Properties

Label 6003.2.a.j.1.3
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.586186\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.586186 q^{2} -1.65639 q^{4} +0.842866 q^{5} +1.23837 q^{7} +2.14332 q^{8} +O(q^{10})\) \(q-0.586186 q^{2} -1.65639 q^{4} +0.842866 q^{5} +1.23837 q^{7} +2.14332 q^{8} -0.494076 q^{10} -1.47020 q^{11} -3.26028 q^{13} -0.725912 q^{14} +2.05639 q^{16} +4.93493 q^{17} +0.260886 q^{19} -1.39611 q^{20} +0.861812 q^{22} -1.00000 q^{23} -4.28958 q^{25} +1.91113 q^{26} -2.05121 q^{28} +1.00000 q^{29} -0.101716 q^{31} -5.49207 q^{32} -2.89279 q^{34} +1.04378 q^{35} -3.33423 q^{37} -0.152928 q^{38} +1.80653 q^{40} +0.732484 q^{41} -5.01063 q^{43} +2.43522 q^{44} +0.586186 q^{46} +8.71377 q^{47} -5.46645 q^{49} +2.51449 q^{50} +5.40027 q^{52} +4.24097 q^{53} -1.23918 q^{55} +2.65422 q^{56} -0.586186 q^{58} -8.76778 q^{59} -1.88602 q^{61} +0.0596245 q^{62} -0.893403 q^{64} -2.74797 q^{65} -3.21685 q^{67} -8.17415 q^{68} -0.611847 q^{70} +1.72395 q^{71} -2.64612 q^{73} +1.95448 q^{74} -0.432129 q^{76} -1.82065 q^{77} +0.465342 q^{79} +1.73326 q^{80} -0.429372 q^{82} +8.34128 q^{83} +4.15949 q^{85} +2.93716 q^{86} -3.15112 q^{88} +5.13593 q^{89} -4.03741 q^{91} +1.65639 q^{92} -5.10789 q^{94} +0.219892 q^{95} -6.87730 q^{97} +3.20436 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.586186 −0.414496 −0.207248 0.978288i \(-0.566451\pi\)
−0.207248 + 0.978288i \(0.566451\pi\)
\(3\) 0 0
\(4\) −1.65639 −0.828193
\(5\) 0.842866 0.376941 0.188471 0.982079i \(-0.439647\pi\)
0.188471 + 0.982079i \(0.439647\pi\)
\(6\) 0 0
\(7\) 1.23837 0.468058 0.234029 0.972230i \(-0.424809\pi\)
0.234029 + 0.972230i \(0.424809\pi\)
\(8\) 2.14332 0.757779
\(9\) 0 0
\(10\) −0.494076 −0.156241
\(11\) −1.47020 −0.443283 −0.221641 0.975128i \(-0.571141\pi\)
−0.221641 + 0.975128i \(0.571141\pi\)
\(12\) 0 0
\(13\) −3.26028 −0.904238 −0.452119 0.891958i \(-0.649332\pi\)
−0.452119 + 0.891958i \(0.649332\pi\)
\(14\) −0.725912 −0.194008
\(15\) 0 0
\(16\) 2.05639 0.514097
\(17\) 4.93493 1.19690 0.598448 0.801161i \(-0.295784\pi\)
0.598448 + 0.801161i \(0.295784\pi\)
\(18\) 0 0
\(19\) 0.260886 0.0598514 0.0299257 0.999552i \(-0.490473\pi\)
0.0299257 + 0.999552i \(0.490473\pi\)
\(20\) −1.39611 −0.312180
\(21\) 0 0
\(22\) 0.861812 0.183739
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.28958 −0.857915
\(26\) 1.91113 0.374803
\(27\) 0 0
\(28\) −2.05121 −0.387643
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.101716 −0.0182688 −0.00913438 0.999958i \(-0.502908\pi\)
−0.00913438 + 0.999958i \(0.502908\pi\)
\(32\) −5.49207 −0.970870
\(33\) 0 0
\(34\) −2.89279 −0.496109
\(35\) 1.04378 0.176430
\(36\) 0 0
\(37\) −3.33423 −0.548144 −0.274072 0.961709i \(-0.588371\pi\)
−0.274072 + 0.961709i \(0.588371\pi\)
\(38\) −0.152928 −0.0248082
\(39\) 0 0
\(40\) 1.80653 0.285638
\(41\) 0.732484 0.114395 0.0571974 0.998363i \(-0.481784\pi\)
0.0571974 + 0.998363i \(0.481784\pi\)
\(42\) 0 0
\(43\) −5.01063 −0.764114 −0.382057 0.924139i \(-0.624784\pi\)
−0.382057 + 0.924139i \(0.624784\pi\)
\(44\) 2.43522 0.367124
\(45\) 0 0
\(46\) 0.586186 0.0864284
\(47\) 8.71377 1.27103 0.635517 0.772087i \(-0.280787\pi\)
0.635517 + 0.772087i \(0.280787\pi\)
\(48\) 0 0
\(49\) −5.46645 −0.780922
\(50\) 2.51449 0.355602
\(51\) 0 0
\(52\) 5.40027 0.748883
\(53\) 4.24097 0.582541 0.291271 0.956641i \(-0.405922\pi\)
0.291271 + 0.956641i \(0.405922\pi\)
\(54\) 0 0
\(55\) −1.23918 −0.167091
\(56\) 2.65422 0.354685
\(57\) 0 0
\(58\) −0.586186 −0.0769700
\(59\) −8.76778 −1.14147 −0.570734 0.821135i \(-0.693341\pi\)
−0.570734 + 0.821135i \(0.693341\pi\)
\(60\) 0 0
\(61\) −1.88602 −0.241481 −0.120740 0.992684i \(-0.538527\pi\)
−0.120740 + 0.992684i \(0.538527\pi\)
\(62\) 0.0596245 0.00757232
\(63\) 0 0
\(64\) −0.893403 −0.111675
\(65\) −2.74797 −0.340844
\(66\) 0 0
\(67\) −3.21685 −0.393001 −0.196500 0.980504i \(-0.562958\pi\)
−0.196500 + 0.980504i \(0.562958\pi\)
\(68\) −8.17415 −0.991262
\(69\) 0 0
\(70\) −0.611847 −0.0731297
\(71\) 1.72395 0.204595 0.102297 0.994754i \(-0.467381\pi\)
0.102297 + 0.994754i \(0.467381\pi\)
\(72\) 0 0
\(73\) −2.64612 −0.309705 −0.154852 0.987938i \(-0.549490\pi\)
−0.154852 + 0.987938i \(0.549490\pi\)
\(74\) 1.95448 0.227203
\(75\) 0 0
\(76\) −0.432129 −0.0495685
\(77\) −1.82065 −0.207482
\(78\) 0 0
\(79\) 0.465342 0.0523551 0.0261775 0.999657i \(-0.491666\pi\)
0.0261775 + 0.999657i \(0.491666\pi\)
\(80\) 1.73326 0.193784
