Properties

Label 6025.2.a.f.1.1
Level $6025$
Weight $2$
Character 6025.1
Self dual yes
Analytic conductor $48.110$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 6025.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73684 q^{2} +2.37146 q^{3} +1.01662 q^{4} -4.11885 q^{6} +2.01025 q^{7} +1.70797 q^{8} +2.62382 q^{9} +O(q^{10})\) \(q-1.73684 q^{2} +2.37146 q^{3} +1.01662 q^{4} -4.11885 q^{6} +2.01025 q^{7} +1.70797 q^{8} +2.62382 q^{9} -3.39618 q^{11} +2.41088 q^{12} -5.63669 q^{13} -3.49150 q^{14} -4.99972 q^{16} -0.866432 q^{17} -4.55716 q^{18} +2.46437 q^{19} +4.76723 q^{21} +5.89863 q^{22} +6.37847 q^{23} +4.05038 q^{24} +9.79005 q^{26} -0.892104 q^{27} +2.04367 q^{28} -4.52212 q^{29} -3.51511 q^{31} +5.26780 q^{32} -8.05390 q^{33} +1.50486 q^{34} +2.66743 q^{36} +5.19315 q^{37} -4.28022 q^{38} -13.3672 q^{39} +1.35422 q^{41} -8.27994 q^{42} -8.49015 q^{43} -3.45264 q^{44} -11.0784 q^{46} +9.44537 q^{47} -11.8566 q^{48} -2.95888 q^{49} -2.05471 q^{51} -5.73040 q^{52} -9.71877 q^{53} +1.54945 q^{54} +3.43345 q^{56} +5.84415 q^{57} +7.85421 q^{58} +6.03110 q^{59} +4.45402 q^{61} +6.10520 q^{62} +5.27454 q^{63} +0.850111 q^{64} +13.9884 q^{66} +10.9216 q^{67} -0.880836 q^{68} +15.1263 q^{69} -3.01063 q^{71} +4.48140 q^{72} +0.255916 q^{73} -9.01969 q^{74} +2.50534 q^{76} -6.82718 q^{77} +23.2167 q^{78} -10.3262 q^{79} -9.98704 q^{81} -2.35207 q^{82} +16.8148 q^{83} +4.84649 q^{84} +14.7461 q^{86} -10.7240 q^{87} -5.80057 q^{88} -17.4574 q^{89} -11.3312 q^{91} +6.48451 q^{92} -8.33594 q^{93} -16.4051 q^{94} +12.4924 q^{96} -0.273223 q^{97} +5.13911 q^{98} -8.91095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} + O(q^{10}) \) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} - 18q^{11} - q^{12} + q^{13} - 6q^{14} + 4q^{16} + 2q^{17} - 8q^{18} - 6q^{19} - 2q^{21} - 10q^{22} + 22q^{23} - 3q^{24} + 8q^{26} - 3q^{27} - 9q^{28} - 16q^{29} - 18q^{31} + 6q^{32} - 4q^{33} + 11q^{34} - 7q^{36} - 8q^{37} - 16q^{38} - 9q^{39} - 15q^{41} - 19q^{42} - 14q^{43} - 4q^{44} + 11q^{46} + 10q^{47} - 31q^{48} + 6q^{49} + 13q^{51} - 27q^{52} - 15q^{53} + 16q^{54} + 13q^{56} - 14q^{57} - 17q^{58} - 18q^{59} + 4q^{61} - 13q^{62} + 16q^{63} + 2q^{64} + 16q^{66} - 18q^{67} + 15q^{68} + 26q^{69} - 50q^{71} - 30q^{72} + 10q^{74} - 20q^{76} - 17q^{77} + 32q^{78} - 15q^{79} - 9q^{81} - 45q^{82} + 24q^{83} + 6q^{84} - 23q^{86} - 12q^{87} - 8q^{88} - 13q^{89} - 12q^{91} + 10q^{92} - 14q^{93} - 32q^{94} - 15q^{96} - q^{97} - 9q^{98} - 20q^{99} + O(q^{100}) \)

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 \(0.710467\pi\)
\(3\) 2.37146 1.36916 0.684581 0.728937i \(-0.259985\pi\)
0.684581 + 0.728937i \(0.259985\pi\)
\(4\) 1.01662 0.508312
\(5\) 0 0
\(6\) −4.11885 −1.68151
\(7\) 2.01025 0.759805 0.379902 0.925027i \(-0.375958\pi\)
0.379902 + 0.925027i \(0.375958\pi\)
\(8\) 1.70797 0.603858
\(9\) 2.62382 0.874605
\(10\) 0 0
\(11\) −3.39618 −1.02399 −0.511993 0.858989i \(-0.671093\pi\)
−0.511993 + 0.858989i \(0.671093\pi\)
\(12\) 2.41088 0.695962
\(13\) −5.63669 −1.56334 −0.781669 0.623694i \(-0.785631\pi\)
−0.781669 + 0.623694i \(0.785631\pi\)
\(14\) −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 \(0.533507\pi\)
\(18\) −4.55716 −1.07413
\(19\) 2.46437 0.565365 0.282682 0.959214i \(-0.408776\pi\)
0.282682 + 0.959214i \(0.408776\pi\)
\(20\) 0 0
\(21\) 4.76723 1.04030
\(22\) 5.89863 1.25759
\(23\) 6.37847 1.33000 0.665001 0.746842i \(-0.268431\pi\)
0.665001 + 0.746842i \(0.268431\pi\)
\(24\) 4.05038 0.826780
\(25\) 0 0
\(26\) 9.79005 1.91999
\(27\) −0.892104 −0.171686
\(28\) 2.04367 0.386218
\(29\) −4.52212 −0.839737 −0.419868 0.907585i \(-0.637924\pi\)
−0.419868 + 0.907585i \(0.637924\pi\)
\(30\) 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
\(33\) −8.05390 −1.40200
\(34\) 1.50486 0.258081
\(35\) 0 0
\(36\) 2.66743 0.444572
\(37\) 5.19315 0.853748 0.426874 0.904311i \(-0.359615\pi\)
0.426874 + 0.904311i \(0.359615\pi\)
\(38\) −4.28022 −0.694344
\(39\) −13.3672 −2.14046
\(40\) 0 0
\(41\) 1.35422 0.211494 0.105747 0.994393i \(-0.466277\pi\)
0.105747 + 0.994393i \(0.466277\pi\)
\(42\) −8.27994 −1.27762
\(43\) −8.49015 −1.29474 −0.647368 0.762178i \(-0.724130\pi\)
−0.647368 + 0.762178i \(0.724130\pi\)
\(44\) −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 \(0.258105\pi\)
\(48\) −11.8566 −1.71136
