Properties

Label 4025.2.a.n.1.3
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.27060\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.15561 q^{2} +2.62393 q^{3} +2.64667 q^{4} +5.65618 q^{6} +1.00000 q^{7} +1.39396 q^{8} +3.88502 q^{9} +O(q^{10})\) \(q+2.15561 q^{2} +2.62393 q^{3} +2.64667 q^{4} +5.65618 q^{6} +1.00000 q^{7} +1.39396 q^{8} +3.88502 q^{9} +3.15561 q^{11} +6.94467 q^{12} +1.72940 q^{13} +2.15561 q^{14} -2.28849 q^{16} +5.38558 q^{17} +8.37459 q^{18} -3.42621 q^{19} +2.62393 q^{21} +6.80228 q^{22} -1.00000 q^{23} +3.65766 q^{24} +3.72792 q^{26} +2.32222 q^{27} +2.64667 q^{28} -2.54119 q^{29} -9.96741 q^{31} -7.72102 q^{32} +8.28011 q^{33} +11.6092 q^{34} +10.2823 q^{36} +5.69681 q^{37} -7.38558 q^{38} +4.53783 q^{39} +7.30285 q^{41} +5.65618 q^{42} -4.93516 q^{43} +8.35185 q^{44} -2.15561 q^{46} +7.42621 q^{47} -6.00484 q^{48} +1.00000 q^{49} +14.1314 q^{51} +4.57715 q^{52} +3.20575 q^{53} +5.00581 q^{54} +1.39396 q^{56} -8.99014 q^{57} -5.47783 q^{58} +8.27060 q^{59} -7.95789 q^{61} -21.4859 q^{62} +3.88502 q^{63} -12.0666 q^{64} +17.8487 q^{66} -5.20723 q^{67} +14.2538 q^{68} -2.62393 q^{69} +13.0308 q^{71} +5.41556 q^{72} -8.81031 q^{73} +12.2801 q^{74} -9.06803 q^{76} +3.15561 q^{77} +9.78181 q^{78} -4.05014 q^{79} -5.56170 q^{81} +15.7421 q^{82} -12.0597 q^{83} +6.94467 q^{84} -10.6383 q^{86} -6.66792 q^{87} +4.39880 q^{88} +5.22011 q^{89} +1.72940 q^{91} -2.64667 q^{92} -26.1538 q^{93} +16.0080 q^{94} -20.2594 q^{96} -4.36621 q^{97} +2.15561 q^{98} +12.2596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 11 q^{4} + 4 q^{7} + 18 q^{9} + 5 q^{11} - 21 q^{12} + 17 q^{13} + q^{14} + 17 q^{16} + 9 q^{17} + 24 q^{18} + 4 q^{19} + 20 q^{22} - 4 q^{23} + 12 q^{24} + 5 q^{26} - 9 q^{27} + 11 q^{28} + 10 q^{29} - 2 q^{31} - 34 q^{32} - 18 q^{34} + 45 q^{36} - 5 q^{37} - 17 q^{38} - 9 q^{39} + 7 q^{41} + 6 q^{43} + 13 q^{44} - q^{46} + 12 q^{47} - 51 q^{48} + 4 q^{49} + 18 q^{51} + 43 q^{52} - 23 q^{53} - 60 q^{54} - 9 q^{57} + 4 q^{58} + 23 q^{59} - 17 q^{61} - 17 q^{62} + 18 q^{63} + 16 q^{64} - 21 q^{66} - 5 q^{67} + 33 q^{68} + 20 q^{71} + 57 q^{72} + 15 q^{73} + 16 q^{74} + 8 q^{76} + 5 q^{77} - 39 q^{78} + 12 q^{79} + 24 q^{81} - 20 q^{82} + 3 q^{83} - 21 q^{84} - 36 q^{86} - 18 q^{87} + 39 q^{88} - 11 q^{89} + 17 q^{91} - 11 q^{92} - 27 q^{93} + 21 q^{94} + 3 q^{96} - q^{97} + q^{98} + 42 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.15561 1.52425 0.762124 0.647431i \(-0.224157\pi\)
0.762124 + 0.647431i \(0.224157\pi\)
\(3\) 2.62393 1.51493 0.757464 0.652877i \(-0.226438\pi\)
0.757464 + 0.652877i \(0.226438\pi\)
\(4\) 2.64667 1.32333
\(5\) 0 0
\(6\) 5.65618 2.30913
\(7\) 1.00000 0.377964
\(8\) 1.39396 0.492840
\(9\) 3.88502 1.29501
\(10\) 0 0
\(11\) 3.15561 0.951453 0.475727 0.879593i \(-0.342185\pi\)
0.475727 + 0.879593i \(0.342185\pi\)
\(12\) 6.94467 2.00475
\(13\) 1.72940 0.479650 0.239825 0.970816i \(-0.422910\pi\)
0.239825 + 0.970816i \(0.422910\pi\)
\(14\) 2.15561 0.576112
\(15\) 0 0
\(16\) −2.28849 −0.572123
\(17\) 5.38558 1.30620 0.653098 0.757274i \(-0.273469\pi\)
0.653098 + 0.757274i \(0.273469\pi\)
\(18\) 8.37459 1.97391
\(19\) −3.42621 −0.786027 −0.393013 0.919533i \(-0.628567\pi\)
−0.393013 + 0.919533i \(0.628567\pi\)
\(20\) 0 0
\(21\) 2.62393 0.572589
\(22\) 6.80228 1.45025
\(23\) −1.00000 −0.208514
\(24\) 3.65766 0.746617
\(25\) 0 0
\(26\) 3.72792 0.731106
\(27\) 2.32222 0.446911
\(28\) 2.64667 0.500173
\(29\) −2.54119 −0.471888 −0.235944 0.971767i \(-0.575818\pi\)
−0.235944 + 0.971767i \(0.575818\pi\)
\(30\) 0 0
\(31\) −9.96741 −1.79020 −0.895099 0.445867i \(-0.852896\pi\)
−0.895099 + 0.445867i \(0.852896\pi\)
\(32\) −7.72102 −1.36490
\(33\) 8.28011 1.44138
\(34\) 11.6092 1.99097
\(35\) 0 0
\(36\) 10.2823 1.71372
\(37\) 5.69681 0.936549 0.468275 0.883583i \(-0.344876\pi\)
0.468275 + 0.883583i \(0.344876\pi\)
\(38\) −7.38558 −1.19810
\(39\) 4.53783 0.726635
\(40\) 0 0
\(41\) 7.30285 1.14051 0.570256 0.821467i \(-0.306844\pi\)
0.570256 + 0.821467i \(0.306844\pi\)
\(42\) 5.65618 0.872767
\(43\) −4.93516 −0.752604 −0.376302 0.926497i \(-0.622805\pi\)
−0.376302 + 0.926497i \(0.622805\pi\)
\(44\) 8.35185 1.25909
\(45\) 0 0
\(46\) −2.15561 −0.317828
\(47\) 7.42621 1.08322 0.541612 0.840629i \(-0.317814\pi\)
0.541612 + 0.840629i \(0.317814\pi\)
\(48\) −6.00484 −0.866724
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 14.1314 1.97879
\(52\) 4.57715 0.634737
\(53\) 3.20575 0.440344 0.220172 0.975461i \(-0.429338\pi\)
0.220172 + 0.975461i \(0.429338\pi\)
\(54\) 5.00581 0.681204
\(55\) 0 0
\(56\) 1.39396 0.186276
\(57\) −8.99014 −1.19077
\(58\) −5.47783 −0.719275
\(59\) 8.27060 1.07674 0.538370 0.842709i \(-0.319040\pi\)
0.538370 + 0.842709i \(0.319040\pi\)
\(60\) 0 0
