Properties

Label 448.6.a.z.1.1
Level $448$
Weight $6$
Character 448.1
Self dual yes
Analytic conductor $71.852$
Analytic rank $0$
Dimension $2$
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] = [448,6,Mod(1,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("448.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.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: \(71.8519512762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.15207\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.304135 q^{3} -35.5207 q^{5} +49.0000 q^{7} -242.908 q^{9} +O(q^{10})\) \(q-0.304135 q^{3} -35.5207 q^{5} +49.0000 q^{7} -242.908 q^{9} -565.825 q^{11} -983.594 q^{13} +10.8031 q^{15} +200.175 q^{17} -828.747 q^{19} -14.9026 q^{21} +4435.43 q^{23} -1863.28 q^{25} +147.781 q^{27} +3717.74 q^{29} +992.462 q^{31} +172.087 q^{33} -1740.51 q^{35} +8359.69 q^{37} +299.145 q^{39} -13473.0 q^{41} -298.798 q^{43} +8628.24 q^{45} -18736.5 q^{47} +2401.00 q^{49} -60.8802 q^{51} -16036.4 q^{53} +20098.5 q^{55} +252.051 q^{57} -12749.2 q^{59} +34975.9 q^{61} -11902.5 q^{63} +34937.9 q^{65} -11978.9 q^{67} -1348.97 q^{69} -12924.9 q^{71} +81177.2 q^{73} +566.689 q^{75} -27725.4 q^{77} +46998.8 q^{79} +58981.6 q^{81} +111544. q^{83} -7110.36 q^{85} -1130.69 q^{87} -34726.8 q^{89} -48196.1 q^{91} -301.842 q^{93} +29437.7 q^{95} +92655.6 q^{97} +137443. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9} - 972 q^{11} - 78 q^{13} + 2576 q^{15} + 560 q^{17} - 2642 q^{19} + 1274 q^{21} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} + 7808 q^{29} + 5444 q^{31} - 10512 q^{33} + 3038 q^{35} - 576 q^{37} + 24120 q^{39} - 16888 q^{41} + 8396 q^{43} + 52406 q^{45} - 4532 q^{47} + 4802 q^{49} + 9404 q^{51} - 1420 q^{53} - 19512 q^{55} - 47444 q^{57} - 34146 q^{59} - 19106 q^{61} + 10094 q^{63} + 123252 q^{65} - 56952 q^{67} - 58256 q^{69} - 7224 q^{71} + 128828 q^{73} + 168526 q^{75} - 47628 q^{77} + 52808 q^{79} + 92366 q^{81} + 84486 q^{83} + 27980 q^{85} + 106460 q^{87} - 130972 q^{89} - 3822 q^{91} + 116792 q^{93} - 147392 q^{95} + 194624 q^{97} - 44892 q^{99}+O(q^{100}) \) 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 0
\(3\) −0.304135 −0.0195102 −0.00975512 0.999952i \(-0.503105\pi\)
−0.00975512 + 0.999952i \(0.503105\pi\)
\(4\) 0 0
\(5\) −35.5207 −0.635413 −0.317707 0.948189i \(-0.602913\pi\)
−0.317707 + 0.948189i \(0.602913\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) −242.908 −0.999619
\(10\) 0 0
\(11\) −565.825 −1.40994 −0.704969 0.709238i \(-0.749039\pi\)
−0.704969 + 0.709238i \(0.749039\pi\)
\(12\) 0 0
\(13\) −983.594 −1.61420 −0.807100 0.590415i \(-0.798964\pi\)
−0.807100 + 0.590415i \(0.798964\pi\)
\(14\) 0 0
\(15\) 10.8031 0.0123971
\(16\) 0 0
\(17\) 200.175 0.167992 0.0839959 0.996466i \(-0.473232\pi\)
0.0839959 + 0.996466i \(0.473232\pi\)
\(18\) 0 0
\(19\) −828.747 −0.526669 −0.263335 0.964705i \(-0.584822\pi\)
−0.263335 + 0.964705i \(0.584822\pi\)
\(20\) 0 0
\(21\) −14.9026 −0.00737418
\(22\) 0 0
\(23\) 4435.43 1.74830 0.874149 0.485657i \(-0.161420\pi\)
0.874149 + 0.485657i \(0.161420\pi\)
\(24\) 0 0
\(25\) −1863.28 −0.596250
\(26\) 0 0
\(27\) 147.781 0.0390131
\(28\) 0 0
\(29\) 3717.74 0.820889 0.410444 0.911886i \(-0.365374\pi\)
0.410444 + 0.911886i \(0.365374\pi\)
\(30\) 0 0
\(31\) 992.462 0.185485 0.0927427 0.995690i \(-0.470437\pi\)
0.0927427 + 0.995690i \(0.470437\pi\)
\(32\) 0 0
\(33\) 172.087 0.0275083
\(34\) 0 0
\(35\) −1740.51 −0.240164
\(36\) 0 0
\(37\) 8359.69 1.00389 0.501945 0.864900i \(-0.332618\pi\)
0.501945 + 0.864900i \(0.332618\pi\)
\(38\) 0 0
\(39\) 299.145 0.0314934
\(40\) 0 0
\(41\) −13473.0 −1.25171 −0.625855 0.779940i \(-0.715250\pi\)
−0.625855 + 0.779940i \(0.715250\pi\)
\(42\) 0 0
\(43\) −298.798 −0.0246437 −0.0123218 0.999924i \(-0.503922\pi\)
−0.0123218 + 0.999924i \(0.503922\pi\)
\(44\) 0 0
\(45\) 8628.24 0.635171
\(46\) 0 0
\(47\) −18736.5 −1.23721 −0.618606 0.785701i \(-0.712302\pi\)
−0.618606 + 0.785701i \(0.712302\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) −60.8802 −0.00327756
\(52\) 0 0
\(53\) −16036.4 −0.784181 −0.392090 0.919927i \(-0.628248\pi\)
−0.392090 + 0.919927i \(0.628248\pi\)
\(54\) 0 0
\(55\) 20098.5 0.895894
\(56\) 0 0
\(57\) 252.051 0.0102754
\(58\) 0 0
\(59\) −12749.2 −0.476817 −0.238408 0.971165i \(-0.576626\pi\)
−0.238408 + 0.971165i \(0.576626\pi\)
\(60\) 0 0
