Properties

Label 2151.2.a.j.1.17
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Root \(-1.88454\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.88454 q^{2} +1.55150 q^{4} +1.57704 q^{5} -2.47195 q^{7} -0.845212 q^{8} +O(q^{10})\) \(q+1.88454 q^{2} +1.55150 q^{4} +1.57704 q^{5} -2.47195 q^{7} -0.845212 q^{8} +2.97201 q^{10} -3.83501 q^{11} -3.18691 q^{13} -4.65849 q^{14} -4.69584 q^{16} -4.51018 q^{17} +4.54355 q^{19} +2.44679 q^{20} -7.22723 q^{22} +1.92415 q^{23} -2.51293 q^{25} -6.00587 q^{26} -3.83524 q^{28} -0.0587099 q^{29} -9.58500 q^{31} -7.15910 q^{32} -8.49962 q^{34} -3.89837 q^{35} +0.789192 q^{37} +8.56252 q^{38} -1.33294 q^{40} -2.07589 q^{41} +2.55270 q^{43} -5.95002 q^{44} +3.62615 q^{46} +11.9701 q^{47} -0.889470 q^{49} -4.73573 q^{50} -4.94450 q^{52} -0.844934 q^{53} -6.04797 q^{55} +2.08932 q^{56} -0.110641 q^{58} -1.09245 q^{59} +2.97957 q^{61} -18.0633 q^{62} -4.09994 q^{64} -5.02589 q^{65} +1.93061 q^{67} -6.99755 q^{68} -7.34665 q^{70} -8.02351 q^{71} -6.42371 q^{73} +1.48727 q^{74} +7.04934 q^{76} +9.47994 q^{77} +8.18519 q^{79} -7.40555 q^{80} -3.91210 q^{82} -9.82889 q^{83} -7.11274 q^{85} +4.81067 q^{86} +3.24139 q^{88} +8.85934 q^{89} +7.87787 q^{91} +2.98533 q^{92} +22.5581 q^{94} +7.16538 q^{95} +16.4977 q^{97} -1.67625 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.88454 1.33257 0.666287 0.745696i \(-0.267883\pi\)
0.666287 + 0.745696i \(0.267883\pi\)
\(3\) 0 0
\(4\) 1.55150 0.775751
\(5\) 1.57704 0.705275 0.352638 0.935760i \(-0.385285\pi\)
0.352638 + 0.935760i \(0.385285\pi\)
\(6\) 0 0
\(7\) −2.47195 −0.934309 −0.467154 0.884176i \(-0.654721\pi\)
−0.467154 + 0.884176i \(0.654721\pi\)
\(8\) −0.845212 −0.298828
\(9\) 0 0
\(10\) 2.97201 0.939831
\(11\) −3.83501 −1.15630 −0.578149 0.815931i \(-0.696225\pi\)
−0.578149 + 0.815931i \(0.696225\pi\)
\(12\) 0 0
\(13\) −3.18691 −0.883889 −0.441945 0.897042i \(-0.645711\pi\)
−0.441945 + 0.897042i \(0.645711\pi\)
\(14\) −4.65849 −1.24503
\(15\) 0 0
\(16\) −4.69584 −1.17396
\(17\) −4.51018 −1.09388 −0.546939 0.837172i \(-0.684207\pi\)
−0.546939 + 0.837172i \(0.684207\pi\)
\(18\) 0 0
\(19\) 4.54355 1.04236 0.521181 0.853446i \(-0.325492\pi\)
0.521181 + 0.853446i \(0.325492\pi\)
\(20\) 2.44679 0.547118
\(21\) 0 0
\(22\) −7.22723 −1.54085
\(23\) 1.92415 0.401213 0.200607 0.979672i \(-0.435709\pi\)
0.200607 + 0.979672i \(0.435709\pi\)
\(24\) 0 0
\(25\) −2.51293 −0.502587
\(26\) −6.00587 −1.17785
\(27\) 0 0
\(28\) −3.83524 −0.724791
\(29\) −0.0587099 −0.0109022 −0.00545108 0.999985i \(-0.501735\pi\)
−0.00545108 + 0.999985i \(0.501735\pi\)
\(30\) 0 0
\(31\) −9.58500 −1.72152 −0.860758 0.509014i \(-0.830010\pi\)
−0.860758 + 0.509014i \(0.830010\pi\)
\(32\) −7.15910 −1.26556
\(33\) 0 0
\(34\) −8.49962 −1.45767
\(35\) −3.89837 −0.658945
\(36\) 0 0
\(37\) 0.789192 0.129742 0.0648712 0.997894i \(-0.479336\pi\)
0.0648712 + 0.997894i \(0.479336\pi\)
\(38\) 8.56252 1.38902
\(39\) 0 0
\(40\) −1.33294 −0.210756
\(41\) −2.07589 −0.324199 −0.162100 0.986774i \(-0.551827\pi\)
−0.162100 + 0.986774i \(0.551827\pi\)
\(42\) 0 0
\(43\) 2.55270 0.389283 0.194641 0.980874i \(-0.437646\pi\)
0.194641 + 0.980874i \(0.437646\pi\)
\(44\) −5.95002 −0.897000
\(45\) 0 0
\(46\) 3.62615 0.534646
\(47\) 11.9701 1.74601 0.873007 0.487708i \(-0.162167\pi\)
0.873007 + 0.487708i \(0.162167\pi\)
\(48\) 0 0
\(49\) −0.889470 −0.127067
\(50\) −4.73573 −0.669734
\(51\) 0 0
\(52\) −4.94450 −0.685678
\(53\) −0.844934 −0.116061 −0.0580303 0.998315i \(-0.518482\pi\)
−0.0580303 + 0.998315i \(0.518482\pi\)
\(54\) 0 0
\(55\) −6.04797 −0.815508
\(56\) 2.08932 0.279197
\(57\) 0 0
\(58\) −0.110641 −0.0145279
\(59\) −1.09245 −0.142225 −0.0711123 0.997468i \(-0.522655\pi\)
−0.0711123 + 0.997468i \(0.522655\pi\)
\(60\) 0 0
\(61\) 2.97957 0.381494 0.190747 0.981639i \(-0.438909\pi\)
0.190747 + 0.981639i \(0.438909\pi\)
\(62\) −18.0633 −2.29405
\(63\) 0 0
\(64\) −4.09994 −0.512492
\(65\) −5.02589 −0.623385
\(66\) 0 0
\(67\) 1.93061 0.235862 0.117931 0.993022i \(-0.462374\pi\)
0.117931 + 0.993022i \(0.462374\pi\)
\(68\) −6.99755 −0.848578
\(69\) 0 0
\(70\) −7.34665 −0.878092
\(71\) −8.02351 −0.952215 −0.476107 0.879387i \(-0.657953\pi\)
−0.476107 + 0.879387i \(0.657953\pi\)
\(72\) 0 0
\(73\) −6.42371 −0.751839 −0.375919 0.926652i \(-0.622673\pi\)
−0.375919 + 0.926652i \(0.622673\pi\)
\(74\) 1.48727 0.172891
\(75\) 0 0
\(76\) 7.04934 0.808614
\(77\) 9.47994 1.08034
\(78\) 0 0
