Properties

Label 325.6.a.j.1.1
Level $325$
Weight $6$
Character 325.1
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $11$
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(11.2252\) 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)
 
\(f(q)\) \(=\) \(q-11.2252 q^{2} -19.1897 q^{3} +94.0044 q^{4} +215.408 q^{6} -70.9909 q^{7} -696.009 q^{8} +125.246 q^{9} -161.025 q^{11} -1803.92 q^{12} -169.000 q^{13} +796.885 q^{14} +4804.68 q^{16} +121.397 q^{17} -1405.90 q^{18} -3111.76 q^{19} +1362.30 q^{21} +1807.53 q^{22} +3100.00 q^{23} +13356.2 q^{24} +1897.05 q^{26} +2259.68 q^{27} -6673.46 q^{28} +3172.08 q^{29} -3515.11 q^{31} -31661.0 q^{32} +3090.02 q^{33} -1362.70 q^{34} +11773.6 q^{36} -6991.30 q^{37} +34930.1 q^{38} +3243.06 q^{39} -19551.6 q^{41} -15292.0 q^{42} -13398.3 q^{43} -15137.0 q^{44} -34798.0 q^{46} -7637.52 q^{47} -92200.5 q^{48} -11767.3 q^{49} -2329.57 q^{51} -15886.7 q^{52} -27750.4 q^{53} -25365.2 q^{54} +49410.3 q^{56} +59713.9 q^{57} -35607.1 q^{58} +33042.3 q^{59} -33195.2 q^{61} +39457.6 q^{62} -8891.30 q^{63} +201651. q^{64} -34686.0 q^{66} -24591.9 q^{67} +11411.8 q^{68} -59488.1 q^{69} +16641.4 q^{71} -87172.0 q^{72} +5252.97 q^{73} +78478.5 q^{74} -292519. q^{76} +11431.3 q^{77} -36403.9 q^{78} -3121.54 q^{79} -73797.2 q^{81} +219470. q^{82} +25125.8 q^{83} +128062. q^{84} +150399. q^{86} -60871.3 q^{87} +112075. q^{88} +634.739 q^{89} +11997.5 q^{91} +291413. q^{92} +67453.9 q^{93} +85732.5 q^{94} +607566. q^{96} -156159. q^{97} +132090. q^{98} -20167.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.2252 −1.98435 −0.992174 0.124864i \(-0.960151\pi\)
−0.992174 + 0.124864i \(0.960151\pi\)
\(3\) −19.1897 −1.23102 −0.615511 0.788129i \(-0.711050\pi\)
−0.615511 + 0.788129i \(0.711050\pi\)
\(4\) 94.0044 2.93764
\(5\) 0 0
\(6\) 215.408 2.44277
\(7\) −70.9909 −0.547593 −0.273796 0.961788i \(-0.588279\pi\)
−0.273796 + 0.961788i \(0.588279\pi\)
\(8\) −696.009 −3.84494
\(9\) 125.246 0.515414
\(10\) 0 0
\(11\) −161.025 −0.401246 −0.200623 0.979669i \(-0.564297\pi\)
−0.200623 + 0.979669i \(0.564297\pi\)
\(12\) −1803.92 −3.61629
\(13\) −169.000 −0.277350
\(14\) 796.885 1.08661
\(15\) 0 0
\(16\) 4804.68 4.69207
\(17\) 121.397 0.101879 0.0509396 0.998702i \(-0.483778\pi\)
0.0509396 + 0.998702i \(0.483778\pi\)
\(18\) −1405.90 −1.02276
\(19\) −3111.76 −1.97753 −0.988764 0.149485i \(-0.952238\pi\)
−0.988764 + 0.149485i \(0.952238\pi\)
\(20\) 0 0
\(21\) 1362.30 0.674099
\(22\) 1807.53 0.796211
\(23\) 3100.00 1.22192 0.610959 0.791662i \(-0.290784\pi\)
0.610959 + 0.791662i \(0.290784\pi\)
\(24\) 13356.2 4.73321
\(25\) 0 0
\(26\) 1897.05 0.550359
\(27\) 2259.68 0.596536
\(28\) −6673.46 −1.60863
\(29\) 3172.08 0.700404 0.350202 0.936674i \(-0.386113\pi\)
0.350202 + 0.936674i \(0.386113\pi\)
\(30\) 0 0
\(31\) −3515.11 −0.656953 −0.328476 0.944512i \(-0.606535\pi\)
−0.328476 + 0.944512i \(0.606535\pi\)
\(32\) −31661.0 −5.46575
\(33\) 3090.02 0.493942
\(34\) −1362.70 −0.202164
\(35\) 0 0
\(36\) 11773.6 1.51410
\(37\) −6991.30 −0.839563 −0.419782 0.907625i \(-0.637893\pi\)
−0.419782 + 0.907625i \(0.637893\pi\)
\(38\) 34930.1 3.92410
\(39\) 3243.06 0.341424
\(40\) 0 0
\(41\) −19551.6 −1.81644 −0.908222 0.418488i \(-0.862560\pi\)
−0.908222 + 0.418488i \(0.862560\pi\)
\(42\) −15292.0 −1.33765
\(43\) −13398.3 −1.10504 −0.552522 0.833498i \(-0.686335\pi\)
−0.552522 + 0.833498i \(0.686335\pi\)
\(44\) −15137.0 −1.17871
\(45\) 0 0
\(46\) −34798.0 −2.42471
\(47\) −7637.52 −0.504322 −0.252161 0.967685i \(-0.581141\pi\)
−0.252161 + 0.967685i \(0.581141\pi\)
\(48\) −92200.5 −5.77604
\(49\) −11767.3 −0.700142
\(50\) 0 0
\(51\) −2329.57 −0.125415
\(52\) −15886.7 −0.814754
\(53\) −27750.4 −1.35700 −0.678500 0.734600i \(-0.737370\pi\)
−0.678500 + 0.734600i \(0.737370\pi\)
\(54\) −25365.2 −1.18374
\(55\) 0 0
\(56\) 49410.3 2.10546
\(57\) 59713.9 2.43438
\(58\) −35607.1 −1.38985
\(59\) 33042.3 1.23578 0.617889 0.786265i \(-0.287988\pi\)
0.617889 + 0.786265i \(0.287988\pi\)
\(60\) 0 0
\(61\) −33195.2 −1.14222 −0.571110 0.820873i \(-0.693487\pi\)
−0.571110 + 0.820873i \(0.693487\pi\)
\(62\) 39457.6 1.30362
\(63\) −8891.30 −0.282237
\(64\) 201651. 6.15389
\(65\) 0 0
\(66\) −34686.0 −0.980153
\(67\) −24591.9 −0.669275 −0.334637 0.942347i \(-0.608614\pi\)
−0.334637 + 0.942347i \(0.608614\pi\)
\(68\) 11411.8 0.299284
\(69\) −59488.1 −1.50421
\(70\) 0 0
\(71\) 16641.4 0.391781 0.195891 0.980626i \(-0.437240\pi\)
0.195891 + 0.980626i \(0.437240\pi\)
\(72\) −87172.0 −1.98174
\(73\) 5252.97 0.115371 0.0576856 0.998335i \(-0.481628\pi\)
0.0576856 + 0.998335i \(0.481628\pi\)
\(74\) 78478.5 1.66599
\(75\) 0 0
\(76\) −292519. −5.80926
\(77\) 11431.3 0.219719
\(78\) −36403.9 −0.677504
\(79\) −3121.54 −0.0562731 −0.0281366 0.999604i \(-0.508957\pi\)
−0.0281366 + 0.999604i \(0.508957\pi\)