\(61\) 34975.9 1.20350 0.601748 0.798686i \(-0.294471\pi\)
0.601748 + 0.798686i \(0.294471\pi\)
\(62\) 0 0
\(63\) −11902.5 −0.377821
\(64\) 0 0
\(65\) 34937.9 1.02568
\(66\) 0 0
\(67\) −11978.9 −0.326009 −0.163004 0.986625i \(-0.552118\pi\)
−0.163004 + 0.986625i \(0.552118\pi\)
\(68\) 0 0
\(69\) −1348.97 −0.0341097
\(70\) 0 0
\(71\) −12924.9 −0.304285 −0.152143 0.988359i \(-0.548617\pi\)
−0.152143 + 0.988359i \(0.548617\pi\)
\(72\) 0 0
\(73\) 81177.2 1.78290 0.891450 0.453119i \(-0.149689\pi\)
0.891450 + 0.453119i \(0.149689\pi\)
\(74\) 0 0
\(75\) 566.689 0.0116330
\(76\) 0 0
\(77\) −27725.4 −0.532907
\(78\) 0 0
\(79\) 46998.8 0.847265 0.423632 0.905834i \(-0.360755\pi\)
0.423632 + 0.905834i \(0.360755\pi\)
\(80\) 0 0
\(81\) 58981.6 0.998858
\(82\) 0 0
\(83\) 111544. 1.77726 0.888632 0.458621i \(-0.151656\pi\)
0.888632 + 0.458621i \(0.151656\pi\)
\(84\) 0 0
\(85\) −7110.36 −0.106744
\(86\) 0 0
\(87\) −1130.69 −0.0160157
\(88\) 0 0
\(89\) −34726.8 −0.464719 −0.232359 0.972630i \(-0.574645\pi\)
−0.232359 + 0.972630i \(0.574645\pi\)
\(90\) 0 0
\(91\) −48196.1 −0.610110
\(92\) 0 0
\(93\) −301.842 −0.00361887
\(94\) 0 0
\(95\) 29437.7 0.334653
\(96\) 0 0
\(97\) 92655.6 0.999867 0.499933 0.866064i \(-0.333358\pi\)
0.499933 + 0.866064i \(0.333358\pi\)
\(98\) 0 0
\(99\) 137443. 1.40940
\(100\) 0 0
\(101\) 154765. 1.50963 0.754814 0.655939i \(-0.227727\pi\)
0.754814 + 0.655939i \(0.227727\pi\)
\(102\) 0 0
\(103\) −197146. −1.83102 −0.915512 0.402290i \(-0.868214\pi\)
−0.915512 + 0.402290i \(0.868214\pi\)
\(104\) 0 0
\(105\) 529.350 0.00468565
\(106\) 0 0
\(107\) −63901.3 −0.539573 −0.269786 0.962920i \(-0.586953\pi\)
−0.269786 + 0.962920i \(0.586953\pi\)
\(108\) 0 0
\(109\) −66910.4 −0.539421 −0.269710 0.962941i \(-0.586928\pi\)
−0.269710 + 0.962941i \(0.586928\pi\)
\(110\) 0 0
\(111\) −2542.47 −0.0195861
\(112\) 0 0
\(113\) −100524. −0.740580 −0.370290 0.928916i \(-0.620742\pi\)
−0.370290 + 0.928916i \(0.620742\pi\)
\(114\) 0 0
\(115\) −157549. −1.11089
\(116\) 0 0
\(117\) 238922. 1.61359
\(118\) 0 0
\(119\) 9808.58 0.0634949
\(120\) 0 0
\(121\) 159107. 0.987928
\(122\) 0 0
\(123\) 4097.60 0.0244212
\(124\) 0 0
\(125\) 177187. 1.01428
\(126\) 0 0
\(127\) −155009. −0.852800 −0.426400 0.904535i \(-0.640218\pi\)
−0.426400 + 0.904535i \(0.640218\pi\)
\(128\) 0 0
\(129\) 90.8747 0.000480805 0
\(130\) 0 0
\(131\) 70396.8 0.358405 0.179203 0.983812i \(-0.442648\pi\)
0.179203 + 0.983812i \(0.442648\pi\)
\(132\) 0 0
\(133\) −40608.6 −0.199062
\(134\) 0 0
\(135\) −5249.29 −0.0247894
\(136\) 0 0
\(137\) 175732. 0.799925 0.399962 0.916532i \(-0.369023\pi\)
0.399962 + 0.916532i \(0.369023\pi\)
\(138\) 0 0
\(139\) −44017.9 −0.193238 −0.0966189 0.995321i \(-0.530803\pi\)
−0.0966189 + 0.995321i \(0.530803\pi\)
\(140\) 0 0
\(141\) 5698.43 0.0241383
\(142\) 0 0
\(143\) 556542. 2.27592
\(144\) 0 0
\(145\) −132057. −0.521603
\(146\) 0 0
\(147\) −730.227 −0.00278718
\(148\) 0 0
\(149\) 121136. 0.447001 0.223500 0.974704i \(-0.428252\pi\)
0.223500 + 0.974704i \(0.428252\pi\)
\(150\) 0 0
\(151\) 398463. 1.42215 0.711075 0.703116i \(-0.248209\pi\)
0.711075 + 0.703116i \(0.248209\pi\)
\(152\) 0 0
\(153\) −48624.1 −0.167928
\(154\) 0 0
\(155\) −35252.9 −0.117860
\(156\) 0 0
\(157\) 136739. 0.442733 0.221367 0.975191i \(-0.428948\pi\)
0.221367 + 0.975191i \(0.428948\pi\)
\(158\) 0 0
\(159\) 4877.21 0.0152996
\(160\) 0 0
\(161\) 217336. 0.660795
\(162\) 0 0
\(163\) −642409. −1.89384 −0.946919 0.321473i \(-0.895822\pi\)
−0.946919 + 0.321473i \(0.895822\pi\)
\(164\) 0 0
\(165\) −6112.64 −0.0174791
\(166\) 0 0
\(167\) −290841. −0.806984 −0.403492 0.914983i \(-0.632204\pi\)
−0.403492 + 0.914983i \(0.632204\pi\)
\(168\) 0 0
\(169\) 596163. 1.60564
\(170\) 0 0
\(171\) 201309. 0.526469
\(172\) 0 0
\(173\) 562234. 1.42824 0.714121 0.700022i \(-0.246827\pi\)
0.714121 + 0.700022i \(0.246827\pi\)
\(174\) 0 0
\(175\) −91300.8 −0.225361
\(176\) 0 0
\(177\) 3877.46 0.00930281
\(178\) 0 0
\(179\) 312167. 0.728207 0.364103 0.931359i \(-0.381375\pi\)
0.364103 + 0.931359i \(0.381375\pi\)
\(180\) 0 0
\(181\) −254232. −0.576811 −0.288405 0.957508i \(-0.593125\pi\)
−0.288405 + 0.957508i \(0.593125\pi\)
\(182\) 0 0
\(183\) −10637.4 −0.0234805
\(184\) 0 0
\(185\) −296942. −0.637884
\(186\) 0 0
\(187\) −113264. −0.236858
\(188\) 0 0
\(189\) 7241.29 0.0147456
\(190\) 0 0
\(191\) −189060. −0.374986 −0.187493 0.982266i \(-0.560036\pi\)
