Properties

Label 75.10.b.f.49.2
Level $75$
Weight $10$
Character 75.49
Analytic conductor $38.628$
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] = [75,10,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.f.49.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.78626i q^{2} +81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} +1839.88i q^{7} -7501.08i q^{8} -6561.00 q^{9} +O(q^{10})\) \(q-7.78626i q^{2} +81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} +1839.88i q^{7} -7501.08i q^{8} -6561.00 q^{9} +44385.9 q^{11} +36561.3i q^{12} -136584. i q^{13} +14325.8 q^{14} +172698. q^{16} -253591. i q^{17} +51085.7i q^{18} -85435.7 q^{19} -149030. q^{21} -345600. i q^{22} +979409. i q^{23} +607588. q^{24} -1.06348e6 q^{26} -531441. i q^{27} +830473. i q^{28} -2.58640e6 q^{29} +8.94787e6 q^{31} -5.18523e6i q^{32} +3.59526e6i q^{33} -1.97452e6 q^{34} -2.96147e6 q^{36} +1.56064e7i q^{37} +665225. i q^{38} +1.10633e7 q^{39} +2.44893e7 q^{41} +1.16039e6i q^{42} -1.27592e7i q^{43} +2.00346e7 q^{44} +7.62593e6 q^{46} -6.16764e7i q^{47} +1.39886e7i q^{48} +3.69685e7 q^{49} +2.05408e7 q^{51} -6.16504e7i q^{52} -5.70418e6i q^{53} -4.13794e6 q^{54} +1.38011e7 q^{56} -6.92029e6i q^{57} +2.01384e7i q^{58} -8.35095e7 q^{59} +1.48622e8 q^{61} -6.96704e7i q^{62} -1.20714e7i q^{63} +4.80479e7 q^{64} +2.79936e7 q^{66} -1.68003e8i q^{67} -1.14464e8i q^{68} -7.93321e7 q^{69} +2.10986e8 q^{71} +4.92146e7i q^{72} +1.43534e8i q^{73} +1.21515e8 q^{74} -3.85635e7 q^{76} +8.16646e7i q^{77} -8.61416e7i q^{78} +4.55960e8 q^{79} +4.30467e7 q^{81} -1.90680e8i q^{82} +3.55106e8i q^{83} -6.72683e7 q^{84} -9.93465e7 q^{86} -2.09498e8i q^{87} -3.32942e8i q^{88} +4.24540e8 q^{89} +2.51297e8 q^{91} +4.42080e8i q^{92} +7.24777e8i q^{93} -4.80228e8 q^{94} +4.20003e8 q^{96} +1.19905e9i q^{97} -2.87846e8i q^{98} -2.91216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9} - 43024 q^{11} - 923328 q^{14} + 774530 q^{16} + 191792 q^{19} - 2286144 q^{21} + 4233546 q^{24} - 10838332 q^{26} + 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34} + 7099002 q^{36} + 3934008 q^{39} + 52120744 q^{41} + 170859944 q^{44} + 34190784 q^{46} - 146565860 q^{49} + 25426872 q^{51} + 32949342 q^{54} + 484908480 q^{56} + 70989328 q^{59} + 682994680 q^{61} + 410240942 q^{64} + 470048184 q^{66} - 119112768 q^{69} + 420572128 q^{71} - 250480468 q^{74} - 437024728 q^{76} + 49510080 q^{79} + 172186884 q^{81} + 1838386368 q^{84} - 1746292744 q^{86} + 855278232 q^{89} - 2253718656 q^{91} - 3295087792 q^{94} + 1037208726 q^{96} + 282280464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 7.78626i − 0.344107i −0.985088 0.172054i \(-0.944960\pi\)
0.985088 0.172054i \(-0.0550403\pi\)
\(3\) 81.0000i 0.577350i
\(4\) 451.374 0.881590
\(5\) 0 0
\(6\) 630.687 0.198671
\(7\) 1839.88i 0.289633i 0.989459 + 0.144816i \(0.0462591\pi\)
−0.989459 + 0.144816i \(0.953741\pi\)
\(8\) − 7501.08i − 0.647469i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 44385.9 0.914066 0.457033 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(12\) 36561.3i 0.508986i
\(13\) − 136584.i − 1.32634i −0.748470 0.663169i \(-0.769211\pi\)
0.748470 0.663169i \(-0.230789\pi\)
\(14\) 14325.8 0.0996648
\(15\) 0 0
\(16\) 172698. 0.658791
\(17\) − 253591.i − 0.736399i −0.929747 0.368199i \(-0.879974\pi\)
0.929747 0.368199i \(-0.120026\pi\)
\(18\) 51085.7i 0.114702i
\(19\) −85435.7 −0.150400 −0.0752001 0.997168i \(-0.523960\pi\)
−0.0752001 + 0.997168i \(0.523960\pi\)
\(20\) 0 0
\(21\) −149030. −0.167220
\(22\) − 345600.i − 0.314537i
\(23\) 979409.i 0.729775i 0.931052 + 0.364887i \(0.118892\pi\)
−0.931052 + 0.364887i \(0.881108\pi\)
\(24\) 607588. 0.373816
\(25\) 0 0
\(26\) −1.06348e6 −0.456403
\(27\) − 531441.i − 0.192450i
\(28\) 830473.i 0.255337i
\(29\) −2.58640e6 −0.679054 −0.339527 0.940596i \(-0.610267\pi\)
−0.339527 + 0.940596i \(0.610267\pi\)
\(30\) 0 0
\(31\) 8.94787e6 1.74017 0.870085 0.492901i \(-0.164064\pi\)
0.870085 + 0.492901i \(0.164064\pi\)
\(32\) − 5.18523e6i − 0.874164i
\(33\) 3.59526e6i 0.527736i
\(34\) −1.97452e6 −0.253400
\(35\) 0 0
\(36\) −2.96147e6 −0.293863
\(37\) 1.56064e7i 1.36897i 0.729026 + 0.684486i \(0.239974\pi\)
−0.729026 + 0.684486i \(0.760026\pi\)
\(38\) 665225.i 0.0517538i
\(39\) 1.10633e7 0.765761
\(40\) 0 0
\(41\) 2.44893e7 1.35347 0.676735 0.736227i \(-0.263394\pi\)
0.676735 + 0.736227i \(0.263394\pi\)
\(42\) 1.16039e6i 0.0575415i
\(43\) − 1.27592e7i − 0.569135i −0.958656 0.284568i \(-0.908150\pi\)
0.958656 0.284568i \(-0.0918500\pi\)
\(44\) 2.00346e7 0.805832
\(45\) 0 0
\(46\) 7.62593e6 0.251121
\(47\) − 6.16764e7i − 1.84365i −0.387607 0.921825i \(-0.626698\pi\)
0.387607 0.921825i \(-0.373302\pi\)
\(48\) 1.39886e7i 0.380353i
\(49\) 3.69685e7 0.916113
\(50\) 0 0
\(51\) 2.05408e7 0.425160
\(52\) − 6.16504e7i − 1.16929i
\(53\) − 5.70418e6i − 0.0993006i −0.998767 0.0496503i \(-0.984189\pi\)
0.998767 0.0496503i \(-0.0158107\pi\)
\(54\) −4.13794e6 −0.0662235
\(55\) 0 0
\(56\) 1.38011e7 0.187528
\(57\) − 6.92029e6i − 0.0868336i
\(58\) 2.01384e7i 0.233668i
\(59\) −8.35095e7 −0.897226 −0.448613 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(60\) 0 0
\(61\) 1.48622e8 1.37435 0.687177 0.726490i \(-0.258850\pi\)
0.687177 + 0.726490i \(0.258850\pi\)
\(62\) − 6.96704e7i − 0.598806i
\(63\) − 1.20714e7i − 0.0965443i
