Properties

Label 475.2.a.i.1.3
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
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] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, 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("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.552409\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.14243 q^{2} +2.87834 q^{3} +2.59002 q^{4} +6.16666 q^{6} -3.10482 q^{7} +1.26409 q^{8} +5.28487 q^{9} +O(q^{10})\) \(q+2.14243 q^{2} +2.87834 q^{3} +2.59002 q^{4} +6.16666 q^{6} -3.10482 q^{7} +1.26409 q^{8} +5.28487 q^{9} -1.10482 q^{11} +7.45498 q^{12} -1.77353 q^{13} -6.65187 q^{14} -2.47182 q^{16} -7.75669 q^{17} +11.3225 q^{18} +1.00000 q^{19} -8.93674 q^{21} -2.36700 q^{22} +6.65187 q^{23} +3.63849 q^{24} -3.79966 q^{26} +6.57664 q^{27} -8.04156 q^{28} +7.75669 q^{29} +6.57664 q^{31} -7.82389 q^{32} -3.18005 q^{33} -16.6182 q^{34} +13.6879 q^{36} +1.40652 q^{37} +2.14243 q^{38} -5.10482 q^{39} +2.81995 q^{41} -19.1464 q^{42} -3.10482 q^{43} -2.86151 q^{44} +14.2512 q^{46} -1.46492 q^{47} -7.11475 q^{48} +2.63990 q^{49} -22.3264 q^{51} -4.59348 q^{52} +2.59348 q^{53} +14.0900 q^{54} -3.92477 q^{56} +2.87834 q^{57} +16.6182 q^{58} -5.38969 q^{59} +4.07523 q^{61} +14.0900 q^{62} -16.4086 q^{63} -11.8185 q^{64} -6.81305 q^{66} +15.8151 q^{67} -20.0900 q^{68} +19.1464 q^{69} +7.14638 q^{71} +6.68055 q^{72} -0.243310 q^{73} +3.01339 q^{74} +2.59002 q^{76} +3.43026 q^{77} -10.9367 q^{78} -9.38969 q^{79} +3.07523 q^{81} +6.04156 q^{82} -8.86151 q^{83} -23.1464 q^{84} -6.65187 q^{86} +22.3264 q^{87} -1.39659 q^{88} +0.813048 q^{89} +5.50648 q^{91} +17.2285 q^{92} +18.9298 q^{93} -3.13849 q^{94} -22.5199 q^{96} -3.19689 q^{97} +5.65581 q^{98} -5.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.14243 1.51493 0.757465 0.652876i \(-0.226438\pi\)
0.757465 + 0.652876i \(0.226438\pi\)
\(3\) 2.87834 1.66181 0.830907 0.556412i \(-0.187822\pi\)
0.830907 + 0.556412i \(0.187822\pi\)
\(4\) 2.59002 1.29501
\(5\) 0 0
\(6\) 6.16666 2.51753
\(7\) −3.10482 −1.17351 −0.586756 0.809764i \(-0.699595\pi\)
−0.586756 + 0.809764i \(0.699595\pi\)
\(8\) 1.26409 0.446923
\(9\) 5.28487 1.76162
\(10\) 0 0
\(11\) −1.10482 −0.333115 −0.166558 0.986032i \(-0.553265\pi\)
−0.166558 + 0.986032i \(0.553265\pi\)
\(12\) 7.45498 2.15207
\(13\) −1.77353 −0.491888 −0.245944 0.969284i \(-0.579098\pi\)
−0.245944 + 0.969284i \(0.579098\pi\)
\(14\) −6.65187 −1.77779
\(15\) 0 0
\(16\) −2.47182 −0.617955
\(17\) −7.75669 −1.88127 −0.940637 0.339415i \(-0.889771\pi\)
−0.940637 + 0.339415i \(0.889771\pi\)
\(18\) 11.3225 2.66874
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −8.93674 −1.95016
\(22\) −2.36700 −0.504647
\(23\) 6.65187 1.38701 0.693505 0.720451i \(-0.256065\pi\)
0.693505 + 0.720451i \(0.256065\pi\)
\(24\) 3.63849 0.742703
\(25\) 0 0
\(26\) −3.79966 −0.745175
\(27\) 6.57664 1.26567
\(28\) −8.04156 −1.51971
\(29\) 7.75669 1.44038 0.720191 0.693776i \(-0.244054\pi\)
0.720191 + 0.693776i \(0.244054\pi\)
\(30\) 0 0
\(31\) 6.57664 1.18120 0.590600 0.806965i \(-0.298891\pi\)
0.590600 + 0.806965i \(0.298891\pi\)
\(32\) −7.82389 −1.38308
\(33\) −3.18005 −0.553576
\(34\) −16.6182 −2.85000
\(35\) 0 0
\(36\) 13.6879 2.28132
\(37\) 1.40652 0.231231 0.115616 0.993294i \(-0.463116\pi\)
0.115616 + 0.993294i \(0.463116\pi\)
\(38\) 2.14243 0.347549
\(39\) −5.10482 −0.817425
\(40\) 0 0
\(41\) 2.81995 0.440402 0.220201 0.975454i \(-0.429329\pi\)
0.220201 + 0.975454i \(0.429329\pi\)
\(42\) −19.1464 −2.95435
\(43\) −3.10482 −0.473480 −0.236740 0.971573i \(-0.576079\pi\)
−0.236740 + 0.971573i \(0.576079\pi\)
\(44\) −2.86151 −0.431389
\(45\) 0 0
\(46\) 14.2512 2.10122
\(47\) −1.46492 −0.213680 −0.106840 0.994276i \(-0.534073\pi\)
−0.106840 + 0.994276i \(0.534073\pi\)
\(48\) −7.11475 −1.02693
\(49\) 2.63990 0.377129
\(50\) 0 0
\(51\) −22.3264 −3.12633
\(52\) −4.59348 −0.637001
\(53\) 2.59348 0.356241 0.178121 0.984009i \(-0.442998\pi\)
0.178121 + 0.984009i \(0.442998\pi\)
\(54\) 14.0900 1.91741
\(55\) 0 0
\(56\) −3.92477 −0.524469
\(57\) 2.87834 0.381246
\(58\) 16.6182 2.18208
\(59\) −5.38969 −0.701678 −0.350839 0.936436i \(-0.614103\pi\)
−0.350839 + 0.936436i \(0.614103\pi\)
\(60\) 0 0
\(61\) 4.07523 0.521780 0.260890 0.965369i \(-0.415984\pi\)
0.260890 + 0.965369i \(0.415984\pi\)
\(62\) 14.0900 1.78943
\(63\) −16.4086 −2.06728
\(64\) −11.8185 −1.47732
\(65\) 0 0
\(66\) −6.81305 −0.838628
\(67\) 15.8151 1.93212 0.966060 0.258318i \(-0.0831681\pi\)
0.966060 + 0.258318i \(0.0831681\pi\)
\(68\) −20.0900 −2.43627
\(69\) 19.1464 2.30495
\(70\) 0 0
\(71\) 7.14638 0.848119 0.424059 0.905634i \(-0.360605\pi\)
0.424059 + 0.905634i \(0.360605\pi\)
\(72\) 6.68055 0.787310
\(73\) −0.243310 −0.0284773 −0.0142387 0.999899i \(-0.504532\pi\)
