Properties

Label 252.4.e.a.71.14
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 71.14
Character \(\chi\) \(=\) 252.71
Dual form 252.4.e.a.71.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.16208 + 2.57868i) q^{2} +(-5.29916 - 5.99324i) q^{4} +0.300247i q^{5} +7.00000i q^{7} +(21.6127 - 6.70021i) q^{8} +O(q^{10})\) \(q+(-1.16208 + 2.57868i) q^{2} +(-5.29916 - 5.99324i) q^{4} +0.300247i q^{5} +7.00000i q^{7} +(21.6127 - 6.70021i) q^{8} +(-0.774240 - 0.348910i) q^{10} +9.71649 q^{11} -76.4714 q^{13} +(-18.0507 - 8.13454i) q^{14} +(-7.83789 + 63.5182i) q^{16} -93.2361i q^{17} -81.8849i q^{19} +(1.79945 - 1.59106i) q^{20} +(-11.2913 + 25.0557i) q^{22} +185.343 q^{23} +124.910 q^{25} +(88.8656 - 197.195i) q^{26} +(41.9527 - 37.0941i) q^{28} -159.193i q^{29} +139.107i q^{31} +(-154.685 - 94.0245i) q^{32} +(240.426 + 108.348i) q^{34} -2.10173 q^{35} +30.5025 q^{37} +(211.155 + 95.1565i) q^{38} +(2.01172 + 6.48914i) q^{40} -173.772i q^{41} -356.320i q^{43} +(-51.4892 - 58.2333i) q^{44} +(-215.382 + 477.939i) q^{46} +346.929 q^{47} -49.0000 q^{49} +(-145.155 + 322.102i) q^{50} +(405.234 + 458.312i) q^{52} +154.800i q^{53} +2.91735i q^{55} +(46.9015 + 151.289i) q^{56} +(410.508 + 184.995i) q^{58} +586.598 q^{59} +167.768 q^{61} +(-358.711 - 161.653i) q^{62} +(422.214 - 289.619i) q^{64} -22.9603i q^{65} -403.531i q^{67} +(-558.787 + 494.073i) q^{68} +(2.44237 - 5.41968i) q^{70} +156.489 q^{71} -565.672 q^{73} +(-35.4462 + 78.6561i) q^{74} +(-490.756 + 433.921i) q^{76} +68.0154i q^{77} -542.718i q^{79} +(-19.0712 - 2.35330i) q^{80} +(448.101 + 201.936i) q^{82} -791.534 q^{83} +27.9939 q^{85} +(918.834 + 414.071i) q^{86} +(209.999 - 65.1025i) q^{88} -809.418i q^{89} -535.300i q^{91} +(-982.160 - 1110.80i) q^{92} +(-403.158 + 894.618i) q^{94} +24.5857 q^{95} -192.401 q^{97} +(56.9418 - 126.355i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 24 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.16208 + 2.57868i −0.410856 + 0.911700i
\(3\) 0 0
\(4\) −5.29916 5.99324i −0.662395 0.749155i
\(5\) 0.300247i 0.0268549i 0.999910 + 0.0134275i \(0.00427422\pi\)
−0.999910 + 0.0134275i \(0.995726\pi\)
\(6\) 0 0
\(7\) 7.00000i 0.377964i
\(8\) 21.6127 6.70021i 0.955154 0.296110i
\(9\) 0 0
\(10\) −0.774240 0.348910i −0.0244836 0.0110335i
\(11\) 9.71649 0.266330 0.133165 0.991094i \(-0.457486\pi\)
0.133165 + 0.991094i \(0.457486\pi\)
\(12\) 0 0
\(13\) −76.4714 −1.63149 −0.815744 0.578413i \(-0.803672\pi\)
−0.815744 + 0.578413i \(0.803672\pi\)
\(14\) −18.0507 8.13454i −0.344590 0.155289i
\(15\) 0 0
\(16\) −7.83789 + 63.5182i −0.122467 + 0.992473i
\(17\) 93.2361i 1.33018i −0.746763 0.665091i \(-0.768393\pi\)
0.746763 0.665091i \(-0.231607\pi\)
\(18\) 0 0
\(19\) 81.8849i 0.988720i −0.869257 0.494360i \(-0.835402\pi\)
0.869257 0.494360i \(-0.164598\pi\)
\(20\) 1.79945 1.59106i 0.0201185 0.0177885i
\(21\) 0 0
\(22\) −11.2913 + 25.0557i −0.109423 + 0.242813i
\(23\) 185.343 1.68029 0.840144 0.542363i \(-0.182470\pi\)
0.840144 + 0.542363i \(0.182470\pi\)
\(24\) 0 0
\(25\) 124.910 0.999279
\(26\) 88.8656 197.195i 0.670307 1.48743i
\(27\) 0 0
\(28\) 41.9527 37.0941i 0.283154 0.250362i
\(29\) 159.193i 1.01936i −0.860364 0.509679i \(-0.829764\pi\)
0.860364 0.509679i \(-0.170236\pi\)
\(30\) 0 0
\(31\) 139.107i 0.805945i 0.915212 + 0.402973i \(0.132023\pi\)
−0.915212 + 0.402973i \(0.867977\pi\)
\(32\) −154.685 94.0245i −0.854521 0.519417i
\(33\) 0 0
\(34\) 240.426 + 108.348i 1.21273 + 0.546513i
\(35\) −2.10173 −0.0101502
\(36\) 0 0
\(37\) 30.5025 0.135529 0.0677646 0.997701i \(-0.478413\pi\)
0.0677646 + 0.997701i \(0.478413\pi\)
\(38\) 211.155 + 95.1565i 0.901417 + 0.406222i
\(39\) 0 0
\(40\) 2.01172 + 6.48914i 0.00795201 + 0.0256506i
\(41\) 173.772i 0.661916i −0.943645 0.330958i \(-0.892628\pi\)
0.943645 0.330958i \(-0.107372\pi\)
\(42\) 0 0
\(43\) 356.320i 1.26368i −0.775099 0.631840i \(-0.782300\pi\)
0.775099 0.631840i \(-0.217700\pi\)
\(44\) −51.4892 58.2333i −0.176416 0.199523i
\(45\) 0 0
\(46\) −215.382 + 477.939i −0.690357 + 1.53192i
\(47\) 346.929 1.07670 0.538349 0.842722i \(-0.319048\pi\)
0.538349 + 0.842722i \(0.319048\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) −145.155 + 322.102i −0.410560 + 0.911043i
\(51\) 0 0
\(52\) 405.234 + 458.312i 1.08069 + 1.22224i
\(53\) 154.800i 0.401197i 0.979674 + 0.200598i \(0.0642886\pi\)
−0.979674 + 0.200598i \(0.935711\pi\)
\(54\) 0 0
\(55\) 2.91735i 0.00715228i
\(56\) 46.9015 + 151.289i 0.111919 + 0.361014i
\(57\) 0 0
\(58\) 410.508 + 184.995i 0.929350 + 0.418810i
\(59\) 586.598 1.29438 0.647192 0.762327i \(-0.275943\pi\)
0.647192 + 0.762327i \(0.275943\pi\)
\(60\) 0 0
\(61\) 167.768 0.352140 0.176070 0.984378i \(-0.443662\pi\)
0.176070 + 0.984378i \(0.443662\pi\)
\(62\) −358.711 161.653i −0.734780 0.331127i
\(63\) 0 0
\(64\) 422.214 289.619i 0.824637 0.565662i
\(65\) 22.9603i 0.0438135i
\(66\) 0 0
\(67\) 403.531i 0.735809i −0.929864 0.367905i \(-0.880075\pi\)
0.929864 0.367905i \(-0.119925\pi\)
\(68\) −558.787 + 494.073i −0.996512 + 0.881105i
\(69\) 0 0
\(70\) 2.44237 5.41968i 0.00417027 0.00925394i
\(71\) 156.489 0.261576 0.130788 0.991410i \(-0.458249\pi\)
0.130788 + 0.991410i \(0.458249\pi\)
\(72\) 0 0
\(73\) −565.672 −0.906944 −0.453472 0.891270i \(-0.649815\pi\)
−0.453472 + 0.891270i \(0.649815\pi\)
\(74\) −35.4462 + 78.6561i −0.0556830 + 0.123562i
\(75\) 0 0
\(76\) −490.756 + 433.921i −0.740705 + 0.654923i
\(77\) 68.0154i 0.100663i
\(78\) 0 0
