Properties

Label 8048.2.a.x.1.4
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
Dimension $33$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, 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("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(33\)
Twist minimal: no (minimal twist has level 4024)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.91492 q^{3} +1.91993 q^{5} -3.81516 q^{7} +5.49674 q^{9} +O(q^{10})\) \(q-2.91492 q^{3} +1.91993 q^{5} -3.81516 q^{7} +5.49674 q^{9} +0.0492775 q^{11} +2.32659 q^{13} -5.59644 q^{15} -6.76001 q^{17} +8.12242 q^{19} +11.1209 q^{21} -6.65471 q^{23} -1.31387 q^{25} -7.27780 q^{27} +9.38336 q^{29} +2.64599 q^{31} -0.143640 q^{33} -7.32485 q^{35} -0.740541 q^{37} -6.78183 q^{39} -10.7106 q^{41} -10.8939 q^{43} +10.5534 q^{45} +8.22628 q^{47} +7.55546 q^{49} +19.7049 q^{51} +8.63256 q^{53} +0.0946094 q^{55} -23.6762 q^{57} -12.7530 q^{59} +6.76090 q^{61} -20.9710 q^{63} +4.46690 q^{65} +3.07357 q^{67} +19.3979 q^{69} +5.82993 q^{71} +8.06072 q^{73} +3.82981 q^{75} -0.188002 q^{77} -3.75511 q^{79} +4.72395 q^{81} +4.63757 q^{83} -12.9787 q^{85} -27.3517 q^{87} +6.74306 q^{89} -8.87634 q^{91} -7.71283 q^{93} +15.5945 q^{95} -9.93673 q^{97} +0.270866 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 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 0
\(3\) −2.91492 −1.68293 −0.841464 0.540313i \(-0.818306\pi\)
−0.841464 + 0.540313i \(0.818306\pi\)
\(4\) 0 0
\(5\) 1.91993 0.858619 0.429310 0.903157i \(-0.358757\pi\)
0.429310 + 0.903157i \(0.358757\pi\)
\(6\) 0 0
\(7\) −3.81516 −1.44200 −0.720998 0.692937i \(-0.756316\pi\)
−0.720998 + 0.692937i \(0.756316\pi\)
\(8\) 0 0
\(9\) 5.49674 1.83225
\(10\) 0 0
\(11\) 0.0492775 0.0148577 0.00742886 0.999972i \(-0.497635\pi\)
0.00742886 + 0.999972i \(0.497635\pi\)
\(12\) 0 0
\(13\) 2.32659 0.645281 0.322641 0.946522i \(-0.395429\pi\)
0.322641 + 0.946522i \(0.395429\pi\)
\(14\) 0 0
\(15\) −5.59644 −1.44499
\(16\) 0 0
\(17\) −6.76001 −1.63954 −0.819771 0.572691i \(-0.805900\pi\)
−0.819771 + 0.572691i \(0.805900\pi\)
\(18\) 0 0
\(19\) 8.12242 1.86341 0.931706 0.363214i \(-0.118321\pi\)
0.931706 + 0.363214i \(0.118321\pi\)
\(20\) 0 0
\(21\) 11.1209 2.42678
\(22\) 0 0
\(23\) −6.65471 −1.38760 −0.693801 0.720166i \(-0.744065\pi\)
−0.693801 + 0.720166i \(0.744065\pi\)
\(24\) 0 0
\(25\) −1.31387 −0.262773
\(26\) 0 0
\(27\) −7.27780 −1.40061
\(28\) 0 0
\(29\) 9.38336 1.74245 0.871224 0.490886i \(-0.163327\pi\)
0.871224 + 0.490886i \(0.163327\pi\)
\(30\) 0 0
\(31\) 2.64599 0.475233 0.237617 0.971359i \(-0.423634\pi\)
0.237617 + 0.971359i \(0.423634\pi\)
\(32\) 0 0
\(33\) −0.143640 −0.0250045
\(34\) 0 0
\(35\) −7.32485 −1.23813
\(36\) 0 0
\(37\) −0.740541 −0.121744 −0.0608721 0.998146i \(-0.519388\pi\)
−0.0608721 + 0.998146i \(0.519388\pi\)
\(38\) 0 0
\(39\) −6.78183 −1.08596
\(40\) 0 0
\(41\) −10.7106 −1.67272 −0.836358 0.548183i \(-0.815320\pi\)
−0.836358 + 0.548183i \(0.815320\pi\)
\(42\) 0 0
\(43\) −10.8939 −1.66131 −0.830653 0.556790i \(-0.812033\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(44\) 0 0
\(45\) 10.5534 1.57320
\(46\) 0 0
\(47\) 8.22628 1.19993 0.599963 0.800028i \(-0.295182\pi\)
0.599963 + 0.800028i \(0.295182\pi\)
\(48\) 0 0
\(49\) 7.55546 1.07935
\(50\) 0 0
\(51\) 19.7049 2.75923
\(52\) 0 0
\(53\) 8.63256 1.18577 0.592887 0.805286i \(-0.297988\pi\)
0.592887 + 0.805286i \(0.297988\pi\)
\(54\) 0 0
\(55\) 0.0946094 0.0127571
\(56\) 0 0
\(57\) −23.6762 −3.13599
\(58\) 0 0
\(59\) −12.7530 −1.66030 −0.830151 0.557539i \(-0.811746\pi\)
−0.830151 + 0.557539i \(0.811746\pi\)
\(60\) 0 0
\(61\) 6.76090 0.865645 0.432822 0.901479i \(-0.357518\pi\)
0.432822 + 0.901479i \(0.357518\pi\)
\(62\) 0 0
\(63\) −20.9710 −2.64209
\(64\) 0 0
\(65\) 4.46690 0.554051
\(66\) 0 0
\(67\) 3.07357 0.375496 0.187748 0.982217i \(-0.439881\pi\)
0.187748 + 0.982217i \(0.439881\pi\)
\(68\) 0 0
\(69\) 19.3979 2.33524
\(70\) 0 0
\(71\) 5.82993 0.691886 0.345943 0.938256i \(-0.387559\pi\)
0.345943 + 0.938256i \(0.387559\pi\)
\(72\) 0 0
\(73\) 8.06072 0.943436 0.471718 0.881750i \(-0.343634\pi\)
0.471718 + 0.881750i \(0.343634\pi\)
