Properties

Label 304.6.i.d.273.8
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: \( x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} - 2546775754 x^{12} + 7879525640 x^{11} + 976140188391 x^{10} + \cdots + 66\!\cdots\!83 \) 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 273.8
Root \(-21.8370 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 304.273
Dual form 304.6.i.d.49.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(11.6685 + 20.2105i) q^{3} +(-25.2470 - 43.7291i) q^{5} +187.942 q^{7} +(-150.808 + 261.208i) q^{9} +O(q^{10})\) \(q+(11.6685 + 20.2105i) q^{3} +(-25.2470 - 43.7291i) q^{5} +187.942 q^{7} +(-150.808 + 261.208i) q^{9} -411.164 q^{11} +(286.794 - 496.742i) q^{13} +(589.191 - 1020.51i) q^{15} +(559.243 + 968.637i) q^{17} +(1048.98 - 1172.92i) q^{19} +(2193.01 + 3798.40i) q^{21} +(72.0483 - 124.791i) q^{23} +(287.675 - 498.268i) q^{25} -1367.94 q^{27} +(-2616.14 + 4531.29i) q^{29} +6724.09 q^{31} +(-4797.68 - 8309.82i) q^{33} +(-4744.99 - 8218.56i) q^{35} +12337.9 q^{37} +13385.8 q^{39} +(-5964.03 - 10330.0i) q^{41} +(2611.95 + 4524.03i) q^{43} +15229.9 q^{45} +(10457.1 - 18112.3i) q^{47} +18515.4 q^{49} +(-13051.1 + 22605.1i) q^{51} +(4550.48 - 7881.66i) q^{53} +(10380.7 + 17979.9i) q^{55} +(35945.2 + 7514.28i) q^{57} +(12030.3 + 20837.0i) q^{59} +(-21555.9 + 37335.8i) q^{61} +(-28343.3 + 49092.0i) q^{63} -28962.8 q^{65} +(8051.46 - 13945.5i) q^{67} +3362.78 q^{69} +(39902.4 + 69113.1i) q^{71} +(-17210.0 - 29808.6i) q^{73} +13427.0 q^{75} -77275.3 q^{77} +(-16379.2 - 28369.7i) q^{79} +(20684.6 + 35826.8i) q^{81} -49610.5 q^{83} +(28238.4 - 48910.4i) q^{85} -122106. q^{87} +(-39238.1 + 67962.3i) q^{89} +(53900.8 - 93359.0i) q^{91} +(78460.1 + 135897. i) q^{93} +(-77774.3 - 16258.6i) q^{95} +(62657.4 + 108526. i) q^{97} +(62007.0 - 107399. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) \) 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
\(3\) 11.6685 + 20.2105i 0.748535 + 1.29650i 0.948525 + 0.316703i \(0.102576\pi\)
−0.199989 + 0.979798i \(0.564091\pi\)
\(4\) 0 0
\(5\) −25.2470 43.7291i −0.451633 0.782251i 0.546855 0.837227i \(-0.315825\pi\)
−0.998488 + 0.0549767i \(0.982492\pi\)
\(6\) 0 0
\(7\) 187.942 1.44971 0.724853 0.688904i \(-0.241908\pi\)
0.724853 + 0.688904i \(0.241908\pi\)
\(8\) 0 0
\(9\) −150.808 + 261.208i −0.620610 + 1.07493i
\(10\) 0 0
\(11\) −411.164 −1.02455 −0.512276 0.858821i \(-0.671197\pi\)
−0.512276 + 0.858821i \(0.671197\pi\)
\(12\) 0 0
\(13\) 286.794 496.742i 0.470665 0.815216i −0.528772 0.848764i \(-0.677347\pi\)
0.999437 + 0.0335480i \(0.0106807\pi\)
\(14\) 0 0
\(15\) 589.191 1020.51i 0.676126 1.17108i
\(16\) 0 0
\(17\) 559.243 + 968.637i 0.469330 + 0.812903i 0.999385 0.0350601i \(-0.0111623\pi\)
−0.530056 + 0.847963i \(0.677829\pi\)
\(18\) 0 0
\(19\) 1048.98 1172.92i 0.666630 0.745388i
\(20\) 0 0
\(21\) 2193.01 + 3798.40i 1.08516 + 1.87955i
\(22\) 0 0
\(23\) 72.0483 124.791i 0.0283991 0.0491886i −0.851477 0.524393i \(-0.824292\pi\)
0.879876 + 0.475204i \(0.157626\pi\)
\(24\) 0 0
\(25\) 287.675 498.268i 0.0920560 0.159446i
\(26\) 0 0
\(27\) −1367.94 −0.361125
\(28\) 0 0
\(29\) −2616.14 + 4531.29i −0.577652 + 1.00052i 0.418096 + 0.908403i \(0.362697\pi\)
−0.995748 + 0.0921194i \(0.970636\pi\)
\(30\) 0 0
\(31\) 6724.09 1.25669 0.628347 0.777934i \(-0.283732\pi\)
0.628347 + 0.777934i \(0.283732\pi\)
\(32\) 0 0
\(33\) −4797.68 8309.82i −0.766913 1.32833i
\(34\) 0 0
\(35\) −4744.99 8218.56i −0.654734 1.13403i
\(36\) 0 0
\(37\) 12337.9 1.48162 0.740811 0.671713i \(-0.234441\pi\)
0.740811 + 0.671713i \(0.234441\pi\)
\(38\) 0 0
\(39\) 13385.8 1.40924
\(40\) 0 0
\(41\) −5964.03 10330.0i −0.554090 0.959712i −0.997974 0.0636278i \(-0.979733\pi\)
0.443884 0.896084i \(-0.353600\pi\)
\(42\) 0 0
\(43\) 2611.95 + 4524.03i 0.215424 + 0.373125i 0.953404 0.301698i \(-0.0975534\pi\)
−0.737980 + 0.674823i \(0.764220\pi\)
\(44\) 0 0
\(45\) 15229.9 1.12115
\(46\) 0 0
\(47\) 10457.1 18112.3i 0.690507 1.19599i −0.281165 0.959659i \(-0.590721\pi\)
0.971672 0.236333i \(-0.0759458\pi\)
\(48\) 0 0
\(49\) 18515.4 1.10165
\(50\) 0 0
\(51\) −13051.1 + 22605.1i −0.702620 + 1.21697i
\(52\) 0 0
\(53\) 4550.48 7881.66i 0.222519 0.385414i −0.733053 0.680171i \(-0.761905\pi\)
0.955572 + 0.294757i \(0.0952387\pi\)
\(54\) 0 0
\(55\) 10380.7 + 17979.9i 0.462721 + 0.801456i
\(56\) 0 0
\(57\) 35945.2 + 7514.28i 1.46539 + 0.306338i
\(58\) 0 0
\(59\) 12030.3 + 20837.0i 0.449930 + 0.779301i 0.998381 0.0568813i \(-0.0181157\pi\)
−0.548451 + 0.836183i \(0.684782\pi\)
\(60\) 0 0
\(61\) −21555.9 + 37335.8i −0.741721 + 1.28470i 0.209990 + 0.977704i \(0.432657\pi\)
−0.951711 + 0.306995i \(0.900676\pi\)
\(62\) 0 0
\(63\) −28343.3 + 49092.0i −0.899703 + 1.55833i
\(64\) 0 0
\(65\) −28962.8 −0.850271
\(66\) 0 0
\(67\) 8051.46 13945.5i 0.219123 0.379532i −0.735417 0.677615i \(-0.763014\pi\)
0.954540 + 0.298083i \(0.0963471\pi\)
\(68\) 0 0
\(69\) 3362.78 0.0850308
\(70\) 0 0
\(71\) 39902.4 + 69113.1i 0.939407 + 1.62710i 0.766581 + 0.642148i \(0.221956\pi\)
0.172826 + 0.984952i \(0.444710\pi\)
\(72\) 0 0
\(73\) −17210.0 29808.6i −0.377985 0.654688i 0.612784 0.790250i \(-0.290049\pi\)
−0.990769 + 0.135562i \(0.956716\pi\)
\(74\) 0 0
\(75\) 13427.0 0.275629
\(76\) 0 0
\(77\) −77275.3 −1.48530
