Properties

Label 25.22.b.a.24.2
Level $25$
Weight $22$
Character 25.24
Analytic conductor $69.869$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,22,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(69.8693360718\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.22.b.a.24.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+288.000i q^{2} -128844. i q^{3} +2.01421e6 q^{4} +3.71071e7 q^{6} +7.68079e8i q^{7} +1.18407e9i q^{8} -6.14042e9 q^{9} +O(q^{10})\) \(q+288.000i q^{2} -128844. i q^{3} +2.01421e6 q^{4} +3.71071e7 q^{6} +7.68079e8i q^{7} +1.18407e9i q^{8} -6.14042e9 q^{9} -9.47249e10 q^{11} -2.59519e11i q^{12} -8.06218e10i q^{13} -2.21207e11 q^{14} +3.88309e12 q^{16} -3.05228e12i q^{17} -1.76844e12i q^{18} +7.92079e12 q^{19} +9.89623e13 q^{21} -2.72808e13i q^{22} -7.38454e13i q^{23} +1.52561e14 q^{24} +2.32191e13 q^{26} -5.56597e14i q^{27} +1.54707e15i q^{28} +4.25303e15 q^{29} +1.90054e15 q^{31} +3.60151e15i q^{32} +1.22047e16i q^{33} +8.79057e14 q^{34} -1.23681e16 q^{36} -2.21914e16i q^{37} +2.28119e15i q^{38} -1.03876e16 q^{39} -2.06228e16 q^{41} +2.85012e16i q^{42} -1.93606e17i q^{43} -1.90796e17 q^{44} +2.12675e16 q^{46} -1.46961e17i q^{47} -5.00313e17i q^{48} -3.13992e16 q^{49} -3.93268e17 q^{51} -1.62389e17i q^{52} +2.03827e18i q^{53} +1.60300e17 q^{54} -9.09460e17 q^{56} -1.02055e18i q^{57} +1.22487e18i q^{58} +5.97588e18 q^{59} +6.19062e18 q^{61} +5.47356e17i q^{62} -4.71633e18i q^{63} +7.10619e18 q^{64} -3.51496e18 q^{66} -1.69613e19i q^{67} -6.14793e18i q^{68} -9.51454e18 q^{69} -5.63276e18 q^{71} -7.27070e18i q^{72} -4.32848e19i q^{73} +6.39113e18 q^{74} +1.59541e19 q^{76} -7.27562e19i q^{77} -2.99164e18i q^{78} +5.12649e19 q^{79} -1.35945e20 q^{81} -5.93937e18i q^{82} +4.89119e19i q^{83} +1.99331e20 q^{84} +5.57585e19 q^{86} -5.47978e20i q^{87} -1.12161e20i q^{88} +5.04303e20 q^{89} +6.19239e19 q^{91} -1.48740e20i q^{92} -2.44873e20i q^{93} +4.23246e19 q^{94} +4.64033e20 q^{96} -8.08275e20i q^{97} -9.04297e18i q^{98} +5.81651e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4028416 q^{4} + 74214144 q^{6} - 12280846266 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4028416 q^{4} + 74214144 q^{6} - 12280846266 q^{9} - 189449858376 q^{11} - 442413393408 q^{14} + 7766175383552 q^{16} + 15841576703480 q^{19} + 197924691875904 q^{21} + 305121063075840 q^{24} + 46438150921344 q^{26} + 85\!\cdots\!20 q^{29}+ \cdots + 11\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 288.000i 0.198874i 0.995044 + 0.0994369i \(0.0317041\pi\)
−0.995044 + 0.0994369i \(0.968296\pi\)
\(3\) − 128844.i − 1.25977i −0.776689 0.629885i \(-0.783102\pi\)
0.776689 0.629885i \(-0.216898\pi\)
\(4\) 2.01421e6 0.960449
\(5\) 0 0
\(6\) 3.71071e7 0.250535
\(7\) 7.68079e8i 1.02772i 0.857873 + 0.513862i \(0.171786\pi\)
−0.857873 + 0.513862i \(0.828214\pi\)
\(8\) 1.18407e9i 0.389882i
\(9\) −6.14042e9 −0.587019
\(10\) 0 0
\(11\) −9.47249e10 −1.10114 −0.550568 0.834790i \(-0.685589\pi\)
−0.550568 + 0.834790i \(0.685589\pi\)
\(12\) − 2.59519e11i − 1.20994i
\(13\) − 8.06218e10i − 0.162199i −0.996706 0.0810993i \(-0.974157\pi\)
0.996706 0.0810993i \(-0.0258431\pi\)
\(14\) −2.21207e11 −0.204387
\(15\) 0 0
\(16\) 3.88309e12 0.882912
\(17\) − 3.05228e12i − 0.367207i −0.983000 0.183604i \(-0.941224\pi\)
0.983000 0.183604i \(-0.0587763\pi\)
\(18\) − 1.76844e12i − 0.116743i
\(19\) 7.92079e12 0.296385 0.148192 0.988959i \(-0.452655\pi\)
0.148192 + 0.988959i \(0.452655\pi\)
\(20\) 0 0
\(21\) 9.89623e13 1.29469
\(22\) − 2.72808e13i − 0.218987i
\(23\) − 7.38454e13i − 0.371690i −0.982579 0.185845i \(-0.940498\pi\)
0.982579 0.185845i \(-0.0595023\pi\)
\(24\) 1.52561e14 0.491161
\(25\) 0 0
\(26\) 2.32191e13 0.0322571
\(27\) − 5.56597e14i − 0.520261i
\(28\) 1.54707e15i 0.987076i
\(29\) 4.25303e15 1.87724 0.938620 0.344954i \(-0.112105\pi\)
0.938620 + 0.344954i \(0.112105\pi\)
\(30\) 0 0
\(31\) 1.90054e15 0.416466 0.208233 0.978079i \(-0.433229\pi\)
0.208233 + 0.978079i \(0.433229\pi\)
\(32\) 3.60151e15i 0.565470i
\(33\) 1.22047e16i 1.38718i
\(34\) 8.79057e14 0.0730279
\(35\) 0 0
\(36\) −1.23681e16 −0.563802
\(37\) − 2.21914e16i − 0.758695i −0.925254 0.379347i \(-0.876149\pi\)
0.925254 0.379347i \(-0.123851\pi\)
\(38\) 2.28119e15i 0.0589431i
\(39\) −1.03876e16 −0.204333
\(40\) 0 0
\(41\) −2.06228e16 −0.239948 −0.119974 0.992777i \(-0.538281\pi\)
−0.119974 + 0.992777i \(0.538281\pi\)
\(42\) 2.85012e16i 0.257481i
\(43\) − 1.93606e17i − 1.36615i −0.730346 0.683077i \(-0.760641\pi\)
0.730346 0.683077i \(-0.239359\pi\)
\(44\) −1.90796e17 −1.05759
\(45\) 0 0
\(46\) 2.12675e16 0.0739195
\(47\) − 1.46961e17i − 0.407543i −0.979019 0.203771i \(-0.934680\pi\)
0.979019 0.203771i \(-0.0653199\pi\)
\(48\) − 5.00313e17i − 1.11227i
\(49\) −3.13992e16 −0.0562160
\(50\) 0 0
\(51\) −3.93268e17 −0.462596
\(52\) − 1.62389e17i − 0.155784i
\(53\) 2.03827e18i 1.60090i 0.599399 + 0.800450i \(0.295406\pi\)
−0.599399 + 0.800450i \(0.704594\pi\)
\(54\) 1.60300e17 0.103466
\(55\) 0 0
\(56\) −9.09460e17 −0.400691
\(57\) − 1.02055e18i − 0.373376i
\(58\) 1.22487e18i 0.373334i
\(59\) 5.97588e18 1.52214 0.761072 0.648667i \(-0.224673\pi\)
0.761072 + 0.648667i \(0.224673\pi\)
\(60\) 0 0
\(61\) 6.19062e18 1.11114 0.555572 0.831468i \(-0.312499\pi\)
0.555572 + 0.831468i \(0.312499\pi\)
\(62\) 5.47356e17i 0.0828242i
