Properties

Label 304.6.i.d.273.2
Level $304$
Weight $6$
Character 304.273
Analytic conductor $48.757$
Analytic rank $0$
Dimension $18$
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] = [304,6,Mod(49,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.7566812231\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} + \cdots + 66\!\cdots\!83 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 273.2
Root \(22.4078 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 304.273
Dual form 304.6.i.d.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-10.4539 - 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} -95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} +O(q^{10})\) \(q+(-10.4539 - 18.1067i) q^{3} +(50.5928 + 87.6293i) q^{5} -95.5451 q^{7} +(-97.0688 + 168.128i) q^{9} +119.504 q^{11} +(297.428 - 515.161i) q^{13} +(1057.79 - 1832.14i) q^{15} +(-459.911 - 796.589i) q^{17} +(806.703 - 1351.05i) q^{19} +(998.821 + 1730.01i) q^{21} +(-2124.58 + 3679.89i) q^{23} +(-3556.76 + 6160.48i) q^{25} -1021.61 q^{27} +(-2214.69 + 3835.95i) q^{29} +4955.30 q^{31} +(-1249.29 - 2163.83i) q^{33} +(-4833.89 - 8372.55i) q^{35} +7651.95 q^{37} -12437.2 q^{39} +(2006.97 + 3476.18i) q^{41} +(5228.80 + 9056.55i) q^{43} -19643.9 q^{45} +(-4318.48 + 7479.83i) q^{47} -7678.13 q^{49} +(-9615.75 + 16655.0i) q^{51} +(18534.0 - 32101.9i) q^{53} +(6046.06 + 10472.1i) q^{55} +(-32896.2 - 482.998i) q^{57} +(20727.5 + 35901.1i) q^{59} +(-2244.12 + 3886.93i) q^{61} +(9274.45 - 16063.8i) q^{63} +60190.9 q^{65} +(-18484.7 + 32016.5i) q^{67} +88840.9 q^{69} +(-23408.1 - 40543.9i) q^{71} +(40666.6 + 70436.6i) q^{73} +148728. q^{75} -11418.1 q^{77} +(-6656.35 - 11529.1i) q^{79} +(34267.5 + 59353.1i) q^{81} +18380.7 q^{83} +(46536.4 - 80603.3i) q^{85} +92608.6 q^{87} +(-7307.86 + 12657.6i) q^{89} +(-28417.8 + 49221.1i) q^{91} +(-51802.3 - 89724.2i) q^{93} +(159205. + 2337.52i) q^{95} +(61844.5 + 107118. i) q^{97} +(-11600.1 + 20092.0i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9} + 320 q^{11} + 227 q^{13} + 101 q^{15} + 179 q^{17} + 868 q^{19} - 5700 q^{21} + 3425 q^{23} - 7054 q^{25} - 14722 q^{27} - 7349 q^{29} + 9960 q^{31} - 2998 q^{33} - 15888 q^{35} + 26444 q^{37} + 30246 q^{39} - 7311 q^{41} + 8283 q^{43} - 62164 q^{45} - 37603 q^{47} + 124738 q^{49} - 47227 q^{51} - 20337 q^{53} - 716 q^{55} - 57555 q^{57} + 74455 q^{59} - 7569 q^{61} + 52544 q^{63} + 188998 q^{65} + 26177 q^{67} + 116282 q^{69} + 53463 q^{71} - 14103 q^{73} - 120912 q^{75} - 31960 q^{77} - 31825 q^{79} - 21137 q^{81} - 82600 q^{83} - 50787 q^{85} + 339766 q^{87} - 155197 q^{89} + 2800 q^{91} - 46460 q^{93} - 49315 q^{95} + 111241 q^{97} + 193544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −10.4539 18.1067i −0.670619 1.16155i −0.977729 0.209872i \(-0.932695\pi\)
0.307110 0.951674i \(-0.400638\pi\)
\(4\) 0 0
\(5\) 50.5928 + 87.6293i 0.905031 + 1.56756i 0.820876 + 0.571107i \(0.193486\pi\)
0.0841554 + 0.996453i \(0.473181\pi\)
\(6\) 0 0
\(7\) −95.5451 −0.736993 −0.368497 0.929629i \(-0.620127\pi\)
−0.368497 + 0.929629i \(0.620127\pi\)
\(8\) 0 0
\(9\) −97.0688 + 168.128i −0.399460 + 0.691885i
\(10\) 0 0
\(11\) 119.504 0.297784 0.148892 0.988853i \(-0.452429\pi\)
0.148892 + 0.988853i \(0.452429\pi\)
\(12\) 0 0
\(13\) 297.428 515.161i 0.488117 0.845443i −0.511790 0.859111i \(-0.671017\pi\)
0.999907 + 0.0136677i \(0.00435069\pi\)
\(14\) 0 0
\(15\) 1057.79 1832.14i 1.21386 2.10247i
\(16\) 0 0
\(17\) −459.911 796.589i −0.385968 0.668517i 0.605935 0.795514i \(-0.292799\pi\)
−0.991903 + 0.126998i \(0.959466\pi\)
\(18\) 0 0
\(19\) 806.703 1351.05i 0.512660 0.858592i
\(20\) 0 0
\(21\) 998.821 + 1730.01i 0.494242 + 0.856052i
\(22\) 0 0
\(23\) −2124.58 + 3679.89i −0.837441 + 1.45049i 0.0545863 + 0.998509i \(0.482616\pi\)
−0.892027 + 0.451981i \(0.850717\pi\)
\(24\) 0 0
\(25\) −3556.76 + 6160.48i −1.13816 + 1.97135i
\(26\) 0 0
\(27\) −1021.61 −0.269696
\(28\) 0 0
\(29\) −2214.69 + 3835.95i −0.489009 + 0.846989i −0.999920 0.0126447i \(-0.995975\pi\)
0.510911 + 0.859634i \(0.329308\pi\)
\(30\) 0 0
\(31\) 4955.30 0.926116 0.463058 0.886328i \(-0.346752\pi\)
0.463058 + 0.886328i \(0.346752\pi\)
\(32\) 0 0
\(33\) −1249.29 2163.83i −0.199700 0.345890i
\(34\) 0 0
\(35\) −4833.89 8372.55i −0.667002 1.15528i
\(36\) 0 0
\(37\) 7651.95 0.918900 0.459450 0.888204i \(-0.348047\pi\)
0.459450 + 0.888204i \(0.348047\pi\)
\(38\) 0 0
\(39\) −12437.2 −1.30936
\(40\) 0 0
\(41\) 2006.97 + 3476.18i 0.186458 + 0.322955i 0.944067 0.329754i \(-0.106966\pi\)
−0.757609 + 0.652709i \(0.773632\pi\)
\(42\) 0 0
\(43\) 5228.80 + 9056.55i 0.431252 + 0.746950i 0.996981 0.0776405i \(-0.0247386\pi\)
−0.565729 + 0.824591i \(0.691405\pi\)
\(44\) 0 0
\(45\) −19643.9 −1.44610
\(46\) 0 0
\(47\) −4318.48 + 7479.83i −0.285159 + 0.493909i −0.972648 0.232286i \(-0.925380\pi\)
0.687489 + 0.726195i \(0.258713\pi\)
\(48\) 0 0
\(49\) −7678.13 −0.456841
\(50\) 0 0
\(51\) −9615.75 + 16655.0i −0.517675 + 0.896640i
\(52\) 0 0
\(53\) 18534.0 32101.9i 0.906318 1.56979i 0.0871791 0.996193i \(-0.472215\pi\)
0.819139 0.573596i \(-0.194452\pi\)
\(54\) 0 0
\(55\) 6046.06 + 10472.1i 0.269504 + 0.466795i
\(56\) 0 0
\(57\) −32896.2 482.998i −1.34109 0.0196906i
\(58\) 0 0
\(59\) 20727.5 + 35901.1i 0.775207 + 1.34270i 0.934678 + 0.355495i \(0.115688\pi\)
−0.159472 + 0.987203i \(0.550979\pi\)
\(60\) 0 0