−0.187493 + 0.982266i \(0.560036\pi\)
\(192\) 0 0
\(193\) 660426. 1.27624 0.638118 0.769939i \(-0.279713\pi\)
0.638118 + 0.769939i \(0.279713\pi\)
\(194\) 0 0
\(195\) −10625.8 −0.0200113
\(196\) 0 0
\(197\) 786619. 1.44410 0.722052 0.691838i \(-0.243199\pi\)
0.722052 + 0.691838i \(0.243199\pi\)
\(198\) 0 0
\(199\) 360530. 0.645370 0.322685 0.946506i \(-0.395414\pi\)
0.322685 + 0.946506i \(0.395414\pi\)
\(200\) 0 0
\(201\) 3643.19 0.00636051
\(202\) 0 0
\(203\) 182169. 0.310267
\(204\) 0 0
\(205\) 478569. 0.795353
\(206\) 0 0
\(207\) −1.07740e6 −1.74763
\(208\) 0 0
\(209\) 468926. 0.742571
\(210\) 0 0
\(211\) 1.13491e6 1.75491 0.877455 0.479658i \(-0.159239\pi\)
0.877455 + 0.479658i \(0.159239\pi\)
\(212\) 0 0
\(213\) 3930.91 0.00593668
\(214\) 0 0
\(215\) 10613.5 0.0156589
\(216\) 0 0
\(217\) 48630.7 0.0701069
\(218\) 0 0
\(219\) −24688.8 −0.0347848
\(220\) 0 0
\(221\) −196891. −0.271172
\(222\) 0 0
\(223\) −783934. −1.05564 −0.527822 0.849355i \(-0.676991\pi\)
−0.527822 + 0.849355i \(0.676991\pi\)
\(224\) 0 0
\(225\) 452605. 0.596023
\(226\) 0 0
\(227\) −600754. −0.773806 −0.386903 0.922120i \(-0.626455\pi\)
−0.386903 + 0.922120i \(0.626455\pi\)
\(228\) 0 0
\(229\) −1.47992e6 −1.86487 −0.932435 0.361337i \(-0.882320\pi\)
−0.932435 + 0.361337i \(0.882320\pi\)
\(230\) 0 0
\(231\) 8432.26 0.0103971
\(232\) 0 0
\(233\) −1.04350e6 −1.25922 −0.629610 0.776911i \(-0.716785\pi\)
−0.629610 + 0.776911i \(0.716785\pi\)
\(234\) 0 0
\(235\) 665534. 0.786141
\(236\) 0 0
\(237\) −14294.0 −0.0165303
\(238\) 0 0
\(239\) 262948. 0.297767 0.148883 0.988855i \(-0.452432\pi\)
0.148883 + 0.988855i \(0.452432\pi\)
\(240\) 0 0
\(241\) −1.79946e6 −1.99571 −0.997857 0.0654284i \(-0.979159\pi\)
−0.997857 + 0.0654284i \(0.979159\pi\)
\(242\) 0 0
\(243\) −53849.2 −0.0585010
\(244\) 0 0
\(245\) −85285.1 −0.0907733
\(246\) 0 0
\(247\) 815150. 0.850149
\(248\) 0 0
\(249\) −33924.5 −0.0346749
\(250\) 0 0
\(251\) −1.06631e6 −1.06831 −0.534156 0.845386i \(-0.679370\pi\)
−0.534156 + 0.845386i \(0.679370\pi\)
\(252\) 0 0
\(253\) −2.50967e6 −2.46499
\(254\) 0 0
\(255\) 2162.51 0.00208261
\(256\) 0 0
\(257\) 675178. 0.637654 0.318827 0.947813i \(-0.396711\pi\)
0.318827 + 0.947813i \(0.396711\pi\)
\(258\) 0 0
\(259\) 409625. 0.379434
\(260\) 0 0
\(261\) −903067. −0.820576
\(262\) 0 0
\(263\) 301937. 0.269170 0.134585 0.990902i \(-0.457030\pi\)
0.134585 + 0.990902i \(0.457030\pi\)
\(264\) 0 0
\(265\) 569622. 0.498279
\(266\) 0 0
\(267\) 10561.6 0.00906678
\(268\) 0 0
\(269\) 817428. 0.688762 0.344381 0.938830i \(-0.388089\pi\)
0.344381 + 0.938830i \(0.388089\pi\)
\(270\) 0 0
\(271\) 896619. 0.741626 0.370813 0.928708i \(-0.379079\pi\)
0.370813 + 0.928708i \(0.379079\pi\)
\(272\) 0 0
\(273\) 14658.1 0.0119034
\(274\) 0 0
\(275\) 1.05429e6 0.840676
\(276\) 0 0
\(277\) 909620. 0.712296 0.356148 0.934430i \(-0.384090\pi\)
0.356148 + 0.934430i \(0.384090\pi\)
\(278\) 0 0
\(279\) −241077. −0.185415
\(280\) 0 0
\(281\) 762090. 0.575758 0.287879 0.957667i \(-0.407050\pi\)
0.287879 + 0.957667i \(0.407050\pi\)
\(282\) 0 0
\(283\) −931060. −0.691054 −0.345527 0.938409i \(-0.612300\pi\)
−0.345527 + 0.938409i \(0.612300\pi\)
\(284\) 0 0
\(285\) −8953.01 −0.00652916
\(286\) 0 0
\(287\) −660175. −0.473102
\(288\) 0 0
\(289\) −1.37979e6 −0.971779
\(290\) 0 0
\(291\) −28179.8 −0.0195076
\(292\) 0 0
\(293\) −204188. −0.138951 −0.0694755 0.997584i \(-0.522133\pi\)
−0.0694755 + 0.997584i \(0.522133\pi\)
\(294\) 0 0
\(295\) 452859. 0.302976
\(296\) 0 0
\(297\) −83618.3 −0.0550060
\(298\) 0 0
\(299\) −4.36266e6 −2.82210
\(300\) 0 0
\(301\) −14641.1 −0.00931444
\(302\) 0 0
\(303\) −47069.5 −0.0294532
\(304\) 0 0
\(305\) −1.24237e6 −0.764717
\(306\) 0 0
\(307\) 1.00786e6 0.610314 0.305157 0.952302i \(-0.401291\pi\)
0.305157 + 0.952302i \(0.401291\pi\)
\(308\) 0 0
\(309\) 59958.8 0.0357238
\(310\) 0 0
\(311\) 1.77736e6 1.04202 0.521009 0.853551i \(-0.325556\pi\)
0.521009 + 0.853551i \(0.325556\pi\)
\(312\) 0 0
\(313\) −1.73100e6 −0.998705 −0.499352 0.866399i \(-0.666429\pi\)
−0.499352 + 0.866399i \(0.666429\pi\)
\(314\) 0 0
\(315\) 422784. 0.240072
\(316\) 0 0
\(317\) 2.22072e6 1.24121 0.620605 0.784123i \(-0.286887\pi\)
0.620605 + 0.784123i \(0.286887\pi\)
\(318\) 0 0
\(319\) −2.10359e6 −1.15740
\(320\) 0 0
\(321\) 19434.6 0.0105272
\(322\) 0 0
\(323\) −165895. −0.0884761
\(324\) 0 0
\(325\) 1.83271e6 0.962467
