Properties

Label 5225.2.a.r.1.3
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
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: \(1\)
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) 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 1.3
Root \(2.40444\) 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)
 
\(f(q)\) \(=\) \(q-2.40444 q^{2} -1.94792 q^{3} +3.78133 q^{4} +4.68365 q^{6} +0.665236 q^{7} -4.28310 q^{8} +0.794388 q^{9} +O(q^{10})\) \(q-2.40444 q^{2} -1.94792 q^{3} +3.78133 q^{4} +4.68365 q^{6} +0.665236 q^{7} -4.28310 q^{8} +0.794388 q^{9} +1.00000 q^{11} -7.36572 q^{12} -3.05467 q^{13} -1.59952 q^{14} +2.73579 q^{16} -4.49220 q^{17} -1.91006 q^{18} +1.00000 q^{19} -1.29583 q^{21} -2.40444 q^{22} -0.205258 q^{23} +8.34313 q^{24} +7.34478 q^{26} +4.29635 q^{27} +2.51548 q^{28} +0.0417920 q^{29} +6.76694 q^{31} +1.98815 q^{32} -1.94792 q^{33} +10.8012 q^{34} +3.00384 q^{36} -7.60762 q^{37} -2.40444 q^{38} +5.95026 q^{39} +5.90733 q^{41} +3.11574 q^{42} +1.77676 q^{43} +3.78133 q^{44} +0.493531 q^{46} +2.44032 q^{47} -5.32910 q^{48} -6.55746 q^{49} +8.75045 q^{51} -11.5507 q^{52} +3.34793 q^{53} -10.3303 q^{54} -2.84927 q^{56} -1.94792 q^{57} -0.100486 q^{58} +7.44003 q^{59} -11.7720 q^{61} -16.2707 q^{62} +0.528456 q^{63} -10.2520 q^{64} +4.68365 q^{66} +7.55329 q^{67} -16.9865 q^{68} +0.399826 q^{69} +2.36041 q^{71} -3.40244 q^{72} +1.57148 q^{73} +18.2921 q^{74} +3.78133 q^{76} +0.665236 q^{77} -14.3070 q^{78} -4.66885 q^{79} -10.7521 q^{81} -14.2038 q^{82} -12.5787 q^{83} -4.89995 q^{84} -4.27211 q^{86} -0.0814074 q^{87} -4.28310 q^{88} +6.06754 q^{89} -2.03208 q^{91} -0.776148 q^{92} -13.1815 q^{93} -5.86759 q^{94} -3.87276 q^{96} -8.44105 q^{97} +15.7670 q^{98} +0.794388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9} + 15 q^{11} - 11 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 17 q^{17} - 22 q^{18} + 15 q^{19} + 6 q^{21} - 5 q^{22} - 26 q^{23} + q^{24} + 3 q^{26} - q^{27} - 46 q^{28} + 9 q^{29} + 14 q^{31} - 18 q^{32} - 4 q^{33} - 13 q^{34} + 12 q^{36} - 9 q^{37} - 5 q^{38} - 22 q^{39} + 4 q^{41} + 6 q^{42} - 28 q^{43} + 17 q^{44} + 27 q^{46} - 14 q^{47} + 4 q^{48} + 32 q^{49} - 40 q^{51} - 14 q^{52} - 3 q^{53} - 39 q^{54} + 34 q^{56} - 4 q^{57} - 26 q^{58} + q^{59} + 2 q^{61} + 3 q^{62} - 45 q^{63} + 5 q^{64} - q^{66} - 37 q^{67} - 26 q^{68} - 7 q^{69} - 7 q^{71} - 16 q^{72} - 42 q^{73} - 43 q^{74} + 17 q^{76} - 21 q^{77} + 64 q^{78} - 10 q^{79} + 31 q^{81} - 22 q^{82} - 14 q^{83} - 32 q^{84} + 37 q^{86} - 29 q^{87} - 9 q^{88} + 15 q^{89} - 22 q^{91} - 26 q^{92} + 18 q^{93} - 44 q^{94} + 71 q^{96} - 8 q^{97} + 10 q^{98} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.40444 −1.70020 −0.850098 0.526625i \(-0.823457\pi\)
−0.850098 + 0.526625i \(0.823457\pi\)
\(3\) −1.94792 −1.12463 −0.562316 0.826923i \(-0.690089\pi\)
−0.562316 + 0.826923i \(0.690089\pi\)
\(4\) 3.78133 1.89066
\(5\) 0 0
\(6\) 4.68365 1.91209
\(7\) 0.665236 0.251436 0.125718 0.992066i \(-0.459877\pi\)
0.125718 + 0.992066i \(0.459877\pi\)
\(8\) −4.28310 −1.51430
\(9\) 0.794388 0.264796
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −7.36572 −2.12630
\(13\) −3.05467 −0.847214 −0.423607 0.905846i \(-0.639236\pi\)
−0.423607 + 0.905846i \(0.639236\pi\)
\(14\) −1.59952 −0.427490
\(15\) 0 0
\(16\) 2.73579 0.683948
\(17\) −4.49220 −1.08952 −0.544759 0.838592i \(-0.683379\pi\)
−0.544759 + 0.838592i \(0.683379\pi\)
\(18\) −1.91006 −0.450205
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.29583 −0.282773
\(22\) −2.40444 −0.512628
\(23\) −0.205258 −0.0427993 −0.0213996 0.999771i \(-0.506812\pi\)
−0.0213996 + 0.999771i \(0.506812\pi\)
\(24\) 8.34313 1.70303
\(25\) 0 0
\(26\) 7.34478 1.44043
\(27\) 4.29635 0.826833
\(28\) 2.51548 0.475381
\(29\) 0.0417920 0.00776057 0.00388029 0.999992i \(-0.498765\pi\)
0.00388029 + 0.999992i \(0.498765\pi\)
\(30\) 0 0
\(31\) 6.76694 1.21538 0.607690 0.794175i \(-0.292097\pi\)
0.607690 + 0.794175i \(0.292097\pi\)
\(32\) 1.98815 0.351459
\(33\) −1.94792 −0.339089
\(34\) 10.8012 1.85240
\(35\) 0 0
\(36\) 3.00384 0.500641
\(37\) −7.60762 −1.25069 −0.625343 0.780350i \(-0.715041\pi\)
−0.625343 + 0.780350i \(0.715041\pi\)
\(38\) −2.40444 −0.390052
\(39\) 5.95026 0.952804
\(40\) 0 0
\(41\) 5.90733 0.922569 0.461285 0.887252i \(-0.347389\pi\)
0.461285 + 0.887252i \(0.347389\pi\)
\(42\) 3.11574 0.480769
\(43\) 1.77676 0.270953 0.135477 0.990781i \(-0.456743\pi\)
0.135477 + 0.990781i \(0.456743\pi\)
\(44\) 3.78133 0.570057
\(45\) 0 0
\(46\) 0.493531 0.0727671
\(47\) 2.44032 0.355957 0.177978 0.984034i \(-0.443044\pi\)
0.177978 + 0.984034i \(0.443044\pi\)
\(48\) −5.32910 −0.769189
\(49\) −6.55746 −0.936780
\(50\) 0 0
\(51\) 8.75045 1.22531
\(52\) −11.5507 −1.60180
\(53\) 3.34793 0.459874 0.229937 0.973205i \(-0.426148\pi\)
