Properties

Label 1344.4.c.f.673.11
Level $1344$
Weight $4$
Character 1344.673
Analytic conductor $79.299$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(673,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.673");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + 1646096 x^{3} + 1391904 x^{2} + 174720 x + 43264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 673.11
Root \(6.03643 - 1.61746i\) of defining polynomial
Character \(\chi\) \(=\) 1344.673
Dual form 1344.4.c.f.673.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +6.24978i q^{5} -7.00000 q^{7} -9.00000 q^{9} +63.4042i q^{11} +82.2923i q^{13} -18.7493 q^{15} +75.4992 q^{17} +125.613i q^{19} -21.0000i q^{21} +155.435 q^{23} +85.9403 q^{25} -27.0000i q^{27} -56.2501i q^{29} +159.142 q^{31} -190.212 q^{33} -43.7484i q^{35} +197.279i q^{37} -246.877 q^{39} +137.974 q^{41} -295.005i q^{43} -56.2480i q^{45} -186.683 q^{47} +49.0000 q^{49} +226.498i q^{51} -409.210i q^{53} -396.262 q^{55} -376.839 q^{57} +311.117i q^{59} -168.801i q^{61} +63.0000 q^{63} -514.308 q^{65} +563.149i q^{67} +466.305i q^{69} -282.950 q^{71} +250.409 q^{73} +257.821i q^{75} -443.829i q^{77} +948.033 q^{79} +81.0000 q^{81} +1289.88i q^{83} +471.853i q^{85} +168.750 q^{87} -655.368 q^{89} -576.046i q^{91} +477.426i q^{93} -785.053 q^{95} +706.813 q^{97} -570.637i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 6.24978i 0.558997i 0.960146 + 0.279499i \(0.0901682\pi\)
−0.960146 + 0.279499i \(0.909832\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 63.4042i 1.73792i 0.494886 + 0.868958i \(0.335210\pi\)
−0.494886 + 0.868958i \(0.664790\pi\)
\(12\) 0 0
\(13\) 82.2923i 1.75567i 0.478959 + 0.877837i \(0.341014\pi\)
−0.478959 + 0.877837i \(0.658986\pi\)
\(14\) 0 0
\(15\) −18.7493 −0.322737
\(16\) 0 0
\(17\) 75.4992 1.07713 0.538566 0.842583i \(-0.318966\pi\)
0.538566 + 0.842583i \(0.318966\pi\)
\(18\) 0 0
\(19\) 125.613i 1.51672i 0.651839 + 0.758358i \(0.273998\pi\)
−0.651839 + 0.758358i \(0.726002\pi\)
\(20\) 0 0
\(21\) − 21.0000i − 0.218218i
\(22\) 0 0
\(23\) 155.435 1.40915 0.704575 0.709629i \(-0.251138\pi\)
0.704575 + 0.709629i \(0.251138\pi\)
\(24\) 0 0
\(25\) 85.9403 0.687522
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) − 56.2501i − 0.360185i −0.983650 0.180093i \(-0.942360\pi\)
0.983650 0.180093i \(-0.0576398\pi\)
\(30\) 0 0
\(31\) 159.142 0.922024 0.461012 0.887394i \(-0.347487\pi\)
0.461012 + 0.887394i \(0.347487\pi\)
\(32\) 0 0
\(33\) −190.212 −1.00339
\(34\) 0 0
\(35\) − 43.7484i − 0.211281i
\(36\) 0 0
\(37\) 197.279i 0.876554i 0.898840 + 0.438277i \(0.144411\pi\)
−0.898840 + 0.438277i \(0.855589\pi\)
\(38\) 0 0
\(39\) −246.877 −1.01364
\(40\) 0 0
\(41\) 137.974 0.525560 0.262780 0.964856i \(-0.415361\pi\)
0.262780 + 0.964856i \(0.415361\pi\)
\(42\) 0 0
\(43\) − 295.005i − 1.04623i −0.852262 0.523115i \(-0.824770\pi\)
0.852262 0.523115i \(-0.175230\pi\)
\(44\) 0 0
\(45\) − 56.2480i − 0.186332i
\(46\) 0 0
\(47\) −186.683 −0.579374 −0.289687 0.957121i \(-0.593551\pi\)
−0.289687 + 0.957121i \(0.593551\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 226.498i 0.621882i
\(52\) 0 0
\(53\) − 409.210i − 1.06055i −0.847824 0.530277i \(-0.822088\pi\)
0.847824 0.530277i \(-0.177912\pi\)
\(54\) 0 0
\(55\) −396.262 −0.971490
\(56\) 0 0
\(57\) −376.839 −0.875676
\(58\) 0 0
\(59\) 311.117i 0.686507i 0.939243 + 0.343254i \(0.111529\pi\)
−0.939243 + 0.343254i \(0.888471\pi\)
\(60\) 0 0
\(61\) − 168.801i − 0.354306i −0.984183 0.177153i \(-0.943311\pi\)
0.984183 0.177153i \(-0.0566888\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −514.308 −0.981417
\(66\) 0 0
\(67\) 563.149i 1.02686i 0.858132 + 0.513430i \(0.171625\pi\)
−0.858132 + 0.513430i \(0.828375\pi\)
\(68\) 0 0
\(69\) 466.305i 0.813573i
\(70\) 0 0
\(71\) −282.950 −0.472958 −0.236479 0.971637i \(-0.575993\pi\)
−0.236479 + 0.971637i \(0.575993\pi\)
\(72\) 0 0
\(73\) 250.409 0.401481 0.200740 0.979644i \(-0.435665\pi\)
0.200740 + 0.979644i \(0.435665\pi\)
\(74\) 0 0
\(75\) 257.821i 0.396941i
\(76\) 0 0
\(77\) − 443.829i − 0.656870i
\(78\) 0 0
\(79\) 948.033 1.35015 0.675076 0.737748i \(-0.264111\pi\)
0.675076 + 0.737748i \(0.264111\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1289.88i 1.70582i 0.522057 + 0.852911i \(0.325165\pi\)
