Properties

Label 354.8.a.b.1.5
Level $354$
Weight $8$
Character 354.1
Self dual yes
Analytic conductor $110.584$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [354,8,Mod(1,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 354.a (trivial)

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: yes
Analytic conductor: \(110.584299021\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6162x^{3} - 12837x^{2} + 3760259x - 17264060 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(76.1395\) of defining polynomial
Character \(\chi\) \(=\) 354.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +322.557 q^{5} +216.000 q^{6} -1226.16 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +322.557 q^{5} +216.000 q^{6} -1226.16 q^{7} +512.000 q^{8} +729.000 q^{9} +2580.46 q^{10} -5358.18 q^{11} +1728.00 q^{12} -10351.4 q^{13} -9809.25 q^{14} +8709.05 q^{15} +4096.00 q^{16} +20353.6 q^{17} +5832.00 q^{18} +29921.3 q^{19} +20643.7 q^{20} -33106.2 q^{21} -42865.5 q^{22} +9894.44 q^{23} +13824.0 q^{24} +25918.3 q^{25} -82811.4 q^{26} +19683.0 q^{27} -78474.0 q^{28} -110415. q^{29} +69672.4 q^{30} -186449. q^{31} +32768.0 q^{32} -144671. q^{33} +162829. q^{34} -395506. q^{35} +46656.0 q^{36} -394202. q^{37} +239370. q^{38} -279489. q^{39} +165149. q^{40} -474592. q^{41} -264850. q^{42} -423064. q^{43} -342924. q^{44} +235144. q^{45} +79155.5 q^{46} +382727. q^{47} +110592. q^{48} +679915. q^{49} +207347. q^{50} +549547. q^{51} -662491. q^{52} -126831. q^{53} +157464. q^{54} -1.72832e6 q^{55} -627792. q^{56} +807874. q^{57} -883318. q^{58} +205379. q^{59} +557379. q^{60} -162477. q^{61} -1.49159e6 q^{62} -893868. q^{63} +262144. q^{64} -3.33893e6 q^{65} -1.15737e6 q^{66} +1.53838e6 q^{67} +1.30263e6 q^{68} +267150. q^{69} -3.16405e6 q^{70} -5.92431e6 q^{71} +373248. q^{72} -2.48019e6 q^{73} -3.15361e6 q^{74} +699795. q^{75} +1.91496e6 q^{76} +6.56997e6 q^{77} -2.23591e6 q^{78} +3.93508e6 q^{79} +1.32120e6 q^{80} +531441. q^{81} -3.79673e6 q^{82} -4.28658e6 q^{83} -2.11880e6 q^{84} +6.56520e6 q^{85} -3.38451e6 q^{86} -2.98120e6 q^{87} -2.74339e6 q^{88} -4.71156e6 q^{89} +1.88116e6 q^{90} +1.26925e7 q^{91} +633244. q^{92} -5.03412e6 q^{93} +3.06181e6 q^{94} +9.65132e6 q^{95} +884736. q^{96} +1.26688e7 q^{97} +5.43932e6 q^{98} -3.90612e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9} + 1312 q^{10} - 15730 q^{11} + 8640 q^{12} - 21854 q^{13} - 608 q^{14} + 4428 q^{15} + 20480 q^{16} - 34548 q^{17} + 29160 q^{18} - 43828 q^{19} + 10496 q^{20} - 2052 q^{21} - 125840 q^{22} - 110582 q^{23} + 69120 q^{24} - 174577 q^{25} - 174832 q^{26} + 98415 q^{27} - 4864 q^{28} - 307558 q^{29} + 35424 q^{30} - 277994 q^{31} + 163840 q^{32} - 424710 q^{33} - 276384 q^{34} - 764338 q^{35} + 233280 q^{36} - 853778 q^{37} - 350624 q^{38} - 590058 q^{39} + 83968 q^{40} + 131342 q^{41} - 16416 q^{42} - 721996 q^{43} - 1006720 q^{44} + 119556 q^{45} - 884656 q^{46} - 358832 q^{47} + 552960 q^{48} - 207141 q^{49} - 1396616 q^{50} - 932796 q^{51} - 1398656 q^{52} + 1006180 q^{53} + 787320 q^{54} + 81944 q^{55} - 38912 q^{56} - 1183356 q^{57} - 2460464 q^{58} + 1026895 q^{59} + 283392 q^{60} + 101158 q^{61} - 2223952 q^{62} - 55404 q^{63} + 1310720 q^{64} - 3138378 q^{65} - 3397680 q^{66} - 6362512 q^{67} - 2211072 q^{68} - 2985714 q^{69} - 6114704 q^{70} - 8877414 q^{71} + 1866240 q^{72} - 4881862 q^{73} - 6830224 q^{74} - 4713579 q^{75} - 2804992 q^{76} - 2205694 q^{77} - 4720464 q^{78} - 2769432 q^{79} + 671744 q^{80} + 2657205 q^{81} + 1050736 q^{82} - 1430156 q^{83} - 131328 q^{84} - 7564814 q^{85} - 5775968 q^{86} - 8304066 q^{87} - 8053760 q^{88} - 17172176 q^{89} + 956448 q^{90} - 932474 q^{91} - 7077248 q^{92} - 7505838 q^{93} - 2870656 q^{94} + 15962708 q^{95} + 4423680 q^{96} + 20863830 q^{97} - 1657128 q^{98} - 11467170 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) 322.557 1.15402 0.577008 0.816738i \(-0.304220\pi\)
0.577008 + 0.816738i \(0.304220\pi\)
\(6\) 216.000 0.408248
\(7\) −1226.16 −1.35115 −0.675573 0.737293i \(-0.736104\pi\)
−0.675573 + 0.737293i \(0.736104\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 2580.46 0.816013
\(11\) −5358.18 −1.21379 −0.606895 0.794782i \(-0.707585\pi\)
−0.606895 + 0.794782i \(0.707585\pi\)
\(12\) 1728.00 0.288675
\(13\) −10351.4 −1.30677 −0.653384 0.757027i \(-0.726651\pi\)
−0.653384 + 0.757027i \(0.726651\pi\)
\(14\) −9809.25 −0.955405
\(15\) 8709.05 0.666272
\(16\) 4096.00 0.250000
\(17\) 20353.6 1.00478 0.502389 0.864642i \(-0.332455\pi\)
0.502389 + 0.864642i \(0.332455\pi\)
\(18\) 5832.00 0.235702
\(19\) 29921.3 1.00079 0.500394 0.865798i \(-0.333188\pi\)
0.500394 + 0.865798i \(0.333188\pi\)
\(20\) 20643.7 0.577008
\(21\) −33106.2 −0.780085
\(22\) −42865.5 −0.858279
\(23\) 9894.44 0.169568 0.0847839 0.996399i \(-0.472980\pi\)
0.0847839 + 0.996399i \(0.472980\pi\)
\(24\) 13824.0 0.204124
\(25\) 25918.3 0.331755
\(26\) −82811.4 −0.924025
\(27\) 19683.0 0.192450
\(28\) −78474.0 −0.675573
\(29\) −110415. −0.840686 −0.420343 0.907365i \(-0.638090\pi\)
−0.420343 + 0.907365i \(0.638090\pi\)
\(30\) 69672.4 0.471125
\(31\) −186449. −1.12407 −0.562036 0.827113i \(-0.689982\pi\)
−0.562036 + 0.827113i \(0.689982\pi\)
\(32\) 32768.0 0.176777
\(33\) −144671. −0.700782
\(34\) 162829. 0.710485
\(35\) −395506. −1.55925
\(36\) 46656.0 0.166667
\(37\) −394202. −1.27942 −0.639709 0.768617i \(-0.720945\pi\)
−0.639709 + 0.768617i \(0.720945\pi\)
\(38\) 239370. 0.707664
\(39\) −279489. −0.754463
\(40\) 165149. 0.408007
\(41\) −474592. −1.07542 −0.537708 0.843131i \(-0.680710\pi\)