\(79\) 8.18519 0.920906 0.460453 0.887684i \(-0.347687\pi\)
0.460453 + 0.887684i \(0.347687\pi\)
\(80\) −7.40555 −0.827966
\(81\) 0 0
\(82\) −3.91210 −0.432019
\(83\) −9.82889 −1.07886 −0.539431 0.842030i \(-0.681360\pi\)
−0.539431 + 0.842030i \(0.681360\pi\)
\(84\) 0 0
\(85\) −7.11274 −0.771485
\(86\) 4.81067 0.518748
\(87\) 0 0
\(88\) 3.24139 0.345534
\(89\) 8.85934 0.939088 0.469544 0.882909i \(-0.344418\pi\)
0.469544 + 0.882909i \(0.344418\pi\)
\(90\) 0 0
\(91\) 7.87787 0.825825
\(92\) 2.98533 0.311242
\(93\) 0 0
\(94\) 22.5581 2.32669
\(95\) 7.16538 0.735153
\(96\) 0 0
\(97\) 16.4977 1.67508 0.837541 0.546374i \(-0.183992\pi\)
0.837541 + 0.546374i \(0.183992\pi\)
\(98\) −1.67625 −0.169326
\(99\) 0 0
\(100\) −3.89882 −0.389882
\(101\) −8.00022 −0.796052 −0.398026 0.917374i \(-0.630305\pi\)
−0.398026 + 0.917374i \(0.630305\pi\)
\(102\) 0 0
\(103\) −4.66612 −0.459766 −0.229883 0.973218i \(-0.573834\pi\)
−0.229883 + 0.973218i \(0.573834\pi\)
\(104\) 2.69361 0.264130
\(105\) 0 0
\(106\) −1.59232 −0.154659
\(107\) −1.59511 −0.154205 −0.0771023 0.997023i \(-0.524567\pi\)
−0.0771023 + 0.997023i \(0.524567\pi\)
\(108\) 0 0
\(109\) 6.67236 0.639096 0.319548 0.947570i \(-0.396469\pi\)
0.319548 + 0.947570i \(0.396469\pi\)
\(110\) −11.3977 −1.08672
\(111\) 0 0
\(112\) 11.6079 1.09684
\(113\) 1.46613 0.137922 0.0689612 0.997619i \(-0.478032\pi\)
0.0689612 + 0.997619i \(0.478032\pi\)
\(114\) 0 0
\(115\) 3.03447 0.282966
\(116\) −0.0910886 −0.00845736
\(117\) 0 0
\(118\) −2.05876 −0.189525
\(119\) 11.1489 1.02202
\(120\) 0 0
\(121\) 3.70727 0.337024
\(122\) 5.61512 0.508369
\(123\) 0 0
\(124\) −14.8712 −1.33547
\(125\) −11.8482 −1.05974
\(126\) 0 0
\(127\) −15.0707 −1.33731 −0.668655 0.743572i \(-0.733130\pi\)
−0.668655 + 0.743572i \(0.733130\pi\)
\(128\) 6.59168 0.582628
\(129\) 0 0
\(130\) −9.47151 −0.830706
\(131\) −10.1307 −0.885122 −0.442561 0.896738i \(-0.645930\pi\)
−0.442561 + 0.896738i \(0.645930\pi\)
\(132\) 0 0
\(133\) −11.2314 −0.973888
\(134\) 3.63832 0.314303
\(135\) 0 0
\(136\) 3.81205 0.326881
\(137\) −1.49257 −0.127519 −0.0637596 0.997965i \(-0.520309\pi\)
−0.0637596 + 0.997965i \(0.520309\pi\)
\(138\) 0 0
\(139\) 22.4652 1.90547 0.952736 0.303800i \(-0.0982555\pi\)
0.952736 + 0.303800i \(0.0982555\pi\)
\(140\) −6.04833 −0.511177
\(141\) 0 0
\(142\) −15.1206 −1.26890
\(143\) 12.2218 1.02204
\(144\) 0 0
\(145\) −0.0925881 −0.00768902
\(146\) −12.1058 −1.00188
\(147\) 0 0
\(148\) 1.22443 0.100648
\(149\) −11.3282 −0.928045 −0.464022 0.885823i \(-0.653594\pi\)
−0.464022 + 0.885823i \(0.653594\pi\)
\(150\) 0 0
\(151\) 10.2512 0.834231 0.417116 0.908853i \(-0.363041\pi\)
0.417116 + 0.908853i \(0.363041\pi\)
\(152\) −3.84027 −0.311487
\(153\) 0 0
\(154\) 17.8653 1.43963
\(155\) −15.1160 −1.21414
\(156\) 0 0
\(157\) −2.23497 −0.178370 −0.0891849 0.996015i \(-0.528426\pi\)
−0.0891849 + 0.996015i \(0.528426\pi\)
\(158\) 15.4253 1.22717
\(159\) 0 0
\(160\) −11.2902 −0.892569
\(161\) −4.75640 −0.374857
\(162\) 0 0
\(163\) −11.1937 −0.876762 −0.438381 0.898789i \(-0.644448\pi\)
−0.438381 + 0.898789i \(0.644448\pi\)
\(164\) −3.22075 −0.251498
\(165\) 0 0
\(166\) −18.5230 −1.43766
\(167\) −9.54997 −0.738999 −0.369499 0.929231i \(-0.620471\pi\)
−0.369499 + 0.929231i \(0.620471\pi\)
\(168\) 0 0
\(169\) −2.84362 −0.218740
\(170\) −13.4043 −1.02806
\(171\) 0 0
\(172\) 3.96052 0.301987
\(173\) 7.16166 0.544491 0.272246 0.962228i \(-0.412234\pi\)
0.272246 + 0.962228i \(0.412234\pi\)
\(174\) 0 0
\(175\) 6.21184 0.469571
\(176\) 18.0086 1.35745
\(177\) 0 0
\(178\) 16.6958 1.25140
\(179\) 0.114586 0.00856454 0.00428227 0.999991i \(-0.498637\pi\)
0.00428227 + 0.999991i \(0.498637\pi\)
\(180\) 0 0
\(181\) −2.24629 −0.166966 −0.0834828 0.996509i \(-0.526604\pi\)
−0.0834828 + 0.996509i \(0.526604\pi\)
\(182\) 14.8462 1.10047
\(183\) 0 0
\(184\) −1.62632 −0.119894
\(185\) 1.24459 0.0915041
\(186\) 0 0
\(187\) 17.2965 1.26485
\(188\) 18.5716 1.35447
\(189\) 0 0
\(190\) 13.5035 0.979645
\(191\) −7.90682 −0.572117 −0.286059 0.958212i \(-0.592345\pi\)
−0.286059 + 0.958212i \(0.592345\pi\)
\(192\) 0 0
\(193\) −10.5321 −0.758120 −0.379060 0.925372i \(-0.623753\pi\)
−0.379060 + 0.925372i \(0.623753\pi\)
\(194\) 31.0905 2.23217
\(195\) 0 0
\(196\) −1.38002 −0.0985726
\(197\) −16.5832 −1.18150 −0.590751 0.806854i \(-0.701168\pi\)
−0.590751 + 0.806854i \(0.701168\pi\)
\(198\) 0 0
\(199\) 12.2741 0.870089 0.435044 0.900409i \(-0.356733\pi\)