\(61\) −7.95789 −1.01890 −0.509452 0.860499i \(-0.670152\pi\)
−0.509452 + 0.860499i \(0.670152\pi\)
\(62\) −21.4859 −2.72871
\(63\) 3.88502 0.489466
\(64\) −12.0666 −1.50832
\(65\) 0 0
\(66\) 17.8487 2.19702
\(67\) −5.20723 −0.636165 −0.318082 0.948063i \(-0.603039\pi\)
−0.318082 + 0.948063i \(0.603039\pi\)
\(68\) 14.2538 1.72853
\(69\) −2.62393 −0.315884
\(70\) 0 0
\(71\) 13.0308 1.54647 0.773234 0.634121i \(-0.218638\pi\)
0.773234 + 0.634121i \(0.218638\pi\)
\(72\) 5.41556 0.638230
\(73\) −8.81031 −1.03117 −0.515585 0.856839i \(-0.672425\pi\)
−0.515585 + 0.856839i \(0.672425\pi\)
\(74\) 12.2801 1.42753
\(75\) 0 0
\(76\) −9.06803 −1.04017
\(77\) 3.15561 0.359615
\(78\) 9.78181 1.10757
\(79\) −4.05014 −0.455677 −0.227838 0.973699i \(-0.573166\pi\)
−0.227838 + 0.973699i \(0.573166\pi\)
\(80\) 0 0
\(81\) −5.56170 −0.617967
\(82\) 15.7421 1.73842
\(83\) −12.0597 −1.32372 −0.661860 0.749628i \(-0.730232\pi\)
−0.661860 + 0.749628i \(0.730232\pi\)
\(84\) 6.94467 0.757726
\(85\) 0 0
\(86\) −10.6383 −1.14716
\(87\) −6.66792 −0.714876
\(88\) 4.39880 0.468914
\(89\) 5.22011 0.553330 0.276665 0.960966i \(-0.410771\pi\)
0.276665 + 0.960966i \(0.410771\pi\)
\(90\) 0 0
\(91\) 1.72940 0.181291
\(92\) −2.64667 −0.275934
\(93\) −26.1538 −2.71202
\(94\) 16.0080 1.65110
\(95\) 0 0
\(96\) −20.2594 −2.06772
\(97\) −4.36621 −0.443321 −0.221661 0.975124i \(-0.571148\pi\)
−0.221661 + 0.975124i \(0.571148\pi\)
\(98\) 2.15561 0.217750
\(99\) 12.2596 1.23214
\(100\) 0 0
\(101\) 4.60958 0.458670 0.229335 0.973348i \(-0.426345\pi\)
0.229335 + 0.973348i \(0.426345\pi\)
\(102\) 30.4618 3.01617
\(103\) −5.73407 −0.564995 −0.282498 0.959268i \(-0.591163\pi\)
−0.282498 + 0.959268i \(0.591163\pi\)
\(104\) 2.41072 0.236391
\(105\) 0 0
\(106\) 6.91036 0.671194
\(107\) −18.6320 −1.80122 −0.900610 0.434628i \(-0.856880\pi\)
−0.900610 + 0.434628i \(0.856880\pi\)
\(108\) 6.14614 0.591413
\(109\) −6.26912 −0.600473 −0.300236 0.953865i \(-0.597066\pi\)
−0.300236 + 0.953865i \(0.597066\pi\)
\(110\) 0 0
\(111\) 14.9480 1.41880
\(112\) −2.28849 −0.216242
\(113\) −6.58330 −0.619305 −0.309653 0.950850i \(-0.600213\pi\)
−0.309653 + 0.950850i \(0.600213\pi\)
\(114\) −19.3793 −1.81503
\(115\) 0 0
\(116\) −6.72569 −0.624465
\(117\) 6.71876 0.621149
\(118\) 17.8282 1.64122
\(119\) 5.38558 0.493696
\(120\) 0 0
\(121\) −1.04211 −0.0947371
\(122\) −17.1541 −1.55306
\(123\) 19.1622 1.72779
\(124\) −26.3804 −2.36903
\(125\) 0 0
\(126\) 8.37459 0.746068
\(127\) 17.4907 1.55205 0.776025 0.630703i \(-0.217233\pi\)
0.776025 + 0.630703i \(0.217233\pi\)
\(128\) −10.5688 −0.934156
\(129\) −12.9495 −1.14014
\(130\) 0 0
\(131\) −1.83453 −0.160283 −0.0801417 0.996783i \(-0.525537\pi\)
−0.0801417 + 0.996783i \(0.525537\pi\)
\(132\) 21.9147 1.90743
\(133\) −3.42621 −0.297090
\(134\) −11.2248 −0.969673
\(135\) 0 0
\(136\) 7.50730 0.643745
\(137\) 16.5865 1.41708 0.708540 0.705671i \(-0.249354\pi\)
0.708540 + 0.705671i \(0.249354\pi\)
\(138\) −5.65618 −0.481486
\(139\) 2.79425 0.237005 0.118502 0.992954i \(-0.462191\pi\)
0.118502 + 0.992954i \(0.462191\pi\)
\(140\) 0 0
\(141\) 19.4859 1.64101
\(142\) 28.0893 2.35720
\(143\) 5.45732 0.456364
\(144\) −8.89082 −0.740902
\(145\) 0 0
\(146\) −18.9916 −1.57176
\(147\) 2.62393 0.216418
\(148\) 15.0775 1.23937
\(149\) −21.7613 −1.78276 −0.891378 0.453261i \(-0.850260\pi\)
−0.891378 + 0.453261i \(0.850260\pi\)
\(150\) 0 0
\(151\) 21.7598 1.77079 0.885395 0.464840i \(-0.153888\pi\)
0.885395 + 0.464840i \(0.153888\pi\)
\(152\) −4.77601 −0.387385
\(153\) 20.9231 1.69153
\(154\) 6.80228 0.548143
\(155\) 0 0
\(156\) 12.0101 0.961580
\(157\) −9.73892 −0.777250 −0.388625 0.921396i \(-0.627050\pi\)
−0.388625 + 0.921396i \(0.627050\pi\)
\(158\) −8.73054 −0.694564
\(159\) 8.41168 0.667089
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −11.9889 −0.941935
\(163\) −10.7712 −0.843663 −0.421831 0.906674i \(-0.638613\pi\)
−0.421831 + 0.906674i \(0.638613\pi\)
\(164\) 19.3282 1.50928
\(165\) 0 0
\(166\) −25.9959 −2.01768
\(167\) 18.0066 1.39339 0.696694 0.717368i \(-0.254653\pi\)
0.696694 + 0.717368i \(0.254653\pi\)
\(168\) 3.65766 0.282195
\(169\) −10.0092 −0.769936
\(170\) 0 0
\(171\) −13.3109 −1.01791
\(172\) −13.0617 −0.995946
\(173\) −14.2380 −1.08250 −0.541248 0.840863i \(-0.682048\pi\)
−0.541248 + 0.840863i \(0.682048\pi\)
\(174\) −14.3735 −1.08965
\(175\) 0 0
\(176\) −7.22159 −0.544348
\(177\) 21.7015 1.63118
\(178\) 11.2525 0.843413
\(179\) 9.07436 0.678249 0.339125 0.940741i \(-0.389869\pi\)
0.339125 + 0.940741i \(0.389869\pi\)
\(180\) 0 0
\(181\) 1.15413 0.0857860 0.0428930 0.999080i \(-0.486343\pi\)
0.0428930 + 0.999080i \(0.486343\pi\)
\(182\) 3.72792 0.276332
\(183\) −20.8810 −1.54357
\(184\) −1.39396 −0.102764
\(185\) 0 0
\(186\) −56.3774 −4.13379
\(187\) 16.9948 1.24278
