Properties

Label 725.6.a.b.1.5
Level $725$
Weight $6$
Character 725.1
Self dual yes
Analytic conductor $116.278$
Analytic rank $0$
Dimension $7$
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] = [725,6,Mod(1,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(116.278269364\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.90786\) of defining polynomial
Character \(\chi\) \(=\) 725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.90786 q^{2} -29.3989 q^{3} -16.7287 q^{4} -114.887 q^{6} +138.793 q^{7} -190.425 q^{8} +621.298 q^{9} +O(q^{10})\) \(q+3.90786 q^{2} -29.3989 q^{3} -16.7287 q^{4} -114.887 q^{6} +138.793 q^{7} -190.425 q^{8} +621.298 q^{9} +557.286 q^{11} +491.805 q^{12} +41.1854 q^{13} +542.382 q^{14} -208.835 q^{16} +1643.99 q^{17} +2427.94 q^{18} +258.134 q^{19} -4080.36 q^{21} +2177.79 q^{22} +2828.17 q^{23} +5598.28 q^{24} +160.946 q^{26} -11121.6 q^{27} -2321.82 q^{28} +841.000 q^{29} -5980.62 q^{31} +5277.49 q^{32} -16383.6 q^{33} +6424.48 q^{34} -10393.5 q^{36} +3327.01 q^{37} +1008.75 q^{38} -1210.81 q^{39} -3895.80 q^{41} -15945.5 q^{42} +3589.20 q^{43} -9322.64 q^{44} +11052.1 q^{46} +7502.91 q^{47} +6139.52 q^{48} +2456.45 q^{49} -48331.6 q^{51} -688.976 q^{52} -9015.58 q^{53} -43461.5 q^{54} -26429.6 q^{56} -7588.86 q^{57} +3286.51 q^{58} +39101.0 q^{59} +3951.11 q^{61} -23371.4 q^{62} +86231.7 q^{63} +27306.4 q^{64} -64024.8 q^{66} -62985.3 q^{67} -27501.8 q^{68} -83145.1 q^{69} +7121.60 q^{71} -118310. q^{72} +13910.9 q^{73} +13001.5 q^{74} -4318.23 q^{76} +77347.2 q^{77} -4731.66 q^{78} -37581.7 q^{79} +175987. q^{81} -15224.2 q^{82} +74905.7 q^{83} +68259.0 q^{84} +14026.1 q^{86} -24724.5 q^{87} -106121. q^{88} +102613. q^{89} +5716.23 q^{91} -47311.5 q^{92} +175824. q^{93} +29320.3 q^{94} -155153. q^{96} +25501.5 q^{97} +9599.44 q^{98} +346241. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 4 q^{2} - 26 q^{3} + 154 q^{4} + 22 q^{6} - 184 q^{7} - 942 q^{8} + 1005 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 4 q^{2} - 26 q^{3} + 154 q^{4} + 22 q^{6} - 184 q^{7} - 942 q^{8} + 1005 q^{9} + 1106 q^{11} - 214 q^{12} - 408 q^{13} - 2008 q^{14} + 242 q^{16} + 874 q^{17} + 5598 q^{18} + 4288 q^{19} - 4200 q^{21} + 6114 q^{22} + 4532 q^{23} - 4318 q^{24} - 19806 q^{26} - 5942 q^{27} + 496 q^{28} + 5887 q^{29} + 7794 q^{31} - 7898 q^{32} - 34410 q^{33} + 20840 q^{34} - 572 q^{36} - 5086 q^{37} - 23732 q^{38} + 33394 q^{39} + 19826 q^{41} + 55440 q^{42} - 19498 q^{43} - 6074 q^{44} - 12404 q^{46} - 14278 q^{47} + 16406 q^{48} + 38431 q^{49} + 23892 q^{51} + 34302 q^{52} + 58644 q^{53} - 31194 q^{54} - 79560 q^{56} + 88540 q^{57} - 3364 q^{58} + 12888 q^{59} + 102866 q^{61} + 42654 q^{62} + 88632 q^{63} - 10170 q^{64} + 7710 q^{66} - 102996 q^{67} - 85100 q^{68} - 107244 q^{69} - 51596 q^{71} - 135568 q^{72} + 17566 q^{73} + 12132 q^{74} + 360740 q^{76} + 94104 q^{77} - 46386 q^{78} + 212058 q^{79} - 128285 q^{81} - 201924 q^{82} + 122928 q^{83} - 12328 q^{84} - 63290 q^{86} - 21866 q^{87} - 136666 q^{88} - 66510 q^{89} + 194368 q^{91} + 110108 q^{92} + 474274 q^{93} + 438926 q^{94} - 117018 q^{96} + 118182 q^{97} + 29132 q^{98} + 300668 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.90786 0.690818 0.345409 0.938452i \(-0.387740\pi\)
0.345409 + 0.938452i \(0.387740\pi\)
\(3\) −29.3989 −1.88594 −0.942972 0.332873i \(-0.891982\pi\)
−0.942972 + 0.332873i \(0.891982\pi\)
\(4\) −16.7287 −0.522771
\(5\) 0 0
\(6\) −114.887 −1.30284
\(7\) 138.793 1.07059 0.535293 0.844666i \(-0.320201\pi\)
0.535293 + 0.844666i \(0.320201\pi\)
\(8\) −190.425 −1.05196
\(9\) 621.298 2.55678
\(10\) 0 0
\(11\) 557.286 1.38866 0.694330 0.719656i \(-0.255701\pi\)
0.694330 + 0.719656i \(0.255701\pi\)
\(12\) 491.805 0.985916
\(13\) 41.1854 0.0675903 0.0337952 0.999429i \(-0.489241\pi\)
0.0337952 + 0.999429i \(0.489241\pi\)
\(14\) 542.382 0.739580
\(15\) 0 0
\(16\) −208.835 −0.203940
\(17\) 1643.99 1.37968 0.689838 0.723964i \(-0.257682\pi\)
0.689838 + 0.723964i \(0.257682\pi\)
\(18\) 2427.94 1.76627
\(19\) 258.134 0.164044 0.0820221 0.996631i \(-0.473862\pi\)
0.0820221 + 0.996631i \(0.473862\pi\)
\(20\) 0 0
\(21\) −4080.36 −2.01907
\(22\) 2177.79 0.959311
\(23\) 2828.17 1.11477 0.557385 0.830254i \(-0.311805\pi\)
0.557385 + 0.830254i \(0.311805\pi\)
\(24\) 5598.28 1.98393
\(25\) 0 0
\(26\) 160.946 0.0466926
\(27\) −11121.6 −2.93600
\(28\) −2321.82 −0.559671
\(29\) 841.000 0.185695
\(30\) 0 0
\(31\) −5980.62 −1.11774 −0.558872 0.829254i \(-0.688766\pi\)
−0.558872 + 0.829254i \(0.688766\pi\)
\(32\) 5277.49 0.911072
\(33\) −16383.6 −2.61894
\(34\) 6424.48 0.953105
\(35\) 0 0
\(36\) −10393.5 −1.33661
\(37\) 3327.01 0.399531 0.199765 0.979844i \(-0.435982\pi\)
0.199765 + 0.979844i \(0.435982\pi\)
\(38\) 1008.75 0.113325
\(39\) −1210.81 −0.127472
\(40\) 0 0
\(41\) −3895.80 −0.361940 −0.180970 0.983489i \(-0.557924\pi\)
−0.180970 + 0.983489i \(0.557924\pi\)
\(42\) −15945.5 −1.39481
\(43\) 3589.20 0.296023 0.148012 0.988986i \(-0.452713\pi\)