0.435044 + 0.900409i \(0.356733\pi\)
\(200\) 2.12396 0.150187
\(201\) 0 0
\(202\) −15.0768 −1.06080
\(203\) 0.145128 0.0101860
\(204\) 0 0
\(205\) −3.27377 −0.228650
\(206\) −8.79350 −0.612672
\(207\) 0 0
\(208\) 14.9652 1.03765
\(209\) −17.4246 −1.20528
\(210\) 0 0
\(211\) −1.24295 −0.0855685 −0.0427842 0.999084i \(-0.513623\pi\)
−0.0427842 + 0.999084i \(0.513623\pi\)
\(212\) −1.31092 −0.0900342
\(213\) 0 0
\(214\) −3.00605 −0.205489
\(215\) 4.02571 0.274551
\(216\) 0 0
\(217\) 23.6936 1.60843
\(218\) 12.5743 0.851642
\(219\) 0 0
\(220\) −9.38344 −0.632632
\(221\) 14.3735 0.966867
\(222\) 0 0
\(223\) 17.4338 1.16745 0.583727 0.811950i \(-0.301594\pi\)
0.583727 + 0.811950i \(0.301594\pi\)
\(224\) 17.6969 1.18243
\(225\) 0 0
\(226\) 2.76299 0.183792
\(227\) −1.91090 −0.126831 −0.0634154 0.997987i \(-0.520199\pi\)
−0.0634154 + 0.997987i \(0.520199\pi\)
\(228\) 0 0
\(229\) −12.7477 −0.842389 −0.421195 0.906970i \(-0.638389\pi\)
−0.421195 + 0.906970i \(0.638389\pi\)
\(230\) 5.71859 0.377073
\(231\) 0 0
\(232\) 0.0496223 0.00325787
\(233\) −9.04128 −0.592314 −0.296157 0.955139i \(-0.595705\pi\)
−0.296157 + 0.955139i \(0.595705\pi\)
\(234\) 0 0
\(235\) 18.8773 1.23142
\(236\) −1.69494 −0.110331
\(237\) 0 0
\(238\) 21.0106 1.36192
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) 21.3266 1.37377 0.686884 0.726767i \(-0.258978\pi\)
0.686884 + 0.726767i \(0.258978\pi\)
\(242\) 6.98650 0.449110
\(243\) 0 0
\(244\) 4.62281 0.295945
\(245\) −1.40273 −0.0896174
\(246\) 0 0
\(247\) −14.4799 −0.921333
\(248\) 8.10135 0.514437
\(249\) 0 0
\(250\) −22.3285 −1.41218
\(251\) −29.4696 −1.86010 −0.930052 0.367427i \(-0.880239\pi\)
−0.930052 + 0.367427i \(0.880239\pi\)
\(252\) 0 0
\(253\) −7.37913 −0.463922
\(254\) −28.4014 −1.78206
\(255\) 0 0
\(256\) 20.6222 1.28889
\(257\) −6.15087 −0.383681 −0.191840 0.981426i \(-0.561446\pi\)
−0.191840 + 0.981426i \(0.561446\pi\)
\(258\) 0 0
\(259\) −1.95084 −0.121219
\(260\) −7.79769 −0.483592
\(261\) 0 0
\(262\) −19.0917 −1.17949
\(263\) 10.2302 0.630823 0.315412 0.948955i \(-0.397857\pi\)
0.315412 + 0.948955i \(0.397857\pi\)
\(264\) 0 0
\(265\) −1.33250 −0.0818547
\(266\) −21.1661 −1.29778
\(267\) 0 0
\(268\) 2.99535 0.182970
\(269\) −27.2980 −1.66439 −0.832193 0.554486i \(-0.812915\pi\)
−0.832193 + 0.554486i \(0.812915\pi\)
\(270\) 0 0
\(271\) 21.5829 1.31107 0.655534 0.755166i \(-0.272444\pi\)
0.655534 + 0.755166i \(0.272444\pi\)
\(272\) 21.1791 1.28417
\(273\) 0 0
\(274\) −2.81282 −0.169929
\(275\) 9.63712 0.581140
\(276\) 0 0
\(277\) −16.8035 −1.00962 −0.504812 0.863230i \(-0.668438\pi\)
−0.504812 + 0.863230i \(0.668438\pi\)
\(278\) 42.3366 2.53918
\(279\) 0 0
\(280\) 3.29495 0.196911
\(281\) −2.74082 −0.163504 −0.0817519 0.996653i \(-0.526052\pi\)
−0.0817519 + 0.996653i \(0.526052\pi\)
\(282\) 0 0
\(283\) 19.1177 1.13643 0.568214 0.822881i \(-0.307635\pi\)
0.568214 + 0.822881i \(0.307635\pi\)
\(284\) −12.4485 −0.738682
\(285\) 0 0
\(286\) 23.0325 1.36194
\(287\) 5.13149 0.302902
\(288\) 0 0
\(289\) 3.34168 0.196569
\(290\) −0.174486 −0.0102462
\(291\) 0 0
\(292\) −9.96641 −0.583240
\(293\) −26.6427 −1.55648 −0.778241 0.627966i \(-0.783888\pi\)
−0.778241 + 0.627966i \(0.783888\pi\)
\(294\) 0 0
\(295\) −1.72284 −0.100307
\(296\) −0.667034 −0.0387706
\(297\) 0 0
\(298\) −21.3485 −1.23669
\(299\) −6.13210 −0.354628
\(300\) 0 0
\(301\) −6.31014 −0.363710
\(302\) 19.3188 1.11167
\(303\) 0 0
\(304\) −21.3358 −1.22369
\(305\) 4.69891 0.269059
\(306\) 0 0
\(307\) 15.0399 0.858373 0.429186 0.903216i \(-0.358800\pi\)
0.429186 + 0.903216i \(0.358800\pi\)
\(308\) 14.7081 0.838075
\(309\) 0 0
\(310\) −28.4867 −1.61793
\(311\) −13.7719 −0.780933 −0.390467 0.920617i \(-0.627686\pi\)
−0.390467 + 0.920617i \(0.627686\pi\)
\(312\) 0 0
\(313\) −1.15686 −0.0653898 −0.0326949 0.999465i \(-0.510409\pi\)
−0.0326949 + 0.999465i \(0.510409\pi\)
\(314\) −4.21189 −0.237691
\(315\) 0 0
\(316\) 12.6993 0.714394
\(317\) 19.8540 1.11511 0.557557 0.830139i \(-0.311739\pi\)
0.557557 + 0.830139i \(0.311739\pi\)
\(318\) 0 0
\(319\) 0.225153 0.0126061
\(320\) −6.46578 −0.361448
\(321\) 0 0
\(322\) −8.96365 −0.499525
\(323\) −20.4922 −1.14022
\(324\) 0 0
\(325\) 8.00849 0.444231
\(326\) −21.0951 −1.16835
\(327\) 0 0
\(328\) 1.75457 0.0968797
\(329\) −29.5894 −1.63132
\(330\) 0 0
\(331\) 14.5712 0.800904 0.400452 0.916318i \(-0.368853\pi\)
0.400452 + 0.916318i \(0.368853\pi\)
