Properties

Label 25.34.b.b.24.8
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $10$
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] = [25,34,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, 34, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 34);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
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: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 2 x^{8} + 325405868686 x^{7} + \cdots + 34\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{10}\cdot 5^{24}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.8
Root \(-8230.22 + 8230.22i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.b.24.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+71937.8i q^{2} -1.37573e8i q^{3} +3.41489e9 q^{4} +9.89671e12 q^{6} -1.07684e14i q^{7} +8.63600e14i q^{8} -1.33673e16 q^{9} +O(q^{10})\) \(q+71937.8i q^{2} -1.37573e8i q^{3} +3.41489e9 q^{4} +9.89671e12 q^{6} -1.07684e14i q^{7} +8.63600e14i q^{8} -1.33673e16 q^{9} -2.33530e17 q^{11} -4.69797e17i q^{12} +2.28395e18i q^{13} +7.74658e18 q^{14} -3.27918e19 q^{16} -2.09885e20i q^{17} -9.61615e20i q^{18} -6.01644e20 q^{19} -1.48145e22 q^{21} -1.67997e22i q^{22} -4.84991e22i q^{23} +1.18808e23 q^{24} -1.64302e23 q^{26} +1.07421e24i q^{27} -3.67730e23i q^{28} +8.15476e22 q^{29} -1.49925e24 q^{31} +5.05930e24i q^{32} +3.21275e25i q^{33} +1.50986e25 q^{34} -4.56479e25 q^{36} +1.78708e25i q^{37} -4.32809e25i q^{38} +3.14210e26 q^{39} -1.94621e26 q^{41} -1.06572e27i q^{42} +9.48110e26i q^{43} -7.97481e26 q^{44} +3.48891e27 q^{46} -6.29157e26i q^{47} +4.51128e27i q^{48} -3.86494e27 q^{49} -2.88745e28 q^{51} +7.79943e27i q^{52} +8.56042e27i q^{53} -7.72760e28 q^{54} +9.29963e28 q^{56} +8.27701e28i q^{57} +5.86635e27i q^{58} -5.43089e27 q^{59} +1.79220e29 q^{61} -1.07853e29i q^{62} +1.43945e30i q^{63} -6.45635e29 q^{64} -2.31118e30 q^{66} -3.33703e29i q^{67} -7.16732e29i q^{68} -6.67217e30 q^{69} -3.44997e30 q^{71} -1.15440e31i q^{72} +7.11823e30i q^{73} -1.28559e30 q^{74} -2.05455e30 q^{76} +2.51476e31i q^{77} +2.26036e31i q^{78} +1.57830e31 q^{79} +7.34723e31 q^{81} -1.40006e31i q^{82} -3.45599e31i q^{83} -5.05898e31 q^{84} -6.82049e31 q^{86} -1.12188e31i q^{87} -2.01677e32i q^{88} +2.35733e32 q^{89} +2.45946e32 q^{91} -1.65619e32i q^{92} +2.06256e32i q^{93} +4.52602e31 q^{94} +6.96024e32 q^{96} -3.55763e32i q^{97} -2.78035e32i q^{98} +3.12168e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2282622720 q^{4} + 25850126231520 q^{6} - 29\!\cdots\!30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2282622720 q^{4} + 25850126231520 q^{6} - 29\!\cdots\!30 q^{9}+ \cdots + 35\!\cdots\!40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 71937.8i 0.776180i 0.921622 + 0.388090i \(0.126865\pi\)
−0.921622 + 0.388090i \(0.873135\pi\)
\(3\) − 1.37573e8i − 1.84516i −0.385811 0.922578i \(-0.626078\pi\)
0.385811 0.922578i \(-0.373922\pi\)
\(4\) 3.41489e9 0.397545
\(5\) 0 0
\(6\) 9.89671e12 1.43217
\(7\) − 1.07684e14i − 1.22472i −0.790581 0.612358i \(-0.790221\pi\)
0.790581 0.612358i \(-0.209779\pi\)
\(8\) 8.63600e14i 1.08475i
\(9\) −1.33673e16 −2.40460
\(10\) 0 0
\(11\) −2.33530e17 −1.53237 −0.766186 0.642619i \(-0.777848\pi\)
−0.766186 + 0.642619i \(0.777848\pi\)
\(12\) − 4.69797e17i − 0.733533i
\(13\) 2.28395e18i 0.951965i 0.879455 + 0.475983i \(0.157908\pi\)
−0.879455 + 0.475983i \(0.842092\pi\)
\(14\) 7.74658e18 0.950599
\(15\) 0 0
\(16\) −3.27918e19 −0.444412
\(17\) − 2.09885e20i − 1.04610i −0.852302 0.523050i \(-0.824794\pi\)
0.852302 0.523050i \(-0.175206\pi\)
\(18\) − 9.61615e20i − 1.86640i
\(19\) −6.01644e20 −0.478526 −0.239263 0.970955i \(-0.576906\pi\)
−0.239263 + 0.970955i \(0.576906\pi\)
\(20\) 0 0
\(21\) −1.48145e22 −2.25979
\(22\) − 1.67997e22i − 1.18940i
\(23\) − 4.84991e22i − 1.64901i −0.565853 0.824506i \(-0.691453\pi\)
0.565853 0.824506i \(-0.308547\pi\)
\(24\) 1.18808e23 2.00153
\(25\) 0 0
\(26\) −1.64302e23 −0.738896
\(27\) 1.07421e24i 2.59171i
\(28\) − 3.67730e23i − 0.486880i
\(29\) 8.15476e22 0.0605124 0.0302562 0.999542i \(-0.490368\pi\)
0.0302562 + 0.999542i \(0.490368\pi\)
\(30\) 0 0
\(31\) −1.49925e24 −0.370174 −0.185087 0.982722i \(-0.559257\pi\)
−0.185087 + 0.982722i \(0.559257\pi\)
\(32\) 5.05930e24i 0.739802i
\(33\) 3.21275e25i 2.82747i
\(34\) 1.50986e25 0.811961
\(35\) 0 0
\(36\) −4.56479e25 −0.955938
\(37\) 1.78708e25i 0.238131i 0.992886 + 0.119066i \(0.0379899\pi\)
−0.992886 + 0.119066i \(0.962010\pi\)
\(38\) − 4.32809e25i − 0.371422i
\(39\) 3.14210e26 1.75652
\(40\) 0 0
\(41\) −1.94621e26 −0.476712 −0.238356 0.971178i \(-0.576608\pi\)
−0.238356 + 0.971178i \(0.576608\pi\)
\(42\) − 1.06572e27i − 1.75400i
\(43\) 9.48110e26i 1.05835i 0.848513 + 0.529175i \(0.177498\pi\)
−0.848513 + 0.529175i \(0.822502\pi\)
\(44\) −7.97481e26 −0.609187
\(45\) 0 0
\(46\) 3.48891e27 1.27993
\(47\) − 6.29157e26i − 0.161862i −0.996720 0.0809309i \(-0.974211\pi\)
0.996720 0.0809309i \(-0.0257893\pi\)
\(48\) 4.51128e27i 0.820010i
\(49\) −3.86494e27 −0.499927
\(50\) 0 0
\(51\) −2.88745e28 −1.93022
\(52\) 7.79943e27i 0.378449i
\(53\) 8.56042e27i 0.303350i 0.988430 + 0.151675i \(0.0484666\pi\)
−0.988430 + 0.151675i \(0.951533\pi\)
\(54\) −7.72760e28 −2.01163
\(55\) 0 0
\(56\) 9.29963e28 1.32851
\(57\) 8.27701e28i 0.882954i
