Properties

Label 2169.2.a.e.1.1
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.73684\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} +O(q^{10})\) \(q-1.73684 q^{2} +1.01662 q^{4} +2.63180 q^{5} -2.01025 q^{7} +1.70797 q^{8} -4.57103 q^{10} +3.39618 q^{11} +5.63669 q^{13} +3.49150 q^{14} -4.99972 q^{16} -0.866432 q^{17} +2.46437 q^{19} +2.67556 q^{20} -5.89863 q^{22} +6.37847 q^{23} +1.92640 q^{25} -9.79005 q^{26} -2.04367 q^{28} +4.52212 q^{29} -3.51511 q^{31} +5.26780 q^{32} +1.50486 q^{34} -5.29060 q^{35} -5.19315 q^{37} -4.28022 q^{38} +4.49504 q^{40} -1.35422 q^{41} +8.49015 q^{43} +3.45264 q^{44} -11.0784 q^{46} +9.44537 q^{47} -2.95888 q^{49} -3.34585 q^{50} +5.73040 q^{52} -9.71877 q^{53} +8.93808 q^{55} -3.43345 q^{56} -7.85421 q^{58} -6.03110 q^{59} +4.45402 q^{61} +6.10520 q^{62} +0.850111 q^{64} +14.8347 q^{65} -10.9216 q^{67} -0.880836 q^{68} +9.18894 q^{70} +3.01063 q^{71} -0.255916 q^{73} +9.01969 q^{74} +2.50534 q^{76} -6.82718 q^{77} -10.3262 q^{79} -13.1583 q^{80} +2.35207 q^{82} +16.8148 q^{83} -2.28028 q^{85} -14.7461 q^{86} +5.80057 q^{88} +17.4574 q^{89} -11.3312 q^{91} +6.48451 q^{92} -16.4051 q^{94} +6.48574 q^{95} +0.273223 q^{97} +5.13911 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73684 −1.22813 −0.614067 0.789254i \(-0.710467\pi\)
−0.614067 + 0.789254i \(0.710467\pi\)
\(3\) 0 0
\(4\) 1.01662 0.508312
\(5\) 2.63180 1.17698 0.588489 0.808505i \(-0.299723\pi\)
0.588489 + 0.808505i \(0.299723\pi\)
\(6\) 0 0
\(7\) −2.01025 −0.759805 −0.379902 0.925027i \(-0.624042\pi\)
−0.379902 + 0.925027i \(0.624042\pi\)
\(8\) 1.70797 0.603858
\(9\) 0 0
\(10\) −4.57103 −1.44549
\(11\) 3.39618 1.02399 0.511993 0.858989i \(-0.328907\pi\)
0.511993 + 0.858989i \(0.328907\pi\)
\(12\) 0 0
\(13\) 5.63669 1.56334 0.781669 0.623694i \(-0.214369\pi\)
0.781669 + 0.623694i \(0.214369\pi\)
\(14\) 3.49150 0.933141
\(15\) 0 0
\(16\) −4.99972 −1.24993
\(17\) −0.866432 −0.210141 −0.105070 0.994465i \(-0.533507\pi\)
−0.105070 + 0.994465i \(0.533507\pi\)
\(18\) 0 0
\(19\) 2.46437 0.565365 0.282682 0.959214i \(-0.408776\pi\)
0.282682 + 0.959214i \(0.408776\pi\)
\(20\) 2.67556 0.598273
\(21\) 0 0
\(22\) −5.89863 −1.25759
\(23\) 6.37847 1.33000 0.665001 0.746842i \(-0.268431\pi\)
0.665001 + 0.746842i \(0.268431\pi\)
\(24\) 0 0
\(25\) 1.92640 0.385279
\(26\) −9.79005 −1.91999
\(27\) 0 0
\(28\) −2.04367 −0.386218
\(29\) 4.52212 0.839737 0.419868 0.907585i \(-0.362076\pi\)
0.419868 + 0.907585i \(0.362076\pi\)
\(30\) 0 0
\(31\) −3.51511 −0.631332 −0.315666 0.948870i \(-0.602228\pi\)
−0.315666 + 0.948870i \(0.602228\pi\)
\(32\) 5.26780 0.931224
\(33\) 0 0
\(34\) 1.50486 0.258081
\(35\) −5.29060 −0.894274
\(36\) 0 0
\(37\) −5.19315 −0.853748 −0.426874 0.904311i \(-0.640385\pi\)
−0.426874 + 0.904311i \(0.640385\pi\)
\(38\) −4.28022 −0.694344
\(39\) 0 0
\(40\) 4.49504 0.710729
\(41\) −1.35422 −0.211494 −0.105747 0.994393i \(-0.533723\pi\)
−0.105747 + 0.994393i \(0.533723\pi\)
\(42\) 0 0
\(43\) 8.49015 1.29474 0.647368 0.762178i \(-0.275870\pi\)
0.647368 + 0.762178i \(0.275870\pi\)
\(44\) 3.45264 0.520505
\(45\) 0 0
\(46\) −11.0784 −1.63342
\(47\) 9.44537 1.37775 0.688875 0.724880i \(-0.258105\pi\)
0.688875 + 0.724880i \(0.258105\pi\)
\(48\) 0 0
\(49\) −2.95888 −0.422697
\(50\) −3.34585 −0.473174
\(51\) 0 0
\(52\) 5.73040 0.794663
\(53\) −9.71877 −1.33498 −0.667488 0.744620i \(-0.732631\pi\)
−0.667488 + 0.744620i \(0.732631\pi\)
\(54\) 0 0
\(55\) 8.93808 1.20521
\(56\) −3.43345 −0.458814
\(57\) 0 0
\(58\) −7.85421 −1.03131
\(59\) −6.03110 −0.785182 −0.392591 0.919713i \(-0.628421\pi\)
−0.392591 + 0.919713i \(0.628421\pi\)
\(60\) 0 0
\(61\) 4.45402 0.570279 0.285140 0.958486i \(-0.407960\pi\)
0.285140 + 0.958486i \(0.407960\pi\)
\(62\) 6.10520 0.775361
\(63\) 0 0
\(64\) 0.850111 0.106264
\(65\) 14.8347 1.84002
\(66\) 0 0
\(67\) −10.9216 −1.33428 −0.667141 0.744932i \(-0.732482\pi\)
−0.667141 + 0.744932i \(0.732482\pi\)
\(68\) −0.880836 −0.106817
\(69\) 0 0
\(70\) 9.18894 1.09829
\(71\) 3.01063 0.357296 0.178648 0.983913i \(-0.442828\pi\)
0.178648 + 0.983913i \(0.442828\pi\)
\(72\) 0 0
\(73\) −0.255916 −0.0299527 −0.0149764 0.999888i \(-0.504767\pi\)
−0.0149764 + 0.999888i \(0.504767\pi\)
\(74\) 9.01969 1.04852
\(75\) 0 0
\(76\) 2.50534 0.287382
\(77\) −6.82718 −0.778030
\(78\) 0 0
\(79\) −10.3262 −1.16179 −0.580893 0.813980i \(-0.697297\pi\)
−0.580893 + 0.813980i \(0.697297\pi\)
