Properties

Label 5225.2.a.u.1.11
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.26824\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.26824 q^{2} +1.37964 q^{3} -0.391576 q^{4} +1.74971 q^{6} +2.74560 q^{7} -3.03308 q^{8} -1.09659 q^{9} +O(q^{10})\) \(q+1.26824 q^{2} +1.37964 q^{3} -0.391576 q^{4} +1.74971 q^{6} +2.74560 q^{7} -3.03308 q^{8} -1.09659 q^{9} -1.00000 q^{11} -0.540234 q^{12} +2.90239 q^{13} +3.48208 q^{14} -3.06352 q^{16} +5.83714 q^{17} -1.39074 q^{18} +1.00000 q^{19} +3.78795 q^{21} -1.26824 q^{22} -1.00708 q^{23} -4.18457 q^{24} +3.68092 q^{26} -5.65182 q^{27} -1.07511 q^{28} +8.92770 q^{29} +5.91308 q^{31} +2.18091 q^{32} -1.37964 q^{33} +7.40287 q^{34} +0.429399 q^{36} +8.15096 q^{37} +1.26824 q^{38} +4.00426 q^{39} -7.98447 q^{41} +4.80401 q^{42} -6.88762 q^{43} +0.391576 q^{44} -1.27722 q^{46} -10.9565 q^{47} -4.22655 q^{48} +0.538344 q^{49} +8.05315 q^{51} -1.13651 q^{52} -5.29137 q^{53} -7.16785 q^{54} -8.32765 q^{56} +1.37964 q^{57} +11.3224 q^{58} -5.09173 q^{59} +7.63872 q^{61} +7.49919 q^{62} -3.01081 q^{63} +8.89294 q^{64} -1.74971 q^{66} +15.6685 q^{67} -2.28568 q^{68} -1.38941 q^{69} -0.653706 q^{71} +3.32605 q^{72} -4.93939 q^{73} +10.3373 q^{74} -0.391576 q^{76} -2.74560 q^{77} +5.07835 q^{78} +12.8034 q^{79} -4.50771 q^{81} -10.1262 q^{82} +12.7259 q^{83} -1.48327 q^{84} -8.73513 q^{86} +12.3170 q^{87} +3.03308 q^{88} +11.9827 q^{89} +7.96882 q^{91} +0.394349 q^{92} +8.15793 q^{93} -13.8954 q^{94} +3.00887 q^{96} -0.982691 q^{97} +0.682748 q^{98} +1.09659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23} - 27 q^{24} + 15 q^{26} + 31 q^{27} + 34 q^{28} + 11 q^{29} - 2 q^{31} + 10 q^{32} - 4 q^{33} - 13 q^{34} + 8 q^{36} + 27 q^{37} - q^{38} + 4 q^{39} + 6 q^{41} + 2 q^{42} + 38 q^{43} - 13 q^{44} - 9 q^{46} + 14 q^{47} + 32 q^{48} + 28 q^{49} - 32 q^{51} + 16 q^{52} + 11 q^{53} - 11 q^{54} + 2 q^{56} + 4 q^{57} - 6 q^{58} + 13 q^{59} + 8 q^{61} + 7 q^{62} + 49 q^{63} + 9 q^{64} + q^{66} + 31 q^{67} - 26 q^{68} + 39 q^{69} - 3 q^{71} - 18 q^{72} + 18 q^{73} + 7 q^{74} + 13 q^{76} - 17 q^{77} + 18 q^{78} + 10 q^{79} + 31 q^{81} + 58 q^{82} + 16 q^{83} + 112 q^{84} - 63 q^{86} + 67 q^{87} - 3 q^{88} - 7 q^{89} - 50 q^{91} + 98 q^{92} + 26 q^{93} + 22 q^{94} - 37 q^{96} + 24 q^{97} - 46 q^{98} - 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.26824 0.896779 0.448389 0.893838i \(-0.351998\pi\)
0.448389 + 0.893838i \(0.351998\pi\)
\(3\) 1.37964 0.796536 0.398268 0.917269i \(-0.369611\pi\)
0.398268 + 0.917269i \(0.369611\pi\)
\(4\) −0.391576 −0.195788
\(5\) 0 0
\(6\) 1.74971 0.714316
\(7\) 2.74560 1.03774 0.518870 0.854853i \(-0.326353\pi\)
0.518870 + 0.854853i \(0.326353\pi\)
\(8\) −3.03308 −1.07236
\(9\) −1.09659 −0.365531
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −0.540234 −0.155952
\(13\) 2.90239 0.804979 0.402489 0.915425i \(-0.368145\pi\)
0.402489 + 0.915425i \(0.368145\pi\)
\(14\) 3.48208 0.930624
\(15\) 0 0
\(16\) −3.06352 −0.765879
\(17\) 5.83714 1.41571 0.707857 0.706356i \(-0.249662\pi\)
0.707857 + 0.706356i \(0.249662\pi\)
\(18\) −1.39074 −0.327800
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.78795 0.826598
\(22\) −1.26824 −0.270389
\(23\) −1.00708 −0.209991 −0.104996 0.994473i \(-0.533483\pi\)
−0.104996 + 0.994473i \(0.533483\pi\)
\(24\) −4.18457 −0.854171
\(25\) 0 0
\(26\) 3.68092 0.721888
\(27\) −5.65182 −1.08769
\(28\) −1.07511 −0.203177
\(29\) 8.92770 1.65783 0.828916 0.559373i \(-0.188958\pi\)
0.828916 + 0.559373i \(0.188958\pi\)
\(30\) 0 0
\(31\) 5.91308 1.06202 0.531011 0.847365i \(-0.321812\pi\)
0.531011 + 0.847365i \(0.321812\pi\)
\(32\) 2.18091 0.385533
\(33\) −1.37964 −0.240165
\(34\) 7.40287 1.26958
\(35\) 0 0
\(36\) 0.429399 0.0715665
\(37\) 8.15096 1.34001 0.670004 0.742357i \(-0.266292\pi\)
0.670004 + 0.742357i \(0.266292\pi\)
\(38\) 1.26824 0.205735
\(39\) 4.00426 0.641195
\(40\) 0 0
\(41\) −7.98447 −1.24696 −0.623482 0.781837i \(-0.714283\pi\)
−0.623482 + 0.781837i \(0.714283\pi\)
\(42\) 4.80401 0.741275
\(43\) −6.88762 −1.05035 −0.525176 0.850994i \(-0.676001\pi\)
−0.525176 + 0.850994i \(0.676001\pi\)
\(44\) 0.391576 0.0590323
\(45\) 0 0
\(46\) −1.27722 −0.188315
\(47\) −10.9565 −1.59816 −0.799082 0.601222i \(-0.794681\pi\)
−0.799082 + 0.601222i \(0.794681\pi\)
\(48\) −4.22655 −0.610050
\(49\) 0.538344 0.0769063
\(50\) 0 0
\(51\) 8.05315 1.12767
\(52\) −1.13651 −0.157605
\(53\) −5.29137 −0.726825 −0.363413 0.931628i \(-0.618388\pi\)
−0.363413 + 0.931628i \(0.618388\pi\)
\(54\) −7.16785 −0.975421
\(55\) 0 0
\(56\) −8.32765 −1.11283
\(57\) 1.37964 0.182738
\(58\) 11.3224 1.48671
\(59\) −5.09173 −0.662886 −0.331443 0.943475i \(-0.607536\pi\)
−0.331443 + 0.943475i \(0.607536\pi\)
\(60\) 0 0
\(61\) 7.63872 0.978038 0.489019 0.872273i \(-0.337355\pi\)