0.229937 + 0.973205i \(0.426148\pi\)
\(54\) −10.3303 −1.40578
\(55\) 0 0
\(56\) −2.84927 −0.380750
\(57\) −1.94792 −0.258008
\(58\) −0.100486 −0.0131945
\(59\) 7.44003 0.968610 0.484305 0.874899i \(-0.339073\pi\)
0.484305 + 0.874899i \(0.339073\pi\)
\(60\) 0 0
\(61\) −11.7720 −1.50725 −0.753623 0.657307i \(-0.771695\pi\)
−0.753623 + 0.657307i \(0.771695\pi\)
\(62\) −16.2707 −2.06638
\(63\) 0.528456 0.0665792
\(64\) −10.2520 −1.28150
\(65\) 0 0
\(66\) 4.68365 0.576518
\(67\) 7.55329 0.922781 0.461390 0.887197i \(-0.347351\pi\)
0.461390 + 0.887197i \(0.347351\pi\)
\(68\) −16.9865 −2.05991
\(69\) 0.399826 0.0481334
\(70\) 0 0
\(71\) 2.36041 0.280129 0.140065 0.990142i \(-0.455269\pi\)
0.140065 + 0.990142i \(0.455269\pi\)
\(72\) −3.40244 −0.400982
\(73\) 1.57148 0.183928 0.0919639 0.995762i \(-0.470686\pi\)
0.0919639 + 0.995762i \(0.470686\pi\)
\(74\) 18.2921 2.12641
\(75\) 0 0
\(76\) 3.78133 0.433748
\(77\) 0.665236 0.0758107
\(78\) −14.3070 −1.61995
\(79\) −4.66885 −0.525287 −0.262643 0.964893i \(-0.584594\pi\)
−0.262643 + 0.964893i \(0.584594\pi\)
\(80\) 0 0
\(81\) −10.7521 −1.19468
\(82\) −14.2038 −1.56855
\(83\) −12.5787 −1.38069 −0.690347 0.723478i \(-0.742542\pi\)
−0.690347 + 0.723478i \(0.742542\pi\)
\(84\) −4.89995 −0.534628
\(85\) 0 0
\(86\) −4.27211 −0.460673
\(87\) −0.0814074 −0.00872779
\(88\) −4.28310 −0.456580
\(89\) 6.06754 0.643158 0.321579 0.946883i \(-0.395786\pi\)
0.321579 + 0.946883i \(0.395786\pi\)
\(90\) 0 0
\(91\) −2.03208 −0.213020
\(92\) −0.776148 −0.0809191
\(93\) −13.1815 −1.36685
\(94\) −5.86759 −0.605196
\(95\) 0 0
\(96\) −3.87276 −0.395262
\(97\) −8.44105 −0.857059 −0.428529 0.903528i \(-0.640968\pi\)
−0.428529 + 0.903528i \(0.640968\pi\)
\(98\) 15.7670 1.59271
\(99\) 0.794388 0.0798390
\(100\) 0 0
\(101\) 13.8688 1.37999 0.689997 0.723813i \(-0.257612\pi\)
0.689997 + 0.723813i \(0.257612\pi\)
\(102\) −21.0399 −2.08326
\(103\) −4.43412 −0.436906 −0.218453 0.975847i \(-0.570101\pi\)
−0.218453 + 0.975847i \(0.570101\pi\)
\(104\) 13.0835 1.28294
\(105\) 0 0
\(106\) −8.04991 −0.781876
\(107\) 16.9555 1.63915 0.819574 0.572973i \(-0.194210\pi\)
0.819574 + 0.572973i \(0.194210\pi\)
\(108\) 16.2459 1.56326
\(109\) −9.27495 −0.888379 −0.444189 0.895933i \(-0.646508\pi\)
−0.444189 + 0.895933i \(0.646508\pi\)
\(110\) 0 0
\(111\) 14.8190 1.40656
\(112\) 1.81995 0.171969
\(113\) −2.93718 −0.276307 −0.138153 0.990411i \(-0.544117\pi\)
−0.138153 + 0.990411i \(0.544117\pi\)
\(114\) 4.68365 0.438664
\(115\) 0 0
\(116\) 0.158029 0.0146726
\(117\) −2.42660 −0.224339
\(118\) −17.8891 −1.64683
\(119\) −2.98838 −0.273944
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 28.3050 2.56261
\(123\) −11.5070 −1.03755
\(124\) 25.5880 2.29787
\(125\) 0 0
\(126\) −1.27064 −0.113198
\(127\) 9.66723 0.857828 0.428914 0.903345i \(-0.358896\pi\)
0.428914 + 0.903345i \(0.358896\pi\)
\(128\) 20.6739 1.82734
\(129\) −3.46098 −0.304722
\(130\) 0 0
\(131\) −9.82740 −0.858624 −0.429312 0.903156i \(-0.641244\pi\)
−0.429312 + 0.903156i \(0.641244\pi\)
\(132\) −7.36572 −0.641104
\(133\) 0.665236 0.0576833
\(134\) −18.1614 −1.56891
\(135\) 0 0
\(136\) 19.2405 1.64986
\(137\) 5.19124 0.443517 0.221759 0.975102i \(-0.428820\pi\)
0.221759 + 0.975102i \(0.428820\pi\)
\(138\) −0.961358 −0.0818362
\(139\) −12.5477 −1.06428 −0.532142 0.846655i \(-0.678613\pi\)
−0.532142 + 0.846655i \(0.678613\pi\)
\(140\) 0 0
\(141\) −4.75354 −0.400320
\(142\) −5.67546 −0.476274
\(143\) −3.05467 −0.255445
\(144\) 2.17328 0.181107
\(145\) 0 0
\(146\) −3.77853 −0.312713
\(147\) 12.7734 1.05353
\(148\) −28.7669 −2.36463
\(149\) −12.9080 −1.05746 −0.528731 0.848790i \(-0.677332\pi\)
−0.528731 + 0.848790i \(0.677332\pi\)
\(150\) 0 0
\(151\) 14.1347 1.15027 0.575134 0.818059i \(-0.304950\pi\)
0.575134 + 0.818059i \(0.304950\pi\)
\(152\) −4.28310 −0.347405
\(153\) −3.56855 −0.288500
\(154\) −1.59952 −0.128893
\(155\) 0 0
\(156\) 22.4999 1.80143
\(157\) 18.3892 1.46762 0.733810 0.679354i \(-0.237740\pi\)
0.733810 + 0.679354i \(0.237740\pi\)
\(158\) 11.2260 0.893090
\(159\) −6.52150 −0.517189
\(160\) 0 0
\(161\) −0.136545 −0.0107613
\(162\) 25.8528 2.03119
\(163\) 8.81855 0.690722 0.345361 0.938470i \(-0.387756\pi\)
0.345361 + 0.938470i \(0.387756\pi\)
\(164\) 22.3375 1.74427
\(165\) 0 0
\(166\) 30.2448 2.34745
\(167\) −1.12696 −0.0872065 −0.0436033 0.999049i \(-0.513884\pi\)
−0.0436033 + 0.999049i \(0.513884\pi\)
\(168\) 5.55015 0.428203
\(169\) −3.66897 −0.282228
\(170\) 0 0
\(171\) 0.794388 0.0607484
\(172\) 6.71851 0.512281
\(173\) 10.7254 0.815439 0.407719 0.913107i \(-0.366324\pi\)
0.407719 + 0.913107i \(0.366324\pi\)
\(174\) 0.195739 0.0148389
\(175\) 0 0
\(176\) 2.73579 0.206218
\(177\) −14.4926 −1.08933
\(178\) −14.5890 −1.09349
\(179\) −7.65684 −0.572299 −0.286150 0.958185i \(-0.592375\pi\)
−0.286150 + 0.958185i \(0.592375\pi\)
\(180\) 0 0