\(74\) 0 0
\(75\) 3.82981 0.442228
\(76\) 0 0
\(77\) −0.188002 −0.0214248
\(78\) 0 0
\(79\) −3.75511 −0.422482 −0.211241 0.977434i \(-0.567751\pi\)
−0.211241 + 0.977434i \(0.567751\pi\)
\(80\) 0 0
\(81\) 4.72395 0.524883
\(82\) 0 0
\(83\) 4.63757 0.509039 0.254520 0.967068i \(-0.418083\pi\)
0.254520 + 0.967068i \(0.418083\pi\)
\(84\) 0 0
\(85\) −12.9787 −1.40774
\(86\) 0 0
\(87\) −27.3517 −2.93241
\(88\) 0 0
\(89\) 6.74306 0.714763 0.357381 0.933959i \(-0.383670\pi\)
0.357381 + 0.933959i \(0.383670\pi\)
\(90\) 0 0
\(91\) −8.87634 −0.930493
\(92\) 0 0
\(93\) −7.71283 −0.799783
\(94\) 0 0
\(95\) 15.5945 1.59996
\(96\) 0 0
\(97\) −9.93673 −1.00892 −0.504461 0.863434i \(-0.668309\pi\)
−0.504461 + 0.863434i \(0.668309\pi\)
\(98\) 0 0
\(99\) 0.270866 0.0272230
\(100\) 0 0
\(101\) −1.33314 −0.132653 −0.0663264 0.997798i \(-0.521128\pi\)
−0.0663264 + 0.997798i \(0.521128\pi\)
\(102\) 0 0
\(103\) 6.78884 0.668924 0.334462 0.942409i \(-0.391445\pi\)
0.334462 + 0.942409i \(0.391445\pi\)
\(104\) 0 0
\(105\) 21.3513 2.08368
\(106\) 0 0
\(107\) 17.4430 1.68628 0.843140 0.537694i \(-0.180705\pi\)
0.843140 + 0.537694i \(0.180705\pi\)
\(108\) 0 0
\(109\) −3.42253 −0.327819 −0.163910 0.986475i \(-0.552411\pi\)
−0.163910 + 0.986475i \(0.552411\pi\)
\(110\) 0 0
\(111\) 2.15862 0.204887
\(112\) 0 0
\(113\) −15.0455 −1.41537 −0.707683 0.706530i \(-0.750260\pi\)
−0.707683 + 0.706530i \(0.750260\pi\)
\(114\) 0 0
\(115\) −12.7766 −1.19142
\(116\) 0 0
\(117\) 12.7887 1.18231
\(118\) 0 0
\(119\) 25.7905 2.36421
\(120\) 0 0
\(121\) −10.9976 −0.999779
\(122\) 0 0
\(123\) 31.2205 2.81506
\(124\) 0 0
\(125\) −12.1222 −1.08424
\(126\) 0 0
\(127\) −1.44797 −0.128486 −0.0642432 0.997934i \(-0.520463\pi\)
−0.0642432 + 0.997934i \(0.520463\pi\)
\(128\) 0 0
\(129\) 31.7548 2.79586
\(130\) 0 0
\(131\) −2.80644 −0.245200 −0.122600 0.992456i \(-0.539123\pi\)
−0.122600 + 0.992456i \(0.539123\pi\)
\(132\) 0 0
\(133\) −30.9884 −2.68703
\(134\) 0 0
\(135\) −13.9729 −1.20259
\(136\) 0 0
\(137\) 13.6495 1.16616 0.583079 0.812415i \(-0.301848\pi\)
0.583079 + 0.812415i \(0.301848\pi\)
\(138\) 0 0
\(139\) −2.09368 −0.177584 −0.0887918 0.996050i \(-0.528301\pi\)
−0.0887918 + 0.996050i \(0.528301\pi\)
\(140\) 0 0
\(141\) −23.9789 −2.01939
\(142\) 0 0
\(143\) 0.114649 0.00958741
\(144\) 0 0
\(145\) 18.0154 1.49610
\(146\) 0 0
\(147\) −22.0235 −1.81647
\(148\) 0 0
\(149\) 15.0301 1.23131 0.615656 0.788015i \(-0.288891\pi\)
0.615656 + 0.788015i \(0.288891\pi\)
\(150\) 0 0
\(151\) 21.8858 1.78104 0.890520 0.454944i \(-0.150341\pi\)
0.890520 + 0.454944i \(0.150341\pi\)
\(152\) 0 0
\(153\) −37.1580 −3.00405
\(154\) 0 0
\(155\) 5.08011 0.408044
\(156\) 0 0
\(157\) −4.55666 −0.363661 −0.181831 0.983330i \(-0.558202\pi\)
−0.181831 + 0.983330i \(0.558202\pi\)
\(158\) 0 0
\(159\) −25.1632 −1.99557
\(160\) 0 0
\(161\) 25.3888 2.00092
\(162\) 0 0
\(163\) −16.8610 −1.32065 −0.660326 0.750979i \(-0.729582\pi\)
−0.660326 + 0.750979i \(0.729582\pi\)
\(164\) 0 0
\(165\) −0.275779 −0.0214693
\(166\) 0 0
\(167\) −2.57459 −0.199228 −0.0996140 0.995026i \(-0.531761\pi\)
−0.0996140 + 0.995026i \(0.531761\pi\)
\(168\) 0 0
\(169\) −7.58696 −0.583612
\(170\) 0 0
\(171\) 44.6469 3.41423
\(172\) 0 0
\(173\) 3.27149 0.248727 0.124363 0.992237i \(-0.460311\pi\)
0.124363 + 0.992237i \(0.460311\pi\)
\(174\) 0 0
\(175\) 5.01261 0.378918
\(176\) 0 0
\(177\) 37.1740 2.79417
\(178\) 0 0
\(179\) 15.3592 1.14800 0.574002 0.818854i \(-0.305390\pi\)
0.574002 + 0.818854i \(0.305390\pi\)
\(180\) 0 0
\(181\) −16.8666 −1.25368 −0.626842 0.779146i \(-0.715653\pi\)
−0.626842 + 0.779146i \(0.715653\pi\)
\(182\) 0 0
\(183\) −19.7075 −1.45682
\(184\) 0 0
\(185\) −1.42179 −0.104532
\(186\) 0 0
\(187\) −0.333116 −0.0243599
\(188\) 0 0
\(189\) 27.7660 2.01968
\(190\) 0 0
\(191\) 2.20431 0.159498 0.0797492 0.996815i \(-0.474588\pi\)
0.0797492 + 0.996815i \(0.474588\pi\)
\(192\) 0 0
\(193\) 7.82115 0.562979 0.281489 0.959564i \(-0.409172\pi\)
0.281489 + 0.959564i \(0.409172\pi\)
\(194\) 0 0
\(195\) −13.0206 −0.932428
\(196\) 0 0
\(197\) 7.75701 0.552664 0.276332 0.961062i \(-0.410881\pi\)
