Properties

Label 207.7.d.e.91.9
Level $207$
Weight $7$
Character 207.91
Analytic conductor $47.621$
Analytic rank $0$
Dimension $24$
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] = [207,7,Mod(91,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.91");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 207.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.6211953093\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 69)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.9
Character \(\chi\) \(=\) 207.91
Dual form 207.7.d.e.91.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.88321 q^{2} -55.6871 q^{4} +133.948i q^{5} +59.3150i q^{7} +345.083 q^{8} +O(q^{10})\) \(q-2.88321 q^{2} -55.6871 q^{4} +133.948i q^{5} +59.3150i q^{7} +345.083 q^{8} -386.201i q^{10} -578.563i q^{11} +943.370 q^{13} -171.018i q^{14} +2569.03 q^{16} -5432.92i q^{17} +3909.12i q^{19} -7459.19i q^{20} +1668.12i q^{22} +(4480.54 - 11312.0i) q^{23} -2317.15 q^{25} -2719.93 q^{26} -3303.08i q^{28} -11255.2 q^{29} +3515.19 q^{31} -29492.4 q^{32} +15664.2i q^{34} -7945.15 q^{35} +8099.73i q^{37} -11270.8i q^{38} +46223.3i q^{40} +13414.0 q^{41} +66345.2i q^{43} +32218.5i q^{44} +(-12918.3 + 32614.8i) q^{46} +25836.4 q^{47} +114131. q^{49} +6680.84 q^{50} -52533.5 q^{52} +60990.7i q^{53} +77497.6 q^{55} +20468.6i q^{56} +32451.1 q^{58} +76808.7 q^{59} +18606.1i q^{61} -10135.0 q^{62} -79385.2 q^{64} +126363. i q^{65} -24517.2i q^{67} +302543. i q^{68} +22907.5 q^{70} +511739. q^{71} -104789. q^{73} -23353.2i q^{74} -217688. i q^{76} +34317.5 q^{77} +699697. i q^{79} +344117. i q^{80} -38675.4 q^{82} +288024. i q^{83} +727730. q^{85} -191287. i q^{86} -199652. i q^{88} +90597.8i q^{89} +55956.0i q^{91} +(-249509. + 629931. i) q^{92} -74491.7 q^{94} -523620. q^{95} +1.69461e6i q^{97} -329063. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 20 q^{2} + 816 q^{4} + 940 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 20 q^{2} + 816 q^{4} + 940 q^{8} + 384 q^{13} + 29544 q^{16} - 29336 q^{23} - 61272 q^{25} - 10088 q^{26} - 64672 q^{29} + 9696 q^{31} + 319620 q^{32} + 225744 q^{35} - 135280 q^{41} + 233232 q^{46} + 74336 q^{47} - 722136 q^{49} - 619324 q^{50} + 1059720 q^{52} - 1019328 q^{55} - 694344 q^{58} - 1057648 q^{59} + 488776 q^{62} - 273888 q^{64} + 2785512 q^{70} + 255392 q^{71} - 322560 q^{73} + 1002960 q^{77} - 5732712 q^{82} - 2704704 q^{85} + 1611444 q^{92} - 147720 q^{94} + 1672656 q^{95} - 9104212 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.88321 −0.360401 −0.180201 0.983630i \(-0.557675\pi\)
−0.180201 + 0.983630i \(0.557675\pi\)
\(3\) 0 0
\(4\) −55.6871 −0.870111
\(5\) 133.948i 1.07159i 0.844349 + 0.535793i \(0.179987\pi\)
−0.844349 + 0.535793i \(0.820013\pi\)
\(6\) 0 0
\(7\) 59.3150i 0.172930i 0.996255 + 0.0864650i \(0.0275571\pi\)
−0.996255 + 0.0864650i \(0.972443\pi\)
\(8\) 345.083 0.673990
\(9\) 0 0
\(10\) 386.201i 0.386201i
\(11\) 578.563i 0.434683i −0.976096 0.217341i \(-0.930261\pi\)
0.976096 0.217341i \(-0.0697385\pi\)
\(12\) 0 0
\(13\) 943.370 0.429390 0.214695 0.976681i \(-0.431124\pi\)
0.214695 + 0.976681i \(0.431124\pi\)
\(14\) 171.018i 0.0623242i
\(15\) 0 0
\(16\) 2569.03 0.627204
\(17\) 5432.92i 1.10582i −0.833240 0.552912i \(-0.813516\pi\)
0.833240 0.552912i \(-0.186484\pi\)
\(18\) 0 0
\(19\) 3909.12i 0.569926i 0.958539 + 0.284963i \(0.0919813\pi\)
−0.958539 + 0.284963i \(0.908019\pi\)
\(20\) 7459.19i 0.932399i
\(21\) 0 0
\(22\) 1668.12i 0.156660i
\(23\) 4480.54 11312.0i 0.368254 0.929725i
\(24\) 0 0
\(25\) −2317.15 −0.148298
\(26\) −2719.93 −0.154753
\(27\) 0 0
\(28\) 3303.08i 0.150468i
\(29\) −11255.2 −0.461487 −0.230743 0.973015i \(-0.574116\pi\)
−0.230743 + 0.973015i \(0.574116\pi\)
\(30\) 0 0
\(31\) 3515.19 0.117995 0.0589975 0.998258i \(-0.481210\pi\)
0.0589975 + 0.998258i \(0.481210\pi\)
\(32\) −29492.4 −0.900035
\(33\) 0 0
\(34\) 15664.2i 0.398540i
\(35\) −7945.15 −0.185310
\(36\) 0 0
\(37\) 8099.73i 0.159906i 0.996799 + 0.0799532i \(0.0254771\pi\)
−0.996799 + 0.0799532i \(0.974523\pi\)
\(38\) 11270.8i 0.205402i
\(39\) 0 0
\(40\) 46223.3i 0.722239i
\(41\) 13414.0 0.194629 0.0973144 0.995254i \(-0.468975\pi\)
0.0973144 + 0.995254i \(0.468975\pi\)
\(42\) 0 0
\(43\) 66345.2i 0.834457i 0.908802 + 0.417229i \(0.136999\pi\)
−0.908802 + 0.417229i \(0.863001\pi\)
\(44\) 32218.5i 0.378222i
\(45\) 0 0
\(46\) −12918.3 + 32614.8i −0.132719 + 0.335074i
\(47\) 25836.4 0.248850 0.124425 0.992229i \(-0.460291\pi\)
0.124425 + 0.992229i \(0.460291\pi\)
\(48\) 0 0
\(49\) 114131. 0.970095
\(50\) 6680.84 0.0534467
\(51\) 0 0
\(52\) −52533.5 −0.373617
\(53\) 60990.7i 0.409672i 0.978796 + 0.204836i \(0.0656661\pi\)
−0.978796 + 0.204836i \(0.934334\pi\)
\(54\) 0 0
\(55\) 77497.6 0.465800
\(56\) 20468.6i 0.116553i
\(57\) 0 0
\(58\) 32451.1 0.166320
\(59\) 76808.7 0.373985 0.186993 0.982361i \(-0.440126\pi\)
