Properties

Label 384.7.l.b.31.18
Level $384$
Weight $7$
Character 384.31
Analytic conductor $88.341$
Analytic rank $0$
Dimension $48$
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] = [384,7,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), 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: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 31.18
Character \(\chi\) \(=\) 384.31
Dual form 384.7.l.b.223.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(11.0227 + 11.0227i) q^{3} +(-45.7781 - 45.7781i) q^{5} -565.544 q^{7} +243.000i q^{9} +O(q^{10})\) \(q+(11.0227 + 11.0227i) q^{3} +(-45.7781 - 45.7781i) q^{5} -565.544 q^{7} +243.000i q^{9} +(-1291.95 + 1291.95i) q^{11} +(2882.93 - 2882.93i) q^{13} -1009.20i q^{15} -662.555 q^{17} +(-1940.19 - 1940.19i) q^{19} +(-6233.83 - 6233.83i) q^{21} -15879.3 q^{23} -11433.7i q^{25} +(-2678.52 + 2678.52i) q^{27} +(-11555.8 + 11555.8i) q^{29} -39097.9i q^{31} -28481.5 q^{33} +(25889.6 + 25889.6i) q^{35} +(51336.5 + 51336.5i) q^{37} +63555.4 q^{39} +66520.0i q^{41} +(-68923.6 + 68923.6i) q^{43} +(11124.1 - 11124.1i) q^{45} -107734. i q^{47} +202191. q^{49} +(-7303.15 - 7303.15i) q^{51} +(108477. + 108477. i) q^{53} +118286. q^{55} -42772.3i q^{57} +(67889.1 - 67889.1i) q^{59} +(99209.6 - 99209.6i) q^{61} -137427. i q^{63} -263951. q^{65} +(105403. + 105403. i) q^{67} +(-175033. - 175033. i) q^{69} +266114. q^{71} +267965. i q^{73} +(126031. - 126031. i) q^{75} +(730653. - 730653. i) q^{77} +557984. i q^{79} -59049.0 q^{81} +(236224. + 236224. i) q^{83} +(30330.5 + 30330.5i) q^{85} -254751. q^{87} +419786. i q^{89} +(-1.63043e6 + 1.63043e6i) q^{91} +(430964. - 430964. i) q^{93} +177637. i q^{95} +393308. q^{97} +(-313943. - 313943. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 2720 q^{11} + 3936 q^{19} + 26240 q^{23} - 66400 q^{29} - 162336 q^{35} + 7200 q^{37} - 340704 q^{43} + 806736 q^{49} - 80352 q^{51} - 443680 q^{53} + 232704 q^{55} + 886144 q^{59} + 326496 q^{61} - 372832 q^{65} + 962112 q^{67} - 541728 q^{69} + 534016 q^{71} + 1073088 q^{75} + 932960 q^{77} - 2834352 q^{81} + 2497760 q^{83} + 372000 q^{85} - 775008 q^{91} + 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 11.0227 + 11.0227i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) −45.7781 45.7781i −0.366225 0.366225i 0.499873 0.866098i \(-0.333380\pi\)
−0.866098 + 0.499873i \(0.833380\pi\)
\(6\) 0 0
\(7\) −565.544 −1.64882 −0.824408 0.565995i \(-0.808492\pi\)
−0.824408 + 0.565995i \(0.808492\pi\)
\(8\) 0 0
\(9\) 243.000i 0.333333i
\(10\) 0 0
\(11\) −1291.95 + 1291.95i −0.970658 + 0.970658i −0.999582 0.0289233i \(-0.990792\pi\)
0.0289233 + 0.999582i \(0.490792\pi\)
\(12\) 0 0
\(13\) 2882.93 2882.93i 1.31221 1.31221i 0.392433 0.919781i \(-0.371634\pi\)
0.919781 0.392433i \(-0.128366\pi\)
\(14\) 0 0
\(15\) 1009.20i 0.299022i
\(16\) 0 0
\(17\) −662.555 −0.134858 −0.0674288 0.997724i \(-0.521480\pi\)
−0.0674288 + 0.997724i \(0.521480\pi\)
\(18\) 0 0
\(19\) −1940.19 1940.19i −0.282868 0.282868i 0.551384 0.834252i \(-0.314100\pi\)
−0.834252 + 0.551384i \(0.814100\pi\)
\(20\) 0 0
\(21\) −6233.83 6233.83i −0.673127 0.673127i
\(22\) 0 0
\(23\) −15879.3 −1.30511 −0.652557 0.757740i \(-0.726304\pi\)
−0.652557 + 0.757740i \(0.726304\pi\)
\(24\) 0 0
\(25\) 11433.7i 0.731758i
\(26\) 0 0
\(27\) −2678.52 + 2678.52i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) −11555.8 + 11555.8i −0.473810 + 0.473810i −0.903145 0.429335i \(-0.858748\pi\)
0.429335 + 0.903145i \(0.358748\pi\)
\(30\) 0 0
\(31\) 39097.9i 1.31241i −0.754585 0.656203i \(-0.772162\pi\)
0.754585 0.656203i \(-0.227838\pi\)
\(32\) 0 0
\(33\) −28481.5 −0.792539
\(34\) 0 0
\(35\) 25889.6 + 25889.6i 0.603838 + 0.603838i
\(36\) 0 0
\(37\) 51336.5 + 51336.5i 1.01349 + 1.01349i 0.999908 + 0.0135861i \(0.00432473\pi\)
0.0135861 + 0.999908i \(0.495675\pi\)
\(38\) 0 0
\(39\) 63555.4 1.07142
\(40\) 0 0
\(41\) 66520.0i 0.965163i 0.875851 + 0.482582i \(0.160301\pi\)
−0.875851 + 0.482582i \(0.839699\pi\)
\(42\) 0 0
\(43\) −68923.6 + 68923.6i −0.866887 + 0.866887i −0.992127 0.125239i \(-0.960030\pi\)
0.125239 + 0.992127i \(0.460030\pi\)
\(44\) 0 0
\(45\) 11124.1 11124.1i 0.122075 0.122075i
\(46\) 0 0
\(47\) 107734.i 1.03767i −0.854873 0.518837i \(-0.826365\pi\)
0.854873 0.518837i \(-0.173635\pi\)
\(48\) 0 0
\(49\) 202191. 1.71860
\(50\) 0 0
\(51\) −7303.15 7303.15i −0.0550554 0.0550554i
\(52\) 0 0
\(53\) 108477. + 108477.i 0.728637 + 0.728637i 0.970348 0.241711i \(-0.0777086\pi\)
−0.241711 + 0.970348i \(0.577709\pi\)
\(54\) 0 0
\(55\) 118286. 0.710959
\(56\) 0 0
\(57\) 42772.3i 0.230961i
\(58\) 0 0
\(59\) 67889.1 67889.1i 0.330555 0.330555i −0.522242 0.852797i \(-0.674904\pi\)
0.852797 + 0.522242i \(0.174904\pi\)
\(60\) 0 0
\(61\) 99209.6 99209.6i 0.437083 0.437083i −0.453946 0.891029i \(-0.649984\pi\)
0.891029 + 0.453946i \(0.149984\pi\)
\(62\) 0 0
\(63\) 137427.i 0.549606i
\(64\) 0 0
\(65\) −263951. −0.961131
\(66\) 0 0
\(67\) 105403. + 105403.i 0.350451 + 0.350451i 0.860277 0.509826i \(-0.170290\pi\)
−0.509826 + 0.860277i \(0.670290\pi\)
\(68\) 0 0
\(69\) −175033. 175033.i −0.532810 0.532810i
\(70\) 0 0