−0.0142387 + 0.999899i \(0.504532\pi\)
\(74\) 3.01339 0.350299
\(75\) 0 0
\(76\) 2.59002 0.297096
\(77\) 3.43026 0.390915
\(78\) −10.9367 −1.23834
\(79\) −9.38969 −1.05642 −0.528211 0.849113i \(-0.677137\pi\)
−0.528211 + 0.849113i \(0.677137\pi\)
\(80\) 0 0
\(81\) 3.07523 0.341692
\(82\) 6.04156 0.667178
\(83\) −8.86151 −0.972677 −0.486338 0.873771i \(-0.661668\pi\)
−0.486338 + 0.873771i \(0.661668\pi\)
\(84\) −23.1464 −2.52548
\(85\) 0 0
\(86\) −6.65187 −0.717290
\(87\) 22.3264 2.39364
\(88\) −1.39659 −0.148877
\(89\) 0.813048 0.0861829 0.0430914 0.999071i \(-0.486279\pi\)
0.0430914 + 0.999071i \(0.486279\pi\)
\(90\) 0 0
\(91\) 5.50648 0.577236
\(92\) 17.2285 1.79620
\(93\) 18.9298 1.96293
\(94\) −3.13849 −0.323711
\(95\) 0 0
\(96\) −22.5199 −2.29842
\(97\) −3.19689 −0.324595 −0.162297 0.986742i \(-0.551890\pi\)
−0.162297 + 0.986742i \(0.551890\pi\)
\(98\) 5.65581 0.571323
\(99\) −5.83882 −0.586824
\(100\) 0 0
\(101\) −3.01887 −0.300389 −0.150195 0.988656i \(-0.547990\pi\)
−0.150195 + 0.988656i \(0.547990\pi\)
\(102\) −47.8329 −4.73616
\(103\) −6.05839 −0.596951 −0.298476 0.954417i \(-0.596478\pi\)
−0.298476 + 0.954417i \(0.596478\pi\)
\(104\) −2.24190 −0.219836
\(105\) 0 0
\(106\) 5.55635 0.539681
\(107\) −20.3848 −1.97068 −0.985338 0.170616i \(-0.945424\pi\)
−0.985338 + 0.170616i \(0.945424\pi\)
\(108\) 17.0337 1.63906
\(109\) 4.72710 0.452774 0.226387 0.974037i \(-0.427309\pi\)
0.226387 + 0.974037i \(0.427309\pi\)
\(110\) 0 0
\(111\) 4.04846 0.384263
\(112\) 7.67456 0.725177
\(113\) 7.98316 0.750993 0.375496 0.926824i \(-0.377472\pi\)
0.375496 + 0.926824i \(0.377472\pi\)
\(114\) 6.16666 0.577561
\(115\) 0 0
\(116\) 20.0900 1.86531
\(117\) −9.37285 −0.866520
\(118\) −11.5471 −1.06299
\(119\) 24.0831 2.20770
\(120\) 0 0
\(121\) −9.77938 −0.889034
\(122\) 8.73091 0.790460
\(123\) 8.11679 0.731866
\(124\) 17.0337 1.52967
\(125\) 0 0
\(126\) −35.1543 −3.13179
\(127\) −4.63503 −0.411293 −0.205646 0.978626i \(-0.565930\pi\)
−0.205646 + 0.978626i \(0.565930\pi\)
\(128\) −9.67265 −0.854950
\(129\) −8.93674 −0.786836
\(130\) 0 0
\(131\) 15.5134 1.35541 0.677705 0.735334i \(-0.262975\pi\)
0.677705 + 0.735334i \(0.262975\pi\)
\(132\) −8.23641 −0.716887
\(133\) −3.10482 −0.269222
\(134\) 33.8828 2.92703
\(135\) 0 0
\(136\) −9.80515 −0.840785
\(137\) 0.813048 0.0694634 0.0347317 0.999397i \(-0.488942\pi\)
0.0347317 + 0.999397i \(0.488942\pi\)
\(138\) 41.0199 3.49184
\(139\) −12.7687 −1.08302 −0.541512 0.840693i \(-0.682148\pi\)
−0.541512 + 0.840693i \(0.682148\pi\)
\(140\) 0 0
\(141\) −4.21654 −0.355097
\(142\) 15.3106 1.28484
\(143\) 1.95942 0.163855
\(144\) −13.0632 −1.08860
\(145\) 0 0
\(146\) −0.521277 −0.0431412
\(147\) 7.59854 0.626717
\(148\) 3.64293 0.299447
\(149\) −13.7982 −1.13040 −0.565198 0.824955i \(-0.691200\pi\)
−0.565198 + 0.824955i \(0.691200\pi\)
\(150\) 0 0
\(151\) 5.02269 0.408740 0.204370 0.978894i \(-0.434485\pi\)
0.204370 + 0.978894i \(0.434485\pi\)
\(152\) 1.26409 0.102531
\(153\) −40.9931 −3.31409
\(154\) 7.34911 0.592208
\(155\) 0 0
\(156\) −13.2216 −1.05858
\(157\) −7.18695 −0.573581 −0.286791 0.957993i \(-0.592588\pi\)
−0.286791 + 0.957993i \(0.592588\pi\)
\(158\) −20.1168 −1.60041
\(159\) 7.46492 0.592007
\(160\) 0 0
\(161\) −20.6529 −1.62767
\(162\) 6.58848 0.517640
\(163\) −7.25528 −0.568277 −0.284139 0.958783i \(-0.591708\pi\)
−0.284139 + 0.958783i \(0.591708\pi\)
\(164\) 7.30374 0.570326
\(165\) 0 0
\(166\) −18.9852 −1.47354
\(167\) 7.33129 0.567312 0.283656 0.958926i \(-0.408453\pi\)
0.283656 + 0.958926i \(0.408453\pi\)
\(168\) −11.2968 −0.871570
\(169\) −9.85461 −0.758047
\(170\) 0 0
\(171\) 5.28487 0.404144
\(172\) −8.04156 −0.613163
\(173\) 5.77353 0.438953 0.219477 0.975618i \(-0.429565\pi\)
0.219477 + 0.975618i \(0.429565\pi\)
\(174\) 47.8329 3.62620
\(175\) 0 0
\(176\) 2.73091 0.205850
\(177\) −15.5134 −1.16606
\(178\) 1.74190 0.130561
\(179\) −4.48379 −0.335134 −0.167567 0.985861i \(-0.553591\pi\)
−0.167567 + 0.985861i \(0.553591\pi\)
\(180\) 0 0
\(181\) −2.73400 −0.203217 −0.101608 0.994824i \(-0.532399\pi\)
−0.101608 + 0.994824i \(0.532399\pi\)
\(182\) 11.7973 0.874471
\(183\) 11.7299 0.867101
\(184\) 8.40856 0.619887
\(185\) 0 0
\(186\) 40.5559 2.97371
\(187\) 8.56974 0.626681
\(188\) −3.79418 −0.276719
\(189\) −20.4193 −1.48528
\(190\) 0 0
\(191\) −17.7230 −1.28239 −0.641196 0.767377i \(-0.721562\pi\)
−0.641196 + 0.767377i \(0.721562\pi\)
\(192\) −34.0178 −2.45502
\(193\) 13.5371 0.974423 0.487212 0.873284i \(-0.338014\pi\)
0.487212 + 0.873284i \(0.338014\pi\)
\(194\) −6.84912 −0.491738
\(195\) 0 0
\(196\) 6.83741 0.488386
\(197\) 21.5134 1.53276 0.766382 0.642385i \(-0.222055\pi\)
0.766382 + 0.642385i \(0.222055\pi\)
\(198\) −12.5093 −0.888997
\(199\) 7.09410 0.502888 0.251444 0.967872i \(-0.419095\pi\)
