Properties

Label 29.6.b.a.28.9
Level $29$
Weight $6$
Character 29.28
Analytic conductor $4.651$
Analytic rank $0$
Dimension $12$
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] = [29,6,Mod(28,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.b (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: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{14}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 28.9
Root \(4.44887i\) of defining polynomial
Character \(\chi\) \(=\) 29.28
Dual form 29.6.b.a.28.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.44887i q^{2} +21.5070i q^{3} +12.2075 q^{4} +90.0922 q^{5} -95.6821 q^{6} -211.678 q^{7} +196.674i q^{8} -219.553 q^{9} +O(q^{10})\) \(q+4.44887i q^{2} +21.5070i q^{3} +12.2075 q^{4} +90.0922 q^{5} -95.6821 q^{6} -211.678 q^{7} +196.674i q^{8} -219.553 q^{9} +400.809i q^{10} -593.900i q^{11} +262.548i q^{12} +445.466 q^{13} -941.727i q^{14} +1937.62i q^{15} -484.336 q^{16} -368.642i q^{17} -976.764i q^{18} +761.926i q^{19} +1099.80 q^{20} -4552.56i q^{21} +2642.19 q^{22} +728.987 q^{23} -4229.87 q^{24} +4991.60 q^{25} +1981.82i q^{26} +504.276i q^{27} -2584.06 q^{28} +(2787.76 + 3569.25i) q^{29} -8620.22 q^{30} -9222.49i q^{31} +4138.81i q^{32} +12773.0 q^{33} +1640.04 q^{34} -19070.5 q^{35} -2680.20 q^{36} -3392.70i q^{37} -3389.71 q^{38} +9580.65i q^{39} +17718.8i q^{40} +1763.84i q^{41} +20253.8 q^{42} +5296.04i q^{43} -7250.04i q^{44} -19780.0 q^{45} +3243.17i q^{46} -12459.7i q^{47} -10416.6i q^{48} +28000.4 q^{49} +22207.0i q^{50} +7928.41 q^{51} +5438.03 q^{52} -14753.2 q^{53} -2243.46 q^{54} -53505.8i q^{55} -41631.4i q^{56} -16386.8 q^{57} +(-15879.1 + 12402.4i) q^{58} -34598.5 q^{59} +23653.5i q^{60} -9455.51i q^{61} +41029.7 q^{62} +46474.5 q^{63} -33911.8 q^{64} +40133.0 q^{65} +56825.6i q^{66} +41093.0 q^{67} -4500.21i q^{68} +15678.4i q^{69} -84842.3i q^{70} -23404.1 q^{71} -43180.3i q^{72} +25457.6i q^{73} +15093.7 q^{74} +107355. i q^{75} +9301.22i q^{76} +125715. i q^{77} -42623.1 q^{78} -45401.0i q^{79} -43634.9 q^{80} -64196.9 q^{81} -7847.11 q^{82} -53492.2 q^{83} -55575.5i q^{84} -33211.8i q^{85} -23561.4 q^{86} +(-76763.9 + 59956.6i) q^{87} +116805. q^{88} -112498. i q^{89} -87998.8i q^{90} -94295.2 q^{91} +8899.12 q^{92} +198349. q^{93} +55431.7 q^{94} +68643.6i q^{95} -89013.5 q^{96} -724.916i q^{97} +124570. i q^{98} +130393. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 172 q^{4} + 46 q^{5} + 24 q^{6} + 20 q^{7} - 1574 q^{9} + 1362 q^{13} + 340 q^{16} - 4508 q^{20} + 11376 q^{22} + 5852 q^{23} - 6292 q^{24} + 12678 q^{25} - 25056 q^{28} + 11328 q^{29} + 14952 q^{30} - 22694 q^{33} - 22504 q^{34} + 4532 q^{35} + 22840 q^{36} - 43408 q^{38} + 8280 q^{42} - 52816 q^{45} + 102836 q^{49} + 58540 q^{51} + 15172 q^{52} + 25650 q^{53} - 89080 q^{54} - 32824 q^{57} + 4960 q^{58} - 3900 q^{59} + 37720 q^{62} - 146616 q^{63} + 252276 q^{64} + 169574 q^{65} - 28264 q^{67} - 286832 q^{71} - 263072 q^{74} + 519072 q^{78} - 230964 q^{80} - 24084 q^{81} - 178008 q^{82} + 85692 q^{83} - 126624 q^{86} - 137716 q^{87} - 83604 q^{88} - 182372 q^{91} - 5664 q^{92} + 377966 q^{93} + 192144 q^{94} - 415284 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.44887i 0.786457i 0.919441 + 0.393229i \(0.128642\pi\)
−0.919441 + 0.393229i \(0.871358\pi\)
\(3\) 21.5070i 1.37968i 0.723963 + 0.689839i \(0.242319\pi\)
−0.723963 + 0.689839i \(0.757681\pi\)
\(4\) 12.2075 0.381485
\(5\) 90.0922 1.61162 0.805809 0.592175i \(-0.201731\pi\)
0.805809 + 0.592175i \(0.201731\pi\)
\(6\) −95.6821 −1.08506
\(7\) −211.678 −1.63279 −0.816394 0.577495i \(-0.804030\pi\)
−0.816394 + 0.577495i \(0.804030\pi\)
\(8\) 196.674i 1.08648i
\(9\) −219.553 −0.903510
\(10\) 400.809i 1.26747i
\(11\) 593.900i 1.47990i −0.672663 0.739949i \(-0.734850\pi\)
0.672663 0.739949i \(-0.265150\pi\)
\(12\) 262.548i 0.526326i
\(13\) 445.466 0.731065 0.365532 0.930799i \(-0.380887\pi\)
0.365532 + 0.930799i \(0.380887\pi\)
\(14\) 941.727i 1.28412i
\(15\) 1937.62i 2.22351i
\(16\) −484.336 −0.472985
\(17\) 368.642i 0.309373i −0.987964 0.154687i \(-0.950563\pi\)
0.987964 0.154687i \(-0.0494368\pi\)
\(18\) 976.764i 0.710572i
\(19\) 761.926i 0.484205i 0.970251 + 0.242102i \(0.0778370\pi\)
−0.970251 + 0.242102i \(0.922163\pi\)
\(20\) 1099.80 0.614808
\(21\) 4552.56i 2.25272i
\(22\) 2642.19 1.16388
\(23\) 728.987 0.287343 0.143671 0.989625i \(-0.454109\pi\)
0.143671 + 0.989625i \(0.454109\pi\)
\(24\) −4229.87 −1.49899
\(25\) 4991.60 1.59731
\(26\) 1981.82i 0.574951i
\(27\) 504.276i 0.133125i
\(28\) −2584.06 −0.622884
\(29\) 2787.76 + 3569.25i 0.615547 + 0.788100i
\(30\) −8620.22 −1.74870
\(31\) 9222.49i 1.72363i −0.507223 0.861815i \(-0.669328\pi\)
0.507223 0.861815i \(-0.330672\pi\)
\(32\) 4138.81i 0.714497i
\(33\) 12773.0 2.04178
\(34\) 1640.04 0.243309
\(35\) −19070.5 −2.63143
\(36\) −2680.20 −0.344675
\(37\) 3392.70i 0.407418i −0.979031 0.203709i \(-0.934700\pi\)
0.979031 0.203709i \(-0.0652997\pi\)
\(38\) −3389.71 −0.380806
\(39\) 9580.65i 1.00863i
\(40\) 17718.8i 1.75099i
\(41\) 1763.84i 0.163870i 0.996638 + 0.0819351i \(0.0261100\pi\)
−0.996638 + 0.0819351i \(0.973890\pi\)
\(42\) 20253.8 1.77167
\(43\) 5296.04i 0.436797i 0.975860 + 0.218399i \(0.0700833\pi\)
−0.975860 + 0.218399i \(0.929917\pi\)
\(44\) 7250.04i 0.564559i
\(45\) −19780.0 −1.45611
\(46\) 3243.17i 0.225983i
\(47\) 12459.7i 0.822741i −0.911468 0.411371i \(-0.865050\pi\)
0.911468 0.411371i \(-0.134950\pi\)
\(48\) 10416.6i 0.652566i
\(49\) 28000.4 1.66600
\(50\) 22207.0i 1.25622i
\(51\) 7928.41 0.426836
\(52\) 5438.03 0.278890