\(71\) 266114. 0.743520 0.371760 0.928329i \(-0.378754\pi\)
0.371760 + 0.928329i \(0.378754\pi\)
\(72\) 0 0
\(73\) 267965.i 0.688826i 0.938818 + 0.344413i \(0.111922\pi\)
−0.938818 + 0.344413i \(0.888078\pi\)
\(74\) 0 0
\(75\) 126031. 126031.i 0.298739 0.298739i
\(76\) 0 0
\(77\) 730653. 730653.i 1.60044 1.60044i
\(78\) 0 0
\(79\) 557984.i 1.13172i 0.824500 + 0.565862i \(0.191456\pi\)
−0.824500 + 0.565862i \(0.808544\pi\)
\(80\) 0 0
\(81\) −59049.0 −0.111111
\(82\) 0 0
\(83\) 236224. + 236224.i 0.413133 + 0.413133i 0.882829 0.469695i \(-0.155636\pi\)
−0.469695 + 0.882829i \(0.655636\pi\)
\(84\) 0 0
\(85\) 30330.5 + 30330.5i 0.0493882 + 0.0493882i
\(86\) 0 0
\(87\) −254751. −0.386864
\(88\) 0 0
\(89\) 419786.i 0.595467i 0.954649 + 0.297733i \(0.0962307\pi\)
−0.954649 + 0.297733i \(0.903769\pi\)
\(90\) 0 0
\(91\) −1.63043e6 + 1.63043e6i −2.16360 + 2.16360i
\(92\) 0 0
\(93\) 430964. 430964.i 0.535787 0.535787i
\(94\) 0 0
\(95\) 177637.i 0.207187i
\(96\) 0 0
\(97\) 393308. 0.430940 0.215470 0.976510i \(-0.430872\pi\)
0.215470 + 0.976510i \(0.430872\pi\)
\(98\) 0 0
\(99\) −313943. 313943.i −0.323553 0.323553i
\(100\) 0 0
\(101\) −670944. 670944.i −0.651212 0.651212i 0.302073 0.953285i \(-0.402321\pi\)
−0.953285 + 0.302073i \(0.902321\pi\)
\(102\) 0 0
\(103\) 456775. 0.418013 0.209007 0.977914i \(-0.432977\pi\)
0.209007 + 0.977914i \(0.432977\pi\)
\(104\) 0 0
\(105\) 570746.i 0.493032i
\(106\) 0 0
\(107\) 498780. 498780.i 0.407153 0.407153i −0.473592 0.880745i \(-0.657043\pi\)
0.880745 + 0.473592i \(0.157043\pi\)
\(108\) 0 0
\(109\) 686021. 686021.i 0.529734 0.529734i −0.390759 0.920493i \(-0.627787\pi\)
0.920493 + 0.390759i \(0.127787\pi\)
\(110\) 0 0
\(111\) 1.13173e6i 0.827514i
\(112\) 0 0
\(113\) 1.76590e6 1.22386 0.611930 0.790912i \(-0.290394\pi\)
0.611930 + 0.790912i \(0.290394\pi\)
\(114\) 0 0
\(115\) 726926. + 726926.i 0.477965 + 0.477965i
\(116\) 0 0
\(117\) 700553. + 700553.i 0.437404 + 0.437404i
\(118\) 0 0
\(119\) 374704. 0.222355
\(120\) 0 0
\(121\) 1.56669e6i 0.884355i
\(122\) 0 0
\(123\) −733230. + 733230.i −0.394026 + 0.394026i
\(124\) 0 0
\(125\) −1.23870e6 + 1.23870e6i −0.634213 + 0.634213i
\(126\) 0 0
\(127\) 2.52519e6i 1.23277i −0.787445 0.616385i \(-0.788596\pi\)
0.787445 0.616385i \(-0.211404\pi\)
\(128\) 0 0
\(129\) −1.51945e6 −0.707810
\(130\) 0 0
\(131\) 2.20532e6 + 2.20532e6i 0.980976 + 0.980976i 0.999822 0.0188466i \(-0.00599941\pi\)
−0.0188466 + 0.999822i \(0.505999\pi\)
\(132\) 0 0
\(133\) 1.09726e6 + 1.09726e6i 0.466398 + 0.466398i
\(134\) 0 0
\(135\) 245235. 0.0996738
\(136\) 0 0
\(137\) 210985.i 0.0820520i −0.999158 0.0410260i \(-0.986937\pi\)
0.999158 0.0410260i \(-0.0130626\pi\)
\(138\) 0 0
\(139\) 678952. 678952.i 0.252810 0.252810i −0.569312 0.822122i \(-0.692790\pi\)
0.822122 + 0.569312i \(0.192790\pi\)
\(140\) 0 0
\(141\) 1.18752e6 1.18752e6i 0.423629 0.423629i
\(142\) 0 0
\(143\) 7.44919e6i 2.54742i
\(144\) 0 0
\(145\) 1.05800e6 0.347042
\(146\) 0 0
\(147\) 2.22869e6 + 2.22869e6i 0.701614 + 0.701614i
\(148\) 0 0
\(149\) 2.62816e6 + 2.62816e6i 0.794497 + 0.794497i 0.982222 0.187724i \(-0.0601111\pi\)
−0.187724 + 0.982222i \(0.560111\pi\)
\(150\) 0 0
\(151\) 2.88045e6 0.836622 0.418311 0.908304i \(-0.362622\pi\)
0.418311 + 0.908304i \(0.362622\pi\)
\(152\) 0 0
\(153\) 161001.i 0.0449525i
\(154\) 0 0
\(155\) −1.78983e6 + 1.78983e6i −0.480636 + 0.480636i
\(156\) 0 0
\(157\) 501498. 501498.i 0.129590 0.129590i −0.639337 0.768927i \(-0.720791\pi\)
0.768927 + 0.639337i \(0.220791\pi\)
\(158\) 0 0
\(159\) 2.39143e6i 0.594930i
\(160\) 0 0
\(161\) 8.98046e6 2.15189
\(162\) 0 0
\(163\) −1.30009e6 1.30009e6i −0.300199 0.300199i 0.540893 0.841092i \(-0.318087\pi\)
−0.841092 + 0.540893i \(0.818087\pi\)
\(164\) 0 0
\(165\) 1.30383e6 + 1.30383e6i 0.290248 + 0.290248i
\(166\) 0 0
\(167\) −8.60262e6 −1.84706 −0.923531 0.383524i \(-0.874710\pi\)
−0.923531 + 0.383524i \(0.874710\pi\)
\(168\) 0 0
\(169\) 1.17958e7i 2.44381i
\(170\) 0 0
\(171\) 471467. 471467.i 0.0942894 0.0942894i
\(172\) 0 0
\(173\) 891920. 891920.i 0.172261 0.172261i −0.615711 0.787972i \(-0.711131\pi\)
0.787972 + 0.615711i \(0.211131\pi\)
\(174\) 0 0
\(175\) 6.46628e6i 1.20654i
\(176\) 0 0
\(177\) 1.49664e6 0.269897
\(178\) 0 0
\(179\) 2.89580e6 + 2.89580e6i 0.504904 + 0.504904i 0.912958 0.408054i \(-0.133792\pi\)
−0.408054 + 0.912958i \(0.633792\pi\)
\(180\) 0 0
\(181\) −2.14643e6 2.14643e6i −0.361977 0.361977i 0.502563 0.864541i \(-0.332390\pi\)
−0.864541 + 0.502563i \(0.832390\pi\)
\(182\) 0 0
\(183\) 2.18712e6 0.356877
\(184\) 0 0
\(185\) 4.70018e6i 0.742334i
\(186\) 0 0
\(187\) 855986. 855986.i 0.130901 0.130901i
\(188\) 0 0
\(189\) 1.51482e6 1.51482e6i 0.224376 0.224376i
\(190\) 0 0
\(191\) 1.12449e6i 0.161382i −0.996739 0.0806912i \(-0.974287\pi\)
0.996739 0.0806912i \(-0.0257128\pi\)
\(192\) 0 0
\(193\) 6.65489e6 0.925697 0.462848 0.886437i \(-0.346827\pi\)
0.462848 + 0.886437i \(0.346827\pi\)
\(194\) 0 0
\(195\) −2.90945e6 2.90945e6i −0.392380 0.392380i
\(196\) 0 0
\(197\) −1.01919e6 1.01919e6i −0.133307 0.133307i 0.637305 0.770612i \(-0.280049\pi\)
−0.770612 + 0.637305i \(0.780049\pi\)
\(198\) 0 0
