Properties

Label 76.5.h.a.65.1
Level $76$
Weight $5$
Character 76.65
Analytic conductor $7.856$
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] = [76,5,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.h (of order \(6\), degree \(2\), 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: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.1
Root \(0.500000 + 14.3199i\) of defining polynomial
Character \(\chi\) \(=\) 76.65
Dual form 76.5.h.a.69.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-13.1514 - 7.59296i) q^{3} +(19.9265 - 34.5138i) q^{5} -53.3663 q^{7} +(74.8062 + 129.568i) q^{9} +O(q^{10})\) \(q+(-13.1514 - 7.59296i) q^{3} +(19.9265 - 34.5138i) q^{5} -53.3663 q^{7} +(74.8062 + 129.568i) q^{9} +9.01156 q^{11} +(-5.39539 + 3.11503i) q^{13} +(-524.123 + 302.603i) q^{15} +(-261.609 + 453.120i) q^{17} +(-329.612 - 147.231i) q^{19} +(701.842 + 405.208i) q^{21} +(204.027 + 353.386i) q^{23} +(-481.633 - 834.213i) q^{25} -1041.94i q^{27} +(1176.08 - 679.011i) q^{29} +214.814i q^{31} +(-118.515 - 68.4244i) q^{33} +(-1063.41 + 1841.87i) q^{35} -1223.68i q^{37} +94.6092 q^{39} +(-666.384 - 384.737i) q^{41} +(-1147.74 + 1987.94i) q^{43} +5962.51 q^{45} +(-1740.86 - 3015.26i) q^{47} +446.963 q^{49} +(6881.05 - 3972.78i) q^{51} +(-4815.71 + 2780.35i) q^{53} +(179.569 - 311.023i) q^{55} +(3216.93 + 4439.03i) q^{57} +(-2130.74 - 1230.18i) q^{59} +(-1471.38 - 2548.51i) q^{61} +(-3992.13 - 6914.57i) q^{63} +248.287i q^{65} +(1474.24 - 851.150i) q^{67} -6196.69i q^{69} +(-2989.25 - 1725.84i) q^{71} +(-4122.11 + 7139.70i) q^{73} +14628.1i q^{75} -480.914 q^{77} +(7356.86 + 4247.48i) q^{79} +(-1852.13 + 3207.98i) q^{81} -2757.20 q^{83} +(10425.9 + 18058.2i) q^{85} -20622.8 q^{87} +(5987.69 - 3457.00i) q^{89} +(287.932 - 166.238i) q^{91} +(1631.07 - 2825.10i) q^{93} +(-11649.5 + 8442.33i) q^{95} +(-10519.1 - 6073.22i) q^{97} +(674.120 + 1167.61i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9} + 6 q^{11} - 93 q^{13} - 741 q^{15} - 483 q^{17} - 533 q^{19} + 972 q^{21} + 531 q^{23} - 217 q^{25} + 2025 q^{29} - 75 q^{33} - 1128 q^{35} - 2250 q^{39} - 1692 q^{41} - 63 q^{43} + 7976 q^{45} - 3471 q^{47} + 420 q^{49} + 6741 q^{51} - 3771 q^{53} - 2014 q^{55} + 7617 q^{57} - 9594 q^{59} + 1229 q^{61} + 1514 q^{63} + 7590 q^{67} + 963 q^{71} - 2838 q^{73} - 15408 q^{77} + 11073 q^{79} + 2086 q^{81} - 14202 q^{83} + 9455 q^{85} - 39510 q^{87} + 6525 q^{89} - 7686 q^{91} - 5316 q^{93} + 1521 q^{95} - 34110 q^{97} + 13220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −13.1514 7.59296i −1.46127 0.843663i −0.462196 0.886778i \(-0.652938\pi\)
−0.999070 + 0.0431151i \(0.986272\pi\)
\(4\) 0 0
\(5\) 19.9265 34.5138i 0.797061 1.38055i −0.124461 0.992225i \(-0.539720\pi\)
0.921522 0.388326i \(-0.126947\pi\)
\(6\) 0 0
\(7\) −53.3663 −1.08911 −0.544554 0.838726i \(-0.683301\pi\)
−0.544554 + 0.838726i \(0.683301\pi\)
\(8\) 0 0
\(9\) 74.8062 + 129.568i 0.923533 + 1.59961i
\(10\) 0 0
\(11\) 9.01156 0.0744757 0.0372379 0.999306i \(-0.488144\pi\)
0.0372379 + 0.999306i \(0.488144\pi\)
\(12\) 0 0
\(13\) −5.39539 + 3.11503i −0.0319254 + 0.0184321i −0.515878 0.856662i \(-0.672534\pi\)
0.483952 + 0.875094i \(0.339201\pi\)
\(14\) 0 0
\(15\) −524.123 + 302.603i −2.32944 + 1.34490i
\(16\) 0 0
\(17\) −261.609 + 453.120i −0.905222 + 1.56789i −0.0846030 + 0.996415i \(0.526962\pi\)
−0.820619 + 0.571476i \(0.806371\pi\)
\(18\) 0 0
\(19\) −329.612 147.231i −0.913052 0.407843i
\(20\) 0 0
\(21\) 701.842 + 405.208i 1.59148 + 0.918840i
\(22\) 0 0
\(23\) 204.027 + 353.386i 0.385685 + 0.668026i 0.991864 0.127302i \(-0.0406319\pi\)
−0.606179 + 0.795328i \(0.707299\pi\)
\(24\) 0 0
\(25\) −481.633 834.213i −0.770613 1.33474i
\(26\) 0 0
\(27\) 1041.94i 1.42928i
\(28\) 0 0
\(29\) 1176.08 679.011i 1.39843 0.807385i 0.404204 0.914669i \(-0.367549\pi\)
0.994228 + 0.107283i \(0.0342152\pi\)
\(30\) 0 0
\(31\) 214.814i 0.223532i 0.993735 + 0.111766i \(0.0356506\pi\)
−0.993735 + 0.111766i \(0.964349\pi\)
\(32\) 0 0
\(33\) −118.515 68.4244i −0.108829 0.0628324i
\(34\) 0 0
\(35\) −1063.41 + 1841.87i −0.868086 + 1.50357i
\(36\) 0 0
\(37\) 1223.68i 0.893851i −0.894571 0.446925i \(-0.852519\pi\)
0.894571 0.446925i \(-0.147481\pi\)
\(38\) 0 0
\(39\) 94.6092 0.0622020
\(40\) 0 0
\(41\) −666.384 384.737i −0.396421 0.228874i 0.288517 0.957475i \(-0.406838\pi\)
−0.684939 + 0.728601i \(0.740171\pi\)
\(42\) 0 0
\(43\) −1147.74 + 1987.94i −0.620733 + 1.07514i 0.368616 + 0.929582i \(0.379832\pi\)
−0.989349 + 0.145560i \(0.953502\pi\)
\(44\) 0 0
\(45\) 5962.51 2.94445
\(46\) 0 0
\(47\) −1740.86 3015.26i −0.788076 1.36499i −0.927144 0.374706i \(-0.877744\pi\)
0.139067 0.990283i \(-0.455590\pi\)
\(48\) 0 0
\(49\) 446.963 0.186157
\(50\) 0 0
\(51\) 6881.05 3972.78i 2.64554 1.52740i
\(52\) 0 0
\(53\) −4815.71 + 2780.35i −1.71438 + 0.989801i −0.785963 + 0.618274i \(0.787832\pi\)
−0.928422 + 0.371527i \(0.878834\pi\)
\(54\) 0 0
\(55\) 179.569 311.023i 0.0593617 0.102817i
\(56\) 0 0
\(57\) 3216.93 + 4439.03i 0.990130 + 1.36628i
\(58\) 0 0
\(59\) −2130.74 1230.18i −0.612105 0.353399i 0.161684 0.986843i \(-0.448307\pi\)
−0.773789 + 0.633444i \(0.781641\pi\)
\(60\) 0 0
\(61\) −1471.38 2548.51i −0.395427 0.684900i 0.597728 0.801699i \(-0.296070\pi\)