0.489019 + 0.872273i \(0.337355\pi\)
\(62\) 7.49919 0.952398
\(63\) −3.01081 −0.379326
\(64\) 8.89294 1.11162
\(65\) 0 0
\(66\) −1.74971 −0.215375
\(67\) 15.6685 1.91421 0.957104 0.289743i \(-0.0935699\pi\)
0.957104 + 0.289743i \(0.0935699\pi\)
\(68\) −2.28568 −0.277180
\(69\) −1.38941 −0.167265
\(70\) 0 0
\(71\) −0.653706 −0.0775806 −0.0387903 0.999247i \(-0.512350\pi\)
−0.0387903 + 0.999247i \(0.512350\pi\)
\(72\) 3.32605 0.391979
\(73\) −4.93939 −0.578112 −0.289056 0.957312i \(-0.593341\pi\)
−0.289056 + 0.957312i \(0.593341\pi\)
\(74\) 10.3373 1.20169
\(75\) 0 0
\(76\) −0.391576 −0.0449168
\(77\) −2.74560 −0.312891
\(78\) 5.07835 0.575010
\(79\) 12.8034 1.44049 0.720245 0.693720i \(-0.244029\pi\)
0.720245 + 0.693720i \(0.244029\pi\)
\(80\) 0 0
\(81\) −4.50771 −0.500857
\(82\) −10.1262 −1.11825
\(83\) 12.7259 1.39685 0.698424 0.715685i \(-0.253885\pi\)
0.698424 + 0.715685i \(0.253885\pi\)
\(84\) −1.48327 −0.161838
\(85\) 0 0
\(86\) −8.73513 −0.941933
\(87\) 12.3170 1.32052
\(88\) 3.03308 0.323328
\(89\) 11.9827 1.27016 0.635081 0.772445i \(-0.280967\pi\)
0.635081 + 0.772445i \(0.280967\pi\)
\(90\) 0 0
\(91\) 7.96882 0.835359
\(92\) 0.394349 0.0411137
\(93\) 8.15793 0.845938
\(94\) −13.8954 −1.43320
\(95\) 0 0
\(96\) 3.00887 0.307091
\(97\) −0.982691 −0.0997771 −0.0498886 0.998755i \(-0.515887\pi\)
−0.0498886 + 0.998755i \(0.515887\pi\)
\(98\) 0.682748 0.0689680
\(99\) 1.09659 0.110212
\(100\) 0 0
\(101\) 0.585289 0.0582384 0.0291192 0.999576i \(-0.490730\pi\)
0.0291192 + 0.999576i \(0.490730\pi\)
\(102\) 10.2133 1.01127
\(103\) 0.163288 0.0160893 0.00804464 0.999968i \(-0.497439\pi\)
0.00804464 + 0.999968i \(0.497439\pi\)
\(104\) −8.80320 −0.863225
\(105\) 0 0
\(106\) −6.71071 −0.651801
\(107\) 8.57415 0.828894 0.414447 0.910073i \(-0.363975\pi\)
0.414447 + 0.910073i \(0.363975\pi\)
\(108\) 2.21312 0.212957
\(109\) 16.1503 1.54692 0.773458 0.633848i \(-0.218525\pi\)
0.773458 + 0.633848i \(0.218525\pi\)
\(110\) 0 0
\(111\) 11.2454 1.06737
\(112\) −8.41120 −0.794784
\(113\) 9.09077 0.855188 0.427594 0.903971i \(-0.359361\pi\)
0.427594 + 0.903971i \(0.359361\pi\)
\(114\) 1.74971 0.163875
\(115\) 0 0
\(116\) −3.49587 −0.324584
\(117\) −3.18274 −0.294244
\(118\) −6.45751 −0.594462
\(119\) 16.0265 1.46914
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 9.68770 0.877083
\(123\) −11.0157 −0.993252
\(124\) −2.31542 −0.207931
\(125\) 0 0
\(126\) −3.81842 −0.340171
\(127\) 13.6931 1.21506 0.607531 0.794296i \(-0.292160\pi\)
0.607531 + 0.794296i \(0.292160\pi\)
\(128\) 6.91654 0.611341
\(129\) −9.50244 −0.836643
\(130\) 0 0
\(131\) 14.7215 1.28623 0.643114 0.765771i \(-0.277642\pi\)
0.643114 + 0.765771i \(0.277642\pi\)
\(132\) 0.540234 0.0470214
\(133\) 2.74560 0.238074
\(134\) 19.8713 1.71662
\(135\) 0 0
\(136\) −17.7045 −1.51815
\(137\) −7.29444 −0.623206 −0.311603 0.950212i \(-0.600866\pi\)
−0.311603 + 0.950212i \(0.600866\pi\)
\(138\) −1.76210 −0.150000
\(139\) 16.5868 1.40687 0.703435 0.710760i \(-0.251649\pi\)
0.703435 + 0.710760i \(0.251649\pi\)
\(140\) 0 0
\(141\) −15.1160 −1.27299
\(142\) −0.829054 −0.0695727
\(143\) −2.90239 −0.242710
\(144\) 3.35943 0.279952
\(145\) 0 0
\(146\) −6.26432 −0.518439
\(147\) 0.742722 0.0612587
\(148\) −3.19172 −0.262358
\(149\) −6.10805 −0.500391 −0.250195 0.968195i \(-0.580495\pi\)
−0.250195 + 0.968195i \(0.580495\pi\)
\(150\) 0 0
\(151\) 7.21136 0.586852 0.293426 0.955982i \(-0.405205\pi\)
0.293426 + 0.955982i \(0.405205\pi\)
\(152\) −3.03308 −0.246016
\(153\) −6.40095 −0.517487
\(154\) −3.48208 −0.280594
\(155\) 0 0
\(156\) −1.56797 −0.125538
\(157\) −8.21699 −0.655787 −0.327894 0.944715i \(-0.606339\pi\)
−0.327894 + 0.944715i \(0.606339\pi\)
\(158\) 16.2377 1.29180
\(159\) −7.30019 −0.578942
\(160\) 0 0
\(161\) −2.76505 −0.217916
\(162\) −5.71685 −0.449158
\(163\) −15.5417 −1.21732 −0.608660 0.793431i \(-0.708292\pi\)
−0.608660 + 0.793431i \(0.708292\pi\)
\(164\) 3.12653 0.244141
\(165\) 0 0
\(166\) 16.1394 1.25266
\(167\) −9.04791 −0.700148 −0.350074 0.936722i \(-0.613844\pi\)
−0.350074 + 0.936722i \(0.613844\pi\)
\(168\) −11.4892 −0.886408
\(169\) −4.57612 −0.352009
\(170\) 0 0
\(171\) −1.09659 −0.0838585
\(172\) 2.69703 0.205646
\(173\) −4.77616 −0.363124 −0.181562 0.983379i \(-0.558115\pi\)
−0.181562 + 0.983379i \(0.558115\pi\)
\(174\) 15.6209 1.18422
\(175\) 0 0
\(176\) 3.06352 0.230921
\(177\) −7.02475 −0.528013
\(178\) 15.1969 1.13905
\(179\) 2.94093 0.219816 0.109908 0.993942i \(-0.464944\pi\)
0.109908 + 0.993942i \(0.464944\pi\)
\(180\) 0 0
\(181\) 1.63416 0.121466 0.0607330 0.998154i \(-0.480656\pi\)
0.0607330 + 0.998154i \(0.480656\pi\)
\(182\) 10.1064 0.749133
\(183\) 10.5387 0.779042
\(184\) 3.05456 0.225185
\(185\) 0 0
\(186\) 10.3462 0.758619
\(187\) −5.83714 −0.426854