\(81\) 0 0
\(82\) −0.429372 −0.0474162
\(83\) 8.34128 0.915575 0.457787 0.889062i \(-0.348642\pi\)
0.457787 + 0.889062i \(0.348642\pi\)
\(84\) 0 0
\(85\) 4.15949 0.451160
\(86\) 2.93716 0.316722
\(87\) 0 0
\(88\) −3.15112 −0.335910
\(89\) 5.13593 0.544408 0.272204 0.962240i \(-0.412247\pi\)
0.272204 + 0.962240i \(0.412247\pi\)
\(90\) 0 0
\(91\) −4.03741 −0.423236
\(92\) 1.65639 0.172690
\(93\) 0 0
\(94\) −5.10789 −0.526838
\(95\) 0.219892 0.0225605
\(96\) 0 0
\(97\) −6.87730 −0.698284 −0.349142 0.937070i \(-0.613527\pi\)
−0.349142 + 0.937070i \(0.613527\pi\)
\(98\) 3.20436 0.323689
\(99\) 0 0
\(100\) 7.10520 0.710520
\(101\) 15.4552 1.53785 0.768923 0.639342i \(-0.220793\pi\)
0.768923 + 0.639342i \(0.220793\pi\)
\(102\) 0 0
\(103\) 9.59885 0.945803 0.472901 0.881115i \(-0.343207\pi\)
0.472901 + 0.881115i \(0.343207\pi\)
\(104\) −6.98782 −0.685212
\(105\) 0 0
\(106\) −2.48599 −0.241461
\(107\) −18.4638 −1.78496 −0.892480 0.451088i \(-0.851036\pi\)
−0.892480 + 0.451088i \(0.851036\pi\)
\(108\) 0 0
\(109\) −13.6145 −1.30403 −0.652017 0.758204i \(-0.726077\pi\)
−0.652017 + 0.758204i \(0.726077\pi\)
\(110\) 0.726391 0.0692587
\(111\) 0 0
\(112\) 2.54656 0.240627
\(113\) 9.49007 0.892750 0.446375 0.894846i \(-0.352715\pi\)
0.446375 + 0.894846i \(0.352715\pi\)
\(114\) 0 0
\(115\) −0.842866 −0.0785976
\(116\) −1.65639 −0.153792
\(117\) 0 0
\(118\) 5.13955 0.473134
\(119\) 6.11125 0.560217
\(120\) 0 0
\(121\) −8.83851 −0.803501
\(122\) 1.10556 0.100093
\(123\) 0 0
\(124\) 0.168481 0.0151301
\(125\) −7.82987 −0.700325
\(126\) 0 0
\(127\) 19.1350 1.69796 0.848980 0.528426i \(-0.177217\pi\)
0.848980 + 0.528426i \(0.177217\pi\)
\(128\) 11.5078 1.01716
\(129\) 0 0
\(130\) 1.61082 0.141279
\(131\) −14.4038 −1.25846 −0.629232 0.777218i \(-0.716630\pi\)
−0.629232 + 0.777218i \(0.716630\pi\)
\(132\) 0 0
\(133\) 0.323073 0.0280140
\(134\) 1.88567 0.162897
\(135\) 0 0
\(136\) 10.5771 0.906983
\(137\) −10.9664 −0.936922 −0.468461 0.883484i \(-0.655191\pi\)
−0.468461 + 0.883484i \(0.655191\pi\)
\(138\) 0 0
\(139\) −10.7820 −0.914519 −0.457259 0.889333i \(-0.651169\pi\)
−0.457259 + 0.889333i \(0.651169\pi\)
\(140\) −1.72890 −0.146118
\(141\) 0 0
\(142\) −1.01055 −0.0848038
\(143\) 4.79326 0.400833
\(144\) 0 0
\(145\) 0.842866 0.0699962
\(146\) 1.55112 0.128371
\(147\) 0 0
\(148\) 5.52277 0.453969
\(149\) 14.9598 1.22555 0.612776 0.790257i \(-0.290053\pi\)
0.612776 + 0.790257i \(0.290053\pi\)
\(150\) 0 0
\(151\) −4.51476 −0.367406 −0.183703 0.982982i \(-0.558808\pi\)
−0.183703 + 0.982982i \(0.558808\pi\)
\(152\) 0.559163 0.0453541
\(153\) 0 0
\(154\) 1.06724 0.0860005
\(155\) −0.0857330 −0.00688624
\(156\) 0 0
\(157\) −16.2289 −1.29521 −0.647604 0.761977i \(-0.724229\pi\)
−0.647604 + 0.761977i \(0.724229\pi\)
\(158\) −0.272777 −0.0217010
\(159\) 0 0
\(160\) −4.62908 −0.365961
\(161\) −1.23837 −0.0975969
\(162\) 0 0
\(163\) −2.54577 −0.199400 −0.0997001 0.995018i \(-0.531788\pi\)
−0.0997001 + 0.995018i \(0.531788\pi\)
\(164\) −1.21328 −0.0947410
\(165\) 0 0
\(166\) −4.88954 −0.379502
\(167\) −19.0688 −1.47559 −0.737794 0.675026i \(-0.764132\pi\)
−0.737794 + 0.675026i \(0.764132\pi\)
\(168\) 0 0
\(169\) −2.37061 −0.182354
\(170\) −2.43823 −0.187004
\(171\) 0 0
\(172\) 8.29954 0.632834
\(173\) 9.37891 0.713065 0.356533 0.934283i \(-0.383959\pi\)
0.356533 + 0.934283i \(0.383959\pi\)
\(174\) 0 0
\(175\) −5.31207 −0.401554
\(176\) −3.02331 −0.227890
\(177\) 0 0
\(178\) −3.01061 −0.225655
\(179\) −4.33101 −0.323715 −0.161857 0.986814i \(-0.551748\pi\)
−0.161857 + 0.986814i \(0.551748\pi\)
\(180\) 0 0
\(181\) 20.4714 1.52163 0.760813 0.648971i \(-0.224801\pi\)
0.760813 + 0.648971i \(0.224801\pi\)
\(182\) 2.36667 0.175430
\(183\) 0 0
\(184\) −2.14332 −0.158008
\(185\) −2.81031 −0.206618
\(186\) 0 0
\(187\) −7.25535 −0.530564
\(188\) −14.4334 −1.05266
\(189\) 0 0
\(190\) −0.128898 −0.00935122
\(191\) −7.10652 −0.514210 −0.257105 0.966383i \(-0.582769\pi\)
−0.257105 + 0.966383i \(0.582769\pi\)
\(192\) 0 0
\(193\) 2.37896 0.171242 0.0856208 0.996328i \(-0.472713\pi\)
0.0856208 + 0.996328i \(0.472713\pi\)
\(194\) 4.03138 0.289436
\(195\) 0 0
\(196\) 9.05455 0.646754
\(197\) −4.61033 −0.328472 −0.164236 0.986421i \(-0.552516\pi\)
−0.164236 + 0.986421i \(0.552516\pi\)
\(198\) 0 0
\(199\) −12.7702 −0.905254 −0.452627 0.891700i \(-0.649513\pi\)
−0.452627 + 0.891700i \(0.649513\pi\)
\(200\) −9.19394 −0.650110
\(201\) 0 0
\(202\) −9.05959 −0.637431