0.148012 + 0.988986i \(0.452713\pi\)
\(44\) −9322.64 −0.725951
\(45\) 0 0
\(46\) 11052.1 0.770103
\(47\) 7502.91 0.495433 0.247717 0.968833i \(-0.420320\pi\)
0.247717 + 0.968833i \(0.420320\pi\)
\(48\) 6139.52 0.384619
\(49\) 2456.45 0.146156
\(50\) 0 0
\(51\) −48331.6 −2.60199
\(52\) −688.976 −0.0353342
\(53\) −9015.58 −0.440863 −0.220432 0.975402i \(-0.570747\pi\)
−0.220432 + 0.975402i \(0.570747\pi\)
\(54\) −43461.5 −2.02824
\(55\) 0 0
\(56\) −26429.6 −1.12621
\(57\) −7588.86 −0.309378
\(58\) 3286.51 0.128282
\(59\) 39101.0 1.46237 0.731187 0.682177i \(-0.238967\pi\)
0.731187 + 0.682177i \(0.238967\pi\)
\(60\) 0 0
\(61\) 3951.11 0.135955 0.0679773 0.997687i \(-0.478345\pi\)
0.0679773 + 0.997687i \(0.478345\pi\)
\(62\) −23371.4 −0.772157
\(63\) 86231.7 2.73726
\(64\) 27306.4 0.833325
\(65\) 0 0
\(66\) −64024.8 −1.80921
\(67\) −62985.3 −1.71416 −0.857082 0.515180i \(-0.827725\pi\)
−0.857082 + 0.515180i \(0.827725\pi\)
\(68\) −27501.8 −0.721254
\(69\) −83145.1 −2.10239
\(70\) 0 0
\(71\) 7121.60 0.167661 0.0838304 0.996480i \(-0.473285\pi\)
0.0838304 + 0.996480i \(0.473285\pi\)
\(72\) −118310. −2.68963
\(73\) 13910.9 0.305525 0.152763 0.988263i \(-0.451183\pi\)
0.152763 + 0.988263i \(0.451183\pi\)
\(74\) 13001.5 0.276003
\(75\) 0 0
\(76\) −4318.23 −0.0857575
\(77\) 77347.2 1.48668
\(78\) −4731.66 −0.0880596
\(79\) −37581.7 −0.677499 −0.338749 0.940877i \(-0.610004\pi\)
−0.338749 + 0.940877i \(0.610004\pi\)
\(80\) 0 0
\(81\) 175987. 2.98035
\(82\) −15224.2 −0.250035
\(83\) 74905.7 1.19349 0.596746 0.802430i \(-0.296460\pi\)
0.596746 + 0.802430i \(0.296460\pi\)
\(84\) 68259.0 1.05551
\(85\) 0 0
\(86\) 14026.1 0.204498
\(87\) −24724.5 −0.350211
\(88\) −106121. −1.46081
\(89\) 102613. 1.37318 0.686590 0.727045i \(-0.259107\pi\)
0.686590 + 0.727045i \(0.259107\pi\)
\(90\) 0 0
\(91\) 5716.23 0.0723613
\(92\) −47311.5 −0.582769
\(93\) 175824. 2.10800
\(94\) 29320.3 0.342254
\(95\) 0 0
\(96\) −155153. −1.71823
\(97\) 25501.5 0.275193 0.137596 0.990488i \(-0.456062\pi\)
0.137596 + 0.990488i \(0.456062\pi\)
\(98\) 9599.44 0.100967
\(99\) 346241. 3.55050
\(100\) 0 0
\(101\) −29590.5 −0.288635 −0.144317 0.989531i \(-0.546099\pi\)
−0.144317 + 0.989531i \(0.546099\pi\)
\(102\) −188873. −1.79750
\(103\) −170152. −1.58032 −0.790159 0.612902i \(-0.790002\pi\)
−0.790159 + 0.612902i \(0.790002\pi\)
\(104\) −7842.71 −0.0711021
\(105\) 0 0
\(106\) −35231.6 −0.304556
\(107\) 200423. 1.69234 0.846169 0.532915i \(-0.178903\pi\)
0.846169 + 0.532915i \(0.178903\pi\)
\(108\) 186049. 1.53486
\(109\) −80642.9 −0.650130 −0.325065 0.945692i \(-0.605386\pi\)
−0.325065 + 0.945692i \(0.605386\pi\)
\(110\) 0 0
\(111\) −97810.6 −0.753492
\(112\) −28984.7 −0.218336
\(113\) −17299.3 −0.127448 −0.0637240 0.997968i \(-0.520298\pi\)
−0.0637240 + 0.997968i \(0.520298\pi\)
\(114\) −29656.2 −0.213724
\(115\) 0 0
\(116\) −14068.8 −0.0970761
\(117\) 25588.4 0.172814
\(118\) 152801. 1.01023
\(119\) 228174. 1.47706
\(120\) 0 0
\(121\) 149516. 0.928378
\(122\) 15440.4 0.0939199
\(123\) 114532. 0.682599
\(124\) 100048. 0.584323
\(125\) 0 0
\(126\) 336981. 1.89095
\(127\) −191838. −1.05542 −0.527711 0.849424i \(-0.676949\pi\)
−0.527711 + 0.849424i \(0.676949\pi\)
\(128\) −62170.3 −0.335396
\(129\) −105519. −0.558283
\(130\) 0 0
\(131\) 112493. 0.572726 0.286363 0.958121i \(-0.407554\pi\)
0.286363 + 0.958121i \(0.407554\pi\)
\(132\) 274076. 1.36910
\(133\) 35827.1 0.175623
\(134\) −246138. −1.18417
\(135\) 0 0
\(136\) −313056. −1.45136
\(137\) 13514.2 0.0615162 0.0307581 0.999527i \(-0.490208\pi\)
0.0307581 + 0.999527i \(0.490208\pi\)
\(138\) −324919. −1.45237
\(139\) 315337. 1.38432 0.692161 0.721743i \(-0.256659\pi\)
0.692161 + 0.721743i \(0.256659\pi\)
\(140\) 0 0
\(141\) −220578. −0.934359
\(142\) 27830.2 0.115823
\(143\) 22952.0 0.0938600
\(144\) −129749. −0.521430
\(145\) 0 0
\(146\) 54361.7 0.211062
\(147\) −72216.9 −0.275642
\(148\) −55656.5 −0.208863
\(149\) −1500.22 −0.00553590 −0.00276795 0.999996i \(-0.500881\pi\)
−0.00276795 + 0.999996i \(0.500881\pi\)
\(150\) 0 0
\(151\) −141060. −0.503457 −0.251729 0.967798i \(-0.580999\pi\)
−0.251729 + 0.967798i \(0.580999\pi\)
\(152\) −49155.0 −0.172567
\(153\) 1.02141e6 3.52753
\(154\) 302262. 1.02703
\(155\) 0 0
\(156\) 20255.2 0.0666384
\(157\) 335858. 1.08744 0.543721 0.839266i \(-0.317015\pi\)
0.543721 + 0.839266i \(0.317015\pi\)
\(158\) −146864. −0.468028
\(159\) 265049. 0.831443
\(160\) 0 0
\(161\) 392529. 1.19346
\(162\) 687732. 2.05888
\(163\) −208039. −0.613303 −0.306651 0.951822i \(-0.599209\pi\)
−0.306651 + 0.951822i \(0.599209\pi\)
\(164\) 65171.5 0.189212
\(165\) 0 0
\(166\) 292721. 0.824486
\(167\) −235392. −0.653131 −0.326565 0.945175i \(-0.605891\pi\)
−0.326565 + 0.945175i \(0.605891\pi\)
\(168\) 777001. 2.12397
\(169\) −369597. −0.995432
\(170\) 0 0
\(171\) 160378. 0.419425
\(172\) −60042.4 −0.154752
\(173\) −126867. −0.322280 −0.161140 0.986932i \(-0.551517\pi\)
−0.161140 + 0.986932i \(0.551517\pi\)
\(174\) −96619.8 −0.241932
\(175\) 0 0
\(176\) −116381. −0.283203