\(199\) 889727. 0.112901 0.0564505 0.998405i \(-0.482022\pi\)
0.0564505 + 0.998405i \(0.482022\pi\)
\(200\) 0 0
\(201\) 2.32365e6i 0.286142i
\(202\) 0 0
\(203\) 6.53529e6 6.53529e6i 0.781226 0.781226i
\(204\) 0 0
\(205\) 3.04516e6 3.04516e6i 0.353467 0.353467i
\(206\) 0 0
\(207\) 3.85867e6i 0.435038i
\(208\) 0 0
\(209\) 5.01325e6 0.549137
\(210\) 0 0
\(211\) 6.10371e6 + 6.10371e6i 0.649750 + 0.649750i 0.952933 0.303182i \(-0.0980491\pi\)
−0.303182 + 0.952933i \(0.598049\pi\)
\(212\) 0 0
\(213\) 2.93330e6 + 2.93330e6i 0.303541 + 0.303541i
\(214\) 0 0
\(215\) 6.31039e6 0.634952
\(216\) 0 0
\(217\) 2.21116e7i 2.16392i
\(218\) 0 0
\(219\) −2.95370e6 + 2.95370e6i −0.281212 + 0.281212i
\(220\) 0 0
\(221\) −1.91010e6 + 1.91010e6i −0.176962 + 0.176962i
\(222\) 0 0
\(223\) 8.07405e6i 0.728077i 0.931384 + 0.364038i \(0.118602\pi\)
−0.931384 + 0.364038i \(0.881398\pi\)
\(224\) 0 0
\(225\) 2.77840e6 0.243919
\(226\) 0 0
\(227\) 4.13774e6 + 4.13774e6i 0.353741 + 0.353741i 0.861500 0.507758i \(-0.169526\pi\)
−0.507758 + 0.861500i \(0.669526\pi\)
\(228\) 0 0
\(229\) 1.19667e7 + 1.19667e7i 0.996480 + 0.996480i 0.999994 0.00351356i \(-0.00111840\pi\)
−0.00351356 + 0.999994i \(0.501118\pi\)
\(230\) 0 0
\(231\) 1.61075e7 1.30675
\(232\) 0 0
\(233\) 1.46761e6i 0.116023i 0.998316 + 0.0580114i \(0.0184760\pi\)
−0.998316 + 0.0580114i \(0.981524\pi\)
\(234\) 0 0
\(235\) −4.93188e6 + 4.93188e6i −0.380022 + 0.380022i
\(236\) 0 0
\(237\) −6.15049e6 + 6.15049e6i −0.462024 + 0.462024i
\(238\) 0 0
\(239\) 1.28489e7i 0.941177i 0.882353 + 0.470588i \(0.155958\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(240\) 0 0
\(241\) −1.87104e6 −0.133670 −0.0668348 0.997764i \(-0.521290\pi\)
−0.0668348 + 0.997764i \(0.521290\pi\)
\(242\) 0 0
\(243\) −650880. 650880.i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) −9.25593e6 9.25593e6i −0.629393 0.629393i
\(246\) 0 0
\(247\) −1.11869e7 −0.742367
\(248\) 0 0
\(249\) 5.20766e6i 0.337322i
\(250\) 0 0
\(251\) 2.06552e7 2.06552e7i 1.30619 1.30619i 0.382052 0.924141i \(-0.375218\pi\)
0.924141 0.382052i \(-0.124782\pi\)
\(252\) 0 0
\(253\) 2.05152e7 2.05152e7i 1.26682 1.26682i
\(254\) 0 0
\(255\) 668649.i 0.0403253i
\(256\) 0 0
\(257\) 960591. 0.0565900 0.0282950 0.999600i \(-0.490992\pi\)
0.0282950 + 0.999600i \(0.490992\pi\)
\(258\) 0 0
\(259\) −2.90331e7 2.90331e7i −1.67107 1.67107i
\(260\) 0 0
\(261\) −2.80805e6 2.80805e6i −0.157937 0.157937i
\(262\) 0 0
\(263\) −2.66308e7 −1.46392 −0.731960 0.681348i \(-0.761394\pi\)
−0.731960 + 0.681348i \(0.761394\pi\)
\(264\) 0 0
\(265\) 9.93178e6i 0.533690i
\(266\) 0 0
\(267\) −4.62717e6 + 4.62717e6i −0.243098 + 0.243098i
\(268\) 0 0
\(269\) −2.00276e7 + 2.00276e7i −1.02890 + 1.02890i −0.0293254 + 0.999570i \(0.509336\pi\)
−0.999570 + 0.0293254i \(0.990664\pi\)
\(270\) 0 0
\(271\) 1.33995e7i 0.673256i −0.941638 0.336628i \(-0.890714\pi\)
0.941638 0.336628i \(-0.109286\pi\)
\(272\) 0 0
\(273\) −3.59434e7 −1.76657
\(274\) 0 0
\(275\) 1.47718e7 + 1.47718e7i 0.710287 + 0.710287i
\(276\) 0 0
\(277\) 1.43027e6 + 1.43027e6i 0.0672942 + 0.0672942i 0.739953 0.672659i \(-0.234848\pi\)
−0.672659 + 0.739953i \(0.734848\pi\)
\(278\) 0 0
\(279\) 9.50078e6 0.437469
\(280\) 0 0
\(281\) 1.75247e7i 0.789825i −0.918719 0.394912i \(-0.870775\pi\)
0.918719 0.394912i \(-0.129225\pi\)
\(282\) 0 0
\(283\) −2.40129e7 + 2.40129e7i −1.05946 + 1.05946i −0.0613460 + 0.998117i \(0.519539\pi\)
−0.998117 + 0.0613460i \(0.980461\pi\)
\(284\) 0 0
\(285\) −1.95804e6 + 1.95804e6i −0.0845837 + 0.0845837i
\(286\) 0 0
\(287\) 3.76200e7i 1.59138i
\(288\) 0 0
\(289\) −2.36986e7 −0.981813
\(290\) 0 0
\(291\) 4.33531e6 + 4.33531e6i 0.175931 + 0.175931i
\(292\) 0 0
\(293\) −2.49680e7 2.49680e7i −0.992617 0.992617i 0.00735615 0.999973i \(-0.497658\pi\)
−0.999973 + 0.00735615i \(0.997658\pi\)
\(294\) 0 0
\(295\) −6.21568e6 −0.242115
\(296\) 0 0
\(297\) 6.92100e6i 0.264180i
\(298\) 0 0
\(299\) −4.57790e7 + 4.57790e7i −1.71259 + 1.71259i
\(300\) 0 0
\(301\) 3.89793e7 3.89793e7i 1.42934 1.42934i
\(302\) 0 0
\(303\) 1.47912e7i 0.531712i
\(304\) 0 0
\(305\) −9.08326e6 −0.320142
\(306\) 0 0
\(307\) −3.13583e6 3.13583e6i −0.108377 0.108377i 0.650839 0.759216i \(-0.274417\pi\)
−0.759216 + 0.650839i \(0.774417\pi\)
\(308\) 0 0
\(309\) 5.03489e6 + 5.03489e6i 0.170653 + 0.170653i
\(310\) 0 0
\(311\) 2.82703e7 0.939830 0.469915 0.882712i \(-0.344285\pi\)
0.469915 + 0.882712i \(0.344285\pi\)
\(312\) 0 0
\(313\) 4.17098e6i 0.136021i −0.997685 0.0680104i \(-0.978335\pi\)
0.997685 0.0680104i \(-0.0216651\pi\)
\(314\) 0 0
\(315\) −6.29116e6 + 6.29116e6i −0.201279 + 0.201279i
\(316\) 0 0
\(317\) −3.49603e7 + 3.49603e7i −1.09748 + 1.09748i −0.102779 + 0.994704i \(0.532773\pi\)
−0.994704 + 0.102779i \(0.967227\pi\)
\(318\) 0 0
\(319\) 2.98588e7i 0.919815i
\(320\) 0 0
\(321\) 1.09958e7 0.332439
\(322\) 0 0
\(323\) 1.28548e6 + 1.28548e6i 0.0381469 + 0.0381469i
\(324\) 0 0
\(325\) −3.29627e7 3.29627e7i −0.960223 0.960223i
\(326\) 0 0
\(327\) 1.51236e7 0.432526
\(328\) 0 0
\(329\) 6.09286e7i 1.71093i
\(330\) 0 0
\(331\) 2.32895e7 2.32895e7i 0.642210 0.642210i −0.308888 0.951098i \(-0.599957\pi\)
0.951098 + 0.308888i \(0.0999569\pi\)