−0.993156 + 0.116799i \(0.962737\pi\)
\(62\) 0 0
\(63\) −3992.13 6914.57i −1.00583 1.74214i
\(64\) 0 0
\(65\) 248.287i 0.0587661i
\(66\) 0 0
\(67\) 1474.24 851.150i 0.328411 0.189608i −0.326725 0.945120i \(-0.605945\pi\)
0.655135 + 0.755512i \(0.272612\pi\)
\(68\) 0 0
\(69\) 6196.69i 1.30155i
\(70\) 0 0
\(71\) −2989.25 1725.84i −0.592988 0.342361i 0.173290 0.984871i \(-0.444560\pi\)
−0.766278 + 0.642509i \(0.777893\pi\)
\(72\) 0 0
\(73\) −4122.11 + 7139.70i −0.773524 + 1.33978i 0.162097 + 0.986775i \(0.448174\pi\)
−0.935620 + 0.353008i \(0.885159\pi\)
\(74\) 0 0
\(75\) 14628.1i 2.60055i
\(76\) 0 0
\(77\) −480.914 −0.0811121
\(78\) 0 0
\(79\) 7356.86 + 4247.48i 1.17879 + 0.680577i 0.955735 0.294227i \(-0.0950623\pi\)
0.223059 + 0.974805i \(0.428396\pi\)
\(80\) 0 0
\(81\) −1852.13 + 3207.98i −0.282293 + 0.488947i
\(82\) 0 0
\(83\) −2757.20 −0.400233 −0.200116 0.979772i \(-0.564132\pi\)
−0.200116 + 0.979772i \(0.564132\pi\)
\(84\) 0 0
\(85\) 10425.9 + 18058.2i 1.44303 + 2.49941i
\(86\) 0 0
\(87\) −20622.8 −2.72464
\(88\) 0 0
\(89\) 5987.69 3457.00i 0.755926 0.436434i −0.0719049 0.997411i \(-0.522908\pi\)
0.827831 + 0.560977i \(0.189574\pi\)
\(90\) 0 0
\(91\) 287.932 166.238i 0.0347702 0.0200746i
\(92\) 0 0
\(93\) 1631.07 2825.10i 0.188585 0.326639i
\(94\) 0 0
\(95\) −11649.5 + 8442.33i −1.29081 + 0.935438i
\(96\) 0 0
\(97\) −10519.1 6073.22i −1.11798 0.645469i −0.177099 0.984193i \(-0.556671\pi\)
−0.940886 + 0.338724i \(0.890005\pi\)
\(98\) 0 0
\(99\) 674.120 + 1167.61i 0.0687808 + 0.119132i
\(100\) 0 0
\(101\) −4704.98 8149.27i −0.461228 0.798870i 0.537795 0.843076i \(-0.319257\pi\)
−0.999022 + 0.0442061i \(0.985924\pi\)
\(102\) 0 0
\(103\) 15630.0i 1.47328i −0.676284 0.736641i \(-0.736411\pi\)
0.676284 0.736641i \(-0.263589\pi\)
\(104\) 0 0
\(105\) 27970.5 16148.8i 2.53701 1.46474i
\(106\) 0 0
\(107\) 1064.07i 0.0929398i −0.998920 0.0464699i \(-0.985203\pi\)
0.998920 0.0464699i \(-0.0147972\pi\)
\(108\) 0 0
\(109\) 2702.69 + 1560.40i 0.227480 + 0.131336i 0.609409 0.792856i \(-0.291407\pi\)
−0.381929 + 0.924192i \(0.624740\pi\)
\(110\) 0 0
\(111\) −9291.37 + 16093.1i −0.754108 + 1.30615i
\(112\) 0 0
\(113\) 4559.56i 0.357081i −0.983933 0.178540i \(-0.942863\pi\)
0.983933 0.178540i \(-0.0571375\pi\)
\(114\) 0 0
\(115\) 16262.2 1.22966
\(116\) 0 0
\(117\) −807.216 466.047i −0.0589683 0.0340453i
\(118\) 0 0
\(119\) 13961.1 24181.4i 0.985885 1.70760i
\(120\) 0 0
\(121\) −14559.8 −0.994453
\(122\) 0 0
\(123\) 5842.59 + 10119.7i 0.386185 + 0.668892i
\(124\) 0 0
\(125\) −13480.9 −0.862781
\(126\) 0 0
\(127\) 16901.8 9758.24i 1.04791 0.605012i 0.125848 0.992050i \(-0.459835\pi\)
0.922064 + 0.387038i \(0.126502\pi\)
\(128\) 0 0
\(129\) 30188.7 17429.4i 1.81411 1.04738i
\(130\) 0 0
\(131\) −9199.29 + 15933.6i −0.536058 + 0.928479i 0.463054 + 0.886330i \(0.346754\pi\)
−0.999111 + 0.0421489i \(0.986580\pi\)
\(132\) 0 0
\(133\) 17590.2 + 7857.20i 0.994413 + 0.444186i
\(134\) 0 0
\(135\) −35961.3 20762.3i −1.97319 1.13922i
\(136\) 0 0
\(137\) 9527.92 + 16502.8i 0.507642 + 0.879261i 0.999961 + 0.00884630i \(0.00281590\pi\)
−0.492319 + 0.870415i \(0.663851\pi\)
\(138\) 0 0
\(139\) 7531.70 + 13045.3i 0.389819 + 0.675187i 0.992425 0.122852i \(-0.0392041\pi\)
−0.602606 + 0.798039i \(0.705871\pi\)
\(140\) 0 0
\(141\) 52873.2i 2.65948i
\(142\) 0 0
\(143\) −48.6209 + 28.0713i −0.00237766 + 0.00137274i
\(144\) 0 0
\(145\) 54121.3i 2.57414i
\(146\) 0 0
\(147\) −5878.19 3393.78i −0.272025 0.157054i
\(148\) 0 0
\(149\) −6317.79 + 10942.7i −0.284572 + 0.492894i −0.972505 0.232880i \(-0.925185\pi\)
0.687933 + 0.725774i \(0.258518\pi\)
\(150\) 0 0
\(151\) 14119.5i 0.619248i −0.950859 0.309624i \(-0.899797\pi\)
0.950859 0.309624i \(-0.100203\pi\)
\(152\) 0 0
\(153\) −78279.9 −3.34401
\(154\) 0 0
\(155\) 7414.03 + 4280.49i 0.308597 + 0.178168i
\(156\) 0 0
\(157\) 6248.53 10822.8i 0.253500 0.439075i −0.710987 0.703205i \(-0.751751\pi\)
0.964487 + 0.264130i \(0.0850848\pi\)
\(158\) 0 0
\(159\) 84444.4 3.34023
\(160\) 0 0
\(161\) −10888.2 18858.9i −0.420053 0.727553i
\(162\) 0 0
\(163\) 29683.1 1.11721 0.558604 0.829434i \(-0.311337\pi\)
0.558604 + 0.829434i \(0.311337\pi\)
\(164\) 0 0
\(165\) −4723.17 + 2726.92i −0.173486 + 0.100162i
\(166\) 0 0
\(167\) −20882.9 + 12056.8i −0.748788 + 0.432313i −0.825256 0.564759i \(-0.808969\pi\)
0.0764680 + 0.997072i \(0.475636\pi\)
\(168\) 0 0
\(169\) −14261.1 + 24700.9i −0.499321 + 0.864848i
\(170\) 0 0
\(171\) −5580.50 53721.0i −0.190845 1.83718i
\(172\) 0 0
\(173\) −26503.6 15301.9i −0.885550 0.511273i −0.0130660 0.999915i \(-0.504159\pi\)
−0.872484 + 0.488642i \(0.837492\pi\)
\(174\) 0 0
\(175\) 25703.0 + 44518.9i 0.839281 + 1.45368i
\(176\) 0 0
\(177\) 18681.4 + 32357.2i 0.596299 + 1.03282i
\(178\) 0 0
\(179\) 31622.5i 0.986940i −0.869763 0.493470i \(-0.835728\pi\)
0.869763 0.493470i \(-0.164272\pi\)
\(180\) 0 0
\(181\) −2932.84 + 1693.28i −0.0895223 + 0.0516857i −0.544093 0.839025i \(-0.683126\pi\)
0.454571 + 0.890711i \(0.349793\pi\)
\(182\) 0 0
\(183\) 44688.7i 1.33443i
\(184\) 0 0
\(185\) −42233.9 24383.7i −1.23401 0.712454i
\(186\) 0 0
\(187\) −2357.51 + 4083.32i −0.0674170 + 0.116770i
\(188\) 0 0
\(189\) 55604.6i 1.55664i
\(190\) 0 0
\(191\) −7140.61 −0.195735 −0.0978675 0.995199i \(-0.531202\pi\)