−0.537708 + 0.843131i \(0.680710\pi\)
\(42\) −264850. −0.551603
\(43\) −423064. −0.811459 −0.405730 0.913993i \(-0.632983\pi\)
−0.405730 + 0.913993i \(0.632983\pi\)
\(44\) −342924. −0.606895
\(45\) 235144. 0.384672
\(46\) 79155.5 0.119903
\(47\) 382727. 0.537707 0.268854 0.963181i \(-0.413355\pi\)
0.268854 + 0.963181i \(0.413355\pi\)
\(48\) 110592. 0.144338
\(49\) 679915. 0.825598
\(50\) 207347. 0.234586
\(51\) 549547. 0.580108
\(52\) −662491. −0.653384
\(53\) −126831. −0.117019 −0.0585097 0.998287i \(-0.518635\pi\)
−0.0585097 + 0.998287i \(0.518635\pi\)
\(54\) 157464. 0.136083
\(55\) −1.72832e6 −1.40073
\(56\) −627792. −0.477703
\(57\) 807874. 0.577805
\(58\) −883318. −0.594455
\(59\) 205379. 0.130189
\(60\) 557379. 0.333136
\(61\) −162477. −0.0916510 −0.0458255 0.998949i \(-0.514592\pi\)
−0.0458255 + 0.998949i \(0.514592\pi\)
\(62\) −1.49159e6 −0.794838
\(63\) −893868. −0.450382
\(64\) 262144. 0.125000
\(65\) −3.33893e6 −1.50803
\(66\) −1.15737e6 −0.495527
\(67\) 1.53838e6 0.624887 0.312444 0.949936i \(-0.398852\pi\)
0.312444 + 0.949936i \(0.398852\pi\)
\(68\) 1.30263e6 0.502389
\(69\) 267150. 0.0979001
\(70\) −3.16405e6 −1.10255
\(71\) −5.92431e6 −1.96442 −0.982208 0.187799i \(-0.939865\pi\)
−0.982208 + 0.187799i \(0.939865\pi\)
\(72\) 373248. 0.117851
\(73\) −2.48019e6 −0.746199 −0.373099 0.927791i \(-0.621705\pi\)
−0.373099 + 0.927791i \(0.621705\pi\)
\(74\) −3.15361e6 −0.904685
\(75\) 699795. 0.191539
\(76\) 1.91496e6 0.500394
\(77\) 6.56997e6 1.64001
\(78\) −2.23591e6 −0.533486
\(79\) 3.93508e6 0.897964 0.448982 0.893541i \(-0.351787\pi\)
0.448982 + 0.893541i \(0.351787\pi\)
\(80\) 1.32120e6 0.288504
\(81\) 531441. 0.111111
\(82\) −3.79673e6 −0.760434
\(83\) −4.28658e6 −0.822883 −0.411441 0.911436i \(-0.634974\pi\)
−0.411441 + 0.911436i \(0.634974\pi\)
\(84\) −2.11880e6 −0.390043
\(85\) 6.56520e6 1.15953
\(86\) −3.38451e6 −0.573788
\(87\) −2.98120e6 −0.485370
\(88\) −2.74339e6 −0.429139
\(89\) −4.71156e6 −0.708435 −0.354217 0.935163i \(-0.615253\pi\)
−0.354217 + 0.935163i \(0.615253\pi\)
\(90\) 1.88116e6 0.272004
\(91\) 1.26925e7 1.76564
\(92\) 633244. 0.0847839
\(93\) −5.03412e6 −0.648983
\(94\) 3.06181e6 0.380217
\(95\) 9.65132e6 1.15493
\(96\) 884736. 0.102062
\(97\) 1.26688e7 1.40940 0.704699 0.709506i \(-0.251082\pi\)
0.704699 + 0.709506i \(0.251082\pi\)
\(98\) 5.43932e6 0.583786
\(99\) −3.90612e6 −0.404596
\(100\) 1.65877e6 0.165877
\(101\) 1.06235e7 1.02599 0.512994 0.858392i \(-0.328536\pi\)
0.512994 + 0.858392i \(0.328536\pi\)
\(102\) 4.39638e6 0.410199
\(103\) 6.29033e6 0.567209 0.283605 0.958941i \(-0.408470\pi\)
0.283605 + 0.958941i \(0.408470\pi\)
\(104\) −5.29993e6 −0.462012
\(105\) −1.06787e7 −0.900231
\(106\) −1.01464e6 −0.0827453
\(107\) −8.52760e6 −0.672951 −0.336476 0.941692i \(-0.609235\pi\)
−0.336476 + 0.941692i \(0.609235\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −7.25070e6 −0.536274 −0.268137 0.963381i \(-0.586408\pi\)
−0.268137 + 0.963381i \(0.586408\pi\)
\(110\) −1.38266e7 −0.990468
\(111\) −1.06434e7 −0.738672
\(112\) −5.02233e6 −0.337787
\(113\) −2.35877e7 −1.53784 −0.768920 0.639345i \(-0.779206\pi\)
−0.768920 + 0.639345i \(0.779206\pi\)
\(114\) 6.46299e6 0.408570
\(115\) 3.19152e6 0.195684
\(116\) −7.06654e6 −0.420343
\(117\) −7.54619e6 −0.435589
\(118\) 1.64303e6 0.0920575
\(119\) −2.49567e7 −1.35760
\(120\) 4.45903e6 0.235563
\(121\) 9.22297e6 0.473284
\(122\) −1.29982e6 −0.0648070
\(123\) −1.28140e7 −0.620892
\(124\) −1.19327e7 −0.562036
\(125\) −1.68397e7 −0.771166
\(126\) −7.15094e6 −0.318468
\(127\) 1.27104e7 0.550613 0.275306 0.961357i \(-0.411221\pi\)
0.275306 + 0.961357i \(0.411221\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.14227e7 −0.468496
\(130\) −2.67114e7 −1.06634
\(131\) −1.26929e6 −0.0493301 −0.0246651 0.999696i \(-0.507852\pi\)
−0.0246651 + 0.999696i \(0.507852\pi\)
\(132\) −9.25894e6 −0.350391
\(133\) −3.66881e7 −1.35221
\(134\) 1.23070e7 0.441862
\(135\) 6.34890e6 0.222091
\(136\) 1.04210e7 0.355242
\(137\) −1.04727e7 −0.347965 −0.173982 0.984749i \(-0.555664\pi\)
−0.173982 + 0.984749i \(0.555664\pi\)
\(138\) 2.13720e6 0.0692258
\(139\) −7.35998e6 −0.232447 −0.116224 0.993223i \(-0.537079\pi\)
−0.116224 + 0.993223i \(0.537079\pi\)
\(140\) −2.53124e7 −0.779623
\(141\) 1.03336e7 0.310446
\(142\) −4.73944e7 −1.38905
\(143\) 5.54649e7 1.58614
\(144\) 2.98598e6 0.0833333
\(145\) −3.56151e7 −0.970166
\(146\) −1.98415e7 −0.527642
\(147\) 1.83577e7 0.476659
\(148\) −2.52289e7 −0.639709
\(149\) −5.23831e6 −0.129730 −0.0648648 0.997894i \(-0.520662\pi\)
−0.0648648 + 0.997894i \(0.520662\pi\)
\(150\) 5.59836e6 0.135438
\(151\) 2.69252e7 0.636415 0.318207 0.948021i \(-0.396919\pi\)
0.318207 + 0.948021i \(0.396919\pi\)
\(152\) 1.53197e7 0.353832
\(153\) 1.48378e7 0.334926
\(154\) 5.25598e7 1.15966
\(155\) −6.01405e7 −1.29720
\(156\) −1.78873e7 −0.377231
\(157\) −4.63248e6 −0.0955355 −0.0477677 0.998858i \(-0.515211\pi\)
−0.0477677 + 0.998858i \(0.515211\pi\)
\(158\) 3.14807e7 0.634957
\(159\) −3.42442e6 −0.0675612
\(160\) 1.05696e7 0.204003
\(161\) −1.21321e7 −0.229111
\(162\) 4.25153e6 0.0785674
\(163\) 4.98385e6 0.0901381 0.0450690 0.998984i \(-0.485649\pi\)
0.0450690 + 0.998984i \(0.485649\pi\)
\(164\) −3.03739e7 −0.537708
\(165\) −4.66647e7 −0.808714
\(166\) −3.42927e7 −0.581866
\(167\) 8.87363e7 1.47433 0.737163 0.675715i \(-0.236165\pi\)
0.737163 + 0.675715i \(0.236165\pi\)
\(168\) −1.69504e7 −0.275802
\(169\) 4.44035e7 0.707643
\(170\) 5.25216e7 0.819911
\(171\) 2.18126e7 0.333596
\(172\) −2.70761e7 −0.405730