0.186993 + 0.982361i \(0.440126\pi\)
\(60\) 0 0
\(61\) 18606.1i 0.0819721i 0.999160 + 0.0409861i \(0.0130499\pi\)
−0.999160 + 0.0409861i \(0.986950\pi\)
\(62\) −10135.0 −0.0425255
\(63\) 0 0
\(64\) −79385.2 −0.302830
\(65\) 126363.i 0.460129i
\(66\) 0 0
\(67\) 24517.2i 0.0815167i −0.999169 0.0407583i \(-0.987023\pi\)
0.999169 0.0407583i \(-0.0129774\pi\)
\(68\) 302543.i 0.962190i
\(69\) 0 0
\(70\) 22907.5 0.0667858
\(71\) 511739. 1.42980 0.714898 0.699229i \(-0.246473\pi\)
0.714898 + 0.699229i \(0.246473\pi\)
\(72\) 0 0
\(73\) −104789. −0.269369 −0.134685 0.990889i \(-0.543002\pi\)
−0.134685 + 0.990889i \(0.543002\pi\)
\(74\) 23353.2i 0.0576304i
\(75\) 0 0
\(76\) 217688.i 0.495899i
\(77\) 34317.5 0.0751698
\(78\) 0 0
\(79\) 699697.i 1.41915i 0.704630 + 0.709575i \(0.251113\pi\)
−0.704630 + 0.709575i \(0.748887\pi\)
\(80\) 344117.i 0.672104i
\(81\) 0 0
\(82\) −38675.4 −0.0701444
\(83\) 288024.i 0.503725i 0.967763 + 0.251863i \(0.0810431\pi\)
−0.967763 + 0.251863i \(0.918957\pi\)
\(84\) 0 0
\(85\) 727730. 1.18499
\(86\) 191287.i 0.300739i
\(87\) 0 0
\(88\) 199652.i 0.292972i
\(89\) 90597.8i 0.128513i 0.997933 + 0.0642566i \(0.0204676\pi\)
−0.997933 + 0.0642566i \(0.979532\pi\)
\(90\) 0 0
\(91\) 55956.0i 0.0742545i
\(92\) −249509. + 629931.i −0.320422 + 0.808964i
\(93\) 0 0
\(94\) −74491.7 −0.0896859
\(95\) −523620. −0.610725
\(96\) 0 0
\(97\) 1.69461e6i 1.85676i 0.371634 + 0.928379i \(0.378798\pi\)
−0.371634 + 0.928379i \(0.621202\pi\)
\(98\) −329063. −0.349623
\(99\) 0 0
\(100\) 129036. 0.129036
\(101\) −219524. −0.213068 −0.106534 0.994309i \(-0.533975\pi\)
−0.106534 + 0.994309i \(0.533975\pi\)
\(102\) 0 0
\(103\) 1.67613e6i 1.53389i 0.641711 + 0.766947i \(0.278225\pi\)
−0.641711 + 0.766947i \(0.721775\pi\)
\(104\) 325541. 0.289405
\(105\) 0 0
\(106\) 175849.i 0.147646i
\(107\) 1.73015e6i 1.41232i −0.708054 0.706159i \(-0.750426\pi\)
0.708054 0.706159i \(-0.249574\pi\)
\(108\) 0 0
\(109\) 1.72020e6i 1.32831i 0.747595 + 0.664155i \(0.231209\pi\)
−0.747595 + 0.664155i \(0.768791\pi\)
\(110\) −223442. −0.167875
\(111\) 0 0
\(112\) 152382.i 0.108462i
\(113\) 226700.i 0.157114i −0.996910 0.0785571i \(-0.974969\pi\)
0.996910 0.0785571i \(-0.0250313\pi\)
\(114\) 0 0
\(115\) 1.51522e6 + 600161.i 0.996281 + 0.394616i
\(116\) 626769. 0.401545
\(117\) 0 0
\(118\) −221456. −0.134785
\(119\) 322253. 0.191230
\(120\) 0 0
\(121\) 1.43683e6 0.811051
\(122\) 53645.3i 0.0295428i
\(123\) 0 0
\(124\) −195751. −0.102669
\(125\) 1.78256e6i 0.912673i
\(126\) 0 0
\(127\) −745176. −0.363787 −0.181894 0.983318i \(-0.558223\pi\)
−0.181894 + 0.983318i \(0.558223\pi\)
\(128\) 2.11639e6 1.00918
\(129\) 0 0
\(130\) 364330.i 0.165831i
\(131\) −721322. −0.320860 −0.160430 0.987047i \(-0.551288\pi\)
−0.160430 + 0.987047i \(0.551288\pi\)
\(132\) 0 0
\(133\) −231870. −0.0985573
\(134\) 70688.2i 0.0293787i
\(135\) 0 0
\(136\) 1.87481e6i 0.745315i
\(137\) 2.81334e6i 1.09411i 0.837097 + 0.547055i \(0.184251\pi\)
−0.837097 + 0.547055i \(0.815749\pi\)
\(138\) 0 0
\(139\) 2.52591e6 0.940533 0.470267 0.882524i \(-0.344158\pi\)
0.470267 + 0.882524i \(0.344158\pi\)
\(140\) 442442. 0.161240
\(141\) 0 0
\(142\) −1.47545e6 −0.515300
\(143\) 545799.i 0.186649i
\(144\) 0 0
\(145\) 1.50762e6i 0.494523i
\(146\) 302129. 0.0970810
\(147\) 0 0
\(148\) 451051.i 0.139136i
\(149\) 4.50361e6i 1.36145i −0.732539 0.680725i \(-0.761665\pi\)
0.732539 0.680725i \(-0.238335\pi\)
\(150\) 0 0
\(151\) −5.14332e6 −1.49387 −0.746935 0.664897i \(-0.768476\pi\)
−0.746935 + 0.664897i \(0.768476\pi\)
\(152\) 1.34897e6i 0.384125i
\(153\) 0 0
\(154\) −98944.5 −0.0270913
\(155\) 470854.i 0.126442i
\(156\) 0 0
\(157\) 6.18385e6i 1.59794i −0.601372 0.798969i \(-0.705379\pi\)
0.601372 0.798969i \(-0.294621\pi\)
\(158\) 2.01737e6i 0.511463i
\(159\) 0 0
\(160\) 3.95045e6i 0.964466i
\(161\) 670970. + 265764.i 0.160777 + 0.0636822i
\(162\) 0 0
\(163\) −1.21979e6 −0.281658 −0.140829 0.990034i \(-0.544977\pi\)
−0.140829 + 0.990034i \(0.544977\pi\)
\(164\) −746987. −0.169349
\(165\) 0 0
\(166\) 830432.i 0.181543i
\(167\) −5.49073e6 −1.17891 −0.589455 0.807801i \(-0.700657\pi\)
−0.589455 + 0.807801i \(0.700657\pi\)
\(168\) 0 0
\(169\) −3.93686e6 −0.815624
\(170\) −2.09820e6 −0.427071
\(171\) 0 0
\(172\) 3.69457e6i 0.726070i
\(173\) 3.24115e6 0.625980 0.312990 0.949756i \(-0.398669\pi\)
0.312990 + 0.949756i \(0.398669\pi\)
\(174\) 0 0
\(175\) 137442.i 0.0256452i
\(176\) 1.48634e6i 0.272635i
\(177\) 0 0
\(178\) 261212.i 0.0463163i
\(179\) −6.13192e6 −1.06915 −0.534573 0.845122i \(-0.679528\pi\)
−0.534573 + 0.845122i \(0.679528\pi\)
\(180\) 0 0
\(181\) 3.65220e6i 0.615912i −0.951401 0.307956i \(-0.900355\pi\)
0.951401 0.307956i \(-0.0996450\pi\)
\(182\) 161333.i 0.0267614i
\(183\) 0 0
\(184\) 1.54616e6 3.90357e6i 0.248199 0.626626i
\(185\) −1.08495e6 −0.171353
\(186\) 0 0
\(187\) −3.14328e6 −0.480683
\(188\) −1.43875e6 −0.216527
