Properties

Label 25.30.b.a.24.4
Level $25$
Weight $30$
Character 25.24
Analytic conductor $133.195$
Analytic rank $0$
Dimension $4$
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] = [25,30,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, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 30 \)
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: \(133.195105958\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25675x^{2} + 164788569 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.4
Root \(113.802i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.30.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)
 
\(f(q)\) \(=\) \(q+26073.9i q^{2} +1.44920e7i q^{3} -1.42978e8 q^{4} -3.77862e11 q^{6} -3.40335e10i q^{7} +1.02703e13i q^{8} -1.41387e14 q^{9} +O(q^{10})\) \(q+26073.9i q^{2} +1.44920e7i q^{3} -1.42978e8 q^{4} -3.77862e11 q^{6} -3.40335e10i q^{7} +1.02703e13i q^{8} -1.41387e14 q^{9} -7.97271e14 q^{11} -2.07203e15i q^{12} -1.10490e16i q^{13} +8.87385e14 q^{14} -3.44548e17 q^{16} -7.47691e17i q^{17} -3.68651e18i q^{18} +2.49040e18 q^{19} +4.93212e17 q^{21} -2.07880e19i q^{22} +1.63712e18i q^{23} -1.48837e20 q^{24} +2.88091e20 q^{26} -1.05439e21i q^{27} +4.86602e18i q^{28} -1.87750e21 q^{29} -4.79819e21 q^{31} -3.46987e21i q^{32} -1.15540e22i q^{33} +1.94952e22 q^{34} +2.02152e22 q^{36} +7.91036e22i q^{37} +6.49345e22i q^{38} +1.60122e23 q^{39} -6.06439e21 q^{41} +1.28600e22i q^{42} -6.72526e23i q^{43} +1.13992e23 q^{44} -4.26862e22 q^{46} -1.90366e24i q^{47} -4.99319e24i q^{48} +3.21875e24 q^{49} +1.08355e25 q^{51} +1.57976e24i q^{52} +1.65597e24i q^{53} +2.74920e25 q^{54} +3.49535e23 q^{56} +3.60909e25i q^{57} -4.89537e25i q^{58} +8.93685e25 q^{59} +4.59924e25 q^{61} -1.25108e26i q^{62} +4.81189e24i q^{63} -9.45048e25 q^{64} +3.01259e26 q^{66} -1.04774e26i q^{67} +1.06903e26i q^{68} -2.37252e25 q^{69} +1.75243e26 q^{71} -1.45209e27i q^{72} +4.56674e26i q^{73} -2.06254e27 q^{74} -3.56072e26 q^{76} +2.71339e25i q^{77} +4.17501e27i q^{78} +3.28137e27 q^{79} +5.57671e27 q^{81} -1.58122e26i q^{82} +4.40086e27i q^{83} -7.05183e25 q^{84} +1.75354e28 q^{86} -2.72087e28i q^{87} -8.18824e27i q^{88} +6.06897e26 q^{89} -3.76036e26 q^{91} -2.34072e26i q^{92} -6.95352e28i q^{93} +4.96357e28 q^{94} +5.02853e28 q^{96} +1.44471e28i q^{97} +8.39253e28i q^{98} +1.12724e29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 179904512 q^{4} - 1087817513472 q^{6} - 326939139047412 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 179904512 q^{4} - 1087817513472 q^{6} - 326939139047412 q^{9} - 41\!\cdots\!32 q^{11}+ \cdots + 28\!\cdots\!96 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 26073.9i 1.12531i 0.826693 + 0.562654i \(0.190220\pi\)
−0.826693 + 0.562654i \(0.809780\pi\)
\(3\) 1.44920e7i 1.74932i 0.484736 + 0.874660i \(0.338916\pi\)
−0.484736 + 0.874660i \(0.661084\pi\)
\(4\) −1.42978e8 −0.266317
\(5\) 0 0
\(6\) −3.77862e11 −1.96852
\(7\) − 3.40335e10i − 0.0189664i −0.999955 0.00948319i \(-0.996981\pi\)
0.999955 0.00948319i \(-0.00301864\pi\)
\(8\) 1.02703e13i 0.825619i
\(9\) −1.41387e14 −2.06012
\(10\) 0 0
\(11\) −7.97271e14 −0.633012 −0.316506 0.948590i \(-0.602510\pi\)
−0.316506 + 0.948590i \(0.602510\pi\)
\(12\) − 2.07203e15i − 0.465873i
\(13\) − 1.10490e16i − 0.778296i −0.921175 0.389148i \(-0.872769\pi\)
0.921175 0.389148i \(-0.127231\pi\)
\(14\) 8.87385e14 0.0213430
\(15\) 0 0
\(16\) −3.44548e17 −1.19539
\(17\) − 7.47691e17i − 1.07699i −0.842628 0.538496i \(-0.818993\pi\)
0.842628 0.538496i \(-0.181007\pi\)
\(18\) − 3.68651e18i − 2.31827i
\(19\) 2.49040e18 0.715060 0.357530 0.933902i \(-0.383619\pi\)
0.357530 + 0.933902i \(0.383619\pi\)
\(20\) 0 0
\(21\) 4.93212e17 0.0331783
\(22\) − 2.07880e19i − 0.712333i
\(23\) 1.63712e18i 0.0294461i 0.999892 + 0.0147231i \(0.00468667\pi\)
−0.999892 + 0.0147231i \(0.995313\pi\)
\(24\) −1.48837e20 −1.44427
\(25\) 0 0
\(26\) 2.88091e20 0.875822
\(27\) − 1.05439e21i − 1.85449i
\(28\) 4.86602e18i 0.00505106i
\(29\) −1.87750e21 −1.17168 −0.585839 0.810427i \(-0.699235\pi\)
−0.585839 + 0.810427i \(0.699235\pi\)
\(30\) 0 0
\(31\) −4.79819e21 −1.13850 −0.569250 0.822165i \(-0.692766\pi\)
−0.569250 + 0.822165i \(0.692766\pi\)
\(32\) − 3.46987e21i − 0.519564i
\(33\) − 1.15540e22i − 1.10734i
\(34\) 1.94952e22 1.21195
\(35\) 0 0
\(36\) 2.02152e22 0.548645
\(37\) 7.91036e22i 1.44302i 0.692406 + 0.721508i \(0.256551\pi\)
−0.692406 + 0.721508i \(0.743449\pi\)
\(38\) 6.49345e22i 0.804662i
\(39\) 1.60122e23 1.36149
\(40\) 0 0
\(41\) −6.06439e21 −0.0249702 −0.0124851 0.999922i \(-0.503974\pi\)
−0.0124851 + 0.999922i \(0.503974\pi\)
\(42\) 1.28600e22i 0.0373358i
\(43\) − 6.72526e23i − 1.38809i −0.719933 0.694044i \(-0.755827\pi\)
0.719933 0.694044i \(-0.244173\pi\)
\(44\) 1.13992e23 0.168582
\(45\) 0 0
\(46\) −4.26862e22 −0.0331359
\(47\) − 1.90366e24i − 1.08186i −0.841069 0.540928i \(-0.818073\pi\)
0.841069 0.540928i \(-0.181927\pi\)
\(48\) − 4.99319e24i − 2.09112i
\(49\) 3.21875e24 0.999640
\(50\) 0 0
\(51\) 1.08355e25 1.88401
\(52\) 1.57976e24i 0.207273i
\(53\) 1.65597e24i 0.164836i 0.996598 + 0.0824181i \(0.0262643\pi\)
−0.996598 + 0.0824181i \(0.973736\pi\)
\(54\) 2.74920e25 2.08688
\(55\) 0 0
\(56\) 3.49535e23 0.0156590
\(57\) 3.60909e25i 1.25087i
\(58\) − 4.89537e25i − 1.31850i
\(59\) 8.93685e25 1.87858 0.939292 0.343120i \(-0.111484\pi\)