\(79\) 542.718i 0.772918i −0.922307 0.386459i \(-0.873698\pi\)
0.922307 0.386459i \(-0.126302\pi\)
\(80\) −19.0712 2.35330i −0.0266528 0.00328884i
\(81\) 0 0
\(82\) 448.101 + 201.936i 0.603469 + 0.271952i
\(83\) −791.534 −1.04677 −0.523386 0.852095i \(-0.675332\pi\)
−0.523386 + 0.852095i \(0.675332\pi\)
\(84\) 0 0
\(85\) 27.9939 0.0357219
\(86\) 918.834 + 414.071i 1.15210 + 0.519191i
\(87\) 0 0
\(88\) 209.999 65.1025i 0.254386 0.0788631i
\(89\) 809.418i 0.964024i −0.876165 0.482012i \(-0.839906\pi\)
0.876165 0.482012i \(-0.160094\pi\)
\(90\) 0 0
\(91\) 535.300i 0.616645i
\(92\) −982.160 1110.80i −1.11301 1.25880i
\(93\) 0 0
\(94\) −403.158 + 894.618i −0.442368 + 0.981626i
\(95\) 24.5857 0.0265520
\(96\) 0 0
\(97\) −192.401 −0.201395 −0.100698 0.994917i \(-0.532107\pi\)
−0.100698 + 0.994917i \(0.532107\pi\)
\(98\) 56.9418 126.355i 0.0586937 0.130243i
\(99\) 0 0
\(100\) −661.917 748.615i −0.661917 0.748615i
\(101\) 39.5093i 0.0389240i −0.999811 0.0194620i \(-0.993805\pi\)
0.999811 0.0194620i \(-0.00619534\pi\)
\(102\) 0 0
\(103\) 1420.94i 1.35932i −0.733528 0.679659i \(-0.762128\pi\)
0.733528 0.679659i \(-0.237872\pi\)
\(104\) −1652.75 + 512.374i −1.55832 + 0.483100i
\(105\) 0 0
\(106\) −399.179 179.889i −0.365771 0.164834i
\(107\) 1455.04 1.31462 0.657310 0.753620i \(-0.271694\pi\)
0.657310 + 0.753620i \(0.271694\pi\)
\(108\) 0 0
\(109\) −1904.97 −1.67397 −0.836984 0.547227i \(-0.815684\pi\)
−0.836984 + 0.547227i \(0.815684\pi\)
\(110\) −7.52290 3.39018i −0.00652073 0.00293856i
\(111\) 0 0
\(112\) −444.628 54.8652i −0.375119 0.0462882i
\(113\) 1236.88i 1.02970i 0.857281 + 0.514848i \(0.172152\pi\)
−0.857281 + 0.514848i \(0.827848\pi\)
\(114\) 0 0
\(115\) 55.6486i 0.0451240i
\(116\) −954.083 + 843.589i −0.763658 + 0.675218i
\(117\) 0 0
\(118\) −681.672 + 1512.65i −0.531805 + 1.18009i
\(119\) 652.653 0.502761
\(120\) 0 0
\(121\) −1236.59 −0.929068
\(122\) −194.960 + 432.620i −0.144679 + 0.321046i
\(123\) 0 0
\(124\) 833.700 737.148i 0.603778 0.533854i
\(125\) 75.0347i 0.0536905i
\(126\) 0 0
\(127\) 2425.86i 1.69497i −0.530823 0.847483i \(-0.678117\pi\)
0.530823 0.847483i \(-0.321883\pi\)
\(128\) 256.188 + 1425.31i 0.176906 + 0.984228i
\(129\) 0 0
\(130\) 59.2073 + 26.6816i 0.0399448 + 0.0180010i
\(131\) −2516.53 −1.67840 −0.839198 0.543826i \(-0.816975\pi\)
−0.839198 + 0.543826i \(0.816975\pi\)
\(132\) 0 0
\(133\) 573.194 0.373701
\(134\) 1040.58 + 468.934i 0.670837 + 0.302312i
\(135\) 0 0
\(136\) −624.702 2015.08i −0.393880 1.27053i
\(137\) 228.393i 0.142430i −0.997461 0.0712149i \(-0.977312\pi\)
0.997461 0.0712149i \(-0.0226876\pi\)
\(138\) 0 0
\(139\) 1316.68i 0.803449i 0.915761 + 0.401724i \(0.131589\pi\)
−0.915761 + 0.401724i \(0.868411\pi\)
\(140\) 11.1374 + 12.5962i 0.00672344 + 0.00760408i
\(141\) 0 0
\(142\) −181.853 + 403.535i −0.107470 + 0.238478i
\(143\) −743.034 −0.434515
\(144\) 0 0
\(145\) 47.7973 0.0273748
\(146\) 657.354 1458.69i 0.372624 0.826861i
\(147\) 0 0
\(148\) −161.638 182.809i −0.0897738 0.101532i
\(149\) 1186.97i 0.652622i 0.945262 + 0.326311i \(0.105806\pi\)
−0.945262 + 0.326311i \(0.894194\pi\)
\(150\) 0 0
\(151\) 3479.34i 1.87513i 0.347812 + 0.937564i \(0.386925\pi\)
−0.347812 + 0.937564i \(0.613075\pi\)
\(152\) −548.646 1769.75i −0.292770 0.944380i
\(153\) 0 0
\(154\) −175.390 79.0392i −0.0917748 0.0413582i
\(155\) −41.7664 −0.0216436
\(156\) 0 0
\(157\) −1045.32 −0.531372 −0.265686 0.964060i \(-0.585599\pi\)
−0.265686 + 0.964060i \(0.585599\pi\)
\(158\) 1399.49 + 630.680i 0.704670 + 0.317558i
\(159\) 0 0
\(160\) 28.2306 46.4437i 0.0139489 0.0229481i
\(161\) 1297.40i 0.635089i
\(162\) 0 0
\(163\) 318.248i 0.152927i −0.997072 0.0764636i \(-0.975637\pi\)
0.997072 0.0764636i \(-0.0243629\pi\)
\(164\) −1041.46 + 920.843i −0.495878 + 0.438450i
\(165\) 0 0
\(166\) 919.823 2041.11i 0.430073 0.954343i
\(167\) −767.396 −0.355586 −0.177793 0.984068i \(-0.556896\pi\)
−0.177793 + 0.984068i \(0.556896\pi\)
\(168\) 0 0
\(169\) 3650.88 1.66176
\(170\) −32.5310 + 72.1872i −0.0146766 + 0.0325677i
\(171\) 0 0
\(172\) −2135.51 + 1888.19i −0.946693 + 0.837055i
\(173\) 933.370i 0.410190i −0.978742 0.205095i \(-0.934250\pi\)
0.978742 0.205095i \(-0.0657503\pi\)
\(174\) 0 0
\(175\) 874.369i 0.377692i
\(176\) −76.1568 + 617.175i −0.0326167 + 0.264325i
\(177\) 0 0
\(178\) 2087.23 + 940.606i 0.878901 + 0.396075i
\(179\) 1371.81 0.572816 0.286408 0.958108i \(-0.407539\pi\)
0.286408 + 0.958108i \(0.407539\pi\)
\(180\) 0 0
\(181\) 1678.90 0.689455 0.344728 0.938703i \(-0.387971\pi\)
0.344728 + 0.938703i \(0.387971\pi\)
\(182\) 1380.37 + 622.059i 0.562195 + 0.253352i
\(183\) 0 0
\(184\) 4005.75 1241.84i 1.60493 0.497551i
\(185\) 9.15829i 0.00363963i
\(186\) 0 0
\(187\) 905.928i 0.354268i
\(188\) −1838.43 2079.23i −0.713199 0.806614i
\(189\) 0 0
\(190\) −28.5705 + 63.3986i −0.0109091 + 0.0242075i
\(191\) −960.981 −0.364053 −0.182026 0.983294i \(-0.558266\pi\)
−0.182026 + 0.983294i \(0.558266\pi\)
\(192\) 0 0
\(193\) 4482.75 1.67189 0.835947 0.548810i \(-0.184919\pi\)
0.835947 + 0.548810i \(0.184919\pi\)
\(194\) 223.585 496.140i 0.0827445 0.183612i
\(195\) 0 0
\(196\) 259.659 + 293.669i 0.0946278 + 0.107022i
\(197\) 4174.06i 1.50959i −0.655961 0.754795i \(-0.727736\pi\)
0.655961 0.754795i \(-0.272264\pi\)
\(198\) 0 0
\(199\) 4302.61i 1.53268i 0.642433 + 0.766342i \(0.277925\pi\)
−0.642433 + 0.766342i \(0.722075\pi\)
\(200\) 2699.63 836.922i 0.954465 0.295897i
\(201\) 0 0
