Properties

Label 6422.2.a.ba.1.4
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.16609.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.18109\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.60502 q^{3} +1.00000 q^{4} +2.78611 q^{5} -2.60502 q^{6} -3.47175 q^{7} -1.00000 q^{8} +3.78611 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.60502 q^{3} +1.00000 q^{4} +2.78611 q^{5} -2.60502 q^{6} -3.47175 q^{7} -1.00000 q^{8} +3.78611 q^{9} -2.78611 q^{10} +0.290659 q^{11} +2.60502 q^{12} +3.47175 q^{14} +7.25786 q^{15} +1.00000 q^{16} -7.54852 q^{17} -3.78611 q^{18} -1.00000 q^{19} +2.78611 q^{20} -9.04397 q^{21} -0.290659 q^{22} -5.86288 q^{23} -2.60502 q^{24} +2.76241 q^{25} +2.04783 q^{27} -3.47175 q^{28} +8.22507 q^{29} -7.25786 q^{30} -5.41002 q^{31} -1.00000 q^{32} +0.757172 q^{33} +7.54852 q^{34} -9.67269 q^{35} +3.78611 q^{36} -1.37609 q^{37} +1.00000 q^{38} -2.78611 q^{40} -4.80115 q^{41} +9.04397 q^{42} -8.94831 q^{43} +0.290659 q^{44} +10.5485 q^{45} +5.86288 q^{46} +6.19613 q^{47} +2.60502 q^{48} +5.05307 q^{49} -2.76241 q^{50} -19.6640 q^{51} -12.8866 q^{53} -2.04783 q^{54} +0.809809 q^{55} +3.47175 q^{56} -2.60502 q^{57} -8.22507 q^{58} -5.57222 q^{59} +7.25786 q^{60} +1.41868 q^{61} +5.41002 q^{62} -13.1444 q^{63} +1.00000 q^{64} -0.757172 q^{66} +8.72438 q^{67} -7.54852 q^{68} -15.2729 q^{69} +9.67269 q^{70} -1.27177 q^{71} -3.78611 q^{72} -4.36219 q^{73} +1.37609 q^{74} +7.19613 q^{75} -1.00000 q^{76} -1.00910 q^{77} -9.88701 q^{79} +2.78611 q^{80} -6.02370 q^{81} +4.80115 q^{82} +12.4968 q^{83} -9.04397 q^{84} -21.0310 q^{85} +8.94831 q^{86} +21.4264 q^{87} -0.290659 q^{88} -11.4351 q^{89} -10.5485 q^{90} -5.86288 q^{92} -14.0932 q^{93} -6.19613 q^{94} -2.78611 q^{95} -2.60502 q^{96} +5.53848 q^{97} -5.05307 q^{98} +1.10047 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 7 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 7 q^{14} + 9 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 3 q^{21} + q^{22} + 5 q^{23} - 2 q^{24} + 2 q^{25} + 5 q^{27} - 7 q^{28} - 5 q^{29} - 9 q^{30} - 9 q^{31} - 4 q^{32} + 2 q^{33} + 8 q^{34} + 7 q^{35} + 2 q^{36} - 5 q^{37} + 4 q^{38} + 2 q^{40} + 15 q^{41} + 3 q^{42} - 9 q^{43} - q^{44} + 20 q^{45} - 5 q^{46} - q^{47} + 2 q^{48} + 9 q^{49} - 2 q^{50} - 16 q^{51} - 19 q^{53} - 5 q^{54} - 14 q^{55} + 7 q^{56} - 2 q^{57} + 5 q^{58} + 4 q^{59} + 9 q^{60} + 10 q^{61} + 9 q^{62} + 4 q^{64} - 2 q^{66} + 16 q^{67} - 8 q^{68} - 20 q^{69} - 7 q^{70} + 6 q^{71} - 2 q^{72} - 8 q^{73} + 5 q^{74} + 3 q^{75} - 4 q^{76} - 26 q^{77} - 12 q^{79} - 2 q^{80} - 20 q^{81} - 15 q^{82} + q^{83} - 3 q^{84} + q^{85} + 9 q^{86} + 14 q^{87} + q^{88} + 9 q^{89} - 20 q^{90} + 5 q^{92} + 18 q^{93} + q^{94} + 2 q^{95} - 2 q^{96} + 21 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.60502 1.50401 0.752003 0.659159i \(-0.229088\pi\)
0.752003 + 0.659159i \(0.229088\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.78611 1.24599 0.622993 0.782227i \(-0.285916\pi\)
0.622993 + 0.782227i \(0.285916\pi\)
\(6\) −2.60502 −1.06349
\(7\) −3.47175 −1.31220 −0.656100 0.754674i \(-0.727795\pi\)
−0.656100 + 0.754674i \(0.727795\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.78611 1.26204
\(10\) −2.78611 −0.881046
\(11\) 0.290659 0.0876370 0.0438185 0.999040i \(-0.486048\pi\)
0.0438185 + 0.999040i \(0.486048\pi\)
\(12\) 2.60502 0.752003
\(13\) 0 0
\(14\) 3.47175 0.927865
\(15\) 7.25786 1.87397
\(16\) 1.00000 0.250000
\(17\) −7.54852 −1.83079 −0.915393 0.402562i \(-0.868120\pi\)
−0.915393 + 0.402562i \(0.868120\pi\)
\(18\) −3.78611 −0.892395
\(19\) −1.00000 −0.229416
\(20\) 2.78611 0.622993
\(21\) −9.04397 −1.97356
\(22\) −0.290659 −0.0619688
\(23\) −5.86288 −1.22250 −0.611248 0.791439i \(-0.709332\pi\)
−0.611248 + 0.791439i \(0.709332\pi\)
\(24\) −2.60502 −0.531747
\(25\) 2.76241 0.552483
\(26\) 0 0
\(27\) 2.04783 0.394105
\(28\) −3.47175 −0.656100
\(29\) 8.22507 1.52736 0.763678 0.645597i \(-0.223391\pi\)
0.763678 + 0.645597i \(0.223391\pi\)
\(30\) −7.25786 −1.32510
\(31\) −5.41002 −0.971668 −0.485834 0.874051i \(-0.661484\pi\)
−0.485834 + 0.874051i \(0.661484\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.757172 0.131807
\(34\) 7.54852 1.29456
\(35\) −9.67269 −1.63498
\(36\) 3.78611 0.631018
\(37\) −1.37609 −0.226228 −0.113114 0.993582i \(-0.536083\pi\)
−0.113114 + 0.993582i \(0.536083\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −2.78611 −0.440523
\(41\) −4.80115 −0.749813 −0.374907 0.927063i \(-0.622325\pi\)
−0.374907 + 0.927063i \(0.622325\pi\)
\(42\) 9.04397 1.39552
\(43\) −8.94831 −1.36460 −0.682302 0.731070i \(-0.739021\pi\)
−0.682302 + 0.731070i \(0.739021\pi\)
\(44\) 0.290659 0.0438185
\(45\) 10.5485 1.57248
\(46\) 5.86288 0.864435
\(47\) 6.19613 0.903798 0.451899 0.892069i \(-0.350747\pi\)
0.451899 + 0.892069i \(0.350747\pi\)
\(48\) 2.60502 0.376002
\(49\) 5.05307 0.721867
\(50\) −2.76241 −0.390664
\(51\) −19.6640 −2.75351
\(52\) 0 0
\(53\) −12.8866 −1.77011 −0.885054 0.465489i \(-0.845879\pi\)
−0.885054 + 0.465489i \(0.845879\pi\)
\(54\) −2.04783 −0.278675
