Properties

Label 300.8.i.c.257.8
Level $300$
Weight $8$
Character 300.257
Analytic conductor $93.716$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,8,Mod(257,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.257");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.i (of order \(4\), degree \(2\), 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: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 5524236 x^{12} - 33135588 x^{11} - 49457570 x^{10} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{12}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.8
Root \(27.4835 + 23.4128i\) of defining polynomial
Character \(\chi\) \(=\) 300.257
Dual form 300.8.i.c.293.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(46.0669 - 8.05216i) q^{3} +(565.187 + 565.187i) q^{7} +(2057.33 - 741.876i) q^{9} +O(q^{10})\) \(q+(46.0669 - 8.05216i) q^{3} +(565.187 + 565.187i) q^{7} +(2057.33 - 741.876i) q^{9} -2361.39i q^{11} +(-732.128 + 732.128i) q^{13} +(-22896.0 + 22896.0i) q^{17} +38729.0i q^{19} +(30587.4 + 21485.4i) q^{21} +(23922.4 + 23922.4i) q^{23} +(88801.0 - 50741.9i) q^{27} +172492. q^{29} -119020. q^{31} +(-19014.3 - 108782. i) q^{33} +(-93692.5 - 93692.5i) q^{37} +(-27831.7 + 39622.1i) q^{39} +673768. i q^{41} +(-265451. + 265451. i) q^{43} +(218777. - 218777. i) q^{47} -184671. i q^{49} +(-870387. + 1.23911e6i) q^{51} +(9795.00 + 9795.00i) q^{53} +(311852. + 1.78413e6i) q^{57} +1.54803e6 q^{59} -394224. q^{61} +(1.58207e6 + 743474. i) q^{63} +(-898446. - 898446. i) q^{67} +(1.29466e6 + 909405. i) q^{69} +2.33054e6i q^{71} +(-2.64459e6 + 2.64459e6i) q^{73} +(1.33463e6 - 1.33463e6i) q^{77} +3.05235e6i q^{79} +(3.68221e6 - 3.05256e6i) q^{81} +(2.34959e6 + 2.34959e6i) q^{83} +(7.94617e6 - 1.38893e6i) q^{87} -3.61447e6 q^{89} -827578. q^{91} +(-5.48287e6 + 958365. i) q^{93} +(3.89118e6 + 3.89118e6i) q^{97} +(-1.75186e6 - 4.85815e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 215904 q^{21} + 872704 q^{31} + 1978560 q^{51} + 12752864 q^{61} + 10696176 q^{81} - 39496704 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 46.0669 8.05216i 0.985065 0.172182i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 565.187 + 565.187i 0.622800 + 0.622800i 0.946247 0.323446i \(-0.104841\pi\)
−0.323446 + 0.946247i \(0.604841\pi\)
\(8\) 0 0
\(9\) 2057.33 741.876i 0.940707 0.339221i
\(10\) 0 0
\(11\) 2361.39i 0.534926i −0.963568 0.267463i \(-0.913815\pi\)
0.963568 0.267463i \(-0.0861852\pi\)
\(12\) 0 0
\(13\) −732.128 + 732.128i −0.0924241 + 0.0924241i −0.751807 0.659383i \(-0.770818\pi\)
0.659383 + 0.751807i \(0.270818\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −22896.0 + 22896.0i −1.13029 + 1.13029i −0.140157 + 0.990129i \(0.544761\pi\)
−0.990129 + 0.140157i \(0.955239\pi\)
\(18\) 0 0
\(19\) 38729.0i 1.29539i 0.761902 + 0.647693i \(0.224266\pi\)
−0.761902 + 0.647693i \(0.775734\pi\)
\(20\) 0 0
\(21\) 30587.4 + 21485.4i 0.720734 + 0.506264i
\(22\) 0 0
\(23\) 23922.4 + 23922.4i 0.409975 + 0.409975i 0.881730 0.471755i \(-0.156379\pi\)
−0.471755 + 0.881730i \(0.656379\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 88801.0 50741.9i 0.868250 0.496128i
\(28\) 0 0
\(29\) 172492. 1.31333 0.656667 0.754181i \(-0.271966\pi\)
0.656667 + 0.754181i \(0.271966\pi\)
\(30\) 0 0
\(31\) −119020. −0.717551 −0.358776 0.933424i \(-0.616806\pi\)
−0.358776 + 0.933424i \(0.616806\pi\)
\(32\) 0 0
\(33\) −19014.3 108782.i −0.0921046 0.526937i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −93692.5 93692.5i −0.304088 0.304088i 0.538523 0.842611i \(-0.318983\pi\)
−0.842611 + 0.538523i \(0.818983\pi\)
\(38\) 0 0
\(39\) −27831.7 + 39622.1i −0.0751300 + 0.106958i
\(40\) 0 0
\(41\) 673768.i 1.52675i 0.645958 + 0.763373i \(0.276458\pi\)
−0.645958 + 0.763373i \(0.723542\pi\)
\(42\) 0 0
\(43\) −265451. + 265451.i −0.509148 + 0.509148i −0.914265 0.405117i \(-0.867231\pi\)
0.405117 + 0.914265i \(0.367231\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 218777. 218777.i 0.307368 0.307368i −0.536520 0.843888i \(-0.680261\pi\)
0.843888 + 0.536520i \(0.180261\pi\)
\(48\) 0 0
\(49\) 184671.i 0.224240i
\(50\) 0 0
\(51\) −870387. + 1.23911e6i −0.918791 + 1.30802i
\(52\) 0 0
\(53\) 9795.00 + 9795.00i 0.00903730 + 0.00903730i 0.711611 0.702574i \(-0.247966\pi\)
−0.702574 + 0.711611i \(0.747966\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 311852. + 1.78413e6i 0.223042 + 1.27604i
\(58\) 0 0
\(59\) 1.54803e6 0.981292 0.490646 0.871359i \(-0.336761\pi\)
0.490646 + 0.871359i \(0.336761\pi\)
\(60\) 0 0
\(61\) −394224. −0.222376 −0.111188 0.993799i \(-0.535466\pi\)
−0.111188 + 0.993799i \(0.535466\pi\)
\(62\) 0 0
\(63\) 1.58207e6 + 743474.i 0.797139 + 0.374605i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −898446. 898446.i −0.364947 0.364947i 0.500683 0.865631i \(-0.333082\pi\)
−0.865631 + 0.500683i \(0.833082\pi\)
\(68\) 0 0
\(69\) 1.29466e6 + 909405.i 0.474442 + 0.333262i
\(70\) 0 0
\(71\) 2.33054e6i 0.772772i 0.922337 + 0.386386i \(0.126277\pi\)
−0.922337 + 0.386386i \(0.873723\pi\)
\(72\) 0 0