\(49\) −2.95888 −0.422697
\(50\) 0 0
\(51\) −2.05471 −0.287717
\(52\) −5.73040 −0.794663
\(53\) −9.71877 −1.33498 −0.667488 0.744620i \(-0.732631\pi\)
−0.667488 + 0.744620i \(0.732631\pi\)
\(54\) 1.54945 0.210853
\(55\) 0 0
\(56\) 3.43345 0.458814
\(57\) 5.84415 0.774076
\(58\) 7.85421 1.03131
\(59\) 6.03110 0.785182 0.392591 0.919713i \(-0.371579\pi\)
0.392591 + 0.919713i \(0.371579\pi\)
\(60\) 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
\(63\) 5.27454 0.664529
\(64\) 0.850111 0.106264
\(65\) 0 0
\(66\) 13.9884 1.72185
\(67\) 10.9216 1.33428 0.667141 0.744932i \(-0.267518\pi\)
0.667141 + 0.744932i \(0.267518\pi\)
\(68\) −0.880836 −0.106817
\(69\) 15.1263 1.82099
\(70\) 0 0
\(71\) −3.01063 −0.357296 −0.178648 0.983913i \(-0.557172\pi\)
−0.178648 + 0.983913i \(0.557172\pi\)
\(72\) 4.48140 0.528138
\(73\) 0.255916 0.0299527 0.0149764 0.999888i \(-0.495233\pi\)
0.0149764 + 0.999888i \(0.495233\pi\)
\(74\) −9.01969 −1.04852
\(75\) 0 0
\(76\) 2.50534 0.287382
\(77\) −6.82718 −0.778030
\(78\) 23.2167 2.62877
\(79\) −10.3262 −1.16179 −0.580893 0.813980i \(-0.697297\pi\)
−0.580893 + 0.813980i \(0.697297\pi\)
\(80\) 0 0
\(81\) −9.98704 −1.10967
\(82\) −2.35207 −0.259743
\(83\) 16.8148 1.84567 0.922833 0.385201i \(-0.125868\pi\)
0.922833 + 0.385201i \(0.125868\pi\)
\(84\) 4.84649 0.528795
\(85\) 0 0
\(86\) 14.7461 1.59011
\(87\) −10.7240 −1.14974
\(88\) −5.80057 −0.618343
\(89\) −17.4574 −1.85048 −0.925241 0.379380i \(-0.876137\pi\)
−0.925241 + 0.379380i \(0.876137\pi\)
\(90\) 0 0
\(91\) −11.3312 −1.18783
\(92\) 6.48451 0.676056
\(93\) −8.33594 −0.864397
\(94\) −16.4051 −1.69206
\(95\) 0 0
\(96\) 12.4924 1.27500
\(97\) −0.273223 −0.0277416 −0.0138708 0.999904i \(-0.504415\pi\)
−0.0138708 + 0.999904i \(0.504415\pi\)
\(98\) 5.13911 0.519128
\(99\) −8.91095 −0.895584
\(100\) 0 0
\(101\) 7.47149 0.743441 0.371721 0.928345i \(-0.378768\pi\)
0.371721 + 0.928345i \(0.378768\pi\)
\(102\) 3.56870 0.353354
\(103\) −14.5145 −1.43015 −0.715077 0.699046i \(-0.753608\pi\)
−0.715077 + 0.699046i \(0.753608\pi\)
\(104\) −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 \(0.800318\pi\)
\(108\) −0.906935 −0.0872698
\(109\) −6.01965 −0.576578 −0.288289 0.957543i \(-0.593086\pi\)
−0.288289 + 0.957543i \(0.593086\pi\)
\(110\) 0 0
\(111\) 12.3153 1.16892
\(112\) −10.0507 −0.949703
\(113\) −18.9246 −1.78028 −0.890139 0.455689i \(-0.849393\pi\)
−0.890139 + 0.455689i \(0.849393\pi\)
\(114\) −10.1504 −0.950669
\(115\) 0 0
\(116\) −4.59730 −0.426848
\(117\) −14.7896 −1.36730
\(118\) −10.4751 −0.964308
\(119\) −1.74175 −0.159666
\(120\) 0 0
\(121\) 0.534035 0.0485487
\(122\) −7.73594 −0.700379
\(123\) 3.21149 0.289570
\(124\) −3.57355 −0.320914
\(125\) 0 0
\(126\) −9.16104 −0.816131
\(127\) 13.7300 1.21834 0.609172 0.793038i \(-0.291502\pi\)
0.609172 + 0.793038i \(0.291502\pi\)
\(128\) −12.0121 −1.06173
\(129\) −20.1340 −1.77270
\(130\) 0 0
\(131\) −14.3193 −1.25108 −0.625542 0.780190i \(-0.715122\pi\)
−0.625542 + 0.780190i \(0.715122\pi\)
\(132\) −8.18779 −0.712656
\(133\) 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 \(0.643119\pi\)
\(138\) −26.2720 −2.23642
\(139\) 8.00345 0.678844 0.339422 0.940634i \(-0.389769\pi\)
0.339422 + 0.940634i \(0.389769\pi\)
\(140\) 0 0
\(141\) 22.3993 1.88636
\(142\) 5.22899 0.438807
\(143\) 19.1432 1.60084
\(144\) −13.1184 −1.09320
\(145\) 0 0
\(146\) −0.444486 −0.0367860
\(147\) −7.01686 −0.578741
\(148\) 5.27948 0.433971
\(149\) −15.5640 −1.27505 −0.637527 0.770428i \(-0.720042\pi\)
−0.637527 + 0.770428i \(0.720042\pi\)
\(150\) 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
\(153\) −2.27336 −0.183790
\(154\) 11.8577 0.955524
\(155\) 0 0
\(156\) −13.5894 −1.08802
\(157\) −12.1633 −0.970735 −0.485367 0.874310i \(-0.661314\pi\)
−0.485367 + 0.874310i \(0.661314\pi\)
\(158\) 17.9350 1.42683
\(159\) −23.0477 −1.82780
\(160\) 0 0
\(161\) 12.8223 1.01054
\(162\) 17.3459 1.36282
\(163\) −4.23026 −0.331340 −0.165670 0.986181i \(-0.552979\pi\)
−0.165670 + 0.986181i \(0.552979\pi\)
\(164\) 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 \(0.687628\pi\)
\(168\) 8.14229 0.628191
\(169\) 18.7723 1.44402
\(170\) 0 0
\(171\) 6.46605 0.494471
\(172\) −8.63129 −0.658130
\(173\) −1.08253 −0.0823030 −0.0411515 0.999153i \(-0.513103\pi\)
−0.0411515 + 0.999153i \(0.513103\pi\)
\(174\) 18.6259 1.41203
