Properties

Label 245.6.a.l.1.5
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $10$
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] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.63335\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.633352 q^{2} -19.5171 q^{3} -31.5989 q^{4} -25.0000 q^{5} +12.3612 q^{6} +40.2804 q^{8} +137.918 q^{9} +O(q^{10})\) \(q-0.633352 q^{2} -19.5171 q^{3} -31.5989 q^{4} -25.0000 q^{5} +12.3612 q^{6} +40.2804 q^{8} +137.918 q^{9} +15.8338 q^{10} +76.2344 q^{11} +616.719 q^{12} -266.495 q^{13} +487.928 q^{15} +985.652 q^{16} -267.518 q^{17} -87.3504 q^{18} +1763.32 q^{19} +789.972 q^{20} -48.2831 q^{22} +4413.22 q^{23} -786.158 q^{24} +625.000 q^{25} +168.785 q^{26} +2050.90 q^{27} +129.019 q^{29} -309.030 q^{30} -5981.45 q^{31} -1913.24 q^{32} -1487.87 q^{33} +169.433 q^{34} -4358.04 q^{36} +8909.64 q^{37} -1116.80 q^{38} +5201.22 q^{39} -1007.01 q^{40} -9262.01 q^{41} -6782.24 q^{43} -2408.92 q^{44} -3447.94 q^{45} -2795.12 q^{46} -25145.7 q^{47} -19237.1 q^{48} -395.845 q^{50} +5221.19 q^{51} +8420.94 q^{52} -3066.58 q^{53} -1298.94 q^{54} -1905.86 q^{55} -34415.0 q^{57} -81.7146 q^{58} +23669.3 q^{59} -15418.0 q^{60} -32287.4 q^{61} +3788.36 q^{62} -30329.1 q^{64} +6662.38 q^{65} +942.348 q^{66} -31965.8 q^{67} +8453.28 q^{68} -86133.2 q^{69} +59587.9 q^{71} +5555.39 q^{72} +24102.7 q^{73} -5642.93 q^{74} -12198.2 q^{75} -55719.0 q^{76} -3294.20 q^{78} +107001. q^{79} -24641.3 q^{80} -73541.7 q^{81} +5866.11 q^{82} -5517.40 q^{83} +6687.96 q^{85} +4295.54 q^{86} -2518.08 q^{87} +3070.75 q^{88} -39714.8 q^{89} +2183.76 q^{90} -139453. q^{92} +116741. q^{93} +15926.0 q^{94} -44083.0 q^{95} +37340.9 q^{96} +131282. q^{97} +10514.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 58 q^{3} + 182 q^{4} - 250 q^{5} - 144 q^{6} + 270 q^{8} + 700 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 58 q^{3} + 182 q^{4} - 250 q^{5} - 144 q^{6} + 270 q^{8} + 700 q^{9} - 250 q^{10} + 794 q^{11} - 2560 q^{12} - 474 q^{13} + 1450 q^{15} + 2394 q^{16} - 802 q^{17} + 3702 q^{18} - 7292 q^{19} - 4550 q^{20} + 3948 q^{22} + 3708 q^{23} - 2092 q^{24} + 6250 q^{25} - 6576 q^{26} - 11818 q^{27} - 8866 q^{29} + 3600 q^{30} - 13292 q^{31} + 2590 q^{32} - 9854 q^{33} - 44468 q^{34} - 10690 q^{36} + 16124 q^{37} - 2180 q^{38} - 24982 q^{39} - 6750 q^{40} - 34836 q^{41} - 28604 q^{43} - 31120 q^{44} - 17500 q^{45} - 39732 q^{46} - 18106 q^{47} - 101788 q^{48} + 6250 q^{50} + 31602 q^{51} + 22480 q^{52} + 36440 q^{53} - 80836 q^{54} - 19850 q^{55} + 126988 q^{57} - 100356 q^{58} - 18644 q^{59} + 64000 q^{60} - 68120 q^{61} - 181052 q^{62} - 59358 q^{64} + 11850 q^{65} - 157780 q^{66} + 92328 q^{67} - 288540 q^{68} - 170888 q^{69} + 5044 q^{71} - 61654 q^{72} - 170160 q^{73} - 216584 q^{74} - 36250 q^{75} - 505180 q^{76} - 158008 q^{78} + 26442 q^{79} - 59850 q^{80} - 56314 q^{81} - 353948 q^{82} - 353360 q^{83} + 20050 q^{85} - 52940 q^{86} + 3190 q^{87} - 114916 q^{88} - 90704 q^{89} - 92550 q^{90} + 183520 q^{92} + 188560 q^{93} + 121388 q^{94} + 182300 q^{95} - 442220 q^{96} - 236382 q^{97} + 109024 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.633352 −0.111962 −0.0559809 0.998432i \(-0.517829\pi\)
−0.0559809 + 0.998432i \(0.517829\pi\)
\(3\) −19.5171 −1.25202 −0.626012 0.779814i \(-0.715314\pi\)
−0.626012 + 0.779814i \(0.715314\pi\)
\(4\) −31.5989 −0.987465
\(5\) −25.0000 −0.447214
\(6\) 12.3612 0.140179
\(7\) 0 0
\(8\) 40.2804 0.222520
\(9\) 137.918 0.567563
\(10\) 15.8338 0.0500708
\(11\) 76.2344 0.189963 0.0949815 0.995479i \(-0.469721\pi\)
0.0949815 + 0.995479i \(0.469721\pi\)
\(12\) 616.719 1.23633
\(13\) −266.495 −0.437352 −0.218676 0.975798i \(-0.570174\pi\)
−0.218676 + 0.975798i \(0.570174\pi\)
\(14\) 0 0
\(15\) 487.928 0.559922
\(16\) 985.652 0.962551
\(17\) −267.518 −0.224508 −0.112254 0.993680i \(-0.535807\pi\)
−0.112254 + 0.993680i \(0.535807\pi\)
\(18\) −87.3504 −0.0635453
\(19\) 1763.32 1.12059 0.560296 0.828292i \(-0.310687\pi\)
0.560296 + 0.828292i \(0.310687\pi\)
\(20\) 789.972 0.441608
\(21\) 0 0
\(22\) −48.2831 −0.0212686
\(23\) 4413.22 1.73954 0.869772 0.493453i \(-0.164266\pi\)
0.869772 + 0.493453i \(0.164266\pi\)
\(24\) −786.158 −0.278600
\(25\) 625.000 0.200000
\(26\) 168.785 0.0489667
\(27\) 2050.90 0.541422
\(28\) 0 0
\(29\) 129.019 0.0284879 0.0142439 0.999899i \(-0.495466\pi\)
0.0142439 + 0.999899i \(0.495466\pi\)
\(30\) −309.030 −0.0626899
\(31\) −5981.45 −1.11790 −0.558949 0.829202i \(-0.688795\pi\)
−0.558949 + 0.829202i \(0.688795\pi\)
\(32\) −1913.24 −0.330289
\(33\) −1487.87 −0.237838
\(34\) 169.433 0.0251363
\(35\) 0 0
\(36\) −4358.04 −0.560448
\(37\) 8909.64 1.06993 0.534966 0.844874i \(-0.320325\pi\)
0.534966 + 0.844874i \(0.320325\pi\)
\(38\) −1116.80 −0.125464
\(39\) 5201.22 0.547575
\(40\) −1007.01 −0.0995140
\(41\) −9262.01 −0.860490 −0.430245 0.902712i \(-0.641573\pi\)
−0.430245 + 0.902712i \(0.641573\pi\)
\(42\) 0 0
\(43\) −6782.24 −0.559374 −0.279687 0.960091i \(-0.590231\pi\)
−0.279687 + 0.960091i \(0.590231\pi\)
\(44\) −2408.92 −0.187582
\(45\) −3447.94 −0.253822
