Properties

Label 1648.2.a.f.1.2
Level $1648$
Weight $2$
Character 1648.1
Self dual yes
Analytic conductor $13.159$
Analytic rank $1$
Dimension $2$
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] = [1648,2,Mod(1,1648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1648, 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("1648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1648 = 2^{4} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1648.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: \(13.1593462531\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1648.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.381966 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.381966 q^{5} +1.00000 q^{7} -2.00000 q^{9} +2.61803 q^{11} -4.85410 q^{13} -0.381966 q^{15} -5.61803 q^{17} -5.85410 q^{19} +1.00000 q^{21} -4.47214 q^{23} -4.85410 q^{25} -5.00000 q^{27} -5.23607 q^{29} +6.70820 q^{31} +2.61803 q^{33} -0.381966 q^{35} +6.70820 q^{37} -4.85410 q^{39} +8.94427 q^{41} +8.70820 q^{43} +0.763932 q^{45} -4.09017 q^{47} -6.00000 q^{49} -5.61803 q^{51} +1.09017 q^{53} -1.00000 q^{55} -5.85410 q^{57} -6.38197 q^{59} +4.14590 q^{61} -2.00000 q^{63} +1.85410 q^{65} -14.4164 q^{67} -4.47214 q^{69} +4.09017 q^{71} -10.8541 q^{73} -4.85410 q^{75} +2.61803 q^{77} +6.56231 q^{79} +1.00000 q^{81} +6.32624 q^{83} +2.14590 q^{85} -5.23607 q^{87} -2.29180 q^{89} -4.85410 q^{91} +6.70820 q^{93} +2.23607 q^{95} -1.70820 q^{97} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} - 4 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{15} - 9 q^{17} - 5 q^{19} + 2 q^{21} - 3 q^{25} - 10 q^{27} - 6 q^{29} + 3 q^{33} - 3 q^{35} - 3 q^{39} + 4 q^{43} + 6 q^{45} + 3 q^{47} - 12 q^{49} - 9 q^{51} - 9 q^{53} - 2 q^{55} - 5 q^{57} - 15 q^{59} + 15 q^{61} - 4 q^{63} - 3 q^{65} - 2 q^{67} - 3 q^{71} - 15 q^{73} - 3 q^{75} + 3 q^{77} - 7 q^{79} + 2 q^{81} - 3 q^{83} + 11 q^{85} - 6 q^{87} - 18 q^{89} - 3 q^{91} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 2.61803 0.789367 0.394683 0.918817i \(-0.370854\pi\)
0.394683 + 0.918817i \(0.370854\pi\)
\(12\) 0 0
\(13\) −4.85410 −1.34629 −0.673143 0.739512i \(-0.735056\pi\)
−0.673143 + 0.739512i \(0.735056\pi\)
\(14\) 0 0
\(15\) −0.381966 −0.0986232
\(16\) 0 0
\(17\) −5.61803 −1.36257 −0.681287 0.732017i \(-0.738579\pi\)
−0.681287 + 0.732017i \(0.738579\pi\)
\(18\) 0 0
\(19\) −5.85410 −1.34302 −0.671512 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) −4.85410 −0.970820
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −5.23607 −0.972313 −0.486157 0.873872i \(-0.661602\pi\)
−0.486157 + 0.873872i \(0.661602\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 0 0
\(33\) 2.61803 0.455741
\(34\) 0 0
\(35\) −0.381966 −0.0645640
\(36\) 0 0
\(37\) 6.70820 1.10282 0.551411 0.834234i \(-0.314090\pi\)
0.551411 + 0.834234i \(0.314090\pi\)
\(38\) 0 0
\(39\) −4.85410 −0.777278
\(40\) 0 0
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) 0 0
\(43\) 8.70820 1.32799 0.663994 0.747738i \(-0.268860\pi\)
0.663994 + 0.747738i \(0.268860\pi\)
\(44\) 0 0
\(45\) 0.763932 0.113880
\(46\) 0 0
\(47\) −4.09017 −0.596613 −0.298306 0.954470i \(-0.596422\pi\)
−0.298306 + 0.954470i \(0.596422\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −5.61803 −0.786682
\(52\) 0 0
\(53\) 1.09017 0.149746 0.0748732 0.997193i \(-0.476145\pi\)
0.0748732 + 0.997193i \(0.476145\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −5.85410 −0.775395
\(58\) 0 0
\(59\) −6.38197 −0.830861 −0.415431 0.909625i \(-0.636369\pi\)
−0.415431 + 0.909625i \(0.636369\pi\)
\(60\) 0 0
\(61\) 4.14590 0.530828 0.265414 0.964135i \(-0.414491\pi\)
0.265414 + 0.964135i \(0.414491\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 1.85410 0.229973
\(66\) 0 0
\(67\) −14.4164 −1.76124 −0.880622 0.473819i \(-0.842875\pi\)
−0.880622 + 0.473819i \(0.842875\pi\)
\(68\) 0 0
\(69\) −4.47214 −0.538382
\(70\) 0 0
\(71\) 4.09017 0.485414 0.242707 0.970100i \(-0.421965\pi\)
0.242707 + 0.970100i \(0.421965\pi\)
