Properties

Label 5225.2.a.t.1.4
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: \( x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 \) 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 1.4
Root \(1.67859\) 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)
 
\(f(q)\) \(=\) \(q-1.67859 q^{2} -2.69106 q^{3} +0.817650 q^{4} +4.51717 q^{6} +2.51534 q^{7} +1.98468 q^{8} +4.24179 q^{9} +O(q^{10})\) \(q-1.67859 q^{2} -2.69106 q^{3} +0.817650 q^{4} +4.51717 q^{6} +2.51534 q^{7} +1.98468 q^{8} +4.24179 q^{9} -1.00000 q^{11} -2.20034 q^{12} +3.42325 q^{13} -4.22221 q^{14} -4.96675 q^{16} +1.55883 q^{17} -7.12021 q^{18} -1.00000 q^{19} -6.76892 q^{21} +1.67859 q^{22} +8.59423 q^{23} -5.34088 q^{24} -5.74623 q^{26} -3.34173 q^{27} +2.05666 q^{28} -0.732149 q^{29} +9.53542 q^{31} +4.36776 q^{32} +2.69106 q^{33} -2.61663 q^{34} +3.46830 q^{36} +7.74223 q^{37} +1.67859 q^{38} -9.21217 q^{39} +3.46517 q^{41} +11.3622 q^{42} -5.41881 q^{43} -0.817650 q^{44} -14.4262 q^{46} +12.0999 q^{47} +13.3658 q^{48} -0.673078 q^{49} -4.19491 q^{51} +2.79902 q^{52} +5.37822 q^{53} +5.60938 q^{54} +4.99213 q^{56} +2.69106 q^{57} +1.22898 q^{58} +7.67030 q^{59} +10.7718 q^{61} -16.0060 q^{62} +10.6695 q^{63} +2.60184 q^{64} -4.51717 q^{66} +3.64371 q^{67} +1.27458 q^{68} -23.1276 q^{69} +5.47291 q^{71} +8.41858 q^{72} +8.79408 q^{73} -12.9960 q^{74} -0.817650 q^{76} -2.51534 q^{77} +15.4634 q^{78} -2.28936 q^{79} -3.73259 q^{81} -5.81659 q^{82} -6.13533 q^{83} -5.53460 q^{84} +9.09593 q^{86} +1.97026 q^{87} -1.98468 q^{88} -11.0612 q^{89} +8.61064 q^{91} +7.02707 q^{92} -25.6604 q^{93} -20.3107 q^{94} -11.7539 q^{96} +16.8289 q^{97} +1.12982 q^{98} -4.24179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 11 q^{7} - 3 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 11 q^{7} - 3 q^{8} + 19 q^{9} - 15 q^{11} + 11 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} - 5 q^{17} + 12 q^{18} - 15 q^{19} + 10 q^{21} + q^{22} + 26 q^{23} - 11 q^{24} - 5 q^{26} + 19 q^{27} + 18 q^{28} + 7 q^{29} - 10 q^{31} - 12 q^{32} - 4 q^{33} + 17 q^{34} + 24 q^{36} + 31 q^{37} + q^{38} + 4 q^{39} + 2 q^{41} + 22 q^{42} + 26 q^{43} - 13 q^{44} - 23 q^{46} + 26 q^{47} + 46 q^{48} + 12 q^{49} + 12 q^{51} + 16 q^{52} + 21 q^{53} + 5 q^{54} - 10 q^{56} - 4 q^{57} + 34 q^{58} - 11 q^{59} + 20 q^{61} + 25 q^{62} + 27 q^{63} - 3 q^{64} + q^{66} + 41 q^{67} - 6 q^{68} + q^{69} + 25 q^{71} + 54 q^{72} - 6 q^{73} - 9 q^{74} - 13 q^{76} - 11 q^{77} + 28 q^{78} - 6 q^{79} + 43 q^{81} + 18 q^{82} - 20 q^{83} - 14 q^{84} + 35 q^{86} + 29 q^{87} + 3 q^{88} - 3 q^{89} + 30 q^{91} + 54 q^{92} + 2 q^{93} - 28 q^{94} - 61 q^{96} + 28 q^{97} - 2 q^{98} - 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.67859 −1.18694 −0.593470 0.804856i \(-0.702242\pi\)
−0.593470 + 0.804856i \(0.702242\pi\)
\(3\) −2.69106 −1.55368 −0.776841 0.629696i \(-0.783179\pi\)
−0.776841 + 0.629696i \(0.783179\pi\)
\(4\) 0.817650 0.408825
\(5\) 0 0
\(6\) 4.51717 1.84413
\(7\) 2.51534 0.950708 0.475354 0.879795i \(-0.342320\pi\)
0.475354 + 0.879795i \(0.342320\pi\)
\(8\) 1.98468 0.701689
\(9\) 4.24179 1.41393
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.20034 −0.635184
\(13\) 3.42325 0.949440 0.474720 0.880137i \(-0.342549\pi\)
0.474720 + 0.880137i \(0.342549\pi\)
\(14\) −4.22221 −1.12843
\(15\) 0 0
\(16\) −4.96675 −1.24169
\(17\) 1.55883 0.378072 0.189036 0.981970i \(-0.439464\pi\)
0.189036 + 0.981970i \(0.439464\pi\)
\(18\) −7.12021 −1.67825
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.76892 −1.47710
\(22\) 1.67859 0.357876
\(23\) 8.59423 1.79202 0.896011 0.444033i \(-0.146453\pi\)
0.896011 + 0.444033i \(0.146453\pi\)
\(24\) −5.34088 −1.09020
\(25\) 0 0
\(26\) −5.74623 −1.12693
\(27\) −3.34173 −0.643116
\(28\) 2.05666 0.388673
\(29\) −0.732149 −0.135957 −0.0679783 0.997687i \(-0.521655\pi\)
−0.0679783 + 0.997687i \(0.521655\pi\)
\(30\) 0 0
\(31\) 9.53542 1.71261 0.856306 0.516470i \(-0.172754\pi\)
0.856306 + 0.516470i \(0.172754\pi\)
\(32\) 4.36776 0.772118
\(33\) 2.69106 0.468453
\(34\) −2.61663 −0.448749
\(35\) 0 0
\(36\) 3.46830 0.578050
\(37\) 7.74223 1.27281 0.636407 0.771353i \(-0.280420\pi\)
0.636407 + 0.771353i \(0.280420\pi\)
\(38\) 1.67859 0.272303
\(39\) −9.21217 −1.47513
\(40\) 0 0
\(41\) 3.46517 0.541169 0.270585 0.962696i \(-0.412783\pi\)
0.270585 + 0.962696i \(0.412783\pi\)
\(42\) 11.3622 1.75323
\(43\) −5.41881 −0.826360 −0.413180 0.910649i \(-0.635582\pi\)
−0.413180 + 0.910649i \(0.635582\pi\)
\(44\) −0.817650 −0.123265
\(45\) 0 0
\(46\) −14.4262 −2.12702
\(47\) 12.0999 1.76495 0.882473 0.470363i \(-0.155877\pi\)
0.882473 + 0.470363i \(0.155877\pi\)
\(48\) 13.3658 1.92919
\(49\) −0.673078 −0.0961540
\(50\) 0 0
\(51\) −4.19491 −0.587404
\(52\) 2.79902 0.388155