\(53\) −14753.2 −0.721432 −0.360716 0.932676i \(-0.617468\pi\)
−0.360716 + 0.932676i \(0.617468\pi\)
\(54\) −2243.46 −0.104697
\(55\) 53505.8i 2.38503i
\(56\) 41631.4i 1.77399i
\(57\) −16386.8 −0.668046
\(58\) −15879.1 + 12402.4i −0.619807 + 0.484101i
\(59\) −34598.5 −1.29398 −0.646989 0.762499i \(-0.723972\pi\)
−0.646989 + 0.762499i \(0.723972\pi\)
\(60\) 23653.5i 0.848237i
\(61\) 9455.51i 0.325357i −0.986679 0.162679i \(-0.947987\pi\)
0.986679 0.162679i \(-0.0520133\pi\)
\(62\) 41029.7 1.35556
\(63\) 46474.5 1.47524
\(64\) −33911.8 −1.03491
\(65\) 40133.0 1.17820
\(66\) 56825.6i 1.60577i
\(67\) 41093.0 1.11836 0.559179 0.829047i \(-0.311116\pi\)
0.559179 + 0.829047i \(0.311116\pi\)
\(68\) 4500.21i 0.118021i
\(69\) 15678.4i 0.396440i
\(70\) 84842.3i 2.06951i
\(71\) −23404.1 −0.550992 −0.275496 0.961302i \(-0.588842\pi\)
−0.275496 + 0.961302i \(0.588842\pi\)
\(72\) 43180.3i 0.981645i
\(73\) 25457.6i 0.559127i 0.960127 + 0.279563i \(0.0901897\pi\)
−0.960127 + 0.279563i \(0.909810\pi\)
\(74\) 15093.7 0.320417
\(75\) 107355.i 2.20378i
\(76\) 9301.22i 0.184717i
\(77\) 125715.i 2.41636i
\(78\) −42623.1 −0.793248
\(79\) 45401.0i 0.818460i −0.912431 0.409230i \(-0.865797\pi\)
0.912431 0.409230i \(-0.134203\pi\)
\(80\) −43634.9 −0.762270
\(81\) −64196.9 −1.08718
\(82\) −7847.11 −0.128877
\(83\) −53492.2 −0.852306 −0.426153 0.904651i \(-0.640131\pi\)
−0.426153 + 0.904651i \(0.640131\pi\)
\(84\) 55575.5i 0.859379i
\(85\) 33211.8i 0.498592i
\(86\) −23561.4 −0.343523
\(87\) −76763.9 + 59956.6i −1.08732 + 0.849256i
\(88\) 116805. 1.60788
\(89\) 112498.i 1.50547i −0.658326 0.752733i \(-0.728735\pi\)
0.658326 0.752733i \(-0.271265\pi\)
\(90\) 87998.8i 1.14517i
\(91\) −94295.2 −1.19367
\(92\) 8899.12 0.109617
\(93\) 198349. 2.37805
\(94\) 55431.7 0.647051
\(95\) 68643.6i 0.780353i
\(96\) −89013.5 −0.985775
\(97\) 724.916i 0.00782273i −0.999992 0.00391137i \(-0.998755\pi\)
0.999992 0.00391137i \(-0.00124503\pi\)
\(98\) 124570.i 1.31024i
\(99\) 130393.i 1.33710i
\(100\) 60935.1 0.609351
\(101\) 55652.7i 0.542854i 0.962459 + 0.271427i \(0.0874955\pi\)
−0.962459 + 0.271427i \(0.912505\pi\)
\(102\) 35272.5i 0.335688i
\(103\) 86986.8 0.807905 0.403953 0.914780i \(-0.367636\pi\)
0.403953 + 0.914780i \(0.367636\pi\)
\(104\) 87611.4i 0.794287i
\(105\) 410150.i 3.63053i
\(106\) 65635.0i 0.567376i
\(107\) −67319.9 −0.568439 −0.284220 0.958759i \(-0.591734\pi\)
−0.284220 + 0.958759i \(0.591734\pi\)
\(108\) 6155.96i 0.0507851i
\(109\) −107618. −0.867595 −0.433797 0.901010i \(-0.642827\pi\)
−0.433797 + 0.901010i \(0.642827\pi\)
\(110\) 238040. 1.87572
\(111\) 72966.9 0.562106
\(112\) 102523. 0.772284
\(113\) 140630.i 1.03605i −0.855365 0.518027i \(-0.826667\pi\)
0.855365 0.518027i \(-0.173333\pi\)
\(114\) 72902.7i 0.525390i
\(115\) 65676.0 0.463087
\(116\) 34031.7 + 43571.6i 0.234822 + 0.300648i
\(117\) −97803.3 −0.660525
\(118\) 153924.i 1.01766i
\(119\) 78033.4i 0.505141i
\(120\) −381078. −2.41580
\(121\) −191666. −1.19010
\(122\) 42066.4 0.255879
\(123\) −37935.0 −0.226088
\(124\) 112584.i 0.657539i
\(125\) 168167. 0.962642
\(126\) 206759.i 1.16021i
\(127\) 210012.i 1.15541i 0.816247 + 0.577703i \(0.196051\pi\)
−0.816247 + 0.577703i \(0.803949\pi\)
\(128\) 18427.5i 0.0994125i
\(129\) −113902. −0.602640
\(130\) 178547.i 0.926602i
\(131\) 92160.7i 0.469210i 0.972091 + 0.234605i \(0.0753797\pi\)
−0.972091 + 0.234605i \(0.924620\pi\)
\(132\) 155927. 0.778909
\(133\) 161283.i 0.790604i
\(134\) 182818.i 0.879541i
\(135\) 45431.3i 0.214546i
\(136\) 72502.3 0.336128
\(137\) 102506.i 0.466602i −0.972405 0.233301i \(-0.925047\pi\)
0.972405 0.233301i \(-0.0749528\pi\)
\(138\) −69751.0 −0.311783
\(139\) −199342. −0.875106 −0.437553 0.899193i \(-0.644155\pi\)
−0.437553 + 0.899193i \(0.644155\pi\)
\(140\) −232803. −1.00385
\(141\) 267972. 1.13512
\(142\) 104122.i 0.433332i
\(143\) 264562.i 1.08190i
\(144\) 106337. 0.427346
\(145\) 251156. + 321561.i 0.992026 + 1.27012i
\(146\) −113258. −0.439729
\(147\) 602207.i 2.29854i
\(148\) 41416.4i 0.155424i
\(149\) −9725.12 −0.0358863 −0.0179432 0.999839i \(-0.505712\pi\)
−0.0179432 + 0.999839i \(0.505712\pi\)
\(150\) −477607. −1.73318
\(151\) 483239. 1.72472 0.862362 0.506292i \(-0.168984\pi\)
0.862362 + 0.506292i \(0.168984\pi\)
\(152\) −149851. −0.526078
\(153\) 80936.5i 0.279522i
\(154\) −559292. −1.90036
\(155\) 830874.i 2.77783i
\(156\) 116956.i 0.384779i
\(157\) 84963.6i 0.275096i −0.990495 0.137548i \(-0.956078\pi\)
0.990495 0.137548i \(-0.0439221\pi\)
\(158\) 201983. 0.643684
\(159\) 317297.i 0.995344i
\(160\) 372874.i 1.15150i
\(161\) −154310. −0.469170
\(162\) 285604.i 0.855020i
\(163\) 531836.i 1.56786i 0.620846 + 0.783932i \(0.286789\pi\)
−0.620846 + 0.783932i \(0.713211\pi\)
\(164\) 21532.1i 0.0625140i
\(165\) 1.15075e6 3.29057
\(166\) 237980.i 0.670302i
\(167\) −505065. −1.40138 −0.700690 0.713466i \(-0.747124\pi\)
−0.700690 + 0.713466i \(0.747124\pi\)
\(168\) 895369. 2.44753
\(169\) −172853. −0.465544
\(170\) 147755. 0.392121
\(171\) 167283.i 0.437484i
\(172\) 64651.5i 0.166632i
\(173\) −187852. −0.477201 −0.238601 0.971118i \(-0.576689\pi\)
−0.238601 + 0.971118i \(0.576689\pi\)
\(174\) −266739. 341513.i −0.667904 0.855134i
\(175\) −1.05661e6 −2.60808
\(176\) 287647.i 0.699969i
\(177\) 744111.i 1.78527i
\(178\) 500491. 1.18398
\(179\) 834374. 1.94638 0.973191 0.229999i \(-0.0738724\pi\)
0.973191 + 0.229999i \(0.0738724\pi\)
\(180\) −241465. −0.555485
\(181\) 84063.7 0.190727 0.0953635 0.995443i \(-0.469599\pi\)
0.0953635 + 0.995443i \(0.469599\pi\)
\(182\) 419507.i 0.938774i
\(183\) 203360. 0.448888
\(184\) 143373.i 0.312192i
\(185\) 305655.i 0.656603i
