Properties

Label 1502.2.a.h.1.3
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
Analytic rank $0$
Dimension $19$
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] = [1502,2,Mod(1,1502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1502.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1502 = 2 \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1502.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: \(11.9935303836\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} - 11338 x^{11} - 56744 x^{10} + 50183 x^{9} + 120237 x^{8} - 102992 x^{7} + \cdots + 4222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.13636\) of defining polynomial
Character \(\chi\) \(=\) 1502.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.13636 q^{3} +1.00000 q^{4} -2.42355 q^{5} +2.13636 q^{6} +4.87888 q^{7} -1.00000 q^{8} +1.56402 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.13636 q^{3} +1.00000 q^{4} -2.42355 q^{5} +2.13636 q^{6} +4.87888 q^{7} -1.00000 q^{8} +1.56402 q^{9} +2.42355 q^{10} +4.17648 q^{11} -2.13636 q^{12} +3.80781 q^{13} -4.87888 q^{14} +5.17758 q^{15} +1.00000 q^{16} -3.58856 q^{17} -1.56402 q^{18} +2.08604 q^{19} -2.42355 q^{20} -10.4230 q^{21} -4.17648 q^{22} +3.33317 q^{23} +2.13636 q^{24} +0.873611 q^{25} -3.80781 q^{26} +3.06776 q^{27} +4.87888 q^{28} -9.07425 q^{29} -5.17758 q^{30} -1.55922 q^{31} -1.00000 q^{32} -8.92246 q^{33} +3.58856 q^{34} -11.8242 q^{35} +1.56402 q^{36} -2.16036 q^{37} -2.08604 q^{38} -8.13485 q^{39} +2.42355 q^{40} +8.24485 q^{41} +10.4230 q^{42} +2.98863 q^{43} +4.17648 q^{44} -3.79049 q^{45} -3.33317 q^{46} -6.93762 q^{47} -2.13636 q^{48} +16.8035 q^{49} -0.873611 q^{50} +7.66645 q^{51} +3.80781 q^{52} -4.05475 q^{53} -3.06776 q^{54} -10.1219 q^{55} -4.87888 q^{56} -4.45652 q^{57} +9.07425 q^{58} +11.1211 q^{59} +5.17758 q^{60} -8.56035 q^{61} +1.55922 q^{62} +7.63068 q^{63} +1.00000 q^{64} -9.22843 q^{65} +8.92246 q^{66} +1.12384 q^{67} -3.58856 q^{68} -7.12085 q^{69} +11.8242 q^{70} +2.87963 q^{71} -1.56402 q^{72} +5.17680 q^{73} +2.16036 q^{74} -1.86634 q^{75} +2.08604 q^{76} +20.3766 q^{77} +8.13485 q^{78} +5.93193 q^{79} -2.42355 q^{80} -11.2459 q^{81} -8.24485 q^{82} -2.08785 q^{83} -10.4230 q^{84} +8.69707 q^{85} -2.98863 q^{86} +19.3858 q^{87} -4.17648 q^{88} +5.27367 q^{89} +3.79049 q^{90} +18.5779 q^{91} +3.33317 q^{92} +3.33104 q^{93} +6.93762 q^{94} -5.05562 q^{95} +2.13636 q^{96} +8.80362 q^{97} -16.8035 q^{98} +6.53211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.13636 −1.23343 −0.616713 0.787188i \(-0.711536\pi\)
−0.616713 + 0.787188i \(0.711536\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.42355 −1.08385 −0.541923 0.840428i \(-0.682304\pi\)
−0.541923 + 0.840428i \(0.682304\pi\)
\(6\) 2.13636 0.872164
\(7\) 4.87888 1.84404 0.922022 0.387137i \(-0.126536\pi\)
0.922022 + 0.387137i \(0.126536\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.56402 0.521341
\(10\) 2.42355 0.766395
\(11\) 4.17648 1.25926 0.629629 0.776896i \(-0.283207\pi\)
0.629629 + 0.776896i \(0.283207\pi\)
\(12\) −2.13636 −0.616713
\(13\) 3.80781 1.05610 0.528048 0.849214i \(-0.322924\pi\)
0.528048 + 0.849214i \(0.322924\pi\)
\(14\) −4.87888 −1.30394
\(15\) 5.17758 1.33684
\(16\) 1.00000 0.250000
\(17\) −3.58856 −0.870354 −0.435177 0.900345i \(-0.643314\pi\)
−0.435177 + 0.900345i \(0.643314\pi\)
\(18\) −1.56402 −0.368643
\(19\) 2.08604 0.478570 0.239285 0.970949i \(-0.423087\pi\)
0.239285 + 0.970949i \(0.423087\pi\)
\(20\) −2.42355 −0.541923
\(21\) −10.4230 −2.27449
\(22\) −4.17648 −0.890429
\(23\) 3.33317 0.695015 0.347507 0.937677i \(-0.387028\pi\)
0.347507 + 0.937677i \(0.387028\pi\)
\(24\) 2.13636 0.436082
\(25\) 0.873611 0.174722
\(26\) −3.80781 −0.746773
\(27\) 3.06776 0.590391
\(28\) 4.87888 0.922022
\(29\) −9.07425 −1.68505 −0.842523 0.538660i \(-0.818931\pi\)
−0.842523 + 0.538660i \(0.818931\pi\)
\(30\) −5.17758 −0.945292
\(31\) −1.55922 −0.280044 −0.140022 0.990148i \(-0.544717\pi\)
−0.140022 + 0.990148i \(0.544717\pi\)
\(32\) −1.00000 −0.176777
\(33\) −8.92246 −1.55320
\(34\) 3.58856 0.615433
\(35\) −11.8242 −1.99866
\(36\) 1.56402 0.260670
\(37\) −2.16036 −0.355161 −0.177580 0.984106i \(-0.556827\pi\)
−0.177580 + 0.984106i \(0.556827\pi\)
\(38\) −2.08604 −0.338400
\(39\) −8.13485 −1.30262
\(40\) 2.42355 0.383197
\(41\) 8.24485 1.28763 0.643815 0.765182i \(-0.277351\pi\)
0.643815 + 0.765182i \(0.277351\pi\)
\(42\) 10.4230 1.60831
\(43\) 2.98863 0.455761 0.227881 0.973689i \(-0.426820\pi\)
0.227881 + 0.973689i \(0.426820\pi\)
\(44\) 4.17648 0.629629
\(45\) −3.79049 −0.565053
\(46\) −3.33317 −0.491450
\(47\) −6.93762 −1.01196 −0.505978 0.862546i \(-0.668868\pi\)
−0.505978 + 0.862546i \(0.668868\pi\)
\(48\) −2.13636 −0.308357
\(49\) 16.8035 2.40050
\(50\) −0.873611 −0.123547
\(51\) 7.66645 1.07352
\(52\) 3.80781 0.528048
\(53\) −4.05475 −0.556962 −0.278481 0.960442i \(-0.589831\pi\)
−0.278481 + 0.960442i \(0.589831\pi\)
\(54\) −3.06776 −0.417470
\(55\) −10.1219 −1.36484
\(56\) −4.87888 −0.651968
\(57\) −4.45652 −0.590280
\(58\) 9.07425 1.19151
\(59\) 11.1211 1.44784 0.723921 0.689883i \(-0.242338\pi\)