\(189\) 0 0
\(190\) 1.50971e6 0.220106
\(191\) 1.83792e6i 0.263771i 0.991265 + 0.131885i \(0.0421031\pi\)
−0.991265 + 0.131885i \(0.957897\pi\)
\(192\) 0 0
\(193\) 649119. 0.0902927 0.0451463 0.998980i \(-0.485625\pi\)
0.0451463 + 0.998980i \(0.485625\pi\)
\(194\) 4.88593e6i 0.669178i
\(195\) 0 0
\(196\) −6.35561e6 −0.844091
\(197\) 1.26065e7 1.64890 0.824450 0.565935i \(-0.191485\pi\)
0.824450 + 0.565935i \(0.191485\pi\)
\(198\) 0 0
\(199\) 3.26083e6i 0.413779i 0.978364 + 0.206889i \(0.0663340\pi\)
−0.978364 + 0.206889i \(0.933666\pi\)
\(200\) −799610. −0.0999513
\(201\) 0 0
\(202\) 632933. 0.0767899
\(203\) 667602.i 0.0798049i
\(204\) 0 0
\(205\) 1.79678e6i 0.208562i
\(206\) 4.83262e6i 0.552817i
\(207\) 0 0
\(208\) 2.42354e6 0.269315
\(209\) 2.26167e6 0.247737
\(210\) 0 0
\(211\) 7.77976e6 0.828169 0.414084 0.910239i \(-0.364102\pi\)
0.414084 + 0.910239i \(0.364102\pi\)
\(212\) 3.39640e6i 0.356460i
\(213\) 0 0
\(214\) 4.98838e6i 0.509001i
\(215\) −8.88683e6 −0.894193
\(216\) 0 0
\(217\) 208504.i 0.0204049i
\(218\) 4.95970e6i 0.478724i
\(219\) 0 0
\(220\) −4.31561e6 −0.405298
\(221\) 5.12525e6i 0.474830i
\(222\) 0 0
\(223\) 1.33038e7 1.19966 0.599832 0.800126i \(-0.295234\pi\)
0.599832 + 0.800126i \(0.295234\pi\)
\(224\) 1.74934e6i 0.155643i
\(225\) 0 0
\(226\) 653622.i 0.0566241i
\(227\) 1.61060e6i 0.137693i 0.997627 + 0.0688464i \(0.0219318\pi\)
−0.997627 + 0.0688464i \(0.978068\pi\)
\(228\) 0 0
\(229\) 1.58708e7i 1.32158i 0.750572 + 0.660789i \(0.229778\pi\)
−0.750572 + 0.660789i \(0.770222\pi\)
\(230\) −4.36869e6 1.73039e6i −0.359061 0.142220i
\(231\) 0 0
\(232\) −3.88398e6 −0.311037
\(233\) 7.28829e6 0.576180 0.288090 0.957603i \(-0.406980\pi\)
0.288090 + 0.957603i \(0.406980\pi\)
\(234\) 0 0
\(235\) 3.46074e6i 0.266665i
\(236\) −4.27726e6 −0.325409
\(237\) 0 0
\(238\) −929124. −0.0689196
\(239\) 7.01963e6 0.514186 0.257093 0.966387i \(-0.417235\pi\)
0.257093 + 0.966387i \(0.417235\pi\)
\(240\) 0 0
\(241\) 2.59499e7i 1.85389i −0.375197 0.926945i \(-0.622425\pi\)
0.375197 0.926945i \(-0.377575\pi\)
\(242\) −4.14267e6 −0.292304
\(243\) 0 0
\(244\) 1.03612e6i 0.0713248i
\(245\) 1.52876e7i 1.03954i
\(246\) 0 0
\(247\) 3.68775e6i 0.244721i
\(248\) 1.21303e6 0.0795275
\(249\) 0 0
\(250\) 5.13950e6i 0.328928i
\(251\) 881038.i 0.0557152i 0.999612 + 0.0278576i \(0.00886849\pi\)
−0.999612 + 0.0278576i \(0.991132\pi\)
\(252\) 0 0
\(253\) −6.54469e6 2.59228e6i −0.404136 0.160074i
\(254\) 2.14850e6 0.131109
\(255\) 0 0
\(256\) −1.02136e6 −0.0608776
\(257\) 5.39464e6 0.317807 0.158903 0.987294i \(-0.449204\pi\)
0.158903 + 0.987294i \(0.449204\pi\)
\(258\) 0 0
\(259\) −480436. −0.0276526
\(260\) 7.03678e6i 0.400363i
\(261\) 0 0
\(262\) 2.07972e6 0.115638
\(263\) 1.83758e7i 1.01013i 0.863080 + 0.505067i \(0.168532\pi\)
−0.863080 + 0.505067i \(0.831468\pi\)
\(264\) 0 0
\(265\) −8.16961e6 −0.438999
\(266\) 668529. 0.0355202
\(267\) 0 0
\(268\) 1.36529e6i 0.0709286i
\(269\) −2.33084e6 −0.119745 −0.0598723 0.998206i \(-0.519069\pi\)
−0.0598723 + 0.998206i \(0.519069\pi\)
\(270\) 0 0
\(271\) −2.55701e7 −1.28477 −0.642384 0.766383i \(-0.722055\pi\)
−0.642384 + 0.766383i \(0.722055\pi\)
\(272\) 1.39573e7i 0.693578i
\(273\) 0 0
\(274\) 8.11145e6i 0.394318i
\(275\) 1.34062e6i 0.0644626i
\(276\) 0 0
\(277\) 2.84046e7 1.33644 0.668220 0.743964i \(-0.267056\pi\)
0.668220 + 0.743964i \(0.267056\pi\)
\(278\) −7.28274e6 −0.338969
\(279\) 0 0
\(280\) −2.74173e6 −0.124897
\(281\) 2.93377e7i 1.32223i 0.750284 + 0.661115i \(0.229917\pi\)
−0.750284 + 0.661115i \(0.770083\pi\)
\(282\) 0 0
\(283\) 9.30377e6i 0.410487i 0.978711 + 0.205244i \(0.0657987\pi\)
−0.978711 + 0.205244i \(0.934201\pi\)
\(284\) −2.84973e7 −1.24408
\(285\) 0 0
\(286\) 1.57365e6i 0.0672684i
\(287\) 795652.i 0.0336572i
\(288\) 0 0
\(289\) −5.37900e6 −0.222848
\(290\) 4.34677e6i 0.178227i
\(291\) 0 0
\(292\) 5.83541e6 0.234381
\(293\) 2.23704e7i 0.889345i 0.895693 + 0.444673i \(0.146680\pi\)
−0.895693 + 0.444673i \(0.853320\pi\)
\(294\) 0 0
\(295\) 1.02884e7i 0.400758i
\(296\) 2.79508e6i 0.107775i
\(297\) 0 0
\(298\) 1.29848e7i 0.490668i
\(299\) 4.22681e6 1.06714e7i 0.158125 0.399215i
\(300\) 0 0
\(301\) −3.93527e6 −0.144303
\(302\) 1.48293e7 0.538393
\(303\) 0 0
\(304\) 1.00426e7i 0.357460i
\(305\) −2.49226e6 −0.0878402
\(306\) 0 0
\(307\) 2.43258e7 0.840723 0.420361 0.907357i \(-0.361903\pi\)
0.420361 + 0.907357i \(0.361903\pi\)
\(308\) −1.91104e6 −0.0654060
\(309\) 0 0
\(310\) 1.35757e6i 0.0455698i
\(311\) 3.22419e7 1.07186 0.535932 0.844261i \(-0.319960\pi\)
0.535932 + 0.844261i \(0.319960\pi\)
\(312\) 0 0
\(313\) 2.99672e7i 0.977265i 0.872490 + 0.488633i \(0.162504\pi\)
−0.872490 + 0.488633i \(0.837496\pi\)
\(314\) 1.78293e7i 0.575899i
\(315\) 0 0
\(316\) 3.89641e7i 1.23482i
\(317\) 3.88340e7 1.21909 0.609543 0.792753i \(-0.291353\pi\)
0.609543 + 0.792753i \(0.291353\pi\)
\(318\) 0 0
