Properties

Label 325.6.a.i.1.9
Level $325$
Weight $6$
Character 325.1
Self dual yes
Analytic conductor $52.125$
Analytic rank $1$
Dimension $9$
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] = [325,6,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, 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("325.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(52.1247414392\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-9.17271\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.1727 q^{2} -23.0461 q^{3} +71.4841 q^{4} -234.441 q^{6} +51.0548 q^{7} +401.660 q^{8} +288.123 q^{9} -512.445 q^{11} -1647.43 q^{12} -169.000 q^{13} +519.366 q^{14} +1798.48 q^{16} -978.964 q^{17} +2930.99 q^{18} +1353.56 q^{19} -1176.61 q^{21} -5212.95 q^{22} -11.3026 q^{23} -9256.71 q^{24} -1719.19 q^{26} -1039.91 q^{27} +3649.61 q^{28} -8050.51 q^{29} +759.478 q^{31} +5442.33 q^{32} +11809.9 q^{33} -9958.72 q^{34} +20596.2 q^{36} +12780.7 q^{37} +13769.3 q^{38} +3894.79 q^{39} -18129.3 q^{41} -11969.4 q^{42} -19533.0 q^{43} -36631.6 q^{44} -114.978 q^{46} -22049.0 q^{47} -41448.1 q^{48} -14200.4 q^{49} +22561.3 q^{51} -12080.8 q^{52} -13861.1 q^{53} -10578.7 q^{54} +20506.7 q^{56} -31194.2 q^{57} -81895.5 q^{58} -18164.6 q^{59} +7623.02 q^{61} +7725.95 q^{62} +14710.1 q^{63} -2188.19 q^{64} +120138. q^{66} -3764.44 q^{67} -69980.4 q^{68} +260.481 q^{69} -75797.0 q^{71} +115728. q^{72} +86605.1 q^{73} +130015. q^{74} +96757.7 q^{76} -26162.8 q^{77} +39620.6 q^{78} +61535.6 q^{79} -46048.0 q^{81} -184425. q^{82} +46849.0 q^{83} -84109.2 q^{84} -198704. q^{86} +185533. q^{87} -205829. q^{88} +1237.02 q^{89} -8628.26 q^{91} -807.955 q^{92} -17503.0 q^{93} -224298. q^{94} -125425. q^{96} -57565.4 q^{97} -144457. q^{98} -147647. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.1727 1.79830 0.899149 0.437642i \(-0.144186\pi\)
0.899149 + 0.437642i \(0.144186\pi\)
\(3\) −23.0461 −1.47841 −0.739204 0.673481i \(-0.764798\pi\)
−0.739204 + 0.673481i \(0.764798\pi\)
\(4\) 71.4841 2.23388
\(5\) 0 0
\(6\) −234.441 −2.65862
\(7\) 51.0548 0.393814 0.196907 0.980422i \(-0.436910\pi\)
0.196907 + 0.980422i \(0.436910\pi\)
\(8\) 401.660 2.21888
\(9\) 288.123 1.18569
\(10\) 0 0
\(11\) −512.445 −1.27692 −0.638462 0.769653i \(-0.720429\pi\)
−0.638462 + 0.769653i \(0.720429\pi\)
\(12\) −1647.43 −3.30258
\(13\) −169.000 −0.277350
\(14\) 519.366 0.708196
\(15\) 0 0
\(16\) 1798.48 1.75633
\(17\) −978.964 −0.821570 −0.410785 0.911732i \(-0.634745\pi\)
−0.410785 + 0.911732i \(0.634745\pi\)
\(18\) 2930.99 2.13223
\(19\) 1353.56 0.860186 0.430093 0.902785i \(-0.358481\pi\)
0.430093 + 0.902785i \(0.358481\pi\)
\(20\) 0 0
\(21\) −1176.61 −0.582218
\(22\) −5212.95 −2.29629
\(23\) −11.3026 −0.00445511 −0.00222755 0.999998i \(-0.500709\pi\)
−0.00222755 + 0.999998i \(0.500709\pi\)
\(24\) −9256.71 −3.28041
\(25\) 0 0
\(26\) −1719.19 −0.498758
\(27\) −1039.91 −0.274528
\(28\) 3649.61 0.879733
\(29\) −8050.51 −1.77758 −0.888788 0.458318i \(-0.848452\pi\)
−0.888788 + 0.458318i \(0.848452\pi\)
\(30\) 0 0
\(31\) 759.478 0.141942 0.0709710 0.997478i \(-0.477390\pi\)
0.0709710 + 0.997478i \(0.477390\pi\)
\(32\) 5442.33 0.939529
\(33\) 11809.9 1.88782
\(34\) −9958.72 −1.47743
\(35\) 0 0
\(36\) 20596.2 2.64869
\(37\) 12780.7 1.53480 0.767399 0.641170i \(-0.221551\pi\)
0.767399 + 0.641170i \(0.221551\pi\)
\(38\) 13769.3 1.54687
\(39\) 3894.79 0.410037
\(40\) 0 0
\(41\) −18129.3 −1.68431 −0.842156 0.539234i \(-0.818714\pi\)
−0.842156 + 0.539234i \(0.818714\pi\)
\(42\) −11969.4 −1.04700
\(43\) −19533.0 −1.61101 −0.805505 0.592589i \(-0.798106\pi\)
−0.805505 + 0.592589i \(0.798106\pi\)
\(44\) −36631.6 −2.85249
\(45\) 0 0
\(46\) −114.978 −0.00801161
\(47\) −22049.0 −1.45594 −0.727971 0.685608i \(-0.759536\pi\)
−0.727971 + 0.685608i \(0.759536\pi\)
\(48\) −41448.1 −2.59658
\(49\) −14200.4 −0.844910
\(50\) 0 0
\(51\) 22561.3 1.21462
\(52\) −12080.8 −0.619566
\(53\) −13861.1 −0.677811 −0.338905 0.940821i \(-0.610057\pi\)
−0.338905 + 0.940821i \(0.610057\pi\)
\(54\) −10578.7 −0.493683
\(55\) 0 0
\(56\) 20506.7 0.873827
\(57\) −31194.2 −1.27171
\(58\) −81895.5 −3.19661
\(59\) −18164.6 −0.679352 −0.339676 0.940542i \(-0.610317\pi\)
−0.339676 + 0.940542i \(0.610317\pi\)
\(60\) 0 0
\(61\) 7623.02 0.262303 0.131151 0.991362i \(-0.458133\pi\)
0.131151 + 0.991362i \(0.458133\pi\)
\(62\) 7725.95 0.255254
\(63\) 14710.1 0.466942
\(64\) −2188.19 −0.0667783
\(65\) 0 0
\(66\) 120138. 3.39486
\(67\) −3764.44 −0.102450 −0.0512252 0.998687i \(-0.516313\pi\)
−0.0512252 + 0.998687i \(0.516313\pi\)
\(68\) −69980.4 −1.83529
\(69\) 260.481 0.00658647
\(70\) 0 0
\(71\) −75797.0 −1.78446 −0.892229 0.451583i \(-0.850859\pi\)
−0.892229 + 0.451583i \(0.850859\pi\)
\(72\) 115728. 2.63091
\(73\) 86605.1 1.90211 0.951057 0.309015i \(-0.0999995\pi\)
0.951057 + 0.309015i \(0.0999995\pi\)