\(53\) 5.37822 0.738755 0.369378 0.929279i \(-0.379571\pi\)
0.369378 + 0.929279i \(0.379571\pi\)
\(54\) 5.60938 0.763340
\(55\) 0 0
\(56\) 4.99213 0.667101
\(57\) 2.69106 0.356439
\(58\) 1.22898 0.161372
\(59\) 7.67030 0.998588 0.499294 0.866433i \(-0.333593\pi\)
0.499294 + 0.866433i \(0.333593\pi\)
\(60\) 0 0
\(61\) 10.7718 1.37918 0.689592 0.724198i \(-0.257790\pi\)
0.689592 + 0.724198i \(0.257790\pi\)
\(62\) −16.0060 −2.03277
\(63\) 10.6695 1.34423
\(64\) 2.60184 0.325230
\(65\) 0 0
\(66\) −4.51717 −0.556025
\(67\) 3.64371 0.445150 0.222575 0.974916i \(-0.428554\pi\)
0.222575 + 0.974916i \(0.428554\pi\)
\(68\) 1.27458 0.154565
\(69\) −23.1276 −2.78423
\(70\) 0 0
\(71\) 5.47291 0.649515 0.324757 0.945797i \(-0.394717\pi\)
0.324757 + 0.945797i \(0.394717\pi\)
\(72\) 8.41858 0.992139
\(73\) 8.79408 1.02927 0.514634 0.857410i \(-0.327928\pi\)
0.514634 + 0.857410i \(0.327928\pi\)
\(74\) −12.9960 −1.51075
\(75\) 0 0
\(76\) −0.817650 −0.0937909
\(77\) −2.51534 −0.286649
\(78\) 15.4634 1.75089
\(79\) −2.28936 −0.257573 −0.128787 0.991672i \(-0.541108\pi\)
−0.128787 + 0.991672i \(0.541108\pi\)
\(80\) 0 0
\(81\) −3.73259 −0.414732
\(82\) −5.81659 −0.642335
\(83\) −6.13533 −0.673440 −0.336720 0.941605i \(-0.609318\pi\)
−0.336720 + 0.941605i \(0.609318\pi\)
\(84\) −5.53460 −0.603875
\(85\) 0 0
\(86\) 9.09593 0.980839
\(87\) 1.97026 0.211234
\(88\) −1.98468 −0.211567
\(89\) −11.0612 −1.17249 −0.586244 0.810135i \(-0.699394\pi\)
−0.586244 + 0.810135i \(0.699394\pi\)
\(90\) 0 0
\(91\) 8.61064 0.902640
\(92\) 7.02707 0.732623
\(93\) −25.6604 −2.66085
\(94\) −20.3107 −2.09488
\(95\) 0 0
\(96\) −11.7539 −1.19963
\(97\) 16.8289 1.70872 0.854360 0.519681i \(-0.173949\pi\)
0.854360 + 0.519681i \(0.173949\pi\)
\(98\) 1.12982 0.114129
\(99\) −4.24179 −0.426316
\(100\) 0 0
\(101\) 13.5575 1.34902 0.674512 0.738264i \(-0.264354\pi\)
0.674512 + 0.738264i \(0.264354\pi\)
\(102\) 7.04151 0.697213
\(103\) 1.44011 0.141898 0.0709492 0.997480i \(-0.477397\pi\)
0.0709492 + 0.997480i \(0.477397\pi\)
\(104\) 6.79405 0.666212
\(105\) 0 0
\(106\) −9.02780 −0.876858
\(107\) −0.862637 −0.0833943 −0.0416971 0.999130i \(-0.513276\pi\)
−0.0416971 + 0.999130i \(0.513276\pi\)
\(108\) −2.73236 −0.262922
\(109\) −6.16384 −0.590389 −0.295194 0.955437i \(-0.595384\pi\)
−0.295194 + 0.955437i \(0.595384\pi\)
\(110\) 0 0
\(111\) −20.8348 −1.97755
\(112\) −12.4930 −1.18048
\(113\) −8.07745 −0.759863 −0.379931 0.925015i \(-0.624052\pi\)
−0.379931 + 0.925015i \(0.624052\pi\)
\(114\) −4.51717 −0.423072
\(115\) 0 0
\(116\) −0.598642 −0.0555825
\(117\) 14.5207 1.34244
\(118\) −12.8753 −1.18526
\(119\) 3.92099 0.359436
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −18.0813 −1.63701
\(123\) −9.32498 −0.840805
\(124\) 7.79663 0.700158
\(125\) 0 0
\(126\) −17.9097 −1.59553
\(127\) 9.60028 0.851887 0.425944 0.904750i \(-0.359942\pi\)
0.425944 + 0.904750i \(0.359942\pi\)
\(128\) −13.1029 −1.15815
\(129\) 14.5823 1.28390
\(130\) 0 0
\(131\) −13.4000 −1.17077 −0.585383 0.810757i \(-0.699056\pi\)
−0.585383 + 0.810757i \(0.699056\pi\)
\(132\) 2.20034 0.191515
\(133\) −2.51534 −0.218107
\(134\) −6.11628 −0.528366
\(135\) 0 0
\(136\) 3.09378 0.265289
\(137\) −16.4284 −1.40358 −0.701788 0.712386i \(-0.747615\pi\)
−0.701788 + 0.712386i \(0.747615\pi\)
\(138\) 38.8216 3.30471
\(139\) 3.28762 0.278852 0.139426 0.990232i \(-0.455474\pi\)
0.139426 + 0.990232i \(0.455474\pi\)
\(140\) 0 0
\(141\) −32.5614 −2.74217
\(142\) −9.18675 −0.770935
\(143\) −3.42325 −0.286267
\(144\) −21.0679 −1.75566
\(145\) 0 0
\(146\) −14.7616 −1.22168
\(147\) 1.81129 0.149393
\(148\) 6.33043 0.520358
\(149\) −12.7310 −1.04297 −0.521483 0.853262i \(-0.674621\pi\)
−0.521483 + 0.853262i \(0.674621\pi\)
\(150\) 0 0
\(151\) 9.48167 0.771608 0.385804 0.922581i \(-0.373924\pi\)
0.385804 + 0.922581i \(0.373924\pi\)
\(152\) −1.98468 −0.160978
\(153\) 6.61224 0.534568
\(154\) 4.22221 0.340235
\(155\) 0 0
\(156\) −7.53233 −0.603069
\(157\) 5.08953 0.406189 0.203095 0.979159i \(-0.434900\pi\)
0.203095 + 0.979159i \(0.434900\pi\)
\(158\) 3.84289 0.305724
\(159\) −14.4731 −1.14779
\(160\) 0 0
\(161\) 21.6174 1.70369
\(162\) 6.26547 0.492261
\(163\) −20.9778 −1.64311 −0.821554 0.570130i \(-0.806893\pi\)
−0.821554 + 0.570130i \(0.806893\pi\)
\(164\) 2.83330 0.221243
\(165\) 0 0
\(166\) 10.2987 0.799333
\(167\) −22.4548 −1.73760 −0.868801 0.495161i \(-0.835109\pi\)
−0.868801 + 0.495161i \(0.835109\pi\)
\(168\) −13.4341 −1.03646
\(169\) −1.28133 −0.0985638
\(170\) 0 0
\(171\) −4.24179 −0.324378
\(172\) −4.43069 −0.337837
\(173\) −17.1327 −1.30258 −0.651288 0.758831i \(-0.725771\pi\)
−0.651288 + 0.758831i \(0.725771\pi\)
\(174\) −3.30724 −0.250721
\(175\) 0 0
\(176\) 4.96675 0.374383
\(177\) −20.6412 −1.55149
\(178\) 18.5672 1.39167
\(179\) 16.0445 1.19922 0.599611 0.800292i \(-0.295322\pi\)