\(78\) 0 0
\(79\) −16379.2 28369.7i −0.295274 0.511430i 0.679774 0.733421i \(-0.262078\pi\)
−0.975049 + 0.221991i \(0.928744\pi\)
\(80\) 0 0
\(81\) 20684.6 + 35826.8i 0.350296 + 0.606730i
\(82\) 0 0
\(83\) −49610.5 −0.790457 −0.395229 0.918583i \(-0.629335\pi\)
−0.395229 + 0.918583i \(0.629335\pi\)
\(84\) 0 0
\(85\) 28238.4 48910.4i 0.423929 0.734267i
\(86\) 0 0
\(87\) −122106. −1.72957
\(88\) 0 0
\(89\) −39238.1 + 67962.3i −0.525089 + 0.909480i 0.474484 + 0.880264i \(0.342635\pi\)
−0.999573 + 0.0292164i \(0.990699\pi\)
\(90\) 0 0
\(91\) 53900.8 93359.0i 0.682326 1.18182i
\(92\) 0 0
\(93\) 78460.1 + 135897.i 0.940679 + 1.62930i
\(94\) 0 0
\(95\) −77774.3 16258.6i −0.884153 0.184830i
\(96\) 0 0
\(97\) 62657.4 + 108526.i 0.676149 + 1.17113i 0.976131 + 0.217180i \(0.0696860\pi\)
−0.299982 + 0.953945i \(0.596981\pi\)
\(98\) 0 0
\(99\) 62007.0 107399.i 0.635847 1.10132i
\(100\) 0 0
\(101\) −64376.2 + 111503.i −0.627945 + 1.08763i 0.360018 + 0.932945i \(0.382770\pi\)
−0.987963 + 0.154688i \(0.950563\pi\)
\(102\) 0 0
\(103\) 185375. 1.72170 0.860851 0.508857i \(-0.169932\pi\)
0.860851 + 0.508857i \(0.169932\pi\)
\(104\) 0 0
\(105\) 110734. 191797.i 0.980184 1.69773i
\(106\) 0 0
\(107\) 51720.8 0.436723 0.218361 0.975868i \(-0.429929\pi\)
0.218361 + 0.975868i \(0.429929\pi\)
\(108\) 0 0
\(109\) −10666.1 18474.1i −0.0859879 0.148935i 0.819824 0.572616i \(-0.194071\pi\)
−0.905812 + 0.423680i \(0.860738\pi\)
\(110\) 0 0
\(111\) 143965. + 249355.i 1.10905 + 1.92093i
\(112\) 0 0
\(113\) 181009. 1.33353 0.666767 0.745266i \(-0.267678\pi\)
0.666767 + 0.745266i \(0.267678\pi\)
\(114\) 0 0
\(115\) −7276.02 −0.0513038
\(116\) 0 0
\(117\) 86501.9 + 149826.i 0.584199 + 1.01186i
\(118\) 0 0
\(119\) 105105. + 182048.i 0.680390 + 1.17847i
\(120\) 0 0
\(121\) 8005.17 0.0497058
\(122\) 0 0
\(123\) 139183. 241072.i 0.829512 1.43676i
\(124\) 0 0
\(125\) −186846. −1.06957
\(126\) 0 0
\(127\) −909.283 + 1574.92i −0.00500253 + 0.00866464i −0.868516 0.495661i \(-0.834926\pi\)
0.863513 + 0.504326i \(0.168259\pi\)
\(128\) 0 0
\(129\) −60955.1 + 105577.i −0.322504 + 0.558594i
\(130\) 0 0
\(131\) 39930.7 + 69162.1i 0.203296 + 0.352119i 0.949589 0.313499i \(-0.101501\pi\)
−0.746292 + 0.665618i \(0.768168\pi\)
\(132\) 0 0
\(133\) 197149. 220441.i 0.966418 1.08059i
\(134\) 0 0
\(135\) 34536.4 + 59818.8i 0.163096 + 0.282490i
\(136\) 0 0
\(137\) 99388.3 172146.i 0.452412 0.783600i −0.546123 0.837705i \(-0.683897\pi\)
0.998535 + 0.0541044i \(0.0172304\pi\)
\(138\) 0 0
\(139\) 188358. 326246.i 0.826890 1.43222i −0.0735761 0.997290i \(-0.523441\pi\)
0.900466 0.434926i \(-0.143225\pi\)
\(140\) 0 0
\(141\) 488077. 2.06748
\(142\) 0 0
\(143\) −117920. + 204243.i −0.482221 + 0.835231i
\(144\) 0 0
\(145\) 264199. 1.04355
\(146\) 0 0
\(147\) 216047. + 374204.i 0.824622 + 1.42829i
\(148\) 0 0
\(149\) −115828. 200620.i −0.427413 0.740300i 0.569230 0.822179i \(-0.307242\pi\)
−0.996642 + 0.0818781i \(0.973908\pi\)
\(150\) 0 0
\(151\) −465504. −1.66143 −0.830714 0.556700i \(-0.812067\pi\)
−0.830714 + 0.556700i \(0.812067\pi\)
\(152\) 0 0
\(153\) −337354. −1.16508
\(154\) 0 0
\(155\) −169763. 294039.i −0.567564 0.983049i
\(156\) 0 0
\(157\) −256177. 443712.i −0.829451 1.43665i −0.898469 0.439037i \(-0.855320\pi\)
0.0690177 0.997615i \(-0.478013\pi\)
\(158\) 0 0
\(159\) 212389. 0.666254
\(160\) 0 0
\(161\) 13540.9 23453.6i 0.0411703 0.0713090i
\(162\) 0 0
\(163\) −551349. −1.62539 −0.812694 0.582690i \(-0.802000\pi\)
−0.812694 + 0.582690i \(0.802000\pi\)
\(164\) 0 0
\(165\) −242254. + 419597.i −0.692726 + 1.19984i
\(166\) 0 0
\(167\) −24265.6 + 42029.2i −0.0673286 + 0.116617i −0.897725 0.440557i \(-0.854781\pi\)
0.830396 + 0.557174i \(0.188114\pi\)
\(168\) 0 0
\(169\) 21144.6 + 36623.6i 0.0569487 + 0.0986380i
\(170\) 0 0
\(171\) 148179. + 450888.i 0.387522 + 1.17918i
\(172\) 0 0
\(173\) −84590.8 146516.i −0.214886 0.372193i 0.738351 0.674416i \(-0.235605\pi\)
−0.953237 + 0.302223i \(0.902271\pi\)
\(174\) 0 0
\(175\) 54066.3 93645.7i 0.133454 0.231149i
\(176\) 0 0
\(177\) −280750. + 486274.i −0.673577 + 1.16667i
\(178\) 0 0
\(179\) 560804. 1.30821 0.654106 0.756403i \(-0.273045\pi\)
0.654106 + 0.756403i \(0.273045\pi\)
\(180\) 0 0
\(181\) 186731. 323428.i 0.423663 0.733806i −0.572631 0.819813i \(-0.694077\pi\)
0.996295 + 0.0860068i \(0.0274107\pi\)
\(182\) 0 0
\(183\) −1.00610e6 −2.22082
\(184\) 0 0
\(185\) −311496. 539526.i −0.669149 1.15900i
\(186\) 0 0
\(187\) −229941. 398269.i −0.480852 0.832861i
\(188\) 0 0
\(189\) −257094. −0.523525
\(190\) 0 0
\(191\) 603335. 1.19667 0.598336 0.801245i \(-0.295829\pi\)
0.598336 + 0.801245i \(0.295829\pi\)
\(192\) 0 0
\(193\) 52.8895 + 91.6073i 0.000102206 + 0.000177026i 0.866077 0.499911i \(-0.166634\pi\)
−0.865974 + 0.500089i \(0.833301\pi\)
\(194\) 0 0
\(195\) −337953. 585352.i −0.636458 1.10238i
\(196\) 0 0
\(197\) −704077. −1.29257 −0.646285 0.763096i \(-0.723678\pi\)
−0.646285 + 0.763096i \(0.723678\pi\)
\(198\) 0 0
\(199\) −81423.6 + 141030.i −0.145753 + 0.252452i −0.929654 0.368435i \(-0.879894\pi\)
0.783901 + 0.620886i \(0.213227\pi\)
\(200\) 0 0
\(201\) 375794. 0.656085
\(202\) 0 0
\(203\) −491684. + 851622.i −0.837425 + 1.45046i
\(204\) 0 0
\(205\) −301148. + 521604.i −0.500490 + 0.866875i
\(206\) 0 0