\(46\) −2795.12 −0.194763
\(47\) −25145.7 −1.66042 −0.830211 0.557450i \(-0.811780\pi\)
−0.830211 + 0.557450i \(0.811780\pi\)
\(48\) −19237.1 −1.20514
\(49\) 0 0
\(50\) −395.845 −0.0223924
\(51\) 5221.19 0.281089
\(52\) 8420.94 0.431869
\(53\) −3066.58 −0.149956 −0.0749782 0.997185i \(-0.523889\pi\)
−0.0749782 + 0.997185i \(0.523889\pi\)
\(54\) −1298.94 −0.0606185
\(55\) −1905.86 −0.0849540
\(56\) 0 0
\(57\) −34415.0 −1.40301
\(58\) −81.7146 −0.00318955
\(59\) 23669.3 0.885230 0.442615 0.896712i \(-0.354051\pi\)
0.442615 + 0.896712i \(0.354051\pi\)
\(60\) −15418.0 −0.552903
\(61\) −32287.4 −1.11099 −0.555493 0.831521i \(-0.687471\pi\)
−0.555493 + 0.831521i \(0.687471\pi\)
\(62\) 3788.36 0.125162
\(63\) 0 0
\(64\) −30329.1 −0.925571
\(65\) 6662.38 0.195590
\(66\) 942.348 0.0266288
\(67\) −31965.8 −0.869960 −0.434980 0.900440i \(-0.643245\pi\)
−0.434980 + 0.900440i \(0.643245\pi\)
\(68\) 8453.28 0.221693
\(69\) −86133.2 −2.17795
\(70\) 0 0
\(71\) 59587.9 1.40285 0.701426 0.712742i \(-0.252547\pi\)
0.701426 + 0.712742i \(0.252547\pi\)
\(72\) 5555.39 0.126294
\(73\) 24102.7 0.529370 0.264685 0.964335i \(-0.414732\pi\)
0.264685 + 0.964335i \(0.414732\pi\)
\(74\) −5642.93 −0.119791
\(75\) −12198.2 −0.250405
\(76\) −55719.0 −1.10655
\(77\) 0 0
\(78\) −3294.20 −0.0613074
\(79\) 107001. 1.92895 0.964474 0.264178i \(-0.0851007\pi\)
0.964474 + 0.264178i \(0.0851007\pi\)
\(80\) −24641.3 −0.430466
\(81\) −73541.7 −1.24544
\(82\) 5866.11 0.0963420
\(83\) −5517.40 −0.0879101 −0.0439551 0.999034i \(-0.513996\pi\)
−0.0439551 + 0.999034i \(0.513996\pi\)
\(84\) 0 0
\(85\) 6687.96 0.100403
\(86\) 4295.54 0.0626285
\(87\) −2518.08 −0.0356675
\(88\) 3070.75 0.0422706
\(89\) −39714.8 −0.531469 −0.265734 0.964046i \(-0.585614\pi\)
−0.265734 + 0.964046i \(0.585614\pi\)
\(90\) 2183.76 0.0284183
\(91\) 0 0
\(92\) −139453. −1.71774
\(93\) 116741. 1.39963
\(94\) 15926.0 0.185904
\(95\) −44083.0 −0.501144
\(96\) 37340.9 0.413530
\(97\) 131282. 1.41670 0.708348 0.705864i \(-0.249441\pi\)
0.708348 + 0.705864i \(0.249441\pi\)
\(98\) 0 0
\(99\) 10514.1 0.107816
\(100\) −19749.3 −0.197493
\(101\) −170783. −1.66587 −0.832935 0.553371i \(-0.813341\pi\)
−0.832935 + 0.553371i \(0.813341\pi\)
\(102\) −3306.85 −0.0314712
\(103\) −41960.6 −0.389716 −0.194858 0.980831i \(-0.562425\pi\)
−0.194858 + 0.980831i \(0.562425\pi\)
\(104\) −10734.5 −0.0973196
\(105\) 0 0
\(106\) 1942.22 0.0167894
\(107\) 126667. 1.06956 0.534779 0.844992i \(-0.320395\pi\)
0.534779 + 0.844992i \(0.320395\pi\)
\(108\) −64806.2 −0.534635
\(109\) 149620. 1.20621 0.603107 0.797661i \(-0.293929\pi\)
0.603107 + 0.797661i \(0.293929\pi\)
\(110\) 1207.08 0.00951161
\(111\) −173890. −1.33958
\(112\) 0 0
\(113\) 84809.5 0.624810 0.312405 0.949949i \(-0.398865\pi\)
0.312405 + 0.949949i \(0.398865\pi\)
\(114\) 21796.8 0.157083
\(115\) −110330. −0.777948
\(116\) −4076.86 −0.0281307
\(117\) −36754.4 −0.248225
\(118\) −14991.0 −0.0991120
\(119\) 0 0
\(120\) 19653.9 0.124594
\(121\) −155239. −0.963914
\(122\) 20449.3 0.124388
\(123\) 180768. 1.07735
\(124\) 189007. 1.10388
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) −248125. −1.36509 −0.682545 0.730844i \(-0.739127\pi\)
−0.682545 + 0.730844i \(0.739127\pi\)
\(128\) 80432.6 0.433918
\(129\) 132370. 0.700349
\(130\) −4219.63 −0.0218986
\(131\) −261050. −1.32906 −0.664532 0.747260i \(-0.731369\pi\)
−0.664532 + 0.747260i \(0.731369\pi\)
\(132\) 47015.1 0.234857
\(133\) 0 0
\(134\) 20245.6 0.0974022
\(135\) −51272.6 −0.242131
\(136\) −10775.8 −0.0499575
\(137\) −16931.3 −0.0770706 −0.0385353 0.999257i \(-0.512269\pi\)
−0.0385353 + 0.999257i \(0.512269\pi\)
\(138\) 54552.6 0.243847
\(139\) 179252. 0.786914 0.393457 0.919343i \(-0.371279\pi\)
0.393457 + 0.919343i \(0.371279\pi\)
\(140\) 0 0
\(141\) 490771. 2.07889
\(142\) −37740.1 −0.157066
\(143\) −20316.1 −0.0830806
\(144\) 135939. 0.546308
\(145\) −3225.48 −0.0127402
\(146\) −15265.5 −0.0592692
\(147\) 0 0
\(148\) −281535. −1.05652
\(149\) −376763. −1.39028 −0.695140 0.718875i \(-0.744658\pi\)
−0.695140 + 0.718875i \(0.744658\pi\)
\(150\) 7725.75 0.0280358
\(151\) −355503. −1.26882 −0.634412 0.772995i \(-0.718758\pi\)
−0.634412 + 0.772995i \(0.718758\pi\)
\(152\) 71027.4 0.249354
\(153\) −36895.5 −0.127422
\(154\) 0 0
\(155\) 149536. 0.499939
\(156\) −164353. −0.540711
\(157\) 491459. 1.59125 0.795624 0.605791i \(-0.207143\pi\)
0.795624 + 0.605791i \(0.207143\pi\)
\(158\) −67769.3 −0.215968
\(159\) 59850.8 0.187749
\(160\) 47831.0 0.147710
\(161\) 0 0
\(162\) 46577.8 0.139441
\(163\) −143009. −0.421593 −0.210796 0.977530i \(-0.567606\pi\)
−0.210796 + 0.977530i \(0.567606\pi\)
\(164\) 292669. 0.849703
\(165\) 37196.9 0.106364
\(166\) 3494.45 0.00984258
\(167\) 313947. 0.871094 0.435547 0.900166i \(-0.356555\pi\)
0.435547 + 0.900166i \(0.356555\pi\)
\(168\) 0 0
\(169\) −300273. −0.808723
\(170\) −4235.83 −0.0112413
\(171\) 243193. 0.636006
\(172\) 214311. 0.552362
\(173\) 437124. 1.11043 0.555213 0.831708i \(-0.312637\pi\)
0.555213 + 0.831708i \(0.312637\pi\)
\(174\) 1594.83 0.00399339
\(175\) 0 0
\(176\) 75140.5 0.182849
\(177\) −461957. −1.10833