0.599611 + 0.800292i \(0.295322\pi\)
\(180\) 0 0
\(181\) −8.22647 −0.611469 −0.305734 0.952117i \(-0.598902\pi\)
−0.305734 + 0.952117i \(0.598902\pi\)
\(182\) −14.4537 −1.07138
\(183\) −28.9874 −2.14281
\(184\) 17.0568 1.25744
\(185\) 0 0
\(186\) 43.0731 3.15827
\(187\) −1.55883 −0.113993
\(188\) 9.89345 0.721554
\(189\) −8.40558 −0.611416
\(190\) 0 0
\(191\) −18.2373 −1.31961 −0.659804 0.751438i \(-0.729360\pi\)
−0.659804 + 0.751438i \(0.729360\pi\)
\(192\) −7.00169 −0.505304
\(193\) 2.94251 0.211806 0.105903 0.994376i \(-0.466227\pi\)
0.105903 + 0.994376i \(0.466227\pi\)
\(194\) −28.2488 −2.02815
\(195\) 0 0
\(196\) −0.550342 −0.0393102
\(197\) 10.5011 0.748169 0.374084 0.927395i \(-0.377957\pi\)
0.374084 + 0.927395i \(0.377957\pi\)
\(198\) 7.12021 0.506011
\(199\) 14.2424 1.00962 0.504809 0.863231i \(-0.331563\pi\)
0.504809 + 0.863231i \(0.331563\pi\)
\(200\) 0 0
\(201\) −9.80544 −0.691622
\(202\) −22.7575 −1.60121
\(203\) −1.84160 −0.129255
\(204\) −3.42996 −0.240146
\(205\) 0 0
\(206\) −2.41735 −0.168425
\(207\) 36.4549 2.53379
\(208\) −17.0024 −1.17891
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −18.0956 −1.24575 −0.622876 0.782321i \(-0.714036\pi\)
−0.622876 + 0.782321i \(0.714036\pi\)
\(212\) 4.39750 0.302021
\(213\) −14.7279 −1.00914
\(214\) 1.44801 0.0989839
\(215\) 0 0
\(216\) −6.63225 −0.451267
\(217\) 23.9848 1.62819
\(218\) 10.3465 0.700755
\(219\) −23.6654 −1.59916
\(220\) 0 0
\(221\) 5.33628 0.358957
\(222\) 34.9729 2.34723
\(223\) 14.5632 0.975226 0.487613 0.873060i \(-0.337868\pi\)
0.487613 + 0.873060i \(0.337868\pi\)
\(224\) 10.9864 0.734059
\(225\) 0 0
\(226\) 13.5587 0.901911
\(227\) 18.1940 1.20758 0.603790 0.797144i \(-0.293657\pi\)
0.603790 + 0.797144i \(0.293657\pi\)
\(228\) 2.20034 0.145721
\(229\) −16.4481 −1.08692 −0.543460 0.839435i \(-0.682886\pi\)
−0.543460 + 0.839435i \(0.682886\pi\)
\(230\) 0 0
\(231\) 6.76892 0.445362
\(232\) −1.45308 −0.0953993
\(233\) −28.3928 −1.86007 −0.930036 0.367468i \(-0.880225\pi\)
−0.930036 + 0.367468i \(0.880225\pi\)
\(234\) −24.3743 −1.59340
\(235\) 0 0
\(236\) 6.27162 0.408248
\(237\) 6.16080 0.400187
\(238\) −6.58172 −0.426629
\(239\) 13.8912 0.898545 0.449272 0.893395i \(-0.351683\pi\)
0.449272 + 0.893395i \(0.351683\pi\)
\(240\) 0 0
\(241\) 5.63681 0.363099 0.181549 0.983382i \(-0.441889\pi\)
0.181549 + 0.983382i \(0.441889\pi\)
\(242\) −1.67859 −0.107904
\(243\) 20.0698 1.28748
\(244\) 8.80753 0.563845
\(245\) 0 0
\(246\) 15.6528 0.997985
\(247\) −3.42325 −0.217816
\(248\) 18.9247 1.20172
\(249\) 16.5105 1.04631
\(250\) 0 0
\(251\) −25.4939 −1.60916 −0.804581 0.593843i \(-0.797610\pi\)
−0.804581 + 0.593843i \(0.797610\pi\)
\(252\) 8.72394 0.549557
\(253\) −8.59423 −0.540315
\(254\) −16.1149 −1.01114
\(255\) 0 0
\(256\) 16.7907 1.04942
\(257\) −8.64725 −0.539401 −0.269700 0.962944i \(-0.586925\pi\)
−0.269700 + 0.962944i \(0.586925\pi\)
\(258\) −24.4777 −1.52391
\(259\) 19.4743 1.21007
\(260\) 0 0
\(261\) −3.10562 −0.192233
\(262\) 22.4931 1.38963
\(263\) 23.3907 1.44233 0.721165 0.692763i \(-0.243607\pi\)
0.721165 + 0.692763i \(0.243607\pi\)
\(264\) 5.34088 0.328708
\(265\) 0 0
\(266\) 4.22221 0.258880
\(267\) 29.7664 1.82167
\(268\) 2.97928 0.181988
\(269\) −11.5163 −0.702160 −0.351080 0.936345i \(-0.614185\pi\)
−0.351080 + 0.936345i \(0.614185\pi\)
\(270\) 0 0
\(271\) 28.9414 1.75807 0.879033 0.476761i \(-0.158189\pi\)
0.879033 + 0.476761i \(0.158189\pi\)
\(272\) −7.74233 −0.469448
\(273\) −23.1717 −1.40242
\(274\) 27.5765 1.66596
\(275\) 0 0
\(276\) −18.9103 −1.13826
\(277\) 8.30493 0.498995 0.249497 0.968375i \(-0.419735\pi\)
0.249497 + 0.968375i \(0.419735\pi\)
\(278\) −5.51855 −0.330980
\(279\) 40.4472 2.42151
\(280\) 0 0
\(281\) 12.5712 0.749937 0.374968 0.927038i \(-0.377654\pi\)
0.374968 + 0.927038i \(0.377654\pi\)
\(282\) 54.6571 3.25478
\(283\) −0.750970 −0.0446405 −0.0223203 0.999751i \(-0.507105\pi\)
−0.0223203 + 0.999751i \(0.507105\pi\)
\(284\) 4.47492 0.265538
\(285\) 0 0
\(286\) 5.74623 0.339781
\(287\) 8.71608 0.514494
\(288\) 18.5271 1.09172
\(289\) −14.5700 −0.857061
\(290\) 0 0
\(291\) −45.2877 −2.65481
\(292\) 7.19048 0.420791
\(293\) 6.93157 0.404947 0.202473 0.979288i \(-0.435102\pi\)
0.202473 + 0.979288i \(0.435102\pi\)
\(294\) −3.04041 −0.177320
\(295\) 0 0
\(296\) 15.3658 0.893120
\(297\) 3.34173 0.193907
\(298\) 21.3701 1.23794
\(299\) 29.4202 1.70142
\(300\) 0 0
\(301\) −13.6301 −0.785627
\(302\) −15.9158 −0.915851
\(303\) −36.4841 −2.09596
\(304\) 4.96675 0.284863
\(305\) 0 0
\(306\) −11.0992 −0.634500
\(307\) 14.9513 0.853315 0.426658 0.904413i \(-0.359691\pi\)
0.426658 + 0.904413i \(0.359691\pi\)
\(308\) −2.05666 −0.117189
\(309\) −3.87543 −0.220465
\(310\) 0 0
\(311\) −17.7728 −1.00780 −0.503901 0.863761i \(-0.668102\pi\)
−0.503901 + 0.863761i \(0.668102\pi\)