\(74\) 130015. 2.76003
\(75\) 0 0
\(76\) 96757.7 1.92155
\(77\) −26162.8 −0.502871
\(78\) 39620.6 0.737368
\(79\) 61535.6 1.10932 0.554662 0.832076i \(-0.312847\pi\)
0.554662 + 0.832076i \(0.312847\pi\)
\(80\) 0 0
\(81\) −46048.0 −0.779827
\(82\) −184425. −3.02890
\(83\) 46849.0 0.746458 0.373229 0.927739i \(-0.378251\pi\)
0.373229 + 0.927739i \(0.378251\pi\)
\(84\) −84109.2 −1.30060
\(85\) 0 0
\(86\) −198704. −2.89708
\(87\) 185533. 2.62798
\(88\) −205829. −2.83334
\(89\) 1237.02 0.0165540 0.00827698 0.999966i \(-0.497365\pi\)
0.00827698 + 0.999966i \(0.497365\pi\)
\(90\) 0 0
\(91\) −8628.26 −0.109224
\(92\) −807.955 −0.00995217
\(93\) −17503.0 −0.209848
\(94\) −224298. −2.61822
\(95\) 0 0
\(96\) −125425. −1.38901
\(97\) −57565.4 −0.621201 −0.310600 0.950541i \(-0.600530\pi\)
−0.310600 + 0.950541i \(0.600530\pi\)
\(98\) −144457. −1.51940
\(99\) −147647. −1.51404
\(100\) 0 0
\(101\) −13535.7 −0.132031 −0.0660155 0.997819i \(-0.521029\pi\)
−0.0660155 + 0.997819i \(0.521029\pi\)
\(102\) 229510. 2.18424
\(103\) 99640.9 0.925432 0.462716 0.886507i \(-0.346875\pi\)
0.462716 + 0.886507i \(0.346875\pi\)
\(104\) −67880.6 −0.615407
\(105\) 0 0
\(106\) −141005. −1.21891
\(107\) 3447.88 0.0291134 0.0145567 0.999894i \(-0.495366\pi\)
0.0145567 + 0.999894i \(0.495366\pi\)
\(108\) −74337.0 −0.613261
\(109\) 54235.5 0.437238 0.218619 0.975810i \(-0.429845\pi\)
0.218619 + 0.975810i \(0.429845\pi\)
\(110\) 0 0
\(111\) −294546. −2.26906
\(112\) 91821.3 0.691669
\(113\) 103423. 0.761937 0.380969 0.924588i \(-0.375591\pi\)
0.380969 + 0.924588i \(0.375591\pi\)
\(114\) −317330. −2.28691
\(115\) 0 0
\(116\) −575483. −3.97089
\(117\) −48692.8 −0.328852
\(118\) −184783. −1.22168
\(119\) −49980.8 −0.323546
\(120\) 0 0
\(121\) 101549. 0.630537
\(122\) 77546.8 0.471698
\(123\) 417811. 2.49010
\(124\) 54290.6 0.317081
\(125\) 0 0
\(126\) 149641. 0.839702
\(127\) 86597.6 0.476427 0.238214 0.971213i \(-0.423438\pi\)
0.238214 + 0.971213i \(0.423438\pi\)
\(128\) −196415. −1.05962
\(129\) 450160. 2.38173
\(130\) 0 0
\(131\) 212824. 1.08354 0.541768 0.840528i \(-0.317755\pi\)
0.541768 + 0.840528i \(0.317755\pi\)
\(132\) 844217. 4.21715
\(133\) 69105.6 0.338754
\(134\) −38294.6 −0.184236
\(135\) 0 0
\(136\) −393211. −1.82297
\(137\) 392531. 1.78678 0.893392 0.449277i \(-0.148319\pi\)
0.893392 + 0.449277i \(0.148319\pi\)
\(138\) 2649.79 0.0118444
\(139\) 64771.0 0.284344 0.142172 0.989842i \(-0.454591\pi\)
0.142172 + 0.989842i \(0.454591\pi\)
\(140\) 0 0
\(141\) 508143. 2.15248
\(142\) −771061. −3.20899
\(143\) 86603.2 0.354155
\(144\) 518185. 2.08247
\(145\) 0 0
\(146\) 881009. 3.42057
\(147\) 327264. 1.24912
\(148\) 913619. 3.42855
\(149\) −277263. −1.02312 −0.511560 0.859248i \(-0.670932\pi\)
−0.511560 + 0.859248i \(0.670932\pi\)
\(150\) 0 0
\(151\) −56639.1 −0.202150 −0.101075 0.994879i \(-0.532228\pi\)
−0.101075 + 0.994879i \(0.532228\pi\)
\(152\) 543670. 1.90865
\(153\) −282062. −0.974128
\(154\) −266146. −0.904313
\(155\) 0 0
\(156\) 278416. 0.915972
\(157\) 229864. 0.744257 0.372128 0.928181i \(-0.378628\pi\)
0.372128 + 0.928181i \(0.378628\pi\)
\(158\) 625984. 1.99490
\(159\) 319445. 1.00208
\(160\) 0 0
\(161\) −577.051 −0.00175449
\(162\) −468433. −1.40236
\(163\) −400206. −1.17982 −0.589908 0.807470i \(-0.700836\pi\)
−0.589908 + 0.807470i \(0.700836\pi\)
\(164\) −1.29596e6 −3.76255
\(165\) 0 0
\(166\) 476582. 1.34235
\(167\) −276559. −0.767355 −0.383677 0.923467i \(-0.625343\pi\)
−0.383677 + 0.923467i \(0.625343\pi\)
\(168\) −472599. −1.29187
\(169\) 28561.0 0.0769231
\(170\) 0 0
\(171\) 389991. 1.01991
\(172\) −1.39630e6 −3.59880
\(173\) 323519. 0.821834 0.410917 0.911673i \(-0.365209\pi\)
0.410917 + 0.911673i \(0.365209\pi\)
\(174\) 1.88737e6 4.72590
\(175\) 0 0
\(176\) −921624. −2.24270
\(177\) 418623. 1.00436
\(178\) 12583.9 0.0297690
\(179\) −757066. −1.76604 −0.883021 0.469333i \(-0.844494\pi\)
−0.883021 + 0.469333i \(0.844494\pi\)
\(180\) 0 0
\(181\) 640530. 1.45326 0.726629 0.687030i \(-0.241086\pi\)
0.726629 + 0.687030i \(0.241086\pi\)
\(182\) −87772.8 −0.196418
\(183\) −175681. −0.387790
\(184\) −4539.80 −0.00988535
\(185\) 0 0
\(186\) −178053. −0.377370
\(187\) 501665. 1.04908
\(188\) −1.57615e6 −3.25239
\(189\) −53092.4 −0.108113
\(190\) 0 0
\(191\) −611932. −1.21372 −0.606862 0.794807i \(-0.707572\pi\)
−0.606862 + 0.794807i \(0.707572\pi\)
\(192\) 50429.3 0.0987256
\(193\) 134676. 0.260254 0.130127 0.991497i \(-0.458461\pi\)
0.130127 + 0.991497i \(0.458461\pi\)
\(194\) −585596. −1.11710
\(195\) 0 0
\(196\) −1.01510e6 −1.88743
\(197\) 435184. 0.798927 0.399463 0.916749i \(-0.369196\pi\)
0.399463 + 0.916749i \(0.369196\pi\)
\(198\) −1.50197e6 −2.72269
\(199\) 179852. 0.321946 0.160973 0.986959i \(-0.448537\pi\)
0.160973 + 0.986959i \(0.448537\pi\)
\(200\) 0 0
\(201\) 86755.8 0.151464
\(202\) −137694. −0.237431