\(332\) −15.2496 −0.836928
\(333\) 0 0
\(334\) −17.9973 −0.984770
\(335\) 3.04466 0.166347
\(336\) 0 0
\(337\) 15.1805 0.826933 0.413466 0.910519i \(-0.364318\pi\)
0.413466 + 0.910519i \(0.364318\pi\)
\(338\) −5.35892 −0.291487
\(339\) 0 0
\(340\) −11.0354 −0.598481
\(341\) 36.7585 1.99059
\(342\) 0 0
\(343\) 19.5024 1.05303
\(344\) −2.15757 −0.116328
\(345\) 0 0
\(346\) 13.4965 0.725574
\(347\) 10.1274 0.543667 0.271834 0.962344i \(-0.412370\pi\)
0.271834 + 0.962344i \(0.412370\pi\)
\(348\) 0 0
\(349\) 14.4379 0.772842 0.386421 0.922322i \(-0.373711\pi\)
0.386421 + 0.922322i \(0.373711\pi\)
\(350\) 11.7065 0.625738
\(351\) 0 0
\(352\) 27.4552 1.46337
\(353\) −24.6191 −1.31034 −0.655171 0.755481i \(-0.727403\pi\)
−0.655171 + 0.755481i \(0.727403\pi\)
\(354\) 0 0
\(355\) −12.6534 −0.671574
\(356\) 13.7453 0.728499
\(357\) 0 0
\(358\) 0.215942 0.0114129
\(359\) 15.5002 0.818070 0.409035 0.912519i \(-0.365865\pi\)
0.409035 + 0.912519i \(0.365865\pi\)
\(360\) 0 0
\(361\) 1.64388 0.0865198
\(362\) −4.23324 −0.222494
\(363\) 0 0
\(364\) 12.2225 0.640635
\(365\) −10.1305 −0.530253
\(366\) 0 0
\(367\) −11.1151 −0.580205 −0.290103 0.956996i \(-0.593689\pi\)
−0.290103 + 0.956996i \(0.593689\pi\)
\(368\) −9.03552 −0.471009
\(369\) 0 0
\(370\) 2.34548 0.121936
\(371\) 2.08863 0.108436
\(372\) 0 0
\(373\) 28.7318 1.48767 0.743837 0.668361i \(-0.233004\pi\)
0.743837 + 0.668361i \(0.233004\pi\)
\(374\) 32.5961 1.68550
\(375\) 0 0
\(376\) −10.1172 −0.521757
\(377\) 0.187103 0.00963630
\(378\) 0 0
\(379\) −23.0688 −1.18496 −0.592481 0.805584i \(-0.701852\pi\)
−0.592481 + 0.805584i \(0.701852\pi\)
\(380\) 11.1171 0.570296
\(381\) 0 0
\(382\) −14.9007 −0.762388
\(383\) 25.1173 1.28343 0.641717 0.766941i \(-0.278222\pi\)
0.641717 + 0.766941i \(0.278222\pi\)
\(384\) 0 0
\(385\) 14.9503 0.761936
\(386\) −19.8483 −1.01025
\(387\) 0 0
\(388\) 25.5962 1.29945
\(389\) −20.2979 −1.02915 −0.514573 0.857447i \(-0.672049\pi\)
−0.514573 + 0.857447i \(0.672049\pi\)
\(390\) 0 0
\(391\) −8.67826 −0.438879
\(392\) 0.751791 0.0379712
\(393\) 0 0
\(394\) −31.2517 −1.57444
\(395\) 12.9084 0.649492
\(396\) 0 0
\(397\) −5.09694 −0.255808 −0.127904 0.991787i \(-0.540825\pi\)
−0.127904 + 0.991787i \(0.540825\pi\)
\(398\) 23.1311 1.15946
\(399\) 0 0
\(400\) 11.8003 0.590017
\(401\) −22.6376 −1.13047 −0.565235 0.824930i \(-0.691214\pi\)
−0.565235 + 0.824930i \(0.691214\pi\)
\(402\) 0 0
\(403\) 30.5465 1.52163
\(404\) −12.4124 −0.617539
\(405\) 0 0
\(406\) 0.273500 0.0135736
\(407\) −3.02656 −0.150021
\(408\) 0 0
\(409\) −12.2093 −0.603712 −0.301856 0.953353i \(-0.597606\pi\)
−0.301856 + 0.953353i \(0.597606\pi\)
\(410\) −6.16955 −0.304693
\(411\) 0 0
\(412\) −7.23950 −0.356665
\(413\) 2.70047 0.132882
\(414\) 0 0
\(415\) −15.5006 −0.760894
\(416\) 22.8154 1.11862
\(417\) 0 0
\(418\) −32.8373 −1.60613
\(419\) 36.9777 1.80648 0.903239 0.429137i \(-0.141182\pi\)
0.903239 + 0.429137i \(0.141182\pi\)
\(420\) 0 0
\(421\) −23.2814 −1.13466 −0.567332 0.823489i \(-0.692024\pi\)
−0.567332 + 0.823489i \(0.692024\pi\)
\(422\) −2.34240 −0.114026
\(423\) 0 0
\(424\) 0.714149 0.0346821
\(425\) 11.3338 0.549769
\(426\) 0 0
\(427\) −7.36534 −0.356434
\(428\) −2.47481 −0.119625
\(429\) 0 0
\(430\) 7.58663 0.365860
\(431\) 34.2064 1.64766 0.823832 0.566834i \(-0.191832\pi\)
0.823832 + 0.566834i \(0.191832\pi\)
\(432\) 0 0
\(433\) 0.581765 0.0279578 0.0139789 0.999902i \(-0.495550\pi\)
0.0139789 + 0.999902i \(0.495550\pi\)
\(434\) 44.6516 2.14335
\(435\) 0 0
\(436\) 10.3522 0.495780
\(437\) 8.74249 0.418210
\(438\) 0 0
\(439\) −15.5628 −0.742771 −0.371386 0.928479i \(-0.621117\pi\)
−0.371386 + 0.928479i \(0.621117\pi\)
\(440\) 5.11182 0.243696
\(441\) 0 0
\(442\) 27.0875 1.28842
\(443\) 37.1224 1.76374 0.881870 0.471493i \(-0.156285\pi\)
0.881870 + 0.471493i \(0.156285\pi\)
\(444\) 0 0
\(445\) 13.9716 0.662316
\(446\) 32.8548 1.55572
\(447\) 0 0
\(448\) 10.1348 0.478826
\(449\) 16.6269 0.784670 0.392335 0.919822i \(-0.371667\pi\)
0.392335 + 0.919822i \(0.371667\pi\)
\(450\) 0 0
\(451\) 7.96104 0.374871
\(452\) 2.27471 0.106993
\(453\) 0 0
\(454\) −3.60117 −0.169011
\(455\) 12.4237 0.582434
\(456\) 0 0
\(457\) 10.3822 0.485659 0.242829 0.970069i \(-0.421924\pi\)
0.242829 + 0.970069i \(0.421924\pi\)
\(458\) −24.0235 −1.12255
\(459\) 0 0
\(460\) 4.70799 0.219511
\(461\) −13.7020 −0.638164 −0.319082 0.947727i \(-0.603375\pi\)
−0.319082 + 0.947727i \(0.603375\pi\)
\(462\) 0 0