0.939292 + 0.343120i \(0.111484\pi\)
\(60\) 0 0
\(61\) 4.59924e25 0.596216 0.298108 0.954532i \(-0.403644\pi\)
0.298108 + 0.954532i \(0.403644\pi\)
\(62\) − 1.25108e26i − 1.28116i
\(63\) 4.81189e24i 0.0390731i
\(64\) −9.45048e25 −0.610723
\(65\) 0 0
\(66\) 3.01259e26 1.24610
\(67\) − 1.04774e26i − 0.348472i −0.984704 0.174236i \(-0.944254\pi\)
0.984704 0.174236i \(-0.0557456\pi\)
\(68\) 1.06903e26i 0.286821i
\(69\) −2.37252e25 −0.0515107
\(70\) 0 0
\(71\) 1.75243e26 0.251417 0.125709 0.992067i \(-0.459880\pi\)
0.125709 + 0.992067i \(0.459880\pi\)
\(72\) − 1.45209e27i − 1.70088i
\(73\) 4.56674e26i 0.437950i 0.975730 + 0.218975i \(0.0702713\pi\)
−0.975730 + 0.218975i \(0.929729\pi\)
\(74\) −2.06254e27 −1.62384
\(75\) 0 0
\(76\) −3.56072e26 −0.190432
\(77\) 2.71339e25i 0.0120059i
\(78\) 4.17501e27i 1.53209i
\(79\) 3.28137e27 1.00107 0.500533 0.865717i \(-0.333137\pi\)
0.500533 + 0.865717i \(0.333137\pi\)
\(80\) 0 0
\(81\) 5.57671e27 1.18398
\(82\) − 1.58122e26i − 0.0280991i
\(83\) 4.40086e27i 0.656002i 0.944677 + 0.328001i \(0.106375\pi\)
−0.944677 + 0.328001i \(0.893625\pi\)
\(84\) −7.05183e25 −0.00883593
\(85\) 0 0
\(86\) 1.75354e28 1.56202
\(87\) − 2.72087e28i − 2.04964i
\(88\) − 8.18824e27i − 0.522627i
\(89\) 6.06897e26 0.0328821 0.0164411 0.999865i \(-0.494766\pi\)
0.0164411 + 0.999865i \(0.494766\pi\)
\(90\) 0 0
\(91\) −3.76036e26 −0.0147615
\(92\) − 2.34072e26i − 0.00784199i
\(93\) − 6.95352e28i − 1.99160i
\(94\) 4.96357e28 1.21742
\(95\) 0 0
\(96\) 5.02853e28 0.908884
\(97\) 1.44471e28i 0.224694i 0.993669 + 0.112347i \(0.0358368\pi\)
−0.993669 + 0.112347i \(0.964163\pi\)
\(98\) 8.39253e28i 1.12490i
\(99\) 1.12724e29 1.30408
\(100\) 0 0
\(101\) −4.90236e28 −0.424371 −0.212185 0.977229i \(-0.568058\pi\)
−0.212185 + 0.977229i \(0.568058\pi\)
\(102\) 2.82524e29i 2.12009i
\(103\) − 1.31966e29i − 0.859654i −0.902911 0.429827i \(-0.858574\pi\)
0.902911 0.429827i \(-0.141426\pi\)
\(104\) 1.13477e29 0.642576
\(105\) 0 0
\(106\) −4.31776e28 −0.185491
\(107\) − 2.31447e29i − 0.867733i −0.900977 0.433867i \(-0.857149\pi\)
0.900977 0.433867i \(-0.142851\pi\)
\(108\) 1.50754e29i 0.493882i
\(109\) −6.73768e29 −1.93119 −0.965597 0.260043i \(-0.916263\pi\)
−0.965597 + 0.260043i \(0.916263\pi\)
\(110\) 0 0
\(111\) −1.14637e30 −2.52430
\(112\) 1.17262e28i 0.0226723i
\(113\) − 5.10156e28i − 0.0867092i −0.999060 0.0433546i \(-0.986195\pi\)
0.999060 0.0433546i \(-0.0138045\pi\)
\(114\) −9.41029e29 −1.40761
\(115\) 0 0
\(116\) 2.68440e29 0.312037
\(117\) 1.56219e30i 1.60339i
\(118\) 2.33019e30i 2.11398i
\(119\) −2.54465e28 −0.0204267
\(120\) 0 0
\(121\) −9.50668e29 −0.599296
\(122\) 1.19920e30i 0.670926i
\(123\) − 8.78850e28i − 0.0436808i
\(124\) 6.86034e29 0.303201
\(125\) 0 0
\(126\) −1.25465e29 −0.0439692
\(127\) 3.46836e30i 1.08385i 0.840426 + 0.541926i \(0.182305\pi\)
−0.840426 + 0.541926i \(0.817695\pi\)
\(128\) − 4.32698e30i − 1.20681i
\(129\) 9.74624e30 2.42821
\(130\) 0 0
\(131\) −1.16207e30 −0.231631 −0.115816 0.993271i \(-0.536948\pi\)
−0.115816 + 0.993271i \(0.536948\pi\)
\(132\) 1.65197e30i 0.294903i
\(133\) − 8.47570e28i − 0.0135621i
\(134\) 2.73186e30 0.392138
\(135\) 0 0
\(136\) 7.67904e30 0.889186
\(137\) − 1.12214e31i − 1.16842i −0.811603 0.584210i \(-0.801404\pi\)
0.811603 0.584210i \(-0.198596\pi\)
\(138\) − 6.18608e29i − 0.0579654i
\(139\) 2.10245e30 0.177423 0.0887117 0.996057i \(-0.471725\pi\)
0.0887117 + 0.996057i \(0.471725\pi\)
\(140\) 0 0
\(141\) 2.75877e31 1.89251
\(142\) 4.56927e30i 0.282922i
\(143\) 8.80906e30i 0.492671i
\(144\) 4.87146e31 2.46265
\(145\) 0 0
\(146\) −1.19073e31 −0.492828
\(147\) 4.66460e31i 1.74869i
\(148\) − 1.13100e31i − 0.384299i
\(149\) 4.28030e30 0.131909 0.0659543 0.997823i \(-0.478991\pi\)
0.0659543 + 0.997823i \(0.478991\pi\)
\(150\) 0 0
\(151\) −9.32731e30 −0.236914 −0.118457 0.992959i \(-0.537795\pi\)
−0.118457 + 0.992959i \(0.537795\pi\)
\(152\) 2.55773e31i 0.590367i
\(153\) 1.05714e32i 2.21874i
\(154\) −7.07486e29 −0.0135104
\(155\) 0 0
\(156\) −2.28939e31 −0.362587
\(157\) 2.58823e31i 0.373644i 0.982394 + 0.186822i \(0.0598188\pi\)
−0.982394 + 0.186822i \(0.940181\pi\)
\(158\) 8.55583e31i 1.12651i
\(159\) −2.39983e31 −0.288351
\(160\) 0 0
\(161\) 5.57170e28 0.000558486 0
\(162\) 1.45407e32i 1.33234i
\(163\) − 1.19680e31i − 0.100300i −0.998742 0.0501501i \(-0.984030\pi\)
0.998742 0.0501501i \(-0.0159700\pi\)
\(164\) 8.67072e29 0.00664997
\(165\) 0 0
\(166\) −1.14748e32 −0.738204
\(167\) 1.07804e32i 0.635692i 0.948142 + 0.317846i \(0.102959\pi\)
−0.948142 + 0.317846i \(0.897041\pi\)
\(168\) 5.06545e30i 0.0273926i
\(169\) 7.94575e31 0.394255
\(170\) 0 0
\(171\) −3.52110e32 −1.47311
\(172\) 9.61562e31i 0.369671i
\(173\) − 3.82198e32i − 1.35089i −0.737410 0.675446i \(-0.763951\pi\)
0.737410 0.675446i \(-0.236049\pi\)
\(174\) 7.09436e32 2.30648
\(175\) 0 0
\(176\) 2.74698e32 0.756698
\(177\) 1.29513e33i 3.28624i
\(178\) 1.58242e31i 0.0370025i
\(179\) 5.24041e32 1.12979 0.564893 0.825164i \(-0.308917\pi\)
0.564893 + 0.825164i \(0.308917\pi\)
\(180\) 0 0
\(181\) −7.54402e32 −1.38441 −0.692203 0.721703i \(-0.743360\pi\)
−0.692203 + 0.721703i \(0.743360\pi\)
\(182\) − 9.80473e30i − 0.0166112i
\(183\) 6.66521e32i 1.04297i
\(184\) −1.68138e31 −0.0243113
\(185\) 0 0
\(186\) 1.81306e33 2.24116
\(187\) 5.96112e32i 0.681750i
