Properties

Label 200.6.c.c.49.1
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
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] = [200,6,Mod(49,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.c (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: \(32.0767639626\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.c.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} +O(q^{10})\) \(q-8.00000i q^{3} +108.000i q^{7} +179.000 q^{9} -604.000 q^{11} -306.000i q^{13} -930.000i q^{17} +1324.00 q^{19} +864.000 q^{21} -852.000i q^{23} -3376.00i q^{27} -5902.00 q^{29} -3320.00 q^{31} +4832.00i q^{33} -10774.0i q^{37} -2448.00 q^{39} -17958.0 q^{41} +9264.00i q^{43} +9796.00i q^{47} +5143.00 q^{49} -7440.00 q^{51} -31434.0i q^{53} -10592.0i q^{57} -33228.0 q^{59} -40210.0 q^{61} +19332.0i q^{63} -58864.0i q^{67} -6816.00 q^{69} -55312.0 q^{71} +27258.0i q^{73} -65232.0i q^{77} -31456.0 q^{79} +16489.0 q^{81} +24552.0i q^{83} +47216.0i q^{87} +90854.0 q^{89} +33048.0 q^{91} +26560.0i q^{93} -154706. i q^{97} -108116. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 358 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 358 q^{9} - 1208 q^{11} + 2648 q^{19} + 1728 q^{21} - 11804 q^{29} - 6640 q^{31} - 4896 q^{39} - 35916 q^{41} + 10286 q^{49} - 14880 q^{51} - 66456 q^{59} - 80420 q^{61} - 13632 q^{69} - 110624 q^{71} - 62912 q^{79} + 32978 q^{81} + 181708 q^{89} + 66096 q^{91} - 216232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 8.00000i − 0.513200i −0.966518 0.256600i \(-0.917398\pi\)
0.966518 0.256600i \(-0.0826023\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 108.000i 0.833065i 0.909121 + 0.416532i \(0.136755\pi\)
−0.909121 + 0.416532i \(0.863245\pi\)
\(8\) 0 0
\(9\) 179.000 0.736626
\(10\) 0 0
\(11\) −604.000 −1.50506 −0.752532 0.658555i \(-0.771168\pi\)
−0.752532 + 0.658555i \(0.771168\pi\)
\(12\) 0 0
\(13\) − 306.000i − 0.502184i −0.967963 0.251092i \(-0.919210\pi\)
0.967963 0.251092i \(-0.0807897\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 930.000i − 0.780478i −0.920714 0.390239i \(-0.872392\pi\)
0.920714 0.390239i \(-0.127608\pi\)
\(18\) 0 0
\(19\) 1324.00 0.841403 0.420701 0.907199i \(-0.361784\pi\)
0.420701 + 0.907199i \(0.361784\pi\)
\(20\) 0 0
\(21\) 864.000 0.427529
\(22\) 0 0
\(23\) − 852.000i − 0.335830i −0.985801 0.167915i \(-0.946297\pi\)
0.985801 0.167915i \(-0.0537035\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3376.00i − 0.891237i
\(28\) 0 0
\(29\) −5902.00 −1.30318 −0.651590 0.758572i \(-0.725898\pi\)
−0.651590 + 0.758572i \(0.725898\pi\)
\(30\) 0 0
\(31\) −3320.00 −0.620489 −0.310244 0.950657i \(-0.600411\pi\)
−0.310244 + 0.950657i \(0.600411\pi\)
\(32\) 0 0
\(33\) 4832.00i 0.772400i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10774.0i − 1.29382i −0.762568 0.646908i \(-0.776062\pi\)
0.762568 0.646908i \(-0.223938\pi\)
\(38\) 0 0
\(39\) −2448.00 −0.257721
\(40\) 0 0
\(41\) −17958.0 −1.66839 −0.834196 0.551467i \(-0.814068\pi\)
−0.834196 + 0.551467i \(0.814068\pi\)
\(42\) 0 0
\(43\) 9264.00i 0.764060i 0.924150 + 0.382030i \(0.124775\pi\)
−0.924150 + 0.382030i \(0.875225\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9796.00i 0.646851i 0.946254 + 0.323425i \(0.104834\pi\)
−0.946254 + 0.323425i \(0.895166\pi\)
\(48\) 0 0
\(49\) 5143.00 0.306003
\(50\) 0 0
\(51\) −7440.00 −0.400541
\(52\) 0 0
\(53\) − 31434.0i − 1.53713i −0.639773 0.768564i \(-0.720972\pi\)
0.639773 0.768564i \(-0.279028\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 10592.0i − 0.431808i
\(58\) 0 0
\(59\) −33228.0 −1.24272 −0.621361 0.783524i \(-0.713420\pi\)
−0.621361 + 0.783524i \(0.713420\pi\)
\(60\) 0 0
\(61\) −40210.0 −1.38360 −0.691798 0.722091i \(-0.743181\pi\)
−0.691798 + 0.722091i \(0.743181\pi\)
\(62\) 0 0
\(63\) 19332.0i 0.613657i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 58864.0i − 1.60200i −0.598664 0.801000i \(-0.704301\pi\)
0.598664 0.801000i \(-0.295699\pi\)
\(68\) 0 0
\(69\) −6816.00 −0.172348
\(70\) 0 0
\(71\) −55312.0 −1.30219 −0.651094 0.758997i \(-0.725690\pi\)
−0.651094 + 0.758997i \(0.725690\pi\)
\(72\) 0 0
\(73\) 27258.0i 0.598669i 0.954148 + 0.299335i \(0.0967647\pi\)
−0.954148 + 0.299335i \(0.903235\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 65232.0i − 1.25382i