\(175\) 0 0
\(176\) 16.9800 1.27991
\(177\) 14.3025 1.07504
\(178\) 30.3208 2.27264
\(179\) −24.0238 −1.79562 −0.897810 0.440382i \(-0.854843\pi\)
−0.897810 + 0.440382i \(0.854843\pi\)
\(180\) 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
\(183\) 10.5625 0.780805
\(184\) 10.8942 0.803133
\(185\) 0 0
\(186\) 14.4782 1.06159
\(187\) 2.94256 0.215181
\(188\) 9.60239 0.700327
\(189\) −1.79336 −0.130447
\(190\) 0 0
\(191\) −24.3499 −1.76190 −0.880948 0.473213i \(-0.843094\pi\)
−0.880948 + 0.473213i \(0.843094\pi\)
\(192\) 2.01600 0.145492
\(193\) −3.22480 −0.232126 −0.116063 0.993242i \(-0.537027\pi\)
−0.116063 + 0.993242i \(0.537027\pi\)
\(194\) 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 \(0.458957\pi\)
\(198\) 15.4769 1.09990
\(199\) 11.6028 0.822499 0.411249 0.911523i \(-0.365093\pi\)
0.411249 + 0.911523i \(0.365093\pi\)
\(200\) 0 0
\(201\) 25.9000 1.82685
\(202\) −12.9768 −0.913045
\(203\) −9.09061 −0.638036
\(204\) −2.08887 −0.146250
\(205\) 0 0
\(206\) 25.2094 1.75642
\(207\) 16.7359 1.16323
\(208\) 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
\(213\) −7.13958 −0.489196
\(214\) 29.0908 1.98861
\(215\) 0 0
\(216\) −1.52369 −0.103674
\(217\) −7.06626 −0.479689
\(218\) 10.4552 0.708115
\(219\) 0.606895 0.0410102
\(220\) 0 0
\(221\) 4.88381 0.328521
\(222\) −21.3898 −1.43559
\(223\) 4.40090 0.294706 0.147353 0.989084i \(-0.452925\pi\)
0.147353 + 0.989084i \(0.452925\pi\)
\(224\) 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 \(0.222307\pi\)
\(228\) 5.94130 0.393472
\(229\) 3.21172 0.212236 0.106118 0.994354i \(-0.466158\pi\)
0.106118 + 0.994354i \(0.466158\pi\)
\(230\) 0 0
\(231\) −16.1904 −1.06525
\(232\) −7.72364 −0.507082
\(233\) −7.13210 −0.467239 −0.233620 0.972328i \(-0.575057\pi\)
−0.233620 + 0.972328i \(0.575057\pi\)
\(234\) 25.6873 1.67923
\(235\) 0 0
\(236\) 6.13136 0.399117
\(237\) −24.4881 −1.59067
\(238\) 3.02514 0.196091
\(239\) 3.66241 0.236902 0.118451 0.992960i \(-0.462207\pi\)
0.118451 + 0.992960i \(0.462207\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) −0.927536 −0.0596243
\(243\) −21.0075 −1.34763
\(244\) 4.52807 0.289880
\(245\) 0 0
\(246\) −5.57785 −0.355630
\(247\) −13.8909 −0.883856
\(248\) −6.00370 −0.381235
\(249\) 39.8756 2.52702
\(250\) 0 0
\(251\) 1.41613 0.0893854 0.0446927 0.999001i \(-0.485769\pi\)
0.0446927 + 0.999001i \(0.485769\pi\)
\(252\) 5.36222 0.337788
\(253\) −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 \(0.387800\pi\)
\(258\) 34.9697 2.17712
\(259\) 10.4395 0.648682
\(260\) 0 0
\(261\) −11.8652 −0.734438
\(262\) 24.8704 1.53650
\(263\) 3.31569 0.204454 0.102227 0.994761i \(-0.467403\pi\)
0.102227 + 0.994761i \(0.467403\pi\)
\(264\) −13.7558 −0.846612
\(265\) 0 0
\(266\) −8.60433 −0.527565
\(267\) −41.3995 −2.53361
\(268\) 11.1031 0.678231
\(269\) −14.3455 −0.874663 −0.437332 0.899300i \(-0.644076\pi\)
−0.437332 + 0.899300i \(0.644076\pi\)
\(270\) 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
\(273\) −26.8714 −1.62633
\(274\) 17.6712 1.06756
\(275\) 0 0
\(276\) 15.3777 0.925631
\(277\) −18.0089 −1.08205 −0.541024 0.841007i \(-0.681963\pi\)
−0.541024 + 0.841007i \(0.681963\pi\)
\(278\) −13.9007 −0.833711
\(279\) −9.22300 −0.552167
\(280\) 0 0
\(281\) −11.5476 −0.688873 −0.344436 0.938810i \(-0.611930\pi\)
−0.344436 + 0.938810i \(0.611930\pi\)
\(282\) −38.9041 −2.31671
\(283\) 1.84755 0.109825 0.0549126 0.998491i \(-0.482512\pi\)
0.0549126 + 0.998491i \(0.482512\pi\)
\(284\) −3.06068 −0.181618
\(285\) 0 0
\(286\) −33.2488 −1.96604
\(287\) 2.72233 0.160694
\(288\) 13.8217 0.814453
\(289\) −16.2493 −0.955841
\(290\) 0 0
\(291\) −0.647937 −0.0379827
\(292\) 0.260171 0.0152253
\(293\) 3.03213 0.177139 0.0885694 0.996070i \(-0.471771\pi\)
0.0885694 + 0.996070i \(0.471771\pi\)
\(294\) 12.1872 0.710771
\(295\) 0 0
\(296\) 8.86974 0.515543
\(297\) 3.02975 0.175804
\(298\) 27.0322 1.56594
\(299\) −35.9535 −2.07924
\(300\) 0 0
\(301\) −17.0674 −0.983746
\(302\) 4.23380 0.243628
\(303\) 17.7183 1.01789
\(304\) −12.3212 −0.706667
\(305\) 0 0
\(306\) 3.94847 0.225719
\(307\) −8.40693 −0.479809 −0.239905 0.970796i \(-0.577116\pi\)
−0.239905 + 0.970796i \(0.577116\pi\)
\(308\) −6.94068 −0.395482
\(309\) −34.4205 −1.95811
\(310\) 0 0
\(311\) 2.94070 0.166752 0.0833759 0.996518i \(-0.473430\pi\)
0.0833759 + 0.996518i \(0.473430\pi\)