\(203\) −411017. −0.700035
\(204\) 1.61277e6 2.71330
\(205\) 0 0
\(206\) 1.01362e6 1.66420
\(207\) −3256.54 −0.00528238
\(208\) −303944. −0.487119
\(209\) −693623. −1.09839
\(210\) 0 0
\(211\) −1.03041e6 −1.59333 −0.796663 0.604424i \(-0.793403\pi\)
−0.796663 + 0.604424i \(0.793403\pi\)
\(212\) −990849. −1.51415
\(213\) 1.74683e6 2.63816
\(214\) 35074.3 0.0523546
\(215\) 0 0
\(216\) −417690. −0.609144
\(217\) 38775.0 0.0558988
\(218\) 551722. 0.786284
\(219\) −1.99591e6 −2.81210
\(220\) 0 0
\(221\) 165445. 0.227862
\(222\) −2.99633e6 −4.08044
\(223\) 1.04356e6 1.40525 0.702626 0.711559i \(-0.252011\pi\)
0.702626 + 0.711559i \(0.252011\pi\)
\(224\) 277857. 0.370000
\(225\) 0 0
\(226\) 1.05209e6 1.37019
\(227\) −1.04811e6 −1.35003 −0.675016 0.737803i \(-0.735863\pi\)
−0.675016 + 0.737803i \(0.735863\pi\)
\(228\) −2.22989e6 −2.84084
\(229\) 718967. 0.905984 0.452992 0.891515i \(-0.350357\pi\)
0.452992 + 0.891515i \(0.350357\pi\)
\(230\) 0 0
\(231\) 602950. 0.743449
\(232\) −3.23357e6 −3.94423
\(233\) 714611. 0.862343 0.431172 0.902270i \(-0.358100\pi\)
0.431172 + 0.902270i \(0.358100\pi\)
\(234\) −495338. −0.591373
\(235\) 0 0
\(236\) −1.29848e6 −1.51759
\(237\) −1.41816e6 −1.64003
\(238\) −508441. −0.581832
\(239\) 114365. 0.129508 0.0647541 0.997901i \(-0.479374\pi\)
0.0647541 + 0.997901i \(0.479374\pi\)
\(240\) 0 0
\(241\) 1.01606e6 1.12688 0.563440 0.826157i \(-0.309478\pi\)
0.563440 + 0.826157i \(0.309478\pi\)
\(242\) 1.03303e6 1.13389
\(243\) 1.31393e6 1.42743
\(244\) 544925. 0.585952
\(245\) 0 0
\(246\) 4.25027e6 4.47794
\(247\) −228751. −0.238573
\(248\) 305052. 0.314952
\(249\) −1.07969e6 −1.10357
\(250\) 0 0
\(251\) −1.13208e6 −1.13421 −0.567105 0.823645i \(-0.691937\pi\)
−0.567105 + 0.823645i \(0.691937\pi\)
\(252\) 1.05154e6 1.04309
\(253\) 5791.95 0.00568884
\(254\) 880933. 0.856758
\(255\) 0 0
\(256\) −1.92805e6 −1.83873
\(257\) −357876. −0.337987 −0.168993 0.985617i \(-0.554052\pi\)
−0.168993 + 0.985617i \(0.554052\pi\)
\(258\) 4.57935e6 4.28306
\(259\) 652518. 0.604426
\(260\) 0 0
\(261\) −2.31954e6 −2.10766
\(262\) 2.16500e6 1.94852
\(263\) −305594. −0.272431 −0.136215 0.990679i \(-0.543494\pi\)
−0.136215 + 0.990679i \(0.543494\pi\)
\(264\) 4.74355e6 4.18884
\(265\) 0 0
\(266\) 702991. 0.609180
\(267\) −28508.5 −0.0244735
\(268\) −269098. −0.228862
\(269\) −667878. −0.562751 −0.281375 0.959598i \(-0.590791\pi\)
−0.281375 + 0.959598i \(0.590791\pi\)
\(270\) 0 0
\(271\) −1.05893e6 −0.875875 −0.437937 0.899005i \(-0.644291\pi\)
−0.437937 + 0.899005i \(0.644291\pi\)
\(272\) −1.76065e6 −1.44295
\(273\) 198848. 0.161478
\(274\) 3.99310e6 3.21317
\(275\) 0 0
\(276\) 18620.2 0.0147134
\(277\) −1.91406e6 −1.49884 −0.749420 0.662095i \(-0.769667\pi\)
−0.749420 + 0.662095i \(0.769667\pi\)
\(278\) 658896. 0.511335
\(279\) 218823. 0.168299
\(280\) 0 0
\(281\) 1.08128e6 0.816908 0.408454 0.912779i \(-0.366068\pi\)
0.408454 + 0.912779i \(0.366068\pi\)
\(282\) 5.16919e6 3.87079
\(283\) 859352. 0.637830 0.318915 0.947783i \(-0.396682\pi\)
0.318915 + 0.947783i \(0.396682\pi\)
\(284\) −5.41828e6 −3.98626
\(285\) 0 0
\(286\) 880989. 0.636877
\(287\) −925590. −0.663306
\(288\) 1.56806e6 1.11399
\(289\) −461486. −0.325023
\(290\) 0 0
\(291\) 1.32666e6 0.918389
\(292\) 6.19089e6 4.24909
\(293\) −2.36378e6 −1.60856 −0.804280 0.594250i \(-0.797449\pi\)
−0.804280 + 0.594250i \(0.797449\pi\)
\(294\) 3.32916e6 2.24630
\(295\) 0 0
\(296\) 5.13351e6 3.40553
\(297\) 532896. 0.350551
\(298\) −2.82052e6 −1.83988
\(299\) 1910.14 0.00123562
\(300\) 0 0
\(301\) −997255. −0.634439
\(302\) −576173. −0.363526
\(303\) 311944. 0.195196
\(304\) 2.43435e6 1.51077
\(305\) 0 0
\(306\) −2.86934e6 −1.75177
\(307\) −1.61376e6 −0.977220 −0.488610 0.872502i \(-0.662496\pi\)
−0.488610 + 0.872502i \(0.662496\pi\)
\(308\) −1.87022e6 −1.12335
\(309\) −2.29633e6 −1.36817
\(310\) 0 0
\(311\) 776229. 0.455081 0.227541 0.973769i \(-0.426932\pi\)
0.227541 + 0.973769i \(0.426932\pi\)
\(312\) 1.56438e6 0.909823
\(313\) 656850. 0.378970 0.189485 0.981884i \(-0.439318\pi\)
0.189485 + 0.981884i \(0.439318\pi\)
\(314\) 2.33835e6 1.33840
\(315\) 0 0
\(316\) 4.39882e6 2.47810
\(317\) 1.41713e6 0.792069 0.396034 0.918236i \(-0.370386\pi\)
0.396034 + 0.918236i \(0.370386\pi\)
\(318\) 3.24962e6 1.80204
\(319\) 4.12544e6 2.26983
\(320\) 0 0
\(321\) −79460.3 −0.0430415
\(322\) −5870.18 −0.00315509
\(323\) −1.32508e6 −0.706703
\(324\) −3.29170e6 −1.74204
\(325\) 0 0
\(326\) −4.07118e6 −2.12166
\(327\) −1.24992e6 −0.646416
\(328\) −7.28184e6 −3.73729
\(329\) −1.12571e6 −0.573371
\(330\) 0 0
\(331\) −2.12709e6 −1.06713 −0.533564 0.845760i \(-0.679148\pi\)
−0.533564 + 0.845760i \(0.679148\pi\)
\(332\) 3.34896e6 1.66750
\(333\) 3.68242e6 1.81980
\(334\) −2.81335e6 −1.37993
\(335\) 0 0
\(336\) −2.11612e6 −1.02257
\(337\) 483390. 0.231859 0.115929 0.993257i \(-0.463015\pi\)