\(61\) −2244.12 + 3886.93i −0.0772186 + 0.133747i −0.902049 0.431634i \(-0.857937\pi\)
0.824830 + 0.565380i \(0.191271\pi\)
\(62\) 0 0
\(63\) 9274.45 16063.8i 0.294399 0.509915i
\(64\) 0 0
\(65\) 60190.9 1.76704
\(66\) 0 0
\(67\) −18484.7 + 32016.5i −0.503067 + 0.871338i 0.496927 + 0.867793i \(0.334462\pi\)
−0.999994 + 0.00354526i \(0.998872\pi\)
\(68\) 0 0
\(69\) 88840.9 2.24642
\(70\) 0 0
\(71\) −23408.1 40543.9i −0.551086 0.954509i −0.998197 0.0600303i \(-0.980880\pi\)
0.447110 0.894479i \(-0.352453\pi\)
\(72\) 0 0
\(73\) 40666.6 + 70436.6i 0.893163 + 1.54700i 0.836062 + 0.548635i \(0.184852\pi\)
0.0571012 + 0.998368i \(0.481814\pi\)
\(74\) 0 0
\(75\) 148728. 3.05309
\(76\) 0 0
\(77\) −11418.1 −0.219465
\(78\) 0 0
\(79\) −6656.35 11529.1i −0.119996 0.207840i 0.799770 0.600307i \(-0.204955\pi\)
−0.919766 + 0.392467i \(0.871622\pi\)
\(80\) 0 0
\(81\) 34267.5 + 59353.1i 0.580323 + 1.00515i
\(82\) 0 0
\(83\) 18380.7 0.292864 0.146432 0.989221i \(-0.453221\pi\)
0.146432 + 0.989221i \(0.453221\pi\)
\(84\) 0 0
\(85\) 46536.4 80603.3i 0.698627 1.21006i
\(86\) 0 0
\(87\) 92608.6 1.31176
\(88\) 0 0
\(89\) −7307.86 + 12657.6i −0.0977946 + 0.169385i −0.910772 0.412911i \(-0.864512\pi\)
0.812977 + 0.582296i \(0.197846\pi\)
\(90\) 0 0
\(91\) −28417.8 + 49221.1i −0.359739 + 0.623086i
\(92\) 0 0
\(93\) −51802.3 89724.2i −0.621071 1.07573i
\(94\) 0 0
\(95\) 159205. + 2337.52i 1.80987 + 0.0265733i
\(96\) 0 0
\(97\) 61844.5 + 107118.i 0.667378 + 1.15593i 0.978635 + 0.205606i \(0.0659166\pi\)
−0.311257 + 0.950326i \(0.600750\pi\)
\(98\) 0 0
\(99\) −11600.1 + 20092.0i −0.118953 + 0.206033i
\(100\) 0 0
\(101\) −24891.8 + 43113.9i −0.242803 + 0.420546i −0.961511 0.274765i \(-0.911400\pi\)
0.718709 + 0.695311i \(0.244733\pi\)
\(102\) 0 0
\(103\) 102447. 0.951496 0.475748 0.879582i \(-0.342177\pi\)
0.475748 + 0.879582i \(0.342177\pi\)
\(104\) 0 0
\(105\) −101066. + 175052.i −0.894608 + 1.54951i
\(106\) 0 0
\(107\) −45310.9 −0.382598 −0.191299 0.981532i \(-0.561270\pi\)
−0.191299 + 0.981532i \(0.561270\pi\)
\(108\) 0 0
\(109\) 75259.9 + 130354.i 0.606733 + 1.05089i 0.991775 + 0.127993i \(0.0408535\pi\)
−0.385042 + 0.922899i \(0.625813\pi\)
\(110\) 0 0
\(111\) −79992.9 138552.i −0.616232 1.06734i
\(112\) 0 0
\(113\) 71068.5 0.523577 0.261789 0.965125i \(-0.415688\pi\)
0.261789 + 0.965125i \(0.415688\pi\)
\(114\) 0 0
\(115\) −429954. −3.03164
\(116\) 0 0
\(117\) 57742.0 + 100012.i 0.389966 + 0.675442i
\(118\) 0 0
\(119\) 43942.3 + 76110.2i 0.284456 + 0.492692i
\(120\) 0 0
\(121\) −146770. −0.911324
\(122\) 0 0
\(123\) 41961.5 72679.4i 0.250085 0.433160i
\(124\) 0 0
\(125\) −403580. −2.31023
\(126\) 0 0
\(127\) −65658.5 + 113724.i −0.361228 + 0.625666i −0.988163 0.153406i \(-0.950976\pi\)
0.626935 + 0.779071i \(0.284309\pi\)
\(128\) 0 0
\(129\) 109323. 189353.i 0.578412 1.00184i
\(130\) 0 0
\(131\) 100412. + 173920.i 0.511222 + 0.885462i 0.999915 + 0.0130067i \(0.00414029\pi\)
−0.488694 + 0.872456i \(0.662526\pi\)
\(132\) 0 0
\(133\) −77076.5 + 129086.i −0.377827 + 0.632776i
\(134\) 0 0
\(135\) −51685.9 89522.6i −0.244083 0.422764i
\(136\) 0 0
\(137\) 181520. 314402.i 0.826272 1.43114i −0.0746720 0.997208i \(-0.523791\pi\)
0.900944 0.433936i \(-0.142876\pi\)
\(138\) 0 0
\(139\) −190140. + 329331.i −0.834710 + 1.44576i 0.0595567 + 0.998225i \(0.481031\pi\)
−0.894266 + 0.447535i \(0.852302\pi\)
\(140\) 0 0
\(141\) 180580. 0.764931
\(142\) 0 0
\(143\) 35544.0 61563.9i 0.145354 0.251760i
\(144\) 0 0
\(145\) −448188. −1.77027
\(146\) 0 0
\(147\) 80266.5 + 139026.i 0.306366 + 0.530642i
\(148\) 0 0
\(149\) −21911.4 37951.7i −0.0808546 0.140044i 0.822763 0.568385i \(-0.192432\pi\)
−0.903617 + 0.428341i \(0.859098\pi\)
\(150\) 0 0
\(151\) 131110. 0.467942 0.233971 0.972244i \(-0.424828\pi\)
0.233971 + 0.972244i \(0.424828\pi\)
\(152\) 0 0
\(153\) 178572. 0.616716
\(154\) 0 0
\(155\) 250702. + 434229.i 0.838164 + 1.45174i
\(156\) 0 0
\(157\) 239102. + 414137.i 0.774166 + 1.34089i 0.935262 + 0.353956i \(0.115164\pi\)
−0.161096 + 0.986939i \(0.551503\pi\)
\(158\) 0 0
\(159\) −775013. −2.43118
\(160\) 0 0
\(161\) 202994. 351595.i 0.617188 1.06900i
\(162\) 0 0
\(163\) 389081. 1.14702 0.573509 0.819199i \(-0.305582\pi\)
0.573509 + 0.819199i \(0.305582\pi\)
\(164\) 0 0
\(165\) 126410. 218948.i 0.361469 0.626083i
\(166\) 0 0
\(167\) 261490. 452914.i 0.725544 1.25668i −0.233206 0.972427i \(-0.574922\pi\)
0.958750 0.284252i \(-0.0917451\pi\)
\(168\) 0 0
\(169\) 8719.47 + 15102.6i 0.0234841 + 0.0406756i
\(170\) 0 0
\(171\) 148843. + 266774.i 0.389260 + 0.697675i
\(172\) 0 0
\(173\) −228433. 395658.i −0.580289 1.00509i −0.995445 0.0953400i \(-0.969606\pi\)
0.415156 0.909750i \(-0.363727\pi\)
\(174\) 0 0
\(175\) 339831. 588604.i 0.838818 1.45288i
\(176\) 0 0
\(177\) 433368. 750615.i 1.03974 1.80088i
\(178\) 0 0
\(179\) −380329. −0.887211 −0.443606 0.896222i \(-0.646301\pi\)
−0.443606 + 0.896222i \(0.646301\pi\)
\(180\) 0 0
\(181\) −172037. + 297977.i −0.390325 + 0.676062i −0.992492 0.122308i \(-0.960971\pi\)
0.602168 + 0.798370i \(0.294304\pi\)
\(182\) 0 0
\(183\) 93839.5 0.207137
\(184\) 0 0
\(185\) 387134. + 670535.i 0.831633 + 1.44043i
\(186\) 0 0
\(187\) −54961.4 95195.9i −0.114935 0.199074i
\(188\) 0 0
\(189\) 97609.5 0.198764
\(190\) 0 0
\(191\) 699156. 1.38673 0.693363 0.720588i \(-0.256128\pi\)
0.693363 + 0.720588i \(0.256128\pi\)
\(192\) 0 0
