Properties

Label 1344.4.c.f.673.7
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 673.7
Root \(-0.0488747 + 0.182403i\) of defining polynomial
Character \(\chi\) \(=\) 1344.673
Dual form 1344.4.c.f.673.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -17.1345i q^{5} -7.00000 q^{7} -9.00000 q^{9} -2.98040i q^{11} +39.8701i q^{13} +51.4036 q^{15} +132.530 q^{17} -83.2004i q^{19} -21.0000i q^{21} -80.0478 q^{23} -168.592 q^{25} -27.0000i q^{27} +139.743i q^{29} +204.541 q^{31} +8.94119 q^{33} +119.942i q^{35} +377.916i q^{37} -119.610 q^{39} +45.3733 q^{41} -540.789i q^{43} +154.211i q^{45} +362.473 q^{47} +49.0000 q^{49} +397.589i q^{51} +668.494i q^{53} -51.0677 q^{55} +249.601 q^{57} -757.392i q^{59} -690.035i q^{61} +63.0000 q^{63} +683.157 q^{65} -370.530i q^{67} -240.143i q^{69} -652.184 q^{71} +229.804 q^{73} -505.777i q^{75} +20.8628i q^{77} +43.3308 q^{79} +81.0000 q^{81} -802.404i q^{83} -2270.84i q^{85} -419.230 q^{87} -868.188 q^{89} -279.091i q^{91} +613.622i q^{93} -1425.60 q^{95} -1181.82 q^{97} +26.8236i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 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
\(5\) − 17.1345i − 1.53256i −0.642507 0.766280i \(-0.722106\pi\)
0.642507 0.766280i \(-0.277894\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) − 2.98040i − 0.0816931i −0.999165 0.0408465i \(-0.986995\pi\)
0.999165 0.0408465i \(-0.0130055\pi\)
\(12\) 0 0
\(13\) 39.8701i 0.850615i 0.905049 + 0.425307i \(0.139834\pi\)
−0.905049 + 0.425307i \(0.860166\pi\)
\(14\) 0 0
\(15\) 51.4036 0.884824
\(16\) 0 0
\(17\) 132.530 1.89077 0.945387 0.325949i \(-0.105684\pi\)
0.945387 + 0.325949i \(0.105684\pi\)
\(18\) 0 0
\(19\) − 83.2004i − 1.00460i −0.864692 0.502302i \(-0.832486\pi\)
0.864692 0.502302i \(-0.167514\pi\)
\(20\) 0 0
\(21\) − 21.0000i − 0.218218i
\(22\) 0 0
\(23\) −80.0478 −0.725701 −0.362850 0.931847i \(-0.618196\pi\)
−0.362850 + 0.931847i \(0.618196\pi\)
\(24\) 0 0
\(25\) −168.592 −1.34874
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 139.743i 0.894817i 0.894330 + 0.447408i \(0.147653\pi\)
−0.894330 + 0.447408i \(0.852347\pi\)
\(30\) 0 0
\(31\) 204.541 1.18505 0.592525 0.805552i \(-0.298131\pi\)
0.592525 + 0.805552i \(0.298131\pi\)
\(32\) 0 0
\(33\) 8.94119 0.0471655
\(34\) 0 0
\(35\) 119.942i 0.579253i
\(36\) 0 0
\(37\) 377.916i 1.67916i 0.543234 + 0.839582i \(0.317200\pi\)
−0.543234 + 0.839582i \(0.682800\pi\)
\(38\) 0 0
\(39\) −119.610 −0.491103
\(40\) 0 0
\(41\) 45.3733 0.172832 0.0864161 0.996259i \(-0.472459\pi\)
0.0864161 + 0.996259i \(0.472459\pi\)
\(42\) 0 0
\(43\) − 540.789i − 1.91790i −0.283582 0.958948i \(-0.591523\pi\)
0.283582 0.958948i \(-0.408477\pi\)
\(44\) 0 0
\(45\) 154.211i 0.510853i
\(46\) 0 0
\(47\) 362.473 1.12494 0.562470 0.826818i \(-0.309851\pi\)
0.562470 + 0.826818i \(0.309851\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 397.589i 1.09164i
\(52\) 0 0
\(53\) 668.494i 1.73254i 0.499575 + 0.866270i \(0.333489\pi\)
−0.499575 + 0.866270i \(0.666511\pi\)
\(54\) 0 0
\(55\) −51.0677 −0.125200
\(56\) 0 0
\(57\) 249.601 0.580009
\(58\) 0 0
\(59\) − 757.392i − 1.67125i −0.549297 0.835627i \(-0.685105\pi\)
0.549297 0.835627i \(-0.314895\pi\)
\(60\) 0 0
\(61\) − 690.035i − 1.44836i −0.689611 0.724180i \(-0.742218\pi\)
0.689611 0.724180i \(-0.257782\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 683.157 1.30362
\(66\) 0 0
\(67\) − 370.530i − 0.675634i −0.941212 0.337817i \(-0.890311\pi\)
0.941212 0.337817i \(-0.109689\pi\)
\(68\) 0 0
\(69\) − 240.143i − 0.418984i
\(70\) 0 0
\(71\) −652.184 −1.09014 −0.545070 0.838390i \(-0.683497\pi\)
−0.545070 + 0.838390i \(0.683497\pi\)
\(72\) 0 0
\(73\) 229.804 0.368445 0.184223 0.982885i \(-0.441023\pi\)
0.184223 + 0.982885i \(0.441023\pi\)
\(74\) 0 0
\(75\) − 505.777i − 0.778695i
\(76\) 0 0
\(77\) 20.8628i 0.0308771i
\(78\) 0 0
\(79\) 43.3308 0.0617101 0.0308550 0.999524i \(-0.490177\pi\)
0.0308550 + 0.999524i \(0.490177\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 802.404i − 1.06115i −0.847639 0.530574i \(-0.821976\pi\)
0.847639 0.530574i \(-0.178024\pi\)
\(84\) 0 0
\(85\) − 2270.84i − 2.89773i
\(86\) 0 0
\(87\) −419.230 −0.516623
\(88\) 0 0
