Properties

Label 2151.2.a.k.1.4
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.4
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 \(-0.732117\pi\)
−0.666287 + 0.745696i \(0.732117\pi\)
\(3\) 0 0
\(4\) 1.55150 0.775751
\(5\) −1.57704 −0.705275 −0.352638 0.935760i \(-0.614715\pi\)
−0.352638 + 0.935760i \(0.614715\pi\)
\(6\) 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 \(-0.303775\pi\)
0.578149 + 0.815931i \(0.303775\pi\)
\(12\) 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 \(-0.315793\pi\)
0.546939 + 0.837172i \(0.315793\pi\)
\(18\) 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 \(-0.564291\pi\)
−0.200607 + 0.979672i \(0.564291\pi\)
\(24\) 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 \(-0.498265\pi\)
0.00545108 + 0.999985i \(0.498265\pi\)
\(30\) 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 \(-0.448173\pi\)
0.162100 + 0.986774i \(0.448173\pi\)
\(42\) 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 \(-0.837833\pi\)
−0.873007 + 0.487708i \(0.837833\pi\)
\(48\) 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 \(-0.481518\pi\)
0.0580303 + 0.998315i \(0.481518\pi\)
\(54\) 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 \(-0.477345\pi\)
0.0711123 + 0.997468i \(0.477345\pi\)
\(60\) 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 \(-0.342047\pi\)
0.476107 + 0.879387i \(0.342047\pi\)
\(72\) 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 \(-0.318640\pi\)
0.539431 + 0.842030i \(0.318640\pi\)
\(84\) 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 \(-0.655582\pi\)
−0.469544 + 0.882909i \(0.655582\pi\)
\(90\) 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 \(-0.369695\pi\)
0.398026 + 0.917374i \(0.369695\pi\)
\(102\) 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 \(-0.475433\pi\)
0.0771023 + 0.997023i \(0.475433\pi\)
\(108\) 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 \(-0.521968\pi\)
−0.0689612 + 0.997619i \(0.521968\pi\)
\(114\) 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 \(-0.354070\pi\)
0.442561 + 0.896738i \(0.354070\pi\)
\(132\) 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 \(-0.479691\pi\)
0.0637596 + 0.997965i \(0.479691\pi\)
\(138\) 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 \(-0.346406\pi\)
0.464022 + 0.885823i \(0.346406\pi\)
\(150\) 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 \(-0.379529\pi\)
0.369499 + 0.929231i \(0.379529\pi\)
\(168\) 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 \(-0.587766\pi\)
−0.272246 + 0.962228i \(0.587766\pi\)
\(174\) 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 \(-0.501363\pi\)
−0.00428227 + 0.999991i \(0.501363\pi\)
\(180\) 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 \(-0.407655\pi\)
0.286059 + 0.958212i \(0.407655\pi\)
\(192\) 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 \(-0.298832\pi\)
0.590751 + 0.806854i \(0.298832\pi\)
\(198\) 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 \(-0.479801\pi\)
0.0634154 + 0.997987i \(0.479801\pi\)
\(228\) 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 \(-0.404295\pi\)
0.296157 + 0.955139i \(0.404295\pi\)
\(234\) 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 \(-0.119761\pi\)
0.930052 + 0.367427i \(0.119761\pi\)
\(252\) 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 \(-0.438554\pi\)
0.191840 + 0.981426i \(0.438554\pi\)
\(258\) 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 \(-0.602143\pi\)
−0.315412 + 0.948955i \(0.602143\pi\)
\(264\) 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 \(-0.187085\pi\)
0.832193 + 0.554486i \(0.187085\pi\)
\(270\) 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 \(-0.473948\pi\)
0.0817519 + 0.996653i \(0.473948\pi\)
\(282\) 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 \(-0.216112\pi\)
0.778241 + 0.627966i \(0.216112\pi\)
\(294\) 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 \(-0.372314\pi\)
0.390467 + 0.920617i \(0.372314\pi\)
\(312\) 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 \(-0.688261\pi\)
−0.557557 + 0.830139i \(0.688261\pi\)
\(318\) 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 \(-0.587630\pi\)
−0.271834 + 0.962344i \(0.587630\pi\)
\(348\) 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 \(-0.272597\pi\)
0.655171 + 0.755481i \(0.272597\pi\)
\(354\) 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 \(-0.634135\pi\)
−0.409035 + 0.912519i \(0.634135\pi\)
\(360\) 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 \(-0.721778\pi\)
−0.641717 + 0.766941i \(0.721778\pi\)
\(384\) 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 \(-0.327951\pi\)
0.514573 + 0.857447i \(0.327951\pi\)
\(390\) 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 \(-0.308786\pi\)
0.565235 + 0.824930i \(0.308786\pi\)
\(402\) 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 \(-0.858818\pi\)
−0.903239 + 0.429137i \(0.858818\pi\)
\(420\) 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 \(-0.808168\pi\)
−0.823832 + 0.566834i \(0.808168\pi\)
\(432\) 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 \(-0.843715\pi\)
−0.881870 + 0.471493i \(0.843715\pi\)
\(444\) 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 \(-0.628333\pi\)