\(178\) 25153.5 0.0595042
\(179\) −391475. −0.913211 −0.456605 0.889669i \(-0.650935\pi\)
−0.456605 + 0.889669i \(0.650935\pi\)
\(180\) 108951. 0.250640
\(181\) 82459.5 0.187087 0.0935437 0.995615i \(-0.470181\pi\)
0.0935437 + 0.995615i \(0.470181\pi\)
\(182\) 0 0
\(183\) 630157. 1.39098
\(184\) 177766. 0.387084
\(185\) −222741. −0.478488
\(186\) −73937.9 −0.156706
\(187\) −20394.1 −0.0426482
\(188\) 794574. 1.63961
\(189\) 0 0
\(190\) 27920.1 0.0561090
\(191\) −849381. −1.68469 −0.842343 0.538942i \(-0.818824\pi\)
−0.842343 + 0.538942i \(0.818824\pi\)
\(192\) 591937. 1.15884
\(193\) 284540. 0.549857 0.274928 0.961465i \(-0.411346\pi\)
0.274928 + 0.961465i \(0.411346\pi\)
\(194\) −83147.8 −0.158616
\(195\) −130030. −0.244883
\(196\) 0 0
\(197\) 554266. 1.01754 0.508772 0.860902i \(-0.330100\pi\)
0.508772 + 0.860902i \(0.330100\pi\)
\(198\) −6659.10 −0.0120713
\(199\) −635373. −1.13735 −0.568677 0.822561i \(-0.692545\pi\)
−0.568677 + 0.822561i \(0.692545\pi\)
\(200\) 25175.3 0.0445040
\(201\) 623881. 1.08921
\(202\) 108166. 0.186514
\(203\) 0 0
\(204\) −164984. −0.277565
\(205\) 231550. 0.384823
\(206\) 26575.8 0.0436333
\(207\) 608661. 0.987301
\(208\) −262671. −0.420973
\(209\) 134426. 0.212871
\(210\) 0 0
\(211\) −280331. −0.433476 −0.216738 0.976230i \(-0.569542\pi\)
−0.216738 + 0.976230i \(0.569542\pi\)
\(212\) 96900.5 0.148077
\(213\) −1.16298e6 −1.75640
\(214\) −80224.8 −0.119750
\(215\) 169556. 0.250160
\(216\) 82611.3 0.120477
\(217\) 0 0
\(218\) −94762.2 −0.135050
\(219\) −470416. −0.662783
\(220\) 60223.0 0.0838891
\(221\) 71292.3 0.0981888
\(222\) 110134. 0.149982
\(223\) 970109. 1.30635 0.653173 0.757208i \(-0.273437\pi\)
0.653173 + 0.757208i \(0.273437\pi\)
\(224\) 0 0
\(225\) 86198.6 0.113513
\(226\) −53714.2 −0.0699549
\(227\) −569413. −0.733437 −0.366719 0.930332i \(-0.619519\pi\)
−0.366719 + 0.930332i \(0.619519\pi\)
\(228\) 1.08747e6 1.38542
\(229\) −77988.6 −0.0982748 −0.0491374 0.998792i \(-0.515647\pi\)
−0.0491374 + 0.998792i \(0.515647\pi\)
\(230\) 69877.9 0.0871005
\(231\) 0 0
\(232\) 5196.95 0.00633912
\(233\) 829373. 1.00083 0.500415 0.865786i \(-0.333181\pi\)
0.500415 + 0.865786i \(0.333181\pi\)
\(234\) 23278.5 0.0277917
\(235\) 628641. 0.742563
\(236\) −747924. −0.874133
\(237\) −2.08835e6 −2.41509
\(238\) 0 0
\(239\) 252273. 0.285678 0.142839 0.989746i \(-0.454377\pi\)
0.142839 + 0.989746i \(0.454377\pi\)
\(240\) 480927. 0.538953
\(241\) 535968. 0.594424 0.297212 0.954812i \(-0.403943\pi\)
0.297212 + 0.954812i \(0.403943\pi\)
\(242\) 98321.1 0.107922
\(243\) 936953. 1.01789
\(244\) 1.02025e6 1.09706
\(245\) 0 0
\(246\) −114490. −0.120622
\(247\) −469917. −0.490093
\(248\) −240935. −0.248755
\(249\) 107684. 0.110066
\(250\) 9896.12 0.0100142
\(251\) 52953.6 0.0530531 0.0265266 0.999648i \(-0.491555\pi\)
0.0265266 + 0.999648i \(0.491555\pi\)
\(252\) 0 0
\(253\) 336439. 0.330449
\(254\) 157150. 0.152838
\(255\) −130530. −0.125707
\(256\) 919589. 0.876989
\(257\) 527109. 0.497814 0.248907 0.968527i \(-0.419929\pi\)
0.248907 + 0.968527i \(0.419929\pi\)
\(258\) −83836.6 −0.0784123
\(259\) 0 0
\(260\) −210524. −0.193138
\(261\) 17794.1 0.0161686
\(262\) 165337. 0.148804
\(263\) −1.26966e6 −1.13187 −0.565935 0.824450i \(-0.691485\pi\)
−0.565935 + 0.824450i \(0.691485\pi\)
\(264\) −59932.2 −0.0529238
\(265\) 76664.6 0.0670625
\(266\) 0 0
\(267\) 775119. 0.665412
\(268\) 1.01008e6 0.859054
\(269\) −863268. −0.727386 −0.363693 0.931519i \(-0.618484\pi\)
−0.363693 + 0.931519i \(0.618484\pi\)
\(270\) 32473.6 0.0271094
\(271\) −1.82397e6 −1.50867 −0.754335 0.656490i \(-0.772040\pi\)
−0.754335 + 0.656490i \(0.772040\pi\)
\(272\) −263680. −0.216100
\(273\) 0 0
\(274\) 10723.5 0.00862896
\(275\) 47646.5 0.0379926
\(276\) 2.72171e6 2.15065
\(277\) −2.09940e6 −1.64398 −0.821990 0.569503i \(-0.807136\pi\)
−0.821990 + 0.569503i \(0.807136\pi\)
\(278\) −113530. −0.0881042
\(279\) −824948. −0.634477
\(280\) 0 0
\(281\) −1.72140e6 −1.30052 −0.650260 0.759712i \(-0.725340\pi\)
−0.650260 + 0.759712i \(0.725340\pi\)
\(282\) −310830. −0.232756
\(283\) −2.53726e6 −1.88321 −0.941606 0.336716i \(-0.890684\pi\)
−0.941606 + 0.336716i \(0.890684\pi\)
\(284\) −1.88291e6 −1.38527
\(285\) 860374. 0.627444
\(286\) 12867.2 0.00930186
\(287\) 0 0
\(288\) −263869. −0.187460
\(289\) −1.34829e6 −0.949596
\(290\) 2042.86 0.00142641
\(291\) −2.56225e6 −1.77374
\(292\) −761619. −0.522734
\(293\) −1.14064e6 −0.776213 −0.388106 0.921615i \(-0.626871\pi\)
−0.388106 + 0.921615i \(0.626871\pi\)
\(294\) 0 0
\(295\) −591734. −0.395887
\(296\) 358884. 0.238081
\(297\) 156349. 0.102850
\(298\) 238623. 0.155658
\(299\) −1.17610e6 −0.760793
\(300\) 385449. 0.247266
\(301\) 0 0
\(302\) 225158. 0.142060
\(303\) 3.33319e6 2.08571
\(304\) 1.73802e6 1.07863
\(305\) 807186. 0.496848
\(306\) 23367.8 0.0142664
\(307\) −1.50913e6 −0.913861 −0.456931 0.889502i \(-0.651051\pi\)
−0.456931 + 0.889502i \(0.651051\pi\)
\(308\) 0 0
\(309\) 818950. 0.487934
\(310\) −94709.0 −0.0559741
\(311\) 3.37087e6 1.97625 0.988124 0.153660i \(-0.0491059\pi\)
0.988124 + 0.153660i \(0.0491059\pi\)
\(312\) 209507. 0.121846