\(173\) −2.76291e7 −0.405700 −0.202850 0.979210i \(-0.565020\pi\)
−0.202850 + 0.979210i \(0.565020\pi\)
\(174\) −2.38496e7 −0.343209
\(175\) −3.17799e7 −0.448249
\(176\) −2.19471e7 −0.303447
\(177\) 5.54523e6 0.0751646
\(178\) −3.76925e7 −0.500939
\(179\) 3.52061e7 0.458809 0.229405 0.973331i \(-0.426322\pi\)
0.229405 + 0.973331i \(0.426322\pi\)
\(180\) 1.50492e7 0.192336
\(181\) 1.78616e7 0.223895 0.111948 0.993714i \(-0.464291\pi\)
0.111948 + 0.993714i \(0.464291\pi\)
\(182\) 1.01540e8 1.24849
\(183\) −4.38688e6 −0.0529147
\(184\) 5.06595e6 0.0599513
\(185\) −1.27153e8 −1.47647
\(186\) −4.02730e7 −0.458900
\(187\) −1.09058e8 −1.21959
\(188\) 2.44945e7 0.268854
\(189\) −2.41344e7 −0.260028
\(190\) 7.72106e7 0.816656
\(191\) 1.81334e8 1.88305 0.941525 0.336943i \(-0.109393\pi\)
0.941525 + 0.336943i \(0.109393\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −1.03710e8 −1.03842 −0.519208 0.854648i \(-0.673773\pi\)
−0.519208 + 0.854648i \(0.673773\pi\)
\(194\) 1.01350e8 0.996595
\(195\) −9.01511e7 −0.870663
\(196\) 4.35146e7 0.412799
\(197\) −3.56650e7 −0.332361 −0.166180 0.986095i \(-0.553143\pi\)
−0.166180 + 0.986095i \(0.553143\pi\)
\(198\) −3.12489e7 −0.286093
\(199\) 1.41306e8 1.27108 0.635542 0.772066i \(-0.280777\pi\)
0.635542 + 0.772066i \(0.280777\pi\)
\(200\) 1.32702e7 0.117293
\(201\) 4.15363e7 0.360779
\(202\) 8.49878e7 0.725483
\(203\) 1.35386e8 1.13589
\(204\) 3.51710e7 0.290054
\(205\) −1.53083e8 −1.24105
\(206\) 5.03227e7 0.401078
\(207\) 7.21304e6 0.0565226
\(208\) −4.23994e7 −0.326692
\(209\) −1.60324e8 −1.21475
\(210\) −8.54292e7 −0.636560
\(211\) −7.74278e7 −0.567425 −0.283712 0.958909i \(-0.591566\pi\)
−0.283712 + 0.958909i \(0.591566\pi\)
\(212\) −8.11715e6 −0.0585097
\(213\) −1.59956e8 −1.13416
\(214\) −6.82208e7 −0.475848
\(215\) −1.36463e8 −0.936438
\(216\) 1.00777e7 0.0680414
\(217\) 2.28615e8 1.51879
\(218\) −5.80056e7 −0.379203
\(219\) −6.69651e7 −0.430818
\(220\) −1.10613e8 −0.700367
\(221\) −2.10689e8 −1.31301
\(222\) −8.51475e7 −0.522320
\(223\) −1.17128e8 −0.707286 −0.353643 0.935381i \(-0.615057\pi\)
−0.353643 + 0.935381i \(0.615057\pi\)
\(224\) −4.01787e7 −0.238851
\(225\) 1.88945e7 0.110585
\(226\) −1.88702e8 −1.08742
\(227\) 1.23610e8 0.701396 0.350698 0.936489i \(-0.385944\pi\)
0.350698 + 0.936489i \(0.385944\pi\)
\(228\) 5.17039e7 0.288903
\(229\) −6.88315e6 −0.0378759 −0.0189380 0.999821i \(-0.506029\pi\)
−0.0189380 + 0.999821i \(0.506029\pi\)
\(230\) 2.55322e7 0.138370
\(231\) 1.77389e8 0.946859
\(232\) −5.65323e7 −0.297227
\(233\) −3.18340e8 −1.64872 −0.824358 0.566068i \(-0.808464\pi\)
−0.824358 + 0.566068i \(0.808464\pi\)
\(234\) −6.03695e7 −0.308008
\(235\) 1.23451e8 0.620523
\(236\) 1.31443e7 0.0650945
\(237\) 1.06247e8 0.518440
\(238\) −1.99653e8 −0.959969
\(239\) 9.13265e7 0.432717 0.216359 0.976314i \(-0.430582\pi\)
0.216359 + 0.976314i \(0.430582\pi\)
\(240\) 3.56723e7 0.166568
\(241\) 3.27837e8 1.50868 0.754342 0.656482i \(-0.227956\pi\)
0.754342 + 0.656482i \(0.227956\pi\)
\(242\) 7.37838e7 0.334663
\(243\) 1.43489e7 0.0641500
\(244\) −1.03985e7 −0.0458255
\(245\) 2.19312e8 0.952754
\(246\) −1.02512e8 −0.439037
\(247\) −3.09728e8 −1.30780
\(248\) −9.54618e7 −0.397419
\(249\) −1.15738e8 −0.475091
\(250\) −1.34717e8 −0.545297
\(251\) −4.81502e8 −1.92194 −0.960970 0.276652i \(-0.910775\pi\)
−0.960970 + 0.276652i \(0.910775\pi\)
\(252\) −5.72075e7 −0.225191
\(253\) −5.30162e7 −0.205820
\(254\) 1.01683e8 0.389342
\(255\) 1.77260e8 0.669455
\(256\) 1.67772e7 0.0625000
\(257\) −6.16203e7 −0.226442 −0.113221 0.993570i \(-0.536117\pi\)
−0.113221 + 0.993570i \(0.536117\pi\)
\(258\) −9.13819e7 −0.331277
\(259\) 4.83353e8 1.72868
\(260\) −2.13692e8 −0.754016
\(261\) −8.04923e7 −0.280229
\(262\) −1.01543e7 −0.0348817
\(263\) 1.61880e8 0.548718 0.274359 0.961627i \(-0.411534\pi\)
0.274359 + 0.961627i \(0.411534\pi\)
\(264\) −7.40715e7 −0.247764
\(265\) −4.09101e7 −0.135042
\(266\) −2.93505e8 −0.956158
\(267\) −1.27212e8 −0.409015
\(268\) 9.84563e7 0.312444
\(269\) 1.19284e8 0.373638 0.186819 0.982394i \(-0.440182\pi\)
0.186819 + 0.982394i \(0.440182\pi\)
\(270\) 5.07912e7 0.157042
\(271\) 4.65783e8 1.42164 0.710822 0.703372i \(-0.248323\pi\)
0.710822 + 0.703372i \(0.248323\pi\)
\(272\) 8.33683e7 0.251194
\(273\) 3.42697e8 1.01939
\(274\) −8.37814e7 −0.246048
\(275\) −1.38875e8 −0.402680
\(276\) 1.70976e7 0.0489500
\(277\) 2.02046e7 0.0571176 0.0285588 0.999592i \(-0.490908\pi\)
0.0285588 + 0.999592i \(0.490908\pi\)
\(278\) −5.88798e7 −0.164365
\(279\) −1.35921e8 −0.374690
\(280\) −2.02499e8 −0.551277
\(281\) −4.07044e8 −1.09438 −0.547192 0.837007i \(-0.684303\pi\)
−0.547192 + 0.837007i \(0.684303\pi\)
\(282\) 8.26689e7 0.219518
\(283\) −2.22689e8 −0.584045 −0.292022 0.956411i \(-0.594328\pi\)
−0.292022 + 0.956411i \(0.594328\pi\)
\(284\) −3.79156e8 −0.982208
\(285\) 2.60586e8 0.666797
\(286\) 4.43719e8 1.12157
\(287\) 5.81924e8 1.45305
\(288\) 2.38879e7 0.0589256
\(289\) 3.92990e6 0.00957722
\(290\) −2.84921e8 −0.686011
\(291\) 3.42057e8 0.813716
\(292\) −1.58732e8 −0.373099
\(293\) −4.37450e8 −1.01600 −0.507998 0.861358i \(-0.669614\pi\)
−0.507998 + 0.861358i \(0.669614\pi\)
\(294\) 1.46862e8 0.337049
\(295\) 6.62465e7 0.150240
\(296\) −2.01831e8 −0.452342
\(297\) −1.05465e8 −0.233594
\(298\) −4.19065e7 −0.0917327
\(299\) −1.02422e8 −0.221586
\(300\) 4.47869e7 0.0957693
\(301\) 5.18743e8 1.09640
\(302\) 2.15402e8 0.450013
\(303\) 2.86834e8 0.592354
\(304\) 1.22557e8 0.250197
\(305\) −5.24081e7 −0.105767
\(306\) 1.18702e8 0.236828
\(307\) 8.33109e8 1.64330 0.821651 0.569990i \(-0.193053\pi\)