\(312\) −22.8307 −1.29254
\(313\) −16.5110 −0.933258 −0.466629 0.884453i \(-0.654532\pi\)
−0.466629 + 0.884453i \(0.654532\pi\)
\(314\) 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 \(0.306041\pi\)
\(318\) 40.0302 2.24478
\(319\) 15.3579 0.859879
\(320\) 0 0
\(321\) −39.7201 −2.21696
\(322\) −22.2704 −1.24108
\(323\) −2.13521 −0.118806
\(324\) −10.1531 −0.564059
\(325\) 0 0
\(326\) 7.34731 0.406930
\(327\) −14.2754 −0.789429
\(328\) 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
\(333\) 13.6259 0.746693
\(334\) 24.9545 1.36545
\(335\) 0 0
\(336\) −23.8349 −1.30030
\(337\) 34.1864 1.86225 0.931125 0.364700i \(-0.118829\pi\)
0.931125 + 0.364700i \(0.118829\pi\)
\(338\) −32.6046 −1.77345
\(339\) −44.8789 −2.43749
\(340\) 0 0
\(341\) 11.9379 0.646476
\(342\) −11.2305 −0.607277
\(343\) −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 \(0.421155\pi\)
\(348\) −10.9023 −0.584425
\(349\) −20.3980 −1.09188 −0.545940 0.837824i \(-0.683827\pi\)
−0.545940 + 0.837824i \(0.683827\pi\)
\(350\) 0 0
\(351\) 5.02852 0.268402
\(352\) −17.8904 −0.953561
\(353\) −13.2353 −0.704444 −0.352222 0.935916i \(-0.614574\pi\)
−0.352222 + 0.935916i \(0.614574\pi\)
\(354\) −24.8412 −1.32029
\(355\) 0 0
\(356\) −17.7476 −0.940622
\(357\) −4.13048 −0.218608
\(358\) 41.7255 2.20526
\(359\) 35.7474 1.88668 0.943339 0.331831i \(-0.107666\pi\)
0.943339 + 0.331831i \(0.107666\pi\)
\(360\) 0 0
\(361\) −12.9269 −0.680363
\(362\) 8.20873 0.431441
\(363\) 1.26644 0.0664710
\(364\) −11.5196 −0.603789
\(365\) 0 0
\(366\) −18.3455 −0.958933
\(367\) 34.7219 1.81247 0.906234 0.422777i \(-0.138945\pi\)
0.906234 + 0.422777i \(0.138945\pi\)
\(368\) −31.8906 −1.66241
\(369\) 3.55323 0.184974
\(370\) 0 0
\(371\) −19.5372 −1.01432
\(372\) −8.47452 −0.439383
\(373\) 34.9987 1.81217 0.906083 0.423100i \(-0.139058\pi\)
0.906083 + 0.423100i \(0.139058\pi\)
\(374\) −5.11076 −0.264271
\(375\) 0 0
\(376\) 16.1324 0.831966
\(377\) 25.4898 1.31279
\(378\) 3.11478 0.160207
\(379\) 17.4032 0.893943 0.446972 0.894548i \(-0.352502\pi\)
0.446972 + 0.894548i \(0.352502\pi\)
\(380\) 0 0
\(381\) 32.5602 1.66811
\(382\) 42.2919 2.16384
\(383\) −7.58514 −0.387582 −0.193791 0.981043i \(-0.562078\pi\)
−0.193791 + 0.981043i \(0.562078\pi\)
\(384\) −28.4862 −1.45368
\(385\) 0 0
\(386\) 5.60097 0.285082
\(387\) −22.2766 −1.13238
\(388\) −0.277765 −0.0141014
\(389\) −9.68216 −0.490905 −0.245452 0.969409i \(-0.578937\pi\)
−0.245452 + 0.969409i \(0.578937\pi\)
\(390\) 0 0
\(391\) −5.52651 −0.279488
\(392\) −5.05368 −0.255249
\(393\) −33.9577 −1.71294
\(394\) −6.26917 −0.315836
\(395\) 0 0
\(396\) −9.05909 −0.455236
\(397\) 8.54563 0.428893 0.214446 0.976736i \(-0.431205\pi\)
0.214446 + 0.976736i \(0.431205\pi\)
\(398\) −20.1522 −1.01014
\(399\) 11.7482 0.588147
\(400\) 0 0
\(401\) 23.0124 1.14918 0.574592 0.818440i \(-0.305161\pi\)
0.574592 + 0.818440i \(0.305161\pi\)
\(402\) −44.9843 −2.24361
\(403\) 19.8136 0.986986
\(404\) 7.59570 0.377900
\(405\) 0 0
\(406\) 15.7890 0.783593
\(407\) −17.6369 −0.874227
\(408\) −3.50938 −0.173740
\(409\) −24.2236 −1.19778 −0.598891 0.800831i \(-0.704392\pi\)
−0.598891 + 0.800831i \(0.704392\pi\)
\(410\) 0 0
\(411\) −24.1280 −1.19015
\(412\) −14.7558 −0.726965
\(413\) 12.1240 0.596585
\(414\) −29.0677 −1.42860
\(415\) 0 0
\(416\) −29.6930 −1.45582
\(417\) 18.9799 0.929448
\(418\) 14.5364 0.710999
\(419\) 27.7565 1.35600 0.677998 0.735064i \(-0.262848\pi\)
0.677998 + 0.735064i \(0.262848\pi\)
\(420\) 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
\(423\) 24.7829 1.20499
\(424\) −16.5994 −0.806137
\(425\) 0 0
\(426\) 12.4003 0.600798
\(427\) 8.95372 0.433301
\(428\) −17.0277 −0.823064
\(429\) 45.3974 2.19181
\(430\) 0 0
\(431\) −5.07371 −0.244392 −0.122196 0.992506i \(-0.538994\pi\)
−0.122196 + 0.992506i \(0.538994\pi\)
\(432\) 4.46027 0.214595
\(433\) −11.6416 −0.559457 −0.279729 0.960079i \(-0.590245\pi\)
−0.279729 + 0.960079i \(0.590245\pi\)
\(434\) 12.2730 0.589122
\(435\) 0 0
\(436\) −6.11973 −0.293082
\(437\) 15.7189 0.751937
\(438\) −1.05408 −0.0503660
\(439\) −14.7945 −0.706105 −0.353052 0.935604i \(-0.614856\pi\)
−0.353052 + 0.935604i \(0.614856\pi\)
\(440\) 0 0
\(441\) −7.76356 −0.369693
\(442\) −8.48241 −0.403467
\(443\) 2.54544 0.120937 0.0604687 0.998170i \(-0.480740\pi\)
0.0604687 + 0.998170i \(0.480740\pi\)