\(80\) −13.1583 −1.47114
\(81\) 0 0
\(82\) 2.35207 0.259743
\(83\) 16.8148 1.84567 0.922833 0.385201i \(-0.125868\pi\)
0.922833 + 0.385201i \(0.125868\pi\)
\(84\) 0 0
\(85\) −2.28028 −0.247331
\(86\) −14.7461 −1.59011
\(87\) 0 0
\(88\) 5.80057 0.618343
\(89\) 17.4574 1.85048 0.925241 0.379380i \(-0.123863\pi\)
0.925241 + 0.379380i \(0.123863\pi\)
\(90\) 0 0
\(91\) −11.3312 −1.18783
\(92\) 6.48451 0.676056
\(93\) 0 0
\(94\) −16.4051 −1.69206
\(95\) 6.48574 0.665423
\(96\) 0 0
\(97\) 0.273223 0.0277416 0.0138708 0.999904i \(-0.495585\pi\)
0.0138708 + 0.999904i \(0.495585\pi\)
\(98\) 5.13911 0.519128
\(99\) 0 0
\(100\) 1.95842 0.195842
\(101\) −7.47149 −0.743441 −0.371721 0.928345i \(-0.621232\pi\)
−0.371721 + 0.928345i \(0.621232\pi\)
\(102\) 0 0
\(103\) 14.5145 1.43015 0.715077 0.699046i \(-0.246392\pi\)
0.715077 + 0.699046i \(0.246392\pi\)
\(104\) 9.62730 0.944035
\(105\) 0 0
\(106\) 16.8800 1.63953
\(107\) −16.7492 −1.61921 −0.809605 0.586976i \(-0.800318\pi\)
−0.809605 + 0.586976i \(0.800318\pi\)
\(108\) 0 0
\(109\) −6.01965 −0.576578 −0.288289 0.957543i \(-0.593086\pi\)
−0.288289 + 0.957543i \(0.593086\pi\)
\(110\) −15.5240 −1.48016
\(111\) 0 0
\(112\) 10.0507 0.949703
\(113\) −18.9246 −1.78028 −0.890139 0.455689i \(-0.849393\pi\)
−0.890139 + 0.455689i \(0.849393\pi\)
\(114\) 0 0
\(115\) 16.7869 1.56539
\(116\) 4.59730 0.426848
\(117\) 0 0
\(118\) 10.4751 0.964308
\(119\) 1.74175 0.159666
\(120\) 0 0
\(121\) 0.534035 0.0485487
\(122\) −7.73594 −0.700379
\(123\) 0 0
\(124\) −3.57355 −0.320914
\(125\) −8.08912 −0.723513
\(126\) 0 0
\(127\) −13.7300 −1.21834 −0.609172 0.793038i \(-0.708498\pi\)
−0.609172 + 0.793038i \(0.708498\pi\)
\(128\) −12.0121 −1.06173
\(129\) 0 0
\(130\) −25.7655 −2.25978
\(131\) 14.3193 1.25108 0.625542 0.780190i \(-0.284878\pi\)
0.625542 + 0.780190i \(0.284878\pi\)
\(132\) 0 0
\(133\) −4.95401 −0.429567
\(134\) 18.9690 1.63868
\(135\) 0 0
\(136\) −1.47984 −0.126895
\(137\) −10.1743 −0.869252 −0.434626 0.900611i \(-0.643119\pi\)
−0.434626 + 0.900611i \(0.643119\pi\)
\(138\) 0 0
\(139\) 8.00345 0.678844 0.339422 0.940634i \(-0.389769\pi\)
0.339422 + 0.940634i \(0.389769\pi\)
\(140\) −5.37855 −0.454570
\(141\) 0 0
\(142\) −5.22899 −0.438807
\(143\) 19.1432 1.60084
\(144\) 0 0
\(145\) 11.9013 0.988352
\(146\) 0.444486 0.0367860
\(147\) 0 0
\(148\) −5.27948 −0.433971
\(149\) 15.5640 1.27505 0.637527 0.770428i \(-0.279958\pi\)
0.637527 + 0.770428i \(0.279958\pi\)
\(150\) 0 0
\(151\) −2.43764 −0.198372 −0.0991862 0.995069i \(-0.531624\pi\)
−0.0991862 + 0.995069i \(0.531624\pi\)
\(152\) 4.20907 0.341400
\(153\) 0 0
\(154\) 11.8577 0.955524
\(155\) −9.25108 −0.743065
\(156\) 0 0
\(157\) 12.1633 0.970735 0.485367 0.874310i \(-0.338686\pi\)
0.485367 + 0.874310i \(0.338686\pi\)
\(158\) 17.9350 1.42683
\(159\) 0 0
\(160\) 13.8638 1.09603
\(161\) −12.8223 −1.01054
\(162\) 0 0
\(163\) 4.23026 0.331340 0.165670 0.986181i \(-0.447021\pi\)
0.165670 + 0.986181i \(0.447021\pi\)
\(164\) −1.37674 −0.107505
\(165\) 0 0
\(166\) −29.2047 −2.26672
\(167\) −14.3677 −1.11181 −0.555903 0.831247i \(-0.687628\pi\)
−0.555903 + 0.831247i \(0.687628\pi\)
\(168\) 0 0
\(169\) 18.7723 1.44402
\(170\) 3.96049 0.303756
\(171\) 0 0
\(172\) 8.63129 0.658130
\(173\) −1.08253 −0.0823030 −0.0411515 0.999153i \(-0.513103\pi\)
−0.0411515 + 0.999153i \(0.513103\pi\)
\(174\) 0 0
\(175\) −3.87255 −0.292737
\(176\) −16.9800 −1.27991
\(177\) 0 0
\(178\) −30.3208 −2.27264
\(179\) 24.0238 1.79562 0.897810 0.440382i \(-0.145157\pi\)
0.897810 + 0.440382i \(0.145157\pi\)
\(180\) 0 0
\(181\) −4.72624 −0.351298 −0.175649 0.984453i \(-0.556202\pi\)
−0.175649 + 0.984453i \(0.556202\pi\)
\(182\) 19.6805 1.45882
\(183\) 0 0
\(184\) 10.8942 0.803133
\(185\) −13.6674 −1.00484
\(186\) 0 0
\(187\) −2.94256 −0.215181
\(188\) 9.60239 0.700327
\(189\) 0 0
\(190\) −11.2647 −0.817228
\(191\) 24.3499 1.76190 0.880948 0.473213i \(-0.156906\pi\)
0.880948 + 0.473213i \(0.156906\pi\)
\(192\) 0 0
\(193\) 3.22480 0.232126 0.116063 0.993242i \(-0.462973\pi\)
0.116063 + 0.993242i \(0.462973\pi\)
\(194\) −0.474545 −0.0340704
\(195\) 0 0
\(196\) −3.00807 −0.214862
\(197\) 3.60952 0.257167 0.128584 0.991699i \(-0.458957\pi\)
0.128584 + 0.991699i \(0.458957\pi\)
\(198\) 0 0
\(199\) 11.6028 0.822499 0.411249 0.911523i \(-0.365093\pi\)
0.411249 + 0.911523i \(0.365093\pi\)
\(200\) 3.29023 0.232654
\(201\) 0 0
\(202\) 12.9768 0.913045