\(463\) −41.4574 −1.92669 −0.963345 0.268265i \(-0.913550\pi\)
−0.963345 + 0.268265i \(0.913550\pi\)
\(464\) 0.275693 0.0127987
\(465\) 0 0
\(466\) −17.0387 −0.789302
\(467\) 19.8888 0.920345 0.460173 0.887829i \(-0.347788\pi\)
0.460173 + 0.887829i \(0.347788\pi\)
\(468\) 0 0
\(469\) −4.77237 −0.220367
\(470\) 35.5751 1.64096
\(471\) 0 0
\(472\) 0.923350 0.0425006
\(473\) −9.78961 −0.450127
\(474\) 0 0
\(475\) −11.4176 −0.523878
\(476\) 17.2976 0.792833
\(477\) 0 0
\(478\) −1.88454 −0.0861970
\(479\) 13.2603 0.605878 0.302939 0.953010i \(-0.402032\pi\)
0.302939 + 0.953010i \(0.402032\pi\)
\(480\) 0 0
\(481\) −2.51508 −0.114678
\(482\) 40.1909 1.83065
\(483\) 0 0
\(484\) 5.75184 0.261447
\(485\) 26.0175 1.18139
\(486\) 0 0
\(487\) 35.1123 1.59109 0.795545 0.605895i \(-0.207185\pi\)
0.795545 + 0.605895i \(0.207185\pi\)
\(488\) −2.51837 −0.114001
\(489\) 0 0
\(490\) −2.64351 −0.119422
\(491\) −6.60916 −0.298267 −0.149134 0.988817i \(-0.547648\pi\)
−0.149134 + 0.988817i \(0.547648\pi\)
\(492\) 0 0
\(493\) 0.264792 0.0119256
\(494\) −27.2880 −1.22774
\(495\) 0 0
\(496\) 45.0097 2.02099
\(497\) 19.8337 0.889663
\(498\) 0 0
\(499\) −22.6811 −1.01534 −0.507672 0.861550i \(-0.669494\pi\)
−0.507672 + 0.861550i \(0.669494\pi\)
\(500\) −18.3826 −0.822093
\(501\) 0 0
\(502\) −55.5367 −2.47873
\(503\) 30.0329 1.33910 0.669549 0.742768i \(-0.266487\pi\)
0.669549 + 0.742768i \(0.266487\pi\)
\(504\) 0 0
\(505\) −12.6167 −0.561436
\(506\) −13.9063 −0.618210
\(507\) 0 0
\(508\) −23.3823 −1.03742
\(509\) −39.2106 −1.73798 −0.868990 0.494829i \(-0.835231\pi\)
−0.868990 + 0.494829i \(0.835231\pi\)
\(510\) 0 0
\(511\) 15.8791 0.702450
\(512\) 25.6800 1.13491
\(513\) 0 0
\(514\) −11.5916 −0.511283
\(515\) −7.35867 −0.324262
\(516\) 0 0
\(517\) −45.9053 −2.01891
\(518\) −3.67645 −0.161534
\(519\) 0 0
\(520\) 4.24794 0.186285
\(521\) −34.0212 −1.49049 −0.745247 0.666789i \(-0.767668\pi\)
−0.745247 + 0.666789i \(0.767668\pi\)
\(522\) 0 0
\(523\) −26.3066 −1.15031 −0.575153 0.818046i \(-0.695058\pi\)
−0.575153 + 0.818046i \(0.695058\pi\)
\(524\) −15.7178 −0.686635
\(525\) 0 0
\(526\) 19.2793 0.840618
\(527\) 43.2300 1.88313
\(528\) 0 0
\(529\) −19.2976 −0.839028
\(530\) −2.51115 −0.109077
\(531\) 0 0
\(532\) −17.4256 −0.755495
\(533\) 6.61567 0.286556
\(534\) 0 0
\(535\) −2.51555 −0.108757
\(536\) −1.63177 −0.0704819
\(537\) 0 0
\(538\) −51.4442 −2.21792
\(539\) 3.41112 0.146928
\(540\) 0 0
\(541\) 16.1226 0.693163 0.346581 0.938020i \(-0.387342\pi\)
0.346581 + 0.938020i \(0.387342\pi\)
\(542\) 40.6739 1.74709
\(543\) 0 0
\(544\) 32.2888 1.38437
\(545\) 10.5226 0.450739
\(546\) 0 0
\(547\) −0.430130 −0.0183910 −0.00919552 0.999958i \(-0.502927\pi\)
−0.00919552 + 0.999958i \(0.502927\pi\)
\(548\) −2.31573 −0.0989232
\(549\) 0 0
\(550\) 18.1616 0.774412
\(551\) −0.266752 −0.0113640
\(552\) 0 0
\(553\) −20.2334 −0.860410
\(554\) −31.6669 −1.34540
\(555\) 0 0
\(556\) 34.8548 1.47817
\(557\) −13.6791 −0.579602 −0.289801 0.957087i \(-0.593589\pi\)
−0.289801 + 0.957087i \(0.593589\pi\)
\(558\) 0 0
\(559\) −8.13521 −0.344083
\(560\) 18.3061 0.773576
\(561\) 0 0
\(562\) −5.16520 −0.217881
\(563\) −13.3807 −0.563931 −0.281965 0.959425i \(-0.590986\pi\)
−0.281965 + 0.959425i \(0.590986\pi\)
\(564\) 0 0
\(565\) 2.31216 0.0972732
\(566\) 36.0281 1.51437
\(567\) 0 0
\(568\) 6.78156 0.284548
\(569\) 5.69249 0.238641 0.119321 0.992856i \(-0.461928\pi\)
0.119321 + 0.992856i \(0.461928\pi\)
\(570\) 0 0
\(571\) 14.9082 0.623889 0.311944 0.950100i \(-0.399020\pi\)
0.311944 + 0.950100i \(0.399020\pi\)
\(572\) 18.9622 0.792848
\(573\) 0 0
\(574\) 9.67051 0.403639
\(575\) −4.83527 −0.201645
\(576\) 0 0
\(577\) −26.0799 −1.08572 −0.542860 0.839823i \(-0.682659\pi\)
−0.542860 + 0.839823i \(0.682659\pi\)
\(578\) 6.29754 0.261943
\(579\) 0 0
\(580\) −0.143651 −0.00596477
\(581\) 24.2965 1.00799
\(582\) 0 0
\(583\) 3.24033 0.134201
\(584\) 5.42940 0.224670
\(585\) 0 0
\(586\) −50.2093 −2.07413
\(587\) −20.0301 −0.826729 −0.413365 0.910566i \(-0.635646\pi\)
−0.413365 + 0.910566i \(0.635646\pi\)
\(588\) 0 0
\(589\) −43.5499 −1.79444
\(590\) −3.24676 −0.133667
\(591\) 0 0
\(592\) −3.70592 −0.152312
\(593\) −28.0351 −1.15126 −0.575632 0.817709i \(-0.695244\pi\)
−0.575632 + 0.817709i \(0.695244\pi\)
\(594\) 0 0
\(595\) 17.5823 0.720805
\(596\) −17.5758 −0.719932
\(597\) 0 0
\(598\) −11.5562 −0.472568
\(599\) −31.5499 −1.28909 −0.644547 0.764564i \(-0.722954\pi\)