−0.522057 + 0.852911i \(0.674835\pi\)
\(84\) 0 0
\(85\) 471.853i 0.602114i
\(86\) 0 0
\(87\) 168.750 0.207953
\(88\) 0 0
\(89\) −655.368 −0.780549 −0.390275 0.920699i \(-0.627620\pi\)
−0.390275 + 0.920699i \(0.627620\pi\)
\(90\) 0 0
\(91\) − 576.046i − 0.663583i
\(92\) 0 0
\(93\) 477.426i 0.532331i
\(94\) 0 0
\(95\) −785.053 −0.847840
\(96\) 0 0
\(97\) 706.813 0.739856 0.369928 0.929061i \(-0.379382\pi\)
0.369928 + 0.929061i \(0.379382\pi\)
\(98\) 0 0
\(99\) − 570.637i − 0.579305i
\(100\) 0 0
\(101\) 916.177i 0.902604i 0.892371 + 0.451302i \(0.149040\pi\)
−0.892371 + 0.451302i \(0.850960\pi\)
\(102\) 0 0
\(103\) −1373.56 −1.31399 −0.656993 0.753897i \(-0.728172\pi\)
−0.656993 + 0.753897i \(0.728172\pi\)
\(104\) 0 0
\(105\) 131.245 0.121983
\(106\) 0 0
\(107\) 1543.06i 1.39414i 0.717001 + 0.697072i \(0.245514\pi\)
−0.717001 + 0.697072i \(0.754486\pi\)
\(108\) 0 0
\(109\) − 1798.08i − 1.58005i −0.613077 0.790023i \(-0.710069\pi\)
0.613077 0.790023i \(-0.289931\pi\)
\(110\) 0 0
\(111\) −591.838 −0.506079
\(112\) 0 0
\(113\) −284.375 −0.236741 −0.118370 0.992970i \(-0.537767\pi\)
−0.118370 + 0.992970i \(0.537767\pi\)
\(114\) 0 0
\(115\) 971.435i 0.787711i
\(116\) 0 0
\(117\) − 740.630i − 0.585225i
\(118\) 0 0
\(119\) −528.494 −0.407117
\(120\) 0 0
\(121\) −2689.09 −2.02035
\(122\) 0 0
\(123\) 413.922i 0.303432i
\(124\) 0 0
\(125\) 1318.33i 0.943320i
\(126\) 0 0
\(127\) 842.514 0.588669 0.294335 0.955702i \(-0.404902\pi\)
0.294335 + 0.955702i \(0.404902\pi\)
\(128\) 0 0
\(129\) 885.016 0.604041
\(130\) 0 0
\(131\) − 2590.38i − 1.72765i −0.503790 0.863826i \(-0.668061\pi\)
0.503790 0.863826i \(-0.331939\pi\)
\(132\) 0 0
\(133\) − 879.291i − 0.573265i
\(134\) 0 0
\(135\) 168.744 0.107579
\(136\) 0 0
\(137\) 1917.81 1.19599 0.597993 0.801502i \(-0.295965\pi\)
0.597993 + 0.801502i \(0.295965\pi\)
\(138\) 0 0
\(139\) 2527.67i 1.54240i 0.636590 + 0.771202i \(0.280344\pi\)
−0.636590 + 0.771202i \(0.719656\pi\)
\(140\) 0 0
\(141\) − 560.050i − 0.334502i
\(142\) 0 0
\(143\) −5217.67 −3.05121
\(144\) 0 0
\(145\) 351.550 0.201343
\(146\) 0 0
\(147\) 147.000i 0.0824786i
\(148\) 0 0
\(149\) − 1340.62i − 0.737099i −0.929608 0.368549i \(-0.879855\pi\)
0.929608 0.368549i \(-0.120145\pi\)
\(150\) 0 0
\(151\) 680.821 0.366917 0.183458 0.983027i \(-0.441271\pi\)
0.183458 + 0.983027i \(0.441271\pi\)
\(152\) 0 0
\(153\) −679.493 −0.359044
\(154\) 0 0
\(155\) 994.602i 0.515409i
\(156\) 0 0
\(157\) − 3025.39i − 1.53791i −0.639301 0.768957i \(-0.720776\pi\)
0.639301 0.768957i \(-0.279224\pi\)
\(158\) 0 0
\(159\) 1227.63 0.612311
\(160\) 0 0
\(161\) −1088.05 −0.532609
\(162\) 0 0
\(163\) − 506.970i − 0.243613i −0.992554 0.121807i \(-0.961131\pi\)
0.992554 0.121807i \(-0.0388687\pi\)
\(164\) 0 0
\(165\) − 1188.79i − 0.560890i
\(166\) 0 0
\(167\) 243.431 0.112798 0.0563991 0.998408i \(-0.482038\pi\)
0.0563991 + 0.998408i \(0.482038\pi\)
\(168\) 0 0
\(169\) −4575.02 −2.08239
\(170\) 0 0
\(171\) − 1130.52i − 0.505572i
\(172\) 0 0
\(173\) − 73.9661i − 0.0325060i −0.999868 0.0162530i \(-0.994826\pi\)
0.999868 0.0162530i \(-0.00517372\pi\)
\(174\) 0 0
\(175\) −601.582 −0.259859
\(176\) 0 0
\(177\) −933.350 −0.396355
\(178\) 0 0
\(179\) − 926.879i − 0.387029i −0.981097 0.193514i \(-0.938011\pi\)
0.981097 0.193514i \(-0.0619887\pi\)
\(180\) 0 0
\(181\) − 2615.68i − 1.07416i −0.843533 0.537078i \(-0.819528\pi\)
0.843533 0.537078i \(-0.180472\pi\)
\(182\) 0 0
\(183\) 506.402 0.204559
\(184\) 0 0
\(185\) −1232.95 −0.489991
\(186\) 0 0
\(187\) 4786.96i 1.87196i
\(188\) 0 0
\(189\) 189.000i 0.0727393i
\(190\) 0 0
\(191\) 4268.06 1.61689 0.808444 0.588573i \(-0.200310\pi\)
0.808444 + 0.588573i \(0.200310\pi\)
\(192\) 0 0
\(193\) −3518.04 −1.31209 −0.656047 0.754720i \(-0.727773\pi\)
−0.656047 + 0.754720i \(0.727773\pi\)
\(194\) 0 0
\(195\) − 1542.93i − 0.566621i
\(196\) 0 0
\(197\) − 3490.01i − 1.26220i −0.775702 0.631099i \(-0.782604\pi\)
0.775702 0.631099i \(-0.217396\pi\)
\(198\) 0 0
\(199\) −1731.09 −0.616652 −0.308326 0.951281i \(-0.599769\pi\)
−0.308326 + 0.951281i \(0.599769\pi\)
\(200\) 0 0
\(201\) −1689.45 −0.592858
\(202\) 0 0
\(203\) 393.750i 0.136137i
\(204\) 0 0