\(313\) 427067. 0.246397 0.123198 0.992382i \(-0.460685\pi\)
0.123198 + 0.992382i \(0.460685\pi\)
\(314\) −311266. −0.178159
\(315\) 0 0
\(316\) −3.38111e6 −1.90477
\(317\) −868486. −0.485417 −0.242708 0.970099i \(-0.578036\pi\)
−0.242708 + 0.970099i \(0.578036\pi\)
\(318\) −37906.6 −0.0210207
\(319\) 9835.70 0.00541164
\(320\) 758228. 0.413928
\(321\) −2.47218e6 −1.33911
\(322\) 0 0
\(323\) −471721. −0.251582
\(324\) 2.32383e6 1.22982
\(325\) −166559. −0.0874703
\(326\) 90574.7 0.0472023
\(327\) −2.92016e6 −1.51021
\(328\) −373078. −0.191476
\(329\) 0 0
\(330\) −23558.7 −0.0119088
\(331\) −1.48917e6 −0.747092 −0.373546 0.927612i \(-0.621858\pi\)
−0.373546 + 0.927612i \(0.621858\pi\)
\(332\) 174343. 0.0868082
\(333\) 1.22880e6 0.607253
\(334\) −198839. −0.0975293
\(335\) 799146. 0.389058
\(336\) 0 0
\(337\) 1.73365e6 0.831547 0.415773 0.909468i \(-0.363511\pi\)
0.415773 + 0.909468i \(0.363511\pi\)
\(338\) 190179. 0.0905461
\(339\) −1.65524e6 −0.782277
\(340\) −211332. −0.0991443
\(341\) −455992. −0.212359
\(342\) −154027. −0.0712084
\(343\) 0 0
\(344\) −273192. −0.124472
\(345\) 2.15333e6 0.974009
\(346\) −276853. −0.124325
\(347\) −3.26650e6 −1.45633 −0.728164 0.685403i \(-0.759626\pi\)
−0.728164 + 0.685403i \(0.759626\pi\)
\(348\) 79568.6 0.0352204
\(349\) −4.01741e6 −1.76556 −0.882779 0.469788i \(-0.844330\pi\)
−0.882779 + 0.469788i \(0.844330\pi\)
\(350\) 0 0
\(351\) −546555. −0.236792
\(352\) −145854. −0.0627427
\(353\) −2.44907e6 −1.04608 −0.523040 0.852308i \(-0.675202\pi\)
−0.523040 + 0.852308i \(0.675202\pi\)
\(354\) 292581. 0.124090
\(355\) −1.48970e6 −0.627375
\(356\) 1.25494e6 0.524807
\(357\) 0 0
\(358\) 247941. 0.102245
\(359\) −177274. −0.0725954 −0.0362977 0.999341i \(-0.511556\pi\)
−0.0362977 + 0.999341i \(0.511556\pi\)
\(360\) −138885. −0.0564804
\(361\) 633205. 0.255727
\(362\) −52225.9 −0.0209466
\(363\) 3.02982e6 1.20684
\(364\) 0 0
\(365\) −602568. −0.236741
\(366\) −399111. −0.155737
\(367\) −3.26871e6 −1.26681 −0.633405 0.773820i \(-0.718343\pi\)
−0.633405 + 0.773820i \(0.718343\pi\)
\(368\) 4.34990e6 1.67440
\(369\) −1.27740e6 −0.488382
\(370\) 141073. 0.0535724
\(371\) 0 0
\(372\) −3.68887e6 −1.38209
\(373\) 3.59046e6 1.33622 0.668110 0.744062i \(-0.267104\pi\)
0.668110 + 0.744062i \(0.267104\pi\)
\(374\) 12916.6 0.00477496
\(375\) 304955. 0.111984
\(376\) −1.01288e6 −0.369477
\(377\) −34383.0 −0.0124592
\(378\) 0 0
\(379\) −444241. −0.158862 −0.0794312 0.996840i \(-0.525310\pi\)
−0.0794312 + 0.996840i \(0.525310\pi\)
\(380\) 1.39297e6 0.494862
\(381\) 4.84268e6 1.70912
\(382\) 537957. 0.188621
\(383\) 4.56035e6 1.58855 0.794275 0.607558i \(-0.207851\pi\)
0.794275 + 0.607558i \(0.207851\pi\)
\(384\) −1.56981e6 −0.543275
\(385\) 0 0
\(386\) −180214. −0.0615629
\(387\) −935391. −0.317480
\(388\) −4.14837e6 −1.39894
\(389\) 2.50465e6 0.839215 0.419608 0.907706i \(-0.362168\pi\)
0.419608 + 0.907706i \(0.362168\pi\)
\(390\) 82354.9 0.0274175
\(391\) −1.18062e6 −0.390541
\(392\) 0 0
\(393\) 5.09495e6 1.66402
\(394\) −351045. −0.113926
\(395\) −2.67503e6 −0.862652
\(396\) −332233. −0.106464
\(397\) −2.38913e6 −0.760787 −0.380394 0.924825i \(-0.624211\pi\)
−0.380394 + 0.924825i \(0.624211\pi\)
\(398\) 402414. 0.127340
\(399\) 0 0
\(400\) 616033. 0.192510
\(401\) 62929.2 0.0195430 0.00977150 0.999952i \(-0.496890\pi\)
0.00977150 + 0.999952i \(0.496890\pi\)
\(402\) −395136. −0.121950
\(403\) 1.59403e6 0.488915
\(404\) 5.39655e6 1.64499
\(405\) 1.83854e6 0.556976
\(406\) 0 0
\(407\) 679221. 0.203247
\(408\) 210312. 0.0625479
\(409\) −404988. −0.119711 −0.0598555 0.998207i \(-0.519064\pi\)
−0.0598555 + 0.998207i \(0.519064\pi\)
\(410\) −146653. −0.0430855
\(411\) 330450. 0.0964942
\(412\) 1.32591e6 0.384831
\(413\) 0 0
\(414\) −385496. −0.110540
\(415\) 137935. 0.0393146
\(416\) 509869. 0.144452
\(417\) −3.49848e6 −0.985234
\(418\) −85138.7 −0.0238334
\(419\) 1.73727e6 0.483429 0.241714 0.970347i \(-0.422290\pi\)
0.241714 + 0.970347i \(0.422290\pi\)
\(420\) 0 0
\(421\) 3.09244e6 0.850347 0.425174 0.905112i \(-0.360213\pi\)
0.425174 + 0.905112i \(0.360213\pi\)
\(422\) 177548. 0.0485328
\(423\) −3.46803e6 −0.942393
\(424\) −123523. −0.0333683
\(425\) −167199. −0.0449015
\(426\) 736577. 0.196650
\(427\) 0 0
\(428\) −4.00254e6 −1.05615
\(429\) 396511. 0.104019
\(430\) −107389. −0.0280083
\(431\) −3.04862e6 −0.790515 −0.395258 0.918570i \(-0.629345\pi\)
−0.395258 + 0.918570i \(0.629345\pi\)
\(432\) 2.02148e6 0.521146
\(433\) 986108. 0.252758 0.126379 0.991982i \(-0.459664\pi\)
0.126379 + 0.991982i \(0.459664\pi\)
\(434\) 0 0
\(435\) 62952.1 0.0159510
\(436\) −4.72783e6 −1.19109
\(437\) 7.78192e6 1.94932
\(438\) 297939. 0.0742064
\(439\) −3.69313e6 −0.914603 −0.457302 0.889312i \(-0.651184\pi\)
−0.457302 + 0.889312i \(0.651184\pi\)
\(440\) −76768.8 −0.0189040
\(441\) 0 0
\(442\) −45153.1 −0.0109934
\(443\) 1.92890e6 0.466983 0.233492 0.972359i \(-0.424985\pi\)
0.233492 + 0.972359i \(0.424985\pi\)
\(444\) 5.49474e6 1.32279
\(445\) 992871. 0.237680
\(446\) −614420. −0.146261
\(447\) 7.35332e6 1.74066
\(448\) 0 0
\(449\) 4.12359e6 0.965295 0.482647 0.875815i \(-0.339675\pi\)