0.251444 + 0.967872i \(0.419095\pi\)
\(200\) 0 0
\(201\) 45.5213 3.21082
\(202\) −6.46774 −0.455068
\(203\) −24.0831 −1.69030
\(204\) −57.8260 −4.04863
\(205\) 0 0
\(206\) −12.9797 −0.904339
\(207\) 35.1543 2.44339
\(208\) 4.38384 0.303964
\(209\) −1.10482 −0.0764219
\(210\) 0 0
\(211\) 11.0604 0.761431 0.380716 0.924692i \(-0.375678\pi\)
0.380716 + 0.924692i \(0.375678\pi\)
\(212\) 6.71717 0.461337
\(213\) 20.5697 1.40942
\(214\) −43.6731 −2.98543
\(215\) 0 0
\(216\) 8.31346 0.565659
\(217\) −20.4193 −1.38615
\(218\) 10.1275 0.685921
\(219\) −0.700331 −0.0473240
\(220\) 0 0
\(221\) 13.7567 0.925375
\(222\) 8.67356 0.582131
\(223\) 12.6350 0.846104 0.423052 0.906105i \(-0.360959\pi\)
0.423052 + 0.906105i \(0.360959\pi\)
\(224\) 24.2918 1.62306
\(225\) 0 0
\(226\) 17.1034 1.13770
\(227\) 6.62813 0.439925 0.219962 0.975508i \(-0.429407\pi\)
0.219962 + 0.975508i \(0.429407\pi\)
\(228\) 7.45498 0.493718
\(229\) −0.0752308 −0.00497139 −0.00248570 0.999997i \(-0.500791\pi\)
−0.00248570 + 0.999997i \(0.500791\pi\)
\(230\) 0 0
\(231\) 9.87348 0.649627
\(232\) 9.80515 0.643740
\(233\) −9.51338 −0.623242 −0.311621 0.950206i \(-0.600872\pi\)
−0.311621 + 0.950206i \(0.600872\pi\)
\(234\) −20.0807 −1.31272
\(235\) 0 0
\(236\) −13.9594 −0.908681
\(237\) −27.0268 −1.75558
\(238\) 51.5965 3.34450
\(239\) −2.92984 −0.189515 −0.0947577 0.995500i \(-0.530208\pi\)
−0.0947577 + 0.995500i \(0.530208\pi\)
\(240\) 0 0
\(241\) −9.42743 −0.607274 −0.303637 0.952788i \(-0.598201\pi\)
−0.303637 + 0.952788i \(0.598201\pi\)
\(242\) −20.9517 −1.34682
\(243\) −10.8783 −0.697846
\(244\) 10.5549 0.675711
\(245\) 0 0
\(246\) 17.3897 1.10873
\(247\) −1.77353 −0.112847
\(248\) 8.31346 0.527905
\(249\) −25.5065 −1.61641
\(250\) 0 0
\(251\) −14.9298 −0.942363 −0.471181 0.882036i \(-0.656172\pi\)
−0.471181 + 0.882036i \(0.656172\pi\)
\(252\) −42.4986 −2.67716
\(253\) −7.34911 −0.462035
\(254\) −9.93026 −0.623080
\(255\) 0 0
\(256\) 2.91405 0.182128
\(257\) 15.1295 0.943755 0.471877 0.881664i \(-0.343576\pi\)
0.471877 + 0.881664i \(0.343576\pi\)
\(258\) −19.1464 −1.19200
\(259\) −4.36700 −0.271352
\(260\) 0 0
\(261\) 40.9931 2.53741
\(262\) 33.2364 2.05335
\(263\) 15.2216 0.938605 0.469302 0.883038i \(-0.344505\pi\)
0.469302 + 0.883038i \(0.344505\pi\)
\(264\) −4.01987 −0.247406
\(265\) 0 0
\(266\) −6.65187 −0.407852
\(267\) 2.34023 0.143220
\(268\) 40.9615 2.50212
\(269\) 11.5203 0.702404 0.351202 0.936300i \(-0.385773\pi\)
0.351202 + 0.936300i \(0.385773\pi\)
\(270\) 0 0
\(271\) 2.16118 0.131282 0.0656411 0.997843i \(-0.479091\pi\)
0.0656411 + 0.997843i \(0.479091\pi\)
\(272\) 19.1731 1.16254
\(273\) 15.8495 0.959258
\(274\) 1.74190 0.105232
\(275\) 0 0
\(276\) 49.5896 2.98494
\(277\) −23.4205 −1.40720 −0.703602 0.710595i \(-0.748426\pi\)
−0.703602 + 0.710595i \(0.748426\pi\)
\(278\) −27.3560 −1.64070
\(279\) 34.7567 2.08083
\(280\) 0 0
\(281\) 25.6824 1.53209 0.766043 0.642789i \(-0.222223\pi\)
0.766043 + 0.642789i \(0.222223\pi\)
\(282\) −9.03366 −0.537947
\(283\) −7.38587 −0.439045 −0.219522 0.975607i \(-0.570450\pi\)
−0.219522 + 0.975607i \(0.570450\pi\)
\(284\) 18.5093 1.09832
\(285\) 0 0
\(286\) 4.19794 0.248229
\(287\) −8.75543 −0.516817
\(288\) −41.3482 −2.43647
\(289\) 43.1662 2.53919
\(290\) 0 0
\(291\) −9.20174 −0.539416
\(292\) −0.630180 −0.0368785
\(293\) 24.1522 1.41099 0.705494 0.708716i \(-0.250725\pi\)
0.705494 + 0.708716i \(0.250725\pi\)
\(294\) 16.2794 0.949433
\(295\) 0 0
\(296\) 1.77797 0.103343
\(297\) −7.26600 −0.421616
\(298\) −29.5618 −1.71247
\(299\) −11.7973 −0.682253
\(300\) 0 0
\(301\) 9.63990 0.555635
\(302\) 10.7608 0.619213
\(303\) −8.68936 −0.499190
\(304\) −2.47182 −0.141769
\(305\) 0 0
\(306\) −87.8250 −5.02062
\(307\) −4.38482 −0.250255 −0.125127 0.992141i \(-0.539934\pi\)
−0.125127 + 0.992141i \(0.539934\pi\)
\(308\) 8.88447 0.506239
\(309\) −17.4381 −0.992022
\(310\) 0 0
\(311\) −15.7123 −0.890963 −0.445481 0.895291i \(-0.646967\pi\)
−0.445481 + 0.895291i \(0.646967\pi\)
\(312\) −6.45295 −0.365326
\(313\) −24.7794 −1.40061 −0.700307 0.713842i \(-0.746953\pi\)
−0.700307 + 0.713842i \(0.746953\pi\)
\(314\) −15.3976 −0.868935
\(315\) 0 0
\(316\) −24.3195 −1.36808
\(317\) 27.3798 1.53780 0.768900 0.639369i \(-0.220804\pi\)
0.768900 + 0.639369i \(0.220804\pi\)
\(318\) 15.9931 0.896848
\(319\) −8.56974 −0.479813
\(320\) 0 0
\(321\) −58.6745 −3.27489
\(322\) −44.2474 −2.46581
\(323\) −7.75669 −0.431594
\(324\) 7.96492 0.442496
\(325\) 0 0
\(326\) −15.5440 −0.860900
\(327\) 13.6062 0.752426
\(328\) 3.56467 0.196826
\(329\) 4.54831 0.250756
\(330\) 0 0
\(331\) 4.70033 0.258354 0.129177 0.991622i \(-0.458767\pi\)
0.129177 + 0.991622i \(0.458767\pi\)
\(332\) −22.9515 −1.25963
\(333\) 7.43329 0.407342
\(334\) 15.7068 0.859439
\(335\) 0 0