\(326\) 0 0
\(327\) 20349.8 0.0105242
\(328\) 0 0
\(329\) −918089. −0.467622
\(330\) 0 0
\(331\) 2.45218e6 1.23022 0.615109 0.788442i \(-0.289112\pi\)
0.615109 + 0.788442i \(0.289112\pi\)
\(332\) 0 0
\(333\) −2.03063e6 −1.00351
\(334\) 0 0
\(335\) 425498. 0.207150
\(336\) 0 0
\(337\) −2.06932e6 −0.992549 −0.496275 0.868166i \(-0.665299\pi\)
−0.496275 + 0.868166i \(0.665299\pi\)
\(338\) 0 0
\(339\) 30572.7 0.0144489
\(340\) 0 0
\(341\) −561560. −0.261523
\(342\) 0 0
\(343\) 117649. 0.0539949
\(344\) 0 0
\(345\) 47916.2 0.0216738
\(346\) 0 0
\(347\) 2.21939e6 0.989487 0.494743 0.869039i \(-0.335262\pi\)
0.494743 + 0.869039i \(0.335262\pi\)
\(348\) 0 0
\(349\) −2.92580e6 −1.28582 −0.642911 0.765941i \(-0.722273\pi\)
−0.642911 + 0.765941i \(0.722273\pi\)
\(350\) 0 0
\(351\) −145357. −0.0629749
\(352\) 0 0
\(353\) −3.03319e6 −1.29558 −0.647788 0.761821i \(-0.724306\pi\)
−0.647788 + 0.761821i \(0.724306\pi\)
\(354\) 0 0
\(355\) 459101. 0.193347
\(356\) 0 0
\(357\) −2983.13 −0.00123880
\(358\) 0 0
\(359\) 3.66596e6 1.50125 0.750623 0.660731i \(-0.229754\pi\)
0.750623 + 0.660731i \(0.229754\pi\)
\(360\) 0 0
\(361\) −1.78928e6 −0.722619
\(362\) 0 0
\(363\) −48389.9 −0.0192747
\(364\) 0 0
\(365\) −2.88347e6 −1.13288
\(366\) 0 0
\(367\) −3.47355e6 −1.34620 −0.673098 0.739553i \(-0.735037\pi\)
−0.673098 + 0.739553i \(0.735037\pi\)
\(368\) 0 0
\(369\) 3.27268e6 1.25123
\(370\) 0 0
\(371\) −785782. −0.296392
\(372\) 0 0
\(373\) −581430. −0.216384 −0.108192 0.994130i \(-0.534506\pi\)
−0.108192 + 0.994130i \(0.534506\pi\)
\(374\) 0 0
\(375\) −53888.8 −0.0197888
\(376\) 0 0
\(377\) −3.65675e6 −1.32508
\(378\) 0 0
\(379\) −1.83859e6 −0.657486 −0.328743 0.944419i \(-0.606625\pi\)
−0.328743 + 0.944419i \(0.606625\pi\)
\(380\) 0 0
\(381\) 47143.6 0.0166383
\(382\) 0 0
\(383\) 139133. 0.0484655 0.0242328 0.999706i \(-0.492286\pi\)
0.0242328 + 0.999706i \(0.492286\pi\)
\(384\) 0 0
\(385\) 984825. 0.338616
\(386\) 0 0
\(387\) 72580.2 0.0246343
\(388\) 0 0
\(389\) −2.21985e6 −0.743788 −0.371894 0.928275i \(-0.621292\pi\)
−0.371894 + 0.928275i \(0.621292\pi\)
\(390\) 0 0
\(391\) 887862. 0.293700
\(392\) 0 0
\(393\) −21410.1 −0.00699258
\(394\) 0 0
\(395\) −1.66943e6 −0.538363
\(396\) 0 0
\(397\) −505998. −0.161129 −0.0805643 0.996749i \(-0.525672\pi\)
−0.0805643 + 0.996749i \(0.525672\pi\)
\(398\) 0 0
\(399\) 12350.5 0.00388375
\(400\) 0 0
\(401\) 2.06663e6 0.641802 0.320901 0.947113i \(-0.396014\pi\)
0.320901 + 0.947113i \(0.396014\pi\)
\(402\) 0 0
\(403\) −976180. −0.299411
\(404\) 0 0
\(405\) −2.09507e6 −0.634688
\(406\) 0 0
\(407\) −4.73012e6 −1.41542
\(408\) 0 0
\(409\) 5.96711e6 1.76383 0.881913 0.471411i \(-0.156255\pi\)
0.881913 + 0.471411i \(0.156255\pi\)
\(410\) 0 0
\(411\) −53446.2 −0.0156067
\(412\) 0 0
\(413\) −624709. −0.180220
\(414\) 0 0
\(415\) −3.96213e6 −1.12930
\(416\) 0 0
\(417\) 13387.4 0.00377012
\(418\) 0 0
\(419\) 569081. 0.158358 0.0791788 0.996860i \(-0.474770\pi\)
0.0791788 + 0.996860i \(0.474770\pi\)
\(420\) 0 0
\(421\) 171377. 0.0471247 0.0235623 0.999722i \(-0.492499\pi\)
0.0235623 + 0.999722i \(0.492499\pi\)
\(422\) 0 0
\(423\) 4.55124e6 1.23674
\(424\) 0 0
\(425\) −372983. −0.100165
\(426\) 0 0
\(427\) 1.71382e6 0.454879
\(428\) 0 0
\(429\) −169264. −0.0444038
\(430\) 0 0
\(431\) 4.75930e6 1.23410 0.617049 0.786924i \(-0.288328\pi\)
0.617049 + 0.786924i \(0.288328\pi\)
\(432\) 0 0
\(433\) 623369. 0.159781 0.0798906 0.996804i \(-0.474543\pi\)
0.0798906 + 0.996804i \(0.474543\pi\)
\(434\) 0 0
\(435\) 40163.0 0.0101766
\(436\) 0 0
\(437\) −3.67585e6 −0.920775
\(438\) 0 0
\(439\) −5.24578e6 −1.29912 −0.649559 0.760311i \(-0.725046\pi\)
−0.649559 + 0.760311i \(0.725046\pi\)
\(440\) 0 0
\(441\) −583221. −0.142803
\(442\) 0 0
\(443\) −5.64926e6 −1.36767 −0.683837 0.729635i \(-0.739690\pi\)
−0.683837 + 0.729635i \(0.739690\pi\)
\(444\) 0 0
\(445\) 1.23352e6 0.295288
\(446\) 0 0
\(447\) −36841.7 −0.00872110
\(448\) 0 0
\(449\) 812156. 0.190118 0.0950591 0.995472i \(-0.469696\pi\)
0.0950591 + 0.995472i \(0.469696\pi\)
\(450\) 0 0
\(451\) 7.62334e6 1.76483
\(452\) 0 0
\(453\) −121186. −0.0277465
\(454\) 0 0
\(455\) 1.71196e6 0.387672
\(456\) 0 0
\(457\) −926920. −0.207612 −0.103806 0.994598i \(-0.533102\pi\)
−0.103806 + 0.994598i \(0.533102\pi\)
\(458\) 0 0
\(459\) 29582.2 0.00655387
\(460\) 0 0
\(461\) 3.67043e6 0.804386 0.402193 0.915555i \(-0.368248\pi\)