\(203\) −9.09061 −0.638036
\(204\) 0 0
\(205\) −3.56405 −0.248924
\(206\) −25.2094 −1.75642
\(207\) 0 0
\(208\) −28.1819 −1.95406
\(209\) 8.36944 0.578926
\(210\) 0 0
\(211\) 25.9439 1.78605 0.893025 0.450007i \(-0.148579\pi\)
0.893025 + 0.450007i \(0.148579\pi\)
\(212\) −9.88034 −0.678585
\(213\) 0 0
\(214\) 29.0908 1.98861
\(215\) 22.3444 1.52388
\(216\) 0 0
\(217\) 7.06626 0.479689
\(218\) 10.4552 0.708115
\(219\) 0 0
\(220\) 9.08667 0.612623
\(221\) −4.88381 −0.328521
\(222\) 0 0
\(223\) −4.40090 −0.294706 −0.147353 0.989084i \(-0.547075\pi\)
−0.147353 + 0.989084i \(0.547075\pi\)
\(224\) −10.5896 −0.707548
\(225\) 0 0
\(226\) 32.8691 2.18642
\(227\) 23.0781 1.53175 0.765873 0.642991i \(-0.222307\pi\)
0.765873 + 0.642991i \(0.222307\pi\)
\(228\) 0 0
\(229\) 3.21172 0.212236 0.106118 0.994354i \(-0.466158\pi\)
0.106118 + 0.994354i \(0.466158\pi\)
\(230\) −29.1562 −1.92250
\(231\) 0 0
\(232\) 7.72364 0.507082
\(233\) −7.13210 −0.467239 −0.233620 0.972328i \(-0.575057\pi\)
−0.233620 + 0.972328i \(0.575057\pi\)
\(234\) 0 0
\(235\) 24.8584 1.62158
\(236\) −6.13136 −0.399117
\(237\) 0 0
\(238\) −3.02514 −0.196091
\(239\) −3.66241 −0.236902 −0.118451 0.992960i \(-0.537793\pi\)
−0.118451 + 0.992960i \(0.537793\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) −0.927536 −0.0596243
\(243\) 0 0
\(244\) 4.52807 0.289880
\(245\) −7.78719 −0.497506
\(246\) 0 0
\(247\) 13.8909 0.883856
\(248\) −6.00370 −0.381235
\(249\) 0 0
\(250\) 14.0495 0.888571
\(251\) −1.41613 −0.0893854 −0.0446927 0.999001i \(-0.514231\pi\)
−0.0446927 + 0.999001i \(0.514231\pi\)
\(252\) 0 0
\(253\) 21.6624 1.36190
\(254\) 23.8469 1.49629
\(255\) 0 0
\(256\) 19.1629 1.19768
\(257\) 11.0690 0.690466 0.345233 0.938517i \(-0.387800\pi\)
0.345233 + 0.938517i \(0.387800\pi\)
\(258\) 0 0
\(259\) 10.4395 0.648682
\(260\) 15.0813 0.935302
\(261\) 0 0
\(262\) −24.8704 −1.53650
\(263\) 3.31569 0.204454 0.102227 0.994761i \(-0.467403\pi\)
0.102227 + 0.994761i \(0.467403\pi\)
\(264\) 0 0
\(265\) −25.5779 −1.57124
\(266\) 8.60433 0.527565
\(267\) 0 0
\(268\) −11.1031 −0.678231
\(269\) 14.3455 0.874663 0.437332 0.899300i \(-0.355924\pi\)
0.437332 + 0.899300i \(0.355924\pi\)
\(270\) 0 0
\(271\) −1.55240 −0.0943014 −0.0471507 0.998888i \(-0.515014\pi\)
−0.0471507 + 0.998888i \(0.515014\pi\)
\(272\) 4.33192 0.262661
\(273\) 0 0
\(274\) 17.6712 1.06756
\(275\) 6.54239 0.394521
\(276\) 0 0
\(277\) 18.0089 1.08205 0.541024 0.841007i \(-0.318037\pi\)
0.541024 + 0.841007i \(0.318037\pi\)
\(278\) −13.9007 −0.833711
\(279\) 0 0
\(280\) −9.03618 −0.540015
\(281\) 11.5476 0.688873 0.344436 0.938810i \(-0.388070\pi\)
0.344436 + 0.938810i \(0.388070\pi\)
\(282\) 0 0
\(283\) −1.84755 −0.109825 −0.0549126 0.998491i \(-0.517488\pi\)
−0.0549126 + 0.998491i \(0.517488\pi\)
\(284\) 3.06068 0.181618
\(285\) 0 0
\(286\) −33.2488 −1.96604
\(287\) 2.72233 0.160694
\(288\) 0 0
\(289\) −16.2493 −0.955841
\(290\) −20.6708 −1.21383
\(291\) 0 0
\(292\) −0.260171 −0.0152253
\(293\) 3.03213 0.177139 0.0885694 0.996070i \(-0.471771\pi\)
0.0885694 + 0.996070i \(0.471771\pi\)
\(294\) 0 0
\(295\) −15.8727 −0.924143
\(296\) −8.86974 −0.515543
\(297\) 0 0
\(298\) −27.0322 −1.56594
\(299\) 35.9535 2.07924
\(300\) 0 0
\(301\) −17.0674 −0.983746
\(302\) 4.23380 0.243628
\(303\) 0 0
\(304\) −12.3212 −0.706667
\(305\) 11.7221 0.671207
\(306\) 0 0
\(307\) 8.40693 0.479809 0.239905 0.970796i \(-0.422884\pi\)
0.239905 + 0.970796i \(0.422884\pi\)
\(308\) −6.94068 −0.395482
\(309\) 0 0
\(310\) 16.0677 0.912583
\(311\) −2.94070 −0.166752 −0.0833759 0.996518i \(-0.526570\pi\)
−0.0833759 + 0.996518i \(0.526570\pi\)
\(312\) 0 0
\(313\) 16.5110 0.933258 0.466629 0.884453i \(-0.345468\pi\)
0.466629 + 0.884453i \(0.345468\pi\)
\(314\) −21.1257 −1.19219
\(315\) 0 0
\(316\) −10.4978 −0.590550
\(317\) 20.3799 1.14465 0.572325 0.820027i \(-0.306041\pi\)
0.572325 + 0.820027i \(0.306041\pi\)
\(318\) 0 0
\(319\) 15.3579 0.859879
\(320\) 2.23733 0.125070
\(321\) 0 0
\(322\) 22.2704 1.24108
\(323\) −2.13521 −0.118806
\(324\) 0 0
\(325\) 10.8585 0.602322
\(326\) −7.34731 −0.406930
\(327\) 0 0
\(328\) −2.31297 −0.127713
\(329\) −18.9876 −1.04682
\(330\) 0 0
\(331\) 9.08522 0.499369 0.249684 0.968327i \(-0.419673\pi\)
0.249684 + 0.968327i \(0.419673\pi\)
\(332\) 17.0943 0.938174
\(333\) 0 0
\(334\) 24.9545 1.36545
\(335\) −28.7434 −1.57042
\(336\) 0 0
\(337\) −34.1864 −1.86225 −0.931125 0.364700i \(-0.881171\pi\)