\(205\) 862.308i 0.293786i
\(206\) 0 0
\(207\) −1398.92 −0.469717
\(208\) 0 0
\(209\) −7964.38 −2.63592
\(210\) 0 0
\(211\) 2021.06i 0.659411i 0.944084 + 0.329706i \(0.106950\pi\)
−0.944084 + 0.329706i \(0.893050\pi\)
\(212\) 0 0
\(213\) − 848.851i − 0.273063i
\(214\) 0 0
\(215\) 1843.72 0.584839
\(216\) 0 0
\(217\) −1113.99 −0.348492
\(218\) 0 0
\(219\) 751.226i 0.231795i
\(220\) 0 0
\(221\) 6213.00i 1.89109i
\(222\) 0 0
\(223\) 4624.54 1.38871 0.694354 0.719634i \(-0.255690\pi\)
0.694354 + 0.719634i \(0.255690\pi\)
\(224\) 0 0
\(225\) −773.462 −0.229174
\(226\) 0 0
\(227\) 4071.85i 1.19056i 0.803517 + 0.595282i \(0.202960\pi\)
−0.803517 + 0.595282i \(0.797040\pi\)
\(228\) 0 0
\(229\) − 2670.03i − 0.770484i −0.922816 0.385242i \(-0.874118\pi\)
0.922816 0.385242i \(-0.125882\pi\)
\(230\) 0 0
\(231\) 1331.49 0.379244
\(232\) 0 0
\(233\) 6613.19 1.85942 0.929709 0.368295i \(-0.120058\pi\)
0.929709 + 0.368295i \(0.120058\pi\)
\(234\) 0 0
\(235\) − 1166.73i − 0.323868i
\(236\) 0 0
\(237\) 2844.10i 0.779511i
\(238\) 0 0
\(239\) −301.154 −0.0815064 −0.0407532 0.999169i \(-0.512976\pi\)
−0.0407532 + 0.999169i \(0.512976\pi\)
\(240\) 0 0
\(241\) −1311.73 −0.350605 −0.175303 0.984515i \(-0.556090\pi\)
−0.175303 + 0.984515i \(0.556090\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 306.239i 0.0798567i
\(246\) 0 0
\(247\) −10337.0 −2.66286
\(248\) 0 0
\(249\) −3869.65 −0.984856
\(250\) 0 0
\(251\) 4893.64i 1.23061i 0.788288 + 0.615307i \(0.210968\pi\)
−0.788288 + 0.615307i \(0.789032\pi\)
\(252\) 0 0
\(253\) 9855.23i 2.44898i
\(254\) 0 0
\(255\) −1415.56 −0.347630
\(256\) 0 0
\(257\) 2818.68 0.684141 0.342070 0.939674i \(-0.388872\pi\)
0.342070 + 0.939674i \(0.388872\pi\)
\(258\) 0 0
\(259\) − 1380.95i − 0.331306i
\(260\) 0 0
\(261\) 506.251i 0.120062i
\(262\) 0 0
\(263\) −1460.94 −0.342529 −0.171265 0.985225i \(-0.554785\pi\)
−0.171265 + 0.985225i \(0.554785\pi\)
\(264\) 0 0
\(265\) 2557.47 0.592847
\(266\) 0 0
\(267\) − 1966.10i − 0.450650i
\(268\) 0 0
\(269\) − 2399.05i − 0.543765i −0.962330 0.271882i \(-0.912354\pi\)
0.962330 0.271882i \(-0.0876462\pi\)
\(270\) 0 0
\(271\) −221.818 −0.0497214 −0.0248607 0.999691i \(-0.507914\pi\)
−0.0248607 + 0.999691i \(0.507914\pi\)
\(272\) 0 0
\(273\) 1728.14 0.383120
\(274\) 0 0
\(275\) 5448.97i 1.19486i
\(276\) 0 0
\(277\) − 8218.84i − 1.78275i −0.453264 0.891376i \(-0.649741\pi\)
0.453264 0.891376i \(-0.350259\pi\)
\(278\) 0 0
\(279\) −1432.28 −0.307341
\(280\) 0 0
\(281\) −4275.73 −0.907718 −0.453859 0.891073i \(-0.649953\pi\)
−0.453859 + 0.891073i \(0.649953\pi\)
\(282\) 0 0
\(283\) − 1576.10i − 0.331059i −0.986205 0.165530i \(-0.947067\pi\)
0.986205 0.165530i \(-0.0529333\pi\)
\(284\) 0 0
\(285\) − 2355.16i − 0.489500i
\(286\) 0 0
\(287\) −965.819 −0.198643
\(288\) 0 0
\(289\) 787.124 0.160213
\(290\) 0 0
\(291\) 2120.44i 0.427156i
\(292\) 0 0
\(293\) − 5078.63i − 1.01262i −0.862353 0.506308i \(-0.831010\pi\)
0.862353 0.506308i \(-0.168990\pi\)
\(294\) 0 0
\(295\) −1944.41 −0.383756
\(296\) 0 0
\(297\) 1711.91 0.334462
\(298\) 0 0
\(299\) 12791.1i 2.47401i
\(300\) 0 0
\(301\) 2065.04i 0.395438i
\(302\) 0 0
\(303\) −2748.53 −0.521119
\(304\) 0 0
\(305\) 1054.97 0.198056
\(306\) 0 0
\(307\) − 10138.3i − 1.88476i −0.334540 0.942382i \(-0.608581\pi\)
0.334540 0.942382i \(-0.391419\pi\)
\(308\) 0 0
\(309\) − 4120.67i − 0.758630i
\(310\) 0 0
\(311\) −2488.16 −0.453667 −0.226833 0.973934i \(-0.572837\pi\)
−0.226833 + 0.973934i \(0.572837\pi\)
\(312\) 0 0
\(313\) −2857.67 −0.516055 −0.258027 0.966138i \(-0.583072\pi\)
−0.258027 + 0.966138i \(0.583072\pi\)
\(314\) 0 0
\(315\) 393.736i 0.0704270i
\(316\) 0 0
\(317\) − 7112.51i − 1.26018i −0.776520 0.630092i \(-0.783017\pi\)
0.776520 0.630092i \(-0.216983\pi\)
\(318\) 0 0
\(319\) 3566.49 0.625972
\(320\) 0 0
\(321\) −4629.18 −0.804909
\(322\) 0 0
\(323\) 9483.67i 1.63370i
\(324\) 0 0
\(325\) 7072.22i 1.20707i
\(326\) 0 0
\(327\) 5394.25 0.912240
\(328\) 0 0
\(329\) 1306.78 0.218983
\(330\) 0 0
\(331\) − 6184.60i − 1.02700i −0.858090 0.513499i \(-0.828349\pi\)
0.858090 0.513499i \(-0.171651\pi\)
\(332\) 0 0
\(333\) − 1775.51i − 0.292185i
\(334\) 0 0
\(335\) −3519.56 −0.574012
\(336\) 0 0
\(337\) −451.244 −0.0729402 −0.0364701 0.999335i \(-0.511611\pi\)