\(64\) 4.80479e7 0.357985
\(65\) 0 0
\(66\) 2.79936e7 0.181598
\(67\) − 1.68003e8i − 1.01854i −0.860606 0.509272i \(-0.829915\pi\)
0.860606 0.509272i \(-0.170085\pi\)
\(68\) − 1.14464e8i − 0.649202i
\(69\) −7.93321e7 −0.421336
\(70\) 0 0
\(71\) 2.10986e8 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(72\) 4.92146e7i 0.215823i
\(73\) 1.43534e8i 0.591566i 0.955255 + 0.295783i \(0.0955805\pi\)
−0.955255 + 0.295783i \(0.904420\pi\)
\(74\) 1.21515e8 0.471074
\(75\) 0 0
\(76\) −3.85635e7 −0.132591
\(77\) 8.16646e7i 0.264744i
\(78\) − 8.61416e7i − 0.263504i
\(79\) 4.55960e8 1.31706 0.658529 0.752555i \(-0.271179\pi\)
0.658529 + 0.752555i \(0.271179\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) − 1.90680e8i − 0.465739i
\(83\) 3.55106e8i 0.821309i 0.911791 + 0.410655i \(0.134700\pi\)
−0.911791 + 0.410655i \(0.865300\pi\)
\(84\) −6.72683e7 −0.147419
\(85\) 0 0
\(86\) −9.93465e7 −0.195844
\(87\) − 2.09498e8i − 0.392052i
\(88\) − 3.32942e8i − 0.591830i
\(89\) 4.24540e8 0.717239 0.358620 0.933484i \(-0.383248\pi\)
0.358620 + 0.933484i \(0.383248\pi\)
\(90\) 0 0
\(91\) 2.51297e8 0.384151
\(92\) 4.42080e8i 0.643362i
\(93\) 7.24777e8i 1.00469i
\(94\) −4.80228e8 −0.634413
\(95\) 0 0
\(96\) 4.20003e8 0.504699
\(97\) 1.19905e9i 1.37520i 0.726092 + 0.687598i \(0.241335\pi\)
−0.726092 + 0.687598i \(0.758665\pi\)
\(98\) − 2.87846e8i − 0.315241i
\(99\) −2.91216e8 −0.304689
\(100\) 0 0
\(101\) −1.77085e9 −1.69331 −0.846654 0.532144i \(-0.821387\pi\)
−0.846654 + 0.532144i \(0.821387\pi\)
\(102\) − 1.59936e8i − 0.146301i
\(103\) − 3.03322e8i − 0.265544i −0.991147 0.132772i \(-0.957612\pi\)
0.991147 0.132772i \(-0.0423878\pi\)
\(104\) −1.02453e9 −0.858763
\(105\) 0 0
\(106\) −4.44142e7 −0.0341701
\(107\) − 1.95414e8i − 0.144121i −0.997400 0.0720607i \(-0.977042\pi\)
0.997400 0.0720607i \(-0.0229575\pi\)
\(108\) − 2.39879e8i − 0.169662i
\(109\) −2.31494e9 −1.57080 −0.785401 0.618988i \(-0.787543\pi\)
−0.785401 + 0.618988i \(0.787543\pi\)
\(110\) 0 0
\(111\) −1.26412e9 −0.790377
\(112\) 3.17743e8i 0.190808i
\(113\) − 1.31945e9i − 0.761270i −0.924725 0.380635i \(-0.875706\pi\)
0.924725 0.380635i \(-0.124294\pi\)
\(114\) −5.38832e7 −0.0298801
\(115\) 0 0
\(116\) −1.16743e9 −0.598648
\(117\) 8.96126e8i 0.442113i
\(118\) 6.50227e8i 0.308742i
\(119\) 4.66576e8 0.213285
\(120\) 0 0
\(121\) −3.87842e8 −0.164483
\(122\) − 1.15721e9i − 0.472925i
\(123\) 1.98363e9i 0.781426i
\(124\) 4.03884e9 1.53412
\(125\) 0 0
\(126\) −9.39914e7 −0.0332216
\(127\) 3.18960e9i 1.08798i 0.839093 + 0.543988i \(0.183086\pi\)
−0.839093 + 0.543988i \(0.816914\pi\)
\(128\) − 3.02895e9i − 0.997349i
\(129\) 1.03350e9 0.328590
\(130\) 0 0
\(131\) −5.40115e9 −1.60238 −0.801190 0.598410i \(-0.795799\pi\)
−0.801190 + 0.598410i \(0.795799\pi\)
\(132\) 1.62281e9i 0.465247i
\(133\) − 1.57191e8i − 0.0435608i
\(134\) −1.30811e9 −0.350488
\(135\) 0 0
\(136\) −1.90220e9 −0.476796
\(137\) 1.19903e7i 0.00290795i 0.999999 + 0.00145398i \(0.000462815\pi\)
−0.999999 + 0.00145398i \(0.999537\pi\)
\(138\) 6.17701e8i 0.144985i
\(139\) −9.15482e8 −0.208010 −0.104005 0.994577i \(-0.533166\pi\)
−0.104005 + 0.994577i \(0.533166\pi\)
\(140\) 0 0
\(141\) 4.99578e9 1.06443
\(142\) − 1.64279e9i − 0.339066i
\(143\) − 6.06239e9i − 1.21236i
\(144\) −1.13307e9 −0.219597
\(145\) 0 0
\(146\) 1.11760e9 0.203562
\(147\) 2.99445e9i 0.528918i
\(148\) 7.04432e9i 1.20687i
\(149\) −5.00462e9 −0.831827 −0.415913 0.909404i \(-0.636538\pi\)
−0.415913 + 0.909404i \(0.636538\pi\)
\(150\) 0 0
\(151\) 3.20554e9 0.501770 0.250885 0.968017i \(-0.419278\pi\)
0.250885 + 0.968017i \(0.419278\pi\)
\(152\) 6.40860e8i 0.0973794i
\(153\) 1.66381e9i 0.245466i
\(154\) 6.35862e8 0.0911002
\(155\) 0 0
\(156\) 4.99368e9 0.675088
\(157\) − 4.63430e8i − 0.0608745i −0.999537 0.0304373i \(-0.990310\pi\)
0.999537 0.0304373i \(-0.00968998\pi\)
\(158\) − 3.55023e9i − 0.453210i
\(159\) 4.62038e8 0.0573312
\(160\) 0 0
\(161\) −1.80199e9 −0.211367
\(162\) − 3.35173e8i − 0.0382342i
\(163\) 1.27948e10i 1.41968i 0.704363 + 0.709840i \(0.251233\pi\)
−0.704363 + 0.709840i \(0.748767\pi\)
\(164\) 1.10538e10 1.19320
\(165\) 0 0
\(166\) 2.76495e9 0.282619
\(167\) − 1.85699e10i − 1.84750i −0.382992 0.923752i \(-0.625106\pi\)
0.382992 0.923752i \(-0.374894\pi\)
\(168\) 1.11789e9i 0.108269i
\(169\) −8.05063e9 −0.759171
\(170\) 0 0
\(171\) 5.60544e8 0.0501334
\(172\) − 5.75917e9i − 0.501744i
\(173\) − 4.90746e9i − 0.416533i −0.978072 0.208266i \(-0.933218\pi\)
0.978072 0.208266i \(-0.0667821\pi\)
\(174\) −1.63121e9 −0.134908
\(175\) 0 0
\(176\) 7.66536e9 0.602179
\(177\) − 6.76427e9i − 0.518014i
\(178\) − 3.30558e9i − 0.246807i
\(179\) 1.28930e10 0.938678 0.469339 0.883018i \(-0.344492\pi\)
0.469339 + 0.883018i \(0.344492\pi\)
\(180\) 0 0
\(181\) 2.66233e10 1.84378 0.921889 0.387453i \(-0.126645\pi\)
0.921889 + 0.387453i \(0.126645\pi\)
\(182\) − 1.95667e9i − 0.132189i
\(183\) 1.20384e10i 0.793483i
\(184\) 7.34663e9 0.472506
\(185\) 0 0
\(186\) 5.64331e9 0.345721
\(187\) − 1.12558e10i − 0.673117i
\(188\) − 2.78391e10i − 1.62534i
\(189\) 9.77786e8 0.0557399
\(190\) 0 0
\(191\) −2.72802e10 −1.48319 −0.741595 0.670848i \(-0.765930\pi\)
−0.741595 + 0.670848i \(0.765930\pi\)
\(192\) 3.89188e9i 0.206683i
\(193\) 9.65442e9i 0.500862i 0.968134 + 0.250431i \(0.0805723\pi\)