0.821651 + 0.569990i \(0.193053\pi\)
\(308\) 4.20478e8 0.820004
\(309\) 1.69839e8 0.327478
\(310\) −4.81124e8 −0.917257
\(311\) −3.29748e8 −0.621614 −0.310807 0.950473i \(-0.600599\pi\)
−0.310807 + 0.950473i \(0.600599\pi\)
\(312\) −1.43098e8 −0.266743
\(313\) 2.41712e8 0.445547 0.222773 0.974870i \(-0.428489\pi\)
0.222773 + 0.974870i \(0.428489\pi\)
\(314\) −3.70598e7 −0.0675538
\(315\) −2.88324e8 −0.519749
\(316\) 2.51845e8 0.448982
\(317\) 5.09923e8 0.899078 0.449539 0.893261i \(-0.351588\pi\)
0.449539 + 0.893261i \(0.351588\pi\)
\(318\) −2.73954e7 −0.0477730
\(319\) 5.91622e8 1.02042
\(320\) 8.45565e7 0.144252
\(321\) −2.30245e8 −0.388529
\(322\) −9.70570e7 −0.162006
\(323\) 6.09005e8 1.00557
\(324\) 3.40122e7 0.0555556
\(325\) −2.68292e8 −0.433526
\(326\) 3.98708e7 0.0637372
\(327\) −1.95769e8 −0.309618
\(328\) −2.42991e8 −0.380217
\(329\) −4.69282e8 −0.726522
\(330\) −3.73318e8 −0.571847
\(331\) −8.63343e8 −1.30854 −0.654268 0.756263i \(-0.727023\pi\)
−0.654268 + 0.756263i \(0.727023\pi\)
\(332\) −2.74341e8 −0.411441
\(333\) −2.87373e8 −0.426472
\(334\) 7.09890e8 1.04251
\(335\) 4.96216e8 0.721130
\(336\) −1.35603e8 −0.195021
\(337\) −7.48761e8 −1.06571 −0.532854 0.846207i \(-0.678881\pi\)
−0.532854 + 0.846207i \(0.678881\pi\)
\(338\) 3.55228e8 0.500379
\(339\) −6.36868e8 −0.887872
\(340\) 4.20173e8 0.579765
\(341\) 9.99028e8 1.36439
\(342\) 1.74501e8 0.235888
\(343\) 1.76110e8 0.235643
\(344\) −2.16609e8 −0.286894
\(345\) 8.61712e7 0.112978
\(346\) −2.21033e8 −0.286873
\(347\) 4.18560e7 0.0537780 0.0268890 0.999638i \(-0.491440\pi\)
0.0268890 + 0.999638i \(0.491440\pi\)
\(348\) −1.90797e8 −0.242685
\(349\) 1.12818e9 1.42065 0.710327 0.703872i \(-0.248547\pi\)
0.710327 + 0.703872i \(0.248547\pi\)
\(350\) −2.54239e8 −0.316960
\(351\) −2.03747e8 −0.251488
\(352\) −1.75577e8 −0.214570
\(353\) 6.63700e8 0.803083 0.401542 0.915841i \(-0.368475\pi\)
0.401542 + 0.915841i \(0.368475\pi\)
\(354\) 4.43619e7 0.0531494
\(355\) −1.91093e9 −2.26697
\(356\) −3.01540e8 −0.354217
\(357\) −6.73830e8 −0.783812
\(358\) 2.81649e8 0.324427
\(359\) 1.27993e9 1.46001 0.730005 0.683441i \(-0.239517\pi\)
0.730005 + 0.683441i \(0.239517\pi\)
\(360\) 1.20394e8 0.136002
\(361\) 1.40957e6 0.00157693
\(362\) 1.42893e8 0.158318
\(363\) 2.49020e8 0.273251
\(364\) 8.12318e8 0.882818
\(365\) −8.00003e8 −0.861126
\(366\) −3.50950e7 −0.0374164
\(367\) −7.43725e8 −0.785382 −0.392691 0.919670i \(-0.628456\pi\)
−0.392691 + 0.919670i \(0.628456\pi\)
\(368\) 4.05276e7 0.0423920
\(369\) −3.45977e8 −0.358472
\(370\) −1.01722e9 −1.04402
\(371\) 1.55514e8 0.158111
\(372\) −3.22184e8 −0.324491
\(373\) −7.18516e8 −0.716895 −0.358448 0.933550i \(-0.616694\pi\)
−0.358448 + 0.933550i \(0.616694\pi\)
\(374\) −8.72466e8 −0.862379
\(375\) −4.54671e8 −0.445233
\(376\) 1.95956e8 0.190108
\(377\) 1.14295e9 1.09858
\(378\) −1.93075e8 −0.183868
\(379\) −2.01382e9 −1.90013 −0.950066 0.312050i \(-0.898984\pi\)
−0.950066 + 0.312050i \(0.898984\pi\)
\(380\) 6.17685e8 0.577463
\(381\) 3.43181e8 0.317897
\(382\) 1.45067e9 1.33152
\(383\) 1.25647e9 1.14276 0.571380 0.820686i \(-0.306408\pi\)
0.571380 + 0.820686i \(0.306408\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 2.11919e9 1.89260
\(386\) −8.29683e8 −0.734271
\(387\) −3.08414e8 −0.270486
\(388\) 8.10802e8 0.704699
\(389\) 8.28001e8 0.713193 0.356597 0.934258i \(-0.383937\pi\)
0.356597 + 0.934258i \(0.383937\pi\)
\(390\) −7.21209e8 −0.615652
\(391\) 2.01387e8 0.170378
\(392\) 3.48117e8 0.291893
\(393\) −3.42709e7 −0.0284808
\(394\) −2.85320e8 −0.235015
\(395\) 1.26929e9 1.03627
\(396\) −2.49991e8 −0.202298
\(397\) 8.08343e8 0.648379 0.324190 0.945992i \(-0.394908\pi\)
0.324190 + 0.945992i \(0.394908\pi\)
\(398\) 1.13045e9 0.898792
\(399\) −9.90579e8 −0.780700
\(400\) 1.06161e8 0.0829387
\(401\) 2.78432e8 0.215632 0.107816 0.994171i \(-0.465614\pi\)
0.107816 + 0.994171i \(0.465614\pi\)
\(402\) 3.32290e8 0.255109
\(403\) 1.93001e9 1.46890
\(404\) 6.79903e8 0.512994
\(405\) 1.71420e8 0.128224
\(406\) 1.08309e9 0.803196
\(407\) 2.11220e9 1.55294
\(408\) 2.81368e8 0.205099
\(409\) 4.22442e8 0.305306 0.152653 0.988280i \(-0.451218\pi\)
0.152653 + 0.988280i \(0.451218\pi\)
\(410\) −1.22467e9 −0.877554
\(411\) −2.82762e8 −0.200898
\(412\) 4.02581e8 0.283605
\(413\) −2.51827e8 −0.175904
\(414\) 5.77043e7 0.0399675
\(415\) −1.38267e9 −0.949620
\(416\) −3.39196e8 −0.231006
\(417\) −1.98719e8 −0.134204
\(418\) −1.28259e9 −0.858955
\(419\) 2.24349e8 0.148996 0.0744982 0.997221i \(-0.476265\pi\)
0.0744982 + 0.997221i \(0.476265\pi\)
\(420\) −6.83434e8 −0.450116
\(421\) 1.77468e9 1.15913 0.579565 0.814926i \(-0.303222\pi\)
0.579565 + 0.814926i \(0.303222\pi\)
\(422\) −6.19422e8 −0.401230
\(423\) 2.79008e8 0.179236
\(424\) −6.49372e7 −0.0413726
\(425\) 5.27531e8 0.333340
\(426\) −1.27965e9 −0.801969
\(427\) 1.99222e8 0.123834
\(428\) −5.45766e8 −0.336476
\(429\) 1.49755e9 0.915759
\(430\) −1.09170e9 −0.662161
\(431\) 3.12710e9 1.88136 0.940679 0.339297i \(-0.110189\pi\)
0.940679 + 0.339297i \(0.110189\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 4.54242e8 0.268893 0.134446 0.990921i \(-0.457074\pi\)
0.134446 + 0.990921i \(0.457074\pi\)
\(434\) 1.82892e9 1.07394
\(435\) −9.61607e8 −0.560126
\(436\) −4.64045e8 −0.268137
\(437\) 2.96054e8 0.169702
\(438\) −5.35720e8 −0.304634
\(439\) 6.63917e8 0.374531 0.187266 0.982309i \(-0.440037\pi\)
0.187266 + 0.982309i \(0.440037\pi\)
\(440\) −8.84901e8 −0.495234
\(441\) 4.95658e8 0.275199
\(442\) −1.68551e9 −0.928439
\(443\) −3.29437e9 −1.80036 −0.900180 0.435517i \(-0.856565\pi\)