\(188\) 2.72180e32i 0.288116i
\(189\) −3.58844e31 −0.0351730
\(190\) 0 0
\(191\) −1.08024e33 −0.908947 −0.454473 0.890760i \(-0.650173\pi\)
−0.454473 + 0.890760i \(0.650173\pi\)
\(192\) − 1.36956e33i − 1.06835i
\(193\) − 1.91169e33i − 1.38305i −0.722354 0.691524i \(-0.756940\pi\)
0.722354 0.691524i \(-0.243060\pi\)
\(194\) −3.76693e32 −0.252849
\(195\) 0 0
\(196\) −4.60209e32 −0.266221
\(197\) − 2.82411e33i − 1.51747i −0.651397 0.758737i \(-0.725817\pi\)
0.651397 0.758737i \(-0.274183\pi\)
\(198\) 2.93915e33i 1.46749i
\(199\) −2.05181e33 −0.952288 −0.476144 0.879367i \(-0.657966\pi\)
−0.476144 + 0.879367i \(0.657966\pi\)
\(200\) 0 0
\(201\) 1.51838e33 0.609590
\(202\) − 1.27824e33i − 0.477547i
\(203\) 6.38978e31i 0.0222225i
\(204\) −1.54924e33 −0.501742
\(205\) 0 0
\(206\) 3.44088e33 0.967375
\(207\) − 2.31468e32i − 0.0606626i
\(208\) 3.80692e33i 0.930369i
\(209\) −1.98553e33 −0.452641
\(210\) 0 0
\(211\) −4.24073e33 −0.842063 −0.421032 0.907046i \(-0.638332\pi\)
−0.421032 + 0.907046i \(0.638332\pi\)
\(212\) − 2.36767e32i − 0.0438986i
\(213\) 2.53962e33i 0.439809i
\(214\) 6.03472e33 0.976466
\(215\) 0 0
\(216\) 1.08289e34 1.53111
\(217\) 1.63299e32i 0.0215932i
\(218\) − 1.75678e34i − 2.17319i
\(219\) −6.61810e33 −0.766115
\(220\) 0 0
\(221\) −8.26125e33 −0.838219
\(222\) − 2.98903e34i − 2.84061i
\(223\) − 1.95826e34i − 1.74362i −0.489848 0.871808i \(-0.662948\pi\)
0.489848 0.871808i \(-0.337052\pi\)
\(224\) −1.18092e32 −0.00985425
\(225\) 0 0
\(226\) 1.33017e33 0.0975745
\(227\) 1.22424e34i 0.842345i 0.906980 + 0.421173i \(0.138381\pi\)
−0.906980 + 0.421173i \(0.861619\pi\)
\(228\) − 5.16018e33i − 0.333127i
\(229\) −5.07275e33 −0.307347 −0.153673 0.988122i \(-0.549110\pi\)
−0.153673 + 0.988122i \(0.549110\pi\)
\(230\) 0 0
\(231\) −3.93224e32 −0.0210023
\(232\) − 1.92825e34i − 0.967361i
\(233\) 2.33456e34i 1.10038i 0.835038 + 0.550192i \(0.185446\pi\)
−0.835038 + 0.550192i \(0.814554\pi\)
\(234\) −4.07323e34 −1.80430
\(235\) 0 0
\(236\) −1.27777e34 −0.500298
\(237\) 4.75536e34i 1.75119i
\(238\) − 6.63490e32i − 0.0229863i
\(239\) 1.69274e34 0.551849 0.275925 0.961179i \(-0.411016\pi\)
0.275925 + 0.961179i \(0.411016\pi\)
\(240\) 0 0
\(241\) −2.26884e34 −0.655476 −0.327738 0.944769i \(-0.606286\pi\)
−0.327738 + 0.944769i \(0.606286\pi\)
\(242\) − 2.47876e34i − 0.674392i
\(243\) 8.45454e33i 0.216670i
\(244\) −6.57588e33 −0.158782
\(245\) 0 0
\(246\) 2.29151e33 0.0491543
\(247\) − 2.75165e34i − 0.556528i
\(248\) − 4.92790e34i − 0.939967i
\(249\) −6.37771e34 −1.14756
\(250\) 0 0
\(251\) −9.82834e34 −1.57475 −0.787374 0.616475i \(-0.788560\pi\)
−0.787374 + 0.616475i \(0.788560\pi\)
\(252\) − 6.87992e32i − 0.0104058i
\(253\) − 1.30523e33i − 0.0186398i
\(254\) −9.04337e34 −1.21967
\(255\) 0 0
\(256\) 6.20845e34 0.747315
\(257\) − 8.53561e34i − 0.970966i −0.874246 0.485483i \(-0.838644\pi\)
0.874246 0.485483i \(-0.161356\pi\)
\(258\) 2.54122e35i 2.73248i
\(259\) 2.69217e33 0.0273688
\(260\) 0 0
\(261\) 2.65454e35 2.41380
\(262\) − 3.02996e34i − 0.260656i
\(263\) − 2.07906e35i − 1.69243i −0.532845 0.846213i \(-0.678877\pi\)
0.532845 0.846213i \(-0.321123\pi\)
\(264\) 1.18664e35 0.914242
\(265\) 0 0
\(266\) 2.20995e33 0.0152615
\(267\) 8.79513e33i 0.0575214i
\(268\) 1.49803e34i 0.0928039i
\(269\) 2.34883e35 1.37862 0.689309 0.724468i \(-0.257914\pi\)
0.689309 + 0.724468i \(0.257914\pi\)
\(270\) 0 0
\(271\) −8.82959e34 −0.465463 −0.232732 0.972541i \(-0.574766\pi\)
−0.232732 + 0.972541i \(0.574766\pi\)
\(272\) 2.57616e35i 1.28743i
\(273\) − 5.44951e33i − 0.0258225i
\(274\) 2.92586e35 1.31483
\(275\) 0 0
\(276\) 3.39217e33 0.0137182
\(277\) − 7.48695e34i − 0.287308i −0.989628 0.143654i \(-0.954115\pi\)
0.989628 0.143654i \(-0.0458853\pi\)
\(278\) 5.48191e34i 0.199656i
\(279\) 6.78402e35 2.34545
\(280\) 0 0
\(281\) 3.79424e35 1.18272 0.591361 0.806407i \(-0.298591\pi\)
0.591361 + 0.806407i \(0.298591\pi\)
\(282\) 7.19320e35i 2.12966i
\(283\) − 5.23811e35i − 1.47323i −0.676311 0.736616i \(-0.736422\pi\)
0.676311 0.736616i \(-0.263578\pi\)
\(284\) −2.50558e34 −0.0669566
\(285\) 0 0
\(286\) −2.29686e35 −0.554406
\(287\) 2.06392e32i 0 0.000473593i
\(288\) 4.90595e35i 1.07037i
\(289\) −7.70734e34 −0.159914
\(290\) 0 0
\(291\) −2.09367e35 −0.393061
\(292\) − 6.52941e34i − 0.116633i
\(293\) 8.91980e35i 1.51626i 0.652102 + 0.758131i \(0.273887\pi\)
−0.652102 + 0.758131i \(0.726113\pi\)
\(294\) −1.21624e36 −1.96782
\(295\) 0 0
\(296\) −8.12420e35 −1.19138
\(297\) 8.40632e35i 1.17392i
\(298\) 1.11604e35i 0.148438i
\(299\) 1.80886e34 0.0229178
\(300\) 0 0
\(301\) −2.28884e34 −0.0263270
\(302\) − 2.43200e35i − 0.266601i
\(303\) − 7.10449e35i − 0.742360i
\(304\) −8.58064e35 −0.854777
\(305\) 0 0
\(306\) −2.75637e36 −2.49676
\(307\) 6.34791e35i 0.548434i 0.961668 + 0.274217i \(0.0884187\pi\)
−0.961668 + 0.274217i \(0.911581\pi\)
\(308\) − 3.87954e33i − 0.00319738i
\(309\) 1.91246e36 1.50381
\(310\) 0 0
\(311\) 1.79660e36 1.28655 0.643273 0.765637i \(-0.277576\pi\)
0.643273 + 0.765637i \(0.277576\pi\)
\(312\) 1.64451e36i 1.12407i
\(313\) 2.68729e36i 1.75357i 0.480886 + 0.876783i \(0.340315\pi\)
−0.480886 + 0.876783i \(0.659685\pi\)
\(314\) −6.74853e35 −0.420465
\(315\) 0 0
\(316\) −4.69163e35 −0.266601
\(317\) 1.03006e36i 0.559119i 0.960128 + 0.279559i \(0.0901884\pi\)
−0.960128 + 0.279559i \(0.909812\pi\)