\(193\) 27621.7 + 47842.2i 0.0533774 + 0.0924524i 0.891480 0.453061i \(-0.149668\pi\)
−0.838102 + 0.545513i \(0.816335\pi\)
\(194\) 0 0
\(195\) −629230. 1.08986e6i −1.18501 2.05250i
\(196\) 0 0
\(197\) −1.00021e6 −1.83623 −0.918113 0.396318i \(-0.870288\pi\)
−0.918113 + 0.396318i \(0.870288\pi\)
\(198\) 0 0
\(199\) −246676. + 427255.i −0.441564 + 0.764811i −0.997806 0.0662094i \(-0.978909\pi\)
0.556242 + 0.831020i \(0.312243\pi\)
\(200\) 0 0
\(201\) 772951. 1.34947
\(202\) 0 0
\(203\) 211602. 366506.i 0.360397 0.624225i
\(204\) 0 0
\(205\) −203077. + 351739.i −0.337501 + 0.584569i
\(206\) 0 0
\(207\) −412462. 714405.i −0.669049 1.15883i
\(208\) 0 0
\(209\) 96404.5 161456.i 0.152662 0.255675i
\(210\) 0 0
\(211\) 309293. + 535711.i 0.478259 + 0.828369i 0.999689 0.0249246i \(-0.00793458\pi\)
−0.521430 + 0.853294i \(0.674601\pi\)
\(212\) 0 0
\(213\) −489412. + 847686.i −0.739138 + 1.28022i
\(214\) 0 0
\(215\) −529079. + 916392.i −0.780593 + 1.35203i
\(216\) 0 0
\(217\) −473455. −0.682541
\(218\) 0 0
\(219\) 850251. 1.47268e6i 1.19794 2.07490i
\(220\) 0 0
\(221\) −547162. −0.753590
\(222\) 0 0
\(223\) 75025.2 + 129947.i 0.101029 + 0.174987i 0.912109 0.409948i \(-0.134453\pi\)
−0.811080 + 0.584935i \(0.801120\pi\)
\(224\) 0 0
\(225\) −690500. 1.19598e6i −0.909301 1.57496i
\(226\) 0 0
\(227\) −1.05214e6 −1.35522 −0.677610 0.735421i \(-0.736984\pi\)
−0.677610 + 0.735421i \(0.736984\pi\)
\(228\) 0 0
\(229\) 766814. 0.966276 0.483138 0.875544i \(-0.339497\pi\)
0.483138 + 0.875544i \(0.339497\pi\)
\(230\) 0 0
\(231\) 119363. + 206744.i 0.147177 + 0.254919i
\(232\) 0 0
\(233\) −722964. 1.25221e6i −0.872423 1.51108i −0.859483 0.511165i \(-0.829214\pi\)
−0.0129403 0.999916i \(-0.504119\pi\)
\(234\) 0 0
\(235\) −873936. −1.03231
\(236\) 0 0
\(237\) −139170. + 241049.i −0.160944 + 0.278763i
\(238\) 0 0
\(239\) 749353. 0.848578 0.424289 0.905527i \(-0.360524\pi\)
0.424289 + 0.905527i \(0.360524\pi\)
\(240\) 0 0
\(241\) −459395. + 795696.i −0.509499 + 0.882479i 0.490440 + 0.871475i \(0.336836\pi\)
−0.999939 + 0.0110040i \(0.996497\pi\)
\(242\) 0 0
\(243\) 592334. 1.02595e6i 0.643504 1.11458i
\(244\) 0 0
\(245\) −388458. 672829.i −0.413455 0.716126i
\(246\) 0 0
\(247\) −456071. 817421.i −0.475652 0.852518i
\(248\) 0 0
\(249\) −192150. 332814.i −0.196400 0.340176i
\(250\) 0 0
\(251\) −158549. + 274616.i −0.158848 + 0.275132i −0.934453 0.356086i \(-0.884111\pi\)
0.775606 + 0.631218i \(0.217444\pi\)
\(252\) 0 0
\(253\) −253897. + 439762.i −0.249377 + 0.431933i
\(254\) 0 0
\(255\) −1.94595e6 −1.87405
\(256\) 0 0
\(257\) −293395. + 508174.i −0.277089 + 0.479932i −0.970660 0.240456i \(-0.922703\pi\)
0.693571 + 0.720388i \(0.256036\pi\)
\(258\) 0 0
\(259\) −731107. −0.677223
\(260\) 0 0
\(261\) −429954. 744702.i −0.390680 0.676677i
\(262\) 0 0
\(263\) 602410. + 1.04341e6i 0.537036 + 0.930173i 0.999062 + 0.0433068i \(0.0137893\pi\)
−0.462026 + 0.886866i \(0.652877\pi\)
\(264\) 0 0
\(265\) 3.75075e6 3.28098
\(266\) 0 0
\(267\) 305583. 0.262332
\(268\) 0 0
\(269\) 154072. + 266860.i 0.129820 + 0.224855i 0.923607 0.383341i \(-0.125227\pi\)
−0.793787 + 0.608196i \(0.791893\pi\)
\(270\) 0 0
\(271\) −476222. 824840.i −0.393900 0.682255i 0.599060 0.800704i \(-0.295541\pi\)
−0.992960 + 0.118449i \(0.962208\pi\)
\(272\) 0 0
\(273\) 1.18831e6 0.964991
\(274\) 0 0
\(275\) −425048. + 736205.i −0.338927 + 0.587039i
\(276\) 0 0
\(277\) −575120. −0.450359 −0.225179 0.974317i \(-0.572297\pi\)
−0.225179 + 0.974317i \(0.572297\pi\)
\(278\) 0 0
\(279\) −481005. + 833125.i −0.369947 + 0.640766i
\(280\) 0 0
\(281\) −917605. + 1.58934e6i −0.693250 + 1.20074i 0.277517 + 0.960721i \(0.410488\pi\)
−0.970767 + 0.240023i \(0.922845\pi\)
\(282\) 0 0
\(283\) 615501. + 1.06608e6i 0.456838 + 0.791267i 0.998792 0.0491413i \(-0.0156485\pi\)
−0.541954 + 0.840408i \(0.682315\pi\)
\(284\) 0 0
\(285\) −1.62199e6 2.90711e6i −1.18287 2.12007i
\(286\) 0 0
\(287\) −191757. 332132.i −0.137419 0.238016i
\(288\) 0 0
\(289\) 286892. 496912.i 0.202057 0.349973i
\(290\) 0 0
\(291\) 1.29303e6 2.23960e6i 0.895113 1.55038i
\(292\) 0 0
\(293\) −923293. −0.628305 −0.314153 0.949373i \(-0.601720\pi\)
−0.314153 + 0.949373i \(0.601720\pi\)
\(294\) 0 0
\(295\) −2.09733e6 + 3.63268e6i −1.40317 + 2.43037i
\(296\) 0 0
\(297\) −122086. −0.0803112
\(298\) 0 0
\(299\) 1.26382e6 + 2.18900e6i 0.817538 + 1.41602i
\(300\) 0 0
\(301\) −499587. 865310.i −0.317830 0.550497i
\(302\) 0 0
\(303\) 1.04087e6 0.651312
\(304\) 0 0
\(305\) −454145. −0.279541
\(306\) 0 0
\(307\) −843091. 1.46028e6i −0.510539 0.884279i −0.999925 0.0122119i \(-0.996113\pi\)
0.489387 0.872067i \(-0.337221\pi\)
\(308\) 0 0
\(309\) −1.07097e6 1.85498e6i −0.638091 1.10521i
\(310\) 0 0
\(311\) 1.06080e6 0.621915 0.310957 0.950424i \(-0.399350\pi\)
0.310957 + 0.950424i \(0.399350\pi\)
\(312\) 0 0
\(313\) 8090.23 14012.7i 0.00466767 0.00808464i −0.863682 0.504037i \(-0.831848\pi\)
0.868350 + 0.495952i \(0.165181\pi\)
\(314\) 0 0
\(315\) 1.87688e6 1.06576
\(316\) 0 0
\(317\) 1.27571e6 2.20960e6i 0.713024 1.23499i −0.250693 0.968067i \(-0.580658\pi\)
0.963717 0.266927i \(-0.0860082\pi\)
\(318\) 0 0
\(319\) −264665. + 458413.i −0.145619 + 0.252220i
\(320\) 0 0
\(321\) 473676. + 820431.i 0.256578 + 0.444406i
\(322\) 0 0
\(323\) −1.44724e6 21249.1i −0.771853 0.0113327i
\(324\) 0 0
\(325\) 2.11576e6 + 3.66460e6i 1.11111 + 1.92450i
\(326\) 0 0