\(332\) 0 0
\(333\) −1.24748e7 + 1.24748e7i −0.337831 + 0.337831i
\(334\) 0 0
\(335\) 9.65029e6i 0.256688i
\(336\) 0 0
\(337\) −1.28256e7 −0.335110 −0.167555 0.985863i \(-0.553587\pi\)
−0.167555 + 0.985863i \(0.553587\pi\)
\(338\) 0 0
\(339\) 1.94650e7 + 1.94650e7i 0.499638 + 0.499638i
\(340\) 0 0
\(341\) 5.05124e7 + 5.05124e7i 1.27390 + 1.27390i
\(342\) 0 0
\(343\) −4.78123e7 −1.18483
\(344\) 0 0
\(345\) 1.60254e7i 0.390257i
\(346\) 0 0
\(347\) 3.91869e7 3.91869e7i 0.937891 0.937891i −0.0602903 0.998181i \(-0.519203\pi\)
0.998181 + 0.0602903i \(0.0192026\pi\)
\(348\) 0 0
\(349\) −4.06672e7 + 4.06672e7i −0.956682 + 0.956682i −0.999100 0.0424175i \(-0.986494\pi\)
0.0424175 + 0.999100i \(0.486494\pi\)
\(350\) 0 0
\(351\) 1.54440e7i 0.357139i
\(352\) 0 0
\(353\) −3.01960e7 −0.686475 −0.343237 0.939249i \(-0.611524\pi\)
−0.343237 + 0.939249i \(0.611524\pi\)
\(354\) 0 0
\(355\) −1.21822e7 1.21822e7i −0.272296 0.272296i
\(356\) 0 0
\(357\) 4.13025e6 + 4.13025e6i 0.0907762 + 0.0907762i
\(358\) 0 0
\(359\) 6.27234e7 1.35565 0.677823 0.735225i \(-0.262924\pi\)
0.677823 + 0.735225i \(0.262924\pi\)
\(360\) 0 0
\(361\) 3.95172e7i 0.839971i
\(362\) 0 0
\(363\) 1.72691e7 1.72691e7i 0.361036 0.361036i
\(364\) 0 0
\(365\) 1.22669e7 1.22669e7i 0.252265 0.252265i
\(366\) 0 0
\(367\) 7.63574e7i 1.54473i −0.635179 0.772365i \(-0.719073\pi\)
0.635179 0.772365i \(-0.280927\pi\)
\(368\) 0 0
\(369\) −1.61644e7 −0.321721
\(370\) 0 0
\(371\) −6.13487e7 6.13487e7i −1.20139 1.20139i
\(372\) 0 0
\(373\) −3.70594e7 3.70594e7i −0.714121 0.714121i 0.253274 0.967395i \(-0.418493\pi\)
−0.967395 + 0.253274i \(0.918493\pi\)
\(374\) 0 0
\(375\) −2.73076e7 −0.517833
\(376\) 0 0
\(377\) 6.66289e7i 1.24348i
\(378\) 0 0
\(379\) 4.65182e7 4.65182e7i 0.854487 0.854487i −0.136195 0.990682i \(-0.543487\pi\)
0.990682 + 0.136195i \(0.0434873\pi\)
\(380\) 0 0
\(381\) 2.78344e7 2.78344e7i 0.503276 0.503276i
\(382\) 0 0
\(383\) 1.38162e7i 0.245919i 0.992412 + 0.122959i \(0.0392385\pi\)
−0.992412 + 0.122959i \(0.960762\pi\)
\(384\) 0 0
\(385\) −6.68958e7 −1.17224
\(386\) 0 0
\(387\) −1.67484e7 1.67484e7i −0.288962 0.288962i
\(388\) 0 0
\(389\) −3.29148e7 3.29148e7i −0.559168 0.559168i 0.369902 0.929071i \(-0.379391\pi\)
−0.929071 + 0.369902i \(0.879391\pi\)
\(390\) 0 0
\(391\) 1.05209e7 0.176004
\(392\) 0 0
\(393\) 4.86172e7i 0.800963i
\(394\) 0 0
\(395\) 2.55435e7 2.55435e7i 0.414465 0.414465i
\(396\) 0 0
\(397\) −9.09263e6 + 9.09263e6i −0.145318 + 0.145318i −0.776023 0.630705i \(-0.782766\pi\)
0.630705 + 0.776023i \(0.282766\pi\)
\(398\) 0 0
\(399\) 2.41896e7i 0.380812i
\(400\) 0 0
\(401\) 5.10274e7 0.791353 0.395677 0.918390i \(-0.370510\pi\)
0.395677 + 0.918390i \(0.370510\pi\)
\(402\) 0 0
\(403\) −1.12717e8 1.12717e8i −1.72216 1.72216i
\(404\) 0 0
\(405\) 2.70315e6 + 2.70315e6i 0.0406917 + 0.0406917i
\(406\) 0 0
\(407\) −1.32648e8 −1.96751
\(408\) 0 0
\(409\) 8.06130e7i 1.17824i −0.808044 0.589122i \(-0.799474\pi\)
0.808044 0.589122i \(-0.200526\pi\)
\(410\) 0 0
\(411\) 2.32562e6 2.32562e6i 0.0334976 0.0334976i
\(412\) 0 0
\(413\) −3.83943e7 + 3.83943e7i −0.545025 + 0.545025i
\(414\) 0 0
\(415\) 2.16278e7i 0.302599i
\(416\) 0 0
\(417\) 1.49678e7 0.206419
\(418\) 0 0
\(419\) 8.05454e7 + 8.05454e7i 1.09496 + 1.09496i 0.994990 + 0.0999711i \(0.0318750\pi\)
0.0999711 + 0.994990i \(0.468125\pi\)
\(420\) 0 0
\(421\) 1.99109e7 + 1.99109e7i 0.266837 + 0.266837i 0.827824 0.560988i \(-0.189578\pi\)
−0.560988 + 0.827824i \(0.689578\pi\)
\(422\) 0 0
\(423\) 2.61795e7 0.345891
\(424\) 0 0
\(425\) 7.57548e6i 0.0986832i
\(426\) 0 0
\(427\) −5.61074e7 + 5.61074e7i −0.720670 + 0.720670i
\(428\) 0 0
\(429\) −8.21102e7 + 8.21102e7i −1.03998 + 1.03998i
\(430\) 0 0
\(431\) 5.56500e7i 0.695078i 0.937666 + 0.347539i \(0.112982\pi\)
−0.937666 + 0.347539i \(0.887018\pi\)
\(432\) 0 0
\(433\) 1.28922e8 1.58804 0.794022 0.607889i \(-0.207984\pi\)
0.794022 + 0.607889i \(0.207984\pi\)
\(434\) 0 0
\(435\) 1.16620e7 + 1.16620e7i 0.141679 + 0.141679i
\(436\) 0 0
\(437\) 3.08089e7 + 3.08089e7i 0.369175 + 0.369175i
\(438\) 0 0
\(439\) −1.16571e8 −1.37784 −0.688918 0.724839i \(-0.741914\pi\)
−0.688918 + 0.724839i \(0.741914\pi\)
\(440\) 0 0
\(441\) 4.91325e7i 0.572865i
\(442\) 0 0
\(443\) 8.56146e7 8.56146e7i 0.984775 0.984775i −0.0151112 0.999886i \(-0.504810\pi\)
0.999886 + 0.0151112i \(0.00481022\pi\)
\(444\) 0 0
\(445\) 1.92170e7 1.92170e7i 0.218075 0.218075i
\(446\) 0 0
\(447\) 5.79388e7i 0.648704i
\(448\) 0 0
\(449\) −1.62296e8 −1.79295 −0.896474 0.443096i \(-0.853880\pi\)
−0.896474 + 0.443096i \(0.853880\pi\)
\(450\) 0 0
\(451\) −8.59403e7 8.59403e7i −0.936844 0.936844i
\(452\) 0 0
\(453\) 3.17503e7 + 3.17503e7i 0.341550 + 0.341550i
\(454\) 0 0
\(455\) 1.49276e8 1.58473
\(456\) 0 0
\(457\) 1.31224e8i 1.37488i −0.726241 0.687440i \(-0.758734\pi\)
0.726241 0.687440i \(-0.241266\pi\)
\(458\) 0 0
\(459\) 1.77467e6 1.77467e6i 0.0183518 0.0183518i
\(460\) 0 0
\(461\) −3.76397e7 + 3.76397e7i −0.384188 + 0.384188i −0.872608 0.488421i \(-0.837573\pi\)
0.488421 + 0.872608i \(0.337573\pi\)
\(462\) 0 0
\(463\) 5.50458e6i 0.0554602i −0.999615 0.0277301i \(-0.991172\pi\)