\(444\) 12.5201 0.594176
\(445\) 0 0
\(446\) −7.64367 −0.361938
\(447\) −36.9094 −1.74575
\(448\) 1.70894 0.0807398
\(449\) −13.1558 −0.620860 −0.310430 0.950596i \(-0.600473\pi\)
−0.310430 + 0.950596i \(0.600473\pi\)
\(450\) 0 0
\(451\) −4.59919 −0.216567
\(452\) −19.2392 −0.904937
\(453\) −5.78077 −0.271604
\(454\) −40.0830 −1.88119
\(455\) 0 0
\(456\) 9.98163 0.467432
\(457\) −5.54809 −0.259528 −0.129764 0.991545i \(-0.541422\pi\)
−0.129764 + 0.991545i \(0.541422\pi\)
\(458\) −5.57825 −0.260655
\(459\) 0.772948 0.0360781
\(460\) 0 0
\(461\) −32.2282 −1.50102 −0.750508 0.660861i \(-0.770191\pi\)
−0.750508 + 0.660861i \(0.770191\pi\)
\(462\) 28.1202 1.30827
\(463\) 18.5042 0.859965 0.429983 0.902837i \(-0.358520\pi\)
0.429983 + 0.902837i \(0.358520\pi\)
\(464\) 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 \(0.660821\pi\)
\(468\) −15.0355 −0.695017
\(469\) 21.9551 1.01379
\(470\) 0 0
\(471\) −28.8447 −1.32909
\(472\) 10.3009 0.474139
\(473\) 28.8341 1.32579
\(474\) 42.5320 1.95356
\(475\) 0 0
\(476\) −1.77070 −0.0811600
\(477\) −25.5003 −1.16758
\(478\) −6.36103 −0.290947
\(479\) −9.86001 −0.450515 −0.225258 0.974299i \(-0.572322\pi\)
−0.225258 + 0.974299i \(0.572322\pi\)
\(480\) 0 0
\(481\) −29.2722 −1.33470
\(482\) 1.73684 0.0791110
\(483\) 30.4077 1.38360
\(484\) 0.542913 0.0246779
\(485\) 0 0
\(486\) 36.4868 1.65507
\(487\) 25.9290 1.17496 0.587478 0.809240i \(-0.300121\pi\)
0.587478 + 0.809240i \(0.300121\pi\)
\(488\) 7.60734 0.344368
\(489\) −10.0319 −0.453658
\(490\) 0 0
\(491\) −22.5594 −1.01809 −0.509046 0.860739i \(-0.670002\pi\)
−0.509046 + 0.860739i \(0.670002\pi\)
\(492\) 3.26487 0.147192
\(493\) 3.91811 0.176463
\(494\) 24.1263 1.08549
\(495\) 0 0
\(496\) 17.5746 0.789122
\(497\) −6.05213 −0.271475
\(498\) −69.2577 −3.10351
\(499\) 39.8682 1.78475 0.892373 0.451298i \(-0.149039\pi\)
0.892373 + 0.451298i \(0.149039\pi\)
\(500\) 0 0
\(501\) −34.0724 −1.52224
\(502\) −2.45960 −0.109777
\(503\) 19.7008 0.878416 0.439208 0.898386i \(-0.355259\pi\)
0.439208 + 0.898386i \(0.355259\pi\)
\(504\) 9.00875 0.401282
\(505\) 0 0
\(506\) 37.6242 1.67260
\(507\) 44.5178 1.97710
\(508\) 13.9583 0.619298
\(509\) −23.4312 −1.03857 −0.519284 0.854602i \(-0.673801\pi\)
−0.519284 + 0.854602i \(0.673801\pi\)
\(510\) 0 0
\(511\) 0.514457 0.0227582
\(512\) −9.25877 −0.409184
\(513\) −2.19847 −0.0970650
\(514\) −19.2251 −0.847985
\(515\) 0 0
\(516\) −20.4687 −0.901086
\(517\) −32.0782 −1.41080
\(518\) −18.1319 −0.796668
\(519\) −2.56717 −0.112686
\(520\) 0 0
\(521\) −5.76181 −0.252430 −0.126215 0.992003i \(-0.540283\pi\)
−0.126215 + 0.992003i \(0.540283\pi\)
\(522\) 20.6080 0.901988
\(523\) −42.0766 −1.83988 −0.919942 0.392056i \(-0.871764\pi\)
−0.919942 + 0.392056i \(0.871764\pi\)
\(524\) −14.5574 −0.635941
\(525\) 0 0
\(526\) −5.75883 −0.251097
\(527\) 3.04560 0.132669
\(528\) 40.2673 1.75241
\(529\) 17.6849 0.768907
\(530\) 0 0
\(531\) 15.8245 0.686724
\(532\) 5.03636 0.218354
\(533\) −7.63334 −0.330637
\(534\) 71.9045 3.11161
\(535\) 0 0
\(536\) 18.6537 0.805717
\(537\) −56.9714 −2.45850
\(538\) 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
\(543\) −11.2081 −0.480984
\(544\) −4.56419 −0.195688
\(545\) 0 0
\(546\) 46.6715 1.99735
\(547\) 27.6574 1.18254 0.591272 0.806472i \(-0.298626\pi\)
0.591272 + 0.806472i \(0.298626\pi\)
\(548\) −10.3435 −0.441851
\(549\) 11.6865 0.498769
\(550\) 0 0
\(551\) −11.1442 −0.474758
\(552\) 25.8352 1.09962
\(553\) −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 \(0.691806\pi\)
\(558\) 16.0189 0.678135
\(559\) 47.8564 2.02411
\(560\) 0 0
\(561\) 6.97816 0.294618
\(562\) 20.0564 0.846028
\(563\) −20.2482 −0.853362 −0.426681 0.904402i \(-0.640317\pi\)
−0.426681 + 0.904402i \(0.640317\pi\)
\(564\) 22.7717 0.958861
\(565\) 0 0
\(566\) −3.20890 −0.134880
\(567\) −20.0765 −0.843133
\(568\) −5.14206 −0.215756
\(569\) 41.1274 1.72415 0.862076 0.506779i \(-0.169164\pi\)
0.862076 + 0.506779i \(0.169164\pi\)
\(570\) 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
\(573\) −57.7448 −2.41232
\(574\) −4.72827 −0.197354
\(575\) 0 0
\(576\) 2.23054 0.0929390
\(577\) −4.90936 −0.204379 −0.102190 0.994765i \(-0.532585\pi\)
−0.102190 + 0.994765i \(0.532585\pi\)
\(578\) 28.2225 1.17390
\(579\) −7.64748 −0.317818
\(580\) 0 0
\(581\) 33.8020 1.40234