\(203\) 1.23837 0.0869162
\(204\) 0 0
\(205\) 0.617386 0.0431201
\(206\) −5.62671 −0.392031
\(207\) 0 0
\(208\) −6.70439 −0.464866
\(209\) −0.383556 −0.0265311
\(210\) 0 0
\(211\) −8.75135 −0.602468 −0.301234 0.953550i \(-0.597398\pi\)
−0.301234 + 0.953550i \(0.597398\pi\)
\(212\) −7.02468 −0.482457
\(213\) 0 0
\(214\) 10.8232 0.739858
\(215\) −4.22329 −0.288026
\(216\) 0 0
\(217\) −0.125962 −0.00855084
\(218\) 7.98064 0.540517
\(219\) 0 0
\(220\) 2.05257 0.138384
\(221\) −16.0892 −1.08228
\(222\) 0 0
\(223\) 29.3753 1.96712 0.983559 0.180586i \(-0.0577993\pi\)
0.983559 + 0.180586i \(0.0577993\pi\)
\(224\) −6.80119 −0.454424
\(225\) 0 0
\(226\) −5.56294 −0.370041
\(227\) −21.4702 −1.42503 −0.712514 0.701658i \(-0.752443\pi\)
−0.712514 + 0.701658i \(0.752443\pi\)
\(228\) 0 0
\(229\) 6.14700 0.406205 0.203103 0.979157i \(-0.434898\pi\)
0.203103 + 0.979157i \(0.434898\pi\)
\(230\) 0.494076 0.0325784
\(231\) 0 0
\(232\) 2.14332 0.140716
\(233\) −21.0310 −1.37778 −0.688892 0.724864i \(-0.741903\pi\)
−0.688892 + 0.724864i \(0.741903\pi\)
\(234\) 0 0
\(235\) 7.34454 0.479105
\(236\) 14.5228 0.945355
\(237\) 0 0
\(238\) −3.58233 −0.232208
\(239\) 22.6235 1.46339 0.731697 0.681630i \(-0.238729\pi\)
0.731697 + 0.681630i \(0.238729\pi\)
\(240\) 0 0
\(241\) 16.1684 1.04150 0.520749 0.853710i \(-0.325653\pi\)
0.520749 + 0.853710i \(0.325653\pi\)
\(242\) 5.18101 0.333048
\(243\) 0 0
\(244\) 3.12398 0.199993
\(245\) −4.60748 −0.294361
\(246\) 0 0
\(247\) −0.850561 −0.0541199
\(248\) −0.218010 −0.0138437
\(249\) 0 0
\(250\) 4.58976 0.290282
\(251\) −4.88555 −0.308373 −0.154186 0.988042i \(-0.549276\pi\)
−0.154186 + 0.988042i \(0.549276\pi\)
\(252\) 0 0
\(253\) 1.47020 0.0924308
\(254\) −11.2167 −0.703797
\(255\) 0 0
\(256\) −4.95892 −0.309933
\(257\) −14.0680 −0.877539 −0.438770 0.898600i \(-0.644586\pi\)
−0.438770 + 0.898600i \(0.644586\pi\)
\(258\) 0 0
\(259\) −4.12899 −0.256563
\(260\) 4.55171 0.282285
\(261\) 0 0
\(262\) 8.44329 0.521628
\(263\) 15.7419 0.970687 0.485344 0.874323i \(-0.338694\pi\)
0.485344 + 0.874323i \(0.338694\pi\)
\(264\) 0 0
\(265\) 3.57456 0.219584
\(266\) −0.189381 −0.0116117
\(267\) 0 0
\(268\) 5.32834 0.325480
\(269\) −5.80954 −0.354214 −0.177107 0.984192i \(-0.556674\pi\)
−0.177107 + 0.984192i \(0.556674\pi\)
\(270\) 0 0
\(271\) 13.5144 0.820944 0.410472 0.911873i \(-0.365364\pi\)
0.410472 + 0.911873i \(0.365364\pi\)
\(272\) 10.1481 0.615321
\(273\) 0 0
\(274\) 6.42834 0.388350
\(275\) 6.30655 0.380299
\(276\) 0 0
\(277\) −14.1794 −0.851956 −0.425978 0.904733i \(-0.640070\pi\)
−0.425978 + 0.904733i \(0.640070\pi\)
\(278\) 6.32027 0.379064
\(279\) 0 0
\(280\) 2.23715 0.133695
\(281\) 0.303631 0.0181131 0.00905656 0.999959i \(-0.497117\pi\)
0.00905656 + 0.999959i \(0.497117\pi\)
\(282\) 0 0
\(283\) 17.4684 1.03839 0.519195 0.854656i \(-0.326232\pi\)
0.519195 + 0.854656i \(0.326232\pi\)
\(284\) −2.85552 −0.169444
\(285\) 0 0
\(286\) −2.80974 −0.166144
\(287\) 0.907084 0.0535434
\(288\) 0 0
\(289\) 7.35356 0.432562
\(290\) −0.494076 −0.0290131
\(291\) 0 0
\(292\) 4.38300 0.256495
\(293\) −18.0182 −1.05263 −0.526317 0.850289i \(-0.676427\pi\)
−0.526317 + 0.850289i \(0.676427\pi\)
\(294\) 0 0
\(295\) −7.39006 −0.430266
\(296\) −7.14632 −0.415371
\(297\) 0 0
\(298\) −8.76920 −0.507986
\(299\) 3.26028 0.188547
\(300\) 0 0
\(301\) −6.20499 −0.357650
\(302\) 2.64649 0.152288
\(303\) 0 0
\(304\) 0.536484 0.0307694
\(305\) −1.58966 −0.0910239
\(306\) 0 0
\(307\) 14.8992 0.850340 0.425170 0.905113i \(-0.360214\pi\)
0.425170 + 0.905113i \(0.360214\pi\)
\(308\) 3.01570 0.171835
\(309\) 0 0
\(310\) 0.0502555 0.00285432
\(311\) 1.83728 0.104183 0.0520914 0.998642i \(-0.483411\pi\)
0.0520914 + 0.998642i \(0.483411\pi\)
\(312\) 0 0
\(313\) 4.46857 0.252579 0.126289 0.991993i \(-0.459693\pi\)
0.126289 + 0.991993i \(0.459693\pi\)
\(314\) 9.51315 0.536858
\(315\) 0 0
\(316\) −0.770786 −0.0433601
\(317\) −15.9563 −0.896194 −0.448097 0.893985i \(-0.647898\pi\)
−0.448097 + 0.893985i \(0.647898\pi\)
\(318\) 0 0
\(319\) −1.47020 −0.0823155
\(320\) −0.753019 −0.0420950
\(321\) 0 0
\(322\) 0.725912 0.0404535
\(323\) 1.28746 0.0716360
\(324\) 0 0
\(325\) 13.9852 0.775759
\(326\) 1.49230 0.0826506
\(327\) 0 0
\(328\) 1.56995 0.0866860
\(329\) 10.7908 0.594918
\(330\) 0 0
\(331\) −35.5195 −1.95233 −0.976164 0.217035i \(-0.930362\pi\)
−0.976164 + 0.217035i \(0.930362\pi\)
\(332\) −13.8164 −0.758273
\(333\) 0 0
\(334\) 11.1779 0.611625