0.482647 + 0.875815i \(0.339675\pi\)
\(450\) −54594.0 −0.0127091
\(451\) −706084. −0.163461
\(452\) −2.67988e6 −0.616978
\(453\) 6.93840e6 1.58860
\(454\) 360639. 0.0821170
\(455\) 0 0
\(456\) −1.38625e6 −0.312197
\(457\) −3.96950e6 −0.889089 −0.444544 0.895757i \(-0.646634\pi\)
−0.444544 + 0.895757i \(0.646634\pi\)
\(458\) 49394.2 0.0110030
\(459\) −548654. −0.121553
\(460\) 3.48632e6 0.768196
\(461\) −3.39581e6 −0.744202 −0.372101 0.928192i \(-0.621362\pi\)
−0.372101 + 0.928192i \(0.621362\pi\)
\(462\) 0 0
\(463\) 1.05461e6 0.228633 0.114317 0.993444i \(-0.463532\pi\)
0.114317 + 0.993444i \(0.463532\pi\)
\(464\) 127168. 0.0274210
\(465\) −2.91852e6 −0.625936
\(466\) −525285. −0.112055
\(467\) −984239. −0.208837 −0.104419 0.994533i \(-0.533298\pi\)
−0.104419 + 0.994533i \(0.533298\pi\)
\(468\) 1.16140e6 0.245113
\(469\) 0 0
\(470\) −398151. −0.0831387
\(471\) −9.59186e6 −1.99228
\(472\) 953412. 0.196982
\(473\) −517040. −0.106260
\(474\) 1.32266e6 0.270398
\(475\) 1.10208e6 0.224118
\(476\) 0 0
\(477\) −422936. −0.0851097
\(478\) −159778. −0.0319850
\(479\) 5.10476e6 1.01657 0.508284 0.861189i \(-0.330280\pi\)
0.508284 + 0.861189i \(0.330280\pi\)
\(480\) −933522. −0.184936
\(481\) −2.37438e6 −0.467936
\(482\) −339456. −0.0665527
\(483\) 0 0
\(484\) 4.90539e6 0.951831
\(485\) −3.28206e6 −0.633566
\(486\) −593420. −0.113965
\(487\) 6.71429e6 1.28286 0.641428 0.767184i \(-0.278342\pi\)
0.641428 + 0.767184i \(0.278342\pi\)
\(488\) −1.30055e6 −0.247217
\(489\) 2.79111e6 0.527844
\(490\) 0 0
\(491\) 8.62514e6 1.61459 0.807295 0.590148i \(-0.200931\pi\)
0.807295 + 0.590148i \(0.200931\pi\)
\(492\) −5.71206e6 −1.06385
\(493\) −34515.0 −0.00639574
\(494\) 297622. 0.0548717
\(495\) −262852. −0.0482167
\(496\) −5.89563e6 −1.07603
\(497\) 0 0
\(498\) −68201.6 −0.0123231
\(499\) −9.39508e6 −1.68908 −0.844538 0.535496i \(-0.820125\pi\)
−0.844538 + 0.535496i \(0.820125\pi\)
\(500\) 493732. 0.0883215
\(501\) −6.12734e6 −1.09063
\(502\) −33538.2 −0.00593993
\(503\) 429394. 0.0756722 0.0378361 0.999284i \(-0.487954\pi\)
0.0378361 + 0.999284i \(0.487954\pi\)
\(504\) 0 0
\(505\) 4.26957e6 0.745000
\(506\) −213084. −0.0369977
\(507\) 5.86047e6 1.01254
\(508\) 7.84047e6 1.34798
\(509\) 248510. 0.0425157 0.0212579 0.999774i \(-0.493233\pi\)
0.0212579 + 0.999774i \(0.493233\pi\)
\(510\) 82671.2 0.0140744
\(511\) 0 0
\(512\) −3.15627e6 −0.532107
\(513\) 3.61640e6 0.606713
\(514\) −333845. −0.0557362
\(515\) 1.04901e6 0.174286
\(516\) −4.18273e6 −0.691570
\(517\) −1.91696e6 −0.315419
\(518\) 0 0
\(519\) −8.53140e6 −1.39028
\(520\) 268364. 0.0435226
\(521\) 4.77060e6 0.769979 0.384990 0.922921i \(-0.374205\pi\)
0.384990 + 0.922921i \(0.374205\pi\)
\(522\) −11269.9 −0.00181027
\(523\) −2.70325e6 −0.432148 −0.216074 0.976377i \(-0.569325\pi\)
−0.216074 + 0.976377i \(0.569325\pi\)
\(524\) 8.24889e6 1.31240
\(525\) 0 0
\(526\) 804139. 0.126726
\(527\) 1.60015e6 0.250977
\(528\) −1.46653e6 −0.228931
\(529\) 1.30401e7 2.02602
\(530\) −48555.6 −0.00750844
\(531\) 3.26442e6 0.502424
\(532\) 0 0
\(533\) 2.46828e6 0.376337
\(534\) −490923. −0.0745007
\(535\) −3.16668e6 −0.478321
\(536\) −1.28760e6 −0.193584
\(537\) 7.64046e6 1.14336
\(538\) 546752. 0.0814395
\(539\) 0 0
\(540\) 1.62015e6 0.239096
\(541\) −5.93736e6 −0.872168 −0.436084 0.899906i \(-0.643635\pi\)
−0.436084 + 0.899906i \(0.643635\pi\)
\(542\) 1.15521e6 0.168913
\(543\) −1.60937e6 −0.234238
\(544\) 511826. 0.0741524
\(545\) −3.74051e6 −0.539435
\(546\) 0 0
\(547\) −1.27658e7 −1.82422 −0.912112 0.409941i \(-0.865549\pi\)
−0.912112 + 0.409941i \(0.865549\pi\)
\(548\) 535010. 0.0761045
\(549\) −4.45301e6 −0.630555
\(550\) −30177.0 −0.00425372
\(551\) 227503. 0.0319233
\(552\) −3.46948e6 −0.484638
\(553\) 0 0
\(554\) 1.32966e6 0.184063
\(555\) 4.34726e6 0.599078
\(556\) −5.66416e6 −0.777049
\(557\) 7.25744e6 0.991163 0.495582 0.868561i \(-0.334955\pi\)
0.495582 + 0.868561i \(0.334955\pi\)
\(558\) 522482. 0.0710372
\(559\) 1.80743e6 0.244643
\(560\) 0 0
\(561\) 398034. 0.0533965
\(562\) 1.09025e6 0.145609
\(563\) −6.52782e6 −0.867955 −0.433977 0.900924i \(-0.642890\pi\)
−0.433977 + 0.900924i \(0.642890\pi\)
\(564\) −1.55078e7 −2.05283
\(565\) −2.12024e6 −0.279424
\(566\) 1.60698e6 0.210848
\(567\) 0 0
\(568\) 2.40023e6 0.312163
\(569\) 3.51551e6 0.455206 0.227603 0.973754i \(-0.426911\pi\)
0.227603 + 0.973754i \(0.426911\pi\)
\(570\) −544919. −0.0702498
\(571\) −41320.8 −0.00530370 −0.00265185 0.999996i \(-0.500844\pi\)
−0.00265185 + 0.999996i \(0.500844\pi\)
\(572\) 641965. 0.0820392
\(573\) 1.65775e7 2.10927
\(574\) 0 0
\(575\) 2.75826e6 0.347909
\(576\) −4.18292e6 −0.525320
\(577\) 919163. 0.114935 0.0574676 0.998347i \(-0.481697\pi\)
0.0574676 + 0.998347i \(0.481697\pi\)
\(578\) 853942. 0.106319
\(579\) −5.55339e6 −0.688434
\(580\) 101922. 0.0125805
\(581\) 0 0
\(582\) 1.62280e6 0.198591
\(583\) −233779. −0.0284862
\(584\) 970869. 0.117795
\(585\) 918860. 0.111009
\(586\) 722428. 0.0869062
\(587\) 182482. 0.0218587 0.0109294 0.999940i \(-0.496521\pi\)
0.0109294 + 0.999940i \(0.496521\pi\)
\(588\) 0 0
