i \(-0.840725\pi\)
−0.877402 + 0.479756i \(0.840725\pi\)
\(654\) 5.90288 0.230821
\(655\) −4.48683 −0.175315
\(656\) −12.2963 −0.480088
\(657\) 6.34209 0.247428
\(658\) 5.83241 0.227371
\(659\) 12.1552 0.473500 0.236750 0.971571i \(-0.423918\pi\)
0.236750 + 0.971571i \(0.423918\pi\)
\(660\) 3.44932 0.134264
\(661\) 25.2866 0.983534 0.491767 0.870727i \(-0.336351\pi\)
0.491767 + 0.870727i \(0.336351\pi\)
\(662\) −13.4718 −0.523598
\(663\) −18.4905 −0.718110
\(664\) −15.8755 −0.616089
\(665\) −20.9231 −0.811364
\(666\) 14.4850 0.561282
\(667\) −20.1123 −0.778752
\(668\) 16.6894 0.645734
\(669\) 2.00753 0.0776155
\(670\) −9.32681 −0.360326
\(671\) −25.2988 −0.976650
\(672\) 3.90247 0.150541
\(673\) 2.92717 0.112834 0.0564170 0.998407i \(-0.482032\pi\)
0.0564170 + 0.998407i \(0.482032\pi\)
\(674\) 12.1885 0.469485
\(675\) 5.40548 0.208057
\(676\) −4.96546 −0.190979
\(677\) −39.1349 −1.50408 −0.752038 0.659120i \(-0.770929\pi\)
−0.752038 + 0.659120i \(0.770929\pi\)
\(678\) 19.1940 0.737142
\(679\) 9.10943 0.349588
\(680\) −5.60775 −0.215047
\(681\) 9.32880 0.357480
\(682\) 23.7167 0.908158
\(683\) 9.61464 0.367894 0.183947 0.982936i \(-0.441113\pi\)
0.183947 + 0.982936i \(0.441113\pi\)
\(684\) 10.2710 0.392720
\(685\) −13.7647 −0.525921
\(686\) 9.21089 0.351673
\(687\) −14.9118 −0.568920
\(688\) −0.840974 −0.0320618
\(689\) −28.9586 −1.10324
\(690\) −3.48473 −0.132662
\(691\) −10.5130 −0.399933 −0.199966 0.979803i \(-0.564083\pi\)
−0.199966 + 0.979803i \(0.564083\pi\)
\(692\) −13.7598 −0.523070
\(693\) 16.3817 0.622291
\(694\) 1.94541 0.0738468
\(695\) −15.8688 −0.601940
\(696\) −7.80997 −0.296036
\(697\) −68.9544 −2.61183
\(698\) −10.2515 −0.388026
\(699\) 1.65425 0.0625695
\(700\) −3.35475 −0.126798
\(701\) 1.33284 0.0503408 0.0251704 0.999683i \(-0.491987\pi\)
0.0251704 + 0.999683i \(0.491987\pi\)
\(702\) 15.3220 0.578290
\(703\) 54.8579 2.06901
\(704\) 2.96520 0.111755
\(705\) −2.02240 −0.0761678
\(706\) −32.3292 −1.21673
\(707\) 46.0367 1.73139
\(708\) −15.3447 −0.576688
\(709\) −10.3253 −0.387774 −0.193887 0.981024i \(-0.562109\pi\)
−0.193887 + 0.981024i \(0.562109\pi\)
\(710\) 9.13221 0.342726
\(711\) −14.8996 −0.558778
\(712\) −4.65541 −0.174469
\(713\) −23.9602 −0.897316
\(714\) 21.8841 0.818990
\(715\) −8.40495 −0.314327
\(716\) −2.09821 −0.0784137
\(717\) 11.3876 0.425277
\(718\) 4.50747 0.168217
\(719\) 11.2934 0.421172 0.210586 0.977575i \(-0.432463\pi\)
0.210586 + 0.977575i \(0.432463\pi\)
\(720\) 1.64681 0.0613732
\(721\) 51.8437 1.93076
\(722\) 19.8984 0.740543
\(723\) −0.765481 −0.0284685
\(724\) 16.1159 0.598942