\(319\) 6.51184e6i 0.200600i
\(320\) 1.06335e7i 0.324509i
\(321\) 0 0
\(322\) −1.93455e6 766252.i −0.0579444 0.0229511i
\(323\) 2.12379e7 0.630238
\(324\) 0 0
\(325\) −2.18593e6 −0.0636776
\(326\) 3.51691e6 0.101510
\(327\) 0 0
\(328\) 4.62895e6 0.131178
\(329\) 1.53249e6i 0.0430337i
\(330\) 0 0
\(331\) 2.73385e7 0.753860 0.376930 0.926242i \(-0.376980\pi\)
0.376930 + 0.926242i \(0.376980\pi\)
\(332\) 1.60392e7i 0.438297i
\(333\) 0 0
\(334\) 1.58309e7 0.424880
\(335\) 3.28404e6 0.0873522
\(336\) 0 0
\(337\) 2.42977e7i 0.634857i 0.948282 + 0.317428i \(0.102819\pi\)
−0.948282 + 0.317428i \(0.897181\pi\)
\(338\) 1.13508e7 0.293952
\(339\) 0 0
\(340\) −4.05252e7 −1.03107
\(341\) 2.03376e6i 0.0512904i
\(342\) 0 0
\(343\) 1.37480e7i 0.340689i
\(344\) 2.28946e7i 0.562416i
\(345\) 0 0
\(346\) −9.34491e6 −0.225604
\(347\) −6.67195e7 −1.59685 −0.798426 0.602093i \(-0.794334\pi\)
−0.798426 + 0.602093i \(0.794334\pi\)
\(348\) 0 0
\(349\) 4.82883e7 1.13597 0.567983 0.823040i \(-0.307724\pi\)
0.567983 + 0.823040i \(0.307724\pi\)
\(350\) 396274.i 0.00924255i
\(351\) 0 0
\(352\) 1.70632e7i 0.391230i
\(353\) −5.50097e7 −1.25059 −0.625295 0.780388i \(-0.715022\pi\)
−0.625295 + 0.780388i \(0.715022\pi\)
\(354\) 0 0
\(355\) 6.85466e7i 1.53215i
\(356\) 5.04513e6i 0.111821i
\(357\) 0 0
\(358\) 1.76796e7 0.385322
\(359\) 2.47715e7i 0.535389i −0.963504 0.267695i \(-0.913738\pi\)
0.963504 0.267695i \(-0.0862619\pi\)
\(360\) 0 0
\(361\) 3.17646e7 0.675184
\(362\) 1.05301e7i 0.221975i
\(363\) 0 0
\(364\) 3.11603e6i 0.0646096i
\(365\) 1.40363e7i 0.288653i
\(366\) 0 0
\(367\) 9.76701e7i 1.97589i 0.154798 + 0.987946i \(0.450527\pi\)
−0.154798 + 0.987946i \(0.549473\pi\)
\(368\) 1.15106e7 2.90608e7i 0.230970 0.583128i
\(369\) 0 0
\(370\) 3.12813e6 0.0617560
\(371\) −3.61767e6 −0.0708446
\(372\) 0 0
\(373\) 7.18558e7i 1.38464i 0.721593 + 0.692318i \(0.243410\pi\)
−0.721593 + 0.692318i \(0.756590\pi\)
\(374\) 9.06275e6 0.173239
\(375\) 0 0
\(376\) 8.91570e6 0.167723
\(377\) −1.06178e7 −0.198158
\(378\) 0 0
\(379\) 8.89305e6i 0.163355i −0.996659 0.0816777i \(-0.973972\pi\)
0.996659 0.0816777i \(-0.0260278\pi\)
\(380\) 2.91589e7 0.531399
\(381\) 0 0
\(382\) 5.29911e6i 0.0950632i
\(383\) 3.07656e7i 0.547607i −0.961786 0.273803i \(-0.911718\pi\)
0.961786 0.273803i \(-0.0882817\pi\)
\(384\) 0 0
\(385\) 4.59677e6i 0.0805509i
\(386\) −1.87155e6 −0.0325416
\(387\) 0 0
\(388\) 9.43681e7i 1.61559i
\(389\) 1.67646e7i 0.284802i −0.989809 0.142401i \(-0.954518\pi\)
0.989809 0.142401i \(-0.0454823\pi\)
\(390\) 0 0
\(391\) −6.14570e7 2.43424e7i −1.02811 0.407224i
\(392\) 3.93846e7 0.653835
\(393\) 0 0
\(394\) −3.63470e7 −0.594265
\(395\) −9.37232e7 −1.52074
\(396\) 0 0
\(397\) 2.13788e7 0.341674 0.170837 0.985299i \(-0.445353\pi\)
0.170837 + 0.985299i \(0.445353\pi\)
\(398\) 9.40164e6i 0.149126i
\(399\) 0 0
\(400\) −5.95283e6 −0.0930130
\(401\) 6.98073e7i 1.08260i 0.840830 + 0.541300i \(0.182068\pi\)
−0.840830 + 0.541300i \(0.817932\pi\)
\(402\) 0 0
\(403\) 3.31612e6 0.0506659
\(404\) 1.22247e7 0.185393
\(405\) 0 0
\(406\) 1.92484e6i 0.0287618i
\(407\) 4.68621e6 0.0695086
\(408\) 0 0
\(409\) −5.80792e7 −0.848888 −0.424444 0.905454i \(-0.639530\pi\)
−0.424444 + 0.905454i \(0.639530\pi\)
\(410\) 5.18050e6i 0.0751658i
\(411\) 0 0
\(412\) 9.33387e7i 1.33466i
\(413\) 4.55591e6i 0.0646733i
\(414\) 0 0
\(415\) −3.85803e7 −0.539785
\(416\) −2.78222e7 −0.386466
\(417\) 0 0
\(418\) −6.52088e6 −0.0892847
\(419\) 2.02760e7i 0.275639i 0.990457 + 0.137819i \(0.0440093\pi\)
−0.990457 + 0.137819i \(0.955991\pi\)
\(420\) 0 0
\(421\) 9.25488e7i 1.24029i −0.784486 0.620147i \(-0.787073\pi\)
0.784486 0.620147i \(-0.212927\pi\)
\(422\) −2.24307e7 −0.298473
\(423\) 0 0
\(424\) 2.10469e7i 0.276115i
\(425\) 1.25889e7i 0.163991i
\(426\) 0 0
\(427\) −1.10362e6 −0.0141754
\(428\) 9.63470e7i 1.22887i
\(429\) 0 0
\(430\) 2.56226e7 0.322268
\(431\) 9.59109e7i 1.19794i −0.800770 0.598971i \(-0.795576\pi\)
0.800770 0.598971i \(-0.204424\pi\)
\(432\) 0 0
\(433\) 1.00838e8i 1.24211i −0.783767 0.621055i \(-0.786705\pi\)
0.783767 0.621055i \(-0.213295\pi\)
\(434\) 601159.i 0.00735394i
\(435\) 0 0
\(436\) 9.57930e7i 1.15578i
\(437\) 4.42199e7 + 1.75150e7i 0.529875 + 0.209877i
\(438\) 0 0
\(439\) 9.86541e7 1.16606 0.583031 0.812450i \(-0.301867\pi\)
0.583031 + 0.812450i \(0.301867\pi\)
\(440\) 2.67431e7 0.313945
\(441\) 0 0
\(442\) 1.47772e7i 0.171129i
\(443\) −5.76044e6 −0.0662589 −0.0331294 0.999451i \(-0.510547\pi\)
−0.0331294 + 0.999451i \(0.510547\pi\)
\(444\) 0 0
\(445\) −1.21354e7 −0.137713
\(446\) −3.83575e7 −0.432361
\(447\) 0 0
\(448\) 4.70873e6i 0.0523685i
\(449\) 1.48365e8 1.63905 0.819523 0.573046i \(-0.194238\pi\)
0.819523 + 0.573046i \(0.194238\pi\)
\(450\) 0 0
\(451\) 7.76085e6i 0.0846018i
\(452\) 1.26242e7i 0.136707i
\(453\) 0 0
\(454\) 4.64371e6i 0.0496246i
\(455\) −7.49521e6 −0.0795701