\(202\) 101.882 + 45.9129i 0.0354870 + 0.0159922i
\(203\) 1114.35 0.385281
\(204\) 0 0
\(205\) 52.1744 0.0177757
\(206\) 3664.16 + 1651.25i 1.23929 + 0.558484i
\(207\) 0 0
\(208\) 599.375 4857.33i 0.199804 1.61921i
\(209\) 795.634i 0.263326i
\(210\) 0 0
\(211\) 1832.00i 0.597725i 0.954296 + 0.298862i \(0.0966072\pi\)
−0.954296 + 0.298862i \(0.903393\pi\)
\(212\) 927.754 820.309i 0.300558 0.265750i
\(213\) 0 0
\(214\) −1690.87 + 3752.09i −0.540119 + 1.19854i
\(215\) 106.984 0.0339360
\(216\) 0 0
\(217\) −973.747 −0.304619
\(218\) 2213.72 4912.29i 0.687760 1.52616i
\(219\) 0 0
\(220\) 17.4844 15.4595i 0.00535816 0.00473763i
\(221\) 7129.90i 2.17018i
\(222\) 0 0
\(223\) 3117.21i 0.936072i 0.883709 + 0.468036i \(0.155038\pi\)
−0.883709 + 0.468036i \(0.844962\pi\)
\(224\) 658.171 1082.79i 0.196321 0.322979i
\(225\) 0 0
\(226\) −3189.51 1437.35i −0.938775 0.423057i
\(227\) −5108.49 −1.49367 −0.746833 0.665011i \(-0.768427\pi\)
−0.746833 + 0.665011i \(0.768427\pi\)
\(228\) 0 0
\(229\) 4443.04 1.28211 0.641057 0.767493i \(-0.278496\pi\)
0.641057 + 0.767493i \(0.278496\pi\)
\(230\) −143.500 64.6679i −0.0411396 0.0185395i
\(231\) 0 0
\(232\) −1066.63 3440.59i −0.301843 0.973645i
\(233\) 241.778i 0.0679801i 0.999422 + 0.0339901i \(0.0108215\pi\)
−0.999422 + 0.0339901i \(0.989179\pi\)
\(234\) 0 0
\(235\) 104.164i 0.0289146i
\(236\) −3108.48 3515.63i −0.857392 0.969694i
\(237\) 0 0
\(238\) −758.433 + 1682.98i −0.206563 + 0.458368i
\(239\) 758.452 0.205273 0.102636 0.994719i \(-0.467272\pi\)
0.102636 + 0.994719i \(0.467272\pi\)
\(240\) 0 0
\(241\) −5755.96 −1.53848 −0.769241 0.638958i \(-0.779366\pi\)
−0.769241 + 0.638958i \(0.779366\pi\)
\(242\) 1437.01 3188.77i 0.381713 0.847032i
\(243\) 0 0
\(244\) −889.030 1005.48i −0.233255 0.263807i
\(245\) 14.7121i 0.00383642i
\(246\) 0 0
\(247\) 6261.85i 1.61309i
\(248\) 932.044 + 3006.47i 0.238649 + 0.769801i
\(249\) 0 0
\(250\) −193.490 87.1961i −0.0489496 0.0220591i
\(251\) −1275.64 −0.320787 −0.160394 0.987053i \(-0.551276\pi\)
−0.160394 + 0.987053i \(0.551276\pi\)
\(252\) 0 0
\(253\) 1800.88 0.447512
\(254\) 6255.52 + 2819.04i 1.54530 + 0.696387i
\(255\) 0 0
\(256\) −3973.13 995.698i −0.970004 0.243090i
\(257\) 7412.37i 1.79911i −0.436810 0.899554i \(-0.643892\pi\)
0.436810 0.899554i \(-0.356108\pi\)
\(258\) 0 0
\(259\) 213.518i 0.0512252i
\(260\) −137.607 + 121.670i −0.0328231 + 0.0290218i
\(261\) 0 0
\(262\) 2924.40 6489.31i 0.689580 1.53019i
\(263\) 5386.10 1.26282 0.631409 0.775450i \(-0.282477\pi\)
0.631409 + 0.775450i \(0.282477\pi\)
\(264\) 0 0
\(265\) −46.4782 −0.0107741
\(266\) −666.096 + 1478.08i −0.153537 + 0.340703i
\(267\) 0 0
\(268\) −2418.46 + 2138.38i −0.551235 + 0.487396i
\(269\) 5903.70i 1.33812i −0.743207 0.669062i \(-0.766696\pi\)
0.743207 0.669062i \(-0.233304\pi\)
\(270\) 0 0
\(271\) 1486.67i 0.333242i −0.986021 0.166621i \(-0.946714\pi\)
0.986021 0.166621i \(-0.0532857\pi\)
\(272\) 5922.20 + 730.774i 1.32017 + 0.162903i
\(273\) 0 0
\(274\) 588.951 + 265.410i 0.129853 + 0.0585182i
\(275\) 1213.69 0.266138
\(276\) 0 0
\(277\) −91.0097 −0.0197410 −0.00987048 0.999951i \(-0.503142\pi\)
−0.00987048 + 0.999951i \(0.503142\pi\)
\(278\) −3395.30 1530.08i −0.732505 0.330102i
\(279\) 0 0
\(280\) −45.4240 + 14.0820i −0.00969501 + 0.00300558i
\(281\) 7462.63i 1.58428i 0.610339 + 0.792140i \(0.291033\pi\)
−0.610339 + 0.792140i \(0.708967\pi\)
\(282\) 0 0
\(283\) 696.270i 0.146251i 0.997323 + 0.0731254i \(0.0232973\pi\)
−0.997323 + 0.0731254i \(0.976703\pi\)
\(284\) −829.261 937.878i −0.173266 0.195961i
\(285\) 0 0
\(286\) 863.462 1916.04i 0.178523 0.396147i
\(287\) 1216.40 0.250181
\(288\) 0 0
\(289\) −3779.98 −0.769382
\(290\) −55.5441 + 123.254i −0.0112471 + 0.0249576i
\(291\) 0 0
\(292\) 2997.59 + 3390.21i 0.600755 + 0.679442i
\(293\) 752.697i 0.150079i −0.997181 0.0750393i \(-0.976092\pi\)
0.997181 0.0750393i \(-0.0239082\pi\)
\(294\) 0 0
\(295\) 176.124i 0.0347606i
\(296\) 659.240 204.373i 0.129451 0.0401316i
\(297\) 0 0
\(298\) −3060.82 1379.35i −0.594996 0.268134i
\(299\) −14173.4 −2.74137
\(300\) 0 0
\(301\) 2494.24 0.477626
\(302\) −8972.09 4043.26i −1.70955 0.770408i
\(303\) 0 0
\(304\) 5201.19 + 641.805i 0.981278 + 0.121086i
\(305\) 50.3719i 0.00945668i
\(306\) 0 0
\(307\) 7832.90i 1.45618i −0.685482 0.728090i \(-0.740408\pi\)
0.685482 0.728090i \(-0.259592\pi\)
\(308\) 407.633 360.424i 0.0754125 0.0666789i
\(309\) 0 0
\(310\) 48.5357 107.702i 0.00889240 0.0197325i
\(311\) 8094.20 1.47582 0.737910 0.674899i \(-0.235813\pi\)
0.737910 + 0.674899i \(0.235813\pi\)
\(312\) 0 0
\(313\) 1194.07 0.215632 0.107816 0.994171i \(-0.465614\pi\)
0.107816 + 0.994171i \(0.465614\pi\)
\(314\) 1214.74 2695.54i 0.218318 0.484452i
\(315\) 0 0
\(316\) −3252.64 + 2875.95i −0.579036 + 0.511977i
\(317\) 558.918i 0.0990283i 0.998773 + 0.0495141i \(0.0157673\pi\)
−0.998773 + 0.0495141i \(0.984233\pi\)
\(318\) 0 0
\(319\) 1546.80i 0.271486i
\(320\) 86.9572 + 126.769i 0.0151908 + 0.0221456i
\(321\) 0 0
\(322\) −3345.57 1507.68i −0.579011 0.260930i
\(323\) −7634.63 −1.31518
\(324\) 0 0
\(325\) −9552.03 −1.63031
\(326\) 820.660 + 369.829i 0.139424 + 0.0628311i
\(327\) 0 0
\(328\) −1164.31 3755.67i −0.196000 0.632232i
\(329\) 2428.50i 0.406954i
\(330\) 0 0
\(331\) 4269.51i 0.708983i 0.935059 + 0.354491i \(0.115346\pi\)
−0.935059 + 0.354491i \(0.884654\pi\)
\(332\) 4194.46 + 4743.85i 0.693376 + 0.784195i
\(333\) 0 0
\(334\) 891.773 1978.87i 0.146095 0.324188i