\(207\) 21731.0 + 37639.1i 0.0352495 + 0.0610539i
\(208\) 0 0
\(209\) −431305. + 482261.i −0.682997 + 0.763689i
\(210\) 0 0
\(211\) −349140. 604728.i −0.539875 0.935091i −0.998910 0.0466728i \(-0.985138\pi\)
0.459035 0.888418i \(-0.348195\pi\)
\(212\) 0 0
\(213\) −931204. + 1.61289e6i −1.40636 + 2.43588i
\(214\) 0 0
\(215\) 131888. 228436.i 0.194585 0.337030i
\(216\) 0 0
\(217\) 1.26374e6 1.82184
\(218\) 0 0
\(219\) 401630. 695644.i 0.565870 0.980115i
\(220\) 0 0
\(221\) 641550. 0.883588
\(222\) 0 0
\(223\) 93703.3 + 162299.i 0.126181 + 0.218551i 0.922194 0.386728i \(-0.126395\pi\)
−0.796013 + 0.605279i \(0.793061\pi\)
\(224\) 0 0
\(225\) 86767.6 + 150286.i 0.114262 + 0.197907i
\(226\) 0 0
\(227\) −606601. −0.781338 −0.390669 0.920531i \(-0.627756\pi\)
−0.390669 + 0.920531i \(0.627756\pi\)
\(228\) 0 0
\(229\) −215431. −0.271469 −0.135734 0.990745i \(-0.543339\pi\)
−0.135734 + 0.990745i \(0.543339\pi\)
\(230\) 0 0
\(231\) −901687. 1.56177e6i −1.11180 1.92569i
\(232\) 0 0
\(233\) −511411. 885789.i −0.617135 1.06891i −0.990006 0.141026i \(-0.954960\pi\)
0.372871 0.927883i \(-0.378373\pi\)
\(234\) 0 0
\(235\) −1.05605e6 −1.24742
\(236\) 0 0
\(237\) 382242. 662063.i 0.442047 0.765647i
\(238\) 0 0
\(239\) 1.31113e6 1.48474 0.742371 0.669989i \(-0.233701\pi\)
0.742371 + 0.669989i \(0.233701\pi\)
\(240\) 0 0
\(241\) 4784.45 8286.91i 0.00530627 0.00919073i −0.863360 0.504588i \(-0.831644\pi\)
0.868666 + 0.495398i \(0.164978\pi\)
\(242\) 0 0
\(243\) −648922. + 1.12397e6i −0.704980 + 1.22106i
\(244\) 0 0
\(245\) −467458. 809662.i −0.497540 0.861764i
\(246\) 0 0
\(247\) −281794. 857460.i −0.293893 0.894276i
\(248\) 0 0
\(249\) −578881. 1.00265e6i −0.591685 1.02483i
\(250\) 0 0
\(251\) 228222. 395293.i 0.228651 0.396036i −0.728757 0.684772i \(-0.759902\pi\)
0.957409 + 0.288736i \(0.0932351\pi\)
\(252\) 0 0
\(253\) −29623.7 + 51309.7i −0.0290963 + 0.0503963i
\(254\) 0 0
\(255\) 1.31800e6 1.26930
\(256\) 0 0
\(257\) 259837. 450050.i 0.245396 0.425038i −0.716847 0.697231i \(-0.754415\pi\)
0.962243 + 0.272192i \(0.0877487\pi\)
\(258\) 0 0
\(259\) 2.31882e6 2.14792
\(260\) 0 0
\(261\) −789072. 1.36671e6i −0.716993 1.24187i
\(262\) 0 0
\(263\) 159146. + 275650.i 0.141876 + 0.245736i 0.928203 0.372074i \(-0.121353\pi\)
−0.786327 + 0.617810i \(0.788020\pi\)
\(264\) 0 0
\(265\) −459544. −0.401987
\(266\) 0 0
\(267\) −1.83140e6 −1.57219
\(268\) 0 0
\(269\) −71668.3 124133.i −0.0603874 0.104594i 0.834251 0.551385i \(-0.185900\pi\)
−0.894639 + 0.446790i \(0.852567\pi\)
\(270\) 0 0
\(271\) −304619. 527616.i −0.251961 0.436410i 0.712104 0.702074i \(-0.247742\pi\)
−0.964066 + 0.265664i \(0.914409\pi\)
\(272\) 0 0
\(273\) 2.51577e6 2.04298
\(274\) 0 0
\(275\) −118282. + 204870.i −0.0943161 + 0.163360i
\(276\) 0 0
\(277\) −1.94146e6 −1.52030 −0.760151 0.649746i \(-0.774875\pi\)
−0.760151 + 0.649746i \(0.774875\pi\)
\(278\) 0 0
\(279\) −1.01405e6 + 1.75638e6i −0.779917 + 1.35086i
\(280\) 0 0
\(281\) −286688. + 496557.i −0.216592 + 0.375149i −0.953764 0.300557i \(-0.902828\pi\)
0.737172 + 0.675706i \(0.236161\pi\)
\(282\) 0 0
\(283\) 1.04413e6 + 1.80849e6i 0.774977 + 1.34230i 0.934808 + 0.355155i \(0.115572\pi\)
−0.159831 + 0.987144i \(0.551095\pi\)
\(284\) 0 0
\(285\) −578917. 1.76157e6i −0.422187 1.28466i
\(286\) 0 0
\(287\) −1.12090e6 1.94145e6i −0.803268 1.39130i
\(288\) 0 0
\(289\) 84423.8 146226.i 0.0594593 0.102987i
\(290\) 0 0
\(291\) −1.46224e6 + 2.53267e6i −1.01224 + 1.75326i
\(292\) 0 0
\(293\) −598199. −0.407077 −0.203538 0.979067i \(-0.565244\pi\)
−0.203538 + 0.979067i \(0.565244\pi\)
\(294\) 0 0
\(295\) 607456. 1.05215e6i 0.406406 0.703916i
\(296\) 0 0
\(297\) 562448. 0.369991
\(298\) 0 0
\(299\) −41326.0 71578.8i −0.0267329 0.0463027i
\(300\) 0 0
\(301\) 490896. + 850257.i 0.312301 + 0.540921i
\(302\) 0 0
\(303\) −3.00470e6 −1.88016
\(304\) 0 0
\(305\) 2.17689e6 1.33994
\(306\) 0 0
\(307\) −71331.2 123549.i −0.0431950 0.0748159i 0.843620 0.536941i \(-0.180420\pi\)
−0.886815 + 0.462125i \(0.847087\pi\)
\(308\) 0 0
\(309\) 2.16305e6 + 3.74651e6i 1.28876 + 2.23219i
\(310\) 0 0
\(311\) 3.33734e6 1.95659 0.978294 0.207219i \(-0.0664414\pi\)
0.978294 + 0.207219i \(0.0664414\pi\)
\(312\) 0 0
\(313\) 1.12350e6 1.94596e6i 0.648205 1.12272i −0.335347 0.942095i \(-0.608854\pi\)
0.983551 0.180628i \(-0.0578131\pi\)
\(314\) 0 0
\(315\) 2.86234e6 1.62534
\(316\) 0 0
\(317\) −29587.8 + 51247.6i −0.0165373 + 0.0286434i −0.874176 0.485610i \(-0.838598\pi\)
0.857638 + 0.514253i \(0.171931\pi\)
\(318\) 0 0
\(319\) 1.07566e6 1.86310e6i 0.591834 1.02509i
\(320\) 0 0
\(321\) 603505. + 1.04530e6i 0.326902 + 0.566212i
\(322\) 0 0
\(323\) 1.72277e6 + 360141.i 0.918798 + 0.192073i
\(324\) 0 0
\(325\) −165007. 285801.i −0.0866551 0.150091i
\(326\) 0 0
\(327\) 248914. 431132.i 0.128730 0.222967i
\(328\) 0 0
\(329\) 1.96534e6 3.40407e6i 1.00103 1.73384i
\(330\) 0 0
\(331\) −2.35196e6 −1.17994 −0.589970 0.807425i \(-0.700860\pi\)
−0.589970 + 0.807425i \(0.700860\pi\)
\(332\) 0 0
\(333\) −1.86066e6 + 3.22276e6i −0.919510 + 1.59264i
\(334\) 0 0
\(335\) −813102. −0.395852
\(336\) 0 0
\(337\) −82853.0 143506.i −0.0397405 0.0688326i 0.845471 0.534021i \(-0.179320\pi\)
−0.885212 + 0.465189i \(0.845986\pi\)
\(338\) 0 0
\(339\) 2.11211e6 + 3.65827e6i 0.998197 + 1.72893i
\(340\) 0 0