\(582\) 1.12536 0.0466479
\(583\) 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 \(0.504819\pi\)
\(588\) −7.13351 −0.294181
\(589\) −8.66253 −0.356933
\(590\) 0 0
\(591\) 8.55982 0.352104
\(592\) −25.9643 −1.06713
\(593\) −43.2540 −1.77623 −0.888115 0.459621i \(-0.847985\pi\)
−0.888115 + 0.459621i \(0.847985\pi\)
\(594\) −5.26219 −0.215910
\(595\) 0 0
\(596\) −15.8227 −0.648125
\(597\) 27.5155 1.12613
\(598\) 62.4455 2.55359
\(599\) −21.1031 −0.862251 −0.431126 0.902292i \(-0.641883\pi\)
−0.431126 + 0.902292i \(0.641883\pi\)
\(600\) 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
\(603\) 28.6562 1.16697
\(604\) −2.47816 −0.100835
\(605\) 0 0
\(606\) −30.7740 −1.25011
\(607\) 32.2628 1.30951 0.654753 0.755843i \(-0.272773\pi\)
0.654753 + 0.755843i \(0.272773\pi\)
\(608\) 12.9818 0.526481
\(609\) −21.5580 −0.873574
\(610\) 0 0
\(611\) −53.2407 −2.15389
\(612\) −2.31115 −0.0934227
\(613\) 44.6321 1.80267 0.901337 0.433119i \(-0.142587\pi\)
0.901337 + 0.433119i \(0.142587\pi\)
\(614\) 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 \(0.633663\pi\)
\(618\) 59.7830 2.40482
\(619\) 23.9533 0.962766 0.481383 0.876510i \(-0.340135\pi\)
0.481383 + 0.876510i \(0.340135\pi\)
\(620\) 0 0
\(621\) −5.69026 −0.228342
\(622\) −5.10754 −0.204794
\(623\) −35.0938 −1.40600
\(624\) 66.8322 2.67543
\(625\) 0 0
\(626\) 28.6770 1.14617
\(627\) −19.8478 −0.792644
\(628\) −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
\(633\) 61.5248 2.44539
\(634\) −35.3967 −1.40578
\(635\) 0 0
\(636\) −23.4308 −0.929092
\(637\) 16.6783 0.660818
\(638\) −26.6743 −1.05605
\(639\) −7.89934 −0.312493
\(640\) 0 0
\(641\) −28.9401 −1.14307 −0.571533 0.820579i \(-0.693651\pi\)
−0.571533 + 0.820579i \(0.693651\pi\)
\(642\) 68.9876 2.72272
\(643\) −18.6657 −0.736103 −0.368052 0.929805i \(-0.619975\pi\)
−0.368052 + 0.929805i \(0.619975\pi\)
\(644\) 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 \(0.661110\pi\)
\(648\) −17.0576 −0.670084
\(649\) −20.4827 −0.804016
\(650\) 0 0
\(651\) −16.7574 −0.656772
\(652\) −4.30059 −0.168424
\(653\) −12.4942 −0.488937 −0.244469 0.969657i \(-0.578614\pi\)
−0.244469 + 0.969657i \(0.578614\pi\)
\(654\) 24.7941 0.969525
\(655\) 0 0
\(656\) −6.77074 −0.264353
\(657\) 0.671477 0.0261968
\(658\) −32.9785 −1.28564
\(659\) 34.3869 1.33952 0.669762 0.742576i \(-0.266396\pi\)
0.669762 + 0.742576i \(0.266396\pi\)
\(660\) 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
\(663\) 11.5818 0.449798
\(664\) 28.7192 1.11452
\(665\) 0 0
\(666\) −23.6660 −0.917039
\(667\) −28.8442 −1.11685
\(668\) −14.6066 −0.565145
\(669\) 10.4365 0.403500
\(670\) 0 0
\(671\) −15.1267 −0.583958
\(672\) 25.1128 0.968748
\(673\) 26.1090 1.00643 0.503215 0.864161i \(-0.332150\pi\)
0.503215 + 0.864161i \(0.332150\pi\)
\(674\) −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 \(0.370641\pi\)
\(678\) 77.9477 2.99356
\(679\) −0.549247 −0.0210782
\(680\) 0 0
\(681\) 54.7287 2.09721
\(682\) −20.7343 −0.793959
\(683\) −23.7002 −0.906864 −0.453432 0.891291i \(-0.649801\pi\)
−0.453432 + 0.891291i \(0.649801\pi\)
\(684\) 6.57354 0.251346
\(685\) 0 0
\(686\) 34.7714 1.32758
\(687\) 7.61646 0.290586
\(688\) 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
\(693\) −17.9133 −0.680469
\(694\) −15.8646 −0.602211
\(695\) 0 0
\(696\) −18.3163 −0.694278
\(697\) −1.17334 −0.0444435
\(698\) 35.4282 1.34098
\(699\) −16.9135 −0.639727
\(700\) 0 0
\(701\) −28.4805 −1.07569 −0.537846 0.843043i \(-0.680762\pi\)
−0.537846 + 0.843043i \(0.680762\pi\)
\(702\) −8.73375 −0.329634
\(703\) 12.7978 0.482679
\(704\) −2.88713 −0.108813
\(705\) 0 0
\(706\) 22.9876 0.865151
\(707\) 15.0196 0.564870
\(708\) 14.5403 0.546457
\(709\) 32.8736 1.23459 0.617297 0.786730i \(-0.288228\pi\)
0.617297 + 0.786730i \(0.288228\pi\)
\(710\) 0 0
\(711\) −27.0940 −1.01610
\(712\) −29.8167 −1.11743
\(713\) −22.4210 −0.839674
\(714\) 7.17400 0.268480
\(715\) 0 0
\(716\) −24.4231 −0.912736
\(717\) 8.68526 0.324357
\(718\) −62.0877 −2.31709
\(719\) −13.3988 −0.499690 −0.249845 0.968286i \(-0.580380\pi\)
−0.249845 + 0.968286i \(0.580380\pi\)
\(720\) 0 0
\(721\) −29.1778 −1.08664
\(722\) 22.4520 0.835576
\(723\) −2.37146 −0.0881955
\(724\) −4.80481 −0.178569
\(725\) 0 0
\(726\) −2.19961 −0.0816353
\(727\) −24.5791 −0.911587 −0.455793 0.890086i \(-0.650644\pi\)
−0.455793 + 0.890086i \(0.650644\pi\)