\(335\) −2.71137 −0.148138
\(336\) 0 0
\(337\) −20.6824 −1.12664 −0.563322 0.826237i \(-0.690477\pi\)
−0.563322 + 0.826237i \(0.690477\pi\)
\(338\) 1.38962 0.0755851
\(339\) 0 0
\(340\) −6.88971 −0.373647
\(341\) 0.149543 0.00809822
\(342\) 0 0
\(343\) −15.4380 −0.833575
\(344\) −10.7394 −0.579029
\(345\) 0 0
\(346\) −5.49778 −0.295563
\(347\) −13.4784 −0.723557 −0.361779 0.932264i \(-0.617830\pi\)
−0.361779 + 0.932264i \(0.617830\pi\)
\(348\) 0 0
\(349\) −32.9095 −1.76160 −0.880801 0.473486i \(-0.842995\pi\)
−0.880801 + 0.473486i \(0.842995\pi\)
\(350\) 3.11386 0.166443
\(351\) 0 0
\(352\) 8.07445 0.430370
\(353\) −23.8249 −1.26807 −0.634035 0.773305i \(-0.718602\pi\)
−0.634035 + 0.773305i \(0.718602\pi\)
\(354\) 0 0
\(355\) 1.45306 0.0771202
\(356\) −8.50709 −0.450875
\(357\) 0 0
\(358\) 2.53877 0.134178
\(359\) −26.2175 −1.38370 −0.691852 0.722039i \(-0.743205\pi\)
−0.691852 + 0.722039i \(0.743205\pi\)
\(360\) 0 0
\(361\) −18.9319 −0.996418
\(362\) −12.0000 −0.630708
\(363\) 0 0
\(364\) 6.68751 0.350521
\(365\) −2.23032 −0.116740
\(366\) 0 0
\(367\) 9.01887 0.470781 0.235391 0.971901i \(-0.424363\pi\)
0.235391 + 0.971901i \(0.424363\pi\)
\(368\) −2.05639 −0.107197
\(369\) 0 0
\(370\) 1.64736 0.0856422
\(371\) 5.25187 0.272663
\(372\) 0 0
\(373\) −8.59391 −0.444976 −0.222488 0.974935i \(-0.571418\pi\)
−0.222488 + 0.974935i \(0.571418\pi\)
\(374\) 4.25298 0.219916
\(375\) 0 0
\(376\) 18.6764 0.963162
\(377\) −3.26028 −0.167913
\(378\) 0 0
\(379\) 8.48972 0.436088 0.218044 0.975939i \(-0.430032\pi\)
0.218044 + 0.975939i \(0.430032\pi\)
\(380\) −0.364226 −0.0186844
\(381\) 0 0
\(382\) 4.16574 0.213138
\(383\) −10.6436 −0.543865 −0.271933 0.962316i \(-0.587663\pi\)
−0.271933 + 0.962316i \(0.587663\pi\)
\(384\) 0 0
\(385\) −1.53456 −0.0782085
\(386\) −1.39451 −0.0709789
\(387\) 0 0
\(388\) 11.3915 0.578314
\(389\) 14.6896 0.744794 0.372397 0.928074i \(-0.378536\pi\)
0.372397 + 0.928074i \(0.378536\pi\)
\(390\) 0 0
\(391\) −4.93493 −0.249570
\(392\) −11.7164 −0.591766
\(393\) 0 0
\(394\) 2.70251 0.136150
\(395\) 0.392221 0.0197348
\(396\) 0 0
\(397\) 7.75333 0.389128 0.194564 0.980890i \(-0.437671\pi\)
0.194564 + 0.980890i \(0.437671\pi\)
\(398\) 7.48570 0.375224
\(399\) 0 0
\(400\) −8.82103 −0.441052
\(401\) 1.17706 0.0587797 0.0293899 0.999568i \(-0.490644\pi\)
0.0293899 + 0.999568i \(0.490644\pi\)
\(402\) 0 0
\(403\) 0.331623 0.0165193
\(404\) −25.5997 −1.27363
\(405\) 0 0
\(406\) −0.725912 −0.0360264
\(407\) 4.90199 0.242982
\(408\) 0 0
\(409\) −23.6328 −1.16857 −0.584283 0.811550i \(-0.698624\pi\)
−0.584283 + 0.811550i \(0.698624\pi\)
\(410\) −0.361903 −0.0178731
\(411\) 0 0
\(412\) −15.8994 −0.783307
\(413\) −10.8577 −0.534273
\(414\) 0 0
\(415\) 7.03058 0.345118
\(416\) 17.9057 0.877897
\(417\) 0 0
\(418\) 0.224835 0.0109970
\(419\) −18.6252 −0.909901 −0.454951 0.890517i \(-0.650343\pi\)
−0.454951 + 0.890517i \(0.650343\pi\)
\(420\) 0 0
\(421\) −31.0986 −1.51565 −0.757827 0.652456i \(-0.773739\pi\)
−0.757827 + 0.652456i \(0.773739\pi\)
\(422\) 5.12992 0.249720
\(423\) 0 0
\(424\) 9.08975 0.441437
\(425\) −21.1688 −1.02684
\(426\) 0 0
\(427\) −2.33559 −0.113027
\(428\) 30.5831 1.47829
\(429\) 0 0
\(430\) 2.47563 0.119386
\(431\) −9.88560 −0.476173 −0.238086 0.971244i \(-0.576520\pi\)
−0.238086 + 0.971244i \(0.576520\pi\)
\(432\) 0 0
\(433\) −8.91141 −0.428255 −0.214127 0.976806i \(-0.568691\pi\)
−0.214127 + 0.976806i \(0.568691\pi\)
\(434\) 0.0738370 0.00354429
\(435\) 0 0
\(436\) 22.5509 1.07999
\(437\) −0.260886 −0.0124799
\(438\) 0 0
\(439\) −31.9814 −1.52639 −0.763193 0.646170i \(-0.776370\pi\)
−0.763193 + 0.646170i \(0.776370\pi\)
\(440\) −2.65597 −0.126618
\(441\) 0 0
\(442\) 9.43128 0.448600
\(443\) −20.8775 −0.991921 −0.495960 0.868345i \(-0.665184\pi\)
−0.495960 + 0.868345i \(0.665184\pi\)
\(444\) 0 0
\(445\) 4.32890 0.205210
\(446\) −17.2194 −0.815363
\(447\) 0 0
\(448\) −1.10636 −0.0522706
\(449\) 13.7130 0.647158 0.323579 0.946201i \(-0.395114\pi\)
0.323579 + 0.946201i \(0.395114\pi\)
\(450\) 0 0
\(451\) −1.07690 −0.0507092
\(452\) −15.7192 −0.739370
\(453\) 0 0
\(454\) 12.5855 0.590668
\(455\) −3.40300 −0.159535
\(456\) 0 0
\(457\) 12.8613 0.601627 0.300813 0.953683i \(-0.402742\pi\)
0.300813 + 0.953683i \(0.402742\pi\)
\(458\) −3.60328 −0.168370
\(459\) 0 0
\(460\) 1.39611 0.0650940
\(461\) 24.1332 1.12400 0.561998 0.827139i \(-0.310033\pi\)
0.561998 + 0.827139i \(0.310033\pi\)
\(462\) 0 0
