Properties

Label 384.7.l.b.31.14
Level $384$
Weight $7$
Character 384.31
Analytic conductor $88.341$
Analytic rank $0$
Dimension $48$
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] = [384,7,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), 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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 31.14
Character \(\chi\) \(=\) 384.31
Dual form 384.7.l.b.223.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(11.0227 + 11.0227i) q^{3} +(-128.299 - 128.299i) q^{5} +76.4178 q^{7} +243.000i q^{9} +O(q^{10})\) \(q+(11.0227 + 11.0227i) q^{3} +(-128.299 - 128.299i) q^{5} +76.4178 q^{7} +243.000i q^{9} +(565.739 - 565.739i) q^{11} +(-2779.09 + 2779.09i) q^{13} -2828.39i q^{15} +5241.26 q^{17} +(46.8386 + 46.8386i) q^{19} +(842.331 + 842.331i) q^{21} -19532.3 q^{23} +17296.1i q^{25} +(-2678.52 + 2678.52i) q^{27} +(16453.9 - 16453.9i) q^{29} +24099.2i q^{31} +12471.9 q^{33} +(-9804.30 - 9804.30i) q^{35} +(38199.7 + 38199.7i) q^{37} -61266.2 q^{39} +66525.2i q^{41} +(24959.9 - 24959.9i) q^{43} +(31176.6 - 31176.6i) q^{45} -120821. i q^{47} -111809. q^{49} +(57772.8 + 57772.8i) q^{51} +(-92727.0 - 92727.0i) q^{53} -145167. q^{55} +1032.58i q^{57} +(160301. - 160301. i) q^{59} +(178988. - 178988. i) q^{61} +18569.5i q^{63} +713107. q^{65} +(274936. + 274936. i) q^{67} +(-215299. - 215299. i) q^{69} +631790. q^{71} -470653. i q^{73} +(-190649. + 190649. i) q^{75} +(43232.5 - 43232.5i) q^{77} -535148. i q^{79} -59049.0 q^{81} +(284269. + 284269. i) q^{83} +(-672446. - 672446. i) q^{85} +362733. q^{87} +272304. i q^{89} +(-212372. + 212372. i) q^{91} +(-265638. + 265638. i) q^{93} -12018.7i q^{95} +499709. q^{97} +(137474. + 137474. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 2720 q^{11} + 3936 q^{19} + 26240 q^{23} - 66400 q^{29} - 162336 q^{35} + 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} - 443680 q^{53} + 232704 q^{55} + 886144 q^{59} + 326496 q^{61} - 372832 q^{65} + 962112 q^{67} - 541728 q^{69} + 534016 q^{71} + 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} + 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 11.0227 + 11.0227i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) −128.299 128.299i −1.02639 1.02639i −0.999642 0.0267466i \(-0.991485\pi\)
−0.0267466 0.999642i \(-0.508515\pi\)
\(6\) 0 0
\(7\) 76.4178 0.222792 0.111396 0.993776i \(-0.464468\pi\)
0.111396 + 0.993776i \(0.464468\pi\)
\(8\) 0 0
\(9\) 243.000i 0.333333i
\(10\) 0 0
\(11\) 565.739 565.739i 0.425048 0.425048i −0.461890 0.886937i \(-0.652828\pi\)
0.886937 + 0.461890i \(0.152828\pi\)
\(12\) 0 0
\(13\) −2779.09 + 2779.09i −1.26495 + 1.26495i −0.316283 + 0.948665i \(0.602435\pi\)
−0.948665 + 0.316283i \(0.897565\pi\)
\(14\) 0 0
\(15\) 2828.39i 0.838043i
\(16\) 0 0
\(17\) 5241.26 1.06681 0.533407 0.845859i \(-0.320912\pi\)
0.533407 + 0.845859i \(0.320912\pi\)
\(18\) 0 0
\(19\) 46.8386 + 46.8386i 0.00682878 + 0.00682878i 0.710513 0.703684i \(-0.248463\pi\)
−0.703684 + 0.710513i \(0.748463\pi\)
\(20\) 0 0
\(21\) 842.331 + 842.331i 0.0909546 + 0.0909546i
\(22\) 0 0
\(23\) −19532.3 −1.60535 −0.802675 0.596417i \(-0.796590\pi\)
−0.802675 + 0.596417i \(0.796590\pi\)
\(24\) 0 0
\(25\) 17296.1i 1.10695i
\(26\) 0 0
\(27\) −2678.52 + 2678.52i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 16453.9 16453.9i 0.674645 0.674645i −0.284138 0.958783i \(-0.591707\pi\)
0.958783 + 0.284138i \(0.0917074\pi\)
\(30\) 0 0
\(31\) 24099.2i 0.808941i 0.914551 + 0.404470i \(0.132544\pi\)
−0.914551 + 0.404470i \(0.867456\pi\)
\(32\) 0 0
\(33\) 12471.9 0.347050
\(34\) 0 0
\(35\) −9804.30 9804.30i −0.228672 0.228672i
\(36\) 0 0
\(37\) 38199.7 + 38199.7i 0.754144 + 0.754144i 0.975250 0.221106i \(-0.0709665\pi\)
−0.221106 + 0.975250i \(0.570967\pi\)
\(38\) 0 0
\(39\) −61266.2 −1.03283
\(40\) 0 0
\(41\) 66525.2i 0.965239i 0.875830 + 0.482620i \(0.160315\pi\)
−0.875830 + 0.482620i \(0.839685\pi\)
\(42\) 0 0
\(43\) 24959.9 24959.9i 0.313934 0.313934i −0.532498 0.846431i \(-0.678747\pi\)
0.846431 + 0.532498i \(0.178747\pi\)
\(44\) 0 0
\(45\) 31176.6 31176.6i 0.342130 0.342130i
\(46\) 0 0
\(47\) 120821.i 1.16372i −0.813288 0.581862i \(-0.802324\pi\)
0.813288 0.581862i \(-0.197676\pi\)
\(48\) 0 0
\(49\) −111809. −0.950364
\(50\) 0 0
\(51\) 57772.8 + 57772.8i 0.435525 + 0.435525i
\(52\) 0 0
\(53\) −92727.0 92727.0i −0.622843 0.622843i 0.323415 0.946257i \(-0.395169\pi\)
−0.946257 + 0.323415i \(0.895169\pi\)
\(54\) 0 0
\(55\) −145167. −0.872529
\(56\) 0 0
\(57\) 1032.58i 0.00557568i
\(58\) 0 0
\(59\) 160301. 160301.i 0.780511 0.780511i −0.199406 0.979917i \(-0.563901\pi\)
0.979917 + 0.199406i \(0.0639013\pi\)
\(60\) 0 0
\(61\) 178988. 178988.i 0.788557 0.788557i −0.192700 0.981258i \(-0.561725\pi\)
0.981258 + 0.192700i \(0.0617245\pi\)
\(62\) 0 0
\(63\) 18569.5i 0.0742642i
\(64\) 0 0
\(65\) 713107. 2.59666
\(66\) 0 0
\(67\) 274936. + 274936.i 0.914127 + 0.914127i 0.996594 0.0824667i \(-0.0262798\pi\)
−0.0824667 + 0.996594i \(0.526280\pi\)
\(68\) 0 0
\(69\) −215299. 215299.i −0.655381 0.655381i
\(70\) 0 0
\(71\) 631790. 1.76521 0.882607 0.470111i \(-0.155786\pi\)