−0.0364701 + 0.999335i \(0.511611\pi\)
\(338\) 0 0
\(339\) − 853.124i − 0.136682i
\(340\) 0 0
\(341\) 10090.3i 1.60240i
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) −2914.30 −0.454785
\(346\) 0 0
\(347\) − 12493.5i − 1.93282i −0.257006 0.966410i \(-0.582736\pi\)
0.257006 0.966410i \(-0.417264\pi\)
\(348\) 0 0
\(349\) − 11188.5i − 1.71606i −0.513596 0.858032i \(-0.671687\pi\)
0.513596 0.858032i \(-0.328313\pi\)
\(350\) 0 0
\(351\) 2221.89 0.337880
\(352\) 0 0
\(353\) 1060.98 0.159972 0.0799859 0.996796i \(-0.474512\pi\)
0.0799859 + 0.996796i \(0.474512\pi\)
\(354\) 0 0
\(355\) − 1768.38i − 0.264382i
\(356\) 0 0
\(357\) − 1585.48i − 0.235049i
\(358\) 0 0
\(359\) −10700.8 −1.57317 −0.786585 0.617482i \(-0.788153\pi\)
−0.786585 + 0.617482i \(0.788153\pi\)
\(360\) 0 0
\(361\) −8919.62 −1.30043
\(362\) 0 0
\(363\) − 8067.26i − 1.16645i
\(364\) 0 0
\(365\) 1565.00i 0.224427i
\(366\) 0 0
\(367\) 10121.9 1.43967 0.719833 0.694147i \(-0.244218\pi\)
0.719833 + 0.694147i \(0.244218\pi\)
\(368\) 0 0
\(369\) −1241.77 −0.175187
\(370\) 0 0
\(371\) 2864.47i 0.400852i
\(372\) 0 0
\(373\) 2099.58i 0.291454i 0.989325 + 0.145727i \(0.0465521\pi\)
−0.989325 + 0.145727i \(0.953448\pi\)
\(374\) 0 0
\(375\) −3954.99 −0.544626
\(376\) 0 0
\(377\) 4628.94 0.632368
\(378\) 0 0
\(379\) − 10101.3i − 1.36905i −0.728989 0.684526i \(-0.760009\pi\)
0.728989 0.684526i \(-0.239991\pi\)
\(380\) 0 0
\(381\) 2527.54i 0.339868i
\(382\) 0 0
\(383\) 8403.09 1.12109 0.560546 0.828124i \(-0.310591\pi\)
0.560546 + 0.828124i \(0.310591\pi\)
\(384\) 0 0
\(385\) 2773.83 0.367189
\(386\) 0 0
\(387\) 2655.05i 0.348743i
\(388\) 0 0
\(389\) − 411.307i − 0.0536095i −0.999641 0.0268048i \(-0.991467\pi\)
0.999641 0.0268048i \(-0.00853324\pi\)
\(390\) 0 0
\(391\) 11735.2 1.51784
\(392\) 0 0
\(393\) 7771.14 0.997461
\(394\) 0 0
\(395\) 5925.00i 0.754731i
\(396\) 0 0
\(397\) 10734.2i 1.35701i 0.734595 + 0.678506i \(0.237372\pi\)
−0.734595 + 0.678506i \(0.762628\pi\)
\(398\) 0 0
\(399\) 2637.87 0.330974
\(400\) 0 0
\(401\) 12415.7 1.54616 0.773080 0.634309i \(-0.218715\pi\)
0.773080 + 0.634309i \(0.218715\pi\)
\(402\) 0 0
\(403\) 13096.1i 1.61877i
\(404\) 0 0
\(405\) 506.232i 0.0621108i
\(406\) 0 0
\(407\) −12508.3 −1.52338
\(408\) 0 0
\(409\) 2349.10 0.283999 0.141999 0.989867i \(-0.454647\pi\)
0.141999 + 0.989867i \(0.454647\pi\)
\(410\) 0 0
\(411\) 5753.44i 0.690502i
\(412\) 0 0
\(413\) − 2177.82i − 0.259475i
\(414\) 0 0
\(415\) −8061.49 −0.953549
\(416\) 0 0
\(417\) −7583.01 −0.890508
\(418\) 0 0
\(419\) − 1665.84i − 0.194228i −0.995273 0.0971138i \(-0.969039\pi\)
0.995273 0.0971138i \(-0.0309611\pi\)
\(420\) 0 0
\(421\) 1154.48i 0.133648i 0.997765 + 0.0668239i \(0.0212866\pi\)
−0.997765 + 0.0668239i \(0.978713\pi\)
\(422\) 0 0
\(423\) 1680.15 0.193125
\(424\) 0 0
\(425\) 6488.42 0.740552
\(426\) 0 0
\(427\) 1181.60i 0.133915i
\(428\) 0 0
\(429\) − 15653.0i − 1.76162i
\(430\) 0 0
\(431\) 237.625 0.0265569 0.0132784 0.999912i \(-0.495773\pi\)
0.0132784 + 0.999912i \(0.495773\pi\)
\(432\) 0 0
\(433\) 4763.79 0.528714 0.264357 0.964425i \(-0.414840\pi\)
0.264357 + 0.964425i \(0.414840\pi\)
\(434\) 0 0
\(435\) 1054.65i 0.116245i
\(436\) 0 0
\(437\) 19524.7i 2.13728i
\(438\) 0 0
\(439\) 1715.57 0.186515 0.0932573 0.995642i \(-0.470272\pi\)
0.0932573 + 0.995642i \(0.470272\pi\)
\(440\) 0 0
\(441\) −441.000 −0.0476190
\(442\) 0 0
\(443\) 6287.39i 0.674318i 0.941448 + 0.337159i \(0.109466\pi\)
−0.941448 + 0.337159i \(0.890534\pi\)
\(444\) 0 0
\(445\) − 4095.90i − 0.436325i
\(446\) 0 0
\(447\) 4021.86 0.425564
\(448\) 0 0
\(449\) 9104.58 0.956952 0.478476 0.878101i \(-0.341189\pi\)
0.478476 + 0.878101i \(0.341189\pi\)
\(450\) 0 0
\(451\) 8748.13i 0.913378i
\(452\) 0 0
\(453\) 2042.46i 0.211839i
\(454\) 0 0
\(455\) 3600.16 0.370941
\(456\) 0 0
\(457\) 9817.29 1.00489 0.502444 0.864610i \(-0.332434\pi\)
0.502444 + 0.864610i \(0.332434\pi\)
\(458\) 0 0
\(459\) − 2038.48i − 0.207294i
\(460\) 0 0
\(461\) 3975.83i 0.401676i 0.979624 + 0.200838i \(0.0643665\pi\)
−0.979624 + 0.200838i \(0.935633\pi\)
\(462\) 0 0
\(463\) −7559.96 −0.758837 −0.379418 0.925225i \(-0.623876\pi\)
−0.379418 + 0.925225i \(0.623876\pi\)
\(464\) 0 0