\(728\) −19.3533 −0.717282
\(729\) −19.8574 −0.735459
\(730\) 0 0
\(731\) 7.35613 0.272076
\(732\) 10.7381 0.396893
\(733\) −2.08207 −0.0769029 −0.0384515 0.999260i \(-0.512242\pi\)
−0.0384515 + 0.999260i \(0.512242\pi\)
\(734\) −60.3065 −2.22595
\(735\) 0 0
\(736\) 33.6005 1.23853
\(737\) −37.0916 −1.36629
\(738\) −6.17141 −0.227173
\(739\) −46.1104 −1.69620 −0.848099 0.529838i \(-0.822253\pi\)
−0.848099 + 0.529838i \(0.822253\pi\)
\(740\) 0 0
\(741\) −32.9417 −1.21014
\(742\) 33.9331 1.24572
\(743\) −16.5456 −0.606998 −0.303499 0.952832i \(-0.598155\pi\)
−0.303499 + 0.952832i \(0.598155\pi\)
\(744\) −14.2375 −0.521973
\(745\) 0 0
\(746\) −60.7873 −2.22558
\(747\) 44.1190 1.61423
\(748\) 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
\(753\) 3.35830 0.122383
\(754\) −44.2718 −1.61228
\(755\) 0 0
\(756\) −1.82317 −0.0663080
\(757\) −16.3953 −0.595898 −0.297949 0.954582i \(-0.596303\pi\)
−0.297949 + 0.954582i \(0.596303\pi\)
\(758\) −30.2267 −1.09788
\(759\) −51.3715 −1.86467
\(760\) 0 0
\(761\) 44.2450 1.60388 0.801940 0.597404i \(-0.203801\pi\)
0.801940 + 0.597404i \(0.203801\pi\)
\(762\) −56.5520 −2.04866
\(763\) −12.1010 −0.438087
\(764\) −24.7547 −0.895593
\(765\) 0 0
\(766\) 13.1742 0.476003
\(767\) −33.9954 −1.22750
\(768\) 45.4441 1.63982
\(769\) −21.3439 −0.769680 −0.384840 0.922983i \(-0.625743\pi\)
−0.384840 + 0.922983i \(0.625743\pi\)
\(770\) 0 0
\(771\) 26.2497 0.945360
\(772\) −3.27841 −0.117993
\(773\) −20.7861 −0.747623 −0.373812 0.927505i \(-0.621949\pi\)
−0.373812 + 0.927505i \(0.621949\pi\)
\(774\) 38.6909 1.39072
\(775\) 0 0
\(776\) −0.466656 −0.0167520
\(777\) 24.7570 0.888151
\(778\) 16.8164 0.602897
\(779\) 3.33731 0.119571
\(780\) 0 0
\(781\) 10.2246 0.365866
\(782\) 9.59868 0.343248
\(783\) 4.03420 0.144171
\(784\) 14.7936 0.528342
\(785\) 0 0
\(786\) 58.9791 2.10372
\(787\) 28.9072 1.03043 0.515215 0.857061i \(-0.327712\pi\)
0.515215 + 0.857061i \(0.327712\pi\)
\(788\) 3.66952 0.130721
\(789\) 7.86302 0.279931
\(790\) 0 0
\(791\) −38.0433 −1.35266
\(792\) −15.2196 −0.540806
\(793\) −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 \(0.625300\pi\)
\(798\) −20.4048 −0.722323
\(799\) −8.18377 −0.289521
\(800\) 0 0
\(801\) −45.8050 −1.61844
\(802\) −39.9689 −1.41135
\(803\) −0.869138 −0.0306712
\(804\) 26.3306 0.928609
\(805\) 0 0
\(806\) −34.4131 −1.21215
\(807\) −34.0199 −1.19756
\(808\) 12.7611 0.448933
\(809\) 4.12546 0.145043 0.0725217 0.997367i \(-0.476895\pi\)
0.0725217 + 0.997367i \(0.476895\pi\)
\(810\) 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
\(813\) −3.68145 −0.129114
\(814\) 30.6325 1.07367
\(815\) 0 0
\(816\) 10.2730 0.359626
\(817\) −20.9228 −0.731998
\(818\) 42.0726 1.47104
\(819\) −29.7309 −1.03888
\(820\) 0 0
\(821\) 26.7943 0.935127 0.467564 0.883959i \(-0.345132\pi\)
0.467564 + 0.883959i \(0.345132\pi\)
\(822\) 41.9066 1.46166
\(823\) −29.5890 −1.03141 −0.515704 0.856767i \(-0.672469\pi\)
−0.515704 + 0.856767i \(0.672469\pi\)
\(824\) −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 \(0.245050\pi\)
\(828\) 17.0142 0.591283
\(829\) −3.90138 −0.135501 −0.0677503 0.997702i \(-0.521582\pi\)
−0.0677503 + 0.997702i \(0.521582\pi\)
\(830\) 0 0
\(831\) −42.7073 −1.48150
\(832\) −4.79182 −0.166126
\(833\) 2.56367 0.0888258
\(834\) −32.9650 −1.14149
\(835\) 0 0
\(836\) −8.50857 −0.294275
\(837\) 3.13584 0.108391
\(838\) −48.2088 −1.66534
\(839\) 18.4300 0.636274 0.318137 0.948045i \(-0.396943\pi\)
0.318137 + 0.948045i \(0.396943\pi\)
\(840\) 0 0
\(841\) −8.55043 −0.294842
\(842\) 57.5252 1.98245
\(843\) −27.3847 −0.943179
\(844\) 26.3752 0.907871
\(845\) 0 0
\(846\) −43.0441 −1.47989
\(847\) 1.07355 0.0368875
\(848\) 48.5912 1.66863
\(849\) 4.38138 0.150368
\(850\) 0 0
\(851\) 33.1243 1.13549
\(852\) −7.25827 −0.248664
\(853\) −37.7897 −1.29389 −0.646947 0.762535i \(-0.723954\pi\)
−0.646947 + 0.762535i \(0.723954\pi\)
\(854\) −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 \(0.0351472\pi\)
\(858\) −78.8481 −2.69183
\(859\) 35.3869 1.20738 0.603692 0.797218i \(-0.293696\pi\)
0.603692 + 0.797218i \(0.293696\pi\)
\(860\) 0 0
\(861\) 6.45590 0.220016
\(862\) 8.81224 0.300146
\(863\) −27.4547 −0.934568 −0.467284 0.884107i \(-0.654767\pi\)
−0.467284 + 0.884107i \(0.654767\pi\)
\(864\) −4.69942 −0.159878
\(865\) 0 0
\(866\) 20.2196 0.687088