\(177\) −1.14953e6 −2.75795
\(178\) 400997. 0.948617
\(179\) −582710. −1.35932 −0.679658 0.733529i \(-0.737872\pi\)
−0.679658 + 0.733529i \(0.737872\pi\)
\(180\) 0 0
\(181\) 469745. 1.06578 0.532888 0.846186i \(-0.321107\pi\)
0.532888 + 0.846186i \(0.321107\pi\)
\(182\) 22338.2 0.0499885
\(183\) −116158. −0.256403
\(184\) −538553. −1.17269
\(185\) 0 0
\(186\) 687095. 1.45624
\(187\) 916172. 1.91590
\(188\) −125514. −0.258998
\(189\) −1.54359e6 −3.14325
\(190\) 0 0
\(191\) 808559. 1.60372 0.801860 0.597512i \(-0.203844\pi\)
0.801860 + 0.597512i \(0.203844\pi\)
\(192\) −802779. −1.57160
\(193\) −397971. −0.769057 −0.384528 0.923113i \(-0.625636\pi\)
−0.384528 + 0.923113i \(0.625636\pi\)
\(194\) 99656.3 0.190108
\(195\) 0 0
\(196\) −41093.1 −0.0764062
\(197\) 821639. 1.50840 0.754198 0.656647i \(-0.228026\pi\)
0.754198 + 0.656647i \(0.228026\pi\)
\(198\) 1.35306e6 2.45275
\(199\) 304702. 0.545435 0.272718 0.962094i \(-0.412078\pi\)
0.272718 + 0.962094i \(0.412078\pi\)
\(200\) 0 0
\(201\) 1.85170e6 3.23282
\(202\) −115635. −0.199394
\(203\) 116725. 0.198803
\(204\) 808523. 1.36024
\(205\) 0 0
\(206\) −664930. −1.09171
\(207\) 1.75714e6 2.85023
\(208\) −8600.93 −0.0137844
\(209\) 143854. 0.227802
\(210\) 0 0
\(211\) 657135. 1.01613 0.508064 0.861319i \(-0.330361\pi\)
0.508064 + 0.861319i \(0.330361\pi\)
\(212\) 150819. 0.230470
\(213\) −209367. −0.316199
\(214\) 783222. 1.16910
\(215\) 0 0
\(216\) 2.11782e6 3.08855
\(217\) −830067. −1.19664
\(218\) −315141. −0.449121
\(219\) −408965. −0.576204
\(220\) 0 0
\(221\) 67708.4 0.0932528
\(222\) −382230. −0.520526
\(223\) 899606. 1.21141 0.605703 0.795690i \(-0.292892\pi\)
0.605703 + 0.795690i \(0.292892\pi\)
\(224\) 732478. 0.975381
\(225\) 0 0
\(226\) −67603.3 −0.0880434
\(227\) −660017. −0.850139 −0.425070 0.905161i \(-0.639750\pi\)
−0.425070 + 0.905161i \(0.639750\pi\)
\(228\) 126951. 0.161734
\(229\) −622380. −0.784273 −0.392136 0.919907i \(-0.628264\pi\)
−0.392136 + 0.919907i \(0.628264\pi\)
\(230\) 0 0
\(231\) −2.27393e6 −2.80380
\(232\) −160147. −0.195344
\(233\) −24182.1 −0.0291813 −0.0145906 0.999894i \(-0.504645\pi\)
−0.0145906 + 0.999894i \(0.504645\pi\)
\(234\) 99995.7 0.119383
\(235\) 0 0
\(236\) −654108. −0.764486
\(237\) 1.10486e6 1.27772
\(238\) 891672. 1.02038
\(239\) 201571. 0.228262 0.114131 0.993466i \(-0.463592\pi\)
0.114131 + 0.993466i \(0.463592\pi\)
\(240\) 0 0
\(241\) −1.01961e6 −1.13081 −0.565407 0.824812i \(-0.691281\pi\)
−0.565407 + 0.824812i \(0.691281\pi\)
\(242\) 584288. 0.641340
\(243\) −2.47129e6 −2.68478
\(244\) −66096.7 −0.0710731
\(245\) 0 0
\(246\) 447576. 0.471552
\(247\) 10631.3 0.0110878
\(248\) 1.13886e6 1.17582
\(249\) −2.20215e6 −2.25086
\(250\) 0 0
\(251\) 1.41854e6 1.42121 0.710604 0.703593i \(-0.248422\pi\)
0.710604 + 0.703593i \(0.248422\pi\)
\(252\) −1.44254e6 −1.43096
\(253\) 1.57610e6 1.54804
\(254\) −749676. −0.729104
\(255\) 0 0
\(256\) −1.11676e6 −1.06502
\(257\) 54526.1 0.0514958 0.0257479 0.999668i \(-0.491803\pi\)
0.0257479 + 0.999668i \(0.491803\pi\)
\(258\) −412351. −0.385672
\(259\) 461765. 0.427732
\(260\) 0 0
\(261\) 522512. 0.474783
\(262\) 439606. 0.395649
\(263\) −2.15754e6 −1.92340 −0.961702 0.274098i \(-0.911621\pi\)
−0.961702 + 0.274098i \(0.911621\pi\)
\(264\) 3.11984e6 2.75501
\(265\) 0 0
\(266\) 140007. 0.121324
\(267\) −3.01672e6 −2.58974
\(268\) 1.05366e6 0.896115
\(269\) −128043. −0.107888 −0.0539442 0.998544i \(-0.517179\pi\)
−0.0539442 + 0.998544i \(0.517179\pi\)
\(270\) 0 0
\(271\) 691071. 0.571610 0.285805 0.958288i \(-0.407739\pi\)
0.285805 + 0.958288i \(0.407739\pi\)
\(272\) −343322. −0.281371
\(273\) −168051. −0.136469
\(274\) 52811.6 0.0424965
\(275\) 0 0
\(276\) 1.39091e6 1.09907
\(277\) −589147. −0.461343 −0.230672 0.973032i \(-0.574092\pi\)
−0.230672 + 0.973032i \(0.574092\pi\)
\(278\) 1.23229e6 0.956315
\(279\) −3.71575e6 −2.85783
\(280\) 0 0
\(281\) −1.20853e6 −0.913044 −0.456522 0.889712i \(-0.650905\pi\)
−0.456522 + 0.889712i \(0.650905\pi\)
\(282\) −861985. −0.645472
\(283\) −1.88980e6 −1.40265 −0.701325 0.712842i \(-0.747408\pi\)
−0.701325 + 0.712842i \(0.747408\pi\)
\(284\) −119135. −0.0876481
\(285\) 0 0
\(286\) 89693.1 0.0648402
\(287\) −540709. −0.387489
\(288\) 3.27890e6 2.32941
\(289\) 1.28285e6 0.903506
\(290\) 0 0
\(291\) −749718. −0.518998
\(292\) −232710. −0.159720
\(293\) 2.43906e6 1.65979 0.829895 0.557920i \(-0.188400\pi\)
0.829895 + 0.557920i \(0.188400\pi\)
\(294\) −282213. −0.190419
\(295\) 0 0
\(296\) −633545. −0.420289
\(297\) −6.19789e6 −4.07711
\(298\) −5862.63 −0.00382430
\(299\) 116479. 0.0753477
\(300\) 0 0
\(301\) 498154. 0.316919
\(302\) −551243. −0.347797
\(303\) 869929. 0.544349
\(304\) −53907.2 −0.0334552
\(305\) 0 0
\(306\) 3.99152e6 2.43688
\(307\) 842448. 0.510149 0.255074 0.966921i \(-0.417900\pi\)
0.255074 + 0.966921i \(0.417900\pi\)
\(308\) −1.29392e6 −0.777194
\(309\) 5.00229e6 2.98039
\(310\) 0 0
\(311\) −1.78495e6 −1.04647 −0.523233 0.852190i \(-0.675274\pi\)
−0.523233 + 0.852190i \(0.675274\pi\)