\(465\) −2983.81 −0.297571
\(466\) 0 0
\(467\) − 524.945i − 0.0520161i −0.999662 0.0260081i \(-0.991720\pi\)
0.999662 0.0260081i \(-0.00827956\pi\)
\(468\) 0 0
\(469\) − 3942.04i − 0.388116i
\(470\) 0 0
\(471\) 9076.17 0.887915
\(472\) 0 0
\(473\) 18704.6 1.81826
\(474\) 0 0
\(475\) 10795.2i 1.04278i
\(476\) 0 0
\(477\) 3682.89i 0.353518i
\(478\) 0 0
\(479\) 78.5348 0.00749133 0.00374567 0.999993i \(-0.498808\pi\)
0.00374567 + 0.999993i \(0.498808\pi\)
\(480\) 0 0
\(481\) −16234.6 −1.53894
\(482\) 0 0
\(483\) − 3264.14i − 0.307502i
\(484\) 0 0
\(485\) 4417.42i 0.413577i
\(486\) 0 0
\(487\) −14048.9 −1.30722 −0.653610 0.756832i \(-0.726746\pi\)
−0.653610 + 0.756832i \(0.726746\pi\)
\(488\) 0 0
\(489\) 1520.91 0.140650
\(490\) 0 0
\(491\) − 14954.2i − 1.37449i −0.726428 0.687243i \(-0.758821\pi\)
0.726428 0.687243i \(-0.241179\pi\)
\(492\) 0 0
\(493\) − 4246.83i − 0.387967i
\(494\) 0 0
\(495\) 3566.36 0.323830
\(496\) 0 0
\(497\) 1980.65 0.178761
\(498\) 0 0
\(499\) − 11687.5i − 1.04851i −0.851562 0.524254i \(-0.824344\pi\)
0.851562 0.524254i \(-0.175656\pi\)
\(500\) 0 0
\(501\) 730.294i 0.0651240i
\(502\) 0 0
\(503\) −20229.8 −1.79324 −0.896620 0.442800i \(-0.853985\pi\)
−0.896620 + 0.442800i \(0.853985\pi\)
\(504\) 0 0
\(505\) −5725.90 −0.504553
\(506\) 0 0
\(507\) − 13725.0i − 1.20227i
\(508\) 0 0
\(509\) 12175.4i 1.06025i 0.847919 + 0.530125i \(0.177855\pi\)
−0.847919 + 0.530125i \(0.822145\pi\)
\(510\) 0 0
\(511\) −1752.86 −0.151746
\(512\) 0 0
\(513\) 3391.55 0.291892
\(514\) 0 0
\(515\) − 8584.43i − 0.734515i
\(516\) 0 0
\(517\) − 11836.5i − 1.00690i
\(518\) 0 0
\(519\) 221.898 0.0187674
\(520\) 0 0
\(521\) −2133.49 −0.179405 −0.0897025 0.995969i \(-0.528592\pi\)
−0.0897025 + 0.995969i \(0.528592\pi\)
\(522\) 0 0
\(523\) − 6979.20i − 0.583516i −0.956492 0.291758i \(-0.905760\pi\)
0.956492 0.291758i \(-0.0942402\pi\)
\(524\) 0 0
\(525\) − 1804.75i − 0.150030i
\(526\) 0 0
\(527\) 12015.1 0.993141
\(528\) 0 0
\(529\) 11993.1 0.985705
\(530\) 0 0
\(531\) − 2800.05i − 0.228836i
\(532\) 0 0
\(533\) 11354.2i 0.922711i
\(534\) 0 0
\(535\) −9643.79 −0.779322
\(536\) 0 0
\(537\) 2780.64 0.223451
\(538\) 0 0
\(539\) 3106.80i 0.248274i
\(540\) 0 0
\(541\) − 5909.97i − 0.469666i −0.972036 0.234833i \(-0.924546\pi\)
0.972036 0.234833i \(-0.0754543\pi\)
\(542\) 0 0
\(543\) 7847.05 0.620164
\(544\) 0 0
\(545\) 11237.6 0.883241
\(546\) 0 0
\(547\) − 16572.1i − 1.29538i −0.761906 0.647688i \(-0.775736\pi\)
0.761906 0.647688i \(-0.224264\pi\)
\(548\) 0 0
\(549\) 1519.20i 0.118102i
\(550\) 0 0
\(551\) 7065.74 0.546299
\(552\) 0 0
\(553\) −6636.23 −0.510310
\(554\) 0 0
\(555\) − 3698.85i − 0.282897i
\(556\) 0 0
\(557\) 9525.34i 0.724599i 0.932062 + 0.362300i \(0.118008\pi\)
−0.932062 + 0.362300i \(0.881992\pi\)
\(558\) 0 0
\(559\) 24276.6 1.83684
\(560\) 0 0
\(561\) −14360.9 −1.08078
\(562\) 0 0
\(563\) 291.473i 0.0218190i 0.999940 + 0.0109095i \(0.00347268\pi\)
−0.999940 + 0.0109095i \(0.996527\pi\)
\(564\) 0 0
\(565\) − 1777.28i − 0.132337i
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) −852.491 −0.0628089 −0.0314045 0.999507i \(-0.509998\pi\)
−0.0314045 + 0.999507i \(0.509998\pi\)
\(570\) 0 0
\(571\) − 13366.5i − 0.979634i −0.871825 0.489817i \(-0.837064\pi\)
0.871825 0.489817i \(-0.162936\pi\)
\(572\) 0 0
\(573\) 12804.2i 0.933511i
\(574\) 0 0
\(575\) 13358.1 0.968822
\(576\) 0 0
\(577\) 704.046 0.0507969 0.0253985 0.999677i \(-0.491915\pi\)
0.0253985 + 0.999677i \(0.491915\pi\)
\(578\) 0 0
\(579\) − 10554.1i − 0.757538i
\(580\) 0 0
\(581\) − 9029.19i − 0.644740i
\(582\) 0 0
\(583\) 25945.6 1.84315
\(584\) 0 0
\(585\) 4628.78 0.327139
\(586\) 0 0
\(587\) 18056.0i 1.26959i 0.772679 + 0.634797i \(0.218916\pi\)
−0.772679 + 0.634797i \(0.781084\pi\)
\(588\) 0 0
\(589\) 19990.3i 1.39845i
\(590\) 0 0
\(591\) 10470.0 0.728730
\(592\) 0 0
\(593\) 19109.4 1.32332 0.661660 0.749804i \(-0.269852\pi\)
0.661660 + 0.749804i \(0.269852\pi\)
\(594\) 0 0
\(595\) − 3302.97i − 0.227578i
\(596\) 0 0
\(597\) − 5193.27i − 0.356024i
\(598\) 0 0
\(599\) −7474.66 −0.509861 −0.254930 0.966959i \(-0.582053\pi\)
−0.254930 + 0.966959i \(0.582053\pi\)
\(600\) 0 0
\(601\) 2875.13 0.195140 0.0975699 0.995229i \(-0.468893\pi\)