\(336\) 22.0900 1.20511
\(337\) 20.6360 1.12412 0.562058 0.827098i \(-0.310010\pi\)
0.562058 + 0.827098i \(0.310010\pi\)
\(338\) −21.1128 −1.14839
\(339\) 22.9783 1.24801
\(340\) 0 0
\(341\) −7.26600 −0.393476
\(342\) 11.3225 0.612250
\(343\) 13.5373 0.730947
\(344\) −3.92477 −0.211609
\(345\) 0 0
\(346\) 12.3694 0.664983
\(347\) −30.0148 −1.61128 −0.805639 0.592407i \(-0.798178\pi\)
−0.805639 + 0.592407i \(0.798178\pi\)
\(348\) 57.8260 3.09980
\(349\) −14.7340 −0.788693 −0.394347 0.918962i \(-0.629029\pi\)
−0.394347 + 0.918962i \(0.629029\pi\)
\(350\) 0 0
\(351\) −11.6638 −0.622570
\(352\) 8.64399 0.460726
\(353\) 29.6302 1.57705 0.788527 0.615000i \(-0.210844\pi\)
0.788527 + 0.615000i \(0.210844\pi\)
\(354\) −33.2364 −1.76649
\(355\) 0 0
\(356\) 2.10581 0.111608
\(357\) 69.3195 3.66878
\(358\) −9.60623 −0.507705
\(359\) 21.7577 1.14833 0.574163 0.818741i \(-0.305328\pi\)
0.574163 + 0.818741i \(0.305328\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −5.85742 −0.307859
\(363\) −28.1484 −1.47741
\(364\) 14.2619 0.747527
\(365\) 0 0
\(366\) 25.1306 1.31360
\(367\) −30.0347 −1.56780 −0.783898 0.620889i \(-0.786772\pi\)
−0.783898 + 0.620889i \(0.786772\pi\)
\(368\) −16.4422 −0.857111
\(369\) 14.9031 0.775823
\(370\) 0 0
\(371\) −8.05227 −0.418053
\(372\) 49.0287 2.54202
\(373\) 37.3055 1.93161 0.965803 0.259277i \(-0.0834843\pi\)
0.965803 + 0.259277i \(0.0834843\pi\)
\(374\) 18.3601 0.949378
\(375\) 0 0
\(376\) −1.85179 −0.0954987
\(377\) −13.7567 −0.708506
\(378\) −43.7470 −2.25010
\(379\) −14.7794 −0.759166 −0.379583 0.925158i \(-0.623932\pi\)
−0.379583 + 0.925158i \(0.623932\pi\)
\(380\) 0 0
\(381\) −13.3412 −0.683492
\(382\) −37.9704 −1.94273
\(383\) 29.0020 1.48193 0.740967 0.671541i \(-0.234367\pi\)
0.740967 + 0.671541i \(0.234367\pi\)
\(384\) −27.8412 −1.42077
\(385\) 0 0
\(386\) 29.0024 1.47618
\(387\) −16.4086 −0.834094
\(388\) −8.28001 −0.420354
\(389\) 21.1987 1.07481 0.537407 0.843323i \(-0.319404\pi\)
0.537407 + 0.843323i \(0.319404\pi\)
\(390\) 0 0
\(391\) −51.5965 −2.60935
\(392\) 3.33707 0.168548
\(393\) 44.6529 2.25244
\(394\) 46.0910 2.32203
\(395\) 0 0
\(396\) −15.1227 −0.759944
\(397\) −10.3856 −0.521238 −0.260619 0.965442i \(-0.583927\pi\)
−0.260619 + 0.965442i \(0.583927\pi\)
\(398\) 15.1987 0.761840
\(399\) −8.93674 −0.447397
\(400\) 0 0
\(401\) 25.6638 1.28159 0.640796 0.767712i \(-0.278605\pi\)
0.640796 + 0.767712i \(0.278605\pi\)
\(402\) 97.5263 4.86417
\(403\) −11.6638 −0.581017
\(404\) −7.81896 −0.389008
\(405\) 0 0
\(406\) −51.5965 −2.56069
\(407\) −1.55395 −0.0770266
\(408\) −28.2226 −1.39723
\(409\) 5.71611 0.282644 0.141322 0.989964i \(-0.454865\pi\)
0.141322 + 0.989964i \(0.454865\pi\)
\(410\) 0 0
\(411\) 2.34023 0.115435
\(412\) −15.6914 −0.773059
\(413\) 16.7340 0.823427
\(414\) 75.3157 3.70156
\(415\) 0 0
\(416\) 13.8759 0.680321
\(417\) −36.7526 −1.79978
\(418\) −2.36700 −0.115774
\(419\) 15.4303 0.753818 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(420\) 0 0
\(421\) −1.18005 −0.0575121 −0.0287561 0.999586i \(-0.509155\pi\)
−0.0287561 + 0.999586i \(0.509155\pi\)
\(422\) 23.6962 1.15352
\(423\) −7.74190 −0.376424
\(424\) 3.27839 0.159213
\(425\) 0 0
\(426\) 44.0693 2.13517
\(427\) −12.6529 −0.612315
\(428\) −52.7972 −2.55205
\(429\) 5.63990 0.272297
\(430\) 0 0
\(431\) −14.0255 −0.675585 −0.337792 0.941221i \(-0.609680\pi\)
−0.337792 + 0.941221i \(0.609680\pi\)
\(432\) −16.2563 −0.782130
\(433\) −25.5894 −1.22975 −0.614874 0.788625i \(-0.710793\pi\)
−0.614874 + 0.788625i \(0.710793\pi\)
\(434\) −43.7470 −2.09992
\(435\) 0 0
\(436\) 12.2433 0.586348
\(437\) 6.65187 0.318202
\(438\) −1.50041 −0.0716925
\(439\) −21.9732 −1.04873 −0.524363 0.851495i \(-0.675696\pi\)
−0.524363 + 0.851495i \(0.675696\pi\)
\(440\) 0 0
\(441\) 13.9515 0.664358
\(442\) 29.4728 1.40188
\(443\) 19.3721 0.920395 0.460197 0.887817i \(-0.347779\pi\)
0.460197 + 0.887817i \(0.347779\pi\)
\(444\) 10.4856 0.497625
\(445\) 0 0
\(446\) 27.0697 1.28179
\(447\) −39.7161 −1.87851
\(448\) 36.6944 1.73365
\(449\) −19.1841 −0.905355 −0.452677 0.891674i \(-0.649531\pi\)
−0.452677 + 0.891674i \(0.649531\pi\)
\(450\) 0 0
\(451\) −3.11553 −0.146705
\(452\) 20.6766 0.972545
\(453\) 14.4570 0.679250
\(454\) 14.2003 0.666455
\(455\) 0 0
\(456\) 3.63849 0.170388
\(457\) −8.36010 −0.391069 −0.195534 0.980697i \(-0.562644\pi\)
−0.195534 + 0.980697i \(0.562644\pi\)
\(458\) −0.161177 −0.00753131
\(459\) −51.0130 −2.38108
\(460\) 0 0
\(461\) −25.0268 −1.16561 −0.582806 0.812611i \(-0.698045\pi\)
−0.582806 + 0.812611i \(0.698045\pi\)
\(462\) 21.1533 0.984140
\(463\) −32.3749 −1.50459 −0.752294 0.658827i \(-0.771053\pi\)
−0.752294 + 0.658827i \(0.771053\pi\)
\(464\) −19.1731 −0.890091
\(465\) 0 0
\(466\) −20.3818 −0.944168