\(63\) − 4.71633e18i − 0.603293i
\(64\) 7.10619e18 0.770455
\(65\) 0 0
\(66\) −3.51496e18 −0.275873
\(67\) − 1.69613e19i − 1.13677i −0.822761 0.568387i \(-0.807568\pi\)
0.822761 0.568387i \(-0.192432\pi\)
\(68\) − 6.14793e18i − 0.352684i
\(69\) −9.51454e18 −0.468244
\(70\) 0 0
\(71\) −5.63276e18 −0.205357 −0.102678 0.994715i \(-0.532741\pi\)
−0.102678 + 0.994715i \(0.532741\pi\)
\(72\) − 7.27070e18i − 0.228868i
\(73\) − 4.32848e19i − 1.17881i −0.807837 0.589407i \(-0.799362\pi\)
0.807837 0.589407i \(-0.200638\pi\)
\(74\) 6.39113e18 0.150885
\(75\) 0 0
\(76\) 1.59541e19 0.284662
\(77\) − 7.27562e19i − 1.13166i
\(78\) − 2.99164e18i − 0.0406365i
\(79\) 5.12649e19 0.609166 0.304583 0.952486i \(-0.401483\pi\)
0.304583 + 0.952486i \(0.401483\pi\)
\(80\) 0 0
\(81\) −1.35945e20 −1.24243
\(82\) − 5.93937e18i − 0.0477194i
\(83\) 4.89119e19i 0.346014i 0.984921 + 0.173007i \(0.0553484\pi\)
−0.984921 + 0.173007i \(0.944652\pi\)
\(84\) 1.99331e20 1.24349
\(85\) 0 0
\(86\) 5.57585e19 0.271692
\(87\) − 5.47978e20i − 2.36489i
\(88\) − 1.12161e20i − 0.429313i
\(89\) 5.04303e20 1.71434 0.857170 0.515034i \(-0.172221\pi\)
0.857170 + 0.515034i \(0.172221\pi\)
\(90\) 0 0
\(91\) 6.19239e19 0.166695
\(92\) − 1.48740e20i − 0.356990i
\(93\) − 2.44873e20i − 0.524651i
\(94\) 4.23246e19 0.0810496
\(95\) 0 0
\(96\) 4.64033e20 0.712362
\(97\) − 8.08275e20i − 1.11290i −0.830881 0.556450i \(-0.812163\pi\)
0.830881 0.556450i \(-0.187837\pi\)
\(98\) − 9.04297e18i − 0.0111799i
\(99\) 5.81651e20 0.646388
\(100\) 0 0
\(101\) −1.00202e21 −0.902612 −0.451306 0.892369i \(-0.649042\pi\)
−0.451306 + 0.892369i \(0.649042\pi\)
\(102\) − 1.13261e20i − 0.0919983i
\(103\) − 5.89747e20i − 0.432389i −0.976350 0.216195i \(-0.930635\pi\)
0.976350 0.216195i \(-0.0693646\pi\)
\(104\) 9.54620e19 0.0632383
\(105\) 0 0
\(106\) −5.87021e20 −0.318377
\(107\) − 1.12210e21i − 0.551445i −0.961237 0.275723i \(-0.911083\pi\)
0.961237 0.275723i \(-0.0889171\pi\)
\(108\) − 1.12110e21i − 0.499684i
\(109\) −1.72394e21 −0.697499 −0.348750 0.937216i \(-0.613394\pi\)
−0.348750 + 0.937216i \(0.613394\pi\)
\(110\) 0 0
\(111\) −2.85923e21 −0.955781
\(112\) 2.98252e21i 0.907389i
\(113\) 4.95810e20i 0.137402i 0.997637 + 0.0687008i \(0.0218854\pi\)
−0.997637 + 0.0687008i \(0.978115\pi\)
\(114\) 2.93917e20 0.0742547
\(115\) 0 0
\(116\) 8.56649e21 1.80299
\(117\) 4.95052e20i 0.0952136i
\(118\) 1.72105e21i 0.302715i
\(119\) 2.34439e21 0.377387
\(120\) 0 0
\(121\) 1.57256e21 0.212501
\(122\) 1.78290e21i 0.220978i
\(123\) 2.65712e21i 0.302279i
\(124\) 3.82809e21 0.399994
\(125\) 0 0
\(126\) 1.35830e21 0.119979
\(127\) − 1.63609e21i − 0.133005i −0.997786 0.0665027i \(-0.978816\pi\)
0.997786 0.0665027i \(-0.0211841\pi\)
\(128\) 9.59949e21i 0.718693i
\(129\) −2.49450e22 −1.72104
\(130\) 0 0
\(131\) −1.38650e22 −0.813898 −0.406949 0.913451i \(-0.633407\pi\)
−0.406949 + 0.913451i \(0.633407\pi\)
\(132\) 2.45829e22i 1.33231i
\(133\) 6.08379e21i 0.304601i
\(134\) 4.88486e21 0.226075
\(135\) 0 0
\(136\) 3.61412e21 0.143167
\(137\) 4.00789e22i 1.47011i 0.678007 + 0.735055i \(0.262844\pi\)
−0.678007 + 0.735055i \(0.737156\pi\)
\(138\) − 2.74019e21i − 0.0931215i
\(139\) −4.47585e22 −1.41000 −0.705001 0.709206i \(-0.749054\pi\)
−0.705001 + 0.709206i \(0.749054\pi\)
\(140\) 0 0
\(141\) −1.89350e22 −0.513410
\(142\) − 1.62223e21i − 0.0408400i
\(143\) 7.63689e21i 0.178603i
\(144\) −2.38438e22 −0.518286
\(145\) 0 0
\(146\) 1.24660e22 0.234435
\(147\) 4.04560e21i 0.0708191i
\(148\) − 4.46982e22i − 0.728688i
\(149\) −4.93289e22 −0.749283 −0.374641 0.927170i \(-0.622234\pi\)
−0.374641 + 0.927170i \(0.622234\pi\)
\(150\) 0 0
\(151\) 5.70415e22 0.753239 0.376620 0.926368i \(-0.377086\pi\)
0.376620 + 0.926368i \(0.377086\pi\)
\(152\) 9.37878e21i 0.115555i
\(153\) 1.87423e22i 0.215557i
\(154\) 2.09538e22 0.225058
\(155\) 0 0
\(156\) −2.09229e22 −0.196251
\(157\) − 6.35623e22i − 0.557511i −0.960362 0.278756i \(-0.910078\pi\)
0.960362 0.278756i \(-0.0899219\pi\)
\(158\) 1.47643e22i 0.121147i
\(159\) 2.62618e23 2.01677
\(160\) 0 0
\(161\) 5.67191e22 0.381995
\(162\) − 3.91522e22i − 0.247086i
\(163\) 8.68484e22i 0.513797i 0.966438 + 0.256899i \(0.0827006\pi\)
−0.966438 + 0.256899i \(0.917299\pi\)
\(164\) −4.15386e22 −0.230458
\(165\) 0 0
\(166\) −1.40866e22 −0.0688132
\(167\) 1.89411e23i 0.868726i 0.900738 + 0.434363i \(0.143026\pi\)
−0.900738 + 0.434363i \(0.856974\pi\)
\(168\) 1.17179e23i 0.504778i
\(169\) 2.40565e23 0.973692
\(170\) 0 0
\(171\) −4.86370e22 −0.173983
\(172\) − 3.89962e23i − 1.31212i
\(173\) − 4.18508e23i − 1.32501i −0.749057 0.662506i \(-0.769493\pi\)
0.749057 0.662506i \(-0.230507\pi\)
\(174\) 1.57818e23 0.470314
\(175\) 0 0
\(176\) −3.67825e23 −0.972206
\(177\) − 7.69957e23i − 1.91755i
\(178\) 1.45239e23i 0.340937i
\(179\) −4.76752e23 −1.05520 −0.527601 0.849493i \(-0.676908\pi\)
−0.527601 + 0.849493i \(0.676908\pi\)
\(180\) 0 0
\(181\) −2.88627e22 −0.0568476 −0.0284238 0.999596i \(-0.509049\pi\)
−0.0284238 + 0.999596i \(0.509049\pi\)
\(182\) 1.78341e22i 0.0331513i
\(183\) − 7.97624e23i − 1.39979i
\(184\) 8.74383e22 0.144915
\(185\) 0 0
\(186\) 7.05235e22 0.104339
\(187\) 2.89127e23i 0.404345i
\(188\) − 2.96009e23i − 0.391424i
\(189\) 4.27510e23 0.534685
\(190\) 0 0
\(191\) 8.86378e23 0.992587 0.496293 0.868155i \(-0.334694\pi\)
0.496293 + 0.868155i \(0.334694\pi\)