0.723921 + 0.689883i \(0.242338\pi\)
\(60\) 5.17758 0.668422
\(61\) −8.56035 −1.09604 −0.548020 0.836465i \(-0.684618\pi\)
−0.548020 + 0.836465i \(0.684618\pi\)
\(62\) 1.55922 0.198021
\(63\) 7.63068 0.961375
\(64\) 1.00000 0.125000
\(65\) −9.22843 −1.14465
\(66\) 8.92246 1.09828
\(67\) 1.12384 0.137299 0.0686495 0.997641i \(-0.478131\pi\)
0.0686495 + 0.997641i \(0.478131\pi\)
\(68\) −3.58856 −0.435177
\(69\) −7.12085 −0.857249
\(70\) 11.8242 1.41327
\(71\) 2.87963 0.341749 0.170875 0.985293i \(-0.445341\pi\)
0.170875 + 0.985293i \(0.445341\pi\)
\(72\) −1.56402 −0.184322
\(73\) 5.17680 0.605898 0.302949 0.953007i \(-0.402029\pi\)
0.302949 + 0.953007i \(0.402029\pi\)
\(74\) 2.16036 0.251137
\(75\) −1.86634 −0.215507
\(76\) 2.08604 0.239285
\(77\) 20.3766 2.32213
\(78\) 8.13485 0.921090
\(79\) 5.93193 0.667394 0.333697 0.942680i \(-0.391704\pi\)
0.333697 + 0.942680i \(0.391704\pi\)
\(80\) −2.42355 −0.270961
\(81\) −11.2459 −1.24954
\(82\) −8.24485 −0.910491
\(83\) −2.08785 −0.229171 −0.114586 0.993413i \(-0.536554\pi\)
−0.114586 + 0.993413i \(0.536554\pi\)
\(84\) −10.4230 −1.13725
\(85\) 8.69707 0.943329
\(86\) −2.98863 −0.322272
\(87\) 19.3858 2.07838
\(88\) −4.17648 −0.445215
\(89\) 5.27367 0.559007 0.279504 0.960145i \(-0.409830\pi\)
0.279504 + 0.960145i \(0.409830\pi\)
\(90\) 3.79049 0.399553
\(91\) 18.5779 1.94749
\(92\) 3.33317 0.347507
\(93\) 3.33104 0.345413
\(94\) 6.93762 0.715561
\(95\) −5.05562 −0.518696
\(96\) 2.13636 0.218041
\(97\) 8.80362 0.893872 0.446936 0.894566i \(-0.352515\pi\)
0.446936 + 0.894566i \(0.352515\pi\)
\(98\) −16.8035 −1.69741
\(99\) 6.53211 0.656502
\(100\) 0.873611 0.0873611
\(101\) −1.40231 −0.139535 −0.0697673 0.997563i \(-0.522226\pi\)
−0.0697673 + 0.997563i \(0.522226\pi\)
\(102\) −7.66645 −0.759091
\(103\) 17.5860 1.73280 0.866401 0.499349i \(-0.166427\pi\)
0.866401 + 0.499349i \(0.166427\pi\)
\(104\) −3.80781 −0.373387
\(105\) 25.2608 2.46520
\(106\) 4.05475 0.393832
\(107\) −2.16657 −0.209450 −0.104725 0.994501i \(-0.533396\pi\)
−0.104725 + 0.994501i \(0.533396\pi\)
\(108\) 3.06776 0.295196
\(109\) −14.8582 −1.42316 −0.711578 0.702607i \(-0.752019\pi\)
−0.711578 + 0.702607i \(0.752019\pi\)
\(110\) 10.1219 0.965088
\(111\) 4.61530 0.438065
\(112\) 4.87888 0.461011
\(113\) 20.4505 1.92382 0.961910 0.273367i \(-0.0881374\pi\)
0.961910 + 0.273367i \(0.0881374\pi\)
\(114\) 4.45652 0.417391
\(115\) −8.07812 −0.753289
\(116\) −9.07425 −0.842523
\(117\) 5.95550 0.550586
\(118\) −11.1211 −1.02378
\(119\) −17.5082 −1.60497
\(120\) −5.17758 −0.472646
\(121\) 6.44301 0.585729
\(122\) 8.56035 0.775017
\(123\) −17.6139 −1.58820
\(124\) −1.55922 −0.140022
\(125\) 10.0005 0.894474
\(126\) −7.63068 −0.679795
\(127\) −14.8677 −1.31930 −0.659648 0.751574i \(-0.729295\pi\)
−0.659648 + 0.751574i \(0.729295\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.38478 −0.562148
\(130\) 9.22843 0.809387
\(131\) −8.50866 −0.743405 −0.371703 0.928352i \(-0.621226\pi\)
−0.371703 + 0.928352i \(0.621226\pi\)
\(132\) −8.92246 −0.776601
\(133\) 10.1775 0.882504
\(134\) −1.12384 −0.0970851
\(135\) −7.43489 −0.639893
\(136\) 3.58856 0.307717
\(137\) 13.1946 1.12729 0.563644 0.826018i \(-0.309399\pi\)
0.563644 + 0.826018i \(0.309399\pi\)
\(138\) 7.12085 0.606167
\(139\) −4.26167 −0.361470 −0.180735 0.983532i \(-0.557848\pi\)
−0.180735 + 0.983532i \(0.557848\pi\)
\(140\) −11.8242 −0.999330
\(141\) 14.8212 1.24817
\(142\) −2.87963 −0.241653
\(143\) 15.9033 1.32990
\(144\) 1.56402 0.130335
\(145\) 21.9919 1.82633
\(146\) −5.17680 −0.428435
\(147\) −35.8983 −2.96084
\(148\) −2.16036 −0.177580
\(149\) −12.4180 −1.01732 −0.508661 0.860967i \(-0.669859\pi\)
−0.508661 + 0.860967i \(0.669859\pi\)
\(150\) 1.86634 0.152386
\(151\) 21.1521 1.72133 0.860666 0.509169i \(-0.170047\pi\)
0.860666 + 0.509169i \(0.170047\pi\)
\(152\) −2.08604 −0.169200
\(153\) −5.61259 −0.453751
\(154\) −20.3766 −1.64199
\(155\) 3.77885 0.303524
\(156\) −8.13485 −0.651309
\(157\) 17.2057 1.37316 0.686582 0.727052i \(-0.259110\pi\)
0.686582 + 0.727052i \(0.259110\pi\)
\(158\) −5.93193 −0.471919
\(159\) 8.66239 0.686972
\(160\) 2.42355 0.191599
\(161\) 16.2622 1.28164
\(162\) 11.2459 0.883561
\(163\) 1.80306 0.141226 0.0706132 0.997504i \(-0.477504\pi\)
0.0706132 + 0.997504i \(0.477504\pi\)
\(164\) 8.24485 0.643815
\(165\) 21.6241 1.68343
\(166\) 2.08785 0.162049
\(167\) −0.507845 −0.0392982 −0.0196491 0.999807i \(-0.506255\pi\)
−0.0196491 + 0.999807i \(0.506255\pi\)
\(168\) 10.4230 0.804155
\(169\) 1.49943 0.115341
\(170\) −8.69707 −0.667035
\(171\) 3.26261 0.249498
\(172\) 2.98863 0.227881
\(173\) 16.6498 1.26586 0.632932 0.774208i \(-0.281851\pi\)
0.632932 + 0.774208i \(0.281851\pi\)
\(174\) −19.3858 −1.46964
\(175\) 4.26225 0.322195
\(176\) 4.17648 0.314814
\(177\) −23.7586 −1.78581
\(178\) −5.27367 −0.395278
\(179\) −11.9707 −0.894732 −0.447366 0.894351i \(-0.647638\pi\)
−0.447366 + 0.894351i \(0.647638\pi\)
\(180\) −3.79049 −0.282526
\(181\) 2.20077 0.163582 0.0817911 0.996649i \(-0.473936\pi\)
0.0817911 + 0.996649i \(0.473936\pi\)
\(182\) −18.5779 −1.37708
\(183\) 18.2880 1.35188
\(184\) −3.33317 −0.245725