\(73\) −2.64459e6 + 2.64459e6i −0.795663 + 0.795663i −0.982408 0.186745i \(-0.940206\pi\)
0.186745 + 0.982408i \(0.440206\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.33463e6 1.33463e6i 0.333152 0.333152i
\(78\) 0 0
\(79\) 3.05235e6i 0.696530i 0.937396 + 0.348265i \(0.113229\pi\)
−0.937396 + 0.348265i \(0.886771\pi\)
\(80\) 0 0
\(81\) 3.68221e6 3.05256e6i 0.769858 0.638215i
\(82\) 0 0
\(83\) 2.34959e6 + 2.34959e6i 0.451044 + 0.451044i 0.895701 0.444657i \(-0.146674\pi\)
−0.444657 + 0.895701i \(0.646674\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.94617e6 1.38893e6i 1.29372 0.226133i
\(88\) 0 0
\(89\) −3.61447e6 −0.543475 −0.271738 0.962371i \(-0.587598\pi\)
−0.271738 + 0.962371i \(0.587598\pi\)
\(90\) 0 0
\(91\) −827578. −0.115124
\(92\) 0 0
\(93\) −5.48287e6 + 958365.i −0.706835 + 0.123549i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.89118e6 + 3.89118e6i 0.432892 + 0.432892i 0.889611 0.456719i \(-0.150975\pi\)
−0.456719 + 0.889611i \(0.650975\pi\)
\(98\) 0 0
\(99\) −1.75186e6 4.85815e6i −0.181458 0.503208i
\(100\) 0 0
\(101\) 1.34688e7i 1.30078i −0.759599 0.650392i \(-0.774605\pi\)
0.759599 0.650392i \(-0.225395\pi\)
\(102\) 0 0
\(103\) −1.12200e7 + 1.12200e7i −1.01172 + 1.01172i −0.0117938 + 0.999930i \(0.503754\pi\)
−0.999930 + 0.0117938i \(0.996246\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −513749. + 513749.i −0.0405423 + 0.0405423i −0.727087 0.686545i \(-0.759126\pi\)
0.686545 + 0.727087i \(0.259126\pi\)
\(108\) 0 0
\(109\) 1.44407e7i 1.06806i −0.845466 0.534030i \(-0.820677\pi\)
0.845466 0.534030i \(-0.179323\pi\)
\(110\) 0 0
\(111\) −5.07056e6 3.56170e6i −0.351905 0.247188i
\(112\) 0 0
\(113\) 1.81341e7 + 1.81341e7i 1.18228 + 1.18228i 0.979152 + 0.203131i \(0.0651118\pi\)
0.203131 + 0.979152i \(0.434888\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −963077. + 2.04937e6i −0.0555918 + 0.118296i
\(118\) 0 0
\(119\) −2.58810e7 −1.40789
\(120\) 0 0
\(121\) 1.39110e7 0.713855
\(122\) 0 0
\(123\) 5.42528e6 + 3.10384e7i 0.262878 + 1.50394i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.98952e7 + 2.98952e7i 1.29506 + 1.29506i 0.931618 + 0.363438i \(0.118397\pi\)
0.363438 + 0.931618i \(0.381603\pi\)
\(128\) 0 0
\(129\) −1.00910e7 + 1.43659e7i −0.413878 + 0.589210i
\(130\) 0 0
\(131\) 1.04475e7i 0.406035i −0.979175 0.203018i \(-0.934925\pi\)
0.979175 0.203018i \(-0.0650748\pi\)
\(132\) 0 0
\(133\) −2.18891e7 + 2.18891e7i −0.806767 + 0.806767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.55355e7 3.55355e7i 1.18070 1.18070i 0.201138 0.979563i \(-0.435536\pi\)
0.979563 0.201138i \(-0.0644640\pi\)
\(138\) 0 0
\(139\) 2.50412e6i 0.0790865i −0.999218 0.0395433i \(-0.987410\pi\)
0.999218 0.0395433i \(-0.0125903\pi\)
\(140\) 0 0
\(141\) 8.31675e6 1.18400e7i 0.249854 0.355701i
\(142\) 0 0
\(143\) 1.72884e6 + 1.72884e6i 0.0494400 + 0.0494400i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.48700e6 8.50723e6i −0.0386100 0.220891i
\(148\) 0 0
\(149\) 7.78790e7 1.92872 0.964358 0.264602i \(-0.0852406\pi\)
0.964358 + 0.264602i \(0.0852406\pi\)
\(150\) 0 0
\(151\) −1.59598e7 −0.377231 −0.188615 0.982051i \(-0.560400\pi\)
−0.188615 + 0.982051i \(0.560400\pi\)
\(152\) 0 0
\(153\) −3.01185e7 + 6.40905e7i −0.679851 + 1.44668i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.04188e7 + 4.04188e7i 0.833555 + 0.833555i 0.988001 0.154446i \(-0.0493593\pi\)
−0.154446 + 0.988001i \(0.549359\pi\)
\(158\) 0 0
\(159\) 530096. + 372355.i 0.0104584 + 0.00734627i
\(160\) 0 0
\(161\) 2.70412e7i 0.510665i
\(162\) 0 0
\(163\) −6.56387e7 + 6.56387e7i −1.18714 + 1.18714i −0.209290 + 0.977854i \(0.567115\pi\)
−0.977854 + 0.209290i \(0.932885\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.17012e7 2.17012e7i 0.360559 0.360559i −0.503459 0.864019i \(-0.667940\pi\)
0.864019 + 0.503459i \(0.167940\pi\)
\(168\) 0 0
\(169\) 6.16765e7i 0.982916i
\(170\) 0 0
\(171\) 2.87322e7 + 7.96782e7i 0.439422 + 1.21858i
\(172\) 0 0
\(173\) 8.65454e7 + 8.65454e7i 1.27082 + 1.27082i 0.945661 + 0.325154i \(0.105416\pi\)
0.325154 + 0.945661i \(0.394584\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.13132e7 1.24650e7i 0.966637 0.168961i
\(178\) 0 0
\(179\) −5.55469e7 −0.723892 −0.361946 0.932199i \(-0.617888\pi\)
−0.361946 + 0.932199i \(0.617888\pi\)
\(180\) 0 0
\(181\) −8.21040e7 −1.02917 −0.514587 0.857438i \(-0.672055\pi\)
−0.514587 + 0.857438i \(0.672055\pi\)
\(182\) 0 0
\(183\) −1.81607e7 + 3.17435e6i −0.219055 + 0.0382892i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.40664e7 + 5.40664e7i 0.604619 + 0.604619i
\(188\) 0 0
\(189\) 7.88678e7 + 2.15105e7i 0.849735 + 0.231758i
\(190\) 0 0
\(191\) 1.27821e8i 1.32735i 0.748021 + 0.663675i \(0.231004\pi\)
−0.748021 + 0.663675i \(0.768996\pi\)
\(192\) 0 0
\(193\) −4.53151e7 + 4.53151e7i −0.453724 + 0.453724i −0.896589 0.442864i \(-0.853962\pi\)
0.442864 + 0.896589i \(0.353962\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.34501e7 + 6.34501e7i −0.591290 + 0.591290i −0.937980 0.346690i \(-0.887306\pi\)
0.346690 + 0.937980i \(0.387306\pi\)
\(198\) 0 0