\(78\) 0 0
\(79\) −31456.0 −0.567069 −0.283534 0.958962i \(-0.591507\pi\)
−0.283534 + 0.958962i \(0.591507\pi\)
\(80\) 0 0
\(81\) 16489.0 0.279243
\(82\) 0 0
\(83\) 24552.0i 0.391194i 0.980684 + 0.195597i \(0.0626644\pi\)
−0.980684 + 0.195597i \(0.937336\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 47216.0i 0.668792i
\(88\) 0 0
\(89\) 90854.0 1.21582 0.607910 0.794006i \(-0.292008\pi\)
0.607910 + 0.794006i \(0.292008\pi\)
\(90\) 0 0
\(91\) 33048.0 0.418352
\(92\) 0 0
\(93\) 26560.0i 0.318435i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 154706.i − 1.66947i −0.550654 0.834733i \(-0.685622\pi\)
0.550654 0.834733i \(-0.314378\pi\)
\(98\) 0 0
\(99\) −108116. −1.10867
\(100\) 0 0
\(101\) −72714.0 −0.709275 −0.354637 0.935004i \(-0.615396\pi\)
−0.354637 + 0.935004i \(0.615396\pi\)
\(102\) 0 0
\(103\) − 129396.i − 1.20179i −0.799329 0.600894i \(-0.794811\pi\)
0.799329 0.600894i \(-0.205189\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 206680.i 1.74518i 0.488458 + 0.872588i \(0.337560\pi\)
−0.488458 + 0.872588i \(0.662440\pi\)
\(108\) 0 0
\(109\) 70146.0 0.565505 0.282753 0.959193i \(-0.408752\pi\)
0.282753 + 0.959193i \(0.408752\pi\)
\(110\) 0 0
\(111\) −86192.0 −0.663987
\(112\) 0 0
\(113\) − 151854.i − 1.11874i −0.828917 0.559371i \(-0.811043\pi\)
0.828917 0.559371i \(-0.188957\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 54774.0i − 0.369922i
\(118\) 0 0
\(119\) 100440. 0.650189
\(120\) 0 0
\(121\) 203765. 1.26522
\(122\) 0 0
\(123\) 143664.i 0.856220i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 336596.i 1.85182i 0.377740 + 0.925912i \(0.376701\pi\)
−0.377740 + 0.925912i \(0.623299\pi\)
\(128\) 0 0
\(129\) 74112.0 0.392116
\(130\) 0 0
\(131\) 275308. 1.40165 0.700827 0.713332i \(-0.252815\pi\)
0.700827 + 0.713332i \(0.252815\pi\)
\(132\) 0 0
\(133\) 142992.i 0.700943i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 228502.i 1.04013i 0.854126 + 0.520066i \(0.174093\pi\)
−0.854126 + 0.520066i \(0.825907\pi\)
\(138\) 0 0
\(139\) −224284. −0.984603 −0.492302 0.870425i \(-0.663844\pi\)
−0.492302 + 0.870425i \(0.663844\pi\)
\(140\) 0 0
\(141\) 78368.0 0.331964
\(142\) 0 0
\(143\) 184824.i 0.755820i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 41144.0i − 0.157041i
\(148\) 0 0
\(149\) 183802. 0.678242 0.339121 0.940743i \(-0.389870\pi\)
0.339121 + 0.940743i \(0.389870\pi\)
\(150\) 0 0
\(151\) 296032. 1.05657 0.528283 0.849069i \(-0.322836\pi\)
0.528283 + 0.849069i \(0.322836\pi\)
\(152\) 0 0
\(153\) − 166470.i − 0.574920i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 134766.i − 0.436346i −0.975910 0.218173i \(-0.929990\pi\)
0.975910 0.218173i \(-0.0700097\pi\)
\(158\) 0 0
\(159\) −251472. −0.788854
\(160\) 0 0
\(161\) 92016.0 0.279768
\(162\) 0 0
\(163\) − 60248.0i − 0.177613i −0.996049 0.0888063i \(-0.971695\pi\)
0.996049 0.0888063i \(-0.0283052\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 62012.0i 0.172062i 0.996292 + 0.0860309i \(0.0274184\pi\)
−0.996292 + 0.0860309i \(0.972582\pi\)
\(168\) 0 0
\(169\) 277657. 0.747811
\(170\) 0 0
\(171\) 236996. 0.619799
\(172\) 0 0
\(173\) − 591682.i − 1.50305i −0.659705 0.751524i \(-0.729319\pi\)
0.659705 0.751524i \(-0.270681\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 265824.i 0.637766i
\(178\) 0 0
\(179\) 241404. 0.563134 0.281567 0.959542i \(-0.409146\pi\)
0.281567 + 0.959542i \(0.409146\pi\)
\(180\) 0 0
\(181\) 187622. 0.425684 0.212842 0.977087i \(-0.431728\pi\)
0.212842 + 0.977087i \(0.431728\pi\)
\(182\) 0 0
\(183\) 321680.i 0.710062i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 561720.i 1.17467i
\(188\) 0 0
\(189\) 364608. 0.742458
\(190\) 0 0
\(191\) 37560.0 0.0744976 0.0372488 0.999306i \(-0.488141\pi\)
0.0372488 + 0.999306i \(0.488141\pi\)
\(192\) 0 0
\(193\) 164434.i 0.317759i 0.987298 + 0.158880i \(0.0507882\pi\)
−0.987298 + 0.158880i \(0.949212\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 360518.i − 0.661853i −0.943657 0.330926i \(-0.892639\pi\)
0.943657 0.330926i \(-0.107361\pi\)
\(198\) 0 0
\(199\) 654168. 1.17100 0.585500 0.810673i \(-0.300898\pi\)
0.585500 + 0.810673i \(0.300898\pi\)
\(200\) 0 0
\(201\) −470912. −0.822147
\(202\) 0 0