0.402193 + 0.915555i \(0.368248\pi\)
\(462\) 0 0
\(463\) −2.70462e6 −0.586345 −0.293173 0.956060i \(-0.594711\pi\)
−0.293173 + 0.956060i \(0.594711\pi\)
\(464\) 0 0
\(465\) 10721.6 0.00229948
\(466\) 0 0
\(467\) 8.24431e6 1.74929 0.874645 0.484763i \(-0.161094\pi\)
0.874645 + 0.484763i \(0.161094\pi\)
\(468\) 0 0
\(469\) −586965. −0.123220
\(470\) 0 0
\(471\) −41587.0 −0.00863784
\(472\) 0 0
\(473\) 169067. 0.0347461
\(474\) 0 0
\(475\) 1.54419e6 0.314027
\(476\) 0 0
\(477\) 3.89535e6 0.783882
\(478\) 0 0
\(479\) −2.55003e6 −0.507816 −0.253908 0.967228i \(-0.581716\pi\)
−0.253908 + 0.967228i \(0.581716\pi\)
\(480\) 0 0
\(481\) −8.22253e6 −1.62048
\(482\) 0 0
\(483\) −66099.4 −0.0128923
\(484\) 0 0
\(485\) −3.29119e6 −0.635328
\(486\) 0 0
\(487\) 7.66815e6 1.46510 0.732552 0.680711i \(-0.238329\pi\)
0.732552 + 0.680711i \(0.238329\pi\)
\(488\) 0 0
\(489\) 195379. 0.0369492
\(490\) 0 0
\(491\) −9.33268e6 −1.74704 −0.873519 0.486790i \(-0.838168\pi\)
−0.873519 + 0.486790i \(0.838168\pi\)
\(492\) 0 0
\(493\) 744200. 0.137903
\(494\) 0 0
\(495\) −4.88207e6 −0.895553
\(496\) 0 0
\(497\) −633320. −0.115009
\(498\) 0 0
\(499\) 1.19065e6 0.214059 0.107030 0.994256i \(-0.465866\pi\)
0.107030 + 0.994256i \(0.465866\pi\)
\(500\) 0 0
\(501\) 88454.9 0.0157445
\(502\) 0 0
\(503\) 2.34213e6 0.412753 0.206377 0.978473i \(-0.433833\pi\)
0.206377 + 0.978473i \(0.433833\pi\)
\(504\) 0 0
\(505\) −5.49736e6 −0.959237
\(506\) 0 0
\(507\) −181314. −0.0313265
\(508\) 0 0
\(509\) 6.69767e6 1.14585 0.572927 0.819606i \(-0.305808\pi\)
0.572927 + 0.819606i \(0.305808\pi\)
\(510\) 0 0
\(511\) 3.97768e6 0.673873
\(512\) 0 0
\(513\) −122473. −0.0205470
\(514\) 0 0
\(515\) 7.00275e6 1.16346
\(516\) 0 0
\(517\) 1.06016e7 1.74439
\(518\) 0 0
\(519\) −170995. −0.0278654
\(520\) 0 0
\(521\) 5.81727e6 0.938911 0.469456 0.882956i \(-0.344450\pi\)
0.469456 + 0.882956i \(0.344450\pi\)
\(522\) 0 0
\(523\) −1.02571e7 −1.63972 −0.819858 0.572568i \(-0.805947\pi\)
−0.819858 + 0.572568i \(0.805947\pi\)
\(524\) 0 0
\(525\) 27767.7 0.00439686
\(526\) 0 0
\(527\) 198666. 0.0311600
\(528\) 0 0
\(529\) 1.32367e7 2.05655
\(530\) 0 0
\(531\) 3.09687e6 0.476635
\(532\) 0 0
\(533\) 1.32519e7 2.02051
\(534\) 0 0
\(535\) 2.26982e6 0.342852
\(536\) 0 0
\(537\) −94940.9 −0.0142075
\(538\) 0 0
\(539\) −1.35855e6 −0.201420
\(540\) 0 0
\(541\) −5.26601e6 −0.773550 −0.386775 0.922174i \(-0.626411\pi\)
−0.386775 + 0.922174i \(0.626411\pi\)
\(542\) 0 0
\(543\) 77320.7 0.0112537
\(544\) 0 0
\(545\) 2.37670e6 0.342755
\(546\) 0 0
\(547\) −4.89900e6 −0.700066 −0.350033 0.936737i \(-0.613830\pi\)
−0.350033 + 0.936737i \(0.613830\pi\)
\(548\) 0 0
\(549\) −8.49592e6 −1.20304
\(550\) 0 0
\(551\) −3.08107e6 −0.432337
\(552\) 0 0
\(553\) 2.30294e6 0.320236
\(554\) 0 0
\(555\) 90310.3 0.0124453
\(556\) 0 0
\(557\) 9.17946e6 1.25366 0.626829 0.779157i \(-0.284352\pi\)
0.626829 + 0.779157i \(0.284352\pi\)
\(558\) 0 0
\(559\) 293895. 0.0397799
\(560\) 0 0
\(561\) 34447.5 0.00462116
\(562\) 0 0
\(563\) 249472. 0.0331704 0.0165852 0.999862i \(-0.494721\pi\)
0.0165852 + 0.999862i \(0.494721\pi\)
\(564\) 0 0
\(565\) 3.57066e6 0.470574
\(566\) 0 0
\(567\) 2.89010e6 0.377533
\(568\) 0 0
\(569\) 1.31498e7 1.70270 0.851351 0.524596i \(-0.175784\pi\)
0.851351 + 0.524596i \(0.175784\pi\)
\(570\) 0 0
\(571\) −1.51038e7 −1.93863 −0.969316 0.245817i \(-0.920944\pi\)
−0.969316 + 0.245817i \(0.920944\pi\)
\(572\) 0 0
\(573\) 57499.6 0.00731608
\(574\) 0 0
\(575\) −8.26445e6 −1.04242
\(576\) 0 0
\(577\) −727822. −0.0910093 −0.0455046 0.998964i \(-0.514490\pi\)
−0.0455046 + 0.998964i \(0.514490\pi\)
\(578\) 0 0
\(579\) −200858. −0.0248997
\(580\) 0 0
\(581\) 5.46567e6 0.671743
\(582\) 0 0
\(583\) 9.07377e6 1.10565
\(584\) 0 0
\(585\) −8.48668e6 −1.02529
\(586\) 0 0
\(587\) −1.24256e7 −1.48841 −0.744207 0.667949i \(-0.767172\pi\)
−0.744207 + 0.667949i \(0.767172\pi\)
\(588\) 0 0
\(589\) −822500. −0.0976895
\(590\) 0 0
\(591\) −239238. −0.0281748
\(592\) 0 0
\(593\) 6.17981e6 0.721669 0.360835 0.932630i \(-0.382492\pi\)
0.360835 + 0.932630i \(0.382492\pi\)
\(594\) 0 0
\(595\) −348408. −0.0403455
\(596\) 0 0
\(597\) −109650. −0.0125913
\(598\) 0 0
\(599\) 1.30603e7 1.48726 0.743630 0.668592i \(-0.233103\pi\)
0.743630 + 0.668592i \(0.233103\pi\)
\(600\) 0 0
\(601\) 9.01833e6 1.01845 0.509225 0.860633i \(-0.329932\pi\)