\(589\) −1.05472e7 −1.25271
\(590\) 374775. 0.0443242
\(591\) −1.08177e7 −1.27399
\(592\) 8.78181e6 1.02986
\(593\) 1.59052e7 1.85739 0.928695 0.370846i \(-0.120932\pi\)
0.928695 + 0.370846i \(0.120932\pi\)
\(594\) −99024.0 −0.0115153
\(595\) 0 0
\(596\) 1.19053e7 1.37285
\(597\) 1.24006e7 1.42399
\(598\) 744885. 0.0851797
\(599\) 935648. 0.106548 0.0532740 0.998580i \(-0.483034\pi\)
0.0532740 + 0.998580i \(0.483034\pi\)
\(600\) −491349. −0.0557201
\(601\) −6.79914e6 −0.767835 −0.383917 0.923367i \(-0.625425\pi\)
−0.383917 + 0.923367i \(0.625425\pi\)
\(602\) 0 0
\(603\) −4.40866e6 −0.493757
\(604\) 1.12335e7 1.25292
\(605\) 3.88098e6 0.431075
\(606\) −2.11108e6 −0.233520
\(607\) −3.15831e6 −0.347923 −0.173962 0.984752i \(-0.555657\pi\)
−0.173962 + 0.984752i \(0.555657\pi\)
\(608\) −3.37366e6 −0.370119
\(609\) 0 0
\(610\) −511232. −0.0556280
\(611\) 6.70119e6 0.726188
\(612\) 1.16586e6 0.125825
\(613\) −2.50952e6 −0.269736 −0.134868 0.990864i \(-0.543061\pi\)
−0.134868 + 0.990864i \(0.543061\pi\)
\(614\) 955809. 0.102318
\(615\) −4.51919e6 −0.481807
\(616\) 0 0
\(617\) 1.18777e7 1.25608 0.628042 0.778179i \(-0.283857\pi\)
0.628042 + 0.778179i \(0.283857\pi\)
\(618\) −518683. −0.0546300
\(619\) −6.97529e6 −0.731705 −0.365852 0.930673i \(-0.619222\pi\)
−0.365852 + 0.930673i \(0.619222\pi\)
\(620\) −4.72518e6 −0.493672
\(621\) 9.05108e6 0.941827
\(622\) −2.13495e6 −0.221264
\(623\) 0 0
\(624\) 5.12659e6 0.527068
\(625\) 390625. 0.0400000
\(626\) −270484. −0.0275870
\(627\) −2.62360e6 −0.266520
\(628\) −1.55295e7 −1.57130
\(629\) −2.38349e6 −0.240208
\(630\) 0 0
\(631\) −1.05579e7 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(632\) 4.31005e6 0.429230
\(633\) 5.47126e6 0.542723
\(634\) 550057. 0.0543481
\(635\) 6.20313e6 0.610487
\(636\) −1.89122e6 −0.185395
\(637\) 0 0
\(638\) −6229.46 −0.000605897 0
\(639\) 8.21822e6 0.796207
\(640\) −2.01082e6 −0.194054
\(641\) −1.01477e7 −0.975488 −0.487744 0.872987i \(-0.662180\pi\)
−0.487744 + 0.872987i \(0.662180\pi\)
\(642\) 1.56576e6 0.149929
\(643\) −9.41430e6 −0.897968 −0.448984 0.893540i \(-0.648214\pi\)
−0.448984 + 0.893540i \(0.648214\pi\)
\(644\) 0 0
\(645\) −3.30924e6 −0.313206
\(646\) 298765. 0.0281675
\(647\) 1.08548e7 1.01943 0.509717 0.860342i \(-0.329750\pi\)
0.509717 + 0.860342i \(0.329750\pi\)
\(648\) −2.96229e6 −0.277134
\(649\) 1.80442e6 0.168161
\(650\) 105491. 0.00979334
\(651\) 0 0
\(652\) 4.51891e6 0.416308
\(653\) 7.47330e6 0.685850 0.342925 0.939363i \(-0.388582\pi\)
0.342925 + 0.939363i \(0.388582\pi\)
\(654\) 1.84948e6 0.169086
\(655\) 6.52626e6 0.594376
\(656\) −9.12912e6 −0.828265
\(657\) 3.32419e6 0.300451
\(658\) 0 0
\(659\) −1.62806e7 −1.46035 −0.730176 0.683259i \(-0.760562\pi\)
−0.730176 + 0.683259i \(0.760562\pi\)
\(660\) −1.17538e6 −0.105031
\(661\) 5.36073e6 0.477222 0.238611 0.971115i \(-0.423308\pi\)
0.238611 + 0.971115i \(0.423308\pi\)
\(662\) 943168. 0.0836458
\(663\) −1.39142e6 −0.122935
\(664\) −222243. −0.0195618
\(665\) 0 0
\(666\) −778261. −0.0679891
\(667\) 569390. 0.0495559
\(668\) −9.92037e6 −0.860175
\(669\) −1.89337e7 −1.63558
\(670\) −506140. −0.0435596
\(671\) −2.46141e6 −0.211046
\(672\) 0 0
\(673\) −1.77331e7 −1.50920 −0.754602 0.656183i \(-0.772170\pi\)
−0.754602 + 0.656183i \(0.772170\pi\)
\(674\) −1.09801e6 −0.0931015
\(675\) 1.28181e6 0.108284
\(676\) 9.48830e6 0.798586
\(677\) −5.82373e6 −0.488348 −0.244174 0.969731i \(-0.578517\pi\)
−0.244174 + 0.969731i \(0.578517\pi\)
\(678\) 1.04835e6 0.0875852
\(679\) 0 0
\(680\) 269394. 0.0223417
\(681\) 1.11133e7 0.918281
\(682\) 288803. 0.0237761
\(683\) −1.56080e7 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(684\) −7.68463e6 −0.628034
\(685\) 423282. 0.0344670
\(686\) 0 0
\(687\) 1.52211e6 0.123042
\(688\) −6.68493e6 −0.538426
\(689\) 817229. 0.0655837
\(690\) −1.36382e6 −0.109052
\(691\) −8.52021e6 −0.678821 −0.339410 0.940638i \(-0.610228\pi\)
−0.339410 + 0.940638i \(0.610228\pi\)
\(692\) −1.38126e7 −1.09651
\(693\) 0 0
\(694\) 2.06884e6 0.163053
\(695\) −4.48130e6 −0.351918
\(696\) −101430. −0.00793673
\(697\) 2.47776e6 0.193187
\(698\) 2.54443e6 0.197675
\(699\) −1.61870e7 −1.25306
\(700\) 0 0
\(701\) −5.48253e6 −0.421392 −0.210696 0.977552i \(-0.567573\pi\)
−0.210696 + 0.977552i \(0.567573\pi\)
\(702\) 346162. 0.0265116
\(703\) 1.57106e7 1.19896
\(704\) −2.31212e6 −0.175824
\(705\) −1.22693e7 −0.929706
\(706\) 1.55112e6 0.117121
\(707\) 0 0
\(708\) 1.45973e7 1.09444
\(709\) −1.75446e7 −1.31078 −0.655388 0.755292i \(-0.727495\pi\)
−0.655388 + 0.755292i \(0.727495\pi\)
\(710\) 943502. 0.0702420
\(711\) 1.47573e7 1.09480
\(712\) −1.59973e6 −0.118263
\(713\) −2.63974e7 −1.94463
\(714\) 0 0
\(715\) 507902. 0.0371548
\(716\) 1.23702e7 0.901763
\(717\) −4.92365e6 −0.357675
\(718\) 112277. 0.00812791
\(719\) −2.90492e6 −0.209562 −0.104781 0.994495i \(-0.533414\pi\)
−0.104781 + 0.994495i \(0.533414\pi\)
\(720\) −3.39847e6 −0.244316
\(721\) 0 0
\(722\) −401042. −0.0286317
\(723\) −1.04605e7 −0.744232
\(724\) −2.60563e6 −0.184742
\(725\) 80637.1 0.00569757
\(726\) −1.91894e6 −0.135120
\(727\) −1.58709e7 −1.11369 −0.556846 0.830616i \(-0.687988\pi\)