\(203\) − 637416.i − 1.08563i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 152508.i − 0.247381i
\(208\) 0 0
\(209\) −799696. −1.26637
\(210\) 0 0
\(211\) −693156. −1.07183 −0.535914 0.844273i \(-0.680033\pi\)
−0.535914 + 0.844273i \(0.680033\pi\)
\(212\) 0 0
\(213\) 442496.i 0.668283i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 358560.i − 0.516907i
\(218\) 0 0
\(219\) 218064. 0.307237
\(220\) 0 0
\(221\) −284580. −0.391944
\(222\) 0 0
\(223\) 494756.i 0.666237i 0.942885 + 0.333119i \(0.108101\pi\)
−0.942885 + 0.333119i \(0.891899\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 907088.i − 1.16838i −0.811616 0.584191i \(-0.801412\pi\)
0.811616 0.584191i \(-0.198588\pi\)
\(228\) 0 0
\(229\) −1.08949e6 −1.37289 −0.686446 0.727181i \(-0.740830\pi\)
−0.686446 + 0.727181i \(0.740830\pi\)
\(230\) 0 0
\(231\) −521856. −0.643459
\(232\) 0 0
\(233\) 499706.i 0.603010i 0.953465 + 0.301505i \(0.0974891\pi\)
−0.953465 + 0.301505i \(0.902511\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 251648.i 0.291020i
\(238\) 0 0
\(239\) −1.62038e6 −1.83495 −0.917473 0.397799i \(-0.869774\pi\)
−0.917473 + 0.397799i \(0.869774\pi\)
\(240\) 0 0
\(241\) 1.00122e6 1.11042 0.555208 0.831711i \(-0.312638\pi\)
0.555208 + 0.831711i \(0.312638\pi\)
\(242\) 0 0
\(243\) − 952280.i − 1.03454i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 405144.i − 0.422539i
\(248\) 0 0
\(249\) 196416. 0.200761
\(250\) 0 0
\(251\) 368980. 0.369674 0.184837 0.982769i \(-0.440824\pi\)
0.184837 + 0.982769i \(0.440824\pi\)
\(252\) 0 0
\(253\) 514608.i 0.505447i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 279010.i − 0.263504i −0.991283 0.131752i \(-0.957940\pi\)
0.991283 0.131752i \(-0.0420602\pi\)
\(258\) 0 0
\(259\) 1.16359e6 1.07783
\(260\) 0 0
\(261\) −1.05646e6 −0.959955
\(262\) 0 0
\(263\) 811740.i 0.723648i 0.932246 + 0.361824i \(0.117846\pi\)
−0.932246 + 0.361824i \(0.882154\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 726832.i − 0.623959i
\(268\) 0 0
\(269\) −353214. −0.297617 −0.148808 0.988866i \(-0.547544\pi\)
−0.148808 + 0.988866i \(0.547544\pi\)
\(270\) 0 0
\(271\) −1.71622e6 −1.41954 −0.709772 0.704432i \(-0.751202\pi\)
−0.709772 + 0.704432i \(0.751202\pi\)
\(272\) 0 0
\(273\) − 264384.i − 0.214698i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 245882.i 0.192543i 0.995355 + 0.0962714i \(0.0306917\pi\)
−0.995355 + 0.0962714i \(0.969308\pi\)
\(278\) 0 0
\(279\) −594280. −0.457068
\(280\) 0 0
\(281\) −1.67618e6 −1.26635 −0.633177 0.774007i \(-0.718250\pi\)
−0.633177 + 0.774007i \(0.718250\pi\)
\(282\) 0 0
\(283\) − 1.25882e6i − 0.934321i −0.884173 0.467161i \(-0.845277\pi\)
0.884173 0.467161i \(-0.154723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 1.93946e6i − 1.38988i
\(288\) 0 0
\(289\) 554957. 0.390854
\(290\) 0 0
\(291\) −1.23765e6 −0.856771
\(292\) 0 0
\(293\) 719158.i 0.489390i 0.969600 + 0.244695i \(0.0786879\pi\)
−0.969600 + 0.244695i \(0.921312\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.03910e6i 1.34137i
\(298\) 0 0
\(299\) −260712. −0.168649
\(300\) 0 0
\(301\) −1.00051e6 −0.636511
\(302\) 0 0
\(303\) 581712.i 0.364000i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.86013e6i − 1.12641i −0.826317 0.563206i \(-0.809568\pi\)
0.826317 0.563206i \(-0.190432\pi\)
\(308\) 0 0
\(309\) −1.03517e6 −0.616758
\(310\) 0 0
\(311\) 278384. 0.163209 0.0816043 0.996665i \(-0.473996\pi\)
0.0816043 + 0.996665i \(0.473996\pi\)
\(312\) 0 0
\(313\) − 474182.i − 0.273580i −0.990600 0.136790i \(-0.956321\pi\)
0.990600 0.136790i \(-0.0436785\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.83738e6i 1.02695i 0.858104 + 0.513476i \(0.171643\pi\)
−0.858104 + 0.513476i \(0.828357\pi\)
\(318\) 0 0
\(319\) 3.56481e6 1.96137
\(320\) 0 0
\(321\) 1.65344e6 0.895624
\(322\) 0 0
\(323\) − 1.23132e6i − 0.656696i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 561168.i − 0.290217i
\(328\) 0 0
\(329\) −1.05797e6 −0.538868
\(330\) 0 0
\(331\) 2.99743e6 1.50376 0.751880 0.659299i \(-0.229147\pi\)
0.751880 + 0.659299i \(0.229147\pi\)
\(332\) 0 0
\(333\) − 1.92855e6i − 0.953058i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.87531e6i − 0.899496i −0.893155 0.449748i \(-0.851514\pi\)
0.893155 0.449748i \(-0.148486\pi\)
\(338\) 0 0