−0.931125 + 0.364700i \(0.881171\pi\)
\(338\) −32.6046 −1.77345
\(339\) 0 0
\(340\) −2.31819 −0.125721
\(341\) −11.9379 −0.646476
\(342\) 0 0
\(343\) 20.0199 1.08097
\(344\) 14.5009 0.781837
\(345\) 0 0
\(346\) 1.88018 0.101079
\(347\) 9.13414 0.490346 0.245173 0.969479i \(-0.421155\pi\)
0.245173 + 0.969479i \(0.421155\pi\)
\(348\) 0 0
\(349\) −20.3980 −1.09188 −0.545940 0.837824i \(-0.683827\pi\)
−0.545940 + 0.837824i \(0.683827\pi\)
\(350\) 6.72601 0.359520
\(351\) 0 0
\(352\) 17.8904 0.953561
\(353\) −13.2353 −0.704444 −0.352222 0.935916i \(-0.614574\pi\)
−0.352222 + 0.935916i \(0.614574\pi\)
\(354\) 0 0
\(355\) 7.92339 0.420530
\(356\) 17.7476 0.940622
\(357\) 0 0
\(358\) −41.7255 −2.20526
\(359\) −35.7474 −1.88668 −0.943339 0.331831i \(-0.892334\pi\)
−0.943339 + 0.331831i \(0.892334\pi\)
\(360\) 0 0
\(361\) −12.9269 −0.680363
\(362\) 8.20873 0.431441
\(363\) 0 0
\(364\) −11.5196 −0.603789
\(365\) −0.673522 −0.0352537
\(366\) 0 0
\(367\) −34.7219 −1.81247 −0.906234 0.422777i \(-0.861055\pi\)
−0.906234 + 0.422777i \(0.861055\pi\)
\(368\) −31.8906 −1.66241
\(369\) 0 0
\(370\) 23.7381 1.23408
\(371\) 19.5372 1.01432
\(372\) 0 0
\(373\) −34.9987 −1.81217 −0.906083 0.423100i \(-0.860942\pi\)
−0.906083 + 0.423100i \(0.860942\pi\)
\(374\) 5.11076 0.264271
\(375\) 0 0
\(376\) 16.1324 0.831966
\(377\) 25.4898 1.31279
\(378\) 0 0
\(379\) 17.4032 0.893943 0.446972 0.894548i \(-0.352502\pi\)
0.446972 + 0.894548i \(0.352502\pi\)
\(380\) 6.59356 0.338242
\(381\) 0 0
\(382\) −42.2919 −2.16384
\(383\) −7.58514 −0.387582 −0.193791 0.981043i \(-0.562078\pi\)
−0.193791 + 0.981043i \(0.562078\pi\)
\(384\) 0 0
\(385\) −17.9678 −0.915725
\(386\) −5.60097 −0.285082
\(387\) 0 0
\(388\) 0.277765 0.0141014
\(389\) 9.68216 0.490905 0.245452 0.969409i \(-0.421063\pi\)
0.245452 + 0.969409i \(0.421063\pi\)
\(390\) 0 0
\(391\) −5.52651 −0.279488
\(392\) −5.05368 −0.255249
\(393\) 0 0
\(394\) −6.26917 −0.315836
\(395\) −27.1765 −1.36740
\(396\) 0 0
\(397\) −8.54563 −0.428893 −0.214446 0.976736i \(-0.568795\pi\)
−0.214446 + 0.976736i \(0.568795\pi\)
\(398\) −20.1522 −1.01014
\(399\) 0 0
\(400\) −9.63145 −0.481573
\(401\) −23.0124 −1.14918 −0.574592 0.818440i \(-0.694839\pi\)
−0.574592 + 0.818440i \(0.694839\pi\)
\(402\) 0 0
\(403\) −19.8136 −0.986986
\(404\) −7.59570 −0.377900
\(405\) 0 0
\(406\) 15.7890 0.783593
\(407\) −17.6369 −0.874227
\(408\) 0 0
\(409\) −24.2236 −1.19778 −0.598891 0.800831i \(-0.704392\pi\)
−0.598891 + 0.800831i \(0.704392\pi\)
\(410\) 6.19020 0.305712
\(411\) 0 0
\(412\) 14.7558 0.726965
\(413\) 12.1240 0.596585
\(414\) 0 0
\(415\) 44.2533 2.17231
\(416\) 29.6930 1.45582
\(417\) 0 0
\(418\) −14.5364 −0.710999
\(419\) −27.7565 −1.35600 −0.677998 0.735064i \(-0.737152\pi\)
−0.677998 + 0.735064i \(0.737152\pi\)
\(420\) 0 0
\(421\) −33.1205 −1.61420 −0.807098 0.590418i \(-0.798963\pi\)
−0.807098 + 0.590418i \(0.798963\pi\)
\(422\) −45.0604 −2.19351
\(423\) 0 0
\(424\) −16.5994 −0.806137
\(425\) −1.66909 −0.0809628
\(426\) 0 0
\(427\) −8.95372 −0.433301
\(428\) −17.0277 −0.823064
\(429\) 0 0
\(430\) −38.8087 −1.87152
\(431\) 5.07371 0.244392 0.122196 0.992506i \(-0.461006\pi\)
0.122196 + 0.992506i \(0.461006\pi\)
\(432\) 0 0
\(433\) 11.6416 0.559457 0.279729 0.960079i \(-0.409755\pi\)
0.279729 + 0.960079i \(0.409755\pi\)
\(434\) −12.2730 −0.589122
\(435\) 0 0
\(436\) −6.11973 −0.293082
\(437\) 15.7189 0.751937
\(438\) 0 0
\(439\) −14.7945 −0.706105 −0.353052 0.935604i \(-0.614856\pi\)
−0.353052 + 0.935604i \(0.614856\pi\)
\(440\) 15.2660 0.727777
\(441\) 0 0
\(442\) 8.48241 0.403467
\(443\) 2.54544 0.120937 0.0604687 0.998170i \(-0.480740\pi\)
0.0604687 + 0.998170i \(0.480740\pi\)
\(444\) 0 0
\(445\) 45.9445 2.17798
\(446\) 7.64367 0.361938
\(447\) 0 0
\(448\) −1.70894 −0.0807398
\(449\) 13.1558 0.620860 0.310430 0.950596i \(-0.399527\pi\)
0.310430 + 0.950596i \(0.399527\pi\)
\(450\) 0 0
\(451\) −4.59919 −0.216567
\(452\) −19.2392 −0.904937
\(453\) 0 0
\(454\) −40.0830 −1.88119
\(455\) −29.8215 −1.39805
\(456\) 0 0
\(457\) 5.54809 0.259528 0.129764 0.991545i \(-0.458578\pi\)
0.129764 + 0.991545i \(0.458578\pi\)
\(458\) −5.57825 −0.260655
\(459\) 0 0
\(460\) 17.0660 0.795704
\(461\) 32.2282 1.50102 0.750508 0.660861i \(-0.229809\pi\)
0.750508 + 0.660861i \(0.229809\pi\)
\(462\) 0 0
\(463\) −18.5042 −0.859965 −0.429983 0.902837i \(-0.641480\pi\)
−0.429983 + 0.902837i \(0.641480\pi\)
