Properties

Label 1280.4.d.b.641.1
Level $1280$
Weight $4$
Character 1280.641
Analytic conductor $75.522$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,4,Mod(641,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1280.641");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not 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: \(75.5224448073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.4.d.b.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0000i q^{3} +5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} +O(q^{10})\) \(q-10.0000i q^{3} +5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} -16.0000i q^{11} -6.00000i q^{13} +50.0000 q^{15} -6.00000 q^{17} +124.000i q^{19} +180.000i q^{21} +42.0000 q^{23} -25.0000 q^{25} +460.000i q^{27} +142.000i q^{29} +188.000 q^{31} -160.000 q^{33} -90.0000i q^{35} -202.000i q^{37} -60.0000 q^{39} -54.0000 q^{41} +66.0000i q^{43} -365.000i q^{45} -38.0000 q^{47} -19.0000 q^{49} +60.0000i q^{51} -738.000i q^{53} +80.0000 q^{55} +1240.00 q^{57} +564.000i q^{59} -262.000i q^{61} +1314.00 q^{63} +30.0000 q^{65} +554.000i q^{67} -420.000i q^{69} +140.000 q^{71} -882.000 q^{73} +250.000i q^{75} +288.000i q^{77} +1160.00 q^{79} +2629.00 q^{81} -642.000i q^{83} -30.0000i q^{85} +1420.00 q^{87} +854.000 q^{89} +108.000i q^{91} -1880.00i q^{93} -620.000 q^{95} -478.000 q^{97} +1168.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 36 q^{7} - 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 36 q^{7} - 146 q^{9} + 100 q^{15} - 12 q^{17} + 84 q^{23} - 50 q^{25} + 376 q^{31} - 320 q^{33} - 120 q^{39} - 108 q^{41} - 76 q^{47} - 38 q^{49} + 160 q^{55} + 2480 q^{57} + 2628 q^{63} + 60 q^{65} + 280 q^{71} - 1764 q^{73} + 2320 q^{79} + 5258 q^{81} + 2840 q^{87} + 1708 q^{89} - 1240 q^{95} - 956 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 10.0000i − 1.92450i −0.272166 0.962250i \(-0.587740\pi\)
0.272166 0.962250i \(-0.412260\pi\)
\(4\) 0 0
\(5\) 5.00000i 0.447214i
\(6\) 0 0
\(7\) −18.0000 −0.971909 −0.485954 0.873984i \(-0.661528\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(8\) 0 0
\(9\) −73.0000 −2.70370
\(10\) 0 0
\(11\) − 16.0000i − 0.438562i −0.975662 0.219281i \(-0.929629\pi\)
0.975662 0.219281i \(-0.0703711\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 0.128008i −0.997950 0.0640039i \(-0.979613\pi\)
0.997950 0.0640039i \(-0.0203870\pi\)
\(14\) 0 0
\(15\) 50.0000 0.860663
\(16\) 0 0
\(17\) −6.00000 −0.0856008 −0.0428004 0.999084i \(-0.513628\pi\)
−0.0428004 + 0.999084i \(0.513628\pi\)
\(18\) 0 0
\(19\) 124.000i 1.49724i 0.663000 + 0.748620i \(0.269283\pi\)
−0.663000 + 0.748620i \(0.730717\pi\)
\(20\) 0 0
\(21\) 180.000i 1.87044i
\(22\) 0 0
\(23\) 42.0000 0.380765 0.190383 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) 460.000i 3.27878i
\(28\) 0 0
\(29\) 142.000i 0.909267i 0.890679 + 0.454633i \(0.150230\pi\)
−0.890679 + 0.454633i \(0.849770\pi\)
\(30\) 0 0
\(31\) 188.000 1.08922 0.544610 0.838690i \(-0.316678\pi\)
0.544610 + 0.838690i \(0.316678\pi\)
\(32\) 0 0
\(33\) −160.000 −0.844013
\(34\) 0 0
\(35\) − 90.0000i − 0.434651i
\(36\) 0 0
\(37\) − 202.000i − 0.897530i −0.893650 0.448765i \(-0.851864\pi\)
0.893650 0.448765i \(-0.148136\pi\)
\(38\) 0 0
\(39\) −60.0000 −0.246351
\(40\) 0 0
\(41\) −54.0000 −0.205692 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(42\) 0 0
\(43\) 66.0000i 0.234068i 0.993128 + 0.117034i \(0.0373386\pi\)
−0.993128 + 0.117034i \(0.962661\pi\)
\(44\) 0 0
\(45\) − 365.000i − 1.20913i
\(46\) 0 0
\(47\) −38.0000 −0.117933 −0.0589667 0.998260i \(-0.518781\pi\)
−0.0589667 + 0.998260i \(0.518781\pi\)
\(48\) 0 0
\(49\) −19.0000 −0.0553936
\(50\) 0 0
\(51\) 60.0000i 0.164739i
\(52\) 0 0
\(53\) − 738.000i − 1.91268i −0.292255 0.956341i \(-0.594405\pi\)
0.292255 0.956341i \(-0.405595\pi\)
\(54\) 0 0
\(55\) 80.0000 0.196131
\(56\) 0 0
\(57\) 1240.00 2.88144
\(58\) 0 0
\(59\) 564.000i 1.24452i 0.782812 + 0.622259i \(0.213785\pi\)
−0.782812 + 0.622259i \(0.786215\pi\)
\(60\) 0 0
\(61\) − 262.000i − 0.549929i −0.961454 0.274964i \(-0.911334\pi\)
0.961454 0.274964i \(-0.0886661\pi\)
\(62\) 0 0
\(63\) 1314.00 2.62775
\(64\) 0 0
\(65\) 30.0000 0.0572468
\(66\) 0 0
\(67\) 554.000i 1.01018i 0.863067 + 0.505089i \(0.168540\pi\)
−0.863067 + 0.505089i \(0.831460\pi\)
\(68\) 0 0
\(69\) − 420.000i − 0.732783i
\(70\) 0 0
\(71\) 140.000 0.234013 0.117007 0.993131i \(-0.462670\pi\)