\(188\) 19.6547 1.43347
\(189\) 2.32222 0.168917
\(190\) 0 0
\(191\) 23.0321 1.66654 0.833271 0.552864i \(-0.186465\pi\)
0.833271 + 0.552864i \(0.186465\pi\)
\(192\) −31.6618 −2.28499
\(193\) −4.27060 −0.307404 −0.153702 0.988117i \(-0.549120\pi\)
−0.153702 + 0.988117i \(0.549120\pi\)
\(194\) −9.41185 −0.675732
\(195\) 0 0
\(196\) 2.64667 0.189048
\(197\) −0.444103 −0.0316410 −0.0158205 0.999875i \(-0.505036\pi\)
−0.0158205 + 0.999875i \(0.505036\pi\)
\(198\) 26.4270 1.87808
\(199\) −2.18171 −0.154657 −0.0773286 0.997006i \(-0.524639\pi\)
−0.0773286 + 0.997006i \(0.524639\pi\)
\(200\) 0 0
\(201\) −13.6634 −0.963744
\(202\) 9.93646 0.699127
\(203\) −2.54119 −0.178357
\(204\) 37.4011 2.61860
\(205\) 0 0
\(206\) −12.3604 −0.861193
\(207\) −3.88502 −0.270027
\(208\) −3.95772 −0.274419
\(209\) −10.8118 −0.747867
\(210\) 0 0
\(211\) 10.7018 0.736744 0.368372 0.929678i \(-0.379915\pi\)
0.368372 + 0.929678i \(0.379915\pi\)
\(212\) 8.48456 0.582722
\(213\) 34.1918 2.34279
\(214\) −40.1633 −2.74551
\(215\) 0 0
\(216\) 3.23708 0.220256
\(217\) −9.96741 −0.676632
\(218\) −13.5138 −0.915269
\(219\) −23.1177 −1.56215
\(220\) 0 0
\(221\) 9.31384 0.626517
\(222\) 32.2222 2.16261
\(223\) −20.9249 −1.40124 −0.700619 0.713535i \(-0.747093\pi\)
−0.700619 + 0.713535i \(0.747093\pi\)
\(224\) −7.72102 −0.515883
\(225\) 0 0
\(226\) −14.1911 −0.943975
\(227\) −16.4346 −1.09080 −0.545401 0.838175i \(-0.683623\pi\)
−0.545401 + 0.838175i \(0.683623\pi\)
\(228\) −23.7939 −1.57579
\(229\) 5.76855 0.381197 0.190598 0.981668i \(-0.438957\pi\)
0.190598 + 0.981668i \(0.438957\pi\)
\(230\) 0 0
\(231\) 8.28011 0.544791
\(232\) −3.54233 −0.232565
\(233\) −24.0692 −1.57682 −0.788412 0.615148i \(-0.789096\pi\)
−0.788412 + 0.615148i \(0.789096\pi\)
\(234\) 14.4830 0.946786
\(235\) 0 0
\(236\) 21.8895 1.42489
\(237\) −10.6273 −0.690317
\(238\) 11.6092 0.752515
\(239\) 14.8308 0.959326 0.479663 0.877453i \(-0.340759\pi\)
0.479663 + 0.877453i \(0.340759\pi\)
\(240\) 0 0
\(241\) −16.8088 −1.08275 −0.541376 0.840781i \(-0.682096\pi\)
−0.541376 + 0.840781i \(0.682096\pi\)
\(242\) −2.24638 −0.144403
\(243\) −21.5602 −1.38309
\(244\) −21.0619 −1.34835
\(245\) 0 0
\(246\) 41.3062 2.63359
\(247\) −5.92530 −0.377018
\(248\) −13.8942 −0.882281
\(249\) −31.6437 −2.00534
\(250\) 0 0
\(251\) 13.3401 0.842020 0.421010 0.907056i \(-0.361676\pi\)
0.421010 + 0.907056i \(0.361676\pi\)
\(252\) 10.2823 0.647726
\(253\) −3.15561 −0.198392
\(254\) 37.7032 2.36571
\(255\) 0 0
\(256\) 1.35093 0.0844331
\(257\) 29.0985 1.81511 0.907556 0.419931i \(-0.137946\pi\)
0.907556 + 0.419931i \(0.137946\pi\)
\(258\) −27.9141 −1.73786
\(259\) 5.69681 0.353982
\(260\) 0 0
\(261\) −9.87258 −0.611097
\(262\) −3.95453 −0.244312
\(263\) 8.98547 0.554068 0.277034 0.960860i \(-0.410649\pi\)
0.277034 + 0.960860i \(0.410649\pi\)
\(264\) 11.5422 0.710371
\(265\) 0 0
\(266\) −7.38558 −0.452839
\(267\) 13.6972 0.838256
\(268\) −13.7818 −0.841858
\(269\) 12.3996 0.756016 0.378008 0.925802i \(-0.376609\pi\)
0.378008 + 0.925802i \(0.376609\pi\)
\(270\) 0 0
\(271\) 1.65055 0.100264 0.0501319 0.998743i \(-0.484036\pi\)
0.0501319 + 0.998743i \(0.484036\pi\)
\(272\) −12.3249 −0.747304
\(273\) 4.53783 0.274642
\(274\) 35.7541 2.15998
\(275\) 0 0
\(276\) −6.94467 −0.418020
\(277\) 8.00951 0.481245 0.240623 0.970619i \(-0.422648\pi\)
0.240623 + 0.970619i \(0.422648\pi\)
\(278\) 6.02331 0.361254
\(279\) −38.7235 −2.31832
\(280\) 0 0
\(281\) −6.90256 −0.411772 −0.205886 0.978576i \(-0.566008\pi\)
−0.205886 + 0.978576i \(0.566008\pi\)
\(282\) 42.0040 2.50130
\(283\) 2.96627 0.176327 0.0881633 0.996106i \(-0.471900\pi\)
0.0881633 + 0.996106i \(0.471900\pi\)
\(284\) 34.4881 2.04649
\(285\) 0 0
\(286\) 11.7639 0.695613
\(287\) 7.30285 0.431073
\(288\) −29.9963 −1.76755
\(289\) 12.0045 0.706147
\(290\) 0 0
\(291\) −11.4566 −0.671600
\(292\) −23.3180 −1.36458
\(293\) 6.83151 0.399101 0.199551 0.979888i \(-0.436052\pi\)
0.199551 + 0.979888i \(0.436052\pi\)
\(294\) 5.65618 0.329875
\(295\) 0 0
\(296\) 7.94113 0.461569
\(297\) 7.32802 0.425215
\(298\) −46.9089 −2.71736
\(299\) −1.72940 −0.100014
\(300\) 0 0
\(301\) −4.93516 −0.284458
\(302\) 46.9058 2.69912
\(303\) 12.0952 0.694852
\(304\) 7.84085 0.449704
\(305\) 0 0
\(306\) 45.1020 2.57831
\(307\) −23.0999 −1.31838 −0.659191 0.751975i \(-0.729101\pi\)
−0.659191 + 0.751975i \(0.729101\pi\)
\(308\) 8.35185 0.475891
\(309\) −15.0458 −0.855927
\(310\) 0 0
\(311\) 1.21208 0.0687305 0.0343653 0.999409i \(-0.489059\pi\)
0.0343653 + 0.999409i \(0.489059\pi\)
\(312\) 6.32557 0.358115
\(313\) −24.2346 −1.36982 −0.684911 0.728626i \(-0.740159\pi\)
−0.684911 + 0.728626i \(0.740159\pi\)
\(314\) −20.9933 −1.18472
\(315\) 0 0
\(316\) −10.7194 −0.603012
\(317\) −17.5337 −0.984793 −0.492396 0.870371i \(-0.663879\pi\)