\(55\) 0.809809 0.109195
\(56\) 3.47175 0.463933
\(57\) −2.60502 −0.345043
\(58\) −8.22507 −1.08000
\(59\) −5.57222 −0.725441 −0.362721 0.931898i \(-0.618152\pi\)
−0.362721 + 0.931898i \(0.618152\pi\)
\(60\) 7.25786 0.936986
\(61\) 1.41868 0.181644 0.0908218 0.995867i \(-0.471051\pi\)
0.0908218 + 0.995867i \(0.471051\pi\)
\(62\) 5.41002 0.687073
\(63\) −13.1444 −1.65604
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.757172 −0.0932014
\(67\) 8.72438 1.06585 0.532926 0.846162i \(-0.321092\pi\)
0.532926 + 0.846162i \(0.321092\pi\)
\(68\) −7.54852 −0.915393
\(69\) −15.2729 −1.83864
\(70\) 9.67269 1.15611
\(71\) −1.27177 −0.150931 −0.0754655 0.997148i \(-0.524044\pi\)
−0.0754655 + 0.997148i \(0.524044\pi\)
\(72\) −3.78611 −0.446197
\(73\) −4.36219 −0.510556 −0.255278 0.966868i \(-0.582167\pi\)
−0.255278 + 0.966868i \(0.582167\pi\)
\(74\) 1.37609 0.159967
\(75\) 7.19613 0.830938
\(76\) −1.00000 −0.114708
\(77\) −1.00910 −0.114997
\(78\) 0 0
\(79\) −9.88701 −1.11238 −0.556188 0.831057i \(-0.687737\pi\)
−0.556188 + 0.831057i \(0.687737\pi\)
\(80\) 2.78611 0.311497
\(81\) −6.02370 −0.669300
\(82\) 4.80115 0.530198
\(83\) 12.4968 1.37171 0.685853 0.727740i \(-0.259429\pi\)
0.685853 + 0.727740i \(0.259429\pi\)
\(84\) −9.04397 −0.986779
\(85\) −21.0310 −2.28113
\(86\) 8.94831 0.964921
\(87\) 21.4264 2.29716
\(88\) −0.290659 −0.0309844
\(89\) −11.4351 −1.21212 −0.606059 0.795420i \(-0.707251\pi\)
−0.606059 + 0.795420i \(0.707251\pi\)
\(90\) −10.5485 −1.11191
\(91\) 0 0
\(92\) −5.86288 −0.611248
\(93\) −14.0932 −1.46140
\(94\) −6.19613 −0.639082
\(95\) −2.78611 −0.285849
\(96\) −2.60502 −0.265873
\(97\) 5.53848 0.562347 0.281174 0.959657i \(-0.409276\pi\)
0.281174 + 0.959657i \(0.409276\pi\)
\(98\) −5.05307 −0.510437
\(99\) 1.10047 0.110601
\(100\) 2.76241 0.276241
\(101\) 4.48566 0.446339 0.223170 0.974780i \(-0.428360\pi\)
0.223170 + 0.974780i \(0.428360\pi\)
\(102\) 19.6640 1.94703
\(103\) 16.2492 1.60108 0.800541 0.599278i \(-0.204546\pi\)
0.800541 + 0.599278i \(0.204546\pi\)
\(104\) 0 0
\(105\) −25.1975 −2.45903
\(106\) 12.8866 1.25166
\(107\) 8.04922 0.778147 0.389074 0.921207i \(-0.372795\pi\)
0.389074 + 0.921207i \(0.372795\pi\)
\(108\) 2.04783 0.197053
\(109\) −10.8248 −1.03683 −0.518416 0.855129i \(-0.673478\pi\)
−0.518416 + 0.855129i \(0.673478\pi\)
\(110\) −0.809809 −0.0772122
\(111\) −3.58474 −0.340248
\(112\) −3.47175 −0.328050
\(113\) 0.847844 0.0797585 0.0398792 0.999205i \(-0.487303\pi\)
0.0398792 + 0.999205i \(0.487303\pi\)
\(114\) 2.60502 0.243982
\(115\) −16.3346 −1.52321
\(116\) 8.22507 0.763678
\(117\) 0 0
\(118\) 5.57222 0.512965
\(119\) 26.2066 2.40236
\(120\) −7.25786 −0.662549
\(121\) −10.9155 −0.992320
\(122\) −1.41868 −0.128441
\(123\) −12.5071 −1.12772
\(124\) −5.41002 −0.485834
\(125\) −6.23417 −0.557601
\(126\) 13.1444 1.17100
\(127\) 7.83875 0.695576 0.347788 0.937573i \(-0.386933\pi\)
0.347788 + 0.937573i \(0.386933\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −23.3105 −2.05238
\(130\) 0 0
\(131\) 15.1396 1.32276 0.661378 0.750053i \(-0.269972\pi\)
0.661378 + 0.750053i \(0.269972\pi\)
\(132\) 0.757172 0.0659034
\(133\) 3.47175 0.301039
\(134\) −8.72438 −0.753671
\(135\) 5.70548 0.491050
\(136\) 7.54852 0.647281
\(137\) 16.4501 1.40543 0.702715 0.711471i \(-0.251971\pi\)
0.702715 + 0.711471i \(0.251971\pi\)
\(138\) 15.2729 1.30012
\(139\) −12.7119 −1.07821 −0.539103 0.842240i \(-0.681237\pi\)
−0.539103 + 0.842240i \(0.681237\pi\)
\(140\) −9.67269 −0.817491
\(141\) 16.1410 1.35932
\(142\) 1.27177 0.106724
\(143\) 0 0
\(144\) 3.78611 0.315509
\(145\) 22.9160 1.90307
\(146\) 4.36219 0.361017
\(147\) 13.1633 1.08569
\(148\) −1.37609 −0.113114
\(149\) −13.7447 −1.12601 −0.563003 0.826455i \(-0.690354\pi\)
−0.563003 + 0.826455i \(0.690354\pi\)
\(150\) −7.19613 −0.587562
\(151\) 1.69682 0.138085 0.0690427 0.997614i \(-0.478006\pi\)
0.0690427 + 0.997614i \(0.478006\pi\)
\(152\) 1.00000 0.0811107
\(153\) −28.5795 −2.31052
\(154\) 1.00910 0.0813154
\(155\) −15.0729 −1.21069
\(156\) 0 0
\(157\) −7.47656 −0.596694 −0.298347 0.954457i \(-0.596435\pi\)
−0.298347 + 0.954457i \(0.596435\pi\)
\(158\) 9.88701 0.786569
\(159\) −33.5697 −2.66225
\(160\) −2.78611 −0.220261
\(161\) 20.3545 1.60416
\(162\) 6.02370 0.473266
\(163\) 5.74465 0.449956 0.224978 0.974364i \(-0.427769\pi\)
0.224978 + 0.974364i \(0.427769\pi\)
\(164\) −4.80115 −0.374907
\(165\) 2.10956 0.164229
\(166\) −12.4968 −0.969943
\(167\) −21.4403 −1.65910 −0.829552 0.558430i \(-0.811404\pi\)
−0.829552 + 0.558430i \(0.811404\pi\)
\(168\) 9.04397 0.697758
\(169\) 0 0
\(170\) 21.0310 1.61301
\(171\) −3.78611 −0.289531
\(172\) −8.94831 −0.682302
\(173\) 2.17243 0.165167 0.0825835 0.996584i \(-0.473683\pi\)
0.0825835 + 0.996584i \(0.473683\pi\)
\(174\) −21.4264 −1.62433
\(175\) −9.59042 −0.724967
\(176\) 0.290659 0.0219093
\(177\) −14.5157 −1.09107
\(178\) 11.4351 0.857097
\(179\) 7.35695 0.549884 0.274942 0.961461i \(-0.411341\pi\)
0.274942 + 0.961461i \(0.411341\pi\)
\(180\) 10.5485 0.786241
\(181\) 18.3734 1.36568 0.682841 0.730567i \(-0.260744\pi\)