0.117007 + 0.993131i \(0.462670\pi\)
\(72\) 0 0
\(73\) −882.000 −1.41411 −0.707057 0.707157i \(-0.749977\pi\)
−0.707057 + 0.707157i \(0.749977\pi\)
\(74\) 0 0
\(75\) 250.000i 0.384900i
\(76\) 0 0
\(77\) 288.000i 0.426242i
\(78\) 0 0
\(79\) 1160.00 1.65203 0.826014 0.563650i \(-0.190603\pi\)
0.826014 + 0.563650i \(0.190603\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 0 0
\(83\) − 642.000i − 0.849020i −0.905423 0.424510i \(-0.860446\pi\)
0.905423 0.424510i \(-0.139554\pi\)
\(84\) 0 0
\(85\) − 30.0000i − 0.0382818i
\(86\) 0 0
\(87\) 1420.00 1.74988
\(88\) 0 0
\(89\) 854.000 1.01712 0.508561 0.861026i \(-0.330178\pi\)
0.508561 + 0.861026i \(0.330178\pi\)
\(90\) 0 0
\(91\) 108.000i 0.124412i
\(92\) 0 0
\(93\) − 1880.00i − 2.09620i
\(94\) 0 0
\(95\) −620.000 −0.669586
\(96\) 0 0
\(97\) −478.000 −0.500346 −0.250173 0.968201i \(-0.580487\pi\)
−0.250173 + 0.968201i \(0.580487\pi\)
\(98\) 0 0
\(99\) 1168.00i 1.18574i
\(100\) 0 0
\(101\) 1794.00i 1.76742i 0.468033 + 0.883711i \(0.344963\pi\)
−0.468033 + 0.883711i \(0.655037\pi\)
\(102\) 0 0
\(103\) 642.000 0.614157 0.307078 0.951684i \(-0.400649\pi\)
0.307078 + 0.951684i \(0.400649\pi\)
\(104\) 0 0
\(105\) −900.000 −0.836486
\(106\) 0 0
\(107\) − 850.000i − 0.767968i −0.923340 0.383984i \(-0.874552\pi\)
0.923340 0.383984i \(-0.125448\pi\)
\(108\) 0 0
\(109\) 666.000i 0.585241i 0.956229 + 0.292620i \(0.0945272\pi\)
−0.956229 + 0.292620i \(0.905473\pi\)
\(110\) 0 0
\(111\) −2020.00 −1.72730
\(112\) 0 0
\(113\) −1446.00 −1.20379 −0.601895 0.798575i \(-0.705587\pi\)
−0.601895 + 0.798575i \(0.705587\pi\)
\(114\) 0 0
\(115\) 210.000i 0.170283i
\(116\) 0 0
\(117\) 438.000i 0.346095i
\(118\) 0 0
\(119\) 108.000 0.0831962
\(120\) 0 0
\(121\) 1075.00 0.807663
\(122\) 0 0
\(123\) 540.000i 0.395855i
\(124\) 0 0
\(125\) − 125.000i − 0.0894427i
\(126\) 0 0
\(127\) 1154.00 0.806307 0.403153 0.915132i \(-0.367914\pi\)
0.403153 + 0.915132i \(0.367914\pi\)
\(128\) 0 0
\(129\) 660.000 0.450463
\(130\) 0 0
\(131\) 368.000i 0.245437i 0.992441 + 0.122719i \(0.0391613\pi\)
−0.992441 + 0.122719i \(0.960839\pi\)
\(132\) 0 0
\(133\) − 2232.00i − 1.45518i
\(134\) 0 0
\(135\) −2300.00 −1.46631
\(136\) 0 0
\(137\) 670.000 0.417825 0.208912 0.977934i \(-0.433008\pi\)
0.208912 + 0.977934i \(0.433008\pi\)
\(138\) 0 0
\(139\) − 572.000i − 0.349039i −0.984654 0.174519i \(-0.944163\pi\)
0.984654 0.174519i \(-0.0558372\pi\)
\(140\) 0 0
\(141\) 380.000i 0.226963i
\(142\) 0 0
\(143\) −96.0000 −0.0561393
\(144\) 0 0
\(145\) −710.000 −0.406636
\(146\) 0 0
\(147\) 190.000i 0.106605i
\(148\) 0 0
\(149\) − 1730.00i − 0.951189i −0.879665 0.475594i \(-0.842233\pi\)
0.879665 0.475594i \(-0.157767\pi\)
\(150\) 0 0
\(151\) 1324.00 0.713547 0.356773 0.934191i \(-0.383877\pi\)
0.356773 + 0.934191i \(0.383877\pi\)
\(152\) 0 0
\(153\) 438.000 0.231439
\(154\) 0 0
\(155\) 940.000i 0.487114i
\(156\) 0 0
\(157\) 2946.00i 1.49756i 0.662820 + 0.748778i \(0.269359\pi\)
−0.662820 + 0.748778i \(0.730641\pi\)
\(158\) 0 0
\(159\) −7380.00 −3.68096
\(160\) 0 0
\(161\) −756.000 −0.370069
\(162\) 0 0
\(163\) − 2098.00i − 1.00815i −0.863661 0.504074i \(-0.831834\pi\)
0.863661 0.504074i \(-0.168166\pi\)
\(164\) 0 0
\(165\) − 800.000i − 0.377454i
\(166\) 0 0
\(167\) −866.000 −0.401276 −0.200638 0.979665i \(-0.564301\pi\)
−0.200638 + 0.979665i \(0.564301\pi\)
\(168\) 0 0
\(169\) 2161.00 0.983614
\(170\) 0 0
\(171\) − 9052.00i − 4.04809i
\(172\) 0 0
\(173\) − 1678.00i − 0.737433i −0.929542 0.368717i \(-0.879797\pi\)
0.929542 0.368717i \(-0.120203\pi\)
\(174\) 0 0
\(175\) 450.000 0.194382
\(176\) 0 0
\(177\) 5640.00 2.39508
\(178\) 0 0
\(179\) 1620.00i 0.676450i 0.941065 + 0.338225i \(0.109826\pi\)
−0.941065 + 0.338225i \(0.890174\pi\)
\(180\) 0 0
\(181\) − 2510.00i − 1.03076i −0.856963 0.515378i \(-0.827652\pi\)
0.856963 0.515378i \(-0.172348\pi\)
\(182\) 0 0
\(183\) −2620.00 −1.05834
\(184\) 0 0
\(185\) 1010.00 0.401387
\(186\) 0 0
\(187\) 96.0000i 0.0375413i
\(188\) 0 0
\(189\) − 8280.00i − 3.18667i
\(190\) 0 0
\(191\) 372.000 0.140927 0.0704633 0.997514i \(-0.477552\pi\)
0.0704633 + 0.997514i \(0.477552\pi\)
\(192\) 0 0
\(193\) 2938.00 1.09576 0.547880 0.836557i \(-0.315435\pi\)
0.547880 + 0.836557i \(0.315435\pi\)
\(194\) 0 0
\(195\) − 300.000i − 0.110172i
\(196\) 0 0
\(197\) − 2234.00i − 0.807949i −0.914770 0.403974i \(-0.867628\pi\)
0.914770 0.403974i \(-0.132372\pi\)
\(198\) 0 0
\(199\) −3048.00 −1.08576 −0.542882 0.839809i \(-0.682667\pi\)