\(181\) 18.9423 1.40797 0.703986 0.710213i \(-0.251402\pi\)
0.703986 + 0.710213i \(0.251402\pi\)
\(182\) 4.88601 0.362175
\(183\) 22.9308 1.69510
\(184\) 0.879140 0.0648111
\(185\) 0 0
\(186\) 31.6940 2.32392
\(187\) −4.49220 −0.328502
\(188\) 9.22763 0.672994
\(189\) 2.85809 0.207895
\(190\) 0 0
\(191\) −7.50113 −0.542763 −0.271382 0.962472i \(-0.587481\pi\)
−0.271382 + 0.962472i \(0.587481\pi\)
\(192\) 19.9700 1.44121
\(193\) −20.2215 −1.45557 −0.727786 0.685804i \(-0.759451\pi\)
−0.727786 + 0.685804i \(0.759451\pi\)
\(194\) 20.2960 1.45717
\(195\) 0 0
\(196\) −24.7959 −1.77114
\(197\) 24.5070 1.74605 0.873024 0.487676i \(-0.162155\pi\)
0.873024 + 0.487676i \(0.162155\pi\)
\(198\) −1.91006 −0.135742
\(199\) 9.96226 0.706206 0.353103 0.935584i \(-0.385127\pi\)
0.353103 + 0.935584i \(0.385127\pi\)
\(200\) 0 0
\(201\) −14.7132 −1.03779
\(202\) −33.3466 −2.34626
\(203\) 0.0278015 0.00195129
\(204\) 33.0883 2.31665
\(205\) 0 0
\(206\) 10.6616 0.742826
\(207\) −0.163055 −0.0113331
\(208\) −8.35695 −0.579450
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 7.15413 0.492511 0.246255 0.969205i \(-0.420800\pi\)
0.246255 + 0.969205i \(0.420800\pi\)
\(212\) 12.6596 0.869468
\(213\) −4.59789 −0.315042
\(214\) −40.7684 −2.78687
\(215\) 0 0
\(216\) −18.4017 −1.25208
\(217\) 4.50162 0.305590
\(218\) 22.3010 1.51042
\(219\) −3.06111 −0.206851
\(220\) 0 0
\(221\) 13.7222 0.923056
\(222\) −35.6315 −2.39143
\(223\) −11.2847 −0.755680 −0.377840 0.925871i \(-0.623333\pi\)
−0.377840 + 0.925871i \(0.623333\pi\)
\(224\) 1.32259 0.0883693
\(225\) 0 0
\(226\) 7.06228 0.469776
\(227\) −11.0481 −0.733291 −0.366646 0.930361i \(-0.619494\pi\)
−0.366646 + 0.930361i \(0.619494\pi\)
\(228\) −7.36572 −0.487807
\(229\) −21.0370 −1.39016 −0.695081 0.718932i \(-0.744631\pi\)
−0.695081 + 0.718932i \(0.744631\pi\)
\(230\) 0 0
\(231\) −1.29583 −0.0852591
\(232\) −0.178999 −0.0117519
\(233\) 4.59419 0.300975 0.150488 0.988612i \(-0.451916\pi\)
0.150488 + 0.988612i \(0.451916\pi\)
\(234\) 5.83461 0.381420
\(235\) 0 0
\(236\) 28.1332 1.83132
\(237\) 9.09454 0.590754
\(238\) 7.18537 0.465758
\(239\) 10.8821 0.703904 0.351952 0.936018i \(-0.385518\pi\)
0.351952 + 0.936018i \(0.385518\pi\)
\(240\) 0 0
\(241\) 3.13644 0.202036 0.101018 0.994885i \(-0.467790\pi\)
0.101018 + 0.994885i \(0.467790\pi\)
\(242\) −2.40444 −0.154563
\(243\) 8.05519 0.516740
\(244\) −44.5137 −2.84970
\(245\) 0 0
\(246\) 27.6679 1.76404
\(247\) −3.05467 −0.194364
\(248\) −28.9835 −1.84045
\(249\) 24.5023 1.55277
\(250\) 0 0
\(251\) 12.9263 0.815899 0.407950 0.913004i \(-0.366244\pi\)
0.407950 + 0.913004i \(0.366244\pi\)
\(252\) 1.99827 0.125879
\(253\) −0.205258 −0.0129045
\(254\) −23.2443 −1.45848
\(255\) 0 0
\(256\) −29.2053 −1.82533
\(257\) 27.5153 1.71635 0.858177 0.513353i \(-0.171597\pi\)
0.858177 + 0.513353i \(0.171597\pi\)
\(258\) 8.32172 0.518088
\(259\) −5.06087 −0.314467
\(260\) 0 0
\(261\) 0.0331991 0.00205497
\(262\) 23.6294 1.45983
\(263\) −26.7289 −1.64818 −0.824088 0.566462i \(-0.808312\pi\)
−0.824088 + 0.566462i \(0.808312\pi\)
\(264\) 8.34313 0.513484
\(265\) 0 0
\(266\) −1.59952 −0.0980729
\(267\) −11.8191 −0.723316
\(268\) 28.5615 1.74467
\(269\) −10.8583 −0.662044 −0.331022 0.943623i \(-0.607393\pi\)
−0.331022 + 0.943623i \(0.607393\pi\)
\(270\) 0 0
\(271\) −6.99688 −0.425030 −0.212515 0.977158i \(-0.568166\pi\)
−0.212515 + 0.977158i \(0.568166\pi\)
\(272\) −12.2897 −0.745174
\(273\) 3.95833 0.239569
\(274\) −12.4820 −0.754066
\(275\) 0 0
\(276\) 1.51187 0.0910041
\(277\) 5.93902 0.356841 0.178420 0.983954i \(-0.442901\pi\)
0.178420 + 0.983954i \(0.442901\pi\)
\(278\) 30.1702 1.80949
\(279\) 5.37558 0.321828
\(280\) 0 0
\(281\) −9.12274 −0.544217 −0.272109 0.962267i \(-0.587721\pi\)
−0.272109 + 0.962267i \(0.587721\pi\)
\(282\) 11.4296 0.680622
\(283\) 9.66343 0.574431 0.287216 0.957866i \(-0.407270\pi\)
0.287216 + 0.957866i \(0.407270\pi\)
\(284\) 8.92549 0.529630
\(285\) 0 0
\(286\) 7.34478 0.434306
\(287\) 3.92977 0.231967
\(288\) 1.57937 0.0930650
\(289\) 3.17988 0.187052
\(290\) 0 0
\(291\) 16.4425 0.963876
\(292\) 5.94228 0.347746
\(293\) −8.88003 −0.518777 −0.259389 0.965773i \(-0.583521\pi\)
−0.259389 + 0.965773i \(0.583521\pi\)
\(294\) −30.7129 −1.79121
\(295\) 0 0
\(296\) 32.5842 1.89392
\(297\) 4.29635 0.249300
\(298\) 31.0364 1.79789
\(299\) 0.626996 0.0362601
\(300\) 0 0
\(301\) 1.18196 0.0681273
\(302\) −33.9861 −1.95568
\(303\) −27.0152 −1.55198
\(304\) 2.73579 0.156908
\(305\) 0 0
\(306\) 8.58037 0.490507
\(307\) 5.48367 0.312969 0.156485 0.987680i \(-0.449984\pi\)
0.156485 + 0.987680i \(0.449984\pi\)
\(308\) 2.51548 0.143333
\(309\) 8.63730 0.491359
\(310\) 0 0
\(311\) −18.4687 −1.04726 −0.523632 0.851945i \(-0.675423\pi\)
−0.523632 + 0.851945i \(0.675423\pi\)
\(312\) −25.4855 −1.44283