\(58\) 5.86635e27i 0.0469685i
\(59\) −5.43089e27 −0.0327954 −0.0163977 0.999866i \(-0.505220\pi\)
−0.0163977 + 0.999866i \(0.505220\pi\)
\(60\) 0 0
\(61\) 1.79220e29 0.624373 0.312186 0.950021i \(-0.398939\pi\)
0.312186 + 0.950021i \(0.398939\pi\)
\(62\) − 1.07853e29i − 0.287321i
\(63\) 1.43945e30i 2.94495i
\(64\) −6.45635e29 −1.01863
\(65\) 0 0
\(66\) −2.31118e30 −2.19462
\(67\) − 3.33703e29i − 0.247244i −0.992329 0.123622i \(-0.960549\pi\)
0.992329 0.123622i \(-0.0394510\pi\)
\(68\) − 7.16732e29i − 0.415872i
\(69\) −6.67217e30 −3.04269
\(70\) 0 0
\(71\) −3.44997e30 −0.981866 −0.490933 0.871197i \(-0.663344\pi\)
−0.490933 + 0.871197i \(0.663344\pi\)
\(72\) − 1.15440e31i − 2.60838i
\(73\) 7.11823e30i 1.28099i 0.767963 + 0.640494i \(0.221270\pi\)
−0.767963 + 0.640494i \(0.778730\pi\)
\(74\) −1.28559e30 −0.184833
\(75\) 0 0
\(76\) −2.05455e30 −0.190236
\(77\) 2.51476e31i 1.87672i
\(78\) 2.26036e31i 1.36338i
\(79\) 1.57830e31 0.771510 0.385755 0.922601i \(-0.373941\pi\)
0.385755 + 0.922601i \(0.373941\pi\)
\(80\) 0 0
\(81\) 7.34723e31 2.37750
\(82\) − 1.40006e31i − 0.370014i
\(83\) − 3.45599e31i − 0.747798i −0.927470 0.373899i \(-0.878021\pi\)
0.927470 0.373899i \(-0.121979\pi\)
\(84\) −5.05898e31 −0.898369
\(85\) 0 0
\(86\) −6.82049e31 −0.821469
\(87\) − 1.12188e31i − 0.111655i
\(88\) − 2.01677e32i − 1.66223i
\(89\) 2.35733e32 1.61244 0.806222 0.591613i \(-0.201509\pi\)
0.806222 + 0.591613i \(0.201509\pi\)
\(90\) 0 0
\(91\) 2.45946e32 1.16589
\(92\) − 1.65619e32i − 0.655557i
\(93\) 2.06256e32i 0.683029i
\(94\) 4.52602e31 0.125634
\(95\) 0 0
\(96\) 6.96024e32 1.36505
\(97\) − 3.55763e32i − 0.588067i −0.955795 0.294033i \(-0.905002\pi\)
0.955795 0.294033i \(-0.0949977\pi\)
\(98\) − 2.78035e32i − 0.388033i
\(99\) 3.12168e33 3.68474
\(100\) 0 0
\(101\) 9.03859e32 0.767004 0.383502 0.923540i \(-0.374718\pi\)
0.383502 + 0.923540i \(0.374718\pi\)
\(102\) − 2.07717e33i − 1.49820i
\(103\) − 2.36642e33i − 1.45304i −0.687146 0.726519i \(-0.741137\pi\)
0.687146 0.726519i \(-0.258863\pi\)
\(104\) −1.97242e33 −1.03264
\(105\) 0 0
\(106\) −6.15818e32 −0.235454
\(107\) 2.84756e33i 0.932482i 0.884658 + 0.466241i \(0.154392\pi\)
−0.884658 + 0.466241i \(0.845608\pi\)
\(108\) 3.66830e33i 1.03032i
\(109\) −3.46217e33 −0.835241 −0.417620 0.908622i \(-0.637136\pi\)
−0.417620 + 0.908622i \(0.637136\pi\)
\(110\) 0 0
\(111\) 2.45855e33 0.439389
\(112\) 3.53117e33i 0.544279i
\(113\) 9.04247e33i 1.20363i 0.798636 + 0.601815i \(0.205556\pi\)
−0.798636 + 0.601815i \(0.794444\pi\)
\(114\) −5.95430e33 −0.685331
\(115\) 0 0
\(116\) 2.78476e32 0.0240564
\(117\) − 3.05303e34i − 2.28910i
\(118\) − 3.90687e32i − 0.0254551i
\(119\) −2.26013e34 −1.28117
\(120\) 0 0
\(121\) 3.13113e34 1.34816
\(122\) 1.28927e34i 0.484626i
\(123\) 2.67746e34i 0.879608i
\(124\) −5.11977e33 −0.147161
\(125\) 0 0
\(126\) −1.03551e35 −2.28581
\(127\) − 4.59458e33i − 0.0890194i −0.999009 0.0445097i \(-0.985827\pi\)
0.999009 0.0445097i \(-0.0141726\pi\)
\(128\) − 2.98648e33i − 0.0508388i
\(129\) 1.30434e35 1.95282
\(130\) 0 0
\(131\) −2.50614e34 −0.291091 −0.145545 0.989352i \(-0.546494\pi\)
−0.145545 + 0.989352i \(0.546494\pi\)
\(132\) 1.09712e35i 1.12405i
\(133\) 6.47877e34i 0.586058i
\(134\) 2.40059e34 0.191906
\(135\) 0 0
\(136\) 1.81256e35 1.13475
\(137\) 2.78859e35i 1.54702i 0.633784 + 0.773510i \(0.281501\pi\)
−0.633784 + 0.773510i \(0.718499\pi\)
\(138\) − 4.79981e35i − 2.36167i
\(139\) −1.40995e35 −0.615827 −0.307914 0.951414i \(-0.599631\pi\)
−0.307914 + 0.951414i \(0.599631\pi\)
\(140\) 0 0
\(141\) −8.65551e34 −0.298660
\(142\) − 2.48183e35i − 0.762104i
\(143\) − 5.33372e35i − 1.45877i
\(144\) 4.38339e35 1.06863
\(145\) 0 0
\(146\) −5.12070e35 −0.994277
\(147\) 5.31711e35i 0.922444i
\(148\) 6.10269e34i 0.0946680i
\(149\) 4.84716e35 0.672844 0.336422 0.941711i \(-0.390783\pi\)
0.336422 + 0.941711i \(0.390783\pi\)
\(150\) 0 0
\(151\) −1.29129e35 −0.143848 −0.0719238 0.997410i \(-0.522914\pi\)
−0.0719238 + 0.997410i \(0.522914\pi\)
\(152\) − 5.19580e35i − 0.519079i
\(153\) 2.80559e36i 2.51545i
\(154\) −1.80906e36 −1.45667
\(155\) 0 0
\(156\) 1.07299e36 0.698298
\(157\) 1.48727e36i 0.871052i 0.900176 + 0.435526i \(0.143438\pi\)
−0.900176 + 0.435526i \(0.856562\pi\)
\(158\) 1.13539e36i 0.598830i
\(159\) 1.17768e36 0.559727
\(160\) 0 0
\(161\) −5.22259e36 −2.01957
\(162\) 5.28543e36i 1.84537i
\(163\) 5.65066e36i 1.78240i 0.453615 + 0.891198i \(0.350134\pi\)
−0.453615 + 0.891198i \(0.649866\pi\)
\(164\) −6.64609e35 −0.189515
\(165\) 0 0
\(166\) 2.48616e36 0.580425
\(167\) − 8.79383e36i − 1.85933i −0.368407 0.929665i \(-0.620097\pi\)
0.368407 0.929665i \(-0.379903\pi\)
\(168\) − 1.27938e37i − 2.45130i
\(169\) 5.39705e35 0.0937617
\(170\) 0 0
\(171\) 8.04237e36 1.15066
\(172\) 3.23769e36i 0.420742i
\(173\) 5.34130e36i 0.630792i 0.948960 + 0.315396i \(0.102137\pi\)
−0.948960 + 0.315396i \(0.897863\pi\)
\(174\) 8.07053e35 0.0866641
\(175\) 0 0
\(176\) 7.65790e36 0.681005
\(177\) 7.47145e35i 0.0605127i
\(178\) 1.69581e37i 1.25155i
\(179\) 1.75097e37 1.17816 0.589080 0.808075i \(-0.299490\pi\)
0.589080 + 0.808075i \(0.299490\pi\)
\(180\) 0 0
\(181\) 1.00404e37 0.562411 0.281205 0.959648i \(-0.409266\pi\)
0.281205 + 0.959648i \(0.409266\pi\)
\(182\) 1.76928e37i 0.904937i
\(183\) − 2.46559e37i − 1.15207i