−0.542882 + 0.839809i \(0.682667\pi\)
\(200\) 0 0
\(201\) 5540.00 1.94409
\(202\) 0 0
\(203\) − 2556.00i − 0.883724i
\(204\) 0 0
\(205\) − 270.000i − 0.0919884i
\(206\) 0 0
\(207\) −3066.00 −1.02948
\(208\) 0 0
\(209\) 1984.00 0.656632
\(210\) 0 0
\(211\) − 4896.00i − 1.59741i −0.601720 0.798707i \(-0.705518\pi\)
0.601720 0.798707i \(-0.294482\pi\)
\(212\) 0 0
\(213\) − 1400.00i − 0.450359i
\(214\) 0 0
\(215\) −330.000 −0.104678
\(216\) 0 0
\(217\) −3384.00 −1.05862
\(218\) 0 0
\(219\) 8820.00i 2.72146i
\(220\) 0 0
\(221\) 36.0000i 0.0109576i
\(222\) 0 0
\(223\) 5302.00 1.59214 0.796072 0.605202i \(-0.206908\pi\)
0.796072 + 0.605202i \(0.206908\pi\)
\(224\) 0 0
\(225\) 1825.00 0.540741
\(226\) 0 0
\(227\) 3778.00i 1.10465i 0.833630 + 0.552323i \(0.186259\pi\)
−0.833630 + 0.552323i \(0.813741\pi\)
\(228\) 0 0
\(229\) 3034.00i 0.875513i 0.899094 + 0.437756i \(0.144227\pi\)
−0.899094 + 0.437756i \(0.855773\pi\)
\(230\) 0 0
\(231\) 2880.00 0.820303
\(232\) 0 0
\(233\) 3478.00 0.977903 0.488951 0.872311i \(-0.337380\pi\)
0.488951 + 0.872311i \(0.337380\pi\)
\(234\) 0 0
\(235\) − 190.000i − 0.0527414i
\(236\) 0 0
\(237\) − 11600.0i − 3.17933i
\(238\) 0 0
\(239\) −1560.00 −0.422209 −0.211105 0.977463i \(-0.567706\pi\)
−0.211105 + 0.977463i \(0.567706\pi\)
\(240\) 0 0
\(241\) −3218.00 −0.860123 −0.430061 0.902800i \(-0.641508\pi\)
−0.430061 + 0.902800i \(0.641508\pi\)
\(242\) 0 0
\(243\) − 13870.0i − 3.66157i
\(244\) 0 0
\(245\) − 95.0000i − 0.0247728i
\(246\) 0 0
\(247\) 744.000 0.191658
\(248\) 0 0
\(249\) −6420.00 −1.63394
\(250\) 0 0
\(251\) 688.000i 0.173013i 0.996251 + 0.0865063i \(0.0275703\pi\)
−0.996251 + 0.0865063i \(0.972430\pi\)
\(252\) 0 0
\(253\) − 672.000i − 0.166989i
\(254\) 0 0
\(255\) −300.000 −0.0736734
\(256\) 0 0
\(257\) 2170.00 0.526696 0.263348 0.964701i \(-0.415173\pi\)
0.263348 + 0.964701i \(0.415173\pi\)
\(258\) 0 0
\(259\) 3636.00i 0.872317i
\(260\) 0 0
\(261\) − 10366.0i − 2.45839i
\(262\) 0 0
\(263\) 2274.00 0.533159 0.266580 0.963813i \(-0.414106\pi\)
0.266580 + 0.963813i \(0.414106\pi\)
\(264\) 0 0
\(265\) 3690.00 0.855377
\(266\) 0 0
\(267\) − 8540.00i − 1.95745i
\(268\) 0 0
\(269\) 7146.00i 1.61970i 0.586637 + 0.809850i \(0.300452\pi\)
−0.586637 + 0.809850i \(0.699548\pi\)
\(270\) 0 0
\(271\) −2604.00 −0.583696 −0.291848 0.956465i \(-0.594270\pi\)
−0.291848 + 0.956465i \(0.594270\pi\)
\(272\) 0 0
\(273\) 1080.00 0.239431
\(274\) 0 0
\(275\) 400.000i 0.0877124i
\(276\) 0 0
\(277\) 5150.00i 1.11709i 0.829475 + 0.558544i \(0.188640\pi\)
−0.829475 + 0.558544i \(0.811360\pi\)
\(278\) 0 0
\(279\) −13724.0 −2.94493
\(280\) 0 0
\(281\) −5270.00 −1.11880 −0.559398 0.828899i \(-0.688968\pi\)
−0.559398 + 0.828899i \(0.688968\pi\)
\(282\) 0 0
\(283\) 3434.00i 0.721308i 0.932700 + 0.360654i \(0.117446\pi\)
−0.932700 + 0.360654i \(0.882554\pi\)
\(284\) 0 0
\(285\) 6200.00i 1.28862i
\(286\) 0 0
\(287\) 972.000 0.199914
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 0 0
\(291\) 4780.00i 0.962916i
\(292\) 0 0
\(293\) 9878.00i 1.96955i 0.173826 + 0.984776i \(0.444387\pi\)
−0.173826 + 0.984776i \(0.555613\pi\)
\(294\) 0 0
\(295\) −2820.00 −0.556565
\(296\) 0 0
\(297\) 7360.00 1.43795
\(298\) 0 0
\(299\) − 252.000i − 0.0487409i
\(300\) 0 0
\(301\) − 1188.00i − 0.227492i
\(302\) 0 0
\(303\) 17940.0 3.40141
\(304\) 0 0
\(305\) 1310.00 0.245936
\(306\) 0 0
\(307\) − 8054.00i − 1.49728i −0.662975 0.748642i \(-0.730706\pi\)
0.662975 0.748642i \(-0.269294\pi\)
\(308\) 0 0
\(309\) − 6420.00i − 1.18195i
\(310\) 0 0
\(311\) 5492.00 1.00136 0.500680 0.865633i \(-0.333083\pi\)
0.500680 + 0.865633i \(0.333083\pi\)
\(312\) 0 0
\(313\) 422.000 0.0762072 0.0381036 0.999274i \(-0.487868\pi\)
0.0381036 + 0.999274i \(0.487868\pi\)
\(314\) 0 0
\(315\) 6570.00i 1.17517i
\(316\) 0 0
\(317\) 6194.00i 1.09744i 0.836005 + 0.548722i \(0.184885\pi\)
−0.836005 + 0.548722i \(0.815115\pi\)
\(318\) 0 0
\(319\) 2272.00 0.398770
\(320\) 0 0
\(321\) −8500.00 −1.47796
\(322\) 0 0
\(323\) − 744.000i − 0.128165i
\(324\) 0 0
\(325\) 150.000i 0.0256015i
\(326\) 0 0
\(327\) 6660.00 1.12630
\(328\) 0 0
\(329\) 684.000 0.114620
\(330\) 0 0
\(331\) 7688.00i 1.27665i 0.769768 + 0.638324i \(0.220372\pi\)
−0.769768 + 0.638324i \(0.779628\pi\)
\(332\) 0 0
\(333\) 14746.0i 2.42665i
\(334\) 0 0
\(335\) −2770.00 −0.451765
\(336\) 0 0
\(337\) −1438.00 −0.232442 −0.116221 0.993223i \(-0.537078\pi\)
