Properties

Label 252.4.e.a.71.28
Level $252$
Weight $4$
Character 252.71
Analytic conductor $14.868$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(71,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.71");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 71.28
Character \(\chi\) \(=\) 252.71
Dual form 252.4.e.a.71.27

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.88294 + 2.11058i) q^{2} +(-0.909099 + 7.94818i) q^{4} -21.8666i q^{5} +7.00000i q^{7} +(-18.4870 + 13.0472i) q^{8} +O(q^{10})\) \(q+(1.88294 + 2.11058i) q^{2} +(-0.909099 + 7.94818i) q^{4} -21.8666i q^{5} +7.00000i q^{7} +(-18.4870 + 13.0472i) q^{8} +(46.1512 - 41.1734i) q^{10} -66.1125 q^{11} -61.5090 q^{13} +(-14.7741 + 13.1806i) q^{14} +(-62.3471 - 14.4514i) q^{16} -11.7720i q^{17} +87.9289i q^{19} +(173.800 + 19.8789i) q^{20} +(-124.486 - 139.536i) q^{22} +13.2343 q^{23} -353.148 q^{25} +(-115.818 - 129.820i) q^{26} +(-55.6372 - 6.36369i) q^{28} +7.53630i q^{29} -45.1117i q^{31} +(-86.8949 - 158.800i) q^{32} +(24.8457 - 22.1659i) q^{34} +153.066 q^{35} +207.904 q^{37} +(-185.581 + 165.565i) q^{38} +(285.298 + 404.249i) q^{40} -172.688i q^{41} -331.004i q^{43} +(60.1028 - 525.474i) q^{44} +(24.9194 + 27.9322i) q^{46} +436.796 q^{47} -49.0000 q^{49} +(-664.956 - 745.347i) q^{50} +(55.9177 - 488.884i) q^{52} -282.237i q^{53} +1445.66i q^{55} +(-91.3303 - 129.409i) q^{56} +(-15.9060 + 14.1904i) q^{58} +553.350 q^{59} -262.257 q^{61} +(95.2119 - 84.9425i) q^{62} +(171.542 - 482.408i) q^{64} +1344.99i q^{65} -89.8443i q^{67} +(93.5658 + 10.7019i) q^{68} +(288.214 + 323.058i) q^{70} -891.802 q^{71} -506.529 q^{73} +(391.470 + 438.798i) q^{74} +(-698.874 - 79.9360i) q^{76} -462.788i q^{77} +836.714i q^{79} +(-316.002 + 1363.32i) q^{80} +(364.472 - 325.161i) q^{82} -114.691 q^{83} -257.413 q^{85} +(698.610 - 623.259i) q^{86} +(1222.22 - 862.583i) q^{88} -162.452i q^{89} -430.563i q^{91} +(-12.0313 + 105.189i) q^{92} +(822.460 + 921.894i) q^{94} +1922.71 q^{95} -1054.85 q^{97} +(-92.2639 - 103.418i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 24 q^{4} + 264 q^{10} - 468 q^{16} + 444 q^{22} - 900 q^{25} - 84 q^{28} - 432 q^{34} - 264 q^{37} + 1416 q^{40} + 180 q^{46} - 1764 q^{49} + 2736 q^{52} + 636 q^{58} - 3960 q^{61} + 1392 q^{64} - 504 q^{70} - 2520 q^{76} + 1032 q^{82} - 3144 q^{85} + 2748 q^{88} + 5496 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.88294 + 2.11058i 0.665719 + 0.746203i
\(3\) 0 0
\(4\) −0.909099 + 7.94818i −0.113637 + 0.993522i
\(5\) 21.8666i 1.95581i −0.209055 0.977904i \(-0.567039\pi\)
0.209055 0.977904i \(-0.432961\pi\)
\(6\) 0 0
\(7\) 7.00000i 0.377964i
\(8\) −18.4870 + 13.0472i −0.817020 + 0.576610i
\(9\) 0 0
\(10\) 46.1512 41.1734i 1.45943 1.30202i
\(11\) −66.1125 −1.81215 −0.906076 0.423115i \(-0.860937\pi\)
−0.906076 + 0.423115i \(0.860937\pi\)
\(12\) 0 0
\(13\) −61.5090 −1.31227 −0.656135 0.754643i \(-0.727810\pi\)
−0.656135 + 0.754643i \(0.727810\pi\)
\(14\) −14.7741 + 13.1806i −0.282038 + 0.251618i
\(15\) 0 0
\(16\) −62.3471 14.4514i −0.974173 0.225802i
\(17\) 11.7720i 0.167949i −0.996468 0.0839743i \(-0.973239\pi\)
0.996468 0.0839743i \(-0.0267614\pi\)
\(18\) 0 0
\(19\) 87.9289i 1.06170i 0.847466 + 0.530849i \(0.178127\pi\)
−0.847466 + 0.530849i \(0.821873\pi\)
\(20\) 173.800 + 19.8789i 1.94314 + 0.222253i
\(21\) 0 0
\(22\) −124.486 139.536i −1.20638 1.35223i
\(23\) 13.2343 0.119981 0.0599903 0.998199i \(-0.480893\pi\)
0.0599903 + 0.998199i \(0.480893\pi\)
\(24\) 0 0
\(25\) −353.148 −2.82518
\(26\) −115.818 129.820i −0.873603 0.979220i
\(27\) 0 0
\(28\) −55.6372 6.36369i −0.375516 0.0429509i
\(29\) 7.53630i 0.0482571i 0.999709 + 0.0241285i \(0.00768110\pi\)
−0.999709 + 0.0241285i \(0.992319\pi\)
\(30\) 0 0
\(31\) 45.1117i 0.261365i −0.991424 0.130682i \(-0.958283\pi\)
0.991424 0.130682i \(-0.0417168\pi\)
\(32\) −86.8949 158.800i −0.480031 0.877252i
\(33\) 0 0
\(34\) 24.8457 22.1659i 0.125324 0.111806i
\(35\) 153.066 0.739226
\(36\) 0 0
\(37\) 207.904 0.923763 0.461881 0.886942i \(-0.347175\pi\)
0.461881 + 0.886942i \(0.347175\pi\)
\(38\) −185.581 + 165.565i −0.792242 + 0.706792i
\(39\) 0 0
\(40\) 285.298 + 404.249i 1.12774 + 1.59793i
\(41\) 172.688i 0.657789i −0.944367 0.328894i \(-0.893324\pi\)
0.944367 0.328894i \(-0.106676\pi\)
\(42\) 0 0
\(43\) 331.004i 1.17390i −0.809624 0.586949i \(-0.800329\pi\)
0.809624 0.586949i \(-0.199671\pi\)
\(44\) 60.1028 525.474i 0.205928 1.80041i
\(45\) 0 0
\(46\) 24.9194 + 27.9322i 0.0798733 + 0.0895298i
\(47\) 436.796 1.35560 0.677801 0.735246i \(-0.262933\pi\)
0.677801 + 0.735246i \(0.262933\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) −664.956 745.347i −1.88078 2.10816i
\(51\) 0 0
\(52\) 55.9177 488.884i 0.149123 1.30377i
\(53\) 282.237i 0.731477i −0.930718 0.365739i \(-0.880816\pi\)
0.930718 0.365739i \(-0.119184\pi\)
\(54\) 0 0
\(55\) 1445.66i 3.54422i
\(56\) −91.3303 129.409i −0.217938 0.308804i
\(57\) 0 0
\(58\) −15.9060 + 14.1904i −0.0360096 + 0.0321256i
\(59\) 553.350 1.22102 0.610509 0.792009i \(-0.290965\pi\)
0.610509 + 0.792009i \(0.290965\pi\)
\(60\) 0 0
\(61\) −262.257 −0.550468 −0.275234 0.961377i \(-0.588755\pi\)
−0.275234 + 0.961377i \(0.588755\pi\)
\(62\) 95.2119 84.9425i 0.195031 0.173995i
\(63\) 0 0
\(64\) 171.542 482.408i 0.335042 0.942203i
\(65\) 1344.99i 2.56655i
\(66\) 0 0
\(67\) 89.8443i 0.163824i −0.996640 0.0819121i \(-0.973897\pi\)
0.996640 0.0819121i \(-0.0261027\pi\)
\(68\) 93.5658 + 10.7019i 0.166861 + 0.0190852i