0.276332 + 0.961062i \(0.410881\pi\)
\(198\) 0 0
\(199\) 23.1998 1.64459 0.822295 0.569062i \(-0.192694\pi\)
0.822295 + 0.569062i \(0.192694\pi\)
\(200\) 0 0
\(201\) −8.95919 −0.631933
\(202\) 0 0
\(203\) −35.7991 −2.51260
\(204\) 0 0
\(205\) −20.5636 −1.43623
\(206\) 0 0
\(207\) −36.5792 −2.54243
\(208\) 0 0
\(209\) 0.400253 0.0276861
\(210\) 0 0
\(211\) 7.11100 0.489542 0.244771 0.969581i \(-0.421287\pi\)
0.244771 + 0.969581i \(0.421287\pi\)
\(212\) 0 0
\(213\) −16.9938 −1.16439
\(214\) 0 0
\(215\) −20.9156 −1.42643
\(216\) 0 0
\(217\) −10.0949 −0.685284
\(218\) 0 0
\(219\) −23.4963 −1.58773
\(220\) 0 0
\(221\) −15.7278 −1.05797
\(222\) 0 0
\(223\) −14.9352 −1.00013 −0.500067 0.865987i \(-0.666691\pi\)
−0.500067 + 0.865987i \(0.666691\pi\)
\(224\) 0 0
\(225\) −7.22198 −0.481465
\(226\) 0 0
\(227\) −8.03714 −0.533444 −0.266722 0.963774i \(-0.585941\pi\)
−0.266722 + 0.963774i \(0.585941\pi\)
\(228\) 0 0
\(229\) −0.403582 −0.0266694 −0.0133347 0.999911i \(-0.504245\pi\)
−0.0133347 + 0.999911i \(0.504245\pi\)
\(230\) 0 0
\(231\) 0.548009 0.0360564
\(232\) 0 0
\(233\) −18.7111 −1.22581 −0.612903 0.790158i \(-0.709998\pi\)
−0.612903 + 0.790158i \(0.709998\pi\)
\(234\) 0 0
\(235\) 15.7939 1.03028
\(236\) 0 0
\(237\) 10.9458 0.711007
\(238\) 0 0
\(239\) −10.1106 −0.654000 −0.327000 0.945024i \(-0.606038\pi\)
−0.327000 + 0.945024i \(0.606038\pi\)
\(240\) 0 0
\(241\) 7.18094 0.462565 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(242\) 0 0
\(243\) 8.06348 0.517272
\(244\) 0 0
\(245\) 14.5060 0.926752
\(246\) 0 0
\(247\) 18.8976 1.20242
\(248\) 0 0
\(249\) −13.5181 −0.856677
\(250\) 0 0
\(251\) 10.0999 0.637500 0.318750 0.947839i \(-0.396737\pi\)
0.318750 + 0.947839i \(0.396737\pi\)
\(252\) 0 0
\(253\) −0.327927 −0.0206166
\(254\) 0 0
\(255\) 37.8320 2.36913
\(256\) 0 0
\(257\) 6.02742 0.375980 0.187990 0.982171i \(-0.439803\pi\)
0.187990 + 0.982171i \(0.439803\pi\)
\(258\) 0 0
\(259\) 2.82528 0.175555
\(260\) 0 0
\(261\) 51.5779 3.19259
\(262\) 0 0
\(263\) −10.2511 −0.632109 −0.316055 0.948741i \(-0.602358\pi\)
−0.316055 + 0.948741i \(0.602358\pi\)
\(264\) 0 0
\(265\) 16.5739 1.01813
\(266\) 0 0
\(267\) −19.6555 −1.20289
\(268\) 0 0
\(269\) −27.9144 −1.70197 −0.850986 0.525188i \(-0.823995\pi\)
−0.850986 + 0.525188i \(0.823995\pi\)
\(270\) 0 0
\(271\) −31.0863 −1.88836 −0.944180 0.329430i \(-0.893143\pi\)
−0.944180 + 0.329430i \(0.893143\pi\)
\(272\) 0 0
\(273\) 25.8738 1.56595
\(274\) 0 0
\(275\) −0.0647440 −0.00390421
\(276\) 0 0
\(277\) −19.9609 −1.19933 −0.599667 0.800250i \(-0.704700\pi\)
−0.599667 + 0.800250i \(0.704700\pi\)
\(278\) 0 0
\(279\) 14.5443 0.870745
\(280\) 0 0
\(281\) 19.7244 1.17666 0.588329 0.808621i \(-0.299786\pi\)
0.588329 + 0.808621i \(0.299786\pi\)
\(282\) 0 0
\(283\) −12.8936 −0.766447 −0.383223 0.923656i \(-0.625186\pi\)
−0.383223 + 0.923656i \(0.625186\pi\)
\(284\) 0 0
\(285\) −45.4566 −2.69262
\(286\) 0 0
\(287\) 40.8627 2.41205
\(288\) 0 0
\(289\) 28.6977 1.68810
\(290\) 0 0
\(291\) 28.9648 1.69794
\(292\) 0 0
\(293\) 22.5403 1.31682 0.658411 0.752659i \(-0.271229\pi\)
0.658411 + 0.752659i \(0.271229\pi\)
\(294\) 0 0
\(295\) −24.4849 −1.42557
\(296\) 0 0
\(297\) −0.358632 −0.0208099
\(298\) 0 0
\(299\) −15.4828 −0.895394
\(300\) 0 0
\(301\) 41.5620 2.39560
\(302\) 0 0
\(303\) 3.88601 0.223245
\(304\) 0 0
\(305\) 12.9805 0.743259
\(306\) 0 0
\(307\) 23.9314 1.36584 0.682918 0.730495i \(-0.260710\pi\)
0.682918 + 0.730495i \(0.260710\pi\)
\(308\) 0 0
\(309\) −19.7889 −1.12575
\(310\) 0 0
\(311\) −1.33343 −0.0756120 −0.0378060 0.999285i \(-0.512037\pi\)
−0.0378060 + 0.999285i \(0.512037\pi\)
\(312\) 0 0
\(313\) −29.3797 −1.66064 −0.830318 0.557289i \(-0.811841\pi\)
−0.830318 + 0.557289i \(0.811841\pi\)
\(314\) 0 0
\(315\) −40.2628 −2.26855
\(316\) 0 0
\(317\) −4.85921 −0.272921 −0.136460 0.990646i \(-0.543573\pi\)
−0.136460 + 0.990646i \(0.543573\pi\)
\(318\) 0 0
\(319\) 0.462389 0.0258888
\(320\) 0 0
\(321\) −50.8449 −2.83789
\(322\) 0 0
\(323\) −54.9076 −3.05514
\(324\) 0 0
\(325\) −3.05683 −0.169562
\(326\) 0 0
\(327\) 9.97640 0.551696
\(328\) 0 0
\(329\) −31.3846 −1.73029