\(312\) 230567. 0.134095
\(313\) −1.79477e6 −1.03550 −0.517749 0.855533i \(-0.673230\pi\)
−0.517749 + 0.855533i \(0.673230\pi\)
\(314\) 1.31248e6 0.751225
\(315\) 0 0
\(316\) 628691. 0.354177
\(317\) 1.00473e6 0.561564 0.280782 0.959772i \(-0.409406\pi\)
0.280782 + 0.959772i \(0.409406\pi\)
\(318\) 1.03577e6 0.574376
\(319\) 468677. 0.257868
\(320\) 0 0
\(321\) −5.89221e6 −3.19165
\(322\) 1.53395e6 0.824463
\(323\) 424369. 0.226328
\(324\) −2.94403e6 −1.55804
\(325\) 0 0
\(326\) −812985. −0.423680
\(327\) 2.37082e6 1.22611
\(328\) 741856. 0.380746
\(329\) 1.04135e6 0.530404
\(330\) 0 0
\(331\) 3.52272e6 1.76729 0.883646 0.468155i \(-0.155081\pi\)
0.883646 + 0.468155i \(0.155081\pi\)
\(332\) −1.25307e6 −0.623923
\(333\) 2.06707e6 1.02151
\(334\) −919878. −0.451195
\(335\) 0 0
\(336\) 852121. 0.411768
\(337\) −2.94676e6 −1.41342 −0.706708 0.707506i \(-0.749820\pi\)
−0.706708 + 0.707506i \(0.749820\pi\)
\(338\) −1.44433e6 −0.687662
\(339\) 508582. 0.240360
\(340\) 0 0
\(341\) −3.33291e6 −1.55217
\(342\) 626734. 0.289746
\(343\) −1.99175e6 −0.914114
\(344\) −683471. −0.311404
\(345\) 0 0
\(346\) −495777. −0.222637
\(347\) 2.31985e6 1.03428 0.517138 0.855902i \(-0.326997\pi\)
0.517138 + 0.855902i \(0.326997\pi\)
\(348\) 413608. 0.183080
\(349\) 1.13664e6 0.499526 0.249763 0.968307i \(-0.419647\pi\)
0.249763 + 0.968307i \(0.419647\pi\)
\(350\) 0 0
\(351\) −458046. −0.198445
\(352\) 2.94107e6 1.26517
\(353\) 942783. 0.402694 0.201347 0.979520i \(-0.435468\pi\)
0.201347 + 0.979520i \(0.435468\pi\)
\(354\) −4.49220e6 −1.90524
\(355\) 0 0
\(356\) −1.71658e6 −0.717858
\(357\) −6.70808e6 −2.78566
\(358\) −2.27715e6 −0.939039
\(359\) −2.78926e6 −1.14223 −0.571113 0.820871i \(-0.693488\pi\)
−0.571113 + 0.820871i \(0.693488\pi\)
\(360\) 0 0
\(361\) −2.40947e6 −0.973090
\(362\) 1.83570e6 0.736257
\(363\) −4.39562e6 −1.75087
\(364\) −95624.9 −0.0378284
\(365\) 0 0
\(366\) −453930. −0.177128
\(367\) 4.20179e6 1.62843 0.814215 0.580564i \(-0.197168\pi\)
0.814215 + 0.580564i \(0.197168\pi\)
\(368\) −590619. −0.227346
\(369\) −2.42045e6 −0.925403
\(370\) 0 0
\(371\) −1.25130e6 −0.471982
\(372\) −2.94130e6 −1.10200
\(373\) −4.83152e6 −1.79809 −0.899045 0.437856i \(-0.855738\pi\)
−0.899045 + 0.437856i \(0.855738\pi\)
\(374\) 3.58027e6 1.32354
\(375\) 0 0
\(376\) −1.42874e6 −0.521174
\(377\) 34636.9 0.0125512
\(378\) −6.03214e6 −2.17141
\(379\) 2.03959e6 0.729366 0.364683 0.931132i \(-0.381177\pi\)
0.364683 + 0.931132i \(0.381177\pi\)
\(380\) 0 0
\(381\) 5.63984e6 1.99046
\(382\) 3.15973e6 1.10788
\(383\) −923917. −0.321837 −0.160919 0.986968i \(-0.551446\pi\)
−0.160919 + 0.986968i \(0.551446\pi\)
\(384\) 1.82774e6 0.632538
\(385\) 0 0
\(386\) −1.55521e6 −0.531278
\(387\) 2.22996e6 0.756867
\(388\) −426606. −0.143863
\(389\) −3.39429e6 −1.13730 −0.568650 0.822580i \(-0.692534\pi\)
−0.568650 + 0.822580i \(0.692534\pi\)
\(390\) 0 0
\(391\) 4.64948e6 1.53802
\(392\) −467768. −0.153750
\(393\) −3.30717e6 −1.08013
\(394\) 3.21085e6 1.04203
\(395\) 0 0
\(396\) −5.79214e6 −1.85610
\(397\) 1.00161e6 0.318949 0.159475 0.987202i \(-0.449020\pi\)
0.159475 + 0.987202i \(0.449020\pi\)
\(398\) 1.19073e6 0.376796
\(399\) −1.05328e6 −0.331216
\(400\) 0 0
\(401\) 5.53448e6 1.71876 0.859381 0.511337i \(-0.170849\pi\)
0.859381 + 0.511337i \(0.170849\pi\)
\(402\) 7.23618e6 2.23329
\(403\) −246314. −0.0755486
\(404\) 495009. 0.150890
\(405\) 0 0
\(406\) 456144. 0.137337
\(407\) 1.85410e6 0.554812
\(408\) 9.20353e6 2.73718
\(409\) 1.93289e6 0.571345 0.285672 0.958327i \(-0.407783\pi\)
0.285672 + 0.958327i \(0.407783\pi\)
\(410\) 0 0
\(411\) −397304. −0.116016
\(412\) 2.84642e6 0.826144
\(413\) 5.42694e6 1.56560
\(414\) 6.86663e6 1.96899
\(415\) 0 0
\(416\) 217355. 0.0615796
\(417\) −9.27057e6 −2.61075
\(418\) 562161. 0.157369
\(419\) 1.80082e6 0.501112 0.250556 0.968102i \(-0.419387\pi\)
0.250556 + 0.968102i \(0.419387\pi\)
\(420\) 0 0
\(421\) −1.32654e6 −0.364766 −0.182383 0.983228i \(-0.558381\pi\)
−0.182383 + 0.983228i \(0.558381\pi\)
\(422\) 2.56799e6 0.701959
\(423\) 4.66154e6 1.26671
\(424\) 1.71679e6 0.463769
\(425\) 0 0
\(426\) −818178. −0.218436
\(427\) 548385. 0.145551
\(428\) −3.35280e6 −0.884705
\(429\) −674765. −0.177015
\(430\) 0 0
\(431\) 584408. 0.151539 0.0757693 0.997125i \(-0.475859\pi\)
0.0757693 + 0.997125i \(0.475859\pi\)
\(432\) 2.32257e6 0.598769
\(433\) 707379. 0.181315 0.0906573 0.995882i \(-0.471103\pi\)
0.0906573 + 0.995882i \(0.471103\pi\)
\(434\) −3.24378e6 −0.826661
\(435\) 0 0
\(436\) 1.34905e6 0.339869
\(437\) 730045. 0.182872
\(438\) −1.59818e6 −0.398052
\(439\) 5.06180e6 1.25356 0.626778 0.779198i \(-0.284373\pi\)
0.626778 + 0.779198i \(0.284373\pi\)
\(440\) 0 0
\(441\) 1.52619e6 0.373689
\(442\) 264595. 0.0644207
\(443\) 3.33111e6 0.806453 0.403227 0.915100i \(-0.367889\pi\)
0.403227 + 0.915100i \(0.367889\pi\)
\(444\) 1.63624e6 0.393904
\(445\) 0 0
\(446\) 3.51553e6 0.836861
\(447\) 44104.8 0.0104404
\(448\) 3.78993e6 0.892146
\(449\) 37802.9 0.00884930 0.00442465 0.999990i \(-0.498592\pi\)