0.882607 + 0.470111i \(0.155786\pi\)
\(72\) 0 0
\(73\) 470653.i 1.20985i −0.796281 0.604926i \(-0.793203\pi\)
0.796281 0.604926i \(-0.206797\pi\)
\(74\) 0 0
\(75\) −190649. + 190649.i −0.451910 + 0.451910i
\(76\) 0 0
\(77\) 43232.5 43232.5i 0.0946974 0.0946974i
\(78\) 0 0
\(79\) 535148.i 1.08541i −0.839924 0.542704i \(-0.817401\pi\)
0.839924 0.542704i \(-0.182599\pi\)
\(80\) 0 0
\(81\) −59049.0 −0.111111
\(82\) 0 0
\(83\) 284269. + 284269.i 0.497159 + 0.497159i 0.910552 0.413394i \(-0.135657\pi\)
−0.413394 + 0.910552i \(0.635657\pi\)
\(84\) 0 0
\(85\) −672446. 672446.i −1.09497 1.09497i
\(86\) 0 0
\(87\) 362733. 0.550846
\(88\) 0 0
\(89\) 272304.i 0.386264i 0.981173 + 0.193132i \(0.0618645\pi\)
−0.981173 + 0.193132i \(0.938135\pi\)
\(90\) 0 0
\(91\) −212372. + 212372.i −0.281821 + 0.281821i
\(92\) 0 0
\(93\) −265638. + 265638.i −0.330249 + 0.330249i
\(94\) 0 0
\(95\) 12018.7i 0.0140180i
\(96\) 0 0
\(97\) 499709. 0.547523 0.273761 0.961798i \(-0.411732\pi\)
0.273761 + 0.961798i \(0.411732\pi\)
\(98\) 0 0
\(99\) 137474. + 137474.i 0.141683 + 0.141683i
\(100\) 0 0
\(101\) 532345. + 532345.i 0.516688 + 0.516688i 0.916568 0.399879i \(-0.130948\pi\)
−0.399879 + 0.916568i \(0.630948\pi\)
\(102\) 0 0
\(103\) −1.15325e6 −1.05538 −0.527691 0.849436i \(-0.676942\pi\)
−0.527691 + 0.849436i \(0.676942\pi\)
\(104\) 0 0
\(105\) 216140.i 0.186710i
\(106\) 0 0
\(107\) 350884. 350884.i 0.286426 0.286426i −0.549239 0.835665i \(-0.685082\pi\)
0.835665 + 0.549239i \(0.185082\pi\)
\(108\) 0 0
\(109\) 555679. 555679.i 0.429086 0.429086i −0.459231 0.888317i \(-0.651875\pi\)
0.888317 + 0.459231i \(0.151875\pi\)
\(110\) 0 0
\(111\) 842127.i 0.615756i
\(112\) 0 0
\(113\) 1.30361e6 0.903465 0.451732 0.892153i \(-0.350806\pi\)
0.451732 + 0.892153i \(0.350806\pi\)
\(114\) 0 0
\(115\) 2.50596e6 + 2.50596e6i 1.64771 + 1.64771i
\(116\) 0 0
\(117\) −675319. 675319.i −0.421649 0.421649i
\(118\) 0 0
\(119\) 400525. 0.237678
\(120\) 0 0
\(121\) 1.13144e6i 0.638669i
\(122\) 0 0
\(123\) −733288. + 733288.i −0.394057 + 0.394057i
\(124\) 0 0
\(125\) 214395. 214395.i 0.109770 0.109770i
\(126\) 0 0
\(127\) 2.55793e6i 1.24875i 0.781123 + 0.624377i \(0.214647\pi\)
−0.781123 + 0.624377i \(0.785353\pi\)
\(128\) 0 0
\(129\) 550252. 0.256326
\(130\) 0 0
\(131\) 28783.5 + 28783.5i 0.0128035 + 0.0128035i 0.713480 0.700676i \(-0.247118\pi\)
−0.700676 + 0.713480i \(0.747118\pi\)
\(132\) 0 0
\(133\) 3579.31 + 3579.31i 0.00152140 + 0.00152140i
\(134\) 0 0
\(135\) 687300. 0.279348
\(136\) 0 0
\(137\) 3.78484e6i 1.47192i −0.677023 0.735962i \(-0.736730\pi\)
0.677023 0.735962i \(-0.263270\pi\)
\(138\) 0 0
\(139\) −331261. + 331261.i −0.123346 + 0.123346i −0.766085 0.642739i \(-0.777798\pi\)
0.642739 + 0.766085i \(0.277798\pi\)
\(140\) 0 0
\(141\) 1.33178e6 1.33178e6i 0.475088 0.475088i
\(142\) 0 0
\(143\) 3.14448e6i 1.07533i
\(144\) 0 0
\(145\) −4.22203e6 −1.38490
\(146\) 0 0
\(147\) −1.23244e6 1.23244e6i −0.387984 0.387984i
\(148\) 0 0
\(149\) 2.00745e6 + 2.00745e6i 0.606855 + 0.606855i 0.942123 0.335268i \(-0.108827\pi\)
−0.335268 + 0.942123i \(0.608827\pi\)
\(150\) 0 0
\(151\) 3.78985e6 1.10076 0.550378 0.834916i \(-0.314484\pi\)
0.550378 + 0.834916i \(0.314484\pi\)
\(152\) 0 0
\(153\) 1.27363e6i 0.355605i
\(154\) 0 0
\(155\) 3.09189e6 3.09189e6i 0.830288 0.830288i
\(156\) 0 0
\(157\) 4.38208e6 4.38208e6i 1.13235 1.13235i 0.142565 0.989785i \(-0.454465\pi\)
0.989785 0.142565i \(-0.0455351\pi\)
\(158\) 0 0
\(159\) 2.04420e6i 0.508549i
\(160\) 0 0
\(161\) −1.49261e6 −0.357660
\(162\) 0 0
\(163\) 502348. + 502348.i 0.115996 + 0.115996i 0.762722 0.646726i \(-0.223862\pi\)
−0.646726 + 0.762722i \(0.723862\pi\)
\(164\) 0 0
\(165\) −1.60013e6 1.60013e6i −0.356208 0.356208i
\(166\) 0 0
\(167\) 7.89436e6 1.69499 0.847496 0.530802i \(-0.178109\pi\)
0.847496 + 0.530802i \(0.178109\pi\)
\(168\) 0 0
\(169\) 1.06199e7i 2.20019i
\(170\) 0 0
\(171\) −11381.8 + 11381.8i −0.00227626 + 0.00227626i
\(172\) 0 0
\(173\) 3.35604e6 3.35604e6i 0.648170 0.648170i −0.304381 0.952550i \(-0.598449\pi\)
0.952550 + 0.304381i \(0.0984495\pi\)
\(174\) 0 0
\(175\) 1.32173e6i 0.246620i
\(176\) 0 0
\(177\) 3.53389e6 0.637284
\(178\) 0 0
\(179\) 1.48022e6 + 1.48022e6i 0.258087 + 0.258087i 0.824276 0.566189i \(-0.191583\pi\)
−0.566189 + 0.824276i \(0.691583\pi\)
\(180\) 0 0
\(181\) 4.91348e6 + 4.91348e6i 0.828616 + 0.828616i 0.987325 0.158709i \(-0.0507332\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(182\) 0 0
\(183\) 3.94585e6 0.643854
\(184\) 0 0
\(185\) 9.80193e6i 1.54809i
\(186\) 0 0
\(187\) 2.96518e6 2.96518e6i 0.453447 0.453447i
\(188\) 0 0
\(189\) −204686. + 204686.i −0.0303182 + 0.0303182i
\(190\) 0 0
\(191\) 1.24457e7i 1.78615i 0.449909 + 0.893074i \(0.351456\pi\)
−0.449909 + 0.893074i \(0.648544\pi\)
\(192\) 0 0
\(193\) 1.03510e7 1.43983 0.719913 0.694064i \(-0.244182\pi\)
0.719913 + 0.694064i \(0.244182\pi\)
\(194\) 0 0
\(195\) 7.86037e6 + 7.86037e6i 1.06008 + 1.06008i
\(196\) 0 0
\(197\) −4.37780e6 4.37780e6i −0.572608 0.572608i 0.360248 0.932856i \(-0.382692\pi\)
−0.932856 + 0.360248i \(0.882692\pi\)
\(198\) 0 0