\(330\) 0 0
\(331\) 14.7431 0.810357 0.405178 0.914238i \(-0.367209\pi\)
0.405178 + 0.914238i \(0.367209\pi\)
\(332\) 0 0
\(333\) −4.07056 −0.223066
\(334\) 0 0
\(335\) 5.90104 0.322408
\(336\) 0 0
\(337\) 8.27405 0.450716 0.225358 0.974276i \(-0.427645\pi\)
0.225358 + 0.974276i \(0.427645\pi\)
\(338\) 0 0
\(339\) 43.8565 2.38196
\(340\) 0 0
\(341\) 0.130388 0.00706088
\(342\) 0 0
\(343\) −2.11917 −0.114424
\(344\) 0 0
\(345\) 37.2427 2.00508
\(346\) 0 0
\(347\) −19.5548 −1.04976 −0.524879 0.851177i \(-0.675889\pi\)
−0.524879 + 0.851177i \(0.675889\pi\)
\(348\) 0 0
\(349\) −35.6012 −1.90569 −0.952843 0.303464i \(-0.901857\pi\)
−0.952843 + 0.303464i \(0.901857\pi\)
\(350\) 0 0
\(351\) −16.9325 −0.903789
\(352\) 0 0
\(353\) −15.9871 −0.850910 −0.425455 0.904980i \(-0.639886\pi\)
−0.425455 + 0.904980i \(0.639886\pi\)
\(354\) 0 0
\(355\) 11.1931 0.594066
\(356\) 0 0
\(357\) −75.1772 −3.97880
\(358\) 0 0
\(359\) −20.6664 −1.09073 −0.545366 0.838198i \(-0.683609\pi\)
−0.545366 + 0.838198i \(0.683609\pi\)
\(360\) 0 0
\(361\) 46.9737 2.47230
\(362\) 0 0
\(363\) 32.0570 1.68256
\(364\) 0 0
\(365\) 15.4760 0.810052
\(366\) 0 0
\(367\) 5.83969 0.304829 0.152415 0.988317i \(-0.451295\pi\)
0.152415 + 0.988317i \(0.451295\pi\)
\(368\) 0 0
\(369\) −58.8735 −3.06483
\(370\) 0 0
\(371\) −32.9346 −1.70988
\(372\) 0 0
\(373\) −10.3104 −0.533853 −0.266927 0.963717i \(-0.586008\pi\)
−0.266927 + 0.963717i \(0.586008\pi\)
\(374\) 0 0
\(375\) 35.3352 1.82470
\(376\) 0 0
\(377\) 21.8313 1.12437
\(378\) 0 0
\(379\) −1.97374 −0.101384 −0.0506921 0.998714i \(-0.516143\pi\)
−0.0506921 + 0.998714i \(0.516143\pi\)
\(380\) 0 0
\(381\) 4.22071 0.216233
\(382\) 0 0
\(383\) 34.7397 1.77512 0.887559 0.460695i \(-0.152400\pi\)
0.887559 + 0.460695i \(0.152400\pi\)
\(384\) 0 0
\(385\) −0.360950 −0.0183957
\(386\) 0 0
\(387\) −59.8810 −3.04392
\(388\) 0 0
\(389\) 7.51098 0.380822 0.190411 0.981704i \(-0.439018\pi\)
0.190411 + 0.981704i \(0.439018\pi\)
\(390\) 0 0
\(391\) 44.9859 2.27503
\(392\) 0 0
\(393\) 8.18054 0.412654
\(394\) 0 0
\(395\) −7.20954 −0.362751
\(396\) 0 0
\(397\) 2.80451 0.140754 0.0703771 0.997520i \(-0.477580\pi\)
0.0703771 + 0.997520i \(0.477580\pi\)
\(398\) 0 0
\(399\) 90.3285 4.52208
\(400\) 0 0
\(401\) −13.7828 −0.688278 −0.344139 0.938919i \(-0.611829\pi\)
−0.344139 + 0.938919i \(0.611829\pi\)
\(402\) 0 0
\(403\) 6.15614 0.306659
\(404\) 0 0
\(405\) 9.06965 0.450674
\(406\) 0 0
\(407\) −0.0364920 −0.00180884
\(408\) 0 0
\(409\) −14.7252 −0.728112 −0.364056 0.931377i \(-0.618608\pi\)
−0.364056 + 0.931377i \(0.618608\pi\)
\(410\) 0 0
\(411\) −39.7873 −1.96256
\(412\) 0 0
\(413\) 48.6548 2.39415
\(414\) 0 0
\(415\) 8.90381 0.437071
\(416\) 0 0
\(417\) 6.10290 0.298860
\(418\) 0 0
\(419\) −16.4707 −0.804649 −0.402324 0.915497i \(-0.631798\pi\)
−0.402324 + 0.915497i \(0.631798\pi\)
\(420\) 0 0
\(421\) −2.08683 −0.101706 −0.0508528 0.998706i \(-0.516194\pi\)
−0.0508528 + 0.998706i \(0.516194\pi\)
\(422\) 0 0
\(423\) 45.2177 2.19856
\(424\) 0 0
\(425\) 8.88174 0.430828
\(426\) 0 0
\(427\) −25.7939 −1.24826
\(428\) 0 0
\(429\) −0.334192 −0.0161349
\(430\) 0 0
\(431\) −33.9574 −1.63567 −0.817834 0.575454i \(-0.804825\pi\)
−0.817834 + 0.575454i \(0.804825\pi\)
\(432\) 0 0
\(433\) −12.2573 −0.589048 −0.294524 0.955644i \(-0.595161\pi\)
−0.294524 + 0.955644i \(0.595161\pi\)
\(434\) 0 0
\(435\) −52.5134 −2.51783
\(436\) 0 0
\(437\) −54.0524 −2.58567
\(438\) 0 0
\(439\) −4.67197 −0.222981 −0.111490 0.993766i \(-0.535562\pi\)
−0.111490 + 0.993766i \(0.535562\pi\)
\(440\) 0 0
\(441\) 41.5304 1.97764
\(442\) 0 0
\(443\) −1.83465 −0.0871670 −0.0435835 0.999050i \(-0.513877\pi\)
−0.0435835 + 0.999050i \(0.513877\pi\)
\(444\) 0 0
\(445\) 12.9462 0.613709
\(446\) 0 0
\(447\) −43.8114 −2.07221
\(448\) 0 0
\(449\) −26.5909 −1.25490 −0.627452 0.778656i \(-0.715902\pi\)
−0.627452 + 0.778656i \(0.715902\pi\)
\(450\) 0 0
\(451\) −0.527792 −0.0248528
\(452\) 0 0
\(453\) −63.7953 −2.99736
\(454\) 0 0
\(455\) −17.0420 −0.798939
\(456\) 0 0
\(457\) −3.41740 −0.159859 −0.0799296 0.996801i \(-0.525470\pi\)
−0.0799296 + 0.996801i \(0.525470\pi\)