\(463\) −40.5906 −1.88640 −0.943202 0.332219i \(-0.892203\pi\)
−0.943202 + 0.332219i \(0.892203\pi\)
\(464\) 2.05639 0.0954654
\(465\) 0 0
\(466\) 12.3281 0.571086
\(467\) 16.4110 0.759413 0.379706 0.925107i \(-0.376025\pi\)
0.379706 + 0.925107i \(0.376025\pi\)
\(468\) 0 0
\(469\) −3.98363 −0.183947
\(470\) −4.30526 −0.198587
\(471\) 0 0
\(472\) −18.7922 −0.864979
\(473\) 7.36664 0.338718
\(474\) 0 0
\(475\) −1.11909 −0.0513475
\(476\) −10.1226 −0.463968
\(477\) 0 0
\(478\) −13.2616 −0.606571
\(479\) 11.5781 0.529017 0.264508 0.964383i \(-0.414790\pi\)
0.264508 + 0.964383i \(0.414790\pi\)
\(480\) 0 0
\(481\) 10.8705 0.495652
\(482\) −9.47768 −0.431697
\(483\) 0 0
\(484\) 14.6400 0.665454
\(485\) −5.79664 −0.263212
\(486\) 0 0
\(487\) −5.57197 −0.252490 −0.126245 0.991999i \(-0.540293\pi\)
−0.126245 + 0.991999i \(0.540293\pi\)
\(488\) −4.04236 −0.182989
\(489\) 0 0
\(490\) 2.70084 0.122012
\(491\) −20.4086 −0.921028 −0.460514 0.887653i \(-0.652335\pi\)
−0.460514 + 0.887653i \(0.652335\pi\)
\(492\) 0 0
\(493\) 4.93493 0.222258
\(494\) 0.498587 0.0224325
\(495\) 0 0
\(496\) −0.209168 −0.00939191
\(497\) 2.13488 0.0957624
\(498\) 0 0
\(499\) −32.6022 −1.45947 −0.729737 0.683728i \(-0.760357\pi\)
−0.729737 + 0.683728i \(0.760357\pi\)
\(500\) 12.9693 0.580004
\(501\) 0 0
\(502\) 2.86384 0.127819
\(503\) −6.33199 −0.282330 −0.141165 0.989986i \(-0.545085\pi\)
−0.141165 + 0.989986i \(0.545085\pi\)
\(504\) 0 0
\(505\) 13.0266 0.579677
\(506\) −0.861812 −0.0383122
\(507\) 0 0
\(508\) −31.6950 −1.40624
\(509\) −27.0573 −1.19929 −0.599646 0.800265i \(-0.704692\pi\)
−0.599646 + 0.800265i \(0.704692\pi\)
\(510\) 0 0
\(511\) −3.27686 −0.144960
\(512\) −20.1088 −0.888693
\(513\) 0 0
\(514\) 8.24647 0.363736
\(515\) 8.09054 0.356512
\(516\) 0 0
\(517\) −12.8110 −0.563427
\(518\) 2.42036 0.106344
\(519\) 0 0
\(520\) −5.88979 −0.258284
\(521\) 5.85670 0.256587 0.128293 0.991736i \(-0.459050\pi\)
0.128293 + 0.991736i \(0.459050\pi\)
\(522\) 0 0
\(523\) 8.29727 0.362814 0.181407 0.983408i \(-0.441935\pi\)
0.181407 + 0.983408i \(0.441935\pi\)
\(524\) 23.8582 1.04225
\(525\) 0 0
\(526\) −9.22768 −0.402346
\(527\) −0.501962 −0.0218658
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −2.09536 −0.0910166
\(531\) 0 0
\(532\) −0.535133 −0.0232010
\(533\) −2.38810 −0.103440
\(534\) 0 0
\(535\) −15.5625 −0.672824
\(536\) −6.89474 −0.297807
\(537\) 0 0
\(538\) 3.40547 0.146820
\(539\) 8.03679 0.346169
\(540\) 0 0
\(541\) 11.0825 0.476473 0.238237 0.971207i \(-0.423431\pi\)
0.238237 + 0.971207i \(0.423431\pi\)
\(542\) −7.92197 −0.340278
\(543\) 0 0
\(544\) −27.1030 −1.16203
\(545\) −11.4752 −0.491544
\(546\) 0 0
\(547\) 21.4790 0.918375 0.459187 0.888339i \(-0.348141\pi\)
0.459187 + 0.888339i \(0.348141\pi\)
\(548\) 18.1646 0.775952
\(549\) 0 0
\(550\) −3.69681 −0.157632
\(551\) 0.260886 0.0111141
\(552\) 0 0
\(553\) 0.576264 0.0245052
\(554\) 8.31175 0.353132
\(555\) 0 0
\(556\) 17.8592 0.757398
\(557\) −38.2929 −1.62252 −0.811262 0.584683i \(-0.801219\pi\)
−0.811262 + 0.584683i \(0.801219\pi\)
\(558\) 0 0
\(559\) 16.3360 0.690940
\(560\) 2.14641 0.0907023
\(561\) 0 0
\(562\) −0.177984 −0.00750781
\(563\) 4.52231 0.190592 0.0952962 0.995449i \(-0.469620\pi\)
0.0952962 + 0.995449i \(0.469620\pi\)
\(564\) 0 0
\(565\) 7.99885 0.336514
\(566\) −10.2397 −0.430409
\(567\) 0 0
\(568\) 3.69497 0.155038
\(569\) −44.8603 −1.88064 −0.940321 0.340289i \(-0.889475\pi\)
−0.940321 + 0.340289i \(0.889475\pi\)
\(570\) 0 0
\(571\) 0.417218 0.0174600 0.00873001 0.999962i \(-0.497221\pi\)
0.00873001 + 0.999962i \(0.497221\pi\)
\(572\) −7.93949 −0.331967
\(573\) 0 0
\(574\) −0.531719 −0.0221935
\(575\) 4.28958 0.178888
\(576\) 0 0
\(577\) −11.8343 −0.492670 −0.246335 0.969185i \(-0.579226\pi\)
−0.246335 + 0.969185i \(0.579226\pi\)
\(578\) −4.31055 −0.179295
\(579\) 0 0
\(580\) −1.39611 −0.0579704
\(581\) 10.3296 0.428542
\(582\) 0 0
\(583\) −6.23508 −0.258230
\(584\) −5.67148 −0.234688
\(585\) 0 0
\(586\) 10.5620 0.436312
\(587\) 18.5466 0.765501 0.382750 0.923852i \(-0.374977\pi\)
0.382750 + 0.923852i \(0.374977\pi\)
\(588\) 0 0
\(589\) −0.0265363 −0.00109341
\(590\) 4.33195 0.178343
\(591\) 0 0
\(592\) −6.85646 −0.281799
\(593\) 29.0821 1.19426 0.597129 0.802145i \(-0.296308\pi\)
0.597129 + 0.802145i \(0.296308\pi\)
\(594\) 0 0
\(595\) 5.15096 0.211169
\(596\) −24.7792 −1.01499
\(597\) 0 0
\(598\) −1.91113 −0.0781518
\(599\) −29.5204 −1.20617 −0.603085 0.797677i \(-0.706062\pi\)