\(867\) −38.5345 −1.30870
\(868\) −7.18373 −0.243832
\(869\) 35.0696 1.18965
\(870\) 0 0
\(871\) −61.5615 −2.08593
\(872\) −10.2814 −0.348172
\(873\) −0.716886 −0.0242629
\(874\) −27.3013 −0.923479
\(875\) 0 0
\(876\) 0.616984 0.0208460
\(877\) 14.8517 0.501505 0.250752 0.968051i \(-0.419322\pi\)
0.250752 + 0.968051i \(0.419322\pi\)
\(878\) 25.6958 0.867191
\(879\) 7.19056 0.242532
\(880\) 0 0
\(881\) 37.8714 1.27592 0.637960 0.770070i \(-0.279779\pi\)
0.637960 + 0.770070i \(0.279779\pi\)
\(882\) 13.4841 0.454033
\(883\) 15.3679 0.517171 0.258586 0.965988i \(-0.416744\pi\)
0.258586 + 0.965988i \(0.416744\pi\)
\(884\) 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 \(0.514585\pi\)
\(888\) 21.0342 0.705862
\(889\) 27.6008 0.925703
\(890\) 0 0
\(891\) 33.9178 1.13629
\(892\) 4.47406 0.149803
\(893\) 23.2769 0.778931
\(894\) 64.1058 2.14402
\(895\) 0 0
\(896\) −24.1474 −0.806707
\(897\) −85.2622 −2.84682
\(898\) 22.8495 0.762499
\(899\) 15.8958 0.530153
\(900\) 0 0
\(901\) 8.42066 0.280533
\(902\) 7.98807 0.265973
\(903\) −40.4745 −1.34691
\(904\) −32.3227 −1.07504
\(905\) 0 0
\(906\) 10.0403 0.333566
\(907\) 6.49334 0.215608 0.107804 0.994172i \(-0.465618\pi\)
0.107804 + 0.994172i \(0.465618\pi\)
\(908\) 23.4617 0.778605
\(909\) 19.6038 0.650218
\(910\) 0 0
\(911\) −32.2049 −1.06699 −0.533497 0.845802i \(-0.679123\pi\)
−0.533497 + 0.845802i \(0.679123\pi\)
\(912\) −29.2191 −0.967542
\(913\) −57.1061 −1.88994
\(914\) 9.63615 0.318736
\(915\) 0 0
\(916\) 3.26511 0.107882
\(917\) −28.7855 −0.950580
\(918\) −1.34249 −0.0443087
\(919\) −25.5609 −0.843177 −0.421588 0.906787i \(-0.638527\pi\)
−0.421588 + 0.906787i \(0.638527\pi\)
\(920\) 0 0
\(921\) −19.9367 −0.656936
\(922\) 55.9753 1.84345
\(923\) 16.9700 0.558574
\(924\) −16.4595 −0.541479
\(925\) 0 0
\(926\) −32.1390 −1.05615
\(927\) −38.0833 −1.25082
\(928\) −23.8216 −0.781983
\(929\) −43.9109 −1.44067 −0.720336 0.693626i \(-0.756012\pi\)
−0.720336 + 0.693626i \(0.756012\pi\)
\(930\) 0 0
\(931\) −7.29177 −0.238978
\(932\) −7.25067 −0.237503
\(933\) 6.97375 0.228310
\(934\) 36.3334 1.18886
\(935\) 0 0
\(936\) −25.2603 −0.825658
\(937\) −37.5543 −1.22684 −0.613422 0.789755i \(-0.710208\pi\)
−0.613422 + 0.789755i \(0.710208\pi\)
\(938\) −38.1326 −1.24507
\(939\) −39.1552 −1.27778
\(940\) 0 0
\(941\) −20.6958 −0.674663 −0.337332 0.941386i \(-0.609524\pi\)
−0.337332 + 0.941386i \(0.609524\pi\)
\(942\) 50.0987 1.63230
\(943\) 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 \(0.426736\pi\)
\(948\) −24.8952 −0.808559
\(949\) −1.44252 −0.0468262
\(950\) 0 0
\(951\) 48.3302 1.56721
\(952\) −2.97485 −0.0964155
\(953\) 32.4866 1.05235 0.526173 0.850378i \(-0.323627\pi\)
0.526173 + 0.850378i \(0.323627\pi\)
\(954\) 44.2900 1.43394
\(955\) 0 0
\(956\) 3.72330 0.120420
\(957\) 36.4207 1.17731
\(958\) 17.1253 0.553293
\(959\) −20.4530 −0.660461
\(960\) 0 0
\(961\) −18.6440 −0.601419
\(962\) 50.8412 1.63919
\(963\) −43.9469 −1.41617
\(964\) −1.01662 −0.0327433
\(965\) 0 0
\(966\) −52.8133 −1.69924
\(967\) 53.7501 1.72849 0.864244 0.503073i \(-0.167797\pi\)
0.864244 + 0.503073i \(0.167797\pi\)
\(968\) 0.912116 0.0293165
\(969\) −5.06356 −0.162665
\(970\) 0 0
\(971\) 34.1747 1.09672 0.548359 0.836243i \(-0.315253\pi\)
0.548359 + 0.836243i \(0.315253\pi\)
\(972\) −21.3568 −0.685019
\(973\) 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 \(0.454513\pi\)
\(978\) 17.4238 0.557153
\(979\) 59.2885 1.89487
\(980\) 0 0
\(981\) −15.7945 −0.504278
\(982\) 39.1822 1.25035
\(983\) −22.5713 −0.719912 −0.359956 0.932969i \(-0.617208\pi\)
−0.359956 + 0.932969i \(0.617208\pi\)
\(984\) 5.48512 0.174859
\(985\) 0 0
\(986\) −6.80514 −0.216720
\(987\) 45.0283 1.43327
\(988\) −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
\(993\) 21.5452 0.683717
\(994\) 10.5116 0.333408
\(995\) 0 0
\(996\) 40.5385 1.28451
\(997\) 20.7833 0.658214 0.329107 0.944293i \(-0.393252\pi\)
0.329107 + 0.944293i \(0.393252\pi\)
\(998\) −69.2448 −2.19191
\(999\) −4.63283 −0.146576
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 6025.2.a.f.1.1 7
5.4 even 2 241.2.a.a.1.7 7
15.14 odd 2 2169.2.a.e.1.1 7
20.19 odd 2 3856.2.a.j.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.7 7 5.4 even 2
2169.2.a.e.1.1 7 15.14 odd 2
3856.2.a.j.1.6 7 20.19 odd 2
6025.2.a.f.1.1 7 1.1 even 1 trivial