\(327\) 1.57352e6 2.72542e6i 0.813773 1.40950i
\(328\) 0 0
\(329\) 412610. 714661.i 0.210160 0.364008i
\(330\) 0 0
\(331\) −1.14830e6 −0.576082 −0.288041 0.957618i \(-0.593004\pi\)
−0.288041 + 0.957618i \(0.593004\pi\)
\(332\) 0 0
\(333\) −742766. + 1.28651e6i −0.367064 + 0.635773i
\(334\) 0 0
\(335\) −3.74077e6 −1.82117
\(336\) 0 0
\(337\) −742181. 1.28549e6i −0.355988 0.616589i 0.631299 0.775540i \(-0.282522\pi\)
−0.987286 + 0.158951i \(0.949189\pi\)
\(338\) 0 0
\(339\) −742944. 1.28682e6i −0.351121 0.608160i
\(340\) 0 0
\(341\) 592180. 0.275783
\(342\) 0 0
\(343\) 2.33943e6 1.07368
\(344\) 0 0
\(345\) 4.49471e6 + 7.78506e6i 2.03308 + 3.52139i
\(346\) 0 0
\(347\) −1.09955e6 1.90448e6i −0.490222 0.849089i 0.509715 0.860343i \(-0.329751\pi\)
−0.999937 + 0.0112544i \(0.996418\pi\)
\(348\) 0 0
\(349\) −3.81500e6 −1.67660 −0.838302 0.545206i \(-0.816451\pi\)
−0.838302 + 0.545206i \(0.816451\pi\)
\(350\) 0 0
\(351\) −303855. + 526292.i −0.131643 + 0.228012i
\(352\) 0 0
\(353\) 550982. 0.235342 0.117671 0.993053i \(-0.462457\pi\)
0.117671 + 0.993053i \(0.462457\pi\)
\(354\) 0 0
\(355\) 2.36856e6 4.10246e6i 0.997500 1.72772i
\(356\) 0 0
\(357\) 918738. 1.59130e6i 0.381523 0.660818i
\(358\) 0 0
\(359\) 110651. + 191653.i 0.0453126 + 0.0784838i 0.887792 0.460245i \(-0.152238\pi\)
−0.842480 + 0.538728i \(0.818905\pi\)
\(360\) 0 0
\(361\) −1.17456e6 2.17979e6i −0.474359 0.880331i
\(362\) 0 0
\(363\) 1.53432e6 + 2.65752e6i 0.611152 + 1.05855i
\(364\) 0 0
\(365\) −4.11487e6 + 7.12717e6i −1.61668 + 2.80017i
\(366\) 0 0
\(367\) 680360. 1.17842e6i 0.263678 0.456704i −0.703538 0.710657i \(-0.748398\pi\)
0.967216 + 0.253954i \(0.0817311\pi\)
\(368\) 0 0
\(369\) −779258. −0.297931
\(370\) 0 0
\(371\) −1.77084e6 + 3.06718e6i −0.667950 + 1.15692i
\(372\) 0 0
\(373\) 4.37071e6 1.62660 0.813298 0.581848i \(-0.197670\pi\)
0.813298 + 0.581848i \(0.197670\pi\)
\(374\) 0 0
\(375\) 4.21899e6 + 7.30751e6i 1.54928 + 2.68344i
\(376\) 0 0
\(377\) 1.31742e6 + 2.28184e6i 0.477387 + 0.826859i
\(378\) 0 0
\(379\) 3.26062e6 1.16601 0.583005 0.812469i \(-0.301877\pi\)
0.583005 + 0.812469i \(0.301877\pi\)
\(380\) 0 0
\(381\) 2.74555e6 0.968986
\(382\) 0 0
\(383\) −1.11542e6 1.93197e6i −0.388546 0.672981i 0.603708 0.797205i \(-0.293689\pi\)
−0.992254 + 0.124224i \(0.960356\pi\)
\(384\) 0 0
\(385\) −577671. 1.00056e6i −0.198623 0.344025i
\(386\) 0 0
\(387\) −2.03021e6 −0.689072
\(388\) 0 0
\(389\) 520809. 902068.i 0.174504 0.302249i −0.765486 0.643453i \(-0.777501\pi\)
0.939989 + 0.341204i \(0.110835\pi\)
\(390\) 0 0
\(391\) 3.90848e6 1.29290
\(392\) 0 0
\(393\) 2.09941e6 3.63628e6i 0.685670 1.18762i
\(394\) 0 0
\(395\) 673527. 1.16658e6i 0.217201 0.376203i
\(396\) 0 0
\(397\) −2.19687e6 3.80509e6i −0.699565 1.21168i −0.968618 0.248556i \(-0.920044\pi\)
0.269053 0.963125i \(-0.413289\pi\)
\(398\) 0 0
\(399\) 3.14308e6 + 46148.1i 0.988377 + 0.0145118i
\(400\) 0 0
\(401\) 393217. + 681071.i 0.122116 + 0.211510i 0.920602 0.390503i \(-0.127699\pi\)
−0.798486 + 0.602013i \(0.794365\pi\)
\(402\) 0 0
\(403\) 1.47385e6 2.55277e6i 0.452053 0.782979i
\(404\) 0 0
\(405\) −3.46738e6 + 6.00567e6i −1.05042 + 1.81938i
\(406\) 0 0
\(407\) 914442. 0.273634
\(408\) 0 0
\(409\) −584641. + 1.01263e6i −0.172815 + 0.299324i −0.939403 0.342815i \(-0.888620\pi\)
0.766588 + 0.642139i \(0.221953\pi\)
\(410\) 0 0
\(411\) −7.59038e6 −2.21645
\(412\) 0 0
\(413\) −1.98041e6 3.43018e6i −0.571322 0.989559i
\(414\) 0 0
\(415\) 929930. + 1.61069e6i 0.265051 + 0.459082i
\(416\) 0 0
\(417\) 7.95081e6 2.23909
\(418\) 0 0
\(419\) 4.14739e6 1.15409 0.577045 0.816712i \(-0.304206\pi\)
0.577045 + 0.816712i \(0.304206\pi\)
\(420\) 0 0
\(421\) 1.59962e6 + 2.77063e6i 0.439858 + 0.761857i 0.997678 0.0681052i \(-0.0216954\pi\)
−0.557820 + 0.829962i \(0.688362\pi\)
\(422\) 0 0
\(423\) −838380. 1.45212e6i −0.227819 0.394594i
\(424\) 0 0
\(425\) 6.54317e6 1.75718
\(426\) 0 0
\(427\) 214415. 371378.i 0.0569096 0.0985703i
\(428\) 0 0
\(429\) −1.48629e6 −0.389908
\(430\) 0 0
\(431\) −2.91306e6 + 5.04557e6i −0.755364 + 1.30833i 0.189830 + 0.981817i \(0.439206\pi\)
−0.945193 + 0.326511i \(0.894127\pi\)
\(432\) 0 0
\(433\) 1.29561e6 2.24407e6i 0.332090 0.575196i −0.650832 0.759222i \(-0.725580\pi\)
0.982921 + 0.184026i \(0.0589130\pi\)
\(434\) 0 0
\(435\) 4.68533e6 + 8.11522e6i 1.18718 + 2.05626i
\(436\) 0 0
\(437\) 3.25779e6 + 5.83899e6i 0.816056 + 1.46263i
\(438\) 0 0
\(439\) −986761. 1.70912e6i −0.244372 0.423264i 0.717583 0.696473i \(-0.245248\pi\)
−0.961955 + 0.273209i \(0.911915\pi\)
\(440\) 0 0
\(441\) 745307. 1.29091e6i 0.182490 0.316082i
\(442\) 0 0
\(443\) −73870.7 + 127948.i −0.0178839 + 0.0309759i −0.874829 0.484432i \(-0.839026\pi\)
0.856945 + 0.515408i \(0.172360\pi\)
\(444\) 0 0
\(445\) −1.47890e6 −0.354029
\(446\) 0 0
\(447\) −458120. + 793487.i −0.108445 + 0.187833i
\(448\) 0 0
\(449\) −6.33075e6 −1.48197 −0.740985 0.671522i \(-0.765641\pi\)
−0.740985 + 0.671522i \(0.765641\pi\)
\(450\) 0 0
\(451\) 239842. + 415419.i 0.0555244 + 0.0961711i
\(452\) 0 0
\(453\) −1.37061e6 2.37396e6i −0.313811 0.543536i
\(454\) 0 0
\(455\) −5.75094e6 −1.30230
\(456\) 0 0
\(457\) −175313. −0.0392667 −0.0196334 0.999807i \(-0.506250\pi\)
−0.0196334 + 0.999807i \(0.506250\pi\)
\(458\) 0 0
\(459\) 469848. + 813801.i 0.104094 + 0.180296i
\(460\) 0 0
\(461\) −3.20957e6 5.55914e6i −0.703388 1.21830i −0.967270 0.253749i \(-0.918336\pi\)