0.115929 + 0.993257i \(0.463015\pi\)
\(338\) 290543. 0.138331
\(339\) −2.38349e6 −1.12645
\(340\) 0 0
\(341\) −389190. −0.181249
\(342\) 3.96726e6 1.83411
\(343\) −1.58308e6 −0.726552
\(344\) −7.84564e6 −3.57464
\(345\) 0 0
\(346\) 3.29106e6 1.47790
\(347\) 1.95372e6 0.871041 0.435521 0.900179i \(-0.356564\pi\)
0.435521 + 0.900179i \(0.356564\pi\)
\(348\) 1.32626e7 5.87060
\(349\) 471784. 0.207338 0.103669 0.994612i \(-0.466942\pi\)
0.103669 + 0.994612i \(0.466942\pi\)
\(350\) 0 0
\(351\) 175745. 0.0761403
\(352\) −2.78890e6 −1.19971
\(353\) 4.45914e6 1.90465 0.952324 0.305090i \(-0.0986865\pi\)
0.952324 + 0.305090i \(0.0986865\pi\)
\(354\) 4.25853e6 1.80614
\(355\) 0 0
\(356\) 88427.3 0.0369795
\(357\) 1.15186e6 0.478333
\(358\) −7.70141e6 −3.17587
\(359\) −22798.5 −0.00933619 −0.00466810 0.999989i \(-0.501486\pi\)
−0.00466810 + 0.999989i \(0.501486\pi\)
\(360\) 0 0
\(361\) −643984. −0.260080
\(362\) 6.51593e6 2.61339
\(363\) −2.34030e6 −0.932192
\(364\) −616784. −0.243994
\(365\) 0 0
\(366\) −1.78715e6 −0.697363
\(367\) 1.45349e6 0.563307 0.281654 0.959516i \(-0.409117\pi\)
0.281654 + 0.959516i \(0.409117\pi\)
\(368\) −20327.5 −0.00782465
\(369\) −5.22348e6 −1.99707
\(370\) 0 0
\(371\) −707676. −0.266932
\(372\) −1.25119e6 −0.468775
\(373\) 1.98547e6 0.738910 0.369455 0.929249i \(-0.379544\pi\)
0.369455 + 0.929249i \(0.379544\pi\)
\(374\) 5.10329e6 1.88656
\(375\) 0 0
\(376\) −8.85620e6 −3.23056
\(377\) 1.36054e6 0.493011
\(378\) −540094. −0.194419
\(379\) 1.09576e6 0.391847 0.195924 0.980619i \(-0.437230\pi\)
0.195924 + 0.980619i \(0.437230\pi\)
\(380\) 0 0
\(381\) −1.99574e6 −0.704354
\(382\) −6.22501e6 −2.18264
\(383\) 2.86567e6 0.998225 0.499113 0.866537i \(-0.333659\pi\)
0.499113 + 0.866537i \(0.333659\pi\)
\(384\) 4.52659e6 1.56655
\(385\) 0 0
\(386\) 1.37002e6 0.468015
\(387\) −5.62791e6 −1.91016
\(388\) −4.11501e6 −1.38769
\(389\) 1.42325e6 0.476878 0.238439 0.971158i \(-0.423364\pi\)
0.238439 + 0.971158i \(0.423364\pi\)
\(390\) 0 0
\(391\) 11064.8 0.00366018
\(392\) −5.70374e6 −1.87476
\(393\) −4.90477e6 −1.60191
\(394\) 4.42700e6 1.43671
\(395\) 0 0
\(396\) −1.05544e7 −3.38218
\(397\) −6.03873e6 −1.92296 −0.961478 0.274882i \(-0.911361\pi\)
−0.961478 + 0.274882i \(0.911361\pi\)
\(398\) 1.82958e6 0.578955
\(399\) −1.59261e6 −0.500816
\(400\) 0 0
\(401\) 5.69168e6 1.76758 0.883791 0.467881i \(-0.154983\pi\)
0.883791 + 0.467881i \(0.154983\pi\)
\(402\) 882541. 0.272377
\(403\) −128352. −0.0393676
\(404\) −967584. −0.294941
\(405\) 0 0
\(406\) −4.18116e6 −1.25887
\(407\) −6.54942e6 −1.95982
\(408\) 9.06198e6 2.69509
\(409\) −972884. −0.287576 −0.143788 0.989609i \(-0.545928\pi\)
−0.143788 + 0.989609i \(0.545928\pi\)
\(410\) 0 0
\(411\) −9.04630e6 −2.64160
\(412\) 7.12274e6 2.06730
\(413\) −927388. −0.267539
\(414\) −33127.8 −0.00949930
\(415\) 0 0
\(416\) −919755. −0.260579
\(417\) −1.49272e6 −0.420376
\(418\) −7.05603e6 −1.97524
\(419\) 3.58406e6 0.997334 0.498667 0.866794i \(-0.333823\pi\)
0.498667 + 0.866794i \(0.333823\pi\)
\(420\) 0 0
\(421\) 1.21460e6 0.333986 0.166993 0.985958i \(-0.446594\pi\)
0.166993 + 0.985958i \(0.446594\pi\)
\(422\) −1.04821e7 −2.86528
\(423\) −6.35282e6 −1.72630
\(424\) −5.56746e6 −1.50398
\(425\) 0 0
\(426\) 1.77700e7 4.74419
\(427\) 389192. 0.103299
\(428\) 246469. 0.0650358
\(429\) −1.99587e6 −0.523586
\(430\) 0 0
\(431\) 3.18545e6 0.825996 0.412998 0.910732i \(-0.364482\pi\)
0.412998 + 0.910732i \(0.364482\pi\)
\(432\) −1.87026e6 −0.482162
\(433\) −5.64608e6 −1.44720 −0.723598 0.690221i \(-0.757513\pi\)
−0.723598 + 0.690221i \(0.757513\pi\)
\(434\) 394447. 0.100523
\(435\) 0 0
\(436\) 3.87698e6 0.976735
\(437\) −15298.7 −0.00383222
\(438\) −2.03038e7 −5.05700
\(439\) 7.65369e6 1.89544 0.947719 0.319108i \(-0.103383\pi\)
0.947719 + 0.319108i \(0.103383\pi\)
\(440\) 0 0
\(441\) −4.09146e6 −1.00180
\(442\) 1.68302e6 0.409765
\(443\) −5.65650e6 −1.36943 −0.684713 0.728813i \(-0.740073\pi\)
−0.684713 + 0.728813i \(0.740073\pi\)
\(444\) −2.10554e7 −5.06880
\(445\) 0 0
\(446\) 1.06158e7 2.52706
\(447\) 6.38984e6 1.51259
\(448\) −111718. −0.0262983
\(449\) −1.93752e6 −0.453556 −0.226778 0.973947i \(-0.572819\pi\)
−0.226778 + 0.973947i \(0.572819\pi\)
\(450\) 0 0
\(451\) 9.29029e6 2.15074
\(452\) 7.39307e6 1.70208
\(453\) 1.30531e6 0.298860
\(454\) −1.06622e7 −2.42776
\(455\) 0 0
\(456\) −1.25295e7 −2.82176
\(457\) −3.22536e6 −0.722417 −0.361208 0.932485i \(-0.617636\pi\)
−0.361208 + 0.932485i \(0.617636\pi\)
\(458\) 7.31385e6 1.62923
\(459\) 1.01803e6 0.225544
\(460\) 0 0
\(461\) −4.53894e6 −0.994723 −0.497362 0.867543i \(-0.665698\pi\)
−0.497362 + 0.867543i \(0.665698\pi\)
\(462\) 6.13364e6 1.33694
\(463\) −4.40536e6 −0.955056 −0.477528 0.878617i \(-0.658467\pi\)
−0.477528 + 0.878617i \(0.658467\pi\)
\(464\) −1.44787e7 −3.12202
\(465\) 0 0