−0.603085 + 0.797677i \(0.706062\pi\)
\(600\) 0 0
\(601\) 3.49768 0.142673 0.0713367 0.997452i \(-0.477274\pi\)
0.0713367 + 0.997452i \(0.477274\pi\)
\(602\) 3.63728 0.148244
\(603\) 0 0
\(604\) 7.47819 0.304283
\(605\) −7.44967 −0.302872
\(606\) 0 0
\(607\) 7.88945 0.320223 0.160111 0.987099i \(-0.448815\pi\)
0.160111 + 0.987099i \(0.448815\pi\)
\(608\) −1.43281 −0.0581079
\(609\) 0 0
\(610\) 0.931839 0.0377291
\(611\) −28.4093 −1.14932
\(612\) 0 0
\(613\) −33.4560 −1.35127 −0.675637 0.737235i \(-0.736131\pi\)
−0.675637 + 0.737235i \(0.736131\pi\)
\(614\) −8.73368 −0.352463
\(615\) 0 0
\(616\) −3.90223 −0.157225
\(617\) 33.2433 1.33832 0.669162 0.743117i \(-0.266653\pi\)
0.669162 + 0.743117i \(0.266653\pi\)
\(618\) 0 0
\(619\) −23.1547 −0.930668 −0.465334 0.885135i \(-0.654066\pi\)
−0.465334 + 0.885135i \(0.654066\pi\)
\(620\) 0.142007 0.00570314
\(621\) 0 0
\(622\) −1.07699 −0.0431834
\(623\) 6.36016 0.254815
\(624\) 0 0
\(625\) 14.8484 0.593934
\(626\) −2.61941 −0.104693
\(627\) 0 0
\(628\) 26.8813 1.07268
\(629\) −16.4542 −0.656071
\(630\) 0 0
\(631\) −17.7854 −0.708027 −0.354014 0.935240i \(-0.615183\pi\)
−0.354014 + 0.935240i \(0.615183\pi\)
\(632\) 0.997378 0.0396735
\(633\) 0 0
\(634\) 9.35334 0.371469
\(635\) 16.1283 0.640030
\(636\) 0 0
\(637\) 17.8221 0.706139
\(638\) 0.861812 0.0341194
\(639\) 0 0
\(640\) 9.69956 0.383409
\(641\) −27.7374 −1.09556 −0.547782 0.836621i \(-0.684528\pi\)
−0.547782 + 0.836621i \(0.684528\pi\)
\(642\) 0 0
\(643\) 15.3293 0.604528 0.302264 0.953224i \(-0.402258\pi\)
0.302264 + 0.953224i \(0.402258\pi\)
\(644\) 2.05121 0.0808291
\(645\) 0 0
\(646\) −0.754689 −0.0296928
\(647\) 41.3220 1.62454 0.812268 0.583284i \(-0.198233\pi\)
0.812268 + 0.583284i \(0.198233\pi\)
\(648\) 0 0
\(649\) 12.8904 0.505993
\(650\) −8.19793 −0.321549
\(651\) 0 0
\(652\) 4.21678 0.165142
\(653\) 31.9364 1.24977 0.624883 0.780718i \(-0.285146\pi\)
0.624883 + 0.780718i \(0.285146\pi\)
\(654\) 0 0
\(655\) −12.1405 −0.474367
\(656\) 1.50627 0.0588100
\(657\) 0 0
\(658\) −6.32543 −0.246591
\(659\) 39.4143 1.53536 0.767681 0.640833i \(-0.221411\pi\)
0.767681 + 0.640833i \(0.221411\pi\)
\(660\) 0 0
\(661\) 16.5791 0.644850 0.322425 0.946595i \(-0.395502\pi\)
0.322425 + 0.946595i \(0.395502\pi\)
\(662\) 20.8210 0.809232
\(663\) 0 0
\(664\) 17.8781 0.693803
\(665\) 0.272307 0.0105596
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 31.5853 1.22207
\(669\) 0 0
\(670\) 1.58937 0.0614026
\(671\) 2.77284 0.107044
\(672\) 0 0
\(673\) −13.0131 −0.501618 −0.250809 0.968037i \(-0.580697\pi\)
−0.250809 + 0.968037i \(0.580697\pi\)
\(674\) 12.1237 0.466989
\(675\) 0 0
\(676\) 3.92664 0.151025
\(677\) −12.2182 −0.469583 −0.234792 0.972046i \(-0.575441\pi\)
−0.234792 + 0.972046i \(0.575441\pi\)
\(678\) 0 0
\(679\) −8.51661 −0.326838
\(680\) 8.91512 0.341879
\(681\) 0 0
\(682\) −0.0876601 −0.00335668
\(683\) −1.56826 −0.0600076 −0.0300038 0.999550i \(-0.509552\pi\)
−0.0300038 + 0.999550i \(0.509552\pi\)
\(684\) 0 0
\(685\) −9.24319 −0.353164
\(686\) 9.04955 0.345513
\(687\) 0 0
\(688\) −10.3038 −0.392829
\(689\) −13.8267 −0.526756
\(690\) 0 0
\(691\) 6.57025 0.249944 0.124972 0.992160i \(-0.460116\pi\)
0.124972 + 0.992160i \(0.460116\pi\)
\(692\) −15.5351 −0.590556
\(693\) 0 0
\(694\) 7.90084 0.299912
\(695\) −9.08779 −0.344720
\(696\) 0 0
\(697\) 3.61476 0.136919
\(698\) 19.2911 0.730177
\(699\) 0 0
\(700\) 8.79883 0.332565
\(701\) −16.6273 −0.628004 −0.314002 0.949422i \(-0.601670\pi\)
−0.314002 + 0.949422i \(0.601670\pi\)
\(702\) 0 0
\(703\) −0.869854 −0.0328072
\(704\) 1.31348 0.0495038
\(705\) 0 0
\(706\) 13.9658 0.525610
\(707\) 19.1391 0.719801
\(708\) 0 0
\(709\) −9.39341 −0.352777 −0.176389 0.984321i \(-0.556442\pi\)
−0.176389 + 0.984321i \(0.556442\pi\)
\(710\) −0.851761 −0.0319660
\(711\) 0 0
\(712\) 11.0080 0.412541
\(713\) 0.101716 0.00380930
\(714\) 0 0
\(715\) 4.04008 0.151090
\(716\) 7.17382 0.268098
\(717\) 0 0
\(718\) 15.3683 0.573540
\(719\) 0.289428 0.0107939 0.00539693 0.999985i \(-0.498282\pi\)
0.00539693 + 0.999985i \(0.498282\pi\)
\(720\) 0 0
\(721\) 11.8869 0.442691
\(722\) 11.0976 0.413011
\(723\) 0 0
\(724\) −33.9085 −1.26020
\(725\) −4.28958 −0.159311
\(726\) 0 0
\(727\) 12.8787 0.477644 0.238822 0.971063i \(-0.423239\pi\)
0.238822 + 0.971063i \(0.423239\pi\)
\(728\) −8.65347 −0.320719
\(729\) 0 0
\(730\) 1.30738 0.0483884
\(731\) −24.7271 −0.914566
\(732\) 0 0