−0.0978675 + 0.995199i \(0.531202\pi\)
\(192\) 0 0
\(193\) −13922.0 8037.86i −0.373754 0.215787i 0.301343 0.953516i \(-0.402565\pi\)
−0.675097 + 0.737729i \(0.735898\pi\)
\(194\) 0 0
\(195\) 1885.23 3265.32i 0.0495788 0.0858729i
\(196\) 0 0
\(197\) −26545.9 −0.684013 −0.342007 0.939698i \(-0.611107\pi\)
−0.342007 + 0.939698i \(0.611107\pi\)
\(198\) 0 0
\(199\) −2357.02 4082.47i −0.0595191 0.103090i 0.834731 0.550659i \(-0.185623\pi\)
−0.894250 + 0.447569i \(0.852290\pi\)
\(200\) 0 0
\(201\) −25851.0 −0.639861
\(202\) 0 0
\(203\) −62763.1 + 36236.3i −1.52304 + 0.879330i
\(204\) 0 0
\(205\) −26557.4 + 15332.9i −0.631944 + 0.364853i
\(206\) 0 0
\(207\) −30525.0 + 52870.9i −0.712385 + 1.23389i
\(208\) 0 0
\(209\) −2970.32 1326.78i −0.0680002 0.0303744i
\(210\) 0 0
\(211\) −7586.62 4380.14i −0.170405 0.0983836i 0.412372 0.911016i \(-0.364701\pi\)
−0.582777 + 0.812632i \(0.698034\pi\)
\(212\) 0 0
\(213\) 26208.5 + 45394.5i 0.577675 + 1.00056i
\(214\) 0 0
\(215\) 45740.8 + 79225.3i 0.989524 + 1.71391i
\(216\) 0 0
\(217\) 11463.8i 0.243450i
\(218\) 0 0
\(219\) 108423. 62598.0i 2.26065 1.30519i
\(220\) 0 0
\(221\) 3259.68i 0.0667406i
\(222\) 0 0
\(223\) 74100.7 + 42782.0i 1.49009 + 0.860304i 0.999936 0.0113323i \(-0.00360726\pi\)
0.490154 + 0.871636i \(0.336941\pi\)
\(224\) 0 0
\(225\) 72058.3 124809.i 1.42337 2.46535i
\(226\) 0 0
\(227\) 52738.6i 1.02347i −0.859142 0.511737i \(-0.829002\pi\)
0.859142 0.511737i \(-0.170998\pi\)
\(228\) 0 0
\(229\) 27552.9 0.525407 0.262704 0.964877i \(-0.415386\pi\)
0.262704 + 0.964877i \(0.415386\pi\)
\(230\) 0 0
\(231\) 6324.69 + 3651.56i 0.118526 + 0.0684313i
\(232\) 0 0
\(233\) 29960.8 51893.6i 0.551876 0.955877i −0.446263 0.894902i \(-0.647246\pi\)
0.998139 0.0609754i \(-0.0194211\pi\)
\(234\) 0 0
\(235\) −138757. −2.51258
\(236\) 0 0
\(237\) −64502.0 111721.i −1.14836 1.98901i
\(238\) 0 0
\(239\) 18546.4 0.324686 0.162343 0.986734i \(-0.448095\pi\)
0.162343 + 0.986734i \(0.448095\pi\)
\(240\) 0 0
\(241\) 51352.4 29648.3i 0.884152 0.510465i 0.0121266 0.999926i \(-0.496140\pi\)
0.872025 + 0.489461i \(0.162807\pi\)
\(242\) 0 0
\(243\) −24374.1 + 14072.4i −0.412777 + 0.238317i
\(244\) 0 0
\(245\) 8906.43 15426.4i 0.148379 0.256999i
\(246\) 0 0
\(247\) 2237.01 232.379i 0.0366669 0.00380893i
\(248\) 0 0
\(249\) 36261.1 + 20935.3i 0.584847 + 0.337661i
\(250\) 0 0
\(251\) −13531.1 23436.6i −0.214776 0.372003i 0.738427 0.674333i \(-0.235569\pi\)
−0.953203 + 0.302330i \(0.902235\pi\)
\(252\) 0 0
\(253\) 1838.60 + 3184.56i 0.0287242 + 0.0497517i
\(254\) 0 0
\(255\) 316655.i 4.86974i
\(256\) 0 0
\(257\) −43689.8 + 25224.3i −0.661475 + 0.381903i −0.792839 0.609431i \(-0.791398\pi\)
0.131364 + 0.991334i \(0.458064\pi\)
\(258\) 0 0
\(259\) 65303.4i 0.973500i
\(260\) 0 0
\(261\) 175956. + 101588.i 2.58300 + 1.49129i
\(262\) 0 0
\(263\) −1642.93 + 2845.63i −0.0237523 + 0.0411403i −0.877657 0.479289i \(-0.840895\pi\)
0.853905 + 0.520429i \(0.174228\pi\)
\(264\) 0 0
\(265\) 221611.i 3.15573i
\(266\) 0 0
\(267\) −104995. −1.47281
\(268\) 0 0
\(269\) −14796.8 8542.92i −0.204486 0.118060i 0.394260 0.918999i \(-0.371001\pi\)
−0.598746 + 0.800939i \(0.704334\pi\)
\(270\) 0 0
\(271\) −10303.0 + 17845.4i −0.140290 + 0.242989i −0.927606 0.373561i \(-0.878137\pi\)
0.787316 + 0.616550i \(0.211470\pi\)
\(272\) 0 0
\(273\) −5048.94 −0.0677447
\(274\) 0 0
\(275\) −4340.27 7517.56i −0.0573919 0.0994058i
\(276\) 0 0
\(277\) −24421.2 −0.318278 −0.159139 0.987256i \(-0.550872\pi\)
−0.159139 + 0.987256i \(0.550872\pi\)
\(278\) 0 0
\(279\) −27833.0 + 16069.4i −0.357562 + 0.206439i
\(280\) 0 0
\(281\) −6075.51 + 3507.70i −0.0769431 + 0.0444231i −0.537978 0.842959i \(-0.680812\pi\)
0.461035 + 0.887382i \(0.347478\pi\)
\(282\) 0 0
\(283\) 47805.3 82801.3i 0.596903 1.03387i −0.396373 0.918090i \(-0.629731\pi\)
0.993275 0.115776i \(-0.0369354\pi\)
\(284\) 0 0
\(285\) 217310. 22574.0i 2.67541 0.277919i
\(286\) 0 0
\(287\) 35562.5 + 20532.0i 0.431746 + 0.249269i
\(288\) 0 0
\(289\) −95118.2 164750.i −1.13885 1.97255i
\(290\) 0 0
\(291\) 92227.4 + 159743.i 1.08912 + 1.88640i
\(292\) 0 0
\(293\) 110120.i 1.28272i 0.767239 + 0.641361i \(0.221630\pi\)
−0.767239 + 0.641361i \(0.778370\pi\)
\(294\) 0 0
\(295\) −84916.4 + 49026.5i −0.975770 + 0.563361i
\(296\) 0 0
\(297\) 9389.52i 0.106446i
\(298\) 0 0
\(299\) −2201.61 1271.10i −0.0246263 0.0142180i
\(300\) 0 0
\(301\) 61250.4 106089.i 0.676046 1.17095i
\(302\) 0 0
\(303\) 142899.i 1.55648i
\(304\) 0 0
\(305\) −117278. −1.26072
\(306\) 0 0
\(307\) 37032.8 + 21380.9i 0.392925 + 0.226855i 0.683427 0.730019i \(-0.260489\pi\)
−0.290502 + 0.956874i \(0.593822\pi\)
\(308\) 0 0
\(309\) −118678. + 205557.i −1.24295 + 2.15286i
\(310\) 0 0
\(311\) −62672.8 −0.647976 −0.323988 0.946061i \(-0.605024\pi\)
−0.323988 + 0.946061i \(0.605024\pi\)
\(312\) 0 0
\(313\) −87372.5 151334.i −0.891838 1.54471i −0.837670 0.546177i \(-0.816083\pi\)
−0.0541679 0.998532i \(-0.517251\pi\)
\(314\) 0 0
\(315\) −318197. −3.20682
\(316\) 0 0
\(317\) 26126.0 15083.9i 0.259989 0.150105i −0.364341 0.931266i \(-0.618706\pi\)
0.624329 + 0.781161i \(0.285372\pi\)
\(318\) 0 0
\(319\) 10598.3 6118.95i 0.104149 0.0601306i
\(320\) 0 0
\(321\) −8079.42 + 13994.0i −0.0784098 + 0.135810i