−0.968134 + 0.250431i \(0.919428\pi\)
\(194\) 9.33612e9 0.473215
\(195\) 0 0
\(196\) 1.66866e10 0.807636
\(197\) 1.21091e10i 0.572812i 0.958108 + 0.286406i \(0.0924606\pi\)
−0.958108 + 0.286406i \(0.907539\pi\)
\(198\) 2.26748e9i 0.104846i
\(199\) −1.89741e10 −0.857673 −0.428837 0.903382i \(-0.641076\pi\)
−0.428837 + 0.903382i \(0.641076\pi\)
\(200\) 0 0
\(201\) 1.36082e10 0.588056
\(202\) 1.37883e10i 0.582680i
\(203\) − 4.75866e9i − 0.196676i
\(204\) 9.27161e9 0.374817
\(205\) 0 0
\(206\) −2.36174e9 −0.0913757
\(207\) − 6.42590e9i − 0.243258i
\(208\) − 2.35878e10i − 0.873779i
\(209\) −3.79214e9 −0.137476
\(210\) 0 0
\(211\) −8.91928e9 −0.309784 −0.154892 0.987931i \(-0.549503\pi\)
−0.154892 + 0.987931i \(0.549503\pi\)
\(212\) − 2.57472e9i − 0.0875424i
\(213\) 1.70898e10i 0.568892i
\(214\) −1.52154e9 −0.0495933
\(215\) 0 0
\(216\) −3.98638e9 −0.124605
\(217\) 1.64630e10i 0.504010i
\(218\) 1.80248e10i 0.540524i
\(219\) −1.16263e10 −0.341541
\(220\) 0 0
\(221\) −3.46364e10 −0.976714
\(222\) 9.84275e9i 0.271975i
\(223\) 4.42755e10i 1.19892i 0.800403 + 0.599462i \(0.204619\pi\)
−0.800403 + 0.599462i \(0.795381\pi\)
\(224\) 9.54018e9 0.253187
\(225\) 0 0
\(226\) −1.02735e10 −0.261959
\(227\) − 5.40045e10i − 1.34994i −0.737847 0.674968i \(-0.764157\pi\)
0.737847 0.674968i \(-0.235843\pi\)
\(228\) − 3.12364e9i − 0.0765516i
\(229\) −5.80250e9 −0.139430 −0.0697149 0.997567i \(-0.522209\pi\)
−0.0697149 + 0.997567i \(0.522209\pi\)
\(230\) 0 0
\(231\) −6.61483e9 −0.152850
\(232\) 1.94008e10i 0.439667i
\(233\) − 5.41865e10i − 1.20445i −0.798326 0.602226i \(-0.794281\pi\)
0.798326 0.602226i \(-0.205719\pi\)
\(234\) 6.97747e9 0.152134
\(235\) 0 0
\(236\) −3.76940e10 −0.790986
\(237\) 3.69328e10i 0.760404i
\(238\) − 3.63288e9i − 0.0733930i
\(239\) −7.91761e10 −1.56965 −0.784826 0.619716i \(-0.787248\pi\)
−0.784826 + 0.619716i \(0.787248\pi\)
\(240\) 0 0
\(241\) −6.14920e10 −1.17420 −0.587100 0.809514i \(-0.699730\pi\)
−0.587100 + 0.809514i \(0.699730\pi\)
\(242\) 3.01984e9i 0.0565998i
\(243\) 3.48678e9i 0.0641500i
\(244\) 6.70841e10 1.21162
\(245\) 0 0
\(246\) 1.54451e10 0.268894
\(247\) 1.16691e10i 0.199481i
\(248\) − 6.71187e10i − 1.12671i
\(249\) −2.87636e10 −0.474183
\(250\) 0 0
\(251\) 2.89319e10 0.460093 0.230046 0.973180i \(-0.426112\pi\)
0.230046 + 0.973180i \(0.426112\pi\)
\(252\) − 5.44873e9i − 0.0851125i
\(253\) 4.34719e10i 0.667062i
\(254\) 2.48350e10 0.374381
\(255\) 0 0
\(256\) 1.01633e9 0.0147896
\(257\) 1.22388e11i 1.75001i 0.484114 + 0.875005i \(0.339142\pi\)
−0.484114 + 0.875005i \(0.660858\pi\)
\(258\) − 8.04706e9i − 0.113070i
\(259\) −2.87139e10 −0.396499
\(260\) 0 0
\(261\) 1.69694e10 0.226351
\(262\) 4.20548e10i 0.551391i
\(263\) 6.24892e10i 0.805386i 0.915335 + 0.402693i \(0.131926\pi\)
−0.915335 + 0.402693i \(0.868074\pi\)
\(264\) 2.69683e10 0.341693
\(265\) 0 0
\(266\) −1.22393e9 −0.0149896
\(267\) 3.43878e10i 0.414098i
\(268\) − 7.58321e10i − 0.897938i
\(269\) −1.27214e11 −1.48132 −0.740659 0.671881i \(-0.765487\pi\)
−0.740659 + 0.671881i \(0.765487\pi\)
\(270\) 0 0
\(271\) 1.54116e10 0.173574 0.0867871 0.996227i \(-0.472340\pi\)
0.0867871 + 0.996227i \(0.472340\pi\)
\(272\) − 4.37946e10i − 0.485133i
\(273\) 2.03551e10i 0.221790i
\(274\) 9.33596e7 0.00100065
\(275\) 0 0
\(276\) −3.58085e10 −0.371445
\(277\) − 9.20867e10i − 0.939806i −0.882718 0.469903i \(-0.844289\pi\)
0.882718 0.469903i \(-0.155711\pi\)
\(278\) 7.12818e9i 0.0715776i
\(279\) −5.87070e10 −0.580057
\(280\) 0 0
\(281\) −7.78782e10 −0.745139 −0.372570 0.928004i \(-0.621523\pi\)
−0.372570 + 0.928004i \(0.621523\pi\)
\(282\) − 3.88985e10i − 0.366279i
\(283\) − 2.92463e10i − 0.271039i −0.990775 0.135520i \(-0.956730\pi\)
0.990775 0.135520i \(-0.0432704\pi\)
\(284\) 9.52334e10 0.868674
\(285\) 0 0
\(286\) −4.72034e10 −0.417182
\(287\) 4.50572e10i 0.392009i
\(288\) 3.40203e10i 0.291388i
\(289\) 5.42796e10 0.457717
\(290\) 0 0
\(291\) −9.71231e10 −0.793970
\(292\) 6.47877e10i 0.521519i
\(293\) 2.27403e11i 1.80257i 0.433227 + 0.901285i \(0.357375\pi\)
−0.433227 + 0.901285i \(0.642625\pi\)
\(294\) 2.33155e10 0.182005
\(295\) 0 0
\(296\) 1.17065e11 0.886368
\(297\) − 2.35885e10i − 0.175912i
\(298\) 3.89673e10i 0.286238i
\(299\) 1.33771e11 0.967927
\(300\) 0 0
\(301\) 2.34754e10 0.164840
\(302\) − 2.49592e10i − 0.172663i
\(303\) − 1.43439e11i − 0.977632i
\(304\) −1.47546e10 −0.0990823
\(305\) 0 0
\(306\) 1.29548e10 0.0844668
\(307\) − 6.02579e10i − 0.387160i −0.981084 0.193580i \(-0.937990\pi\)
0.981084 0.193580i \(-0.0620100\pi\)
\(308\) 3.68613e10i 0.233395i
\(309\) 2.45691e10 0.153312
\(310\) 0 0
\(311\) 1.36816e11 0.829308 0.414654 0.909979i \(-0.363903\pi\)
0.414654 + 0.909979i \(0.363903\pi\)
\(312\) − 8.29866e10i − 0.495807i
\(313\) − 2.25535e11i − 1.32820i −0.747643 0.664101i \(-0.768815\pi\)
0.747643 0.664101i \(-0.231185\pi\)
\(314\) −3.60839e9 −0.0209474
\(315\) 0 0
\(316\) 2.05809e11 1.16111
\(317\) − 1.28386e11i − 0.714089i −0.934087 0.357045i \(-0.883784\pi\)
0.934087 0.357045i \(-0.116216\pi\)
\(318\) − 3.59755e9i − 0.0197281i
\(319\) −1.14800e11 −0.620701
\(320\) 0 0
\(321\) 1.58285e10 0.0832086
\(322\) 1.40308e10i 0.0727328i
\(323\) 2.16657e10i 0.110755i
\(324\) 1.94302e10 0.0979545
\(325\) 0 0
\(326\) 9.96239e10 0.488522
\(327\) − 1.87510e11i − 0.906902i
\(328\) − 1.83696e11i − 0.876329i
