Properties

Label 160.6.f.a.49.10
Level $160$
Weight $6$
Character 160.49
Analytic conductor $25.661$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,6,Mod(49,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.f (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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.10
Character \(\chi\) \(=\) 160.49
Dual form 160.6.f.a.49.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.5561 q^{3} +(-52.7686 + 18.4521i) q^{5} +47.9937i q^{7} -131.568 q^{9} +O(q^{10})\) \(q-10.5561 q^{3} +(-52.7686 + 18.4521i) q^{5} +47.9937i q^{7} -131.568 q^{9} -690.713i q^{11} -743.139 q^{13} +(557.032 - 194.783i) q^{15} +2042.04i q^{17} +1446.84i q^{19} -506.628i q^{21} -1433.47i q^{23} +(2444.04 - 1947.38i) q^{25} +3953.99 q^{27} -4388.18i q^{29} +5825.62 q^{31} +7291.26i q^{33} +(-885.584 - 2532.56i) q^{35} -632.966 q^{37} +7844.67 q^{39} +9131.48 q^{41} +1992.47 q^{43} +(6942.65 - 2427.70i) q^{45} +1291.21i q^{47} +14503.6 q^{49} -21556.1i q^{51} +7767.93 q^{53} +(12745.1 + 36447.9i) q^{55} -15273.1i q^{57} -7014.55i q^{59} -5005.26i q^{61} -6314.44i q^{63} +(39214.4 - 13712.5i) q^{65} -14569.6 q^{67} +15131.9i q^{69} +59737.3 q^{71} +37773.3i q^{73} +(-25799.6 + 20556.8i) q^{75} +33149.9 q^{77} -78197.2 q^{79} -9767.83 q^{81} +8704.60 q^{83} +(-37679.9 - 107756. i) q^{85} +46322.2i q^{87} -78835.9 q^{89} -35666.0i q^{91} -61496.0 q^{93} +(-26697.3 - 76347.9i) q^{95} -5863.79i q^{97} +90875.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 1940 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −10.5561 −0.677176 −0.338588 0.940935i \(-0.609949\pi\)
−0.338588 + 0.940935i \(0.609949\pi\)
\(4\) 0 0
\(5\) −52.7686 + 18.4521i −0.943953 + 0.330081i
\(6\) 0 0
\(7\) 47.9937i 0.370203i 0.982719 + 0.185101i \(0.0592613\pi\)
−0.982719 + 0.185101i \(0.940739\pi\)
\(8\) 0 0
\(9\) −131.568 −0.541432
\(10\) 0 0
\(11\) 690.713i 1.72114i −0.509332 0.860570i \(-0.670108\pi\)
0.509332 0.860570i \(-0.329892\pi\)
\(12\) 0 0
\(13\) −743.139 −1.21958 −0.609792 0.792562i \(-0.708747\pi\)
−0.609792 + 0.792562i \(0.708747\pi\)
\(14\) 0 0
\(15\) 557.032 194.783i 0.639222 0.223523i
\(16\) 0 0
\(17\) 2042.04i 1.71373i 0.515542 + 0.856864i \(0.327591\pi\)
−0.515542 + 0.856864i \(0.672409\pi\)
\(18\) 0 0
\(19\) 1446.84i 0.919470i 0.888056 + 0.459735i \(0.152056\pi\)
−0.888056 + 0.459735i \(0.847944\pi\)
\(20\) 0 0
\(21\) 506.628i 0.250692i
\(22\) 0 0
\(23\) 1433.47i 0.565026i −0.959263 0.282513i \(-0.908832\pi\)
0.959263 0.282513i \(-0.0911680\pi\)
\(24\) 0 0
\(25\) 2444.04 1947.38i 0.782093 0.623161i
\(26\) 0 0
\(27\) 3953.99 1.04382
\(28\) 0 0
\(29\) 4388.18i 0.968923i −0.874813 0.484462i \(-0.839016\pi\)
0.874813 0.484462i \(-0.160984\pi\)
\(30\) 0 0
\(31\) 5825.62 1.08877 0.544387 0.838834i \(-0.316762\pi\)
0.544387 + 0.838834i \(0.316762\pi\)
\(32\) 0 0
\(33\) 7291.26i 1.16552i
\(34\) 0 0
\(35\) −885.584 2532.56i −0.122197 0.349454i
\(36\) 0 0
\(37\) −632.966 −0.0760110 −0.0380055 0.999278i \(-0.512100\pi\)
−0.0380055 + 0.999278i \(0.512100\pi\)
\(38\) 0 0
\(39\) 7844.67 0.825873
\(40\) 0 0
\(41\) 9131.48 0.848363 0.424182 0.905577i \(-0.360562\pi\)
0.424182 + 0.905577i \(0.360562\pi\)
\(42\) 0 0
\(43\) 1992.47 0.164332 0.0821658 0.996619i \(-0.473816\pi\)
0.0821658 + 0.996619i \(0.473816\pi\)
\(44\) 0 0
\(45\) 6942.65 2427.70i 0.511086 0.178716i
\(46\) 0 0
\(47\) 1291.21i 0.0852616i 0.999091 + 0.0426308i \(0.0135739\pi\)
−0.999091 + 0.0426308i \(0.986426\pi\)
\(48\) 0 0
\(49\) 14503.6 0.862950
\(50\) 0 0
\(51\) 21556.1i 1.16050i
\(52\) 0 0
\(53\) 7767.93 0.379853 0.189927 0.981798i \(-0.439175\pi\)
0.189927 + 0.981798i \(0.439175\pi\)
\(54\) 0 0
\(55\) 12745.1 + 36447.9i 0.568115 + 1.62467i
\(56\) 0 0
\(57\) 15273.1i 0.622644i
\(58\) 0 0
\(59\) 7014.55i 0.262343i −0.991360 0.131172i \(-0.958126\pi\)
0.991360 0.131172i \(-0.0418739\pi\)
\(60\) 0 0
\(61\) 5005.26i 0.172227i −0.996285 0.0861137i \(-0.972555\pi\)
0.996285 0.0861137i \(-0.0274448\pi\)
\(62\) 0 0
\(63\) 6314.44i 0.200440i
\(64\) 0 0
\(65\) 39214.4 13712.5i 1.15123 0.402561i
\(66\) 0 0
\(67\) −14569.6 −0.396516 −0.198258 0.980150i \(-0.563528\pi\)
−0.198258 + 0.980150i \(0.563528\pi\)
\(68\) 0 0
\(69\) 15131.9i 0.382622i
\(70\) 0 0
\(71\) 59737.3 1.40637 0.703186 0.711006i \(-0.251760\pi\)
0.703186 + 0.711006i \(0.251760\pi\)
\(72\) 0 0
\(73\) 37773.3i 0.829616i 0.909909 + 0.414808i \(0.136151\pi\)
−0.909909 + 0.414808i \(0.863849\pi\)
\(74\) 0 0
\(75\) −25799.6 + 20556.8i −0.529615 + 0.421990i
\(76\) 0 0
\(77\) 33149.9 0.637170
\(78\) 0 0
\(79\) −78197.2 −1.40969 −0.704844 0.709362i \(-0.748983\pi\)
−0.704844 + 0.709362i \(0.748983\pi\)