\(456\) 0 0
\(457\) 4.09189e7i 0.428721i −0.976755 0.214361i \(-0.931233\pi\)
0.976755 0.214361i \(-0.0687668\pi\)
\(458\) 4.57589e7i 0.476298i
\(459\) 0 0
\(460\) −8.43782e7 3.34213e7i −0.866875 0.343360i
\(461\) 1.53056e8 1.56223 0.781117 0.624384i \(-0.214650\pi\)
0.781117 + 0.624384i \(0.214650\pi\)
\(462\) 0 0
\(463\) −1.76680e7 −0.178010 −0.0890051 0.996031i \(-0.528369\pi\)
−0.0890051 + 0.996031i \(0.528369\pi\)
\(464\) −2.89149e7 −0.289446
\(465\) 0 0
\(466\) −2.10137e7 −0.207656
\(467\) 1.15744e7i 0.113645i 0.998384 + 0.0568224i \(0.0180969\pi\)
−0.998384 + 0.0568224i \(0.981903\pi\)
\(468\) 0 0
\(469\) 1.45424e6 0.0140967
\(470\) 9.97804e6i 0.0961062i
\(471\) 0 0
\(472\) 2.65054e7 0.252062
\(473\) 3.83849e7 0.362724
\(474\) 0 0
\(475\) 9.05804e6i 0.0845188i
\(476\) −1.79454e7 −0.166392
\(477\) 0 0
\(478\) −2.02391e7 −0.185313
\(479\) 7.45450e7i 0.678285i 0.940735 + 0.339142i \(0.110137\pi\)
−0.940735 + 0.339142i \(0.889863\pi\)
\(480\) 0 0
\(481\) 7.64105e6i 0.0686622i
\(482\) 7.48189e7i 0.668144i
\(483\) 0 0
\(484\) −8.00127e7 −0.705704
\(485\) −2.26991e8 −1.98968
\(486\) 0 0
\(487\) −1.21827e8 −1.05477 −0.527384 0.849627i \(-0.676827\pi\)
−0.527384 + 0.849627i \(0.676827\pi\)
\(488\) 6.42065e6i 0.0552484i
\(489\) 0 0
\(490\) 4.40774e7i 0.374652i
\(491\) −5.85887e6 −0.0494960 −0.0247480 0.999694i \(-0.507878\pi\)
−0.0247480 + 0.999694i \(0.507878\pi\)
\(492\) 0 0
\(493\) 6.11485e7i 0.510323i
\(494\) 1.06326e7i 0.0881976i
\(495\) 0 0
\(496\) 9.03062e6 0.0740070
\(497\) 3.03538e7i 0.247255i
\(498\) 0 0
\(499\) 1.83935e8 1.48034 0.740171 0.672419i \(-0.234744\pi\)
0.740171 + 0.672419i \(0.234744\pi\)
\(500\) 9.92658e7i 0.794126i
\(501\) 0 0
\(502\) 2.54022e6i 0.0200798i
\(503\) 2.26141e8i 1.77695i −0.458922 0.888477i \(-0.651764\pi\)
0.458922 0.888477i \(-0.348236\pi\)
\(504\) 0 0
\(505\) 2.94049e7i 0.228321i
\(506\) 1.88697e7 + 7.47408e6i 0.145651 + 0.0576907i
\(507\) 0 0
\(508\) 4.14967e7 0.316535
\(509\) 1.34998e8 1.02370 0.511851 0.859074i \(-0.328960\pi\)
0.511851 + 0.859074i \(0.328960\pi\)
\(510\) 0 0
\(511\) 6.21558e6i 0.0465821i
\(512\) −1.32504e8 −0.987235
\(513\) 0 0
\(514\) −1.55539e7 −0.114538
\(515\) −2.24514e8 −1.64370
\(516\) 0 0
\(517\) 1.49480e7i 0.108171i
\(518\) 1.38520e6 0.00996603
\(519\) 0 0
\(520\) 4.36057e7i 0.310122i
\(521\) 3.97398e7i 0.281004i −0.990080 0.140502i \(-0.955128\pi\)
0.990080 0.140502i \(-0.0448716\pi\)
\(522\) 0 0
\(523\) 9.74182e7i 0.680981i 0.940248 + 0.340491i \(0.110593\pi\)
−0.940248 + 0.340491i \(0.889407\pi\)
\(524\) 4.01683e7 0.279184
\(525\) 0 0
\(526\) 5.29813e7i 0.364054i
\(527\) 1.90977e7i 0.130482i
\(528\) 0 0
\(529\) −1.07885e8 1.01368e8i −0.728778 0.684750i
\(530\) 2.35547e7 0.158216
\(531\) 0 0
\(532\) 1.29122e7 0.0857558
\(533\) 1.26544e7 0.0835717
\(534\) 0 0
\(535\) 2.31751e8 1.51342
\(536\) 8.46047e6i 0.0549414i
\(537\) 0 0
\(538\) 6.72031e6 0.0431561
\(539\) 6.60318e7i 0.421684i
\(540\) 0 0
\(541\) −5.61803e7 −0.354807 −0.177404 0.984138i \(-0.556770\pi\)
−0.177404 + 0.984138i \(0.556770\pi\)
\(542\) 7.37240e7 0.463032
\(543\) 0 0
\(544\) 1.60229e8i 0.995281i
\(545\) −2.30418e8 −1.42340
\(546\) 0 0
\(547\) −2.26908e8 −1.38640 −0.693200 0.720745i \(-0.743800\pi\)
−0.693200 + 0.720745i \(0.743800\pi\)
\(548\) 1.56667e8i 0.951996i
\(549\) 0 0
\(550\) 3.86529e6i 0.0232324i
\(551\) 4.39980e7i 0.263013i
\(552\) 0 0
\(553\) −4.15025e7 −0.245414
\(554\) −8.18964e7 −0.481654
\(555\) 0 0
\(556\) −1.40661e8 −0.818368
\(557\) 1.89140e7i 0.109450i −0.998501 0.0547252i \(-0.982572\pi\)
0.998501 0.0547252i \(-0.0174283\pi\)
\(558\) 0 0
\(559\) 6.25881e7i 0.358308i
\(560\) −2.04113e7 −0.116227
\(561\) 0 0
\(562\) 8.45868e7i 0.476534i
\(563\) 2.57543e8i 1.44320i −0.692312 0.721598i \(-0.743408\pi\)
0.692312 0.721598i \(-0.256592\pi\)
\(564\) 0 0
\(565\) 3.03660e7 0.168361
\(566\) 2.68247e7i 0.147940i
\(567\) 0 0
\(568\) 1.76593e8 0.963668
\(569\) 1.72019e8i 0.933769i 0.884318 + 0.466884i \(0.154624\pi\)
−0.884318 + 0.466884i \(0.845376\pi\)
\(570\) 0 0
\(571\) 1.82638e8i 0.981031i −0.871432 0.490516i \(-0.836808\pi\)
0.871432 0.490516i \(-0.163192\pi\)
\(572\) 3.03940e7i 0.162405i
\(573\) 0 0
\(574\) 2.29403e6i 0.0121301i
\(575\) −1.03821e7 + 2.62116e7i −0.0546113 + 0.137876i
\(576\) 0 0
\(577\) −4.64583e6 −0.0241844 −0.0120922 0.999927i \(-0.503849\pi\)
−0.0120922 + 0.999927i \(0.503849\pi\)
\(578\) 1.55088e7 0.0803145
\(579\) 0 0
\(580\) 8.39547e7i 0.430290i
\(581\) −1.70841e7 −0.0871093
\(582\) 0 0
\(583\) 3.52870e7 0.178077
\(584\) −3.61610e7 −0.181552
\(585\) 0 0
\(586\) 6.44985e7i 0.320521i
\(587\) 3.52293e8 1.74177 0.870883 0.491490i \(-0.163548\pi\)
0.870883 + 0.491490i \(0.163548\pi\)
\(588\) 0 0
\(589\) 1.37413e7i 0.0672484i
\(590\) 2.96636e7i 0.144434i
\(591\) 0 0
\(592\) 2.08084e7i 0.100294i
\(593\) −2.14097e7 −0.102671 −0.0513353 0.998681i \(-0.516348\pi\)