0.509225 + 0.860633i \(0.329932\pi\)
\(602\) 0 0
\(603\) 2.90976e6 0.325884
\(604\) 0 0
\(605\) −5.65158e6 −0.627742
\(606\) 0 0
\(607\) −1.36835e7 −1.50739 −0.753694 0.657226i \(-0.771730\pi\)
−0.753694 + 0.657226i \(0.771730\pi\)
\(608\) 0 0
\(609\) −55404.0 −0.00605338
\(610\) 0 0
\(611\) 1.84291e7 1.99711
\(612\) 0 0
\(613\) 4.23266e6 0.454949 0.227474 0.973784i \(-0.426953\pi\)
0.227474 + 0.973784i \(0.426953\pi\)
\(614\) 0 0
\(615\) −145549. −0.0155175
\(616\) 0 0
\(617\) 3.32584e6 0.351713 0.175856 0.984416i \(-0.443731\pi\)
0.175856 + 0.984416i \(0.443731\pi\)
\(618\) 0 0
\(619\) 1.19987e7 1.25866 0.629328 0.777140i \(-0.283330\pi\)
0.629328 + 0.777140i \(0.283330\pi\)
\(620\) 0 0
\(621\) 655473. 0.0682065
\(622\) 0 0
\(623\) −1.70162e6 −0.175647
\(624\) 0 0
\(625\) −471051. −0.0482356
\(626\) 0 0
\(627\) −142617. −0.0144878
\(628\) 0 0
\(629\) 1.67340e6 0.168645
\(630\) 0 0
\(631\) 9.99962e6 0.999793 0.499896 0.866085i \(-0.333371\pi\)
0.499896 + 0.866085i \(0.333371\pi\)
\(632\) 0 0
\(633\) −345165. −0.0342387
\(634\) 0 0
\(635\) 5.50602e6 0.541880
\(636\) 0 0
\(637\) −2.36161e6 −0.230600
\(638\) 0 0
\(639\) 3.13955e6 0.304170
\(640\) 0 0
\(641\) −3.16624e6 −0.304368 −0.152184 0.988352i \(-0.548631\pi\)
−0.152184 + 0.988352i \(0.548631\pi\)
\(642\) 0 0
\(643\) −950578. −0.0906693 −0.0453346 0.998972i \(-0.514435\pi\)
−0.0453346 + 0.998972i \(0.514435\pi\)
\(644\) 0 0
\(645\) −3227.93 −0.000305510 0
\(646\) 0 0
\(647\) −1.14671e7 −1.07694 −0.538470 0.842645i \(-0.680997\pi\)
−0.538470 + 0.842645i \(0.680997\pi\)
\(648\) 0 0
\(649\) 7.21379e6 0.672282
\(650\) 0 0
\(651\) −14790.3 −0.00136780
\(652\) 0 0
\(653\) 2.80936e6 0.257824 0.128912 0.991656i \(-0.458851\pi\)
0.128912 + 0.991656i \(0.458851\pi\)
\(654\) 0 0
\(655\) −2.50054e6 −0.227735
\(656\) 0 0
\(657\) −1.97186e7 −1.78222
\(658\) 0 0
\(659\) 4.23794e6 0.380138 0.190069 0.981771i \(-0.439129\pi\)
0.190069 + 0.981771i \(0.439129\pi\)
\(660\) 0 0
\(661\) −1.70903e6 −0.152141 −0.0760703 0.997102i \(-0.524237\pi\)
−0.0760703 + 0.997102i \(0.524237\pi\)
\(662\) 0 0
\(663\) 59881.4 0.00529064
\(664\) 0 0
\(665\) 1.44244e6 0.126487
\(666\) 0 0
\(667\) 1.64898e7 1.43516
\(668\) 0 0
\(669\) 238421. 0.0205959
\(670\) 0 0
\(671\) −1.97903e7 −1.69686
\(672\) 0 0
\(673\) 1.11371e7 0.947836 0.473918 0.880569i \(-0.342839\pi\)
0.473918 + 0.880569i \(0.342839\pi\)
\(674\) 0 0
\(675\) −275358. −0.0232616
\(676\) 0 0
\(677\) −8.09873e6 −0.679118 −0.339559 0.940585i \(-0.610278\pi\)
−0.339559 + 0.940585i \(0.610278\pi\)
\(678\) 0 0
\(679\) 4.54012e6 0.377914
\(680\) 0 0
\(681\) 182710. 0.0150971
\(682\) 0 0
\(683\) 1.49864e7 1.22926 0.614632 0.788814i \(-0.289305\pi\)
0.614632 + 0.788814i \(0.289305\pi\)
\(684\) 0 0
\(685\) −6.24211e6 −0.508283
\(686\) 0 0
\(687\) 450094. 0.0363841
\(688\) 0 0
\(689\) 1.57733e7 1.26582
\(690\) 0 0
\(691\) −2.18569e7 −1.74138 −0.870691 0.491831i \(-0.836328\pi\)
−0.870691 + 0.491831i \(0.836328\pi\)
\(692\) 0 0
\(693\) 6.73471e6 0.532704
\(694\) 0 0
\(695\) 1.56354e6 0.122786
\(696\) 0 0
\(697\) −2.69695e6 −0.210277
\(698\) 0 0
\(699\) 317364. 0.0245677
\(700\) 0 0
\(701\) 1.21819e7 0.936307 0.468154 0.883647i \(-0.344919\pi\)
0.468154 + 0.883647i \(0.344919\pi\)
\(702\) 0 0
\(703\) −6.92807e6 −0.528718
\(704\) 0 0
\(705\) −202412. −0.0153378
\(706\) 0 0
\(707\) 7.58349e6 0.570585
\(708\) 0 0
\(709\) 8.03040e6 0.599959 0.299980 0.953946i \(-0.403020\pi\)
0.299980 + 0.953946i \(0.403020\pi\)
\(710\) 0 0
\(711\) −1.14164e7 −0.846942
\(712\) 0 0
\(713\) 4.40199e6 0.324284
\(714\) 0 0
\(715\) −1.97687e7 −1.44615
\(716\) 0 0
\(717\) −79971.8 −0.00580950
\(718\) 0 0
\(719\) −1.76681e7 −1.27458 −0.637290 0.770624i \(-0.719945\pi\)
−0.637290 + 0.770624i \(0.719945\pi\)
\(720\) 0 0
\(721\) −9.66014e6 −0.692062
\(722\) 0 0
\(723\) 547277. 0.0389369
\(724\) 0 0
\(725\) −6.92720e6 −0.489455
\(726\) 0 0
\(727\) 1.71905e7 1.20629 0.603146 0.797631i \(-0.293914\pi\)
0.603146 + 0.797631i \(0.293914\pi\)
\(728\) 0 0
\(729\) −1.43161e7 −0.997717
\(730\) 0 0
\(731\) −59811.9 −0.00413994
\(732\) 0 0
\(733\) −9.30219e6 −0.639478 −0.319739 0.947506i \(-0.603595\pi\)
−0.319739 + 0.947506i \(0.603595\pi\)
\(734\) 0 0
\(735\) 25938.2 0.00177101
\(736\) 0 0
\(737\) 6.77794e6 0.459652
\(738\) 0 0
\(739\) 1.29625e7 0.873129 0.436564 0.899673i \(-0.356195\pi\)
0.436564 + 0.899673i \(0.356195\pi\)
\(740\) 0 0