\(184\) 4.18838e37 1.78876
\(185\) 0 0
\(186\) −1.48376e37 −0.530153
\(187\) 4.90144e37i 1.60301i
\(188\) − 2.14850e36i − 0.0643474i
\(189\) 1.15675e38 3.17410
\(190\) 0 0
\(191\) −1.96610e37 −0.453477 −0.226739 0.973956i \(-0.572806\pi\)
−0.226739 + 0.973956i \(0.572806\pi\)
\(192\) 8.88220e37i 1.87953i
\(193\) − 3.92456e37i − 0.762246i −0.924524 0.381123i \(-0.875537\pi\)
0.924524 0.381123i \(-0.124463\pi\)
\(194\) 2.55928e37 0.456446
\(195\) 0 0
\(196\) −1.31983e37 −0.198744
\(197\) − 1.18069e38i − 1.63471i −0.576131 0.817357i \(-0.695438\pi\)
0.576131 0.817357i \(-0.304562\pi\)
\(198\) 2.24566e38i 2.86002i
\(199\) −1.43693e38 −1.68407 −0.842035 0.539423i \(-0.818642\pi\)
−0.842035 + 0.539423i \(0.818642\pi\)
\(200\) 0 0
\(201\) −4.59086e37 −0.456204
\(202\) 6.50216e37i 0.595333i
\(203\) − 8.78140e36i − 0.0741104i
\(204\) −9.86031e37 −0.767349
\(205\) 0 0
\(206\) 1.70235e38 1.12782
\(207\) 6.48302e38i 3.96522i
\(208\) − 7.48949e37i − 0.423065i
\(209\) 1.40502e38 0.733279
\(210\) 0 0
\(211\) −2.97825e38 −1.32831 −0.664157 0.747593i \(-0.731210\pi\)
−0.664157 + 0.747593i \(0.731210\pi\)
\(212\) 2.92329e37i 0.120595i
\(213\) 4.74623e38i 1.81170i
\(214\) −2.04847e38 −0.723773
\(215\) 0 0
\(216\) −9.27685e38 −2.81134
\(217\) 1.61446e38i 0.453358i
\(218\) − 2.49061e38i − 0.648297i
\(219\) 9.79278e38 2.36362
\(220\) 0 0
\(221\) 4.79366e38 0.995851
\(222\) 1.76863e38i 0.341045i
\(223\) 1.05983e38i 0.189761i 0.995489 + 0.0948804i \(0.0302469\pi\)
−0.995489 + 0.0948804i \(0.969753\pi\)
\(224\) 5.44808e38 0.906047
\(225\) 0 0
\(226\) −6.50496e38 −0.934233
\(227\) 7.63192e38i 1.01908i 0.860448 + 0.509538i \(0.170184\pi\)
−0.860448 + 0.509538i \(0.829816\pi\)
\(228\) 2.82651e38i 0.351014i
\(229\) −2.74424e38 −0.317057 −0.158528 0.987354i \(-0.550675\pi\)
−0.158528 + 0.987354i \(0.550675\pi\)
\(230\) 0 0
\(231\) 3.45963e39 3.46284
\(232\) 7.04245e37i 0.0656406i
\(233\) 7.50623e38i 0.651702i 0.945421 + 0.325851i \(0.105651\pi\)
−0.945421 + 0.325851i \(0.894349\pi\)
\(234\) 2.19628e39 1.77675
\(235\) 0 0
\(236\) −1.85459e37 −0.0130377
\(237\) − 2.17131e39i − 1.42356i
\(238\) − 1.62589e39i − 0.994422i
\(239\) 9.41651e38 0.537434 0.268717 0.963219i \(-0.413400\pi\)
0.268717 + 0.963219i \(0.413400\pi\)
\(240\) 0 0
\(241\) −4.90507e38 −0.243985 −0.121993 0.992531i \(-0.538928\pi\)
−0.121993 + 0.992531i \(0.538928\pi\)
\(242\) 2.25247e39i 1.04642i
\(243\) − 4.13623e39i − 1.79515i
\(244\) 6.12016e38 0.248217
\(245\) 0 0
\(246\) −1.92611e39 −0.682734
\(247\) − 1.37412e39i − 0.455540i
\(248\) − 1.29475e39i − 0.401545i
\(249\) −4.75452e39 −1.37980
\(250\) 0 0
\(251\) 5.96650e39 1.51741 0.758705 0.651434i \(-0.225832\pi\)
0.758705 + 0.651434i \(0.225832\pi\)
\(252\) 4.91557e39i 1.17075i
\(253\) 1.13260e40i 2.52690i
\(254\) 3.30524e38 0.0690950
\(255\) 0 0
\(256\) −5.33112e39 −0.979172
\(257\) − 8.25525e39i − 1.42178i −0.703301 0.710892i \(-0.748291\pi\)
0.703301 0.710892i \(-0.251709\pi\)
\(258\) 9.38317e39i 1.51574i
\(259\) 1.92441e39 0.291643
\(260\) 0 0
\(261\) −1.09007e39 −0.145508
\(262\) − 1.80286e39i − 0.225939i
\(263\) − 6.74712e39i − 0.794051i −0.917808 0.397026i \(-0.870042\pi\)
0.917808 0.397026i \(-0.129958\pi\)
\(264\) −2.77454e40 −3.06708
\(265\) 0 0
\(266\) −4.66068e39 −0.454886
\(267\) − 3.24305e40i − 2.97521i
\(268\) − 1.13956e39i − 0.0982907i
\(269\) 2.97390e39 0.241220 0.120610 0.992700i \(-0.461515\pi\)
0.120610 + 0.992700i \(0.461515\pi\)
\(270\) 0 0
\(271\) −7.89524e39 −0.566725 −0.283362 0.959013i \(-0.591450\pi\)
−0.283362 + 0.959013i \(0.591450\pi\)
\(272\) 6.88250e39i 0.464900i
\(273\) − 3.38355e40i − 2.15124i
\(274\) −2.00605e40 −1.20076
\(275\) 0 0
\(276\) −2.27847e40 −1.20961
\(277\) 2.12171e40i 1.06113i 0.847644 + 0.530566i \(0.178020\pi\)
−0.847644 + 0.530566i \(0.821980\pi\)
\(278\) − 1.01429e40i − 0.477993i
\(279\) 2.00409e40 0.890120
\(280\) 0 0
\(281\) 4.91397e40 1.93989 0.969947 0.243317i \(-0.0782355\pi\)
0.969947 + 0.243317i \(0.0782355\pi\)
\(282\) − 6.22658e39i − 0.231814i
\(283\) 4.22019e40i 1.48203i 0.671491 + 0.741013i \(0.265654\pi\)
−0.671491 + 0.741013i \(0.734346\pi\)
\(284\) −1.17813e40 −0.390336
\(285\) 0 0
\(286\) 3.83696e40 1.13226
\(287\) 2.09576e40i 0.583836i
\(288\) − 6.76292e40i − 1.77893i
\(289\) −3.79702e39 −0.0943253
\(290\) 0 0
\(291\) −4.89434e40 −1.08508
\(292\) 2.43080e40i 0.509251i
\(293\) − 2.96004e40i − 0.586115i −0.956095 0.293057i \(-0.905327\pi\)
0.956095 0.293057i \(-0.0946727\pi\)
\(294\) −3.82502e40 −0.715982
\(295\) 0 0
\(296\) −1.54333e40 −0.258312
\(297\) − 2.50860e41i − 3.97146i
\(298\) 3.48694e40i 0.522248i
\(299\) 1.10769e41 1.56980
\(300\) 0 0
\(301\) 1.02097e41 1.29618
\(302\) − 9.28922e39i − 0.111652i
\(303\) − 1.24347e41i − 1.41524i
\(304\) 1.97290e40 0.212663
\(305\) 0 0
\(306\) −2.01828e41 −1.95244
\(307\) − 1.70501e41i − 1.56295i −0.623938 0.781474i \(-0.714468\pi\)
0.623938 0.781474i \(-0.285532\pi\)
\(308\) 8.58762e40i 0.746081i
\(309\) −3.25556e41 −2.68108
\(310\) 0 0
\(311\) 1.09238e41 0.808774 0.404387 0.914588i \(-0.367485\pi\)
0.404387 + 0.914588i \(0.367485\pi\)
\(312\) 2.71352e41i 1.90538i
\(313\) 2.55151e41i 1.69948i 0.527205 + 0.849738i \(0.323240\pi\)
−0.527205 + 0.849738i \(0.676760\pi\)
\(314\) −1.06991e41 −0.676093
\(315\) 0 0