\(329\) 1.13477e11 0.533981
\(330\) 0 0
\(331\) 1.15018e10 0.0526673 0.0263337 0.999653i \(-0.491617\pi\)
0.0263337 + 0.999653i \(0.491617\pi\)
\(332\) 1.60286e11i 0.724058i
\(333\) − 1.02394e11i − 0.456324i
\(334\) −1.44590e11 −0.635740
\(335\) 0 0
\(336\) −2.57372e10 −0.110163
\(337\) − 2.79631e11i − 1.18100i −0.807036 0.590502i \(-0.798930\pi\)
0.807036 0.590502i \(-0.201070\pi\)
\(338\) 6.26843e10i 0.261236i
\(339\) 1.06875e11 0.439519
\(340\) 0 0
\(341\) 3.97159e11 1.59063
\(342\) − 4.36454e9i − 0.0172513i
\(343\) 1.42263e11i 0.554969i
\(344\) −9.57078e10 −0.368497
\(345\) 0 0
\(346\) −3.82107e10 −0.143332
\(347\) 9.07088e10i 0.335867i 0.985798 + 0.167933i \(0.0537093\pi\)
−0.985798 + 0.167933i \(0.946291\pi\)
\(348\) − 9.45621e10i − 0.345629i
\(349\) 3.98825e11 1.43902 0.719511 0.694481i \(-0.244366\pi\)
0.719511 + 0.694481i \(0.244366\pi\)
\(350\) 0 0
\(351\) −7.25862e10 −0.255254
\(352\) − 2.30151e11i − 0.799044i
\(353\) 4.56192e11i 1.56373i 0.623448 + 0.781865i \(0.285731\pi\)
−0.623448 + 0.781865i \(0.714269\pi\)
\(354\) −5.26684e10 −0.178252
\(355\) 0 0
\(356\) 1.91627e11 0.632311
\(357\) 3.77926e10i 0.123140i
\(358\) − 1.00389e11i − 0.323006i
\(359\) −1.89176e11 −0.601094 −0.300547 0.953767i \(-0.597169\pi\)
−0.300547 + 0.953767i \(0.597169\pi\)
\(360\) 0 0
\(361\) −3.15388e11 −0.977380
\(362\) − 2.07296e11i − 0.634458i
\(363\) − 3.14152e10i − 0.0949643i
\(364\) 1.13429e11 0.338664
\(365\) 0 0
\(366\) 9.37339e10 0.273044
\(367\) − 3.17920e10i − 0.0914787i −0.998953 0.0457394i \(-0.985436\pi\)
0.998953 0.0457394i \(-0.0145644\pi\)
\(368\) 1.69142e11i 0.480769i
\(369\) −1.60674e11 −0.451156
\(370\) 0 0
\(371\) 1.04950e10 0.0287607
\(372\) 3.27146e11i 0.885723i
\(373\) 4.46388e11i 1.19405i 0.802222 + 0.597025i \(0.203651\pi\)
−0.802222 + 0.597025i \(0.796349\pi\)
\(374\) −8.76409e10 −0.231625
\(375\) 0 0
\(376\) −4.62639e11 −1.19371
\(377\) 3.53260e11i 0.900655i
\(378\) − 7.61330e9i − 0.0191805i
\(379\) −3.53467e11 −0.879978 −0.439989 0.898003i \(-0.645018\pi\)
−0.439989 + 0.898003i \(0.645018\pi\)
\(380\) 0 0
\(381\) −2.58357e11 −0.628143
\(382\) 2.12411e11i 0.510377i
\(383\) − 3.14974e11i − 0.747963i −0.927436 0.373981i \(-0.877992\pi\)
0.927436 0.373981i \(-0.122008\pi\)
\(384\) 2.45345e11 0.575820
\(385\) 0 0
\(386\) 7.51718e10 0.172350
\(387\) 8.37131e10i 0.189712i
\(388\) 5.41220e11i 1.21236i
\(389\) 4.20729e11 0.931600 0.465800 0.884890i \(-0.345767\pi\)
0.465800 + 0.884890i \(0.345767\pi\)
\(390\) 0 0
\(391\) 2.48369e11 0.537405
\(392\) − 2.77303e11i − 0.593155i
\(393\) − 4.37493e11i − 0.925134i
\(394\) 9.42843e10 0.197109
\(395\) 0 0
\(396\) −1.31447e11 −0.268611
\(397\) 5.63093e11i 1.13769i 0.822446 + 0.568844i \(0.192609\pi\)
−0.822446 + 0.568844i \(0.807391\pi\)
\(398\) 1.47737e11i 0.295132i
\(399\) 1.27325e10 0.0251498
\(400\) 0 0
\(401\) 2.75897e11 0.532841 0.266420 0.963857i \(-0.414159\pi\)
0.266420 + 0.963857i \(0.414159\pi\)
\(402\) − 1.05957e11i − 0.202355i
\(403\) − 1.22213e12i − 2.30805i
\(404\) −7.99317e11 −1.49280
\(405\) 0 0
\(406\) −3.70521e10 −0.0676778
\(407\) 6.92703e11i 1.25133i
\(408\) − 1.54079e11i − 0.275278i
\(409\) 6.54552e10 0.115662 0.0578308 0.998326i \(-0.481582\pi\)
0.0578308 + 0.998326i \(0.481582\pi\)
\(410\) 0 0
\(411\) −9.71214e8 −0.00167891
\(412\) − 1.36912e11i − 0.234101i
\(413\) − 1.53647e11i − 0.259866i
\(414\) −5.00337e10 −0.0837069
\(415\) 0 0
\(416\) −7.08218e11 −1.15944
\(417\) − 7.41541e10i − 0.120094i
\(418\) 2.95266e10i 0.0473064i
\(419\) −2.70250e10 −0.0428354 −0.0214177 0.999771i \(-0.506818\pi\)
−0.0214177 + 0.999771i \(0.506818\pi\)
\(420\) 0 0
\(421\) −4.29698e11 −0.666644 −0.333322 0.942813i \(-0.608170\pi\)
−0.333322 + 0.942813i \(0.608170\pi\)
\(422\) 6.94478e10i 0.106599i
\(423\) 4.04659e11i 0.614550i
\(424\) −4.27875e10 −0.0642940
\(425\) 0 0
\(426\) 1.33066e11 0.195760
\(427\) 2.73446e11i 0.398058i
\(428\) − 8.82048e10i − 0.127056i
\(429\) 4.91054e11 0.699957
\(430\) 0 0
\(431\) −9.44700e11 −1.31870 −0.659350 0.751836i \(-0.729169\pi\)
−0.659350 + 0.751836i \(0.729169\pi\)
\(432\) − 9.17789e10i − 0.126784i
\(433\) − 2.10762e11i − 0.288136i −0.989568 0.144068i \(-0.953982\pi\)
0.989568 0.144068i \(-0.0460184\pi\)
\(434\) 1.28185e11 0.173434
\(435\) 0 0
\(436\) −1.04491e12 −1.38480
\(437\) − 8.36765e10i − 0.109758i
\(438\) 9.05253e10i 0.117527i
\(439\) −7.69079e11 −0.988282 −0.494141 0.869382i \(-0.664517\pi\)
−0.494141 + 0.869382i \(0.664517\pi\)
\(440\) 0 0
\(441\) −2.42550e11 −0.305371
\(442\) 2.69688e11i 0.336094i
\(443\) 6.49894e11i 0.801726i 0.916138 + 0.400863i \(0.131290\pi\)
−0.916138 + 0.400863i \(0.868710\pi\)
\(444\) −5.70590e11 −0.696788
\(445\) 0 0
\(446\) 3.44741e11 0.412559
\(447\) − 4.05374e11i − 0.480255i
\(448\) 8.84023e10i 0.103684i
\(449\) 4.51858e11 0.524679 0.262339 0.964976i \(-0.415506\pi\)
0.262339 + 0.964976i \(0.415506\pi\)
\(450\) 0 0
\(451\) 1.08698e12 1.23716
\(452\) − 5.95564e11i − 0.671128i
\(453\) 2.59649e11i 0.289697i
\(454\) −4.20493e11 −0.464523
\(455\) 0 0
\(456\) −5.19097e10 −0.0562220
\(457\) − 6.18434e11i − 0.663240i −0.943413 0.331620i \(-0.892405\pi\)
0.943413 0.331620i \(-0.107595\pi\)
\(458\) 4.51798e10i 0.0479788i
\(459\) −1.34768e11 −0.141720
\(460\) 0 0
\(461\) −2.76450e11 −0.285078 −0.142539 0.989789i \(-0.545527\pi\)