0.999615 0.0277301i \(-0.00882789\pi\)
\(464\) 0 0
\(465\) −3.94575e7 −0.392438
\(466\) 0 0
\(467\) 9.63536e6 + 9.63536e6i 0.0946057 + 0.0946057i 0.752826 0.658220i \(-0.228690\pi\)
−0.658220 + 0.752826i \(0.728690\pi\)
\(468\) 0 0
\(469\) −5.96099e7 5.96099e7i −0.577830 0.577830i
\(470\) 0 0
\(471\) 1.10557e7 0.105809
\(472\) 0 0
\(473\) 1.78091e8i 1.68290i
\(474\) 0 0
\(475\) −2.21836e7 + 2.21836e7i −0.206991 + 0.206991i
\(476\) 0 0
\(477\) −2.63600e7 + 2.63600e7i −0.242879 + 0.242879i
\(478\) 0 0
\(479\) 3.68794e6i 0.0335566i 0.999859 + 0.0167783i \(0.00534095\pi\)
−0.999859 + 0.0167783i \(0.994659\pi\)
\(480\) 0 0
\(481\) 2.95999e8 2.65984
\(482\) 0 0
\(483\) 9.89889e7 + 9.89889e7i 0.878507 + 0.878507i
\(484\) 0 0
\(485\) −1.80049e7 1.80049e7i −0.157821 0.157821i
\(486\) 0 0
\(487\) −6.06471e7 −0.525077 −0.262539 0.964921i \(-0.584560\pi\)
−0.262539 + 0.964921i \(0.584560\pi\)
\(488\) 0 0
\(489\) 2.86609e7i 0.245112i
\(490\) 0 0
\(491\) −2.63117e7 + 2.63117e7i −0.222282 + 0.222282i −0.809459 0.587177i \(-0.800239\pi\)
0.587177 + 0.809459i \(0.300239\pi\)
\(492\) 0 0
\(493\) 7.65633e6 7.65633e6i 0.0638969 0.0638969i
\(494\) 0 0
\(495\) 2.87434e7i 0.236986i
\(496\) 0 0
\(497\) −1.50499e8 −1.22593
\(498\) 0 0
\(499\) 1.17267e8 + 1.17267e8i 0.943788 + 0.943788i 0.998502 0.0547144i \(-0.0174248\pi\)
−0.0547144 + 0.998502i \(0.517425\pi\)
\(500\) 0 0
\(501\) −9.48241e7 9.48241e7i −0.754060 0.754060i
\(502\) 0 0
\(503\) 8.84245e7 0.694814 0.347407 0.937714i \(-0.387062\pi\)
0.347407 + 0.937714i \(0.387062\pi\)
\(504\) 0 0
\(505\) 6.14291e7i 0.476980i
\(506\) 0 0
\(507\) 1.30022e8 1.30022e8i 0.997681 0.997681i
\(508\) 0 0
\(509\) 9.88133e7 9.88133e7i 0.749311 0.749311i −0.225039 0.974350i \(-0.572251\pi\)
0.974350 + 0.225039i \(0.0722509\pi\)
\(510\) 0 0
\(511\) 1.51546e8i 1.13575i
\(512\) 0 0
\(513\) 1.03937e7 0.0769870
\(514\) 0 0
\(515\) −2.09103e7 2.09103e7i −0.153087 0.153087i
\(516\) 0 0
\(517\) 1.39187e8 + 1.39187e8i 1.00723 + 1.00723i
\(518\) 0 0
\(519\) 1.96627e7 0.140651
\(520\) 0 0
\(521\) 1.96978e8i 1.39285i 0.717629 + 0.696425i \(0.245227\pi\)
−0.717629 + 0.696425i \(0.754773\pi\)
\(522\) 0 0
\(523\) −7.96274e7 + 7.96274e7i −0.556618 + 0.556618i −0.928343 0.371725i \(-0.878767\pi\)
0.371725 + 0.928343i \(0.378767\pi\)
\(524\) 0 0
\(525\) −7.12758e7 + 7.12758e7i −0.492566 + 0.492566i
\(526\) 0 0
\(527\) 2.59045e7i 0.176988i
\(528\) 0 0
\(529\) 1.04117e8 0.703322
\(530\) 0 0
\(531\) 1.64971e7 + 1.64971e7i 0.110185 + 0.110185i
\(532\) 0 0
\(533\) 1.91773e8 + 1.91773e8i 1.26650 + 1.26650i
\(534\) 0 0
\(535\) −4.56664e7 −0.298219
\(536\) 0 0
\(537\) 6.38390e7i 0.412252i
\(538\) 0 0
\(539\) −2.61220e8 + 2.61220e8i −1.66817 + 1.66817i
\(540\) 0 0
\(541\) 1.92069e8 1.92069e8i 1.21302 1.21302i 0.242986 0.970030i \(-0.421873\pi\)
0.970030 0.242986i \(-0.0781268\pi\)
\(542\) 0 0
\(543\) 4.73190e7i 0.295553i
\(544\) 0 0
\(545\) −6.28095e7 −0.388004
\(546\) 0 0
\(547\) 7.48181e7 + 7.48181e7i 0.457135 + 0.457135i 0.897714 0.440579i \(-0.145227\pi\)
−0.440579 + 0.897714i \(0.645227\pi\)
\(548\) 0 0
\(549\) 2.41079e7 + 2.41079e7i 0.145694 + 0.145694i
\(550\) 0 0
\(551\) 4.48408e7 0.268052
\(552\) 0 0
\(553\) 3.15564e8i 1.86600i
\(554\) 0 0
\(555\) 5.18087e7 5.18087e7i 0.303056 0.303056i
\(556\) 0 0
\(557\) 1.38814e8 1.38814e8i 0.803281 0.803281i −0.180326 0.983607i \(-0.557715\pi\)
0.983607 + 0.180326i \(0.0577152\pi\)
\(558\) 0 0
\(559\) 3.97404e8i 2.27508i
\(560\) 0 0
\(561\) 1.88706e7 0.106880
\(562\) 0 0
\(563\) 1.37839e8 + 1.37839e8i 0.772411 + 0.772411i 0.978527 0.206117i \(-0.0660827\pi\)
−0.206117 + 0.978527i \(0.566083\pi\)
\(564\) 0 0
\(565\) −8.08397e7 8.08397e7i −0.448208 0.448208i
\(566\) 0 0
\(567\) 3.33948e7 0.183202
\(568\) 0 0
\(569\) 1.94110e8i 1.05368i 0.849963 + 0.526842i \(0.176624\pi\)
−0.849963 + 0.526842i \(0.823376\pi\)
\(570\) 0 0
\(571\) −5.50758e7 + 5.50758e7i −0.295837 + 0.295837i −0.839381 0.543544i \(-0.817082\pi\)
0.543544 + 0.839381i \(0.317082\pi\)
\(572\) 0 0
\(573\) 1.23949e7 1.23949e7i 0.0658841 0.0658841i
\(574\) 0 0
\(575\) 1.81560e8i 0.955028i
\(576\) 0 0
\(577\) −1.62355e8 −0.845158 −0.422579 0.906326i \(-0.638875\pi\)
−0.422579 + 0.906326i \(0.638875\pi\)
\(578\) 0 0
\(579\) 7.33548e7 + 7.33548e7i 0.377914 + 0.377914i
\(580\) 0 0
\(581\) −1.33595e8 1.33595e8i −0.681181 0.681181i
\(582\) 0 0
\(583\) −2.80294e8 −1.41452
\(584\) 0 0
\(585\) 6.41400e7i 0.320377i
\(586\) 0 0
\(587\) −1.11238e8 + 1.11238e8i −0.549968 + 0.549968i −0.926431 0.376464i \(-0.877140\pi\)
0.376464 + 0.926431i \(0.377140\pi\)
\(588\) 0 0
\(589\) −7.58574e7 + 7.58574e7i −0.371238 + 0.371238i
\(590\) 0 0
\(591\) 2.24683e7i 0.108845i
\(592\) 0 0
\(593\) 3.77045e8 1.80813 0.904064 0.427397i \(-0.140569\pi\)
0.904064 + 0.427397i \(0.140569\pi\)
\(594\) 0 0
\(595\) −1.71533e7 1.71533e7i −0.0814321 0.0814321i
\(596\) 0 0
\(597\) 9.80720e6 + 9.80720e6i 0.0460916 + 0.0460916i
\(598\) 0 0
\(599\) 3.55681e8 1.65493 0.827465 0.561517i \(-0.189782\pi\)
0.827465 + 0.561517i \(0.189782\pi\)
\(600\) 0 0
\(601\) 1.48769e8i 0.685313i 0.939461 + 0.342656i \(0.111327\pi\)
−0.939461 + 0.342656i \(0.888673\pi\)