\(186\) 882428.i 1.87024i
\(187\) −218937. −0.457841
\(188\) 152102.i 0.313863i
\(189\) 106744.i 0.217365i
\(190\) −305387. −0.613714
\(191\) 454281.i 0.901034i 0.892768 + 0.450517i \(0.148760\pi\)
−0.892768 + 0.450517i \(0.851240\pi\)
\(192\) 729342.i 1.42784i
\(193\) 461063.i 0.890979i −0.895287 0.445489i \(-0.853030\pi\)
0.895287 0.445489i \(-0.146970\pi\)
\(194\) 3225.06 0.00615225
\(195\) 863142.i 1.62553i
\(196\) 341816. 0.635553
\(197\) −401536. −0.737154 −0.368577 0.929597i \(-0.620155\pi\)
−0.368577 + 0.929597i \(0.620155\pi\)
\(198\) −580100. −1.05157
\(199\) 107914. 0.193173 0.0965863 0.995325i \(-0.469208\pi\)
0.0965863 + 0.995325i \(0.469208\pi\)
\(200\) 981717.i 1.73545i
\(201\) 883789.i 1.54297i
\(202\) −247592. −0.426931
\(203\) −590107. 755530.i −1.00506 1.28680i
\(204\) 96786.2 0.162831
\(205\) 158908.i 0.264096i
\(206\) 386994.i 0.635383i
\(207\) −160051. −0.259617
\(208\) −215755. −0.345782
\(209\) 452508. 0.716573
\(210\) 1.82471e6 2.85526
\(211\) 296058.i 0.457794i −0.973451 0.228897i \(-0.926488\pi\)
0.973451 0.228897i \(-0.0735120\pi\)
\(212\) −180099. −0.275215
\(213\) 503352.i 0.760192i
\(214\) 299498.i 0.447053i
\(215\) 477132.i 0.703951i
\(216\) −99177.8 −0.144637
\(217\) 1.95220e6i 2.81432i
\(218\) 478777.i 0.682326i
\(219\) −547518. −0.771415
\(220\) 653172.i 0.909853i
\(221\) 164218.i 0.226172i
\(222\) 324620.i 0.442072i
\(223\) 857377. 1.15454 0.577271 0.816552i \(-0.304118\pi\)
0.577271 + 0.816552i \(0.304118\pi\)
\(224\) 876093.i 1.16662i
\(225\) −1.09592e6 −1.44319
\(226\) 625646. 0.814812
\(227\) −265585. −0.342089 −0.171045 0.985263i \(-0.554714\pi\)
−0.171045 + 0.985263i \(0.554714\pi\)
\(228\) −200042. −0.254849
\(229\) 1.13602e6i 1.43152i 0.698349 + 0.715758i \(0.253919\pi\)
−0.698349 + 0.715758i \(0.746081\pi\)
\(230\) 292184.i 0.364198i
\(231\) −2.70377e6 −3.33380
\(232\) −701977. + 548280.i −0.856254 + 0.668779i
\(233\) −341111. −0.411629 −0.205814 0.978591i \(-0.565984\pi\)
−0.205814 + 0.978591i \(0.565984\pi\)
\(234\) 435115.i 0.519475i
\(235\) 1.12252e6i 1.32594i
\(236\) −422361. −0.493633
\(237\) 976441. 1.12921
\(238\) −347161. −0.397272
\(239\) 1.46515e6 1.65916 0.829578 0.558391i \(-0.188581\pi\)
0.829578 + 0.558391i \(0.188581\pi\)
\(240\) 938458.i 1.05169i
\(241\) 31493.9 0.0349288 0.0174644 0.999847i \(-0.494441\pi\)
0.0174644 + 0.999847i \(0.494441\pi\)
\(242\) 852700.i 0.935961i
\(243\) 1.25815e6i 1.36683i
\(244\) 115428.i 0.124119i
\(245\) 2.52262e6 2.68495
\(246\) 168768.i 0.177809i
\(247\) 339412.i 0.353985i
\(248\) 1.81382e6 1.87269
\(249\) 1.15046e6i 1.17591i
\(250\) 748152.i 0.757076i
\(251\) 974972.i 0.976805i 0.872618 + 0.488402i \(0.162420\pi\)
−0.872618 + 0.488402i \(0.837580\pi\)
\(252\) 567338. 0.562782
\(253\) 432945.i 0.425238i
\(254\) −934317. −0.908678
\(255\) 714288. 0.687896
\(256\) −1.00320e6 −0.956722
\(257\) −127188. −0.120119 −0.0600596 0.998195i \(-0.519129\pi\)
−0.0600596 + 0.998195i \(0.519129\pi\)
\(258\) 506736.i 0.473950i
\(259\) 718158.i 0.665228i
\(260\) 489924. 0.449465
\(261\) −612062. 783639.i −0.556153 0.712057i
\(262\) −410011. −0.369014
\(263\) 644306.i 0.574385i 0.957873 + 0.287192i \(0.0927219\pi\)
−0.957873 + 0.287192i \(0.907278\pi\)
\(264\) 2.51212e6i 2.21835i
\(265\) −1.32914e6 −1.16267
\(266\) 717527. 0.621776
\(267\) 2.41951e6 2.07706
\(268\) 501643. 0.426637
\(269\) 2.20790e6i 1.86037i 0.367095 + 0.930183i \(0.380352\pi\)
−0.367095 + 0.930183i \(0.619648\pi\)
\(270\) −202118. −0.168731
\(271\) 1.15746e6i 0.957374i 0.877986 + 0.478687i \(0.158887\pi\)
−0.877986 + 0.478687i \(0.841113\pi\)
\(272\) 178547.i 0.146329i
\(273\) 2.02801e6i 1.64689i
\(274\) 456035. 0.366963
\(275\) 2.96451e6i 2.36386i
\(276\) 191394.i 0.151236i
\(277\) −699513. −0.547768 −0.273884 0.961763i \(-0.588308\pi\)
−0.273884 + 0.961763i \(0.588308\pi\)
\(278\) 886846.i 0.688234i
\(279\) 2.02483e6i 1.55732i
\(280\) 3.75067e6i 2.85900i
\(281\) −2.48281e6 −1.87576 −0.937881 0.346956i \(-0.887215\pi\)
−0.937881 + 0.346956i \(0.887215\pi\)
\(282\) 1.19217e6i 0.892722i
\(283\) 169883. 0.126091 0.0630456 0.998011i \(-0.479919\pi\)
0.0630456 + 0.998011i \(0.479919\pi\)
\(284\) −285705. −0.210195
\(285\) −1.47632e6 −1.07664
\(286\) 1.17700e6 0.850869
\(287\) 373366.i 0.267565i
\(288\) 908688.i 0.645555i
\(289\) 1.28396e6 0.904288
\(290\) −1.43059e6 + 1.11736e6i −0.998893 + 0.780187i
\(291\) 15590.8 0.0107929
\(292\) 310774.i 0.213298i
\(293\) 1.09182e6i 0.742985i 0.928436 + 0.371493i \(0.121154\pi\)
−0.928436 + 0.371493i \(0.878846\pi\)
\(294\) −2.67914e6 −1.80770
\(295\) −3.11705e6 −2.08540
\(296\) 667254. 0.442651
\(297\) 299490. 0.197011
\(298\) 43265.8i 0.0282231i
\(299\) 324739. 0.210066
\(300\) 1.31053e6i 0.840708i
\(301\) 1.12105e6i 0.713198i
\(302\) 2.14987e6i 1.35642i
\(303\) −1.19692e6 −0.748963
\(304\) 369028.i 0.229021i
\(305\) 851867.i 0.524351i
\(306\) −360076. −0.219832
\(307\) 2.17551e6i 1.31739i 0.752410 + 0.658696i \(0.228891\pi\)
−0.752410 + 0.658696i \(0.771109\pi\)
\(308\) 1.53467e6i 0.921805i
\(309\) 1.87083e6i 1.11465i
\(310\) 3.69646e6 2.18465
\(311\) 891275.i 0.522529i 0.965267 + 0.261265i \(0.0841395\pi\)
−0.965267 + 0.261265i \(0.915860\pi\)
\(312\) −1.88426e6 −1.09586
\(313\) 1.33851e6 0.772252 0.386126 0.922446i \(-0.373813\pi\)
0.386126 + 0.922446i \(0.373813\pi\)
\(314\) 377992. 0.216351
\(315\) 4.18699e6 2.37753
\(316\) 554233.i 0.312230i
\(317\) 1.37267e6i 0.767219i −0.923495 0.383610i \(-0.874681\pi\)
0.923495 0.383610i \(-0.125319\pi\)
\(318\) 1.41161e6 0.782795
\(319\) 2.11978e6 1.65565e6i 1.16631 0.910946i
\(320\) −3.05519e6 −1.66787
\(321\) 1.44785e6i 0.784263i