−0.142539 + 0.989789i \(0.545527\pi\)
\(462\) 5.15048e10i 0.0525967i
\(463\) − 6.09969e11i − 0.616870i −0.951245 0.308435i \(-0.900195\pi\)
0.951245 0.308435i \(-0.0998052\pi\)
\(464\) −4.46666e11 −0.447355
\(465\) 0 0
\(466\) −4.21910e11 −0.414461
\(467\) 2.83260e11i 0.275587i 0.990461 + 0.137794i \(0.0440011\pi\)
−0.990461 + 0.137794i \(0.955999\pi\)
\(468\) 4.04488e11i 0.389762i
\(469\) 3.09104e11 0.295004
\(470\) 0 0
\(471\) 3.75378e10 0.0351459
\(472\) 6.26412e11i 0.580926i
\(473\) − 5.66328e11i − 0.520227i
\(474\) 2.87568e11 0.261661
\(475\) 0 0
\(476\) 2.10600e11 0.188030
\(477\) 3.74251e10i 0.0331002i
\(478\) 6.16485e11i 0.540129i
\(479\) 1.39149e12 1.20773 0.603865 0.797087i \(-0.293627\pi\)
0.603865 + 0.797087i \(0.293627\pi\)
\(480\) 0 0
\(481\) 2.13158e12 1.81572
\(482\) 4.78793e11i 0.404051i
\(483\) − 1.45961e11i − 0.122033i
\(484\) −1.75062e11 −0.145007
\(485\) 0 0
\(486\) 2.71490e10 0.0220745
\(487\) − 1.41532e11i − 0.114018i −0.998374 0.0570092i \(-0.981844\pi\)
0.998374 0.0570092i \(-0.0181564\pi\)
\(488\) − 1.11483e12i − 0.889851i
\(489\) −1.03638e12 −0.819652
\(490\) 0 0
\(491\) −4.66246e11 −0.362034 −0.181017 0.983480i \(-0.557939\pi\)
−0.181017 + 0.983480i \(0.557939\pi\)
\(492\) 8.95359e11i 0.688897i
\(493\) 6.55887e11i 0.500055i
\(494\) 9.08589e10 0.0686430
\(495\) 0 0
\(496\) 1.54528e12 1.14641
\(497\) 3.88188e11i 0.285389i
\(498\) 2.23961e11i 0.163170i
\(499\) 2.21254e12 1.59749 0.798746 0.601668i \(-0.205497\pi\)
0.798746 + 0.601668i \(0.205497\pi\)
\(500\) 0 0
\(501\) 1.50416e12 1.06666
\(502\) − 2.25271e11i − 0.158321i
\(503\) − 3.39574e11i − 0.236526i −0.992982 0.118263i \(-0.962267\pi\)
0.992982 0.118263i \(-0.0377326\pi\)
\(504\) −9.05488e10 −0.0625094
\(505\) 0 0
\(506\) 3.38484e11 0.229541
\(507\) − 6.52101e11i − 0.438308i
\(508\) 1.43970e12i 0.959149i
\(509\) 5.66691e11 0.374211 0.187106 0.982340i \(-0.440089\pi\)
0.187106 + 0.982340i \(0.440089\pi\)
\(510\) 0 0
\(511\) −2.64086e11 −0.171337
\(512\) − 1.55874e12i − 1.00244i
\(513\) 4.54040e10i 0.0289445i
\(514\) 9.52947e11 0.602191
\(515\) 0 0
\(516\) 4.66493e11 0.289682
\(517\) − 2.73756e12i − 1.68522i
\(518\) 2.23574e11i 0.136438i
\(519\) 3.97504e11 0.240485
\(520\) 0 0
\(521\) 4.42970e11 0.263393 0.131697 0.991290i \(-0.457958\pi\)
0.131697 + 0.991290i \(0.457958\pi\)
\(522\) − 1.32128e11i − 0.0778892i
\(523\) 1.44683e11i 0.0845591i 0.999106 + 0.0422796i \(0.0134620\pi\)
−0.999106 + 0.0422796i \(0.986538\pi\)
\(524\) −2.43794e12 −1.41264
\(525\) 0 0
\(526\) 4.86557e11 0.277139
\(527\) − 2.26910e12i − 1.28146i
\(528\) 6.20894e11i 0.347668i
\(529\) 8.41911e11 0.467429
\(530\) 0 0
\(531\) 5.47906e11 0.299075
\(532\) − 7.09520e10i − 0.0384028i
\(533\) − 3.34484e12i − 1.79516i
\(534\) 2.67752e11 0.142494
\(535\) 0 0
\(536\) −1.26020e12 −0.659475
\(537\) 1.04434e12i 0.541946i
\(538\) 9.90519e11i 0.509733i
\(539\) 1.64088e12 0.837388
\(540\) 0 0
\(541\) −3.10308e11 −0.155742 −0.0778710 0.996963i \(-0.524812\pi\)
−0.0778710 + 0.996963i \(0.524812\pi\)
\(542\) − 1.19999e11i − 0.0597281i
\(543\) 2.15649e12i 1.06451i
\(544\) −1.31493e12 −0.643733
\(545\) 0 0
\(546\) 1.58490e11 0.0763194
\(547\) 1.68502e12i 0.804750i 0.915475 + 0.402375i \(0.131815\pi\)
−0.915475 + 0.402375i \(0.868185\pi\)
\(548\) 5.41211e9i 0.00256362i
\(549\) −9.75108e11 −0.458118
\(550\) 0 0
\(551\) 2.20971e11 0.102130
\(552\) 5.95077e11i 0.272802i
\(553\) 8.38911e11i 0.381463i
\(554\) −7.17011e11 −0.323394
\(555\) 0 0
\(556\) −4.13225e11 −0.183379
\(557\) − 2.31328e12i − 1.01831i −0.860674 0.509156i \(-0.829958\pi\)
0.860674 0.509156i \(-0.170042\pi\)
\(558\) 4.57108e11i 0.199602i
\(559\) −1.74270e12 −0.754865
\(560\) 0 0
\(561\) 9.11723e11 0.388624
\(562\) 6.06380e11i 0.256408i
\(563\) − 7.03648e11i − 0.295167i −0.989050 0.147583i \(-0.952850\pi\)
0.989050 0.147583i \(-0.0471495\pi\)
\(564\) 2.25497e12 0.938392
\(565\) 0 0
\(566\) −2.27720e11 −0.0932666
\(567\) 7.92007e10i 0.0321814i
\(568\) − 1.58262e12i − 0.637983i
\(569\) −3.21997e12 −1.28779 −0.643896 0.765113i \(-0.722683\pi\)
−0.643896 + 0.765113i \(0.722683\pi\)
\(570\) 0 0
\(571\) −1.21116e12 −0.476801 −0.238401 0.971167i \(-0.576623\pi\)
−0.238401 + 0.971167i \(0.576623\pi\)
\(572\) − 2.73641e12i − 1.06880i
\(573\) − 2.20969e12i − 0.856320i
\(574\) 3.50827e11 0.134893
\(575\) 0 0
\(576\) −3.15242e11 −0.119328
\(577\) 7.30673e11i 0.274430i 0.990541 + 0.137215i \(0.0438151\pi\)
−0.990541 + 0.137215i \(0.956185\pi\)
\(578\) − 4.22635e11i − 0.157504i
\(579\) −7.82008e11 −0.289173
\(580\) 0 0
\(581\) −6.53352e11 −0.237878
\(582\) 7.56226e11i 0.273211i
\(583\) − 2.53185e11i − 0.0907673i
\(584\) 1.07666e12 0.383021
\(585\) 0 0
\(586\) 1.77062e12 0.620278
\(587\) 4.55331e12i 1.58291i 0.611230 + 0.791453i \(0.290675\pi\)
−0.611230 + 0.791453i \(0.709325\pi\)
\(588\) 1.35162e12i 0.466289i
\(589\) −7.64467e11 −0.261722
\(590\) 0 0
\(591\) −9.80833e11 −0.330713
\(592\) 2.69520e12i 0.901867i
\(593\) − 3.00074e12i − 0.996510i −0.867030 0.498255i \(-0.833974\pi\)
0.867030 0.498255i \(-0.166026\pi\)
\(594\) −1.83666e11 −0.0605327
\(595\) 0 0
\(596\) −2.25896e12 −0.733330
\(597\) − 1.53690e12i − 0.495178i
\(598\) − 1.04158e12i − 0.333071i
\(599\) 4.03514e12 1.28067 0.640336 0.768095i \(-0.278795\pi\)
0.640336 + 0.768095i \(0.278795\pi\)
\(600\) 0 0