\(89\) −868.188 −1.03402 −0.517010 0.855980i \(-0.672955\pi\)
−0.517010 + 0.855980i \(0.672955\pi\)
\(90\) 0 0
\(91\) − 279.091i − 0.321502i
\(92\) 0 0
\(93\) 613.622i 0.684189i
\(94\) 0 0
\(95\) −1425.60 −1.53962
\(96\) 0 0
\(97\) −1181.82 −1.23707 −0.618537 0.785756i \(-0.712274\pi\)
−0.618537 + 0.785756i \(0.712274\pi\)
\(98\) 0 0
\(99\) 26.8236i 0.0272310i
\(100\) 0 0
\(101\) − 215.909i − 0.212710i −0.994328 0.106355i \(-0.966082\pi\)
0.994328 0.106355i \(-0.0339180\pi\)
\(102\) 0 0
\(103\) 1204.93 1.15267 0.576334 0.817214i \(-0.304483\pi\)
0.576334 + 0.817214i \(0.304483\pi\)
\(104\) 0 0
\(105\) −359.825 −0.334432
\(106\) 0 0
\(107\) 327.101i 0.295533i 0.989022 + 0.147767i \(0.0472085\pi\)
−0.989022 + 0.147767i \(0.952792\pi\)
\(108\) 0 0
\(109\) − 668.438i − 0.587383i −0.955900 0.293692i \(-0.905116\pi\)
0.955900 0.293692i \(-0.0948839\pi\)
\(110\) 0 0
\(111\) −1133.75 −0.969465
\(112\) 0 0
\(113\) −725.333 −0.603837 −0.301919 0.953334i \(-0.597627\pi\)
−0.301919 + 0.953334i \(0.597627\pi\)
\(114\) 0 0
\(115\) 1371.58i 1.11218i
\(116\) 0 0
\(117\) − 358.831i − 0.283538i
\(118\) 0 0
\(119\) −927.708 −0.714646
\(120\) 0 0
\(121\) 1322.12 0.993326
\(122\) 0 0
\(123\) 136.120i 0.0997847i
\(124\) 0 0
\(125\) 746.937i 0.534465i
\(126\) 0 0
\(127\) −672.231 −0.469692 −0.234846 0.972033i \(-0.575459\pi\)
−0.234846 + 0.972033i \(0.575459\pi\)
\(128\) 0 0
\(129\) 1622.37 1.10730
\(130\) 0 0
\(131\) 2082.36i 1.38883i 0.719575 + 0.694414i \(0.244336\pi\)
−0.719575 + 0.694414i \(0.755664\pi\)
\(132\) 0 0
\(133\) 582.403i 0.379705i
\(134\) 0 0
\(135\) −462.633 −0.294941
\(136\) 0 0
\(137\) 1221.50 0.761749 0.380875 0.924627i \(-0.375623\pi\)
0.380875 + 0.924627i \(0.375623\pi\)
\(138\) 0 0
\(139\) − 2590.38i − 1.58067i −0.612674 0.790336i \(-0.709906\pi\)
0.612674 0.790336i \(-0.290094\pi\)
\(140\) 0 0
\(141\) 1087.42i 0.649484i
\(142\) 0 0
\(143\) 118.829 0.0694893
\(144\) 0 0
\(145\) 2394.44 1.37136
\(146\) 0 0
\(147\) 147.000i 0.0824786i
\(148\) 0 0
\(149\) − 3406.10i − 1.87274i −0.351015 0.936370i \(-0.614163\pi\)
0.351015 0.936370i \(-0.385837\pi\)
\(150\) 0 0
\(151\) −1795.43 −0.967619 −0.483809 0.875173i \(-0.660747\pi\)
−0.483809 + 0.875173i \(0.660747\pi\)
\(152\) 0 0
\(153\) −1192.77 −0.630258
\(154\) 0 0
\(155\) − 3504.71i − 1.81616i
\(156\) 0 0
\(157\) − 2968.25i − 1.50887i −0.656377 0.754433i \(-0.727912\pi\)
0.656377 0.754433i \(-0.272088\pi\)
\(158\) 0 0
\(159\) −2005.48 −1.00028
\(160\) 0 0
\(161\) 560.335 0.274289
\(162\) 0 0
\(163\) − 1529.60i − 0.735016i −0.930020 0.367508i \(-0.880211\pi\)
0.930020 0.367508i \(-0.119789\pi\)
\(164\) 0 0
\(165\) − 153.203i − 0.0722840i
\(166\) 0 0
\(167\) −252.934 −0.117201 −0.0586007 0.998282i \(-0.518664\pi\)
−0.0586007 + 0.998282i \(0.518664\pi\)
\(168\) 0 0
\(169\) 607.371 0.276455
\(170\) 0 0
\(171\) 748.804i 0.334868i
\(172\) 0 0
\(173\) − 1196.52i − 0.525838i −0.964818 0.262919i \(-0.915315\pi\)
0.964818 0.262919i \(-0.0846853\pi\)
\(174\) 0 0
\(175\) 1180.15 0.509776
\(176\) 0 0
\(177\) 2272.17 0.964899
\(178\) 0 0
\(179\) − 3863.73i − 1.61335i −0.590998 0.806673i \(-0.701266\pi\)
0.590998 0.806673i \(-0.298734\pi\)
\(180\) 0 0
\(181\) − 1166.62i − 0.479085i −0.970886 0.239542i \(-0.923003\pi\)
0.970886 0.239542i \(-0.0769974\pi\)
\(182\) 0 0
\(183\) 2070.11 0.836211
\(184\) 0 0
\(185\) 6475.42 2.57342
\(186\) 0 0
\(187\) − 394.991i − 0.154463i
\(188\) 0 0
\(189\) 189.000i 0.0727393i
\(190\) 0 0
\(191\) 1734.96 0.657263 0.328631 0.944458i \(-0.393413\pi\)
0.328631 + 0.944458i \(0.393413\pi\)
\(192\) 0 0
\(193\) −308.801 −0.115171 −0.0575854 0.998341i \(-0.518340\pi\)
−0.0575854 + 0.998341i \(0.518340\pi\)
\(194\) 0 0
\(195\) 2049.47i 0.752644i
\(196\) 0 0
\(197\) 1297.96i 0.469420i 0.972065 + 0.234710i \(0.0754140\pi\)
−0.972065 + 0.234710i \(0.924586\pi\)
\(198\) 0 0
\(199\) 1750.53 0.623576 0.311788 0.950152i \(-0.399072\pi\)
0.311788 + 0.950152i \(0.399072\pi\)
\(200\) 0 0
\(201\) 1111.59 0.390078
\(202\) 0 0
\(203\) − 978.203i − 0.338209i
\(204\) 0 0
\(205\) − 777.450i − 0.264876i
\(206\) 0 0
\(207\) 720.430 0.241900
\(208\) 0 0
\(209\) −247.970 −0.0820692
\(210\) 0 0
\(211\) 1022.70i 0.333676i 0.985984 + 0.166838i \(0.0533557\pi\)