\(322\) 0 0
\(323\) 152943. 110837.i 1.46597 1.06238i
\(324\) 0 0
\(325\) 5197.19 + 3000.60i 0.0492042 + 0.0284081i
\(326\) 0 0
\(327\) −23696.1 41042.9i −0.221606 0.383833i
\(328\) 0 0
\(329\) 92903.3 + 160913.i 0.858301 + 1.48662i
\(330\) 0 0
\(331\) 63185.8i 0.576718i −0.957522 0.288359i \(-0.906890\pi\)
0.957522 0.288359i \(-0.0931096\pi\)
\(332\) 0 0
\(333\) 158550. 91538.9i 1.42981 0.825501i
\(334\) 0 0
\(335\) 67841.9i 0.604517i
\(336\) 0 0
\(337\) −125600. 72515.1i −1.10593 0.638511i −0.168160 0.985760i \(-0.553783\pi\)
−0.937773 + 0.347249i \(0.887116\pi\)
\(338\) 0 0
\(339\) −34620.6 + 59964.6i −0.301256 + 0.521790i
\(340\) 0 0
\(341\) 1935.81i 0.0166477i
\(342\) 0 0
\(343\) 104280. 0.886363
\(344\) 0 0
\(345\) −213871. 123478.i −1.79686 1.03742i
\(346\) 0 0
\(347\) 1089.43 1886.95i 0.00904776 0.0156712i −0.861466 0.507815i \(-0.830453\pi\)
0.870514 + 0.492144i \(0.163787\pi\)
\(348\) 0 0
\(349\) 88628.2 0.727647 0.363824 0.931468i \(-0.381471\pi\)
0.363824 + 0.931468i \(0.381471\pi\)
\(350\) 0 0
\(351\) 3245.68 + 5621.68i 0.0263446 + 0.0456302i
\(352\) 0 0
\(353\) 197267. 1.58309 0.791545 0.611111i \(-0.209277\pi\)
0.791545 + 0.611111i \(0.209277\pi\)
\(354\) 0 0
\(355\) −119131. + 68780.2i −0.945295 + 0.545766i
\(356\) 0 0
\(357\) −367216. + 212012.i −2.88128 + 1.66351i
\(358\) 0 0
\(359\) −69978.8 + 121207.i −0.542972 + 0.940455i 0.455760 + 0.890103i \(0.349368\pi\)
−0.998732 + 0.0503521i \(0.983966\pi\)
\(360\) 0 0
\(361\) 86966.8 + 97058.4i 0.667328 + 0.744764i
\(362\) 0 0
\(363\) 191482. + 110552.i 1.45316 + 0.838983i
\(364\) 0 0
\(365\) 164279. + 284539.i 1.23309 + 2.13578i
\(366\) 0 0
\(367\) 16609.2 + 28768.0i 0.123315 + 0.213588i 0.921073 0.389390i \(-0.127314\pi\)
−0.797758 + 0.602978i \(0.793981\pi\)
\(368\) 0 0
\(369\) 115123.i 0.845491i
\(370\) 0 0
\(371\) 256997. 148377.i 1.86715 1.07800i
\(372\) 0 0
\(373\) 23103.2i 0.166056i −0.996547 0.0830281i \(-0.973541\pi\)
0.996547 0.0830281i \(-0.0264591\pi\)
\(374\) 0 0
\(375\) 177293. + 102360.i 1.26075 + 0.727896i
\(376\) 0 0
\(377\) −4230.28 + 7327.06i −0.0297636 + 0.0515521i
\(378\) 0 0
\(379\) 32221.6i 0.224321i 0.993690 + 0.112160i \(0.0357770\pi\)
−0.993690 + 0.112160i \(0.964223\pi\)
\(380\) 0 0
\(381\) −296376. −2.04170
\(382\) 0 0
\(383\) −195003. 112585.i −1.32936 0.767509i −0.344163 0.938910i \(-0.611837\pi\)
−0.985201 + 0.171401i \(0.945171\pi\)
\(384\) 0 0
\(385\) −9582.94 + 16598.1i −0.0646513 + 0.111979i
\(386\) 0 0
\(387\) −343431. −2.29307
\(388\) 0 0
\(389\) 14122.8 + 24461.4i 0.0933302 + 0.161653i 0.908910 0.416991i \(-0.136915\pi\)
−0.815580 + 0.578644i \(0.803582\pi\)
\(390\) 0 0
\(391\) −213502. −1.39652
\(392\) 0 0
\(393\) 241967. 139700.i 1.56665 0.904504i
\(394\) 0 0
\(395\) 293193. 169275.i 1.87914 1.08492i
\(396\) 0 0
\(397\) 97341.9 168601.i 0.617616 1.06974i −0.372303 0.928111i \(-0.621432\pi\)
0.989919 0.141632i \(-0.0452349\pi\)
\(398\) 0 0
\(399\) −171676. 236895.i −1.07836 1.48802i
\(400\) 0 0
\(401\) −20526.9 11851.2i −0.127654 0.0737012i 0.434813 0.900521i \(-0.356814\pi\)
−0.562467 + 0.826820i \(0.690148\pi\)
\(402\) 0 0
\(403\) −669.151 1159.00i −0.00412016 0.00713633i
\(404\) 0 0
\(405\) 73812.9 + 127848.i 0.450010 + 0.779441i
\(406\) 0 0
\(407\) 11027.3i 0.0665702i
\(408\) 0 0
\(409\) 142037. 82004.9i 0.849090 0.490222i −0.0112539 0.999937i \(-0.503582\pi\)
0.860344 + 0.509715i \(0.170249\pi\)
\(410\) 0 0
\(411\) 289381.i 1.71311i
\(412\) 0 0
\(413\) 113710. + 65650.3i 0.666649 + 0.384890i
\(414\) 0 0
\(415\) −54941.5 + 95161.5i −0.319010 + 0.552542i
\(416\) 0 0
\(417\) 228752.i 1.31550i
\(418\) 0 0
\(419\) 60552.8 0.344910 0.172455 0.985017i \(-0.444830\pi\)
0.172455 + 0.985017i \(0.444830\pi\)
\(420\) 0 0
\(421\) 116808. + 67439.2i 0.659036 + 0.380494i 0.791909 0.610638i \(-0.209087\pi\)
−0.132874 + 0.991133i \(0.542420\pi\)
\(422\) 0 0
\(423\) 260454. 451120.i 1.45563 2.52122i
\(424\) 0 0
\(425\) 503999. 2.79030
\(426\) 0 0
\(427\) 78522.4 + 136005.i 0.430663 + 0.745931i
\(428\) 0 0
\(429\) 852.576 0.00463253
\(430\) 0 0
\(431\) −93228.7 + 53825.6i −0.501874 + 0.289757i −0.729487 0.683994i \(-0.760241\pi\)
0.227613 + 0.973752i \(0.426908\pi\)
\(432\) 0 0
\(433\) 82985.3 47911.6i 0.442614 0.255543i −0.262092 0.965043i \(-0.584412\pi\)
0.704706 + 0.709500i \(0.251079\pi\)
\(434\) 0 0
\(435\) −410941. + 711771.i −2.17171 + 3.76151i
\(436\) 0 0
\(437\) −15220.3 146519.i −0.0797005 0.767241i
\(438\) 0 0
\(439\) −220263. 127169.i −1.14291 0.659859i −0.195760 0.980652i \(-0.562717\pi\)
−0.947149 + 0.320793i \(0.896051\pi\)
\(440\) 0 0
\(441\) 33435.6 + 57912.2i 0.171922 + 0.297778i
\(442\) 0 0
\(443\) −146103. 253058.i −0.744479 1.28947i −0.950438 0.310914i \(-0.899365\pi\)
0.205959 0.978561i \(-0.433969\pi\)
\(444\) 0 0
\(445\) 275544.i 1.39146i
\(446\) 0 0
\(447\) 166176. 95941.5i 0.831672 0.480166i
\(448\) 0 0
\(449\) 123714.i 0.613657i 0.951765 + 0.306828i \(0.0992677\pi\)
−0.951765 + 0.306828i \(0.900732\pi\)
\(450\) 0 0
\(451\) −6005.16 3467.08i −0.0295238 0.0170455i
\(452\) 0 0
\(453\) −107209. + 185691.i −0.522437 + 0.904887i
\(454\) 0 0
\(455\) 13250.2i 0.0640027i
\(456\) 0 0
\(457\) −249320. −1.19378 −0.596890 0.802323i \(-0.703597\pi\)
−0.596890 + 0.802323i \(0.703597\pi\)