\(601\) 2.04760e12 0.640192 0.320096 0.947385i \(-0.396285\pi\)
0.320096 + 0.947385i \(0.396285\pi\)
\(602\) − 1.82785e11i − 0.0567227i
\(603\) 1.10227e12i 0.339514i
\(604\) 1.44690e12 0.442355
\(605\) 0 0
\(606\) −1.11685e12 −0.336410
\(607\) 3.15792e12i 0.944175i 0.881552 + 0.472088i \(0.156499\pi\)
−0.881552 + 0.472088i \(0.843501\pi\)
\(608\) 4.43004e11i 0.131474i
\(609\) 3.85451e11 0.113551
\(610\) 0 0
\(611\) −8.42399e12 −2.44530
\(612\) 7.51000e11i 0.216401i
\(613\) 2.89302e12i 0.827520i 0.910386 + 0.413760i \(0.135785\pi\)
−0.910386 + 0.413760i \(0.864215\pi\)
\(614\) −4.69183e11 −0.133225
\(615\) 0 0
\(616\) 6.12573e11 0.171413
\(617\) 6.03603e12i 1.67675i 0.545094 + 0.838375i \(0.316494\pi\)
−0.545094 + 0.838375i \(0.683506\pi\)
\(618\) − 1.91301e11i − 0.0527558i
\(619\) −4.05606e12 −1.11044 −0.555222 0.831702i \(-0.687367\pi\)
−0.555222 + 0.831702i \(0.687367\pi\)
\(620\) 0 0
\(621\) 5.20498e11 0.140445
\(622\) − 1.06529e12i − 0.285371i
\(623\) 7.81102e11i 0.207736i
\(624\) 1.91061e12 0.504477
\(625\) 0 0
\(626\) −1.75607e12 −0.457044
\(627\) − 3.07163e11i − 0.0793716i
\(628\) − 2.09180e11i − 0.0536664i
\(629\) 3.95764e12 1.00811
\(630\) 0 0
\(631\) −5.34498e12 −1.34219 −0.671095 0.741372i \(-0.734176\pi\)
−0.671095 + 0.741372i \(0.734176\pi\)
\(632\) − 3.42020e12i − 0.852755i
\(633\) − 7.22461e11i − 0.178854i
\(634\) −9.99651e11 −0.245723
\(635\) 0 0
\(636\) 2.08552e11 0.0505426
\(637\) − 5.04929e12i − 1.21507i
\(638\) 8.93859e11i 0.213588i
\(639\) −1.38428e12 −0.328450
\(640\) 0 0
\(641\) 4.81739e12 1.12707 0.563534 0.826093i \(-0.309441\pi\)
0.563534 + 0.826093i \(0.309441\pi\)
\(642\) − 1.23245e11i − 0.0286327i
\(643\) 5.85505e11i 0.135077i 0.997717 + 0.0675385i \(0.0215145\pi\)
−0.997717 + 0.0675385i \(0.978485\pi\)
\(644\) −8.13372e11 −0.186339
\(645\) 0 0
\(646\) 1.68695e11 0.0381114
\(647\) 4.54903e12i 1.02059i 0.860001 + 0.510293i \(0.170463\pi\)
−0.860001 + 0.510293i \(0.829537\pi\)
\(648\) − 3.22897e11i − 0.0719410i
\(649\) −3.70664e12 −0.820124
\(650\) 0 0
\(651\) −1.33350e12 −0.290991
\(652\) 5.77526e12i 1.25158i
\(653\) 7.04350e12i 1.51593i 0.652295 + 0.757965i \(0.273806\pi\)
−0.652295 + 0.757965i \(0.726194\pi\)
\(654\) −1.46001e12 −0.312072
\(655\) 0 0
\(656\) 4.22925e12 0.891653
\(657\) − 9.41729e11i − 0.197189i
\(658\) − 8.83561e11i − 0.183747i
\(659\) −5.23559e12 −1.08139 −0.540694 0.841219i \(-0.681838\pi\)
−0.540694 + 0.841219i \(0.681838\pi\)
\(660\) 0 0
\(661\) −5.79017e12 −1.17974 −0.589868 0.807500i \(-0.700820\pi\)
−0.589868 + 0.807500i \(0.700820\pi\)
\(662\) − 8.95563e10i − 0.0181232i
\(663\) − 2.80555e12i − 0.563906i
\(664\) 2.66368e12 0.531772
\(665\) 0 0
\(666\) −7.97263e11 −0.157025
\(667\) − 2.53314e12i − 0.495557i
\(668\) − 8.38197e12i − 1.62874i
\(669\) −3.58632e12 −0.692199
\(670\) 0 0
\(671\) 6.59671e12 1.25625
\(672\) 7.72755e11i 0.146177i
\(673\) 5.92328e10i 0.0111300i 0.999985 + 0.00556499i \(0.00177140\pi\)
−0.999985 + 0.00556499i \(0.998229\pi\)
\(674\) −2.17728e12 −0.406392
\(675\) 0 0
\(676\) −3.63385e12 −0.669278
\(677\) 4.57177e12i 0.836440i 0.908346 + 0.418220i \(0.137346\pi\)
−0.908346 + 0.418220i \(0.862654\pi\)
\(678\) − 8.32157e11i − 0.151242i
\(679\) −2.20611e12 −0.398302
\(680\) 0 0
\(681\) 4.37436e12 0.779386
\(682\) − 3.09238e12i − 0.547348i
\(683\) 9.71286e12i 1.70787i 0.520382 + 0.853934i \(0.325790\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(684\) 2.53015e11 0.0441971
\(685\) 0 0
\(686\) 1.10770e12 0.190969
\(687\) − 4.70003e11i − 0.0804998i
\(688\) − 2.20349e12i − 0.374941i
\(689\) −7.79098e11 −0.131706
\(690\) 0 0
\(691\) 2.23726e12 0.373306 0.186653 0.982426i \(-0.440236\pi\)
0.186653 + 0.982426i \(0.440236\pi\)
\(692\) − 2.21510e12i − 0.367211i
\(693\) − 5.35801e11i − 0.0882478i
\(694\) 7.06282e11 0.115574
\(695\) 0 0
\(696\) −1.57146e12 −0.253842
\(697\) − 6.21025e12i − 0.996693i
\(698\) − 3.10535e12i − 0.495178i
\(699\) 4.38911e12 0.695391
\(700\) 0 0
\(701\) −2.29477e10 −0.00358929 −0.00179464 0.999998i \(-0.500571\pi\)
−0.00179464 + 0.999998i \(0.500571\pi\)
\(702\) 5.65175e11i 0.0878347i
\(703\) − 1.33334e12i − 0.205894i
\(704\) 2.13265e12 0.327222
\(705\) 0 0
\(706\) 3.55203e12 0.538091
\(707\) − 3.25815e12i − 0.490438i
\(708\) − 3.05322e12i − 0.456676i
\(709\) 9.57637e12 1.42329 0.711644 0.702540i \(-0.247951\pi\)
0.711644 + 0.702540i \(0.247951\pi\)
\(710\) 0 0
\(711\) −2.99156e12 −0.439020
\(712\) − 3.18451e12i − 0.464390i
\(713\) 8.76362e12i 1.26993i
\(714\) 2.94263e11 0.0423735
\(715\) 0 0
\(716\) 5.81959e12 0.827529
\(717\) − 6.41326e12i − 0.906239i
\(718\) 1.47298e12i 0.206841i
\(719\) 5.46390e12 0.762469 0.381235 0.924478i \(-0.375499\pi\)
0.381235 + 0.924478i \(0.375499\pi\)
\(720\) 0 0
\(721\) 5.58075e11 0.0769102
\(722\) 2.45570e12i 0.336324i
\(723\) − 4.98085e12i − 0.677925i
\(724\) 1.20171e13 1.62546
\(725\) 0 0
\(726\) −2.44607e11 −0.0326779
\(727\) 6.59842e12i 0.876062i 0.898960 + 0.438031i \(0.144324\pi\)
−0.898960 + 0.438031i \(0.855676\pi\)
\(728\) − 1.88500e12i − 0.248726i
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) −3.23561e12 −0.419111
\(732\) 5.43381e12i 0.699527i
\(733\) − 6.91041e11i − 0.0884170i −0.999022 0.0442085i \(-0.985923\pi\)
0.999022 0.0442085i \(-0.0140766\pi\)
\(734\) −2.47541e11 −0.0314785
\(735\) 0 0
\(736\) 5.07846e12 0.637943