\(69\) 0 0
\(70\) 288.214 + 323.058i 0.492116 + 0.551612i
\(71\) −891.802 −1.49067 −0.745334 0.666691i \(-0.767710\pi\)
−0.745334 + 0.666691i \(0.767710\pi\)
\(72\) 0 0
\(73\) −506.529 −0.812119 −0.406059 0.913847i \(-0.633097\pi\)
−0.406059 + 0.913847i \(0.633097\pi\)
\(74\) 391.470 + 438.798i 0.614966 + 0.689314i
\(75\) 0 0
\(76\) −698.874 79.9360i −1.05482 0.120649i
\(77\) 462.788i 0.684929i
\(78\) 0 0
\(79\) 836.714i 1.19162i 0.803127 + 0.595808i \(0.203168\pi\)
−0.803127 + 0.595808i \(0.796832\pi\)
\(80\) −316.002 + 1363.32i −0.441626 + 1.90530i
\(81\) 0 0
\(82\) 364.472 325.161i 0.490844 0.437902i
\(83\) −114.691 −0.151674 −0.0758369 0.997120i \(-0.524163\pi\)
−0.0758369 + 0.997120i \(0.524163\pi\)
\(84\) 0 0
\(85\) −257.413 −0.328475
\(86\) 698.610 623.259i 0.875965 0.781485i
\(87\) 0 0
\(88\) 1222.22 862.583i 1.48056 1.04490i
\(89\) 162.452i 0.193482i −0.995310 0.0967410i \(-0.969158\pi\)
0.995310 0.0967410i \(-0.0308419\pi\)
\(90\) 0 0
\(91\) 430.563i 0.495992i
\(92\) −12.0313 + 105.189i −0.0136343 + 0.119203i
\(93\) 0 0
\(94\) 822.460 + 921.894i 0.902449 + 1.01155i
\(95\) 1922.71 2.07648
\(96\) 0 0
\(97\) −1054.85 −1.10416 −0.552082 0.833790i \(-0.686167\pi\)
−0.552082 + 0.833790i \(0.686167\pi\)
\(98\) −92.2639 103.418i −0.0951027 0.106600i
\(99\) 0 0
\(100\) 321.046 2806.88i 0.321046 2.80688i
\(101\) 170.869i 0.168337i −0.996452 0.0841687i \(-0.973177\pi\)
0.996452 0.0841687i \(-0.0268235\pi\)
\(102\) 0 0
\(103\) 75.9945i 0.0726986i 0.999339 + 0.0363493i \(0.0115729\pi\)
−0.999339 + 0.0363493i \(0.988427\pi\)
\(104\) 1137.12 802.519i 1.07215 0.756668i
\(105\) 0 0
\(106\) 595.685 531.435i 0.545830 0.486958i
\(107\) −1082.89 −0.978385 −0.489193 0.872176i \(-0.662709\pi\)
−0.489193 + 0.872176i \(0.662709\pi\)
\(108\) 0 0
\(109\) −701.186 −0.616160 −0.308080 0.951360i \(-0.599686\pi\)
−0.308080 + 0.951360i \(0.599686\pi\)
\(110\) −3051.17 + 2722.08i −2.64471 + 2.35945i
\(111\) 0 0
\(112\) 101.159 436.430i 0.0853453 0.368203i
\(113\) 804.271i 0.669553i 0.942298 + 0.334776i \(0.108661\pi\)
−0.942298 + 0.334776i \(0.891339\pi\)
\(114\) 0 0
\(115\) 289.390i 0.234659i
\(116\) −59.8998 6.85124i −0.0479445 0.00548381i
\(117\) 0 0
\(118\) 1041.92 + 1167.89i 0.812854 + 0.911127i
\(119\) 82.4039 0.0634786
\(120\) 0 0
\(121\) 3039.86 2.28389
\(122\) −493.813 553.514i −0.366457 0.410761i
\(123\) 0 0
\(124\) 358.556 + 41.0110i 0.259672 + 0.0297008i
\(125\) 4988.82i 3.56971i
\(126\) 0 0
\(127\) 1464.86i 1.02351i 0.859133 + 0.511753i \(0.171004\pi\)
−0.859133 + 0.511753i \(0.828996\pi\)
\(128\) 1341.16 546.291i 0.926118 0.377233i
\(129\) 0 0
\(130\) −2838.71 + 2532.54i −1.91517 + 1.70860i
\(131\) −2092.35 −1.39549 −0.697747 0.716344i \(-0.745814\pi\)
−0.697747 + 0.716344i \(0.745814\pi\)
\(132\) 0 0
\(133\) −615.502 −0.401284
\(134\) 189.624 169.171i 0.122246 0.109061i
\(135\) 0 0
\(136\) 153.591 + 217.629i 0.0968408 + 0.137217i
\(137\) 578.141i 0.360539i 0.983617 + 0.180270i \(0.0576970\pi\)
−0.983617 + 0.180270i \(0.942303\pi\)
\(138\) 0 0
\(139\) 1529.70i 0.933433i −0.884407 0.466716i \(-0.845437\pi\)
0.884407 0.466716i \(-0.154563\pi\)
\(140\) −139.152 + 1216.60i −0.0840037 + 0.734437i
\(141\) 0 0
\(142\) −1679.21 1882.22i −0.992366 1.11234i
\(143\) 4066.51 2.37803
\(144\) 0 0
\(145\) 164.793 0.0943816
\(146\) −953.761 1069.07i −0.540643 0.606005i
\(147\) 0 0
\(148\) −189.005 + 1652.46i −0.104974 + 0.917779i
\(149\) 1177.79i 0.647571i −0.946130 0.323786i \(-0.895044\pi\)
0.946130 0.323786i \(-0.104956\pi\)
\(150\) 0 0
\(151\) 274.592i 0.147987i 0.997259 + 0.0739933i \(0.0235744\pi\)
−0.997259 + 0.0739933i \(0.976426\pi\)
\(152\) −1147.22 1625.55i −0.612186 0.867428i
\(153\) 0 0
\(154\) 976.750 871.400i 0.511096 0.455970i
\(155\) −986.440 −0.511179
\(156\) 0 0
\(157\) 803.908 0.408655 0.204328 0.978903i \(-0.434499\pi\)
0.204328 + 0.978903i \(0.434499\pi\)
\(158\) −1765.95 + 1575.48i −0.889187 + 0.793281i
\(159\) 0 0
\(160\) −3472.40 + 1900.09i −1.71574 + 0.938848i
\(161\) 92.6404i 0.0453484i
\(162\) 0 0
\(163\) 1909.17i 0.917410i −0.888589 0.458705i \(-0.848314\pi\)
0.888589 0.458705i \(-0.151686\pi\)
\(164\) 1372.56 + 156.990i 0.653528 + 0.0747494i
\(165\) 0 0
\(166\) −215.955 242.064i −0.100972 0.113179i
\(167\) 531.691 0.246368 0.123184 0.992384i \(-0.460689\pi\)
0.123184 + 0.992384i \(0.460689\pi\)
\(168\) 0 0
\(169\) 1586.35 0.722055
\(170\) −484.693 543.291i −0.218672 0.245109i
\(171\) 0 0
\(172\) 2630.88 + 300.915i 1.16629 + 0.133399i
\(173\) 3717.57i 1.63377i −0.576804 0.816883i \(-0.695700\pi\)
0.576804 0.816883i \(-0.304300\pi\)
\(174\) 0 0
\(175\) 2472.04i 1.06782i
\(176\) 4121.92 + 955.415i 1.76535 + 0.409188i
\(177\) 0 0
\(178\) 342.868 305.887i 0.144377 0.128805i
\(179\) 2173.06 0.907387 0.453694 0.891158i \(-0.350106\pi\)
0.453694 + 0.891158i \(0.350106\pi\)
\(180\) 0 0
\(181\) 2449.11 1.00575 0.502876 0.864358i \(-0.332275\pi\)
0.502876 + 0.864358i \(0.332275\pi\)
\(182\) 908.738 810.723i 0.370110 0.330191i
\(183\) 0 0
\(184\) −244.664 + 172.671i −0.0980265 + 0.0691820i
\(185\) 4546.15i 1.80670i
\(186\) 0 0
\(187\) 778.275i 0.304348i
\(188\) −397.091 + 3471.73i −0.154047 + 1.34682i
\(189\) 0 0
\(190\) 3620.33 + 4058.02i 1.38235 + 1.54947i
\(191\) −3122.32 −1.18284 −0.591421 0.806363i \(-0.701433\pi\)
−0.591421 + 0.806363i \(0.701433\pi\)
\(192\) 0 0
\(193\) −686.947 −0.256205 −0.128102 0.991761i \(-0.540889\pi\)
−0.128102 + 0.991761i \(0.540889\pi\)
\(194\) −1986.22 2226.35i −0.735063 0.823931i