\(318\) − 6.25729e35i − 0.324484i
\(319\) 1.49687e36 0.741687
\(320\) 0 0
\(321\) 3.35412e36 1.51794
\(322\) 1.45276e33i 0 0.000628469i
\(323\) − 1.86205e36i − 0.770114i
\(324\) −7.97344e35 −0.315314
\(325\) 0 0
\(326\) 3.12053e35 0.112869
\(327\) − 9.76422e36i − 3.37828i
\(328\) − 6.22833e34i − 0.0206158i
\(329\) −6.47880e34 −0.0205189
\(330\) 0 0
\(331\) 1.57492e36 0.456827 0.228414 0.973564i \(-0.426646\pi\)
0.228414 + 0.973564i \(0.426646\pi\)
\(332\) − 6.29224e35i − 0.174704i
\(333\) − 1.11842e37i − 2.97279i
\(334\) −2.81087e36 −0.715348
\(335\) 0 0
\(336\) −1.69935e35 −0.0396611
\(337\) − 3.02595e36i − 0.676438i −0.941067 0.338219i \(-0.890176\pi\)
0.941067 0.338219i \(-0.109824\pi\)
\(338\) 2.07177e36i 0.443658i
\(339\) 7.39316e35 0.151682
\(340\) 0 0
\(341\) 3.82546e36 0.720684
\(342\) − 9.18090e36i − 1.65770i
\(343\) − 2.19130e35i − 0.0379259i
\(344\) 6.90707e36 1.14603
\(345\) 0 0
\(346\) 9.96541e36 1.52017
\(347\) 4.44778e36i 0.650678i 0.945597 + 0.325339i \(0.105478\pi\)
−0.945597 + 0.325339i \(0.894522\pi\)
\(348\) 3.89023e36i 0.545854i
\(349\) 9.68519e35 0.130359 0.0651793 0.997874i \(-0.479238\pi\)
0.0651793 + 0.997874i \(0.479238\pi\)
\(350\) 0 0
\(351\) −1.16499e37 −1.44335
\(352\) 2.76643e36i 0.328890i
\(353\) 9.40660e36i 1.07325i 0.843822 + 0.536623i \(0.180300\pi\)
−0.843822 + 0.536623i \(0.819700\pi\)
\(354\) −3.37690e37 −3.69804
\(355\) 0 0
\(356\) −8.67726e34 −0.00875705
\(357\) − 3.68770e35i − 0.0357328i
\(358\) 1.36638e37i 1.27136i
\(359\) −2.90920e36 −0.259958 −0.129979 0.991517i \(-0.541491\pi\)
−0.129979 + 0.991517i \(0.541491\pi\)
\(360\) 0 0
\(361\) −5.92772e36 −0.488690
\(362\) − 1.96702e37i − 1.55788i
\(363\) − 1.37771e37i − 1.04836i
\(364\) 5.37647e34 0.00393122
\(365\) 0 0
\(366\) −1.73788e37 −1.17366
\(367\) 3.00435e36i 0.195026i 0.995234 + 0.0975131i \(0.0310888\pi\)
−0.995234 + 0.0975131i \(0.968911\pi\)
\(368\) − 5.64069e35i − 0.0351997i
\(369\) 8.57426e35 0.0514416
\(370\) 0 0
\(371\) 5.63584e34 0.00312634
\(372\) 9.94198e36i 0.530396i
\(373\) − 2.82375e37i − 1.44893i −0.689310 0.724466i \(-0.742086\pi\)
0.689310 0.724466i \(-0.257914\pi\)
\(374\) −1.55430e37 −0.767178
\(375\) 0 0
\(376\) 1.95512e37 0.893201
\(377\) 2.07445e37i 0.911913i
\(378\) − 9.35647e35i − 0.0395805i
\(379\) −1.51814e37 −0.618079 −0.309039 0.951049i \(-0.600007\pi\)
−0.309039 + 0.951049i \(0.600007\pi\)
\(380\) 0 0
\(381\) −5.02634e37 −1.89601
\(382\) − 2.81662e37i − 1.02284i
\(383\) 2.59896e37i 0.908692i 0.890825 + 0.454346i \(0.150127\pi\)
−0.890825 + 0.454346i \(0.849873\pi\)
\(384\) 6.27065e37 2.11111
\(385\) 0 0
\(386\) 4.98453e37 1.55635
\(387\) 9.50865e37i 2.85963i
\(388\) − 2.06561e36i − 0.0598397i
\(389\) 4.80732e37 1.34163 0.670816 0.741623i \(-0.265944\pi\)
0.670816 + 0.741623i \(0.265944\pi\)
\(390\) 0 0
\(391\) 1.22406e36 0.0317133
\(392\) 3.30576e37i 0.825322i
\(393\) − 1.68406e37i − 0.405197i
\(394\) 7.36356e37 1.70762
\(395\) 0 0
\(396\) −1.61170e37 −0.347299
\(397\) − 4.33277e37i − 0.900126i −0.892997 0.450063i \(-0.851401\pi\)
0.892997 0.450063i \(-0.148599\pi\)
\(398\) − 5.34988e37i − 1.07162i
\(399\) 1.22830e36 0.0237245
\(400\) 0 0
\(401\) −2.87417e37 −0.516320 −0.258160 0.966102i \(-0.583116\pi\)
−0.258160 + 0.966102i \(0.583116\pi\)
\(402\) 3.95901e37i 0.685976i
\(403\) 5.30153e37i 0.886090i
\(404\) 7.00928e36 0.113017
\(405\) 0 0
\(406\) −1.66606e36 −0.0250072
\(407\) − 6.30670e37i − 0.913446i
\(408\) 1.11284e38i 1.55547i
\(409\) −8.54677e37 −1.15296 −0.576480 0.817112i \(-0.695574\pi\)
−0.576480 + 0.817112i \(0.695574\pi\)
\(410\) 0 0
\(411\) 1.62620e38 2.04394
\(412\) 1.88683e37i 0.228940i
\(413\) − 3.04152e36i − 0.0356299i
\(414\) 6.03528e36 0.0682641
\(415\) 0 0
\(416\) −3.83387e37 −0.404375
\(417\) 3.04686e37i 0.310370i
\(418\) − 5.17704e37i − 0.509361i
\(419\) 5.26548e37 0.500420 0.250210 0.968192i \(-0.419500\pi\)
0.250210 + 0.968192i \(0.419500\pi\)
\(420\) 0 0
\(421\) 9.94249e37 0.881870 0.440935 0.897539i \(-0.354647\pi\)
0.440935 + 0.897539i \(0.354647\pi\)
\(422\) − 1.10572e38i − 0.947580i
\(423\) 2.69152e38i 2.22876i
\(424\) −1.70074e37 −0.136092
\(425\) 0 0
\(426\) −6.62178e37 −0.494921
\(427\) − 1.56528e36i − 0.0113081i
\(428\) 3.30917e37i 0.231092i
\(429\) −1.27661e38 −0.861839
\(430\) 0 0
\(431\) −1.96737e38 −1.24155 −0.620777 0.783987i \(-0.713183\pi\)
−0.620777 + 0.783987i \(0.713183\pi\)
\(432\) 3.63287e38i 2.21685i
\(433\) − 8.93739e37i − 0.527396i −0.964605 0.263698i \(-0.915058\pi\)
0.964605 0.263698i \(-0.0849422\pi\)
\(434\) −4.25784e36 −0.0242990
\(435\) 0 0
\(436\) 9.63337e37 0.514309
\(437\) 4.07710e36i 0.0210557i
\(438\) − 1.72560e38i − 0.862115i
\(439\) 3.31662e38 1.60310 0.801548 0.597930i \(-0.204010\pi\)
0.801548 + 0.597930i \(0.204010\pi\)
\(440\) 0 0
\(441\) −4.55089e38 −2.05938
\(442\) − 2.15403e38i − 0.943254i
\(443\) − 2.62823e38i − 1.11380i −0.830578 0.556902i \(-0.811990\pi\)
0.830578 0.556902i \(-0.188010\pi\)
\(444\) 1.63905e38 0.672262
\(445\) 0 0
\(446\) 5.10596e38 1.96210
\(447\) 6.20299e37i 0.230750i
\(448\) 3.21632e36i 0.0115832i
\(449\) −1.08508e38 −0.378348 −0.189174 0.981944i \(-0.560581\pi\)
−0.189174 + 0.981944i \(0.560581\pi\)
\(450\) 0 0
\(451\) 4.83496e36 0.0158064
\(452\) 7.29408e36i 0.0230921i
\(453\) − 1.35171e38i − 0.414439i
\(454\) −3.19206e38 −0.947897