−0.900180 + 0.435517i \(0.856565\pi\)
\(444\) −6.81180e8 −0.369336
\(445\) −1.51975e9 −0.817546
\(446\) −9.37026e8 −0.500126
\(447\) −1.41434e8 −0.0748994
\(448\) −3.21429e8 −0.168893
\(449\) 1.96325e9 1.02356 0.511779 0.859117i \(-0.328987\pi\)
0.511779 + 0.859117i \(0.328987\pi\)
\(450\) 1.51156e8 0.0781953
\(451\) 2.54295e9 1.30533
\(452\) −1.50961e9 −0.768920
\(453\) 7.26981e8 0.367434
\(454\) 9.88880e8 0.495962
\(455\) 4.09405e9 2.03757
\(456\) 4.13631e8 0.204285
\(457\) 1.93867e9 0.950160 0.475080 0.879943i \(-0.342419\pi\)
0.475080 + 0.879943i \(0.342419\pi\)
\(458\) −5.50652e7 −0.0267823
\(459\) 4.00620e8 0.193369
\(460\) 2.04258e8 0.0978421
\(461\) −1.69548e9 −0.806009 −0.403005 0.915198i \(-0.632034\pi\)
−0.403005 + 0.915198i \(0.632034\pi\)
\(462\) 1.41911e9 0.669530
\(463\) −5.76366e8 −0.269876 −0.134938 0.990854i \(-0.543084\pi\)
−0.134938 + 0.990854i \(0.543084\pi\)
\(464\) −4.52259e8 −0.210172
\(465\) −1.62379e9 −0.748937
\(466\) −2.54672e9 −1.16582
\(467\) −1.46753e9 −0.666771 −0.333385 0.942791i \(-0.608191\pi\)
−0.333385 + 0.942791i \(0.608191\pi\)
\(468\) −4.82956e8 −0.217795
\(469\) −1.88629e9 −0.844315
\(470\) 9.87610e8 0.438776
\(471\) −1.25077e8 −0.0551574
\(472\) 1.05154e8 0.0460287
\(473\) 2.26686e9 0.984941
\(474\) 8.49978e8 0.366592
\(475\) 7.75509e8 0.332016
\(476\) −1.59723e9 −0.678801
\(477\) −9.24594e7 −0.0390065
\(478\) 7.30612e8 0.305977
\(479\) 3.66657e9 1.52435 0.762177 0.647369i \(-0.224131\pi\)
0.762177 + 0.647369i \(0.224131\pi\)
\(480\) 2.85378e8 0.117781
\(481\) 4.08055e9 1.67190
\(482\) 2.62270e9 1.06680
\(483\) −3.27567e8 −0.132277
\(484\) 5.90270e8 0.236642
\(485\) 4.08641e9 1.62647
\(486\) 1.14791e8 0.0453609
\(487\) −3.75489e9 −1.47315 −0.736573 0.676358i \(-0.763557\pi\)
−0.736573 + 0.676358i \(0.763557\pi\)
\(488\) −8.31882e7 −0.0324035
\(489\) 1.34564e8 0.0520412
\(490\) 1.75449e9 0.673699
\(491\) 4.00343e9 1.52633 0.763163 0.646206i \(-0.223645\pi\)
0.763163 + 0.646206i \(0.223645\pi\)
\(492\) −8.20095e8 −0.310446
\(493\) −2.24734e9 −0.844702
\(494\) −2.47782e9 −0.924753
\(495\) −1.25995e9 −0.466911
\(496\) −7.63695e8 −0.281018
\(497\) 7.26412e9 2.65421
\(498\) −9.25902e8 −0.335940
\(499\) −3.91834e9 −1.41173 −0.705863 0.708348i \(-0.749441\pi\)
−0.705863 + 0.708348i \(0.749441\pi\)
\(500\) −1.07774e9 −0.385583
\(501\) 2.39588e9 0.851202
\(502\) −3.85201e9 −1.35902
\(503\) −3.88903e9 −1.36255 −0.681276 0.732027i \(-0.738575\pi\)
−0.681276 + 0.732027i \(0.738575\pi\)
\(504\) −4.57660e8 −0.159234
\(505\) 3.42668e9 1.18401
\(506\) −4.24130e8 −0.145536
\(507\) 1.19890e9 0.408558
\(508\) 8.13466e8 0.275306
\(509\) −3.25729e9 −1.09482 −0.547411 0.836864i \(-0.684387\pi\)
−0.547411 + 0.836864i \(0.684387\pi\)
\(510\) 1.41808e9 0.473376
\(511\) 3.04110e9 1.00822
\(512\) 1.34218e8 0.0441942
\(513\) 5.88940e8 0.192602
\(514\) −4.92962e8 −0.160119
\(515\) 2.02899e9 0.654569
\(516\) −7.31055e8 −0.234248
\(517\) −2.05072e9 −0.652664
\(518\) 3.86682e9 1.22236
\(519\) −7.45985e8 −0.234231
\(520\) −1.70953e9 −0.533170
\(521\) −2.44271e9 −0.756727 −0.378363 0.925657i \(-0.623513\pi\)
−0.378363 + 0.925657i \(0.623513\pi\)
\(522\) −6.43939e8 −0.198152
\(523\) 4.46703e9 1.36541 0.682705 0.730695i \(-0.260804\pi\)
0.682705 + 0.730695i \(0.260804\pi\)
\(524\) −8.12348e7 −0.0246651
\(525\) −8.58058e8 −0.258797
\(526\) 1.29504e9 0.388002
\(527\) −3.79490e9 −1.12944
\(528\) −5.92572e8 −0.175195
\(529\) −3.30693e9 −0.971247
\(530\) −3.27281e8 −0.0954894
\(531\) 1.49721e8 0.0433963
\(532\) −2.34804e9 −0.676106
\(533\) 4.91270e9 1.40532
\(534\) −1.01770e9 −0.289217
\(535\) −2.75064e9 −0.776597
\(536\) 7.87650e8 0.220931
\(537\) 9.50564e8 0.264894
\(538\) 9.54276e8 0.264202
\(539\) −3.64311e9 −1.00210
\(540\) 4.06330e8 0.111045
\(541\) −2.69368e9 −0.731402 −0.365701 0.930732i \(-0.619171\pi\)
−0.365701 + 0.930732i \(0.619171\pi\)
\(542\) 3.72626e9 1.00525
\(543\) 4.82263e8 0.129266
\(544\) 6.66946e8 0.177621
\(545\) −2.33877e9 −0.618870
\(546\) 2.74157e9 0.720818
\(547\) 5.19290e9 1.35661 0.678304 0.734782i \(-0.262715\pi\)
0.678304 + 0.734782i \(0.262715\pi\)
\(548\) −6.70251e8 −0.173982
\(549\) −1.18446e8 −0.0305503
\(550\) −1.11100e9 −0.284738
\(551\) −3.30375e9 −0.841349
\(552\) 1.36781e8 0.0346129
\(553\) −4.82502e9 −1.21328
\(554\) 1.61636e8 0.0403882
\(555\) −3.43312e9 −0.852440
\(556\) −4.71039e8 −0.116224
\(557\) 1.72219e9 0.422269 0.211134 0.977457i \(-0.432284\pi\)
0.211134 + 0.977457i \(0.432284\pi\)
\(558\) −1.08737e9 −0.264946
\(559\) 4.37932e9 1.06039
\(560\) −1.61999e9 −0.389812
\(561\) −2.94457e9 −0.704129
\(562\) −3.25635e9 −0.773846
\(563\) 1.40294e9 0.331329 0.165664 0.986182i \(-0.447023\pi\)
0.165664 + 0.986182i \(0.447023\pi\)
\(564\) 6.61351e8 0.155223
\(565\) −7.60839e9 −1.77469
\(566\) −1.78151e9 −0.412982
\(567\) −6.51630e8 −0.150127
\(568\) −3.03324e9 −0.694526
\(569\) −4.92673e9 −1.12115 −0.560577 0.828102i \(-0.689421\pi\)
−0.560577 + 0.828102i \(0.689421\pi\)
\(570\) 2.08469e9 0.471497
\(571\) 4.78084e8 0.107468 0.0537338 0.998555i \(-0.482888\pi\)
0.0537338 + 0.998555i \(0.482888\pi\)
\(572\) 3.54975e9 0.793071
\(573\) 4.89601e9 1.08718
\(574\) 4.65539e9 1.02746
\(575\) 2.56447e8 0.0562549
\(576\) 1.91103e8 0.0416667
\(577\) −3.05951e9 −0.663034 −0.331517 0.943449i \(-0.607560\pi\)
−0.331517 + 0.943449i \(0.607560\pi\)
\(578\) 3.14392e7 0.00677212
\(579\) −2.80018e9 −0.599530
\(580\) −2.27937e9 −0.485083
\(581\) 5.25602e9 1.11184
\(582\) 2.73646e9 0.575384
\(583\) 6.79581e8 0.142037
\(584\) −1.26986e9 −0.263821
\(585\) −2.43408e9 −0.502677