\(341\) −2.76471e6 −1.28755
\(342\) 0 0
\(343\) 321077. 0.147358
\(344\) 0 0
\(345\) −84900.3 147052.i −0.0384027 0.0665154i
\(346\) 0 0
\(347\) −407820. 706365.i −0.181821 0.314924i 0.760679 0.649128i \(-0.224866\pi\)
−0.942501 + 0.334204i \(0.891533\pi\)
\(348\) 0 0
\(349\) 1.07255e6 0.471359 0.235680 0.971831i \(-0.424268\pi\)
0.235680 + 0.971831i \(0.424268\pi\)
\(350\) 0 0
\(351\) −392317. + 679513.i −0.169969 + 0.294395i
\(352\) 0 0
\(353\) −3.70760e6 −1.58364 −0.791819 0.610756i \(-0.790866\pi\)
−0.791819 + 0.610756i \(0.790866\pi\)
\(354\) 0 0
\(355\) 2.01484e6 3.48980e6i 0.848533 1.46970i
\(356\) 0 0
\(357\) −2.45285e6 + 4.24846e6i −1.01859 + 1.76425i
\(358\) 0 0
\(359\) 440125. + 762318.i 0.180235 + 0.312176i 0.941961 0.335724i \(-0.108981\pi\)
−0.761725 + 0.647900i \(0.775647\pi\)
\(360\) 0 0
\(361\) −275361. 2.46074e6i −0.111208 0.993797i
\(362\) 0 0
\(363\) 93408.4 + 161788.i 0.0372066 + 0.0644437i
\(364\) 0 0
\(365\) −869003. + 1.50516e6i −0.341420 + 0.591357i
\(366\) 0 0
\(367\) −1.32405e6 + 2.29332e6i −0.513143 + 0.888789i 0.486741 + 0.873546i \(0.338186\pi\)
−0.999884 + 0.0152431i \(0.995148\pi\)
\(368\) 0 0
\(369\) 3.59770e6 1.37550
\(370\) 0 0
\(371\) 855228. 1.48130e6i 0.322587 0.558737i
\(372\) 0 0
\(373\) 1.42105e6 0.528856 0.264428 0.964405i \(-0.414817\pi\)
0.264428 + 0.964405i \(0.414817\pi\)
\(374\) 0 0
\(375\) −2.18021e6 3.77624e6i −0.800609 1.38670i
\(376\) 0 0
\(377\) 1.50059e6 + 2.59909e6i 0.543761 + 0.941822i
\(378\) 0 0
\(379\) −4.14092e6 −1.48081 −0.740405 0.672161i \(-0.765366\pi\)
−0.740405 + 0.672161i \(0.765366\pi\)
\(380\) 0 0
\(381\) −42439.9 −0.0149783
\(382\) 0 0
\(383\) −932604. 1.61532e6i −0.324863 0.562679i 0.656622 0.754220i \(-0.271985\pi\)
−0.981485 + 0.191541i \(0.938652\pi\)
\(384\) 0 0
\(385\) 1.95097e6 + 3.37918e6i 0.670809 + 1.16188i
\(386\) 0 0
\(387\) −1.57561e6 −0.534777
\(388\) 0 0
\(389\) −225886. + 391246.i −0.0756859 + 0.131092i −0.901384 0.433020i \(-0.857448\pi\)
0.825698 + 0.564112i \(0.190781\pi\)
\(390\) 0 0
\(391\) 161170. 0.0533141
\(392\) 0 0
\(393\) −931865. + 1.61404e6i −0.304349 + 0.527147i
\(394\) 0 0
\(395\) −827054. + 1.43250e6i −0.266711 + 0.461957i
\(396\) 0 0
\(397\) 387696. + 671509.i 0.123457 + 0.213833i 0.921129 0.389258i \(-0.127269\pi\)
−0.797672 + 0.603092i \(0.793935\pi\)
\(398\) 0 0
\(399\) 6.75564e6 + 1.41225e6i 2.12439 + 0.444099i
\(400\) 0 0
\(401\) −3.11762e6 5.39987e6i −0.968193 1.67696i −0.700781 0.713376i \(-0.747165\pi\)
−0.267412 0.963582i \(-0.586168\pi\)
\(402\) 0 0
\(403\) 1.92843e6 3.34014e6i 0.591482 1.02448i
\(404\) 0 0
\(405\) 1.04445e6 1.80904e6i 0.316410 0.548038i
\(406\) 0 0
\(407\) −5.07291e6 −1.51800
\(408\) 0 0
\(409\) −2.43708e6 + 4.22114e6i −0.720379 + 1.24773i 0.240468 + 0.970657i \(0.422699\pi\)
−0.960848 + 0.277077i \(0.910634\pi\)
\(410\) 0 0
\(411\) 4.63886e6 1.35459
\(412\) 0 0
\(413\) 2.26100e6 + 3.91616e6i 0.652266 + 1.12976i
\(414\) 0 0
\(415\) 1.25252e6 + 2.16942e6i 0.356996 + 0.618336i
\(416\) 0 0
\(417\) 8.79144e6 2.47583
\(418\) 0 0
\(419\) −3.11548e6 −0.866943 −0.433471 0.901167i \(-0.642711\pi\)
−0.433471 + 0.901167i \(0.642711\pi\)
\(420\) 0 0
\(421\) −2.17883e6 3.77384e6i −0.599125 1.03771i −0.992951 0.118529i \(-0.962182\pi\)
0.393826 0.919185i \(-0.371151\pi\)
\(422\) 0 0
\(423\) 3.15405e6 + 5.46297e6i 0.857071 + 1.48449i
\(424\) 0 0
\(425\) 643520. 0.172818
\(426\) 0 0
\(427\) −4.05126e6 + 7.01699e6i −1.07528 + 1.86244i
\(428\) 0 0
\(429\) −5.50378e6 −1.44384
\(430\) 0 0
\(431\) −3.39109e6 + 5.87353e6i −0.879317 + 1.52302i −0.0272262 + 0.999629i \(0.508667\pi\)
−0.852091 + 0.523393i \(0.824666\pi\)
\(432\) 0 0
\(433\) −16420.8 + 28441.7i −0.00420896 + 0.00729014i −0.868122 0.496351i \(-0.834673\pi\)
0.863913 + 0.503641i \(0.168006\pi\)
\(434\) 0 0
\(435\) 3.08281e6 + 5.33958e6i 0.781131 + 1.35296i
\(436\) 0 0
\(437\) −70792.0 215411.i −0.0177329 0.0539589i
\(438\) 0 0
\(439\) 2.63969e6 + 4.57208e6i 0.653720 + 1.13228i 0.982213 + 0.187770i \(0.0601259\pi\)
−0.328493 + 0.944506i \(0.606541\pi\)
\(440\) 0 0
\(441\) −2.79227e6 + 4.83636e6i −0.683694 + 1.18419i
\(442\) 0 0
\(443\) 3.86131e6 6.68798e6i 0.934813 1.61914i 0.159846 0.987142i \(-0.448900\pi\)
0.774967 0.632002i \(-0.217767\pi\)
\(444\) 0 0
\(445\) 3.96258e6 0.948589
\(446\) 0 0
\(447\) 2.70308e6 4.68187e6i 0.639867 1.10828i
\(448\) 0 0
\(449\) 6.68110e6 1.56398 0.781992 0.623288i \(-0.214204\pi\)
0.781992 + 0.623288i \(0.214204\pi\)
\(450\) 0 0
\(451\) 2.45220e6 + 4.24733e6i 0.567694 + 0.983275i
\(452\) 0 0
\(453\) −5.43174e6 9.40806e6i −1.24364 2.15404i
\(454\) 0 0
\(455\) −5.44334e6 −1.23264
\(456\) 0 0
\(457\) −1.15454e6 −0.258593 −0.129297 0.991606i \(-0.541272\pi\)
−0.129297 + 0.991606i \(0.541272\pi\)
\(458\) 0 0
\(459\) −765010. 1.32504e6i −0.169487 0.293559i
\(460\) 0 0
\(461\) 3.99643e6 + 6.92202e6i 0.875830 + 1.51698i 0.855876 + 0.517181i \(0.173019\pi\)
0.0199544 + 0.999801i \(0.493648\pi\)
\(462\) 0 0
\(463\) 4.73362e6 1.02622 0.513111 0.858322i \(-0.328493\pi\)
0.513111 + 0.858322i \(0.328493\pi\)
\(464\) 0 0
\(465\) 3.96177e6 6.86199e6i 0.849683 1.47169i
\(466\) 0 0
\(467\) 3.37127e6 0.715322 0.357661 0.933851i \(-0.383574\pi\)
0.357661 + 0.933851i \(0.383574\pi\)
\(468\) 0 0
\(469\) 1.51321e6 2.62096e6i 0.317664 0.550210i