\(80\) 0 0
\(81\) −9767.83 −0.165419
\(82\) 0 0
\(83\) 8704.60 0.138693 0.0693463 0.997593i \(-0.477909\pi\)
0.0693463 + 0.997593i \(0.477909\pi\)
\(84\) 0 0
\(85\) −37679.9 107756.i −0.565669 1.61768i
\(86\) 0 0
\(87\) 46322.2i 0.656132i
\(88\) 0 0
\(89\) −78835.9 −1.05499 −0.527496 0.849558i \(-0.676869\pi\)
−0.527496 + 0.849558i \(0.676869\pi\)
\(90\) 0 0
\(91\) 35666.0i 0.451493i
\(92\) 0 0
\(93\) −61496.0 −0.737292
\(94\) 0 0
\(95\) −26697.3 76347.9i −0.303500 0.867937i
\(96\) 0 0
\(97\) 5863.79i 0.0632775i −0.999499 0.0316387i \(-0.989927\pi\)
0.999499 0.0316387i \(-0.0100726\pi\)
\(98\) 0 0
\(99\) 90875.8i 0.931880i
\(100\) 0 0
\(101\) 189963.i 1.85296i 0.376343 + 0.926481i \(0.377182\pi\)
−0.376343 + 0.926481i \(0.622818\pi\)
\(102\) 0 0
\(103\) 115497.i 1.07270i −0.843996 0.536349i \(-0.819803\pi\)
0.843996 0.536349i \(-0.180197\pi\)
\(104\) 0 0
\(105\) 9348.35 + 26734.0i 0.0827488 + 0.236642i
\(106\) 0 0
\(107\) 167020. 1.41030 0.705148 0.709060i \(-0.250881\pi\)
0.705148 + 0.709060i \(0.250881\pi\)
\(108\) 0 0
\(109\) 18122.4i 0.146100i −0.997328 0.0730499i \(-0.976727\pi\)
0.997328 0.0730499i \(-0.0232732\pi\)
\(110\) 0 0
\(111\) 6681.68 0.0514728
\(112\) 0 0
\(113\) 24447.2i 0.180108i −0.995937 0.0900541i \(-0.971296\pi\)
0.995937 0.0900541i \(-0.0287040\pi\)
\(114\) 0 0
\(115\) 26450.5 + 75642.0i 0.186504 + 0.533358i
\(116\) 0 0
\(117\) 97773.3 0.660322
\(118\) 0 0
\(119\) −98005.2 −0.634427
\(120\) 0 0
\(121\) −316034. −1.96232
\(122\) 0 0
\(123\) −96393.2 −0.574491
\(124\) 0 0
\(125\) −93035.3 + 147858.i −0.532565 + 0.846389i
\(126\) 0 0
\(127\) 274865.i 1.51220i −0.654456 0.756101i \(-0.727102\pi\)
0.654456 0.756101i \(-0.272898\pi\)
\(128\) 0 0
\(129\) −21032.8 −0.111281
\(130\) 0 0
\(131\) 171066.i 0.870934i −0.900205 0.435467i \(-0.856583\pi\)
0.900205 0.435467i \(-0.143417\pi\)
\(132\) 0 0
\(133\) −69439.5 −0.340390
\(134\) 0 0
\(135\) −208646. + 72959.4i −0.985318 + 0.344546i
\(136\) 0 0
\(137\) 208744.i 0.950194i −0.879934 0.475097i \(-0.842413\pi\)
0.879934 0.475097i \(-0.157587\pi\)
\(138\) 0 0
\(139\) 236729.i 1.03924i 0.854399 + 0.519618i \(0.173926\pi\)
−0.854399 + 0.519618i \(0.826074\pi\)
\(140\) 0 0
\(141\) 13630.2i 0.0577372i
\(142\) 0 0
\(143\) 513296.i 2.09907i
\(144\) 0 0
\(145\) 80971.0 + 231558.i 0.319823 + 0.914618i
\(146\) 0 0
\(147\) −153102. −0.584369
\(148\) 0 0
\(149\) 157593.i 0.581530i −0.956794 0.290765i \(-0.906090\pi\)
0.956794 0.290765i \(-0.0939098\pi\)
\(150\) 0 0
\(151\) −59264.9 −0.211522 −0.105761 0.994392i \(-0.533728\pi\)
−0.105761 + 0.994392i \(0.533728\pi\)
\(152\) 0 0
\(153\) 268667.i 0.927868i
\(154\) 0 0
\(155\) −307410. + 107495.i −1.02775 + 0.359384i
\(156\) 0 0
\(157\) 152525. 0.493845 0.246923 0.969035i \(-0.420581\pi\)
0.246923 + 0.969035i \(0.420581\pi\)
\(158\) 0 0
\(159\) −81999.3 −0.257228
\(160\) 0 0
\(161\) 68797.5 0.209174
\(162\) 0 0
\(163\) −272338. −0.802859 −0.401430 0.915890i \(-0.631487\pi\)
−0.401430 + 0.915890i \(0.631487\pi\)
\(164\) 0 0
\(165\) −134539. 384749.i −0.384714 1.10019i
\(166\) 0 0
\(167\) 195117.i 0.541383i −0.962666 0.270692i \(-0.912748\pi\)
0.962666 0.270692i \(-0.0872524\pi\)
\(168\) 0 0
\(169\) 180962. 0.487383
\(170\) 0 0
\(171\) 190358.i 0.497831i
\(172\) 0 0
\(173\) 213434. 0.542185 0.271093 0.962553i \(-0.412615\pi\)
0.271093 + 0.962553i \(0.412615\pi\)
\(174\) 0 0
\(175\) 93462.0 + 117299.i 0.230696 + 0.289533i
\(176\) 0 0
\(177\) 74046.5i 0.177653i
\(178\) 0 0
\(179\) 464341.i 1.08319i −0.840640 0.541595i \(-0.817821\pi\)
0.840640 0.541595i \(-0.182179\pi\)
\(180\) 0 0
\(181\) 601344.i 1.36435i 0.731188 + 0.682176i \(0.238966\pi\)
−0.731188 + 0.682176i \(0.761034\pi\)
\(182\) 0 0
\(183\) 52836.2i 0.116628i
\(184\) 0 0
\(185\) 33400.7 11679.5i 0.0717507 0.0250898i
\(186\) 0 0
\(187\) 1.41046e6 2.94957
\(188\) 0 0
\(189\) 189767.i 0.386425i
\(190\) 0 0
\(191\) 618397. 1.22655 0.613273 0.789871i \(-0.289852\pi\)
0.613273 + 0.789871i \(0.289852\pi\)
\(192\) 0 0
\(193\) 24646.1i 0.0476271i −0.999716 0.0238136i \(-0.992419\pi\)
0.999716 0.0238136i \(-0.00758081\pi\)
\(194\) 0 0
\(195\) −413952. + 144751.i −0.779585 + 0.272605i
\(196\) 0 0
\(197\) 791174. 1.45247 0.726233 0.687448i \(-0.241269\pi\)
0.726233 + 0.687448i \(0.241269\pi\)
\(198\) 0 0
\(199\) 787051. 1.40887 0.704434 0.709770i \(-0.251201\pi\)
0.704434 + 0.709770i \(0.251201\pi\)
\(200\) 0 0
\(201\) 153799. 0.268511
\(202\) 0 0
\(203\) 210605. 0.358698
\(204\) 0 0
\(205\) −481855. + 168495.i −0.800815 + 0.280028i
\(206\) 0 0
\(207\) 188598.i 0.305923i
\(208\) 0 0
\(209\) 999355. 1.58254
\(210\) 0 0
\(211\) 100172.i 0.154896i 0.996996 + 0.0774482i \(0.0246772\pi\)