0.263882 0.964555i \(-0.414997\pi\)
\(462\) 0 0
\(463\) −2.17798e6 −0.472174 −0.236087 0.971732i \(-0.575865\pi\)
−0.236087 + 0.971732i \(0.575865\pi\)
\(464\) 0 0
\(465\) 5.24164e6 9.07879e6i 1.12418 1.94713i
\(466\) 0 0
\(467\) −742392. −0.157522 −0.0787610 0.996894i \(-0.525096\pi\)
−0.0787610 + 0.996894i \(0.525096\pi\)
\(468\) 0 0
\(469\) 1.76613e6 3.05902e6i 0.370757 0.642170i
\(470\) 0 0
\(471\) 4.99910e6 8.65870e6i 1.03834 1.79846i
\(472\) 0 0
\(473\) 624865. + 1.08230e6i 0.128420 + 0.222430i
\(474\) 0 0
\(475\) 5.45386e6 + 9.77503e6i 1.10910 + 1.98785i
\(476\) 0 0
\(477\) 3.59815e6 + 6.23219e6i 0.724076 + 1.25414i
\(478\) 0 0
\(479\) 2.42943e6 4.20789e6i 0.483799 0.837965i −0.516028 0.856572i \(-0.672590\pi\)
0.999827 + 0.0186069i \(0.00592311\pi\)
\(480\) 0 0
\(481\) 2.27591e6 3.94199e6i 0.448530 0.776877i
\(482\) 0 0
\(483\) −8.48832e6 −1.65559
\(484\) 0 0
\(485\) −6.25777e6 + 1.08388e7i −1.20800 + 2.09231i
\(486\) 0 0
\(487\) −3.50290e6 −0.669275 −0.334638 0.942347i \(-0.608614\pi\)
−0.334638 + 0.942347i \(0.608614\pi\)
\(488\) 0 0
\(489\) −4.06742e6 7.04497e6i −0.769213 1.33232i
\(490\) 0 0
\(491\) −368324. 637956.i −0.0689488 0.119423i 0.829490 0.558521i \(-0.188631\pi\)
−0.898439 + 0.439099i \(0.855298\pi\)
\(492\) 0 0
\(493\) 4.07424e6 0.754968
\(494\) 0 0
\(495\) −2.34753e6 −0.430625
\(496\) 0 0
\(497\) 2.23653e6 + 3.87378e6i 0.406147 + 0.703467i
\(498\) 0 0
\(499\) −9216.83 15964.0i −0.00165703 0.00287006i 0.865196 0.501434i \(-0.167194\pi\)
−0.866853 + 0.498564i \(0.833861\pi\)
\(500\) 0 0
\(501\) −1.09344e7 −1.94625
\(502\) 0 0
\(503\) −249975. + 432970.i −0.0440532 + 0.0763024i −0.887211 0.461363i \(-0.847360\pi\)
0.843158 + 0.537666i \(0.180694\pi\)
\(504\) 0 0
\(505\) −5.03738e6 −0.878975
\(506\) 0 0
\(507\) 182305. 315762.i 0.0314977 0.0545557i
\(508\) 0 0
\(509\) −2.32529e6 + 4.02752e6i −0.397816 + 0.689038i −0.993456 0.114213i \(-0.963565\pi\)
0.595640 + 0.803252i \(0.296899\pi\)
\(510\) 0 0
\(511\) −3.88550e6 6.72988e6i −0.658255 1.14013i
\(512\) 0 0
\(513\) −824133. + 1.38024e6i −0.138262 + 0.231559i
\(514\) 0 0
\(515\) 5.18309e6 + 8.97737e6i 0.861133 + 1.49153i
\(516\) 0 0
\(517\) −516077. + 893872.i −0.0849158 + 0.147078i
\(518\) 0 0
\(519\) −4.77605e6 + 8.27236e6i −0.778306 + 1.34807i
\(520\) 0 0
\(521\) 7.64507e6 1.23392 0.616960 0.786994i \(-0.288364\pi\)
0.616960 + 0.786994i \(0.288364\pi\)
\(522\) 0 0
\(523\) −794941. + 1.37688e6i −0.127081 + 0.220111i −0.922544 0.385891i \(-0.873894\pi\)
0.795463 + 0.606002i \(0.207227\pi\)
\(524\) 0 0
\(525\) −1.42103e7 −2.25011
\(526\) 0 0
\(527\) −2.27900e6 3.94734e6i −0.357451 0.619124i
\(528\) 0 0
\(529\) −5.80954e6 1.00624e7i −0.902615 1.56338i
\(530\) 0 0
\(531\) −8.04799e6 −1.23866
\(532\) 0 0
\(533\) 2.38772e6 0.364054
\(534\) 0 0
\(535\) −2.29240e6 3.97056e6i −0.346263 0.599746i
\(536\) 0 0
\(537\) 3.97593e6 + 6.88651e6i 0.594981 + 1.03054i
\(538\) 0 0
\(539\) −917570. −0.136040
\(540\) 0 0
\(541\) −2.63644e6 + 4.56645e6i −0.387280 + 0.670789i −0.992083 0.125587i \(-0.959919\pi\)
0.604803 + 0.796375i \(0.293252\pi\)
\(542\) 0 0
\(543\) 7.19385e6 1.04704
\(544\) 0 0
\(545\) −7.61521e6 + 1.31899e7i −1.09822 + 1.90218i
\(546\) 0 0
\(547\) 1.94792e6 3.37389e6i 0.278357 0.482128i −0.692620 0.721303i \(-0.743544\pi\)
0.970977 + 0.239175i \(0.0768769\pi\)
\(548\) 0 0
\(549\) −435669. 754600.i −0.0616915 0.106853i
\(550\) 0 0
\(551\) 3.39596e6 + 6.08662e6i 0.476522 + 0.854077i
\(552\) 0 0
\(553\) 635982. + 1.10155e6i 0.0884366 + 0.153177i
\(554\) 0 0
\(555\) 8.09413e6 1.40194e7i 1.11542 1.93196i
\(556\) 0 0
\(557\) 3.90618e6 6.76570e6i 0.533475 0.924006i −0.465761 0.884911i \(-0.654219\pi\)
0.999235 0.0390950i \(-0.0124475\pi\)
\(558\) 0 0
\(559\) 6.22077e6 0.842005
\(560\) 0 0
\(561\) −1.14912e6 + 1.99034e6i −0.154156 + 0.267005i
\(562\) 0 0
\(563\) −7.01256e6 −0.932407 −0.466204 0.884677i \(-0.654379\pi\)
−0.466204 + 0.884677i \(0.654379\pi\)
\(564\) 0 0
\(565\) 3.59555e6 + 6.22768e6i 0.473854 + 0.820739i
\(566\) 0 0
\(567\) −3.27409e6 5.67090e6i −0.427694 0.740788i
\(568\) 0 0
\(569\) 6.70622e6 0.868355 0.434177 0.900827i \(-0.357039\pi\)
0.434177 + 0.900827i \(0.357039\pi\)
\(570\) 0 0
\(571\) −5.59306e6 −0.717893 −0.358946 0.933358i \(-0.616864\pi\)
−0.358946 + 0.933358i \(0.616864\pi\)
\(572\) 0 0
\(573\) −7.30892e6 1.26594e7i −0.929965 1.61075i
\(574\) 0 0
\(575\) −1.51133e7 2.61769e7i −1.90629 3.30179i
\(576\) 0 0
\(577\) 1.19869e7 1.49888 0.749442 0.662070i \(-0.230322\pi\)
0.749442 + 0.662070i \(0.230322\pi\)
\(578\) 0 0
\(579\) 577510. 1.00028e6i 0.0715918 0.124001i
\(580\) 0 0
\(581\) −1.75619e6 −0.215839
\(582\) 0 0
\(583\) 2.21490e6 3.83632e6i 0.269887 0.467458i
\(584\) 0 0
\(585\) −5.84266e6 + 1.01198e7i −0.705863 + 1.22259i
\(586\) 0 0
\(587\) −7.13321e6 1.23551e7i −0.854456 1.47996i −0.877149 0.480219i \(-0.840557\pi\)
0.0226925 0.999742i \(-0.492776\pi\)
\(588\) 0 0
\(589\) 3.99745e6 6.69484e6i 0.474783 0.795156i
\(590\) 0 0
\(591\) 1.04561e7 + 1.81105e7i 1.23141 + 2.13286i
\(592\) 0 0
\(593\) 3.81201e6 6.60260e6i 0.445161 0.771042i −0.552902 0.833246i \(-0.686480\pi\)
0.998063 + 0.0622043i \(0.0198130\pi\)
\(594\) 0 0
\(595\) −4.44632e6 + 7.70126e6i −0.514883 + 0.891803i
\(596\) 0 0
\(597\) 1.03149e7 1.18448
\(598\) 0 0
\(599\) 5.20469e6 9.01479e6i 0.592691 1.02657i −0.401178 0.916000i \(-0.631399\pi\)