\(470\) 0 0
\(471\) 5.97841e6 1.03549e7i 1.24175 2.15077i
\(472\) 0 0
\(473\) −1.07394e6 1.86012e6i −0.220713 0.382285i
\(474\) 0 0
\(475\) −282659. 860093.i −0.0574816 0.174909i
\(476\) 0 0
\(477\) 1.37250e6 + 2.37724e6i 0.276195 + 0.478384i
\(478\) 0 0
\(479\) −3.45446e6 + 5.98330e6i −0.687925 + 1.19152i 0.284583 + 0.958652i \(0.408145\pi\)
−0.972508 + 0.232870i \(0.925188\pi\)
\(480\) 0 0
\(481\) 3.53844e6 6.12876e6i 0.697348 1.20784i
\(482\) 0 0
\(483\) 632010. 0.123270
\(484\) 0 0
\(485\) 3.16382e6 5.47990e6i 0.610742 1.05784i
\(486\) 0 0
\(487\) 3.84736e6 0.735089 0.367545 0.930006i \(-0.380199\pi\)
0.367545 + 0.930006i \(0.380199\pi\)
\(488\) 0 0
\(489\) −6.43342e6 1.11430e7i −1.21666 2.10732i
\(490\) 0 0
\(491\) −4.95587e6 8.58382e6i −0.927719 1.60686i −0.787129 0.616788i \(-0.788433\pi\)
−0.140590 0.990068i \(-0.544900\pi\)
\(492\) 0 0
\(493\) −5.85223e6 −1.08444
\(494\) 0 0
\(495\) −6.26197e6 −1.14868
\(496\) 0 0
\(497\) 7.49937e6 + 1.29893e7i 1.36186 + 2.35882i
\(498\) 0 0
\(499\) 3.50894e6 + 6.07766e6i 0.630847 + 1.09266i 0.987379 + 0.158376i \(0.0506259\pi\)
−0.356532 + 0.934283i \(0.616041\pi\)
\(500\) 0 0
\(501\) −1.13257e6 −0.201591
\(502\) 0 0
\(503\) −1.11612e6 + 1.93318e6i −0.196694 + 0.340684i −0.947455 0.319890i \(-0.896354\pi\)
0.750760 + 0.660575i \(0.229687\pi\)
\(504\) 0 0
\(505\) 6.50123e6 1.13440
\(506\) 0 0
\(507\) −493453. + 854686.i −0.0852562 + 0.147668i
\(508\) 0 0
\(509\) 787950. 1.36477e6i 0.134804 0.233488i −0.790718 0.612180i \(-0.790293\pi\)
0.925523 + 0.378692i \(0.123626\pi\)
\(510\) 0 0
\(511\) −3.23449e6 5.60231e6i −0.547966 0.949106i
\(512\) 0 0
\(513\) −1.43495e6 + 1.60448e6i −0.240737 + 0.269178i
\(514\) 0 0
\(515\) −4.68017e6 8.10629e6i −0.777577 1.34680i
\(516\) 0 0
\(517\) −4.29960e6 + 7.44713e6i −0.707460 + 1.22536i
\(518\) 0 0
\(519\) 1.97410e6 3.41924e6i 0.321700 0.557200i
\(520\) 0 0
\(521\) −4.35615e6 −0.703086 −0.351543 0.936172i \(-0.614343\pi\)
−0.351543 + 0.936172i \(0.614343\pi\)
\(522\) 0 0
\(523\) 2.64694e6 4.58464e6i 0.423146 0.732911i −0.573099 0.819486i \(-0.694259\pi\)
0.996245 + 0.0865752i \(0.0275923\pi\)
\(524\) 0 0
\(525\) 2.52350e6 0.399580
\(526\) 0 0
\(527\) 3.76040e6 + 6.51320e6i 0.589803 + 1.02157i
\(528\) 0 0
\(529\) 3.20779e6 + 5.55605e6i 0.498387 + 0.863232i
\(530\) 0 0
\(531\) −7.25705e6 −1.11692
\(532\) 0 0
\(533\) −6.84180e6 −1.04316
\(534\) 0 0
\(535\) −1.30580e6 2.26171e6i −0.197238 0.341627i
\(536\) 0 0
\(537\) 6.54374e6 + 1.13341e7i 0.979243 + 1.69610i
\(538\) 0 0
\(539\) −7.61286e6 −1.12869
\(540\) 0 0
\(541\) −5.45983e6 + 9.45670e6i −0.802021 + 1.38914i 0.116263 + 0.993218i \(0.462909\pi\)
−0.918284 + 0.395923i \(0.870425\pi\)
\(542\) 0 0
\(543\) 8.71551e6 1.26851
\(544\) 0 0
\(545\) −538572. + 932835.i −0.0776699 + 0.134528i
\(546\) 0 0
\(547\) 1.49890e6 2.59617e6i 0.214192 0.370992i −0.738830 0.673892i \(-0.764621\pi\)
0.953022 + 0.302900i \(0.0979548\pi\)
\(548\) 0 0
\(549\) −6.50161e6 1.12611e7i −0.920640 1.59460i
\(550\) 0 0
\(551\) 2.57053e6 + 7.82176e6i 0.360697 + 1.09755i
\(552\) 0 0
\(553\) −3.07835e6 5.33186e6i −0.428061 0.741423i
\(554\) 0 0
\(555\) 7.26938e6 1.25909e7i 1.00176 1.73510i
\(556\) 0 0
\(557\) 197093. 341375.i 0.0269174 0.0466223i −0.852253 0.523130i \(-0.824764\pi\)
0.879170 + 0.476508i \(0.158098\pi\)
\(558\) 0 0
\(559\) 2.99637e6 0.405569
\(560\) 0 0
\(561\) 5.36613e6 9.29441e6i 0.719870 1.24685i
\(562\) 0 0
\(563\) −1.10461e7 −1.46872 −0.734361 0.678760i \(-0.762518\pi\)
−0.734361 + 0.678760i \(0.762518\pi\)
\(564\) 0 0
\(565\) −4.56994e6 7.91537e6i −0.602267 1.04316i
\(566\) 0 0
\(567\) 3.88752e6 + 6.73338e6i 0.507826 + 0.879580i
\(568\) 0 0
\(569\) −7.21256e6 −0.933918 −0.466959 0.884279i \(-0.654651\pi\)
−0.466959 + 0.884279i \(0.654651\pi\)
\(570\) 0 0
\(571\) −3.09113e6 −0.396760 −0.198380 0.980125i \(-0.563568\pi\)
−0.198380 + 0.980125i \(0.563568\pi\)
\(572\) 0 0
\(573\) 7.04003e6 + 1.21937e7i 0.895752 + 1.55149i
\(574\) 0 0
\(575\) −41453.0 71798.6i −0.00522861 0.00905621i
\(576\) 0 0
\(577\) −2.16381e6 −0.270570 −0.135285 0.990807i \(-0.543195\pi\)
−0.135285 + 0.990807i \(0.543195\pi\)
\(578\) 0 0
\(579\) −1234.28 + 2137.84i −0.000153009 + 0.000265020i
\(580\) 0 0
\(581\) −9.32392e6 −1.14593
\(582\) 0 0
\(583\) −1.87099e6 + 3.24066e6i −0.227982 + 0.394877i
\(584\) 0 0
\(585\) 4.36783e6 7.56531e6i 0.527687 0.913981i
\(586\) 0 0
\(587\) 632835. + 1.09610e6i 0.0758045 + 0.131297i 0.901436 0.432913i \(-0.142514\pi\)
−0.825631 + 0.564210i \(0.809181\pi\)
\(588\) 0 0
\(589\) 7.05347e6 7.88679e6i 0.837750 0.936725i
\(590\) 0 0
\(591\) −8.21553e6 1.42297e7i −0.967535 1.67582i
\(592\) 0 0
\(593\) −8.38325e6 + 1.45202e7i −0.978984 + 1.69565i −0.312878 + 0.949793i \(0.601293\pi\)
−0.666106 + 0.745857i \(0.732040\pi\)
\(594\) 0 0
\(595\) 5.30720e6 9.19234e6i 0.614573 1.06447i
\(596\) 0 0
\(597\) −3.80037e6 −0.436405
\(598\) 0 0
\(599\) −2.31238e6 + 4.00516e6i −0.263325 + 0.456092i −0.967123 0.254308i \(-0.918152\pi\)
0.703798 + 0.710400i \(0.251486\pi\)
\(600\) 0 0
\(601\) −568699. −0.0642238 −0.0321119 0.999484i \(-0.510223\pi\)
−0.0321119 + 0.999484i \(0.510223\pi\)
\(602\) 0 0
\(603\) 2.42846e6 + 4.20621e6i 0.271980 + 0.471083i
\(604\) 0 0
\(605\) −202107. 350059.i −0.0224488 0.0388824i