\(72\) 0 0
\(73\) −10.8541 −1.27038 −0.635188 0.772357i \(-0.719077\pi\)
−0.635188 + 0.772357i \(0.719077\pi\)
\(74\) 0 0
\(75\) −4.85410 −0.560503
\(76\) 0 0
\(77\) 2.61803 0.298353
\(78\) 0 0
\(79\) 6.56231 0.738317 0.369159 0.929366i \(-0.379646\pi\)
0.369159 + 0.929366i \(0.379646\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.32624 0.694395 0.347197 0.937792i \(-0.387133\pi\)
0.347197 + 0.937792i \(0.387133\pi\)
\(84\) 0 0
\(85\) 2.14590 0.232755
\(86\) 0 0
\(87\) −5.23607 −0.561365
\(88\) 0 0
\(89\) −2.29180 −0.242930 −0.121465 0.992596i \(-0.538759\pi\)
−0.121465 + 0.992596i \(0.538759\pi\)
\(90\) 0 0
\(91\) −4.85410 −0.508848
\(92\) 0 0
\(93\) 6.70820 0.695608
\(94\) 0 0
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) −1.70820 −0.173442 −0.0867209 0.996233i \(-0.527639\pi\)
−0.0867209 + 0.996233i \(0.527639\pi\)
\(98\) 0 0
\(99\) −5.23607 −0.526245
\(100\) 0 0
\(101\) 1.90983 0.190035 0.0950176 0.995476i \(-0.469709\pi\)
0.0950176 + 0.995476i \(0.469709\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329
\(104\) 0 0
\(105\) −0.381966 −0.0372761
\(106\) 0 0
\(107\) 7.09017 0.685433 0.342716 0.939439i \(-0.388653\pi\)
0.342716 + 0.939439i \(0.388653\pi\)
\(108\) 0 0
\(109\) −9.56231 −0.915903 −0.457951 0.888977i \(-0.651417\pi\)
−0.457951 + 0.888977i \(0.651417\pi\)
\(110\) 0 0
\(111\) 6.70820 0.636715
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 1.70820 0.159291
\(116\) 0 0
\(117\) 9.70820 0.897524
\(118\) 0 0
\(119\) −5.61803 −0.515004
\(120\) 0 0
\(121\) −4.14590 −0.376900
\(122\) 0 0
\(123\) 8.94427 0.806478
\(124\) 0 0
\(125\) 3.76393 0.336656
\(126\) 0 0
\(127\) −15.2705 −1.35504 −0.677519 0.735505i \(-0.736945\pi\)
−0.677519 + 0.735505i \(0.736945\pi\)
\(128\) 0 0
\(129\) 8.70820 0.766715
\(130\) 0 0
\(131\) 2.23607 0.195366 0.0976831 0.995218i \(-0.468857\pi\)
0.0976831 + 0.995218i \(0.468857\pi\)
\(132\) 0 0
\(133\) −5.85410 −0.507615
\(134\) 0 0
\(135\) 1.90983 0.164372
\(136\) 0 0
\(137\) −12.7082 −1.08574 −0.542868 0.839818i \(-0.682661\pi\)
−0.542868 + 0.839818i \(0.682661\pi\)
\(138\) 0 0
\(139\) 11.1459 0.945383 0.472691 0.881228i \(-0.343283\pi\)
0.472691 + 0.881228i \(0.343283\pi\)
\(140\) 0 0
\(141\) −4.09017 −0.344454
\(142\) 0 0
\(143\) −12.7082 −1.06271
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) −7.47214 −0.612141 −0.306071 0.952009i \(-0.599014\pi\)
−0.306071 + 0.952009i \(0.599014\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) 11.2361 0.908382
\(154\) 0 0
\(155\) −2.56231 −0.205809
\(156\) 0 0
\(157\) 16.7082 1.33346 0.666730 0.745299i \(-0.267693\pi\)
0.666730 + 0.745299i \(0.267693\pi\)
\(158\) 0 0
\(159\) 1.09017 0.0864561
\(160\) 0 0
\(161\) −4.47214 −0.352454
\(162\) 0 0
\(163\) −2.70820 −0.212123 −0.106061 0.994360i \(-0.533824\pi\)
−0.106061 + 0.994360i \(0.533824\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 10.5623 0.812485
\(170\) 0 0
\(171\) 11.7082 0.895349
\(172\) 0 0
\(173\) 16.0344 1.21908 0.609538 0.792757i \(-0.291355\pi\)
0.609538 + 0.792757i \(0.291355\pi\)
\(174\) 0 0
\(175\) −4.85410 −0.366936
\(176\) 0 0
\(177\) −6.38197 −0.479698
\(178\) 0 0
\(179\) 7.85410 0.587043 0.293522 0.955952i \(-0.405173\pi\)
0.293522 + 0.955952i \(0.405173\pi\)
\(180\) 0 0
\(181\) 3.85410 0.286473 0.143237 0.989688i \(-0.454249\pi\)
0.143237 + 0.989688i \(0.454249\pi\)
\(182\) 0 0
\(183\) 4.14590 0.306474
\(184\) 0 0
\(185\) −2.56231 −0.188384
\(186\) 0 0
\(187\) −14.7082 −1.07557
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −5.61803 −0.406507 −0.203253 0.979126i \(-0.565152\pi\)
−0.203253 + 0.979126i \(0.565152\pi\)
\(192\) 0 0
\(193\) 20.1246 1.44860 0.724301 0.689484i \(-0.242163\pi\)
0.724301 + 0.689484i \(0.242163\pi\)
\(194\) 0 0
\(195\) 1.85410 0.132775
\(196\) 0 0
\(197\) −16.4164 −1.16962 −0.584810 0.811170i \(-0.698831\pi\)
−0.584810 + 0.811170i \(0.698831\pi\)
\(198\) 0 0
\(199\) 23.4164 1.65995 0.829973 0.557804i \(-0.188356\pi\)
0.829973 + 0.557804i \(0.188356\pi\)