\(188\) 4.29029 0.312901
\(189\) −15.5177 −1.12874
\(190\) 0 0
\(191\) −6.07300 −0.439427 −0.219714 0.975564i \(-0.570512\pi\)
−0.219714 + 0.975564i \(0.570512\pi\)
\(192\) 12.2691 0.885443
\(193\) 0.167517 0.0120582 0.00602908 0.999982i \(-0.498081\pi\)
0.00602908 + 0.999982i \(0.498081\pi\)
\(194\) −1.24628 −0.0894780
\(195\) 0 0
\(196\) −0.210803 −0.0150573
\(197\) −17.0070 −1.21170 −0.605850 0.795579i \(-0.707167\pi\)
−0.605850 + 0.795579i \(0.707167\pi\)
\(198\) 1.39074 0.0988354
\(199\) −19.1932 −1.36057 −0.680286 0.732947i \(-0.738144\pi\)
−0.680286 + 0.732947i \(0.738144\pi\)
\(200\) 0 0
\(201\) 21.6169 1.52474
\(202\) 0.742285 0.0522270
\(203\) 24.5119 1.72040
\(204\) −3.15342 −0.220784
\(205\) 0 0
\(206\) 0.207088 0.0144285
\(207\) 1.10436 0.0767581
\(208\) −8.89153 −0.616516
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 16.0927 1.10786 0.553932 0.832562i \(-0.313127\pi\)
0.553932 + 0.832562i \(0.313127\pi\)
\(212\) 2.07197 0.142304
\(213\) −0.901880 −0.0617958
\(214\) 10.8740 0.743334
\(215\) 0 0
\(216\) 17.1425 1.16640
\(217\) 16.2350 1.10210
\(218\) 20.4824 1.38724
\(219\) −6.81459 −0.460487
\(220\) 0 0
\(221\) 16.9417 1.13962
\(222\) 14.2618 0.957190
\(223\) −24.8789 −1.66601 −0.833006 0.553263i \(-0.813382\pi\)
−0.833006 + 0.553263i \(0.813382\pi\)
\(224\) 5.98790 0.400084
\(225\) 0 0
\(226\) 11.5292 0.766914
\(227\) 27.5594 1.82918 0.914591 0.404381i \(-0.132513\pi\)
0.914591 + 0.404381i \(0.132513\pi\)
\(228\) −0.540234 −0.0357779
\(229\) −17.5798 −1.16171 −0.580854 0.814008i \(-0.697281\pi\)
−0.580854 + 0.814008i \(0.697281\pi\)
\(230\) 0 0
\(231\) −3.78795 −0.249229
\(232\) −27.0785 −1.77779
\(233\) 4.90663 0.321444 0.160722 0.987000i \(-0.448618\pi\)
0.160722 + 0.987000i \(0.448618\pi\)
\(234\) −4.03647 −0.263872
\(235\) 0 0
\(236\) 1.99380 0.129785
\(237\) 17.6640 1.14740
\(238\) 20.3254 1.31750
\(239\) 4.67250 0.302239 0.151119 0.988516i \(-0.451712\pi\)
0.151119 + 0.988516i \(0.451712\pi\)
\(240\) 0 0
\(241\) −24.3107 −1.56599 −0.782994 0.622029i \(-0.786308\pi\)
−0.782994 + 0.622029i \(0.786308\pi\)
\(242\) 1.26824 0.0815253
\(243\) 10.7365 0.688744
\(244\) −2.99114 −0.191488
\(245\) 0 0
\(246\) −13.9705 −0.890727
\(247\) 2.90239 0.184675
\(248\) −17.9349 −1.13887
\(249\) 17.5571 1.11264
\(250\) 0 0
\(251\) −17.6394 −1.11339 −0.556694 0.830718i \(-0.687930\pi\)
−0.556694 + 0.830718i \(0.687930\pi\)
\(252\) 1.17896 0.0742675
\(253\) 1.00708 0.0633147
\(254\) 17.3660 1.08964
\(255\) 0 0
\(256\) −9.01407 −0.563379
\(257\) 7.03977 0.439129 0.219564 0.975598i \(-0.429536\pi\)
0.219564 + 0.975598i \(0.429536\pi\)
\(258\) −12.0513 −0.750284
\(259\) 22.3793 1.39058
\(260\) 0 0
\(261\) −9.79004 −0.605988
\(262\) 18.6704 1.15346
\(263\) −9.19090 −0.566735 −0.283368 0.959011i \(-0.591452\pi\)
−0.283368 + 0.959011i \(0.591452\pi\)
\(264\) 4.18457 0.257542
\(265\) 0 0
\(266\) 3.48208 0.213500
\(267\) 16.5318 1.01173
\(268\) −6.13540 −0.374779
\(269\) −16.7645 −1.02215 −0.511075 0.859536i \(-0.670753\pi\)
−0.511075 + 0.859536i \(0.670753\pi\)
\(270\) 0 0
\(271\) −0.336276 −0.0204273 −0.0102137 0.999948i \(-0.503251\pi\)
−0.0102137 + 0.999948i \(0.503251\pi\)
\(272\) −17.8822 −1.08427
\(273\) 10.9941 0.665394
\(274\) −9.25108 −0.558878
\(275\) 0 0
\(276\) 0.544060 0.0327486
\(277\) 0.773144 0.0464537 0.0232269 0.999730i \(-0.492606\pi\)
0.0232269 + 0.999730i \(0.492606\pi\)
\(278\) 21.0359 1.26165
\(279\) −6.48424 −0.388201
\(280\) 0 0
\(281\) −13.2396 −0.789806 −0.394903 0.918723i \(-0.629222\pi\)
−0.394903 + 0.918723i \(0.629222\pi\)
\(282\) −19.1706 −1.14159
\(283\) 3.64509 0.216678 0.108339 0.994114i \(-0.465447\pi\)
0.108339 + 0.994114i \(0.465447\pi\)
\(284\) 0.255976 0.0151894
\(285\) 0 0
\(286\) −3.68092 −0.217657
\(287\) −21.9222 −1.29403
\(288\) −2.39156 −0.140924
\(289\) 17.0722 1.00424
\(290\) 0 0
\(291\) −1.35576 −0.0794761
\(292\) 1.93415 0.113187
\(293\) −9.46530 −0.552969 −0.276484 0.961018i \(-0.589169\pi\)
−0.276484 + 0.961018i \(0.589169\pi\)
\(294\) 0.941947 0.0549355
\(295\) 0 0
\(296\) −24.7225 −1.43697
\(297\) 5.65182 0.327952
\(298\) −7.74645 −0.448740
\(299\) −2.92295 −0.169038
\(300\) 0 0
\(301\) −18.9107 −1.08999
\(302\) 9.14571 0.526276
\(303\) 0.807489 0.0463890
\(304\) −3.06352 −0.175705
\(305\) 0 0
\(306\) −8.11793 −0.464071
\(307\) 34.1985 1.95181 0.975906 0.218190i \(-0.0700154\pi\)
0.975906 + 0.218190i \(0.0700154\pi\)
\(308\) 1.07511 0.0612602
\(309\) 0.225279 0.0128157
\(310\) 0 0
\(311\) −31.2004 −1.76921 −0.884607 0.466337i \(-0.845573\pi\)
−0.884607 + 0.466337i \(0.845573\pi\)
\(312\) −12.1453 −0.687590
\(313\) 8.52890 0.482082 0.241041 0.970515i \(-0.422511\pi\)
0.241041 + 0.970515i \(0.422511\pi\)
\(314\) −10.4211 −0.588096
\(315\) 0 0
\(316\) −5.01349 −0.282031