−0.0513353 + 0.998681i \(0.516348\pi\)
\(594\) 0 0
\(595\) 4.31653e7i 0.204920i
\(596\) 2.50793e8i 1.18461i
\(597\) 0 0
\(598\) −1.21868e7 + 3.07678e7i −0.0569883 + 0.143877i
\(599\) 2.67976e8 1.24686 0.623428 0.781881i \(-0.285740\pi\)
0.623428 + 0.781881i \(0.285740\pi\)
\(600\) 0 0
\(601\) −1.13179e8 −0.521365 −0.260682 0.965425i \(-0.583947\pi\)
−0.260682 + 0.965425i \(0.583947\pi\)
\(602\) 1.13462e7 0.0520069
\(603\) 0 0
\(604\) 2.86417e8 1.29983
\(605\) 1.92460e8i 0.869111i
\(606\) 0 0
\(607\) 3.82018e8 1.70812 0.854059 0.520176i \(-0.174134\pi\)
0.854059 + 0.520176i \(0.174134\pi\)
\(608\) 1.15289e8i 0.512954i
\(609\) 0 0
\(610\) 7.18570e6 0.0316577
\(611\) 2.43733e7 0.106854
\(612\) 0 0
\(613\) 7.07824e7i 0.307287i 0.988126 + 0.153643i \(0.0491007\pi\)
−0.988126 + 0.153643i \(0.950899\pi\)
\(614\) −7.01365e7 −0.302997
\(615\) 0 0
\(616\) 1.18424e7 0.0506637
\(617\) 1.84723e8i 0.786441i 0.919444 + 0.393221i \(0.128639\pi\)
−0.919444 + 0.393221i \(0.871361\pi\)
\(618\) 0 0
\(619\) 5.75005e7i 0.242438i 0.992626 + 0.121219i \(0.0386803\pi\)
−0.992626 + 0.121219i \(0.961320\pi\)
\(620\) 2.62205e7i 0.110018i
\(621\) 0 0
\(622\) −9.29601e7 −0.386301
\(623\) −5.37381e6 −0.0222238
\(624\) 0 0
\(625\) −2.74977e8 −1.12631
\(626\) 8.64016e7i 0.352208i
\(627\) 0 0
\(628\) 3.44361e8i 1.39038i
\(629\) 4.40052e7 0.176828
\(630\) 0 0
\(631\) 1.50039e8i 0.597195i −0.954379 0.298597i \(-0.903481\pi\)
0.954379 0.298597i \(-0.0965188\pi\)
\(632\) 2.41453e8i 0.956493i
\(633\) 0 0
\(634\) −1.11967e8 −0.439360
\(635\) 9.98150e7i 0.389830i
\(636\) 0 0
\(637\) 1.07668e8 0.416549
\(638\) 1.87750e7i 0.0722966i
\(639\) 0 0
\(640\) 2.83488e8i 1.08142i
\(641\) 4.57545e8i 1.73724i 0.495481 + 0.868619i \(0.334992\pi\)
−0.495481 + 0.868619i \(0.665008\pi\)
\(642\) 0 0
\(643\) 2.92745e8i 1.10118i 0.834777 + 0.550588i \(0.185596\pi\)
−0.834777 + 0.550588i \(0.814404\pi\)
\(644\) −3.73643e7 1.47996e7i −0.139894 0.0554106i
\(645\) 0 0
\(646\) −6.12334e7 −0.227139
\(647\) 3.52954e8 1.30318 0.651592 0.758570i \(-0.274102\pi\)
0.651592 + 0.758570i \(0.274102\pi\)
\(648\) 0 0
\(649\) 4.44387e7i 0.162565i
\(650\) 6.30250e6 0.0229495
\(651\) 0 0
\(652\) 6.79265e7 0.245074
\(653\) −2.79192e8 −1.00268 −0.501342 0.865249i \(-0.667160\pi\)
−0.501342 + 0.865249i \(0.667160\pi\)
\(654\) 0 0
\(655\) 9.66198e7i 0.343829i
\(656\) 3.44610e7 0.122072
\(657\) 0 0
\(658\) 4.41848e6i 0.0155094i
\(659\) 3.74057e8i 1.30702i 0.756919 + 0.653509i \(0.226704\pi\)
−0.756919 + 0.653509i \(0.773296\pi\)
\(660\) 0 0
\(661\) 1.99796e8i 0.691802i −0.938271 0.345901i \(-0.887573\pi\)
0.938271 0.345901i \(-0.112427\pi\)
\(662\) −7.88226e7 −0.271692
\(663\) 0 0
\(664\) 9.93920e7i 0.339506i
\(665\) 3.10586e7i 0.105613i
\(666\) 0 0
\(667\) −5.04294e7 + 1.27318e8i −0.169944 + 0.429056i
\(668\) 3.05763e8 1.02578
\(669\) 0 0
\(670\) −9.46857e6 −0.0314818
\(671\) 1.07648e7 0.0356319
\(672\) 0 0
\(673\) 4.50528e8 1.47801 0.739004 0.673701i \(-0.235296\pi\)
0.739004 + 0.673701i \(0.235296\pi\)
\(674\) 7.00554e7i 0.228803i
\(675\) 0 0
\(676\) 2.19232e8 0.709684
\(677\) 3.07185e8i 0.989999i 0.868893 + 0.494999i \(0.164832\pi\)
−0.868893 + 0.494999i \(0.835168\pi\)
\(678\) 0 0
\(679\) −1.00516e8 −0.321089
\(680\) 2.51127e8 0.798669
\(681\) 0 0
\(682\) 5.86375e6i 0.0184851i
\(683\) −3.65060e8 −1.14578 −0.572890 0.819632i \(-0.694178\pi\)
−0.572890 + 0.819632i \(0.694178\pi\)
\(684\) 0 0
\(685\) −3.76842e8 −1.17243
\(686\) 3.96384e7i 0.122785i
\(687\) 0 0
\(688\) 1.70443e8i 0.523375i
\(689\) 5.75368e7i 0.175909i
\(690\) 0 0
\(691\) −1.40121e8 −0.424688 −0.212344 0.977195i \(-0.568110\pi\)
−0.212344 + 0.977195i \(0.568110\pi\)
\(692\) −1.80490e8 −0.544673
\(693\) 0 0
\(694\) 1.92366e8 0.575507
\(695\) 3.38342e8i 1.00786i
\(696\) 0 0
\(697\) 7.28772e7i 0.215225i
\(698\) −1.39225e8 −0.409404
\(699\) 0 0
\(700\) 7.65375e6i 0.0223141i
\(701\) 1.00190e8i 0.290850i 0.989369 + 0.145425i \(0.0464550\pi\)
−0.989369 + 0.145425i \(0.953545\pi\)
\(702\) 0 0
\(703\) −3.16629e7 −0.0911348
\(704\) 4.59293e7i 0.131635i
\(705\) 0 0
\(706\) 1.58605e8 0.450714
\(707\) 1.30211e7i 0.0368458i
\(708\) 0 0
\(709\) 2.00481e8i 0.562517i 0.959632 + 0.281258i \(0.0907518\pi\)
−0.959632 + 0.281258i \(0.909248\pi\)
\(710\) 1.97634e8i 0.552188i
\(711\) 0 0
\(712\) 3.12637e7i 0.0866166i
\(713\) 1.57500e7 3.97637e7i 0.0434521 0.109703i
\(714\) 0 0
\(715\) 7.31089e7 0.200010
\(716\) 3.41469e8 0.930276
\(717\) 0 0
\(718\) 7.14215e7i 0.192955i
\(719\) −5.03409e8 −1.35436 −0.677180 0.735817i \(-0.736798\pi\)
−0.677180 + 0.735817i \(0.736798\pi\)
\(720\) 0 0
\(721\) −9.94195e7 −0.265256
\(722\) −9.15841e7 −0.243337
\(723\) 0 0
\(724\) 2.03380e8i 0.535912i
\(725\) 2.60800e7 0.0684375
\(726\) 0 0
\(727\) 2.36133e8i 0.614546i −0.951621 0.307273i \(-0.900584\pi\)
0.951621 0.307273i \(-0.0994164\pi\)
\(728\) 1.93095e7i 0.0500468i
\(729\) 0 0
\(730\) 4.04697e7i 0.104031i