−0.492396 + 0.870371i \(0.663879\pi\)
\(318\) 18.1323 1.01681
\(319\) −8.01903 −0.448979
\(320\) 0 0
\(321\) −48.8890 −2.72872
\(322\) −2.15561 −0.120128
\(323\) −18.4521 −1.02670
\(324\) −14.7200 −0.817776
\(325\) 0 0
\(326\) −23.2185 −1.28595
\(327\) −16.4497 −0.909672
\(328\) 10.1799 0.562090
\(329\) 7.42621 0.409420
\(330\) 0 0
\(331\) 16.2555 0.893486 0.446743 0.894662i \(-0.352584\pi\)
0.446743 + 0.894662i \(0.352584\pi\)
\(332\) −31.9179 −1.75172
\(333\) 22.1322 1.21284
\(334\) 38.8152 2.12387
\(335\) 0 0
\(336\) −6.00484 −0.327591
\(337\) 11.1765 0.608824 0.304412 0.952540i \(-0.401540\pi\)
0.304412 + 0.952540i \(0.401540\pi\)
\(338\) −21.5759 −1.17357
\(339\) −17.2741 −0.938202
\(340\) 0 0
\(341\) −31.4533 −1.70329
\(342\) −28.6931 −1.55155
\(343\) 1.00000 0.0539949
\(344\) −6.87942 −0.370913
\(345\) 0 0
\(346\) −30.6916 −1.64999
\(347\) −11.4812 −0.616343 −0.308171 0.951331i \(-0.599717\pi\)
−0.308171 + 0.951331i \(0.599717\pi\)
\(348\) −17.6478 −0.946019
\(349\) 16.6091 0.889062 0.444531 0.895763i \(-0.353370\pi\)
0.444531 + 0.895763i \(0.353370\pi\)
\(350\) 0 0
\(351\) 4.01605 0.214361
\(352\) −24.3646 −1.29864
\(353\) 14.1601 0.753666 0.376833 0.926281i \(-0.377013\pi\)
0.376833 + 0.926281i \(0.377013\pi\)
\(354\) 46.7800 2.48633
\(355\) 0 0
\(356\) 13.8159 0.732241
\(357\) 14.1314 0.747913
\(358\) 19.5608 1.03382
\(359\) −11.2009 −0.591162 −0.295581 0.955318i \(-0.595513\pi\)
−0.295581 + 0.955318i \(0.595513\pi\)
\(360\) 0 0
\(361\) −7.26108 −0.382162
\(362\) 2.48786 0.130759
\(363\) −2.73442 −0.143520
\(364\) 4.57715 0.239908
\(365\) 0 0
\(366\) −45.0113 −2.35278
\(367\) 23.2104 1.21157 0.605787 0.795627i \(-0.292858\pi\)
0.605787 + 0.795627i \(0.292858\pi\)
\(368\) 2.28849 0.119296
\(369\) 28.3717 1.47697
\(370\) 0 0
\(371\) 3.20575 0.166434
\(372\) −69.2203 −3.58891
\(373\) −4.68843 −0.242758 −0.121379 0.992606i \(-0.538732\pi\)
−0.121379 + 0.992606i \(0.538732\pi\)
\(374\) 36.6342 1.89431
\(375\) 0 0
\(376\) 10.3519 0.533856
\(377\) −4.39475 −0.226341
\(378\) 5.00581 0.257471
\(379\) −11.4540 −0.588351 −0.294175 0.955751i \(-0.595045\pi\)
−0.294175 + 0.955751i \(0.595045\pi\)
\(380\) 0 0
\(381\) 45.8944 2.35124
\(382\) 49.6482 2.54022
\(383\) −27.2946 −1.39469 −0.697346 0.716735i \(-0.745636\pi\)
−0.697346 + 0.716735i \(0.745636\pi\)
\(384\) −27.7317 −1.41518
\(385\) 0 0
\(386\) −9.20575 −0.468561
\(387\) −19.1732 −0.974626
\(388\) −11.5559 −0.586662
\(389\) 28.2460 1.43213 0.716065 0.698033i \(-0.245941\pi\)
0.716065 + 0.698033i \(0.245941\pi\)
\(390\) 0 0
\(391\) −5.38558 −0.272361
\(392\) 1.39396 0.0704057
\(393\) −4.81367 −0.242818
\(394\) −0.957315 −0.0482288
\(395\) 0 0
\(396\) 32.4471 1.63053
\(397\) −30.9383 −1.55275 −0.776376 0.630271i \(-0.782944\pi\)
−0.776376 + 0.630271i \(0.782944\pi\)
\(398\) −4.70292 −0.235736
\(399\) −8.99014 −0.450070
\(400\) 0 0
\(401\) −22.2509 −1.11116 −0.555578 0.831464i \(-0.687503\pi\)
−0.555578 + 0.831464i \(0.687503\pi\)
\(402\) −29.4531 −1.46898
\(403\) −17.2377 −0.858669
\(404\) 12.2000 0.606973
\(405\) 0 0
\(406\) −5.47783 −0.271860
\(407\) 17.9769 0.891083
\(408\) 19.6986 0.975227
\(409\) 5.84941 0.289234 0.144617 0.989488i \(-0.453805\pi\)
0.144617 + 0.989488i \(0.453805\pi\)
\(410\) 0 0
\(411\) 43.5218 2.14677
\(412\) −15.1762 −0.747677
\(413\) 8.27060 0.406969
\(414\) −8.37459 −0.411589
\(415\) 0 0
\(416\) −13.3528 −0.654673
\(417\) 7.33191 0.359045
\(418\) −23.3060 −1.13994
\(419\) −25.7082 −1.25593 −0.627964 0.778242i \(-0.716111\pi\)
−0.627964 + 0.778242i \(0.716111\pi\)
\(420\) 0 0
\(421\) 19.3849 0.944762 0.472381 0.881395i \(-0.343395\pi\)
0.472381 + 0.881395i \(0.343395\pi\)
\(422\) 23.0690 1.12298
\(423\) 28.8509 1.40278
\(424\) 4.46870 0.217019
\(425\) 0 0
\(426\) 73.7044 3.57099
\(427\) −7.95789 −0.385109
\(428\) −49.3126 −2.38361
\(429\) 14.3196 0.691359
\(430\) 0 0
\(431\) −2.90474 −0.139916 −0.0699581 0.997550i \(-0.522287\pi\)
−0.0699581 + 0.997550i \(0.522287\pi\)
\(432\) −5.31438 −0.255688
\(433\) −24.8312 −1.19331 −0.596656 0.802497i \(-0.703504\pi\)
−0.596656 + 0.802497i \(0.703504\pi\)
\(434\) −21.4859 −1.03135
\(435\) 0 0
\(436\) −16.5923 −0.794625
\(437\) 3.42621 0.163898
\(438\) −49.8327 −2.38110
\(439\) 18.6918 0.892110 0.446055 0.895005i \(-0.352828\pi\)
0.446055 + 0.895005i \(0.352828\pi\)
\(440\) 0 0
\(441\) 3.88502 0.185001
\(442\) 20.0770 0.954967
\(443\) 21.0951 1.00226 0.501129 0.865373i \(-0.332918\pi\)
0.501129 + 0.865373i \(0.332918\pi\)
\(444\) 39.5624 1.87755
\(445\) 0 0
\(446\) −45.1061 −2.13584
\(447\) −57.1002 −2.70075
\(448\) −12.0666 −0.570091
\(449\) 3.43961 0.162325 0.0811626 0.996701i \(-0.474137\pi\)
0.0811626 + 0.996701i \(0.474137\pi\)
\(450\) 0 0
\(451\) 23.0450 1.08514
\(452\) −17.4238 −0.819547
\(453\) 57.0963 2.68262