\(606\) 0 0
\(607\) −9.98344e6 −1.09979 −0.549893 0.835235i \(-0.685332\pi\)
−0.549893 + 0.835235i \(0.685332\pi\)
\(608\) 0 0
\(609\) −2.29489e7 −2.50737
\(610\) 0 0
\(611\) −5.99809e6 1.03890e7i −0.649995 1.12582i
\(612\) 0 0
\(613\) −2.93664e6 5.08641e6i −0.315645 0.546714i 0.663929 0.747796i \(-0.268888\pi\)
−0.979574 + 0.201082i \(0.935554\pi\)
\(614\) 0 0
\(615\) −1.40558e7 −1.49854
\(616\) 0 0
\(617\) −4.31577e6 + 7.47513e6i −0.456400 + 0.790508i −0.998767 0.0496336i \(-0.984195\pi\)
0.542368 + 0.840141i \(0.317528\pi\)
\(618\) 0 0
\(619\) 6.29725e6 0.660578 0.330289 0.943880i \(-0.392854\pi\)
0.330289 + 0.943880i \(0.392854\pi\)
\(620\) 0 0
\(621\) −98557.6 + 170707.i −0.0102556 + 0.0177632i
\(622\) 0 0
\(623\) −7.37450e6 + 1.27730e7i −0.761224 + 1.31848i
\(624\) 0 0
\(625\) 3.81831e6 + 6.61351e6i 0.390995 + 0.677224i
\(626\) 0 0
\(627\) −1.47794e7 3.08961e6i −1.50137 0.313859i
\(628\) 0 0
\(629\) 6.89989e6 + 1.19510e7i 0.695369 + 1.20441i
\(630\) 0 0
\(631\) −2.14448e6 + 3.71434e6i −0.214411 + 0.371371i −0.953090 0.302686i \(-0.902117\pi\)
0.738679 + 0.674057i \(0.235450\pi\)
\(632\) 0 0
\(633\) 8.14788e6 1.41125e7i 0.808231 1.39990i
\(634\) 0 0
\(635\) 91826.8 0.00903723
\(636\) 0 0
\(637\) 5.31010e6 9.19737e6i 0.518507 0.898080i
\(638\) 0 0
\(639\) −2.40705e7 −2.33202
\(640\) 0 0
\(641\) −5.61057e6 9.71779e6i −0.539339 0.934163i −0.998940 0.0460372i \(-0.985341\pi\)
0.459600 0.888126i \(-0.347993\pi\)
\(642\) 0 0
\(643\) 2.20567e6 + 3.82032e6i 0.210384 + 0.364395i 0.951835 0.306612i \(-0.0991953\pi\)
−0.741451 + 0.671007i \(0.765862\pi\)
\(644\) 0 0
\(645\) 6.15574e6 0.582614
\(646\) 0 0
\(647\) −7.20279e6 −0.676457 −0.338229 0.941064i \(-0.609828\pi\)
−0.338229 + 0.941064i \(0.609828\pi\)
\(648\) 0 0
\(649\) −4.94641e6 8.56744e6i −0.460976 0.798434i
\(650\) 0 0
\(651\) 1.47460e7 + 2.55408e7i 1.36371 + 2.36201i
\(652\) 0 0
\(653\) −4.23537e6 −0.388695 −0.194347 0.980933i \(-0.562259\pi\)
−0.194347 + 0.980933i \(0.562259\pi\)
\(654\) 0 0
\(655\) 2.01627e6 3.49227e6i 0.183630 0.318057i
\(656\) 0 0
\(657\) 1.03817e7 0.938325
\(658\) 0 0
\(659\) 3.24491e6 5.62036e6i 0.291065 0.504139i −0.682997 0.730421i \(-0.739324\pi\)
0.974062 + 0.226282i \(0.0726572\pi\)
\(660\) 0 0
\(661\) −5.60734e6 + 9.71219e6i −0.499175 + 0.864597i −1.00000 0.000952002i \(-0.999697\pi\)
0.500824 + 0.865549i \(0.333030\pi\)
\(662\) 0 0
\(663\) 7.48594e6 + 1.29660e7i 0.661397 + 1.14557i
\(664\) 0 0
\(665\) −1.46171e7 3.05568e6i −1.28176 0.267950i
\(666\) 0 0
\(667\) 376977. + 652943.i 0.0328095 + 0.0568278i
\(668\) 0 0
\(669\) −2.18676e6 + 3.78757e6i −0.188901 + 0.327187i
\(670\) 0 0
\(671\) 8.86300e6 1.53512e7i 0.759932 1.31624i
\(672\) 0 0
\(673\) 301881. 0.0256920 0.0128460 0.999917i \(-0.495911\pi\)
0.0128460 + 0.999917i \(0.495911\pi\)
\(674\) 0 0
\(675\) −393522. + 681600.i −0.0332437 + 0.0575798i
\(676\) 0 0
\(677\) −470872. −0.0394849 −0.0197424 0.999805i \(-0.506285\pi\)
−0.0197424 + 0.999805i \(0.506285\pi\)
\(678\) 0 0
\(679\) 1.17760e7 + 2.03966e7i 0.980218 + 1.69779i
\(680\) 0 0
\(681\) −7.07813e6 1.22597e7i −0.584859 1.01301i
\(682\) 0 0
\(683\) 1.01992e7 0.836589 0.418295 0.908311i \(-0.362628\pi\)
0.418295 + 0.908311i \(0.362628\pi\)
\(684\) 0 0
\(685\) −1.00370e7 −0.817296
\(686\) 0 0
\(687\) −2.51376e6 4.35396e6i −0.203204 0.351960i
\(688\) 0 0
\(689\) −2.61010e6 4.52083e6i −0.209464 0.362802i
\(690\) 0 0
\(691\) −4.81669e6 −0.383754 −0.191877 0.981419i \(-0.561458\pi\)
−0.191877 + 0.981419i \(0.561458\pi\)
\(692\) 0 0
\(693\) 1.16538e7 2.01849e7i 0.921792 1.59659i
\(694\) 0 0
\(695\) −1.90220e7 −1.49380
\(696\) 0 0
\(697\) 6.67068e6 1.15540e7i 0.520102 0.900843i
\(698\) 0 0
\(699\) 1.19348e7 2.06717e7i 0.923895 1.60023i
\(700\) 0 0
\(701\) 2.09052e6 + 3.62088e6i 0.160679 + 0.278304i 0.935112 0.354351i \(-0.115298\pi\)
−0.774434 + 0.632655i \(0.781965\pi\)
\(702\) 0 0
\(703\) 1.29423e7 1.44713e7i 0.987694 1.10438i
\(704\) 0 0
\(705\) −1.23225e7 2.13432e7i −0.933739 1.61728i
\(706\) 0 0
\(707\) −1.20990e7 + 2.09561e7i −0.910336 + 1.57675i
\(708\) 0 0
\(709\) −4.41436e6 + 7.64589e6i −0.329801 + 0.571232i −0.982472 0.186409i \(-0.940315\pi\)
0.652671 + 0.757641i \(0.273648\pi\)
\(710\) 0 0
\(711\) 9.88050e6 0.733001
\(712\) 0 0
\(713\) 484459. 839107.i 0.0356889 0.0618150i
\(714\) 0 0
\(715\) 1.19085e7 0.871146
\(716\) 0 0
\(717\) 1.52989e7 + 2.64985e7i 1.11138 + 1.92497i
\(718\) 0 0
\(719\) 2.62278e6 + 4.54279e6i 0.189208 + 0.327718i 0.944986 0.327109i \(-0.106075\pi\)
−0.755778 + 0.654828i \(0.772741\pi\)
\(720\) 0 0
\(721\) 3.48398e7 2.49596
\(722\) 0 0
\(723\) 223310. 0.0158877
\(724\) 0 0
\(725\) 1.50520e6 + 2.60708e6i 0.106353 + 0.184208i
\(726\) 0 0
\(727\) −9.18888e6 1.59156e7i −0.644802 1.11683i −0.984347 0.176241i \(-0.943606\pi\)
0.339545 0.940590i \(-0.389727\pi\)
\(728\) 0 0
\(729\) −2.02351e7 −1.41022
\(730\) 0 0
\(731\) −2.92142e6 + 5.06006e6i −0.202209 + 0.350237i
\(732\) 0 0
\(733\) 2.42837e7 1.66938 0.834689 0.550721i \(-0.185647\pi\)
0.834689 + 0.550721i \(0.185647\pi\)
\(734\) 0 0
\(735\) 1.09091e7 1.88951e7i 0.744852 1.29012i
\(736\) 0 0
\(737\) −3.31048e6 + 5.73391e6i −0.224503 + 0.388850i
\(738\) 0 0
\(739\) 1.29261e6 + 2.23886e6i 0.0870674 + 0.150805i 0.906270 0.422699i \(-0.138917\pi\)