\(192\) − 9.15590e23i − 0.970595i
\(193\) 8.63509e22i 0.0866792i 0.999060 + 0.0433396i \(0.0137997\pi\)
−0.999060 + 0.0433396i \(0.986200\pi\)
\(194\) 2.32783e23 0.221327
\(195\) 0 0
\(196\) −6.32445e22 −0.0539926
\(197\) 6.99008e23i 0.565701i 0.959164 + 0.282850i \(0.0912800\pi\)
−0.959164 + 0.282850i \(0.908720\pi\)
\(198\) 1.67516e23i 0.128550i
\(199\) 1.24542e24 0.906483 0.453242 0.891388i \(-0.350268\pi\)
0.453242 + 0.891388i \(0.350268\pi\)
\(200\) 0 0
\(201\) −2.18536e24 −1.43207
\(202\) − 2.88581e23i − 0.179506i
\(203\) 3.26666e24i 1.92928i
\(204\) −7.92124e23 −0.444300
\(205\) 0 0
\(206\) 1.69847e23 0.0859909
\(207\) 4.53442e23i 0.218189i
\(208\) − 3.13061e23i − 0.143207i
\(209\) −7.50296e23 −0.326360
\(210\) 0 0
\(211\) 3.50841e24 1.38085 0.690423 0.723406i \(-0.257425\pi\)
0.690423 + 0.723406i \(0.257425\pi\)
\(212\) 4.10549e24i 1.53758i
\(213\) 7.25747e23i 0.258702i
\(214\) 3.23165e23 0.109668
\(215\) 0 0
\(216\) 6.59051e23 0.202840
\(217\) 1.45977e24i 0.428012i
\(218\) − 4.96495e23i − 0.138714i
\(219\) −5.57698e24 −1.48503
\(220\) 0 0
\(221\) −2.46081e23 −0.0595605
\(222\) − 8.23459e23i − 0.190080i
\(223\) − 4.72350e24i − 1.04007i −0.854145 0.520035i \(-0.825919\pi\)
0.854145 0.520035i \(-0.174081\pi\)
\(224\) −2.76624e24 −0.581147
\(225\) 0 0
\(226\) −1.42793e23 −0.0273256
\(227\) 5.44317e24i 0.994444i 0.867623 + 0.497222i \(0.165647\pi\)
−0.867623 + 0.497222i \(0.834353\pi\)
\(228\) − 2.05559e24i − 0.358609i
\(229\) −6.90677e24 −1.15081 −0.575403 0.817870i \(-0.695155\pi\)
−0.575403 + 0.817870i \(0.695155\pi\)
\(230\) 0 0
\(231\) −9.37420e24 −1.42564
\(232\) 5.03589e24i 0.731902i
\(233\) 4.53650e24i 0.630208i 0.949057 + 0.315104i \(0.102039\pi\)
−0.949057 + 0.315104i \(0.897961\pi\)
\(234\) −1.42575e23 −0.0189355
\(235\) 0 0
\(236\) 1.20367e25 1.46194
\(237\) − 6.60518e24i − 0.767409i
\(238\) 6.75185e23i 0.0750525i
\(239\) 2.73493e24 0.290916 0.145458 0.989364i \(-0.453534\pi\)
0.145458 + 0.989364i \(0.453534\pi\)
\(240\) 0 0
\(241\) −8.08907e24 −0.788351 −0.394175 0.919035i \(-0.628970\pi\)
−0.394175 + 0.919035i \(0.628970\pi\)
\(242\) 4.52898e23i 0.0422609i
\(243\) 1.16935e25i 1.04491i
\(244\) 1.24692e25 1.06720
\(245\) 0 0
\(246\) −7.65252e23 −0.0601154
\(247\) − 6.38588e23i − 0.0480732i
\(248\) 2.25038e24i 0.162373i
\(249\) 6.30200e24 0.435898
\(250\) 0 0
\(251\) 6.63927e24 0.422227 0.211113 0.977462i \(-0.432291\pi\)
0.211113 + 0.977462i \(0.432291\pi\)
\(252\) − 9.49967e24i − 0.579432i
\(253\) 6.99500e24i 0.409282i
\(254\) 4.71195e23 0.0264513
\(255\) 0 0
\(256\) 1.21381e25 0.627526
\(257\) − 1.57278e24i − 0.0780497i −0.999238 0.0390249i \(-0.987575\pi\)
0.999238 0.0390249i \(-0.0124252\pi\)
\(258\) − 7.18415e24i − 0.342270i
\(259\) 1.70448e25 0.779729
\(260\) 0 0
\(261\) −2.61154e25 −1.10197
\(262\) − 3.99311e24i − 0.161863i
\(263\) − 3.40077e25i − 1.32447i −0.749296 0.662235i \(-0.769608\pi\)
0.749296 0.662235i \(-0.230392\pi\)
\(264\) −1.44513e25 −0.540836
\(265\) 0 0
\(266\) −1.75213e24 −0.0605772
\(267\) − 6.49765e25i − 2.15967i
\(268\) − 3.41636e25i − 1.09181i
\(269\) 3.57975e25 1.10015 0.550077 0.835114i \(-0.314598\pi\)
0.550077 + 0.835114i \(0.314598\pi\)
\(270\) 0 0
\(271\) 2.46104e25 0.699746 0.349873 0.936797i \(-0.386225\pi\)
0.349873 + 0.936797i \(0.386225\pi\)
\(272\) − 1.18523e25i − 0.324212i
\(273\) − 7.97852e24i − 0.209998i
\(274\) −1.15427e25 −0.292366
\(275\) 0 0
\(276\) −1.91643e25 −0.449725
\(277\) 6.11679e25i 1.38193i 0.722888 + 0.690965i \(0.242814\pi\)
−0.722888 + 0.690965i \(0.757186\pi\)
\(278\) − 1.28904e25i − 0.280413i
\(279\) −1.16701e25 −0.244473
\(280\) 0 0
\(281\) 1.73710e25 0.337605 0.168802 0.985650i \(-0.446010\pi\)
0.168802 + 0.985650i \(0.446010\pi\)
\(282\) − 5.45327e24i − 0.102104i
\(283\) 7.57237e25i 1.36607i 0.730383 + 0.683037i \(0.239341\pi\)
−0.730383 + 0.683037i \(0.760659\pi\)
\(284\) −1.13455e25 −0.197235
\(285\) 0 0
\(286\) −2.19943e24 −0.0355194
\(287\) − 1.58399e25i − 0.246600i
\(288\) − 2.21148e25i − 0.331941i
\(289\) 5.97755e25 0.865159
\(290\) 0 0
\(291\) −1.04141e26 −1.40200
\(292\) − 8.71845e25i − 1.13219i
\(293\) 4.88684e25i 0.612235i 0.951994 + 0.306118i \(0.0990301\pi\)
−0.951994 + 0.306118i \(0.900970\pi\)
\(294\) −1.16513e24 −0.0140841
\(295\) 0 0
\(296\) 2.62762e25 0.295801
\(297\) 5.27236e25i 0.572878i
\(298\) − 1.42067e25i − 0.149013i
\(299\) −5.95355e24 −0.0602877
\(300\) 0 0
\(301\) 1.48705e26 1.40403
\(302\) 1.64279e25i 0.149800i
\(303\) 1.29104e26i 1.13708i
\(304\) 3.07571e25 0.261681
\(305\) 0 0
\(306\) −5.39778e24 −0.0428687
\(307\) 2.17987e26i 1.67293i 0.548019 + 0.836466i \(0.315382\pi\)
−0.548019 + 0.836466i \(0.684618\pi\)
\(308\) − 1.46546e26i − 1.08691i
\(309\) −7.59854e25 −0.544711
\(310\) 0 0
\(311\) −4.04644e25 −0.271075 −0.135538 0.990772i \(-0.543276\pi\)
−0.135538 + 0.990772i \(0.543276\pi\)
\(312\) − 1.22997e25i − 0.0796657i
\(313\) − 8.74174e24i − 0.0547498i −0.999625 0.0273749i \(-0.991285\pi\)
0.999625 0.0273749i \(-0.00871478\pi\)
\(314\) 1.83060e25 0.110874
\(315\) 0 0
\(316\) 1.03258e26 0.585073
\(317\) 3.19758e25i 0.175267i 0.996153 + 0.0876334i \(0.0279304\pi\)
−0.996153 + 0.0876334i \(0.972070\pi\)
\(318\) 7.56341e25i 0.401082i
\(319\) −4.02868e26 −2.06710
\(320\) 0 0
\(321\) −1.44576e26 −0.694694
\(322\) 1.63351e25i 0.0759688i
\(323\) − 2.41765e25i − 0.108835i