\(200\) 0 0
\(201\) −14.4164 −1.01686
\(202\) 0 0
\(203\) −5.23607 −0.367500
\(204\) 0 0
\(205\) −3.41641 −0.238612
\(206\) 0 0
\(207\) 8.94427 0.621670
\(208\) 0 0
\(209\) −15.3262 −1.06014
\(210\) 0 0
\(211\) −14.8541 −1.02260 −0.511299 0.859403i \(-0.670836\pi\)
−0.511299 + 0.859403i \(0.670836\pi\)
\(212\) 0 0
\(213\) 4.09017 0.280254
\(214\) 0 0
\(215\) −3.32624 −0.226848
\(216\) 0 0
\(217\) 6.70820 0.455383
\(218\) 0 0
\(219\) −10.8541 −0.733452
\(220\) 0 0
\(221\) 27.2705 1.83441
\(222\) 0 0
\(223\) −7.70820 −0.516180 −0.258090 0.966121i \(-0.583093\pi\)
−0.258090 + 0.966121i \(0.583093\pi\)
\(224\) 0 0
\(225\) 9.70820 0.647214
\(226\) 0 0
\(227\) −2.94427 −0.195418 −0.0977091 0.995215i \(-0.531151\pi\)
−0.0977091 + 0.995215i \(0.531151\pi\)
\(228\) 0 0
\(229\) 6.70820 0.443291 0.221645 0.975127i \(-0.428857\pi\)
0.221645 + 0.975127i \(0.428857\pi\)
\(230\) 0 0
\(231\) 2.61803 0.172254
\(232\) 0 0
\(233\) −26.8885 −1.76153 −0.880764 0.473556i \(-0.842970\pi\)
−0.880764 + 0.473556i \(0.842970\pi\)
\(234\) 0 0
\(235\) 1.56231 0.101914
\(236\) 0 0
\(237\) 6.56231 0.426268
\(238\) 0 0
\(239\) −8.67376 −0.561059 −0.280530 0.959845i \(-0.590510\pi\)
−0.280530 + 0.959845i \(0.590510\pi\)
\(240\) 0 0
\(241\) −8.27051 −0.532750 −0.266375 0.963869i \(-0.585826\pi\)
−0.266375 + 0.963869i \(0.585826\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 2.29180 0.146417
\(246\) 0 0
\(247\) 28.4164 1.80809
\(248\) 0 0
\(249\) 6.32624 0.400909
\(250\) 0 0
\(251\) −11.2361 −0.709214 −0.354607 0.935015i \(-0.615385\pi\)
−0.354607 + 0.935015i \(0.615385\pi\)
\(252\) 0 0
\(253\) −11.7082 −0.736088
\(254\) 0 0
\(255\) 2.14590 0.134381
\(256\) 0 0
\(257\) −13.4721 −0.840369 −0.420184 0.907439i \(-0.638035\pi\)
−0.420184 + 0.907439i \(0.638035\pi\)
\(258\) 0 0
\(259\) 6.70820 0.416828
\(260\) 0 0
\(261\) 10.4721 0.648209
\(262\) 0 0
\(263\) 20.6180 1.27136 0.635681 0.771952i \(-0.280719\pi\)
0.635681 + 0.771952i \(0.280719\pi\)
\(264\) 0 0
\(265\) −0.416408 −0.0255797
\(266\) 0 0
\(267\) −2.29180 −0.140256
\(268\) 0 0
\(269\) 12.3262 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(270\) 0 0
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) −4.85410 −0.293784
\(274\) 0 0
\(275\) −12.7082 −0.766334
\(276\) 0 0
\(277\) 4.70820 0.282889 0.141444 0.989946i \(-0.454825\pi\)
0.141444 + 0.989946i \(0.454825\pi\)
\(278\) 0 0
\(279\) −13.4164 −0.803219
\(280\) 0 0
\(281\) −31.4721 −1.87747 −0.938735 0.344640i \(-0.888001\pi\)
−0.938735 + 0.344640i \(0.888001\pi\)
\(282\) 0 0
\(283\) 19.7082 1.17153 0.585766 0.810481i \(-0.300794\pi\)
0.585766 + 0.810481i \(0.300794\pi\)
\(284\) 0 0
\(285\) 2.23607 0.132453
\(286\) 0 0
\(287\) 8.94427 0.527964
\(288\) 0 0
\(289\) 14.5623 0.856606
\(290\) 0 0
\(291\) −1.70820 −0.100137
\(292\) 0 0
\(293\) −21.6525 −1.26495 −0.632476 0.774580i \(-0.717961\pi\)
−0.632476 + 0.774580i \(0.717961\pi\)
\(294\) 0 0
\(295\) 2.43769 0.141928
\(296\) 0 0
\(297\) −13.0902 −0.759569
\(298\) 0 0
\(299\) 21.7082 1.25542
\(300\) 0 0
\(301\) 8.70820 0.501933
\(302\) 0 0
\(303\) 1.90983 0.109717
\(304\) 0 0
\(305\) −1.58359 −0.0906762
\(306\) 0 0
\(307\) 2.85410 0.162892 0.0814461 0.996678i \(-0.474046\pi\)
0.0814461 + 0.996678i \(0.474046\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −2.88854 −0.163794 −0.0818971 0.996641i \(-0.526098\pi\)
−0.0818971 + 0.996641i \(0.526098\pi\)
\(312\) 0 0
\(313\) 16.2918 0.920867 0.460433 0.887694i \(-0.347694\pi\)
0.460433 + 0.887694i \(0.347694\pi\)
\(314\) 0 0
\(315\) 0.763932 0.0430427
\(316\) 0 0
\(317\) 28.4164 1.59602 0.798012 0.602641i \(-0.205885\pi\)
0.798012 + 0.602641i \(0.205885\pi\)
\(318\) 0 0
\(319\) −13.7082 −0.767512
\(320\) 0 0
\(321\) 7.09017 0.395735
\(322\) 0 0
\(323\) 32.8885 1.82997
\(324\) 0 0
\(325\) 23.5623 1.30700
\(326\) 0 0
\(327\) −9.56231 −0.528797
\(328\) 0 0
\(329\) −4.09017 −0.225498
\(330\) 0 0
\(331\) 16.1459 0.887459 0.443729 0.896161i \(-0.353655\pi\)
0.443729 + 0.896161i \(0.353655\pi\)
\(332\) 0 0