\(602\) 0 0
\(603\) −2.56129e7 + 2.56129e7i −0.116817 + 0.116817i
\(604\) 0 0
\(605\) −7.17201e7 + 7.17201e7i −0.323873 + 0.323873i
\(606\) 0 0
\(607\) 2.97217e7i 0.132894i 0.997790 + 0.0664472i \(0.0211664\pi\)
−0.997790 + 0.0664472i \(0.978834\pi\)
\(608\) 0 0
\(609\) 1.44073e8 0.637868
\(610\) 0 0
\(611\) −3.10591e8 3.10591e8i −1.36165 1.36165i
\(612\) 0 0
\(613\) −1.04355e8 1.04355e8i −0.453034 0.453034i 0.443326 0.896360i \(-0.353798\pi\)
−0.896360 + 0.443326i \(0.853798\pi\)
\(614\) 0 0
\(615\) 6.71318e7 0.288605
\(616\) 0 0
\(617\) 1.67053e8i 0.711213i −0.934636 0.355606i \(-0.884274\pi\)
0.934636 0.355606i \(-0.115726\pi\)
\(618\) 0 0
\(619\) 3.33498e7 3.33498e7i 0.140612 0.140612i −0.633297 0.773909i \(-0.718299\pi\)
0.773909 + 0.633297i \(0.218299\pi\)
\(620\) 0 0
\(621\) 4.25330e7 4.25330e7i 0.177603 0.177603i
\(622\) 0 0
\(623\) 2.37407e8i 0.981816i
\(624\) 0 0
\(625\) −6.52414e7 −0.267229
\(626\) 0 0
\(627\) 5.52596e7 + 5.52596e7i 0.224184 + 0.224184i
\(628\) 0 0
\(629\) −3.40133e7 3.40133e7i −0.136677 0.136677i
\(630\) 0 0
\(631\) 1.26476e8 0.503408 0.251704 0.967804i \(-0.419009\pi\)
0.251704 + 0.967804i \(0.419009\pi\)
\(632\) 0 0
\(633\) 1.34559e8i 0.530519i
\(634\) 0 0
\(635\) −1.15598e8 + 1.15598e8i −0.451471 + 0.451471i
\(636\) 0 0
\(637\) 5.82904e8 5.82904e8i 2.25517 2.25517i
\(638\) 0 0
\(639\) 6.46657e7i 0.247840i
\(640\) 0 0
\(641\) −3.79771e8 −1.44194 −0.720971 0.692965i \(-0.756304\pi\)
−0.720971 + 0.692965i \(0.756304\pi\)
\(642\) 0 0
\(643\) −2.19573e8 2.19573e8i −0.825937 0.825937i 0.161015 0.986952i \(-0.448523\pi\)
−0.986952 + 0.161015i \(0.948523\pi\)
\(644\) 0 0
\(645\) 6.95575e7 + 6.95575e7i 0.259218 + 0.259218i
\(646\) 0 0
\(647\) 1.63625e8 0.604139 0.302069 0.953286i \(-0.402323\pi\)
0.302069 + 0.953286i \(0.402323\pi\)
\(648\) 0 0
\(649\) 1.75418e8i 0.641713i
\(650\) 0 0
\(651\) −2.43729e8 + 2.43729e8i −0.883415 + 0.883415i
\(652\) 0 0
\(653\) 2.17072e8 2.17072e8i 0.779587 0.779587i −0.200173 0.979761i \(-0.564151\pi\)
0.979761 + 0.200173i \(0.0641505\pi\)
\(654\) 0 0
\(655\) 2.01911e8i 0.718516i
\(656\) 0 0
\(657\) −6.51155e7 −0.229609
\(658\) 0 0
\(659\) 3.75232e8 + 3.75232e8i 1.31112 + 1.31112i 0.920588 + 0.390535i \(0.127710\pi\)
0.390535 + 0.920588i \(0.372290\pi\)
\(660\) 0 0
\(661\) 9.25679e7 + 9.25679e7i 0.320521 + 0.320521i 0.848967 0.528446i \(-0.177225\pi\)
−0.528446 + 0.848967i \(0.677225\pi\)
\(662\) 0 0
\(663\) −4.21090e7 −0.144489
\(664\) 0 0
\(665\) 1.00461e8i 0.341613i
\(666\) 0 0
\(667\) 1.83498e8 1.83498e8i 0.618376 0.618376i
\(668\) 0 0
\(669\) −8.89979e7 + 8.89979e7i −0.297236 + 0.297236i
\(670\) 0 0
\(671\) 2.56347e8i 0.848517i
\(672\) 0 0
\(673\) 5.35988e8 1.75837 0.879185 0.476481i \(-0.158088\pi\)
0.879185 + 0.476481i \(0.158088\pi\)
\(674\) 0 0
\(675\) 3.06254e7 + 3.06254e7i 0.0995797 + 0.0995797i
\(676\) 0 0
\(677\) 3.17164e8 + 3.17164e8i 1.02216 + 1.02216i 0.999749 + 0.0224072i \(0.00713303\pi\)
0.0224072 + 0.999749i \(0.492867\pi\)
\(678\) 0 0
\(679\) −2.22433e8 −0.710542
\(680\) 0 0
\(681\) 9.12182e7i 0.288829i
\(682\) 0 0
\(683\) 2.38997e8 2.38997e8i 0.750121 0.750121i −0.224381 0.974502i \(-0.572036\pi\)
0.974502 + 0.224381i \(0.0720359\pi\)
\(684\) 0 0
\(685\) −9.65848e6 + 9.65848e6i −0.0300495 + 0.0300495i
\(686\) 0 0
\(687\) 2.63811e8i 0.813623i
\(688\) 0 0
\(689\) 6.25466e8 1.91226
\(690\) 0 0
\(691\) 1.14977e8 + 1.14977e8i 0.348478 + 0.348478i 0.859542 0.511065i \(-0.170749\pi\)
−0.511065 + 0.859542i \(0.670749\pi\)
\(692\) 0 0
\(693\) 1.77549e8 + 1.77549e8i 0.533479 + 0.533479i
\(694\) 0 0
\(695\) −6.21623e7 −0.185171
\(696\) 0 0
\(697\) 4.40732e7i 0.130160i
\(698\) 0 0
\(699\) −1.61770e7 + 1.61770e7i −0.0473661 + 0.0473661i
\(700\) 0 0
\(701\) −1.52614e8 + 1.52614e8i −0.443036 + 0.443036i −0.893031 0.449995i \(-0.851426\pi\)
0.449995 + 0.893031i \(0.351426\pi\)
\(702\) 0 0
\(703\) 1.99205e8i 0.573370i
\(704\) 0 0
\(705\) −1.08725e8 −0.310287
\(706\) 0 0
\(707\) 3.79448e8 + 3.79448e8i 1.07373 + 1.07373i
\(708\) 0 0
\(709\) −2.20534e8 2.20534e8i −0.618780 0.618780i 0.326439 0.945218i \(-0.394151\pi\)
−0.945218 + 0.326439i \(0.894151\pi\)
\(710\) 0 0
\(711\) −1.35590e8 −0.377241
\(712\) 0 0
\(713\) 6.20848e8i 1.71284i
\(714\) 0 0
\(715\) 3.41010e8 3.41010e8i 0.932930 0.932930i
\(716\) 0 0
\(717\) −1.41629e8 + 1.41629e8i −0.384234 + 0.384234i
\(718\) 0 0
\(719\) 1.10928e8i 0.298439i 0.988804 + 0.149220i \(0.0476762\pi\)
−0.988804 + 0.149220i \(0.952324\pi\)
\(720\) 0 0
\(721\) −2.58326e8 −0.689228
\(722\) 0 0
\(723\) −2.06240e7 2.06240e7i −0.0545704 0.0545704i
\(724\) 0 0
\(725\) 1.32125e8 + 1.32125e8i 0.346714 + 0.346714i
\(726\) 0 0
\(727\) 4.40388e8 1.14613 0.573063 0.819512i \(-0.305755\pi\)
0.573063 + 0.819512i \(0.305755\pi\)
\(728\) 0 0
\(729\) 1.43489e7i 0.0370370i
\(730\) 0 0
\(731\) 4.56657e7 4.56657e7i 0.116906 0.116906i
\(732\) 0 0
\(733\) −8.77336e7 + 8.77336e7i −0.222769 + 0.222769i −0.809663 0.586895i \(-0.800350\pi\)
0.586895 + 0.809663i \(0.300350\pi\)
\(734\) 0 0
\(735\) 2.04051e8i 0.513897i
\(736\) 0 0
\(737\) −2.72349e8 −0.680337
\(738\) 0 0
\(739\) −1.75595e8 1.75595e8i −0.435089 0.435089i 0.455266 0.890355i \(-0.349544\pi\)