0.993868 0.110570i \(-0.0352676\pi\)
\(600\) 0 0
\(601\) 1.96843e6 0.222297 0.111149 0.993804i \(-0.464547\pi\)
0.111149 + 0.993804i \(0.464547\pi\)
\(602\) 0 0
\(603\) −3.58858e6 6.21560e6i −0.401911 0.696129i
\(604\) 0 0
\(605\) −7.42549e6 1.28613e7i −0.824777 1.42856i
\(606\) 0 0
\(607\) 1.79114e7 1.97314 0.986569 0.163347i \(-0.0522290\pi\)
0.986569 + 0.163347i \(0.0522290\pi\)
\(608\) 0 0
\(609\) −8.84830e6 −0.966755
\(610\) 0 0
\(611\) 2.56888e6 + 4.44942e6i 0.278381 + 0.482171i
\(612\) 0 0
\(613\) −206107. 356988.i −0.0221535 0.0383709i 0.854736 0.519063i \(-0.173719\pi\)
−0.876890 + 0.480692i \(0.840386\pi\)
\(614\) 0 0
\(615\) 8.49179e6 0.905339
\(616\) 0 0
\(617\) 6.97181e6 1.20755e7i 0.737281 1.27701i −0.216435 0.976297i \(-0.569443\pi\)
0.953715 0.300711i \(-0.0972238\pi\)
\(618\) 0 0
\(619\) 1.59596e7 1.67415 0.837075 0.547088i \(-0.184264\pi\)
0.837075 + 0.547088i \(0.184264\pi\)
\(620\) 0 0
\(621\) 2.17049e6 3.75940e6i 0.225854 0.391191i
\(622\) 0 0
\(623\) 698230. 1.20937e6i 0.0720740 0.124836i
\(624\) 0 0
\(625\) −9.30336e6 1.61139e7i −0.952664 1.65006i
\(626\) 0 0
\(627\) −3.93124e6 57720.4i −0.399357 0.00586354i
\(628\) 0 0
\(629\) −3.51922e6 6.09547e6i −0.354666 0.614300i
\(630\) 0 0
\(631\) −3.10790e6 + 5.38304e6i −0.310737 + 0.538213i −0.978522 0.206141i \(-0.933909\pi\)
0.667785 + 0.744354i \(0.267243\pi\)
\(632\) 0 0
\(633\) 6.46664e6 1.12005e7i 0.641460 1.11104i
\(634\) 0 0
\(635\) −1.32874e7 −1.30769
\(636\) 0 0
\(637\) −2.28369e6 + 3.95547e6i −0.222992 + 0.386233i
\(638\) 0 0
\(639\) 9.08877e6 0.880548
\(640\) 0 0
\(641\) 4.32033e6 + 7.48302e6i 0.415309 + 0.719337i 0.995461 0.0951719i \(-0.0303401\pi\)
−0.580152 + 0.814508i \(0.697007\pi\)
\(642\) 0 0
\(643\) 8.07254e6 + 1.39820e7i 0.769985 + 1.33365i 0.937570 + 0.347796i \(0.113070\pi\)
−0.167585 + 0.985858i \(0.553597\pi\)
\(644\) 0 0
\(645\) 2.21238e7 2.09392
\(646\) 0 0
\(647\) −8.16344e6 −0.766677 −0.383338 0.923608i \(-0.625226\pi\)
−0.383338 + 0.923608i \(0.625226\pi\)
\(648\) 0 0
\(649\) 2.47703e6 + 4.29034e6i 0.230844 + 0.399834i
\(650\) 0 0
\(651\) 4.94946e6 + 8.57271e6i 0.457725 + 0.792803i
\(652\) 0 0
\(653\) 1.08567e7 0.996359 0.498179 0.867074i \(-0.334002\pi\)
0.498179 + 0.867074i \(0.334002\pi\)
\(654\) 0 0
\(655\) −1.01603e7 + 1.75981e7i −0.925343 + 1.60274i
\(656\) 0 0
\(657\) −1.57898e7 −1.42713
\(658\) 0 0
\(659\) −2.49623e6 + 4.32359e6i −0.223908 + 0.387821i −0.955991 0.293395i \(-0.905215\pi\)
0.732083 + 0.681215i \(0.238548\pi\)
\(660\) 0 0
\(661\) −7.54002e6 + 1.30597e7i −0.671226 + 1.16260i 0.306330 + 0.951925i \(0.400899\pi\)
−0.977557 + 0.210673i \(0.932435\pi\)
\(662\) 0 0
\(663\) 5.71999e6 + 9.90731e6i 0.505372 + 0.875330i
\(664\) 0 0
\(665\) −1.52112e7 223338.i −1.33386 0.0195844i
\(666\) 0 0
\(667\) −9.41057e6 1.62996e7i −0.819033 1.41861i
\(668\) 0 0
\(669\) 1.56862e6 2.71692e6i 0.135504 0.234699i
\(670\) 0 0
\(671\) −268182. + 464505.i −0.0229945 + 0.0398276i
\(672\) 0 0
\(673\) −1.89452e7 −1.61236 −0.806180 0.591671i \(-0.798469\pi\)
−0.806180 + 0.591671i \(0.798469\pi\)
\(674\) 0 0
\(675\) 3.63361e6 6.29359e6i 0.306958 0.531666i
\(676\) 0 0
\(677\) −4.05825e6 −0.340304 −0.170152 0.985418i \(-0.554426\pi\)
−0.170152 + 0.985418i \(0.554426\pi\)
\(678\) 0 0
\(679\) −5.90894e6 1.02346e7i −0.491853 0.851914i
\(680\) 0 0
\(681\) 1.09990e7 + 1.90508e7i 0.908836 + 1.57415i
\(682\) 0 0
\(683\) −2.85503e6 −0.234185 −0.117092 0.993121i \(-0.537357\pi\)
−0.117092 + 0.993121i \(0.537357\pi\)
\(684\) 0 0
\(685\) 3.67344e7 2.99121
\(686\) 0 0
\(687\) −8.01621e6 1.38845e7i −0.648003 1.12237i
\(688\) 0 0
\(689\) −1.10251e7 1.90960e7i −0.884778 1.53248i
\(690\) 0 0
\(691\) −4.38220e6 −0.349138 −0.174569 0.984645i \(-0.555853\pi\)
−0.174569 + 0.984645i \(0.555853\pi\)
\(692\) 0 0
\(693\) 1.10834e6 1.91970e6i 0.0876675 0.151845i
\(694\) 0 0
\(695\) −3.84788e7 −3.02175
\(696\) 0 0
\(697\) 1.84606e6 3.19747e6i 0.143934 0.249301i
\(698\) 0 0
\(699\) −1.51156e7 + 2.61810e7i −1.17013 + 2.02672i
\(700\) 0 0
\(701\) −2.33852e6 4.05044e6i −0.179741 0.311320i 0.762051 0.647517i \(-0.224193\pi\)
−0.941792 + 0.336197i \(0.890859\pi\)
\(702\) 0 0
\(703\) 6.17285e6 1.03382e7i 0.471083 0.788959i
\(704\) 0 0
\(705\) 9.13605e6 + 1.58241e7i 0.692287 + 1.19908i
\(706\) 0 0
\(707\) 2.37829e6 4.11932e6i 0.178944 0.309940i
\(708\) 0 0
\(709\) −177992. + 308291.i −0.0132980 + 0.0230328i −0.872598 0.488439i \(-0.837566\pi\)
0.859300 + 0.511472i \(0.170900\pi\)
\(710\) 0 0
\(711\) 2.58450e6 0.191735
\(712\) 0 0
\(713\) −1.05279e7 + 1.82349e7i −0.775568 + 1.34332i
\(714\) 0 0
\(715\) 7.19307e6 0.526198
\(716\) 0 0
\(717\) −7.83368e6 1.35683e7i −0.569073 0.985663i
\(718\) 0 0
\(719\) 705747. + 1.22239e6i 0.0509128 + 0.0881835i 0.890359 0.455260i \(-0.150454\pi\)
−0.839446 + 0.543443i \(0.817120\pi\)
\(720\) 0 0
\(721\) −9.78833e6 −0.701246
\(722\) 0 0
\(723\) 1.92099e7 1.36672
\(724\) 0 0
\(725\) −1.57542e7 2.72871e7i −1.11314 1.92802i
\(726\) 0 0
\(727\) −6.82049e6 1.18134e7i −0.478608 0.828973i 0.521091 0.853501i \(-0.325525\pi\)
−0.999699 + 0.0245278i \(0.992192\pi\)
\(728\) 0 0
\(729\) −8.11485e6 −0.565538
\(730\) 0 0
\(731\) 4.80957e6 8.33042e6i 0.332899 0.576598i
\(732\) 0 0
\(733\) −2.66079e7 −1.82915 −0.914577 0.404412i \(-0.867476\pi\)
−0.914577 + 0.404412i \(0.867476\pi\)
\(734\) 0 0
\(735\) −8.12181e6 + 1.40674e7i −0.554542 + 0.960495i
\(736\) 0 0