0.00442465 + 0.999990i \(0.498592\pi\)
\(450\) 0 0
\(451\) −2.17107e6 −0.502612
\(452\) 289395. 0.0666261
\(453\) 4.14702e6 0.949491
\(454\) −2.57925e6 −0.587292
\(455\) 0 0
\(456\) 1.44511e6 0.325452
\(457\) 7.55109e6 1.69129 0.845647 0.533743i \(-0.179215\pi\)
0.845647 + 0.533743i \(0.179215\pi\)
\(458\) −2.43217e6 −0.541790
\(459\) −1.82838e7 −4.05073
\(460\) 0 0
\(461\) 1.97564e6 0.432967 0.216484 0.976286i \(-0.430541\pi\)
0.216484 + 0.976286i \(0.430541\pi\)
\(462\) −8.88618e6 −1.93691
\(463\) 8.35199e6 1.81066 0.905331 0.424707i \(-0.139623\pi\)
0.905331 + 0.424707i \(0.139623\pi\)
\(464\) −175630. −0.0378707
\(465\) 0 0
\(466\) −94500.2 −0.0201590
\(467\) 856159. 0.181661 0.0908306 0.995866i \(-0.471048\pi\)
0.0908306 + 0.995866i \(0.471048\pi\)
\(468\) −428060. −0.0903420
\(469\) −8.74191e6 −1.83516
\(470\) 0 0
\(471\) −9.87387e6 −2.05086
\(472\) −7.44580e6 −1.53835
\(473\) 2.00021e6 0.411076
\(474\) 4.31764e6 0.882675
\(475\) 0 0
\(476\) −3.81705e6 −0.772165
\(477\) −5.60136e6 −1.12719
\(478\) 787709. 0.157687
\(479\) 3.97374e6 0.791336 0.395668 0.918394i \(-0.370513\pi\)
0.395668 + 0.918394i \(0.370513\pi\)
\(480\) 0 0
\(481\) 137024. 0.0270044
\(482\) −3.98449e6 −0.781186
\(483\) −1.15399e7 −2.25080
\(484\) −2.50121e6 −0.485329
\(485\) 0 0
\(486\) −9.65745e6 −1.85469
\(487\) −2.73024e6 −0.521649 −0.260824 0.965386i \(-0.583994\pi\)
−0.260824 + 0.965386i \(0.583994\pi\)
\(488\) −752388. −0.143018
\(489\) 6.11611e6 1.15665
\(490\) 0 0
\(491\) 2.52027e6 0.471785 0.235892 0.971779i \(-0.424199\pi\)
0.235892 + 0.971779i \(0.424199\pi\)
\(492\) −1.91597e6 −0.356843
\(493\) 1.38260e6 0.256199
\(494\) 41545.7 0.00765965
\(495\) 0 0
\(496\) 1.24896e6 0.227953
\(497\) 988426. 0.179495
\(498\) −8.60568e6 −1.55493
\(499\) 8.63079e6 1.55167 0.775834 0.630937i \(-0.217329\pi\)
0.775834 + 0.630937i \(0.217329\pi\)
\(500\) 0 0
\(501\) 6.92027e6 1.23177
\(502\) 5.54345e6 0.981795
\(503\) 8.02315e6 1.41392 0.706960 0.707254i \(-0.250066\pi\)
0.706960 + 0.707254i \(0.250066\pi\)
\(504\) −1.64206e7 −2.87948
\(505\) 0 0
\(506\) 6.15916e6 1.06941
\(507\) 1.08658e7 1.87733
\(508\) 3.20920e6 0.551743
\(509\) 5.81312e6 0.994523 0.497261 0.867601i \(-0.334339\pi\)
0.497261 + 0.867601i \(0.334339\pi\)
\(510\) 0 0
\(511\) 1.93073e6 0.327092
\(512\) −2.37467e6 −0.400340
\(513\) −2.87085e6 −0.481634
\(514\) 213080. 0.0355742
\(515\) 0 0
\(516\) 1.76518e6 0.291854
\(517\) 4.18126e6 0.687988
\(518\) 1.80451e6 0.295485
\(519\) 3.72975e6 0.607801
\(520\) 0 0
\(521\) 381523. 0.0615781 0.0307891 0.999526i \(-0.490198\pi\)
0.0307891 + 0.999526i \(0.490198\pi\)
\(522\) 2.04190e6 0.327988
\(523\) −1.85496e6 −0.296538 −0.148269 0.988947i \(-0.547370\pi\)
−0.148269 + 0.988947i \(0.547370\pi\)
\(524\) −1.88186e6 −0.299404
\(525\) 0 0
\(526\) −8.43137e6 −1.32872
\(527\) −9.83209e6 −1.54212
\(528\) 3.42146e6 0.534106
\(529\) 1.56219e6 0.242713
\(530\) 0 0
\(531\) 2.42934e7 3.73897
\(532\) −599340. −0.0918108
\(533\) −160450. −0.0244637
\(534\) −1.17889e7 −1.78904
\(535\) 0 0
\(536\) 1.19940e7 1.80323
\(537\) 1.71311e7 2.56359
\(538\) −500373. −0.0745312
\(539\) 1.36894e6 0.202961
\(540\) 0 0
\(541\) −4.92237e6 −0.723071 −0.361535 0.932358i \(-0.617747\pi\)
−0.361535 + 0.932358i \(0.617747\pi\)
\(542\) 2.70061e6 0.394878
\(543\) −1.38100e7 −2.00999
\(544\) 8.67615e6 1.25698
\(545\) 0 0
\(546\) −656720. −0.0942755
\(547\) 1.35722e6 0.193947 0.0969735 0.995287i \(-0.469084\pi\)
0.0969735 + 0.995287i \(0.469084\pi\)
\(548\) −226075. −0.0321589
\(549\) 2.45481e6 0.347607
\(550\) 0 0
\(551\) 217090. 0.0304622
\(552\) 1.58329e7 2.21163
\(553\) −5.21607e6 −0.725321
\(554\) −2.30230e6 −0.318704
\(555\) 0 0
\(556\) −5.27516e6 −0.723684
\(557\) 629428. 0.0859623 0.0429811 0.999076i \(-0.486314\pi\)
0.0429811 + 0.999076i \(0.486314\pi\)
\(558\) −1.45206e7 −1.97424
\(559\) 147822. 0.0200083
\(560\) 0 0
\(561\) −2.69345e7 −3.61328
\(562\) −4.72276e6 −0.630747
\(563\) 706922. 0.0939941 0.0469971 0.998895i \(-0.485035\pi\)
0.0469971 + 0.998895i \(0.485035\pi\)
\(564\) 3.68997e6 0.488455
\(565\) 0 0
\(566\) −7.38506e6 −0.968975
\(567\) 2.44257e7 3.19073
\(568\) −1.35613e6 −0.176372
\(569\) 1.26921e7 1.64343 0.821716 0.569898i \(-0.193017\pi\)
0.821716 + 0.569898i \(0.193017\pi\)
\(570\) 0 0
\(571\) −8.06713e6 −1.03545 −0.517725 0.855547i \(-0.673221\pi\)
−0.517725 + 0.855547i \(0.673221\pi\)
\(572\) −383956. −0.0490673
\(573\) −2.37708e7 −3.02452
\(574\) −2.11301e6 −0.267684
\(575\) 0 0
\(576\) 1.69654e7 2.13063
\(577\) −5.23139e6 −0.654150 −0.327075 0.944998i \(-0.606063\pi\)
−0.327075 + 0.944998i \(0.606063\pi\)
\(578\) 5.01319e6 0.624158
\(579\) 1.16999e7 1.45040
\(580\) 0 0
\(581\) 1.03964e7 1.27774
\(582\) −2.92979e6 −0.358533
\(583\) −5.02425e6 −0.612209
\(584\) −2.64897e6 −0.321400
\(585\) 0 0
\(586\) 9.53149e6 1.14661
\(587\) 1.22738e7 1.47022 0.735111 0.677947i \(-0.237130\pi\)
0.735111 + 0.677947i \(0.237130\pi\)
\(588\) 1.20809e6 0.144098
\(589\) −1.54380e6 −0.183359
\(590\) 0 0
\(591\) −2.41553e7 −2.84475