\(333\) −13.4164 −0.735215
\(334\) 0 0
\(335\) 5.50658 0.300856
\(336\) 0 0
\(337\) −2.43769 −0.132790 −0.0663948 0.997793i \(-0.521150\pi\)
−0.0663948 + 0.997793i \(0.521150\pi\)
\(338\) 0 0
\(339\) −15.0000 −0.814688
\(340\) 0 0
\(341\) 17.5623 0.951052
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 1.70820 0.0919666
\(346\) 0 0
\(347\) 1.47214 0.0790284 0.0395142 0.999219i \(-0.487419\pi\)
0.0395142 + 0.999219i \(0.487419\pi\)
\(348\) 0 0
\(349\) 15.4164 0.825221 0.412611 0.910907i \(-0.364617\pi\)
0.412611 + 0.910907i \(0.364617\pi\)
\(350\) 0 0
\(351\) 24.2705 1.29546
\(352\) 0 0
\(353\) −25.0344 −1.33245 −0.666224 0.745751i \(-0.732091\pi\)
−0.666224 + 0.745751i \(0.732091\pi\)
\(354\) 0 0
\(355\) −1.56231 −0.0829186
\(356\) 0 0
\(357\) −5.61803 −0.297338
\(358\) 0 0
\(359\) −14.6738 −0.774452 −0.387226 0.921985i \(-0.626567\pi\)
−0.387226 + 0.921985i \(0.626567\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) 0 0
\(363\) −4.14590 −0.217603
\(364\) 0 0
\(365\) 4.14590 0.217006
\(366\) 0 0
\(367\) −16.4377 −0.858041 −0.429020 0.903295i \(-0.641141\pi\)
−0.429020 + 0.903295i \(0.641141\pi\)
\(368\) 0 0
\(369\) −17.8885 −0.931240
\(370\) 0 0
\(371\) 1.09017 0.0565988
\(372\) 0 0
\(373\) −22.6869 −1.17468 −0.587342 0.809339i \(-0.699826\pi\)
−0.587342 + 0.809339i \(0.699826\pi\)
\(374\) 0 0
\(375\) 3.76393 0.194369
\(376\) 0 0
\(377\) 25.4164 1.30901
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −15.2705 −0.782332
\(382\) 0 0
\(383\) −0.819660 −0.0418827 −0.0209413 0.999781i \(-0.506666\pi\)
−0.0209413 + 0.999781i \(0.506666\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −17.4164 −0.885326
\(388\) 0 0
\(389\) 7.41641 0.376027 0.188013 0.982166i \(-0.439795\pi\)
0.188013 + 0.982166i \(0.439795\pi\)
\(390\) 0 0
\(391\) 25.1246 1.27061
\(392\) 0 0
\(393\) 2.23607 0.112795
\(394\) 0 0
\(395\) −2.50658 −0.126120
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 0 0
\(399\) −5.85410 −0.293072
\(400\) 0 0
\(401\) −23.8885 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(402\) 0 0
\(403\) −32.5623 −1.62204
\(404\) 0 0
\(405\) −0.381966 −0.0189800
\(406\) 0 0
\(407\) 17.5623 0.870531
\(408\) 0 0
\(409\) 36.7082 1.81510 0.907552 0.419940i \(-0.137949\pi\)
0.907552 + 0.419940i \(0.137949\pi\)
\(410\) 0 0
\(411\) −12.7082 −0.626849
\(412\) 0 0
\(413\) −6.38197 −0.314036
\(414\) 0 0
\(415\) −2.41641 −0.118617
\(416\) 0 0
\(417\) 11.1459 0.545817
\(418\) 0 0
\(419\) 4.09017 0.199818 0.0999089 0.994997i \(-0.468145\pi\)
0.0999089 + 0.994997i \(0.468145\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) 0 0
\(423\) 8.18034 0.397742
\(424\) 0 0
\(425\) 27.2705 1.32281
\(426\) 0 0
\(427\) 4.14590 0.200634
\(428\) 0 0
\(429\) −12.7082 −0.613558
\(430\) 0 0
\(431\) −34.3607 −1.65510 −0.827548 0.561395i \(-0.810265\pi\)
−0.827548 + 0.561395i \(0.810265\pi\)
\(432\) 0 0
\(433\) 14.4164 0.692808 0.346404 0.938085i \(-0.387403\pi\)
0.346404 + 0.938085i \(0.387403\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) 26.1803 1.25238
\(438\) 0 0
\(439\) −29.5623 −1.41093 −0.705466 0.708744i \(-0.749262\pi\)
−0.705466 + 0.708744i \(0.749262\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 15.4377 0.733467 0.366733 0.930326i \(-0.380476\pi\)
0.366733 + 0.930326i \(0.380476\pi\)
\(444\) 0 0
\(445\) 0.875388 0.0414974
\(446\) 0 0
\(447\) −7.47214 −0.353420
\(448\) 0 0
\(449\) −13.3607 −0.630529 −0.315265 0.949004i \(-0.602093\pi\)
−0.315265 + 0.949004i \(0.602093\pi\)
\(450\) 0 0
\(451\) 23.4164 1.10264
\(452\) 0 0
\(453\) 19.0000 0.892698
\(454\) 0 0
\(455\) 1.85410 0.0869216
\(456\) 0 0
\(457\) −9.85410 −0.460955 −0.230478 0.973078i \(-0.574029\pi\)
−0.230478 + 0.973078i \(0.574029\pi\)
\(458\) 0 0
\(459\) 28.0902 1.31114
\(460\) 0 0
\(461\) 12.2148 0.568899 0.284450 0.958691i \(-0.408189\pi\)
0.284450 + 0.958691i \(0.408189\pi\)
\(462\) 0 0
\(463\) 21.4164 0.995305 0.497652 0.867377i \(-0.334196\pi\)
0.497652 + 0.867377i \(0.334196\pi\)
\(464\) 0 0
\(465\) −2.56231 −0.118824