\(464\) −22.6094 −1.04961
\(465\) 0 0
\(466\) 12.3873 0.573832
\(467\) −20.9192 −0.968026 −0.484013 0.875061i \(-0.660821\pi\)
−0.484013 + 0.875061i \(0.660821\pi\)
\(468\) 0 0
\(469\) 21.9551 1.01379
\(470\) −43.1751 −1.99152
\(471\) 0 0
\(472\) −10.3009 −0.474139
\(473\) 28.8341 1.32579
\(474\) 0 0
\(475\) 4.74735 0.217823
\(476\) 1.77070 0.0811600
\(477\) 0 0
\(478\) 6.36103 0.290947
\(479\) 9.86001 0.450515 0.225258 0.974299i \(-0.427678\pi\)
0.225258 + 0.974299i \(0.427678\pi\)
\(480\) 0 0
\(481\) −29.2722 −1.33470
\(482\) 1.73684 0.0791110
\(483\) 0 0
\(484\) 0.542913 0.0246779
\(485\) 0.719069 0.0326512
\(486\) 0 0
\(487\) −25.9290 −1.17496 −0.587478 0.809240i \(-0.699879\pi\)
−0.587478 + 0.809240i \(0.699879\pi\)
\(488\) 7.60734 0.344368
\(489\) 0 0
\(490\) 13.5251 0.611003
\(491\) 22.5594 1.01809 0.509046 0.860739i \(-0.329998\pi\)
0.509046 + 0.860739i \(0.329998\pi\)
\(492\) 0 0
\(493\) −3.91811 −0.176463
\(494\) −24.1263 −1.08549
\(495\) 0 0
\(496\) 17.5746 0.789122
\(497\) −6.05213 −0.271475
\(498\) 0 0
\(499\) 39.8682 1.78475 0.892373 0.451298i \(-0.149039\pi\)
0.892373 + 0.451298i \(0.149039\pi\)
\(500\) −8.22360 −0.367771
\(501\) 0 0
\(502\) 2.45960 0.109777
\(503\) 19.7008 0.878416 0.439208 0.898386i \(-0.355259\pi\)
0.439208 + 0.898386i \(0.355259\pi\)
\(504\) 0 0
\(505\) −19.6635 −0.875015
\(506\) −37.6242 −1.67260
\(507\) 0 0
\(508\) −13.9583 −0.619298
\(509\) 23.4312 1.03857 0.519284 0.854602i \(-0.326199\pi\)
0.519284 + 0.854602i \(0.326199\pi\)
\(510\) 0 0
\(511\) 0.514457 0.0227582
\(512\) −9.25877 −0.409184
\(513\) 0 0
\(514\) −19.2251 −0.847985
\(515\) 38.1993 1.68326
\(516\) 0 0
\(517\) 32.0782 1.41080
\(518\) −18.1319 −0.796668
\(519\) 0 0
\(520\) 25.3372 1.11111
\(521\) 5.76181 0.252430 0.126215 0.992003i \(-0.459717\pi\)
0.126215 + 0.992003i \(0.459717\pi\)
\(522\) 0 0
\(523\) 42.0766 1.83988 0.919942 0.392056i \(-0.128236\pi\)
0.919942 + 0.392056i \(0.128236\pi\)
\(524\) 14.5574 0.635941
\(525\) 0 0
\(526\) −5.75883 −0.251097
\(527\) 3.04560 0.132669
\(528\) 0 0
\(529\) 17.6849 0.768907
\(530\) 44.4248 1.92969
\(531\) 0 0
\(532\) −5.03636 −0.218354
\(533\) −7.63334 −0.330637
\(534\) 0 0
\(535\) −44.0807 −1.90578
\(536\) −18.6537 −0.805717
\(537\) 0 0
\(538\) −24.9160 −1.07420
\(539\) −10.0489 −0.432836
\(540\) 0 0
\(541\) 7.25557 0.311941 0.155971 0.987762i \(-0.450149\pi\)
0.155971 + 0.987762i \(0.450149\pi\)
\(542\) 2.69627 0.115815
\(543\) 0 0
\(544\) −4.56419 −0.195688
\(545\) −15.8426 −0.678620
\(546\) 0 0
\(547\) −27.6574 −1.18254 −0.591272 0.806472i \(-0.701374\pi\)
−0.591272 + 0.806472i \(0.701374\pi\)
\(548\) −10.3435 −0.441851
\(549\) 0 0
\(550\) −11.3631 −0.484524
\(551\) 11.1442 0.474758
\(552\) 0 0
\(553\) 20.7582 0.882730
\(554\) −31.2786 −1.32890
\(555\) 0 0
\(556\) 8.13650 0.345065
\(557\) −26.7523 −1.13353 −0.566766 0.823878i \(-0.691806\pi\)
−0.566766 + 0.823878i \(0.691806\pi\)
\(558\) 0 0
\(559\) 47.8564 2.02411
\(560\) 26.4515 1.11778
\(561\) 0 0
\(562\) −20.0564 −0.846028
\(563\) −20.2482 −0.853362 −0.426681 0.904402i \(-0.640317\pi\)
−0.426681 + 0.904402i \(0.640317\pi\)
\(564\) 0 0
\(565\) −49.8059 −2.09535
\(566\) 3.20890 0.134880
\(567\) 0 0
\(568\) 5.14206 0.215756
\(569\) −41.1274 −1.72415 −0.862076 0.506779i \(-0.830836\pi\)
−0.862076 + 0.506779i \(0.830836\pi\)
\(570\) 0 0
\(571\) 25.2419 1.05634 0.528170 0.849139i \(-0.322878\pi\)
0.528170 + 0.849139i \(0.322878\pi\)
\(572\) 19.4615 0.813725
\(573\) 0 0
\(574\) −4.72827 −0.197354
\(575\) 12.2875 0.512423
\(576\) 0 0
\(577\) 4.90936 0.204379 0.102190 0.994765i \(-0.467415\pi\)
0.102190 + 0.994765i \(0.467415\pi\)
\(578\) 28.2225 1.17390
\(579\) 0 0
\(580\) 12.0992 0.502391
\(581\) −33.8020 −1.40234
\(582\) 0 0
\(583\) −33.0067 −1.36700
\(584\) −0.437097 −0.0180872
\(585\) 0 0
\(586\) −5.26633 −0.217550
\(587\) −0.733495 −0.0302746 −0.0151373 0.999885i \(-0.504819\pi\)
−0.0151373 + 0.999885i \(0.504819\pi\)
\(588\) 0 0
\(589\) −8.66253 −0.356933
\(590\) 27.5683 1.13497
\(591\) 0 0
\(592\) 25.9643 1.06713
\(593\) −43.2540 −1.77623 −0.888115 0.459621i \(-0.847985\pi\)
−0.888115 + 0.459621i \(0.847985\pi\)
\(594\) 0 0
\(595\) 4.58394 0.187923
\(596\) 15.8227 0.648125
\(597\) 0 0
\(598\) −62.4455 −2.55359
\(599\) 21.1031 0.862251 0.431126 0.902292i \(-0.358117\pi\)
0.431126 + 0.902292i \(0.358117\pi\)
\(600\) 0 0
\(601\) −16.1436 −0.658512 −0.329256 0.944241i \(-0.606798\pi\)