\(199\) 2.01928e8i 1.81640i −0.418538 0.908199i \(-0.637457\pi\)
0.418538 0.908199i \(-0.362543\pi\)
\(200\) 0 0
\(201\) −4.86231e7 3.41542e7i −0.422334 0.296659i
\(202\) 0 0
\(203\) 9.74900e7 + 9.74900e7i 0.817945 + 0.817945i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.69636e7 + 3.14687e7i 0.524738 + 0.246594i
\(208\) 0 0
\(209\) 9.14544e7 0.692935
\(210\) 0 0
\(211\) −6.55270e6 −0.0480210 −0.0240105 0.999712i \(-0.507644\pi\)
−0.0240105 + 0.999712i \(0.507644\pi\)
\(212\) 0 0
\(213\) 1.87658e7 + 1.07361e8i 0.133057 + 0.761231i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.72683e7 6.72683e7i −0.446891 0.446891i
\(218\) 0 0
\(219\) −1.00534e8 + 1.43123e8i −0.646781 + 0.920779i
\(220\) 0 0
\(221\) 3.35256e7i 0.208931i
\(222\) 0 0
\(223\) −1.30061e8 + 1.30061e8i −0.785380 + 0.785380i −0.980733 0.195353i \(-0.937415\pi\)
0.195353 + 0.980733i \(0.437415\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.84611e8 1.84611e8i 1.04753 1.04753i 0.0487216 0.998812i \(-0.484485\pi\)
0.998812 0.0487216i \(-0.0155147\pi\)
\(228\) 0 0
\(229\) 3.37952e7i 0.185965i 0.995668 + 0.0929825i \(0.0296401\pi\)
−0.995668 + 0.0929825i \(0.970360\pi\)
\(230\) 0 0
\(231\) 5.07355e7 7.22288e7i 0.270813 0.385539i
\(232\) 0 0
\(233\) −2.26237e8 2.26237e8i −1.17170 1.17170i −0.981804 0.189898i \(-0.939184\pi\)
−0.189898 0.981804i \(-0.560816\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.45780e7 + 1.40612e8i 0.119930 + 0.686127i
\(238\) 0 0
\(239\) −2.17366e8 −1.02991 −0.514954 0.857218i \(-0.672191\pi\)
−0.514954 + 0.857218i \(0.672191\pi\)
\(240\) 0 0
\(241\) 1.52613e8 0.702316 0.351158 0.936316i \(-0.385788\pi\)
0.351158 + 0.936316i \(0.385788\pi\)
\(242\) 0 0
\(243\) 1.45048e8 1.70272e8i 0.648471 0.761239i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.83546e7 2.83546e7i −0.119725 0.119725i
\(248\) 0 0
\(249\) 1.27158e8 + 8.93191e7i 0.521969 + 0.366646i
\(250\) 0 0
\(251\) 3.18158e8i 1.26994i −0.772535 0.634972i \(-0.781012\pi\)
0.772535 0.634972i \(-0.218988\pi\)
\(252\) 0 0
\(253\) 5.64901e7 5.64901e7i 0.219306 0.219306i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.37997e7 + 2.37997e7i −0.0874591 + 0.0874591i −0.749483 0.662024i \(-0.769698\pi\)
0.662024 + 0.749483i \(0.269698\pi\)
\(258\) 0 0
\(259\) 1.05908e8i 0.378772i
\(260\) 0 0
\(261\) 3.54872e8 1.27968e8i 1.23546 0.445511i
\(262\) 0 0
\(263\) 8.33888e7 + 8.33888e7i 0.282659 + 0.282659i 0.834169 0.551510i \(-0.185948\pi\)
−0.551510 + 0.834169i \(0.685948\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.66508e8 + 2.91043e7i −0.535358 + 0.0935766i
\(268\) 0 0
\(269\) 5.61377e6 0.0175842 0.00879208 0.999961i \(-0.497201\pi\)
0.00879208 + 0.999961i \(0.497201\pi\)
\(270\) 0 0
\(271\) −3.57656e8 −1.09162 −0.545812 0.837907i \(-0.683779\pi\)
−0.545812 + 0.837907i \(0.683779\pi\)
\(272\) 0 0
\(273\) −3.81240e7 + 6.66379e6i −0.113404 + 0.0198222i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.65642e8 2.65642e8i −0.750962 0.750962i 0.223697 0.974659i \(-0.428187\pi\)
−0.974659 + 0.223697i \(0.928187\pi\)
\(278\) 0 0
\(279\) −2.44862e8 + 8.82979e7i −0.675005 + 0.243408i
\(280\) 0 0
\(281\) 4.57465e8i 1.22995i −0.788548 0.614973i \(-0.789167\pi\)
0.788548 0.614973i \(-0.210833\pi\)
\(282\) 0 0
\(283\) 3.81936e8 3.81936e8i 1.00170 1.00170i 0.00170342 0.999999i \(-0.499458\pi\)
0.999999 0.00170342i \(-0.000542216\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.80805e8 + 3.80805e8i −0.950858 + 0.950858i
\(288\) 0 0
\(289\) 6.38115e8i 1.55509i
\(290\) 0 0
\(291\) 2.10587e8 + 1.47922e8i 0.500963 + 0.351891i
\(292\) 0 0
\(293\) 7.69493e6 + 7.69493e6i 0.0178718 + 0.0178718i 0.715986 0.698114i \(-0.245977\pi\)
−0.698114 + 0.715986i \(0.745977\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.19821e8 2.09694e8i −0.265391 0.464449i
\(298\) 0 0
\(299\) −3.50285e7 −0.0757832
\(300\) 0 0
\(301\) −3.00058e8 −0.634195
\(302\) 0 0
\(303\) −1.08453e8 6.20468e8i −0.223971 1.28136i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.12115e8 + 2.12115e8i 0.418396 + 0.418396i 0.884651 0.466254i \(-0.154397\pi\)
−0.466254 + 0.884651i \(0.654397\pi\)
\(308\) 0 0
\(309\) −4.26525e8 + 6.07216e8i −0.822414 + 1.17082i
\(310\) 0 0
\(311\) 8.81144e8i 1.66106i 0.556973 + 0.830531i \(0.311963\pi\)
−0.556973 + 0.830531i \(0.688037\pi\)
\(312\) 0 0
\(313\) 5.14996e8 5.14996e8i 0.949289 0.949289i −0.0494862 0.998775i \(-0.515758\pi\)
0.998775 + 0.0494862i \(0.0157584\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.34448e8 + 3.34448e8i −0.589687 + 0.589687i −0.937547 0.347860i \(-0.886909\pi\)
0.347860 + 0.937547i \(0.386909\pi\)
\(318\) 0 0
\(319\) 4.07320e8i 0.702536i
\(320\) 0 0
\(321\) −1.95301e7 + 2.78036e7i −0.0329561 + 0.0469174i
\(322\) 0 0
\(323\) −8.86740e8 8.86740e8i −1.46416 1.46416i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −1.16279e8 6.65239e8i −0.183901 1.05211i
\(328\) 0 0
\(329\) 2.47299e8 0.382858
\(330\) 0 0
\(331\) −1.49153e8 −0.226065 −0.113033 0.993591i \(-0.536056\pi\)
−0.113033 + 0.993591i \(0.536056\pi\)
\(332\) 0 0