\(80\) 0 0
\(81\) −73797.2 −1.24976
\(82\) 219470. 3.60446
\(83\) 25125.8 0.400336 0.200168 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(84\) 128062. 1.98026
\(85\) 0 0
\(86\) 150399. 2.19279
\(87\) −60871.3 −0.862213
\(88\) 112075. 1.54277
\(89\) 634.739 0.00849416 0.00424708 0.999991i \(-0.498648\pi\)
0.00424708 + 0.999991i \(0.498648\pi\)
\(90\) 0 0
\(91\) 11997.5 0.151875
\(92\) 291413. 3.58955
\(93\) 67453.9 0.808723
\(94\) 85732.5 1.00075
\(95\) 0 0
\(96\) 607566. 6.72846
\(97\) −156159. −1.68514 −0.842572 0.538584i \(-0.818959\pi\)
−0.842572 + 0.538584i \(0.818959\pi\)
\(98\) 132090. 1.38933
\(99\) −20167.6 −0.206808
\(100\) 0 0
\(101\) −165081. −1.61026 −0.805128 0.593101i \(-0.797903\pi\)
−0.805128 + 0.593101i \(0.797903\pi\)
\(102\) 26149.9 0.248868
\(103\) −75861.6 −0.704578 −0.352289 0.935891i \(-0.614597\pi\)
−0.352289 + 0.935891i \(0.614597\pi\)
\(104\) 117626. 1.06640
\(105\) 0 0
\(106\) 311503. 2.69276
\(107\) 141728. 1.19673 0.598366 0.801223i \(-0.295817\pi\)
0.598366 + 0.801223i \(0.295817\pi\)
\(108\) 212419. 1.75241
\(109\) 11820.9 0.0952981 0.0476490 0.998864i \(-0.484827\pi\)
0.0476490 + 0.998864i \(0.484827\pi\)
\(110\) 0 0
\(111\) 134161. 1.03352
\(112\) −341089. −2.56934
\(113\) 43552.7 0.320862 0.160431 0.987047i \(-0.448712\pi\)
0.160431 + 0.987047i \(0.448712\pi\)
\(114\) −670298. −4.83065
\(115\) 0 0
\(116\) 298189. 2.05753
\(117\) −21166.5 −0.142950
\(118\) −370906. −2.45222
\(119\) −8618.08 −0.0557883
\(120\) 0 0
\(121\) −135122. −0.839002
\(122\) 372621. 2.26656
\(123\) 375189. 2.23608
\(124\) −330435. −1.92989
\(125\) 0 0
\(126\) 99806.3 0.560056
\(127\) −229412. −1.26214 −0.631068 0.775727i \(-0.717383\pi\)
−0.631068 + 0.775727i \(0.717383\pi\)
\(128\) −1.25041e6 −6.74570
\(129\) 257110. 1.36033
\(130\) 0 0
\(131\) −161719. −0.823347 −0.411674 0.911331i \(-0.635056\pi\)
−0.411674 + 0.911331i \(0.635056\pi\)
\(132\) 290475. 1.45102
\(133\) 220907. 1.08288
\(134\) 276048. 1.32807
\(135\) 0 0
\(136\) −84493.4 −0.391720
\(137\) −233597. −1.06332 −0.531662 0.846956i \(-0.678433\pi\)
−0.531662 + 0.846956i \(0.678433\pi\)
\(138\) 667764. 2.98487
\(139\) 97992.9 0.430187 0.215094 0.976593i \(-0.430994\pi\)
0.215094 + 0.976593i \(0.430994\pi\)
\(140\) 0 0
\(141\) 146562. 0.620831
\(142\) −186802. −0.777430
\(143\) 27213.1 0.111286
\(144\) 601765. 2.41836
\(145\) 0 0
\(146\) −58965.4 −0.228937
\(147\) 225811. 0.861890
\(148\) −657212. −2.46633
\(149\) 267805. 0.988219 0.494110 0.869400i \(-0.335494\pi\)
0.494110 + 0.869400i \(0.335494\pi\)
\(150\) 0 0
\(151\) −125041. −0.446283 −0.223141 0.974786i \(-0.571631\pi\)
−0.223141 + 0.974786i \(0.571631\pi\)
\(152\) 2.16582e6 7.60348
\(153\) 15204.4 0.0525099
\(154\) −128318. −0.436000
\(155\) 0 0
\(156\) 304862. 1.00298
\(157\) 323793. 1.04838 0.524189 0.851602i \(-0.324368\pi\)
0.524189 + 0.851602i \(0.324368\pi\)
\(158\) 35039.8 0.111665
\(159\) 532523. 1.67050
\(160\) 0 0
\(161\) −220072. −0.669113
\(162\) 828386. 2.47996
\(163\) 483015. 1.42394 0.711970 0.702210i \(-0.247803\pi\)
0.711970 + 0.702210i \(0.247803\pi\)
\(164\) −1.83793e6 −5.33605
\(165\) 0 0
\(166\) −282041. −0.794406
\(167\) −272660. −0.756538 −0.378269 0.925696i \(-0.623480\pi\)
−0.378269 + 0.925696i \(0.623480\pi\)
\(168\) −948171. −2.59187
\(169\) 28561.0 0.0769231
\(170\) 0 0
\(171\) −389734. −1.01924
\(172\) −1.25950e6 −3.24622
\(173\) −248584. −0.631477 −0.315738 0.948846i \(-0.602252\pi\)
−0.315738 + 0.948846i \(0.602252\pi\)
\(174\) 683290. 1.71093
\(175\) 0 0
\(176\) −773671. −1.88267
\(177\) −634073. −1.52127
\(178\) −7125.05 −0.0168554
\(179\) 29132.9 0.0679597 0.0339798 0.999423i \(-0.489182\pi\)
0.0339798 + 0.999423i \(0.489182\pi\)
\(180\) 0 0
\(181\) 583931. 1.32485 0.662423 0.749130i \(-0.269528\pi\)
0.662423 + 0.749130i \(0.269528\pi\)
\(182\) −134674. −0.301373
\(183\) 637006. 1.40610
\(184\) −2.15763e6 −4.69820
\(185\) 0 0
\(186\) −757181. −1.60479
\(187\) −19547.9 −0.0408786
\(188\) −717960. −1.48151
\(189\) −160417. −0.326659
\(190\) 0 0
\(191\) 708228. 1.40472 0.702359 0.711822i \(-0.252130\pi\)
0.702359 + 0.711822i \(0.252130\pi\)
\(192\) −3.86962e6 −7.57557
\(193\) 596277. 1.15227 0.576136 0.817354i \(-0.304560\pi\)
0.576136 + 0.817354i \(0.304560\pi\)
\(194\) 1.75291e6 3.34391
\(195\) 0 0
\(196\) −1.10618e6 −2.05676
\(197\) 23822.5 0.0437342 0.0218671 0.999761i \(-0.493039\pi\)
0.0218671 + 0.999761i \(0.493039\pi\)
\(198\) 226385. 0.410378
\(199\) 470656. 0.842503 0.421251 0.906944i \(-0.361591\pi\)
0.421251 + 0.906944i \(0.361591\pi\)
\(200\) 0 0
\(201\) 471911. 0.823892
\(202\) 1.85307e6 3.19531
\(203\) −225189. −0.383536
\(204\) −218990. −0.368425
\(205\) 0 0
\(206\) 851560. 1.39813
\(207\) 388261. 0.629793
\(208\) −811991. −1.30135
\(209\) 501070. 0.793475
\(210\) 0 0