\(317\) −13.9541 −0.783741 −0.391870 0.920020i \(-0.628172\pi\)
−0.391870 + 0.920020i \(0.628172\pi\)
\(318\) −9.25836 −0.519183
\(319\) −8.92770 −0.499855
\(320\) 0 0
\(321\) 11.8292 0.660244
\(322\) −3.50673 −0.195423
\(323\) 5.83714 0.324787
\(324\) 1.76511 0.0980618
\(325\) 0 0
\(326\) −19.7105 −1.09167
\(327\) 22.2816 1.23217
\(328\) 24.2176 1.33719
\(329\) −30.0821 −1.65848
\(330\) 0 0
\(331\) −20.4997 −1.12676 −0.563382 0.826196i \(-0.690500\pi\)
−0.563382 + 0.826196i \(0.690500\pi\)
\(332\) −4.98315 −0.273486
\(333\) −8.93827 −0.489814
\(334\) −11.4749 −0.627878
\(335\) 0 0
\(336\) −11.6044 −0.633074
\(337\) 31.9046 1.73795 0.868977 0.494853i \(-0.164778\pi\)
0.868977 + 0.494853i \(0.164778\pi\)
\(338\) −5.80360 −0.315674
\(339\) 12.5420 0.681188
\(340\) 0 0
\(341\) −5.91308 −0.320211
\(342\) −1.39074 −0.0752025
\(343\) −17.7412 −0.957932
\(344\) 20.8907 1.12635
\(345\) 0 0
\(346\) −6.05730 −0.325642
\(347\) 14.1093 0.757426 0.378713 0.925514i \(-0.376367\pi\)
0.378713 + 0.925514i \(0.376367\pi\)
\(348\) −4.82305 −0.258542
\(349\) −28.3982 −1.52012 −0.760060 0.649853i \(-0.774831\pi\)
−0.760060 + 0.649853i \(0.774831\pi\)
\(350\) 0 0
\(351\) −16.4038 −0.875571
\(352\) −2.18091 −0.116243
\(353\) −23.4620 −1.24875 −0.624377 0.781123i \(-0.714647\pi\)
−0.624377 + 0.781123i \(0.714647\pi\)
\(354\) −8.90905 −0.473511
\(355\) 0 0
\(356\) −4.69213 −0.248683
\(357\) 22.1108 1.17023
\(358\) 3.72980 0.197126
\(359\) 11.4832 0.606061 0.303031 0.952981i \(-0.402002\pi\)
0.303031 + 0.952981i \(0.402002\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.07250 0.108928
\(363\) 1.37964 0.0724124
\(364\) −3.12040 −0.163553
\(365\) 0 0
\(366\) 13.3655 0.698628
\(367\) −25.0513 −1.30767 −0.653833 0.756639i \(-0.726840\pi\)
−0.653833 + 0.756639i \(0.726840\pi\)
\(368\) 3.08521 0.160828
\(369\) 8.75570 0.455804
\(370\) 0 0
\(371\) −14.5280 −0.754256
\(372\) −3.19445 −0.165625
\(373\) −23.5796 −1.22090 −0.610451 0.792054i \(-0.709012\pi\)
−0.610451 + 0.792054i \(0.709012\pi\)
\(374\) −7.40287 −0.382793
\(375\) 0 0
\(376\) 33.2319 1.71380
\(377\) 25.9117 1.33452
\(378\) −19.6801 −1.01223
\(379\) −26.7205 −1.37254 −0.686269 0.727348i \(-0.740753\pi\)
−0.686269 + 0.727348i \(0.740753\pi\)
\(380\) 0 0
\(381\) 18.8915 0.967841
\(382\) −7.70200 −0.394069
\(383\) −24.2328 −1.23824 −0.619119 0.785297i \(-0.712510\pi\)
−0.619119 + 0.785297i \(0.712510\pi\)
\(384\) 9.54234 0.486955
\(385\) 0 0
\(386\) 0.212452 0.0108135
\(387\) 7.55290 0.383936
\(388\) 0.384798 0.0195352
\(389\) 5.28290 0.267854 0.133927 0.990991i \(-0.457241\pi\)
0.133927 + 0.990991i \(0.457241\pi\)
\(390\) 0 0
\(391\) −5.87847 −0.297287
\(392\) −1.63284 −0.0824711
\(393\) 20.3104 1.02453
\(394\) −21.5689 −1.08663
\(395\) 0 0
\(396\) −0.429399 −0.0215781
\(397\) 2.09703 0.105247 0.0526235 0.998614i \(-0.483242\pi\)
0.0526235 + 0.998614i \(0.483242\pi\)
\(398\) −24.3415 −1.22013
\(399\) 3.78795 0.189635
\(400\) 0 0
\(401\) 20.2889 1.01318 0.506590 0.862187i \(-0.330906\pi\)
0.506590 + 0.862187i \(0.330906\pi\)
\(402\) 27.4153 1.36735
\(403\) 17.1621 0.854905
\(404\) −0.229185 −0.0114024
\(405\) 0 0
\(406\) 31.0869 1.54282
\(407\) −8.15096 −0.404028
\(408\) −24.4259 −1.20926
\(409\) 33.3628 1.64969 0.824843 0.565362i \(-0.191263\pi\)
0.824843 + 0.565362i \(0.191263\pi\)
\(410\) 0 0
\(411\) −10.0637 −0.496406
\(412\) −0.0639398 −0.00315009
\(413\) −13.9799 −0.687904
\(414\) 1.40059 0.0688350
\(415\) 0 0
\(416\) 6.32984 0.310346
\(417\) 22.8838 1.12062
\(418\) −1.26824 −0.0620315
\(419\) 11.4626 0.559983 0.279992 0.960002i \(-0.409668\pi\)
0.279992 + 0.960002i \(0.409668\pi\)
\(420\) 0 0
\(421\) 21.4615 1.04597 0.522986 0.852341i \(-0.324818\pi\)
0.522986 + 0.852341i \(0.324818\pi\)
\(422\) 20.4093 0.993509
\(423\) 12.0148 0.584177
\(424\) 16.0492 0.779416
\(425\) 0 0
\(426\) −1.14380 −0.0554171
\(427\) 20.9729 1.01495
\(428\) −3.35743 −0.162287
\(429\) −4.00426 −0.193327
\(430\) 0 0
\(431\) −1.86277 −0.0897266 −0.0448633 0.998993i \(-0.514285\pi\)
−0.0448633 + 0.998993i \(0.514285\pi\)
\(432\) 17.3145 0.833042
\(433\) −16.5406 −0.794890 −0.397445 0.917626i \(-0.630103\pi\)
−0.397445 + 0.917626i \(0.630103\pi\)
\(434\) 20.5898 0.988342
\(435\) 0 0
\(436\) −6.32406 −0.302868
\(437\) −1.00708 −0.0481752
\(438\) −8.64251 −0.412955
\(439\) 13.4407 0.641488 0.320744 0.947166i \(-0.396067\pi\)
0.320744 + 0.947166i \(0.396067\pi\)
\(440\) 0 0
\(441\) −0.590344 −0.0281116
\(442\) 21.4860 1.02199
\(443\) 15.6710 0.744551 0.372276 0.928122i \(-0.378578\pi\)
0.372276 + 0.928122i \(0.378578\pi\)
\(444\) −4.40342 −0.208977
\(445\) 0 0
\(446\) −31.5523 −1.49404
\(447\) −8.42691 −0.398579
\(448\) 24.4165 1.15357
\(449\) −39.1867 −1.84933 −0.924667 0.380778i \(-0.875656\pi\)
−0.924667 + 0.380778i \(0.875656\pi\)
\(450\) 0 0
\(451\) 7.98447 0.375974