\(339\) −1.21483e6 −0.574139
\(340\) 0 0
\(341\) 2.00528e6 0.933876
\(342\) 0 0
\(343\) 2.37060e6i 1.08799i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 180312.i − 0.0803898i −0.999192 0.0401949i \(-0.987202\pi\)
0.999192 0.0401949i \(-0.0127979\pi\)
\(348\) 0 0
\(349\) 87058.0 0.0382600 0.0191300 0.999817i \(-0.493910\pi\)
0.0191300 + 0.999817i \(0.493910\pi\)
\(350\) 0 0
\(351\) −1.03306e6 −0.447565
\(352\) 0 0
\(353\) 2.65901e6i 1.13575i 0.823114 + 0.567876i \(0.192235\pi\)
−0.823114 + 0.567876i \(0.807765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 803520.i − 0.333677i
\(358\) 0 0
\(359\) 2.14937e6 0.880186 0.440093 0.897952i \(-0.354945\pi\)
0.440093 + 0.897952i \(0.354945\pi\)
\(360\) 0 0
\(361\) −723123. −0.292041
\(362\) 0 0
\(363\) − 1.63012e6i − 0.649311i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.08258e6i 1.19467i 0.801991 + 0.597337i \(0.203774\pi\)
−0.801991 + 0.597337i \(0.796226\pi\)
\(368\) 0 0
\(369\) −3.21448e6 −1.22898
\(370\) 0 0
\(371\) 3.39487e6 1.28053
\(372\) 0 0
\(373\) 2.28727e6i 0.851227i 0.904905 + 0.425613i \(0.139942\pi\)
−0.904905 + 0.425613i \(0.860058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.80601e6i 0.654436i
\(378\) 0 0
\(379\) 1.30154e6 0.465435 0.232718 0.972544i \(-0.425238\pi\)
0.232718 + 0.972544i \(0.425238\pi\)
\(380\) 0 0
\(381\) 2.69277e6 0.950356
\(382\) 0 0
\(383\) − 2.03276e6i − 0.708093i −0.935228 0.354046i \(-0.884806\pi\)
0.935228 0.354046i \(-0.115194\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.65826e6i 0.562826i
\(388\) 0 0
\(389\) −94230.0 −0.0315730 −0.0157865 0.999875i \(-0.505025\pi\)
−0.0157865 + 0.999875i \(0.505025\pi\)
\(390\) 0 0
\(391\) −792360. −0.262108
\(392\) 0 0
\(393\) − 2.20246e6i − 0.719329i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.55551e6i 1.76908i 0.466465 + 0.884540i \(0.345527\pi\)
−0.466465 + 0.884540i \(0.654473\pi\)
\(398\) 0 0
\(399\) 1.14394e6 0.359724
\(400\) 0 0
\(401\) −784814. −0.243728 −0.121864 0.992547i \(-0.538887\pi\)
−0.121864 + 0.992547i \(0.538887\pi\)
\(402\) 0 0
\(403\) 1.01592e6i 0.311600i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.50750e6i 1.94728i
\(408\) 0 0
\(409\) 4.59401e6 1.35795 0.678974 0.734162i \(-0.262425\pi\)
0.678974 + 0.734162i \(0.262425\pi\)
\(410\) 0 0
\(411\) 1.82802e6 0.533796
\(412\) 0 0
\(413\) − 3.58862e6i − 1.03527i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.79427e6i 0.505299i
\(418\) 0 0
\(419\) −1.41301e6 −0.393198 −0.196599 0.980484i \(-0.562990\pi\)
−0.196599 + 0.980484i \(0.562990\pi\)
\(420\) 0 0
\(421\) 5.94556e6 1.63489 0.817443 0.576010i \(-0.195391\pi\)
0.817443 + 0.576010i \(0.195391\pi\)
\(422\) 0 0
\(423\) 1.75348e6i 0.476487i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.34268e6i − 1.15263i
\(428\) 0 0
\(429\) 1.47859e6 0.387887
\(430\) 0 0
\(431\) −6.48114e6 −1.68058 −0.840289 0.542139i \(-0.817615\pi\)
−0.840289 + 0.542139i \(0.817615\pi\)
\(432\) 0 0
\(433\) 4.05597e6i 1.03962i 0.854282 + 0.519810i \(0.173997\pi\)
−0.854282 + 0.519810i \(0.826003\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.12805e6i − 0.282569i
\(438\) 0 0
\(439\) 1.21450e6 0.300772 0.150386 0.988627i \(-0.451948\pi\)
0.150386 + 0.988627i \(0.451948\pi\)
\(440\) 0 0
\(441\) 920597. 0.225410
\(442\) 0 0
\(443\) − 5.53154e6i − 1.33917i −0.742734 0.669586i \(-0.766472\pi\)
0.742734 0.669586i \(-0.233528\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1.47042e6i − 0.348074i
\(448\) 0 0
\(449\) −2.20111e6 −0.515258 −0.257629 0.966244i \(-0.582941\pi\)
−0.257629 + 0.966244i \(0.582941\pi\)
\(450\) 0 0
\(451\) 1.08466e7 2.51104
\(452\) 0 0
\(453\) − 2.36826e6i − 0.542229i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.29835e6i − 0.738764i −0.929278 0.369382i \(-0.879569\pi\)
0.929278 0.369382i \(-0.120431\pi\)
\(458\) 0 0
\(459\) −3.13968e6 −0.695591
\(460\) 0 0
\(461\) −3.94266e6 −0.864046 −0.432023 0.901863i \(-0.642200\pi\)
−0.432023 + 0.901863i \(0.642200\pi\)
\(462\) 0 0
\(463\) 8.82040e6i 1.91221i 0.293021 + 0.956106i \(0.405339\pi\)
−0.293021 + 0.956106i \(0.594661\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.28709e6i 0.273096i 0.990633 + 0.136548i \(0.0436009\pi\)
−0.990633 + 0.136548i \(0.956399\pi\)
\(468\) 0 0