−0.116221 + 0.993223i \(0.537078\pi\)
\(338\) 0 0
\(339\) 14460.0i 2.31669i
\(340\) 0 0
\(341\) − 3008.00i − 0.477690i
\(342\) 0 0
\(343\) 6516.00 1.02575
\(344\) 0 0
\(345\) 2100.00 0.327711
\(346\) 0 0
\(347\) 8838.00i 1.36729i 0.729816 + 0.683644i \(0.239606\pi\)
−0.729816 + 0.683644i \(0.760394\pi\)
\(348\) 0 0
\(349\) − 7810.00i − 1.19788i −0.800794 0.598939i \(-0.795589\pi\)
0.800794 0.598939i \(-0.204411\pi\)
\(350\) 0 0
\(351\) 2760.00 0.419709
\(352\) 0 0
\(353\) 5906.00 0.890495 0.445247 0.895408i \(-0.353116\pi\)
0.445247 + 0.895408i \(0.353116\pi\)
\(354\) 0 0
\(355\) 700.000i 0.104654i
\(356\) 0 0
\(357\) − 1080.00i − 0.160111i
\(358\) 0 0
\(359\) 8904.00 1.30901 0.654506 0.756057i \(-0.272877\pi\)
0.654506 + 0.756057i \(0.272877\pi\)
\(360\) 0 0
\(361\) −8517.00 −1.24173
\(362\) 0 0
\(363\) − 10750.0i − 1.55435i
\(364\) 0 0
\(365\) − 4410.00i − 0.632411i
\(366\) 0 0
\(367\) 7370.00 1.04826 0.524129 0.851639i \(-0.324391\pi\)
0.524129 + 0.851639i \(0.324391\pi\)
\(368\) 0 0
\(369\) 3942.00 0.556131
\(370\) 0 0
\(371\) 13284.0i 1.85895i
\(372\) 0 0
\(373\) 734.000i 0.101890i 0.998701 + 0.0509451i \(0.0162234\pi\)
−0.998701 + 0.0509451i \(0.983777\pi\)
\(374\) 0 0
\(375\) −1250.00 −0.172133
\(376\) 0 0
\(377\) 852.000 0.116393
\(378\) 0 0
\(379\) 10300.0i 1.39598i 0.716109 + 0.697989i \(0.245921\pi\)
−0.716109 + 0.697989i \(0.754079\pi\)
\(380\) 0 0
\(381\) − 11540.0i − 1.55174i
\(382\) 0 0
\(383\) −2682.00 −0.357817 −0.178908 0.983866i \(-0.557257\pi\)
−0.178908 + 0.983866i \(0.557257\pi\)
\(384\) 0 0
\(385\) −1440.00 −0.190621
\(386\) 0 0
\(387\) − 4818.00i − 0.632849i
\(388\) 0 0
\(389\) − 6114.00i − 0.796895i −0.917191 0.398447i \(-0.869549\pi\)
0.917191 0.398447i \(-0.130451\pi\)
\(390\) 0 0
\(391\) −252.000 −0.0325938
\(392\) 0 0
\(393\) 3680.00 0.472345
\(394\) 0 0
\(395\) 5800.00i 0.738809i
\(396\) 0 0
\(397\) − 7174.00i − 0.906934i −0.891273 0.453467i \(-0.850187\pi\)
0.891273 0.453467i \(-0.149813\pi\)
\(398\) 0 0
\(399\) −22320.0 −2.80050
\(400\) 0 0
\(401\) 10498.0 1.30734 0.653672 0.756778i \(-0.273228\pi\)
0.653672 + 0.756778i \(0.273228\pi\)
\(402\) 0 0
\(403\) − 1128.00i − 0.139428i
\(404\) 0 0
\(405\) 13145.0i 1.61279i
\(406\) 0 0
\(407\) −3232.00 −0.393622
\(408\) 0 0
\(409\) 1810.00 0.218823 0.109412 0.993997i \(-0.465103\pi\)
0.109412 + 0.993997i \(0.465103\pi\)
\(410\) 0 0
\(411\) − 6700.00i − 0.804104i
\(412\) 0 0
\(413\) − 10152.0i − 1.20956i
\(414\) 0 0
\(415\) 3210.00 0.379693
\(416\) 0 0
\(417\) −5720.00 −0.671726
\(418\) 0 0
\(419\) 3396.00i 0.395956i 0.980206 + 0.197978i \(0.0634374\pi\)
−0.980206 + 0.197978i \(0.936563\pi\)
\(420\) 0 0
\(421\) 14974.0i 1.73346i 0.498775 + 0.866732i \(0.333784\pi\)
−0.498775 + 0.866732i \(0.666216\pi\)
\(422\) 0 0
\(423\) 2774.00 0.318857
\(424\) 0 0
\(425\) 150.000 0.0171202
\(426\) 0 0
\(427\) 4716.00i 0.534481i
\(428\) 0 0
\(429\) 960.000i 0.108040i
\(430\) 0 0
\(431\) 13540.0 1.51322 0.756611 0.653865i \(-0.226854\pi\)
0.756611 + 0.653865i \(0.226854\pi\)
\(432\) 0 0
\(433\) 15426.0 1.71207 0.856035 0.516918i \(-0.172921\pi\)
0.856035 + 0.516918i \(0.172921\pi\)
\(434\) 0 0
\(435\) 7100.00i 0.782572i
\(436\) 0 0
\(437\) 5208.00i 0.570097i
\(438\) 0 0
\(439\) −10472.0 −1.13850 −0.569250 0.822165i \(-0.692766\pi\)
−0.569250 + 0.822165i \(0.692766\pi\)
\(440\) 0 0
\(441\) 1387.00 0.149768
\(442\) 0 0
\(443\) 722.000i 0.0774340i 0.999250 + 0.0387170i \(0.0123271\pi\)
−0.999250 + 0.0387170i \(0.987673\pi\)
\(444\) 0 0
\(445\) 4270.00i 0.454871i
\(446\) 0 0
\(447\) −17300.0 −1.83056
\(448\) 0 0
\(449\) −11898.0 −1.25056 −0.625280 0.780401i \(-0.715015\pi\)
−0.625280 + 0.780401i \(0.715015\pi\)
\(450\) 0 0
\(451\) 864.000i 0.0902088i
\(452\) 0 0
\(453\) − 13240.0i − 1.37322i
\(454\) 0 0
\(455\) −540.000 −0.0556387
\(456\) 0 0
\(457\) 790.000 0.0808635 0.0404318 0.999182i \(-0.487127\pi\)
0.0404318 + 0.999182i \(0.487127\pi\)
\(458\) 0 0
\(459\) − 2760.00i − 0.280666i
\(460\) 0 0
\(461\) − 3418.00i − 0.345319i −0.984982 0.172660i \(-0.944764\pi\)
0.984982 0.172660i \(-0.0552360\pi\)
\(462\) 0 0
\(463\) 7534.00 0.756230 0.378115 0.925759i \(-0.376572\pi\)
0.378115 + 0.925759i \(0.376572\pi\)
\(464\) 0 0
\(465\) 9400.00 0.937451
\(466\) 0 0
\(467\) 14314.0i 1.41836i 0.705029 + 0.709179i \(0.250934\pi\)
−0.705029 + 0.709179i \(0.749066\pi\)
\(468\) 0 0
\(469\) − 9972.00i − 0.981800i
\(470\) 0 0
\(471\) 29460.0 2.88205
\(472\) 0 0