−0.985984 + 0.166838i \(0.946644\pi\)
\(212\) 0 0
\(213\) − 1956.55i − 0.629393i
\(214\) 0 0
\(215\) −9266.17 −2.93929
\(216\) 0 0
\(217\) −1431.78 −0.447907
\(218\) 0 0
\(219\) 689.411i 0.212722i
\(220\) 0 0
\(221\) 5283.98i 1.60832i
\(222\) 0 0
\(223\) 255.416 0.0766992 0.0383496 0.999264i \(-0.487790\pi\)
0.0383496 + 0.999264i \(0.487790\pi\)
\(224\) 0 0
\(225\) 1517.33 0.449580
\(226\) 0 0
\(227\) 1041.15i 0.304422i 0.988348 + 0.152211i \(0.0486393\pi\)
−0.988348 + 0.152211i \(0.951361\pi\)
\(228\) 0 0
\(229\) 2592.03i 0.747975i 0.927434 + 0.373988i \(0.122010\pi\)
−0.927434 + 0.373988i \(0.877990\pi\)
\(230\) 0 0
\(231\) −62.5884 −0.0178269
\(232\) 0 0
\(233\) −1830.07 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(234\) 0 0
\(235\) − 6210.82i − 1.72404i
\(236\) 0 0
\(237\) 129.992i 0.0356283i
\(238\) 0 0
\(239\) 135.227 0.0365989 0.0182994 0.999833i \(-0.494175\pi\)
0.0182994 + 0.999833i \(0.494175\pi\)
\(240\) 0 0
\(241\) −312.681 −0.0835748 −0.0417874 0.999127i \(-0.513305\pi\)
−0.0417874 + 0.999127i \(0.513305\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 839.592i − 0.218937i
\(246\) 0 0
\(247\) 3317.21 0.854531
\(248\) 0 0
\(249\) 2407.21 0.612654
\(250\) 0 0
\(251\) − 4385.72i − 1.10289i −0.834213 0.551443i \(-0.814078\pi\)
0.834213 0.551443i \(-0.185922\pi\)
\(252\) 0 0
\(253\) 238.574i 0.0592847i
\(254\) 0 0
\(255\) 6812.51 1.67300
\(256\) 0 0
\(257\) 6255.49 1.51831 0.759157 0.650908i \(-0.225612\pi\)
0.759157 + 0.650908i \(0.225612\pi\)
\(258\) 0 0
\(259\) − 2645.41i − 0.634664i
\(260\) 0 0
\(261\) − 1257.69i − 0.298272i
\(262\) 0 0
\(263\) 5952.39 1.39559 0.697795 0.716298i \(-0.254165\pi\)
0.697795 + 0.716298i \(0.254165\pi\)
\(264\) 0 0
\(265\) 11454.3 2.65522
\(266\) 0 0
\(267\) − 2604.56i − 0.596991i
\(268\) 0 0
\(269\) − 2997.81i − 0.679479i −0.940520 0.339739i \(-0.889661\pi\)
0.940520 0.339739i \(-0.110339\pi\)
\(270\) 0 0
\(271\) 681.179 0.152689 0.0763444 0.997082i \(-0.475675\pi\)
0.0763444 + 0.997082i \(0.475675\pi\)
\(272\) 0 0
\(273\) 837.273 0.185619
\(274\) 0 0
\(275\) 502.473i 0.110183i
\(276\) 0 0
\(277\) − 1466.94i − 0.318194i −0.987263 0.159097i \(-0.949142\pi\)
0.987263 0.159097i \(-0.0508583\pi\)
\(278\) 0 0
\(279\) −1840.86 −0.395017
\(280\) 0 0
\(281\) −6558.87 −1.39242 −0.696209 0.717840i \(-0.745131\pi\)
−0.696209 + 0.717840i \(0.745131\pi\)
\(282\) 0 0
\(283\) 8554.70i 1.79690i 0.439071 + 0.898452i \(0.355308\pi\)
−0.439071 + 0.898452i \(0.644692\pi\)
\(284\) 0 0
\(285\) − 4276.80i − 0.888898i
\(286\) 0 0
\(287\) −317.613 −0.0653244
\(288\) 0 0
\(289\) 12651.1 2.57503
\(290\) 0 0
\(291\) − 3545.47i − 0.714225i
\(292\) 0 0
\(293\) − 1262.01i − 0.251629i −0.992054 0.125815i \(-0.959846\pi\)
0.992054 0.125815i \(-0.0401544\pi\)
\(294\) 0 0
\(295\) −12977.6 −2.56130
\(296\) 0 0
\(297\) −80.4707 −0.0157218
\(298\) 0 0
\(299\) − 3191.52i − 0.617292i
\(300\) 0 0
\(301\) 3785.52i 0.724896i
\(302\) 0 0
\(303\) 647.726 0.122808
\(304\) 0 0
\(305\) −11823.4 −2.21970
\(306\) 0 0
\(307\) − 234.049i − 0.0435111i −0.999763 0.0217555i \(-0.993074\pi\)
0.999763 0.0217555i \(-0.00692555\pi\)
\(308\) 0 0
\(309\) 3614.78i 0.665493i
\(310\) 0 0
\(311\) −182.019 −0.0331876 −0.0165938 0.999862i \(-0.505282\pi\)
−0.0165938 + 0.999862i \(0.505282\pi\)
\(312\) 0 0
\(313\) 3515.22 0.634799 0.317399 0.948292i \(-0.397190\pi\)
0.317399 + 0.948292i \(0.397190\pi\)
\(314\) 0 0
\(315\) − 1079.48i − 0.193084i
\(316\) 0 0
\(317\) 3339.39i 0.591667i 0.955239 + 0.295834i \(0.0955974\pi\)
−0.955239 + 0.295834i \(0.904403\pi\)
\(318\) 0 0
\(319\) 416.491 0.0731003
\(320\) 0 0
\(321\) −981.304 −0.170626
\(322\) 0 0
\(323\) − 11026.5i − 1.89948i
\(324\) 0 0
\(325\) − 6721.81i − 1.14726i
\(326\) 0 0
\(327\) 2005.32 0.339126
\(328\) 0 0
\(329\) −2537.31 −0.425187
\(330\) 0 0
\(331\) − 1594.18i − 0.264725i −0.991201 0.132363i \(-0.957744\pi\)
0.991201 0.132363i \(-0.0422563\pi\)
\(332\) 0 0
\(333\) − 3401.25i − 0.559721i
\(334\) 0 0
\(335\) −6348.87 −1.03545
\(336\) 0 0
\(337\) −12224.7 −1.97603 −0.988017 0.154342i \(-0.950674\pi\)
−0.988017 + 0.154342i \(0.950674\pi\)
\(338\) 0 0
\(339\) − 2176.00i − 0.348626i
\(340\) 0 0
\(341\) − 609.612i − 0.0968104i