0.682841 + 0.730567i \(0.260744\pi\)
\(182\) 0 0
\(183\) 3.69569 0.273193
\(184\) 5.86288 0.432217
\(185\) −3.83394 −0.281877
\(186\) 14.0932 1.03336
\(187\) −2.19405 −0.160445
\(188\) 6.19613 0.451899
\(189\) −7.10956 −0.517145
\(190\) 2.78611 0.202126
\(191\) 18.1923 1.31635 0.658173 0.752866i \(-0.271329\pi\)
0.658173 + 0.752866i \(0.271329\pi\)
\(192\) 2.60502 0.188001
\(193\) 13.9734 1.00583 0.502913 0.864337i \(-0.332261\pi\)
0.502913 + 0.864337i \(0.332261\pi\)
\(194\) −5.53848 −0.397639
\(195\) 0 0
\(196\) 5.05307 0.360934
\(197\) −27.2855 −1.94401 −0.972004 0.234963i \(-0.924503\pi\)
−0.972004 + 0.234963i \(0.924503\pi\)
\(198\) −1.10047 −0.0782068
\(199\) 14.9773 1.06171 0.530855 0.847463i \(-0.321871\pi\)
0.530855 + 0.847463i \(0.321871\pi\)
\(200\) −2.76241 −0.195332
\(201\) 22.7271 1.60305
\(202\) −4.48566 −0.315610
\(203\) −28.5554 −2.00420
\(204\) −19.6640 −1.37676
\(205\) −13.3765 −0.934257
\(206\) −16.2492 −1.13214
\(207\) −22.1975 −1.54283
\(208\) 0 0
\(209\) −0.290659 −0.0201053
\(210\) 25.1975 1.73879
\(211\) −0.456717 −0.0314417 −0.0157209 0.999876i \(-0.505004\pi\)
−0.0157209 + 0.999876i \(0.505004\pi\)
\(212\) −12.8866 −0.885054
\(213\) −3.31297 −0.227001
\(214\) −8.04922 −0.550233
\(215\) −24.9310 −1.70028
\(216\) −2.04783 −0.139337
\(217\) 18.7823 1.27502
\(218\) 10.8248 0.733151
\(219\) −11.3636 −0.767879
\(220\) 0.809809 0.0545973
\(221\) 0 0
\(222\) 3.58474 0.240592
\(223\) −7.60640 −0.509362 −0.254681 0.967025i \(-0.581971\pi\)
−0.254681 + 0.967025i \(0.581971\pi\)
\(224\) 3.47175 0.231966
\(225\) 10.4588 0.697253
\(226\) −0.847844 −0.0563978
\(227\) −23.9923 −1.59242 −0.796212 0.605018i \(-0.793166\pi\)
−0.796212 + 0.605018i \(0.793166\pi\)
\(228\) −2.60502 −0.172521
\(229\) 7.00252 0.462739 0.231370 0.972866i \(-0.425679\pi\)
0.231370 + 0.972866i \(0.425679\pi\)
\(230\) 16.3346 1.07707
\(231\) −2.62871 −0.172957
\(232\) −8.22507 −0.540002
\(233\) −26.6731 −1.74741 −0.873707 0.486452i \(-0.838291\pi\)
−0.873707 + 0.486452i \(0.838291\pi\)
\(234\) 0 0
\(235\) 17.2631 1.12612
\(236\) −5.57222 −0.362721
\(237\) −25.7558 −1.67302
\(238\) −26.2066 −1.69872
\(239\) 2.88082 0.186345 0.0931725 0.995650i \(-0.470299\pi\)
0.0931725 + 0.995650i \(0.470299\pi\)
\(240\) 7.25786 0.468493
\(241\) −15.7344 −1.01354 −0.506772 0.862080i \(-0.669161\pi\)
−0.506772 + 0.862080i \(0.669161\pi\)
\(242\) 10.9155 0.701676
\(243\) −21.8353 −1.40074
\(244\) 1.41868 0.0908218
\(245\) 14.0784 0.899437
\(246\) 12.5071 0.797421
\(247\) 0 0
\(248\) 5.41002 0.343537
\(249\) 32.5545 2.06306
\(250\) 6.23417 0.394283
\(251\) 14.0957 0.889710 0.444855 0.895603i \(-0.353255\pi\)
0.444855 + 0.895603i \(0.353255\pi\)
\(252\) −13.1444 −0.828022
\(253\) −1.70410 −0.107136
\(254\) −7.83875 −0.491847
\(255\) −54.7862 −3.43084
\(256\) 1.00000 0.0625000
\(257\) −6.35738 −0.396563 −0.198281 0.980145i \(-0.563536\pi\)
−0.198281 + 0.980145i \(0.563536\pi\)
\(258\) 23.3105 1.45125
\(259\) 4.77745 0.296856
\(260\) 0 0
\(261\) 31.1410 1.92758
\(262\) −15.1396 −0.935330
\(263\) −17.5360 −1.08132 −0.540658 0.841242i \(-0.681825\pi\)
−0.540658 + 0.841242i \(0.681825\pi\)
\(264\) −0.757172 −0.0466007
\(265\) −35.9034 −2.20553
\(266\) −3.47175 −0.212867
\(267\) −29.7886 −1.82303
\(268\) 8.72438 0.532926
\(269\) −21.8688 −1.33337 −0.666683 0.745342i \(-0.732286\pi\)
−0.666683 + 0.745342i \(0.732286\pi\)
\(270\) −5.70548 −0.347225
\(271\) 27.7606 1.68634 0.843169 0.537648i \(-0.180687\pi\)
0.843169 + 0.537648i \(0.180687\pi\)
\(272\) −7.54852 −0.457696
\(273\) 0 0
\(274\) −16.4501 −0.993789
\(275\) 0.802921 0.0484179
\(276\) −15.2729 −0.919321
\(277\) 22.6693 1.36206 0.681032 0.732254i \(-0.261531\pi\)
0.681032 + 0.732254i \(0.261531\pi\)
\(278\) 12.7119 0.762407
\(279\) −20.4829 −1.22628
\(280\) 9.67269 0.578054
\(281\) −14.3786 −0.857756 −0.428878 0.903362i \(-0.641091\pi\)
−0.428878 + 0.903362i \(0.641091\pi\)
\(282\) −16.1410 −0.961184
\(283\) 1.33350 0.0792684 0.0396342 0.999214i \(-0.487381\pi\)
0.0396342 + 0.999214i \(0.487381\pi\)
\(284\) −1.27177 −0.0754655
\(285\) −7.25786 −0.429919
\(286\) 0 0
\(287\) 16.6684 0.983904
\(288\) −3.78611 −0.223099
\(289\) 39.9802 2.35178
\(290\) −22.9160 −1.34567
\(291\) 14.4278 0.845774
\(292\) −4.36219 −0.255278
\(293\) −4.26172 −0.248972 −0.124486 0.992221i \(-0.539728\pi\)
−0.124486 + 0.992221i \(0.539728\pi\)
\(294\) −13.1633 −0.767701
\(295\) −15.5248 −0.903890
\(296\) 1.37609 0.0799837
\(297\) 0.595221 0.0345382
\(298\) 13.7447 0.796206
\(299\) 0 0
\(300\) 7.19613 0.415469
\(301\) 31.0663 1.79063
\(302\) −1.69682 −0.0976412
\(303\) 11.6852 0.671298
\(304\) −1.00000 −0.0573539
\(305\) 3.95260 0.226325
\(306\) 28.5795 1.63378
\(307\) −15.7809 −0.900662 −0.450331 0.892862i \(-0.648694\pi\)
−0.450331 + 0.892862i \(0.648694\pi\)
\(308\) −1.00910 −0.0574986
\(309\) 42.3294 2.40804
\(310\) 15.0729 0.856084
\(311\) 0.504549 0.0286103 0.0143052 0.999898i \(-0.495446\pi\)
0.0143052 + 0.999898i \(0.495446\pi\)
\(312\) 0 0
\(313\) −20.5871 −1.16365 −0.581825 0.813314i \(-0.697661\pi\)
−0.581825 + 0.813314i \(0.697661\pi\)