0.0975699 + 0.995229i \(0.468893\pi\)
\(602\) 0 0
\(603\) − 5068.34i − 0.342287i
\(604\) 0 0
\(605\) − 16806.2i − 1.12937i
\(606\) 0 0
\(607\) 24346.2 1.62797 0.813987 0.580883i \(-0.197293\pi\)
0.813987 + 0.580883i \(0.197293\pi\)
\(608\) 0 0
\(609\) −1181.25 −0.0785989
\(610\) 0 0
\(611\) − 15362.6i − 1.01719i
\(612\) 0 0
\(613\) 19444.9i 1.28120i 0.767877 + 0.640598i \(0.221313\pi\)
−0.767877 + 0.640598i \(0.778687\pi\)
\(614\) 0 0
\(615\) −2586.92 −0.169618
\(616\) 0 0
\(617\) −9935.81 −0.648299 −0.324150 0.946006i \(-0.605078\pi\)
−0.324150 + 0.946006i \(0.605078\pi\)
\(618\) 0 0
\(619\) − 19396.4i − 1.25946i −0.776813 0.629731i \(-0.783165\pi\)
0.776813 0.629731i \(-0.216835\pi\)
\(620\) 0 0
\(621\) − 4196.75i − 0.271191i
\(622\) 0 0
\(623\) 4587.58 0.295020
\(624\) 0 0
\(625\) 2503.26 0.160209
\(626\) 0 0
\(627\) − 23893.2i − 1.52185i
\(628\) 0 0
\(629\) 14894.4i 0.944164i
\(630\) 0 0
\(631\) 1391.17 0.0877681 0.0438840 0.999037i \(-0.486027\pi\)
0.0438840 + 0.999037i \(0.486027\pi\)
\(632\) 0 0
\(633\) −6063.19 −0.380711
\(634\) 0 0
\(635\) 5265.53i 0.329065i
\(636\) 0 0
\(637\) 4032.32i 0.250811i
\(638\) 0 0
\(639\) 2546.55 0.157653
\(640\) 0 0
\(641\) 17848.0 1.09977 0.549886 0.835240i \(-0.314671\pi\)
0.549886 + 0.835240i \(0.314671\pi\)
\(642\) 0 0
\(643\) 4152.29i 0.254666i 0.991860 + 0.127333i \(0.0406417\pi\)
−0.991860 + 0.127333i \(0.959358\pi\)
\(644\) 0 0
\(645\) 5531.15i 0.337657i
\(646\) 0 0
\(647\) −8018.93 −0.487259 −0.243630 0.969868i \(-0.578338\pi\)
−0.243630 + 0.969868i \(0.578338\pi\)
\(648\) 0 0
\(649\) −19726.1 −1.19309
\(650\) 0 0
\(651\) − 3341.98i − 0.201202i
\(652\) 0 0
\(653\) 639.444i 0.0383206i 0.999816 + 0.0191603i \(0.00609929\pi\)
−0.999816 + 0.0191603i \(0.993901\pi\)
\(654\) 0 0
\(655\) 16189.3 0.965753
\(656\) 0 0
\(657\) −2253.68 −0.133827
\(658\) 0 0
\(659\) 24156.4i 1.42792i 0.700187 + 0.713960i \(0.253100\pi\)
−0.700187 + 0.713960i \(0.746900\pi\)
\(660\) 0 0
\(661\) − 8195.25i − 0.482236i −0.970496 0.241118i \(-0.922486\pi\)
0.970496 0.241118i \(-0.0775141\pi\)
\(662\) 0 0
\(663\) −18639.0 −1.09182
\(664\) 0 0
\(665\) 5495.37 0.320453
\(666\) 0 0
\(667\) − 8743.23i − 0.507555i
\(668\) 0 0
\(669\) 13873.6i 0.801771i
\(670\) 0 0
\(671\) 10702.7 0.615755
\(672\) 0 0
\(673\) −26794.9 −1.53472 −0.767360 0.641217i \(-0.778430\pi\)
−0.767360 + 0.641217i \(0.778430\pi\)
\(674\) 0 0
\(675\) − 2320.39i − 0.132314i
\(676\) 0 0
\(677\) − 24832.7i − 1.40975i −0.709333 0.704874i \(-0.751004\pi\)
0.709333 0.704874i \(-0.248996\pi\)
\(678\) 0 0
\(679\) −4947.69 −0.279639
\(680\) 0 0
\(681\) −12215.5 −0.687373
\(682\) 0 0
\(683\) 22982.8i 1.28758i 0.765204 + 0.643788i \(0.222638\pi\)
−0.765204 + 0.643788i \(0.777362\pi\)
\(684\) 0 0
\(685\) 11985.9i 0.668552i
\(686\) 0 0
\(687\) 8010.10 0.444839
\(688\) 0 0
\(689\) 33674.8 1.86199
\(690\) 0 0
\(691\) − 2558.77i − 0.140868i −0.997516 0.0704342i \(-0.977562\pi\)
0.997516 0.0704342i \(-0.0224385\pi\)
\(692\) 0 0
\(693\) 3994.46i 0.218957i
\(694\) 0 0
\(695\) −15797.4 −0.862200
\(696\) 0 0
\(697\) 10416.9 0.566097
\(698\) 0 0
\(699\) 19839.6i 1.07354i
\(700\) 0 0
\(701\) 33784.3i 1.82028i 0.414302 + 0.910140i \(0.364026\pi\)
−0.414302 + 0.910140i \(0.635974\pi\)
\(702\) 0 0
\(703\) −24780.8 −1.32948
\(704\) 0 0
\(705\) 3500.19 0.186985
\(706\) 0 0
\(707\) − 6413.24i − 0.341152i
\(708\) 0 0
\(709\) − 3291.19i − 0.174334i −0.996194 0.0871672i \(-0.972219\pi\)
0.996194 0.0871672i \(-0.0277814\pi\)
\(710\) 0 0
\(711\) −8532.30 −0.450051
\(712\) 0 0
\(713\) 24736.2 1.29927
\(714\) 0 0
\(715\) − 32609.3i − 1.70562i
\(716\) 0 0
\(717\) − 903.462i − 0.0470577i
\(718\) 0 0
\(719\) 19459.7 1.00935 0.504676 0.863309i \(-0.331612\pi\)
0.504676 + 0.863309i \(0.331612\pi\)
\(720\) 0 0
\(721\) 9614.90 0.496640
\(722\) 0 0
\(723\) − 3935.19i − 0.202422i
\(724\) 0 0
\(725\) − 4834.15i − 0.247635i
\(726\) 0 0
\(727\) 15508.8 0.791184 0.395592 0.918426i \(-0.370539\pi\)
0.395592 + 0.918426i \(0.370539\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 22272.6i − 1.12693i
\(732\) 0 0
\(733\) − 26507.3i − 1.33570i −0.744295 0.667851i \(-0.767214\pi\)
0.744295 0.667851i \(-0.232786\pi\)
\(734\) 0 0
\(735\) −918.717 −0.0461053
\(736\) 0 0