\(199\) 3.15098e6 0.399840 0.199920 0.979812i \(-0.435932\pi\)
0.199920 + 0.979812i \(0.435932\pi\)
\(200\) 0 0
\(201\) 6.06107e6i 0.746382i
\(202\) 0 0
\(203\) 1.25737e6 1.25737e6i 0.150306 0.150306i
\(204\) 0 0
\(205\) 8.53510e6 8.53510e6i 0.990711 0.990711i
\(206\) 0 0
\(207\) 4.74635e6i 0.535116i
\(208\) 0 0
\(209\) 52996.8 0.00580512
\(210\) 0 0
\(211\) −5.31096e6 5.31096e6i −0.565361 0.565361i 0.365465 0.930825i \(-0.380910\pi\)
−0.930825 + 0.365465i \(0.880910\pi\)
\(212\) 0 0
\(213\) 6.96403e6 + 6.96403e6i 0.720646 + 0.720646i
\(214\) 0 0
\(215\) −6.40465e6 −0.644436
\(216\) 0 0
\(217\) 1.84160e6i 0.180226i
\(218\) 0 0
\(219\) 5.18787e6 5.18787e6i 0.493920 0.493920i
\(220\) 0 0
\(221\) −1.45659e7 + 1.45659e7i −1.34946 + 1.34946i
\(222\) 0 0
\(223\) 1.27496e7i 1.14969i −0.818261 0.574847i \(-0.805062\pi\)
0.818261 0.574847i \(-0.194938\pi\)
\(224\) 0 0
\(225\) −4.20294e6 −0.368983
\(226\) 0 0
\(227\) −9.00100e6 9.00100e6i −0.769508 0.769508i 0.208512 0.978020i \(-0.433138\pi\)
−0.978020 + 0.208512i \(0.933138\pi\)
\(228\) 0 0
\(229\) −1.27977e7 1.27977e7i −1.06568 1.06568i −0.997686 0.0679892i \(-0.978342\pi\)
−0.0679892 0.997686i \(-0.521658\pi\)
\(230\) 0 0
\(231\) 953078. 0.0773201
\(232\) 0 0
\(233\) 1.59408e7i 1.26021i −0.776511 0.630103i \(-0.783012\pi\)
0.776511 0.630103i \(-0.216988\pi\)
\(234\) 0 0
\(235\) −1.55012e7 + 1.55012e7i −1.19443 + 1.19443i
\(236\) 0 0
\(237\) 5.89878e6 5.89878e6i 0.443116 0.443116i
\(238\) 0 0
\(239\) 1.36617e7i 1.00072i 0.865818 + 0.500359i \(0.166799\pi\)
−0.865818 + 0.500359i \(0.833201\pi\)
\(240\) 0 0
\(241\) 3.16851e6 0.226362 0.113181 0.993574i \(-0.463896\pi\)
0.113181 + 0.993574i \(0.463896\pi\)
\(242\) 0 0
\(243\) −650880. 650880.i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) 1.43450e7 + 1.43450e7i 0.975442 + 0.975442i
\(246\) 0 0
\(247\) −260338. −0.0172761
\(248\) 0 0
\(249\) 6.26682e6i 0.405928i
\(250\) 0 0
\(251\) −1.38815e7 + 1.38815e7i −0.877838 + 0.877838i −0.993311 0.115473i \(-0.963162\pi\)
0.115473 + 0.993311i \(0.463162\pi\)
\(252\) 0 0
\(253\) −1.10502e7 + 1.10502e7i −0.682350 + 0.682350i
\(254\) 0 0
\(255\) 1.48243e7i 0.894036i
\(256\) 0 0
\(257\) 8.16993e6 0.481304 0.240652 0.970612i \(-0.422639\pi\)
0.240652 + 0.970612i \(0.422639\pi\)
\(258\) 0 0
\(259\) 2.91914e6 + 2.91914e6i 0.168018 + 0.168018i
\(260\) 0 0
\(261\) 3.99830e6 + 3.99830e6i 0.224882 + 0.224882i
\(262\) 0 0
\(263\) 1.07117e7 0.588834 0.294417 0.955677i \(-0.404875\pi\)
0.294417 + 0.955677i \(0.404875\pi\)
\(264\) 0 0
\(265\) 2.37935e7i 1.27856i
\(266\) 0 0
\(267\) −3.00153e6 + 3.00153e6i −0.157691 + 0.157691i
\(268\) 0 0
\(269\) 3.98822e6 3.98822e6i 0.204891 0.204891i −0.597201 0.802092i \(-0.703721\pi\)
0.802092 + 0.597201i \(0.203721\pi\)
\(270\) 0 0
\(271\) 573671.i 0.0288240i 0.999896 + 0.0144120i \(0.00458765\pi\)
−0.999896 + 0.0144120i \(0.995412\pi\)
\(272\) 0 0
\(273\) −4.68183e6 −0.230106
\(274\) 0 0
\(275\) 9.78505e6 + 9.78505e6i 0.470506 + 0.470506i
\(276\) 0 0
\(277\) −1.83692e7 1.83692e7i −0.864275 0.864275i 0.127556 0.991831i \(-0.459287\pi\)
−0.991831 + 0.127556i \(0.959287\pi\)
\(278\) 0 0
\(279\) −5.85609e6 −0.269647
\(280\) 0 0
\(281\) 3.16574e7i 1.42678i 0.700769 + 0.713388i \(0.252840\pi\)
−0.700769 + 0.713388i \(0.747160\pi\)
\(282\) 0 0
\(283\) −659467. + 659467.i −0.0290960 + 0.0290960i −0.721505 0.692409i \(-0.756549\pi\)
0.692409 + 0.721505i \(0.256549\pi\)
\(284\) 0 0
\(285\) 132478. 132478.i 0.00572281 0.00572281i
\(286\) 0 0
\(287\) 5.08371e6i 0.215048i
\(288\) 0 0
\(289\) 3.33319e6 0.138091
\(290\) 0 0
\(291\) 5.50814e6 + 5.50814e6i 0.223525 + 0.223525i
\(292\) 0 0
\(293\) 4.40668e6 + 4.40668e6i 0.175190 + 0.175190i 0.789255 0.614065i \(-0.210467\pi\)
−0.614065 + 0.789255i \(0.710467\pi\)
\(294\) 0 0
\(295\) −4.11327e7 −1.60222
\(296\) 0 0
\(297\) 3.03068e6i 0.115683i
\(298\) 0 0
\(299\) 5.42820e7 5.42820e7i 2.03068 2.03068i
\(300\) 0 0
\(301\) 1.90738e6 1.90738e6i 0.0699421 0.0699421i
\(302\) 0 0
\(303\) 1.17358e7i 0.421874i
\(304\) 0 0
\(305\) −4.59277e7 −1.61873
\(306\) 0 0
\(307\) 1.31828e7 + 1.31828e7i 0.455611 + 0.455611i 0.897212 0.441601i \(-0.145589\pi\)
−0.441601 + 0.897212i \(0.645589\pi\)
\(308\) 0 0
\(309\) −1.27119e7 1.27119e7i −0.430858 0.430858i
\(310\) 0 0
\(311\) −5.31930e7 −1.76837 −0.884185 0.467137i \(-0.845286\pi\)
−0.884185 + 0.467137i \(0.845286\pi\)
\(312\) 0 0
\(313\) 5.25409e6i 0.171342i −0.996323 0.0856711i \(-0.972697\pi\)
0.996323 0.0856711i \(-0.0273034\pi\)
\(314\) 0 0
\(315\) 2.38244e6 2.38244e6i 0.0762239 0.0762239i
\(316\) 0 0
\(317\) −1.15447e7 + 1.15447e7i −0.362415 + 0.362415i −0.864702 0.502286i \(-0.832492\pi\)
0.502286 + 0.864702i \(0.332492\pi\)
\(318\) 0 0
\(319\) 1.86172e7i 0.573513i
\(320\) 0 0
\(321\) 7.73539e6 0.233866
\(322\) 0 0
\(323\) 245493. + 245493.i 0.00728504 + 0.00728504i
\(324\) 0 0
\(325\) −4.80673e7 4.80673e7i −1.40023 1.40023i
\(326\) 0 0
\(327\) 1.22502e7 0.350347
\(328\) 0 0
\(329\) 9.23290e6i 0.259269i
\(330\) 0 0
\(331\) −3.33618e7 + 3.33618e7i −0.919954 + 0.919954i −0.997026 0.0770720i \(-0.975443\pi\)
0.0770720 + 0.997026i \(0.475443\pi\)
\(332\) 0 0