\(458\) 0 0
\(459\) 472125. + 272582.i 2.24095 + 1.29381i
\(460\) 0 0
\(461\) 63180.9 109432.i 0.297292 0.514925i −0.678223 0.734856i \(-0.737250\pi\)
0.975516 + 0.219931i \(0.0705831\pi\)
\(462\) 0 0
\(463\) −86701.1 −0.404448 −0.202224 0.979339i \(-0.564817\pi\)
−0.202224 + 0.979339i \(0.564817\pi\)
\(464\) 0 0
\(465\) −65003.3 112589.i −0.300628 0.520703i
\(466\) 0 0
\(467\) 226847. 1.04016 0.520078 0.854119i \(-0.325903\pi\)
0.520078 + 0.854119i \(0.325903\pi\)
\(468\) 0 0
\(469\) −78674.5 + 45422.7i −0.357675 + 0.206504i
\(470\) 0 0
\(471\) −164354. + 94889.6i −0.740863 + 0.427737i
\(472\) 0 0
\(473\) −10342.9 + 17914.4i −0.0462295 + 0.0800719i
\(474\) 0 0
\(475\) 35929.6 + 345878.i 0.159245 + 1.53298i
\(476\) 0 0
\(477\) −720489. 415975.i −3.16658 1.82823i
\(478\) 0 0
\(479\) 130496. + 226026.i 0.568757 + 0.985117i 0.996689 + 0.0813056i \(0.0259090\pi\)
−0.427932 + 0.903811i \(0.640758\pi\)
\(480\) 0 0
\(481\) 3811.80 + 6602.24i 0.0164756 + 0.0285365i
\(482\) 0 0
\(483\) 330694.i 1.41753i
\(484\) 0 0
\(485\) −419219. + 242036.i −1.78220 + 1.02896i
\(486\) 0 0
\(487\) 402444.i 1.69687i 0.529303 + 0.848433i \(0.322453\pi\)
−0.529303 + 0.848433i \(0.677547\pi\)
\(488\) 0 0
\(489\) −390374. 225383.i −1.63254 0.942547i
\(490\) 0 0
\(491\) 8446.75 14630.2i 0.0350370 0.0606858i −0.847975 0.530036i \(-0.822178\pi\)
0.883012 + 0.469350i \(0.155512\pi\)
\(492\) 0 0
\(493\) 710542.i 2.92345i
\(494\) 0 0
\(495\) 53731.5 0.219290
\(496\) 0 0
\(497\) 159525. + 92101.9i 0.645828 + 0.372869i
\(498\) 0 0
\(499\) 96752.2 167580.i 0.388562 0.673009i −0.603695 0.797216i \(-0.706305\pi\)
0.992256 + 0.124207i \(0.0396387\pi\)
\(500\) 0 0
\(501\) 366186. 1.45890
\(502\) 0 0
\(503\) −7000.74 12125.6i −0.0276699 0.0479257i 0.851859 0.523771i \(-0.175475\pi\)
−0.879529 + 0.475846i \(0.842142\pi\)
\(504\) 0 0
\(505\) −375016. −1.47051
\(506\) 0 0
\(507\) 375107. 216568.i 1.45928 0.842516i
\(508\) 0 0
\(509\) 68947.5 39806.9i 0.266123 0.153646i −0.361001 0.932565i \(-0.617565\pi\)
0.627125 + 0.778919i \(0.284232\pi\)
\(510\) 0 0
\(511\) 219982. 381020.i 0.842451 1.45917i
\(512\) 0 0
\(513\) −153407. + 343436.i −0.582921 + 1.30500i
\(514\) 0 0
\(515\) −539451. 311452.i −2.03394 1.17430i
\(516\) 0 0
\(517\) −15687.9 27172.2i −0.0586925 0.101658i
\(518\) 0 0
\(519\) 232373. + 402482.i 0.862683 + 1.49421i
\(520\) 0 0
\(521\) 358410.i 1.32040i 0.751090 + 0.660200i \(0.229528\pi\)
−0.751090 + 0.660200i \(0.770472\pi\)
\(522\) 0 0
\(523\) 194918. 112536.i 0.712606 0.411423i −0.0994192 0.995046i \(-0.531698\pi\)
0.812025 + 0.583622i \(0.198365\pi\)
\(524\) 0 0
\(525\) 780647.i 2.83228i
\(526\) 0 0
\(527\) −97336.5 56197.3i −0.350473 0.202346i
\(528\) 0 0
\(529\) 56666.2 98148.8i 0.202494 0.350731i
\(530\) 0 0
\(531\) 368101.i 1.30550i
\(532\) 0 0
\(533\) 4793.87 0.0168745
\(534\) 0 0
\(535\) −36725.0 21203.2i −0.128308 0.0740787i
\(536\) 0 0
\(537\) −240109. + 415881.i −0.832644 + 1.44218i
\(538\) 0 0
\(539\) 4027.84 0.0138642
\(540\) 0 0
\(541\) 122595. + 212341.i 0.418870 + 0.725504i 0.995826 0.0912713i \(-0.0290930\pi\)
−0.576956 + 0.816775i \(0.695760\pi\)
\(542\) 0 0
\(543\) 51427.9 0.174421
\(544\) 0 0
\(545\) 107711. 62186.8i 0.362631 0.209365i
\(546\) 0 0
\(547\) 381981. 220537.i 1.27664 0.737067i 0.300409 0.953811i \(-0.402877\pi\)
0.976229 + 0.216744i \(0.0695436\pi\)
\(548\) 0 0
\(549\) 220137. 381289.i 0.730380 1.26506i
\(550\) 0 0
\(551\) −487622. + 50653.8i −1.60613 + 0.166843i
\(552\) 0 0
\(553\) −392608. 226673.i −1.28384 0.741223i
\(554\) 0 0
\(555\) 370290. + 641360.i 1.20214 + 2.08217i
\(556\) 0 0
\(557\) −161284. 279353.i −0.519855 0.900415i −0.999734 0.0230804i \(-0.992653\pi\)
0.479879 0.877335i \(-0.340681\pi\)
\(558\) 0 0
\(559\) 14300.9i 0.0457657i
\(560\) 0 0
\(561\) 62009.0 35800.9i 0.197029 0.113754i
\(562\) 0 0
\(563\) 35055.3i 0.110595i 0.998470 + 0.0552977i \(0.0176108\pi\)
−0.998470 + 0.0552977i \(0.982389\pi\)
\(564\) 0 0
\(565\) −157368. 90856.2i −0.492968 0.284615i
\(566\) 0 0
\(567\) 98841.2 171198.i 0.307448 0.532516i
\(568\) 0 0
\(569\) 510214.i 1.57590i 0.615742 + 0.787948i \(0.288856\pi\)
−0.615742 + 0.787948i \(0.711144\pi\)
\(570\) 0 0
\(571\) −191075. −0.586047 −0.293023 0.956105i \(-0.594661\pi\)
−0.293023 + 0.956105i \(0.594661\pi\)
\(572\) 0 0
\(573\) 93909.0 + 54218.4i 0.286021 + 0.165134i
\(574\) 0 0
\(575\) 196533. 340404.i 0.594428 1.02958i
\(576\) 0 0
\(577\) −397878. −1.19508 −0.597542 0.801838i \(-0.703856\pi\)
−0.597542 + 0.801838i \(0.703856\pi\)
\(578\) 0 0
\(579\) 122062. + 211418.i 0.364103 + 0.630645i
\(580\) 0 0
\(581\) 147142. 0.435897
\(582\) 0 0
\(583\) −43397.0 + 25055.3i −0.127680 + 0.0737161i
\(584\) 0 0
\(585\) −32170.0 + 18573.4i −0.0940026 + 0.0542724i
\(586\) 0 0
\(587\) −294687. + 510414.i −0.855235 + 1.48131i 0.0211920 + 0.999775i \(0.493254\pi\)
−0.876427 + 0.481535i \(0.840079\pi\)
\(588\) 0 0
\(589\) 31627.3 70805.2i 0.0911658 0.204096i
\(590\) 0 0
\(591\) 349115. + 201562.i 0.999526 + 0.577077i
\(592\) 0 0
\(593\) −153979. 266700.i −0.437878 0.758426i 0.559648 0.828730i \(-0.310936\pi\)
−0.997526 + 0.0703041i \(0.977603\pi\)
\(594\) 0 0
\(595\) −556393. 963701.i −1.57162 2.72213i
\(596\) 0 0
\(597\) 71587.0i 0.200856i
\(598\) 0 0