\(454\) −35.4266 −1.66265
\(455\) 0 0
\(456\) −12.5319 −0.586861
\(457\) 3.47356 0.162486 0.0812432 0.996694i \(-0.474111\pi\)
0.0812432 + 0.996694i \(0.474111\pi\)
\(458\) 12.4348 0.581038
\(459\) 12.5065 0.583754
\(460\) 0 0
\(461\) −35.4226 −1.64979 −0.824896 0.565284i \(-0.808767\pi\)
−0.824896 + 0.565284i \(0.808767\pi\)
\(462\) 17.8487 0.830397
\(463\) 41.0574 1.90810 0.954050 0.299647i \(-0.0968688\pi\)
0.954050 + 0.299647i \(0.0968688\pi\)
\(464\) 5.81550 0.269978
\(465\) 0 0
\(466\) −51.8838 −2.40347
\(467\) −26.1893 −1.21190 −0.605949 0.795503i \(-0.707206\pi\)
−0.605949 + 0.795503i \(0.707206\pi\)
\(468\) 17.7823 0.821987
\(469\) −5.20723 −0.240448
\(470\) 0 0
\(471\) −25.5542 −1.17748
\(472\) 11.5289 0.530661
\(473\) −15.5734 −0.716068
\(474\) −22.9083 −1.05221
\(475\) 0 0
\(476\) 14.2538 0.653324
\(477\) 12.4544 0.570248
\(478\) 31.9695 1.46225
\(479\) −28.7194 −1.31222 −0.656111 0.754665i \(-0.727800\pi\)
−0.656111 + 0.754665i \(0.727800\pi\)
\(480\) 0 0
\(481\) 9.85207 0.449216
\(482\) −36.2333 −1.65038
\(483\) −2.62393 −0.119393
\(484\) −2.75811 −0.125369
\(485\) 0 0
\(486\) −46.4754 −2.10817
\(487\) 10.1974 0.462087 0.231044 0.972943i \(-0.425786\pi\)
0.231044 + 0.972943i \(0.425786\pi\)
\(488\) −11.0930 −0.502156
\(489\) −28.2628 −1.27809
\(490\) 0 0
\(491\) −11.4829 −0.518216 −0.259108 0.965848i \(-0.583429\pi\)
−0.259108 + 0.965848i \(0.583429\pi\)
\(492\) 50.7159 2.28645
\(493\) −13.6858 −0.616378
\(494\) −12.7726 −0.574668
\(495\) 0 0
\(496\) 22.8103 1.02421
\(497\) 13.0308 0.584510
\(498\) −68.2116 −3.05663
\(499\) −3.89636 −0.174425 −0.0872124 0.996190i \(-0.527796\pi\)
−0.0872124 + 0.996190i \(0.527796\pi\)
\(500\) 0 0
\(501\) 47.2480 2.11088
\(502\) 28.7561 1.28345
\(503\) 39.2844 1.75161 0.875803 0.482668i \(-0.160332\pi\)
0.875803 + 0.482668i \(0.160332\pi\)
\(504\) 5.41556 0.241228
\(505\) 0 0
\(506\) −6.80228 −0.302398
\(507\) −26.2634 −1.16640
\(508\) 46.2921 2.05388
\(509\) −42.4548 −1.88178 −0.940888 0.338718i \(-0.890007\pi\)
−0.940888 + 0.338718i \(0.890007\pi\)
\(510\) 0 0
\(511\) −8.81031 −0.389745
\(512\) 24.0496 1.06285
\(513\) −7.95641 −0.351284
\(514\) 62.7250 2.76668
\(515\) 0 0
\(516\) −34.2730 −1.50879
\(517\) 23.4342 1.03064
\(518\) 12.2801 0.539557
\(519\) −37.3595 −1.63990
\(520\) 0 0
\(521\) 37.7986 1.65599 0.827993 0.560739i \(-0.189483\pi\)
0.827993 + 0.560739i \(0.189483\pi\)
\(522\) −21.2815 −0.931464
\(523\) 10.8004 0.472268 0.236134 0.971720i \(-0.424119\pi\)
0.236134 + 0.971720i \(0.424119\pi\)
\(524\) −4.85538 −0.212108
\(525\) 0 0
\(526\) 19.3692 0.844537
\(527\) −53.6803 −2.33835
\(528\) −18.9490 −0.824647
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 32.1314 1.39438
\(532\) −9.06803 −0.393149
\(533\) 12.6296 0.547047
\(534\) 29.5259 1.27771
\(535\) 0 0
\(536\) −7.25869 −0.313527
\(537\) 23.8105 1.02750
\(538\) 26.7287 1.15236
\(539\) 3.15561 0.135922
\(540\) 0 0
\(541\) 29.8362 1.28276 0.641380 0.767223i \(-0.278362\pi\)
0.641380 + 0.767223i \(0.278362\pi\)
\(542\) 3.55795 0.152827
\(543\) 3.02836 0.129960
\(544\) −41.5822 −1.78282
\(545\) 0 0
\(546\) 9.78181 0.418623
\(547\) 31.8433 1.36152 0.680762 0.732505i \(-0.261649\pi\)
0.680762 + 0.732505i \(0.261649\pi\)
\(548\) 43.8989 1.87527
\(549\) −30.9165 −1.31949
\(550\) 0 0
\(551\) 8.70667 0.370917
\(552\) −3.65766 −0.155680
\(553\) −4.05014 −0.172230
\(554\) 17.2654 0.733537
\(555\) 0 0
\(556\) 7.39544 0.313636
\(557\) −43.2395 −1.83212 −0.916059 0.401044i \(-0.868648\pi\)
−0.916059 + 0.401044i \(0.868648\pi\)
\(558\) −83.4729 −3.53369
\(559\) −8.53487 −0.360987
\(560\) 0 0
\(561\) 44.5932 1.88273
\(562\) −14.8792 −0.627643
\(563\) 6.58034 0.277328 0.138664 0.990339i \(-0.455719\pi\)
0.138664 + 0.990339i \(0.455719\pi\)
\(564\) 51.5726 2.17160
\(565\) 0 0
\(566\) 6.39413 0.268765
\(567\) −5.56170 −0.233570
\(568\) 18.1644 0.762161
\(569\) 37.6191 1.57708 0.788538 0.614986i \(-0.210838\pi\)
0.788538 + 0.614986i \(0.210838\pi\)
\(570\) 0 0
\(571\) 0.913726 0.0382382 0.0191191 0.999817i \(-0.493914\pi\)
0.0191191 + 0.999817i \(0.493914\pi\)
\(572\) 14.4437 0.603922
\(573\) 60.4346 2.52469
\(574\) 15.7421 0.657063
\(575\) 0 0
\(576\) −46.8787 −1.95328
\(577\) −22.0692 −0.918751 −0.459376 0.888242i \(-0.651927\pi\)
−0.459376 + 0.888242i \(0.651927\pi\)
\(578\) 25.8770 1.07634
\(579\) −11.2058 −0.465695
\(580\) 0 0
\(581\) −12.0597 −0.500319
\(582\) −24.6961 −1.02368
\(583\) 10.1161 0.418967
\(584\) −12.2812 −0.508201
\(585\) 0 0
\(586\) 14.7261 0.608329
\(587\) −42.9058 −1.77091 −0.885455 0.464725i \(-0.846153\pi\)
−0.885455 + 0.464725i \(0.846153\pi\)
\(588\) 6.94467 0.286393
\(589\) 34.1504 1.40714
\(590\) 0 0
\(591\) −1.16530 −0.0479339
\(592\) −13.0371 −0.535821
\(593\) 29.4911 1.21105 0.605527 0.795825i \(-0.292962\pi\)