\(185\) 5.23574 0.384939
\(186\) −3.33104 −0.244244
\(187\) −14.9876 −1.09600
\(188\) −6.93762 −0.505978
\(189\) 14.9673 1.08871
\(190\) 5.05562 0.366773
\(191\) −5.90750 −0.427452 −0.213726 0.976894i \(-0.568560\pi\)
−0.213726 + 0.976894i \(0.568560\pi\)
\(192\) −2.13636 −0.154178
\(193\) −13.4643 −0.969178 −0.484589 0.874742i \(-0.661031\pi\)
−0.484589 + 0.874742i \(0.661031\pi\)
\(194\) −8.80362 −0.632063
\(195\) 19.7152 1.41184
\(196\) 16.8035 1.20025
\(197\) 12.7144 0.905867 0.452933 0.891544i \(-0.350378\pi\)
0.452933 + 0.891544i \(0.350378\pi\)
\(198\) −6.53211 −0.464217
\(199\) 26.8112 1.90059 0.950296 0.311347i \(-0.100780\pi\)
0.950296 + 0.311347i \(0.100780\pi\)
\(200\) −0.873611 −0.0617736
\(201\) −2.40093 −0.169348
\(202\) 1.40231 0.0986659
\(203\) −44.2722 −3.10730
\(204\) 7.66645 0.536759
\(205\) −19.9818 −1.39559
\(206\) −17.5860 −1.22528
\(207\) 5.21316 0.362339
\(208\) 3.80781 0.264024
\(209\) 8.71230 0.602642
\(210\) −25.2608 −1.74316
\(211\) −10.9857 −0.756286 −0.378143 0.925747i \(-0.623437\pi\)
−0.378143 + 0.925747i \(0.623437\pi\)
\(212\) −4.05475 −0.278481
\(213\) −6.15192 −0.421523
\(214\) 2.16657 0.148104
\(215\) −7.24310 −0.493975
\(216\) −3.06776 −0.208735
\(217\) −7.60724 −0.516413
\(218\) 14.8582 1.00632
\(219\) −11.0595 −0.747331
\(220\) −10.1219 −0.682420
\(221\) −13.6646 −0.919178
\(222\) −4.61530 −0.309758
\(223\) 27.6057 1.84861 0.924307 0.381650i \(-0.124644\pi\)
0.924307 + 0.381650i \(0.124644\pi\)
\(224\) −4.87888 −0.325984
\(225\) 1.36635 0.0910898
\(226\) −20.4505 −1.36035
\(227\) −26.2090 −1.73955 −0.869777 0.493444i \(-0.835738\pi\)
−0.869777 + 0.493444i \(0.835738\pi\)
\(228\) −4.45652 −0.295140
\(229\) −22.1584 −1.46427 −0.732135 0.681159i \(-0.761476\pi\)
−0.732135 + 0.681159i \(0.761476\pi\)
\(230\) 8.07812 0.532656
\(231\) −43.5316 −2.86417
\(232\) 9.07425 0.595754
\(233\) 14.4276 0.945182 0.472591 0.881282i \(-0.343319\pi\)
0.472591 + 0.881282i \(0.343319\pi\)
\(234\) −5.95550 −0.389323
\(235\) 16.8137 1.09680
\(236\) 11.1211 0.723921
\(237\) −12.6727 −0.823181
\(238\) 17.5082 1.13489
\(239\) 7.47287 0.483380 0.241690 0.970354i \(-0.422298\pi\)
0.241690 + 0.970354i \(0.422298\pi\)
\(240\) 5.17758 0.334211
\(241\) −5.81160 −0.374358 −0.187179 0.982326i \(-0.559934\pi\)
−0.187179 + 0.982326i \(0.559934\pi\)
\(242\) −6.44301 −0.414173
\(243\) 14.8220 0.950830
\(244\) −8.56035 −0.548020
\(245\) −40.7242 −2.60177
\(246\) 17.6139 1.12302
\(247\) 7.94323 0.505416
\(248\) 1.55922 0.0990104
\(249\) 4.46039 0.282666
\(250\) −10.0005 −0.632489
\(251\) 31.3685 1.97996 0.989982 0.141194i \(-0.0450941\pi\)
0.989982 + 0.141194i \(0.0450941\pi\)
\(252\) 7.63068 0.480688
\(253\) 13.9209 0.875202
\(254\) 14.8677 0.932884
\(255\) −18.5800 −1.16353
\(256\) 1.00000 0.0625000
\(257\) 1.79913 0.112227 0.0561133 0.998424i \(-0.482129\pi\)
0.0561133 + 0.998424i \(0.482129\pi\)
\(258\) 6.38478 0.397499
\(259\) −10.5401 −0.654932
\(260\) −9.22843 −0.572323
\(261\) −14.1923 −0.878483
\(262\) 8.50866 0.525667
\(263\) −14.5641 −0.898063 −0.449032 0.893516i \(-0.648231\pi\)
−0.449032 + 0.893516i \(0.648231\pi\)
\(264\) 8.92246 0.549139
\(265\) 9.82690 0.603662
\(266\) −10.1775 −0.624024
\(267\) −11.2664 −0.689494
\(268\) 1.12384 0.0686495
\(269\) 9.22945 0.562729 0.281365 0.959601i \(-0.409213\pi\)
0.281365 + 0.959601i \(0.409213\pi\)
\(270\) 7.43489 0.452473
\(271\) −21.8990 −1.33027 −0.665134 0.746724i \(-0.731625\pi\)
−0.665134 + 0.746724i \(0.731625\pi\)
\(272\) −3.58856 −0.217588
\(273\) −39.6890 −2.40209
\(274\) −13.1946 −0.797114
\(275\) 3.64862 0.220020
\(276\) −7.12085 −0.428625
\(277\) 27.9450 1.67905 0.839525 0.543321i \(-0.182833\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(278\) 4.26167 0.255598
\(279\) −2.43865 −0.145998
\(280\) 11.8242 0.706633
\(281\) 11.7874 0.703177 0.351588 0.936155i \(-0.385642\pi\)
0.351588 + 0.936155i \(0.385642\pi\)
\(282\) −14.8212 −0.882592
\(283\) 24.8420 1.47670 0.738352 0.674415i \(-0.235604\pi\)
0.738352 + 0.674415i \(0.235604\pi\)
\(284\) 2.87963 0.170875
\(285\) 10.8006 0.639773
\(286\) −15.9033 −0.940380
\(287\) 40.2257 2.37445
\(288\) −1.56402 −0.0921609
\(289\) −4.12223 −0.242484
\(290\) −21.9919 −1.29141
\(291\) −18.8077 −1.10252
\(292\) 5.17680 0.302949
\(293\) −6.33434 −0.370056 −0.185028 0.982733i \(-0.559238\pi\)
−0.185028 + 0.982733i \(0.559238\pi\)
\(294\) 35.8983 2.09363
\(295\) −26.9525 −1.56924
\(296\) 2.16036 0.125568
\(297\) 12.8125 0.743454
\(298\) 12.4180 0.719355
\(299\) 12.6921 0.734003
\(300\) −1.86634 −0.107753
\(301\) 14.5812 0.840444
\(302\) −21.1521 −1.21717
\(303\) 2.99583 0.172106
\(304\) 2.08604 0.119642
\(305\) 20.7465 1.18794
\(306\) 5.61259 0.320850
\(307\) −6.64633 −0.379326 −0.189663 0.981849i \(-0.560740\pi\)
−0.189663 + 0.981849i \(0.560740\pi\)
\(308\) 20.3766 1.16106
\(309\) −37.5700 −2.13728
\(310\) −3.77885 −0.214624
\(311\) 1.93843 0.109918 0.0549592 0.998489i \(-0.482497\pi\)
0.0549592 + 0.998489i \(0.482497\pi\)
\(312\) 8.13485 0.460545
\(313\) 12.7744 0.722054 0.361027 0.932555i \(-0.382426\pi\)
0.361027 + 0.932555i \(0.382426\pi\)
\(314\) −17.2057 −0.970974
\(315\) −18.4934 −1.04198