\(335\) 121.159 0.0197601
\(336\) 0 0
\(337\) 5238.96 0.846837 0.423419 0.905934i \(-0.360830\pi\)
0.423419 + 0.905934i \(0.360830\pi\)
\(338\) −4242.60 + 9414.44i −0.682742 + 1.51502i
\(339\) 0 0
\(340\) −148.344 167.774i −0.0236620 0.0267613i
\(341\) 1351.63i 0.214648i
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) −2387.42 7701.02i −0.374189 1.20701i
\(345\) 0 0
\(346\) 2406.86 + 1084.65i 0.373970 + 0.168529i
\(347\) 765.526 0.118431 0.0592156 0.998245i \(-0.481140\pi\)
0.0592156 + 0.998245i \(0.481140\pi\)
\(348\) 0 0
\(349\) 12201.9 1.87149 0.935747 0.352672i \(-0.114727\pi\)
0.935747 + 0.352672i \(0.114727\pi\)
\(350\) −2254.72 1016.08i −0.344342 0.155177i
\(351\) 0 0
\(352\) −1502.99 913.588i −0.227585 0.138336i
\(353\) 4748.81i 0.716017i 0.933718 + 0.358008i \(0.116544\pi\)
−0.933718 + 0.358008i \(0.883456\pi\)
\(354\) 0 0
\(355\) 46.9855i 0.00702459i
\(356\) −4851.04 + 4289.23i −0.722204 + 0.638564i
\(357\) 0 0
\(358\) −1594.15 + 3537.46i −0.235345 + 0.522236i
\(359\) 2985.96 0.438977 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(360\) 0 0
\(361\) 153.862 0.0224321
\(362\) −1951.01 + 4329.33i −0.283267 + 0.628577i
\(363\) 0 0
\(364\) −3208.18 + 2836.64i −0.461963 + 0.408462i
\(365\) 169.841i 0.0243559i
\(366\) 0 0
\(367\) 8144.25i 1.15838i 0.815192 + 0.579191i \(0.196632\pi\)
−0.815192 + 0.579191i \(0.803368\pi\)
\(368\) −1452.70 + 11772.6i −0.205780 + 1.66764i
\(369\) 0 0
\(370\) −23.6163 10.6426i −0.00331825 0.00149536i
\(371\) −1083.60 −0.151638
\(372\) 0 0
\(373\) −12379.6 −1.71847 −0.859236 0.511579i \(-0.829061\pi\)
−0.859236 + 0.511579i \(0.829061\pi\)
\(374\) 2336.10 + 1052.76i 0.322986 + 0.145553i
\(375\) 0 0
\(376\) 7498.06 2324.50i 1.02841 0.318821i
\(377\) 12173.7i 1.66307i
\(378\) 0 0
\(379\) 12503.2i 1.69458i −0.531129 0.847291i \(-0.678232\pi\)
0.531129 0.847291i \(-0.321768\pi\)
\(380\) −130.283 147.348i −0.0175879 0.0198916i
\(381\) 0 0
\(382\) 1116.73 2478.06i 0.149573 0.331907i
\(383\) 246.082 0.0328308 0.0164154 0.999865i \(-0.494775\pi\)
0.0164154 + 0.999865i \(0.494775\pi\)
\(384\) 0 0
\(385\) −20.4214 −0.00270331
\(386\) −5209.30 + 11559.6i −0.686908 + 1.52427i
\(387\) 0 0
\(388\) 1019.56 + 1153.10i 0.133403 + 0.150876i
\(389\) 10288.3i 1.34097i 0.741923 + 0.670485i \(0.233914\pi\)
−0.741923 + 0.670485i \(0.766086\pi\)
\(390\) 0 0
\(391\) 17280.6i 2.23509i
\(392\) −1059.02 + 328.310i −0.136451 + 0.0423015i
\(393\) 0 0
\(394\) 10763.5 + 4850.57i 1.37629 + 0.620224i
\(395\) 162.949 0.0207567
\(396\) 0 0
\(397\) −11462.6 −1.44909 −0.724547 0.689225i \(-0.757951\pi\)
−0.724547 + 0.689225i \(0.757951\pi\)
\(398\) −11095.1 4999.97i −1.39735 0.629713i
\(399\) 0 0
\(400\) −979.030 + 7934.05i −0.122379 + 0.991757i
\(401\) 9125.36i 1.13641i −0.822888 0.568203i \(-0.807639\pi\)
0.822888 0.568203i \(-0.192361\pi\)
\(402\) 0 0
\(403\) 10637.7i 1.31489i
\(404\) −236.789 + 209.366i −0.0291601 + 0.0257831i
\(405\) 0 0
\(406\) −1294.96 + 2873.55i −0.158295 + 0.351261i
\(407\) 296.377 0.0360955
\(408\) 0 0
\(409\) −5019.52 −0.606844 −0.303422 0.952856i \(-0.598129\pi\)
−0.303422 + 0.952856i \(0.598129\pi\)
\(410\) −60.6307 + 134.541i −0.00730326 + 0.0162061i
\(411\) 0 0
\(412\) −8516.06 + 7529.81i −1.01834 + 0.900405i
\(413\) 4106.19i 0.489231i
\(414\) 0 0
\(415\) 237.656i 0.0281110i
\(416\) 11829.0 + 7190.18i 1.39414 + 0.847422i
\(417\) 0 0
\(418\) 2051.68 + 924.588i 0.240074 + 0.108189i
\(419\) 10727.8 1.25081 0.625403 0.780302i \(-0.284935\pi\)
0.625403 + 0.780302i \(0.284935\pi\)
\(420\) 0 0
\(421\) 8669.15 1.00358 0.501792 0.864989i \(-0.332674\pi\)
0.501792 + 0.864989i \(0.332674\pi\)
\(422\) −4724.13 2128.92i −0.544946 0.245579i
\(423\) 0 0
\(424\) 1037.19 + 3345.64i 0.118798 + 0.383204i
\(425\) 11646.1i 1.32922i
\(426\) 0 0
\(427\) 1174.38i 0.133096i
\(428\) −7710.50 8720.42i −0.870797 0.984854i
\(429\) 0 0
\(430\) −124.324 + 275.877i −0.0139428 + 0.0309395i
\(431\) 4819.01 0.538569 0.269285 0.963061i \(-0.413213\pi\)
0.269285 + 0.963061i \(0.413213\pi\)
\(432\) 0 0
\(433\) −1274.56 −0.141458 −0.0707290 0.997496i \(-0.522533\pi\)
−0.0707290 + 0.997496i \(0.522533\pi\)
\(434\) 1131.57 2510.98i 0.125154 0.277721i
\(435\) 0 0
\(436\) 10094.7 + 11416.9i 1.10883 + 1.25406i
\(437\) 15176.8i 1.66134i
\(438\) 0 0
\(439\) 6189.10i 0.672869i 0.941707 + 0.336434i \(0.109221\pi\)
−0.941707 + 0.336434i \(0.890779\pi\)
\(440\) 19.5468 + 63.0517i 0.00211786 + 0.00683152i
\(441\) 0 0
\(442\) −18385.7 8285.49i −1.97855 0.891630i
\(443\) −10421.5 −1.11770 −0.558848 0.829270i \(-0.688757\pi\)
−0.558848 + 0.829270i \(0.688757\pi\)
\(444\) 0 0
\(445\) 243.025 0.0258888
\(446\) −8038.29 3622.44i −0.853417 0.384591i
\(447\) 0 0
\(448\) 2027.33 + 2955.50i 0.213800 + 0.311684i
\(449\) 4319.61i 0.454020i −0.973892 0.227010i \(-0.927105\pi\)
0.973892 0.227010i \(-0.0728949\pi\)
\(450\) 0 0
\(451\) 1688.45i 0.176288i
\(452\) 7412.91 6554.41i 0.771403 0.682065i
\(453\) 0 0
\(454\) 5936.45 13173.1i 0.613682 1.36178i
\(455\) 160.722 0.0165599
\(456\) 0 0
\(457\) −6228.62 −0.637554 −0.318777 0.947830i \(-0.603272\pi\)
−0.318777 + 0.947830i \(0.603272\pi\)
\(458\) −5163.15 + 11457.2i −0.526764 + 1.16890i
\(459\) 0 0
\(460\) 333.516 294.891i 0.0338049 0.0298899i
\(461\) 3665.71i 0.370345i −0.982706 0.185172i \(-0.940716\pi\)
0.982706 0.185172i \(-0.0592844\pi\)
\(462\) 0 0
\(463\) 1640.57i 0.164673i 0.996605 + 0.0823367i \(0.0262383\pi\)
−0.996605 + 0.0823367i \(0.973762\pi\)