\(314\) 7.47656 0.421927
\(315\) −36.6219 −2.06341
\(316\) −9.88701 −0.556188
\(317\) −2.43826 −0.136946 −0.0684732 0.997653i \(-0.521813\pi\)
−0.0684732 + 0.997653i \(0.521813\pi\)
\(318\) 33.5697 1.88250
\(319\) 2.39069 0.133853
\(320\) 2.78611 0.155748
\(321\) 20.9683 1.17034
\(322\) −20.3545 −1.13431
\(323\) 7.54852 0.420011
\(324\) −6.02370 −0.334650
\(325\) 0 0
\(326\) −5.74465 −0.318167
\(327\) −28.1989 −1.55940
\(328\) 4.80115 0.265099
\(329\) −21.5114 −1.18596
\(330\) −2.10956 −0.116128
\(331\) 13.7558 0.756089 0.378044 0.925787i \(-0.376597\pi\)
0.378044 + 0.925787i \(0.376597\pi\)
\(332\) 12.4968 0.685853
\(333\) −5.21003 −0.285508
\(334\) 21.4403 1.17316
\(335\) 24.3071 1.32804
\(336\) −9.04397 −0.493389
\(337\) −10.0909 −0.549684 −0.274842 0.961489i \(-0.588626\pi\)
−0.274842 + 0.961489i \(0.588626\pi\)
\(338\) 0 0
\(339\) 2.20865 0.119957
\(340\) −21.0310 −1.14057
\(341\) −1.57247 −0.0851541
\(342\) 3.78611 0.204729
\(343\) 6.75925 0.364965
\(344\) 8.94831 0.482461
\(345\) −42.5520 −2.29092
\(346\) −2.17243 −0.116791
\(347\) −10.6335 −0.570837 −0.285419 0.958403i \(-0.592133\pi\)
−0.285419 + 0.958403i \(0.592133\pi\)
\(348\) 21.4264 1.14858
\(349\) −11.7572 −0.629347 −0.314673 0.949200i \(-0.601895\pi\)
−0.314673 + 0.949200i \(0.601895\pi\)
\(350\) 9.59042 0.512629
\(351\) 0 0
\(352\) −0.290659 −0.0154922
\(353\) 36.3071 1.93243 0.966215 0.257736i \(-0.0829764\pi\)
0.966215 + 0.257736i \(0.0829764\pi\)
\(354\) 14.5157 0.771502
\(355\) −3.54328 −0.188058
\(356\) −11.4351 −0.606059
\(357\) 68.2687 3.61316
\(358\) −7.35695 −0.388827
\(359\) 18.5736 0.980277 0.490139 0.871644i \(-0.336946\pi\)
0.490139 + 0.871644i \(0.336946\pi\)
\(360\) −10.5485 −0.555956
\(361\) 1.00000 0.0526316
\(362\) −18.3734 −0.965683
\(363\) −28.4351 −1.49246
\(364\) 0 0
\(365\) −12.1535 −0.636145
\(366\) −3.69569 −0.193177
\(367\) 4.42873 0.231178 0.115589 0.993297i \(-0.463124\pi\)
0.115589 + 0.993297i \(0.463124\pi\)
\(368\) −5.86288 −0.305624
\(369\) −18.1777 −0.946292
\(370\) 3.83394 0.199317
\(371\) 44.7390 2.32273
\(372\) −14.0932 −0.730698
\(373\) 22.8590 1.18360 0.591798 0.806087i \(-0.298418\pi\)
0.591798 + 0.806087i \(0.298418\pi\)
\(374\) 2.19405 0.113452
\(375\) −16.2401 −0.838635
\(376\) −6.19613 −0.319541
\(377\) 0 0
\(378\) 7.10956 0.365677
\(379\) 7.55263 0.387953 0.193976 0.981006i \(-0.437862\pi\)
0.193976 + 0.981006i \(0.437862\pi\)
\(380\) −2.78611 −0.142924
\(381\) 20.4201 1.04615
\(382\) −18.1923 −0.930798
\(383\) 13.8900 0.709746 0.354873 0.934915i \(-0.384524\pi\)
0.354873 + 0.934915i \(0.384524\pi\)
\(384\) −2.60502 −0.132937
\(385\) −2.81146 −0.143285
\(386\) −13.9734 −0.711227
\(387\) −33.8793 −1.72218
\(388\) 5.53848 0.281174
\(389\) −20.8952 −1.05943 −0.529715 0.848175i \(-0.677701\pi\)
−0.529715 + 0.848175i \(0.677701\pi\)
\(390\) 0 0
\(391\) 44.2561 2.23813
\(392\) −5.05307 −0.255219
\(393\) 39.4390 1.98943
\(394\) 27.2855 1.37462
\(395\) −27.5463 −1.38601
\(396\) 1.10047 0.0553006
\(397\) 12.5175 0.628238 0.314119 0.949384i \(-0.398291\pi\)
0.314119 + 0.949384i \(0.398291\pi\)
\(398\) −14.9773 −0.750742
\(399\) 9.04397 0.452765
\(400\) 2.76241 0.138121
\(401\) 12.1517 0.606828 0.303414 0.952859i \(-0.401873\pi\)
0.303414 + 0.952859i \(0.401873\pi\)
\(402\) −22.7271 −1.13353
\(403\) 0 0
\(404\) 4.48566 0.223170
\(405\) −16.7827 −0.833939
\(406\) 28.5554 1.41718
\(407\) −0.399974 −0.0198260
\(408\) 19.6640 0.973514
\(409\) −27.5549 −1.36250 −0.681251 0.732050i \(-0.738564\pi\)
−0.681251 + 0.732050i \(0.738564\pi\)
\(410\) 13.3765 0.660620
\(411\) 42.8529 2.11378
\(412\) 16.2492 0.800541
\(413\) 19.3454 0.951924
\(414\) 22.1975 1.09095
\(415\) 34.8176 1.70913
\(416\) 0 0
\(417\) −33.1146 −1.62163
\(418\) 0.290659 0.0142166
\(419\) 18.1109 0.884778 0.442389 0.896823i \(-0.354131\pi\)
0.442389 + 0.896823i \(0.354131\pi\)
\(420\) −25.1975 −1.22951
\(421\) −10.8526 −0.528926 −0.264463 0.964396i \(-0.585195\pi\)
−0.264463 + 0.964396i \(0.585195\pi\)
\(422\) 0.456717 0.0222327
\(423\) 23.4592 1.14063
\(424\) 12.8866 0.625828
\(425\) −20.8521 −1.01148
\(426\) 3.31297 0.160514
\(427\) −4.92531 −0.238353
\(428\) 8.04922 0.389074
\(429\) 0 0
\(430\) 24.9310 1.20228
\(431\) 27.1681 1.30864 0.654322 0.756216i \(-0.272954\pi\)
0.654322 + 0.756216i \(0.272954\pi\)
\(432\) 2.04783 0.0985263
\(433\) 1.63490 0.0785685 0.0392842 0.999228i \(-0.487492\pi\)
0.0392842 + 0.999228i \(0.487492\pi\)
\(434\) −18.7823 −0.901577
\(435\) 59.6964 2.86222
\(436\) −10.8248 −0.518416
\(437\) 5.86288 0.280460
\(438\) 11.3636 0.542973
\(439\) −37.8206 −1.80508 −0.902538 0.430610i \(-0.858299\pi\)
−0.902538 + 0.430610i \(0.858299\pi\)
\(440\) −0.809809 −0.0386061
\(441\) 19.1315 0.911023
\(442\) 0 0
\(443\) 9.70049 0.460884 0.230442 0.973086i \(-0.425983\pi\)
0.230442 + 0.973086i \(0.425983\pi\)
\(444\) −3.58474 −0.170124
\(445\) −31.8595 −1.51028
\(446\) 7.60640 0.360174
\(447\) −35.8050 −1.69352
\(448\) −3.47175 −0.164025
\(449\) 22.4381 1.05892 0.529459 0.848335i \(-0.322395\pi\)
0.529459 + 0.848335i \(0.322395\pi\)