\(333\) −2.62264e8 1.23248e8i −0.389210 0.182904i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.81331e8 6.81331e8i −0.969736 0.969736i 0.0298192 0.999555i \(-0.490507\pi\)
−0.999555 + 0.0298192i \(0.990507\pi\)
\(338\) 0 0
\(339\) 9.81401e8 + 6.89364e8i 1.36819 + 0.961058i
\(340\) 0 0
\(341\) 2.81052e8i 0.383836i
\(342\) 0 0
\(343\) 5.69829e8 5.69829e8i 0.762457 0.762457i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.34509e8 2.34509e8i 0.301305 0.301305i −0.540219 0.841524i \(-0.681659\pi\)
0.841524 + 0.540219i \(0.181659\pi\)
\(348\) 0 0
\(349\) 6.82894e8i 0.859932i −0.902845 0.429966i \(-0.858526\pi\)
0.902845 0.429966i \(-0.141474\pi\)
\(350\) 0 0
\(351\) −2.78641e7 + 1.02163e8i −0.0343931 + 0.126101i
\(352\) 0 0
\(353\) −3.59877e7 3.59877e7i −0.0435454 0.0435454i 0.684999 0.728544i \(-0.259803\pi\)
−0.728544 + 0.684999i \(0.759803\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1.19226e9 + 2.08398e8i −1.38686 + 0.242413i
\(358\) 0 0
\(359\) 7.87990e8 0.898856 0.449428 0.893317i \(-0.351628\pi\)
0.449428 + 0.893317i \(0.351628\pi\)
\(360\) 0 0
\(361\) −6.06066e8 −0.678024
\(362\) 0 0
\(363\) 6.40838e8 1.12014e8i 0.703193 0.122913i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.88227e8 + 2.88227e8i 0.304371 + 0.304371i 0.842721 0.538350i \(-0.180952\pi\)
−0.538350 + 0.842721i \(0.680952\pi\)
\(368\) 0 0
\(369\) 4.99853e8 + 1.38616e9i 0.517904 + 1.43622i
\(370\) 0 0
\(371\) 1.10720e7i 0.0112569i
\(372\) 0 0
\(373\) 9.75486e8 9.75486e8i 0.973285 0.973285i −0.0263674 0.999652i \(-0.508394\pi\)
0.999652 + 0.0263674i \(0.00839398\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.26286e8 + 1.26286e8i −0.121384 + 0.121384i
\(378\) 0 0
\(379\) 2.44236e7i 0.0230448i −0.999934 0.0115224i \(-0.996332\pi\)
0.999934 0.0115224i \(-0.00366777\pi\)
\(380\) 0 0
\(381\) 1.61790e9 + 1.13646e9i 1.49870 + 1.05273i
\(382\) 0 0
\(383\) −8.30215e8 8.30215e8i −0.755083 0.755083i 0.220340 0.975423i \(-0.429283\pi\)
−0.975423 + 0.220340i \(0.929283\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.49187e8 + 7.43050e8i −0.306245 + 0.651673i
\(388\) 0 0
\(389\) −8.85170e8 −0.762435 −0.381218 0.924485i \(-0.624495\pi\)
−0.381218 + 0.924485i \(0.624495\pi\)
\(390\) 0 0
\(391\) −1.09545e9 −0.926778
\(392\) 0 0
\(393\) −8.41250e7 4.81285e8i −0.0699119 0.399971i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −4.72503e8 4.72503e8i −0.378999 0.378999i 0.491742 0.870741i \(-0.336360\pi\)
−0.870741 + 0.491742i \(0.836360\pi\)
\(398\) 0 0
\(399\) −8.32111e8 + 1.18462e9i −0.655807 + 0.933628i
\(400\) 0 0
\(401\) 1.26513e9i 0.979784i 0.871783 + 0.489892i \(0.162964\pi\)
−0.871783 + 0.489892i \(0.837036\pi\)
\(402\) 0 0
\(403\) 8.71377e7 8.71377e7i 0.0663190 0.0663190i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.21245e8 + 2.21245e8i −0.162664 + 0.162664i
\(408\) 0 0
\(409\) 2.15154e9i 1.55495i −0.628911 0.777477i \(-0.716499\pi\)
0.628911 0.777477i \(-0.283501\pi\)
\(410\) 0 0
\(411\) 1.35087e9 1.92315e9i 0.959772 1.36636i
\(412\) 0 0
\(413\) 8.74928e8 + 8.74928e8i 0.611149 + 0.611149i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.01635e7 1.15357e8i −0.0136173 0.0779054i
\(418\) 0 0
\(419\) 7.48977e8 0.497415 0.248708 0.968579i \(-0.419994\pi\)
0.248708 + 0.968579i \(0.419994\pi\)
\(420\) 0 0
\(421\) −2.49357e9 −1.62867 −0.814337 0.580392i \(-0.802899\pi\)
−0.814337 + 0.580392i \(0.802899\pi\)
\(422\) 0 0
\(423\) 2.87790e8 6.12400e8i 0.184877 0.393409i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.22810e8 2.22810e8i −0.138496 0.138496i
\(428\) 0 0
\(429\) 9.35633e7 + 6.57215e7i 0.0572143 + 0.0401890i
\(430\) 0 0
\(431\) 2.00871e9i 1.20850i −0.796794 0.604251i \(-0.793473\pi\)
0.796794 0.604251i \(-0.206527\pi\)
\(432\) 0 0
\(433\) −2.15215e9 + 2.15215e9i −1.27399 + 1.27399i −0.330011 + 0.943977i \(0.607052\pi\)
−0.943977 + 0.330011i \(0.892948\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.26492e8 + 9.26492e8i −0.531076 + 0.531076i
\(438\) 0 0
\(439\) 3.52076e9i 1.98615i −0.117499 0.993073i \(-0.537488\pi\)
0.117499 0.993073i \(-0.462512\pi\)
\(440\) 0 0
\(441\) −1.37003e8 3.79928e8i −0.0760668 0.210944i
\(442\) 0 0
\(443\) −4.05454e8 4.05454e8i −0.221579 0.221579i 0.587584 0.809163i \(-0.300079\pi\)
−0.809163 + 0.587584i \(0.800079\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 3.58764e9 6.27094e8i 1.89991 0.332090i
\(448\) 0 0
\(449\) −3.04161e9 −1.58578 −0.792888 0.609367i \(-0.791424\pi\)
−0.792888 + 0.609367i \(0.791424\pi\)
\(450\) 0 0
\(451\) 1.59103e9 0.816695
\(452\) 0 0
\(453\) −7.35217e8 + 1.28510e8i −0.371597 + 0.0649523i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.04947e9 1.04947e9i −0.514355 0.514355i 0.401502 0.915858i \(-0.368488\pi\)
−0.915858 + 0.401502i \(0.868488\pi\)
\(458\) 0 0
\(459\) −8.71401e8 + 3.19497e9i −0.420604 + 1.54214i
\(460\) 0 0
\(461\) 1.42379e9i 0.676851i −0.940993 0.338425i \(-0.890106\pi\)
0.940993 0.338425i \(-0.109894\pi\)
\(462\) 0 0
\(463\) 1.31932e9 1.31932e9i 0.617754 0.617754i −0.327201 0.944955i \(-0.606105\pi\)