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) 0 0
\(345\) −4114.75 −0.642117
\(346\) 0 0
\(347\) 3116.35i 0.482117i 0.970511 + 0.241058i \(0.0774945\pi\)
−0.970511 + 0.241058i \(0.922505\pi\)
\(348\) 0 0
\(349\) 7826.76i 1.20045i 0.799831 + 0.600225i \(0.204922\pi\)
−0.799831 + 0.600225i \(0.795078\pi\)
\(350\) 0 0
\(351\) 1076.49 0.163701
\(352\) 0 0
\(353\) 4326.14 0.652287 0.326143 0.945320i \(-0.394251\pi\)
0.326143 + 0.945320i \(0.394251\pi\)
\(354\) 0 0
\(355\) 11174.9i 1.67071i
\(356\) 0 0
\(357\) − 2783.12i − 0.412601i
\(358\) 0 0
\(359\) −3871.77 −0.569204 −0.284602 0.958646i \(-0.591861\pi\)
−0.284602 + 0.958646i \(0.591861\pi\)
\(360\) 0 0
\(361\) −63.3136 −0.00923073
\(362\) 0 0
\(363\) 3966.35i 0.573497i
\(364\) 0 0
\(365\) − 3937.58i − 0.564664i
\(366\) 0 0
\(367\) 8206.26 1.16720 0.583601 0.812041i \(-0.301643\pi\)
0.583601 + 0.812041i \(0.301643\pi\)
\(368\) 0 0
\(369\) −408.360 −0.0576107
\(370\) 0 0
\(371\) − 4679.46i − 0.654839i
\(372\) 0 0
\(373\) 2149.21i 0.298343i 0.988811 + 0.149172i \(0.0476607\pi\)
−0.988811 + 0.149172i \(0.952339\pi\)
\(374\) 0 0
\(375\) −2240.81 −0.308573
\(376\) 0 0
\(377\) −5571.59 −0.761144
\(378\) 0 0
\(379\) 658.658i 0.0892690i 0.999003 + 0.0446345i \(0.0142123\pi\)
−0.999003 + 0.0446345i \(0.985788\pi\)
\(380\) 0 0
\(381\) − 2016.69i − 0.271177i
\(382\) 0 0
\(383\) −3084.93 −0.411573 −0.205786 0.978597i \(-0.565975\pi\)
−0.205786 + 0.978597i \(0.565975\pi\)
\(384\) 0 0
\(385\) 357.474 0.0473210
\(386\) 0 0
\(387\) 4867.10i 0.639299i
\(388\) 0 0
\(389\) − 4413.01i − 0.575189i −0.957752 0.287595i \(-0.907144\pi\)
0.957752 0.287595i \(-0.0928556\pi\)
\(390\) 0 0
\(391\) −10608.7 −1.37214
\(392\) 0 0
\(393\) −6247.08 −0.801840
\(394\) 0 0
\(395\) − 742.454i − 0.0945744i
\(396\) 0 0
\(397\) − 3900.51i − 0.493101i −0.969130 0.246551i \(-0.920703\pi\)
0.969130 0.246551i \(-0.0792972\pi\)
\(398\) 0 0
\(399\) −1747.21 −0.219223
\(400\) 0 0
\(401\) −8593.07 −1.07012 −0.535059 0.844814i \(-0.679711\pi\)
−0.535059 + 0.844814i \(0.679711\pi\)
\(402\) 0 0
\(403\) 8155.06i 1.00802i
\(404\) 0 0
\(405\) − 1387.90i − 0.170284i
\(406\) 0 0
\(407\) 1126.34 0.137176
\(408\) 0 0
\(409\) 11212.2 1.35552 0.677758 0.735285i \(-0.262952\pi\)
0.677758 + 0.735285i \(0.262952\pi\)
\(410\) 0 0
\(411\) 3664.49i 0.439796i
\(412\) 0 0
\(413\) 5301.74i 0.631675i
\(414\) 0 0
\(415\) −13748.8 −1.62627
\(416\) 0 0
\(417\) 7771.15 0.912601
\(418\) 0 0
\(419\) − 7948.88i − 0.926798i −0.886150 0.463399i \(-0.846630\pi\)
0.886150 0.463399i \(-0.153370\pi\)
\(420\) 0 0
\(421\) − 11113.8i − 1.28658i −0.765622 0.643291i \(-0.777568\pi\)
0.765622 0.643291i \(-0.222432\pi\)
\(422\) 0 0
\(423\) −3262.26 −0.374980
\(424\) 0 0
\(425\) −22343.5 −2.55016
\(426\) 0 0
\(427\) 4830.25i 0.547429i
\(428\) 0 0
\(429\) 356.487i 0.0401197i
\(430\) 0 0
\(431\) −3571.39 −0.399137 −0.199568 0.979884i \(-0.563954\pi\)
−0.199568 + 0.979884i \(0.563954\pi\)
\(432\) 0 0
\(433\) −13245.8 −1.47010 −0.735050 0.678013i \(-0.762841\pi\)
−0.735050 + 0.678013i \(0.762841\pi\)
\(434\) 0 0
\(435\) 7183.31i 0.791755i
\(436\) 0 0
\(437\) 6660.01i 0.729043i
\(438\) 0 0
\(439\) −7028.36 −0.764112 −0.382056 0.924139i \(-0.624784\pi\)
−0.382056 + 0.924139i \(0.624784\pi\)
\(440\) 0 0
\(441\) −441.000 −0.0476190
\(442\) 0 0
\(443\) 133.981i 0.0143694i 0.999974 + 0.00718469i \(0.00228698\pi\)
−0.999974 + 0.00718469i \(0.997713\pi\)
\(444\) 0 0
\(445\) 14876.0i 1.58470i
\(446\) 0 0
\(447\) 10218.3 1.08123
\(448\) 0 0
\(449\) 2783.94 0.292611 0.146306 0.989239i \(-0.453262\pi\)
0.146306 + 0.989239i \(0.453262\pi\)
\(450\) 0 0
\(451\) − 135.230i − 0.0141192i
\(452\) 0 0
\(453\) − 5386.30i − 0.558655i
\(454\) 0 0
\(455\) −4782.10 −0.492721
\(456\) 0 0
\(457\) 19117.6 1.95686 0.978428 0.206590i \(-0.0662367\pi\)
0.978428 + 0.206590i \(0.0662367\pi\)
\(458\) 0 0
\(459\) − 3578.30i − 0.363880i
\(460\) 0 0
\(461\) − 6684.09i − 0.675291i −0.941273 0.337645i \(-0.890370\pi\)
0.941273 0.337645i \(-0.109630\pi\)
\(462\) 0 0
\(463\) 12556.5 1.26037 0.630185 0.776445i \(-0.282979\pi\)
0.630185 + 0.776445i \(0.282979\pi\)
\(464\) 0 0
\(465\) 10514.1 1.04856
\(466\) 0 0
\(467\) − 5380.24i − 0.533122i −0.963818 0.266561i \(-0.914113\pi\)