\(450\) −10.4588 −0.493033
\(451\) −1.39550 −0.0657114
\(452\) 0.847844 0.0398792
\(453\) 4.42025 0.207681
\(454\) 23.9923 1.12601
\(455\) 0 0
\(456\) 2.60502 0.121991
\(457\) 2.85746 0.133666 0.0668331 0.997764i \(-0.478711\pi\)
0.0668331 + 0.997764i \(0.478711\pi\)
\(458\) −7.00252 −0.327206
\(459\) −15.4581 −0.721522
\(460\) −16.3346 −0.761606
\(461\) −27.5356 −1.28246 −0.641230 0.767349i \(-0.721575\pi\)
−0.641230 + 0.767349i \(0.721575\pi\)
\(462\) 2.62871 0.122299
\(463\) −17.6784 −0.821583 −0.410792 0.911729i \(-0.634748\pi\)
−0.410792 + 0.911729i \(0.634748\pi\)
\(464\) 8.22507 0.381839
\(465\) −39.2652 −1.82088
\(466\) 26.6731 1.23561
\(467\) 29.7663 1.37742 0.688710 0.725037i \(-0.258177\pi\)
0.688710 + 0.725037i \(0.258177\pi\)
\(468\) 0 0
\(469\) −30.2889 −1.39861
\(470\) −17.2631 −0.796288
\(471\) −19.4766 −0.897432
\(472\) 5.57222 0.256482
\(473\) −2.60091 −0.119590
\(474\) 25.7558 1.18300
\(475\) −2.76241 −0.126748
\(476\) 26.2066 1.20118
\(477\) −48.7900 −2.23394
\(478\) −2.88082 −0.131766
\(479\) 29.1907 1.33376 0.666878 0.745167i \(-0.267630\pi\)
0.666878 + 0.745167i \(0.267630\pi\)
\(480\) −7.25786 −0.331275
\(481\) 0 0
\(482\) 15.7344 0.716683
\(483\) 53.0237 2.41266
\(484\) −10.9155 −0.496160
\(485\) 15.4308 0.700677
\(486\) 21.8353 0.990471
\(487\) −23.4246 −1.06147 −0.530735 0.847538i \(-0.678084\pi\)
−0.530735 + 0.847538i \(0.678084\pi\)
\(488\) −1.41868 −0.0642207
\(489\) 14.9649 0.676737
\(490\) −14.0784 −0.635998
\(491\) 26.5137 1.19655 0.598273 0.801292i \(-0.295854\pi\)
0.598273 + 0.801292i \(0.295854\pi\)
\(492\) −12.5071 −0.563862
\(493\) −62.0871 −2.79626
\(494\) 0 0
\(495\) 3.06603 0.137808
\(496\) −5.41002 −0.242917
\(497\) 4.41526 0.198051
\(498\) −32.5545 −1.45880
\(499\) 25.0203 1.12006 0.560031 0.828472i \(-0.310789\pi\)
0.560031 + 0.828472i \(0.310789\pi\)
\(500\) −6.23417 −0.278800
\(501\) −55.8524 −2.49530
\(502\) −14.0957 −0.629120
\(503\) −23.4669 −1.04634 −0.523170 0.852229i \(-0.675251\pi\)
−0.523170 + 0.852229i \(0.675251\pi\)
\(504\) 13.1444 0.585500
\(505\) 12.4975 0.556133
\(506\) 1.70410 0.0757565
\(507\) 0 0
\(508\) 7.83875 0.347788
\(509\) 33.5449 1.48685 0.743426 0.668818i \(-0.233199\pi\)
0.743426 + 0.668818i \(0.233199\pi\)
\(510\) 54.7862 2.42597
\(511\) 15.1444 0.669951
\(512\) −1.00000 −0.0441942
\(513\) −2.04783 −0.0904140
\(514\) 6.35738 0.280412
\(515\) 45.2721 1.99493
\(516\) −23.3105 −1.02619
\(517\) 1.80096 0.0792062
\(518\) −4.77745 −0.209909
\(519\) 5.65922 0.248412
\(520\) 0 0
\(521\) 19.6253 0.859800 0.429900 0.902877i \(-0.358549\pi\)
0.429900 + 0.902877i \(0.358549\pi\)
\(522\) −31.1410 −1.36301
\(523\) 13.7276 0.600265 0.300133 0.953897i \(-0.402969\pi\)
0.300133 + 0.953897i \(0.402969\pi\)
\(524\) 15.1396 0.661378
\(525\) −24.9832 −1.09036
\(526\) 17.5360 0.764606
\(527\) 40.8377 1.77892
\(528\) 0.757172 0.0329517
\(529\) 11.3734 0.494494
\(530\) 35.9034 1.55955
\(531\) −21.0970 −0.915534
\(532\) 3.47175 0.150520
\(533\) 0 0
\(534\) 29.7886 1.28908
\(535\) 22.4260 0.969561
\(536\) −8.72438 −0.376836
\(537\) 19.1650 0.827030
\(538\) 21.8688 0.942832
\(539\) 1.46872 0.0632623
\(540\) 5.70548 0.245525
\(541\) 13.1985 0.567446 0.283723 0.958906i \(-0.408430\pi\)
0.283723 + 0.958906i \(0.408430\pi\)
\(542\) −27.7606 −1.19242
\(543\) 47.8629 2.05399
\(544\) 7.54852 0.323640
\(545\) −30.1592 −1.29188
\(546\) 0 0
\(547\) 34.7060 1.48392 0.741960 0.670444i \(-0.233896\pi\)
0.741960 + 0.670444i \(0.233896\pi\)
\(548\) 16.4501 0.702715
\(549\) 5.37129 0.229241
\(550\) −0.802921 −0.0342367
\(551\) −8.22507 −0.350400
\(552\) 15.2729 0.650058
\(553\) 34.3253 1.45966
\(554\) −22.6693 −0.963125
\(555\) −9.98748 −0.423945
\(556\) −12.7119 −0.539103
\(557\) 18.3939 0.779374 0.389687 0.920947i \(-0.372583\pi\)
0.389687 + 0.920947i \(0.372583\pi\)
\(558\) 20.4829 0.867112
\(559\) 0 0
\(560\) −9.67269 −0.408746
\(561\) −5.71553 −0.241310
\(562\) 14.3786 0.606525
\(563\) −14.5543 −0.613390 −0.306695 0.951808i \(-0.599223\pi\)
−0.306695 + 0.951808i \(0.599223\pi\)
\(564\) 16.1410 0.679660
\(565\) 2.36219 0.0993780
\(566\) −1.33350 −0.0560512
\(567\) 20.9128 0.878255
\(568\) 1.27177 0.0533621
\(569\) −1.47175 −0.0616991 −0.0308496 0.999524i \(-0.509821\pi\)
−0.0308496 + 0.999524i \(0.509821\pi\)
\(570\) 7.25786 0.303998
\(571\) −34.2772 −1.43446 −0.717229 0.696838i \(-0.754590\pi\)
−0.717229 + 0.696838i \(0.754590\pi\)
\(572\) 0 0
\(573\) 47.3912 1.97979
\(574\) −16.6684 −0.695725
\(575\) −16.1957 −0.675407
\(576\) 3.78611 0.157755
\(577\) −13.1722 −0.548368 −0.274184 0.961677i \(-0.588408\pi\)
−0.274184 + 0.961677i \(0.588408\pi\)
\(578\) −39.9802 −1.66296
\(579\) 36.4009 1.51277
\(580\) 22.9160 0.951533
\(581\) −43.3859 −1.79995
\(582\) −14.4278 −0.598053
\(583\) −3.74560 −0.155127
\(584\) 4.36219 0.180509
\(585\) 0 0
\(586\) 4.26172 0.176050
\(587\) −44.4228 −1.83353 −0.916763 0.399431i \(-0.869208\pi\)
−0.916763 + 0.399431i \(0.869208\pi\)
\(588\) 13.1633 0.542847
\(589\) 5.41002 0.222916
\(590\) 15.5248 0.639147
\(591\) −71.0791 −2.92380
\(592\) −1.37609 −0.0565570