\(599\) 529620. 305776.i 1.47608 0.852217i 0.476447 0.879203i \(-0.341924\pi\)
0.999636 + 0.0269863i \(0.00859106\pi\)
\(600\) 0 0
\(601\) 374521.i 1.03688i 0.855115 + 0.518439i \(0.173487\pi\)
−0.855115 + 0.518439i \(0.826513\pi\)
\(602\) 0 0
\(603\) 220564. + 127343.i 0.606596 + 0.350218i
\(604\) 0 0
\(605\) −290126. + 502513.i −0.792640 + 1.37289i
\(606\) 0 0
\(607\) 66013.3i 0.179165i 0.995979 + 0.0895827i \(0.0285533\pi\)
−0.995979 + 0.0895827i \(0.971447\pi\)
\(608\) 0 0
\(609\) 1.10056e6 2.96743
\(610\) 0 0
\(611\) 18785.2 + 10845.7i 0.0503193 + 0.0290518i
\(612\) 0 0
\(613\) 18704.7 32397.6i 0.0497772 0.0862167i −0.840063 0.542489i \(-0.817482\pi\)
0.889840 + 0.456272i \(0.150816\pi\)
\(614\) 0 0
\(615\) 465690. 1.23125
\(616\) 0 0
\(617\) 56603.6 + 98040.4i 0.148687 + 0.257534i 0.930743 0.365675i \(-0.119162\pi\)
−0.782055 + 0.623209i \(0.785829\pi\)
\(618\) 0 0
\(619\) 313267. 0.817586 0.408793 0.912627i \(-0.365950\pi\)
0.408793 + 0.912627i \(0.365950\pi\)
\(620\) 0 0
\(621\) 368207. 212585.i 0.954793 0.551250i
\(622\) 0 0
\(623\) −319541. + 184487.i −0.823286 + 0.475324i
\(624\) 0 0
\(625\) 32392.2 56105.0i 0.0829241 0.143629i
\(626\) 0 0
\(627\) 28989.6 + 40002.6i 0.0737406 + 0.101754i
\(628\) 0 0
\(629\) 554475. + 320126.i 1.40146 + 0.809133i
\(630\) 0 0
\(631\) −246290. 426586.i −0.618568 1.07139i −0.989747 0.142830i \(-0.954380\pi\)
0.371180 0.928561i \(-0.378953\pi\)
\(632\) 0 0
\(633\) 66516.4 + 115210.i 0.166005 + 0.287529i
\(634\) 0 0
\(635\) 777791.i 1.92893i
\(636\) 0 0
\(637\) −2411.54 + 1392.30i −0.00594314 + 0.00343127i
\(638\) 0 0
\(639\) 516415.i 1.26473i
\(640\) 0 0
\(641\) −168128. 97068.9i −0.409190 0.236246i 0.281252 0.959634i \(-0.409250\pi\)
−0.690441 + 0.723388i \(0.742584\pi\)
\(642\) 0 0
\(643\) −43796.5 + 75857.7i −0.105930 + 0.183475i −0.914118 0.405449i \(-0.867115\pi\)
0.808188 + 0.588925i \(0.200448\pi\)
\(644\) 0 0
\(645\) 1.38923e6i 3.33930i
\(646\) 0 0
\(647\) −636027. −1.51938 −0.759690 0.650285i \(-0.774650\pi\)
−0.759690 + 0.650285i \(0.774650\pi\)
\(648\) 0 0
\(649\) −19201.3 11085.9i −0.0455869 0.0263196i
\(650\) 0 0
\(651\) −87044.4 + 150765.i −0.205390 + 0.355745i
\(652\) 0 0
\(653\) 46609.6 0.109307 0.0546536 0.998505i \(-0.482595\pi\)
0.0546536 + 0.998505i \(0.482595\pi\)
\(654\) 0 0
\(655\) 366620. + 635004.i 0.854542 + 1.48011i
\(656\) 0 0
\(657\) −1.23344e6 −2.85750
\(658\) 0 0
\(659\) −577704. + 333537.i −1.33025 + 0.768022i −0.985338 0.170612i \(-0.945426\pi\)
−0.344915 + 0.938634i \(0.612092\pi\)
\(660\) 0 0
\(661\) 387846. 223923.i 0.887681 0.512503i 0.0144975 0.999895i \(-0.495385\pi\)
0.873183 + 0.487392i \(0.162052\pi\)
\(662\) 0 0
\(663\) −24750.6 + 42869.3i −0.0563066 + 0.0975258i
\(664\) 0 0
\(665\) 621692. 450536.i 1.40583 1.01879i
\(666\) 0 0
\(667\) 479906. + 277074.i 1.07871 + 0.622793i
\(668\) 0 0
\(669\) −649685. 1.12529e6i −1.45161 2.51427i
\(670\) 0 0
\(671\) −13259.5 22966.1i −0.0294497 0.0510084i
\(672\) 0 0
\(673\) 146080.i 0.322522i 0.986912 + 0.161261i \(0.0515561\pi\)
−0.986912 + 0.161261i \(0.948444\pi\)
\(674\) 0 0
\(675\) −869202. + 501834.i −1.90771 + 1.10142i
\(676\) 0 0
\(677\) 383372.i 0.836456i 0.908342 + 0.418228i \(0.137349\pi\)
−0.908342 + 0.418228i \(0.862651\pi\)
\(678\) 0 0
\(679\) 561367. + 324105.i 1.21761 + 0.702986i
\(680\) 0 0
\(681\) −400442. + 693586.i −0.863466 + 1.49557i
\(682\) 0 0
\(683\) 828560.i 1.77616i −0.459687 0.888081i \(-0.652038\pi\)
0.459687 0.888081i \(-0.347962\pi\)
\(684\) 0 0
\(685\) 759434. 1.61849
\(686\) 0 0
\(687\) −362359. 209208.i −0.767760 0.443266i
\(688\) 0 0
\(689\) 17321.7 30002.1i 0.0364882 0.0631995i
\(690\) 0 0
\(691\) −734197. −1.53765 −0.768824 0.639461i \(-0.779158\pi\)
−0.768824 + 0.639461i \(0.779158\pi\)
\(692\) 0 0
\(693\) −35975.3 62311.1i −0.0749097 0.129747i
\(694\) 0 0
\(695\) 600322. 1.24284
\(696\) 0 0
\(697\) 348664. 201301.i 0.717699 0.414363i
\(698\) 0 0
\(699\) −788053. + 454982.i −1.61288 + 0.931194i
\(700\) 0 0
\(701\) −86328.7 + 149526.i −0.175679 + 0.304284i −0.940396 0.340082i \(-0.889545\pi\)
0.764717 + 0.644366i \(0.222879\pi\)
\(702\) 0 0
\(703\) −180164. + 403340.i −0.364551 + 0.816132i
\(704\) 0 0
\(705\) 1.82485e6 + 1.05358e6i 3.67155 + 2.11977i
\(706\) 0 0
\(707\) 251088. + 434896.i 0.502327 + 0.870056i
\(708\) 0 0
\(709\) 81982.8 + 141998.i 0.163091 + 0.282482i 0.935976 0.352064i \(-0.114520\pi\)
−0.772885 + 0.634547i \(0.781187\pi\)
\(710\) 0 0
\(711\) 1.27095e6i 2.51414i
\(712\) 0 0
\(713\) −75912.1 + 43827.9i −0.149325 + 0.0862127i
\(714\) 0 0
\(715\) 2237.45i 0.00437665i
\(716\) 0 0
\(717\) −243911. 140822.i −0.474453 0.273925i
\(718\) 0 0
\(719\) −309781. + 536556.i −0.599234 + 1.03790i 0.393700 + 0.919239i \(0.371195\pi\)
−0.992934 + 0.118665i \(0.962138\pi\)
\(720\) 0 0
\(721\) 834118.i 1.60456i
\(722\) 0 0
\(723\) −900475. −1.72264
\(724\) 0 0
\(725\) −1.13288e6 654069.i −2.15530 1.24436i
\(726\) 0 0
\(727\) −95900.8 + 166105.i −0.181449 + 0.314278i −0.942374 0.334561i \(-0.891412\pi\)
0.760925 + 0.648839i \(0.224745\pi\)
\(728\) 0 0
\(729\) 727449. 1.36882
\(730\) 0 0
\(731\) −600516. 1.04012e6i −1.12380 1.94648i
\(732\) 0 0
\(733\) 589533. 1.09724 0.548618 0.836073i \(-0.315154\pi\)
0.548618 + 0.836073i \(0.315154\pi\)