0.605527 + 0.795825i \(0.292962\pi\)
\(594\) 15.7964 0.648133
\(595\) 0 0
\(596\) −57.5949 −2.35918
\(597\) −5.72465 −0.234295
\(598\) −3.72792 −0.152446
\(599\) 5.76536 0.235566 0.117783 0.993039i \(-0.462421\pi\)
0.117783 + 0.993039i \(0.462421\pi\)
\(600\) 0 0
\(601\) −12.5587 −0.512282 −0.256141 0.966639i \(-0.582451\pi\)
−0.256141 + 0.966639i \(0.582451\pi\)
\(602\) −10.6383 −0.433584
\(603\) −20.2302 −0.823837
\(604\) 57.5910 2.34334
\(605\) 0 0
\(606\) 26.0726 1.05913
\(607\) −4.76165 −0.193269 −0.0966347 0.995320i \(-0.530808\pi\)
−0.0966347 + 0.995320i \(0.530808\pi\)
\(608\) 26.4538 1.07285
\(609\) −6.66792 −0.270198
\(610\) 0 0
\(611\) 12.8429 0.519568
\(612\) 55.3764 2.23846
\(613\) −11.1622 −0.450836 −0.225418 0.974262i \(-0.572375\pi\)
−0.225418 + 0.974262i \(0.572375\pi\)
\(614\) −49.7945 −2.00954
\(615\) 0 0
\(616\) 4.39880 0.177233
\(617\) −13.5572 −0.545793 −0.272896 0.962043i \(-0.587982\pi\)
−0.272896 + 0.962043i \(0.587982\pi\)
\(618\) −32.4330 −1.30464
\(619\) −7.23539 −0.290815 −0.145407 0.989372i \(-0.546449\pi\)
−0.145407 + 0.989372i \(0.546449\pi\)
\(620\) 0 0
\(621\) −2.32222 −0.0931874
\(622\) 2.61277 0.104762
\(623\) 5.22011 0.209139
\(624\) −10.3848 −0.415724
\(625\) 0 0
\(626\) −52.2405 −2.08795
\(627\) −28.3694 −1.13296
\(628\) −25.7757 −1.02856
\(629\) 30.6806 1.22332
\(630\) 0 0
\(631\) −13.6513 −0.543451 −0.271726 0.962375i \(-0.587594\pi\)
−0.271726 + 0.962375i \(0.587594\pi\)
\(632\) −5.64574 −0.224576
\(633\) 28.0809 1.11611
\(634\) −37.7960 −1.50107
\(635\) 0 0
\(636\) 22.2629 0.882782
\(637\) 1.72940 0.0685214
\(638\) −17.2859 −0.684356
\(639\) 50.6247 2.00268
\(640\) 0 0
\(641\) −24.9739 −0.986410 −0.493205 0.869913i \(-0.664175\pi\)
−0.493205 + 0.869913i \(0.664175\pi\)
\(642\) −105.386 −4.15924
\(643\) 34.5414 1.36218 0.681090 0.732199i \(-0.261506\pi\)
0.681090 + 0.732199i \(0.261506\pi\)
\(644\) −2.64667 −0.104293
\(645\) 0 0
\(646\) −39.7757 −1.56495
\(647\) 40.9669 1.61057 0.805287 0.592885i \(-0.202011\pi\)
0.805287 + 0.592885i \(0.202011\pi\)
\(648\) −7.75280 −0.304559
\(649\) 26.0988 1.02447
\(650\) 0 0
\(651\) −26.1538 −1.02505
\(652\) −28.5077 −1.11645
\(653\) 38.9770 1.52529 0.762643 0.646819i \(-0.223901\pi\)
0.762643 + 0.646819i \(0.223901\pi\)
\(654\) −35.4593 −1.38657
\(655\) 0 0
\(656\) −16.7125 −0.652513
\(657\) −34.2282 −1.33537
\(658\) 16.0080 0.624058
\(659\) 36.2460 1.41195 0.705973 0.708239i \(-0.250510\pi\)
0.705973 + 0.708239i \(0.250510\pi\)
\(660\) 0 0
\(661\) 4.05310 0.157647 0.0788237 0.996889i \(-0.474884\pi\)
0.0788237 + 0.996889i \(0.474884\pi\)
\(662\) 35.0407 1.36189
\(663\) 24.4389 0.949127
\(664\) −16.8107 −0.652382
\(665\) 0 0
\(666\) 47.7084 1.84866
\(667\) 2.54119 0.0983955
\(668\) 47.6573 1.84392
\(669\) −54.9056 −2.12277
\(670\) 0 0
\(671\) −25.1120 −0.969439
\(672\) −20.2594 −0.781525
\(673\) 43.0790 1.66057 0.830287 0.557336i \(-0.188176\pi\)
0.830287 + 0.557336i \(0.188176\pi\)
\(674\) 24.0922 0.927999
\(675\) 0 0
\(676\) −26.4909 −1.01888
\(677\) −38.0161 −1.46108 −0.730538 0.682872i \(-0.760731\pi\)
−0.730538 + 0.682872i \(0.760731\pi\)
\(678\) −37.2363 −1.43005
\(679\) −4.36621 −0.167560
\(680\) 0 0
\(681\) −43.1232 −1.65249
\(682\) −67.8011 −2.59624
\(683\) −46.7213 −1.78774 −0.893870 0.448327i \(-0.852020\pi\)
−0.893870 + 0.448327i \(0.852020\pi\)
\(684\) −35.2295 −1.34703
\(685\) 0 0
\(686\) 2.15561 0.0823017
\(687\) 15.1363 0.577485
\(688\) 11.2941 0.430582
\(689\) 5.54404 0.211211
\(690\) 0 0
\(691\) −17.9309 −0.682125 −0.341062 0.940041i \(-0.610787\pi\)
−0.341062 + 0.940041i \(0.610787\pi\)
\(692\) −37.6832 −1.43250
\(693\) 12.2596 0.465704
\(694\) −24.7490 −0.939459
\(695\) 0 0
\(696\) −9.29483 −0.352320
\(697\) 39.3301 1.48973
\(698\) 35.8027 1.35515
\(699\) −63.1558 −2.38877
\(700\) 0 0
\(701\) 12.5215 0.472930 0.236465 0.971640i \(-0.424011\pi\)
0.236465 + 0.971640i \(0.424011\pi\)
\(702\) 8.65705 0.326739
\(703\) −19.5185 −0.736153
\(704\) −38.0774 −1.43509
\(705\) 0 0
\(706\) 30.5237 1.14877
\(707\) 4.60958 0.173361
\(708\) 57.4366 2.15860
\(709\) 19.5766 0.735216 0.367608 0.929981i \(-0.380177\pi\)
0.367608 + 0.929981i \(0.380177\pi\)
\(710\) 0 0
\(711\) −15.7349 −0.590103
\(712\) 7.27663 0.272703
\(713\) 9.96741 0.373282
\(714\) 30.4618 1.14000
\(715\) 0 0
\(716\) 24.0168 0.897550
\(717\) 38.9151 1.45331
\(718\) −24.1448 −0.901077
\(719\) −1.11818 −0.0417009 −0.0208505 0.999783i \(-0.506637\pi\)
−0.0208505 + 0.999783i \(0.506637\pi\)
\(720\) 0 0
\(721\) −5.73407 −0.213548
\(722\) −15.6521 −0.582510
\(723\) −44.1052 −1.64029
\(724\) 3.05460 0.113523
\(725\) 0 0
\(726\) −5.89435 −0.218760
\(727\) −11.1732 −0.414390 −0.207195 0.978300i \(-0.566433\pi\)
−0.207195 + 0.978300i \(0.566433\pi\)
\(728\) 2.41072 0.0893473