\(333\) −9.28252e6 + 9.28252e6i −0.251381 + 0.251381i
\(334\) 0 0
\(335\) 7.05477e7i 1.87650i
\(336\) 0 0
\(337\) −3.53546e7 −0.923753 −0.461877 0.886944i \(-0.652824\pi\)
−0.461877 + 0.886944i \(0.652824\pi\)
\(338\) 0 0
\(339\) 1.43693e7 + 1.43693e7i 0.368838 + 0.368838i
\(340\) 0 0
\(341\) 1.36338e7 + 1.36338e7i 0.343838 + 0.343838i
\(342\) 0 0
\(343\) −1.75347e7 −0.434526
\(344\) 0 0
\(345\) 5.52450e7i 1.34535i
\(346\) 0 0
\(347\) 2.72087e7 2.72087e7i 0.651207 0.651207i −0.302077 0.953284i \(-0.597680\pi\)
0.953284 + 0.302077i \(0.0976799\pi\)
\(348\) 0 0
\(349\) 1.75646e7 1.75646e7i 0.413201 0.413201i −0.469651 0.882852i \(-0.655620\pi\)
0.882852 + 0.469651i \(0.155620\pi\)
\(350\) 0 0
\(351\) 1.48877e7i 0.344275i
\(352\) 0 0
\(353\) 4.31019e7 0.979878 0.489939 0.871757i \(-0.337019\pi\)
0.489939 + 0.871757i \(0.337019\pi\)
\(354\) 0 0
\(355\) −8.10577e7 8.10577e7i −1.81180 1.81180i
\(356\) 0 0
\(357\) 4.41487e6 + 4.41487e6i 0.0970317 + 0.0970317i
\(358\) 0 0
\(359\) 5.07426e7 1.09670 0.548352 0.836248i \(-0.315255\pi\)
0.548352 + 0.836248i \(0.315255\pi\)
\(360\) 0 0
\(361\) 4.70415e7i 0.999907i
\(362\) 0 0
\(363\) −1.24715e7 + 1.24715e7i −0.260735 + 0.260735i
\(364\) 0 0
\(365\) −6.03842e7 + 6.03842e7i −1.24178 + 1.24178i
\(366\) 0 0
\(367\) 5.29504e7i 1.07120i 0.844471 + 0.535601i \(0.179915\pi\)
−0.844471 + 0.535601i \(0.820085\pi\)
\(368\) 0 0
\(369\) −1.61656e7 −0.321746
\(370\) 0 0
\(371\) −7.08599e6 7.08599e6i −0.138765 0.138765i
\(372\) 0 0
\(373\) 3.19127e7 + 3.19127e7i 0.614946 + 0.614946i 0.944231 0.329285i \(-0.106808\pi\)
−0.329285 + 0.944231i \(0.606808\pi\)
\(374\) 0 0
\(375\) 4.72642e6 0.0896270
\(376\) 0 0
\(377\) 9.14539e7i 1.70678i
\(378\) 0 0
\(379\) −5.31956e7 + 5.31956e7i −0.977142 + 0.977142i −0.999745 0.0226024i \(-0.992805\pi\)
0.0226024 + 0.999745i \(0.492805\pi\)
\(380\) 0 0
\(381\) −2.81953e7 + 2.81953e7i −0.509802 + 0.509802i
\(382\) 0 0
\(383\) 1.34659e7i 0.239685i −0.992793 0.119842i \(-0.961761\pi\)
0.992793 0.119842i \(-0.0382389\pi\)
\(384\) 0 0
\(385\) −1.10933e7 −0.194393
\(386\) 0 0
\(387\) 6.06526e6 + 6.06526e6i 0.104645 + 0.104645i
\(388\) 0 0
\(389\) 6.53726e7 + 6.53726e7i 1.11057 + 1.11057i 0.993073 + 0.117499i \(0.0374878\pi\)
0.117499 + 0.993073i \(0.462512\pi\)
\(390\) 0 0
\(391\) −1.02374e8 −1.71261
\(392\) 0 0
\(393\) 634545.i 0.0104541i
\(394\) 0 0
\(395\) −6.86587e7 + 6.86587e7i −1.11405 + 1.11405i
\(396\) 0 0
\(397\) −2.77960e7 + 2.77960e7i −0.444233 + 0.444233i −0.893432 0.449199i \(-0.851709\pi\)
0.449199 + 0.893432i \(0.351709\pi\)
\(398\) 0 0
\(399\) 78907.3i 0.00124222i
\(400\) 0 0
\(401\) 6.42149e7 0.995871 0.497935 0.867214i \(-0.334092\pi\)
0.497935 + 0.867214i \(0.334092\pi\)
\(402\) 0 0
\(403\) −6.69737e7 6.69737e7i −1.02327 1.02327i
\(404\) 0 0
\(405\) 7.57590e6 + 7.57590e6i 0.114043 + 0.114043i
\(406\) 0 0
\(407\) 4.32221e7 0.641095
\(408\) 0 0
\(409\) 6.68143e7i 0.976562i −0.872687 0.488281i \(-0.837624\pi\)
0.872687 0.488281i \(-0.162376\pi\)
\(410\) 0 0
\(411\) 4.17191e7 4.17191e7i 0.600910 0.600910i
\(412\) 0 0
\(413\) 1.22498e7 1.22498e7i 0.173892 0.173892i
\(414\) 0 0
\(415\) 7.29426e7i 1.02056i
\(416\) 0 0
\(417\) −7.30279e6 −0.100712
\(418\) 0 0
\(419\) −7.40014e7 7.40014e7i −1.00600 1.00600i −0.999982 0.00601716i \(-0.998085\pi\)
−0.00601716 0.999982i \(-0.501915\pi\)
\(420\) 0 0
\(421\) 6.36241e6 + 6.36241e6i 0.0852659 + 0.0852659i 0.748453 0.663187i \(-0.230797\pi\)
−0.663187 + 0.748453i \(0.730797\pi\)
\(422\) 0 0
\(423\) 2.93596e7 0.387908
\(424\) 0 0
\(425\) 9.06531e7i 1.18091i
\(426\) 0 0
\(427\) 1.36778e7 1.36778e7i 0.175685 0.175685i
\(428\) 0 0
\(429\) −3.46606e7 + 3.46606e7i −0.439000 + 0.439000i
\(430\) 0 0
\(431\) 3.69859e7i 0.461960i −0.972959 0.230980i \(-0.925807\pi\)
0.972959 0.230980i \(-0.0741931\pi\)
\(432\) 0 0
\(433\) −5.09232e7 −0.627266 −0.313633 0.949544i \(-0.601546\pi\)
−0.313633 + 0.949544i \(0.601546\pi\)
\(434\) 0 0
\(435\) −4.65382e7 4.65382e7i −0.565382 0.565382i
\(436\) 0 0
\(437\) −914865. 914865.i −0.0109626 0.0109626i
\(438\) 0 0
\(439\) −1.15508e8 −1.36527 −0.682634 0.730760i \(-0.739166\pi\)
−0.682634 + 0.730760i \(0.739166\pi\)
\(440\) 0 0
\(441\) 2.71697e7i 0.316788i
\(442\) 0 0
\(443\) 5.85528e7 5.85528e7i 0.673499 0.673499i −0.285022 0.958521i \(-0.592001\pi\)
0.958521 + 0.285022i \(0.0920009\pi\)
\(444\) 0 0
\(445\) 3.49362e7 3.49362e7i 0.396457 0.396457i
\(446\) 0 0
\(447\) 4.42550e7i 0.495495i
\(448\) 0 0
\(449\) −8.93732e7 −0.987344 −0.493672 0.869648i \(-0.664346\pi\)
−0.493672 + 0.869648i \(0.664346\pi\)
\(450\) 0 0
\(451\) 3.76359e7 + 3.76359e7i 0.410273 + 0.410273i
\(452\) 0 0
\(453\) 4.17744e7 + 4.17744e7i 0.449382 + 0.449382i
\(454\) 0 0
\(455\) 5.44941e7 0.578516
\(456\) 0 0
\(457\) 8.96514e7i 0.939309i −0.882850 0.469654i \(-0.844378\pi\)
0.882850 0.469654i \(-0.155622\pi\)
\(458\) 0 0
\(459\) −1.40388e7 + 1.40388e7i −0.145175 + 0.145175i
\(460\) 0 0
\(461\) 3.72692e6 3.72692e6i 0.0380406 0.0380406i −0.687831 0.725871i \(-0.741437\pi\)
0.725871 + 0.687831i \(0.241437\pi\)
\(462\) 0 0
\(463\) 1.18637e8i 1.19530i 0.801757 + 0.597650i \(0.203899\pi\)
−0.801757 + 0.597650i \(0.796101\pi\)