\(725\) 6.71384 0.249346
\(726\) 2.56799 0.0953070
\(727\) 8.06008 0.298932 0.149466 0.988767i \(-0.452245\pi\)
0.149466 + 0.988767i \(0.452245\pi\)
\(728\) −9.50914 −0.352432
\(729\) 21.0832 0.780858
\(730\) 3.85112 0.142536
\(731\) −4.71597 −0.174427
\(732\) 9.92486 0.366833
\(733\) 31.9707 1.18087 0.590433 0.807087i \(-0.298957\pi\)
0.590433 + 0.807087i \(0.298957\pi\)
\(734\) −4.62156 −0.170585
\(735\) 4.94897 0.182545
\(736\) −2.99565 −0.110421
\(737\) 27.6559 1.01872
\(738\) 20.2497 0.745401
\(739\) 20.0705 0.738304 0.369152 0.929369i \(-0.379648\pi\)
0.369152 + 0.929369i \(0.379648\pi\)
\(740\) 8.79576 0.323339
\(741\) 20.5648 0.755468
\(742\) 34.2735 1.25822
\(743\) 26.0838 0.956920 0.478460 0.878109i \(-0.341195\pi\)
0.478460 + 0.878109i \(0.341195\pi\)
\(744\) −9.30417 −0.341107
\(745\) 22.9427 0.840555
\(746\) −23.2544 −0.851405
\(747\) 26.1440 0.956559
\(748\) 16.6281 0.607984
\(749\) 8.11669 0.296577
\(750\) 1.16326 0.0424764
\(751\) −51.9115 −1.89428 −0.947139 0.320824i \(-0.896040\pi\)
−0.947139 + 0.320824i \(0.896040\pi\)
\(752\) −1.73855 −0.0633984
\(753\) −16.6159 −0.605516
\(754\) 19.0306 0.693052
\(755\) 12.5535 0.456867
\(756\) −18.1340 −0.659529
\(757\) −24.6306 −0.895215 −0.447608 0.894230i \(-0.647724\pi\)
−0.447608 + 0.894230i \(0.647724\pi\)
\(758\) −18.0541 −0.655753
\(759\) 10.3329 0.375062
\(760\) 6.23686 0.226235
\(761\) −43.7807 −1.58705 −0.793525 0.608538i \(-0.791756\pi\)
−0.793525 + 0.608538i \(0.791756\pi\)
\(762\) −14.8208 −0.536902
\(763\) 17.0234 0.616288
\(764\) 6.77893 0.245253
\(765\) 9.23492 0.333889
\(766\) 5.71349 0.206437
\(767\) 37.3903 1.35009
\(768\) −1.16326 −0.0419757
\(769\) 5.27100 0.190077 0.0950385 0.995474i \(-0.469703\pi\)
0.0950385 + 0.995474i \(0.469703\pi\)
\(770\) 9.94753 0.358484
\(771\) 1.00735 0.0362787
\(772\) 20.3721 0.733207
\(773\) 18.0530 0.649321 0.324660 0.945831i \(-0.394750\pi\)
0.324660 + 0.945831i \(0.394750\pi\)
\(774\) 1.38493 0.0497802
\(775\) 7.99833 0.287308
\(776\) −2.71538 −0.0974764
\(777\) −34.3252 −1.23141
\(778\) 26.0785 0.934960
\(779\) 76.6901 2.74771
\(780\) 3.29731 0.118062
\(781\) −27.0789 −0.968958
\(782\) −16.7988 −0.600725
\(783\) 36.2915 1.29695
\(784\) 4.25438 0.151942
\(785\) 7.44522 0.265731
\(786\) −5.21937 −0.186169
\(787\) −1.12359 −0.0400515 −0.0200258 0.999799i \(-0.506375\pi\)
−0.0200258 + 0.999799i \(0.506375\pi\)
\(788\) −18.4708 −0.657994
\(789\) 16.1354 0.574437
\(790\) −9.04751 −0.321896
\(791\) 55.3539 1.96816
\(792\) −4.88314 −0.173515
\(793\) −24.1839 −0.858795
\(794\) 4.13836 0.146865