\(452\) −3.55973 −0.167435
\(453\) 9.94908 0.467449
\(454\) 34.9518 1.64037
\(455\) 0 0
\(456\) −4.18457 −0.195960
\(457\) −16.2611 −0.760662 −0.380331 0.924850i \(-0.624190\pi\)
−0.380331 + 0.924850i \(0.624190\pi\)
\(458\) −22.2954 −1.04179
\(459\) −32.9905 −1.53986
\(460\) 0 0
\(461\) 30.0292 1.39860 0.699299 0.714829i \(-0.253496\pi\)
0.699299 + 0.714829i \(0.253496\pi\)
\(462\) −4.80401 −0.223503
\(463\) 34.3384 1.59584 0.797921 0.602762i \(-0.205933\pi\)
0.797921 + 0.602762i \(0.205933\pi\)
\(464\) −27.3501 −1.26970
\(465\) 0 0
\(466\) 6.22276 0.288264
\(467\) −32.9295 −1.52380 −0.761898 0.647697i \(-0.775732\pi\)
−0.761898 + 0.647697i \(0.775732\pi\)
\(468\) 1.24628 0.0576095
\(469\) 43.0194 1.98645
\(470\) 0 0
\(471\) −11.3365 −0.522358
\(472\) 15.4436 0.710851
\(473\) 6.88762 0.316693
\(474\) 22.4022 1.02897
\(475\) 0 0
\(476\) −6.27558 −0.287641
\(477\) 5.80247 0.265677
\(478\) 5.92583 0.271041
\(479\) −33.5237 −1.53174 −0.765869 0.642997i \(-0.777691\pi\)
−0.765869 + 0.642997i \(0.777691\pi\)
\(480\) 0 0
\(481\) 23.6573 1.07868
\(482\) −30.8317 −1.40434
\(483\) −3.81477 −0.173578
\(484\) −0.391576 −0.0177989
\(485\) 0 0
\(486\) 13.6164 0.617651
\(487\) 20.6826 0.937216 0.468608 0.883406i \(-0.344756\pi\)
0.468608 + 0.883406i \(0.344756\pi\)
\(488\) −23.1689 −1.04881
\(489\) −21.4419 −0.969638
\(490\) 0 0
\(491\) −22.7879 −1.02840 −0.514202 0.857669i \(-0.671912\pi\)
−0.514202 + 0.857669i \(0.671912\pi\)
\(492\) 4.31348 0.194467
\(493\) 52.1122 2.34702
\(494\) 3.68092 0.165612
\(495\) 0 0
\(496\) −18.1148 −0.813380
\(497\) −1.79482 −0.0805086
\(498\) 22.2666 0.997791
\(499\) −18.7580 −0.839723 −0.419861 0.907588i \(-0.637921\pi\)
−0.419861 + 0.907588i \(0.637921\pi\)
\(500\) 0 0
\(501\) −12.4829 −0.557693
\(502\) −22.3709 −0.998463
\(503\) 24.1464 1.07663 0.538317 0.842742i \(-0.319060\pi\)
0.538317 + 0.842742i \(0.319060\pi\)
\(504\) 9.13203 0.406773
\(505\) 0 0
\(506\) 1.27722 0.0567792
\(507\) −6.31340 −0.280388
\(508\) −5.36187 −0.237895
\(509\) −10.5884 −0.469322 −0.234661 0.972077i \(-0.575398\pi\)
−0.234661 + 0.972077i \(0.575398\pi\)
\(510\) 0 0
\(511\) −13.5616 −0.599931
\(512\) −25.2650 −1.11657
\(513\) −5.65182 −0.249534
\(514\) 8.92809 0.393801
\(515\) 0 0
\(516\) 3.72093 0.163805
\(517\) 10.9565 0.481864
\(518\) 28.3822 1.24704
\(519\) −6.58938 −0.289242
\(520\) 0 0
\(521\) 4.39548 0.192570 0.0962848 0.995354i \(-0.469304\pi\)
0.0962848 + 0.995354i \(0.469304\pi\)
\(522\) −12.4161 −0.543437
\(523\) 31.2908 1.36825 0.684125 0.729364i \(-0.260184\pi\)
0.684125 + 0.729364i \(0.260184\pi\)
\(524\) −5.76461 −0.251828
\(525\) 0 0
\(526\) −11.6562 −0.508236
\(527\) 34.5155 1.50352
\(528\) 4.22655 0.183937
\(529\) −21.9858 −0.955904
\(530\) 0 0
\(531\) 5.58355 0.242305
\(532\) −1.07511 −0.0466121
\(533\) −23.1741 −1.00378
\(534\) 20.9662 0.907298
\(535\) 0 0
\(536\) −47.5238 −2.05272
\(537\) 4.05743 0.175091
\(538\) −21.2614 −0.916643
\(539\) −0.538344 −0.0231881
\(540\) 0 0
\(541\) 12.9098 0.555035 0.277518 0.960721i \(-0.410488\pi\)
0.277518 + 0.960721i \(0.410488\pi\)
\(542\) −0.426477 −0.0183188
\(543\) 2.25455 0.0967521
\(544\) 12.7302 0.545805
\(545\) 0 0
\(546\) 13.9431 0.596711
\(547\) 17.1687 0.734081 0.367041 0.930205i \(-0.380371\pi\)
0.367041 + 0.930205i \(0.380371\pi\)
\(548\) 2.85633 0.122016
\(549\) −8.37655 −0.357503
\(550\) 0 0
\(551\) 8.92770 0.380333
\(552\) 4.21420 0.179368
\(553\) 35.1530 1.49486
\(554\) 0.980529 0.0416587
\(555\) 0 0
\(556\) −6.49497 −0.275448
\(557\) 19.6073 0.830788 0.415394 0.909642i \(-0.363644\pi\)
0.415394 + 0.909642i \(0.363644\pi\)
\(558\) −8.22355 −0.348131
\(559\) −19.9906 −0.845511
\(560\) 0 0
\(561\) −8.05315 −0.340004
\(562\) −16.7909 −0.708281
\(563\) −3.06142 −0.129023 −0.0645117 0.997917i \(-0.520549\pi\)
−0.0645117 + 0.997917i \(0.520549\pi\)
\(564\) 5.91905 0.249237
\(565\) 0 0
\(566\) 4.62284 0.194312
\(567\) −12.3764 −0.519760
\(568\) 1.98275 0.0831942
\(569\) −35.8142 −1.50141 −0.750704 0.660639i \(-0.770285\pi\)
−0.750704 + 0.660639i \(0.770285\pi\)
\(570\) 0 0
\(571\) −33.6489 −1.40816 −0.704081 0.710120i \(-0.748641\pi\)
−0.704081 + 0.710120i \(0.748641\pi\)
\(572\) 1.13651 0.0475198
\(573\) −8.37856 −0.350019
\(574\) −27.8025 −1.16046
\(575\) 0 0
\(576\) −9.75192 −0.406330
\(577\) −29.8985 −1.24469 −0.622345 0.782743i \(-0.713820\pi\)
−0.622345 + 0.782743i \(0.713820\pi\)
\(578\) 21.6515 0.900586
\(579\) 0.231114 0.00960475
\(580\) 0 0
\(581\) 34.9402 1.44957
\(582\) −1.71942 −0.0712724
\(583\) 5.29137 0.219146
\(584\) 14.9816 0.619943
\(585\) 0 0
\(586\) −12.0042 −0.495891
\(587\) 33.6644 1.38948 0.694740 0.719261i \(-0.255520\pi\)
0.694740 + 0.719261i \(0.255520\pi\)
\(588\) −0.290832 −0.0119937
\(589\) 5.91308 0.243644
\(590\) 0 0
\(591\) −23.4636 −0.965163