−0.996996 + 0.0774482i \(0.975323\pi\)
\(212\) 0 0
\(213\) −630596. −0.952362
\(214\) 0 0
\(215\) −105140. + 36765.2i −0.155121 + 0.0542427i
\(216\) 0 0
\(217\) 279593.i 0.403067i
\(218\) 0 0
\(219\) 398740.i 0.561797i
\(220\) 0 0
\(221\) 1.51752e6i 2.09004i
\(222\) 0 0
\(223\) 641949.i 0.864447i −0.901766 0.432224i \(-0.857729\pi\)
0.901766 0.432224i \(-0.142271\pi\)
\(224\) 0 0
\(225\) −321558. + 256213.i −0.423450 + 0.337400i
\(226\) 0 0
\(227\) 168375. 0.216876 0.108438 0.994103i \(-0.465415\pi\)
0.108438 + 0.994103i \(0.465415\pi\)
\(228\) 0 0
\(229\) 847070.i 1.06741i −0.845671 0.533704i \(-0.820800\pi\)
0.845671 0.533704i \(-0.179200\pi\)
\(230\) 0 0
\(231\) −349935. −0.431477
\(232\) 0 0
\(233\) 1.62230e6i 1.95768i 0.204623 + 0.978841i \(0.434403\pi\)
−0.204623 + 0.978841i \(0.565597\pi\)
\(234\) 0 0
\(235\) −23825.6 68135.5i −0.0281432 0.0804829i
\(236\) 0 0
\(237\) 825460. 0.954608
\(238\) 0 0
\(239\) 755195. 0.855193 0.427597 0.903970i \(-0.359360\pi\)
0.427597 + 0.903970i \(0.359360\pi\)
\(240\) 0 0
\(241\) 203535. 0.225734 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(242\) 0 0
\(243\) −857709. −0.931804
\(244\) 0 0
\(245\) −765334. + 267622.i −0.814584 + 0.284843i
\(246\) 0 0
\(247\) 1.07521e6i 1.12137i
\(248\) 0 0
\(249\) −91886.9 −0.0939194
\(250\) 0 0
\(251\) 1.07763e6i 1.07966i 0.841775 + 0.539829i \(0.181511\pi\)
−0.841775 + 0.539829i \(0.818489\pi\)
\(252\) 0 0
\(253\) −990115. −0.972488
\(254\) 0 0
\(255\) 397754. + 1.13748e6i 0.383058 + 1.09545i
\(256\) 0 0
\(257\) 1.85261e6i 1.74965i −0.484439 0.874825i \(-0.660976\pi\)
0.484439 0.874825i \(-0.339024\pi\)
\(258\) 0 0
\(259\) 30378.4i 0.0281394i
\(260\) 0 0
\(261\) 577344.i 0.524606i
\(262\) 0 0
\(263\) 914131.i 0.814927i 0.913221 + 0.407464i \(0.133587\pi\)
−0.913221 + 0.407464i \(0.866413\pi\)
\(264\) 0 0
\(265\) −409903. + 143335.i −0.358563 + 0.125382i
\(266\) 0 0
\(267\) 832202. 0.714415
\(268\) 0 0
\(269\) 767239.i 0.646473i 0.946318 + 0.323236i \(0.104771\pi\)
−0.946318 + 0.323236i \(0.895229\pi\)
\(270\) 0 0
\(271\) −1.13627e6 −0.939848 −0.469924 0.882707i \(-0.655719\pi\)
−0.469924 + 0.882707i \(0.655719\pi\)
\(272\) 0 0
\(273\) 376495.i 0.305740i
\(274\) 0 0
\(275\) −1.34508e6 1.68813e6i −1.07255 1.34609i
\(276\) 0 0
\(277\) 1.36392e6 1.06805 0.534024 0.845469i \(-0.320679\pi\)
0.534024 + 0.845469i \(0.320679\pi\)
\(278\) 0 0
\(279\) −766465. −0.589498
\(280\) 0 0
\(281\) 1.48233e6 1.11990 0.559951 0.828526i \(-0.310820\pi\)
0.559951 + 0.828526i \(0.310820\pi\)
\(282\) 0 0
\(283\) 767342. 0.569538 0.284769 0.958596i \(-0.408083\pi\)
0.284769 + 0.958596i \(0.408083\pi\)
\(284\) 0 0
\(285\) 281820. + 805939.i 0.205523 + 0.587746i
\(286\) 0 0
\(287\) 438254.i 0.314066i
\(288\) 0 0
\(289\) −2.75007e6 −1.93687
\(290\) 0 0
\(291\) 61899.0i 0.0428500i
\(292\) 0 0
\(293\) 318861. 0.216986 0.108493 0.994097i \(-0.465397\pi\)
0.108493 + 0.994097i \(0.465397\pi\)
\(294\) 0 0
\(295\) 129433. + 370147.i 0.0865944 + 0.247639i
\(296\) 0 0
\(297\) 2.73107e6i 1.79656i
\(298\) 0 0
\(299\) 1.06527e6i 0.689096i
\(300\) 0 0
\(301\) 95626.1i 0.0608359i
\(302\) 0 0
\(303\) 2.00528e6i 1.25478i
\(304\) 0 0
\(305\) 92357.6 + 264121.i 0.0568490 + 0.162575i
\(306\) 0 0
\(307\) 209398. 0.126802 0.0634010 0.997988i \(-0.479805\pi\)
0.0634010 + 0.997988i \(0.479805\pi\)
\(308\) 0 0
\(309\) 1.21920e6i 0.726406i
\(310\) 0 0
\(311\) −663890. −0.389220 −0.194610 0.980881i \(-0.562344\pi\)
−0.194610 + 0.980881i \(0.562344\pi\)
\(312\) 0 0
\(313\) 960429.i 0.554121i −0.960853 0.277060i \(-0.910640\pi\)
0.960853 0.277060i \(-0.0893602\pi\)
\(314\) 0 0
\(315\) 116515. + 333204.i 0.0661613 + 0.189205i
\(316\) 0 0
\(317\) 1.05328e6 0.588704 0.294352 0.955697i \(-0.404896\pi\)
0.294352 + 0.955697i \(0.404896\pi\)
\(318\) 0 0
\(319\) −3.03097e6 −1.66765
\(320\) 0 0
\(321\) −1.76309e6 −0.955019
\(322\) 0 0
\(323\) −2.95451e6 −1.57572
\(324\) 0 0
\(325\) −1.81626e6 + 1.44717e6i −0.953828 + 0.759997i
\(326\) 0 0
\(327\) 191303.i 0.0989353i
\(328\) 0 0
\(329\) −61970.2 −0.0315641
\(330\) 0 0
\(331\) 1.28525e6i 0.644788i 0.946606 + 0.322394i \(0.104488\pi\)
−0.946606 + 0.322394i \(0.895512\pi\)
\(332\) 0 0
\(333\) 83278.1 0.0411548
\(334\) 0 0
\(335\) 768817. 268840.i 0.374292 0.130882i
\(336\) 0 0
\(337\) 2.75075e6i 1.31940i 0.751529 + 0.659700i \(0.229317\pi\)
−0.751529 + 0.659700i \(0.770683\pi\)
\(338\) 0 0
\(339\) 258068.i 0.121965i
\(340\) 0 0
\(341\) 4.02383e6i 1.87393i
\(342\) 0 0
\(343\) 1.50271e6i 0.689669i
\(344\) 0 0
\(345\) −279215. 798487.i −0.126296 0.361177i
\(346\) 0 0
\(347\) 2.93898e6 1.31031 0.655153 0.755496i \(-0.272604\pi\)
0.655153 + 0.755496i \(0.272604\pi\)