\(322\) 686507.i 0.368982i
\(323\) 280878. 0.149800
\(324\) −783684. −0.414742
\(325\) 2.22359e6 1.16774
\(326\) −2.36607e6 −1.23306
\(327\) 2.31454e6i 1.19700i
\(328\) −346901. −0.178041
\(329\) 2.63744e6i 1.34336i
\(330\) 5.11955e6i 2.58790i
\(331\) 3.39178e6i 1.70160i −0.525488 0.850801i \(-0.676117\pi\)
0.525488 0.850801i \(-0.323883\pi\)
\(332\) −653007. −0.325142
\(333\) 744876.i 0.368107i
\(334\) 2.24697e6i 1.10213i
\(335\) 3.70216e6 1.80237
\(336\) 2.20497e6i 1.06550i
\(337\) 3.55335e6i 1.70437i −0.523243 0.852184i \(-0.675278\pi\)
0.523243 0.852184i \(-0.324722\pi\)
\(338\) 769002.i 0.366130i
\(339\) 3.02454e6 1.42942
\(340\) 405434.i 0.190205i
\(341\) −5.47724e6 −2.55080
\(342\) 744222. 0.344062
\(343\) −2.36940e6 −1.08743
\(344\) −1.04159e6 −0.474571
\(345\) 1.41250e6i 0.638910i
\(346\) 835732.i 0.375298i
\(347\) 657758. 0.293253 0.146626 0.989192i \(-0.453158\pi\)
0.146626 + 0.989192i \(0.453158\pi\)
\(348\) −937097. + 731921.i −0.414798 + 0.323978i
\(349\) −2.16941e6 −0.953408 −0.476704 0.879064i \(-0.658169\pi\)
−0.476704 + 0.879064i \(0.658169\pi\)
\(350\) 4.70073e6i 2.05114i
\(351\) 224638.i 0.0973228i
\(352\) 2.45804e6 1.05738
\(353\) 420369. 0.179554 0.0897768 0.995962i \(-0.471385\pi\)
0.0897768 + 0.995962i \(0.471385\pi\)
\(354\) 3.31046e6 1.40404
\(355\) −2.10852e6 −0.887989
\(356\) 1.37332e6i 0.574312i
\(357\) −1.67827e6 −0.696932
\(358\) 3.71202e6i 1.53075i
\(359\) 4.02963e6i 1.65017i −0.565006 0.825087i \(-0.691126\pi\)
0.565006 0.825087i \(-0.308874\pi\)
\(360\) 3.89021e6i 1.58204i
\(361\) 1.89557e6 0.765546
\(362\) 373989.i 0.149999i
\(363\) 4.12218e6i 1.64195i
\(364\) −1.15111e6 −0.455369
\(365\) 2.29353e6i 0.901099i
\(366\) 904723.i 0.353031i
\(367\) 4.42973e6i 1.71677i 0.513006 + 0.858385i \(0.328532\pi\)
−0.513006 + 0.858385i \(0.671468\pi\)
\(368\) −353075. −0.135909
\(369\) 387257.i 0.148058i
\(370\) 1.35982e6 0.516390
\(371\) 3.12292e6 1.17795
\(372\) 2.42134e6 0.907191
\(373\) −185657. −0.0690939 −0.0345470 0.999403i \(-0.510999\pi\)
−0.0345470 + 0.999403i \(0.510999\pi\)
\(374\) 974022.i 0.360072i
\(375\) 3.61677e6i 1.32813i
\(376\) 2.45050e6 0.893891
\(377\) 1.24185e6 + 1.58998e6i 0.450005 + 0.576153i
\(378\) 474890. 0.170948
\(379\) 2.45032e6i 0.876245i −0.898915 0.438123i \(-0.855644\pi\)
0.898915 0.438123i \(-0.144356\pi\)
\(380\) 837968.i 0.297693i
\(381\) −4.51674e6 −1.59409
\(382\) −2.02104e6 −0.708625
\(383\) −3.91750e6 −1.36462 −0.682310 0.731063i \(-0.739025\pi\)
−0.682310 + 0.731063i \(0.739025\pi\)
\(384\) 396320. 0.137157
\(385\) 1.13260e7i 3.89425i
\(386\) 2.05121e6 0.700717
\(387\) 1.16276e6i 0.394651i
\(388\) 8849.43i 0.00298425i
\(389\) 571898.i 0.191622i −0.995400 0.0958109i \(-0.969456\pi\)
0.995400 0.0958109i \(-0.0305444\pi\)
\(390\) −3.84001e6 −1.27841
\(391\) 268735.i 0.0888962i
\(392\) 5.50695e6i 1.81007i
\(393\) −1.98210e6 −0.647359
\(394\) 1.78638e6i 0.579740i
\(395\) 4.09028e6i 1.31905i
\(396\) 1.59177e6i 0.510084i
\(397\) 1.49161e6 0.474984 0.237492 0.971389i \(-0.423675\pi\)
0.237492 + 0.971389i \(0.423675\pi\)
\(398\) 480096.i 0.151922i
\(399\) 3.46872e6 1.09078
\(400\) −2.41761e6 −0.755505
\(401\) 1.92589e6 0.598096 0.299048 0.954238i \(-0.403331\pi\)
0.299048 + 0.954238i \(0.403331\pi\)
\(402\) −3.93187e6 −1.21348
\(403\) 4.10830e6i 1.26009i
\(404\) 679381.i 0.207090i
\(405\) −5.78364e6 −1.75212
\(406\) 3.36126e6 2.62531e6i 1.01201 0.790435i
\(407\) −2.01492e6 −0.602938
\(408\) 1.55931e6i 0.463748i
\(409\) 2.15032e6i 0.635615i 0.948155 + 0.317808i \(0.102947\pi\)
−0.948155 + 0.317808i \(0.897053\pi\)
\(410\) −706963. −0.207700
\(411\) 2.20460e6 0.643761
\(412\) 1.06189e6 0.308204
\(413\) 7.32373e6 2.11279
\(414\) 712048.i 0.204178i
\(415\) −4.81923e6 −1.37359
\(416\) 1.84370e6i 0.522344i
\(417\) 4.28725e6i 1.20736i
\(418\) 2.01315e6i 0.563554i
\(419\) −4.31814e6 −1.20161 −0.600803 0.799397i \(-0.705152\pi\)
−0.600803 + 0.799397i \(0.705152\pi\)
\(420\) 5.00692e6i 1.38499i
\(421\) 3.57350e6i 0.982627i 0.870983 + 0.491314i \(0.163483\pi\)
−0.870983 + 0.491314i \(0.836517\pi\)
\(422\) 1.31712e6 0.360036
\(423\) 2.73557e6i 0.743355i
\(424\) 2.90156e6i 0.783821i
\(425\) 1.84012e6i 0.494166i
\(426\) 2.23935e6 0.597858
\(427\) 2.00152e6i 0.531239i
\(428\) −821809. −0.216851
\(429\) 5.68995e6 1.49268
\(430\) −2.12270e6 −0.553627
\(431\) 3.79128e6 0.983088 0.491544 0.870853i \(-0.336433\pi\)
0.491544 + 0.870853i \(0.336433\pi\)
\(432\) 244239.i 0.0629659i
\(433\) 2.38236e6i 0.610644i −0.952249 0.305322i \(-0.901236\pi\)
0.952249 0.305322i \(-0.0987641\pi\)
\(434\) −8.68507e6 −2.21334
\(435\) −6.91583e6 + 5.40162e6i −1.75235 + 1.36868i
\(436\) −1.31374e6 −0.330974
\(437\) 555434.i 0.139133i
\(438\) 2.43584e6i 0.606685i
\(439\) 7.23282e6 1.79121 0.895605 0.444851i \(-0.146743\pi\)
0.895605 + 0.444851i \(0.146743\pi\)
\(440\) 1.05232e7 2.59129
\(441\) −6.14758e6 −1.50525
\(442\) 730583. 0.177875
\(443\) 2.05198e6i 0.496780i 0.968660 + 0.248390i \(0.0799015\pi\)
−0.968660 + 0.248390i \(0.920099\pi\)
\(444\) 890744. 0.214435
\(445\) 1.01352e7i 2.42624i
\(446\) 3.81436e6i 0.907998i
\(447\) 209159.i 0.0495116i
\(448\) 7.17837e6 1.68978
\(449\) 3.05219e6i 0.714489i 0.934011 + 0.357245i \(0.116284\pi\)
−0.934011 + 0.357245i \(0.883716\pi\)
\(450\) 4.87562e6i 1.13501i
\(451\) 1.04755e6 0.242511
\(452\) 1.71674e6i 0.395239i
\(453\) 1.03930e7i 2.37956i
\(454\) 1.18156e6i 0.269039i
\(455\) −8.49526e6 −1.92375
\(456\) 3.22285e6i 0.725818i
\(457\) 2.06974e6 0.463581 0.231790 0.972766i \(-0.425542\pi\)
0.231790 + 0.972766i \(0.425542\pi\)
\(458\) −5.05400e6 −1.12583
\(459\) 185897. 0.0411853