\(586\) −3.49960e9 −0.718417
\(587\) 6.44785e8 0.131578 0.0657888 0.997834i \(-0.479044\pi\)
0.0657888 + 0.997834i \(0.479044\pi\)
\(588\) 1.17489e9 0.238330
\(589\) −5.57878e9 −1.12496
\(590\) 5.29972e8 0.106236
\(591\) −9.62954e8 −0.191889
\(592\) −1.61465e9 −0.319854
\(593\) 6.38051e9 1.25650 0.628252 0.778010i \(-0.283771\pi\)
0.628252 + 0.778010i \(0.283771\pi\)
\(594\) −8.43721e8 −0.165176
\(595\) −8.04996e9 −1.56669
\(596\) −3.35252e8 −0.0648648
\(597\) 3.81526e9 0.733861
\(598\) −8.19372e8 −0.156685
\(599\) −2.32167e9 −0.441373 −0.220687 0.975345i \(-0.570830\pi\)
−0.220687 + 0.975345i \(0.570830\pi\)
\(600\) 3.58295e8 0.0677191
\(601\) −5.96090e9 −1.12009 −0.560043 0.828464i \(-0.689215\pi\)
−0.560043 + 0.828464i \(0.689215\pi\)
\(602\) 4.14994e9 0.775272
\(603\) 1.12148e9 0.208296
\(604\) 1.72321e9 0.318207
\(605\) 2.97494e9 0.546178
\(606\) 2.29467e9 0.418858
\(607\) −7.59139e9 −1.37772 −0.688860 0.724895i \(-0.741888\pi\)
−0.688860 + 0.724895i \(0.741888\pi\)
\(608\) 9.80460e8 0.176916
\(609\) 3.65541e9 0.655807
\(610\) −4.19265e8 −0.0747884
\(611\) −3.96177e9 −0.702659
\(612\) 9.49617e8 0.167463
\(613\) −6.19336e9 −1.08596 −0.542981 0.839745i \(-0.682704\pi\)
−0.542981 + 0.839745i \(0.682704\pi\)
\(614\) 6.66487e9 1.16199
\(615\) −4.13324e9 −0.716520
\(616\) 3.36382e9 0.579830
\(617\) 9.17538e9 1.57263 0.786314 0.617827i \(-0.211987\pi\)
0.786314 + 0.617827i \(0.211987\pi\)
\(618\) 1.35871e9 0.231562
\(619\) 1.05282e9 0.178417 0.0892084 0.996013i \(-0.471566\pi\)
0.0892084 + 0.996013i \(0.471566\pi\)
\(620\) −3.84899e9 −0.648599
\(621\) 1.94752e8 0.0326334
\(622\) −2.63798e9 −0.439548
\(623\) 5.77711e9 0.957200
\(624\) −1.14479e9 −0.188616
\(625\) −7.45663e9 −1.22169
\(626\) 1.93370e9 0.315049
\(627\) −4.32874e9 −0.701334
\(628\) −2.96479e8 −0.0477677
\(629\) −8.02342e9 −1.28553
\(630\) −2.30659e9 −0.367518
\(631\) −7.36983e9 −1.16776 −0.583882 0.811839i \(-0.698467\pi\)
−0.583882 + 0.811839i \(0.698467\pi\)
\(632\) 2.01476e9 0.317478
\(633\) −2.09055e9 −0.327603
\(634\) 4.07939e9 0.635744
\(635\) 4.09984e9 0.635417
\(636\) −2.19163e8 −0.0337806
\(637\) −7.03810e9 −1.07887
\(638\) 4.73298e9 0.721543
\(639\) −4.31882e9 −0.654805
\(640\) 6.76452e8 0.102002
\(641\) 1.57655e9 0.236431 0.118216 0.992988i \(-0.462283\pi\)
0.118216 + 0.992988i \(0.462283\pi\)
\(642\) −1.84196e9 −0.274731
\(643\) −8.94286e9 −1.32659 −0.663297 0.748356i \(-0.730843\pi\)
−0.663297 + 0.748356i \(0.730843\pi\)
\(644\) −7.76456e8 −0.114556
\(645\) −3.68449e9 −0.540652
\(646\) 4.87204e9 0.711045
\(647\) 1.24615e10 1.80886 0.904432 0.426617i \(-0.140295\pi\)
0.904432 + 0.426617i \(0.140295\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −1.10046e9 −0.158022
\(650\) −2.14633e9 −0.306549
\(651\) 6.17262e9 0.876871
\(652\) 3.18966e8 0.0450690
\(653\) −2.44833e9 −0.344091 −0.172045 0.985089i \(-0.555038\pi\)
−0.172045 + 0.985089i \(0.555038\pi\)
\(654\) −1.56615e9 −0.218933
\(655\) −4.09420e8 −0.0569278
\(656\) −1.94393e9 −0.268854
\(657\) −1.80806e9 −0.248733
\(658\) −3.75426e9 −0.513728
\(659\) −1.99119e8 −0.0271028 −0.0135514 0.999908i \(-0.504314\pi\)
−0.0135514 + 0.999908i \(0.504314\pi\)
\(660\) −2.98654e9 −0.404357
\(661\) 1.28846e10 1.73526 0.867630 0.497211i \(-0.165642\pi\)
0.867630 + 0.497211i \(0.165642\pi\)
\(662\) −6.90674e9 −0.925274
\(663\) −5.68860e9 −0.758067
\(664\) −2.19473e9 −0.290933
\(665\) −1.18340e10 −1.56048
\(666\) −2.29898e9 −0.301562
\(667\) −1.09249e9 −0.142553
\(668\) 5.67912e9 0.737163
\(669\) −3.16246e9 −0.408352
\(670\) 3.96973e9 0.509916
\(671\) 8.70581e8 0.111245
\(672\) −1.08482e9 −0.137901
\(673\) −3.87189e8 −0.0489633 −0.0244816 0.999700i \(-0.507794\pi\)
−0.0244816 + 0.999700i \(0.507794\pi\)
\(674\) −5.99009e9 −0.753570
\(675\) 5.10151e8 0.0638462
\(676\) 2.84183e9 0.353821
\(677\) 4.34195e8 0.0537805 0.0268902 0.999638i \(-0.491440\pi\)
0.0268902 + 0.999638i \(0.491440\pi\)
\(678\) −5.09494e9 −0.627820
\(679\) −1.55339e10 −1.90430
\(680\) 3.36138e9 0.409956
\(681\) 3.33747e9 0.404951
\(682\) 7.99222e9 0.964766
\(683\) −1.36648e10 −1.64109 −0.820545 0.571582i \(-0.806330\pi\)
−0.820545 + 0.571582i \(0.806330\pi\)
\(684\) 1.39601e9 0.166798
\(685\) −3.37804e9 −0.401557
\(686\) 1.40888e9 0.166625
\(687\) −1.85845e8 −0.0218677
\(688\) −1.73287e9 −0.202865
\(689\) 1.31288e9 0.152917
\(690\) 6.89369e8 0.0798877
\(691\) 7.90278e9 0.911185 0.455593 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(692\) −1.76826e9 −0.202850
\(693\) 4.78951e9 0.546669
\(694\) 3.34848e8 0.0380268
\(695\) −2.37402e9 −0.268248
\(696\) −1.52637e9 −0.171604
\(697\) −9.65965e9 −1.08055
\(698\) 9.02541e9 1.00455
\(699\) −8.59519e9 −0.951887
\(700\) −2.03391e9 −0.224125
\(701\) −1.58623e9 −0.173922 −0.0869610 0.996212i \(-0.527716\pi\)
−0.0869610 + 0.996212i \(0.527716\pi\)
\(702\) −1.62998e9 −0.177829
\(703\) −1.17950e10 −1.28043
\(704\) −1.40462e9 −0.151724
\(705\) 3.33319e9 0.358259
\(706\) 5.30960e9 0.567866
\(707\) −1.30260e10 −1.38626
\(708\) 3.54895e8 0.0375823
\(709\) −1.59906e10 −1.68501 −0.842505 0.538688i \(-0.818920\pi\)
−0.842505 + 0.538688i \(0.818920\pi\)
\(710\) −1.52874e10 −1.60299
\(711\) 2.86868e9 0.299321
\(712\) −2.41232e9 −0.250470
\(713\) −1.84481e9 −0.190606
\(714\) −5.39064e9 −0.554239
\(715\) 1.78906e10 1.83043
\(716\) 2.25319e9 0.229405
\(717\) 2.46581e9 0.249829
\(718\) 1.02395e10 1.03238
\(719\) −1.65500e10 −1.66053 −0.830264 0.557371i \(-0.811810\pi\)
−0.830264 + 0.557371i \(0.811810\pi\)
\(720\) 9.63151e8 0.0961681
\(721\) −7.71293e9 −0.766383
\(722\) 1.12766e7 0.00111506