\(316\) 5.93193 0.333697
\(317\) 4.51399 0.253531 0.126765 0.991933i \(-0.459540\pi\)
0.126765 + 0.991933i \(0.459540\pi\)
\(318\) −8.66239 −0.485763
\(319\) −37.8985 −2.12191
\(320\) −2.42355 −0.135481
\(321\) 4.62857 0.258341
\(322\) −16.2622 −0.906255
\(323\) −7.48587 −0.416525
\(324\) −11.2459 −0.624772
\(325\) 3.32655 0.184524
\(326\) −1.80306 −0.0998621
\(327\) 31.7424 1.75536
\(328\) −8.24485 −0.455246
\(329\) −33.8478 −1.86609
\(330\) −21.6241 −1.19037
\(331\) −17.9117 −0.984516 −0.492258 0.870449i \(-0.663828\pi\)
−0.492258 + 0.870449i \(0.663828\pi\)
\(332\) −2.08785 −0.114586
\(333\) −3.37885 −0.185160
\(334\) 0.507845 0.0277880
\(335\) −2.72369 −0.148811
\(336\) −10.4230 −0.568623
\(337\) −25.7627 −1.40338 −0.701691 0.712482i \(-0.747571\pi\)
−0.701691 + 0.712482i \(0.747571\pi\)
\(338\) −1.49943 −0.0815583
\(339\) −43.6895 −2.37289
\(340\) 8.69707 0.471665
\(341\) −6.51204 −0.352647
\(342\) −3.26261 −0.176422
\(343\) 47.8302 2.58259
\(344\) −2.98863 −0.161136
\(345\) 17.2578 0.929126
\(346\) −16.6498 −0.895101
\(347\) 9.33419 0.501086 0.250543 0.968106i \(-0.419391\pi\)
0.250543 + 0.968106i \(0.419391\pi\)
\(348\) 19.3858 1.03919
\(349\) −23.3987 −1.25250 −0.626251 0.779621i \(-0.715412\pi\)
−0.626251 + 0.779621i \(0.715412\pi\)
\(350\) −4.26225 −0.227827
\(351\) 11.6815 0.623510
\(352\) −4.17648 −0.222607
\(353\) 16.7459 0.891296 0.445648 0.895208i \(-0.352973\pi\)
0.445648 + 0.895208i \(0.352973\pi\)
\(354\) 23.7586 1.26276
\(355\) −6.97894 −0.370404
\(356\) 5.27367 0.279504
\(357\) 37.4037 1.97961
\(358\) 11.9707 0.632671
\(359\) 11.2319 0.592796 0.296398 0.955065i \(-0.404215\pi\)
0.296398 + 0.955065i \(0.404215\pi\)
\(360\) 3.79049 0.199776
\(361\) −14.6485 −0.770971
\(362\) −2.20077 −0.115670
\(363\) −13.7646 −0.722453
\(364\) 18.5779 0.973745
\(365\) −12.5462 −0.656701
\(366\) −18.2880 −0.955927
\(367\) −36.8380 −1.92293 −0.961464 0.274931i \(-0.911345\pi\)
−0.961464 + 0.274931i \(0.911345\pi\)
\(368\) 3.33317 0.173754
\(369\) 12.8951 0.671293
\(370\) −5.23574 −0.272193
\(371\) −19.7826 −1.02706
\(372\) 3.33104 0.172707
\(373\) 17.1113 0.885991 0.442995 0.896524i \(-0.353916\pi\)
0.442995 + 0.896524i \(0.353916\pi\)
\(374\) 14.9876 0.774988
\(375\) −21.3647 −1.10327
\(376\) 6.93762 0.357780
\(377\) −34.5530 −1.77957
\(378\) −14.9673 −0.769833
\(379\) 22.7500 1.16859 0.584293 0.811543i \(-0.301372\pi\)
0.584293 + 0.811543i \(0.301372\pi\)
\(380\) −5.05562 −0.259348
\(381\) 31.7628 1.62726
\(382\) 5.90750 0.302254
\(383\) 14.2435 0.727806 0.363903 0.931437i \(-0.381444\pi\)
0.363903 + 0.931437i \(0.381444\pi\)
\(384\) 2.13636 0.109021
\(385\) −49.3837 −2.51683
\(386\) 13.4643 0.685313
\(387\) 4.67428 0.237607
\(388\) 8.80362 0.446936
\(389\) −14.3431 −0.727226 −0.363613 0.931550i \(-0.618457\pi\)
−0.363613 + 0.931550i \(0.618457\pi\)
\(390\) −19.7152 −0.998320
\(391\) −11.9613 −0.604909
\(392\) −16.8035 −0.848705
\(393\) 18.1775 0.916935
\(394\) −12.7144 −0.640544
\(395\) −14.3763 −0.723352
\(396\) 6.53211 0.328251
\(397\) −17.1101 −0.858731 −0.429366 0.903131i \(-0.641263\pi\)
−0.429366 + 0.903131i \(0.641263\pi\)
\(398\) −26.8112 −1.34392
\(399\) −21.7428 −1.08850
\(400\) 0.873611 0.0436805
\(401\) −5.89718 −0.294491 −0.147246 0.989100i \(-0.547041\pi\)
−0.147246 + 0.989100i \(0.547041\pi\)
\(402\) 2.40093 0.119747
\(403\) −5.93720 −0.295753
\(404\) −1.40231 −0.0697673
\(405\) 27.2550 1.35431
\(406\) 44.2722 2.19719
\(407\) −9.02270 −0.447239
\(408\) −7.66645 −0.379546
\(409\) 8.66087 0.428253 0.214126 0.976806i \(-0.431310\pi\)
0.214126 + 0.976806i \(0.431310\pi\)
\(410\) 19.9818 0.986832
\(411\) −28.1883 −1.39043
\(412\) 17.5860 0.866401
\(413\) 54.2585 2.66989
\(414\) −5.21316 −0.256213
\(415\) 5.06002 0.248386
\(416\) −3.80781 −0.186693
\(417\) 9.10445 0.445847
\(418\) −8.71230 −0.426132
\(419\) −15.7433 −0.769112 −0.384556 0.923102i \(-0.625645\pi\)
−0.384556 + 0.923102i \(0.625645\pi\)
\(420\) 25.2608 1.23260
\(421\) 27.7975 1.35477 0.677383 0.735631i \(-0.263114\pi\)
0.677383 + 0.735631i \(0.263114\pi\)
\(422\) 10.9857 0.534775
\(423\) −10.8506 −0.527574
\(424\) 4.05475 0.196916
\(425\) −3.13501 −0.152070
\(426\) 6.15192 0.298061
\(427\) −41.7649 −2.02115
\(428\) −2.16657 −0.104725
\(429\) −33.9751 −1.64033
\(430\) 7.24310 0.349293
\(431\) −39.1696 −1.88673 −0.943366 0.331754i \(-0.892360\pi\)
−0.943366 + 0.331754i \(0.892360\pi\)
\(432\) 3.06776 0.147598
\(433\) 21.6849 1.04211 0.521055 0.853523i \(-0.325539\pi\)
0.521055 + 0.853523i \(0.325539\pi\)
\(434\) 7.60724 0.365159
\(435\) −46.9826 −2.25264
\(436\) −14.8582 −0.711578
\(437\) 6.95312 0.332613
\(438\) 11.0595 0.528443
\(439\) −13.8473 −0.660895 −0.330447 0.943824i \(-0.607200\pi\)
−0.330447 + 0.943824i \(0.607200\pi\)
\(440\) 10.1219 0.482544
\(441\) 26.2811 1.25148
\(442\) 13.6646 0.649957
\(443\) 30.0846 1.42936 0.714681 0.699451i \(-0.246572\pi\)
0.714681 + 0.699451i \(0.246572\pi\)
\(444\) 4.61530 0.219032
\(445\) −12.7810 −0.605878
\(446\) −27.6057 −1.30717
\(447\) 26.5293 1.25479
\(448\) 4.87888 0.230506
\(449\) 34.8294 1.64370 0.821850 0.569704i \(-0.192942\pi\)