\(737\) − 7.45694e12i − 0.931016i
\(738\) 1.25105e12i 0.155246i
\(739\) −7.60189e12 −0.937608 −0.468804 0.883302i \(-0.655315\pi\)
−0.468804 + 0.883302i \(0.655315\pi\)
\(740\) 0 0
\(741\) −9.45200e11 −0.115171
\(742\) − 8.17167e10i − 0.00989677i
\(743\) 2.23002e12i 0.268448i 0.990951 + 0.134224i \(0.0428541\pi\)
−0.990951 + 0.134224i \(0.957146\pi\)
\(744\) 5.43661e12 0.650504
\(745\) 0 0
\(746\) 3.47569e12 0.410882
\(747\) − 2.32985e12i − 0.273770i
\(748\) − 5.08060e12i − 0.593414i
\(749\) 3.59538e11 0.0417423
\(750\) 0 0
\(751\) 1.65708e12 0.190092 0.0950459 0.995473i \(-0.469700\pi\)
0.0950459 + 0.995473i \(0.469700\pi\)
\(752\) − 1.06514e13i − 1.21458i
\(753\) 2.34348e12i 0.265635i
\(754\) 2.75058e12 0.309922
\(755\) 0 0
\(756\) 4.41347e11 0.0491397
\(757\) − 1.28420e13i − 1.42135i −0.703521 0.710675i \(-0.748390\pi\)
0.703521 0.710675i \(-0.251610\pi\)
\(758\) 2.75218e12i 0.302807i
\(759\) −3.52122e12 −0.385129
\(760\) 0 0
\(761\) 1.61852e13 1.74939 0.874694 0.484676i \(-0.161063\pi\)
0.874694 + 0.484676i \(0.161063\pi\)
\(762\) 2.01164e12i 0.216149i
\(763\) − 4.25921e12i − 0.454955i
\(764\) −1.23136e13 −1.30757
\(765\) 0 0
\(766\) −2.45247e12 −0.257380
\(767\) 1.14060e13i 1.19002i
\(768\) 8.23228e10i 0.00853875i
\(769\) −5.30462e12 −0.546998 −0.273499 0.961872i \(-0.588181\pi\)
−0.273499 + 0.961872i \(0.588181\pi\)
\(770\) 0 0
\(771\) −9.91345e12 −1.01037
\(772\) 4.35775e12i 0.441555i
\(773\) 1.26188e13i 1.27119i 0.772022 + 0.635596i \(0.219245\pi\)
−0.772022 + 0.635596i \(0.780755\pi\)
\(774\) 6.51812e11 0.0652812
\(775\) 0 0
\(776\) 8.99418e12 0.890397
\(777\) − 2.32582e12i − 0.228919i
\(778\) − 3.27591e12i − 0.320571i
\(779\) −2.09226e12 −0.203562
\(780\) 0 0
\(781\) 9.36478e12 0.900675
\(782\) − 1.93387e12i − 0.184925i
\(783\) 1.37452e12i 0.130684i
\(784\) 6.38438e12 0.603527
\(785\) 0 0
\(786\) −3.40644e12 −0.318346
\(787\) 1.22887e13i 1.14188i 0.820991 + 0.570941i \(0.193421\pi\)
−0.820991 + 0.570941i \(0.806579\pi\)
\(788\) 5.46571e12i 0.504985i
\(789\) −5.06162e12 −0.464990
\(790\) 0 0
\(791\) 2.42762e12 0.220489
\(792\) 2.18443e12i 0.197277i
\(793\) − 2.02993e13i − 1.82286i
\(794\) 4.38439e12 0.391487
\(795\) 0 0
\(796\) −8.56441e12 −0.756116
\(797\) − 3.56495e12i − 0.312962i −0.987681 0.156481i \(-0.949985\pi\)
0.987681 0.156481i \(-0.0500150\pi\)
\(798\) − 9.91385e10i − 0.00865425i
\(799\) −1.56405e13 −1.35766
\(800\) 0 0
\(801\) −2.78541e12 −0.239080
\(802\) − 2.14821e12i − 0.183354i
\(803\) 6.37090e12i 0.540730i
\(804\) 6.14240e12 0.518425
\(805\) 0 0
\(806\) −9.51585e12 −0.794218
\(807\) − 1.03043e13i − 0.855239i
\(808\) 1.32833e13i 1.09636i
\(809\) 3.11624e12 0.255778 0.127889 0.991789i \(-0.459180\pi\)
0.127889 + 0.991789i \(0.459180\pi\)
\(810\) 0 0
\(811\) −2.30256e12 −0.186904 −0.0934518 0.995624i \(-0.529790\pi\)
−0.0934518 + 0.995624i \(0.529790\pi\)
\(812\) − 2.14793e12i − 0.173388i
\(813\) 1.24834e12i 0.100213i
\(814\) 5.39357e12 0.430593
\(815\) 0 0
\(816\) 3.54737e12 0.280092
\(817\) 1.09009e12i 0.0855980i
\(818\) − 5.09651e11i − 0.0398000i
\(819\) −1.64876e12 −0.128050
\(820\) 0 0
\(821\) −6.28720e12 −0.482962 −0.241481 0.970406i \(-0.577633\pi\)
−0.241481 + 0.970406i \(0.577633\pi\)
\(822\) 7.56212e9i 0 0.000577724i
\(823\) − 3.76056e12i − 0.285728i −0.989742 0.142864i \(-0.954369\pi\)
0.989742 0.142864i \(-0.0456312\pi\)
\(824\) −2.27524e12 −0.171932
\(825\) 0 0
\(826\) −1.19634e12 −0.0894219
\(827\) 1.75054e13i 1.30136i 0.759353 + 0.650679i \(0.225516\pi\)
−0.759353 + 0.650679i \(0.774484\pi\)
\(828\) − 2.90049e12i − 0.214454i
\(829\) 2.42570e12 0.178378 0.0891891 0.996015i \(-0.471572\pi\)
0.0891891 + 0.996015i \(0.471572\pi\)
\(830\) 0 0
\(831\) 7.45902e12 0.542597
\(832\) − 6.56257e12i − 0.474809i
\(833\) − 9.37486e12i − 0.674625i
\(834\) −5.77383e11 −0.0413254
\(835\) 0 0
\(836\) −1.71167e12 −0.121197
\(837\) − 4.75526e12i − 0.334896i
\(838\) 2.10424e11i 0.0147400i
\(839\) −9.69596e11 −0.0675557 −0.0337778 0.999429i \(-0.510754\pi\)
−0.0337778 + 0.999429i \(0.510754\pi\)
\(840\) 0 0
\(841\) −7.81769e12 −0.538885
\(842\) 3.34574e12i 0.229397i
\(843\) − 6.30813e12i − 0.430206i
\(844\) −4.02593e12 −0.273102
\(845\) 0 0
\(846\) 3.15078e12 0.211471
\(847\) − 7.13582e11i − 0.0476397i
\(848\) − 9.85101e11i − 0.0654183i
\(849\) 2.36895e12 0.156485
\(850\) 0 0
\(851\) −1.52850e13 −0.999042
\(852\) 7.71391e12i 0.501529i
\(853\) 2.11898e13i 1.37043i 0.728340 + 0.685215i \(0.240292\pi\)
−0.728340 + 0.685215i \(0.759708\pi\)
\(854\) 2.12912e12 0.136975
\(855\) 0 0
\(856\) −1.46582e12 −0.0933142
\(857\) 8.34904e12i 0.528717i 0.964425 + 0.264358i \(0.0851602\pi\)
−0.964425 + 0.264358i \(0.914840\pi\)
\(858\) − 3.82347e12i − 0.240860i
\(859\) 8.24621e12 0.516755 0.258378 0.966044i \(-0.416812\pi\)
0.258378 + 0.966044i \(0.416812\pi\)
\(860\) 0 0
\(861\) −3.64964e12 −0.226326
\(862\) 7.35568e12i 0.453775i
\(863\) − 4.35663e12i − 0.267364i −0.991024 0.133682i \(-0.957320\pi\)
0.991024 0.133682i \(-0.0426800\pi\)
\(864\) −2.75564e12 −0.168233
\(865\) 0 0
\(866\) −1.64105e12 −0.0991497
\(867\) 4.39665e12i 0.264263i
\(868\) 7.43096e12i 0.444331i
\(869\) 2.02382e13 1.20388
\(870\) 0 0
\(871\) −2.29464e13 −1.35093
\(872\) 1.73646e13i 1.01705i
\(873\) − 7.86697e12i − 0.458399i
\(874\) −6.51527e11 −0.0377686
\(875\) 0 0
\(876\) −5.24780e12 −0.301099