−0.329256 + 0.944241i \(0.606798\pi\)
\(602\) 29.6433 1.20817
\(603\) 0 0
\(604\) −2.47816 −0.100835
\(605\) 1.40548 0.0571408
\(606\) 0 0
\(607\) −32.2628 −1.30951 −0.654753 0.755843i \(-0.727227\pi\)
−0.654753 + 0.755843i \(0.727227\pi\)
\(608\) 12.9818 0.526481
\(609\) 0 0
\(610\) −20.3595 −0.824331
\(611\) 53.2407 2.15389
\(612\) 0 0
\(613\) −44.6321 −1.80267 −0.901337 0.433119i \(-0.857413\pi\)
−0.901337 + 0.433119i \(0.857413\pi\)
\(614\) −14.6015 −0.589270
\(615\) 0 0
\(616\) −11.6606 −0.469820
\(617\) −20.2532 −0.815363 −0.407682 0.913124i \(-0.633663\pi\)
−0.407682 + 0.913124i \(0.633663\pi\)
\(618\) 0 0
\(619\) 23.9533 0.962766 0.481383 0.876510i \(-0.340135\pi\)
0.481383 + 0.876510i \(0.340135\pi\)
\(620\) −9.40488 −0.377709
\(621\) 0 0
\(622\) 5.10754 0.204794
\(623\) −35.0938 −1.40600
\(624\) 0 0
\(625\) −30.9210 −1.23684
\(626\) −28.6770 −1.14617
\(627\) 0 0
\(628\) 12.3655 0.493436
\(629\) 4.49951 0.179407
\(630\) 0 0
\(631\) −3.46481 −0.137932 −0.0689659 0.997619i \(-0.521970\pi\)
−0.0689659 + 0.997619i \(0.521970\pi\)
\(632\) −17.6368 −0.701554
\(633\) 0 0
\(634\) −35.3967 −1.40578
\(635\) −36.1348 −1.43396
\(636\) 0 0
\(637\) −16.6783 −0.660818
\(638\) −26.6743 −1.05605
\(639\) 0 0
\(640\) −31.6135 −1.24963
\(641\) 28.9401 1.14307 0.571533 0.820579i \(-0.306349\pi\)
0.571533 + 0.820579i \(0.306349\pi\)
\(642\) 0 0
\(643\) 18.6657 0.736103 0.368052 0.929805i \(-0.380025\pi\)
0.368052 + 0.929805i \(0.380025\pi\)
\(644\) −13.0355 −0.513671
\(645\) 0 0
\(646\) 3.70852 0.145910
\(647\) −24.6633 −0.969615 −0.484808 0.874621i \(-0.661110\pi\)
−0.484808 + 0.874621i \(0.661110\pi\)
\(648\) 0 0
\(649\) −20.4827 −0.804016
\(650\) −18.8595 −0.739731
\(651\) 0 0
\(652\) 4.30059 0.168424
\(653\) −12.4942 −0.488937 −0.244469 0.969657i \(-0.578614\pi\)
−0.244469 + 0.969657i \(0.578614\pi\)
\(654\) 0 0
\(655\) 37.6856 1.47250
\(656\) 6.77074 0.264353
\(657\) 0 0
\(658\) 32.9785 1.28564
\(659\) −34.3869 −1.33952 −0.669762 0.742576i \(-0.733604\pi\)
−0.669762 + 0.742576i \(0.733604\pi\)
\(660\) 0 0
\(661\) 19.4957 0.758295 0.379147 0.925336i \(-0.376217\pi\)
0.379147 + 0.925336i \(0.376217\pi\)
\(662\) −15.7796 −0.613292
\(663\) 0 0
\(664\) 28.7192 1.11452
\(665\) −13.0380 −0.505591
\(666\) 0 0
\(667\) 28.8442 1.11685
\(668\) −14.6066 −0.565145
\(669\) 0 0
\(670\) 49.9228 1.92869
\(671\) 15.1267 0.583958
\(672\) 0 0
\(673\) −26.1090 −1.00643 −0.503215 0.864161i \(-0.667850\pi\)
−0.503215 + 0.864161i \(0.667850\pi\)
\(674\) 59.3764 2.28709
\(675\) 0 0
\(676\) 19.0844 0.734015
\(677\) 20.5707 0.790597 0.395298 0.918553i \(-0.370641\pi\)
0.395298 + 0.918553i \(0.370641\pi\)
\(678\) 0 0
\(679\) −0.549247 −0.0210782
\(680\) −3.89465 −0.149353
\(681\) 0 0
\(682\) 20.7343 0.793959
\(683\) −23.7002 −0.906864 −0.453432 0.891291i \(-0.649801\pi\)
−0.453432 + 0.891291i \(0.649801\pi\)
\(684\) 0 0
\(685\) −26.7768 −1.02309
\(686\) −34.7714 −1.32758
\(687\) 0 0
\(688\) −42.4484 −1.61833
\(689\) −54.7818 −2.08702
\(690\) 0 0
\(691\) −7.36209 −0.280067 −0.140034 0.990147i \(-0.544721\pi\)
−0.140034 + 0.990147i \(0.544721\pi\)
\(692\) −1.10052 −0.0418356
\(693\) 0 0
\(694\) −15.8646 −0.602211
\(695\) 21.0635 0.798985
\(696\) 0 0
\(697\) 1.17334 0.0444435
\(698\) 35.4282 1.34098
\(699\) 0 0
\(700\) −3.93692 −0.148802
\(701\) 28.4805 1.07569 0.537846 0.843043i \(-0.319238\pi\)
0.537846 + 0.843043i \(0.319238\pi\)
\(702\) 0 0
\(703\) −12.7978 −0.482679
\(704\) 2.88713 0.108813
\(705\) 0 0
\(706\) 22.9876 0.865151
\(707\) 15.0196 0.564870
\(708\) 0 0
\(709\) 32.8736 1.23459 0.617297 0.786730i \(-0.288228\pi\)
0.617297 + 0.786730i \(0.288228\pi\)
\(710\) −13.7617 −0.516467
\(711\) 0 0
\(712\) 29.8167 1.11743
\(713\) −22.4210 −0.839674
\(714\) 0 0
\(715\) 50.3812 1.88415
\(716\) 24.4231 0.912736
\(717\) 0 0
\(718\) 62.0877 2.31709
\(719\) 13.3988 0.499690 0.249845 0.968286i \(-0.419620\pi\)
0.249845 + 0.968286i \(0.419620\pi\)
\(720\) 0 0
\(721\) −29.1778 −1.08664
\(722\) 22.4520 0.835576
\(723\) 0 0
\(724\) −4.80481 −0.178569
\(725\) 8.71140 0.323533
\(726\) 0 0
\(727\) 24.5791 0.911587 0.455793 0.890086i \(-0.349356\pi\)
0.455793 + 0.890086i \(0.349356\pi\)
\(728\) −19.3533 −0.717282
\(729\) 0 0
\(730\) 1.16980 0.0432963
\(731\) −7.35613 −0.272076
\(732\) 0 0
\(733\) 2.08207 0.0769029 0.0384515 0.999260i \(-0.487758\pi\)
0.0384515 + 0.999260i \(0.487758\pi\)
\(734\) 60.3065 2.22595
\(735\) 0 0