\(469\) 6.35731e6 1.33457
\(470\) 0 0
\(471\) −1.07813e6 −0.223933
\(472\) 0 0
\(473\) − 5.59546e6i − 1.14996i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 5.62669e6i − 1.13229i
\(478\) 0 0
\(479\) −6.51179e6 −1.29677 −0.648383 0.761314i \(-0.724555\pi\)
−0.648383 + 0.761314i \(0.724555\pi\)
\(480\) 0 0
\(481\) −3.29684e6 −0.649734
\(482\) 0 0
\(483\) − 736128.i − 0.143577i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.79523e6i 1.10726i 0.832764 + 0.553628i \(0.186757\pi\)
−0.832764 + 0.553628i \(0.813243\pi\)
\(488\) 0 0
\(489\) −481984. −0.0911508
\(490\) 0 0
\(491\) 990276. 0.185376 0.0926878 0.995695i \(-0.470454\pi\)
0.0926878 + 0.995695i \(0.470454\pi\)
\(492\) 0 0
\(493\) 5.48886e6i 1.01710i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.97370e6i − 1.08481i
\(498\) 0 0
\(499\) −2.91500e6 −0.524067 −0.262033 0.965059i \(-0.584393\pi\)
−0.262033 + 0.965059i \(0.584393\pi\)
\(500\) 0 0
\(501\) 496096. 0.0883022
\(502\) 0 0
\(503\) 2.47872e6i 0.436824i 0.975857 + 0.218412i \(0.0700877\pi\)
−0.975857 + 0.218412i \(0.929912\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.22126e6i − 0.383777i
\(508\) 0 0
\(509\) 6.75807e6 1.15619 0.578093 0.815971i \(-0.303797\pi\)
0.578093 + 0.815971i \(0.303797\pi\)
\(510\) 0 0
\(511\) −2.94386e6 −0.498730
\(512\) 0 0
\(513\) − 4.46982e6i − 0.749889i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.91678e6i − 0.973552i
\(518\) 0 0
\(519\) −4.73346e6 −0.771365
\(520\) 0 0
\(521\) −6.33903e6 −1.02312 −0.511562 0.859246i \(-0.670933\pi\)
−0.511562 + 0.859246i \(0.670933\pi\)
\(522\) 0 0
\(523\) 231920.i 0.0370752i 0.999828 + 0.0185376i \(0.00590105\pi\)
−0.999828 + 0.0185376i \(0.994099\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.08760e6i 0.484278i
\(528\) 0 0
\(529\) 5.71044e6 0.887218
\(530\) 0 0
\(531\) −5.94781e6 −0.915421
\(532\) 0 0
\(533\) 5.49515e6i 0.837841i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 1.93123e6i − 0.289001i
\(538\) 0 0
\(539\) −3.10637e6 −0.460555
\(540\) 0 0
\(541\) −9.44440e6 −1.38733 −0.693667 0.720295i \(-0.744006\pi\)
−0.693667 + 0.720295i \(0.744006\pi\)
\(542\) 0 0
\(543\) − 1.50098e6i − 0.218461i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3.10162e6i − 0.443220i −0.975135 0.221610i \(-0.928869\pi\)
0.975135 0.221610i \(-0.0711312\pi\)
\(548\) 0 0
\(549\) −7.19759e6 −1.01919
\(550\) 0 0
\(551\) −7.81425e6 −1.09650
\(552\) 0 0
\(553\) − 3.39725e6i − 0.472405i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.22330e6i 0.167068i 0.996505 + 0.0835342i \(0.0266208\pi\)
−0.996505 + 0.0835342i \(0.973379\pi\)
\(558\) 0 0
\(559\) 2.83478e6 0.383699
\(560\) 0 0
\(561\) 4.49376e6 0.602841
\(562\) 0 0
\(563\) − 1.40896e7i − 1.87339i −0.350151 0.936693i \(-0.613870\pi\)
0.350151 0.936693i \(-0.386130\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.78081e6i 0.232627i
\(568\) 0 0
\(569\) −1.48468e6 −0.192244 −0.0961220 0.995370i \(-0.530644\pi\)
−0.0961220 + 0.995370i \(0.530644\pi\)
\(570\) 0 0
\(571\) −2.86470e6 −0.367696 −0.183848 0.982955i \(-0.558855\pi\)
−0.183848 + 0.982955i \(0.558855\pi\)
\(572\) 0 0
\(573\) − 300480.i − 0.0382322i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 4.21728e6i − 0.527343i −0.964613 0.263671i \(-0.915067\pi\)
0.964613 0.263671i \(-0.0849335\pi\)
\(578\) 0 0
\(579\) 1.31547e6 0.163074
\(580\) 0 0
\(581\) −2.65162e6 −0.325889
\(582\) 0 0
\(583\) 1.89861e7i 2.31348i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.01047e6i 0.240826i 0.992724 + 0.120413i \(0.0384218\pi\)
−0.992724 + 0.120413i \(0.961578\pi\)
\(588\) 0 0
\(589\) −4.39568e6 −0.522081
\(590\) 0 0
\(591\) −2.88414e6 −0.339663
\(592\) 0 0
\(593\) 7.33691e6i 0.856795i 0.903590 + 0.428397i \(0.140922\pi\)
−0.903590 + 0.428397i \(0.859078\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 5.23334e6i − 0.600957i
\(598\) 0 0
\(599\) −1.14884e6 −0.130826 −0.0654128 0.997858i \(-0.520836\pi\)
−0.0654128 + 0.997858i \(0.520836\pi\)
\(600\) 0 0
\(601\) 1.16409e7 1.31462 0.657312 0.753618i \(-0.271693\pi\)
0.657312 + 0.753618i \(0.271693\pi\)
\(602\) 0 0
\(603\) − 1.05367e7i − 1.18007i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 155540.i 0.0171345i 0.999963 + 0.00856723i \(0.00272707\pi\)
−0.999963 + 0.00856723i \(0.997273\pi\)