0.944955 + 0.327201i \(0.106105\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.20143e9 2.20143e9i 1.00022 1.00022i 0.000219918 1.00000i \(-0.499930\pi\)
1.00000 0.000219918i \(-7.00021e-5\pi\)
\(468\) 0 0
\(469\) 1.01558e9i 0.454578i
\(470\) 0 0
\(471\) 2.18743e9 + 1.53651e9i 0.964629 + 0.677583i
\(472\) 0 0
\(473\) 6.26832e8 + 6.26832e8i 0.272356 + 0.272356i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.74182e7 + 1.28848e7i 0.0115671 + 0.00543581i
\(478\) 0 0
\(479\) 1.19523e9 0.496910 0.248455 0.968643i \(-0.420077\pi\)
0.248455 + 0.968643i \(0.420077\pi\)
\(480\) 0 0
\(481\) 1.37190e8 0.0562101
\(482\) 0 0
\(483\) 2.17740e8 + 1.24571e9i 0.0879274 + 0.503038i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.43704e9 + 2.43704e9i 0.956117 + 0.956117i 0.999077 0.0429600i \(-0.0136788\pi\)
−0.0429600 + 0.999077i \(0.513679\pi\)
\(488\) 0 0
\(489\) −2.49524e9 + 3.55231e9i −0.965009 + 1.37382i
\(490\) 0 0
\(491\) 3.69275e9i 1.40788i −0.710261 0.703938i \(-0.751423\pi\)
0.710261 0.703938i \(-0.248577\pi\)
\(492\) 0 0
\(493\) −3.94937e9 + 3.94937e9i −1.48444 + 1.48444i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.31719e9 + 1.31719e9i −0.481283 + 0.481283i
\(498\) 0 0
\(499\) 3.10681e9i 1.11934i −0.828715 0.559671i \(-0.810928\pi\)
0.828715 0.559671i \(-0.189072\pi\)
\(500\) 0 0
\(501\) 8.24968e8 1.17445e9i 0.293093 0.417256i
\(502\) 0 0
\(503\) 3.71686e9 + 3.71686e9i 1.30223 + 1.30223i 0.926885 + 0.375345i \(0.122476\pi\)
0.375345 + 0.926885i \(0.377524\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.96629e8 + 2.84125e9i 0.169240 + 0.968236i
\(508\) 0 0
\(509\) 1.98710e9 0.667894 0.333947 0.942592i \(-0.391619\pi\)
0.333947 + 0.942592i \(0.391619\pi\)
\(510\) 0 0
\(511\) −2.98938e9 −0.991078
\(512\) 0 0
\(513\) 1.96518e9 + 3.43918e9i 0.642677 + 1.12472i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −5.16617e8 5.16617e8i −0.164419 0.164419i
\(518\) 0 0
\(519\) 4.68376e9 + 3.29000e9i 1.47065 + 1.03302i
\(520\) 0 0
\(521\) 2.68557e9i 0.831963i 0.909373 + 0.415982i \(0.136562\pi\)
−0.909373 + 0.415982i \(0.863438\pi\)
\(522\) 0 0
\(523\) −1.83283e9 + 1.83283e9i −0.560231 + 0.560231i −0.929373 0.369142i \(-0.879652\pi\)
0.369142 + 0.929373i \(0.379652\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.72508e9 2.72508e9i 0.811038 0.811038i
\(528\) 0 0
\(529\) 2.26026e9i 0.663841i
\(530\) 0 0
\(531\) 3.18481e9 1.14845e9i 0.923108 0.332875i
\(532\) 0 0
\(533\) −4.93284e8 4.93284e8i −0.141108 0.141108i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.55887e9 + 4.47272e8i −0.713081 + 0.124641i
\(538\) 0 0
\(539\) −4.36080e8 −0.119951
\(540\) 0 0
\(541\) −3.09705e9 −0.840925 −0.420463 0.907310i \(-0.638132\pi\)
−0.420463 + 0.907310i \(0.638132\pi\)
\(542\) 0 0
\(543\) −3.78228e9 + 6.61114e8i −1.01380 + 0.177205i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.08212e9 + 5.08212e9i 1.32767 + 1.32767i 0.907401 + 0.420267i \(0.138064\pi\)
0.420267 + 0.907401i \(0.361936\pi\)
\(548\) 0 0
\(549\) −8.11047e8 + 2.92466e8i −0.209191 + 0.0754348i
\(550\) 0 0
\(551\) 6.68044e9i 1.70127i
\(552\) 0 0
\(553\) −1.72515e9 + 1.72515e9i −0.433799 + 0.433799i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.22298e9 2.22298e9i 0.545059 0.545059i −0.379949 0.925008i \(-0.624058\pi\)
0.925008 + 0.379949i \(0.124058\pi\)
\(558\) 0 0
\(559\) 3.88688e8i 0.0941151i
\(560\) 0 0
\(561\) 2.92602e9 + 2.05532e9i 0.699694 + 0.491485i
\(562\) 0 0
\(563\) −3.57065e8 3.57065e8i −0.0843273 0.0843273i 0.663685 0.748012i \(-0.268992\pi\)
−0.748012 + 0.663685i \(0.768992\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.80640e9 + 3.55867e8i 0.876948 + 0.0819874i
\(568\) 0 0
\(569\) 2.15255e8 0.0489848 0.0244924 0.999700i \(-0.492203\pi\)
0.0244924 + 0.999700i \(0.492203\pi\)
\(570\) 0 0
\(571\) −6.21269e9 −1.39654 −0.698270 0.715834i \(-0.746047\pi\)
−0.698270 + 0.715834i \(0.746047\pi\)
\(572\) 0 0
\(573\) 1.02924e9 + 5.88832e9i 0.228546 + 1.30753i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.09785e9 + 1.09785e9i 0.237918 + 0.237918i 0.815988 0.578069i \(-0.196194\pi\)
−0.578069 + 0.815988i \(0.696194\pi\)
\(578\) 0 0
\(579\) −1.72264e9 + 2.45241e9i −0.368825 + 0.525071i
\(580\) 0 0
\(581\) 2.65591e9i 0.561820i
\(582\) 0 0
\(583\) 2.31298e7 2.31298e7i 0.00483428 0.00483428i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.05457e9 3.05457e9i 0.623329 0.623329i −0.323052 0.946381i \(-0.604709\pi\)
0.946381 + 0.323052i \(0.104709\pi\)
\(588\) 0 0
\(589\) 4.60952e9i 0.929505i
\(590\) 0 0
\(591\) −2.41204e9 + 3.43386e9i −0.480650 + 0.684269i
\(592\) 0 0
\(593\) 5.59943e9 + 5.59943e9i 1.10269 + 1.10269i 0.994085 + 0.108602i \(0.0346374\pi\)
0.108602 + 0.994085i \(0.465363\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.62596e9 9.30221e9i −0.312751 1.78927i
\(598\) 0 0
\(599\) −7.22831e8 −0.137418 −0.0687088 0.997637i \(-0.521888\pi\)
−0.0687088 + 0.997637i \(0.521888\pi\)
\(600\) 0 0
\(601\) −1.31712e9 −0.247493 −0.123747 0.992314i \(-0.539491\pi\)
−0.123747 + 0.992314i \(0.539491\pi\)
\(602\) 0 0