\(316\) 5.38971e40 0.306710
\(317\) 1.40033e41i 0.756400i 0.925724 + 0.378200i \(0.123457\pi\)
−0.925724 + 0.378200i \(0.876543\pi\)
\(318\) 8.47200e40i 0.434449i
\(319\) −1.90438e40 −0.0927275
\(320\) 0 0
\(321\) 3.91748e41 1.72057
\(322\) − 3.75702e41i − 1.56755i
\(323\) 1.26276e41i 0.500586i
\(324\) 2.50900e41 0.945164
\(325\) 0 0
\(326\) −4.06496e41 −1.38346
\(327\) 4.76302e41i 1.54115i
\(328\) − 1.68075e41i − 0.517111i
\(329\) −6.77504e40 −0.198235
\(330\) 0 0
\(331\) −2.36103e41 −0.625086 −0.312543 0.949904i \(-0.601181\pi\)
−0.312543 + 0.949904i \(0.601181\pi\)
\(332\) − 1.18018e41i − 0.297283i
\(333\) − 2.38885e41i − 0.572611i
\(334\) 6.32609e41 1.44317
\(335\) 0 0
\(336\) 4.85794e41 1.00428
\(337\) 5.72695e41i 1.12728i 0.826022 + 0.563638i \(0.190599\pi\)
−0.826022 + 0.563638i \(0.809401\pi\)
\(338\) 3.88252e40i 0.0727759i
\(339\) 1.24400e42 2.22088
\(340\) 0 0
\(341\) 3.50120e41 0.567244
\(342\) 5.78550e41i 0.893121i
\(343\) − 4.16314e41i − 0.612446i
\(344\) −8.18788e41 −1.14804
\(345\) 0 0
\(346\) −3.84241e41 −0.489608
\(347\) 7.23409e41i 0.878917i 0.898263 + 0.439458i \(0.144830\pi\)
−0.898263 + 0.439458i \(0.855170\pi\)
\(348\) − 3.83108e40i − 0.0443878i
\(349\) 1.72145e42 1.90228 0.951140 0.308761i \(-0.0999145\pi\)
0.951140 + 0.308761i \(0.0999145\pi\)
\(350\) 0 0
\(351\) −2.45343e42 −2.46721
\(352\) − 1.18150e42i − 1.13365i
\(353\) 1.10466e42i 1.01145i 0.862695 + 0.505725i \(0.168775\pi\)
−0.862695 + 0.505725i \(0.831225\pi\)
\(354\) −5.37480e40 −0.0469687
\(355\) 0 0
\(356\) 8.05001e41 0.641020
\(357\) 3.10933e42i 2.36397i
\(358\) 1.25961e42i 0.914464i
\(359\) −1.24115e42 −0.860528 −0.430264 0.902703i \(-0.641580\pi\)
−0.430264 + 0.902703i \(0.641580\pi\)
\(360\) 0 0
\(361\) −1.21880e42 −0.771013
\(362\) 7.22283e41i 0.436532i
\(363\) − 4.30760e42i − 2.48757i
\(364\) 8.39877e41 0.463493
\(365\) 0 0
\(366\) 1.77369e42 0.894210
\(367\) 7.83869e41i 0.377792i 0.981997 + 0.188896i \(0.0604910\pi\)
−0.981997 + 0.188896i \(0.939509\pi\)
\(368\) 1.59037e42i 0.732842i
\(369\) 2.60156e42 1.14630
\(370\) 0 0
\(371\) 9.21824e41 0.371517
\(372\) 7.04343e41i 0.271535i
\(373\) 4.48131e42i 1.65276i 0.563116 + 0.826378i \(0.309603\pi\)
−0.563116 + 0.826378i \(0.690397\pi\)
\(374\) −3.52599e42 −1.24423
\(375\) 0 0
\(376\) 5.43340e41 0.175579
\(377\) 1.86251e41i 0.0576057i
\(378\) 8.32142e42i 2.46367i
\(379\) −5.32753e42 −1.51001 −0.755003 0.655721i \(-0.772365\pi\)
−0.755003 + 0.655721i \(0.772365\pi\)
\(380\) 0 0
\(381\) −6.32091e41 −0.164255
\(382\) − 1.41437e42i − 0.351980i
\(383\) − 7.66000e42i − 1.82578i −0.408203 0.912891i \(-0.633844\pi\)
0.408203 0.912891i \(-0.366156\pi\)
\(384\) −4.10860e41 −0.0938055
\(385\) 0 0
\(386\) 2.82324e42 0.591640
\(387\) − 1.26737e43i − 2.54491i
\(388\) − 1.21489e42i − 0.233783i
\(389\) −2.42415e42 −0.447085 −0.223543 0.974694i \(-0.571762\pi\)
−0.223543 + 0.974694i \(0.571762\pi\)
\(390\) 0 0
\(391\) −1.01792e43 −1.72503
\(392\) − 3.33776e42i − 0.542294i
\(393\) 3.44777e42i 0.537108i
\(394\) 8.49360e42 1.26883
\(395\) 0 0
\(396\) 1.06602e43 1.46485
\(397\) 4.43937e42i 0.585164i 0.956240 + 0.292582i \(0.0945145\pi\)
−0.956240 + 0.292582i \(0.905486\pi\)
\(398\) − 1.03370e43i − 1.30714i
\(399\) 8.91304e42 1.08137
\(400\) 0 0
\(401\) 6.39786e41 0.0714747 0.0357374 0.999361i \(-0.488622\pi\)
0.0357374 + 0.999361i \(0.488622\pi\)
\(402\) − 3.30256e42i − 0.354096i
\(403\) − 3.42421e42i − 0.352393i
\(404\) 3.08658e42 0.304919
\(405\) 0 0
\(406\) 6.31715e41 0.0575230
\(407\) − 4.17339e42i − 0.364906i
\(408\) − 2.49360e43i − 2.09380i
\(409\) 1.16096e43 0.936227 0.468114 0.883668i \(-0.344934\pi\)
0.468114 + 0.883668i \(0.344934\pi\)
\(410\) 0 0
\(411\) 3.83636e43 2.85449
\(412\) − 8.08105e42i − 0.577649i
\(413\) 5.84823e41i 0.0401650i
\(414\) −4.66374e43 −3.07772
\(415\) 0 0
\(416\) −1.15552e43 −0.704266
\(417\) 1.93971e43i 1.13630i
\(418\) 1.01074e43i 0.569156i
\(419\) 7.07188e42 0.382828 0.191414 0.981509i \(-0.438693\pi\)
0.191414 + 0.981509i \(0.438693\pi\)
\(420\) 0 0
\(421\) −2.32219e43 −1.16210 −0.581050 0.813868i \(-0.697358\pi\)
−0.581050 + 0.813868i \(0.697358\pi\)
\(422\) − 2.14249e43i − 1.03101i
\(423\) 8.41014e42i 0.389213i
\(424\) −7.39279e42 −0.329057
\(425\) 0 0
\(426\) −3.41433e43 −1.40620
\(427\) − 1.92992e43i − 0.764679i
\(428\) 9.72410e42i 0.370704i
\(429\) −7.33777e43 −2.69165
\(430\) 0 0
\(431\) −3.01488e43 −1.02422 −0.512112 0.858919i \(-0.671137\pi\)
−0.512112 + 0.858919i \(0.671137\pi\)
\(432\) − 3.52252e43i − 1.15179i
\(433\) − 3.63888e43i − 1.14530i −0.819802 0.572648i \(-0.805916\pi\)
0.819802 0.572648i \(-0.194084\pi\)
\(434\) −1.16141e43 −0.351887
\(435\) 0 0
\(436\) −1.18229e43 −0.332046
\(437\) 2.91792e43i 0.789095i
\(438\) 7.04471e43i 1.83460i
\(439\) 3.62391e42 0.0908893 0.0454446 0.998967i \(-0.485530\pi\)
0.0454446 + 0.998967i \(0.485530\pi\)
\(440\) 0 0
\(441\) 5.16638e43 1.20213
\(442\) 3.44845e43i 0.772959i
\(443\) − 2.84653e43i − 0.614686i −0.951599 0.307343i \(-0.900560\pi\)
0.951599 0.307343i \(-0.0994400\pi\)
\(444\) 8.39567e42 0.174677
\(445\) 0 0
\(446\) −7.62419e42 −0.147288
\(447\) − 6.66840e43i − 1.24150i
\(448\) 6.95248e43i 1.24753i
\(449\) 3.05597e43 0.528548 0.264274 0.964448i \(-0.414868\pi\)
0.264274 + 0.964448i \(0.414868\pi\)
\(450\) 0 0
\(451\) 4.54499e43 0.730500