\(313\) 19.9848 1.12961 0.564805 0.825225i \(-0.308951\pi\)
0.564805 + 0.825225i \(0.308951\pi\)
\(314\) −44.2158 −2.49524
\(315\) 0 0
\(316\) −17.6545 −0.993141
\(317\) −7.52151 −0.422450 −0.211225 0.977437i \(-0.567745\pi\)
−0.211225 + 0.977437i \(0.567745\pi\)
\(318\) 15.6806 0.879322
\(319\) 0.0417920 0.00233990
\(320\) 0 0
\(321\) −33.0279 −1.84344
\(322\) 0.328315 0.0182963
\(323\) −4.49220 −0.249953
\(324\) −40.6573 −2.25874
\(325\) 0 0
\(326\) −21.2037 −1.17436
\(327\) 18.0668 0.999099
\(328\) −25.3017 −1.39705
\(329\) 1.62339 0.0895002
\(330\) 0 0
\(331\) 24.7743 1.36172 0.680860 0.732414i \(-0.261606\pi\)
0.680860 + 0.732414i \(0.261606\pi\)
\(332\) −47.5643 −2.61043
\(333\) −6.04341 −0.331177
\(334\) 2.70970 0.148268
\(335\) 0 0
\(336\) −3.54511 −0.193402
\(337\) −20.7859 −1.13228 −0.566140 0.824309i \(-0.691564\pi\)
−0.566140 + 0.824309i \(0.691564\pi\)
\(338\) 8.82182 0.479844
\(339\) 5.72139 0.310743
\(340\) 0 0
\(341\) 6.76694 0.366451
\(342\) −1.91006 −0.103284
\(343\) −9.01891 −0.486976
\(344\) −7.61003 −0.410305
\(345\) 0 0
\(346\) −25.7886 −1.38640
\(347\) 5.29461 0.284230 0.142115 0.989850i \(-0.454610\pi\)
0.142115 + 0.989850i \(0.454610\pi\)
\(348\) −0.307828 −0.0165013
\(349\) −5.79324 −0.310105 −0.155053 0.987906i \(-0.549555\pi\)
−0.155053 + 0.987906i \(0.549555\pi\)
\(350\) 0 0
\(351\) −13.1240 −0.700505
\(352\) 1.98815 0.105969
\(353\) 24.4881 1.30337 0.651686 0.758489i \(-0.274062\pi\)
0.651686 + 0.758489i \(0.274062\pi\)
\(354\) 34.8465 1.85207
\(355\) 0 0
\(356\) 22.9434 1.21600
\(357\) 5.82111 0.308086
\(358\) 18.4104 0.973021
\(359\) −1.29578 −0.0683884 −0.0341942 0.999415i \(-0.510886\pi\)
−0.0341942 + 0.999415i \(0.510886\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −45.5457 −2.39383
\(363\) −1.94792 −0.102239
\(364\) −7.68396 −0.402749
\(365\) 0 0
\(366\) −55.1358 −2.88199
\(367\) 1.48217 0.0773688 0.0386844 0.999251i \(-0.487683\pi\)
0.0386844 + 0.999251i \(0.487683\pi\)
\(368\) −0.561543 −0.0292725
\(369\) 4.69271 0.244293
\(370\) 0 0
\(371\) 2.22717 0.115629
\(372\) −49.8434 −2.58426
\(373\) 21.1434 1.09477 0.547383 0.836882i \(-0.315624\pi\)
0.547383 + 0.836882i \(0.315624\pi\)
\(374\) 10.8012 0.558518
\(375\) 0 0
\(376\) −10.4521 −0.539026
\(377\) −0.127661 −0.00657487
\(378\) −6.87210 −0.353463
\(379\) −7.09981 −0.364693 −0.182346 0.983234i \(-0.558369\pi\)
−0.182346 + 0.983234i \(0.558369\pi\)
\(380\) 0 0
\(381\) −18.8310 −0.964740
\(382\) 18.0360 0.922803
\(383\) 0.626205 0.0319976 0.0159988 0.999872i \(-0.494907\pi\)
0.0159988 + 0.999872i \(0.494907\pi\)
\(384\) −40.2712 −2.05508
\(385\) 0 0
\(386\) 48.6213 2.47476
\(387\) 1.41144 0.0717473
\(388\) −31.9184 −1.62041
\(389\) −0.689943 −0.0349815 −0.0174907 0.999847i \(-0.505568\pi\)
−0.0174907 + 0.999847i \(0.505568\pi\)
\(390\) 0 0
\(391\) 0.922061 0.0466306
\(392\) 28.0862 1.41857
\(393\) 19.1430 0.965636
\(394\) −58.9255 −2.96862
\(395\) 0 0
\(396\) 3.00384 0.150949
\(397\) −27.4815 −1.37926 −0.689628 0.724164i \(-0.742226\pi\)
−0.689628 + 0.724164i \(0.742226\pi\)
\(398\) −23.9537 −1.20069
\(399\) −1.29583 −0.0648725
\(400\) 0 0
\(401\) 27.6174 1.37915 0.689574 0.724215i \(-0.257798\pi\)
0.689574 + 0.724215i \(0.257798\pi\)
\(402\) 35.3770 1.76444
\(403\) −20.6708 −1.02969
\(404\) 52.4423 2.60910
\(405\) 0 0
\(406\) −0.0668471 −0.00331757
\(407\) −7.60762 −0.377096
\(408\) −37.4790 −1.85549
\(409\) −8.03757 −0.397432 −0.198716 0.980057i \(-0.563677\pi\)
−0.198716 + 0.980057i \(0.563677\pi\)
\(410\) 0 0
\(411\) −10.1121 −0.498793
\(412\) −16.7669 −0.826043
\(413\) 4.94938 0.243543
\(414\) 0.392055 0.0192685
\(415\) 0 0
\(416\) −6.07316 −0.297761
\(417\) 24.4419 1.19693
\(418\) −2.40444 −0.117605
\(419\) −17.7286 −0.866101 −0.433050 0.901370i \(-0.642563\pi\)
−0.433050 + 0.901370i \(0.642563\pi\)
\(420\) 0 0
\(421\) −15.5510 −0.757908 −0.378954 0.925416i \(-0.623716\pi\)
−0.378954 + 0.925416i \(0.623716\pi\)
\(422\) −17.2017 −0.837364
\(423\) 1.93856 0.0942559
\(424\) −14.3395 −0.696389
\(425\) 0 0
\(426\) 11.0553 0.535633
\(427\) −7.83114 −0.378975
\(428\) 64.1142 3.09908
\(429\) 5.95026 0.287281
\(430\) 0 0
\(431\) −34.1423 −1.64458 −0.822288 0.569072i \(-0.807303\pi\)
−0.822288 + 0.569072i \(0.807303\pi\)
\(432\) 11.7539 0.565511
\(433\) −5.06729 −0.243519 −0.121759 0.992560i \(-0.538854\pi\)
−0.121759 + 0.992560i \(0.538854\pi\)
\(434\) −10.8239 −0.519562
\(435\) 0 0
\(436\) −35.0716 −1.67963
\(437\) −0.205258 −0.00981883
\(438\) 7.36027 0.351687
\(439\) −23.7809 −1.13500 −0.567500 0.823374i \(-0.692089\pi\)
−0.567500 + 0.823374i \(0.692089\pi\)
\(440\) 0 0
\(441\) −5.20917 −0.248056
\(442\) −32.9942 −1.56938
\(443\) 13.0487 0.619961 0.309981 0.950743i \(-0.399677\pi\)
0.309981 + 0.950743i \(0.399677\pi\)
\(444\) 56.0356 2.65933
\(445\) 0 0
\(446\) 27.1334 1.28480
\(447\) 25.1437 1.18925
\(448\) −6.81998 −0.322214