0.821850 + 0.569704i \(0.192942\pi\)
\(450\) −1.36635 −0.0644102
\(451\) 34.4345 1.62146
\(452\) 20.4505 0.961910
\(453\) −45.1884 −2.12314
\(454\) 26.2090 1.23005
\(455\) −45.0245 −2.11078
\(456\) 4.45652 0.208696
\(457\) 0.0780659 0.00365177 0.00182588 0.999998i \(-0.499419\pi\)
0.00182588 + 0.999998i \(0.499419\pi\)
\(458\) 22.1584 1.03540
\(459\) −11.0089 −0.513849
\(460\) −8.07812 −0.376644
\(461\) 20.0398 0.933345 0.466673 0.884430i \(-0.345453\pi\)
0.466673 + 0.884430i \(0.345453\pi\)
\(462\) 43.5316 2.02528
\(463\) 23.2743 1.08165 0.540824 0.841136i \(-0.318113\pi\)
0.540824 + 0.841136i \(0.318113\pi\)
\(464\) −9.07425 −0.421262
\(465\) −8.07296 −0.374375
\(466\) −14.4276 −0.668345
\(467\) 2.21624 0.102555 0.0512776 0.998684i \(-0.483671\pi\)
0.0512776 + 0.998684i \(0.483671\pi\)
\(468\) 5.95550 0.275293
\(469\) 5.48309 0.253186
\(470\) −16.8137 −0.775558
\(471\) −36.7575 −1.69370
\(472\) −11.1211 −0.511890
\(473\) 12.4820 0.573921
\(474\) 12.6727 0.582077
\(475\) 1.82238 0.0836167
\(476\) −17.5082 −0.802486
\(477\) −6.34171 −0.290367
\(478\) −7.47287 −0.341801
\(479\) 7.59253 0.346912 0.173456 0.984842i \(-0.444507\pi\)
0.173456 + 0.984842i \(0.444507\pi\)
\(480\) −5.17758 −0.236323
\(481\) −8.22624 −0.375084
\(482\) 5.81160 0.264711
\(483\) −34.7418 −1.58081
\(484\) 6.44301 0.292864
\(485\) −21.3360 −0.968819
\(486\) −14.8220 −0.672338
\(487\) 6.15247 0.278795 0.139397 0.990237i \(-0.455483\pi\)
0.139397 + 0.990237i \(0.455483\pi\)
\(488\) 8.56035 0.387509
\(489\) −3.85197 −0.174192
\(490\) 40.7242 1.83973
\(491\) −37.0886 −1.67378 −0.836892 0.547369i \(-0.815630\pi\)
−0.836892 + 0.547369i \(0.815630\pi\)
\(492\) −17.6139 −0.794098
\(493\) 32.5635 1.46659
\(494\) −7.94323 −0.357383
\(495\) −15.8309 −0.711547
\(496\) −1.55922 −0.0700109
\(497\) 14.0494 0.630201
\(498\) −4.46039 −0.199875
\(499\) −33.3384 −1.49243 −0.746217 0.665703i \(-0.768132\pi\)
−0.746217 + 0.665703i \(0.768132\pi\)
\(500\) 10.0005 0.447237
\(501\) 1.08494 0.0484714
\(502\) −31.3685 −1.40005
\(503\) 19.4007 0.865036 0.432518 0.901625i \(-0.357625\pi\)
0.432518 + 0.901625i \(0.357625\pi\)
\(504\) −7.63068 −0.339898
\(505\) 3.39856 0.151234
\(506\) −13.9209 −0.618861
\(507\) −3.20332 −0.142264
\(508\) −14.8677 −0.659648
\(509\) 4.81858 0.213580 0.106790 0.994282i \(-0.465943\pi\)
0.106790 + 0.994282i \(0.465943\pi\)
\(510\) 18.5800 0.822738
\(511\) 25.2570 1.11730
\(512\) −1.00000 −0.0441942
\(513\) 6.39946 0.282543
\(514\) −1.79913 −0.0793562
\(515\) −42.6207 −1.87809
\(516\) −6.38478 −0.281074
\(517\) −28.9749 −1.27431
\(518\) 10.5401 0.463107
\(519\) −35.5700 −1.56135
\(520\) 9.22843 0.404694
\(521\) −0.578491 −0.0253441 −0.0126721 0.999920i \(-0.504034\pi\)
−0.0126721 + 0.999920i \(0.504034\pi\)
\(522\) 14.1923 0.621181
\(523\) 13.8406 0.605209 0.302604 0.953116i \(-0.402144\pi\)
0.302604 + 0.953116i \(0.402144\pi\)
\(524\) −8.50866 −0.371703
\(525\) −9.10568 −0.397404
\(526\) 14.5641 0.635027
\(527\) 5.59534 0.243737
\(528\) −8.92246 −0.388300
\(529\) −11.8900 −0.516955
\(530\) −9.82690 −0.426853
\(531\) 17.3936 0.754819
\(532\) 10.1775 0.441252
\(533\) 31.3948 1.35986
\(534\) 11.2664 0.487546
\(535\) 5.25080 0.227012
\(536\) −1.12384 −0.0485426
\(537\) 25.5737 1.10359
\(538\) −9.22945 −0.397910
\(539\) 70.1796 3.02285
\(540\) −7.43489 −0.319947
\(541\) −25.9841 −1.11714 −0.558571 0.829457i \(-0.688650\pi\)
−0.558571 + 0.829457i \(0.688650\pi\)
\(542\) 21.8990 0.940641
\(543\) −4.70164 −0.201767
\(544\) 3.58856 0.153858
\(545\) 36.0096 1.54248
\(546\) 39.6890 1.69853
\(547\) −11.4985 −0.491639 −0.245819 0.969316i \(-0.579057\pi\)
−0.245819 + 0.969316i \(0.579057\pi\)
\(548\) 13.1946 0.563644
\(549\) −13.3886 −0.571410
\(550\) −3.64862 −0.155578
\(551\) −18.9292 −0.806412
\(552\) 7.12085 0.303083
\(553\) 28.9412 1.23070
\(554\) −27.9450 −1.18727
\(555\) −11.1854 −0.474795
\(556\) −4.26167 −0.180735
\(557\) −40.1044 −1.69928 −0.849638 0.527366i \(-0.823180\pi\)
−0.849638 + 0.527366i \(0.823180\pi\)
\(558\) 2.43865 0.103236
\(559\) 11.3801 0.481328
\(560\) −11.8242 −0.499665
\(561\) 32.0188 1.35183
\(562\) −11.7874 −0.497221
\(563\) 38.1782 1.60902 0.804510 0.593940i \(-0.202428\pi\)
0.804510 + 0.593940i \(0.202428\pi\)
\(564\) 14.8212 0.624087
\(565\) −49.5628 −2.08512
\(566\) −24.8420 −1.04419
\(567\) −54.8674 −2.30422
\(568\) −2.87963 −0.120827
\(569\) 9.33767 0.391456 0.195728 0.980658i \(-0.437293\pi\)
0.195728 + 0.980658i \(0.437293\pi\)
\(570\) −10.8006 −0.452388
\(571\) 37.0912 1.55222 0.776110 0.630597i \(-0.217190\pi\)
0.776110 + 0.630597i \(0.217190\pi\)
\(572\) 15.9033 0.664949
\(573\) 12.6205 0.527231
\(574\) −40.2257 −1.67899
\(575\) 2.91190 0.121434
\(576\) 1.56402 0.0651676
\(577\) −16.1394 −0.671893 −0.335947 0.941881i \(-0.609056\pi\)
−0.335947 + 0.941881i \(0.609056\pi\)
\(578\) 4.12223 0.171462
\(579\) 28.7645 1.19541
\(580\) 21.9919 0.913165
\(581\) −10.1864 −0.422602
\(582\) 18.8077 0.779603
\(583\) −16.9346 −0.701359
\(584\) −5.17680 −0.214217
\(585\) −14.4335 −0.596751
\(586\) 6.33434 0.261669
\(587\) 27.8662 1.15016 0.575081 0.818097i \(-0.304971\pi\)
0.575081 + 0.818097i \(0.304971\pi\)