\(452\) 3.08790e43i 0.478497i
\(453\) 1.77646e43i 0.265421i
\(454\) −5.49023e43 −0.790987
\(455\) 0 0
\(456\) −7.14803e43 −0.957781
\(457\) − 8.70256e42i − 0.112468i −0.998418 0.0562341i \(-0.982091\pi\)
0.998418 0.0562341i \(-0.0179093\pi\)
\(458\) − 1.97415e43i − 0.246093i
\(459\) 2.25459e44 2.71118
\(460\) 0 0
\(461\) 7.28964e43 0.815907 0.407954 0.913003i \(-0.366243\pi\)
0.407954 + 0.913003i \(0.366243\pi\)
\(462\) 2.48878e44i 2.68779i
\(463\) − 1.54940e44i − 1.61465i −0.590108 0.807324i \(-0.700915\pi\)
0.590108 0.807324i \(-0.299085\pi\)
\(464\) −2.67410e42 −0.0268924
\(465\) 0 0
\(466\) −5.39982e43 −0.505838
\(467\) − 9.15725e43i − 0.828011i −0.910274 0.414006i \(-0.864129\pi\)
0.910274 0.414006i \(-0.135871\pi\)
\(468\) − 1.04257e44i − 0.910020i
\(469\) −3.59346e43 −0.302804
\(470\) 0 0
\(471\) 2.04608e44 1.60723
\(472\) − 4.69012e42i − 0.0355747i
\(473\) − 2.21413e44i − 1.62179i
\(474\) 1.56199e44 1.10494
\(475\) 0 0
\(476\) −7.71809e43 −0.509325
\(477\) − 1.14430e44i − 0.729434i
\(478\) 6.77403e43i 0.417145i
\(479\) −3.67745e43 −0.218782 −0.109391 0.993999i \(-0.534890\pi\)
−0.109391 + 0.993999i \(0.534890\pi\)
\(480\) 0 0
\(481\) −4.08161e43 −0.226693
\(482\) − 3.52860e43i − 0.189376i
\(483\) 7.18488e44i 3.72642i
\(484\) 1.06925e44 0.535957
\(485\) 0 0
\(486\) 2.97552e44 1.39336
\(487\) 2.46290e44i 1.11485i 0.830226 + 0.557427i \(0.188211\pi\)
−0.830226 + 0.557427i \(0.811789\pi\)
\(488\) 1.54774e44i 0.677286i
\(489\) 7.77380e44 3.28880
\(490\) 0 0
\(491\) 1.21128e44 0.479071 0.239535 0.970888i \(-0.423005\pi\)
0.239535 + 0.970888i \(0.423005\pi\)
\(492\) 9.14324e43i 0.349684i
\(493\) − 1.71156e43i − 0.0633020i
\(494\) 9.88515e43 0.353581
\(495\) 0 0
\(496\) 4.91632e43 0.164510
\(497\) 3.71508e44i 1.20251i
\(498\) − 3.42029e44i − 1.07097i
\(499\) −4.07835e44 −1.23545 −0.617726 0.786393i \(-0.711946\pi\)
−0.617726 + 0.786393i \(0.711946\pi\)
\(500\) 0 0
\(501\) −1.20980e45 −3.43075
\(502\) 4.29217e44i 1.17778i
\(503\) 5.88791e44i 1.56347i 0.623612 + 0.781734i \(0.285665\pi\)
−0.623612 + 0.781734i \(0.714335\pi\)
\(504\) −1.24311e45 −3.19452
\(505\) 0 0
\(506\) −8.14768e44 −1.96133
\(507\) − 7.42489e43i − 0.173005i
\(508\) − 1.56900e43i − 0.0353892i
\(509\) −7.13633e44 −1.55823 −0.779115 0.626881i \(-0.784331\pi\)
−0.779115 + 0.626881i \(0.784331\pi\)
\(510\) 0 0
\(511\) 7.66523e44 1.56885
\(512\) − 4.09163e44i − 0.810852i
\(513\) − 6.46290e44i − 1.24020i
\(514\) 5.93864e44 1.10356
\(515\) 0 0
\(516\) 4.45419e44 0.776335
\(517\) 1.46927e44i 0.248033i
\(518\) 1.38438e44i 0.226367i
\(519\) 7.34820e44 1.16391
\(520\) 0 0
\(521\) −1.39893e44 −0.207957 −0.103978 0.994580i \(-0.533157\pi\)
−0.103978 + 0.994580i \(0.533157\pi\)
\(522\) − 7.84174e43i − 0.112940i
\(523\) 4.12677e44i 0.575880i 0.957648 + 0.287940i \(0.0929704\pi\)
−0.957648 + 0.287940i \(0.907030\pi\)
\(524\) −8.55818e43 −0.115722
\(525\) 0 0
\(526\) 4.85373e44 0.616326
\(527\) 3.14669e44i 0.387239i
\(528\) − 1.05352e45i − 1.25656i
\(529\) −1.48715e45 −1.71924
\(530\) 0 0
\(531\) 7.25965e43 0.0788599
\(532\) 2.21243e44i 0.232984i
\(533\) − 4.44504e44i − 0.453813i
\(534\) 2.33298e45 2.30930
\(535\) 0 0
\(536\) 2.88186e44 0.268197
\(537\) − 2.40887e45i − 2.17389i
\(538\) 2.13936e44i 0.187230i
\(539\) 9.02580e44 0.766075
\(540\) 0 0
\(541\) −2.35683e45 −1.88180 −0.940899 0.338687i \(-0.890017\pi\)
−0.940899 + 0.338687i \(0.890017\pi\)
\(542\) − 5.67966e44i − 0.439880i
\(543\) − 1.38129e45i − 1.03774i
\(544\) 1.06187e45 0.773907
\(545\) 0 0
\(546\) 2.43405e45 1.66975
\(547\) 2.11321e45i 1.40654i 0.710923 + 0.703270i \(0.248277\pi\)
−0.710923 + 0.703270i \(0.751723\pi\)
\(548\) 9.52274e44i 0.615010i
\(549\) −2.39569e45 −1.50137
\(550\) 0 0
\(551\) −4.90626e43 −0.0289567
\(552\) − 5.76209e45i − 3.30054i
\(553\) − 1.69958e45i − 0.944880i
\(554\) −1.52631e45 −0.823628
\(555\) 0 0
\(556\) −4.81482e44 −0.244819
\(557\) 2.27438e45i 1.12267i 0.827589 + 0.561335i \(0.189712\pi\)
−0.827589 + 0.561335i \(0.810288\pi\)
\(558\) 1.44170e45i 0.690893i
\(559\) −2.16544e45 −1.00751
\(560\) 0 0
\(561\) 6.74307e45 2.95781
\(562\) 3.53500e45i 1.50571i
\(563\) 1.75752e45i 0.726962i 0.931602 + 0.363481i \(0.118412\pi\)
−0.931602 + 0.363481i \(0.881588\pi\)
\(564\) −2.95576e44 −0.118731
\(565\) 0 0
\(566\) −3.03591e45 −1.15032
\(567\) − 7.91182e45i − 2.91176i
\(568\) − 2.97939e45i − 1.06508i
\(569\) −9.61177e44 −0.333773 −0.166886 0.985976i \(-0.553371\pi\)
−0.166886 + 0.985976i \(0.553371\pi\)
\(570\) 0 0
\(571\) −5.28842e45 −1.73313 −0.866563 0.499067i \(-0.833676\pi\)
−0.866563 + 0.499067i \(0.833676\pi\)
\(572\) − 1.82141e45i − 0.579925i
\(573\) 2.70483e45i 0.836737i
\(574\) −1.50765e45 −0.453162
\(575\) 0 0
\(576\) 8.63040e45 2.44940
\(577\) − 1.11538e45i − 0.307626i −0.988100 0.153813i \(-0.950845\pi\)
0.988100 0.153813i \(-0.0491554\pi\)
\(578\) − 2.73149e44i − 0.0732134i
\(579\) −5.39914e45 −1.40646
\(580\) 0 0
\(581\) −3.72156e45 −0.915839
\(582\) − 3.52088e45i − 0.842213i
\(583\) − 1.99912e45i − 0.464845i
\(584\) −6.14731e45 −1.38955
\(585\) 0 0
\(586\) 2.12939e45 0.454930
\(587\) 3.02679e45i 0.628714i 0.949305 + 0.314357i \(0.101789\pi\)
−0.949305 + 0.314357i \(0.898211\pi\)
\(588\) 1.81574e45i 0.366713i
\(589\) 9.02014e44 0.177138
\(590\) 0 0
\(591\) −1.62431e46 −3.01630