\(729\) −39.8873 −1.47731
\(730\) 0 0
\(731\) −26.5787 −0.983048
\(732\) −55.2649 −2.04265
\(733\) 16.9333 0.625445 0.312722 0.949845i \(-0.398759\pi\)
0.312722 + 0.949845i \(0.398759\pi\)
\(734\) 50.0327 1.84674
\(735\) 0 0
\(736\) 7.72102 0.284601
\(737\) −16.4320 −0.605281
\(738\) 61.1583 2.25127
\(739\) 18.4339 0.678102 0.339051 0.940768i \(-0.389894\pi\)
0.339051 + 0.940768i \(0.389894\pi\)
\(740\) 0 0
\(741\) −15.5476 −0.571154
\(742\) 6.91036 0.253687
\(743\) −6.08445 −0.223217 −0.111608 0.993752i \(-0.535600\pi\)
−0.111608 + 0.993752i \(0.535600\pi\)
\(744\) −36.4574 −1.33659
\(745\) 0 0
\(746\) −10.1064 −0.370023
\(747\) −46.8519 −1.71422
\(748\) 44.9796 1.64462
\(749\) −18.6320 −0.680797
\(750\) 0 0
\(751\) 13.6886 0.499504 0.249752 0.968310i \(-0.419651\pi\)
0.249752 + 0.968310i \(0.419651\pi\)
\(752\) −16.9948 −0.619737
\(753\) 35.0035 1.27560
\(754\) −9.47338 −0.345000
\(755\) 0 0
\(756\) 6.14614 0.223533
\(757\) 27.9301 1.01514 0.507569 0.861611i \(-0.330544\pi\)
0.507569 + 0.861611i \(0.330544\pi\)
\(758\) −24.6903 −0.896792
\(759\) −8.28011 −0.300549
\(760\) 0 0
\(761\) −11.7557 −0.426143 −0.213071 0.977037i \(-0.568347\pi\)
−0.213071 + 0.977037i \(0.568347\pi\)
\(762\) 98.9306 3.58388
\(763\) −6.26912 −0.226957
\(764\) 60.9582 2.20539
\(765\) 0 0
\(766\) −58.8367 −2.12586
\(767\) 14.3032 0.516458
\(768\) 3.54475 0.127910
\(769\) −36.1868 −1.30493 −0.652464 0.757820i \(-0.726265\pi\)
−0.652464 + 0.757820i \(0.726265\pi\)
\(770\) 0 0
\(771\) 76.3523 2.74976
\(772\) −11.3028 −0.406798
\(773\) 29.4969 1.06093 0.530464 0.847707i \(-0.322018\pi\)
0.530464 + 0.847707i \(0.322018\pi\)
\(774\) −41.3299 −1.48557
\(775\) 0 0
\(776\) −6.08633 −0.218486
\(777\) 14.9480 0.536258
\(778\) 60.8875 2.18292
\(779\) −25.0211 −0.896473
\(780\) 0 0
\(781\) 41.1201 1.47139
\(782\) −11.6092 −0.415145
\(783\) −5.90121 −0.210892
\(784\) −2.28849 −0.0817318
\(785\) 0 0
\(786\) −10.3764 −0.370114
\(787\) 26.9176 0.959509 0.479755 0.877403i \(-0.340726\pi\)
0.479755 + 0.877403i \(0.340726\pi\)
\(788\) −1.17539 −0.0418716
\(789\) 23.5773 0.839372
\(790\) 0 0
\(791\) −6.58330 −0.234075
\(792\) 17.0894 0.607246
\(793\) −13.7624 −0.488717
\(794\) −66.6911 −2.36678
\(795\) 0 0
\(796\) −5.77426 −0.204663
\(797\) 39.8228 1.41060 0.705299 0.708910i \(-0.250813\pi\)
0.705299 + 0.708910i \(0.250813\pi\)
\(798\) −19.3793 −0.686018
\(799\) 39.9945 1.41490
\(800\) 0 0
\(801\) 20.2802 0.716566
\(802\) −47.9643 −1.69368
\(803\) −27.8019 −0.981109
\(804\) −36.1625 −1.27535
\(805\) 0 0
\(806\) −37.1577 −1.30882
\(807\) 32.5357 1.14531
\(808\) 6.42557 0.226051
\(809\) −14.2056 −0.499442 −0.249721 0.968318i \(-0.580339\pi\)
−0.249721 + 0.968318i \(0.580339\pi\)
\(810\) 0 0
\(811\) 9.06992 0.318488 0.159244 0.987239i \(-0.449094\pi\)
0.159244 + 0.987239i \(0.449094\pi\)
\(812\) −6.72569 −0.236026
\(813\) 4.33093 0.151892
\(814\) 38.7513 1.35823
\(815\) 0 0
\(816\) −32.3396 −1.13211
\(817\) 16.9089 0.591567
\(818\) 12.6091 0.440865
\(819\) 6.71876 0.234772
\(820\) 0 0
\(821\) 39.8448 1.39059 0.695297 0.718722i \(-0.255273\pi\)
0.695297 + 0.718722i \(0.255273\pi\)
\(822\) 93.8162 3.27222
\(823\) 13.6308 0.475141 0.237570 0.971370i \(-0.423649\pi\)
0.237570 + 0.971370i \(0.423649\pi\)
\(824\) −7.99308 −0.278452
\(825\) 0 0
\(826\) 17.8282 0.620323
\(827\) 27.3750 0.951921 0.475961 0.879467i \(-0.342100\pi\)
0.475961 + 0.879467i \(0.342100\pi\)
\(828\) −10.2823 −0.357336
\(829\) −34.1484 −1.18602 −0.593012 0.805194i \(-0.702061\pi\)
−0.593012 + 0.805194i \(0.702061\pi\)
\(830\) 0 0
\(831\) 21.0164 0.729051
\(832\) −20.8679 −0.723465
\(833\) 5.38558 0.186599
\(834\) 15.8048 0.547274
\(835\) 0 0
\(836\) −28.6152 −0.989678
\(837\) −23.1465 −0.800060
\(838\) −55.4169 −1.91435
\(839\) 39.9352 1.37872 0.689358 0.724421i \(-0.257893\pi\)
0.689358 + 0.724421i \(0.257893\pi\)
\(840\) 0 0
\(841\) −22.5423 −0.777322
\(842\) 41.7863 1.44005
\(843\) −18.1118 −0.623805
\(844\) 28.3242 0.974958
\(845\) 0 0
\(846\) 62.1915 2.13819
\(847\) −1.04211 −0.0358073
\(848\) −7.33634 −0.251931
\(849\) 7.78329 0.267122
\(850\) 0 0
\(851\) −5.69681 −0.195284
\(852\) 90.4944 3.10029
\(853\) −0.378296 −0.0129526 −0.00647631 0.999979i \(-0.502061\pi\)
−0.00647631 + 0.999979i \(0.502061\pi\)
\(854\) −17.1541 −0.587002
\(855\) 0 0
\(856\) −25.9722 −0.887713
\(857\) −21.7667 −0.743535 −0.371768 0.928326i \(-0.621248\pi\)
−0.371768 + 0.928326i \(0.621248\pi\)
\(858\) 30.8676 1.05380
\(859\) −11.5199 −0.393055 −0.196528 0.980498i \(-0.562967\pi\)
−0.196528 + 0.980498i \(0.562967\pi\)
\(860\) 0 0
\(861\) 19.1622 0.653045
\(862\) −6.26148 −0.213267
\(863\) −6.32314 −0.215242 −0.107621 0.994192i \(-0.534323\pi\)
−0.107621 + 0.994192i \(0.534323\pi\)
\(864\) −17.9299 −0.609988
\(865\) 0 0
\(866\) −53.5265 −1.81890