\(464\) 10111.7 + 1247.74i 1.01169 + 0.124838i
\(465\) 0 0
\(466\) −623.466 280.964i −0.0619775 0.0279301i
\(467\) 15141.3 1.50034 0.750168 0.661247i \(-0.229972\pi\)
0.750168 + 0.661247i \(0.229972\pi\)
\(468\) 0 0
\(469\) 2824.72 0.278110
\(470\) −268.607 121.047i −0.0263615 0.0118798i
\(471\) 0 0
\(472\) 12678.0 3930.33i 1.23634 0.383280i
\(473\) 3462.18i 0.336556i
\(474\) 0 0
\(475\) 10228.2i 0.988007i
\(476\) −3458.51 3911.51i −0.333026 0.376646i
\(477\) 0 0
\(478\) −881.379 + 1955.80i −0.0843375 + 0.187147i
\(479\) −430.774 −0.0410910 −0.0205455 0.999789i \(-0.506540\pi\)
−0.0205455 + 0.999789i \(0.506540\pi\)
\(480\) 0 0
\(481\) −2332.57 −0.221114
\(482\) 6688.87 14842.8i 0.632095 1.40263i
\(483\) 0 0
\(484\) 6552.88 + 7411.18i 0.615410 + 0.696016i
\(485\) 57.7678i 0.00540845i
\(486\) 0 0
\(487\) 4431.53i 0.412344i −0.978516 0.206172i \(-0.933899\pi\)
0.978516 0.206172i \(-0.0661007\pi\)
\(488\) 3625.92 1124.08i 0.336348 0.104272i
\(489\) 0 0
\(490\) 37.9378 + 17.0966i 0.00349766 + 0.00157622i
\(491\) 3768.32 0.346358 0.173179 0.984890i \(-0.444596\pi\)
0.173179 + 0.984890i \(0.444596\pi\)
\(492\) 0 0
\(493\) −14842.5 −1.35593
\(494\) −16147.3 7276.75i −1.47065 0.662746i
\(495\) 0 0
\(496\) −8835.81 1090.30i −0.799878 0.0987017i
\(497\) 1095.42i 0.0988663i
\(498\) 0 0
\(499\) 10794.4i 0.968384i 0.874962 + 0.484192i \(0.160886\pi\)
−0.874962 + 0.484192i \(0.839114\pi\)
\(500\) 449.701 397.621i 0.0402225 0.0355643i
\(501\) 0 0
\(502\) 1482.39 3289.46i 0.131797 0.292462i
\(503\) 18367.1 1.62812 0.814062 0.580777i \(-0.197251\pi\)
0.814062 + 0.580777i \(0.197251\pi\)
\(504\) 0 0
\(505\) 11.8626 0.00104530
\(506\) −2092.76 + 4643.89i −0.183863 + 0.407996i
\(507\) 0 0
\(508\) −14538.8 + 12855.0i −1.26979 + 1.12274i
\(509\) 4154.17i 0.361749i −0.983506 0.180875i \(-0.942107\pi\)
0.983506 0.180875i \(-0.0578928\pi\)
\(510\) 0 0
\(511\) 3959.71i 0.342793i
\(512\) 7184.67 9088.36i 0.620157 0.784477i
\(513\) 0 0
\(514\) 19114.1 + 8613.74i 1.64025 + 0.739174i
\(515\) 426.634 0.0365044
\(516\) 0 0
\(517\) 3370.93 0.286757
\(518\) −550.593 248.124i −0.0467020 0.0210462i
\(519\) 0 0
\(520\) −153.839 496.234i −0.0129736 0.0418486i
\(521\) 21474.8i 1.80581i −0.429839 0.902905i \(-0.641430\pi\)
0.429839 0.902905i \(-0.358570\pi\)
\(522\) 0 0
\(523\) 16488.9i 1.37860i −0.724474 0.689302i \(-0.757917\pi\)
0.724474 0.689302i \(-0.242083\pi\)
\(524\) 13335.5 + 15082.2i 1.11176 + 1.25738i
\(525\) 0 0
\(526\) −6259.06 + 13889.0i −0.518836 + 1.15131i
\(527\) 12969.8 1.07205
\(528\) 0 0
\(529\) 22184.9 1.82337
\(530\) 54.0113 119.852i 0.00442660 0.00982275i
\(531\) 0 0
\(532\) −3037.45 3435.29i −0.247538 0.279960i
\(533\) 13288.6i 1.07991i
\(534\) 0 0
\(535\) 436.872i 0.0353040i
\(536\) −2703.75 8721.39i −0.217881 0.702811i
\(537\) 0 0
\(538\) 15223.8 + 6860.56i 1.21997 + 0.549776i
\(539\) −476.108 −0.0380472
\(540\) 0 0
\(541\) −3208.96 −0.255017 −0.127508 0.991837i \(-0.540698\pi\)
−0.127508 + 0.991837i \(0.540698\pi\)
\(542\) 3833.64 + 1727.62i 0.303817 + 0.136915i
\(543\) 0 0
\(544\) −8766.48 + 14422.2i −0.690918 + 1.13667i
\(545\) 571.960i 0.0449543i
\(546\) 0 0
\(547\) 6448.31i 0.504040i −0.967722 0.252020i \(-0.918905\pi\)
0.967722 0.252020i \(-0.0810949\pi\)
\(548\) −1368.81 + 1210.29i −0.106702 + 0.0943448i
\(549\) 0 0
\(550\) −1410.40 + 3129.70i −0.109344 + 0.242638i
\(551\) −13035.5 −1.00786
\(552\) 0 0
\(553\) 3799.03 0.292136
\(554\) 105.760 234.685i 0.00811069 0.0179978i
\(555\) 0 0
\(556\) 7891.19 6977.30i 0.601908 0.532200i
\(557\) 1897.44i 0.144340i 0.997392 + 0.0721698i \(0.0229923\pi\)
−0.997392 + 0.0721698i \(0.977008\pi\)
\(558\) 0 0
\(559\) 27248.3i 2.06168i
\(560\) 16.4731 133.498i 0.00124307 0.0100738i
\(561\) 0 0
\(562\) −19243.7 8672.14i −1.44439 0.650911i
\(563\) −9998.80 −0.748488 −0.374244 0.927330i \(-0.622098\pi\)
−0.374244 + 0.927330i \(0.622098\pi\)
\(564\) 0 0
\(565\) −371.369 −0.0276524
\(566\) −1795.46 809.119i −0.133337 0.0600880i
\(567\) 0 0
\(568\) 3382.15 1048.51i 0.249845 0.0774552i
\(569\) 4882.81i 0.359750i −0.983689 0.179875i \(-0.942431\pi\)
0.983689 0.179875i \(-0.0575694\pi\)
\(570\) 0 0
\(571\) 4778.99i 0.350253i −0.984546 0.175126i \(-0.943967\pi\)
0.984546 0.175126i \(-0.0560334\pi\)
\(572\) 3937.45 + 4453.18i 0.287820 + 0.325519i
\(573\) 0 0
\(574\) −1413.55 + 3136.71i −0.102788 + 0.228090i
\(575\) 23151.1 1.67908
\(576\) 0 0
\(577\) −24393.3 −1.75998 −0.879990 0.474993i \(-0.842451\pi\)
−0.879990 + 0.474993i \(0.842451\pi\)
\(578\) 4392.62 9747.34i 0.316105 0.701446i
\(579\) 0 0
\(580\) −253.285 286.460i −0.0181329 0.0205080i
\(581\) 5540.74i 0.395643i
\(582\) 0 0
\(583\) 1504.11i 0.106851i
\(584\) −12225.7 + 3790.12i −0.866271 + 0.268555i
\(585\) 0 0
\(586\) 1940.96 + 874.691i 0.136827 + 0.0616607i
\(587\) −24922.9 −1.75243 −0.876217 0.481917i \(-0.839941\pi\)
−0.876217 + 0.481917i \(0.839941\pi\)
\(588\) 0 0
\(589\) 11390.7 0.796854
\(590\) −454.168 204.670i −0.0316912 0.0142816i
\(591\) 0 0
\(592\) −239.075 + 1937.47i −0.0165979 + 0.134509i
\(593\) 17621.1i 1.22025i 0.792304 + 0.610127i \(0.208882\pi\)
−0.792304 + 0.610127i \(0.791118\pi\)
\(594\) 0 0
\(595\) 195.957i 0.0135016i
\(596\) 7113.82 6289.96i 0.488915 0.432293i
\(597\) 0 0
\(598\) 16470.6 36548.7i 1.12631 2.49931i
\(599\) 26506.9 1.80809 0.904044 0.427440i \(-0.140585\pi\)
0.904044 + 0.427440i \(0.140585\pi\)
\(600\) 0 0
\(601\) 4960.07 0.336648 0.168324 0.985732i \(-0.446165\pi\)