\(473\) 1056.00 0.102653
\(474\) 0 0
\(475\) − 3100.00i − 0.299448i
\(476\) 0 0
\(477\) 53874.0i 5.17132i
\(478\) 0 0
\(479\) 7016.00 0.669247 0.334623 0.942352i \(-0.391391\pi\)
0.334623 + 0.942352i \(0.391391\pi\)
\(480\) 0 0
\(481\) −1212.00 −0.114891
\(482\) 0 0
\(483\) 7560.00i 0.712199i
\(484\) 0 0
\(485\) − 2390.00i − 0.223761i
\(486\) 0 0
\(487\) 15190.0 1.41340 0.706699 0.707515i \(-0.250184\pi\)
0.706699 + 0.707515i \(0.250184\pi\)
\(488\) 0 0
\(489\) −20980.0 −1.94018
\(490\) 0 0
\(491\) − 12624.0i − 1.16031i −0.814505 0.580156i \(-0.802992\pi\)
0.814505 0.580156i \(-0.197008\pi\)
\(492\) 0 0
\(493\) − 852.000i − 0.0778340i
\(494\) 0 0
\(495\) −5840.00 −0.530280
\(496\) 0 0
\(497\) −2520.00 −0.227440
\(498\) 0 0
\(499\) 2492.00i 0.223562i 0.993733 + 0.111781i \(0.0356555\pi\)
−0.993733 + 0.111781i \(0.964345\pi\)
\(500\) 0 0
\(501\) 8660.00i 0.772256i
\(502\) 0 0
\(503\) 11714.0 1.03837 0.519186 0.854661i \(-0.326235\pi\)
0.519186 + 0.854661i \(0.326235\pi\)
\(504\) 0 0
\(505\) −8970.00 −0.790415
\(506\) 0 0
\(507\) − 21610.0i − 1.89297i
\(508\) 0 0
\(509\) − 5618.00i − 0.489221i −0.969621 0.244610i \(-0.921340\pi\)
0.969621 0.244610i \(-0.0786601\pi\)
\(510\) 0 0
\(511\) 15876.0 1.37439
\(512\) 0 0
\(513\) −57040.0 −4.90912
\(514\) 0 0
\(515\) 3210.00i 0.274659i
\(516\) 0 0
\(517\) 608.000i 0.0517211i
\(518\) 0 0
\(519\) −16780.0 −1.41919
\(520\) 0 0
\(521\) −13770.0 −1.15792 −0.578958 0.815357i \(-0.696541\pi\)
−0.578958 + 0.815357i \(0.696541\pi\)
\(522\) 0 0
\(523\) 6986.00i 0.584085i 0.956405 + 0.292042i \(0.0943349\pi\)
−0.956405 + 0.292042i \(0.905665\pi\)
\(524\) 0 0
\(525\) − 4500.00i − 0.374088i
\(526\) 0 0
\(527\) −1128.00 −0.0932380
\(528\) 0 0
\(529\) −10403.0 −0.855018
\(530\) 0 0
\(531\) − 41172.0i − 3.36481i
\(532\) 0 0
\(533\) 324.000i 0.0263302i
\(534\) 0 0
\(535\) 4250.00 0.343446
\(536\) 0 0
\(537\) 16200.0 1.30183
\(538\) 0 0
\(539\) 304.000i 0.0242935i
\(540\) 0 0
\(541\) 11958.0i 0.950304i 0.879904 + 0.475152i \(0.157607\pi\)
−0.879904 + 0.475152i \(0.842393\pi\)
\(542\) 0 0
\(543\) −25100.0 −1.98369
\(544\) 0 0
\(545\) −3330.00 −0.261728
\(546\) 0 0
\(547\) 4194.00i 0.327829i 0.986475 + 0.163915i \(0.0524121\pi\)
−0.986475 + 0.163915i \(0.947588\pi\)
\(548\) 0 0
\(549\) 19126.0i 1.48684i
\(550\) 0 0
\(551\) −17608.0 −1.36139
\(552\) 0 0
\(553\) −20880.0 −1.60562
\(554\) 0 0
\(555\) − 10100.0i − 0.772470i
\(556\) 0 0
\(557\) − 5382.00i − 0.409412i −0.978823 0.204706i \(-0.934376\pi\)
0.978823 0.204706i \(-0.0656239\pi\)
\(558\) 0 0
\(559\) 396.000 0.0299625
\(560\) 0 0
\(561\) 960.000 0.0722482
\(562\) 0 0
\(563\) − 15418.0i − 1.15416i −0.816688 0.577079i \(-0.804192\pi\)
0.816688 0.577079i \(-0.195808\pi\)
\(564\) 0 0
\(565\) − 7230.00i − 0.538351i
\(566\) 0 0
\(567\) −47322.0 −3.50500
\(568\) 0 0
\(569\) 5778.00 0.425705 0.212853 0.977084i \(-0.431725\pi\)
0.212853 + 0.977084i \(0.431725\pi\)
\(570\) 0 0
\(571\) 6024.00i 0.441500i 0.975330 + 0.220750i \(0.0708505\pi\)
−0.975330 + 0.220750i \(0.929149\pi\)
\(572\) 0 0
\(573\) − 3720.00i − 0.271213i
\(574\) 0 0
\(575\) −1050.00 −0.0761531
\(576\) 0 0
\(577\) 554.000 0.0399711 0.0199855 0.999800i \(-0.493638\pi\)
0.0199855 + 0.999800i \(0.493638\pi\)
\(578\) 0 0
\(579\) − 29380.0i − 2.10879i
\(580\) 0 0
\(581\) 11556.0i 0.825170i
\(582\) 0 0
\(583\) −11808.0 −0.838829
\(584\) 0 0
\(585\) −2190.00 −0.154778
\(586\) 0 0
\(587\) − 2386.00i − 0.167770i −0.996475 0.0838848i \(-0.973267\pi\)
0.996475 0.0838848i \(-0.0267328\pi\)
\(588\) 0 0
\(589\) 23312.0i 1.63082i
\(590\) 0 0
\(591\) −22340.0 −1.55490
\(592\) 0 0
\(593\) −846.000 −0.0585853 −0.0292926 0.999571i \(-0.509325\pi\)
−0.0292926 + 0.999571i \(0.509325\pi\)
\(594\) 0 0
\(595\) 540.000i 0.0372065i
\(596\) 0 0
\(597\) 30480.0i 2.08955i
\(598\) 0 0
\(599\) 22304.0 1.52140 0.760698 0.649105i \(-0.224857\pi\)
0.760698 + 0.649105i \(0.224857\pi\)
\(600\) 0 0
\(601\) −5510.00 −0.373973 −0.186986 0.982363i \(-0.559872\pi\)
−0.186986 + 0.982363i \(0.559872\pi\)
\(602\) 0 0
\(603\) − 40442.0i − 2.73122i
\(604\) 0 0
\(605\) 5375.00i 0.361198i
\(606\) 0 0
\(607\) 8234.00 0.550589 0.275295 0.961360i \(-0.411225\pi\)
0.275295 + 0.961360i \(0.411225\pi\)
\(608\) 0 0
\(609\) −25560.0 −1.70073
\(610\) 0 0
\(611\) 228.000i 0.0150964i
\(612\) 0 0
\(613\) 1046.00i 0.0689193i 0.999406 + 0.0344597i \(0.0109710\pi\)
−0.999406 + 0.0344597i \(0.989029\pi\)
\(614\) 0 0
\(615\) −2700.00 −0.177032