\(324\) −2.73822e26 −1.19329
\(325\) 0 0
\(326\) −2.50123e25 −0.102181
\(327\) 2.22119e26i 0.878688i
\(328\) − 2.44189e25i − 0.0935514i
\(329\) 1.12877e26 0.418841
\(330\) 0 0
\(331\) −2.52215e26 −0.878166 −0.439083 0.898446i \(-0.644697\pi\)
−0.439083 + 0.898446i \(0.644697\pi\)
\(332\) 9.85186e25i 0.332329i
\(333\) 1.36265e26i 0.445368i
\(334\) −5.45504e25 −0.172767
\(335\) 0 0
\(336\) 3.84279e26 1.14310
\(337\) 1.53127e26i 0.441507i 0.975330 + 0.220753i \(0.0708516\pi\)
−0.975330 + 0.220753i \(0.929148\pi\)
\(338\) 6.92826e25i 0.193642i
\(339\) 6.38821e25 0.173094
\(340\) 0 0
\(341\) −1.80029e26 −0.458586
\(342\) − 1.40075e25i − 0.0346007i
\(343\) 4.04890e26i 0.969949i
\(344\) 2.29243e26 0.532639
\(345\) 0 0
\(346\) 1.20530e26 0.263510
\(347\) − 3.23436e26i − 0.686008i −0.939334 0.343004i \(-0.888556\pi\)
0.939334 0.343004i \(-0.111444\pi\)
\(348\) − 1.10374e27i − 2.27136i
\(349\) 6.77854e26 1.35353 0.676767 0.736197i \(-0.263380\pi\)
0.676767 + 0.736197i \(0.263380\pi\)
\(350\) 0 0
\(351\) −4.48739e25 −0.0843857
\(352\) − 3.41153e26i − 0.622660i
\(353\) 6.84291e26i 1.21229i 0.795354 + 0.606145i \(0.207285\pi\)
−0.795354 + 0.606145i \(0.792715\pi\)
\(354\) 2.21748e26 0.381351
\(355\) 0 0
\(356\) 1.01577e27 1.64654
\(357\) − 3.02061e26i − 0.475421i
\(358\) − 1.37304e26i − 0.209852i
\(359\) 6.85500e25 0.101746 0.0508728 0.998705i \(-0.483800\pi\)
0.0508728 + 0.998705i \(0.483800\pi\)
\(360\) 0 0
\(361\) −6.51471e26 −0.912156
\(362\) − 8.31247e24i − 0.0113055i
\(363\) − 2.02615e26i − 0.267703i
\(364\) 1.24728e26 0.160102
\(365\) 0 0
\(366\) 2.29716e26 0.278381
\(367\) − 1.13575e27i − 1.33749i −0.743492 0.668745i \(-0.766832\pi\)
0.743492 0.668745i \(-0.233168\pi\)
\(368\) − 2.86748e26i − 0.328170i
\(369\) 1.26633e26 0.140854
\(370\) 0 0
\(371\) −1.56555e27 −1.64528
\(372\) − 4.93226e26i − 0.503901i
\(373\) 3.82975e26i 0.380389i 0.981746 + 0.190195i \(0.0609119\pi\)
−0.981746 + 0.190195i \(0.939088\pi\)
\(374\) −8.32687e25 −0.0804136
\(375\) 0 0
\(376\) 1.74012e26 0.158894
\(377\) − 3.42887e26i − 0.304486i
\(378\) 1.23123e26i 0.106335i
\(379\) −7.14767e25 −0.0600417 −0.0300208 0.999549i \(-0.509557\pi\)
−0.0300208 + 0.999549i \(0.509557\pi\)
\(380\) 0 0
\(381\) −2.10801e26 −0.167556
\(382\) 2.55277e26i 0.197400i
\(383\) − 1.38425e27i − 1.04142i −0.853732 0.520712i \(-0.825666\pi\)
0.853732 0.520712i \(-0.174334\pi\)
\(384\) 1.23684e27 0.905388
\(385\) 0 0
\(386\) −2.48690e25 −0.0172382
\(387\) 1.18882e27i 0.801958i
\(388\) − 1.62803e27i − 1.06888i
\(389\) −1.63213e27 −1.04300 −0.521500 0.853251i \(-0.674627\pi\)
−0.521500 + 0.853251i \(0.674627\pi\)
\(390\) 0 0
\(391\) −2.25397e26 −0.136487
\(392\) − 3.71789e25i − 0.0219176i
\(393\) 1.78642e27i 1.02532i
\(394\) −2.01314e26 −0.112503
\(395\) 0 0
\(396\) 1.17157e27 0.620823
\(397\) − 1.21029e27i − 0.624581i −0.949987 0.312291i \(-0.898904\pi\)
0.949987 0.312291i \(-0.101096\pi\)
\(398\) 3.58682e26i 0.180276i
\(399\) 7.83860e26 0.383728
\(400\) 0 0
\(401\) −6.51358e26 −0.302555 −0.151277 0.988491i \(-0.548339\pi\)
−0.151277 + 0.988491i \(0.548339\pi\)
\(402\) − 6.29385e26i − 0.284802i
\(403\) − 1.53225e26i − 0.0675502i
\(404\) −2.01827e27 −0.866913
\(405\) 0 0
\(406\) −9.40799e26 −0.383684
\(407\) 2.10208e27i 0.835427i
\(408\) − 4.65658e26i − 0.180358i
\(409\) 4.45251e27 1.68078 0.840389 0.541984i \(-0.182327\pi\)
0.840389 + 0.541984i \(0.182327\pi\)
\(410\) 0 0
\(411\) 5.16393e27 1.85200
\(412\) − 1.18787e27i − 0.415288i
\(413\) 4.58995e27i 1.56434i
\(414\) −1.30591e26 −0.0433921
\(415\) 0 0
\(416\) 2.90360e26 0.0917185
\(417\) 5.76686e27i 1.77628i
\(418\) − 2.16085e26i − 0.0649044i
\(419\) −4.92912e27 −1.44385 −0.721925 0.691972i \(-0.756742\pi\)
−0.721925 + 0.691972i \(0.756742\pi\)
\(420\) 0 0
\(421\) 1.50145e27 0.418359 0.209180 0.977877i \(-0.432921\pi\)
0.209180 + 0.977877i \(0.432921\pi\)
\(422\) 1.01042e27i 0.274614i
\(423\) 9.02400e26i 0.239235i
\(424\) −2.41345e27 −0.624162
\(425\) 0 0
\(426\) −2.09015e26 −0.0514490
\(427\) 4.75488e27i 1.14195i
\(428\) − 2.26014e27i − 0.529635i
\(429\) 9.83968e26 0.224998
\(430\) 0 0
\(431\) −7.19221e27 −1.56621 −0.783107 0.621887i \(-0.786366\pi\)
−0.783107 + 0.621887i \(0.786366\pi\)
\(432\) − 2.16132e27i − 0.459345i
\(433\) 4.23104e27i 0.877656i 0.898571 + 0.438828i \(0.144606\pi\)
−0.898571 + 0.438828i \(0.855394\pi\)
\(434\) −4.20412e26 −0.0851203
\(435\) 0 0
\(436\) −3.47237e27 −0.669913
\(437\) − 5.84914e26i − 0.110163i
\(438\) − 1.60617e27i − 0.295334i
\(439\) −4.53235e27 −0.813665 −0.406832 0.913503i \(-0.633367\pi\)
−0.406832 + 0.913503i \(0.633367\pi\)
\(440\) 0 0
\(441\) 1.92804e26 0.0329998
\(442\) − 7.08712e25i − 0.0118450i
\(443\) − 4.61186e27i − 0.752726i −0.926472 0.376363i \(-0.877174\pi\)
0.926472 0.376363i \(-0.122826\pi\)
\(444\) −5.75909e27 −0.917979
\(445\) 0 0
\(446\) 1.36037e27 0.206843
\(447\) 6.35573e27i 0.943923i
\(448\) 5.45811e27i 0.791815i
\(449\) 1.25692e26 0.0178123 0.00890615 0.999960i \(-0.497165\pi\)
0.00890615 + 0.999960i \(0.497165\pi\)
\(450\) 0 0
\(451\) 1.95349e27 0.264216
\(452\) 9.98664e26i 0.131967i
\(453\) − 7.34945e27i − 0.948907i
\(454\) −1.56763e27 −0.197769
\(455\) 0 0
\(456\) 1.20840e27 0.145573
\(457\) − 1.15924e28i − 1.36475i −0.731003 0.682374i \(-0.760948\pi\)
0.731003 0.682374i \(-0.239052\pi\)
\(458\) − 1.98915e27i − 0.228865i