\(211\) −619805. −0.958405 −0.479202 0.877704i \(-0.659074\pi\)
−0.479202 + 0.877704i \(0.659074\pi\)
\(212\) −2.60866e6 −3.98637
\(213\) −319344. −0.482291
\(214\) −1.59092e6 −2.37473
\(215\) 0 0
\(216\) −1.57276e6 −2.29365
\(217\) 249541. 0.359743
\(218\) −132692. −0.189105
\(219\) −100803. −0.142024
\(220\) 0 0
\(221\) −20516.1 −0.0282562
\(222\) −1.50598e6 −2.05086
\(223\) 310997. 0.418787 0.209394 0.977831i \(-0.432851\pi\)
0.209394 + 0.977831i \(0.432851\pi\)
\(224\) 2.24765e6 2.99301
\(225\) 0 0
\(226\) −488886. −0.636702
\(227\) 450336. 0.580059 0.290030 0.957018i \(-0.406335\pi\)
0.290030 + 0.957018i \(0.406335\pi\)
\(228\) 5.61337e6 7.15132
\(229\) 855126. 1.07756 0.538780 0.842447i \(-0.318886\pi\)
0.538780 + 0.842447i \(0.318886\pi\)
\(230\) 0 0
\(231\) −219363. −0.270479
\(232\) −2.20779e6 −2.69301
\(233\) 1.09514e6 1.32153 0.660767 0.750591i \(-0.270231\pi\)
0.660767 + 0.750591i \(0.270231\pi\)
\(234\) 237597. 0.283663
\(235\) 0 0
\(236\) 3.10612e6 3.63027
\(237\) 59901.4 0.0692734
\(238\) 96739.4 0.110703
\(239\) 917921. 1.03947 0.519733 0.854329i \(-0.326031\pi\)
0.519733 + 0.854329i \(0.326031\pi\)
\(240\) 0 0
\(241\) −642632. −0.712721 −0.356361 0.934349i \(-0.615982\pi\)
−0.356361 + 0.934349i \(0.615982\pi\)
\(242\) 1.51677e6 1.66487
\(243\) 867047. 0.941948
\(244\) −3.12049e6 −3.35543
\(245\) 0 0
\(246\) −4.21156e6 −4.43717
\(247\) 525888. 0.548468
\(248\) 2.44655e6 2.52595
\(249\) −482157. −0.492822
\(250\) 0 0
\(251\) −96744.1 −0.0969260 −0.0484630 0.998825i \(-0.515432\pi\)
−0.0484630 + 0.998825i \(0.515432\pi\)
\(252\) −835821. −0.829109
\(253\) −499176. −0.490289
\(254\) 2.57518e6 2.50452
\(255\) 0 0
\(256\) 7.58322e6 7.23192
\(257\) −1.06681e6 −1.00752 −0.503759 0.863844i \(-0.668050\pi\)
−0.503759 + 0.863844i \(0.668050\pi\)
\(258\) −2.88611e6 −2.69937
\(259\) 496319. 0.459739
\(260\) 0 0
\(261\) 397288. 0.360998
\(262\) 1.81532e6 1.63381
\(263\) −503116. −0.448517 −0.224258 0.974530i \(-0.571996\pi\)
−0.224258 + 0.974530i \(0.571996\pi\)
\(264\) −2.15068e6 −1.89918
\(265\) 0 0
\(266\) −2.47972e6 −2.14881
\(267\) −12180.5 −0.0104565
\(268\) −2.31174e6 −1.96609
\(269\) −104820. −0.0883212 −0.0441606 0.999024i \(-0.514061\pi\)
−0.0441606 + 0.999024i \(0.514061\pi\)
\(270\) 0 0
\(271\) −168792. −0.139614 −0.0698069 0.997561i \(-0.522238\pi\)
−0.0698069 + 0.997561i \(0.522238\pi\)
\(272\) 583273. 0.478024
\(273\) −230228. −0.186961
\(274\) 2.62217e6 2.11001
\(275\) 0 0
\(276\) −5.59214e6 −4.41881
\(277\) 863313. 0.676034 0.338017 0.941140i \(-0.390244\pi\)
0.338017 + 0.941140i \(0.390244\pi\)
\(278\) −1.09999e6 −0.853641
\(279\) −440251. −0.338602
\(280\) 0 0
\(281\) −1.56951e6 −1.18577 −0.592883 0.805289i \(-0.702010\pi\)
−0.592883 + 0.805289i \(0.702010\pi\)
\(282\) −1.64518e6 −1.23194
\(283\) −1.10553e6 −0.820546 −0.410273 0.911963i \(-0.634567\pi\)
−0.410273 + 0.911963i \(0.634567\pi\)
\(284\) 1.56436e6 1.15091
\(285\) 0 0
\(286\) −305472. −0.220829
\(287\) 1.38799e6 0.994672
\(288\) −3.96540e6 −2.81712
\(289\) −1.40512e6 −0.989621
\(290\) 0 0
\(291\) 2.99664e6 2.07445
\(292\) 493802. 0.338919
\(293\) 116407. 0.0792155 0.0396077 0.999215i \(-0.487389\pi\)
0.0396077 + 0.999215i \(0.487389\pi\)
\(294\) −2.53477e6 −1.71029
\(295\) 0 0
\(296\) 4.86601e6 3.22807
\(297\) −363863. −0.239358
\(298\) −3.00616e6 −1.96097
\(299\) −523900. −0.338899
\(300\) 0 0
\(301\) 951161. 0.605115
\(302\) 1.40361e6 0.885580
\(303\) 3.16787e6 1.98226
\(304\) −1.49510e7 −9.27870
\(305\) 0 0
\(306\) −170672. −0.104198
\(307\) 830234. 0.502753 0.251376 0.967889i \(-0.419117\pi\)
0.251376 + 0.967889i \(0.419117\pi\)
\(308\) 1.07459e6 0.645455
\(309\) 1.45576e6 0.867351
\(310\) 0 0
\(311\) 204863. 0.120105 0.0600526 0.998195i \(-0.480873\pi\)
0.0600526 + 0.998195i \(0.480873\pi\)
\(312\) −2.25720e6 −1.31276
\(313\) 1.96760e6 1.13521 0.567604 0.823302i \(-0.307871\pi\)
0.567604 + 0.823302i \(0.307871\pi\)
\(314\) −3.63463e6 −2.08035
\(315\) 0 0
\(316\) −293438. −0.165310
\(317\) 2.47859e6 1.38534 0.692670 0.721255i \(-0.256434\pi\)
0.692670 + 0.721255i \(0.256434\pi\)
\(318\) −5.97766e6 −3.31485
\(319\) −510782. −0.281034
\(320\) 0 0
\(321\) −2.71973e6 −1.47320
\(322\) 2.47034e6 1.32775
\(323\) −377759. −0.201469
\(324\) −6.93726e6 −3.67135
\(325\) 0 0
\(326\) −5.42193e6 −2.82559
\(327\) −226840. −0.117314
\(328\) 1.36081e7 6.98413
\(329\) 542195. 0.276163
\(330\) 0 0
\(331\) −3.17720e6 −1.59395 −0.796974 0.604013i \(-0.793567\pi\)
−0.796974 + 0.604013i \(0.793567\pi\)
\(332\) 2.36193e6 1.17604
\(333\) −875629. −0.432722
\(334\) 3.06066e6 1.50123
\(335\) 0 0
\(336\) 6.54540e6 3.16292
\(337\) −1.04972e6 −0.503500 −0.251750 0.967792i \(-0.581006\pi\)
−0.251750 + 0.967792i \(0.581006\pi\)
\(338\) −320602. −0.152642
\(339\) −835764. −0.394988
\(340\) 0 0
\(341\) 566018. 0.263599