\(467\) −15.2216 −0.704372 −0.352186 0.935930i \(-0.614562\pi\)
−0.352186 + 0.935930i \(0.614562\pi\)
\(468\) −24.2759 −1.12215
\(469\) −49.1030 −2.26736
\(470\) 0 0
\(471\) −20.6865 −0.953185
\(472\) −6.81305 −0.313596
\(473\) 3.43026 0.157724
\(474\) −57.9031 −2.65958
\(475\) 0 0
\(476\) 62.3759 2.85899
\(477\) 13.7062 0.627563
\(478\) −6.27698 −0.287103
\(479\) −20.0485 −0.916038 −0.458019 0.888943i \(-0.651441\pi\)
−0.458019 + 0.888943i \(0.651441\pi\)
\(480\) 0 0
\(481\) −2.49451 −0.113740
\(482\) −20.1977 −0.919978
\(483\) −59.4460 −2.70489
\(484\) −25.3288 −1.15131
\(485\) 0 0
\(486\) −23.3061 −1.05719
\(487\) 31.6508 1.43424 0.717118 0.696952i \(-0.245461\pi\)
0.717118 + 0.696952i \(0.245461\pi\)
\(488\) 5.15146 0.233195
\(489\) −20.8832 −0.944371
\(490\) 0 0
\(491\) 24.8033 1.11936 0.559679 0.828710i \(-0.310924\pi\)
0.559679 + 0.828710i \(0.310924\pi\)
\(492\) 21.0227 0.947776
\(493\) −60.1662 −2.70975
\(494\) −3.79966 −0.170955
\(495\) 0 0
\(496\) −16.2563 −0.729928
\(497\) −22.1882 −0.995277
\(498\) −54.6460 −2.44874
\(499\) −33.0237 −1.47834 −0.739171 0.673518i \(-0.764783\pi\)
−0.739171 + 0.673518i \(0.764783\pi\)
\(500\) 0 0
\(501\) 21.1020 0.942767
\(502\) −31.9862 −1.42761
\(503\) 3.54396 0.158017 0.0790087 0.996874i \(-0.474825\pi\)
0.0790087 + 0.996874i \(0.474825\pi\)
\(504\) −20.7419 −0.923917
\(505\) 0 0
\(506\) −15.7450 −0.699950
\(507\) −28.3650 −1.25973
\(508\) −12.0049 −0.532629
\(509\) −2.11679 −0.0938250 −0.0469125 0.998899i \(-0.514938\pi\)
−0.0469125 + 0.998899i \(0.514938\pi\)
\(510\) 0 0
\(511\) 0.755435 0.0334185
\(512\) 25.5885 1.13086
\(513\) 6.57664 0.290366
\(514\) 32.4140 1.42972
\(515\) 0 0
\(516\) −23.1464 −1.01896
\(517\) 1.61847 0.0711802
\(518\) −9.35601 −0.411080
\(519\) 16.6182 0.729458
\(520\) 0 0
\(521\) −8.60748 −0.377101 −0.188550 0.982064i \(-0.560379\pi\)
−0.188550 + 0.982064i \(0.560379\pi\)
\(522\) 87.8250 3.84400
\(523\) −36.8349 −1.61068 −0.805340 0.592814i \(-0.798017\pi\)
−0.805340 + 0.592814i \(0.798017\pi\)
\(524\) 40.1800 1.75527
\(525\) 0 0
\(526\) 32.6113 1.42192
\(527\) −51.0130 −2.22216
\(528\) 7.86051 0.342085
\(529\) 21.2474 0.923799
\(530\) 0 0
\(531\) −28.4838 −1.23609
\(532\) −8.04156 −0.348646
\(533\) −5.00125 −0.216628
\(534\) 5.01379 0.216968
\(535\) 0 0
\(536\) 19.9917 0.863509
\(537\) −12.9059 −0.556931
\(538\) 24.6814 1.06409
\(539\) −2.91661 −0.125627
\(540\) 0 0
\(541\) −32.0079 −1.37613 −0.688063 0.725651i \(-0.741539\pi\)
−0.688063 + 0.725651i \(0.741539\pi\)
\(542\) 4.63018 0.198883
\(543\) −7.86941 −0.337709
\(544\) 60.6875 2.60196
\(545\) 0 0
\(546\) 33.9566 1.45321
\(547\) −17.6240 −0.753550 −0.376775 0.926305i \(-0.622967\pi\)
−0.376775 + 0.926305i \(0.622967\pi\)
\(548\) 2.10581 0.0899559
\(549\) 21.5371 0.919179
\(550\) 0 0
\(551\) 7.75669 0.330446
\(552\) 24.2027 1.03014
\(553\) 29.1533 1.23972
\(554\) −50.1769 −2.13181
\(555\) 0 0
\(556\) −33.0711 −1.40253
\(557\) 34.6865 1.46972 0.734858 0.678221i \(-0.237249\pi\)
0.734858 + 0.678221i \(0.237249\pi\)
\(558\) 74.4639 3.15231
\(559\) 5.50648 0.232899
\(560\) 0 0
\(561\) 24.6667 1.04143
\(562\) 55.0229 2.32100
\(563\) −13.5073 −0.569263 −0.284632 0.958637i \(-0.591871\pi\)
−0.284632 + 0.958637i \(0.591871\pi\)
\(564\) −10.9209 −0.459855
\(565\) 0 0
\(566\) −15.8238 −0.665122
\(567\) −9.54803 −0.400980
\(568\) 9.03366 0.379044
\(569\) 32.3588 1.35655 0.678276 0.734807i \(-0.262727\pi\)
0.678276 + 0.734807i \(0.262727\pi\)
\(570\) 0 0
\(571\) −8.93293 −0.373831 −0.186916 0.982376i \(-0.559849\pi\)
−0.186916 + 0.982376i \(0.559849\pi\)
\(572\) 5.07496 0.212195
\(573\) −51.0130 −2.13110
\(574\) −18.7579 −0.782941
\(575\) 0 0
\(576\) −62.4594 −2.60248
\(577\) −14.9059 −0.620541 −0.310270 0.950648i \(-0.600420\pi\)
−0.310270 + 0.950648i \(0.600420\pi\)
\(578\) 92.4808 3.84669
\(579\) 38.9645 1.61931
\(580\) 0 0
\(581\) 27.5134 1.14145
\(582\) −19.7141 −0.817177
\(583\) −2.86532 −0.118669
\(584\) −0.307566 −0.0127272
\(585\) 0 0
\(586\) 51.7446 2.13755
\(587\) −7.37207 −0.304278 −0.152139 0.988359i \(-0.548616\pi\)
−0.152139 + 0.988359i \(0.548616\pi\)
\(588\) 19.6804 0.811607
\(589\) 6.57664 0.270986
\(590\) 0 0
\(591\) 61.9229 2.54717
\(592\) −3.47668 −0.142890
\(593\) 17.6853 0.726247 0.363124 0.931741i \(-0.381710\pi\)
0.363124 + 0.931741i \(0.381710\pi\)
\(594\) −15.5669 −0.638718
\(595\) 0 0
\(596\) −35.7378 −1.46388
\(597\) 20.4193 0.835705
\(598\) −25.2749 −1.03357
\(599\) 5.30657 0.216821 0.108410 0.994106i \(-0.465424\pi\)
0.108410 + 0.994106i \(0.465424\pi\)
\(600\) 0 0
\(601\) 43.7875 1.78613 0.893065 0.449927i \(-0.148550\pi\)
0.893065 + 0.449927i \(0.148550\pi\)
\(602\) 20.6529 0.841747
\(603\) 83.5806 3.40367
\(604\) 13.0089 0.529324
\(605\) 0 0
\(606\) −18.6164 −0.756239
\(607\) 7.24535 0.294080 0.147040 0.989131i \(-0.453025\pi\)