0.963818 0.266561i \(-0.0858873\pi\)
\(468\) 0 0
\(469\) 2593.71i 0.255366i
\(470\) 0 0
\(471\) 8904.75 0.871145
\(472\) 0 0
\(473\) −1611.77 −0.156679
\(474\) 0 0
\(475\) 14027.0i 1.35495i
\(476\) 0 0
\(477\) − 6016.44i − 0.577514i
\(478\) 0 0
\(479\) 9401.83 0.896828 0.448414 0.893826i \(-0.351989\pi\)
0.448414 + 0.893826i \(0.351989\pi\)
\(480\) 0 0
\(481\) −15067.6 −1.42832
\(482\) 0 0
\(483\) 1681.00i 0.158361i
\(484\) 0 0
\(485\) 20250.0i 1.89589i
\(486\) 0 0
\(487\) 6836.64 0.636135 0.318067 0.948068i \(-0.396966\pi\)
0.318067 + 0.948068i \(0.396966\pi\)
\(488\) 0 0
\(489\) 4588.80 0.424362
\(490\) 0 0
\(491\) − 44.2581i − 0.00406790i −0.999998 0.00203395i \(-0.999353\pi\)
0.999998 0.00203395i \(-0.000647427\pi\)
\(492\) 0 0
\(493\) 18520.1i 1.69190i
\(494\) 0 0
\(495\) 459.610 0.0417332
\(496\) 0 0
\(497\) 4565.29 0.412034
\(498\) 0 0
\(499\) − 7884.29i − 0.707313i −0.935375 0.353657i \(-0.884938\pi\)
0.935375 0.353657i \(-0.115062\pi\)
\(500\) 0 0
\(501\) − 758.803i − 0.0676663i
\(502\) 0 0
\(503\) −14818.8 −1.31360 −0.656798 0.754066i \(-0.728090\pi\)
−0.656798 + 0.754066i \(0.728090\pi\)
\(504\) 0 0
\(505\) −3699.50 −0.325991
\(506\) 0 0
\(507\) 1822.11i 0.159611i
\(508\) 0 0
\(509\) − 5035.72i − 0.438515i −0.975667 0.219258i \(-0.929637\pi\)
0.975667 0.219258i \(-0.0703635\pi\)
\(510\) 0 0
\(511\) −1608.63 −0.139259
\(512\) 0 0
\(513\) −2246.41 −0.193336
\(514\) 0 0
\(515\) − 20645.8i − 1.76653i
\(516\) 0 0
\(517\) − 1080.31i − 0.0918998i
\(518\) 0 0
\(519\) 3589.57 0.303593
\(520\) 0 0
\(521\) −13373.1 −1.12454 −0.562269 0.826954i \(-0.690071\pi\)
−0.562269 + 0.826954i \(0.690071\pi\)
\(522\) 0 0
\(523\) 7349.16i 0.614448i 0.951637 + 0.307224i \(0.0994000\pi\)
−0.951637 + 0.307224i \(0.900600\pi\)
\(524\) 0 0
\(525\) 3540.44i 0.294319i
\(526\) 0 0
\(527\) 27107.7 2.24066
\(528\) 0 0
\(529\) −5759.35 −0.473358
\(530\) 0 0
\(531\) 6816.52i 0.557085i
\(532\) 0 0
\(533\) 1809.04i 0.147014i
\(534\) 0 0
\(535\) 5604.73 0.452923
\(536\) 0 0
\(537\) 11591.2 0.931465
\(538\) 0 0
\(539\) − 146.039i − 0.0116704i
\(540\) 0 0
\(541\) − 8183.14i − 0.650315i −0.945660 0.325158i \(-0.894583\pi\)
0.945660 0.325158i \(-0.105417\pi\)
\(542\) 0 0
\(543\) 3499.87 0.276600
\(544\) 0 0
\(545\) −11453.4 −0.900200
\(546\) 0 0
\(547\) − 11819.9i − 0.923916i −0.886902 0.461958i \(-0.847147\pi\)
0.886902 0.461958i \(-0.152853\pi\)
\(548\) 0 0
\(549\) 6210.32i 0.482787i
\(550\) 0 0
\(551\) 11626.7 0.898937
\(552\) 0 0
\(553\) −303.316 −0.0233242
\(554\) 0 0
\(555\) 19426.3i 1.48576i
\(556\) 0 0
\(557\) 22731.0i 1.72916i 0.502491 + 0.864582i \(0.332417\pi\)
−0.502491 + 0.864582i \(0.667583\pi\)
\(558\) 0 0
\(559\) 21561.3 1.63139
\(560\) 0 0
\(561\) 1184.97 0.0891794
\(562\) 0 0
\(563\) − 7860.21i − 0.588399i −0.955744 0.294199i \(-0.904947\pi\)
0.955744 0.294199i \(-0.0950530\pi\)
\(564\) 0 0
\(565\) 12428.3i 0.925417i
\(566\) 0 0
\(567\) −567.000 −0.0419961
\(568\) 0 0
\(569\) −11633.8 −0.857146 −0.428573 0.903507i \(-0.640983\pi\)
−0.428573 + 0.903507i \(0.640983\pi\)
\(570\) 0 0
\(571\) 8319.09i 0.609707i 0.952399 + 0.304854i \(0.0986076\pi\)
−0.952399 + 0.304854i \(0.901392\pi\)
\(572\) 0 0
\(573\) 5204.87i 0.379471i
\(574\) 0 0
\(575\) 13495.5 0.978782
\(576\) 0 0
\(577\) −12339.9 −0.890324 −0.445162 0.895450i \(-0.646854\pi\)
−0.445162 + 0.895450i \(0.646854\pi\)
\(578\) 0 0
\(579\) − 926.402i − 0.0664939i
\(580\) 0 0
\(581\) 5616.83i 0.401076i
\(582\) 0 0
\(583\) 1992.38 0.141537
\(584\) 0 0
\(585\) −6148.41 −0.434539
\(586\) 0 0
\(587\) 22785.2i 1.60212i 0.598584 + 0.801060i \(0.295730\pi\)
−0.598584 + 0.801060i \(0.704270\pi\)
\(588\) 0 0
\(589\) − 17017.9i − 1.19051i
\(590\) 0 0
\(591\) −3893.87 −0.271020
\(592\) 0 0
\(593\) 6786.01 0.469929 0.234965 0.972004i \(-0.424503\pi\)
0.234965 + 0.972004i \(0.424503\pi\)
\(594\) 0 0
\(595\) 15895.8i 1.09524i
\(596\) 0 0
\(597\) 5251.59i 0.360022i
\(598\) 0 0
\(599\) −10498.4 −0.716114 −0.358057 0.933700i \(-0.616561\pi\)
−0.358057 + 0.933700i \(0.616561\pi\)
\(600\) 0 0
\(601\) 11360.3 0.771045 0.385522 0.922698i \(-0.374021\pi\)
0.385522 + 0.922698i \(0.374021\pi\)
\(602\) 0 0
\(603\) 3334.77i 0.225211i