−0.644547 + 0.764564i \(0.722954\pi\)
\(600\) 0 0
\(601\) 2.70265 0.110243 0.0551217 0.998480i \(-0.482445\pi\)
0.0551217 + 0.998480i \(0.482445\pi\)
\(602\) −11.8917 −0.484670
\(603\) 0 0
\(604\) 15.9048 0.647156
\(605\) 5.84652 0.237695
\(606\) 0 0
\(607\) −39.4724 −1.60213 −0.801067 0.598575i \(-0.795734\pi\)
−0.801067 + 0.598575i \(0.795734\pi\)
\(608\) −32.5277 −1.31917
\(609\) 0 0
\(610\) 8.85529 0.358540
\(611\) −38.1475 −1.54328
\(612\) 0 0
\(613\) 17.3060 0.698982 0.349491 0.936940i \(-0.386354\pi\)
0.349491 + 0.936940i \(0.386354\pi\)
\(614\) 28.3434 1.14384
\(615\) 0 0
\(616\) −8.01256 −0.322835
\(617\) −42.7513 −1.72110 −0.860551 0.509365i \(-0.829880\pi\)
−0.860551 + 0.509365i \(0.829880\pi\)
\(618\) 0 0
\(619\) 2.29106 0.0920853 0.0460427 0.998939i \(-0.485339\pi\)
0.0460427 + 0.998939i \(0.485339\pi\)
\(620\) −23.4525 −0.941873
\(621\) 0 0
\(622\) −25.9538 −1.04065
\(623\) −21.8998 −0.877398
\(624\) 0 0
\(625\) −6.12049 −0.244820
\(626\) −2.18016 −0.0871367
\(627\) 0 0
\(628\) −3.46756 −0.138371
\(629\) −3.55939 −0.141922
\(630\) 0 0
\(631\) 40.3021 1.60440 0.802201 0.597054i \(-0.203662\pi\)
0.802201 + 0.597054i \(0.203662\pi\)
\(632\) −6.91822 −0.275192
\(633\) 0 0
\(634\) 37.4158 1.48597
\(635\) −23.7672 −0.943172
\(636\) 0 0
\(637\) 2.83466 0.112313
\(638\) 0.424310 0.0167986
\(639\) 0 0
\(640\) 10.3954 0.410913
\(641\) −12.3428 −0.487511 −0.243756 0.969837i \(-0.578379\pi\)
−0.243756 + 0.969837i \(0.578379\pi\)
\(642\) 0 0
\(643\) 6.36474 0.251001 0.125501 0.992094i \(-0.459946\pi\)
0.125501 + 0.992094i \(0.459946\pi\)
\(644\) −7.37958 −0.290796
\(645\) 0 0
\(646\) −38.6185 −1.51942
\(647\) 13.9746 0.549399 0.274700 0.961530i \(-0.411422\pi\)
0.274700 + 0.961530i \(0.411422\pi\)
\(648\) 0 0
\(649\) 4.18954 0.164454
\(650\) 15.0923 0.591970
\(651\) 0 0
\(652\) −17.3671 −0.680149
\(653\) 0.742205 0.0290447 0.0145224 0.999895i \(-0.495377\pi\)
0.0145224 + 0.999895i \(0.495377\pi\)
\(654\) 0 0
\(655\) −15.9765 −0.624255
\(656\) 9.74805 0.380597
\(657\) 0 0
\(658\) −55.7625 −2.17385
\(659\) 41.9051 1.63239 0.816196 0.577776i \(-0.196079\pi\)
0.816196 + 0.577776i \(0.196079\pi\)
\(660\) 0 0
\(661\) −30.6618 −1.19261 −0.596304 0.802759i \(-0.703365\pi\)
−0.596304 + 0.802759i \(0.703365\pi\)
\(662\) 27.4600 1.06726
\(663\) 0 0
\(664\) 8.30750 0.322393
\(665\) −17.7125 −0.686859
\(666\) 0 0
\(667\) −0.112967 −0.00437409
\(668\) −14.8168 −0.573279
\(669\) 0 0
\(670\) 5.73778 0.221670
\(671\) −11.4267 −0.441121
\(672\) 0 0
\(673\) 41.2232 1.58904 0.794519 0.607240i \(-0.207723\pi\)
0.794519 + 0.607240i \(0.207723\pi\)
\(674\) 28.6082 1.10195
\(675\) 0 0
\(676\) −4.41188 −0.169688
\(677\) 42.5360 1.63479 0.817395 0.576078i \(-0.195417\pi\)
0.817395 + 0.576078i \(0.195417\pi\)
\(678\) 0 0
\(679\) −40.7814 −1.56504
\(680\) 6.01178 0.230541
\(681\) 0 0
\(682\) 69.2730 2.65260
\(683\) −32.3571 −1.23811 −0.619055 0.785348i \(-0.712484\pi\)
−0.619055 + 0.785348i \(0.712484\pi\)
\(684\) 0 0
\(685\) −2.35385 −0.0899361
\(686\) 36.7530 1.40324
\(687\) 0 0
\(688\) −11.9871 −0.457003
\(689\) 2.69273 0.102585
\(690\) 0 0
\(691\) 29.4308 1.11960 0.559801 0.828627i \(-0.310878\pi\)
0.559801 + 0.828627i \(0.310878\pi\)
\(692\) 11.1113 0.422390
\(693\) 0 0
\(694\) 19.0855 0.724476
\(695\) 35.4286 1.34388
\(696\) 0 0
\(697\) 9.36262 0.354635
\(698\) 27.2088 1.02987
\(699\) 0 0
\(700\) 9.63769 0.364271
\(701\) −27.9008 −1.05380 −0.526899 0.849928i \(-0.676646\pi\)
−0.526899 + 0.849928i \(0.676646\pi\)
\(702\) 0 0
\(703\) 3.58574 0.135239
\(704\) 15.7233 0.592594
\(705\) 0 0
\(706\) −46.3957 −1.74613
\(707\) 19.7761 0.743758
\(708\) 0 0
\(709\) −13.6158 −0.511351 −0.255676 0.966763i \(-0.582298\pi\)
−0.255676 + 0.966763i \(0.582298\pi\)
\(710\) −23.8459 −0.894921
\(711\) 0 0
\(712\) −7.48802 −0.280625
\(713\) −18.4430 −0.690695
\(714\) 0 0
\(715\) 19.2743 0.720819
\(716\) 0.177780 0.00664395
\(717\) 0 0
\(718\) 29.2108 1.09014
\(719\) −42.0015 −1.56639 −0.783195 0.621776i \(-0.786411\pi\)
−0.783195 + 0.621776i \(0.786411\pi\)
\(720\) 0 0
\(721\) 11.5344 0.429564
\(722\) 3.09796 0.115294
\(723\) 0 0
\(724\) −3.48513 −0.129524
\(725\) 0.147534 0.00547928
\(726\) 0 0
\(727\) −45.9245 −1.70324 −0.851622 0.524156i \(-0.824381\pi\)
−0.851622 + 0.524156i \(0.824381\pi\)
\(728\) −6.65847 −0.246779
\(729\) 0 0
\(730\) −19.0913 −0.706601
\(731\) −11.5131 −0.425828
\(732\) 0 0
\(733\) 2.25611 0.0833315 0.0416657 0.999132i \(-0.486734\pi\)