\(455\) 0 0
\(456\) −3.70665e38 −1.03274
\(457\) − 3.49142e38i − 0.942360i −0.882037 0.471180i \(-0.843828\pi\)
0.882037 0.471180i \(-0.156172\pi\)
\(458\) − 1.32266e38i − 0.345860i
\(459\) −7.88356e38 −1.99728
\(460\) 0 0
\(461\) 5.84500e38 1.39034 0.695171 0.718845i \(-0.255329\pi\)
0.695171 + 0.718845i \(0.255329\pi\)
\(462\) − 1.02529e37i − 0.0236340i
\(463\) − 2.77160e38i − 0.619164i −0.950873 0.309582i \(-0.899811\pi\)
0.950873 0.309582i \(-0.100189\pi\)
\(464\) 6.46889e38 1.40062
\(465\) 0 0
\(466\) −6.08712e38 −1.23827
\(467\) − 9.01377e38i − 1.77751i −0.458384 0.888754i \(-0.651571\pi\)
0.458384 0.888754i \(-0.348429\pi\)
\(468\) − 2.23358e38i − 0.427008i
\(469\) −3.56582e36 −0.00660926
\(470\) 0 0
\(471\) −3.75086e38 −0.653623
\(472\) 9.17845e38i 1.55099i
\(473\) 5.36186e38i 0.878676i
\(474\) −1.23991e39 −1.97062
\(475\) 0 0
\(476\) 3.63828e36 0.00543996
\(477\) − 2.34133e38i − 0.339583i
\(478\) 4.41363e38i 0.621000i
\(479\) −5.62143e38 −0.767330 −0.383665 0.923472i \(-0.625338\pi\)
−0.383665 + 0.923472i \(0.625338\pi\)
\(480\) 0 0
\(481\) 8.74016e38 1.12309
\(482\) − 5.91575e38i − 0.737612i
\(483\) 8.07450e35i 0 0.000976972i
\(484\) 1.35924e38 0.159602
\(485\) 0 0
\(486\) −2.20443e38 −0.243820
\(487\) − 1.05085e39i − 1.12816i −0.825720 0.564081i \(-0.809231\pi\)
0.825720 0.564081i \(-0.190769\pi\)
\(488\) 4.72358e38i 0.492247i
\(489\) 1.73440e38 0.175457
\(490\) 0 0
\(491\) 7.99399e38 0.762223 0.381112 0.924529i \(-0.375541\pi\)
0.381112 + 0.924529i \(0.375541\pi\)
\(492\) 1.25656e37i 0.0116329i
\(493\) 1.40379e39i 1.26189i
\(494\) 7.17462e38 0.626265
\(495\) 0 0
\(496\) 1.65321e39 1.36095
\(497\) − 5.96413e36i − 0.00476848i
\(498\) − 1.66292e39i − 1.29136i
\(499\) −1.14067e39 −0.860401 −0.430201 0.902733i \(-0.641557\pi\)
−0.430201 + 0.902733i \(0.641557\pi\)
\(500\) 0 0
\(501\) −1.56229e39 −1.11203
\(502\) − 2.56263e39i − 1.77208i
\(503\) − 3.45164e38i − 0.231894i −0.993255 0.115947i \(-0.963010\pi\)
0.993255 0.115947i \(-0.0369903\pi\)
\(504\) −4.94197e37 −0.0322595
\(505\) 0 0
\(506\) 3.40325e37 0.0209755
\(507\) 1.15150e39i 0.689679i
\(508\) − 4.95898e38i − 0.288648i
\(509\) 7.06121e37 0.0399458 0.0199729 0.999801i \(-0.493642\pi\)
0.0199729 + 0.999801i \(0.493642\pi\)
\(510\) 0 0
\(511\) 1.55422e37 0.00830633
\(512\) − 7.04246e38i − 0.365856i
\(513\) − 2.62585e39i − 1.32607i
\(514\) 2.22557e39 1.09264
\(515\) 0 0
\(516\) −1.39349e39 −0.646672
\(517\) 1.51773e39i 0.684828i
\(518\) 7.01953e37i 0.0307983i
\(519\) 5.53881e39 2.36314
\(520\) 0 0
\(521\) 3.46255e39 1.39717 0.698585 0.715528i \(-0.253814\pi\)
0.698585 + 0.715528i \(0.253814\pi\)
\(522\) 6.92142e39i 2.71627i
\(523\) 1.15392e39i 0.440453i 0.975449 + 0.220227i \(0.0706797\pi\)
−0.975449 + 0.220227i \(0.929320\pi\)
\(524\) 1.66149e38 0.0616872
\(525\) 0 0
\(526\) 5.42093e39 1.90450
\(527\) 3.58756e39i 1.22616i
\(528\) 3.98092e39i 1.32371i
\(529\) 3.08838e39 0.999133
\(530\) 0 0
\(531\) −1.26355e40 −3.87011
\(532\) 1.21184e37i 0.00361181i
\(533\) 6.70055e37i 0.0194342i
\(534\) −2.29323e38 −0.0647292
\(535\) 0 0
\(536\) 1.07606e39 0.287705
\(537\) 7.59439e39i 1.97636i
\(538\) 6.12432e39i 1.55137i
\(539\) −2.56621e39 −0.632784
\(540\) 0 0
\(541\) −2.74672e39 −0.641879 −0.320940 0.947100i \(-0.603999\pi\)
−0.320940 + 0.947100i \(0.603999\pi\)
\(542\) − 2.30222e39i − 0.523789i
\(543\) − 1.09328e40i − 2.42177i
\(544\) −2.59439e39 −0.559567
\(545\) 0 0
\(546\) 1.42090e38 0.0290583
\(547\) − 5.50487e39i − 1.09630i −0.836379 0.548152i \(-0.815331\pi\)
0.836379 0.548152i \(-0.184669\pi\)
\(548\) 1.60441e39i 0.311170i
\(549\) −6.50273e39 −1.22828
\(550\) 0 0
\(551\) −4.67573e39 −0.837820
\(552\) − 2.43666e38i − 0.0425282i
\(553\) − 1.11677e38i − 0.0189866i
\(554\) 1.95214e39 0.323310
\(555\) 0 0
\(556\) −3.00603e38 −0.0472508
\(557\) − 4.48537e39i − 0.686908i −0.939170 0.343454i \(-0.888403\pi\)
0.939170 0.343454i \(-0.111597\pi\)
\(558\) 1.76886e40i 2.63935i
\(559\) −7.43075e39 −1.08034
\(560\) 0 0
\(561\) −8.63885e39 −1.19260
\(562\) 9.89305e39i 1.33093i
\(563\) − 6.55822e38i − 0.0859834i −0.999075 0.0429917i \(-0.986311\pi\)
0.999075 0.0429917i \(-0.0136889\pi\)
\(564\) −3.94443e39 −0.504008
\(565\) 0 0
\(566\) 1.36578e40 1.65784
\(567\) − 1.89795e38i − 0.0224559i
\(568\) 1.79981e39i 0.207575i
\(569\) 1.62256e40 1.82420 0.912099 0.409970i \(-0.134461\pi\)
0.912099 + 0.409970i \(0.134461\pi\)
\(570\) 0 0
\(571\) 9.28706e39 0.992327 0.496163 0.868229i \(-0.334742\pi\)
0.496163 + 0.868229i \(0.334742\pi\)
\(572\) − 1.25950e39i − 0.131206i
\(573\) − 1.56549e40i − 1.59004i
\(574\) −5.38145e36 −0.000532938 0
\(575\) 0 0
\(576\) 1.33617e40 1.25816
\(577\) − 2.84801e39i − 0.261513i −0.991415 0.130756i \(-0.958259\pi\)
0.991415 0.130756i \(-0.0417406\pi\)
\(578\) − 2.00960e39i − 0.179952i
\(579\) 2.77042e40 2.41939
\(580\) 0 0
\(581\) 1.49776e38 0.0124420
\(582\) − 5.45902e39i − 0.442315i
\(583\) − 1.32026e39i − 0.104343i
\(584\) −4.69019e39 −0.361580
\(585\) 0 0
\(586\) −2.32574e40 −1.70626
\(587\) − 5.47078e39i − 0.391559i −0.980648 0.195779i \(-0.937276\pi\)
0.980648 0.195779i \(-0.0627236\pi\)
\(588\) − 6.66934e39i − 0.465705i
\(589\) −1.19494e40 −0.814095
\(590\) 0 0
\(591\) 4.09269e40 2.65455
\(592\) − 2.72550e40i − 1.72497i
\(593\) − 2.22596e40i − 1.37475i −0.726302 0.687376i \(-0.758762\pi\)
0.726302 0.687376i \(-0.241238\pi\)