0.168324 + 0.985732i \(0.446165\pi\)
\(602\) −2898.50 + 6431.84i −0.196236 + 0.435452i
\(603\) 0 0
\(604\) 20852.5 18437.5i 1.40476 1.24207i
\(605\) 371.282i 0.0249500i
\(606\) 0 0
\(607\) 22066.5i 1.47554i 0.675052 + 0.737770i \(0.264121\pi\)
−0.675052 + 0.737770i \(0.735879\pi\)
\(608\) −7699.18 + 12666.4i −0.513558 + 0.844882i
\(609\) 0 0
\(610\) −129.893 58.5360i −0.00862166 0.00388534i
\(611\) −26530.2 −1.75662
\(612\) 0 0
\(613\) 26585.0 1.75165 0.875823 0.482632i \(-0.160319\pi\)
0.875823 + 0.482632i \(0.160319\pi\)
\(614\) 20198.5 + 9102.43i 1.32760 + 0.598280i
\(615\) 0 0
\(616\) 455.718 + 1469.99i 0.0298075 + 0.0961490i
\(617\) 10612.8i 0.692472i 0.938148 + 0.346236i \(0.112540\pi\)
−0.938148 + 0.346236i \(0.887460\pi\)
\(618\) 0 0
\(619\) 16881.1i 1.09614i 0.836433 + 0.548068i \(0.184637\pi\)
−0.836433 + 0.548068i \(0.815363\pi\)
\(620\) 221.327 + 250.316i 0.0143366 + 0.0162144i
\(621\) 0 0
\(622\) −9406.08 + 20872.3i −0.606350 + 1.34551i
\(623\) 5665.93 0.364367
\(624\) 0 0
\(625\) 15591.2 0.997837
\(626\) −1387.60 + 3079.12i −0.0885936 + 0.196591i
\(627\) 0 0
\(628\) 5539.30 + 6264.84i 0.351978 + 0.398080i
\(629\) 2843.94i 0.180278i
\(630\) 0 0
\(631\) 20426.2i 1.28868i 0.764741 + 0.644338i \(0.222867\pi\)
−0.764741 + 0.644338i \(0.777133\pi\)
\(632\) −3636.32 11729.6i −0.228869 0.738256i
\(633\) 0 0
\(634\) −1441.27 649.505i −0.0902841 0.0406864i
\(635\) 728.358 0.0455182
\(636\) 0 0
\(637\) 3747.10 0.233070
\(638\) 3988.69 + 1797.50i 0.247514 + 0.111542i
\(639\) 0 0
\(640\) −427.946 + 76.9197i −0.0264314 + 0.00475081i
\(641\) 21909.2i 1.35002i 0.737809 + 0.675010i \(0.235861\pi\)
−0.737809 + 0.675010i \(0.764139\pi\)
\(642\) 0 0
\(643\) 9377.07i 0.575110i −0.957764 0.287555i \(-0.907158\pi\)
0.957764 0.287555i \(-0.0928424\pi\)
\(644\) 7775.63 6875.12i 0.475780 0.420680i
\(645\) 0 0
\(646\) 8872.03 19687.3i 0.540349 1.19905i
\(647\) −23771.8 −1.44446 −0.722231 0.691651i \(-0.756883\pi\)
−0.722231 + 0.691651i \(0.756883\pi\)
\(648\) 0 0
\(649\) 5699.68 0.344733
\(650\) 11100.2 24631.6i 0.669824 1.48636i
\(651\) 0 0
\(652\) −1907.34 + 1686.45i −0.114566 + 0.101298i
\(653\) 19631.7i 1.17649i −0.808682 0.588246i \(-0.799819\pi\)
0.808682 0.588246i \(-0.200181\pi\)
\(654\) 0 0
\(655\) 755.580i 0.0450732i
\(656\) 11037.7 + 1362.00i 0.656934 + 0.0810629i
\(657\) 0 0
\(658\) −6262.33 2822.11i −0.371020 0.167199i
\(659\) −10907.1 −0.644732 −0.322366 0.946615i \(-0.604478\pi\)
−0.322366 + 0.946615i \(0.604478\pi\)
\(660\) 0 0
\(661\) −9092.18 −0.535015 −0.267507 0.963556i \(-0.586200\pi\)
−0.267507 + 0.963556i \(0.586200\pi\)
\(662\) −11009.7 4961.49i −0.646380 0.291290i
\(663\) 0 0
\(664\) −17107.2 + 5303.44i −0.999829 + 0.309960i
\(665\) 172.100i 0.0100357i
\(666\) 0 0
\(667\) 29505.3i 1.71282i
\(668\) 4066.55 + 4599.19i 0.235538 + 0.266389i
\(669\) 0 0
\(670\) −140.796 + 312.430i −0.00811855 + 0.0180153i
\(671\) 1630.12 0.0937855
\(672\) 0 0
\(673\) 1978.31 0.113311 0.0566554 0.998394i \(-0.481956\pi\)
0.0566554 + 0.998394i \(0.481956\pi\)
\(674\) −6088.07 + 13509.6i −0.347928 + 0.772062i
\(675\) 0 0
\(676\) −19346.6 21880.6i −1.10074 1.24491i
\(677\) 8522.24i 0.483805i 0.970301 + 0.241903i \(0.0777715\pi\)
−0.970301 + 0.241903i \(0.922229\pi\)
\(678\) 0 0
\(679\) 1346.81i 0.0761203i
\(680\) 605.022 187.565i 0.0341199 0.0105776i
\(681\) 0 0
\(682\) −3485.42 1570.70i −0.195694 0.0881893i
\(683\) −35190.8 −1.97150 −0.985752 0.168208i \(-0.946202\pi\)
−0.985752 + 0.168208i \(0.946202\pi\)
\(684\) 0 0
\(685\) 68.5742 0.00382494
\(686\) 884.486 + 398.592i 0.0492272 + 0.0221841i
\(687\) 0 0
\(688\) 22632.8 + 2792.79i 1.25417 + 0.154759i
\(689\) 11837.8i 0.654548i
\(690\) 0 0
\(691\) 16471.4i 0.906802i 0.891307 + 0.453401i \(0.149790\pi\)
−0.891307 + 0.453401i \(0.850210\pi\)
\(692\) −5593.91 + 4946.07i −0.307296 + 0.271707i
\(693\) 0 0
\(694\) −889.600 + 1974.05i −0.0486582 + 0.107974i
\(695\) −395.330 −0.0215766
\(696\) 0 0
\(697\) −16201.8 −0.880469
\(698\) −14179.5 + 31464.7i −0.768915 + 1.70624i
\(699\) 0 0
\(700\) 5240.30 4633.42i 0.282950 0.250181i
\(701\) 25256.3i 1.36080i 0.732843 + 0.680398i \(0.238193\pi\)
−0.732843 + 0.680398i \(0.761807\pi\)
\(702\) 0 0
\(703\) 2497.69i 0.134000i
\(704\) 4102.44 2814.08i 0.219626 0.150653i
\(705\) 0 0
\(706\) −12245.7 5518.49i −0.652792 0.294180i
\(707\) 276.565 0.0147119
\(708\) 0 0
\(709\) −19712.2 −1.04416 −0.522079 0.852897i \(-0.674843\pi\)
−0.522079 + 0.852897i \(0.674843\pi\)
\(710\) −121.160 54.6007i −0.00640432 0.00288609i
\(711\) 0 0
\(712\) −5423.27 17493.7i −0.285457 0.920791i
\(713\) 25782.4i 1.35422i
\(714\) 0 0
\(715\) 223.094i 0.0116689i
\(716\) −7269.44 8221.60i −0.379430 0.429128i
\(717\) 0 0
\(718\) −3469.91 + 7699.83i −0.180357 + 0.400216i
\(719\) 31049.2 1.61049 0.805244 0.592944i \(-0.202034\pi\)
0.805244 + 0.592944i \(0.202034\pi\)
\(720\) 0 0
\(721\) 9946.61 0.513774
\(722\) −178.799 + 396.760i −0.00921638 + 0.0204514i
\(723\) 0 0
\(724\) −8896.74 10062.0i −0.456691 0.516509i
\(725\) 19884.8i 1.01862i
\(726\) 0 0
\(727\) 21491.9i 1.09641i 0.836343 + 0.548207i \(0.184689\pi\)
−0.836343 + 0.548207i \(0.815311\pi\)
\(728\) −3586.62 11569.3i −0.182595 0.588991i
\(729\) 0 0
\(730\) 437.966 + 197.369i 0.0222053 + 0.0100068i
\(731\) −33221.9 −1.68092
\(732\) 0 0
\(733\) −28172.7 −1.41962 −0.709811 0.704392i \(-0.751220\pi\)
−0.709811 + 0.704392i \(0.751220\pi\)
\(734\) −21001.4 9464.24i −1.05610 0.475928i
\(735\) 0 0