\(460\) 801741. 0.176661
\(461\) 3.36651e6i 0.737781i −0.929473 0.368890i \(-0.879738\pi\)
0.929473 0.368890i \(-0.120262\pi\)
\(462\) 1.20287e7i 2.62189i
\(463\) −81095.6 −0.0175811 −0.00879053 0.999961i \(-0.502798\pi\)
−0.00879053 + 0.999961i \(0.502798\pi\)
\(464\) −1.35022e6 1.72871e6i −0.291144 0.372759i
\(465\) 1.78697e7 3.83251
\(466\) 1.51756e6i 0.323729i
\(467\) 1.95508e6i 0.414833i 0.978253 + 0.207416i \(0.0665055\pi\)
−0.978253 + 0.207416i \(0.933495\pi\)
\(468\) −1.19394e6 −0.251980
\(469\) −8.69847e6 −1.82604
\(470\) 4.99396e6 1.04280
\(471\) 1.82732e6 0.379543
\(472\) 6.80461e6i 1.40588i
\(473\) 3.14532e6 0.646416
\(474\) 4.34406e6i 0.888077i
\(475\) 3.80323e6i 0.773427i
\(476\) 952593.i 0.192704i
\(477\) 3.23910e6 0.651821
\(478\) 6.51827e6i 1.30486i
\(479\) 5.02666e6i 1.00102i −0.865732 0.500508i \(-0.833147\pi\)
0.865732 0.500508i \(-0.166853\pi\)
\(480\) −8.01942e6 −1.58869
\(481\) 1.51133e6i 0.297849i
\(482\) 140112.i 0.0274700i
\(483\) 3.31876e6i 0.647303i
\(484\) −2.33977e6 −0.454004
\(485\) 65309.3i 0.0126073i
\(486\) 5.59733e6 1.07496
\(487\) −7.01534e6 −1.34038 −0.670188 0.742192i \(-0.733786\pi\)
−0.670188 + 0.742192i \(0.733786\pi\)
\(488\) 1.85965e6 0.353494
\(489\) −1.14382e7 −2.16315
\(490\) 1.12228e7i 2.11160i
\(491\) 6.04666e6i 1.13191i 0.824436 + 0.565954i \(0.191492\pi\)
−0.824436 + 0.565954i \(0.808508\pi\)
\(492\) −463092. −0.0862491
\(493\) 1.31578e6 1.02769e6i 0.243817 0.190434i
\(494\) −1.51000e6 −0.278394
\(495\) 1.17474e7i 2.15490i
\(496\) 4.46679e6i 0.815250i
\(497\) 4.95412e6 0.899654
\(498\) 5.11825e6 0.924801
\(499\) −5.31655e6 −0.955825 −0.477912 0.878408i \(-0.658606\pi\)
−0.477912 + 0.878408i \(0.658606\pi\)
\(500\) 2.05290e6 0.367233
\(501\) 1.08625e7i 1.93345i
\(502\) −4.33753e6 −0.768215
\(503\) 667123.i 0.117567i −0.998271 0.0587835i \(-0.981278\pi\)
0.998271 0.0587835i \(-0.0187222\pi\)
\(504\) 9.14030e6i 1.60282i
\(505\) 5.01387e6i 0.874873i
\(506\) 1.92612e6 0.334431
\(507\) 3.71756e6i 0.642301i
\(508\) 2.56372e6i 0.440770i
\(509\) 852016. 0.145765 0.0728825 0.997341i \(-0.476780\pi\)
0.0728825 + 0.997341i \(0.476780\pi\)
\(510\) 3.17778e6i 0.541001i
\(511\) 5.38881e6i 0.912936i
\(512\) 5.05277e6i 0.851834i
\(513\) −384221. −0.0644596
\(514\) 565842.i 0.0944686i
\(515\) 7.83684e6 1.30204
\(516\) −1.39046e6 −0.229898
\(517\) −7.39982e6 −1.21757
\(518\) −3.19499e6 −0.523173
\(519\) 4.04015e6i 0.658384i
\(520\) 7.89310e6i 1.28009i
\(521\) 8.30897e6 1.34107 0.670537 0.741876i \(-0.266064\pi\)
0.670537 + 0.741876i \(0.266064\pi\)
\(522\) 3.48631e6 2.72299e6i 0.560002 0.437390i
\(523\) −5.37728e6 −0.859625 −0.429812 0.902918i \(-0.641420\pi\)
−0.429812 + 0.902918i \(0.641420\pi\)
\(524\) 1.12505e6i 0.178997i
\(525\) 2.27246e7i 3.59830i
\(526\) −2.86644e6 −0.451729
\(527\) −3.39980e6 −0.533245
\(528\) −6.18644e6 −0.965731
\(529\) −5.90492e6 −0.917434
\(530\) 5.91320e6i 0.914393i
\(531\) 7.59620e6 1.16912
\(532\) 1.96886e6i 0.301603i
\(533\) 785731.i 0.119800i
\(534\) 1.07641e7i 1.63352i
\(535\) −6.06500e6 −0.916107
\(536\) 8.08191e6i 1.21507i
\(537\) 1.79449e7i 2.68538i
\(538\) −9.82267e6 −1.46310
\(539\) 1.66295e7i 2.46551i
\(540\) 554604.i 0.0818461i
\(541\) 7.17375e6i 1.05379i 0.849931 + 0.526894i \(0.176643\pi\)
−0.849931 + 0.526894i \(0.823357\pi\)
\(542\) −5.14938e6 −0.752934
\(543\) 1.80796e6i 0.263142i
\(544\) 1.52574e6 0.221046
\(545\) −9.69550e6 −1.39823
\(546\) 9.02236e6 1.29521
\(547\) 8.04876e6 1.15017 0.575083 0.818095i \(-0.304970\pi\)
0.575083 + 0.818095i \(0.304970\pi\)
\(548\) 1.25134e6i 0.178002i
\(549\) 2.07598e6i 0.293963i
\(550\) 1.31888e7 1.85908
\(551\) −2.71950e6 + 2.12407e6i −0.381602 + 0.298051i
\(552\) −3.08352e6 −0.430724
\(553\) 9.61038e6i 1.33637i
\(554\) 3.11205e6i 0.430796i
\(555\) 6.57374e6 0.905900
\(556\) −2.43347e6 −0.333840
\(557\) 1.01144e6 0.138134 0.0690669 0.997612i \(-0.477998\pi\)
0.0690669 + 0.997612i \(0.477998\pi\)
\(558\) −9.00819e6 −1.22476
\(559\) 2.35920e6i 0.319327i
\(560\) 9.23654e6 1.24463
\(561\) 4.70868e6i 0.631673i
\(562\) 1.10457e7i 1.47521i
\(563\) 1.26692e6i 0.168453i 0.996447 + 0.0842266i \(0.0268419\pi\)
−0.996447 + 0.0842266i \(0.973158\pi\)
\(564\) 3.27127e6 0.433030
\(565\) 1.26697e7i 1.66972i
\(566\) 755789.i 0.0991653i
\(567\) 1.35890e7 1.77513
\(568\) 4.60296e6i 0.598641i
\(569\) 8.25094e6i 1.06837i −0.845367 0.534187i \(-0.820618\pi\)
0.845367 0.534187i \(-0.179382\pi\)
\(570\) 6.56797e6i 0.846728i
\(571\) 1.12215e7 1.44032 0.720160 0.693808i \(-0.244068\pi\)
0.720160 + 0.693808i \(0.244068\pi\)
\(572\) 3.22965e6i 0.412729i
\(573\) −9.77025e6 −1.24314
\(574\) 1.66106e6 0.210429
\(575\) 3.63881e6 0.458976
\(576\) 7.44544e6 0.935048
\(577\) 7.06718e6i 0.883703i 0.897088 + 0.441852i \(0.145678\pi\)
−0.897088 + 0.441852i \(0.854322\pi\)
\(578\) 5.71218e6i 0.711184i
\(579\) 9.91611e6 1.22926
\(580\) 3.06599e6 + 3.92546e6i 0.378443 + 0.484530i
\(581\) 1.13231e7 1.39164
\(582\) 69361.5i 0.00848812i
\(583\) 8.76191e6i 1.06765i
\(584\) −5.00684e6 −0.607480
\(585\) −8.81132e6 −1.06451
\(586\) −4.85735e6 −0.584326
\(587\) −9.89439e6 −1.18521 −0.592603 0.805495i \(-0.701900\pi\)
−0.592603 + 0.805495i \(0.701900\pi\)
\(588\) 7.35145e6i 0.876858i
\(589\) 7.02686e6 0.834589
\(590\) 1.38674e7i 1.64008i
\(591\) 8.63584e6i 1.01704i
\(592\) 1.64321e6i 0.192703i
\(593\) 3.83383e6 0.447709 0.223854 0.974623i \(-0.428136\pi\)
0.223854 + 0.974623i \(0.428136\pi\)
\(594\) 1.33239e6i 0.154941i
\(595\) 7.03020e6i 0.814095i
\(596\) −118720. −0.0136901
\(597\) 2.32091e6i 0.266516i
\(598\) 1.44472e6i 0.165208i
\(599\) 1.47715e7i 1.68212i 0.540938 + 0.841062i \(0.318069\pi\)