\(466\) 7.26954e6 1.55075
\(467\) 7.89872e6 1.67596 0.837982 0.545698i \(-0.183735\pi\)
0.837982 + 0.545698i \(0.183735\pi\)
\(468\) −3.48076e6 −0.734614
\(469\) −192193. −0.0403464
\(470\) 0 0
\(471\) −5.29748e6 −1.10032
\(472\) −7.29599e6 −1.50740
\(473\) 1.00096e7 2.05714
\(474\) −1.44265e7 −2.94927
\(475\) 0 0
\(476\) −3.57283e6 −0.722762
\(477\) −3.99370e6 −0.803674
\(478\) 1.16340e6 0.232895
\(479\) 1.62018e6 0.322644 0.161322 0.986902i \(-0.448424\pi\)
0.161322 + 0.986902i \(0.448424\pi\)
\(480\) 0 0
\(481\) −2.15994e6 −0.425676
\(482\) 1.03361e7 2.02647
\(483\) 13298.8 0.00259385
\(484\) 7.25911e6 1.40854
\(485\) 0 0
\(486\) 1.33662e7 2.56695
\(487\) −4.09572e6 −0.782543 −0.391272 0.920275i \(-0.627965\pi\)
−0.391272 + 0.920275i \(0.627965\pi\)
\(488\) 3.06187e6 0.582018
\(489\) 9.22319e6 1.74425
\(490\) 0 0
\(491\) −7.09337e6 −1.32785 −0.663925 0.747799i \(-0.731111\pi\)
−0.663925 + 0.747799i \(0.731111\pi\)
\(492\) 2.98668e7 5.56258
\(493\) 7.88116e6 1.46040
\(494\) −2.32702e6 −0.429025
\(495\) 0 0
\(496\) 1.36591e6 0.249297
\(497\) −3.86980e6 −0.702745
\(498\) −1.09834e7 −1.98455
\(499\) −1.60514e6 −0.288577 −0.144289 0.989536i \(-0.546089\pi\)
−0.144289 + 0.989536i \(0.546089\pi\)
\(500\) 0 0
\(501\) 6.37360e6 1.13446
\(502\) −1.15164e7 −2.03965
\(503\) −5.32213e6 −0.937920 −0.468960 0.883219i \(-0.655371\pi\)
−0.468960 + 0.883219i \(0.655371\pi\)
\(504\) 5.90845e6 1.03609
\(505\) 0 0
\(506\) 58919.9 0.0102302
\(507\) −658220. −0.113724
\(508\) 6.19035e6 1.06428
\(509\) 2.38983e6 0.408857 0.204429 0.978881i \(-0.434466\pi\)
0.204429 + 0.978881i \(0.434466\pi\)
\(510\) 0 0
\(511\) 4.42161e6 0.749080
\(512\) −1.33282e7 −2.24697
\(513\) −1.40758e6 −0.236145
\(514\) −3.64057e6 −0.607801
\(515\) 0 0
\(516\) 3.21793e7 5.32050
\(517\) 1.12989e7 1.85913
\(518\) 6.63787e6 1.08694
\(519\) −7.45584e6 −1.21501
\(520\) 0 0
\(521\) −970922. −0.156708 −0.0783538 0.996926i \(-0.524966\pi\)
−0.0783538 + 0.996926i \(0.524966\pi\)
\(522\) −2.35960e7 −3.79020
\(523\) −7.93187e6 −1.26801 −0.634003 0.773330i \(-0.718590\pi\)
−0.634003 + 0.773330i \(0.718590\pi\)
\(524\) 1.52136e7 2.42049
\(525\) 0 0
\(526\) −3.10872e6 −0.489912
\(527\) −743501. −0.116615
\(528\) 2.12398e7 3.31563
\(529\) −6.43622e6 −0.999980
\(530\) 0 0
\(531\) −5.23363e6 −0.805502
\(532\) 4.93995e6 0.756734
\(533\) 3.06386e6 0.467144
\(534\) −290009. −0.0440107
\(535\) 0 0
\(536\) −1.51203e6 −0.227325
\(537\) 1.74474e7 2.61093
\(538\) −6.79413e6 −1.01199
\(539\) 7.27692e6 1.07889
\(540\) 0 0
\(541\) −6.96364e6 −1.02292 −0.511462 0.859306i \(-0.670896\pi\)
−0.511462 + 0.859306i \(0.670896\pi\)
\(542\) −1.07721e7 −1.57508
\(543\) −1.47617e7 −2.14851
\(544\) −5.32785e6 −0.771889
\(545\) 0 0
\(546\) 2.02282e6 0.290386
\(547\) −1.01847e7 −1.45539 −0.727693 0.685903i \(-0.759407\pi\)
−0.727693 + 0.685903i \(0.759407\pi\)
\(548\) 2.80597e7 3.99146
\(549\) 2.19637e6 0.311010
\(550\) 0 0
\(551\) −1.08968e7 −1.52905
\(552\) 104625. 0.0146146
\(553\) 3.14169e6 0.436868
\(554\) −1.94711e7 −2.69536
\(555\) 0 0
\(556\) 4.63009e6 0.635189
\(557\) −1.17433e7 −1.60381 −0.801903 0.597454i \(-0.796179\pi\)
−0.801903 + 0.597454i \(0.796179\pi\)
\(558\) 2.22602e6 0.302652
\(559\) 3.30108e6 0.446814
\(560\) 0 0
\(561\) −1.15614e7 −1.55097
\(562\) 1.09996e7 1.46904
\(563\) 1.14861e7 1.52722 0.763611 0.645677i \(-0.223425\pi\)
0.763611 + 0.645677i \(0.223425\pi\)
\(564\) 3.63241e7 4.80837
\(565\) 0 0
\(566\) 8.74194e6 1.14701
\(567\) −2.35097e6 −0.307107
\(568\) −3.04447e7 −3.95950
\(569\) 1.51914e7 1.96706 0.983532 0.180734i \(-0.0578473\pi\)
0.983532 + 0.180734i \(0.0578473\pi\)
\(570\) 0 0
\(571\) −7.26504e6 −0.932498 −0.466249 0.884654i \(-0.654395\pi\)
−0.466249 + 0.884654i \(0.654395\pi\)
\(572\) 6.19075e6 0.791140
\(573\) 1.41027e7 1.79438
\(574\) −9.41576e6 −1.19282
\(575\) 0 0
\(576\) −630468. −0.0791784
\(577\) −5.66789e6 −0.708731 −0.354366 0.935107i \(-0.615303\pi\)
−0.354366 + 0.935107i \(0.615303\pi\)
\(578\) −4.69457e6 −0.584489
\(579\) −3.10377e6 −0.384762
\(580\) 0 0
\(581\) 2.39187e6 0.293966
\(582\) 1.34957e7 1.65154
\(583\) 7.10305e6 0.865513
\(584\) 3.47859e7 4.22056
\(585\) 0 0
\(586\) −2.40460e7 −2.89267
\(587\) −1.09050e6 −0.130627 −0.0653133 0.997865i \(-0.520805\pi\)
−0.0653133 + 0.997865i \(0.520805\pi\)
\(588\) 2.33942e7 2.79039
\(589\) 1.02800e6 0.122096
\(590\) 0 0
\(591\) −1.00293e7 −1.18114
\(592\) 2.29859e7 2.69562
\(593\) −4.95413e6 −0.578536 −0.289268 0.957248i \(-0.593412\pi\)
−0.289268 + 0.957248i \(0.593412\pi\)
\(594\) 5.42100e6 0.630396
\(595\) 0 0
\(596\) −1.98199e7 −2.28553
\(597\) −4.14489e6 −0.475968
\(598\) 19431.3 0.00222202
\(599\) 6.54475e6 0.745292 0.372646 0.927974i \(-0.378451\pi\)
0.372646 + 0.927974i \(0.378451\pi\)
\(600\) 0 0
\(601\) −2.16617e6 −0.244628 −0.122314 0.992491i \(-0.539031\pi\)
−0.122314 + 0.992491i \(0.539031\pi\)