\(588\) −35.8983 −1.48042
\(589\) −3.25258 −0.134020
\(590\) 26.9525 1.10962
\(591\) −27.1626 −1.11732
\(592\) −2.16036 −0.0887902
\(593\) −39.1529 −1.60782 −0.803908 0.594754i \(-0.797249\pi\)
−0.803908 + 0.594754i \(0.797249\pi\)
\(594\) −12.8125 −0.525702
\(595\) 42.4320 1.73954
\(596\) −12.4180 −0.508661
\(597\) −57.2782 −2.34424
\(598\) −12.6921 −0.519018
\(599\) 9.56015 0.390617 0.195309 0.980742i \(-0.437429\pi\)
0.195309 + 0.980742i \(0.437429\pi\)
\(600\) 1.86634 0.0761932
\(601\) 31.3941 1.28059 0.640295 0.768129i \(-0.278812\pi\)
0.640295 + 0.768129i \(0.278812\pi\)
\(602\) −14.5812 −0.594284
\(603\) 1.75771 0.0715796
\(604\) 21.1521 0.860666
\(605\) −15.6150 −0.634840
\(606\) −2.99583 −0.121697
\(607\) 5.95832 0.241841 0.120920 0.992662i \(-0.461415\pi\)
0.120920 + 0.992662i \(0.461415\pi\)
\(608\) −2.08604 −0.0845999
\(609\) 94.5813 3.83263
\(610\) −20.7465 −0.839999
\(611\) −26.4172 −1.06872
\(612\) −5.61259 −0.226875
\(613\) −7.81119 −0.315491 −0.157746 0.987480i \(-0.550423\pi\)
−0.157746 + 0.987480i \(0.550423\pi\)
\(614\) 6.64633 0.268224
\(615\) 42.6883 1.72136
\(616\) −20.3766 −0.820996
\(617\) 4.71909 0.189983 0.0949916 0.995478i \(-0.469718\pi\)
0.0949916 + 0.995478i \(0.469718\pi\)
\(618\) 37.5700 1.51129
\(619\) −16.0858 −0.646543 −0.323271 0.946306i \(-0.604783\pi\)
−0.323271 + 0.946306i \(0.604783\pi\)
\(620\) 3.77885 0.151762
\(621\) 10.2254 0.410331
\(622\) −1.93843 −0.0777241
\(623\) 25.7296 1.03083
\(624\) −8.13485 −0.325654
\(625\) −28.6049 −1.14419
\(626\) −12.7744 −0.510569
\(627\) −18.6126 −0.743315
\(628\) 17.2057 0.686582
\(629\) 7.75258 0.309115
\(630\) 18.4934 0.736793
\(631\) 17.1134 0.681273 0.340636 0.940195i \(-0.389357\pi\)
0.340636 + 0.940195i \(0.389357\pi\)
\(632\) −5.93193 −0.235959
\(633\) 23.4694 0.932824
\(634\) −4.51399 −0.179273
\(635\) 36.0327 1.42991
\(636\) 8.66239 0.343486
\(637\) 63.9846 2.53516
\(638\) 37.8985 1.50041
\(639\) 4.50380 0.178168
\(640\) 2.42355 0.0957994
\(641\) 21.9533 0.867102 0.433551 0.901129i \(-0.357260\pi\)
0.433551 + 0.901129i \(0.357260\pi\)
\(642\) −4.62857 −0.182675
\(643\) 39.4200 1.55457 0.777287 0.629147i \(-0.216596\pi\)
0.777287 + 0.629147i \(0.216596\pi\)
\(644\) 16.2622 0.640819
\(645\) 15.4738 0.609282
\(646\) 7.48587 0.294528
\(647\) 25.1145 0.987354 0.493677 0.869645i \(-0.335653\pi\)
0.493677 + 0.869645i \(0.335653\pi\)
\(648\) 11.2459 0.441781
\(649\) 46.4470 1.82321
\(650\) −3.32655 −0.130478
\(651\) 16.2518 0.636957
\(652\) 1.80306 0.0706132
\(653\) −28.4238 −1.11231 −0.556154 0.831079i \(-0.687724\pi\)
−0.556154 + 0.831079i \(0.687724\pi\)
\(654\) −31.7424 −1.24123
\(655\) 20.6212 0.805737
\(656\) 8.24485 0.321907
\(657\) 8.09663 0.315879
\(658\) 33.8478 1.31953
\(659\) −27.9099 −1.08722 −0.543608 0.839339i \(-0.682942\pi\)
−0.543608 + 0.839339i \(0.682942\pi\)
\(660\) 21.6241 0.841715
\(661\) −4.83388 −0.188016 −0.0940081 0.995571i \(-0.529968\pi\)
−0.0940081 + 0.995571i \(0.529968\pi\)
\(662\) 17.9117 0.696158
\(663\) 29.1924 1.13374
\(664\) 2.08785 0.0810243
\(665\) −24.6658 −0.956498
\(666\) 3.37885 0.130928
\(667\) −30.2461 −1.17113
\(668\) −0.507845 −0.0196491
\(669\) −58.9756 −2.28013
\(670\) 2.72369 0.105225
\(671\) −35.7521 −1.38020
\(672\) 10.4230 0.402077
\(673\) 22.1093 0.852253 0.426126 0.904664i \(-0.359878\pi\)
0.426126 + 0.904664i \(0.359878\pi\)
\(674\) 25.7627 0.992340
\(675\) 2.68003 0.103154
\(676\) 1.49943 0.0576704
\(677\) 14.1275 0.542964 0.271482 0.962444i \(-0.412486\pi\)
0.271482 + 0.962444i \(0.412486\pi\)
\(678\) 43.6895 1.67789
\(679\) 42.9518 1.64834
\(680\) −8.69707 −0.333517
\(681\) 55.9919 2.14561
\(682\) 6.51204 0.249359
\(683\) −5.65598 −0.216420 −0.108210 0.994128i \(-0.534512\pi\)
−0.108210 + 0.994128i \(0.534512\pi\)
\(684\) 3.26261 0.124749
\(685\) −31.9778 −1.22181
\(686\) −47.8302 −1.82616
\(687\) 47.3383 1.80607
\(688\) 2.98863 0.113940
\(689\) −15.4397 −0.588206
\(690\) −17.2578 −0.656992
\(691\) 12.8809 0.490013 0.245006 0.969521i \(-0.421210\pi\)
0.245006 + 0.969521i \(0.421210\pi\)
\(692\) 16.6498 0.632932
\(693\) 31.8694 1.21062
\(694\) −9.33419 −0.354321
\(695\) 10.3284 0.391778
\(696\) −19.3858 −0.734818
\(697\) −29.5871 −1.12069
\(698\) 23.3987 0.885653
\(699\) −30.8225 −1.16581
\(700\) 4.26225 0.161098
\(701\) −14.0994 −0.532529 −0.266264 0.963900i \(-0.585789\pi\)
−0.266264 + 0.963900i \(0.585789\pi\)
\(702\) −11.6815 −0.440888
\(703\) −4.50659 −0.169969
\(704\) 4.17648 0.157407
\(705\) −35.9201 −1.35283
\(706\) −16.7459 −0.630242
\(707\) −6.84169 −0.257308
\(708\) −23.7586 −0.892903
\(709\) 19.9917 0.750806 0.375403 0.926862i \(-0.377504\pi\)
0.375403 + 0.926862i \(0.377504\pi\)
\(710\) 6.97894 0.261915
\(711\) 9.27766 0.347940
\(712\) −5.27367 −0.197639
\(713\) −5.19714 −0.194634
\(714\) −37.4037 −1.39980
\(715\) −38.5424 −1.44140
\(716\) −11.9707 −0.447366
\(717\) −15.9647 −0.596214
\(718\) −11.2319 −0.419170
\(719\) −41.5863 −1.55091 −0.775454 0.631404i \(-0.782479\pi\)
−0.775454 + 0.631404i \(0.782479\pi\)
\(720\) −3.79049 −0.141263
\(721\) 85.8002 3.19536
\(722\) 14.6485 0.545159
\(723\) 12.4156 0.461743
\(724\) 2.20077 0.0817911