\(342\) 4.37483e6 2.02254
\(343\) 2.02852e6 0.930986
\(344\) 9.32537e6 4.24883
\(345\) 0 0
\(346\) 2.79039e6 1.25307
\(347\) −502238. −0.223916 −0.111958 0.993713i \(-0.535712\pi\)
−0.111958 + 0.993713i \(0.535712\pi\)
\(348\) −5.72217e6 −2.53287
\(349\) −1.28824e6 −0.566152 −0.283076 0.959098i \(-0.591355\pi\)
−0.283076 + 0.959098i \(0.591355\pi\)
\(350\) 0 0
\(351\) −381885. −0.165449
\(352\) 5.09820e6 2.19311
\(353\) −3.80443e6 −1.62500 −0.812499 0.582962i \(-0.801893\pi\)
−0.812499 + 0.582962i \(0.801893\pi\)
\(354\) 7.11758e6 3.01873
\(355\) 0 0
\(356\) 59668.2 0.0249527
\(357\) 165379. 0.0686766
\(358\) −327022. −0.134856
\(359\) 3.18303e6 1.30348 0.651740 0.758442i \(-0.274039\pi\)
0.651740 + 0.758442i \(0.274039\pi\)
\(360\) 0 0
\(361\) 7.20697e6 2.91062
\(362\) −6.55472e6 −2.62895
\(363\) 2.59296e6 1.03283
\(364\) 1.12781e6 0.446153
\(365\) 0 0
\(366\) −7.15050e6 −2.79019
\(367\) −1.51403e6 −0.586770 −0.293385 0.955994i \(-0.594782\pi\)
−0.293385 + 0.955994i \(0.594782\pi\)
\(368\) 1.48945e7 5.73332
\(369\) −2.44875e6 −0.936221
\(370\) 0 0
\(371\) 1.97003e6 0.743084
\(372\) 6.34096e6 2.37573
\(373\) −1.59920e6 −0.595156 −0.297578 0.954698i \(-0.596179\pi\)
−0.297578 + 0.954698i \(0.596179\pi\)
\(374\) 219428. 0.0811173
\(375\) 0 0
\(376\) 5.31579e6 1.93909
\(377\) −536081. −0.194257
\(378\) 1.80070e6 0.648205
\(379\) 2.52547e6 0.903116 0.451558 0.892242i \(-0.350868\pi\)
0.451558 + 0.892242i \(0.350868\pi\)
\(380\) 0 0
\(381\) 4.40235e6 1.55372
\(382\) −7.94997e6 −2.78745
\(383\) 3.16883e6 1.10383 0.551915 0.833900i \(-0.313897\pi\)
0.551915 + 0.833900i \(0.313897\pi\)
\(384\) 2.39950e7 8.30410
\(385\) 0 0
\(386\) −6.69331e6 −2.28651
\(387\) −1.67808e6 −0.569555
\(388\) −1.46796e7 −4.95034
\(389\) 1.03129e6 0.345545 0.172773 0.984962i \(-0.444727\pi\)
0.172773 + 0.984962i \(0.444727\pi\)
\(390\) 0 0
\(391\) 376330. 0.124488
\(392\) 8.19014e6 2.69201
\(393\) 3.10334e6 1.01356
\(394\) −267411. −0.0867838
\(395\) 0 0
\(396\) −1.89584e6 −0.607525
\(397\) −634862. −0.202164 −0.101082 0.994878i \(-0.532230\pi\)
−0.101082 + 0.994878i \(0.532230\pi\)
\(398\) −5.28320e6 −1.67182
\(399\) −4.23915e6 −1.33305
\(400\) 0 0
\(401\) 2.19913e6 0.682952 0.341476 0.939890i \(-0.389073\pi\)
0.341476 + 0.939890i \(0.389073\pi\)
\(402\) −5.29728e6 −1.63489
\(403\) 594053. 0.182206
\(404\) −1.55184e7 −4.73035
\(405\) 0 0
\(406\) 2.52778e6 0.761070
\(407\) 1.12577e6 0.336871
\(408\) 1.62140e6 0.482215
\(409\) −4.25223e6 −1.25692 −0.628461 0.777841i \(-0.716315\pi\)
−0.628461 + 0.777841i \(0.716315\pi\)
\(410\) 0 0
\(411\) 4.48266e6 1.30898
\(412\) −7.13132e6 −2.06979
\(413\) −2.34571e6 −0.676704
\(414\) −4.35829e6 −1.24973
\(415\) 0 0
\(416\) 5.35071e6 1.51593
\(417\) −1.88046e6 −0.529570
\(418\) −5.62460e6 −1.57453
\(419\) 3.54751e6 0.987161 0.493581 0.869700i \(-0.335688\pi\)
0.493581 + 0.869700i \(0.335688\pi\)
\(420\) 0 0
\(421\) 1.40038e6 0.385071 0.192535 0.981290i \(-0.438329\pi\)
0.192535 + 0.981290i \(0.438329\pi\)
\(422\) 6.95742e6 1.90181
\(423\) −956565. −0.259934
\(424\) 1.93146e7 5.21759
\(425\) 0 0
\(426\) 3.58469e6 0.957033
\(427\) 2.35656e6 0.625472
\(428\) 1.33231e7 3.51556
\(429\) −522213. −0.136995
\(430\) 0 0
\(431\) 5.41665e6 1.40455 0.702276 0.711905i \(-0.252167\pi\)
0.702276 + 0.711905i \(0.252167\pi\)
\(432\) 1.08570e7 2.79899
\(433\) −5.58069e6 −1.43043 −0.715217 0.698902i \(-0.753672\pi\)
−0.715217 + 0.698902i \(0.753672\pi\)
\(434\) −2.80113e6 −0.713854
\(435\) 0 0
\(436\) 1.11122e6 0.279951
\(437\) −9.64646e6 −2.41638
\(438\) 1.13153e6 0.281826
\(439\) −1.76748e6 −0.437716 −0.218858 0.975757i \(-0.570233\pi\)
−0.218858 + 0.975757i \(0.570233\pi\)
\(440\) 0 0
\(441\) −1.47380e6 −0.360863
\(442\) 230296. 0.0560701
\(443\) 2.11511e6 0.512062 0.256031 0.966669i \(-0.417585\pi\)
0.256031 + 0.966669i \(0.417585\pi\)
\(444\) 1.26117e7 3.03611
\(445\) 0 0
\(446\) −3.49099e6 −0.831019
\(447\) −5.13911e6 −1.21652
\(448\) −1.43154e7 −3.36982
\(449\) 2.74804e6 0.643290 0.321645 0.946860i \(-0.395764\pi\)
0.321645 + 0.946860i \(0.395764\pi\)
\(450\) 0 0
\(451\) 3.14828e6 0.728841
\(452\) 4.09414e6 0.942577
\(453\) 2.39950e6 0.549384
\(454\) −5.05510e6 −1.15104
\(455\) 0 0
\(456\) −4.15614e7 −9.36005
\(457\) −3.46825e6 −0.776820 −0.388410 0.921487i \(-0.626976\pi\)
−0.388410 + 0.921487i \(0.626976\pi\)
\(458\) −9.59893e6 −2.13825
\(459\) 274318. 0.0607746
\(460\) 0 0
\(461\) 5.16521e6 1.13197 0.565986 0.824415i \(-0.308496\pi\)
0.565986 + 0.824415i \(0.308496\pi\)
\(462\) 2.46239e6 0.536725
\(463\) −6.87934e6 −1.49140 −0.745700 0.666282i \(-0.767885\pi\)
−0.745700 + 0.666282i \(0.767885\pi\)
\(464\) 1.52408e7 3.28634
\(465\) 0 0
\(466\) −1.22931e7 −2.62238
\(467\) −569700. −0.120880 −0.0604400 0.998172i \(-0.519250\pi\)
−0.0604400 + 0.998172i \(0.519250\pi\)