\(458\) 0 0
\(459\) 49.1980 2.29636
\(460\) 0 0
\(461\) 14.1968 0.661213 0.330607 0.943769i \(-0.392747\pi\)
0.330607 + 0.943769i \(0.392747\pi\)
\(462\) 0 0
\(463\) −40.8064 −1.89643 −0.948217 0.317623i \(-0.897115\pi\)
−0.948217 + 0.317623i \(0.897115\pi\)
\(464\) 0 0
\(465\) −14.8081 −0.686709
\(466\) 0 0
\(467\) −9.42953 −0.436346 −0.218173 0.975910i \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(468\) 0 0
\(469\) −11.7262 −0.541464
\(470\) 0 0
\(471\) 13.2823 0.612015
\(472\) 0 0
\(473\) −0.536825 −0.0246832
\(474\) 0 0
\(475\) −10.6718 −0.489654
\(476\) 0 0
\(477\) 47.4510 2.17263
\(478\) 0 0
\(479\) −31.6403 −1.44568 −0.722842 0.691014i \(-0.757164\pi\)
−0.722842 + 0.691014i \(0.757164\pi\)
\(480\) 0 0
\(481\) −1.72294 −0.0785593
\(482\) 0 0
\(483\) −74.0062 −3.36740
\(484\) 0 0
\(485\) −19.0778 −0.866280
\(486\) 0 0
\(487\) −13.9987 −0.634339 −0.317170 0.948369i \(-0.602732\pi\)
−0.317170 + 0.948369i \(0.602732\pi\)
\(488\) 0 0
\(489\) 49.1483 2.22256
\(490\) 0 0
\(491\) −19.8085 −0.893944 −0.446972 0.894548i \(-0.647498\pi\)
−0.446972 + 0.894548i \(0.647498\pi\)
\(492\) 0 0
\(493\) −63.4316 −2.85682
\(494\) 0 0
\(495\) 0.520043 0.0233742
\(496\) 0 0
\(497\) −22.2421 −0.997696
\(498\) 0 0
\(499\) −12.2517 −0.548459 −0.274230 0.961664i \(-0.588423\pi\)
−0.274230 + 0.961664i \(0.588423\pi\)
\(500\) 0 0
\(501\) 7.50473 0.335286
\(502\) 0 0
\(503\) −1.00000 −0.0445878
\(504\) 0 0
\(505\) −2.55955 −0.113898
\(506\) 0 0
\(507\) 22.1154 0.982177
\(508\) 0 0
\(509\) 26.7007 1.18349 0.591744 0.806126i \(-0.298440\pi\)
0.591744 + 0.806126i \(0.298440\pi\)
\(510\) 0 0
\(511\) −30.7529 −1.36043
\(512\) 0 0
\(513\) −59.1133 −2.60992
\(514\) 0 0
\(515\) 13.0341 0.574351
\(516\) 0 0
\(517\) 0.405370 0.0178282
\(518\) 0 0
\(519\) −9.53613 −0.418590
\(520\) 0 0
\(521\) 19.1798 0.840283 0.420141 0.907459i \(-0.361980\pi\)
0.420141 + 0.907459i \(0.361980\pi\)
\(522\) 0 0
\(523\) 0.290547 0.0127047 0.00635236 0.999980i \(-0.497978\pi\)
0.00635236 + 0.999980i \(0.497978\pi\)
\(524\) 0 0
\(525\) −14.6113 −0.637691
\(526\) 0 0
\(527\) −17.8869 −0.779165
\(528\) 0 0
\(529\) 21.2852 0.925441
\(530\) 0 0
\(531\) −70.1000 −3.04208
\(532\) 0 0
\(533\) −24.9193 −1.07937
\(534\) 0 0
\(535\) 33.4894 1.44787
\(536\) 0 0
\(537\) −44.7709 −1.93201
\(538\) 0 0
\(539\) 0.372314 0.0160367
\(540\) 0 0
\(541\) −18.5726 −0.798497 −0.399249 0.916843i \(-0.630729\pi\)
−0.399249 + 0.916843i \(0.630729\pi\)
\(542\) 0 0
\(543\) 49.1647 2.10986
\(544\) 0 0
\(545\) −6.57103 −0.281472
\(546\) 0 0
\(547\) 15.5449 0.664650 0.332325 0.943165i \(-0.392167\pi\)
0.332325 + 0.943165i \(0.392167\pi\)
\(548\) 0 0
\(549\) 37.1629 1.58608
\(550\) 0 0
\(551\) 76.2157 3.24690
\(552\) 0 0
\(553\) 14.3263 0.609218
\(554\) 0 0
\(555\) 4.14439 0.175920
\(556\) 0 0
\(557\) −34.0341 −1.44207 −0.721036 0.692897i \(-0.756334\pi\)
−0.721036 + 0.692897i \(0.756334\pi\)
\(558\) 0 0
\(559\) −25.3457 −1.07201
\(560\) 0 0
\(561\) 0.971006 0.0409959
\(562\) 0 0
\(563\) −29.3438 −1.23669 −0.618346 0.785906i \(-0.712197\pi\)
−0.618346 + 0.785906i \(0.712197\pi\)
\(564\) 0 0
\(565\) −28.8864 −1.21526
\(566\) 0 0
\(567\) −18.0226 −0.756879
\(568\) 0 0
\(569\) −37.6423 −1.57805 −0.789025 0.614362i \(-0.789414\pi\)
−0.789025 + 0.614362i \(0.789414\pi\)
\(570\) 0 0
\(571\) −30.6908 −1.28437 −0.642185 0.766550i \(-0.721972\pi\)
−0.642185 + 0.766550i \(0.721972\pi\)
\(572\) 0 0
\(573\) −6.42538 −0.268424
\(574\) 0 0
\(575\) 8.74339 0.364625
\(576\) 0 0
\(577\) −18.5886 −0.773853 −0.386927 0.922111i \(-0.626463\pi\)
−0.386927 + 0.922111i \(0.626463\pi\)
\(578\) 0 0
\(579\) −22.7980 −0.947453
\(580\) 0 0
\(581\) −17.6931 −0.734033
\(582\) 0 0
\(583\) 0.425391 0.0176179
\(584\) 0 0
\(585\) 24.5534 1.01516
\(586\) 0 0
\(587\) −25.5566 −1.05484 −0.527418 0.849606i \(-0.676840\pi\)
−0.527418 + 0.849606i \(0.676840\pi\)
\(588\) 0 0
\(589\) 21.4918 0.885555
\(590\) 0 0
\(591\) −22.6110 −0.930094
\(592\) 0 0
\(593\) −5.02195 −0.206227 −0.103113 0.994670i \(-0.532880\pi\)
−0.103113 + 0.994670i \(0.532880\pi\)
\(594\) 0 0
\(595\) 49.5160 2.02996
\(596\) 0 0
\(597\) −67.6255 −2.76773