−0.392335 + 0.919822i \(0.628333\pi\)
\(450\) 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 \(-0.396625\pi\)
0.319082 + 0.947727i \(0.396625\pi\)
\(462\) 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 \(-0.652212\pi\)
−0.460173 + 0.887829i \(0.652212\pi\)
\(468\) 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 \(-0.597968\pi\)
−0.302939 + 0.953010i \(0.597968\pi\)
\(480\) 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 \(-0.452352\pi\)
0.149134 + 0.988817i \(0.452352\pi\)
\(492\) 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 \(-0.733513\pi\)
−0.669549 + 0.742768i \(0.733513\pi\)
\(504\) 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 \(-0.164769\pi\)
0.868990 + 0.494829i \(0.164769\pi\)
\(510\) 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 \(-0.232332\pi\)
0.745247 + 0.666789i \(0.232332\pi\)
\(522\) 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 \(-0.406411\pi\)
0.289801 + 0.957087i \(0.406411\pi\)
\(558\) 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 \(-0.409014\pi\)
0.281965 + 0.959425i \(0.409014\pi\)
\(564\) 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 \(-0.538072\pi\)
−0.119321 + 0.992856i \(0.538072\pi\)
\(570\) 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 \(-0.364354\pi\)
0.413365 + 0.910566i \(0.364354\pi\)
\(588\) 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 \(-0.304756\pi\)
0.575632 + 0.817709i \(0.304756\pi\)
\(594\) 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 \(-0.277046\pi\)
0.644547 + 0.764564i \(0.277046\pi\)
\(600\) 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 \(-0.170120\pi\)
0.860551 + 0.509365i \(0.170120\pi\)
\(618\) 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 \(-0.421621\pi\)
0.243756 + 0.969837i \(0.421621\pi\)
\(642\) 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 \(-0.588578\pi\)
−0.274700 + 0.961530i \(0.588578\pi\)
\(648\) 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 \(-0.504623\pi\)
−0.0145224 + 0.999895i \(0.504623\pi\)
\(654\) 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 \(-0.803921\pi\)
−0.816196 + 0.577776i \(0.803921\pi\)
\(660\) 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 \(-0.804583\pi\)
−0.817395 + 0.576078i \(0.804583\pi\)
\(678\) 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 \(-0.287516\pi\)
0.619055 + 0.785348i \(0.287516\pi\)
\(684\) 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 \(-0.323354\pi\)
0.526899 + 0.849928i \(0.323354\pi\)
\(702\) 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 \(-0.213589\pi\)
0.783195 + 0.621776i \(0.213589\pi\)
\(720\) 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 \(-0.553937\pi\)
−0.168638 + 0.985678i \(0.553937\pi\)
\(744\) 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 \(-0.442117\pi\)
0.180845 + 0.983512i \(0.442117\pi\)
\(762\) 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 \(-0.483064\pi\)
0.0531809 + 0.998585i \(0.483064\pi\)
\(774\) 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 \(-0.466977\pi\)
0.103559 + 0.994623i \(0.466977\pi\)
\(798\) 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 \(-0.493804\pi\)
0.0194653 + 0.999811i \(0.493804\pi\)
\(810\) 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 \(-0.778814\pi\)
−0.768133 + 0.640290i \(0.778814\pi\)
\(822\) 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 \(-0.308034\pi\)
0.567180 + 0.823594i \(0.308034\pi\)
\(828\) 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 \(-0.339943\pi\)
0.481911 + 0.876220i \(0.339943\pi\)
\(840\) 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 \(-0.393889\pi\)
0.327219 + 0.944949i \(0.393889\pi\)
\(858\) 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 \(-0.628837\pi\)
−0.393791 + 0.919200i \(0.628837\pi\)
\(864\) 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 \(-0.480429\pi\)
0.0614467 + 0.998110i \(0.480429\pi\)
\(882\) 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 \(-0.201620\pi\)
0.806014 + 0.591896i \(0.201620\pi\)
\(888\) 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 \(-0.517908\pi\)
−0.0562294 + 0.998418i \(0.517908\pi\)
\(912\) 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 \(-0.257522\pi\)
0.690202 + 0.723617i \(0.257522\pi\)
\(930\) 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 \(-0.520387\pi\)
−0.0640047 + 0.997950i \(0.520387\pi\)
\(942\) 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 \(-0.544607\pi\)
−0.139680 + 0.990197i \(0.544607\pi\)
\(948\) 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 \(-0.447505\pi\)
0.164171 + 0.986432i \(0.447505\pi\)
\(954\) 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 \(-0.625965\pi\)
−0.385483 + 0.922715i \(0.625965\pi\)
\(972\) 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 \(-0.832132\pi\)
−0.864132 + 0.503265i \(0.832132\pi\)
\(978\) 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 \(-0.644693\pi\)
−0.439073 + 0.898451i \(0.644693\pi\)
\(984\) 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.k.1.4 yes 20
3.2 odd 2 2151.2.a.j.1.17 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.17 20 3.2 odd 2
2151.2.a.k.1.4 yes 20 1.1 even 1 trivial