\(195\) 0 0
\(196\) 44.5458 389.461i 0.0162339 0.141932i
\(197\) 644.412i 0.233058i −0.993187 0.116529i \(-0.962823\pi\)
0.993187 0.116529i \(-0.0371768\pi\)
\(198\) 0 0
\(199\) 2749.76i 0.979523i 0.871856 + 0.489761i \(0.162916\pi\)
−0.871856 + 0.489761i \(0.837084\pi\)
\(200\) 6528.66 4607.59i 2.30823 1.62903i
\(201\) 0 0
\(202\) 360.632 321.735i 0.125614 0.112065i
\(203\) −52.7541 −0.0182395
\(204\) 0 0
\(205\) −3776.10 −1.28651
\(206\) −160.392 + 143.093i −0.0542479 + 0.0483968i
\(207\) 0 0
\(208\) 3834.91 + 888.888i 1.27838 + 0.296314i
\(209\) 5813.20i 1.92396i
\(210\) 0 0
\(211\) 15.7611i 0.00514235i −0.999997 0.00257118i \(-0.999182\pi\)
0.999997 0.00257118i \(-0.000818432\pi\)
\(212\) 2243.27 + 256.582i 0.726739 + 0.0831231i
\(213\) 0 0
\(214\) −2039.02 2285.53i −0.651329 0.730074i
\(215\) −7237.92 −2.29592
\(216\) 0 0
\(217\) 315.782 0.0987866
\(218\) −1320.29 1479.91i −0.410189 0.459780i
\(219\) 0 0
\(220\) −11490.3 1314.24i −3.52126 0.402756i
\(221\) 724.083i 0.220394i
\(222\) 0 0
\(223\) 3774.33i 1.13340i 0.823925 + 0.566699i \(0.191780\pi\)
−0.823925 + 0.566699i \(0.808220\pi\)
\(224\) 1111.60 608.264i 0.331570 0.181435i
\(225\) 0 0
\(226\) −1697.48 + 1514.39i −0.499622 + 0.445734i
\(227\) −1031.00 −0.301452 −0.150726 0.988576i \(-0.548161\pi\)
−0.150726 + 0.988576i \(0.548161\pi\)
\(228\) 0 0
\(229\) 2221.01 0.640910 0.320455 0.947264i \(-0.396164\pi\)
0.320455 + 0.947264i \(0.396164\pi\)
\(230\) 610.781 544.903i 0.175103 0.156217i
\(231\) 0 0
\(232\) −98.3275 139.324i −0.0278255 0.0394270i
\(233\) 4787.61i 1.34612i −0.739586 0.673062i \(-0.764979\pi\)
0.739586 0.673062i \(-0.235021\pi\)
\(234\) 0 0
\(235\) 9551.25i 2.65130i
\(236\) −503.050 + 4398.13i −0.138753 + 1.21311i
\(237\) 0 0
\(238\) 155.161 + 173.920i 0.0422589 + 0.0473679i
\(239\) −5520.83 −1.49420 −0.747098 0.664714i \(-0.768554\pi\)
−0.747098 + 0.664714i \(0.768554\pi\)
\(240\) 0 0
\(241\) 611.900 0.163552 0.0817758 0.996651i \(-0.473941\pi\)
0.0817758 + 0.996651i \(0.473941\pi\)
\(242\) 5723.87 + 6415.88i 1.52043 + 1.70425i
\(243\) 0 0
\(244\) 238.417 2084.47i 0.0625538 0.546903i
\(245\) 1071.46i 0.279401i
\(246\) 0 0
\(247\) 5408.42i 1.39324i
\(248\) 588.581 + 833.982i 0.150705 + 0.213540i
\(249\) 0 0
\(250\) −10529.3 + 9393.64i −2.66373 + 2.37642i
\(251\) −5063.20 −1.27325 −0.636626 0.771173i \(-0.719671\pi\)
−0.636626 + 0.771173i \(0.719671\pi\)
\(252\) 0 0
\(253\) −874.956 −0.217423
\(254\) −3091.70 + 2758.24i −0.763743 + 0.681367i
\(255\) 0 0
\(256\) 3678.32 + 1802.00i 0.898027 + 0.439941i
\(257\) 6434.13i 1.56167i 0.624735 + 0.780837i \(0.285207\pi\)
−0.624735 + 0.780837i \(0.714793\pi\)
\(258\) 0 0
\(259\) 1455.33i 0.349149i
\(260\) −10690.2 1222.73i −2.54992 0.291656i
\(261\) 0 0
\(262\) −3939.77 4416.08i −0.929006 1.04132i
\(263\) −3816.92 −0.894911 −0.447455 0.894306i \(-0.647670\pi\)
−0.447455 + 0.894306i \(0.647670\pi\)
\(264\) 0 0
\(265\) −6171.57 −1.43063
\(266\) −1158.95 1299.07i −0.267142 0.299439i
\(267\) 0 0
\(268\) 714.099 + 81.6773i 0.162763 + 0.0186166i
\(269\) 2855.07i 0.647126i −0.946207 0.323563i \(-0.895119\pi\)
0.946207 0.323563i \(-0.104881\pi\)
\(270\) 0 0
\(271\) 5572.22i 1.24903i 0.781011 + 0.624517i \(0.214704\pi\)
−0.781011 + 0.624517i \(0.785296\pi\)
\(272\) −170.121 + 733.949i −0.0379232 + 0.163611i
\(273\) 0 0
\(274\) −1220.21 + 1088.60i −0.269035 + 0.240018i
\(275\) 23347.5 5.11966
\(276\) 0 0
\(277\) −1433.15 −0.310865 −0.155433 0.987846i \(-0.549677\pi\)
−0.155433 + 0.987846i \(0.549677\pi\)
\(278\) 3228.55 2880.32i 0.696530 0.621404i
\(279\) 0 0
\(280\) −2829.74 + 1997.08i −0.603962 + 0.426245i
\(281\) 1560.76i 0.331342i 0.986181 + 0.165671i \(0.0529789\pi\)
−0.986181 + 0.165671i \(0.947021\pi\)
\(282\) 0 0
\(283\) 557.943i 0.117195i 0.998282 + 0.0585976i \(0.0186629\pi\)
−0.998282 + 0.0585976i \(0.981337\pi\)
\(284\) 810.736 7088.20i 0.169396 1.48101i
\(285\) 0 0
\(286\) 7656.99 + 8582.70i 1.58310 + 1.77450i
\(287\) 1208.82 0.248621
\(288\) 0 0
\(289\) 4774.42 0.971793
\(290\) 310.295 + 347.809i 0.0628316 + 0.0704278i
\(291\) 0 0
\(292\) 460.484 4025.98i 0.0922870 0.806858i
\(293\) 1422.60i 0.283649i −0.989892 0.141824i \(-0.954703\pi\)
0.989892 0.141824i \(-0.0452968\pi\)
\(294\) 0 0
\(295\) 12099.9i 2.38808i
\(296\) −3843.53 + 2712.56i −0.754732 + 0.532651i
\(297\) 0 0
\(298\) 2485.82 2217.70i 0.483219 0.431100i
\(299\) −814.031 −0.157447
\(300\) 0 0
\(301\) 2317.03 0.443691
\(302\) −579.549 + 517.040i −0.110428 + 0.0985175i
\(303\) 0 0
\(304\) 1270.69 5482.11i 0.239734 1.03428i
\(305\) 5734.67i 1.07661i
\(306\) 0 0
\(307\) 7600.14i 1.41291i −0.707758 0.706455i \(-0.750293\pi\)
0.707758 0.706455i \(-0.249707\pi\)
\(308\) 3678.32 + 420.720i 0.680492 + 0.0778335i
\(309\) 0 0
\(310\) −1857.40 2081.96i −0.340301 0.381443i
\(311\) −4016.60 −0.732349 −0.366174 0.930546i \(-0.619333\pi\)
−0.366174 + 0.930546i \(0.619333\pi\)
\(312\) 0 0
\(313\) 6789.33 1.22606 0.613028 0.790061i \(-0.289951\pi\)
0.613028 + 0.790061i \(0.289951\pi\)
\(314\) 1513.71 + 1696.71i 0.272049 + 0.304940i
\(315\) 0 0
\(316\) −6650.35 760.655i −1.18390 0.135412i
\(317\) 4680.61i 0.829303i −0.909980 0.414651i \(-0.863904\pi\)
0.909980 0.414651i \(-0.136096\pi\)
\(318\) 0 0
\(319\) 498.244i 0.0874492i
\(320\) −10548.6 3751.03i −1.84277 0.655278i
\(321\) 0 0
\(322\) −195.525 + 174.436i −0.0338391 + 0.0301893i
\(323\) 1035.10 0.178311
\(324\) 0 0
\(325\) 21721.8 3.70741
\(326\) 4029.46 3594.85i 0.684574 0.610737i
\(327\) 0 0