\(616\) 0 0
\(617\) 3862.00 0.251991 0.125995 0.992031i \(-0.459788\pi\)
0.125995 + 0.992031i \(0.459788\pi\)
\(618\) 0 0
\(619\) 13964.0i 0.906721i 0.891327 + 0.453361i \(0.149775\pi\)
−0.891327 + 0.453361i \(0.850225\pi\)
\(620\) 0 0
\(621\) 19320.0i 1.24845i
\(622\) 0 0
\(623\) −15372.0 −0.988549
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) − 19840.0i − 1.26369i
\(628\) 0 0
\(629\) 1212.00i 0.0768293i
\(630\) 0 0
\(631\) −14884.0 −0.939022 −0.469511 0.882927i \(-0.655570\pi\)
−0.469511 + 0.882927i \(0.655570\pi\)
\(632\) 0 0
\(633\) −48960.0 −3.07423
\(634\) 0 0
\(635\) 5770.00i 0.360591i
\(636\) 0 0
\(637\) 114.000i 0.00709081i
\(638\) 0 0
\(639\) −10220.0 −0.632703
\(640\) 0 0
\(641\) 17838.0 1.09916 0.549578 0.835443i \(-0.314789\pi\)
0.549578 + 0.835443i \(0.314789\pi\)
\(642\) 0 0
\(643\) 7814.00i 0.479244i 0.970866 + 0.239622i \(0.0770236\pi\)
−0.970866 + 0.239622i \(0.922976\pi\)
\(644\) 0 0
\(645\) 3300.00i 0.201453i
\(646\) 0 0
\(647\) 774.000 0.0470310 0.0235155 0.999723i \(-0.492514\pi\)
0.0235155 + 0.999723i \(0.492514\pi\)
\(648\) 0 0
\(649\) 9024.00 0.545798
\(650\) 0 0
\(651\) 33840.0i 2.03732i
\(652\) 0 0
\(653\) − 23422.0i − 1.40364i −0.712357 0.701818i \(-0.752372\pi\)
0.712357 0.701818i \(-0.247628\pi\)
\(654\) 0 0
\(655\) −1840.00 −0.109763
\(656\) 0 0
\(657\) 64386.0 3.82334
\(658\) 0 0
\(659\) 13508.0i 0.798478i 0.916847 + 0.399239i \(0.130726\pi\)
−0.916847 + 0.399239i \(0.869274\pi\)
\(660\) 0 0
\(661\) 6222.00i 0.366124i 0.983101 + 0.183062i \(0.0586009\pi\)
−0.983101 + 0.183062i \(0.941399\pi\)
\(662\) 0 0
\(663\) 360.000 0.0210878
\(664\) 0 0
\(665\) 11160.0 0.650776
\(666\) 0 0
\(667\) 5964.00i 0.346217i
\(668\) 0 0
\(669\) − 53020.0i − 3.06408i
\(670\) 0 0
\(671\) −4192.00 −0.241178
\(672\) 0 0
\(673\) −15566.0 −0.891568 −0.445784 0.895141i \(-0.647075\pi\)
−0.445784 + 0.895141i \(0.647075\pi\)
\(674\) 0 0
\(675\) − 11500.0i − 0.655756i
\(676\) 0 0
\(677\) − 2234.00i − 0.126824i −0.997987 0.0634118i \(-0.979802\pi\)
0.997987 0.0634118i \(-0.0201982\pi\)
\(678\) 0 0
\(679\) 8604.00 0.486290
\(680\) 0 0
\(681\) 37780.0 2.12589
\(682\) 0 0
\(683\) 13282.0i 0.744102i 0.928212 + 0.372051i \(0.121345\pi\)
−0.928212 + 0.372051i \(0.878655\pi\)
\(684\) 0 0
\(685\) 3350.00i 0.186857i
\(686\) 0 0
\(687\) 30340.0 1.68492
\(688\) 0 0
\(689\) −4428.00 −0.244838
\(690\) 0 0
\(691\) 27416.0i 1.50934i 0.656105 + 0.754670i \(0.272203\pi\)
−0.656105 + 0.754670i \(0.727797\pi\)
\(692\) 0 0
\(693\) − 21024.0i − 1.15243i
\(694\) 0 0
\(695\) 2860.00 0.156095
\(696\) 0 0
\(697\) 324.000 0.0176074
\(698\) 0 0
\(699\) − 34780.0i − 1.88197i
\(700\) 0 0
\(701\) 25626.0i 1.38071i 0.723469 + 0.690357i \(0.242547\pi\)
−0.723469 + 0.690357i \(0.757453\pi\)
\(702\) 0 0
\(703\) 25048.0 1.34382
\(704\) 0 0
\(705\) −1900.00 −0.101501
\(706\) 0 0
\(707\) − 32292.0i − 1.71777i
\(708\) 0 0
\(709\) − 11702.0i − 0.619856i −0.950760 0.309928i \(-0.899695\pi\)
0.950760 0.309928i \(-0.100305\pi\)
\(710\) 0 0
\(711\) −84680.0 −4.46659
\(712\) 0 0
\(713\) 7896.00 0.414737
\(714\) 0 0
\(715\) − 480.000i − 0.0251063i
\(716\) 0 0
\(717\) 15600.0i 0.812542i
\(718\) 0 0
\(719\) 28008.0 1.45274 0.726371 0.687302i \(-0.241205\pi\)
0.726371 + 0.687302i \(0.241205\pi\)
\(720\) 0 0
\(721\) −11556.0 −0.596904
\(722\) 0 0
\(723\) 32180.0i 1.65531i
\(724\) 0 0
\(725\) − 3550.00i − 0.181853i
\(726\) 0 0
\(727\) −7682.00 −0.391898 −0.195949 0.980614i \(-0.562779\pi\)
−0.195949 + 0.980614i \(0.562779\pi\)
\(728\) 0 0
\(729\) −67717.0 −3.44038
\(730\) 0 0
\(731\) − 396.000i − 0.0200364i
\(732\) 0 0
\(733\) − 14270.0i − 0.719065i −0.933133 0.359532i \(-0.882936\pi\)
0.933133 0.359532i \(-0.117064\pi\)
\(734\) 0 0
\(735\) −950.000 −0.0476752
\(736\) 0 0
\(737\) 8864.00 0.443025
\(738\) 0 0
\(739\) − 29324.0i − 1.45968i −0.683620 0.729838i \(-0.739595\pi\)
0.683620 0.729838i \(-0.260405\pi\)
\(740\) 0 0
\(741\) − 7440.00i − 0.368846i
\(742\) 0 0
\(743\) 29258.0 1.44465 0.722323 0.691556i \(-0.243074\pi\)
0.722323 + 0.691556i \(0.243074\pi\)
\(744\) 0 0
\(745\) 8650.00 0.425385
\(746\) 0 0
\(747\) 46866.0i 2.29550i
\(748\) 0 0
\(749\) 15300.0i 0.746395i
\(750\) 0 0
\(751\) −19076.0 −0.926888 −0.463444 0.886126i \(-0.653387\pi\)
−0.463444 + 0.886126i \(0.653387\pi\)
\(752\) 0 0
\(753\) 6880.00 0.332963
\(754\) 0 0
\(755\) 6620.00i 0.319108i
\(756\) 0 0
\(757\) 22670.0i 1.08845i 0.838940 + 0.544224i \(0.183176\pi\)