\(723\) 8.85160e9 0.871039
\(724\) 1.14314e9 0.111948
\(725\) −2.86176e9 −0.278902
\(726\) 1.99216e9 0.193218
\(727\) 1.62431e10 1.56783 0.783915 0.620869i \(-0.213220\pi\)
0.783915 + 0.620869i \(0.213220\pi\)
\(728\) 6.49854e9 0.624246
\(729\) 3.87420e8 0.0370370
\(730\) −6.40002e9 −0.608908
\(731\) −8.61088e9 −0.815336
\(732\) −2.80760e8 −0.0264574
\(733\) 1.38947e10 1.30312 0.651561 0.758596i \(-0.274115\pi\)
0.651561 + 0.758596i \(0.274115\pi\)
\(734\) −5.94980e9 −0.555349
\(735\) 5.92142e9 0.550073
\(736\) 3.24221e8 0.0299756
\(737\) −8.24292e9 −0.758482
\(738\) −2.76782e9 −0.253478
\(739\) −8.35461e9 −0.761501 −0.380750 0.924678i \(-0.624334\pi\)
−0.380750 + 0.924678i \(0.624334\pi\)
\(740\) −8.13777e9 −0.738235
\(741\) −8.36265e9 −0.755058
\(742\) 1.24411e9 0.111801
\(743\) 1.96752e10 1.75978 0.879889 0.475179i \(-0.157617\pi\)
0.879889 + 0.475179i \(0.157617\pi\)
\(744\) −2.57747e9 −0.229450
\(745\) −1.68966e9 −0.149710
\(746\) −5.74813e9 −0.506922
\(747\) −3.12492e9 −0.274294
\(748\) −6.97973e9 −0.609794
\(749\) 1.04562e10 0.909256
\(750\) −3.63736e9 −0.314827
\(751\) −7.23538e9 −0.623335 −0.311668 0.950191i \(-0.600888\pi\)
−0.311668 + 0.950191i \(0.600888\pi\)
\(752\) 1.56765e9 0.134427
\(753\) −1.30006e10 −1.10963
\(754\) 9.14360e9 0.776815
\(755\) 8.68493e9 0.734433
\(756\) −1.54460e9 −0.130014
\(757\) 5.60569e9 0.469671 0.234836 0.972035i \(-0.424545\pi\)
0.234836 + 0.972035i \(0.424545\pi\)
\(758\) −1.61106e10 −1.34360
\(759\) −1.43144e9 −0.118830
\(760\) 4.94148e9 0.408328
\(761\) 1.97229e10 1.62228 0.811139 0.584853i \(-0.198848\pi\)
0.811139 + 0.584853i \(0.198848\pi\)
\(762\) 2.74545e9 0.224787
\(763\) 8.89049e9 0.724586
\(764\) 1.16054e10 0.941525
\(765\) 4.78603e9 0.386510
\(766\) 1.00517e10 0.808053
\(767\) −2.12597e9 −0.170127
\(768\) 4.52985e8 0.0360844
\(769\) −4.89997e8 −0.0388554 −0.0194277 0.999811i \(-0.506184\pi\)
−0.0194277 + 0.999811i \(0.506184\pi\)
\(770\) 1.69535e10 1.33827
\(771\) −1.66375e9 −0.130737
\(772\) −6.63746e9 −0.519208
\(773\) 2.14868e10 1.67319 0.836593 0.547825i \(-0.184544\pi\)
0.836593 + 0.547825i \(0.184544\pi\)
\(774\) −2.46731e9 −0.191263
\(775\) −4.83244e9 −0.372916
\(776\) 6.48642e9 0.498298
\(777\) 1.30505e10 0.998054
\(778\) 6.62401e9 0.504304
\(779\) −1.42004e10 −1.07626
\(780\) −5.76967e9 −0.435331
\(781\) 3.17435e10 2.38439
\(782\) 1.61110e9 0.120475
\(783\) −2.17329e9 −0.161790
\(784\) 2.78493e9 0.206399
\(785\) −1.49424e9 −0.110250
\(786\) −2.74167e8 −0.0201389
\(787\) 1.39897e9 0.102305 0.0511524 0.998691i \(-0.483711\pi\)
0.0511524 + 0.998691i \(0.483711\pi\)
\(788\) −2.28256e9 −0.166180
\(789\) 4.37077e9 0.316803
\(790\) 1.01543e10 0.732751
\(791\) 2.89222e10 2.07785
\(792\) −1.99993e9 −0.143046
\(793\) 1.68187e9 0.119767
\(794\) 6.46675e9 0.458473
\(795\) −1.10457e9 −0.0779668
\(796\) 9.04357e9 0.635542
\(797\) −1.79920e10 −1.25885 −0.629425 0.777061i \(-0.716710\pi\)
−0.629425 + 0.777061i \(0.716710\pi\)
\(798\) −7.92463e9 −0.552038
\(799\) 7.78986e9 0.540276
\(800\) 8.49292e8 0.0586465
\(801\) −3.43473e9 −0.236145
\(802\) 2.22745e9 0.152475
\(803\) 1.32893e10 0.905728
\(804\) 2.65832e9 0.180389
\(805\) −3.91331e9 −0.264398
\(806\) 1.54401e10 1.03867
\(807\) 3.22068e9 0.215720
\(808\) 5.43922e9 0.362741
\(809\) −2.09906e10 −1.39382 −0.696909 0.717160i \(-0.745442\pi\)
−0.696909 + 0.717160i \(0.745442\pi\)
\(810\) 1.37136e9 0.0906681
\(811\) −9.94254e8 −0.0654522 −0.0327261 0.999464i \(-0.510419\pi\)
−0.0327261 + 0.999464i \(0.510419\pi\)
\(812\) 8.66468e9 0.567945
\(813\) 1.25761e10 0.820786
\(814\) 1.68976e10 1.09810
\(815\) 1.60758e9 0.104021
\(816\) 2.25094e9 0.145027
\(817\) −1.26586e10 −0.812099
\(818\) 3.37954e9 0.215884
\(819\) 9.25281e9 0.588545
\(820\) −9.79732e9 −0.620524
\(821\) −1.54313e10 −0.973197 −0.486598 0.873626i \(-0.661762\pi\)
−0.486598 + 0.873626i \(0.661762\pi\)
\(822\) −2.26210e9 −0.142056
\(823\) −9.46903e9 −0.592115 −0.296057 0.955170i \(-0.595672\pi\)
−0.296057 + 0.955170i \(0.595672\pi\)
\(824\) 3.22065e9 0.200539
\(825\) −3.74963e9 −0.232488
\(826\) −2.01461e9 −0.124383
\(827\) 2.36563e10 1.45438 0.727190 0.686436i \(-0.240826\pi\)
0.727190 + 0.686436i \(0.240826\pi\)
\(828\) 4.61635e8 0.0282613
\(829\) −1.69387e10 −1.03262 −0.516308 0.856403i \(-0.672694\pi\)
−0.516308 + 0.856403i \(0.672694\pi\)
\(830\) −1.10614e10 −0.671483
\(831\) 5.45523e8 0.0329769
\(832\) −2.71356e9 −0.163346
\(833\) 1.38387e10 0.829542
\(834\) −1.58976e9 −0.0948962
\(835\) 2.86225e10 1.70140
\(836\) −1.02607e10 −0.607373
\(837\) −3.66987e9 −0.216328
\(838\) 1.79479e9 0.105356
\(839\) −2.22357e10 −1.29982 −0.649912 0.760010i \(-0.725194\pi\)
−0.649912 + 0.760010i \(0.725194\pi\)
\(840\) −5.46747e9 −0.318280
\(841\) −5.05847e9 −0.293247
\(842\) 1.41974e10 0.819629
\(843\) −1.09902e10 −0.631842
\(844\) −4.95538e9 −0.283712
\(845\) 1.43227e10 0.816632
\(846\) 2.23206e9 0.126739
\(847\) −1.13088e10 −0.639477
\(848\) −5.19498e8 −0.0292549
\(849\) −6.01260e9 −0.337198
\(850\) 4.22025e9 0.235707
\(851\) −3.90040e9 −0.216948
\(852\) −1.02372e10 −0.567078
\(853\) 1.84144e10 1.01587 0.507933 0.861397i \(-0.330410\pi\)
0.507933 + 0.861397i \(0.330410\pi\)
\(854\) 1.59378e9 0.0875638
\(855\) 7.03582e9 0.384975
\(856\) −4.36613e9 −0.237924
\(857\) −2.00335e10 −1.08723 −0.543617 0.839333i \(-0.682946\pi\)
−0.543617 + 0.839333i \(0.682946\pi\)
\(858\) 1.19804e10 0.647539
\(859\) −1.92592e10 −1.03672 −0.518360 0.855162i \(-0.673457\pi\)
−0.518360 + 0.855162i \(0.673457\pi\)
\(860\) −8.73360e9 −0.468219
\(861\) 1.57119e10 0.838916