0.147040 + 0.989131i \(0.453025\pi\)
\(608\) −7.82389 −0.317301
\(609\) −69.3195 −2.80897
\(610\) 0 0
\(611\) 2.59807 0.105107
\(612\) −106.173 −4.29179
\(613\) −20.8962 −0.843988 −0.421994 0.906599i \(-0.638670\pi\)
−0.421994 + 0.906599i \(0.638670\pi\)
\(614\) −9.39419 −0.379119
\(615\) 0 0
\(616\) 4.33616 0.174709
\(617\) 22.0494 0.887677 0.443839 0.896107i \(-0.353616\pi\)
0.443839 + 0.896107i \(0.353616\pi\)
\(618\) −37.3601 −1.50284
\(619\) −0.0484607 −0.00194780 −0.000973900 1.00000i \(-0.500310\pi\)
−0.000973900 1.00000i \(0.500310\pi\)
\(620\) 0 0
\(621\) 43.7470 1.75550
\(622\) −33.6626 −1.34975
\(623\) −2.52437 −0.101137
\(624\) 12.6182 0.505132
\(625\) 0 0
\(626\) −53.0882 −2.12183
\(627\) −3.18005 −0.126999
\(628\) −18.6144 −0.742795
\(629\) −10.9100 −0.435009
\(630\) 0 0
\(631\) −32.6182 −1.29851 −0.649255 0.760571i \(-0.724919\pi\)
−0.649255 + 0.760571i \(0.724919\pi\)
\(632\) −11.8694 −0.472140
\(633\) 31.8357 1.26536
\(634\) 58.6593 2.32966
\(635\) 0 0
\(636\) 19.3343 0.766656
\(637\) −4.68193 −0.185505
\(638\) −18.3601 −0.726883
\(639\) 37.7677 1.49407
\(640\) 0 0
\(641\) −13.4136 −0.529806 −0.264903 0.964275i \(-0.585340\pi\)
−0.264903 + 0.964275i \(0.585340\pi\)
\(642\) −125.706 −4.96123
\(643\) −23.8052 −0.938783 −0.469392 0.882990i \(-0.655527\pi\)
−0.469392 + 0.882990i \(0.655527\pi\)
\(644\) −53.4914 −2.10786
\(645\) 0 0
\(646\) −16.6182 −0.653834
\(647\) 48.4580 1.90508 0.952540 0.304412i \(-0.0984601\pi\)
0.952540 + 0.304412i \(0.0984601\pi\)
\(648\) 3.88737 0.152710
\(649\) 5.95463 0.233740
\(650\) 0 0
\(651\) −58.7737 −2.30352
\(652\) −18.7914 −0.735926
\(653\) −22.2351 −0.870128 −0.435064 0.900399i \(-0.643274\pi\)
−0.435064 + 0.900399i \(0.643274\pi\)
\(654\) 29.1504 1.13987
\(655\) 0 0
\(656\) −6.97041 −0.272149
\(657\) −1.28586 −0.0501663
\(658\) 9.74445 0.379878
\(659\) −10.7072 −0.417095 −0.208547 0.978012i \(-0.566874\pi\)
−0.208547 + 0.978012i \(0.566874\pi\)
\(660\) 0 0
\(661\) −44.7980 −1.74244 −0.871220 0.490893i \(-0.836670\pi\)
−0.871220 + 0.490893i \(0.836670\pi\)
\(662\) 10.0702 0.391388
\(663\) 39.5965 1.53780
\(664\) −11.2017 −0.434712
\(665\) 0 0
\(666\) 15.9253 0.617095
\(667\) 51.5965 1.99782
\(668\) 18.9882 0.734677
\(669\) 36.3680 1.40607
\(670\) 0 0
\(671\) −4.50239 −0.173813
\(672\) 69.9201 2.69723
\(673\) 43.2772 1.66821 0.834106 0.551604i \(-0.185984\pi\)
0.834106 + 0.551604i \(0.185984\pi\)
\(674\) 44.2113 1.70296
\(675\) 0 0
\(676\) −25.5237 −0.981680
\(677\) −11.5330 −0.443251 −0.221625 0.975132i \(-0.571136\pi\)
−0.221625 + 0.975132i \(0.571136\pi\)
\(678\) 49.2295 1.89065
\(679\) 9.92575 0.380915
\(680\) 0 0
\(681\) 19.0780 0.731072
\(682\) −15.5669 −0.596088
\(683\) −0.916090 −0.0350532 −0.0175266 0.999846i \(-0.505579\pi\)
−0.0175266 + 0.999846i \(0.505579\pi\)
\(684\) 13.6879 0.523372
\(685\) 0 0
\(686\) 29.0028 1.10733
\(687\) −0.216540 −0.00826153
\(688\) 7.67456 0.292590
\(689\) −4.59960 −0.175231
\(690\) 0 0
\(691\) 14.8278 0.564077 0.282039 0.959403i \(-0.408989\pi\)
0.282039 + 0.959403i \(0.408989\pi\)
\(692\) 14.9536 0.568450
\(693\) 18.1285 0.688644
\(694\) −64.3047 −2.44097
\(695\) 0 0
\(696\) 28.2226 1.06978
\(697\) −21.8735 −0.828517
\(698\) −31.5666 −1.19481
\(699\) −27.3828 −1.03571
\(700\) 0 0
\(701\) 36.1722 1.36620 0.683102 0.730323i \(-0.260631\pi\)
0.683102 + 0.730323i \(0.260631\pi\)
\(702\) −24.9890 −0.943150
\(703\) 1.40652 0.0530481
\(704\) 13.0573 0.492117
\(705\) 0 0
\(706\) 63.4807 2.38913
\(707\) 9.37305 0.352510
\(708\) −40.1800 −1.51006
\(709\) 10.4331 0.391823 0.195911 0.980622i \(-0.437234\pi\)
0.195911 + 0.980622i \(0.437234\pi\)
\(710\) 0 0
\(711\) −49.6233 −1.86102
\(712\) 1.02777 0.0385171
\(713\) 43.7470 1.63834
\(714\) 148.513 5.55794
\(715\) 0 0
\(716\) −11.6131 −0.434003
\(717\) −8.43308 −0.314939
\(718\) 46.6144 1.73963
\(719\) 7.73373 0.288420 0.144210 0.989547i \(-0.453936\pi\)
0.144210 + 0.989547i \(0.453936\pi\)
\(720\) 0 0
\(721\) 18.8102 0.700529
\(722\) 2.14243 0.0797331
\(723\) −27.1354 −1.00918
\(724\) −7.08114 −0.263168
\(725\) 0 0
\(726\) −60.3061 −2.23817
\(727\) 39.5241 1.46587 0.732934 0.680300i \(-0.238151\pi\)
0.732934 + 0.680300i \(0.238151\pi\)
\(728\) 6.96068 0.257980
\(729\) −40.5373 −1.50138
\(730\) 0 0
\(731\) 24.0831 0.890746
\(732\) 30.3808 1.12291
\(733\) −50.7498 −1.87449 −0.937243 0.348677i \(-0.886631\pi\)
−0.937243 + 0.348677i \(0.886631\pi\)
\(734\) −64.3473 −2.37510
\(735\) 0 0
\(736\) −52.0435 −1.91835
\(737\) −17.4728 −0.643619
\(738\) 31.9288 1.17532
\(739\) 0.419276 0.0154233 0.00771165 0.999970i \(-0.497545\pi\)
0.00771165 + 0.999970i \(0.497545\pi\)
\(740\) 0 0
\(741\) −5.10482 −0.187530
\(742\) −17.2515 −0.633321
\(743\) −19.5511 −0.717259 −0.358630 0.933480i \(-0.616756\pi\)