\(592\) − 5.86018e44i − 0.105829i
\(593\) − 2.02049e45i − 0.354857i −0.984134 0.177428i \(-0.943222\pi\)
0.984134 0.177428i \(-0.0567778\pi\)
\(594\) 1.80463e46 3.08256
\(595\) 0 0
\(596\) 1.65525e45 0.267486
\(597\) 1.97683e46i 3.10737i
\(598\) 7.96851e45i 1.21845i
\(599\) 7.17593e45 1.06742 0.533710 0.845667i \(-0.320797\pi\)
0.533710 + 0.845667i \(0.320797\pi\)
\(600\) 0 0
\(601\) −8.79572e44 −0.123835 −0.0619174 0.998081i \(-0.519722\pi\)
−0.0619174 + 0.998081i \(0.519722\pi\)
\(602\) 7.34461e45i 1.00607i
\(603\) 4.46071e45i 0.594523i
\(604\) −4.40960e44 −0.0571859
\(605\) 0 0
\(606\) 8.94523e45 1.09848
\(607\) − 1.35040e46i − 1.61380i −0.590686 0.806902i \(-0.701143\pi\)
0.590686 0.806902i \(-0.298857\pi\)
\(608\) − 3.04390e45i − 0.354014i
\(609\) −1.20809e45 −0.136745
\(610\) 0 0
\(611\) 1.43696e45 0.154087
\(612\) 9.58079e45i 1.00001i
\(613\) 1.09063e45i 0.110810i 0.998464 + 0.0554052i \(0.0176450\pi\)
−0.998464 + 0.0554052i \(0.982355\pi\)
\(614\) 1.22655e46 1.21313
\(615\) 0 0
\(616\) −2.17175e46 −2.03576
\(617\) 1.20780e46i 1.10227i 0.834415 + 0.551137i \(0.185806\pi\)
−0.834415 + 0.551137i \(0.814194\pi\)
\(618\) − 2.34197e46i − 2.08100i
\(619\) 3.00533e45 0.260014 0.130007 0.991513i \(-0.458500\pi\)
0.130007 + 0.991513i \(0.458500\pi\)
\(620\) 0 0
\(621\) 5.20980e46 4.27376
\(622\) 7.85834e45i 0.627754i
\(623\) − 2.53847e46i − 1.97478i
\(624\) −1.03035e46 −0.780621
\(625\) 0 0
\(626\) −1.83550e46 −1.31910
\(627\) − 1.93293e46i − 1.35301i
\(628\) 5.07885e45i 0.346283i
\(629\) 3.75081e45 0.249109
\(630\) 0 0
\(631\) −2.08614e46 −1.31480 −0.657400 0.753542i \(-0.728344\pi\)
−0.657400 + 0.753542i \(0.728344\pi\)
\(632\) 1.36302e46i 0.836892i
\(633\) 4.09728e46i 2.45095i
\(634\) −1.00736e46 −0.587102
\(635\) 0 0
\(636\) 4.02166e45 0.222517
\(637\) − 8.82732e45i − 0.475914i
\(638\) − 1.36997e45i − 0.0719732i
\(639\) 4.61168e46 2.36099
\(640\) 0 0
\(641\) −1.91329e46 −0.930299 −0.465149 0.885232i \(-0.653999\pi\)
−0.465149 + 0.885232i \(0.653999\pi\)
\(642\) 2.81815e46i 1.33547i
\(643\) 3.18460e46i 1.47087i 0.677597 + 0.735434i \(0.263022\pi\)
−0.677597 + 0.735434i \(0.736978\pi\)
\(644\) −1.78346e46 −0.802871
\(645\) 0 0
\(646\) −9.08400e45 −0.388544
\(647\) 3.25342e46i 1.35650i 0.734831 + 0.678250i \(0.237261\pi\)
−0.734831 + 0.678250i \(0.762739\pi\)
\(648\) 6.34507e46i 2.57898i
\(649\) 1.26828e45 0.0502548
\(650\) 0 0
\(651\) 2.22106e46 0.836516
\(652\) 1.92964e46i 0.708583i
\(653\) − 1.98304e46i − 0.710009i −0.934865 0.355005i \(-0.884479\pi\)
0.934865 0.355005i \(-0.115521\pi\)
\(654\) −3.42641e46 −1.19621
\(655\) 0 0
\(656\) 6.38198e45 0.211857
\(657\) − 9.51517e46i − 3.08026i
\(658\) − 4.87381e45i − 0.153866i
\(659\) −2.96542e45 −0.0913011 −0.0456506 0.998957i \(-0.514536\pi\)
−0.0456506 + 0.998957i \(0.514536\pi\)
\(660\) 0 0
\(661\) 1.20833e46 0.353886 0.176943 0.984221i \(-0.443379\pi\)
0.176943 + 0.984221i \(0.443379\pi\)
\(662\) − 1.69847e46i − 0.485179i
\(663\) − 6.59479e46i − 1.83750i
\(664\) 2.98460e46 0.811170
\(665\) 0 0
\(666\) 1.71849e46 0.444449
\(667\) − 3.95498e45i − 0.0997857i
\(668\) − 3.00300e46i − 0.739168i
\(669\) 1.45804e46 0.350138
\(670\) 0 0
\(671\) −4.18533e46 −0.956772
\(672\) − 7.49509e46i − 1.67180i
\(673\) − 2.52503e46i − 0.549565i −0.961506 0.274782i \(-0.911394\pi\)
0.961506 0.274782i \(-0.0886058\pi\)
\(674\) −4.11984e46 −0.874968
\(675\) 0 0
\(676\) 1.84303e45 0.0372745
\(677\) 3.98633e46i 0.786793i 0.919369 + 0.393396i \(0.128700\pi\)
−0.919369 + 0.393396i \(0.871300\pi\)
\(678\) 8.94908e46i 1.72381i
\(679\) −3.83101e46 −0.720215
\(680\) 0 0
\(681\) 1.04995e47 1.88036
\(682\) 2.51869e46i 0.440284i
\(683\) 3.89128e46i 0.663973i 0.943284 + 0.331986i \(0.107719\pi\)
−0.943284 + 0.331986i \(0.892281\pi\)
\(684\) 2.74638e46 0.457441
\(685\) 0 0
\(686\) 2.99487e46 0.475368
\(687\) 3.77534e46i 0.585019i
\(688\) − 3.10903e46i − 0.470344i
\(689\) −1.95516e46 −0.288778
\(690\) 0 0
\(691\) −3.16293e46 −0.445350 −0.222675 0.974893i \(-0.571479\pi\)
−0.222675 + 0.974893i \(0.571479\pi\)
\(692\) 1.82399e46i 0.250769i
\(693\) − 3.36156e47i − 4.51276i
\(694\) −5.20405e46 −0.682197
\(695\) 0 0
\(696\) 9.68853e45 0.121117
\(697\) 4.08479e46i 0.498688i
\(698\) 1.23837e47i 1.47651i
\(699\) 1.03266e47 1.20249
\(700\) 0 0
\(701\) −5.26072e46 −0.584384 −0.292192 0.956360i \(-0.594385\pi\)
−0.292192 + 0.956360i \(0.594385\pi\)
\(702\) − 1.76495e47i − 1.91500i
\(703\) − 1.07519e46i − 0.113952i
\(704\) 1.50775e47 1.56092
\(705\) 0 0
\(706\) −7.94665e46 −0.785066
\(707\) − 9.73315e46i − 0.939362i
\(708\) 2.55142e45i 0.0240565i
\(709\) −1.31306e47 −1.20954 −0.604771 0.796399i \(-0.706735\pi\)
−0.604771 + 0.796399i \(0.706735\pi\)
\(710\) 0 0
\(711\) −2.10976e47 −1.85517
\(712\) 2.03579e47i 1.74909i
\(713\) 7.27122e46i 0.610422i
\(714\) −2.23678e47 −1.83486
\(715\) 0 0
\(716\) 5.97938e46 0.468372
\(717\) − 1.29546e47i − 0.991649i
\(718\) − 8.92855e46i − 0.667925i
\(719\) −1.58298e47 −1.15731 −0.578653 0.815574i \(-0.696421\pi\)
−0.578653 + 0.815574i \(0.696421\pi\)
\(720\) 0 0
\(721\) −2.54826e47 −1.77956
\(722\) − 8.76774e46i − 0.598445i
\(723\) 6.74806e46i 0.450191i
\(724\) 3.42868e46 0.223584
\(725\) 0 0
\(726\) 3.09879e47 1.93080
\(727\) − 2.56013e47i − 1.55935i −0.626184 0.779676i \(-0.715384\pi\)