\(731\) 3.60448e8 0.922763
\(732\) 0 0
\(733\) 5.20274e8i 1.32105i 0.750803 + 0.660526i \(0.229667\pi\)
−0.750803 + 0.660526i \(0.770333\pi\)
\(734\) 2.81603e8i 0.712114i
\(735\) 0 0
\(736\) −1.32142e8 + 3.33617e8i −0.331441 + 0.836786i
\(737\) −1.41847e7 −0.0354339
\(738\) 0 0
\(739\) −4.39693e8 −1.08947 −0.544736 0.838607i \(-0.683370\pi\)
−0.544736 + 0.838607i \(0.683370\pi\)
\(740\) 6.04175e7 0.149097
\(741\) 0 0
\(742\) 1.04305e7 0.0255325
\(743\) 3.38209e8i 0.824554i −0.911059 0.412277i \(-0.864734\pi\)
0.911059 0.412277i \(-0.135266\pi\)
\(744\) 0 0
\(745\) 6.03251e8 1.45891
\(746\) 2.07175e8i 0.499024i
\(747\) 0 0
\(748\) 1.75040e8 0.418248
\(749\) 1.02624e8 0.244232
\(750\) 0 0
\(751\) 8.04152e8i 1.89853i −0.314472 0.949267i \(-0.601827\pi\)
0.314472 0.949267i \(-0.398173\pi\)
\(752\) 6.63744e7 0.156080
\(753\) 0 0
\(754\) 3.06134e7 0.0714163
\(755\) 6.88940e8i 1.60081i
\(756\) 0 0
\(757\) 4.21509e7i 0.0971670i 0.998819 + 0.0485835i \(0.0154707\pi\)
−0.998819 + 0.0485835i \(0.984529\pi\)
\(758\) 2.56405e7i 0.0588734i
\(759\) 0 0
\(760\) −1.80692e8 −0.411623
\(761\) −5.67488e8 −1.28766 −0.643832 0.765167i \(-0.722656\pi\)
−0.643832 + 0.765167i \(0.722656\pi\)
\(762\) 0 0
\(763\) −1.02034e8 −0.229705
\(764\) 1.02348e8i 0.229510i
\(765\) 0 0
\(766\) 8.87036e7i 0.197358i
\(767\) 7.24591e7 0.160586
\(768\) 0 0
\(769\) 4.44661e7i 0.0977800i 0.998804 + 0.0488900i \(0.0155684\pi\)
−0.998804 + 0.0488900i \(0.984432\pi\)
\(770\) 1.32534e7i 0.0290306i
\(771\) 0 0
\(772\) −3.61476e7 −0.0785646
\(773\) 6.81005e8i 1.47439i −0.675681 0.737194i \(-0.736150\pi\)
0.675681 0.737194i \(-0.263850\pi\)
\(774\) 0 0
\(775\) −8.14524e6 −0.0174984
\(776\) 5.84782e8i 1.25144i
\(777\) 0 0
\(778\) 4.83358e7i 0.102643i
\(779\) 5.24370e7i 0.110924i
\(780\) 0 0
\(781\) 2.96073e8i 0.621508i
\(782\) 1.77193e8 + 7.01843e7i 0.370533 + 0.146764i
\(783\) 0 0
\(784\) 2.93205e8 0.608448
\(785\) 8.28316e8 1.71233
\(786\) 0 0
\(787\) 5.36571e8i 1.10079i −0.834905 0.550394i \(-0.814478\pi\)
0.834905 0.550394i \(-0.185522\pi\)
\(788\) −7.02017e8 −1.43473
\(789\) 0 0
\(790\) 2.70224e8 0.548077
\(791\) 1.34467e7 0.0271698
\(792\) 0 0
\(793\) 1.75524e7i 0.0351980i
\(794\) −6.16397e7 −0.123140
\(795\) 0 0
\(796\) 1.81586e8i 0.360034i
\(797\) 7.42677e8i 1.46698i −0.679699 0.733491i \(-0.737890\pi\)
0.679699 0.733491i \(-0.262110\pi\)
\(798\) 0 0
\(799\) 1.40367e8i 0.275185i
\(800\) 6.83383e7 0.133473
\(801\) 0 0
\(802\) 2.01269e8i 0.390170i
\(803\) 6.06272e7i 0.117090i
\(804\) 0 0
\(805\) −3.55986e7 + 8.98752e7i −0.0682410 + 0.172287i
\(806\) −9.56108e6 −0.0182600
\(807\) 0 0
\(808\) −7.57540e7 −0.143606
\(809\) −6.69319e8 −1.26412 −0.632059 0.774920i \(-0.717790\pi\)
−0.632059 + 0.774920i \(0.717790\pi\)
\(810\) 0 0
\(811\) −1.60228e8 −0.300384 −0.150192 0.988657i \(-0.547989\pi\)
−0.150192 + 0.988657i \(0.547989\pi\)
\(812\) 3.71768e7i 0.0694391i
\(813\) 0 0
\(814\) −1.35113e7 −0.0250510
\(815\) 1.63389e8i 0.301821i
\(816\) 0 0
\(817\) −2.59351e8 −0.475579
\(818\) 1.67454e8 0.305940
\(819\) 0 0
\(820\) 1.00058e8i 0.181472i
\(821\) 7.01161e7 0.126703 0.0633517 0.997991i \(-0.479821\pi\)
0.0633517 + 0.997991i \(0.479821\pi\)
\(822\) 0 0
\(823\) −7.75298e8 −1.39081 −0.695407 0.718616i \(-0.744776\pi\)
−0.695407 + 0.718616i \(0.744776\pi\)
\(824\) 5.78403e8i 1.03383i
\(825\) 0 0
\(826\) 1.31356e7i 0.0233083i
\(827\) 7.09941e8i 1.25518i −0.778544 0.627590i \(-0.784042\pi\)
0.778544 0.627590i \(-0.215958\pi\)
\(828\) 0 0
\(829\) 5.98520e8 1.05055 0.525273 0.850934i \(-0.323963\pi\)
0.525273 + 0.850934i \(0.323963\pi\)
\(830\) 1.11235e8 0.194539
\(831\) 0 0
\(832\) −7.48896e7 −0.130032
\(833\) 6.20063e8i 1.07275i
\(834\) 0 0
\(835\) 7.35474e8i 1.26330i
\(836\) −1.25946e8 −0.215559
\(837\) 0 0
\(838\) 5.84599e7i 0.0993405i
\(839\) 7.07760e8i 1.19840i 0.800601 + 0.599198i \(0.204514\pi\)
−0.800601 + 0.599198i \(0.795486\pi\)
\(840\) 0 0
\(841\) −4.68144e8 −0.787030
\(842\) 2.66837e8i 0.447003i
\(843\) 0 0
\(844\) −4.33232e8 −0.720599
\(845\) 5.27336e8i 0.874012i
\(846\) 0 0
\(847\) 8.52253e7i 0.140255i
\(848\) 1.56687e8i 0.256948i
\(849\) 0 0
\(850\) 3.62964e7i 0.0591027i
\(851\) 9.16239e7 + 3.62912e7i 0.148669 + 0.0588861i
\(852\) 0 0
\(853\) 5.67257e8 0.913972 0.456986 0.889474i \(-0.348929\pi\)
0.456986 + 0.889474i \(0.348929\pi\)
\(854\) 3.18197e6 0.00510885
\(855\) 0 0
\(856\) 5.97045e8i 0.951888i
\(857\) −6.81847e8 −1.08329 −0.541645 0.840607i \(-0.682198\pi\)
−0.541645 + 0.840607i \(0.682198\pi\)
\(858\) 0 0
\(859\) −2.53119e8 −0.399342 −0.199671 0.979863i \(-0.563987\pi\)
−0.199671 + 0.979863i \(0.563987\pi\)
\(860\) 4.94882e8 0.778047
\(861\) 0 0
\(862\) 2.76531e8i 0.431740i
\(863\) −2.00680e8 −0.312228 −0.156114 0.987739i \(-0.549897\pi\)
−0.156114 + 0.987739i \(0.549897\pi\)
\(864\) 0 0
\(865\) 4.34147e8i 0.670792i
\(866\) 2.90737e8i 0.447658i
\(867\) 0 0