\(459\) −1.69889e27 −0.191044
\(460\) 0 0
\(461\) 1.02514e28 1.10134 0.550671 0.834723i \(-0.314372\pi\)
0.550671 + 0.834723i \(0.314372\pi\)
\(462\) − 2.69977e27i − 0.283522i
\(463\) − 5.72801e27i − 0.588035i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949924\pi\)
\(464\) 1.65149e28 1.65744
\(465\) 0 0
\(466\) −1.30651e27 −0.125332
\(467\) − 1.12658e28i − 1.05666i −0.849040 0.528329i \(-0.822819\pi\)
0.849040 0.528329i \(-0.177181\pi\)
\(468\) 9.97138e26i 0.0914479i
\(469\) 1.30276e28 1.16829
\(470\) 0 0
\(471\) −8.18963e27 −0.702335
\(472\) 7.07587e27i 0.593457i
\(473\) 1.83393e28i 1.50432i
\(474\) 1.90229e27 0.152618
\(475\) 0 0
\(476\) 4.72210e27 0.362462
\(477\) − 1.25158e28i − 0.939759i
\(478\) 7.87660e26i 0.0578557i
\(479\) 1.17373e28 0.843422 0.421711 0.906730i \(-0.361430\pi\)
0.421711 + 0.906730i \(0.361430\pi\)
\(480\) 0 0
\(481\) −1.78911e27 −0.123059
\(482\) − 2.32965e27i − 0.156782i
\(483\) − 7.30792e27i − 0.481226i
\(484\) 3.16747e27 0.204097
\(485\) 0 0
\(486\) −3.36773e27 −0.207805
\(487\) − 4.75272e27i − 0.287004i −0.989650 0.143502i \(-0.954164\pi\)
0.989650 0.143502i \(-0.0458363\pi\)
\(488\) 7.33013e27i 0.433215i
\(489\) 1.11899e28 0.647266
\(490\) 0 0
\(491\) −2.59837e28 −1.43994 −0.719971 0.694004i \(-0.755845\pi\)
−0.719971 + 0.694004i \(0.755845\pi\)
\(492\) 5.35200e27i 0.290324i
\(493\) − 1.29815e28i − 0.689336i
\(494\) 1.83913e26 0.00956049
\(495\) 0 0
\(496\) 7.37997e27 0.367703
\(497\) − 4.32640e27i − 0.211050i
\(498\) 1.81498e27i 0.0866887i
\(499\) −3.84508e28 −1.79825 −0.899124 0.437695i \(-0.855795\pi\)
−0.899124 + 0.437695i \(0.855795\pi\)
\(500\) 0 0
\(501\) 2.44045e28 1.09439
\(502\) 1.91211e27i 0.0839699i
\(503\) − 3.27446e28i − 1.40824i −0.710083 0.704118i \(-0.751343\pi\)
0.710083 0.704118i \(-0.248657\pi\)
\(504\) 5.58447e27 0.235213
\(505\) 0 0
\(506\) −2.01456e27 −0.0813954
\(507\) − 3.09953e28i − 1.22663i
\(508\) − 3.29543e27i − 0.127745i
\(509\) 2.02472e28 0.768826 0.384413 0.923161i \(-0.374404\pi\)
0.384413 + 0.923161i \(0.374404\pi\)
\(510\) 0 0
\(511\) 3.32461e28 1.21149
\(512\) 2.36274e28i 0.843492i
\(513\) − 4.40869e27i − 0.154197i
\(514\) 4.52962e26 0.0155220
\(515\) 0 0
\(516\) −5.02443e28 −1.65297
\(517\) 1.39208e28i 0.448760i
\(518\) 4.90889e27i 0.155068i
\(519\) −5.39222e28 −1.66921
\(520\) 0 0
\(521\) 4.75207e27 0.141282 0.0706411 0.997502i \(-0.477496\pi\)
0.0706411 + 0.997502i \(0.477496\pi\)
\(522\) − 7.52124e27i − 0.219154i
\(523\) − 1.28180e26i − 0.00366061i −0.999998 0.00183030i \(-0.999417\pi\)
0.999998 0.00183030i \(-0.000582604\pi\)
\(524\) −2.79269e28 −0.781708
\(525\) 0 0
\(526\) 9.79422e27 0.263402
\(527\) − 5.80099e27i − 0.152929i
\(528\) 4.73921e28i 1.22476i
\(529\) 3.40184e28 0.861846
\(530\) 0 0
\(531\) −3.66944e28 −0.893527
\(532\) 1.22540e28i 0.292554i
\(533\) 1.66265e27i 0.0389192i
\(534\) 1.87132e28 0.429502
\(535\) 0 0
\(536\) 2.00834e28 0.443208
\(537\) 6.14266e28i 1.32931i
\(538\) 1.03097e28i 0.218792i
\(539\) 2.97429e27 0.0619014
\(540\) 0 0
\(541\) 3.42747e28 0.686123 0.343061 0.939313i \(-0.388536\pi\)
0.343061 + 0.939313i \(0.388536\pi\)
\(542\) 7.08778e27i 0.139161i
\(543\) 3.71879e27i 0.0716149i
\(544\) 1.09928e28 0.207645
\(545\) 0 0
\(546\) 2.29781e27 0.0417630
\(547\) 7.30329e28i 1.30212i 0.759026 + 0.651061i \(0.225676\pi\)
−0.759026 + 0.651061i \(0.774324\pi\)
\(548\) 8.07273e28i 1.41197i
\(549\) −3.80130e28 −0.652263
\(550\) 0 0
\(551\) 3.36874e28 0.556385
\(552\) − 1.12659e28i − 0.182560i
\(553\) 3.93755e28i 0.626055i
\(554\) −1.76164e28 −0.274830
\(555\) 0 0
\(556\) −9.01529e28 −1.35424
\(557\) 3.32597e28i 0.490274i 0.969488 + 0.245137i \(0.0788330\pi\)
−0.969488 + 0.245137i \(0.921167\pi\)
\(558\) − 3.36100e27i − 0.0486193i
\(559\) −1.56089e28 −0.221588
\(560\) 0 0
\(561\) 3.72523e28 0.509382
\(562\) 5.00285e27i 0.0671407i
\(563\) − 8.28332e28i − 1.09110i −0.838077 0.545552i \(-0.816320\pi\)
0.838077 0.545552i \(-0.183680\pi\)
\(564\) −3.81390e28 −0.493104
\(565\) 0 0
\(566\) −2.18084e28 −0.271676
\(567\) − 1.04417e29i − 1.27687i
\(568\) − 6.66959e27i − 0.0800648i
\(569\) −7.35414e28 −0.866669 −0.433335 0.901233i \(-0.642663\pi\)
−0.433335 + 0.901233i \(0.642663\pi\)
\(570\) 0 0
\(571\) 1.09131e29 1.23956 0.619780 0.784776i \(-0.287222\pi\)
0.619780 + 0.784776i \(0.287222\pi\)
\(572\) 1.53823e28i 0.171539i
\(573\) − 1.14204e29i − 1.25043i
\(574\) 4.56190e27 0.0490423
\(575\) 0 0
\(576\) −4.36350e28 −0.452271
\(577\) − 1.30727e28i − 0.133051i −0.997785 0.0665257i \(-0.978809\pi\)
0.997785 0.0665257i \(-0.0211914\pi\)
\(578\) 1.72153e28i 0.172057i
\(579\) 1.11258e28 0.109196
\(580\) 0 0
\(581\) −3.75682e28 −0.355607
\(582\) − 2.99927e28i − 0.278820i
\(583\) − 1.93075e29i − 1.76281i
\(584\) 5.12523e28 0.459598
\(585\) 0 0
\(586\) −1.40741e28 −0.121758
\(587\) 1.71459e29i 1.45701i 0.685043 + 0.728503i \(0.259784\pi\)
−0.685043 + 0.728503i \(0.740216\pi\)
\(588\) 8.14867e27i 0.0680182i
\(589\) 1.50538e28 0.123434
\(590\) 0 0
\(591\) 9.00630e28 0.712652
\(592\) − 8.61713e28i − 0.669861i
\(593\) 2.35277e29i 1.79682i 0.439153 + 0.898412i \(0.355279\pi\)
−0.439153 + 0.898412i \(0.644721\pi\)
\(594\) −1.51844e28 −0.113931
\(595\) 0 0
\(596\) −9.93587e28 −0.719648
\(597\) − 1.60465e29i − 1.14196i
\(598\) − 1.71462e27i − 0.0119896i
\(599\) 1.33384e29 0.916481 0.458240 0.888828i \(-0.348480\pi\)