\(608\) 0 0
\(609\) −5.09933e6 −0.557147
\(610\) 0 0
\(611\) 2.99758e6 0.324838
\(612\) 0 0
\(613\) − 1.18137e7i − 1.26980i −0.772595 0.634899i \(-0.781042\pi\)
0.772595 0.634899i \(-0.218958\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.42252e6i − 0.679192i −0.940571 0.339596i \(-0.889710\pi\)
0.940571 0.339596i \(-0.110290\pi\)
\(618\) 0 0
\(619\) −3.85252e6 −0.404128 −0.202064 0.979372i \(-0.564765\pi\)
−0.202064 + 0.979372i \(0.564765\pi\)
\(620\) 0 0
\(621\) −2.87635e6 −0.299304
\(622\) 0 0
\(623\) 9.81223e6i 1.01286i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.39757e6i 0.649899i
\(628\) 0 0
\(629\) −1.00198e7 −1.00980
\(630\) 0 0
\(631\) −6.75136e6 −0.675022 −0.337511 0.941322i \(-0.609585\pi\)
−0.337511 + 0.941322i \(0.609585\pi\)
\(632\) 0 0
\(633\) 5.54525e6i 0.550062i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.57376e6i − 0.153670i
\(638\) 0 0
\(639\) −9.90085e6 −0.959224
\(640\) 0 0
\(641\) −7.35493e6 −0.707022 −0.353511 0.935430i \(-0.615012\pi\)
−0.353511 + 0.935430i \(0.615012\pi\)
\(642\) 0 0
\(643\) 1.59694e7i 1.52322i 0.648036 + 0.761610i \(0.275591\pi\)
−0.648036 + 0.761610i \(0.724409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.72667e7i − 1.62162i −0.585311 0.810809i \(-0.699028\pi\)
0.585311 0.810809i \(-0.300972\pi\)
\(648\) 0 0
\(649\) 2.00697e7 1.87038
\(650\) 0 0
\(651\) −2.86848e6 −0.265277
\(652\) 0 0
\(653\) − 1.36251e6i − 0.125043i −0.998044 0.0625213i \(-0.980086\pi\)
0.998044 0.0625213i \(-0.0199141\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.87918e6i 0.440995i
\(658\) 0 0
\(659\) 8.81808e6 0.790971 0.395485 0.918472i \(-0.370576\pi\)
0.395485 + 0.918472i \(0.370576\pi\)
\(660\) 0 0
\(661\) −1.52035e6 −0.135344 −0.0676720 0.997708i \(-0.521557\pi\)
−0.0676720 + 0.997708i \(0.521557\pi\)
\(662\) 0 0
\(663\) 2.27664e6i 0.201146i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.02850e6i 0.437647i
\(668\) 0 0
\(669\) 3.95805e6 0.341913
\(670\) 0 0
\(671\) 2.42868e7 2.08240
\(672\) 0 0
\(673\) − 315086.i − 0.0268158i −0.999910 0.0134079i \(-0.995732\pi\)
0.999910 0.0134079i \(-0.00426800\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.74092e6i 0.145985i 0.997332 + 0.0729924i \(0.0232549\pi\)
−0.997332 + 0.0729924i \(0.976745\pi\)
\(678\) 0 0
\(679\) 1.67082e7 1.39077
\(680\) 0 0
\(681\) −7.25670e6 −0.599614
\(682\) 0 0
\(683\) − 1.98935e7i − 1.63177i −0.578214 0.815885i \(-0.696250\pi\)
0.578214 0.815885i \(-0.303750\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.71595e6i 0.704568i
\(688\) 0 0
\(689\) −9.61880e6 −0.771921
\(690\) 0 0
\(691\) 2.01519e7 1.60554 0.802770 0.596289i \(-0.203359\pi\)
0.802770 + 0.596289i \(0.203359\pi\)
\(692\) 0 0
\(693\) − 1.16765e7i − 0.923593i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.67009e7i 1.30214i
\(698\) 0 0
\(699\) 3.99765e6 0.309465
\(700\) 0 0
\(701\) 8.10766e6 0.623161 0.311581 0.950220i \(-0.399142\pi\)
0.311581 + 0.950220i \(0.399142\pi\)
\(702\) 0 0
\(703\) − 1.42648e7i − 1.08862i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 7.85311e6i − 0.590872i
\(708\) 0 0
\(709\) 1.35613e7 1.01317 0.506587 0.862189i \(-0.330907\pi\)
0.506587 + 0.862189i \(0.330907\pi\)
\(710\) 0 0
\(711\) −5.63062e6 −0.417717
\(712\) 0 0
\(713\) 2.82864e6i 0.208379i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.29631e7i 0.941695i
\(718\) 0 0
\(719\) −4.28314e6 −0.308987 −0.154493 0.987994i \(-0.549375\pi\)
−0.154493 + 0.987994i \(0.549375\pi\)
\(720\) 0 0
\(721\) 1.39748e7 1.00117
\(722\) 0 0
\(723\) − 8.00974e6i − 0.569866i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.12084e7i − 0.786515i −0.919428 0.393258i \(-0.871348\pi\)
0.919428 0.393258i \(-0.128652\pi\)
\(728\) 0 0
\(729\) −3.61141e6 −0.251686
\(730\) 0 0
\(731\) 8.61552e6 0.596332
\(732\) 0 0
\(733\) 4.70549e6i 0.323478i 0.986834 + 0.161739i \(0.0517102\pi\)
−0.986834 + 0.161739i \(0.948290\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.55539e7i 2.41112i
\(738\) 0 0
\(739\) 2.31099e7 1.55663 0.778317 0.627872i \(-0.216074\pi\)
0.778317 + 0.627872i \(0.216074\pi\)
\(740\) 0 0
\(741\) −3.24115e6 −0.216847
\(742\) 0 0
\(743\) 5.75294e6i 0.382312i 0.981560 + 0.191156i \(0.0612236\pi\)