\(737\) −35706.0 −1.78460
\(738\) 0 0
\(739\) − 13802.6i − 0.687057i −0.939142 0.343529i \(-0.888378\pi\)
0.939142 0.343529i \(-0.111622\pi\)
\(740\) 0 0
\(741\) − 31010.9i − 1.53740i
\(742\) 0 0
\(743\) −22406.2 −1.10633 −0.553165 0.833072i \(-0.686580\pi\)
−0.553165 + 0.833072i \(0.686580\pi\)
\(744\) 0 0
\(745\) 8378.57 0.412036
\(746\) 0 0
\(747\) − 11609.0i − 0.568607i
\(748\) 0 0
\(749\) − 10801.4i − 0.526937i
\(750\) 0 0
\(751\) 27470.5 1.33477 0.667384 0.744713i \(-0.267414\pi\)
0.667384 + 0.744713i \(0.267414\pi\)
\(752\) 0 0
\(753\) −14680.9 −0.710495
\(754\) 0 0
\(755\) 4254.98i 0.205105i
\(756\) 0 0
\(757\) − 10421.8i − 0.500379i −0.968197 0.250189i \(-0.919507\pi\)
0.968197 0.250189i \(-0.0804929\pi\)
\(758\) 0 0
\(759\) −29565.7 −1.41392
\(760\) 0 0
\(761\) 32815.4 1.56315 0.781576 0.623810i \(-0.214416\pi\)
0.781576 + 0.623810i \(0.214416\pi\)
\(762\) 0 0
\(763\) 12586.6i 0.597201i
\(764\) 0 0
\(765\) − 4246.68i − 0.200705i
\(766\) 0 0
\(767\) −25602.5 −1.20528
\(768\) 0 0
\(769\) 32970.9 1.54611 0.773057 0.634336i \(-0.218727\pi\)
0.773057 + 0.634336i \(0.218727\pi\)
\(770\) 0 0
\(771\) 8456.03i 0.394989i
\(772\) 0 0
\(773\) 19379.4i 0.901720i 0.892595 + 0.450860i \(0.148883\pi\)
−0.892595 + 0.450860i \(0.851117\pi\)
\(774\) 0 0
\(775\) 13676.7 0.633912
\(776\) 0 0
\(777\) 4142.86 0.191280
\(778\) 0 0
\(779\) 17331.3i 0.797124i
\(780\) 0 0
\(781\) − 17940.2i − 0.821961i
\(782\) 0 0
\(783\) −1518.75 −0.0693177
\(784\) 0 0
\(785\) 18908.0 0.859689
\(786\) 0 0
\(787\) − 39440.8i − 1.78642i −0.449640 0.893210i \(-0.648448\pi\)
0.449640 0.893210i \(-0.351552\pi\)
\(788\) 0 0
\(789\) − 4382.81i − 0.197759i
\(790\) 0 0
\(791\) 1990.62 0.0894797
\(792\) 0 0
\(793\) 13891.0 0.622047
\(794\) 0 0
\(795\) 7672.42i 0.342280i
\(796\) 0 0
\(797\) 1593.15i 0.0708058i 0.999373 + 0.0354029i \(0.0112714\pi\)
−0.999373 + 0.0354029i \(0.988729\pi\)
\(798\) 0 0
\(799\) −14094.4 −0.624062
\(800\) 0 0
\(801\) 5898.31 0.260183
\(802\) 0 0
\(803\) 15876.9i 0.697740i
\(804\) 0 0
\(805\) − 6800.04i − 0.297727i
\(806\) 0 0
\(807\) 7197.15 0.313943
\(808\) 0 0
\(809\) −24472.3 −1.06354 −0.531768 0.846890i \(-0.678472\pi\)
−0.531768 + 0.846890i \(0.678472\pi\)
\(810\) 0 0
\(811\) 12522.8i 0.542215i 0.962549 + 0.271107i \(0.0873898\pi\)
−0.962549 + 0.271107i \(0.912610\pi\)
\(812\) 0 0
\(813\) − 665.455i − 0.0287067i
\(814\) 0 0
\(815\) 3168.45 0.136179
\(816\) 0 0
\(817\) 37056.5 1.58683
\(818\) 0 0
\(819\) 5184.41i 0.221194i
\(820\) 0 0
\(821\) 42179.5i 1.79303i 0.443016 + 0.896514i \(0.353909\pi\)
−0.443016 + 0.896514i \(0.646091\pi\)
\(822\) 0 0
\(823\) −19287.4 −0.816909 −0.408454 0.912779i \(-0.633932\pi\)
−0.408454 + 0.912779i \(0.633932\pi\)
\(824\) 0 0
\(825\) −16346.9 −0.689850
\(826\) 0 0
\(827\) − 25707.3i − 1.08093i −0.841367 0.540465i \(-0.818248\pi\)
0.841367 0.540465i \(-0.181752\pi\)
\(828\) 0 0
\(829\) − 23979.6i − 1.00464i −0.864682 0.502320i \(-0.832480\pi\)
0.864682 0.502320i \(-0.167520\pi\)
\(830\) 0 0
\(831\) 24656.5 1.02927
\(832\) 0 0
\(833\) 3699.46 0.153876
\(834\) 0 0
\(835\) 1521.39i 0.0630539i
\(836\) 0 0
\(837\) − 4296.83i − 0.177444i
\(838\) 0 0
\(839\) 17884.0 0.735907 0.367953 0.929844i \(-0.380059\pi\)
0.367953 + 0.929844i \(0.380059\pi\)
\(840\) 0 0
\(841\) 21224.9 0.870267
\(842\) 0 0
\(843\) − 12827.2i − 0.524071i
\(844\) 0 0
\(845\) − 28592.8i − 1.16405i
\(846\) 0 0
\(847\) 18823.6 0.763621
\(848\) 0 0
\(849\) 4728.31 0.191137
\(850\) 0 0
\(851\) 30664.1i 1.23520i
\(852\) 0 0
\(853\) 38267.4i 1.53605i 0.640421 + 0.768024i \(0.278760\pi\)
−0.640421 + 0.768024i \(0.721240\pi\)
\(854\) 0 0
\(855\) 7065.48 0.282613
\(856\) 0 0
\(857\) −15924.9 −0.634753 −0.317376 0.948300i \(-0.602802\pi\)
−0.317376 + 0.948300i \(0.602802\pi\)
\(858\) 0 0
\(859\) 5525.23i 0.219462i 0.993961 + 0.109731i \(0.0349990\pi\)
−0.993961 + 0.109731i \(0.965001\pi\)
\(860\) 0 0
\(861\) − 2897.46i − 0.114686i
\(862\) 0 0
\(863\) 6293.58 0.248245 0.124123 0.992267i \(-0.460388\pi\)
0.124123 + 0.992267i \(0.460388\pi\)
\(864\) 0 0
\(865\) 462.272 0.0181708
\(866\) 0 0
\(867\) 2361.37i 0.0924988i
\(868\) 0 0
\(869\) 60109.2i 2.34645i
\(870\) 0 0
\(871\) −46342.8 −1.80283
\(872\) 0 0
\(873\) −6361.32 −0.246619