\(737\) −2.20900e6 + 3.82611e6i −0.149806 + 0.259471i
\(738\) 0 0
\(739\) −4.22169e6 7.31218e6i −0.284364 0.492533i 0.688091 0.725625i \(-0.258449\pi\)
−0.972455 + 0.233091i \(0.925116\pi\)
\(740\) 0 0
\(741\) −1.00331e7 + 1.68032e7i −0.671258 + 1.12421i
\(742\) 0 0
\(743\) 2.70095e6 + 4.67819e6i 0.179492 + 0.310889i 0.941707 0.336435i \(-0.109221\pi\)
−0.762215 + 0.647324i \(0.775888\pi\)
\(744\) 0 0
\(745\) 2.21712e6 3.84016e6i 0.146352 0.253489i
\(746\) 0 0
\(747\) −1.78419e6 + 3.09031e6i −0.116988 + 0.202629i
\(748\) 0 0
\(749\) 4.32923e6 0.281972
\(750\) 0 0
\(751\) −6.66101e6 + 1.15372e7i −0.430963 + 0.746450i −0.996956 0.0779599i \(-0.975159\pi\)
0.565994 + 0.824410i \(0.308493\pi\)
\(752\) 0 0
\(753\) 6.62985e6 0.426105
\(754\) 0 0
\(755\) 6.63319e6 + 1.14890e7i 0.423502 + 0.733527i
\(756\) 0 0
\(757\) 1.61706e6 + 2.80083e6i 0.102562 + 0.177642i 0.912739 0.408542i \(-0.133963\pi\)
−0.810178 + 0.586185i \(0.800629\pi\)
\(758\) 0 0
\(759\) 1.06169e7 0.668948
\(760\) 0 0
\(761\) 602035. 0.0376843 0.0188421 0.999822i \(-0.494002\pi\)
0.0188421 + 0.999822i \(0.494002\pi\)
\(762\) 0 0
\(763\) −7.19072e6 1.24547e7i −0.447158 0.774500i
\(764\) 0 0
\(765\) 9.03446e6 + 1.56481e7i 0.558147 + 0.966739i
\(766\) 0 0
\(767\) 2.46598e7 1.51357
\(768\) 0 0
\(769\) −9.81675e6 + 1.70031e7i −0.598621 + 1.03684i 0.394404 + 0.918937i \(0.370951\pi\)
−0.993025 + 0.117904i \(0.962382\pi\)
\(770\) 0 0
\(771\) 1.22685e7 0.743285
\(772\) 0 0
\(773\) 2.03679e6 3.52782e6i 0.122602 0.212353i −0.798191 0.602404i \(-0.794210\pi\)
0.920793 + 0.390052i \(0.127543\pi\)
\(774\) 0 0
\(775\) −1.76248e7 + 3.05270e7i −1.05407 + 1.82570i
\(776\) 0 0
\(777\) 7.64293e6 + 1.32379e7i 0.454159 + 0.786626i
\(778\) 0 0
\(779\) 6.31552e6 + 92727.4i 0.372877 + 0.00547475i
\(780\) 0 0
\(781\) −2.79736e6 4.84518e6i −0.164105 0.284238i
\(782\) 0 0
\(783\) 2.26254e6 3.91883e6i 0.131884 0.228429i
\(784\) 0 0
\(785\) −2.41937e7 + 4.19047e7i −1.40129 + 2.42710i
\(786\) 0 0
\(787\) −4.13033e6 −0.237710 −0.118855 0.992912i \(-0.537922\pi\)
−0.118855 + 0.992912i \(0.537922\pi\)
\(788\) 0 0
\(789\) 1.25951e7 2.18154e7i 0.720293 1.24758i
\(790\) 0 0
\(791\) −6.79025e6 −0.385873
\(792\) 0 0
\(793\) 1.33493e6 + 2.31217e6i 0.0753834 + 0.130568i
\(794\) 0 0
\(795\) −3.92101e7 6.79138e7i −2.20029 3.81101i
\(796\) 0 0
\(797\) −3.62791e6 −0.202307 −0.101154 0.994871i \(-0.532253\pi\)
−0.101154 + 0.994871i \(0.532253\pi\)
\(798\) 0 0
\(799\) 7.94447e6 0.440249
\(800\) 0 0
\(801\) −1.41873e6 2.45731e6i −0.0781301 0.135325i
\(802\) 0 0
\(803\) 4.85983e6 + 8.41748e6i 0.265970 + 0.460674i
\(804\) 0 0
\(805\) 4.10800e7 2.23430
\(806\) 0 0
\(807\) 3.22131e6 5.57947e6i 0.174120 0.301585i
\(808\) 0 0
\(809\) 5.98904e6 0.321726 0.160863 0.986977i \(-0.448572\pi\)
0.160863 + 0.986977i \(0.448572\pi\)
\(810\) 0 0
\(811\) 8.65166e6 1.49851e7i 0.461899 0.800033i −0.537156 0.843483i \(-0.680501\pi\)
0.999056 + 0.0434495i \(0.0138348\pi\)
\(812\) 0 0
\(813\) −9.95677e6 + 1.72456e7i −0.528314 + 0.915066i
\(814\) 0 0
\(815\) 1.96847e7 + 3.40948e7i 1.03809 + 1.79802i
\(816\) 0 0
\(817\) 1.64539e7 + 241584.i 0.862411 + 0.0126623i
\(818\) 0 0
\(819\) −5.51697e6 9.55567e6i −0.287403 0.497796i
\(820\) 0 0
\(821\) −7.04373e6 + 1.22001e7i −0.364708 + 0.631692i −0.988729 0.149715i \(-0.952164\pi\)
0.624022 + 0.781407i \(0.285498\pi\)
\(822\) 0 0
\(823\) −1.76700e6 + 3.06053e6i −0.0909361 + 0.157506i −0.907905 0.419175i \(-0.862319\pi\)
0.816969 + 0.576681i \(0.195653\pi\)
\(824\) 0 0
\(825\) 1.77737e7 0.909164
\(826\) 0 0
\(827\) 5.25927e6 9.10933e6i 0.267400 0.463151i −0.700789 0.713368i \(-0.747169\pi\)
0.968190 + 0.250217i \(0.0805020\pi\)
\(828\) 0 0
\(829\) 3.82603e7 1.93358 0.966790 0.255571i \(-0.0822636\pi\)
0.966790 + 0.255571i \(0.0822636\pi\)
\(830\) 0 0
\(831\) 6.01225e6 + 1.04135e7i 0.302019 + 0.523113i
\(832\) 0 0
\(833\) 3.53126e6 + 6.11632e6i 0.176326 + 0.305406i
\(834\) 0 0
\(835\) 5.29180e7 2.62656
\(836\) 0 0
\(837\) −5.06236e6 −0.249770
\(838\) 0 0
\(839\) −1.04200e7 1.80480e7i −0.511050 0.885164i −0.999918 0.0128063i \(-0.995924\pi\)
0.488868 0.872358i \(-0.337410\pi\)
\(840\) 0 0
\(841\) 445904. + 772328.i 0.0217396 + 0.0376541i
\(842\) 0 0
\(843\) 3.83703e7 1.85963
\(844\) 0 0
\(845\) −882284. + 1.52816e6i −0.0425076 + 0.0736253i
\(846\) 0 0
\(847\) 1.40231e7 0.671640
\(848\) 0 0
\(849\) 1.28688e7 2.22894e7i 0.612729 1.06128i
\(850\) 0 0
\(851\) −1.62572e7 + 2.81583e7i −0.769524 + 1.33286i
\(852\) 0 0
\(853\) 1.06809e7 + 1.84999e7i 0.502617 + 0.870558i 0.999995 + 0.00302417i \(0.000962625\pi\)
−0.497379 + 0.867534i \(0.665704\pi\)
\(854\) 0 0
\(855\) −1.58468e7 + 2.65399e7i −0.741355 + 1.24161i
\(856\) 0 0
\(857\) −1.36791e7 2.36929e7i −0.636216 1.10196i −0.986256 0.165224i \(-0.947165\pi\)
0.350040 0.936735i \(-0.386168\pi\)
\(858\) 0 0
\(859\) −3.27009e6 + 5.66397e6i −0.151209 + 0.261901i −0.931672 0.363300i \(-0.881650\pi\)
0.780463 + 0.625202i \(0.214983\pi\)
\(860\) 0 0
\(861\) −4.00921e6 + 6.94416e6i −0.184311 + 0.319236i
\(862\) 0 0
\(863\) 3.18151e7 1.45414 0.727069 0.686564i \(-0.240882\pi\)
0.727069 + 0.686564i \(0.240882\pi\)
\(864\) 0 0
\(865\) 2.31142e7 4.00349e7i 1.05036 1.81928i
\(866\) 0 0
\(867\) −1.19966e7 −0.542013
\(868\) 0 0
\(869\) −795463. 1.37778e6i −0.0357331 0.0618915i
\(870\) 0 0
\(871\) 1.09958e7 + 1.90452e7i 0.491111 + 0.850629i
\(872\) 0 0