−0.890355 + 0.455266i \(0.849544\pi\)
\(740\) 0 0
\(741\) −1.23310e8 1.23310e8i −0.303070 0.303070i
\(742\) 0 0
\(743\) 1.99950e8 0.487478 0.243739 0.969841i \(-0.421626\pi\)
0.243739 + 0.969841i \(0.421626\pi\)
\(744\) 0 0
\(745\) 2.40624e8i 0.581930i
\(746\) 0 0
\(747\) −5.74025e7 + 5.74025e7i −0.137711 + 0.137711i
\(748\) 0 0
\(749\) −2.82082e8 + 2.82082e8i −0.671320 + 0.671320i
\(750\) 0 0
\(751\) 4.31676e7i 0.101915i −0.998701 0.0509575i \(-0.983773\pi\)
0.998701 0.0509575i \(-0.0162273\pi\)
\(752\) 0 0
\(753\) 4.55351e8 1.06650
\(754\) 0 0
\(755\) −1.31862e8 1.31862e8i −0.306392 0.306392i
\(756\) 0 0
\(757\) 3.04827e8 + 3.04827e8i 0.702692 + 0.702692i 0.964988 0.262295i \(-0.0844794\pi\)
−0.262295 + 0.964988i \(0.584479\pi\)
\(758\) 0 0
\(759\) 4.52267e8 1.03435
\(760\) 0 0
\(761\) 4.91137e8i 1.11442i −0.830372 0.557209i \(-0.811872\pi\)
0.830372 0.557209i \(-0.188128\pi\)
\(762\) 0 0
\(763\) −3.87975e8 + 3.87975e8i −0.873434 + 0.873434i
\(764\) 0 0
\(765\) −7.37032e6 + 7.37032e6i −0.0164627 + 0.0164627i
\(766\) 0 0
\(767\) 3.91440e8i 0.867519i
\(768\) 0 0
\(769\) −7.18352e8 −1.57964 −0.789820 0.613338i \(-0.789826\pi\)
−0.789820 + 0.613338i \(0.789826\pi\)
\(770\) 0 0
\(771\) 1.05883e7 + 1.05883e7i 0.0231028 + 0.0231028i
\(772\) 0 0
\(773\) −4.46865e8 4.46865e8i −0.967471 0.967471i 0.0320160 0.999487i \(-0.489807\pi\)
−0.999487 + 0.0320160i \(0.989807\pi\)
\(774\) 0 0
\(775\) −4.47034e8 −0.960364
\(776\) 0 0
\(777\) 6.40046e8i 1.36442i
\(778\) 0 0
\(779\) 1.29062e8 1.29062e8i 0.273014 0.273014i
\(780\) 0 0
\(781\) −3.43805e8 + 3.43805e8i −0.721704 + 0.721704i
\(782\) 0 0
\(783\) 6.19046e7i 0.128955i
\(784\) 0 0
\(785\) −4.59153e7 −0.0949179
\(786\) 0 0
\(787\) −1.27217e8 1.27217e8i −0.260989 0.260989i 0.564467 0.825456i \(-0.309082\pi\)
−0.825456 + 0.564467i \(0.809082\pi\)
\(788\) 0 0
\(789\) −2.93544e8 2.93544e8i −0.597643 0.597643i
\(790\) 0 0
\(791\) −9.98696e8 −2.01792
\(792\) 0 0
\(793\) 5.72029e8i 1.14709i
\(794\) 0 0
\(795\) 1.09475e8 1.09475e8i 0.217878 0.217878i
\(796\) 0 0
\(797\) −5.09748e7 + 5.09748e7i −0.100689 + 0.100689i −0.755657 0.654968i \(-0.772682\pi\)
0.654968 + 0.755657i \(0.272682\pi\)
\(798\) 0 0
\(799\) 7.13800e7i 0.139938i
\(800\) 0 0
\(801\) −1.02008e8 −0.198489
\(802\) 0 0
\(803\) −3.46196e8 3.46196e8i −0.668615 0.668615i
\(804\) 0 0
\(805\) −4.11109e8 4.11109e8i −0.788077 0.788077i
\(806\) 0 0
\(807\) −4.41516e8 −0.840090
\(808\) 0 0
\(809\) 9.47571e8i 1.78964i −0.446424 0.894822i \(-0.647303\pi\)
0.446424 0.894822i \(-0.352697\pi\)
\(810\) 0 0
\(811\) −9.22998e7 + 9.22998e7i −0.173037 + 0.173037i −0.788312 0.615275i \(-0.789045\pi\)
0.615275 + 0.788312i \(0.289045\pi\)
\(812\) 0 0
\(813\) 1.47699e8 1.47699e8i 0.274856 0.274856i
\(814\) 0 0
\(815\) 1.19031e8i 0.219881i
\(816\) 0 0
\(817\) 2.67450e8 0.490429
\(818\) 0 0
\(819\) −3.96193e8 3.96193e8i −0.721200 0.721200i
\(820\) 0 0
\(821\) 5.05480e8 + 5.05480e8i 0.913429 + 0.913429i 0.996540 0.0831113i \(-0.0264857\pi\)
−0.0831113 + 0.996540i \(0.526486\pi\)
\(822\) 0 0
\(823\) −3.69057e7 −0.0662055 −0.0331027 0.999452i \(-0.510539\pi\)
−0.0331027 + 0.999452i \(0.510539\pi\)
\(824\) 0 0
\(825\) 3.25649e8i 0.579947i
\(826\) 0 0
\(827\) 1.45689e8 1.45689e8i 0.257579 0.257579i −0.566490 0.824069i \(-0.691699\pi\)
0.824069 + 0.566490i \(0.191699\pi\)
\(828\) 0 0
\(829\) −6.21154e8 + 6.21154e8i −1.09027 + 1.09027i −0.0947757 + 0.995499i \(0.530213\pi\)
−0.995499 + 0.0947757i \(0.969787\pi\)
\(830\) 0 0
\(831\) 3.15308e7i 0.0549455i
\(832\) 0 0
\(833\) −1.33963e8 −0.231766
\(834\) 0 0
\(835\) 3.93812e8 + 3.93812e8i 0.676440 + 0.676440i
\(836\) 0 0
\(837\) 1.04724e8 + 1.04724e8i 0.178596 + 0.178596i
\(838\) 0 0
\(839\) −7.28215e8 −1.23303 −0.616515 0.787343i \(-0.711456\pi\)
−0.616515 + 0.787343i \(0.711456\pi\)
\(840\) 0 0
\(841\) 3.27752e8i 0.551008i
\(842\) 0 0
\(843\) 1.93169e8 1.93169e8i 0.322445 0.322445i
\(844\) 0 0
\(845\) −5.39990e8 + 5.39990e8i −0.894984 + 0.894984i
\(846\) 0 0
\(847\) 8.86032e8i 1.45814i
\(848\) 0 0
\(849\) −5.29375e8 −0.865048
\(850\) 0 0
\(851\) −8.15189e8 8.15189e8i −1.32272 1.32272i
\(852\) 0 0
\(853\) 3.95707e8 + 3.95707e8i 0.637568 + 0.637568i 0.949955 0.312387i \(-0.101128\pi\)
−0.312387 + 0.949955i \(0.601128\pi\)
\(854\) 0 0
\(855\) −4.31657e7 −0.0690623
\(856\) 0 0
\(857\) 7.60780e8i 1.20870i 0.796721 + 0.604348i \(0.206566\pi\)
−0.796721 + 0.604348i \(0.793434\pi\)
\(858\) 0 0
\(859\) −2.52493e8 + 2.52493e8i −0.398354 + 0.398354i −0.877652 0.479298i \(-0.840891\pi\)
0.479298 + 0.877652i \(0.340891\pi\)
\(860\) 0 0
\(861\) 4.14674e8 4.14674e8i 0.649677 0.649677i
\(862\) 0 0
\(863\) 7.25728e7i 0.112912i 0.998405 + 0.0564562i \(0.0179801\pi\)
−0.998405 + 0.0564562i \(0.982020\pi\)
\(864\) 0 0
\(865\) −8.16609e7 −0.126173
\(866\) 0 0
\(867\) −2.61223e8 2.61223e8i −0.400824 0.400824i
\(868\) 0 0
\(869\) −7.20885e8 7.20885e8i −1.09852 1.09852i
\(870\) 0 0
\(871\) 6.07738e8 0.919734
\(872\) 0 0
\(873\) 9.55738e7i 0.143647i
\(874\) 0 0
\(875\) 7.00538e8 7.00538e8i 1.04570 1.04570i
\(876\) 0 0
\(877\) −5.64497e8 + 5.64497e8i −0.836879 + 0.836879i −0.988447 0.151568i \(-0.951568\pi\)