0.458240 + 0.888828i \(0.348480\pi\)
\(600\) 0 0
\(601\) −3.64474e28 −0.241816 −0.120908 0.992664i \(-0.538581\pi\)
−0.120908 + 0.992664i \(0.538581\pi\)
\(602\) 4.28269e28i 0.279225i
\(603\) 1.04150e29i 0.667308i
\(604\) 1.14893e29 0.723448
\(605\) 0 0
\(606\) −3.71820e28 −0.226136
\(607\) − 2.71228e29i − 1.62126i −0.585559 0.810630i \(-0.699125\pi\)
0.585559 0.810630i \(-0.300875\pi\)
\(608\) 2.85268e28i 0.167597i
\(609\) 4.20890e29 2.43045
\(610\) 0 0
\(611\) −1.18482e28 −0.0661029
\(612\) 3.77509e28i 0.207032i
\(613\) − 2.02699e29i − 1.09274i −0.837544 0.546370i \(-0.816009\pi\)
0.837544 0.546370i \(-0.183991\pi\)
\(614\) −6.27803e28 −0.332702
\(615\) 0 0
\(616\) 8.61486e28 0.441215
\(617\) − 3.04169e29i − 1.53151i −0.643130 0.765757i \(-0.722364\pi\)
0.643130 0.765757i \(-0.277636\pi\)
\(618\) − 2.18838e28i − 0.108329i
\(619\) −1.21487e29 −0.591262 −0.295631 0.955302i \(-0.595530\pi\)
−0.295631 + 0.955302i \(0.595530\pi\)
\(620\) 0 0
\(621\) −4.11022e28 −0.193376
\(622\) − 1.16538e28i − 0.0539097i
\(623\) 3.87345e29i 1.76187i
\(624\) −4.03361e28 −0.180408
\(625\) 0 0
\(626\) 2.51762e27 0.0108883
\(627\) 9.66712e28i 0.411138i
\(628\) − 1.28028e29i − 0.535461i
\(629\) −6.77345e28 −0.278598
\(630\) 0 0
\(631\) −1.79118e29 −0.712577 −0.356288 0.934376i \(-0.615958\pi\)
−0.356288 + 0.934376i \(0.615958\pi\)
\(632\) 6.07014e28i 0.237503i
\(633\) − 4.52038e29i − 1.73955i
\(634\) −9.20904e27 −0.0348560
\(635\) 0 0
\(636\) 5.28968e29 1.93700
\(637\) 2.53146e27i 0.00911815i
\(638\) − 1.16026e29i − 0.411091i
\(639\) 3.45875e28 0.120548
\(640\) 0 0
\(641\) −3.41742e29 −1.15263 −0.576315 0.817228i \(-0.695510\pi\)
−0.576315 + 0.817228i \(0.695510\pi\)
\(642\) − 4.16379e28i − 0.138156i
\(643\) 7.57582e28i 0.247294i 0.992326 + 0.123647i \(0.0394591\pi\)
−0.992326 + 0.123647i \(0.960541\pi\)
\(644\) 1.14244e29 0.366887
\(645\) 0 0
\(646\) 6.96283e27 0.0216443
\(647\) 2.99088e29i 0.914755i 0.889273 + 0.457377i \(0.151211\pi\)
−0.889273 + 0.457377i \(0.848789\pi\)
\(648\) − 1.60969e29i − 0.484400i
\(649\) −5.66065e29 −1.67609
\(650\) 0 0
\(651\) 1.88082e29 0.539196
\(652\) 1.74931e29i 0.493476i
\(653\) 5.40507e28i 0.150042i 0.997182 + 0.0750210i \(0.0239024\pi\)
−0.997182 + 0.0750210i \(0.976098\pi\)
\(654\) −6.39703e28 −0.174748
\(655\) 0 0
\(656\) −8.00802e28 −0.211853
\(657\) 2.65787e29i 0.691985i
\(658\) 3.25086e28i 0.0832966i
\(659\) −2.26606e28 −0.0571446 −0.0285723 0.999592i \(-0.509096\pi\)
−0.0285723 + 0.999592i \(0.509096\pi\)
\(660\) 0 0
\(661\) −1.87483e29 −0.457980 −0.228990 0.973429i \(-0.573542\pi\)
−0.228990 + 0.973429i \(0.573542\pi\)
\(662\) − 7.26379e28i − 0.174644i
\(663\) 3.17060e28i 0.0750325i
\(664\) −5.79151e28 −0.134905
\(665\) 0 0
\(666\) −3.92443e28 −0.0885720
\(667\) − 3.14067e29i − 0.697752i
\(668\) 3.81513e29i 0.834367i
\(669\) −6.08595e29 −1.31025
\(670\) 0 0
\(671\) −5.86406e29 −1.22352
\(672\) 3.56414e29i 0.732111i
\(673\) 7.53001e29i 1.52278i 0.648294 + 0.761390i \(0.275483\pi\)
−0.648294 + 0.761390i \(0.724517\pi\)
\(674\) −4.41005e28 −0.0878041
\(675\) 0 0
\(676\) 4.84547e29 0.935181
\(677\) 8.12052e29i 1.54313i 0.636150 + 0.771565i \(0.280526\pi\)
−0.636150 + 0.771565i \(0.719474\pi\)
\(678\) 1.83981e28i 0.0344239i
\(679\) 6.20819e29 1.14375
\(680\) 0 0
\(681\) 7.01320e29 1.25277
\(682\) − 5.18482e28i − 0.0912007i
\(683\) 4.69973e29i 0.814058i 0.913415 + 0.407029i \(0.133435\pi\)
−0.913415 + 0.407029i \(0.866565\pi\)
\(684\) −9.79650e28 −0.167102
\(685\) 0 0
\(686\) −1.16608e29 −0.192897
\(687\) 8.89896e29i 1.44975i
\(688\) − 7.51789e29i − 1.20619i
\(689\) 1.64329e29 0.259664
\(690\) 0 0
\(691\) 3.58806e29 0.549971 0.274985 0.961448i \(-0.411327\pi\)
0.274985 + 0.961448i \(0.411327\pi\)
\(692\) − 8.42962e29i − 1.27261i
\(693\) 4.46754e29i 0.664308i
\(694\) 9.31496e28 0.136429
\(695\) 0 0
\(696\) 6.48845e29 0.922027
\(697\) 6.29466e28i 0.0881106i
\(698\) 1.95222e29i 0.269183i
\(699\) 5.84501e29 0.793917
\(700\) 0 0
\(701\) −7.33110e29 −0.966339 −0.483170 0.875527i \(-0.660515\pi\)
−0.483170 + 0.875527i \(0.660515\pi\)
\(702\) − 1.29237e28i − 0.0167821i
\(703\) − 1.75774e29i − 0.224865i
\(704\) −6.73133e29 −0.848376
\(705\) 0 0
\(706\) −1.97076e29 −0.241093
\(707\) − 7.69629e29i − 0.927636i
\(708\) − 1.55085e30i − 1.84171i
\(709\) −1.20909e30 −1.41473 −0.707365 0.706848i \(-0.750116\pi\)
−0.707365 + 0.706848i \(0.750116\pi\)
\(710\) 0 0
\(711\) −3.14788e29 −0.357592
\(712\) 5.97131e29i 0.668390i
\(713\) − 1.40346e29i − 0.154796i
\(714\) 8.69936e28 0.0945488
\(715\) 0 0
\(716\) −9.60277e29 −1.01347
\(717\) − 3.52379e29i − 0.366488i
\(718\) 1.97424e28i 0.0202345i
\(719\) 5.25189e29 0.530472 0.265236 0.964183i \(-0.414550\pi\)
0.265236 + 0.964183i \(0.414550\pi\)
\(720\) 0 0
\(721\) 4.52972e29 0.444377
\(722\) − 1.87624e29i − 0.181404i
\(723\) 1.04223e30i 0.993140i
\(724\) −5.81356e28 −0.0545993
\(725\) 0 0
\(726\) 5.83532e28 0.0532390
\(727\) − 7.23318e29i − 0.650456i −0.945636 0.325228i \(-0.894559\pi\)
0.945636 0.325228i \(-0.105441\pi\)
\(728\) 7.33223e28i 0.0649915i
\(729\) 8.46052e28 0.0739193
\(730\) 0 0
\(731\) −5.90940e29 −0.501662
\(732\) − 1.60658e30i − 1.34442i
\(733\) − 2.11409e30i − 1.74394i −0.489560 0.871969i \(-0.662843\pi\)
0.489560 0.871969i \(-0.337157\pi\)
\(734\) 3.27097e29 0.265992
\(735\) 0 0