−0.981560 + 0.191156i \(0.938776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.39481e6i 0.288163i
\(748\) 0 0
\(749\) −2.23214e7 −1.45384
\(750\) 0 0
\(751\) −1.92424e7 −1.24497 −0.622485 0.782632i \(-0.713877\pi\)
−0.622485 + 0.782632i \(0.713877\pi\)
\(752\) 0 0
\(753\) − 2.95184e6i − 0.189717i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 4.49210e6i − 0.284911i −0.989801 0.142456i \(-0.954500\pi\)
0.989801 0.142456i \(-0.0454999\pi\)
\(758\) 0 0
\(759\) 4.11686e6 0.259395
\(760\) 0 0
\(761\) 7.33500e6 0.459133 0.229567 0.973293i \(-0.426269\pi\)
0.229567 + 0.973293i \(0.426269\pi\)
\(762\) 0 0
\(763\) 7.57577e6i 0.471102i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.01678e7i 0.624076i
\(768\) 0 0
\(769\) 5.85526e6 0.357051 0.178526 0.983935i \(-0.442867\pi\)
0.178526 + 0.983935i \(0.442867\pi\)
\(770\) 0 0
\(771\) −2.23208e6 −0.135230
\(772\) 0 0
\(773\) 1.34558e7i 0.809952i 0.914327 + 0.404976i \(0.132720\pi\)
−0.914327 + 0.404976i \(0.867280\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.30874e6i − 0.553144i
\(778\) 0 0
\(779\) −2.37764e7 −1.40379
\(780\) 0 0
\(781\) 3.34084e7 1.95988
\(782\) 0 0
\(783\) 1.99252e7i 1.16144i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.00706e7i − 0.579587i −0.957089 0.289794i \(-0.906413\pi\)
0.957089 0.289794i \(-0.0935867\pi\)
\(788\) 0 0
\(789\) 6.49392e6 0.371377
\(790\) 0 0
\(791\) 1.64002e7 0.931985
\(792\) 0 0
\(793\) 1.23043e7i 0.694820i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.18844e7i 0.662723i 0.943504 + 0.331362i \(0.107508\pi\)
−0.943504 + 0.331362i \(0.892492\pi\)
\(798\) 0 0
\(799\) 9.11028e6 0.504853
\(800\) 0 0
\(801\) 1.62629e7 0.895604
\(802\) 0 0
\(803\) − 1.64638e7i − 0.901036i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.82571e6i 0.152737i
\(808\) 0 0
\(809\) −1.06053e7 −0.569705 −0.284852 0.958571i \(-0.591945\pi\)
−0.284852 + 0.958571i \(0.591945\pi\)
\(810\) 0 0
\(811\) −1.38944e6 −0.0741799 −0.0370900 0.999312i \(-0.511809\pi\)
−0.0370900 + 0.999312i \(0.511809\pi\)
\(812\) 0 0
\(813\) 1.37297e7i 0.728510i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.22655e7i 0.642882i
\(818\) 0 0
\(819\) 5.91559e6 0.308169
\(820\) 0 0
\(821\) −1.12661e7 −0.583334 −0.291667 0.956520i \(-0.594210\pi\)
−0.291667 + 0.956520i \(0.594210\pi\)
\(822\) 0 0
\(823\) − 2.77093e7i − 1.42602i −0.701152 0.713011i \(-0.747331\pi\)
0.701152 0.713011i \(-0.252669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.23662e7i − 0.628740i −0.949300 0.314370i \(-0.898207\pi\)
0.949300 0.314370i \(-0.101793\pi\)
\(828\) 0 0
\(829\) −1.23182e7 −0.622530 −0.311265 0.950323i \(-0.600753\pi\)
−0.311265 + 0.950323i \(0.600753\pi\)
\(830\) 0 0
\(831\) 1.96706e6 0.0988130
\(832\) 0 0
\(833\) − 4.78299e6i − 0.238829i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.12083e7i 0.553002i
\(838\) 0 0
\(839\) 1.17277e7 0.575183 0.287592 0.957753i \(-0.407145\pi\)
0.287592 + 0.957753i \(0.407145\pi\)
\(840\) 0 0
\(841\) 1.43225e7 0.698277
\(842\) 0 0
\(843\) 1.34095e7i 0.649894i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.20066e7i 1.05401i
\(848\) 0 0
\(849\) −1.00705e7 −0.479494
\(850\) 0 0
\(851\) −9.17945e6 −0.434503
\(852\) 0 0
\(853\) 1.57059e7i 0.739077i 0.929215 + 0.369538i \(0.120484\pi\)
−0.929215 + 0.369538i \(0.879516\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.52390e7i − 1.17387i −0.809634 0.586935i \(-0.800334\pi\)
0.809634 0.586935i \(-0.199666\pi\)
\(858\) 0 0
\(859\) −3.66248e6 −0.169353 −0.0846763 0.996409i \(-0.526986\pi\)
−0.0846763 + 0.996409i \(0.526986\pi\)
\(860\) 0 0
\(861\) −1.55157e7 −0.713286
\(862\) 0 0
\(863\) − 4.17938e7i − 1.91023i −0.296243 0.955113i \(-0.595734\pi\)
0.296243 0.955113i \(-0.404266\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4.43966e6i − 0.200586i
\(868\) 0 0
\(869\) 1.89994e7 0.853475
\(870\) 0 0
\(871\) −1.80124e7 −0.804500
\(872\) 0 0
\(873\) − 2.76924e7i − 1.22977i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.08990e7i − 0.478505i −0.970957 0.239253i \(-0.923098\pi\)
0.970957 0.239253i \(-0.0769024\pi\)
\(878\) 0 0
\(879\) 5.75326e6 0.251155
\(880\) 0 0
\(881\) −3.04336e7 −1.32103 −0.660517 0.750811i \(-0.729663\pi\)
−0.660517 + 0.750811i \(0.729663\pi\)
\(882\) 0 0
\(883\) 6.09028e6i 0.262867i 0.991325 + 0.131433i \(0.0419579\pi\)