\(312\) −18.2832 −1.03508
\(313\) −6.99172 −0.395195 −0.197598 0.980283i \(-0.563314\pi\)
−0.197598 + 0.980283i \(0.563314\pi\)
\(314\) −8.54322 −0.482122
\(315\) 0 0
\(316\) −1.87189 −0.105302
\(317\) 22.1516 1.24416 0.622079 0.782955i \(-0.286288\pi\)
0.622079 + 0.782955i \(0.286288\pi\)
\(318\) 24.2943 1.36236
\(319\) 0.732149 0.0409925
\(320\) 0 0
\(321\) 2.32141 0.129568
\(322\) −36.2866 −2.02218
\(323\) −1.55883 −0.0867357
\(324\) −3.05195 −0.169553
\(325\) 0 0
\(326\) 35.2130 1.95027
\(327\) 16.5872 0.917277
\(328\) 6.87725 0.379732
\(329\) 30.4352 1.67795
\(330\) 0 0
\(331\) 30.1394 1.65661 0.828306 0.560276i \(-0.189305\pi\)
0.828306 + 0.560276i \(0.189305\pi\)
\(332\) −5.01655 −0.275319
\(333\) 32.8409 1.79967
\(334\) 37.6922 2.06243
\(335\) 0 0
\(336\) 33.6195 1.83409
\(337\) −11.1211 −0.605806 −0.302903 0.953021i \(-0.597956\pi\)
−0.302903 + 0.953021i \(0.597956\pi\)
\(338\) 2.15082 0.116989
\(339\) 21.7369 1.18059
\(340\) 0 0
\(341\) −9.53542 −0.516372
\(342\) 7.12021 0.385017
\(343\) −19.3004 −1.04212
\(344\) −10.7546 −0.579848
\(345\) 0 0
\(346\) 28.7587 1.54608
\(347\) −30.2831 −1.62568 −0.812841 0.582486i \(-0.802080\pi\)
−0.812841 + 0.582486i \(0.802080\pi\)
\(348\) 1.61098 0.0863575
\(349\) −23.0063 −1.23150 −0.615749 0.787942i \(-0.711146\pi\)
−0.615749 + 0.787942i \(0.711146\pi\)
\(350\) 0 0
\(351\) −11.4396 −0.610600
\(352\) −4.36776 −0.232802
\(353\) 15.0380 0.800393 0.400196 0.916429i \(-0.368942\pi\)
0.400196 + 0.916429i \(0.368942\pi\)
\(354\) 34.6481 1.84152
\(355\) 0 0
\(356\) −9.04421 −0.479342
\(357\) −10.5516 −0.558450
\(358\) −26.9320 −1.42340
\(359\) −18.6503 −0.984324 −0.492162 0.870504i \(-0.663793\pi\)
−0.492162 + 0.870504i \(0.663793\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 13.8088 0.725777
\(363\) −2.69106 −0.141244
\(364\) 7.04049 0.369022
\(365\) 0 0
\(366\) 48.6579 2.54339
\(367\) 19.4659 1.01611 0.508057 0.861324i \(-0.330364\pi\)
0.508057 + 0.861324i \(0.330364\pi\)
\(368\) −42.6854 −2.22513
\(369\) 14.6985 0.765175
\(370\) 0 0
\(371\) 13.5280 0.702341
\(372\) −20.9812 −1.08782
\(373\) −30.6816 −1.58863 −0.794317 0.607503i \(-0.792171\pi\)
−0.794317 + 0.607503i \(0.792171\pi\)
\(374\) 2.61663 0.135303
\(375\) 0 0
\(376\) 24.0143 1.23844
\(377\) −2.50633 −0.129083
\(378\) 14.1095 0.725713
\(379\) −0.614145 −0.0315465 −0.0157733 0.999876i \(-0.505021\pi\)
−0.0157733 + 0.999876i \(0.505021\pi\)
\(380\) 0 0
\(381\) −25.8349 −1.32356
\(382\) 30.6129 1.56629
\(383\) 10.0988 0.516026 0.258013 0.966141i \(-0.416932\pi\)
0.258013 + 0.966141i \(0.416932\pi\)
\(384\) 35.2607 1.79939
\(385\) 0 0
\(386\) −4.93925 −0.251401
\(387\) −22.9854 −1.16842
\(388\) 13.7602 0.698567
\(389\) −20.1964 −1.02400 −0.511998 0.858987i \(-0.671095\pi\)
−0.511998 + 0.858987i \(0.671095\pi\)
\(390\) 0 0
\(391\) 13.3970 0.677514
\(392\) −1.33584 −0.0674702
\(393\) 36.0602 1.81900
\(394\) −17.6269 −0.888031
\(395\) 0 0
\(396\) −3.46830 −0.174289
\(397\) −17.2043 −0.863458 −0.431729 0.902003i \(-0.642096\pi\)
−0.431729 + 0.902003i \(0.642096\pi\)
\(398\) −23.9071 −1.19836
\(399\) 6.76892 0.338870
\(400\) 0 0
\(401\) 24.6530 1.23111 0.615555 0.788094i \(-0.288932\pi\)
0.615555 + 0.788094i \(0.288932\pi\)
\(402\) 16.4593 0.820914
\(403\) 32.6422 1.62602
\(404\) 11.0853 0.551515
\(405\) 0 0
\(406\) 3.09129 0.153418
\(407\) −7.74223 −0.383768
\(408\) −8.32553 −0.412175
\(409\) −16.7084 −0.826179 −0.413089 0.910690i \(-0.635550\pi\)
−0.413089 + 0.910690i \(0.635550\pi\)
\(410\) 0 0
\(411\) 44.2099 2.18071
\(412\) 1.17751 0.0580116
\(413\) 19.2934 0.949366
\(414\) −61.1927 −3.00746
\(415\) 0 0
\(416\) 14.9520 0.733080
\(417\) −8.84717 −0.433248
\(418\) −1.67859 −0.0821023
\(419\) −24.4684 −1.19536 −0.597679 0.801736i \(-0.703910\pi\)
−0.597679 + 0.801736i \(0.703910\pi\)
\(420\) 0 0
\(421\) 0.669036 0.0326068 0.0163034 0.999867i \(-0.494810\pi\)
0.0163034 + 0.999867i \(0.494810\pi\)
\(422\) 30.3750 1.47863
\(423\) 51.3251 2.49551
\(424\) 10.6740 0.518376
\(425\) 0 0
\(426\) 24.7221 1.19779
\(427\) 27.0946 1.31120
\(428\) −0.705335 −0.0340937
\(429\) 9.21217 0.444768
\(430\) 0 0
\(431\) −5.91908 −0.285112 −0.142556 0.989787i \(-0.545532\pi\)
−0.142556 + 0.989787i \(0.545532\pi\)
\(432\) 16.5975 0.798549
\(433\) 21.3626 1.02662 0.513310 0.858203i \(-0.328419\pi\)
0.513310 + 0.858203i \(0.328419\pi\)
\(434\) −40.2605 −1.93257
\(435\) 0 0
\(436\) −5.03986 −0.241366
\(437\) −8.59423 −0.411118
\(438\) 39.7243 1.89810
\(439\) 7.64851 0.365044 0.182522 0.983202i \(-0.441574\pi\)
0.182522 + 0.983202i \(0.441574\pi\)
\(440\) 0 0
\(441\) −2.85506 −0.135955
\(442\) −8.95740 −0.426060
\(443\) 29.0913 1.38217 0.691084 0.722774i \(-0.257133\pi\)
0.691084 + 0.722774i \(0.257133\pi\)
\(444\) −17.0355 −0.808471
\(445\) 0 0
\(446\) −24.4456 −1.15753