\(736\) −28669.7 17426.7i −1.43584 0.872770i
\(737\) 3920.91i 0.195968i
\(738\) 0 0
\(739\) 15944.0i 0.793651i 0.917894 + 0.396825i \(0.129888\pi\)
−0.917894 + 0.396825i \(0.870112\pi\)
\(740\) 54.8878 48.5312i 0.00272664 0.00241087i
\(741\) 0 0
\(742\) 1259.23 2794.26i 0.0623014 0.138248i
\(743\) 19257.0 0.950837 0.475419 0.879760i \(-0.342297\pi\)
0.475419 + 0.879760i \(0.342297\pi\)
\(744\) 0 0
\(745\) −356.385 −0.0175261
\(746\) 14386.0 31922.9i 0.706045 1.56673i
\(747\) 0 0
\(748\) −5429.45 + 4800.65i −0.265401 + 0.234665i
\(749\) 10185.3i 0.496880i
\(750\) 0 0
\(751\) 3970.69i 0.192933i 0.995336 + 0.0964663i \(0.0307540\pi\)
−0.995336 + 0.0964663i \(0.969246\pi\)
\(752\) −2719.19 + 22036.3i −0.131860 + 1.06859i
\(753\) 0 0
\(754\) −31392.1 14146.8i −1.51622 0.683284i
\(755\) −1044.66 −0.0503564
\(756\) 0 0
\(757\) 20546.9 0.986511 0.493255 0.869885i \(-0.335807\pi\)
0.493255 + 0.869885i \(0.335807\pi\)
\(758\) 32241.7 + 14529.7i 1.54495 + 0.696229i
\(759\) 0 0
\(760\) 531.363 164.729i 0.0253612 0.00786232i
\(761\) 12620.6i 0.601176i 0.953754 + 0.300588i \(0.0971829\pi\)
−0.953754 + 0.300588i \(0.902817\pi\)
\(762\) 0 0
\(763\) 13334.8i 0.632701i
\(764\) 5092.39 + 5759.39i 0.241147 + 0.272732i
\(765\) 0 0
\(766\) −285.966 + 634.567i −0.0134888 + 0.0299319i
\(767\) −44858.0 −2.11177
\(768\) 0 0
\(769\) 10838.7 0.508261 0.254130 0.967170i \(-0.418211\pi\)
0.254130 + 0.967170i \(0.418211\pi\)
\(770\) 23.7313 52.6603i 0.00111067 0.00246460i
\(771\) 0 0
\(772\) −23754.8 26866.2i −1.10745 1.25251i
\(773\) 6610.44i 0.307582i 0.988103 + 0.153791i \(0.0491482\pi\)
−0.988103 + 0.153791i \(0.950852\pi\)
\(774\) 0 0
\(775\) 17375.8i 0.805364i
\(776\) −4158.30 + 1289.13i −0.192364 + 0.0596352i
\(777\) 0 0
\(778\) −26530.2 11955.8i −1.22256 0.550946i
\(779\) −14229.3 −0.654450
\(780\) 0 0
\(781\) 1520.53 0.0696655
\(782\) 44561.2 + 20081.4i 2.03773 + 0.918300i
\(783\) 0 0
\(784\) 384.057 3112.39i 0.0174953 0.141782i
\(785\) 313.854i 0.0142700i
\(786\) 0 0
\(787\) 2544.84i 0.115265i −0.998338 0.0576326i \(-0.981645\pi\)
0.998338 0.0576326i \(-0.0183552\pi\)
\(788\) −25016.1 + 22119.0i −1.13092 + 0.999944i
\(789\) 0 0
\(790\) −189.360 + 420.194i −0.00852800 + 0.0189238i
\(791\) −8658.15 −0.389189
\(792\) 0 0
\(793\) −12829.5 −0.574512
\(794\) 13320.4 29558.3i 0.595369 1.32114i
\(795\) 0 0
\(796\) 25786.6 22800.2i 1.14822 1.01524i
\(797\) 24488.2i 1.08835i 0.838971 + 0.544176i \(0.183157\pi\)
−0.838971 + 0.544176i \(0.816843\pi\)
\(798\) 0 0
\(799\) 32346.3i 1.43220i
\(800\) −19321.7 11744.6i −0.853905 0.519042i
\(801\) 0 0
\(802\) 23531.4 + 10604.4i 1.03606 + 0.466899i
\(803\) −5496.35 −0.241547
\(804\) 0 0
\(805\) −389.540 −0.0170553
\(806\) 27431.2 + 12361.8i 1.19879 + 0.540231i
\(807\) 0 0
\(808\) −264.721 853.902i −0.0115258 0.0371784i
\(809\) 13881.7i 0.603280i −0.953422 0.301640i \(-0.902466\pi\)
0.953422 0.301640i \(-0.0975341\pi\)
\(810\) 0 0
\(811\) 26250.2i 1.13658i 0.822828 + 0.568291i \(0.192395\pi\)
−0.822828 + 0.568291i \(0.807605\pi\)
\(812\) −5905.12 6678.58i −0.255208 0.288636i
\(813\) 0 0
\(814\) −344.413 + 764.262i −0.0148301 + 0.0329083i
\(815\) 95.5532 0.00410685
\(816\) 0 0
\(817\) −29177.2 −1.24943
\(818\) 5833.07 12943.7i 0.249326 0.553260i
\(819\) 0 0
\(820\) −276.480 312.694i −0.0117745 0.0133168i
\(821\) 17579.8i 0.747307i 0.927568 + 0.373653i \(0.121895\pi\)
−0.927568 + 0.373653i \(0.878105\pi\)
\(822\) 0 0
\(823\) 1245.89i 0.0527691i 0.999652 + 0.0263846i \(0.00839944\pi\)
−0.999652 + 0.0263846i \(0.991601\pi\)
\(824\) −9520.63 30710.4i −0.402508 1.29836i
\(825\) 0 0
\(826\) −10588.5 4771.71i −0.446032 0.201004i
\(827\) 30172.3 1.26867 0.634337 0.773057i \(-0.281273\pi\)
0.634337 + 0.773057i \(0.281273\pi\)
\(828\) 0 0
\(829\) 21765.8 0.911891 0.455945 0.890008i \(-0.349301\pi\)
0.455945 + 0.890008i \(0.349301\pi\)
\(830\) 612.838 + 276.174i 0.0256288 + 0.0115496i
\(831\) 0 0
\(832\) −32287.3 + 22147.6i −1.34539 + 0.922871i
\(833\) 4568.57i 0.190026i
\(834\) 0 0
\(835\) 230.408i 0.00954923i
\(836\) −4768.43 + 4216.19i −0.197272 + 0.174426i
\(837\) 0 0
\(838\) −12466.5 + 27663.6i −0.513901 + 1.14036i
\(839\) −5120.96 −0.210721 −0.105361 0.994434i \(-0.533600\pi\)
−0.105361 + 0.994434i \(0.533600\pi\)
\(840\) 0 0
\(841\) −953.433 −0.0390927
\(842\) −10074.2 + 22354.9i −0.412328 + 0.914967i
\(843\) 0 0
\(844\) 10979.6 9708.04i 0.447789 0.395930i
\(845\) 1096.17i 0.0446263i
\(846\) 0 0
\(847\) 8656.13i 0.351155i
\(848\) −9832.62 1213.31i −0.398177 0.0491333i
\(849\) 0 0
\(850\) 30031.6 + 13533.7i 1.21185 + 0.546119i
\(851\) 5653.42 0.227728
\(852\) 0 0
\(853\) −29483.3 −1.18346 −0.591729 0.806137i \(-0.701555\pi\)
−0.591729 + 0.806137i \(0.701555\pi\)
\(854\) −3028.34 1364.72i −0.121344 0.0546834i
\(855\) 0 0
\(856\) 31447.3 9749.09i 1.25566 0.389272i
\(857\) 47834.8i 1.90666i −0.301936 0.953328i \(-0.597633\pi\)
0.301936 0.953328i \(-0.402367\pi\)
\(858\) 0 0
\(859\) 6519.42i 0.258952i −0.991583 0.129476i \(-0.958671\pi\)
0.991583 0.129476i \(-0.0413295\pi\)
\(860\) −566.925 641.181i −0.0224790 0.0254233i
\(861\) 0 0
\(862\) −5600.05 + 12426.7i −0.221274 + 0.491014i
\(863\) 3991.43 0.157439 0.0787195 0.996897i \(-0.474917\pi\)
0.0787195 + 0.996897i \(0.474917\pi\)
\(864\) 0 0
\(865\) 280.242 0.0110156
\(866\) 1481.13 3286.67i 0.0581189 0.128967i
\(867\) 0 0
\(868\) 5160.04 + 5835.90i 0.201778 + 0.228207i
\(869\) 5273.31i 0.205852i
\(870\) 0 0
\(871\) 30858.6i 1.20046i