\(348\) 0 0
\(349\) 817288.i 0.359179i 0.983742 + 0.179590i \(0.0574770\pi\)
−0.983742 + 0.179590i \(0.942523\pi\)
\(350\) 0 0
\(351\) −2.93836e6 −1.27303
\(352\) 0 0
\(353\) 1.83106e6i 0.782106i −0.920368 0.391053i \(-0.872111\pi\)
0.920368 0.391053i \(-0.127889\pi\)
\(354\) 0 0
\(355\) −3.15225e6 + 1.10228e6i −1.32755 + 0.464216i
\(356\) 0 0
\(357\) 1.03456e6 0.429619
\(358\) 0 0
\(359\) 1.94811e6 0.797768 0.398884 0.917001i \(-0.369398\pi\)
0.398884 + 0.917001i \(0.369398\pi\)
\(360\) 0 0
\(361\) 382741. 0.154574
\(362\) 0 0
\(363\) 3.33610e6 1.32884
\(364\) 0 0
\(365\) −696995. 1.99324e6i −0.273841 0.783119i
\(366\) 0 0
\(367\) 4.04360e6i 1.56712i −0.621314 0.783562i \(-0.713401\pi\)
0.621314 0.783562i \(-0.286599\pi\)
\(368\) 0 0
\(369\) −1.20141e6 −0.459331
\(370\) 0 0
\(371\) 372812.i 0.140623i
\(372\) 0 0
\(373\) −1.18472e6 −0.440903 −0.220451 0.975398i \(-0.570753\pi\)
−0.220451 + 0.975398i \(0.570753\pi\)
\(374\) 0 0
\(375\) 982093. 1.56081e6i 0.360641 0.573155i
\(376\) 0 0
\(377\) 3.26103e6i 1.18168i
\(378\) 0 0
\(379\) 983110.i 0.351564i −0.984429 0.175782i \(-0.943755\pi\)
0.984429 0.175782i \(-0.0562453\pi\)
\(380\) 0 0
\(381\) 2.90151e6i 1.02403i
\(382\) 0 0
\(383\) 2.10213e6i 0.732256i 0.930564 + 0.366128i \(0.119317\pi\)
−0.930564 + 0.366128i \(0.880683\pi\)
\(384\) 0 0
\(385\) −1.74927e6 + 611685.i −0.601459 + 0.210318i
\(386\) 0 0
\(387\) −262145. −0.0889744
\(388\) 0 0
\(389\) 2.32916e6i 0.780414i 0.920727 + 0.390207i \(0.127596\pi\)
−0.920727 + 0.390207i \(0.872404\pi\)
\(390\) 0 0
\(391\) 2.92720e6 0.968301
\(392\) 0 0
\(393\) 1.80579e6i 0.589776i
\(394\) 0 0
\(395\) 4.12635e6 1.44290e6i 1.33068 0.465311i
\(396\) 0 0
\(397\) −4.36029e6 −1.38848 −0.694239 0.719745i \(-0.744259\pi\)
−0.694239 + 0.719745i \(0.744259\pi\)
\(398\) 0 0
\(399\) 733012. 0.230504
\(400\) 0 0
\(401\) 3.73757e6 1.16072 0.580361 0.814359i \(-0.302911\pi\)
0.580361 + 0.814359i \(0.302911\pi\)
\(402\) 0 0
\(403\) −4.32924e6 −1.32785
\(404\) 0 0
\(405\) 515434. 180237.i 0.156148 0.0546017i
\(406\) 0 0
\(407\) 437198.i 0.130825i
\(408\) 0 0
\(409\) 90602.1 0.0267812 0.0133906 0.999910i \(-0.495738\pi\)
0.0133906 + 0.999910i \(0.495738\pi\)
\(410\) 0 0
\(411\) 2.20353e6i 0.643449i
\(412\) 0 0
\(413\) 336654. 0.0971201
\(414\) 0 0
\(415\) −459329. + 160618.i −0.130919 + 0.0457798i
\(416\) 0 0
\(417\) 2.49894e6i 0.703746i
\(418\) 0 0
\(419\) 4.75171e6i 1.32225i −0.750274 0.661127i \(-0.770078\pi\)
0.750274 0.661127i \(-0.229922\pi\)
\(420\) 0 0
\(421\) 2.47064e6i 0.679367i −0.940540 0.339684i \(-0.889680\pi\)
0.940540 0.339684i \(-0.110320\pi\)
\(422\) 0 0
\(423\) 169882.i 0.0461634i
\(424\) 0 0
\(425\) 3.97663e6 + 4.99083e6i 1.06793 + 1.34030i
\(426\) 0 0
\(427\) 240221. 0.0637591
\(428\) 0 0
\(429\) 5.41842e6i 1.42144i
\(430\) 0 0
\(431\) 6.40187e6 1.66002 0.830010 0.557748i \(-0.188334\pi\)
0.830010 + 0.557748i \(0.188334\pi\)
\(432\) 0 0
\(433\) 2.21518e6i 0.567792i −0.958855 0.283896i \(-0.908373\pi\)
0.958855 0.283896i \(-0.0916271\pi\)
\(434\) 0 0
\(435\) −854741. 2.44436e6i −0.216577 0.619357i
\(436\) 0 0
\(437\) 2.07400e6 0.519525
\(438\) 0 0
\(439\) −3.50374e6 −0.867702 −0.433851 0.900985i \(-0.642845\pi\)
−0.433851 + 0.900985i \(0.642845\pi\)
\(440\) 0 0
\(441\) −1.90821e6 −0.467229
\(442\) 0 0
\(443\) 3.78808e6 0.917085 0.458543 0.888672i \(-0.348372\pi\)
0.458543 + 0.888672i \(0.348372\pi\)
\(444\) 0 0
\(445\) 4.16006e6 1.45469e6i 0.995862 0.348233i
\(446\) 0 0
\(447\) 1.66358e6i 0.393799i
\(448\) 0 0
\(449\) 4.77118e6 1.11689 0.558444 0.829542i \(-0.311398\pi\)
0.558444 + 0.829542i \(0.311398\pi\)
\(450\) 0 0
\(451\) 6.30724e6i 1.46015i
\(452\) 0 0
\(453\) 625609. 0.143238
\(454\) 0 0
\(455\) 658112. + 1.88204e6i 0.149029 + 0.426188i
\(456\) 0 0
\(457\) 2.15919e6i 0.483617i −0.970324 0.241808i \(-0.922259\pi\)
0.970324 0.241808i \(-0.0777405\pi\)
\(458\) 0 0
\(459\) 8.07421e6i 1.78883i
\(460\) 0 0
\(461\) 4.80473e6i 1.05297i 0.850184 + 0.526486i \(0.176491\pi\)
−0.850184 + 0.526486i \(0.823509\pi\)
\(462\) 0 0
\(463\) 7.27794e6i 1.57781i 0.614512 + 0.788907i \(0.289353\pi\)
−0.614512 + 0.788907i \(0.710647\pi\)
\(464\) 0 0
\(465\) 3.24506e6 1.13473e6i 0.695969 0.243366i
\(466\) 0 0
\(467\) −6.04375e6 −1.28237 −0.641186 0.767385i \(-0.721557\pi\)
−0.641186 + 0.767385i \(0.721557\pi\)
\(468\) 0 0
\(469\) 699250.i 0.146791i
\(470\) 0 0
\(471\) −1.61007e6 −0.334420
\(472\) 0 0
\(473\) 1.37623e6i 0.282838i
\(474\) 0 0
\(475\) 2.81756e6 + 3.53615e6i 0.572979 + 0.719112i
\(476\) 0 0
\(477\) −1.02201e6 −0.205665
\(478\) 0 0
\(479\) 9.37213e6 1.86638 0.933188 0.359388i \(-0.117014\pi\)
0.933188 + 0.359388i \(0.117014\pi\)
\(480\) 0 0