\(447\) 34.2599 1.62044
\(448\) 6.54450 0.309199
\(449\) 25.3358 1.19567 0.597835 0.801619i \(-0.296028\pi\)
0.597835 + 0.801619i \(0.296028\pi\)
\(450\) 0 0
\(451\) −3.46517 −0.163169
\(452\) −6.60453 −0.310651
\(453\) −25.5157 −1.19883
\(454\) −30.5402 −1.43332
\(455\) 0 0
\(456\) 5.34088 0.250110
\(457\) 0.335009 0.0156711 0.00783553 0.999969i \(-0.497506\pi\)
0.00783553 + 0.999969i \(0.497506\pi\)
\(458\) 27.6095 1.29011
\(459\) −5.20919 −0.243144
\(460\) 0 0
\(461\) 7.47620 0.348201 0.174101 0.984728i \(-0.444298\pi\)
0.174101 + 0.984728i \(0.444298\pi\)
\(462\) −11.3622 −0.528618
\(463\) −6.86115 −0.318865 −0.159432 0.987209i \(-0.550966\pi\)
−0.159432 + 0.987209i \(0.550966\pi\)
\(464\) 3.63640 0.168816
\(465\) 0 0
\(466\) 47.6597 2.20779
\(467\) −34.6722 −1.60444 −0.802218 0.597031i \(-0.796347\pi\)
−0.802218 + 0.597031i \(0.796347\pi\)
\(468\) 11.8729 0.548824
\(469\) 9.16516 0.423208
\(470\) 0 0
\(471\) −13.6962 −0.631089
\(472\) 15.2231 0.700699
\(473\) 5.41881 0.249157
\(474\) −10.3414 −0.474998
\(475\) 0 0
\(476\) 3.20600 0.146947
\(477\) 22.8133 1.04455
\(478\) −23.3175 −1.06652
\(479\) −39.0116 −1.78249 −0.891243 0.453527i \(-0.850166\pi\)
−0.891243 + 0.453527i \(0.850166\pi\)
\(480\) 0 0
\(481\) 26.5036 1.20846
\(482\) −9.46186 −0.430976
\(483\) −58.1736 −2.64699
\(484\) 0.817650 0.0371659
\(485\) 0 0
\(486\) −33.6889 −1.52816
\(487\) 20.8684 0.945637 0.472818 0.881160i \(-0.343237\pi\)
0.472818 + 0.881160i \(0.343237\pi\)
\(488\) 21.3785 0.967758
\(489\) 56.4525 2.55287
\(490\) 0 0
\(491\) 7.91384 0.357146 0.178573 0.983927i \(-0.442852\pi\)
0.178573 + 0.983927i \(0.442852\pi\)
\(492\) −7.62457 −0.343742
\(493\) −1.14130 −0.0514015
\(494\) 5.74623 0.258535
\(495\) 0 0
\(496\) −47.3600 −2.12653
\(497\) 13.7662 0.617499
\(498\) −27.7143 −1.24191
\(499\) 8.30479 0.371773 0.185887 0.982571i \(-0.440484\pi\)
0.185887 + 0.982571i \(0.440484\pi\)
\(500\) 0 0
\(501\) 60.4271 2.69968
\(502\) 42.7937 1.90998
\(503\) 9.04940 0.403493 0.201746 0.979438i \(-0.435338\pi\)
0.201746 + 0.979438i \(0.435338\pi\)
\(504\) 21.1756 0.943235
\(505\) 0 0
\(506\) 14.4262 0.641321
\(507\) 3.44813 0.153137
\(508\) 7.84967 0.348273
\(509\) −29.7251 −1.31754 −0.658771 0.752344i \(-0.728923\pi\)
−0.658771 + 0.752344i \(0.728923\pi\)
\(510\) 0 0
\(511\) 22.1201 0.978534
\(512\) −1.97879 −0.0874510
\(513\) 3.34173 0.147541
\(514\) 14.5151 0.640236
\(515\) 0 0
\(516\) 11.9232 0.524891
\(517\) −12.0999 −0.532151
\(518\) −32.6893 −1.43629
\(519\) 46.1051 2.02379
\(520\) 0 0
\(521\) 12.5266 0.548800 0.274400 0.961616i \(-0.411521\pi\)
0.274400 + 0.961616i \(0.411521\pi\)
\(522\) 5.21305 0.228169
\(523\) −2.60523 −0.113919 −0.0569594 0.998376i \(-0.518141\pi\)
−0.0569594 + 0.998376i \(0.518141\pi\)
\(524\) −10.9565 −0.478638
\(525\) 0 0
\(526\) −39.2632 −1.71196
\(527\) 14.8641 0.647491
\(528\) −13.3658 −0.581672
\(529\) 50.8608 2.21134
\(530\) 0 0
\(531\) 32.5358 1.41193
\(532\) −2.05666 −0.0891677
\(533\) 11.8622 0.513808
\(534\) −49.9654 −2.16222
\(535\) 0 0
\(536\) 7.23159 0.312357
\(537\) −43.1766 −1.86321
\(538\) 19.3311 0.833421
\(539\) 0.673078 0.0289915
\(540\) 0 0
\(541\) −27.5043 −1.18250 −0.591251 0.806488i \(-0.701366\pi\)
−0.591251 + 0.806488i \(0.701366\pi\)
\(542\) −48.5807 −2.08672
\(543\) 22.1379 0.950029
\(544\) 6.80860 0.291917
\(545\) 0 0
\(546\) 38.8957 1.66458
\(547\) 26.7101 1.14204 0.571021 0.820936i \(-0.306548\pi\)
0.571021 + 0.820936i \(0.306548\pi\)
\(548\) −13.4327 −0.573817
\(549\) 45.6916 1.95007
\(550\) 0 0
\(551\) 0.732149 0.0311906
\(552\) −45.9007 −1.95367
\(553\) −5.75851 −0.244877
\(554\) −13.9405 −0.592276
\(555\) 0 0
\(556\) 2.68812 0.114002
\(557\) −7.56102 −0.320371 −0.160185 0.987087i \(-0.551209\pi\)
−0.160185 + 0.987087i \(0.551209\pi\)
\(558\) −67.8941 −2.87419
\(559\) −18.5500 −0.784579
\(560\) 0 0
\(561\) 4.19491 0.177109
\(562\) −21.1019 −0.890129
\(563\) −18.1554 −0.765158 −0.382579 0.923923i \(-0.624964\pi\)
−0.382579 + 0.923923i \(0.624964\pi\)
\(564\) −26.6238 −1.12107
\(565\) 0 0
\(566\) 1.26057 0.0529856
\(567\) −9.38871 −0.394289
\(568\) 10.8620 0.455757
\(569\) −7.77097 −0.325776 −0.162888 0.986645i \(-0.552081\pi\)
−0.162888 + 0.986645i \(0.552081\pi\)
\(570\) 0 0
\(571\) 17.3236 0.724972 0.362486 0.931989i \(-0.381928\pi\)
0.362486 + 0.931989i \(0.381928\pi\)
\(572\) −2.79902 −0.117033
\(573\) 49.0777 2.05025
\(574\) −14.6307 −0.610673
\(575\) 0 0
\(576\) 11.0364 0.459852
\(577\) −31.3795 −1.30635 −0.653174 0.757208i \(-0.726563\pi\)
−0.653174 + 0.757208i \(0.726563\pi\)
\(578\) 24.4571 1.01728
\(579\) −7.91846 −0.329080
\(580\) 0 0
\(581\) −15.4324 −0.640245
\(582\) 76.0192 3.15110
\(583\) −5.37822 −0.222743
\(584\) 17.4534 0.722227
\(585\) 0 0
\(586\) −11.6352 −0.480647
\(587\) 6.86741 0.283448 0.141724 0.989906i \(-0.454735\pi\)