\(592\) −24.9706 −1.02628
\(593\) 2.13964 0.0878646 0.0439323 0.999035i \(-0.486011\pi\)
0.0439323 + 0.999035i \(0.486011\pi\)
\(594\) 7.16785 0.294100
\(595\) 0 0
\(596\) 2.39176 0.0979705
\(597\) −26.4798 −1.08374
\(598\) −3.70699 −0.151590
\(599\) 25.3896 1.03739 0.518696 0.854959i \(-0.326418\pi\)
0.518696 + 0.854959i \(0.326418\pi\)
\(600\) 0 0
\(601\) −16.6132 −0.677666 −0.338833 0.940846i \(-0.610032\pi\)
−0.338833 + 0.940846i \(0.610032\pi\)
\(602\) −23.9832 −0.977483
\(603\) −17.1819 −0.699702
\(604\) −2.82379 −0.114899
\(605\) 0 0
\(606\) 1.02409 0.0416007
\(607\) −15.5197 −0.629927 −0.314963 0.949104i \(-0.601992\pi\)
−0.314963 + 0.949104i \(0.601992\pi\)
\(608\) 2.18091 0.0884474
\(609\) 33.8177 1.37036
\(610\) 0 0
\(611\) −31.7999 −1.28649
\(612\) 2.50646 0.101318
\(613\) −28.8792 −1.16642 −0.583210 0.812322i \(-0.698203\pi\)
−0.583210 + 0.812322i \(0.698203\pi\)
\(614\) 43.3718 1.75034
\(615\) 0 0
\(616\) 8.32765 0.335531
\(617\) −44.7531 −1.80169 −0.900846 0.434138i \(-0.857053\pi\)
−0.900846 + 0.434138i \(0.857053\pi\)
\(618\) 0.285707 0.0114928
\(619\) −31.6774 −1.27322 −0.636610 0.771186i \(-0.719664\pi\)
−0.636610 + 0.771186i \(0.719664\pi\)
\(620\) 0 0
\(621\) 5.69185 0.228406
\(622\) −39.5695 −1.58659
\(623\) 32.8997 1.31810
\(624\) −12.2671 −0.491077
\(625\) 0 0
\(626\) 10.8167 0.432321
\(627\) −1.37964 −0.0550975
\(628\) 3.21758 0.128395
\(629\) 47.5782 1.89707
\(630\) 0 0
\(631\) −22.3735 −0.890677 −0.445338 0.895362i \(-0.646917\pi\)
−0.445338 + 0.895362i \(0.646917\pi\)
\(632\) −38.8337 −1.54472
\(633\) 22.2021 0.882454
\(634\) −17.6971 −0.702842
\(635\) 0 0
\(636\) 2.85858 0.113350
\(637\) 1.56249 0.0619080
\(638\) −11.3224 −0.448259
\(639\) 0.716849 0.0283581
\(640\) 0 0
\(641\) 5.11199 0.201911 0.100956 0.994891i \(-0.467810\pi\)
0.100956 + 0.994891i \(0.467810\pi\)
\(642\) 15.0023 0.592093
\(643\) 19.5913 0.772606 0.386303 0.922372i \(-0.373752\pi\)
0.386303 + 0.922372i \(0.373752\pi\)
\(644\) 1.08273 0.0426654
\(645\) 0 0
\(646\) 7.40287 0.291262
\(647\) 22.5387 0.886089 0.443045 0.896500i \(-0.353898\pi\)
0.443045 + 0.896500i \(0.353898\pi\)
\(648\) 13.6723 0.537098
\(649\) 5.09173 0.199868
\(650\) 0 0
\(651\) 22.3985 0.877865
\(652\) 6.08575 0.238337
\(653\) 19.5580 0.765364 0.382682 0.923880i \(-0.375001\pi\)
0.382682 + 0.923880i \(0.375001\pi\)
\(654\) 28.2583 1.10499
\(655\) 0 0
\(656\) 24.4606 0.955024
\(657\) 5.41649 0.211318
\(658\) −38.1512 −1.48729
\(659\) −32.6875 −1.27332 −0.636662 0.771143i \(-0.719685\pi\)
−0.636662 + 0.771143i \(0.719685\pi\)
\(660\) 0 0
\(661\) 15.4629 0.601437 0.300718 0.953713i \(-0.402774\pi\)
0.300718 + 0.953713i \(0.402774\pi\)
\(662\) −25.9984 −1.01046
\(663\) 23.3734 0.907748
\(664\) −38.5987 −1.49792
\(665\) 0 0
\(666\) −11.3358 −0.439255
\(667\) −8.99092 −0.348130
\(668\) 3.54294 0.137081
\(669\) −34.3239 −1.32704
\(670\) 0 0
\(671\) −7.63872 −0.294889
\(672\) 8.26115 0.318681
\(673\) 24.8554 0.958105 0.479053 0.877786i \(-0.340980\pi\)
0.479053 + 0.877786i \(0.340980\pi\)
\(674\) 40.4626 1.55856
\(675\) 0 0
\(676\) 1.79190 0.0689192
\(677\) −48.0537 −1.84685 −0.923427 0.383773i \(-0.874624\pi\)
−0.923427 + 0.383773i \(0.874624\pi\)
\(678\) 15.9062 0.610875
\(679\) −2.69808 −0.103543
\(680\) 0 0
\(681\) 38.0221 1.45701
\(682\) −7.49919 −0.287159
\(683\) −10.6837 −0.408799 −0.204400 0.978888i \(-0.565524\pi\)
−0.204400 + 0.978888i \(0.565524\pi\)
\(684\) 0.429399 0.0164185
\(685\) 0 0
\(686\) −22.5000 −0.859053
\(687\) −24.2538 −0.925342
\(688\) 21.1003 0.804443
\(689\) −15.3576 −0.585079
\(690\) 0 0
\(691\) 15.2228 0.579101 0.289551 0.957163i \(-0.406494\pi\)
0.289551 + 0.957163i \(0.406494\pi\)
\(692\) 1.87023 0.0710954
\(693\) 3.01081 0.114371
\(694\) 17.8939 0.679243
\(695\) 0 0
\(696\) −37.3585 −1.41607
\(697\) −46.6064 −1.76534
\(698\) −36.0156 −1.36321
\(699\) 6.76938 0.256042
\(700\) 0 0
\(701\) 2.56955 0.0970504 0.0485252 0.998822i \(-0.484548\pi\)
0.0485252 + 0.998822i \(0.484548\pi\)
\(702\) −20.8039 −0.785193
\(703\) 8.15096 0.307419
\(704\) −8.89294 −0.335165
\(705\) 0 0
\(706\) −29.7553 −1.11986
\(707\) 1.60697 0.0604364
\(708\) 2.75073 0.103379
\(709\) −0.432067 −0.0162266 −0.00811331 0.999967i \(-0.502583\pi\)
−0.00811331 + 0.999967i \(0.502583\pi\)
\(710\) 0 0
\(711\) −14.0401 −0.526543
\(712\) −36.3445 −1.36207
\(713\) −5.95496 −0.223015
\(714\) 28.0417 1.04943
\(715\) 0 0
\(716\) −1.15160 −0.0430373
\(717\) 6.44637 0.240744
\(718\) 14.5634 0.543503
\(719\) 32.4602 1.21056 0.605281 0.796012i \(-0.293061\pi\)
0.605281 + 0.796012i \(0.293061\pi\)
\(720\) 0 0
\(721\) 0.448325 0.0166965
\(722\) 1.26824 0.0471989
\(723\) −33.5400 −1.24737
\(724\) −0.639897 −0.0237816
\(725\) 0 0
\(726\) 1.74971 0.0649379
\(727\) 17.6543 0.654763 0.327381 0.944892i \(-0.393834\pi\)