\(862\) 2.50168e10 1.33032
\(863\) −2.49169e10 −1.31964 −0.659820 0.751424i \(-0.729368\pi\)
−0.659820 + 0.751424i \(0.729368\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −8.91197e9 −0.468185
\(866\) 3.63393e9 0.190136
\(867\) 1.06107e8 0.00552941
\(868\) 1.46314e10 0.759393
\(869\) −2.10849e10 −1.08994
\(870\) −7.69286e9 −0.396069
\(871\) −1.59244e10 −0.816583
\(872\) −3.71236e9 −0.189602
\(873\) 9.23554e9 0.469799
\(874\) 2.36843e9 0.119997
\(875\) 2.06480e10 1.04196
\(876\) −4.28576e9 −0.215409
\(877\) −7.87530e9 −0.394247 −0.197124 0.980379i \(-0.563160\pi\)
−0.197124 + 0.980379i \(0.563160\pi\)
\(878\) 5.31134e9 0.264834
\(879\) −1.18111e10 −0.586585
\(880\) −7.07921e9 −0.350183
\(881\) 1.25524e9 0.0618460 0.0309230 0.999522i \(-0.490155\pi\)
0.0309230 + 0.999522i \(0.490155\pi\)
\(882\) 3.96527e9 0.194595
\(883\) −1.18621e10 −0.579826 −0.289913 0.957053i \(-0.593626\pi\)
−0.289913 + 0.957053i \(0.593626\pi\)
\(884\) −1.34841e10 −0.656505
\(885\) 1.78866e9 0.0867412
\(886\) −2.63550e10 −1.27305
\(887\) 3.32958e10 1.60198 0.800988 0.598680i \(-0.204308\pi\)
0.800988 + 0.598680i \(0.204308\pi\)
\(888\) −5.44944e9 −0.261160
\(889\) −1.55849e10 −0.743959
\(890\) −1.21580e10 −0.578092
\(891\) −2.84756e9 −0.134865
\(892\) −7.49621e9 −0.353643
\(893\) 1.14517e10 0.538131
\(894\) −1.13147e9 −0.0529619
\(895\) 1.13560e10 0.529474
\(896\) −2.57144e9 −0.119426
\(897\) −2.76538e9 −0.127933
\(898\) 1.57060e10 0.723765
\(899\) 2.05867e10 0.944991
\(900\) 1.20925e9 0.0552924
\(901\) −2.58146e9 −0.117579
\(902\) 2.03436e10 0.923007
\(903\) 1.40061e10 0.633007
\(904\) −1.20769e10 −0.543708
\(905\) 5.76139e9 0.258379
\(906\) 5.81585e9 0.259815
\(907\) 3.41140e10 1.51813 0.759063 0.651017i \(-0.225658\pi\)
0.759063 + 0.651017i \(0.225658\pi\)
\(908\) 7.91104e9 0.350698
\(909\) 7.74452e9 0.341996
\(910\) 3.27524e10 1.44078
\(911\) −9.72303e9 −0.426076 −0.213038 0.977044i \(-0.568336\pi\)
−0.213038 + 0.977044i \(0.568336\pi\)
\(912\) 3.30905e9 0.144451
\(913\) 2.29683e10 0.998806
\(914\) 1.55093e10 0.671865
\(915\) −1.41502e9 −0.0610645
\(916\) −4.40522e8 −0.0189380
\(917\) 1.55635e9 0.0666523
\(918\) 3.20496e9 0.136733
\(919\) −2.18969e10 −0.930632 −0.465316 0.885145i \(-0.654059\pi\)
−0.465316 + 0.885145i \(0.654059\pi\)
\(920\) 1.63406e9 0.0691848
\(921\) 2.24939e10 0.948761
\(922\) −1.35639e10 −0.569935
\(923\) 6.13250e10 2.56703
\(924\) 1.13529e10 0.473429
\(925\) −1.02170e10 −0.424453
\(926\) −4.61093e9 −0.190831
\(927\) 4.58565e9 0.189070
\(928\) −3.61807e9 −0.148614
\(929\) 2.68885e9 0.110030 0.0550152 0.998486i \(-0.482479\pi\)
0.0550152 + 0.998486i \(0.482479\pi\)
\(930\) −1.29903e10 −0.529579
\(931\) 2.03439e10 0.826249
\(932\) −2.03738e10 −0.824358
\(933\) −8.90320e9 −0.358889
\(934\) −1.17402e10 −0.471478
\(935\) −3.51776e10 −1.40742
\(936\) −3.86365e9 −0.154004
\(937\) −3.69169e10 −1.46601 −0.733004 0.680224i \(-0.761882\pi\)
−0.733004 + 0.680224i \(0.761882\pi\)
\(938\) −1.50903e10 −0.597021
\(939\) 6.52623e9 0.257237
\(940\) 7.90088e9 0.310262
\(941\) −2.24060e10 −0.876597 −0.438298 0.898830i \(-0.644419\pi\)
−0.438298 + 0.898830i \(0.644419\pi\)
\(942\) −1.00062e9 −0.0390022
\(943\) −4.69582e9 −0.182356
\(944\) 8.41232e8 0.0325472
\(945\) −7.78474e9 −0.300077
\(946\) 1.81349e10 0.696458
\(947\) 3.09991e10 1.18611 0.593053 0.805163i \(-0.297922\pi\)
0.593053 + 0.805163i \(0.297922\pi\)
\(948\) 6.79982e9 0.259220
\(949\) 2.56735e10 0.975109
\(950\) 6.20407e9 0.234771
\(951\) 1.37679e10 0.519083
\(952\) −1.27778e10 −0.479985
\(953\) −2.38381e10 −0.892169 −0.446084 0.894991i \(-0.647182\pi\)
−0.446084 + 0.894991i \(0.647182\pi\)
\(954\) −7.39676e8 −0.0275818
\(955\) 5.84906e10 2.17307
\(956\) 5.84489e9 0.216359
\(957\) 1.59738e10 0.589137
\(958\) 2.93326e10 1.07788
\(959\) 1.28411e10 0.470152
\(960\) 2.28303e9 0.0832840
\(961\) 7.25057e9 0.263536
\(962\) 3.26444e10 1.18221
\(963\) −6.21662e9 −0.224317
\(964\) 2.09816e10 0.754342
\(965\) −3.34525e10 −1.19835
\(966\) −2.62054e9 −0.0935342
\(967\) 2.26067e10 0.803980 0.401990 0.915644i \(-0.368319\pi\)
0.401990 + 0.915644i \(0.368319\pi\)
\(968\) 4.72216e9 0.167331
\(969\) 1.64431e10 0.580566
\(970\) 3.26913e10 1.15009
\(971\) 1.48779e10 0.521524 0.260762 0.965403i \(-0.416026\pi\)
0.260762 + 0.965403i \(0.416026\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 9.02448e9 0.314071
\(974\) −3.00391e10 −1.04167
\(975\) −7.24388e9 −0.250297
\(976\) −6.65505e8 −0.0229127
\(977\) −5.80549e10 −1.99163 −0.995814 0.0914033i \(-0.970865\pi\)
−0.995814 + 0.0914033i \(0.970865\pi\)
\(978\) 1.07651e9 0.0367987
\(979\) 2.52454e10 0.859891
\(980\) 1.40360e10 0.476377
\(981\) −5.28576e9 −0.178758
\(982\) 3.20275e10 1.07928
\(983\) 3.18931e10 1.07093 0.535463 0.844559i \(-0.320137\pi\)
0.535463 + 0.844559i \(0.320137\pi\)
\(984\) −6.56076e9 −0.219519
\(985\) −1.15040e10 −0.383550
\(986\) −1.79787e10 −0.597295
\(987\) −1.26706e10 −0.419458
\(988\) −1.98226e10 −0.653899
\(989\) −4.18598e9 −0.137597
\(990\) −1.00796e10 −0.330156
\(991\) 3.64453e10 1.18955 0.594777 0.803891i \(-0.297240\pi\)
0.594777 + 0.803891i \(0.297240\pi\)
\(992\) −6.10956e9 −0.198710
\(993\) −2.33103e10 −0.755483
\(994\) 5.81130e10 1.87681
\(995\) 4.55793e10 1.46685
\(996\) −7.40721e9 −0.237546
\(997\) −6.66993e9 −0.213151 −0.106576 0.994305i \(-0.533989\pi\)
−0.106576 + 0.994305i \(0.533989\pi\)
\(998\) −3.13467e10 −0.998241
\(999\) −7.75907e9 −0.246224
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 354.8.a.b.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.8.a.b.1.5 5 1.1 even 1 trivial