0.151568 + 0.988447i \(0.451568\pi\)
\(878\) 0 0
\(879\) 5.50431e8i 0.810468i
\(880\) 0 0
\(881\) 9.73736e7 0.142401 0.0712006 0.997462i \(-0.477317\pi\)
0.0712006 + 0.997462i \(0.477317\pi\)
\(882\) 0 0
\(883\) 6.90645e7 + 6.90645e7i 0.100317 + 0.100317i 0.755484 0.655167i \(-0.227402\pi\)
−0.655167 + 0.755484i \(0.727402\pi\)
\(884\) 0 0
\(885\) −6.85136e7 6.85136e7i −0.0988432 0.0988432i
\(886\) 0 0
\(887\) −4.66487e8 −0.668450 −0.334225 0.942493i \(-0.608474\pi\)
−0.334225 + 0.942493i \(0.608474\pi\)
\(888\) 0 0
\(889\) 1.42810e9i 2.03261i
\(890\) 0 0
\(891\) 7.62881e7 7.62881e7i 0.107851 0.107851i
\(892\) 0 0
\(893\) −2.09026e8 + 2.09026e8i −0.293525 + 0.293525i
\(894\) 0 0
\(895\) 2.65128e8i 0.369817i
\(896\) 0 0
\(897\) −1.00922e9 −1.39832
\(898\) 0 0
\(899\) 4.51805e8 + 4.51805e8i 0.621831 + 0.621831i
\(900\) 0 0
\(901\) −7.18722e7 7.18722e7i −0.0982623 0.0982623i
\(902\) 0 0
\(903\) 8.59315e8 1.16705
\(904\) 0 0
\(905\) 1.96519e8i 0.265130i
\(906\) 0 0
\(907\) 1.99657e8 1.99657e8i 0.267585 0.267585i −0.560541 0.828126i \(-0.689407\pi\)
0.828126 + 0.560541i \(0.189407\pi\)
\(908\) 0 0
\(909\) 1.63039e8 1.63039e8i 0.217071 0.217071i
\(910\) 0 0
\(911\) 7.71797e8i 1.02082i −0.859932 0.510409i \(-0.829494\pi\)
0.859932 0.510409i \(-0.170506\pi\)
\(912\) 0 0
\(913\) −6.10378e8 −0.802022
\(914\) 0 0
\(915\) −1.00122e8 1.00122e8i −0.130697 0.130697i
\(916\) 0 0
\(917\) −1.24721e9 1.24721e9i −1.61745 1.61745i
\(918\) 0 0
\(919\) 8.94727e8 1.15277 0.576387 0.817177i \(-0.304462\pi\)
0.576387 + 0.817177i \(0.304462\pi\)
\(920\) 0 0
\(921\) 6.91307e7i 0.0884896i
\(922\) 0 0
\(923\) 7.67189e8 7.67189e8i 0.975658 0.975658i
\(924\) 0 0
\(925\) 5.86967e8 5.86967e8i 0.741633 0.741633i
\(926\) 0 0
\(927\) 1.10996e8i 0.139338i
\(928\) 0 0
\(929\) 5.16188e8 0.643814 0.321907 0.946771i \(-0.395676\pi\)
0.321907 + 0.946771i \(0.395676\pi\)
\(930\) 0 0
\(931\) −3.92290e8 3.92290e8i −0.486136 0.486136i
\(932\) 0 0
\(933\) 3.11615e8 + 3.11615e8i 0.383684 + 0.383684i
\(934\) 0 0
\(935\) −7.83709e7 −0.0958782
\(936\) 0 0
\(937\) 5.89366e8i 0.716418i 0.933641 + 0.358209i \(0.116612\pi\)
−0.933641 + 0.358209i \(0.883388\pi\)
\(938\) 0 0
\(939\) 4.59755e7 4.59755e7i 0.0555303 0.0555303i
\(940\) 0 0
\(941\) −4.11063e7 + 4.11063e7i −0.0493333 + 0.0493333i −0.731343 0.682010i \(-0.761106\pi\)
0.682010 + 0.731343i \(0.261106\pi\)
\(942\) 0 0
\(943\) 1.05629e9i 1.25965i
\(944\) 0 0
\(945\) −1.38691e8 −0.164344
\(946\) 0 0
\(947\) 1.82225e8 + 1.82225e8i 0.214565 + 0.214565i 0.806203 0.591639i \(-0.201519\pi\)
−0.591639 + 0.806203i \(0.701519\pi\)
\(948\) 0 0
\(949\) 7.72525e8 + 7.72525e8i 0.903887 + 0.903887i
\(950\) 0 0
\(951\) −7.70715e8 −0.896091
\(952\) 0 0
\(953\) 1.41084e9i 1.63004i 0.579433 + 0.815020i \(0.303274\pi\)
−0.579433 + 0.815020i \(0.696726\pi\)
\(954\) 0 0
\(955\) −5.14771e7 + 5.14771e7i −0.0591023 + 0.0591023i
\(956\) 0 0
\(957\) 3.29125e8 3.29125e8i 0.375513 0.375513i
\(958\) 0 0
\(959\) 1.19321e8i 0.135289i
\(960\) 0 0
\(961\) −6.41141e8 −0.722409
\(962\) 0 0
\(963\) 1.21203e8 + 1.21203e8i 0.135718 + 0.135718i
\(964\) 0 0
\(965\) −3.04648e8 3.04648e8i −0.339013 0.339013i
\(966\) 0 0
\(967\) 9.62310e8 1.06423 0.532115 0.846672i \(-0.321397\pi\)
0.532115 + 0.846672i \(0.321397\pi\)
\(968\) 0 0
\(969\) 2.83390e7i 0.0311468i
\(970\) 0 0
\(971\) −8.97286e8 + 8.97286e8i −0.980106 + 0.980106i −0.999806 0.0196995i \(-0.993729\pi\)
0.0196995 + 0.999806i \(0.493729\pi\)
\(972\) 0 0
\(973\) −3.83977e8 + 3.83977e8i −0.416838 + 0.416838i
\(974\) 0 0
\(975\) 7.26675e8i 0.784019i
\(976\) 0 0
\(977\) −1.56733e8 −0.168065 −0.0840324 0.996463i \(-0.526780\pi\)
−0.0840324 + 0.996463i \(0.526780\pi\)
\(978\) 0 0
\(979\) −5.42341e8 5.42341e8i −0.577995 0.577995i
\(980\) 0 0
\(981\) 1.66703e8 + 1.66703e8i 0.176578 + 0.176578i
\(982\) 0 0
\(983\) 2.49727e7 0.0262909 0.0131454 0.999914i \(-0.495816\pi\)
0.0131454 + 0.999914i \(0.495816\pi\)
\(984\) 0 0
\(985\) 9.33128e7i 0.0976411i
\(986\) 0 0
\(987\) −6.71598e8 + 6.71598e8i −0.698486 + 0.698486i
\(988\) 0 0
\(989\) 1.09446e9 1.09446e9i 1.13139 1.13139i
\(990\) 0 0
\(991\) 4.63587e8i 0.476333i −0.971224 0.238166i \(-0.923454\pi\)
0.971224 0.238166i \(-0.0765463\pi\)
\(992\) 0 0
\(993\) 5.13428e8 0.524362
\(994\) 0 0
\(995\) −4.07300e7 4.07300e7i −0.0413472 0.0413472i
\(996\) 0 0
\(997\) 9.19997e7 + 9.19997e7i 0.0928327 + 0.0928327i 0.751998 0.659165i \(-0.229090\pi\)
−0.659165 + 0.751998i \(0.729090\pi\)
\(998\) 0 0
\(999\) −2.75011e8 −0.275838
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 384.7.l.b.31.18 48
4.3 odd 2 384.7.l.a.31.7 48
8.3 odd 2 192.7.l.a.79.18 48
8.5 even 2 48.7.l.a.43.18 yes 48
16.3 odd 4 inner 384.7.l.b.223.18 48
16.5 even 4 192.7.l.a.175.18 48
16.11 odd 4 48.7.l.a.19.18 48
16.13 even 4 384.7.l.a.223.7 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.7.l.a.19.18 48 16.11 odd 4
48.7.l.a.43.18 yes 48 8.5 even 2
192.7.l.a.79.18 48 8.3 odd 2
192.7.l.a.175.18 48 16.5 even 4
384.7.l.a.31.7 48 4.3 odd 2
384.7.l.a.223.7 48 16.13 even 4
384.7.l.b.31.18 48 1.1 even 1 trivial
384.7.l.b.223.18 48 16.3 odd 4 inner