\(481\) 470382. 0.0927017
\(482\) 0 0
\(483\) −726235. −0.141648
\(484\) 0 0
\(485\) 108199. + 309424.i 0.0208867 + 0.0597310i
\(486\) 0 0
\(487\) 3.12157e6i 0.596418i 0.954501 + 0.298209i \(0.0963892\pi\)
−0.954501 + 0.298209i \(0.903611\pi\)
\(488\) 0 0
\(489\) 2.87484e6 0.543677
\(490\) 0 0
\(491\) 7.79083e6i 1.45841i −0.684294 0.729206i \(-0.739890\pi\)
0.684294 0.729206i \(-0.260110\pi\)
\(492\) 0 0
\(493\) 8.96084e6 1.66047
\(494\) 0 0
\(495\) −1.67685e6 4.79538e6i −0.307596 0.879651i
\(496\) 0 0
\(497\) 2.86702e6i 0.520642i
\(498\) 0 0
\(499\) 2.49895e6i 0.449268i 0.974443 + 0.224634i \(0.0721187\pi\)
−0.974443 + 0.224634i \(0.927881\pi\)
\(500\) 0 0
\(501\) 2.05969e6i 0.366612i
\(502\) 0 0
\(503\) 1.06363e7i 1.87444i −0.348734 0.937222i \(-0.613388\pi\)
0.348734 0.937222i \(-0.386612\pi\)
\(504\) 0 0
\(505\) −3.50522e6 1.00241e7i −0.611627 1.74911i
\(506\) 0 0
\(507\) −1.91026e6 −0.330045
\(508\) 0 0
\(509\) 5.62277e6i 0.961958i −0.876732 0.480979i \(-0.840281\pi\)
0.876732 0.480979i \(-0.159719\pi\)
\(510\) 0 0
\(511\) −1.81288e6 −0.307126
\(512\) 0 0
\(513\) 5.72081e6i 0.959763i
\(514\) 0 0
\(515\) 2.13116e6 + 6.09461e6i 0.354077 + 1.01258i
\(516\) 0 0
\(517\) 891859. 0.146747
\(518\) 0 0
\(519\) −2.25304e6 −0.367155
\(520\) 0 0
\(521\) −2.57801e6 −0.416093 −0.208047 0.978119i \(-0.566711\pi\)
−0.208047 + 0.978119i \(0.566711\pi\)
\(522\) 0 0
\(523\) 4.31533e6 0.689859 0.344929 0.938629i \(-0.387903\pi\)
0.344929 + 0.938629i \(0.387903\pi\)
\(524\) 0 0
\(525\) −986598. 1.23822e6i −0.156222 0.196065i
\(526\) 0 0
\(527\) 1.18962e7i 1.86586i
\(528\) 0 0
\(529\) 4.38151e6 0.680746
\(530\) 0 0
\(531\) 922890.i 0.142041i
\(532\) 0 0
\(533\) −6.78596e6 −1.03465
\(534\) 0 0
\(535\) −8.81343e6 + 3.08188e6i −1.33125 + 0.465512i
\(536\) 0 0
\(537\) 4.90165e6i 0.733511i
\(538\) 0 0
\(539\) 1.00178e7i 1.48526i
\(540\) 0 0
\(541\) 6.44548e6i 0.946808i 0.880845 + 0.473404i \(0.156975\pi\)
−0.880845 + 0.473404i \(0.843025\pi\)
\(542\) 0 0
\(543\) 6.34787e6i 0.923907i
\(544\) 0 0
\(545\) 334396. + 956293.i 0.0482247 + 0.137911i
\(546\) 0 0
\(547\) −1.12757e7 −1.61130 −0.805650 0.592392i \(-0.798184\pi\)
−0.805650 + 0.592392i \(0.798184\pi\)
\(548\) 0 0
\(549\) 658533.i 0.0932495i
\(550\) 0 0
\(551\) 6.34901e6 0.890896
\(552\) 0 0
\(553\) 3.75297e6i 0.521870i
\(554\) 0 0
\(555\) −352582. + 123291.i −0.0485879 + 0.0169902i
\(556\) 0 0
\(557\) −1.41719e7 −1.93549 −0.967743 0.251938i \(-0.918932\pi\)
−0.967743 + 0.251938i \(0.918932\pi\)
\(558\) 0 0
\(559\) −1.48068e6 −0.200416
\(560\) 0 0
\(561\) −1.48891e7 −1.99738
\(562\) 0 0
\(563\) 2.87590e6 0.382387 0.191194 0.981552i \(-0.438764\pi\)
0.191194 + 0.981552i \(0.438764\pi\)
\(564\) 0 0
\(565\) 451102. + 1.29004e6i 0.0594503 + 0.170014i
\(566\) 0 0
\(567\) 468795.i 0.0612386i
\(568\) 0 0
\(569\) −5.05148e6 −0.654091 −0.327046 0.945009i \(-0.606053\pi\)
−0.327046 + 0.945009i \(0.606053\pi\)
\(570\) 0 0
\(571\) 7.13766e6i 0.916148i 0.888914 + 0.458074i \(0.151461\pi\)
−0.888914 + 0.458074i \(0.848539\pi\)
\(572\) 0 0
\(573\) −6.52788e6 −0.830588
\(574\) 0 0
\(575\) −2.79151e6 3.50345e6i −0.352102 0.441903i
\(576\) 0 0
\(577\) 3.58999e6i 0.448904i 0.974485 + 0.224452i \(0.0720591\pi\)
−0.974485 + 0.224452i \(0.927941\pi\)
\(578\) 0 0
\(579\) 260167.i 0.0322520i
\(580\) 0 0
\(581\) 417766.i 0.0513444i
\(582\) 0 0
\(583\) 5.36542e6i 0.653780i
\(584\) 0 0
\(585\) −5.15935e6 + 1.80412e6i −0.623312 + 0.217960i
\(586\) 0 0
\(587\) 4.54658e6 0.544615 0.272307 0.962210i \(-0.412213\pi\)
0.272307 + 0.962210i \(0.412213\pi\)
\(588\) 0 0
\(589\) 8.42877e6i 1.00110i
\(590\) 0 0
\(591\) −8.35173e6 −0.983576
\(592\) 0 0
\(593\) 6.35478e6i 0.742102i 0.928613 + 0.371051i \(0.121002\pi\)
−0.928613 + 0.371051i \(0.878998\pi\)
\(594\) 0 0
\(595\) 5.17159e6 1.80840e6i 0.598869 0.209412i
\(596\) 0 0
\(597\) −8.30822e6 −0.954052
\(598\) 0 0
\(599\) −5.19156e6 −0.591195 −0.295597 0.955313i \(-0.595519\pi\)
−0.295597 + 0.955313i \(0.595519\pi\)
\(600\) 0 0
\(601\) −6.30782e6 −0.712349 −0.356174 0.934419i \(-0.615919\pi\)
−0.356174 + 0.934419i \(0.615919\pi\)
\(602\) 0 0
\(603\) 1.91689e6 0.214687
\(604\) 0 0
\(605\) 1.66767e7 5.83149e6i 1.85234 0.647725i
\(606\) 0 0
\(607\) 1.24819e7i 1.37502i −0.726176 0.687509i \(-0.758704\pi\)
0.726176 0.687509i \(-0.241296\pi\)
\(608\) 0 0
\(609\) −2.22318e6 −0.242902
\(610\) 0 0
\(611\) 959551.i 0.103984i
\(612\) 0 0
\(613\) 1.20806e7 1.29848 0.649241 0.760583i \(-0.275087\pi\)
0.649241 + 0.760583i \(0.275087\pi\)
\(614\) 0 0
\(615\) 5.08653e6 1.77866e6i 0.542293 0.189629i
\(616\) 0 0
\(617\) 4.45314e6i 0.470927i −0.971883 0.235464i \(-0.924339\pi\)