−0.540938 + 0.841062i \(0.681931\pi\)
\(600\) −2.11138e7 −2.39436
\(601\) 1.70212e7i 1.92223i −0.276155 0.961113i \(-0.589060\pi\)
0.276155 0.961113i \(-0.410940\pi\)
\(602\) 4.98743e6 0.560900
\(603\) −9.02209e6 −1.01045
\(604\) 5.89915e6 0.657956
\(605\) −1.72676e7 −1.91798
\(606\) 5.32497e6i 0.589027i
\(607\) 1.99520e6i 0.219794i −0.993943 0.109897i \(-0.964948\pi\)
0.993943 0.109897i \(-0.0350520\pi\)
\(608\) −3.15347e6 −0.345963
\(609\) 1.62492e7 1.26915e7i 1.77537 1.38666i
\(610\) 3.78985e6 0.412380
\(611\) 5.55037e6i 0.601477i
\(612\) 988034.i 0.106633i
\(613\) 8.31000e6 0.893202 0.446601 0.894733i \(-0.352634\pi\)
0.446601 + 0.894733i \(0.352634\pi\)
\(614\) −9.67856e6 −1.03607
\(615\) −3.41765e6 −0.364368
\(616\) −2.47249e7 −2.62532
\(617\) 3.70169e6i 0.391460i −0.980658 0.195730i \(-0.937292\pi\)
0.980658 0.195730i \(-0.0627076\pi\)
\(618\) −8.32309e6 −0.876624
\(619\) 1.29198e7i 1.35528i −0.735395 0.677639i \(-0.763003\pi\)
0.735395 0.677639i \(-0.236997\pi\)
\(620\) 1.01429e7i 1.05970i
\(621\) 367611.i 0.0382524i
\(622\) −3.96517e6 −0.410947
\(623\) 2.38134e7i 2.45811i
\(624\) 4.64026e6i 0.477068i
\(625\) −448271. −0.0459029
\(626\) 5.95484e6i 0.607344i
\(627\) 9.73211e6i 0.988640i
\(628\) 1.03719e6i 0.104945i
\(629\) −1.25069e6 −0.126044
\(630\) 1.86274e7i 1.86982i
\(631\) 1.52718e6 0.152692 0.0763459 0.997081i \(-0.475675\pi\)
0.0763459 + 0.997081i \(0.475675\pi\)
\(632\) 8.92918e6 0.889240
\(633\) 6.36733e6 0.631609
\(634\) 6.10686e6 0.603385
\(635\) 1.89204e7i 1.86207i
\(636\) 3.87341e6i 0.379709i
\(637\) 1.24732e7 1.21795
\(638\) 7.36580e6 + 9.43062e6i 0.716420 + 0.917251i
\(639\) 5.13843e6 0.497827
\(640\) 1.66017e6i 0.160215i
\(641\) 9.43372e6i 0.906855i −0.891293 0.453427i \(-0.850201\pi\)
0.891293 0.453427i \(-0.149799\pi\)
\(642\) 6.44131e6 0.616789
\(643\) 1.37293e7 1.30954 0.654772 0.755826i \(-0.272765\pi\)
0.654772 + 0.755826i \(0.272765\pi\)
\(644\) −1.88374e6 −0.178981
\(645\) −1.02617e7 −0.971225
\(646\) 1.24959e6i 0.117811i
\(647\) −1.73477e7 −1.62922 −0.814611 0.580007i \(-0.803050\pi\)
−0.814611 + 0.580007i \(0.803050\pi\)
\(648\) 1.26258e7i 1.18120i
\(649\) 2.05480e7i 1.91496i
\(650\) 9.89247e6i 0.918378i
\(651\) −4.19860e7 −3.88286
\(652\) 6.49239e6i 0.598117i
\(653\) 6.34963e6i 0.582728i −0.956612 0.291364i \(-0.905891\pi\)
0.956612 0.291364i \(-0.0941090\pi\)
\(654\) 1.02971e7 0.941390
\(655\) 8.30296e6i 0.756188i
\(656\) 854292.i 0.0775081i
\(657\) 5.58929e6i 0.505177i
\(658\) −1.17336e7 −1.05650
\(659\) 4.42658e6i 0.397058i −0.980095 0.198529i \(-0.936384\pi\)
0.980095 0.198529i \(-0.0636165\pi\)
\(660\) 1.40478e7 1.25530
\(661\) 1.51522e7 1.34887 0.674437 0.738332i \(-0.264386\pi\)
0.674437 + 0.738332i \(0.264386\pi\)
\(662\) 1.50896e7 1.33824
\(663\) 3.53183e6 0.312044
\(664\) 1.05205e7i 0.926012i
\(665\) 1.45303e7i 1.27415i
\(666\) −3.31386e6 −0.289500
\(667\) 2.03224e6 + 2.60193e6i 0.176873 + 0.226455i
\(668\) −6.16559e6 −0.534605
\(669\) 1.84397e7i 1.59290i
\(670\) 1.64704e7i 1.41748i
\(671\) −5.61563e6 −0.481495
\(672\) 1.88422e7 1.60956
\(673\) −1.21002e7 −1.02980 −0.514901 0.857249i \(-0.672171\pi\)
−0.514901 + 0.857249i \(0.672171\pi\)
\(674\) 1.58084e7 1.34041
\(675\) 2.51715e6i 0.212642i
\(676\) −2.11011e6 −0.177598
\(677\) 1.94913e7i 1.63444i 0.576326 + 0.817220i \(0.304486\pi\)
−0.576326 + 0.817220i \(0.695514\pi\)
\(678\) 1.34558e7i 1.12418i
\(679\) 153449.i 0.0127729i
\(680\) 6.53189e6 0.541709
\(681\) 5.71196e6i 0.471973i
\(682\) 2.43675e7i 2.00609i
\(683\) −1.58984e6 −0.130407 −0.0652034 0.997872i \(-0.520770\pi\)
−0.0652034 + 0.997872i \(0.520770\pi\)
\(684\) 2.04211e6i 0.166893i
\(685\) 9.23497e6i 0.751985i
\(686\) 1.05412e7i 0.855221i
\(687\) −2.44324e7 −1.97503
\(688\) 2.56506e6i 0.206598i
\(689\) −6.57203e6 −0.527414
\(690\) −6.28402e6 −0.502476
\(691\) −478964. −0.0381599 −0.0190800 0.999818i \(-0.506074\pi\)
−0.0190800 + 0.999818i \(0.506074\pi\)
\(692\) −2.29321e6 −0.182045
\(693\) 2.76012e7i 2.18321i
\(694\) 2.92628e6i 0.230631i
\(695\) −1.79591e7 −1.41034
\(696\) −1.17919e7 1.50974e7i −0.922699 1.18136i
\(697\) 650227. 0.0506971
\(698\) 9.65145e6i 0.749815i
\(699\) 7.33629e6i 0.567915i
\(700\) −1.28986e7 −0.994941
\(701\) 4.60638e6 0.354050 0.177025 0.984206i \(-0.443353\pi\)
0.177025 + 0.984206i \(0.443353\pi\)
\(702\) −999385. −0.0765403
\(703\) 2.58498e6 0.197274
\(704\) 2.01402e7i 1.53155i
\(705\) 2.41421e7 1.82938
\(706\) 1.87017e6i 0.141211i
\(707\) 1.17804e7i 0.886365i
\(708\) 9.08375e6i 0.681055i
\(709\) 9.47025e6 0.707531 0.353766 0.935334i \(-0.384901\pi\)
0.353766 + 0.935334i \(0.384901\pi\)
\(710\) 9.38056e6i 0.698366i
\(711\) 9.96792e6i 0.739487i
\(712\) 2.21254e7 1.63566
\(713\) 6.72307e6i 0.495272i
\(714\) 7.46640e6i 0.548107i
\(715\) 2.38350e7i 1.74361i
\(716\) 1.01856e7 0.742515
\(717\) 3.15110e7i 2.28910i
\(718\) 1.79273e7 1.29779
\(719\) −1.63490e7 −1.17942 −0.589711 0.807614i \(-0.700759\pi\)
−0.589711 + 0.807614i \(0.700759\pi\)
\(720\) 9.58017e6 0.688719
\(721\) −1.84132e7 −1.31914
\(722\) 8.43314e6i 0.602069i
\(723\) 677341.i 0.0481905i
\(724\) 1.02621e6 0.0727594
\(725\) 1.39154e7 + 1.78163e7i 0.983221 + 1.25884i
\(726\) 1.83391e7 1.29132
\(727\) 1.79177e7i 1.25732i −0.777679 0.628661i \(-0.783603\pi\)
0.777679 0.628661i \(-0.216397\pi\)
\(728\) 1.85454e7i 1.29690i
\(729\) 1.14592e7 0.798609
\(730\) −1.02036e7 −0.708676
\(731\) 1.95234e6 0.135133
\(732\) 2.48252e6 0.171244
\(733\) 1.78468e7i 1.22687i −0.789744 0.613436i \(-0.789787\pi\)
0.789744 0.613436i \(-0.210213\pi\)
\(734\) −1.97073e7 −1.35017
\(735\) 5.42541e7i 3.70437i