\(604\) 0 0
\(605\) − 22653.9i − 1.52233i
\(606\) 0 0
\(607\) 5460.38 0.365124 0.182562 0.983194i \(-0.441561\pi\)
0.182562 + 0.983194i \(0.441561\pi\)
\(608\) 0 0
\(609\) 2934.61 0.195265
\(610\) 0 0
\(611\) 14451.9i 0.956890i
\(612\) 0 0
\(613\) − 17628.3i − 1.16150i −0.814082 0.580750i \(-0.802759\pi\)
0.814082 0.580750i \(-0.197241\pi\)
\(614\) 0 0
\(615\) 2332.35 0.152926
\(616\) 0 0
\(617\) −17543.6 −1.14470 −0.572351 0.820009i \(-0.693968\pi\)
−0.572351 + 0.820009i \(0.693968\pi\)
\(618\) 0 0
\(619\) 27868.2i 1.80956i 0.425880 + 0.904779i \(0.359964\pi\)
−0.425880 + 0.904779i \(0.640036\pi\)
\(620\) 0 0
\(621\) 2161.29i 0.139661i
\(622\) 0 0
\(623\) 6077.31 0.390823
\(624\) 0 0
\(625\) −8275.64 −0.529641
\(626\) 0 0
\(627\) − 743.911i − 0.0473827i
\(628\) 0 0
\(629\) 50085.1i 3.17492i
\(630\) 0 0
\(631\) 21948.0 1.38469 0.692343 0.721569i \(-0.256579\pi\)
0.692343 + 0.721569i \(0.256579\pi\)
\(632\) 0 0
\(633\) −3068.11 −0.192648
\(634\) 0 0
\(635\) 11518.4i 0.719830i
\(636\) 0 0
\(637\) 1953.64i 0.121516i
\(638\) 0 0
\(639\) 5869.65 0.363380
\(640\) 0 0
\(641\) 5186.49 0.319585 0.159793 0.987151i \(-0.448917\pi\)
0.159793 + 0.987151i \(0.448917\pi\)
\(642\) 0 0
\(643\) − 21931.1i − 1.34507i −0.740066 0.672534i \(-0.765206\pi\)
0.740066 0.672534i \(-0.234794\pi\)
\(644\) 0 0
\(645\) − 27798.5i − 1.69700i
\(646\) 0 0
\(647\) −19475.1 −1.18338 −0.591690 0.806166i \(-0.701539\pi\)
−0.591690 + 0.806166i \(0.701539\pi\)
\(648\) 0 0
\(649\) −2257.33 −0.136530
\(650\) 0 0
\(651\) − 4295.35i − 0.258599i
\(652\) 0 0
\(653\) 29761.6i 1.78355i 0.452477 + 0.891776i \(0.350540\pi\)
−0.452477 + 0.891776i \(0.649460\pi\)
\(654\) 0 0
\(655\) 35680.3 2.12846
\(656\) 0 0
\(657\) −2068.23 −0.122815
\(658\) 0 0
\(659\) 12257.3i 0.724547i 0.932072 + 0.362274i \(0.117999\pi\)
−0.932072 + 0.362274i \(0.882001\pi\)
\(660\) 0 0
\(661\) 9737.88i 0.573010i 0.958079 + 0.286505i \(0.0924935\pi\)
−0.958079 + 0.286505i \(0.907506\pi\)
\(662\) 0 0
\(663\) −15851.9 −0.928564
\(664\) 0 0
\(665\) 9979.21 0.581921
\(666\) 0 0
\(667\) − 11186.1i − 0.649369i
\(668\) 0 0
\(669\) 766.248i 0.0442823i
\(670\) 0 0
\(671\) −2056.58 −0.118321
\(672\) 0 0
\(673\) 33809.6 1.93650 0.968250 0.249984i \(-0.0804252\pi\)
0.968250 + 0.249984i \(0.0804252\pi\)
\(674\) 0 0
\(675\) 4552.00i 0.259565i
\(676\) 0 0
\(677\) 29402.3i 1.66916i 0.550887 + 0.834580i \(0.314289\pi\)
−0.550887 + 0.834580i \(0.685711\pi\)
\(678\) 0 0
\(679\) 8272.77 0.467570
\(680\) 0 0
\(681\) −3123.46 −0.175758
\(682\) 0 0
\(683\) 5540.58i 0.310402i 0.987883 + 0.155201i \(0.0496025\pi\)
−0.987883 + 0.155201i \(0.950398\pi\)
\(684\) 0 0
\(685\) − 20929.8i − 1.16743i
\(686\) 0 0
\(687\) −7776.10 −0.431844
\(688\) 0 0
\(689\) −26652.9 −1.47372
\(690\) 0 0
\(691\) 33967.2i 1.87000i 0.354642 + 0.935002i \(0.384603\pi\)
−0.354642 + 0.935002i \(0.615397\pi\)
\(692\) 0 0
\(693\) − 187.765i − 0.0102924i
\(694\) 0 0
\(695\) −44385.0 −2.42247
\(696\) 0 0
\(697\) 6013.31 0.326787
\(698\) 0 0
\(699\) − 5490.22i − 0.297080i
\(700\) 0 0
\(701\) − 11444.0i − 0.616597i −0.951290 0.308299i \(-0.900240\pi\)
0.951290 0.308299i \(-0.0997596\pi\)
\(702\) 0 0
\(703\) 31442.8 1.68690
\(704\) 0 0
\(705\) 18632.4 0.995374
\(706\) 0 0
\(707\) 1511.36i 0.0803968i
\(708\) 0 0
\(709\) − 30373.3i − 1.60888i −0.594036 0.804438i \(-0.702466\pi\)
0.594036 0.804438i \(-0.297534\pi\)
\(710\) 0 0
\(711\) −389.977 −0.0205700
\(712\) 0 0
\(713\) −16373.0 −0.859992
\(714\) 0 0
\(715\) − 2036.08i − 0.106497i
\(716\) 0 0
\(717\) 405.682i 0.0211304i
\(718\) 0 0
\(719\) −36362.7 −1.88609 −0.943046 0.332663i \(-0.892053\pi\)
−0.943046 + 0.332663i \(0.892053\pi\)
\(720\) 0 0
\(721\) −8434.48 −0.435668
\(722\) 0 0
\(723\) − 938.042i − 0.0482519i
\(724\) 0 0
\(725\) − 23559.7i − 1.20688i
\(726\) 0 0
\(727\) −34240.1 −1.74676 −0.873379 0.487041i \(-0.838076\pi\)
−0.873379 + 0.487041i \(0.838076\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) − 71670.6i − 3.62631i
\(732\) 0 0
\(733\) − 30023.1i − 1.51286i −0.654074 0.756431i \(-0.726941\pi\)
0.654074 0.756431i \(-0.273059\pi\)
\(734\) 0 0
\(735\) 2518.78 0.126403
\(736\) 0 0
\(737\) −1104.33 −0.0551946
\(738\) 0 0