\(449\) −26.5900 −1.25486 −0.627430 0.778673i \(-0.715893\pi\)
−0.627430 + 0.778673i \(0.715893\pi\)
\(450\) 0 0
\(451\) 5.90733 0.278165
\(452\) −11.1064 −0.522403
\(453\) −27.5333 −1.29363
\(454\) 26.5646 1.24674
\(455\) 0 0
\(456\) 8.34313 0.390703
\(457\) 0.764032 0.0357399 0.0178700 0.999840i \(-0.494312\pi\)
0.0178700 + 0.999840i \(0.494312\pi\)
\(458\) 50.5821 2.36355
\(459\) −19.3001 −0.900851
\(460\) 0 0
\(461\) −5.17462 −0.241006 −0.120503 0.992713i \(-0.538451\pi\)
−0.120503 + 0.992713i \(0.538451\pi\)
\(462\) 3.11574 0.144957
\(463\) −14.9993 −0.697078 −0.348539 0.937294i \(-0.613322\pi\)
−0.348539 + 0.937294i \(0.613322\pi\)
\(464\) 0.114334 0.00530783
\(465\) 0 0
\(466\) −11.0465 −0.511717
\(467\) 8.46795 0.391850 0.195925 0.980619i \(-0.437229\pi\)
0.195925 + 0.980619i \(0.437229\pi\)
\(468\) −9.17576 −0.424150
\(469\) 5.02472 0.232020
\(470\) 0 0
\(471\) −35.8207 −1.65053
\(472\) −31.8664 −1.46677
\(473\) 1.77676 0.0816954
\(474\) −21.8673 −1.00440
\(475\) 0 0
\(476\) −11.3000 −0.517936
\(477\) 2.65956 0.121773
\(478\) −26.1653 −1.19677
\(479\) 11.4343 0.522446 0.261223 0.965278i \(-0.415874\pi\)
0.261223 + 0.965278i \(0.415874\pi\)
\(480\) 0 0
\(481\) 23.2388 1.05960
\(482\) −7.54139 −0.343501
\(483\) 0.265979 0.0121025
\(484\) 3.78133 0.171879
\(485\) 0 0
\(486\) −19.3682 −0.878560
\(487\) −34.9001 −1.58148 −0.790738 0.612155i \(-0.790303\pi\)
−0.790738 + 0.612155i \(0.790303\pi\)
\(488\) 50.4205 2.28243
\(489\) −17.1778 −0.776808
\(490\) 0 0
\(491\) 15.5534 0.701917 0.350958 0.936391i \(-0.385856\pi\)
0.350958 + 0.936391i \(0.385856\pi\)
\(492\) −43.5117 −1.96166
\(493\) −0.187738 −0.00845529
\(494\) 7.34478 0.330457
\(495\) 0 0
\(496\) 18.5129 0.831256
\(497\) 1.57023 0.0704345
\(498\) −58.9144 −2.64002
\(499\) −29.1290 −1.30399 −0.651996 0.758222i \(-0.726068\pi\)
−0.651996 + 0.758222i \(0.726068\pi\)
\(500\) 0 0
\(501\) 2.19522 0.0980752
\(502\) −31.0804 −1.38719
\(503\) −5.76237 −0.256931 −0.128466 0.991714i \(-0.541005\pi\)
−0.128466 + 0.991714i \(0.541005\pi\)
\(504\) −2.26343 −0.100821
\(505\) 0 0
\(506\) 0.493531 0.0219401
\(507\) 7.14686 0.317403
\(508\) 36.5550 1.62186
\(509\) 20.3839 0.903502 0.451751 0.892144i \(-0.350799\pi\)
0.451751 + 0.892144i \(0.350799\pi\)
\(510\) 0 0
\(511\) 1.04541 0.0462460
\(512\) 28.8745 1.27608
\(513\) 4.29635 0.189689
\(514\) −66.1588 −2.91814
\(515\) 0 0
\(516\) −13.0871 −0.576128
\(517\) 2.44032 0.107325
\(518\) 12.1685 0.534655
\(519\) −20.8922 −0.917068
\(520\) 0 0
\(521\) 21.9184 0.960262 0.480131 0.877197i \(-0.340589\pi\)
0.480131 + 0.877197i \(0.340589\pi\)
\(522\) −0.0798251 −0.00349385
\(523\) −3.65894 −0.159994 −0.0799972 0.996795i \(-0.525491\pi\)
−0.0799972 + 0.996795i \(0.525491\pi\)
\(524\) −37.1606 −1.62337
\(525\) 0 0
\(526\) 64.2681 2.80222
\(527\) −30.3985 −1.32418
\(528\) −5.32910 −0.231919
\(529\) −22.9579 −0.998168
\(530\) 0 0
\(531\) 5.91027 0.256484
\(532\) 2.51548 0.109060
\(533\) −18.0450 −0.781614
\(534\) 28.4182 1.22978
\(535\) 0 0
\(536\) −32.3515 −1.39737
\(537\) 14.9149 0.643626
\(538\) 26.1082 1.12560
\(539\) −6.55746 −0.282450
\(540\) 0 0
\(541\) −29.6166 −1.27332 −0.636659 0.771145i \(-0.719684\pi\)
−0.636659 + 0.771145i \(0.719684\pi\)
\(542\) 16.8236 0.722635
\(543\) −36.8981 −1.58345
\(544\) −8.93118 −0.382921
\(545\) 0 0
\(546\) −9.51756 −0.407314
\(547\) −0.893755 −0.0382142 −0.0191071 0.999817i \(-0.506082\pi\)
−0.0191071 + 0.999817i \(0.506082\pi\)
\(548\) 19.6298 0.838542
\(549\) −9.35151 −0.399113
\(550\) 0 0
\(551\) 0.0417920 0.00178040
\(552\) −1.71249 −0.0728886
\(553\) −3.10589 −0.132076
\(554\) −14.2800 −0.606699
\(555\) 0 0
\(556\) −47.4470 −2.01220
\(557\) 2.52940 0.107174 0.0535870 0.998563i \(-0.482935\pi\)
0.0535870 + 0.998563i \(0.482935\pi\)
\(558\) −12.9253 −0.547170
\(559\) −5.42742 −0.229555
\(560\) 0 0
\(561\) 8.75045 0.369444
\(562\) 21.9351 0.925276
\(563\) −26.4031 −1.11276 −0.556379 0.830929i \(-0.687810\pi\)
−0.556379 + 0.830929i \(0.687810\pi\)
\(564\) −17.9747 −0.756871
\(565\) 0 0
\(566\) −23.2351 −0.976645
\(567\) −7.15270 −0.300385
\(568\) −10.1099 −0.424201
\(569\) 25.7456 1.07931 0.539656 0.841886i \(-0.318554\pi\)
0.539656 + 0.841886i \(0.318554\pi\)
\(570\) 0 0
\(571\) −32.1904 −1.34712 −0.673562 0.739131i \(-0.735237\pi\)
−0.673562 + 0.739131i \(0.735237\pi\)
\(572\) −11.5507 −0.482960
\(573\) 14.6116 0.610408
\(574\) −9.44889 −0.394389
\(575\) 0 0
\(576\) −8.14405 −0.339335
\(577\) −44.7621 −1.86347 −0.931735 0.363138i \(-0.881705\pi\)
−0.931735 + 0.363138i \(0.881705\pi\)
\(578\) −7.64582 −0.318024
\(579\) 39.3898 1.63698
\(580\) 0 0
\(581\) −8.36783 −0.347156
\(582\) −39.5350 −1.63878
\(583\) 3.34793 0.138657
\(584\) −6.73080 −0.278522
\(585\) 0 0
\(586\) 21.3515 0.882022
\(587\) −20.5841 −0.849597 −0.424799 0.905288i \(-0.639655\pi\)
