Properties

Label 5225.2.a.y.1.13
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) 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 1.13
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 \(-0.176543\pi\)
0.850098 + 0.526625i \(0.176543\pi\)
\(3\) 1.94792 1.12463 0.562316 0.826923i \(-0.309911\pi\)
0.562316 + 0.826923i \(0.309911\pi\)
\(4\) 3.78133 1.89066
\(5\) 0 0
\(6\) 4.68365 1.91209
\(7\) −0.665236 −0.251436 −0.125718 0.992066i \(-0.540123\pi\)
−0.125718 + 0.992066i \(0.540123\pi\)
\(8\) 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 \(-0.360764\pi\)
0.423607 + 0.905846i \(0.360764\pi\)
\(14\) −1.59952 −0.427490
\(15\) 0 0
\(16\) 2.73579 0.683948
\(17\) 4.49220 1.08952 0.544759 0.838592i \(-0.316621\pi\)
0.544759 + 0.838592i \(0.316621\pi\)
\(18\) 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 \(-0.493188\pi\)
0.0213996 + 0.999771i \(0.493188\pi\)
\(24\) 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 \(-0.284959\pi\)
0.625343 + 0.780350i \(0.284959\pi\)
\(38\) 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 \(-0.543257\pi\)
−0.135477 + 0.990781i \(0.543257\pi\)
\(44\) 3.78133 0.570057
\(45\) 0 0
\(46\) 0.493531 0.0727671
\(47\) −2.44032 −0.355957 −0.177978 0.984034i \(-0.556956\pi\)
−0.177978 + 0.984034i \(0.556956\pi\)
\(48\) 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 \(-0.573852\pi\)
−0.229937 + 0.973205i \(0.573852\pi\)
\(54\) −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 \(-0.652649\pi\)
−0.461390 + 0.887197i \(0.652649\pi\)
\(68\) 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 \(-0.529314\pi\)
−0.0919639 + 0.995762i \(0.529314\pi\)
\(74\) 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 \(-0.257458\pi\)
0.690347 + 0.723478i \(0.257458\pi\)
\(84\) −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 \(-0.359032\pi\)
0.428529 + 0.903528i \(0.359032\pi\)
\(98\) −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 \(-0.429899\pi\)
0.218453 + 0.975847i \(0.429899\pi\)
\(104\) 13.0835 1.28294
\(105\) 0 0
\(106\) −8.04991 −0.781876
\(107\) −16.9555 −1.63915 −0.819574 0.572973i \(-0.805790\pi\)
−0.819574 + 0.572973i \(0.805790\pi\)
\(108\) −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 \(-0.455883\pi\)
0.138153 + 0.990411i \(0.455883\pi\)
\(114\) 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 \(-0.641104\pi\)
−0.428914 + 0.903345i \(0.641104\pi\)
\(128\) −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 \(-0.571180\pi\)
−0.221759 + 0.975102i \(0.571180\pi\)
\(138\) 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 \(-0.762260\pi\)
−0.733810 + 0.679354i \(0.762260\pi\)
\(158\) −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 \(-0.612244\pi\)
−0.345361 + 0.938470i \(0.612244\pi\)
\(164\) 22.3375 1.74427
\(165\) 0 0
\(166\) 30.2448 2.34745
\(167\) 1.12696 0.0872065 0.0436033 0.999049i \(-0.486116\pi\)
0.0436033 + 0.999049i \(0.486116\pi\)
\(168\) −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 \(-0.633676\pi\)
−0.407719 + 0.913107i \(0.633676\pi\)
\(174\) 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 \(-0.240549\pi\)
0.727786 + 0.685804i \(0.240549\pi\)
\(194\) 20.2960 1.45717
\(195\) 0 0
\(196\) −24.7959 −1.77114
\(197\) −24.5070 −1.74605 −0.873024 0.487676i \(-0.837845\pi\)
−0.873024 + 0.487676i \(0.837845\pi\)
\(198\) 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 \(-0.376667\pi\)
0.377840 + 0.925871i \(0.376667\pi\)
\(224\) 1.32259 0.0883693
\(225\) 0 0
\(226\) 7.06228 0.469776
\(227\) 11.0481 0.733291 0.366646 0.930361i \(-0.380506\pi\)
0.366646 + 0.930361i \(0.380506\pi\)
\(228\) 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 \(-0.548084\pi\)
−0.150488 + 0.988612i \(0.548084\pi\)
\(234\) 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 \(-0.828403\pi\)
−0.858177 + 0.513353i \(0.828403\pi\)