−0.991325 + 0.131433i \(0.958042\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.77908e7i − 1.18602i −0.805195 0.593010i \(-0.797940\pi\)
0.805195 0.593010i \(-0.202060\pi\)
\(888\) 0 0
\(889\) −3.63524e7 −1.54269
\(890\) 0 0
\(891\) −9.95936e6 −0.420278
\(892\) 0 0
\(893\) 1.29699e7i 0.544262i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.08570e6i 0.0865506i
\(898\) 0 0
\(899\) 1.95946e7 0.808608
\(900\) 0 0
\(901\) −2.92336e7 −1.19969
\(902\) 0 0
\(903\) 8.00410e6i 0.326658i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 3.71510e7i − 1.49952i −0.661709 0.749761i \(-0.730169\pi\)
0.661709 0.749761i \(-0.269831\pi\)
\(908\) 0 0
\(909\) −1.30158e7 −0.522470
\(910\) 0 0
\(911\) −7.85959e6 −0.313765 −0.156882 0.987617i \(-0.550144\pi\)
−0.156882 + 0.987617i \(0.550144\pi\)
\(912\) 0 0
\(913\) − 1.48294e7i − 0.588772i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.97333e7i 1.16767i
\(918\) 0 0
\(919\) 1.62693e7 0.635448 0.317724 0.948183i \(-0.397081\pi\)
0.317724 + 0.948183i \(0.397081\pi\)
\(920\) 0 0
\(921\) −1.48810e7 −0.578074
\(922\) 0 0
\(923\) 1.69255e7i 0.653938i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.31619e7i − 0.885268i
\(928\) 0 0
\(929\) 3.69365e7 1.40416 0.702079 0.712099i \(-0.252255\pi\)
0.702079 + 0.712099i \(0.252255\pi\)
\(930\) 0 0
\(931\) 6.80933e6 0.257472
\(932\) 0 0
\(933\) − 2.22707e6i − 0.0837587i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 4.89705e6i − 0.182216i −0.995841 0.0911078i \(-0.970959\pi\)
0.995841 0.0911078i \(-0.0290408\pi\)
\(938\) 0 0
\(939\) −3.79346e6 −0.140401
\(940\) 0 0
\(941\) −6.83943e6 −0.251794 −0.125897 0.992043i \(-0.540181\pi\)
−0.125897 + 0.992043i \(0.540181\pi\)
\(942\) 0 0
\(943\) 1.53002e7i 0.560297i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.03790e7i − 0.376082i −0.982161 0.188041i \(-0.939786\pi\)
0.982161 0.188041i \(-0.0602137\pi\)
\(948\) 0 0
\(949\) 8.34095e6 0.300642
\(950\) 0 0
\(951\) 1.46990e7 0.527032
\(952\) 0 0
\(953\) 2.59587e7i 0.925873i 0.886391 + 0.462937i \(0.153204\pi\)
−0.886391 + 0.462937i \(0.846796\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 2.85185e7i − 1.00658i
\(958\) 0 0
\(959\) −2.46782e7 −0.866497
\(960\) 0 0
\(961\) −1.76068e7 −0.614994
\(962\) 0 0
\(963\) 3.69957e7i 1.28554i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.92120e7i 1.34851i 0.738501 + 0.674253i \(0.235534\pi\)
−0.738501 + 0.674253i \(0.764466\pi\)
\(968\) 0 0
\(969\) −9.85056e6 −0.337017
\(970\) 0 0
\(971\) −1.06876e7 −0.363774 −0.181887 0.983319i \(-0.558221\pi\)
−0.181887 + 0.983319i \(0.558221\pi\)
\(972\) 0 0
\(973\) − 2.42227e7i − 0.820238i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 2.77266e7i − 0.929308i −0.885492 0.464654i \(-0.846179\pi\)
0.885492 0.464654i \(-0.153821\pi\)
\(978\) 0 0
\(979\) −5.48758e7 −1.82989
\(980\) 0 0
\(981\) 1.25561e7 0.416566
\(982\) 0 0
\(983\) − 9.49272e6i − 0.313334i −0.987652 0.156667i \(-0.949925\pi\)
0.987652 0.156667i \(-0.0500749\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.46374e6i 0.276547i
\(988\) 0 0
\(989\) 7.89293e6 0.256595
\(990\) 0 0
\(991\) −2.03243e7 −0.657403 −0.328702 0.944434i \(-0.606611\pi\)
−0.328702 + 0.944434i \(0.606611\pi\)
\(992\) 0 0
\(993\) − 2.39794e7i − 0.771730i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 4.70508e7i − 1.49909i −0.661951 0.749547i \(-0.730271\pi\)
0.661951 0.749547i \(-0.269729\pi\)
\(998\) 0 0
\(999\) −3.63730e7 −1.15310
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 200.6.c.c.49.1 2
4.3 odd 2 400.6.c.h.49.2 2
5.2 odd 4 40.6.a.b.1.1 1
5.3 odd 4 200.6.a.c.1.1 1
5.4 even 2 inner 200.6.c.c.49.2 2
15.2 even 4 360.6.a.b.1.1 1
20.3 even 4 400.6.a.f.1.1 1
20.7 even 4 80.6.a.f.1.1 1
20.19 odd 2 400.6.c.h.49.1 2
40.27 even 4 320.6.a.e.1.1 1
40.37 odd 4 320.6.a.l.1.1 1
60.47 odd 4 720.6.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.a.b.1.1 1 5.2 odd 4
80.6.a.f.1.1 1 20.7 even 4
200.6.a.c.1.1 1 5.3 odd 4
200.6.c.c.49.1 2 1.1 even 1 trivial
200.6.c.c.49.2 2 5.4 even 2 inner
320.6.a.e.1.1 1 40.27 even 4
320.6.a.l.1.1 1 40.37 odd 4
360.6.a.b.1.1 1 15.2 even 4
400.6.a.f.1.1 1 20.3 even 4
400.6.c.h.49.1 2 20.19 odd 2
400.6.c.h.49.2 2 4.3 odd 2
720.6.a.h.1.1 1 60.47 odd 4