\(466\) 0 0
\(467\) 3.65248 0.169016 0.0845082 0.996423i \(-0.473068\pi\)
0.0845082 + 0.996423i \(0.473068\pi\)
\(468\) 0 0
\(469\) −14.4164 −0.665688
\(470\) 0 0
\(471\) 16.7082 0.769873
\(472\) 0 0
\(473\) 22.7984 1.04827
\(474\) 0 0
\(475\) 28.4164 1.30383
\(476\) 0 0
\(477\) −2.18034 −0.0998309
\(478\) 0 0
\(479\) −8.18034 −0.373769 −0.186885 0.982382i \(-0.559839\pi\)
−0.186885 + 0.982382i \(0.559839\pi\)
\(480\) 0 0
\(481\) −32.5623 −1.48471
\(482\) 0 0
\(483\) −4.47214 −0.203489
\(484\) 0 0
\(485\) 0.652476 0.0296274
\(486\) 0 0
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 0 0
\(489\) −2.70820 −0.122469
\(490\) 0 0
\(491\) −35.7771 −1.61460 −0.807299 0.590143i \(-0.799071\pi\)
−0.807299 + 0.590143i \(0.799071\pi\)
\(492\) 0 0
\(493\) 29.4164 1.32485
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 4.09017 0.183469
\(498\) 0 0
\(499\) −13.2705 −0.594070 −0.297035 0.954867i \(-0.595998\pi\)
−0.297035 + 0.954867i \(0.595998\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) −13.3607 −0.595723 −0.297862 0.954609i \(-0.596273\pi\)
−0.297862 + 0.954609i \(0.596273\pi\)
\(504\) 0 0
\(505\) −0.729490 −0.0324619
\(506\) 0 0
\(507\) 10.5623 0.469088
\(508\) 0 0
\(509\) 26.6180 1.17982 0.589912 0.807468i \(-0.299163\pi\)
0.589912 + 0.807468i \(0.299163\pi\)
\(510\) 0 0
\(511\) −10.8541 −0.480157
\(512\) 0 0
\(513\) 29.2705 1.29232
\(514\) 0 0
\(515\) −0.381966 −0.0168314
\(516\) 0 0
\(517\) −10.7082 −0.470946
\(518\) 0 0
\(519\) 16.0344 0.703834
\(520\) 0 0
\(521\) 17.1803 0.752684 0.376342 0.926481i \(-0.377182\pi\)
0.376342 + 0.926481i \(0.377182\pi\)
\(522\) 0 0
\(523\) 10.5836 0.462788 0.231394 0.972860i \(-0.425671\pi\)
0.231394 + 0.972860i \(0.425671\pi\)
\(524\) 0 0
\(525\) −4.85410 −0.211850
\(526\) 0 0
\(527\) −37.6869 −1.64167
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 0 0
\(531\) 12.7639 0.553907
\(532\) 0 0
\(533\) −43.4164 −1.88057
\(534\) 0 0
\(535\) −2.70820 −0.117086
\(536\) 0 0
\(537\) 7.85410 0.338930
\(538\) 0 0
\(539\) −15.7082 −0.676600
\(540\) 0 0
\(541\) 9.14590 0.393213 0.196606 0.980482i \(-0.437008\pi\)
0.196606 + 0.980482i \(0.437008\pi\)
\(542\) 0 0
\(543\) 3.85410 0.165395
\(544\) 0 0
\(545\) 3.65248 0.156455
\(546\) 0 0
\(547\) 2.27051 0.0970800 0.0485400 0.998821i \(-0.484543\pi\)
0.0485400 + 0.998821i \(0.484543\pi\)
\(548\) 0 0
\(549\) −8.29180 −0.353885
\(550\) 0 0
\(551\) 30.6525 1.30584
\(552\) 0 0
\(553\) 6.56231 0.279058
\(554\) 0 0
\(555\) −2.56231 −0.108764
\(556\) 0 0
\(557\) −31.6869 −1.34262 −0.671309 0.741178i \(-0.734268\pi\)
−0.671309 + 0.741178i \(0.734268\pi\)
\(558\) 0 0
\(559\) −42.2705 −1.78785
\(560\) 0 0
\(561\) −14.7082 −0.620981
\(562\) 0 0
\(563\) 1.20163 0.0506425 0.0253213 0.999679i \(-0.491939\pi\)
0.0253213 + 0.999679i \(0.491939\pi\)
\(564\) 0 0
\(565\) 5.72949 0.241041
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −27.1591 −1.13857 −0.569283 0.822141i \(-0.692779\pi\)
−0.569283 + 0.822141i \(0.692779\pi\)
\(570\) 0 0
\(571\) 22.5623 0.944203 0.472102 0.881544i \(-0.343496\pi\)
0.472102 + 0.881544i \(0.343496\pi\)
\(572\) 0 0
\(573\) −5.61803 −0.234697
\(574\) 0 0
\(575\) 21.7082 0.905295
\(576\) 0 0
\(577\) −36.8328 −1.53337 −0.766685 0.642023i \(-0.778095\pi\)
−0.766685 + 0.642023i \(0.778095\pi\)
\(578\) 0 0
\(579\) 20.1246 0.836350
\(580\) 0 0
\(581\) 6.32624 0.262457
\(582\) 0 0
\(583\) 2.85410 0.118205
\(584\) 0 0
\(585\) −3.70820 −0.153315
\(586\) 0 0
\(587\) 47.0689 1.94274 0.971370 0.237570i \(-0.0763510\pi\)
0.971370 + 0.237570i \(0.0763510\pi\)
\(588\) 0 0
\(589\) −39.2705 −1.61811
\(590\) 0 0
\(591\) −16.4164 −0.675281
\(592\) 0 0
\(593\) 2.18034 0.0895358 0.0447679 0.998997i \(-0.485745\pi\)
0.0447679 + 0.998997i \(0.485745\pi\)
\(594\) 0 0
\(595\) 2.14590 0.0879732
\(596\) 0 0
\(597\) 23.4164 0.958370
\(598\) 0 0
\(599\) 35.4508 1.44848 0.724241 0.689547i \(-0.242190\pi\)
0.724241 + 0.689547i \(0.242190\pi\)
\(600\) 0 0
\(601\) 3.56231 0.145309 0.0726547 0.997357i \(-0.476853\pi\)