0.327381 + 0.944892i \(0.393834\pi\)
\(728\) −24.1701 −0.895804
\(729\) 28.3356 1.04947
\(730\) 0 0
\(731\) −40.2040 −1.48700
\(732\) −4.12670 −0.152527
\(733\) 12.7227 0.469925 0.234962 0.972004i \(-0.424503\pi\)
0.234962 + 0.972004i \(0.424503\pi\)
\(734\) −31.7710 −1.17269
\(735\) 0 0
\(736\) −2.19635 −0.0809585
\(737\) −15.6685 −0.577156
\(738\) 11.1043 0.408755
\(739\) 5.08759 0.187150 0.0935750 0.995612i \(-0.470171\pi\)
0.0935750 + 0.995612i \(0.470171\pi\)
\(740\) 0 0
\(741\) 4.00426 0.147100
\(742\) −18.4249 −0.676401
\(743\) −38.3289 −1.40615 −0.703075 0.711116i \(-0.748190\pi\)
−0.703075 + 0.711116i \(0.748190\pi\)
\(744\) −24.7437 −0.907148
\(745\) 0 0
\(746\) −29.9044 −1.09488
\(747\) −13.9551 −0.510590
\(748\) 2.28568 0.0835728
\(749\) 23.5412 0.860177
\(750\) 0 0
\(751\) −39.9351 −1.45725 −0.728626 0.684911i \(-0.759841\pi\)
−0.728626 + 0.684911i \(0.759841\pi\)
\(752\) 33.5653 1.22400
\(753\) −24.3360 −0.886854
\(754\) 32.8621 1.19677
\(755\) 0 0
\(756\) 6.07635 0.220995
\(757\) 44.8404 1.62975 0.814876 0.579636i \(-0.196805\pi\)
0.814876 + 0.579636i \(0.196805\pi\)
\(758\) −33.8879 −1.23086
\(759\) 1.38941 0.0504324
\(760\) 0 0
\(761\) 43.9069 1.59163 0.795813 0.605543i \(-0.207044\pi\)
0.795813 + 0.605543i \(0.207044\pi\)
\(762\) 23.9589 0.867939
\(763\) 44.3423 1.60530
\(764\) 2.37804 0.0860346
\(765\) 0 0
\(766\) −30.7329 −1.11043
\(767\) −14.7782 −0.533609
\(768\) −12.4362 −0.448752
\(769\) −11.9549 −0.431106 −0.215553 0.976492i \(-0.569155\pi\)
−0.215553 + 0.976492i \(0.569155\pi\)
\(770\) 0 0
\(771\) 9.71235 0.349782
\(772\) −0.0655957 −0.00236084
\(773\) −4.78825 −0.172221 −0.0861107 0.996286i \(-0.527444\pi\)
−0.0861107 + 0.996286i \(0.527444\pi\)
\(774\) 9.57887 0.344305
\(775\) 0 0
\(776\) 2.98058 0.106997
\(777\) 30.8754 1.10765
\(778\) 6.69997 0.240206
\(779\) −7.98447 −0.286073
\(780\) 0 0
\(781\) 0.653706 0.0233914
\(782\) −7.45529 −0.266601
\(783\) −50.4578 −1.80321
\(784\) −1.64923 −0.0589010
\(785\) 0 0
\(786\) 25.7585 0.918774
\(787\) −35.2409 −1.25620 −0.628100 0.778132i \(-0.716167\pi\)
−0.628100 + 0.778132i \(0.716167\pi\)
\(788\) 6.65955 0.237236
\(789\) −12.6801 −0.451425
\(790\) 0 0
\(791\) 24.9597 0.887463
\(792\) −3.32605 −0.118186
\(793\) 22.1706 0.787300
\(794\) 2.65953 0.0943833
\(795\) 0 0
\(796\) 7.51561 0.266384
\(797\) 45.1345 1.59875 0.799373 0.600835i \(-0.205165\pi\)
0.799373 + 0.600835i \(0.205165\pi\)
\(798\) 4.80401 0.170060
\(799\) −63.9543 −2.26254
\(800\) 0 0
\(801\) −13.1401 −0.464283
\(802\) 25.7312 0.908599
\(803\) 4.93939 0.174307
\(804\) −8.46465 −0.298525
\(805\) 0 0
\(806\) 21.7656 0.766660
\(807\) −23.1290 −0.814180
\(808\) −1.77523 −0.0624524
\(809\) −42.3579 −1.48922 −0.744612 0.667498i \(-0.767365\pi\)
−0.744612 + 0.667498i \(0.767365\pi\)
\(810\) 0 0
\(811\) 32.2926 1.13395 0.566973 0.823736i \(-0.308114\pi\)
0.566973 + 0.823736i \(0.308114\pi\)
\(812\) −9.59828 −0.336834
\(813\) −0.463940 −0.0162711
\(814\) −10.3373 −0.362324
\(815\) 0 0
\(816\) −24.6710 −0.863656
\(817\) −6.88762 −0.240967
\(818\) 42.3120 1.47940
\(819\) −8.73854 −0.305349
\(820\) 0 0
\(821\) 1.70149 0.0593824 0.0296912 0.999559i \(-0.490548\pi\)
0.0296912 + 0.999559i \(0.490548\pi\)
\(822\) −12.7632 −0.445167
\(823\) 2.68188 0.0934844 0.0467422 0.998907i \(-0.485116\pi\)
0.0467422 + 0.998907i \(0.485116\pi\)
\(824\) −0.495267 −0.0172535
\(825\) 0 0
\(826\) −17.7298 −0.616898
\(827\) −11.9898 −0.416928 −0.208464 0.978030i \(-0.566846\pi\)
−0.208464 + 0.978030i \(0.566846\pi\)
\(828\) −0.432440 −0.0150283
\(829\) −40.3322 −1.40079 −0.700397 0.713753i \(-0.746994\pi\)
−0.700397 + 0.713753i \(0.746994\pi\)
\(830\) 0 0
\(831\) 1.06666 0.0370021
\(832\) 25.8108 0.894828
\(833\) 3.14239 0.108877
\(834\) 29.0220 1.00495
\(835\) 0 0
\(836\) 0.391576 0.0135429
\(837\) −33.4197 −1.15515
\(838\) 14.5372 0.502181
\(839\) 19.1737 0.661950 0.330975 0.943639i \(-0.392622\pi\)
0.330975 + 0.943639i \(0.392622\pi\)
\(840\) 0 0
\(841\) 50.7038 1.74841
\(842\) 27.2183 0.938005
\(843\) −18.2658 −0.629109
\(844\) −6.30150 −0.216907
\(845\) 0 0
\(846\) 15.2376 0.523878
\(847\) 2.74560 0.0943401
\(848\) 16.2102 0.556660
\(849\) 5.02892 0.172592
\(850\) 0 0
\(851\) −8.20868 −0.281390
\(852\) 0.353154 0.0120989
\(853\) 22.8894 0.783717 0.391858 0.920026i \(-0.371832\pi\)
0.391858 + 0.920026i \(0.371832\pi\)
\(854\) 26.5986 0.910185
\(855\) 0 0
\(856\) −26.0061 −0.888870
\(857\) −39.5732 −1.35180 −0.675898 0.736995i \(-0.736244\pi\)
−0.675898 + 0.736995i \(0.736244\pi\)
\(858\) −5.07835 −0.173372
\(859\) −26.7629 −0.913138 −0.456569 0.889688i \(-0.650922\pi\)
−0.456569 + 0.889688i \(0.650922\pi\)
\(860\) 0 0
\(861\) −30.2448 −1.03074
\(862\) −2.36244 −0.0804649
\(863\) 6.34654 0.216039 0.108019 0.994149i \(-0.465549\pi\)