\(464\) 0 0
\(465\) 6.81619e7 0.677927
\(466\) 0 0
\(467\) −3.26400e6 3.26400e6i −0.0320479 0.0320479i 0.690901 0.722949i \(-0.257214\pi\)
−0.722949 + 0.690901i \(0.757214\pi\)
\(468\) 0 0
\(469\) 2.10100e7 + 2.10100e7i 0.203661 + 0.203661i
\(470\) 0 0
\(471\) 9.66047e7 0.924561
\(472\) 0 0
\(473\) 2.82416e7i 0.266874i
\(474\) 0 0
\(475\) −810124. + 810124.i −0.00755911 + 0.00755911i
\(476\) 0 0
\(477\) 2.25327e7 2.25327e7i 0.207614 0.207614i
\(478\) 0 0
\(479\) 7.56324e7i 0.688179i 0.938937 + 0.344089i \(0.111812\pi\)
−0.938937 + 0.344089i \(0.888188\pi\)
\(480\) 0 0
\(481\) −2.12321e8 −1.90791
\(482\) 0 0
\(483\) −1.64526e7 1.64526e7i −0.146014 0.146014i
\(484\) 0 0
\(485\) −6.41120e7 6.41120e7i −0.561971 0.561971i
\(486\) 0 0
\(487\) 2.10626e7 0.182358 0.0911790 0.995835i \(-0.470936\pi\)
0.0911790 + 0.995835i \(0.470936\pi\)
\(488\) 0 0
\(489\) 1.10745e7i 0.0947101i
\(490\) 0 0
\(491\) −4.57469e7 + 4.57469e7i −0.386471 + 0.386471i −0.873427 0.486955i \(-0.838107\pi\)
0.486955 + 0.873427i \(0.338107\pi\)
\(492\) 0 0
\(493\) 8.62392e7 8.62392e7i 0.719721 0.719721i
\(494\) 0 0
\(495\) 3.52756e7i 0.290843i
\(496\) 0 0
\(497\) 4.82800e7 0.393277
\(498\) 0 0
\(499\) 1.47954e8 + 1.47954e8i 1.19076 + 1.19076i 0.976853 + 0.213912i \(0.0686206\pi\)
0.213912 + 0.976853i \(0.431379\pi\)
\(500\) 0 0
\(501\) 8.70172e7 + 8.70172e7i 0.691977 + 0.691977i
\(502\) 0 0
\(503\) 1.13896e8 0.894962 0.447481 0.894293i \(-0.352321\pi\)
0.447481 + 0.894293i \(0.352321\pi\)
\(504\) 0 0
\(505\) 1.36598e8i 1.06065i
\(506\) 0 0
\(507\) 1.17060e8 1.17060e8i 0.898223 0.898223i
\(508\) 0 0
\(509\) 6.31902e7 6.31902e7i 0.479177 0.479177i −0.425691 0.904868i \(-0.639969\pi\)
0.904868 + 0.425691i \(0.139969\pi\)
\(510\) 0 0
\(511\) 3.59663e7i 0.269546i
\(512\) 0 0
\(513\) −250916. −0.00185856
\(514\) 0 0
\(515\) 1.47960e8 + 1.47960e8i 1.08323 + 1.08323i
\(516\) 0 0
\(517\) −6.83533e7 6.83533e7i −0.494638 0.494638i
\(518\) 0 0
\(519\) 7.39853e7 0.529228
\(520\) 0 0
\(521\) 1.69906e7i 0.120142i −0.998194 0.0600710i \(-0.980867\pi\)
0.998194 0.0600710i \(-0.0191327\pi\)
\(522\) 0 0
\(523\) 9.27719e7 9.27719e7i 0.648502 0.648502i −0.304129 0.952631i \(-0.598365\pi\)
0.952631 + 0.304129i \(0.0983654\pi\)
\(524\) 0 0
\(525\) −1.45690e7 + 1.45690e7i −0.100682 + 0.100682i
\(526\) 0 0
\(527\) 1.26310e8i 0.862989i
\(528\) 0 0
\(529\) 2.33474e8 1.57715
\(530\) 0 0
\(531\) 3.89530e7 + 3.89530e7i 0.260170 + 0.260170i
\(532\) 0 0
\(533\) −1.84880e8 1.84880e8i −1.22098 1.22098i
\(534\) 0 0
\(535\) −9.00360e7 −0.587969
\(536\) 0 0
\(537\) 3.26320e7i 0.210727i
\(538\) 0 0
\(539\) −6.32548e7 + 6.32548e7i −0.403950 + 0.403950i
\(540\) 0 0
\(541\) 3.86819e7 3.86819e7i 0.244296 0.244296i −0.574329 0.818625i \(-0.694737\pi\)
0.818625 + 0.574329i \(0.194737\pi\)
\(542\) 0 0
\(543\) 1.08320e8i 0.676562i
\(544\) 0 0
\(545\) −1.42586e8 −0.880818
\(546\) 0 0
\(547\) −2.18514e8 2.18514e8i −1.33511 1.33511i −0.900728 0.434384i \(-0.856966\pi\)
−0.434384 0.900728i \(-0.643034\pi\)
\(548\) 0 0
\(549\) 4.34940e7 + 4.34940e7i 0.262852 + 0.262852i
\(550\) 0 0
\(551\) 1.54136e6 0.00921401
\(552\) 0 0
\(553\) 4.08948e7i 0.241821i
\(554\) 0 0
\(555\) 1.08044e8 1.08044e8i 0.632005 0.632005i
\(556\) 0 0
\(557\) 2.29193e7 2.29193e7i 0.132628 0.132628i −0.637676 0.770304i \(-0.720104\pi\)
0.770304 + 0.637676i \(0.220104\pi\)
\(558\) 0 0
\(559\) 1.38732e8i 0.794219i
\(560\) 0 0
\(561\) 6.53686e7 0.370238
\(562\) 0 0
\(563\) 2.43461e8 + 2.43461e8i 1.36428 + 1.36428i 0.868379 + 0.495901i \(0.165162\pi\)
0.495901 + 0.868379i \(0.334838\pi\)
\(564\) 0 0
\(565\) −1.67251e8 1.67251e8i −0.927306 0.927306i
\(566\) 0 0
\(567\) −4.51240e6 −0.0247547
\(568\) 0 0
\(569\) 5.46262e7i 0.296527i 0.988948 + 0.148263i \(0.0473684\pi\)
−0.988948 + 0.148263i \(0.952632\pi\)
\(570\) 0 0
\(571\) −1.62311e8 + 1.62311e8i −0.871848 + 0.871848i −0.992674 0.120826i \(-0.961446\pi\)
0.120826 + 0.992674i \(0.461446\pi\)
\(572\) 0 0
\(573\) −1.37185e8 + 1.37185e8i −0.729192 + 0.729192i
\(574\) 0 0
\(575\) 3.37832e8i 1.77704i
\(576\) 0 0
\(577\) 1.65063e8 0.859257 0.429629 0.903006i \(-0.358644\pi\)
0.429629 + 0.903006i \(0.358644\pi\)
\(578\) 0 0
\(579\) 1.14096e8 + 1.14096e8i 0.587807 + 0.587807i
\(580\) 0 0
\(581\) 2.17232e7 + 2.17232e7i 0.110763 + 0.110763i
\(582\) 0 0
\(583\) −1.04918e8 −0.529476
\(584\) 0 0
\(585\) 1.73285e8i 0.865552i
\(586\) 0 0
\(587\) −1.58258e8 + 1.58258e8i −0.782441 + 0.782441i −0.980242 0.197801i \(-0.936620\pi\)
0.197801 + 0.980242i \(0.436620\pi\)
\(588\) 0 0
\(589\) −1.12877e6 + 1.12877e6i −0.00552408 + 0.00552408i
\(590\) 0 0
\(591\) 9.65104e7i 0.467532i
\(592\) 0 0
\(593\) 1.27415e8 0.611022 0.305511 0.952189i \(-0.401173\pi\)
0.305511 + 0.952189i \(0.401173\pi\)
\(594\) 0 0
\(595\) −5.13868e7 5.13868e7i −0.243950 0.243950i
\(596\) 0 0
\(597\) 3.47323e7 + 3.47323e7i 0.163234 + 0.163234i
\(598\) 0 0
\(599\) −1.41336e8 −0.657615 −0.328807 0.944397i \(-0.606647\pi\)
−0.328807 + 0.944397i \(0.606647\pi\)
\(600\) 0 0
\(601\) 3.36592e8i 1.55053i −0.631636 0.775265i \(-0.717616\pi\)
0.631636 0.775265i \(-0.282384\pi\)
\(602\) 0 0