0.0726547 + 0.997357i \(0.476853\pi\)
\(602\) 0 0
\(603\) 28.8328 1.17416
\(604\) 0 0
\(605\) 1.58359 0.0643822
\(606\) 0 0
\(607\) −5.70820 −0.231689 −0.115844 0.993267i \(-0.536957\pi\)
−0.115844 + 0.993267i \(0.536957\pi\)
\(608\) 0 0
\(609\) −5.23607 −0.212176
\(610\) 0 0
\(611\) 19.8541 0.803211
\(612\) 0 0
\(613\) 24.4164 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(614\) 0 0
\(615\) −3.41641 −0.137763
\(616\) 0 0
\(617\) 3.27051 0.131666 0.0658329 0.997831i \(-0.479030\pi\)
0.0658329 + 0.997831i \(0.479030\pi\)
\(618\) 0 0
\(619\) 28.6869 1.15302 0.576512 0.817088i \(-0.304413\pi\)
0.576512 + 0.817088i \(0.304413\pi\)
\(620\) 0 0
\(621\) 22.3607 0.897303
\(622\) 0 0
\(623\) −2.29180 −0.0918189
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 0 0
\(627\) −15.3262 −0.612071
\(628\) 0 0
\(629\) −37.6869 −1.50268
\(630\) 0 0
\(631\) −42.2705 −1.68276 −0.841381 0.540442i \(-0.818257\pi\)
−0.841381 + 0.540442i \(0.818257\pi\)
\(632\) 0 0
\(633\) −14.8541 −0.590398
\(634\) 0 0
\(635\) 5.83282 0.231468
\(636\) 0 0
\(637\) 29.1246 1.15396
\(638\) 0 0
\(639\) −8.18034 −0.323609
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −3.32624 −0.130970
\(646\) 0 0
\(647\) −1.25735 −0.0494317 −0.0247158 0.999695i \(-0.507868\pi\)
−0.0247158 + 0.999695i \(0.507868\pi\)
\(648\) 0 0
\(649\) −16.7082 −0.655854
\(650\) 0 0
\(651\) 6.70820 0.262915
\(652\) 0 0
\(653\) −5.23607 −0.204903 −0.102452 0.994738i \(-0.532669\pi\)
−0.102452 + 0.994738i \(0.532669\pi\)
\(654\) 0 0
\(655\) −0.854102 −0.0333725
\(656\) 0 0
\(657\) 21.7082 0.846918
\(658\) 0 0
\(659\) −36.5967 −1.42561 −0.712803 0.701364i \(-0.752575\pi\)
−0.712803 + 0.701364i \(0.752575\pi\)
\(660\) 0 0
\(661\) −31.5623 −1.22763 −0.613816 0.789449i \(-0.710366\pi\)
−0.613816 + 0.789449i \(0.710366\pi\)
\(662\) 0 0
\(663\) 27.2705 1.05910
\(664\) 0 0
\(665\) 2.23607 0.0867110
\(666\) 0 0
\(667\) 23.4164 0.906687
\(668\) 0 0
\(669\) −7.70820 −0.298016
\(670\) 0 0
\(671\) 10.8541 0.419018
\(672\) 0 0
\(673\) −24.2918 −0.936380 −0.468190 0.883628i \(-0.655094\pi\)
−0.468190 + 0.883628i \(0.655094\pi\)
\(674\) 0 0
\(675\) 24.2705 0.934172
\(676\) 0 0
\(677\) 40.0344 1.53865 0.769324 0.638858i \(-0.220593\pi\)
0.769324 + 0.638858i \(0.220593\pi\)
\(678\) 0 0
\(679\) −1.70820 −0.0655549
\(680\) 0 0
\(681\) −2.94427 −0.112825
\(682\) 0 0
\(683\) −47.2361 −1.80744 −0.903719 0.428126i \(-0.859174\pi\)
−0.903719 + 0.428126i \(0.859174\pi\)
\(684\) 0 0
\(685\) 4.85410 0.185466
\(686\) 0 0
\(687\) 6.70820 0.255934
\(688\) 0 0
\(689\) −5.29180 −0.201601
\(690\) 0 0
\(691\) −7.85410 −0.298784 −0.149392 0.988778i \(-0.547732\pi\)
−0.149392 + 0.988778i \(0.547732\pi\)
\(692\) 0 0
\(693\) −5.23607 −0.198902
\(694\) 0 0
\(695\) −4.25735 −0.161491
\(696\) 0 0
\(697\) −50.2492 −1.90333
\(698\) 0 0
\(699\) −26.8885 −1.01702
\(700\) 0 0
\(701\) −25.7984 −0.974391 −0.487196 0.873293i \(-0.661980\pi\)
−0.487196 + 0.873293i \(0.661980\pi\)
\(702\) 0 0
\(703\) −39.2705 −1.48112
\(704\) 0 0
\(705\) 1.56231 0.0588398
\(706\) 0 0
\(707\) 1.90983 0.0718266
\(708\) 0 0
\(709\) −1.02129 −0.0383552 −0.0191776 0.999816i \(-0.506105\pi\)
−0.0191776 + 0.999816i \(0.506105\pi\)
\(710\) 0 0
\(711\) −13.1246 −0.492211
\(712\) 0 0
\(713\) −30.0000 −1.12351
\(714\) 0 0
\(715\) 4.85410 0.181533
\(716\) 0 0
\(717\) −8.67376 −0.323928
\(718\) 0 0
\(719\) −39.3262 −1.46662 −0.733311 0.679894i \(-0.762026\pi\)
−0.733311 + 0.679894i \(0.762026\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 0 0
\(723\) −8.27051 −0.307584
\(724\) 0 0
\(725\) 25.4164 0.943942
\(726\) 0 0
\(727\) 4.72949 0.175407 0.0877035 0.996147i \(-0.472047\pi\)
0.0877035 + 0.996147i \(0.472047\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −48.9230 −1.80948
\(732\) 0 0
\(733\) 14.7082 0.543260 0.271630 0.962402i \(-0.412437\pi\)
0.271630 + 0.962402i \(0.412437\pi\)
\(734\) 0 0
\(735\) 2.29180 0.0845342
\(736\) 0 0
\(737\) −37.7426 −1.39027
\(738\) 0 0
\(739\) 16.8328 0.619205 0.309603 0.950866i \(-0.399804\pi\)