\(795\) −11.8844 −0.421495
\(796\) −24.5092 −0.868704
\(797\) −7.42536 −0.263020 −0.131510 0.991315i \(-0.541983\pi\)
−0.131510 + 0.991315i \(0.541983\pi\)
\(798\) −24.3391 −0.861597
\(799\) −9.74936 −0.344907
\(800\) 1.00000 0.0353553
\(801\) 7.66660 0.270886
\(802\) 1.00000 0.0353112
\(803\) −11.4194 −0.402980
\(804\) −10.8495 −0.382634
\(805\) −10.0497 −0.354204
\(806\) 22.6715 0.798568
\(807\) −31.9577 −1.12497
\(808\) −13.7228 −0.482767
\(809\) 43.4719 1.52839 0.764195 0.644985i \(-0.223137\pi\)
0.764195 + 0.644985i \(0.223137\pi\)
\(810\) 1.34756 0.0473483
\(811\) 45.7453 1.60633 0.803167 0.595754i \(-0.203147\pi\)
0.803167 + 0.595754i \(0.203147\pi\)
\(812\) −22.5233 −0.790412
\(813\) −36.8169 −1.29122
\(814\) −26.0812 −0.914146
\(815\) 12.7158 0.445416
\(816\) −6.52330 −0.228361
\(817\) 5.24504 0.183501
\(818\) 10.4102 0.363985
\(819\) 15.6598 0.547198
\(820\) 12.2963 0.429404
\(821\) 24.0502 0.839359 0.419680 0.907672i \(-0.362142\pi\)
0.419680 + 0.907672i \(0.362142\pi\)
\(822\) −16.0119 −0.558481
\(823\) −2.22191 −0.0774509 −0.0387255 0.999250i \(-0.512330\pi\)
−0.0387255 + 0.999250i \(0.512330\pi\)
\(824\) −15.4538 −0.538359
\(825\) −3.44932 −0.120090
\(826\) −44.2527 −1.53975
\(827\) −18.7569 −0.652240 −0.326120 0.945328i \(-0.605741\pi\)
−0.326120 + 0.945328i \(0.605741\pi\)
\(828\) 4.93328 0.171443
\(829\) −30.1510 −1.04719 −0.523593 0.851968i \(-0.675409\pi\)
−0.523593 + 0.851968i \(0.675409\pi\)
\(830\) 15.8755 0.551046
\(831\) 2.89497 0.100425
\(832\) 2.83453 0.0982695
\(833\) 23.8575 0.826612
\(834\) −18.4597 −0.639206
\(835\) −16.6894 −0.577562
\(836\) −18.4936 −0.639613
\(837\) 43.2348 1.49441
\(838\) 2.10590 0.0727471
\(839\) 23.4703 0.810284 0.405142 0.914254i \(-0.367222\pi\)
0.405142 + 0.914254i \(0.367222\pi\)
\(840\) −3.90247 −0.134648
\(841\) 16.0756 0.554333
\(842\) 4.89803 0.168797
\(843\) −8.94960 −0.308241
\(844\) 0.410059 0.0141148
\(845\) 4.96546 0.170817
\(846\) 2.86307 0.0984344
\(847\) 7.40586 0.254468
\(848\) −10.2164 −0.350832
\(849\) 12.7391 0.437206
\(850\) 5.60775 0.192344
\(851\) 26.3490 0.903232
\(852\) 10.6232 0.363944
\(853\) 27.3501 0.936449 0.468225 0.883609i \(-0.344894\pi\)
0.468225 + 0.883609i \(0.344894\pi\)
\(854\) 28.6224 0.979439
\(855\) −10.2710 −0.351259
\(856\) −2.41946 −0.0826954
\(857\) 2.53798 0.0866958 0.0433479 0.999060i \(-0.486198\pi\)
0.0433479 + 0.999060i \(0.486198\pi\)
\(858\) −9.77718 −0.333787
\(859\) 28.0654 0.957581 0.478790 0.877929i \(-0.341075\pi\)
0.478790 + 0.877929i \(0.341075\pi\)
\(860\) 0.840974 0.0286770
\(861\) −47.9858 −1.63535
\(862\) 5.86490 0.199759