\(328\) 2253.09 + 3192.49i 0.379288 + 0.537426i
\(329\) 3057.57i 0.512369i
\(330\) 0 0
\(331\) 5224.33i 0.867538i 0.901024 + 0.433769i \(0.142817\pi\)
−0.901024 + 0.433769i \(0.857183\pi\)
\(332\) 104.265 911.582i 0.0172358 0.150691i
\(333\) 0 0
\(334\) 1001.14 + 1122.18i 0.164012 + 0.183841i
\(335\) −1964.59 −0.320409
\(336\) 0 0
\(337\) −11037.2 −1.78408 −0.892038 0.451961i \(-0.850725\pi\)
−0.892038 + 0.451961i \(0.850725\pi\)
\(338\) 2987.01 + 3348.13i 0.480685 + 0.538799i
\(339\) 0 0
\(340\) 234.014 2045.97i 0.0373270 0.326347i
\(341\) 2982.45i 0.473633i
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) 4318.67 + 6119.28i 0.676881 + 0.959097i
\(345\) 0 0
\(346\) 7846.23 6999.95i 1.21912 1.08763i
\(347\) −4831.11 −0.747399 −0.373699 0.927550i \(-0.621911\pi\)
−0.373699 + 0.927550i \(0.621911\pi\)
\(348\) 0 0
\(349\) −4994.72 −0.766079 −0.383039 0.923732i \(-0.625123\pi\)
−0.383039 + 0.923732i \(0.625123\pi\)
\(350\) 5217.43 4654.69i 0.796810 0.710867i
\(351\) 0 0
\(352\) 5744.84 + 10498.6i 0.869889 + 1.58971i
\(353\) 7932.09i 1.19598i 0.801502 + 0.597992i \(0.204035\pi\)
−0.801502 + 0.597992i \(0.795965\pi\)
\(354\) 0 0
\(355\) 19500.7i 2.91546i
\(356\) 1291.20 + 147.685i 0.192229 + 0.0219868i
\(357\) 0 0
\(358\) 4091.74 + 4586.42i 0.604065 + 0.677095i
\(359\) 2834.37 0.416692 0.208346 0.978055i \(-0.433192\pi\)
0.208346 + 0.978055i \(0.433192\pi\)
\(360\) 0 0
\(361\) −872.488 −0.127203
\(362\) 4611.53 + 5169.05i 0.669548 + 0.750495i
\(363\) 0 0
\(364\) 3422.19 + 391.424i 0.492779 + 0.0563632i
\(365\) 11076.1i 1.58835i
\(366\) 0 0
\(367\) 5182.11i 0.737068i 0.929614 + 0.368534i \(0.120140\pi\)
−0.929614 + 0.368534i \(0.879860\pi\)
\(368\) −825.123 191.254i −0.116882 0.0270919i
\(369\) 0 0
\(370\) 9595.02 8560.12i 1.34817 1.20276i
\(371\) 1975.66 0.276472
\(372\) 0 0
\(373\) −5714.61 −0.793274 −0.396637 0.917976i \(-0.629823\pi\)
−0.396637 + 0.917976i \(0.629823\pi\)
\(374\) −1642.61 + 1465.44i −0.227106 + 0.202610i
\(375\) 0 0
\(376\) −8075.07 + 5698.96i −1.10755 + 0.781653i
\(377\) 463.550i 0.0633264i
\(378\) 0 0
\(379\) 4307.45i 0.583796i 0.956449 + 0.291898i \(0.0942868\pi\)
−0.956449 + 0.291898i \(0.905713\pi\)
\(380\) −1747.93 + 15282.0i −0.235965 + 2.06303i
\(381\) 0 0
\(382\) −5879.13 6589.90i −0.787440 0.882640i
\(383\) −6878.76 −0.917724 −0.458862 0.888508i \(-0.651743\pi\)
−0.458862 + 0.888508i \(0.651743\pi\)
\(384\) 0 0
\(385\) −10119.6 −1.33959
\(386\) −1293.48 1449.86i −0.170560 0.191181i
\(387\) 0 0
\(388\) 958.965 8384.15i 0.125474 1.09701i
\(389\) 9982.91i 1.30117i −0.759435 0.650583i \(-0.774524\pi\)
0.759435 0.650583i \(-0.225476\pi\)
\(390\) 0 0
\(391\) 155.795i 0.0201506i
\(392\) 905.865 639.312i 0.116717 0.0823728i
\(393\) 0 0
\(394\) 1360.08 1213.39i 0.173909 0.155151i
\(395\) 18296.1 2.33057
\(396\) 0 0
\(397\) 591.572 0.0747862 0.0373931 0.999301i \(-0.488095\pi\)
0.0373931 + 0.999301i \(0.488095\pi\)
\(398\) −5803.58 + 5177.62i −0.730923 + 0.652087i
\(399\) 0 0
\(400\) 22017.8 + 5103.47i 2.75222 + 0.637934i
\(401\) 8963.04i 1.11619i −0.829776 0.558096i \(-0.811532\pi\)
0.829776 0.558096i \(-0.188468\pi\)
\(402\) 0 0
\(403\) 2774.78i 0.342981i
\(404\) 1358.10 + 155.337i 0.167247 + 0.0191294i
\(405\) 0 0
\(406\) −99.3326 111.342i −0.0121424 0.0136103i
\(407\) −13745.1 −1.67400
\(408\) 0 0
\(409\) 1445.03 0.174700 0.0873500 0.996178i \(-0.472160\pi\)
0.0873500 + 0.996178i \(0.472160\pi\)
\(410\) −7110.16 7969.76i −0.856453 0.959997i
\(411\) 0 0
\(412\) −604.018 69.0865i −0.0722277 0.00826128i
\(413\) 3873.45i 0.461501i
\(414\) 0 0
\(415\) 2507.89i 0.296645i
\(416\) 5344.81 + 9767.60i 0.629930 + 1.15119i
\(417\) 0 0
\(418\) 12269.2 10945.9i 1.43566 1.28082i
\(419\) 4787.43 0.558190 0.279095 0.960264i \(-0.409966\pi\)
0.279095 + 0.960264i \(0.409966\pi\)
\(420\) 0 0
\(421\) −13140.8 −1.52124 −0.760619 0.649198i \(-0.775104\pi\)
−0.760619 + 0.649198i \(0.775104\pi\)
\(422\) 33.2650 29.6771i 0.00383724 0.00342336i
\(423\) 0 0
\(424\) 3682.41 + 5217.74i 0.421777 + 0.597631i
\(425\) 4157.25i 0.474486i
\(426\) 0 0
\(427\) 1835.80i 0.208057i
\(428\) 984.457 8607.03i 0.111181 0.972048i
\(429\) 0 0
\(430\) −13628.5 15276.2i −1.52843 1.71322i
\(431\) 8691.74 0.971384 0.485692 0.874130i \(-0.338568\pi\)
0.485692 + 0.874130i \(0.338568\pi\)
\(432\) 0 0
\(433\) 17570.6 1.95009 0.975043 0.222016i \(-0.0712639\pi\)
0.975043 + 0.222016i \(0.0712639\pi\)
\(434\) 594.598 + 666.483i 0.0657641 + 0.0737148i
\(435\) 0 0
\(436\) 637.447 5573.15i 0.0700188 0.612169i
\(437\) 1163.68i 0.127383i
\(438\) 0 0
\(439\) 1150.20i 0.125048i 0.998043 + 0.0625238i \(0.0199149\pi\)
−0.998043 + 0.0625238i \(0.980085\pi\)
\(440\) −18861.7 26725.9i −2.04363 2.89570i
\(441\) 0 0
\(442\) −1528.23 + 1363.40i −0.164459 + 0.146720i
\(443\) 863.626 0.0926232 0.0463116 0.998927i \(-0.485253\pi\)
0.0463116 + 0.998927i \(0.485253\pi\)
\(444\) 0 0
\(445\) −3552.28 −0.378414
\(446\) −7966.02 + 7106.82i −0.845744 + 0.754524i
\(447\) 0 0
\(448\) 3376.86 + 1200.79i 0.356119 + 0.126634i
\(449\) 10187.4i 1.07076i −0.844611 0.535381i \(-0.820168\pi\)
0.844611 0.535381i \(-0.179832\pi\)
\(450\) 0 0
\(451\) 11416.8i 1.19201i
\(452\) −6392.49 731.162i −0.665216 0.0760862i
\(453\) 0 0
\(454\) −1941.30 2176.00i −0.200682 0.224944i
\(455\) −9414.95 −0.970065
\(456\) 0 0
\(457\) 1094.43 0.112025 0.0560123 0.998430i \(-0.482161\pi\)
0.0560123 + 0.998430i \(0.482161\pi\)
\(458\) 4182.02 + 4687.62i 0.426666 + 0.478249i
\(459\) 0 0
\(460\) 2300.12 + 263.084i 0.233139 + 0.0266660i