\(734\) 0 0
\(735\) −234264. + 135252.i −0.433642 + 0.250363i
\(736\) 0 0
\(737\) 13285.2 7670.19i 0.0244586 0.0141212i
\(738\) 0 0
\(739\) 415929. 720409.i 0.761605 1.31914i −0.180418 0.983590i \(-0.557745\pi\)
0.942023 0.335549i \(-0.108922\pi\)
\(740\) 0 0
\(741\) −31184.3 13929.4i −0.0567936 0.0253686i
\(742\) 0 0
\(743\) 79675.6 + 46000.7i 0.144327 + 0.0833273i 0.570425 0.821350i \(-0.306779\pi\)
−0.426098 + 0.904677i \(0.640112\pi\)
\(744\) 0 0
\(745\) 251783. + 436101.i 0.453643 + 0.785733i
\(746\) 0 0
\(747\) −206256. 357246.i −0.369628 0.640215i
\(748\) 0 0
\(749\) 56785.4i 0.101221i
\(750\) 0 0
\(751\) −807256. + 466069.i −1.43130 + 0.826362i −0.997220 0.0745111i \(-0.976260\pi\)
−0.434082 + 0.900874i \(0.642927\pi\)
\(752\) 0 0
\(753\) 410965.i 0.724794i
\(754\) 0 0
\(755\) −487316. 281352.i −0.854904 0.493579i
\(756\) 0 0
\(757\) 286555. 496329.i 0.500054 0.866119i −0.499946 0.866057i \(-0.666647\pi\)
1.00000 6.22515e-5i \(-1.98153e-5\pi\)
\(758\) 0 0
\(759\) 55841.8i 0.0969340i
\(760\) 0 0
\(761\) 250379. 0.432342 0.216171 0.976355i \(-0.430643\pi\)
0.216171 + 0.976355i \(0.430643\pi\)
\(762\) 0 0
\(763\) −144233. 83272.8i −0.247751 0.143039i
\(764\) 0 0
\(765\) −1.55985e6 + 2.70173e6i −2.66538 + 4.61657i
\(766\) 0 0
\(767\) 15328.2 0.0260556
\(768\) 0 0
\(769\) −330979. 573272.i −0.559690 0.969411i −0.997522 0.0703546i \(-0.977587\pi\)
0.437832 0.899057i \(-0.355746\pi\)
\(770\) 0 0
\(771\) 766109. 1.28879
\(772\) 0 0
\(773\) −519862. + 300143.i −0.870020 + 0.502306i −0.867355 0.497690i \(-0.834182\pi\)
−0.00266520 + 0.999996i \(0.500848\pi\)
\(774\) 0 0
\(775\) 179201. 103461.i 0.298357 0.172256i
\(776\) 0 0
\(777\) 495846. 858831.i 0.821306 1.42254i
\(778\) 0 0
\(779\) 163003. + 224927.i 0.268609 + 0.370652i
\(780\) 0 0
\(781\) −26937.8 15552.5i −0.0441632 0.0254976i
\(782\) 0 0
\(783\) −707490. 1.22541e6i −1.15398 1.99875i
\(784\) 0 0
\(785\) −249023. 431320.i −0.404110 0.699940i
\(786\) 0 0
\(787\) 51876.0i 0.0837563i 0.999123 + 0.0418781i \(0.0133341\pi\)
−0.999123 + 0.0418781i \(0.986666\pi\)
\(788\) 0 0
\(789\) 43213.5 24949.3i 0.0694170 0.0400779i
\(790\) 0 0
\(791\) 243327.i 0.388899i
\(792\) 0 0
\(793\) 15877.4 + 9166.81i 0.0252483 + 0.0145771i
\(794\) 0 0
\(795\) 1.68268e6 2.91449e6i 2.66237 4.61136i
\(796\) 0 0
\(797\) 532921.i 0.838969i −0.907762 0.419485i \(-0.862211\pi\)
0.907762 0.419485i \(-0.137789\pi\)
\(798\) 0 0
\(799\) 1.82170e6 2.85354
\(800\) 0 0
\(801\) 895833. + 517209.i 1.39625 + 0.806123i
\(802\) 0 0
\(803\) −37146.6 + 64339.8i −0.0576087 + 0.0997812i
\(804\) 0 0
\(805\) −867855. −1.33923
\(806\) 0 0
\(807\) 129732. + 224703.i 0.199205 + 0.345034i
\(808\) 0 0
\(809\) −253863. −0.387884 −0.193942 0.981013i \(-0.562127\pi\)
−0.193942 + 0.981013i \(0.562127\pi\)
\(810\) 0 0
\(811\) 45687.4 26377.7i 0.0694632 0.0401046i −0.464866 0.885381i \(-0.653898\pi\)
0.534329 + 0.845276i \(0.320564\pi\)
\(812\) 0 0
\(813\) 270998. 156461.i 0.410002 0.236715i
\(814\) 0 0
\(815\) 591481. 1.02448e6i 0.890483 1.54236i
\(816\) 0 0
\(817\) 670994. 486264.i 1.00525 0.728498i
\(818\) 0 0
\(819\) 43078.2 + 24871.2i 0.0642228 + 0.0370791i
\(820\) 0 0
\(821\) −432787. 749609.i −0.642078 1.11211i −0.984968 0.172737i \(-0.944739\pi\)
0.342890 0.939376i \(-0.388594\pi\)
\(822\) 0 0
\(823\) 564220. + 977258.i 0.833008 + 1.44281i 0.895642 + 0.444775i \(0.146716\pi\)
−0.0626344 + 0.998037i \(0.519950\pi\)
\(824\) 0 0
\(825\) 131822.i 0.193678i
\(826\) 0 0
\(827\) −878302. + 507088.i −1.28420 + 0.741434i −0.977614 0.210409i \(-0.932521\pi\)
−0.306587 + 0.951843i \(0.599187\pi\)
\(828\) 0 0
\(829\) 561580.i 0.817152i −0.912724 0.408576i \(-0.866025\pi\)
0.912724 0.408576i \(-0.133975\pi\)
\(830\) 0 0
\(831\) 321172. + 185429.i 0.465089 + 0.268519i
\(832\) 0 0
\(833\) −116930. + 202528.i −0.168514 + 0.291874i
\(834\) 0 0
\(835\) 960998.i 1.37832i
\(836\) 0 0
\(837\) 223824. 0.319488
\(838\) 0 0
\(839\) −647385. 373768.i −0.919685 0.530980i −0.0361502 0.999346i \(-0.511509\pi\)
−0.883534 + 0.468366i \(0.844843\pi\)
\(840\) 0 0
\(841\) 568472. 984622.i 0.803742 1.39212i
\(842\) 0 0
\(843\) 106535. 0.149913
\(844\) 0 0
\(845\) 568348. + 984408.i 0.795978 + 1.37867i
\(846\) 0 0
\(847\) 777002. 1.08307
\(848\) 0 0
\(849\) −1.25741e6 + 725968.i −1.74447 + 1.00717i
\(850\) 0 0
\(851\) 432432. 249664.i 0.597115 0.344745i
\(852\) 0 0
\(853\) −190927. + 330696.i −0.262404 + 0.454496i −0.966880 0.255231i \(-0.917849\pi\)
0.704477 + 0.709727i \(0.251182\pi\)
\(854\) 0 0
\(855\) −1.96531e6 877869.i −2.68844 1.20087i
\(856\) 0 0
\(857\) 1.10306e6 + 636854.i 1.50189 + 0.867119i 0.999998 + 0.00219068i \(0.000697317\pi\)
0.501896 + 0.864928i \(0.332636\pi\)
\(858\) 0 0
\(859\) 235569. + 408017.i 0.319250 + 0.552957i 0.980332 0.197357i \(-0.0632357\pi\)
−0.661082 + 0.750314i \(0.729902\pi\)
\(860\) 0 0
\(861\) −311797. 540049.i −0.420597 0.728495i
\(862\) 0 0
\(863\) 25220.7i 0.0338638i 0.999857 + 0.0169319i \(0.00538985\pi\)
−0.999857 + 0.0169319i \(0.994610\pi\)
\(864\) 0 0
\(865\) −1.05625e6 + 609827.i −1.41168 + 0.815031i
\(866\) 0 0
\(867\) 2.88892e6i 3.84323i
\(868\) 0 0
\(869\) 66296.8 + 38276.5i 0.0877916 + 0.0506865i
\(870\) 0 0
\(871\) −5302.71 + 9184.57i −0.00698975 + 0.0121066i