0.141724 + 0.989906i \(0.454735\pi\)
\(588\) 1.48100 0.0610755
\(589\) −9.53542 −0.392900
\(590\) 0 0
\(591\) −28.2589 −1.16242
\(592\) −38.4537 −1.58044
\(593\) −33.7885 −1.38753 −0.693763 0.720203i \(-0.744048\pi\)
−0.693763 + 0.720203i \(0.744048\pi\)
\(594\) −5.60938 −0.230156
\(595\) 0 0
\(596\) −10.4095 −0.426390
\(597\) −38.3272 −1.56863
\(598\) −49.3844 −2.01948
\(599\) −39.9085 −1.63062 −0.815308 0.579028i \(-0.803432\pi\)
−0.815308 + 0.579028i \(0.803432\pi\)
\(600\) 0 0
\(601\) −35.6108 −1.45259 −0.726297 0.687381i \(-0.758760\pi\)
−0.726297 + 0.687381i \(0.758760\pi\)
\(602\) 22.8793 0.932492
\(603\) 15.4559 0.629411
\(604\) 7.75269 0.315452
\(605\) 0 0
\(606\) 61.2417 2.48777
\(607\) −12.0761 −0.490153 −0.245077 0.969504i \(-0.578813\pi\)
−0.245077 + 0.969504i \(0.578813\pi\)
\(608\) −4.36776 −0.177136
\(609\) 4.95586 0.200821
\(610\) 0 0
\(611\) 41.4209 1.67571
\(612\) 5.40650 0.218545
\(613\) −12.3728 −0.499732 −0.249866 0.968280i \(-0.580387\pi\)
−0.249866 + 0.968280i \(0.580387\pi\)
\(614\) −25.0970 −1.01283
\(615\) 0 0
\(616\) −4.99213 −0.201139
\(617\) 9.96161 0.401039 0.200520 0.979690i \(-0.435737\pi\)
0.200520 + 0.979690i \(0.435737\pi\)
\(618\) 6.50523 0.261679
\(619\) 11.3261 0.455236 0.227618 0.973750i \(-0.426906\pi\)
0.227618 + 0.973750i \(0.426906\pi\)
\(620\) 0 0
\(621\) −28.7196 −1.15248
\(622\) 29.8331 1.19620
\(623\) −27.8227 −1.11469
\(624\) 45.7546 1.83165
\(625\) 0 0
\(626\) 11.7362 0.469073
\(627\) −2.69106 −0.107470
\(628\) 4.16146 0.166060
\(629\) 12.0688 0.481216
\(630\) 0 0
\(631\) 11.8141 0.470311 0.235155 0.971958i \(-0.424440\pi\)
0.235155 + 0.971958i \(0.424440\pi\)
\(632\) −4.54364 −0.180736
\(633\) 48.6963 1.93550
\(634\) −37.1833 −1.47674
\(635\) 0 0
\(636\) −11.8339 −0.469246
\(637\) −2.30412 −0.0912925
\(638\) −1.22898 −0.0486556
\(639\) 23.2149 0.918368
\(640\) 0 0
\(641\) −3.74179 −0.147792 −0.0738959 0.997266i \(-0.523543\pi\)
−0.0738959 + 0.997266i \(0.523543\pi\)
\(642\) −3.89668 −0.153790
\(643\) 27.5870 1.08793 0.543963 0.839109i \(-0.316923\pi\)
0.543963 + 0.839109i \(0.316923\pi\)
\(644\) 17.6755 0.696510
\(645\) 0 0
\(646\) 2.61663 0.102950
\(647\) 26.7701 1.05244 0.526220 0.850348i \(-0.323609\pi\)
0.526220 + 0.850348i \(0.323609\pi\)
\(648\) −7.40798 −0.291013
\(649\) −7.67030 −0.301086
\(650\) 0 0
\(651\) −64.5444 −2.52970
\(652\) −17.1525 −0.671744
\(653\) −15.6170 −0.611142 −0.305571 0.952169i \(-0.598847\pi\)
−0.305571 + 0.952169i \(0.598847\pi\)
\(654\) −27.8431 −1.08875
\(655\) 0 0
\(656\) −17.2106 −0.671963
\(657\) 37.3026 1.45531
\(658\) −51.0881 −1.99162
\(659\) 4.43183 0.172639 0.0863197 0.996267i \(-0.472489\pi\)
0.0863197 + 0.996267i \(0.472489\pi\)
\(660\) 0 0
\(661\) 18.4256 0.716675 0.358337 0.933592i \(-0.383344\pi\)
0.358337 + 0.933592i \(0.383344\pi\)
\(662\) −50.5916 −1.96630
\(663\) −14.3602 −0.557705
\(664\) −12.1766 −0.472546
\(665\) 0 0
\(666\) −55.1263 −2.13610
\(667\) −6.29226 −0.243637
\(668\) −18.3601 −0.710375
\(669\) −39.1905 −1.51519
\(670\) 0 0
\(671\) −10.7718 −0.415840
\(672\) −29.5650 −1.14049
\(673\) −3.98251 −0.153515 −0.0767573 0.997050i \(-0.524457\pi\)
−0.0767573 + 0.997050i \(0.524457\pi\)
\(674\) 18.6678 0.719055
\(675\) 0 0
\(676\) −1.04768 −0.0402953
\(677\) −13.9230 −0.535105 −0.267553 0.963543i \(-0.586215\pi\)
−0.267553 + 0.963543i \(0.586215\pi\)
\(678\) −36.4872 −1.40128
\(679\) 42.3305 1.62449
\(680\) 0 0
\(681\) −48.9611 −1.87620
\(682\) 16.0060 0.612902
\(683\) −7.79713 −0.298349 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(684\) −3.46830 −0.132614
\(685\) 0 0
\(686\) 32.3973 1.23694
\(687\) 44.2628 1.68873
\(688\) 26.9138 1.02608
\(689\) 18.4110 0.701404
\(690\) 0 0
\(691\) −18.9983 −0.722728 −0.361364 0.932425i \(-0.617689\pi\)
−0.361364 + 0.932425i \(0.617689\pi\)
\(692\) −14.0085 −0.532525
\(693\) −10.6695 −0.405302
\(694\) 50.8328 1.92958
\(695\) 0 0
\(696\) 3.91032 0.148220
\(697\) 5.40162 0.204601
\(698\) 38.6180 1.46171
\(699\) 76.4066 2.88996
\(700\) 0 0
\(701\) 12.6719 0.478612 0.239306 0.970944i \(-0.423080\pi\)
0.239306 + 0.970944i \(0.423080\pi\)
\(702\) 19.2023 0.724745
\(703\) −7.74223 −0.292004
\(704\) −2.60184 −0.0980604
\(705\) 0 0
\(706\) −25.2426 −0.950017
\(707\) 34.1018 1.28253
\(708\) −16.8773 −0.634288
\(709\) 16.0710 0.603559 0.301779 0.953378i \(-0.402419\pi\)
0.301779 + 0.953378i \(0.402419\pi\)
\(710\) 0 0
\(711\) −9.71098 −0.364190
\(712\) −21.9529 −0.822721
\(713\) 81.9496 3.06904
\(714\) 17.7118 0.662846
\(715\) 0 0
\(716\) 13.1188 0.490271
\(717\) −37.3819 −1.39605
\(718\) 31.3061 1.16833
\(719\) 19.3327 0.720987 0.360494 0.932762i \(-0.382608\pi\)
0.360494 + 0.932762i \(0.382608\pi\)
\(720\) 0 0
\(721\) 3.62237 0.134904
\(722\) −1.67859 −0.0624705
\(723\) −15.1690 −0.564140
\(724\) −6.72637 −0.249984