\(461\) 5352.20i 0.540731i 0.962758 + 0.270365i \(0.0871445\pi\)
−0.962758 + 0.270365i \(0.912855\pi\)
\(462\) 0 0
\(463\) 14115.5i 1.41685i 0.705785 + 0.708426i \(0.250595\pi\)
−0.705785 + 0.708426i \(0.749405\pi\)
\(464\) 108.910 469.866i 0.0108966 0.0470108i
\(465\) 0 0
\(466\) 10104.6 9014.77i 1.00448 0.896140i
\(467\) 10004.7 0.991357 0.495678 0.868506i \(-0.334919\pi\)
0.495678 + 0.868506i \(0.334919\pi\)
\(468\) 0 0
\(469\) 628.910 0.0619198
\(470\) 20158.7 17984.4i 1.97840 1.76502i
\(471\) 0 0
\(472\) −10229.8 + 7219.67i −0.997596 + 0.704051i
\(473\) 21883.5i 2.12728i
\(474\) 0 0
\(475\) 31051.9i 2.99949i
\(476\) −74.9133 + 654.961i −0.00721354 + 0.0630674i
\(477\) 0 0
\(478\) −10395.4 11652.2i −0.994714 1.11497i
\(479\) 5471.20 0.521891 0.260945 0.965354i \(-0.415966\pi\)
0.260945 + 0.965354i \(0.415966\pi\)
\(480\) 0 0
\(481\) −12788.0 −1.21223
\(482\) 1152.17 + 1291.46i 0.108879 + 0.122043i
\(483\) 0 0
\(484\) −2763.54 + 24161.4i −0.259536 + 2.26910i
\(485\) 23066.0i 2.15953i
\(486\) 0 0
\(487\) 14865.7i 1.38323i −0.722268 0.691613i \(-0.756900\pi\)
0.722268 0.691613i \(-0.243100\pi\)
\(488\) 4848.36 3421.72i 0.449743 0.317405i
\(489\) 0 0
\(490\) −2261.41 + 2017.50i −0.208490 + 0.186003i
\(491\) 19778.0 1.81786 0.908930 0.416950i \(-0.136901\pi\)
0.908930 + 0.416950i \(0.136901\pi\)
\(492\) 0 0
\(493\) 88.7172 0.00810471
\(494\) 11414.9 10183.7i 1.03964 0.927503i
\(495\) 0 0
\(496\) −651.926 + 2812.58i −0.0590168 + 0.254614i
\(497\) 6242.62i 0.563420i
\(498\) 0 0
\(499\) 14497.9i 1.30063i 0.759663 + 0.650317i \(0.225364\pi\)
−0.759663 + 0.650317i \(0.774636\pi\)
\(500\) −39652.1 4535.33i −3.54659 0.405652i
\(501\) 0 0
\(502\) −9533.69 10686.3i −0.847628 0.950105i
\(503\) −11520.0 −1.02118 −0.510589 0.859825i \(-0.670573\pi\)
−0.510589 + 0.859825i \(0.670573\pi\)
\(504\) 0 0
\(505\) −3736.32 −0.329236
\(506\) −1647.49 1846.66i −0.144743 0.162242i
\(507\) 0 0
\(508\) −11643.0 1331.70i −1.01688 0.116309i
\(509\) 15531.3i 1.35248i −0.736683 0.676239i \(-0.763609\pi\)
0.736683 0.676239i \(-0.236391\pi\)
\(510\) 0 0
\(511\) 3545.70i 0.306952i
\(512\) 3122.77 + 11156.4i 0.269548 + 0.962987i
\(513\) 0 0
\(514\) −13579.8 + 12115.1i −1.16533 + 1.03964i
\(515\) 1661.74 0.142185
\(516\) 0 0
\(517\) −28877.7 −2.45656
\(518\) −3071.59 + 2740.29i −0.260536 + 0.232435i
\(519\) 0 0
\(520\) −17548.4 24864.9i −1.47990 2.09692i
\(521\) 17911.8i 1.50620i −0.657906 0.753100i \(-0.728557\pi\)
0.657906 0.753100i \(-0.271443\pi\)
\(522\) 0 0
\(523\) 15173.0i 1.26859i −0.773093 0.634293i \(-0.781291\pi\)
0.773093 0.634293i \(-0.218709\pi\)
\(524\) 1902.15 16630.4i 0.158580 1.38645i
\(525\) 0 0
\(526\) −7187.02 8055.92i −0.595759 0.667785i
\(527\) −531.055 −0.0438958
\(528\) 0 0
\(529\) −11991.9 −0.985605
\(530\) −11620.7 13025.6i −0.952397 1.06754i
\(531\) 0 0
\(532\) 559.552 4892.12i 0.0456009 0.398685i
\(533\) 10621.9i 0.863197i
\(534\) 0 0
\(535\) 23679.2i 1.91353i
\(536\) 1172.22 + 1660.96i 0.0944627 + 0.133848i
\(537\) 0 0
\(538\) 6025.86 5375.92i 0.482887 0.430804i
\(539\) 3239.51 0.258879
\(540\) 0 0
\(541\) −8514.18 −0.676623 −0.338312 0.941034i \(-0.609856\pi\)
−0.338312 + 0.941034i \(0.609856\pi\)
\(542\) −11760.6 + 10492.1i −0.932033 + 0.831505i
\(543\) 0 0
\(544\) −1869.39 + 1022.92i −0.147333 + 0.0806205i
\(545\) 15332.6i 1.20509i
\(546\) 0 0
\(547\) 8321.92i 0.650493i 0.945629 + 0.325246i \(0.105447\pi\)
−0.945629 + 0.325246i \(0.894553\pi\)
\(548\) −4595.16 525.587i −0.358204 0.0409707i
\(549\) 0 0
\(550\) 43961.9 + 49276.8i 3.40826 + 3.82031i
\(551\) −662.658 −0.0512345
\(552\) 0 0
\(553\) −5856.99 −0.450388
\(554\) −2698.53 3024.78i −0.206949 0.231969i
\(555\) 0 0
\(556\) 12158.3 + 1390.65i 0.927386 + 0.106073i
\(557\) 2082.84i 0.158443i −0.996857 0.0792213i \(-0.974757\pi\)
0.996857 0.0792213i \(-0.0252434\pi\)
\(558\) 0 0
\(559\) 20359.7i 1.54047i
\(560\) −9543.23 2212.01i −0.720134 0.166919i
\(561\) 0 0
\(562\) −3294.11 + 2938.81i −0.247248 + 0.220580i
\(563\) 8425.49 0.630714 0.315357 0.948973i \(-0.397876\pi\)
0.315357 + 0.948973i \(0.397876\pi\)
\(564\) 0 0
\(565\) 17586.7 1.30952
\(566\) −1177.58 + 1050.57i −0.0874514 + 0.0780191i
\(567\) 0 0
\(568\) 16486.8 11635.5i 1.21791 0.859534i
\(569\) 22690.3i 1.67175i −0.548919 0.835875i \(-0.684961\pi\)
0.548919 0.835875i \(-0.315039\pi\)
\(570\) 0 0
\(571\) 23862.8i 1.74891i −0.485109 0.874454i \(-0.661220\pi\)
0.485109 0.874454i \(-0.338780\pi\)
\(572\) −3696.86 + 32321.4i −0.270233 + 2.36263i
\(573\) 0 0
\(574\) 2276.13 + 2551.30i 0.165512 + 0.185522i
\(575\) −4673.68 −0.338967
\(576\) 0 0
\(577\) 13667.8 0.986131 0.493066 0.869992i \(-0.335876\pi\)
0.493066 + 0.869992i \(0.335876\pi\)
\(578\) 8989.93 + 10076.8i 0.646941 + 0.725155i
\(579\) 0 0
\(580\) −149.813 + 1309.81i −0.0107253 + 0.0937702i
\(581\) 802.834i 0.0573273i
\(582\) 0 0
\(583\) 18659.4i 1.32555i
\(584\) 9364.22 6608.77i 0.663517 0.468276i
\(585\) 0 0
\(586\) 3002.51 2678.66i 0.211659 0.188830i
\(587\) −18696.1 −1.31460 −0.657301 0.753628i \(-0.728302\pi\)
−0.657301 + 0.753628i \(0.728302\pi\)
\(588\) 0 0
\(589\) 3966.62 0.277490
\(590\) 25537.8 22783.3i 1.78199 1.58979i
\(591\) 0 0
\(592\) −12962.2 3004.50i −0.899905 0.208588i
\(593\) 12602.7i 0.872735i −0.899769 0.436367i \(-0.856265\pi\)
0.899769 0.436367i \(-0.143735\pi\)
\(594\) 0 0
\(595\) 1801.89i 0.124152i
\(596\) 9361.27 + 1070.73i 0.643376 + 0.0735883i
\(597\) 0 0
\(598\) −1532.77 1718.08i −0.104815 0.117487i