\(868\) 1.16110e7i 0.0177545i
\(869\) 4.04819e8 0.616881
\(870\) 0 0
\(871\) 2.31288e7i 0.0350024i
\(872\) 5.93612e8i 0.895268i
\(873\) 0 0
\(874\) −1.27495e8 5.04994e7i −0.190967 0.0756401i
\(875\) −1.05733e8 −0.157829
\(876\) 0 0
\(877\) −5.93741e8 −0.880234 −0.440117 0.897940i \(-0.645063\pi\)
−0.440117 + 0.897940i \(0.645063\pi\)
\(878\) −2.84440e8 −0.420250
\(879\) 0 0
\(880\) 1.99093e8 0.292152
\(881\) 2.69314e8i 0.393851i 0.980418 + 0.196925i \(0.0630957\pi\)
−0.980418 + 0.196925i \(0.936904\pi\)
\(882\) 0 0
\(883\) 5.61291e8 0.815278 0.407639 0.913143i \(-0.366352\pi\)
0.407639 + 0.913143i \(0.366352\pi\)
\(884\) 2.85410e8i 0.413155i
\(885\) 0 0
\(886\) 1.66085e7 0.0238798
\(887\) −8.59983e8 −1.23231 −0.616154 0.787626i \(-0.711310\pi\)
−0.616154 + 0.787626i \(0.711310\pi\)
\(888\) 0 0
\(889\) 4.42001e7i 0.0629097i
\(890\) 3.49889e7 0.0496319
\(891\) 0 0
\(892\) −7.40848e8 −1.04384
\(893\) 1.00998e8i 0.141826i
\(894\) 0 0
\(895\) 8.21360e8i 1.14568i
\(896\) 1.25534e8i 0.174517i
\(897\) 0 0
\(898\) −4.27766e8 −0.590714
\(899\) −3.95642e7 −0.0544531
\(900\) 0 0
\(901\) 3.31357e8 0.453025
\(902\) 2.23762e7i 0.0304906i
\(903\) 0 0
\(904\) 7.82302e7i 0.105893i
\(905\) 4.89206e8 0.660003
\(906\) 0 0
\(907\) 7.14901e8i 0.958129i −0.877780 0.479065i \(-0.840976\pi\)
0.877780 0.479065i \(-0.159024\pi\)
\(908\) 8.96899e7i 0.119808i
\(909\) 0 0
\(910\) 2.16103e7 0.0286771
\(911\) 5.76443e8i 0.762432i 0.924486 + 0.381216i \(0.124495\pi\)
−0.924486 + 0.381216i \(0.875505\pi\)
\(912\) 0 0
\(913\) 1.66640e8 0.218961
\(914\) 1.17978e8i 0.154512i
\(915\) 0 0
\(916\) 8.83799e8i 1.14992i
\(917\) 4.27852e7i 0.0554863i
\(918\) 0 0
\(919\) 1.49428e9i 1.92524i −0.270858 0.962619i \(-0.587307\pi\)
0.270858 0.962619i \(-0.412693\pi\)
\(920\) 5.22876e8 + 2.07106e8i 0.671484 + 0.265967i
\(921\) 0 0
\(922\) −4.41291e8 −0.563031
\(923\) 4.82760e8 0.613940
\(924\) 0 0
\(925\) 1.87683e7i 0.0237138i
\(926\) 5.09406e7 0.0641551
\(927\) 0 0
\(928\) 3.31942e8 0.415354
\(929\) 8.50023e8 1.06019 0.530095 0.847938i \(-0.322156\pi\)
0.530095 + 0.847938i \(0.322156\pi\)
\(930\) 0 0
\(931\) 4.46151e8i 0.552882i
\(932\) −4.05864e8 −0.501340
\(933\) 0 0
\(934\) 3.33715e7i 0.0409577i
\(935\) 4.21038e8i 0.515094i
\(936\) 0 0
\(937\) 6.39691e8i 0.777592i 0.921324 + 0.388796i \(0.127109\pi\)
−0.921324 + 0.388796i \(0.872891\pi\)
\(938\) −4.19287e6 −0.00508046
\(939\) 0 0
\(940\) 1.92719e8i 0.232028i
\(941\) 4.61469e8i 0.553826i −0.960895 0.276913i \(-0.910689\pi\)
0.960895 0.276913i \(-0.0893114\pi\)
\(942\) 0 0
\(943\) 6.01021e7 1.51739e8i 0.0716728 0.180951i
\(944\) 1.97324e8 0.234565
\(945\) 0 0
\(946\) −1.10672e8 −0.130726
\(947\) −6.26119e8 −0.737237 −0.368619 0.929581i \(-0.620169\pi\)
−0.368619 + 0.929581i \(0.620169\pi\)
\(948\) 0 0
\(949\) −9.88551e7 −0.115665
\(950\) 2.61162e7i 0.0304607i
\(951\) 0 0
\(952\) 1.11204e8 0.128887
\(953\) 4.99599e8i 0.577222i −0.957446 0.288611i \(-0.906807\pi\)
0.957446 0.288611i \(-0.0931934\pi\)
\(954\) 0 0
\(955\) −2.46186e8 −0.282653
\(956\) −3.90903e8 −0.447399
\(957\) 0 0
\(958\) 2.14929e8i 0.244455i
\(959\) −1.66873e8 −0.189204
\(960\) 0 0
\(961\) −8.75147e8 −0.986077
\(962\) 2.20307e7i 0.0247459i
\(963\) 0 0
\(964\) 1.44507e9i 1.61309i
\(965\) 8.69484e7i 0.0967564i
\(966\) 0 0
\(967\) 5.93301e8 0.656138 0.328069 0.944654i \(-0.393602\pi\)
0.328069 + 0.944654i \(0.393602\pi\)
\(968\) 4.95824e8 0.546640
\(969\) 0 0
\(970\) 6.54462e8 0.717082
\(971\) 8.71183e8i 0.951594i −0.879555 0.475797i \(-0.842160\pi\)
0.879555 0.475797i \(-0.157840\pi\)
\(972\) 0 0
\(973\) 1.49825e8i 0.162647i
\(974\) 3.51253e8 0.380140
\(975\) 0 0
\(976\) 4.77996e7i 0.0514133i
\(977\) 1.19611e9i 1.28259i −0.767296 0.641293i \(-0.778398\pi\)
0.767296 0.641293i \(-0.221602\pi\)
\(978\) 0 0
\(979\) 5.24165e7 0.0558625
\(980\) 8.51323e8i 0.904516i
\(981\) 0 0
\(982\) 1.68924e7 0.0178384
\(983\) 6.05873e7i 0.0637854i 0.999491 + 0.0318927i \(0.0101535\pi\)
−0.999491 + 0.0318927i \(0.989847\pi\)
\(984\) 0 0
\(985\) 1.68861e9i 1.76694i
\(986\) 1.76304e8i 0.183921i
\(987\) 0 0
\(988\) 2.05360e8i 0.212934i
\(989\) 7.50494e8 + 2.97263e8i 0.775816 + 0.307292i
\(990\) 0 0
\(991\) −2.72089e7 −0.0279570 −0.0139785 0.999902i \(-0.504450\pi\)
−0.0139785 + 0.999902i \(0.504450\pi\)
\(992\) −1.03671e8 −0.106200
\(993\) 0 0
\(994\) 8.75164e7i 0.0891108i
\(995\) −4.36782e8 −0.443400
\(996\) 0 0
\(997\) 1.32840e9 1.34043 0.670215 0.742167i \(-0.266202\pi\)
0.670215 + 0.742167i \(0.266202\pi\)
\(998\) −5.30322e8 −0.533517
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 207.7.d.e.91.9 24
3.2 odd 2 69.7.d.a.22.15 24
23.22 odd 2 inner 207.7.d.e.91.10 24
69.68 even 2 69.7.d.a.22.16 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.7.d.a.22.15 24 3.2 odd 2
69.7.d.a.22.16 yes 24 69.68 even 2
207.7.d.e.91.9 24 1.1 even 1 trivial
207.7.d.e.91.10 24 23.22 odd 2 inner