\(602\) −1.01448e7 −1.14091
\(603\) −1.08462e6 −0.121475
\(604\) −4.04879e6 −0.451578
\(605\) 0 0
\(606\) 3.17332e6 0.351020
\(607\) 986604. 0.108685 0.0543427 0.998522i \(-0.482694\pi\)
0.0543427 + 0.998522i \(0.482694\pi\)
\(608\) 7.36651e6 0.808170
\(609\) 9.47235e6 1.03494
\(610\) 0 0
\(611\) 3.72628e6 0.403805
\(612\) −2.01630e7 −2.17608
\(613\) −1.50041e7 −1.61272 −0.806360 0.591425i \(-0.798565\pi\)
−0.806360 + 0.591425i \(0.798565\pi\)
\(614\) −1.64163e7 −1.75733
\(615\) 0 0
\(616\) −1.05085e7 −1.11581
\(617\) −1.27376e6 −0.134703 −0.0673514 0.997729i \(-0.521455\pi\)
−0.0673514 + 0.997729i \(0.521455\pi\)
\(618\) −2.33599e7 −2.46037
\(619\) −2.59865e6 −0.272597 −0.136299 0.990668i \(-0.543521\pi\)
−0.136299 + 0.990668i \(0.543521\pi\)
\(620\) 0 0
\(621\) 11753.7 0.00122305
\(622\) 7.89635e6 0.818372
\(623\) 63155.8 0.00651919
\(624\) 7.00472e6 0.720161
\(625\) 0 0
\(626\) 6.68195e6 0.681502
\(627\) 1.59853e7 1.62387
\(628\) 1.64317e7 1.66258
\(629\) −1.25119e7 −1.26094
\(630\) 0 0
\(631\) −1.92424e7 −1.92392 −0.961959 0.273194i \(-0.911920\pi\)
−0.961959 + 0.273194i \(0.911920\pi\)
\(632\) 2.47164e7 2.46146
\(633\) 2.37470e7 2.35559
\(634\) 1.44161e7 1.42438
\(635\) 0 0
\(636\) 2.28352e7 2.23853
\(637\) 2.39987e6 0.234336
\(638\) 4.19669e7 4.08184
\(639\) −2.18389e7 −2.11582
\(640\) 0 0
\(641\) 1.83373e7 1.76275 0.881375 0.472417i \(-0.156618\pi\)
0.881375 + 0.472417i \(0.156618\pi\)
\(642\) −808326. −0.0774015
\(643\) 1.43215e6 0.136603 0.0683015 0.997665i \(-0.478242\pi\)
0.0683015 + 0.997665i \(0.478242\pi\)
\(644\) −41250.0 −0.00391931
\(645\) 0 0
\(646\) −1.34797e7 −1.27086
\(647\) −1.04396e7 −0.980440 −0.490220 0.871599i \(-0.663084\pi\)
−0.490220 + 0.871599i \(0.663084\pi\)
\(648\) −1.84957e7 −1.73034
\(649\) 9.30834e6 0.867482
\(650\) 0 0
\(651\) −893612. −0.0826412
\(652\) −2.86084e7 −2.63557
\(653\) 1.32094e7 1.21227 0.606137 0.795360i \(-0.292718\pi\)
0.606137 + 0.795360i \(0.292718\pi\)
\(654\) −1.27151e7 −1.16245
\(655\) 0 0
\(656\) −3.26053e7 −2.95821
\(657\) 2.49529e7 2.25532
\(658\) −1.14515e7 −1.03109
\(659\) 3.20415e6 0.287408 0.143704 0.989621i \(-0.454099\pi\)
0.143704 + 0.989621i \(0.454099\pi\)
\(660\) 0 0
\(661\) −2.22430e7 −1.98011 −0.990057 0.140667i \(-0.955075\pi\)
−0.990057 + 0.140667i \(0.955075\pi\)
\(662\) −2.16383e7 −1.91901
\(663\) −3.81286e6 −0.336874
\(664\) 1.88174e7 1.65630
\(665\) 0 0
\(666\) 3.74602e7 3.27254
\(667\) 90991.6 0.00791930
\(668\) −1.97696e7 −1.71418
\(669\) −2.40500e7 −2.07754
\(670\) 0 0
\(671\) −3.90638e6 −0.334941
\(672\) −6.40353e6 −0.547011
\(673\) −1.08386e7 −0.922434 −0.461217 0.887287i \(-0.652587\pi\)
−0.461217 + 0.887287i \(0.652587\pi\)
\(674\) 4.91739e6 0.416951
\(675\) 0 0
\(676\) 2.04166e6 0.171837
\(677\) −1.21301e7 −1.01717 −0.508585 0.861012i \(-0.669831\pi\)
−0.508585 + 0.861012i \(0.669831\pi\)
\(678\) −2.42465e7 −2.02570
\(679\) −2.93899e6 −0.244638
\(680\) 0 0
\(681\) 2.41550e7 1.99590
\(682\) −3.95912e6 −0.325940
\(683\) −1.86818e7 −1.53238 −0.766191 0.642613i \(-0.777851\pi\)
−0.766191 + 0.642613i \(0.777851\pi\)
\(684\) 2.78781e7 2.27837
\(685\) 0 0
\(686\) −1.61042e7 −1.30656
\(687\) −1.65694e7 −1.33941
\(688\) −3.51298e7 −2.82947
\(689\) 2.34253e6 0.187991
\(690\) 0 0
\(691\) −1.25804e7 −1.00230 −0.501151 0.865360i \(-0.667090\pi\)
−0.501151 + 0.865360i \(0.667090\pi\)
\(692\) 2.31264e7 1.83588
\(693\) −7.53810e6 −0.596250
\(694\) 1.98746e7 1.56639
\(695\) 0 0
\(696\) 7.45212e7 5.83118
\(697\) 1.77480e7 1.38378
\(698\) 4.79933e6 0.372856
\(699\) −1.64690e7 −1.27490
\(700\) 0 0
\(701\) 8.22912e6 0.632497 0.316249 0.948676i \(-0.397577\pi\)
0.316249 + 0.948676i \(0.397577\pi\)
\(702\) 1.78780e6 0.136923
\(703\) 1.72994e7 1.32021
\(704\) 1.12133e6 0.0852709
\(705\) 0 0
\(706\) 4.53616e7 3.42512
\(707\) −691060. −0.0519957
\(708\) 2.99248e7 2.24362
\(709\) 2.08389e7 1.55690 0.778449 0.627708i \(-0.216007\pi\)
0.778449 + 0.627708i \(0.216007\pi\)
\(710\) 0 0
\(711\) 1.77298e7 1.31532
\(712\) 496862. 0.0367313
\(713\) −8584.06 −0.000632367 0
\(714\) 1.17176e7 0.860186
\(715\) 0 0
\(716\) −5.41182e7 −3.94512
\(717\) −2.63566e6 −0.191466
\(718\) −231922. −0.0167893
\(719\) 1.98244e7 1.43014 0.715068 0.699055i \(-0.246396\pi\)
0.715068 + 0.699055i \(0.246396\pi\)
\(720\) 0 0
\(721\) 5.08715e6 0.364448
\(722\) −6.55107e6 −0.467702
\(723\) −2.34163e7 −1.66599
\(724\) 4.57877e7 3.24640
\(725\) 0 0
\(726\) −2.38072e7 −1.67636
\(727\) 9.64612e6 0.676888 0.338444 0.940987i \(-0.390099\pi\)
0.338444 + 0.940987i \(0.390099\pi\)
\(728\) −3.46563e6 −0.242356
\(729\) −1.90912e7 −1.33050
\(730\) 0 0
\(731\) 1.91221e7 1.32356
\(732\) −1.25584e7 −0.866276
\(733\) 1.74331e7 1.19844 0.599219 0.800585i \(-0.295478\pi\)
0.599219 + 0.800585i \(0.295478\pi\)
\(734\) 1.47859e7 1.01299
\(735\) 0 0
\(736\) −61512.5 −0.00418570