\(593\) 3.58955 0.147405 0.0737025 0.997280i \(-0.476518\pi\)
0.0737025 + 0.997280i \(0.476518\pi\)
\(594\) −0.595221 −0.0244222
\(595\) 73.0145 2.99330
\(596\) −13.7447 −0.563003
\(597\) 39.0160 1.59682
\(598\) 0 0
\(599\) 18.9253 0.773267 0.386634 0.922233i \(-0.373638\pi\)
0.386634 + 0.922233i \(0.373638\pi\)
\(600\) −7.19613 −0.293781
\(601\) −38.5333 −1.57181 −0.785903 0.618350i \(-0.787801\pi\)
−0.785903 + 0.618350i \(0.787801\pi\)
\(602\) −31.0663 −1.26617
\(603\) 33.0315 1.34514
\(604\) 1.69682 0.0690427
\(605\) −30.4118 −1.23642
\(606\) −11.6852 −0.474679
\(607\) −27.0515 −1.09799 −0.548994 0.835826i \(-0.684989\pi\)
−0.548994 + 0.835826i \(0.684989\pi\)
\(608\) 1.00000 0.0405554
\(609\) −74.3873 −3.01433
\(610\) −3.95260 −0.160036
\(611\) 0 0
\(612\) −28.5795 −1.15526
\(613\) −13.6592 −0.551691 −0.275845 0.961202i \(-0.588958\pi\)
−0.275845 + 0.961202i \(0.588958\pi\)
\(614\) 15.7809 0.636864
\(615\) −34.8461 −1.40513
\(616\) 1.00910 0.0406577
\(617\) −24.2233 −0.975193 −0.487597 0.873069i \(-0.662126\pi\)
−0.487597 + 0.873069i \(0.662126\pi\)
\(618\) −42.3294 −1.70274
\(619\) 19.0506 0.765708 0.382854 0.923809i \(-0.374941\pi\)
0.382854 + 0.923809i \(0.374941\pi\)
\(620\) −15.0729 −0.605343
\(621\) −12.0062 −0.481792
\(622\) −0.504549 −0.0202306
\(623\) 39.6999 1.59054
\(624\) 0 0
\(625\) −31.1811 −1.24725
\(626\) 20.5871 0.822825
\(627\) −0.757172 −0.0302385
\(628\) −7.47656 −0.298347
\(629\) 10.3875 0.414175
\(630\) 36.6219 1.45905
\(631\) −22.6358 −0.901116 −0.450558 0.892747i \(-0.648775\pi\)
−0.450558 + 0.892747i \(0.648775\pi\)
\(632\) 9.88701 0.393284
\(633\) −1.18976 −0.0472886
\(634\) 2.43826 0.0968357
\(635\) 21.8396 0.866679
\(636\) −33.5697 −1.33113
\(637\) 0 0
\(638\) −2.39069 −0.0946484
\(639\) −4.81505 −0.190480
\(640\) −2.78611 −0.110131
\(641\) 34.9057 1.37869 0.689347 0.724432i \(-0.257898\pi\)
0.689347 + 0.724432i \(0.257898\pi\)
\(642\) −20.9683 −0.827554
\(643\) −15.7725 −0.622005 −0.311003 0.950409i \(-0.600665\pi\)
−0.311003 + 0.950409i \(0.600665\pi\)
\(644\) 20.3545 0.802079
\(645\) −64.9456 −2.55723
\(646\) −7.54852 −0.296993
\(647\) −38.6151 −1.51812 −0.759058 0.651023i \(-0.774340\pi\)
−0.759058 + 0.651023i \(0.774340\pi\)
\(648\) 6.02370 0.236633
\(649\) −1.61962 −0.0635755
\(650\) 0 0
\(651\) 48.9281 1.91764
\(652\) 5.74465 0.224978
\(653\) −50.6059 −1.98036 −0.990182 0.139785i \(-0.955359\pi\)
−0.990182 + 0.139785i \(0.955359\pi\)
\(654\) 28.1989 1.10266
\(655\) 42.1807 1.64814
\(656\) −4.80115 −0.187453
\(657\) −16.5157 −0.644340
\(658\) 21.5114 0.838603
\(659\) −31.4982 −1.22700 −0.613498 0.789696i \(-0.710238\pi\)
−0.613498 + 0.789696i \(0.710238\pi\)
\(660\) 2.10956 0.0821147
\(661\) −20.2665 −0.788277 −0.394138 0.919051i \(-0.628957\pi\)
−0.394138 + 0.919051i \(0.628957\pi\)
\(662\) −13.7558 −0.534636
\(663\) 0 0
\(664\) −12.4968 −0.484971
\(665\) 9.67269 0.375091
\(666\) 5.21003 0.201885
\(667\) −48.2226 −1.86719
\(668\) −21.4403 −0.829552
\(669\) −19.8148 −0.766084
\(670\) −24.3071 −0.939064
\(671\) 0.412353 0.0159187
\(672\) 9.04397 0.348879
\(673\) −21.5018 −0.828834 −0.414417 0.910087i \(-0.636015\pi\)
−0.414417 + 0.910087i \(0.636015\pi\)
\(674\) 10.0909 0.388685
\(675\) 5.65695 0.217736
\(676\) 0 0
\(677\) −17.9882 −0.691344 −0.345672 0.938355i \(-0.612349\pi\)
−0.345672 + 0.938355i \(0.612349\pi\)
\(678\) −2.20865 −0.0848226
\(679\) −19.2282 −0.737912
\(680\) 21.0310 0.806503
\(681\) −62.5003 −2.39502
\(682\) 1.57247 0.0602131
\(683\) 3.39909 0.130063 0.0650313 0.997883i \(-0.479285\pi\)
0.0650313 + 0.997883i \(0.479285\pi\)
\(684\) −3.78611 −0.144766
\(685\) 45.8319 1.75115
\(686\) −6.75925 −0.258070
\(687\) 18.2417 0.695963
\(688\) −8.94831 −0.341151
\(689\) 0 0
\(690\) 42.5520 1.61993
\(691\) −3.51802 −0.133832 −0.0669158 0.997759i \(-0.521316\pi\)
−0.0669158 + 0.997759i \(0.521316\pi\)
\(692\) 2.17243 0.0825835
\(693\) −3.82055 −0.145131
\(694\) 10.6335 0.403643
\(695\) −35.4166 −1.34343
\(696\) −21.4264 −0.812167
\(697\) 36.2416 1.37275
\(698\) 11.7572 0.445016
\(699\) −69.4839 −2.62812
\(700\) −9.59042 −0.362484
\(701\) −5.97219 −0.225567 −0.112783 0.993620i \(-0.535977\pi\)
−0.112783 + 0.993620i \(0.535977\pi\)
\(702\) 0 0
\(703\) 1.37609 0.0519003
\(704\) 0.290659 0.0109546
\(705\) 44.9707 1.69369
\(706\) −36.3071 −1.36643
\(707\) −15.5731 −0.585686
\(708\) −14.5157 −0.545534
\(709\) −5.39138 −0.202477 −0.101239 0.994862i \(-0.532281\pi\)
−0.101239 + 0.994862i \(0.532281\pi\)
\(710\) 3.54328 0.132977
\(711\) −37.4333 −1.40386
\(712\) 11.4351 0.428549
\(713\) 31.7183 1.18786
\(714\) −68.2687 −2.55489
\(715\) 0 0
\(716\) 7.35695 0.274942
\(717\) 7.50459 0.280264
\(718\) −18.5736 −0.693161
\(719\) 28.3167 1.05603 0.528017 0.849234i \(-0.322936\pi\)
0.528017 + 0.849234i \(0.322936\pi\)
\(720\) 10.5485 0.393120
\(721\) −56.4132 −2.10094
\(722\) −1.00000 −0.0372161
\(723\) −40.9884 −1.52438
\(724\) 18.3734 0.682841
\(725\) 22.7210 0.843838
\(726\) 28.4351 1.05533
\(727\) −28.8814 −1.07115 −0.535576 0.844487i \(-0.679905\pi\)
−0.535576 + 0.844487i \(0.679905\pi\)
\(728\) 0 0