\(863\) 27.7232 0.943708 0.471854 0.881677i \(-0.343585\pi\)
0.471854 + 0.881677i \(0.343585\pi\)
\(864\) 5.40548 0.183898
\(865\) 13.7598 0.467848
\(866\) 7.60619 0.258469
\(867\) −16.8055 −0.570745
\(868\) −26.8324 −0.910752
\(869\) 26.8277 0.910067
\(870\) 7.80997 0.264783
\(871\) 26.4371 0.895787
\(872\) −5.07441 −0.171841
\(873\) 4.47173 0.151345
\(874\) 18.6834 0.631977
\(875\) 3.35475 0.113411
\(876\) 4.47987 0.151361
\(877\) −7.57190 −0.255685 −0.127842 0.991794i \(-0.540805\pi\)
−0.127842 + 0.991794i \(0.540805\pi\)
\(878\) −2.73186 −0.0921957
\(879\) −19.2920 −0.650703
\(880\) −2.96520 −0.0999570
\(881\) 35.2934 1.18907 0.594533 0.804072i \(-0.297337\pi\)
0.594533 + 0.804072i \(0.297337\pi\)
\(882\) −7.00617 −0.235910
\(883\) 26.6693 0.897494 0.448747 0.893659i \(-0.351870\pi\)
0.448747 + 0.893659i \(0.351870\pi\)
\(884\) 15.8953 0.534617
\(885\) 15.3447 0.515805
\(886\) −21.0264 −0.706397
\(887\) 30.6394 1.02877 0.514385 0.857559i \(-0.328020\pi\)
0.514385 + 0.857559i \(0.328020\pi\)
\(888\) 10.2318 0.343357
\(889\) −42.7419 −1.43352
\(890\) 4.65541 0.156050
\(891\) −3.99578 −0.133864
\(892\) −1.72577 −0.0577830
\(893\) 10.8431 0.362851
\(894\) 26.6884 0.892594
\(895\) 2.09821 0.0701354
\(896\) −3.35475 −0.112074
\(897\) 9.87757 0.329802
\(898\) 26.2757 0.876830
\(899\) 53.6995 1.79098
\(900\) −1.64681 −0.0548938
\(901\) −57.2909 −1.90864
\(902\) −36.4609 −1.21402
\(903\) −3.28188 −0.109214
\(904\) −16.5001 −0.548786
\(905\) −16.1159 −0.535710
\(906\) 14.6030 0.485152
\(907\) 18.1586 0.602946 0.301473 0.953475i \(-0.402522\pi\)
0.301473 + 0.953475i \(0.402522\pi\)
\(908\) −8.01949 −0.266136
\(909\) 22.5989 0.749560
\(910\) 9.50914 0.315225
\(911\) −41.3697 −1.37064 −0.685319 0.728243i \(-0.740337\pi\)
−0.685319 + 0.728243i \(0.740337\pi\)
\(912\) 7.25512 0.240241
\(913\) −47.0741 −1.55792
\(914\) −20.4868 −0.677643
\(915\) −9.92486 −0.328106
\(916\) 12.8189 0.423549
\(917\) −15.0522 −0.497067
\(918\) 30.3126 1.00046
\(919\) −55.5649 −1.83292 −0.916459 0.400128i \(-0.868966\pi\)
−0.916459 + 0.400128i \(0.868966\pi\)
\(920\) 2.99565 0.0987636
\(921\) 20.4203 0.672871
\(922\) −6.83475 −0.225090
\(923\) −25.8855 −0.852031
\(924\) 11.5716 0.380678
\(925\) −8.79576 −0.289203
\(926\) −13.1377 −0.431731
\(927\) 25.4496 0.835873
\(928\) 6.71384 0.220393
\(929\) 14.4873 0.475314 0.237657 0.971349i \(-0.423621\pi\)
0.237657 + 0.971349i \(0.423621\pi\)
\(930\) 9.30417 0.305096
\(931\) −26.5340 −0.869615
\(932\) −1.42208 −0.0465817
\(933\) −16.3699 −0.535928
\(934\) −9.73693 −0.318602
\(935\) −16.6281 −0.543797