\(603\) −6.68094e7 + 6.68094e7i −0.304709 + 0.304709i
\(604\) 0 0
\(605\) 1.45162e8 1.45162e8i 0.655523 0.655523i
\(606\) 0 0
\(607\) 1.16255e8i 0.519810i 0.965634 + 0.259905i \(0.0836912\pi\)
−0.965634 + 0.259905i \(0.916309\pi\)
\(608\) 0 0
\(609\) 2.77193e7 0.122724
\(610\) 0 0
\(611\) 3.35773e8 + 3.35773e8i 1.47205 + 1.47205i
\(612\) 0 0
\(613\) 1.25208e8 + 1.25208e8i 0.543563 + 0.543563i 0.924571 0.381009i \(-0.124423\pi\)
−0.381009 + 0.924571i \(0.624423\pi\)
\(614\) 0 0
\(615\) 1.88160e8 0.808912
\(616\) 0 0
\(617\) 3.41424e8i 1.45358i 0.686861 + 0.726789i \(0.258988\pi\)
−0.686861 + 0.726789i \(0.741012\pi\)
\(618\) 0 0
\(619\) −5.23598e7 + 5.23598e7i −0.220763 + 0.220763i −0.808820 0.588057i \(-0.799893\pi\)
0.588057 + 0.808820i \(0.299893\pi\)
\(620\) 0 0
\(621\) 5.23176e7 5.23176e7i 0.218460 0.218460i
\(622\) 0 0
\(623\) 2.08089e7i 0.0860566i
\(624\) 0 0
\(625\) 2.15238e8 0.881614
\(626\) 0 0
\(627\) 584168. + 584168.i 0.00236993 + 0.00236993i
\(628\) 0 0
\(629\) 2.00214e8 + 2.00214e8i 0.804531 + 0.804531i
\(630\) 0 0
\(631\) 3.73769e8 1.48770 0.743850 0.668347i \(-0.232998\pi\)
0.743850 + 0.668347i \(0.232998\pi\)
\(632\) 0 0
\(633\) 1.17082e8i 0.461615i
\(634\) 0 0
\(635\) 3.28179e8 3.28179e8i 1.28171 1.28171i
\(636\) 0 0
\(637\) 3.10728e8 3.10728e8i 1.20216 1.20216i
\(638\) 0 0
\(639\) 1.53525e8i 0.588405i
\(640\) 0 0
\(641\) −1.16876e7 −0.0443763 −0.0221881 0.999754i \(-0.507063\pi\)
−0.0221881 + 0.999754i \(0.507063\pi\)
\(642\) 0 0
\(643\) −7.95708e7 7.95708e7i −0.299310 0.299310i 0.541434 0.840743i \(-0.317882\pi\)
−0.840743 + 0.541434i \(0.817882\pi\)
\(644\) 0 0
\(645\) −7.05965e7 7.05965e7i −0.263090 0.263090i
\(646\) 0 0
\(647\) 1.53020e8 0.564981 0.282491 0.959270i \(-0.408839\pi\)
0.282491 + 0.959270i \(0.408839\pi\)
\(648\) 0 0
\(649\) 1.81376e8i 0.663509i
\(650\) 0 0
\(651\) −2.02995e7 + 2.02995e7i −0.0735769 + 0.0735769i
\(652\) 0 0
\(653\) −4.21038e7 + 4.21038e7i −0.151210 + 0.151210i −0.778658 0.627448i \(-0.784099\pi\)
0.627448 + 0.778658i \(0.284099\pi\)
\(654\) 0 0
\(655\) 7.38578e6i 0.0262828i
\(656\) 0 0
\(657\) 1.14369e8 0.403284
\(658\) 0 0
\(659\) −1.81805e8 1.81805e8i −0.635256 0.635256i 0.314125 0.949382i \(-0.398289\pi\)
−0.949382 + 0.314125i \(0.898289\pi\)
\(660\) 0 0
\(661\) −2.69937e8 2.69937e8i −0.934669 0.934669i 0.0633241 0.997993i \(-0.479830\pi\)
−0.997993 + 0.0633241i \(0.979830\pi\)
\(662\) 0 0
\(663\) −3.21112e8 −1.10183
\(664\) 0 0
\(665\) 918440.i 0.00312310i
\(666\) 0 0
\(667\) −3.21383e8 + 3.21383e8i −1.08304 + 1.08304i
\(668\) 0 0
\(669\) 1.40535e8 1.40535e8i 0.469360 0.469360i
\(670\) 0 0
\(671\) 2.02520e8i 0.670349i
\(672\) 0 0
\(673\) 94641.0 0.000310480 0.000155240 1.00000i \(-0.499951\pi\)
0.000155240 1.00000i \(0.499951\pi\)
\(674\) 0 0
\(675\) −4.63278e7 4.63278e7i −0.150637 0.150637i
\(676\) 0 0
\(677\) −1.76652e8 1.76652e8i −0.569314 0.569314i 0.362622 0.931936i \(-0.381881\pi\)
−0.931936 + 0.362622i \(0.881881\pi\)
\(678\) 0 0
\(679\) 3.81867e7 0.121984
\(680\) 0 0
\(681\) 1.98431e8i 0.628301i
\(682\) 0 0
\(683\) 3.36247e7 3.36247e7i 0.105535 0.105535i −0.652368 0.757903i \(-0.726224\pi\)
0.757903 + 0.652368i \(0.226224\pi\)
\(684\) 0 0
\(685\) −4.85589e8 + 4.85589e8i −1.51077 + 1.51077i
\(686\) 0 0
\(687\) 2.82130e8i 0.870120i
\(688\) 0 0
\(689\) 5.15393e8 1.57573
\(690\) 0 0
\(691\) 2.67951e8 + 2.67951e8i 0.812122 + 0.812122i 0.984952 0.172830i \(-0.0552912\pi\)
−0.172830 + 0.984952i \(0.555291\pi\)
\(692\) 0 0
\(693\) 1.05055e7 + 1.05055e7i 0.0315658 + 0.0315658i
\(694\) 0 0
\(695\) 8.50007e7 0.253203
\(696\) 0 0
\(697\) 3.48676e8i 1.02973i
\(698\) 0 0
\(699\) 1.75711e8 1.75711e8i 0.514477 0.514477i
\(700\) 0 0
\(701\) 2.07036e8 2.07036e8i 0.601023 0.601023i −0.339561 0.940584i \(-0.610279\pi\)
0.940584 + 0.339561i \(0.110279\pi\)
\(702\) 0 0
\(703\) 3.57844e6i 0.0102998i
\(704\) 0 0
\(705\) −3.41730e8 −0.975250
\(706\) 0 0
\(707\) 4.06806e7 + 4.06806e7i 0.115114 + 0.115114i
\(708\) 0 0
\(709\) −2.86413e8 2.86413e8i −0.803626 0.803626i 0.180034 0.983660i \(-0.442379\pi\)
−0.983660 + 0.180034i \(0.942379\pi\)
\(710\) 0 0
\(711\) 1.30041e8 0.361802
\(712\) 0 0
\(713\) 4.70712e8i 1.29863i
\(714\) 0 0
\(715\) 4.03432e8 4.03432e8i 1.10370 1.10370i
\(716\) 0 0
\(717\) −1.50589e8 + 1.50589e8i −0.408541 + 0.408541i
\(718\) 0 0
\(719\) 2.52364e8i 0.678954i −0.940614 0.339477i \(-0.889750\pi\)
0.940614 0.339477i \(-0.110250\pi\)
\(720\) 0 0
\(721\) −8.81285e7 −0.235131
\(722\) 0 0
\(723\) 3.49255e7 + 3.49255e7i 0.0924120 + 0.0924120i
\(724\) 0 0
\(725\) 2.84588e8 + 2.84588e8i 0.746797 + 0.746797i
\(726\) 0 0
\(727\) −1.87607e8 −0.488253 −0.244127 0.969743i \(-0.578501\pi\)
−0.244127 + 0.969743i \(0.578501\pi\)
\(728\) 0 0
\(729\) 1.43489e7i 0.0370370i
\(730\) 0 0
\(731\) 1.30821e8 1.30821e8i 0.334909 0.334909i
\(732\) 0 0
\(733\) 2.14523e8 2.14523e8i 0.544705 0.544705i −0.380199 0.924905i \(-0.624145\pi\)
0.924905 + 0.380199i \(0.124145\pi\)
\(734\) 0 0
\(735\) 3.16241e8i 0.796445i
\(736\) 0 0
\(737\) 3.11083e8 0.777095
\(738\) 0 0
\(739\) 1.89247e8 + 1.89247e8i 0.468916 + 0.468916i 0.901563 0.432647i \(-0.142420\pi\)