0.971883 0.235464i \(-0.0756608\pi\)
\(618\) 0 0
\(619\) 1.30525e7i 1.36920i 0.728920 + 0.684599i \(0.240023\pi\)
−0.728920 + 0.684599i \(0.759977\pi\)
\(620\) 0 0
\(621\) 5.66792e6i 0.589786i
\(622\) 0 0
\(623\) 3.78363e6i 0.390561i
\(624\) 0 0
\(625\) 2.18105e6 9.51895e6i 0.223340 0.974741i
\(626\) 0 0
\(627\) −1.05493e7 −1.07166
\(628\) 0 0
\(629\) 1.29254e6i 0.130262i
\(630\) 0 0
\(631\) −134264. −0.0134242 −0.00671208 0.999977i \(-0.502137\pi\)
−0.00671208 + 0.999977i \(0.502137\pi\)
\(632\) 0 0
\(633\) 1.05743e6i 0.104892i
\(634\) 0 0
\(635\) 5.07182e6 + 1.45042e7i 0.499149 + 1.42745i
\(636\) 0 0
\(637\) −1.07782e7 −1.05244
\(638\) 0 0
\(639\) −7.85952e6 −0.761455
\(640\) 0 0
\(641\) 1.52563e7 1.46657 0.733286 0.679921i \(-0.237986\pi\)
0.733286 + 0.679921i \(0.237986\pi\)
\(642\) 0 0
\(643\) 4.40110e6 0.419792 0.209896 0.977724i \(-0.432688\pi\)
0.209896 + 0.977724i \(0.432688\pi\)
\(644\) 0 0
\(645\) 1.10987e6 388099.i 0.105044 0.0367319i
\(646\) 0 0
\(647\) 157813.i 0.0148211i −0.999973 0.00741057i \(-0.997641\pi\)
0.999973 0.00741057i \(-0.00235888\pi\)
\(648\) 0 0
\(649\) −4.84504e6 −0.451529
\(650\) 0 0
\(651\) 2.95142e6i 0.272948i
\(652\) 0 0
\(653\) 8.08352e6 0.741853 0.370926 0.928662i \(-0.379040\pi\)
0.370926 + 0.928662i \(0.379040\pi\)
\(654\) 0 0
\(655\) 3.15652e6 + 9.02690e6i 0.287479 + 0.822120i
\(656\) 0 0
\(657\) 4.96975e6i 0.449181i
\(658\) 0 0
\(659\) 3.03509e6i 0.272244i 0.990692 + 0.136122i \(0.0434639\pi\)
−0.990692 + 0.136122i \(0.956536\pi\)
\(660\) 0 0
\(661\) 9.53992e6i 0.849261i 0.905367 + 0.424630i \(0.139596\pi\)
−0.905367 + 0.424630i \(0.860404\pi\)
\(662\) 0 0
\(663\) 1.60191e7i 1.41532i
\(664\) 0 0
\(665\) 3.66422e6 1.28130e6i 0.321312 0.112356i
\(666\) 0 0
\(667\) −6.29031e6 −0.547467
\(668\) 0 0
\(669\) 6.77650e6i 0.585383i
\(670\) 0 0
\(671\) −3.45720e6 −0.296428
\(672\) 0 0
\(673\) 1.03245e7i 0.878680i −0.898321 0.439340i \(-0.855212\pi\)
0.898321 0.439340i \(-0.144788\pi\)
\(674\) 0 0
\(675\) 9.66372e6 7.69992e6i 0.816366 0.650469i
\(676\) 0 0
\(677\) 7.69059e6 0.644893 0.322447 0.946588i \(-0.395495\pi\)
0.322447 + 0.946588i \(0.395495\pi\)
\(678\) 0 0
\(679\) 281425. 0.0234255
\(680\) 0 0
\(681\) −1.77739e6 −0.146864
\(682\) 0 0
\(683\) −7.58453e6 −0.622124 −0.311062 0.950390i \(-0.600685\pi\)
−0.311062 + 0.950390i \(0.600685\pi\)
\(684\) 0 0
\(685\) 3.85176e6 + 1.10151e7i 0.313641 + 0.896938i
\(686\) 0 0
\(687\) 8.94178e6i 0.722823i
\(688\) 0 0
\(689\) −5.77265e6 −0.463263
\(690\) 0 0
\(691\) 2.18130e6i 0.173788i −0.996218 0.0868942i \(-0.972306\pi\)
0.996218 0.0868942i \(-0.0276942\pi\)
\(692\) 0 0
\(693\) −4.36147e6 −0.344985
\(694\) 0 0
\(695\) −4.36814e6 1.24918e7i −0.343032 0.980990i
\(696\) 0 0
\(697\) 1.86469e7i 1.45386i
\(698\) 0 0
\(699\) 1.71252e7i 1.32570i
\(700\) 0 0
\(701\) 1.10413e7i 0.848640i −0.905512 0.424320i \(-0.860513\pi\)
0.905512 0.424320i \(-0.139487\pi\)
\(702\) 0 0
\(703\) 915803.i 0.0698898i
\(704\) 0 0
\(705\) 251506. + 719248.i 0.0190579 + 0.0545011i
\(706\) 0 0
\(707\) −9.11705e6 −0.685971
\(708\) 0 0
\(709\) 1.52020e7i 1.13575i −0.823114 0.567876i \(-0.807765\pi\)
0.823114 0.567876i \(-0.192235\pi\)
\(710\) 0 0
\(711\) 1.02882e7 0.763251
\(712\) 0 0
\(713\) 8.35084e6i 0.615186i
\(714\) 0 0
\(715\) −9.47138e6 2.70859e7i −0.692864 1.98143i
\(716\) 0 0
\(717\) −7.97194e6 −0.579117
\(718\) 0 0
\(719\) 1.01762e7 0.734115 0.367058 0.930198i \(-0.380365\pi\)
0.367058 + 0.930198i \(0.380365\pi\)
\(720\) 0 0
\(721\) 5.54313e6 0.397116
\(722\) 0 0
\(723\) −2.14855e6 −0.152862
\(724\) 0 0
\(725\) −8.54545e6 1.07249e7i −0.603796 0.757788i
\(726\) 0 0
\(727\) 1.61142e7i 1.13077i 0.824828 + 0.565384i \(0.191272\pi\)
−0.824828 + 0.565384i \(0.808728\pi\)
\(728\) 0 0
\(729\) 1.14277e7 0.796414
\(730\) 0 0
\(731\) 4.06871e6i 0.281620i
\(732\) 0 0
\(733\) 205074. 0.0140978 0.00704888 0.999975i \(-0.497756\pi\)
0.00704888 + 0.999975i \(0.497756\pi\)
\(734\) 0 0
\(735\) 8.07897e6 2.82505e6i 0.551617 0.192889i
\(736\) 0 0
\(737\) 1.00634e7i 0.682460i
\(738\) 0 0
\(739\) 2.90886e6i 0.195935i −0.995190 0.0979673i \(-0.968766\pi\)
0.995190 0.0979673i \(-0.0312341\pi\)
\(740\) 0 0
\(741\) 1.13500e7i 0.759366i
\(742\) 0 0
\(743\) 1.95734e7i 1.30075i 0.759612 + 0.650376i \(0.225389\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(744\) 0 0
\(745\) 2.90793e6 + 8.31598e6i 0.191952 + 0.548937i
\(746\) 0 0
\(747\) −1.14525e6 −0.0750927
\(748\) 0 0
\(749\) 8.01594e6i 0.522095i
\(750\) 0 0
\(751\) −1.78300e6 −0.115359 −0.0576795 0.998335i \(-0.518370\pi\)
−0.0576795 + 0.998335i \(0.518370\pi\)
\(752\) 0 0
\(753\) 1.13756e7i 0.731119i
\(754\) 0 0
\(755\) 3.12733e6 1.09356e6i 0.199667 0.0698194i