−0.358630 + 0.933480i \(0.616756\pi\)
\(744\) 23.9290 0.877280
\(745\) 0 0
\(746\) 79.9246 2.92625
\(747\) −46.8319 −1.71349
\(748\) 22.1958 0.811560
\(749\) 63.2912 2.31261
\(750\) 0 0
\(751\) 8.76768 0.319937 0.159969 0.987122i \(-0.448861\pi\)
0.159969 + 0.987122i \(0.448861\pi\)
\(752\) 3.62102 0.132045
\(753\) −42.9732 −1.56603
\(754\) −29.4728 −1.07334
\(755\) 0 0
\(756\) −52.8864 −1.92346
\(757\) 48.0535 1.74653 0.873267 0.487241i \(-0.161997\pi\)
0.873267 + 0.487241i \(0.161997\pi\)
\(758\) −31.6638 −1.15008
\(759\) −21.1533 −0.767815
\(760\) 0 0
\(761\) 29.3947 1.06556 0.532779 0.846254i \(-0.321148\pi\)
0.532779 + 0.846254i \(0.321148\pi\)
\(762\) −28.5827 −1.03544
\(763\) −14.6768 −0.531336
\(764\) −45.9031 −1.66071
\(765\) 0 0
\(766\) 62.1350 2.24503
\(767\) 9.55875 0.345146
\(768\) 8.38765 0.302663
\(769\) −13.3790 −0.482458 −0.241229 0.970468i \(-0.577551\pi\)
−0.241229 + 0.970468i \(0.577551\pi\)
\(770\) 0 0
\(771\) 43.5480 1.56834
\(772\) 35.0615 1.26189
\(773\) −0.133625 −0.00480617 −0.00240309 0.999997i \(-0.500765\pi\)
−0.00240309 + 0.999997i \(0.500765\pi\)
\(774\) −35.1543 −1.26359
\(775\) 0 0
\(776\) −4.04115 −0.145069
\(777\) −12.5697 −0.450937
\(778\) 45.4167 1.62827
\(779\) 2.81995 0.101035
\(780\) 0 0
\(781\) −7.89545 −0.282522
\(782\) −110.542 −3.95298
\(783\) 51.0130 1.82305
\(784\) −6.52536 −0.233049
\(785\) 0 0
\(786\) 95.6658 3.41229
\(787\) 4.84060 0.172549 0.0862744 0.996271i \(-0.472504\pi\)
0.0862744 + 0.996271i \(0.472504\pi\)
\(788\) 55.7202 1.98495
\(789\) 43.8130 1.55979
\(790\) 0 0
\(791\) −24.7863 −0.881299
\(792\) −7.38079 −0.262265
\(793\) −7.22753 −0.256657
\(794\) −22.2505 −0.789640
\(795\) 0 0
\(796\) 18.3739 0.651246
\(797\) −11.6993 −0.414410 −0.207205 0.978298i \(-0.566437\pi\)
−0.207205 + 0.978298i \(0.566437\pi\)
\(798\) −19.1464 −0.677774
\(799\) 11.3629 0.401991
\(800\) 0 0
\(801\) 4.29685 0.151822
\(802\) 54.9831 1.94152
\(803\) 0.268814 0.00948624
\(804\) 117.901 4.15806
\(805\) 0 0
\(806\) −24.9890 −0.880200
\(807\) 33.1593 1.16726
\(808\) −3.81613 −0.134251
\(809\) −33.1987 −1.16720 −0.583601 0.812040i \(-0.698357\pi\)
−0.583601 + 0.812040i \(0.698357\pi\)
\(810\) 0 0
\(811\) −26.5766 −0.933232 −0.466616 0.884460i \(-0.654527\pi\)
−0.466616 + 0.884460i \(0.654527\pi\)
\(812\) −62.3759 −2.18896
\(813\) 6.22061 0.218166
\(814\) −3.32924 −0.116690
\(815\) 0 0
\(816\) 55.1869 1.93193
\(817\) −3.10482 −0.108624
\(818\) 12.2464 0.428185
\(819\) 29.1010 1.01687
\(820\) 0 0
\(821\) 25.5807 0.892773 0.446387 0.894840i \(-0.352711\pi\)
0.446387 + 0.894840i \(0.352711\pi\)
\(822\) 5.01379 0.174876
\(823\) −12.9783 −0.452395 −0.226198 0.974081i \(-0.572629\pi\)
−0.226198 + 0.974081i \(0.572629\pi\)
\(824\) −7.65835 −0.266791
\(825\) 0 0
\(826\) 35.8515 1.24743
\(827\) 12.5167 0.435248 0.217624 0.976033i \(-0.430169\pi\)
0.217624 + 0.976033i \(0.430169\pi\)
\(828\) 91.0504 3.16422
\(829\) −19.4391 −0.675149 −0.337574 0.941299i \(-0.609606\pi\)
−0.337574 + 0.941299i \(0.609606\pi\)
\(830\) 0 0
\(831\) −67.4124 −2.33851
\(832\) 20.9605 0.726674
\(833\) −20.4769 −0.709482
\(834\) −78.7400 −2.72654
\(835\) 0 0
\(836\) −2.86151 −0.0989673
\(837\) 43.2522 1.49501
\(838\) 33.0583 1.14198
\(839\) −43.9972 −1.51895 −0.759475 0.650536i \(-0.774544\pi\)
−0.759475 + 0.650536i \(0.774544\pi\)
\(840\) 0 0
\(841\) 31.1662 1.07470
\(842\) −2.52818 −0.0871268
\(843\) 73.9229 2.54604
\(844\) 28.6468 0.986063
\(845\) 0 0
\(846\) −16.5865 −0.570256
\(847\) 30.3632 1.04329
\(848\) −6.41061 −0.220141
\(849\) −21.2591 −0.729610
\(850\) 0 0
\(851\) 9.35601 0.320720
\(852\) 53.2761 1.82521
\(853\) −3.30374 −0.113118 −0.0565590 0.998399i \(-0.518013\pi\)
−0.0565590 + 0.998399i \(0.518013\pi\)
\(854\) −27.1079 −0.927614
\(855\) 0 0
\(856\) −25.7682 −0.880740
\(857\) −6.02374 −0.205767 −0.102883 0.994693i \(-0.532807\pi\)
−0.102883 + 0.994693i \(0.532807\pi\)
\(858\) 12.0831 0.412511
\(859\) −36.0158 −1.22884 −0.614421 0.788978i \(-0.710610\pi\)
−0.614421 + 0.788978i \(0.710610\pi\)
\(860\) 0 0
\(861\) −25.2012 −0.858853
\(862\) −30.0487 −1.02346
\(863\) −29.9250 −1.01866 −0.509329 0.860572i \(-0.670106\pi\)
−0.509329 + 0.860572i \(0.670106\pi\)
\(864\) −51.4549 −1.75053
\(865\) 0 0
\(866\) −54.8236 −1.86298
\(867\) 124.247 4.21966
\(868\) −52.8864 −1.79508
\(869\) 10.3739 0.351911
\(870\) 0 0
\(871\) −28.0485 −0.950386
\(872\) 5.97548 0.202355
\(873\) −16.8951 −0.571813
\(874\) 14.2512 0.482054
\(875\) 0 0
\(876\) −1.81388 −0.0612852
\(877\) −0.618980 −0.0209015 −0.0104507 0.999945i \(-0.503327\pi\)
−0.0104507 + 0.999945i \(0.503327\pi\)
\(878\) −47.0762 −1.58874
\(879\) 69.5184 2.34480
\(880\) 0 0
\(881\) −21.2423 −0.715672 −0.357836 0.933784i \(-0.616485\pi\)
−0.357836 + 0.933784i \(0.616485\pi\)
\(882\) 29.8902 1.00646