\(737\) 1.92907e6 0.130821
\(738\) −5.31370e7 −3.59133
\(739\) −3.56942e6 −0.240429 −0.120214 0.992748i \(-0.538358\pi\)
−0.120214 + 0.992748i \(0.538358\pi\)
\(740\) 0 0
\(741\) 5.27182e6 0.352708
\(742\) −7.19899e6 −0.480023
\(743\) 9.04400e6 0.601019 0.300510 0.953779i \(-0.402843\pi\)
0.300510 + 0.953779i \(0.402843\pi\)
\(744\) −7.03026e6 −0.465628
\(745\) 0 0
\(746\) 2.01976e7 1.32878
\(747\) 1.34983e7 0.885069
\(748\) 3.58611e7 2.34352
\(749\) 176031. 0.0114653
\(750\) 0 0
\(751\) −2.27403e7 −1.47128 −0.735642 0.677371i \(-0.763119\pi\)
−0.735642 + 0.677371i \(0.763119\pi\)
\(752\) −3.96547e7 −2.55712
\(753\) 2.60901e7 1.67683
\(754\) 1.38403e7 0.886581
\(755\) 0 0
\(756\) −3.79526e6 −0.241511
\(757\) 1.51736e7 0.962385 0.481192 0.876615i \(-0.340204\pi\)
0.481192 + 0.876615i \(0.340204\pi\)
\(758\) 1.11468e7 0.704659
\(759\) −133482. −0.00841043
\(760\) 0 0
\(761\) 1.74364e7 1.09143 0.545714 0.837971i \(-0.316258\pi\)
0.545714 + 0.837971i \(0.316258\pi\)
\(762\) −2.03021e7 −1.26664
\(763\) 2.76898e6 0.172190
\(764\) −4.37434e7 −2.71131
\(765\) 0 0
\(766\) 2.91516e7 1.79511
\(767\) 3.06981e6 0.188418
\(768\) 4.44340e7 2.71839
\(769\) 1.64867e7 1.00535 0.502675 0.864476i \(-0.332349\pi\)
0.502675 + 0.864476i \(0.332349\pi\)
\(770\) 0 0
\(771\) 8.24764e6 0.499682
\(772\) 9.62722e6 0.581377
\(773\) −1.70607e7 −1.02695 −0.513474 0.858105i \(-0.671642\pi\)
−0.513474 + 0.858105i \(0.671642\pi\)
\(774\) −5.72512e7 −3.43504
\(775\) 0 0
\(776\) −2.31217e7 −1.37837
\(777\) −1.50380e7 −0.893588
\(778\) 1.44783e7 0.857569
\(779\) −2.45391e7 −1.44882
\(780\) 0 0
\(781\) 3.88418e7 2.27862
\(782\) 112559. 0.00658210
\(783\) 8.37180e6 0.487994
\(784\) −2.55392e7 −1.48394
\(785\) 0 0
\(786\) −4.98949e7 −2.88071
\(787\) −2.38884e6 −0.137483 −0.0687417 0.997634i \(-0.521898\pi\)
−0.0687417 + 0.997634i \(0.521898\pi\)
\(788\) 3.11087e7 1.78471
\(789\) 7.04276e6 0.402764
\(790\) 0 0
\(791\) 5.28022e6 0.300062
\(792\) −5.93040e7 −3.35947
\(793\) −1.28829e6 −0.0727496
\(794\) −6.14303e7 −3.45805
\(795\) 0 0
\(796\) 1.28566e7 0.719188
\(797\) −7.46487e6 −0.416272 −0.208136 0.978100i \(-0.566740\pi\)
−0.208136 + 0.978100i \(0.566740\pi\)
\(798\) −1.62012e7 −0.900617
\(799\) 2.15852e7 1.19616
\(800\) 0 0
\(801\) 356414. 0.0196279
\(802\) 5.78999e7 3.17864
\(803\) −4.43804e7 −2.42886
\(804\) 6.20166e6 0.338351
\(805\) 0 0
\(806\) −1.30569e6 −0.0707947
\(807\) 1.53920e7 0.831976
\(808\) −5.43674e6 −0.292961
\(809\) −1.91578e7 −1.02914 −0.514569 0.857449i \(-0.672048\pi\)
−0.514569 + 0.857449i \(0.672048\pi\)
\(810\) 0 0
\(811\) 2.86526e6 0.152972 0.0764859 0.997071i \(-0.475630\pi\)
0.0764859 + 0.997071i \(0.475630\pi\)
\(812\) −2.93812e7 −1.56379
\(813\) 2.44041e7 1.29490
\(814\) −6.66253e7 −3.52435
\(815\) 0 0
\(816\) 4.05762e7 2.13327
\(817\) −2.64391e7 −1.38577
\(818\) −9.89687e6 −0.517147
\(819\) −2.48600e6 −0.129507
\(820\) 0 0
\(821\) −1.42195e7 −0.736254 −0.368127 0.929775i \(-0.620001\pi\)
−0.368127 + 0.929775i \(0.620001\pi\)
\(822\) −9.20255e7 −4.75038
\(823\) −8.97879e6 −0.462081 −0.231041 0.972944i \(-0.574213\pi\)
−0.231041 + 0.972944i \(0.574213\pi\)
\(824\) 4.00218e7 2.05342
\(825\) 0 0
\(826\) −9.43406e6 −0.481114
\(827\) −1.46294e7 −0.743813 −0.371907 0.928270i \(-0.621296\pi\)
−0.371907 + 0.928270i \(0.621296\pi\)
\(828\) −232790. −0.0118002
\(829\) 6.70195e6 0.338700 0.169350 0.985556i \(-0.445833\pi\)
0.169350 + 0.985556i \(0.445833\pi\)
\(830\) 0 0
\(831\) 4.41115e7 2.21590
\(832\) 369804. 0.0185210
\(833\) 1.39017e7 0.694153
\(834\) −1.51850e7 −0.755961
\(835\) 0 0
\(836\) −4.95830e7 −2.45368
\(837\) −789788. −0.0389670
\(838\) 3.64597e7 1.79350
\(839\) −1.00902e7 −0.494876 −0.247438 0.968904i \(-0.579589\pi\)
−0.247438 + 0.968904i \(0.579589\pi\)
\(840\) 0 0
\(841\) 4.42995e7 2.15978
\(842\) 1.23558e7 0.600607
\(843\) −2.49193e7 −1.20772
\(844\) −7.36580e7 −3.55930
\(845\) 0 0
\(846\) −6.46254e7 −3.10440
\(847\) 5.18455e6 0.248315
\(848\) −2.49290e7 −1.19046
\(849\) −1.98047e7 −0.942973
\(850\) 0 0
\(851\) −144455. −0.00683769
\(852\) 1.24870e8 5.89332
\(853\) −8.26689e6 −0.389018 −0.194509 0.980901i \(-0.562311\pi\)
−0.194509 + 0.980901i \(0.562311\pi\)
\(854\) 3.95914e6 0.185762
\(855\) 0 0
\(856\) 1.38488e6 0.0645992
\(857\) 1.34533e6 0.0625714 0.0312857 0.999510i \(-0.490040\pi\)
0.0312857 + 0.999510i \(0.490040\pi\)
\(858\) −2.03034e7 −0.941564
\(859\) −2.57272e7 −1.18962 −0.594812 0.803865i \(-0.702774\pi\)
−0.594812 + 0.803865i \(0.702774\pi\)
\(860\) 0 0
\(861\) 2.13313e7 0.980637
\(862\) 3.24047e7 1.48539
\(863\) 9.60150e6 0.438846 0.219423 0.975630i \(-0.429583\pi\)
0.219423 + 0.975630i \(0.429583\pi\)
\(864\) −5.65954e6 −0.257927
\(865\) 0 0
\(866\) −5.74360e7 −2.60249
\(867\) 1.06355e7 0.480517
\(868\) 2.77179e6 0.124871
\(869\) −3.15336e7 −1.41652
\(870\) 0 0
\(871\) 636191. 0.0284146