\(598\) 0 0
\(599\) 28.8801 1.18001 0.590004 0.807400i \(-0.299126\pi\)
0.590004 + 0.807400i \(0.299126\pi\)
\(600\) 0 0
\(601\) −33.3107 −1.35877 −0.679386 0.733781i \(-0.737754\pi\)
−0.679386 + 0.733781i \(0.737754\pi\)
\(602\) 0 0
\(603\) 16.8946 0.688002
\(604\) 0 0
\(605\) −21.1146 −0.858430
\(606\) 0 0
\(607\) −33.7256 −1.36888 −0.684441 0.729069i \(-0.739954\pi\)
−0.684441 + 0.729069i \(0.739954\pi\)
\(608\) 0 0
\(609\) 104.351 4.22853
\(610\) 0 0
\(611\) 19.1392 0.774290
\(612\) 0 0
\(613\) −31.6932 −1.28008 −0.640039 0.768343i \(-0.721082\pi\)
−0.640039 + 0.768343i \(0.721082\pi\)
\(614\) 0 0
\(615\) 59.9413 2.41707
\(616\) 0 0
\(617\) 38.7409 1.55965 0.779824 0.625998i \(-0.215308\pi\)
0.779824 + 0.625998i \(0.215308\pi\)
\(618\) 0 0
\(619\) 24.8896 1.00040 0.500200 0.865910i \(-0.333260\pi\)
0.500200 + 0.865910i \(0.333260\pi\)
\(620\) 0 0
\(621\) 48.4316 1.94349
\(622\) 0 0
\(623\) −25.7259 −1.03068
\(624\) 0 0
\(625\) −16.7044 −0.668177
\(626\) 0 0
\(627\) −1.16670 −0.0465936
\(628\) 0 0
\(629\) 5.00606 0.199605
\(630\) 0 0
\(631\) −17.8569 −0.710871 −0.355435 0.934701i \(-0.615667\pi\)
−0.355435 + 0.934701i \(0.615667\pi\)
\(632\) 0 0
\(633\) −20.7280 −0.823863
\(634\) 0 0
\(635\) −2.78000 −0.110321
\(636\) 0 0
\(637\) 17.5785 0.696485
\(638\) 0 0
\(639\) 32.0456 1.26771
\(640\) 0 0
\(641\) −41.5304 −1.64035 −0.820177 0.572110i \(-0.806125\pi\)
−0.820177 + 0.572110i \(0.806125\pi\)
\(642\) 0 0
\(643\) −10.8973 −0.429748 −0.214874 0.976642i \(-0.568934\pi\)
−0.214874 + 0.976642i \(0.568934\pi\)
\(644\) 0 0
\(645\) 60.9671 2.40058
\(646\) 0 0
\(647\) −8.49716 −0.334058 −0.167029 0.985952i \(-0.553417\pi\)
−0.167029 + 0.985952i \(0.553417\pi\)
\(648\) 0 0
\(649\) −0.628437 −0.0246683
\(650\) 0 0
\(651\) 29.4257 1.15328
\(652\) 0 0
\(653\) 26.2695 1.02801 0.514003 0.857789i \(-0.328162\pi\)
0.514003 + 0.857789i \(0.328162\pi\)
\(654\) 0 0
\(655\) −5.38817 −0.210533
\(656\) 0 0
\(657\) 44.3077 1.72861
\(658\) 0 0
\(659\) −0.748762 −0.0291676 −0.0145838 0.999894i \(-0.504642\pi\)
−0.0145838 + 0.999894i \(0.504642\pi\)
\(660\) 0 0
\(661\) 6.21166 0.241606 0.120803 0.992677i \(-0.461453\pi\)
0.120803 + 0.992677i \(0.461453\pi\)
\(662\) 0 0
\(663\) 45.8452 1.78048
\(664\) 0 0
\(665\) −59.4955 −2.30714
\(666\) 0 0
\(667\) −62.4436 −2.41782
\(668\) 0 0
\(669\) 43.5348 1.68315
\(670\) 0 0
\(671\) 0.333160 0.0128615
\(672\) 0 0
\(673\) 30.2232 1.16502 0.582510 0.812823i \(-0.302071\pi\)
0.582510 + 0.812823i \(0.302071\pi\)
\(674\) 0 0
\(675\) 9.56204 0.368043
\(676\) 0 0
\(677\) −0.0861188 −0.00330982 −0.00165491 0.999999i \(-0.500527\pi\)
−0.00165491 + 0.999999i \(0.500527\pi\)
\(678\) 0 0
\(679\) 37.9102 1.45486
\(680\) 0 0
\(681\) 23.4276 0.897747
\(682\) 0 0
\(683\) −21.0287 −0.804640 −0.402320 0.915499i \(-0.631796\pi\)
−0.402320 + 0.915499i \(0.631796\pi\)
\(684\) 0 0
\(685\) 26.2062 1.00129
\(686\) 0 0
\(687\) 1.17641 0.0448827
\(688\) 0 0
\(689\) 20.0845 0.765157
\(690\) 0 0
\(691\) 27.1755 1.03380 0.516902 0.856045i \(-0.327085\pi\)
0.516902 + 0.856045i \(0.327085\pi\)
\(692\) 0 0
\(693\) −1.03340 −0.0392555
\(694\) 0 0
\(695\) −4.01972 −0.152477
\(696\) 0 0
\(697\) 72.4038 2.74249
\(698\) 0 0
\(699\) 54.5413 2.06294
\(700\) 0 0
\(701\) −6.85723 −0.258994 −0.129497 0.991580i \(-0.541336\pi\)
−0.129497 + 0.991580i \(0.541336\pi\)
\(702\) 0 0
\(703\) −6.01499 −0.226860
\(704\) 0 0
\(705\) −46.0379 −1.73389
\(706\) 0 0
\(707\) 5.08616 0.191285
\(708\) 0 0
\(709\) 8.04815 0.302254 0.151127 0.988514i \(-0.451710\pi\)
0.151127 + 0.988514i \(0.451710\pi\)
\(710\) 0 0
\(711\) −20.6408 −0.774092
\(712\) 0 0
\(713\) −17.6083 −0.659435
\(714\) 0 0
\(715\) 0.220118 0.00823194
\(716\) 0 0
\(717\) 29.4715 1.10064
\(718\) 0 0
\(719\) 8.57264 0.319705 0.159853 0.987141i \(-0.448898\pi\)
0.159853 + 0.987141i \(0.448898\pi\)
\(720\) 0 0
\(721\) −25.9005 −0.964586
\(722\) 0 0
\(723\) −20.9318 −0.778464
\(724\) 0 0
\(725\) −12.3285 −0.457868
\(726\) 0 0
\(727\) 20.6702 0.766617 0.383308 0.923620i \(-0.374785\pi\)
0.383308 + 0.923620i \(0.374785\pi\)
\(728\) 0 0
\(729\) −37.6762 −1.39541
\(730\) 0 0