\(736\) 3.01714e6i 0.205305i
\(737\) 2.44051e7i 1.65506i
\(738\) 1.72286e6 0.116442
\(739\) 2.55426e7i 1.72050i −0.509876 0.860248i \(-0.670309\pi\)
0.509876 0.860248i \(-0.329691\pi\)
\(740\) 3.73129e6i 0.250484i
\(741\) −7.29975e6 −0.488385
\(742\) 1.38935e7i 0.926404i
\(743\) 1.50256e7i 0.998528i 0.866450 + 0.499264i \(0.166396\pi\)
−0.866450 + 0.499264i \(0.833604\pi\)
\(744\) 3.90099e7i 2.58370i
\(745\) −876157. −0.0578351
\(746\) 825966.i 0.0543394i
\(747\) 1.17444e7 0.770067
\(748\) −2.67267e6 −0.174659
\(749\) 1.42501e7 0.928141
\(750\) −1.60905e7 −1.04452
\(751\) 2.09579e6i 0.135597i −0.997699 0.0677983i \(-0.978403\pi\)
0.997699 0.0677983i \(-0.0215974\pi\)
\(752\) 6.03469e6i 0.389144i
\(753\) −2.09688e7 −1.34768
\(754\) −7.07361e6 + 5.52485e6i −0.453119 + 0.353910i
\(755\) 4.35361e7 2.77960
\(756\) 1.30308e6i 0.0829213i
\(757\) 2.93991e7i 1.86464i 0.361639 + 0.932318i \(0.382217\pi\)
−0.361639 + 0.932318i \(0.617783\pi\)
\(758\) 1.09012e7 0.689129
\(759\) 9.31138e6 0.586691
\(760\) −1.35004e7 −0.847837
\(761\) −9.58603e6 −0.600036 −0.300018 0.953934i \(-0.596993\pi\)
−0.300018 + 0.953934i \(0.596993\pi\)
\(762\) 2.00944e7i 1.25368i
\(763\) 2.27802e7 1.41660
\(764\) 5.54564e6i 0.343731i
\(765\) 7.29175e6i 0.450483i
\(766\) 1.74285e7i 1.07322i
\(767\) −1.54124e7 −0.945982
\(768\) 2.15758e7i 1.31997i
\(769\) 1.57832e7i 0.962453i 0.876596 + 0.481226i \(0.159808\pi\)
−0.876596 + 0.481226i \(0.840192\pi\)
\(770\) −5.03879e7 −3.06266
\(771\) 2.73543e6i 0.165726i
\(772\) 5.62844e6i 0.339895i
\(773\) 1.49255e7i 0.898423i 0.893425 + 0.449212i \(0.148295\pi\)
−0.893425 + 0.449212i \(0.851705\pi\)
\(774\) 5.17298e6 0.310376
\(775\) 4.60350e7i 2.75318i
\(776\) 142572. 0.00849923
\(777\) −1.54455e7 −0.917800
\(778\) 2.54430e6 0.150702
\(779\) −1.34392e6 −0.0793467
\(780\) 1.05368e7i 0.620116i
\(781\) 1.38997e7i 0.815412i
\(782\) 1.19557e6 0.0699130
\(783\) −1.79988e6 + 1.40580e6i −0.104916 + 0.0819445i
\(784\) −1.35616e7 −0.787991
\(785\) 7.65456e6i 0.443349i
\(786\) 8.81813e6i 0.509120i
\(787\) 3.13562e7 1.80462 0.902312 0.431083i \(-0.141868\pi\)
0.902312 + 0.431083i \(0.141868\pi\)
\(788\) −4.90175e6 −0.281213
\(789\) −1.38571e7 −0.792466
\(790\) 1.81971e7 1.03737
\(791\) 2.97682e7i 1.69166i
\(792\) −2.56448e7 −1.45273
\(793\) 4.21210e6i 0.237857i
\(794\) 6.63599e6i 0.373555i
\(795\) 2.85860e7i 1.60411i
\(796\) 1.31736e6 0.0736924
\(797\) 2.64771e6i 0.147647i −0.997271 0.0738235i \(-0.976480\pi\)
0.997271 0.0738235i \(-0.0235202\pi\)
\(798\) 1.54319e7i 0.857851i
\(799\) −4.59318e6 −0.254534
\(800\) 2.06593e7i 1.14128i
\(801\) 2.46993e7i 1.36020i
\(802\) 8.56805e6i 0.470377i
\(803\) 1.51193e7 0.827451
\(804\) 1.07889e7i 0.588621i
\(805\) −1.39021e7 −0.756123
\(806\) 1.82773e7 0.991003
\(807\) −4.74854e7 −2.56671
\(808\) −1.09454e7 −0.589799
\(809\) 9.80290e6i 0.526603i −0.964714 0.263301i \(-0.915189\pi\)
0.964714 0.263301i \(-0.0848114\pi\)
\(810\) 2.57307e7i 1.37797i
\(811\) −6.02035e6 −0.321417 −0.160709 0.987002i \(-0.551378\pi\)
−0.160709 + 0.987002i \(0.551378\pi\)
\(812\) −7.20375e6 9.22314e6i −0.383414 0.490895i
\(813\) −2.48935e7 −1.32087
\(814\) 8.96414e6i 0.474185i
\(815\) 4.79143e7i 2.52680i
\(816\) −3.84001e6 −0.201887
\(817\) −4.03519e6 −0.211499
\(818\) −9.56650e6 −0.499884
\(819\) 2.07028e7 1.07850
\(820\) 1.93988e6i 0.100749i
\(821\) 3.70550e7 1.91862 0.959309 0.282359i \(-0.0911171\pi\)
0.959309 + 0.282359i \(0.0911171\pi\)
\(822\) 9.80797e6i 0.506290i
\(823\) 2.15600e7i 1.10956i −0.831998 0.554779i \(-0.812803\pi\)
0.831998 0.554779i \(-0.187197\pi\)
\(824\) 1.71080e7i 0.877772i
\(825\) 6.37580e7 3.26137
\(826\) 3.25823e7i 1.66162i
\(827\) 1.21823e7i 0.619391i −0.950836 0.309696i \(-0.899773\pi\)
0.950836 0.309696i \(-0.100227\pi\)
\(828\) −1.95383e6 −0.0990400
\(829\) 1.49741e6i 0.0756756i −0.999284 0.0378378i \(-0.987953\pi\)
0.999284 0.0378378i \(-0.0120470\pi\)
\(830\) 2.14402e7i 1.08027i
\(831\) 1.50445e7i 0.755743i
\(832\) −1.51065e7 −0.756583
\(833\) 1.03221e7i 0.515416i
\(834\) 1.90734e7 0.949541
\(835\) −4.55024e7 −2.25849
\(836\) 5.52400e6 0.273362
\(837\) 4.65068e6 0.229458
\(838\) 1.92109e7i 0.945012i
\(839\) 2.46412e7i 1.20853i −0.796785 0.604263i \(-0.793468\pi\)
0.796785 0.604263i \(-0.206532\pi\)
\(840\) 8.06658e7 3.94449
\(841\) −4.96789e6 + 1.99004e7i −0.242204 + 0.970225i
\(842\) −1.58981e7 −0.772794
\(843\) 5.33979e7i 2.58795i
\(844\) 3.61413e6i 0.174642i
\(845\) −1.55727e7 −0.750279
\(846\) −1.21702e7 −0.584617
\(847\) 4.05715e7 1.94318
\(848\) 7.14549e6 0.341226
\(849\) 3.65369e6i 0.173965i
\(850\) 8.18645e6 0.388641
\(851\) 2.47323e6i 0.117069i
\(852\) 6.14468e6i 0.290002i
\(853\) 2.54111e7i 1.19578i 0.801578 + 0.597890i \(0.203994\pi\)
−0.801578 + 0.597890i \(0.796006\pi\)
\(854\) −8.90451e6 −0.417797
\(855\) 1.50709e7i 0.705057i
\(856\) 1.32401e7i 0.617597i
\(857\) −1.32471e7 −0.616127 −0.308063 0.951366i \(-0.599681\pi\)
−0.308063 + 0.951366i \(0.599681\pi\)
\(858\) 2.53139e7i 1.17393i
\(859\) 8.05724e6i 0.372566i −0.982496 0.186283i \(-0.940356\pi\)
0.982496 0.186283i \(-0.0596441\pi\)
\(860\) 5.82459e6i 0.268547i
\(861\) 8.03000e6 0.369154
\(862\) 1.68669e7i 0.773157i
\(863\) 2.46723e7 1.12767 0.563836 0.825887i \(-0.309325\pi\)
0.563836 + 0.825887i \(0.309325\pi\)
\(864\) −2.08710e6 −0.0951172
\(865\) −1.69240e7 −0.769066
\(866\) 1.05988e7 0.480246
\(867\) 2.76142e7i 1.24763i
\(868\) 2.38315e7i 1.07362i
\(869\) −2.69637e7 −1.21124
\(870\) −2.40311e7 3.07677e7i −1.07641 1.37815i
\(871\) 1.83055e7 0.817592
\(872\) 2.11655e7i 0.942623i
\(873\) 159158.i 0.00706792i
\(874\) −2.47106e6 −0.109422