−0.424799 + 0.905288i \(0.639655\pi\)
\(588\) 48.3004 1.99188
\(589\) 6.76694 0.278827
\(590\) 0 0
\(591\) −47.7376 −1.96366
\(592\) −20.8129 −0.855403
\(593\) −37.6866 −1.54760 −0.773802 0.633428i \(-0.781647\pi\)
−0.773802 + 0.633428i \(0.781647\pi\)
\(594\) −10.3303 −0.423858
\(595\) 0 0
\(596\) −48.8092 −1.99930
\(597\) −19.4057 −0.794222
\(598\) −1.50758 −0.0616493
\(599\) −29.3217 −1.19805 −0.599027 0.800729i \(-0.704446\pi\)
−0.599027 + 0.800729i \(0.704446\pi\)
\(600\) 0 0
\(601\) −17.4596 −0.712190 −0.356095 0.934450i \(-0.615892\pi\)
−0.356095 + 0.934450i \(0.615892\pi\)
\(602\) −2.84196 −0.115830
\(603\) 6.00024 0.244349
\(604\) 53.4481 2.17477
\(605\) 0 0
\(606\) 64.9565 2.63868
\(607\) −39.4241 −1.60017 −0.800087 0.599884i \(-0.795213\pi\)
−0.800087 + 0.599884i \(0.795213\pi\)
\(608\) 1.98815 0.0806302
\(609\) −0.0541551 −0.00219448
\(610\) 0 0
\(611\) −7.45437 −0.301571
\(612\) −13.4939 −0.545457
\(613\) −24.8034 −1.00180 −0.500901 0.865505i \(-0.666998\pi\)
−0.500901 + 0.865505i \(0.666998\pi\)
\(614\) −13.1851 −0.532109
\(615\) 0 0
\(616\) −2.84927 −0.114800
\(617\) 44.9932 1.81136 0.905678 0.423966i \(-0.139362\pi\)
0.905678 + 0.423966i \(0.139362\pi\)
\(618\) −20.7679 −0.835406
\(619\) 4.48348 0.180206 0.0901031 0.995932i \(-0.471280\pi\)
0.0901031 + 0.995932i \(0.471280\pi\)
\(620\) 0 0
\(621\) −0.881861 −0.0353879
\(622\) 44.4069 1.78055
\(623\) 4.03635 0.161713
\(624\) 16.2787 0.651668
\(625\) 0 0
\(626\) −48.0523 −1.92056
\(627\) −1.94792 −0.0777924
\(628\) 69.5358 2.77478
\(629\) 34.1750 1.36265
\(630\) 0 0
\(631\) 41.6458 1.65789 0.828947 0.559327i \(-0.188941\pi\)
0.828947 + 0.559327i \(0.188941\pi\)
\(632\) 19.9971 0.795444
\(633\) −13.9357 −0.553893
\(634\) 18.0850 0.718247
\(635\) 0 0
\(636\) −24.6600 −0.977831
\(637\) 20.0309 0.793653
\(638\) −0.100486 −0.00397829
\(639\) 1.87508 0.0741771
\(640\) 0 0
\(641\) −28.0569 −1.10818 −0.554090 0.832457i \(-0.686934\pi\)
−0.554090 + 0.832457i \(0.686934\pi\)
\(642\) 79.4136 3.13420
\(643\) −22.4021 −0.883453 −0.441727 0.897150i \(-0.645634\pi\)
−0.441727 + 0.897150i \(0.645634\pi\)
\(644\) −0.516322 −0.0203459
\(645\) 0 0
\(646\) 10.8012 0.424969
\(647\) 25.5651 1.00507 0.502533 0.864558i \(-0.332401\pi\)
0.502533 + 0.864558i \(0.332401\pi\)
\(648\) 46.0523 1.80911
\(649\) 7.44003 0.292047
\(650\) 0 0
\(651\) −8.76879 −0.343676
\(652\) 33.3458 1.30592
\(653\) −11.9470 −0.467523 −0.233761 0.972294i \(-0.575103\pi\)
−0.233761 + 0.972294i \(0.575103\pi\)
\(654\) −43.4406 −1.69866
\(655\) 0 0
\(656\) 16.1612 0.630989
\(657\) 1.24837 0.0487034
\(658\) −3.90333 −0.152168
\(659\) 17.3208 0.674724 0.337362 0.941375i \(-0.390465\pi\)
0.337362 + 0.941375i \(0.390465\pi\)
\(660\) 0 0
\(661\) 18.6524 0.725495 0.362748 0.931887i \(-0.381839\pi\)
0.362748 + 0.931887i \(0.381839\pi\)
\(662\) −59.5683 −2.31519
\(663\) −26.7298 −1.03810
\(664\) 53.8759 2.09079
\(665\) 0 0
\(666\) 14.5310 0.563065
\(667\) −0.00857814 −0.000332147 0
\(668\) −4.26139 −0.164878
\(669\) 21.9817 0.849862
\(670\) 0 0
\(671\) −11.7720 −0.454452
\(672\) −2.57630 −0.0993830
\(673\) −47.6707 −1.83757 −0.918785 0.394758i \(-0.870829\pi\)
−0.918785 + 0.394758i \(0.870829\pi\)
\(674\) 49.9784 1.92510
\(675\) 0 0
\(676\) −13.8736 −0.533599
\(677\) −4.64806 −0.178639 −0.0893197 0.996003i \(-0.528469\pi\)
−0.0893197 + 0.996003i \(0.528469\pi\)
\(678\) −13.7567 −0.528324
\(679\) −5.61529 −0.215495
\(680\) 0 0
\(681\) 21.5209 0.824683
\(682\) −16.2707 −0.623038
\(683\) −25.0058 −0.956820 −0.478410 0.878137i \(-0.658787\pi\)
−0.478410 + 0.878137i \(0.658787\pi\)
\(684\) 3.00384 0.114855
\(685\) 0 0
\(686\) 21.6854 0.827954
\(687\) 40.9783 1.56342
\(688\) 4.86084 0.185318
\(689\) −10.2268 −0.389612
\(690\) 0 0
\(691\) −19.9866 −0.760328 −0.380164 0.924919i \(-0.624132\pi\)
−0.380164 + 0.924919i \(0.624132\pi\)
\(692\) 40.5563 1.54172
\(693\) 0.528456 0.0200744
\(694\) −12.7306 −0.483246
\(695\) 0 0
\(696\) 0.348676 0.0132165
\(697\) −26.5369 −1.00516
\(698\) 13.9295 0.527239
\(699\) −8.94911 −0.338487
\(700\) 0 0
\(701\) −28.2573 −1.06726 −0.533632 0.845717i \(-0.679173\pi\)
−0.533632 + 0.845717i \(0.679173\pi\)
\(702\) 31.5558 1.19100
\(703\) −7.60762 −0.286927
\(704\) −10.2520 −0.386386
\(705\) 0 0
\(706\) −58.8803 −2.21599
\(707\) 9.22600 0.346980
\(708\) −54.8012 −2.05956
\(709\) −20.5903 −0.773287 −0.386643 0.922229i \(-0.626366\pi\)
−0.386643 + 0.922229i \(0.626366\pi\)
\(710\) 0 0
\(711\) −3.70888 −0.139094
\(712\) −25.9879 −0.973936
\(713\) −1.38897 −0.0520173
\(714\) −13.9965 −0.523806
\(715\) 0 0
\(716\) −28.9530 −1.08203
\(717\) −21.1974 −0.791632
\(718\) 3.11561 0.116274
\(719\) 36.7987 1.37236 0.686179 0.727433i \(-0.259287\pi\)
0.686179 + 0.727433i \(0.259287\pi\)
\(720\) 0 0
\(721\) −2.94973 −0.109854
\(722\) −2.40444 −0.0894840
\(723\) −6.10954 −0.227216