\(874\) 0 0
\(875\) − 9228.31i − 0.356541i
\(876\) 0 0
\(877\) 20722.2i 0.797879i 0.916977 + 0.398940i \(0.130622\pi\)
−0.916977 + 0.398940i \(0.869378\pi\)
\(878\) 0 0
\(879\) 15235.9 0.584634
\(880\) 0 0
\(881\) 776.203 0.0296832 0.0148416 0.999890i \(-0.495276\pi\)
0.0148416 + 0.999890i \(0.495276\pi\)
\(882\) 0 0
\(883\) 28411.0i 1.08279i 0.840767 + 0.541397i \(0.182104\pi\)
−0.840767 + 0.541397i \(0.817896\pi\)
\(884\) 0 0
\(885\) − 5833.23i − 0.221561i
\(886\) 0 0
\(887\) −8007.06 −0.303101 −0.151551 0.988449i \(-0.548427\pi\)
−0.151551 + 0.988449i \(0.548427\pi\)
\(888\) 0 0
\(889\) −5897.60 −0.222496
\(890\) 0 0
\(891\) 5135.74i 0.193102i
\(892\) 0 0
\(893\) − 23449.9i − 0.878745i
\(894\) 0 0
\(895\) 5792.79 0.216348
\(896\) 0 0
\(897\) −38373.3 −1.42837
\(898\) 0 0
\(899\) − 8951.74i − 0.332099i
\(900\) 0 0
\(901\) − 30895.0i − 1.14236i
\(902\) 0 0
\(903\) −6195.11 −0.228306
\(904\) 0 0
\(905\) 16347.4 0.600450
\(906\) 0 0
\(907\) − 5330.38i − 0.195140i −0.995229 0.0975701i \(-0.968893\pi\)
0.995229 0.0975701i \(-0.0311070\pi\)
\(908\) 0 0
\(909\) − 8245.59i − 0.300868i
\(910\) 0 0
\(911\) −43468.8 −1.58089 −0.790443 0.612536i \(-0.790150\pi\)
−0.790443 + 0.612536i \(0.790150\pi\)
\(912\) 0 0
\(913\) −81784.0 −2.96457
\(914\) 0 0
\(915\) 3164.90i 0.114348i
\(916\) 0 0
\(917\) 18132.7i 0.652991i
\(918\) 0 0
\(919\) 35514.2 1.27476 0.637381 0.770549i \(-0.280018\pi\)
0.637381 + 0.770549i \(0.280018\pi\)
\(920\) 0 0
\(921\) 30414.8 1.08817
\(922\) 0 0
\(923\) − 23284.6i − 0.830361i
\(924\) 0 0
\(925\) 16954.2i 0.602651i
\(926\) 0 0
\(927\) 12362.0 0.437995
\(928\) 0 0
\(929\) −38350.8 −1.35441 −0.677206 0.735793i \(-0.736810\pi\)
−0.677206 + 0.735793i \(0.736810\pi\)
\(930\) 0 0
\(931\) 6155.04i 0.216674i
\(932\) 0 0
\(933\) − 7464.47i − 0.261925i
\(934\) 0 0
\(935\) −29917.4 −1.04642
\(936\) 0 0
\(937\) −39488.4 −1.37676 −0.688382 0.725348i \(-0.741679\pi\)
−0.688382 + 0.725348i \(0.741679\pi\)
\(938\) 0 0
\(939\) − 8573.01i − 0.297944i
\(940\) 0 0
\(941\) 46022.8i 1.59437i 0.603737 + 0.797184i \(0.293678\pi\)
−0.603737 + 0.797184i \(0.706322\pi\)
\(942\) 0 0
\(943\) 21446.0 0.740592
\(944\) 0 0
\(945\) −1181.21 −0.0406611
\(946\) 0 0
\(947\) 34780.0i 1.19345i 0.802446 + 0.596725i \(0.203532\pi\)
−0.802446 + 0.596725i \(0.796468\pi\)
\(948\) 0 0
\(949\) 20606.7i 0.704870i
\(950\) 0 0
\(951\) 21337.5 0.727568
\(952\) 0 0
\(953\) −1554.63 −0.0528431 −0.0264215 0.999651i \(-0.508411\pi\)
−0.0264215 + 0.999651i \(0.508411\pi\)
\(954\) 0 0
\(955\) 26674.4i 0.903836i
\(956\) 0 0
\(957\) 10699.5i 0.361405i
\(958\) 0 0
\(959\) −13424.7 −0.452040
\(960\) 0 0
\(961\) −4464.85 −0.149872
\(962\) 0 0
\(963\) − 13887.5i − 0.464714i
\(964\) 0 0
\(965\) − 21987.0i − 0.733457i
\(966\) 0 0
\(967\) −32939.5 −1.09541 −0.547706 0.836671i \(-0.684499\pi\)
−0.547706 + 0.836671i \(0.684499\pi\)
\(968\) 0 0
\(969\) −28451.0 −0.943218
\(970\) 0 0
\(971\) 10059.9i 0.332480i 0.986085 + 0.166240i \(0.0531627\pi\)
−0.986085 + 0.166240i \(0.946837\pi\)
\(972\) 0 0
\(973\) − 17693.7i − 0.582974i
\(974\) 0 0
\(975\) −21216.7 −0.696899
\(976\) 0 0
\(977\) −25682.5 −0.840999 −0.420500 0.907293i \(-0.638145\pi\)
−0.420500 + 0.907293i \(0.638145\pi\)
\(978\) 0 0
\(979\) − 41553.1i − 1.35653i
\(980\) 0 0
\(981\) 16182.7i 0.526682i
\(982\) 0 0
\(983\) 35827.0 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(984\) 0 0
\(985\) 21811.8 0.705565
\(986\) 0 0
\(987\) 3920.35i 0.126430i
\(988\) 0 0
\(989\) − 45854.2i − 1.47429i
\(990\) 0 0
\(991\) −28383.9 −0.909832 −0.454916 0.890534i \(-0.650331\pi\)
−0.454916 + 0.890534i \(0.650331\pi\)
\(992\) 0 0
\(993\) 18553.8 0.592938
\(994\) 0 0
\(995\) − 10818.9i − 0.344707i
\(996\) 0 0
\(997\) 22391.1i 0.711268i 0.934625 + 0.355634i \(0.115735\pi\)
−0.934625 + 0.355634i \(0.884265\pi\)
\(998\) 0 0
\(999\) 5326.54 0.168693
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 1344.4.c.f.673.11 yes 12
4.3 odd 2 1344.4.c.g.673.5 yes 12
8.3 odd 2 1344.4.c.g.673.8 yes 12
8.5 even 2 inner 1344.4.c.f.673.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.c.f.673.2 12 8.5 even 2 inner
1344.4.c.f.673.11 yes 12 1.1 even 1 trivial
1344.4.c.g.673.5 yes 12 4.3 odd 2
1344.4.c.g.673.8 yes 12 8.3 odd 2