\(258\) −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 \(-0.191688\pi\)
0.824088 + 0.566462i \(0.191688\pi\)
\(264\) 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 \(-0.557099\pi\)
−0.178420 + 0.983954i \(0.557099\pi\)
\(278\) −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 \(-0.592730\pi\)
−0.287216 + 0.957866i \(0.592730\pi\)
\(284\) 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 \(-0.416479\pi\)
0.259389 + 0.965773i \(0.416479\pi\)
\(294\) −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 \(-0.550016\pi\)
−0.156485 + 0.987680i \(0.550016\pi\)
\(308\) −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 \(-0.691049\pi\)
−0.564805 + 0.825225i \(0.691049\pi\)
\(314\) −44.2158 −2.49524
\(315\) 0 0
\(316\) −17.6545 −0.993141
\(317\) 7.52151 0.422450 0.211225 0.977437i \(-0.432255\pi\)
0.211225 + 0.977437i \(0.432255\pi\)
\(318\) −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 \(-0.308436\pi\)
0.566140 + 0.824309i \(0.308436\pi\)
\(338\) −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 \(-0.545390\pi\)
−0.142115 + 0.989850i \(0.545390\pi\)
\(348\) 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 \(-0.725938\pi\)
−0.651686 + 0.758489i \(0.725938\pi\)
\(354\) 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 \(-0.512317\pi\)
−0.0386844 + 0.999251i \(0.512317\pi\)
\(368\) 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 \(-0.684376\pi\)
−0.547383 + 0.836882i \(0.684376\pi\)
\(374\) 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 \(-0.505093\pi\)
−0.0159988 + 0.999872i \(0.505093\pi\)
\(384\) −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 \(-0.257774\pi\)
0.689628 + 0.724164i \(0.257774\pi\)
\(398\) 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 \(-0.461146\pi\)
0.121759 + 0.992560i \(0.461146\pi\)
\(434\) −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 \(-0.600323\pi\)
−0.309981 + 0.950743i \(0.600323\pi\)
\(444\) 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 \(-0.505688\pi\)
−0.0178700 + 0.999840i \(0.505688\pi\)
\(458\) −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 \(-0.386678\pi\)
0.348539 + 0.937294i \(0.386678\pi\)
\(464\) 0.114334 0.00530783
\(465\) 0 0
\(466\) −11.0465 −0.511717
\(467\) −8.46795 −0.391850 −0.195925 0.980619i \(-0.562771\pi\)
−0.195925 + 0.980619i \(0.562771\pi\)
\(468\) 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 \(-0.209697\pi\)
0.790738 + 0.612155i \(0.209697\pi\)
\(488\) −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 \(-0.458995\pi\)
0.128466 + 0.991714i \(0.458995\pi\)
\(504\) −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 \(-0.474509\pi\)
0.0799972 + 0.996795i \(0.474509\pi\)
\(524\) −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 \(-0.493918\pi\)
0.0191071 + 0.999817i \(0.493918\pi\)
\(548\) −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 \(-0.517065\pi\)
−0.0535870 + 0.998563i \(0.517065\pi\)
\(558\) 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 \(-0.312190\pi\)
0.556379 + 0.830929i \(0.312190\pi\)
\(564\) −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 \(-0.118295\pi\)
0.931735 + 0.363138i \(0.118295\pi\)
\(578\) 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 \(-0.360345\pi\)
0.424799 + 0.905288i \(0.360345\pi\)
\(588\) −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 \(-0.218353\pi\)
0.773802 + 0.633428i \(0.218353\pi\)
\(594\) −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 \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(608\) −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 \(-0.333002\pi\)
0.500901 + 0.865505i \(0.333002\pi\)
\(614\) −13.1851 −0.532109
\(615\) 0 0
\(616\) −2.84927 −0.114800
\(617\) −44.9932 −1.81136 −0.905678 0.423966i \(-0.860638\pi\)
−0.905678 + 0.423966i \(0.860638\pi\)