\(468\) −1.98974e6 −0.419935
\(469\) 1.74580e6 0.366490
\(470\) 0 0
\(471\) −6.21350e6 −1.29058
\(472\) −2.29978e7 −4.75150
\(473\) 2.15746e6 0.443394
\(474\) −672404. −0.137463
\(475\) 0 0
\(476\) −810137. −0.163886
\(477\) −3.47562e6 −0.699417
\(478\) −1.03038e7 −2.06266
\(479\) −3.33256e6 −0.663649 −0.331825 0.943341i \(-0.607664\pi\)
−0.331825 + 0.943341i \(0.607664\pi\)
\(480\) 0 0
\(481\) 1.18153e6 0.232853
\(482\) 7.21365e6 1.41429
\(483\) 4.22312e6 0.823693
\(484\) −1.27021e7 −2.46468
\(485\) 0 0
\(486\) −9.73275e6 −1.86915
\(487\) −5.24387e6 −1.00191 −0.500956 0.865473i \(-0.667018\pi\)
−0.500956 + 0.865473i \(0.667018\pi\)
\(488\) 2.31041e7 4.39177
\(489\) −9.26893e6 −1.75290
\(490\) 0 0
\(491\) −4.52706e6 −0.847447 −0.423724 0.905792i \(-0.639277\pi\)
−0.423724 + 0.905792i \(0.639277\pi\)
\(492\) 3.52694e7 6.56880
\(493\) 385080. 0.0713566
\(494\) −5.90318e6 −1.08835
\(495\) 0 0
\(496\) −1.68890e7 −3.08247
\(497\) −1.18139e6 −0.214537
\(498\) 5.41229e6 0.977930
\(499\) 8.06232e6 1.44947 0.724734 0.689029i \(-0.241963\pi\)
0.724734 + 0.689029i \(0.241963\pi\)
\(500\) 0 0
\(501\) 5.23227e6 0.931314
\(502\) 1.08597e6 0.192335
\(503\) 3.91235e6 0.689474 0.344737 0.938699i \(-0.387968\pi\)
0.344737 + 0.938699i \(0.387968\pi\)
\(504\) 6.18842e6 1.08518
\(505\) 0 0
\(506\) 5.60333e6 0.972904
\(507\) −548078. −0.0946940
\(508\) −2.15657e7 −3.70770
\(509\) −1.01584e7 −1.73792 −0.868959 0.494884i \(-0.835210\pi\)
−0.868959 + 0.494884i \(0.835210\pi\)
\(510\) 0 0
\(511\) −372913. −0.0631765
\(512\) −4.51098e7 −7.60495
\(513\) −7.03158e6 −1.17967
\(514\) 1.19751e7 1.99927
\(515\) 0 0
\(516\) 2.41695e7 3.99616
\(517\) 1.22983e6 0.202357
\(518\) −5.57126e6 −0.912282
\(519\) 4.77025e6 0.777361
\(520\) 0 0
\(521\) 1.44957e6 0.233962 0.116981 0.993134i \(-0.462678\pi\)
0.116981 + 0.993134i \(0.462678\pi\)
\(522\) −4.45963e6 −0.716345
\(523\) 203871. 0.0325912 0.0162956 0.999867i \(-0.494813\pi\)
0.0162956 + 0.999867i \(0.494813\pi\)
\(524\) −1.52023e7 −2.41869
\(525\) 0 0
\(526\) 5.64756e6 0.890013
\(527\) −426723. −0.0669298
\(528\) 1.48465e7 2.31761
\(529\) 3.17365e6 0.493082
\(530\) 0 0
\(531\) 4.13841e6 0.636937
\(532\) 2.07662e7 3.18111
\(533\) 3.30422e6 0.503791
\(534\) 136728. 0.0207493
\(535\) 0 0
\(536\) 1.71162e7 2.57332
\(537\) −559052. −0.0836598
\(538\) 1.17663e6 0.175260
\(539\) 1.89482e6 0.280929
\(540\) 0 0
\(541\) 7.28883e6 1.07069 0.535346 0.844633i \(-0.320181\pi\)
0.535346 + 0.844633i \(0.320181\pi\)
\(542\) 1.89472e6 0.277042
\(543\) −1.12055e7 −1.63091
\(544\) −3.84355e6 −0.556846
\(545\) 0 0
\(546\) 2.58435e6 0.370996
\(547\) 5.64096e6 0.806092 0.403046 0.915180i \(-0.367951\pi\)
0.403046 + 0.915180i \(0.367951\pi\)
\(548\) −2.19591e7 −3.12366
\(549\) −4.15754e6 −0.588716
\(550\) 0 0
\(551\) −9.87075e6 −1.38507
\(552\) 4.14043e7 5.78359
\(553\) 221601. 0.0308148
\(554\) −9.69083e6 −1.34149
\(555\) 0 0
\(556\) 9.21176e6 1.26373
\(557\) −910831. −0.124394 −0.0621971 0.998064i \(-0.519811\pi\)
−0.0621971 + 0.998064i \(0.519811\pi\)
\(558\) 4.94189e6 0.671905
\(559\) 2.26432e6 0.306484
\(560\) 0 0
\(561\) 375119. 0.0503224
\(562\) 1.76180e7 2.35297
\(563\) −7.96586e6 −1.05916 −0.529580 0.848260i \(-0.677651\pi\)
−0.529580 + 0.848260i \(0.677651\pi\)
\(564\) 1.37775e7 1.82378
\(565\) 0 0
\(566\) 1.24097e7 1.62825
\(567\) 5.23893e6 0.684361
\(568\) −1.15826e7 −1.50638
\(569\) −3.26286e6 −0.422492 −0.211246 0.977433i \(-0.567752\pi\)
−0.211246 + 0.977433i \(0.567752\pi\)
\(570\) 0 0
\(571\) −7.89949e6 −1.01393 −0.506966 0.861966i \(-0.669233\pi\)
−0.506966 + 0.861966i \(0.669233\pi\)
\(572\) 2.55815e6 0.326916
\(573\) −1.35907e7 −1.72924
\(574\) −1.55804e7 −1.97378
\(575\) 0 0
\(576\) 2.52558e7 3.17180
\(577\) 1.91866e6 0.239916 0.119958 0.992779i \(-0.461724\pi\)
0.119958 + 0.992779i \(0.461724\pi\)
\(578\) 1.57727e7 1.96375
\(579\) −1.14424e7 −1.41847
\(580\) 0 0
\(581\) −1.78370e6 −0.219221
\(582\) −3.36378e7 −4.11642
\(583\) 4.46850e6 0.544491
\(584\) −3.65611e6 −0.443596
\(585\) 0 0
\(586\) −1.30669e6 −0.157191
\(587\) −1.98302e6 −0.237537 −0.118769 0.992922i \(-0.537895\pi\)
−0.118769 + 0.992922i \(0.537895\pi\)
\(588\) 2.12272e7 2.53192
\(589\) 1.09382e7 1.29914
\(590\) 0 0
\(591\) −457146. −0.0538377
\(592\) −3.35909e7 −3.93929
\(593\) −9.43563e6 −1.10188 −0.550940 0.834545i \(-0.685731\pi\)
−0.550940 + 0.834545i \(0.685731\pi\)
\(594\) 4.08443e6 0.474969
\(595\) 0 0
\(596\) 2.51749e7 2.90303
\(597\) −9.03177e6 −1.03714
\(598\) 5.88086e6 0.672493
\(599\) 6.57359e6 0.748576 0.374288 0.927313i \(-0.377887\pi\)
0.374288 + 0.927313i \(0.377887\pi\)
\(600\) 0 0
\(601\) 9.04817e6 1.02182 0.510910 0.859634i \(-0.329308\pi\)
0.510910 + 0.859634i \(0.329308\pi\)
\(602\) −1.06769e7 −1.20076
\(603\) −3.08002e6 −0.344953
\(604\) −1.17544e7 −1.31102
\(605\) 0 0