0.0416657 + 0.999132i \(0.486734\pi\)
\(734\) −20.9469 −0.773166
\(735\) 0 0
\(736\) −13.7752 −0.507760
\(737\) −7.40390 −0.272726
\(738\) 0 0
\(739\) −11.6417 −0.428248 −0.214124 0.976806i \(-0.568690\pi\)
−0.214124 + 0.976806i \(0.568690\pi\)
\(740\) 1.93098 0.0709844
\(741\) 0 0
\(742\) 3.93612 0.144500
\(743\) 9.19350 0.337277 0.168638 0.985678i \(-0.446063\pi\)
0.168638 + 0.985678i \(0.446063\pi\)
\(744\) 0 0
\(745\) −17.8651 −0.654527
\(746\) 54.1463 1.98244
\(747\) 0 0
\(748\) 26.8356 0.981208
\(749\) 3.94302 0.144075
\(750\) 0 0
\(751\) 53.6899 1.95917 0.979586 0.201027i \(-0.0644280\pi\)
0.979586 + 0.201027i \(0.0644280\pi\)
\(752\) −56.2096 −2.04975
\(753\) 0 0
\(754\) 0.352604 0.0128411
\(755\) 16.1666 0.588363
\(756\) 0 0
\(757\) −16.7983 −0.610546 −0.305273 0.952265i \(-0.598748\pi\)
−0.305273 + 0.952265i \(0.598748\pi\)
\(758\) −43.4741 −1.57905
\(759\) 0 0
\(760\) −6.05627 −0.219684
\(761\) −9.97766 −0.361690 −0.180845 0.983512i \(-0.557883\pi\)
−0.180845 + 0.983512i \(0.557883\pi\)
\(762\) 0 0
\(763\) −16.4937 −0.597113
\(764\) −12.2675 −0.443821
\(765\) 0 0
\(766\) 47.3347 1.71027
\(767\) 3.48153 0.125711
\(768\) 0 0
\(769\) −49.8609 −1.79803 −0.899016 0.437917i \(-0.855716\pi\)
−0.899016 + 0.437917i \(0.855716\pi\)
\(770\) 28.1744 1.01534
\(771\) 0 0
\(772\) −16.3406 −0.588112
\(773\) −2.95716 −0.106362 −0.0531809 0.998585i \(-0.516936\pi\)
−0.0531809 + 0.998585i \(0.516936\pi\)
\(774\) 0 0
\(775\) 24.0865 0.865211
\(776\) −13.9440 −0.500561
\(777\) 0 0
\(778\) −38.2523 −1.37141
\(779\) −9.43191 −0.337933
\(780\) 0 0
\(781\) 30.7702 1.10104
\(782\) −16.3546 −0.584838
\(783\) 0 0
\(784\) 4.17681 0.149172
\(785\) −3.52464 −0.125800
\(786\) 0 0
\(787\) −25.9607 −0.925399 −0.462699 0.886515i \(-0.653119\pi\)
−0.462699 + 0.886515i \(0.653119\pi\)
\(788\) −25.7288 −0.916552
\(789\) 0 0
\(790\) 24.3264 0.865496
\(791\) −3.62421 −0.128862
\(792\) 0 0
\(793\) −9.49560 −0.337199
\(794\) −9.60541 −0.340883
\(795\) 0 0
\(796\) 19.0433 0.674973
\(797\) −5.84720 −0.207119 −0.103559 0.994623i \(-0.533023\pi\)
−0.103559 + 0.994623i \(0.533023\pi\)
\(798\) 0 0
\(799\) −53.9871 −1.90993
\(800\) 17.9903 0.636055
\(801\) 0 0
\(802\) −42.6616 −1.50643
\(803\) 24.6350 0.869349
\(804\) 0 0
\(805\) −7.50106 −0.264378
\(806\) 57.5662 2.02768
\(807\) 0 0
\(808\) 6.76189 0.237882
\(809\) −1.10730 −0.0389306 −0.0194653 0.999811i \(-0.506196\pi\)
−0.0194653 + 0.999811i \(0.506196\pi\)
\(810\) 0 0
\(811\) −30.9703 −1.08751 −0.543757 0.839243i \(-0.682999\pi\)
−0.543757 + 0.839243i \(0.682999\pi\)
\(812\) 0.225166 0.00790179
\(813\) 0 0
\(814\) −5.70367 −0.199914
\(815\) −17.6530 −0.618358
\(816\) 0 0
\(817\) 11.5983 0.405774
\(818\) −23.0090 −0.804491
\(819\) 0 0
\(820\) −5.07926 −0.177375
\(821\) 44.0188 1.53627 0.768133 0.640290i \(-0.221186\pi\)
0.768133 + 0.640290i \(0.221186\pi\)
\(822\) 0 0
\(823\) 3.54554 0.123590 0.0617949 0.998089i \(-0.480318\pi\)
0.0617949 + 0.998089i \(0.480318\pi\)
\(824\) 3.94386 0.137391
\(825\) 0 0
\(826\) 5.08916 0.177075
\(827\) −32.6215 −1.13436 −0.567180 0.823594i \(-0.691966\pi\)
−0.567180 + 0.823594i \(0.691966\pi\)
\(828\) 0 0
\(829\) 43.5811 1.51363 0.756817 0.653627i \(-0.226754\pi\)
0.756817 + 0.653627i \(0.226754\pi\)
\(830\) −29.2115 −1.01395
\(831\) 0 0
\(832\) 13.0661 0.452987
\(833\) 4.01167 0.138996
\(834\) 0 0
\(835\) −15.0607 −0.521197
\(836\) −27.0342 −0.934999
\(837\) 0 0
\(838\) 69.6861 2.40726
\(839\) −27.9176 −0.963822 −0.481911 0.876220i \(-0.660057\pi\)
−0.481911 + 0.876220i \(0.660057\pi\)
\(840\) 0 0
\(841\) −28.9966 −0.999881
\(842\) −43.8747 −1.51202
\(843\) 0 0
\(844\) −1.92845 −0.0663799
\(845\) −4.48451 −0.154272
\(846\) 0 0
\(847\) −9.16417 −0.314885
\(848\) 3.96768 0.136251
\(849\) 0 0
\(850\) 21.3590 0.732607
\(851\) 1.51853 0.0520544
\(852\) 0 0
\(853\) 15.6342 0.535303 0.267652 0.963516i \(-0.413752\pi\)
0.267652 + 0.963516i \(0.413752\pi\)
\(854\) −13.8803 −0.474974
\(855\) 0 0
\(856\) 1.34820 0.0460806
\(857\) −19.1584 −0.654438 −0.327219 0.944949i \(-0.606111\pi\)
−0.327219 + 0.944949i \(0.606111\pi\)
\(858\) 0 0
\(859\) 4.26808 0.145625 0.0728125 0.997346i \(-0.476803\pi\)
0.0728125 + 0.997346i \(0.476803\pi\)
\(860\) 6.24591 0.212984
\(861\) 0 0
\(862\) 64.4635 2.19563
\(863\) 23.1367 0.787582 0.393791 0.919200i \(-0.371163\pi\)
0.393791 + 0.919200i \(0.371163\pi\)
\(864\) 0 0
\(865\) 11.2943 0.384016