\(741\) −247915. −0.0165866
\(742\) 0 0
\(743\) 1.45466e7 0.966693 0.483347 0.875429i \(-0.339421\pi\)
0.483347 + 0.875429i \(0.339421\pi\)
\(744\) 0 0
\(745\) −4.30284e6 −0.284030
\(746\) 0 0
\(747\) −2.70949e7 −1.77659
\(748\) 0 0
\(749\) −3.13116e6 −0.203939
\(750\) 0 0
\(751\) 1.44950e7 0.937816 0.468908 0.883247i \(-0.344648\pi\)
0.468908 + 0.883247i \(0.344648\pi\)
\(752\) 0 0
\(753\) 324301. 0.0208430
\(754\) 0 0
\(755\) −1.41537e7 −0.903653
\(756\) 0 0
\(757\) −2.49809e7 −1.58441 −0.792207 0.610252i \(-0.791068\pi\)
−0.792207 + 0.610252i \(0.791068\pi\)
\(758\) 0 0
\(759\) 763279. 0.0480927
\(760\) 0 0
\(761\) 1.56594e7 0.980196 0.490098 0.871667i \(-0.336961\pi\)
0.490098 + 0.871667i \(0.336961\pi\)
\(762\) 0 0
\(763\) −3.27861e6 −0.203882
\(764\) 0 0
\(765\) 1.72716e6 0.106704
\(766\) 0 0
\(767\) 1.25400e7 0.769677
\(768\) 0 0
\(769\) 1.30420e7 0.795292 0.397646 0.917539i \(-0.369827\pi\)
0.397646 + 0.917539i \(0.369827\pi\)
\(770\) 0 0
\(771\) −205345. −0.0124408
\(772\) 0 0
\(773\) 1.46084e7 0.879336 0.439668 0.898160i \(-0.355096\pi\)
0.439668 + 0.898160i \(0.355096\pi\)
\(774\) 0 0
\(775\) −1.84924e6 −0.110596
\(776\) 0 0
\(777\) −124581. −0.00740286
\(778\) 0 0
\(779\) 1.11657e7 0.659237
\(780\) 0 0
\(781\) 7.31323e6 0.429024
\(782\) 0 0
\(783\) 549413. 0.0320254
\(784\) 0 0
\(785\) −4.85705e6 −0.281319
\(786\) 0 0
\(787\) 2.62167e7 1.50883 0.754416 0.656396i \(-0.227920\pi\)
0.754416 + 0.656396i \(0.227920\pi\)
\(788\) 0 0
\(789\) −91829.4 −0.00525157
\(790\) 0 0
\(791\) −4.92565e6 −0.279913
\(792\) 0 0
\(793\) −3.44021e7 −1.94268
\(794\) 0 0
\(795\) −173242. −0.00972154
\(796\) 0 0
\(797\) −3.41107e6 −0.190215 −0.0951075 0.995467i \(-0.530319\pi\)
−0.0951075 + 0.995467i \(0.530319\pi\)
\(798\) 0 0
\(799\) −3.75059e6 −0.207841
\(800\) 0 0
\(801\) 8.43541e6 0.464542
\(802\) 0 0
\(803\) −4.59321e7 −2.51378
\(804\) 0 0
\(805\) −7.71992e6 −0.419878
\(806\) 0 0
\(807\) −248608. −0.0134379
\(808\) 0 0
\(809\) −1.72606e7 −0.927225 −0.463612 0.886038i \(-0.653447\pi\)
−0.463612 + 0.886038i \(0.653447\pi\)
\(810\) 0 0
\(811\) −2.42616e7 −1.29529 −0.647647 0.761941i \(-0.724247\pi\)
−0.647647 + 0.761941i \(0.724247\pi\)
\(812\) 0 0
\(813\) −272693. −0.0144693
\(814\) 0 0
\(815\) 2.28188e7 1.20337
\(816\) 0 0
\(817\) 247628. 0.0129791
\(818\) 0 0
\(819\) 1.17072e7 0.609878
\(820\) 0 0
\(821\) 3.29927e6 0.170829 0.0854143 0.996346i \(-0.472779\pi\)
0.0854143 + 0.996346i \(0.472779\pi\)
\(822\) 0 0
\(823\) 2.15209e7 1.10754 0.553772 0.832669i \(-0.313188\pi\)
0.553772 + 0.832669i \(0.313188\pi\)
\(824\) 0 0
\(825\) −320646. −0.0164018
\(826\) 0 0
\(827\) 7.31588e6 0.371966 0.185983 0.982553i \(-0.440453\pi\)
0.185983 + 0.982553i \(0.440453\pi\)
\(828\) 0 0
\(829\) 2.33238e7 1.17873 0.589364 0.807867i \(-0.299378\pi\)
0.589364 + 0.807867i \(0.299378\pi\)
\(830\) 0 0
\(831\) −276647. −0.0138971
\(832\) 0 0
\(833\) 480621. 0.0239988
\(834\) 0 0
\(835\) 1.03309e7 0.512768
\(836\) 0 0
\(837\) 146667. 0.00723636
\(838\) 0 0
\(839\) −2.69417e7 −1.32136 −0.660678 0.750669i \(-0.729731\pi\)
−0.660678 + 0.750669i \(0.729731\pi\)
\(840\) 0 0
\(841\) −6.68954e6 −0.326142
\(842\) 0 0
\(843\) −231778. −0.0112332
\(844\) 0 0
\(845\) −2.11761e7 −1.02025
\(846\) 0 0
\(847\) 7.79623e6 0.373402
\(848\) 0 0
\(849\) 283168. 0.0134826
\(850\) 0 0
\(851\) 3.70788e7 1.75510
\(852\) 0 0
\(853\) 1.20124e7 0.565270 0.282635 0.959228i \(-0.408792\pi\)
0.282635 + 0.959228i \(0.408792\pi\)
\(854\) 0 0
\(855\) −7.15063e6 −0.334525
\(856\) 0 0
\(857\) −2.71627e6 −0.126334 −0.0631672 0.998003i \(-0.520120\pi\)
−0.0631672 + 0.998003i \(0.520120\pi\)
\(858\) 0 0
\(859\) 7.59580e6 0.351229 0.175615 0.984459i \(-0.443809\pi\)
0.175615 + 0.984459i \(0.443809\pi\)
\(860\) 0 0
\(861\) 200782. 0.00923033
\(862\) 0 0
\(863\) −1.72463e6 −0.0788260 −0.0394130 0.999223i \(-0.512549\pi\)
−0.0394130 + 0.999223i \(0.512549\pi\)
\(864\) 0 0
\(865\) −1.99709e7 −0.907524
\(866\) 0 0
\(867\) 419641. 0.0189596
\(868\) 0 0
\(869\) −2.65931e7 −1.19459
\(870\) 0 0
\(871\) 1.17823e7 0.526243
\(872\) 0 0
\(873\) −2.25067e7 −0.999486
\(874\) 0 0
\(875\) 8.68217e6 0.383361
\(876\) 0 0
\(877\) −7.20923e6 −0.316512 −0.158256 0.987398i \(-0.550587\pi\)
−0.158256 + 0.987398i \(0.550587\pi\)
\(878\) 0 0
\(879\) 62100.7 0.00271097
\(880\) 0 0
\(881\) −3.88596e7 −1.68678 −0.843391 0.537301i \(-0.819444\pi\)