\(725\) −7.92736 −0.294415
\(726\) 13.7646 0.510852
\(727\) −1.73876 −0.0644869 −0.0322435 0.999480i \(-0.510265\pi\)
−0.0322435 + 0.999480i \(0.510265\pi\)
\(728\) −18.5779 −0.688542
\(729\) 2.07267 0.0767656
\(730\) 12.5462 0.464357
\(731\) −10.7249 −0.396674
\(732\) 18.2880 0.675942
\(733\) −13.9364 −0.514754 −0.257377 0.966311i \(-0.582858\pi\)
−0.257377 + 0.966311i \(0.582858\pi\)
\(734\) 36.8380 1.35972
\(735\) 87.0014 3.20910
\(736\) −3.33317 −0.122862
\(737\) 4.69371 0.172895
\(738\) −12.8951 −0.474676
\(739\) −3.48282 −0.128117 −0.0640587 0.997946i \(-0.520404\pi\)
−0.0640587 + 0.997946i \(0.520404\pi\)
\(740\) 5.23574 0.192470
\(741\) −16.9696 −0.623393
\(742\) 19.7826 0.726244
\(743\) −42.2131 −1.54865 −0.774323 0.632790i \(-0.781910\pi\)
−0.774323 + 0.632790i \(0.781910\pi\)
\(744\) −3.33104 −0.122122
\(745\) 30.0957 1.10262
\(746\) −17.1113 −0.626490
\(747\) −3.26544 −0.119476
\(748\) −14.9876 −0.548000
\(749\) −10.5704 −0.386236
\(750\) 21.3647 0.780128
\(751\) 1.00000 0.0364905
\(752\) −6.93762 −0.252989
\(753\) −67.0144 −2.44214
\(754\) 34.5530 1.25835
\(755\) −51.2632 −1.86566
\(756\) 14.9673 0.544354
\(757\) 26.0282 0.946010 0.473005 0.881060i \(-0.343169\pi\)
0.473005 + 0.881060i \(0.343169\pi\)
\(758\) −22.7500 −0.826315
\(759\) −29.7401 −1.07950
\(760\) 5.05562 0.183387
\(761\) 15.2514 0.552864 0.276432 0.961033i \(-0.410848\pi\)
0.276432 + 0.961033i \(0.410848\pi\)
\(762\) −31.7628 −1.15064
\(763\) −72.4913 −2.62436
\(764\) −5.90750 −0.213726
\(765\) 13.6024 0.491796
\(766\) −14.2435 −0.514637
\(767\) 42.3470 1.52906
\(768\) −2.13636 −0.0770891
\(769\) 8.24274 0.297241 0.148620 0.988894i \(-0.452517\pi\)
0.148620 + 0.988894i \(0.452517\pi\)
\(770\) 49.3837 1.77967
\(771\) −3.84358 −0.138423
\(772\) −13.4643 −0.484589
\(773\) 0.616153 0.0221615 0.0110807 0.999939i \(-0.496473\pi\)
0.0110807 + 0.999939i \(0.496473\pi\)
\(774\) −4.67428 −0.168013
\(775\) −1.36215 −0.0489298
\(776\) −8.80362 −0.316031
\(777\) 22.5175 0.807811
\(778\) 14.3431 0.514226
\(779\) 17.1991 0.616220
\(780\) 19.7152 0.705919
\(781\) 12.0267 0.430350
\(782\) 11.9613 0.427735
\(783\) −27.8376 −0.994836
\(784\) 16.8035 0.600125
\(785\) −41.6989 −1.48830
\(786\) −18.1775 −0.648371
\(787\) −40.1242 −1.43027 −0.715137 0.698985i \(-0.753636\pi\)
−0.715137 + 0.698985i \(0.753636\pi\)
\(788\) 12.7144 0.452933
\(789\) 31.1142 1.10770
\(790\) 14.3763 0.511487
\(791\) 99.7755 3.54761
\(792\) −6.53211 −0.232108
\(793\) −32.5962 −1.15752
\(794\) 17.1101 0.607215
\(795\) −20.9938 −0.744572
\(796\) 26.8112 0.950296
\(797\) 31.9726 1.13253 0.566263 0.824225i \(-0.308389\pi\)
0.566263 + 0.824225i \(0.308389\pi\)
\(798\) 21.7428 0.769688
\(799\) 24.8961 0.880760
\(800\) −0.873611 −0.0308868
\(801\) 8.24813 0.291433
\(802\) 5.89718 0.208237
\(803\) 21.6208 0.762982
\(804\) −2.40093 −0.0846742
\(805\) −39.4122 −1.38910
\(806\) 5.93720 0.209129
\(807\) −19.7174 −0.694085
\(808\) 1.40231 0.0493330
\(809\) 7.06495 0.248390 0.124195 0.992258i \(-0.460365\pi\)
0.124195 + 0.992258i \(0.460365\pi\)
\(810\) −27.2550 −0.957645
\(811\) 28.2760 0.992904 0.496452 0.868064i \(-0.334636\pi\)
0.496452 + 0.868064i \(0.334636\pi\)
\(812\) −44.2722 −1.55365
\(813\) 46.7840 1.64079
\(814\) 9.02270 0.316245
\(815\) −4.36981 −0.153068
\(816\) 7.66645 0.268379
\(817\) 6.23439 0.218114
\(818\) −8.66087 −0.302820
\(819\) 29.0562 1.01531
\(820\) −19.9818 −0.697796
\(821\) 30.3585 1.05952 0.529760 0.848148i \(-0.322282\pi\)
0.529760 + 0.848148i \(0.322282\pi\)
\(822\) 28.1883 0.983181
\(823\) −56.4647 −1.96824 −0.984118 0.177513i \(-0.943195\pi\)
−0.984118 + 0.177513i \(0.943195\pi\)
\(824\) −17.5860 −0.612638
\(825\) −7.79476 −0.271379
\(826\) −54.2585 −1.88789
\(827\) 41.0454 1.42729 0.713644 0.700508i \(-0.247043\pi\)
0.713644 + 0.700508i \(0.247043\pi\)
\(828\) 5.21316 0.181170
\(829\) −29.9069 −1.03871 −0.519354 0.854559i \(-0.673827\pi\)
−0.519354 + 0.854559i \(0.673827\pi\)
\(830\) −5.06002 −0.175636
\(831\) −59.7004 −2.07099
\(832\) 3.80781 0.132012
\(833\) −60.3004 −2.08929
\(834\) −9.10445 −0.315261
\(835\) 1.23079 0.0425932
\(836\) 8.71230 0.301321
\(837\) −4.78331 −0.165335
\(838\) 15.7433 0.543844
\(839\) 43.0763 1.48716 0.743579 0.668648i \(-0.233127\pi\)
0.743579 + 0.668648i \(0.233127\pi\)
\(840\) −25.2608 −0.871580
\(841\) 53.3420 1.83938
\(842\) −27.7975 −0.957964
\(843\) −25.1821 −0.867317
\(844\) −10.9857 −0.378143
\(845\) −3.63395 −0.125012
\(846\) 10.8506 0.373051
\(847\) 31.4347 1.08011
\(848\) −4.05475 −0.139241
\(849\) −53.0714 −1.82141
\(850\) 3.13501 0.107530
\(851\) −7.20085 −0.246842
\(852\) −6.15192 −0.210761
\(853\) −43.1557 −1.47762 −0.738812 0.673912i \(-0.764613\pi\)
−0.738812 + 0.673912i \(0.764613\pi\)
\(854\) 41.7649 1.42917
\(855\) −7.90710 −0.270417
\(856\) 2.16657 0.0740518
\(857\) 46.1759 1.57734 0.788670 0.614817i \(-0.210770\pi\)
0.788670 + 0.614817i \(0.210770\pi\)
\(858\) 33.9751 1.15989
\(859\) 9.73190 0.332048 0.166024 0.986122i \(-0.446907\pi\)
0.166024 + 0.986122i \(0.446907\pi\)
\(860\) −7.24310 −0.246988
\(861\) −85.9364 −2.92870
\(862\) 39.1696 1.33412
\(863\) 5.84488 0.198962 0.0994810 0.995039i \(-0.468282\pi\)