\(731\) 73.6429 2.72378
\(732\) 0 0
\(733\) 21.4859 0.793600 0.396800 0.917905i \(-0.370121\pi\)
0.396800 + 0.917905i \(0.370121\pi\)
\(734\) 0 0
\(735\) −42.2837 −1.55966
\(736\) 0 0
\(737\) 0.151458 0.00557902
\(738\) 0 0
\(739\) 36.4769 1.34182 0.670912 0.741537i \(-0.265903\pi\)
0.670912 + 0.741537i \(0.265903\pi\)
\(740\) 0 0
\(741\) −55.0849 −2.02359
\(742\) 0 0
\(743\) −24.7307 −0.907282 −0.453641 0.891185i \(-0.649875\pi\)
−0.453641 + 0.891185i \(0.649875\pi\)
\(744\) 0 0
\(745\) 28.8567 1.05723
\(746\) 0 0
\(747\) 25.4915 0.932686
\(748\) 0 0
\(749\) −66.5479 −2.43161
\(750\) 0 0
\(751\) −38.4930 −1.40463 −0.702315 0.711867i \(-0.747850\pi\)
−0.702315 + 0.711867i \(0.747850\pi\)
\(752\) 0 0
\(753\) −29.4404 −1.07287
\(754\) 0 0
\(755\) 42.0192 1.52924
\(756\) 0 0
\(757\) 0.613093 0.0222832 0.0111416 0.999938i \(-0.496453\pi\)
0.0111416 + 0.999938i \(0.496453\pi\)
\(758\) 0 0
\(759\) 0.955881 0.0346963
\(760\) 0 0
\(761\) −49.4532 −1.79268 −0.896338 0.443372i \(-0.853782\pi\)
−0.896338 + 0.443372i \(0.853782\pi\)
\(762\) 0 0
\(763\) 13.0575 0.472714
\(764\) 0 0
\(765\) −71.3408 −2.57933
\(766\) 0 0
\(767\) −29.6711 −1.07136
\(768\) 0 0
\(769\) −38.3625 −1.38339 −0.691694 0.722190i \(-0.743135\pi\)
−0.691694 + 0.722190i \(0.743135\pi\)
\(770\) 0 0
\(771\) −17.5694 −0.632747
\(772\) 0 0
\(773\) 13.0048 0.467749 0.233875 0.972267i \(-0.424859\pi\)
0.233875 + 0.972267i \(0.424859\pi\)
\(774\) 0 0
\(775\) −3.47647 −0.124878
\(776\) 0 0
\(777\) −8.23547 −0.295446
\(778\) 0 0
\(779\) −86.9961 −3.11696
\(780\) 0 0
\(781\) 0.287285 0.0102798
\(782\) 0 0
\(783\) −68.2902 −2.44049
\(784\) 0 0
\(785\) −8.74847 −0.312246
\(786\) 0 0
\(787\) −39.9722 −1.42486 −0.712428 0.701745i \(-0.752404\pi\)
−0.712428 + 0.701745i \(0.752404\pi\)
\(788\) 0 0
\(789\) 29.8811 1.06379
\(790\) 0 0
\(791\) 57.4012 2.04095
\(792\) 0 0
\(793\) 15.7299 0.558584
\(794\) 0 0
\(795\) −48.3116 −1.71344
\(796\) 0 0
\(797\) −0.903441 −0.0320015 −0.0160008 0.999872i \(-0.505093\pi\)
−0.0160008 + 0.999872i \(0.505093\pi\)
\(798\) 0 0
\(799\) −55.6097 −1.96733
\(800\) 0 0
\(801\) 37.0649 1.30962
\(802\) 0 0
\(803\) 0.397212 0.0140173
\(804\) 0 0
\(805\) 48.7447 1.71803
\(806\) 0 0
\(807\) 81.3682 2.86430
\(808\) 0 0
\(809\) −12.8707 −0.452510 −0.226255 0.974068i \(-0.572648\pi\)
−0.226255 + 0.974068i \(0.572648\pi\)
\(810\) 0 0
\(811\) −34.8684 −1.22440 −0.612198 0.790705i \(-0.709714\pi\)
−0.612198 + 0.790705i \(0.709714\pi\)
\(812\) 0 0
\(813\) 90.6141 3.17797
\(814\) 0 0
\(815\) −32.3719 −1.13394
\(816\) 0 0
\(817\) −88.4850 −3.09570
\(818\) 0 0
\(819\) −48.7909 −1.70489
\(820\) 0 0
\(821\) 38.5797 1.34644 0.673220 0.739443i \(-0.264911\pi\)
0.673220 + 0.739443i \(0.264911\pi\)
\(822\) 0 0
\(823\) −10.5735 −0.368571 −0.184285 0.982873i \(-0.558997\pi\)
−0.184285 + 0.982873i \(0.558997\pi\)
\(824\) 0 0
\(825\) 0.188723 0.00657050
\(826\) 0 0
\(827\) 42.6187 1.48200 0.740998 0.671507i \(-0.234353\pi\)
0.740998 + 0.671507i \(0.234353\pi\)
\(828\) 0 0
\(829\) 15.1668 0.526766 0.263383 0.964691i \(-0.415162\pi\)
0.263383 + 0.964691i \(0.415162\pi\)
\(830\) 0 0
\(831\) 58.1844 2.01839
\(832\) 0 0
\(833\) −51.0750 −1.76964
\(834\) 0 0
\(835\) −4.94304 −0.171061
\(836\) 0 0
\(837\) −19.2569 −0.665617
\(838\) 0 0
\(839\) −31.3287 −1.08159 −0.540793 0.841156i \(-0.681876\pi\)
−0.540793 + 0.841156i \(0.681876\pi\)
\(840\) 0 0
\(841\) 59.0475 2.03612
\(842\) 0 0
\(843\) −57.4950 −1.98023
\(844\) 0 0
\(845\) −14.5664 −0.501101
\(846\) 0 0
\(847\) 41.9575 1.44168
\(848\) 0 0
\(849\) 37.5839 1.28988
\(850\) 0 0
\(851\) 4.92809 0.168933
\(852\) 0 0
\(853\) 33.3531 1.14199 0.570993 0.820955i \(-0.306558\pi\)
0.570993 + 0.820955i \(0.306558\pi\)
\(854\) 0 0
\(855\) 85.7189 2.93152
\(856\) 0 0
\(857\) 31.9922 1.09283 0.546417 0.837513i \(-0.315991\pi\)
0.546417 + 0.837513i \(0.315991\pi\)
\(858\) 0 0
\(859\) −5.83878 −0.199217 −0.0996083 0.995027i \(-0.531759\pi\)
−0.0996083 + 0.995027i \(0.531759\pi\)
\(860\) 0 0
\(861\) −119.111 −4.05931
\(862\) 0 0
\(863\) −39.9498 −1.35991 −0.679953 0.733256i \(-0.738000\pi\)
−0.679953 + 0.733256i \(0.738000\pi\)
\(864\) 0 0