\(594\) −2.19186e40 −1.32102
\(595\) 0 0
\(596\) −6.11986e38 −0.0351294
\(597\) − 2.97348e40i − 1.66586i
\(598\) 4.71641e38i 0.0257896i
\(599\) −2.08204e39 −0.111122 −0.0555611 0.998455i \(-0.517695\pi\)
−0.0555611 + 0.998455i \(0.517695\pi\)
\(600\) 0 0
\(601\) −1.47384e40 −0.749498 −0.374749 0.927126i \(-0.622271\pi\)
−0.374749 + 0.927126i \(0.622271\pi\)
\(602\) − 5.96790e38i − 0.0296260i
\(603\) 1.48137e40i 0.717895i
\(604\) 1.33360e39 0.0630941
\(605\) 0 0
\(606\) 1.85242e40 0.835383
\(607\) 1.37809e40i 0.606796i 0.952864 + 0.303398i \(0.0981211\pi\)
−0.952864 + 0.303398i \(0.901879\pi\)
\(608\) − 8.64138e39i − 0.371519i
\(609\) −9.26005e38 −0.0388743
\(610\) 0 0
\(611\) −2.10335e40 −0.842004
\(612\) − 1.51147e40i − 0.590886i
\(613\) − 1.47390e40i − 0.562717i −0.959603 0.281358i \(-0.909215\pi\)
0.959603 0.281358i \(-0.0907850\pi\)
\(614\) −1.65515e40 −0.617157
\(615\) 0 0
\(616\) −2.78674e38 −0.00991234
\(617\) 4.06519e39i 0.141236i 0.997503 + 0.0706181i \(0.0224971\pi\)
−0.997503 + 0.0706181i \(0.977503\pi\)
\(618\) 4.98652e40i 1.69225i
\(619\) −3.64687e40 −1.20894 −0.604472 0.796627i \(-0.706616\pi\)
−0.604472 + 0.796627i \(0.706616\pi\)
\(620\) 0 0
\(621\) 1.72616e39 0.0546077
\(622\) 4.68444e40i 1.44776i
\(623\) − 2.06548e37i 0 0.000623655i
\(624\) −5.51698e40 −1.62751
\(625\) 0 0
\(626\) −7.00682e40 −1.97330
\(627\) − 2.87742e40i − 0.791815i
\(628\) − 3.70059e39i − 0.0995076i
\(629\) 5.91450e40 1.55412
\(630\) 0 0
\(631\) −6.88296e40 −1.72723 −0.863613 0.504155i \(-0.831804\pi\)
−0.863613 + 0.504155i \(0.831804\pi\)
\(632\) 3.37008e40i 0.826500i
\(633\) − 6.14565e40i − 1.47304i
\(634\) −2.68577e40 −0.629181
\(635\) 0 0
\(636\) 3.43122e39 0.0767927
\(637\) − 3.55640e40i − 0.778016i
\(638\) 3.90294e40i 0.834626i
\(639\) −2.47771e40 −0.517950
\(640\) 0 0
\(641\) 5.79208e40 1.15716 0.578580 0.815626i \(-0.303607\pi\)
0.578580 + 0.815626i \(0.303607\pi\)
\(642\) 8.74550e40i 1.70815i
\(643\) 6.13559e40i 1.17165i 0.810439 + 0.585824i \(0.199229\pi\)
−0.810439 + 0.585824i \(0.800771\pi\)
\(644\) −7.96629e36 −0.000148734 0
\(645\) 0 0
\(646\) 4.85510e40 0.866615
\(647\) − 1.10620e40i − 0.193074i −0.995329 0.0965369i \(-0.969223\pi\)
0.995329 0.0965369i \(-0.0307766\pi\)
\(648\) 5.72747e40i 0.977518i
\(649\) −7.12509e40 −1.18917
\(650\) 0 0
\(651\) −2.36652e39 −0.0377735
\(652\) 1.71116e39i 0.0267116i
\(653\) − 9.61992e40i − 1.46869i −0.678776 0.734346i \(-0.737489\pi\)
0.678776 0.734346i \(-0.262511\pi\)
\(654\) 2.54591e41 3.80160
\(655\) 0 0
\(656\) 2.08948e39 0.0298491
\(657\) − 6.45677e40i − 0.902231i
\(658\) − 1.68928e39i − 0.0230901i
\(659\) −2.42717e40 −0.324535 −0.162268 0.986747i \(-0.551881\pi\)
−0.162268 + 0.986747i \(0.551881\pi\)
\(660\) 0 0
\(661\) −8.66824e40 −1.10920 −0.554600 0.832117i \(-0.687129\pi\)
−0.554600 + 0.832117i \(0.687129\pi\)
\(662\) 4.10643e40i 0.514071i
\(663\) − 1.19722e41i − 1.46631i
\(664\) −4.51983e40 −0.541608
\(665\) 0 0
\(666\) 2.91616e41 3.34530
\(667\) − 3.07370e39i − 0.0345014i
\(668\) − 1.54136e40i − 0.169295i
\(669\) 2.83791e41 3.05014
\(670\) 0 0
\(671\) −3.66684e40 −0.377412
\(672\) − 1.71138e39i − 0.0172382i
\(673\) 8.20694e40i 0.809027i 0.914532 + 0.404513i \(0.132559\pi\)
−0.914532 + 0.404513i \(0.867441\pi\)
\(674\) 7.88983e40 0.761201
\(675\) 0 0
\(676\) −1.13606e40 −0.104997
\(677\) 1.82761e41i 1.65328i 0.562730 + 0.826640i \(0.309751\pi\)
−0.562730 + 0.826640i \(0.690249\pi\)
\(678\) 1.92769e40i 0.170689i
\(679\) 4.91685e38 0.00426163
\(680\) 0 0
\(681\) −1.77416e41 −1.47353
\(682\) 9.97446e40i 0.810991i
\(683\) 3.66780e39i 0.0291948i 0.999893 + 0.0145974i \(0.00464666\pi\)
−0.999893 + 0.0145974i \(0.995353\pi\)
\(684\) 5.03439e40 0.392314
\(685\) 0 0
\(686\) 5.71356e39 0.0426783
\(687\) − 7.35141e40i − 0.537648i
\(688\) 2.31718e41i 1.65931i
\(689\) 1.82968e40 0.128291
\(690\) 0 0
\(691\) 5.62493e40 0.378169 0.189084 0.981961i \(-0.439448\pi\)
0.189084 + 0.981961i \(0.439448\pi\)
\(692\) 5.46458e40i 0.359765i
\(693\) − 3.83638e39i − 0.0247337i
\(694\) −1.15971e41 −0.732213
\(695\) 0 0
\(696\) 2.79442e41 1.69222
\(697\) 4.53429e39i 0.0268927i
\(698\) 2.52531e40i 0.146693i
\(699\) −3.38324e41 −1.92493
\(700\) 0 0
\(701\) 4.31831e40 0.235724 0.117862 0.993030i \(-0.462396\pi\)
0.117862 + 0.993030i \(0.462396\pi\)
\(702\) − 3.03759e41i − 1.62421i
\(703\) 1.97000e41i 1.03184i
\(704\) 7.53459e40 0.386595
\(705\) 0 0
\(706\) −2.45267e41 −1.20773
\(707\) 1.66844e39i 0.00804878i
\(708\) − 1.85174e41i − 0.875181i
\(709\) −3.70730e41 −1.71667 −0.858336 0.513088i \(-0.828501\pi\)
−0.858336 + 0.513088i \(0.828501\pi\)
\(710\) 0 0
\(711\) −4.63944e41 −2.06232
\(712\) 6.23303e39i 0.0271481i
\(713\) − 7.85524e39i − 0.0335244i
\(714\) 9.61528e39 0.0402104
\(715\) 0 0
\(716\) −7.49262e40 −0.300881
\(717\) 2.45311e41i 0.965361i
\(718\) − 7.58541e40i − 0.292533i
\(719\) 3.14942e40 0.119031 0.0595157 0.998227i \(-0.481044\pi\)
0.0595157 + 0.998227i \(0.481044\pi\)
\(720\) 0 0
\(721\) −4.49128e39 −0.0163045
\(722\) − 1.54559e41i − 0.549926i
\(723\) − 3.28800e41i − 1.14664i
\(724\) 1.07863e41 0.368690
\(725\) 0 0
\(726\) 3.59222e41 1.17973
\(727\) 2.91745e41i 0.939192i 0.882881 + 0.469596i \(0.155600\pi\)
−0.882881 + 0.469596i \(0.844400\pi\)
\(728\) − 3.86202e39i − 0.0121873i
\(729\) 2.60209e41 0.804957
\(730\) 0 0