\(599\) −28096.8 −1.91653 −0.958267 0.285876i \(-0.907715\pi\)
−0.958267 + 0.285876i \(0.907715\pi\)
\(600\) 0 0
\(601\) 12077.8 0.819737 0.409869 0.912145i \(-0.365575\pi\)
0.409869 + 0.912145i \(0.365575\pi\)
\(602\) 4362.81 + 4890.27i 0.295374 + 0.331084i
\(603\) 0 0
\(604\) −2182.51 249.631i −0.147028 0.0168168i
\(605\) 66471.5i 4.46686i
\(606\) 0 0
\(607\) 8916.88i 0.596252i −0.954527 0.298126i \(-0.903638\pi\)
0.954527 0.298126i \(-0.0963616\pi\)
\(608\) 13963.1 7640.57i 0.931377 0.509648i
\(609\) 0 0
\(610\) −12103.5 + 10798.0i −0.803370 + 0.716720i
\(611\) −26866.9 −1.77892
\(612\) 0 0
\(613\) −22083.1 −1.45502 −0.727510 0.686098i \(-0.759322\pi\)
−0.727510 + 0.686098i \(0.759322\pi\)
\(614\) 16040.7 14310.6i 1.05432 0.940600i
\(615\) 0 0
\(616\) 6038.08 + 8555.57i 0.394937 + 0.559600i
\(617\) 11370.8i 0.741930i 0.928647 + 0.370965i \(0.120973\pi\)
−0.928647 + 0.370965i \(0.879027\pi\)
\(618\) 0 0
\(619\) 9687.07i 0.629009i −0.949256 0.314504i \(-0.898162\pi\)
0.949256 0.314504i \(-0.101838\pi\)
\(620\) 896.771 7840.40i 0.0580890 0.507868i
\(621\) 0 0
\(622\) −7563.00 8477.35i −0.487538 0.546481i
\(623\) 1137.17 0.0731293
\(624\) 0 0
\(625\) 64945.1 4.15648
\(626\) 12783.9 + 14329.4i 0.816209 + 0.914887i
\(627\) 0 0
\(628\) −730.832 + 6389.61i −0.0464385 + 0.406008i
\(629\) 2447.44i 0.155145i
\(630\) 0 0
\(631\) 25566.3i 1.61296i −0.591259 0.806482i \(-0.701369\pi\)
0.591259 0.806482i \(-0.298631\pi\)
\(632\) −10916.8 15468.4i −0.687097 0.973573i
\(633\) 0 0
\(634\) 9878.79 8813.28i 0.618828 0.552082i
\(635\) 32031.5 2.00178
\(636\) 0 0
\(637\) 3013.94 0.187467
\(638\) 1051.58 938.161i 0.0652548 0.0582165i
\(639\) 0 0
\(640\) −11945.5 29326.7i −0.737795 1.81131i
\(641\) 7551.94i 0.465341i −0.972556 0.232671i \(-0.925254\pi\)
0.972556 0.232671i \(-0.0747464\pi\)
\(642\) 0 0
\(643\) 11807.7i 0.724182i 0.932143 + 0.362091i \(0.117937\pi\)
−0.932143 + 0.362091i \(0.882063\pi\)
\(644\) −736.323 84.2193i −0.0450546 0.00515327i
\(645\) 0 0
\(646\) 1949.02 + 2184.66i 0.118705 + 0.133056i
\(647\) 16918.7 1.02804 0.514020 0.857778i \(-0.328156\pi\)
0.514020 + 0.857778i \(0.328156\pi\)
\(648\) 0 0
\(649\) −36583.4 −2.21267
\(650\) 40900.7 + 45845.6i 2.46809 + 2.76648i
\(651\) 0 0
\(652\) 15174.4 + 1735.62i 0.911467 + 0.104252i
\(653\) 26856.9i 1.60948i 0.593628 + 0.804740i \(0.297695\pi\)
−0.593628 + 0.804740i \(0.702305\pi\)
\(654\) 0 0
\(655\) 45752.6i 2.72932i
\(656\) −2495.58 + 10766.6i −0.148530 + 0.640800i
\(657\) 0 0
\(658\) −6453.25 + 5757.22i −0.382331 + 0.341094i
\(659\) −17861.3 −1.05581 −0.527904 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(660\) 0 0
\(661\) 17592.6 1.03521 0.517603 0.855621i \(-0.326824\pi\)
0.517603 + 0.855621i \(0.326824\pi\)
\(662\) −11026.4 + 9837.08i −0.647359 + 0.577536i
\(663\) 0 0
\(664\) 2120.29 1496.39i 0.123921 0.0874567i
\(665\) 13458.9i 0.784835i
\(666\) 0 0
\(667\) 99.7380i 0.00578991i
\(668\) −483.359 + 4225.97i −0.0279966 + 0.244772i
\(669\) 0 0
\(670\) −3699.20 4146.42i −0.213302 0.239090i
\(671\) 17338.5 0.997532
\(672\) 0 0
\(673\) 1930.27 0.110559 0.0552795 0.998471i \(-0.482395\pi\)
0.0552795 + 0.998471i \(0.482395\pi\)
\(674\) −20782.3 23294.9i −1.18769 1.33128i
\(675\) 0 0
\(676\) −1442.15 + 12608.6i −0.0820524 + 0.717378i
\(677\) 7085.47i 0.402240i −0.979567 0.201120i \(-0.935542\pi\)
0.979567 0.201120i \(-0.0644582\pi\)
\(678\) 0 0
\(679\) 7383.96i 0.417335i
\(680\) 4758.81 3358.52i 0.268371 0.189402i
\(681\) 0 0
\(682\) −6294.70 + 5615.76i −0.353426 + 0.315306i
\(683\) 6132.11 0.343541 0.171770 0.985137i \(-0.445051\pi\)
0.171770 + 0.985137i \(0.445051\pi\)
\(684\) 0 0
\(685\) 12642.0 0.705146
\(686\) 723.929 645.847i 0.0402912 0.0359454i
\(687\) 0 0
\(688\) −4783.45 + 20637.1i −0.265069 + 1.14358i
\(689\) 17360.1i 0.959896i
\(690\) 0 0
\(691\) 32430.8i 1.78542i −0.450633 0.892710i \(-0.648802\pi\)
0.450633 0.892710i \(-0.351198\pi\)
\(692\) 29547.9 + 3379.64i 1.62318 + 0.185657i
\(693\) 0 0
\(694\) −9096.67 10196.4i −0.497557 0.557711i
\(695\) −33449.3 −1.82562
\(696\) 0 0
\(697\) −2032.88 −0.110475
\(698\) −9404.75 10541.8i −0.509993 0.571650i
\(699\) 0 0
\(700\) 19648.2 + 2247.33i 1.06090 + 0.121344i
\(701\) 36023.2i 1.94091i 0.241283 + 0.970455i \(0.422432\pi\)
−0.241283 + 0.970455i \(0.577568\pi\)
\(702\) 0 0
\(703\) 18280.8i 0.980757i
\(704\) −11341.0 + 31893.2i −0.607147 + 1.70742i
\(705\) 0 0
\(706\) −16741.3 + 14935.6i −0.892447 + 0.796189i
\(707\) 1196.08 0.0636256
\(708\) 0 0
\(709\) 2852.85 0.151116 0.0755579 0.997141i \(-0.475926\pi\)
0.0755579 + 0.997141i \(0.475926\pi\)
\(710\) −41157.8 + 36718.5i −2.17553 + 1.94088i
\(711\) 0 0
\(712\) 2119.54 + 3003.26i 0.111564 + 0.158079i
\(713\) 597.024i 0.0313587i
\(714\) 0 0
\(715\) 88920.8i 4.65098i
\(716\) −1975.53 + 17271.9i −0.103113 + 0.901510i
\(717\) 0 0
\(718\) 5336.94 + 5982.17i 0.277400 + 0.310937i
\(719\) 18135.0 0.940643 0.470322 0.882495i \(-0.344138\pi\)
0.470322 + 0.882495i \(0.344138\pi\)
\(720\) 0 0
\(721\) −531.961 −0.0274775
\(722\) −1642.84 1841.46i −0.0846816 0.0949195i
\(723\) 0 0
\(724\) −2226.49 + 19466.0i −0.114291 + 0.999238i
\(725\) 2661.43i 0.136335i
\(726\) 0 0
\(727\) 2295.77i 0.117119i 0.998284 + 0.0585595i \(0.0186507\pi\)
−0.998284 + 0.0585595i \(0.981349\pi\)
\(728\) 5617.64 + 7959.83i 0.285994 + 0.405235i
\(729\) 0 0
\(730\) −23376.9 + 20855.5i −1.18523 + 1.05739i
\(731\) −3896.57 −0.197154
\(732\) 0 0
\(733\) 37706.1 1.90001 0.950005 0.312234i \(-0.101077\pi\)