−0.843391 + 0.537301i \(0.819444\pi\)
\(882\) 0 0
\(883\) −1.52433e7 −0.657927 −0.328964 0.944343i \(-0.606699\pi\)
−0.328964 + 0.944343i \(0.606699\pi\)
\(884\) 0 0
\(885\) −137730. −0.00591113
\(886\) 0 0
\(887\) −2.66999e6 −0.113947 −0.0569733 0.998376i \(-0.518145\pi\)
−0.0569733 + 0.998376i \(0.518145\pi\)
\(888\) 0 0
\(889\) −7.59543e6 −0.322328
\(890\) 0 0
\(891\) −3.33732e7 −1.40833
\(892\) 0 0
\(893\) 1.55278e7 0.651602
\(894\) 0 0
\(895\) −1.10884e7 −0.462712
\(896\) 0 0
\(897\) 1.32684e6 0.0550600
\(898\) 0 0
\(899\) 3.68972e6 0.152263
\(900\) 0 0
\(901\) −3.21008e6 −0.131736
\(902\) 0 0
\(903\) 4452.86 0.000181727 0
\(904\) 0 0
\(905\) 9.03048e6 0.366513
\(906\) 0 0
\(907\) 8.48123e6 0.342327 0.171163 0.985243i \(-0.445247\pi\)
0.171163 + 0.985243i \(0.445247\pi\)
\(908\) 0 0
\(909\) −3.75936e7 −1.50905
\(910\) 0 0
\(911\) −3.89967e7 −1.55680 −0.778399 0.627770i \(-0.783968\pi\)
−0.778399 + 0.627770i \(0.783968\pi\)
\(912\) 0 0
\(913\) −6.31145e7 −2.50583
\(914\) 0 0
\(915\) 377847. 0.0149198
\(916\) 0 0
\(917\) 3.44944e6 0.135464
\(918\) 0 0
\(919\) −2.65313e7 −1.03626 −0.518130 0.855302i \(-0.673372\pi\)
−0.518130 + 0.855302i \(0.673372\pi\)
\(920\) 0 0
\(921\) −306525. −0.0119074
\(922\) 0 0
\(923\) 1.27128e7 0.491177
\(924\) 0 0
\(925\) −1.55765e7 −0.598569
\(926\) 0 0
\(927\) 4.78882e7 1.83033
\(928\) 0 0
\(929\) 9.74701e6 0.370537 0.185269 0.982688i \(-0.440684\pi\)
0.185269 + 0.982688i \(0.440684\pi\)
\(930\) 0 0
\(931\) −1.98982e6 −0.0752385
\(932\) 0 0
\(933\) −540557. −0.0203300
\(934\) 0 0
\(935\) 4.02322e6 0.150503
\(936\) 0 0
\(937\) 3.21792e7 1.19736 0.598682 0.800987i \(-0.295691\pi\)
0.598682 + 0.800987i \(0.295691\pi\)
\(938\) 0 0
\(939\) 526458. 0.0194850
\(940\) 0 0
\(941\) −2.89802e7 −1.06691 −0.533454 0.845829i \(-0.679106\pi\)
−0.533454 + 0.845829i \(0.679106\pi\)
\(942\) 0 0
\(943\) −5.97583e7 −2.18836
\(944\) 0 0
\(945\) −257215. −0.00936952
\(946\) 0 0
\(947\) 3.99329e6 0.144696 0.0723479 0.997379i \(-0.476951\pi\)
0.0723479 + 0.997379i \(0.476951\pi\)
\(948\) 0 0
\(949\) −7.98454e7 −2.87796
\(950\) 0 0
\(951\) −675398. −0.0242163
\(952\) 0 0
\(953\) 3.26669e7 1.16513 0.582566 0.812783i \(-0.302049\pi\)
0.582566 + 0.812783i \(0.302049\pi\)
\(954\) 0 0
\(955\) 6.71553e6 0.238271
\(956\) 0 0
\(957\) 639775. 0.0225812
\(958\) 0 0
\(959\) 8.61086e6 0.302343
\(960\) 0 0
\(961\) −2.76442e7 −0.965595
\(962\) 0 0
\(963\) 1.55221e7 0.539367
\(964\) 0 0
\(965\) −2.34588e7 −0.810937
\(966\) 0 0
\(967\) −9.51237e6 −0.327132 −0.163566 0.986532i \(-0.552300\pi\)
−0.163566 + 0.986532i \(0.552300\pi\)
\(968\) 0 0
\(969\) 50454.3 0.00172619
\(970\) 0 0
\(971\) −2.99431e7 −1.01917 −0.509587 0.860419i \(-0.670202\pi\)
−0.509587 + 0.860419i \(0.670202\pi\)
\(972\) 0 0
\(973\) −2.15688e6 −0.0730370
\(974\) 0 0
\(975\) −557391. −0.0187780
\(976\) 0 0
\(977\) 4.93182e7 1.65299 0.826496 0.562943i \(-0.190331\pi\)
0.826496 + 0.562943i \(0.190331\pi\)
\(978\) 0 0
\(979\) 1.96493e7 0.655225
\(980\) 0 0
\(981\) 1.62530e7 0.539215
\(982\) 0 0
\(983\) 4.39536e7 1.45081 0.725405 0.688322i \(-0.241653\pi\)
0.725405 + 0.688322i \(0.241653\pi\)
\(984\) 0 0
\(985\) −2.79412e7 −0.917603
\(986\) 0 0
\(987\) 279223. 0.00912343
\(988\) 0 0
\(989\) −1.32529e6 −0.0430845
\(990\) 0 0
\(991\) 2.45885e7 0.795331 0.397665 0.917531i \(-0.369820\pi\)
0.397665 + 0.917531i \(0.369820\pi\)
\(992\) 0 0
\(993\) −745793. −0.0240019
\(994\) 0 0
\(995\) −1.28063e7 −0.410077
\(996\) 0 0
\(997\) 2.71682e7 0.865612 0.432806 0.901487i \(-0.357524\pi\)
0.432806 + 0.901487i \(0.357524\pi\)
\(998\) 0 0
\(999\) 1.23541e6 0.0391648
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 448.6.a.z.1.1 2
4.3 odd 2 448.6.a.q.1.2 2
8.3 odd 2 112.6.a.k.1.1 2
8.5 even 2 56.6.a.c.1.2 2
24.5 odd 2 504.6.a.s.1.1 2
24.11 even 2 1008.6.a.bt.1.1 2
56.5 odd 6 392.6.i.g.361.2 4
56.13 odd 2 392.6.a.f.1.1 2
56.27 even 2 784.6.a.p.1.2 2
56.37 even 6 392.6.i.l.361.1 4
56.45 odd 6 392.6.i.g.177.2 4
56.53 even 6 392.6.i.l.177.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.c.1.2 2 8.5 even 2
112.6.a.k.1.1 2 8.3 odd 2
392.6.a.f.1.1 2 56.13 odd 2
392.6.i.g.177.2 4 56.45 odd 6
392.6.i.g.361.2 4 56.5 odd 6
392.6.i.l.177.1 4 56.53 even 6
392.6.i.l.361.1 4 56.37 even 6
448.6.a.q.1.2 2 4.3 odd 2
448.6.a.z.1.1 2 1.1 even 1 trivial
504.6.a.s.1.1 2 24.5 odd 2
784.6.a.p.1.2 2 56.27 even 2
1008.6.a.bt.1.1 2 24.11 even 2