\(603\) −2.51493e9 1.18186e9i −0.467106 0.219511i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.78474e9 + 1.78474e9i 0.323903 + 0.323903i 0.850262 0.526360i \(-0.176443\pi\)
−0.526360 + 0.850262i \(0.676443\pi\)
\(608\) 0 0
\(609\) 5.27607e9 + 3.70606e9i 0.946564 + 0.664894i
\(610\) 0 0
\(611\) 3.20345e8i 0.0568164i
\(612\) 0 0
\(613\) −5.53669e9 + 5.53669e9i −0.970819 + 0.970819i −0.999586 0.0287671i \(-0.990842\pi\)
0.0287671 + 0.999586i \(0.490842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.29818e9 + 6.29818e9i −1.07949 + 1.07949i −0.0829309 + 0.996555i \(0.526428\pi\)
−0.996555 + 0.0829309i \(0.973572\pi\)
\(618\) 0 0
\(619\) 4.02532e9i 0.682155i −0.940035 0.341078i \(-0.889208\pi\)
0.940035 0.341078i \(-0.110792\pi\)
\(620\) 0 0
\(621\) 3.33820e9 + 9.10465e8i 0.559361 + 0.152561i
\(622\) 0 0
\(623\) −2.04285e9 2.04285e9i −0.338476 0.338476i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.21302e9 7.36405e8i 0.682586 0.119311i
\(628\) 0 0
\(629\) 4.29037e9 0.687413
\(630\) 0 0
\(631\) 1.00012e9 0.158471 0.0792357 0.996856i \(-0.474752\pi\)
0.0792357 + 0.996856i \(0.474752\pi\)
\(632\) 0 0
\(633\) −3.01863e8 + 5.27633e7i −0.0473038 + 0.00826836i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.35203e8 + 1.35203e8i 0.0207252 + 0.0207252i
\(638\) 0 0
\(639\) 1.72897e9 + 4.79467e9i 0.262141 + 0.726952i
\(640\) 0 0
\(641\) 5.84409e9i 0.876423i −0.898872 0.438211i \(-0.855612\pi\)
0.898872 0.438211i \(-0.144388\pi\)
\(642\) 0 0
\(643\) 7.70724e9 7.70724e9i 1.14330 1.14330i 0.155458 0.987842i \(-0.450315\pi\)
0.987842 0.155458i \(-0.0496854\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.26457e9 + 1.26457e9i −0.183560 + 0.183560i −0.792905 0.609345i \(-0.791432\pi\)
0.609345 + 0.792905i \(0.291432\pi\)
\(648\) 0 0
\(649\) 3.65551e9i 0.524918i
\(650\) 0 0
\(651\) −3.64050e9 2.55719e9i −0.517163 0.363270i
\(652\) 0 0
\(653\) 4.06128e9 + 4.06128e9i 0.570777 + 0.570777i 0.932345 0.361569i \(-0.117759\pi\)
−0.361569 + 0.932345i \(0.617759\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.47883e9 + 7.40275e9i −0.478580 + 1.01839i
\(658\) 0 0
\(659\) −4.05577e9 −0.552045 −0.276022 0.961151i \(-0.589016\pi\)
−0.276022 + 0.961151i \(0.589016\pi\)
\(660\) 0 0
\(661\) 3.61381e9 0.486698 0.243349 0.969939i \(-0.421754\pi\)
0.243349 + 0.969939i \(0.421754\pi\)
\(662\) 0 0
\(663\) −2.69954e8 1.54442e9i −0.0359742 0.205811i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.12642e9 + 4.12642e9i 0.538434 + 0.538434i
\(668\) 0 0
\(669\) −4.94424e9 + 7.03878e9i −0.638422 + 0.908879i
\(670\) 0 0
\(671\) 9.30917e8i 0.118955i
\(672\) 0 0
\(673\) −2.83270e9 + 2.83270e9i −0.358218 + 0.358218i −0.863156 0.504938i \(-0.831515\pi\)
0.504938 + 0.863156i \(0.331515\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.30153e9 + 1.30153e9i −0.161211 + 0.161211i −0.783103 0.621892i \(-0.786364\pi\)
0.621892 + 0.783103i \(0.286364\pi\)
\(678\) 0 0
\(679\) 4.39848e9i 0.539211i
\(680\) 0 0
\(681\) 7.01796e9 9.99100e9i 0.851523 1.21226i
\(682\) 0 0
\(683\) −6.66517e9 6.66517e9i −0.800458 0.800458i 0.182709 0.983167i \(-0.441513\pi\)
−0.983167 + 0.182709i \(0.941513\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.72124e8 + 1.55684e9i 0.0320198 + 0.183188i
\(688\) 0 0
\(689\) −1.43424e7 −0.00167053
\(690\) 0 0
\(691\) 8.08878e9 0.932631 0.466316 0.884618i \(-0.345581\pi\)
0.466316 + 0.884618i \(0.345581\pi\)
\(692\) 0 0
\(693\) 1.75563e9 3.73589e9i 0.200386 0.426410i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.54266e10 1.54266e10i −1.72566 1.72566i
\(698\) 0 0
\(699\) −1.22437e10 8.60034e9i −1.35595 0.952457i
\(700\) 0 0
\(701\) 9.03327e9i 0.990448i −0.868765 0.495224i \(-0.835086\pi\)
0.868765 0.495224i \(-0.164914\pi\)
\(702\) 0 0
\(703\) 3.62862e9 3.62862e9i 0.393911 0.393911i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.61240e9 7.61240e9i 0.810128 0.810128i
\(708\) 0 0
\(709\) 4.76635e9i 0.502255i 0.967954 + 0.251127i \(0.0808013\pi\)
−0.967954 + 0.251127i \(0.919199\pi\)
\(710\) 0 0
\(711\) 2.26447e9 + 6.27968e9i 0.236278 + 0.655230i
\(712\) 0 0
\(713\) −2.84724e9 2.84724e9i −0.294178 0.294178i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.00134e10 + 1.75026e9i −1.01453 + 0.177332i
\(718\) 0 0
\(719\) −3.15153e8 −0.0316206 −0.0158103 0.999875i \(-0.505033\pi\)
−0.0158103 + 0.999875i \(0.505033\pi\)
\(720\) 0 0
\(721\) −1.26828e10 −1.26020
\(722\) 0 0
\(723\) 7.03043e9 1.22887e9i 0.691827 0.120926i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −9.64502e8 9.64502e8i −0.0930963 0.0930963i 0.659025 0.752121i \(-0.270969\pi\)
−0.752121 + 0.659025i \(0.770969\pi\)
\(728\) 0 0
\(729\) 5.31088e9 9.01186e9i 0.507715 0.861525i
\(730\) 0 0
\(731\) 1.21555e10i 1.15097i
\(732\) 0 0
\(733\) 7.00690e9 7.00690e9i 0.657147 0.657147i −0.297557 0.954704i \(-0.596172\pi\)
0.954704 + 0.297557i \(0.0961720\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.12158e9 + 2.12158e9i −0.195220 + 0.195220i
\(738\) 0 0
\(739\) 1.63138e10i 1.48696i −0.668757 0.743481i \(-0.733174\pi\)
0.668757 0.743481i \(-0.266826\pi\)
\(740\) 0 0
\(741\) −1.53453e9 1.07789e9i −0.138551 0.0973223i
\(742\) 0 0