−0.432647 + 0.901563i \(0.642420\pi\)
\(740\) 0 0
\(741\) −2.86962e6 2.86962e6i −0.00705294 0.00705294i
\(742\) 0 0
\(743\) 2.98525e8 0.727804 0.363902 0.931437i \(-0.381444\pi\)
0.363902 + 0.931437i \(0.381444\pi\)
\(744\) 0 0
\(745\) 5.15105e8i 1.24574i
\(746\) 0 0
\(747\) −6.90774e7 + 6.90774e7i −0.165720 + 0.165720i
\(748\) 0 0
\(749\) 2.68138e7 2.68138e7i 0.0638136 0.0638136i
\(750\) 0 0
\(751\) 3.52347e8i 0.831860i 0.909397 + 0.415930i \(0.136544\pi\)
−0.909397 + 0.415930i \(0.863456\pi\)
\(752\) 0 0
\(753\) −3.06023e8 −0.716752
\(754\) 0 0
\(755\) −4.86232e8 4.86232e8i −1.12980 1.12980i
\(756\) 0 0
\(757\) 1.50447e8 + 1.50447e8i 0.346814 + 0.346814i 0.858921 0.512107i \(-0.171135\pi\)
−0.512107 + 0.858921i \(0.671135\pi\)
\(758\) 0 0
\(759\) −2.43605e8 −0.557137
\(760\) 0 0
\(761\) 1.52589e8i 0.346233i 0.984901 + 0.173117i \(0.0553837\pi\)
−0.984901 + 0.173117i \(0.944616\pi\)
\(762\) 0 0
\(763\) 4.24637e7 4.24637e7i 0.0955971 0.0955971i
\(764\) 0 0
\(765\) 1.63404e8 1.63404e8i 0.364989 0.364989i
\(766\) 0 0
\(767\) 8.90979e8i 1.97461i
\(768\) 0 0
\(769\) −4.19596e8 −0.922682 −0.461341 0.887223i \(-0.652631\pi\)
−0.461341 + 0.887223i \(0.652631\pi\)
\(770\) 0 0
\(771\) 9.00548e7 + 9.00548e7i 0.196491 + 0.196491i
\(772\) 0 0
\(773\) 3.63380e8 + 3.63380e8i 0.786723 + 0.786723i 0.980956 0.194232i \(-0.0622216\pi\)
−0.194232 + 0.980956i \(0.562222\pi\)
\(774\) 0 0
\(775\) −4.16820e8 −0.895455
\(776\) 0 0
\(777\) 6.43535e7i 0.137186i
\(778\) 0 0
\(779\) −3.11595e6 + 3.11595e6i −0.00659141 + 0.00659141i
\(780\) 0 0
\(781\) 3.57428e8 3.57428e8i 0.750300 0.750300i
\(782\) 0 0
\(783\) 8.81442e7i 0.183615i
\(784\) 0 0
\(785\) −1.12443e9 −2.32446
\(786\) 0 0
\(787\) −2.05644e8 2.05644e8i −0.421884 0.421884i 0.463968 0.885852i \(-0.346425\pi\)
−0.885852 + 0.463968i \(0.846425\pi\)
\(788\) 0 0
\(789\) 1.18072e8 + 1.18072e8i 0.240391 + 0.240391i
\(790\) 0 0
\(791\) 9.96188e7 0.201285
\(792\) 0 0
\(793\) 9.94845e8i 1.99497i
\(794\) 0 0
\(795\) −2.62268e8 + 2.62268e8i −0.521969 + 0.521969i
\(796\) 0 0
\(797\) −1.45649e8 + 1.45649e8i −0.287695 + 0.287695i −0.836168 0.548473i \(-0.815209\pi\)
0.548473 + 0.836168i \(0.315209\pi\)
\(798\) 0 0
\(799\) 6.33255e8i 1.24148i
\(800\) 0 0
\(801\) −6.61699e7 −0.128755
\(802\) 0 0
\(803\) −2.66267e8 2.66267e8i −0.514245 0.514245i
\(804\) 0 0
\(805\) 1.91500e8 + 1.91500e8i 0.367098 + 0.367098i
\(806\) 0 0
\(807\) 8.79219e7 0.167292
\(808\) 0 0
\(809\) 2.70877e8i 0.511595i −0.966730 0.255798i \(-0.917662\pi\)
0.966730 0.255798i \(-0.0823381\pi\)
\(810\) 0 0
\(811\) −2.66755e8 + 2.66755e8i −0.500092 + 0.500092i −0.911467 0.411374i \(-0.865049\pi\)
0.411374 + 0.911467i \(0.365049\pi\)
\(812\) 0 0
\(813\) −6.32340e6 + 6.32340e6i −0.0117674 + 0.0117674i
\(814\) 0 0
\(815\) 1.28901e8i 0.238113i
\(816\) 0 0
\(817\) 2.33818e6 0.00428757
\(818\) 0 0
\(819\) −5.16064e7 5.16064e7i −0.0939403 0.0939403i
\(820\) 0 0
\(821\) 1.82685e8 + 1.82685e8i 0.330121 + 0.330121i 0.852632 0.522511i \(-0.175005\pi\)
−0.522511 + 0.852632i \(0.675005\pi\)
\(822\) 0 0
\(823\) 7.27338e8 1.30478 0.652389 0.757884i \(-0.273767\pi\)
0.652389 + 0.757884i \(0.273767\pi\)
\(824\) 0 0
\(825\) 2.15715e8i 0.384166i
\(826\) 0 0
\(827\) 3.81741e8 3.81741e8i 0.674919 0.674919i −0.283927 0.958846i \(-0.591637\pi\)
0.958846 + 0.283927i \(0.0916372\pi\)
\(828\) 0 0
\(829\) −5.84827e8 + 5.84827e8i −1.02651 + 1.02651i −0.0268724 + 0.999639i \(0.508555\pi\)
−0.999639 + 0.0268724i \(0.991445\pi\)
\(830\) 0 0
\(831\) 4.04957e8i 0.705677i
\(832\) 0 0
\(833\) −5.86021e8 −1.01386
\(834\) 0 0
\(835\) −1.01284e9 1.01284e9i −1.73972 1.73972i
\(836\) 0 0
\(837\) −6.45500e7 6.45500e7i −0.110083 0.110083i
\(838\) 0 0
\(839\) −5.40695e8 −0.915517 −0.457759 0.889076i \(-0.651348\pi\)
−0.457759 + 0.889076i \(0.651348\pi\)
\(840\) 0 0
\(841\) 5.33601e7i 0.0897074i
\(842\) 0 0
\(843\) −3.48950e8 + 3.48950e8i −0.582479 + 0.582479i
\(844\) 0 0
\(845\) −1.36252e9 + 1.36252e9i −2.25825 + 2.25825i
\(846\) 0 0
\(847\) 8.64622e7i 0.142291i
\(848\) 0 0
\(849\) −1.45382e7 −0.0237568
\(850\) 0 0
\(851\) −7.46127e8 7.46127e8i −1.21066 1.21066i
\(852\) 0 0
\(853\) 8.16023e8 + 8.16023e8i 1.31479 + 1.31479i 0.917843 + 0.396944i \(0.129929\pi\)
0.396944 + 0.917843i \(0.370071\pi\)
\(854\) 0 0
\(855\) 2.92053e6 0.00467266
\(856\) 0 0
\(857\) 8.99758e8i 1.42950i 0.699382 + 0.714748i \(0.253459\pi\)
−0.699382 + 0.714748i \(0.746541\pi\)
\(858\) 0 0
\(859\) −5.05566e8 + 5.05566e8i −0.797624 + 0.797624i −0.982720 0.185096i \(-0.940740\pi\)
0.185096 + 0.982720i \(0.440740\pi\)
\(860\) 0 0
\(861\) −5.60363e7 + 5.60363e7i −0.0877930 + 0.0877930i
\(862\) 0 0
\(863\) 3.65472e8i 0.568619i 0.958733 + 0.284310i \(0.0917643\pi\)
−0.958733 + 0.284310i \(0.908236\pi\)
\(864\) 0 0
\(865\) −8.61150e8 −1.33055
\(866\) 0 0
\(867\) 3.67407e7 + 3.67407e7i 0.0563755 + 0.0563755i
\(868\) 0 0
\(869\) −3.02754e8 3.02754e8i −0.461350 0.461350i
\(870\) 0 0
\(871\) −1.52814e9 −2.31265
\(872\) 0 0
\(873\) 1.21429e8i 0.182508i
\(874\) 0 0
\(875\) 1.63836e7 1.63836e7i 0.0244560 0.0244560i
\(876\) 0 0
\(877\) −1.17148e8 + 1.17148e8i −0.173674 + 0.173674i −0.788592 0.614917i \(-0.789189\pi\)