\(739\) 32000.0i 1.59288i 0.604717 + 0.796440i \(0.293286\pi\)
−0.604717 + 0.796440i \(0.706714\pi\)
\(740\) 0 0
\(741\) 9951.64i 0.493364i
\(742\) 0 0
\(743\) −8412.86 −0.415394 −0.207697 0.978193i \(-0.566597\pi\)
−0.207697 + 0.978193i \(0.566597\pi\)
\(744\) 0 0
\(745\) −58361.9 −2.87009
\(746\) 0 0
\(747\) 7221.64i 0.353716i
\(748\) 0 0
\(749\) − 2289.71i − 0.111701i
\(750\) 0 0
\(751\) 3889.62 0.188994 0.0944969 0.995525i \(-0.469876\pi\)
0.0944969 + 0.995525i \(0.469876\pi\)
\(752\) 0 0
\(753\) 13157.2 0.636751
\(754\) 0 0
\(755\) 30763.9i 1.48293i
\(756\) 0 0
\(757\) 17814.2i 0.855309i 0.903942 + 0.427654i \(0.140660\pi\)
−0.903942 + 0.427654i \(0.859340\pi\)
\(758\) 0 0
\(759\) −715.723 −0.0342281
\(760\) 0 0
\(761\) 9356.95 0.445715 0.222857 0.974851i \(-0.428462\pi\)
0.222857 + 0.974851i \(0.428462\pi\)
\(762\) 0 0
\(763\) 4679.07i 0.222010i
\(764\) 0 0
\(765\) 20437.5i 0.965909i
\(766\) 0 0
\(767\) 30197.3 1.42159
\(768\) 0 0
\(769\) 32968.7 1.54601 0.773005 0.634400i \(-0.218753\pi\)
0.773005 + 0.634400i \(0.218753\pi\)
\(770\) 0 0
\(771\) 18766.5i 0.876598i
\(772\) 0 0
\(773\) − 11914.1i − 0.554359i −0.960818 0.277180i \(-0.910600\pi\)
0.960818 0.277180i \(-0.0893997\pi\)
\(774\) 0 0
\(775\) −34484.0 −1.59832
\(776\) 0 0
\(777\) 7936.24 0.366423
\(778\) 0 0
\(779\) − 3775.08i − 0.173628i
\(780\) 0 0
\(781\) 1943.77i 0.0890569i
\(782\) 0 0
\(783\) 3773.07 0.172208
\(784\) 0 0
\(785\) −50859.6 −2.31243
\(786\) 0 0
\(787\) − 29231.8i − 1.32402i −0.749496 0.662009i \(-0.769704\pi\)
0.749496 0.662009i \(-0.230296\pi\)
\(788\) 0 0
\(789\) 17857.2i 0.805744i
\(790\) 0 0
\(791\) 5077.33 0.228229
\(792\) 0 0
\(793\) 27511.8 1.23200
\(794\) 0 0
\(795\) 34363.0i 1.53299i
\(796\) 0 0
\(797\) 13241.6i 0.588507i 0.955727 + 0.294254i \(0.0950710\pi\)
−0.955727 + 0.294254i \(0.904929\pi\)
\(798\) 0 0
\(799\) 48038.5 2.12701
\(800\) 0 0
\(801\) 7813.69 0.344673
\(802\) 0 0
\(803\) − 684.906i − 0.0300994i
\(804\) 0 0
\(805\) − 9601.08i − 0.420365i
\(806\) 0 0
\(807\) 8993.44 0.392297
\(808\) 0 0
\(809\) −10389.0 −0.451491 −0.225746 0.974186i \(-0.572482\pi\)
−0.225746 + 0.974186i \(0.572482\pi\)
\(810\) 0 0
\(811\) 10637.7i 0.460593i 0.973120 + 0.230297i \(0.0739697\pi\)
−0.973120 + 0.230297i \(0.926030\pi\)
\(812\) 0 0
\(813\) 2043.54i 0.0881549i
\(814\) 0 0
\(815\) −26209.0 −1.12646
\(816\) 0 0
\(817\) −44993.9 −1.92673
\(818\) 0 0
\(819\) 2511.82i 0.107167i
\(820\) 0 0
\(821\) − 4915.12i − 0.208939i −0.994528 0.104470i \(-0.966686\pi\)
0.994528 0.104470i \(-0.0333145\pi\)
\(822\) 0 0
\(823\) 22017.8 0.932555 0.466278 0.884638i \(-0.345595\pi\)
0.466278 + 0.884638i \(0.345595\pi\)
\(824\) 0 0
\(825\) −1507.42 −0.0636140
\(826\) 0 0
\(827\) 4114.43i 0.173002i 0.996252 + 0.0865010i \(0.0275686\pi\)
−0.996252 + 0.0865010i \(0.972431\pi\)
\(828\) 0 0
\(829\) 3132.86i 0.131253i 0.997844 + 0.0656264i \(0.0209046\pi\)
−0.997844 + 0.0656264i \(0.979095\pi\)
\(830\) 0 0
\(831\) 4400.82 0.183710
\(832\) 0 0
\(833\) 6493.96 0.270111
\(834\) 0 0
\(835\) 4333.91i 0.179618i
\(836\) 0 0
\(837\) − 5522.59i − 0.228063i
\(838\) 0 0
\(839\) 11892.0 0.489341 0.244670 0.969606i \(-0.421320\pi\)
0.244670 + 0.969606i \(0.421320\pi\)
\(840\) 0 0
\(841\) 4860.80 0.199303
\(842\) 0 0
\(843\) − 19676.6i − 0.803912i
\(844\) 0 0
\(845\) − 10407.0i − 0.423684i
\(846\) 0 0
\(847\) −9254.82 −0.375442
\(848\) 0 0
\(849\) −25664.1 −1.03744
\(850\) 0 0
\(851\) − 30251.4i − 1.21857i
\(852\) 0 0
\(853\) − 28626.0i − 1.14905i −0.818488 0.574524i \(-0.805187\pi\)
0.818488 0.574524i \(-0.194813\pi\)
\(854\) 0 0
\(855\) 12830.4 0.513206
\(856\) 0 0
\(857\) −41513.8 −1.65471 −0.827353 0.561683i \(-0.810154\pi\)
−0.827353 + 0.561683i \(0.810154\pi\)
\(858\) 0 0
\(859\) 23085.1i 0.916944i 0.888709 + 0.458472i \(0.151603\pi\)
−0.888709 + 0.458472i \(0.848397\pi\)
\(860\) 0 0
\(861\) − 952.839i − 0.0377151i
\(862\) 0 0
\(863\) −12260.9 −0.483622 −0.241811 0.970323i \(-0.577741\pi\)
−0.241811 + 0.970323i \(0.577741\pi\)
\(864\) 0 0
\(865\) −20501.9 −0.805879
\(866\) 0 0
\(867\) 37953.4i 1.48669i
\(868\) 0 0
\(869\) − 129.143i − 0.00504129i
\(870\) 0 0
\(871\) 14773.1 0.574704
\(872\) 0 0
\(873\) 10636.4 0.412358
\(874\) 0 0
\(875\) − 5228.56i − 0.202009i