\(592\) −694795. −0.0814803
\(593\) 1.56297e7 1.82521 0.912607 0.408838i \(-0.134066\pi\)
0.912607 + 0.408838i \(0.134066\pi\)
\(594\) −2.42205e7 −2.81654
\(595\) 0 0
\(596\) 25096.6 0.00289401
\(597\) −8.95793e6 −1.02866
\(598\) 455184. 0.0520515
\(599\) −909739. −0.103598 −0.0517988 0.998658i \(-0.516495\pi\)
−0.0517988 + 0.998658i \(0.516495\pi\)
\(600\) 0 0
\(601\) −1.28807e7 −1.45463 −0.727315 0.686303i \(-0.759232\pi\)
−0.727315 + 0.686303i \(0.759232\pi\)
\(602\) 1.94672e6 0.218933
\(603\) −3.91327e7 −4.38274
\(604\) 2.35975e6 0.263193
\(605\) 0 0
\(606\) 3.39956e6 0.376046
\(607\) −1.39560e7 −1.53741 −0.768706 0.639602i \(-0.779099\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(608\) 1.36230e6 0.149456
\(609\) −3.43159e6 −0.374931
\(610\) 0 0
\(611\) 309010. 0.0334865
\(612\) −1.70868e7 −1.84409
\(613\) 4.77628e6 0.513380 0.256690 0.966494i \(-0.417368\pi\)
0.256690 + 0.966494i \(0.417368\pi\)
\(614\) 3.29216e6 0.352420
\(615\) 0 0
\(616\) −1.47288e7 −1.56393
\(617\) 5.08319e6 0.537556 0.268778 0.963202i \(-0.413380\pi\)
0.268778 + 0.963202i \(0.413380\pi\)
\(618\) 1.95482e7 2.05891
\(619\) 2.45717e6 0.257755 0.128878 0.991660i \(-0.458863\pi\)
0.128878 + 0.991660i \(0.458863\pi\)
\(620\) 0 0
\(621\) −3.14537e7 −3.27297
\(622\) −6.97533e6 −0.722917
\(623\) 1.42420e7 1.47011
\(624\) 252858. 0.0259965
\(625\) 0 0
\(626\) −7.01372e6 −0.715340
\(627\) −4.22916e6 −0.429621
\(628\) −5.61845e6 −0.568483
\(629\) 5.46958e6 0.551223
\(630\) 0 0
\(631\) −1.11683e7 −1.11664 −0.558322 0.829624i \(-0.688555\pi\)
−0.558322 + 0.829624i \(0.688555\pi\)
\(632\) 7.15648e6 0.712700
\(633\) −1.93191e7 −1.91636
\(634\) 3.92632e6 0.387938
\(635\) 0 0
\(636\) −4.43391e6 −0.434654
\(637\) 101170. 0.00987874
\(638\) 1.83152e6 0.178140
\(639\) 4.42463e6 0.428672
\(640\) 0 0
\(641\) −9.24222e6 −0.888446 −0.444223 0.895916i \(-0.646520\pi\)
−0.444223 + 0.895916i \(0.646520\pi\)
\(642\) −2.30259e7 −2.20485
\(643\) −1.65662e6 −0.158014 −0.0790069 0.996874i \(-0.525175\pi\)
−0.0790069 + 0.996874i \(0.525175\pi\)
\(644\) −6.56649e6 −0.623905
\(645\) 0 0
\(646\) 1.65837e6 0.156351
\(647\) 1.26223e6 0.118544 0.0592720 0.998242i \(-0.481122\pi\)
0.0592720 + 0.998242i \(0.481122\pi\)
\(648\) −3.35122e7 −3.13521
\(649\) 2.17904e7 2.03074
\(650\) 0 0
\(651\) 2.44031e7 2.25680
\(652\) 3.48021e6 0.320617
\(653\) −1.64066e7 −1.50569 −0.752843 0.658200i \(-0.771318\pi\)
−0.752843 + 0.658200i \(0.771318\pi\)
\(654\) 9.26481e6 0.847017
\(655\) 0 0
\(656\) 813578. 0.0738141
\(657\) 8.64280e6 0.781162
\(658\) 4.06944e6 0.366413
\(659\) −2.46135e6 −0.220780 −0.110390 0.993888i \(-0.535210\pi\)
−0.110390 + 0.993888i \(0.535210\pi\)
\(660\) 0 0
\(661\) −1.29962e7 −1.15694 −0.578471 0.815703i \(-0.696350\pi\)
−0.578471 + 0.815703i \(0.696350\pi\)
\(662\) 1.37663e7 1.22088
\(663\) −1.99055e6 −0.175869
\(664\) −1.42639e7 −1.25550
\(665\) 0 0
\(666\) 8.07780e6 0.705679
\(667\) 2.37849e6 0.207008
\(668\) 3.93779e6 0.341438
\(669\) −2.64475e7 −2.28464
\(670\) 0 0
\(671\) 2.20189e6 0.188795
\(672\) −2.15341e7 −1.83951
\(673\) 1.00711e7 0.857119 0.428559 0.903514i \(-0.359021\pi\)
0.428559 + 0.903514i \(0.359021\pi\)
\(674\) −1.15155e7 −0.976413
\(675\) 0 0
\(676\) 6.18286e6 0.520382
\(677\) −9.82242e6 −0.823658 −0.411829 0.911261i \(-0.635110\pi\)
−0.411829 + 0.911261i \(0.635110\pi\)
\(678\) 1.98747e6 0.166045
\(679\) 3.53943e6 0.294618
\(680\) 0 0
\(681\) 1.94038e7 1.60331
\(682\) −1.30245e7 −1.07226
\(683\) 1.66547e7 1.36611 0.683053 0.730369i \(-0.260652\pi\)
0.683053 + 0.730369i \(0.260652\pi\)
\(684\) −2.68291e6 −0.219263
\(685\) 0 0
\(686\) −7.78349e6 −0.631486
\(687\) 1.82973e7 1.47909
\(688\) −749548. −0.0603710
\(689\) −371310. −0.0297981
\(690\) 0 0
\(691\) 6.35915e6 0.506645 0.253323 0.967382i \(-0.418477\pi\)
0.253323 + 0.967382i \(0.418477\pi\)
\(692\) 2.12231e6 0.168478
\(693\) 4.80557e7 3.80112
\(694\) 9.06565e6 0.714497
\(695\) 0 0
\(696\) 4.70816e6 0.368407
\(697\) −6.40466e6 −0.499360
\(698\) 4.44181e6 0.345081
\(699\) 710929. 0.0550342
\(700\) 0 0
\(701\) −167796. −0.0128969 −0.00644846 0.999979i \(-0.502053\pi\)
−0.00644846 + 0.999979i \(0.502053\pi\)
\(702\) −1.78998e6 −0.137090
\(703\) 858814. 0.0655406
\(704\) 1.52175e7 1.15721
\(705\) 0 0
\(706\) 3.68426e6 0.278188
\(707\) −4.10695e6 −0.309009
\(708\) 1.92301e7 1.44178
\(709\) −2.01558e7 −1.50586 −0.752932 0.658099i \(-0.771361\pi\)
−0.752932 + 0.658099i \(0.771361\pi\)
\(710\) 0 0
\(711\) −2.33494e7 −1.73222
\(712\) −1.95400e7 −1.44453
\(713\) −1.69142e7 −1.24603
\(714\) −2.62142e7 −1.92438
\(715\) 0 0
\(716\) 9.74797e6 0.710610
\(717\) −5.92597e6 −0.430488
\(718\) −1.09000e7 −0.789071
\(719\) −1.06393e7 −0.767523 −0.383761 0.923432i \(-0.625372\pi\)
−0.383761 + 0.923432i \(0.625372\pi\)
\(720\) 0 0
\(721\) −2.36159e7 −1.69187
\(722\) −9.41585e6 −0.672228
\(723\) 2.99754e7 2.13265
\(724\) −7.85821e6 −0.557157
\(725\) 0 0
\(726\) −1.71774e7 −1.20953
\(727\) −1.07639e7 −0.755323 −0.377661 0.925944i \(-0.623272\pi\)
−0.377661 + 0.925944i \(0.623272\pi\)