−0.838940 + 0.544224i \(0.816824\pi\)
\(758\) 0 0
\(759\) −6720.00 −0.321371
\(760\) 0 0
\(761\) 23206.0 1.10541 0.552705 0.833377i \(-0.313596\pi\)
0.552705 + 0.833377i \(0.313596\pi\)
\(762\) 0 0
\(763\) − 11988.0i − 0.568800i
\(764\) 0 0
\(765\) 2190.00i 0.103503i
\(766\) 0 0
\(767\) 3384.00 0.159308
\(768\) 0 0
\(769\) −1854.00 −0.0869401 −0.0434701 0.999055i \(-0.513841\pi\)
−0.0434701 + 0.999055i \(0.513841\pi\)
\(770\) 0 0
\(771\) − 21700.0i − 1.01363i
\(772\) 0 0
\(773\) − 6474.00i − 0.301234i −0.988592 0.150617i \(-0.951874\pi\)
0.988592 0.150617i \(-0.0481260\pi\)
\(774\) 0 0
\(775\) −4700.00 −0.217844
\(776\) 0 0
\(777\) 36360.0 1.67877
\(778\) 0 0
\(779\) − 6696.00i − 0.307971i
\(780\) 0 0
\(781\) − 2240.00i − 0.102629i
\(782\) 0 0
\(783\) −65320.0 −2.98129
\(784\) 0 0
\(785\) −14730.0 −0.669728
\(786\) 0 0
\(787\) 20354.0i 0.921908i 0.887424 + 0.460954i \(0.152493\pi\)
−0.887424 + 0.460954i \(0.847507\pi\)
\(788\) 0 0
\(789\) − 22740.0i − 1.02607i
\(790\) 0 0
\(791\) 26028.0 1.16997
\(792\) 0 0
\(793\) −1572.00 −0.0703952
\(794\) 0 0
\(795\) − 36900.0i − 1.64617i
\(796\) 0 0
\(797\) − 1886.00i − 0.0838213i −0.999121 0.0419106i \(-0.986656\pi\)
0.999121 0.0419106i \(-0.0133445\pi\)
\(798\) 0 0
\(799\) 228.000 0.0100952
\(800\) 0 0
\(801\) −62342.0 −2.75000
\(802\) 0 0
\(803\) 14112.0i 0.620176i
\(804\) 0 0
\(805\) − 3780.00i − 0.165500i
\(806\) 0 0
\(807\) 71460.0 3.11711
\(808\) 0 0
\(809\) 9462.00 0.411207 0.205603 0.978635i \(-0.434084\pi\)
0.205603 + 0.978635i \(0.434084\pi\)
\(810\) 0 0
\(811\) 24512.0i 1.06132i 0.847584 + 0.530661i \(0.178056\pi\)
−0.847584 + 0.530661i \(0.821944\pi\)
\(812\) 0 0
\(813\) 26040.0i 1.12332i
\(814\) 0 0
\(815\) 10490.0 0.450857
\(816\) 0 0
\(817\) −8184.00 −0.350455
\(818\) 0 0
\(819\) − 7884.00i − 0.336373i
\(820\) 0 0
\(821\) − 36242.0i − 1.54063i −0.637666 0.770313i \(-0.720100\pi\)
0.637666 0.770313i \(-0.279900\pi\)
\(822\) 0 0
\(823\) −17718.0 −0.750438 −0.375219 0.926936i \(-0.622433\pi\)
−0.375219 + 0.926936i \(0.622433\pi\)
\(824\) 0 0
\(825\) 4000.00 0.168803
\(826\) 0 0
\(827\) 6726.00i 0.282812i 0.989952 + 0.141406i \(0.0451624\pi\)
−0.989952 + 0.141406i \(0.954838\pi\)
\(828\) 0 0
\(829\) 41722.0i 1.74797i 0.485955 + 0.873984i \(0.338472\pi\)
−0.485955 + 0.873984i \(0.661528\pi\)
\(830\) 0 0
\(831\) 51500.0 2.14984
\(832\) 0 0
\(833\) 114.000 0.00474174
\(834\) 0 0
\(835\) − 4330.00i − 0.179456i
\(836\) 0 0
\(837\) 86480.0i 3.57131i
\(838\) 0 0
\(839\) 16720.0 0.688008 0.344004 0.938968i \(-0.388217\pi\)
0.344004 + 0.938968i \(0.388217\pi\)
\(840\) 0 0
\(841\) 4225.00 0.173234
\(842\) 0 0
\(843\) 52700.0i 2.15313i
\(844\) 0 0
\(845\) 10805.0i 0.439886i
\(846\) 0 0
\(847\) −19350.0 −0.784975
\(848\) 0 0
\(849\) 34340.0 1.38816
\(850\) 0 0
\(851\) − 8484.00i − 0.341748i
\(852\) 0 0
\(853\) 33286.0i 1.33610i 0.744118 + 0.668049i \(0.232870\pi\)
−0.744118 + 0.668049i \(0.767130\pi\)
\(854\) 0 0
\(855\) 45260.0 1.81036
\(856\) 0 0
\(857\) −38978.0 −1.55363 −0.776816 0.629727i \(-0.783167\pi\)
−0.776816 + 0.629727i \(0.783167\pi\)
\(858\) 0 0
\(859\) − 1916.00i − 0.0761037i −0.999276 0.0380518i \(-0.987885\pi\)
0.999276 0.0380518i \(-0.0121152\pi\)
\(860\) 0 0
\(861\) − 9720.00i − 0.384735i
\(862\) 0 0
\(863\) 2374.00 0.0936407 0.0468203 0.998903i \(-0.485091\pi\)
0.0468203 + 0.998903i \(0.485091\pi\)
\(864\) 0 0
\(865\) 8390.00 0.329790
\(866\) 0 0
\(867\) 48770.0i 1.91040i
\(868\) 0 0
\(869\) − 18560.0i − 0.724517i
\(870\) 0 0
\(871\) 3324.00 0.129310
\(872\) 0 0
\(873\) 34894.0 1.35279
\(874\) 0 0
\(875\) 2250.00i 0.0869302i
\(876\) 0 0
\(877\) 32722.0i 1.25991i 0.776631 + 0.629956i \(0.216927\pi\)
−0.776631 + 0.629956i \(0.783073\pi\)
\(878\) 0 0
\(879\) 98780.0 3.79041
\(880\) 0 0
\(881\) 5390.00 0.206122 0.103061 0.994675i \(-0.467136\pi\)
0.103061 + 0.994675i \(0.467136\pi\)
\(882\) 0 0
\(883\) 43238.0i 1.64788i 0.566680 + 0.823938i \(0.308228\pi\)
−0.566680 + 0.823938i \(0.691772\pi\)
\(884\) 0 0
\(885\) 28200.0i 1.07111i
\(886\) 0 0
\(887\) −11010.0 −0.416775 −0.208388 0.978046i \(-0.566822\pi\)
−0.208388 + 0.978046i \(0.566822\pi\)
\(888\) 0 0
\(889\) −20772.0 −0.783656
\(890\) 0 0
\(891\) − 42064.0i − 1.58159i
\(892\) 0 0
\(893\) − 4712.00i − 0.176575i
\(894\) 0 0
\(895\) −8100.00 −0.302517
\(896\) 0 0
\(897\) −2520.00 −0.0938020
\(898\) 0 0
\(899\) 26696.0i 0.990391i
\(900\) 0 0
\(901\) 4428.00i 0.163727i
\(902\) 0 0