\(725\) 0 0
\(726\) 4.51717 0.167648
\(727\) 25.3868 0.941544 0.470772 0.882255i \(-0.343975\pi\)
0.470772 + 0.882255i \(0.343975\pi\)
\(728\) 17.0893 0.633373
\(729\) −42.8112 −1.58560
\(730\) 0 0
\(731\) −8.44701 −0.312424
\(732\) −23.7016 −0.876036
\(733\) 29.5750 1.09238 0.546189 0.837662i \(-0.316078\pi\)
0.546189 + 0.837662i \(0.316078\pi\)
\(734\) −32.6752 −1.20606
\(735\) 0 0
\(736\) 37.5375 1.38365
\(737\) −3.64371 −0.134218
\(738\) −24.6728 −0.908217
\(739\) −37.4736 −1.37849 −0.689244 0.724529i \(-0.742057\pi\)
−0.689244 + 0.724529i \(0.742057\pi\)
\(740\) 0 0
\(741\) 9.21217 0.338418
\(742\) −22.7080 −0.833636
\(743\) −28.1970 −1.03445 −0.517223 0.855850i \(-0.673034\pi\)
−0.517223 + 0.855850i \(0.673034\pi\)
\(744\) −50.9275 −1.86709
\(745\) 0 0
\(746\) 51.5017 1.88561
\(747\) −26.0248 −0.952197
\(748\) −1.27458 −0.0466032
\(749\) −2.16982 −0.0792836
\(750\) 0 0
\(751\) −21.4710 −0.783487 −0.391743 0.920074i \(-0.628128\pi\)
−0.391743 + 0.920074i \(0.628128\pi\)
\(752\) −60.0970 −2.19151
\(753\) 68.6056 2.50013
\(754\) 4.20709 0.153213
\(755\) 0 0
\(756\) −6.87282 −0.249962
\(757\) −41.2342 −1.49868 −0.749342 0.662184i \(-0.769630\pi\)
−0.749342 + 0.662184i \(0.769630\pi\)
\(758\) 1.03089 0.0374438
\(759\) 23.1276 0.839478
\(760\) 0 0
\(761\) 9.51070 0.344762 0.172381 0.985030i \(-0.444854\pi\)
0.172381 + 0.985030i \(0.444854\pi\)
\(762\) 43.3661 1.57099
\(763\) −15.5041 −0.561287
\(764\) −14.9117 −0.539488
\(765\) 0 0
\(766\) −16.9518 −0.612492
\(767\) 26.2574 0.948100
\(768\) −45.1848 −1.63046
\(769\) 39.1284 1.41101 0.705503 0.708707i \(-0.250721\pi\)
0.705503 + 0.708707i \(0.250721\pi\)
\(770\) 0 0
\(771\) 23.2702 0.838057
\(772\) 2.40594 0.0865917
\(773\) 14.0241 0.504411 0.252205 0.967674i \(-0.418844\pi\)
0.252205 + 0.967674i \(0.418844\pi\)
\(774\) 38.5830 1.38684
\(775\) 0 0
\(776\) 33.4000 1.19899
\(777\) −52.4065 −1.88007
\(778\) 33.9013 1.21542
\(779\) −3.46517 −0.124153
\(780\) 0 0
\(781\) −5.47291 −0.195836
\(782\) −22.4880 −0.804167
\(783\) 2.44664 0.0874359
\(784\) 3.34301 0.119393
\(785\) 0 0
\(786\) −60.5302 −2.15904
\(787\) −22.8071 −0.812985 −0.406492 0.913654i \(-0.633248\pi\)
−0.406492 + 0.913654i \(0.633248\pi\)
\(788\) 8.58618 0.305870
\(789\) −62.9456 −2.24092
\(790\) 0 0
\(791\) −20.3175 −0.722408
\(792\) −8.41858 −0.299141
\(793\) 36.8745 1.30945
\(794\) 28.8788 1.02487
\(795\) 0 0
\(796\) 11.6453 0.412757
\(797\) −48.4687 −1.71685 −0.858424 0.512940i \(-0.828556\pi\)
−0.858424 + 0.512940i \(0.828556\pi\)
\(798\) −11.3622 −0.402218
\(799\) 18.8617 0.667277
\(800\) 0 0
\(801\) −46.9194 −1.65782
\(802\) −41.3821 −1.46125
\(803\) −8.79408 −0.310336
\(804\) −8.01741 −0.282752
\(805\) 0 0
\(806\) −54.7926 −1.92999
\(807\) 30.9910 1.09093
\(808\) 26.9073 0.946596
\(809\) −32.4414 −1.14058 −0.570289 0.821444i \(-0.693169\pi\)
−0.570289 + 0.821444i \(0.693169\pi\)
\(810\) 0 0
\(811\) 18.0017 0.632123 0.316062 0.948739i \(-0.397639\pi\)
0.316062 + 0.948739i \(0.397639\pi\)
\(812\) −1.50579 −0.0528427
\(813\) −77.8830 −2.73148
\(814\) 12.9960 0.455509
\(815\) 0 0
\(816\) 20.8350 0.729373
\(817\) 5.41881 0.189580
\(818\) 28.0465 0.980624
\(819\) 36.5245 1.27627
\(820\) 0 0
\(821\) −50.2613 −1.75413 −0.877066 0.480369i \(-0.840503\pi\)
−0.877066 + 0.480369i \(0.840503\pi\)
\(822\) −74.2100 −2.58837
\(823\) 17.4401 0.607924 0.303962 0.952684i \(-0.401690\pi\)
0.303962 + 0.952684i \(0.401690\pi\)
\(824\) 2.85816 0.0995686
\(825\) 0 0
\(826\) −32.3856 −1.12684
\(827\) 38.7966 1.34909 0.674544 0.738234i \(-0.264340\pi\)
0.674544 + 0.738234i \(0.264340\pi\)
\(828\) 29.8074 1.03588
\(829\) −5.49506 −0.190851 −0.0954257 0.995437i \(-0.530421\pi\)
−0.0954257 + 0.995437i \(0.530421\pi\)
\(830\) 0 0
\(831\) −22.3490 −0.775279
\(832\) 8.90675 0.308786
\(833\) −1.04922 −0.0363532
\(834\) 14.8507 0.514239
\(835\) 0 0
\(836\) 0.817650 0.0282790
\(837\) −31.8648 −1.10141
\(838\) 41.0722 1.41882
\(839\) 4.11116 0.141933 0.0709665 0.997479i \(-0.477392\pi\)
0.0709665 + 0.997479i \(0.477392\pi\)
\(840\) 0 0
\(841\) −28.4640 −0.981516
\(842\) −1.12303 −0.0387023
\(843\) −33.8299 −1.16516
\(844\) −14.7959 −0.509294
\(845\) 0 0
\(846\) −86.1535 −2.96202
\(847\) 2.51534 0.0864280
\(848\) −26.7123 −0.917303
\(849\) 2.02090 0.0693572
\(850\) 0 0
\(851\) 66.5385 2.28091
\(852\) −12.0423 −0.412561
\(853\) 27.6529 0.946818 0.473409 0.880843i \(-0.343023\pi\)
0.473409 + 0.880843i \(0.343023\pi\)
\(854\) −45.4807 −1.55632
\(855\) 0 0
\(856\) −1.71206 −0.0585168
\(857\) −35.5049 −1.21282 −0.606412 0.795151i \(-0.707392\pi\)
−0.606412 + 0.795151i \(0.707392\pi\)
\(858\) −15.4634 −0.527913
\(859\) −3.07721 −0.104993 −0.0524966 0.998621i \(-0.516718\pi\)
−0.0524966 + 0.998621i \(0.516718\pi\)
\(860\) 0 0
\(861\) −23.4555 −0.799360
\(862\) 9.93569 0.338411