\(724\) 71.6272 2.66200
\(725\) 0 0
\(726\) 4.68365 0.173827
\(727\) −7.19540 −0.266862 −0.133431 0.991058i \(-0.542600\pi\)
−0.133431 + 0.991058i \(0.542600\pi\)
\(728\) 8.70360 0.322577
\(729\) 16.5655 0.613537
\(730\) 0 0
\(731\) −7.98156 −0.295208
\(732\) 86.7090 3.20486
\(733\) −23.2371 −0.858282 −0.429141 0.903237i \(-0.641184\pi\)
−0.429141 + 0.903237i \(0.641184\pi\)
\(734\) −3.56379 −0.131542
\(735\) 0 0
\(736\) −0.408084 −0.0150422
\(737\) 7.55329 0.278229
\(738\) −11.2833 −0.415345
\(739\) 8.05244 0.296214 0.148107 0.988971i \(-0.452682\pi\)
0.148107 + 0.988971i \(0.452682\pi\)
\(740\) 0 0
\(741\) 5.95026 0.218588
\(742\) −5.35509 −0.196592
\(743\) −28.2801 −1.03750 −0.518748 0.854927i \(-0.673602\pi\)
−0.518748 + 0.854927i \(0.673602\pi\)
\(744\) 56.4575 2.06983
\(745\) 0 0
\(746\) −50.8381 −1.86132
\(747\) −9.99239 −0.365603
\(748\) −16.9865 −0.621088
\(749\) 11.2794 0.412140
\(750\) 0 0
\(751\) 22.2029 0.810195 0.405098 0.914273i \(-0.367238\pi\)
0.405098 + 0.914273i \(0.367238\pi\)
\(752\) 6.67619 0.243456
\(753\) −25.1793 −0.917586
\(754\) 0.306953 0.0111786
\(755\) 0 0
\(756\) 10.8074 0.393061
\(757\) 8.75295 0.318131 0.159066 0.987268i \(-0.449152\pi\)
0.159066 + 0.987268i \(0.449152\pi\)
\(758\) 17.0711 0.620049
\(759\) 0.399826 0.0145128
\(760\) 0 0
\(761\) −13.4713 −0.488334 −0.244167 0.969733i \(-0.578514\pi\)
−0.244167 + 0.969733i \(0.578514\pi\)
\(762\) 45.2780 1.64025
\(763\) −6.17003 −0.223370
\(764\) −28.3643 −1.02618
\(765\) 0 0
\(766\) −1.50567 −0.0544022
\(767\) −22.7269 −0.820620
\(768\) 56.8896 2.05282
\(769\) 23.4526 0.845723 0.422862 0.906194i \(-0.361026\pi\)
0.422862 + 0.906194i \(0.361026\pi\)
\(770\) 0 0
\(771\) −53.5975 −1.93027
\(772\) −76.4640 −2.75200
\(773\) 36.4566 1.31125 0.655626 0.755086i \(-0.272405\pi\)
0.655626 + 0.755086i \(0.272405\pi\)
\(774\) −3.39371 −0.121984
\(775\) 0 0
\(776\) 36.1538 1.29785
\(777\) 9.85816 0.353660
\(778\) 1.65893 0.0594754
\(779\) 5.90733 0.211652
\(780\) 0 0
\(781\) 2.36041 0.0844621
\(782\) −2.21704 −0.0792812
\(783\) 0.179553 0.00641670
\(784\) −17.9398 −0.640708
\(785\) 0 0
\(786\) −46.0281 −1.64177
\(787\) −0.310043 −0.0110519 −0.00552593 0.999985i \(-0.501759\pi\)
−0.00552593 + 0.999985i \(0.501759\pi\)
\(788\) 92.6689 3.30119
\(789\) 52.0658 1.85359
\(790\) 0 0
\(791\) −1.95392 −0.0694734
\(792\) −3.40244 −0.120901
\(793\) 35.9595 1.27696
\(794\) 66.0775 2.34500
\(795\) 0 0
\(796\) 37.6706 1.33520
\(797\) −39.9134 −1.41381 −0.706903 0.707311i \(-0.749908\pi\)
−0.706903 + 0.707311i \(0.749908\pi\)
\(798\) 3.11574 0.110296
\(799\) −10.9624 −0.387821
\(800\) 0 0
\(801\) 4.81998 0.170306
\(802\) −66.4044 −2.34482
\(803\) 1.57148 0.0554563
\(804\) −55.6354 −1.96211
\(805\) 0 0
\(806\) 49.7017 1.75067
\(807\) 21.1511 0.744555
\(808\) −59.4012 −2.08973
\(809\) 35.5265 1.24904 0.624522 0.781007i \(-0.285294\pi\)
0.624522 + 0.781007i \(0.285294\pi\)
\(810\) 0 0
\(811\) −5.44459 −0.191185 −0.0955927 0.995421i \(-0.530475\pi\)
−0.0955927 + 0.995421i \(0.530475\pi\)
\(812\) 0.105127 0.00368923
\(813\) 13.6294 0.478003
\(814\) 18.2921 0.641137
\(815\) 0 0
\(816\) 23.9394 0.838046
\(817\) 1.77676 0.0621609
\(818\) 19.3258 0.675712
\(819\) −1.61426 −0.0564068
\(820\) 0 0
\(821\) −5.04790 −0.176173 −0.0880865 0.996113i \(-0.528075\pi\)
−0.0880865 + 0.996113i \(0.528075\pi\)
\(822\) 24.3139 0.848046
\(823\) 43.4928 1.51607 0.758033 0.652217i \(-0.226161\pi\)
0.758033 + 0.652217i \(0.226161\pi\)
\(824\) 18.9918 0.661609
\(825\) 0 0
\(826\) −11.9005 −0.414071
\(827\) 14.5718 0.506711 0.253356 0.967373i \(-0.418466\pi\)
0.253356 + 0.967373i \(0.418466\pi\)
\(828\) −0.616563 −0.0214271
\(829\) 42.0994 1.46217 0.731086 0.682285i \(-0.239014\pi\)
0.731086 + 0.682285i \(0.239014\pi\)
\(830\) 0 0
\(831\) −11.5687 −0.401315
\(832\) 31.3164 1.08570
\(833\) 29.4574 1.02064
\(834\) −58.7691 −2.03501
\(835\) 0 0
\(836\) 3.78133 0.130780
\(837\) 29.0732 1.00492
\(838\) 42.6274 1.47254
\(839\) 39.2894 1.35642 0.678211 0.734868i \(-0.262756\pi\)
0.678211 + 0.734868i \(0.262756\pi\)
\(840\) 0 0
\(841\) −28.9983 −0.999940
\(842\) 37.3914 1.28859
\(843\) 17.7704 0.612044
\(844\) 27.0521 0.931172
\(845\) 0 0
\(846\) −4.66115 −0.160253
\(847\) 0.665236 0.0228578
\(848\) 9.15925 0.314530
\(849\) −18.8236 −0.646023
\(850\) 0 0
\(851\) 1.56153 0.0535284
\(852\) −17.3861 −0.595639
\(853\) −40.0833 −1.37243 −0.686213 0.727401i \(-0.740728\pi\)
−0.686213 + 0.727401i \(0.740728\pi\)
\(854\) 18.8295 0.644332
\(855\) 0 0
\(856\) −72.6220 −2.48217
\(857\) −8.21405 −0.280587 −0.140293 0.990110i \(-0.544805\pi\)
−0.140293 + 0.990110i \(0.544805\pi\)
\(858\) −14.3070 −0.488434
\(859\) −47.6682 −1.62642 −0.813210 0.581971i \(-0.802282\pi\)
−0.813210 + 0.581971i \(0.802282\pi\)
\(860\) 0 0
\(861\) −7.65487 −0.260877
\(862\) 82.0930 2.79610