0.950005 + 0.312234i \(0.101077\pi\)
\(734\) −10937.3 + 9757.58i −0.550002 + 0.490680i
\(735\) 0 0
\(736\) −1150.00 2101.61i −0.0575944 0.105253i
\(737\) 5939.83i 0.296875i
\(738\) 0 0
\(739\) 30540.4i 1.52023i −0.649791 0.760113i \(-0.725144\pi\)
0.649791 0.760113i \(-0.274856\pi\)
\(740\) 36133.6 + 4132.90i 1.79500 + 0.205309i
\(741\) 0 0
\(742\) 3720.05 + 4169.79i 0.184053 + 0.206305i
\(743\) −3816.22 −0.188430 −0.0942150 0.995552i \(-0.530034\pi\)
−0.0942150 + 0.995552i \(0.530034\pi\)
\(744\) 0 0
\(745\) −25754.2 −1.26652
\(746\) −10760.2 12061.1i −0.528097 0.591943i
\(747\) 0 0
\(748\) −6185.87 707.529i −0.302377 0.0345853i
\(749\) 7580.25i 0.369795i
\(750\) 0 0
\(751\) 3291.17i 0.159916i 0.996798 + 0.0799578i \(0.0254786\pi\)
−0.996798 + 0.0799578i \(0.974521\pi\)
\(752\) −27233.0 6312.30i −1.32059 0.306098i
\(753\) 0 0
\(754\) 978.360 872.835i 0.0472543 0.0421575i
\(755\) 6004.39 0.289434
\(756\) 0 0
\(757\) 4954.49 0.237878 0.118939 0.992902i \(-0.462051\pi\)
0.118939 + 0.992902i \(0.462051\pi\)
\(758\) −9091.21 + 8110.65i −0.435630 + 0.388644i
\(759\) 0 0
\(760\) −35545.1 + 25085.9i −1.69652 + 1.19732i
\(761\) 8155.76i 0.388497i 0.980952 + 0.194248i \(0.0622268\pi\)
−0.980952 + 0.194248i \(0.937773\pi\)
\(762\) 0 0
\(763\) 4908.30i 0.232887i
\(764\) 2838.49 24816.7i 0.134415 1.17518i
\(765\) 0 0
\(766\) −12952.3 14518.2i −0.610946 0.684808i
\(767\) −34036.0 −1.60231
\(768\) 0 0
\(769\) −19910.5 −0.933668 −0.466834 0.884345i \(-0.654605\pi\)
−0.466834 + 0.884345i \(0.654605\pi\)
\(770\) −19054.5 21358.2i −0.891790 0.999606i
\(771\) 0 0
\(772\) 624.503 5459.98i 0.0291144 0.254545i
\(773\) 13405.6i 0.623759i 0.950122 + 0.311880i \(0.100959\pi\)
−0.950122 + 0.311880i \(0.899041\pi\)
\(774\) 0 0
\(775\) 15931.1i 0.738404i
\(776\) 19501.1 13762.9i 0.902124 0.636672i
\(777\) 0 0
\(778\) 21069.7 18797.2i 0.970934 0.866211i
\(779\) 15184.3 0.698373
\(780\) 0 0
\(781\) 58959.3 2.70132
\(782\) 328.817 293.351i 0.0150364 0.0134146i
\(783\) 0 0
\(784\) 3055.01 + 708.116i 0.139168 + 0.0322575i
\(785\) 17578.7i 0.799251i
\(786\) 0 0
\(787\) 14037.3i 0.635800i −0.948124 0.317900i \(-0.897022\pi\)
0.948124 0.317900i \(-0.102978\pi\)
\(788\) 5121.90 + 585.834i 0.231549 + 0.0264841i
\(789\) 0 0
\(790\) 34450.4 + 38615.3i 1.55150 + 1.73908i
\(791\) −5629.90 −0.253067
\(792\) 0 0
\(793\) 16131.2 0.722364
\(794\) 1113.89 + 1248.56i 0.0497866 + 0.0558057i
\(795\) 0 0
\(796\) −21855.6 2499.80i −0.973178 0.111310i
\(797\) 36858.2i 1.63813i 0.573704 + 0.819063i \(0.305506\pi\)
−0.573704 + 0.819063i \(0.694494\pi\)
\(798\) 0 0
\(799\) 5141.96i 0.227671i
\(800\) 30686.8 + 56079.7i 1.35618 + 2.47840i
\(801\) 0 0
\(802\) 18917.2 16876.8i 0.832906 0.743070i
\(803\) 33487.9 1.47168
\(804\) 0 0
\(805\) 2025.73 0.0886927
\(806\) −5856.39 + 5224.73i −0.255934 + 0.228329i
\(807\) 0 0
\(808\) 2229.36 + 3158.86i 0.0970650 + 0.137535i
\(809\) 12429.2i 0.540156i 0.962839 + 0.270078i \(0.0870494\pi\)
−0.962839 + 0.270078i \(0.912951\pi\)
\(810\) 0 0
\(811\) 15370.2i 0.665501i 0.943015 + 0.332751i \(0.107977\pi\)
−0.943015 + 0.332751i \(0.892023\pi\)
\(812\) 47.9587 419.299i 0.00207268 0.0181213i
\(813\) 0 0
\(814\) −25881.1 29010.0i −1.11441 1.24914i
\(815\) −41747.1 −1.79428
\(816\) 0 0
\(817\) 29104.8 1.24632
\(818\) 2720.91 + 3049.86i 0.116301 + 0.130362i
\(819\) 0 0
\(820\) 3432.85 30013.1i 0.146195 1.27818i
\(821\) 18477.9i 0.785486i −0.919648 0.392743i \(-0.871526\pi\)
0.919648 0.392743i \(-0.128474\pi\)
\(822\) 0 0
\(823\) 10598.5i 0.448895i 0.974486 + 0.224448i \(0.0720578\pi\)
−0.974486 + 0.224448i \(0.927942\pi\)
\(824\) −991.515 1404.91i −0.0419188 0.0593962i
\(825\) 0 0
\(826\) −8175.23 + 7293.46i −0.344374 + 0.307230i
\(827\) −25115.2 −1.05603 −0.528017 0.849234i \(-0.677064\pi\)
−0.528017 + 0.849234i \(0.677064\pi\)
\(828\) 0 0
\(829\) 25579.4 1.07166 0.535832 0.844324i \(-0.319998\pi\)
0.535832 + 0.844324i \(0.319998\pi\)
\(830\) −5293.11 + 4722.21i −0.221357 + 0.197482i
\(831\) 0 0
\(832\) −10551.3 + 29672.4i −0.439666 + 1.23643i
\(833\) 576.827i 0.0239927i
\(834\) 0 0
\(835\) 11626.3i 0.481849i
\(836\) 46204.3 + 5284.77i 1.91150 + 0.218634i
\(837\) 0 0
\(838\) 9014.44 + 10104.3i 0.371597 + 0.416523i
\(839\) −6978.90 −0.287173 −0.143587 0.989638i \(-0.545864\pi\)
−0.143587 + 0.989638i \(0.545864\pi\)
\(840\) 0 0
\(841\) 24332.2 0.997671
\(842\) −24743.2 27734.6i −1.01272 1.13515i
\(843\) 0 0
\(844\) 125.272 + 14.3284i 0.00510904 + 0.000584363i
\(845\) 34688.2i 1.41220i
\(846\) 0 0
\(847\) 21279.0i 0.863231i
\(848\) −4078.71 + 17596.7i −0.165169 + 0.712586i
\(849\) 0 0
\(850\) −8774.22 + 7827.85i −0.354063 + 0.315874i
\(851\) 2751.47 0.110834
\(852\) 0 0
\(853\) −44677.6 −1.79335 −0.896677 0.442685i \(-0.854026\pi\)
−0.896677 + 0.442685i \(0.854026\pi\)
\(854\) 3874.60 3456.69i 0.155253 0.138508i
\(855\) 0 0
\(856\) 20019.5 14128.7i 0.799360 0.564147i
\(857\) 25565.0i 1.01900i 0.860470 + 0.509501i \(0.170170\pi\)
−0.860470 + 0.509501i \(0.829830\pi\)
\(858\) 0 0
\(859\) 6461.94i 0.256669i −0.991731 0.128334i \(-0.959037\pi\)
0.991731 0.128334i \(-0.0409631\pi\)
\(860\) 6579.99 57528.3i 0.260902 2.28105i
\(861\) 0 0
\(862\) 16366.0 + 18344.6i 0.646668 + 0.724849i
\(863\) 27097.5 1.06884 0.534420 0.845219i \(-0.320530\pi\)
0.534420 + 0.845219i \(0.320530\pi\)
\(864\) 0 0
\(865\) −81290.6 −3.19533
\(866\) 33084.3 + 37084.1i 1.29821 + 1.45516i
\(867\) 0 0
\(868\) −287.077 + 2509.89i −0.0112258 + 0.0981467i
\(869\) 55317.2i 2.15939i