\(606\) −3.55598e7 −3.93349
\(607\) −2.54414e6 −0.280266 −0.140133 0.990133i \(-0.544753\pi\)
−0.140133 + 0.990133i \(0.544753\pi\)
\(608\) 9.85216e7 10.8087
\(609\) 4.32131e6 0.472141
\(610\) 0 0
\(611\) 1.29074e6 0.139874
\(612\) 1.42928e6 0.154255
\(613\) −7.53921e6 −0.810354 −0.405177 0.914238i \(-0.632790\pi\)
−0.405177 + 0.914238i \(0.632790\pi\)
\(614\) −9.31951e6 −0.997636
\(615\) 0 0
\(616\) −7.95628e6 −0.844808
\(617\) −1.44546e6 −0.152860 −0.0764300 0.997075i \(-0.524352\pi\)
−0.0764300 + 0.997075i \(0.524352\pi\)
\(618\) −1.63412e7 −1.72113
\(619\) 832032. 0.0872797 0.0436399 0.999047i \(-0.486105\pi\)
0.0436399 + 0.999047i \(0.486105\pi\)
\(620\) 0 0
\(621\) 7.00499e6 0.728918
\(622\) −2.29962e6 −0.238330
\(623\) −45060.7 −0.00465134
\(624\) 1.55819e7 1.60198
\(625\) 0 0
\(626\) −2.20866e7 −2.25265
\(627\) −9.61540e6 −0.976784
\(628\) 3.04379e7 3.07975
\(629\) −848722. −0.0855340
\(630\) 0 0
\(631\) 1.42941e6 0.142916 0.0714582 0.997444i \(-0.477235\pi\)
0.0714582 + 0.997444i \(0.477235\pi\)
\(632\) 2.17262e6 0.216367
\(633\) 1.18939e7 1.17982
\(634\) −2.78226e7 −2.74900
\(635\) 0 0
\(636\) 5.00595e7 4.90731
\(637\) 1.98867e6 0.194184
\(638\) 5.73362e6 0.557669
\(639\) 2.08426e6 0.201929
\(640\) 0 0
\(641\) −1.54071e7 −1.48107 −0.740537 0.672015i \(-0.765429\pi\)
−0.740537 + 0.672015i \(0.765429\pi\)
\(642\) 3.05294e7 2.92335
\(643\) 3.71125e6 0.353992 0.176996 0.984212i \(-0.443362\pi\)
0.176996 + 0.984212i \(0.443362\pi\)
\(644\) −2.06877e7 −1.96561
\(645\) 0 0
\(646\) 4.24040e6 0.399784
\(647\) 8.31438e6 0.780853 0.390426 0.920634i \(-0.372328\pi\)
0.390426 + 0.920634i \(0.372328\pi\)
\(648\) 5.13635e7 4.80527
\(649\) −5.32063e6 −0.495851
\(650\) 0 0
\(651\) −4.78862e6 −0.442851
\(652\) 4.54055e7 4.18302
\(653\) −2.58516e6 −0.237249 −0.118624 0.992939i \(-0.537848\pi\)
−0.118624 + 0.992939i \(0.537848\pi\)
\(654\) 2.54631e6 0.232792
\(655\) 0 0
\(656\) −9.39391e7 −8.52289
\(657\) 657911. 0.0594639
\(658\) −6.08623e6 −0.548004
\(659\) 1.77052e7 1.58813 0.794065 0.607833i \(-0.207961\pi\)
0.794065 + 0.607833i \(0.207961\pi\)
\(660\) 0 0
\(661\) 2.01104e6 0.179027 0.0895134 0.995986i \(-0.471469\pi\)
0.0895134 + 0.995986i \(0.471469\pi\)
\(662\) 3.56646e7 3.16295
\(663\) 393698. 0.0347840
\(664\) −1.74878e7 −1.53927
\(665\) 0 0
\(666\) 9.82908e6 0.858672
\(667\) 9.83343e6 0.855836
\(668\) −2.56312e7 −2.22243
\(669\) −5.96794e6 −0.515536
\(670\) 0 0
\(671\) 5.34523e6 0.458311
\(672\) −4.31317e7 −3.68446
\(673\) 5.48300e6 0.466639 0.233319 0.972400i \(-0.425041\pi\)
0.233319 + 0.972400i \(0.425041\pi\)
\(674\) 1.17833e7 0.999119
\(675\) 0 0
\(676\) 2.68486e6 0.225972
\(677\) 1.58127e7 1.32597 0.662987 0.748631i \(-0.269288\pi\)
0.662987 + 0.748631i \(0.269288\pi\)
\(678\) 9.38159e6 0.783794
\(679\) 1.10859e7 0.922772
\(680\) 0 0
\(681\) −8.64183e6 −0.714065
\(682\) −6.35365e6 −0.523073
\(683\) −2.24599e7 −1.84229 −0.921143 0.389225i \(-0.872743\pi\)
−0.921143 + 0.389225i \(0.872743\pi\)
\(684\) −3.66367e7 −2.99417
\(685\) 0 0
\(686\) −2.27704e7 −1.84740
\(687\) −1.64096e7 −1.32650
\(688\) −6.43747e7 −5.18495
\(689\) 4.68982e6 0.376364
\(690\) 0 0
\(691\) −8.90456e6 −0.709443 −0.354721 0.934972i \(-0.615424\pi\)
−0.354721 + 0.934972i \(0.615424\pi\)
\(692\) −2.33679e7 −1.85505
\(693\) 1.43172e6 0.113246
\(694\) 5.63770e6 0.444328
\(695\) 0 0
\(696\) 4.23670e7 3.31516
\(697\) −2.37350e6 −0.185058
\(698\) 1.44607e7 1.12344
\(699\) −2.10154e7 −1.62684
\(700\) 0 0
\(701\) −1.83550e7 −1.41078 −0.705392 0.708818i \(-0.749229\pi\)
−0.705392 + 0.708818i \(0.749229\pi\)
\(702\) 4.28673e6 0.328309
\(703\) 2.17553e7 1.66026
\(704\) −3.24707e7 −2.46922
\(705\) 0 0
\(706\) 4.27054e7 3.22456
\(707\) 1.17193e7 0.881765
\(708\) −5.96057e7 −4.46894
\(709\) −2.57581e7 −1.92441 −0.962207 0.272318i \(-0.912210\pi\)
−0.962207 + 0.272318i \(0.912210\pi\)
\(710\) 0 0
\(711\) −390959. −0.0290039
\(712\) −441784. −0.0326596
\(713\) −1.08968e7 −0.802742
\(714\) −1.85640e6 −0.136278
\(715\) 0 0
\(716\) 2.73862e6 0.199641
\(717\) −1.76146e7 −1.27961
\(718\) −3.57300e7 −2.58656
\(719\) 1.69710e6 0.122429 0.0612146 0.998125i \(-0.480503\pi\)
0.0612146 + 0.998125i \(0.480503\pi\)
\(720\) 0 0
\(721\) 5.38549e6 0.385822
\(722\) −8.08995e7 −5.77568
\(723\) 1.23319e7 0.877375
\(724\) 5.48921e7 3.89191
\(725\) 0 0
\(726\) −2.91064e7 −2.04949
\(727\) 1.84315e7 1.29338 0.646689 0.762754i \(-0.276153\pi\)
0.646689 + 0.762754i \(0.276153\pi\)
\(728\) −8.35035e6 −0.583951
\(729\) 1.29433e6 0.0902041
\(730\) 0 0
\(731\) −1.62652e6 −0.112581
\(732\) 5.98813e7 4.13060
\(733\) 1.79395e7 1.23325 0.616624 0.787258i \(-0.288500\pi\)
0.616624 + 0.787258i \(0.288500\pi\)
\(734\) 1.69952e7 1.16436
\(735\) 0 0
\(736\) −9.81491e7 −6.67870
\(737\) 3.95989e6 0.268544
\(738\) 2.74876e7 1.85779