\(865\) 6.28104 0.213562
\(866\) 0 0
\(867\) −83.6514 −2.84095
\(868\) 0 0
\(869\) −0.185042 −0.00627713
\(870\) 0 0
\(871\) 7.15095 0.242301
\(872\) 0 0
\(873\) −54.6197 −1.84860
\(874\) 0 0
\(875\) 46.2481 1.56347
\(876\) 0 0
\(877\) 25.4088 0.857993 0.428997 0.903306i \(-0.358867\pi\)
0.428997 + 0.903306i \(0.358867\pi\)
\(878\) 0 0
\(879\) −65.7033 −2.21612
\(880\) 0 0
\(881\) −54.3860 −1.83231 −0.916155 0.400825i \(-0.868724\pi\)
−0.916155 + 0.400825i \(0.868724\pi\)
\(882\) 0 0
\(883\) −3.88025 −0.130581 −0.0652905 0.997866i \(-0.520797\pi\)
−0.0652905 + 0.997866i \(0.520797\pi\)
\(884\) 0 0
\(885\) 71.3715 2.39913
\(886\) 0 0
\(887\) −45.1782 −1.51693 −0.758467 0.651711i \(-0.774051\pi\)
−0.758467 + 0.651711i \(0.774051\pi\)
\(888\) 0 0
\(889\) 5.52423 0.185277
\(890\) 0 0
\(891\) 0.232784 0.00779856
\(892\) 0 0
\(893\) 66.8173 2.23596
\(894\) 0 0
\(895\) 29.4887 0.985698
\(896\) 0 0
\(897\) 45.1311 1.50688
\(898\) 0 0
\(899\) 24.8283 0.828069
\(900\) 0 0
\(901\) −58.3562 −1.94413
\(902\) 0 0
\(903\) −121.150 −4.03162
\(904\) 0 0
\(905\) −32.3827 −1.07644
\(906\) 0 0
\(907\) 45.3498 1.50582 0.752908 0.658126i \(-0.228650\pi\)
0.752908 + 0.658126i \(0.228650\pi\)
\(908\) 0 0
\(909\) −7.32795 −0.243053
\(910\) 0 0
\(911\) 34.3520 1.13813 0.569066 0.822292i \(-0.307305\pi\)
0.569066 + 0.822292i \(0.307305\pi\)
\(912\) 0 0
\(913\) 0.228528 0.00756317
\(914\) 0 0
\(915\) −37.8370 −1.25085
\(916\) 0 0
\(917\) 10.7070 0.353577
\(918\) 0 0
\(919\) 6.66121 0.219733 0.109866 0.993946i \(-0.464958\pi\)
0.109866 + 0.993946i \(0.464958\pi\)
\(920\) 0 0
\(921\) −69.7580 −2.29860
\(922\) 0 0
\(923\) 13.5639 0.446461
\(924\) 0 0
\(925\) 0.972971 0.0319911
\(926\) 0 0
\(927\) 37.3165 1.22563
\(928\) 0 0
\(929\) 26.0988 0.856275 0.428138 0.903713i \(-0.359170\pi\)
0.428138 + 0.903713i \(0.359170\pi\)
\(930\) 0 0
\(931\) 61.3686 2.01128
\(932\) 0 0
\(933\) 3.88685 0.127250
\(934\) 0 0
\(935\) −0.639560 −0.0209159
\(936\) 0 0
\(937\) 24.4264 0.797975 0.398987 0.916956i \(-0.369362\pi\)
0.398987 + 0.916956i \(0.369362\pi\)
\(938\) 0 0
\(939\) 85.6393 2.79473
\(940\) 0 0
\(941\) 33.8986 1.10506 0.552532 0.833492i \(-0.313662\pi\)
0.552532 + 0.833492i \(0.313662\pi\)
\(942\) 0 0
\(943\) 71.2760 2.32107
\(944\) 0 0
\(945\) 53.3087 1.73413
\(946\) 0 0
\(947\) 28.1887 0.916010 0.458005 0.888950i \(-0.348564\pi\)
0.458005 + 0.888950i \(0.348564\pi\)
\(948\) 0 0
\(949\) 18.7540 0.608781
\(950\) 0 0
\(951\) 14.1642 0.459306
\(952\) 0 0
\(953\) −21.3674 −0.692157 −0.346079 0.938206i \(-0.612487\pi\)
−0.346079 + 0.938206i \(0.612487\pi\)
\(954\) 0 0
\(955\) 4.23212 0.136948
\(956\) 0 0
\(957\) −1.34782 −0.0435690
\(958\) 0 0
\(959\) −52.0752 −1.68160
\(960\) 0 0
\(961\) −23.9988 −0.774153
\(962\) 0 0
\(963\) 95.8797 3.08968
\(964\) 0 0
\(965\) 15.0161 0.483385
\(966\) 0 0
\(967\) −51.7347 −1.66367 −0.831837 0.555020i \(-0.812711\pi\)
−0.831837 + 0.555020i \(0.812711\pi\)
\(968\) 0 0
\(969\) 160.051 5.14159
\(970\) 0 0
\(971\) −4.78933 −0.153697 −0.0768484 0.997043i \(-0.524486\pi\)
−0.0768484 + 0.997043i \(0.524486\pi\)
\(972\) 0 0
\(973\) 7.98773 0.256075
\(974\) 0 0
\(975\) 8.91041 0.285362
\(976\) 0 0
\(977\) 53.9391 1.72566 0.862832 0.505491i \(-0.168689\pi\)
0.862832 + 0.505491i \(0.168689\pi\)
\(978\) 0 0
\(979\) 0.332281 0.0106198
\(980\) 0 0
\(981\) −18.8128 −0.600646
\(982\) 0 0
\(983\) −17.4039 −0.555098 −0.277549 0.960712i \(-0.589522\pi\)
−0.277549 + 0.960712i \(0.589522\pi\)
\(984\) 0 0
\(985\) 14.8929 0.474528
\(986\) 0 0
\(987\) 91.4834 2.91195
\(988\) 0 0
\(989\) 72.4958 2.30523
\(990\) 0 0
\(991\) 44.5225 1.41430 0.707152 0.707062i \(-0.249980\pi\)
0.707152 + 0.707062i \(0.249980\pi\)
\(992\) 0 0
\(993\) −42.9751 −1.36377
\(994\) 0 0
\(995\) 44.5420 1.41208
\(996\) 0 0
\(997\) −39.2457 −1.24292 −0.621462 0.783444i \(-0.713461\pi\)
−0.621462 + 0.783444i \(0.713461\pi\)
\(998\) 0 0
\(999\) 5.38951 0.170516
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 8048.2.a.x.1.4 33
4.3 odd 2 4024.2.a.g.1.30 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.g.1.30 33 4.3 odd 2
8048.2.a.x.1.4 33 1.1 even 1 trivial