\(731\) −5.02842e41 −1.49496
\(732\) − 9.52976e40i − 0.277761i
\(733\) 3.81345e41i 1.08971i 0.838531 + 0.544854i \(0.183415\pi\)
−0.838531 + 0.544854i \(0.816585\pi\)
\(734\) −7.83352e40 −0.219464
\(735\) 0 0
\(736\) 5.68062e39 0.0152991
\(737\) 8.35331e40i 0.220587i
\(738\) 2.23564e40i 0.0578876i
\(739\) −5.69450e41 −1.44581 −0.722905 0.690947i \(-0.757194\pi\)
−0.722905 + 0.690947i \(0.757194\pi\)
\(740\) 0 0
\(741\) 3.98768e41 0.973546
\(742\) 1.46948e39i 0.00351810i
\(743\) − 2.24951e41i − 0.528140i −0.964503 0.264070i \(-0.914935\pi\)
0.964503 0.264070i \(-0.0850650\pi\)
\(744\) 7.14150e41 1.64430
\(745\) 0 0
\(746\) 7.36262e41 1.63049
\(747\) − 6.22224e41i − 1.35144i
\(748\) − 8.52307e40i − 0.181561i
\(749\) −7.87693e39 −0.0164578
\(750\) 0 0
\(751\) −6.45261e41 −1.29705 −0.648524 0.761194i \(-0.724614\pi\)
−0.648524 + 0.761194i \(0.724614\pi\)
\(752\) 6.55901e41i 1.29324i
\(753\) − 1.42432e42i − 2.75474i
\(754\) −5.40890e41 −1.02618
\(755\) 0 0
\(756\) 5.13067e39 0.00936716
\(757\) − 8.68640e41i − 1.55579i −0.628397 0.777893i \(-0.716289\pi\)
0.628397 0.777893i \(-0.283711\pi\)
\(758\) − 3.95839e41i − 0.695528i
\(759\) 1.89154e40 0.0326069
\(760\) 0 0
\(761\) 4.39407e41 0.729104 0.364552 0.931183i \(-0.381222\pi\)
0.364552 + 0.931183i \(0.381222\pi\)
\(762\) − 1.31056e42i − 2.13359i
\(763\) 2.29306e40i 0.0366278i
\(764\) 1.54451e41 0.242068
\(765\) 0 0
\(766\) −6.77649e41 −1.02256
\(767\) − 9.87434e41i − 1.46209i
\(768\) 8.99727e41i 1.30729i
\(769\) 3.86450e41 0.551012 0.275506 0.961299i \(-0.411155\pi\)
0.275506 + 0.961299i \(0.411155\pi\)
\(770\) 0 0
\(771\) 1.23698e42 1.69853
\(772\) 2.73329e41i 0.368328i
\(773\) − 3.89700e41i − 0.515380i −0.966228 0.257690i \(-0.917039\pi\)
0.966228 0.257690i \(-0.0829614\pi\)
\(774\) −2.47928e42 −3.21796
\(775\) 0 0
\(776\) −1.48377e41 −0.185511
\(777\) 3.90148e40i 0.0478768i
\(778\) 1.25346e42i 1.50975i
\(779\) −1.51028e40 −0.0178552
\(780\) 0 0
\(781\) −1.39716e41 −0.159150
\(782\) 3.19161e40i 0.0356872i
\(783\) 1.97961e42i 2.17287i
\(784\) −1.10901e42 −1.19496
\(785\) 0 0
\(786\) 4.39101e41 0.455972
\(787\) 8.84928e41i 0.902140i 0.892488 + 0.451070i \(0.148958\pi\)
−0.892488 + 0.451070i \(0.851042\pi\)
\(788\) 4.03784e41i 0.404128i
\(789\) 3.01297e42 2.96059
\(790\) 0 0
\(791\) −1.73624e39 −0.00164456
\(792\) 1.15771e42i 1.07668i
\(793\) − 5.08171e41i − 0.464032i
\(794\) 1.12972e42 1.01292
\(795\) 0 0
\(796\) 2.93363e41 0.253610
\(797\) − 6.26660e41i − 0.531969i −0.963977 0.265985i \(-0.914303\pi\)
0.963977 0.265985i \(-0.0856971\pi\)
\(798\) 3.20265e40i 0.0266973i
\(799\) −1.42335e42 −1.16515
\(800\) 0 0
\(801\) −8.58073e40 −0.0677412
\(802\) − 7.49409e41i − 0.581019i
\(803\) − 3.64093e41i − 0.277228i
\(804\) −2.17094e41 −0.162344
\(805\) 0 0
\(806\) −1.38231e42 −0.997123
\(807\) 3.40392e42i 2.41164i
\(808\) − 5.03489e41i − 0.350369i
\(809\) 8.71861e41 0.595928 0.297964 0.954577i \(-0.403692\pi\)
0.297964 + 0.954577i \(0.403692\pi\)
\(810\) 0 0
\(811\) 9.09697e41 0.599922 0.299961 0.953952i \(-0.403026\pi\)
0.299961 + 0.953952i \(0.403026\pi\)
\(812\) − 9.13595e39i − 0.00591822i
\(813\) − 1.27958e42i − 0.814245i
\(814\) 1.64440e42 1.02791
\(815\) 0 0
\(816\) −3.73336e42 −2.25213
\(817\) − 1.67486e42i − 0.992565i
\(818\) − 2.22848e42i − 1.29743i
\(819\) 5.31666e40 0.0304104
\(820\) 0 0
\(821\) 1.52548e42 0.842231 0.421115 0.907007i \(-0.361639\pi\)
0.421115 + 0.907007i \(0.361639\pi\)
\(822\) 4.24015e42i 2.30006i
\(823\) 1.67746e42i 0.894037i 0.894525 + 0.447019i \(0.147514\pi\)
−0.894525 + 0.447019i \(0.852486\pi\)
\(824\) 1.35534e42 0.709747
\(825\) 0 0
\(826\) 7.93043e40 0.0400946
\(827\) − 3.12102e42i − 1.55048i −0.631665 0.775242i \(-0.717628\pi\)
0.631665 0.775242i \(-0.282372\pi\)
\(828\) 3.30948e40i 0.0161555i
\(829\) 1.01143e42 0.485174 0.242587 0.970130i \(-0.422004\pi\)
0.242587 + 0.970130i \(0.422004\pi\)
\(830\) 0 0
\(831\) 1.08501e42 0.502595
\(832\) 1.04418e42i 0.475323i
\(833\) − 2.40663e42i − 1.07661i
\(834\) −7.94436e41 −0.349262
\(835\) 0 0
\(836\) 2.83886e41 0.120546
\(837\) 5.05915e42i 2.11134i
\(838\) 1.37292e42i 0.563126i
\(839\) −1.10356e42 −0.444887 −0.222443 0.974946i \(-0.571403\pi\)
−0.222443 + 0.974946i \(0.571403\pi\)
\(840\) 0 0
\(841\) 9.57314e41 0.372832
\(842\) 2.59240e42i 0.992375i
\(843\) 5.49860e42i 2.06896i
\(844\) 6.06329e41 0.224255
\(845\) 0 0
\(846\) −7.01785e42 −2.50804
\(847\) 3.23545e40i 0.0113665i
\(848\) − 5.70562e41i − 0.197044i
\(849\) 7.59106e42 2.57716
\(850\) 0 0
\(851\) −1.29502e41 −0.0424912
\(852\) − 3.63109e41i − 0.117129i
\(853\) − 1.80886e42i − 0.573646i −0.957984 0.286823i \(-0.907401\pi\)
0.957984 0.286823i \(-0.0925992\pi\)
\(854\) 4.08130e40 0.0127250
\(855\) 0 0
\(856\) 2.37704e42 0.716417
\(857\) 3.58273e42i 1.06168i 0.847473 + 0.530838i \(0.178123\pi\)
−0.847473 + 0.530838i \(0.821877\pi\)
\(858\) − 3.32861e42i − 0.969834i
\(859\) −3.55534e42 −1.01855 −0.509273 0.860605i \(-0.670086\pi\)
−0.509273 + 0.860605i \(0.670086\pi\)
\(860\) 0 0
\(861\) −2.99103e39 −0.000828467 0
\(862\) − 5.12970e42i − 1.39713i
\(863\) 5.78959e42i 1.55057i 0.631612 + 0.775285i \(0.282394\pi\)
−0.631612 + 0.775285i \(0.717606\pi\)
\(864\) −3.65859e42 −0.963528
\(865\) 0 0
\(866\) 2.33033e42 0.593482
\(867\) − 1.11695e42i − 0.279740i
\(868\) − 2.33481e40i − 0.00575063i
\(869\) −2.61614e42 −0.633687