\(873\) −2.40127e7 −1.06636
\(874\) 0 0
\(875\) 3.85601e7 1.70262
\(876\) 0 0
\(877\) 1.41774e7 + 2.45560e7i 0.622442 + 1.07810i 0.989030 + 0.147717i \(0.0471925\pi\)
−0.366588 + 0.930383i \(0.619474\pi\)
\(878\) 0 0
\(879\) 9.65203e6 + 1.67178e7i 0.421353 + 0.729806i
\(880\) 0 0
\(881\) 4.86162e6 0.211029 0.105514 0.994418i \(-0.466351\pi\)
0.105514 + 0.994418i \(0.466351\pi\)
\(882\) 0 0
\(883\) 412959. 715267.i 0.0178240 0.0308721i −0.856976 0.515357i \(-0.827659\pi\)
0.874800 + 0.484485i \(0.160993\pi\)
\(884\) 0 0
\(885\) 8.77011e7 3.76398
\(886\) 0 0
\(887\) −6.55015e6 + 1.13452e7i −0.279539 + 0.484176i −0.971270 0.237979i \(-0.923515\pi\)
0.691731 + 0.722155i \(0.256848\pi\)
\(888\) 0 0
\(889\) 6.27335e6 1.08658e7i 0.266223 0.461111i
\(890\) 0 0
\(891\) 4.09512e6 + 7.09295e6i 0.172811 + 0.299318i
\(892\) 0 0
\(893\) 6.62188e6 + 1.18685e7i 0.277877 + 0.498042i
\(894\) 0 0
\(895\) −1.92419e7 3.33280e7i −0.802954 1.39076i
\(896\) 0 0
\(897\) 2.64238e7 4.57673e7i 1.09651 1.89922i
\(898\) 0 0
\(899\) −1.09744e7 + 1.90083e7i −0.452880 + 0.784410i
\(900\) 0 0
\(901\) −3.40960e7 −1.39924
\(902\) 0 0
\(903\) −1.04453e7 + 1.80918e7i −0.426286 + 0.738348i
\(904\) 0 0
\(905\) −3.48154e7 −1.41302
\(906\) 0 0
\(907\) −4.54799e6 7.87734e6i −0.183570 0.317952i 0.759524 0.650479i \(-0.225432\pi\)
−0.943094 + 0.332527i \(0.892099\pi\)
\(908\) 0 0
\(909\) −4.83244e6 8.37003e6i −0.193980 0.335983i
\(910\) 0 0
\(911\) −4.66286e6 −0.186147 −0.0930737 0.995659i \(-0.529669\pi\)
−0.0930737 + 0.995659i \(0.529669\pi\)
\(912\) 0 0
\(913\) 2.19657e6 0.0872105
\(914\) 0 0
\(915\) 4.74760e6 + 8.22308e6i 0.187466 + 0.324700i
\(916\) 0 0
\(917\) −9.59392e6 1.66172e7i −0.376767 0.652580i
\(918\) 0 0
\(919\) 3.34522e7 1.30658 0.653291 0.757107i \(-0.273388\pi\)
0.653291 + 0.757107i \(0.273388\pi\)
\(920\) 0 0
\(921\) −1.76272e7 + 3.05312e7i −0.684754 + 1.18603i
\(922\) 0 0
\(923\) −2.78489e7 −1.07598
\(924\) 0 0
\(925\) −2.72161e7 + 4.71397e7i −1.04586 + 1.81148i
\(926\) 0 0
\(927\) −9.94443e6 + 1.72243e7i −0.380085 + 0.658326i
\(928\) 0 0
\(929\) −4.29234e6 7.43456e6i −0.163176 0.282628i 0.772830 0.634613i \(-0.218840\pi\)
−0.936006 + 0.351984i \(0.885507\pi\)
\(930\) 0 0
\(931\) −6.19397e6 + 1.03735e7i −0.234204 + 0.392240i
\(932\) 0 0
\(933\) −1.10895e7 1.92075e7i −0.417068 0.722383i
\(934\) 0 0
\(935\) 5.56130e6 9.63245e6i 0.208040 0.360336i
\(936\) 0 0
\(937\) −2.35809e7 + 4.08434e7i −0.877429 + 1.51975i −0.0232757 + 0.999729i \(0.507410\pi\)
−0.854153 + 0.520022i \(0.825924\pi\)
\(938\) 0 0
\(939\) −338298. −0.0125209
\(940\) 0 0
\(941\) 1.05661e7 1.83010e7i 0.388992 0.673754i −0.603322 0.797498i \(-0.706157\pi\)
0.992314 + 0.123744i \(0.0394900\pi\)
\(942\) 0 0
\(943\) −1.70559e7 −0.624592
\(944\) 0 0
\(945\) 4.93834e6 + 8.55345e6i 0.179888 + 0.311574i
\(946\) 0 0
\(947\) 1.04875e7 + 1.81649e7i 0.380012 + 0.658199i 0.991063 0.133391i \(-0.0425867\pi\)
−0.611052 + 0.791591i \(0.709253\pi\)
\(948\) 0 0
\(949\) 4.83816e7 1.74387
\(950\) 0 0
\(951\) −5.33447e7 −1.91267
\(952\) 0 0
\(953\) −2.41521e7 4.18327e7i −0.861435 1.49205i −0.870544 0.492091i \(-0.836233\pi\)
0.00910824 0.999959i \(-0.497101\pi\)
\(954\) 0 0
\(955\) 3.53723e7 + 6.12665e7i 1.25503 + 2.17378i
\(956\) 0 0
\(957\) 1.10671e7 0.390621
\(958\) 0 0
\(959\) −1.73433e7 + 3.00396e7i −0.608957 + 1.05474i
\(960\) 0 0
\(961\) −4.07418e6 −0.142309
\(962\) 0 0
\(963\) 4.39827e6 7.61803e6i 0.152833 0.264714i
\(964\) 0 0
\(965\) −2.79492e6 + 4.84094e6i −0.0966164 + 0.167345i
\(966\) 0 0
\(967\) −2.71370e7 4.70027e7i −0.933247 1.61643i −0.777731 0.628597i \(-0.783629\pi\)
−0.155516 0.987833i \(-0.549704\pi\)
\(968\) 0 0
\(969\) 1.47446e7 + 2.64269e7i 0.504456 + 0.904143i
\(970\) 0 0
\(971\) −1.62480e7 2.81423e7i −0.553033 0.957882i −0.998054 0.0623611i \(-0.980137\pi\)
0.445021 0.895520i \(-0.353196\pi\)
\(972\) 0 0
\(973\) 1.81669e7 3.14660e7i 0.615175 1.06552i
\(974\) 0 0
\(975\) 4.42360e7 7.66189e7i 1.49027 2.58122i
\(976\) 0 0
\(977\) −6.98320e6 −0.234055 −0.117028 0.993129i \(-0.537337\pi\)
−0.117028 + 0.993129i \(0.537337\pi\)
\(978\) 0 0
\(979\) −873321. + 1.51264e6i −0.0291217 + 0.0504403i
\(980\) 0 0
\(981\) −2.92216e7 −0.969462
\(982\) 0 0
\(983\) 2.88200e6 + 4.99177e6i 0.0951285 + 0.164767i 0.909662 0.415349i \(-0.136340\pi\)
−0.814534 + 0.580116i \(0.803007\pi\)
\(984\) 0 0
\(985\) −5.06035e7 8.76478e7i −1.66184 2.87839i
\(986\) 0 0
\(987\) −1.72536e7 −0.563749
\(988\) 0 0
\(989\) −4.44361e7 −1.44459
\(990\) 0 0
\(991\) −3.74277e6 6.48266e6i −0.121062 0.209686i 0.799125 0.601165i \(-0.205297\pi\)
−0.920187 + 0.391480i \(0.871963\pi\)
\(992\) 0 0
\(993\) 1.20042e7 + 2.07919e7i 0.386331 + 0.669146i
\(994\) 0 0
\(995\) −4.99200e7 −1.59852
\(996\) 0 0
\(997\) −1.00119e7 + 1.73412e7i −0.318992 + 0.552510i −0.980278 0.197624i \(-0.936678\pi\)
0.661286 + 0.750134i \(0.270011\pi\)
\(998\) 0 0
\(999\) −7.81729e6 −0.247823
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 304.6.i.d.273.2 18
4.3 odd 2 76.6.e.a.45.8 18
12.11 even 2 684.6.k.f.577.1 18
19.11 even 3 inner 304.6.i.d.49.2 18
76.11 odd 6 76.6.e.a.49.8 yes 18
228.11 even 6 684.6.k.f.505.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.8 18 4.3 odd 2
76.6.e.a.49.8 yes 18 76.11 odd 6
304.6.i.d.49.2 18 19.11 even 3 inner
304.6.i.d.273.2 18 1.1 even 1 trivial
684.6.k.f.505.1 18 228.11 even 6
684.6.k.f.577.1 18 12.11 even 2