\(863\) −1.03040 −0.0350752 −0.0175376 0.999846i \(-0.505583\pi\)
−0.0175376 + 0.999846i \(0.505583\pi\)
\(864\) 8.54180 0.290598
\(865\) 0 0
\(866\) 12.1840 0.414029
\(867\) −6.19414 −0.210364
\(868\) 17.0221 0.577768
\(869\) −4.66885 −0.158380
\(870\) 0 0
\(871\) −23.0728 −0.781793
\(872\) 39.7255 1.34528
\(873\) −6.70547 −0.226946
\(874\) 0.493531 0.0166939
\(875\) 0 0
\(876\) −11.5751 −0.391086
\(877\) −25.9529 −0.876366 −0.438183 0.898886i \(-0.644378\pi\)
−0.438183 + 0.898886i \(0.644378\pi\)
\(878\) 57.1797 1.92972
\(879\) 17.2976 0.583433
\(880\) 0 0
\(881\) −14.8144 −0.499109 −0.249554 0.968361i \(-0.580284\pi\)
−0.249554 + 0.968361i \(0.580284\pi\)
\(882\) 12.5251 0.421743
\(883\) −1.57133 −0.0528794 −0.0264397 0.999650i \(-0.508417\pi\)
−0.0264397 + 0.999650i \(0.508417\pi\)
\(884\) 51.8882 1.74519
\(885\) 0 0
\(886\) −31.3748 −1.05406
\(887\) −39.1857 −1.31573 −0.657864 0.753137i \(-0.728540\pi\)
−0.657864 + 0.753137i \(0.728540\pi\)
\(888\) −63.4714 −2.12996
\(889\) 6.43099 0.215689
\(890\) 0 0
\(891\) −10.7521 −0.360209
\(892\) −42.6712 −1.42874
\(893\) 2.44032 0.0816620
\(894\) −60.4564 −2.02196
\(895\) 0 0
\(896\) 13.7531 0.459457
\(897\) −1.22134 −0.0407793
\(898\) 63.9340 2.13351
\(899\) 0.282804 0.00943204
\(900\) 0 0
\(901\) −15.0396 −0.501042
\(902\) −14.2038 −0.472935
\(903\) −2.30237 −0.0766181
\(904\) 12.5802 0.418412
\(905\) 0 0
\(906\) 66.2022 2.19942
\(907\) 51.6485 1.71496 0.857481 0.514516i \(-0.172028\pi\)
0.857481 + 0.514516i \(0.172028\pi\)
\(908\) −41.7767 −1.38641
\(909\) 11.0172 0.365417
\(910\) 0 0
\(911\) 20.7905 0.688822 0.344411 0.938819i \(-0.388079\pi\)
0.344411 + 0.938819i \(0.388079\pi\)
\(912\) −5.32910 −0.176464
\(913\) −12.5787 −0.416295
\(914\) −1.83707 −0.0607649
\(915\) 0 0
\(916\) −79.5477 −2.62833
\(917\) −6.53754 −0.215889
\(918\) 46.4059 1.53162
\(919\) 11.0153 0.363363 0.181681 0.983357i \(-0.441846\pi\)
0.181681 + 0.983357i \(0.441846\pi\)
\(920\) 0 0
\(921\) −10.6817 −0.351975
\(922\) 12.4421 0.409758
\(923\) −7.21028 −0.237329
\(924\) −4.89995 −0.161196
\(925\) 0 0
\(926\) 36.0650 1.18517
\(927\) −3.52241 −0.115691
\(928\) 0.0830888 0.00272752
\(929\) 22.8406 0.749375 0.374687 0.927151i \(-0.377750\pi\)
0.374687 + 0.927151i \(0.377750\pi\)
\(930\) 0 0
\(931\) −6.55746 −0.214912
\(932\) 17.3721 0.569044
\(933\) 35.9755 1.17779
\(934\) −20.3607 −0.666221
\(935\) 0 0
\(936\) 10.3934 0.339717
\(937\) −13.4463 −0.439270 −0.219635 0.975582i \(-0.570487\pi\)
−0.219635 + 0.975582i \(0.570487\pi\)
\(938\) −12.0816 −0.394479
\(939\) −38.9288 −1.27039
\(940\) 0 0
\(941\) −15.2843 −0.498253 −0.249126 0.968471i \(-0.580143\pi\)
−0.249126 + 0.968471i \(0.580143\pi\)
\(942\) 86.1288 2.80623
\(943\) −1.21253 −0.0394853
\(944\) 20.3544 0.662478
\(945\) 0 0
\(946\) −4.27211 −0.138898
\(947\) −31.6259 −1.02770 −0.513852 0.857879i \(-0.671782\pi\)
−0.513852 + 0.857879i \(0.671782\pi\)
\(948\) 34.3895 1.11692
\(949\) −4.80036 −0.155826
\(950\) 0 0
\(951\) 14.6513 0.475100
\(952\) 12.7995 0.414834
\(953\) 16.2564 0.526596 0.263298 0.964715i \(-0.415190\pi\)
0.263298 + 0.964715i \(0.415190\pi\)
\(954\) −6.39475 −0.207038
\(955\) 0 0
\(956\) 41.1488 1.33085
\(957\) −0.0814074 −0.00263153
\(958\) −27.4931 −0.888261
\(959\) 3.45340 0.111516
\(960\) 0 0
\(961\) 14.7915 0.477146
\(962\) −55.8763 −1.80152
\(963\) 13.4692 0.434040
\(964\) 11.8599 0.381983
\(965\) 0 0
\(966\) −0.639530 −0.0205765
\(967\) −43.6663 −1.40421 −0.702106 0.712072i \(-0.747757\pi\)
−0.702106 + 0.712072i \(0.747757\pi\)
\(968\) −4.28310 −0.137664
\(969\) 8.75045 0.281105
\(970\) 0 0
\(971\) −32.5513 −1.04462 −0.522311 0.852755i \(-0.674930\pi\)
−0.522311 + 0.852755i \(0.674930\pi\)
\(972\) 30.4593 0.976983
\(973\) −8.34720 −0.267599
\(974\) 83.9153 2.68882
\(975\) 0 0
\(976\) −32.2056 −1.03088
\(977\) −37.0156 −1.18423 −0.592117 0.805852i \(-0.701708\pi\)
−0.592117 + 0.805852i \(0.701708\pi\)
\(978\) 41.3030 1.32073
\(979\) 6.06754 0.193919
\(980\) 0 0
\(981\) −7.36791 −0.235239
\(982\) −37.3973 −1.19340
\(983\) −10.3708 −0.330777 −0.165389 0.986228i \(-0.552888\pi\)
−0.165389 + 0.986228i \(0.552888\pi\)
\(984\) 49.2856 1.57117
\(985\) 0 0
\(986\) 0.451405 0.0143757
\(987\) −3.16223 −0.100655
\(988\) −11.5507 −0.367478
\(989\) −0.364694 −0.0115966
\(990\) 0 0
\(991\) −20.4472 −0.649525 −0.324763 0.945796i \(-0.605284\pi\)
−0.324763 + 0.945796i \(0.605284\pi\)
\(992\) 13.4537 0.427156
\(993\) −48.2583 −1.53143
\(994\) −3.77552 −0.119752
\(995\) 0 0
\(996\) 92.6514 2.93577
\(997\) 51.6575 1.63601 0.818004 0.575212i \(-0.195080\pi\)
0.818004 + 0.575212i \(0.195080\pi\)
\(998\) 70.0389 2.21704
\(999\) −32.6850 −1.03411
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.r.1.3 15
5.4 even 2 5225.2.a.y.1.13 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.3 15 1.1 even 1 trivial
5225.2.a.y.1.13 yes 15 5.4 even 2