−0.819203 + 0.573504i \(0.805584\pi\)
\(740\) 0 0
\(741\) 1.40415e7 1.57005e7i 0.939441 1.05043i
\(742\) 0 0
\(743\) −9.08526e6 1.57361e7i −0.603761 1.04575i −0.992246 0.124290i \(-0.960335\pi\)
0.388485 0.921455i \(-0.372999\pi\)
\(744\) 0 0
\(745\) −5.84862e6 + 1.01301e7i −0.386067 + 0.668688i
\(746\) 0 0
\(747\) 7.48168e6 1.29586e7i 0.490566 0.849685i
\(748\) 0 0
\(749\) 9.72054e6 0.633120
\(750\) 0 0
\(751\) 7.21562e6 1.24978e7i 0.466846 0.808601i −0.532437 0.846470i \(-0.678724\pi\)
0.999283 + 0.0378689i \(0.0120569\pi\)
\(752\) 0 0
\(753\) 1.06521e7 0.684615
\(754\) 0 0
\(755\) 1.17526e7 + 2.03561e7i 0.750355 + 1.29965i
\(756\) 0 0
\(757\) −1.51265e6 2.61999e6i −0.0959399 0.166173i 0.814061 0.580780i \(-0.197252\pi\)
−0.910001 + 0.414607i \(0.863919\pi\)
\(758\) 0 0
\(759\) −1.38266e6 −0.0871184
\(760\) 0 0
\(761\) 1.91756e7 1.20029 0.600147 0.799890i \(-0.295109\pi\)
0.600147 + 0.799890i \(0.295109\pi\)
\(762\) 0 0
\(763\) −2.00460e6 3.47208e6i −0.124657 0.215913i
\(764\) 0 0
\(765\) 8.51718e6 + 1.47522e7i 0.526190 + 0.911387i
\(766\) 0 0
\(767\) 1.38008e7 0.847065
\(768\) 0 0
\(769\) −1.08868e7 + 1.88565e7i −0.663872 + 1.14986i 0.315718 + 0.948853i \(0.397755\pi\)
−0.979590 + 0.201007i \(0.935579\pi\)
\(770\) 0 0
\(771\) 1.21276e7 0.734750
\(772\) 0 0
\(773\) 1.04601e6 1.81174e6i 0.0629631 0.109055i −0.832826 0.553536i \(-0.813278\pi\)
0.895789 + 0.444480i \(0.146612\pi\)
\(774\) 0 0
\(775\) 1.93435e6 3.35040e6i 0.115686 0.200374i
\(776\) 0 0
\(777\) 2.70572e7 + 4.68644e7i 1.60779 + 2.78478i
\(778\) 0 0
\(779\) −1.83724e7 3.84072e6i −1.08473 0.226761i
\(780\) 0 0
\(781\) −1.64065e7 2.84168e7i −0.962471 1.66705i
\(782\) 0 0
\(783\) 3.57872e6 6.19852e6i 0.208604 0.361313i
\(784\) 0 0
\(785\) −1.29354e7 + 2.24048e7i −0.749215 + 1.29768i
\(786\) 0 0
\(787\) −3.41901e7 −1.96772 −0.983860 0.178942i \(-0.942733\pi\)
−0.983860 + 0.178942i \(0.942733\pi\)
\(788\) 0 0
\(789\) −3.71400e6 + 6.43284e6i −0.212398 + 0.367884i
\(790\) 0 0
\(791\) 3.40193e7 1.93323
\(792\) 0 0
\(793\) 1.23642e7 + 2.14154e7i 0.698205 + 1.20933i
\(794\) 0 0
\(795\) −5.36220e6 9.28759e6i −0.300902 0.521177i
\(796\) 0 0
\(797\) −1.39978e7 −0.780573 −0.390287 0.920693i \(-0.627624\pi\)
−0.390287 + 0.920693i \(0.627624\pi\)
\(798\) 0 0
\(799\) 2.33923e7 1.29630
\(800\) 0 0
\(801\) −1.18349e7 2.04986e7i −0.651751 1.12887i
\(802\) 0 0
\(803\) 7.07614e6 + 1.22562e7i 0.387265 + 0.670762i
\(804\) 0 0
\(805\) −1.36747e6 −0.0743754
\(806\) 0 0
\(807\) 1.67253e6 2.89690e6i 0.0904043 0.156585i
\(808\) 0 0
\(809\) −7.41687e6 −0.398428 −0.199214 0.979956i \(-0.563839\pi\)
−0.199214 + 0.979956i \(0.563839\pi\)
\(810\) 0 0
\(811\) −1.16311e7 + 2.01456e7i −0.620965 + 1.07554i 0.368341 + 0.929691i \(0.379926\pi\)
−0.989306 + 0.145852i \(0.953408\pi\)
\(812\) 0 0
\(813\) 7.10890e6 1.23130e7i 0.377204 0.653337i
\(814\) 0 0
\(815\) 1.39199e7 + 2.41100e7i 0.734079 + 1.27146i
\(816\) 0 0
\(817\) 8.04619e6 + 1.68204e6i 0.421731 + 0.0881620i
\(818\) 0 0
\(819\) 1.62574e7 + 2.81586e7i 0.846917 + 1.46690i
\(820\) 0 0
\(821\) 4.30361e6 7.45407e6i 0.222831 0.385954i −0.732836 0.680406i \(-0.761804\pi\)
0.955666 + 0.294452i \(0.0951370\pi\)
\(822\) 0 0
\(823\) −7.89709e6 + 1.36782e7i −0.406413 + 0.703928i −0.994485 0.104881i \(-0.966554\pi\)
0.588072 + 0.808809i \(0.299887\pi\)
\(824\) 0 0
\(825\) −5.52069e6 −0.282396
\(826\) 0 0
\(827\) −6.47559e6 + 1.12161e7i −0.329242 + 0.570264i −0.982362 0.186990i \(-0.940127\pi\)
0.653119 + 0.757255i \(0.273460\pi\)
\(828\) 0 0
\(829\) −1.28769e7 −0.650765 −0.325383 0.945582i \(-0.605493\pi\)
−0.325383 + 0.945582i \(0.605493\pi\)
\(830\) 0 0
\(831\) −2.26540e7 3.92379e7i −1.13800 1.97107i
\(832\) 0 0
\(833\) 1.03546e7 + 1.79347e7i 0.517036 + 0.895532i
\(834\) 0 0
\(835\) 2.45054e6 0.121631
\(836\) 0 0
\(837\) −9.19814e6 −0.453823
\(838\) 0 0
\(839\) 1.39980e7 + 2.42452e7i 0.686530 + 1.18911i 0.972953 + 0.231002i \(0.0742003\pi\)
−0.286423 + 0.958103i \(0.592466\pi\)
\(840\) 0 0
\(841\) −3.43281e6 5.94580e6i −0.167363 0.289881i
\(842\) 0 0
\(843\) −1.33809e7 −0.648508
\(844\) 0 0
\(845\) 1.06768e6 1.84927e6i 0.0514398 0.0890963i
\(846\) 0 0
\(847\) 1.50451e6 0.0720588
\(848\) 0 0
\(849\) −2.43669e7 + 4.22047e7i −1.16020 + 2.00952i
\(850\) 0 0
\(851\) 888925. 1.53966e6i 0.0420767 0.0728789i
\(852\) 0 0
\(853\) 7.33528e6 + 1.27051e7i 0.345179 + 0.597867i 0.985386 0.170335i \(-0.0544850\pi\)
−0.640208 + 0.768202i \(0.721152\pi\)
\(854\) 0 0
\(855\) 1.59759e7 1.78633e7i 0.747394 0.835693i
\(856\) 0 0
\(857\) −1.31078e7 2.27033e7i −0.609645 1.05594i −0.991299 0.131631i \(-0.957979\pi\)
0.381653 0.924306i \(-0.375355\pi\)
\(858\) 0 0
\(859\) −1.60178e7 + 2.77437e7i −0.740664 + 1.28287i 0.211530 + 0.977372i \(0.432156\pi\)
−0.952193 + 0.305496i \(0.901178\pi\)
\(860\) 0 0
\(861\) 2.61584e7 4.53076e7i 1.20255 2.08288i
\(862\) 0 0
\(863\) 2.79175e6 0.127600 0.0637998 0.997963i \(-0.479678\pi\)
0.0637998 + 0.997963i \(0.479678\pi\)
\(864\) 0 0
\(865\) −4.27133e6 + 7.39817e6i −0.194099 + 0.336189i
\(866\) 0 0
\(867\) 3.94040e6 0.178030
\(868\) 0 0
\(869\) 6.73456e6 + 1.16646e7i 0.302524 + 0.523986i
\(870\) 0 0
\(871\) −4.61823e6 7.99900e6i −0.206267 0.357265i
\(872\) 0 0
\(873\) −3.77970e7 −1.67850
\(874\) 0 0
\(875\) −3.51162e7 −1.55056
\(876\) 0 0