\(729\) −38.8103 −1.43742
\(730\) 12.1535 0.449823
\(731\) 67.5465 2.49830
\(732\) 3.69569 0.136597
\(733\) −35.3948 −1.30734 −0.653668 0.756781i \(-0.726771\pi\)
−0.653668 + 0.756781i \(0.726771\pi\)
\(734\) −4.42873 −0.163467
\(735\) 36.6745 1.35276
\(736\) 5.86288 0.216109
\(737\) 2.53582 0.0934081
\(738\) 18.1777 0.669129
\(739\) −35.6747 −1.31231 −0.656157 0.754624i \(-0.727819\pi\)
−0.656157 + 0.754624i \(0.727819\pi\)
\(740\) −3.83394 −0.140939
\(741\) 0 0
\(742\) −44.7390 −1.64242
\(743\) 4.75604 0.174482 0.0872411 0.996187i \(-0.472195\pi\)
0.0872411 + 0.996187i \(0.472195\pi\)
\(744\) 14.0932 0.516681
\(745\) −38.2941 −1.40299
\(746\) −22.8590 −0.836928
\(747\) 47.3144 1.73114
\(748\) −2.19405 −0.0802223
\(749\) −27.9449 −1.02108
\(750\) 16.2401 0.593005
\(751\) 7.69796 0.280902 0.140451 0.990088i \(-0.455145\pi\)
0.140451 + 0.990088i \(0.455145\pi\)
\(752\) 6.19613 0.225950
\(753\) 36.7194 1.33813
\(754\) 0 0
\(755\) 4.72753 0.172053
\(756\) −7.10956 −0.258572
\(757\) −1.28610 −0.0467443 −0.0233721 0.999727i \(-0.507440\pi\)
−0.0233721 + 0.999727i \(0.507440\pi\)
\(758\) −7.55263 −0.274324
\(759\) −4.43921 −0.161133
\(760\) 2.78611 0.101063
\(761\) 9.65827 0.350112 0.175056 0.984558i \(-0.443989\pi\)
0.175056 + 0.984558i \(0.443989\pi\)
\(762\) −20.4201 −0.739741
\(763\) 37.5812 1.36053
\(764\) 18.1923 0.658173
\(765\) −79.6258 −2.87888
\(766\) −13.8900 −0.501866
\(767\) 0 0
\(768\) 2.60502 0.0940004
\(769\) 21.6051 0.779098 0.389549 0.921006i \(-0.372631\pi\)
0.389549 + 0.921006i \(0.372631\pi\)
\(770\) 2.81146 0.101318
\(771\) −16.5611 −0.596433
\(772\) 13.9734 0.502913
\(773\) −4.56376 −0.164147 −0.0820736 0.996626i \(-0.526154\pi\)
−0.0820736 + 0.996626i \(0.526154\pi\)
\(774\) 33.8793 1.21777
\(775\) −14.9447 −0.536830
\(776\) −5.53848 −0.198820
\(777\) 12.4453 0.446474
\(778\) 20.8952 0.749131
\(779\) 4.80115 0.172019
\(780\) 0 0
\(781\) −0.369651 −0.0132271
\(782\) −44.2561 −1.58259
\(783\) 16.8436 0.601939
\(784\) 5.05307 0.180467
\(785\) −20.8305 −0.743473
\(786\) −39.4390 −1.40674
\(787\) −0.533922 −0.0190323 −0.00951614 0.999955i \(-0.503029\pi\)
−0.00951614 + 0.999955i \(0.503029\pi\)
\(788\) −27.2855 −0.972004
\(789\) −45.6816 −1.62631
\(790\) 27.5463 0.980054
\(791\) −2.94351 −0.104659
\(792\) −1.10047 −0.0391034
\(793\) 0 0
\(794\) −12.5175 −0.444231
\(795\) −93.5290 −3.31713
\(796\) 14.9773 0.530855
\(797\) 25.6716 0.909333 0.454667 0.890662i \(-0.349758\pi\)
0.454667 + 0.890662i \(0.349758\pi\)
\(798\) −9.04397 −0.320153
\(799\) −46.7716 −1.65466
\(800\) −2.76241 −0.0976660
\(801\) −43.2946 −1.52974
\(802\) −12.1517 −0.429092
\(803\) −1.26791 −0.0447436
\(804\) 22.7271 0.801525
\(805\) 56.7098 1.99876
\(806\) 0 0
\(807\) −56.9686 −2.00539
\(808\) −4.48566 −0.157805
\(809\) −12.2404 −0.430348 −0.215174 0.976576i \(-0.569032\pi\)
−0.215174 + 0.976576i \(0.569032\pi\)
\(810\) 16.7827 0.589684
\(811\) 13.6378 0.478888 0.239444 0.970910i \(-0.423035\pi\)
0.239444 + 0.970910i \(0.423035\pi\)
\(812\) −28.5554 −1.00210
\(813\) 72.3169 2.53627
\(814\) 0.399974 0.0140191
\(815\) 16.0052 0.560639
\(816\) −19.6640 −0.688379
\(817\) 8.94831 0.313062
\(818\) 27.5549 0.963434
\(819\) 0 0
\(820\) −13.3765 −0.467129
\(821\) 1.60520 0.0560219 0.0280109 0.999608i \(-0.491083\pi\)
0.0280109 + 0.999608i \(0.491083\pi\)
\(822\) −42.8529 −1.49467
\(823\) −56.3737 −1.96506 −0.982531 0.186098i \(-0.940416\pi\)
−0.982531 + 0.186098i \(0.940416\pi\)
\(824\) −16.2492 −0.566068
\(825\) 2.09162 0.0728209
\(826\) −19.3454 −0.673112
\(827\) 3.73537 0.129892 0.0649458 0.997889i \(-0.479313\pi\)
0.0649458 + 0.997889i \(0.479313\pi\)
\(828\) −22.1975 −0.771417
\(829\) −13.3946 −0.465213 −0.232607 0.972571i \(-0.574725\pi\)
−0.232607 + 0.972571i \(0.574725\pi\)
\(830\) −34.8176 −1.20854
\(831\) 59.0538 2.04855
\(832\) 0 0
\(833\) −38.1432 −1.32158
\(834\) 33.1146 1.14666
\(835\) −59.7352 −2.06722
\(836\) −0.290659 −0.0100527
\(837\) −11.0788 −0.382940
\(838\) −18.1109 −0.625632
\(839\) 35.6369 1.23032 0.615161 0.788401i \(-0.289091\pi\)
0.615161 + 0.788401i \(0.289091\pi\)
\(840\) 25.1975 0.869397
\(841\) 38.6518 1.33282
\(842\) 10.8526 0.374007
\(843\) −37.4565 −1.29007
\(844\) −0.456717 −0.0157209
\(845\) 0 0
\(846\) −23.4592 −0.806545
\(847\) 37.8960 1.30212
\(848\) −12.8866 −0.442527
\(849\) 3.47379 0.119220
\(850\) 20.8521 0.715222
\(851\) 8.06786 0.276563
\(852\) −3.31297 −0.113501
\(853\) 43.8953 1.50295 0.751473 0.659764i \(-0.229344\pi\)
0.751473 + 0.659764i \(0.229344\pi\)
\(854\) 4.92531 0.168541
\(855\) −10.5485 −0.360752
\(856\) −8.04922 −0.275117
\(857\) 48.5026 1.65682 0.828408 0.560125i \(-0.189247\pi\)
0.828408 + 0.560125i \(0.189247\pi\)
\(858\) 0 0
\(859\) 2.44787 0.0835203 0.0417601 0.999128i \(-0.486703\pi\)
0.0417601 + 0.999128i \(0.486703\pi\)
\(860\) −24.9310 −0.850140
\(861\) 43.4214 1.47980
\(862\) −27.1681 −0.925351
\(863\) −8.84127 −0.300960 −0.150480 0.988613i \(-0.548082\pi\)
−0.150480 + 0.988613i \(0.548082\pi\)
\(864\) −2.04783 −0.0696686
\(865\) 6.05264 0.205796
\(866\) −1.63490 −0.0555563
\(867\) 104.149 3.53709