0.0994810 + 0.995039i \(0.468282\pi\)
\(864\) −3.06776 −0.104367
\(865\) −40.3518 −1.37200
\(866\) −21.6849 −0.736883
\(867\) 8.80656 0.299087
\(868\) −7.60724 −0.258206
\(869\) 24.7746 0.840420
\(870\) 46.9826 1.59286
\(871\) 4.27938 0.145001
\(872\) 14.8582 0.503161
\(873\) 13.7690 0.466012
\(874\) −6.95312 −0.235193
\(875\) 48.7914 1.64945
\(876\) −11.0595 −0.373666
\(877\) −42.5042 −1.43526 −0.717632 0.696422i \(-0.754774\pi\)
−0.717632 + 0.696422i \(0.754774\pi\)
\(878\) 13.8473 0.467323
\(879\) 13.5324 0.456437
\(880\) −10.1219 −0.341210
\(881\) −33.2111 −1.11891 −0.559456 0.828860i \(-0.688990\pi\)
−0.559456 + 0.828860i \(0.688990\pi\)
\(882\) −26.2811 −0.884929
\(883\) −29.9671 −1.00847 −0.504236 0.863566i \(-0.668226\pi\)
−0.504236 + 0.863566i \(0.668226\pi\)
\(884\) −13.6646 −0.459589
\(885\) 57.5803 1.93554
\(886\) −30.0846 −1.01071
\(887\) −32.1452 −1.07933 −0.539665 0.841880i \(-0.681449\pi\)
−0.539665 + 0.841880i \(0.681449\pi\)
\(888\) −4.61530 −0.154879
\(889\) −72.5379 −2.43284
\(890\) 12.7810 0.428420
\(891\) −46.9683 −1.57350
\(892\) 27.6057 0.924307
\(893\) −14.4721 −0.484291
\(894\) −26.5293 −0.887271
\(895\) 29.0116 0.969752
\(896\) −4.87888 −0.162992
\(897\) −27.1149 −0.905339
\(898\) −34.8294 −1.16227
\(899\) 14.1487 0.471886
\(900\) 1.36635 0.0455449
\(901\) 14.5507 0.484754
\(902\) −34.4345 −1.14654
\(903\) −31.1506 −1.03663
\(904\) −20.4505 −0.680173
\(905\) −5.33369 −0.177298
\(906\) 45.1884 1.50128
\(907\) 57.2241 1.90010 0.950048 0.312105i \(-0.101034\pi\)
0.950048 + 0.312105i \(0.101034\pi\)
\(908\) −26.2090 −0.869777
\(909\) −2.19324 −0.0727451
\(910\) 45.0245 1.49255
\(911\) −49.9403 −1.65460 −0.827299 0.561762i \(-0.810124\pi\)
−0.827299 + 0.561762i \(0.810124\pi\)
\(912\) −4.45652 −0.147570
\(913\) −8.71987 −0.288586
\(914\) −0.0780659 −0.00258219
\(915\) −44.3218 −1.46523
\(916\) −22.1584 −0.732135
\(917\) −41.5128 −1.37087
\(918\) 11.0089 0.363346
\(919\) −31.4234 −1.03656 −0.518282 0.855210i \(-0.673428\pi\)
−0.518282 + 0.855210i \(0.673428\pi\)
\(920\) 8.07812 0.266328
\(921\) 14.1989 0.467871
\(922\) −20.0398 −0.659975
\(923\) 10.9651 0.360920
\(924\) −43.5316 −1.43209
\(925\) −1.88731 −0.0620544
\(926\) −23.2743 −0.764840
\(927\) 27.5049 0.903380
\(928\) 9.07425 0.297877
\(929\) −31.7034 −1.04015 −0.520077 0.854119i \(-0.674097\pi\)
−0.520077 + 0.854119i \(0.674097\pi\)
\(930\) 8.07296 0.264723
\(931\) 35.0527 1.14881
\(932\) 14.4276 0.472591
\(933\) −4.14118 −0.135576
\(934\) −2.21624 −0.0725175
\(935\) 36.3232 1.18789
\(936\) −5.95550 −0.194662
\(937\) −35.4887 −1.15937 −0.579683 0.814842i \(-0.696823\pi\)
−0.579683 + 0.814842i \(0.696823\pi\)
\(938\) −5.48309 −0.179029
\(939\) −27.2908 −0.890600
\(940\) 16.8137 0.548402
\(941\) −12.6223 −0.411474 −0.205737 0.978607i \(-0.565959\pi\)
−0.205737 + 0.978607i \(0.565959\pi\)
\(942\) 36.7575 1.19762
\(943\) 27.4815 0.894921
\(944\) 11.1211 0.361961
\(945\) −36.2739 −1.17999
\(946\) −12.4820 −0.405823
\(947\) −53.0576 −1.72414 −0.862070 0.506789i \(-0.830832\pi\)
−0.862070 + 0.506789i \(0.830832\pi\)
\(948\) −12.6727 −0.411591
\(949\) 19.7123 0.639887
\(950\) −1.82238 −0.0591259
\(951\) −9.64349 −0.312712
\(952\) 17.5082 0.567443
\(953\) −9.79368 −0.317248 −0.158624 0.987339i \(-0.550706\pi\)
−0.158624 + 0.987339i \(0.550706\pi\)
\(954\) 6.34171 0.205321
\(955\) 14.3171 0.463292
\(956\) 7.47287 0.241690
\(957\) 80.9647 2.61722
\(958\) −7.59253 −0.245304
\(959\) 64.3748 2.07877
\(960\) 5.17758 0.167106
\(961\) −28.5688 −0.921576
\(962\) 8.22624 0.265225
\(963\) −3.38856 −0.109195
\(964\) −5.81160 −0.187179
\(965\) 32.6313 1.05044
\(966\) 34.7418 1.11780
\(967\) −16.6778 −0.536321 −0.268161 0.963374i \(-0.586416\pi\)
−0.268161 + 0.963374i \(0.586416\pi\)
\(968\) −6.44301 −0.207086
\(969\) 15.9925 0.513753
\(970\) 21.3360 0.685059
\(971\) 0.0825230 0.00264829 0.00132414 0.999999i \(-0.499579\pi\)
0.00132414 + 0.999999i \(0.499579\pi\)
\(972\) 14.8220 0.475415
\(973\) −20.7922 −0.666567
\(974\) −6.15247 −0.197138
\(975\) −7.10669 −0.227596
\(976\) −8.56035 −0.274010
\(977\) −26.7169 −0.854750 −0.427375 0.904075i \(-0.640562\pi\)
−0.427375 + 0.904075i \(0.640562\pi\)
\(978\) 3.85197 0.123173
\(979\) 22.0254 0.703934
\(980\) −40.7242 −1.30089
\(981\) −23.2385 −0.741949
\(982\) 37.0886 1.18354
\(983\) 14.2752 0.455307 0.227653 0.973742i \(-0.426895\pi\)
0.227653 + 0.973742i \(0.426895\pi\)
\(984\) 17.6139 0.561512
\(985\) −30.8141 −0.981820
\(986\) −32.5635 −1.03703
\(987\) 72.3111 2.30169
\(988\) 7.94323 0.252708
\(989\) 9.96161 0.316761
\(990\) 15.8309 0.503140
\(991\) 42.8136 1.36002 0.680009 0.733203i \(-0.261976\pi\)
0.680009 + 0.733203i \(0.261976\pi\)
\(992\) 1.55922 0.0495052
\(993\) 38.2658 1.21433
\(994\) −14.0494 −0.445619
\(995\) −64.9783 −2.05995
\(996\) 4.46039 0.141333
\(997\) 5.28690 0.167438 0.0837189 0.996489i \(-0.473320\pi\)
0.0837189 + 0.996489i \(0.473320\pi\)
\(998\) 33.3384 1.05531
\(999\) −6.62746 −0.209684
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 1502.2.a.h.1.3 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.h.1.3 19 1.1 even 1 trivial