\(728\) −1.08851e6 −0.0761210
\(729\) 2.98885e7 2.08298
\(730\) 0 0
\(731\) 5.90060e6 0.408416
\(732\) 1.94317e6 0.134040
\(733\) −8.86456e6 −0.609393 −0.304696 0.952450i \(-0.598555\pi\)
−0.304696 + 0.952450i \(0.598555\pi\)
\(734\) 1.64200e7 1.12495
\(735\) 0 0
\(736\) 1.49256e7 1.01564
\(737\) −3.51008e7 −2.38039
\(738\) −9.45878e6 −0.639285
\(739\) 1.34276e7 0.904457 0.452229 0.891902i \(-0.350629\pi\)
0.452229 + 0.891902i \(0.350629\pi\)
\(740\) 0 0
\(741\) −312550. −0.0209110
\(742\) −4.88989e6 −0.326054
\(743\) 3.76228e6 0.250023 0.125011 0.992155i \(-0.460103\pi\)
0.125011 + 0.992155i \(0.460103\pi\)
\(744\) −3.34812e7 −2.21753
\(745\) 0 0
\(746\) −1.88809e7 −1.24215
\(747\) 4.65388e7 3.05150
\(748\) −1.53263e7 −1.00158
\(749\) 2.78172e7 1.81179
\(750\) 0 0
\(751\) 1.97179e7 1.27574 0.637870 0.770144i \(-0.279816\pi\)
0.637870 + 0.770144i \(0.279816\pi\)
\(752\) −1.56687e6 −0.101039
\(753\) −4.17036e7 −2.68032
\(754\) 135356. 0.00867060
\(755\) 0 0
\(756\) 2.58223e7 1.64320
\(757\) 1.91175e7 1.21253 0.606264 0.795264i \(-0.292668\pi\)
0.606264 + 0.795264i \(0.292668\pi\)
\(758\) 7.97043e6 0.503859
\(759\) −4.63356e7 −2.91951
\(760\) 0 0
\(761\) −4.51417e6 −0.282563 −0.141282 0.989969i \(-0.545122\pi\)
−0.141282 + 0.989969i \(0.545122\pi\)
\(762\) 2.20397e7 1.37505
\(763\) −1.11927e7 −0.696020
\(764\) −1.35261e7 −0.838378
\(765\) 0 0
\(766\) −3.61054e6 −0.222331
\(767\) 1.61039e6 0.0988423
\(768\) 3.28315e7 2.00857
\(769\) 2.08251e7 1.26990 0.634952 0.772551i \(-0.281020\pi\)
0.634952 + 0.772551i \(0.281020\pi\)
\(770\) 0 0
\(771\) −1.60301e6 −0.0971182
\(772\) 6.65753e6 0.402040
\(773\) 1.68649e7 1.01516 0.507581 0.861604i \(-0.330540\pi\)
0.507581 + 0.861604i \(0.330540\pi\)
\(774\) 8.71436e6 0.522857
\(775\) 0 0
\(776\) −4.85612e6 −0.289491
\(777\) −1.35754e7 −0.806679
\(778\) −1.32644e7 −0.785667
\(779\) −1.00564e6 −0.0593742
\(780\) 0 0
\(781\) 3.96876e6 0.232824
\(782\) 1.81695e7 1.06249
\(783\) −9.35324e6 −0.545202
\(784\) −512991. −0.0298071
\(785\) 0 0
\(786\) −1.29240e7 −0.746172
\(787\) 532946. 0.0306723 0.0153361 0.999882i \(-0.495118\pi\)
0.0153361 + 0.999882i \(0.495118\pi\)
\(788\) −1.37449e7 −0.788546
\(789\) 6.34295e7 3.62743
\(790\) 0 0
\(791\) −2.40102e6 −0.136444
\(792\) −6.59327e7 −3.73498
\(793\) 162728. 0.00918922
\(794\) 3.91414e6 0.220336
\(795\) 0 0
\(796\) −5.09726e6 −0.285137
\(797\) −2.74340e6 −0.152983 −0.0764916 0.997070i \(-0.524372\pi\)
−0.0764916 + 0.997070i \(0.524372\pi\)
\(798\) −4.11606e6 −0.228810
\(799\) 1.23347e7 0.683537
\(800\) 0 0
\(801\) 6.37533e7 3.51092
\(802\) 2.16279e7 1.18735
\(803\) 7.75233e6 0.424271
\(804\) −3.09765e7 −1.69002
\(805\) 0 0
\(806\) −962560. −0.0521903
\(807\) 3.76433e6 0.203471
\(808\) 5.63476e6 0.303631
\(809\) 3.60884e6 0.193864 0.0969318 0.995291i \(-0.469097\pi\)
0.0969318 + 0.995291i \(0.469097\pi\)
\(810\) 0 0
\(811\) −2.42636e7 −1.29540 −0.647698 0.761897i \(-0.724268\pi\)
−0.647698 + 0.761897i \(0.724268\pi\)
\(812\) −1.95265e6 −0.103928
\(813\) −2.03168e7 −1.07802
\(814\) 7.24554e6 0.383274
\(815\) 0 0
\(816\) 1.00933e7 0.530650
\(817\) 926492. 0.0485609
\(818\) 7.55344e6 0.394695
\(819\) 3.55149e6 0.185012
\(820\) 0 0
\(821\) −2.65430e6 −0.137433 −0.0687167 0.997636i \(-0.521890\pi\)
−0.0687167 + 0.997636i \(0.521890\pi\)
\(822\) −1.55261e6 −0.0801460
\(823\) 1.45761e7 0.750137 0.375068 0.926997i \(-0.377619\pi\)
0.375068 + 0.926997i \(0.377619\pi\)
\(824\) 3.24012e7 1.66243
\(825\) 0 0
\(826\) 2.12077e7 1.08154
\(827\) 2.56966e7 1.30651 0.653254 0.757139i \(-0.273403\pi\)
0.653254 + 0.757139i \(0.273403\pi\)
\(828\) −2.93945e7 −1.49001
\(829\) 5.04257e6 0.254839 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(830\) 0 0
\(831\) 1.73203e7 0.870067
\(832\) 1.12462e6 0.0563247
\(833\) 4.03838e6 0.201648
\(834\) −3.62280e7 −1.80356
\(835\) 0 0
\(836\) −2.40649e6 −0.119088
\(837\) 6.65139e7 3.28170
\(838\) 7.03734e6 0.346177
\(839\) −6.91282e6 −0.339039 −0.169520 0.985527i \(-0.554222\pi\)
−0.169520 + 0.985527i \(0.554222\pi\)
\(840\) 0 0
\(841\) 707281. 0.0344828
\(842\) −5.18392e6 −0.251987
\(843\) 3.55295e7 1.72195
\(844\) −1.09930e7 −0.531202
\(845\) 0 0
\(846\) 1.82166e7 0.875069
\(847\) 2.07518e7 0.993909
\(848\) 1.88276e6 0.0899097
\(849\) 5.55581e7 2.64532
\(850\) 0 0
\(851\) 9.40934e6 0.445385
\(852\) 3.50244e6 0.165299
\(853\) 2.16826e7 1.02033 0.510164 0.860077i \(-0.329585\pi\)
0.510164 + 0.860077i \(0.329585\pi\)
\(854\) 2.14301e6 0.100549
\(855\) 0 0
\(856\) −3.81654e7 −1.78027
\(857\) 2.22637e7 1.03549 0.517745 0.855535i \(-0.326772\pi\)
0.517745 + 0.855535i \(0.326772\pi\)
\(858\) −2.63688e6 −0.122285
\(859\) −1.56538e7 −0.723828 −0.361914 0.932211i \(-0.617877\pi\)
−0.361914 + 0.932211i \(0.617877\pi\)
\(860\) 0 0
\(861\) 1.58963e7 0.730781
\(862\) 2.28378e6 0.104685
\(863\) −3.14396e7 −1.43698 −0.718489 0.695538i \(-0.755166\pi\)
−0.718489 + 0.695538i \(0.755166\pi\)
\(864\) −5.86940e7 −2.67491
\(865\) 0 0
\(866\) 2.76434e6 0.125255
\(867\) −3.77144e7 −1.70396