\(618\) 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 \(-0.354366\pi\)
0.441727 + 0.897150i \(0.354366\pi\)
\(644\) −0.516322 −0.0203459
\(645\) 0 0
\(646\) 10.8012 0.424969
\(647\) −25.5651 −1.00507 −0.502533 0.864558i \(-0.667599\pi\)
−0.502533 + 0.864558i \(0.667599\pi\)
\(648\) −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 \(-0.424897\pi\)
0.233761 + 0.972294i \(0.424897\pi\)
\(654\) −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 \(-0.129171\pi\)
0.918785 + 0.394758i \(0.129171\pi\)
\(674\) 49.9784 1.92510
\(675\) 0 0
\(676\) −13.8736 −0.533599
\(677\) 4.64806 0.178639 0.0893197 0.996003i \(-0.471531\pi\)
0.0893197 + 0.996003i \(0.471531\pi\)
\(678\) 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 \(-0.341213\pi\)
0.478410 + 0.878137i \(0.341213\pi\)
\(684\) 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 \(-0.457400\pi\)
0.133431 + 0.991058i \(0.457400\pi\)
\(728\) −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 \(-0.358816\pi\)
0.429141 + 0.903237i \(0.358816\pi\)
\(734\) −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 \(-0.326398\pi\)
0.518748 + 0.854927i \(0.326398\pi\)
\(744\) 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 \(-0.550848\pi\)
−0.159066 + 0.987268i \(0.550848\pi\)
\(758\) −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 \(-0.727595\pi\)
−0.655626 + 0.755086i \(0.727595\pi\)
\(774\) −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 \(-0.498241\pi\)
0.00552593 + 0.999985i \(0.498241\pi\)
\(788\) −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 \(-0.250092\pi\)
0.706903 + 0.707311i \(0.250092\pi\)
\(798\) −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 \(-0.773839\pi\)
−0.758033 + 0.652217i \(0.773839\pi\)
\(824\) 18.9918 0.661609
\(825\) 0 0
\(826\) −11.9005 −0.414071
\(827\) −14.5718 −0.506711 −0.253356 0.967373i \(-0.581534\pi\)
−0.253356 + 0.967373i \(0.581534\pi\)
\(828\) 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 \(-0.259272\pi\)
0.686213 + 0.727401i \(0.259272\pi\)
\(854\) 18.8295 0.644332
\(855\) 0 0
\(856\) −72.6220 −2.48217
\(857\) 8.21405 0.280587 0.140293 0.990110i \(-0.455195\pi\)
0.140293 + 0.990110i \(0.455195\pi\)
\(858\) 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 \(-0.494417\pi\)
0.0175376 + 0.999846i \(0.494417\pi\)
\(864\) 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 \(-0.355622\pi\)
0.438183 + 0.898886i \(0.355622\pi\)
\(878\) −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 \(-0.491583\pi\)
0.0264397 + 0.999650i \(0.491583\pi\)
\(884\) 51.8882 1.74519
\(885\) 0 0
\(886\) −31.3748 −1.05406
\(887\) 39.1857 1.31573 0.657864 0.753137i \(-0.271460\pi\)
0.657864 + 0.753137i \(0.271460\pi\)
\(888\) 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 \(-0.827972\pi\)
−0.857481 + 0.514516i \(0.827972\pi\)
\(908\) 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 \(-0.429513\pi\)
0.219635 + 0.975582i \(0.429513\pi\)
\(938\) 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 \(-0.328218\pi\)
0.513852 + 0.857879i \(0.328218\pi\)
\(948\) −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 \(-0.584810\pi\)
−0.263298 + 0.964715i \(0.584810\pi\)
\(954\) −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 \(-0.252243\pi\)
0.702106 + 0.712072i \(0.252243\pi\)
\(968\) 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 \(-0.298292\pi\)
0.592117 + 0.805852i \(0.298292\pi\)
\(978\) −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 \(-0.447112\pi\)
0.165389 + 0.986228i \(0.447112\pi\)
\(984\) 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 \(-0.804920\pi\)
−0.818004 + 0.575212i \(0.804920\pi\)
\(998\) −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.y.1.13 yes 15
5.4 even 2 5225.2.a.r.1.3 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.3 15 5.4 even 2
5225.2.a.y.1.13 yes 15 1.1 even 1 trivial