0.626184 0.779676i \(-0.284616\pi\)
\(728\) 2.12399e47i 1.26469i
\(729\) −1.60598e47 −0.934838
\(730\) 0 0
\(731\) 1.98994e47 1.10714
\(732\) − 8.41970e46i − 0.457998i
\(733\) 5.58232e46i 0.296893i 0.988920 + 0.148446i \(0.0474272\pi\)
−0.988920 + 0.148446i \(0.952573\pi\)
\(734\) −5.63898e46 −0.293235
\(735\) 0 0
\(736\) 2.45371e47 1.21994
\(737\) 7.79298e46i 0.378870i
\(738\) 1.87150e47i 0.889736i
\(739\) −1.40164e47 −0.651634 −0.325817 0.945433i \(-0.605639\pi\)
−0.325817 + 0.945433i \(0.605639\pi\)
\(740\) 0 0
\(741\) −1.89043e47 −0.840542
\(742\) 6.63140e46i 0.288364i
\(743\) − 3.74285e46i − 0.159180i −0.996828 0.0795898i \(-0.974639\pi\)
0.996828 0.0795898i \(-0.0253610\pi\)
\(744\) −1.78123e47 −0.740913
\(745\) 0 0
\(746\) −3.22375e47 −1.28284
\(747\) 4.61973e47i 1.79815i
\(748\) 1.67379e47i 0.637271i
\(749\) 3.06638e47 1.14202
\(750\) 0 0
\(751\) 2.45331e47 0.874365 0.437182 0.899373i \(-0.355976\pi\)
0.437182 + 0.899373i \(0.355976\pi\)
\(752\) 2.06312e46i 0.0719334i
\(753\) − 8.20830e47i − 2.79986i
\(754\) −1.33985e46 −0.0447124
\(755\) 0 0
\(756\) 3.95018e47 1.26185
\(757\) 3.89117e47i 1.21618i 0.793868 + 0.608090i \(0.208064\pi\)
−0.793868 + 0.608090i \(0.791936\pi\)
\(758\) − 3.83251e47i − 1.17204i
\(759\) 1.55815e48 4.66253
\(760\) 0 0
\(761\) −2.57108e47 −0.736662 −0.368331 0.929695i \(-0.620071\pi\)
−0.368331 + 0.929695i \(0.620071\pi\)
\(762\) − 4.54712e46i − 0.127491i
\(763\) 3.72822e47i 1.02293i
\(764\) −6.71402e46 −0.180278
\(765\) 0 0
\(766\) 5.51044e47 1.41714
\(767\) − 1.24039e46i − 0.0312201i
\(768\) 7.33419e47i 1.80672i
\(769\) −1.74753e47 −0.421348 −0.210674 0.977556i \(-0.567566\pi\)
−0.210674 + 0.977556i \(0.567566\pi\)
\(770\) 0 0
\(771\) −1.13570e48 −2.62341
\(772\) − 1.34019e47i − 0.303027i
\(773\) − 1.07838e47i − 0.238676i −0.992854 0.119338i \(-0.961923\pi\)
0.992854 0.119338i \(-0.0380772\pi\)
\(774\) 9.11717e47 1.97531
\(775\) 0 0
\(776\) 3.07237e47 0.637903
\(777\) − 2.64747e47i − 0.538127i
\(778\) − 1.74388e47i − 0.347019i
\(779\) 1.17093e47 0.228119
\(780\) 0 0
\(781\) 8.05672e47 1.50458
\(782\) − 7.32269e47i − 1.33893i
\(783\) 8.75990e46i 0.156830i
\(784\) 1.26738e47 0.222174
\(785\) 0 0
\(786\) −2.48025e47 −0.416892
\(787\) − 2.21953e47i − 0.365324i −0.983176 0.182662i \(-0.941529\pi\)
0.983176 0.182662i \(-0.0584714\pi\)
\(788\) − 4.03191e47i − 0.649873i
\(789\) −9.28223e47 −1.46515
\(790\) 0 0
\(791\) 9.73733e47 1.47410
\(792\) 2.69588e48i 3.99701i
\(793\) 4.09329e47i 0.594382i
\(794\) −3.19358e47 −0.454192
\(795\) 0 0
\(796\) −4.90696e47 −0.669494
\(797\) 9.77934e47i 1.30691i 0.756963 + 0.653457i \(0.226682\pi\)
−0.756963 + 0.653457i \(0.773318\pi\)
\(798\) 6.41185e47i 0.839335i
\(799\) −1.32050e47 −0.169324
\(800\) 0 0
\(801\) −3.15111e48 −3.87728
\(802\) 4.60248e46i 0.0554772i
\(803\) − 1.66232e48i − 1.96295i
\(804\) −1.56773e47 −0.181362
\(805\) 0 0
\(806\) 2.46330e47 0.273520
\(807\) − 4.09129e47i − 0.445089i
\(808\) 7.80573e47i 0.832005i
\(809\) 6.43989e47 0.672555 0.336278 0.941763i \(-0.390832\pi\)
0.336278 + 0.941763i \(0.390832\pi\)
\(810\) 0 0
\(811\) −3.92035e47 −0.393080 −0.196540 0.980496i \(-0.562971\pi\)
−0.196540 + 0.980496i \(0.562971\pi\)
\(812\) − 2.99875e46i − 0.0294623i
\(813\) 1.08617e48i 1.04570i
\(814\) 3.00224e47 0.283232
\(815\) 0 0
\(816\) 9.46847e47 0.857813
\(817\) − 5.70425e47i − 0.506447i
\(818\) 8.35166e47i 0.726680i
\(819\) −3.28763e48 −2.80349
\(820\) 0 0
\(821\) −3.66301e47 −0.300038 −0.150019 0.988683i \(-0.547934\pi\)
−0.150019 + 0.988683i \(0.547934\pi\)
\(822\) 2.75979e48i 2.21560i
\(823\) − 6.74278e47i − 0.530569i −0.964170 0.265284i \(-0.914534\pi\)
0.964170 0.265284i \(-0.0854658\pi\)
\(824\) 2.04364e48 1.57618
\(825\) 0 0
\(826\) −4.20708e46 −0.0311753
\(827\) − 1.83329e48i − 1.33165i −0.746106 0.665827i \(-0.768079\pi\)
0.746106 0.665827i \(-0.231921\pi\)
\(828\) 2.21388e48i 1.57635i
\(829\) −1.99989e48 −1.39591 −0.697954 0.716142i \(-0.745906\pi\)
−0.697954 + 0.716142i \(0.745906\pi\)
\(830\) 0 0
\(831\) 2.91890e48 1.95795
\(832\) − 1.47460e48i − 0.969702i
\(833\) 8.11190e47i 0.522974i
\(834\) −1.39539e48 −0.881971
\(835\) 0 0
\(836\) 4.79799e47 0.291512
\(837\) − 1.61050e48i − 0.959382i
\(838\) 5.08735e47i 0.297143i
\(839\) 7.68878e47 0.440337 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(840\) 0 0
\(841\) −1.80943e48 −0.996338
\(842\) − 1.67053e48i − 0.901998i
\(843\) − 6.76030e48i − 3.57941i
\(844\) −1.01704e48 −0.528065
\(845\) 0 0
\(846\) −6.05007e47 −0.302099
\(847\) − 3.37174e48i − 1.65112i
\(848\) − 2.80712e47i − 0.134812i
\(849\) 5.80585e48 2.73457
\(850\) 0 0
\(851\) 8.66719e47 0.392682
\(852\) 1.62078e48i 0.720231i
\(853\) − 1.63399e47i − 0.0712180i −0.999366 0.0356090i \(-0.988663\pi\)
0.999366 0.0356090i \(-0.0113371\pi\)
\(854\) 1.38834e48 0.593528
\(855\) 0 0
\(856\) −2.45915e48 −1.01151
\(857\) − 5.35960e47i − 0.216247i −0.994137 0.108123i \(-0.965516\pi\)
0.994137 0.108123i \(-0.0344841\pi\)
\(858\) − 5.27863e48i − 2.08920i
\(859\) 3.45793e48 1.34255 0.671273 0.741210i \(-0.265748\pi\)
0.671273 + 0.741210i \(0.265748\pi\)
\(860\) 0 0
\(861\) 2.88321e48 1.07727
\(862\) − 2.16884e48i − 0.794982i
\(863\) − 1.95932e48i − 0.704574i −0.935892 0.352287i \(-0.885404\pi\)
0.935892 0.352287i \(-0.114596\pi\)
\(864\) −5.43473e48 −1.91735