\(870\) 0 0
\(871\) −1.15765e42 −0.271215
\(872\) − 6.91982e42i − 1.59443i
\(873\) − 2.04263e42i − 0.462897i
\(874\) −1.06306e41 −0.0236942
\(875\) 0 0
\(876\) 9.46241e41 0.204029
\(877\) − 1.74620e42i − 0.370340i −0.982707 0.185170i \(-0.940716\pi\)
0.982707 0.185170i \(-0.0592835\pi\)
\(878\) 8.64771e42i 1.80398i
\(879\) −1.29265e43 −2.65243
\(880\) 0 0
\(881\) 6.11305e42 1.21369 0.606844 0.794821i \(-0.292435\pi\)
0.606844 + 0.794821i \(0.292435\pi\)
\(882\) − 1.18659e43i − 2.31744i
\(883\) 5.67511e42i 1.09029i 0.838340 + 0.545147i \(0.183526\pi\)
−0.838340 + 0.545147i \(0.816474\pi\)
\(884\) 1.18117e42 0.223232
\(885\) 0 0
\(886\) 6.85281e42 1.25337
\(887\) 1.28948e42i 0.232019i 0.993248 + 0.116009i \(0.0370102\pi\)
−0.993248 + 0.116009i \(0.962990\pi\)
\(888\) − 1.17736e43i − 2.08411i
\(889\) 1.18040e41 0.0205568
\(890\) 0 0
\(891\) −4.44615e42 −0.749475
\(892\) 2.79988e42i 0.464354i
\(893\) − 4.74087e42i − 0.773592i
\(894\) −1.61736e42 −0.259665
\(895\) 0 0
\(896\) −1.47262e41 −0.0228889
\(897\) 2.62140e41i 0.0400906i
\(898\) − 2.82924e42i − 0.425758i
\(899\) 9.00859e42 1.33396
\(900\) 0 0
\(901\) 1.23815e42 0.177527
\(902\) 1.26066e41i 0.0177871i
\(903\) − 3.31698e41i − 0.0460543i
\(904\) 5.23947e41 0.0715888
\(905\) 0 0
\(906\) 3.52444e42 0.466371
\(907\) − 6.85671e42i − 0.892915i −0.894805 0.446457i \(-0.852685\pi\)
0.894805 0.446457i \(-0.147315\pi\)
\(908\) − 1.75038e42i − 0.224331i
\(909\) 6.93130e42 0.874255
\(910\) 0 0
\(911\) −6.18262e42 −0.755363 −0.377682 0.925936i \(-0.623279\pi\)
−0.377682 + 0.925936i \(0.623279\pi\)
\(912\) − 1.24350e43i − 1.49528i
\(913\) − 3.50868e42i − 0.415257i
\(914\) 9.10348e42 1.06044
\(915\) 0 0
\(916\) 7.25289e41 0.0818516
\(917\) 3.95491e40i 0.00439321i
\(918\) − 2.05555e43i − 2.24755i
\(919\) −7.15141e42 −0.769691 −0.384846 0.922981i \(-0.625745\pi\)
−0.384846 + 0.922981i \(0.625745\pi\)
\(920\) 0 0
\(921\) −9.19938e42 −0.959387
\(922\) 1.52402e43i 1.56456i
\(923\) − 1.93626e42i − 0.195677i
\(924\) 5.62222e40 0.00559325
\(925\) 0 0
\(926\) 7.22663e42 0.696749
\(927\) 1.86583e43i 1.77099i
\(928\) 6.51468e42i 0.608762i
\(929\) −1.19347e43 −1.09795 −0.548975 0.835839i \(-0.684982\pi\)
−0.548975 + 0.835839i \(0.684982\pi\)
\(930\) 0 0
\(931\) 8.01598e42 0.714803
\(932\) − 3.33790e42i − 0.293051i
\(933\) 2.60363e43i 2.25058i
\(934\) 2.35024e43 2.00024
\(935\) 0 0
\(936\) −1.60442e43 −1.32379
\(937\) − 4.02931e42i − 0.327345i −0.986515 0.163673i \(-0.947666\pi\)
0.986515 0.163673i \(-0.0523341\pi\)
\(938\) − 9.29747e40i − 0.00743744i
\(939\) −3.89442e43 −3.06755
\(940\) 0 0
\(941\) 9.40592e42 0.718376 0.359188 0.933265i \(-0.383054\pi\)
0.359188 + 0.933265i \(0.383054\pi\)
\(942\) − 9.77996e42i − 0.735527i
\(943\) − 9.92816e39i 0 0.000735274i
\(944\) −3.07918e43 −2.24564
\(945\) 0 0
\(946\) −1.39805e43 −0.988781
\(947\) − 1.85611e43i − 1.29279i −0.763001 0.646397i \(-0.776275\pi\)
0.763001 0.646397i \(-0.223725\pi\)
\(948\) − 6.79910e42i − 0.466370i
\(949\) 5.04579e42 0.340855
\(950\) 0 0
\(951\) −1.49276e43 −0.978078
\(952\) − 2.61344e41i − 0.0168646i
\(953\) 3.20332e42i 0.203588i 0.994805 + 0.101794i \(0.0324583\pi\)
−0.994805 + 0.101794i \(0.967542\pi\)
\(954\) 6.10475e42 0.382135
\(955\) 0 0
\(956\) −2.42024e42 −0.146967
\(957\) 2.16927e43i 1.29745i
\(958\) − 1.46573e43i − 0.863482i
\(959\) −3.81903e41 −0.0221607
\(960\) 0 0
\(961\) 5.26074e42 0.296181
\(962\) 2.27890e43i 1.26383i
\(963\) 3.27235e43i 1.78764i
\(964\) 3.24393e42 0.174564
\(965\) 0 0
\(966\) −2.10534e40 −0.00109939
\(967\) 5.23025e42i 0.269053i 0.990910 + 0.134527i \(0.0429514\pi\)
−0.990910 + 0.134527i \(0.957049\pi\)
\(968\) − 9.76368e42i − 0.494790i
\(969\) 2.69848e43 1.34718
\(970\) 0 0
\(971\) −2.58906e43 −1.25448 −0.627239 0.778827i \(-0.715815\pi\)
−0.627239 + 0.778827i \(0.715815\pi\)
\(972\) − 1.20881e42i − 0.0577028i
\(973\) − 7.15536e40i − 0.00336508i
\(974\) 2.73998e43 1.26953
\(975\) 0 0
\(976\) −1.58466e43 −0.712712
\(977\) 6.80879e42i 0.301716i 0.988555 + 0.150858i \(0.0482037\pi\)
−0.988555 + 0.150858i \(0.951796\pi\)
\(978\) 4.52226e42i 0.197443i
\(979\) −4.83861e41 −0.0208148
\(980\) 0 0
\(981\) 9.52620e43 3.97850
\(982\) 2.08435e43i 0.857735i
\(983\) 4.18545e43i 1.69713i 0.529089 + 0.848566i \(0.322534\pi\)
−0.529089 + 0.848566i \(0.677466\pi\)
\(984\) 9.02608e41 0.0360637
\(985\) 0 0
\(986\) −3.66023e43 −1.42001
\(987\) − 9.38906e41i − 0.0358941i
\(988\) 3.93424e42i 0.148213i
\(989\) 1.10101e42 0.0408738
\(990\) 0 0
\(991\) 2.63339e43 0.949397 0.474699 0.880148i \(-0.342557\pi\)
0.474699 + 0.880148i \(0.342557\pi\)
\(992\) 1.66491e43i 0.591523i
\(993\) 2.28237e43i 0.799138i
\(994\) 1.55508e41 0.00536600
\(995\) 0 0
\(996\) 9.11870e42 0.305614
\(997\) − 1.61216e43i − 0.532511i −0.963903 0.266255i \(-0.914214\pi\)
0.963903 0.266255i \(-0.0857865\pi\)
\(998\) − 2.97416e43i − 0.968216i
\(999\) 8.34058e43 2.67606
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 25.30.b.a.24.4 4
5.2 odd 4 25.30.a.a.1.1 2
5.3 odd 4 1.30.a.a.1.2 2
5.4 even 2 inner 25.30.b.a.24.1 4
15.8 even 4 9.30.a.a.1.1 2
20.3 even 4 16.30.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.30.a.a.1.2 2 5.3 odd 4
9.30.a.a.1.1 2 15.8 even 4
16.30.a.c.1.2 2 20.3 even 4
25.30.a.a.1.1 2 5.2 odd 4
25.30.b.a.24.1 4 5.4 even 2 inner
25.30.b.a.24.4 4 1.1 even 1 trivial