\(736\) 33.6005 1.23853
\(737\) −37.0916 −1.36629
\(738\) 0 0
\(739\) −46.1104 −1.69620 −0.848099 0.529838i \(-0.822253\pi\)
−0.848099 + 0.529838i \(0.822253\pi\)
\(740\) −13.8946 −0.510774
\(741\) 0 0
\(742\) −33.9331 −1.24572
\(743\) −16.5456 −0.606998 −0.303499 0.952832i \(-0.598155\pi\)
−0.303499 + 0.952832i \(0.598155\pi\)
\(744\) 0 0
\(745\) 40.9614 1.50071
\(746\) 60.7873 2.22558
\(747\) 0 0
\(748\) −2.99148 −0.109379
\(749\) 33.6702 1.23028
\(750\) 0 0
\(751\) 36.4121 1.32870 0.664349 0.747423i \(-0.268709\pi\)
0.664349 + 0.747423i \(0.268709\pi\)
\(752\) −47.2243 −1.72209
\(753\) 0 0
\(754\) −44.2718 −1.61228
\(755\) −6.41540 −0.233480
\(756\) 0 0
\(757\) 16.3953 0.595898 0.297949 0.954582i \(-0.403697\pi\)
0.297949 + 0.954582i \(0.403697\pi\)
\(758\) −30.2267 −1.09788
\(759\) 0 0
\(760\) 11.0774 0.401821
\(761\) −44.2450 −1.60388 −0.801940 0.597404i \(-0.796199\pi\)
−0.801940 + 0.597404i \(0.796199\pi\)
\(762\) 0 0
\(763\) 12.1010 0.438087
\(764\) 24.7547 0.895593
\(765\) 0 0
\(766\) 13.1742 0.476003
\(767\) −33.9954 −1.22750
\(768\) 0 0
\(769\) −21.3439 −0.769680 −0.384840 0.922983i \(-0.625743\pi\)
−0.384840 + 0.922983i \(0.625743\pi\)
\(770\) 31.2073 1.12463
\(771\) 0 0
\(772\) 3.27841 0.117993
\(773\) −20.7861 −0.747623 −0.373812 0.927505i \(-0.621949\pi\)
−0.373812 + 0.927505i \(0.621949\pi\)
\(774\) 0 0
\(775\) −6.77150 −0.243239
\(776\) 0.466656 0.0167520
\(777\) 0 0
\(778\) −16.8164 −0.602897
\(779\) −3.33731 −0.119571
\(780\) 0 0
\(781\) 10.2246 0.365866
\(782\) 9.59868 0.343248
\(783\) 0 0
\(784\) 14.7936 0.528342
\(785\) 32.0114 1.14253
\(786\) 0 0
\(787\) −28.9072 −1.03043 −0.515215 0.857061i \(-0.672288\pi\)
−0.515215 + 0.857061i \(0.672288\pi\)
\(788\) 3.66952 0.130721
\(789\) 0 0
\(790\) 47.2013 1.67935
\(791\) 38.0433 1.35266
\(792\) 0 0
\(793\) 25.1060 0.891539
\(794\) 14.8424 0.526738
\(795\) 0 0
\(796\) 11.7957 0.418086
\(797\) −21.6563 −0.767106 −0.383553 0.923519i \(-0.625300\pi\)
−0.383553 + 0.923519i \(0.625300\pi\)
\(798\) 0 0
\(799\) −8.18377 −0.289521
\(800\) 10.1479 0.358781
\(801\) 0 0
\(802\) 39.9689 1.41135
\(803\) −0.869138 −0.0306712
\(804\) 0 0
\(805\) −33.7459 −1.18939
\(806\) 34.4131 1.21215
\(807\) 0 0
\(808\) −12.7611 −0.448933
\(809\) −4.12546 −0.145043 −0.0725217 0.997367i \(-0.523105\pi\)
−0.0725217 + 0.997367i \(0.523105\pi\)
\(810\) 0 0
\(811\) −27.4598 −0.964243 −0.482121 0.876104i \(-0.660134\pi\)
−0.482121 + 0.876104i \(0.660134\pi\)
\(812\) −9.24173 −0.324321
\(813\) 0 0
\(814\) 30.6325 1.07367
\(815\) 11.1332 0.389980
\(816\) 0 0
\(817\) 20.9228 0.731998
\(818\) 42.0726 1.47104
\(819\) 0 0
\(820\) −3.62330 −0.126531
\(821\) −26.7943 −0.935127 −0.467564 0.883959i \(-0.654868\pi\)
−0.467564 + 0.883959i \(0.654868\pi\)
\(822\) 0 0
\(823\) 29.5890 1.03141 0.515704 0.856767i \(-0.327531\pi\)
0.515704 + 0.856767i \(0.327531\pi\)
\(824\) 24.7903 0.863611
\(825\) 0 0
\(826\) −21.0575 −0.732686
\(827\) 41.2969 1.43603 0.718017 0.696025i \(-0.245050\pi\)
0.718017 + 0.696025i \(0.245050\pi\)
\(828\) 0 0
\(829\) −3.90138 −0.135501 −0.0677503 0.997702i \(-0.521582\pi\)
−0.0677503 + 0.997702i \(0.521582\pi\)
\(830\) −76.8610 −2.66789
\(831\) 0 0
\(832\) 4.79182 0.166126
\(833\) 2.56367 0.0888258
\(834\) 0 0
\(835\) −37.8130 −1.30857
\(836\) 8.50857 0.294275
\(837\) 0 0
\(838\) 48.2088 1.66534
\(839\) −18.4300 −0.636274 −0.318137 0.948045i \(-0.603057\pi\)
−0.318137 + 0.948045i \(0.603057\pi\)
\(840\) 0 0
\(841\) −8.55043 −0.294842
\(842\) 57.5252 1.98245
\(843\) 0 0
\(844\) 26.3752 0.907871
\(845\) 49.4051 1.69959
\(846\) 0 0
\(847\) −1.07355 −0.0368875
\(848\) 48.5912 1.66863
\(849\) 0 0
\(850\) 2.89895 0.0994332
\(851\) −33.1243 −1.13549
\(852\) 0 0
\(853\) 37.7897 1.29389 0.646947 0.762535i \(-0.276046\pi\)
0.646947 + 0.762535i \(0.276046\pi\)
\(854\) 15.5512 0.532151
\(855\) 0 0
\(856\) −28.6072 −0.977773
\(857\) 58.1926 1.98782 0.993910 0.110194i \(-0.0351472\pi\)
0.993910 + 0.110194i \(0.0351472\pi\)
\(858\) 0 0
\(859\) 35.3869 1.20738 0.603692 0.797218i \(-0.293696\pi\)
0.603692 + 0.797218i \(0.293696\pi\)
\(860\) 22.7159 0.774605
\(861\) 0 0
\(862\) −8.81224 −0.300146
\(863\) −27.4547 −0.934568 −0.467284 0.884107i \(-0.654767\pi\)
−0.467284 + 0.884107i \(0.654767\pi\)
\(864\) 0 0
\(865\) −2.84900 −0.0968689
\(866\) −20.2196 −0.687088
\(867\) 0 0
\(868\) 7.18373 0.243832
\(869\) −35.0696 −1.18965
\(870\) 0 0
\(871\) −61.5615 −2.08593