\(739\) 1.84597e7 1.24341 0.621703 0.783253i \(-0.286441\pi\)
0.621703 + 0.783253i \(0.286441\pi\)
\(740\) 0 0
\(741\) −1.00916e7 −0.675175
\(742\) −2.21139e7 −1.47454
\(743\) 3.97606e6 0.264229 0.132115 0.991234i \(-0.457823\pi\)
0.132115 + 0.991234i \(0.457823\pi\)
\(744\) −4.69485e7 −3.10949
\(745\) 0 0
\(746\) 1.79513e7 1.18100
\(747\) 3.14689e6 0.206339
\(748\) −1.83759e6 −0.120086
\(749\) −1.00614e7 −0.655322
\(750\) 0 0
\(751\) 2.25945e7 1.46185 0.730925 0.682458i \(-0.239089\pi\)
0.730925 + 0.682458i \(0.239089\pi\)
\(752\) −3.66958e7 −2.36631
\(753\) 1.85649e6 0.119318
\(754\) 6.01760e6 0.385474
\(755\) 0 0
\(756\) −1.50799e7 −0.959605
\(757\) 1.40367e7 0.890279 0.445139 0.895461i \(-0.353154\pi\)
0.445139 + 0.895461i \(0.353154\pi\)
\(758\) −2.83488e7 −1.79210
\(759\) 9.57905e6 0.603556
\(760\) 0 0
\(761\) −6.05735e6 −0.379159 −0.189579 0.981865i \(-0.560712\pi\)
−0.189579 + 0.981865i \(0.560712\pi\)
\(762\) −4.94171e7 −3.08311
\(763\) −839177. −0.0521846
\(764\) 6.65765e7 4.12655
\(765\) 0 0
\(766\) −3.55707e7 −2.19038
\(767\) −5.58416e6 −0.342743
\(768\) −1.45520e8 −8.90265
\(769\) 1.07705e7 0.656778 0.328389 0.944543i \(-0.393494\pi\)
0.328389 + 0.944543i \(0.393494\pi\)
\(770\) 0 0
\(771\) 2.04717e7 1.24028
\(772\) 5.60527e7 3.38496
\(773\) 8.92164e6 0.537027 0.268514 0.963276i \(-0.413468\pi\)
0.268514 + 0.963276i \(0.413468\pi\)
\(774\) 1.88367e7 1.13020
\(775\) 0 0
\(776\) 1.08688e8 6.47928
\(777\) −9.52422e6 −0.565949
\(778\) −1.15764e7 −0.685682
\(779\) 6.08399e7 3.59207
\(780\) 0 0
\(781\) −2.67967e6 −0.157201
\(782\) −4.22437e6 −0.247027
\(783\) 7.16787e6 0.417816
\(784\) −5.65380e7 −3.28511
\(785\) 0 0
\(786\) −3.48356e7 −2.01125
\(787\) 542225. 0.0312063 0.0156032 0.999878i \(-0.495033\pi\)
0.0156032 + 0.999878i \(0.495033\pi\)
\(788\) 2.23941e6 0.128475
\(789\) 9.65466e6 0.552134
\(790\) 0 0
\(791\) −3.09185e6 −0.175702
\(792\) 1.40368e7 0.795163
\(793\) 5.60998e6 0.316795
\(794\) 7.12643e6 0.401163
\(795\) 0 0
\(796\) 4.42437e7 2.47497
\(797\) −1.82250e7 −1.01630 −0.508149 0.861269i \(-0.669670\pi\)
−0.508149 + 0.861269i \(0.669670\pi\)
\(798\) 4.75851e7 2.64523
\(799\) −927172. −0.0513799
\(800\) 0 0
\(801\) 79498.2 0.00437800
\(802\) −2.46856e7 −1.35522
\(803\) −845857. −0.0462922
\(804\) 4.43617e7 2.42029
\(805\) 0 0
\(806\) −6.66834e6 −0.361560
\(807\) 2.01147e6 0.108725
\(808\) 1.14898e8 6.19134
\(809\) −3.56238e7 −1.91368 −0.956840 0.290615i \(-0.906140\pi\)
−0.956840 + 0.290615i \(0.906140\pi\)
\(810\) 0 0
\(811\) 9.68050e6 0.516828 0.258414 0.966034i \(-0.416800\pi\)
0.258414 + 0.966034i \(0.416800\pi\)
\(812\) −2.11687e7 −1.12669
\(813\) 3.23907e6 0.171868
\(814\) −1.26370e7 −0.668470
\(815\) 0 0
\(816\) −1.11929e7 −0.588458
\(817\) 4.16925e7 2.18526
\(818\) 4.77320e7 2.49417
\(819\) 1.50263e6 0.0782784
\(820\) 0 0
\(821\) 2.15879e7 1.11777 0.558885 0.829245i \(-0.311229\pi\)
0.558885 + 0.829245i \(0.311229\pi\)
\(822\) −5.03186e7 −2.59746
\(823\) 2.95621e7 1.52137 0.760686 0.649120i \(-0.224863\pi\)
0.760686 + 0.649120i \(0.224863\pi\)
\(824\) 5.28004e7 2.70906
\(825\) 0 0
\(826\) 2.63309e7 1.34282
\(827\) 2.85332e7 1.45073 0.725364 0.688365i \(-0.241671\pi\)
0.725364 + 0.688365i \(0.241671\pi\)
\(828\) 3.64982e7 1.85010
\(829\) −2.02156e7 −1.02165 −0.510823 0.859686i \(-0.670659\pi\)
−0.510823 + 0.859686i \(0.670659\pi\)
\(830\) 0 0
\(831\) −1.65667e7 −0.832213
\(832\) −3.40789e7 −1.70678
\(833\) −1.42851e6 −0.0713299
\(834\) 2.11084e7 1.05085
\(835\) 0 0
\(836\) 4.71028e7 2.33094
\(837\) −7.94300e6 −0.391896
\(838\) −3.98214e7 −1.95887
\(839\) −6.64057e6 −0.325687 −0.162843 0.986652i \(-0.552067\pi\)
−0.162843 + 0.986652i \(0.552067\pi\)
\(840\) 0 0
\(841\) −1.04491e7 −0.509434
\(842\) −1.57195e7 −0.764114
\(843\) 3.01185e7 1.45970
\(844\) −5.82644e7 −2.81544
\(845\) 0 0
\(846\) 1.07376e7 0.515800
\(847\) 9.59244e6 0.459432
\(848\) −1.33332e8 −6.36714
\(849\) 2.12148e7 1.01011
\(850\) 0 0
\(851\) −2.16730e7 −1.02588
\(852\) −3.00197e7 −1.41680
\(853\) −1.92258e7 −0.904716 −0.452358 0.891836i \(-0.649417\pi\)
−0.452358 + 0.891836i \(0.649417\pi\)
\(854\) −2.64527e7 −1.24115
\(855\) 0 0
\(856\) −9.86442e7 −4.60137
\(857\) −1.98896e6 −0.0925068 −0.0462534 0.998930i \(-0.514728\pi\)
−0.0462534 + 0.998930i \(0.514728\pi\)
\(858\) 5.86193e6 0.271845
\(859\) −7.32029e6 −0.338490 −0.169245 0.985574i \(-0.554133\pi\)
−0.169245 + 0.985574i \(0.554133\pi\)
\(860\) 0 0
\(861\) −2.66351e7 −1.22446
\(862\) −6.08028e7 −2.78712
\(863\) 3.13431e7 1.43257 0.716283 0.697809i \(-0.245842\pi\)
0.716283 + 0.697809i \(0.245842\pi\)
\(864\) −7.15437e7 −3.26052
\(865\) 0 0
\(866\) 6.26441e7 2.83848
\(867\) 2.69639e7 1.21824
\(868\) 2.34579e7 1.05679
\(869\) 502644. 0.0225793
\(870\) 0 0
\(871\) 4.15603e6 0.185623
\(872\) −8.22745e6 −0.366416
\(873\) −1.95582e7 −0.868546