\(883\) 48.8971 1.64552 0.822760 0.568389i \(-0.192433\pi\)
0.822760 + 0.568389i \(0.192433\pi\)
\(884\) 35.6302 1.19837
\(885\) 0 0
\(886\) 41.5034 1.39433
\(887\) −58.2797 −1.95684 −0.978421 0.206622i \(-0.933753\pi\)
−0.978421 + 0.206622i \(0.933753\pi\)
\(888\) 5.11762 0.171736
\(889\) 14.3909 0.482657
\(890\) 0 0
\(891\) −3.39757 −0.113823
\(892\) 32.7251 1.09572
\(893\) −1.46492 −0.0490216
\(894\) −85.0892 −2.84581
\(895\) 0 0
\(896\) 30.0318 1.00329
\(897\) −33.9566 −1.13378
\(898\) −41.1007 −1.37155
\(899\) 51.0130 1.70138
\(900\) 0 0
\(901\) −20.1168 −0.670187
\(902\) −6.67483 −0.222247
\(903\) 27.7470 0.923361
\(904\) 10.0914 0.335636
\(905\) 0 0
\(906\) 30.9732 1.02902
\(907\) 36.5154 1.21247 0.606237 0.795284i \(-0.292678\pi\)
0.606237 + 0.795284i \(0.292678\pi\)
\(908\) 17.1670 0.569708
\(909\) −15.9543 −0.529172
\(910\) 0 0
\(911\) 6.62735 0.219574 0.109787 0.993955i \(-0.464983\pi\)
0.109787 + 0.993955i \(0.464983\pi\)
\(912\) −7.11475 −0.235593
\(913\) 9.79036 0.324014
\(914\) −17.9110 −0.592442
\(915\) 0 0
\(916\) −0.194850 −0.00643802
\(917\) −48.1662 −1.59059
\(918\) −109.292 −3.60717
\(919\) 7.91688 0.261154 0.130577 0.991438i \(-0.458317\pi\)
0.130577 + 0.991438i \(0.458317\pi\)
\(920\) 0 0
\(921\) −12.6210 −0.415877
\(922\) −53.6182 −1.76582
\(923\) −12.6743 −0.417179
\(924\) 25.5726 0.841275
\(925\) 0 0
\(926\) −69.3611 −2.27935
\(927\) −32.0178 −1.05160
\(928\) −60.6875 −1.99217
\(929\) 35.8597 1.17652 0.588259 0.808673i \(-0.299814\pi\)
0.588259 + 0.808673i \(0.299814\pi\)
\(930\) 0 0
\(931\) 2.63990 0.0865192
\(932\) −24.6399 −0.807106
\(933\) −45.2254 −1.48061
\(934\) −32.6113 −1.06707
\(935\) 0 0
\(936\) −11.8481 −0.387268
\(937\) 11.4420 0.373793 0.186896 0.982380i \(-0.440157\pi\)
0.186896 + 0.982380i \(0.440157\pi\)
\(938\) −105.200 −3.43490
\(939\) −71.3236 −2.32756
\(940\) 0 0
\(941\) −0.360100 −0.0117389 −0.00586945 0.999983i \(-0.501868\pi\)
−0.00586945 + 0.999983i \(0.501868\pi\)
\(942\) −44.3195 −1.44401
\(943\) 18.7579 0.610843
\(944\) 13.3223 0.433605
\(945\) 0 0
\(946\) 7.34911 0.238940
\(947\) −2.63807 −0.0857256 −0.0428628 0.999081i \(-0.513648\pi\)
−0.0428628 + 0.999081i \(0.513648\pi\)
\(948\) −70.0000 −2.27349
\(949\) 0.431517 0.0140076
\(950\) 0 0
\(951\) 78.8084 2.55554
\(952\) 30.4432 0.986670
\(953\) −10.4430 −0.338282 −0.169141 0.985592i \(-0.554099\pi\)
−0.169141 + 0.985592i \(0.554099\pi\)
\(954\) 29.3646 0.950714
\(955\) 0 0
\(956\) −7.58835 −0.245425
\(957\) −24.6667 −0.797360
\(958\) −42.9525 −1.38773
\(959\) −2.52437 −0.0815160
\(960\) 0 0
\(961\) 12.2522 0.395232
\(962\) −5.34432 −0.172308
\(963\) −107.731 −3.47159
\(964\) −24.4173 −0.786428
\(965\) 0 0
\(966\) −127.359 −4.09772
\(967\) −33.2732 −1.06999 −0.534996 0.844854i \(-0.679687\pi\)
−0.534996 + 0.844854i \(0.679687\pi\)
\(968\) −12.3620 −0.397330
\(969\) −22.3264 −0.717228
\(970\) 0 0
\(971\) 23.5657 0.756258 0.378129 0.925753i \(-0.376568\pi\)
0.378129 + 0.925753i \(0.376568\pi\)
\(972\) −28.1752 −0.903719
\(973\) 39.6444 1.27094
\(974\) 67.8098 2.17277
\(975\) 0 0
\(976\) −10.0732 −0.322437
\(977\) −0.402439 −0.0128752 −0.00643759 0.999979i \(-0.502049\pi\)
−0.00643759 + 0.999979i \(0.502049\pi\)
\(978\) −44.7409 −1.43066
\(979\) −0.898271 −0.0287089
\(980\) 0 0
\(981\) 24.9821 0.797617
\(982\) 53.1395 1.69575
\(983\) −11.4927 −0.366561 −0.183281 0.983061i \(-0.558672\pi\)
−0.183281 + 0.983061i \(0.558672\pi\)
\(984\) 10.2603 0.327088
\(985\) 0 0
\(986\) −128.902 −4.10508
\(987\) 13.0916 0.416710
\(988\) −4.59348 −0.146138
\(989\) −20.6529 −0.656723
\(990\) 0 0
\(991\) 6.05761 0.192426 0.0962132 0.995361i \(-0.469327\pi\)
0.0962132 + 0.995361i \(0.469327\pi\)
\(992\) −51.4549 −1.63370
\(993\) 13.5292 0.429335
\(994\) −47.5368 −1.50777
\(995\) 0 0
\(996\) −66.0624 −2.09327
\(997\) 48.1086 1.52362 0.761808 0.647803i \(-0.224312\pi\)
0.761808 + 0.647803i \(0.224312\pi\)
\(998\) −70.7510 −2.23959
\(999\) 9.25020 0.292663
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 475.2.a.i.1.3 4
3.2 odd 2 4275.2.a.bo.1.2 4
4.3 odd 2 7600.2.a.cf.1.1 4
5.2 odd 4 475.2.b.e.324.7 8
5.3 odd 4 475.2.b.e.324.2 8
5.4 even 2 95.2.a.b.1.2 4
15.14 odd 2 855.2.a.m.1.3 4
19.18 odd 2 9025.2.a.bf.1.2 4
20.19 odd 2 1520.2.a.t.1.4 4
35.34 odd 2 4655.2.a.y.1.2 4
40.19 odd 2 6080.2.a.ch.1.1 4
40.29 even 2 6080.2.a.cc.1.4 4
95.94 odd 2 1805.2.a.p.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.2 4 5.4 even 2
475.2.a.i.1.3 4 1.1 even 1 trivial
475.2.b.e.324.2 8 5.3 odd 4
475.2.b.e.324.7 8 5.2 odd 4
855.2.a.m.1.3 4 15.14 odd 2
1520.2.a.t.1.4 4 20.19 odd 2
1805.2.a.p.1.3 4 95.94 odd 2
4275.2.a.bo.1.2 4 3.2 odd 2
4655.2.a.y.1.2 4 35.34 odd 2
6080.2.a.cc.1.4 4 40.29 even 2
6080.2.a.ch.1.1 4 40.19 odd 2
7600.2.a.cf.1.1 4 4.3 odd 2
9025.2.a.bf.1.2 4 19.18 odd 2