−0.556846 + 0.830616i \(0.687988\pi\)
\(728\) 0 0
\(729\) −415975. −0.0289900
\(730\) 381638. 0.0265060
\(731\) 1.81437e6 0.125584
\(732\) −1.99123e7 −1.37354
\(733\) −2.22308e7 −1.52825 −0.764127 0.645065i \(-0.776830\pi\)
−0.764127 + 0.645065i \(0.776830\pi\)
\(734\) 2.07025e6 0.141834
\(735\) 0 0
\(736\) −8.44353e6 −0.574552
\(737\) −2.43690e6 −0.165260
\(738\) 809041. 0.0546801
\(739\) 2.25210e7 1.51697 0.758484 0.651691i \(-0.225940\pi\)
0.758484 + 0.651691i \(0.225940\pi\)
\(740\) 7.03836e6 0.472490
\(741\) 9.17142e6 0.613608
\(742\) 0 0
\(743\) −1.71099e7 −1.13704 −0.568521 0.822669i \(-0.692484\pi\)
−0.568521 + 0.822669i \(0.692484\pi\)
\(744\) 4.70236e6 0.311447
\(745\) 9.41907e6 0.621752
\(746\) −2.27402e6 −0.149606
\(747\) −760947. −0.0498945
\(748\) 644430. 0.0421135
\(749\) 0 0
\(750\) −193144. −0.0125380
\(751\) 1.52459e7 0.986401 0.493201 0.869916i \(-0.335827\pi\)
0.493201 + 0.869916i \(0.335827\pi\)
\(752\) −2.47849e7 −1.59824
\(753\) −1.03350e6 −0.0664238
\(754\) 21776.5 0.00139496
\(755\) 8.88758e6 0.567435
\(756\) 0 0
\(757\) −1.18251e7 −0.750004 −0.375002 0.927024i \(-0.622358\pi\)
−0.375002 + 0.927024i \(0.622358\pi\)
\(758\) 281361. 0.0177865
\(759\) −6.56631e6 −0.413730
\(760\) −1.77568e6 −0.111515
\(761\) 4.31523e6 0.270111 0.135056 0.990838i \(-0.456879\pi\)
0.135056 + 0.990838i \(0.456879\pi\)
\(762\) −3.06712e6 −0.191357
\(763\) 0 0
\(764\) 2.68395e7 1.66357
\(765\) 922388. 0.0569849
\(766\) −2.88830e6 −0.177857
\(767\) −6.30776e6 −0.387157
\(768\) −1.79477e7 −1.09801
\(769\) 1.74124e7 1.06180 0.530900 0.847434i \(-0.321854\pi\)
0.530900 + 0.847434i \(0.321854\pi\)
\(770\) 0 0
\(771\) −1.02876e7 −0.623275
\(772\) −8.99113e6 −0.542964
\(773\) 2.39416e7 1.44113 0.720566 0.693386i \(-0.243882\pi\)
0.720566 + 0.693386i \(0.243882\pi\)
\(774\) 592431. 0.0355456
\(775\) −3.73841e6 −0.223580
\(776\) 5.28811e6 0.315243
\(777\) 0 0
\(778\) −1.58632e6 −0.0939600
\(779\) −1.63319e7 −0.964258
\(780\) 4.10881e6 0.241813
\(781\) 4.54264e6 0.266490
\(782\) 747745. 0.0437257
\(783\) 264606. 0.0154239
\(784\) 0 0
\(785\) −1.22865e7 −0.711628
\(786\) −3.22689e6 −0.186307
\(787\) −3.01391e7 −1.73458 −0.867288 0.497807i \(-0.834139\pi\)
−0.867288 + 0.497807i \(0.834139\pi\)
\(788\) −1.75142e7 −1.00479
\(789\) 2.47800e7 1.41713
\(790\) 1.69423e6 0.0965840
\(791\) 0 0
\(792\) 423511. 0.0239912
\(793\) 8.60444e6 0.485892
\(794\) 1.51316e6 0.0851791
\(795\) −1.49627e6 −0.0839639
\(796\) 2.00771e7 1.12310
\(797\) 2.19395e7 1.22344 0.611718 0.791076i \(-0.290479\pi\)
0.611718 + 0.791076i \(0.290479\pi\)
\(798\) 0 0
\(799\) 6.72692e6 0.372777
\(800\) −1.19577e6 −0.0660578
\(801\) −5.47738e6 −0.301642
\(802\) −39856.3 −0.00218807
\(803\) 1.83746e6 0.100561
\(804\) −1.97139e7 −1.07556
\(805\) 0 0
\(806\) −1.00958e6 −0.0547398
\(807\) 1.68485e7 0.910705
\(808\) −6.87921e6 −0.370690
\(809\) −1.09650e7 −0.589029 −0.294514 0.955647i \(-0.595158\pi\)
−0.294514 + 0.955647i \(0.595158\pi\)
\(810\) −1.16444e6 −0.0623600
\(811\) 1.80078e7 0.961409 0.480705 0.876883i \(-0.340381\pi\)
0.480705 + 0.876883i \(0.340381\pi\)
\(812\) 0 0
\(813\) 3.55986e7 1.88889
\(814\) −430185. −0.0227559
\(815\) 3.57521e6 0.188542
\(816\) 5.14627e6 0.270562
\(817\) −1.19593e7 −0.626830
\(818\) 256500. 0.0134030
\(819\) 0 0
\(820\) −7.31673e6 −0.379999
\(821\) 3.23751e7 1.67630 0.838152 0.545437i \(-0.183636\pi\)
0.838152 + 0.545437i \(0.183636\pi\)
\(822\) −209291. −0.0108037
\(823\) 3.24695e7 1.67100 0.835500 0.549491i \(-0.185178\pi\)
0.835500 + 0.549491i \(0.185178\pi\)
\(824\) −1.69019e6 −0.0867197
\(825\) −929922. −0.0475676
\(826\) 0 0
\(827\) 3.10708e7 1.57975 0.789876 0.613267i \(-0.210145\pi\)
0.789876 + 0.613267i \(0.210145\pi\)
\(828\) −1.92330e7 −0.974924
\(829\) −2.82139e6 −0.142586 −0.0712931 0.997455i \(-0.522713\pi\)
−0.0712931 + 0.997455i \(0.522713\pi\)
\(830\) −87361.3 −0.00440173
\(831\) 4.09743e7 2.05830
\(832\) 8.08256e6 0.404800
\(833\) 0 0
\(834\) 2.21577e6 0.110309
\(835\) −7.84867e6 −0.389565
\(836\) −4.24770e6 −0.210203
\(837\) −1.22674e7 −0.605254
\(838\) −1.10030e6 −0.0541255
\(839\) −2.48604e6 −0.121928 −0.0609639 0.998140i \(-0.519417\pi\)
−0.0609639 + 0.998140i \(0.519417\pi\)
\(840\) 0 0
\(841\) −2.04945e7 −0.999188
\(842\) −1.95860e6 −0.0952064
\(843\) 3.35968e7 1.62828
\(844\) 8.85815e6 0.428043
\(845\) 7.50683e6 0.361672
\(846\) 2.19648e6 0.105512
\(847\) 0 0
\(848\) −3.02258e6 −0.144341
\(849\) 4.95201e7 2.35783
\(850\) 105896. 0.00502726
\(851\) 3.93202e7 1.86119
\(852\) 3.67489e7 1.73439
\(853\) 2.80284e6 0.131894 0.0659472 0.997823i \(-0.478993\pi\)
0.0659472 + 0.997823i \(0.478993\pi\)
\(854\) 0 0
\(855\) −6.07983e6 −0.284431
\(856\) 5.10220e6 0.237998
\(857\) −8.57829e6 −0.398978 −0.199489 0.979900i \(-0.563928\pi\)
−0.199489 + 0.979900i \(0.563928\pi\)
\(858\) −251131. −0.0116461
\(859\) −7.92542e6 −0.366471 −0.183235 0.983069i \(-0.558657\pi\)
−0.183235 + 0.983069i \(0.558657\pi\)
\(860\) −5.35778e6 −0.247024
\(861\) 0 0
\(862\) 1.93085e6 0.0885075
\(863\) −3.64157e7 −1.66441 −0.832207 0.554466i \(-0.812923\pi\)