\(868\) 1.38859e7 0.625569
\(869\) −2.09437e7 −0.940816
\(870\) 0 0
\(871\) −2.59407e6 −0.115861
\(872\) 1.53564e7 0.683909
\(873\) 1.58440e7 0.703608
\(874\) 2.85291e6 0.126331
\(875\) 0 0
\(876\) 6.84144e6 0.301222
\(877\) −9.99092e6 −0.438638 −0.219319 0.975653i \(-0.570384\pi\)
−0.219319 + 0.975653i \(0.570384\pi\)
\(878\) 1.97808e7 0.865978
\(879\) −7.17057e7 −3.13027
\(880\) 0 0
\(881\) −2.37367e6 −0.103034 −0.0515169 0.998672i \(-0.516406\pi\)
−0.0515169 + 0.998672i \(0.516406\pi\)
\(882\) 5.96411e6 0.258151
\(883\) 686405. 0.0296264 0.0148132 0.999890i \(-0.495285\pi\)
0.0148132 + 0.999890i \(0.495285\pi\)
\(884\) −1.13267e6 −0.0487498
\(885\) 0 0
\(886\) 1.30175e7 0.557112
\(887\) −1.54019e7 −0.657302 −0.328651 0.944451i \(-0.606594\pi\)
−0.328651 + 0.944451i \(0.606594\pi\)
\(888\) 1.86256e7 0.792641
\(889\) −2.66258e7 −1.12992
\(890\) 0 0
\(891\) 9.80750e7 4.13870
\(892\) −1.50492e7 −0.633288
\(893\) 1.93675e6 0.0812729
\(894\) 172355. 0.00721242
\(895\) 0 0
\(896\) −8.62879e6 −0.359071
\(897\) −3.42436e6 −0.142102
\(898\) 147728. 0.00611325
\(899\) −5.02970e6 −0.207560
\(900\) 0 0
\(901\) −1.48215e7 −0.608248
\(902\) −8.48424e6 −0.347213
\(903\) −1.46452e7 −0.597691
\(904\) 3.29422e6 0.134070
\(905\) 0 0
\(906\) 1.62060e7 0.655926
\(907\) 2.67004e7 1.07770 0.538852 0.842401i \(-0.318858\pi\)
0.538852 + 0.842401i \(0.318858\pi\)
\(908\) 1.10412e7 0.444428
\(909\) −1.83845e7 −0.737976
\(910\) 0 0
\(911\) 1.90851e7 0.761902 0.380951 0.924595i \(-0.375597\pi\)
0.380951 + 0.924595i \(0.375597\pi\)
\(912\) 1.58482e6 0.0630946
\(913\) 4.17439e7 1.65736
\(914\) 2.95086e7 1.16838
\(915\) 0 0
\(916\) 1.04116e7 0.409995
\(917\) 1.56132e7 0.613153
\(918\) −7.14503e7 −2.79832
\(919\) 3.04815e7 1.19055 0.595274 0.803523i \(-0.297043\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(920\) 0 0
\(921\) −2.47671e7 −0.962112
\(922\) 7.72051e6 0.299101
\(923\) 293306. 0.0113322
\(924\) 3.80398e7 1.46574
\(925\) 0 0
\(926\) 3.26384e7 1.25084
\(927\) −1.05715e8 −4.04053
\(928\) 4.43837e6 0.169182
\(929\) −2.44730e7 −0.930352 −0.465176 0.885218i \(-0.654009\pi\)
−0.465176 + 0.885218i \(0.654009\pi\)
\(930\) 0 0
\(931\) 634092. 0.0239761
\(932\) 404534. 0.0152551
\(933\) 5.24757e7 1.97358
\(934\) 3.34575e6 0.125495
\(935\) 0 0
\(936\) −4.87266e6 −0.181793
\(937\) 3.03413e7 1.12898 0.564489 0.825441i \(-0.309073\pi\)
0.564489 + 0.825441i \(0.309073\pi\)
\(938\) −3.41621e7 −1.26776
\(939\) 5.27645e7 1.95289
\(940\) 0 0
\(941\) 4.32048e7 1.59059 0.795294 0.606224i \(-0.207317\pi\)
0.795294 + 0.606224i \(0.207317\pi\)
\(942\) −3.85857e7 −1.41677
\(943\) −1.10180e7 −0.403480
\(944\) −8.16565e6 −0.298236
\(945\) 0 0
\(946\) 7.81652e6 0.283979
\(947\) 2.50483e7 0.907620 0.453810 0.891099i \(-0.350065\pi\)
0.453810 + 0.891099i \(0.350065\pi\)
\(948\) −1.84829e7 −0.667957
\(949\) 572925. 0.0206506
\(950\) 0 0
\(951\) −2.95379e7 −1.05908
\(952\) −4.34500e7 −1.55381
\(953\) −4.30580e7 −1.53576 −0.767878 0.640596i \(-0.778687\pi\)
−0.767878 + 0.640596i \(0.778687\pi\)
\(954\) −2.18893e7 −0.778684
\(955\) 0 0
\(956\) −3.37201e6 −0.119328
\(957\) −1.37786e7 −0.486324
\(958\) 1.55288e7 0.546669
\(959\) 1.87568e6 0.0658584
\(960\) 0 0
\(961\) 7.13868e6 0.249350
\(962\) 535471. 0.0186551
\(963\) 1.24522e8 4.32694
\(964\) 1.70567e7 0.591156
\(965\) 0 0
\(966\) −4.50965e7 −1.55489
\(967\) 3.26447e7 1.12266 0.561328 0.827593i \(-0.310290\pi\)
0.561328 + 0.827593i \(0.310290\pi\)
\(968\) −2.84716e7 −0.976614
\(969\) −1.24760e7 −0.426841
\(970\) 0 0
\(971\) −9.02885e6 −0.307315 −0.153658 0.988124i \(-0.549105\pi\)
−0.153658 + 0.988124i \(0.549105\pi\)
\(972\) 4.13414e7 1.40352
\(973\) 4.37665e7 1.48204
\(974\) −1.06694e7 −0.360364
\(975\) 0 0
\(976\) −825128. −0.0277266
\(977\) −8.87743e6 −0.297544 −0.148772 0.988872i \(-0.547532\pi\)
−0.148772 + 0.988872i \(0.547532\pi\)
\(978\) 2.39009e7 0.799037
\(979\) 5.71848e7 1.90688
\(980\) 0 0
\(981\) −5.01033e7 −1.66224
\(982\) 9.84886e6 0.325917
\(983\) 4.56481e7 1.50674 0.753371 0.657596i \(-0.228427\pi\)
0.753371 + 0.657596i \(0.228427\pi\)
\(984\) −2.18098e7 −0.718065
\(985\) 0 0
\(986\) 5.40299e6 0.176987
\(987\) −3.06146e7 −1.00031
\(988\) −177848. −0.00579638
\(989\) 1.01508e7 0.329998
\(990\) 0 0
\(991\) −7.74121e6 −0.250395 −0.125197 0.992132i \(-0.539956\pi\)
−0.125197 + 0.992132i \(0.539956\pi\)
\(992\) −3.15627e7 −1.01834
\(993\) −1.03564e8 −3.33301
\(994\) 3.86263e6 0.123999
\(995\) 0 0
\(996\) 3.68390e7 1.17668
\(997\) −5.64996e7 −1.80014 −0.900072 0.435740i \(-0.856487\pi\)
−0.900072 + 0.435740i \(0.856487\pi\)
\(998\) 3.37279e7 1.07192
\(999\) −3.70016e7 −1.17302
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 725.6.a.b.1.5 7
5.4 even 2 29.6.a.b.1.3 7
15.14 odd 2 261.6.a.e.1.5 7
20.19 odd 2 464.6.a.k.1.1 7
145.144 even 2 841.6.a.b.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.b.1.3 7 5.4 even 2
261.6.a.e.1.5 7 15.14 odd 2
464.6.a.k.1.1 7 20.19 odd 2
725.6.a.b.1.5 7 1.1 even 1 trivial
841.6.a.b.1.5 7 145.144 even 2