\(903\) −11880.0 −0.437809
\(904\) 0 0
\(905\) 12550.0 0.460968
\(906\) 0 0
\(907\) − 74.0000i − 0.00270907i −0.999999 0.00135454i \(-0.999569\pi\)
0.999999 0.00135454i \(-0.000431163\pi\)
\(908\) 0 0
\(909\) − 130962.i − 4.77859i
\(910\) 0 0
\(911\) −17460.0 −0.634990 −0.317495 0.948260i \(-0.602842\pi\)
−0.317495 + 0.948260i \(0.602842\pi\)
\(912\) 0 0
\(913\) −10272.0 −0.372348
\(914\) 0 0
\(915\) − 13100.0i − 0.473303i
\(916\) 0 0
\(917\) − 6624.00i − 0.238543i
\(918\) 0 0
\(919\) 17072.0 0.612789 0.306395 0.951905i \(-0.400877\pi\)
0.306395 + 0.951905i \(0.400877\pi\)
\(920\) 0 0
\(921\) −80540.0 −2.88152
\(922\) 0 0
\(923\) − 840.000i − 0.0299555i
\(924\) 0 0
\(925\) 5050.00i 0.179506i
\(926\) 0 0
\(927\) −46866.0 −1.66050
\(928\) 0 0
\(929\) −14826.0 −0.523601 −0.261800 0.965122i \(-0.584316\pi\)
−0.261800 + 0.965122i \(0.584316\pi\)
\(930\) 0 0
\(931\) − 2356.00i − 0.0829375i
\(932\) 0 0
\(933\) − 54920.0i − 1.92712i
\(934\) 0 0
\(935\) −480.000 −0.0167890
\(936\) 0 0
\(937\) −3354.00 −0.116937 −0.0584687 0.998289i \(-0.518622\pi\)
−0.0584687 + 0.998289i \(0.518622\pi\)
\(938\) 0 0
\(939\) − 4220.00i − 0.146661i
\(940\) 0 0
\(941\) − 15434.0i − 0.534680i −0.963602 0.267340i \(-0.913855\pi\)
0.963602 0.267340i \(-0.0861447\pi\)
\(942\) 0 0
\(943\) −2268.00 −0.0783205
\(944\) 0 0
\(945\) 41400.0 1.42512
\(946\) 0 0
\(947\) 9306.00i 0.319329i 0.987171 + 0.159664i \(0.0510412\pi\)
−0.987171 + 0.159664i \(0.948959\pi\)
\(948\) 0 0
\(949\) 5292.00i 0.181017i
\(950\) 0 0
\(951\) 61940.0 2.11203
\(952\) 0 0
\(953\) −12202.0 −0.414755 −0.207378 0.978261i \(-0.566493\pi\)
−0.207378 + 0.978261i \(0.566493\pi\)
\(954\) 0 0
\(955\) 1860.00i 0.0630243i
\(956\) 0 0
\(957\) − 22720.0i − 0.767433i
\(958\) 0 0
\(959\) −12060.0 −0.406087
\(960\) 0 0
\(961\) 5553.00 0.186399
\(962\) 0 0
\(963\) 62050.0i 2.07636i
\(964\) 0 0
\(965\) 14690.0i 0.490039i
\(966\) 0 0
\(967\) 17478.0 0.581235 0.290618 0.956839i \(-0.406139\pi\)
0.290618 + 0.956839i \(0.406139\pi\)
\(968\) 0 0
\(969\) −7440.00 −0.246653
\(970\) 0 0
\(971\) 10920.0i 0.360906i 0.983584 + 0.180453i \(0.0577563\pi\)
−0.983584 + 0.180453i \(0.942244\pi\)
\(972\) 0 0
\(973\) 10296.0i 0.339234i
\(974\) 0 0
\(975\) 1500.00 0.0492702
\(976\) 0 0
\(977\) 10834.0 0.354770 0.177385 0.984142i \(-0.443236\pi\)
0.177385 + 0.984142i \(0.443236\pi\)
\(978\) 0 0
\(979\) − 13664.0i − 0.446071i
\(980\) 0 0
\(981\) − 48618.0i − 1.58232i
\(982\) 0 0
\(983\) −36862.0 −1.19605 −0.598024 0.801478i \(-0.704047\pi\)
−0.598024 + 0.801478i \(0.704047\pi\)
\(984\) 0 0
\(985\) 11170.0 0.361326
\(986\) 0 0
\(987\) − 6840.00i − 0.220587i
\(988\) 0 0
\(989\) 2772.00i 0.0891248i
\(990\) 0 0
\(991\) −5380.00 −0.172453 −0.0862267 0.996276i \(-0.527481\pi\)
−0.0862267 + 0.996276i \(0.527481\pi\)
\(992\) 0 0
\(993\) 76880.0 2.45691
\(994\) 0 0
\(995\) − 15240.0i − 0.485568i
\(996\) 0 0
\(997\) − 31266.0i − 0.993184i −0.867984 0.496592i \(-0.834585\pi\)
0.867984 0.496592i \(-0.165415\pi\)
\(998\) 0 0
\(999\) 92920.0 2.94280
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 1280.4.d.b.641.1 2
4.3 odd 2 1280.4.d.o.641.2 2
8.3 odd 2 1280.4.d.o.641.1 2
8.5 even 2 inner 1280.4.d.b.641.2 2
16.3 odd 4 320.4.a.a.1.1 1
16.5 even 4 80.4.a.a.1.1 1
16.11 odd 4 40.4.a.c.1.1 1
16.13 even 4 320.4.a.n.1.1 1
48.5 odd 4 720.4.a.ba.1.1 1
48.11 even 4 360.4.a.i.1.1 1
80.19 odd 4 1600.4.a.ca.1.1 1
80.27 even 4 200.4.c.a.49.1 2
80.29 even 4 1600.4.a.a.1.1 1
80.37 odd 4 400.4.c.a.49.2 2
80.43 even 4 200.4.c.a.49.2 2
80.53 odd 4 400.4.c.a.49.1 2
80.59 odd 4 200.4.a.a.1.1 1
80.69 even 4 400.4.a.u.1.1 1
112.27 even 4 1960.4.a.a.1.1 1
240.59 even 4 1800.4.a.bd.1.1 1
240.107 odd 4 1800.4.f.n.649.1 2
240.203 odd 4 1800.4.f.n.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.c.1.1 1 16.11 odd 4
80.4.a.a.1.1 1 16.5 even 4
200.4.a.a.1.1 1 80.59 odd 4
200.4.c.a.49.1 2 80.27 even 4
200.4.c.a.49.2 2 80.43 even 4
320.4.a.a.1.1 1 16.3 odd 4
320.4.a.n.1.1 1 16.13 even 4
360.4.a.i.1.1 1 48.11 even 4
400.4.a.u.1.1 1 80.69 even 4
400.4.c.a.49.1 2 80.53 odd 4
400.4.c.a.49.2 2 80.37 odd 4
720.4.a.ba.1.1 1 48.5 odd 4
1280.4.d.b.641.1 2 1.1 even 1 trivial
1280.4.d.b.641.2 2 8.5 even 2 inner
1280.4.d.o.641.1 2 8.3 odd 2
1280.4.d.o.641.2 2 4.3 odd 2
1600.4.a.a.1.1 1 80.29 even 4
1600.4.a.ca.1.1 1 80.19 odd 4
1800.4.a.bd.1.1 1 240.59 even 4
1800.4.f.n.649.1 2 240.107 odd 4
1800.4.f.n.649.2 2 240.203 odd 4
1960.4.a.a.1.1 1 112.27 even 4