\(872\) 0 0
\(873\) 1.81726e6i 2.38445i
\(874\) 0 0
\(875\) 719428. 0.939662
\(876\) 0 0
\(877\) 883407. + 510035.i 1.14858 + 0.663134i 0.948541 0.316654i \(-0.102559\pi\)
0.200040 + 0.979788i \(0.435893\pi\)
\(878\) 0 0
\(879\) 836140. 1.44824e6i 1.08218 1.87440i
\(880\) 0 0
\(881\) −1.23116e6 −1.58621 −0.793107 0.609082i \(-0.791538\pi\)
−0.793107 + 0.609082i \(0.791538\pi\)
\(882\) 0 0
\(883\) −225170. 390006.i −0.288794 0.500207i 0.684728 0.728799i \(-0.259921\pi\)
−0.973522 + 0.228592i \(0.926588\pi\)
\(884\) 0 0
\(885\) 1.48903e6 1.90115
\(886\) 0 0
\(887\) 82480.8 47620.3i 0.104835 0.0605264i −0.446666 0.894701i \(-0.647389\pi\)
0.551501 + 0.834174i \(0.314055\pi\)
\(888\) 0 0
\(889\) −901985. + 520761.i −1.14129 + 0.658924i
\(890\) 0 0
\(891\) −16690.6 + 28908.9i −0.0210240 + 0.0364146i
\(892\) 0 0
\(893\) 129867. + 1.25017e6i 0.162853 + 1.56772i
\(894\) 0 0
\(895\) −1.09141e6 630128.i −1.36252 0.786652i
\(896\) 0 0
\(897\) 19302.9 + 33433.5i 0.0239904 + 0.0415525i
\(898\) 0 0
\(899\) 145861. + 252639.i 0.180476 + 0.312594i
\(900\) 0 0
\(901\) 2.90946e6i 3.58396i
\(902\) 0 0
\(903\) −1.61106e6 + 930144.i −1.97577 + 1.14071i
\(904\) 0 0
\(905\) 134964.i 0.164787i
\(906\) 0 0
\(907\) −780267. 450487.i −0.948481 0.547606i −0.0558723 0.998438i \(-0.517794\pi\)
−0.892609 + 0.450832i \(0.851127\pi\)
\(908\) 0 0
\(909\) 703923. 1.21923e6i 0.851918 1.47556i
\(910\) 0 0
\(911\) 317691.i 0.382797i 0.981512 + 0.191398i \(0.0613022\pi\)
−0.981512 + 0.191398i \(0.938698\pi\)
\(912\) 0 0
\(913\) −24846.7 −0.0298076
\(914\) 0 0
\(915\) 1.54237e6 + 890490.i 1.84225 + 1.06362i
\(916\) 0 0
\(917\) 490932. 850319.i 0.583825 1.01121i
\(918\) 0 0
\(919\) −371224. −0.439547 −0.219774 0.975551i \(-0.570532\pi\)
−0.219774 + 0.975551i \(0.570532\pi\)
\(920\) 0 0
\(921\) −324689. 562377.i −0.382779 0.662993i
\(922\) 0 0
\(923\) 21504.2 0.0252418
\(924\) 0 0
\(925\) −1.02081e6 + 589366.i −1.19306 + 0.688813i
\(926\) 0 0
\(927\) 2.02515e6 1.16922e6i 2.35667 1.36062i
\(928\) 0 0
\(929\) −31626.7 + 54779.0i −0.0366456 + 0.0634721i −0.883767 0.467928i \(-0.845001\pi\)
0.847121 + 0.531400i \(0.178334\pi\)
\(930\) 0 0
\(931\) −147324. 65807.1i −0.169971 0.0759230i
\(932\) 0 0
\(933\) 824235. + 475873.i 0.946865 + 0.546673i
\(934\) 0 0
\(935\) 93953.8 + 162733.i 0.107471 + 0.186145i
\(936\) 0 0
\(937\) −578958. 1.00279e6i −0.659429 1.14216i −0.980764 0.195199i \(-0.937465\pi\)
0.321335 0.946966i \(-0.395869\pi\)
\(938\) 0 0
\(939\) 2.65366e6i 3.00964i
\(940\) 0 0
\(941\) −295283. + 170482.i −0.333472 + 0.192530i −0.657381 0.753558i \(-0.728336\pi\)
0.323910 + 0.946088i \(0.395003\pi\)
\(942\) 0 0
\(943\) 313987.i 0.353093i
\(944\) 0 0
\(945\) 1.91912e6 + 1.10801e6i 2.14901 + 1.24073i
\(946\) 0 0
\(947\) −236902. + 410326.i −0.264161 + 0.457541i −0.967344 0.253469i \(-0.918428\pi\)
0.703182 + 0.711010i \(0.251762\pi\)
\(948\) 0 0
\(949\) 51361.9i 0.0570307i
\(950\) 0 0
\(951\) −458125. −0.506550
\(952\) 0 0
\(953\) 305054. + 176123.i 0.335885 + 0.193923i 0.658451 0.752624i \(-0.271212\pi\)
−0.322566 + 0.946547i \(0.604545\pi\)
\(954\) 0 0
\(955\) −142288. + 246449.i −0.156013 + 0.270222i
\(956\) 0 0
\(957\) −185844. −0.202920
\(958\) 0 0
\(959\) −508470. 880696.i −0.552877 0.957611i
\(960\) 0 0
\(961\) 877376. 0.950034
\(962\) 0 0
\(963\) 137869. 79598.8i 0.148667 0.0858329i
\(964\) 0 0
\(965\) −554833. + 320333.i −0.595810 + 0.343991i
\(966\) 0 0
\(967\) −342618. + 593431.i −0.366401 + 0.634626i −0.989000 0.147916i \(-0.952744\pi\)
0.622599 + 0.782541i \(0.286077\pi\)
\(968\) 0 0
\(969\) −2.85299e6 + 296367.i −3.03846 + 0.315633i
\(970\) 0 0
\(971\) −114126. 65890.5i −0.121045 0.0698851i 0.438255 0.898851i \(-0.355597\pi\)
−0.559300 + 0.828965i \(0.688930\pi\)
\(972\) 0 0
\(973\) −401939. 696179.i −0.424555 0.735352i
\(974\) 0 0
\(975\) −45566.9 78924.2i −0.0479336 0.0830235i
\(976\) 0 0
\(977\) 853791.i 0.894463i 0.894418 + 0.447231i \(0.147590\pi\)
−0.894418 + 0.447231i \(0.852410\pi\)
\(978\) 0 0
\(979\) 53958.5 31152.9i 0.0562981 0.0325038i
\(980\) 0 0
\(981\) 466910.i 0.485172i
\(982\) 0 0
\(983\) −63978.8 36938.2i −0.0662108 0.0382268i 0.466529 0.884506i \(-0.345504\pi\)
−0.532740 + 0.846279i \(0.678838\pi\)
\(984\) 0 0
\(985\) −528967. + 916198.i −0.545201 + 0.944315i
\(986\) 0 0
\(987\) 2.82165e6i 2.89646i
\(988\) 0 0
\(989\) −936677. −0.957629
\(990\) 0 0
\(991\) −1.57948e6 911914.i −1.60830 0.928553i −0.989750 0.142808i \(-0.954387\pi\)
−0.618550 0.785745i \(-0.712280\pi\)
\(992\) 0 0
\(993\) −479767. + 830981.i −0.486555 + 0.842738i
\(994\) 0 0
\(995\) −187869. −0.189762
\(996\) 0 0
\(997\) 639728. + 1.10804e6i 0.643583 + 1.11472i 0.984627 + 0.174672i \(0.0558864\pi\)
−0.341043 + 0.940048i \(0.610780\pi\)
\(998\) 0 0
\(999\) −1.27501e6 −1.27756
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 76.5.h.a.65.1 12
3.2 odd 2 684.5.y.c.217.1 12
4.3 odd 2 304.5.r.b.65.6 12
19.12 odd 6 inner 76.5.h.a.69.1 yes 12
57.50 even 6 684.5.y.c.145.1 12
76.31 even 6 304.5.r.b.145.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.h.a.65.1 12 1.1 even 1 trivial
76.5.h.a.69.1 yes 12 19.12 odd 6 inner
304.5.r.b.65.6 12 4.3 odd 2
304.5.r.b.145.6 12 76.31 even 6
684.5.y.c.145.1 12 57.50 even 6
684.5.y.c.217.1 12 3.2 odd 2