0.309603 + 0.950866i \(0.399804\pi\)
\(740\) 0 0
\(741\) 28.4164 1.04390
\(742\) 0 0
\(743\) −30.7082 −1.12657 −0.563287 0.826261i \(-0.690464\pi\)
−0.563287 + 0.826261i \(0.690464\pi\)
\(744\) 0 0
\(745\) 2.85410 0.104566
\(746\) 0 0
\(747\) −12.6525 −0.462930
\(748\) 0 0
\(749\) 7.09017 0.259069
\(750\) 0 0
\(751\) 44.1246 1.61013 0.805065 0.593187i \(-0.202130\pi\)
0.805065 + 0.593187i \(0.202130\pi\)
\(752\) 0 0
\(753\) −11.2361 −0.409465
\(754\) 0 0
\(755\) −7.25735 −0.264122
\(756\) 0 0
\(757\) −21.2918 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(758\) 0 0
\(759\) −11.7082 −0.424981
\(760\) 0 0
\(761\) −7.47214 −0.270865 −0.135432 0.990787i \(-0.543242\pi\)
−0.135432 + 0.990787i \(0.543242\pi\)
\(762\) 0 0
\(763\) −9.56231 −0.346179
\(764\) 0 0
\(765\) −4.29180 −0.155170
\(766\) 0 0
\(767\) 30.9787 1.11858
\(768\) 0 0
\(769\) 31.3951 1.13214 0.566069 0.824358i \(-0.308464\pi\)
0.566069 + 0.824358i \(0.308464\pi\)
\(770\) 0 0
\(771\) −13.4721 −0.485187
\(772\) 0 0
\(773\) −16.5279 −0.594466 −0.297233 0.954805i \(-0.596064\pi\)
−0.297233 + 0.954805i \(0.596064\pi\)
\(774\) 0 0
\(775\) −32.5623 −1.16967
\(776\) 0 0
\(777\) 6.70820 0.240655
\(778\) 0 0
\(779\) −52.3607 −1.87602
\(780\) 0 0
\(781\) 10.7082 0.383170
\(782\) 0 0
\(783\) 26.1803 0.935609
\(784\) 0 0
\(785\) −6.38197 −0.227782
\(786\) 0 0
\(787\) −43.4164 −1.54763 −0.773814 0.633413i \(-0.781653\pi\)
−0.773814 + 0.633413i \(0.781653\pi\)
\(788\) 0 0
\(789\) 20.6180 0.734021
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) −20.1246 −0.714646
\(794\) 0 0
\(795\) −0.416408 −0.0147685
\(796\) 0 0
\(797\) −50.1246 −1.77550 −0.887752 0.460321i \(-0.847734\pi\)
−0.887752 + 0.460321i \(0.847734\pi\)
\(798\) 0 0
\(799\) 22.9787 0.812928
\(800\) 0 0
\(801\) 4.58359 0.161953
\(802\) 0 0
\(803\) −28.4164 −1.00279
\(804\) 0 0
\(805\) 1.70820 0.0602063
\(806\) 0 0
\(807\) 12.3262 0.433904
\(808\) 0 0
\(809\) 2.94427 0.103515 0.0517575 0.998660i \(-0.483518\pi\)
0.0517575 + 0.998660i \(0.483518\pi\)
\(810\) 0 0
\(811\) −21.5410 −0.756408 −0.378204 0.925722i \(-0.623458\pi\)
−0.378204 + 0.925722i \(0.623458\pi\)
\(812\) 0 0
\(813\) −1.00000 −0.0350715
\(814\) 0 0
\(815\) 1.03444 0.0362349
\(816\) 0 0
\(817\) −50.9787 −1.78352
\(818\) 0 0
\(819\) 9.70820 0.339232
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) −3.43769 −0.119830 −0.0599152 0.998203i \(-0.519083\pi\)
−0.0599152 + 0.998203i \(0.519083\pi\)
\(824\) 0 0
\(825\) −12.7082 −0.442443
\(826\) 0 0
\(827\) −6.70820 −0.233267 −0.116634 0.993175i \(-0.537210\pi\)
−0.116634 + 0.993175i \(0.537210\pi\)
\(828\) 0 0
\(829\) 14.2705 0.495635 0.247818 0.968807i \(-0.420287\pi\)
0.247818 + 0.968807i \(0.420287\pi\)
\(830\) 0 0
\(831\) 4.70820 0.163326
\(832\) 0 0
\(833\) 33.7082 1.16792
\(834\) 0 0
\(835\) 3.43769 0.118966
\(836\) 0 0
\(837\) −33.5410 −1.15935
\(838\) 0 0
\(839\) −23.6180 −0.815385 −0.407693 0.913119i \(-0.633666\pi\)
−0.407693 + 0.913119i \(0.633666\pi\)
\(840\) 0 0
\(841\) −1.58359 −0.0546066
\(842\) 0 0
\(843\) −31.4721 −1.08396
\(844\) 0 0
\(845\) −4.03444 −0.138789
\(846\) 0 0
\(847\) −4.14590 −0.142455
\(848\) 0 0
\(849\) 19.7082 0.676384
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) 0 0
\(853\) −44.2705 −1.51579 −0.757897 0.652375i \(-0.773773\pi\)
−0.757897 + 0.652375i \(0.773773\pi\)
\(854\) 0 0
\(855\) −4.47214 −0.152944
\(856\) 0 0
\(857\) 8.23607 0.281339 0.140669 0.990057i \(-0.455075\pi\)
0.140669 + 0.990057i \(0.455075\pi\)
\(858\) 0 0
\(859\) −10.5623 −0.360381 −0.180191 0.983632i \(-0.557671\pi\)
−0.180191 + 0.983632i \(0.557671\pi\)
\(860\) 0 0
\(861\) 8.94427 0.304820
\(862\) 0 0
\(863\) 21.7082 0.738956 0.369478 0.929240i \(-0.379537\pi\)
0.369478 + 0.929240i \(0.379537\pi\)
\(864\) 0 0
\(865\) −6.12461 −0.208243
\(866\) 0 0
\(867\) 14.5623 0.494562
\(868\) 0 0
\(869\) 17.1803 0.582803
\(870\) 0 0
\(871\) 69.9787 2.37114
\(872\) 0 0
\(873\) 3.41641 0.115628
\(874\) 0 0
\(875\) 3.76393 0.127244