\(736\) 2.65955e29 0.210180
\(737\) 1.60666e30i 1.25174i
\(738\) 3.64702e28i 0.0280122i
\(739\) 7.42069e29 0.561924 0.280962 0.959719i \(-0.409347\pi\)
0.280962 + 0.959719i \(0.409347\pi\)
\(740\) 0 0
\(741\) −8.22782e28 −0.0605611
\(742\) − 4.50878e29i − 0.327204i
\(743\) 1.69968e30i 1.21614i 0.793882 + 0.608071i \(0.208057\pi\)
−0.793882 + 0.608071i \(0.791943\pi\)
\(744\) 2.89948e29 0.204552
\(745\) 0 0
\(746\) −1.10297e29 −0.0756494
\(747\) − 3.00339e29i − 0.203117i
\(748\) 5.82362e29i 0.388353i
\(749\) 8.61862e29 0.566733
\(750\) 0 0
\(751\) 5.90225e29 0.377397 0.188698 0.982035i \(-0.439573\pi\)
0.188698 + 0.982035i \(0.439573\pi\)
\(752\) − 5.70661e29i − 0.359824i
\(753\) − 8.55430e29i − 0.531909i
\(754\) 9.87515e28 0.0605542
\(755\) 0 0
\(756\) 8.61095e29 0.513537
\(757\) − 1.33308e30i − 0.784058i −0.919953 0.392029i \(-0.871773\pi\)
0.919953 0.392029i \(-0.128227\pi\)
\(758\) − 2.05853e28i − 0.0119407i
\(759\) 9.01264e29 0.515601
\(760\) 0 0
\(761\) −7.86217e29 −0.437526 −0.218763 0.975778i \(-0.570202\pi\)
−0.218763 + 0.975778i \(0.570202\pi\)
\(762\) − 6.07106e28i − 0.0333225i
\(763\) − 1.32412e30i − 0.716837i
\(764\) 1.78535e30 0.953329
\(765\) 0 0
\(766\) 3.98665e29 0.207112
\(767\) − 4.81786e29i − 0.246890i
\(768\) − 1.56392e30i − 0.790537i
\(769\) 1.65083e30 0.823142 0.411571 0.911378i \(-0.364980\pi\)
0.411571 + 0.911378i \(0.364980\pi\)
\(770\) 0 0
\(771\) −2.02644e29 −0.0983247
\(772\) 1.73929e29i 0.0832510i
\(773\) 3.77930e30i 1.78454i 0.451499 + 0.892271i \(0.350889\pi\)
−0.451499 + 0.892271i \(0.649111\pi\)
\(774\) −3.42381e29 −0.159488
\(775\) 0 0
\(776\) 9.57056e29 0.433899
\(777\) − 2.19612e30i − 0.982278i
\(778\) − 4.70054e29i − 0.207425i
\(779\) −1.63349e29 −0.0711169
\(780\) 0 0
\(781\) 5.33563e29 0.226126
\(782\) − 6.49144e28i − 0.0271438i
\(783\) − 2.36723e30i − 0.976655i
\(784\) −1.21926e29 −0.0496337
\(785\) 0 0
\(786\) −5.14488e29 −0.203910
\(787\) 3.58219e30i 1.40092i 0.713691 + 0.700461i \(0.247022\pi\)
−0.713691 + 0.700461i \(0.752978\pi\)
\(788\) 1.40795e30i 0.543327i
\(789\) −4.38169e30 −1.66853
\(790\) 0 0
\(791\) −3.80821e29 −0.141211
\(792\) 6.88717e29i 0.252015i
\(793\) − 4.99099e29i − 0.180226i
\(794\) 3.48564e29 0.124213
\(795\) 0 0
\(796\) 2.50854e30 0.870631
\(797\) 4.45901e30i 1.52731i 0.645626 + 0.763654i \(0.276596\pi\)
−0.645626 + 0.763654i \(0.723404\pi\)
\(798\) 2.25752e29i 0.0763134i
\(799\) −4.48565e29 −0.149653
\(800\) 0 0
\(801\) −3.09664e30 −1.00635
\(802\) − 1.87591e29i − 0.0601702i
\(803\) 4.10015e30i 1.29803i
\(804\) −4.40178e30 −1.37543
\(805\) 0 0
\(806\) 4.41288e28 0.0134340
\(807\) − 4.61229e30i − 1.38594i
\(808\) − 1.18646e30i − 0.351912i
\(809\) −3.09156e30 −0.905144 −0.452572 0.891728i \(-0.649493\pi\)
−0.452572 + 0.891728i \(0.649493\pi\)
\(810\) 0 0
\(811\) −9.85711e29 −0.281210 −0.140605 0.990066i \(-0.544905\pi\)
−0.140605 + 0.990066i \(0.544905\pi\)
\(812\) 6.57974e30i 1.85298i
\(813\) − 3.17090e30i − 0.881518i
\(814\) −6.05400e29 −0.166144
\(815\) 0 0
\(816\) −1.52710e30 −0.408432
\(817\) − 1.53351e30i − 0.404907i
\(818\) 1.28232e30i 0.334262i
\(819\) −3.80239e29 −0.0978533
\(820\) 0 0
\(821\) −5.81930e30 −1.45971 −0.729856 0.683601i \(-0.760413\pi\)
−0.729856 + 0.683601i \(0.760413\pi\)
\(822\) 1.48721e30i 0.368314i
\(823\) 1.82848e30i 0.447086i 0.974694 + 0.223543i \(0.0717624\pi\)
−0.974694 + 0.223543i \(0.928238\pi\)
\(824\) 6.98303e29 0.168581
\(825\) 0 0
\(826\) −1.32191e30 −0.311107
\(827\) 3.80997e30i 0.885347i 0.896683 + 0.442673i \(0.145970\pi\)
−0.896683 + 0.442673i \(0.854030\pi\)
\(828\) 9.13327e29i 0.209560i
\(829\) −9.24247e29 −0.209395 −0.104697 0.994504i \(-0.533387\pi\)
−0.104697 + 0.994504i \(0.533387\pi\)
\(830\) 0 0
\(831\) 7.88112e30 1.74091
\(832\) − 5.72914e29i − 0.124967i
\(833\) 9.58392e28i 0.0206429i
\(834\) −1.66086e30 −0.353255
\(835\) 0 0
\(836\) −1.51125e30 −0.313452
\(837\) − 1.05784e30i − 0.216671i
\(838\) − 1.41959e30i − 0.287144i
\(839\) −2.18057e30 −0.435582 −0.217791 0.975995i \(-0.569885\pi\)
−0.217791 + 0.975995i \(0.569885\pi\)
\(840\) 0 0
\(841\) 1.29554e31 2.52403
\(842\) 4.32418e29i 0.0832007i
\(843\) − 2.23815e30i − 0.425304i
\(844\) 7.06668e30 1.32623
\(845\) 0 0
\(846\) −2.59891e29 −0.0475776
\(847\) 1.20785e30i 0.218393i
\(848\) 7.91477e30i 1.41345i
\(849\) 9.75655e30 1.72094
\(850\) 0 0
\(851\) −1.63874e30 −0.282000
\(852\) 1.46181e30i 0.248470i
\(853\) − 4.27754e30i − 0.718173i −0.933304 0.359087i \(-0.883088\pi\)
0.933304 0.359087i \(-0.116912\pi\)
\(854\) −1.36941e30 −0.227104
\(855\) 0 0
\(856\) 1.32865e30 0.214998
\(857\) 1.76322e29i 0.0281843i 0.999901 + 0.0140922i \(0.00448582\pi\)
−0.999901 + 0.0140922i \(0.995514\pi\)
\(858\) 2.83383e29i 0.0447463i
\(859\) −4.74009e30 −0.739365 −0.369682 0.929158i \(-0.620533\pi\)
−0.369682 + 0.929158i \(0.620533\pi\)
\(860\) 0 0
\(861\) −2.04088e30 −0.310659
\(862\) − 2.07136e30i − 0.311479i
\(863\) 8.10824e29i 0.120452i 0.998185 + 0.0602259i \(0.0191821\pi\)
−0.998185 + 0.0602259i \(0.980818\pi\)
\(864\) 2.00459e30 0.294192
\(865\) 0 0
\(866\) −1.21854e30 −0.174543
\(867\) − 7.70171e30i − 1.08990i
\(868\) 2.94027e30i 0.411084i
\(869\) −4.85607e30 −0.670775
\(870\) 0 0
\(871\) −1.36745e30 −0.184383
\(872\) − 2.04127e30i − 0.271942i
\(873\) 4.96315e30i 0.653293i
\(874\) 1.68455e29 0.0219086
\(875\) 0 0