\(872\) −41171.4 + 12763.7i −1.59890 + 0.495679i
\(873\) 0 0
\(874\) 39136.0 + 17636.6i 1.51464 + 0.682570i
\(875\) −525.243 −0.0202931
\(876\) 0 0
\(877\) 40731.8 1.56832 0.784158 0.620561i \(-0.213095\pi\)
0.784158 + 0.620561i \(0.213095\pi\)
\(878\) −15959.7 7192.20i −0.613455 0.276452i
\(879\) 0 0
\(880\) −185.305 22.8659i −0.00709844 0.000875918i
\(881\) 36733.9i 1.40476i −0.711800 0.702382i \(-0.752120\pi\)
0.711800 0.702382i \(-0.247880\pi\)
\(882\) 0 0
\(883\) 8461.49i 0.322482i −0.986915 0.161241i \(-0.948450\pi\)
0.986915 0.161241i \(-0.0515497\pi\)
\(884\) 42731.2 37782.4i 1.62580 1.43751i
\(885\) 0 0
\(886\) 12110.6 26873.6i 0.459213 1.01900i
\(887\) 38063.3 1.44086 0.720428 0.693529i \(-0.243945\pi\)
0.720428 + 0.693529i \(0.243945\pi\)
\(888\) 0 0
\(889\) 16981.0 0.640637
\(890\) −282.414 + 626.684i −0.0106366 + 0.0236028i
\(891\) 0 0
\(892\) 18682.2 16518.6i 0.701264 0.620049i
\(893\) 28408.3i 1.06455i
\(894\) 0 0
\(895\) 411.882i 0.0153829i
\(896\) −9977.20 + 1793.32i −0.372003 + 0.0668643i
\(897\) 0 0
\(898\) 11138.9 + 5019.71i 0.413930 + 0.186537i
\(899\) 22144.8 0.821547
\(900\) 0 0
\(901\) 14433.0 0.533664
\(902\) 4353.97 + 1962.11i 0.160722 + 0.0724291i
\(903\) 0 0
\(904\) 8287.34 + 26732.2i 0.304904 + 0.983519i
\(905\) 504.084i 0.0185153i
\(906\) 0 0
\(907\) 21480.7i 0.786387i −0.919456 0.393194i \(-0.871370\pi\)
0.919456 0.393194i \(-0.128630\pi\)
\(908\) 27070.7 + 30616.4i 0.989396 + 1.11899i
\(909\) 0 0
\(910\) −186.772 + 414.451i −0.00680375 + 0.0150977i
\(911\) −40098.6 −1.45832 −0.729158 0.684346i \(-0.760088\pi\)
−0.729158 + 0.684346i \(0.760088\pi\)
\(912\) 0 0
\(913\) −7690.93 −0.278787
\(914\) 7238.13 16061.6i 0.261943 0.581258i
\(915\) 0 0
\(916\) −23544.3 26628.2i −0.849265 0.960502i
\(917\) 17615.7i 0.634374i
\(918\) 0 0
\(919\) 30820.2i 1.10627i 0.833091 + 0.553136i \(0.186569\pi\)
−0.833091 + 0.553136i \(0.813431\pi\)
\(920\) 372.857 + 1202.71i 0.0133617 + 0.0431004i
\(921\) 0 0
\(922\) 9452.67 + 4259.83i 0.337643 + 0.152158i
\(923\) −11967.0 −0.426758
\(924\) 0 0
\(925\) 3810.06 0.135431
\(926\) −4230.50 1906.47i −0.150133 0.0676571i
\(927\) 0 0
\(928\) −14968.0 + 24624.8i −0.529472 + 0.871064i
\(929\) 32211.7i 1.13760i 0.822475 + 0.568801i \(0.192593\pi\)
−0.822475 + 0.568801i \(0.807407\pi\)
\(930\) 0 0
\(931\) 4012.36i 0.141246i
\(932\) 1449.03 1281.22i 0.0509277 0.0450297i
\(933\) 0 0
\(934\) −17595.4 + 39044.6i −0.616422 + 1.36786i
\(935\) 272.002 0.00951382
\(936\) 0 0
\(937\) 26750.4 0.932653 0.466327 0.884613i \(-0.345577\pi\)
0.466327 + 0.884613i \(0.345577\pi\)
\(938\) −3282.54 + 7284.04i −0.114263 + 0.253553i
\(939\) 0 0
\(940\) 624.283 551.984i 0.0216616 0.0191529i
\(941\) 30346.4i 1.05129i 0.850704 + 0.525645i \(0.176176\pi\)
−0.850704 + 0.525645i \(0.823824\pi\)
\(942\) 0 0
\(943\) 32207.3i 1.11221i
\(944\) −4597.69 + 37259.7i −0.158519 + 1.28464i
\(945\) 0 0
\(946\) 8927.84 + 4023.32i 0.306838 + 0.138276i
\(947\) 19938.9 0.684189 0.342095 0.939665i \(-0.388864\pi\)
0.342095 + 0.939665i \(0.388864\pi\)
\(948\) 0 0
\(949\) 43257.8 1.47967
\(950\) 26375.3 + 11886.0i 0.900766 + 0.405929i
\(951\) 0 0
\(952\) 14105.6 4372.91i 0.480214 0.148873i
\(953\) 33363.8i 1.13406i 0.823697 + 0.567031i \(0.191908\pi\)
−0.823697 + 0.567031i \(0.808092\pi\)
\(954\) 0 0
\(955\) 288.532i 0.00977661i
\(956\) −4019.15 4545.58i −0.135971 0.153781i
\(957\) 0 0
\(958\) 500.593 1110.83i 0.0168825 0.0374627i
\(959\) 1598.75 0.0538334
\(960\) 0 0
\(961\) 10440.3 0.350453
\(962\) 2710.62 6014.94i 0.0908462 0.201590i
\(963\) 0 0
\(964\) 30501.8 + 34496.9i 1.01908 + 1.15256i
\(965\) 1345.93i 0.0448986i
\(966\) 0 0
\(967\) 20192.7i 0.671513i 0.941949 + 0.335756i \(0.108992\pi\)
−0.941949 + 0.335756i \(0.891008\pi\)
\(968\) −26726.0 + 8285.41i −0.887403 + 0.275107i
\(969\) 0 0
\(970\) 148.965 + 67.1306i 0.00493089 + 0.00222210i
\(971\) 23232.6 0.767837 0.383918 0.923367i \(-0.374574\pi\)
0.383918 + 0.923367i \(0.374574\pi\)
\(972\) 0 0
\(973\) −9216.76 −0.303675
\(974\) 11427.5 + 5149.78i 0.375935 + 0.169414i
\(975\) 0 0
\(976\) −1314.95 + 10656.3i −0.0431255 + 0.349489i
\(977\) 42470.8i 1.39075i 0.718648 + 0.695374i \(0.244761\pi\)
−0.718648 + 0.695374i \(0.755239\pi\)
\(978\) 0 0
\(979\) 7864.70i 0.256749i
\(980\) −88.1732 + 77.9617i −0.00287407 + 0.00254122i
\(981\) 0 0
\(982\) −4379.07 + 9717.28i −0.142303 + 0.315775i
\(983\) 53713.8 1.74283 0.871417 0.490544i \(-0.163202\pi\)
0.871417 + 0.490544i \(0.163202\pi\)
\(984\) 0 0
\(985\) 1253.25 0.0405399
\(986\) 17248.2 38274.1i 0.557093 1.23620i
\(987\) 0 0
\(988\) 37528.8 33182.5i 1.20845 1.06850i
\(989\) 66041.3i 2.12335i
\(990\) 0 0
\(991\) 1994.68i 0.0639385i −0.999489 0.0319693i \(-0.989822\pi\)
0.999489 0.0319693i \(-0.0101779\pi\)
\(992\) 13079.4 21517.7i 0.418621 0.688697i
\(993\) 0 0
\(994\) −2824.75 1272.97i −0.0901364 0.0406198i
\(995\) −1291.85 −0.0411601
\(996\) 0 0
\(997\) −48379.4 −1.53680 −0.768401 0.639969i \(-0.778947\pi\)
−0.768401 + 0.639969i \(0.778947\pi\)
\(998\) −27835.3 12543.9i −0.882876 0.397866i
\(999\) 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.71.14 yes 36
3.2 odd 2 inner 252.4.e.a.71.23 yes 36
4.3 odd 2 inner 252.4.e.a.71.24 yes 36
12.11 even 2 inner 252.4.e.a.71.13 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.e.a.71.13 36 12.11 even 2 inner
252.4.e.a.71.14 yes 36 1.1 even 1 trivial
252.4.e.a.71.23 yes 36 3.2 odd 2 inner
252.4.e.a.71.24 yes 36 4.3 odd 2 inner