\(875\) −3.55971e7 −1.57179
\(876\) −6.68383e6 −0.294283
\(877\) −4.30471e6 −0.188993 −0.0944963 0.995525i \(-0.530124\pi\)
−0.0944963 + 0.995525i \(0.530124\pi\)
\(878\) 3.21779e7i 1.40871i
\(879\) −2.34817e7 −1.02508
\(880\) 2.59148e7i 1.12808i
\(881\) 7.17894e6i 0.311617i 0.987787 + 0.155808i \(0.0497982\pi\)
−0.987787 + 0.155808i \(0.950202\pi\)
\(882\) 2.73498e7i 1.18381i
\(883\) −2.27258e6 −0.0980884 −0.0490442 0.998797i \(-0.515618\pi\)
−0.0490442 + 0.998797i \(0.515618\pi\)
\(884\) 2.00469e6i 0.0862812i
\(885\) 6.70386e7i 2.87718i
\(886\) −9.12901e6 −0.390696
\(887\) 3.19149e7i 1.36202i −0.732273 0.681012i \(-0.761540\pi\)
0.732273 0.681012i \(-0.238460\pi\)
\(888\) 1.43507e7i 0.610716i
\(889\) 4.44548e7i 1.88653i
\(890\) 4.50903e7 1.90813
\(891\) 3.81265e7i 1.60891i
\(892\) 1.04664e7 0.440440
\(893\) 9.49338e6 0.398375
\(894\) 930520. 0.0389388
\(895\) 7.51706e7 3.13682
\(896\) 3.90068e6i 0.162320i
\(897\) 6.98417e6i 0.289824i
\(898\) −1.35788e7 −0.561915
\(899\) 3.29173e7 2.57101e7i 1.35839 1.06097i
\(900\) −1.33785e7 −0.550555
\(901\) 5.43864e6i 0.223192i
\(902\) 4.66040e6i 0.190725i
\(903\) 2.41105e7 0.983983
\(904\) 2.76582e7 1.12565
\(905\) 7.57348e6 0.307379
\(906\) −4.62373e7 −1.87142
\(907\) 2.84496e7i 1.14831i 0.818748 + 0.574153i \(0.194669\pi\)
−0.818748 + 0.574153i \(0.805331\pi\)
\(908\) −3.24214e6 −0.130502
\(909\) 1.22187e7i 0.490474i
\(910\) 3.77943e7i 1.51295i
\(911\) 1.64758e7i 0.657734i 0.944376 + 0.328867i \(0.106667\pi\)
−0.944376 + 0.328867i \(0.893333\pi\)
\(912\) 7.93671e6 0.315976
\(913\) 3.17690e7i 1.26133i
\(914\) 9.20802e6i 0.364587i
\(915\) 1.83211e7 0.723436
\(916\) 1.38679e7i 0.546101i
\(917\) 1.95084e7i 0.766121i
\(918\) 827034.i 0.0323904i
\(919\) −2.11310e7 −0.825336 −0.412668 0.910881i \(-0.635403\pi\)
−0.412668 + 0.910881i \(0.635403\pi\)
\(920\) 1.29167e7i 0.503134i
\(921\) −4.67887e7 −1.81757
\(922\) 1.49772e7 0.580233
\(923\) −1.04257e7 −0.402811
\(924\) −3.30063e7 −1.27179
\(925\) 1.69350e7i 0.650775i
\(926\) 360784.i 0.0138268i
\(927\) −1.90982e7 −0.729951
\(928\) −1.47724e7 + 1.15380e7i −0.563095 + 0.439806i
\(929\) −1.53237e7 −0.582538 −0.291269 0.956641i \(-0.594077\pi\)
−0.291269 + 0.956641i \(0.594077\pi\)
\(930\) 7.94999e7i 3.01411i
\(931\) 2.13343e7i 0.806684i
\(932\) −4.16412e6 −0.157030
\(933\) −1.91687e7 −0.720922
\(934\) −8.69792e6 −0.326248
\(935\) −1.97245e7 −0.737865
\(936\) 1.92353e7i 0.717646i
\(937\) −2.82423e7 −1.05087 −0.525437 0.850832i \(-0.676098\pi\)
−0.525437 + 0.850832i \(0.676098\pi\)
\(938\) 3.86984e7i 1.43610i
\(939\) 2.87873e7i 1.06546i
\(940\) 1.37032e7i 0.505828i
\(941\) −1.91232e7 −0.704022 −0.352011 0.935996i \(-0.614502\pi\)
−0.352011 + 0.935996i \(0.614502\pi\)
\(942\) 8.12950e6i 0.298495i
\(943\) 1.28582e6i 0.0470869i
\(944\) 1.67573e7 0.612032
\(945\) 9.61680e6i 0.350309i
\(946\) 1.39931e7i 0.508378i
\(947\) 916050.i 0.0331928i 0.999862 + 0.0165964i \(0.00528305\pi\)
−0.999862 + 0.0165964i \(0.994717\pi\)
\(948\) 1.19199e7 0.430777
\(949\) 1.13405e7i 0.408758i
\(950\) −1.69201e7 −0.608267
\(951\) 2.95222e7 1.05852
\(952\) −1.53471e7 −0.548825
\(953\) 7.31338e6 0.260847 0.130423 0.991458i \(-0.458366\pi\)
0.130423 + 0.991458i \(0.458366\pi\)
\(954\) 1.44104e7i 0.512630i
\(955\) 4.09272e7i 1.45212i
\(956\) 1.78858e7 0.632943
\(957\) 3.56082e7 + 4.55901e7i 1.25681 + 1.60913i
\(958\) 2.23630e7 0.787257
\(959\) 2.16982e7i 0.761863i
\(960\) 6.57081e7i 2.30113i
\(961\) −5.64252e7 −1.97090
\(962\) 6.72372e6 0.234246
\(963\) 1.47803e7 0.513591
\(964\) 384462. 0.0133248
\(965\) 4.15382e7i 1.43592i
\(966\) 1.47647e7 0.509076
\(967\) 5.71496e6i 0.196538i −0.995160 0.0982692i \(-0.968669\pi\)
0.995160 0.0982692i \(-0.0313306\pi\)
\(968\) 3.76957e7i 1.29302i
\(969\) 6.04086e6i 0.206676i
\(970\) 290553. 0.00991507
\(971\) 3.76567e7i 1.28172i −0.767657 0.640861i \(-0.778578\pi\)
0.767657 0.640861i \(-0.221422\pi\)
\(972\) 1.53588e7i 0.521426i
\(973\) 4.21962e7 1.42886
\(974\) 3.12104e7i 1.05415i
\(975\) 4.78228e7i 1.61110i
\(976\) 4.57964e6i 0.153889i
\(977\) −1.50840e7 −0.505567 −0.252784 0.967523i \(-0.581346\pi\)
−0.252784 + 0.967523i \(0.581346\pi\)
\(978\) 5.08872e7i 1.70122i
\(979\) −6.68127e7 −2.22793
\(980\) 3.07949e7 1.02427
\(981\) 2.36278e7 0.783881
\(982\) −2.69008e7 −0.890198
\(983\) 4.82887e7i 1.59390i 0.604043 + 0.796951i \(0.293555\pi\)
−0.604043 + 0.796951i \(0.706445\pi\)
\(984\) 7.46082e6i 0.245640i
\(985\) −3.61752e7 −1.18801
\(986\) 4.57206e6 + 5.85372e6i 0.149768 + 0.191752i
\(987\) −5.67236e7 −1.85341
\(988\) 4.14338e6i 0.135040i
\(989\) 3.86074e6i 0.125511i
\(990\) −5.22625e7 −1.69474
\(991\) 4.54332e7 1.46957 0.734783 0.678303i \(-0.237284\pi\)
0.734783 + 0.678303i \(0.237284\pi\)
\(992\) 3.81701e7 1.23153
\(993\) 7.29472e7 2.34766
\(994\) 2.20403e7i 0.707539i
\(995\) 9.72222e6 0.311321
\(996\) 1.40443e7i 0.448591i
\(997\) 6.29767e6i 0.200651i 0.994955 + 0.100326i \(0.0319884\pi\)
−0.994955 + 0.100326i \(0.968012\pi\)
\(998\) 2.36527e7i 0.751715i
\(999\) 1.71085e6 0.0542375
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 29.6.b.a.28.9 yes 12
3.2 odd 2 261.6.c.b.28.4 12
4.3 odd 2 464.6.e.c.289.3 12
29.12 odd 4 841.6.a.d.1.4 12
29.17 odd 4 841.6.a.d.1.9 12
29.28 even 2 inner 29.6.b.a.28.4 12
87.86 odd 2 261.6.c.b.28.9 12
116.115 odd 2 464.6.e.c.289.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.b.a.28.4 12 29.28 even 2 inner
29.6.b.a.28.9 yes 12 1.1 even 1 trivial
261.6.c.b.28.4 12 3.2 odd 2
261.6.c.b.28.9 12 87.86 odd 2
464.6.e.c.289.3 12 4.3 odd 2
464.6.e.c.289.10 12 116.115 odd 2
841.6.a.d.1.4 12 29.12 odd 4
841.6.a.d.1.9 12 29.17 odd 4