\(877\) 1.02849e7 + 1.78139e7i 0.451543 + 0.782096i 0.998482 0.0550767i \(-0.0175403\pi\)
−0.546939 + 0.837173i \(0.684207\pi\)
\(878\) 0 0
\(879\) −6.98009e6 1.20899e7i −0.304711 0.527776i
\(880\) 0 0
\(881\) −884571. −0.0383966 −0.0191983 0.999816i \(-0.506111\pi\)
−0.0191983 + 0.999816i \(0.506111\pi\)
\(882\) 0 0
\(883\) −2.71324e6 + 4.69946e6i −0.117108 + 0.202837i −0.918620 0.395141i \(-0.870696\pi\)
0.801513 + 0.597978i \(0.204029\pi\)
\(884\) 0 0
\(885\) 2.83524e7 1.21684
\(886\) 0 0
\(887\) −7.62216e6 + 1.32020e7i −0.325289 + 0.563417i −0.981571 0.191099i \(-0.938795\pi\)
0.656282 + 0.754516i \(0.272128\pi\)
\(888\) 0 0
\(889\) −170893. + 295995.i −0.00725220 + 0.0125612i
\(890\) 0 0
\(891\) −8.50478e6 1.47307e7i −0.358896 0.621626i
\(892\) 0 0
\(893\) −1.02748e7 3.12648e7i −0.431166 1.31198i
\(894\) 0 0
\(895\) −1.41586e7 2.45235e7i −0.590831 1.02335i
\(896\) 0 0
\(897\) 964427. 1.67044e6i 0.0400210 0.0693185i
\(898\) 0 0
\(899\) −1.75912e7 + 3.04688e7i −0.725931 + 1.25735i
\(900\) 0 0
\(901\) 1.01793e7 0.417739
\(902\) 0 0
\(903\) −1.14560e7 + 1.98425e7i −0.467536 + 0.809797i
\(904\) 0 0
\(905\) −1.88576e7 −0.765360
\(906\) 0 0
\(907\) −3.62809e6 6.28404e6i −0.146440 0.253642i 0.783469 0.621431i \(-0.213448\pi\)
−0.929909 + 0.367789i \(0.880115\pi\)
\(908\) 0 0
\(909\) −1.94169e7 3.36311e7i −0.779419 1.34999i
\(910\) 0 0
\(911\) −3.21717e6 −0.128433 −0.0642166 0.997936i \(-0.520455\pi\)
−0.0642166 + 0.997936i \(0.520455\pi\)
\(912\) 0 0
\(913\) 2.03981e7 0.809864
\(914\) 0 0
\(915\) 2.54010e7 + 4.39959e7i 1.00299 + 1.73724i
\(916\) 0 0
\(917\) 7.50468e6 + 1.29985e7i 0.294720 + 0.510469i
\(918\) 0 0
\(919\) 5.78840e6 0.226084 0.113042 0.993590i \(-0.463941\pi\)
0.113042 + 0.993590i \(0.463941\pi\)
\(920\) 0 0
\(921\) 1.66466e6 2.88327e6i 0.0646660 0.112005i
\(922\) 0 0
\(923\) 4.57752e7 1.76858
\(924\) 0 0
\(925\) 3.54931e6 6.14758e6i 0.136392 0.236238i
\(926\) 0 0
\(927\) −2.79561e7 + 4.84214e7i −1.06851 + 1.85071i
\(928\) 0 0
\(929\) −1.26109e7 2.18428e7i −0.479411 0.830364i 0.520310 0.853977i \(-0.325816\pi\)
−0.999721 + 0.0236130i \(0.992483\pi\)
\(930\) 0 0
\(931\) 1.94223e7 2.17170e7i 0.734391 0.821155i
\(932\) 0 0
\(933\) 3.89418e7 + 6.74492e7i 1.46458 + 2.53672i
\(934\) 0 0
\(935\) −1.16106e7 + 2.01102e7i −0.434337 + 0.752294i
\(936\) 0 0
\(937\) 1.25018e7 2.16537e7i 0.465182 0.805719i −0.534027 0.845467i \(-0.679322\pi\)
0.999210 + 0.0397478i \(0.0126555\pi\)
\(938\) 0 0
\(939\) 5.24383e7 1.94082
\(940\) 0 0
\(941\) −1.25003e7 + 2.16511e7i −0.460198 + 0.797087i −0.998970 0.0453646i \(-0.985555\pi\)
0.538772 + 0.842452i \(0.318888\pi\)
\(942\) 0 0
\(943\) −1.71879e6 −0.0629425
\(944\) 0 0
\(945\) 6.49085e6 + 1.12425e7i 0.236441 + 0.409527i
\(946\) 0 0
\(947\) 155598. + 269504.i 0.00563806 + 0.00976540i 0.868831 0.495109i \(-0.164872\pi\)
−0.863193 + 0.504875i \(0.831539\pi\)
\(948\) 0 0
\(949\) −1.97429e7 −0.711617
\(950\) 0 0
\(951\) −1.38098e6 −0.0495150
\(952\) 0 0
\(953\) −2.37486e7 4.11338e7i −0.847045 1.46712i −0.883834 0.467802i \(-0.845046\pi\)
0.0367888 0.999323i \(-0.488287\pi\)
\(954\) 0 0
\(955\) −1.52324e7 2.63833e7i −0.540456 0.936098i
\(956\) 0 0
\(957\) 5.02056e7 1.77203
\(958\) 0 0
\(959\) 1.86793e7 3.23535e7i 0.655864 1.13599i
\(960\) 0 0
\(961\) 1.65842e7 0.579278
\(962\) 0 0
\(963\) −7.79993e6 + 1.35099e7i −0.271035 + 0.469446i
\(964\) 0 0
\(965\) 2670.60 4625.62i 9.23190e−5 0.000159901i
\(966\) 0 0
\(967\) 1.00225e7 + 1.73594e7i 0.344674 + 0.596992i 0.985294 0.170865i \(-0.0546562\pi\)
−0.640621 + 0.767857i \(0.721323\pi\)
\(968\) 0 0
\(969\) 1.28235e7 + 3.90202e7i 0.438730 + 1.33500i
\(970\) 0 0
\(971\) −1.33282e7 2.30850e7i −0.453651 0.785747i 0.544959 0.838463i \(-0.316545\pi\)
−0.998610 + 0.0527164i \(0.983212\pi\)
\(972\) 0 0
\(973\) 3.54005e7 6.13155e7i 1.19875 2.07629i
\(974\) 0 0
\(975\) 3.85077e6 6.66973e6i 0.129729 0.224697i
\(976\) 0 0
\(977\) −2.83585e7 −0.950487 −0.475243 0.879854i \(-0.657640\pi\)
−0.475243 + 0.879854i \(0.657640\pi\)
\(978\) 0 0
\(979\) 1.61333e7 2.79437e7i 0.537980 0.931809i
\(980\) 0 0
\(981\) 6.43412e6 0.213460
\(982\) 0 0
\(983\) −8.09422e6 1.40196e7i −0.267172 0.462756i 0.700958 0.713202i \(-0.252756\pi\)
−0.968130 + 0.250446i \(0.919423\pi\)
\(984\) 0 0
\(985\) 1.77758e7 + 3.07887e7i 0.583767 + 1.01111i
\(986\) 0 0
\(987\) 9.17303e7 2.99723
\(988\) 0 0
\(989\) 752745. 0.0244713
\(990\) 0 0
\(991\) 1.72685e7 + 2.99099e7i 0.558561 + 0.967456i 0.997617 + 0.0689965i \(0.0219797\pi\)
−0.439056 + 0.898460i \(0.644687\pi\)
\(992\) 0 0
\(993\) −2.74439e7 4.75342e7i −0.883227 1.52979i
\(994\) 0 0
\(995\) 8.22282e6 0.263307
\(996\) 0 0
\(997\) 1.12806e7 1.95385e7i 0.359412 0.622521i −0.628450 0.777850i \(-0.716310\pi\)
0.987863 + 0.155329i \(0.0496438\pi\)
\(998\) 0 0
\(999\) −1.68775e7 −0.535050
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.273.8 18
4.3 odd 2 76.6.e.a.45.2 18
12.11 even 2 684.6.k.f.577.8 18
19.11 even 3 inner 304.6.i.d.49.8 18
76.11 odd 6 76.6.e.a.49.2 yes 18
228.11 even 6 684.6.k.f.505.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.2 18 4.3 odd 2
76.6.e.a.49.2 yes 18 76.11 odd 6
304.6.i.d.49.8 18 19.11 even 3 inner
304.6.i.d.273.8 18 1.1 even 1 trivial
684.6.k.f.505.8 18 228.11 even 6
684.6.k.f.577.8 18 12.11 even 2