\(733\) −33.0973 −1.22248 −0.611239 0.791446i \(-0.709328\pi\)
−0.611239 + 0.791446i \(0.709328\pi\)
\(734\) −5.28673 −0.195137
\(735\) 0 0
\(736\) 5.49207 0.202440
\(737\) 4.72942 0.174210
\(738\) 0 0
\(739\) 17.1818 0.632043 0.316022 0.948752i \(-0.397653\pi\)
0.316022 + 0.948752i \(0.397653\pi\)
\(740\) 4.65495 0.171119
\(741\) 0 0
\(742\) −3.07857 −0.113018
\(743\) 21.8990 0.803395 0.401697 0.915772i \(-0.368420\pi\)
0.401697 + 0.915772i \(0.368420\pi\)
\(744\) 0 0
\(745\) 12.6091 0.461961
\(746\) 5.03763 0.184441
\(747\) 0 0
\(748\) 12.0177 0.439409
\(749\) −22.8649 −0.835465
\(750\) 0 0
\(751\) 50.1136 1.82867 0.914336 0.404956i \(-0.132713\pi\)
0.914336 + 0.404956i \(0.132713\pi\)
\(752\) 17.9189 0.653435
\(753\) 0 0
\(754\) 1.91113 0.0695991
\(755\) −3.80534 −0.138490
\(756\) 0 0
\(757\) 49.4866 1.79862 0.899311 0.437309i \(-0.144068\pi\)
0.899311 + 0.437309i \(0.144068\pi\)
\(758\) −4.97655 −0.180757
\(759\) 0 0
\(760\) 0.471300 0.0170958
\(761\) 18.5155 0.671186 0.335593 0.942007i \(-0.391063\pi\)
0.335593 + 0.942007i \(0.391063\pi\)
\(762\) 0 0
\(763\) −16.8598 −0.610364
\(764\) 11.7711 0.425865
\(765\) 0 0
\(766\) 6.23916 0.225430
\(767\) 28.5854 1.03216
\(768\) 0 0
\(769\) 9.36922 0.337863 0.168931 0.985628i \(-0.445968\pi\)
0.168931 + 0.985628i \(0.445968\pi\)
\(770\) 0.899538 0.0324171
\(771\) 0 0
\(772\) −3.94048 −0.141821
\(773\) −3.24015 −0.116540 −0.0582700 0.998301i \(-0.518558\pi\)
−0.0582700 + 0.998301i \(0.518558\pi\)
\(774\) 0 0
\(775\) 0.436319 0.0156730
\(776\) −14.7403 −0.529145
\(777\) 0 0
\(778\) −8.61085 −0.308714
\(779\) 0.191095 0.00684670
\(780\) 0 0
\(781\) −2.53455 −0.0906934
\(782\) 2.89279 0.103446
\(783\) 0 0
\(784\) −11.2411 −0.401469
\(785\) −13.6788 −0.488217
\(786\) 0 0
\(787\) 34.2199 1.21981 0.609905 0.792475i \(-0.291208\pi\)
0.609905 + 0.792475i \(0.291208\pi\)
\(788\) 7.63648 0.272038
\(789\) 0 0
\(790\) −0.229914 −0.00817998
\(791\) 11.7522 0.417859
\(792\) 0 0
\(793\) 6.14896 0.218356
\(794\) −4.54489 −0.161292
\(795\) 0 0
\(796\) 21.1524 0.749725
\(797\) 9.08039 0.321644 0.160822 0.986983i \(-0.448585\pi\)
0.160822 + 0.986983i \(0.448585\pi\)
\(798\) 0 0
\(799\) 43.0019 1.52130
\(800\) 23.5587 0.832924
\(801\) 0 0
\(802\) −0.689978 −0.0243640
\(803\) 3.89033 0.137287
\(804\) 0 0
\(805\) −1.04378 −0.0367883
\(806\) −0.194392 −0.00684718
\(807\) 0 0
\(808\) 33.1254 1.16535
\(809\) 2.55646 0.0898804 0.0449402 0.998990i \(-0.485690\pi\)
0.0449402 + 0.998990i \(0.485690\pi\)
\(810\) 0 0
\(811\) 1.90277 0.0668153 0.0334077 0.999442i \(-0.489364\pi\)
0.0334077 + 0.999442i \(0.489364\pi\)
\(812\) −2.05121 −0.0719834
\(813\) 0 0
\(814\) −2.87348 −0.100715
\(815\) −2.14574 −0.0751621
\(816\) 0 0
\(817\) −1.30720 −0.0457333
\(818\) 13.8532 0.484366
\(819\) 0 0
\(820\) −1.02263 −0.0357118
\(821\) 13.7415 0.479581 0.239790 0.970825i \(-0.422921\pi\)
0.239790 + 0.970825i \(0.422921\pi\)
\(822\) 0 0
\(823\) −10.1853 −0.355039 −0.177519 0.984117i \(-0.556807\pi\)
−0.177519 + 0.984117i \(0.556807\pi\)
\(824\) 20.5734 0.716709
\(825\) 0 0
\(826\) 6.36464 0.221454
\(827\) −26.7391 −0.929809 −0.464905 0.885361i \(-0.653911\pi\)
−0.464905 + 0.885361i \(0.653911\pi\)
\(828\) 0 0
\(829\) −51.5489 −1.79037 −0.895184 0.445697i \(-0.852956\pi\)
−0.895184 + 0.445697i \(0.852956\pi\)
\(830\) −4.12123 −0.143050
\(831\) 0 0
\(832\) 2.91274 0.100981
\(833\) −26.9766 −0.934683
\(834\) 0 0
\(835\) −16.0724 −0.556210
\(836\) 0.635316 0.0219729
\(837\) 0 0
\(838\) 10.9178 0.377150
\(839\) −4.54360 −0.156862 −0.0784312 0.996920i \(-0.524991\pi\)
−0.0784312 + 0.996920i \(0.524991\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 18.2296 0.628232
\(843\) 0 0
\(844\) 14.4956 0.498960
\(845\) −1.99810 −0.0687368
\(846\) 0 0
\(847\) −10.9453 −0.376085
\(848\) 8.72107 0.299483
\(849\) 0 0
\(850\) 12.4088 0.425620
\(851\) 3.33423 0.114296
\(852\) 0 0
\(853\) −14.2400 −0.487568 −0.243784 0.969830i \(-0.578389\pi\)
−0.243784 + 0.969830i \(0.578389\pi\)
\(854\) 1.36909 0.0468492
\(855\) 0 0
\(856\) −39.5738 −1.35260
\(857\) 2.90828 0.0993451 0.0496726 0.998766i \(-0.484182\pi\)
0.0496726 + 0.998766i \(0.484182\pi\)
\(858\) 0 0
\(859\) 0.525329 0.0179240 0.00896200 0.999960i \(-0.497147\pi\)
0.00896200 + 0.999960i \(0.497147\pi\)
\(860\) 6.99540 0.238541
\(861\) 0 0
\(862\) 5.79480 0.197372
\(863\) 35.2839 1.20108 0.600539 0.799595i \(-0.294953\pi\)
0.600539 + 0.799595i \(0.294953\pi\)
\(864\) 0 0
\(865\) 7.90516 0.268784