−0.832207 + 0.554466i \(0.812923\pi\)
\(864\) −3.92387e6 −0.178826
\(865\) −1.09281e7 −0.496598
\(866\) −624553. −0.0282992
\(867\) 2.63147e7 1.18892
\(868\) 0 0
\(869\) 8.15716e6 0.366429
\(870\) −39870.8 −0.00178590
\(871\) 8.51874e6 0.380478
\(872\) 6.02677e6 0.268407
\(873\) 1.81061e7 0.804064
\(874\) −4.92869e6 −0.218249
\(875\) 0 0
\(876\) 1.48646e7 0.654475
\(877\) −1.69244e7 −0.743045 −0.371523 0.928424i \(-0.621164\pi\)
−0.371523 + 0.928424i \(0.621164\pi\)
\(878\) 2.33905e6 0.102401
\(879\) 2.22621e7 0.971837
\(880\) −1.87851e6 −0.0817726
\(881\) 4.18273e7 1.81560 0.907799 0.419405i \(-0.137761\pi\)
0.907799 + 0.419405i \(0.137761\pi\)
\(882\) 0 0
\(883\) 1.08396e7 0.467857 0.233928 0.972254i \(-0.424842\pi\)
0.233928 + 0.972254i \(0.424842\pi\)
\(884\) −2.25276e6 −0.0969580
\(885\) 1.15489e7 0.495660
\(886\) −1.22167e6 −0.0522843
\(887\) −1.10623e7 −0.472104 −0.236052 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(888\) −7.00438e6 −0.298083
\(889\) 0 0
\(890\) −628836. −0.0266111
\(891\) −5.60640e6 −0.236587
\(892\) −3.06544e7 −1.28997
\(893\) −4.43399e7 −1.86066
\(894\) −4.65724e6 −0.194888
\(895\) 9.78687e6 0.408400
\(896\) 0 0
\(897\) 2.29541e7 0.952530
\(898\) −2.61168e6 −0.108076
\(899\) −771723. −0.0318465
\(900\) −2.72378e6 −0.112090
\(901\) 820367. 0.0336664
\(902\) 447199. 0.0183014
\(903\) 0 0
\(904\) 3.41616e6 0.139033
\(905\) −2.06149e6 −0.0836680
\(906\) −4.39444e6 −0.177862
\(907\) 9.50001e6 0.383447 0.191724 0.981449i \(-0.438592\pi\)
0.191724 + 0.981449i \(0.438592\pi\)
\(908\) 1.79928e7 0.724243
\(909\) −2.35540e7 −0.945486
\(910\) 0 0
\(911\) 1.64397e7 0.656295 0.328147 0.944627i \(-0.393576\pi\)
0.328147 + 0.944627i \(0.393576\pi\)
\(912\) −3.39212e7 −1.35047
\(913\) −420615. −0.0166997
\(914\) 2.51409e6 0.0995440
\(915\) −1.57539e7 −0.622066
\(916\) 2.46435e6 0.0970429
\(917\) 0 0
\(918\) 347491. 0.0136093
\(919\) −3.69090e7 −1.44160 −0.720799 0.693144i \(-0.756225\pi\)
−0.720799 + 0.693144i \(0.756225\pi\)
\(920\) −4.44416e6 −0.173109
\(921\) 2.94538e7 1.14418
\(922\) 2.15074e6 0.0833222
\(923\) −1.58799e7 −0.613540
\(924\) 0 0
\(925\) 5.56853e6 0.213986
\(926\) −667939. −0.0255982
\(927\) −5.78711e6 −0.221188
\(928\) −246845. −0.00940922
\(929\) 3.53275e7 1.34299 0.671497 0.741007i \(-0.265652\pi\)
0.671497 + 0.741007i \(0.265652\pi\)
\(930\) 1.84845e6 0.0700809
\(931\) 0 0
\(932\) −2.62073e7 −0.988284
\(933\) −6.57897e7 −2.47431
\(934\) 623369. 0.0233818
\(935\) 509852. 0.0190728
\(936\) −1.48048e6 −0.0552349
\(937\) 4.24339e7 1.57893 0.789467 0.613793i \(-0.210357\pi\)
0.789467 + 0.613793i \(0.210357\pi\)
\(938\) 0 0
\(939\) −8.33512e6 −0.308495
\(940\) −1.98644e7 −0.733255
\(941\) 3.71877e6 0.136907 0.0684534 0.997654i \(-0.478194\pi\)
0.0684534 + 0.997654i \(0.478194\pi\)
\(942\) 6.07502e6 0.223059
\(943\) −4.08753e7 −1.49686
\(944\) 2.33297e7 0.852079
\(945\) 0 0
\(946\) 327468. 0.0118971
\(947\) 3.42458e7 1.24089 0.620444 0.784250i \(-0.286952\pi\)
0.620444 + 0.784250i \(0.286952\pi\)
\(948\) 6.59896e7 2.38481
\(949\) −6.42326e6 −0.231521
\(950\) −698002. −0.0250927
\(951\) 1.69503e7 0.607753
\(952\) 0 0
\(953\) −4.77379e7 −1.70267 −0.851336 0.524621i \(-0.824207\pi\)
−0.851336 + 0.524621i \(0.824207\pi\)
\(954\) 267867. 0.00952903
\(955\) 2.12345e7 0.753415
\(956\) −7.97155e6 −0.282097
\(957\) −191965. −0.00677550
\(958\) −3.23311e6 −0.113817
\(959\) 0 0
\(960\) −1.47984e7 −0.518248
\(961\) 7.14860e6 0.249696
\(962\) 1.50381e6 0.0523910
\(963\) 1.74696e7 0.607041
\(964\) −1.69360e7 −0.586972
\(965\) −7.11349e6 −0.245903
\(966\) 0 0
\(967\) 4.11582e6 0.141543 0.0707717 0.997493i \(-0.477454\pi\)
0.0707717 + 0.997493i \(0.477454\pi\)
\(968\) −6.25311e6 −0.214490
\(969\) 9.20663e6 0.314986
\(970\) 2.07869e6 0.0709351
\(971\) −3.11715e7 −1.06099 −0.530493 0.847689i \(-0.677993\pi\)
−0.530493 + 0.847689i \(0.677993\pi\)
\(972\) −2.96066e7 −1.00513
\(973\) 0 0
\(974\) −4.25251e6 −0.143631
\(975\) 3.25076e6 0.109515
\(976\) −3.18242e7 −1.06938
\(977\) −1.40483e7 −0.470857 −0.235428 0.971892i \(-0.575649\pi\)
−0.235428 + 0.971892i \(0.575649\pi\)
\(978\) −1.76776e6 −0.0590984
\(979\) −3.02764e6 −0.100959
\(980\) 0 0
\(981\) 2.06353e7 0.684602
\(982\) −5.46274e6 −0.180772
\(983\) 4.87717e6 0.160985 0.0804923 0.996755i \(-0.474351\pi\)
0.0804923 + 0.996755i \(0.474351\pi\)
\(984\) 7.28141e6 0.239733
\(985\) −1.38567e7 −0.455059
\(986\) 21860.1 0.000716079 0
\(987\) 0 0
\(988\) 1.48488e7 0.483949
\(989\) −2.99315e7 −0.973056
\(990\) 166478. 0.00539843
\(991\) 2.73913e7 0.885990 0.442995 0.896524i \(-0.353916\pi\)
0.442995 + 0.896524i \(0.353916\pi\)
\(992\) 1.14439e7 0.369230
\(993\) 2.90643e7 0.935377
\(994\) 0 0
\(995\) 1.58843e7 0.508641
\(996\) −3.40268e6 −0.108686
\(997\) 3.59221e7 1.14452 0.572261 0.820072i \(-0.306067\pi\)
0.572261 + 0.820072i \(0.306067\pi\)
\(998\) 5.95039e6 0.189112
\(999\) 1.82728e7 0.579284
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 245.6.a.l.1.5 10
7.6 odd 2 245.6.a.m.1.5 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.6.a.l.1.5 10 1.1 even 1 trivial
245.6.a.m.1.5 yes 10 7.6 odd 2