\(936\) −4.66794 −0.152576
\(937\) −17.9644 −0.586870 −0.293435 0.955979i \(-0.594798\pi\)
−0.293435 + 0.955979i \(0.594798\pi\)
\(938\) −31.2892 −1.02163
\(939\) 22.3298 0.728707
\(940\) 1.73855 0.0567053
\(941\) −20.0750 −0.654427 −0.327213 0.944951i \(-0.606110\pi\)
−0.327213 + 0.944951i \(0.606110\pi\)
\(942\) 8.66077 0.282183
\(943\) 36.8353 1.19952
\(944\) 13.1910 0.429331
\(945\) 18.1340 0.589901
\(946\) −2.49366 −0.0810759
\(947\) 42.6727 1.38668 0.693338 0.720613i \(-0.256139\pi\)
0.693338 + 0.720613i \(0.256139\pi\)
\(948\) −10.5246 −0.341825
\(949\) −10.9161 −0.354352
\(950\) −6.23686 −0.202350
\(951\) 29.7843 0.965821
\(952\) −18.8126 −0.609720
\(953\) −36.2127 −1.17304 −0.586522 0.809933i \(-0.699503\pi\)
−0.586522 + 0.809933i \(0.699503\pi\)
\(954\) 16.8245 0.544713
\(955\) −6.77893 −0.219361
\(956\) −9.78933 −0.316609
\(957\) −23.1582 −0.748597
\(958\) −32.0677 −1.03606
\(959\) −46.1770 −1.49113
\(960\) 1.16326 0.0375442
\(961\) 32.9732 1.06365
\(962\) −24.9318 −0.803834
\(963\) 3.98440 0.128396
\(964\) 0.658045 0.0211942
\(965\) −20.3721 −0.655800
\(966\) −11.6904 −0.376133
\(967\) −3.17166 −0.101994 −0.0509969 0.998699i \(-0.516240\pi\)
−0.0509969 + 0.998699i \(0.516240\pi\)
\(968\) −2.20757 −0.0709540
\(969\) 40.6849 1.30699
\(970\) 2.71538 0.0871856
\(971\) −21.1298 −0.678088 −0.339044 0.940771i \(-0.610104\pi\)
−0.339044 + 0.940771i \(0.610104\pi\)
\(972\) −14.6489 −0.469863
\(973\) −53.2361 −1.70667
\(974\) 1.95740 0.0627193
\(975\) −3.29731 −0.105598
\(976\) −8.53190 −0.273099
\(977\) −47.6806 −1.52544 −0.762718 0.646731i \(-0.776136\pi\)
−0.762718 + 0.646731i \(0.776136\pi\)
\(978\) 14.7919 0.472992
\(979\) −13.8042 −0.441185
\(980\) −4.25438 −0.135901
\(981\) 8.35661 0.266806
\(982\) 30.0347 0.958447
\(983\) 22.2103 0.708400 0.354200 0.935170i \(-0.384753\pi\)
0.354200 + 0.935170i \(0.384753\pi\)
\(984\) 14.3038 0.455989
\(985\) 18.4708 0.588528
\(986\) 37.6495 1.19901
\(987\) −6.78464 −0.215957
\(988\) −17.6785 −0.562430
\(989\) 2.51926 0.0801079
\(990\) 4.88314 0.155196
\(991\) −40.0552 −1.27240 −0.636198 0.771526i \(-0.719494\pi\)
−0.636198 + 0.771526i \(0.719494\pi\)
\(992\) 7.99833 0.253947
\(993\) 15.6713 0.497315
\(994\) 30.6363 0.971725
\(995\) 24.5092 0.776993
\(996\) 18.4674 0.585162
\(997\) −20.6799 −0.654939 −0.327470 0.944862i \(-0.606196\pi\)
−0.327470 + 0.944862i \(0.606196\pi\)
\(998\) −12.0810 −0.382418
\(999\) −47.5453 −1.50427
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 4010.2.a.j.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.j.1.4 12 1.1 even 1 trivial