\(866\) 1.09636 0.0372559
\(867\) 0 0
\(868\) 36.7607 1.24774
\(869\) −31.3902 −1.06484
\(870\) 0 0
\(871\) −6.15268 −0.208475
\(872\) −5.63956 −0.190980
\(873\) 0 0
\(874\) 16.4756 0.557295
\(875\) 29.2882 0.990122
\(876\) 0 0
\(877\) 0.378032 0.0127652 0.00638262 0.999980i \(-0.497968\pi\)
0.00638262 + 0.999980i \(0.497968\pi\)
\(878\) −29.3287 −0.989797
\(879\) 0 0
\(880\) 28.4003 0.957375
\(881\) −3.64768 −0.122893 −0.0614467 0.998110i \(-0.519571\pi\)
−0.0614467 + 0.998110i \(0.519571\pi\)
\(882\) 0 0
\(883\) −34.3198 −1.15495 −0.577477 0.816407i \(-0.695963\pi\)
−0.577477 + 0.816407i \(0.695963\pi\)
\(884\) 22.3005 0.750049
\(885\) 0 0
\(886\) 69.9588 2.35031
\(887\) −48.0103 −1.61203 −0.806014 0.591896i \(-0.798380\pi\)
−0.806014 + 0.591896i \(0.798380\pi\)
\(888\) 0 0
\(889\) 37.2541 1.24946
\(890\) 26.3300 0.882584
\(891\) 0 0
\(892\) 27.0486 0.905654
\(893\) 54.3866 1.81998
\(894\) 0 0
\(895\) 0.180707 0.00604036
\(896\) −16.2943 −0.544354
\(897\) 0 0
\(898\) 31.3340 1.04563
\(899\) 0.562734 0.0187682
\(900\) 0 0
\(901\) 3.81080 0.126956
\(902\) 15.0029 0.499543
\(903\) 0 0
\(904\) −1.23919 −0.0412150
\(905\) −3.54250 −0.117757
\(906\) 0 0
\(907\) −18.2463 −0.605857 −0.302929 0.953013i \(-0.597964\pi\)
−0.302929 + 0.953013i \(0.597964\pi\)
\(908\) −2.96476 −0.0983891
\(909\) 0 0
\(910\) 23.4131 0.776136
\(911\) 3.39432 0.112459 0.0562294 0.998418i \(-0.482092\pi\)
0.0562294 + 0.998418i \(0.482092\pi\)
\(912\) 0 0
\(913\) 37.6939 1.24748
\(914\) 19.5657 0.647176
\(915\) 0 0
\(916\) −19.7780 −0.653485
\(917\) 25.0425 0.826977
\(918\) 0 0
\(919\) −58.2249 −1.92066 −0.960332 0.278860i \(-0.910044\pi\)
−0.960332 + 0.278860i \(0.910044\pi\)
\(920\) −2.56477 −0.0845580
\(921\) 0 0
\(922\) −25.8220 −0.850401
\(923\) 25.5702 0.841653
\(924\) 0 0
\(925\) −1.98319 −0.0652068
\(926\) −78.1283 −2.56746
\(927\) 0 0
\(928\) 0.420310 0.0137974
\(929\) −42.0740 −1.38040 −0.690202 0.723617i \(-0.742478\pi\)
−0.690202 + 0.723617i \(0.742478\pi\)
\(930\) 0 0
\(931\) −4.04136 −0.132450
\(932\) −14.0276 −0.459489
\(933\) 0 0
\(934\) 37.4814 1.22643
\(935\) 27.2774 0.892067
\(936\) 0 0
\(937\) 23.9489 0.782375 0.391188 0.920311i \(-0.372064\pi\)
0.391188 + 0.920311i \(0.372064\pi\)
\(938\) −8.99373 −0.293656
\(939\) 0 0
\(940\) 29.2882 0.955276
\(941\) 3.92678 0.128009 0.0640047 0.997950i \(-0.479613\pi\)
0.0640047 + 0.997950i \(0.479613\pi\)
\(942\) 0 0
\(943\) −3.99433 −0.130073
\(944\) 5.12996 0.166966
\(945\) 0 0
\(946\) −18.4489 −0.599827
\(947\) 8.59682 0.279359 0.139680 0.990197i \(-0.455393\pi\)
0.139680 + 0.990197i \(0.455393\pi\)
\(948\) 0 0
\(949\) 20.4718 0.664542
\(950\) −21.5171 −0.698105
\(951\) 0 0
\(952\) −9.42320 −0.305408
\(953\) −10.1361 −0.328342 −0.164171 0.986432i \(-0.552495\pi\)
−0.164171 + 0.986432i \(0.552495\pi\)
\(954\) 0 0
\(955\) −12.4694 −0.403500
\(956\) −1.55150 −0.0501792
\(957\) 0 0
\(958\) 24.9896 0.807377
\(959\) 3.68957 0.119142
\(960\) 0 0
\(961\) 60.8722 1.96362
\(962\) −4.73978 −0.152817
\(963\) 0 0
\(964\) 33.0883 1.06570
\(965\) −16.6096 −0.534683
\(966\) 0 0
\(967\) 37.1978 1.19620 0.598100 0.801421i \(-0.295922\pi\)
0.598100 + 0.801421i \(0.295922\pi\)
\(968\) −3.13343 −0.100712
\(969\) 0 0
\(970\) 49.0311 1.57429
\(971\) 24.0240 0.770967 0.385483 0.922715i \(-0.374035\pi\)
0.385483 + 0.922715i \(0.374035\pi\)
\(972\) 0 0
\(973\) −55.5328 −1.78030
\(974\) 66.1706 2.12024
\(975\) 0 0
\(976\) −13.9916 −0.447860
\(977\) 54.0203 1.72826 0.864132 0.503265i \(-0.167868\pi\)
0.864132 + 0.503265i \(0.167868\pi\)
\(978\) 0 0
\(979\) −33.9756 −1.08587
\(980\) −2.17635 −0.0695208
\(981\) 0 0
\(982\) −12.4552 −0.397463
\(983\) 27.5324 0.878146 0.439073 0.898451i \(-0.355307\pi\)
0.439073 + 0.898451i \(0.355307\pi\)
\(984\) 0 0
\(985\) −26.1524 −0.833284
\(986\) 0.499012 0.0158918
\(987\) 0 0
\(988\) −22.4656 −0.714726
\(989\) 4.91178 0.156185
\(990\) 0 0
\(991\) −0.915480 −0.0290812 −0.0145406 0.999894i \(-0.504629\pi\)
−0.0145406 + 0.999894i \(0.504629\pi\)
\(992\) 68.6199 2.17868
\(993\) 0 0
\(994\) 37.3774 1.18554
\(995\) 19.3568 0.613652
\(996\) 0 0
\(997\) 59.5160 1.88489 0.942446 0.334359i \(-0.108520\pi\)
0.942446 + 0.334359i \(0.108520\pi\)
\(998\) −42.7435 −1.35302
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2151.2.a.j.1.17 20
3.2 odd 2 2151.2.a.k.1.4 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.17 20 1.1 even 1 trivial
2151.2.a.k.1.4 yes 20 3.2 odd 2