\(872\) −10.2814 −0.348172
\(873\) 0 0
\(874\) −27.3013 −0.923479
\(875\) 16.2612 0.549729
\(876\) 0 0
\(877\) −14.8517 −0.501505 −0.250752 0.968051i \(-0.580678\pi\)
−0.250752 + 0.968051i \(0.580678\pi\)
\(878\) 25.6958 0.867191
\(879\) 0 0
\(880\) −44.6879 −1.50643
\(881\) −37.8714 −1.27592 −0.637960 0.770070i \(-0.720221\pi\)
−0.637960 + 0.770070i \(0.720221\pi\)
\(882\) 0 0
\(883\) −15.3679 −0.517171 −0.258586 0.965988i \(-0.583256\pi\)
−0.258586 + 0.965988i \(0.583256\pi\)
\(884\) −4.96500 −0.166991
\(885\) 0 0
\(886\) −4.42102 −0.148527
\(887\) −2.72841 −0.0916111 −0.0458056 0.998950i \(-0.514585\pi\)
−0.0458056 + 0.998950i \(0.514585\pi\)
\(888\) 0 0
\(889\) 27.6008 0.925703
\(890\) −79.7984 −2.67485
\(891\) 0 0
\(892\) −4.47406 −0.149803
\(893\) 23.2769 0.778931
\(894\) 0 0
\(895\) 63.2259 2.11341
\(896\) 24.1474 0.806707
\(897\) 0 0
\(898\) −22.8495 −0.762499
\(899\) −15.8958 −0.530153
\(900\) 0 0
\(901\) 8.42066 0.280533
\(902\) 7.98807 0.265973
\(903\) 0 0
\(904\) −32.3227 −1.07504
\(905\) −12.4385 −0.413471
\(906\) 0 0
\(907\) −6.49334 −0.215608 −0.107804 0.994172i \(-0.534382\pi\)
−0.107804 + 0.994172i \(0.534382\pi\)
\(908\) 23.4617 0.778605
\(909\) 0 0
\(910\) 51.7952 1.71699
\(911\) 32.2049 1.06699 0.533497 0.845802i \(-0.320877\pi\)
0.533497 + 0.845802i \(0.320877\pi\)
\(912\) 0 0
\(913\) 57.1061 1.88994
\(914\) −9.63615 −0.318736
\(915\) 0 0
\(916\) 3.26511 0.107882
\(917\) −28.7855 −0.950580
\(918\) 0 0
\(919\) −25.5609 −0.843177 −0.421588 0.906787i \(-0.638527\pi\)
−0.421588 + 0.906787i \(0.638527\pi\)
\(920\) 28.6715 0.945271
\(921\) 0 0
\(922\) −55.9753 −1.84345
\(923\) 16.9700 0.558574
\(924\) 0 0
\(925\) −10.0041 −0.328932
\(926\) 32.1390 1.05615
\(927\) 0 0
\(928\) 23.8216 0.781983
\(929\) 43.9109 1.44067 0.720336 0.693626i \(-0.243988\pi\)
0.720336 + 0.693626i \(0.243988\pi\)
\(930\) 0 0
\(931\) −7.29177 −0.238978
\(932\) −7.25067 −0.237503
\(933\) 0 0
\(934\) 36.3334 1.18886
\(935\) −7.74424 −0.253264
\(936\) 0 0
\(937\) 37.5543 1.22684 0.613422 0.789755i \(-0.289792\pi\)
0.613422 + 0.789755i \(0.289792\pi\)
\(938\) −38.1326 −1.24507
\(939\) 0 0
\(940\) 25.2716 0.824270
\(941\) 20.6958 0.674663 0.337332 0.941386i \(-0.390476\pi\)
0.337332 + 0.941386i \(0.390476\pi\)
\(942\) 0 0
\(943\) −8.63787 −0.281288
\(944\) 30.1538 0.981423
\(945\) 0 0
\(946\) −50.0802 −1.62825
\(947\) 14.0412 0.456276 0.228138 0.973629i \(-0.426736\pi\)
0.228138 + 0.973629i \(0.426736\pi\)
\(948\) 0 0
\(949\) −1.44252 −0.0468262
\(950\) −8.24540 −0.267516
\(951\) 0 0
\(952\) 2.97485 0.0964155
\(953\) 32.4866 1.05235 0.526173 0.850378i \(-0.323627\pi\)
0.526173 + 0.850378i \(0.323627\pi\)
\(954\) 0 0
\(955\) 64.0842 2.07371
\(956\) −3.72330 −0.120420
\(957\) 0 0
\(958\) −17.1253 −0.553293
\(959\) 20.4530 0.660461
\(960\) 0 0
\(961\) −18.6440 −0.601419
\(962\) 50.8412 1.63919
\(963\) 0 0
\(964\) −1.01662 −0.0327433
\(965\) 8.48704 0.273208
\(966\) 0 0
\(967\) −53.7501 −1.72849 −0.864244 0.503073i \(-0.832203\pi\)
−0.864244 + 0.503073i \(0.832203\pi\)
\(968\) 0.912116 0.0293165
\(969\) 0 0
\(970\) −1.24891 −0.0401001
\(971\) −34.1747 −1.09672 −0.548359 0.836243i \(-0.684747\pi\)
−0.548359 + 0.836243i \(0.684747\pi\)
\(972\) 0 0
\(973\) −16.0890 −0.515789
\(974\) 45.0346 1.44300
\(975\) 0 0
\(976\) −22.2689 −0.712810
\(977\) 8.90298 0.284832 0.142416 0.989807i \(-0.454513\pi\)
0.142416 + 0.989807i \(0.454513\pi\)
\(978\) 0 0
\(979\) 59.2885 1.89487
\(980\) −7.91665 −0.252888
\(981\) 0 0
\(982\) −39.1822 −1.25035
\(983\) −22.5713 −0.719912 −0.359956 0.932969i \(-0.617208\pi\)
−0.359956 + 0.932969i \(0.617208\pi\)
\(984\) 0 0
\(985\) 9.49955 0.302681
\(986\) 6.80514 0.216720
\(987\) 0 0
\(988\) 14.1218 0.449275
\(989\) 54.1541 1.72200
\(990\) 0 0
\(991\) 46.7740 1.48583 0.742913 0.669388i \(-0.233444\pi\)
0.742913 + 0.669388i \(0.233444\pi\)
\(992\) −18.5169 −0.587912
\(993\) 0 0
\(994\) 10.5116 0.333408
\(995\) 30.5362 0.968063
\(996\) 0 0
\(997\) −20.7833 −0.658214 −0.329107 0.944293i \(-0.606748\pi\)
−0.329107 + 0.944293i \(0.606748\pi\)
\(998\) −69.2448 −2.19191
\(999\) 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 2169.2.a.e.1.1 7
3.2 odd 2 241.2.a.a.1.7 7
12.11 even 2 3856.2.a.j.1.6 7
15.14 odd 2 6025.2.a.f.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.7 7 3.2 odd 2
2169.2.a.e.1.1 7 1.1 even 1 trivial
3856.2.a.j.1.6 7 12.11 even 2
6025.2.a.f.1.1 7 15.14 odd 2