\(877\) − 1.21873e12i − 0.0695680i −0.999395 0.0347840i \(-0.988926\pi\)
0.999395 0.0347840i \(-0.0110743\pi\)
\(878\) 5.98825e12i 0.340075i
\(879\) −1.84197e13 −1.04071
\(880\) 0 0
\(881\) 3.98577e12 0.222905 0.111453 0.993770i \(-0.464450\pi\)
0.111453 + 0.993770i \(0.464450\pi\)
\(882\) 1.88856e12i 0.105080i
\(883\) − 2.19963e13i − 1.21766i −0.793300 0.608831i \(-0.791639\pi\)
0.793300 0.608831i \(-0.208361\pi\)
\(884\) −1.56340e13 −0.861061
\(885\) 0 0
\(886\) 5.06025e12 0.275880
\(887\) 1.39860e13i 0.758640i 0.925266 + 0.379320i \(0.123842\pi\)
−0.925266 + 0.379320i \(0.876158\pi\)
\(888\) 9.48226e12i 0.511745i
\(889\) −5.86847e12 −0.315113
\(890\) 0 0
\(891\) 1.91067e12 0.101563
\(892\) 1.99848e13i 1.05696i
\(893\) 5.26936e12i 0.277285i
\(894\) −3.15635e12 −0.165259
\(895\) 0 0
\(896\) 5.57290e12 0.288865
\(897\) 1.08355e13i 0.558833i
\(898\) − 3.51829e12i − 0.180546i
\(899\) −2.31428e13 −1.18167
\(900\) 0 0
\(901\) −1.44653e12 −0.0731248
\(902\) − 8.46349e12i − 0.425716i
\(903\) 1.90150e12i 0.0951705i
\(904\) −9.89727e12 −0.492899
\(905\) 0 0
\(906\) 2.02169e12 0.0996869
\(907\) − 8.00749e12i − 0.392884i −0.980515 0.196442i \(-0.937061\pi\)
0.980515 0.196442i \(-0.0629387\pi\)
\(908\) − 2.43762e13i − 1.19009i
\(909\) 1.16186e13 0.564436
\(910\) 0 0
\(911\) −4.09248e13 −1.96858 −0.984291 0.176555i \(-0.943505\pi\)
−0.984291 + 0.176555i \(0.943505\pi\)
\(912\) − 1.19512e12i − 0.0572052i
\(913\) 1.57617e13i 0.750731i
\(914\) −4.81529e12 −0.228226
\(915\) 0 0
\(916\) −2.61910e12 −0.122920
\(917\) − 9.93745e12i − 0.464102i
\(918\) 1.04934e12i 0.0487669i
\(919\) −8.12730e12 −0.375860 −0.187930 0.982182i \(-0.560178\pi\)
−0.187930 + 0.982182i \(0.560178\pi\)
\(920\) 0 0
\(921\) 4.88089e12 0.223527
\(922\) 2.15252e12i 0.0980973i
\(923\) − 2.88172e13i − 1.30691i
\(924\) −2.98576e12 −0.134751
\(925\) 0 0
\(926\) −4.74938e12 −0.212269
\(927\) 1.99010e12i 0.0885147i
\(928\) 1.34111e13i 0.593605i
\(929\) 4.43436e13 1.95326 0.976631 0.214923i \(-0.0689500\pi\)
0.976631 + 0.214923i \(0.0689500\pi\)
\(930\) 0 0
\(931\) −3.15843e12 −0.137783
\(932\) − 2.44584e13i − 1.06183i
\(933\) 1.10821e13i 0.478801i
\(934\) 2.20554e12 0.0948317
\(935\) 0 0
\(936\) 6.72192e12 0.286254
\(937\) − 3.56882e13i − 1.51250i −0.654281 0.756251i \(-0.727029\pi\)
0.654281 0.756251i \(-0.272971\pi\)
\(938\) − 2.40677e12i − 0.101513i
\(939\) 1.82683e13 0.766838
\(940\) 0 0
\(941\) 3.76351e13 1.56473 0.782366 0.622818i \(-0.214012\pi\)
0.782366 + 0.622818i \(0.214012\pi\)
\(942\) − 2.92279e11i − 0.0120940i
\(943\) 2.39850e13i 0.987727i
\(944\) −1.44219e13 −0.591085
\(945\) 0 0
\(946\) −4.40958e12 −0.179014
\(947\) − 1.01014e13i − 0.408139i −0.978956 0.204070i \(-0.934583\pi\)
0.978956 0.204070i \(-0.0654169\pi\)
\(948\) 1.66705e13i 0.670365i
\(949\) 1.96045e13 0.784616
\(950\) 0 0
\(951\) 1.03993e13 0.412280
\(952\) − 3.49982e12i − 0.138096i
\(953\) − 1.49469e13i − 0.586993i −0.955960 0.293497i \(-0.905181\pi\)
0.955960 0.293497i \(-0.0948190\pi\)
\(954\) 2.91402e11 0.0113900
\(955\) 0 0
\(956\) −3.57380e13 −1.38379
\(957\) − 9.29876e12i − 0.358362i
\(958\) − 1.08345e13i − 0.415589i
\(959\) −2.20607e10 −0.000842238 0
\(960\) 0 0
\(961\) 5.36247e13 2.02819
\(962\) − 1.65970e13i − 0.624803i
\(963\) 1.28211e12i 0.0480405i
\(964\) −2.77559e13 −1.03516
\(965\) 0 0
\(966\) −1.13649e12 −0.0419923
\(967\) 2.42942e13i 0.893475i 0.894665 + 0.446738i \(0.147414\pi\)
−0.894665 + 0.446738i \(0.852586\pi\)
\(968\) 2.90924e12i 0.106498i
\(969\) −1.75492e12 −0.0639441
\(970\) 0 0
\(971\) −1.07032e13 −0.386392 −0.193196 0.981160i \(-0.561885\pi\)
−0.193196 + 0.981160i \(0.561885\pi\)
\(972\) 1.57384e12i 0.0565540i
\(973\) − 1.68437e12i − 0.0602464i
\(974\) −1.10201e12 −0.0392346
\(975\) 0 0
\(976\) 2.56667e13 0.905412
\(977\) − 1.53038e13i − 0.537372i −0.963228 0.268686i \(-0.913411\pi\)
0.963228 0.268686i \(-0.0865894\pi\)
\(978\) 8.06954e12i 0.282048i
\(979\) 1.88436e13 0.655604
\(980\) 0 0
\(981\) 1.51883e13 0.523600
\(982\) 3.63032e12i 0.124578i
\(983\) − 3.27251e13i − 1.11787i −0.829212 0.558934i \(-0.811210\pi\)
0.829212 0.558934i \(-0.188790\pi\)
\(984\) 1.48794e13 0.505949
\(985\) 0 0
\(986\) 5.10691e12 0.172073
\(987\) 9.19163e12i 0.308294i
\(988\) 5.26714e12i 0.175861i
\(989\) 1.24965e13 0.415340
\(990\) 0 0
\(991\) 3.05488e13 1.00615 0.503075 0.864243i \(-0.332202\pi\)
0.503075 + 0.864243i \(0.332202\pi\)
\(992\) − 4.63967e13i − 1.52119i
\(993\) 9.31649e11i 0.0304075i
\(994\) 3.02253e12 0.0982046
\(995\) 0 0
\(996\) −1.29831e13 −0.418035
\(997\) 4.87857e13i 1.56374i 0.623441 + 0.781870i \(0.285734\pi\)
−0.623441 + 0.781870i \(0.714266\pi\)
\(998\) − 1.72274e13i − 0.549709i
\(999\) 8.29388e12 0.263459
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 75.10.b.f.49.2 4
3.2 odd 2 225.10.b.i.199.3 4
5.2 odd 4 75.10.a.f.1.2 2
5.3 odd 4 15.10.a.d.1.1 2
5.4 even 2 inner 75.10.b.f.49.3 4
15.2 even 4 225.10.a.k.1.1 2
15.8 even 4 45.10.a.d.1.2 2
15.14 odd 2 225.10.b.i.199.2 4
20.3 even 4 240.10.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.1 2 5.3 odd 4
45.10.a.d.1.2 2 15.8 even 4
75.10.a.f.1.2 2 5.2 odd 4
75.10.b.f.49.2 4 1.1 even 1 trivial
75.10.b.f.49.3 4 5.4 even 2 inner
225.10.a.k.1.1 2 15.2 even 4
225.10.b.i.199.2 4 15.14 odd 2
225.10.b.i.199.3 4 3.2 odd 2
240.10.a.r.1.2 2 20.3 even 4