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 0 0
\(879\) −21.6525 −0.730320
\(880\) 0 0
\(881\) 29.8885 1.00697 0.503485 0.864004i \(-0.332051\pi\)
0.503485 + 0.864004i \(0.332051\pi\)
\(882\) 0 0
\(883\) −5.87539 −0.197723 −0.0988613 0.995101i \(-0.531520\pi\)
−0.0988613 + 0.995101i \(0.531520\pi\)
\(884\) 0 0
\(885\) 2.43769 0.0819422
\(886\) 0 0
\(887\) −40.1935 −1.34957 −0.674783 0.738016i \(-0.735763\pi\)
−0.674783 + 0.738016i \(0.735763\pi\)
\(888\) 0 0
\(889\) −15.2705 −0.512156
\(890\) 0 0
\(891\) 2.61803 0.0877074
\(892\) 0 0
\(893\) 23.9443 0.801265
\(894\) 0 0
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 21.7082 0.724816
\(898\) 0 0
\(899\) −35.1246 −1.17147
\(900\) 0 0
\(901\) −6.12461 −0.204040
\(902\) 0 0
\(903\) 8.70820 0.289791
\(904\) 0 0
\(905\) −1.47214 −0.0489355
\(906\) 0 0
\(907\) 7.12461 0.236569 0.118284 0.992980i \(-0.462261\pi\)
0.118284 + 0.992980i \(0.462261\pi\)
\(908\) 0 0
\(909\) −3.81966 −0.126690
\(910\) 0 0
\(911\) 10.0344 0.332456 0.166228 0.986087i \(-0.446841\pi\)
0.166228 + 0.986087i \(0.446841\pi\)
\(912\) 0 0
\(913\) 16.5623 0.548132
\(914\) 0 0
\(915\) −1.58359 −0.0523519
\(916\) 0 0
\(917\) 2.23607 0.0738415
\(918\) 0 0
\(919\) 27.9787 0.922933 0.461466 0.887158i \(-0.347324\pi\)
0.461466 + 0.887158i \(0.347324\pi\)
\(920\) 0 0
\(921\) 2.85410 0.0940459
\(922\) 0 0
\(923\) −19.8541 −0.653506
\(924\) 0 0
\(925\) −32.5623 −1.07064
\(926\) 0 0
\(927\) −2.00000 −0.0656886
\(928\) 0 0
\(929\) 14.9443 0.490306 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(930\) 0 0
\(931\) 35.1246 1.15116
\(932\) 0 0
\(933\) −2.88854 −0.0945667
\(934\) 0 0
\(935\) 5.61803 0.183729
\(936\) 0 0
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) 0 0
\(939\) 16.2918 0.531663
\(940\) 0 0
\(941\) −23.3951 −0.762659 −0.381330 0.924439i \(-0.624534\pi\)
−0.381330 + 0.924439i \(0.624534\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 0 0
\(945\) 1.90983 0.0621268
\(946\) 0 0
\(947\) 41.0132 1.33275 0.666374 0.745617i \(-0.267845\pi\)
0.666374 + 0.745617i \(0.267845\pi\)
\(948\) 0 0
\(949\) 52.6869 1.71029
\(950\) 0 0
\(951\) 28.4164 0.921465
\(952\) 0 0
\(953\) 13.3607 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(954\) 0 0
\(955\) 2.14590 0.0694396
\(956\) 0 0
\(957\) −13.7082 −0.443123
\(958\) 0 0
\(959\) −12.7082 −0.410369
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) −14.1803 −0.456955
\(964\) 0 0
\(965\) −7.68692 −0.247451
\(966\) 0 0
\(967\) 14.4164 0.463600 0.231800 0.972763i \(-0.425538\pi\)
0.231800 + 0.972763i \(0.425538\pi\)
\(968\) 0 0
\(969\) 32.8885 1.05653
\(970\) 0 0
\(971\) 53.0132 1.70127 0.850637 0.525754i \(-0.176217\pi\)
0.850637 + 0.525754i \(0.176217\pi\)
\(972\) 0 0
\(973\) 11.1459 0.357321
\(974\) 0 0
\(975\) 23.5623 0.754598
\(976\) 0 0
\(977\) 46.7426 1.49543 0.747715 0.664020i \(-0.231151\pi\)
0.747715 + 0.664020i \(0.231151\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 19.1246 0.610602
\(982\) 0 0
\(983\) −18.6525 −0.594922 −0.297461 0.954734i \(-0.596140\pi\)
−0.297461 + 0.954734i \(0.596140\pi\)
\(984\) 0 0
\(985\) 6.27051 0.199795
\(986\) 0 0
\(987\) −4.09017 −0.130192
\(988\) 0 0
\(989\) −38.9443 −1.23836
\(990\) 0 0
\(991\) −24.2705 −0.770978 −0.385489 0.922712i \(-0.625967\pi\)
−0.385489 + 0.922712i \(0.625967\pi\)
\(992\) 0 0
\(993\) 16.1459 0.512375
\(994\) 0 0
\(995\) −8.94427 −0.283552
\(996\) 0 0
\(997\) −39.2918 −1.24438 −0.622192 0.782865i \(-0.713758\pi\)
−0.622192 + 0.782865i \(0.713758\pi\)
\(998\) 0 0
\(999\) −33.5410 −1.06119
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 1648.2.a.f.1.2 2
4.3 odd 2 103.2.a.a.1.1 2
8.3 odd 2 6592.2.a.t.1.1 2
8.5 even 2 6592.2.a.h.1.1 2
12.11 even 2 927.2.a.b.1.2 2
20.19 odd 2 2575.2.a.g.1.2 2
28.27 even 2 5047.2.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 4.3 odd 2
927.2.a.b.1.2 2 12.11 even 2
1648.2.a.f.1.2 2 1.1 even 1 trivial
2575.2.a.g.1.2 2 20.19 odd 2
5047.2.a.a.1.1 2 28.27 even 2
6592.2.a.h.1.1 2 8.5 even 2
6592.2.a.t.1.1 2 8.3 odd 2