\(867\) 31.4990 1.06976
\(868\) −26.3804 −0.895409
\(869\) −12.7807 −0.433555
\(870\) 0 0
\(871\) −9.00540 −0.305136
\(872\) −8.73891 −0.295937
\(873\) −16.9628 −0.574103
\(874\) 7.38558 0.249821
\(875\) 0 0
\(876\) −61.1847 −2.06724
\(877\) −50.6948 −1.71184 −0.855921 0.517107i \(-0.827009\pi\)
−0.855921 + 0.517107i \(0.827009\pi\)
\(878\) 40.2923 1.35980
\(879\) 17.9254 0.604609
\(880\) 0 0
\(881\) −17.9010 −0.603101 −0.301550 0.953450i \(-0.597504\pi\)
−0.301550 + 0.953450i \(0.597504\pi\)
\(882\) 8.37459 0.281987
\(883\) 42.2971 1.42341 0.711706 0.702478i \(-0.247923\pi\)
0.711706 + 0.702478i \(0.247923\pi\)
\(884\) 24.6506 0.829090
\(885\) 0 0
\(886\) 45.4729 1.52769
\(887\) 34.6437 1.16322 0.581612 0.813467i \(-0.302422\pi\)
0.581612 + 0.813467i \(0.302422\pi\)
\(888\) 20.8370 0.699243
\(889\) 17.4907 0.586619
\(890\) 0 0
\(891\) −17.5506 −0.587966
\(892\) −55.3814 −1.85430
\(893\) −25.4438 −0.851443
\(894\) −123.086 −4.11661
\(895\) 0 0
\(896\) −10.5688 −0.353078
\(897\) −4.53783 −0.151514
\(898\) 7.41446 0.247424
\(899\) 25.3291 0.844773
\(900\) 0 0
\(901\) 17.2649 0.575176
\(902\) 49.6760 1.65403
\(903\) −12.9495 −0.430933
\(904\) −9.17687 −0.305218
\(905\) 0 0
\(906\) 123.077 4.08898
\(907\) −12.2650 −0.407252 −0.203626 0.979049i \(-0.565273\pi\)
−0.203626 + 0.979049i \(0.565273\pi\)
\(908\) −43.4969 −1.44349
\(909\) 17.9083 0.593980
\(910\) 0 0
\(911\) 29.6665 0.982894 0.491447 0.870907i \(-0.336468\pi\)
0.491447 + 0.870907i \(0.336468\pi\)
\(912\) 20.5738 0.681268
\(913\) −38.0556 −1.25946
\(914\) 7.48765 0.247670
\(915\) 0 0
\(916\) 15.2674 0.504450
\(917\) −1.83453 −0.0605814
\(918\) 26.9592 0.889785
\(919\) 21.5442 0.710678 0.355339 0.934738i \(-0.384365\pi\)
0.355339 + 0.934738i \(0.384365\pi\)
\(920\) 0 0
\(921\) −60.6126 −1.99725
\(922\) −76.3573 −2.51469
\(923\) 22.5354 0.741763
\(924\) 21.9147 0.720940
\(925\) 0 0
\(926\) 88.5039 2.90842
\(927\) −22.2770 −0.731672
\(928\) 19.6206 0.644079
\(929\) 49.0601 1.60961 0.804805 0.593540i \(-0.202270\pi\)
0.804805 + 0.593540i \(0.202270\pi\)
\(930\) 0 0
\(931\) −3.42621 −0.112290
\(932\) −63.7031 −2.08666
\(933\) 3.18041 0.104122
\(934\) −56.4541 −1.84723
\(935\) 0 0
\(936\) 9.36569 0.306127
\(937\) −50.1197 −1.63734 −0.818670 0.574265i \(-0.805288\pi\)
−0.818670 + 0.574265i \(0.805288\pi\)
\(938\) −11.2248 −0.366502
\(939\) −63.5900 −2.07518
\(940\) 0 0
\(941\) −5.29965 −0.172764 −0.0863819 0.996262i \(-0.527531\pi\)
−0.0863819 + 0.996262i \(0.527531\pi\)
\(942\) −55.0851 −1.79477
\(943\) −7.30285 −0.237813
\(944\) −18.9272 −0.616027
\(945\) 0 0
\(946\) −33.5703 −1.09146
\(947\) 26.0370 0.846088 0.423044 0.906109i \(-0.360962\pi\)
0.423044 + 0.906109i \(0.360962\pi\)
\(948\) −28.1269 −0.913519
\(949\) −15.2366 −0.494600
\(950\) 0 0
\(951\) −46.0073 −1.49189
\(952\) 7.50730 0.243313
\(953\) 24.8848 0.806099 0.403050 0.915178i \(-0.367950\pi\)
0.403050 + 0.915178i \(0.367950\pi\)
\(954\) 26.8469 0.869200
\(955\) 0 0
\(956\) 39.2522 1.26951
\(957\) −21.0414 −0.680171
\(958\) −61.9078 −2.00015
\(959\) 16.5865 0.535606
\(960\) 0 0
\(961\) 68.3492 2.20481
\(962\) 21.2373 0.684717
\(963\) −72.3855 −2.33259
\(964\) −44.4874 −1.43284
\(965\) 0 0
\(966\) −5.65618 −0.181985
\(967\) 26.8969 0.864947 0.432474 0.901647i \(-0.357641\pi\)
0.432474 + 0.901647i \(0.357641\pi\)
\(968\) −1.45266 −0.0466902
\(969\) −48.4171 −1.55538
\(970\) 0 0
\(971\) 27.0213 0.867153 0.433577 0.901117i \(-0.357251\pi\)
0.433577 + 0.901117i \(0.357251\pi\)
\(972\) −57.0626 −1.83028
\(973\) 2.79425 0.0895794
\(974\) 21.9816 0.704336
\(975\) 0 0
\(976\) 18.2116 0.582938
\(977\) 56.9300 1.82135 0.910676 0.413120i \(-0.135561\pi\)
0.910676 + 0.413120i \(0.135561\pi\)
\(978\) −60.9236 −1.94812
\(979\) 16.4726 0.526468
\(980\) 0 0
\(981\) −24.3556 −0.777615
\(982\) −24.7527 −0.789890
\(983\) −58.9417 −1.87995 −0.939974 0.341245i \(-0.889151\pi\)
−0.939974 + 0.341245i \(0.889151\pi\)
\(984\) 26.7113 0.851526
\(985\) 0 0
\(986\) −29.5013 −0.939513
\(987\) 19.4859 0.620242
\(988\) −15.6823 −0.498920
\(989\) 4.93516 0.156929
\(990\) 0 0
\(991\) 53.4471 1.69780 0.848902 0.528551i \(-0.177264\pi\)
0.848902 + 0.528551i \(0.177264\pi\)
\(992\) 76.9586 2.44344
\(993\) 42.6534 1.35357
\(994\) 28.0893 0.890938
\(995\) 0 0
\(996\) −83.7503 −2.65373
\(997\) 3.09915 0.0981510 0.0490755 0.998795i \(-0.484373\pi\)
0.0490755 + 0.998795i \(0.484373\pi\)
\(998\) −8.39903 −0.265867
\(999\) 13.2292 0.418555
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 4025.2.a.n.1.3 4
5.4 even 2 805.2.a.h.1.2 4
15.14 odd 2 7245.2.a.be.1.3 4
35.34 odd 2 5635.2.a.t.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
805.2.a.h.1.2 4 5.4 even 2
4025.2.a.n.1.3 4 1.1 even 1 trivial
5635.2.a.t.1.2 4 35.34 odd 2
7245.2.a.be.1.3 4 15.14 odd 2