\(866\) 5.22374 0.177510
\(867\) 0 0
\(868\) 0.208641 0.00708175
\(869\) −0.684147 −0.0232081
\(870\) 0 0
\(871\) 10.4878 0.355366
\(872\) −29.1803 −0.988170
\(873\) 0 0
\(874\) 0.152928 0.00517286
\(875\) −9.69624 −0.327793
\(876\) 0 0
\(877\) −32.9988 −1.11429 −0.557145 0.830415i \(-0.688103\pi\)
−0.557145 + 0.830415i \(0.688103\pi\)
\(878\) 18.7470 0.632681
\(879\) 0 0
\(880\) −2.54824 −0.0859012
\(881\) 8.43036 0.284026 0.142013 0.989865i \(-0.454642\pi\)
0.142013 + 0.989865i \(0.454642\pi\)
\(882\) 0 0
\(883\) −26.7930 −0.901656 −0.450828 0.892611i \(-0.648871\pi\)
−0.450828 + 0.892611i \(0.648871\pi\)
\(884\) 26.6500 0.896336
\(885\) 0 0
\(886\) 12.2381 0.411147
\(887\) −18.0659 −0.606594 −0.303297 0.952896i \(-0.598087\pi\)
−0.303297 + 0.952896i \(0.598087\pi\)
\(888\) 0 0
\(889\) 23.6962 0.794744
\(890\) −2.53754 −0.0850585
\(891\) 0 0
\(892\) −48.6569 −1.62915
\(893\) 2.27330 0.0760732
\(894\) 0 0
\(895\) −3.65046 −0.122021
\(896\) 14.2509 0.476089
\(897\) 0 0
\(898\) −8.03838 −0.268244
\(899\) −0.101716 −0.00339242
\(900\) 0 0
\(901\) 20.9289 0.697242
\(902\) 0.631264 0.0210188
\(903\) 0 0
\(904\) 20.3403 0.676507
\(905\) 17.2546 0.573563
\(906\) 0 0
\(907\) 50.2869 1.66975 0.834874 0.550441i \(-0.185540\pi\)
0.834874 + 0.550441i \(0.185540\pi\)
\(908\) 35.5630 1.18020
\(909\) 0 0
\(910\) 1.99479 0.0661266
\(911\) 25.2794 0.837544 0.418772 0.908091i \(-0.362461\pi\)
0.418772 + 0.908091i \(0.362461\pi\)
\(912\) 0 0
\(913\) −12.2634 −0.405858
\(914\) −7.53912 −0.249372
\(915\) 0 0
\(916\) −10.1818 −0.336416
\(917\) −17.8371 −0.589034
\(918\) 0 0
\(919\) −48.0884 −1.58629 −0.793146 0.609032i \(-0.791558\pi\)
−0.793146 + 0.609032i \(0.791558\pi\)
\(920\) −1.80653 −0.0595596
\(921\) 0 0
\(922\) −14.1465 −0.465891
\(923\) −5.62054 −0.185002
\(924\) 0 0
\(925\) 14.3024 0.470261
\(926\) 23.7936 0.781907
\(927\) 0 0
\(928\) −5.49207 −0.180286
\(929\) 32.8357 1.07731 0.538653 0.842528i \(-0.318934\pi\)
0.538653 + 0.842528i \(0.318934\pi\)
\(930\) 0 0
\(931\) −1.42612 −0.0467393
\(932\) 34.8354 1.14107
\(933\) 0 0
\(934\) −9.61992 −0.314773
\(935\) −6.11528 −0.199991
\(936\) 0 0
\(937\) −28.5880 −0.933930 −0.466965 0.884276i \(-0.654653\pi\)
−0.466965 + 0.884276i \(0.654653\pi\)
\(938\) 2.33515 0.0762453
\(939\) 0 0
\(940\) −12.1654 −0.396791
\(941\) 13.1403 0.428361 0.214181 0.976794i \(-0.431292\pi\)
0.214181 + 0.976794i \(0.431292\pi\)
\(942\) 0 0
\(943\) −0.732484 −0.0238530
\(944\) −18.0299 −0.586825
\(945\) 0 0
\(946\) −4.31822 −0.140397
\(947\) 35.6856 1.15963 0.579813 0.814750i \(-0.303126\pi\)
0.579813 + 0.814750i \(0.303126\pi\)
\(948\) 0 0
\(949\) 8.62708 0.280047
\(950\) 0.655996 0.0212833
\(951\) 0 0
\(952\) 13.0984 0.424521
\(953\) 14.8964 0.482541 0.241270 0.970458i \(-0.422436\pi\)
0.241270 + 0.970458i \(0.422436\pi\)
\(954\) 0 0
\(955\) −5.98985 −0.193827
\(956\) −37.4733 −1.21197
\(957\) 0 0
\(958\) −6.78691 −0.219275
\(959\) −13.5804 −0.438534
\(960\) 0 0
\(961\) −30.9897 −0.999666
\(962\) −6.37213 −0.205446
\(963\) 0 0
\(964\) −26.7811 −0.862561
\(965\) 2.00515 0.0645480
\(966\) 0 0
\(967\) −15.5500 −0.500053 −0.250026 0.968239i \(-0.580439\pi\)
−0.250026 + 0.968239i \(0.580439\pi\)
\(968\) −18.9438 −0.608876
\(969\) 0 0
\(970\) 3.39791 0.109100
\(971\) 40.0985 1.28682 0.643410 0.765522i \(-0.277519\pi\)
0.643410 + 0.765522i \(0.277519\pi\)
\(972\) 0 0
\(973\) −13.3521 −0.428048
\(974\) 3.26621 0.104656
\(975\) 0 0
\(976\) −3.87840 −0.124144
\(977\) 12.6678 0.405277 0.202639 0.979254i \(-0.435048\pi\)
0.202639 + 0.979254i \(0.435048\pi\)
\(978\) 0 0
\(979\) −7.55086 −0.241326
\(980\) 7.63177 0.243788
\(981\) 0 0
\(982\) 11.9632 0.381762
\(983\) 41.8116 1.33358 0.666792 0.745244i \(-0.267667\pi\)
0.666792 + 0.745244i \(0.267667\pi\)
\(984\) 0 0
\(985\) −3.88589 −0.123815
\(986\) −2.89279 −0.0921251
\(987\) 0 0
\(988\) 1.40886 0.0448217
\(989\) 5.01063 0.159329
\(990\) 0 0
\(991\) 6.44884 0.204854 0.102427 0.994741i \(-0.467339\pi\)
0.102427 + 0.994741i \(0.467339\pi\)
\(992\) 0.558632 0.0177366
\(993\) 0 0
\(994\) −1.25143 −0.0396931
\(995\) −10.7635 −0.341227
\(996\) 0 0
\(997\) 28.3269 0.897123 0.448562 0.893752i \(-0.351936\pi\)
0.448562 + 0.893752i \(0.351936\pi\)
\(998\) 19.1109 0.604946
\(999\) 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 6003.2.a.j.1.3 7
3.2 odd 2 2001.2.a.i.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.5 7 3.2 odd 2
6003.2.a.j.1.3 7 1.1 even 1 trivial