\(756\) 0 0
\(757\) −201802. −0.0127993 −0.00639965 0.999980i \(-0.502037\pi\)
−0.00639965 + 0.999980i \(0.502037\pi\)
\(758\) 0 0
\(759\) 1.04518e7 0.658546
\(760\) 0 0
\(761\) 2.89830e7 1.81419 0.907093 0.420929i \(-0.138296\pi\)
0.907093 + 0.420929i \(0.138296\pi\)
\(762\) 0 0
\(763\) 869762. 0.0540865
\(764\) 0 0
\(765\) 4.95747e6 + 1.41772e7i 0.306271 + 0.875863i
\(766\) 0 0
\(767\) 5.21278e6i 0.319949i
\(768\) 0 0
\(769\) 1.39742e6 0.0852140 0.0426070 0.999092i \(-0.486434\pi\)
0.0426070 + 0.999092i \(0.486434\pi\)
\(770\) 0 0
\(771\) 1.95564e7i 1.18482i
\(772\) 0 0
\(773\) 7.09015e6 0.426783 0.213391 0.976967i \(-0.431549\pi\)
0.213391 + 0.976967i \(0.431549\pi\)
\(774\) 0 0
\(775\) 1.42381e7 1.13447e7i 0.851523 0.678482i
\(776\) 0 0
\(777\) 320679.i 0.0190554i
\(778\) 0 0
\(779\) 1.32118e7i 0.780045i
\(780\) 0 0
\(781\) 4.12614e7i 2.42056i
\(782\) 0 0
\(783\) 1.73508e7i 1.01138i
\(784\) 0 0
\(785\) −8.04851e6 + 2.81440e6i −0.466167 + 0.163009i
\(786\) 0 0
\(787\) 2.17815e7 1.25358 0.626790 0.779188i \(-0.284368\pi\)
0.626790 + 0.779188i \(0.284368\pi\)
\(788\) 0 0
\(789\) 9.64969e6i 0.551850i
\(790\) 0 0
\(791\) 1.17331e6 0.0666765
\(792\) 0 0
\(793\) 3.71961e6i 0.210046i
\(794\) 0 0
\(795\) 4.32699e6 1.51306e6i 0.242811 0.0849059i
\(796\) 0 0
\(797\) −619753. −0.0345599 −0.0172800 0.999851i \(-0.505501\pi\)
−0.0172800 + 0.999851i \(0.505501\pi\)
\(798\) 0 0
\(799\) −2.63671e6 −0.146115
\(800\) 0 0
\(801\) 1.03723e7 0.571206
\(802\) 0 0
\(803\) 2.60905e7 1.42789
\(804\) 0 0
\(805\) −3.63034e6 + 1.26946e6i −0.197450 + 0.0690443i
\(806\) 0 0
\(807\) 8.09908e6i 0.437776i
\(808\) 0 0
\(809\) −1.35484e7 −0.727807 −0.363904 0.931437i \(-0.618556\pi\)
−0.363904 + 0.931437i \(0.618556\pi\)
\(810\) 0 0
\(811\) 6.84579e6i 0.365486i −0.983161 0.182743i \(-0.941502\pi\)
0.983161 0.182743i \(-0.0584977\pi\)
\(812\) 0 0
\(813\) 1.19946e7 0.636443
\(814\) 0 0
\(815\) 1.43709e7 5.02520e6i 0.757861 0.265008i
\(816\) 0 0
\(817\) 2.88280e6i 0.151098i
\(818\) 0 0
\(819\) 4.69250e6i 0.244453i
\(820\) 0 0
\(821\) 1.82656e7i 0.945748i −0.881130 0.472874i \(-0.843217\pi\)
0.881130 0.472874i \(-0.156783\pi\)
\(822\) 0 0
\(823\) 7.70732e6i 0.396647i −0.980137 0.198323i \(-0.936450\pi\)
0.980137 0.198323i \(-0.0635496\pi\)
\(824\) 0 0
\(825\) 1.41989e7 + 1.78201e7i 0.726304 + 0.911542i
\(826\) 0 0
\(827\) −1.91726e7 −0.974804 −0.487402 0.873178i \(-0.662055\pi\)
−0.487402 + 0.873178i \(0.662055\pi\)
\(828\) 0 0
\(829\) 8.76259e6i 0.442839i 0.975179 + 0.221420i \(0.0710690\pi\)
−0.975179 + 0.221420i \(0.928931\pi\)
\(830\) 0 0
\(831\) −1.43978e7 −0.723257
\(832\) 0 0
\(833\) 2.96169e7i 1.47886i
\(834\) 0 0
\(835\) 3.60032e6 + 1.02961e7i 0.178700 + 0.511040i
\(836\) 0 0
\(837\) 2.30344e7 1.13649
\(838\) 0 0
\(839\) −1.21316e6 −0.0594992 −0.0297496 0.999557i \(-0.509471\pi\)
−0.0297496 + 0.999557i \(0.509471\pi\)
\(840\) 0 0
\(841\) 1.25504e6 0.0611880
\(842\) 0 0
\(843\) −1.56477e7 −0.758371
\(844\) 0 0
\(845\) −9.54911e6 + 3.33913e6i −0.460067 + 0.160876i
\(846\) 0 0
\(847\) 1.51676e7i 0.726457i
\(848\) 0 0
\(849\) −8.10016e6 −0.385678
\(850\) 0 0
\(851\) 907336.i 0.0429481i
\(852\) 0 0
\(853\) −3.29210e7 −1.54917 −0.774586 0.632468i \(-0.782042\pi\)
−0.774586 + 0.632468i \(0.782042\pi\)
\(854\) 0 0
\(855\) 3.51251e6 + 1.00449e7i 0.164324 + 0.469929i
\(856\) 0 0
\(857\) 3.37640e7i 1.57037i −0.619261 0.785185i \(-0.712568\pi\)
0.619261 0.785185i \(-0.287432\pi\)
\(858\) 0 0
\(859\) 2.06943e7i 0.956905i −0.878114 0.478452i \(-0.841198\pi\)
0.878114 0.478452i \(-0.158802\pi\)
\(860\) 0 0
\(861\) 4.62627e6i 0.212678i
\(862\) 0 0
\(863\) 1.64284e7i 0.750878i 0.926847 + 0.375439i \(0.122508\pi\)
−0.926847 + 0.375439i \(0.877492\pi\)
\(864\) 0 0
\(865\) −1.12626e7 + 3.93830e6i −0.511797 + 0.178965i
\(866\) 0 0
\(867\) 2.90301e7 1.31160
\(868\) 0 0
\(869\) 5.40118e7i 2.42627i
\(870\) 0 0
\(871\) 1.08272e7 0.483585
\(872\) 0 0
\(873\) 771488.i 0.0342605i
\(874\) 0 0
\(875\) −7.09626e6 4.46511e6i −0.313335 0.197157i
\(876\) 0 0
\(877\) −1.39642e7 −0.613080 −0.306540 0.951858i \(-0.599171\pi\)
−0.306540 + 0.951858i \(0.599171\pi\)
\(878\) 0 0
\(879\) −3.36594e6 −0.146938
\(880\) 0 0
\(881\) −3.00733e7 −1.30540 −0.652698 0.757619i \(-0.726363\pi\)
−0.652698 + 0.757619i \(0.726363\pi\)
\(882\) 0 0
\(883\) 1.25984e7 0.543768 0.271884 0.962330i \(-0.412353\pi\)
0.271884 + 0.962330i \(0.412353\pi\)
\(884\) 0 0
\(885\) −1.36631e6 3.90733e6i −0.0586397 0.167696i
\(886\) 0 0
\(887\) 3.03915e7i 1.29701i −0.761210 0.648505i \(-0.775394\pi\)
0.761210 0.648505i \(-0.224606\pi\)
\(888\) 0 0
\(889\) 1.31918e7 0.559821
\(890\) 0 0
\(891\) 6.74677e6i 0.284709i