\(876\) −1.12332e31 −1.42630
\(877\) − 1.55255e30i − 0.194782i −0.995246 0.0973912i \(-0.968950\pi\)
0.995246 0.0973912i \(-0.0310498\pi\)
\(878\) − 1.30532e30i − 0.161817i
\(879\) 6.29641e30 0.771275
\(880\) 0 0
\(881\) −5.75722e30 −0.688598 −0.344299 0.938860i \(-0.611883\pi\)
−0.344299 + 0.938860i \(0.611883\pi\)
\(882\) 5.55276e28i 0.00656280i
\(883\) − 1.29134e31i − 1.50818i −0.656774 0.754088i \(-0.728079\pi\)
0.656774 0.754088i \(-0.271921\pi\)
\(884\) −4.95657e29 −0.0572048
\(885\) 0 0
\(886\) 1.32822e30 0.149698
\(887\) 4.22752e30i 0.470856i 0.971892 + 0.235428i \(0.0756492\pi\)
−0.971892 + 0.235428i \(0.924351\pi\)
\(888\) − 3.38554e30i − 0.372642i
\(889\) 1.25665e30 0.136693
\(890\) 0 0
\(891\) 1.28774e31 1.36808
\(892\) − 9.51411e30i − 0.998935i
\(893\) − 1.16404e30i − 0.120789i
\(894\) −1.83045e30 −0.187722
\(895\) 0 0
\(896\) −7.37317e30 −0.738618
\(897\) 7.67079e29i 0.0759486i
\(898\) 3.61992e28i 0.00354240i
\(899\) 8.08306e30 0.781806
\(900\) 0 0
\(901\) 6.22137e30 0.587862
\(902\) 5.62606e29i 0.0525455i
\(903\) − 1.91597e31i − 1.76875i
\(904\) −5.87074e29 −0.0535704
\(905\) 0 0
\(906\) 2.11664e30 0.188713
\(907\) − 1.34786e31i − 1.18787i −0.804513 0.593935i \(-0.797574\pi\)
0.804513 0.593935i \(-0.202426\pi\)
\(908\) 1.09637e31i 0.955113i
\(909\) 6.15281e30 0.529850
\(910\) 0 0
\(911\) −1.28886e31 −1.08458 −0.542292 0.840190i \(-0.682443\pi\)
−0.542292 + 0.840190i \(0.682443\pi\)
\(912\) − 3.96287e30i − 0.329658i
\(913\) − 4.63317e30i − 0.381009i
\(914\) 3.33860e30 0.271413
\(915\) 0 0
\(916\) −1.39117e31 −1.10529
\(917\) − 1.06494e31i − 0.836463i
\(918\) − 4.89281e29i − 0.0379936i
\(919\) −1.25018e30 −0.0959753 −0.0479876 0.998848i \(-0.515281\pi\)
−0.0479876 + 0.998848i \(0.515281\pi\)
\(920\) 0 0
\(921\) 2.80864e31 2.10751
\(922\) 2.95239e30i 0.219028i
\(923\) 4.54123e29i 0.0333086i
\(924\) −1.88816e31 −1.36925
\(925\) 0 0
\(926\) 1.64967e30 0.116945
\(927\) 3.62130e30i 0.253821i
\(928\) 1.53173e31i 1.06152i
\(929\) 1.41245e31 0.967850 0.483925 0.875109i \(-0.339211\pi\)
0.483925 + 0.875109i \(0.339211\pi\)
\(930\) 0 0
\(931\) −2.48706e29 −0.0166615
\(932\) 9.13746e30i 0.605283i
\(933\) 5.21360e30i 0.341492i
\(934\) 3.24455e30 0.210142
\(935\) 0 0
\(936\) −5.86177e29 −0.0371221
\(937\) 7.12851e30i 0.446409i 0.974772 + 0.223204i \(0.0716518\pi\)
−0.974772 + 0.223204i \(0.928348\pi\)
\(938\) 3.75196e30i 0.232342i
\(939\) −1.12632e30 −0.0689721
\(940\) 0 0
\(941\) −1.07254e29 −0.00642275 −0.00321138 0.999995i \(-0.501022\pi\)
−0.00321138 + 0.999995i \(0.501022\pi\)
\(942\) − 2.35861e30i − 0.139676i
\(943\) 1.52290e30i 0.0891864i
\(944\) 2.32049e31 1.34392
\(945\) 0 0
\(946\) −5.28172e30 −0.299170
\(947\) − 2.92935e31i − 1.64095i −0.571680 0.820477i \(-0.693708\pi\)
0.571680 0.820477i \(-0.306292\pi\)
\(948\) − 1.33042e31i − 0.737058i
\(949\) −3.48969e30 −0.191202
\(950\) 0 0
\(951\) 4.11989e30 0.220796
\(952\) 2.77593e30i 0.147137i
\(953\) 3.04320e31i 1.59535i 0.603088 + 0.797675i \(0.293937\pi\)
−0.603088 + 0.797675i \(0.706063\pi\)
\(954\) 3.60456e30 0.186893
\(955\) 0 0
\(956\) 5.50872e30 0.279410
\(957\) 5.19071e31i 2.60406i
\(958\) 3.38034e30i 0.167735i
\(959\) −3.07838e31 −1.51087
\(960\) 0 0
\(961\) −1.72134e31 −0.826556
\(962\) − 5.15264e29i − 0.0244733i
\(963\) 6.89017e30i 0.323709i
\(964\) −1.62931e31 −0.757171
\(965\) 0 0
\(966\) 2.10468e30 0.0957032
\(967\) 3.78485e31i 1.70244i 0.524813 + 0.851218i \(0.324135\pi\)
−0.524813 + 0.851218i \(0.675865\pi\)
\(968\) 1.86203e30i 0.0828504i
\(969\) −3.11500e30 −0.137106
\(970\) 0 0
\(971\) −1.62334e31 −0.699212 −0.349606 0.936897i \(-0.613684\pi\)
−0.349606 + 0.936897i \(0.613684\pi\)
\(972\) 2.35532e31i 1.00358i
\(973\) − 3.43780e31i − 1.44909i
\(974\) 1.36878e30 0.0570776
\(975\) 0 0
\(976\) 2.40387e31 0.981043
\(977\) − 2.98858e30i − 0.120662i −0.998178 0.0603312i \(-0.980784\pi\)
0.998178 0.0603312i \(-0.0192157\pi\)
\(978\) 3.22269e30i 0.128724i
\(979\) −4.77701e31 −1.88772
\(980\) 0 0
\(981\) 1.05857e31 0.409445
\(982\) − 7.48330e30i − 0.286367i
\(983\) − 2.41371e31i − 0.913846i −0.889506 0.456923i \(-0.848951\pi\)
0.889506 0.456923i \(-0.151049\pi\)
\(984\) −3.14623e30 −0.117853
\(985\) 0 0
\(986\) 3.73866e30 0.137091
\(987\) − 1.45436e31i − 0.527643i
\(988\) − 1.28625e30i − 0.0461718i
\(989\) −1.42969e31 −0.507786
\(990\) 0 0
\(991\) −2.29436e31 −0.797790 −0.398895 0.916997i \(-0.630606\pi\)
−0.398895 + 0.916997i \(0.630606\pi\)
\(992\) 6.84481e30i 0.235499i
\(993\) 3.24964e31i 1.10629i
\(994\) 1.24600e30 0.0419723
\(995\) 0 0
\(996\) 1.26935e31 0.418658
\(997\) − 3.55036e31i − 1.15871i −0.815076 0.579354i \(-0.803305\pi\)
0.815076 0.579354i \(-0.196695\pi\)
\(998\) − 1.10738e31i − 0.357624i
\(999\) −1.23517e31 −0.394719
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 25.22.b.a.24.2 2
5.2 odd 4 1.22.a.a.1.1 1
5.3 odd 4 25.22.a.a.1.1 1
5.4 even 2 inner 25.22.b.a.24.1 2
15.2 even 4 9.22.a.c.1.1 1
20.7 even 4 16.22.a.c.1.1 1
35.27 even 4 49.22.a.a.1.1 1
40.27 even 4 64.22.a.a.1.1 1
40.37 odd 4 64.22.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 1 5.2 odd 4
9.22.a.c.1.1 1 15.2 even 4
16.22.a.c.1.1 1 20.7 even 4
25.22.a.a.1.1 1 5.3 odd 4
25.22.b.a.24.1 2 5.4 even 2 inner
25.22.b.a.24.2 2 1.1 even 1 trivial
49.22.a.a.1.1 1 35.27 even 4
64.22.a.a.1.1 1 40.27 even 4
64.22.a.g.1.1 1 40.37 odd 4