\(743\) −9.77965e9 9.77965e9i −0.874707 0.874707i 0.118274 0.992981i \(-0.462264\pi\)
−0.992981 + 0.118274i \(0.962264\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.57697e9 + 3.09076e9i 0.577303 + 0.271296i
\(748\) 0 0
\(749\) −5.80729e8 −0.0504995
\(750\) 0 0
\(751\) −1.45113e9 −0.125016 −0.0625082 0.998044i \(-0.519910\pi\)
−0.0625082 + 0.998044i \(0.519910\pi\)
\(752\) 0 0
\(753\) −2.56186e9 1.46566e10i −0.218662 1.25098i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.94867e9 8.94867e9i −0.749761 0.749761i 0.224673 0.974434i \(-0.427869\pi\)
−0.974434 + 0.224673i \(0.927869\pi\)
\(758\) 0 0
\(759\) 2.14746e9 3.05719e9i 0.178270 0.253791i
\(760\) 0 0
\(761\) 3.51046e8i 0.0288747i 0.999896 + 0.0144374i \(0.00459572\pi\)
−0.999896 + 0.0144374i \(0.995404\pi\)
\(762\) 0 0
\(763\) 8.16169e9 8.16169e9i 0.665188 0.665188i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.13336e9 + 1.13336e9i −0.0906951 + 0.0906951i
\(768\) 0 0
\(769\) 1.06835e10i 0.847170i 0.905856 + 0.423585i \(0.139229\pi\)
−0.905856 + 0.423585i \(0.860771\pi\)
\(770\) 0 0
\(771\) −9.04739e8 + 1.28802e9i −0.0710940 + 0.101212i
\(772\) 0 0
\(773\) −9.62442e9 9.62442e9i −0.749457 0.749457i 0.224921 0.974377i \(-0.427788\pi\)
−0.974377 + 0.224921i \(0.927788\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −8.52784e8 4.87884e9i −0.0652177 0.373115i
\(778\) 0 0
\(779\) −2.60944e10 −1.97772
\(780\) 0 0
\(781\) 5.50330e9 0.413376
\(782\) 0 0
\(783\) 1.53174e10 8.75255e9i 1.14030 0.651581i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.14739e9 3.14739e9i −0.230165 0.230165i 0.582597 0.812762i \(-0.302037\pi\)
−0.812762 + 0.582597i \(0.802037\pi\)
\(788\) 0 0
\(789\) 4.51293e9 + 3.17001e9i 0.327106 + 0.229769i
\(790\) 0 0
\(791\) 2.04983e10i 1.47265i
\(792\) 0 0
\(793\) 2.88623e8 2.88623e8i 0.0205529 0.0205529i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.69169e10 + 1.69169e10i −1.18363 + 1.18363i −0.204835 + 0.978797i \(0.565666\pi\)
−0.978797 + 0.204835i \(0.934334\pi\)
\(798\) 0 0
\(799\) 1.00182e10i 0.694828i
\(800\) 0 0
\(801\) −7.43614e9 + 2.68149e9i −0.511251 + 0.184358i
\(802\) 0 0
\(803\) 6.24492e9 + 6.24492e9i 0.425620 + 0.425620i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.58609e8 4.52030e7i 0.0173215 0.00302768i
\(808\) 0 0
\(809\) 1.86006e10 1.23511 0.617556 0.786527i \(-0.288123\pi\)
0.617556 + 0.786527i \(0.288123\pi\)
\(810\) 0 0
\(811\) 1.00449e10 0.661263 0.330632 0.943760i \(-0.392738\pi\)
0.330632 + 0.943760i \(0.392738\pi\)
\(812\) 0 0
\(813\) −1.64761e10 + 2.87991e9i −1.07532 + 0.187958i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.02806e10 1.02806e10i −0.659543 0.659543i
\(818\) 0 0
\(819\) −1.70260e9 + 6.13961e8i −0.108297 + 0.0390523i
\(820\) 0 0
\(821\) 1.37544e10i 0.867444i 0.901047 + 0.433722i \(0.142800\pi\)
−0.901047 + 0.433722i \(0.857200\pi\)
\(822\) 0 0
\(823\) 1.26443e10 1.26443e10i 0.790669 0.790669i −0.190934 0.981603i \(-0.561152\pi\)
0.981603 + 0.190934i \(0.0611517\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.86210e10 + 1.86210e10i −1.14481 + 1.14481i −0.157252 + 0.987559i \(0.550263\pi\)
−0.987559 + 0.157252i \(0.949737\pi\)
\(828\) 0 0
\(829\) 1.21215e10i 0.738949i 0.929241 + 0.369475i \(0.120462\pi\)
−0.929241 + 0.369475i \(0.879538\pi\)
\(830\) 0 0
\(831\) −1.43763e10 1.00983e10i −0.869049 0.610444i
\(832\) 0 0
\(833\) 4.22823e9 + 4.22823e9i 0.253455 + 0.253455i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.05691e10 + 6.03928e9i −0.623014 + 0.355997i
\(838\) 0 0
\(839\) 3.12593e10 1.82731 0.913656 0.406488i \(-0.133247\pi\)
0.913656 + 0.406488i \(0.133247\pi\)
\(840\) 0 0
\(841\) 1.25035e10 0.724847
\(842\) 0 0
\(843\) −3.68358e9 2.10740e10i −0.211775 1.21158i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.86232e9 + 7.86232e9i 0.444589 + 0.444589i
\(848\) 0 0
\(849\) 1.45192e10 2.06701e10i 0.814267 1.15922i
\(850\) 0 0
\(851\) 4.48270e9i 0.249337i
\(852\) 0 0
\(853\) 1.01070e10 1.01070e10i 0.557570 0.557570i −0.371045 0.928615i \(-0.621000\pi\)
0.928615 + 0.371045i \(0.121000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.22083e9 5.22083e9i 0.283339 0.283339i −0.551100 0.834439i \(-0.685792\pi\)
0.834439 + 0.551100i \(0.185792\pi\)
\(858\) 0 0
\(859\) 1.06595e10i 0.573798i 0.957961 + 0.286899i \(0.0926244\pi\)
−0.957961 + 0.286899i \(0.907376\pi\)
\(860\) 0 0
\(861\) −1.44762e10 + 2.06088e10i −0.772936 + 1.10038i
\(862\) 0 0
\(863\) −1.81636e10 1.81636e10i −0.961978 0.961978i 0.0373248 0.999303i \(-0.488116\pi\)
−0.999303 + 0.0373248i \(0.988116\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −5.13820e9 2.93960e10i −0.267759 1.53187i
\(868\) 0 0
\(869\) 7.20779e9 0.372592
\(870\) 0 0
\(871\) 1.31555e9 0.0674599
\(872\) 0 0
\(873\) 1.08922e10 + 5.11864e9i 0.554071 + 0.260378i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.91061e9 + 4.91061e9i 0.245831 + 0.245831i 0.819257 0.573426i \(-0.194386\pi\)
−0.573426 + 0.819257i \(0.694386\pi\)
\(878\) 0 0
\(879\) 4.16443e8 + 2.92521e8i 0.0206821 + 0.0145277i
\(880\) 0 0