0.614917 + 0.788592i \(0.289189\pi\)
\(878\) 0 0
\(879\) 9.71471e7i 0.143042i
\(880\) 0 0
\(881\) 8.93109e8 1.30610 0.653050 0.757314i \(-0.273489\pi\)
0.653050 + 0.757314i \(0.273489\pi\)
\(882\) 0 0
\(883\) 9.66301e8 + 9.66301e8i 1.40356 + 1.40356i 0.788420 + 0.615137i \(0.210899\pi\)
0.615137 + 0.788420i \(0.289101\pi\)
\(884\) 0 0
\(885\) −4.53393e8 4.53393e8i −0.654102 0.654102i
\(886\) 0 0
\(887\) −2.37358e8 −0.340120 −0.170060 0.985434i \(-0.554396\pi\)
−0.170060 + 0.985434i \(0.554396\pi\)
\(888\) 0 0
\(889\) 1.95471e8i 0.278213i
\(890\) 0 0
\(891\) −3.34063e7 + 3.34063e7i −0.0472275 + 0.0472275i
\(892\) 0 0
\(893\) 5.65910e6 5.65910e6i 0.00794682 0.00794682i
\(894\) 0 0
\(895\) 3.79820e8i 0.529796i
\(896\) 0 0
\(897\) 1.19667e9 1.65805
\(898\) 0 0
\(899\) 3.96526e8 + 3.96526e8i 0.545748 + 0.545748i
\(900\) 0 0
\(901\) −4.86006e8 4.86006e8i −0.664457 0.664457i
\(902\) 0 0
\(903\) 4.20490e7 0.0571074
\(904\) 0 0
\(905\) 1.26079e9i 1.70097i
\(906\) 0 0
\(907\) 3.13327e8 3.13327e8i 0.419930 0.419930i −0.465250 0.885179i \(-0.654035\pi\)
0.885179 + 0.465250i \(0.154035\pi\)
\(908\) 0 0
\(909\) −1.29360e8 + 1.29360e8i −0.172229 + 0.172229i
\(910\) 0 0
\(911\) 3.43893e8i 0.454849i 0.973796 + 0.227425i \(0.0730306\pi\)
−0.973796 + 0.227425i \(0.926969\pi\)
\(912\) 0 0
\(913\) 3.21644e8 0.422632
\(914\) 0 0
\(915\) −5.06247e8 5.06247e8i −0.660845 0.660845i
\(916\) 0 0
\(917\) 2.19958e6 + 2.19958e6i 0.00285253 + 0.00285253i
\(918\) 0 0
\(919\) 1.23387e9 1.58973 0.794866 0.606785i \(-0.207541\pi\)
0.794866 + 0.606785i \(0.207541\pi\)
\(920\) 0 0
\(921\) 2.90621e8i 0.372005i
\(922\) 0 0
\(923\) −1.75580e9 + 1.75580e9i −2.23290 + 2.23290i
\(924\) 0 0
\(925\) −6.60704e8 + 6.60704e8i −0.834799 + 0.834799i
\(926\) 0 0
\(927\) 2.80239e8i 0.351794i
\(928\) 0 0
\(929\) 1.33194e8 0.166125 0.0830627 0.996544i \(-0.473530\pi\)
0.0830627 + 0.996544i \(0.473530\pi\)
\(930\) 0 0
\(931\) −5.23699e6 5.23699e6i −0.00648983 0.00648983i
\(932\) 0 0
\(933\) −5.86331e8 5.86331e8i −0.721934 0.721934i
\(934\) 0 0
\(935\) −7.60857e8 −0.930825
\(936\) 0 0
\(937\) 1.04820e9i 1.27416i 0.770798 + 0.637079i \(0.219858\pi\)
−0.770798 + 0.637079i \(0.780142\pi\)
\(938\) 0 0
\(939\) 5.79143e7 5.79143e7i 0.0699502 0.0699502i
\(940\) 0 0
\(941\) 4.82891e8 4.82891e8i 0.579536 0.579536i −0.355239 0.934775i \(-0.615601\pi\)
0.934775 + 0.355239i \(0.115601\pi\)
\(942\) 0 0
\(943\) 1.29939e9i 1.54955i
\(944\) 0 0
\(945\) 5.25220e7 0.0622366
\(946\) 0 0
\(947\) −1.94344e8 1.94344e8i −0.228835 0.228835i 0.583371 0.812206i \(-0.301733\pi\)
−0.812206 + 0.583371i \(0.801733\pi\)
\(948\) 0 0
\(949\) 1.30799e9 + 1.30799e9i 1.53040 + 1.53040i
\(950\) 0 0
\(951\) −2.54509e8 −0.295911
\(952\) 0 0
\(953\) 1.33145e9i 1.53832i −0.639057 0.769159i \(-0.720675\pi\)
0.639057 0.769159i \(-0.279325\pi\)
\(954\) 0 0
\(955\) 1.59676e9 1.59676e9i 1.83328 1.83328i
\(956\) 0 0
\(957\) 2.05212e8 2.05212e8i 0.234136 0.234136i
\(958\) 0 0
\(959\) 2.89229e8i 0.327934i
\(960\) 0 0
\(961\) 3.06734e8 0.345615
\(962\) 0 0
\(963\) 8.52649e7 + 8.52649e7i 0.0954754 + 0.0954754i
\(964\) 0 0
\(965\) −1.32802e9 1.32802e9i −1.47782 1.47782i
\(966\) 0 0
\(967\) 1.17954e9 1.30447 0.652235 0.758017i \(-0.273831\pi\)
0.652235 + 0.758017i \(0.273831\pi\)
\(968\) 0 0
\(969\) 5.41200e6i 0.00594821i
\(970\) 0 0
\(971\) 1.62523e8 1.62523e8i 0.177524 0.177524i −0.612752 0.790276i \(-0.709937\pi\)
0.790276 + 0.612752i \(0.209937\pi\)
\(972\) 0 0
\(973\) −2.53143e7 + 2.53143e7i −0.0274806 + 0.0274806i
\(974\) 0 0
\(975\) 1.05966e9i 1.14328i
\(976\) 0 0
\(977\) −5.55049e8 −0.595179 −0.297589 0.954694i \(-0.596183\pi\)
−0.297589 + 0.954694i \(0.596183\pi\)
\(978\) 0 0
\(979\) 1.54053e8 + 1.54053e8i 0.164181 + 0.164181i
\(980\) 0 0
\(981\) 1.35030e8 + 1.35030e8i 0.143029 + 0.143029i
\(982\) 0 0
\(983\) −1.19931e9 −1.26262 −0.631309 0.775531i \(-0.717482\pi\)
−0.631309 + 0.775531i \(0.717482\pi\)
\(984\) 0 0
\(985\) 1.12333e9i 1.17544i
\(986\) 0 0
\(987\) 1.01772e8 1.01772e8i 0.105846 0.105846i
\(988\) 0 0
\(989\) −4.87524e8 + 4.87524e8i −0.503973 + 0.503973i
\(990\) 0 0
\(991\) 4.74382e7i 0.0487424i −0.999703 0.0243712i \(-0.992242\pi\)
0.999703 0.0243712i \(-0.00775837\pi\)
\(992\) 0 0
\(993\) −7.35475e8 −0.751139
\(994\) 0 0
\(995\) −4.04266e8 4.04266e8i −0.410391 0.410391i
\(996\) 0 0
\(997\) −5.04216e8 5.04216e8i −0.508781 0.508781i 0.405371 0.914152i \(-0.367142\pi\)
−0.914152 + 0.405371i \(0.867142\pi\)
\(998\) 0 0
\(999\) −2.04637e8 −0.205252
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 384.7.l.b.31.14 48
4.3 odd 2 384.7.l.a.31.11 48
8.3 odd 2 192.7.l.a.79.14 48
8.5 even 2 48.7.l.a.43.24 yes 48
16.3 odd 4 inner 384.7.l.b.223.14 48
16.5 even 4 192.7.l.a.175.14 48
16.11 odd 4 48.7.l.a.19.24 48
16.13 even 4 384.7.l.a.223.11 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.7.l.a.19.24 48 16.11 odd 4
48.7.l.a.43.24 yes 48 8.5 even 2
192.7.l.a.79.14 48 8.3 odd 2
192.7.l.a.175.14 48 16.5 even 4
384.7.l.a.31.11 48 4.3 odd 2
384.7.l.a.223.11 48 16.13 even 4
384.7.l.b.31.14 48 1.1 even 1 trivial
384.7.l.b.223.14 48 16.3 odd 4 inner