\(874\) 1.08283e8 4.79493
\(875\) 0 0
\(876\) −9.47592e6 −0.417216
\(877\) −2.47672e7 −1.08737 −0.543686 0.839289i \(-0.682972\pi\)
−0.543686 + 0.839289i \(0.682972\pi\)
\(878\) 1.98402e7 0.868581
\(879\) −2.23382e6 −0.0975160
\(880\) 0 0
\(881\) 1.22752e7 0.532831 0.266415 0.963858i \(-0.414161\pi\)
0.266415 + 0.963858i \(0.414161\pi\)
\(882\) 1.65436e7 0.716077
\(883\) 2.56013e7 1.10500 0.552498 0.833514i \(-0.313675\pi\)
0.552498 + 0.833514i \(0.313675\pi\)
\(884\) −1.92860e6 −0.0830064
\(885\) 0 0
\(886\) −2.37424e7 −1.01611
\(887\) −5.01026e6 −0.213821 −0.106911 0.994269i \(-0.534096\pi\)
−0.106911 + 0.994269i \(0.534096\pi\)
\(888\) −9.33773e7 −3.97383
\(889\) 1.62862e7 0.691137
\(890\) 0 0
\(891\) 1.18832e7 0.501462
\(892\) 2.92350e7 1.23024
\(893\) 2.37662e7 0.997311
\(894\) 5.76873e7 2.41400
\(895\) 0 0
\(896\) 8.87676e7 3.69390
\(897\) 1.00535e7 0.417192
\(898\) −3.08472e7 −1.27651
\(899\) −1.11502e7 −0.460132
\(900\) 0 0
\(901\) −3.36882e6 −0.138250
\(902\) −3.53400e7 −1.44627
\(903\) −1.82525e7 −0.744909
\(904\) −3.03131e7 −1.23370
\(905\) 0 0
\(906\) −2.69348e7 −1.09017
\(907\) −4.85441e7 −1.95938 −0.979688 0.200527i \(-0.935734\pi\)
−0.979688 + 0.200527i \(0.935734\pi\)
\(908\) 4.23336e7 1.70400
\(909\) −2.06757e7 −0.829948
\(910\) 0 0
\(911\) −3.03805e7 −1.21283 −0.606413 0.795150i \(-0.707392\pi\)
−0.606413 + 0.795150i \(0.707392\pi\)
\(912\) 2.86906e8 11.4223
\(913\) −4.04587e6 −0.160633
\(914\) 3.89317e7 1.54148
\(915\) 0 0
\(916\) 8.03855e7 3.16548
\(917\) 1.14806e7 0.450859
\(918\) −3.07926e6 −0.120598
\(919\) −3.70802e7 −1.44828 −0.724141 0.689652i \(-0.757764\pi\)
−0.724141 + 0.689652i \(0.757764\pi\)
\(920\) 0 0
\(921\) −1.59320e7 −0.618899
\(922\) −5.79803e7 −2.24623
\(923\) −2.81239e6 −0.108661
\(924\) −2.06211e7 −0.794569
\(925\) 0 0
\(926\) 7.72217e7 2.95946
\(927\) −9.50133e6 −0.363149
\(928\) −1.00431e8 −3.82824
\(929\) 2.82561e6 0.107417 0.0537084 0.998557i \(-0.482896\pi\)
0.0537084 + 0.998557i \(0.482896\pi\)
\(930\) 0 0
\(931\) 3.66170e7 1.38455
\(932\) 1.02948e8 3.88219
\(933\) −3.93126e6 −0.147852
\(934\) 6.39498e6 0.239868
\(935\) 0 0
\(936\) 1.47321e7 0.549635
\(937\) 3.19406e6 0.118849 0.0594243 0.998233i \(-0.481074\pi\)
0.0594243 + 0.998233i \(0.481074\pi\)
\(938\) −1.95969e7 −0.727244
\(939\) −3.77576e7 −1.39746
\(940\) 0 0
\(941\) −2.22120e7 −0.817738 −0.408869 0.912593i \(-0.634077\pi\)
−0.408869 + 0.912593i \(0.634077\pi\)
\(942\) 6.97475e7 2.56095
\(943\) −6.06099e7 −2.21955
\(944\) 1.58758e8 5.79836
\(945\) 0 0
\(946\) −2.42179e7 −0.879849
\(947\) 2.24064e7 0.811889 0.405945 0.913898i \(-0.366943\pi\)
0.405945 + 0.913898i \(0.366943\pi\)
\(948\) 5.63100e6 0.203500
\(949\) −887751. −0.0319982
\(950\) 0 0
\(951\) −4.75634e7 −1.70538
\(952\) 5.99826e6 0.214503
\(953\) 1.48416e7 0.529358 0.264679 0.964337i \(-0.414734\pi\)
0.264679 + 0.964337i \(0.414734\pi\)
\(954\) 3.90144e7 1.38789
\(955\) 0 0
\(956\) 8.62886e7 3.05357
\(957\) 9.80177e6 0.345959
\(958\) 3.74085e7 1.31691
\(959\) 1.65833e7 0.582269
\(960\) 0 0
\(961\) −1.62732e7 −0.568413
\(962\) −1.32629e7 −0.462061
\(963\) 1.77508e7 0.616812
\(964\) −6.04102e7 −2.09371
\(965\) 0 0
\(966\) −4.74052e7 −1.63449
\(967\) 3.06919e7 1.05550 0.527749 0.849401i \(-0.323036\pi\)
0.527749 + 0.849401i \(0.323036\pi\)
\(968\) 9.40462e7 3.22591
\(969\) 7.24908e6 0.248013
\(970\) 0 0
\(971\) −4.18932e7 −1.42592 −0.712961 0.701203i \(-0.752646\pi\)
−0.712961 + 0.701203i \(0.752646\pi\)
\(972\) 8.15062e7 2.76710
\(973\) −6.95661e6 −0.235568
\(974\) 5.88633e7 1.98814
\(975\) 0 0
\(976\) −1.59492e8 −5.35938
\(977\) −2.54822e7 −0.854084 −0.427042 0.904232i \(-0.640444\pi\)
−0.427042 + 0.904232i \(0.640444\pi\)
\(978\) 1.04045e8 3.47837
\(979\) −102209. −0.00340824
\(980\) 0 0
\(981\) 1.48051e6 0.0491179
\(982\) 5.08170e7 1.68163
\(983\) −5.91830e7 −1.95350 −0.976749 0.214388i \(-0.931224\pi\)
−0.976749 + 0.214388i \(0.931224\pi\)
\(984\) −2.61135e8 −8.59761
\(985\) 0 0
\(986\) −4.32259e6 −0.141596
\(987\) −1.04046e7 −0.339963
\(988\) 4.94358e7 1.61120
\(989\) −4.15348e7 −1.35027
\(990\) 0 0
\(991\) −625464. −0.0202310 −0.0101155 0.999949i \(-0.503220\pi\)
−0.0101155 + 0.999949i \(0.503220\pi\)
\(992\) 1.11292e8 3.59074
\(993\) 6.09695e7 1.96218
\(994\) 1.32613e7 0.425715
\(995\) 0 0
\(996\) −4.53249e7 −1.44773
\(997\) 5.76822e7 1.83783 0.918913 0.394461i \(-0.129069\pi\)
0.918913 + 0.394461i \(0.129069\pi\)
\(998\) −9.05009e7 −2.87625
\(999\) −1.57981e7 −0.500830
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.j.1.1 11
5.2 odd 4 325.6.b.i.274.1 22
5.3 odd 4 325.6.b.i.274.22 22
5.4 even 2 325.6.a.k.1.11 yes 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.1 11 1.1 even 1 trivial
325.6.a.k.1.11 yes 11 5.4 even 2
325.6.b.i.274.1 22 5.2 odd 4
325.6.b.i.274.22 22 5.3 odd 4