\(881\) 6.36282e9i 0.313497i 0.987639 + 0.156749i \(0.0501013\pi\)
−0.987639 + 0.156749i \(0.949899\pi\)
\(882\) 0 0
\(883\) −9.25527e8 + 9.25527e8i −0.0452404 + 0.0452404i −0.729365 0.684125i \(-0.760184\pi\)
0.684125 + 0.729365i \(0.260184\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.78070e10 2.78070e10i 1.33789 1.33789i 0.439794 0.898099i \(-0.355051\pi\)
0.898099 0.439794i \(-0.144949\pi\)
\(888\) 0 0
\(889\) 3.37928e10i 1.61312i
\(890\) 0 0
\(891\) −7.20829e9 8.69513e9i −0.341398 0.411817i
\(892\) 0 0
\(893\) 8.47301e9 + 8.47301e9i 0.398160 + 0.398160i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.61366e9 + 2.82055e8i −0.0746514 + 0.0130485i
\(898\) 0 0
\(899\) −2.05299e10 −0.942385
\(900\) 0 0
\(901\) −4.48533e8 −0.0204295
\(902\) 0 0
\(903\) −1.38228e10 + 2.41612e9i −0.624723 + 0.109197i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.17395e9 + 4.17395e9i 0.185747 + 0.185747i 0.793855 0.608108i \(-0.208071\pi\)
−0.608108 + 0.793855i \(0.708071\pi\)
\(908\) 0 0
\(909\) −9.99220e9 2.77098e10i −0.441253 1.22366i
\(910\) 0 0
\(911\) 3.10265e10i 1.35962i 0.733386 + 0.679812i \(0.237939\pi\)
−0.733386 + 0.679812i \(0.762061\pi\)
\(912\) 0 0
\(913\) 5.54830e9 5.54830e9i 0.241275 0.241275i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.90480e9 5.90480e9i 0.252879 0.252879i
\(918\) 0 0
\(919\) 1.00910e10i 0.428876i 0.976738 + 0.214438i \(0.0687920\pi\)
−0.976738 + 0.214438i \(0.931208\pi\)
\(920\) 0 0
\(921\) 1.14795e10 + 8.06352e9i 0.484188 + 0.340107i
\(922\) 0 0
\(923\) −1.70625e9 1.70625e9i −0.0714228 0.0714228i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.47593e10 + 3.14070e10i −0.608538 + 1.29493i
\(928\) 0 0
\(929\) −3.63018e10 −1.48550 −0.742751 0.669568i \(-0.766479\pi\)
−0.742751 + 0.669568i \(0.766479\pi\)
\(930\) 0 0
\(931\) 7.15213e9 0.290477
\(932\) 0 0
\(933\) 7.09511e9 + 4.05916e10i 0.286005 + 1.63625i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.89509e9 6.89509e9i −0.273811 0.273811i 0.556821 0.830632i \(-0.312021\pi\)
−0.830632 + 0.556821i \(0.812021\pi\)
\(938\) 0 0
\(939\) 1.95774e10 2.78711e10i 0.771661 1.09856i
\(940\) 0 0
\(941\) 4.05738e10i 1.58739i −0.608319 0.793693i \(-0.708156\pi\)
0.608319 0.793693i \(-0.291844\pi\)
\(942\) 0 0
\(943\) −1.61181e10 + 1.61181e10i −0.625928 + 0.625928i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.16883e9 + 6.16883e9i −0.236036 + 0.236036i −0.815206 0.579171i \(-0.803377\pi\)
0.579171 + 0.815206i \(0.303377\pi\)
\(948\) 0 0
\(949\) 3.87236e9i 0.147077i
\(950\) 0 0
\(951\) −1.27140e10 + 1.81000e10i −0.479346 + 0.682413i
\(952\) 0 0
\(953\) −1.73175e10 1.73175e10i −0.648125 0.648125i 0.304415 0.952540i \(-0.401539\pi\)
−0.952540 + 0.304415i \(0.901539\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −3.27981e9 1.87640e10i −0.120964 0.692044i
\(958\) 0 0
\(959\) 4.01683e10 1.47068
\(960\) 0 0
\(961\) −1.33469e10 −0.485120
\(962\) 0 0
\(963\) −6.75811e8 + 1.43809e9i −0.0243856 + 0.0518912i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.11183e10 + 2.11183e10i 0.751045 + 0.751045i 0.974674 0.223630i \(-0.0717906\pi\)
−0.223630 + 0.974674i \(0.571791\pi\)
\(968\) 0 0
\(969\) −4.79896e10 3.37092e10i −1.69439 1.19019i
\(970\) 0 0
\(971\) 3.83642e10i 1.34480i −0.740186 0.672402i \(-0.765263\pi\)
0.740186 0.672402i \(-0.234737\pi\)
\(972\) 0 0
\(973\) 1.41529e9 1.41529e9i 0.0492551 0.0492551i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.73015e9 + 5.73015e9i −0.196578 + 0.196578i −0.798531 0.601953i \(-0.794389\pi\)
0.601953 + 0.798531i \(0.294389\pi\)
\(978\) 0 0
\(979\) 8.53517e9i 0.290719i
\(980\) 0 0
\(981\) −1.07132e10 2.97092e10i −0.362308 1.00473i
\(982\) 0 0
\(983\) 2.73867e10 + 2.73867e10i 0.919607 + 0.919607i 0.997001 0.0773933i \(-0.0246597\pi\)
−0.0773933 + 0.997001i \(0.524660\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.13923e10 1.99129e9i 0.377140 0.0659212i
\(988\) 0 0
\(989\) −1.27004e10 −0.417476
\(990\) 0 0
\(991\) 4.97037e10 1.62230 0.811149 0.584840i \(-0.198843\pi\)
0.811149 + 0.584840i \(0.198843\pi\)
\(992\) 0 0
\(993\) −6.87102e9 + 1.20100e9i −0.222689 + 0.0389244i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.87057e10 + 2.87057e10i 0.917349 + 0.917349i 0.996836 0.0794873i \(-0.0253283\pi\)
−0.0794873 + 0.996836i \(0.525328\pi\)
\(998\) 0 0
\(999\) −1.30741e10 3.56585e9i −0.414890 0.113158i
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 300.8.i.c.257.8 yes 16
3.2 odd 2 inner 300.8.i.c.257.5 yes 16
5.2 odd 4 inner 300.8.i.c.293.4 yes 16
5.3 odd 4 inner 300.8.i.c.293.5 yes 16
5.4 even 2 inner 300.8.i.c.257.1 16
15.2 even 4 inner 300.8.i.c.293.1 yes 16
15.8 even 4 inner 300.8.i.c.293.8 yes 16
15.14 odd 2 inner 300.8.i.c.257.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.8.i.c.257.1 16 5.4 even 2 inner
300.8.i.c.257.4 yes 16 15.14 odd 2 inner
300.8.i.c.257.5 yes 16 3.2 odd 2 inner
300.8.i.c.257.8 yes 16 1.1 even 1 trivial
300.8.i.c.293.1 yes 16 15.2 even 4 inner
300.8.i.c.293.4 yes 16 5.2 odd 4 inner
300.8.i.c.293.5 yes 16 5.3 odd 4 inner
300.8.i.c.293.8 yes 16 15.8 even 4 inner