\(868\) 18.7823 0.637511
\(869\) −2.87375 −0.0974853
\(870\) −59.6964 −2.02390
\(871\) 0 0
\(872\) 10.8248 0.366575
\(873\) 20.9693 0.709703
\(874\) −5.86288 −0.198315
\(875\) 21.6435 0.731683
\(876\) −11.3636 −0.383940
\(877\) −11.2932 −0.381343 −0.190672 0.981654i \(-0.561067\pi\)
−0.190672 + 0.981654i \(0.561067\pi\)
\(878\) 37.8206 1.27638
\(879\) −11.1019 −0.374456
\(880\) 0.809809 0.0272986
\(881\) −13.5880 −0.457790 −0.228895 0.973451i \(-0.573511\pi\)
−0.228895 + 0.973451i \(0.573511\pi\)
\(882\) −19.1315 −0.644191
\(883\) 17.5315 0.589983 0.294991 0.955500i \(-0.404683\pi\)
0.294991 + 0.955500i \(0.404683\pi\)
\(884\) 0 0
\(885\) −40.4424 −1.35946
\(886\) −9.70049 −0.325894
\(887\) −30.0684 −1.00960 −0.504798 0.863237i \(-0.668433\pi\)
−0.504798 + 0.863237i \(0.668433\pi\)
\(888\) 3.58474 0.120296
\(889\) −27.2142 −0.912735
\(890\) 31.8595 1.06793
\(891\) −1.75084 −0.0586555
\(892\) −7.60640 −0.254681
\(893\) −6.19613 −0.207346
\(894\) 35.8050 1.19750
\(895\) 20.4973 0.685148
\(896\) 3.47175 0.115983
\(897\) 0 0
\(898\) −22.4381 −0.748769
\(899\) −44.4978 −1.48408
\(900\) 10.4588 0.348627
\(901\) 97.2746 3.24069
\(902\) 1.39550 0.0464650
\(903\) 80.9283 2.69313
\(904\) −0.847844 −0.0281989
\(905\) 51.1902 1.70162
\(906\) −4.42025 −0.146853
\(907\) −35.0422 −1.16356 −0.581778 0.813348i \(-0.697643\pi\)
−0.581778 + 0.813348i \(0.697643\pi\)
\(908\) −23.9923 −0.796212
\(909\) 16.9832 0.563297
\(910\) 0 0
\(911\) 0.714766 0.0236813 0.0118406 0.999930i \(-0.496231\pi\)
0.0118406 + 0.999930i \(0.496231\pi\)
\(912\) −2.60502 −0.0862607
\(913\) 3.63232 0.120212
\(914\) −2.85746 −0.0945162
\(915\) 10.2966 0.340395
\(916\) 7.00252 0.231370
\(917\) −52.5611 −1.73572
\(918\) 15.4581 0.510193
\(919\) 37.4522 1.23543 0.617717 0.786400i \(-0.288058\pi\)
0.617717 + 0.786400i \(0.288058\pi\)
\(920\) 16.3346 0.538537
\(921\) −41.1094 −1.35460
\(922\) 27.5356 0.906836
\(923\) 0 0
\(924\) −2.62871 −0.0864784
\(925\) −3.80133 −0.124987
\(926\) 17.6784 0.580947
\(927\) 61.5213 2.02062
\(928\) −8.22507 −0.270001
\(929\) −10.9426 −0.359016 −0.179508 0.983756i \(-0.557451\pi\)
−0.179508 + 0.983756i \(0.557451\pi\)
\(930\) 39.2652 1.28756
\(931\) −5.05307 −0.165608
\(932\) −26.6731 −0.873707
\(933\) 1.31436 0.0430301
\(934\) −29.7663 −0.973984
\(935\) −6.11286 −0.199912
\(936\) 0 0
\(937\) −18.1216 −0.592008 −0.296004 0.955187i \(-0.595654\pi\)
−0.296004 + 0.955187i \(0.595654\pi\)
\(938\) 30.2889 0.988967
\(939\) −53.6297 −1.75014
\(940\) 17.2631 0.563060
\(941\) 39.9721 1.30305 0.651526 0.758626i \(-0.274129\pi\)
0.651526 + 0.758626i \(0.274129\pi\)
\(942\) 19.4766 0.634581
\(943\) 28.1485 0.916643
\(944\) −5.57222 −0.181360
\(945\) −19.8080 −0.644355
\(946\) 2.60091 0.0845629
\(947\) −14.1216 −0.458892 −0.229446 0.973321i \(-0.573691\pi\)
−0.229446 + 0.973321i \(0.573691\pi\)
\(948\) −25.7558 −0.836511
\(949\) 0 0
\(950\) 2.76241 0.0896245
\(951\) −6.35171 −0.205968
\(952\) −26.2066 −0.849361
\(953\) −60.3061 −1.95351 −0.976753 0.214370i \(-0.931230\pi\)
−0.976753 + 0.214370i \(0.931230\pi\)
\(954\) 48.7900 1.57963
\(955\) 50.6857 1.64015
\(956\) 2.88082 0.0931725
\(957\) 6.22779 0.201316
\(958\) −29.1907 −0.943108
\(959\) −57.1108 −1.84420
\(960\) 7.25786 0.234247
\(961\) −1.73169 −0.0558609
\(962\) 0 0
\(963\) 30.4752 0.982050
\(964\) −15.7344 −0.506772
\(965\) 38.9314 1.25325
\(966\) −53.0237 −1.70601
\(967\) −1.06633 −0.0342910 −0.0171455 0.999853i \(-0.505458\pi\)
−0.0171455 + 0.999853i \(0.505458\pi\)
\(968\) 10.9155 0.350838
\(969\) 19.6640 0.631700
\(970\) −15.4308 −0.495453
\(971\) 41.1051 1.31912 0.659562 0.751650i \(-0.270742\pi\)
0.659562 + 0.751650i \(0.270742\pi\)
\(972\) −21.8353 −0.700368
\(973\) 44.1324 1.41482
\(974\) 23.4246 0.750573
\(975\) 0 0
\(976\) 1.41868 0.0454109
\(977\) −29.1331 −0.932049 −0.466025 0.884772i \(-0.654314\pi\)
−0.466025 + 0.884772i \(0.654314\pi\)
\(978\) −14.9649 −0.478525
\(979\) −3.32372 −0.106226
\(980\) 14.0784 0.449719
\(981\) −40.9841 −1.30852
\(982\) −26.5137 −0.846086
\(983\) −45.1403 −1.43975 −0.719877 0.694102i \(-0.755802\pi\)
−0.719877 + 0.694102i \(0.755802\pi\)
\(984\) 12.5071 0.398711
\(985\) −76.0203 −2.42221
\(986\) 62.0871 1.97726
\(987\) −56.0376 −1.78370
\(988\) 0 0
\(989\) 52.4629 1.66822
\(990\) −3.06603 −0.0974447
\(991\) −26.4501 −0.840216 −0.420108 0.907474i \(-0.638008\pi\)
−0.420108 + 0.907474i \(0.638008\pi\)
\(992\) 5.41002 0.171768
\(993\) 35.8342 1.13716
\(994\) −4.41526 −0.140044
\(995\) 41.7283 1.32288
\(996\) 32.5545 1.03153
\(997\) 49.1290 1.55593 0.777966 0.628307i \(-0.216252\pi\)
0.777966 + 0.628307i \(0.216252\pi\)
\(998\) −25.0203 −0.792003
\(999\) −2.81800 −0.0891576
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 6422.2.a.ba.1.4 4
13.12 even 2 494.2.a.h.1.4 4
39.38 odd 2 4446.2.a.bl.1.4 4
52.51 odd 2 3952.2.a.u.1.1 4
247.246 odd 2 9386.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.a.h.1.4 4 13.12 even 2
3952.2.a.u.1.1 4 52.51 odd 2
4446.2.a.bl.1.4 4 39.38 odd 2
6422.2.a.ba.1.4 4 1.1 even 1 trivial
9386.2.a.bf.1.1 4 247.246 odd 2