\(863\) 49.3729 1.68067 0.840336 0.542066i \(-0.182358\pi\)
0.840336 + 0.542066i \(0.182358\pi\)
\(864\) −14.5959 −0.496562
\(865\) 0 0
\(866\) −35.8589 −1.21853
\(867\) 39.2088 1.33160
\(868\) 19.6112 0.665646
\(869\) 2.28936 0.0776612
\(870\) 0 0
\(871\) 12.4734 0.422643
\(872\) −12.2332 −0.414269
\(873\) 71.3848 2.41601
\(874\) 14.4262 0.487972
\(875\) 0 0
\(876\) −19.3500 −0.653775
\(877\) 17.3893 0.587195 0.293597 0.955929i \(-0.405147\pi\)
0.293597 + 0.955929i \(0.405147\pi\)
\(878\) −12.8387 −0.433285
\(879\) −18.6533 −0.629159
\(880\) 0 0
\(881\) −6.19205 −0.208616 −0.104308 0.994545i \(-0.533263\pi\)
−0.104308 + 0.994545i \(0.533263\pi\)
\(882\) 4.79246 0.161370
\(883\) 48.6770 1.63811 0.819055 0.573715i \(-0.194498\pi\)
0.819055 + 0.573715i \(0.194498\pi\)
\(884\) 4.36321 0.146751
\(885\) 0 0
\(886\) −48.8322 −1.64055
\(887\) −18.8432 −0.632692 −0.316346 0.948644i \(-0.602456\pi\)
−0.316346 + 0.948644i \(0.602456\pi\)
\(888\) −41.3503 −1.38762
\(889\) 24.1479 0.809896
\(890\) 0 0
\(891\) 3.73259 0.125046
\(892\) 11.9076 0.398697
\(893\) −12.0999 −0.404906
\(894\) −57.5082 −1.92336
\(895\) 0 0
\(896\) −32.9583 −1.10106
\(897\) −79.1716 −2.64346
\(898\) −42.5283 −1.41919
\(899\) −6.98135 −0.232841
\(900\) 0 0
\(901\) 8.38374 0.279303
\(902\) 5.81659 0.193671
\(903\) 36.6794 1.22062
\(904\) −16.0311 −0.533187
\(905\) 0 0
\(906\) 42.8303 1.42294
\(907\) 42.2094 1.40154 0.700770 0.713387i \(-0.252840\pi\)
0.700770 + 0.713387i \(0.252840\pi\)
\(908\) 14.8763 0.493688
\(909\) 57.5082 1.90743
\(910\) 0 0
\(911\) −44.8926 −1.48736 −0.743679 0.668537i \(-0.766921\pi\)
−0.743679 + 0.668537i \(0.766921\pi\)
\(912\) −13.3658 −0.442586
\(913\) 6.13533 0.203050
\(914\) −0.562341 −0.0186006
\(915\) 0 0
\(916\) −13.4488 −0.444360
\(917\) −33.7056 −1.11306
\(918\) 8.74408 0.288598
\(919\) 16.1540 0.532872 0.266436 0.963853i \(-0.414154\pi\)
0.266436 + 0.963853i \(0.414154\pi\)
\(920\) 0 0
\(921\) −40.2348 −1.32578
\(922\) −12.5494 −0.413294
\(923\) 18.7352 0.616675
\(924\) 5.53460 0.182075
\(925\) 0 0
\(926\) 11.5170 0.378473
\(927\) 6.10865 0.200635
\(928\) −3.19785 −0.104975
\(929\) 4.93374 0.161871 0.0809353 0.996719i \(-0.474209\pi\)
0.0809353 + 0.996719i \(0.474209\pi\)
\(930\) 0 0
\(931\) 0.673078 0.0220593
\(932\) −23.2153 −0.760444
\(933\) 47.8276 1.56580
\(934\) 58.2002 1.90437
\(935\) 0 0
\(936\) 28.8189 0.941977
\(937\) 35.4716 1.15881 0.579403 0.815041i \(-0.303286\pi\)
0.579403 + 0.815041i \(0.303286\pi\)
\(938\) −15.3845 −0.502322
\(939\) 18.8151 0.614008
\(940\) 0 0
\(941\) 12.1011 0.394486 0.197243 0.980355i \(-0.436801\pi\)
0.197243 + 0.980355i \(0.436801\pi\)
\(942\) 22.9903 0.749064
\(943\) 29.7805 0.969786
\(944\) −38.0965 −1.23993
\(945\) 0 0
\(946\) −9.09593 −0.295734
\(947\) −36.0079 −1.17010 −0.585050 0.810997i \(-0.698925\pi\)
−0.585050 + 0.810997i \(0.698925\pi\)
\(948\) 5.03738 0.163606
\(949\) 30.1044 0.977229
\(950\) 0 0
\(951\) −59.6112 −1.93303
\(952\) 7.78189 0.252213
\(953\) 30.1837 0.977746 0.488873 0.872355i \(-0.337408\pi\)
0.488873 + 0.872355i \(0.337408\pi\)
\(954\) −38.2940 −1.23982
\(955\) 0 0
\(956\) 11.3581 0.367347
\(957\) −1.97026 −0.0636893
\(958\) 65.4843 2.11570
\(959\) −41.3230 −1.33439
\(960\) 0 0
\(961\) 59.9241 1.93304
\(962\) −44.4886 −1.43437
\(963\) −3.65913 −0.117914
\(964\) 4.60893 0.148444
\(965\) 0 0
\(966\) 97.6494 3.14182
\(967\) 35.7177 1.14860 0.574301 0.818644i \(-0.305274\pi\)
0.574301 + 0.818644i \(0.305274\pi\)
\(968\) 1.98468 0.0637899
\(969\) 4.19491 0.134760
\(970\) 0 0
\(971\) −24.4683 −0.785225 −0.392612 0.919704i \(-0.628429\pi\)
−0.392612 + 0.919704i \(0.628429\pi\)
\(972\) 16.4101 0.526353
\(973\) 8.26947 0.265107
\(974\) −35.0294 −1.12241
\(975\) 0 0
\(976\) −53.5007 −1.71251
\(977\) −34.8532 −1.11505 −0.557527 0.830159i \(-0.688250\pi\)
−0.557527 + 0.830159i \(0.688250\pi\)
\(978\) −94.7603 −3.03010
\(979\) 11.0612 0.353518
\(980\) 0 0
\(981\) −26.1457 −0.834768
\(982\) −13.2841 −0.423911
\(983\) −15.4143 −0.491640 −0.245820 0.969316i \(-0.579057\pi\)
−0.245820 + 0.969316i \(0.579057\pi\)
\(984\) −18.5071 −0.589984
\(985\) 0 0
\(986\) 1.91577 0.0610104
\(987\) −81.9030 −2.60700
\(988\) −2.79902 −0.0890488
\(989\) −46.5705 −1.48085
\(990\) 0 0
\(991\) −51.3988 −1.63274 −0.816368 0.577533i \(-0.804016\pi\)
−0.816368 + 0.577533i \(0.804016\pi\)
\(992\) 41.6484 1.32234
\(993\) −81.1069 −2.57385
\(994\) −23.1078 −0.732934
\(995\) 0 0
\(996\) 13.4998 0.427759
\(997\) 3.19123 0.101067 0.0505336 0.998722i \(-0.483908\pi\)
0.0505336 + 0.998722i \(0.483908\pi\)
\(998\) −13.9403 −0.441273
\(999\) −25.8724 −0.818567
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.t.1.4 15
5.4 even 2 5225.2.a.w.1.12 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.t.1.4 15 1.1 even 1 trivial
5225.2.a.w.1.12 yes 15 5.4 even 2