0.108019 + 0.994149i \(0.465549\pi\)
\(864\) −12.3261 −0.419342
\(865\) 0 0
\(866\) −20.9774 −0.712840
\(867\) 23.5535 0.799917
\(868\) −6.35723 −0.215779
\(869\) −12.8034 −0.434324
\(870\) 0 0
\(871\) 45.4761 1.54090
\(872\) −48.9851 −1.65885
\(873\) 1.07761 0.0364716
\(874\) −1.27722 −0.0432025
\(875\) 0 0
\(876\) 2.66843 0.0901578
\(877\) −47.2905 −1.59689 −0.798444 0.602070i \(-0.794343\pi\)
−0.798444 + 0.602070i \(0.794343\pi\)
\(878\) 17.0460 0.575273
\(879\) −13.0587 −0.440460
\(880\) 0 0
\(881\) −4.11832 −0.138750 −0.0693748 0.997591i \(-0.522100\pi\)
−0.0693748 + 0.997591i \(0.522100\pi\)
\(882\) −0.748696 −0.0252099
\(883\) −9.63872 −0.324369 −0.162184 0.986760i \(-0.551854\pi\)
−0.162184 + 0.986760i \(0.551854\pi\)
\(884\) −6.63395 −0.223124
\(885\) 0 0
\(886\) 19.8745 0.667698
\(887\) −46.9180 −1.57535 −0.787677 0.616089i \(-0.788716\pi\)
−0.787677 + 0.616089i \(0.788716\pi\)
\(888\) −34.1082 −1.14460
\(889\) 37.5957 1.26092
\(890\) 0 0
\(891\) 4.50771 0.151014
\(892\) 9.74197 0.326185
\(893\) −10.9565 −0.366644
\(894\) −10.6873 −0.357437
\(895\) 0 0
\(896\) 18.9901 0.634414
\(897\) −4.03261 −0.134645
\(898\) −49.6980 −1.65844
\(899\) 52.7902 1.76065
\(900\) 0 0
\(901\) −30.8864 −1.02898
\(902\) 10.1262 0.337165
\(903\) −26.0899 −0.868219
\(904\) −27.5731 −0.917067
\(905\) 0 0
\(906\) 12.6178 0.419198
\(907\) 24.3781 0.809462 0.404731 0.914436i \(-0.367365\pi\)
0.404731 + 0.914436i \(0.367365\pi\)
\(908\) −10.7916 −0.358132
\(909\) −0.641823 −0.0212879
\(910\) 0 0
\(911\) 55.1931 1.82863 0.914315 0.405005i \(-0.132730\pi\)
0.914315 + 0.405005i \(0.132730\pi\)
\(912\) −4.22655 −0.139955
\(913\) −12.7259 −0.421165
\(914\) −20.6229 −0.682145
\(915\) 0 0
\(916\) 6.88384 0.227448
\(917\) 40.4196 1.33477
\(918\) −41.8397 −1.38092
\(919\) −36.4646 −1.20286 −0.601428 0.798927i \(-0.705401\pi\)
−0.601428 + 0.798927i \(0.705401\pi\)
\(920\) 0 0
\(921\) 47.1817 1.55469
\(922\) 38.0841 1.25423
\(923\) −1.89731 −0.0624508
\(924\) 1.48327 0.0487960
\(925\) 0 0
\(926\) 43.5493 1.43112
\(927\) −0.179061 −0.00588112
\(928\) 19.4705 0.639149
\(929\) 13.8456 0.454259 0.227130 0.973865i \(-0.427066\pi\)
0.227130 + 0.973865i \(0.427066\pi\)
\(930\) 0 0
\(931\) 0.538344 0.0176435
\(932\) −1.92132 −0.0629348
\(933\) −43.0454 −1.40924
\(934\) −41.7624 −1.36651
\(935\) 0 0
\(936\) 9.65351 0.315535
\(937\) 6.21716 0.203106 0.101553 0.994830i \(-0.467619\pi\)
0.101553 + 0.994830i \(0.467619\pi\)
\(938\) 54.5588 1.78141
\(939\) 11.7668 0.383995
\(940\) 0 0
\(941\) 8.07984 0.263395 0.131698 0.991290i \(-0.457957\pi\)
0.131698 + 0.991290i \(0.457957\pi\)
\(942\) −14.3774 −0.468439
\(943\) 8.04101 0.261851
\(944\) 15.5986 0.507691
\(945\) 0 0
\(946\) 8.73513 0.284004
\(947\) −33.0342 −1.07347 −0.536734 0.843752i \(-0.680342\pi\)
−0.536734 + 0.843752i \(0.680342\pi\)
\(948\) −6.91681 −0.224648
\(949\) −14.3361 −0.465368
\(950\) 0 0
\(951\) −19.2517 −0.624278
\(952\) −48.6096 −1.57545
\(953\) 1.71006 0.0553943 0.0276972 0.999616i \(-0.491183\pi\)
0.0276972 + 0.999616i \(0.491183\pi\)
\(954\) 7.35890 0.238253
\(955\) 0 0
\(956\) −1.82964 −0.0591747
\(957\) −12.3170 −0.398153
\(958\) −42.5160 −1.37363
\(959\) −20.0277 −0.646727
\(960\) 0 0
\(961\) 3.96456 0.127889
\(962\) 30.0030 0.967336
\(963\) −9.40234 −0.302986
\(964\) 9.51947 0.306602
\(965\) 0 0
\(966\) −4.83803 −0.155661
\(967\) −43.3980 −1.39558 −0.697792 0.716300i \(-0.745834\pi\)
−0.697792 + 0.716300i \(0.745834\pi\)
\(968\) −3.03308 −0.0974870
\(969\) 8.05315 0.258704
\(970\) 0 0
\(971\) −45.2884 −1.45337 −0.726687 0.686969i \(-0.758941\pi\)
−0.726687 + 0.686969i \(0.758941\pi\)
\(972\) −4.20414 −0.134848
\(973\) 45.5407 1.45997
\(974\) 26.2304 0.840476
\(975\) 0 0
\(976\) −23.4013 −0.749059
\(977\) −13.2323 −0.423340 −0.211670 0.977341i \(-0.567890\pi\)
−0.211670 + 0.977341i \(0.567890\pi\)
\(978\) −27.1935 −0.869551
\(979\) −11.9827 −0.382968
\(980\) 0 0
\(981\) −17.7103 −0.565445
\(982\) −28.9004 −0.922250
\(983\) 49.0140 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(984\) 33.4115 1.06512
\(985\) 0 0
\(986\) 66.0906 2.10475
\(987\) −41.5025 −1.32104
\(988\) −1.13651 −0.0361571
\(989\) 6.93639 0.220564
\(990\) 0 0
\(991\) 24.0457 0.763836 0.381918 0.924196i \(-0.375264\pi\)
0.381918 + 0.924196i \(0.375264\pi\)
\(992\) 12.8959 0.409444
\(993\) −28.2822 −0.897508
\(994\) −2.27625 −0.0721984
\(995\) 0 0
\(996\) −6.87496 −0.217841
\(997\) 36.6608 1.16106 0.580530 0.814239i \(-0.302845\pi\)
0.580530 + 0.814239i \(0.302845\pi\)
\(998\) −23.7896 −0.753045
\(999\) −46.0678 −1.45752
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 5225.2.a.u.1.11 15
5.4 even 2 5225.2.a.v.1.5 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.u.1.11 15 1.1 even 1 trivial
5225.2.a.v.1.5 yes 15 5.4 even 2