\(870\) 0 0
\(871\) 5526.23i 0.214982i
\(872\) 12962.9 9148.51i 0.503415 0.355284i
\(873\) 0 0
\(874\) −2456.04 + 2191.14i −0.0950537 + 0.0848013i
\(875\) −34921.8 −1.34922
\(876\) 0 0
\(877\) −40674.5 −1.56611 −0.783055 0.621952i \(-0.786340\pi\)
−0.783055 + 0.621952i \(0.786340\pi\)
\(878\) −2427.58 + 2165.75i −0.0933109 + 0.0832465i
\(879\) 0 0
\(880\) 20891.7 90132.4i 0.800294 3.45268i
\(881\) 36098.7i 1.38047i −0.723584 0.690237i \(-0.757506\pi\)
0.723584 0.690237i \(-0.242494\pi\)
\(882\) 0 0
\(883\) 36446.8i 1.38905i 0.719467 + 0.694526i \(0.244386\pi\)
−0.719467 + 0.694526i \(0.755614\pi\)
\(884\) −5755.14 658.263i −0.218966 0.0250450i
\(885\) 0 0
\(886\) 1626.15 + 1822.75i 0.0616610 + 0.0691157i
\(887\) 9088.36 0.344033 0.172016 0.985094i \(-0.444972\pi\)
0.172016 + 0.985094i \(0.444972\pi\)
\(888\) 0 0
\(889\) −10254.0 −0.386849
\(890\) −6688.71 7497.37i −0.251917 0.282373i
\(891\) 0 0
\(892\) −29999.0 3431.23i −1.12606 0.128796i
\(893\) 38407.0i 1.43924i
\(894\) 0 0
\(895\) 47517.5i 1.77468i
\(896\) 3824.04 + 9388.14i 0.142581 + 0.350040i
\(897\) 0 0
\(898\) 21501.3 19182.2i 0.799005 0.712826i
\(899\) 339.975 0.0126127
\(900\) 0 0
\(901\) −3322.50 −0.122851
\(902\) −24096.2 + 21497.2i −0.889484 + 0.793546i
\(903\) 0 0
\(904\) −10493.5 14868.6i −0.386071 0.547038i
\(905\) 53553.8i 1.96706i
\(906\) 0 0
\(907\) 38031.8i 1.39231i 0.717892 + 0.696155i \(0.245107\pi\)
−0.717892 + 0.696155i \(0.754893\pi\)
\(908\) 937.277 8194.54i 0.0342562 0.299499i
\(909\) 0 0
\(910\) −17727.7 19871.0i −0.645790 0.723865i
\(911\) 24648.4 0.896419 0.448209 0.893929i \(-0.352062\pi\)
0.448209 + 0.893929i \(0.352062\pi\)
\(912\) 0 0
\(913\) 7582.49 0.274856
\(914\) 2060.74 + 2309.88i 0.0745769 + 0.0835931i
\(915\) 0 0
\(916\) −2019.12 + 17653.0i −0.0728313 + 0.636759i
\(917\) 14646.5i 0.527447i
\(918\) 0 0
\(919\) 21102.2i 0.757451i 0.925509 + 0.378726i \(0.123638\pi\)
−0.925509 + 0.378726i \(0.876362\pi\)
\(920\) 3775.73 + 5349.97i 0.135307 + 0.191721i
\(921\) 0 0
\(922\) −11296.3 + 10077.9i −0.403495 + 0.359975i
\(923\) 54853.8 1.95616
\(924\) 0 0
\(925\) −73420.9 −2.60980
\(926\) −29791.9 + 26578.6i −1.05726 + 0.943225i
\(927\) 0 0
\(928\) 1196.76 654.866i 0.0423336 0.0231649i
\(929\) 13010.0i 0.459466i 0.973254 + 0.229733i \(0.0737854\pi\)
−0.973254 + 0.229733i \(0.926215\pi\)
\(930\) 0 0
\(931\) 4308.52i 0.151671i
\(932\) 38052.8 + 4352.41i 1.33740 + 0.152970i
\(933\) 0 0
\(934\) 18838.3 + 21115.8i 0.659965 + 0.739753i
\(935\) 17018.2 0.595247
\(936\) 0 0
\(937\) 6116.05 0.213237 0.106618 0.994300i \(-0.465998\pi\)
0.106618 + 0.994300i \(0.465998\pi\)
\(938\) 1184.20 + 1327.37i 0.0412211 + 0.0462047i
\(939\) 0 0
\(940\) 75915.0 + 8683.03i 2.63412 + 0.301286i
\(941\) 15615.5i 0.540969i −0.962724 0.270485i \(-0.912816\pi\)
0.962724 0.270485i \(-0.0871839\pi\)
\(942\) 0 0
\(943\) 2285.41i 0.0789219i
\(944\) −34499.8 7996.66i −1.18948 0.275709i
\(945\) 0 0
\(946\) −46186.8 + 41205.2i −1.58738 + 1.41617i
\(947\) −33651.5 −1.15473 −0.577364 0.816487i \(-0.695919\pi\)
−0.577364 + 0.816487i \(0.695919\pi\)
\(948\) 0 0
\(949\) 31156.1 1.06572
\(950\) 65537.6 58468.8i 2.23823 1.99682i
\(951\) 0 0
\(952\) −1523.40 + 1075.14i −0.0518633 + 0.0366024i
\(953\) 13028.5i 0.442847i −0.975178 0.221424i \(-0.928930\pi\)
0.975178 0.221424i \(-0.0710703\pi\)
\(954\) 0 0
\(955\) 68274.5i 2.31341i
\(956\) 5018.98 43880.6i 0.169796 1.48452i
\(957\) 0 0
\(958\) 10301.9 + 11547.4i 0.347432 + 0.389436i
\(959\) −4046.98 −0.136271
\(960\) 0 0
\(961\) 27755.9 0.931689
\(962\) −24078.9 26990.0i −0.807002 0.904567i
\(963\) 0 0
\(964\) −556.277 + 4863.49i −0.0185856 + 0.162492i
\(965\) 15021.2i 0.501088i
\(966\) 0 0
\(967\) 28818.4i 0.958363i 0.877716 + 0.479182i \(0.159066\pi\)
−0.877716 + 0.479182i \(0.840934\pi\)
\(968\) −56198.1 + 39661.7i −1.86599 + 1.31692i
\(969\) 0 0
\(970\) −48682.7 + 43431.9i −1.61145 + 1.43764i
\(971\) −57593.1 −1.90345 −0.951725 0.306952i \(-0.900691\pi\)
−0.951725 + 0.306952i \(0.900691\pi\)
\(972\) 0 0
\(973\) 10707.9 0.352804
\(974\) 31375.3 27991.3i 1.03217 0.920839i
\(975\) 0 0
\(976\) 16351.0 + 3789.97i 0.536251 + 0.124297i
\(977\) 43479.4i 1.42378i −0.702293 0.711888i \(-0.747841\pi\)
0.702293 0.711888i \(-0.252159\pi\)
\(978\) 0 0
\(979\) 10740.1i 0.350619i
\(980\) −8516.18 974.066i −0.277591 0.0317504i
\(981\) 0 0
\(982\) 37240.7 + 41743.1i 1.21018 + 1.35649i
\(983\) 1724.27 0.0559468 0.0279734 0.999609i \(-0.491095\pi\)
0.0279734 + 0.999609i \(0.491095\pi\)
\(984\) 0 0
\(985\) −14091.1 −0.455817
\(986\) 167.049 + 187.245i 0.00539546 + 0.00604776i
\(987\) 0 0
\(988\) 42987.1 + 4916.78i 1.38421 + 0.158324i
\(989\) 4380.62i 0.140845i
\(990\) 0 0
\(991\) 45591.3i 1.46141i −0.682695 0.730703i \(-0.739192\pi\)
0.682695 0.730703i \(-0.260808\pi\)
\(992\) −7163.72 + 3919.98i −0.229283 + 0.125463i
\(993\) 0 0
\(994\) 13175.5 11754.5i 0.420425 0.375079i
\(995\) 60127.8 1.91576
\(996\) 0 0
\(997\) 58305.6 1.85211 0.926057 0.377384i \(-0.123176\pi\)
0.926057 + 0.377384i \(0.123176\pi\)
\(998\) −30599.0 + 27298.7i −0.970537 + 0.865857i
\(999\) 0 0
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 252.4.e.a.71.28 yes 36
3.2 odd 2 inner 252.4.e.a.71.9 36
4.3 odd 2 inner 252.4.e.a.71.10 yes 36
12.11 even 2 inner 252.4.e.a.71.27 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.e.a.71.9 36 3.2 odd 2 inner
252.4.e.a.71.10 yes 36 4.3 odd 2 inner
252.4.e.a.71.27 yes 36 12.11 even 2 inner
252.4.e.a.71.28 yes 36 1.1 even 1 trivial