\(876\) 0 0
\(877\) 15969.9i 0.614896i 0.951565 + 0.307448i \(0.0994750\pi\)
−0.951565 + 0.307448i \(0.900525\pi\)
\(878\) 0 0
\(879\) 3786.03 0.145278
\(880\) 0 0
\(881\) 42590.6 1.62873 0.814367 0.580350i \(-0.197084\pi\)
0.814367 + 0.580350i \(0.197084\pi\)
\(882\) 0 0
\(883\) 3087.03i 0.117652i 0.998268 + 0.0588261i \(0.0187357\pi\)
−0.998268 + 0.0588261i \(0.981264\pi\)
\(884\) 0 0
\(885\) − 38932.7i − 1.47877i
\(886\) 0 0
\(887\) −43586.1 −1.64992 −0.824960 0.565191i \(-0.808803\pi\)
−0.824960 + 0.565191i \(0.808803\pi\)
\(888\) 0 0
\(889\) 4705.61 0.177527
\(890\) 0 0
\(891\) − 241.412i − 0.00907701i
\(892\) 0 0
\(893\) − 30158.0i − 1.13012i
\(894\) 0 0
\(895\) −66203.3 −2.47255
\(896\) 0 0
\(897\) 9574.55 0.356394
\(898\) 0 0
\(899\) 28583.2i 1.06040i
\(900\) 0 0
\(901\) 88595.3i 3.27584i
\(902\) 0 0
\(903\) −11356.6 −0.418519
\(904\) 0 0
\(905\) −19989.5 −0.734226
\(906\) 0 0
\(907\) − 38325.2i − 1.40305i −0.712645 0.701525i \(-0.752503\pi\)
0.712645 0.701525i \(-0.247497\pi\)
\(908\) 0 0
\(909\) 1943.18i 0.0709033i
\(910\) 0 0
\(911\) 40999.9 1.49109 0.745547 0.666454i \(-0.232189\pi\)
0.745547 + 0.666454i \(0.232189\pi\)
\(912\) 0 0
\(913\) −2391.48 −0.0866884
\(914\) 0 0
\(915\) − 35470.3i − 1.28154i
\(916\) 0 0
\(917\) − 14576.5i − 0.524928i
\(918\) 0 0
\(919\) −51974.3 −1.86559 −0.932793 0.360414i \(-0.882636\pi\)
−0.932793 + 0.360414i \(0.882636\pi\)
\(920\) 0 0
\(921\) 702.148 0.0251211
\(922\) 0 0
\(923\) − 26002.7i − 0.927289i
\(924\) 0 0
\(925\) − 63713.8i − 2.26475i
\(926\) 0 0
\(927\) −10844.3 −0.384223
\(928\) 0 0
\(929\) 1073.91 0.0379266 0.0189633 0.999820i \(-0.493963\pi\)
0.0189633 + 0.999820i \(0.493963\pi\)
\(930\) 0 0
\(931\) − 4076.82i − 0.143515i
\(932\) 0 0
\(933\) − 546.057i − 0.0191609i
\(934\) 0 0
\(935\) −6767.99 −0.236724
\(936\) 0 0
\(937\) 5315.51 0.185326 0.0926629 0.995698i \(-0.470462\pi\)
0.0926629 + 0.995698i \(0.470462\pi\)
\(938\) 0 0
\(939\) 10545.7i 0.366501i
\(940\) 0 0
\(941\) 24770.6i 0.858127i 0.903274 + 0.429064i \(0.141156\pi\)
−0.903274 + 0.429064i \(0.858844\pi\)
\(942\) 0 0
\(943\) −3632.03 −0.125424
\(944\) 0 0
\(945\) 3238.43 0.111477
\(946\) 0 0
\(947\) 1529.99i 0.0525005i 0.999655 + 0.0262502i \(0.00835667\pi\)
−0.999655 + 0.0262502i \(0.991643\pi\)
\(948\) 0 0
\(949\) 9162.31i 0.313405i
\(950\) 0 0
\(951\) −10018.2 −0.341599
\(952\) 0 0
\(953\) 29634.5 1.00730 0.503649 0.863909i \(-0.331991\pi\)
0.503649 + 0.863909i \(0.331991\pi\)
\(954\) 0 0
\(955\) − 29727.7i − 1.00729i
\(956\) 0 0
\(957\) 1249.47i 0.0422045i
\(958\) 0 0
\(959\) −8550.49 −0.287914
\(960\) 0 0
\(961\) 12045.8 0.404344
\(962\) 0 0
\(963\) − 2943.91i − 0.0985112i
\(964\) 0 0
\(965\) 5291.16i 0.176506i
\(966\) 0 0
\(967\) 25369.8 0.843680 0.421840 0.906670i \(-0.361384\pi\)
0.421840 + 0.906670i \(0.361384\pi\)
\(968\) 0 0
\(969\) 33079.6 1.09667
\(970\) 0 0
\(971\) − 42309.9i − 1.39834i −0.714954 0.699171i \(-0.753553\pi\)
0.714954 0.699171i \(-0.246447\pi\)
\(972\) 0 0
\(973\) 18132.7i 0.597438i
\(974\) 0 0
\(975\) 20165.4 0.662370
\(976\) 0 0
\(977\) 37869.9 1.24009 0.620044 0.784567i \(-0.287115\pi\)
0.620044 + 0.784567i \(0.287115\pi\)
\(978\) 0 0
\(979\) 2587.54i 0.0844722i
\(980\) 0 0
\(981\) 6015.95i 0.195794i
\(982\) 0 0
\(983\) 20941.4 0.679477 0.339739 0.940520i \(-0.389661\pi\)
0.339739 + 0.940520i \(0.389661\pi\)
\(984\) 0 0
\(985\) 22239.9 0.719414
\(986\) 0 0
\(987\) − 7611.94i − 0.245482i
\(988\) 0 0
\(989\) 43289.0i 1.39182i
\(990\) 0 0
\(991\) −21465.4 −0.688065 −0.344033 0.938958i \(-0.611793\pi\)
−0.344033 + 0.938958i \(0.611793\pi\)
\(992\) 0 0
\(993\) 4782.54 0.152839
\(994\) 0 0
\(995\) − 29994.5i − 0.955668i
\(996\) 0 0
\(997\) 8105.46i 0.257475i 0.991679 + 0.128737i \(0.0410925\pi\)
−0.991679 + 0.128737i \(0.958908\pi\)
\(998\) 0 0
\(999\) 10203.7 0.323155
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.673.7 yes 12
4.3 odd 2 1344.4.c.g.673.1 yes 12
8.3 odd 2 1344.4.c.g.673.12 yes 12
8.5 even 2 inner 1344.4.c.f.673.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.c.f.673.6 12 8.5 even 2 inner
1344.4.c.f.673.7 yes 12 1.1 even 1 trivial
1344.4.c.g.673.1 yes 12 4.3 odd 2
1344.4.c.g.673.12 yes 12 8.3 odd 2