\(872\) 2.17843e7 0.970178
\(873\) −1.65859e7 −0.736552
\(874\) −155629. −0.00689148
\(875\) 0 0
\(876\) −1.42676e8 −6.28189
\(877\) 3.88143e6 0.170409 0.0852046 0.996363i \(-0.472846\pi\)
0.0852046 + 0.996363i \(0.472846\pi\)
\(878\) 7.78588e7 3.40856
\(879\) 5.44759e7 2.37811
\(880\) 0 0
\(881\) −1.47028e7 −0.638204 −0.319102 0.947720i \(-0.603381\pi\)
−0.319102 + 0.947720i \(0.603381\pi\)
\(882\) −4.16213e7 −1.80154
\(883\) 4.37837e7 1.88978 0.944889 0.327392i \(-0.106170\pi\)
0.944889 + 0.327392i \(0.106170\pi\)
\(884\) 1.18267e7 0.509017
\(885\) 0 0
\(886\) −5.75420e7 −2.46264
\(887\) −9.31739e6 −0.397636 −0.198818 0.980036i \(-0.563710\pi\)
−0.198818 + 0.980036i \(0.563710\pi\)
\(888\) −1.18307e8 −5.03477
\(889\) 4.42122e6 0.187624
\(890\) 0 0
\(891\) 2.35971e7 0.995781
\(892\) 7.45978e7 3.13916
\(893\) −2.98445e7 −1.25238
\(894\) 6.50020e7 2.72009
\(895\) 0 0
\(896\) −1.00279e7 −0.417292
\(897\) −44021.2 −0.00182676
\(898\) −1.97098e7 −0.815628
\(899\) −6.11418e6 −0.252313
\(900\) 0 0
\(901\) 1.35695e7 0.556869
\(902\) 9.45074e7 3.86767
\(903\) 2.29828e7 0.937960
\(904\) 4.15407e7 1.69065
\(905\) 0 0
\(906\) 1.32785e7 0.537440
\(907\) −1.32431e7 −0.534529 −0.267265 0.963623i \(-0.586120\pi\)
−0.267265 + 0.963623i \(0.586120\pi\)
\(908\) −7.49235e7 −3.01581
\(909\) −3.89993e6 −0.156548
\(910\) 0 0
\(911\) 2.78672e7 1.11249 0.556246 0.831018i \(-0.312241\pi\)
0.556246 + 0.831018i \(0.312241\pi\)
\(912\) −5.61023e7 −2.23354
\(913\) −2.40075e7 −0.953171
\(914\) −3.28107e7 −1.29912
\(915\) 0 0
\(916\) 5.13947e7 2.02386
\(917\) 1.08657e7 0.426712
\(918\) 1.03562e7 0.405595
\(919\) −1.81646e7 −0.709474 −0.354737 0.934966i \(-0.615430\pi\)
−0.354737 + 0.934966i \(0.615430\pi\)
\(920\) 0 0
\(921\) 3.71908e7 1.44473
\(922\) −4.61733e7 −1.78881
\(923\) 1.28097e7 0.494920
\(924\) 4.31013e7 1.66077
\(925\) 0 0
\(926\) −4.48145e7 −1.71748
\(927\) 2.87088e7 1.09728
\(928\) −4.38136e7 −1.67009
\(929\) 3.97985e7 1.51296 0.756480 0.654017i \(-0.226917\pi\)
0.756480 + 0.654017i \(0.226917\pi\)
\(930\) 0 0
\(931\) −1.92210e7 −0.726780
\(932\) 5.10833e7 1.92637
\(933\) −1.78891e7 −0.672796
\(934\) 8.03515e7 3.01388
\(935\) 0 0
\(936\) −1.95580e7 −0.729683
\(937\) 3.23323e7 1.20306 0.601531 0.798850i \(-0.294558\pi\)
0.601531 + 0.798850i \(0.294558\pi\)
\(938\) −1.95512e6 −0.0725550
\(939\) −1.51378e7 −0.560273
\(940\) 0 0
\(941\) −1.24433e7 −0.458102 −0.229051 0.973414i \(-0.573562\pi\)
−0.229051 + 0.973414i \(0.573562\pi\)
\(942\) −5.38898e7 −1.97870
\(943\) 204908. 0.00750379
\(944\) −3.26687e7 −1.19317
\(945\) 0 0
\(946\) 1.01825e8 3.69935
\(947\) 3.32452e7 1.20463 0.602316 0.798258i \(-0.294245\pi\)
0.602316 + 0.798258i \(0.294245\pi\)
\(948\) −1.01376e8 −3.66364
\(949\) −1.46363e7 −0.527552
\(950\) 0 0
\(951\) −3.26594e7 −1.17100
\(952\) −2.00753e7 −0.717910
\(953\) 4.55359e7 1.62413 0.812066 0.583566i \(-0.198343\pi\)
0.812066 + 0.583566i \(0.198343\pi\)
\(954\) −4.06268e7 −1.44525
\(955\) 0 0
\(956\) 8.17526e6 0.289306
\(957\) −9.50754e7 −3.35574
\(958\) 1.64816e7 0.580210
\(959\) 2.00406e7 0.703661
\(960\) 0 0
\(961\) −2.80523e7 −0.979852
\(962\) −2.19725e7 −0.765493
\(963\) 993414. 0.0345195
\(964\) 7.26323e7 2.51731
\(965\) 0 0
\(966\) 135285. 0.00466451
\(967\) 1.52863e7 0.525697 0.262849 0.964837i \(-0.415338\pi\)
0.262849 + 0.964837i \(0.415338\pi\)
\(968\) 4.07881e7 1.39909
\(969\) 3.05380e7 1.04480
\(970\) 0 0
\(971\) −1.92581e7 −0.655490 −0.327745 0.944766i \(-0.606289\pi\)
−0.327745 + 0.944766i \(0.606289\pi\)
\(972\) 9.39248e7 3.18871
\(973\) 3.30687e6 0.111979
\(974\) −4.16646e7 −1.40725
\(975\) 0 0
\(976\) 1.37099e7 0.460690
\(977\) 795487. 0.0266622 0.0133311 0.999911i \(-0.495756\pi\)
0.0133311 + 0.999911i \(0.495756\pi\)
\(978\) 9.38248e7 3.13668
\(979\) −633905. −0.0211382
\(980\) 0 0
\(981\) 1.56265e7 0.518429
\(982\) −7.21588e7 −2.38787
\(983\) 7.51037e6 0.247901 0.123950 0.992288i \(-0.460444\pi\)
0.123950 + 0.992288i \(0.460444\pi\)
\(984\) 1.67818e8 5.52524
\(985\) 0 0
\(986\) 8.01728e7 2.62624
\(987\) 2.59432e7 0.847676
\(988\) −1.63521e7 −0.532942
\(989\) 220774. 0.00717723
\(990\) 0 0
\(991\) 5.90076e7 1.90864 0.954319 0.298788i \(-0.0965824\pi\)
0.954319 + 0.298788i \(0.0965824\pi\)
\(992\) 4.13333e6 0.133359
\(993\) 4.90212e7 1.57765
\(994\) −3.93664e7 −1.26375
\(995\) 0 0
\(996\) −7.71805e7 −2.46524
\(997\) 4.32622e7 1.37839 0.689193 0.724577i \(-0.257965\pi\)
0.689193 + 0.724577i \(0.257965\pi\)
\(998\) −1.63287e7 −0.518948
\(999\) −1.32908e7 −0.421345
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 325.6.a.i.1.9 yes 9
5.2 odd 4 325.6.b.h.274.18 18
5.3 odd 4 325.6.b.h.274.1 18
5.4 even 2 325.6.a.h.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.h.1.1 9 5.4 even 2
325.6.a.i.1.9 yes 9 1.1 even 1 trivial
325.6.b.h.274.1 18 5.3 odd 4
325.6.b.h.274.18 18 5.2 odd 4