\(892\) 0 0
\(893\) −1.86819e6 −0.0783955
\(894\) 0 0
\(895\) 8.56806e6 + 2.45026e7i 0.357540 + 1.02248i
\(896\) 0 0
\(897\) 1.12451e7i 0.466640i
\(898\) 0 0
\(899\) 2.55639e7i 1.05494i
\(900\) 0 0
\(901\) 1.58624e7i 0.650965i
\(902\) 0 0
\(903\) 1.00944e6i 0.0411967i
\(904\) 0 0
\(905\) −1.10961e7 3.17321e7i −0.450347 1.28788i
\(906\) 0 0
\(907\) −2.34696e7 −0.947302 −0.473651 0.880713i \(-0.657064\pi\)
−0.473651 + 0.880713i \(0.657064\pi\)
\(908\) 0 0
\(909\) 2.49931e7i 1.00325i
\(910\) 0 0
\(911\) −2.87754e7 −1.14875 −0.574375 0.818592i \(-0.694755\pi\)
−0.574375 + 0.818592i \(0.694755\pi\)
\(912\) 0 0
\(913\) 6.01238e6i 0.238709i
\(914\) 0 0
\(915\) −974939. 2.78809e6i −0.0384968 0.110092i
\(916\) 0 0
\(917\) 8.21009e6 0.322422
\(918\) 0 0
\(919\) −1.30665e7 −0.510352 −0.255176 0.966895i \(-0.582133\pi\)
−0.255176 + 0.966895i \(0.582133\pi\)
\(920\) 0 0
\(921\) −2.21043e6 −0.0858673
\(922\) 0 0
\(923\) −4.43931e7 −1.71519
\(924\) 0 0
\(925\) −1.54700e6 + 1.23263e6i −0.0594476 + 0.0473671i
\(926\) 0 0
\(927\) 1.51957e7i 0.580793i
\(928\) 0 0
\(929\) −2.53760e7 −0.964682 −0.482341 0.875984i \(-0.660213\pi\)
−0.482341 + 0.875984i \(0.660213\pi\)
\(930\) 0 0
\(931\) 2.09845e7i 0.793457i
\(932\) 0 0
\(933\) 7.00811e6 0.263570
\(934\) 0 0
\(935\) −7.44282e7 + 2.60260e7i −2.78425 + 0.973596i
\(936\) 0 0
\(937\) 1.55239e7i 0.577634i 0.957384 + 0.288817i \(0.0932619\pi\)
−0.957384 + 0.288817i \(0.906738\pi\)
\(938\) 0 0
\(939\) 1.01384e7i 0.375237i
\(940\) 0 0
\(941\) 4.21463e7i 1.55162i 0.630967 + 0.775810i \(0.282658\pi\)
−0.630967 + 0.775810i \(0.717342\pi\)
\(942\) 0 0
\(943\) 1.30897e7i 0.479347i
\(944\) 0 0
\(945\) −3.50159e6 1.00137e7i −0.127552 0.364767i
\(946\) 0 0
\(947\) 4.64274e7 1.68228 0.841142 0.540815i \(-0.181884\pi\)
0.841142 + 0.540815i \(0.181884\pi\)
\(948\) 0 0
\(949\) 2.80708e7i 1.01179i
\(950\) 0 0
\(951\) −1.11186e7 −0.398656
\(952\) 0 0
\(953\) 8.05749e6i 0.287387i 0.989622 + 0.143694i \(0.0458980\pi\)
−0.989622 + 0.143694i \(0.954102\pi\)
\(954\) 0 0
\(955\) −3.26319e7 + 1.14107e7i −1.15780 + 0.404860i
\(956\) 0 0
\(957\) 3.19954e7 1.12929
\(958\) 0 0
\(959\) 1.00184e7 0.351764
\(960\) 0 0
\(961\) 5.30870e6 0.185430
\(962\) 0 0
\(963\) −2.19746e7 −0.763580
\(964\) 0 0
\(965\) 454771. + 1.30054e6i 0.0157208 + 0.0449577i
\(966\) 0 0
\(967\) 7.92356e6i 0.272492i 0.990675 + 0.136246i \(0.0435038\pi\)
−0.990675 + 0.136246i \(0.956496\pi\)
\(968\) 0 0
\(969\) 3.11883e7 1.06704
\(970\) 0 0
\(971\) 1.11594e7i 0.379831i 0.981800 + 0.189916i \(0.0608215\pi\)
−0.981800 + 0.189916i \(0.939179\pi\)
\(972\) 0 0
\(973\) −1.13615e7 −0.384728
\(974\) 0 0
\(975\) 1.91727e7 1.52766e7i 0.645910 0.514652i
\(976\) 0 0
\(977\) 3.69540e7i 1.23858i −0.785162 0.619291i \(-0.787420\pi\)
0.785162 0.619291i \(-0.212580\pi\)
\(978\) 0 0
\(979\) 5.44530e7i 1.81579i
\(980\) 0 0
\(981\) 2.38433e6i 0.0791031i
\(982\) 0 0
\(983\) 3.73269e7i 1.23208i −0.787716 0.616038i \(-0.788737\pi\)
0.787716 0.616038i \(-0.211263\pi\)
\(984\) 0 0
\(985\) −4.17491e7 + 1.45988e7i −1.37106 + 0.479432i
\(986\) 0 0
\(987\) 654166. 0.0213744
\(988\) 0 0
\(989\) 2.85614e6i 0.0928515i
\(990\) 0 0
\(991\) 2.74561e6 0.0888084 0.0444042 0.999014i \(-0.485861\pi\)
0.0444042 + 0.999014i \(0.485861\pi\)
\(992\) 0 0
\(993\) 1.35672e7i 0.436635i
\(994\) 0 0
\(995\) −4.15316e7 + 1.45227e7i −1.32990 + 0.465040i
\(996\) 0 0
\(997\) 3.93328e7 1.25319 0.626595 0.779345i \(-0.284448\pi\)
0.626595 + 0.779345i \(0.284448\pi\)
\(998\) 0 0
\(999\) −2.50274e6 −0.0793419
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 160.6.f.a.49.10 28
4.3 odd 2 40.6.f.a.29.5 28
5.2 odd 4 800.6.d.e.401.10 28
5.3 odd 4 800.6.d.e.401.19 28
5.4 even 2 inner 160.6.f.a.49.19 28
8.3 odd 2 40.6.f.a.29.23 yes 28
8.5 even 2 inner 160.6.f.a.49.20 28
20.3 even 4 200.6.d.e.101.10 28
20.7 even 4 200.6.d.e.101.19 28
20.19 odd 2 40.6.f.a.29.24 yes 28
40.3 even 4 200.6.d.e.101.9 28
40.13 odd 4 800.6.d.e.401.20 28
40.19 odd 2 40.6.f.a.29.6 yes 28
40.27 even 4 200.6.d.e.101.20 28
40.29 even 2 inner 160.6.f.a.49.9 28
40.37 odd 4 800.6.d.e.401.9 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.f.a.29.5 28 4.3 odd 2
40.6.f.a.29.6 yes 28 40.19 odd 2
40.6.f.a.29.23 yes 28 8.3 odd 2
40.6.f.a.29.24 yes 28 20.19 odd 2
160.6.f.a.49.9 28 40.29 even 2 inner
160.6.f.a.49.10 28 1.1 even 1 trivial
160.6.f.a.49.19 28 5.4 even 2 inner
160.6.f.a.49.20 28 8.5 even 2 inner
200.6.d.e.101.9 28 40.3 even 4
200.6.d.e.101.10 28 20.3 even 4
200.6.d.e.101.19 28 20.7 even 4
200.6.d.e.101.20 28 40.27 even 4
800.6.d.e.401.9 28 40.37 odd 4
800.6.d.e.401.10 28 5.2 odd 4
800.6.d.e.401.19 28 5.3 odd 4
800.6.d.e.401.20 28 40.13 odd 4