\(865\) 0 0
\(866\) 2.61773e48 0.888955
\(867\) 5.22368e47i 0.174045i
\(868\) 5.51319e47i 0.180230i
\(869\) −3.68580e48 −1.18224
\(870\) 0 0
\(871\) 7.62161e47 0.235368
\(872\) − 2.98994e48i − 0.906024i
\(873\) 4.75559e48i 1.41407i
\(874\) −2.09908e48 −0.612479
\(875\) 0 0
\(876\) 3.34413e48 0.939647
\(877\) 1.31812e48i 0.363464i 0.983348 + 0.181732i \(0.0581703\pi\)
−0.983348 + 0.181732i \(0.941830\pi\)
\(878\) 2.60696e47i 0.0705464i
\(879\) −4.07223e48 −1.08147
\(880\) 0 0
\(881\) 6.02351e48 1.54080 0.770401 0.637559i \(-0.220056\pi\)
0.770401 + 0.637559i \(0.220056\pi\)
\(882\) 3.71658e48i 0.933065i
\(883\) 5.51914e48i 1.35994i 0.733240 + 0.679970i \(0.238007\pi\)
−0.733240 + 0.679970i \(0.761993\pi\)
\(884\) 1.63698e48 0.395896
\(885\) 0 0
\(886\) 2.04773e48 0.477107
\(887\) 2.46908e48i 0.564672i 0.959316 + 0.282336i \(0.0911093\pi\)
−0.959316 + 0.282336i \(0.908891\pi\)
\(888\) 2.12320e48i 0.476626i
\(889\) −4.94765e47 −0.109023
\(890\) 0 0
\(891\) −1.71580e49 −3.64322
\(892\) 3.61921e47i 0.0754385i
\(893\) 3.78528e47i 0.0774550i
\(894\) 4.79710e48 0.963628
\(895\) 0 0
\(896\) −3.21597e47 −0.0622631
\(897\) − 1.52389e49i − 2.89653i
\(898\) 2.19840e48i 0.410248i
\(899\) −1.22260e47 −0.0224001
\(900\) 0 0
\(901\) 1.79670e48 0.317334
\(902\) 3.26957e48i 0.566999i
\(903\) − 1.40458e49i − 2.39165i
\(904\) −7.80909e48 −1.30563
\(905\) 0 0
\(906\) −1.27795e48 −0.206015
\(907\) 1.76568e48i 0.279507i 0.990186 + 0.139753i \(0.0446310\pi\)
−0.990186 + 0.139753i \(0.955369\pi\)
\(908\) 2.60621e48i 0.405129i
\(909\) −1.20822e49 −1.84434
\(910\) 0 0
\(911\) −4.90626e48 −0.722266 −0.361133 0.932514i \(-0.617610\pi\)
−0.361133 + 0.932514i \(0.617610\pi\)
\(912\) − 2.71418e48i − 0.392396i
\(913\) 8.07079e48i 1.14590i
\(914\) 6.26043e47 0.0872955
\(915\) 0 0
\(916\) −9.37128e47 −0.126044
\(917\) 2.69872e48i 0.356503i
\(918\) 1.62190e49i 2.10437i
\(919\) 7.03091e48 0.895994 0.447997 0.894035i \(-0.352137\pi\)
0.447997 + 0.894035i \(0.352137\pi\)
\(920\) 0 0
\(921\) −2.34564e49 −2.88388
\(922\) 5.24400e48i 0.633290i
\(923\) − 7.87955e48i − 0.934702i
\(924\) 1.18143e49 1.37664
\(925\) 0 0
\(926\) 1.11461e49 1.25326
\(927\) 3.16327e49i 3.49398i
\(928\) 4.12574e47i 0.0447672i
\(929\) −3.42900e48 −0.365517 −0.182759 0.983158i \(-0.558503\pi\)
−0.182759 + 0.983158i \(0.558503\pi\)
\(930\) 0 0
\(931\) 2.32532e48 0.239228
\(932\) 2.56329e48i 0.259081i
\(933\) − 1.50282e49i − 1.49231i
\(934\) 6.58752e48 0.642686
\(935\) 0 0
\(936\) 2.63660e49 2.48309
\(937\) − 2.88611e48i − 0.267060i −0.991045 0.133530i \(-0.957369\pi\)
0.991045 0.133530i \(-0.0426313\pi\)
\(938\) − 2.58506e48i − 0.235030i
\(939\) 3.51019e49 3.13580
\(940\) 0 0
\(941\) −1.90690e49 −1.64475 −0.822373 0.568949i \(-0.807350\pi\)
−0.822373 + 0.568949i \(0.807350\pi\)
\(942\) 1.47190e49i 1.24750i
\(943\) 9.43893e48i 0.786104i
\(944\) 1.78089e47 0.0145747
\(945\) 0 0
\(946\) 1.59279e49 1.25880
\(947\) 7.16410e48i 0.556400i 0.960523 + 0.278200i \(0.0897378\pi\)
−0.960523 + 0.278200i \(0.910262\pi\)
\(948\) − 7.41479e48i − 0.565928i
\(949\) −1.62577e49 −1.21946
\(950\) 0 0
\(951\) 1.92647e49 1.39568
\(952\) − 1.95185e49i − 1.38975i
\(953\) − 3.01662e48i − 0.211100i −0.994414 0.105550i \(-0.966340\pi\)
0.994414 0.105550i \(-0.0336603\pi\)
\(954\) 8.23183e48 0.566172
\(955\) 0 0
\(956\) 3.21563e48 0.213654
\(957\) 2.61992e48i 0.171097i
\(958\) − 2.64548e48i − 0.169814i
\(959\) 3.00288e49 1.89466
\(960\) 0 0
\(961\) −1.41557e49 −0.862971
\(962\) − 2.93622e48i − 0.175954i
\(963\) − 3.80642e49i − 2.24225i
\(964\) −1.67503e48 −0.0969952
\(965\) 0 0
\(966\) −5.16865e49 −2.89237
\(967\) − 9.56534e48i − 0.526216i −0.964766 0.263108i \(-0.915252\pi\)
0.964766 0.263108i \(-0.0847475\pi\)
\(968\) 2.70405e49i 1.46242i
\(969\) 1.73722e49 0.923658
\(970\) 0 0
\(971\) 3.02376e49 1.55392 0.776961 0.629548i \(-0.216760\pi\)
0.776961 + 0.629548i \(0.216760\pi\)
\(972\) − 1.41248e49i − 0.713655i
\(973\) 1.51829e49i 0.754213i
\(974\) −1.77175e49 −0.865327
\(975\) 0 0
\(976\) −5.87695e48 −0.277479
\(977\) 5.10941e48i 0.237198i 0.992942 + 0.118599i \(0.0378403\pi\)
−0.992942 + 0.118599i \(0.962160\pi\)
\(978\) 5.59230e49i 2.55270i
\(979\) −5.50508e49 −2.47086
\(980\) 0 0
\(981\) 4.62800e49 2.00842
\(982\) 8.71368e48i 0.371845i
\(983\) 1.15145e49i 0.483183i 0.970378 + 0.241591i \(0.0776694\pi\)
−0.970378 + 0.241591i \(0.922331\pi\)
\(984\) −2.31226e49 −0.954151
\(985\) 0 0
\(986\) 1.23126e48 0.0491337
\(987\) 9.32064e48i 0.365774i
\(988\) − 4.69248e48i − 0.181098i
\(989\) 4.59824e49 1.74523
\(990\) 0 0
\(991\) 6.06784e48 0.222751 0.111375 0.993778i \(-0.464474\pi\)
0.111375 + 0.993778i \(0.464474\pi\)
\(992\) − 7.58515e48i − 0.273856i
\(993\) 3.24815e49i 1.15338i
\(994\) −2.67254e49 −0.933361
\(995\) 0 0
\(996\) −1.62361e49 −0.548534
\(997\) 3.14175e49i 1.04400i 0.852945 + 0.522000i \(0.174814\pi\)
−0.852945 + 0.522000i \(0.825186\pi\)
\(998\) − 2.93388e49i − 0.958932i
\(999\) −1.91970e49 −0.617166
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 25.34.b.b.24.8 10
5.2 odd 4 25.34.a.b.1.2 5
5.3 odd 4 5.34.a.a.1.4 5
5.4 even 2 inner 25.34.b.b.24.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.a.1.4 5 5.3 odd 4
25.34.a.b.1.2 5 5.2 odd 4
25.34.b.b.24.3 10 5.4 even 2 inner
25.34.b.b.24.8 10 1.1 even 1 trivial